Gas Turbine Theory

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4TH EDITION
·GASTU
T EORY
By G. F. C. Rogers and Y. R. Mayhew
Engineering Thermodynamics Work and Heat Transfer (Longman)
Thermodynamic and Transport Properties of Fluids: SI
By G. F. C. Rogers
The Nature of Engineering (Macmillan)
4TH EDITION
GAS TURBINE
THEORY
H
U
K<l\'c
f i 1"
Lately Fellow, Queen's College, Cambridge
GFC Rogers
Professor Emeritus, University of Bristol
HIH Saravanamuttoo
Professor, Department of Mechanical and Aerospace
Engineering, Carleton University
.-, rt
-j--i
t :
Longman Group Limited .. .
Longman House,.Bunit Mill, ijarlow
Essex CM20 2JE, England..' . . ..' .
and Associated Companies throughout the world
© LonSman Group Limited 198?, 1?96
All rights reserved; no part of this pubhcatlOn may be
reproduced, stored in any retrieval system, or transmitted in
any form or by any means, mechanical,
photocopying, recording or otherwise without either the prior
written permission of the Publishers or a licence permitting
restricted copying in the United Kingdom issued by the
Copyright Licensing Agency Ltd, 90 Tottenham Court Road,
London WIP 9HE.
First published 1951
Eighth impression 1971
Second edition 1972
Tenth impression 1985
Third edition 1987
Ninth impression 1995
Fourth edition 1996
British Library Cataloguing-ill-Publication Data
A catalogue entry for this title is available from the British Library.
ISBN 0-582-23632-0
Library of Congress Cataloging-ill-Publication Data
A catalog entry for this title is available from the Library of Congress.
Printed by T. J. Press, Padstow, Cornwall
·1
I
I
Prefaces
1 Introduction
2
3
-r
+-
-'-
1.1 Open cycle single-shaft and twin-shaft arrangements
1.2 Multi-spool arrangements
1.3 Closed cycles
1.4 Aircraft propulsion
1.5 Industrial applications
1.6 Environmental issues
1. 7 Some future possibilities
1.8 Gas turbine desigu procedure
Shaft power cycles
2.1 Ideal cycles
2.2 Methods of accounting for component losses
2.3 Design point performance calculations
2.4 Comparative performance of practical cycles
2.5 Combined cycles and cogeneration schemes
2.6 Closed-cycle gas turbines
Gas turbine cycles for aircraft propulsion
3.1 Criteria of performance
3.2 Intake and propelling nozzle efficiencies
3.3 Simple turbojet cycle
3.4 The turbofan engine
3.5 The turboprop engine
3.6 Thrust augmentation
Contents
ix
1
5
8
9
12
17
26
29
33
37
37
45
63
70
76
81
86
87
91
99
106
117
120
vi CONTENTS CONTENTS vii
4 Centrifugal compressors lUi
<>
PreciictiOltJ. of performance of simple gas turbine§ 336
"
4.1 Principle of operation 127 8.1 Component characteristics 338
4.2 Work done and pressure rise 128 8.2 Off-design operation of the single-shaft gas turbine 340
4.3 The diffuser 136 8.3 Equilibrium running of a gas generator 345
4.4 Compressibility effects 141 8.4 Off-design operation of free turbine engine 346
4.5 Non-dimensional quantities for plotting compressor 8.5 Off-design operation of the jet engine 358
characteristics 146 8.6 Methods of displacing the equilibrium running line 366
4.6 Compressor characteristics 148 8.7 Incorporation of variable pressure losses 369
4.7 Computerized design procedures 151
Axial flow compressors
II) Prediction of pel1'formalllce---further topics 371
5 154
9.1 Methods of improving part-load performance 371
5.1 Basic operation 155
9.2 Matching procedures for twin-spool engines 375
5.2 Elementary theory 157
9.3 Some notes on the behaviour of twin-spool engines 379
5.3 Factors affecting stage pressure ratio 160
9.4 Matching procedures for turbofan engines 383
5.4 Blockage in the compressor annulus 165
9.5 Transient behaviour of gas turbines 385
5.5 Degree of reaction 167
9.6 Principles of control systems 392
5.6 Three-dimensional flow 169
5.7 Design process 178
5.8 Blade design 199
Appendix A Some notes 0111 gas dynamics 396
5.9 Calculation of stage performance 208
A.1 Compressibility effects (qualitative treatment) 396
5.10 Compressibility effects 216
A.2 Basic equations for steady one-dimensional compressible
5.11 Off-design performance 222
flow of a perfect gas in a duct 400
5.12 Axial compressor characteristics 225
AJ Isentropic flow in a duct of varying area 402
5.13 Closure 231
A.4 Frictionless flow in a constant area duct with heat transfer 403
A.S Adiabatic flow in a constant area duct with friction 405
6 Combustion systems 233
A.6 Plane normal shock waves 406
6.1 Operational requirements 234
A.7 Oblique shock waves 410
6.2 Types of combustion system 235
A.S Isentropic two-dimensional supersonic expansion and
6.3 Some important factors affecting combustor design 237
compression 413
6.4 The combustion process 239
6.5 Combustion chamber performance 243
6.6 Some practical problems 250
Appendix B Problems 416
6.7 Gas turbine emissions 257
6.8 Coal gasification 267
Appendix C References 428
7 Axial and radial flow turbines 271
7.1 Elementary theory of axial flow turbine 271 Index 435
7.2 Vortex theory 287
7.3 Choice of blade profile, pitch and chord 293
7.4 Estimation of stage performance 309
7.5 Overall turbine performance 319
7.6 The cooled turbine 320
7.7 The radial flow turbine 328
Preface to fourth edition
It is exactly 40 years since I was introduced to the study of gas turbines, using a
book simply referred to as Cohen and Rogers. After a decade in the gas turbine
industry I had the good fortune to join Professor Rogers at the University of
Bristol and was greatly honoured to be invited to join the original authors in the
preparation of the second edition in 1971. Sadly, Dr Cohen died in 1987, but I am
sure he would be delighted to know that the book he initiated in 1951 is still
going strong. Professor Rogers, although n:tired, has remained fully active as a
critic and has continued to keep his junior faculty member in line; his comments
have been very helpful to me.
Since the third edition in 1987, the gas turbine has been found to be suitable for
an increasing number of applications, and this is reflected in the new Introduc-
tion. Gas turbines are becoming widely used for base-load electricity generation,
as part of combined-cycle plant, and combined cycles receive more attention in
this edition. With the increased use of gas turbines, control of harmful emissions
has become ever more important. The chapter on combustion has been enlarged
to include a substantial discussion of the factors affecting emissions and de-
scriptions of current methods of attacking the problem. A section on coal gasi-
fication has also been added. Finally, the opportunity has been taken to make
many small but significant improvements and additions to the text and examples.
As in the third edition, no attempt has been made to introduce computational
methods for aerodynamic design of turbomachinery and the treatment is restricted
to the fundamentals; those wishing to specialize must refer to the current, rapidly
changing, literature. Readers wishing access to a suite of perf6nnance programs
suitable for use on a Personal Computer will find the commercially available
program GASTURB by Dr Joachim Kurzke very useful; this deals with both
design and off-design calculations. The way to obtain this program is described in
the rubric of Appendix B.
I would like to express my deepest gratitude to Gordon Rogers, who
introduced me to a university career and for over 30 years has been boss, mentor,
colleague and friend. Together with Henry Cohen, his early work led me to a
career in gas turbines and it has been a great privilege to work with him over the
years.
Thanks are once again due to the Natural Sciences and Engineering Research
Council of Canada for financial support of my work for many years. Finally,
sincere thanks to Lois Whillans for her patience and good humour in preparing
the manuscript.
September 1995 H.I.H.S.
to the third edition
The continued use of this book since the appearance of the first edition in 1951
suggests that the objectives of the original authors were sound. To the great regret
of Professors Rogers and Saravanamuttoo, Dr Cohen was unable to join them in
the preparation of this new edition. They would like to express their appreciation
of his earlier contribution, however, and in particular for initiating the book in the
first place.
Since the second edition was published in 1972 considerable advances have
been made and the gas turbine has become well established in a valiety of
applications. This has required considerable updating, particularly of the Intro-
duction. Sub-sections have been added to the chapter on shaft power cycles to
include combined gas and steam cyclles, cogeneration schemes and closed cycles,
and many minor modifications have been made throughout the book to take
account of recent developments. The chapter on axial flow compressors has been
rewritten and enlarged, and describes in detail the design of a compressor to meet
a particular aerodynamic specification which matches the worked example on
turbine design. A section on radial flow turbines has been added to the turbine
chapter.
In keeping with the introductory nature of the book, and to avoid losing sight
of physical principles, no attempt has been made to introduce modem
computational methods for predicting the flow in turbomachinery. Appropriate
references to such matters have been added, however, to encourage the student to
pursue these more advanced topics.
As a result of many requests from overseas readers, Professor Saravanamuttoo
has agreed to produce a manual of solutions to the problems in Appendix B. The
manual can be purchased by writing to: The Bookstore, Carleton University,
Ottawa, Ontario KlS-5B6, Canada.
Finally, this edition would not have been possible without the generous support
of Professor Saravanamuttoo's research by the Natural Sciences and Engineering
Research Council of Canada. He would particularly like to express his thanks to
Dr Bernard MacIsaac of GasTOPS and the many engineers from Pratt and
Whitney Canada, Rolls-Royce, and the Royal Navy with whom he has
collaborated for many years.
March 1986 G.F.C.R.
H.I.H.S.
Preface to the second edition
At the suggestion ofthe publishers the authors agreed to produce a new edition of
Gas Turbine Theory in SI units. The continued use of the first edition encouraged
the authors to believe that the general plan and scope of the book was basically
correct for an introduction to the subject, and the object of the book remains as
stated in the original Preface. So much development has taken place during the
intervening twenty-one years, however, that the book has had to be rewritten
completely, and even some changes in the general plan have been required. For
example, because gas dynamics now forms part of most undergraduate courses in
fluid mechanics a chapter devoted to this is out of place. Instead, the authors have
provided a sununary of the relevant aspects in an Appendix. Other changes of
plan include the extension of the section on aircraft propUlsion to a complete
chapter, and the addition of a chapter on the prediction of performance of more
complex jet engines and transient behaviour. Needless to say the nomenclature
has been changed throughout to confonn with international standards and current
usage.
The original authors are glad to be joined by Dr Saravanamuttoo in the author-
ship of the new edition. He is actively engaged on aspects of the gas turbine with
which they were not familiar and his contribution has been substantial. They are
glad too that the publisher decided to use a wider page so that the third author's
name could be added without abbreviation. Dr Saravanamuttoo in his turn would
like to acknowledge his debt to the many members of staff of Rolls-Royce, the
National Gas Turbine Establislmlent and Orenda Engines with whom he has been
associated in the course of his work. The authors' sincere appreciation goes to
Miss G. M. Davis for her expert typing of the manuscript.
October 1971
H.C.
G.F.C.R.
HUl.S.
Preface to the first edition
This book is confined to a presentation of the thermodynamic and aerodynamic
theory forming the basis of gas turbine design, avoiding as far as possible the
many controversial topics associated with this new form of prime-mover. While it
is perhaps too early in the development of the gas turbine to say with complete
assurance what are the basic principles involved, it is thought that some attempt
must now be made if only to fill the gap between the necessarily scanty outline of
a lecture course and the many articles which appear in the technical journals.
Such articles are usually written by specialists for specialists and presuppose a
general knowledge of the subject. Since this book is primarily intended for
students, for the sake of clarity some parts of the work have been given a simpler
treatment than the latest refinements of method would permit. Care has been
taken to ensure that no fundamental principle has been misrepresented by this
approach.
Practising engineers who have been engaged on the design and development of
other types of power plant, but who now find themselves called upon to undertake
work on gas turbines, may also find this book helpful. Although they will prob-
ably concentrate their efforts on one particular component, it is always more
satisfactory if one's specialized knowledge is buttressed by a general under-
standing of the theory underlying the design of the complete unit.
Practical aspects, such as control systems and mechanical design features,
have not been described as they do not form part of the basic theory and are
subject to continuous development. Methods of stressing the various components
have also been omitted since the basic principles are considered to have been
adequately treated elsewhere. For a similar reason, the theory of heat-exchangers
has not been included. Although heat-exchangers will undoubtedly be widely
used in gas turbine plant, it is felt that at their present stage of development the
appropriate theory may well be drawn from standard works on heat transmission.
Owing to the rate at which articles and papers on this subject now appear, and
since the authors have been concerned purely with an attempt to define the basic
structure of gas turbine theory, a comprehensive bibliography has not been
included. A few references have been selected, however, and are appended to
appropriate chapters. These are given not only as suggestions for further reading,
PREFACES
xiii
but also as acknowledgements of sources of information. Some problems, with
answers, are given at the end of the book. In most cases these have been chosen to
illustrate various points which have not appeared in the worked examples in the
text. Acknowledgements are due to the Universities of Cambridge, Bristol and
Durham for permission to use problems which have appeared in their
eXaInination papers.
The authors have obtained such knowledge as they possess from contact with
the work of a large number of people. It would therefore be invidious to
acknowledge their debt by mention of one or two names which the authors
associate in their minds with particular aspects of the work. They would, however,
like to express their gratitude in a corporate way to their former colleagues in the
research teams of what were once the Turbine Division of the Royal Aircraft
Establishment and Power Jets (R. & D.) Ltd, now the National Gas Turbine
Establislunent.
In conclusion, the authors will welcome any criticisms both of detail and of the
general scheme of the book. Only from such criticism can they hope to discover
whether the right approach has been adopted in teaching the fundamentals of this
new subject.
April 1950 H.C.
G.F.C.R.
1 Introduction
Of the various means of producing mechanical power the turbine is in many
respects the most satisfactory. The absence of reciprocating and rubbing members
means that balancing problems are few, that the lubricating oil consumption is
exceptionally low, and that reliability can be high. The inherent advantages of the
turbine were first realized using water as the working fluid, and hydro-electric
power is still a significant contributor to the world's energy resources. Around the
turn of the twentieth century the steam turbine began its career and, quite apart
from its wide use as a marine power plant, it has become the most important
prime mover for electricity generation. Steam turbine plant producing well over
1000 MW of shaft power with an efficiency of 40 per cent are now being used. In
spite of its successful development, the steam turbine does have an inherent
disadvantage. It is that the production of high-pressure high-temperature steam
involves the installation of bullcy and expensive steam generating equipment,
whether it be a conventional boiler or nuclear reactor. The significant feature is
that the hot gases produced in the boiler furnace or reactor core never reach the
turbine; they are merely used indirectly to produce an intermediate fluid, namely
steam. A much more compact power plant results when the water to steam step is
eliminated and the hot gases themselves are used to drive the turbine. Serious
development of the gas turbine began not long before the Second World War with
shaftpower in mind, but attention was soon transferred to the turbojet engine for
aircraft propulsion. The gas turbine began to compete successfully in other fields
only in the mid nineteen fifties, but since then it has made a progressively greater
impact in an increasing variety of applications.
In order to produce an expansion through a turbine a pressure ratio must be
provided, and the first necessary step in the cycle of a gas turbine plant must
therefore be compression of the working fluid. If after compression the working
fluid were to be expanded directly in the turbine, and there were no losses in
either component, the power developed by the turbine would just equal that
absorbed by the compressor. Thus if the two were coupled together the
combination would do no more than turn itself round. But the power developed
by the turbine can be increased by the addition of energy to raise the temperature
of the working fluid prior to expansion. When the working fluid is air a very
2 INTRODUCTION
suitable means of doing this is by combustion of fuel in the air which has been
compressed. Expansion of the hot working fluid then produces a greater power
output froin the turbine, so that it is able to provide a useful output in addition to
driving the compressor. This represents the gas turbine or internal-combustion
turbine in its simplest fonn. The three main components are a compressor,
combustion chamber and turbine, connected together as shown diagrammatically
in Fig. 1.1.
In practice, losses occur in both the compressor and turbine which increase the
power absorbed by the compressor and decrease the power output of the turbine.
A certain addition to the energy of the working fluid, and hence a certain fuel
supply, will therefore be required before the one component can drive the other.
This fuel produces no useful power, so that the component losses contribute to a
lowering of the efficiency of the machine. Further addition of fuel will result in a
useful power output, although for a given flow of air there is a limit to the rate at
which fuel can be supplied and therefore to the net power output. The maximum
fuel/air ratio that may be used is governed by the working temperature of the
highly stressed turbine blades, which temperature must not be allowed to exceed a
certain critical value. This value depends upon the creep strength of the materials
used in the construction of the turbine and the working life required.
These then are the two main factors affecting the performance of gas turbines:
component efficiencies and turbine working temperature. The higher they can be
made, the better the all-round perfonnance of the plant. It was, in fact, low
efficiencies and poor turbine materials which brought about the failure of a
number of early attempts to construct a gas turbine engine. For example, in 1904
two French ,engineers, Armengaud and Lemale, built a unit which did little more
than tum itself over: the compressor efficiency was probably no more than 60 per
cent and the maximum gas temperature that could be used was about 740 K.
It will be shown in Chapter 2 that the overall efficiency of the gas turbine cycle
depends also upon the pressure ratio of the compressor. The difficulty of
obtaining a sufficently high pressure ratio with an adequate compressor efficiency
was not resolved until the science of aerodynamics could be applied to the
problem. The development of the gas turbine has gone hand in hand with the
development of this science, and that of metallurgy, with the result that it is now
possible to find advanced engines using pressure ratios of up to 35:1, component
efficiencies of 85-90 per cent, and turbine inlet temperatures exceeding 1650 K.
Fuel
Compressor Turbine
Exhaust gas
r-=ll---+- Power output
FIG. 1.1 Simple gas turbine system
INTRODUCTION 3
In the earliest days of the gas turbine, two possible systems of combustion
were proposed: one· at constant pressure, the other at constant volume.
Theoretically, the thermal efficiency of the constant volume cycle is higher than
that of the constant pressure cycle, but the mecha..'lical difficulties are very much
greater. With heat addition at constant volume, valves are necessary to isolate the
combustion chamber from the compressor and turbine. Combustion is therefore
intermittent, which impairs the smooth running of the machine. It is difficult to
design a turbine to operate efficiently under such conditions and, although several
fairly successful attempts were made in Gennany during the period 1908-1930 to
construct gas turbines operating on this system, the development of the constant
volume type has been discontinued. In the constant pressure gas turbine,
combustion is a continuous process in which valves are unnecessary and it was
soon accepted that the constant pressure cycle had the greater possibilities for
future development.
It is important to realize that in the gas turbine the process of compression,
combustion and expansion do not occur in a single component as they do in a
reciprocating engine. They occur in compoJl1ents which are separate in the sense
that they can be designed, tested and developed individually, and these com-
ponents can be linked together to fonn a gas turbine unit in a variety of ways. The
possible number of components is not limited to the three already mentioned.
Other compressors and turbines can be added, with intercoolers between the com-
pressors, and reheat combustion chambers between the turbines. A heat-
exchanger which uses some of the energy in the turbine exhaust gas to preheat the
air entering the combustion chamber may also be introduced. These refinements
may be used to increase the power output and efficiency of the plant at the
expense of added complexity, weight and cost. The way in which these com-
ponents are linked together not only affects the maximum overall thennal effici-
ency, but also the variation of efficiency with power output and of output torque
with rotational speed. One arrangement may be suitable for driving an alternator
under varying load at constant speed, while another may be more suitable for
driving a ship's propeller where the power varies as the cube of the speed.
Apart from variations of the simple cycle obtained by the addition of these
other components, consideration must be given to two systems distinguished by
the use of open and closed cycles. In the much more common open cycle gas
turbine which we have been considering up to this point, fresh atmospheric air is
drawn into the circuit continuously and energy is added by the combustion of fuel
in the working fluid itself. In this case the products of combustion are expanded
through the turbine and exhausted to atmosphere. In the alternative closed cycle
shown in Fig. 1.2 the same working fluid, be it air or some other gas, is repeatedly
circulated through the machine. Clearly in tins type of plant the fuel cannot be
burnt in the working fluid and the necessary energy must be added in a heater or
'gas-boiler' wherein the fuel is burnt in a separate air stream supplied by an
auxiliary fan. The closed cycle is more akin to that of steam turbine plant in that
the combustion gases do not themselves pass through the turbine. In the gas
turbine the 'condenser' takes the fonn of a precooler for cooling of the gas before
)
4
INTRODUCTION
Heater
Precooler
FIG. 1.2 Simple dosed cycle
it re-enters the compressor. Although little used, numerous advantages are
claimed for the closed cycle and these will be put forward in section 1.3.
Finally, various combined steam and gas cycles have been proposed, with the
gas turbine eyillaust supplying energy to the steam boiler. Figure 1.3 shows such a
system. It makes the best use of the comparatively low grade heat by employing a
dual-pressure steam cycle. This is similar to that used in nuclear power stations
fuelled with natural uranium which also operate at a comparatively low
temperature. AJtematively, because of the unused oxygen in the turbine's exhaust
gas, it is possible to burn additional fuel in the steam boiler. This permits use of a
single-pressure steam cycle, but at the expense of the added complexity of a
combustion system in the boiler. With increasing cycle temperatures the exhaust
gas entering the boiler is hot enough to permit the use of a triple-pressure steam
cycle incorporating a stage of reheat. Although the characteristic compactness of
Exhaust
FIG. 1.3 Combined steam and gas cycle
OPEN CYCLE SINGLE-SHAFT A<'ill TWIN-SHAFT ARRANGEMENTS 5
the gas turbine is sacrificed in binary! cycle plant, the efficiency is so much higher
than is attainable with the simple cycle that they are becoming widely used for
large electricity generating stations (see section 1.5).
The gas turbine has proved itself to be an extremely adaptable source of power
and has been used for a wide variety of functions, ranging from electric power
generation and jet propulsion to the supply of compressed air and process heat,
and the rerilainder of this Introduction is intended to emphasize this adaptability. t
We shall commence, however, by discussing the various ways in which the
components can be linked together when the object is the production of shaft
power. In other words, we shall first have in mind gas turbines for electric power
generation, pump drives for gas or liquid pipe lines, and land and sea transport.
The vast majority of land-based gas turbines are in use for the first two of these,
and applications to land and sea transport are still in their infancy, although the
gas turbine is widely used in naval applications.
1,1 Open cycle single-shaft ami twin-shaft arrangements
If the gas turbine is required to operate at a fixed speed and fixed load condition
such as in base-load power generation schemes, the single-shaft arrangement
shown in Fig. 1.1 is the most suitable. Flexibility of operation, i.e. the rapidity
with which the machine can accorrnnodate itself to changes of load and rotational
speed, is unimportant in this application. Indeed the effectively high inertia due to
the drag of the compressor is an advantage because it reduces the danger of
overspeeding in the event of a loss of electrical load. A heat-exchanger might be
added as in Fig. 1.4( a) to improve the thermal efficiency, although for a given size
of plant the power output could be reduced by as much as 10 per cent owing to
frictional pressure loss in the heat-exchanger. As we will see in Chapter 2, a heat-
exchanger is essential for high efficiency when the cycle pressure ratio is low, but
becomes less advantageous as the pressure ratio is increased. Aerodynamic de-

)Ombustion chamber
. ::
Power
Compressor Turbine output
(a) (b)
FIG. 1.4 Single-slJaft open cycle gas turbines with lIeat-exchanger
t Some of the remarks about the 'stability of operation' and 'part-load perfonnance' will be more fully
understood when the rest of the book, and Chapter 8 in particular, have been studied. It is suggested
therefore that the remainder of the hltroduction be given a second reading at that stage.
6
INTRODUCTION
velopments in compressor design have permitted the use of such high pressure
ratios that efficiencies of over 40 per cent can now be achieved with the simple
cycle. The basic heat-exchange cycle is seldom considered for current gas turbine
designs.
Figure 1.4(b) shows a modified form proposed for use when the fuel, e.g.
pulverized coal, is such that. the products of combustion contain constituents
which corrode or erode the turbine blades. It is much less efficient than the
normal cycle because the heat -exchanger, inevitably less than perfect, is
transferring the whole of the energy input instead of merely a small part of it
Such a cycle would be considered only if a supply of 'dirty' fuel was available at
very low cost. A serious effort was made to develop a coal burning gas turbine in
the early nineteen fifties but with little success. More success has been achieved
with residual oil, and provided that the maximum temperature is kept at a
sufficiently low level the straightforward cycle can be used.
When flexibility in operation is of paramount importance, e.g. when driving a
variable speed load such as a pipeline compressor, marine propeller or road
vehicle, the use of a mechanically independent (or free) power turbine is
desirable. In this twin-shaft arrangement, Fig. 1.5, the high-pressure turbine
drives the compressor and the combination acts as a gas generator for the low-
pressure power turbine. Twin-shaft arrangements may be used for electricity
generating units, with the power turbine designed to run at the alternator speed
without the need for an expensive reduction gearbox; these would normally be
derived from jet engines, with the exhaust expanded through a power turbine
rather than the original exhaust nozzle. A significant advantage is that the starter
unit needs only to be sized to tum over the gas generator. The starter may be
electric, a hydraulic motor, an expansion turbine operated from a supply of
pipeline gas or even a steam turbine or diesel. A disadvantage of a separate power
turbine, however, is that a shedding of electrical load can lead to rapid
overspeeding of the turbine, and the control system must be designed to prevent
this.
Variation of power for both single- and twin-shaft units is obtained by
controlling the fuel flow supplied to the combustion chamber. Although they
behave in rather different ways as will be explained in Chapter 8, in both cases the
cycle pressure ratio and maximum temperature decrease as the power is reduced
from the design value with the result that the thermal efficiency deteriorates
considerably at part load.
I ~ Gas generator ~ J Power turbine
FIG. 1.5 Gas turbine witll separate power turbine
OPEN CYCLE SINGLE-SHAFT AND TWIN-SHAFT ARRANGEMENTS 7
The performance of a gas turbine may be improved substantially by reducing
the work of compression and/or increasing the work of expansion. For any given
compressor pressure ratio, the power required per Ulut quantity of working fluid is
directly proportional to the inlet temperature. If therefore the compression process
is carried out in two or more stages with intercooling, the work of compression
will be reduced. Similarly, the turbine output can be increased by dividing the
expansion' into two or more stages, and reheating the gas to the maximum
permissible temperature between the stages. Although the power output is
improved the cost in additional fuel will be heavy unless a heat -exchanger is also
employed. One arrangement of a plant incorporating intercooling, heat-exchange
and reheat is shown in Fig. 1.6. Complex cycles of this type offer the possibility
of varying the power output by controlling the fuel supply to the reheat chamber,
leaving the gas generator operating closer to its optimum conditions.
Complex cycles were proposed in the early days of gas turbines, when they
were necessary to obtain a reasonable thermal efficiency at the low turbine
temperatures and pressure ratios then possible. It can readily be seen, however,
that the inherent simplicity and compactness of the gas turbine have been lost. In
many applications low capital cost and small size are more important than
thermal efficiency (e.g. electrical peaking, with low running hours), and it is
significant that the gas turbine did not staDt to be widely used (apart from aircraft
applications) until higher turbine inlet temperatures and pressure ratios made the
simple cycle economically viable.
The quest for higher efficiency as gas turbines become more widely used in
base-load applications has led to a revival of interest in more complex cycles in
the mid nineteen nineties. One example is the re-introduction of the reheat cycle,
using a very high cycle pressure ratio with no intercooling or heat exchange; this
can give a thermal efficiency of about 36 per cent. The use of reheat also results in
an exhaust gas temperature exceeding 6001
0
C, permitting the use of reheat steam
cycles which can result in a combined cycle efficiency approaching 60 per cent.
Another example is the intercooled regenerativet (ICR) cycle proposed for naval
propulsion, giving both a high thermal efficiency at the design point and excellent
Coolant Reheat chamber
LP compo HP compo HPturb. LP turbo
FIG. 1.6 Complex plant witll intercooling, heat exchange and reheat
t 'Regenerative' merely refers to the nse ofheat-excbange. It does not imply the nse of a 'regenerator',
which is a particnlar type of heat-exchanger wherein a rotating matrix is alternatively heated and
cooled by the gas streams exchanging heat.
8
INTRODUCTION
, . . " .
efficiency at part load, a very important feature for ships which generally cruise at
much lower power levels thail the design value.
1.2 Multi-spool arrangemelllts
To obtain a high thermal efficiency without using a heat-exchanger, a high pres-
sure ratio is essential. A certain difficulty then arises which follows from the
nature of the compression process.
Because of the high air mass flow rates involved, non-positive displacement
compressors are always used in gas turbines. Although the multi-stage centrifugal
compressor is capable of producing a high pressure ratio for moderate powers, its
efficiency is appreciably lower than that of the axial flow compressor. For this
reason the axial compressor is normally preferred, particularly for large units.
Unfortunately this type of compressor is more prone to instability when operating
at conditions widely removed from its design operating point. When such a
compressor operates at rotational speeds well below the design value, the air
density in the last few stages is much too low, the axial flow velocity becomes
excessive, and the blades stalL The unstable region, manifested by violent
aerodynamic vibration, is likely to be encountered when a gas turbine is started
up or operated at low power.
The problem is particularly sevel1e if an attempt is made to obtain a pressure
ratio of more than about 8:1 in one compressor. One way of overcoming this
difficulty is to divide the compressor into two or more sections. In this context
division means mechanical separation, permitting each section to run at a
different rotational speed, unlike the intercooled compressor shown in Fig. 1.6.
When the compressors are mechanically independent, each will require its own
turbine, a· suitable arrangement being shown in Fig. 1.7. The low-pressure
compressor is driven by the low-pressure turbine and the high-pressure
compressor by the high-pressure turbine. Power is normally taken either from
the low-pressure turbine shaft, or from an additional free power turbine. The
configuration shown in Fig. 1.7 is usually referred to as a twin-spool engine. It
should be noted that although the two spools are mechanically independent,
their speeds are related aerodynamically and this will be discussed further in
Chapter 9.
The twin-spool layout was primarily developed for the aircraft engines
discussed in section lA, but there are many examples of shaft power derivatives
LP HP HP LP
FIG. 1.7 Twin-spool engine
CLOSED CYCLES . 9
of these; a free power turbine is common, but it is also possible to use the low-
pressure turbine to drive both the low-pressure compressor and the driven load. In
some cases, especially with engines of small air flow, the high-pressure
compressor is of the centrifugal type; this is because at the high pressures
involved the volume flow rate is low and the blading required for an axial
compressor would be too small for good efficiency. Twin-spool units were first
introduced at a pressure ratio of about 10: 1 and are suitable for cycle pressure
ratios of at least 35:1. Triple-spool arrangements can also be used in large
turbofan engines, where there is a requirement for both very high pressure ratio
and a low rotational speed for the fan.
As an alternative to multiple spools, a high pressure ratio can be safely
employed with a single compressor if several stages of variable stator blades are
used. This approach was pioneered by General Electric and pressure ratios in
excess of 20: 1 have been obtained in this manner. It may also be necessary to use
blow-off valves at intermediate locations in the compressor to handle the serious
flow mismatch occurring during start up. The single-spool variable geometry
compressor is almost universally used in large electric power generation units.
Advanced technology engines usually employ combinations of multiple-
spools, blow-off valves and variable stators. This is particularly true for the high
bypass ratio turbofans discussed in section 104.
1.3 Closed cycles
Outstanding among the many advantages claimed for the closed cycle is the
possibility of using a high pressure (and hence a high gas density) throughout the
cycle, which would result in a reduced size of turbomachinery for a given output
and enable the power output to be altered by a change of pressure level in the
circuit. This form of control means that a wide range of load can be accommo-
dated without alteration of the maximum cycle temperature and hence with little
variation of overall efficiency. The chief disadvantage of the closed cycle is the
need for an external heating system, which involves the use of an auxiliary cycle
and introduces a temperature difference between the combustion gases and the
working fluid. The allowable working temperature of the surfaces in the heater
will therefore impose an upper limit on the maximum temperature of the main
cycle. A typical arrangement of a closed cycle gas turbine is shown in Fig. 1.8.
The cycle includes a water cooled pre-cooler for the main cycle fluid between the
heat-exchanger and compressor. In this particular arr&llgement the gas heater
forms part of the cycle of an auxiliary gas turbine set, and power is controlled by
means of a blow-off valve and an auxiliary supply of compressed gas as shewn.
Besides the advantages of a smaller ,cfjmpressor and turbine, and efficient
control, the closed cycle also avoids erosion of the turbine blades and other
detrimental effects due to the products of combustion. Also, the need for filtration
of the incoming air, which is a severe problem in the use of open cycle units
operating in contaminated atmospheres, is eliminated. The high density of the
10 INTRODUCTION
__ ?-_D, CL ___ _
r ~ e l :
L __ -
V
- lAuxiliary
I cycle
I
FIG. 1.8 Simple closed-cycle gas turbille
working fluid improves heat transfer, so that more effective heat-exchange is
possible. Finally, the closed circuit opens up the field for the use of gases other
than air having more desirable thermal properties. As will be seen in the next
chapter, the marked difference in the values of the specific heats for air and a
monatomic gas such as helium does not affect the efficiency as much as might be
supposed. But higher fluid velocities can be used with helium and optimum cycle
pressure ratios are lower, so that in spite of the lower density the turbomachinery
may not be much larger. On the credit side, the be1ter heat transfer characteristics
of helium mean that the size of the heat-exchanger and precooler can be about
half that of units designed for use with air. The capital cost of the plant should
therefore be less when helium is the working fluid.
At the time of writing only a small number of closed-cycle plants have been
built, mostly by Escher-Wyss, and few are still in service. They were within the
2-20 MW power range. All used air as the working fluid, with a variety of fuels
such as coal, natural gas, blast furnace gas, and oil. A pilot plant of 25 MW using
helium was built in Germany, and it was thought that with this working fluid large
sets of up to 250 MW would be feasible. They might have been required for use
in nuclear power plant, if efforts to develop a reactor capable of operating at a
sufficiently high core temperature had been successful. Considerable advantage
accrues when the working fluid of the power cycle can be passed directly through
the reactor core, because the reactor coolant circulating pumps are no longer
required and the unwanted temperature drop associated with an intermediate fluid
(e.g. CO2 temperature-steam temperature) is eliminated. Helium is a particularly
suitable working fluid in this application because it absorbs neutrons only weakly
(i.e. it has a low neutron absorption cross-section). Attempts to develop the high
temperature reactor (HTR) have been discontinued, however, and conventional
nuclear reactors operate at much too Iowa temperature to be a possible source of
heat for a gas turbine. It follows that gas turbines are unlikely to be used in any
nuclear power plant in the foreseeable future.
CLOSED CYCLES 11
12 INTRODUCTION
A variety of small closed-cycle gas turbines kWelectrical output)
have been· considered for use in both aerospace and underwater applications.
Possible heat sources include a radioactive isotope such as plutonium 238,
combustion of hydrogen, and solar radiation. To date, none have been built.
1.4 Aircraft propulsion
Without any doubt the greatest impaet of the gas turbine has been in the field of
aircraft propulsion. The most important landmark in this development was the
first experimental Whittle engine in 1937. Since then the gas turbine has com-
pletely supplanted the reciprocating engine, for all but light aircraft, because of its
much higher power/weight ratio. The cycle for the simple turbojet engine is
virtually that shown in Fig. 1.1 except that the turbine is designed to produce just
sufficient power to chive the compressor. The exhaust gas is then expanded to
ahnospheric pressure in a propelling nozzle to produce a high velocity jet. Figure
1.9 shows a sectional view of a Rolls-Royce Olympus jet engine. This engine is
of historical importance, being the first twin-spool engine in production; early
versions were used in the Vulcan bomber and the advanced derivative shown is
used to power the Concorde supersonic transport. (The Olympus has also been
used widely as a gas generator to drive a power turbine, both for electricity
generation and marine propulsion.)
For low-speed aircraft a combination of propeller and exhaust jet provides the
best propulsive efficiency. Figure 1.1 0 shows a single-shaft turboprop engine
(Rolls-Royce Dart) chosen to illustrate the use of a centrifugal compressor (two
stage) and 'can' type combustion chambers. It is notable that this engine entered
airline service around 1953 at a power of about 800 kW, and was still in
production in 1985 with the latest version producing about 2500 kW with an
FIG. 1.111 Single-shaft turboprop engine Iby courtesy IIf Rolls-Royce]
AIRCRAFT PROPULSION 13
improvement in specific fuel consumption of about 20 per cent. Turboprops are
also designed with a· free turbine driving the propeller or propeller plus LP
compressor. The Pratt and Whitney Canada PT-6, shown in Fig. 1.11, uses a free
turbine; the use of a combined compressor and a reverse-flow
aIUlular combustor can also be seen. This engine is available in versions ranging
from 450 to 1200 kW for aircraft ranging in size from single-engined trainers to
four-engined STOL transport aircraft. Another variant using a free turbine is the
turboshaft engine for use in helicopters; in this case the power turbine drives the
helicopter and tail rotors through a complex gearbox and frequently two engines
are coupled to a single rotor.
At high subsonic speeds a propulsive jet of smaller mass flow but higher
velocity is required. This was originally provided by the turbojet engine, but these
have now largely been superseded by turbofan (or bypass) engines in which pa,r(
of the air delivered by an LP compressor or fan by-passes the core of the engine
(HP compressor, combustion chamber and turbines) to form an annnlar
propUlsive jet of cooler air surrounding the hot jet. This results in a jet of lower
mean velocity which provides not only a better propulsive efficiency but also
significantly reduces exhaust noise. Figure 1.12(a) is an example of a small
turbofan engine, the Pratt and Whitney Canada IT-15D. This is an extremely
simple mechanical design giving good performance, intended for small business
aircraft where capital cost is critical. A twin-spool arTangement is used, again
with a centrifugal HP compressor arId a reverse flow annular combustor. The
reverse flow combustor is well suited for use with the centrifugal compressor,
where the flow must be diffused from a very high tangential velocity to a low
axial velocity at enhy to the combustor, and this configuration is widely used.
Figure 1.l2(b) shows an advanced turbofan designed for use with large civil
aircraft, the V2500 engine designed by a consortium of five nations. In this
application the fuel consumption is of paramount importance, requiring the use of
both high bypass ratio and high pressure ratio. It can be seen that all of the
turbomachinery is of the axial type and a straight-through annular combustion
chamber is used.
Heat-exchangers have as yet found no place in aircraft engines for reasons of
bulk and weight, although they remain feasible for turboprop engines. This is
because, with much of the net power output delivered to the propeller, the
velocity of the gas leaving the turbine is relatively low and the frictional pressure
loss need not be prohibitively high in a heat-exchanger of acceptable size. Around
1965 Allison developed a regenerative turboprop for the U.S. Navy, the object
being to obtain an engine of exceptionally low specific fuel consumption for use
on long endurance anti-submarine patrols. In this kind of application it is the total
engine plus fuel weight which is critical, and it was thought that the extra weight
of the heat-exchanger would be more than compensated by the low fuel
consumption. It was proposed that the heat-exchanger should be by-passed on
take-off to give maximum power. The engine did not go into production, but it is
not impossible that regenerative units will appear in the future, perhaps in the
fonn of turboshaft engines for long endurance helicopters.
Exhaust outlet
Free power
turbine
Diffuser
Combustion
chamber
FIG. 1.11 Turboprop engine (by courtesy of Pratt and Whitney Canada)
3-stage axial
flow
Compressor
turbine
Air inlet
Centrifugal
compressor
Outer
flow

Fan
FIG. 1.12(a) Sman turbofan engine (by courtesey of Pratt and Whitney Canada)
I=xhaust
---I-
duCI


13



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13
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Ul
16 INTRODUCTION INDUSTRIAL APPLICATIONS 17
1,5 Industrial! applications
Sometimes in this book we shall find it necessary to use the distinguishing
tenns 'aircraft gas turbine' and 'industrial gas turbine'. The first tenn is self-
explanatory, while the second is intended to include all gas turbines not included
in the first category. This broad distinction has to be made for three main reasons.
Firstly, the 'life required of an industrial plant is of the order of 100 000 hours
without major overhaul, whereas this is not expected of an aircraft gas turbine.
Secondly, limitation of the size and weight of an aircraft power plant is much
more inlportant than in the case of most other applications of the gas turbine.
Thirdly, the aircraft power plant can make use of the kinetic energy of the gases
leaving the turbine, whereas it is wasted in other types and consequently must be
kept as low as possible. These three differences in the requirements can have a
considerable effect on design and, in spite of the fact that the fundamental theory
applies to both categories, it will be necessary to make the distinction from time
to time. Turbomachinery of gas turbines designed specifically for industrial pur-
poses tends to look more like that of traditional steam turbines in mechanical
appearance than the familiar lightweight constructions used in aircraft practice.
Figure 1.13 shows the rugged construction employed in the Ruston Tornado
designed for long life and to operate on either liquid or gaseous fuel; a separate
power turbine is used and both variable and fixed speed loads can be accom-
modated. Figure 1.14 shows a large single-shaft machine, the Siemens V94,
designed specifically for driving a constant speed generator. This machine is
capable of around 150 Mw, and uses two large 'off-board' combustors.
When gas turbines were originally proposed for industrial applications, lmit
sizes tended to be 10 M\V or less and, even with heat exchangers, the cycle
efficiency was only about 28--29 per cent. The availability of fully developed
aircraft engines offered the attractive possibility of higher powers; the fact that a
large part of the expensive research and development cost was borne by a military
budget rather than an industrial user gave a significant. advantage to the
manufacturers of aircraft engines. The early aero-derivative engines, produced by
substituting a power turbine for the exhaust nozzle, produced about 1:5 MW with
a cycle efficiency of some 25 per cent. Modifications required included
strengthening of the bearings, changes to the combustion system to enable it to
burn natural gas or diesel fuel, the addition of a power turbine and a de-rating of
the engine to give it a longer life; in some cases a reduction gearbox was required
to match the power turbine speed to that of the driven load, e.g. a marine
propeller. For other types ofload, such as alternators or pipeline compressors, the
power turbine could be designed to drive the load directly. The Olympus, for
example, had a single-stage power turbine for naval applications resulting in a
very compact and lightweight design. For electric power generation a larger
diameter two- or three-stage power turbine running at 3000 or 3600 rev/min was
directly connected to the generator, requiring an increased length of ducting
between the two turbines to allow for the change in diameter. Figure 1.15 shows a
typical installation for a small power station using a single aero-delivative gas
18 INTRODUCTION INDUSTRIAL APPLICATIONS
FIG. 1.14 Large single-shaft gas turbine IClIllIlrtesy Siemensj
Compressor
cleaning
Silenced exhaust
Controls Coolers
FIG. 1.15 Compact generatillg set Icourtesy Rolls-Royce)
19
AC generator
20 INTRODUCTION
turbine; it should be noted that the air intake is well above ground level, to
prevent the ingestion of debris into the engine. The aircraft and industrial versions
of the Rolls-Royce Trent are shown in Fig. 1.16. The Trent is a large three-spool
turbofan with the single-stage fan driven by a five-stage low-pressure turbine. The
industrial version, designed to drive a generator, replaces the fan with a two-stage
compressor of similar p r e ~ s u r e ratio but much lower flow; as a result, the low-
pressure turbine can provide a large excess of power which can be used to drive
the generator. The low-pressure rotor speed of the aircraft engine is restricted by
the tip speed of the fan to about 3600 rpm; this pemlits the shaft of the industrial
version to be directly connected to a 60 Hz generator, avoiding the need for a
gearbox. The industrial version of the Trent is capable of 50 MW at a thermal
efficiency of 42 per cent, resulting from the high pressure ratio and turbine inlet
temperature. Figure 1.16 also shows the major changes in the design of the
combustion system; the aircraft version uses a conventional fully annular
combustor while the industria! engine uses separate radial cans. This radical
change is caused by the need for low emissions of oxides of nitrogen, which will
be dealt with further in Chapter 6.
The widest applications of the aero-derivative gas turbine have been in
pumping sets for gas and oil transnlission pipelines, electricity generation and
naval propulsion. In the case of natural gas pipelines, the turbines use the fluid
being pumped as fuel and a typical pipeline right consume 7-10 per cent of the
throughput for compression purposes. In recent years the value of gas has
increased dramatically and this has led to a demand for high-efficiency pumping
units. A major pipeline nlight have as much as 1500 MW of installed power and
the fuel bills are comparable to those of a mediunl-sized airline. Pumping stations
may be about 100 km apart and the gas turbines used range in power from 5 to
25 rvrw. Many compressor stations are in remote locations and aircraft derivative
units of 15 .to 25 MW are widely used. Other operators may prefer the use of
industrial gas turbines and in rece:nt years a number of heat-exchangers have been
retrofitted to simple cycle units. With oil pipelines the oil is often not suitable for
burning in a gas turbine without expensive fuel treatment and it becomes
necessary to bring a suitable liquid fuel in by road.
The use of gas turbines for electrical power generation has changed
dramatically in recent years. In the nineteen seventies, gas turbines (particularly
in Great Britain and North America) were primalily used for peaking and
emergency applications; aero-detivative units with a heavy-duty power turbine
were widely used. One of the outstanding advantages of this type was its ability to
produce full power from cold in under two minutes, although this capability
should be used only for emergencies because thermal shock will greatly reduce
the time between overhauls. In the nlid nineteen sixties, a major blackout of the
eastern seaboard of the USA resulted in investment in gas turbines capable of a
'black' start, i.e. completely independent of the main electricity supply. In Great
Britain, over 3000 MW of emergency and peak-load plant based on the Rolls-
Royce Avon and Olympus engines was installed; these formed a key part of the
overall electricity generating system, but only ran for a very small number of
INDUSTRIAL APPLICATIONS
21
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22
INTRODUCTION
hours. Similar plant were built in large numbers in North America using the Pratt
and Whitney FT-4. The aero-derivative units had a maximum rating of about
35 MW: their efficiency was about 28 percent and they burned expensive fuel so
they were not considered for applications involving long running hours.
In marked contrast, countries like Saudi Arabia, with a very rapidly expanding
electrical system and abunqant cheap fuel, used heavy-duty gas turbines for base-
load duty; a particular advantage of the gas turbine in desert conditions is its
freedom from any requirement for cooling water. Initially, the ratings of the aero-
derivative and heavy-duty units were similar, but as cycle conditions improved
over the years, the designers of industrial gas turbines were able to scale up their
designs to give much higher power. The principal manufacturers are ABB,
General Electric, Siemens and Westinghouse, all of whom design single-shaft
engines which are capable of delivering over 200 MW per unit; the upper limit is
fixed by considerations such as the size of disc forging and the maximum width to
permit transport by rail. Unlike steam turbines, gas turbines are not often erected
on site but are delivered as complete packages ready to run. Single-shaft units
running at 3000 and 3600 rpm respectively can drive 50 or 60 Hz generators
directly without the need for an expensive gearbox. Compressor designs suitable
for operation at these speeds result in 60 Hz machines of around 150 MW and
50 Hz machines of around 225 MW, with the power largely determined by the air
flow· North America is standardized on 60 Hz, while Europe and much of Asia
o p e r ~ t e s at 50 Hz. Smaller machines may be designed to run at around 5-
6000 rpm with ratings of about 50-60 MW, with gearboxes capable of either
3000 or 3600 rpm output speeds to meet market requirements. Many heavy-duty
units have run well in excess of 150000 hours and a substantial number have
exceeded 200 000 hours.
Another major market for electricity generation is the provision of power for
off-shore platforms, where gas turbines are used to provide base-load power.
Many Solar and Ruston units of 1-5 MW have been used, but for larger powers
aero-derivatives such as the Rolls-Royce RB-211 and General Electric LM 2500
have been installed at ratings of 20-25 MW; a big platform may require as much
as 125 MW and both surface area and volunle are at a premium. The installed
weight is also of prime importance because of cranage requirements, and
considerable savings accrue if the rig's own cranes can handle the complete
machinery package. The aero-derivative dominates this market because of its
compact nature.
The availability of gas turbines with an output of 100-200 MW has made
large combined cycle plant a major factor in thermal power generation. Japan,
because of its total dependence on imported fuel, was the first large-scale user of
combined cycles, building several 2000 MW stations burning imported liquefied
natural gas (LNG). A typical installation may be made up of 'blocks' consisting
of two gas turbines with their own waste heat boilers and a single steam turbine;
in general, using an unfired boiler the steam turbine power is about half that ofthe
gas turbine. Thus, a single block of two 200 MW gas turbines and a 200 MW
steam turbine provides 600 MW; a complete station may use three or four blocks.
INDUSTRIAL APPLICATIONS 23
At the time of writing several 2000 MW plant have been built, yielding effici-
encies of about 55 per cent, and the largest on order has a rating of 2800 l\1W.
Privatization of the electricity supply in Great Britain led to the installation of
a large number of combined cycle plant of 225-1850 MW burning natural gas. In
the longer term it is quite feasible that lmits burning natural gas could be
converted to gas obtained from the gasification of coal.
Gas turbme power stations are remarkably compact. Figure 1.17(a) shows the
comparison between the size of a 1950s era steam turbine station of 128 MW and
a peak-load gas turbine plant of 160 MW shown ringed; the latter used 8
Olympus 20 MW units. The steam plant required three cooling towers to cope
with the heat rejected from the condensers. This plant has now been
decorrunissioned and replaced by a 700 MW combined cycle base-load plant,
shown in Fig. 1.17(b). The new plant consists of three blocks, each comprising a
Siemens V.94 gas turbine of 150 MW and a waste heat boiler, and a single steam
turbine of250 MW. An air-cooled condenser is used in place of the three cooling
towers, because of restrictions on the use of river cooling water; the condenser is
the large rectangular structure to the left of the picture, and can be seen to be
much less visually intrusive than the cooling towers. A small performance penalty
is paid, because the condenser temperature (and hence back pressure on the steam
turbine) is higher than could be obtained with a river-cooled condenser. The
station, however, has a thermal efficiency of 51 per cent, which is much higher
than that of a conventional steam turbine plant.
Gas turbines were, used successfully in a few high-speed container ships, but
the rapid increase in fuel prices in the mid seventies led to these ships being re-
engined with diesels; the converted ships suffered a major loss in both speed and
cargo capacity, but high speed could no longer be justified. The picture with
respect to naval operations is quite different, however, and many navies (e.g.
Britain, U.S.A., Canada, Netherlands) have now accumulated a large amount of
gas turbine experience. A gas turbine was first used in a Motor Gun Boat in 1947,
and aero-type engines (Rolls-Royce Proteus) were first used in fast patrol boats in
1958. The potential of aero-derivative engines for main propulsion of warships
was soon realized and the Canadian DDH-280 class were the first all gas turbine
powered warships in the western world, using a combination of Pratt and Whitney
FT-4's for 'boost' power and FT-12's for 'cruise'. The Royal Navy selected the
Olympus as boost engine and the Rolls-Royce Tyne for cruise duties; this
machinery arrangement was also selected by the Royal Netherlands Navy. The
Olympus and Tyne are the only naval gas turbines to have been proved in battle,
operating with great success in the Falklands War. The U.S. Navy adopted the GE
LM 2500, derived from the TF39 advanced turbofan, and this engine has been
widely used around the world. With the increasing electrical needs of warships,
and the absence of steam for use in turbo generators, gas turbine driven generators
also offer a very compact source of electricity.
A major disadvantage of the gas turbine in naval use is its poor specific fuel
consumption at part load. If we consider a naval vessel having a maximum speed
of, say, 36 lmots and a cruise speed of 1I8 Imots, with the power required
24
INTRODUCTION
FIG. 1.17(a) Comparative size olfsteam and gas rurbin€ power statiolls [collrtesy
Rolls-Royce]
FIG. 'l.17(IJ) Combined cycle plallt witil air-cooled condenser Icolllrtesy Siemensl
INDUSTRIAL APPLICATIONS 25
proportional to the cube of the speed, the cruise power will be only one eighth of
the maximum power; indeed, much time would be spent at speeds less than 18
knots. To overcome this problem, combined power plant consisting of gas
tnrbines in conjunction with steam turbines, diesel engines and other gas tnrbines
have been used. These go by names such as COSAG, CODOG, COGOG,
COGAG, etc. CO stands for 'combined'; S, D, G for 'steam', 'diesel' and 'gas
tnrbine'; the final A or 0 for 'and', 'or'. The 'or' requires explanation. The
earliest arrangement used by the Royal Navy was COSAG where a ship's shaft
was driven by both steam and gas tnrbines; the gearing was such that the ship
could use either steam or gas turbine or both. The gas tnrbines were originally
intended only for boost purposes, or for rapid starts, but in practice proved so
versatile and popular with the operators that they became used for much longer
periods. Another alternative is to combine a boost gas tnrbine with a cruise diesel;
in this case the diesel power is so small relative to the gas turbine power that there
is little advantage in adding the powers. The vessel therefore operates in either the
diesel or gas tnrbine mode, i.e. as CODOG. For naval use the diesel engine has
the advantage of very good cruise fuel consumption, with the disadvantages of
large bulk for the power available and a loud undelwater noise level. The
COGOG arrangement, with a small cruise gas turbine and a large boost gas
tnrbine, has been widely used; the aim is to keep any gas tnrbine running near full
power where its efficiency is a maximum. The small (4-5 Ivrw) cruise gas
turbines, however, are not competitive with diesels regarding specific fuel
consumption and there appears to be a trend away from COGOG to CODOG.
The COGAG arrangement, using gas tnrbines of the same size, has been used by
the u.s. Navy, using four LM 2500 engines in large destroyers; this arrangement
has also been used in the Royal Navy with four Olympus engines in the Invincible
class of aircraft carrier.
For many years the gas tnrbine has been considered for road and rail transport,
without making any real impact. Union Pacific successfully operated large freight
trains with gas tnrbine power Iii-om about 1955 for 15-20 years, but these have
now given way to diesels. Several high-speed passenger trains were built USLllg
helicopter-type gas turbines, but only with limited success; the most successful
were those built by the French, but the high-speed TGV (Train a Grand Vitesse)
has electric traction. A considerable amount of work was done on gas tnrbines for
long haul trucks, with engines of 200-300 leW being developed. AU used the
low-pressure ratio cycle with a centrifugal compressor, free tnrbine and rotary
heat -exchanger. Similar efforts were expended on gas tnrbines for cars and,
although these continue under U.S. govenllllent sponsorship, the automobile gas
turbine still appears to be far over the horizon and may never appear. There is no
doubt that the cost of gas turbines could be significantly reduced if they were built
in numbers approaching those of piston engines. The major problem is still that of
part-load fuel consumption. The one breakthough achieved by the gas turbine is
its choice for propUlsion of the Ml tank built for the U.S. Army, but it has still not
been proved that the gas tnrbine is superior to the diesel in this application. Ml
tanlcs obtained considerable battlefield experience in deseli conditions in the Gulf
26 INTRODUCTION
War, and appear to have been quite successful. The fact remains, however, that no
other nation has selected gas turbine propulsion for the most recent generation of
tanles.
Another concept of growing importance is that of the combined production of
heat and power, in what are variously Imown as cogeneration or CHP plant. The
gas turbine drives a generi\tor and the exhaust gases are used as a source of low
grade heat. Heat at a relatively low temperature is required for heating buildings
and operating air-conditioning systems. It is also required in many process
industries-paper drying for example. Chemical industries often need large
quantities of hot gas containing a high propOltion of free oxygen at a pressure
sufficient to overcome the pressure loss in chemical reactors. The temperature
limitation in the gas turbine cycle means that high air/fuel ratios must be
employed resulting in a large proportion of unused oxygen in the exhaust. The
exhaust from a gas turbine is therefore often suitable. The unit can be designed to
meet the hot gas requirement, with or without shaft power for other purposes, and
sometimes to burn a fuel which is a by-product of the chemical process. Figure
1.18 illustrates an application of a cogeneration plant using Ruston gas turbines.
It provides the whole of tbe electricity, process steam, heating steam, and chilled
water required for a factory. The use of eight gas turbines feeding four auxiliary-
fired waste-heat boilers enables the changing power and heat demands during the
day to be met by running the necessary number of units at substantially full power
and therefore at peak efficiency.
1.6 Environmental issues
The first major application of the gas turbine was for jet propulsion for military
aircraft, towards the end of World War n. The jet engine led to much higher
aircraft speeds, and this was sufficiently important that serious deficiencies in fuel
consumption and engine life could be ignored; the jet exhaust was noisy but this
mattered little in military applications. When jet engines were considered for use
in civil transport aircraft both fuel consumption and longer overhaul lives became
of great importance, although noise was not then an issue. The appearance of
significant numbers of jet propelled aircraft all civil airpOlts in the late nineteen
fifties rapidly resulted in noise becoming a problem that would severely inhibit
the growth of air transport. The need to reduce engine noise was originally met by
adding silencers, which were not very effective and caused serious loss of per-
formance; it was clear that engine noise had to be properly understood and that
engine design had to cater for noise reduction from the start of an aircraft design
project. Mathematicians deduced that jet noise was proportional to (jet velocity)8,
so the basic requirement was immediately recognized to be provision of the
necessary thrust at a reduced jet velocity with a resultant increase in airflow. This
was precisely what the turbofan did to obtain high propulsive efficiency. It was
fortunate indeed that the search for higher effieiency also resulted in lower noise.
The bypass ratio on early turbofans was restricted by both lack of knowledge of
ENVIRONMENTAL ISSUES
27
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
t
I
I
I
I
I
I
I
I
I
I
I
I
:D:D
I I
rOrO
I I
I I
I I I I
_ 1. ____ 1. ____ _ _ L ____ ..l ____ _
Exhaust with
heat aUXiliary
recovery firing
boiler
i i
,..--stacki -,-_«_ __ 1- ____ . __ L __ _
I
- I Steam at 12 bar
I----l
ill
i L__ I
t I
i '1_>- _______ -.-1 I
I I P.R.V. I
t-- l ;----...... ---, 1
t
- I : 1
I I Absorption I --1
.----"] I Steam at 1 bar water
I - I chiller I 1 I
1 : -.-- : I
I I --------: I I
i I
I I I
I L ___ -------J ! I
I
Chilled water return I
Electrical Steam for Steam for
power process heating Factory services
FIG. 1.18 Ruston cogeneration plant for II flll:tOry
28 INTRODUCTION
three-dimensional flow effects in the longer fan blades and installation problems,
particularly for engines buried in the wing; it was the advent of the pod mounted
engine that pelmitted the move Ito steadily increasing the bypass ratio. As bypass
ratios increased, however, it was soon found that the high fluid velocities at the
fan blade tips was another source of noise that was particularly troublesome
during the landing approach, with the noise spread over a very wide area. This
problem was attacked by the U S I ~ of sound absorbing materials in the intake duct
and careful choice of spacing between rotor and stator blades. Aircraft noise
reduction has required a vast amount of research and capital expenditure, but it
can safely be said that current and future designs of aero engine will result in
airport noise being greatly reduced. The exhaust of industrial gas turbines leaves
at a low velocity and is discharged throngh a chinmey, so the main source of noise
occurring in the jet engine is absent. Owing to their compact nature and ease of
installation, however, gas turbines may be located close to industrial areas and a
low noise level is commonly specified; the requirement may be met by acoustic
treatment of the intake system :md baffles in the exhaust dueting.
When gas turbines were first considered for non-aircraft applications, the
combination of rotating machinery with steady flow combustion and large
amounts of excess air seemed tG offer a relatively clean-burning power plant. In
the late nineteen sixties it was discovered that the notorious 'smog' occurring in
Los Angeles was caused by a photo-chemical reaction between sunlight and
oxides of nitrogen produced by automobile exhausts. This led to major research
programs to reduce oxides of nitrogen (referred to as NOx) and unburned
hydrocarbons CURC) for piston engines. Vllhen gas turbines began to enter the
market in applications such as pipelines, electricity generation and mechanical
drives, they soon became subject to regulations limiting emissions and these have
become steadily more sulligent. The same is now true of emissions from aircraft
engines. It will be apparent from section 6.7, where methods of reducing
emissions are described, that different approaches are used for indusu-ial plant and
aircraft engines because of the variation in operational requirements. Oxides of
nitrogen occur at very hi.gh combustion temperatures, and also increase wi.th
combustion inlet temperature; in other words, the very factors needed for high
efficiency cause increased NOx formation. In the early nineteen nineties the
design of low NOx combustion systems was one of the key drivers in producing
competitive gas turbin.es. The easiest approach for industrial plant was to use
water or steam injection to reduce the peak combustion temperature, but this
introduced a host of other problems and costs related to engine durability and
provision of treated water. Emphasis was placed on the development of dry low
NOx systems, and the principal manufacturers have developed various solutions
which are due to enter service in the mid nineteen nineties. Owing to the large
amount of excess air used in combustion the production ofUBC was less critical,
but still subject to stringent resuictions. The main combustion product of any
hydrocarbon fuel is carbon dioxide, which is believed to cono-ibute significantly
to global warming due to its 'greenhouse effect'. CO2 emissions can be reduced
only by improving engine efficiencies so that less fuel is bumed, or by developing
SOME FUTURE POSSIBILITIES 29
sources of power that do not entail the combustion of fossil fuels. To encourage
this some countries, notably in Scandinavia, have introduced a CO2 tax.
With the successful taclding of the noise and emission problems, the gas
turbine once again offers the same promise of an enviromnentally benign power
plant. Figure 1.19 shows an installation of a 70 rYfW combined cycle base-load
plant located between two hospitals and innnediately adjacent to a prime
residential area.
The increasing scarcity and therefore cost of high-grade fossil fuels will neces-
sitate the wider use of poor-quality coal and heavy fuel oil with a high sulphur
content. Such fuel can be bumt in steam power stations, but only with expensive
boiler maintenance and the costly cleaning of stack gases to meet ever more
stringent anti-pollution regulations. Two other quite distinct approaches exist,
both involving the use of gas turbines. The first makes use of the idea ofjiuidized
bed combustion, and the second involves the transformation of the low-quality
solid or liquid fuel into a clean gaseous fuel.
FIG. 1.19 70 MW combined cycle plant [collrtesy Stewart and Stevellisollil
30 INTRODUCTION
A fluidized bed combustor consists essentially of a refractory-brick lined
cylinder containing sand-sized refractory particles kept in suspension by an
upward flow of air. When used in conjunction with a gas turbine, the required air
can be bled from the compressor. If coal is being burnt, the oxides of sulphur
fonned are trapped in the ash, and if oil is the fuel they can be trapped by particles
of limestone or dolomite in the bed. Figure 1.20 shows one possible scheme. It
makes use of the fact that heat is transferred between a fluidized bed and any solid
surface immersed in it with very high heat transfer coefficients. In this scheme,
most of the compressor air is heated in a tubular heat-exchanger in the bed, and
only the small amount of air bled for fluidization need be cleaned of dust in
cyclone separators before being passed to the turbine. Corrosion and erosion
problems are holding up development, but if they can be overcome the fluidized
bed combustor opens up the possibility of buming coal mined by remote-
controUed methods, or even the material in colliery spoil heaps. In the latter case
not only would useful power be developed from hitherto unusable fuel, but
valuable land would be reclaimed.
A prototype combined cycle plant using fluidized bed combustion was brought
into service in Sweden in 1991. It was built to provide both power and heat, with
a capacity of 135 MW of power and a district heating load of 224 MW Two gas
turbines generated 34 MW of power, the balance being provided by the steam
turbine. This design did not use a heat-exchanger in the bed as shown in Fig.
1.20, and all the compressor air passed through the combustion system before
being cleaned in cyclone separators prior to entry to the turbine.
Before leaving the subject of fluidized bed combustion it is worth mentioning
another possible application: the incineration of municipal waste. The waste is
shredded and useful recoverable materials (steel, tin, aluminium etc.) are
Cyclone separators
FIG. 1.20 Fluidized bed combustion
SOME FUTURE POSSIBILITIES 31
separated out using magnetic and flotation techniques, and vibrating screens. The
remainder, about 85 per cent, is burnt in a fluidized bed combustor. Combustion
. of the waste maintains the temperature somewhere between 700°C and 800 °C,
which is high enough to consume the material without causing the bed to
agglomerate. Supplementary oil burners an: provided for starting the unit. The hot
gases pass through several stages of cleaning to prevent erosion of the turbine and
to satisfy air pollution standards. Income from the electricity generated and sale
of recyclable materials is expected to reduce substantially the cost of waste
disposal compared wit.h the conventional land-fill method. It must be emphasized
that the maximum temperature that can be used in a fluidized bed is unlikely to be
very high so that the gas turbine efficiency will be low. Fluidized bed combustors
are liIcely to be used only for burning cheap or otherwise unusable fuels.
The second approach to the problem of using poor-quality coal or heavy oil is
its transfonnation into a clean gaseous fhe!' Figure 1.21 illustrates a possible
scheme in which a gasification plant is integrated with a combined cycle. The
gasification process removes troublesome vanadium and sodium impurities which
cause corrosion in the turbine, and also the sulphur which produces polluting
oxides in the stack gases. Compressed air required for the process is bled from the
gas turbine compressor. To overcome the pressure loss in the gasification plant the
pressure is boosted in a separate compressor driven by a steam turbine. This
would use some of the steam from the waste heat boiler, the major fraction
supplying the power steam turbine. The gas produced by such a plant would have
a very low calorific value-perhaps only 5000 kJ 1m
3
compared with about
39000 kJ 1m
3
for natural gas. This is because of dilution by the nitrogen in the air
supplied by the gasifier. The low calorific values carries little penalty, however,
because all gas turbines operate with a weak mixture to limit the turbine inlet
Coal or
heavy oil
Boost
compressor
Process
steam
FIG. 1.21 Gasificatioll plant with combined cycle
Gaseous fuel
32 lNTRODUCTION
. . - .
temperature. It siniply means that the nitrogen normally fed directly to the
combustion chamber in the compressor delivery air, now enters as part of the fuel
already at the pressure requITed. Fllirther discussion of this type of plant is given in
section 6.8.
One other possible future application of the gas turbine should be mentioned:
its use as an energy storage, device. The overall efficiency of a country's electricity
generating system can be improved if sufficient energy storage capacity is
provided to enable the most efficie:nt base-load stations to run night and day under
conditions yielding peak efficiency. This provision will become of particular
importance as the contribution of the capital-intensive nuclear power stations
increases. So far hydro-electric pumped storage plant have been built to meet the
need, but suitable sites in Great Britain have virtually all been used. A possible
alternative is illustrated in Fig. 1.22. Here a reversible motor/generator is coupled
either to the compressor or turbine. During the uight, off-peak power is used to
drive the compressor which delivers air to an underground cavern via a 'pebble
bed' regenerator. The regenerator stores the heat in pebbles of alumina or silica.
During the day the compressed air is released through the regenerator, picking up
stored energy on its way to the turbine. To satisfy peak demand it may prove
desirable also to bum some fuel in a combustion chamber to make up for heat
losses in the regenerator. If the cavern is to be sufficiently small to make such a
scheme economic, the pressure must be high-perhaps as high as 100 bar. This
implies a high compressor delivery temperature of about 900°C. By cooling the
air in the regenerator the volume is further reduced, and at the same time the walls
of the cavern are protected from the high temperature. Salt caverns excavated by
washing have been proposed, and disused mine workings are another possibility
if economic meaus can be found of sealing them adequately.
The first air-storage gas turbine plant was built by Brown Boveri and
commissioned in Germany in 1978. It has no regenerator, but it does incorporate
a heat-exchanger and has two-stage compression with intercooling. An after-
cooler protects the salt cavern wans from high temperature. The plant is for peak
load generation and produces up to 290 MW for periods of 1-1·5 hours three
times a day, using about 12 hours for pumping the reservoir up to pressure.
NIGHT
DAY
FIG. 1.22 Energy storage scheme
GAS TURBINE DESIGN PROCEDURE 33
. 1.8 Gas turbine design procedure
It must be emphasized that this book provides an introduction to the theory of the
gas turbine, and not to the design of gas turbines. To place the contents of the
book in proper perspective, a schematic diagram representing a complete design
procedure is shown in Fig. 1.23. This gives some idea of the interrelationships
between thermodynamic, aerodynamic, mechanical and control system design
and emphasizes the need for feedback between the various specialists. The dotted
lines enclose the areas covered in subsequent chapters: where they cut a block it
indicates that the topic has received attention, but only in a cursory manner. Thus
when dealing with the thermodynamic and aerodynamic theory which forms the
core of this book, the reader will be reminded only of those mechanical aspects
which interact directly with it.
The design process is shown as starting from a specification, resulting from
either market research or a customer requirement. The development of a high-
performance gas turbine is extremely costly, and so expensive that most large aero
engines are developed by multi-national consortia. There are very few customers
who are powerful enough to have an engine built to their requirement, and the
specification usually results from market research. Successful engines are those
which find a variety of applications, and their life-cycle from design to final
service use may be in excess of 50 years. When the first edition of this book was
written in 1950, the Rolls-Royce Dart was in the design stage and remained in
production until 1986; in late 1993 there were still nearly 2000 Darts in service
and the engine can certainly expect to continue into the 21st Century.
The specification is rarely a simple statement of required power and efficiency.
Other factors of major importance, which vary with the application, include
weight, cost, volume, life and noise, and many of these criteria act in opposition.
For example, high efficiency inevitably incurs high capital cost, and a simple
engine of lower efficiency may be perfectly acceptable if the running hours are
low. An important decision facing the designer is the choice of cycle, and this
aspect will be covered in Chapters 2 and 3. It is essential to consider at an early
stage what type of turbomachinery to use, and this will in large measure depend
on the size of the engine: turbomachinery and combustor design will be dealt
with in Chapters 4-7. The layout of the engine must also be considered, for
example, whether a single or multi-shaft design should be used, and the behaviour
of these different types of engine will be covered in Chapters 8 and 9.
The first major design step is to carry out thermodynamic design point studies.
These are detailed calculations taking into account all important factors such as
expected component efficiencies, air-bleeds, variable fluid properties and pressure
losses, and would be carried out over a range of pressure ratio and turbine inlet
temperature. A value for the specific output (i.e. power per unit mass flow of air)
and specific fuel consumption will be determined for various values of the cycle
parameters listed above. Although in industry these calculations would be done
on a digital computer, it should be clearly understood that there is not a
mathematically defined optimum. For example, at a given turbine inlet
34
INTRODUCTION
r-------- ---.------,
I I
I I
i Preliminary studies: I
I choice of cycle, I
I type of turbo machinery, I
I layout i
I I
, - - - - - - - - - - - ~ - - - - - - ~ I
I
I
I
I
I
I
I
~ - - - - - - - - - - - - - - I
I I
, - - - ~ - . i
r - ~ ~ ~ - - - - ~ I
Component test
rigs: compressor,
turbine 1-4---'
combustion, etc.
Uprated and
modified
versions
Production
FIG. 1.1.3 Typical gas turbine desigll procedllre
I
I
____ __J
GAS TURBINE DESIGN PROCEDURE 35
temperature a large increase in pressure ratio may give a minimal improvement in
thermal efficiency, and the resulting engine would be too complex and expensive
to be practical. Once the designer has settled on a suitable choice of cycle
parameters, he can make use of the specific output to determine the airflow
required to give the specified power.
It should be clearly understood that the choice of cycle parameters is strongly
influenced 'by the engine size, and in particular by the air flow required. The
turbine of a 500 leW engine, for example, would have very small blades which
could not be cooled for reasons of manufacturing complexity and cost; the
pressure ratio would be restricted to allow blading to be of a reasonable size, and
it might be necessary to use a centrifugal compressor of somewhat reduced
efficiency. A 50 MW unit, on the other ha:o.d, could use sophisticated air-cooled
blades and operate at a turbine inlet temperature of over 1500 K, some 300 K
higher than the uncooled to.rbine in the 500 kWengine. The large unit would also
use an axial compressor with a pressure ratio that could be as high as 30: l.
Knowing the airflow, pressure ratio and turbine inlet temperature, attention can
be turned to the aerodynamic design of the turbomachinery. It is now possible to
determine annulns dimensions, rotational speeds and number of stages. At this
point it may well be found that difficulties arise which may cause the
aerodynamacist to consult with the thermodynamacist to see if a change in the
design point could be considered, perhaps a slight increase in temperature or
decrease in pressure ratio. The aerodynamic design of the turbomachinery must
take into account manufacturing feasibility from the outset. In the case of a
centrifugal impeller for a small engine, for example, the space required for
milling cutters between adjacent passag<:s is of prime importance. In la:rge
turbofans, in contrast, the weight of the fan blade and the imbalance caused by
loss of a blade plays an important role in th,e design of the bearing support system
and the structure required for fan containment.
The mechanical design can start only after aerodynamic and thermodynamic
designs are well advanced. It will then likely be found that stress or vibration
problems may lead to further changes, the requirements of the stress and
aerodynamics groups often being in opposition. At the same time as these sto.dies
are proceeding, off-design performance and control system design must be
considered; off-design operation will include the effects of varying ambient
conditions, as well as reduced power operation. When designing a control system
to ensure the safe and automatic operation of the engine, it is necessary to be able
to predict temperature and pressure levels throughout the engine and to select
some of these for use as control parameters.
Once the engine has entered service, there will be demands from customers for
more powerful or more efficient versions, leading to the development of uprated
engines. Indeed, such demands may often arise before the design process has
been completed. VI/hen engines have to be uprated, the designer must consider
such methods as increasing the mass flow, turbine inlet temperature and
component efficiencies, while maintaining the same basic engine design. A
successful engine may have its power tripled during a long development cycle.
36
INTRODUCTION
Eventually, however, the engine· will become· dated· and no longer competitive.
The timing of a decision to start a new design is of critical importance for the
economic well-being of the manufacturer. References (1) and (2) describe the
choice of design for an industrial gas turbine and turboprop respectively.
References (3) and (4) trace the design evolution of industrial gas turbines over an
extended period, showing how the power and efficiency are continuously
improved by increasing ratio, turbine inlet temperature and airflow. The
evolution of the Westinghouse 501 gas turbine from 1968 to 1993 is described in
Ref. (4), and the table below shows how the power increased from 42 to 160 MW.
Aerodynamic development allowed the pressure ratio to be raised from 7.5 to
14.6, while improvements in and blade cooling penrutted a significant
increase in cycle temperature; the net result was an improvement in thennal
efficiency from 27.1 to 35.6 per cent. The 1968 engine had only the first nozzle
row cooled, whereas the 1993 version required the cooling of six rows. The
steady increase in exhaust gas temperature should be noted, this being an
important factor in obtaining high thennal efficiency in a combined cycle appli-
cation.
Year 1968 1971 1973 1975 1981 1993
Power (MW) 42 60 80 95 107 160
Thennal efficiency (%) 27.1 29.4 30.5 31.2 33.2 35.6
Pressure ratio 7.5 10.5 11.2 12.6 14.0 14.6
Turbine inlet temp. (K) 1153 1161 1266 1369 1406 1533
Air flow (kg/s) 249 337 338 354 354 435
Exhaust gas temp. (0C) 474 426 486 528 531 584
No. of compo stages 17 17 17 19 19 16
No. of turbine stages 4 4 4 4 4 4
No. of cooled rows 1 I 3 4 4 6
The foregoing should give reader an overall, if superficial, view of the
design process and may lead to the realization that the gas turbine industry can
provide an exciting and rewarding technical career for a wide variety of highly
skilled engineers.
Shaft power cycles
Enough has been said in the foregoing chapter for the reader to realize how great
is the number of possible varieties of gas turbine cycle when multi-stage com-
pression and expansion, heat-exchange, reheat and intercooling are incorporated.
A comprehensive study of the perfonnance of all such cycles, allowing for the
inefficiencies of the various components, would result in a very large number of
perfonnance curves. We shall here concentrate mainly on describing methods of
calculating cycle perfonnance. For convenience the cycles are treated in two
groups-shaft power cycles (this chapter) and aircraft propulsion cycles (Chapter
3). An important distinction between the two groups arises from the fact that the
perfonnance of aircraft propulsion cycles depends very significantly upon for-
ward speed and altitude. These two variables do not enter into performance
calculations for marine and land-based power plant to which this chapter is
confined.
Before proceeding with the main task, it will be useful to review the
perfonnance of ideal gas turbine cycles in which perfection of the individual
components is assumed. The specific work output and cycle efficiency then
depend only on the pressure ratio and maximum cycle temperature. The limited
number of perfonnance curves so obtained enables the major effects of various
additions to the simple cycle to be seen clearly. Such curves also show the upper
limit of perfonnance which can be expected of real cycles as the efficiency of gas
turbine components is improved.
2.1 Ideal cycles
Analyses of ideal gas turbine cycles can be found in texts on engineering ther-
modynamics [e.g. Ref. (1)] and only a brief resume will be given here. The
assumption of ideal conditions will be taken to imply the following:
(a) Compression and expansion processes are reversible and adiabatic, i.e.
isentropic.
(b) The change ofkinetic energy of the working fluid between inlet and outlet
of each component is negligible.
38 SHAFT POWER CYCLES
(c) There are no pressure losses in the inlet ducting, combustion chambers,
heat-exchangers, intercoolers, exhaust ducting, and ducts connecting the
components.
(d) The working fluid has the same composition throughout the cycle and is a
perfect gas with constant specific heats.
(e) The mass flow of gas. is constant throughout the cycle.
(I) Heat transfer in a heat-exchanger (assumed counterflow) is 'complete', so that
in conjunction with (d) and (e) the temperature rise on the cold side is the
maximum possible and exactly equal to the temperature drop on the hot side.
Assumptions (d) and (e) imply that the combustion chamber, in which fuel is
introduced and burned, is considered as being replaced by a heater with an
external heat source. For this reason, as far as the calculation of performance of
ideal cycles is concerned, it makes no difference whether one is thinking of them
as 'open' or 'closed' cycles. The diagrammatic sketches of plant will be drawn
for the former, more usual, case.
Simple gas turbine cycle
The ideal cycle for the simple gas turbine is the Joule (or Brayton) cycle, i.e. cycle
1234 in Fig. 2.1. The relevant steady flow energy equation is
Q = (h2 - hI) - CD + W
where Q and Ware the heat and work transfers per unit mass flow. Applying this
to each component, bearing in mind assumption (b), we have
W12 = -(h2 - hI) = -cp(T2 - T1)
Q23 = (h3 - h2) = cp(T3 - T2)
W34 = (h3 - 114) = ciT3 - T4)
Fuel,: Heat
Compressor Turbine
FIG. 2.1 Simple cycle
IDEAL CYCLES
The cycle efficiency is
net work output CpCT3 - T4) - cpCTz - TI )
11 = heat supplied = Cp(T3 - T2)
Making use of the isentropic P - T relation, we have
T2lTI = r(l,-I)/y = T3/T4
39
where r is the pressure ratio P2/PI =r='P3/P4' The cycle efficiency is then
readily shown to be given by
(
l)(Y-l)/J>
'1 = 1- -
r
(2.1)
The efficiency thus depends only on the pressure ratio and the nature of the gas.
Figure 2.2(a) shows the relation between i7 and r when the working fluid is air
(')' = 1·4), or a monatomic gas such as argon (')' = 1·66). For the remaining curves
in this section air will be assumed to be the working fluid. The effect of using
helium instead of air in a closed cycle is studied in section 2.6, where it is shown
that the theoretical advantage indicated in Fig. 2.2(a) is not realized when com-
ponent losses are taken into account. .
The specific work output W, upon which the size of plant for a gIven power
depends, is found to be a function not only of pressure ratio but also of maximum
cycle temperature T3. Thus
W = Cp(T3 - T4) - ciT2 - T1)
which can be expressed as
..!.. = t(1 - _1_) _ (r(l,-I)/y -- 1)
C
p
Tl r(y-l)/Y /
100
80
1.6
1.4
1.2
§;
"S 1.0
Cl.
"S
(2.2)
60
i;-
0.8

() 0.6 t:
40
iE
w
20
10 12
Pressure ratio r
(a)

'"
$ 0.4
0.2
OLL-L __ L--L __
o
Pressure ratio r
(b)
FIG. 2.2 Efficiency and specific work output-simple cycle
40 SHAFT POWER CYCLES
where T3/h TJ is nonnally and is not a major
independent variable. It is therefore convenient to plot the specific work in
non-dirilensional fonn (WI cpTr) as a function of r and t as in Fig. 2.2(b). The
value of T3, and hence t, that can be used in practice, dePends upon the maximum
temperature which the highly stressed parts of the turbine can stand for the
required working life: it is often called the 'metallurgical limit' . Early gas turbines
used values of t between 3·5 and 4, but the introduction of air-cooled turbine
blades allowed t to be raised to between 5 and 6.
A glance at the T - s diagranl of Fig. 2.1 will show why a constant t curve
exhibits a maximum at a certain pressure ratio: W is zero at r = 1 and also at the
value of r for which the compression and expansion processes coincide, namely
r=ty/(y-I). For any given value of t the optimum pressure ratio for maximum
specific work output can be found by differentiating equation (2.2) with respect to
r(y-I)/y and equating to zero: the: result is
r
(y-I)/y - It
opt - v
Since r(y-I)/Y = T21TJ = T31T4, tlus is equivalent to writing
T2 T3
-x-=t
TJ T4
But t = T31 TJ and consequently it follows that T2 = T4• Thus the specific work
output is a maximum when the pressure ratio is such that the compressor and
turbine outlet temperatures are equal. For all values of r between 1 and tY /2(1' -I),
T4 will be greater than T2 and a heat-exchanger can be incorporated to reduce the
heat transfer from the external source and so increase the efficiency.
Heat-exchange cycle
Using the nomenclature of Fig. 2.3, the cycle efficiency now becomes
c/T3 - T4 ) - Cp(T2 -- TJ)
'1=
C/T3 - Ts)
With ideal heat-exchange Ts = T4, and on substituting the isentropic p - T
relations the expression reduces to
r(y-Il/y
'1 = 1--- (2.3)
t
Thus the efficiency of the heat-exchange cycle is not independent of the maxi-
mum cycle temperature, and clearly it increases as t is increased. Furthennore it is
evident that, for a given value of t, the efficiency increases with decrease in
pressure ratio and not with increa.se in pressure ratio as for the simple cycle. The
full lines in Fig. 2.4 represent the equation, each constant t curve starting at r= 1
with a value of '1 = 1 - l/t, i.e. the Carnot efficiency, This is to be expected
because in this limiting case the Carnot requirement of complete external heat
reception and rejection at the upper and lower cycle temperature is satisfied.
IDEAL CYCLES
41
T 3
2
FIG. 2.3 Simple cycle with heat-exchange
The curves fall with increasing pressure ratio until a value corresponding to
/y-J)/y =./t is reached, and at this point equation (2.3) reduces to (2.1). This is
the pressure ratio for which the specific work output curves of Fig. 2.2(b) reach a
maximum and for which it was shown that T4 = Tz. For higher values of r a heat-
exchanger would cool the air leaving the compressor and so reduce the efficiency,
and therefore the constant t lines have not been extended beyond the point where
they meet the efficiency curve for the simple cycle which is shown dotted in
Fig. 2.4.
The specific work output is unchanged by the addition of a heat-exchanger and
the curves of Fig. 2.2(b) are still applicable. From these curves and those in Fig.
2.4 it can be concluded that to obtain an appreciable improvement in efficiency by
heat-exchange, (a) a value of r appreciably less than the' optimum for maximum
specific work output should be used and (b) it is not necessary to use a higher
cycle pressure ratio as the maximum cycle temperature is increased. Later it will
be shown that for real cycles conclusion (a) remains true but conclusion (b)
requires modification.
100
Pressure ratio r
FIG. 2.4 Efficiency-simple cycle with heat-exchange
42 SHAFT POWER CYCLES
Reheat cycle
A substantial increase in specific .work output can be obtained by splitting the
expansion and reheating the gas between the high-pressure and low-pressure
turbines. Figure 2.5(a) shows the relevant portion of the reheat cycle on the T - s
diagram. That the turbine work is increased is obvious when one remembers that
the vertical distance between any pair of constant pressure lines increases as the
entropy increases: thus (T3 - T4) + (Ts - T6) > (T3 -
Assuming that the gas is reheated to a temperature equal to T3, differentiation
of the expression for specific work output shows that the optimum point in the
expansion at which to reheat is when the pressure ratios (and hence temperature
drops and work transfers) for the HP and LP turbines are equal. With this
optimum division, it is then possible to derive expressions for the specific output
and efficiency in tenns of rand t as before. Writing c=r(y-JJ/Y they become
W 2t
-=2t-c+l--
cpTj -,/c
2t - c + 1 - 2t / -,/ c
1)=
2t - c - t/-,/c
(2.4)
(2.5)
Comparison ofthe W/cpTJ curves of Fig. 2.6 with those of Fig. 2.2(b) shows that
reheat markedly increases the specific output. Figure 2.5(b), however, indicates
that this is achieved at the expense of efficiency. TIlls is to be expected because
one is adding a less efficient cycle (4'456 in Fig. 2.5(a)) to the simple cycle-less
efficient because it operates over a smaller temperature range. Note that the
reduction in efficiency becomes less severe as the maximum cycle temperature is
increased.
T
5
/
4'
(a)
FIG. 2.5 Reheat cycle
100
80
20
I I I I I I I
4 6 8 10 12 14 16
Pressure ratio r
(b)
IDEAL CYCLES
)-C
,,"

"5
c.
"5
0
-t:
0
;:
" -=
·u
Q)
c.
(J)
2.2
2.0
1.8
1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0
4
0 2 4

Pressure ratio r
Cycle with reheat and heat-exchange
43
The reduction in efficiency due to reheat can be overcome by adding heat-
exchange as in Fig. 2.7. The higher exhaust gas temperature is now fully utilized
in the heat-exchanger and the increase in work output is no longer offset by the
increase in heat supplied. In fact, when a heat-exchanger is employed, the effi-
ciency is higher with reheat than without as shown by a comparison of Figs. 2.8
and 2.4. The family of constant t lines exhibit the same features as those for the
simple cycle with heat-exchange-each curve having the Camot value at r = 1
3 5
FIG. 2.7 Reheat cycle with heat-exchange
44
100
80
#-
:::- 60
[;'
c:
" '0 40
m
20
Pressure ratio r
FIG. 2.8 cycle witil ileat-exchalllge
SHAFT POWER CYCLES
and falling with increasing r to meet the corresponding efficiency curve of the
reheat cycle without heat-exchang.e at the value of r con'esponding to maximum
specific work output.
Cycles with intercooled compression
A similar improvement in specific work output to that obtained by reheat can be
achieved by splitting the compression and intercooling the gas between the LP
and HP compressors; and, assuming that the air is intercooled to TI, it can be
shown that the specific work output is a maximum when the pressure ratios of the
LP and HP compressors are equal. The use of intercoolers is seldom contem-
plated in practice because they are bulky and need large quantities of cooling
water. The main advantage of the gas turbine, that it is compact and self-
contained, is then lost. For this reason no perfornlance curves for cycles with
inter-cooling are included: suffice i:t to say that they are similar to Figs 2.5(b) and
2.6, although the increase in specific work output and reduction in efficiency with
respect to the simple cycle are not so marked. (A modification to the low
temperature region of a cycle is normally less significant than a comparable
modification to the high temperature region.) As with reheat, intercooling in-
creases the cycle efficiency only when a heat-exchanger is also incorporated, and
then curves almost identical to those of Fig. 2.8 are obtained.
This discussion of ideal cycles should be sufficient to indicate the main effects
of additions to the simple gas turbine. We have seen that the choice of pressure
ratio will depend on whether it is high efficiency or high specific work output (i.e.
small size) which is required; and that in cycles without heat-exchange a higher
pressure ratio must be used to talce advantage of a higher pennissible turbine inlet
temperature. It will be evident from what follows that these conclusions are also
broadly true of practical cycles in which component losses are taken into account.
METHODS OF ACCOUNTING FOR COMPONENT LOSSES 45
2,2 Meth.ods ofacco1J!nting for component Im;ses
The performance of real cycles differs from that of ideal cycles for the following
reasons.
(a) Because fluid velocities are high in turbomachinery the change in kinetic
energy between inlet and outlet of each component cannot necessarily be
ignored. A further consequence is that the compression and expansion
processes are irreversible adiabatics and therefore involve an increase in
entropy.
(b) Fluid friction results in pressure losses in conibustion chambers and heat-
exchangers, and also in the inlet and exhaust ducts. (Losses in the ducts
connecting components are usually included in the associated component
losses.)
(c) If a heat-exchanger is to be of economic size, terminal temperature
differences are inevitable; i.e. the compressed air cannot be heated to the
temperature of the gas leaving the turbine.
td) Slightly more work than that required for the compression process will be
necessary to overcome bearing and 'windage' friction in the transmission
between compressor and turbine, and to drive ancillary components such as
fuel and oil pumps.
(e) The values of cp and'}' of the working fluid vary throughout the cycle due to
changes of temperature and, with internal combustion, due to changes in
chemical composition.
(f) The definition of the efficiency of an ideal cycle is unambiguous, but this is
not the case for an open cycle with internal combustion. Knowing the
compressor delivery temperature, composition of the fuel, and turbine inlet
temperature required, a straightforward combustion calculation yields the
fuel! air ratio necessary; and a combustion efficiency can also be included to
allow for incomplete combustion. Thus it will be possible to express the
cycle performance unambiguously in terms of fuel consumption per unit net
work output, i.e in ternlS of the specific fuel consumption. To convert this to
a cycle efficiency it is necessary to adopt some convention for expressing
the heating value of the fuel.
(g) With internal combustion, the mass flow through the turbine might be
thought to be greater than that through the compressor by virtue of the fuel
added. In practice, about 1-2 per cent of the compressed air is bled off for
cooling turbine discs and blade roots, and we shall see later that the fuel! air
ratio employed is in the region 0·01-0·02. For many cycle calculations it is
sufficiently accurate to assume that the fuel added merely compensates for
this loss. We will assume in the numerical examples which follow that the
mass flows through the compressor and turbine are equal. When turbine
inlet temperatures higher than about 1350 K are used, the turbine blading
must be internally cooled as well as the disc and blade roots. We then have
what is called an air-cooled turbine. Up to 15 per cent of the compressor
46
SHAFT POWER CYCLES
delivery air might be bled fot cooling purposes, ilnd for an accurate estimate
of cycle performance it is necessary to account explicitly for the of
.mass flow through the engine: .
Methods of accounting for factors (a) to (f) must be discussed before giving
examples of cycle calculations. We will also say a little more in general terms
about cooling bleed flows.
Stagnation properties
The kinetic energy terms in the steady flow energy equation can be accounted for
implicitly by making use of the concept of stagnation (or total) enthalpy. Phy-
sically, the stagnation enthalpy ho is the enthalpy which a gas stream of enthalpy h
and velocity C would possess when brought to rest adiabatically and without
work transfer. The energy equation then reduces to
(ho - h) +!(O - C
z
) = 0
and thus ho is defined by
ho = h + C
2
/2 (2.6)
When the fluid is a perfect gas, CpT can be substituted for h, and the corres-
ponding concept of stagnation (or total) temperature To is defined by
To = T + C
Z
/2cp (2.7)
(!2 /2cp is called the dynamic temperature and, when it is necessary to emphasize
the distinction, T is referred to as the static temperature. An idea of the order of
magnitude of the difference between To and T is obtained by considering air at
atmospheric temperature, for which cp = 1·005 kJ/kg K, flowing at 100 m/s.
Then
To -T- 100
2

o - 2 x 1.005 x 103 - 5 K
It follows from the energy equation that if there is no heat or work transfer To will
remain constant. If the duct is varying in cross-sectional area, or friction is
degrading directed kinetic energy into random molecular energy, the static tem-
perature will change-but To will not. Applying the concept to an adiabatic
compression, the energy equation becomes
W = -cp(Tz - Tl ) - - Cn = -cp(Toz - TOl )
Similarly for a heating process without work transfer,
Q = ciTo2 - TOl )
Thus if stagnation temperatures are employed there is no need to refer explicitly
to the kinetic energy term. A practical advantage is that it is easier to measure the
stagnation temperature of a high-velocity stream than the static temperature (see
section 6.5).
METHODS OF ACCOUNTING FOR COMPONENT LOSSES 47
When.a gas is slowed down and the temperature rises there is a simultaneous
rise in pressure. The stagnation (or total) pressure Po is defined in a similar way
to To but with the added restriction iliat the gas is imagined to be brought to rest
not only adiabatically but also reversibly, i.e. isentropically. The stagnation
pressure is thus defined by
==
(2.8)
Stagnation pressure, unlike stagnation temperature, is constant in a stream flow-
ing without-heat or work transfer only iffric:tion is absent: the drop in stagnation
pressure can be used as a measure of the fluid friction.
It is worfu noting that Po is not identieal with the usual pitot pressure
defined for incompressible flow by
=p+pC
z
/2
Substituting equation (2.7) in (2.8), and making use of cp=yR/(y - 1) and
p = pRT, we have
(
pCz y _ I)Y/(Y-l)
po=p l+-x--
2p Y
is seen to be given by the first two terms of the binomial expansion. Thus Po
approaches as the velo"City is decreased and compressibility effects become
negligible. As an example of the difference at high velocities, for air moving with
sonic velocity (Mach number M = 1), Po/p = 1·89 whereas Pff /p = 1·7. Thus by
assuming the flow to be incompressible the stagnation pressure would be under-
estimated by about 11 per cent.
Applying equation (2.8) to an isentropic compression between inlet 1 and
outlet 2 we have the stagnation pressure ratio given by:
Poz =Poz x!2.l.. x pz = (T02 X IL x Tz)Y!(Y-l) = (T02 )Y/(Y-I)
POI P2 POI PI Tz TOl TI TOl
Similarly, if required,
P
(
To )Y/(Y-I)
02 02
p;= r;
Thus Po and To can be used in the same way as static values in isentropic p-T
relations. Stagnation pressure and temperature are properties of the gas stream
which can be used with static values to determine the combined thermodynamic
and mechanical state of the stream. Such state points can be represented on the
T-s diagram, as shown in Fig. 2.9 which dt:picts a compression process between
'static' states 1 and 2; the differences between the constant P and Po lines have
been exaggerated for clarity. The ideal stagnation state which would be reached
after isentropic compression to the same, actual, outlet stagnation pressure is
indicated by 02'. Primes will be attached to symbols to denote such ideal states
throughout this book.
48
s
FIG. 2.9 Stagllatioll states
Compressor and turbine efficiencies
(aJ Isentropic efficiency
SHAFT POWER CYCLES
The .efficiency of any machine, the of which is the absorption or pro-
duction of work, IS normally expressed III terms of the ratio of actual and ideal
:vork trans.fers. Because tlJrbomachines are essentially adiabatic, the ideal process
IS IsentropIC and the efficiency is called an isentropic efficiency. Making use of the
of stagnation enthalpy or temperatlJre to take account of any change in
lanetJc energy of the fluid between inlet and outlet we have, for the compressor,
WI Ah'
I] - - 0
e-W-Ah
o
For a perfect gas Aho = cpATo and this relation is usually sufficiently accurate for
real gases under conditions encountered in gas tlJrbines if a mean c over the
relevant range of temperatlJre is used-see under heading 'Variation specific
heat' below for further discussion of this point. Furthermore, because the ideal
and actual temperatlJre changes are not very different, the mean cp can be as-
sumed the same for both so that the compressor isentropic efficiency is normally
defined in tenus of temperature as
T62 - TOI
I] =
e T02 - TOl
Similarly the tlJrbine isentropic efficiency is defined as
W T03 - T04
rlt = W' = T. - T.'
03 04
(2.9)
(2.10)
When performing cycle calculations, values of 'Ie and 1], will be assumed and
the temperature equivalents of the work transfers for a given pressure ratio are
then found as follows:
METHODS OF ACCOUNTING FOR COMPONENT LOSSES
and finally
T. T.
01 P02 I
T. [( \ (]'-1)/]' ]
02 - 01 = - - I -1
I]e PoJ)
Similarly
. [( 1 ) (]'-1)/1']
T03 - T04 = I],T03 1 - ---
POJiP04
49
(2.11 )
(2.12)
When the compressor is part of a stationary gas turbine, having a short intake
fairing that can be regarded as part of the compressor, POI and TO! in equation
(2.11) will be equal to Pa and Ta respectively because the velocity of the ambient
air is zero. This will be assumed to be the case throughout this chapter. Industrial
gas tlJrbine plant usually incorporate a long inlet duct and air filter, however,
in which case an inlet pressure loss (Api) must be deducted, i.e. POI will be
(Pa - Ap;). Inlet losses vary with installation configurations and engine manu-
facturers often quote performance figures for zero inlet loss, with corrections for
different levels of loss. The situation is rather different when the compressor is
part of an aircraft propulsion unit because then there will be an intake duct of
significant length in which ram compression takes place due to the forward speed
of the aircraft. In this situation POI and Tal would differ from Pa and Ta even if
there were no friction losses, and it is always necessary to consider the intake and
compressor as separate components. A discussion of how intake losses are then
taken into account will be deferred until the next chapter.
In defining IJt according to (2.10) and thus taking the ideal work as
proportional to (T03 - T64)' we are implying that the kinetic energy in the exhaust
gas is going to be utilized, e.g. in a subsequent turbine or in the propelling nozzle
of a jet engine. But if the tlJrbille is part of an industrial plant exhausting directly
to a.tmosphere this kinetic energy is wasted. The ideal qua.ntity of tlJrbine work
would then seem to be taken more appropriately as that produced by an isentropic
expansion from P03 to the static outlet pressure P4, with P4 equal to the ambient
pressure Pa' Thus IJ, would be defined by
T03 - T04
IJ, = [ ( 1 ) (]'-l)/]']
T03 1- --
P03/Pal
(2.13)
In practice, even in such a case the kinetic energy of the gas immediately leaving
the tlJrbine is largely recovered in an exhaust diffuser which in effect increases the
pressure ratio across the tlJrbine: Fig. 2.1 0 indicates this for a diffuser which
reduces the final velocity to a negligible value so that P04 = P4 = Pa. The turbine
pressure ratio is seen to be increased from P03/Pa to P03/Px. The temperatlJre
equivalent of the turbine work (T03 - Tox) is still given by (To3 - T04), because
no work is done in the diffuser and Tox = T04, but T04 is less than it would be if a
diffuser were not fitted and Px was equal to Pa. For ordinary cycle calculations
50 SHAFT POIV.ER CYCLES
T
Diffuser
FIG. 2.10 Thrbille with exhaust diffuser
there is no need to consider the turbine expansion 3 ---+ x and diffusion process
x ---+ 4 separately. We may put P04 = Pa in equation (2.12) and regard 1'/1 as ac-
counting also for the friction pressure loss in the diffuser (Pox - P a). We then
have equation (2.13), but must interpret it as applying to the turbine and exhaust
diffuser combined rather than to the turbine alone. In this book, equation (2.12)
with P04 put equal to Pa will be used for any turbine exhausting direct to the
atmosphere: for any turbine delivering gas to a propelling nozzle, or to a second
turbine in series, equation (2.12) will be employed as it stands.
(b) Polytropic efficiency
So far we have been referring to overall efficiencies applied to the compressor or
turbine as a whole. When perfonning cycle calculations covering a range of
pressure ratio, say to detennine the optimum pressure ratio for a particular ap-
plication, the question arises as to whether it is reasonable to assume fixed typical
values of I'/e and 1'/,. In fact it is found that 'Ie tends to decrease and I), to increase as
the pressure ratio for which the compressor and turbine are designed increases.
The reason for this should be clear from the following argument based on Fig.
2.11. P and T are used instead of Po and To to avoid a multiplicity of suffixes.
Consider an axial flow compressor consisting of a number of successive
stages. If the blade design is similar in successive blade rows it is reasonable to
assume that the isentropic efficiency of a single stage, I)" remains the same
through the compressor. Then the overall temperature rise can be expressed by
I1T = .L AT; = ~ L.AT;
I)s IJs
Also, I1T= 11T' /I)c by definition of I)e, and thus
METHODS OF ACCOUNTING FOR COMPONENT LOSSES 51
T
FIG.2.B
But, because the vertical distance between a pair of constant pressure lines in the
T -s diagram increases as the entropy increases, it is clear from Fig. 2.11 that
L./lT; > AT'. It follows that I)c < lis and that the difference will increase with the
number of stages, i.e. with increase of pres;:ure ratio. A physical explanation is
that the increase in temperature due to friction in one stage results in more work
being required in the next stage: it might be tenned the 'preheat' effect. A similar
argument can be used to show that for a tm-bine '11 > lis. In this case frictional
'reheating' in one stage is partially recovered as work in the next.
These considerations have led to the concept of polytropic (or small-stage)
efficiency 1]00' which is defined as the isentropic efficiency of an elemental stage
in the process such that it is constant throughout the whole process. For a
compression,
dT'
I]ooe = dT = constant
But T/ll'-J)/)' = constant for an isentropic process, which in differential fonn is
dT' 1'-1 dp
T l' P
Substitution of dT' from the previous equation gives
dT y - I dp
IJ -=---
ooc T y P
Integrating between inlet 1 and outlet 2, with I) ooe constant by definition, we have
1n(P2/pd1,-J)/1'
I]coe = In(T2/T])
(2.14)
This equation enables I) we to be calculated from measm-ed values of P and Tat
inlet and outlet of a compressor. Equation (2.14) can also be written in the fonn
T2 __ (P_2)(Y-J)/YQOO,
(2.15)
TI PI
52 SFMT POWER CYCLES
Finally, the relation between '1ooc and 11c is given by, ..
. T' /T1 - 1 (P2/pd
r
-
1l
/
Y
- 1
'1 2 - (2.16)
c = T2/T1 - 1 - - 1
Note that if we write (1' - l)fl'll,Xlc as (n - l)/n, equation (2.15) is the familiar
relation between P and T for a polytropic process, and thus the definition of '100
implies that the non-isentropic process is polytropic. This is the origin ofthe term
polytropic efficiency.
Similarly, since l/ oot is dT/dT', it can be shown that for an expansion between
inlet 3 and outlet 4,
T3 =
T4 P4
(2.17)
and
(2.18)
'It = (1 )(Y-ll/Y
1- --
P3/P4
Making use of equations (2.16) and (2.18) with l' = 1·4, Fig. 2.12 has been drawn
to show how 'Ie and 'It vary with pressure ratio for a fixed value of polytropic
efficiency of 85 per cent in each case.
In practice, as with 'Ie and '11> it is normal to define the polytropic efficiencies
in terms of stagnation temperatures and pressures. Furthermore, when employing
them in cycle calculations, the most convenient equations to use will be shown to
be those corresponding to (2.l1) and (2.12), i.e. from equations (2.15) and (2.17),
[
r p )("-l
ll
n ]
T02 - T01 = T01 -1
(2.19)
where (n - l)jn=(y - 1)l1'l1ooe
95
Turbine
7SL.._.J.1 __ L-_-L-_-L __ ---1.I__ LI_---1.1 __ 1
o 2 4 6 8 10 12 14 16
Pressure ratio r
FIG. 2.12 Variation of tIJlrbille alid compressor iselltropic efficiency witil pressure
ratio for polytropic effidellcy of 85%
METHODS OF ACCOUNTING FOR COMPONENT LOSSES 53
. .[ ( 1 )(n-1
l
/n]
T03 - T04 = T03 1 - 1 __ -.
IIP03/P04
(2.20)
where (n - l)/n = IlootCl' - 1)/1'.
And again, for a compressor of an industrial gas turbine we shall take POI =Pa
and TO! = Ta, while for turbines exhausting to atmosphere P04 will be put equal
to Pa-
We end this sub-section with a reminder that the isentropic and polytropic
efficiencies present the same information in different forms. When performing
calculations over a range of pressure ratio, it is reasonable to assume constant
polytropic efficiency; this automatically allows for a variation of isentropic
efficiency with pressure ratio. In simple terms, the polytropic efficiency may be
interpreted as representing the current state-of-the-art for a particular design
organization. When determiJning the performance of a single cycle of interest, or
analyzing engine test data, it is more appropriate to use isentropic efficiencies.
Pressure losses
Pressure losses in the intake and exhaust ducting have been dealt with in the
previous sub-section. In the combustion chamber a loss in stagnation pressure
(flPb) occurs due to the aerodynamic resistance of flame-stabilizing and mixing
devices, and also due to momentum changes produced by the exothermic reac-
tion. These sources of loss are referred to in detail in Chapter 6. When a heat-
exchanger is included in the plant there will also be frictional pressure losses in
the passages on the air-side (flPha) and gas-side (flPhg). As shown in Fig. 2.13,
the pressure losses have the effect of decreasing the turbine pressure ratio relative
to the compressor pressure ratio and thus reduce the net work output from the
T
P04

FIG. 2.13 Pressllre losses
54 SHAFT POWER CYCLES
plant. The gas turbme cycle is very sellsitiveto irreversibilities, because the net
output is the difference of two large quantities (Le. the 'work ratio' is low), so that
the pressure losses have a significant effect on the cycle performance.
Fixed values of the losses can be fed into the cycle calculation directly. For
example, for a simple cycle with heat-exchange we may determine the turbine
pressure ratio P03/P04 from.
P03 = P02 - APb - Apha and P04 = Pa + APhg
But again the question arises as to whether it is reasonable to assume constant
values for the pressure drops when cycles of different pressure ratio are being
compared. The frictional pressure losses will be roughly proportional to the local
dynamic head of the flow (!pC
2
for incompressible flow), as with ordinary pipe
flow. It might therefore be expected that the pressure drops APha and APb will
increase with cycle pressure ratio because the density of the fluid in the air-side of
the heat-exchanger and in the combustion chamber is increased. Even though p is
not proportional to P because Tincreases also, a better approximation might be to
take APha and APb as fixed proportions of the compressor delivery pressure. The
turbine inlet pressure is then found from
(
Apb APha)
P03 =P02 1-----
P02 P02
We shall express the pressure loss data in this way in subsequent numerical
examples.
The pressure loss in the combustor can be minimized by using a large
combustion chamber with consequent low velocities, which is feasible in an
industrial unit where size is not critical. For an aero engine, where weight, volume
and frontal area are all important, higher pressure losses are inevitable. The
designer also has to consider pressure losses at a typical flight condition at high
altitude, where the air density is low. Typical values of Apb/P02 would be about
2-3 per cent for a large industrial unit and 6-8 per cent for an aero engine.
Heat-exchanger effectiveness
Heat-exchangers for gas turbines can take many forms, including counter-flow
and cross-flow recuperators (where the hot and cold streams exchange heat
through a separating Wall) or regenerators (where the streams are brought cyc-
lically into contact with a matrix which alternately absorbs and rejects heat). In all
cases, using the notation of Fig. 2.13, the fundamental process is that the turbine
exhaust gases reject heat at the rate of mtCp46 (T04 - T06) while the compressor
delivery air receives heat at the rate of mecp2s(Tos - T02)' For conservation of
energy, assuming that the mass flows mt and me are equal,
Cp46(To4 - T06 ) = cp2s(Tos - T02 ) (2.21)
But both Tos and T06 are unknown and a second equation is required for their
evaluation. This is provided by the equation expressing the efficiency of the heat-
exchanger.
METHODS OF ACCOUNTING FOR COMPONENT LOSSES 55
Now the maximum possible value of Tos is when the 'cold' air attains the
temperature of the incoming hot gas T04, and one possible measure of perform-
ance is the ratio of actual energy received by 1he cold air to the maximum possible
value, i.e.
mecp2s(Tos - T02)
mecp2iTo4 - T02)
The mean specific heat of air will not be very different over the two temperature
ranges, and it is usual to define the efficiency in terms oftemperature alone and to
call it the effectiveness (or thermal-ratio) of the heat-exchanger. Thus
effectiveness = Tos - T02
T04 - T02
(2.22)
When a value of effectiveness is specified, equation (2.22) enables the tempera-
ture at inlet to the combustion chamber To:; to be determined. Equation (2.21)
then yields T06 ifrequired. Note that the mean specific heats cp46 and cp2S are not
approximately equal and cannot be cancelled, because the former is for turbine
exhaust gas and the latter for air.
In general, the larger the volume of the heat-exchanger the higher can be the
effectiveness, but the increase in effectiveness becomes asymptotic beyond a
certain value of heat transfer surface area; the cost of heat-exchangers is largely
determined by their surface area, and with tlris in mind modem heat-exchangers
are designed to have an effectiveness of about 0·90. The maximum permissible
turbine exit temperature is fixed by the materials used for the construction of the
heat-exchanger. With stainless steel, this temperature should not be much above
900 K. It follows that the turbine inlet temperature of a regenerative cycle is
limited for this reason. Heat-exchangers are subjected to severe thermal stresses
during start up and are not used where frequent starts are required, e.g. for peak-
load electricity generation. They are best slrited for applications where the gas
turbine operates for extended periods at a steady power, as may occur on a
pipeline. It should be noted, however, that many pipeline stations are located in
remote areas where transportation and installation of a heat-exchanger present
major problems; for this reason, some modem heat-exchangers are built up from
a series of identical modules. In recent years heat-exchangers have seldom been
used, because they no longer offer an advantage relative to simple cycles of high
pressure ratio or, in the case of base-load applications, relative to combined cycle
plant. In the future, however, the search for cycle efficiencies exceeding 60 per
cent may see the return of the heat-exchanger in complex cycles. A survey of the
possibilities is given in Ref. (2).
Mechanical losses
In all gas turbines, the power necessary to drive the compressor is transmitted
directly from the turbine without any intermediate gearing. Any loss that occurs is
56
SHAFT POWER CYCLES
therefore due only to bearing friction and windage. This loss is very small and it
is normal to assume that it amounts to about 1 per cent of the power necessary to
drive the compressor. Ifthe transmission efficiency is denoted by 11m, we have the
work output required to drive the compressor given by
1 )
W = -cplz(Toz - .TOI
11m
We shall take 1)m to be 99 per ce;nt for all numerical examples.
Any power used to drive ancillary components such as fuel and oil pumps can
often be accOlmted for simply by subtracting it from the net output of the unit.
Power absorbed in any gearing between the gas turbine and the load can be dealt
with similarly. Except to say that such losses can be significant, especially for
small gas turbines of low power, we shall not consider them further. Their
consideration would not require the method of cycle performance estimation to be
modified except when the gas turbine has a separate power turbine. Power for fuel
and oil pumps will then be taken from the compressor turbine (because under
some operating conditions the power turbine is stationary).
Variation of specific heat
The properties cp and l' play an important part in the estimation of cycle per-
formance, and it is necessary to take account of variations in values due to
changing conditions through the cycle. In general, for real gases over normal
working ranges of pressure and temperature, cp is a function of temperature alone.
The same is true of l' because it is related to cp by
)' Mcp
(2.23)
l' -1 R
where R is the molar (universal} gas constant and M the molecular weight. The
variation of cp and l' with temperature for air is shown in Fig. 2.14 by the curves
marked zero fuel/air ratio. Only the left-hand portion is of interest because even
with a pressure ratio as high as 35 the compressor delivery temperature will be no
more than about 800 K.
In the turbine of an open cycle plant the working fluid will be a mixture of
combustion gases. Most gas turbines run on kerosene which has a composition to
which the fonnula CnH2n is a good approximation. If some such composition is
assumed, the products analysis can be calculated for various fuel/air ratios.
Knowing the specific heats and molecular weights of the constituents it is then a
simple matter to calculate the mean values of cp and t' for the mixture. Figure 2.14
shows that cp increases and y with increase in fuel/air ratio. It is worth
noting that the mean molecular weight of the combustion products from typical
hydrocarbon fuels is little different from that of air, and therefore cp and l' are
closely related by equation (2.23) with R/M =Rair= 0·287 kI/kg K.
METHODS OF ACCOUNTING FOR COMPONENT LOSSES 57
The calculation of the products analyses is very lengthy when dissociation is
taken into account and then, because pressure has a significant effect on the
amount of dissociation, cp and l' become a function of pressure as well as
temperature. Accurate calculations of this kind have been made, and the results
are tabulated in Ref. (3). Dissociation begins to have a significant effect on cp and
y at a temperature of about 1500 K, and above this temperature the curves of Fig.
2. J 4 are strictly speaking applicable only to a pressure of 1 bar. In fact at 1800 K,
both for air and products 'of combustion corresponding to low values of fuel/ air
ratio, a reduction of pressure to 0·0 I bar increases cp by only about 4 per cent and
an increase to 100 bar decreases cp by only about 1 per cent: the corresponding
changes in l' are even smaller. In this book we will ignore any effect of pressure,
although many aircraft and industrial gas turbines are designed to use turbine inlet
temperatures in excess of 1500 K.
Now compressor temperature rises and turbine temperature drops will be
calculated using equations such as (2.11) and (2.l2) or (2.19) and (2.20). For
accurate calculations a method of successive approximation would be required,
i.e. it would be necessary to guess a value of y, calculate the temperature change,
take a more accurate mean value of y and recalculate the temperature change. In
fact, if this degree of accuracy is required it is better to use tables or curves of
enthalpy and entropy as described, for example, in Ref. (1). For preliminary
design calculations and comparative cycle calculations, however, it has been
found to be sufficiently accurate to assume the following fixed values of c and l'
for the compression and expansion processes respectively, p
air:
combustion gases:
Cpa = 1·005 kJ/kg K, 1'a= 1-40 or = 3·5
l' - 1 a
Cpg = 1·148 kJ/lcg K, 1'g = 1·333 or (-. _1'_) = 4·0
} - 1
g
The reason why this does not lead to much inaccuracy is that C and y vary in
opposing senses with T. For cycle analysis we are interested in Zalculating com-
58 sHAFT POWER CYCLES
. .' .
pressor and turbine work from the product c/1T. Suppose. that the temperature for
which the above values of cp and yare the tiue values is lower than the actual
mean temperature. y is then higher than it should be and I1T will be overestim-
ated. This will be compensated in the product cp l1T by the fact that cp will be
lower than it should be. The actual temperatures at various points in the cycle will
not be very accurate, however\ and for the detailed design of the components it is
necessary to know the exact conditions of the working fluid: the more accurate
approaches mentioned above must then be employed.
Fuel/air ratio, combustion efficiency and cycle efficiency
The performance of real cycles can be unambiguously expressed in terms of the
specific fuel consumption, i.e. fuel mass flow per unit net power output. To obtain
this the fuel/air ratio must be found. In the course of calculating the net output
per unit mass flow of air the temperature at inlet to the combustion chamber (Toz)
will have been obtained; and the temperature at outlet (Tm), which is the max-
imum cycle temperature, will normally be specified. The problem is therefore to
calculate the fuel/air ratio / required to transform unit mass of air at Toz and /kg
of fuel at the fuel temperature tr to (l + f) kg of products at Tm·
Since the process is adiabatic with no work transfer, the energy equation is
simply
L:(mihi03 ) - (haOz + /hf ) = 0
where mi is the mass of product i per unit mass of air and hi its specific enthalpy.
Making use of the enthalpy of reaction at a reference temperature of25 DC, i1Hz5,
the equation can be expanded in the usual way [see Ref. (1)] to become
(1 + /)cpiT03 - 298) + / i1HZ5 + cpa(298 - Toz) + /cptC298 - Tf ) = 0
where cpg is the mean specific heat of the products over the temperature range
298 K to T03' i1H
Z5
should be the enthalpy of reaction per unit mass of fuel with
the H20 in the products in the vapour phase, because Tm is high and above the
dew point. For common fuels i1HZ5 may be taken from tables, or alternatively it
may be evaluated from the enthalpies of formation of the reactants. It is usual to
assume that the fuel temperature is the same as the reference temperature, so that
the fourth term on the L.H.S. of the equation is zero. The term will certainly be
small because/is low and cpf for liquid hydrocarbon fuels is only about
2 kJ /kg K. Finally, we have already discussed in the previous section, 'Variation
of specific heat', the calculation of the mean specific heat of the products cpg as a
function of/and T, so we are left with an equation from which/can be obtained
for any given values of T02 and T03·
Such calculations are too lengthy to be undertaken for every individual cycle
calculation-particularly if dissociation is significant because then the /i1HZ5 term
must be modified to allow for the incompletely burnt carbon and hydrogen arising
from the dissociated CO2 and H20. It is usually sufficiently accurate to use tables
METHODS OF ACCOUNTING FOR COMPONENT LOSSES 59
or charts which have been for al typical fuel composition. Figure 2.15
shows the combustion temperature rise (T03 - Toz) plotted against fuel/air ratio
for various values of inlet temperature (Toz), and these curves will be used for all
numerical examples in this book. It is a small-scale version of larger and more
accurate graphs given in Ref. (4). The reference fuel for which the data have been
calculated is a hypothetical liquid hydrocarbon containing 13·92 per cent Hand
86·08 per cent C, for which the stoichiometric fuel/air ratio is 0·068 and i1H25 is
-43 100 kJ/kg. The curves are certainly :adequate for any kerosene burnt in dry
Fuel/air ratio
0.Q16 0.Q18
_
_ 950
900
850
800

] 750
E
8
500

.a 450
i
400
"

il 350
E
8 300
250
200
150
0.004 0.006 0.008 0.010 0.012
Fuel/air ratio
FIG. 2.15 Combustion temperature rise v. fuel/air ratio
'2
o
';;
750 il
700
650
550
§
*
C.
E
8
Cl
" "E
:::J
Ul
.!
'"
5i
500 ";:

450
IIII
!


:::J
I
IIII .c

300 u
0.014
60
SHAFT POWER CYCLES
air. Methods are given in Ref. (4) whereby the data can be used for hydrocarbon
fuels which differ widely in composition from the reference fuel, or where the
fuel is burnt in a reheat chamber, i.e: not in air but in the products of combustion
from a previous chamber in the cycle.
The data for Fig. 2.15 have been calculated on the assumption that the fuel is
completely burnt, and thus the abscissa could be labelled 'theoretical fuel/air
ratio'. The most convenient method of allowing for combustion loss is by
introducing a combustion efficiency defined by
theoretical/ for given IJ.T
lib = actual/for given IJ..T
This is the definition used in this book. An alternative method is to regard the
ordinate as the theoreticallJ.T for a given/ and define the efficiency in terms of
the ratio: actual IJ.T/theoretical IJ.T. Neither definition is quite the same as the
fundamental definition based on the ratio of actual energy released to the theo-
retical quantity obtainable. ('1b differs from it because of the small additional heat
capacity of the products arising from the increase in fuel needed to produce the
given temperature.) But in practice combustion is so nearly complete-98-99 p ~ r
cent-that the efficiency is difficult to measure accurately and the three defini-
tions of efficiency yield virtually the same result.
Once the fuel/air ratio is knovm, the fuel consumption fly is simply / x m
where m is the air mass flow, and the specific fuel consumption can be found
directly from
SFc=L
WN
Since the fuel consumption is normally measured in kg/h, while W N is in kW per
kg/s of air flow, the SFC in kg/leW h is given by the following numerical
equation:t
SFC f [s] 3600/
-WN-/=-'[kW=-s/-=-kg'] x [h] = WN/[kW s/kg]
[kg/kWh]
If the thennal efficiency of the cycle is required, it must be defined in the fonn
'work outp)lt/heat supplied' even though the combustion process is adiabatic and
in the thennodynamic sense no heat is supplied. We know that if the fuel is bumt
under ideal conditions, such that the products and reactants are virtually at the
same temperature (the reference temperature 25°C), the rate of energy release in
the form of heat will be
t The fundamental principle employed here: is that a physical quantity, or the symbol representing it: is
equal to (pure number x unit) and never the number alone. Consequently an equanon relating
numbers only will, in addition to pure numbers and dimensional rallos, contam only quottents of
symbols and units, e.g. WN /[kW s/kgJ.
METHODS OF ACCOUNTING FOR COMPONENT LOSSES 61
where mj is the fuel flow and Qgr,p is the gross (or higher) calorific value at
constant pressure. b. the gas turbine it is not possible to utilize the latent heat of
the H20 vapour in the products and the convention of using the net calorific value
has been adopted in most countries. Thus the cycle efficiency may be defined as
WN
'1=--
./Qnet,p
With the units used here, the equivalent numelical equation becomes
WN/[kW s/kg]
'1 = -=-----=--'-'-'--'=-=--'-=-::::----c:--::
/ x Qnet,p/[kJ or leW s/kg]
or, using the above numerical equation for SFC
3600
'1 = -:.,----,,,----:-------,---
SFC/[kg/kW h] x Qnet,p/[kJ/kg]
Qnet,p is sensibly equal in magnitude but opposite in sign to the enthalpy of
reaction AH25 referred to earlier, and the value of 43 100 kI/kg will be used for
all numerical examples.
When referring to the thermal efficiency of actual gas turbines, manufacturers
often prefer to use the concept of heat rate rather than efficiency. The reason is
that fuel prices are normally quoted in terms of pounds sterling (or dollars) per
megajoule, and the heat rate can be used to evaluate fuel cost directly. Heat rate is
defined as (SFC x Qnet,p), and thus expresses the heat input required to produce a
unit quantity of power, It is nonnally expressed in leJ /kW h, in which case the
corresponding thermal efficiency can be found from 3600/(heat rate).
Bleed flows
In section 2. I the importance of high turbine inlet temperature was demonstrated.
Turbine inlet temperatures, however, are limited by metallurgical considerations
and many modem engines make use of air-cooled blades to pennit operation at
elevated temperatures. It is possible to operate with uncooled blades up to about
1350-1400 K; and then it can be assumed that the mass flow remains constant
throughout as explained earlier. At higher temperatures it is necessary to extract
air to cool both stator and rotor blades. The required bleeds may amount to 15 per
cent or more of the compressor delivery flow in an advanced engine, and must be
properly accounted for in accurate calculations. The overall air cooling system for
such an engine is complex, but the methods of dealing with bleeds can be
illustrated using a simplified example. A single-stage turbine with cooling of both
the stator and rotor blades will be considered, as shown schematically in Fig.
2.16; PD, fis and fiR denote bleeds for cooling the disc, stator and rotor respec-
tively. Bleeds are normally specified as a percentage of the compressor delivery
flow.
Note that bleed f3D acts to prevent the flow of hot gases dovm the face of the
turbine disc, but it does pass through the rotor. The stator bleed also passes
62
SHAFT POWER CYCLES
FIG. 2.16 Cooling air schematic
through the rotor and both contribute to the power developed. The rotor bleed,
however, does not contribute to the work output and the reduction in mass flow
results in an increase in both temperature drop and pressure ratio for the specified
power. Useful work would be done by the rotor bleed only jf there was another
turbine downstream.
lfthe airflow at compressor delivery is mm then the flow available to the rotor,
mR, is given by
The fuel flow is found from the fuel/air ratio required for the given combustion
temperature rise and combustion inlet temperature, and the air available for
combustion, i.e.
Hence the fuel flow is given by
It should be noted that the cooling flows from 1he stator and disc, at the com-
pressor delivery temperature, will cause some reduction in the effective temper-
ature at entry to the rotor. This effect can be estimated by carrying out an enthalpy
balance assuming complete mixing of the flows. Such perfect mixing does not
occur in practice, but accurate cycle calculations would require a good estimate to
be made of this cooling of the main flow.
In a typical cooled stator with a fJs of 6 per cent, the stator outlet temperature
may be reduced by about 100 K. Since the rotor inlet temperature is equal to the
stator outlet temperature this implies a reduction in turbine power output. In
addition there will be a small drop in efficiency due to the mixing of the bleeds
with the main stream.
DESIGN POINT PERFORMANCE CALCULATIONS 63
Before assessing the effect of component losses on the general performance
curves for the various cycles considered in section 2.1, it is necessary to outline
the method of calculating the performance in any particular case for specified
values of the design parameters. These parameters will include the compressor
pressure ratio, turbine inlet temperature, component efficiencies and pressure
losses.
Several examples will be used to show how these effects are incorporated in
realistic cycle calculations. For the first example we consider a single-shaft unit
with a heat-exchanger, using modest values of pressure ratio and turbine inlet
temperature, and typical isentropic efficiences. A second example shows how to
calculate the perfonnance of a simple cycle of high pressure ratio and turbine
temperature, which incorporates a free power turbine. Finally, we will evaluate the
pelformance of an advanced cycle using reheat, suitable for use in both simple
and combined cycle applications: this example will illustrate the use of polytropic
efficiencies.
EXAMPLE 2.1
Determine the specific work output, specific fuel consumption and cycle effici-
ency for a heat-exchange cycle, Fig. 2.17, having the following specification:
Compressor pressure ratio
Turbine inlet temperature
Isentropic efficiency of compressor, rye
Isentropic efficiency of turbine, I],
Mechanical transmission efficiency, '1m
Combustion efficiency, ryb
Heat-exchanger effectiveness
Pressure losses-
Combustion chamber, IlPb
Heat-exchanger air-side, IlPha
Heat-exchanger gas-side, IlPhg
Ambient conditions, Pm Ta
FIG. 2.17 Heat-exchallge cycie
4·0
llOO K
0·85
0·87
0·99
0·98
0·80
2% compo deliv. press.
3% compo deliv. press.
0·04 bar
1 bar. 288 K
64
SHAFT POWER CYCLES
Since TO! = Ta and POI = Pa,and 'Y == 1·4, the temperature equivalent of the
compressor work from equation (2.11) is
r: - T == -'!. P02 --1
T [( )(1'-1)/1' ]
02 a I'/e Pa
= == 164·7 K
0·85
Tmbine work required to drive compressor per unit mass flow is
cpaCTo2 - Ta) 1·005 x 164·7 _ 16721 J/l
w: - =------. ( cg
Ie - I'/m 0·99
P03 = P02 1 - - - - = 4·0(1 - 0·02 - 0·0 = ·8 ar (
!J,Pb !J,Pha) 3) 3 b
P02 P02
P04 = Pa + !J,Phg = 1·04 bar" and hence P03/P04 == 3·654
Since 'Y = 1·333 for the expanding gases, the temperature equivalent of the total
turbine work from equation (2.13) is
[
(_ 1 )(1'-I)/1'J
T03 - T04 = I'/IT03 1 - \P-;;;/P04
= 0.87 x 1100[1 - (_1_)1
/
4J = 264·8 K
3·654
Total turbine work per unit mass flow is
WI = CpiT03 - T04) = 1·148 x 264·8 = 304·0 kJ/kg
Remembering that the mass flow is to be assumed the same throughout the unit,
the specific work output is simply
WI - W
IC
= 304 -167·2 = 136·8 kJ/leg (or leW s/kg)t
(It follows that for a 1000 kW plant an air mass flow of 7·3 kg/s would be
required.) To find the fuel/air ratio we must first calculate the combustion tem-,\
perature rise (T03 - T05)'
From equation (2.22),
• . T05 - T02
heat-exchanger effectIveness == 0·80 = r: r:
04 - 02
T02 == 164·7 + 288 == 452·7 K, and T04 == 1100 - 264·8 = 835·2 K
t Note: I kW s/lcg == 0·6083 hp s/Ib.
DESIGN POINT PERFORMANCE CA.LCULA.TIONS 65
Hence
Tos == 0·80 x 382·5 + 452·7 = 758·7 K
From Fig. 2.15, for a combustion chamber inlet air temperature of 759 K and a
combustion temperature rise of (llOO - 759) = 341 K, the theoretical fuel/air
ratio required is 0 ·0094 and thus
f
iheoretical f 0·0094
= = --= 0·0096
I1b 0·98
The specific fuel consumption is therefore
3600 x 0·0096
136.8 = 0·253 kg/leW ht
Finally, the cycle efficiency is
3600 3600
1'/= = =0·331
SFC x Qnet,p 0·253 x 43 100
EXAMPLE 2,2
Determine the specific work output, specific fuel consumption and cycle effici-
ency for a simple cycle gas turbine with a free power turbine (Fig. 2.18) given the
following specification:
Compressor pressme ratio
Turbine inlet temperatllre
Isentropic efficiency of compressor, I1c
Isentropic efficiency of each turbine, 1'/1
Mechanical efficiency of each shaft, 11m
Combustion efficiency
Combustion chamber pressme loss
Exhaust pressme loss
Ambient conditions, Pa, Ta

-+-- _wtp
I w
tc
'" G t > I Power turbine
as genera or
FIG. 2.111 Free turbine nnit
t Note: 1 kg/leW h '" 1·644 Ib/hp h.
12·0
1350 K
0·86
0·89
0·99
0·99
6% compo deliv. press
0·03 bar
1 bar, 288 K
66
SHAFT POWER CYCLES
Proceeding as in 0 the previous example,
T02 - TOI = 288 [121i3oS - 1] = 346·3 K
0·86
= 1·005 x 346·3 = 351.5 kJ/k
W;c 0.99. g
P02 = 12·0(1 - 0·06) = 11·28 bar
The intermediate pressure between the two turbines, P04, is unknown, but can be
determined from the fact that the compressor turbine produces just sufficient
work to drive the compressor. The temperature equivalent of the compressor
turbine work is therefore
W;c 351·5
T03 - T04 = -- = --= 306·2 K
cpg 1·148
The corresponding pressure ratio can be found wing equation (2.12)
T03 - T04 = 1'/t T03 [1 - Y-IIY]
[
( 1 )0.25]
306·2 = 0·89 x 1350 1 - \P03/P04
P03 = 3.243
P04
T04 = 1350 - 306·2 = 1043-8 K
The pressure at entry to the power turbine, P04, is then found to be 11.28/
3.243 = 3·478 bar and the power turbine pressure ratio is 3.478/(1 + 0·03) =
3·377.
The temperature drop in the power turbine can now be obtained from equation
(2.12),
[ (
1 )0.25]
T04 - Tos = 0·89 x 1043·8 1 - 3.377 = 243·7 K
and the specific work output, i.e. power turbine work per unit air mass flow, is
Wtp = Cpg(T04 - TOS)1'/m
Wtp = 1·148(243·7)0·99 = 277-0 kJ/kg(or kW s/kg)
The compressor delivery temperature is 288 + 346·3 = 634·3 K and the com-
bustion temperature rise is 1350 - 634·3 = 715·7 K; from Fig. 2.15, the
theoretical fuel/air ratio required is 0·0202 giving an actual fuel/air ratio of
0·0202/0·99 = 0·0204.
DESIGN POINT PERFORMANCE CALCULATIONS
The SFC and cycle efficiency, 1'/, are then given by
SFC = L = 3600 x 0·0204 = 0.265 leg/leW h
Wtp 277-9
_ 3600 _ 0.315
1'/ - 0.265 x 43100 -
67
It should be noted that the cycle calculations have been carried out as above to
determine the overall performance. It is important to realize, however, that they
also provide information that is needed by other groups such as the aerodynamic
and control design groups, as discussed in section 1.8. The temperature at entry to
the power turbine, T04, for example, may be required as a control parameter to
prevent operation above the metallurgical limiting temperature of the compressor
turbine. The exhaust gas temperature (EG'l),Tos, would be important if the gas
turbine were to be considered for combined cycle or cogeneration plant, which
were first mentioned in Chapter 1, and will also be discussed in section 2.5. For
the cycle of Example 2.2, Tos= 1043·8 - 243·7 =800·1 K or 527°C, which is
suitable for use with a waste heat boiler. When thinking of combined cycle plant,
a higher TIT might be desirable because thf:re would be a consequential increase
in EGT, permitting the use of a higher steam temperature and a more efficient
steam cycle. If the cycle p(essure ratio were increased to increase the efficiency of
the gas cycle, however, the EGTwould be decreased resulting in a lower steam
cycle efficiency. The next example will illustrate how the need to make a gas
turbine suitable for more than one applic:ation can affect the choice of cycle
parameters.
EXAMPLE 2.3
Consider the design of a high pressure ratio, single-shaft cycle with reheat at
some point in the expansion when used either as a separate unit, or as part of a
combined cycle. The power required is 240 MW at 288 K and 1·01 bar.
Compressor pressure ratio
Polytropic efficiency
(compressor and turbines)
Turbine inlet temperature
(both turbines)
/)"P/P02 (1st combustor)
/)"P/P04 (2nd combustor)
Exhaust pressure
30
0·89'
1525 K
0·02
0·04
1·02 bar
The plant is shown in Fig. 2.19. A heat ex,changer is not used because it would
result in an exhaust temperature that would be too low for use with a high-
efficiency steam cycle.
----------------------------------
To simplify the it will be assumed that the mass flow is constant
throughout, ignoring the effect of the sub8tantial cooling bleeds that would be
68
SHAFT POWER CYCLES
Reheat chamber
2
1
5
.
6
Fuel
I
!===!
HPturb.
LPturb.
FIG. 2.19 Reheat cycle
required with the high turbine inlet temperatures specified. The reheat is
not specified, but as a starting point it is reasonable to use the value glYmg the
same pressure ratio in each turbine. (As shown in section 2.1, this division of the
expansion leads to equal work in each turbine and a maximum net work output
for the ideal reheat cycle).
It is convenient to start by evaluating (n - 1 )/n for the polytropic compression
and expansion. Referring to equations (2.19) and (2.20), we have:
for compression, n: 1 = e 1) = G:!) = 0·3210
for expansion, n -. 1 = 11 (y - 1) = 0.89(0.333) = 0.2223
n oot y 1·333
Making the usual assumption that POI = Pa and TOl = Tao we have
T02 = 858·1 K
T02 - TO! = 570·1 K
P02 = 30 x 1·01 = 30·3 bar
P03 = 30·3(1·00 - 0·02) = 29·69 bar
P06 = 1·02 bar, so Pm = 29·11
P06
Theoretically, the optimum pressure ratio for each turbine would be ')(29.11) =
5.395. The 4 per cent pressure 10s8 in the reheat combustor has to be considered,
so a value of 5.3 for P03/P04 could be assumed. Then
T03 = (5.3)0.2223 T04 ,= 1052·6 K
T04
P04 = 29-69/5·3 = 5·602 bar
Pos = 5·602(1·00 - 0·04) = 5·378 bar
POS/P06 = 5·378/1·02 = 5·272
T06 = 1053-8 K
DESIGN POINT PERFORMANCE CALCULATIONS
Assuming unit flow of 1·0 kg/s, and a mechanical efficiency of 0·99
Turbine output, Wt = 1·0 x 1·148{(1525 -1052·6)
+ (1525 - 1053-8)} x 0·99
= 1072-3 kJ/kg
Compressor input, We = 1·0 x 1·005 x 570·1
= 573-0kJ/kg
Net work output, WN = 1072-3 - 573·0 = 499·3 kJ/kg
Flow required for 240 MW is given by
240000
m = 499.3 = 480·6 kg/s
69
The combustion temperature rise in the first combustor is (1525 - 858) = 667
K, and with an inlet temperature of 858 K the fuel/air ratio is found from
Fig. 2.15 to be 0·0197. For the second combustor the temperature rise is
(1525 - 1052·6) =472-4 K giving a fuel/air ratio of 0·0142. The total fuel/air
ratio is then
f = 0·0197 + 0·0142 =: 0.0342
0·99
and the thennal efficiency is given by
499·3
11 = 0.0342 x 43 100
This is a reasonable efficiency for simple cycle operation, and the specific output
is excellent. Examination of the temperature at turbine exit, however, shows that
T06 = 1053·8 K or 780·8 DC. This temperature is too high for efficient use in a
combined cycle plant. A reheat steam cycle using conventional steam tempera-
tures of about 550-575 DC would require a turbine exit temperature of about
600 DC.
The turbine exit temperature could be reduced by increasing the reheat
pressure, and if the calculations are repeated for a range of reheat pressure the
results shown in Fig. 2.20 are obtained. It can be seen that a reheat pressure of 13
bar gives an exhaust gas temperature (EGT) of 605 DC; the specific output is
about 10 per cent lower than the optimum value, but the thennal efficiency is
substantially improved to 37·7 per cent. Further increases in reheat pressure
would give slightly higher efficiencies, but the EGT would be reduced below
600 DC resulting in a less efficient steam cycle. With a reheat pressure of 13 bar,
the first turbine has a pressure ratio of 2·284 while the second has a pressure ratio
of 12·23, differing markedly from the equal pressure ratios we assumed at the
outset. This example illustrat'es some of the problems that arise when a gas
turbine must be designed for more than one application.
70
EGT
['C]
WN
[kWs/k91
5 0 0 ~ 480
460
440
420L-_...L_-L_---1 __ .L-_--l
4 6 8 10 12 14
Reheat pressure
[barl
FIG. 2.20 Effect of varying reheat pressure
SHAFT POWER CYCLES
1]%
2.4 Comparative performance of practical cycles
The large number of variables involved make it impracticable to derive algebraic
expressions for the specific output and efficiency of real cycles. On the other
hand, the type of step-by-step calculation illustrated in the previous section is
ideally suited for computer programming, each of the design parameters being
given a set of values in turn to elicit their effect upon the performance.
Some performance curves will now be presented to show the main differences
between practical and ideal cycles, and the relative importance of some of the
parameters. The curves are definitely not a comprehensive set from which
designers can make a choice of cycle for a particular application. To emphasize
that too much importance should not be attached to the values of specific output
and efficiency the full specification of the parameters has not been given: it is
sufficient to note that those parameters which are not specified· on the curves are
kept constant. All the curves use compressor pressure ratio rc as abscissa, the
turbine pressure ratio being less than rc by virtue of the pressure losses. The cycle
efficiency has been evaluated to facilitate comparison with the ideal curves of
section 2.1. In practice it is usual to quote SFC rather than efficiency, not only
because its definition is unambiguous, but also because it provides both a direct
indication of fuel consumption and a measure of cycle efficiency to which it is
inversely proportional.
Simple gas turbine cycle
When component losses are taken into account the efficiency of the simple cycle
becomes dependent upon the maximum cycle temperature T03 as well as pressure
COMPARATIVE PERFORMANCE OF PRACTICAL CYCLES 71
ratio, Fig. 2.21. Furthermore, for each temperature the efficiency has a peak value
at a particular pressure ratio. The fall in efficiency at higher pressure ratios is due
to the fact that the reduction in fuel supply to give the fixed turbine inlet temp-
erature, resulting from the higher compressor delivery temperature, is outweighed
by the increased work necessary to drive the compressor. Although the optimum
pressure ratio for maximum efficiency differs from that for maximum specific
output, the curves are fairly flat near the peak and a pressure ratio between the two
optima can be used without much loss in efficiency. It is perhaps worth pointing
out that the lowest pressure ratio which willI give an acceptable performance is
always chosen: it might even be slightly lower than either optimum value. Me-
chanical design considerations beyond the scope of this book may affect the
choice: such considerations include the number of compressor and turbine stages
required, the avoidance of excessively small blades at the high pressure end of the
compressor, and whirling speed and bearing problems associated with the length
of the compressor-turbine combination.
The advantage of using as high a value of T03 as possible, and the need to use a
higher pressure ratio to take advantage of II higher permissible temperature, is
evident from the curves. The efficiency increases with T03 because the component
losses become relatively less important as Ithe ratio of positive turbine work to
negative compressor work increases, although the gain in efficiency becomes
marginal as T03 is increased beyond 1200 K (particularly if a higher temperature
requires a complex turbine blade cooling system which incurs additional losses).
There is nothing marginal, however, about the gain in specific work output with
increase in T03. The consequent reduction in size of plant for a given power is
very marked, and this is particularly import:amt for aircraft gas turbines as will be
emphasized in the next chapter.
The following figures illustrate the relative importance of some of the other
parameters. Changes in efficiency are quoted as simple differences in percentages.
With a T03 of 1500 K, and a pressure ratio n c ~ a r the optimum value, an increase of
5 per cent in the polytropic efficiency of either the compressor or turbine would
T03 = 1500 K
T03 = 1500 K
30 Ol 300 -
~
"# '"
"'"
~
(; 20 ::::- 200 -
"
"5
.!lI a.
~
"5
1)00,0.87
0
'"
10 ~ 100-
~
"0 1)001 0.85 0
f5
'"
T.288 K
a.
en
0 0 I I I I I I I I
0 2 4 6 8 10 12 14 16 () 2 4 6 8 10 12 14 16
Compressor pressure ratio fc fc
FIG. 2.21 Cycle efficiency and specific output of simple gas turbine
72
SHAFT POWER CYCLES
increase the cycle efficiency by about 4 per cent and the specific output by about
65. kW s/kg. (If isentropic efficiences had been used, the turbine loss would have
been seen to be more important than the compressor loss but the use of polytropic
efficiencies obscures this fact.) A re:duction in combustion chamber pressure loss
from 5 per cent of the compressor delivery pressure to zero would increase the
cycle efficiency by about 1·5 per cent and the specific output by about
12 kW s/kg. The remaining parameter of importance is the ambient temperature,
to which the performance of gas turbines is particularly sensitive.
The ambient temperature affects both the compressor work (proportional to Ta)
and the fuel consumption (a function of T03 - T02)' An increase in Ta reduces
both specific output and cycle efficiency, although the latter is less affected than
the former because for a given To:; the combustion temperature rise is reduced.
Considering again the case of 1(J3 = 1500 K and a pressure ratio near the
optimum, an increase in Ta from 15 to 40 DC reduces the efficiency by about 2·5
per cent and the specific output by about 62 kW s/kg. The latter is nearly 20 per
cent of the output, which emphasizes the importance of designjng a gas turbine to
give the required power output at the highest ambient temperature likely to be
encountered.
Heat-exchange (or regenerative) cycle
As far as the specific work output is concerned, the addition of a heat-exchanger
merely causes a slight reduction due to the additional pressure losses: the curves
retain essentially the same form as those in Fig. 2.21. The efficiency curves are
very different, however, as shown in Fig. 2.22. Heat-exchange increases the
efficiency substantially and mark,edly reduces the optimum pressure ratio for
maximum efficiency. Unlike the corresponding curves for the ideal cycle, they do
not rise to the Carnot value at rc = 1 but fall to zero at the pressure ratio at which
the turbine provides just sufficient work to drive the compressor: at this point
50
40
10
T03 = 1500 K
__ -----..:11 Effectiveness
/ .... , ... 0.875
f
I
: 1200 K
f(/\1000K
f /;\ck
y ..... "dl.""
,-
0.87
0.85
Ta 288 K
Effectiveness 0.75
o L..O.....1..2 ':-0 Ei
lFIG. 2.22 Heat-exchange cycle
COMPARATIVE PERFORMANCE OF PRACTICAL CYCLES 73
there will be positive heat input with zero net work output. The spacing of the
constant TOJ curves in Fig. 2.22 indicate that when a heat-exchanger is used the
gain in efficiency is no longer merely marginal as T03 is raised above 1200 K.
This is a most important feature of the heat-exchange cycle, because progress in
materials science and blade cooling techniques has enabled the permissible
temperature to be increased at an average rate of about 10K per year and there is
every hope that this will continue. It should also be noted that the optimum
pressure ratio for maximum efficiency increases as T03 is increased. Our study of
the ideal heat-exchange cycle suggested that no such increase in pressure ratio
was required: this was the conclusion under the heading 'Heat-exchange cycle' in
section 2.1 which it was state:d would have to be modified. Nevertheless, it
remains true to say that no very high pressure ratio is ever required for a heat-
exchange cycle, and the increase in weight and cost due to a heat-exchanger is
partially offset by the reduction in size of the compressor.
The dotted curves have been added to show the effect of heat-exchanger
effectiveness. Not only does an increase in effectiveness raise the cycle efficiency
appreciably, but it also reduces the value of the optimum pressure ratio still
further. Since the optimum rc for maximum efficiency is below that for maximum
specific output, it is inevitable that a plant designed for high efficiency will suffer
a space and weight penalty. For example, with a T03 of 1500 K. and an
effectivenes of 0·75, the optimum rc for maximum efficiency is about 10 at which
(from Fig. 2.21) the specific output is not far short of the peak value. But with the
effectiveness increased to 0·875 the optimum rc is reduced to about 6, at which
the specific output is only about 90 per cent of the peale value. The position is
similar at the lower (and at present more realistic) values of To3 .
An alternative method of presenting performance characteristics is to plot the
variation of specific fuel consumption and specific output on a single figure for a
range of values of pressure ratio and turbine inlet temperature. Examples for
realistic values of polytropic efficiency, pressure losses and heat-exchanger
effectiveness are shown in Figs 2.23(a) and (b) for simple and heat-exchange
cycles respectively. The marked effect of increasing T03 on specific output in both
cases is clearly evident. Such plots also contrast clearly the small effect of T03 on
SFC for the simple cycle with the much greater effect when a heat-exchanger is
used.
Heat-exchange (regenerative) cycle with reheat or intercooling
Our study of ideal cycles suggested that there is no virtue in employing reheat
without heat-exchange because of the deleterious effect on efficiency. Tllis is
generally true of practical cyclles also, and so we will not include performance
curves for this case here. (Exceptions to the rule will be discussed at the end of
this section.) With heat-exchange, addition of reheat improves the specific output
considerably without loss of efficiency (cf. Figs 2.24 and 2.22). The curves of
Fig. 2.24 are based on the assumption that the gas is reheated to the maximum
74
SFC
[kg 1
kWh
SFC
SHAFT POWER CYCLES
0.35 (a) Simple cycle
Turbine inlet
temperature Tosl[KJ
) .....
1100
0.30
12
0.25
16
0.20L _____ -'-____ ----::l-:--_____ :;'.
100 200 300 400
0.25
Specific work output
[kW s/kgJ
(b) Heat-exchange cycle
Compressor 9 1100
pressure I 8
ratio r 7
4 6
Turbine inlet
0.20
1500
O. 15 l.-______ L-______ -:-:-:1_____ -:::1
100 200 300 400
Specific work output
[kW s/kg]
FIG. 2.23 Cyde performance cllrves
COMPARATIVE PERFORMANCE OF PRACTICAL CYCLES 75
T03 = 1500 K
T03 =1500K
40 400 -
1200 K
;R
0.,. 30
-
>-
"' 1200 K '-'
I
c:
CD
·13
20 i5. 200 -
CD
::;
#""""--- ...... _---
u
0
>- 0.87
"
.-
u

",..-----
10
0.85
1i 100-
r; 1000K
Ta 288K rJ)
Effectiveness 0.75
0 I I 0 I I I I I I I 1
0 4 6 8 10 12 14 16 (I 2 4 6 8 10 12 14 16
rc rc
FIG. 2.24 Cycle with heat-exchalll.ge amI lI"ehealt
cycle temperature at the point in the expansion giving equal pressure ratios for the
two turbines. The gain in efficiency due to reheat obtained with the ideal cycle is
not realized in practice, partly because of the additional pressure loss in the reheat
chamber and the inefficiency of the expansion process, but primarily because the
effectiveness of the heat-exchanger is well short of unity and the additional
energy in the exhaust gas is not wholly recovered. It is important to use a pressure
ratio not less than the optimum value for maximum efficiency, because at lower
pressure ratios the addition of reheat can actually reduce the efficiency as indi-
cated by the curves.
Reheat has not been widely used in practice because the additional combustion
chamber, and the associated control problems, can offset the advantage gained
from the decrease in size of the main components consequent upon the increase
in specific output. With the exception of the application mentioned at the end of
this section, reheat would certainly be considered only (a) if the expansion had to
be split between two turbines for other reasons and (b) if the additional :flexibility
of control provided by the reheat fuel supply was thought to be desirable. With
regard to (a), it must be noted that the natural division of expansion between a
compressor turbine and power turbine may not be the optimum point at which to
reheat and if so the full advantage of reheat will not be realized. Finally, readers
familiar with steam turbine design will understand that reheat also introduces
additional mechanical problems arising from the decrease in gas density, and
hence the need for longer blading, in the low-pressure stages.
Intercooling, which has a similar effect upon the performance of the ideal heat-
exchange cycle as reheat, does not suffer from the same defects. When
incorporated in a practical cycle, even allowing for the additional pressure loss
there is, in addition to the marked increa8e in specific output, a worthwhile
improvement in efficiency. Nevertheless, as pointed out in section 2.1,
76 SHAFT POWER CYCLES
intercoolers tend to be bulky, and if they require cooling water the self-contained
nature of the gas turbine.is lost. The cycle is attractive for naval applications,
however, because cooling water is readily available-the ocean. Not only does the
intercooled cycle with heat-exchange yield a thennal efficiency in excess of 40
per cent, but it also has a good part-lload efficiency. The latter feature means that it
might be possible to use a single engine rather than a COGOG arrangement, thus
offsetting the greater bulk and cost of the engine.
We have already met one exception to the rule that there is little advantage in
incorporating reheat without heat-exchange--in example 2.3. In general, when a
gas turbine is part of a combined cycle plant, or cogeneration scheme, a heat-
exchanger would be superfluous. The increase in the gas turbine exhaust
temperature due to reheat is utilized by the bottoming steam cycle or waste-heat
boiler. The addition of reheat to the gas cycle can then result in an increase in
efficiency as well as specific output. Because of the need to reduce the
enviromnental impact of bruning fossil fuels, there is increasing pressure to seek
the highest possible efficiency from our large base-load power stations. It may be
that gas turbines with reheat will be found to be an economic solution in
combined cycle plant, and ABB iintroduced machines of this type in the mid
1990s. Studies suggest efficiencies of over 60 per cent may be achieved by using
complex cycles involving intercooling, heat-exchange and reheat (Ref. (5).
We may conclude this section by emphasizing that in practice most gas
turbines utilize a high-pressure ratio simple cycle. The only other widely used
cycle is the low-pressure ratio heat-exchange variety, but even so the number of
engines built with heat-exchangers is only a small fraction of the output of the gas
turbine industry. The other modifications mentioned do not normally show
sufficient advantage to offset the increased complexity and cost.
2.5 Combined cycles and cogeneration schemes
In the gas turbine, practically all the energy not converted to shaft power is
available in the exhaust gases for 'Other uses. The only limitation is that the final
exhaust temperature, i.e. the stack temperature, should not be reduced below the
dewpoint to avoid corrosion problems arising from sulphur in the fuel. The
exhaust heat may be used in a v8,riety of ways. If it is wholly used to produce
steam in a waste heat boiler for a steam turbine, with the object of augmenting the
shaft power produced., the system is referred to as a combined gas/steam cycle
powerplant or simply a combined cycle plant (Fig. 1.3). Alternatively, the exhaust
heat may be used to produce hot water or steam for district or factory heating, hot
gas or steam for some chemical process, hot gas for distillation plant, or steam for
operating an absorption refrigerator in water chilling or air-conditioning plant.
The shaft power will nonnally be used to produce electricity. In such circum-
stances the system is referred to lIIS a cogeneration or CHP plant (Fig. 1.18). We
shall consider briefly the main characteristics of these two types of system in tum.
COMBINED CYCLES AND COGENERATION SCHEMES
77
Combined cycle plant
The optimization of binary cycles is too complex to be discussed in detail in this
book,. but the more important decisions that have to be taken when designing
combmed cycle plant can be described in broad tenns.
Figure 2.25(a) shows the gas and steam conditions in the boiler on a T-H
diagram. The enthalpy rise between feed water inlet and steam outlet must equal
the enthalpy drop of the exhaust gases in the waste heat boiler (WHB), and the
pinch point and terminal temperature differences cannot be less than about 20°C
if the boiler is to be of economic size. It follows that a reduction in gas turbine
exhaust temperature will lead to a reduction in the steam pressure that can be used
for the steam cycle. In the combined cycle, therefore, selection of a higher
compressor pressure ratio to improve the gas turbine efficiency may lead to a fall
in cycle. efficiency and a reduction in overall thennal efficiency. In practice,
heavy mdustnal gas turbines must be acceptable both in simple and combined
cycle markets and it appears that a pressure ratio of around 15, giving an exhaust
gas temperature (EGT) of between 550 and 600°C, is suitable for both. Aircraft
derivative engines with design pressure ratios of 25-35, yield lower values of
EGT of a?out 450°C which result in the use of less efficient steam cycles.
Combmed cycle plant are used for large base-load generating stations and the
overall thennal efficiency, while very important, is not the ultimate criterion
which is the cost of a unit, of electricity sent out. This will depend on both
efficiency and the capital cost of the plant. For eXaInple, if a dual-pressure steam
was used., as in the plant shown in Fig. 1.3, a higher efficiency would be
obtamed because the average temperature at which heat is transferred to the steam
is 2.25(b) will make this clear. The additional complication
would certamly mcrease the cost of the boiler and steam turbine, and a detailed
study would be required to see: if this would lead to a reduction in the cost of
electricity produced. The results of such a study can be found in Ref. (6). Dual-
T
120'C
Terminal temp. diff '-U
TT4
Enthalpy H
(a)

o

a..
:r:
Enthalpy H
(b)
FIG. 2.25 T-H diagrams for single and dual-pressure combined cycle plant
78 SHAFT POIVER CYCLES
pressure cycles have, in fact, been widely used; with modern gas turbines yielding
exhaust gas temperatures close to 600 DC, triple-pressure cycles with reheat have
been found to be economic.
Two other decisions have to be made at an early stage in the design process.
There is the possibility of associating more than one gas turbine with a single
WHB and steam turbine. Such an arrangement would pennit each gas turbine to
be fitted with its own diverter valve and stack so that any particular gas turbine
could be shut down for maintenance. A second option requiring a decision is
whether or not to arrange for extra fuel to be burnt in the WHB, referred to as
supplementmy firing. This could be used for increasing the peak load for short
periods, but would increase the capital cost of the WHB substantially. The
exhaust gases from the supplementary firing do not pass tln·ough a turbine, which
opens the possibility of burning heavy oil or coal in the WHB. This is usually
ruled out by the logistic difficulty and cost of supplying a large plant with more
than one fuel. It may become economic in the future, however, if the prices of oil
and gas increase sharply relative to coaL
We will conclude this subsection by showing how to estimate the improvement
in perfonuance resulting from the addition of an exhaust-heated Rankine cycle to
a gas cycle. Only the simple case of a single-pressure superheat cycle (Fig.
2.25(a)) will be dealt with. The heat transferred from the exhaust gas is mgCpg
(T4 - Tstack) where T4 is the temperature at exit from the turbine. This is equal to
ms(h - hw) where ms is the steam flow and hand hw are the specific enthalpies at
steam turbine inlet and WHB inlet respectively. The superheat temperature is
fixed by T4 and the terminal temperature difference, and the enthalpy will depend
on the pressure. The pinch point temperature, Tp, is fixed by the pinch point
temperature difference and the saturation temperature of the steam. The mass flow
of steam is obtained from
mih - hf ) = mgcpiT4 - Tp)
where h
f
is the saturated liquid enthalpy. The stack temperature can then be
determined fl:om
mgcpg(Tp - Tstack) = ms(h
f
- hw)
The total power from the combined cycle is Wgt + Wst but the heat input (from the
fuel burned in the gas turbine) is unchanged. The overall efficiency is then given
by
It is useful to express this in tenus of the gas turbine and steam turbine
efficiencies to illustrate their individual effects. IiVst = I]st Qst, where QSI is the
notional heat supplied to the steam turbine:
Qst = mgCpg(T4 - Tstack)
This, in turn, can be expressed in tenus of the heat rejection from the gas turbine
assuming the turbine exhaust gas at T4 is cooled without useful energy extraction
COMBINED CYCLES AND COGENERA.TJON SCHEMES
79
from Tstack to the ambient temperature Ta, i.e. mgCpg (T4 - Ta) or Q (1 - I]gt).
Thus
Q = m C (T - T ) (T4 - Tstack) = Q(l _ ) (T4 - Tstack)
sl g pg 4 a (T4 _ Ta) I]gl (T4 - Ta)
The overall efficiency is then given by
W IiV W
11 = ~ + - E = ~ + I]stQst
Q Q Q Q
or
n = I] + I] (1 -I] ) (T4 - Tsack)
./ gt st gl (T4 - Ta)
It can now be seen that the overall efficiency lis influenced by both the gas turbine
exhaust temperature and the stack temperature. A typical modem gas turbine used
in combined cycle applications would have an exhaust gas temperature of around
600 DC and a thenual efficiency of about ~ , 4 per cent. The stack temperature
could be around 140 DC ifliquid fuel were being burned, or 120 DC when the fuel
was natural gas, which has a very low sulphur content. A single-pressure steam
cycle might give around 32 per cent thenual efficiency. Hence
[
600 - 120J
I] = 0·34 + 0·32(1 - 0·34) --;:-
. 600- b
= 0·513
Using a more complex steam cycle with I]st:= 0·36, the overall efficiency would
be raised to 53·5 per cent, which is typical of modem combined cycles.
Cogeneration plant
In a cogeneration plant, such as that in Fig. 1.18, the steam generated in the
WHBs is used for several different purposes such as a steam turbine drive for a
centrifugal water chiller, a heat source for absorption chillers, and to meet process
and heating requirements. Each user, such as a hospital or paper making factory,
will have different requirements for steam, and even though the same gas turbine
may be suitable for a variety of cogeneration plant, the WHBs would be tailored
to the specific requirements of the user.
A major problem in the thenuodynamic design of a cogeneration plant is the
balancing of the electric power and steam loads. The steam requirement, for
example, will depend on the heating and cooling loads which may be subject to
major seasonal variations: the electrical demand may peak in either winter or
summer depending on the geographical 10ca1lion. In some cases, the system may
be designed to meet the primary heating load and the gas turbine selected may
provide significantly more power than demanded by the plant; in this case,
surplus electricity would be exported to the local utility. Another possibility is for
electricity generation to be the prime requirement, with steam sold as a by-
product to an industrial complex for process use. The economics and feasibility
80 SHAFT POWER CYCLES
of a cogeneration plant are closely related to the commercial arrangements that
can be made with the local.electric utility or the steam host. Itwould be necessary,
for to negotiate rates at which the utility would purchase power and also
to ensure that back-up power would be available during periods when the gas
turbine was out of service; these could be planned outages, for maintenance, or
forced outages resulting from system failure. Such considerations are crucial
when the cogeneration plant fonns the basis of a district heating system.
When designing a cogeneration plant care must be taken in specifYing the
ambient conditions defining the site rating. In cold climates, for example, the gas
turbine exhaust temperature would he significantly reduced, resulting in a loss of
steam generating capacity; this may require the introduction of supplementary
firing in the WHB. If the design were based on standard day conditions, it could
be quite unsatisfactory on a very cold day. The site rating must take into account
the seasonal variation of temperature and ensure that the critical requirements
can be met. The off-design performance of gas turbines will be discussed in
Chapter 8.
The overall efficiency of a cogeneration plant may be defined as the sum of the
net work and useful heat output per unit mass flow, divided by the product of the
fuel/ air ratio and Qnet.p of the fuel. For the purpose of preliminary cycle
calculations it is sufficiently accurate to evaluate the useful heat output per unit
mass flow as cpg (Tin - 393) bearing in mind the need to keep the stack
temperature from falling much below 120°C. Tin would be the temperature at
entry to the WHB and would nomlally be the exhaust gas temperature of the
turbine. It is unlikely that a regenerative cycle would be used in cogeneration
plant because the low exhaust temperature would limit steam production.
Repowering
Another way in which gas turbines can be used with steam turbines is the re-
powering of old power stations. Steam turbines have much longer lives than
boilers, so it is possible to create an inexpensive combined cycle plant by
replacing an existing fired boiler with a gas turbine and WHB. The steam con-
ditions for which power stations were designed in the 19508 enable a good match
to be achieved between the steam and gas cycles. The power output can be
trebled, while the: thennal efficiency is increased from about 25 per cent to better
than 40 per cent. The result is a vastly superior power plant installed on an
existing site. The problems, and cost, of finding a suitable location for a new
power station are avoided, and this contributes to the reduction in cost of elec-
tricity sent out. A typical repoweriag project is described in Ref. (7).
An interesting example of repowering is the conversion of an uncompleted
nuclear plant to a base-load combined cycle plant, using the steam turbines and
generators of the existing scheme but replacing the nuclear reactor as a heat
source by twelve 85 MW gas turbines and WHBs. The net result in this instance
is a total output of 1380 Mw, plus a substantial supply of steam to a chemical
plant.
CLOSED-CYCLE GAS TURBINES
81
2,,6 Closed-cycle gas turbines
The main features of the closed-cycle gas turbine have been described in section
1.3, where it was suggested that a monatomic gas would have advantages over air
as the working fluid. Although, as we pointed out, closed-cycle gas turbines are
unlikely to be widely used, it will be instructive to quantifY some of these
advantages. To this end, the pe:rfonnance of a particular closed cycle will be
evaluated for air and helium.
Figure 2.26 is a sketch of the plant, annotated with some of the assumed
operating conditions, viz. LP compressor inlet temperature and pressure, HP
compressor inlet temperature, and turbine inlet temperature. The compressor inlet
temperatures would be fixed by the temperature of the available cooling water and
the required temperature difference between water and gas. A modest turbine inlet
temperature of 1100 K has been chosen. (This might have been achievable in the
core of the high temperature nuclear reactor (HTR) discussed in section 1.3). The
higher we can make the pressure in the circuit the smaller the plant becomes, and
we shall assume that a minimum pressure of 20 bar is practicable. Typical values
of component efficiencies and pressure loss are: 11ooc=0·89, 11001=0·88, heat
exchanger effectiveness 0·7; pressure loss as percentage of component inlet
pressure: in precooler and intercooler 1·0 per cent each, in hot and cold sides of
heat exchanger 2·5 per cent each, in gas heater 3 per cent.
The perfonnance will be evaluated over a range of compressor pressure ratio
P04/POI' It will be assumed that the compression is split between LP and HP
compressors such that POl/POl = (P04/POl)O.5 which leads to an approximately
equal division of work input. The usual values of l' = 1·4 and c
p
= 1·005 kJ /kg K
are assumed to hold throughout the cycle when air is the working fluid, and for
LP
reservoir
20 bar
300 K 1
Precooler
2
Transfer
camp.
Intercooler
8
HP
reservoir
4
Make-up
Gas
heater
5
Heat exchanger
FIG. 2.26 Example of a closed-cycle plallt
7
82 SHAFT POWER CYCLES
helium y = 1·666 and cp = 5·193 kJ Ikg K. The cyete efficiency is given by:
[(T06 - T07 ) - (T04 - T03 ) - (T02 - TO])]
1)=
(T06 - T05 )
When comparing the specific work output of cycles using fluids of different
density, it is more useful to express it in terms of output per unit volume flow than
per unit mass flow, because tlie size of plant is detennined by the fonner. Here we
shall evaluate the specific output from the product of the output per unit mass
flow and the density at inlet to the compressor where T = 300 K and p = 20 bar.
The density of air at this state is 23·23 leg/m
3
whereas for helium (molar mass
4 kg/lemol) it is only 3·207 leg/m3.
Figure 2.27 shows the results of the calculations and we will consider first the
efficiency curves. They suggest that the helium cycle has a slightly lower
efficiency. In fact more accurate calculations, allowing for the variation of cp and
y with temperature in the case of air, lead to almost identical maximum
efficiencies. (With helium, cp and y do not vary significantly with temperature
over the range of interest.) As will be shown below, however, the heat transfer
characteristics of helium are better than those of air so that it is probable that a
higher heat-exchanger effectiveness can be used with helium without making the
heat-exchanger excessively large. The portion of a dotted curve in Fig. 2.27
indicates the benefit obtained by raising the effectiveness from 0·7 to 0·8 for the
helium cycle: the efficiency is then 39·5 per cent compared with about 38 per cent
for the air cycle.
That the heat transfer characteristics are better for helium can be deduced quite
simply from the accepted correlation for heat transfer in turbulent flow in tubes
[Ref. (1)], viz.
40
30
10
Nu = 0.023 Reo. s Pr°-4
Air

. /" '-'-.

I
i
i
i
i
:l
/'''''
;/
-.-
-'-
He .-.......
Air
---?>-
,,·-·-·----·-'''''''-H;-·-..·-
8000
6000
4000


i
c.
"5
o
"

CD
c.
2000 (/)

Compressor pressure ratio (P04/P01)
FIG. 2.27 Comparison of ail' and belinm closed-cycle performance
CLOSED-CYCLE GAS TURBINES
83
where .Nu is the Nusselt number (hdlk). The Prandtl number (cplllk) is
approxlIDately the same for air and helium, 80 that the heat transfer coefficients
will be in the ratio
(2.24)
Suffixes h and a refer to helium and air respectively. From tables of properties,
e.g. Ref. (8), it may be seen that the thennal conductivities vary with temperature,
but the ratio k,,1 ka is of the order of 5 over the relevant range. of temperature. The
Reynolds number (pCdlll) is a function of velocity as well as properties of the
fluid, and the flow velocity in the heat-exchanger (and coolers) is determined by
the pressure drop entailed. As may be found in any text on fluid mechanics, or
Ref. (1), the pressure loss for turbulent flow in tubes is given by
tlp = =
where r is the wall shear stress and f the friction factor given by the Blasius law
f = 0·0791 _ 0·0791
Reo.25 - (pCd I pf25
It follows that for equal pressure losses,
(p0.75 C1.75 flO.25)" = (pO.75 C]·75 flO.25)a
p "I Pais 0·13 8 and Ilhl Ila is approximately 1·10, so that
Ch ( 1 ) ]/1·75
C
a
= 0·1380.75 X 1.100·25 = 2·3
for similar pressure losses the flow velocity of helium can be double that of
alT .
Using this result we have
Finally, from equation (2.24)
hh = 0.2908 X 5 = 1.86
ha
and the heat transfer coefficient for helium is therefore almost twice that for air.
This implies that the heat-exchanger need have only half the surface area of
tubing for the same temperature difference, or that a higher effectiveness can be
used economically.
Turning now to the specific work output curves of Fig. 2.27, from which the
comparative size of plant can be deduced, it will be appropriate to make the
84 SHAn POWER CYCLES
comparison at the pressure ratio yielding maxiinum efficiency in each case, i.e.
around 4 for helium and 7 for air. Evidently about 45 percent greater volume
flow is required with helium than with air for a given power output. The reason is
that under conditions of maximum efficiency, in each case the compressor
temperature rises and turbine temperature drops are similar for helium and air, so
that the specific outputs on a mass flow basis are in the ratio Cph/ cpa f'>j 5. The
density of helium is only 0·13 8 times that of air, however, so that on a volume
flow basis the specific output with helium is about 0·7 of that with ai!.
The increase in size of the heat-exchanger and coolers due to this higher
volume flow will be more than offset by the reduction arising from the better heat
transfer coefficients and the higher now velocities that can be used with helium.
Higher velocities will also reduce the diameter of the ducts connecting the
components of the plant. A 45 per cent increase in volume flow with a doubling
of the velocity would imply a 15 per cent reduction in diameter.
The effect of using helium instead of air on the size of the turbomachinery will
only be fully appreciated after the chapters on compressors and turbines have
been studied. We have already noted that the work done per unit mass flow in
these components is about five times as large with helium as with air. One might
expect that five times the number of stages of blading will be required. This is not
so, however, because the Mach number limitation on flow velocity and peripheral
speed (which determine the work that can be done in each stage) is virtually
removed when helium is the working fluid. The reason is that the sonic velocity,
(yRT)°'s, is much higher in helium. The gas constant is inversely proportional to
the molar mass so that the ratio of sonic velocities at any given temperature
becomes
The work per stage of blading can probably be quadrupled with helium, so that
the number of stages is only increased in the ratio 5/4. On top of this we have a
reduction in annulus area, and therefore height of blading, consequent upon the
higher flow velocities through the turbo machinery. Only a detailed study would
show whether the turbomachinery will in fact be much larger for the helium
cycle.
Because helium is a relatively scarce resource, the decision to use it in large
power plant will probably rest not so much on thermodynamic considerations as
the satisfactory solution of the practical problem of sealing the system at the high
pressures requir,ed. Experience with sealing high-pressure carbon dioxide in gas-
cooled reactors has shown that leakage is substantial, and helium, being a lighter
gas, will be even more difficult to contain. Although in section 1.6 it was stated
that closed-cycle gas turbines are unlikely to be used in nuclear power plant, in-
depth studies of the teclmological requirements for future applications continue
(Ref. (9)).
NOMENCLATURE
85
NOMENCLATURE
The most widely used symbols in the book are introduced here and they will not
be repeated in the lists at the end of other chapters.
c
II!
M
n
p
Q
Q n e ~ p
R,ll.
r
s
T
t
W
Y
1]
p
Suffixes
o
1, 2, 3, etc.
CIJ
a
b
c
f
g
h
rn
N
s
velocity
specific heat at constant pressure
fuel/air ratio by weight
specific enthalpy
enthalpy of reaction
mass flow
molecular weight, Mach number
polytropic index
absolute pressure
heat transfer per unit mass flow
net calorific value at constant p
specific, molar (universal), gas constant
pressure rati 0
specific entropy
absolute temperature
temperature ratio
specific work (power) output
ratio of specific heats
efficiency
density
stagnation value
reference planes
polytropic
ambient, air
combustion chamber
compressor
fuel
gas
heat-exchanger
intake, mixture constituent
mechanical
net
stage
turbine
3
Gas turbine cycles
for aircraft propulsion
Aircraft gas turbine cycles differ from shaft power cycles in that the useful power
output is in the form of thrnst: the whole of the thrust of the turbojet and turbofan
is generated in propelling nozzles, whereas with the turboprop most is produced
by a propeller with only a small contribution from the exhaust nozzle. A second
distinguishing feature is the need to consider the effect of forward speed and
altitude on the performance. It was the beneficial aspect of these parameters,
together with a vastly superior power/weight ratio, that enabled the gas turbine to
so rapidly supplant the reciprocating engine for aircraft propulsion except for
low-powered light aircraft.
The designer of aircraft engines must recognize the differing requirements for
take-off, climb, cruise and manoeuvring, the relative importance of these being
different for civil and military applications and for long- and short-haul aircraft.
In the early days it was common practice to focus on the take-off thrust, but this is
no longer adequate. Engines for long-range civil aircraft, for example, require low
SFC at cruise speed and altitude, while the thrust level may be determin.ed either
by take-off thrust on the hottest day likely to be encountered or by the thrust
required at top of climb. Evidently the selection of design conditions is much
more complex than for a land-based unit. As examples, 'design point'
calculations will be shown for take-off (static) and cruise conditions.
The chapter opens with a discussion of the criteria appropriate for evaluating
the performance of jet propulsion cycles, and of the additional parameters
required to allow the losses in the intake and propelling nozzle to be taken into
account. The cycle performance of turbojet, turbofan and turboprop are then
discussed in turn. Methods for calculating the variation of actual performance
with altitude and forward speed for any specific engine will be presented in
Chapter 8. For the analysis of other forms of jet power plant, such as ramjets and
rockets, the reader must turn to specialized texts on aircraft propulsion such as
Ref. (1).
CRITERIA OF PERFORMANCE 87
·3.1 Criteria of performance
Consider the schematic diagram of a propulsive duct shown in Fig. 3.1. Relative
to the engine, the air enters the intake with a velocity Ca equal and opposite to the
forward speed of the aircraft, and the power unit accelerates the air so that it
leaves with !he jet velocity C
f
The 'power unit' may consist of a gas turbine in
which the turbine merely drives the compressor, one in which part of the expan-
sion is carried out in a power turbine driving a propeller, or simply a combustion
chamber as in ramjet engines. For simplicity we shall assume here that the mass
flow m is constant (i.e. that the fuel flow is negligible), and thus the net thrust F
due to the rate of change of momentum is
(3.1)
mCj is called the gross momentum thrnst arid mCa the intake momentum drag.
When the exhaust gases are not expanded completely to Pa in the propulsive duct,
the pressure Pj in the plane of the exit will be greater than Pa and there will be an
additional pressure thrust exerted over the jell exit area A· equal to A {p. - Pa) as
indicated in Fig. 3.1. The net thrust is then the sum of the m o m e n t u ~ :hrust and
the pressure thrust, namely
(3.2)
When the aircraft is flying at a uniform speed! Ca in level flight the thrust must be
equal and opposite to the drag of the aircraft at that speed. .
In what follows we shall assume there is complete expansion to Pa in the
propelling nozzle and therefore that equation (3.1) is applicable. From this
equation it is clear that the required thrust can be obtained by designing the
engine to produce either a high velocity jet of small mass flow or a low velocity
jet of high mass flow. The question arises as to what is the most efficient
combination of these two variables and a qualitative answer is provided by the
following simple analysis.
The propulsion efficiency 1'/P can be defined as the ratio of the useful propulsive
energy or thrust power (FCa) to the sum of that energy and the unused kinetic
energy of the jet. The latter is the kinetic energy of the jet relative to the earth,
Ambient pressure Pa
FIG. 3.1 Propulsive duct
88
GAS TURBINE CYCLES FOR AIRCRAFT PROPULSION
namely m(C. - cai /2. Thus
. J (C) 2
mCa Cj - a _ (3.3)
IJp = m[Ca(C
j
_ Ca) + (C
j
- Ca)2/2] - 1 + (Cj/Ca)
11 is often called the Froude efficiency. Note that it is in ll? sense an overall
plant efficiency, because the unused enthalpy in the jet is Ignored. From equatIOns
(3.1) and (3.3) it is evident that
(a) F is a maximum when C
a
= 0, i.e. under static conditions, but I1p is then
.
(b) I1p is a maximum when C)Ca = 1 but then the thrust IS zero.
We may conclude that although Cj must be greater than Ca the difference
not be too great. This is the reason for the development of the fannly. of
propulsion units shown in Fig. 3..2. in sho:m they proVide
propulsive jets of decreasing mass flow and mcr.easmg therefore
in that order they will be suitable for aircraft of mcreasmg deSIgn
In practice the choice of power plant can be made only th.e speCIficatIOn of
the aircraft is known: it will depend not only on the reqmred speed but .also
on such factors as the desired range of the aircraft and rate cl1ll1b.
Because the thrust and fuel consumption of a jet propulsion urnt vary wIth both
cruise speed and altitude (air density), the latter is also an important parameter.


._,-'
--
I .
(a) piston engine
(b) Turboprop engine
(c) Turbofan engine
(d) Turbojet engine
(e) Ramjet engine
FIG. 3.2 Propulsion engines
CRITERiA OF PERFORMANCE 89
Figure 3.3 indicates the flight regimes found to be suitable for the broad
categories of power plant installed in civil aircraft .
The propulsion efficiency is a measure of the effectiveness with which the
propulsive duct is being used for propelling the aircraft, and it is not the efficiency
of energy conversion within the power plant itself that we will symbolize by l1e'
The rate of energy supplied in the fuel may be regarded as m£}net.p where mj is
the fuel mas's flow. This is converted into potentially useful kinetic energy for
propulsion m(C] - together with unusable enthalpy in the jet
mCp(Tj - Ta). l1e is therefore defined by
m(C] -
11 - (3.4)
e - mjQnet.p
The overall efficiency 110 the ratio of the useful work done in overcoming drag
to the energy in the fuel supplied, i.e.
mCa(Cj - Ca) FCa
110 = =----
mjQnet.p mjQnet.p
(3.5)
It is easy to see that the denominator of equation (3.3), namely
m[Ca(Cj - Ca) + (Cj - cj /2] is equal to the nU11lerator of equation (3.4), and
hence that
(3.6)
The object ofthe simple analysis leading to equation (3.6) is to make the point
that the efficiency of an aircraft power plant is inextricably li.nked to the aircraft
speed. A crude comparison of different engines can be made, however, if the
engine performance is quoted at two operating conditions: the sea-level static
performance at maximum power (i.e. at maximU11l turbine inlet temperature) that
must meet the aircraft take-off requirements, and the cruise performance at the
optimum cruise speed and altitude of the aircraft for which it is intended. The
ambiguous concept of efficiency is discarded in favour of the specific foe!
consumption which, for aircraft engines, is usually defined as the fuel
20000
15000
E
-i!l 10 000
'"

5000

IfJ Turbofan

o I I I I I I
o 0.5 1.0 1.5 2.0 2.5 3.0
Mach number Ma
FIG. 3.3 Fligbt regimes
90
GAS TURBINE CYCLES FOR AIRCRAFT PROPULSION
consumption per unit thrust (e.g. kg/h N), The overall efficiency given by
equation (3.5) can be written in the form
11 == Ca x _1_ (3.7)
o SFC Qnet,p
With a given fuel, the value of Qnet,p will be constant and it can be seen that the
overall efficiency is proportional to CalSFC, rather than I/SFC as for shaft power
units.
Another important performance parameter is the specific thrust F" namely the
thrust per unit mass flow of air (e.g. N s/kg). This provides an of the
relative size of engines producing the same thrust because the dlllenslOns of
the engine are primarily detennined by the air flow. Size is important because
of its association not only with weight but also with frontal area and the
consequent drag. Note that the SFC and specific thrust are related by
SFC=f
Fs
(3.8)
where f is the fuel/air ratio.
When estimating the cycle performance at altitude we shall need to know the
way in which ambient pressure and temperature vary with height above .se.a level.
The variation depends to some extent upon the season and latItude, but It IS usual
to work with an average or 'standard' atmosphere. The International Standard
Atmosphere corresponds to average values at middling latitudes and yields a
temperature decreasing by about 3·2 K per 500 m to 11000 m after
which it is constant at 216·7 K until 20 000 m. Above this heIght the temperature
starts to increase again slowly. Once the temperature is fixed, the variation of
pressure follows according to the laws of hydrostatics. An tabulated
version of the 1. S.A. is given at the end of this chapter, and thiS Will be used for
subsequent numerical examples; it should be appreciated, however, that
ambient conditions can vary widely from LS.A. values both at sea level and high
altitude. For high-subsonic and supersonic aircraft it is more appropriate to use
Ca/[m/s]
FIG. 3.4
INTAKE AND PROPELLING NOZZLE EFFICIENC[ES 91
Mach number rather than mls for aircraft speed, because the drag is more a
function of the former. It must be remembered that for a given speed in mls the
Mach number will rise with altitude up to 11 000 m because the temperature is
falling. Figure 3.4 shows a plot of Ma versus Ca for sea level and 11000 m
obtained from I.S.A. data and Ma == Cala, where a is the sonic velocity (yRTa)l!2.
3.2 Intake and propelling nozzle efficiencies
The nomenclature to be adopted is indicated in Fig. 3.5, which illustrates a simple
turbojet engine and the ideal cycle upon which it operates. The turbine produces
just sufficient work to drive the compressor, and the remaining part of the ex-
pansion is carried out in the propelling nozzle. Because of the significant effect of
forward speed, the intake must be considered as a separate component: it cannot
be regarded as part of the compressor as it often was in Chapter 2. Before
proceeding to discuss the performance of aircraft propulsion cycles, it is neces-
sary to describe how the losses in the two additional components-intake and
propelling nozzle-are to be taken into account.
Intakes
The intake is a critical pa.rt of an aircraft engine installation, having a significant
effect on both engine efficiency and aircraft safety. The prime requirement is to
minimize the pressure loss up to the compressor face while ensuring that the flow
enters the compressor with a nnifonn pressure and velocity, at all flight condi-
tions. Non-uniform, or distorted, flow may cause compressor surge which can
result in either engine flame-out or sever,o mechanical damage due to blade
vibration induced by unsteady aerodynamiG effects. Even with a well designed
intake, it is difficult to avoid some flow distortion during rapid manoeuvring.
Successful engines will find applications in a variety of aircraft having widely
different installations and inlet systems. Engines may be installed in the fuselage,
in pods (wing or rear fuselage mounted), or buried in the wing root. Three-
engined aircraft will require a different arrangement for the centre engine than for
the wing or tail mOlllted engines. The L-lOll and DC-IO both have engines
T
a 1 2 Combustion t f
I Intake I Compressor I chamber T Nozzle I
-[

s
FIG. 3.5 Simple turbojet engine and ideal
92 GAS Turu3lNE CYCLES FOR AIRCRAFT PROPULSION
mounted at the rear, but use radically different forms of intake: the former has the
engine mounted in the rear fuseJagebehind a long S-bend duct, while the latter
has a straight-through duct with the engine mounted in the vertical fin as shown in
Fig. 3.6. The design of the intake involves a compromise between aerodynamic
and structural requirements.
Current designs of compressor require the flow to enter the first stage at an
axial Mach number in the region of 0·4-0·5. Subsonic aircraft will typically
cruise at M 0·8-0·85, while supersonic aircraft may operate at speeds from M 2-
2·5. At take-off, with zero forward speed, the engine will operate at maximum
power and airflow. The intake must therefore satisfY a wide range of operating
conditions and the design of the intake requires close collaboration between the
aircraft and engine designers. The aerodynamic design of intakes is covered in
Ref. (2).
Ways of accounting for the effect of friction in the intake, suitable for use in
cycle calculations, can be found by treating the intake as an adiabatic duct. Since
there is no heat or work transfer, the stagnation temperature is constant although
there will be a loss of stagnation pressure due to friction and due to shock waves
at supersonic flight speeds. Under static conditions or at very low forward speeds
the intake acts a nozzle in which the air accelerates from zero velocity or low
Ca to CI at the compressor inlet. At normal forward speeds, however, the intake
performs as a diffuser with the air decelerating from Ca to C1 and the static
pressure rising frompa to PI. Since it is the stagnation pressure at the compressor
inlet which is required for cycle calculations, it is the pressure rise (POI - Pa) that
is of interest and which is referred to as the ram pressure rise. At supersonic
speeds it will comprise the pressure rise across a system of shock waves at the
inlet (see Appendix A.7) followed by that due to subsonic diffusion in the
remainder of the duct.
The intake efficiency can be expressed in a variety of ways, but the two most
commonly used are the isentropiC efficiency I]i (defined in tenns of temperature
rises) and the ram efficiency I]r (defined in terms of pressure rises). Referring to
FIG. 3.6 and straight-through intakes
INTAKE AND PROPELLING NOZZLE EFFICIENCIES
93
Fig. 3.7,we have
and
01' 101
P
(
7'1 )y/(r-11
Pa == Ta
where Thl is temperatu:e which would have been reached after an isentropic
ram .compresslOn to POI. TOl can be related to TOl by introducing an isentropic
effiCiency I]i defined by
- Ta
1)'=--
, TOl - Ta
It follows that
7'1 T
101 - =Yj.-
a '2c
p
(3.9)
that 1]; be as the fraction of the inlet dynamic temperature which
IS available for Isentropic compression in the intake. The intake pressure
ratIO can then be found fi:om
POI _ [ - Ta]Y/(l'-11 [ , C2 J1'/(j'-11
- 1 +--- = 1 + Yj._a_
Pa Ta . '2c T
p a
(3. lOa)
that M = C/(1,RT)1I2 and yR = cp(Y .,... 1), this equation can be
wntten as
-= 1+1].--M2
POI [ l' - 1 J
1'
/(1'-
1
1
Pa '2 a
(3.10b)
T
Poa
P01
,a 1

Pa , F ,
c:-'i--- ,--
,
s
FIG. 3.7 Intake loss
94
GAS TURBINE CYCLES FOR AIRCRAFT PROPULSION
The stagnation temperature can aiso be expressed in terrils of Ma as
TOI == [1 + Y - 1 M2]
Ta 2 a
(3.11)
The ram efficiency I'/r is defined by the ratio of the ram pressure rise to the inlet
dynamic head, namely
I'/r ==POl - Pa
POa -Pa
I'/r can be shown to be almost identical in magnitude to I'/i and the two quantities
are interchangeable: curves relating them can be found in Ref. (2). Apart from the
fact that I'/r is easier to measure experimentally, it has no advantage over I'/i and we
shall use the latter in this book. For subsonic intakes, both I'/i and I'/r are found to
be independent of inlet Mach number up to a value of about 0·8 and thence their
suitability for cycle calculations. They both suffer equally from the drawback of
implying zero stagnation pressure loss when Ca is zero, because then POI/Pa == 1
and Pa ==POa' This is not serious because under these conditions the average
velocity in the intake is low, and the flow is accelerating, so that the effect of
friction is very small.
The intake efficiency will depend upon the location of the engine in the
aircraft (in wing, pod or fuselage), but we shall assume a value of 0·93 for I'/i in
the. numerical examples relating to subsonic aircraft that follow. It would be less
than this for supersonic intakes, the value decreasing with increase in inlet Mach
number. In practice, neither I'/i nor I'/r is used for supersonic intakes and it is more
usual to quote values of stagnation pressure ratio POl/POa as a function of Mach
number. POI/POa is called the pressure recovery factor of the intake. Knowing the
pressure recovery factor, the pressure ratio POI/Pa can be found from
POI == POI X POa
Pa POa Pa
where POa/Pa is given in terms of Ma by the isentropic relation (8) of Appendix A,
namely
POa == 1 +L=-M2 [
1 ]Y/(l'-I)
Pa 2 a
Some data on the performance of supersonic intakes can be found in Ref. (3);
their design is the province of highly specialized aerodynamicists and much of
the information is classified. A rough working rule adopted by the American
Department of Defense for the pressure recovery factor relating to the shock
system itself is
(EOI) == 1.0 _ 0.075(Ma _1)1.35
\Poa shock
which is valid when 1 < Ma < 5. To obtain the overall pressure recovery factor,
(PollPoa)shock must be multiplied by the pressure recovery factor for the subsonic
part of the intake.
INTAKE AND PROPELLING NozzLE EFFICIENCIES 95
Propelling nozzles·
We shall.here u.se .the tei-m 'propelling nozzk' to refer to the component in which
the flUId IS to give a high velocity jet. With a simple jet engine,
as m Fig. 3.5, there Will be a single nozzle downstream of the turbine. The
turbofan may have two separate nozzles for the hot and cold streams, as in Fig.
3.15, or the,flows may be mixed and leave from a single nozzle. Between the
turbine exit and propelling nozzle there will be a jet pipe of a length determined
by the location of the engine in the aircraft. In the transition from the turbine
to circular jet pipe some increase in area is provided to reduce the
velocIty, and hence friction loss, in the jet pipe. When thrust boosting is required,
an afterburner may be incorporated in the jet pipe as indicated in Fig. 3.8.
in a jet engine is sometimes referred to as 'reheating', although it is
not eqU1valent to the reheating between turbines sometimes proposed for indus-
trial .gas turbines which would be in constant operation.) Depending on the
locatIOn of the in the aircraft, and on whether reheat is to be incorporated
for thrust boosting, the 'propelling nozzle' will comprise some or all of the items
shown in Fig. 3.8.
The question immediately arises as to whether a simple convergent nozzle is
adequate or whether a convergent-divergerrt nozzle should be employed. As we
see from cycle calculations, even with moderate cycle pressure
ratIOs the pressure ratio P04!Pa will be greater than the critical pressure ratiot over
at least part of the operational range of forward speed and altitude. Although a
convergent-divergent nozzle might therefore appear to be necessary, it must be
i
i
.--t-._._.
____ __ ____
I
d Diffuser ,II Jet pipe [-::: I

FIG. 3.8 'Propelling nozzle' system
: An of the of the . critical pressure ratio can be obtained by assuming isentropic flow,
I.e. by puttmg y = 1·333 m equation (12) of Appendix A,
P04 (Y + l)Y!(Y-ll
p, = -2- = 1·853
96 GAS TURBIl'<"'E CYCLES FOR PROPULSION
remembered that it is· thrust which is required and not maximum possible jet
velocity. Certainly it can be shown that for an isentropic expansion the thrust
produced is a maximum when complete expansion to Paocc:rrs in the nozzle: the
pressure thrust As(Ps - Pa) arising Jrom incomplete does
compensate for the loss of momentum thrust due to a smaller Jet velocIty.
is no longer true when friction is taken into account because the theoretIcal Jet
velocity is not achieved. Furthermore, the use of a
would result in significant increases in engine weight, length and dIameter which
would in turn result in major installation difficulties and a penalty in aircraft
weight.
For values of poJPa up to about 3 there is no doubt, from experiments
described in Ref. (4), that the thmst produced by a convergent nozzle is as large
as that from a convergent-divergent nozzle even when the latter has an exit/throat
area ratio suited precisely to the pressure ratio. At operating pressure ratios less
than the design value, a convergent-divergent nozzle of fixed proportions would
certainly be less efficient because ofthe loss incurred by the formation of a shock
wave in the divergent portion. For these reasons aircraft gas turbines normally
employ a convergent propelling nozzle. A secondary advantage of this type is the
relative ease with which the following desirable features can be incorporated:
(a) Variable area, which is essential when an afterburner is incorporated, for
reasons discussed in section 3.6. Earlier engines sometimes used variable
area to improve starting pen[)nnance, but this is seldom necessary today.
Figure 3.9(a) illustrates the 'iris' and 'central plug' methods of achieving
variation of propelling nozzle area.
(b) Thrust reverser to reduce the length of runway required for landing, used
almost universally in civil transport aircraft.
(c) Noise suppression. Most of the jet noise is due to the mixing of the high
velocity hot stream with the c:old atmosphere, and the intensity decreases as
the jet velocity is reduced. For this reason the jet noise of the turbofan is less
than that of the simple turbojet. In any given case, the noise level can be
(a)
'Clam' shut
for reversal
(b)
FIG. 3.9 Variable area, thmst ami lIoise suppressioll
Noise
suppressor
/
TI\fTi'.KE AND PROPELLING NOZZLE EFFICIENCIES
97
reduced by accelerating the mixing process and this is normally achieved by
increasing the surface area of the jet stream as shown in Fig. 3.9(b).
It should not be thought that convergent-divergent nozzles are never used. At
high supersonic speeds the large ram pressure rise in the intake results in a very
high nozzle pressure ratio. The value of P04/Pa is then many times larger than the
critical ratio and may be as high as 10-20 for flight Mach numbers in the
range 2-3. Variable exit/throat area is essential to avoid shock losses over as
much of the operating range as possible, and the additional mechanical
complexity must be accepted. The main limitations on the design are:
(a) the exit diameter must be within the overall diameter of the engirle-
otherwise the additional thrust is offset by the increased extemal drag.
(b) in spite ofthe weight penalty, the included angle of divergence must be kept
below about 30° because the loss in thrust associated with divergence (non-
axiality) of the jet increases sharply at greater angles.
The design of the convergent-divergent nozzle is especially critical for
supersonic transport aircraft, such as the Concorde, which spend extended
periods at high supersonic speeds. The range and payload are so strongly affected
by the performance of the propUlsion system as a whole, that it is necessary to use
both a continuously variable intake, and a nozzle capable of continuous variation
of both the throat and exit areas, as discussed ir! Ref. (5). Military aircraft
normally operate with a supersonic 'dash' capability for only short
periods, and supersonic fuel consumption is not as critical.
After these opening remarks, we may now tum our attention to methods of
allowing for propelling nozzle loss in cycle calculations. We shall restrict our
attention to the almost tmiversally used convergent nozzle. Two approaches are
commonly used: one via an isentropic efficiency IV and the other via a specific
thrust coefficient KF. The latter is defined as the ratio of the actual specific gross
thrust, namely [mCs + As(Ps - Pa)]/m, to that which would have resulted from
isentropic flow. When the expansion to Pa is completed in the nozzie, i.e. when
P04/Pa < P04/Pe, KF bec()mes simply the ratio of actnal to isentropic jet velocity
which is the 'velocity coefficient' often used by steam turbine designers. Under
these conditions it will be easy to see from the following that l/j = K'j. Although
KF is the easier to measure on nozzie test rigs, it is not so useful for our present
purpose as rtf.
Figure 3.10 illustrates the real and isentropic processes on the T -s diagraln;
and rtf is defined by
T04 - Ts

04 - s
It follows that for given inlet conditions (P04, T04) and an assumed value of , Ii' Ts
is given by
T04 - T5 = rtjT04 1 - -- [
G
I ) (Y-IJ/Y]
04/PS
(3.12)
98
GAS TURBINE CYCLES FOR AIRCRAFT PROPULSION
P05
05
1
C§/2 Cp
{
5
(a) When P04 < P04
Pa Pc
T
s
, c'
(b) When P04
Pa
FIG. 3.10 Nozzle loss fol' unchoked and choked lIows
Pos
Pc for isen. flow
./ _
- P04
Pc Y+'1
P04
>-
Pc
5
This is also the temperature equivalent of the jet velocity (CV2cp) because
T05 = T04' For pressure ratios up to the critical value, P5 will be put equal to Pa in
equation (3.12), and the pressure thrust is zero. Above the critical pressure ratio
the nozzle is choked, P5 remains at the critical value pc, and C5 remains at the
sonic value (yRT5)1I2. The outstanding question is how the critical pressure should
be evaluated for non-isentropic flow.
The critical pressure ratio P041pc is the pressure ratio P041pS which yields
Ms = 1. Consider the cOlTesponding critical temperature ratio T04ITe: it is the
same for both isentropic and irreversible adiabatic flow. Thi.s follows from the fact
that T04 = Tos in all cases of adiabatic flow with no work transfer, and thus
T04 = Tos = 1 + = 1 + l' - 1 111§
Ts Ts 2cpTs 2
Putting 1115 = 1 we have the familiar expression
T04 _ l' + 1
Te - 2
(3.13)
Having found Te from equation (3.13), we see from Fig. 3.1O(b) that the value of
1Jj yields which is the temperature reached after an isentropic expansion to the
real critical pressure Pc, namely
Pc is then given by
T' y/(y-l) [ 1 ( T )]Y/CY-IJ
Pc = P04 (T :J = P04 1 - \
1
- T :4
SIMPLE TURBOJET CYCLE 99
On substituting for TelTo4 from equation (3.13), we have the critical pressure ratio
in the convenient form
P04
Pc = [1 _
l'lj l' + 1
(3.14)
This method of using 1Jj to dete111line the critical pressure ratio yields results
consistent with a more detailed lli.alysis involving the momentum equation given
in Ref. (6). In particular, it shows that the effect of friction is to increase the
pressure drop required to achieve a Mach number of unity.
The remaining quantity necessary fiJT evaluating the pressure thrust
As(Pe - Pa), is the nozzle area A5. For a given mass flow m, this is given
approximately by
m
As =- (3.15)
peec
1 1
with Pc obtained fromp/RTe and Ce from [2cp(To4 - TJP or (1'RTc)'. It is only
an approximate value of the exit area because allowance must be made for the
thickness of the boundary layer in the real flow. Furthermore, for conical nozzles,
the more complete analysis in Ref. (6) shows that the condition M = 1 is reached
just downstream ofthe plane of the exit when the flow is iITeversible. ill practice,
the exit area necessary to give the required engine operating conditions is found
by trial and enor during development tests .. Furthermore, it is found that indiv-
idual engines of the same type will require slightly different nozzle areas, due to
tolerance build-ups and small changes in component efficiencies. Minor changes
to the nozzle area can be made nsing nozzle trimmers, which are small tabs used
to block a portion of the nozzle area.
The value of 1'/j is obviously dependent on a wide variety of factors, such as the
length of the jet pipe, and whether the various auxiliary features mentioned earlier
are incorporated because they inevitably ililtroduce additional frictional losses.
Another factor is the amount of swirl in the gases leaving the turbine, which
should be as low as possible (see Chapter 7). We shall assume a value of 0·95 for
11J in the following cycle calculations.
3.3 Simple turbojet cycle
Figure 3.11 shows the real turbojet cycle on the T-s diagram for comparison with
the ideal cycle of Fig. 3.5. The method of calculating the design point perfor-
mance at any given aircraft speed and altitude is illustrated in the following
example.
EXAMPLE 3.1
Determination of the specific thrust and SFC for a simple turbojet engine, having
the following component performance at the design point at which the cruise
100 GAS.TURBINE CYCLES FOR AIRCRAFT PROPULSION
T
5
FIG. 3.11 Thrbojet cycle with losses
speed and altitude are M 0·8 and 10000 m.
Compressor pressure ratio
Turbine inlet temperature
Isentropic efficiency:
of compressor, '1c
of turbine, '11
of intake, '1i
of propelling nozzle, iYJj
Mechanical transmission e:fficiency '1m
Combustion efficiency '1b
Combustion pressure loss I1Pb
From the I.S.A. table, at 10 000 m
8·0
1200 K
0·87
0·90
0·93
0·95
0·99
0·98
4% compo deliv. press.
Pa = 0·2650 bar, Ta = 223·3 K and a = 299·5 m/s
The stagnation conditions after the intake may be obtained as follows:
= (0·8 x 299.5/ == 28.6 K
2cp 2 x 1·005 x 1000
C2
TO! = Ta +-2 a = 223·3 +28·6 = 251·9 K
cp
[
2 ]Y/(Y-IJ 3·S
POI = = [1 0·93 x 28.6] = 1.48
+'1'2 T + 223.3 2
Pa Cp a
POI = 0·2650 x 1·482 = 0·393 bar
SIMPLE TURBOJET CYCLE
At outlet from the compressor,
P02 = 0::)pOI = 8·0 x 0·393 = 3·144 bar
TOl [002)(Y-IJ/Y ] 251·9
T02 - TO! = - - -1 = --[8.0
1
/
3
.
5
- IJ = 234·9 K
'1c 01 0·87
T02 = 251·9 + 234·9 = 486·8 K
J1't= WJ'1m and hence
T. T. cpiT02 - ToI ) 1·005 x 234·9
03 - 04 = Cpgl'lm = 1.148 x 0.99 = 207·7 K
T04 = 1200 - 207·7 = 992·3 K
P03 =P02(1 - tlpb) = 3·144(1- 0·04) = 3·018 bar
P02
Tri4 = T03 - ..!..(T03 - T04) = 1200 - 207·7 = 969·2 K
'11 0·90
(
Tri4)Y/(Y-IJ (969.2)4
P04 = P03 -T. = 3·018 -- = 1·284 bar
03 1200
The nozzle pressure ratio is therefore
P04 1·284 -
Pa = 0.265 = 4·845
The critical pressure ratio, from equation (3.14), is
P04 1 1
-= = = 1·914
Pc [1 _..!.. (L=J.)]Y/(Y-IJ [1 __ 1 (0.333)J4
'1j l' + 1 0·95 2·333
Since PoJPa > P04/Pc the nozzle is choking.t
(
2) 2 x 992·3
Ts = Tc = --1 T 04 = = 850·7 K
l' + 2·333
(_ 1 ) 1·284
Ps = Pc = P04 \P04/Pc = 1.914 = 0·671 bar
100 x 0·671 _ 3
Ps RTc 0.287 x 850.7 - 0·275 kg/m
Cs = = (1·333 x 0·287 x 850·7 x 1000)! = 570·5 m/s
As 1 1
-;:;; = PsC
s
= 0·275 x 570.5 0·006374 m
2
s/kg
101
t The next example in this chapter shows how the calculation is perfonned when the nozzle is
unchoked.
102
GAS TURBINE CYCLES FOR AIRCRAFT PROPULSION
The specific thrust is
I As
Fs = (C5 - Ca) T -(Pc - Pa)
m
= (570·5 - 239·6) + 0·006347(0·671 - 0.265)10
5
= 330·9 + 258·8 = 589·7 N sjkg
From Fig. 2.15, with T02 =486·8 K and T03 - To2 = 1200 - 486·8=713·2 K,
we find that the theoretical fuelJair ratio required is 0·0194. Thus the actual fuel/
air ratio is
0·0194
f = --= 0·0198
0·98
The specific fuel conslUnption is therefore
SFC = f = 0·0198 x 3600 = 0.121 kgjh N
Fs 589·7
Optimization of the turbojet cycle
When considering the design of a turbojet the basic thermodynamic parameters at
the disposal of the designer are the turbine inlet temperature and the compressor
pressure ratio. It is common practice to carry out a series of design point cal-
culations covering a suitable range of these two variables, using fixed polytropic
efficiencies for the compressor and turbine, and to pIlOt SFC versus specific thrust
with turbine inlet temperature T03 and compressor pressure ratio fc as parameters.
Such calculations may be made for several appropriate flight conditions of for-
ward speed and altitude. Typical results applying to a subsonic cruise condition
are shown in Fig. 3.l2. The effects of turbine inlet temperature and compressor
pressure ratio will be considered in tum.
It can be seen that specific thrust is strongly dependent on the value of To3 , and
utilization of the highest possible temperature is desirable in order to keep the
engine as small as possible for a given thrust. At a constant pressure ratio,
however, an increase in T03 will cause some increase in SFC. This is in contrast to
the effect of T03 on shaft power cycle performance, where increasing T03
improves both specific power and SFC as discussed in section 2.4. Nevertheless
the gain in specific thrust with increasing temperature is invariably more
important than the penalty in increased SFC, particularly at high flight speeds
where small engine size is essential to reduce both weight and drag.
The effect of increasing the pressure ratio fc is clearly to reduce the SFC. At a
fixed value of T03, increasing the pressure ratio initially results in an increase in
specific thrust but eventually leads to a decrease; and the optimum pressure ratio
for maximum specific thrust increases as the value of T03 is increased. Evidently
the effects of pressure ratio follow the pattern already observed to hold for shaft
power cycles and need no further comment.
SIMPLE TURBOJET CYCLE
0.16,
SFC

0.14
0.12
0.10
10
15
Ma=O.B
all. =9000 m
Compressor
pressure
ratio
0.08::: __ -:::::-__ --:::-::-____ :-L1___ ---LI___ .--JI
500 600 700 1300 900 1000
Specific thrust![N s/kg]
FIG. 3.12 Typical turbojet cycle performance
103
Figure 3.12 applies to.a particular subsonic cmise condition. When such
calculations are repeated for a higher cruising speed at the same altitude it is
found that in general, for any given values of fc and To3, the SFC is increased and
the specific thrust is reduced. These effects are due to the combination of an
increase in inlet momentum drag and an incr·ease in compressor work consequent
upon the rise in inlet temperature. Corresponding curves for different altitudes
show an increase in specific thrust and a decrease in SFC with increasing altitude,
due to the fall in temperature and the resulting reduction in compressor work.
Perhaps the most notable effect of an increase in the design cmise speed is that
the optimum compressor pressure ratio for maximmn specific thrust is reduced.
This is because of the larger ram compression in the intake. The higher
temperature at the compressor inlet and the need for a higher jet velocity make
the use of a high turbine inlet temperature desirable-and indeed essential for
economic operation of supersonic aircraft.
The thermodynamic optimization of the turbojet cycle cannot be isolated from
mechanical design considerations, and the of cycle parameters depends
very much on the type of aircraft. While high turbine temperatures are
thermodynamically desirable they mean the use of expensive alloys and cooled
turbine blades leading to an increase in complexity and cost, or to the acceptance
of a decrease in engine life. The thermodynamic gains of increased pressure ratio
must be considered in the light of increased weight, complexity and cost due to
the need for more compressor and turbine stages and perhaps the need for a
multispool configuration. Figure 3.13 illustrates the relation between performance
and design considerations. A small business jet or trainer, for example, needs a
104
GAS TURBINE c:YCLES FOR AiRCRAFT PROPULSION
SFC
Business jet
SpecHic thrust
FIG. 3.13 Performance and design (:onsiderations
Lifting engine'
Long range subsonic
simple, reliable engine of low cost: SFC is not critical because of the
small amount of flying done, and low pressure ratio turbojet of modest turbme
inlet temperature would be satisfactory. In recent years, however, noise
restrictions have led to the displa.cement of the turbojet by the turbofan. For
the business jet, this change has also been driven by the need for longer range.
Another example of interest was the development of specialized lifting engines
for Vertical Take Off and Landing (VTOL), where the prime requirement was for
maximum thrust per unit weight and volume, with SFC less critical because of
the very low running times: these requirements were met using a low pressure
ratio unit with a very high turbine inlet temperature (permissible because of the
short life required). The compressor pressure ratio was governed by the maximum
that could be handled by a single-stage turbine. This type of engine was not
widely used because of the inereased aircraft complexity, but engines of
exceptionally high thrust/weight ratio were built in the early 1960s. Lastly,
turbojets of high pressure ratio were used in early commercial aircraft and
bombers because of the need for long range and hence low SFC. The increased
engine weight was acceptable because of the large reduction in (engine plus fuel)
weight for a long range. Turbojets have now been superseded by turbofans for
commercial subsonic aircraft. Tw:bojets are no longer suitable for commercial
supersouic aircraft because of the take-off noise. In future, such aircraft will need
an engine with take-off noise comparable to that of conventional turbofans. This
will require the design and development of variable cycle engines capable of
operating as turbofans during take-off and as turbojets (or very low bypass
turbofans) at supersonic cruise conditions. The subsonic fuel consumption of a
supersonic transport is important because a considerable portion of any journey is
SIMPLE TURBOJEr CYCLE 105
flown at subsonic speeds, and it follows that the optimization procedure cannot be
carried out around a single cruise condition.
Variation of thrust and SFC with flight conditions for a given engine
The reader is reminded that we have been discussing the results of design point
cycle calculations. Curves such as those of Fig. 3.l2 do not represent what
happens to the performance of a particular engine when the turbine inlet tem-
perature, forward speed or altitude differ from the design values. The method of
arriving at such data is described in Chapter 8: here we will merely note some of
the more important aspects of the behaviour of a turbojet.
At different flight conditions both the thrust and SFC will vary, due to the
change in air mass flow with density and the variation of momentum drag with
forward speed. Furthermore, even if the engine were run at a fixed rotational
speed, the compressor pressure ratio and turbine inlet temperature will change
with intake conditions. Typical variations of thrust and SFC with change in
altitude and Mach number, for a simple turbojet operating at its maximum
rotational speed, are shown in Fig. 3.l4. It can be seen that thrust decreases
significantly with increasing altitude, due to the decrease in ambient pressure and
density, even though the specific thrust increases with altitude due to the
favourable effect of the lower intake temperature. Specific fuel consumption,
however, shows some improvement with increasing altitude. It will be shown in
Chapter 8 that SFC is dependent upon ambient temperature, but not pressure, and
hence its change with altitude is not so marked as that of thrust. It is obvious from
the variation in thrust and SFC that the fuel consumption will be greatly reduced
at high altitudes. Reference to Fig. 3.14 shows that with increase of Mach number
at a fixed altitude the thrust initially decreases, due to increasing momentum drag,
and then starts to increase due to the beneficial effects of the ram pressure ratio; at
supersonic Mach numbers this increase in thrust is substantial.
15000
5000
Sea -level
3000m
0.15
Z 0.10
.c

fi
____ --___ --- (J) 0.05
11 DOOm


3000m
OL-__ ____ _L ____ ____
OL-__
o 0.2 0.4 0.6 0.8 o 0.2 0.4 0.6 0.8
Flight Mach number Flight Mach number
FIG. 3.14 Variation of thrust and SFC with Mach number and altitude for typical
turbojet
106
GAS TURBINE CYCLES FOR AIRCRAFT PROPULSION
3.4 The turbofan engine
The turbofan engine was originally conceived as a method of improving the
propulsion efficiency of the jet engine by reducing the mean jet velocity, par-
ticularly for operation at high subsonic speeds. It was soon apparent, however,
that a lower jet velocity also resulted in less jet noise, an important factor when
large numbers of jet propelled aircraft entered commercial service. In the turbofan
a portion of the total flow by-passes part of the compressor, combustion
turbine and nozzle before being ejected through a separate nozzle as shown m
Fig. 3.15. Thus the thrust is made up components, the (orfan)
thrust and the hot stream thrust. Figure 3.15 shows an engme WIth separate
exhausts, but it is sometimes desirable to mix the two streams and eject them as a
single jet of reduced velocity.
Turbofan engines are usually described in terms of bypass ratio, defined as the
ratio of the flow through the bypass duct (cold stream) to the flow at entry to the
high-pressure compressor (hot stream).t With the notation of Fig. 3.15 the bypass
ratio B is given by
B=mc
mh
It immediately follows that
mB m
m =-- mil =-- andm =mc+m"
c B+l' B+l
For the particular case where both streams are expanded to atmospheric pressure
in the propelling nozzles, the net thrust is given by
m
F = (mcCjc + mhCjh) - mCa

qc


.-._------ .-------- -----------_.----
FIG. 3.15 Twin-spool turbofan engine
t The tenns turbofan and bypass engine may both be encountered, often referring to the same engine.
Early engines with a small portion of the flow bypassing the combustion (low value
ratio) were initially referred to as bypass engines. As the bypass ratio. IS the optimmn
pressure ratio for the bypass stream is reduced and can eventually be prOVIded bf a
stage. The term turbofan was originally used for engmes of high bypass ratio but IS mcreasmgly
employed for all bypass engines.
,
. !
THE TURBOFAN ENGINE 107
The design .point calculations for the turbofan are similar to those for the
turbojet; in view of this, only the differences in calculation will be outlined.
(a) Overall pressure ratio and turbine inlet temperature are specified as before,
but it is also necessary to specify the bypass ratio (B) and the fan pressure
ratio (FPR).
(b) From the inlet conditions and FPR, tht: pressure and temperature of the flow
leaving the fan and entering the bypass duct can be calculated. The mass
flow down the bypass duct can be established from the total flow and the
bypass ratio. The cold stream thrust can then be calculated as for the jet
engine, noting that air is the working fluid. It is necessary to check whether
the fan nozzle is choked or unchoked; if choked the pressure thrust must be
calculated.
(c) In the two-spool confignration shown in Fig. 3.15 the fan is driven by the LP
turbine. Calculations for the HP compressor and turbine are quite standard,
and conditions at entry to the LP turbine can then be found. Considering the
work requirement of the low pressure rotor,
mCpallTo12 = '1mmhcpgllTo56
and hence
m cpa Cpa A
IlT056 = - X --X IlTo12 == (B + 1) X --X UT012
mh '1mCpg '1mCpg
The value of B may range from 0·3 to 8 or more, and its value has a major
effect on the temperature drop and pressure ratio required from the LP
turbine. Knowing T05, '11 and IlTo56, the LP turbine pressure ratio can be
found and conditions at entry to the hot stream nozzle can be established;
the calculation of the hot stream thrust is then quite straightforward.
(d) If the two streams are mixed it is necessary to find the conditions after
mixing by means of an enthalpy and momentum balance; this will be
considered following an example on the performance of an engine similar to
that shown in Fig. 3.15.
EXAMPLE 3.2
The following data apply to a twin-spool turbofan engine, with the fan driven by
the LP turbine and the compressor by the HP turbine. Separate cold and hot
nozzles are used.
Overall pressure ratio
Fan pressure ratio
Bypass ratio me/mh
Turbine inlet temperature
Fan, compressor and turbine polytropic efficiency
Isentropic efficiency of each propelling nozzle
Mechanical efficiency of each spool
Combustion pressure loss
Total air mass flow
25·0
1·65
5·0
1550 K
0·90
0·95
0·99
1·50 bar
215 kgls
108 GAS TURBINE CYCLES FOR AIRCRAFT PROPULSION
It is required to find the thrust andSFC under sea level static conditions where the
ambient pressure and temperature are 1·0 bar and 288 K.
The values of (n - l)ln for the polytropic compression and expansion are:
for compression, II - 1 = _1_ (1' - 1) = 1 = 0.3175
II . 1]ooe l' a 0·9 X .J·5
. n - 1 (1' - 1) 0·9
for expanslOn, -n- = 1]oot --1'- g = 4 = 0·225
Under static conditions TOl = Ta and POI = Pa so that, using the nomenclature of
Fig. 3.14,
T
02
= P02 yields T02 = 288 X 1·650.317S = 337·6 K G
)
(n-I)/n
T01 01
T02 - TOl = 337·6 - 288 = 49·6 K
P03 = 25·0 = 15.15
P02 1·65
(p )(n-1)/n
T03 = = 337·6 x 15.15
0
-3175 = 800·1 K
T03 - T02 = 800·1 - 337·6 = 462·5 K
The cold nozzle flressure ratio is
P02 =FPR = 1.65
Pa
and the critical pressure ratio for this: nozzle is
P02 _ 1 _ _ I = 1.965
Pc - [1 -i G D f(Y-l) -[1 - G::) fS .
Thus the cold nozzle is not choking, so that pg = Pa and the cold thrust Fe is given
simply by
Fe = meCg
The nozzle temperature drop, from equation (3.l2), is
T02 - Ts = I1j T02 [I - ()'-1)h]
[ (
1 ) l/3,SJ
== 0·95)( 337·6 1 - T6s = 42·8 K
and hence
1 1
Cg = [2c/To2 - Tg)]'= (2)( 1·005)( 42·8)( 1000)2 = 293·2 mls
THE TURBOFAN ENGINE
Since the bypass ratio B is 5·0,
mB 215 x 5·0
me = B + 1 = 6.0 = 179·2 kg/s
Fe = 179·2 X 293·2 = 52532 N
Considering the work requirement of the HP rotor,
'cpa 1·005 X 462·5
T04 - Tos = --(T03 - TO?) = = 409·0 K
YimCpg - 0·99 x 1·148
and for the LP rotor
109
cpa 6·0 X 1·005 X 49·6
Tos -- T06 = (B + l)--(T02 - T01 ) = = 263.2 K
1]mCpg 0·99 X 1·148
Hence
ToS = T04 - (T04 - Tos) = 1550 - 409·0 = 1141.0
T06 = Tos - (Tos - T06 ) = J 141·0 - 263·2 = 877.8
P06 may then be found as follows.
P04 (T04) n/(n-1) (1550) 1/0·225
Pos == Tos == 1141.0 = 3·902
Pos (Tos) n/(n-l) (1141.0) 1/0·225
-= - = --- =3·208
P06 T06 877·8
P04 = P03 -I1Pb =25·0 X ]·0 - 1·50 == 23·5 bar
P04 23·5
P06 = == == 1·878 bar
(P04/Pos)(PoslP06) 3·902 X 3·208
Thus the hot nozzle pressure ratio is
P06 == 1.878
Pa
while the critical pressure ratio is
, 1
P06 = 4 == 1.914
Pc [1 1 (0.333)J
0·95 2·333
This nozzle is also unchoked, and hence P7 = Pa.
T06 - T7 = 1]j T06 1 - -- [

1 )(Y-I)IYJ
06/Pa
= 0·95 X 877-8[1- (_I_)*J == 121·6 K
1·878
C7 == [2cp CT06 - == (2)( 1·148 x 121·6 x
== 528·3 mls
m 215
mh == B + 1 == 6.0 == 35·83 kg/s
Fh == 35·83 x 528·3 == 18931 N
110
GAS tlJRBINE CYCLES FOR AIRCRAFT PROPULSION
Thus the total thrust is
Fe +Fh = 52532 + 18931 = 11463 N or 71·5 kN
The fuel flow can readily be calculated from the known temperatures in the
combustor and the airflow through the combustor, i.e. mho The combustion
temperature rise is (1550 - 800) = 750 K and the combustion inlet temperature
is 800 K. From Fig. 2.15 the ideal fuel/air ratio is found to be 0·0221 and the
actual fuel/air ratio is then (0.0221/0·99) = 0·0223. Hence the fuel flow is given
by
and
mf = 0·0223 x 35·83 x 3600 = 2876·4 kg/h
2876-4
SFC = --= 0·0403 kg/h N
71463
Because both nozzles were unchoked, the thrust could be evaluated without
calculating the nozzle areas. It is always good practice, however, to calculate key
pieces of information which may be required for other purposes. In both cases the
area can be calculated from continuity, i.e. m = pAC. The density is obtained from
p = p/RT, where p and T are the static values in the plane of the nozzle; for both
nozzles p will equal Pa' The following results are obtained for the two streams:
Static pressure (bar)
Static temperature (K)
Density (kg/m3)
Mass flow (kg/s)
Velocity (mls)
Nozzle area (mz)
Cold Hot
1·0 1·0
192·6 749·8
1·191 0·4647
179·2 35·83
293·2 528·3
0·5132 0·1459
The cold nozzle area is much larger than the hot one, and Fig. 1.12(b) shows the
physical appearance of an engine of similar cycle and bypass ratio; Fig. 1.12(a)
shows an engine of lower bypass ratio, around 2.5.
This example illustrated the method followed when a propelling nozzle is un-
choked, while the previous example showed how a choked nozzle may be dealt
with.
Note that at static conditions the bypass stream contributes approximately
74 per cent of the total thrust. At a forward speed of 60 mis, which is
approaching a normal take-off speed, the momentum drag mCa will be 215 x 60
or 12900 N; the ram pressure ratio and temperature rise will be negligible and
thus the net thrust is reduced to 58563 N. The drop in thrust during take-off is
even more marked for engines of higher bypass ratio and for this reason it is
preferable to quote turbofan thrusts at a typical take-off speed rather than at static
conditions.
THE TURBOFAN ENGINE 111
Mixing of hot and cold streams
Mixing is essential for an after burning turbofan when maximum thrust boosting
is required, to avoid the need for two reheat combustion systems. In certain cases
mixing may also be advantageous in subsonic transport applications, giving a
small but significant gain in SFC. We shall present a simple method of dealing
with mixing. in a constant area duct, with no losses and assuming adiabatic flow.
The duct is shown schematically in Fig. 3.16, with the hot and cold flows be-
ginning to mix at plane A and with complete mixing achieved by plane B.
Starting from the enthalpy balance, with suffix m denoting the properties of the
mixed stream,
mccpeToz + mhcphT06 = mCpmT07
where m=mc+mh
We also have the following equations that relate the properties of a mixture of
gases to those of its constituents:
mccpc + mhcph
(me + mh)
meRe +mhRh
Rm (me+mh)

From the momentum balance,
(mcCc + P2A2) + (mhCh + P6A6) = mC7 + P7A7
If there is no swirl in the jet pipe downstream of plane A, the static pressure will
be uniform across the duct, and so pz = P6'
From continuity,
m = P7C?A?
It is the pressure after mixing, PO?, that is. required for the cycle calculation,
because this is the stagnation pressure at entry to the propelling nozzle. When this
has been found, the calculation for the thrust is the same as described previously.
A B
I I
T02 P02 Ii
me I I T07
7" I I nl_
'06 P06 I I P07
nlh I I
__ ._
FIG. 3.16 Mixing in a constant area duct
112 GAS TURBINE CYCLES FOR AIRCRAFT PROPULSION
The calculation ofPo7 is simplified if we work Mach numbers. in
the hot and cold streams: the hot stream Mach number IS fixed' by the turbme
design, and typically M6 will be about 0·5. Having selected a of M6, the
procedure is as follows. We will malce: use of the standard relatIOns between M
and static and stagnation P and T, equations (8) in Appendix A.2.
(a) Knowing M6, T06 and P06,' we can determine P6 and Ch: P6 and T6 give P6,
hence A6 follows from continuity. We now know (mhCh + PI06). .
(b) With P6 = P2, P2/P02 yields the value of M2· With M2, P02 and T02 known we
can now find Ce and A2 so that (meCe + P2A2) is Imown.
(c) (mC7 +A7P7) is now obtained from the momentum balance.
(d) A7 =A6 + A2, and m = pC7A7 = C7A7
(e) T. is Imown from the enthalpy balance, but neither P7 nor P07 is known.
a value of M7, and then find T7 and C7; continuity then gives P7.
(f) It is now necessary to check that (mC7 +A7P7) is equal to the value obtained
from the momentum balance in (c).
(g) Iterate on M7 until correct P7 is found.
(h) P07 is then obtained fromp7, M,.
The fan outlet pressure (Poz) should be only slightly higher than the turbine
outlet pressure (P06) to keep mixing losses to a minimum; typically, P02/P06
should be about 1·05-1·07. In practice, quite small changes in cycle parameters
can cause significant changes in the ratio POZ/P06 and negate the benefits of
mixing. No hard and fast rules can given and the decision on whetlrer to use
mixing or not will also be influenced by installation and engine weight
considerations, combined with a detailed investigation of the pressure losses
caused by the mixer; Ref. (7) describes an experimental investigation of mixing
losses.
Optimization of the turbofan cycle
DesiQTIers of turbofans have four thermodynamic parameters at their disposal;
over:1I pressure ratio and turbine inlet temperature (as for the simple turbojet),
and also bypass ratio and fan pressure ratio. Optimization of tire cycle is some-
what complex but the basic principles are easily understood.
Let us consider an engine with the: overall pressure ratio and bypass ratio both
specified. If we select a value of turbine inlet temperature the energy input is
because the combustion chamber air flow and entry temperature are determmed
by the chosen operating conditions. The remaining variable is the fan pressure
ratio and as a first step it is necessary to consider the variation of specific tlrrust
and specific fuel consumption with FPR. Ifwe start with a low value of FPR, the
fan thrust will be small and the work extracted from the LP turbine will also be
small; thus little energy will be extracted from the hot stream and a large value ?f
hot thrust will result. As the FPR is raised it is evident that the fan thrust will
increase and the hot thrust will decrease. A typical variation of specific thrust and
THE TURBOFAN ENGINE
113
Overall pressure ratio and bypass ratio fixed
SFC


Fs
------ Turbine inlet
temperature increasing
Fan pressure ratio
FIG. 3.17 Optimization of fan pressure ratio
SFC with FPR, for a range of turbine temperature, is shown in Fig. 3.17. It can be
seen that for any value of turbine inlet temperature there will be an optimum value
of FPR; optimum values of FPR for minimum SFC and maximum specific thrust
coincide because of the fixed energy input. Taking the values of SFC and specific
thrust for each of these values of FPR in turn, a curve of SFC against specific
thrust may be plotted as shown in Fig. 3.18(a). Note that each point on this curve
is the result of a previous optimization, and is associated with a particular value of
FPR and turbine inlet temperature.
The foregoing calculations may be repeated for a series of bypass ratios, still at
the same overall pressure ratio, to give a family of curves as shown in Fig.
3 .18(b). This plot yields the optimum variation of SFC with specific thrust for the
SFC
Bypass ratio fixed
Specific thrust Fs
(a)
FIG. 3.18 TlIrbofan optimization
SFC
Increasing
bypass ratio
Optimum
Specific thrust Fs
(b)
/
/
(/
114 GAS TURBINE CYCLES FOR AIRCRAFT PROPULSION
selected overall pressure ratio as shown by the dotted envelope curve. The
procedure can then be repeated for a range of overall pressure ratio. It will be
clear that the optimization procedure is lengthy and that a large amount of
detailed calculation is necessary. The qualitative results of such a series of
calculations can be sununarized as follows:
(a) Increasing bypass ratio' improves SFC at the expense of a significant
reduction in specific thrust.
(b) The optimum fan pressure ratio increases with turbine inlet temperature.
(c) The optimum fan pressure ratio decreases with increase of bypass ratio. (At
a bypass ratio of about 5 the FPR may be low enough to pennit the use of a
single-stage fan.)
The choice of cycle parameters is dependent on the aircraft application, and
both high and low bypass ratios have their place. Specific fuel consumpti?n is of
major importance for long range subsonic transport aircraft and the
can best be met by using a bypass ratio of 4-6 and a high overall pressure ratIO,
combined with a high turbine inlet temperature. A higher bypass ratio of 8-9 was
chosen for the GE90, which entered service in 1995, representing the first major
change in this parameter for many years. Military aircraft with a supersonic dash
capability and a requirement for good subsonic SFC would use a much lo,:"er
bypass ratio, perhaps 0·5-1, to keep the frontal area down, and afterburnmg
would be used for supersonic operation. Engines currently under development,
with significantly higher values of TIT, will permit operation up to speeds of
about M 1-4 without the use of afterburning. Short-haul connnercial aircraft are
not as critical as long-haul aircraft regarding SFC and for many years bypass
ratios of around 1 were used; modem designs, however, use higher bypass ratios
similar to those used in long-haul aircraft. The prime reason for this change is the
significant decrease in engine noise resulting from increased bypass ratio.
Another reason for concentrating on engines oflow SFC, so that they are suitable
for a wide range of aircraft, is the escalating cost of developing new engines.
Figure 3 .18(b) showed that optimizing the SFC required the use of high values
of BPR resulting in engines of low specific thrust. These curves apply only to
uninstalled engine performance, and installation effects must be considered for
evaluation in a specific aircraft. High BPR engines would have a large diameter
fan and both diameter and weight would increase with BPR; ground clearance
effects would cause increases in undercarriage length and weight, increasing
aircraft weight and thrust required.
The combination of the nacelle and its support system is normally refened to
as the 'pod' and it is instructive to do a simple analysis on the effect of pod drag.
The propulsion efficiency will be modified to give
_ _ 2_ [net thrust - pod drag]
'1p - Cj net thrust
1+-
Ca
THE TURBOFAN ENGINE
The ideal propulsion efficiency can be expressed in a different fmID as
2
11 - -----,---
p - 1 + gross thrust
momentum drag
2 )( momentum drag
momentum drag + gross thrust
2 )( momentum drag _ 2
'2 x momentum drag + net thrust - 2 + net thiust
momentum drag
Thus, including the effect of pod drag
1] = 2 [net thrust - pod drag]
p 2 + net thrust net thrust
momentmn drag
- r1 -
2 net thlUst (net thrust )
+ momentum drag momentum drag
115
The ratio of (net thrust/momentum drag) is directly related to the bypass ratio,
decreasing with increase in BPR. Using IX to denote (pod drag/momentum drag),
the effect of pod drag on '1p can be readily evaluated giving the results shown in
Fig. 3.19. It can be seen that at high values of BPR the effect of pod drag is
significant and the installation effects must be carefully evaluated by the aircraft
manufacturer, working in conjunction with the engine manufacturer.
It was mentioned earlier that, because of their reduced mean jet velocity,
turbofans produce less exhaust noise than turbojets. At first sight it would appear
that noise considerations would demand the highest possible bypass ratio,
1.00

Pod drag
Momentum drag
0.50
...,------c .... ____ ------; ...
High BPR Jets
__ -'-_--1 __
o 0.5 1.5 2.0
Net thrust/momentum drag
FIG. 3.19 Effect of pod drag Uill propulsioll efficiency
116
GAS TURBINE CYCLES FOR AIRCRAFT PROPULSION
resulting in a low jet velocity. Unfortunately, however, as bypass ratio is increased
the resulthig high tip speed of the fan leads to a large increase in fan noise. Indeed
at approach conditions, with the engine operating at a low thrust setting, the fan
noise predominates; fan noise is essentially produced at discrete frequencies
which can be much more irritating than the broad-bandjet noise. The problem
can be alleviated by acoustic treatmeilt of the intake duct, avoiding the use of inlet
guide vanes, and careful choice of axial spacing between the fan rotor and stator
blades.
Turbofan configurations
The cycle parameters for a turbofan have a much greater effect on the mechanical
design of the engine than in the case of the turbojet. This is because variation in
bypass ratio implies variation in component diameters and rotational speeds, and
the configuration of engines of low and high bypass ratio may be completely
different.
Some early turbofans were directly developed from existing turbojets, and this
led to the 'aft fan' configuration shown in Fig. 3.20. A combined turbine-fan was
mounted downstream of the gas-generator turbine. Two major problems arise
with this configuration. Firstly, the blading of the turbine-fan unit must be
designed to give turbine blade sections for the hot stream and compressor blade
sections for the cold stream. This obviously leads to blading of high cost and,
because the entire blade must be made from the turbine material, of high weight.
The other problem is that of sealing between the two streams. The aft-fan
configuration has not been used in a new design for many years, but is a possible
contender for ultra high bypass (UHB) engines. In this case, however, the turbine
and fan sections would probably be connected by a gearbox and the complex
blades used in earlier engines would not be required.
For moderate bypass and overall pressure ratios the simple two-spool
arrangement of Fig. 3.15 is adequate. At very high bypass ratios, especially
when combined with high overall pressure ratios, design problems arise because
the fan rotational speed must be much lower than that of the high-pressure rotor;
the important limitations on blade tip speed will be discussed in Chapter 5.
Four different configurations which may be used to obtain high bypass ratio
and high overall pressure ratio are shown in Fig. 3.21. The configuration of Fig.
Fan
--------------------------------------
FIG. 3.20 Aft-fan configuration
THE TURBOPROP ENGINE 117
(\ll Two·spool (b) Two·spool
(cl Three·spool ------------
(dl Two-spool geared fan
FIG. 3.21 Configurations for high bypass ratio turbofans
; .21 (a) suffers from the fact that the later stages on the LP rotor usuall called
booster stages', contribute little because of their low blade speed ShY (b)'
more attractive, but requires a very hi h . . c erne IS
h' hid . . . g pressure ratio from the HP compressor
w IC ea s to mstabiijty problems referred to in section 1.2. The three-s 001
arrangement of FIg: 3.2l(c) is in many ways the most attractive conce t wfth a
ratio each compressor. All of these have been useJ for large
Trent (FIg. 1.12(b» is an example of (a) while the Rolls-Royce
. g.. IS an example of (c). A geared-fan arran ement as . .
:ssible for smaller engines, and units of this sort have bee! develope: 1:
boPbroPbbackground. The power requirements for the fan of a large turbofan
may e a out 60 MW and the de . f l'gh .
uld' . 0 a 1 twelght gearbox to handle this
be deSIgn studIes, however, are being carried out for ultra
g . ypass ratIO (UHB) engines with variable pitch fans where the low
of the large diameter fan requires a gearbox to using a large
rome stages.
3.5 The turboprop engine
The turboprop engine differs from the shaft power units dealt with in Cha ter 2 in
that some of the useful output appears as J' et thrust In thi th P ..
nec t b' . s case, erefore It IS
essary 0 com me shaft power and J' et thrust Thi b d' '
b . I . s can e one m a number of
ways, ut m al cases a knowledge of the aircraft speed is involved
. be delivered to the aircraft in the form of thrust ower
driving a propeller. thrust power (TP)
power (SP), propeller effiCIency IJpr and jet thrust F by
TP = (SP)IJpr + FCa
118 GAS TURBINE CYCLES FOR AIRCRAFT PROPULSION
In practice the shaft power will· account for a proportion of the enthalpy
drop available at the gas-generator exit, and thrust power is therefore largely
dependent on the propeller efficiency which may vary significantly with flight
conditions.
It is desirable to find some way of expressing the power so that it can be readily
compared with that ofa piston· engine, and so that it is not quite so dependent on
propeller efficiency. The same basic engine may be used in conjunction with a
variety of propellers for different applications, and it is the performance of the
engine itself which is then of most interest to us. A more suitable way of
expressing the power is to quote the equivalent (or effective) power EP defined as
TP Fea
EP=-=SP+-
'1pr '1pr
'1pr now affects only the smaller term. The equivalent power is an arbitrarily
defined quantity and it should not be quoted without reference to the flight speed.
that by definition the EP and SP are equal at static conditions, although
there is some beneficial jet thrust. Allowance must be made for this when com-
paring engines under take-off conditions. Experiments have shown that an aver-
age propeller produces a thrust of about 8·5 N per kW of power input under static
conditions, so that the take-offEP is conventionally taken as SP + (F/8.5) with SP
in kW and F in newtons.
Turboprops are usually rated on the basis of equivalent power at take-off
conditions, and the specific fuel consumption and specific power are often
expressed in terms of that equivalent power. It is nevertheless desirable to quote
both the shaft power and jet thrust available at any condition of interest and it
should be recognized that the equivalent power is merely a useful, but artificial,
concept. In view of the similarity between the turboprop and the shaft power uuits
discussed in Chapter 2 it is not necessary to elaborate on cycle requirements. The
only basic difference is that the designer can choose the proportions of the
available enthalpy drop used to produce shaft power and jet thrust. It can be
shown that there is an optimum division for any given flight speed and altitude; a
simple rule of thumb is to design so that the turbine exit pressure is equal to the
inlet pressure. The equivalent power is not particularly sensitive to
turbine exit pressure in this range. Turboprops generally operate with the nozzle
unchoked and use a simple straight through tailpipe rather than a convergent
nozzle.
The combined efficiency of the power turbine, propeller, and the necessary
reduction gear, is well below that of an equivalent propelling nozzle. It follows
that 'Ie for a turboprop engine is lower than that of a turbojet or turbofan engine.
The turboprop has held its position for speeds up to M 0·6 because the propulsion
efficiency is so much higher than that of the turbojet; the turboprop is widely used
in business aircraft and regional airliners, mostly at power levels of 500-
2000 kW. The propeller efficiency, using conventional propeller design methods,
decreases drastically at flight speeds above M 0·6 and for this reason the
THE TURBOPROP ENGINE
119
did not widely used for longer haul aircraft, being superseded
y d tur ofans of eqUIvalent propulsion efficiency; an exception was long
en urance patrol aircraft, which fly to the search area at M 0·6 and fuen 1o"t
at much lower flight speeds In th I . h. I er
fi .. e ear y eig tIes, however, considerable attention
was ocused on the deSIgn of propellers with the oa1 of obt . .
of 0·80 at fli?ht Mach numbers of 0·8. Th!
would give large savings in fuel compared to existing turbofans
.e of the propeller was markedly different from conventional desi s·
su?ersouic blading and 8-10 blades; these devices were c:ci
to them from conventional propellers. Studies showed that
prop aIrcraft would require much higher powers fu . 1
expenenced wifu turb d an prevIOUS y
b. oprops, an at power levels in excess of 8000 kW fue
gear ox deSl?n becomes difficult. A further major problem is the transruission of
propeller nOIse to the passenger cabin and it was wI·del . d h
bt ·1· .. ' yrecogIllZe tata
su s antia ill noIse over existing turbofan levels would not be acceptable
to It probable that this can be overcome only by the use of 'pusher'
b guranons . Wlfu fue propellers mounted behind fue passenger cabin
ur oprop are surveyed in Ref. (8). .
An alternative configuration, a comproruise between the turbopro and th
was. the 'unducted fan' (UDF), developed by General in th:
eIghties. In tillS two c?unter-rotating variable pitch fans were directI
coupled to counter-rotating turbmes with no stators elinIin tin th d fi Y
gearbox Th h . , a g enee ora
. e sc ematic arrangement depicted in Fig. 3.22 shows a similarity to
Stationary support structure
---t-----{jt:-------.-..-.--_._.
-----------
FIG. 3.22 Dnducted fan engine
120
GAS TURBINE CYCLES FOR AIRCRAFT PROPULSION
the aft-fan configuration described earlier. This revolutionary approach demon-
strated a significant reduction in fuel consumption in flight tests, but
problems with noise and cabin vibration; airlines were also to pIOneer
this new technology, and not all engine manufacturers were convmced. that the
gearbox could be eliminated. The UDr concept, however, was also to
be a promising teclmology for the development of long range mIssIles,
where its excellent SFC would improve range for a fixed quantIty of fuel on
board.
The status of advanced turboprop8, prop fans and unducted fans in the mid
1980s was very similar to that of turbofans in the mid 1950s. At that time there
was a heated controversy regarding the relative merits of turboprops and
turbofans, with the latter being the clear winner. In the mid 1990s the turbofan
has once again emerged as the winner; fuel prices, however, have been stable over
a long period, but ifthis were to change significantly engines having ultra low fuel
consumption would be essential.
The conventional turboprop will certainly continue to dominate the market for
smaller aircraft of 10-60 seats with flight speeds in the range of 400-600 kmIh;
these aircraft would be used on flights with a duration of 60-90 minutes with a
range of perhaps 400-500 km. The turboshaft engine, in which the output power
drives a helicopter rotor is also of great importance and is virtually universally
used because of its low weight and high power. In the helicopter application, free
turbine engines are used. The helicopter rotor is designed to operate at constant
speed by changing the pitch and the power is varied by changing the gas
generator speed. While at first sight the design requirements of. turboprops. and
turboshafts appear identical, the former may be optinuzed for crUIse at an altItude
of 6-10 000 m while the latter is optimized for operation at very low altitudes. A
number of engines are available in both turboprop and turboshaft versions, but
invariably any given engine is found to be much more successful in one
application than the other.
3.6 Thru.st augmentation
If the thrust of an engine has to increased above the original design value
several alternatives are available. Inc:rease of turbine inlet temperature, for exam-
ple, will increase the specific thrust and hence the thrust for a size.
Alternatively the mass flow through the engine could be mcreased Without
altering the cycle parameters. Both of these methods imply some redesign of the
engine, and either or both may be used to uprate an existing engine.
Frequently, however, there will be a requirement for a temporary increase in
thrust, e.g. for talce-off, for acceleration from subsonic to supersonic speed or
during combat manoeuvres; the problem then becomes one of thrust
augmentation. Numerous schemes for thrust augmentation have been proposed,
but the two methods most widely used are liquid injection and afterburning (or
reheat).
THRUST AUGMENTATION 121
inje.cti?n is primarily useful for increasing take-off thrust. Substantial
of hqUld required, but if the liquid is consumed during take-off and
lll1t1al the weight penalty is not significant. Spraying water into the
compressor inlet causes evaporation of the water droplets, resulting in extraction
of heat from the air; the effect of this is equivalent to a drop in compressor inlet
8 will show that reducing the temperature at entry to a
turbOjet mcrease the thrust, due to the increase in pressUI'e ratio and mass
flow resultmg from the effective increase in rotational speed. In practice a mixture
of is used; the methanol lowers the freezirIg point of water,
and It Will ?urn wh.en it reaches the combustion chamber. Liquid is
ll1Jected dIrectly mto the combustion chamber. The resulting
blockage forces the compressor to operate at a higher pressure ratio causing
the thrust to mcrease. In both cases the mass of liquid injected adds to the useful
mass flow, but this is a secondary effect. Liquid irIjection is now seldom used irI
aircraft engines.
. Afterburning, as .the name implies, involves burning additional fuel in the jet
pIpe as shown m FIg. 3.8. In the absence of highly stressed rotating blades the
allowable following afterburning is much higher than the turbine
inlet temperature. Stoichiometric combustion is desirable for maximum thrust
augmentation fmal temperatures of around 2000 K are possible. Figure 3.23
shows the T-s for a turbojet with the addition of afterburning to
2000 K. The large mcrease m fuel flow required is evident from the relative
in the combustion chamber and afterburner, and the penalty in
IS heavy. Assunllng that a choked convergent nozzle is used, the jet
velOCity WIll to the sonic velocity at the appropriate temperature in the
of nozzle, I.e. T7 or T5 depending on whether the engine is operated
WIth or WIthout afterburning. Thus the jet velocity can be found from (yRT )1/2
with Tc given either by T06ITc= (y -/- 1)/2 or T04ITc= (y -/- 1)/2. It follows th:t th;
a
s
FIG. 3.23 Cycle of turbojet with aftel'burnillg
122 GAS TURBINE CYCLES FOR AIRCRAFT PROPULSION
jet velocity is proportional to JTo at inlet to the propelling nozzle, and that the
gross momentum thrust, relative to that of the simple turbojet, will be increased in
the ratio J(TorJT04). For the temperatures shown in Fig. 3.23 this amounts to
J(2000/959) or 1·44. As an approximation, the increase in fuel would be in the
ratio (2000 - 959)+ (1200 - 565) with afterburning, to (1200 - 565) without
afterburning, i.e. 2·64. Thus a '44 per cent increase in thrust is obtained at the
expense of a 164 per cent increase in fuel flow, and clearly afterburning should be
used only for short periods. This might be the thrust augmentation under take-off
conditions where the gross thrust is equal to the net thrust. At high forward
speeds, however, the gain is much greater and is often well over 100 per cent.
This is because for a fixed momentum drag an increase in gross thrust represents
a considerably greater increase in net thrust. The Concorde makes use of
afterburning for transonic acceleration from M 0·9 to M 1-4; the significant
increase in net thrust provides faster acceleration through the high drag regime
near M 1·0, resulting in a reduction in fuel consumption in spite of the short term
increase in fuel flow. Afterburning offers even greater gains for low bypass ratio
turbofans, because of the relatively low temperature after mixing of the hot and
cold streams and the larger quantities of excess air available for combustion;
military turbofans use after burning for take-off and combat manoeuvring.
It is essential for engines fitted with an afterburner to incorporate a variable
area nozzle because of the large change in density of the flow approaching the
nozzle resulting from the large change in temperature. Afterburning will normally
be brought into operation when the engine is running at its maximum rotational
speed, corresponding to its maximum unaugmented thrust. The afterburner
should be designed so that the engine will continue to operate at the same speed
when it is in use, and hence the nozzle must pass the same mass flow at a much
reduced density. This can be achieved only if a variable nozzle is fitted permitting
a significant increase in nozzle area. Note that the pressure thrust will also be
increased owing to the enlarged nozzle area.
The pressure loss in the afterburner can be significant. Combustion pressure
losses are discussed in Chapter 6, where it is shown that the pressure loss is due to
both fluid friction and momentum changes resulting from heat addition. In
combustion chambers the former predominates, but in afterburners the losses due
to momentum changes are much more important. The temperature rise is
determined by the turbine outlet temperature and the fuel/gas ratio in the
afterburner, and the pressure loss due to momentum changes can be determined
using the Rayleigh functions and the method outlined in Appendix A.4. This
pressure loss is found to be a function of both the temperature ratio across the
afterburner and the Mach number at inlet to the duct. If the inlet Mach number is
too high, heat release can result in the downstream Mach number reaching 1·0
and this places an upper limit on the allowable heat release: the phenomenon is
referred to as thermal choking. Figure 3.24 shows values of the pressure loss due
to momentum changes and emphasizes the need for a low Mach number.
Typically, the exit Mach number from the turbine of a jet engine will be about 0·5
and it is necessary to introduce a diffuser between the turbine and afterburner to
NOMENCLATURE
Inlet Mach no.
1.5
Afterburner temperature ratio (TosITo4)
FIG. 3.24 Momentum pressure loss
123
reduce the Mach number to about 0·25-0·30 before introducing the afterburner
fuel.
Even when not in use, an afterburner incurs some penalty in pressure loss due
the presence ?f the burners and flame stabilizing devices. Another
this method of thrust augmentation is that the very high jet
velOCIties from a degree of afterburning result in a noisy exhaust.
The Olympus engmes used on Concorde provide about 15-20 per
cent mcrease. m take-off !hrust, resulting in an exhaust temperature of around
1400 The mcreased nOIse level, though a serious concern, is significantly less
than mIght be expected from with military aircraft. It is most unlikely,
how.ever, any supersomc transport will use after burning for take-off, as
a.pnme WIll be take-off noise levels comparable to current subsonic
arrcraft; mentioned earlier, this will necessitate the development of variable
cycle engmes.
NOMENCLATURE
a sonic velocity
A cross-sectional area
B bypass ratio (mJmh)
F net thrust
Fs specific thrust
Kp specific thrust coefficient
M Mach number
l1e efficiency of energy conversion
Iii intake efficiency
I1j nozzle efficiency
11m mechanical efficiency
THRUST AUGMENTATION 125
124
GAS TURBINE CYCLES FOR AIRCRAFT PROPULSION
Internatiollal Standard Atmosphere
110
overall efficiency
z p T a
I1p
propulsion (Froude ) efficiency
[m] [bar] [K]
p/Po
[m/s]
I1pr
propeller efficiency
I1r
ram efficiency
0 1·01325 288·15 1·0000 340·3
1100
polytropic efficiency
500 0·9546 284·9 0·9529 338-4
Suffixes
1000 0·8988 281·7 0·9075 336·4
critical condition, cold stream
1500 0·8456 278·4 0·8638 334·5
c
2000 0·7950 275·2 0·8217 332·5
h hot stream
2500 0·7469 271·9 0·7812 330·6
j jet
3000 0·7012 268·7 0·7423 328·6
m mixed 3500 0·6578 265·4 0·7048 326·6
4000 0·6166 262·2 0·6689 324·6
4500 0·5775 258·9 0·6343 322·6
5000 0·5405 255·7 0·6012 320·5
5500 0·5054 252·4 0·5694 318·5
6000 0-4722 249·2 0·5389 316·5
6500 0-4408 245·9 0·5096 314-4
7000 0·4111 242·7 0·4817 312·3
7500 0·3830 239·5 0-4549 310·2
8000 0·3565 236·2 0·4292 308·1
8500 0·3315 233·0 0·4047 306·0
9000 0·3080 229·7 0·3813 303·8
9500 0,2858 226·5 0·3589 301·7
10000 0·2650 223·3 0·3376 299·5
10500 0·2454 220·0 0·3172 297-4
11 000 0·2270 216·8 0·2978 295·2
11500 0·2098 216·7 0·2755 295·1
12000 0·1940 216·7 0·2546 295·1
12500 0·1793 216·7 0·2354 295·1
13 000 0·1658 216·7 0·2176 295·1
13 500 0·1533 216·7 0·2012 295·1
14000 0·1417 216·7 0·1860 295·1
14500 0·1310 216·7 0·1720 295·1
15000 0·1211 216·7 0·1590 295·1
15500 0·1120 216·7 0·1470 295·1
16000 0·1035 216·7 0·1359 295·1
16500 0·09572 216·7 0·1256 295·1
17000 0·08850 216·7 0·1162 295·1
17 500 0·08182 216·7 0·1074 295·1
18000 0·07565 216·7 0·09930 295·1
18500 0·06995 216·7 0·09182 295·1
19000 0·06467 216·7 0·08489 295·1
19500 0·05980 216·7 0·07850 295·1
20000 0·05529 216·7 0·07258 295·1
Density at sea level Po = 1·2250 kg/m3
Extracted from: ROGERS G F C and MAYHEW Y R
Thermodynamic and Transport Properties of Fluids (Blackwell
1995)
4
Centrifugal compressors
Very rapid progress in the development of gas turbines was made during the
Second World War, where attention was focused on the simple turbojet unit.
German efforts were based on the axial flow compressor, but British develop-
ments used the centrifugal compressor-Refs (1) and (2). It was recognized in
Britain that development time was critical and much experience had already been
gained on the design of small high-speed centrifugal compressors for super-
charging reciprocating engines. Centrifugal compressors were used in early
British and American fighter aircraft and also in the original Comet airliners
which were the first gas turbine powered civil aircraft in regular service. As power
requirements grew, however, it became clear that the axial flow compressor was
more suitable for large engines. The result was that a very high proportion of
development funding was diverted to the axial type leading to the availability of
axial compressors with an appreciably higher isentropic efficiency than could be
achieved by their centrifugal counterparts.
By the late fifties, however, it became clear that smaller gas turbines would
have to use centrifugal compressors, and serious research and development work
started again. Small turboprops, turboshafts and auxiliary power units (APUs)
have been made in very large numbers and have nearly all used centrifugal
compressors; notable examples include the Pratt and Whituey Canada PT-6, the
Garrett 331 and the large stable of APUs built by the latter organization. They are
also used for the high-pressure spools in small turbofans, see Fig. 1.12(a).
Centrifugals were used primarily for their suitability for handling small volume
flows, but other advantages include a shorter length than an equivalent axial
compressor, better resistance to Foreign Object Damage (FOD), less suscepti
bility to loss of performance by build-up of deposits on the blade surfaces and th
ability to operate over a wider range of mass flow at a particular rotational speed
The importance of the latter feature, in alleviating problems of matchin
operating conditions with those of the associated turbine, will be made clear .
Chapter 8.
A pressure ratio of around 4 : I can readily be obtained from a single-stag
compressor made of aluminium alloys, and in section 2.4 it was shown that this i
adequate for a heat-exchange cycle when the turbine inlet temperature is in th
PRINCIPLE OF OPERATION 127
region of 1000-1200 Ie Many proposals for vehicular gas turbines were based on
this arrangement and manufacturers such as Leyland, Ford, General Motors and
Chrysler built development engines which never went into production. The
advent of titanium alloys, permitting much higher tip speeds, combined with
advances in aerodynamics now permit pressure ratios of greater than 8 : 1 to be
achieved in a single stage. When higher pressure ratios are required, the
centrifugal·compressor may be used in conjunction with an axial flow compressor
(Fig. 1.11), or as a two-stage centrifugal (Fig. 1.10). Even though the latter
arrangement involves rather complex ducting between stages, it is still regarded
as a practical proposition. Reference (2) of Chapter I describes the design process
leading to the choice of a twin-spool all-centrifugal compressor for the Pratt and
Whituey Canada PWlOO turboprop which entered service in 1984.
4.1 Principle of operation
The compressor consists essentially of a stationary casing containing a
rotatmg impeller which imparts a high velocity to the air, and a number of fixed
diverging passages in which the air is decelerated with a consequent rise in static
pressure. The latter process is one of diffusion, and consequently the part of the
compressor containing the diverging passages is known as the difJit.ser. Figure
4.1(a) is a diagrammatic sketch of a centrifugal compressor. The impeller may be
single- or double-sided as in 1 (b) or I ( c), but the fundamental theory is the same
for both. The double-sided impeller was required in early aero-engines because of
the relatively small flow capacity of the centrifugal compressor for a given overall
diameter. .
Air is sucked into the impeller eye and whirled round at high speed by the
vanes on the impeller disc. At any point in the flow of air through the impeller, the
centripetal acceleration is obtained by a pressure head, so that the static pressure
of the air increases from the eye to the tip of the impeller. The remainder of the
static pressure rise is obtained in the diffuser, where the very high velocity of the
air leaving the impeller tip is reduced to somewhere in the region of the velocity
with which the air enters the impeller eye; it should be appreciated that friction in
the diffuser will cause some loss in stagnation pressure. The normal practice is to
design the compressor so that about half the pressure rise occurs in the impeller
and half in the diffuser.
It will be appreciated that owing to the action of the vanes in carrying the air
around with the impeller, there will be a slightly higher static pressure on the
forward face of a vane than on the trailing face. The air will thus tend to flow
round the edges of the vanes in the clearance space between the impeller and the
casing. This naturally results in a loss of efficiency, and the clearance must be
kept as small as possible. A shroud attached to the vanes, Fig. 4.1 (d), would
eliminate such a loss, but the manufacturing difficulties are vastly increased and
there would be a disc friction or 'windage' loss associated with the shroud.
128
Vaneless
seace
Impeller eye
(a)
CENTRIFUGAL COMPRESSORS
or" diffuser channel
Impeller shroud

Ij·-'M

(b) (c)
FIG. 4.1 Diagrammatic sketches of centrifugal compressors
Although shrouds have been used on superchargers, they are not used on
impellers for gas turbines. . .,
The impellers of modern centrifugal compressors operate WIth very high tIp
speeds resulting in very high stress levels. It will be shown the next that
backswept curved vanes are desirable for compressors of high pressure ratio, but
for many years designers were forced to use radial vanes because of the ten?ency
for curved vanes to straighten out under the action of the considerable centrIfugal
force involved, setting up undesirable bending stresses in the vanes. Modern
methods of stress analysis combined with stronger materials, however, now
permit backswept vanes to be used in high-performance compressors.
4.2 Work done and pressure rise
Since no work is done on the air in the diffuser, the energy absorbed by the
compressor will be determined by the conditions of the air at the inlet and outlet
of the impeller. Figure 4.2 shows the nomenclature employed.
WORK DONE AND PRESSURE RISE 129
In the, first instance itwill be assUmed that the air enters the impeller eye in the
axial direction; so that· the initial angular momentum of the air is zero. The axial
. portion of the vanes must be curved so that the air can pass smoothly into the eye.
The angle which the leading edge of a vane makes with the tangential direction ex
will be given by the direction of the relative velocity of the air at inlet, Vb as
shown in Fig. 4.2.
If the air leaves the impeller tip with an absolute velocity C2, it will have a
tangential or whirl component Cw2, and a comparatively small radial component
Cr2• Under ideal conditions C2 would be such that the whirl component is equal
to the impeller tip speed U, as shown by the velocity triangle at the top of Fig. 4.2.
Due to its inertia, the air trapped between the impeller vanes is reluctant to move
round with the impeller, and we have already noted that this results in a higher
static pressure on the leading face of a vane than on the trailing face. It also
prevents the air from acquiring a whirl velocity equal to the impeller speed. This
effect is known as slip. How far the whirl velocity at the impeller tip falls short of
the tip speed depends largely upon the number of vanes on the impeller. The
greater the number of vanes, the smaller the slip, i.e. the more nearly Cw2
approaches U. It is necessary in desigu to assume a value for the slip factor (1,
where (1 is defined as the ratio Cw21U. Various approximate analyses of the flow in
an impeller channel have led to formulae for (1: the one appropriate to radial-
vaned impellers which seems to agree best with experiment is that due to Stanitz,
Ref. (4):
(1 = 1- 0·63n
n
where n is the number of vanes.
Ideal conditions
at impeller tip
FIG. 4.2 Nomenclature
Velocity relative
to impeller
!C/u.

Section through eye
at radius r1
130 CENTRIFUGAL COMPRESSORS
As explained in .any elementalY text on appliedtherrnodynamics, the
theoreticall torque which must be applied to the impeller will be equal to the
rate of change of angular momentum experienced by the air. Considering unit
mass flow of air, this torque is given by
theoretical torque = Cw2r2
If w is the angular velocity, the work done on the air will be
theoretical work done = Cw2r2w = Cw2 U
Or, introducing the slip factor,
theoretical work done = eJU
2
(4.1 )
(4.2)
For convenience, in both the chapters on compressors we shall treat the work
done on the air as a positive quantity.
Due to friction between the casing and the air carried round by the vanes, and
other losses which have a braking effect such as disc friction or 'windage', the
applied torque and therefore the actual work input is greater than this theoretical
value. A power inputfactor tjJ can be introduced to take account of this, so that
the actual work done on the air becomes
work done = '/leJU
2 (4.3)
If (T03 - TOI ) is the stagnation temperature rise across the whole compressor
then, since no energy is added in the diffuser, this must be equal to the stagnation
temperature rise (T02 - ToI ) across the alone. It will therefore be equal
to the temperature equivalent of the work done on the air given by equation (4.3),
namely

T03 - TOl = -- (4.4)
cp
where cp is the mean specific heat over this temperature range. Typical values for
the power input factor lie in the region of 1·035-1·04.
So far we have merely considered the work which must be put into the
compressor. If a value for the overall isentropic efficiency lIe be assumed, then it
is known how much of the work is usefully employed in raising the pressure of
the air. The overall stagnation pressure ratio follows as
= = 1 + 'Ie 03 0] = 1 +_',e_'I' __
P
(
Tol )1'/(1'-1) [ n (To - T )J1' /(1'-I) [ n ,1'eJU2]J'/(1'-1)
PO] TOI TOI CpTOl
(4.5)
The distinction b":tween the power input factor and the slip factor should be
clearly understood: they are neither independent of one another nor of lie. The
power input factor represents an increase in the work input, the whole of which is
absorbed in overcoming frictional loss and therefore degraded into thennal
energy. The fact that the outlet temperature is raised by this loss, and incidentally
WORK DONE AND PRESSURE RISE 131
by other frictional losses as well, enables the maximum cycle temperature to be
reached without burning so much fuel, so that as far as the efficiency of the whole
gas turbine unit is concerned these losses are not entirely wasteful. Nevertheless,
this effect is outweighed by the fact that more turbine work is used in driving the
compressor and isentropic (i.e. frictionless adiabatic) compression is the ideal at
which to ainl, It follows that the power input factor should be as low as possible, a
low value of tjJ implying simultaneously a high value of 17e. It should be
appreciated that 11e depends also upon the friction loss in the diffuser which does
not affect the argument up to equation (4.4). For this reason it is not helpful to
consider tjJ implicitly as part of lJe'
The slip factor, on the other hand, is a factor limiting the work capacity of the
compressor even under isentropic conditions, and this quantity should be as great
as possible. Clearly the more nearly Cw2 approaches U, the greater becomes the
rate at which work can usefully be put into a compressor of given size,
Unfortunately an increase in the number of vanes, which would increase (J, entails
an increase in the solidity of the impeller eye, i.e. a decrease in the effective flow
area. Additional friction losses arise because, for the same mass flow or
'throughput', the inlet velocity must be increased. Thus the additional work input
that can be employed by increasing the number of vanes may not result in an
increase in that portion which is usefully employed in raising the pressure of the
air; it may only increase the thennal energy produced by friction resulting in an
increase in tjJ and reduction in lie' A suitable compromise must be found, and
present-day practice is to use the number of vanes which give a slip factor of
about 0·9, i.e. about 19 or 21 vanes (see also under heading 'Mach nUl1lber in the
diffuser', in section 4.4).
From equation (4.5) it will be seen that the remaining factors influencing the
pressure ratio for a given working fluid are the impeller tip speed U, and the inlet
temperature Tol . Any lowering of the inlet temperature TOl will clearly increase
the pressure ratio of the compressor for a given work input, but it is not a variable
under the control of the designer, Reference to texts on strength of materials will
show that the centrifugal stresses in a rotating disc are proportional to the square
of the rim speed. For single-sided impellers of light alloy, U is limited to about
460 mls by the maximum allowable centrifugal stresses in the impeller: such a
speed yields a pressure ratio of about 4 : 1. Higher speeds can be used with more
expensive materials such as titanium and pressure ratios of over 8: 1 are now
possible. Because of the additional disc loading, lower speeds must be used for
double-sided impellers.
EXAMPLE 4.1
The following data are suggested as a basis for the design of a single-sided
centrifugal compressor:
power input factor tjJ
slip factor (J
rotational speed N
1·04
0·9
290 rev/s
132
overall diameter of impeller
eye tip diameter
eye root diameter
air mass flow m
inlet stagnation temperature TOI
inlet stagnation POl
isentropic efficiency 11e
CENTRIFUGAL COMPRESSORS
0·5 m
0·3 m
0·15 m
9 legis
295 K
1·1 bar
0·78
Requirements are (a) to determine the pressure ratio of the compressor and the
power required to drive it assuming that the velocity of the air at inlet is axial; (b)
to calculate the inlet angle of the illlpeller vanes at the root and tip radii of the eye,
assuming that the axial inlet velocity is constant across the eye annulus; and C c) to
estimate the axial depth of the impeller channels at the periphery of the impeller.
(a) Impeller tip speed U=n x 0·5 x 290=455·5 mls.
Temperature equivalent of the work done on unit mass flow of air is,
_ _ IjJ(JU
2
_ 1·04 x 0·9 x 455.5
2
_ 193K
T03 TOI - c - 1.005 X 103 -
p
P03 = [1 + 11c(To3 - TOI )]V/(V-ll = (1 + 0·78 x 193)3.5=4.23
POI TOI 295
Powerrequired = mCpCT03 - TOI ) = 9 x 1·005 x 193 = 1746 leW
(b) To find the inlet angle of the vanes it is necessary to detennine the inlet
velocity which in this case is axial, i.e. Cal = CI . Cal must satisfy the continuity
equation m=PIAtCa], where Al is the flow area at inlet. Since the density PI
depends upon C], and both are unlmown, a trial and error process is required.
The iterative procedure is not critically dependent on the initial value assumed
for the axial velocity, but clearly it is desirable to have some rational basis for
obtaining an estimated value for starting the iteration. The simplest way of
obtaining a reasonable estimate of the axial velocity is to calculate the density on
the basis of the Imown stagnation temperature and pressure; in practice this will
give a density that is too high and a velocity that is too low. Having obtained an
initial estimate of the axial velocity, the density can be recalculated and thence the
actual velocity from the continuity equation; if the assumed and calculated
velocities do not agree it is necessary to iterate until agreement is reached (only
the final trial is shown below). Note that it is normal to design for an axial
velocity of about 150 mis, this providing a suitable compromise between high
flow per unit frontal area and low frictional losses in the intake.
n(0·3
2
- 0.15
2
)
Annulus area of impeller eye, A] = 4 = 0·053 m
2
Based on stagnation conditions:
POI 1·1 x 100 3
PI ::::: RTOI = 0.287 x 295 = 1·30 kg/m
m 9
Cal = PIAl = 1.30 x 0.053 = 131 m/s
WORK DONE AND PRESSURE RISE
Since CI = Ca], the equivalent dynamic temperature is
C? 131
2
1.31
2
-= =--=8·5K
2cp 2 x 1·005 x 103 0·201
C
2
TI = TOI - _I = 295 - 8·5 = 286·5 K
2cp
. POI 1·1
PI = (TodTIy/()'-I) = (295/286.5)35 = 0·992 bar
_ PI _ 0·992 x 100 3
PI - RTI - 0.287 x 286.5 = 1·21 kg/m
Check on Cal:
C 9 _
al PIAl 1.21 x 0.053 - 140 m/s
Final trial:
Try Cal = CI = 145 m/s
Equivalent dynamic is
Cf 145
2
1.452
2c
p
= 2)( 1.005 X 103 = 0.201 = 10·5 K
C2 .
TI = TOI - -2 1 = 295 - 10·5 = 284·5 K
cp
POI 1·1
PI = (TOI/TIY/(V-I) = = 0·968 bar
0·968 x 100 _ 3
PI - RTI - 0.287 x 284.5 - 1·185 leg/m
Check on Cal:
C 9 _
01 PIAl 1·185 x 0.053 - 143 m/s
133
This is a good agreement and a further trial using Cal = 143 mls is unnecessary
because a small change in C has little effect upon p. For this reason it is more
accurate to use the final value 143 mis, rather than the mean of 145 mls (the trial
value) and 143 mls. The vane angles can now be calculated as follows:
Peripheral speed at the impeller eye tip radius
= n x 0·3 x 290 = 273m/s
and at eye root radius = 136·5 m/s
r:t. at root = tan-
l
143/136·5 = 46.33°
IX at tip = tan-I 143/273 = 27.65°
(c) . The shape of the impeller channel between eye and tip is very much a matter
of tnal and The to obtain as uniform a change of flow velocity up the
channel as pOSSIble, aVOldmg local decelerations up the trailing face of the vane
134 CENTRlFUGAL COMPRESSORS
which might lead to flow separation. Only tests on the machine can show whether
this has been achieved: the flow analyses already referred to [Ref. (4)] are for
inviscid flow and are not sufficiently realistic to be of direct use in design. To
calculate the required depth of the impeller channel at the periphery we must
make some assumptions regarding both the ramal component of velocity at the
tip, and the division of losses between the impeller and the diffuser so that the
density can be evaluated. The radial component of velocity will be relatively
small and can be chosen by the designer; a suitable value is obtained by making it
approximately equal to the axial velocity at inlet to the eye.
To estimate the density at the impeller tip, the static pressure and temperature
are found by calculating the absolute velocity at this point and using it in
conjunction with the stagnation pressure which is calculated ii'om the assumed
loss up to this point. Figure 4.3 may help the reader to follow the calculation.
Making the choice Cr2 = Cal' we have Cr2 = 143 mls
Cwz = (JU = 0·9 x 455·5 = 410 m/s
ci _ C;2 + C ~ 2 _ 1.43
2
+ 4·10
z
- - 0.201 = 93·8 K
2cp 2cp
Assuming that half the total loss, i.e. o· 5(1 -. '1e) = 0·11, occurs in the impeller,
the effective efficiency of compression from POI to POl will be 0·89 so that
P02 = (1 + 0·89 x 193)3.5 = 1-5823.5
POl 295
T
TOl --------
FIG. 4.3 Divisioll of loss between impeller alld diffuser
WORl( DONE AND PRESSURE RlSE
Now (P2lpoz) = (T2ITo2l
5
, and Toz = T03 = 193 + 295 =488 K, so that
C
2
T2 == Toz _.-.1.. = 488 - 93·8 = 394·2 K
2cp
pz = (394.2)3.5
P02 488
Hence, since (P2IpOI) = (P2IP02)(P02IpOl),
pz ( 394.2)3.5
- = 1·582 x -- = 2·35
POI 488
pz = 2·35 x 1·1 = 2·58 bar
pz 2·58 x 100 3
P2 = RT
z
= 0.287 x 394.2 = 2·28 kg/m
135
The required area of cross-section of flow in the radial direction at the impeller tip
IS
A = ~ = 9 _. 2
P2CrZ 2·28 x 143 - 0 0276 m
Hence the depth of impeller channel
0·0276
= --= 0·0176 m or 1·76 em
n x 0·5
This result will be used when discussing the design of the diffuser in the next
section.
Before leaving the subject of the impeller, it is worth noting the effect of using
backswept curved vanes, which we said at the end of section 4.1 are increasingly
being used for high-perfonnance compressors. The velocity triangle at the tip
section for a backswept impeller is shown in Fig. 4.4, drawn for the ideal case of
zero slip for ease of nnderstanding. The corresponding triangle for the radial-
vaned impeller is shown by dotted lines. Assuming the radial component of
velocity to be the same, implying the same mass flow, it can be seen that the
velocity relative to the tip, V2, is increased while the absolute velocity of the fluid,
Cz, is reduced. These changes imply less stringent diffusion requirements in both
the impeller and diffuser, tending to increase the efficiency of both components.
The backsweep angle f3 may be in the region of 30-40 degrees. The work-
absorbing ca:pacity of the rotor is reduced, however, because Cwz is lower and the
temperature rise will be less than would be obtained with the ramal-vaned im-
peller. This effect is countered by the increased efficiency and it must be re-
membered that the ultimate goal is high pressure ratio and efficiency rather than
high temperature rise. The use of backswept vanes also gives the compressor
a wider operating range of air-flow at a given rotational speed, which is im-
portant for matching the compressor to its driving turbine; this will be discussed
in Chapter 8.
136
\-<
i

I
i
----1--
i

FIG. 4.4 Impeller witl; backl;wept vanes
4.3 The diffuser
CENTRIFUGAL COMPRESSORS
In Chapter 6 it will be seen that the problem of designing an combustion
system is eased if the velocity of the air entering the combustIOn chamber IS as
low as possible. It is necessary, therefore, to design the diffuser so that only a
small part of the stagnation temperature at the compressor outlet to
kinetic energy. Usually the velocity of the air at the compressor outlet IS m the
region of 90 mls. .
As will be emphasized throughout this book, it is much dlfficul: to
arrange for an efficient deceleration of flow than it is to obtam effiCient
acceleration. There is a natural tendency in a diffusing process for the aIr to break
away from the walls of the diverging passage, reverse its and flow. back
in the direction of the pressure gradient-see Fig. 4.S(a). If the divergence 18 too
rapid, this may result in the formation of eddies with consequent trans.fer of some
kinetic energy into internal energy and a reduction in useful nse. A
angle of divergence, however, implies a long diffuser .and a value of skin
friction loss. Experiments have shown that the optImum mcluded angle of
divergence is about 7 degrees, although angles of up to twice. this be
used with diffusers of low length/width (or radius) ratio WIthout mcumng a
serious increase in stagnation pressure loss. Empirical design curves for diffus.ers
of both circular and rectangular cross-section can be found in Ref. (8). Dunng
acceleration in a converging passage, on the other hand, the gas naturally tends to
fill the passage and follow the boundary walls closely, as in Fig. 4.S(b), however
THE DIFFUSER
:
Pressure
increasing


(a)
FIG. 4.5 Diffusing and acceleratillg l!Iow
Pressure
decreasing
137
rapid the rate of convergence. Only the normal frictional losses will be incurred in
this case.
In order to control the flow of air effectively and carry out the diffusion process
in as short a length as possible, the air leaving the impeller is divided into a
number of separate streams by fixed diffuser vanes. Usually the passages formed
by the vanes are of constant depth, the width diverging in accordance with the
shape of the vanes, as shown in Fig. 4.1. The angle of the diffuser vanes at the
leading edge must be designed to suit the direction of the absolute velocity of
the air at the radius of the leading edges, so that the air will flow smoothly over
the vanes. As there is always a radial gap between the impeller tip and the leading
edges of the vanes, this direction will not be that with which the air leaves the
impeller tip. The reason for the vaneless space after the impeller will be explained
in section 4.4, when the effects of compressibility in this region are discussed.
To find the correct inlet angle for the diffuser vanes, the flow in the vaneless
space must be considered. No further energy is supplied to the air after it leaves
the impeller so that, neglecting the effect of friction, the angular momentum Cwr
must be constant. Hence Cw decreases from impeller tip to diffuser vane ideally in
inverse proportion to the radius. For a chalIDel of constant depth, the area of flow
in the radial direction is directly proportional to the radius. The radiaJ velocity Cr
will therefore also decrease from impeller tip to diffuser vane, in accordance with
the equation of continuity. Ifboth Cr and Cw decrease, then the resultant velocity
C will decrease from the impeller tip, and some diffusion evidently takes place in
the vaneless space. The consequent increase in density means that Cr will not
decrease in inverse proportion to the radius as does Cw , and the way in which Cr
varies must be found from the continuity equation. An example at the end of this
section will show how this may be done. When Cr and Cw have been calculated at
the radius of the leading edges of the diffuser vanes, then the direction of the
resultant velocity can be found and hence the inlet angle of the vanes.
It will be apparent that the direction of the air flow in the vaneless space will
vary with mass flow and pressure ratio, so that when the compressor is operating
under conditions other than those of the design point the air may not flow
smoothly into the diffuser passages in which event some loss of efficiency will
result. In gas turbines where weight and complexity are flot so important as high
138 CENTRIFUGAL COMPRESSORS
part-load efficiency, it is possible to incorporate adjustable diffuser vanes so that
the inlet <mgle is correct over a wide range of operating conditions.
For a given pressure and temperature at the leading edge of the diffuser vanes,
the mass flow passed will depend upon the total throat area of the diffuser
passages (see Fig. 4.l). Once the number of vanes and the depth of passage have
been decided upon, the throat width can be calculated to suit the mass flow
required under given conditions of temperature and pressure. For reasons given
later in section 4.6, the number of diffuser vanes is appreciably less than the
number of impeller vanes. The length of the diffuser passages will of course be
determined by the maximum angle of divergence permissible, and the amonnt of
diffusion required, and use would be made of the data in Ref. (8) to arrive at an
optimum design. Although up to the throat the vanes must be curved to suit the
changing direction of flow, after the throat the air flow is fully controlled and the
walls of the passage may be straight. Note that diffusion can be carried out in a
much shorter flow path once the air is controlled than it can be in a vaneless space
where the air follows an approximately logarithmic spiral path (for an
incompressible fluid tan -Ie C;Cw) = constant).
After leaving the diffuser vanes, the air may be passed into a volute (or scroll)
and thence to a single combustion chamber (via a heat-exchanger if one is
employed). This would be done only in an industrial gas turbine unit: indeed, in
some small industrial units the diffuser vanes are omitted and the volute alone is
used. For aircraft gas turbines, where volume and frontal area are important, the
individual streams of air may be retained, each diffuser passage being connected
to a separate combustion chamber. Alternatively the streams can be fed into an
annular combustion chamber surrounding the shaft connecting the compressor
and turbine.
EXAMPLE 4.2
Consider the design of a diffuser for the compressor dealt with in the previous
example. The following additional data will be assumed:
radial width of vaneless space
approximate mean radius of diffuser throat
depth of diffuser passages
number of diffuser vanes
5 em
0·33 m
1·76 em
12
Required are (a) the inlet angle of the diffuser vanes, and (b) the throat width of
the diffuser passages which are assumed to be of constant depth. For simplicity it
will be assumed that the additional friction loss in the short distance between
impeller tip and diffuser throat is small and therefore that 50 per cent of the
overall loss can be considered to have occurred up to the diffuser throat. For
convenience suffix 2 will be used to denote any plane in the flow after the impeller
tip, the context making it clear which plane is under consideration.
THE DIFFUSER
139
(a) Consider conditions at the radius of the diffuser vane leading edges, i.e. at
r2 = 0·25 + 0·05 = 0·3 m ~ Since in the vaneless space C ~ . I · = constant for constant
angular momentum,
0·25
Cw2 = 410 x - = 342 mls
0·30
The radial coinponent of velocity can be found by trial and error. The iteration
may be started by assuming that the temperature equivalent of the resultant
velocity is that corresponding to the whirl velocity, but only the final trial is given
here.
Try Cr2 = 97 mls
Ci 3-42
2
+ 0-97
2
---'--:c::--:-- = 62·9 K
2cp 0·201
Ignoring any additional loss between the impeller tip and the diffuser vane lead-
ing edges at 0·3 m radius, the stagnation pressure will be that calculated for the
impeller tip, namely it will be that given by
P02 = 1.5823.5
POI
Proceeding as before we have:
T2 = 488 - 62·9 = 425·1 K
P2 = (425.1)3.5
P02 488
P2 = (1.582 X 425.1)3
0
5 = 3.07
POI 488
P2 = 3·07 x 1·1 = 3·38 bar
3·38 x 100 3
P2 = 0.287 )( 425.1 = 2· 77 kg/m
Area of cross-section of flow in radial direction
= 2n x 0·3 x 0·0176 = O·0332m2
Check on C,.2:
9
Cr2 = 2.77 )( 0.0332 = 97·9 m/s
Taking Cr as 97·9 mis, the angle of the diffuser vane leading edge for zero
incidence should be
140
CENTRIFUGAL COMPRESSORS
(b) The throat width of the diffuser channels may be found by asimilar calc-
ulation for the flow at the assumed throat radius of 0·33 m.
0·25
Cwz == 410 x 0.33 == 311 mls
Try CrZ == 83 mls
ci 3.11
z
+0.83
z
==51.5K
2cp 0·201
T2 == 488 - 51·5 == 436·5 K
P2 = 1.582 x -- == 3·37 (
436.5)3.5
POl 488
pz == 3·37 x 1·1 == 3·71 bar
== 3·71 x 100 = 2.96 k 1m3
P2 0.287 x 436·5 g
As a first approximation we may neglect the thickness of the diffuser vanes, so
that the area of flow in the radial direction
= 2n x 0·33 x 0·0176 == 0·0365 m
2
Check on C,'2:
9
Cr2 = 2.96 x 0.0365 == 83·3 mls
1 83.3 0
Direction of flow = tan - ill == 15
Now m == P2ArZCr2 == P2A2CZ' or A2 ==Ar2Cr2/Cz, and hence area of flow in the
direction of resultant velocity, i.e. total throat area of the diffuser passages, is
0.0365 sin 15° = 0·00945 m
2
With 12 diffuser vanes, the width of the throat in each passage of depth 0·0176 m
is therefore
0·00945
12 x 0·0176
0·0440 m or 4·40 cm
For any required outlet velocity, knowing the total loss for the whole compressor,
it is a simple matter to calculate the final density and hence the flow area required
at outlet: the length of the passage after the throat then depends on the chosen
angle of divergence.
On the basis of such calculations as these it is possible to produce a
preliminary layout of the diffuser. Some idea of the thickness of the vanes at
various radii can then be obtained enabling more accurate estimates of flow areas
and velocities to be made. When arriving at a final value of the throat area, an
empirical contraction coefficient will also be introduced to allow for the blockage
effect of the boundary layer. The design of the diffuser is essentially a process of
COMPRESSIBILITY EFFECTS 141
successive' approximation: sufficient information has been given to indicate the
method of approach.
It has been shown that a diffuser comprising channels of circular cross-section
can be more efficient than the conventional type with rectangular channels. The
passages are formed by drilling holes tangentially through a ring surrounding the
impeller, and after the throats the passages are conical. Overall isentropic
efficiencies of over 85 per cent are claimed for compressors using this 'pipe'
diffuser-see Ref. (5).
Earlier it was stated that the air leaving the diffuser passages might be
colleGted in a volute. The simplest method of volute design assmnes that the
angular momentum of the flow remains constant, i.e. friction in the volute is
neglected. Using the nomenclature of Fig. 4.6, the cross-sectional areaAe at any
angle 0 can then be shown to be given by
Ae == V_O_
r 2nK
where K is the constant angular momentum C".r, V is the volume flow, and r is the
radius of the centroid of the cross··section of the volute. The shape of the cross-
section has been shown to have little effect upon the friction loss: the various
shapes shown in Fig. 4.6 all yield volutes of similar efficiency. For further in-
formation and useful references the reader may turn to Ref. (6).
4.4 CompressibilitY effects
It is well known (see Appendix A) that breakdown of flow and excessive pressure
loss can be incurred if the velocity of a compressible fluid, relative to the surface
over which it is moving, reaches the speed of sound in the fluid. This is a most
important phenomenon in a diffusing process where there is always a tendency
for the flow to brealc away from the boundary even at low speeds. When an effort
is made to obtain the maximum possible mass flow from the smallest possible
compressor, as is done most particularly in the design of aircraft gas turbines, the
air speeds are very high. It is of the utmost importance that the Mach numbers at
FIG. 4.6 Volutes
142
CENTRIFUGAL COMPRESSORS
certain points in the flow do not exceed the value beyond which the losses
increase rapidly due to the formation of shock waves. The critical Mach number
is usually less than unity when calculated on the basis of the mean velocity of the
fluid relative to the botmdary, because the actual relative velocity near the surface
of a curved boundary may be in excess of the mean velocity. As a general rule,
unless actual tests indicate otllerwise, the Mach numbers are restricted to about
0·8.
Consider now the Mach numbers at vital points in the compressor, beginning
with the intake.
Inlet Mach number for impeller and diffuser
At the intake, the air is deflected through a certain angle before it passes into the
radial challl1els on the impeller. There is always a tendency for the air to break
away from the convex face of the curved part of the impeller vane. Here then is a
point at which the Mach number will be extremely important; a shock wave might
occur as shown in Fig. 4.7(a).
The inlet velocity triangle for the impeller eye is shown in Fig. 4.7(b). The full
lines represent the case of axial inlet velocity which we have considered up to
now. It has also been assumed that the axial velocity is uniform from the root to
the tip of the eye. In tlms case, the velocity of the air relative to the vane, Vb will
reach a maximum at the eye tip where the vane speed is greatest. The inlet Mach
number will be given by
M=_V_l_
J(yRT1)
where Tl is the static temperature at the inlet.
0BreakaWay;'C?/
commencing
at rear of Cw1
shockwave n
~ I I
, , ~ tj'
Angle of ~ "1
(a)
prewhirl i
Fixed inlet .
guide vane
t
(b)
FIG. 4.7 Effect of prewhirl
COMPRESSIBILITY EFFECTS 143
Even though this Mach number be satisfactory under ground level
atmospheric conditions, if the compressor is part of an aircraft gas turbine the
Mach number may well be too high at altitude. It is possible to reduce the relative
velocity, and hence the Mach number, by introducing prewhirl at the intake. This
is achieved by allowing the air to be drawn into the impeller eye over curved inlet
guide vanes attached to the compressor casing. For the same axial velocity, and
hence approximately the same mass flow, the relative velocity is reduced as
shown by the dotted triangle in Fig. 4.7(b). An additional advantage of using
prewhirl is that the curvature of the impeller vanes at the inlet is reduced, i.e. the
inlet angle (1. is increased.
Tllis method of reducing the Mach number unfortunately reduces the work
capacity of the compressor. The air now has an initial whirl component Cw]' so
that the rate of change of angular momentum per unit mass flow of air is
If Cw1 is constant over the eye of the impeller, then the initial angular momentum
will increase from root to tip of the eye. The amount of work done on each
kilogramme of air will therefore depend upon the radius at which it enters the eye
of the impeller. A mean value of the work done per kg can be found by using the
initial angular momentum of the air at the mean radius of the eye. There is,
however, no point in reducing the work capacity of the compressor UlIDecessarily,
and the Mach number isonly lligh at the tip. It is clearly preferable to vary the
prewhirl, gradually reducing it from a maximum at the tip to zero at the root of
the eye. This may be done if the inlet guide vanes are suitably twisted.
We will now check the relative Mach numbers in the compressor dealt with in
Examples 4.1 and 4.2. Consider first the inlet Mach number for the tip radius of
the impeller eye.
Inlet velocity
Eye tip speed
Relative velocity at tip
Velocity of sound
= 143 mls and is axial
=273 mls
=J(143
2
+273
2
)=308 mls
= '-/(1·4 x 0·287 x 284·5 x 10
3
)
=338 mls
Maximum Mach number at inlet = 308/338 = 0·91
This would not be considered satisfactory even if it were actually the maximum
value likely to be reached. But if the compressor is part of an aircraft engine
required to operate at an altitude of 11 000 m, where the atmospheric temperature
is only about 217 K, we must calculate the Mach number under these conditions.
Since there will be a temperature rise due to the ram effect in the intake when the
aircraft has a forward speed, the effect of drop in atmospheric temperature will not
be quite so great as might be expected. We will take 90 mls as being the
minimum speed likely to be reached at high altitude.
144
CENTRIFUGAL COMPRESSORS
Temperature equivalent· of
forward speed = 4 K
Inlet stagnation temperature = 217 + 4 = 221 K
Temperature equivalent of axial
inlet velocity from the first
examplet = 10·5 K
Inlet static temperature at altitude = 210·5 K 1
(
284.5)'
Inlet Mach number at altitude :=: 0·91 210.5 = 1·06
This is clearly too high and we will find the Mach number when 30 degrees of
prewhirl are introduced. In this case the absolute inlet :vill be shghtly
higher than before, so that the inlet static temperature wlll be lower. A
new value for the axial velocity must be found by the usual tnal and error
process. Reverting to the original sea-level static case:
Try Cal = 150 mls
150
C
I
=--= 173·2m/s
cos 30
Temperature equivalent of C1 = 14·9 K
T1 = 295 - 14·9 = 280·1 K
PI = 0·918 bar and PI = 1·14 kg/m
3
9
Check on Ca1 = 1.14 x 0.053 :=: 149 mls
Whirl velocity at inlet, Cwl = 149 tan 30
= 86 mls
Maximum relative velocity = )(149
2
+ (273 - 86)2]
= 239 mls
Hence maximum inlet Mach number when TOl = 295 K is
239 = 0.71
.J(1·4 x 0·287 x 280·1 x 1Q3)
Under altitude conditions this would rise to little more than 0·8 and 30 degrees of
prewhirl can be regarded as adequate. . .
To show the effect of the 30 degrees of pre whirl on the pressure ratIO, we Will
take the worst case and assume that the prewhirl is constant over the eye of the
impeller.
t This assumes that the mass flow through the compressor at altitude will be such as to give correct
flow direction into the eye. Then if the rotatIOnal speed IS unaltered the aXial velOCity must be the
same.
COMPRESSIBILITY EFFECTS
. . 273 + 136·5
Speed of impeller eye at mean radIUS, Ue = --,-, -
Actual temperature rise
".
= 204·8 m/s
= !f(aU
2
- Cwl Ue)
cp
145
= 1·04(0·9 )( 455.5
2
- 86 x 204·8) = 175 K
1·005 )( 103
P03 = [1 + 0·78 x l75J3.5 = 3.79
P01 295
This pressure ratio may be compared with the figure of 4·23 obtained with no
prewhirl. It is sometimes advantageous to use adjustable inlet guide vanes to
improve the performance under off-design conditions.
We next consider the relevant Mach numbers in the diffuser. The maximum
value will occur at the entry to the diffuser, that is, at the impeller tip. Once again
the values calculated in the previous examples of this chapter will be used.
In Example 4.1 the temperature equivalent of the resultant velocity of the air
leaving the impeller was found to be
ci = 93.8 K
2cp
and hence
C2 = 434m/s
T2 was found to be 394·2 K, and thus the Mach number at the impeller tip equals
434 = 1.09
.J(l.4 x 0·287 x 394·2 x 1Q3)
Now consider the leading edge of the diffuser vanes. In Example 4.2 the whirl
velocity was found to be 342 mls and the radial component 97·9 mls. The re-
sultamt velocity at this radius is therefore 356 mls. The static temperature was
425·1 K at this radius so that the Mach number is
356 = 0.86
.J(l-4 x 0·287 x 425·1 x 103)
In the particular design under consideration, the Mach number is 1·09 at the
impeller tip and 0·86 at the leading edges of the diffuser vanes. It has been found
that as long as the radial velocity component is subsonic, Mach numbers greater
than unity can be used at the impeller tip without loss of efficiency: it appears that
supersonic diffusion can occur without the formation of shock waves if it is
carried out at constant angular momentum with vortex motion in the vaneless
space, Ref. (7). But the Mach number at the leading edge of the diffuser vanes is
rather high and it would probably be advisable to increase the radial width of the
vaneless space or the depth of the diffuser to reduce the velocity at this radius.
146
CENTRIFUGAL COMPRESSORS
. . .
High Mach n u m b e r ~ at the leading edges of the diffuser varies are undesirable,
not only because of the danger of shock losses; but because they imply high air
speeds' and comparatively large pressures at the stagnation points where the air is
brought to rest locally at the leading edges of the vanes. This causes a
circumferential variation in static pressure, which is transmitted upstream in a
radial direction through the vaneless space to the impeller tip. Although the
variation will have been considerably reduced by the time it reaches the impeller,
it may well be large enough to excite the impeller vanes and cause mechanical
failure due to vibrational fatigue cracks in the vanes. This will occur when the
exciting frequency, which depends on the rotational speed and relative number of
impeller and diffuser vanes, is of the same order as one of the natural frequencies
of the impeller vanes. To reduce the likelihood of this, care is taken to see that the
number of vanes in the impeller is not an exact multiple of the number in the
diffuser; it is common practice to use a prime number for the impeller vanes and
an even number for the diffuser vanes.
The reason for the vaneless space will now be apparent: both the dangers of
shock losses and excessive circumferential variation in static pressure would be
considerably increased if the leading edges ofthe diffuser vanes were too near the
impeller tip where the Mach numbers are very high.
4.5 Non-dimensional quantities for plotting compressor
characteristics
The performance of a compressor may be specified by curves of delivery pressure
and temperature plotted against mass flow for various fixed values of rotational
speed. These characteristics, however, are dependent on other variables such as
the conditions of pressure and temperature at entry to the compressor and the
physical properties of the working fluid. Any attempt to allow for full variations
of all these quantities over the working range would involve an excessive num-
ber of experiments and malm a concise presentation of the results impossible.
Much of this complication may be eliminated by using the technique of dimen-
sional analysis, by which the variables involved may be combined to form a
smaller and more manageable number of dimensionless groups. As will be
shown, the complete characteristics of any particular compressor may then be
specified by only two sets of curves.
Before embarking on the dimensional analysis of the behaviour of a
compressor, the following special points should be noted.
(a) When considering the dimensions of temperature, it is convenient always to
associate with it the gas constant R so that the combined variable RT, being
equaltoplp,hasthedimensionsML -IT-
2
/ML -3=L
2
T-
2
. Thus they are
the same as those of (velocityt When the same gas, e.g. air, is being
employed during both the testing and subsequent use of the compressor, R
can finally be eliminated, but if for any reason there is a change from one gas
to another it must be retained in the final expressions.
NON-DIMENSIONAL QUANTITIES FOR PLOTIING COMPRESSOR 147
(b) A physical property of the gas which undoubtedly influences the behaviour
of the compressor is its density p but, if pressure P and the RT product are
also cited, its inclusion is superfluous because p = piRT.
(c) A further physical property of the gas which in theory would also influence
the problem would be its viscosity. The presence of this variable would
ultimately result in the emergence of a dimensionless group having the
character of a Reynolds number. In the highly turbulent conditions which
generally prevail in machines of this type, it is found from experience that
the influence of this group is negligibly small over the normal operating
ranget and it is customary to exclude it from turbomachinery analysis.
Bearing the above points in mind, it is now possible to consider the various
quantities which will both influence the behaviour of the compressor and depend
upon it. The solution of the problem may then be stated in the form of an equation
in which a function of all these variables is equated to zero. Thus in the present
instance, it would be reasonable to state that
Function (D, N, m, POI' P02' RToI , RTo2 ) = 0 (4.6)
where D is a characteristic linear dimension of the machine (usually taken as the
impeller diameter) and N is the rotational speed. Note that now we are simply
considering the perform!!Dce of the compressor as a whole, suffix 2 will be used to
refer to conditions at the compressor outlet for the remainder of this chapter,
By the principle of dimensional analysis, often referred to as the Pi theorem, it
is known that the function of seven variables expressed by equation (4.6) is
reducible to a different function of 7 - 3 = 4 non-dimensional groups formed
from these variables. The reduction by 3 is due to the presence of the three
fundamental units, M, L, T, in the dimensions of the original variables. Various
techniques exist for the formation of these non-dimensional groups and it is
possible in theory to obtain an infinite variety of self-consistent sets of these
groups. Generally there is little difficulty in deriving them by inspection, having
decided on the most suitable quantities to 'non-dimensionalize' by using the
others. In this case, it is most useful to have the non-dimensional forms of P02,
T02, m and N and the groups emerge as
POl T02 mJ(RTOl ) ND
PO! ' TO!' D2pOI 'J(RTo1 )
This now means that the performance of the machine in regard to the
variations of delivery pressure and temperature with mass flow, rotational speed
and inlet conditions can be expressed in the form of a function of these groups.
t To reduce the power requirements when testing large compressors, it is often necessary to throttle
the flow at inlet, giving a drop in inlet pressure and hence mass flow. The resultant drop in density
lowers the Reynolds number, and it is known that the perfonnance of compressors drops off with a
substantial reduction in Reynolds number.
148
CENTRIFUGAL COMPRESSORS
When weare concerned With the performance ofa.machine .of fixed size
compressing a specified g!lS, R and D may be omitted from the groups so that
. . (P02 T02 m.JTOI = 0 (4.7)
TOl' POI '.JT
Ol
The quantities m.JTol/poi anc;l N/.J.TOI are uSually termed the
mass flow and rotational speed respectively, although they are not truly dImen-
sionless.
A function of this form can be expressed graphically by plotting one group
against another for various fixed values of a third. In this particular case,
experience has shown that the most useful plots are those of the pressure and
temperature ratios P02/POI and T02/Tol against the non-dimensional mass flow
m.JTOI/POI using the non-dimensional rotational speed N/.JTol as a parameter.
As will be shown in the next section, it is also possible to replace the temperature
ratio by a derivative of it, namely the isentropic efficiency.
Finally, it is worth noting one particular physical interpretation of the non-
dimensional mass flow and rotational speed parameters. The former can be
written as
m.J(RT) pAC.J(RT) _pAC.J(RT) _C_ M
--rJfP = D2p - RTD2p ex: .J(RT) ex: F
and the latter as
ND
.J(RT) .J(RT) R
Thus the parameters can be regarded as a flow Mach number MF and a rotational
speed Mach number MR' All operating conditions covered by a pair of values of
m.JT/p and N/.JT should give rise to similar velocity triangles: so the vane
angles and air flow directions Will match and the compressor WIll Yield the same
performance in terms of pressure ratio, temperature ratio and isentropic effici-
ency. This is what the non-dimensional method of plotting the characteristics
implies.
4.6 Compressor characteristics
The value of the dimensional analysis is now evident. It is only necessary to plot
two sets of curves in order to describe the performance of a compressor com-
pletely. The stagnation pressure and temperature ratios are plotted separately
against 'non-dimensional' mass flow in the form of a family of curves, each curve
being drawn for a fixed value of the 'non-dimensional' rotational speed. From
these two sets of curves it is possible to construct constant speed curves of
isentropic efficiency plotted against 'non-dimensional' mass flow because this
efficiency is given by
COMPRESSOR CHARACTERISTICS 149
Before. describing a typical set of characteristics, it Will be as well to consider
what might be expected· to occur when a valve, placed in the delivery line of a
compressor running at constant speed, is slowly opened. The variation in pressure
ratio is shown in Fig. 4.8. When the valve is shut and the mass flow is zero, the
pressure ratio Will have some value A, corresponding to the centrifugal pressure
head produced by the action of the impeller on the air trapped between the vanes.
As the valve'is opened and flow commences, the diffuser begins to contribute its
quota of pressure rise, and the pressure ratio increases. At some point B, where
the efficiency approaches its maximum value, the pressure ratio will reach a
maximum, and any further increase in mass flow will result in a fall of pressure
ratio. For mass flows greatly in excess of that corresponding to the design mass
flow, the air angles will be widely different from the vane angles, breakaway of
the air will occur, and the efficiency will falloff rapidly. In this hypothetical case
the pressure ratio drops to unity at C, when the valve is fully opened and all the
power is absorbed in overcoming internal frictional resistance.
In actual practice, though point A could be obtained if desired, most of the
curve between A and B could not be obtained owing to the phenomenon of
surging. Surging is associated With a sudden drop in delivery pressure, and with
violent aerodynamic pulsation which is transmitted throughout the whole
machine. It may be explained as follows. If we suppose that the compressor is
operating at some point D on the part of the characteristic having positive slope,
then a decrease in mass flow will be accompanied by a fall of delivery pressure. If
the pressure of the air downstream of the compressor does not fall quickly
enough, the air will tend to reverse its direction and flow back in the direction of
the resulting pressure gradient. When this occurs, the pressure ratio drops rapidly.
Meanwhile, the pressure downstream of the compressor has fallen also, so that
the compressor will now be able to pick up again to repeat the cycle of events
which occurs at high frequency.
This surging ofthe air may not happen inImediately the operating point moves
to the left of B in Fig. 4.8, because the pressure downstream of the compressor
________________
o
Mass flow
FIG. 4.8 Theoretical characteristic
150
CENTRIFUGAL COMPRESSORS
may first of all fall at a greater rate. than the delivery pressure. Sooner or later, as
the mass flow is reduced, the reverse will apply and the conditions are inherently
unstable between A :md B. As long as the operating point is on the part of the
characteristic having a negative slope, however, decrease of mass flow is
accompanied by a rise of delivery pressure and stability of operation is assured. In
a gas turbine, the actual point at which surging occurs depends upon the
swallowing capacity of the components downstream of the compressor, e.g. the
turbine, and the way in which this swallowing capacity varies over the range of
operating conditions.
Surging probably starts to occur in the diffuser passages, where the flow is
retarded by frictional forces near the vanes: certainly the tendency to surge
appears to increase with the number of diffuser vanes. This is due to the fact that
it is very difficult to split the flow of air so that the mass flow is the same in each
passage. When there are several diffuser channels to every impeller channel, and
these deliver into a common outlet pipe, there is a tendency for the air to flow up
one channel and down another when the conditions are conducive to surging. If
this occurs in oPJy one pair of channels the delivery pressure will fall, and thus
increase the likelihood of surging. For this reason, the number of diffuser vanes is
usually less than the number of impeller vanes. The conditions of flow are then
approximately the same in each diffuser passage, because if each is supplied with
air from several impeller channels the variations in pressure and velocity between
the channels will be evened out by the time the air reaches the diffuser. Surging is
therefore not likely to occur until the instability has reached a point at which
reversal of flow will. occur in most of the diffuser passages simultaneously.
There is one other important cause of instability and poor performance, which
may contlibute to surge but can exist in the nominally stable operating range: this
is the rotating stall. When there is any non-uniformity in the flow or geometry of
the channels between vanes or blades, breakdown in the flow in one channel, say
B in Fig. 4.9, causes the air to be deflected in such a way that channel C receives
fluid at a reduced angle of incidence and channel A at an increased incidence.
Channel A then staUs, resulting in a reduction of incidence to channel B enabling
the flow in that charmel to recover. Thus the stall passes from channel to channel:
at the impeller eye it would rotate in a direction opposite to the direction of
FIG. 4,9 Rotating stall
COMPUTERIZED DESIGN PROCEDURES l51
rotation of the impeller. Rotating stall may lead to aerodynamically induced
vibrations resulting in fatigue failures in other parts of the gas turbine.
Returning now to consider the hypothetical constant speed curve ABC in Fig.
4.8, there is an additional limitation to the operating range, between Band C. As
the mass flow increases and the pressure decreases, the density is reduced and the
radial component of velocity must increase. At constant rotational speed this must
mean an increase in resultant velocity and hence in angle of incidence at the
diffuser vane leading edge. Sooner or later, at some point E say, the position is
reached where no further increase in mass flow can be obtained and choking is
said to have occurred. This point represents the maximum delivery obtainable at
the particular rotational speed for which the curve is drawn. Other curves may be
obtained for different speeds, so that the actual variation of pressure ratio over the
complete range of mass flow and rotational speed will be shown by curves such as
those in Fig. 4.l0(a). The left-hand extremities ofthe constant speed curves may
be joined up to form what is Imown as the surge line, while the right-hand
extremities represent the points where choking occurs.
The temperature ratio is a simple function of the pressure ratio and isentropic
efficiency, so that the form of the curves for temperature ratio plotted on the same
basis will be similar to Fig. 4.1O(a); there is no need to give a separate diagram
here. From these two sets of curves the isentropic efficiency may be plotted as in
Fig. 4.1 O(b) or, alternatively, contour lines for various values of the efficiency
may be superimposed npon Fig. 4.10(a). The efficiency varies with mass flow at a
givell speed in a similar manner to the pressure ratio, but the maximum value is
approximately the same at all speeds. A curve representing the locus of operating
points for maximum efficiency can be obtained as shown by the dotted curve in
Fig. 4.1 O(a). Ideally, the gas turbine should be so designed that the compressor
will always be operating on this curve. Chapter 8 will describe a method for
estimating the position of the operating line on the compressor charactelistics.
In conclusion, mention must be made of two other parameters sometimes used
in preference to m.jTOI/POI and N/ .jTOI when plotting compressor character-
istics. These are the equivalentfiow m.j8/b and equivalent speed N/.j8, where
8 = TOI/Tref and b = POI/Pref' The reference ambient state is llOlmally that
corresponding to the I.S.A. at sea level, namely 288 K and 1·013 bar. VVhen the
compressor is operating with the reference intake condition, m.j8/b and N/.je
are equal to the actual mass flow and rotational speed respectively. With this
method of plotting the characteristics, the numbers on the axes are recognizable
quantities.
4.7 Computerized desigu procedures
Although the approach to the design of centrifugal compressors given here is
adequate as an introduction to the subject, the reader should be aware that more
sophisticated methods are available. For example, Ref. (9) outlines a computer-
152

6:
0


:J
C/)
C/)

a.
*
0
.,.
,.,
C)
t:
m

m
.2
c.

Ql
.!!1
5
4
3
2
CENTRIFUGAL COMPRESSORS
Surge line / 1.0
. ,
I.. I
0.9
maxImum effiCIency 17' . \
1\
/ I 0.8
N/-J TOI relative to
///' design value
. ,
//-, '\ 0.7
./
._././ 0.6
lL-__ __ __ __ __ __ -7.
a 0.2 0.8
100
80
60
40
20
a
a 0.2
T01/POl (relative to design value)
(a)

0.6 0.7 0.8 0.9 1.0
N/-rr;;; relative to design value
0.4 0.6 0.8
m-J Tal/Pal (relative to design value)
(b)
FIG. 4.10 Centrifugal compressor characteristies
NOMENCLATURJ;l 153
ized design procedure developed at the National Gas Turbine Establishment (now
Aircraft Establishment). This makes use of the Marsh 'matrix throughflow'
aerodynamic analysis for determining the shape of the impeller channels. It in-
cludes a program for checking the stresses in the impeller, and concludes with
one which predicts the performance characteristics of the compressor. The pro-
cedure leads to a numerical output of co-ordinates for defining the shape of
the impeller' and diffuser vanes, in such a form that it can be fed directly to
numerically controlled machine tools used for manufacturing these components.
Tests have suggested that compressors designed in this way have an improved
performance,
Reference (10) describes a method of performance prediction which has been
compared with test results from seven different compressors, including one
designed on the basis of Ref. (9). The form of the pressure ratio v. mass flow
characteristics, and the choking flow, were predicted satisfactorily, The efficiency
was within ± 1-2 per cent of the experimental value at the design speed, with
somewhat larger discrepancies at low speeds. No general means have yet been
devised for predicting the surge line .
NOMENCLATURE
n number of vanes
N rotational speed,
r radius
U impeller speed at tip
Ue impeller speed at mean radius of eye
V relative velocity, volume flow
O! vane angle
(J' slip factor
'" power input factor
QJ angular velocity'
Suffixes
a axial component, ambient
r radial component
w whirl component
Axial flow compressors
The importance of a high overall pressure ratio in reducing specific fuel consump-
tion was referred to in sections 2.4 and 3.3, and the difficulties of obtaining a high
pressure ratio with the centrifugal compressor were pointed out in Chapter 4.
From an early stage in the history of the gas turbine, it was recognized that the
axial flow compressor had the potential for both higher pressure ratio and higher
efficiency than the centrifugal compressor. Another major advantage, especially
for jet engines, was the much larger flow rate possible for a given frontal area.
These potential gains have now been fully realized as the result of intensive
research into the aerodynamics of axial compressors: the axial flow machine
dominates the field for large powers and the centrifugal compressor is restricted to
the lower end of the power spectrum where the flow is too small to be handled
efficiently by axial blading. I
Early axial flow units had pressure ratios of around 5: 1 and required a b ~ ) U t 10
stages. Over the years the overall pressure ratios available have risen dramatically,
and some turbofan engines have pressure ratios exceeding 30: 1. Continued
aerodynamic development has resulted in a steady increase in stage pressure ratio,
with the result that the number of stages for a given overall pressure ratio has
been greatly reduced. There has been in consequence a reduction in engine
weight for a specified level of performance, which is particularly important for
aircraft engines. It should be realized, however, that high stage pressure ratios
imply high Mach numbers and large gas deflections in the blading which would
not generally be justifiable in an industrial gas turbine where weight is not critical;
industrial units, built on a much more restricted budget than an aircraft engine,
will inevitably use more conservative design techniques resulting in more stages.
In accordance with the introductory nature of this book, attention will be
focused on the design of subsonic compressors. True supersonic compressors, i.e.
those for which the velocity at entry is everywhere supersonic, have not
proceeded beyond the experimental stage. Transonic compressors, however, in
which the velocity relative to a moving row of blades is supersonic over part of
the blade height, are now successfully used in both aircraft and industrial gas
turbines. The reader must tum to more advanced texts for a full discussion of
these topics.
BASIC OPERATION
155
S.l Basic operation
The axial flow compressor consists of a series of stages, each stage comprising a
row of rotor blades followed by a row of stator blades: the individual stages can be
seen in Fig. 5.1. The working fluid is initially accelerated by the rotor blades, and
then decelerated in the stator blade passages wherein the kinetic energy trans-
ferred in the rotor is converted to static pressure. The process is repeated in as
many stages as are necessary to yield the required overall pressure ratio.
The flow is always subject to an adverse pressure gradient, and the higher the
pressure ratio the more difficult becomes the design of the compressor. The
process consists of a series of diffusions, both in the rotor and stator blade
passages: although the absolute velocity of the fluid is increased in the rotor, it
will be shown that the fluid velocity relative to the rotor is decreased, i.e. there is
diffusion within the rotor passages. The need for moderate rates of change of
cross-sectional area in a diffusing flow has already been emphasized in the
previous chapter. This limit on the diffusion in each stage means that a single
compressor stage can provide only a relatively small pressure ratio, and very
much less than can be used by a turbine with its advantageous pressure gradient,
converging blade passages, and accelerating flow (Fig. 5.2). This is why a single
turbine stage can drive a large number of compressor stages.
Careful design of compressor blading based on both aerodynamic theory and
experiment is necessary, not only to prevent wasteful losses, but also to ensure a
minimum of stalling troubles which are all too prevalent in axial compressors,
especially if the pressure ratio is high. Stalling, as in the case of isolated aerofoils,
arises when the difference between the flow direction and the blade angle (i.e. the
angle of incidence) becomes excessive. The fact that the pressure gradient is
acting against the flow direction is always a danger to the stability ofthe flow, and
flow reversals may easily occur at conditions of mass flow and rotational speed
which are different from those for which the blades were designed.
The compressor shown in Fig. 5.1 malces use of inlet guide vanes (IGVs),
which guide the flow into the first stage. Many industrial units have variable
FIG. 5.1 Hi-stage high-pressure ratio compressor Iby courtesy of GeuerallElectril:J
156
Turbine
blades
AXIAL FLOW COMPRESSORS
FIG. 5.2 Comparisoll of typical forms of turbine and compressor rotor blades
IGV s, pennitting the flow angle entering the first stage to vary with rotational
speed to improve the off-design performance. Most aircraft engines have now
dispensed with IGVs, however, mainly to obtain the maximum possible flow per
unit area and minimum engine weight. Other benefits include an easing of noise
and icing problems.
Figure 5.1 shows the marked change in blade size from front to rear in a high
pressure ratio compressor. For reasons which will appear later, it is desira15le'to
keep the axial velocity approximately constant throughout the compressor. With
the density increasing as the flow progresses through the machine, it is therefore
necessary to reduce the flow area and hence the blade height When the machine
is running at a lower speed than design, the density in the rear stages will be far
from the design value, resulting in incorrect axial velocities which will cause
blade stalling and compressor surge. Several methods may be used to overcome
this problem, all of which entail increased mechanical complexity. The approach
of Rolls-Royce and Pratt and Whitney has been to use multi-spool configurations,
whereas General Electric has favoured the use of variable stator blades; the IGV s
and first six stator rows of a GE compressor can be seen to be pivoted in Fig. 5.1.
For turbofan engines the large difference in diameter between the fan and the rest
of the compressor requires the use of multi-spool units; Pratt and Whitney and
General Electric have used two spools, but Rolls-Royce have used three spools on
the RB211. Another possibility is the use of blow-off valves, and on advanced
engines it is sometimes necessary to include ali these schemes. It is important to
realize that the designer must consider at the outset the performance of the
compressor at conditions far from design, although detailed discussion of these
matters is left to the end of the chapter.
In early axial compressors the flow was entirely subsonic, and it was found
necessary to use aerofoi! section blading to obtain a high efficiency. The need to
pass higher flow rates at high pressure ratios increased the Mach numbers, which
became especially critical at the tip of the first row of rotor blades. Eventually it
became necessary to design blading for transonic compressors, where the flow
over part of the blade is supersonic. It was found that the most effective blading
for transonic stages consisted of sections based on circular arcs, often referred to
as biconvex blading. As Mach numbers were further increased it was found that
ELEMENTARY THEORY
157
blade sections based on parabolas became more effective, and most high-
performance stages no longer use aero foil sections.
Before looking at the basic theory of the axial flow compressor it should be
emphasized that successful compressor design is very much an art, and all the
major engine manufacturers have developed a body of Imowledge which is kept
proprietary for competitive reasons. Bearing in mind the introductory nature of
this text, the aim of this chapter will be to present only the fundamentals of
compressor design.
5.2 Elementary theory
The working fluid in an axial flow compressor is normally air, but for closed-
cycle gas turbines other gases such as helium or carbon dioxide might be used.
The analysis which follows is applicable to any gas, but unless otherwise noted it
will be assumed that the working fluid is air.
A sketch of a typical stage is shown in Fig. 5.3. Applying the steady flow
energy equation to the rotor, and recognizing that the process can be assumed to
be adiabatic, it can readily be seen that the power input is given by
W = mCp (T02 - Tal) (5.1)
Repeating with the stator, where the process can again be assumed adiabatic and
there is zero work input, it follows that T02 = T03. All the power is absorbed in the
rotor, and the stator merely transforms kinetic energy to an increase in static
pressure with the stagnation temperature remaining constant. The increase in
stagnation pressure is accomplished wholly within the rotor and, in practice, there
will be some decrease in stagnation pressure in the stator due to fluid friction.
Losses will also occur in the rotor and the stagnation pressure rise will be less
than would be obtained with an isentropic compression and the same power input.
A T-s diagram for the stage, showing the effect oflosses in both rotor and stator
is also shown in Fig. 5.3. '
Obtaining the power input to the stage from simple thennodynamics is no help
in designing the blading. For this purpose we need to relate the power input to the
stage velocity triangles. Initially attention will be focused on a simple analysis of
the flow at the mean height of a blade where the peripheral speed is U, assuming
the flow to occur in a tangential plane at the mean radius. This two-dimensional
a p ~ r o a c h means that in general the flow velocity will have two components, one
aXial (denoted by subscript a) and one tangential (denoted by subscript w,
implying a whirl velocity). This simplified analysis is reasonable for the later
stages of an axial flow compressor where the blade height is small and the blade
speeds at root and tip are similar. At the front end of the compressor, however, the
blades are much longer, there are marked variations in blade speed from root to
tip, and it becomes essential to consider three-dimensional effects in analysing the
flow: these will be treated in a later section.
The velocity vectors and associated velocity diagram for a typical stage are
shown in Fig. 5.4. The air approaches the rotor with a velocity C
1
at an angle exl
158 AXIAL FLOW COMPRESSORS
Po, P03
_______________ .9
2

' 2
/
/ P2
---- ----- -----
Rotor
T,
----t----
Entropy 5
FIG. 5.3 Compressor stage IIml T-s diagram

I \c,
f ? 'oro< f-
FIG. 5.'1 Velocity triangles for one stage
ELEMENTARY THEORY 159
from the axial direction; combiningC1 vectorially with the blade speed U gives
the velocity relative to the blade, VI. at an angle /31 from the axial direction. After
passing through the rotor, which increases the absolute velocity of the air, the
fluid leaves the rotor with a relative velocity V2 at an angle Ih detennined by the
rotor blade outlet angle. Assuming that the design is such that the axial velocity
Ca is kept constant, the value of V2 can be obtained and the outlet velocity
triangle constructed by combining Vz and U vectorially to give Cz at angle IXz.
The air leaving the rotor at ()(2 then passes to the stator where it is diffused to a
velocity C3 at angle 1X3; typically the design is such that C3 ::::; C1 and 1X3::::; ()(I so
that the air is prepared for entry to another similar stage.
Assuming that Ca = Cal = CaZ, two basic equations follow immediately from
the geometry of the velocity triangles. These are
U
C = tan ()(j + tan /3] (5.2)
a
U
C = tan ()(2 + tan /32
a
(5.3)
By considering the change in angular momentum of the air in passing through the
rotor, the following expression for the power input to the stage can be deduced:
W == mU(Cw2 - Cwl )
where Cwl and Cw2 are the tangential components of fluid velocity before and
after the rotor. This expression can be put in terms of the axial velocity and air
angles to give
W = mUCaCtan ()(2 - tan IX])
It is more useful, however, to express the power in terms of the rotor blade air
angles, PI and /32. It can readily be seen from equations (5.2) and (5.3) that
(tan ()(z - tan ()(j) = (tan /31 - tan /32). Thus the power input is given by
W = mUCa(tan /31 - tan /32) (5.4)
This input energy will be absorbed usefully in raising the pressure of the air
and wastefully in overcoming various frictional losses. But regardless of the
losses, or in otner words of the efficiency of compression, the whole of this input
will reveal itself as a rise in stagnation temperature of the air. Equating (5.1) and
(5.4), the stagnation temperature rise in the stage, ATos, is given by
ATos = T03 - TOl = T02 - TOl = UC
a
(tan /31 - tan {32) (5.5)
cp
The pressure obtained will be strongly dependent on the efficiency of the
compression process" as should be clear from Fig. 5.3. Denoting the isentropic
efficiency of the stage by 1)s, where lis = (T03 - T01 )/(T03 - T{)]), the stage
pressure ratio is then given by
[
AT. JY/(Y-l)
R
S
=P03,= 1+1)s os
POI TOI
(5.6)
160
AXIAL FLOW COMPRESSORS
It can now be seen that to obtain a high temperature rise in a stage, which is
desirable to minimize the number of stages for a given overall pressure ratio, the
designer must combine
(i) high blade speed
(ii) high axial velocity
(iii) high fluid deflection (j3\ - P2) in the rotor blades.
Blade stresses will obviously limit the blade speed, and it will be seen in the next
section that aerodynamic considerations and the previously mentioned adverse
pressure gradient combine to limit (ii) and (iii).
5.3 Factors affecting stage preSSl1llJre ratio
Tip speed
The centrifugal stress in the rotor blades depends on the rotational speed, the
blade material and the length of the blade. The maximum centrifugal tensile
stress, which occurs at the blade root, can be seen to be given by
where Pb is the density of the blade material, OJ is the angular velocity, a is the
cross-sectional area of the blade at any radius, and suffixed r and t refer to root
and tip of the blade. If, for simplicity, the blade cross-section is assumed to be
constant from root to tip
where N is the rotational speed and A the annulus area. The tip speed, U" is given
by 2nNrt so the equation for centrifugal tensile stress can be written also as
The ratio rrlr, is nonnally referred to as the hub-tip ratio. It is immediately
apparent that the centrifugal stress is proportional to the square of the tip speed,
and that a reduction of hub-tip ratio increases the blade stress. It will be seen later
that the hub-tip ratio is also very important with regard to aerodynamic con-
siderations.
In practice, the blade sectional area will be decreased with radius to relieve the
blade root stress and the loading on the disc carrying the blades, so that the
integral would be evaluated numerically or graphically. A simple analytical
expression can be deduced, however, for a linear variation of cross-sectional area
FACTORS AFFECTING STAGE PRESSURE RATIO 161
from root to tip. If the hub-tip ratio is b and the ratio ofthe cross-sectional area at
the tip to that at the root is d, the stress in a tapered blade is given by
(O"c,)max = Ut
2
(l - b
2
)K
where [( = 1 - [(1 - d)(2 - b - b
2
)j3(l - b
2
)]
Typical values of [( would range from 0·55 to 0·65 for tapered blades.
Direct centrifugal tensile stresses are not often of major ,concern in compressor
blades. The first-stage blades, being the longest, are the most highly stressed, but
the later stages often appear to be very moderately stressed. The designer should
not be lulled into a false sense of security by this, because the blades are also
subject to fluctuating gas bending stresses which may cause fatigue failure.
Although the problem is significantly more difficult for compressors than
turbines, because of the high probability of aerodynamic vibration resulting from
flow instability when some stages are operating in a stalled condition, a
discussion of methods of predicting gas bending stresses is left to Chapter 7.
For tip speeds of around 350 mis, stress problems are not usually critical in the
sizing of the annulus. Tip speeds of 450 m/s, however, are common in the fans of
high bypass ratio turbofans, and with their low hub-tip ratios the design of the
disc to retain the long heavy blades becomes critical. The inner radius of the
annulus may then be dictated by disc stressing constraints.
Axial velocity
The expression for stage temperature rise, which in conjunction with an isen-
tropic efficiency determines the stage pressure ratio, showed the desirability of
using a high value of axial velocity. A high axial velocity is also required to
provide a high flow rate per unit IlIontal area, which is important for turbojet and
turbofan engines.
The axial velocity at inlet, however, must be limited for aerodynamic reasons.
Considering a first stage with no IGVs, the entry velocity will be purely axial in
direction and the velocity diagram at entry to the rotor will be as shown by the
right-angled triangle in Fig. 5.5. The velocity relative to the rotor is given by
vl = C? + U
2
and, assuming the axial velocity to be constant over the blade
height, the maximum relative velocity will occur at the tip, i.e. Vit. The static
temperature at rotor entry, Tb is given by TOI - (CU2cp) and the local acoustic
velocity by a = )CyRTI ). The Mach number relative to the rotor tip is VItia, and
for a given speed T.l.t it is therefore determined by the axial velocity at entry to the
stage as indicated by the curves in Fig. 5.5. Axial for an industrial gas
turbine will usually be of the order of 150 mls whereas for advanced aero engines
they could be up to 200 mls.
On early compressors the design had to be such that the Mach number at the
rotor tip was subsonic, but in the early fifties it became possible to use transonic
Mach numbers up to about 1·1 without introducing excessive losses. With fans of
162
T01 =288 K
1.4
Cw1 =0
:;;
.0
E
::l
1.2 c
J::
"
co
:;;

"iii
a::
0.8
250 300
I
350
AXIAL FLOW COMPRESSORS
Reduction in V1
withlGV '"
\ // \
... YV1 \

U
I I
400 450 500
Blade speed U/[m 5]
FIG. 5.5 Relative Macl! number at rlltllr en1hry
large bypass ratio the Mach number at the rotor tip may be of the order of 1·5.
The dotted velocity triangle in Fig. 5.5 shows how the Mach number at entry may
be slightly reduced by means ofIGVs, and IGVs were aerodynamically necessary
until the ability to operate at transonic Mach numbers was demonstrated. As will
be explained in section 5.12, the twin-spool compressor goes some way towards
alleviating the tip Mach number problem, because the LP compressor runs at a
lower speed than the HP compressor, thereby reducing the tip speed at entry. The
acoustic velocity irlcreases in successive stages because of the progressive
increase in static temperature, so the Mach number problem diminishes for the
later stages. We have already seen that the later stages are also not so critical from
the mechanical point of view, short blades implying low stresses.
For a given frontal diameter, the flow area can be increased by decreasing the
hub-tip ratio. As this ratio is reduced to very small values, however, the
incremental gain in flow area becomes less and Jess, and as mentioned previously
the mechanical design of the first-stage disc hecomes difficult; it will be seen later
that the Mach number at the tip of the first-stage stator blades also becomes
important. For these reasons, hub-tip ratios much below 0-4 are not used for aero
engines and significantly higher values are normal for industrial gas turbines.
When air is the working fluid it is generally found that compressibility effects
become critical before stress considerations, i.e. Ut and Ca are limited by the need
to keep the relative gas velocity to an acceptable level. Closed-cycle gas turbines,
with external combustion, are not restricted to using air as the working fluid and
both helium and carbon dioxide are possible working fluids. With a light gas such
as helium, the gas constant is much higher than for air and the acoustic velocity is
correspondingly greater; in this case, it is found that the Mach numbers are low
and stress limitations predominate. For a heavier gas such as carbon dioxide, the
opposite is true.
FACTORS AFFECTING STAGE PRESSURE RATIO 163
High fluid deflections in the rotor blades
Recalling equation (5.5) for convenience, the stage temperature rise was given by
!lTos= UCaCtan PI - tan P2)/Cp- For most compressor stages it can be assumed
with little error that the value of U is the same at inlet and outlet of a rotor blade
for any particular streamline, and the velocity triangles can conveniently be drawn
on a commqn base as in Fig. 5.6. The amount of deflection required in the rotor is
shown by the directions of the relative velocity vectors VI and V2, and the change
in whirl velocity is !lCw Considering a fixed value of Ph it is obvious that
increasing the deflection by reducing pz entails a reduction in V2. In other words,
high fluid deflection implies a high rate of diffusion. The designer must have
some method for assessing the allowable diffusion, and one of the earliest criteria
used was the de Haller number, defined as V2/VI; a limit of V2/VI 1:. 0·72 was set,
lower values leading to excessive losses. Because of its extreme simplicity, the de
Haller number is still used in preliminary design work, but for final design
calculations a criterion called the diffusion factor is preferred. The latter concept
was developed by NACA (the precursor of NASA), and it is widely used on both
sides of the Atlantic.
To explain the diffusion factor it is necessary to make a small foray into
cascade testing, which will be discussed more fully in II later section. Figure 5.7
shows a pair of typical blades which have a pitch s and a chord c. The air passing
over an aerofoi! will accelerate to a higher velocity on the convex surface, and in a
stationary row this will give lise to a drop in static pressure; for this reason the
convex surface is known as the suction side of the blade. On the concave surface,
the pressure side, the fluid will be decelerated. The velocity distribution through
the blade passage will be of the form shown: the maximum velocity on the
suction surface will occur at around 10-15 per cent of the chord from the leading
edge and will then fall steadily until the outlet velocity is reached. The losses in a
blade row arise primarily from the growth of boundary layers on the suction and
pressure sides of the blade. These surface boundary layers come together at the
blade trailing edge to form a wake, giving lise to a local drop in stagnation
pressure. Relatively thick surface b01ll1dary layers, resulting in high losses, have

u
FIG. 5.6 Effect of increasing lIuid dell.ectillll
164
AXIAL FLOW COMPRESSORS
Chord - per cent
FIG. 5.7 Blade spacing and velocity distribution tlJirough passage
been found to occur in regions where rapid changes of velocity are occurring, i.e.
in regions of high velocity gradient. Figure 5.7 would suggest that these w?uld.be
most likely to occur on the suction surface. The derivati?n of the dlffuslOn
factor is based on the establishment ofthe velocity gradient on the suction surface
in terms of VI, V2 and V max in conjunction with results cascade.
the cascade tests it was deduced that the maximum VelOCity T max
VI + 0·5(.6.C
w
slc). In simplified form,t the diffusion factor, D, can be expressed
as
.6.Cw s V
v: - V
2
VI 2
max
VI VI
V2 .6.Cw s
:::,;1--+-.-
VI 2VI c
(5.7)
The variation of friction loss with D obtained from a large number ofNACA tests
[Ref. (6)] is shown in Fig. 5.8. These tests were carried out over a wide range of
cascade geometries for a particular aerofoil section, and were found to be gen-
ro
.,.;
0.40
c
0
"n
'"
"'
"0 0.30
IS
""
Rotor tip region

"'
0.20
"' .2
c
0
n
:E 0.10
0

::>
"'
ctl
'"
0 ::;;
Rotor hub and stator
I I
0.4
0.6 0.8 1.0
Diffusion factor 0
FIG. 5.8 Variation of frictioll Joss witl! diffusion factor
t The full derivation of the diffusion factor is given in Ref. (4).
BLOCKAGE IN THE COMPRESSOR ANNULUS 165
erally applicable as long as the maximum local Mach numbers were subsonic or
only slightly supersonic. It can be seen that for the rotor hub region and stators
the losses are unaffected by variation in D up to 0·6; in the rotor tip region,
however, the losses increase rapidly at values of D above 0-4. The great merit of
the diffusion factor, as a criterion for placing a limit on the permissible gas
deflection, is its relative simplicity and the fact that it can be established as soon
as the velocity diagram has been constructed and a value of the pitch/chord ratio
has been chosen. In American practice the term solidity, the inverse of pitch/chord
ratio, is used.
5.4 Blockage in the compressor annulus
Because of the adverse pressure gradient in compressors, the boundary layers
along the annulus walls thicken as the flow progresses. The main effect is to
reduce the area available for flow below the geometric area of the annulus. This
will have a considerable effect on the axial velocity through the compressor and
must be allowed for in the design process. The flow is extremely complex, with
successive accelerations and decelerations combined with changes in tangential
flow direction; the effects oftip clearance are also significant, making the calcula-
tion of boundary layer growth extremely difficult. For this reason, compressor
designers normally make use of empirical correction factors based on experi-
mental data from compressor tests.
Early British experiments revealed that the stage temperarure rise was always
less than would be given by equation (5.5). The explanation ofthis is based on the
fact that the radial distribution of axial velocity is not constant across the annulus,
but becomes increasingly 'peaky' as the flow proceeds, settling down to a fixed
profile at about the fourth stage. This is illustrated in Fig. 5.9 which shows typical
axial velocity profiles in the first and fourth stages. To show how the change in
axial velocity affects the work-absorbing capacity of the stage, equation (5.4) can
be recast using equation (5.2) as
W = m U[(U - Ca tan IX]) - Ca tan /32l
= mU[U - CaCtan IXI + tan /32)]
I. ..I
Ca mean
(a)


I
I
I
Blade I
h,;g. I ')
Ca ----
I. .j
Camean
(b)
FIG. 5.9 Axial velocity distributions: (a) at !first stage, (b) at fourth stage
(5.8)
166 AY,JAL FLOW COMPRESSORS
The outlet angles of the stator and rotor blades deternline the values of 0(1 and
/32 which can therefore be regarded as fixed (unlike 0(2 which varies with Ca).
Equation (5.8) shows that an increase in Ca will result in a decrease of W. If it is
assumed that the compressor has been designed for a constant radial distribution
of axial velocity as shown by the dotted line in Fig. 5.9, the effect of an increase
in Ca in the central region of *e annulus will be to reduce the work capacity of
the blading in that area. The reduction in Ca at the root and tip might be expected
to compensate for this effect by increasing the work capacity of the blading close
to the annulus walls. Unfortunately the influence of both the boundary layers on
the annulus walls and the blade tip clearance has an adverse effect on this com-
pensation and the net result is a decrease in total work capacity. This effect
becomes more pronounced as the number of stages is increased.
The reduction in work capacity can be accOlmted for by use of the work-done
factor A, which is a number less than unity. The actual stage temperature rise is
then given by
" A
L\Tos = - UCaCtan /3, - tan fJ2) (5.9)
cp
Because of the variation of axial velocity profile through the compressor, the
mean work-done factor will vary as shown in lFig. 5.10.
Care should be taken to avoid confusion of the work-done factor with the idea
of an efficiency. If W is the value of the work input calculated from equation (5.4),
then AW is lIhe measure of lIhe actual work which can be supplied to the stage.
Having established the actual temperature rise, application of the isentropic
efficiency of the stage yields the pressure ratio in accordance with equation (5.6).
It was mentioned earlier that a high axial velocity was required for a high stage
temperature rise, and at first sight the explanation of work -done factor, on the
basis of a lower temperature rise in regions of high axial velocity, appears to be
contradictory. This is not so for the following reason. At the design stage, where
the values of /3, and can be selected to give a satisfactory L\Tos within the
limits set by the de Haller number or diffusion factor, a high axial velocity is
required. Once the design has been specified, however, and the angles IXh /31 and
/32 are fixed, then equa.tion (5.8) correctly shows that an increase in axial velocity
will reduce the stage temperature rise.
'"
1.0
0
u
\
.Z!
Q)
c:

0
'? 0.9
i:
0
s:
c:
'"
Q)
0.8
I I I :2
0 4 8 12 16 20
Number of stages
FIG. 5.10 Variatioll of meall work-dolle factol" with llIumber of stages
DEGREE OF REACTION
167
An alternative approach is to assign 'blockage factors' to reduce the effective
annulus area to allow for the growth in boundary layer thiclmess, and this is the
American design practice. Both the 'work-done factor' and 'blockage factor'
represent empirical corrections based on a particular organization'8 experience of
compressor development.
5.5 Degree of reaction
It was shown in an earlier section that in an axial compressor stage, diffusion
takes place in both rotor and stator, and there will be an increase in static pressure
through both rows. The degree of reaction J\. provides a measure of the extent to
which the rotor contributes to the overall static pressure rise in the stage. It is
normally defined in terms of enthalpy rise as follows,
J\. = static enthalpy rise in the rotor
static enthalpy rise in the stage
but, because the variation of cp over the relevant temperature ranges is negligible,
the degree of reaction can be expressed more conveniently in terms of temper-
ature rises.
The degree of reacti.on is a useful concept in compressor design, and it is
possible to obtain a formula for it in terms ofthe various velocities and air angles
associated with the stage. This will be done for the most common case in which it
is assumed that (a) Ca is constant through the stage, and (b) the air leaves the
stage with the same absolute velocity with which it enters, i.e. C3 = C1 and hence
L\Ts = ATos. If L\TA and L\TB denote the static temperature rises in the rotor and
stator respectively, then equation (5.5) gives, per unit mass flow,
W = cp(ATA + ATB) = cpATs = UCa(tan p, - tan /32)
= UCaCtan 1X2 - tan 0(1) (5.10)
Since all the work input to the stage takes place in the rotor, the steady flow
energy equation yields
W = CpL\TA +!(C! - CD
so that with equation (5.1 0)
cpATA = UCaCtan 0(2 - tan 0(1) - !(Ci - cD
CpL\TA = UCa(tan!X2 - tan 0(1) -! 0(2 - sec
2
IX,)
= UCa(tan 0(2 - tan 0(,) - !C;(tau2 0(2 - tan
2
IXI )
168
AXIAL FLOW COMPRESSORS
From the definition of A,
I:lTA
I:lTA +I:lTs
UCa(tan (X2 - tan IXJ) - 1X2 - tan
2
IXJ)
UCa(tiID (X2 - tan (Xl)
Ca )
= 1 - 2U (tan (X2 + tan IXJ
By the addition of equations (5.2) and (5.3),
2U
- = tan IXJ + tan Ih + tan 1X2 + tan P2
Ca
Hence
(5.11)
Because the case of 50 per cent reaction is important in design, it is of interest
to see the consequences of putting A = 0·5. In this case, from equation (5.11)
U
tan PI + tan P2 = c-
a
and it immediately follows from equations (5.2) and (5.3) that
tan IXJ = tan /32,
tan PI = tan 1X2 ,
i.e. IXJ = P2
i.e. 1X2 = /3J
Furthermore, since Ca is constant through the stage,
Ca = CJ cos IXJ = C3 cos (X3
It was initially assumed that CI = C3 and hence IXJ = (X3' Because of this equality
of angles, namely 1X1 = /32 = 1X3 and PI = 1X2, the velocity diagranl of Fig. 5.6
becomes symmetrical and blading designed on this basis is sometimes referred to
as symmetrical blading. From the symmetry of the velocity diagram, it follows
that C1 = V2 and V1 = C2·
It must be pointed out that in deriving equation (5.11) for A, a work-done
factor A of unity has implicitly been assumed in making use of equation (5.10). A
stage designed with symmetrical blading will always be referred to as a 50 per
cent reaction stage, a1iliough the value of A actually achieved will differ slightly
from 0·5 because of the influence of A.
It is of interest to consider the significance of the limiting values of A, viz. 0
and 1·0. The degree of reaction was defined in terms of enthalpy rises but could
also have been expressed in terms of changes in static pressure. Making use of
the thermodynamic relation T ds = dh - v dp and assuming incompressible
THREE-DIMENSIONAL FLOW 169
isentropic flow
O=dh-dp/p
which on integration from state 1 to state 2 becomes
showing that enthalpy and static pressure changes are related. By putting A = 0 in
equation (5.11) it can readily be seen that PI = - pz; the rotor blades are then of
impulse type (i.e. passage area the same at inlet and outlet) and all the static
pressure rise occurs in the stator. Conversely, for A = 1·0 the stators are of im-
pulse type. In the interests of obtaining the most efficient overall diffusion, it is
desirable to share the diffusion between both components of the stage, and for
this reason the use of 50 per cent reaction is attractive. We are not suggesting,
however, that the efficiency is very sensitive to the precise degree of reaction
chosen. In practice, as will appear later, the degree of reaction may show con-
siderable variation across the annulus (i.e. along the blade span), especially for
stages of low hub-tip ratio, and the designer will primarily be concerned to
satisfY more stringent conditions set by a limiting diffusion factor and Mach
. number.
At this point the reader may wish to see an example of a preliminary mean
diameter design to consolidate the previous material. If so, he or she can tum to
section 5.7. To enable the full procedure to be illustrated, however, we
need to consider three-dimensional effects, and these will be introduced in the
next section.
5.6 Th.ree-dimensional flow
The elementary theory presented in section 5.2 assumed that the flow in the
compressor annulus is two-dimensional, meaning that any effect due to radial
movement of the fluid is ignored. The assumption of two-dimensional flow is
quite reasonable for stages in which the blade height is small relative to the mean
diameter of the annulus, i.e. those of hub-tip ratio greater than about 0·8, which
would be typical of the later stages of a compressor. The front stages, however,
have lower values of hub-tip ratio, and values as low as 0·4 are used for the first
stage of aero-engine compressors so that a high mass flow can be passed through
a machine of low frontal area. When the compressor has a low hub-tip ratio in ilie
first stage and a high hub-tip ratio in the later stages, the annulus will have a
substantial taper [see Fig. 5.1 ] and the streamlines will not lie on a surface of
revolution parallel to the axis of the rotor as previously assumed. Under these
conditions, the flow must have a radial component of velocity, although it will
generally be small compared with the axial and whirl components.
A second cause of radial movement occurs as follows. Because the flow has a
whirl component, the pressure must increase with radius, i.e. up the blade height,
to provide the force associated with the centripetal acceleration of the fluid. In the
170 AXIAL FLOW COMPRESSORS
course of adjusting itseIfto provide a balance.between the pressure forces and the
inertia forces, the flow will undergo:some movement in the radial direction.
With a low hub-tip ratio, the variation in blade speed from root to tip is large
and this will have a major effect on the shape of the veiocity triangles and the
resulting air angles. Furthermore, the aforementioned change of pressure, and
hence density, with radius >yill cause the fluid velocity vectors to change in
magnitude and these too affect the shape of the velocity triangles. It follows that
the air angles at the mean diameter will be far from representative of those at the
root and tip of a blade row. For high efficiency it is essential that the blade angles
match the air angles closely at all radii, and the blade must therefore be twisted
from root to tip to suit the changing air angles.
The basic equation expressing the balance between pressure forces and inertia
forces can be derived by considering the forces acting on the fluid element shown
in Fig. 5.11. The whirl component of velocity is shown in Fig. 5.ll(a), and the
axial component and much smaller radial component resulting from the curvature
of the streamline in Fig. 5.11(b). The inertia forces in the radial direction arise
from
(i) the centripetal force associated with circumferential flow;
(ii) the radial component of the centripetal force associated with the flow along
the curved streamline;
(iii) the radial component of the force required to produce the linear
acceleration along the streamline.
The total inertia force, Flo must be produced by the pressure forces acting on the
element in the radial direction. (The acceleration in the radial direction may
amount to several thousand times the acceleration due to gravity so that
gravitational forces can be neglected.)

dp
p+ - \ I dp
2 \ I p+_
\ p! 2
\ I
\ I
\ I
\ I
\ I
\ II

\.1
Unit width

(a) (b)
FIG. 5.11 Radial equilibrium of fluid element
THREE-DIMENSIONAL FLOW 171
Considering a fluid element of unit width, having a, density p, the three terms
of the mertia force can be expressed as follows. The centripetal force associated
with the circumferential flow is
mC
2
C
2
F(" = _W = (pr dr dO)
'} r r
For the flow along the curved streamline, the radial force is given by
mC
2
C
2
F(iO = __ s cos (J.s = (pr dr dO)...l: cos (J.s
rs rs
where the suffix S refers to the component along the streamline and rs is the
radius of curvature of the streamline. For the acceleration along the streamline,
the radial component of force is
dCs . (dr dll) dCs '
F(iiiJ = md[ sm (J.s = pr U d[ sm (J.s
With the curvature shown in Fig. 5.11 (b) the forces F (ii) and F (iii) are in the same
direction as F(i). Thus the total inertia force, Flo is given by
dr d
ll[C; dCs . ]
FJ = pr U - + - cos (J.s + -d sm (J.s
r rs t
The pressure force, F p, producing this inertia force is obtained by resolving in
the radial direction to give
(
dp) dO
Fp = (p + dp)(r + dr)dO -pr dO - 2 p +2 &2
The third term in the equation results from the resolution of the pressure forces on
the two sides of the element in the radial-axial plane, on which it is assumed that
the pressure is the average of the two extremes, namely p + (dp/2). Equating the
forces FJ and Fp , and neglecting second-order terms, we are left with
1 dp dCs .
--=-+- sm(J.s+- sm(J.s
p dr r rs dt
(5.12)
This is the complete radial equilibrium equation which includes all contributory
factors.
For most design purposes it can be assumed that rs is so large, and (J.s so small,
that the last two terms .of equation (5.12) can be ignored. Thus, finally, we have
1 dp C;
--= (5.13)
pdr r
and this is what will be referred to as the radial equilibrium equation. In effect,
the radial component of velocity is being neglected. Certainly in the spaces
between the blade rows Cr is very much smaller than either Ca or Cw and can
safely be assumed to be negligible.
172
AXIAL FLOW COMPRESSORS
Equation (5.13) enables us to deduce an energy equation which expresses the
variation of enthalpy with radius. The stagnation enthalpy ho at any radius r
where the absolute velocity is C is given by
C
2
I I(C2 C2 )
ho=h+2:= 1+2: a+ w
and the variation of enthalpy with radius is therefore
dho _ dh C dCa + C
dr-dr+adr wdr
From the thermodynamic relation T ds = dh - dpl p,
dh ds dT 1 dp 1 dp
dr = T dr +ds dr +'oill· - p2 drdp
Dropping second-order terms

dr dl" pdr
Substituting for dh/dr in equation (5.14)
dh
o
ds 1 dp dCa dCw
-=T-+--+C -+Cw -
dr dr pill' a dr dr
(5.14)
Using the radial equilibrium equation (5.13), the second term of right-hand side
of the above equation can be replaced by C;/r, leaving the basic equation for the
analysis of flow in the compressor annulus as
dho _ ds C dCa C dCw C;
dr-Tdr+ a ill' + wdr+r
The term T ds/dr represents the radial variation of loss across the annulus, and
may be significant in detailed design calculations; this is especially true if Mach
numbers relative to the blade are supersonic and shock losses occur. For our
purposes, however, it will be assumed that the entropy gradient term can be
ignored and the final form of the equation is given by
dho _ dCa C dCw
dr - Ca dr + W dr + r
This will be referred to as the vortex energy equation.
(5.15)
Apart from regions near the walls of the annulus, the stagnation enthalpy (and
temperature) will be uniform across the armulus at entry to the compressor. If the
frequently used design condition of constant specific work at all radii is applied,
then although ho will increase progressively through the compressor in the axial
direction, its radial distribution will remain uniform. Thus dho/dr = 0 in any plane
between a pair of blade rows. Equation (5.15) then reduces to
C dCa C dCw C; = 0
a d + w ". +
r ur r
(5.16)
THREE-DIMENSIONAL FLOW 173
A special case may now be considered in which Ca is maintained constant
across the armulus, so that dCjdr=O. Equation (5.16) then reduces to
dCw Cw dCw dr
(i; = - -; or C
w
r
which on integration gives
Cwr = constant (5.17)
Thus the whirl velocity varies inversely with radius, this being known as the free
vortex condition.
It can therefore be seen that the three conditions of (a) constant specific work,
(b) constant axial velocity, and (c) free vortex variation of whirl velocity, naturally
satisfY the radial equilibrium equation (5.13) and are therefore conducive to the
design flow conditions being achieved. It would at first appear that they constitute
an ideal basis for design. Unfortunately, there are certain diE:advantages associated
with the resultant 'free vortex blading', described in later sections, which
influence the designer in considering other combinations of basic conditions. Free
vortex designs, however, are widely used in axial flow turbines and will be
discussed further in Chapter 7.
One disadvantage of free vortex blading will be dealt with here: the marked
variation of degree of reaction with radius. In section 5.5 it was shown that for the
case where C3 = CI and Cal = Ca2 = Coo the degree of reaction can be ,expressed
by
A = 1 - (tan a2 + tan ad
which can readily be written in terms of whirl velocities as
A = 1 _ Cw2 + Cwl
2U
(5.18)
Remembering that U = U mrlr m, where U m is the blade speed at the mean radius of
the annulus rm, equation (5.18) can be written
A = 1 _ Cwzl' + Cwll'
2Umr
z
/l'm
For a free vortex design Cwr = constant, so that
A = 1 _ constant
1'2
(5.19)
Evidently the degree of reaction increases markedly from root to tip oflthe blade.
Even if the stage has the desirable value of 50 per cent at the mean radius, it may
well be too low at the root and too high at the tip for good efficiency. Because of
the lower blade speed at the root section, more fluid deflection is required for a
given work input, i.e. a greater rate of diffusion is required at the root section. It
is, therefore, particularly undesirable to have a low degree of reaction in this
region, and the problem is aggravated as the hub-tip ratio is reduced.
174 AXIAL FLOW COMPRESSORS
When considering other possible sets of design conditions, it is usually
desirable .to retain the constant specific work -input condition to provide a constant
stage pressure ratio up the blade height. It would be possible, however, to choose
a variation of one of the other variables, say Cw, and detennine the variation of Ca
from equation (5.16). The radial equilibrium requirement would still be satisfied.
[It should be clear that in general a design Gan be based on arbitrarily chosen
radial vmiations of any two variables and the appropriate variation of the third can
be detennined by inserting them into equation (S.1S).]
As an illustration we shall use the nOIDlal design condition (a) constant
specific work inpnt at all radii, together with (b) an arbitrary whirl velocity
distribution which is compatible with (a). To obtain constant work input,
U(Cw2 - Cw1) must remain constant across the annulus. Let us consider
distributions of whirl velocity at inlet to and outlet from the rotor blade given by
n b n b
Cwl = aR - R and Cw2 = aR + R
(5.20)
where a band n are constants and R is the radius ratio rlr m. At any radius r the
blade is given by U = UmR. It is immediately seen that (Cw2 - Cwl ) = 2blR
and hence that
U(Cw2 - Cw1 ) = 2bUrn
which is independent of radius. The two design conditions (a) and (b) are there-
fore compatible. We shall consider three special cases: where n = -1, 1 and 0
respectively. (It will be shown later that a and b are not arbitrary constal1ts but
depend upon the chosen values of degree of reaction and stage temperature rise.)
When n=-l
The whirl distributions become
a b
C ----.
-wI - R R
which are of free vortex fonn, Cwl" = constant. From what has gone before it
should be clear that Ca = constal1t must be the third design condition required to
ensure radial equilibrium, i.e. to satisfy equation (S.16). It also follows that the
variation of A is given by equation (5.19), which since (Cw2r+ Cwlr) =2arm, can
be written
2arm a
A=1---=1---2
2UmRr UmR
When n= 1
b
C =aR--
wI R
b
and Cw2 = aR +11.
(S.21 )
THREE-DIMENSIONAL FLOW
Rewriting equation (5.16) in tenns of the dimensionless R we have
C
2
Ca dCa + Cw dCw + ; dR = 0
Integrating from the mean radius (R = 1) to any other radius,
_1[C2]R = 1 [C2]R + -'-"dR
. IR c
2
2al 2·vl IR
175
At exit from the rotor, where Cw = aR + (bIR), the right-hand side of the equation
becomes
[a2R2 + 2ab + '!...]R + JR[a2R + 2ab + b:]dR
2 R 1 1 R R
and, finally, the radial distribution of Ca is given by
C;2 - = -2[a
2
R2 + 2ab In R - a
2
]
Similarly, at inlet to the rotor,
- = -2[a
2
R2 - 2ab In R - a
2
l
(5.22)
(5.23)
Note that Ca2 cannot equal Cal except at the mean radius (R = 1). It is not
possible, therefore, to use the simple equation (5.1S) for the degree of reaction
when finding the variation of A with radius. To do this it is necessary to revert to
the original definition of A [ = IJ.TAI(IJ.T.,i + IJ.TB)] and work from first principles
as follows. We shall retain the assumption that the stage is designed to give
C3 = C1· Then IJ.Ts=IJ.Tos so that
Cp(IJ.TA + IJ.Ts) = W = U(Cw2 - Cwl )
CplJ.TA = - CD + U(Cw2 - Cw1 )
= + - + C;;'2)] + U(Cw2 - Cwl )
A = I + - Cw2 + Cw1
2U(Cw2 - Cwl ) 2U
From equations (S.22) and (S.23), assuming we choose to design with (C
a2
)m =
(Cal),n,
- = Sab InR
From the whirl distributions,
2b
Cw2 - Cw1 = R al1d Cw2 + Cwl = 2aR
Substituting in the equation for A, al1d writing U = UmR, we get finally

Um Urn
(5.24)
176
AXIAL FLOW COMPRESSORS
Designs with the exponent n = 1 have been referred to as first power designs.
At this point it can easily be made clear that the constants a and b in the whirl
distributions are not arbitrary. Equation (5.24) shows that a is fixed once the
degree of reaction at the mean radius is chosen, i.e. at R = 1
a = Um(l- Am)
And b is fixed when the stage temperature rise is chosen because
c/);.Tos = U(Cw2 - Cwl ) = 2bUm
and hence b = c/lTos/2Um·
When n=O
Proceeding as in the previous case, but with
b
Cwl =a-Ii
we find that
b
and Cw2 = a+Ii
- = -2[ a
2
In R - + ab]
- = -2[ a
2
In R + - ab J
(5.25)
(5.26)
Blading designed on this basis is usually referred to as exponential blading.
Again proceeding as before, and assuming (Ca2)m = (Cal)m, it can be shown that
a 2a
A = 1 +--- (5.27)
Um UmR
It should be noted that for all three cases, at the mean radius (R = I), the
expression a = Um(l - Am) holds. If a value is specified for the reaction at the
mean radius, Am, the variation of reaction with radius can readily be obtained by
substituting for a in equations (5.21), (5.24) and (5.27). The results can be
summarized conveniently in the following table.
n A Blading
1
-1 l--(l-A) Free vortex
R2 m
0
Exponential
1+(2 In R - 1)(1 - Am) First power
It is instructive to evaluate the variation of A for all three cases assuming that
Am = 0·5, and the results are given in Fig. 5.12. It can be seen that the free vortex
THREE-DIMENSIONAL FLOW
1.0
n

Free vortex
o

. 1.5
R= rlrm
FIG. 5.12 Radial variatioll of of reaction
177
design gives the greatest reduction in A for low values of R while the first-power
design gives the least reduction. Furthermore, for all designs there is a lower limit
R. below which the degree of reaction becomes negative. A negative value
a reduction in static pressure in the rotor. For the free vortex design, this
l1lTIltmg value of R is given by 0 = 1 - (l/R2)(1 - Am), from which R = 0·707
when Am = 0·50. As mentioned earlier, however, hub-tip ratios as low as 0-4 are
common at entry to the compressor of a jet engine. Noting that
1'/1'/ = (r/rm)(rm/r/) = R(rm/I'/) = R[l + (rr/r,)]/2
the variation in reaction can readily be evaluated in terms of rlrt for a specified
value of hub-tip ratio rr/I'/. Figure 5.13 shows the radial distribution of A for a
free vortex design of hUb-tip ratio 0·4, for a series of values of Am. It is evident
that a stage of low hub-tip ratio must have a high value of Am to provide
satisfactory conditions at the root section.
So far the discussion has been based upon a choice of Ca, Cw and ho (i.e. W)
as the primary variables. This choice is not essential, however, and another
approach could be to specify a variation of degree of reaction with radius rather
than Cwo One such design method was based on (a) axial velocity, (b) work input
and (c) degree of reaction, all being independent of radius; this was referred to as
a constant reaction design. It led to less twisted blading than the free vortex
but consideration of the analysis leading to equation (5.17) and thence to
equatlOn (5.19) shows that radial equilibrium cannot be satisfied: conditions (a)
and (b) involve a varying A when equation (5.16) is satisfied. The fact is that the
flow will adjust itself to satisfy radial equilibrium between the blade rows. If the
178
""
"
~
i!: 0.5
'0
'" i!:
C>
'" o
AXIAL FLOW COMPRESSORS
-
flrl
FIG. 5.13 Free vortex variation of reaction fOil" r,ir, = ()·4
design does not allow for this correctly, the aclual air angles will not agree with
the design values on which the blade angles are based and the efficiency is likely
to be reduced.
While it is nonnally desirable to retain the constant specific work input
condition, a fan for a turbofan engine of high bypass ratio might provide an
exception to this rule. In this case, the inner stream, feeding the high-pressure
core, may be designed for a lower pressure ratio than the outer stream which is
supplying air to the bypass duct and where the blade speed is high. The work then
varies with radius, dho/dr is not zero, and equation (5.15) must be used instead of
equation (5.16).
Further discussion of three-dimensional design procedures will follow in
section 5.7 after an example is given to show how the material presented in the
preceding sections can be used in the design of an axial flow compressor.
5.7 Design process
The theory presented in the previous sections will now be applied to the design of
an axial flow compressor, and it will be seen that the process requires continuous
judgement by the designer. The design of a compressor suitable for a simple low-
cost turboj et will be considered.
A typical gas turbine design procedure was outlined in Fig. 1.20. Assuming
that market research has shown there is a need for a low-cost turbojet with a take-
off thrust of about 12000 N, preliminary studies will show that a single-spool all-
axial flow arrangement is satisfactory, using a low pressure ratio and a modest
turbine inlet temperature to keep the cost down, as discussed in section 3.3. From
DESIGN PROCESS
179
cycle calculations, a suitable design point under sea-level static conditions (with
Pa= 1·01 bar and Ta=288 K) may emerge as follows:
Compressor pressure ratio
Air-mass flow
Turbine inlet temperature
4·15
20 legis
1100 K
With these data specified, it is now necessary to investigate the aerodynamic
design of the compressor, turbine and other components of the engine. It will be
assumed that the compressor has no inlet guide vanes, to keep both weight and
noise down. The design of the turbine for this engine will be considered in
Chapter 7.
The complete design process for the compressor will encompass the following
steps:
(i) choice of rotational speed and annulus dimensions;
(ii) determination of number of stages, using an assumed efficiency;
(iii) calculation of the air angles for each stage at the mean radius;
(iv) detennination of the variation of the air angles from root to tip;
(v) investigation of compressibility effects;
(vi) selection of compressor blading, using experimentally obtained cascade
data;
(vii) check on efficiency previously assumed, using the cascade data;
(viii) estimation of off-design perfonnance;
(ix) rig testing.
Steps (i) to (v) will be outlined in this section and the remaining steps will be
covered in later sections. In practice, the process will be one of continued
refinement, coupled with feedback from other groups such as the designers of the
combustion system and turbine, metallurgists and stress analysts, and those
concerned with mechanical problems associated with whirling speeds, bearings,
stiifuess of structural members and so on.
Determination of rotational speed and annulus dimensions
Reviewing the theory presented earlier, it is seen that there is no equation which
enables the designer to select a suitable value of rotational speed. This can be
found, however, by assuming values for the blade tip speed, and the axial velocity
and hub-tip ratio at inlet to the first stage. The required annulus area at entry is
obtained from the specified mass flow, assumed axial velocity, and ambient con-
ditions, using the continuity equation.
Previous experience will suggest that a tip speed, Ut , of around 350 mls will
lead to acceptable stresses and that the axial velocity, em could range from 150 to
200 mls. Without IGVs there will be no whirl component of velocity at inlet, and
this will increase the Mach number relative to the blade (see Fig. 5.5) so it may be
advisable to use a modest value of 150 mls for Ca. The hub-tip ratio at entry may
180
AXIAL FLOW COMPRESSORS
vary betweenO.4 and 0·6, andfor a specified annulus area the tip radius willbe a
function of the hub-tip ratio. For a fixed blade speed, then, the rotational speed
will also be a function of hub-tip ratio. Thus the designer will, in very short order,
be presented with a wide range of solutions and must use "his judgement to select
the most promising. At the same time the turbine designer will be examining a
suitable turbine and the COmpressor and turbine designers must keep in close
contact while establishing preliminary designs.
To satisfy continuity
m = PIACal = Plnr;[l-
2 m
= 2
nPI Cal [1 - (rrlrt) 1
At sea-level static conditions, To! = Ta = 288 K. Assuming no loss in the intake,
POI=Pa=I·OI bar. With
CI = Cal = 150 mls (Cwl = 0)
150
2
TI = 288 - 2 X 1.005 X 103 = 276·8 K
[
TI ]1/(1-
1
) [276.8]3.5
PI = POI - = 1·01 -- = 0·879 bar
TOl 288
= 100 X 0.879 = 1.106 k 1m3
PI 0 . .287 X 27(>·8 g
? _ ) 20 _ 0·03837
t - n X 1·106 x 150[1 - (rrlril- [1- (rrlril
The tip speed, Ut, is related to rt by Ut =2nrN, and hence if Ut is chosen to be
350 mis,
350
N=-
2nrt
Evaluating rt and N over a range of hub-tip ratios the following table results:
rr/r,
rt N
(m) (revs/s)
0·40 0·2137 260·6
0·45 0·2194 253·9
0·50 0·2262 246·3
0·55 0·2346 237·5
0·60 0·2449 227·5
At this point it would be pertinent to consider the turbine design. The example
in Chapter 7 shows that a speed of 250 revls results in quite an adequate single-
DESIGN PROCESS 181
stage unit, and the outer radius at the turbine inlet is found to be 0·239m.
Referring to the table above, a hub-tip ratio of 0·50 would give a compatible
compressor tip radius of 0·2262 m although the rotational speed is 246·3 rev/so
There was nothing sacrosanct about the choice of 350 mls for the tip speed, and
the design could be adjusted for a rotational speed of 250 rev/s. With the speed
slightly altered then,
Ut = 2n x 0·2262 x 250 = 355·3 mls
For a simple engine of the type under consideration, there would be no merit in
using a low hub-tip ratio; this would merely increase the mismatch between the
compressor and turbine diameters, and also complicate both the mechanical and
aerodynamic design of the first stage. On the other hand, using a high hub-tip
ratio would unnecessarily increase" the compressor diameter and weight. But it
should be realized that the choice of 0·50 for hub-tip ratio is arbitrary, and merely
provides a sensible starting point; later considerations following detailed analysis
could cause an adjustment, and a considerable amount of design optimization is
called for.
At this stage it is appropriate to check the Mach number relative to the rotor tip
at inlet to the compressor. Assuming the axial velocity to be constant across the
annulus, which will be the case where there are no inlet guide vanes,
Vrt = Urt + = 355.3
2
+ 150
2
, and Vit = 385·7 mls
a = .j(yRTI) = .Jl.4 x 0·287 x 1000 x 276·8 = 331·0 mls
Vlt 385·7
Mit 331.0 = 1·165
Thus the Mach number relative to the rotor tip is 1·165 and the first stage is
transonic; this level of Mach number should not present any problem, and
methods of dealing with shock losses will be covered in a later section.
With the geometry selected, i.e. a hub-tip ratio of 0·50 and a tip radius of
0·2262 m, it follows that the root radius is 0·1131 m and the mean radius is
0·1697 m. It is instructive now to estimate the annulus dimensions at exit from
the compressor, and for these preliminary calculations it will be assumed that the
mean radius is kept constant for all stages. The compressor delivery pressure,
P02 = 4·15 x 1·01 = 4·19 bar. To estimate the compressor delivery temperature it
will be assumed that the polytropic efficiency of the compressor is 0·90. Thus
r J7 ](n-I)/n
T02 = To! ,
(n - 1) 1 0·4
where --=-x-=0·3175
n 0·90 1·4
so that
T02 = 288.0(4.15)°·3175 = 452·5 K
Assuming that the air leaving the stator of the last stage has an axial velocity of
150 mls and no swirl, the static temperature, pressure and density at exit can
182 AXIAL FLOW COMPRESSORS
readily be calculated as follows:
150
2
Tz = 452·5 - 2 x 1.005 x 1Q3 = 441·3 K
[
T2 ]Y/(Y-I) [441.3]3'5
P2 = POl T02 4·19 452.5 = 3·838 bar
_ 100 x 3·838 _ . 3 k 3
P2 - 0.287 x 441.3 - 3 0 g/m
The exit annulus area is thus given by
20 2
A2 = 3.031 x 150 = 0·0440 m
With rm = 0·1697 m, the blade height at exit, h, is then given by
h
0·044 0·044
=--= =0·0413m
2nrm 2n x 0·1697
The radii at exit from the last stator are then
r l = 0·1697 -I- (0·0413/2) = 0·1903 m
rr = 0·1697 - (0·0413/2) = 0·1491 m
At this point we have established the rotational speed and the annulus dimensions
at inlet and outlet, on the basis of a constant mean diameter. To summarize:
N = 250 rev/s
UI = 355·3 mls
Ca = 150 mls
r m = 0·1697 m (constant)
Estimation of number of stages
/'1 = 0·2262 m}.
mlet
rr = 0·1131 m
1'1 = 0·1903 m}
outlet
/'r=0·1491m
With the assumed polytropic efficiency of 0·90, the overall stagnation tempera-
ture rise through the compressor is 452·5 - 288 = 164·5 K. The stage tempera-
ture rise !'J.Tos can vary widely in different compressor designs, depending on the
application and the inlportance or othelwise oflow weight: values may vary from
10 to 30 K for subsonic stages and may be 45 K or higher in high-performance
transonic stages. Rather than choosing a value at random, it is instructive to
estimate a suitable !'J.Tos based on the mean blade speed
U = 2 x "IT x 0·1697 x 250 = 266·6 m/s
We will adopt the simple design condition Cal = Ca2 = Ca throughout the com-
pressor, so the temperature rise from equation (5.9) is given by
!'J.Tos = lUCa(tan PI - tan pz) = lU(Cw2 - Cwl )
cp c
p
DESIGN PROCESS 183
With a purely axial velocity at entry to the first stage, in the absence ofIGV s,
U 266·6
tanf3! =-=--
Ca 150
PI = 60·64°
, Ca 150
PI == --= = 305·9 m/s
cos PI cos 60·64
In order to estimate the maximum possible deflection in the rotor, we will apply
the de Haller critelion V2IV\ <f. 0·72. On this basis the minimum allowable value
of Vz = 305·9 x 0·72 = 220 mis, and the corresponding rotor blade outlet angle
is given by
Using this deflection and neglecting the work-done factor for this crude estimate
1lT. = 266·6 x 150(tan 60·64 - tan 47·01) 28 K
os 1.005 X 103
A temperature rise of 28 K per stage implies 164·5/28 = 5·9 stages. It is likely,
then, that the compressor will require six or seven stages and, in view of the
influence of the work-done factor, seven is more likely. An attempt will therefore
be made to design a compressor.
With seven stages and an overall temperature lise of 164·5 K the average
temperature rise is 23·5 K per stage. It is normal to design for a somewhat lower
temperature rise in the first and last stages, for reasons which will be discussed at
the end of this section. A good starting point would be to assume !'J.To 20 K for
the first and last stages, leaving a requirement for !'J.To 25 K in the remaining
stages.
Stage-by-stage design
Having detemlined the rotational speed and annulus dimensions, and estimated
the number of stages required, the next step is to evaluate the air angles for each
stage at the mean radius. It will then be possible to check that the estimated
number of stages is likely to result in an acceptable design.
From the velocity diagram, Fig. 5 .. 6, it is seen that Cw ! = Ca tan IXI and
Cw2 = Cwl -I- !'J.Cw Fot the first stage IXI = 0, because these are no inlet guide
vanes. The stator outlet angle for each stage, 1X3, will be the inlet angle IXI for the
following rotor. Calculations of stage temperature rise are based on rotor
considerations only, but care must be taken to ensure that the diffusion in the
stator is kept to a reasonable leveL The work-done factors will vary through the
compressor and reasonable values for the seven stages would be 0·98 for the first
stage, 0·93 for the second, 0·88 for the third and 0·83 for the remaining four
stages.
184
AXIAL FLOW COMPRESSORS
Stages 1 and 2
Recalling the equation for the stage temperature rise in terms of change in whirl
velocity IJ.Cw = Cw2 - CwI , we have
= cplJ.To = 1·005 )( 10
3
X 20 = 76.9 mls
IJ.Cw AU 0.98 )( 266.6
Since CwI = 0, Cw2 = 76·9 mJs and hence
U 266·6
tan PI = C
a
= 150 = 1·7773,
_ U - Cw2 _ 266·6 - 76·9 _ 1.264
tan pz - C - 150 - ,
a
Cw2 76·9 ,
tan 1X2 = -c: = 150 = 0·5b,
The velocity diagram for the first stage therefore appears as in Fig. 5.14(a).
The deflection in the rotor blades is PI - P2 = 8.98°, which is modest. The
diffusion can readily be checked using the de Haller number as follows:
V2 = Calcos /32 = cos PI = 0·490 = 0.790
V1 Calcos P1 cos P2 0·260
This value of de Haller number indicates a relatively light aerodynamic loading,
i.e. a low rate of diffusion. It is not necessary to calculate the diffusion factor at
this stage, because the de Haller number gives an adequate preliminary check.
After the pitch chord ratio (s/ c) is determined from cascade data, as explained in a
later section, the diffusion factor can be calculated readily from the known vel-
ocities.
At this point it is convenient to calculate the pressure ratio of the stage
(Po3IPoI)J, the suffix outside the parenthesis denoting the number of the stage,
~
I I
I.
u
.1 I.
u
, I
(a) 1st stage (b) 2nd stage
FIG. 5.14 Effect of degree of reaction on shape of velocity diagram
DESIGN PROCESS 185
and then the pressure and temperature at exit which will also be the values at inlet
to the second stage. The isentropic: efficiency of the stage is approxinJately equal
to the polytropic efficiency of the compressor, which has been assumed to be
0·90, so we have
(
P03) = (1 + 0·90 x 20)3.5 = 1.236
PCIJ 1 288
(P03)I = 1,01 x 1·236 =: 1·249 bar
(To3 )1 = 288 + 20 = 308 K
We have finally to choose a value for the air angle at outl,et from the stator row,
1X3, which will also be the direction of flow, IXb into the second stage. Here it is
useful to consider the degree of rc:action. For this first stage, with the prescribed
axial inlet velocity, C3 will not equal C1 (tmless 1X3 is made zero) whereas our
equations for A were derived on the assumption of this equality of inlet and outlet
velocities. Nevertheless, C3 will not differ markedly from Cj, and we can arrive at
an approximate value of A by using equation (5.18).
A ::;,; I _ Cw2 -I- Cw1 = I __ 76·9 = 0.856
2U 2)( 266·6
The degree of reaction is high, but we have seen from Fig, 5.13 that this is
necessary witll low hub-tip ratios to avoid a negative value at the root radius, We
shall hope to be able to use 50 per cent reaction stages from the third or fourth
stage onwards, and an appropriate value of A for the second stage may be about
0·70.
For the second stage IJ.Tos= 25 K and}, = 0·93, and we can determine PI and
P2 using equations (5.9) and (5.11). From (5.9)
0·93 x 266·2 x 150
25 = 1.005 x 1Q3 (tan P1 - tan P2)
(tan /31 - tan P2) = 0·6756
And from (5.11)
150
0·70 ::;,; 2 x 266.6 (tan P1 + tan P2)
(tan P1 + tan P2) ::;,; 2·4883
Solving these sinJultaneous equations we get
P1 = 57·70° and P2 = 42·rgo
Finally, using (5.2) and (5.3)
and
The whirl velocities at inlet and outlet are readily found from the velocity dia-
gram,
CwI = Ca tan IX1 = 150 tan 11·06 = 29·3 mls
Cw2 = Ca tan IX2 = 150 tan 41·05 = 130·6 mls
186 AXIAL FWW COMPRESSORS
The required change in whirl velocity islOl-3 mls,compared with 76·9m1s
for the first stage; this is due to the higher stage temperature rise and the lower
work-done factor. The fluid deflection in the rotor blades has increased to 15·51
degrees. It appears that 1X3 for the first stage should be 11·06 degrees. This design
gives a de Haller number for the second-stage rotor blades of cos 57·701
cos 42·19 = 0·721, which is. satisfactory. With the stator outlet angle for the first-
stage stator now known, the de Haller number for the first-stage stator would be
C3 = cos 1X2 = cos 27·15 = 0.907
Cz cos 1X3 cos 11·06
implying a small amount of diffusion. This is a consequence of the high degree of
reaction in the first stage.
The velocity diagram for the second stage appears as in Fig. 5.14(b) and the
outlet pressure and temperature become
(1203) = (1 + 0·90 x 25)3.3 = 1.280
\POI 2 308
(Po3h = 1·249 x 1·280 = 1·599 bar
(T03 )z = 308 + 25 = 333 K
At this point we do not know 1X3 for the .second stage, but it will be determined
from the fact that it is equal to IXI for the third stage, which we will now proceed
to consider.
Before doing so, it is useful to point out that the degree of reaction is directly
related to the shape of the velocity diagram. It was previously shown that for 50
per cent reaction the velocity diagram is symmetrical. Writing Cwm =
(Cwl + Cwz)/2, equation (5.18) can be rewritten in the form A = I - (CwmlU).
Referring to Fig. 5. 14(a) and (b) it can be seen that when CwmlU is small, and the
corresponding reaction is high, the velocity diagram is highly skewed; the high
degree of reaction in the first stage is a direct consequence of the decision to
dispense with inlet guide vanes and use a purely axial inlet velocity. The degree of
reaction is reduced in the stage, and we would eventually like to achieve
50 per cent reaction in the later stages where the hub-tip ratios are higher.
Stage 3
Using a stage temperature rise of25 K and a work-done factor of 0·88, an attempt
will be made to use a 50 per cent reaction design for the third stage.
Proceeding as before
AToscp 25 x 1·005 x 10
3
tan PI - tan pz = AUC = 0.88 x 266.6 x 150 = 0·7140
a
2U 0·5 x 2 x 266·6
tan PI +tanpz = A C
a
= 150 = 1·7773
DESIGN PROCESS
187
yielding PI=51·24° and P2=28.00°. The corresponding value of de Haller
number is given by cos 51·24/cos 28·00 = 0·709. This is rather low, but could be
deemed satisfactory for a preliminary design. It is instructive, however, to in-
vestigate the possibilities available to the designer for reducing the diffusion. One
possibility is to consider changing the degree of reaction, but it is found that the
de Haller number is not strongly influenced by the degree of reaction chosen; as
A had a value of R:j O· 70 for the second stage it might appear that a suitable value
for the third stage might be between 0·70 and 0·50. Repeating the above calcu-
lations for a range of A, however, shows that A=0·55 results in a further de-
crease of de Haller number to 0·706; referring again to Fig. 5.14 it can be
observed that for a specified axial velocity, the required diffusion increases with
reaction. A de Haller number of 0·725 can be achieved for A = 0·40, but it is
undesirable to use such a low degree of reaction. A more useful approach might
be to accept a slightly lower temperature rise in the stage, and reducing ATos from
25 K to 24 K while keepirig A = 0·50 gives
tan PI - tan P2 = 0·6854
yielding PI = 50.92°, P2 = 28·63° and a de Haller number of 0·718, which is
satisfactory for this preliminary design. Other methods of reducing the aerody-
namic loading include increases in blade speed or axial velocity, which could
readily be accommodated.
With a stage rise of 24 K, the performance of the third stage is
then given by
(
P03) = (1 + 0·90 x 24)3.5 = 1.246
POI 3 333
{P03)3 = 1·599 x 1·246 = 1·992 bar
(T03 )3 = 333 +24 = 357 K
From the symmetry of the velocity diagram IXI = P2 = 28·63° and 1X2 = PI =
50.92°. The whirl velocities are given by
Cwl = 150 tan 28·63 = 81·9 mls
Cwz = 150 tan 50·92 = 184·7 mls
Stages 4, 5 and 6
A work-done factor of 0·83 is appropriate for all stages from the fourth onwards,
and 50 per cent reaction can be used. The design can be simplified by using the
same mean diameter velocity diagrams for stages four to six, although each blade
will have a different length due to the continuous increase in density. The seventh,
and final stage can then be designed to give the required overall pressure ratio. It
is not necessary to repeat all the calculations for stages 4-6, but it should be noted
that the reduction in work-done factor to 0·83, combined with the desired stage
188
AXIAL FLO'WCOMPRESSORS
temperature rise of 25 K, results in an unacceptably low de Haller of
0.695. Reducing the stage temperature rise to 24 K increases the de Haller num-
ber to 0·705, which will be considered to be just acceptable for the preliminary
design.
Proceeding as before,
24'x 1·005 x 10
3
tan PI - tan P2 = 0.83 x 266.6 x 150 = 0·7267
266·6
tan PI + tan P2 = 0·5 x 2 x 150 = 1·7773
yielding PI=51·38° (=0(2) and P2=27·7lo (=IXI). The performance of the
three stages can be summarized below:
Stage 4 5 6
POI (bar) 1·992 2·447 2·968
TOI (K) 357 381 405
(Poi POI)
1·228 1·213 1·199
POJ (bar) 2·447 2·968 3·560
T03 (K) 381 405 429
pOJ - POI (bar) 0-455 0·521 0·592
It should be noted that although each stage is designed for the same temperature
rise, the pressure ratio decreases with stage number; this is a direct consequence
of the increasing inlet temperature as flow progresses through the compressor.
The pressure rise, however, increases steadily.
Stage 7
At entry to the final stage the pressure and temperature are 3·560 bar and 429 K.
The required compressor delivery pressure is 4·15 x 1·01 =4·192 bar. The
pressure ratio of the seventh stage is thus given by
(l!03) = 4·192 = 1.177
\POI 7 3·560
The temperature rise required to give this pressure ratio can be determined from
(
1 + 0.90ATos)3.5 = 1.177
429
giving ATos=22·8 K.
The corresponding air angles, assuming 50 per cent reaction, are then
PI = 50·98° (= 0(2), P2 = 28·52° (= 0(1) with a satisfactory de Haller number of
0·717.
With a 50 per cent reaction design used for the final stage, the fluid will leave
the last stator with an angle 1X3 = IXI = 28.52°, whereas ideally the flow should be
DESIGN PROCESS· 189
axial at entry to the combustion chamber. The flow can be straightened by
incorporating vanes after the final compressor stage and these can form part of the
necessary diffuser at entry to the combustion chamber.
All the preliminary calculations have been carried out on the basis of a
constant mean diameter. Another problem now arises: a sketch, approxnnately to
scale, of the compressor and turbine annuli (Fig. 5.15) shows that the comb)lstor
will be an· awkward shape, the required changes in flow direction causing
additional pressure losses. A more satisfactory solution might be to design the
compressor for a constant outer diameter; both solutions are shown in the figure.
The use of a constant outer diameter results in the mean blade speed increasing
stage number, and this in turn implies that for a given temperature rise ACw
IS reduced. The fluid deflection is correspondingly reduced with a beneficial
increase in de Haller number. Alternatively, because of the higher blade speed a
higher temperature rise could be achieved in the later stages; this might permit the
required pressure ratio to be obtained in six stages rather than seven. The reader
should be aware that the simple equations derived on the basis of U = constant are
then not valid, and it would be necessary to use the appropriate values of Uj and
U2; the stage temperature rise would then be given by 1(U2Cw2 - UICwl)/cp-
Compressors which use constant inner diameter, constant mean diameter or
constant outer diameter will all be found in service. The use of a constant inner
diameter is often found in industrial units, permitting the use of rotor discs of the
same diameter, which lowers the cost. It would be important to minimize the
number of turbine stages, again for reasons of cost, and with a subsonic com-
pressor it is very probable that the turbine diameter would be noticeably larger
than the compressor diameter. The difference in turbine and compressor diameter
is not critical, however, because frontal area is unimportant and with reverse flow
combustion chambers large differences in diameter can be easily accommodated.
Coustant outer diameter compressors are used where the minimum number of
stages is required, and these are commouly found in aircraft engines.
The compressor annulus of the Olympus 593 engine used in Concorde
employs a combination of these approaches; the LP compressor annulus has a
I
Compressor I Combustion I Turbine
chamber

Constant mean diameter
(=0.2262 m (=0.2545 m
FIG. 5.15 Annulus shape
190 AXIAL FLOW COMPRESSORS
virtually constant inner diameter, while the IfP compressor has a constant outer
diameter. The access0l1es are packed around the HP compressor annulus and the
engine. when fully equipped is almost cylindrical in shape, with the compressor
inlet and turbine exit diameters almost equal. In this application, frontal area is of
critical importance because of the high supersonic speed.
variation of air angles from root to tip
In section 5.6 various distributions of wlllrl velocity with radius were considered,
and it was shown that the designer had quite a wide choice. In the case of the first
stage, however, the choice is restricted because of the absence of IGVs; this
means that there is no whirl component at enlry to the compressor and the inlet
velocity will be constant across the ammlus. For all other stages the whirl velocity
at entry to the rotor blades will be detennined by the axial velocity and the stator
outlet angle from the previous stage, giving more freedom in the aerodynamic
design of the stage.
The material developed in section 5.6 will be applied to the first stage, which
is a special case because of the axial inlet velocity, and the third stage, which is
typical of the later stages. The first stage will be investigated using a free vortex
design, noting that the condition C"r = constant is satisfied for Cw = O. Attention
will then be turned to the design of the tlllrd stage, recalling that the mean radius
design was based on Am = 0·50. The tlllrd stage will be investigated for three
different design approaches, viz. (i) free vortex, Am = 0·50, (ii) constant reaction,
Am = 0·50 with radial equilibrium ignored and (iii) exponential blading,
Am =0·50.
Considering the first stage, the rotor blade angle at inlet (P d is obtained
directly from the axial velocity (150 mls) and the blade speed. The blade speeds
at root, mean and tip, corresponding to radii of 0·1131, 0·1697 and 0·2262 m, are
177·7,266·6 and 355·3 mls respectively. Thus
177-7
tan PI}" = 150 '
266·6
tan PIm = 150 '
355·3
tan Plt = 150'
Air angles at any radius can be calculated as above. For our purposes the cal-
culations will be restricted to root, mean and tip radii, although for detailed
design of the blading it would be necessary to calculate the angles at intennediate
radii also.
To calculate the air angles P2 and (X2 it is necessary to detennine the radial
variation of Cw2 ' For the free vortex condition Cw2 r = constant, and the value of
DESIGN PROCESS
191
Cw2m WaS previouslydetennined to be 76·9 mls. Because of the reduction of
annulus area through the compressor, the blade height at exit from the rotor will
be slightly less than at inlet, and it is necessary to calculate the tip and root radii at
exit from the rotor blades to find the relevant variation of Cw2 ' The stagnation
pressure and temperature at exit from the first stage were found to be 1·249 bar
and 308 K. Recalling that a stator exit angle of 11·06 degrees was established,
150
C3 = = 152·8 m/s
cos 11·06
T3 = 308 - (152.8)2/(2 x 1·005 x 10
3
) = 296-4 K
(
296.4)3.5
P3 = 1·249 308 = 1·092 bar
100 x 1·092 3
P3 = 296-4 x 0.287 = 1·283 kg/m
m 20 3
A3 = P3 Ca3 = 1.283 x 150 = 0·1039 m
0·1039
h = = 0·0974 m
2n x 0·1697
0·0974
I"t = 0·1697 + -2- = 0·2184 m
0·0974
r = 0·1697 - --= 0·1210 m
}" 2
These radii refer to conditions at the stator exit. With negligible error it can be
assumed that the radii at exit from the rotor blades are the mean of those at rotor
inlet and stator exit. t Thus at exit from the rotor,
0·2262 + 0·2184 2
r t = 2 = 0·2 23 m,
= 0·1131 +0·1210 = 0.11 1
r,. 27m,
From the free vortex condition,
0·1697
CW2r=76.9xD.1171 = 111·4m/s
0·1697
Cw2t = 76·9 x 0.2223 = 58·7 m/s
Ut = 349·2 m/s
Ur = 183·9 m/s
t For turbines, where there is considerably greater flare in the annulus, it is necessary to make an
allowance for the spacing between the rotor and stator; this is done in the turbine design example
presented in Chapter 7.
192
AXIAL FLOW COMPRESSORS
The stator· inlet ~ g l e is given by tan OCz = Cwzj Ca, and the· rotor .exitangle by
tan P2=(U - Cw2)/Ca• Hence, ..
111·4
tan oc2r = 150' !XZr = 36·60°
76·9
tan oczm = 150' qZm = 27·14°
58·7
tan!XZt =15O' ocZt =21·37°
183·9 -111-4
tan PZr = 150 ' PZr = 25·80°
266·6 -76·9
tan PZm = 150 ,132m = 51·67°
349·2 - 58·7
tan PZt = 150 '
P2t = 62·69°
The radial variation of air angles is shown in Fig. 5.16, which shows both the
increased deflection (PI - pz) at the root and the requirement for considerable
blade twist along the height of the blade to ensure that the blade angles are in
agreement with the air angles.
The degree of reaction at the root can be approximated (approximate because
C3 ;t: CI ) by Ar = 1 - (Cwz/2U) giving
111·4
Ar ~ 1 - 2 x 183.9 ~ 0·697
As expected, with the high value of Am of 0·856 there is no problem with too low
a degree of reaction at the root.
When calculating the annulns area at exit from the stage it was assumed that
the density at the mean radius could be used in the continuity equation. Although
:i(
20
CX2
10
0 CX1=0
Root Mean Tip
FIG. 5.16 Radial variation of air angles, lst stage
DESIGN PROCESS 193
the stagn{ltion pressure and temperature are assumed to be constant across the
height of the blade, the· static pressure and temperature, and hence the density,
will vary. This effect is small in compressor stages, with their low pressure ratio,
but is more pronounced in turbine stages. The method of dealing with radial
variation of density will be described in the turbine example of Chapter 7. The
density variation from root to tip is about 4 per cent for the first stage of the
compressor 1lI1d would be even less for later stages of higher hub-tip ratio.
The velocity diagrams for the first stage are shown (to scale) in Fig. 5.17. The
increase in fluid deflection and diffusion at the root section are readily visible
from the vectors VI and V2• Since VI is a maximum at the tip and C2 a maximum
at the root, it is apparent that the maximum relative Mach numbers occur at rt for
the rotor blade and at rr for the stator blade. Compressibility effects will be
discussed in the next sub-section.
Turning our attention to the third stage, it is useful to summarize conditions at
inlet and outlet before examining different distributions of whirl velocity. From
the mean radius design, with Am = 0·50
POI = 1·599 bar, P031PoJ = 1·246, P03 = 1·992 bar
TOl = 333 K, To3 (= Toz) = 357 K
III = 26·63° (= 132)' PI = 50·92° (= IlZ)
Cwl = 81·9 mis, CwZ = 184·7 mis, L\Cw = 102·8 mls
It is not necessary to repeat the calculations for rotor radii and blade speed at root
and tip for this stage, and it will be sufficient to summarize the results as follows:
Inlet
Exit
(m)
0·1279
0·1341
r,
(m)
0·2115
0·2053
r,lr,
0·605
0·653
Ur
(mjs)
200·9
205·8
~
(mjs)
332·2
322·5
The resulting air angles for afree vortex design are obtained from the calculations
tabulated below, and the values already obtained at the mean radius are included
for completeness.
~ ; } ~ ~ ~ ~
Root Mean Tip
FIG. 5.17 Velocity diagrams, lst stage
194 AXIAL FLOW COMPRESSORS
Root Mean" Tip
Cwl = CWI"rm/r (mls) 10g·7 81·9 65·7
tan !XI = Cwl/Ca 108·7/150 65·7/150
!XI 35·93 28·63 23·65
tan PI = (U - Cwl)/Ca
(200·9 - 108·7) (332·2 - 65·7)
150 150
PI
31·58 50·92 60·63 "
Cw2 = Cw2rm/r (mls) 233·7 184·7 152-7
tan !X2 = Cw2/Ca 233·7/150 152·7/150
!X2 57·31 50·92 45·51
tan P2 = (U - Cw2)/Ca
(205·8 - 233·7) (322·5 - 152· 7)
150 150
P2 -10·54 28·63 48·54
PJ - P2 42·12 22·29 12·09
I
A=l--(l-A)
R2 m
0·161 0·50 0·668
The radial variation of air angles shown in Fig. 5.18 clearly indicates the very
large fluid deflection (fJ 1 - P2) required at the root and the substantial blade twist
from root to tip.
It is instructive to consider a constant reaction design, with radial equilibrium
ignored. With Am = 0·50, conditions at the mean radius will be the same as those
previously considered for the free vortex design. As before, it will be assumed
that 50 per cent reaction gives a symmetrical velocity diagram and this will hold
across the annulus. From symmetry, U = dCw + 2Cw!> and from constant work
input at all radii U dCw = UmdCwm = constant.
-20L..... ___ ,.,-L ___ ---::,
Root Mean Tip
FIG. 5.18 Radial variation of air angles: free vortex, 3rd stage
DESIGN PROCESS 195
Considering the root section at inlet to the blade,
rm 0·1697
dC = dC - = 102·8 x --= 136-4 mls
wr wm rr 0.1279
C = Ur - dCwr = 205·8 - 136·40 = 34.7 mls
wlr 2 2
. Cwl .. 34· 70 03° P
tan IXlr = C = '15()' IXI .. = 13· = 2r
a
Cw2r = Cwlr + dCwr = 34·70 + 136-4 = 171·1 mls
Cw2r 171·1
tan 1X2r = C = '15()' 1X2r = 48·76° = Plr
a
Repeating these calculations at the tip (rt = 0·2115m, U = 332·2 mls) yields
IXlt = 39· 77° = P2t and 1X2t = 54·12° = Pit. The radial variations of the air angles
are shown in Fig. 5.19, which shows the greatly reduced twist compared with the
free vortex variation shown in Fig. 5.18. It will be recalled, however, that the
radial equilibrium condition is not satisfied by the constant reaction design, and
with the relatively low hub-tip ratio for this stage the constant reaction design is
not lilcely to be the most efficient. It should be pointed out, also, that a small error
is introduced by assuming a symmetrical velocity diagram based on the blade
speeds at rotor inlet, because at hub and tip radii the blade speed is not constant
throughout the stage due to the taper of the annulus. The reader should recognize
that reaction is a useful20ncept in design but it does not matter if the actual value
of reaction is not quite equal to the theoretical vruue.
To conclude this subsection, a third type of design will be examined, namely
exponential blading with Am = 0·5. The value of R at the root section is given by
R=0·1279!0·1697 =0·754. The reaction at the root is then given by
A = 1 + (1 - _2_)(1 -0·50)
I' 0.754
AI' = 0·174
60
U>
~ 40
~
cD 30
~
20
:a:
10
OL..... ___ -L ___ ~
Root Mean TIp
FIG. 5.19 Radial variation of air angles: constant 50 per cent reaction, 3rd stage
196 AXIAL FLOW COMPRESSORS
This is slightly higher than the value obtained for the free vortex design (0·164),
as would be expected from Fig. 5.12. With exponentiaL blading n = 0,
C
wl
= a - (bIR) and Cw2 = a + (bIR). Evaluating the constants a and b,
a = Um(1 - Am) = 266·6(1 - 0·50) = 133·3 m/s
b = cplJ.T = 1·005 X 1000 X 24 = 5104 mls
2UmA 2 X 266·6 X 0·88
The following table gives the resulting whirl velocity distribution,
Root Mean
lPlet R 0·754 1·00
b
Cwl = a - R (mls) 65·1 81·9
ExitR 0·790 1·00
b
Cw2 = a + R (mls) 198·4 184·7
Setting Calm = 150 mls as before, from (5.26), at inlet
2 2 [ 2 ab ]
(Cal) -(Calm) = -2 a InR+'R-ab
Hence
(
Cal )2 2 [2 ab ]
-- = 1--
2
- a lnR+--ab
Calm Calm R
Thus, at the root
2
(
Cal) = 1.247, Calr = 167·5 m/s
Calm,
at the tip (CadCalmi = 0·773, Calt = 131·9 mls.
From equation (5.25), at exit from the rotor,
(
Ca2 )2 2 [2 ab ]
-- = 1--
2
- a lnR--+ab
Ca2m Ca2m R
giving
Ca2r = 185·8 m/s
• Ca2t = 115·5 m/s
Tip
1·246
92·0
1·210
175·8
From the velocities calculated above, the rotor air angles can readily be cal-
culated to give Plr = 39.03°, P2r = 2.28°, Plt= 61·23° and P2t= 51.79°. The rotor
air angles for the free vortex, constant reaction and exponential designs are
DESIGN PROCESS 197
compared in Fig. 5.20, both at inlet and exit from the rotor. It can be seen that the
free vortex exhibits the most marked twist over the blade span, with the constant
reaction showing the least; the exponential design gives a. compromise between
the two. The fluid deflection, PI - P2, for the three designs can be summarized as
below.
Root Mean Tip
Free vortex 42·12 22·29 12·09
Constant reaction 35·73 22·29 14-35
Exponential 37·75 22·29 9-44
The aerodynamic loading at the root section of the free vortex is substantially
higher than that for either of the other two designs.
It was pointed out earlier that the maximum relative inlet Mach numbers occur
at the rotor blade tip and the stator blade root. Without detailing the calculations
we may sununarize the results as follows.
At rotor tip (i.e. at I't):
Cal Cwl VI TI
M
(m/s) (m/s) (m/s) (K)
Free vortex 150·0 65·7 305·8 319·7 0·853
Constant reaction 150·0 82·5 291·3 314·0 0·820
Exponential l31·9 92·0 274·0 320·1 0·764
P1
'l
~ 2
70
60 60
50 50
gj 40 Constant gj 40
k-
!£ reaction !£
'" '"
~ ~ 30 Free vortex C!5 30 Exponential
Constant
20 20 reaction
10
0
-10 J
Root Mean Tip Root Mean Tip
(a) (b)
FIG. 5.20 Comparison of mtor air angles, 3rll stage
198 AXIAL FLOW COMPRESSORS
At stator tip (i.e. at rr):
Ca2 Cw2 C2 T2
M
(m/s) (m/s) (m/s) (K)
Free vortex 150·0 233·7 277-7 318·6 0·776
Constant reaction 150·0. 171·1 227·5 331·2 0·624
Exponential 185-8 198·4 271·8 320·2 0·758
Clearly the rotor tip value is the critical one, and it can be seen that the
exponential design yields an appreciably lower Mach number. The importance of
this will be discussed further in section 5.lD. We have been considering the third
stage here as an example, and the Mach numbers in the later stages will be lower
because of the increased temperature and hence increased acoustic velocity.
The exponential design results in a substantial variation in axial velocity, both
across the annulus and through the stage. We have based the annulus area on the
premise of constant axial velocity, and this would have to be recalculated for the
case of non-constant axial velocity. At any radius r, the rate of flow 15m through an
element of width i5r is given by
15m = p2nri5rCa
and
J
r,
m = 2n prCa dr
r,
The annulus area required to pass the known flow could be determined by nu-
merical integration and would be slightly different to the value assumed for the
previous cases of free vortex and constant reaetion.
The three design methods considered all have advantages and the final choice
of design would depend to a large extent on the design team's previous ex-
perience. The constant reaction design looks quite attractive, but it must be borne
in mind that radial equilibrium has been ignored; ignoring radial equilibrium will
result in flow velocities not being in agreement with the predicted air angles,
leading to some loss in efficiency.
It is appropriate at this point to return to the question of the reduced loading in
the first and last stages. The first stage is the most critical because of the high
Mach number at the tip of the rotor; in addition, at certain flight conditions (e.g.
yaw, high angle of attack or rapid turns of the aircraft) the inlet flow may become
distorted with significant variations in axial velocity across the compressor
annulus. By somewhat reducing the design temperature rise, and hence the
aerodynamic loading, these problems can be partially alleviated. In the case of the
last stage, the flow is delivered to the diffuser at entry to the combustion chamber;
it would be desirable to have a purely axial velocity at exit, and certainly the swirl
should be kept as low as possible. Once again, a slight reduction in the required
temperature rise eases the aerodynamic design. problem.
BLADE DESIGN 199
In this section we have been deciding on the air angles likely to lead to a
satisfactory design of compressor. It is now necessary to discuss methods of
obtaining the blade shapes which will lead to these air angles being achieved, and
this will be done in the next section.
5.8 Blade design
Having determined the air angle distributions that will give the required stage
work, it is now necessary to convert these into blade angle distributions from
which the correct geometry of the blade forms may be determined.
The most obvious requirements of any particular blade row are: firstly, that it
should turn the air through the required angle [(PI - P2) in the case of the rotor
and ((Xl - (,(3) in the case of the stator]; and secondly that it should carry out its
diffusing process with optimum efficiency, i.e. with a minimum loss of stagnation
pressure. With regard to the first requirement, we shall see that due allowance
must be made for the fact that the air will not leave a blade precisely in the
direction indicated by the blade outlet angle. As far as the second requirement is
concerned, certainly the air and blade angles at inlet to a blade row must be
similar to minimize l o s s e s ~ But when choosing the blade angle, the designer must
remember that the compressor has to operate over a wide range of speed and
pressure ratio. The air angles have been calculated for the design speed and
pressure ratio, and under different operating conditions both the fluid velocities
and blade speed may change with resulting changes in the air angles. On the other
hand, the blade angles, once chosen, are fixed. It follows that to obtain the best
performance over a range of operating conditions it may not be the best policy to
make the blade inlet angle equal to the design value of the relative air angle.
The number of variables involved in the geometry of a compressor blade row
is so large that the design becomes to a certain extent dependent on the particular
preference and previous experience of the designer. He will have at his disposal,
however, correlated experimental results from wind tunnel tests on single blades
or rows of blades. In the former case, the effects of adjacent blades in the row
have to be accounted for by the application of empirical factors. The second type
of data, from tests on rows or cascades of blades, is more widely used. Although
it might appear desirable to test a cascade of blades in an annular form of wind
tunnel, in an attempt to simulate conditions in an actual compressor, the blades
are generally tested in the form of a straight cascade. An annular tunnel would not
satisfactorily reproduce conditions in a real compressor and a considerable range
of hub-tip ratios would have to be tested. By using a straight cascade, mechanical
complication of the test rig is considerably reduced, and the two-dimensional flow
conditions obtained in a tunnel of rectangular section greatly simplifies the
interpretation of the test results.
In both Britain and the United States, much experimental research has been
devoted to cascade tests on compressor blading. It is proposed to give a broad
outline of this work and to show how the results obtained may be correlated in a
200 AXIAL FLOW COMPRESSORS
fashion suitable for direct use by the compressor designer. Cascade tests result in
two main items of infonnation. These. are (a) the angle through which the air is
turned fora minimum loss, and (b) the corresponding profile drag coefficient
from which the cascade efficiency may be estimated. When high velocities in the
region of the sonic velocity are used, the tests also yield valuable infonnation on
compressibility effects. A typical cascade· tunnel and the type of test result
obtainable from it will now be ·described.
Compressor blade cascade tunnel and typical test results
The tunnel consists essentially of an arrangement whereby a stream of air can be
sucked or blown through a number of blades set out in the fonn of a straight
cascade (Fig. 5.21). Means are provided for traversing pressure and flow direction
measuring instruments over two planes usually a distance of one blade chord
upstream and downstream of the cascade. The height and length of the cascade
are made as large as the available air supply will allow, in an attempt to eliminate
interference effects due to the tunnel walls. Boundary layer suction on the walls is
frequently applied to prevent contraction of the air stream as it passes through the
tunnel.
:
Cascade It .
Adjustable side walls
/\
Air supply )00
Pitot tube To manometer
Slots for traversing
instruments
I
Turn·table
\\,
/
/
FIG. 5.21 Elevation alid plan of simple cascade millie!. (Traversillg paths indicated
by dotted lines)
BLADE DESIGN 201
The cascade is mounted on a tum-table so that its angular direction relative to
the inflow duct can be set to any desired value. This device enables tests to be
made on the cascade over a range of incidence angles of the air entering the
cascade. In other more complex tunnels provision is also ma.de for variation of the
geometry of the blade row, such as the spacing between the blades and their
setting angle, without removing the cascade from the tunnel. Pressure and
velocity measurements are made by the usual form of L-shaped pitot and pitot-
static tubes. Air directions are fotmd by various types of instruments, the most
common being the claw and cylindrical yawmeters illustrated in Fig. 5.22. The
principle of operation is the same in both and consists of rotating the instrument
about its axis until a balance of pressure from the two holes is obtained. The
bisector of the angle between the two holes then gives the air direction. t
A cross-section of three blades forming a part of a typical cascade is shown in
Fig. 5.23 which also includes details of the various angles, lengths and velocities
associated with the cascade experiments. For any particular test, the blade camber
angle 8, its chord c and the pitch (or space) s will be fixed and the blade inlet and
outlet angles IX] and IX; determined by the chosen setting or stagger angle (. The
angle of incidence, i, is then fixed by the choice of a suitable air inlet angle IX1
since i = IXI - IX]. This can ,be done by an appropriate setting of the turn-table on
Axis Axis
TaU-tube
t:
I
¢
t;: To U-tube
I
. I
I I

o
80'
(a) (b)
FIG. 5.22 Yawmeters: (a) cylindrical, (b) claw
t For a full description of cascade tunnel testing see Todd, K. W. [Ref. (7)] and Gostelow, J. P. [Ref.
(8)].
202
FIG. 5.23 Cascade lIotatioll
AXIAL FLOW COIvlPRESSORS
maximum camber
.a', = blade inlet angle
a2 = blade outlet angle.
e = blade camber angle

S = setting or stagger angle
s = pitch (or space)
E = deflection
= a, a2
a, = air inlet angle
a2 = air outlet angle
v, = air inlet velocity
V2 = air outlet velocity
i = incidence angle
=0:1-0:;
ij = deviation angle
= a2 a2
c = chord
which the cascade is mounted. With the cascade in this position, the pressure and
direction-measuring instruments are then traversed along the blade row in the
upstream and downstream positions, and the results plotted as in Fig. 5.24.
This shows the variation of loss of stagnation pressure, and air deflection
e = (X1 - 0(2, covering two blades at the centre of the cascade. As the loss will be
0.4
'"
'" .Q

0.2 ::>
lil
'" Q.
<=
0
:;
0.1
c:
Cll
'" iii
Position of
blade trailing edge
/ "
/ "
"
"
Distance along cascade
45
40
35
30
25
'"
Q)
'" 0,
OJ
'0
oS
c
0
"f5
OJ
'a3
0
FIG. 5.24 Variations of stagnation pressure loss and deliection fol' cascade at fixed
incidence
BLADE DESIGN 203
dependent on the magnitude of the velocity of the air enteling the cascade, it is
convenient to express it in a dimensionless form by dividing by the inlet dynamic
head, i.e.
loss =POl - P02 =
!pV?
(5.28)
This facilitates correlation of test results covering a range of values of V1•
The curves of Fig. 5.24 can now be repeated for different values of incidence
angle, and the whole set of results condensed to the form shown in Fig. 5.25 in
which the mean loss tv / p Vl and mean deflection 8 are plotted against incidence
for a cascade of fixed geometrical fonn. These curves show that the mean loss
remains fairly constant over a wide range of incidence, rising rapidly when the
incidence has a large positive or negative value. At these extreme incidences the
flow of air around the blades breaks down in a manner similar to the stalling of an
isolated aerofoil. The mean deflection rises with increasing incidence, reaching a
maximum value in the region of the positive stalling incidence.
Test results in the form given in Fig. 5.25 are obtained for a wide range of
geometrical fonns of the cascade, by varying the camber, pitch/chord ratio, etc.
Reduction of the resulting data to a set of design curves is achieved by the
following method.
From curves of the form given in Fig. 5.25, a single value of deflection which
is most suitable for use with that particular form of cascade is selected. As the
object of the cascade is to'°turn the air through as large an angle as possible with a
minimum loss, the use of the maximum deflection is not possible because of the
high loss due to stalling. Furthermore, remembering that in a compressor the
blade will have to operate over a range of incidence under off-design conditions,
the design incidence should be in the region of the flat portion of the loss curve.
I;:: itO.075

'"
'" .Q
c:
.Q
1ii
§, 0.025
<1l
40 '"
CIJ
e:
Cll
Q)
'0
oS
c
.12
t3
'"
1ii
'0
t:
'"
Q)
:;;
Ui
minimum
C
'" OJ
:;;
o

Incidence i, degrees
FIG. 5.25 Mean deflectioll amI meall stagllatioll pressm'c loss for cascade of fixed
geometrical form
204
AXIAL FLOW COMPRESSORS
The practice has been to select a deflection which corresponds to a definite
proportion of the stalling deflection. The proportion found to be most suitable is
eight-tenths, so that the selected or nominal deflection E* = 0·8Es where os is the
stalling deflection. Difficulty is sometimes experienced in deciding on the exact
position of the stall, so its position is standardized by assuming that the stall
occurs when the loss has reached twice its minimum value.
Analysis of the values of the nominal deflection E* determined from a large
number of tests covering different forms of cascade, has shown that its value is
mainly dependent on the pitch/chord ratio and air outlet angle (Xz· Its variation
with change of other factors determining the geometrical form of the cascade,
such as blade camber angle, is comparatively small. On this basis, the whole set
of test results is reducible to the form shown in Fig. 5.26 in which the nominal
deflection is plotted against air outlet angle with pitch/chord ratio as parameter.
This set of master curves, as they may well be called, is of great value to the
designer because, having fixed any two of the three variables involved, an
appropriate value for the third may be determined. For instance, if the air inlet and
outlet angles have been fixed by the air angle design, a suitable pitch/chord ratio
can be read from the diagram.
As an example, consider the design of the third-stage rotor blade required for
the free vortex design carried out in the previous section. At the mean radius of
0.1697 m, where PI = 50·92° and P2 = 28·63°, e* = PI - P2 = 22·29° and from
Fig. 5.26, with an air outlet angle of 28·63°, s/c=0·9.
o I
-10 0 10 20 30 40 50 60 70
Air outlet angle (X2' degrees
FIG. 5.26 Design deflection curves
BLADE DESIGN 205
Detennination of the chord length will now depend on the pitch, which itself is
clearly dependent on the number of blades in the row. When making a choice for
this number, the aspect ratio of the blade, i.e. the ratio of length to chord, has to
be considered because of its effect on secondary losses. This will be discussed in
greater detail in the next section on stage performance. For the purpose of our
example it will be assumed that an aspect ratio hlc of about three will be suitable.
The blade height can be obtained from the previously determined annulus
dimensions as 0·0836 m, so that the chord becomes
0·0836
c = --= 0·0279 m
3
and the pitch is
s = 0·9 x 0·0279 = 0·0251 m
The number of blades 11 is then given by
2:n: x 0·1697
n = 0.0251 = 42·5
It is desirable to avoid numbers with COl11.'Ilon multiples for the blades in suc-
cessive rows to reduce the likelihood of introducing resonant forcing frequencies.
One method of doing this is to choose an even number for the stator blades and a
prime number for the rotor blades. An appropriate number for the rotor blades in
this stage would therefore be 43, and recalculation in the reverse order gives
s = 0·0248 m, c = 0·0276 m, and hlc = 3·03
A similar procedure could also be carried out for the stator blades.
The use of prime numbers for rotor blades is less common than used to be the
case. This is a result of developments in mechanical design which have resulted in
the ability to replace damaged rotor blades in the field without the need for re-
balancing the rotor system. In the case of a fan for a high bypass turbofan engine,
for example, an even number of rotor blades is frequently used and airlines keep a
stock of replacement blades in balanced pairs. In the event of a blade failure, the
defective blade and the one diametrically opposite are replaced.
In the above example, the chord was determined simply from aerodynamic
considerations. The chord chosen for the first stage, however, may well be
determined by stringent requirements for resistance to Foreign Object Damage
(FOD). This is often the case for aircraft engines, and it may result in a somewhat
lower than optimum aspect ratio.
One further item of information is necessary before the design of the blade
fomls at this radius can be completed. Whereas the blade inlet angle (Xl will be
known from the air inlet angle and chosen incidence (usually taken as zero so that
IX] = (Xl), the blade outlet angle IX2 cannot be determined from the air outlet angle
IX2 nntil the deviation angle () = (X2 - 1X2 has been detennined. Ideally, the mean
direction of the air leaving the cascade would be that of the outlet angle of the
blades, but in practice it is found that there is a deviation which is due to the
206 AXiAL FLOW COMPRESSORS
, ' .
reluctance of the air to· turn the full' 'angle required by, the shape, of
the blades. This will be seenfrom Fig. 5.23. Ali analysis of the relation between
the air aiJ.d blade outlet angles from cascade tests shows that their difference is
mainly dependent on the blade camber and the pitch/chord ratio. It is also
dependent on the shape of the camber-line of the blade section and on the air
outlet angle itself The whole, of this may be summed up in the following
empirical rule for deviation:
0= m(JJ(s/c) (5.29)
where
m = 0. 23 ec
a
)
2
+
a is the distance of the point of maximum camber from the leading edge of the
blade, which feature is illustrated in Fig. 5.23, and 1X2 is in degrees. Frequently a
circular arc camber-line is chosen so that 2alc = 1, thereby simplifYing the for-
mula for m, but its general form as given above embraces all shapes including a
parabolic arc which is sometimes used. (For inlet gnide vanes, which are essen-
tially nozzle vanes giving accelerating flow, the power of sic in equation (5.29) is
taken as unity instead of 0·5 and m is given a constant value of 0·19.)
Construction of blade shape
On the assumption of a circular arc camber-line, the deviation in the current
example will be
[
28.63]
0= 0·23 + 0·1 x 50 J(0·9)(J = 0·273(J
With this information it is now possible to fix the main geometrical parameters of
the rotor blade row of our example. The procedure is as follows.
Since -1X2 and =IXI - 0
(J=IXI - 1X2+ 0
= IX; - 1X2 + 0·273(J
0·727(J = -1X2
= 50·92 - 28·63
(assuming zero incidence, = 1(1)
It follows that (J=30·64° and Ct2 - (J=20·28°. The deviation angle is
8·63°.
The position of the blade chord can be fixed relative to the axial direction by
the stagger angle , given by
,= IX;
2
= 50·92 - (30·64/2)
= 35·60°
BLADE DESIGN 207
Referring to Fig. 5.27, the chord line AB is drawn 0·0276 m long at 35·60° to the
axial direction. The lines AC and BD at angles Ct
1
and 1X2 are then added and a
circular arc constructed tangential to these lines and having AB as chord. This arc
will now be the camber-line of the blade around which an aerofoil section can be
built up.
The of specifYing the base profile is shown in Fig. 5.27, ordinates
being given at definite positions along the camber-line. The RAF 27 profile and
the so-called 'C series' of profiles have been widely used in British practice, and
the NACA series in America. For moderately loaded stages, in which the
velocities are well removed from sonic values, small variations in form are found
to have little effect on the final performance of the compressor. Further details of
base profiles, together with the geometrical relations necessary when parabolic
camber-lines are employed, can be found in Ref (9). For stages operating with
relative Mach numbers in the transonic range, it has been found that double
circular arc blade sections (see Fig. 5.35) offer the best performance.
The method outlined could now be applied to a selected number of points
along the blade length, bearing in mind that having once fixed the pitch at the
mean diameter by the choice of a particular number of blades, the pitch at all
other radii is determined. As the sic ratio is derived from the air angles, the length
of the chord of the blade, at any particular radius will be determined from the
Rotor
c
-.J
-5
4.76- -4.79 C>
c:

/
-4.31
CD
4.30- .5
Centre of camber arc
":':
Q)
3.70- -3.72
.0
E
'"
()
2.91- -3.00
Stator
2.02- -2.15
1.05- -1.20
0.60- -0.68
0
0- --0
Upper Lower
surface surface
X
y
%L %L
FIG. 5.27 Blade design and base profile
208 AXIAL FLOW COMPRESSORS
pitch. By this means a complete picture of the blade can be built up. The blade
stresses can then be accurately assessed.
Now that the reader understands the basis for selecting the pitch/chord ratio, it
is appropriate to check the diffusion factor for a couple of stages. Recalling for
convenience equation (5.7),
V? ACws
D = 1--"-+--
VI 2VI c
the velocity terms are all available from the calculated velocity triangles. Con-
sidering the free vortex design of the third stage, at the tip section Ca = 150 mis,
f31 =60·63°, f32=48·54°, f31 - f32= 12·09° and ACw =87 m/s. From Fig. 5.26 a
suitable value of sic would be about 1·1 and the values of V2 and VI can be readily
calculated from Ca/cos f3 to be 226·5 and 305·8 mls respectively. Hence
226·5 87xl·l 4
D = 1 - 305 8 + 2 305 8 = O· 2
. x .
Referring back to Fig. 5.8, it can be seen that this value is quite satisfactory.
Repeating the calculations at the root results in a slightly higher value of 0·45,
primarily due to the increase in ACw' It is worth noting that the de Haller numbers
for the third stage were close to the limit of O· 72, and the diffusion factors are also
near the point at which losses increase rapidly. At the tip of the first-stage rotor,
the diffusion factor can be calculated to be about 0·23, and this again correlates
well with the light aerodynamic loading of this stage; the transonic relative Mach
number, however, leads to further losses which will be discussed in section 5.10.
The experienced compressor designer will have a good idea of the expected pitch/
chord ratios at the outset of the design process, and the diffusion factor is an
excellent preliminary check on aerodynamic loading.
5.9 Calculation of stage performance
After completion of the stage design, it will now be necessary to check over the
performance, partiCUlarly in regard to the efficiency which for a given work input
will completely govern the final pressure ratio. This efficiency is dependent on the
total drag coefficient for each of the blade rows comprising the stage, and in order
to evaluate these quantities it will be necessary to revert to the loss measurements
in cascade tests. From the measured values of mean loss W, two coefficients can
be obtained. These are the lift and profile drag coefficients CL and CDp, formulae
for which may be obtained as follows.
Referring to the diagram of forces acting on the cascade as shown in Fig. 5.28,
the static pressure rise across the blades is given by
Ap = pz - PI
= (P02 - !pVi) - (POI - !pV?)
CALCULATION OF STAGE PERFORMANCE
209
>f!l-al
FiG. 5.28 Applied and effective forces acting on cascade

: ...... a2
I
I
iVa
I
I
I
The incompressible flow formula is used because the change of density is neg-
ligible. Hence, still using cascade notation for velocities and angles,
Ap = ! (V? - vi) - w
= 1pvi(tan
2
(XI- tan
2
1J(2) - w (5.30)
the axial velocity being assumed the same at inlet and outlet.
The axial force per unit length of each blade is sAp and, from consideration of
momentum changes, the force acting along the cascade per unit length is given by
F = spVa x change in velocity component along cascade
= spV;(tan IX] - tan 1X2)
(5.31)
The coefficients CL and CDp are based on a vector mean velocity Vm defined by
the velocity triangles in Fig. 5.28. Thus
where IXm is given by
tan IJ(m = [!(Va tan IJ(] - Va tan 1X2) + Va tan 1J(2l/Va = 1 (tan IX] + tan 1X
2
)
If D and L are the drag and lift forces along and perpendicular to the direction of
the vector mean velocity, then resolving along the vector mean gives
D = !pV;,cCDp (by defmition of CDp)
= F sin IJ(m - sAp cos IJ(m
Hence from equations (5.30) and (5.31),
! p = sp V;(tan IX] -- tan (X2) sin IJ(m
- !pVis(tan
2
IJ(] - tan
2
!X2) cos IXm + lVS cos IXm
210
AXIAL FLOW COMPRESSORS
Since
.taIi2 IXI - tan
2
1X2 = (tan: IX\ - tan 1(2)(tan IXI + tan: 1(2)
= 2( tan IXI - tan: 1(2) tan: IXm
the first two terms in the expressions for CDp are equal and the equation reduces
to
i.e.
CDp = (;) (!: ) eO;;m)
=
CDp = (;)(!;Vl)G:::
Also, resolving perpendicularly to the vector mean,
L = !pV;cCL (by definition of CL)
= F cos !Xm + sllp sin !Xm
Therefore
!pV;cCL = - tan: !X2) cos IXm
+ ! p V; s(tan:
2
IX\ - tan
2
!X2) sin IXm - WS sin IXm
which on reduction finally gives
C
L
= 2(sjc)(tan: IX\ - tan !X2) cos IXm - CDp tan: rx.,
(5.32)
(5.33)
Using these formulae the values of CL and CDp may be calculated from the data
given by the curves of Fig. 5.25. Since !X'\ is known from the geometry of the
blade row, the following data may be found for any incidence angle i:
IX\ = IX; +i
1X2 = IXI - €*
!Xm = tan:-l[!(tan!Xl + tan: !X2)]
Then by using values of w j ! p Vf read from the curve and the known value of sic
for the cascade, CDp and CL may be calculated from equations (5.32) and (5.33)
and plotted against incidence as in Fig. 5.29.
Because the value of the term CDp tan: !Xm in equation (5.33) is negligibly
small, it is usual to use the more convenient 'theoretical' value of CL given by
(5.34)
in which the effect of profile drag is ignored. Using this formula, curves of CL can
be plotted for nominal (or design) conditions to correspond with the curves of
CALCULATION OF STAGE PERFORMANCE
0.050

o
0.025 GDp

Cl
r.S
1.0 5l
'0

0.5 8
:5
__ __ __ __ -L __
-20 -15 -10 -5 o 5 10
Incidence i degrees
FIG. 5.29 Lift and drag coefficient for cascade of fixed geometrical form
211
deflection given in Fig. 5.26. These curves, which are again plotted against air
outlet angle !X2 for fixed values of pitch/chord ratio sic, are given in Fig. 5.30.
Before these coefficients can be applied to the blade rows of the compressor
stage, two additional factors must be taken into account. These are the additional
drag effects due to the walls of the compressor annulus, and the secondary loss
due to trailing vortices and tip clearance. The flow effects which give rise to these
losses are illustrated in Fig. 5.31. Analysis of compressor performance figures
have shown that the secondary loss is of major importance and that its magnitude
is of the same order as that incurred by the profile drag of the blades. It is greatly
influenced by tip clearance which should consequently be kept as small as
possible-in the region of 1-2 per cent of the blade height. For typical axial
2.0
Air outlet angle U2, degrees
FIG. 5.30 Design lift coefficients
212
AXIAL FLOW COMPRESSORS
(a) Annulus drag
(b) Secondary losses
FIG. 5.31 'flil!"ee-dimensional J!low effects in compre§sor annulus
compressor designs, the following empirical formula for the additional drag
coefficient arising from secondary losses has been derived,
CDS = 0.OI8Ci
(5.35)
where C
L
is the lift coefficient as given by equation (5.34) and the curves of Fig.
5.30.
The loss due to annulus drag is naturally dependent on the relative proportions
of the blade row, its influence increasing as the blades become shorter relative to
their chord length. It has been found convenient to relate the drag coefficient,
resulting from this loss, to the blade row dimensions by the empirical formula,
CDA = O·020(s/h)
(5.36)
where s and h are the pitch and height of the blades respectively. Therefore, on
inclusion of these factors, an overall drag coefficient can be determined which is
given by
(5.37)
The argument used in deriving equation (5.32) for the profile drag coefficient
in the case of a straight cascade will apply equally well to the annular case if CD is
substituted for CDp' Hence for the armular case
CD = ( ~ ) (!;Vl) C ~ ~ :;')
(5.38)
This enables the loss coefficient w / ~ p Vf for the blade row to be detennined. The
theoretical static pressure rise thr01;gh the blade row is found by putting the loss
w equal to zero in equation (5.30), which gives
I1pth = !pV1(tan2 !Xl - tan
2
!X2)
= ! p V1(sec
2
(Xl - sec
2
0(2)
Therefore
CALCULATION OF STAGE PERFORMANCE 213
so that the theoretical pressure rise in terms of the inlet dynamic head and cascade
air angles becomes
I1pth _ 1 COS
2
!Xl
!pVf - - cos2 !X2
The efficiency of the blade row, Yfb, which is defined as the ratio of the actual
pressure rise to the theoretical pressure rise, can then be found from
lib = (l1pth - i1,)/l1pth' Or in non-dimensional terms,
w!tpVf
lib=l- -12
I1pth/'iPVj
(5.39)
When dealing with cascade data, efficiency is evaluated from pressure rises,
whereas compressor stage efficiency is defined in tenus of temperature rises.
Furthermore, the stage efficiency encompasses both a rotor and a stator row. It is
therefore not obvious how the efficiency of a blade row obtained from cascade
tests can be related to the stage efficiency. Initially we will consider this problem
in the context of a stage designed for 50 per cent reaction at the mean diameter. In
such a case, because of the symmetrical blading, Yfb [evaluated from equation
(5.39) for conditions obtaining at the mean diameter] will be virtually the same
for both the rotor and stator blade row. t What we shall now proceed to show is
that this value of '1b can be regarded as being equal to the isentropic efficiency of
the whole stage, Yfs.
If PI and P2 are ..the static pressures at inlet to and outlet from the rotor,
Yfb =P; -PI
P2 -PI
where P; denotes the ideal pressure at outlet corresponding to no losses. If the
stage efficiency Yf s is defined as the ratio of the isentropic static temperature rise
to the actual static temperature rise,
-1= l + ~
P [ '1 I1T ]Y/(Y-I)
PI 2Tl
because I1Ts /2 will be the temperature rise in the rotor for 50 per cent reaction.
Also
2= l+_s
pi [ flT ]Y/(Y-l)
PI 2Tl
Therefore
t Owing to slight differences in ?lade pitch and height between the rotor and stator rows of a stage,
CDA from equation (5.36) may dIffer margmally. But as the following numerical example will show,
C DA IS only a small proportion of CD and this difference will have little effect on '7b.
214 AXIAL FLOW COMPRESSORS
After expanding and neglecting second-order terms, this reduces to
'1b = '1S[ 1 - Y 1 x (1- '1S)J
But /J.Ts is of the order 20 K and Tj about 300 K, so that the second term in the
bracket is negligible and
'Ib = '1s
For cases other than 50 per cent reaction at the mean diameter, an approximate
stage efficiency can be deduced by taking the arithmetic mean of the efficiencies
of the two blade rows, i.e.
rls = 1 ('1b rotor + I'/b stator)
If the degree of reaction is far removed from the 50 per cent condition, then a
more accurate expression for the stage efficiency may be taken as
I'/s = AI'/b rotor + (I - A)I'/b stator
where A is the degree of reaction as given in section 5.5.
Using this theory, we will estimate the peliormance of the third stage of the
compressor used as an example of compressor design, for which the degree of
reaction was 50 per cent at the mean diameter. From the blade design given in the
previous section, and remembering that /31 and /32 are the air inlet and outlet
angles (XI and (X2 respectively in cascade terminology,
tan (Xm =! (tan (XI + tan (X2) = 1(tan 50·92 + tan 28·63) = 0·889
whence
(Xm = 41·63°
From Fig. 5.30, CL at sic = 0·9 and (X2 = 28·63 is equal to 0·875. Hence from
equation (5.33)
CDS = 0·018(0.875)2 = 0·0138
Recalling that s = 0·0248 m and h = 0·0836 m we have from equation (5.36)
C = 0·020 )( 0·0248 = 0.0059
DA 0.0836
From Fig. 5.29, at zero incidence CDP = 0·018; hence the total drag coefficient is
CD = CDP + CDA + CDS = 0·018 + 0·0059 + 0·0138 = 0·0377
Therefore, from equation (5.38)
= C j[(:) cos
3
(Xm] = 0·0377 cos
2
50·92° = 0.0399
!pVl D c cos2 (X] 0.9cos341.630
We also have
/J.Pth = 1 _. cos
2
(X] = 1 _ cos
2
50·92° = 0-4842
! p Vl cos
2
(X2 cos
2
28·63°
CALCULATION OF STAGE PERFORMANCE
Therefore, from equation (5.39)
0·0399
'ib = 1·0 - --= 0·918
0·4842
The efficiency of the stator row will be similar, and hence we may take
1]s = 0·92
215
The static temperature at entry to the stage can be found to be 318·5 K. and with
a stage temperature rise of 24 K, the static pressure ratio of the stage found to
be
[
I'/sI1Ts]Y/(),-jJ [ 0·92 x 24]3-5
Rs= 1+-- = 1+ =1.264
Tj 318.5
'!'Ie have seen how the design point performance of a stage may be
estimated. It will have been noticed that the results are in terms of an isentropic
stage efficiency based on static temperatures and a stage pressure ratio based on
static pressures. To the compressor designer, however, it is the efficiency based on
stagnation temperature rise and the stagnation pressure ratio which are of inter-
est. Making use of the relations Po/p = (To/T)y/(y-I) and To = T + cZ /2c it is of
'b P
course POSSI Ie to transform a static pressure ratio into a stagnation pressure ratio
when the inlet and-outlet velocities are known. But for the common case where
these velocities are equal (i.e. C3 = Cj ) there is no need for this elaboration.
/J.Tos = I1Ts, and it is easy to show as follows that '1s is virtually the same whether
based on static or stagnation temperatures. Then merely by substituting TOl for T
j
in the foregoing equation for Rs the stagnation pressure ratio is obtained directly.
Referring to Fig. 5.32, where the stage inlet and outlet states are denoted by 1
and 3 (static) and 01 and 03 (stagnation), we have
T
T3-T]-X x
1]s---= =1--
T3 - TI T3 - T] I'lTs
yL :--- 03
03'
x l 3
3' /
I
I
I
01 I
s
/
/',Ts
FIG. 5.32
216
AXIAL FLOW COMPRESSORS
And when based on stagnation temperatures,
TiJ3 - TOllY
lis = = ---
T03 - TOI !iTos
With C
3
= Cl> !iTos= !iTs. Furthermore, since the pressure ratio per stage is
small the constant P03 and P3 lines are virtually parallel between 3' and 3 so that
Y "" x. It follows that lis has the same value on either basis. For the stage of our
example, TOI was 333 K, and hence the stagnation pressure ratio is
[
0·92 X 20J3.5
Rs = 1 + = 1·252
333
It will be remembered that it was necessary to make use of two assumed
efficiencies at the start of the design process: a polytropic efficiency for the
compressor as a whole, and a stage efficiency. As a first approximation these were
taken to be equal and a value of 0·90 was assumed. The estimated value of 11s for
the third stage, i.e. 0·92, is in sufficient agreement, bearing in mind the uncer-
tainties in predicting the secondary and annulus losses. If similar agreement were
to be obtained for all the stages, we might conclude that the design had been
conservative and that the compressor should have no difficulty in achieving the
specified performance. To obtain an estimate of the overall efficiency it would be
necessary to repeat the foregoing for all the stages. The product of the stage
pressure ratios would then yield the overall pressure ratio from which the overali
isentropic temperature rise, and hence the overall efficiency, could be calculated.
Before continuing, it may be helpful to summarize the main steps in the design
procedure described in the previous sections. Having made appropriate
assumptions about the efficiency, tip speed, axial velocity, and so on, it was
possible to size the annulus at inlet and outlet ofthe compressor and calculate the
air angles required for each stage at the mean diameter. A choice was then made
of a suitable vortex theory to enable the air angles to be calculated at various radii
from root to tip. Throughout this work it was necessary to ensure that linlitations
on blade stresses, rates of diffusion and Mach mnnber were not exceeded.
Cascade test data were used to determine a blade geometry which would give
these air angles, and also the lift and drag coefficients for a two-dimensional row
of blades of this form. Finally, empirical correction factors were evaluated to
enable these coefficients to be applied to the annular row at the mean diameter so
that the stage efficiency and pressure ratio could be estimated.
5.10 O!J;mpressibility effeds
Over a long period of time, the development of high-performance gas turbines
has led to the use of much higher flow rates per unit frontal area and blade tip
speeds, resulting in higher velocities and Mach numbers in compressors. While
early units had to be designed with subsonic velocities throughout, Mach num·
COMPRESSffiILITY EFFECTS 217
bers exceeding unity are.now found in the compressors of industrial gas turbines
Mach num?ers as high as 1·5 are used in the design of fans for turbofans of
ratIO. It is not wit?in the scope of this book to GOver transonic design
m detall, and the treatment confined to a brief introduction and provision
some. key. references; agam, It should be realized that much of the relevant
mfor:natlOn IS of a proprietary nature and is not available in the open literature.
cascade testing is required to provide experimental data on
compresslbillty effects, and in particular to determine the values of the Mach
numbers, con-esponding to entry velocities relative to the blades, which bring
about cascade p.erformance. The first high velocity of interest is that
correspond:ug to what IS called the 'critical' Mach number Me; at entry velocities
lower than .his, the performance of the cascade differs very little from that at low
Above this velocity, the losses begin to show a marked increase until a
pomt IS reached where the losses completely cancel the pressure rise and the
blade ceases to be any use as a diffuser. The corresponding Mach nunlber is then
referred to as the 'maximum' al M F . . . . v, ue m. or a typical subsomc compressor
cascade at zero the values of these Mach numbers are in the region of
0·7 and 0.8.5 respe.ctively. A typical variation of fluid deflection and pressure loss
for subsomc bladmg was shown in Fig. 5.25, representing flow at low Mach
numbers. AB the Mach increases, two important effects talce place: first,
overall level. of losses lllcrease substantially, and second, the range of
mClden.ce for losses are acceptable is drastically reduced. This means that
off-deSIgn performance of the .compressor may be seriously affected. Figure 5.33
shows test results for a subsomc blade section over a range of Mach number from
0·5 to 0·8; the usable range of incidence at M 0·8 can be seen to be very narrow
0.35
0.30 - Mach number
0.75 0.8
0.25
C
'"
." 0.20
;;:
'"
0
c.>
(I)
(I)
0.15
0
....J
0.10
0.05
0 I
-20 -15 -10 -5 0 15
Incidence, degrees
FIG. 5.33 Effect of Mach !lumber Oil losses
218 AXIAL FLOW COMPRESSORS
and clearly compressor blading ofthis type could not be.used at Mach numbers
approaching or exceeding unity. It should be pointed out that for compressible
flow the denominator in the loss coefficient is (POl - Pl) rather than pVl/2.
Compressibility effects will be most important at the front of the compressor
where the inlet temperature, and hence the acoustic velocity, are lowest. The
Mach number corresponding to the velocity relative to the tip of the rotor is the
highest encountered and is important both from the viewpoint of shock losses and
noise. The stator Mach number is generally highest at the hub radius because of
the increased whirl velocity normally required to give constant work input at all
radii in the rotor. The rotor and stator Mach numbers corresponding to the first
stage of the compressor designed in section 5.7 are shown in Fig. 5.34, which
shows that the flow relative to the rotor is supersonic over a considerable length of
the blade. The stator Mach numbers can be seen to be significantly lower, this
being due to the fact that they are unaffected by the blade speed.
Analysis of a large amount of compressor tests by NACA [Refs (10, 11)]
showed that losses for subsonic conditions correlated well on the basis of
diffusion factor, as shown in Fig. 5.8, but were significantly higher for transonic
conditions. It was deduced that this increase in loss must be due to shock losses,
but it was also found that the spacing of the blades had a considerable effect; a
reduction in solidity (i.e. an increase in pitch/chord ratio) caused a rapid increase
in loss. A simple method for predicting the loss was developed and can be
explained with reference to Fig. 5.3 5.
A pair of double circular arc (DCA) blades are shown, with a supersonic
velocity entering in a direction aligned with the leading edge. The supersonic
expansion along the uncovered portion of the suction side can be analysed by
means of the Prandtl-Meyer relations discussed in Appendix A.S, and the Mach
number will increase as the flow progresses along the suction surface. A shock
structure is assumed in which the shock stands near the entrance to the blade
passage, striking the suction surface at B, extending in front of the blade at C and
1.2
:;; 1.0
Q;
.c
E
::l
c:
J::
<)
0.8
'" :2
C
E
0.6 ill
0.4
Root Mean
Radius
R o t o r ~
Stator
Tip
FIG. 5.34 Variation of entry Mach numbers
COMPRESSIBILITY EFFECTS
c
Mid-channel
stream line
/ LIe
/ / ~
/ //
. / A
/,
FIG_ 5.35 Shock loss model
219
then bending back similar to a bow wave. It is then assumed that the loss across
the shock can be approximated by the normal shock loss taken for the average of
the Mach numbers at A and B. At A the Mach number is taken to be the inlet
relative Mach number. The value at B can be calculated from the inlet Mach
number and the angle of supersonic turning to the point B, where the flow is
tangential to the surface, using the Prandtl-Meyer relations. The location of the
point B is taken to be at the intersection of the suction surface and a line drawn
normal to the mean passage camber-line and through the leading edge ofthe next
blade. The requirement, then, is to establish the supersonic turning from inlet to
point B and this clearly depends on the blade spacing; moving the blades further
apart (reducing solidity) results in point B moving back, increasing both the
turning and Mach number, resulting in an increased shock loss. Figure 5.36
shows the relationship between inlet Mach number, supersonic turning and shock
loss. If we consider the tip of our first-stage rotor, the relative Mach number was
1·165 and the rotor air angles were 67·11° at inlet and 62·69° at exit. The total
turning was therefore 4.42°, and the supersonic turning to point B would depend
on the blade spacing; making a reasonable assumption of 2°, from Fig. 5.36 it is
found that MB = 1·22 and w=0·015. The additional loss due to shocks can be
significant and would increase rapidly with increased supersonic turning; this is
another reason for the designer to select a slightly reduced temperature lise in the
first stage.
The fans of high bypass turbofans require a large diameter, and even though
they run at a lower rotational speed than a conventional jet engine, the tip speeds
are significantly higher; typical Mach numbers at the tip would be of the order of
1·4-1·6 and DCA blading is not satisfactory. Blade sections have been specially
developed to 'meet this high Mach number requirement and are not based on
aerofoil sections. To keep shock losses to an acceptable level, it is necessary to
provide supersonic diffusion to a Mach number of about 1·2 prior to the normal
shock at entry to the blade passage. It is shown in Appendix A.2 that supersonic
diffusion requires a decrease in flow area. This may be accomplished in two ways,
either by decreasing the annulus area or by making the suction side slightly
220
10
~
Q)
c,
c
'" C)j
c
"E
,;;!
"
.1:
0
~
Q)
a.
::J
C1J
E
Ql
·13
25
20
15
10
~ 0.2
o
" <n
~
-'" 0.1
"
o
.c
Ul
FIG. 5.36 Shock loss coefficient
Ma
Ma
AXIAL FLOW COMPRESSORS
1.4 1.5
1.4
1.3
1.2 MA
1.1
1.0
concave. Both methods may be used together. A typical blade profile is shown in
Fig. 5.37, where it can be seen that the amount of turning is very small. Until
recently it was thought that shock losses explained the decrease in efficiency with
Mach number which has been observed, but it is now realized that the shocks
only account for part of the additional losses. There are also significant losses
resulting from shock-boundary layer interaction and the net effect appears to be a
magnification of the viscous losses. Kerrebrock [Ref. (12)1 gives a comprehensive
review of the aerodynamic problems of transonic compressors and fans.
~ Inlet relative velocity
L7
Direction of rotation
FIG. 5.37 Blade section for slipersonic flow
COMPRESSIBILITY EFFECTS 221
Fan blades tend to be long and flexible, and most require part-span dampers to
prevent excessive vibration and torsional movement. These are located in a region
of high Mach number and lead to considerable local decreases in efficiency.
Manufacturers can now design fan blades witllOut dampers, giving a considerable
improvement in aerodynamic perfonnance; they normally have significantly
wider chords and tills helps to meet bird strike certification requirements. Figure
5.38(a) shows the effect of a seagull shike on a large turbofan during take-off and
FIG. 5.3!J(a)
FIG. 5.38(b)
222 AXIAL FLOW COMPRESSORS
gives some idea of the impact forces involved. The blade shown has two pan:-
span dampers, but later versions have only one. Figure 5.38(b) shows the shape of
the tip section and the use of a concave portion on the suction side can be
observed.
5.11 Off-design performance
Attention so far has focused on the aerothennodynamic design of axial flow
compressors to meet a particular design point of mass flow, pressure ratio and
efficiency. It must be realized at the outset, however, that any compressor will be
required to operate at conditions far removed from the design point including
engine starting, idling, reduced power, maximum power, acceleration and de-
celeration. Thus it is clear that the compressor must be capable of satisfactory
operation over a wide range of rotational speeds and inlet conditions. With the
compressor blading and annulus dimensions chosen to satisfy the design point
condition, it is obvious that these will not be correct for conditions far removed
from design. Previous sections showed that compressor blading had a limited
range of incidence before losses became unacceptably high, resulting in low
compressor efficiency. What is even more important, however, is that the prob-
lems of blade stalling may lead to surging of the whole compressor, preventing
operation of the engine at particular conditions; this may lead to severe engine
damage or, in the case of aircraft, a critical safety hazard.
An overall picture of compressor off-design perfonnance can be built up from
consideration of the behaviour of individual stages and the interaction between a
series of stages. Recalling equation (5.5), the temperature rise in a stage is given
by
1Wos = UC
a
(tan /31 - tan /32)
cp
which is expressed in tenns of the rotor air angles. This can be recast to give
U
I1Tos = - [U - Ca(tan 0(1 + tan /32)]
cp
The angle 0(1 is the outlet air angle from the preceding stator, /32 is the rotor outlet
air angle, and these can be considered essentially constant, being detennined by
the blading geometry; {3], on the other hand, will vary widely as Ca and U change.
Dividing the previous equation throughout by U
2
and rearranging we get
cpl1Tos Ca
--= 1 - -(tan 0(1 + tan /32)
U2 U
The tenn Ca/U is !mown as the flow coefficient (c/» and (cpI1Tos/U
2
) as the
temperature coefficient (t{!). With the stage operating at the design value of c/> the
incidence will be at its design value and a high efficiency will be achieved. With
the assumption that 0(1 and /32 are constant,
OFF-DESIGN PERFORMANCE 223
where k = tan 0( 1 + tan /32. This relationship is shown by the dotted straight line in
Fig. 5.39. It is obvious that the temperature coefficient increases as the flow
coefficient decreases and reaches a value of 1 when c/> = O. The temperature
coefficient can be interpreted from the shape of the velocity diagram. Recalling
that
it immediately follows that
cpl1Tos I1Cw
U2 U
Thus, ift{! = 1, ACw = U and from earlier examination of velocity diagrams it will
be recognized that this results in excessive diffusion in the blade passage and
efficiency will decrease. For satisfactory operation, I1Cw/ U, and hence Ij!, should
be around 0·3-0·4.
The pressure rise across the stage, I1pos, is detennined by the temperature rise
and the isentropic efficiency of the stage.
POI + I1pos = 1 + /).pos = (1 + rtsATos)Y/(l'-ll
POl POI TO!
Expanding by means of the binomial theorem, assuming that I1Tos« To], we
have
Apos Y I1Tos
--=--rts--
POI Y - 1 TOJ
......... Actual V'
N ............ /
:::::, Stalled ........,.(
t-9 I ............
<l
0"
" ~
C
'"
.(3
iE
OJ
0
U
OJ
:;
ro
ill
Cl.
E
~
1J s
¢s ¢d
Flow coefficient ¢ = CalU
FIG. 5.39 Stage characteristic
1)
224
AXIAL FLOW COMPRESSORS
Making use of the equation of state and the relationship(y - l)/y =R/cp, it
follows that· . ...
. !!.pOS AT.
--='1sCpLl. OS
POI
and hence
IJ.pos cpIJ.Tos
--='1s--
POI rJ2 rJ2
The term IJ.pOS/ POI U
2
is referred to as the pressure coefficient, which can be
seen to be the product of the stage efficiency and temperature coefficient. Thus
the overall performance of a stage can be expressed in terms of flow coefficient,
temperature coefficient and either a pressure coefficient or the stage efficiency.
The resulting form of the stage characteristic is shown in Fig. 5.39; in practice, IXI
and P2 will not remain constant due to increased deviation as conditions change
from the design point. In regions of blade stalling, both at positive and negative
incidence, there will be a considerable departure from the linear relationship
giving the shape shown. Choking will occur at a high value of flow coefficient,
leading to a very rapid drop in efficiency and placing an upper limit on the flow
which can be passed at a given blade speed. Stage characteristics may be obtained
from single-stage tests, by analysis of interstage data on a complete compressor
or by prediction using cascade data. In practice, not all the constant speed lines
would collapse into a single curve as shown, but for the following brief discus-
sion a single line characteristic will be assumed.
The 1/1-4> curve in Fig. 5.39 is drawn for the case where the efficiency is a
maximum at the design flow coefficient, 4>d. Moving away from 4>d results in a
change in incidence and increased losses. Reducing 4> results in increased
positive incidence and stall at 4>8; increasing 4> eventually results in choking of the
stage and a severe drop in efficiency. It is essential that all individual stages of a
compressor operate in the region of high efficiency without encountering either
stall or choke at normal operating conditions; at conditions far removed from
design it may not be possible to achieve this without remedial action involving
changes in compressor geometry. The difficulties involved in achieving correct
matching of the stages can be understood by considering the operation of several
identical stages in series. This procedure is known as 'stage stacking' and is an
invaluable tool to the aerodynamicist who is concerned with the overall
performance of a compressor and determining the reasons for sub-standard
performance; stage stacking can be used to produce overall compressor
characteristics, showing the regions in which individual stages stall. In this
elementary treatment we will only touch on the use of this procedure. An
idealized stage characteristic is shown in Fig. 5.40 and we will consider that at the
design point all stages operate at 4>d. If the compressor should be SUbjected to a
decrease in mass flow, this will result in a reduction in the flow coefficient
entering the first stage to 4>10 which will result in an increase in pressure ratio
causing the density at entry to the second stage to be increased. The axial velocity
AXIAL COMPRESSOR CHARACTERISTICS 225
¢n ¢z ¢1 ¢d
Flow coefficient, ¢
FIG. 5.40 Effect of reduction in rfJ at inlet
at entry to the second stage is determined by the continuity equation and both
effects combine to give a further decrease in flow coefficient for that stage to 4>2.
This effect is propagated through the compressor and eventually some stage will
stall at 4>n. Increasing the flow coefficient has the opposite effect and will drive
some stage into choke.
5.12 Axial compressor characteristics
The characteristic curves of an axial compressor take a form similar to those of
the centrifugal type, being plotted on the same non-dimensional basis, i.e. pres-
sure ratio P02/POI and isentropic efficiency '1c against the non-dimensional mass
flow m.,J"fo; / POI for fixed values of the non-dimensional speed N / .,J"fo;.
A typical set of such curves is shown in Fig. 5.41 and in comparison with Fig.
4.10 it will be observed that the characteristics for fixed values of N /.,J"fo; cover a
much narrower range of mass flow than in the case of the centrifugal compressor.
At high rotational speeds the constant speed lines become very steep and
ultimately may be vertical. The same limitatious occur at either end of the
N /.,J"fo; lines due to surging and choking. The surge points, however, are
normally reached before the curves reach a maximum value and, because of this,
the design operating point, which is always near the peak of the characteristic, is
also very near the surge line. Consequently the range of stable operation of axial
compressors is narrow and gas turbine plant incorporating this type of
compressor call for great care in matching the individual components if
instability is to be avoided at operating conditions removed from the design point.
Chapter 8 deals with this aspect in detail. The mechanism of surging in axial
compressors is complex and is still not yet fully understood. It is often difficult to
distinguish between surging and stalling and one phenomenon may easily lead to
226
AXIAL FLOW COMPRESSORS
I
i/\/
, i
Surge line /\
i l \ 1.0
// 09
/:\ '\ 0 9 d"'9" .""",
_ ....... 1
0
.
6
07 I I

° 0.2 0.4 0.6 0,8 1.0 1.2
4
m,fTo;/P01 (relative to design value)
'"1
80
;f.
if
c
60
'" '5

0,5 0.6 1.0
Nlv T01 (relative to design value)
;;:
'" ()
'5.
40
2
C
'" !!!.
20 -
0 0.2 0,6 0.8
m,fTo;/P01 (relative to design value)
FIG. 5.41 Axial compressor characteristics
1,0 1.2
the other. The phenomenon of rotating stall referred to in section 4.6 may also be
encountered: it can lead to loss of performance and severe blade vibration without
actually causing surge. A comprehensive review of compressor stall phenomena
is given by Greitzer in Ref. (13).
The complete compressor characteristic can be obtained only if the
compressor is driven by an external power source; it will be shown in Chapter 8
AXIAL COMPRESSOR CHARACTERISTICS 227
that the running range. of the compressor is severely restricted when It IS
combined with a combustion chamber and turbine. The driving unit must be
capable of variable speed operation, with the speed continuously variable and
closely controlled. Compressor test rigs in the past have been driven by electric
. motors, steam turbines or gas turbines. One of the major problems of testing
compressors is the very large power requirement. As an example, at take-off
condition the compressor of an Olympus 593 in Concorde absorbs about 75 Mw,
with approximately 25 MW going to the LP compressor and 50 MW to the HP
compressor. A typical fan stage from a large turbofan will compress about
700 legis through a pressure ratio of about 1·5, requiring about 30 MW. Two
methods of reducing the total power requirement are to throttle the intalee to the
compressor or to test a scaled-down model. Throttling the intake reduces the
density and hence the mass flow, thereby reducing the power input; a major
problem that occurs, however, is the large drop in Reynolds number with reduced
density. The Reynolds number may well be reduced by a factor of 3 or 4, which
has the effect of increasing the relative magnitude of the viscous losses giving
unrealistically low values of compressor efficiency. Model testing of single-stage
fans is more common than throttled intake testing. Another problem with
compressor testing is that the operating enviromnent of the compressor in the test
rig is not the same as that encountered in the engine, due to causes such as
changes in tip clearance as a result of different casing temperatures and shaft
expansions. Some manufacturers prefer to carry out compressor tests on a gas
generator rig, using the actual combustion system and turbine, but tms requires
the use of a variable exhaust nozzle to cover a reasonable range of the compressor
characteristic; the compressor, however, is exposed to realistic engine operating
conditions. Compressor testing is both complex and expensive, but is essential for
the development of a high-performance engine. A typical compressor develop-
ment programme for an industrial gas turbine is described in Ref. (15).
Some deductions from the compressor characteristics
The axial flow compressor consists of a series of stages, each of which has its
own characteristic: stage characteristics are similar to the overall characteristic but
have much lower pressure ratios. The mass· flow through the compressor is
limited by choking in the various stages and under some conditions tms will
occur in the early stages and under others in the rear stages.
We have noted that if an axial flow compressor is designed for a constant axial
velocity through all stages, the annulus area must progressively decrease as the
flow proceeds through the compressor, because of the increasing density. The
annulus area required for each stage will be determined for the design condition
and clearly at any other operating condition the fixed area will result in a variation
of axial velocity through the compressor. When the compressor is run at a speed
lower than design, the temperature rise and pressure ratio will be lower than the
design value. The effect of the reduction in density will be to increase the axial
velocity in the rear stages, where choking will eventually occur and limit the mass
228
AXIAL FLOW COMPRESSORS
flow. Thus at low speeds the mass flow will be deternlinedbychoking of the rear
stages, as indicated in Fig. 5.42. As the speed is increased, the density in the
stages is mcreased to the design value and the rear stages of the
pass all the flow provided by the early stages. Eventually, however, cholang wIll
occur at the inlet; the vertical constant speed line in Fig. 5.42 is due to choking at
the inlet of the compressor.
When the compressor is at its design condition, all stages are
operating at the correct value of Cal U and hence at the correct .incidence. If .we
consider moving from design point A to point B on the surge Ime at the desIgn
speed it can be seen that the density at the compressor exit will be increased due
to the increase in delivery pressure, but the mass flow is slightly reduced. Both of
these effects will reduce the axial velocity in the last stage and hence increase the
incidence as shown by the velocity triangle (a) in Fig. 5.42. A relatively small
increase in incidence will cause the rotor blade to stall, and it is thought that surge
at high speeds is caused by stalling of the last stage.
When speed is reduced from A to C, the mass flow generally falls off more
rapidly than the speed, and the effect is to decrease the axial velocity at inlet and
cause the incidence on the first-stage blade to increase as shown in Fig. 5.42(b).
The axial velocity in the latter stages, however, is increased because of the lower
pressure and density, so causing the incidence to decrease as shown. Thus at low
speeds, surging is probably due to stalling of the first stage. It is possible for axial
compressors to operate with several of the early stages stalled, and this is thought
to account for the 'kink' in the surge line which is often encountered with high-
performance compressors (see Fig. 5.42). A detailed discussion of the relation
between stage stalling and the surge line using stage stacking teclmiques is given
in Ref. (14).
P02
POi
Last stage stalling ./ C k' t' I t
\ 'r"'- ho Ing a In e
/B / tJ'
. Y c CaA
./ aB
First stage stalling /./\. Low rye due A --; B
\ ./ to stalling at
\
./ -ve incidence

\ ..../ \
X controlled by
Several front '" choking in rear stages
First stage
stages stalled
mnmlpOi (b)A--;C
FIG. 5.42 Phenomella at off-design operatioll
AXIAL COMPRESSOR CHARACTERISTICS 229
At conditions far removed from surge the density will be much lower than
required and the reSUlting high axial velocities will give a large decrease in
incidence, which will eventually result in stalling at negative incidences. The
efficiency will be very low in these regions.
The twin-spool compressor
It will be evident that as the design pressure ratio is increased, the difference in
density between design and off-design conditions will be increased, and the
probability of blades stalling due to incorrect axial velocities will be much higher.
The effects of increased axial velocity towards the rear of the compressor can be
alleviated by means of blow-off, where air is discharged from the compressor at
some intermediate stage through a valve to reduce the mass flow through the later
stages. Blow-off is wasteful, but is sometimes necessary to prevent the engine
running line intersecting the surge line; this will be discussed in Chapter 8. A
more satisfactory solution is to use a twin-spool compressor.
We have seen that reduction of compressor speed from the design value will
cause an increase of incidence in the first stage and a decrease of incidence in the
last stage; clearly the effect will increase with pressure ratio. The incidence could
be maintained at the design value by increasing the speed of the last stage and
decreasing the speed of the first stage, as indicated in Fig. 5.43. These conflicting
requirements can be met by splitting the compressor into two (or more) sections,
each being driven by a separate turbine, as shown in Fig. 1.7 for example. In the
common twin-spool configuration the LP compressor is driven by the LP turbine
and the HP compressor by the HP turbine. The speeds of the two spools are
mechanically independent but a strong aerodynamic coupling exists which has
the desired effect on the relative speeds when the gas turbine is operating off the
design point. This will be discussed in Chapter 9.
The variable geomeuJl compressor
An alternative approach to satisfying the off-design performance of high-pressure
ratio compressors is to use several rows of variable stators at the front of the
compressor, permitting pressure ratios of over 16: 1 to be achieved in a single
FIG. 5.43
230
AXIAL FLOW COMPRESSORS
spool. If the stators are rotated away from the axial direction, increasing IX) in Fig.
S.4, the effect is to decrease the axial velocity and mass flow for a given rotational
speed. This delays stalling of the first few stages and choking in the last stages, at
low rotational speeds. The earlier discussion on stage characteristics showed that
ideally
1ft = 1 - k¢
where k= tan IX) + tan /32' Using variable stators, it is possible to increase IX) with
fJ2 remaining constant. The effect is to decrease the temperature coefficient for a
given flow coefficient; the pressure coefficient will also be reduced because ofthe
reduction in temperatnre coefficient. The stage characteristic will be shifted to the
left as shown in Fig. 5.44 because of the reduction in mass flow. The stage
stacking techniques mentioned in section 5.11 can be used to predict the effects
of incorporating variable geometry on the overall compressor characteristics. One
of the major benefrts is an improvement in the surge margin at low engine speeds,
which is particularly important during starting and operation at idle conditions:
these topics will be discussed further in ChapIer 8.
The compressor of the Ruston Tornado, shown in Fig. 1.13 and described in
Ref. (15), uses variable IGVs followed by four rows of variable stators to achieve
a pressure ratio of 12: 1. The design speed of the compressor is 11085 rev/min
and at speeds below 10000 rev/min the stators start to close progressively until
they are in the fully closed position for all speeds below 8000 rev/min: the IGVs
are rotated through 35° and the stators through 32°, 25°, 25° and 10° from the
first to the fourth stage. While the effect of variable stators can be predicted with
reasonable accuracy, it is necessary to optimize the settings of each row by rig-
testing the compressor.
Ideal 1f! ( ~ 1 - k¢)
Flow coefficient ¢
FIG. 5.44 Effect of variable stators
CLOSURE 231
5.13 Closure
It must be emphasized that the foregoing treatment of axial compressor theory
and design has been kept at an elementary level to serve as an introduction to
what has now become an extremely complex field of study. Since the early
pioneering work ofNGTE and NACA in establishing rational design techniques
based on empirical data, much effort has gone into the development of sophisti-
cated methods making use of digital computers. While digital computers have
been used in compressor development for many years, the enormous improve-
ments in computing speed and storage capacity in recent years have had a major
impact on turbomachinery design; this effort is truly international in scope. Two
of these methods are refened to as 'streamline curvature' and 'matrix through
flow' techniques. These attempt to determine the flow pattern in the so-called
'meridional plane' which is the plane containing the axial and radial directions.
This is in contrast to earlier methods which were essentially concerned with flow
patterns in the peripheral-axial planes at various positions along the blade span.
For a summary of these techniques the reader is referred to Ref. (16) which
compares them from the user's point of view. A third approach involving ad-
vanced numerical analysis is the so-called 'time marching' method, in which an
initial estimate of the flow pattern is made and unsteady aerodynamic theory is
used to predict succeeding solutions until equilibrium is finally achieved, Ref.
(17). Reviews of computational methods and their application are given in Refs
(18) and (19). Modem methods of numerical analysis have resulted in develop-
ments such as 'controlled diffusion' and 'end bend' blading; the first of these
makes use of supercriticai airfoil sections developed from high subsonic wing
sections and the second alters the shape of the blades at both inner and outer
annulus walls to allow for the decreased axial velocity in the boundary layer.
Computational methods will play an ever-increasing role in compressor design in
the future, but it is unlilcely that empiricism or experimental tests will ever be
completely displaced.
Finally, it should be recognized that the very limited number of references
supplied are intended to be central to the foundation of compressor design. The
literature is expanding very rapidly and those actively involved in compressor
design have to follow this closely, maldng it all too easy to forget the source of
this technology. The books by Horlock [Ref. (5)], updated in 1982, and Gostelow
[Ref. (8)] both contain extensive bibliographies.
NOMENCLATURE
For velocity triangle notation (u, C, V, IX, fJ) see Fig. 5.4
For cascade notation (IX', 15, 8, C e, i, s, c) see Fig. 5.23
CD overall drag coefficient
CDA alIDulus drag coefficient
C Dp profile drag coefficient
232
n
N
r
secondary loss coefficient
lift coefficient
diffusion factor
blade height, specific enthalpy
number of blades
number of stages
radius
AXIAL FLOW COMPRESSORS
R
rlr m as well as pressure ratio and gas constant
w
hlc
sic
),
A
¢
i/J
stagnation pressure loss
aspect ratio
pitch/chord ratio
work-done factor
degree of reaction (/:;.Trotorl/:;.Tstage)
flow coefficient (Cal u)
temperature coefficient (cp/:;.Tosl U
2
)
Suffixes
a, w axial, whirl, component
b blade row
m
s
r
mean, vector mean
stage
root radius, radial component
tip radius
Combustion systems
The design of a gas turbine combustion system is a complex process involving
fluid dynamics, combustion and mechanical design. For many years the
combustion system was much less amenable to theoretical treatment than other
components of the gas turbine, and any development program required a
considerable amount of trial and en"Of. With the very high cycle temperatures of
modem gas turbines, mechanical design remains difficult and a mechanical
development program is inevitable. The rapidly increasing use of Computational
Fluid Dynamics (CFD) in recent years has had a major impact on the design
process, greatly increasing the understanding of the complex flow and so
reducing the amount of trial and error required. CFD methods are beyond the
scope of this introductory text, but their importance should be recognized.
The main purpose of this chapter is to show how the problem of design is
basically one of reaching the best compromise between a number of conflicting
requirements, which will vary widely with different applications. Aircraft and
ground-based gas turbine combustion systems differ in some respects and are
similar in others. The most common fuels for gas turbines are liquid petroleum
distillates and natural gas, and attention will be focussed on combustion systems
suitable for these fuels. In the mid 1990s there is considerable interest in the
development of systems to burn gas produced from coal. Gasification requires
large amounts of steam so that it is a 10ng-telID option for modifying combined
cycle plant currently buming natural gas.
In the early days of gas turbine: design, major goals included high combustion
efficiency and the reduction of visible smoke, b(jth of which were largely solved
by the early 1970s. A much more demanding problem has been the reduction of
oxides of nitrogen, and on-going research programs are essential to meet the ever
more stringent pollution limits while maintaining existing levels of reliability and
keeping costs affordable. The chapter will close with a brief treatment of methods
currently under development for low-emissions combustion systems.
234 COMBUSTION SYSTEMS
6.1 Operational requirements
Chapter 2 discussed the thermodynamic design of gas turbines, emphasizing the
importance of high cycle temperature and high component efficiencies. For com-
bustion systems the latter implies the need for high combustion efficiency and
low pressure loss, typical assumed for cycle calculations being 99 per cent
and 2-8 per cent of the compressor delivery pressure. Although the effect of these
losses on cycle efficiency and specific output is not so pronounced as that of
inefficiencies in the turbomachinery, the combustor is a critical component be-
cause it must operate reliably at extreme temperatures, provide a suitable tem-
perature distribution at entry to the turbine (to be discussed in section 7.3), and
create the minimum amount of pollutants over a long operating life.
Aircraft (and ships) must carry the fuel required for their missions, and this
has resulted in the universal use of liquid fuels. Proposals have been made for
hydrogen powered aircraft, and a few experimental flights have made
hydrogen fuelled engines, but it is very unlikely that hydrogen wIll be WIdely
used. Aircraft gas turbines face the problems associated with operating over a
wide range of inlet pressure and temperature within the flight envelope of Mach
number and altitude. A typical subsonic airliner will operate at a cruise altitude of
11 000 m, where the ambient pressure and temperature for I.S.A. conditions are
0·2270 bar and 216·8 K, compared with the sea level values of 1·013 bar and
288·15 K. Thus the combustor has to operate with a greatly reduced air density
and mass flow at altitude, while using approximately the same fueIJair ratio as at
sea level to maintain an appropriate value of turbine inlet temperature.
Atmospheric conditions will change quite rapidly during climb and descent,
and the combustor must deal with a continuously varying fuel flow Without
allowing the engine to flame-out or exceed temperature limits; high performance
fighters, for example, may climb from sea level to 11 000 m in less than 2
minutes. In the event of a flame-out in flight, the combustor must be capable of
relighting over a wide range of flight conditions. Supersonic aircraft operate
under very different conditions; at high Mach nmnber and altitude, because of the
large ram pressure rise, the compressor inlet pressure may be almost
the Sea Level Static value but the inlet temperature is significantly higher. It IS
clear that a supersonic aircraft will operate over a much wider range of conditions
than its subsonic counterpart, and emissions of oxides of nitrogen at high
altitudes proposed for future supersonic transports could be a serious enough
problem to terminate proposed ventures.
Stationary land-based gas turbines have a wider choice of fuel. It should be
recognized, however, that altitude effects can still be significant: some .engines a:e
operated at altitudes of nearly 4000 m in regions of South Amenca, ill
Western Canada many engines operate at altitudes of lOOO m. Natural gas IS the
preferred fuel for applications such as pipeline compression, utility
cogeneration. If natural gas is not available, the most widely used fuel IS a hqmd
distillate. A relatively small number of gas turbines burn residual fuels, but such
fuels generally require pre-treatment which is costly. Units for continuous
TYPES OF COMBUSTION SYSTEM 235
operation would normally use natural gas, but peak load applications may use
liquid fuels requiring the storage of substantial quantities. Gas turbines may be
designed for dual-fuel operation, with normal operation on natural gas with an
option to switch to liquid fuels for short periods. The combustion system may
have to be designed to bum one or more fuels, sometimes simultaneously, and
also to accommodate water or steam injection for emissions control.
There are velY few commercial applications of gas turbines in ships, and
nearly all marine gas turbines are aircraft derivatives operating in warships. They
universally use marine diesel as fuel. Propulsion plant for warships are also
becoming liable for emissions produced in harbour, and are subject to similar
requirements as land based units.
6.2 Types of combustion system
Combustion in the normal, open cycle, gas turbine is a continuous process in
which fuel is burned in the air supplied by the compressor; an electric spark is
required only for initiating the combustion process, and thereafter the flame must
be self-sustaining. The designer has considerable latitude in choosing a combus-
tor configuration and the different requirements of aircraft and ground-based units
with respect to weight, volume and frontal area can result in widely different
solutions. In recent years the effect of stringent restrictions on emissions of oxides
of nitrogen (NOx}"has had a major impact on combustion design, both for in-
dustrial and aircraft applications.
The earliest aircraft engines made use of can (or tubular) combustors, as
shown in Fig. 6.1, in which the air leaving the compressor is split into a number
of separate streams, each supplying a separate chamber. These chambers are
spaced around the shaft connecting the compressor and turbine, each chamber
having its own fuel jet fed from a common supply line. This arrangement was
well suited to engines with centrifugal compressors, where the flow was divided
into separate streams in the diffuser. The Rolls-Royce Dart, shown in Fig. 1.10, is
an example. A major advantage of can type combustors was that development
could be carried out on a single can using only a fraction of the overall airflow
and fuel flow. In the aircraft application, however, the can type of combustor is
undesirable in tenns of weight, volume and frontal area and is no longer used in
current designs. Small gas turbines, such as auxiliary power units (APUs) and
those proposed for vehicles, have often been designed with a single combustion
can.
Separate combustion cans are still widely used in industrial engines, but recent
designs make use of a cannular (or tuba-annular) system, where individual flame
tubes are unifonnly spaced around an annular casing. The Ruston Tornado,
shown in Fig. 1.13, uses this system; the General Electric and Westinghouse
families of industrial gas turbines also use this arrangement. Fignre 1.13 shows
the reverse flow nature of the airflow after leaving the diffuser downstream of the
axial compressor; the use of a reverse flow arrangement allows a significant
236
Primary
air scoop
Primary fuel manifold
Main fuel manifold
COMBUSTION SYSTEMS
Combustion chamber
Interconnector
FIG. 6.1 Can type combustor [courtesy Rolls-Roycel
reduction in the overall length of the compressor-turbine shaft and also permits
easy access to the fuel nozzles and combustion cans for maintenance.
The ideal configuration in terms of compact dimensions is the annular
combustor, in which maximum use is made of the space available within a
specified diameter; this should reduce the pressure loss and result in an engine of
minimum diameter. Annular combustors presented some disadvantages, which
led to the development of cannular combustors initially. Firstly, although a large
number of fuel jets can be employed, it is more difficult to obtain an even fuel/air
distribution and an even outlet temperature distribution. Secondly, the annular
chamber is inevitably weaker structurally and it is difficult to avoid buckling of
the hot flame tube walls. Thirdly, most of the development work must be carried
out on the complete chamber, requiring a test facility capable of supplying the full
engine air mass flow, compared with the testing of a single can in the multi-
chamber layout. These problems were vigorously attacked and annular
combustors are universally used in modem aircraft engines. The Olympus 593
(Fig. 1.9), PT-6 (Fig. 1.11), IT-15D (Fig. 1.12(a)) and V-2500 (Fig. 1.12(b)) all
use annular combustors. The most recent designs by ABB and Siemens have
introduced annular combustors in units of over 150 MW
Large industrial gas turbines, where the space required by the combustion
system is less critical, have used one or two large cylindrical combustion
SOME IMPORTANT FACTORS AFFECTING COMBUSTOR DESIGN 237
chambers;· these were mounted vertically and were often referred to as silo type
combustors because of their size and physical resemblance to silos. ABB designs
used a single combustor, while Siemens used two; Fig. 1.14 shows a typical
Siemens arrangement. These large combustors allowed lower fluid velocities and
hence pressure losses, and were capable of burning lower quality fuels. Some
later Siemens engines use two large combustion chambers arranged horizontally
rather than vertically, and they no longer resemble silos.
Figure 1.16 shows the configurations of both the aero and industrial versions
of the Rolls-Royce Trent, and the major differences in the design of the
combustion system are clearly shown. The industrial engine uses separate
combustion cans arranged radially, this arrangement being used to provide dlY
low emissions (DLE), i.e. without the added complexity of steam or water
injection. The aircraft engine uses a conventional annular combustor.
For the remainder of this chapter we shall concentrate mainly on the way in
which combustion is arranged to talce place inside a flame tube, and need not be
concerned with the overall configuration.
6.3 Some important facton affecting combustor desigJlJl
Over a period of :(:ve decades, the basic factors influencing the design of com-
bustion systems for gas turbines have not changed, although recently some new
requirements have evolved. The key issues may be summarized as follows.
(a) The temperature of the gases after combustion must be comparatively low to
suit the highly stressed turbine materials. Development of improved
materials and methods of blade cooling, however, has enabled permissible
combustor outlet temperatures to rise from about 1100 K to as much as
1850 K for aircraft applications.
(b) At the end of the combustion space the temperature distribution must be of
lmown form if the turbine blades are not to suffer from local overheating. In
practice, the temperature can increase with radius over the turbine annulus,
because of tlle strong influence of temperature on allowable stress and the
decrease of blade centrifugal stress from root to tip.
(c) Combustion must be maintained in a stream of air moving with a high
velocity in the region of 30-60 mis, and stable operation is required over a
wide range of air/fuel ratio from full load to idling conditions. The air/fuel
ratio might vary from about 60: 1 to 120: 1 for simple cycle gas turbines and
from 100: 1 to 200: I if a heat-exchanger is used. Considering that the
stoichiometric ratio is approximately 15: 1 it is clear that a high dilution is
required to maintain the temperature level dictated by turbine stresses.
(d) The fomlation of carbon deposits ('coking') must be avoided, particularly
the hard brittle variety. Small particles carried into the turbine in the high-
velocity gas stream can erode the blades and block cooling air passages;
238 COMBUSTION SYSTEMS
furthennore, aerodynamically excited vibration in the combustion chamber
might cause sizeable pieces of carbon to break free resulting in even worse
damage to the turbine.
(e) In aircraft gas turbines, combustion must also be stable over a wide range of
chamber pressure because of the substantial change in this parameter with
altitude and forward speed. Another impOliant requirement is the capability
of relighting at high altitude in the event of an engine flame-out.
if) Avoidance of smoke in the exhaust is of major importance for all types of
gas turbine; early jet engines had very smoky exhausts, and this became a
serious problem around airports when jet transport aircraft started to operate
in large numbers. Smoke trails in flight were a problem for military aircraft,
permitting them to be seen from a great distance. Stationary gas turbines are
now found in urban locations, sometimes close to residential areas.
(g) Although gas turbine combustion systems operate at extremely high
efficiencies, they produce pollutants such as oxides of nitrogen (NOx),
carbon monoxide (CO) and unburned hydrocarbons (UHC) and these must
be controlled to very low levels. Over the years, the performance of the gas
turbine has been improved mainly by increasing the compressor pressure
ratio and turbine inlet temperature (TIT). Unfortunately this results in
increased production ofNOx ' Ever more stringent emissions legislation has
led to significant changes in combustor design to cope with the problem.
Probably the only feature of the gas turbine that eases the combustion
designer's problem is the peculiar interdependence of compressor delivery air
density and mass flow which leads to the velocity of the air at entry to the
combustion system being reasonably constant over the operating range.
For aircraft applications there are the additional limitations of small space and
low weight, which are, however, slightly offset by somewhat shorter endurance
requirements. Aircraft engin.e combustion chambers are normally constructed of
light-gauge, heat-resisting alloy sheet (approx. 0.8 mm thick), but are only
expected to have a life of some 10 000 hours. Combustion chambers for industrial
gas turbine plant may be constructed on much sturdier lines but, on the other
hand, a life of about 100 000 hours is required. Refractory linings are sometimes
used in heavy chambers, although the remarks made in (d) about the effects of
hard carbon deposits breaking free apply with even greater force to refractor;
material.
We have seen that the gas turbine cycle is very sensitive to component
inefficiencies, and it is important that the aforementioned requirements should be
met without sacrificing combustion efficiency. That is, it is essential that over
most of the operating range all the fuel injected should be completely burnt and
the full calorific value realized. Any pressure drop between inlet and outlet of the
combustor leads to both an increase in SFC and a reduction in specific power
output, so it is essential to keep the pressure loss to a minimum. It will be
appreciated from the following discussion, that the smaller the space available for
combustion, and hence the shorter the time available for the necessary chemical
THE COMBUSTION PROCESS
239
reactions,. the more difficult it is to meet all the requirements and still obtain a
high combustion efficiency with low pressure loss. Clearly in this respect
designers of combustion systems for industrial gas turbines have an easier task
than their counterparts in the aircraft field.
6.4 The 'combustion process
Combustion of a liquid fuel involves the mixing of a fine spray of droplets with
air, vaporization of the droplets, the brealdng down of heavy hydrocarbons into
lighter fractions, the intimate mixing of molecules of these hydrocarbons with
oxygen molecules, and finally the chemical reactions themselves. A high tem-
perature, such as is provided by the combustion of an approximately stoichio-
metric mixture, is necessary if all these processes are to occur sufficiently rapidly
for combustion in a moving air stream to be completed in a small space. Com-
bustion of a gaseous fuel involves fewer processes, but much of what follows is
still applicable.
Since the overall air/fuel ratio is in the region of 100: 1, while the
stoichiometric ratio is approximately 15: I, the first essential is that the air
should be introduced in stages. Three such stages can be distinguished, About
15-20 per cent of the air is introduced around the jet of fuel in the primm)' zone
to provide the necessary high temperature for rapid combustion. Some 30 per
cent of the total air is then introduced through holes in the flame-tube in the
secondalJ! zone to complete the combustion, For high combustion efficiency, this
air must be injected carefully at the right points in the process, to avoid chilling
the flame locally and drastically reducing the reaction rate in that neighbourhood.
Finally, in the tertiary! or dilution zone the remaining air is mixed with the
products of combustion to cool them down to the temperature required at inlet to
the turbine. Sufficient turbulence must be promoted so that the hot and cold
streams are thoroughly mixed to give the desired outlet temperature distribution,
with no hot streaks which would damage the turbine blades.
The zonal method of introducing the air cannot by itself give a self-piloting
flame in an air stream which is moving an order of magnitude faster than the
flame speed in a burning mixture. The second essential feature is therefore a
recirculating flow pattern which directs some of the burning mixture in the
primary zone back on to the incoming fuel and air. One way of achieving this is
shown in Fig. 6.2, which is typical of British practice. The fuel is injected in the
same direction as the air stream, and the primary air is introduced through twisted
radial vanes, known as swirl vanes, so that the resulting vortex motion will induce
a region of low pressure along the axis of the chamber. This vortex motion is
sometimes enhanced by injecting the secondary air through short tangential
chutes in the flame-tube, instead of through plain holes as in the figure. The net
result is that the burning gases tend to flow towards the region of low pressure,
and some portion of them is swept round towards the jet of fuel as indicated by
the arrows.
240
COMBUSTION SYSTEMS
FIG. 6.2 Combustiou chamiJer with swirl vanes
Many other solutions to the problem of obtaining a stable flame are possible.
One American practice is to dispense with the swirl vanes and achieve the
recirculation by a careful positioning of holes in the flame-tube downstream of a
hemispherical baffle as shown in Fig. 6.3(a). Figure 6.3(b) shows a possible
solution using upstream injection which gives good mixing of the fuel and
primary air. It is difficult to avoid overheating the fuel injector, however, and
upstream injection is employed more for afterburners (or 'reheat') in the jet-pipe
of aircraft engines than in main combustion systems. Afterburners operate only
for short periods of thrust-boosting. Finally, Fig. 6.3( c) illustrates a vaporizer
system wherein the fuel is injected at low pressure into walking -stick shaped
tubes placed in the primary zone. A rich mixture of fuel vapour and air issues
from the vaporizer tubes in the upstream direction to mix with the remaining
primary air passing through holes in a baffle around the fuel supply pipes. The
fuel system is much simpler, and the difficulty of arranging for an adequate
distribution of fine droplets over the whole operating range of fuel flow is
overcome (see 'Fuel injection' in section 6.6). The problem in this case is to avoid
local 'cracking' of the fuel in the vaporizer tubes with the formation of deposits of
low thermal conductivity leading to overheating and burn-out. Vaporizer schemes
are particularly well suited for annular combustors where it is inherently more
difficult to obtain a satisfactory fuel-air distribution with sprays of droplets from
high pressure injectors, and they have been used in several successful aircraft
engines. The original walking-stick shaped tubes have been replaced in modem
engines by more compact and mechanically rugged T-shape vaporizers as shown

o 0
o 0
o 0
(a) (b) (c)
FIG. 6.3 Methods of i1ame stabilization
THE COMBUSTION PROCESS
Secondary
air holes
Dilution air holes
FIG. 6.4 Vaporizer comiJustor IClllllrlesy Rolls-Roycel
241
in Fig. 6.4. Sotheran (1) describes the history of vaporizer development at Rolls-
Royce.
Having described the way in which the combustion process is accomplished, it
is now possible to see how incomplete combustion and pressure losses arise.
When not due simply to poor fuel injector design leading to fuel droplets being
carried along the flame-tube wall, incomplete combustion may be caused by local
chilling of the flame at points of secondary air entry. This can easily reduce the
reaction rate to the point where some of the products into which the fuel has
decomposed are left in their partially burnt state, and the temperature at the
downstream end of the chamber is normally below that at which the burning of
these products can be expected to take place. Since the lighter hydrocarbons into
which the fuel has decomposed have a higher ignition temperature than the
original fuel, it is clearly difficult to prevent some chilling from taking place,
particularly if space is limited and the secondary air cannot be introduced
gradually enough. If devices are used to increase large-scale turbulence and so
distribute the secondary air more uniformly throughout the burning gases, the
combustion efficiency will be improved but at the expense of increased pressure
loss. A satisfactory compromise must somehow be reached.
Combustion chamber pressure loss is due to two distinct causes: (i) skin
friction and turbulence and (ii) the rise in temperature due to combustion. The
stagnation pressure drop associated with the latter, often called the fondamental
loss, arises because an increase in temperature implies a decrease in density and
consequently an increase in velocity and momentum of the stream. A pressure
force (i'lp x A) must be present to impart the increase in momentum. One of the
standard idealized cases considered in gas dynamics is that of a heated gas stream
flowing without friction in a duct of constant cross-sectional area. The stagnation
pressure drop in this situation, for any given temperature rise, can be predicted
242
COMBUSTION SYSTEMS
with the aid ofthe Rayleigh-line functions (see Appendix A.4). When the velocity
is low and the flnid flow can be treated as . incompressible (in the sense that
althongh P is a function of T it is independent of p), a simple equation for the
pressure drop can be found as follows. .
The momentum equation for one-dimensional frictionless flow m a duct of
constant cross-sectional area A is
A(pz - PI) + m(Cz - CI ) = 0
For incompressible flow the stagnation pressure po is simply (p + pC /2), and
Paz - POI = (pz - PI) + - PI cD
Combining these equations, remembering that m = PIACI = pzACz,
Poz - POI = -(pzCi - PlcD - PlcD
= -!(P2ci - PICD
The stagnation pressure loss as a fraction of the inlet dynamic head then becomes
POI -toz = _ 1) = (PI - 1)
PICI/2 PICI pz
Finally, since P oc liT for incompressible flow,
POI -t02 = (T2 _ 1)
PICd2 TI
This will be seen from the Appendix to be the same as the compressible flow
value of (POI - POZ)/(POI - PI) in the limiting case of zero inlet Mach number.
At this condition Tz/TI = Toz/TO!.
Although the assumptions of incompressible flow and constant cross-sectional
area are not quite true for a combustion chamber, the result is sufficiently accurate
to provide us with the order of magnitude of the fundamental loss. Thus, since the
outlet/inlet temperature ratio is in the region of 2-3, it is clear that the
fundamental loss is only about 1-2 inlet dynamic heads. The pressure loss due to
friction is found to be very much higher-of the order of 20 inlet dynamic heads.
When measured by pitot traverses at inlet and outlet with no combustion taking
place, it is known as the cold loss. That the friction loss is so high, is due to. the
need for large-scale turbulence. Turbulence of this kind is created by the deVIces
used to stabilize the flame, e.g. the swirl vanes in Fig. 6.2. In addition, there is the
turbulence induced by the jets of secondary and dilution air. The need for good
mixing of the secondary air with the burning gases to avoid chilling has been
emphasized. Similarly, good mixing of the dilution air to avoid hot streaks in the
turbine is essential. In general, the more effective the mixing the higher the
pressure loss. Here again a compromise must be reached: this time between
uniformity of outlet temperature distribution and low pressure loss.
Usually it is found that adequate mixing is obtained merely by injecting air
through circular or elongated holes in the flame-tube. Sufficient penetration of the
COMBUSTION CHAMBER PERFORMANCE 243
cold air jets into the hot stream is achieved as a result of the cold air having the
greater density. The pressure loss produced by such a mixing process is
associated with the change in momentum of the streams before and after mixing.
In aircraft gas turbines the duct between combustion chamber outlet and turbine
inlet is very short, and the compromise reached between good temperature
distribution and low pressure loss is nonnally such that the temperature non-
uniformity is up to ± 10 per cent of the mean value. The length of duct is often
greater in an industrial gas turbine and the temperature distribution at the turbine
inlet may be more unifonn, although at the expense of increased pressure drop
due to skin friction in the ducting. The paper by Lefebvre and Norster in Ref. (2)
outlines a method of proportioning a tubular combustion chamber to give the
most effective mixing for a given pressure loss. Making use of empirical data
from mixing experinlents, such as dilution hole discharge coefficients, the authors
show how to estimate the optimum ratio offiame-tube to casing diameter, and the
optimum pitch/diameter ratio and number of dilution holes.
6.5 Combustion chamber performance
The main factors of importance in assessing combustion chamber perfonnance
are (a) pressure loss, (b) combustion efficiency, (c) outlet temperature distribu-
tion, (d) stability limits and (e) combustion intensity. We need say no more of (c),
but (a) and (b) require further connnent, and (d) and (e) have not yet received
attention.
Pressure loss
We have seen in section 6.4 that the overall stagnation pressure loss can be
regarded as the Stun of the fundamental loss (a small component which is a
function of T02/ TOl ) and the frictional loss. Our lmowledge of friction in ordinary
turbulent pipe flow at mgh Reynolds number would suggest that when the pres-
sure loss is expressed non-dimensionally in tenns of the dynamic head it will not
ValY much over the range of Reynolds number under which combustion systems
operate. Expeliments have shown, in fact, that the overall pressure loss can often
be expressed adequately by an equation of the fonn
pressurelossfactor,PLF= z =KI+K2(T02_1) (6.1)
m" / -PIAm Tal
Note that rather than PI Ci /2, a conventional dynamic head is used based on a
velocity calculated from the inlet density, air mass flow m, and maximum cross-
sectional area Am of the chamber. This velocity-sometimes lmown as the refer-
ence velocity-is more representative of conditions in the chamber, and the
convention is useful when comparing results from chambers of different shape.
Equation (6.1) is illustrated in Fig. 6.5. If KI and K2 are determined from a
combustion chamber on a test rig from a cold run and a hot run, then equation
244
COMBUSTION SYSTEMS
40
Fudamental
is
pressure loss
"
30
J!'
ill
.Q
20
'" :;
'"
'"

10 CL
0
2
Temperature ratio T021To1
FIG. 6.5 V!Ilriatilllll of pressllIre loss factor
(6.l) enables the pressure loss to be estimated when the chamber is operating
part of a gas turbine over a wide range of conditions of mass flow, pressure raho
and fuel input. .
To give an idea of relative orders of magnitude, typical values of at desIgn
operating conditions for tubular, tubo-annular and annular combustIon chambers
are 35 25 and 18 respectively. There are two points which must be remembered
when pressure loss data. Firstly, the velocity of the air leaving the last
stage of an axial compressor is quite high-say 150 mis-and form of
diffusing section is introduced between the compressor and combustlOn cham?er
to reduce the velocity to about 60 mls. It is a matter of conventIOn, dependmg
upon the layout of the gas turbine, as to how much of the stagnation pressure los.s
in this diffuser is included in the PLF of the combustion system. In other words, It
depends on where the compressor is deemed to end and the combustion chamber
.
Secondly, it should be appreciated from Chapters 2 and 3 that from the pomt of
view of cycle performance calculations it is I1po as a fraction of the compressor
delivery pressure (POI in the notation of this chapter) which is the useful
parameter. This is related to the PLF as follows.
I'!.po = I1po x m
2
/2PIA;" = PLF xl!: (m.JTol)2
POI m
2
/2PIA;" POI 2 AnzPOl
(6.2)
where the difference between PI and POI has been ignored because the velocity is
low. By combining equations (6.1) and (6.2) it can be seen that I'!.PO/POI can.be
expressed as a function of non-dimensional mass flow at entry to the combustlOn
chamber and combustion temperature ratio: such a relation is useful when pre-
dicting pressure losses at conditions other than design, as discussed in Chapter 8.
Consider now the two extreme cases of tubular and annular designs. If the values
of I'!.PO/POI are to be similar, it follows from equation (6.2) and the values of PLF
given above that the chamber cross-sectional area per unit mass flow (Am/m) .can
be smaller for the annular design. For aircraft engines, where space and weIght
COMBUSTION CHAMBER PERFORMANCE 245
are vital, the value of Am/m is normally chosen to yield a value of I1PO/POI
between 4 and 7 per cent. For industrial gas turbine chambers, Am/m is usually
such that I1pO/POl is little more than 2 per cent.
Combustion efficiency
The efficiency of a combustion process may be found from a chemical analysis of
the combustion products. Knowing the air/fuel ratio used and the proportion of
incompletely burnt constituents, it is possible to calculate the ratio of the actual
energy released to the theoretical quantity available. This approach via chemical
analysis is not easy, because not only is it difficult to obtain truly representative
samples from the high velocity st{eam, but also, owing to the high air/fuel ratios
employed in gas turbines, the unburnt constituents to be measured are a very
small proportion of the whole sample. Ordinary gas analysis apparatus, such as
the Orsat, is not adequate and much more elaborate techniques have had to be
developed.
If an overall combustion efficiency is all that is required, however, and not an
investigation of the state ofthe combustion process at different stages, it is easier
to conduct development work on a test rig on the basis of the combustion
efficiency which was defined in Chapter 2, namely
theoretical r for actuall'!.T
17b = "
ac.!Ual f for actuall1T
For this purpose, the only measurements required are those necessary for deter-
mining the fuel/air ratio and the mean stagnation temperatures at iulet and outlet
of the chamber. The theoretical f can be obtained from curves such as those in
Fig. 2.15.
It is worth describing how the mean stagnation temperature may be measured:
there are two aspects, associated with the adjectives 'mean' and 'stagnation'.
Firstly, it should be realized fi·om the discussion in section 2.2 under heading
'Fuel/air ratio, combustion efficiency and cycle efficiency', that the expression
for l)b arises from the energy equation which consists of such terms as mCpTo.
Since in practice there is always a variation in velocity as well as temperature over
the cross-section, it is necessary to' determine, not the ordinary arithmetic mean of
a number of temperature readings, but what is known as the 'weighted mean'. If
the cross-section is divided into a number of elemental areas AI, A2, •.. ,
A;, ... , Am at which the stagnation temperatures are Toj, To2,· .• , Toi, ... , TOm
and the mass flows are Inl, mb ... , mi, ... , mm then the weighted mean
temperature Tow is defined by
I:l!1iTOi I:m;Toi

L." mi In
where the summations are from I to n. We may assume that cp is effectively
constant over the cross-section. It follows that the product mCpTow will be a true
measure of the energy passing the section per unit time.
246 COMBUSTION SYSTEMS
A simple expression for Tow in terms of measured quantities can be as
follows. The velocity at the centre of each elemental area may be found usmg a
pitot-static tube. Denoting the dynamic head pe
2
/2 by Pd,the mass flow for area
Ai is then
1
rIl i = p,Ai(2PdJpJ'
If the static pressure is constant over the cross-section, as it will be when there is
simple axial flow with no swirl,
1
p; 0( 1/ Ti and rIl i 0( A;(p dd T;)2
and thus
T.
_ LA;1'o;(PdJT;)!
Ow - 1
'LA;(PdJTi)2
Since Ti has only a second order effect on Tow, we can write Ti = To;.
more, it is usual to divide the cross-section into equal areas. The expreSSIOn then
finally reduces to
1
'1' _ :l)PdiTOi)2
lOw - 1
'L(PdJToY
Thus the weighted mean temperature may be determined directly from measure-
ments of dynamic head and stagnation temperature at the centre of each elemen-
tal area. It remains to describe how stagnation temperatnres may be measured.
In gas turbine work, temperatures are usually measured by thernlOcouples. The
high accuracy of a pitot-static tube is well known, but considerable difficulty has
been experienced in designillg thermocouples to operate a high-temperature
fast-moving gas stream with a similar order of accuracy. Smce the combustion
efficiency rarely falls below 98 per cent over much of the operating range,
accurate measurements are essential. Chromel-alumel thermocouples have been
found tD withstand the arduous requirements Df combustion chamber testing
satisfactorily, and give accurate results up to about 1300 K if special precautions
are taken. It must be remembered that the temperature recorded is that Df the hot
junction of the thermocouple which for various reas?ns not be a: the
temperature Dfthe gas stream in which it is situated, particularly If the velOCIty of
the stream is high.
If it be imagined that the thermocouple junction is moving with the stream of
gas, then the temperature of the junction may differ from the static temperature of
the gas by an amount depending upon the conduction of heat along the
thermocouple wires, the convection between the junction and the gas stream, the
radiation from the hot flame to the junction, and the radiation from the junction to
the walls of the containing boundary if these are cooler than the junction. There is
an additional possible error, because in practice the thermocouple is stationary,
and the gas velocity will be reduced by friction in the boundary layer around the
thermocouple junction. Kinetic energy is transformed intD internal energy, some
of which raises the temperature of the junction, while some is carried away by
COMBUSTION CHAMBER PERFORMANCE
247
convection. In a high-speed gas stream it is obviously important to know how
much of the velocity energy is being measured as temperature. The temperature
con'esponding to the velocity energy, i.e. the dynamic temperature, is about 40 K
for a velocity of300 m/s.
Since it is the stagnation temperature which is of interest, it is usual to place
the thennocouple wires and junction in a metal tnbe in which the gas stream can
be brought' to rest adiabatically so that almost the whole Df the dynamic
temperatnre is measured, on the same principle as a pitot tube measuring
stagnation pressure. Figure 6.6(a) shows one form of stagnation thermocouple
which will measure about 98 per cent of the dynamic temperature as against the
60-70 per cent measured by a simple junction placed directly in the gas stream. A
large hole facing upstream allows the gas to enter the tube, while a small hole, not
more than 5 per cent of the area of the inlet orifice, provides sufficient ventilation
without spoiling the pitot effect. This fonn of thermocouple is excellent for all
work where radiation effects are small, such as the measurement Df cDmpressor
delivery temperatnre. Where radiation effects are appreciable, as at the outlet of a
combustiDn chamber, it is preferable tD use a thennocouple of the type shown in
Fig. 6.6(b). A radiation errDr of the order of 60 Kin 1300 K is quite possible with
a completely unshielded thennocouple. A short length of polished stainless steel
sheet, twisted into a helix and placed in front ofthe junction, provides an effective
radiation shield without impeding the flow of gas into the thermocouple tube. One
or more concentric cylinchical shields may also be included. The bending of the
wires so that about two or three centimetres run parallel with the direction of the
stream, i.e. pm'aIlel with an isothennal, reduces the error due to conduction of
heat away from the junction along the wires. This is ill: most cases quite a small
error if the wires are of small diameter. If all these precautions are taken,
stagnation temperature measurement up to 1300 K is possible to within ± 5 K.
While these few remarks do not by any means exhaust all the possibilities Df
thennocouple design, they at least indicate the extreme care necessary when
choosing thernlocouples for gas tnrbine temperature measurement.
(a)
(b)
FIG. 6.6 Stagllatioll thermocouples
COMBUSTION SYSTEMS
248
ture at outlet from the combustion chamber is n?t measured
The h th . s the added problem that mechamcal faIlure of the
on an engme, were s lead to major damage in the turbine. For this
thermocouple support. I the temperature downstream of the
, . e It IS norma to measure
jet engine the temperature at exit from the turbine is lmown
r me:. .' (JPT) but the nomenclature exhaust gas temperature
as the Jet pipe tempelatwe d!" . d tI1'al gas turbines When a twin-spool or
) . Imuoilly use lor m us .
is used it is more to measure the
downstream of the high pressure turbine, thIS bemg referred to as tb' 'ill t
T)
T' combustion outlet temperature (l.e. tur me 1 e
turbine temperature (IT . ne . - . t f ITT or EGT
.. be calculated from the mdrrect measuremen s 0
described in Chapter 8. Either ITT or EGT is used as ahcontr0jl
usm l . r h turb' and these are t e on y
tem limit to prevent overheatmg or t e me
sys ents that will normally be available to the user of gas
temperature measurem
turbines.
Stability limits
. combustion chamber there is both a rich and a weak limit to the
which the flame is unstable. Usually.the is taken as the
aIr fuel ratio at which the flame blows out, although mstablhty often
this limit is reached. Such instability takes the form of
which not only indicates poor combustion, but sets up VI ra on
h' h reduces the life of the chamber and causes blade VIbratIOn The
of air/fuel ratio between the l1ch and weale limits is reduced wIth
• 7 i and if the air mass flow is increased beyond a certam va ue I .IS
combustion at aIL A typical loop pis sh07r 111
6.7, where the limiting air/fuel ratio is plotted agamst aIr maso flow. a com
200 \

50
o 0.25
Weal< limit
/
Stable !:egion
'y
--
____ Rich limit
0.50 0.75 1.00
Air mass flow kg/s
FIG. 6.7 Stability loop
1.25
COMBUSTION CHAMBER PERFORMANCE 249
bustion chamber is to be suitable, its operating range defined by the stability loop
must obviously cover the required rallge of air/fuel ratio and mass flow of the gas
turbine for which it is intended. Furthermore, allowance must be made for con-
ditions which prevail when the engine is accelerated or decelerated. For example,
on acceleration there will be a rapid increase in fuel flow as the 'throttle' is
opened while the air flow will not reach its new equilibrium value until the engine
has reached"its new speed. Momentarily the combustion system will be operating
with a very low air/fuel ratio. Most control systems have a built-in device which
places an upper limit on the rate of change of fuel flow: not oilly to avoid blow-
out, but also to avoid transient high temperatures in the turbine.
The stability loop is a function of the pressure in the chamber: a decrease in
pressure reduces the rate at which the chemical reactions proceed, and
consequently it narrows the stability limits. For aircraft engines it is important
to check that the limits are sufficiently wide with a chaluber pressure equal to the
compressor delivery pressure which exists at the highest operating altitude.
Engines of high pressure ratio present less of a problem to the combustion
chamber designer than those of low pressure ratio. If the stability limits are too
narrow, changes must be made to improve the recirculation pattem in the primary
zone.
Combustion intr:nsity
The size of combustion chamber is determined primarily by the rate of heat
release required. The nominal heat release rate can be found from m!Qnet,p where
m is the air mass flow,! the fuelj air ratio and Qnet,p tlle net calorific value of the
fuel. Enough has been said for the reader to appreciate that the larger the volume
which can be provided the easier it will be to achieve a low pressure drop, high
efficiency, good outlet temperature distribution and satisfactory stability charac-
teristics.
The design problem is also eased by an increase in the pressure and
temperature of the air entering the chamber, for two reasons. Firstly, an increase
will reduce the tinle necessary for the 'preparation' of the fuel and air mixture
(evaporation of droplets, etc.) making more time available for the combustion
process itself. Note that since the compressor delivery temperature is a function of
the compressor delivery pressure, the pressure (usually expressed in atmospheres)
is an adequate measure of both.
Secondly, we have already observed under the heading 'Stability limits' above,
that the combustion chamber pressure is important because of its effect on the
rate at which the chemical reactions proceed. An indication of the nature of this
dependence can be obtained from chemical kinetics, i.e. kinetic theory applied to
reacting gases. By calculating the number of molecular collisions per unit time
and unit volume which have an energy exceeding a certain activation value E, it is
possible to obtain the following expression for the rate r at which a simple
bimolecular gas reaction proceeds, Ref. (3).
250
COMBUSTION SYSTEMS
2 2TI/2M.c.3/2l"IRT
r ()( mjm/cp (f
p and T have their usual meaning and the other symbols denote:
mj, mk
(J
M
local concentrations of molecules j and k
mean molecular diameter
mean molecUlar weight
E activation energy
R molar (universal) gas constant
Substituting for p in tenns of p and T we can for our purpose simplify the
expression to
r cx:lf(T)
Now T is maintained at a high value by having an approximately
mixture in the primary zone: we are concerned here with independent vanable
p. It is not to be expected that the theoretical exponent will ap?ly the
set of reactions occurring when a hydrocarbon fuel IS burnt. m aIr, and
ments with homogeneous mixtures in stoichiometric propomons suggest that It
should be 1.8. At first sight it appears therefore that the problem be
eased as the pressure is increased according to the law p : In fact IS
to believe that lll1der design operating conditions the chemIcal IS not
a limiting factor in an actual combustion chamber where mIXmg .pr.o-
cesses play such an important role, and that an exponent of more realIstIc.
This is not to say that lll1der extreme conditions-say at hIgh altitude-the per-
·th h 1-81
fonnance will not fall off more in accordance WI t e p a:v.
A quantity known as the combustion been mtroduced to take
aCCOlll1t of the foregoing effects. One definition used IS
.. . _ heat release rate kW 1m3 atm
combustion mtenslty - comb. vol. x pressure
3 ]·8 . .
Another definition employs pI.S, with the units kW/m atm . IS
defined, certainly the lower the value of the combustion eaSIer It IS t.o
design a combustion system which will meet all deSIred requIrements. It
quite inappropriate to compare the perfonnance o.f the baSIS
of efficiency, pressure loss, etc., if they are operating WIth ?rders
of combustion intensity. In aircraft systems the IS m the
region of 2-5 x 104 kW 1m3 atm,t while in industrIal gas turbmes the can
be much lower because of the larger volume of combustion .a
further reduction would result if a heat-exchanger were used, reqUlnng a SIgnI-
ficantly smaller heat release in the combustor.
6.6 Some practical problems
We will briefly describe some of the problems which have not so men-
tioned but which are none the less important. These are concerned WIth (1) flame-
t Note that 1 kW/m
3
aim = 96·62 Btoih W aim
SOME PRACTICAL PROBLEMS 251
tube cooling, (ii) fuel injection, (iii) starting and ignition and (iv) the use of
cheaper fuels.
Flame-tube cooling
One problem which has assumed greater importance as permissible turbine inlet
temperatures have increased is that of cooling the flame-tube. The tube receives
.energy by convection from the hot gases and by radiation from the flame. It loses
energy by convection to the cooler air flowing along the outside surface and by
radiation to the outer casing, but this loss is not sufficient to maintain the tube
wall at a safe temperature. A common practice is to leave narrow amlular gaps
between overlapping sections of the flame tube so that a :film of cooling air is
swept along the inner surface; corrugated 'wigglestrip', spot welded to successive
lengths of flame-tube, provides adequate stiffiless with amlular gaps which do not
vary too much with thermal expansion, as shown in Fig. 6.8(a). Another method
is to use a ring of small holes with an internal splash ring to deflect the jets along
the inner surface, as shown in Fig. 6.8(b). A more recent development is the use
of transpiration cooling, allowing cooling air to enter a network of passages
within the flame tube wall before exiting to fonn an insulating :film of air; this
method may permit a reduction in cooling flow of up to 50 per cent.
Although empirical relations are available from which it is possible to predict
convective heat transfer rates when film cooling a plate of known temperature, the
emissivities of tho: flame and flame-tube can vary so widely that prediction of the
flame-tube temperature from an energy balance is not possible with any accuracy.
Even in this limited aspect of combustion chamber design, final development is
still a matter of trial and error on the test rig. The emissivity of the flame varies
with the type of fuel, tending to increase with the specific gravity. Carbon dioxide
and water vapour are the principal radiating components in non-lUlllinous flames,
and soot particles in lUlllinous flames. It is worth noting that vaporizer systems
ease the problem, because flames from pre-mixed fuel vapour-air mixtures have a
lower lUlllinosity than those from sprays of droplets.
Higher turbine inlet temperatures imply the use of lower air/fuel ratios, with
consequently less air available for :film cooling. Furthennore, the use of a higher
cycle temperature is usually accompanied by the use of a higher cycle pressure
ratio to obtain the full benefit in terms of cycle efficiency. Thus the temperature of
the air leaving the compressor is increased and its cooling potential is reduced. At
projected levels of turbine inlet temperature, up to 1850 K, the use of
transpiration cooling to reduce the required cooling flow may become essential.
+·--74
i
(a)
FIG. 6.8 Film cooling of flame-tube
(b)
252 COMBUSTION SYSTEMS
Fuel injection
Most combustion chambers employ· high-pressure fuel systems in which the
liquid fuel is forced through a small orifice to form a conical spray of fine droplets
in the primary zone. The fuel is said to be 'atomized' and the burner is often
referred to as an 'atomizer'. An alternative is the vaporizer system, but it should
be realized that even this requires an auxiliary starting burner of the atomizing
type.
In the simplest form of atomizing burner, fuel is fed to a conical vortex
chamber via tangential ports which impart a swirling action to the flow. The
vortex chamber does not run full but has a vapour/air core. The combination of
axial and tangential components of velocity causes a hollow conical sheet of fuel
to issue from the orifice, the ratio of the components determining the cone angle.
This conical sheet then breaks up in the air stream into a spray of droplets, and the
higher the fuel pressure the closer to the orifice does this break up occur. There
will be a certain minimum fuel pressure at which a fully developed spray will
issue from the orifice, although for the following reason the effective minimum
pressure may well be higher than this.
The spray will consist of droplets having a wide range of diameter, and the
degree of atomization is usually expressed in tenns of a mean droplet diameter. In
common use is the Sauter mean diameter, which is the diameter of a drop having
the same surface/volume ratio as the mean value for the spray: 50-100 microns is
the order of magnitude used in practice. The higher the supply pressure, the
smaller the mean diameter. If the droplets are too small they will not penetrate far
enough into the air stream, and if too large the evaporation time may be too long.
The effective minimum supply pressure is that which will provide the required
degree of atomization.
The object is to produce an approximately stoichiometric mixture of air and
fuel uniformly distributed across the primary zone, and to achieve this over the
whole range offuel flow from idling to full load conditions. Herein lies the main
problem of burner design. If the fuel is metered by varying the pressure in the fuel
supply line, the simple type of burner just described (sometimes referred to as the
simplex) will have widely differing atomizing properties over the range of fuel
flow. It must be remembered that the flow tlu·ough an orifice is proportional to the
square root of the pressure difference across it, and thus a fuel flow range of 10: 1
implies a supply pressure range of 100: 1. If the burner is designed to give
adequate atomization at full load, the atomization will be inadequate at low load.
This problem has been overcome in a variety of ways.
Perhaps the most commonly used solution to the problem is that employed in
the duplex burner, an example of which is shown in Fig. 6.9(a). Two fuel
manifolds are required, each supplying independent orifices. The small central
orifice is used alone for the lower flows, while the larger annular orifice
surrounding it is additionally brought into operation for the higher flows. The
sketch also shows a third annulus formed by a shroud through which air passes to
prevent carbon deposits building up on the face of the burner. This feature is
SOME PRACTICAL PROBLEMS
Air
Main _
fuel
(a)
FIG. 6.9 ])\lplex !ll1II1I spill !iJllrl1lers

in
Fuel
Qui
(b)
253
incorporated in most burners. An alternate form of duplex burner employs a
single vortex chamber and final orifice, with the two fuel supply lines feeding two
sets of tangential ports in the vortex chamber.
Figure 6.9tb) illustrates a second practical method of obtaining good
atomization over a wide range of fuel flow: the spill burner. It is virtually a
simplex burner with a passage from the vortex chamber through which excess
fuel can be spilled off. The supply pressure can remain at the high value
necessary for good atomization while the flow through the orifice is reduced bv
reducing the pressure in the spill line. One disadvantage of this system is
when a large quantity of fuel is being recirculated to the plll11p inlet there may be
undesirable heating and consequent deterioration of the fuel.
Dual-fuel b1111!ers are used in industrial gas turbines where gas is the normal
fuel, but oil is required for short periods when the gas supply may be intenupted.
The gas and liquid fuels would be supplied through separate concentric annuli· an
additional annulus may also be provided for water or steam injection for emis;ion
control. Such engines can operate on either fuel separately or both simulta-
neously. A typical dual-fuel nozzle for an aero-derivative gas turbine is shown in
Fig. 6.10.
Burners of the type described here by no means exhaust all the possibilities.
For example, for small gas turbines having a single chamber, it has- been fOlLlld
possible to use rotary atomizers. Here fuel is fed to a spinning disc or cup and
flung int.o the air stream from the rim. High tip speeds are required for good
atol111ZatJon, but only a low-pressure fuel supply is required.
Starting and ignition
Under nornlal operating conditions, gas turbine combustion is continuous and
self-sustaining. An ignition system, however, is required for starting, and the
ignition and starting systems must be closely integrated. The first step in starting
a gas turbine is to accelerate the compressor to a speed that gives an air flow
capable of sustaining combustion; during the period of acceleration the ignition
system is switched on and fuel is fed to the burners when the rotational speed
reaches about 15-20 per cent of normal. An igniter plug is situated near the
primary zone in one or two of the flame-tubes or cans. Once the flame is estab-
254 COJVlBUSTION SYSTEMS
FIG, 6,10 Dual-fuel I:mrner IClmrte§y Roils-Roycej
lished, suitably placed intercolUlecting tubes between the cans permit 'light
round', i.e. flame propagation from one flame-tube to the other. Light round
presents few problems in alUlular combustors. When the engine has achieved self-
sustaining operation, the ignition system is turned off. Aircraft gas turbines have
two additional requirements to meet: (i) re-ignition must be possible under wind-
milling conditions if for any reason the flame is extinguished at altitude and (ii)
operation at idle power must be demonstrated while ingesting large amounts of
water. This latter requirement is to prevent flame-out during final approach to an
airport in very heavy rain, and it is nonnal operating procedure to turn on the
ignition system in adverse weather at low altitude during both climb and descent.
Engine shut down normally requires the engine to be brought back to idle fol-
lowed by shutting off the fuel; shut downs from full power should be avoided
because of the possibility of differential expansion/contractions leading to seal
rubs or seiznre of the rotor.
The starting systems for aircraft and industrial gas turbines are quite different,
with compact size and low weight being critical for the former and very large
powers sometimes required for the latter. Starting devices inclnde electric motors,
compressed air or hydraulic starters, diesel engines, steam turbines or gas
expansion turbines. Early civil aircraft were dependent on ground power supplies
for starting and some engines used direct air impingement on the turbine blades;
modem aircraft normally use an air turbine starter, which is cOlUlected to the main
rotor by a reduction gear-box and clutch. The supply of compressed air may be
from a ground cart, auxiliary power unit (APU) or bled from the compressor of an
engine already running. Military aircraft use similar systems, but early aircraft
nsed cartridge-type starters which provided a flow of hot, high-pressure gas for up
to 30 seconds; this was expanded through a small turbine geared to the main rotor
via a clutch.
The type of starting system required for an industrial gas turbine depends on
the configuration; for units with a free power turbine it is only necessary to
SOME PRACTICAL PROBLEMS
255
accelerate the gas generator. A single-shaft unit for electrical power generation,
however, requires that the gas turbine and electric generator be accelerated as a
single train. Power requirements for alSO MW unit can be as high as 5 MW; for
the largest units it is now common for the generator to be wound so that it can
also be used as a motor, which is then used as the starting device. Diesel or steam
turbine starting units with a power of 400-500 kW may be used for 60-80 MW
units. The starter requirements for a free turbine engi\le are much less, and this is
even more pronounced for a twin-spool gas generator where only the high-
pressure rotor has to be turned over; a 30 MW unit may then require as little as
20 leW for starting.
The ignition perfonnance can be expressed by an ignition loop which is
similar to the stability loop of Fig. 6.6 but lying inside it. That is, at any given air
mass flow the range of air/fuel ratio within which the mixture can be ignited is
smaller than that for which stable combustion is possible once ignition has
OCCUlTed. The ignition loop is very dependent on combnstion chamber pressure,
and the lower the pressure the more difficult the problem of ignition, Re-light of
an aircraft engine at altitude is thus the most stringent requirement. Although
high-tension sparking plugs similar to those used in piston engines are adequate
for ground starting, a spark of much greater energy is necessary to ensure ignition
under adverse conditions. A surface-discharge igniter, yielding a spark having an
energy of about three joules at the rate of one per second, is probably the most
widely used type for aircraft gas turbines in which fuel is injected as a spray of
droplets.
One example of a surface-discharge igniter is shown in Fig. 6.11. It consists of
a central and outer electrode separated by a ceramic insulator except near the tip
where the separation is by a layer of semiconductor material. When a condenser
voltage is applied, current flows through the semiconductor which becomes
incandescent and provides an ionized path of low resistance for the energy stored
in the capacitor. Once ionization has occurred, the main discharge takes place as
an intense flashover. To obtain good performance and long life the location of the
igniter is critical: it must protrude through the layer of cooling air on the inside of
Semiconductor
Ceramic
Earthed electrode
Central electrode
FIG, 6,11 Surface-disciltarge igniter
256 COMBUSTION SYSTEMS
the flame-tube wall to the outer edge of the fuel spray, but not so far as to be
seriously wetted by the fuel.
For vaporizing combustors, somefonn of torch igniter is necessary. This
comprises a spark plug and auxiliary spray burner in' a COlllnon housing,
resulting in a bulkier and heavier system than the surface discharge type.
The normal spark rate of " typical ignition system is between 60 and 100
sparks per minute. Each discharge causes progressive erosion of the igniter
electrodes, making periodic replacement of the igniter plug necessary; it is for
this reason that the ignition system is switched off in normal operation. The pilot,
however, must be provided with the capability of re-engaging the ignition system
in the event of flame-out or extremely heavy rain.
Use of cheap fuels
Gas turbines rapidly supplanted piston engines in aircraft because of their major
advantages in power and weight, pennitting much higher flight speeds. Pene-
tration of industrial markets was much slower, initially because of uncompetitive
thennal efficiency and later, as performance improved, because of the need for
expensive fuel. Gas turbines became established in applications such as peak-load
or emergency electricity generation, where the running hours were short; the oil
crises of the 1970s, resulting in soaring oil costs, saw a large number of units
mothballed because of high fuel costs.
Natural gas, although a relatively expensive fuel, is ideal for use in stationary
gas turbines, containing very few impurities such as sulphur and not requiring
atomization or vaporization as do liquid fuels. As a result, gas turbines rapidly
became the prime mover of choice for gas compression duties on pipelines,
making use of the high-pressure gas flowing through the pipeline as fuel. Natural
gas is now used widely for base-load electrical power generation, using combined
cycle plant with capacities in excess of 2000 MW and thermal efficiencies of
around 55 per cent. The long term availability of natural gas is somewhat
controversial, but these stations could eventually be converted to burning gas
obtained from coal.
Market penetration would be greatly enhanced if gas turbines could burn
residual oil. This cheap fuel is the residue from crude oil following the extraction
of profitable light fractions. Some of its undesirable characteristics are:
(a) high viscosity requiring heating before delivery to the atomizers;
(b) tendency to polymerize to form tar or sludge when overheated;
(c) incompatability with other oils with which it might come into contact,
leading to jelly-like substances which can clog the fuel system;
(d) high carbon content leading to excessive carbon deposits in the combustion
chamber;
(e) presence of vanadium, the vanadium compounds formed during combustion
causing corrosion in the turbine;
(f) presence of alkali metals, such as sodium, which combine with sulphur in
the fuel to fonn corrosive sulphates;
GAS TURBINE EMISSIONS
257
(g) relatively large amount of ash, causing build up of deposits on the nozzle
blades with consequent reduction in air mass flow and power output.
The problems arising from characteristics (a), (b), (c) and (d) can be overcome
Without excessive difficulty. A typical residual fuel may require heating to about
140°C, and for a large station this would require steam heating both for the
tanks and prior to delivery to the atomizers. It is the major problems
ansmg from ( e), (f) and (g) that have greatly restricted the use of residual oil.
The rate of corrosi.on from (e) and (f) increases with turbine inlet temperature,
and early mdustnal gas turbines designed for residual oil operated with
around 900 K to avoid the problem. Such a low cycle temperature
mevltably meant a low cycle efficiency. It has now been found that the allcali
metals can. be removed,. and that fuel additives such as magnesium compounds
can neutrahze the vanadiUm; sulphur cannot be removed at this stage and would
need to be done as part of the refinery process. There are two basic 'methods of
alkali The first is to wash the fuel oil with water, followed by
centnfugmg of the llllXture to separate the heavier liquids containing the alkali
The second IS a process in which the fuel is mixed with water, pmnped to
a high and then subjected to a high-voltage static discharge resulting in
the .reqUIred Both methods require a sophisticated fuel treatment plant
which means a conSiderable capital expense plus additional operating costs, so
actual cost of the apparently cheap fuel is significantly increased. Residual oil
IS not used but may be used as a back-up fuel with natural gas as the
fuel. In a typical modem large plant using residual oil as back up, the site
ratmg of the gas turbine was reduced from 139 MW to 116 MW owing to the
need to operate at reduced turbine inlet temperature. It is important to realize that
this dual-fuel type of operation is often dictated by the supply of natural gas at
cost on an interruptible basis; this means that the gas supplier can cut off
gas m peak penods, so the customer must have a capability of switching fuels. It
IS to find gas turbines designed to switch automaticallv from gas to oil
while continuing to operate at full power. .
In the early days of gas turbines, much effort went into experiments with
bummg coal; used coal in the form of a pulverized solid, and were
unsuccessful owm.g to the severe erosion caused by hard particles. For many years
coal was not conSIdered as a fuel, but recently a major effort has been expended
on de:elopmg schemes for producing gas from coal, i.e. coal gasification. The
estabhshmentofthe combined cycle has made this feasible, as many gasification
schemes. requITe large quantities of steam. This will be discussed further in the
last sectIOn of this chapter.
6.7 Gas turbine emissions
Gas combustion is a steady flow process in which a hydrocarbon fuel is
burned a large of air, to keep the turbine inlet temperature at
an appropnate value. This IS essentially a clean and efficient process and for many
258
COMBUSTION SYSTEMS
years there was no concern about emissions,· with the· exception of the need to
eliminate smoke from the exhaust. Recently, however, control of emissions has
become probably the most important factor in the design of industrial gas tur-
bines, as the causes and effects of industrial pollution become better understood
and the population of gas turbines increases. Emissions from aero-engines are
also important, but the problfims and solutions are quite different to those for
land-based gas turbines.
Combustion equations express conservation of mass in molecular terms
following the rearrangement of molecules during the combustion process. The
basic principles of combustion are described in standard texts on thermo-
dynamics, e.g. Ref. (4). The oxygen required for stoichiometric combustion can
be found fi'om the general equation
where
a =X, b = (YI2) andn =x+(V/4)
Each kilogram of oxygen will be accompanied by (76·7123·3) kg of nitrogen,
which is normally considered to be inert and to appear unchanged in the exhaust;
at the temperatures in the primary zone, however, small amounts of oxides of
nitrogen are formed. The combustion equation assumes complete combustion of
the carbon to CO
2
, but incomplete combustion can result in small amounts of
carbon monoxide (CO) and m1bumed hydrocarbons (UHC) being present in the
ey"haust. The gas turbine uses a large quantity of excess air, resulting in consider-
able oxygen in the exhaust; the amount can be deduced from the total oxygen in
the incoming air less that required for combustion. Thus the exhaust of any gas
turbine consists primarily of CO2, H20, O2 and N2 and the composition can be
expressed in terms of either gravimetric (by mass) or molar (by volmne) com-
position. Concern about the possible harmful effects of CO2 as a 'greenhouse'
gas leading to global warming has been mentioned in section 1.6.
The pollutants appearing in the exhaust will include oxides of nitrogen (NOJ,
carbon monoxide (CO) and unbmned hydrocarbons (UHC); any sulphur in the
fuel will result in oxides of sulphur (SO,,), the most common of which is S
0
2'
Although these all represent a very small proportion of the exhaust, the large flow
of exhaust gases produces significant quantities of pollutants which can become
concentrated in the area close to the plant. The oxides of mtrogen, in particular,
can react in the presence of sunlight to produce 'smog' which can be seen as a
brownish cloud; this problem was originally identified in the Los Angeles area,
where the combination of vehicle exhausts, strong sunlight, local geography and
temperature inversions resulted in severe smog. This led to major efforts to clean
up the emissions from vehicle exhausts and stringent restrictions on emissions
from all types of power plant. Oxides of nitrogen also cause acid rain, in
combination with moisture in the atmosphere, and ozone depletion at high
altitudes, which may result in a reduction in the protection from ultra violet rays
GAS TURBINE EMISSIONS 259
provided by the ozone layer, leading to increases in the incidence of skin cancer.
UHC may also contain. carcinogens, and CO is fatal if inhaled in significant
amounts. \1I71th an worldwide demand for power and transporta-
tIOn, control of emISSIOns IS becoming increasingly important.
Operational considerations
The problem of controlling emissions is complicated by the fact that gas turbines
be operated over a wide range of power and ambient conditions. The off-
performance of gas turbines will be discussed in Chapters 8 and 9, where
It WIll. be shown that. different engine configurations will have widely varying
As a example, power changes on a single-shaft
gas tmbllle dnvmg a wIll occur with the compressor operating at COll-
speed and constant airflow, while an engine with a free power
turbllle at dIfferent compressor speeds, and hence airflows, as the
settlllg 18 changed. Multi-spool and variable geometry compressor systems
mtlOduce problems. In the past, the function of the engine control system
was to prOVIde the correct .amount of fuel for all operating conditions, both steady
and enUSSlOns control was a function of the combustor design.
WIth engm.es, however, control system plays a major role in adjusting
fuelj arr ratIOS t.o mm11ll1ze over the complete operating range; this may
supplymg fuel to dIfferent zones of the combustor at different operating
condItIons, and IS only possible with the advent of sophisticated digital fuel
control systems. "0
gas turbines being increasingly used for large base-load electricity
generatlOn, they consunle very large quantities offuel; they are normally operated
at fixed ratmg for long periods, but may be operated at lower loads for shorter
penods. installations are generally designed to operate at full load
on a contl11uous baSIS, but are more likely to be located close to residential areas
than 18 the case for power stations. Pipeline gas turbines are often located in
regions, and will mos:ly operate at steady power settings; at the present
tune they ha:e not been subjected to the same emissions standards as power
statlOns, but It IS likely that this will Challge in the near future.
. Aircraft engines have two quite different requirements. The first is for very
hIgh at low power, because of the large amounts of fuel
burned dmlllg taXll11g and ground manoeuvring. The primary problem here is the
reductIOn ofUHC. At take-off power, climb and cruise the main concern is NO
The Civil Aviation Organization (ICAO) sets standards on
basis, both for the take-off alld landing cycle and also for cmise at
hIgh altItude; the first is concerned with air quality in the vicinity of airports and
the second wlth ozone depletion in the upper atmosphere.
Pollutant formation
For many years the attention of combustion engineers was focussed on the design
and development of high efficiency combustors that were rugged and durable,
260
COMBUSTION SYSTEMS
followed by relatively simple solutions to the problem of smoke. When the re-
quirements for emission control emerged, much basIc research was necessary to
establish the fundamentals of pollutant formatIOn.
The single most important factor affecting the of is the jlam.
e
temperature; this is theoretically a maximum at conditions and Will
fall off at both rich and lean IJ1i?ctures. Unfortunately, while NOx could be
by operating well away from stoichiometric, this results in formatIon.of
both CO and UHC, as shown in Fig. 6.12. The rate of formatIOn of.NOx var:
es
exponentially with the flame temperature, so the key.to NOx IS reductIOn
of the flame temperature. The formation of NOx IS slIghtly dependent the
residence time of the fluid in the combustor, decreasing in a linear fashIOn as
residence time is reduced; an increase in residence time, however, has a
favourable effect on reducing both CO and UHC emissions. Increasing the
residence tinle implies an increase in combustor area or volume.
It is important to understand the relationship between and the key
cycle parameters of pressure ratio and turbine inlet temperature. InltIal.attempts to
correlate emissions were based on the pollutants produced for a given power
level but it was soon realized that the rate of fornmtion of pollutants depends on
the conditions in the combustor, which are functions of the basic cycle
parameters. One of the most widely used correlations was due to Lippfert, Ref.
(5), who found that NOx emissions increased with combustor mlet temperature
(i.e. compressor delivery temperature). His correlation was based on results fr?m
a range of engines from small APUs to the high bypass turbofans of the penod
such as the JT-9D. This work led to the unwelcome conclusion that the use of a
high pressure ratio to obtain high efficiency will have a deleterious effect. on
emissions. Fortunately this is no longer true. Once deSigners of combustIOn
systems began to understand the problem and incorporate necessary measures
to minimize NO
x
, it was found that cycle pressure ratio did not have a ma!or
effect: it is the flame temperature which is important. Figure 6.13, a correlatIOn
t
'" c
o
'w
'"
E
w
Stoichiometric mixture
i/
i
I
i
/\
I
i
I
'-"i co?/

\
Rich Lean Air/fuel ratio
,
FIG. 6.12 Depelldellce of emissions 011 fileR/ai, ratio
GAS TiJRBlNE EMISSIONS
'0
>
E
0.
Eo
0'
Z
"0
Q)
to
i"
"
o
10
/
/
/
/
/
1650 1700 1750 1800 1850
Flame temperature![KJ
./
/'
I
1900 1950
FIG. 6.13 Effect IIf lfIiame tempemihue 0111 NOx emissiol1ls
261
from Leonard and Stegmaier, Ref. (6), shows that NOx emissions can be more
than halved by reducing the flame temperature from 1900 K to 1800 K. In this
figure NOx is measured in units of 'corrected ppmvd', which means 'parts per
million by volmne of dry exhaust gas corrected to standard pressure and
temperature'. In the next subsection we will consider some of the methods used to
l1U11ll11lZe enusslOns.
Methods for reducing emissions
Emissions control requirements for NOx were first applied to stationary gas
turbines in the Los Angeles area in the early 1970s, where it was found
that an emissions level of about 7.5 parts per million by volume of eLry exhaust
(75 ppmvd) when burning oil could be achieved by injecting water into the
combustor to lower the flame temperature. Because of this the Environmental
Protection Agency (EPA) set this level as a standard for new installations; with
the increasing use of gas turbines and the longer running hours associated
with base-load or cogeneration plant these limits have become ever more restric-
tive. Areas which are particularly environmentally sensitive, such as Southern
California and Japan, have promulgated even lower levels. Em-ope also began to
introduce their own standards, in this case specifying pollutants in terms of
mg/m
3
of exhaust flow. Thus there are many varied requirements to be met in
different parts of the world and even in different areas of the same country.
The picture is rather more coherent with respect to emissions from civil
aircraft, where standards are set on a world-wide basis by the International Civil
Aviation Organization (ICAO) with limits being set following extensive
deliberations by multinational committees. Limits are specified in terms of the
amount of pollutants produced per unit thrust or unit mass of fuel blL111ed, with
emphasis on both the take-offllanding cycle and the high altitude cruise
condition.
262
COMBUSTION SYSTEMS
The emission of pollutants into the atmosphere may be tackled either during
the combustion process, or post combustion by exhaust clean-up; the latter
method is widely used in coal bnrning steam plant, e.g. as flue gas
desulphurization. Gas turbine designers have chosen to attack the problem by
focusing on new combustor designs, although sometimes exhaust clean-up is also
used to obtain very low emissions. There are basically three major methods of
minimizing emissions: (i) water or steam injection into the combustor, (ii)
selective catalytic reduction (SCR) and (iii) dry low NOx (so called because no
water is involved).
(i) Water or steam injection
As stated earlier, the purpose of water injection is to provide a substantial de-
crease in flame temperature. In the first installations where this was used it was
found that to obtain 75 ppmvd it was necessruy to use half as much water as fuel,
resulting in approximately 40 per cent reduction in NOx. The atnounts of water
required are substantial, and demineralized water must be used to prevent cor-
rosive deposits in the turbine. For lower levels of NO x the water/fuel ratio may be
1.0 or even higher. Unfortunately, the small increase in power due to the higher
mass flow through the turbine is offset by a decrease in thermal efficiency. It is
also found that increasing the water/fuel ratio, while continuing to decrease NOx ,
increases both CO and UHC emissions. In many locations water is scarce or
expensive, and is obviously difficult to use on a year round basis in countries
where the atnbient temperature is well below freezing for months on end. To give
some indication of the amounts of water required, a 4 MW gas turbine needs
about 4 million litres of water annually. It is clear that water injection introduces
many new problems for the operator, but it has been successful in allowing
operation at low levels of NOx while better methods are developed.
Steatn injection operates on the srune principle, and is often available at high
pressure from the waste heat boiler (WHB) in combined cycle or cogeneration
installations. With high-efficiency engines operating at compressor delivery
pressures in excess of 30 bar, the availability of steam at even higher pressures is
necessary. As an example, a 40 M\V aero-derivative gas turbine may use about 25
per cent of the steam produced in the WHB for NOx control.
(ii) Selective Catalytic Reduction
SCR has been used in situations where extremely low « I 0 ppmvd) limits of
NO
x
have been specified. This is a system for exhaust clean-up, where a catalyst
is used together with injection of controlled atnounts of ammonia (NH3) resulting
in the conversion of NOx to N2 and H20. The catalytic reaction only occurs in
a limited temperature range (285-400°C), and the system is installed midway
through the Vi/1IB; because of the limited temperature range, SCR can be used
only with waste heat recovery appiications. The use of SCR introduces a whole
range of new problems including increased capital cost, handling and storage of a
noxious fluid, control of NH3 flow and difficulty in dealing with variable loads.
SCR systems have been used in gas turbines burning natural gas, and although
GAS TURBINE EMISSIONS 263
proposed for units burning oil it appears that none have actually been built at the
time of writing. The need arose because of the installation of dual-fuel engines
which may be required to operate for relatively short periods on oil when the gas
supply is interrupted. It appears likely that SCR will be a method which is
superseded as dry low NOx systems enter the market.
(iiZ) Dry liJw NOx
Currently all designers of gas turbines are heavily involved in research and de-
velopment into combustor designs capable of operation at low levels of NOx
without any requirement for water, i.e. dry systems. Remembering the exponen-
tial variation of NOx emissions with flrune temperqture, it is possible to consider
either lean bnrning or rich bnrning in the primary zone to achieve the necessaty
reduction in flame temperature. Both approaches have been investigated; in the
case of rich burning, there is the probability of smoke being produced in the
primary zone. This led to investigation of the rich burn/quick quench concept in
which the combustor is designed with two axial 'stages', with rich combustion in
the first followed by large amounts of dilution air in the second. Most manu-
facturers, however, have decided to use the lean bnrn approach, with the major
modification of pre-mixing the air and fuel prior to combustion. The use of lean
bum, however, leads to problems of maintaining stable combustion as power is
reduced; this leads to significant increase in the complexity of the engine control
system. Many different design approaches have been used by the principal engine
manufacturers, and some of these will be considered in the next subsection.
Design of dry low NOx systems
There are three approaches to the design of dry low NOx systems, depending on
the type of gas turbine. With the industrial gas turbine, considerable space is
available and the fuel is usually natural gas. For aircraft engines volume and
frontal area must be kept to a minimum, and liquid fuel is always used. The third
type requiring special consideration is the aero-engine which has been modified
for use as a land-based unit: it is then necessalY for the new combustor to be
capable of being retrofitted in place of the original. There are a large number of
these aero-derivative gas turbines in use which may be subject to more severe
emissions limits than when they entered service. This is also true for industriai
engines, where the problem is eased by the less stringent demands on space.
H is only possible to give a very brief overview of some of the approaches
taken, but the wide range of solutions arrived at gives some indication of the
amount of on-going research and development. Pen important concept for
emissions reduction is fuel staging, where the total engine fuel flow is divided
into two parts which are supplied separately to two distinct combustion zones.
One of these is fuelled continuously, providing fuel for starting and idling, acting
as a pilot stage. The bull, of the fuel is bnrned in the second zone, which serves as
the main stage of combustion. Fuel staging is widely used in modem combustors,
and many different arrangements are found in practice.
264 COMBUSTION SYSTEMS
(i) Industrial gas turbines
Genera! Electric have always used multiple combustors in a cannular configura-
tion in their heavy industlial gas turbines, so their approach is based on modifi-
cations to the individual cans. The original cans had a single bU111er, but this has
been replaced by a ring of six primary dual-fuel burners surrounding a single
secondary dual-fuel burner. The combustor is shown schematically in Fig. 6.14.
A convergent-divergent at the end of the primary zone serves to accel-
erate the flow to prevent upstream propagation of the flame from the second stage
to the first, that could happen under some modes of operation. The venturi also
produces a recirculation zone on its downstream face to stabilize the flame. The
split of fuel between the primary and secondary burners changes with load. At
start up, and up to 20 per cent load, all the fuel is supplied to the primary burners
and combustion takes place in the first stage. From about 20-40 per cent load,
about 30 per cent of the fuel is supplied to the secondary burner and lean
combustion takes place in both stages. At about 40 per cent load, all fuel is
supplied to the secondary burner with no combustion in the first stage; this is a
transient situation leading to fully pre-mixed operation. From 40-100 per cent
load, fuel is supplied to all burners (approximately 83 per cent to the primary set),
the fuel is pre-mixed in the primary stage and then combustion takes place only in
the second stage. The development of this combustor is described by Davis and
Washam Ref. (7).
Siemens and ABB have adopted the approach of developing single burners,
which can then be used in the numbers required for the specified power output.
The Siemens development is refelTed to as a hybrid burner, operating in different
modes at low and high power. At low power only a pilot burner is lit, providing a
diffusion flame. At about 40 per cent load, the mode is changed to pre-mixed lean
burning, giving very low emissions of both NO., and Co.. The burners are
designed to burn gas, liquids, or gas and liquids simultaneously, and also to use
water or steam for No.x reduction. The variation of emission levels with load is
shown. in Fig. 6.15. A rotating shutter ring is used to control fuel/air ratio at low
powers to keep emissions at an acceptable level, and the engine air flow is
modified by use of variable inlet guide vanes (VIGVs) in the compressor.
Maghon et al. (8) describe the development of this burner. The ABB approach is
Primary
burners (6)
Secondary
burner
Lean and pre-mixing
primary zone
/
Venturi
Dilution
zone
FIG. 6.14 Can for General Electric low NOx comiJustor
GAS TURBINE EMISSIONS
E
Q.
C.
100
80
"=:" 60
<J)
c:
o
·w
<J)
40
o
z
20
. Diffusion I Pre·mix
burning i burning
:( I >-
I
I
I

__ ____ ____
a 20 40 60 80 100
Generator outputl[%)
FIG. 6.15 Variation of emissiolls witilloaG (hybrid Immel')
265
novel, using a double-cone burner where flame stabilization is achieved by vortex
breakdown at exit from the burner as shown in Fig. 6.16. This is another exanlple
of a pre-mixed lean burn system, and the burner is used in both silo type and fully
annular combustors. The burners are alTanged in concentlic circles, and various
segments are lit in sequence as power is increased. Development of this concept is
described by SatteJp:1eyer et al. (9).
The methods described above are all based on the requirements of single-shaft
units which operate at constant speed. Solar Gas Turbines build smaller units, up
to 10 MW, most of which use free turbines requiring variation in compressor
speed, and hence airflow, as load is changed. Their solution also uses the pre-
mixed lean burn concept, but requires the use of overboard bleed from
compressor discharge at low power to provide the correct fuel/air ratio to
maintain the specified emission levels. This is a simple and reliable method, but a
penalty in thermal efficiency is inculTed when using bleed. Etheridge (l0)
FIG. 6.16 ABE Dual-fllel double-cone bumer [courtesy ABE]
Air
Fuel and air are mixed
at the inlet slots
266 COMBUSTION SYSTEMS
(a) Parallel staging (b) Axial staging
FIG. 6.17 lLow-emissimn aero-engine combllstors
describes field experience with this system, including the effects of ambient
temperatures on emissions; NOx increases at low' temperatures, whereas CO
increases at high temperatures. This once again illustrates the many problems
facing the combustion designer.
(ii) Aircraft gas turbines
ICAO sets standards for emissions for both the take-off/landing cycle and the
cruise condition. The take-off/landing cycle prescribes standard times at take-off
and approach power, and the limits are specified in telms of g/kN of take-off
thrust. At typical subsonic cruise conditions a modem turbofan produces about
12 g of NO x per kg of fuel burned, compared to about 34 g/kg at take-off; this is
primarily due to the reduction in turbine inlet temperature at cruise. Emissions
of UHC and CO are negligible in comparison, as low as 0·1 and 0·6 g/kg re-
spectively, both at take-off and cruise. These data are from Bahr Ref. (11), who
also shows that for a modem twin engine transport operating over an 800 km
range approximately 25 per cent of the emissions are produced during the take-
off/landing cycle, with the remainder during climb/cruise/descent; approxi-
mately 86 per cent of the total emissions are NOx.
The aircraft problem can be solved by using lean primary zone mixtures, but
great care must be taken that this does not result in unacceptable losses in ignition
and blowout performance or unacceptable increases in UHC and CO emissions at
idle. These conflicting requirements led to the development of combustors using
two or more combustion zones which may be staged, either in parallel or in series.
General Electric developed a double annular combustor for civil aircraft
engines, resulting in parallel staging. The arrangement is shown schematically in
Fig. 6.l7(a); during starting, idle and relighting at altitude only the outer ring of
burners is fuelled. At nOlmal operating conditions both annuli are fuelled and the
fuel flow split can be adjusted to provide lean fuel/ air ratios in both zones at high
powers. Future requirements for lower emissions may require pre-mixing as well,
but this adds considerably to the complexity of the combustor design; Bahr, Ref.
(11), discusses some of the problems which must be overcome.
The alternative approach of axially staged combustion was adopted by
International Aero Engines for the design of a low NOx combustor for the V2500,
COAL GASIFICATION 267
as desclibed by Segelman et at Ref. (12). The combustor layout is shown in Fig.
6.17(b), and results in alonger combustor than could be achieved with a double
annular arrangement. This layout, however, was chosen because it was considered
to have greater potential for emissions reduction, and the length penalty could be
reduced by using the staggered inline arrangement shown.
(iii) Aero-derivative engines
Aero-derivative units in service in the mid 1990s were based on aero-engines
developed in the early 1970s which incorporated very compact annular combus-
tors. These engines were not designed with low emissions in mind, and having
high design pressure ratios the level of emissions was high. This was overcome
by using water or steam injection, especially in cogeneration applications. As
pointed out earlier, water or steam injection introduces many operational probl-
ems and is particularly inappropriate for pipeline applications. Manufacturers of
aero-derivative engines have therefore been forced to develop dry low NOx
systems, which have to be retrofittable to engines in the field.
The well proven concept of lean pre-mixed combustion has been adapted. It is
interesting to contrast the solutions arrived at by General Electric and Rolls-
Royce. The General Electric system increases the combustor volume to get longer
residence tinles for low UHC and CO, and this is done by increasing the depth of
the annulus; three concentric circles of burners are used, with a centre-body
between each circle. The outer two circles each have 30 burners, while the iuner
has 15; the burners .. are mounted on a stallc which may have either 2 or 3 burners,
to allow for the reduced number in the inner circle. As in the case of the ABB
design, they can be lit in sequence as power is increased. Leonard and Stegmaier,
Ref. (6), discuss the development of the low emissions combustor for the
LM6000, and the same technology will be applied to the other aero derivatives in
the General Electric stable (LM2500 and LM1600).
Rolls-Royce have moved away from the fuHy annular combustor to separate
combustion cans ananged radially inwards; this gives the increased volume
required for increased residence time, without the need for any increase in length.
TIns system was originally developed for the RB 211, Ref. (11), but a similar
system is used on the industrial version of the Trent. Figure 1.16 shows the aero
and industrial versions on the same centreline and clearly shows the differences
between the two combustors. Corbett and Lines, Ref. (14), show that the need for
introducing the fuel in stages results in a very sophisticated control system; the
problems are more complex than on the single-shaft machines described earlier,
because the compressor speeds and airflow are changing continually on the multi-
spool engine as load is increased.
6.8 Coal gasification
The prospect of future shortages of natural gas has led to a resurgence of interest
in the use of coal. Several closed-cycle gas turbines, with external combustion,
268 COMBUSTION SYSTEMS
. - .
were built in· Germany and ran successfully for over 120 QOO hours . .All but one
have been decommissioned. They were small units of 2-17 MW and were not
economic with the current relative price of coal and gas. This German experience
is described in Ref. (15).
The approach receiving most attention at the present time is integration of a
coal gasification process with a,combined cycle plant, described by the acronym
!GCC. The principle was discussed in section 1.7, and Fig. 1.21 showed a
diagrammatic sketcb of a possible scheme. The actual gasification takes place in a
pressure vessel where coal is reacted with an oxidant (steam, oxygen or air)
yielding a 'dirty' gas and slag or ash. The dirty gas may contain particulates
which must be removed by cyclone separators to avoid damage to the turbine and
also undesirable chemicals which cause corrosion and pollution. In particular,
sulphur is likely to be present in the form of either HzS or S02; coal with up to 7
per cent sulphur may have to be used. In many systems the sulphur is reduced to
its elemental form and is then sold as a valuable by-product. The cleaned fuel gas
is then delivered to the combustor. It should be noted that the purification (clean-
up) process often requires cooling of the dirty gas, and the heat removed is
transferred to the steam raising process.
Three types of gasifier are under development: (i) moving bed, (ii) fluidized
bed and (iii) entrained bed in which the coal and oxidant are introduced together.
The oxidant preferred seems to be either air or oxygen, rather than steam. Some
relevant characteristics of four gasifiers are given in the following table:
Bed Type Feed Oxidant O2 flow Gas Temp
(kglkg coal) (K)
Texaco Entrained Slurry O2 0·9 1480
Shell Entrained Dry O2 0·85 1750
Combustion Engineering Entrained Dry Air 0·7 1280
British GaslLurgi Moving Dry O2 0·5 1000
The gas temperature is that following the gasification process, and a high value
requires more complex integration into the steam raising process. The quantity of
oxygen required per unit mass of coal is important, as an oxygen blown gasifier
requires the incorporation of an air separation unit (ASU), which separates the
oxygen from the nitrogen in some of the compressor delivery air. The ASU is a
complex and expensive component with a considerable power demand; the
chemical process required for separation is beyond the scope of this book. With
air blown gasifiers the resulting gas has a low calorific value (4500-5500 leJ/m3)
because of the large nitrogen content; oxygen blown units give gas with a
significantly higher calorific value (9000-13000 kJ/m
3
).
Two major demonstration projects have been the Cool Water plant in
California and the Buggenum plant in Holland. Cool Water used a Texaco gasifier
mated to a General Electric gas turbine, which demonstrated that a standard
80 MW gas turbine could operate in IGCC configuration on an electric utility
COAL GASIFICATION. 269
system. The plant was operated from 1984 to 1989, and completed 27000 hours
using four different c o a l s ~ A dual-fuel system was used, permitting operation on
distillate fuel during periods when gasifier maintenance was required. The
emissions achieved were excellent, with NOx values of20 ppm; the ash produced
was non-hazardous and saleable. The Cool Water plant was purchased by Texaco
and will continue to be used to provide electricity while gaining further
operational experience on the gasifier. The coal consumption was about 1100
tonne/day and the maximum output was 118 MW; this is too small for a
commercial plant, but successfully demonstrated the concept at large scale. It is
anticipated that a modern gas turbine and improved gasifier could give an rGCe
efficiency of 40-42 per cent, compared with conventional co!!1-fired steam plant
efficiencies of 36-38 per cent.
A larger demonstration unit, with an output of 250 MW, started operations at
Buggenum in Holland at the end of 1993. The !GCC uses a Siemens gas turbine
combined with a Shell gasifier; the gas turbine has to be capable of operating on
both natural gas and coal gas, with natural gas used for start up and operation
when the gasifier requires maintenance. The steam turbine is also supplied by
Siemens, and the heat to the steam cycle is transferred both from the gas turbine
exhaust and the heat rejection from the gas clean-up process. The gasification unit
has a capacity of 2000 tonne/day and oxygen for the gasifier is supplied by an
ASU with a capacity of 1700 tonne/day. The air supply for the ASU is bled from
the compressor discharge, saving the need for a separate compressor. The coal gas
is desulphurized to' produce sulphur as a saleable commodity. Figure 6.18 shows
the plant layout, where it can be seen that the gas turbine/steam turbine/generator
is dwarfed by the gasification plant, the ASU and the sulphur processing plant.
The clean coal gas, after sulphur removal, has a composition of approximately
65% CO, 30% Hz, I % CO2, I % H20 and 3% Nz + Argon. Both CO and Hz have
adiabatic flame temperatures that are higher than for natural gas, which would
result in a substantial increase in NOx' The technology of pre-mixing the fuel and
air to minimize NOx cannot be used because of the high concentration of Hz,
which would form an explosive mixture. The adiabatic flame temperature is
lowered by another method, dilution with inert constituents; 16 per cent of the
compressor delivery air is used for the ASU and the nitrogen resulting from the
separation process is used to dilute the coal gas. Further dilution is provided by
water vapour resulting from saturating the gas with warm water, and the low
flame temperature results in very favourable levels of NOx'
Although designed as a demonstration plant, to prove the technology at large
scale, a very creditable thermal efficiency of 43 per cent is obtained. It is predicted
that with developments in both the gas turbine and the gas purification process the
efficiency could be raised to 48 per cent. It also appears that the technology used
would be capable of being scaled up to 400 MW, and possibly even to 600 Mw.
While coal gasification has still to be fully proved in commercial service,
progress to date is extremely encouraging and suggests that it may be
economically viable for large power stations. Many of the large natural gas-
fired combined cycles commissioned in the early 1990s could eventually be
270 COMBUSTION SYSTEMS
FIG. 6.18 IGCC plant, Buggenum [courtesy Demkolec)
converted to coal gasification if the supply of natural gas becomes inadequate to
meet growing demand. When planning expansion of an electric power system, by
adding power stations, utilities may do this in phases. In the first phase, simple
cycle units may be installed quicldy to provide the initial increase in capacity;
they do not, of course, have a high enough thermal efficiency for long term use as
base-load plant. In the second phase, these could be converted to combined cycle
operation; steam turbines take longer to manufacture and install, largely because
of the extensive civil engineering work required. The second phase may be
completed about one or two years after the first phase. Finally, if the supply of
natural gas becomes a problem the combined cycle plants could be converted to
an IGee system.
7 Axial and radial flow turbines
As with the compressor, there are two basic types of turbine-radial flow and
axial flow. The vast majority of gas turbines employ the axial flow turbine, so
most of the chapter will be devoted to the theory of this type. The simple mean-
diameter treatment is described first, and the essential differences between steam
and gas turbine design are pointed out. The application of vortex theory is then
discussed, followed by a description of the method of applying cascade test data
to complete the design procedure and provide an estimate of the isentropic ef-
ficiency. Recent developments in the calculation of blade profiles to give specified
velocity distributions are also mentioned. Blade stresses are considered briefly
because they have" a direct impact upon the aerodynamic design. The chapter
closes with a section on the cooled turbine, followed by some material on the
radial flow turbine.
The radial turbine can handle low mass flows more efficiently than the axial
flow machine and has been widely used in the cryogenic industry as a
turboexpander, and in turbochargers for reciprocating engines. Although for all
but the lowest powers the axial flow turbine is normally the more efficient, when
mounted back-to-back with a centrifugal compressor the radial turbine offers the
benefit of a very short and rigid rotor. This configuration is eminently suitable for
gas turbines where compactness is more important than low fuel consumption.
Auxiliary power units for aircraft (APUs), generating sets of up to 3 Mw, and
mobile power plants are typical applications.
7.1 Elementary theory of axial flow turbine
Figure 7.1 shows the velocity triangles for one axial flow turbine stage and the
nomenclature employed. The gas enters the row of nozzle bladest with a static
pressure and temperature PbT1 and a velocity Cj, is expanded to P2,T2 and leaves
t These are also known as 'stator blades' and 'nozzle guide vanes'. We shall use suffix N when
necessary to denote quantities associated with the nozzle row, but will often use the term 'stator' in the
text. Suffix s will be used to denote 'stage'.
272
Annulus t
area A
2 3
FIG. 7.1 Axiaillow turiJine stage
AXIAL AND RADIAL FLOW TURBINES
Nozzle blades I
~
with an increased velocity C2 at an angle (X2' t The rotor blade inlet angle will be
chosen to suit the direction P2 of the gas velocity V2 relative to the blade at inlet.
P2 and V2 are found by vectorial subtraction of the blade speed U from the
absolute velocity C2. After being deflected, and usually further expanded, in the
rotor blade passages, the gas leaves at P3,T3 with relative velocity V3 at angle P3.
Vectorial addition of U yields the magnitude and direction of the gas velocity at
exit from the stage, C3 and 0(3. 0(3 is Imown as the swirl angle.
In a single-stage turbine CI will be axial, i.e. (XI = 0 and C! = Cal' If on the
other hand the stage is typical of many in a multi-stage turbine, C! and (X! will
probably be equal to C3 and (X3 so that the same blade shapes can be used in
successive stages: it is then sometimes called a repeating stage. Because the blade
speed U increases with increasing radius, the shape of the velocity triangles varies
from root to tip of the blade. We shall assume in this section that we are talking
about conditions at the mean diameter of the annulus, and that this represents an
average picture of what happens to the total mass flow m as it passes through the
stage. This approach is valid when the ratio of the tip radius to the root radius is
low, i.e. for short blades, but for long blades it is essential to account for three-
dimensional effects as shown in subsequent sections.
(Cw2 + Cw3) represents the change in whirl (or tangential) component of
momentum per unit mass flow which produces the nseful torque. The change in
axial component (Ca2 - Ca3 ) produces an axial thrust on the rotor which may
supplement or offset the pressure thrust arising from the pressure drop (P2 - P3)'
In a gas turbine the net thrust on the turbine rotor will be partially balanced by the
thrust on the compressor rotor, so easing the design of the thrust bearing. In what
t In the early days of gas turbines the blade angles were measured from the tangential direction
following steam turbine practice. It is now usual to measure angles from the axial direction as for axial
compressor blading.
ELEMENTARY THEORY OF AXIAL FLOW TURBINE
273
follows we shall largely restrict our attention to designs in which the axial flow
velocity Ca is constant through the rotor. This will imply an annulus flared as in
Fig. 7.1 to accommodate the decrease in density as the gas expands through the
stage. With this restriction, when the velocity triangles are superimposed in the
usual way we have the velocity diagram for the stage shown in Fig. 7.2.
The geometry of the diagram gives immediately the relations
U
C = tan (X2 - tan P2 = tan P3 - tan (X3 (7.1)
a
Applying the principle of angular momentum to the rotor, the stage work output
per unit mass flow is
~ = U(Cw2 + Cw3 ) = UCa(tan (X2 + tan (X3)
Combining with (7.1) we have Ws in tenns of the gas angles associated with the
rotor blade, namely
~ = UCa(tan P2 + tan P3) (7.2)
Note that the 'work-done factor' required in the case of the axial compressor is
unnecessary here. This is because in an accelerating flow the effect ofthe growth
of the boundary layer along the aunulus walls is much less than when there is a
decelerating flow with an adverse pressure gradient.
From the steady flow energy equation we have Ws = c/1Toso where !'lTos is the
stagnation temperat,ure drop in the stage, and hence
cp!'lTos = UCa(tan P2 + tan P3) (7.3)
When the stage inlet and outlet velocities are equal, i.e. CI = C3, (7.3) also gives
the static temperature drop in the stage I'lTs. In Chapter 2 we used as typical of
combustion gases the average values
cp = 1·148 leJ/kg K; l' = )·333 or 1'/()! - 1) = 4
and noted that t ~ e y were consistent with a gas constant R of 0·287 kJ/kg K. For
prelll1llnary deSIgn calculations these values are quite adequate and we shall use
them throughout this chapter. With velocities in mis, !'lTo", in Kelvin units is
conveniently given by
!'lTos = 8· 71 C ~ o ) G;o) (tan P2 + tan P3) (7.4)
u
FIG. 7.2 Velocity diagram
274 AXIAL AND RADIAl FLOW TURBINES
The stagnation pressme ratio of the stage POJ/P03 can be found from
[ (
1 )(l'-l)/Y]
!'lTos = 11sTol 1 - -_--
POJiP03
(7.5)
where l1s is the isentropiy stage efficiency based on stagnation (or 'total') tem-
peratme. Equation (7.5) is simply equation (2.12) applied to a stage, and 1]s is
often called the total-to-total stage efficiency. It is the appropriate efficiency if the
stage is followed by others in a multi-stage turbine because the leaving kinetic
energy /2) is utilized in the next stage. It is certainly also relevant if the stage
is part of a tmbojet engine, because the leaving kinetic energy is used in the
propelling nozzle. Even if it is the last stage of an industrial plant exhausting to
atmosphere, the leaving kinetic energy is substantially recovered in a diffuser or
volute and, as explained under the heading 'Compressor and turbine efficiencies',
section 2.2, we can put P03 = Pa and regard 1]s as Lhe combined efficiency of the
last stage and diffuser. [Although we shall not use it, it should be noted that a
total-to-static isentropic efficiency is sometimes quoted for a turbine as a whole
and for a stage, and it would be used where it is desirable to separate the tmbine
and exhaust diffuser losses. Applied to the stage, we would have
total-to-static efficiency = Tal _. T03
Tal --
where is the static temperatme reached after an isentropic expansion from POI
to P3' It assumes that as far as the tmbine is concerned all the leaving kinetic
energy is wasted, and its value is somewhat less than the total-to-total efficiency
which we shall use here.]
There are three dimensionless parameters found to be useful in tmbine design.
One, which expresses the work capacity of a stage, is called the blade loading
coefficient or temperature drop coefficient t/I. We shall adopt NGTE practice and
define it as cp!'lTos/ U
2
, although cp!'lTos/U
2
is also used. Thus from equation
(7.3),
(7.6)
Another useful parameter is the degree of reaction or simply the reaction A.
This expresses the fraction of the stage expansion which occms in the rotor, and it
is usual to define it in terms of static temperature (or enthalpy) drops rather than
pressure drops, namely
A=Tz -T3
T1 - T3
For the type of stage we are considering, where Ca2 = Ca3 = Ca and C3 = Cl , a
simple expression for A can be derived as follows. From (7.4),
CpCTl - T3) = CpCTOl - T03 ) = UCactan (32 + tan (33)
ELEMENTARY THEORY OF AXIAL FLOW TURBIl\'E 275
Relative to the rotor blades the flow does no work and the steady flow energy
equation yields
cpCTz - T3) = !cvl- Vi)
And thus
= ! C;(sec
2
(33 - sec
2
{32)
= ! /33 - tan
Z
{32)
A = (tan (33 - tan (32)
(7.7)
The third dimensionless parameter often referred to in gas tmbine design
appears in both equations (7.6) and (7.7): it is the ratio Ca/U called the flow
coefficient ¢. (It plays the same part as the blade speed ratio U/Cj used by steam
turbine designers.) Thus (7.6) and (7.7) can be written as
t/I = 2¢(tan /32 + tan (33) (7.8)
1>
A = "2 (tan (33 - tan (32) (7.9)
The gas angles can now be expressed in terms of t/I, A and ¢ as follows. Adding
and subtracting (7.8) and (7.9) in tum we get '
1 1
tan (33 = 24/21/1 + 2A) (7.10)
1 1
tan (32 = 2¢ (2'" - 2A) (7.11)
Then using relations (7.J),
1
tan 1X3 = tan (33 - ¢
1
tan 1X2 = tan (32 + ¢
(7.12)
(7.13)
Even with the restrictions we have already introduced (that is, Ca3 = Ca2 and
C3 = Cl ), and remembering that stressing considerations will place a limit on the
blade speed U, there is still an infinite choice facing the designer. For example,
although the overall tmbine temperatme drop will be fixed by cycle calculations,
it is open to the designer to choose one or two stages of large ljI or a larger
number of smaller ljI. To limit still further our discussion at this point, we may
observe that any turbine for a gas tmbine power plant is essentially a low pressure
ratio machine by steam turbine standards (e.g. in the region of 10: 1 compared
with over 1000: 1 even for cycles operating with subcritical steam pressures).
Thus there is little case for adopting impulse stages (A = 0) which find a place at
the high pressure end of steam turbines. hnpulse designs are the most efficient
type for that duty, because under such conditions the 'leakage losses' associated
with rotor blade tip clearances are excessive in reaction stages. It must be
276 AXIAL AND RADIAL FLOW TURBINES
remembered that at the high pressure end of an expansion of large pressure ratio
the stage pressure differences are considerable even though the stage pressure
ratios are modest. Let us then rule out values of A near zero, and for the moment
consider 50 per cent reaction designs. Our generallmowledge of the -:vay nature
behaves would suggest that the most efficient design is likely to be achieved when
the expansion is reasonably ev,enly divided between th.e stator and rotor rows. We
shall see later that the reaction will vary from root to up of the blade, but here we
are thinking of 50 per cent reaction at the mean diameter.
Putting A = 0·5 in equation (7.9) we have
1
- = tan /33 - tan /32
cp
Direct comparison with relations (7.1) then shows that
/33 = (Xz and /33 = (X3
(7.14)
(7.15)
and the velocity diagram becomes symmetrical. Further, if we are considering
a repeating stage with C3 = CI in direction as well as magnitude, we have (Xl =
(J(3 = /32 also, and the stator and rotor blades then have the same inlet and outlet
angles. Finally, from (7.10) and (7.15) we have for A=0·5,
I}! = 4cp tan /33 - 2 = 4cp tan (Xz - 2 (7.16)
and from (7.11) and (7.15) we get
I}! = 4cp tan /3z + 2 = 4cp tan 0:3 + 2 (7.17)
Equations (7.15), (7.16) and (7.17) give all the gas angles in terms of I}! and cp.
Figure 7.3 shows the result of plotting nozzle outlet angle (xz and stage outlet
swirl angle (X3 on a I}!-cp basis.
Because the blade shapes are detennined within close limits by the gas angles,
it is possible from results of cascade tests on of blades to predict the
losses in the blade rows and estimate the stage effiCiency of the range of 50 per
cent reaction designs covered by Fig. 7.3. One such estimate is shown by the
efficiency contours superimposed on the I}!-cp plot. The of IJs on the
contours represent an average of detailed estimates quoted ill Refs (1) and (2.)'
Many assumptions have to be made about blade profile, blade aspect rano
(height/chord), tip clearance and so on, and no reliance should be placed upon the
absolute values of efficiency shown. Nevertheless a Imowledge of the general
trend is valuable and even essential to the designer. Similar curves for other
values of reaction are given in Ref. (2).
We may note that designs having. a low and low cp yield. the
efficiencies. Referring to the comparative velOCity diagrams also given ill Flo. 7.J
(drawn for a constant blade speed U), we can see that low values of I}! and cp
inlply low gas velocities and hence reduced friction losses. But a low l/t
more stages for a given overall turbine output, and a low 4> mea.ns a larger
annulus area for a given mass flow. For an indusnial gas turbme when and
weight are of little consequence and a low SFC is vital, it would be sensible to
ELEMENTARY THEORY OF AXIAL FLOW TURBINE
277
6.0
5.0
60'
N
:;:,
--",
f.2
<l
4.0
,,"-
N
" "'- U
E
.!!!

3.0
Q)
C><J
0
Q
0.
"
"0
a3(=fJ21 = 10'
2.0

.,
0.
E
" (swirl)
)
rys= 0.94

1.0
lfi=1.5, ¢=0.4
0 ______
0.6 0.8 1.0
Flow coefficient ¢ = CalU
-1.0 lfi=-2.0
FIG. 7.3 50 pel' cellt reactioll designs
design with a low I}! and low 4;. Certainly in the last stage a low axial veloc'ity and
a small swirl angle (X3 are desirable to keep down the losses in the exhaust
For an aircraft propulsion unit, however, it is impOliant to keep the
weight and frontal area to a minimum, and this means using higher values of ifi
and cp. The most efficient stage design is one which leads to the most efficient
power plant for its particular purpose, and strictly speaking the optimum I}! and cp
be determined without detailed calculations of the perfonnance of the
arrcraft as a whole. It would appear from current aircraft practice that the
optimum values for I}! range from 3 to 5, with cp ranging from O.g to 1.0. A low
swirl angle ((X3 < 20 degrees) is desirable because swirl increases the losses in the
jet pipe and propelling nozzle; to maintain the required high value of I}! and low
value of (X3 it might be necessary to use a degree of reaction somewhat less than
50 per cent. The dotted lines in the velocity diagram for I}! = 4 indicate what
happens when the proportion of the expansion earned out in the rotor is reduced
and V3 becomes more equal to Vz, while maintaining U, I}! and cp constant.
We will close this section with a worked example showing how a first tentative
'mean-diameter' design may be arrived at. To do this we need some method of
accounting for the losses in the blade rows. Two principal parameters are used,
based upon temperature drops and pressure drops respectively. These parameters
278
P01 P02
P1
__ ________ __
T01 , T02 -- --fl--
T1 -- ---- () ---
1\
To2reJ'
T03 re!
-________ L\-. .,,_f"-+-7I''---.---,
\
T2
\
T2 --------7 2'

T3 -----
T3 ------------ ---- 3"
T!J --------7 3'
Entropy s
FIG. 7.4 T-s diagram flir a reaction stage
AXIAL AND RADIAL FLOW TURBINES
can best be described by sketching the processes in the nozzle rotor
passages on the T-s diagram as in Fig. 7.4. The full and cham dotted
connect stagnation and static states respectively. T02 ::= TOI no work IS
done in the nozzles; and the short horizontal portion of the fulllme represents the
stagnation pressure drop (POI - Poz) due to friction in the nozzles. The are
of course exaggerated in the figure. When obtaining the eqU1valent of
the velocity of the gas leaving the nozzle row, we may say that Ideally the gas
would be expanded from TOI to but that due to friction the temperature at the
nozzle exit is Tz, somewhat higher than The loss coefficient for the nozzle
blades may be defined either by
A ::= T1 - or Y
N
::=POI - Poz (7.18)
N C?!2cp Poz - P2
Both A and Yexpress the proportion of the leaving energy which is degraded by
friction. YN can be measured relatively easily in cascade (the bemg
modified to allow for three-dimensional effects on the same Imes as descnbed for
axial compressors), whereas AN is the more easily used in design. It will be shown
that }oN and YN are not very different numerically. .
Returning to Fig. 7.4, we see that further in. the movmg blade
passages reduces the pressure to P3' expansIOn m the whole
would result in a final temperature and m the rotor blade passages alone T3 .
Expansion with friction leads to a final temperature T3. The rotor blade loss can
be expressed by
T3 - T!j
AR::= V2/2c
31 p
ELEMENTARY THEORY OF AXIAL FLOW TURBINE
279
Note thatit is defined as a proportion of the leaving kinetic energy relative to the
row so that it can be related to cascade test results. As no work is done by the gas
relative to the blades, T03rel::= T02reI. The loss coefficient in terms of pressure drop
for the rotor blades is defined by
] T _ POZrel - P03reI
R-
P03reI - P3
We may show that A and Yare not very different numerically by the following
argument (which applies equally to the stator and rotor rows of blades although
given only for the former).
J
T _POI -Paz (POt/P02) - 1
N-
Poz - P2 1 - (P2/POZ)
Now
Pal ::=POI P2 ::= (TO})Y/(1'-I)(T2)1'!(1'-I)
P02 pz Paz T2 Toz
(
T
2
)Y/(1'-I)
= because TOI = Tn
Hence
I) - I
Y---'--'",--",-----,-,.-,.,.
N - 1 - (Tz/Toz)1'/(r-l)
[
., T - T1J1'!(Y-I)
1 +_Z __ 2 -1

1- [
T - T. JY/(Y-I)
T02
Expanding the bracketed expressions binomially and using the first terms only
(although not very accurate for the denominator) we have
YN = T2 - X TO,2::= }'N(To;) ::: AN (Toz) (7.19)
T02 - T2 T2 T2 T2
From Appendix A, equation (8), we have the general relation
Toz ::= (1 + l' - 1 Mi)
T2 2
Even if the Mach number at the blade exit is unity, as it might be for the nozzle
blades of a highly loaded stage, A = 0·86Y and thus A is only 14 per cent less
than Y.
The type of infonnation which is available for predicting values of A or Y is
described briefly in section 7.4. AN and AR can be related to the stage isentropic
efficiency 118 as follows.
TOI - T03
118 = Tal - 1 + (T03 - - T
03
)
280
AXIAL AND RADIAL FLOW TURBINES
Now a glance at Fig. 7.4 shows that
T03 - T63 :::::: (T3 -T3) = (T3 - T{) + (T3' - T3)
But = (Tz/TD because both equal (P2IP3)(Y-I)ly. Rearranging and sub-
tracting one from both sides we get
Hence
T3' - T3 = (Tz - 'or (T{ - T
3
) :::::: (T
z
- T2) i
T3 z
I'/s :::::: 1 + [(T3 - T{) + (T3/Tz)(T2 - TD]/(Tol - T03 )
1
:::::: 1 + [A'R(V; /2cp) + (T3I Tz)AN(Ci!2cp)]/(TOI - T03 )
Alternatively, substituting V3 = Ca sec th, C2 = Ca sec (X2, and
cp(TOI - T03 ) = UCaCtan P3 + tan P2)
= UCaltan P3 + tan (X2 - (U ICa)]
equation (7.20) can be written in the form
1

1 + 2 U tan P3 + tan (Xl - (U ICa)
(7.20)
(7.21)
Because Y:::::: }. loss coefficients YR and YN may replace AR and }'N in equations
(7.20) and (7.21) if desired.
For the purpose of the following example we shall assume that AN = 0·05 and
118=0·9. In suggesting }w=0·05 we are assuming that convergent nozzles are
employed and that they are operating with a pressure (POllp2) less
critical pressure ratio [(1' + 1)/2t(Y-I). Convergent-cl!vergent nozzles as m F.lg.
7.5(a) are not used, partly because they tend to be inefficient at. pressure ratios
other than the design value (i.e. at part load), and palily because high values of Cz
usually imply high values of Vl . If the Mach number relative to moving blades
at inlet, MV2, exceeds about 0·75, additional losses may be by the
formation of shock waves in the rotor blade passages (see section A.I of
Appendix A). If Vl is in fact not too high, perhaps a low .value of the flow
coefficient <p is being used as in an industrial gas turbme, there IS no reason why
convergent nozzles should not be operated at pressure ratios an
velocity which is slightly supersonic (namely 1 < Mz < 1.2): very little. additIOnal
loss seems to be incurred. The pressure at the throats of the nozzles IS then the
critical pressure, and there is further semi-controlled expansion. P2 after
the throat. As depicted in Fig. 7 .5(b), the flow is controlled by the trallmg edge on
the convex side. On the other side the supersonic stream expands as though
turning a comer: it is possible to obtain some idea of the deflection of the stream
by treating it as a Prandtl-Meyer expansion and using the method. of
characteristics (see section A.8 of Appendix A, and Ref. (2) for further detaIls).
ELEMENTARY THEORY OF AXIAL FLOW TURBINE
281
(a)
(b)
FIG. 7.5 Convergent-divergellt nozzle, alld convergellt nozzle o]perating at a pressure
ratio greater than the critic:al value
As an example of turbine design procedure, we will now consider a possible
'mean-diameter' design for the turbine of a small, cheap, turbojet unit, which
should be a single-stage turbine if possible. From cycle calculations the following
design-point specification is proposed for the turbine.
Mass flow m
Isentropic efficiency 1Jt
Inlet temperature Tal
Temperature drop TOI-T03
Pressure ratio POJ/P03
Inlet pressure POI
20 kg/s
0·9
l100K
145 K
1·873
4 bar
In addition to this information, we are likely to have the rotational speed fixed by
the compressor, the design of which is always more critical than the turbine
because of the decelerating flow. t Also, experience will suggest an upper limit to
the blade speed above which stressing difficulties will be severe. Accordingly, we
will assume
Rotational speed N
Mean blade speed U
250 rev/s
340 m/s
Finally, we shall assume a nozzle loss coefficient AN of 0·05 as a reasonable first
guess.
We will start by assuming (a) CaZ = Ca3 and (b) Cl = C
3
. As it is to be a
single-stage turbine the inlet velocity will be axial, i.e. (Xl = O. From the data, the
temperature drop coefficient is
./, = 2c/1Tos = 2 x 1·148 )( 145 x lO3 _ .
'I' U2 3401 - 2 88
This is a modest value and there is no difficulty about obtaining the required
output from a single stage in a turbojet unit wherein high values of C
a
can be
used. We will try a flow coefficient <P of 0·8 and, because swirl increases the
losses in the jet pipe, an 0:3 of zero. The degree of reaction that these conditions
imply can be found as follows, remembering that because of assumptions (a) and
t In practice the compressor and turbine designs interact at this point: a small increase in rotational
speed to avoid the use of more than one stage in the turbine would normally be worthwhile even if it
meant modifying the compressor design.
282 AXIAL AND RADIAL FLOW TURBINES
(b) equations (7.8) to (7.13) are applicable. From equation (7.l2)
1
. tan (;(3 = 0 = tan P3 - ¢
tan f33 = 1·25
From equation (7.1 0)
1
tan f33 = 24> (il/l + 2A)
I
1·25 = 1.6(1-44 + 2A)
A = 0·28
We shall see from section 7.2 that when three-dimensional effects are included
the reaction will increase from root to tip of the blades, and a degree of reaction of
only 0·28 at the mean diameter might mean too Iowa value at the root. Negative
values must certainly be avoided because this would imply expansion in the
nozzles followed by recompression in the rotor and the losses would be large.
Perhaps a modest amount of swirl will bring the. reaction to a more reasonable
value: we will try 0(3 = 10 degrees.
tan (;(3 = 0·1763; tan f33 = 0·1763 + 1·25 = 1-426
1
1·426 = 1.'6 (1·44 + 2A)
A = 0·421
This is acceptable: the reaction at the root will be checked when the example is
continued in section 7.2. The gas angles can now be established. So far we have
From equations (7.11) and (7.13)
1 .
tan f32 = 1.6(1-44 - 0·842) = 0·37.37
1
tan (X2 = 0·3737 + 0.8 = 1·624
f32 = 20·49°; (X2 = 58·38°
The velocity diagram can now be sketched as in Fig. 7.6, and the next task is to
calculate the density at stations 1, 2 and 3 so that the blade height h and tip Iroot
radius ratio (rtf'"r) can be estimated. We shall commence with station 2 because
some modifications will be required if the pressure ratio pQ]lp2 across the con-
vergent nozzles is much above the critical value, or if the Mach number relative to
the rotor blades at inlet (Mn) exceeds about 0·75.
ELEMENTARY THEORY OF AXIAL FLOW TURBINE
340 mls
FIG. 7.6
From the geometry of the velocity diagram,
Ca2 = U¢ = 340 x 0·8 = 272 mls
C = ~ = ~ -
2 0 "242 - 519 m/s
cos (X2 .)
The temperature equivalent of the outlet velocity is
ci 519
2
T02 - T2 = 2c
p
= 2296 = 117·3 K
Since T02 = TO! == 11 00 K, T2 == 982· 7 K
T T' 'I ci
2 - 2 = "'N- = 0·05)( 117·3 = 5·9 K
'. 2cp
T2 = 982·7 - 5·9 = 976·8 K
P2 can be found from the isentropic relation
POI = (TOl)Y/(Y-l)_ (1100)4_ .
P2 T
z
- 976.8 - 1 607
4·0
P2 = 1.607 = 2-49 bar
283
Ignoring the effect of friction on the critical pressure ratio, and putting ), = J. 3 3 3
in equation (A. 12) we have
PO! (y + l)Y/(Y-l)
-= -- = J·853
Pc 2
The actual pressure ratio is 1·607, well below the critical value. The nozzles are
not choking and the pressure in the plane of the throat is equal to Pz.
-l!2.. _ 100)( 2·49 _ 3
P2 - RT2 - 0.287 x 982.7 - 0·883 kglm
Annulus area at plane 2 is
A =-..!!'!:..--.= 20 _. 2
2 P2 Ca2 0.883)( 272 - 00833 m
284
AXIAL MID RADIAL FLOW TURBINES
Throat area of nozzles required is
. A = or A? cos CX2 = 0·0883 x 0·524 = 0·0437 d
2N P2 C2 -
Note that if the pressure ratio had been slightly above the critical value it would
be acceptable if a check (given later) on M V2 proved satisfactory. P2 and A 2 woul.d
be unchanged, but the throat area would then be given by m/ PeCO' where Pe 18
obtained fromPe and Te, and Ce corresponds to a Mach number of unity so that it
can be found from ,j()IRTJ.
We may now calculate the annulus area required in planes 1 and 3 as follows.
Because it is not a repeating stage, we are assuming that C1 is axial and this,
together with assumptions (a) and (b) that C1 = C3 and Ca3 = Ca2 yields
Ca3 272 /
C
a1
= C] = C3 = --= ---= 276·4 m s
cos cx3 cos 10°
Temperature equivalent of the inlet (and outlet) kinetic energy is
Ci = 276-4
2
= 33.3 K
2cp 2296
C
2
T1 = T01 - = 1100 - 33·3 = 1067 K
- 2c
p
!!J... = orpl = 4 = 3·54 bar
(
T )1/(1-
1
) 4·0
POI TO! (1100/1067)
= 100 x 3·54 = 1.155 k /m3
PI 0.287 x 1067 g
A = = 20 = 0.0626 m
2
I PICal 1·155 x 276·4
Similarly, at outlet from the stage we have
T03 = TOI - 8Tos = 1100 - 145 = 955 K
C
2
T3 = T03 _...1.. = 955 - 33·3 = 922 K
2cp
P03 is given in the data by POl(P03/POI) and hence
P3 = Poo = (_4_) (922)4 = 1.856 bar
o T03 j·873 955
= 100 )( 1·856 = 0.702 k 1m3
P3 0.287 x 922 g,
A 20 = 0.1047 m
2
3 P3 Ca3 0·702 x 272
The blade height and annulus radius ratio at stations 1, 2 and 3 can now be
established. At the mean diameter, which we shall now begin to emphasize by the
ELEMENTARY THEORY OF AXIAL FLOW TURBINE
use of suffix m,
. 340
Um = 2nNr m' so that rm = -2 0 = 0·216 m
n25
Since the annulus area is given by
Umh
A =,2nrmh = --
N
the height and radius ratio of the annulus can be found from
AN (250) I't I'm + (h/2)
h = U
m
= 340 A, and = I'm - (h/2)
and the results are as given in the following table.
Station
A/[m
2
j
h/[m]
rt/rr
0·0626
0·046
1·24
2
0·0833
0·0612
1·33
3
0·1047
0·077
1-43
285
Although we shall leave the discussion of the effects of high and low annulus
radius ratio to a later section, we may note here that values in the region of 1·2-
1·4 would be re&arded as satisfactory. If the rotational speed, which we have
assumed to be fixed by the compressor, had led to an ill-proportioned annulus, it
would be necessary to rework this preliminary design. For example rtfrr Icould be
reduced by increasing the axial velocity, i.e. by using a higher value of the flow
coefficient ¢. This would also increase the nozzle efflux velocity, but we noted
that it was comfortably subsonic and so could be increased if necessary.
The turbine annulus we have arrived at in this example is flared as shown in
Fig. 7.7. In sketching this we have assumed a value of blade height/width ratio of
about 3·0 and a space between the stator and rotor blades of about 0·25 of the
blade width. The included angle of divergence of the walls then becomes
approximately 29 degrees. This might be regarded as rather high, involving a risk
2
3
I
I
! 1
N R i
j i
--, i-
t
I I 1
I I
rm I :
I
I I
= O.2Sw Blade Width w = h!3
FIG. 7.7
286 A.XIAL AND RADIAL FLOW TURBINES
of flow separation from the inner wall where· the reaction, and therefore. the
acceleration, is not large. 25 degrees has been suggested as asafe limit, Ref. (3).
We shall not pause for adjustment here, because the blade height/width ratio of
3·0 is merely a rough guess to be justified or altered later when the effect of blade
stresses on the design has to be considered. Furthermore, the choice of 0·25 for
the space/blade width ratio is ratherlow: a low value is desirable only to reduce
the axial length and weight of the turbine. Vibrational stresses are induced in the
rotor blades as they pass through the wakes of the nozzle blades, and these
stresses increase sharply with decrease in axial space between the blade rows. 0·2
is the lowest value of space/blade width ratio considered to be safe, but a value
nearer 0·5 is often used and this would reduce both the vibrational stresses and
the annulus flare.
If it was thought desirable to reduce the flare without increasing the axial
length of the turbine, then it would be necessary to repeat the calculations
allowing the axial velocity to increase through the stage. It would be necessary to
check the Mach number at exit from the stage, M3, because if this is too high the
friction losses in the jet pipe become unduly large. For the present design we have
_ ~ _ 276·4 = 0.47
M3 - J(yRT3) - J(l.333 x 0·287 x 922 x 1000)
This could be safely increased to reduce the flare if desired. [It must be remem-
bered that most of the equations we have derived, and in particular (7.8)-(7.l3)
relating l/t, ¢ and the gas angles, are not valid when Ca3 i= Ca2. It is necessary to
revert to first principles or derive new relations. For example, relations (7.1)
become
and the temperature drop coefficient becomes
Additionally, if the flare is not symmetrical U must be replaced by Um2 and Urn3
as appropriate.]
Finally, for this preliminary design we have taken losses into account via }'N
and 1]s rather than )wand AR. The value of ;lR implied by the design can be found
by determining (T3 - T ~ ' ) . Thus
T ~ , (j'-ll/l' 982.7
-2. = l!2) or T; = - - - - ~ J = 913 K
T; 3 (2·49/1·856)4
VORTEX THEORY
287
We also require the temperature equivalent of the outlet kinetic energy relative to
the blading.
Ca3 272
V3 = --= = 473·5 m/s
cos f33 cos 54·96°
Vt 473.5
2
2e
p
;= 2296 = 97·8 K
Then
A _ T3 - Ti' _ 922 - 913 _
R - Vt /2e
p
- 97.8 - 0·092
Had we used the approximate relation between lis, }wand AR, equation (7.20), we
would have found }'R to be 0·108 (which is a useful check on the arithmetic), Note
that AR > AN, which it should be by virtue of the tip leakage loss in the rotor
blades.
The next steps in the design are
(a) to consider the three-dimensional nature of the flow in so far as it affects the
variation of the gas angles with radius;
(b) to consider the blade shapes necessary to achieve the required gas angles,
and the effect of centrifugal and gas bending stresses on the design;
(e) to check the design by estimating AN and AR from the results of cascade tests
suitably modified to take account of three-dimensional flows.
7.2 Vortex theory
Early in the previous section it was pointed out that the shape of the velocity
triangles must vary from root to tip of the blade because the blade speed U
increases with radius. Another reason is that the whirl component in the flow at
outlet from the nozzles causes the static pressure and temperature to vary across
the annulus. With a uniform pressure at inlet, or at least with a much smaller
variation because the whirl component is smaller, it is clear that the pressure drop
across the nozzle will vary giving rise to a corresponding variation in efflux
velocity C2• Twisted blading designed to take account of the changing gas angles
is called vortex blading.
It has been common steam turbine practice, except in low-pressure blading
where the blades are very long, to design on conditions at the mean diameter,
keep the blade angles constant from root to tip, and assume that no additional loss
is incurred by the variation in incidence along the blade caused by the changing
gas angles. Comparative tests, Ref. (4), have been conducted on a single-stage gas
turbine of radius ratio 1·37, using in turn blades of constant angle and vortex
blading. The results showed that any improvement in efficiency obtained with
vortex blading was within the margin of experimental error. This contrasts with
similar tests on a 6-stage axial compressor, Ref. (5), which showed a distinct
288 AXIAL AND RADIAL FLOW TURBINES
. . . ".
improvement from the use of vortex: blading. This was, however, not so much an
improvelllent in efficiency(of about 1·5 per cent) as in the delay of the onset of
surging which of course does not arise in accelerating flow. It appears, therefore,
that steam turbine designers have been correct in not applying vortex theory
except when absolutely necessary at the LP end. They have to consider the
additional cost of twisted blades for the very large number of rows of blading
required, and they know that the Rankine cycle is relatively insensitive to
component losses. Conversely, it is not surprising that the gas turbine'designer,
struggling to achieve the highest possible component efficiency, has consistently
used some form of vortex blading which it is felt intuitively must give a better
performance however small.
Vortex theory has been outlined in section 5.4 where it was shown that if the
elements of fluid are to be in radial equilibrium, an increase in static pressure
from root to tip is necessary whenever there is a whirl component of velocity.
Figure 7.8 shows why the gas turbine designer cannot talk of impulse or 50 per
cent reaction stages. The proportion of the stage pressure or temperature drop
which occurs in the rotor must increase from root to tip. Although Fig. 7.8 refers
to a single-stage turbine with axial inlet velocity and no swirl at outlet, the whirl
component at inlet and outlet of a repeating stage will be small compared with
Cw2: the reaction will therefore still increase from root to tip, if somewhat less
markedly.
Free vortex design
Referring to section 5.6, it was shown that if
(a) the stagnation enthalpy ho is constant over the annulus (i.e. dho/dr=O),
(b) the axial velocity is constant over the annulus,
(c) the whirl velocity is inversely proportional to the radius,
then the condition for radial equilibrium of the fluid elements, namely equation
(5.13), is satisfied. A stage designed in accordance with (a), (b) and (c) is called a
free vortex stage. Applying this to the stage in Fig. 7.8, we can see that with
i I
~ I
i I
. I
I Pl I ~
I I
. I
/
Rotor
blades
Conditions constant Conditions constant
across annulus n across annulus if
inlet velocity is axial outlet velocity is axial
FIG. 7.8 Changes in pressure and velocity across the annulus
VORTEX THEORY 289
uniform .inlet conditions to the nozzles then, since no work is done by the gas in
the nozzles, ho must also be constant over the annulus at outlet. Thus condition
'(a) is fulfilled in the space between the nozzles and rotor blades. Furthermore, if
the nozzles are designed to give Ca2 = constant and Cw2r = constant, all three
conditions are fulfilled and the condition for radial equilibrium is satisfied in
plane 2. Similarly, if the rotor blades are designed so that Ca3 = constant and
Cw3r = conStant, it is easy to show as follows that condition (a) will be fulfilled,
and thus radial equilibrium will be achieved in plane 3 also. Writing w for the
angular velocity we have
Ws = U(Cw2 + Cw3 ) = w(Cw2r + Cw3 r) = constant
But when the work done per unit mass of gas is constant over the annulus, and ho
is constant at inlet, ho must be constant at outlet also: thus condition (a) is met.
It is apparent that a free vortex design is one in which the work done per unit
mass of gas is constant over the annulus, and to obtain the total work output this
specific value need only be calculated at one convenient radius and multiplied by
the mass flow.
In contrast, we may note that because the density varies from root to tip at exit
from the nozzles and the axial velocity is constant, an integration over the annulus
will be necessary if the continuity equation is to be used in plane 2. Thus,
considering a flow Om through an annular element of radius r and width or,
om = P22nrorCa2
m = 2 n ~ a 2 J
r
! P2rdr
r,
(7.22)
With the radial variation of density detennined from vortex theory, the integration
can be performed although the algebra is lengthy. For detailed calculations it
would be normal to use a digital computer, permitting ready calculation of the
density at a series of radii and numerical integration of equation (7.22) to obtain
the mass flow. For preliminary calculations, however, it is sufficiently accurate to
take the intensity of mass flow at the mean diameter as being the mean intensity
of mass flow. In other words, the total mass flow is equal to the mass flow per unit
area calculated using the density at the mean diameter (P2mCa2) multiplied by the
annulus area (A2). This is one reason why it is convenient to design the turbine on
conditions at mean diameter (as was done in the previous example) and use the
relations which will now be derived for obtaining the gas angles at other radii.
Using suffix m to denote quantities at mean diameter, the free vortex variation
of nozzle angle (1(2 may be found as follows
Cw2r = rCa2 tan (1(2 = constant
Ca2 = constant
Hence (1(2 at any radius r is related to (l(2m at the mean radius r m by
tan (1(2 = C;)2 tan (l(2m (7.23)
290 JI.xIAL AND RADIAL FLOW TURBINES
Similarly, when there is swirl at outlet from the stage,
tan 1X3 = C; \ tan 1X3m (7.24)
The gas angles at inlet to the rotor blade, /32, can then be found using equation
(7.1), namely
U
tan /32 = tan C ~ 2 - C-
a2
(
rm) (r) Urn
= - tanlX2m- - -
r 12 rm 2Ca2
and similarly /33 is given by
(
rm) (r) Urn
tan /33 = - tan 1X3m + - -C
r3 rm3a3
(7.25)
(7.26)
To obtain some idea of the free vortex variation of gas angles with radius,
equations (7.23}-(7.26) will be applied to the turbine designed in the previous
section. We will merely calculate the angles at the root and tip, although in
practice they would be determined at several stations up the blade to define the
twist more precisely. We will at the same time clear up two loose ends: we have to
check that there is some positive reaction at the root radius, and that the Mach
number relative to the rotor blade at inlet, Mn , is nowhere higher than say 0·75.
From the velocity triangles at root and tip it will be seen that this Mach number is
greatest at the root and it is only at this radius that it need be calculated.
From the mean diameter calculation we found that
1X2m = 58.38
0
• /32m = 20.49
0
, 1X3m = 10
0
, /33m = 54.96
0
From the calculated values of h and rm we have rr = r m - (hI2) and rt =
rm+(hI2), and thus
(
rm) = 1.164,
rr 2
(
rm) = 0.877,
r t 2
Also
Um = Urn =.!.= 1.25
Ca2 Ca3 4>
Applying equations (7.23)-(7.26) we get
Tip
Root
(X2
54·93°
62·15°
1X3
t'
m
) = 1·217,
\J"r 3
(
rm) = 0.849
r t 3
The variation of gas angles with radius appears as in Fig. 7.9, which also includes
the velocity triangles at root and tip drawn to scale. That Mn = V2IJ(yRT2) is
VORTEX THEORY
70,...--'--.,---__
U)
~ 30
Cl
'"
Cl
20
0-
-10'--__ '--_-'
Root Mean Tip
-Root
---- Tip
FIG. 7.9 Variation of gas angles with radius
291
greatest at the root is clear from the velocity triangles: V2 is then a maximum, and
J(yRT2) is a minimum because the temperature drop across the nozzles is
greatest at the root That there is some positive reaction at the root is also clear
because V3r > V2/". Although there is no need literally to calculate the degree of
reaction at the root, we must calculate (Mdr to ensure that the design implies a
safe value. Using data from the example in section 7.1 we have
Vz,. = Ca2 sec /32r = 272 sec 39·32° = 352 mls
C2,. = Ca2 sec IXZ,. = 272 sec 62·15° = 583 mls
C ~ , . 583
2
952 '(
TZr = Toz - 2c
p
= 1100 - 2294 = x
V2r 352 = 0.58
(Mvz)r = J(yRT2r) .1(1·333 x 0·287 x 952 x 1000)
This is a modest value and certainly from this point of view a higher value of the
flow coefficient 4> could safely have been used in the design, perhaps instead of
introducing swirl at exit from the stage.
Constant nozzle angle design
As in the case of the axial compressor, it is not essential to design for free vortex
flow. Conditions other than constant Ca and Cwr may be used to give some other
form of vortex flow, which can still satisfy the requirement for radial equilibrium
of the fluid elements. In particular, it may be desirable to make a constant nozzle
angle one of the conditions determining the type of vortex, to avoid having to
292
AXIAL AND RADIAL FLOW TURBINES
manufacture nozzles of varying outlet angle. This, as will now be shown, requires
particular variations of Ca and CWo
The vortex flow equation (5.15) states that
dCa dCw dho

Consider the flow in the space between the nozzles and blades. As before, we
assume that the flow is unifonn across the annulus at inlet to the nozzles, and so
the stagnation enthalpy at outlet must also be unifonn, i.e. dho/ dr = 0 in plane 2.
Also, if ()(z is to be constant we have
C
= cot ()(z = constant
Cw2
dCa2 dCw2
--=-- cot()(?
dr dr -
The vortex flow equation therefore becomes
2 dCw2 dCwz
Cw2 cot ()(2T+ CwzT+-r- = 0
(1 + r ) dC
w2
+ C
w2
0
co ()(2 T -,-. =
dCw2 . 2 dr
--=-sm ()(,-
C
w2
L. r
Integrating this gives
C or
sin
'", = constant
w_
(7.27)
(7.28)
And with constant G(z, Ca2 oc Cw2 so that the variation of Ca2 must be the same,
namely
(7.29)
Normally nozzle angles are greater than 60 degrees, and quite a good approxi-
mation to the flow satisfYing the equilibrium condition is obtained by designing
with a constant nozzle angle and constant angular momentum, i.e. ()(2 = constant
and C
w2
r = constant. If this approximation is made and the rotor blades are
twisted to give constant angular momentum at outlet also, then, as for free vortex
flow, the work output per unit mass flow is the same at all radii. On the other
hand, if equation (7.28) were used it would be necessary to integrate from root to
tip to obtain the work output. We observed early in section 7.2 that there is little
difference in efficiency between turbines oflow radius ratio designed with twisted
and untwisted blading. It follows that the sort of approximation referred to here is
certainly unlikely to result in a significant deterioration of perfonnance.
The free vortex and constant nozzle angle types of design do not exhaust the
possibilities. For example, one other type of vortex design aims to satisfY the
radial equilibrium condition and at the same time meet a condition of constant
mass flow per unit area at all radii. That is, the axial and whirl velocity
distributions are chosen so that the product P2Ca2 is constant at all radii. The
CHOICE OF BLADE PROFILE, PITCH AND CHORD 293
FIG.7.HI
of t':is approach correctly point out that the simple vortex theory
outlmed m sectIOn 5.6 assumes no radial component of velocity, and yet even if
the turbine is designed with no flare there must be a radial shift of the streamlines
shown in Fig. 7.lO. This shift is due to the increase in density from root to tip
m plane 2. The ass.umption that the radial component is zero would undoubtedly
be true for a turbme of constant armulus area if the stage were designed for
constant per unit area. It is argued that the flow is then more likely to
behave as mtended, so that the gas angles will more closely match the blade
angles. Further details can be found in Ref. (2). In view of what has just been said
about the dubious benefits of vortex blading for turbines of modest radius ratio it
is very doubtful indeed whether such refinements are more than an
exercise.
7.3 Choice of blade profile, pitch and dumd
So our example we have shown how to establish the gas angles at
all blade heIghts. The next step is to choose stator and rotor blade shapes
whIch WIll the gas incident upon the leading edge, and deflect the gas
through the reqUIred angle with the minimum loss. An overall blade loss coeff-
icient Y (or Il) must account for the following sources of friction loss.
(a) with boundary layer growth over the blade profile
loss under adverse conditions of extreme angles of
mCldence or high inlet Mach number).
(b) Annulus loss-associated with boundary layer growth on the inner and outer
walls of the annulus.
(c) Secondary flow loss-arising from secondary flows which are always
pr:sent when a wall boundary layer is turned through an angle by an
adjacent curved surface.
(d) Tip clearance loss-near the rotor blade tip the gas does not follow the
intended path, fails to contribute its quota of work output, and interacts with
the outer wall bounda.ry layer.
The profile loss coefficient Yp is measured directly in cascade tests similar to
described for compressor blading in section 5.8. Losses (b) and (c) cannot
easIly be separated, and they are accounted for by a secondary loss coefficient Y".
294
AXIAL AND RADIAL FLOW TURBINES
The tip clearance loss coefficient, which normally arises. only for rotor blades,
will be denoted by Yk. Thus the total loss coefficient Y comprises the accurately
measured two-dimensional loss Yp , plus the three-dimensional loss (Ys + Yk)
which must be deduced from turbine stage test results. A description of one
important compilation of such data will be given in section 7.4; all that is
necessary for our present purpose is a knowledge of the sources of loss.
Conventional blading
Figure 7.11 shows a conventional steam turbine blade profile constructed from
circular arcs and straight lines. Gas turbines have until recently used profiles
closely resembling this, although specified by aero foil terminology. One example
is shown: the T6 base profile which is symmetrical about the centre line. It has a
thickness/chord ratio (t/c) of 0·1, a leading edge radius of 12 per cent t and a
trailing edge radius of 6 per cent t. When scaled up to a t/c of 0·2 and used in
conjunction with a parabolic camber line having the point of maximum camber a
distance of about 40 per cent c from the leading edge, the T6 profile leads to a
blade section similar to that shown but with a radiused trailing edge. In particular,
the back of the blade after the throat is virtually straight. Other shapes used in
British practice have been RAF 27 and C7 base profiles on both circular and
parabolic arc camber lines. All such blading may be referred to as conventional
blading.

Angle of incidence i
" p
6 /",,"
,-

::: cos-1 (oIs) Pitch s
xII
0.025-
0.05-
0.10-
0.15-
0.20-
'i
f--->- Y
ylL
_ 0.0154
- 0.0199
- 0.0274
- 0.0340
- 0.0395
..J 0.30 - - 0.0472
040-1"';' -0.05 (tlL=0.10)
2 II
0.50- - 0.0467
tr5
-'"
E
'"
(,)
0.60-
0.70-
O.BO-
0.90--
0.95-
-0.0370
- 0.0251
- 0.0142
- 0.0085
- 0.0072
Leading edge radius 0.121
Trailing edge radius 0.601
Symmetrical about <i
FIG. 7.11 'Conventional' blade prolil.es: steam turbine section and T6 aerofoil section
CHOICE OF BLADE PROFILE, PITCH AND CHORD 295
0.18
0.16
Impulse blade
>.
/32/ /33
" c
" '13
0.12
;;:
"
0
" U)
0.08
U)
.Q


0.. 0.04
0
-30' -20' -10' 0' 10' 20' 30'
Incidence i
FIG. 7.12 Effect on incidence upon Yp
It is inlportant to remember that the velocity triangles yield the gas angles, not
the blade angles. Typical cascade results showing the effect of incidence on the
profile loss coefficient Yp for impulse (A = 0 and /32 :::: /33) and reaction type
blading are given in Fig. 7.l2. Evidently, with reaction blading the angle of
incidence can vary from -15° to +15° without increase in Yp' The picture is not
very different even when three-dimensional losses are taken into account. This
means that a rotqr blade could be designed to have an inlet angle 82 equal to say
(/32r - 5°) at the root and (/32t + 10°) at the tip to reduce the twist required by a
vortex design. It must be remembered, however, that a substantial margin of safe
incidence range must be left to cope with part-load operating conditions of
pressure ratio, mass flow and rotational speed.
With regard to the outlet angle, it has been common steam turbine practice
to take the gas angle as being equal to the blade angle defined by cos-
I
(opening/pitch). This takes account of the bending of the flow as it fills up the
narrow space in the wake of the trailing edge; there is no 'deviation' in the sense
of that obtained with decelerating flow in a compressor cascade. Tests on gas
turbine cascades have shown, however, that the cos-
I
(o/s) rule is an over-
correction for blades of small outlet angle operating with low gas velocities, i.e
for some rotor blades. Figure 7.13 shows the relation between the relative gas
outlet angle, /33 say, and the blade angle defined by cos-
1
(o/s). The relation does
not seem to be affected by incidence within the working range of ± 15 degrees.
This curve is applicable to 'straight-backed' conventional blades operating with a
relative outlet Mach number below 0·5. With a Mach number of unity the cos-
1
(o/s) rule is good for all outlet angles, and at Mach numbers intermediate
betvveen 0·5 and 1·0 it can be assumed that [cos-I(o/s) - P3] varies linearly with
Mach number. Reference (3) gives an additional correction for blades with a
curved-back trailing edge.
Note that until the pitch and chord have been established it is not possible to
draw a blade section to scale, determine the 'opening', and proceed by trial and
296
OJ
OJ
"
Ql
Qi
'5
0
'" co
0>
(])
>
"fd
ill
cc
80'
70'

60'
06:
..:
50'

§'
40'
3
L
O-'-,
COS-
1
(a/s)
AXIAL AND RADIAL FLOW TURBINES
FIG. 7.13 Relation between gas and blade outlet angles
error to make adjustments until the required gas outlet angle (X2 or 133 is obtained.
Furthennore, this process must be carried out at a number of radii from root to tip
to specify the shape of the blade as a whole. Now the pitch and chord have to be
chosen with due regard to (a) the effect of the pitch/chord ratio (sic) on the blade
loss coefficient, (b) the effect of chord upon the aspect ratio (hlc), remembering
that h has already been determined, (c) the effect of rotor blade chord on the blade
stresses, and (d) the effect of rotor blade pitch upon the stresses at the point of
attachment of the blades to the turbine disc. We will consider each effect in turn.
(a) 'Optimum' pitchlchord ratio
In section 7.4 (Fig. 7.24) are presented cascade data on profile loss coefficients
and from such data it is possible to obtain the useful design curves in Fig.
7.14. These curves suggest, as might be expected, that the greater the gas deflec-
tion required [( G( I + G(2) for a stator blade and (132 + 133) for a rotor blade] the
smaller must be the 'optimum'slc ratio to control the gas adequately. The ad-
jective 'optimum' is in inverted commas because it is an OptimUlll with respect to
Y not to the overall loss Y. The true optimum value of sic could be found only by
a detailed estimate of stage perfonnance (e.g. on the lines described in
section 7.4) for several stage designs differing in sic but otherwise similar. In fact
the sic value is not very critical.
For the nozzle and rotor blade of our example turbine we have established that
IXlm = 0
0
, G(2m = 58·38°; 132m = 20·49°, 133m = 54·96°
From Fig. 7.14 we therefore have at mean diameter
(slc)N := 0·86 and (S/C)R = 0·83
(b) Aspect ratio (hie)
The influence of aspect ratio is open to conjecture, but for our purpose it is
sufficient to note that, although not critical, too low a value is likely to lead to
CHOICE OF BLADE PROFILE, PITCH AND CHORD
1.1
Relative inlet
1.0
gas angle (001 or

0.8
0.7
0.6
0.5'--:-:::-___ ---:::::--____ =-___ ---=:
40' 50' 60' 70'
Relative efflux angle (002 or
FIG. 7.14 'Optimum' pitcli/ci:lOrd ratio
297
secondary flow and tip clearance effects occupying an unduly large proportion of
the blade height and so increasing Ys for the nozzle row and (Y, + Yk) for the rotor
row. On the other hand, too high a value of hie will increase the likelihood of
vibration trouble: vibration characteristics are difficult to predict and they depend
on the damping"provided by the method of attaching the blades to the turbine
disc., A value of hlc between 3 and 4 would certainly be very satisfactory, and it
would be unwise to use a value below 1.
For our turbine, which is flared, we have the mean heights of the nozzle and
rotor blades given by
hN = !(0.046 + 0·0612) = 0·0536 m
hR = HO.0612 + 0·077) = 0·0691 m
Adopting an aspect ratio (hie) of 3 we then have
CN = 0·0175 m and CR = 0·023 m
Using these values of chord, in conjunction with the chosen sic values, gives the
blade pitches at the mean radius of 0·216 m as
3N = 0·01 506 m and SR == 0·0191 m
and the numbers of blades, from 2nr mis, as
nN = 90 and nR = 71
t Vibration problems with high aspect ratio blading can be significantly reduced by using tip shrouds
which prevent vibrations in the cantilever mode. Shrouds are also sometimes used to reduce tip
leakage loss.
298 AXIAL ft.ND RADIAL FLOW TURBINES
It is usual to avoid numbers with cornmon multiples to reduce the probability of
introducing resonant forcing frequencies. A cornmon practice is to use an even
number for the nozzle blades and a prime number for the rotor blades. As it
happens the foregoing numbers are satisfactory and there is no need to modify
them and re-evaluate the pitch s.
(c) Rotor blade stresses
The next step is to check that the stage design is consistent with a permissible
level of stress in the rotor blades. The final design must be checked by laying out
the blade cross-sections at several radii between root and tip, and performing an
accurate stress analysis on the lines indicated by Sternlicht in Ref. (6). Although
we are not concerned with mechanical design problems in this book, simple
approximate methods adequate for preliminarj design calculations must be men-
tioned because blade stresses have a direct impact upon the stage design. There
are three main sources of stress: (i) centrifugal tensile stress (the largest, but not
necessarily the most important because it is a steady stress), (ii) gas bending
stress (fluctuating as the rotor hlades pass by the trailing edges of the nozzles) and
(iii) centrifugal bending stress when the centroids of the blade cross-sections at
different radii do not lie on a radial line (any torsional stress arising from this
source is small enough to be neglected).
When the rotational speed is specified, the allowable centrifugal tensile stress
places a limit on the annulus area but does not affect the choice of blade chord.
This somewhat surprising result can easily be shown to be the case as follows.
The maximum value of this stress occurs at the root and is readily seen to be
given by
where P b is the density of blade material, w is the angular velocity, a is the cross-
sectional area of the blade and ar its value at the root radius. In practice the
integration is performed graphically or numerically, but if the blade were of
uniform cross-section the equation would reduce directly to
(crct)max = 2nN2p0
where A is the annulus area and N is the rotational speed in rev / s. A rotor blade is
usually tapered in chord and thickness from root to tip such that at! ar is between
1/4 and 1/3. For preliminary design calculations it is sufficiently accurate (and
on the safe side) to assume that the taper reduces the stress to 2/3 of the value for
an untapered blade. Thus
(7.30)
For the flared turbine of our example we have
A = !(A2 +A3) = 0·094 m
2
andN = 250 rev/s
CHOICE OF BLADE PROFILE, PITCH AND CHORD
299
The density of the Ni-Cr-Co alloys used for gas turbine blading is about
8000 kg/m
3
, and so equation (7.30) gives
(crct)max ::::: 200 MN/m2 (or 2000 bar)
Judgement as to whether or not this stress is satisfactory must await the evaluation
of the other stresses.
The force arising from the change in angular momentum of the gas in the
tangential direction, which produces the useful torque, also produces a gas
bending moment about the axial direction, namely Mw in Fig. 7.15. There may be
a change of momentum in the axial direction (i.e. when Ca3 =1= Ca2), and with
reaction blading there will certainly be a pressure force in the axial direction
[(P2 - P3)2nr/n per unit height], so that there will also be a gas bending moment
Ma about the tangential direction. Resolving these bending moments into
components acting about the principal axes of the blade cross-section, the
maximum stresses can be calculated by the method appropriate to asymmetrical
sections. A twisted and tapered blade must be divided into strips of height oh and
the bending moments calculated from the average force acting on each strip. The
gas bending stress (f gb will be tensile in the leading and trailing edges and
compressive in the back of the blade, and even with tapered twisted blades the
maximum value usually occurs at either the leading or trailing edge of the root
section. Because Mw is by far the greater bending moment, and the principal axis
XX does not deviate widely from the axial direction (angle (j) is small), a useful
approximation for preliminary design purposes is provided by
( )
~ m(Cw2m + Cw3m) h 1
(fgbmax- n x
2
x
zc3
(7.31 )
C.G.
\
.- <l>
r-
y
Axial direction
FIG. 7.15 Gas bending stress
300
AXIAL AND RADIAL FLOW TURBINES
n is the number of blades, the.whlrl velocities are evaluated at the mean diameter,
and z is the smallest value of the root section modulus (lxx/Y) of.ablade of unit
chord. Clearly 0' gb is directly proportional to the stage work output and blade
height, and inversely proportional to the number of blades and section modulus. It
is convenient to treat the section modulus as the product zc
3
because z is largely a
function of blade camber angle (:::gas deflection) and thickness/chord ratio ..
An unpublished rule for z due to Ainley, useful for approximate calculations, is
given in Fig. 7.16. We shall apply this, together with equation (7.31), to our
example turbine. Assuming the angle of incidence is zero at the design operating
condition, the blade camber angle is virtually equal to the gas deflection, namely
at the root
fJ2r + fJ3r = 39·32° + 51-13° ::::: 90 degrees
Then from Fig. 7.16, assuming a blade of tic = 0·2,
(10 ~ 0.2)1.27
(z) t = ~ = 0·00423 mm
3
jmm chord
roo 570
m( Cw2 + Cw3 ) = mCa(tan 1X2 + tan 1X3)
which at mean diameter yields
20 x 272(1·624 + 0·176) = 9800 kN
For the chosen value of CR = 0·023 m, nR was found to be 71, while
hR = ~ ( h 2 + h3) = 0·0691 m. Equation (7.31) can now be evaluated to give
~ 9800 0·0691 1 ~ 3 MN m2
(O'gb)max - 71 x 2 x 0.00423 X 0.0233 - 9 /
1000
Ixx 1 ( T 3
Z= - = - 10- c
B Ymax B C
800
600
B
400
2
n
200
40' 60' 80' 100' 120'
Blade camber angle
FIG. 7.1Ii Approximate mle for section modlili
CHOICE OF BLADE PROFILE, PITCH AND CHORD
t
i
Gas Cl ) Centrigugal
bending bending
moment moment
I
----i--....
!
FIG. 7.17
301
By designing the blade with the centroids of the cross-sections slightly off a
radial line, as indicated in Fig. 7.17, it is theoretically possible to design for a
centrifugal bending stress which will cancel the gas bending stress. It must be
remembered, however, that (a) these two stresses would only cancel each other at
the design operating condition, (b) O'gb is only a quasi-steady stress and (c) the
centrifugal bending stress is very sensitive to manufacturing errors in the blade
and blade root fixing. 0' gb is often not regarded as being offset by any centrifugal
bending stress and usually the latter is merely calculated using the extreme values
of manufacturing tolerances to check that it is small and that at least it does not
reinforce 0' gb'
We have now e.stablished a steady centrifugal stress of 200 MN/m
2
and a gas
bending stress of 93 MN/m2 which is subject to periodic fluctuation with a
frequency dependent on N, nR and nN. Creep strength data for possible blade
materials will be available: perhaps in the form of Fig. 7.18(a) which shows the
time of application of a steady stress at various temperatures required to produce
0·2 per cent creep strain. Fatigue data (e.g. Gerber diagrams) will also be
available from which it is possible to assess the relative capacity of the materials
to withstand fluctuating stresses. Such data, together with experience from other
turbines in service, will indicate how the fluctuating gas bending stress and the
steady centrifugal stress can be combined safely. The designer would hope to
have a set of curves of the type shown in Fig. 7 .18(b) for several safe working
lives. The values of temperature on this plot might refer to turbine inlet stagnation
temperature TOb allowance having been made for the fact that
(i) only the leading edge of the rotor blade could theoretically reach stagnation
temperature and chordwise conduction in the metal would prevent even the
local temperature there from reaching TOJ ;
(ii) even in 'uncooled' turbines (those with no cooling passages in the blades),
some cooling air is bled from the compressor and passed over the turbine
disc and blade roots: the metal temperature will therefore be appreciably less
than 1100 K near the root radius for which the stresses have been estimated.
Furthermore, the values of permissible 0' gb and 0' ct will be conservative, and
include a safety factor to allow for local hot streaks of gas from the combustion
302
400
300
1
e. 200-
II)
II)
£!
US
100
o "-__ LI ---"I'--___ -'---___ ..JI
AXIAL AND RADlALFLOW TURBINES
100
Turbine inlet temperature T01
10 50 100 10 000
Time for 0.2% creep strain/h
(a)
Permissible (Jc/[MN/m2]
(b)
FIG. 7.18 Creep data alld designer's aid for assessing preliminary desiglls
system and for the fact that there will be additional thermal stresses due to
chordwise and spanwise temperature gradients in the blade. In our example Uet
and U gb were found to be 200 and 93 lIAN/m
2
respectively. If a life of 10 000 h
was required, the curve relating to our inlet temperature of 1100 K suggests that
the stresses are rather too large. The blade chord could be increased slightly to
reduce O'gb if the need for reduced stresses is confinned by more detailed
calculations. As stated earlier, the final design would be subjected to a complete
stress analysis, which would include an estimate of the temperature field in the
blade and the consequential thermal stresses.
(d) Effect of pitch on the blade root fixing
The blade pitch s at mean diameter has been chosen primarily to be compatible
with required values of sic and hie, and (via the chord) of permissible Ugb' A
check must be made to see that the pitch is not so small that the blades cannot be
attached safely to the turbine disc rim. Only in small turbines is it practicable to
machine the blades and disc from a single forging, cast them integrally, or weld
the blades to the rim, and Fig. 7.19 shows the commonly used fir tree root fixing
which permits replacement of blades. The fir trees are made an easy fit in the rim,
being prevented from axial movement only (e.g. by a lug on one side and peening
on the other). When the turbine is running, the blades are held firmly in the
serrations by centripetal force, but the slight freedom to move can provide a
useful source of damping for unwanted vibration. The designer must take into
account stress concentrations at the individual serrations, and manufacturing
tolerances are extremely important; inaccurate matching can result in some of the
serrations being unloaded at the expense of others. Failure may occur by the disc
rim yielding at the base of the stubs left on the disc after broaching (at section x);
by shearing or crushing of the serrations; or by tensile stress in the fir tree root
itself. The pitch would be regarded as satisfactory when the root stresses can be
optimized at a safe level. This need not detain us here because calculation of these
CHOICE OF BLADE PROFILE. PITCH AND CHORD
303
FIG. 7.19 'Firtree' mot
centrifugal stresses is straightforward once the size of the blade, and therefore its
mass, have been established by the design procedure we have outlined.
Finally, the tOtftl centrifugal blade loading on the disc and the disc rim
diameter both being mown, the disc stresses can be determined to see if the
original assumption of a mean blade speed of 340 mj s is satisfactory. Centrifugal
hoop and radial stresses in a disc are proportional to the square of the rim speed.
Disc design charts (e.g. the Donath chart) are available to permit the nominal
stresses to be estimated rapidly for any disc of arbitrary shape; see Ref. (6). They
will be 'nominal' because the real stress pattern will be affected substantially by
thermal stresses arising from the large temperature gradient between rim and hub
or shaft.
Before proceeding to make a critical assessment of the stage design used as an
example, which we' shall do in section 7.4, it is logical to end this section by
outlining briefly recent developments in the prediction and construction of more
efficient blade profiles. This will be a digression from the main theme. howevel;
and the reader may prefer to omit it at the first reading.
Theoretical approach to the determination of blade profiles
and pitchlchord ratio
There is little doubt that the approach to turbine design via cascade test results,
which has been outlined here, is satisfactory for moderately loaded turbine stages.
Recently, however, advanced aircraft propulsion units have required the use of
high blade loadings (i.e. high If; and q;) and cooled blades, which takes the
designer into regions of flow involving ever-increasing extrapolation from exist-
304
AXIAL AND RADIAL FLOW TURBINES
ing cascade data. Furthermore it is found that under conditions even millor
changes in blade profile, such as the movement of the point of maximum camber
towards -the leading edge from say 40 to 37 per cent chord, can make a substantial
difference to the blade loss coefficient particularly when the turbine is operating
away from the design point, i.e. at part load. Rather than repeat the vast number of
cascade tests to cover the more arduous range of conditions, the approach now is
to run a few such tests to check the adequacy of theoretical predictions and then
apply the theory.
The digital computer has made it possible to develop methods of solving the
equations of compressible flow through a blade row, even taking radial
components into account to cover the case of a flared annulus. (Earlier
incompressible flow solutions, adequate for the small pressure changes in
compressor blading, are of little use in turbine design.) At first the approach was
towards predicting the potential flow for a blade of given profile. That is, the
pressure and velocity distributions outside the boundary layer are calculated in a
passage bounded by the prescribed concave surface of one blade and the convex
surface of the adjacent blade. Conformal transformation theory, methods of
distributed sources and sinks, and stream filament theory have all been applied in
one form or another and with varying degrees of success. Once the potential flow
pattern has been established, boundary layer theory can be applied to predict the
profile loss coefficient. The chief difficulties in this part of the analysis are
the determination of the point of transition between laminar and turbulent flow on
the convex surface, and the determination of a satisfactory mathematical model
for the wake downstream of the trailing edge.
Some idea of the complexity of the calculations involved can be formed from
Ref. (2), which also gives detailed references to this work. A not unimportant
question is which method takes the least computer time consistent with providing
solutions of adequate accuracy. While there is still so much controversy as to the
best method, it is not practicable to attempt a simple introductory exposition in a
book of this kind. One important outcome of the work, however, can be stated in
simple terms with reference to the typical pressure and velocity distributions for
conventional blading shown in Fig. 7.20.t The vital feature is the magnitude of
the opposing pressure gradient on the convex (suction) surface. If too great it will
lead to separation of the boundary layer somewhere on the back of the blade, a
large wake, and a substantial increase in the profile loss coefficient. When trying
to design with high aerodynamic blade loadings, which imply low suction surface
pressures, it is desirable to know what limiting pressure or velocity distributions
on the suction surface will just give separation at the trailing edge of the blade.
One guide is provided by Smith, Ref. (7), who also makes a useful comparison of
six different criteria for the prediction of separation. (He sidesteps the difficulty of
determining the point of transition from a laminar to turbulent boundary layer,
however, by assuming a turbulent layer over the whole of the suction surface.)
t It is the flow relative to the blade which is referred to in this section: if a rotor blade is under
consideration the parameters would be (p - P3)f(P03rel - P3) and VIV3.
CHOICE OF BLADE PROFILE, PITCH AND CHORD
0.4 0.6 0.8 1.0
Fraction of width w I
1.5
c
G.!
Leading edge Trailing edge
305
Convex surface
0.2 0.4 0.6 0.8 1.0
Fraction of width w
FIG. 7.20 Pressure and velocity distributions on a conventional turbine blade
Using the most conservative criterion for separation, namely that of Stratford, he
constructed two extreme families of simplified, limiting, velocity distributions as
shown by the two inset sketches in Fig. 7.21. Within each family the distribution
is defined by a value of (Cmax/ C2) and the position of the point of maximum
velocity, A, along the blade surface. The curves for various Reynolds numbers,
only two per family being reproduced here, represent the loci of point A. Any
velocity falling in the region below the relevant curve should imply
freedom from separation. (Note that, helpfully, separation is delayed by an
increase in Reynolds number.) When this type of information is available the
value of being able to predict the velocity distribution is apparent: small changes
in a proposed profile or camber line shape can be made until the velocity
distribution falls within the safe region.
What we have been referring to so far are approaches that have been made to
the direct problem: the prediction of the velocity distribution around a given blade
in cascade. Attention is now being focused on the indirect problem: the
theoretical determination of a blade shape which will give a prescribed blade
surface velocity distribution. What the ideal velocity distribution should be in
various circumstances to give the minimum loss is certainly not yet established,
but at least enough is known to avoid such obvious weaknesses as boundary layer
separation (e.g. near the blade root where the degree of reaction is low) or the
formation of unwanted shock waves. Ultimately it is hoped to be able to build
into a computer program such restraints on the possible blade shape as those
provided by stressing considerations (blade section area and section moduli) and
by a minimum trailing edge thickness (dictated by manufacturing necessity or the
need to accommodate a cooling passage). The profile loss coefficient of the
restricted range of possible profiles can then be evaluated to enable a final choice
to be made. The method of solving the indirect problem which seems to have
proved the most capable of useful development is that due to Stanitz, although the
approach via stream filament theory initiated by Wu is also receiving attention. A
306 AXIAL AND RADIAL FLOW TURBINES
1.0
I I
0.8 1.0
Fractional distance from leading edge of point A along convex surface
FIG. 7.21 Limiting velocity profiles 011 suction surface for 1:lOumlary layer separation
at trailing edge
helpful summary of the main steps in these solutions is given by Horlock in
Ref. (2).
It is to be hoped that enough has been said here to warn the student who
wishes to know more of these topics that a first essential is a thorough grounding
in aerodynamics and turbulent boundary layer theory. This should be followed by
a study of Sections Band C of Ref. (1), on the 'theory of two-dimensional flow
through cascades' and 'three-dimensional :tlow in turbomachines'. To give the
reader a feel for the indirect problem, however, we will end this section with a
brief description of an approximate solution due to Stanitz which Horlock has put
into terms comprehensible to readers of section 5.9.
The case considered is the relatively simple one of compressible flow in a two-
dimensional cascade as depicted in Fig. 7.22. We shall find it convenient to refer
to the whirl component of the force acting on unit height of blade which was
denoted by F in section 5.9. Unlike the treatment in that section, we shall here be
concerned with th(: way F per unit width (w) changes with x, and moreover we
cannot make the incompressible assumption that P is constant. For the indirect
problem the following data will be specified:
CHOICE OF BLADE PROFILE, PITCH AND CHORD
307
At a given x
I
pCcos a
Linear P. Ca

1 0 Ca2 C
2
t p - Y I
I
Position of
C"" streamline
-<------- Given specification ------+- dividing flow
(a)
(b) (c)
FIG. 7.22 The 'indirect' problem for two-dimensional compressible lfioVl' in a rurhine
cascade
(a) upstream conditions (Po], To!' Cj, IXd and downstream conditions (Cl , IXl);
(b) surface velocity distributions as a function of x between x = 0 and x = 1,
namely C,(x) and Cp(x) for the suction and pressure surfaces respectively.
The directions of Cs and Cp will be unImown because the shape of the
surface is unknown, indicating the need for a process of iteration.
Because we are dealing only with the potential flow, the expansion will be
isentropic so that Po and To will be constant through the passage. For any local
value of C, therefore, local values of p and P can be found from the isentropic
relations
We note that the blade profile will be completely determined when (a) the
pitch/width ratio (s/w) is established, and (b) both the camber line angle IX' and
blade thic1mess/pitch ratio have been calculated for various values of x between
o and 1. The procedure is as follows ..
First approximation
(i) Determine PI from the given inlet conditions Po)' TO! and CI ; and note
that for continuity in two-dimensional flow
m = PIsCI cos 0(1 = PlSCZ cos IXl
where m is the mass flow per unit height of passage.
(ii) Using the isentropic relations, calculate the pressures Pp and Ps for a series
of values of x between 0 and 1 from the given surface velocity
distributions of Cp and C.
(iii) Integrate numerically pix) and pix) frornx= 0 to x= 1, and so determine
the mean force per unit height of blade in the whirl direction from
Fm=APm x w.
308
(iv)
(v)
(vi)
(vii)
(viii)
AXIAL AND RADIAl FLOW TURBINES
Detelmine slw by equating F m to the overall change of momentum in the
whirl direction, i.e.
I1Pm x w = m(Cw1 + Cw2 ) = P1SCl cos 0:1(C1 sin 0:1 + C2 sin 0:2)
s I1Pm
w P1 C1 COS 0: 1 (C1 sin 0:1 + C2 sin 0:2)
For various values of x between 0 and 1, integrate the pressure distribu-
tions from the leading edge to x, and so obtain the pressure differences I1p
which act over the series of areas (xw) )( 1.
Determine the mean whirl velocity Cw at the values of x used in (v) from
I1p x xw = PIsC1 cos 0:1 (C1 sin 0:1 ± Cw)
(The negative sign refers to values near the leading edge where Cw is in
the same direction as Cw1 .)
Finding the corresponding mean values of C from (Cp + Cs)/2, the values
of 0:' at the various values of x are given by
sin 0:' = CwlC
For convenience the blade thickness is measured in the whirl direction.
Thus tw at the various values of x can be found as a fraction of the pitch s
from the continuity equation:
P1CI cos 0:1 =P[l-
where the mean density p at any x is found from the corresponding mean
values of p and C by using the isentropic relations.
In this first approximation the directions of the surface velocities Cp and Cs
have been assumed the same and equal to that of the mean velocity C, namely 0:'.
Furthermore the properties have been assumed constant across the passage at the
mean value. Having established an approximate blade profile it is possible to
refine these assumptions and obtain a better approximation to the true profile. The
value of s Iw did not depend on the assumptions and remains unchanged, but the
calculation of 0:' and twl s as functions of x must be repeated.
Second approximation
(i) The flow directions O:p and O:s of the surface velocities Cp and Cs
determined by the geometry of the concave and convex surfaces resultmg
from the first approximation, and initially the properties are considered to
vary linearly across the passage from one surface to the other.
(ii) The position of the streamline which divides the flow equally between the
surfaces, is determined by assuming that the axial velocity C cos 0: varies
linearly across the passage from Cp cos fXp to Cs cos rxs. [See Fig. 7.22(c).]
ESTIMATION OF STAGE PERFORMANCE 309
(iii) The velocity on the central streamline, Cn" is determined from the fact that
potential flow is ilTotational. The criterion of irrotationality is expressed by
a a
-(C cos 0:) - -(C sin rx) = 0
ay ax
where y is the co-ordinate in the whirl direction. The assumption in (ii)
implies that the first term is constant. With the additional assumption that
aCjBx varies linearly with y across the passage, the equation can be
integrated numerically to give C as a function of y and in particular the
value of C on the central streamline (Cm).
(iv) The value of the whirl velodty on the central streamline, Cwm, is again
determined by equating the change of momentum, from the leading edge to
x, to the corresponding pressure force; and the canIber line angle is then
obtained from tan 0:' = CwmlCm.
(v) The thickness in terms of t".is is determined from the continuity equation
as before.
When integrating across the passage for steps (iv) and (v), parabolic
variations of the relevant properties are used instead of linear variations.
The parabolas pass through the values at the two surfaces and at the central
streamline.
A third approximation on the same lines as the second can be made if neces-
sary, and the final shape is then determined by rounding off the leading and
trailing edges, arid subtracting an estimated displacement thickness of the
boundary layer which may be appreciable on the back of the blade where the
flow is decelerating.
7.4 Estimation of stage performance
The last step in the process of arriving at the preliminary design of a turbine stage
is to check that the design is likely to result in values of nozzle loss coefficient
and stage efficiency which were assumed at the outset. If not, the design calcu-
lations may be repeated with more probable values of loss coefficient and effi-
ciency. When satisfactory agreement has been reached, the final design may be
laid out on the drawing board and accurate stressing calculations can be per-
formed.
Before proceeding to describe a method of estimating the design point
performance of a stage, however, the main factors limiting the choice of design,
which we have noted during the course of the worked example, will be
sununarized. The reason we considered a turbine for a turbojet engine was simply
that we would thereby be working near those limits to keep size and weight to a
minimum. The designer of an industrial gas turbine has a somewhat easier task:
he will be using lower temperatures and stresses to obtain a longer working life,
and this means lower mean blade speeds, more stages, and much less stringent
310 A.."TIAL AND RADIAL FLOW TURBINES
aerodynamic limitations. A power turbine, not mechanically coupled to the gas
generator, is another case where much less difficulty will be encountered in
arriving at a satisfactory solution. The choice of gear ratio between the power
turbine and driven component is normally at the disposal of the turbine designer,
and thus the rotational speed can be varied to suit the turbine, instead of the
compressor as we have assumed here.
Limiting factors in turbine design
(a) Centrifugal stresses in the blades are proportional to the square of the
rotational speed N and the annulus area: when N is fixed they place an upper
limit on the annulus area.
(b) Gas bending stresses are (1) inversely proportional to the number of blades
and blade section moduli, while being (2) directly proportional to the blade
height and specific work output.
(1) The number of blades cannot be increased beyond a point set by blade
fixing considerations, but the section moduli are roughly proportional
to the cube of the blade chord which might be increased to reduce f5 gb.
There is an aerodynamic limit on the pitch/chord ratio, however,
which if too small will incur a high loss coefficient (friction losses
increase because a reduction in sic increases the blade surface area
swept by the gas).
(2) There remains the blade height: but reducing this while maintaining
the same annulus area (and therefore the same axial velocity for the
given mass flow), implies an increase in the mean diameter of the
annulus. For a fixed N, the mean diameter cannot be increased without
increasing the centrifugal disc stresses. There will also be an
aerodynamic limit set by the need to keep the blade aspect ratio
(h/c) and annulus radius ratio (rt/f/") at values which do not imply
disproportionate losses due to secondalY flows, tip clearance and
friction on the annulus walls (say not less than 2 and 1·2 respectively).
The blade height might be reduced by reducing the annulus area (with
the added benefit of reducing the centrifugal blade stresses) but, for a
given mass flow, only by increasing the axial velocity. An aerodynamic
limit on Co will be set by the need to keep the maximum relative Mach
number at the blade inlet (namely at the root radius), and the Mach
number at outlet from the stage, below the levels which mean high
friction losses in the blading and jet pipe respectively.
( c) Optimizing the design, so that it just falls within the limits set by all these
conflicting mechanical and aerodynamic requirements, will lead to an
efficient turb:ine of minimum weight. If it proves to be impossible to meet
one or more of the limiting conditions, the required work output must be
split between two stages. The second design attempt would be commenced
on the assumption that the efficiency is likely to be a maximum when
ESTIMATION OF STAGE PERFORMANCE
311
the work, and hence the temperature drop, is divided equally between the
stages.
(d) The. velocity upon which the rotor blade section depends, are
partially determrned by the to work with an average degree of
of 50 per cent to obtarn low blade loss coefficients and zero swirl
for n:mrnllun loss .in the jet pipe. To avoid the need for two stages in a
case, partICularly If It means adding a bealing on the downstream
SIde, . It would be preferable to design with a lower degree of
react:on and some swrrl. An aerodynamic limit on the minimum value of the
reactIOn at mean diameter is set by the need to ensure some positive reaction
at the blade root radius.
For what follo,:s in the next section, it will be helpful to have a summary of the
results of :he for the tID'bine of our worked example: such a
summary IS gIVen ill FIg. 7.23 and over the page.
T(K)
1100
1067
Gas angles CX1
root 0'
mean 0'
tip 0'
4bar
01
VW2Cp
97.8 K
Entropy
,3
!
,
,
I
,
,
,
Mc,
.-l>-
0.47
Mv,= 0.58
cx2
62.15'
58.38'
54.93'
Plane
p
T
P
A
rm
rrlrr
h
Blade row
sic
h (mean)
hlc
n
CX3
fh /33
12.12' 39.32' 51.13'
10' 20.49' 54.96'
8.52' 0' 58.33'
3.54 2.49 1.856 bar
1067 982.7 922
K
1.155 0.883 0.702 kg/m
3
0.0626 0.0833 0.1047 m2
f- 0.216 .,.)
m
1.24 1.33 1.43
0.046 0.0612 0.077 m
Nozzle Rotor
0.86 0.83
0.0536 0.0691 m
3.0 3.0
0.0175 0.023 m
0.01506 0.0191 m
90 71
FIG. 7.23 Summary of data from preliminary design calculations of sillgle-stage
turbine
312 AXIAL AND RADIJiL FLOW TURBINES
Mean diameter stage parameters
V' == 2c/)'Toslu
2
== 2·88, ¢ == CalU = 0·8, A =0·421
U = 340 m/s, Cal = C] = C3 = 276-4 m/s, Ca2 = Ca3 = Ca
= 272 m/s
C2 == 519 m/s, V3 473·5 m/s
Rotor blade designed for i = 0° with conventional profile having tic = 0·2. At the
root section; camber fJ2r + fJ3r 90° and hence Zr 0·00423 mm
3
/mm chord
giving (O"gb)max 93 MN/mz. (O"ct)max f(N,A) 200 MN/mz.
Estimation of design point pelformance
The method to be outlined here is that due to Ainley and Mathieson, Ref. (8),
which estimates the performance on flow conditions at the mean diameter of the
annulus. Reference (8) describes how to calculate the performance of a turbine
over a range of operating conditions, but we shall be concemed here only to find
the efficiency at the design point. A start is made using the two correlations for
profile loss coefficient Yp obtained from cascade data, which are shown in Fig.
7.24. These refer to nozzle-type blades (j3z = 0) and impulse-type blades (fJz =
fJ3) of conventional profile (e.g. T6) having a thickness/chord ratio (tic) of 0·2
and a trailing edge thickness/pitch ratio (tel s) of 0·02. Rotor blade notation is
used in Fig. 7.24 and in what follows, to emphasize that we are thinking of the
flow relative to any blade row. When the nozzle row is being considered fJ2
becomes IX] and fh becomes 1X2: there is no need to duplicate equations which
apply equally to both rows. The values of Yp in Fig. 7.24 refer to blades operating
at zero incidence, i.e. when the gas inlet angle fJz is also the blade inlet angle.
Step 1
Estimate (Yp)N and (Yph from the gas angles of the proposed design by using Fig.
7.24 in conjunction with the interpolation formula
T { (fJ2)
2
_ ' } (tIC)P2/P3
lp = + /33 [Yp(P2=P3) }P(P,=Ol] 0.2
(7.32)
This equation represents a correction for a change in inlet angle at a constant
outlet angle, so that Yp(P, = 0) and Yp(P2 = P3) are the values for a nozzle and
impulse-type blade having the same outlet gas angle fJ3 as the actual blade.
Equation (7.32) also includes a correction for t/c if it differs from 0·2, a reduction
in t/ c leading to reduced profile loss for all blades other than nozzle-type blades
(f3z = 0). The degree of acceleration of the flow in the blading decreases with the
degree of reaction as fJ21fJ3 -+ 1, and the influence of blade thickness becomes
more marked as the acceleration is diminished. The correction is considered
reliable only for 0·15 < tic < 0·25.
ESTIMATION OF STAGE PERFORMANCE

0.08
0.04
Nozzle blades = 0
Outlet gas angle
/65'
........... 60'
. 50'

-40'

0.2
>- 0.20
Impulse blades =
" c
Q)
·13

o
"
'"
'" .9
0.16
0.12
2
"-
50'
0.08 40'
0.04 Re(= P3 V3cl!l3) = 2 x 105
MV3 < 0.6; incidence = 0'
Pitch/chord (sic)
FIG. 7.24 Profile loss coefficiellt for cOllvelltional blading witll tic = O';W
313
For nozzle blades of our example, IXI = 0 and so (Yp)N can be read straight
from Fig. 7.24. 1X2=58·38° and (slc)N=0·86, and hence
(Yp)N = 0·024.
For the rotor blades, fJ2 = 20·49°, fJ3 = 54·96°, (S/C)R = 0·83 and (tIc) = 0.2,
so that
= (0.023 + 2 [0.087 - 0.023]} = 0.032
Step 2
If it had been decided to design the blades to operate with some incidence at the
point, a correction to would be required. As this correction is really
only Important when estimating performance at part load, we shall refer the reader
to Ref. (8) for the details. Briefly, it involves using correlations of cascade data to
the stalling incidence is for the given blade (i.e. incidence at which is
twICe the. loss i = 0); and then using a curve of versus i/i
s
to find Yp
for the gIVen I and the value of Yp(i = 0) calculated in Step 1.
314 AXIAL AND RADIAL FLOW TURBINES
Step 3
Secondary and tip clearance loss data for 1's and 1'k have been cOlTelated using the
concepts of lift and drag coefficient which were introduced.in section 5.9 for axial
compressors. Without repeating the whole argument leading to equation (5.34), it
should be possible to see by glancing at section 5.9 that for a turbine cascade
(with rotor blade notation) .
where
Now as stated at the beginning of section 7.3, it is convenient to treat 1's and 1'k
simultaneously. The proposed colTelation is
(7.33)
Reference back to equations (5.35) and (5.38) will make the appearance of the
last two bracketed terms seem logical: they occur in the expression for the com-
bined secondary and tip clearance loss coefficient for a compressor blade row.
Considering now the first bracketed term, the tip clearance component is pro-
portional to kl h where k is the clearance and h the blade height. The constant B is
0·5 for a radial tip clearance, and 0·25 for a shrouded blade with side clearance;
see Fig. 7.25. The secondary loss component A is more complex. We have sug-
gested that secondary flow and annulus wall friction might be affected by the
aspect ratio (hlc) andlor annulus radius ratio (rt/rr)' As we shall see, rtJrr is
thought by Ainley and Mathieson to be the more relevant parameter. (They argue
that hI c is only important in so far as there is a change in h, not in c.) Also, like
the profile loss, 1's is considerably affected by the amount of acceleration of the
flow in the blade passage. In general terms, the larger the acceleration the thinner
and more stable are the boundary layers, the smaller is the chance of boundary
layer separation, and the smaller is the effect of a curved neighbouring surface in
setting up the secondary flows. The degree of acceleration is conveniently indi-
cated by the ratio of the area normal to the flow at outlet to that at inlet, i.e.
A3 cos {hI A2 cos P2 where A is the annulus area. It is found that the quantity A in
k


8=0.5 8=0.25
FIG. 7.25
ESTIMATION OF STAGE PERFORMANCE
315
0.03
0.02
0.01
(
A3 COSfi3)2/ (1 +.IL)
A2 cos fi2 r,
FIG. 7.26 Secondary loss parameter
equation (7.33) is given approximately by
A = f{ (A3 cos /13)2/(1 +.':!:) 1
A2 cos /12 rt I
(7.34)
where the function f is given by the curve in Fig. 7.26.
Let us now eyaluate (1's + Yk) for the blades in our example.
Nozzle blades: We shall assume that the nozzles are shrouded, with seals
supporte.d by a at the shaft radius so that the leakage loss is very small.
Then B ill equatIOn (7.33) can be assumed zero. }, is found as follows_
A2 = 0·0833 m
2
, A1 = 0·0626 m
2
cos az = cos 58·38° = 0·524, cos a1 = cos 0 = 1-0
meanrtlrr between planes 1 and 2 = 1·29
G: / (
1
= / (1 + = 0-274
From Fig. 7.26, A = 0·012_
CL
sic = 2(tan cx1 + tan ( 2) cos am
cx., = tan-
1
[(tan a2 - tan 1X1)/2]
= tan-
1
[(tan 58·38° - tan 0°)/2] = 39.08°
CL
-I = 2(tan 58·38° + tan 0°) cos 39·08° - 2·52
sc -
cos
2
a2 0.5242
cos3 IXm = 0-7763 = 0·589
[Ys + 1'klN = 0-012 X 2.52
2
x 0·589 = 0-0448
316
AXIAL AND RADIAL FLOW TURBINES
. . .
Rotor blades: We will assumem1shrouded tOtor blades with radial tip clearance
equal to 2 per cent of the mean blade height,so that
. B(k/h) = 0·5 x 0·02 = 0·01
For these blades, using the data from Fig. 7.23,
(
A3 cos 133)2/(1 + r ..) _ (0.1047 x cos 54.960)2/(1 +_1 )
A2 cos 132 -;:; - 0·0833 x cos 20-49° 1·38
= 0·334
From Fig. 7.26, A=0·015
Step 4
13m = tan-I [(tan 54·96° - tan 20·49°)/2] = 27·74°
CL = 2(tan 54.96° + tan 20·49°) cos 27·74° = 3·18
sic
cos
2
133 = 0.574
2
= 0.475
cos3 13m 0.885
3
[Ys + Yk]R = (0·015 + 0.01)3.18
2
x 0·475 = 0·120
The total loss coefficients become
YN = ( ~ ) N + [Ys + Yk]N = 0·024 + 0·0448 = 0·0688
YR = (Yp)R + [Ys + Ykk = 0·032 + 0·120 = 0·152
lfthe trailing edge thickness/pitch ratio (te/s) differs from 0·02, it is at this point
that a correction is made for the effect on the losses. 0·02 was the value for the
blading to which Fig. 7.24 relates, but trailing edge thickness affects all the
losses, not merely the profile loss. The correction curve in Fig. 7.27 has been
deduced from turbine test results. There is no reason to suppose that the normal
value of 0·02 would be unsuitable for the nozzle and rotor blades of the turbine of
our example and no correction is required. The modest turbine inlet temperature
indicates a low-cost, long-life, application by aircraft standards and certainly there
will be no need to thicken the trailing edge to accommodate cooling passages.
Early development tests may indicate vibration troubles which might be overcome
1.B
N 1.6
o
ci
II 1.4
5
.E 1.2
>-
~ 1.0
tels
FIG. 7.27 Correction factor for trailing edge thickness
ESTIMATION OF STAGE PERFORMANCE 317
by increasing the thickness, and Fig. 7.27 enables the penalty to be paid in loss of
performance to be estimated.
Step 5
The stage efficiency can now be calculated using equations (7.19) and (7.20). We
first calculate the equivalent loss coefficients defined in terms oftemperature. For
the nozzles, .
A - ~ - 0·0688 - 0.0611
N - (T02/T2) - (1100/976.8) -
For the rotor,
YR
AR = ( / ")
T03rei T3
We have previously calculated T3 to be 913 K, but have not found the value of
T03rel ' We did, however, find that (V; /2cp ) = 97·8 K and T3 = 922 K., so that
T03rei = T3 + (Vi /2cp ) = 1020 K
0·152
AR = 1020/913 0·136
Equation (7.20) now becomes
11 = ~ = 0·88
s 1 + [0.136 x 97-8 + ( 9 ~ ~ ~ 7 )0.0611 x 117.3]/145
Thus the design yields AN= 0·061 and I1s = 0·88, in comparison with the values of
0·05 and 0·9 assumed at the outset. This can be regarded as satisfactory agree-
ment, but minor changes would be looked for to improve the efficiency: perhaps a
slight increase in degree of reaction with the reduction in work due to this
compensated by designing with some progressive increase in Ca through the
stage. The latter would have the added advantage of reducing the flare of the
annulus.
Step 6
In conclusion, it must be emphasized that the cascade data and other loss cor-
relations are strictly applicable only to designs where the Mach numbers are such
that no shock losses are incurred in the blade passages. It has been suggested,
Ref. (13), that the additional loss incurred by designing with a blade outlet
relative Mach number greater than unity can be accounted for by adjusting the
profile loss coefficient Yp of the blade row concerned. The correction is given by
~ = [ ~ from eqn. (7.32)] x [1 + 60(M - 1)2]
where Mis MV3 for the rotor blades and Mcz for the nozzles. There is another
restriction on the applicability of the data not yet mentioned: the Reynolds
318 AXIAL AND RADIAL FLOW TURBINES
number of the flow should be in the region of J )( 10
5
to 3 X 10
5
, withRe
defined in terms of blade chord, and density and relative velocity at outlet of a
blade row. If the mean Reynolds number for a turbine, taken as the arithmetic
mean of Re for the first nozzle row and the last rotor row (to cover multistage
turbines), differs much from 2 x 105, an approximate correction can be made to
the overall isentropic by using the expression
(
Re )-0.1
(1 -1]t) = 2 x 105 (1 - 11t)Re=2 x 10'
(7.35)
To calculate Re fot the nozzle and rotor rows of our example, we need the
viscosity of the gas at temperatures T2 = 982·7 K and T3 = 922 K. Using data for
air, which will be sufficiently accurate for this purpose,
f..I1 = 4·11 X 10-
5
kg/m sand /13 = 3·95)( 10-
5
kg/m s
(Re) = P2 C1CN = 0·883)( 519 x 0·0175 = 1.95 )( 105
N /12 4·11 )( 10-5
(Re) = P3 V3CR = 0·702 x 473·5 )( 0·023 = 1.93 )( 105
R /13 3 ·95 X 10-5
Thus no Reynolds number correction is required.
The Ainley-Mathieson method outlined here has been found to predict
efficiencies to within ± 3 per cent of the measured values for aircraft turbines, but
to be not so accurate for small turbines which tend to have blades of rather low
aspect ratio. Dunham and Came, in Ref. (13), suggest that the method becomes
applicable to a wider range of turbines if the secondary and tip clearance loss
correlation, equation (7.33), is modified as follows.
(a) /I., instead of being given by the function expressed in Fig. 7.26, is replaced
by
(cos fi3)
h cos fi2
(with fi3 = IJ.z and fiz = 1J.1 for nozzles).
(b) B(k/h) is replaced by
C (k)0.78
B(-) -
h C
with B equal to 0·47 for radial tip clearances and 0·37 for side clearances on
shrouded blades.
When this modification is applied to the turbine of our example there is, as might
be expected, very little difference in the predicted efficiency, namely 0·89 as
compared with 0·88. Larger differences of up to 5 per cent are obtained with low
aspect ratios of about unity. Further refinements to the method are described in
Ref. (14).
OVERALL TURBINE PERFORMANCE 319
7.5 Overall turbine performance
In the previous section we have described a method of estimating the stage
efficiency 1]s' If the whole turbine comprises a large number of similar stages, it
would be a reasonable approximation to treat the stage efficiency as being equal
to the polytropic efficiency l100t and obtain the overall isentropic efficiency from
equation (2.18). This was the approach suggested in section (5.7) for the pre-
liminary design of multi-stage axial compressors. Turbines have few stages,
however, and it is preferable to work through the turbine stage by stage, with the
outlet conditions from one stage becoming the inlet conditions of the next, until
the outlet temperature is established. The overall efficiency 1)t is then obtained
from the ratio of the actual to isentropic overall temperature drop.
As stated under the heading 'Estimation of design point performance', it is
possible to calculate the performance of a turbine over a range of operating
conditions. Whether calculated, or measured on a test rig, the perfonnance is
normally expressed by plotting 'It and M .jT03 /P03 against pressure ratio P03/P04
for various values of N/.jT03 as ill Fig. 7.28. We are here reverting to cycle
notation, using suffixes 3 and 4 to denote turbine inlet and outlet conditions
respectively. The efficiency plot shows that 1], is sensibly constant over a wide
range of rotational speed and pressure ratio. This is because the accelerating
nature of the flow permits turbine blading to operate over a wide range of
incidence without much increase in the loss coefficient.
0.6
W 1.2
'" co
1.0
en
'w
Q)
"0
.8
Q)
>
__ __ _ 1.0
0.6

Turbine choking
// NI{f,3(rel. to design value)
B 0.4 ./
Ei; 0.4
'-" 0.2
"?
E OL-____ ____ ____ ____
1.0 2.0 3.0 4.0 5.0
Pressure ratio P03/P04
FIG. 7.28 Turbille characteristics
320 AXiAL MID RADIAL FLOW TURBINES
The maximum value of mJT03/P03 is reached itt a pressure ratio which
produces choking conditions at some point in the turbine. Choking may occur in
the nozzle throats or, say, in the annulus at outlet from the turbine depending on
the design. The former is the more normal situation and then the constant speed
lines merge into a single horizontal line as indicated on the mass flow plot of Fig.
7.28. (If choking occurs in the rotor blade passages or outlet annulus' the
maximum mass flow will vary slightly with N / JTQ3.) Even in the unchoked
region of operation the separation of the N / J T 03 lines is not great, and the larger
the number of stages the more nearly can the mass flow characteristics be
represented by a single curve independent of N / J T 03' Such an approximation is
very convenient when predicting the part-load perfonnance of a complete gas
turbine unit as will be apparent from Chapter 8. Furthermore, the approximation
then yields little error because when the turbine is linked to the other components
the whole operating range shown in Fig. 7.28 is not used. Normally both the
pressure ratio and mass flow increase simultaneously as the rotational speed is
increased, as indicated by the dotted curve.
7.6 The cooled turbine
It has always been the practice to pass a quantity of cooling air over the turbine
disc and blade roots. When speaking of the cooled turbine, however, we mean the
application of a substantial quantity of coolant to the nozzle and rotor blades
themselves. Chapters 2 and 3 should have left the reader in no doubt as to the
benefits in reduced SFC and increased specific power output (or increased
specific thrust in the case of aircraft propulsion units) which follow from an in-
crease in permissible turbine inlet temperature. The benefits are still substantial
even when the additional losses introduced by the cooling system are taken into
account.
Figure 7.29 illustrates the methods of blade cooling that have received serious
attention and research effort. Apart from the use of spray cooling for thrust
boosting in turbojet engines, the liquid systems have not proved to be practicable.
There are difficulties associated with channelling the liquid to and from the
blades-whether as primary coolant for forced convection or free convection
open thermo syphon systems, or as secondary coolant for closed thermo syphon
systems. It is impossible to eliminate corrosion or the formation of deposits in
open systems, and very difficult to provide adequate secondary surface cooling
area at the base of the blades for closed systems. The only method used
successfully in production engines has been internal, forced convection, air
cooling. With 1·5-2 per cent of the air mass flow used for cooling per blade row,
the blade temperature can be reduced by between 200 and 300 DC. Using current
alloys, tbis permits turbine inlet temperatures of more than 1650 K to be used.
The blades are either cast, using cores to form the cooling passages, or forged
with holes of any desired shape produced by electrochemical or laser drilling.
Figure 7.30 shows the type of turbine rotor blade introduced in the 1980s. The
THE COOLED TURBINE
I
AIR Goaling .
I
EXTERNAL
I
Film


Transpiration
(effusion)

Porous well
I
I
EXTERNAL
I
Spray
Sweat cooling
(liquid through
porous wall)
FIG. 7.29 Methods of blade cooling
I
LIQUID cooling
I
I

I
Forced conv. Free conv.
I I
321
I THERMOSYPHON
tHt. """
7
Compressed liquid
or evaporating
liquid Liquid metal
or evaporating
liquid
next step forward is likely to be achieved by transpiration cooling, where the
cooling air is forced through a porous blade wall. This method is by far the most
economical in cooling air, because not only does it remove heat from the wall
more uniformly, but the effusing layer of air insulates the outer surface from the
hot gas stream and so reduces the rate of heat transfer to the blade. Successful
application awaits further development of suitable porous materials and
techniques of blade manufacture.
We are here speaking mainly of rotor blade cooling because this presents the
most difficult problem. Nevertheless it should not be forgotten that, with high gas
temperatures, oxidation becomes as significant a limiting factor as creep, and it is
therefore equally important to cool even relatively unstressed components such as
nozzle blades and annulus walls. A typical distribution of cooling air required for
Pressure side concave
cooling holes
/
FIG. 7.30 Cooled tllrbine rotor blade !courtesy Genera! Electric]
322 AXIAL AND RADIAL FLOW TURBINES
a turbine stage designed to operate at 1500 K might be as follows. The are
expressed as fractions of the entry gas mass flow.
annulus walls
nozzle blades
rotor blades
rotor disc
0·016
0·025
0·0,19
0·005
0·065
Figure 7.31(a) illustrates the principal features of nozzle blade c?oling. The, ru: is
introduced in such a way as to provide jet impingement cooling of the mslde
surface of the very hot leading edge. The spent air leaves through slots or holes in
the blade surface (to provide some film cooling) or in the trailing edge. Figure
7.31(b) depicts a modern cast nozzle blade with intricate inserts forming the
cooling passages. It also shows the way the annulus walls are cooled.
It was stated earlier that liquid cooling had not proved to be practical. In 1995,
however General Electric announced the introduction of closed-loop steam
cooling 'in their heavy industrial gas turbines designed for combined cycle
operation; these can be expected to enter service in the late 1990s. Both and
nozzle blades are steam cooled, using steam extracted from the steam turbme at
high pressure; the steam undergoes a pressure loss during the cooling process, but
absorbs heat. The heated steam is then returned at a lower pressure to the steam
turbine contributing to the power output and being contained in the steam circuit.
This is suitable for use with a combined cycle, where steam is readily
available, but requires the use of sophisticated sealing technology to prevent loss
of steam; the losses due to bleeding high pressure air from the compressor for use
in an air-cooled turbine are eliminated. Steam cooling is at an early stage of
development, and attention will now be focussed on the simple analysis of the
conventional air-cooled turbine.
(a) Platform film cooling holes
(b)
FIG. 7.31 Turbine nozzle cooling [(b) courtesy RoDs-Royce]
.. Cooling air
TIlE COOLED TURBINE
323
There are two distinct aspects of cooled turbine design. Firstly, there is the
problem of choosing an: aerodynamic design which requires the least amount of
cooling air for a given cooling performance. One cooling performance parameter
in common use is the blade relative temperature defrned by
Tb - Tcr
bladl? relative temperature = T _ T
g cr
where Tb = mean blade temperature
Tcr = coolant temperature at inlet (i.e. at the root radius rr)
Tg = mean effective gas temperature relative to the blade
temperature+0·85 x dynamic temperature)
The coolant temperature, Ten will usually be the compressor delivery
temperature, and will increase significantly as pressure ratio is raised to reduce
specific fuel consumption. Some industrial turbines pass the cooling flow through
a water-cooled heat-exchanger to reduce T cr and hence the blade relative
temperature. At current levels of turbine inlet temperature three or four stages of a
turbine rotor may be cooled, and air would be bled from earlier stages of the
compressor to cool the later stages of the turbine, where the cooling air discharges
to lower pressures in the gas stream. Bleeding air from earlier stages reduces the
work input required to pressurize the cooling air, with beneficial effects on the net
output.
Relative to an Uncooled turbine, the optimum design might well involve the
use of a higher blade loading coefficient IjJ (to keep the number of stages to a
minimum), a higher pitch/chord ratio (to reduce the number of blades in a row),
and a higher flow coefficient ¢ (which implies a blade of smaller camber and
hence smaller surface area). The importance of these, and other parameters such
as gas flow number, are discussed in detail in Ref. (9).
The second aspect is the effect on the cycle efficiency oflosses incurred by the
cooling process: a pertinent question is whether it is advantageous overall to
sacrifice some aerodynamic efficiency to reduce such losses. The sources of loss
are as follows.
(a) There is a direct loss of turbine work due to the reduction in turbine mass
flow.
(b) The expansion is no longer adiabatic; and furthermore there will be a
negative reheat effect in multi-stage turbines.
(c) There is a pressure loss, and a reduction in enthalpy, due to the mixing of
spent cooling air with the main gas stream at the blade tips. (This has been
found to be partially offset by a reduction in the normal tip leakage loss.)
(d) Some 'pumping' work is done by the blades on the cooling air as it passes
radially outwards through the cooling passages .
(e) When considering cooled turbines for cycles with heat-exchange, account
must be taken of the reduced temperature of the gas leaving the turbine
which makes the heat-exchanger less effective.
324 AXIAL AND RADIAL FLOW TURBINES
Losses (a) and (e) can be incorporated directly into any cycle calculation, while
the effect of (b), (c) and (d) can be taken into account by using a reduced value of
turbine efficiency. One assessment of the latter, Ref. (10), suggests that the
turbine efficiency is likely to be reduced by from 1 to 3 per cent of the uncooled
efficiency, the lower value referring to near-impulse designs and the higher to 50
per cent reaction designs. 1:he estimate for reaction designs is substantially
confinned by the tests on an experimental cooled turbine reported in Ref. (11).
Cycle calculations have shown that even when all these losses are accounted
for, there is a substantial advantage to be gained from using a cooled turbine. t
Before either of these two aspects of cooled turbine design can be investigated
it is necessary to be able to estimate the cooling air flow required to achieve a
specified blade relative temperature for any given aerodynamic stage design. We
will end this section with an outline of an approximate one-dimensional
treatment, and further refinements can be found in Ref. (9). Fignre 7.32 shows the
notation employed and the simplifying assumptions made.
Consider the heat flow to and from an elemental length of blade (j[ a distance l
from the root. As the cooling air passes up the blade it increases in temperature
and becomes less effective as a coolant, so that the blade temperature increases
from root to tip. There must therefore be some conduction of heat along the blade
to and from the element Ol due to this spanwise temperature gradient. Because
turbine blade alloys have a low thermal conductivity, the conduction term will be
small and we shall neglect it here. The heat balance for the elemental length Ol is
then simply
(7.36)
iii
.L
"I
c
Constant ill
" 0
()
I
.r::;"
l'
Root
0
Ter
FIG. 7.32 Forced cOllvectioll air cooling
t It is worth remembering that there may be special applications where a cooled turbine might be
employed not to raise the cycle temperature but to enable cheaper material to be used at ordinary
temperatures: the first researches in blade cooling were carried out in Germany during the Second
World War with this aim in mind.
THE COOLED TURBINE
325
where hg and he are the gas-side and coolant-side heat transfer coefficients, and Sg
and Sc are the wetted perimeters of the blade profile and combined coolant
passages respectively. For the internal air flow me we also have
dTe
mccpcdz = heSeCTb - TJ (7.37)
We may ·first fuld the variation of Tb with I by eliminating Tc between the two
equations. From (7.36) we have
hgSg
Te = Tb - h S (Tg - Tb) (7.38)
e e
and hence
dTe = (1 + hgSg) dTb
dl heSe dl
Substituting in (7.37), remembering that dTb/dl = - d(Tg - Tb)/dl, we get
(
h S ) d(T - T ) h S
1 +..L.! g b + ~ ( T g - T
b
) = 0
heSe dl mecpe
The solution of this differential equation, with Tb = Tbr at l = 0, is
Tg - Tb = (Tg - Tbr)e-kl/L
where
k = hgSgL
mecpe[l + (hgSg/heSJ]
To obtain the variation of Te with I, we may write (7.38) in the form
T - T = (T - T) 1 + Ig g
[
IS J
g e g b heSe
and substitute (7.39) for (Tg - Tb) to give
T - T = (T - T.) 1 +..L.! e-kl/L
l
h S J
g c g h, heSe
When 1=0, Te = Tel' and hence
[
hgSg]
Tg - Ter = (Tg - Tbr) 1 + heSe
Combining (7.40) and (7 AI) we have the variation of Te given by
Tg - Te = (Tg - Ter)e-kl/L
Finally, subtracting (7.39) from (7.41),
T - T = (T - T )[1 + hgS
g
_ e-kl/L]
b er g br h S
e e
(7.39)
(7.40)
(7.41)
(7.42)
326 AXIAL AND FLOW TURBINES
and dividing this by (7.41) we hiwe the blade relative temperature given by
. Tb - Ter = 1 _ . e-
kJ1L
. (7.43)
Tg - Ter [1 + (hgSg/hcSe)]
We may note that he will be a function of coolant flow Reynolds number and
hence of me, and that me also appears in the parameter k. Thus equation (7.43) is
not explicit in me, and it is' convenient to calculate values of blade relative
temperature for various values of me rather than vice versa.
The next step is the evaluation of the heat transfer coefficients: we will
consider he first. For straight cooling passages of uniform cross-section, pipe flow
formulae may be used. For this application one recommended correlation is
Nu = 0·034 (L/D)-o.! (Prt\Re)0'8(TclTb)055 (7.44)
with fluid properties calculated at the mean bulk temperature. The LID term
accounts for the entrance length effect. The Te/Tb term is necessary when the
difference in temperature between fluid and wall is large, to allow for the effect of
variation in fluid properties with temperature. The characteristic dimension D can
be taken as the equivalent diameter (4 x area/periphery) if the cooling passages
are of non-circular cross-section. For air (Pr ::::: 0·71) and practicable values of
L/D between 30 and 100, the equation reduces to the simpler form
Nu = 0.020(Re)08(T
c
/T
b
)0.55 (7.45)
The correlation is for turbulent flow, with the accuracy decreasing at Reynolds
numbers below 8000. Note that the mean value of Te at which the fluid properties
should be evaluated, and the mean value of Tb for the Te/Tb term, are unknown at
this stage in the calculations. Guessed values must be used, to be checked later by
evaluating Te and lb at 1IL=0,5 from equaltions (7.42) and (7.43).
Data for mean blade heat transfer coefficients hg are available both from
cascade tests and turbine tests on a wide variety of blades. The latter yield higher
values than the former, presumably because of the greater intensity of the
turbulence in a turbine. The full line in Fig. 7.33, taken from Ref. (9), is a useful
design curve for the mean value of Nusselt number round the blade surface in
terms of the most significant blade shape parameter which is the ratio of
inlet/outlet angle /32//33 (or rxdrx2 for nozzle blades). NUg decreases as the degree
of acceleration of the flow increases, because the point of transition from a
laminar to a turbulent layer on the convex surface is delayed by accelerating flow.
The curve applies to conventional turbine blade profiles, and gives nominal
values denoted by Nu; for operating conditions of Reg = 2 x 10
5
and Tg/ Tb 1.
NUg can then be found from
NUg = NU;(2 (it
where the exponent x is given by the supplementary curve, and y is given by
y=O·14 . (
R )-0,.
2 X 100
THE COOLED TURBINE
0,8
Turbine
x 0.7 Cascade _------
0,6 _------------
0.5 '-----"-_----L __ -L..._---'--_---l
*",,,, 600
<
a;
.0
§ 500
c:
'16

400
ffi
E
(ij 300
c:
'E
o
z
t
Nozzle
blade
Reg = 2 x 10
5
TglTb -71.0
f33 (or U2) 45' -7 70'
t
Impulse
blade
FIG. 7.33 Heat transfer data for cOllvectiollal blade profiles
327
The characteristic dimension in NUg and Re is the blade chord, and the fluid
. g
propertIes should be evaluated at the temperature Tg. The velocity in Reg is the
gas velocity relative to the blade at outlet (V3 or C2 as the case may be). The
quantities required will be known for any given stage design, with the exception
of the mean value of Tb for which the guessed value must be used.
All the information has now been obtained to permit the calculation of
(Tb - Ter)/(Tg - Ter), for various values of coolant flow me, from equation
(7.43). Typical curves of spanwise variation, for values of me of 1 and 2 per cent
of the gas mass flow per blade row, are shown in Fig. 7.34. Note that the
distribution matches requirements because the blade stresses decrease from root
to tip. The quantity of heat extracted from the blade row can be found by
calculating Tet, i.e, Tc at 1 = L, from equation (7.42), and then evaluating
mecpeCTet - Ter) ,
1.0
'"""
1--" 0.8
I
1--'"
:::::- 0.6
;:?
0.4
0.2 ':--L_-'-----'_--'-----l
o 0.2 0.4 0.6 0.8 1.0
ilL
FIG. 7.34 Typical spallwise temperature distrilmtioll.§
328 AXIAL AND RADIAL FLOW TURBINES
1100
1300
FIG. 7.35 Typical temperature distribution (from Ref. (121)
To conclude, we may note that the final design calculation for a cooled blade
will involve an estimation of the two-dimensional temperature distribution over
the blade cross-section at several values of 1/ L. Finite difference methods are used
to solve the differential equations, and conduction within the blade is taken into
account. Figure 7.35 shows a typical temperature distribution at the mid-span ofa
blade designed to operate with Tg = 1500 K and Tcr = 320 K. It emphasizes one
of the main problems of blade cooling, i.e. that of obtaining adequate cooling at
the trailing edge. Finally, an estimation will be made of the thermal stresses
incurred with due allowance for redistribution of stress by creep: with cooled
blades the thermal stresses can dominate the gas bending stresses and be
comparable with the centrifugal tensile stresses. References to the literature
dealing with these more advanced aspects can be found in the paper by Barnes
and Dunham in Ref. (12).
Finally, mention must be made of an alternative approach to the high-
temperature turbine-the use of ceramic materials which obviates the need for
elaborate cooling passages. Much effort has been expended on the development
of silicon nitride and silicon carbide materials for small turbine rotors (both axial
and radial) in which it would be difficult to incorporate cooling passages.
Adequate reliability and life are difficult to achieve, but demonstrator engines
have been run for short periods. Ceramic rotor blades are being investigated for
use in stationary gas turbines for powers up to about 5 IvIW, and experimental
trials are expected in the late 1990s. The use of ceramic turbines in production
engines, however, remains an elusive goal tbree decades after optinlistic forecasts
about their introduction.
7.7 The radial flow turbine
Figure 7.36 illustrates a rotor having the back-to-back configuration mentioned in
the introduction to tIns chapter, and Ref. (15) discusses the development of a
successful family of radial industrial gas turbines. In a radial flow turbine, gas
flow with a high tangential velocity is directed inwards and leaves the rotor with
as small a whirl velocity as practicable near the axis of rotation. The result is that
the turbine looks very sinnlar to the centrifugal compressor, but with a ring of
nozzle vanes replacing the diffuser vanes as in Fig. 7.37. Also, as shown there
THE RADIAL FLOW TURBINE
329
FIG. 7.36 Back-to-back rotor [courtesy of Kongsberg Ltdl
would normally be a diffuser at the outlet to reduce the exhaust velocity to a
negligible value.
The velocity triangles are drawn for the normal design condition in which the
relative velocity at the rotor tip is radial (i.e. the incidence is zero) and the
absolute velocity at exit is axial. Because Cw3 is zero, the specific work output W
becomes simply"
W = Cp(TOl - To3 ) = Cw2 UZ = ui (7.46)
In the ideal isentropic turbine with pelfect diffuser the specific work output would
be
Volute
I
Nozzle vanes
~ ' - ' - - ' - - 2
Diffuser
FIG. 7.37 Radial inflow turbine
330 AXIAL AND RADIAL FLOW TURBINES
where the velocity equivalent of the isentropic enthalpy drop, Co, is sometimes
called the 'spouting velocity' by analogy with hydraulic turbine practice. For this
ideal case it follows that ul = C ~ / 2 or u2/CO = 0·707. In practice, it is found that
a good overall efficiency is obtained if this velocity ratio lies between 0·68 and
O· 71, Ref. (16). In terms of the turbine pressure ratio, Co is given by
C ~ _ [ ( I ) (Y-I)IYJ
- - cpToI 1 - --
2 PoJiPa
Figure 7.38 depicts the processes in the turbine and exhaust diffuser on the T-s
diagram. The overall isentropic efficiency of the turbine and diffuser may be
expressed by
101 - T03
'10 = To T'
01 - 4
(7.47)
because (To I - T4) is the temperature equivalent of the maximum work that
could be produced by an isentropic expansion from the inlet state (Po], ToI ) to Pa.
Considering the turbine alone, however, the efficiency is more suitably expressed by
TOl - T03
'It = TOI _ T ~ (7.48)
which is the 'total-to-static' isentropic efficiency referred to in section 7.l.
Following axial flow turbine practice, 1he nozzle loss coefficient may be
defined in the usual way by
)'N = T22 - T ~ (7.49)
C2 /2cp
/' P01 P02
-._.--'7r"---,;027[ P 1
f
VY2cp
C§/:<cp t
4'
3
3"
3'
Entropy s
FIG. 7.38 T-s diagram for a radiaillow turbine
THE RADIAL FLOW TURBINE 331
Similarly, the rotor loss coefficient is given by
.Ie _T3- T!j
R - Vl/2c
p
(7.50)
Plane 2 is located at the periphery of the rotor, so that the nozzle loss includes not
only the loss in the inlet volute but also any friction loss in the vaneless space
between noizle vane trailing edge and rotor tip. Bearing in mind that for the small
constant pressure processes denoted by 2' - 2 and 3' - 3" we can Wlite c/jT =
TDS, AN can be alternatively expressed by
(7.51)
A useful expression for 'It in terms of the nozzle and rotor loss coefficients can
be found as follows. The denominator in equation (7.48) may be expanded to
yield
Consequently lIt becomes
From the velocity triangles,
C2 = U2 cosec ()(z, V3 = U3 cosec P3' C3 = U3 cot P3
()(2 is the gas angle at inlet to the impeller and hence the effective outlet angle of
the nozzle vanes, while P3 is the outlet angle of the impeller vanes. Furthermore,
U3 = U2r3/r2 and cp(ToI - T03 ) = ui, so that the expression for efficiency finally
becomes
Here lit is expressed in terms of the nozzle and impeller outlet angles, radius ratio
of impeller, loss coefficients, and the temperature ratio T ~ / T ~ . The temperature
ratio T ~ / T ~ can in turn be expressed in tenns of the major design variables,
although it is usually ignored because it is sufficiently near unity for it to have
little effect on YJ (. Thus we can write
T ~ T!j T2-T!{ 1 "
-, ~ - = 1 - --= 1 - -:-[(T2 - T3) + (T3 - T3 )]
T2 T2 T2 T2
332
AXIAL AND RADIAL FLOw TURBINES
. .
Now (T2 - T3) may be found by expanding equation (7.46) and making use of
the ve1ocio/ triangles in Fig. 7.36. Thus, since TOl =T02,'
'u1 = c/T2 - T3) +!(ci - cD
= c/T2 - T3) +!(Vi + ui) - !(V}- uf)
1 (2 . 2) . (2 2)]
(T2 - T3) = -2 [V3 - V2 + U2 - U3
cp
(Note that TOrel = T + V
2
/2cp is not the same at inlet and outlet of the rotor as it is
in the axial flow machine, because U3 #- U2• This is the main difference between
Fig. 7.37 and Fig. 7.4.) It follows that
T3 Tf 122222
--; R! - = 1---[(V3 - V2 ) + (U2 - U3 ) + ARV3]
T2 T2 2cp T2
= 1 - ui [1 + ( ~ ) 2 {(1 + AR) cosec
2
P3 - I} - cof 1X2]
2cp T2 r2
And, finally, T2 may be expressed as
ui 2
T2 = TOl - -2 cosec 1X2
cp
(7.53)
AN is usually obtained from separate tests on the inlet volute and nozzle vane
assembly, enabling AR to be deduced from overall efficiency measurements with
the aid of equation (7.52) as in the following example.
EXAMPLE 7.1
A radial flow turbine of the following geometry is tested under design conditions
with a pressure ratio POr/P3 of2·0 and an inlet temperature of 1000 K. The work
output, after allowance has been made for mechanical losses, is 45·9 kW when
the rotational speed is 1000 rev/s and the gas mass flow is 0·322 kg/so
Rotor inlet tip diameter
Rotor exit tip diameter
Hub/tip ratio at exit
Nozzle efflux angle 1X2
Rotor vane outlet angle P3
12·7 cm
7·85 cm
0·30
70°
40°
Separate tests on the volute/nozzle combination showed the nozzle loss
coefficient AN to be 0·070. The turbine isentropic efficiency 'l'ft and the rotor
loss coefficient AR are to be evaluated from the results.
[ (
1 )(Y-l)IY] [( 1 )1]
TOl - T3 = To 1 - POr/P3 = 1000 1 - 2.0 = 159·1 K
THE RADIAL FLOW TURBINE
Since W = mcp(TOl - T03),
To - T. - 45·9 - 1242 K
01 03 - 0.322 x 1.148 - .
124·2
'l'ft = 159.1 = 0·781
Putting T 3 / T ~ = 1·0 in equation (7.52), with r3/rz given by
~ = 0·3 x 7·85 + 7·85 = 0.402
r2 2 x 12·7
we get
333
0·781 R! [1 + HO.402
2
(cof 40 + AR cosec
2
40) + 0·07 cosec
2
70}r
l
1·280 = 1 + 0·1148 + 0·1956AR + 0·0396
AR = 0·64
It can be seen from the foregoing that the term containing T3 / T ~ is small. From
the data, U2 =1t X 1000 x 12·7 = 390 m/s. Using this in equation (7.53) we get
T 3 / T ~ equal to (0·921 - 0·028)AR' Substituting this result in equation (7.52) and
recalculating, we have AR = 0·66. The approximate value of 0·64 is within the
margin of experinrental uncertainty.
One of the more comprehensive research programmes on radial flow turbines
was that carried out by Ricardo and Co. and reported in Ref. (17). Turbine
efficiencies of up to 90 per cent were obtained under optimum running conditions
with a 12·5 cm diameter rotor having 12 blades. The optimum number of nozzle
vanes was found to be 17. Values of AN varied from 0·1 to 0·05, decreasing
steadily with increase of nozzle angle from 60 to 80 degrees. AR varied more
widely, from 0·5 to 1·7, increasing rapidly as the nozzle angle was increased
above 70 degrees. In spite of this variation in AR the overall turbine efficiency was
relatively insensitive to nozzle angle: it fell by ouly 2 per cent as 1X2 was increased
from 70 to 80 degrees. Also studied in the programme were the effects of varying
the axial width of the vanes at the rotor tip, radial width of the vaneless space, and
clearance between vanes and casing. The optimum width of vane at the rotor tip
was about 10 per cent of the rotor diameter, and the performance seemed insen-
sitive to the radial width of the vaneless space. There was a fall in efficiency of 1
per cent for every increase in clearance of 1 per cent of the rotor vane width
(averaged between rotor inlet and outlet). This implies that the radial flow turbine
is rather less sensitive to clearance loss than its axial flow counterpart. Because
clearances cannot be reduced in proportion to blade height as the size is reduced,
this lower sensitivity is probably the 1Jasic reason why radial flow has the advan-
tage for very small turbines.
Finally, various major alterations to the geometry of the rotor were made,
including scalloping the disc between the blades to reduce disc stresses, weight
334 AXIAL AND RADIAL FLOW TURBINES
and inertia: .the reader· must tuni to Ref. (17) for thecptisequences of such
changes. All we can do here is to give a rough guide to the major dimensions
determiniIig the basic shape of the rotor, viz. the hub/tip diameter ratio of the
vanes at the rotor exit and the ratio of the vane tip diameter at the exit to the rotor
disc diameter. The former should not be much less than.0·3 to avoid excessive
blade blockage at the hub, and J;he latter should be limited to a maximum of about
O· 7 to avoid excessive curvature of the rotor passages.
Methods of dealing with the losses other than by the simple use of AN and AR
have been devised, and Ref. (18) provides a useful comparison of the various
coefficients that have been used. In particular, if the relative velocity is not radial
at inlet to the rotor there is an additional loss, variously described as an 'incidence
loss' or 'shock loss', for which a separate loss coefficient is desirable. The use of
such a coefficient becomes essential when trying to predict off-design
performance because then the flow relative to the rotor vanes may depart
substantially from the radial direction. There is no shock in the gas dynamic
sense, but there is a shock in the ordinary sense of the word when the fluid
impinging on the rotor vanes at some incidence {J2 (Fig. 7.39) is suddenly
constrained to move in the radial direction. There is a resulting drop of stagnation
pressure and increase in entropy which moves line 2-3 to the right on the T-s
diagram. Bridle and Boulter [Ref. (19)] suggest that in the incidence range
{J2 = ± 65 degrees, this stagnation pressure loss can be accounted for by the
expression
Ilpo 2 = (tan {J2 + O.li
(P02 - P2) cos {J2
Making use of the test results in Ref. (17), Bridle and Boulter have also deduced
equations for the other components of the rotor loss, i.e. friction loss in the rotor
passages and clearance loss, in an attempt to make possible the prediction of off-
design performance. Benson in Ref. (20) describes one method of tackling this
problem, and includes in an Appendix the Fortran programs necessary for the
calculation.
T
Pl
FIG. 7.39 Effect of incidence loss
NOMENCLATURE .
335
NOMENCLATURE
For velocity triangle notation (u, C, V, 0:, {J) see Fig. 7.1 and Fig. 7.37. For blade
geometry notation (s, c, 0, t, e, i) see Fig. 7.11
CL blade lift coefficient
h blade height, heat transfer coefficient
n number of blades
S perimeter
z section modulus of blade
h/c aspect ratio
k/h tip clearance/blade height ratio
sic pitch/chord ratio
tjc thickness/chord ratio
tel s trailing edge thickness/pitch ratio
YN nozzle blade loss coefficient [(POI - P02)/(P02 - pz)]
YR rotor blade loss coefficient [(P02rel - P03rel)f(P03rel - P3)]
Yk tip clearance loss coefficient
Yp profile loss coefficient
Y" secondary loss coefficient
AN nozzle blade loss coefficient [(T2 - TD/(CV2cp)]
AR rotor blade loss coefficient [(T3 - T!j)/(Vl/2cp)]
(J ct centrifugal tensile stress
(J gb gas bending stress
A degree of reaction [(Tz - T3)/(TI - T3)]
cp flow coefficient (Ca/ U)
l/t stage temperature drop coefficient (2cpIlTos/U
2
)
Suffixes
a,w
b
c
m, r, t
N,R
P, S
axial, whirl, component
blade
coolant
at mean, root, tip, radius
nozzle, rotor, blades
pressure, suction, surface of blade
Prediction of pe7jormance of
simple gas turbines
From cycle calculations such as those of Chapter 2, it is possible to determine the
pressure ratio which for any given maximmn cycle temperature will give the
greatest overall efficiency, and the mass flow required to give the desired power.
When such preliminary calculations have been made, the most suitable design
data for any particular application may be chosen. It is then possible to design the
individual components of a gas turbine so that the complete unit will give the
required performance when running at the design point, that is, when it is running
at the particular speed, pressure ratio and mass flow for which the components
were designed. The problem then remains to find the variation of performance of
the gas turbine over the complete operating range of speed and power output,
which is normally referred to as off-design performance.
The performance characteristics of the individual components may be
estimated on the basis of previous experience or obtained from actual tests.
When the components are linked together in an engine, the range of possible
operating conditions for each component is considerably reduced. The problem is
to find corresponding operating points on the characteristics of each component
when the engine is running at a steady speed, or in equilibrium as it is frequently
termed. The equilibrium running points for a series of speeds may be plotted on
the compressor characteristic and joined up to form an equilibrium running line
(or zone, depending upon the type of gas turbine and load), the whole forming an
equilibrium running diagram. 'When once the operating conditions have been
determined, it is a relatively simple matter to obtain performance curves of power
output or thrust, and specific fuel consumption.
The equilibrium running diagram also shows the proximity of the operating
line or zone to the compressor surge line. If it intersects the surge line the gas
turbine will not be capable of being brought up to full speed without some
remedial action being taken. It is this phenomenon which was referred to when
speaking of 'stability of operation' in the Introduction. Finally, it shows whether
the engine is operating in a region of adequate compressor efficiency; ideally the
operating line or zone should lie near the locus of points of maximum compressor
efficiency shown in Fig. 4.1O(a).
PREDICTION OF PERFORMJI.l'fCE OF SIMPLE GAS TURBINES 337
The va.-riation of specific fuel consmnption with reduction in power, sometimes
referred to as part-load peljormance, is of major importance in applications
where considerable running at low power settings is required. This would be the
case for any vehicular gas turbine, and the poor specific fuel consumption at part
load is probably the biggest disadvantage of the gas turbine for vehicular use. The
fuel consumption of aircraft gas turbines at reduced power is of critical
importance when extensive delays occur either at the point of departure or the
destination. In one case the idling fuel consumption when taxiing is important,
and in the other the fuel flow at low flight speeds and medium altitudes is critical.
When determining the off-design performance it is important to be able to
predict not only the effect on specific fuel consumption of operation at part load,
but also the effect of ambient conditions on maximum output. The effects of high
and low ambient temperatures and pressures must all be considered. Land based
gas turbines may operate between ambient temperatures of -60°C in the Arctic
and 50°C in the tropics, and at altitudes from sea level to about 3000 metres,
while aircraft gas turbines have to operate over much wider ranges of inlet
temperature and pressure. The variation of maximum power with ambient
conditions is clearly of prime importance to the customer, and the manufacturer
must be prepared to guarantee the performance available at any specified
condition. If, for example, we consider the use of gas turbines for peale load
generation of electricity, the peak loads will occur in the coldest days of winter in
Europe, but are much more likely to occur in the summer in the United States
because of the hei,lVY demand for air conditioning and refrigeration systems. The
effect of ambient conditions on the pelformance of aircraft gas turbines has a
critical effect on the runway length required and the payload that can be carried,
and hence on both safety and economics.
The basic methods for determining equilibrium off-design performance of
simple gas turbines will be described here, while more complex arrangements and
transient operation are dealt with in Chapter 9. The types of gas turbine discussed
in this chapter will be (a) the single-shaft unit delivering shaft power, (b) the free
turbine engine, where the gas-generator turbine drives the compressor and the
power turbine drives the load and (c) the simple jet engine where the useful output
is a high velocity jet produced in the propelling nozzle. Schematics of these
layouts are shown in Fig. 8.1, and it can readily be observed that the gas generator
performs exactly the same function for both the free turbine engine and the jet
engine. The flow characteristics of a free turbine and a nozzle are similar, and
they impose the same restrictions on the operation of the gas generator, with the
result that the free turbine engine and the jet engine are thermodynamically
similar and differ onJy in the manner in which the output is utilized. Several
successful jet engines have been converted to shaft power use by substituting a
free power turbine for the propelling nozzle, and tlus approach has been widely
used for peale-load electricity generation and propulsion of naval vessels.
All off-design calculations depend on satisfying the essential conditions of
compatibility of mass flow, work and rotational speed between the various
components. It is logical to deal with the single-shaft engine first, and then
338 PREDICTION OF PERFORMANCE OF SIMPLE GAS TURBINES
~ - - ~ ~ ..
~ t = F t t ~
I I I II 2 3 I I'
I 2 3 ,I I I
1 4 L<--- Gas generator --....I 5
(a) 1 4
t
1 I
: 2 3:
k--Gas generator -----.-I
1 4
(c)
FIG. 8.1 Simple ga§ turbine units
(b)
proceed to the free turbine engine where there is the added complication of flow
compatibility between the gas-generator turbine and the power turbine. Lastly, we
will deal with the jet engine where there is the further complication of forward
speed and altitude effects.
Itt Component characteristics
The variation of mass flow, pressure ratio and efficiency with rotational speed of
the compressor and turbine is obtained from the compressor and turbine charac-
teristics, examples of which are given in Chapters 4, 5 and 7. It is convenient to
represent the compressor characteristic as shown in Fig. 8.2, with the variation of
efficiency along each constant speed line plotted against both mass flow and
pressure ratio. With high performance axial compressors the constant speed lines
become vertical on a mass flow basis when the inlet is choked, and in this region
it is essential to plot the efficiency as a function of pressure ratio. The turbine
characteristic can be used in the form given in Fig. 7.28, It is often found in
practice, however, that turbines do not exhibit any significant variation in non-
dimensional flow with non-dimensional speed, and in most cases the turbine
operating region is severely restricted by another component downstream of it. In
explaining the method used for off-design performance calculations it will in-
itially be assmned that the mass flow funl;tion can be represented by a single
curve as in Fig. 8.3. The modification necessary to account for a family of
constant speed curves will be discussed in section 8.3.
For accurate calculations it is necessary to consider the variation of pressure
losses in the inlet ducting, the combustion chamber and the exhaust ducting.
COMPONENT CHARACTERISTICS
P02
P01
rrrJTo;lP01
FIG. 8.2 Compressor characteristics
1)1
miff;
P03
Note: parameters will be
P03/P04
mfT;;, P04 Np
, JOT and 1)lp for a power turbine
P04 Pa' T04
FIG. 11.3 Turbine characteristics
339
These are essentially secondary effects, however, and the off-design calculations
will be introduced on the basis of negligible inlet and exhaust losses and a
combustion chamber pressure loss which is a fixed percentage of the compressor
delivery pressure. Such approximations are quite adequate for many purposes.
For detailed calculations it would be normal to use a digital computer, when
methods of allowing for variable pressure losses are easily introduced. Further
discussion of this matter will be deferred until section 8.7.
340
PREDICTION OF PERFORMANCE OF SIMPLE GAS TURBINES
8.2 Off-design operation of the single-shan. gas turbine
Referring to the single-shaft gas turbine shown in Fig; 8.l(a) it can readily be
seen that when inlet and exhaust pressure losses are ignored, the pressure ratio
across the turbine is determined by the compressor pressure ratio and the pressure
loss in the combustion cham1=1er. The mass flow through the turbine will be equal
to the mass flow delivered by the compressor, less any bleeds, but supplemented
by the fuel flow; it has been pointed out earlier that the bleeds are often approxi-
mately equal to the fuel flow. In general terms, the procedure for obtaining an
equilibrimn running point is as follows.
(a) Select a constant speed line on the compressor characteristic and choose any
point on this line; the values ofmJTodpo)'Po2/pob I1c andNJTol are then
determined.
(b) The cOlTesponding point on the turbine characteristic is obtained from
consideration of compatibility of rotational speed and flow.
(c) Having matched the compressor and turbine characteristics, it is necessary
to ascertain whether the work output corresponding to the selected operating
point is compatible with that required by the driven load; this requires
knowledge of the variation of power with speed, which depends on the
manner in which the power is absorbed.
The compressor and turbine are directly coupled together, so that compatibility
of rotational speed requires
-----x -
N N )TOI
JT03 - JTol T03
(8.1)
Compatibility of flow between the compressor and turbine can be expressed in
tenns of the non-dimensional flows by the identity
m3J To3 = mlJTol x P01 x P02 x IT03 x m3
P03 POI P02 Pm Y TOI ml
The pressure ratio P03/P02 can be obtained directly from the combustion pressure
loss, i.e. P03/P02 = 1 - (I1Pb/P02)' It will nonnally be assumed that ml = m3 = m,
but variation in mass flow at different points in the engine can easily be included
if required. Rewriting the previous equation in terms of m, we get
(8.2)
Now mJTodpOl and P02/POI are fixed by the chosen operating point on the
compressor characteristic, P03/P02 is assumed to be constant and mJT03 /P03 is a
function of the turbine pressure ratio P03/P04. Neglecting inlet and exhaust pres-
sure losses pa = POI = P04, so that the turbine pressure ratio can be calculated
from P03/P04 = (P03/P02)(P02/POl)· Thus all the tenns of equation (8.2) with the
OFF-DESIGN OPERATION OF THE SINGLE-SHAFT GAS TURBINE 341
exception of J(T03/TO!) can be obtained from the compressor and turbine
characteristics. The turbine inlet temperature T03 can therefore be obtained from
equation (8.2) when the ambient temperature, which is equal to TOb is specified.
Having determined the turbine inlet temperature, the turbine non-dimensional
speed N/ JT03 is obtained from equation (8.1). The turbine efficiency can then be
obtained from the turbine characteristic using the known values of N/ JT03 and
P03/P04, and the turbine temperature drop can be calculated from
[ (
1 ) (Y-Il/T]
I1To34 = I1fT03 1 - ---
P03/P04
(8.3)
TIle compressor temperature rise for the point selected on the compressor charac-
teristic can be similarly calculated as
I1T012 = ~ P02 -1
T. [( ) (y-ll/T ]
11e POI
(8.4)
The net power output corresponding to the selected operating point is then found
from
I
net power output = mCpgl1T034 - -mcpaI1T012 l' (8.5)
11m
where 11m is the mechanical efficiency of the compressor-turbine combination,
and m is given by (mJTodPOl)(Pa/ JTa) for prescribed ambient conditions.
Finally, it is necessary to consider the characteristics of the load to determine
whether the compressor operating point selected represents a valid solution. If, for
example, the engine were run on a test bed coupled to a hydraulic or electrical
dynamometer, the load could be set independently of the speed and it would be
possible to operate at any point on the compressor characteristic within the
temperature limit imposed by safety considerations. With a propeller load,
however, the power absorbed varies as the cube of the rotational speed of the
propeller. When the transmission efficiency and gear ratio are known, the load
characteristic in tenns of the net power actually required from the turbine and the
turbine speed can be plotted as in Fig. 8.4. The problem then becomes one of
finding the single point on each constant speed line of the compressor
characteristic which will give the required net power output at that speed; this
can only be done by trial and elTor, taking several operating points on the
compressor characteristic and establishing the power output cOlTesponding to
each one. If the calculated net power output for any point on the compressor
characteristic is not equal to the power required at the selected speed, the engine
will not be in equilibrium and will either accelerate or decelerate depending on
whether there is a surplus or deficiency of power. Repeating this procedure for a
t. Distinguishing suffixes will be. added to cp only in equations where both cpa and cpg appear
sunultaneously. In other cases It WIll be clear from the context which mean C should be used. y will be
treated similarly p
342
PREDICTION OF PERFORMANCE OF SIMPLE GAS TURBINES
Power oc N'
o
Turbine speed
FIG. 8.4 Load characteristics
series of constant speed lines, a series of points are obtained, which can be joined
up to form the equilibrium running line as shown in Fig. 8S
The most common type of load used with a single-shaft gas turbine is the
electric generator which runs at a constant rotational speed with the load varied
electrically. Thus the equilibrium running line for a generator set would
correspond to a particular line of constant non-dimensional speed, as shown in
Fig. 8.5, and each point on this line would represent a different value of turbine
inlet temperature and power output. At each speed it is possible to find, by trial
and error, the compressor operating point corresponding to zero net output and
the no-load running line for a generator set is also shown in Fig. 8.5.
The equilibrium running lines depicted show that a propeller load implies
operation in a zone of high compressor efficiency over a wide range of output,
P02
P01
FIG. 8.5 ml1ming lines
/
Generator
OFF-DESIGN OPERATION OF THE Sfl\lGLE-SHAFT GAS TURBINE 343
whereas the generator load results in a rapid drop in compressor efficiency as load
is reduced. The location of the equilibrium running line relative to the surge line
indicates whether the engine can be brought up to full power without any
complications. In the case of the propeller, the equilibrium running line lies close
to the surge line, and may even intersect it, in which case the engine could not be
accelerated to full power. This can be overcome by incorporating a blow-off valve
towards the rear of the compressor. Section 8.6 deals with the matter in more
detaiL The running line for the generator at no-load can be seen to be well away
from surge, and the generator could be accelerated to full speed before applying
the load without any surge problem being encountered.
The calculations described above determine the values of all the pammeters
required for a complete performance calculation for any point within the
operating range. T03 is Imown, and T02 is found from (LiToI2 + TOl). Thus the
combustion temperature rise is Imown and it is possible to obtain the fuel/air ratio
jfrom the curves of Fig. 2.15 and an assumed value of combustion efficiency. The
fuel flow is then given by mf From the fuel flow and power output at each
operating point the variation in specific fuel consumption (or thermal efficiency)
with load can be detemlined. The results refer to operation at the assumed value
of TOI (= Ta) and POI (= Pa), but the process could be repeated over the range of
values of ambient temperature and pressure likely to be encountered.
The matching calculations for a single-shaft gas turbine are illustrated in the
following example,
lEXAMPLlE S.lI.
The following data refer to a single-shaft gas turbine operating at its design speed.
Compressor characteristic
P02/POI
5·0
4·5
4·0
329·0
339·0
342·0
0·84
0·79
0·75
Turbine characteristic
139·0 0·87
(both constant over
range of pressure ratio
considered)
Assuming ambient conditions of 1·0 13 bar and 288 K, a mechanical efficiency
of 98 per cent, and neglecting all pressure losses, calculate the turbine inlet
temperature required for a power output of 3800 leW The 'non-dimensional'
flows are expressed in terms of kg/s, K and bar.
The method of solution is to establish, for each point given on the compressor
characteristic, the turbine inlet temperature from equation (8.2), the compressor
temperature rise from equation (8.4) and the turbine temperature drop from equa-
tion (8.3). Once these have been established the power output can be obtained
344
PREDICTION OF PERFoRMANCE OF SIMPLE GAS TURBINES
from equation (8.5); it is then necessary to plot turbine inlet temperature against
power output to find the required temperature for an output of 3800 kW
Taking the compressor operating point at a pressure ratio of 5·0,
IT03 _ (mJT03 /P03)(P03/POI)
V TO! - mJTodpol
With pressure losses neglected P03 = POl, and hence
IT03 = 139·0 x 5·0 = 2.11
V TOI 329·0
giving T03 = 1285 K
The compressor temperature rise is given by
.1.To12 = 288 [5.0
1
/
3
.
5
- 1] = 200·5 K
0·84
The turbine temperature drop is given by
.1.T034= 0·87 x 1285[1 - = 370·0 K
The air mass flow is obtained from the non-dimensional flow entering the com-
pressor,
1·013
m = 329 x ---;- = 19·64 kg/s
,,288
The power output can now be obtained, giving
(
19.64 x 1·005 x 200.5)
power output = 19·64 x 1·148 x 370·0 - 0.98
= 8340 - 4035
= 4305 kW
Thus for a power output of 4305 kW the turbine inlet temperature is 1285 K.
Repeating the calculation for the three points given on the compressor
characteristic yields the following results
P02/POl T03 Ll. T012 Ll. To34
m Power output
[K] [K] [K] [kg/s] [kW]
5·0 1285 200·5 370·0 19·64 4305
4·5 982 196·1 267·0 20·25 2130
4·0 761 186·7 194·0 20-4 635
Plotting the value of T03 against power output it is found that for an output of
3800 kW the required turbine inlet temperature is 1215 K.
EQUILIBRIUM RUNNING OF A GAS GENERATOR 345
8.3 Equilibrium running of a gas genentor
It was pointed out at the start of this chapter that the gas generator perfOims the
same function for both the free turbine engine and the jet engine, namely the
generation of a continuous flow of gas at high pressure and temperature which
can be expanded to a lower pressure to produce either shaft work or a high
velocity propulsive jet. Before considering either type of engine, it is appropriate
to consider the behaviour of the gas generator alone.
Considerations of compatibility of speed and flow are the same as for the
single-shaft engine described in the previous section, and equations (8.1) and
(8.2) are applicable. This time, however, the pressure ratio across the turbine is
not known, and it must be determined by equating the turbine work to the
compressor work. The required turbine temperature drop, in conjunction with the
turbine inlet temperature and efficiency, determines the mrbine pressure ratio .
Thus, instead of equation (8.5) the work requirement is expressed by
I1mCpg.1.To34 = cpa.1.To12
Re-writing in tenus of non-dimensional groups
.1.T034 .1. To 12 TOI cpa
--=--x-x--
T03 TO! T03 Cpgl1m
(8.6)
Equations (8.1), (8.2) and (8.6) are all linked by the temperature ratio T03/TO!
and it is necessary to determine, by trial and error, the turbine inlet temperature
required for operation at any arbitrary point on the compressor characteristic. The
procedure is as follows:
(a) Having selected a point on the compressor characteristic, the values of
N/ JTOb P02/POb mJTOJ/POb and I1c are determined, and .1.TOI2/Tol can be
calculated from equation (8.4).
(b) If a value of P03/P04 is guessed, the value of my'T03/P03 can be obtained
from the turbine characteristic, enabling the temperature ratio T03/TiJ! to be
obtained from the flow compatibility equation (8.2).
(c) This value of T03/Tol can now be used to calculate N/ JT03 from equation
(8.1).
(d) With N/ JT03 and P03/P04 known, the turbine efficiency can be obtained
from the turbine characteristic.
(e) The non-dimensional temperature drop .1.T034/To3 can be calculated from
equation (8.3) and used in with equation (8.6) to calculate
another value of T03/TOI.
(j) This second value of T03/ TOI will not, in general, agree with the first value
obtained from equation (8.2), indicating that the guessed value of the
pressure ratio P03/P04 is not valid for an equilibrium running point.
(g) A new value of P03/P04 must now be assumed and the above calculations
repeated until the same value of T03/Tol is obtained from both equation
(8.2) and (8.6).
346 PREDICTION OF PERFORMANCE OF SIMPLE GAS TURBINES
(h) Agreement signifies that the turbine operating point.is compatible with the
originally chosen compressor operating point when the fuel flow is such as
to produce the iterated value of T03 /Tol .
The procedure is summarized in the infonnation flow chart of Fig. 8.6(a).
It would be possible to carry out this calculation for a large number of points
on the compressor characteristic, and express the results by joining up points of
constant T03/Tol on the compressor characteristic as indicated by the dotted lines
in Fig. 8.7. In practice this is unnecessary. The further requirement for flow
compatibility with the component downstream of the gas generator, whether it be
a power turbine or propelling nozzle, seriously restricts the operating zone on the
compressor characteristic. The foregoing procedure thus comprises only one part
of the calculation for the whole unit, and when the whole unit is considered only
relatively few points may be needed. The free turbine engine will be considered
first.
Before doing so, it is necessary to emphasize that the matching procedure just
outlined has been developed on the assumption that the turbine non-dimensional
flow is independent of the non-dimensional speed and is a fimction only of
pressure ratio. If the turbine characteristic does exhibit a variation of mJT03/P03
witlJ N/ JT03 , as in Fig. 7.28, the procedure must be modified. There is no change
for any turbine operating point falling on the choking point of the mass flow
curve, but for otlJers the process would be as follows. Immediately after step (a) it
is necessary to guess a value of T03/TOb which permits calculation of N/ JT03
from equation (8.1) and m/ JT03/P03 from equation (8.2). P03/P04 and 1'/t can then
be obtained from the turbine characteristic, enabling I:lT034/T03 to be calculated
from equation (8.3). The work compatibility equation (8.6) is then used to provide
a value of T03/TOI for comparison with the original guess. To avoid obscuring the
main principles involved, the use of multi-line turbine flow characteristics will not
be discussed further.1" Enough has been said to show that they can be
accommodated without undue difficulty.
8.4 Off-design operation of free turbine engine
Matching of gas generator with free turbine
The gas generator is matched to the power turbine by the fact that the mass flow
leaving tlJe gas generator must equal that at entry to the power turbine, coupled
with the fact that the pressure ratio available to the power turbine is fixed by the
compressor and gas-generator turbine pressure ratios. The power turbine charac-
teristic will have the same fonn as Fig. 8.3, but tlJe parameters will be
mJT04/P04, P04/Pu. Np / J T04 and 1'/tp.
t The characteristic shown in Fig. 7.28 is typical of that for a turbine in which choking occurs in the
stator passages. In some designs choking may occur initially at exit from the rotor passages; this will
result in a small variation of the choking value ofm-./T03 /P03 with N!-./T03. The modification to the
calculation would then apply also to the choking region.
OFF-DESIGN OPERATION OF FREE TURBINE ENGINE
NO
Choose compressor
operating point
P21P1, rye
T3/P3 from
turbine characteristic
calculate Ts!T, from
flow compatibility
eqn. (8.2)
Calculate T3!T, from
work compatibility
eqn. (8.6)
Do eqns (8.2), (8.6)
yield same values
of T3!T,?
(a)
I Select Nd T,
t
Choose compressor
operating point
P2/P1, rrrJT,/P1, 1)e

Iteration for gas
generator turbine
pressure ratio PsfP4
$
Calculate m..,f T4!P4
at exit from gas
generator from
eqn. (8.7)
t
Calculate P4!Pa
from eqn. (8.8)
f
Obtain m-v T4!P 4 at
entry to power
turbine from P4!Pa
and power turbine
characteristic
t
Are the two values
of m-v T4!P4 equal?

Matching completed
(b)
347
Select another
compressor
operating point
on same Nlff,
line
NO
FIG.8.6(a) Iteratioll for gas gellerator (b) Overall iteratioll procedure for free powel"
turbille Ullit
348
PREDICTION OF PERFORMANCE OF SIMPLE GAS TURBINES

P01
increasing
"" Equilibrium running line
N/fTo;
FIG. 8.7 Equilibrium runllilllg for free rurbillle
The preceding section described how the gas generator operating conditions
can be detennined for any point on the compressor characteristic. The value of
m,JT
o4
/P04 at exit from the gas generator can then be calculated from
m,JT04 = m,JT03 xP03 x j
T
04
P04 P03 P04 V T03
(8.7)
where
jT04 = 1(1 _ LlT034) and Ll.
T
034 = /11[1 _
II T03 II T03 T03 \P03/P04
The corresponding pressure ratio across the power turbine can also be established
from
P04 = Poz X P03 X P04
Pa POI P02 P03
(8.8)
(It must be remembered that in the case of a stationary gas turbine with inlet and
exit duct losses ignored POI = P m and the power turbine outlet pressure is also
equal to Pa.)
Having found the pressure ratio across the power turbine, the value of
m,JT
04
/P04 can be found from the power turbine characteristic for comparison
with the value obtained from equation (8.7). If agreement is not reached it is
necessary to choose another point on the same constant speed line of the
compressor characteristic and repeat the procedure until the requirement of flow
compatibility between the two turbines is satisfied. The overall procedure for the
free turbine engine, including the iteration for the gas generator, is summarized in
the information flow chart of Fig. 8.6(b).
For each constant N/ ,JTOl line on the compressor characteristic there will be
only one point which will satisfy both the work requirement of the gas generator
and flow compatibility with the power turbine. If the foregoing calculations are
OFF-DESIGN OPERATION OF FREE TURBINE ENGINE 349
carried out for each constant speed line, the points obtained can be joined up to
fonn the equilibrium running line as shown in Fig. 8.7. The running line for the
free turbine engine is independent of the load and is determined by the
swallowing capacity of the power turbine. This is in contrast to the behaviour of
the single-shaft unit, where the running line depends on the characteristic of the
load as indicated in Fig. 8.5.
The next step is to calculate the power output and specific fuel consumption
for the equilibrium running points. Before discussing this, however, a useful
approximation which simplifies the foregoing procedure will be mentioned. It
arises from the behaviour of two turbines in series. We introduce it here because it
facilitates a better physical understanding of some of the phenomena discussed in
subsequent sections.
Matching of two turbines in series
The iterative procedure required for the matching of a gas generator and a free
turbine can be considerably simplified if the behaviour of two turbines in series is
considered; this approach is also valuable for the analysis of more complex gas
turbines considered in Chapter 9. It was shown in the previous sub-section that by
using equation (8.7) the value of m,JT04/P04 at exit from the gas-generator
turbine can be obtained for any gas-generator operating point, and in particular
that it is a function of m,JT03 /P03, P03/P04 and 1]" The value of 1], could be read
from the characteristic because N/ ,JT03 had been determined for the
operating point in the course of the gas-generator calculation. Now in practice the
variation of 11, at any given pressure ratio is not large (see Fig. 8.3), particularly
over the restricted range of operation of the gas-generator turbine. Furthermore,
such a variation has little effect on m,J T04/ P04 because the resulting change in
,J(T03/T04) is very small. It is often sufficiently accurate to take a mean value of
111 at any given pressure ratio, so that m,JT04/P04 becomes a function only of
m,JT03/P03 and P03/P04' If this is done, a single curve representing the turbine
outlet flow characteristic can readily be obtained by applying equation (8.7) to
points on the single curve of the inlet flow characteristic. The result is shown by
the dotted curve in Fig. 8.8.
The effect of operating two turbines in series is also shown in Fig. 8.8 where it
can be seen that the requirement for flow compatibility between the two turbines
places a major restriction on the operation of the gas-generator turbine. In
particular, as long as the power turbine is choked the gas-generator turbine will
operate at afixed non-dimensional point, i.e. at the pressure ratio marked (a).
With the power turbine unchoked, the gas-generator will be restrained to operate
at a fixed pressure ratio for each power-turbine pressure ratio, e.g. (b) and (c).
Thus the maximum pressure ratio across the gas-generator turbine is controlled
by choking of the power turbine, and at all times the pressure ratio is controlled
by the swallowing capacity of the power turbine.
A further consequence of the fixed relation between the turbine pressure ratios
is that it is possible to plot the gas-generator pressure ratio P03/P04 against
350
PREDICTION OF PERFORJlftANCE OF SIMPLE GAS TURBINES
Gas-generator turbine Power turbine
mffo .
Po
-- m,rt;;
"'''' .....m-rF;;;
'" Po
// P04
,-
-----------------.....,,......----
/: ------
m-rF;;;
P04
./1--:------------------------- ----------
jJ.-_L _________________________ -------
/1 1 \
/'
I': 03
I ,
I :
l :
,
,
(c) (b) (a)
POiP04
FIG. 8.8 Operatioll of turbilles ill
(c) (b)
compressor pressure ratio P02/POI by using the identity
P03 =P03 )( P02 )( Pa
P04 P02 POI P04
(a)
(8.9)
P03/P02 is determined by the assumed combustion pressure loss, and P04/Pa is
obtained from Fig. 8.8 for any value of P03/P04' Such a curve is shown in Fig. 8.9.
From it, for any value of compressor pressure ratio, the pressure ratio of the gas-
generator turbine can be determined; this in tum fixes the values of m.jT03/P03
and fY..T034/T03 which are required for use with equations (8.2) and (8.6). Thus it
is no longer necessary to carry out the iteration for the pressure ratio of the gas-
generator turbine and, for each constant speed line considered, only a single
iteration is required to find the correct equilibrium running point.
Variation of power output and SFC with output speed
The net power output of the free turbine engine is simply the output from the
power turbine, namely
P03
P04
FIG. 11.9
Choking of
power turbine
(8.10)
OFF-DESIGN OPERATION OF FREE TURBINE ENGINE
where
[ (
1 ) (Y-
1
ll1']
fY..T045 = IJlpT04 1 - -;-
P04 Pa
351
For each equilibrium running point established (one for each compressor speed
line) P04/Pq will be Imown, and T04 can be readily calculated from
(8.11)
The mass flow is obtained from m.jToI!pol for assumed values ofPa and Ta. The
outstanding unlmown is the power turbine efficiency IJtp. It can be obtained from
the power turbine characteristic, but it depends not only on the pressure ratio
P04/Pa but also upon Np/ .jT04, i.e. upon the power turbine speed Np- Free turbine
engines are used to drive a variety of loads such as pumps, propellers and electric
generators, each with a different power versus speed characteristic. For this
reason it is usual to calculate the power output over a range of power turbine
speed for each equilibrium running point (i.e. for each compressor speed). The
results could be plotted as in Fig. 8.10. Any curve corresponding to a given
compressor speed will be fairly flat in the useful upper half of the output speed
range where 'lip does not vary much with Np/ .jT04 (see Fig. 8.3).
The fuel consumption can also be calculated for each equilibrium running
point, in the same manner as described in section 8.2 for the single-shaft unit.
Since it depends only on the gas generator parameters there will be only one value
for each compressor speed. When combined with the power output data to give
the SFC, however, it is clear that the SFC, like the power output, will be a
function of both compressor speed and power turbine speed. It is convenient to
express the off-design perfonnance by plotting SFC against power output for
several power turbine speeds as shown in Fig. 8.1 L This type of presentation
pennits the customer to evaluate the perfonnance of the unit when coupled to the
specified load by superimposing the load characteristic upon it. The dotted curve
in Fig. 8.11 indicates a particular vru1ation of power and speed imposed by a load,
"5
c.
"5
o

a..
Constant compressor
speed lines
\
\
Output speed Np
FIG. lUll
352
PREDICTION OF PERFORMANCE OF SIMPLE GAS TURBINES
SFC
Power output
FIG.Il.B
and the points of intersection with the Np curves give the SFC versus power
output curve for the free turbine engine driving that particular load. Figure 8.11
relates to operation at one ambient condition, and nonnally the perfonnance
calculations would be repeated for extreme values of Pa and Ta·
n should be appreciated that although for convenience of calculation the
compressor speed has been chosen as the independent variable, in practice the
fuel flow is the independent variable. A chosen value of fuel flow (and hence T03)
detennines the compressor speed and ultimately the power output. The SFC
curves of Fig. 8.11 exhibit an increase in SFC as the power is reduced because the
reduction in fuel flow leads to a reduction in compressor speed and gas-generator
turbine inlet temperature. It will be remembered from Chapter 2 that the
efficiency of a real cycle falls as the turbine inlet temperature is reduced. This
poor part-load economy is a major disadvantage of the simple gas turbine.
Consideration of the substantial improvement arising from the use of more
complex cycles will be deferred until Chapter 9. Referring again to Fig. 8.11, the
change in SFC with Np at any given power output is not very marked, because the
gas-generator parameters change only slightly under this operating condition.
It is useful to consider the variation of the key variables (power, turbine inlet
temperature and fuel flow) with gas-generator speed as shown in Fig. 8.12. It can
be seen that all increase rapidly as the gas generator approaches its maximum
pernlissible speed. The change in turbine inlet temperature is especially critical
because of the effect on creep life of the first-stage turbine blades and, in general,
operation at the maximum speed would only be for limited periods. In electricity
generation, for example, this maximum speed would be used for emergency peale
duty, and base-load operation would utilize a reduced gas-generator speed with
correspondingly increased blade life.
Single-shaft versus twin-shaft engines
the choice of whether to use a single-shaft or twin-shaft (free turbine) power
plant is largely determined by the characteristics of the driven load. If the load
OFF-DESIGN OPE:RATION OF FREE TURBTh'E ENGINE 353

"
'"
Qj
0..
E ;;:
Qj 2 0
;;: 1i)
'"
0 w
"-
:s
" OJ
I.J..
C
:e
,/
"
Max
I-
Max
speed speed
Gas-generator speed
FIG. 8.12 Variation of key parameters witll gas-generator speed
speed is constant, as in the case of an electric generator, a single-shaft unit is often
specified; an engine specifically designed for electric power generation would
malce use of a single-shaft configuration. An alternative, however, is the use of an
aircraft derivative with a free power turbine in the place of the propelling nozzle.
With arrangement it is possible to design a power turbine of substantially
larger dIameter than the gas generator, using an elongated duct between the gas
generator and the power turbine; this then permits the power turbine to operate at
the required ele,ctric generator speed without the need for a reduction gear box.
Turboprops may use either configuration, as shown in Figs. 1.10 and 1.11.
The running lines for single-shaft and twin-shaft units were shown in Fig. 8.5
and 8.7. It should be noted that in the case of the single-shaft engine driving a
generator, reduction in output power results in a slight increase in compressor
flow; although there is some reduction in compressor pressure ratio, there is
little change in compressor temperature rise because the efficiency is also
reduced. This means that the compressor power remains essentially fixed. With a
twin-shaft engine, however, as Fig. 8.7 shows, reducing net power output involves
a reduction in compressor speed and hence in air flow, pressure ratio and
temperature rise. The compressor power needed is therefore appreciably lower
for the single-shaft engine. It should also be evident from a comparison of
8:5 and 8.7 the compressor operates over a smaller range of efficiency
m a twm-shaft engme. For these reasons the part-load fuel consumption of a twin-
shaft engine is superior when driving a constant speed load.
The two types also have different characteristics regarding the supply of waste
heat to a cogeneration plant, primarily due to the differences in exhaust flow as
load is reduced; the essentially constant air flow and compressor power in a
single-shaft unit results in a larger decrease of exhaust temperature for a given
reduction in power, which might necessitate the burning of supplementary fuel in
waste. heat boiler under operating conditions where it would be unnecessary
With a In cases, the exhaust temperature may be increased by the
use of vanable inlet gnlde vanes. Cogeneration systems have been successfully
354 PREDICTION OF PERFORMANCE OF SIMPLE GAS TURBINES
built using both single"shaft and twin-shaft units, the latter often beirig aITcraft
derivative engines. .
Most gas turbines do not spend long periods at low power and the SFC
differences may not be critical in practice. The torque characteristics, however,
are very different and the variation of torque with output speed at a given power
may well determine the engine's suitability for certain applications; e.g. a'high
starting torque is particularly 'important for traction purposes.
The compressor of a single-shaft engine is constrained to turn at some multiple
of the load speed, fixed by the transmission gear ratio, so that a reduction in load
speed implies a reduction in compressor speed. This results in a reduction in mass
flow and hence of output torque as shown by curve (a) of Fig. 8.13. This type of
turbine is clearly unsuitable for traction purposes. The normal flat torque curve of
an internal combustion engine is shown dotted for comparison.
The free power turbine unit, however, has a torque characteristic even more
favourable than the internal combustion engine. The variation of power output
with load speed, at any given compressor speed determined by the fuel flow, is
shown in Fig. 8.10. It can be seen that the output power remains relatively
constant over a wide load speed range for a fixed compressor speed. This is due to
the fact that the compressor can supply an essentially constant flow at a given
compressor speed, irrespective of the free turbine speed. Thus at fixed gas
generator operating conditions, reduction in output speed results in an increase in
torque as shown by curve (b) in Fig. 8.13. It is quite possible to obtain a stall
torque of two to three times the torque delivered at full speed.
The actual range of speed over which the torque conversion is efficient
depends on the efficiency characteristic of the power turbine. The typical turbine
efficiency characteristic shown in Fig. 8.3 suggests that the fall in efficiency will
not be greater than about 5 or 6 per cent over a speed range from half to full
speed. Thus quite a large increase in torque can be obtained efficiently when the
output speed is reduced to 50 per cent of its maximum value. The efficiency of
0.5 Full speed
OFF-DESIGN OPERATION OF FREE TURBINE ENGINE 355
the torque conversion at low speeds, for example when accelerating a vehicle
from rest, will be very low. In some applications a simple two-speed gearbox
might be sufficient to overcome this defect, but it is probable that gas turbines for
heavy road vehicles will incorporate some form of automatic transmission with
five or six speeds.
Further differences between single- and twin-shaft units are brought out in
section 9.5 where the transient behaviour of gas turbines is discussed.
EXAMPLE 8.2
A gas turbine with a free power turbine is designed for the following conditions:
air mass flow
compressor pressure ratio
compressor isentropic efficiency
turbine inlet temperature
turbine isentropic efficiency (both turbines)
combustion pressure loss
mechanical efficiency
(applies to gas generator and load)
ambient conditions
30 kg/s
6·0
0·84
1200 K
0·87
0,20 bar
0·99
1·01 bar, 288 K
Calculate the power developed and the turbine non-dimensional flows
mJT03/P03 and mJT04/P04 at the design point.
If the engine is run at the same mechanical speed at an ambient temperature of
268 K calculate the values of turbine inlet temperature, pressure ratio and power
output, assuming the following:
(a) combustion pressure loss remains constant at 0·20 bar;
(b) both turbines choking, with values of mJT03/P03 and mJT04/P04 as
calculated above and no change in turbine efficiency;
(c) at 268 K and the same N, the N/ JT01 line on the compressor characteristic
is a vertical line with a non-dimensional flow 5 per cent greater than the
design value;
(d) variation of compressor efficiency with pressure ratio at the relevant value of
N/JTOI is
P02/POI 6·0 6·2 6·4 6·6
I'fc
0·837 0·843 0·845 0·840
Output speed The design point calculation is straightforward and only the salient results are
FIG. 8.13 Torque characteristics presented:
356
PREDICTION OF PERFORMANCE OF SIMPLE GAS TURBINES
Gas-generator turbine
pressure ratio
inlet pressure
temperature drop
Power turbine
pressure ratio
inlet pressure
temperature drop
inlet temperature
2·373
5·86 bar
203 K
2-442
2·47 bar
173·5 K
997 K
The power output is then (30 x 1·148 x 173·5 x 0·99) kWor 5910 kW.
The design point values of m,jTo3/po3 and m,jTo4/po4 are
30 x ,,11200 = 177-4 and 30 x ,,1997 = 383.5 respectively
5.86 2-47
At 268 K the value of m,jTollpol is
. [30 x ,,1288] = 529.5
1 05 x 1.01
If the power turbine remains choked, the gas-generator turbine will be constrained
to operate at a fixed non-dimensional point, and thus the value of I1T034/T03 =
203/1200=0·169, as for the design condition.
For work compatibility
I1TOl2 = I1To34 X T03 [Cpgl'/m] = 0·169 x 1·148 x 0·99 x T03
TOI T03 TOI cpa 1·005 TOI
and hence
T03 = 5.23I1To12
TO! TOI
For flow compatibility
m,jT03 = m,jT01 x POl x /T03
P03 POI P03 V TOI
177-4 = 529.5
POI
x /T03
P03 VTOI
/T03 = O.335P03
V TOI POI
(A)
(B)
The problem is thus to find the compressor operating point that will give the
same value of T03/TOI for both equations (A) and (B). With the variation in
efficiency prescribed the value of I1T0\2/Tol can readily be calculated from
equation (8.4), and with the constant value of combustion pressure loss the value
of P03 can also be calculated. The solution is best carried out in tabular fonn
as shown.
OFF-DESIGN OPERATION OF FREE TURBINE ENGINE 357
P02 [( . r-
l
)/)' J
LlTol2
(T03)
Pm
JT
03
(T03)
, P02 -1
l1e
TOI TOI A
Poz P03
POI TO! TOI B PO! PO!
6·0 0·669 0·837 0·799 4·18 6·06 5·86 5·80 1,943 3·78
6·2 0·684 0·843 0·812 4·25 6·26 6·06 6·00 2·010 4·05
6-4 0·700 0·845 0·828 4·33 6·46 6·26 6·20 2·078 4·32
6·6 0,715 0·840 0·851 4·45 6·66 6·46 6·40 2·144 4·60
Solving graphically the required pressure ratio is found to be 6-41, with the value
of T03/Tol =4·34; the corresponding turbine inlet temperature is 4·34 x 268 =
1163 K.
Having established the compressor pressure ratio and turbine inlet
temperature, it is a straightforward matter to calculate the power developed.
Remembering that the gas generator turbine will still operate at the same non-
dimensional point (/).T034/T03 = 0·169, P03/P04 = 2·373) the power turbine entry
conditions and temperature drop can readily be calculated; the resulting
temperature drop is 179·6 K. The mass flow is obtained from the non-
dimensional flow and the ambient conditions:
m,jTOl 529·5 x 1·01
--= 529·5, hence m = = 32·7 legis
POI ,,1268
and power output=32·7 x 1·148 )( 179·6 x 0·99=6680 leW.
From this example, it can be seen that operation at the design mechanical
speed on a cold day results in a decrease of maximum cyde temperature from
1200 to 1163 K, even though the value of T03/TO! has increased from 4·17 to
4·34 due to the increase in N/ ,jTOl ' The power has increased from 5910 to
6680 leW and this can be seen to be due to the simultaneous increase in air mass
flow and overall pressure ratio. The beneficial effect of low ambient temperature
on gas turbine operation is evident; conversely, high ambient temperatures result
in significant penalties. The effect of increased ambient temperature on turbojet
operation is discussed under the heading 'Variation of thrust with rotational
speed, forward speed and altitude' ..
Industrial gas turbines used to generate electricity in regions with high summer
temperatures may have to meet very high peak demands due to large air-
conditioning loads. It is possible to increase power by decreasing the temperature
of the air entering the compressor. In regions oflow relative humidity, cooling can
be achieved using an evaporative cooler; the incoming air is passed through a
wetted filter, and the heat required to evaporate the water leads to a reduction in
inlet temperature. This approach, however, is not effective in areas of high relative
humidity. An alternative approach, successfuliy introduced in the rnid 1990s is ice
harvesting. At off-peak times, primarily overnight, electricity is used to drive
refrigeration chillers which produce and store large quantities of ice. \Vhen
maximum power is needed during the heat of the day, ice is melted and the chilled
358 PREDICTIoN OF PERFORMANCE OF SIMPLE GAS TURBINES
water is used in heat-exchangerS to lower the inlet This method has
proved to be economically attractive, producing substantial increases in power at
a low cost per kilowatt.
8.5 Off-design operation of the jet engine
Propelling nozzle characteristics
The propelling nozzle area for a jet engine is determined from the design point
calculations as described in Chapter 3, and once the nozzle size has been fixed it
has a major influence on the off-design operation. The characteristic for a nozzle,
in terms of 'non-dimensional' flow m.jT 04/P04 and pressure ratio P04/POS, can
readily be calculated as follows.
The mass flow parameter is given by
m../T04 C A .jT04
--= S sPs-P
P04 04
Cs As Ps T04
=--x-x-x-
.jT04 R P04 Ts
(8.12)
where As is the effective nozzle area Making use of equation (3.12), Cs/ .jT04
can be found from
[( 1 )(Y-I)/Y]
-=2cp1'fj 1- --
T04 P04/PS
and Ts/T04 from
T. T04 - Ts [( 1 )(Y-I)/Y]
-2..= 1----= 1-1'fj 1- --
T04 T04 P04/PS
(8.13)
(8.14)
It follows that for a nozzle of given area and efficiency, m.jr04/P04 can be
calculated as a function of the pressure ratio P04/PS' But equations (8.13) and
(8.14) are valid for pressure ratios only up to the critical value, given by equation
(3.14), namely
P04 = 1/[1 _ (y - l)]Y/(Y-I) (8.15)
Pc 1'fj Y + 1
and up to this point Ps = Pa. For pressure ratios P04/Pa greater than the critical,
m.jT04/P04 remains constant at the maximum (choking) value, i.e. it is indepen-
dent of P04/Pa (and incidentally Ps = Pc> Pa). Thus m.jT04/P04 can be plotted
against the overall nozzle pressure ratio P04/Pa as in Fig. 8.14, and the similarity
between the nozzle flow characteristic and that of a turbine is evident.
Up to the choking condition, Ts/To4 is given by equation (8.14), whereas when
the nozzle is choking it is given by equation (3.13), namely
Tc 2 (8.16)
T
04
=y+1
OFF-DESIGN OPERATION OF TII:E JET ENGINE
IT/ffo;.
P04
1-
Choking
mass flow
FIG. 8.14 Nozzle characteristics
359
Likewise, with the nozzle unchoked Cs/.jT 04 is given by equation (8.13),
whereas when it is choked Cs is the sonic velocity and the Mach number Ms is
unity. Now, recalling that C=M.j(yRT) and To = T[l + (y - l)M 2/2], we have
the general relation
(8.17)
Thus when the nozzle is choked (i.e. Ms = 1) we have
= 2yR (8.18)
T04 Tos y+1
We shall make :u.e of equations (8.15) to (8.18) later when discussing the de-
termination of the thrust.
Matching of gas generator with nozzle
The similarity between the flow characteristic of a nozzle and a turbine means that
the nozzle will exert the same restriction on the operation of the gas generator as a
free power turbine. Thus if the operation of a jet engine under static conditions is
considered, there can be no difference between its behaviour and that of a free
power turbine unit. The equilibrium running line can be determined according to
the procedure specified in the flow chart of Fig. 8.6(b), with the nozzle charac-
teristic replacing the power turbine characteristic. It follows that Fig. 8.7 can be
taken as representative of a typical equilibrium running diagram for a jet engine
under static conditions.
The jet engine, of course, is intended for flight at high speeds and it is
necessary to consider the effect of forward speed on the equilibrium running line.
It is most convenient to express the forward speed in terms of Mach number for
the matching calculations, and this can readily be converted to velocity for
calculation of the momentum drag and thrust as shown later in the next sub-
section.
Forward speed produces a ram pressure ratio which is a function of both flight
Mach number and intake efficiency. This ram effect will give rise to an increase in
360 PREDICTION OF PERFORMANCE OF SIMPLE GAS TURBINES
compressor delivery pressure, which will in tum lead to a higher pressure before
the propelling nozzle, thereby increasing the nozzle pressure ratio. Once the
nozzle is choked, however, the nozzle non-dimensional flow will reach its
maximum value and will then be independent of the nozzle pressure ratio and
therefore of forward speed. The significance of this is that the turbine operating
point will then also be because of the requirement for compatibility of
flow between the turbine and the nozzle. It follows that as long as the nozzle is
choking the equilibrium running line will be uniquely determined by the fixed
turbine operating point and will be independent of flight speed.1"
At current levels of cycle pressure ratio, virtually all jet engines operate with
the nozzle choked during take-off, climb and cruise, and the nozzle only becomes
unchoked when thrust is significantly reduced. Thus the nozzle is liable to be
unchoked only when preparing to land or when taxiing. Nevertheless, it is
important to consider the effect of forward speed on the running line under these
conditions because it is at low rotational engine speeds that the running line is in
close proximity to the surge line.
The nozzle pressure ratio P041Pa is linked to the ram pressure ratio by the
identity
P04 = P04 X P03 X P02 X POl
Pa P03 P02 POl Pa
(8.19)
This differs from the corresponding equation for the free turbine unit, equation
(8.8), only by the inclusion of the ram pressure ratio Poi/Pa. The ram pressure
ratio in terms of intake efficiency 1]i and flight Mach number Ma is given by
equation (3.1O(b)), namely
P
[
(1'
- 1) ]Y!(Y-1l
-'!.!.= 1 +1] -- M2
Pa I 2 a
(8.20)
It follows from equations (8.19) and (8.20) that for a given intake efficiency,
P041Pa is a function of the gas generator parameters and flight Mach number. The
procedure of Fig. 8.6(b) can be followed, with equation (8.19) substituted for
equation (8.8), but for each compressor speed line the calculation is repeated for
several values of Ma covering the desired range of flight speed. The result is a fan
of equilibrium running lines of constant Ma. As shown in Fig. 8.15, they merge
into the single running line obtained with the higher compressor speeds for which
the nozzle is choked.
It should be noted that increasing Mach number pushes the equilibrium
running line away from the surge line at low compressor speeds. Fundamentally,
this is because the ram pressure rise allows the compressor to utilize a lower
pressure ratio for pushing the required flow through the nozzle.
t It should be uoted that the argument based on Fig. 8.8 is equally applicable here. Thus when the
nozzle is choked the turbine operating point, and hence m.jTO'/P03, P03/P04 and I1T034/T03 are fixed
and any increase in the overalI expansion ratio P03/Pa due to increasing ram effect will therefore result
in an increase in overall nozzle pressure ratio P04/Pa with the turbine pressure ratio unaffected.
OFF-DESIGN OPERATION OF THE JET ENGINE
P02
POj
FIG. 8.15 Jet engine mnning lines
361
variation of thrust with rotational speed, forward speed and altitude
We have seen in section 3.1 that when the whole of the expansion frompo4 to Pa
occurs in the propelling nozzle, the net thrust of the jet unit is simply the overall
rate of change of momentum of the working fluid, namely F = m( Cs - Ca) where
Ca is the aircraft speed. On the other hand, when part of the expansion occurs
outside the propelling nozzle, i.e. when the nozzle is choking and Ps is greater
than P a, there vvill be an additional pressure thrust. In this case the net thrust is
given by the more general expression
(8.21)
To obtain the curves which show the thrust delivered by the jet engine over the
complete operating range of inlet pressure and temperature, flight speed and
rotational speed, information obtained from the points used to establish the equi-
librium running diagram can be used. For each running point on this diagram the
values of all the thermodynamic variables such as
POl my'TOl P02 T03 T04 P03 d P04
Pa ' POl' TO! ' 1'03' P04 at-:t Pa
are determined for specified values of NI y'TOl and Ma (and NI y'TOl alone in the
choked nozzle range). The thrust IJan ultimately be expressed in terms of these
non-dimensional quantities. Thus equation (8.21) can immediately be written in
the form
(8.22)
(A dimensional check will show that the true non-dimensional thrust is FI(PaD2),
but for an engine with fixed geometry the characteristic dimension can be
362 PREDICTION OF PERFORMANCE OF SIMPLE GAS TURBINES
omitted.) Making use of equation (8.17),
Ca Ca Ma.j(yR)
. .ITO! == .jToa= j(1+1';lMJ)
When the nozzle is unchoked, (:s/ .jT04 can be obtained from equation (8.13)" with
P04/PS put equal to P04/Pa, and the pressure thrust is zero because PS/Pa = 1.
When the nozzle is choked, Cs/ .jTo4 is given by equation (8.18), imd PS/Pa
can be calculated from
l!J. = Ps X P04 = !!.£. X P04
Pa P04 Pa P04 Pa
Pc/P04 is the reciprocal of the critical pressure ratio given by equation (8.15), i.e.
it is a function only of l' for the gas and the nozzle efficiency 1Jj. P04/ Pais one of
the known parameters having been calculated using equation (8.l9).
A typical variation of thrust with engine speed and flight speed is illustrated in
Fig. 8.16, which shows separate curves of thrust for each Mach number
considered. The abscissa can be left as N/ .ITo! as shown, or be put in terms of
N/ .ITa by making use of the relations
~ = ~ X ITol and TOl = (I + Y - 1 M;)
.ITa .ITo Y Ta Ta 2
It should be noted that although a unique running line is obtained on the com-
pressor characteristic when the propelling nozzle is choked, the thrust for a given
value of N/ .ITo! does depend on the flight Mach number. There is a direct
dependence due to the increase in momentum drag (mCa) with increasing flight
speed, and an indirect dependence due to the increase in compressor inlet pres-
sure arising from the ram compression. At low rotational speeds the effect of
momentum drag predominates and an increase in Ma causes the thrust to
F
Pa
FIG. 8.16 Thrllst curves
OFF-DESIGN OPERATION OF THE JET ENGINE 363
decrease; whereas at high values of N/ .ITO! the beneficial effect of ram pressure
rise predominates.
Although it is convenient to express the perfonnance in terms of non-
dimensional engine speed, it is actual mechanical speed upon which a limit is set
by the turbine stresses and which must be controlled by a governor. The strong
dependence of thrust upon engine speed indicates that accurate control is
essential. t If the speed is controlled at a value below the limit, talce-off thrust will
be substantially reduced. The situation is more serious if the speed exceeds the
correct limit: not only do the centrifugal stresses increase with the square of the
speed but there is also a rapid increase in turbine inlet temperature. The latter can
be seen from the way the running line crosses the T03/TOI lines on Fig. 8.15.
Typically, an increase in rotor speed of only 2 per cent above the limit may result
in an increase in T03 of 50 K. With blade life detennined by creep, the time for
which high speeds are pennitted must be strictly controlled. The maximum
permissible speed is normally restricted to periods of less than 5 min, giving the
take-off rating. The climb rating is obtained with a small reduction in fuel flow
and hence in rotor speed, and can usually be maintained for a period of 30 min.
The cruise rating requires a further reduction in fuel flow and rotor speed,
resulting in stress and temperature conditions permitting unrestricted operation.
Ratings for the Rolls-Royce Viper 20 at Sea Level Static conditions are shown in
the following table; this is a simple turbojet with a low pressure ratio. The very
rapid drop in thrust with reduction of speed is evident.
Engine speed Thrust SFC
(% Nmax) [leN] [leg/kN h]
Take off (5 min) 100 ]3·35 100-4
Climb (30 min) 98 12-30 98·2
Cruise 95 10·90 95·1
The effect of ambient temperature on the talee-off rating is important both to
the manufacturer and the user. With the engine running at its maximum mech-
anical speed an increase in ambient temperature will cause a decrease in N/ .ITa
and hence N/ .ITo!. This will cause the operating point on the compressor
characteristic to move along the equilibrimn rumling line to lower values of both
111/ .jTOl/POl andpo2/po]' and it can readily be seen to be equivalent to a decrease
in mechanical speed. The actual mass flow entering the engine will be further
reduced due to the increase in ambient temperature, because m is given by
(m.jTodpOl)/(.jTodpo!). The wellimown result of all these effects is that an
increase in ambient temperature results in an appreciable loss of thrust as indi-
cated in Fig. 8.16. But the picture is not yet complete. T03/TOI also decreases with
increase in ambient temperature when N is held constant. The actual temperature
t It should be realized that the same arguments are applicable in the case of a free turbine engine
delivering shaft power.
364 PREDICTION OF PERFORMANCE OF SllvlPLE GAS TURBINES
T03 is given by (T03jT01)Tob and the decrease in (To3/Tod will be more than
offset by the increase in TO!. In general, for a fixed mechanical speed T03 will
increase with increase in ambient temperature, and the allowable turbine inlet
temperature may be exceeded on a hot day. To keep within the limit it
necessary to reduce the mechanical speed, giving an even greater reductIon m
N/ JT01 and hence thrust. .
Turning our attention next to variation in ambient pressure, Fig. 8.16 indicates
that the thrust will change in direct proportion to the ambient pressure; no change
in the engine operating point is involved, but the mass flow is reduced as the
ambient pressure is reduced. Both the pressure and temperature decrease with
increasing altitude, the latter levelling off at 11 000 m. Because of the
dependence of the thrust on the first power of the pressure, the decrease in
thrust due to the decrease in Pa more than outweighs the increase due to the
reduction in Ta. Thus the thrust of an engine decreases with increase in altitude.
Airports at high altitudes in the tropics are often critical with regard to take-off
perfonnance, and it may be necessary for aircraft using them to accept
significantly reduced payloads. Mexico City is a wellimown example.
Variation of fuel consumption and SFC with rotational speed,
forward speed and altitude
The fuel consumption of a jet engine together with the fuel capacity of the aircraft
determine the range, and the specific fuel consumption (fuel flow per mlit thrust
for a jet engine) is a convenient indication of the economy of the unit. From the
argmnents of the previous section it is apparent that both fuel consumption and
specific fuel consmnption can be evaluated as functions of N/ JTOI (or N/ JTa)
and Ma.
When a value of the combustion efficiency is assumed, the fuel consumption
can readily be determined from the air flow, the combustion temperature rise and
the fuel/air ratio curves of Fig. 2.15, in the manner described for the other types
of gas turbine engine. To do this, values of Pa and Ta have to be assumed to obtain
m, T03 and T02 from the dimensionless parameters. The fuel flow is therefore a
function of N/ JTa> Ma, Po and The dependence of the fuel flow on ambient
conditions can in fact be virtually elinlinated by plotting the results in terms of the
non-dimensional fuel flow (mf Qnet,p/ D2PaJTa). For a given fuel and engine the
calorific value and linear dimension can be dropped, yielding in practice
nY/PaJTa. Figure 8. 17(a) shows a typical set of fuel consumption curves. Unlike
F/Pa, the fuel parameter depends only slightly on Ma because it is due simply to
the variation of compressor inlet conditions and not also to momentum drag.
Indeed, if mf/POIJTo1 is plotted against N/JTOI as in Fig. 8.l7(b), the curves
merge into a single line for the region where the nozzle is choking. It must be
mentioned that although the combustion efficiency is high and constant over most
of the working range, it can fall drastically at very high altitudes due to low
combustion chamber pressures. Such curves as those of Fig. 8.17 may well
underestimate the fuel consumption when Pa is low.
OFF-DES1GN OPERATION OF THE JET ENGINE
365

(a)
(b)
F][G. 8.17 Fuel cOlIsllImptiolli cllIrves
Curves of 'non-dimensional' SFC may be obtained by combining the data of
Figs. 8.16 and 8.17(a). Figure 8.18 shows SFC/JTa plotted againstN/JTol for
various values of Ma. It is apparent that the SFC will improve with increase in
altitude due to the decrease in Ta. As the SFC is a function only of JTa and not of
P a, however, the effect of altitude is not so marked as in the case of the thrust.
(The actual variation in SFC with altitude and Mach number for a simple turbojet,
operating at its maximum rotational speed, was shown in Fig. 3.14.) It should be
noted that SFC increases with Mach number. Thrust initially drops with
increasing Mach number and then rises when the ram effect overcomes the
increase of momentum drag; the fuel flow, however, will steadily increase with
Mach number because of the increased air flow due to the higher stagnation
pressure at entry to the compressor and this effect predominates.
SFC
ff.
,
FIG. 11.18 SFC curves
366 PREDICTION OF PERFORMANCE OF SIMPLE GAS TURBThTES
8.6 Methods of displacing the. e<luilibdum. mnning line
It was stated earlier that if the equilibrium running line intersects the surge line it
will not be possible to bring the engine up to full power without taking some
remedial action. As will be shown in the next chapter, even when clear of the
surge line, if the running line, approaches it too closely the compressor may 'surge
when the engine is accelerated rapidly.
With most modem compressors, surge is likely to be encountered at low
values of N/ .ITO! and is less of a problem at high rotational speeds. Many high
performance axial compressors exhibit a kink in the surge line as shown in Fig.
8.19. A running line, intersecting the surge line at low speeds and at the kink, is
also shown. To overcome the problem it is necessary to lower the running line
locally in dangerous regions of operation. 1-
One common method of achieving this is blow-off, where air is bled from
some intermediate stage of the compressor. Blow-off clearly involves a waste of
turbine work, so that the blow-off valve must be designed to operate only over the
essential part of the running range. Furthelmore, it may be difficult to discharge
the air when space around the engine is at a premiUlll as in a nacelle. As an
alternative to blow-off a variable area propelling nozzle could be used for a jet
engine. It will be shown in the next chapter that variable area propelling nozzles
serve other useful purposes also.
It can be deduced that either of these methods will produce a reduction in
pressure ratio at a given compressor speed and hence will lower the running line,
P02
P01
Surge line
\/
Normal equilibrium .
running line / / /
X/
1/
Kink //j
\ f/
),,1
<'/
m{7;lp01
FIG. 8.19 Effect of blow-off and increased nozzle area
t An alternative method is to raise the surge line using variable stators in the compressor.
METHODS OF DISPLACING THE EQUILIBRIUM RUNNING LfNE
367
This is most easily demonstrated if we consider operation at the high speed end of
the running range, where the constant N/ .ITO! lines on the compressor
characteristic are almost vertical (i.e. mJTodpol c::::: constant) and where both
the nozzle and turbine are choked.
If we consider the use of a variable area nozzle first, Fig. 8.20 shows that
increasing the nozzle area will cause an increase in turbine pressure ratio and
hence non-dimensional temperature drop !'J.To34/To3' From equation (8.2),
satisfying compatibility of flow,
P02 = mJTol x P02 x IT03 X ~ = KI IT03
POl POI P03 V Tal mJT03 V TOl
(8.23)
where KI is a constant because under the assumed conditions the values of
mJTodpob P02/P03 and mJTo3 /P03 are all fixed. From equation (8.6), which
satisfies compatibility of work,
T03 !'J.T012 T03 Cpa
-=--x--x--
TOI TOI !'J.T034 Cpg1Jm
Now, at a fixed value of compressor speed N the compressor temperature rise is <
approximately constant; Fig. 8.21 shows a typical compressor characteristic re-
plotted on a basis of temperature rise vs. pressure ratio. The wide range of
pressure ratio obtained at almost constant temperature rise is due to the significant
variation of efficiency when operating at conditions away from the design point.
Thus if we assUllle !'J.TOI2/ TOI is constant,
T03 K2
TOI (!'J.T034/T03)
(8.24)
where K2 is another constant. Combining equations (8.23) and (8.24) gives
P02 K3
POI J(!'J.T034 /T03 )
where K3 is a constant, obtained from KI and K2.
mffo
Po
I
I
(8.25)
Increased nozzle area
/
---1-- _________________________ ______ _
I
I
I
I
- - + , ~ I - - _ - - - m ~
I
I
I
P03
I I
I I
~ Increased pressure ratio
I I
FIG. 8.20 Effect of variable-area propelliilg ilozzle
/
Standard nozzle area
368
PREDICTION OF PERFORMANCE OF SIMPLE GAS TURBINES
FIG. 8.21 Compressor characteristics
If the engine speed N/..JTOI is held constant and the nozzle area is increased,
t1T034/T03 will be increased and equation (8.25) shows that P02/POI will be
decreased, hence the running line will be moved away from the surge line. In
order to maintain the speed at the required level, it is necessary to reduce the fuel
flow; opening the nozzle without reducing the fuel flow would cause the engine to
accelerate to a higher speed, as discussed in Chapter 9. It should also be realized
that decreasing the nozzle area will move the operating line towards the surge
line; reasons for doing this will be discussed in the next chapter.
When considering the effect of bleed, ml is no longer equal to m3 and equation
(8.23) is modified to
P02 = Kl I
T
o3 x m3
POI -V TOI ml
(8.26)
With the nozzle, or power turbine in the case of a shaft power unit, remaining
choked, the turbine operating point remains unchanged and equation (8.26) be-
comes
T03 = ml x t1TOI2 x X = K4 mj
To] m3 TO! t1T034 cpgftm m3
(8.27)
where K4 is a constant, assuming as before that t1To12/Tol is constant. Combining
equations (8.26) and (8.27)
P02 _ K 1
m3
- 5
PO] ,ml
(8.28)
where Ks is a constant.
With the use of bleed, m3 is always less than mj, and the result will once again
be to reduce the pressure ratio and lower the operating line. Equation (8.27) also
shows that the use of bleed will increase the turbine inlet temperature. The
reduction in turbine mass flow necessitates an increase in temperature drop to
INCORPORATION OF VARIABLE PRESSURE LOSSES 369
provide the compressor work, and with the non-dimensional temperature drop
fixed this· implies an increase in turbine inlet temperature. Both the effects of
bleed and increase in nozzle area can be interpreted physically in terms of a
decrease in the restriction to the flow, permitting the compressor to operate at a
lower pressure ratio for any given rotational speed.
8.7 Incorporation of variable preS!Hm, losses
The size of the ducting for components such as the intalce and exhaust of a gas
turbine will be determined primarily by the requirement for low pressure losses at
maximum power, which will normally correspond to maximum mass flow. The
designer must compromise between low pressure losses and small duct sizes and
the choice will be largely determined by the application envisaged. It be
shown by cycle calculations that the effect of a given pressure loss is dependent
on the pressure level at which it occurs, and it is essential to keep the terms
t1po/ Po at all stations as low as possible. With pressure losses in the intake and
of gas turbines occurring at essentially atmospheric pressure, it
WIll be realIzed that these losses are much more critical than pressure losses in the
combustion chamber or the air side of a heat exchanger where the air is entering
at the compressor delivery pressure. In a typical installation a pressure loss of
2·5 cm.H20 in. the intake may cause about 1 per cent reduction in power output.
Havmg deCIded on the size of ducting required to give the desired pressure
loss at the design point, the variation of loss with changing operating conditions
can be predicted for incorporation in the off-design calculation procedures
outlined in this chapter. The velocities in such components as the intalce and
exhaust ducts will be sufficiently low for the flow to be treated as incompressible,
so that t1po will be proportional to the inlet dynamic head. Expressed in non-
dimensional terms this implies, as shown by equation (6.2), that
I1po ex (m..JTo )2
Po Po
Assuming the appropriate values at the design condition to be (I1Po/Po)D and
(m..JTO/PO)D, then at any off-design condition
t1pO = (l1po) [(m..JTo)/(m..J
To
) ]2
Po Po D Po Po / D
(8.29)
Values of m..JTo/po for each component are obtained in the course of the off-
design calculations. The pressure loss from equation (8.29) can be incorporated in
same manner as was the combustion pressure loss I1Pb/Po2, namely by mak-
mg use of such equations as
P03 = 1 _ I1Pb
P02 P02
370
PREDICTION OF PERFORMANCE OF SIMPLE GAS TURBINES
Now, however, the pressure ratio across the component is not constant but
changes with m.JTo/po.
In the particular case ofthe combustion chamber, themore accurate expression
of equation (6.1) can be used. This includes the fundamental loss due to heat
addition. Combining (6.1) and (6.2) we would have
!:J.Pb IX (m.JT02)2[K\ +K2(T03 -1)J
P02 P02 T02
Such refinement would not normally be worthwhile because the fundamental loss
is a small proportion of the total.
9
Prediction of performance-
further topics
Evaluation of the off-design performance of complex plant incorporating inter-
cooling, heat-exchange and reheat is inevitably more complicated than for the
simple gas turbine, although the basic principles described in Chapter 8 are still
applicable. The components of such gas turbines can be arranged in a wide
variety of ways, about forty different layouts being possible even with no more
than two compressor rotors. A comprehensive survey of the part-load perform-
ance and operating stability of many of these possibilities can be found in Ref.
(1), wherein use was made of stylized component characteristics. It is as a result
of such calculatipns that it was possible to dismiss certain arrangements as im-
practical because the running line runs into the surge line under some operating
conditions. No attempt will be made to repeat this kind of survey here, and we
shall restrict our attention mainly to discussing the prediction of off-design per-
formance in the practical cases of (a) high pressure ratio twin-spool engines and
(b) turbofans. Before discussing the matching procedures for twin-spool engines
and turbofans, however, methods of improving the part-load performance of gas
turbines will be considered briefly.
The chapter ends with an introduction to transient performance, including a
brief description of methods for predicting the acceleration or deceleration of
rotor systems. Acceleration rates of gas turbine rotors are obviously dependent on
mechanical considerations such as the polar moment of inertia and the maximum
temperature which the turbine blades can withstand for short periods, but the
limiting factor is usually the proximity of the equilibrium running line to the
surge line. Thus a thorough understanding of off-design performance is essential
before transient behaviour can be investigated and a suitable control system can
be designed.
9.1 Methods of improving part-load performance
It was pointed out in the Introduction that the part-load performance of gas
turbines intended for vehicular or naval use was of great importance because of
372 PREDICTION OF PERJ.>ORMANCE-FURTHER lDPICS
the considerabie portioh of the running time spent at low power. Early studies for
both applications resulted iri the· consideration of complex· arrangements incor-
porating intercooling, heat-exchange and reheat. The sole justification for the
marked increase in complexity was the great improvement in part-load specific
fuel consumption indicated in Fig. 9.1. Further details regarding the choice of
cycle parameters, mechanical layout and development problems can be found in
Refs (2) and (3). Such complex arrangements did not prove successful for either
of these applications despite their undoubted thermodynamic merit, the main
reason being the mechanical complexity involved. It is interesting to note, never-
theless, that the Ford engine described in Ref. (3) was an extremely compact unit
of about 225 kW. As a result of the problems with the complex cycle the navies of
the world have concentrated their attention on simple gas turbines, using such
arrangements as CODOG, COSAG and COGOG to overcome the part-load
problems as discussed in the Introduction. Virtually all proponents of the
vehicular gas turbine have now reverted to a low pressure ratio unit with a heat-
exchanger: Ref. (4) discusses performance of vehicular gas turbines.
In practice it is found that the vast majority of applications make use of either
the high pressure ratio simple cycle or the low pressure ratio heat-exchange cycle.
Detailed off-design performance calculations for gas turbines with heat-
exchangers would have to account for variation in heat-exchanger effectiveness
with engine operating conditions. Heat-exchanger effectiveness depends on the
fixed parameters of heat transfer area and configuration (e.g. counter-flow, cross-
flow or parallel-flow), and the parameters varying with engine operating
conditions such as overall heat transfer coefficient between the two fluids and
their thermal capacities (mcp). Methods for estimating the effectiveness of gas
turbine heat-exchangers (and intercoolers) are described in Ref. (5) and will not
be dealt with here.
200
(,) 150
lli
t:
.2'
l3
;; 100

a;

Q)
D.. 50-
/
Simple heat-exchange
cycle
Complex cycle shown --
(Both engines have same design SFC)
OL-__ __ __ __ __
o 20 40 60
Percentage design power
Heat-exchanger
FIG. 9.1 Part-load SFC for simple and complex cycles
METHOPS OF IMPROVING PART-LOAD PERFORMANCE 373
NI-ff;;;
FIG. 9.2
The incorporation of a heat-exchanger will cause an increase in pressure loss
between compressor delivery and turbine inlet, and also an increase in turbine
outlet pressure. Although the additional pressure losses will result in a reduction
in power, they will have little effect on the equilibrium running line and the part
load behaviour of engines with and without heat-exchanger will be similar. A
typical running line for a free turbine engine was shown in Fig. 8.7; the variation
of T03/TOI with N/ JT01 can be deduced from this but is more clearly indicated in
Fig. 9.2. It was shown in Chapter 2 that the thermal efficiency of a real gas turbine
cycle was dependent on the turbine inlet temperature, and the rapid drop in T03
with decreasing power is the basic cause of the poor part-load performance of the
gas turbine. The variation of thermal efficiency with power output for a
hypothetical engffie with and without heat-exchanger is shown in Fig. 9.3. It can
be seen that although the design point thermal efficiency is significantly improved
by incorporating a heat-exchanger, the shape of the efficiency curve remains
virtually unchanged. t This is fundamentally due to the fact that in each case there
is a similar drop in turbine inlet temperature as power is reduced.
To improve the part-load efficiency of gas turbines, therefore, some means
must be found. of raising the turbine inlet temperature at low powers. In the
majority of applications where good part-load economy is required, e.g. vehicular
and marine, a free would be used. If we focus our attention on the free
turbine engine, the turbine inlet temperature at part-load can be increased by
using variable-area power-turbine stators.
Variable-area power-turbine stators
Variation of area is accomplished by rotating the nozzle blades, and this permits
the effective throat area to be reduced or increased as shown in Fig. 9.4. In
t It is possible that for certain values of the cycle parameters the incorporation of a heat-exchanger
will alter the sbape of the efficiency curve slightly, but it is unlikely that addition of a heat-exchanger
alone will give any significant improvement in part-load economy. Curve B in Fig. 9.3 was obtained on
the assumption of a constant heat-exchanger effectiveness but estimates have shown that the
effectiveness does not increase much with reduction in engine power output.
t Single-shaft engines normally operate at constant speed and high power. fu practice it may be
necessary to operate at fixed exhaust gas temperature for different electrical loads when used in
combined cycle or cogeneration applications; this may be achieved by using variable inlet guide vanes
at entry to the compressor, permitting some variation in airflow at fixed rotational speed.
PREDICTION OF PERFORMANCE-FURTHER TOPICS
_B
/
rc T03/ [K]
Heat-exchanger
eliecliveness
IA 6.0 1220 0
la 6.0 1220 0.90
Percentage design power
FIG. 9.3 Variation of thermal efficiency WJlth power output
Chapter 8 it was emphasized that free turbine engines and turbojets ther-
modynamically similar because the flow characteristics of a free turbme and a
propelling nozzle impose the same operating restrictions on the gas
thus variable power-turbine stators will have the same effect as a vanable-area
propelling nozzle. Section 8.6 showed that increasing the nozzle a:ea
of a turbojet moved the running line away from the surge lme, while decreasmg
the area displaced the running line towards surge. Reference to Fig. 8.7
that the latter will cause an increase in turbine inlet temperature at low powers; It
is also likely that the compressor efficiency will be improved as the surge line is
approached. Both of these effects will improve the part-load SFC.
Ideally, the area variation of the power-turbine stators can be controlled so
the turbine inlet temperature is maintained at its maximum value as power IS
reduced as indicated in Fig. 9.5. If the running line at maximum temperature
moves to intersect the surge line as shown, it then becomes necessary to reopen
FIG. 9.4 Variable-geometry power-turbine stators
MATCHING PROCEDURES FOR TWIN-SPOOL ENGINES
P02
POl
Running line with variable-area
power-turbine stators
Stator area increasing
to avoid surge
\
\
\
Normal running line
T03/ TOl
increasing
lFJG. 9.5 Effect of val'iable power-wl'lJillie §tatillirs Olll rlllllllillig lillie
375
the power turbine nozzles for this pati of the rmming range. It should be noted
that operation at constant gas-generator turbine inlet temperature with reducing
power will cause an increase in temperature at entry to the power turbine, because
of the reduced compressor power, and the temperature of the hot gases entering
the heat -exchanger will also be raised. Temperature limitations in either of these
components may restrict operation in this mode. The use of a variable geometry
power turbine is particularly advantageous when combined with a heat-
exchanger, because the increased turbine outlet temperature is utilized.
The efficiency of the power turbine will obviously be affected by the position
of the variable stators, but with careful design the drop in turbine efficiency can
be more than offset by maintaining a higher turbine inlet temperature at part load.
Area variations of ± 20 per cent can be obtained with acceptable losses in turbine
efficiency, see Ref. (6). The variable stators are liable to be exposed to
temperatures in the 1000--11 00 K region, but the development problems
associated with this do not appeat· to have been too difficult. A facility for
increasing the stator area is also advantageous with respect to starting and
accelerating the gas generator. If the stators are rotated still further, the gas
generator flow can be directed against the direction of rotation so that the flow
impinges on the back of the power turbine blades. This can result in a substantial
degree of engine braking which is extremely important for heavy vehicles.
9.2 Matching procedures for twillll-spool eJrngiJrnes
The basic methods used for twin-spool matching are similar to those described in
Chapter 8. They differ only from those for single-spool engines in that it is also
376
PREDICTION OF PERFORl\1ANCE-FURTHER TOPICS
necessary to satisfy compatibility of flow between the spools. This compatibility
requirement gives rise to the phenomenon of aerodynamic coupling which de-
termines the ratio of the rotor speeds even though the rotors are mechanically
independent of each other. The station numbering used for twin-spool turbojets is
shown in Fig. 9.6 and the rotational speeds will be referred to as NL and NH for
the LP and HP rotors respectively. The corresponding shaft power unit would
have a power turbine between 6 and 7 in place of the propelling nozzle.
In outlining the procedures we will assume single line turbine characteristics,
constant turbine efficiencies and constant percentage combustion pressure loss;
these are often close approximations in practice and their use makes it easier to
understand what is happening in physical terms. Considering work compatibility,
the equations for the LP and HP rotors are
and
Flow compatibility must be satisfied between the compressors, the turbines and
finally between the LP turbine and nozzle (or power turbine). The problem of
flow compatibility between two turbines was dealt with under heading 'Matching
of two turbines in series', section 8.4, and Fig. 9.7 illustrates the further restric-
tion in operating conditions resulting from the introduction of a nozzle (or power
turbine).
Calculations for a twin-spool engine can be lengthy but they may be
considerably simplified once it is appreciated that the HP rotor of a twin-spool
engine is equivalent to a single-spool turbojet with a fixed nozzle the area of
which is defined by the throat area of the LP turbine stators. It was shown in
Chapter 8 that for a single-spool turbojet a unique running line is defined when
the nozzle is choked, and it follows that a similarly unique running line will be
defined on the HP compressor characteristic of a twin-spool engine when the LP
turbine stators are choked. (It should be noted that in practice the LP turbine of a
twin-spool unit will be choked over most of the useful running range, but may
become unchoked at idle conditions.) The position of the unique running line on
2 3 4 5 6 7
~ ~ L
'Rg
'"
FIG. 9.6 Statio!l !Iumberi!lg for twin-spool turbojet
MATCHING PROCEDURES FOR TWIN-SPOOL ENGINES
HI" turbine LP turbine Nozzle (fixed area)
or power turbine
377
mf'F;;s
/./"
=- ;'
m': T06 /( _______________________ .:; __ , . . . . . . ~ ~
Pa6/ I mIT;,
Pos / . , # ~ / l
,../ / j P06
,/" /:
/f----:------------- ---1- .--... -\1---
1
/ I, m{fc;
miT;.
/0.._-+ ___ - , Pos
,. P04 I
,
,
FIG. 9.7 Flow compatibility for twill-spool ellgine
the HP characteristic can be detemrined by considering the HP rotor alone, for
which compatibility of flow and work yields
mJ
T
04 = mJT02 x P02 )( l!2i X /T04
P04 P02 P03 P04 'if T02
(9.1)
.<1 T045 AT023 T02 cpa
--=--x-x--
T04 T02 T04 Cpg1JmH
(9.2)
It can be seen that equations (9.1) and (9.2) are the same as equations (8.2) and
(8.6) apart from,the difference in station numbering, and they can be solved by
the trial and error procedure described in section 8.3 to yield a value of T04/T02
for each point on a constant NH/ JT02 line on the HP compressor characteristic.
For the particular case when the LP turbine is choked, the HP turbine will be
restricted to operation at a fixed non-dimensional point with the values of P04/ Pos,
mJT04/P04 and .<1T04S/To4 all fixed. Substituting the relevant values of mJTo4-
/P04 and .<1T04S/T04 in equations (9.1) and (9.2), the ruuning line defined by LP
turbine stator choking can be established on the HP characteristic, and it will be
similar to that shown in Fig. 8.7. The importance of establishing the LF choking
line is that it greatly facilitates satisfYing the requirement for compatibility of flow
between the two compressors as will be apparent in what follows.
Although much of the procedure to be described in this section is equally
applicable to turbojets and shaft power units with a separate power turbine, it is
most easily explained for the case of a turbojet with a fully variable propelling
nozzle. Any effect resulting from variation of nozzle area will directly influence
the LP turbine, and hence the LP compressor; the HP turbine, however, is
separated from the nozzle by the LP turbine and, if the LP turbine is choked, the
HP rotor will be shielded from disturbances caused by the variable nozzle. The
use of a fully variable nozzle penmits operation over a wide area of the LP
compressor characteristic as in the case of a single-spool engine, even though the
operating region on the HP characteristic is a single line determined by choking
of the LP turbine. Thus if we assume a fully variable nozzle we can start with any
point on the LP characteristic, and the final step in the calculations will be the
378
PREDICTION OF TOPICS
determination of the required nozzle area. No iteration is required in this
procedure, which can be summarized as foHows.
(a) Detern:llne the inlet conditions TOI and POI from the ambient and flight
conditions; for a land based unit TO! = Ta and POI = Pa-
(b) Select any point on a line of constant NdToI on the LP compressor
characteristic, which will specify values of m.jTOJ/POb P02/POI and 1}cL·
The compressor temperature rise !l.T012 and temperature ratio T02/TOJ can
then be calculated using equation (8.4).
(c) The non-dimensional flow at LP compressor exit, m.jT02/P02, is obtained
from the identity
m.jT02 = m.jTOl x POl x jT02
P02 POI P02 V TOI
and for compatibility of flow between the compressors this is the nOll-
dimensional flow at entry to the HP compressor.
(d) With the operating line on the HP compressor characteristic determined by
LP turbine choking, the known value of m.jT02/P02 defines the operating
point on the HP compressor characteristic. This gives the values of P03/P02,
NH/ .jT02 and 1}cH, and the values of !l.T023 and T03/T02 can be found.
( e) The overall compressor pressure ratio is now given by
P03 = P02 X P03
POI POI P02
cn The turbine inlet pressure P04 can be fOUJId from
P03 P04
P04 =-X-XPOI
POI P03
where P04/P03 is obtained from the combustion pressure loss.
(g) The value of m.jT04/P04 is lmown, having been determined by flow
compatibility between the two turbines, and m can be fOUJId from
m.jTOJ/POl, TOI and POI.
(h) The turbine inlet temperature T04 can now be fOUJId from
2
T04 = (m.j T 04 x P04)
P04 m
(i) The HP turbine is operating at a fixed non-dimensional point with P04/POS
and !l.T04S/T04 determined by LP turbine choking. Thus conditions at entry
to the LP turbine, Pos and Tos, can be obtained from
P05
P05 =P04 x-
P04
and T05 = T04 - !l.T045
(j) We must now satisfy the work requirement for the LP compressor. This is
given by
l1mLmcpgtiTOS6 = mCpatiTOl2
!l.T012 is known, so I1Tos6 can be calculated.
SOME NOTES ON THE BEHAVIOUR OF TWIN-SPOOL ENGINES 379
(k) With I1To56 and Tos knmvn, the temperature at exit from the LP turbine and
entry to the nozzle is given by
T06 = T05 - !l.T056
(l) The LP turbine pressure ratio POS/P06 is found from
[
G
1 )(Y-1lIYJ
> !l.TOS6 = l1tLT05 I - ---
OS/P06
Once POS/P06 is Imown P06 is given by
P06
P06 =P05)(-
P05
(m) m, T06 and P06 are now established. The overall nozzle pressure ratio P06/Pa
is then lmown, and the calculation of the nozzle area given the inlet
conditions, pressure ratio and mass flow is quite straightforward (see
Chapter 3). Once the required nozzle area has been calculated, the matching
procedure is complete and an equilibrium running point has been obtained
for that particular nozzle area and initial value of Nd .jTOl .
(n) The procedure can be repeated for points on other Nd .jTOl lines. All the
data required for a complete performance calculation, e.g. of the thrust, fuel
flow and SFC at each rUJJ11ing point, are now available.
Note that if a fixed nozzle were used, the area calculated in step (m) would not
in general be equal to the specified area; it would then be necessary to return to
step (b), try another point on the same Nd .jTOl line on the LP characteristic and
iterate until the correct nozzle area was obtained.
The effect of the LP turbine operating on the unchoked part of its
characteristic is to raise the running line on the HP characteristic (i.e. displace
it towards surge), because of the reduced non-dimensional flow at exit from the
HP turbine. The calculation described above must be modified somewhat to deal
with an unchoked LP turbine, but this condition is only liable to occur at low
powers and the modification will not be dealt with here.
Finally, if a shaft power UJIit were being considered, step (m) yields
m.jT06/P06 at inlet to the power turbine and also the power turbine pressure
ratio P06/Pa. If this value of m.jT06/P06 does not agree with the value from the
power turbine characteristic for the known pressure ratio (see Fig. 9.7), again it is
necessary to return to step (b) and repeat the calculation.
9.3 Some notes 011 the behavionr of twin-spool engines
Complete off-design performance calculations for twin-spool engines are obvi-
ously time consuming and in practice would be carried out using digital com-
puters. Some aspects of twin-spool behaviour, however, can be deduced from an
UJIderstanding of the matching procedure and the more important of these will be
briefly described.
380 PREDICTION. OF PERFORMANCE-FURTHER TOPICS
Maximum
nozzle\a!
A
Minimum
nozzle area
FIG. 9.8 Speed relationship for twin-spool engine
Aerodynamic coupling of rotor speeds
From step (c) of the matching procedure described in the previous section it can
be seen that once an operating point on the LP characteristic has been chosen, the
corresponding operating point on the HP characteristic is fixed. Thus for a fixed
value of Nd .JT01 and nozzle area, the value of NHI.JTo2 is determined by flow
compatibility between the compressors. The HP compressor non-dimensional
speed can be expressed in terms of the temperature at engine inlet by
NH _ NH X jTc2
.JTO! - .JT02 V TO!
Nd .JT01 and NHI.JTol are then directly proportionalt to the actual mechanical
speeds of the two rotors.
A typical variation of Nd .JT01 with NHI.JTol for a turbojet is shown in Fig.
9.8; note that the relation between the speeds is dependent on nozzle area, and
only for a fixed nozzle will it be unique. At a fixed value of N H any increase in
nozzle area will cause an increase in NL, the physical reason being that opening
the nozzle will increase the pressure ratio across the LP turbine so causing an
increase in LP rotor torque.
Effect of variable-area propelling nozzle
We have seen that the use of a fully variable nozzle permits operation over a wide
range of the LP compressor characteristic, although the HP running line is not
affected by nozzle variation as long as the LP turbine is choked. Typical running
lines are shown in Fig. 9.9. It is important to note that increasing the nozzle area
moves the LP nnming line towards surge, which is the opposite effect to that
t If the 'equivalent speed' NI,Jeo were used instead of NI,JTo (see section 4.6) the numbers would be
equal to the mechanical speeds under reference entry conditions.
SOME NOTES ON THE BEHAVIOUR OF TWIN-SPOOL ENGINES
P02
P01
Po,
P02
Minimum nozzle area
\.
(a)
(b)
A
FIG. 9.9 Running lilies for twill-spooll turbojet (a) LP compressor characteristic,
(b) HI' compressor characteristic
381
obtained on a single-spool engine. The reason for this is the increase in LP
turbine power resulting from the redistribution of pressure ratio between the LP
turbine and the nozzle. One of the major advantages of the variable nozzle is that
permits selection of rotor speed and turbine inlet temperature independently; or,
m terms more relevant to an aircraft, of air flow and thrust. This is a valuable
feature for an engine that has to operate over a wide range of intake temperature
which produces significant changes in non-dimensional speed and mass flow.
(:) gives an excellent picture of the problems involved in matching the
engme and mtake for a supersonic transport.
382 PREDICTION OF PERFORMANCE-FURTHER TOPICS
It was shown in Chapter 8 that lines of constant T03/TOl can be drawn on the
compressor characteristic of a single-spool engine and these represent possible
equilibrium running lines when a fully variable nozzle is used. As can be seen
from Fig. 8.7, the value of T03/Tol increases as lines of constant temperature ratio
move towards surge. For a twin-spool f:ngine lines of constant T04/T02 can be
drawn on the HP compre&sor characteristic and they behave similarly. If we
consider the LP compressor characteIistic, however, and plot lines of constant
T04/Tol as shown in Fig. 9.9(a), it will be seen that they move away from surge as
the value of T04/Tol is increased. Thus a.gain we have behaviour opposite to that
found for the single-spool turbojet. This can be explained if we consider
operation at a constant value of Nd JT01 ' With the nozzle fully open the LP
compressor operates at point A on Fig. 9. 9( a) and closing the nozzle moves the
operating point to B. The value of mJTOl/POl is constant for the vertical
Nd JT01 line shown and the temperature rise across the LP compressor will be
approximately fixed by the rotational speed. Thus if the LP compressor pressure
ratio is decreased from A to B the value of mJT02/P02 will increase because
mJTo2 = mJTol x POl X /T02
P02 POl P02 V TOl
The increase in mJT02/P02 results in an increase in HP compressor non-
dimensional speed, pressure ratio and T04/To2 as shown in Fig. 9.9(b). With the
value of T02 approximately constant for a fixed rotational speed, it can be seen
that T04, and hence T04/Tob must increase as we move the LP operating point
away from surge. It is also evident that a decrease in LP pressure ratio due to
nozzle variation will be compensated by an increase in HP pressure ratio.
It should be noted that variable-area power-turbine stators may also be used
with a free turbine following a twin-spool gas generator. This is done in the Ml
tanle application and also the intercooled regenerative (ICR) marine engine. The
area changes would be less than could be achieved with a variable-area propelling
nozzle, but the same results apply.
Presentation of peiformance
In Chapter 8 it was shown that the performance of a single-spool engine could be
presented as a function of N/ JT01' The performance of a twin-spool fixed
geometry turbojet can be presented in terms of either the LP or HP rotor speeds
because of the fixed relation between the speeds. For an engine with vaIiable
nozzle the performance would have to be presented in terms of any two of NL , N H
andA7. An example of a 'carpet' plot showing the vaIiation of thrust with NL and
NH for a vaIiable nozzle turbojet is shown in Fig. 9.10; other quantities such as
fuel flow and SFC could be presented in a similar manner.
For fixed geometry engines the vaIiation of turbine inlet temperature and fuel
flow could be plotted versus either NL or N H, but because of the rapid variation of
both with N H the latter may be more suitable as abscissa. It should be noted that
MATCHING PROCEDURES FOR TURBOFAN ENGINES
383
FIG.9.HI
although the HP rotor speed will always be higher than the LP rotor speed and the
HP turbine blades are at the higher temperature, they may not be the critical
components because the LP turbine blades are usually substantially longer.
The mass flow through a twin-spool engine is prirnaIily determined by the LP
compressor speed, and consequently the performance of a twin-spool shaft power
unit with a free power turbine is conveniently presented in a manner similar to
that of Fig. 8.10 with lines of constant LP compressor speed replacing lines of
constant gas-generator speed.
9.4 Matching procedures for turbofan engines
The approach descIibed in section 9.2 is also applicable to turbofans, but in this
case we must take into account the division of flow between the bypass duct and
the gas generator, which will vary with off-design operating conditions. Only the
simplest case, that of the twin-spool turbofan with separate exhausts shown in
Fig. 3.15, will be considered here. The additional information required is the
bypass or 'cold' nozzle characteIistic, as indicated in Fig. 9.11; the turbine
characteristics and the hot stream nozzle characteristics will be similar to those
of the twin-spool turbojet.
Let us consider a turbofan with both nozzles fixed. The running line
corresponding to LP turbine choking can be established on the HP characteristic
in exactly the same manner as for the twin-spool turbojet. The problem now is to
establish the proportion of the total flow (m) passing through the bypass duct (me)
and the gas generator (mh)' Suffixes h and c refer to 'hot' and 'cold' streams
respectively. One possible procedure is as follows.
(a) Select ambient and flight conditions, giving values of POI and T01 '
(b) Select a value of Nd JTOl and guess any point on this line. The value of m
enteIing the fan is then known, and the value of mJTOZ/P02 at exit from the
fan can be calculated.
(c) From the known value of Poz, the pressure ratio across the bypass nozzle,
P02/Pa, can be calculated and the value of meJT02/P02 can be found from
the bypass nozzle characteIistic. The value of me can also be found, and
mh=m - me'
384
PREDICTION OF PERFORMANCE-FURTHER TOPICS
P02
POl
LP compressor (fan)
//
<\\
P03
P02
FIG. 9.11 Chal"!lcteristics required for turbofan matching
Bypass nozzle
HP compressor
(d) The gas-generator non-dimensional flow can then be found from
mhJTo2
P02
mJT02 mc JTo2
-------
Poz P02
U) Knowing the gas-generator non-dimensional flow we can enter the HP
characteristic, and establish the overall pressure ratio and turbine inlet
temperature as for the twin-spool turbojet [see steps (e) to (h) of section
9.2]. The HP turbine pressure ratio and temperature drop are known as a
result of flow compatibility between the two turbines, so that the pressure
and temperature at entry to the LP turbine, Pos and T05, can be determined as
before.
(g) The LP turbine temperature drop is given by
(h) The LP turbine pressure ratio is then found from T05, ATo56 and I1tL. We now
know P06 and T06'
(i) Knowing mh, P06, T06 and P06/Pa the hot stream nozzle area can be
calculated; in general this will not agree with the specified value and it is
necessary to return to step (b), select a new point on the LP characteristic,
and repeat the process until agreement is reached.
The close similarity between the twin-spool and turbofan matching procedures
can be seen. The method described may readily be extended to deal with three-
spool engines, and the configuration shown in Fig. 3.21(c) is equivalent to a
simple turbofan with a twin-spool gas generator. The use of mixed exhausts
TRANSIENT BEHAVIOUR OF GAS TURBINES 385
presents a further complication because it is then necessary to include equations
satisfying conservations of energy and momentum for the mixing process.
Neither of these topics will be pursued further here.
9,5 Transient behavionr OIf gas turbines
In certain applications the transient response of gas turbines following a demand
for a change in output can be critical; in other applications good response may
merely be desirable. Aircraft engines are an obvious example of an application
where the transient behaviour is critical; the prime requirement for civil aircraft is
for a rapid thrust response to cope with a baulked landing when the aircraft is
close to touchdown but is forced to overshoot the runway. Lifting engines for
VTOL aircraft provide another example, where an engine failure on a multi-
engined installation may result in serious unbalanced forces which have to be
adjusted rapidly. Rapid response is also required from gas turbines used for
emergency generation of electricity which must often deliver their rated output
within two minutes of the start signal. The response rate of the gas turbine itself
from idle to maximum power must be less than about ten seconds to allow time
for the starting and synchronization sequences. The transient response of a
vehicle gas turbine to changes in throttle setting is not critical. Although the
relatively slow response of a gas turbine has ft-equently been regarded as a major
disadvantage, for heavy vehicles where it is likely to be competitive the vehicle
acceleration will be largely independent of the engine acceleration.
In the early days of gas turbines little attention was paid to the prediction of
transient behaviour, and the response rate of an engine was established
empirically during development testing; this required extensive test bed running,
and frequently engines were damaged. Now, however, the transient behaviour is
predicted from a knowledge of the off-design performance, and this can be done
during the design phase using estimated component characte:ristics. An in-depth
understanding of the dynamic behaviour at the design stage is essential for the
design and development of control systems. Requirements for higher perfor-
mance have led to multi-spool rotor systems, variable geometry in the compressor
and blow-off valves, which complicate prediction of both steady-state and
transient performance.
The acceleration of gas turbines is obviously dependent on such factors as the
polar moment of inertia of the rotor system and the maximum temperature which
the turbine blades can withstand for short periods. Usually the limiting factor on
acceleration is the proximity of the surge line to the equilibrium running line, and
this is particularly critical at the start of an acceleration from low powers. The
configuration of the gas turbine will have a major effect on its transient behaviour;
for example, the behaviour of single-shaft and free turbine engines delivering
shaft power will be quite different, and a twin-spool engine will respond very
differently from a single-spool engine. Only a brief introduction to the problem of
engine transients will be given in this book, principally via the single-spool
386
PREDICTION OF PERFORMANCE-FURTHER TOPICS
engine with· and. without free power turbine,· and the interested reader must then
turn to the specialized literature.
Prediction of transient performance
It will by now be clear that all off-design equilibrium running calculations are
based on satisfying the requirements for compatibility of flow and work between
the components. During transient operation a gas turbine can be considered to
satisfy compatibility of flowt but not of work, and the excess or deficiency of
power applied to a rotor can be used to calculate its acceleration or deceleration.
The problem then becomes one of calculating increments of net torque associated
with increments of fuel flow and integrating to find the change of rotor speed.
The acceleration of the compressor rotor and the excess torque AG are related
by Newton's Second Law of Motion, namely
AG=Jw
where J is the polar moment of inertia of the rotor and w is its angular acceler-
ation. The excess torque AG is given by
AG = Gt - Gc (free turbine engine)
or
AG = Gt - (Gc + G/) (single-shaft engine)
where suffixes t, c and I refer to turbine, compressor and load respectively. It
should be noted that the net torque is the difference between two quantities of
similar magnitude, and a small change in either may result in a much larger
change in the torque available for acceleration. The problem now is that of
obtaining the torques during transient operation. The turbine torque may be
greater or less than the compressor torque, corresponding to acceleration or
deceleration of the rotor.
To obtain the torques we may use the methods for off-design performance
described in Chapter 8. If we consider any point on the compressor characteristic,
which will not in general be an equilibrium running point, we can satisfy flow
compatibility between compressor and turbine by using the identity
m.jT03 = m.jTol x POI x P02 x IT03
P03 POI POl P03 V TOl
Consider first the turbine non-dimensional flow m.jT03/P03: it can normally be
assumed to be independent of speed and a function of the pressure ratio alone.
t During rapid transients, pressures cannot change instantaneously because of the finite volume
between the components, and the assumption of flow compatibility at all times is not exactly true
although it is a good approximation. Methods for dealing with non-instantaneous pressure changes are
discussed in Ref. (10), and are subject to further research; the effects are confined to a very short
period immediately following the start of a transient.
i
TRANSIENT BEHAVIOUR OF GAS TURBINES 387
The value of the gas generator turbine pressure ratio P03/P04 corresponding to the
selected value of P02/POI on the compressor characteristic can be easily obtained
for both single-shaft and free turbine engines; for a single-shaft engine it is fixed
by the compressor pressure ratio and the combustion pressure loss, and for a free
turbine engine it can be found from Fig. 8.9. Thus it can be seen that m.jT03 /P03
is a function of P02/POI for both types of engine. Rearranging the equation for
flow compatibility given above
)
T03 = m.jT03 xP02 xP03/m.jTol
TOI P03 POI P02 POI
With constant percentage combustion pressure loss P03/P02 will be constant, and
it can be seen that the value of T03/TOI satisfying flow compatibility can be
obtained for any point on the compressor characteristic without reference to work
compatibility. The fuel flow required to produce this temperature ratio can be
calculated because the air flow, compressor delivery temperature and combustion
temperature rise are all Imown. If the process is repeated for a series of points,
lines of constant T03/TOI can be drawn on the compressor characteristic. These
lines will have a similar fonn to the constant temperature lines shown for a free
turbine engine in Fig. 8.7, but it must be emphasized they are not the same lines,
because they represent non-equilibrium running conditions. (Constant tempera-
ture lines of this kind drawn on the compressor characteristic for a single-shaft
engine would, however, represent a series of equilibrium running points for an
engine driving a load which can be set independently of speed, e.g. a hydraulic
dynamometer.)
With the turbine operating point and turbine inlet temperature known the
power developed by the turbine can be calculated. Thus
turbine power = l1mmcpgATo34
where
AT034 T03 ~
AT034 = --X - x 101
T03 TOI
The compressor power is given by IIlcpaATol2 and for an engine with a free
turbine the net torque on the compressor rotor is given by
AG = (1JmlllcpgAT034 - mCpaAToI2)/2rr.N
For a single-shaft engine the load torque would also have to be included.
Once the net torque has been obtained, the angular acceleration of the
compressor rotor can be calculated; it may be assumed that this will be constant
for a small interval of time and the resulting change in speed can be found. The
process can then be repeated many times to provide a transient running line
starting from some convenient equilibrium running point. The shape of this
transient running line will be determined either by the choice of a limiting T03
(which might be 50 K higher than the design point value) or the location of the
388
PREDICTION OF PERFORMANCE-FURTHER TOPICS
surge line. The fuel flow required for operation along the desired running line can
be calculated and the provision· of this fuel during the transient is the
responsibility of the designer of the fuel control system. Typical acceleration
and deceleration trajectories on the compressor characteristic for a free turbine
engine or a single-shaft engine with a propeller load are. shown in Fig. 9.12. With
an acceleration the initial mqvement towards surge shown in Fig. 9.12 is due to
the rise in temperature following an increase in fuel flow, which takes place
before the rotor has had time to increase its speed and provide an increase in mass
flow. Too large an initial increase in fuel flow will cause the compressor to surge,
resulting in very high temperatures which could destroy the turbine. The
acceleration procedure for a single-shaft engine driving an electric generator
would be to bring the rotor system up to full speed before applying the load and
there should be no problem with regard to surge.
During decelerations the operating point is moving away from surge as shown
and the turbine inlet temperature is decreasing; the only problem that may arise is
'flame-out' of the combustion chamber because of very weak mixtures. This can
be overcome by scheduling the deceleration fuel flow as a function of rotor speed
to prevent too rapid reduction of fuel flow.
We have seen in section 8.6 that if the equilibrium running line intersects the
surge line, either blow-off or variable geometry can be used to lower the running
line, but it should be noted that neither would have much effect on the transient
running line. Rates of acceleration will be slowed down by blow-off, because of
the reduction in mass flow through the turbine, but can be improved by a facility
for increasing the area of the propelling nozzle (or power turbine stators).
We pointed out under the heading 'Variable-area power-turbine stators', that
gas turbines for vehicular applications may well incorporate a variable-geometry
P02
P01
FIG. 9.12 Transient trajectories !Ill compressor characteristic
TRANSIEl'-.'T BEHAVIOUR OF GAS TURBINES 389
power turbine to improve the part-load economy, achieving this by decreasing the
stator area as power is reduced. Conversely, on starting, the effect of increasing
the stator area is to decrease the pressure ratio across the power turbine and
increase that across the gas generator turbine. The resulting transient increase in
gas generator turbine torque provides a substantially greater increase in net torque
available for acceleration.
Single-shaft versus free turbine engines
The choice of a single-shaft or free turbine configuration will normally be made
on the basis of some primary requirement such as low cost or good low speed
torque characteristics, but the transient behaviour might also be a deciding factor.
With a free turbine engine, power reduction must be obtained by reducing the
gas generator speed, because of the fixed running lLrle determined by the flow
characteristic of the power turbine. Thus to restore power it is necessary to
accelerate the gas generator up to its maximum speed. With a single-shaft engine,
however, the load may be varied at constant speed, e.g. as with an electric
generator or a variable pitch propeller. This mea.llS that there is no need to
accelerate the gas generator and power may be restored very quickly by
increasing the fuel flow. Such a feature can be very useful for turboprops where
the power can be altered by changing the pitch of the propeller. The single-shaft
engine has the obvious additional advantage that in the event of load being shed
the compressOI"acts as a very efficient brake and for this reason regulation of
output speed is easier than with a free turbine engine.
The starting power requirements for large gas turbines must be carefully
considered; this is partiCUlarly important for emergency generating plant where
one of the prime requirements is to achieve maximum output quic1dy. With a
single-shaft unit the entire rotating assembly must be brought up to speed, and
this may require the use of a steam turbine or diesel engine with a power of the
order of 3 MW for a 100 MW unit. The free turbine engine is in a much more
advantageous position, because neither the power turbine nor the load are driven
by the starter. A twin-spool gas generator is even more favourable: only the HP
rotor need be motored and the starter power will be less than 100 kWeven for a
large unit.
At low powers, with a generator as load, the part-load fuel consumption of a
free turbine engine will be superior to that of a single-shaft engine. Figs 8.7 and
8.5 show the equilibrium running lines in each case. The running line of the free
turbine engine follows the locus of maximum compressor efficiency more closely
than that of the single-shaft engine, and in general T03 does not fall off so rapidly.
Single-shaft units are normally used for electric power generation. An
exception is where an aero-derivative engine is specified, as on off-shore oil rigs
where compactness is essential. The free turbine units can provide power at either
50 or 60 Hz for the European or North American markets, by minor changes in
the speed of the power turbine with very small changes in efficiency; their power,
however, is restricted to about 50 MW because of the relatively small flow
390 .1'REDICTION OF l'ERFORMANCE-FURTHER TOPICS
through the gas generator. Single-shaft units. cali be· designed to p r o ~ d ~ up to
250 Mw, and manufacturers offer different machines for. the 50 or 60 Hz
markets; machines of around 50-75 MW can be designed with output gearboxes
suitable for either 50 or 60 Hz.
Twin-spool transient running lines
A full discussion of twin-spool transient behaviour cannot be covered in a book
of this nature, but some important differences from single-spool behaviour must
be mentioned. The transient behaviour of the HP rotor is similar to that of a
simple jet engine and the trajectories on the HP characteristic will be similar to
those shown in Fig. 9.12. Consideration of the lines of constant T 04/ TOI on the LP
characteristic would suggest that trajectories for accelerations and decelerations
would be as shown in Fig. 9.13, with the likelihood ofLP surge following a rapid
deceleration. The lines of T04/TOh however, apply only to steady state operation
and are irrelevant to transient behaviour. Detailed calculations and experimental
investigations, described in Refs (8) and (9), have shown that the LP transient
runuing lines follow the equilibrium runuing line very closely, both for acceler-
ations and decelerations.
We have noted that the acceleration of a single-spool turbojet engine can be
improved by increasing the nozzle area. Although care must be taken when
extending this idea to twin-spool turbojets, considerable advantage can result
from judicious manipulation of the nozzle area during transients. Increasing the
nozzle area during an acceleration will improve the LP rotor acceleration, because
of the increase in LP turbine torque, but it should be realized that the LP runuing
line will be moved towards surge. The mass flow is increased more rapidly
because of the improved LP rotor acceleration, and this in turn permits the fuel
P02
POl
AntiCipated
acceleration
AntiCipated
deceleration
I
T04IT01
increasing
m{'fo;lpOl
FIG. 9.13 Transient trajectories on LP compressor of twin-spool nnit
i
TRANSIENT BEHAVIOUR OF GAS TURBINES 391
flow to be increased more rapidly. In practice it is found that the thrust response is
primarily detennined· by the rate of increase of fuel flow, so that the thrust
response is improved at the expense of LP surge margin. Detailed calculations
have shown that the HP surge margin is improved, however, and it is possible to
trade surge margins between the two compressors to get the best response.
When operating at high forward speeds the nozzle will usually be open and the
LP operating point will be close to surge. Emergency decelerations from this
condition could result in LP surge, which may lead to a flame-out. A study of
emergency decelerations in Refs (8) and (9) showed that the best method of
avoiding surge was to close the nozzle fully before reducing the fuel flow; this
moved the LP operating point down to the runuing line associated with minimum
nozzle area and also reduced the LP rotor speed. Immediate reduction of fuel
flow, on the other hand, resulted in a runuing line very close to surge, as did
simultaneous reduction of fuel flow and closing of the nozzle.
Simulation of transient response
The development of a suitable control system requires a deep understanding of
the transient behaviour of the gas turbine to be controlled, particularly for new
types of engine where no previous experience exists. If a mathematical model or
'simulation' describing the engine dynamics is constructed and stored in a suit-
able computer, it can provide designers of gas turbines with an extremely versatile
tool with whicq to investigate a wide variety of problems. A prime requirement
for the simulation is that it should be capable of covering the entire runuing range
of the engine and also of incorporating such modifications as blow-off and vari-
able geometry as necessary. This can best be achieved by basing the simulation
on the methods of off-design performance calculation presented in Chapters 8
and 9; the use of the normal component characteristics makes the simulation
flexible in use and easy to understand, and has the further advantage that the
component characteristics can be modified in the light of rig testing as the de-
velopment programme proceeds. Typical problems which can readily be investi-
gated include optimization of acceleration fuel schedules, operation of variable
geometry devices and overspeed protection. Incorrectly chosen operating pro-
cedures, particularly under emergency conditions, may be hazardous to both
engine and operator.
The information flow chart required for a mathematical model of a simple
turbojet engine is shown in Fig. 9.14. It can be seen that the pressure ratios across
each component are governed by flow compatibility, the pressure ratios in turn
determine the temperature ratios, and the turbine temperature drop (proportional
to the power developed by the turbine) is determined by the pressure ratio and
turbine inlet temperature (controlled by the fuel flow).
Successful simulations have been carried out using analogne, digital or hybrid
computers and appropriate techuiques are described in Refs (8) to (11). Early
digital computers did not have the computing speed essential for the continuous
integration of the net torque, and real time simulation could only be achieved
392 PREDICTION OF PERFOR.MA-l\!CE-FURTHER TOPICS
N N
fI T012
I flTo3• t
I Compressor L
"I
Rotor
I I
Turbine
e-
m
I
dynamics m
~
~ Combustion I
T03
chamber I
WORK COMPATIBILITY
----- ---------- ---------------------------------------- - ------ --
FLOW COMPATIBILITY
Po,
P03
-
POl
rrrff;
f!i
P04
--
r
Po, To,
I
I i
I
rnff;;;
I I r
P04
I
--
-
Po.
t
Po,
L
P03 P03 m{fr;.
~
- -
I
P04
To.
Po, Po,
P 0 1 ~
Nozzle
I
~
Pa
Pm
FIG. 9.14 Information fiow cllart for single-spool turbojet
using analogue computers; the hybrid computer was an inteJTIlediate solution
which combined the speed of the analogue with the data processing capability of
the digital. Now, however, digital computer capabilities are so fast that they are
almost universally used for simulation. The judicious use of simulations can give
an invaluable insight into transient response and control problems without
endangering an engine, resulting in significant savings in test bed development
and hence cost.
9.6 Principles of control systems
Figure 9.15 shows the main features of a gas turbine control system. The funda-
mental requirement is to maintain the safety of the engine, regardless of how the
PRINCIPLES OF CONTROL SYSTEMS 393
Throttle --1l
setting V>L ____ --.J
FIG. 9.15 Control system cllmponents
operator moves the throttle lever or how the inlet conditions ( e.g. altitude) are
changing. The control system must ensure that the critical operating limits of
rotational speed and turbine inlet temperature are never exceeded, and that com-
pressor surge is avoided. The sensing of rotational speed presents no problems
and a variety of frequency measuring devices are available. Turbine temperature,
on the other hand, is very difficult to measure. It is not nOJTIlally practicable to
locate temperatwe probes at inlet to the turbine, where the temperature is very
high, and the temperature is measured at some downstream location, typically at
inlet to the power turbine on shaft power units and in the jet pipe on jet engines.
Thus the temperatures used to protect the turbines are indirect measures of the
critical temperatures. The temperatures are usually measured by theJTIlocouples
spaced around the annulus: typically six to eight probes are used. In the case of
turbines with highly cooled blades, it is the actual blade temperature which is
important and this may be sensed by radiation pyrometry in advanced engines.
The methods described in Chapters 8 and 9 can be used to provide the control
system designer with infoJTIlation about the fuel flow required for steady-state
operation over the entire range of operating conditions. Analysis of transient
perfoJTIlance can predict the maximum fuel flow which can be used for
acceleration without encountering surge or exceeding temperature limits. A
typical schedule for fuel flow for a simple jet engine is shown in Fig. 9.16.
Mathematical models for simulating the transient behaviour are an essential tool
for optimization of fuel schedules, which must be experimentally verified during
the engine development programme.
The perfoJTIlance calculations described in Chapters 8 and 9 showed that the
variations of all the key parameters are wholly determined by the matching of
the compressor, turbine and nozzle characteristics. It is important to realize that if
the engine has fixed geometry, the steady-state perfoJTIlance cannot be altered in
any way by the control system. If, however, devices such as variable IGVs,
variable compressor stators or variable nozzles (turbine or propelling) are
394 PREDICTION OF PERFORMANCE-FURTHER TOPICS
Compressor speed
FIG. 9.16 1)'picai fuel schedule
included, the operation of these can be integrated with the control system to
modify the performance. The level of sophistication required of the control
system is strongly dependent on the complexity of the engine.
Figure 9.15 shows that the control system must incorporate both a computing
section and a fuel metering section. For many years both of these functions were
met by hydromechanical systems, in which fuel passing through the unit provided
the necessary hydraulic actuation of a variety of pistons, bellows and levers which
metered the required fuel to the combustion system. In recent years much
development of digital control systems has taken place; the increasing
computational capacity and rapidly decreasing cost of small digital computers
has made them quite feasible for use in control systems. It should be realized,
however, that a fuel metering system is still required; the function of the digital
computer is limited to computation and the control system must open and close a
tap to control fuel flow. Digital control systems with a comprehensive capability
for controlling all modes of engine operation, !mown as Full Authority Digital
Engine Controls (FADECs) are becoming widely used on large aero-engines;
although the cost of the actual computer is small, the need for certification of the
software leads to very high software development costs. For this reason, FADECs
are seldom used on small engines of lower cost.
Digital control systems will increasingly be used for data acquisition, using the
data measured by the sensors, and unusual changes may be used in diagnostic
systems for examining the mechanical condition of the engine. Engine Health
Monitoring (EHM) systems use themlOdynamic measurements, vibration
analysis and chemical analysis of the lubricating oil to assess engine health.
Ref. (12) discusses the use of thermodynamic models in diagnostic systems.
EHM systems have demonstrated significant reductions in maintenance costs and
PRINCIPLES OF CONTROL SYSTEMS
Max. turbine
temperature
Ambient temperature
FIG. 9.17 Power and temperature limiting
395
savings due to early detection of engine deterioration before severe damage
occurs.
To close this section, a very simple example of the use of the control system
for maintaining an engine within operating limits will be considered. Let us
consider a turboshaft engine with a free power turbine, used in a helicopter; the
gearbox will be very highly stressed to keep weight to a minimum and the
maximum power permissible will be strictly limited. The worked example in
Chapter 8 showed the significant increase in power on a cold day, and conversely
the power will decrease on a hot day. The variation in power would be as shown
by the full line in Fig. 9.17; the maximum power limit is shown dotted, and at
high ambient temperatures it will also be necessary to limit turbine temperature
causing an even more rapid decline in power, as previously discussed with respect
to jet engines. One method of achieving these limits would be by controlling gas
generator speed as shown; the gas generator could only be run at maximum speed
where neither limit was exceeded and at both low and high temperatures its value
would be reduced.
Control system design is a specialized field which is changing rapidly and the
interested reader must turn to the current literature. The control designer, in turn,
must have a full understanding of the system to be controlled which necessitates
an appreciation of gas turbine performance.
Appendix A
Some notes on gas dynamics
Owing to the increasing tendency towards specialization even at first degree and diploma
level, it may be that some readers will not have been exposed to a course in gas dynamics.
It is hoped that this Appendix will provide them with an adequate surrnnary of those
aspects which are relevant to gas turbine theory, and that it will serve others as useful
revision material.
A.l Compressibility effeds (qualitative treatment)
It is well known that when the relative velocity between a gas and a solid body reaches a
certain value, the flow behaves in a quite different manner to that expected from a study of
hydrodynamics. The effects produced, which manifest themselves as additional loss of
stagnation pressure in the stream, do not arise when the fluid is a liquid. This suggests that
the phenomena are due to the change in density which accompanies a change in pressure
of a gas. The idea is strengthened by the fact that the phenomena only occur at high speeds
when the pressure changes set up by the relative motion, and therefore the density changes,
become considerable. In consequence, the phenomena here described are Imown as
compressibility effects.
When, in a mass of gas at rest, a small disturbance results in a slight local rise of
pressure, it can be shown that a pressure wave is propagated throughout the gas with a
velocity which depends upon the pressure and density ofthe gas. This velocity is the speed
of sound in the gas, or sonic velocity a, given by
a = J(l'P I p) or J(yRT)
where p, p and T are the local pressure, density and temperature of the gas.
In all processes related to the propagation of pressure waves, the changes take place so
rapidly that there is no time for any heat transfer between adjacent layers of fluid; the
processes are therefore adiabatic. Also, when the amplitude of the pressure wave is small
and there is no material alteration in the pressure and temperature of the gas, as is true of an
ordinary sound wave, there is no increase of entropy. The propagation of a sound wave is
therefore not only adiabatic but isentropic.
Now consider what happens when a similar disturbance occurs in a gas flowing in one
direction with a velocity C. The velocity of propagation of the pressure wave relative to the
gas will still be equal to the speed of sound, a. Relative to a fixed point, however, say the
walls of the passage confining the gas, the speed of propagation will be (a + C)
COMPRESSIBILITY EFFECTS (QUALITATIVE TREATMENT) 397
Ct Ct
Disturbance
(a)
FIG. Al Sound waves in II movillg stream
downstream and (a - C) upstream. It follows that if the velocity of the gas is greater than
the sonic velocity, i.e. supersonic, there can be no propagation of the pressure wave
upstream at all. This is the usual physical explanation given for the existence of a critical
condition in nozzle flow. When once the pressure drop across a nozzle is great enough to
cause the gas velocity to reach the local sonic value, no further decrease in outlet pressure
will be propagated upstream and no fiJrther increase in mass flow will be obtained.
Figure Al illustrates the effects just described, and a useful picture to have in mind is
that of the ever-widening circles of ripples formed by a stone thrown into a pond. When a
disturbance, such as an intermittent electric spark, is placed in a gas stream moving with
subsonic velocity (C < a), the radius of a spherical pressure wave after time t will be at,
while the centre of this wave will have moved downstream a distance Ct. All waves emitted
subsequently will lie within the spherical wave front of this wave, as shown in Fig. Al(a).
On the other hand, when C> a as in Fig. Al (b), the spherical wave fronts will move
downstream at a greater rate than the radii of the waves increase. All the spherical waves
will therefore lie within a cone having its apex at the point of the disturbance.
The effect of a small solid particle placed in a stream of gas is that of a disturbance
emitting pressure waves continuously, so that the spherical wave fronts of Fig. Al(b)
appear as a single conical wave front of semi-angle fl, where fl is given by
. -1 at . -1 a
fl=sm Et=sm C
The ratio Cia frequently arises in the mathematical treatment of compressible flow, and it is
the well known Mach number M. The angle fl is commonly called the Mach angle, and the
conical wave front a Mach wave. Thus
Mach angle fl = sin-
1
(11M)
So far we have been considering pressure impulses of very small amplitude, such that there
is no permanent change in the pressure and temperature of the gas as the wave moves
through it, and consequently such that there is no change in entropy. In many practical
cases of gas flow relative to a solid body these conditions are not fulfilled; there is a
marked pressure and temperature difference across the wave, and there is an. increase in
entropy indicating an irreversible dissipation of kinetic energy which is manifested by a
loss of stagnation pressure. The wave fi·ont represents a discontinuity in the flow, and as the
change of pressure is to all intents and purposes instantaneous, the wave is termed a shock
398
APPENDlX A SOME NOTES ON GAS DYNAMICS
wave. The Mach wave previously discussed can be regarded as the weakest possible fonn
of shock wave. The shock wave fonned by a projectile travelling at supersonic speed, for
example, is analogous to the bow wave set up by a ship: the water, imable to escape rapidly
enough past the sides of the ship piles up to fonna wave whIch travels along
with the ship. In the case ofthe projectile, the air outside the regIOn enclosed by the comcal
wave front does not receive a signal warning it of the approach of the sohd object creatmg
the disturbance. and hence the fonnation of the shock wave at the nose of the projectile. It
must be stressed that it is the motion which is important; it does not matter whether
the body or the fluid or both are moving.
We have said that there is a pressure difference across a shock wave. We must now ask
whether it is a pressure rise or pressure drop in the direction of gas flow relative to the
body, that is, through the shock wave. Both experiment and theoD; indicate a shock
wave can only be fonned when a supersonic flow is decelerated. The velOCItIes m the
divergent part of a convergent-divergent nozzle are supersonic, but if the nozzle is
operating at the pressure ratio for which it is designed no shock waves WIll. be fonned
because the flow is accelerating under the influence of the pressure drop. ConSIder, on the
other hand, what happens when the outlet pressure is appreciably above the value which
would give just the light amount of expansion to suit the outlet area of the n?zzle. Under
these conditions the nozzle over-expands the gas so that before the gas can dIscharge mto
the surroundings some fe-compression and deceleration of the gas must occur. This re-
compression can only be brought about by a shock wave in the divergent part of the nozzle,
because a convergent duct is necessary for isentropic diffusion of a supersomc strean:.
Figure A1 shows typical pressure distributions along a nozzle when the outlet pressure IS
above the design value. As the outlet pressure is reduced, the plane nonnal shock wave
moves towards the exit, and further reduction towards the design outlet pressure is
accompanied by a sudden change to a complex system of oblique shock waves
downstream of the exit.
To sum up so far:
(a) A shock wave only occurs at supersonic speeds in a decelerating flow.
(b) There is a rise in static pressure and temperature through the wave in the direction of
relative motion of the gas, i.e. a shock wave is a compression wave.
(c) There is a drop in stagnation pressure through the wave, some of the kinetic energy
being dissipated with a consequent increase in entropy.
Throat
Distance along nozzle
FIG. A2 Sbock Wllve in lIozzle lII.ow
COMPRESSIBILITY EFFECTS (QUALITATIVE TREATMENT)
Region of high velocity
and low pressure
/
FIG. A3 Streamline How over aeIrofoil!
399
Finally, we may now turn to consider the effect of shock waves on the flow of air over
an aerofoil. This will be directly applicable to problems associated with the design of axial
compressor and turbine blading which are simply rows of aerofoils. First, consider the
changes of pressure and velocity which occur around the aero foil of Fig. A3, assuming that
an air stream is flowing past it with subsonic velocity. As indicated by the convergence and
divergence of the streamlines, the air moving over the curved top surface must frrst
accelerate and then decelerate, the outer streamlines effectively acting as the wall of a duct.
This produces a region of increased velocity and reduced static pressure on the top surface.
It is this region of low pressure or suction on the top surface which produces most of the
lift of an aero foil. The amount by which the velocity is increased along the top surface will
depend upon the shape and camber of the aerofoil; it is quite possible for the local velocity,
somewhere near the point of maximum camber, to be 1·5 times the velocity of the main
stream. Furthennore, the acceleration will be accompanied by a fall in static temperature,
so that the local sonic velocity .j (yRT) will be reduced below that of the free stream. Both
effects contribute to an increase in Mach number, and there may well be a region on the top
surface where the flow is supersonic when the Mach number of the main stream is only
about O· 7. But we have seen that when a deceleration occurs at supersonic speeds in a
diverging passage a shock wave will be formed. The effect ofthis is illustrated in Fig. A4.
If under these conditions a pitot tube is traversed across the flow in front of and behind the
aero foil, the loss of stagnation pressure will be found to vary in tile manner shown. The
loss of head due to the shock wave itself is represented by the part of the curveAB; the loss
of head represented by the peak of the curve may be very much greater than that across the
shock wave, however, and requires some explanation.
When a fluid flows along a surface there is a thin layer of fluid, known as a boundary
layer, in which there is a steep velocity gradient due to viscous friction, the velocity
dropping to zero at the solid surface. Now the pressure gradient across the shock wave
opposes the direction of flow and consequently, in the boundary layer where the kinetic
energy is small, the shock wave may arrest the motion altogether. The boundary layer will
thicken just in front of the shock wave, and may break away from the surface at the rear of
it. If this breakaway of the boundary layer occurs, it will result in the initiation of a vortex
trail involving considerable dissipation of energy. This, then, is the reason for the large loss
of stagnation pressure in the wake of the aero foil, and the reason why the Mach number of
the main stream should be kept below the value likely to cause the fonnation of shock
waves with this shape of aero foil.
FIG. A4 Shock wave Oil aerofoil
400
APPENDIX A SOME NOTES ON GAS DYNAMICS
We may now tum to the mathematical analysis of compressible flow in a few simple,
classical, flow situations. Much of the algebra is too lengthy 'to be here, but by its
omission we hope to enable the reader to see the wood: for the trees he can tum to the
many excellent standard texts on gas dynamics, e.g. Refs (3) and (4).
A.2 Basic equations, for steady one-dimensional compressible
flow of a perfect gas in a duct
A flow can be regarded as one-dimensional if
(a) changes in flow area and curvature of the axis are gradual,
(b) all properties are uniform across planes normal to the axis,
(c) any heat transfer per unit mass flow (dQ), across surface area of duct (dS), changes
the properties uniformly over the cross-section,
(d) the effect of friction can be represented by a shear stress r at the wall.
The flow is steady if there is no change in the mass flowing per unit time at successive
planes along the duct, and if the properties of the gas at any plane do not change with time.
Firstly, because we are dealing with a perfect gas we have the equation of state
P dpdpdT
-=RTor-=-+- (1)
p P P T
Secondly, application of the conservation laws yields the following equations in integral
and differential form (see Fig. A5):
Conservation of mass (continuity equation)
dp dC dA
m=PlAlCl=P2AzCzOf-P+C+A=O (2)
Conservation of momentum
- PlC?Al ) -I- (P2A2 - PIAl) - !(Pz + Pl)(Az - Al )+ ,S = O} (3)
pAC de +A dp +dS = 0
Conservation of energy
Q = Cp (T2 - Tl ) - cD = Cp(T02 - TOl ) 1 (4)
dQ = cp dT + C dC = cp dTo
Thirdly, the Second Law of Thermodynamics must be satisfied, i.e.
for adiabatic processes I'1s ;" 0
Heat transfer per
unit mass flow Q
\ I
dQ
:\ I
(5)
p+dp
BASIC EQUATIONS FOR STEADY ONE-DIMENSIONAL COMPRESSIBLE FLOW 401
For a perfect gas with constant specific heats we have the specific entropy s = f(p, T) given
by
T2 P2
I'1s=cln--Rln-
p Tl PI
and for the special case of a reversible adiabatic (isentropic) process we can put I'1s = 0 to
gIve
P P
Tyl(y-l) = constant or pi = constant (6)
no additional physical principle is introduced, the algebra is often sim-
are (a) expressed in terms of Mach number M, or (b) taken into account
by making of the concepts of stagnation pressure Po, temperature To and
denSIty Po, By defimtlOn M = Cia. For any fluid, the local sonic velocity a is given by
.J(dp/dp)" while for a perfect gas it can be expressed variously as
a (or C/M) = .J(yp/p) = .J(yRT) = .J[(y -1)cpTJ (7)
By (see under heading 'Stagnation properties' in section 2.2), the stagnation
propertIes m terms of M and the static values become
= 1+ = [1 + Y - 1M2]
T 2cp T 2
-= - = 1+ __ M2
Po (To)YI(Y-l) [ y - 1 ]YI(),-l)
P T 2
(8)
= (;j lly = [I + Y ; 1M2 rr-
1
)
We may obtain an important flow equation by combining the differential forms of
equations (1) to (4) and relations (7), 11ms combining (1) and (2) to eliminate dplp we
have
dC + dA + dp _ dT = 0
CAp T
From (3) and (7) we get
dp M2 dC J'rS
-=-y ---
P C pAa2
From (4) and (7) we get
dT _ dQ 2dC
----(y-l)M -
T CpT C
Combining these three equations we have finally
(M2 _ 1) dC = dA = _ dQ _ yrdS
C A CpT pAa2
(9)
Equation (9) expresses the effect on the velocity of (a) changes in flow area, (b) heat
transferred and (c) viscous friction, Considering each of these factors acting in isolation we
can draw the following qualitative conclusions.
When M < 1 (subsonic flow) the flow will accelerate (+ve dC) if (a) the duct converges
(-ve dA) or (b) heat is transferred to the gas (+ve dQ). Conversely, the flow will
decelerate if the duct diverges or is cooled. The effect of friction is always to accelerate
subsonic flow: the physical reason is that the transformation of directed kinetic energy into
402 APPENDIX Ii. SOME NOTES ON GAS DYNAMICS
internal energy is such that the decrease in density, consequent upon the rise in tem-
perature, predominates and the flow velocity increases to satisfy the continuity equation.
When M> 1 (supersonicflow) the flow will accelerate if (a) the duct diverges or (b) the
duct is cooled. Conversely, it will decelerate if the duct converges or is heated. The effect
of friction is always to decelerate supersonic flow.
When M = 1 we have the condition where the flow velocity C equals the local velocity
of sound a. Values of all quantities at this condition will be denoted by an asterisk, e.g. T*,
a*, A*,p*. Equation (9) shows t11atM may be unity when the duct has a throat (dA = 0); or
when sufficient heat has been added in a duct of constant flow area (supersonic flow
decelerates towards M = 1 and subsonic flow accelerates towards M = 1); or when a
constant area adiabatic duct is sufficiently long for friction to decelerate a supersonic flow
or accelerate a subsonic flow to the condition M = 1. Under all such conditions the duct is
said to be choked because the mass flow is the maximum which the duct can pass with the
given inlet conditions: further heat transfer or friction in a constant area duct merely causes
a reduction of mass flow in the duct.
Note that the presence of a throat does not necessarily imply M = 1 at the throat,
because the duct may be acting as a venturi with subsonic expansion followed by subsonic
diffusion. But we are here considering only a continuous expansion or compression along
the duct (i.e. dC is not changing sign during the process). For this reason also, we cannot
expect equation (9) to yield information about discontinuities such as shock waves which
will be considered later. Before doing so we will summarize the analysis which gives us
quantitative information about the behaviour of a gas undergoing a continnous expansion
or compression under the separate actions of changes in flow area (section A.3), heat
transfer (A.4), and friction (A.5). This will involve algebraic manipulation of the integrated
forms of equations (1)-(4) with appropriate tenns omitted, together with relations (5)-(8)
as appropriate. In what follows we shall often find it convenient to consider changes which
occur when a fluid flowing with an arbitrary Mach number M through area A and with
properties denoted by p, Po, T, To, etc., is brought to a reference state where M = I and the
relevant quantities are denoted by A*, p*, T*, etc.
A.3 Isentropic flow in a dud of varying area
For reversible adiabatic flow "[ and Q are zero, and the isentropic relations (6) are applic-
able. The relevant equations become:
pdpITI =P2/P2T2
m = PIAICI = P2A2C2
Making use of p/ p" = constant, the momentum and energy equations (3) and (4) can be
shown to be identical, namely
Cp (T2 - TI) +t(Ci - cD = 0 or cp (lo2 - ToI ) = 0
We shall consider state 1 to be 'reservoir' or stagnation conditions, and state 2 to be any
arbitrary state denoted by suffixless quantities. Then, combining the equations, and making
use of relations (6)-(8), we arrive at the well known equations for flow in nozzles (or
diffusers) :
_ [ 2y {
C- --RTo 1- -
l' -I Po
(10)
mJTo = [...'!L!.J!!...)2/Y{1_
Apo y - 1 R \Po \Po
(11)
FRICTIONLESS FLOW IN A CONSTANT AREA DUCT WITH HEAT TRANSFER
B

.E.
Po
Distance along axis
AlA' > 1 -<--j-.- AlA' > 1
lFlG. A6
o
403
mJTojApo is a maximum at the throat, where C=a (i.e. M= I) and the other quantities
are denoted by asterisks. Differentiating (11) with respect to p and equating to zero gives
p* =
Po Y + 1
(12)
= (y 1)
(13)
Substitution of (12) in (11) then yields
mJTo = [r
A*po R y+ 1
(14)
Dividing (14) by (11), and introducing plpo = f(M) from (8), we obtain the value of A at
any M in terms of the throat area A*, namely
= [2 +(1' -
A* M (y + 1)
(15)
Values of Polp, TolT, Pol P from relations (8) and A/A* from equation (IS) are tabu-
lated in the isentropic flow table of Ref. (1), with M as the argument. The table covers the
range M = 0·01-4, and is for dry air with l' = 1-403. Reference (2) is a much fuller set of
tables covering a range of values of l' and a wider range of M. Given the passage shape, i.e.
values of AIA* at successive distances along the duct, corresponding values of M and
thence pi Po can be read from the isentropic flow table. Two values of M and pi Po will be
obtained from the table for each value of A/A *, corresponding to the two real roots of
equation (15). The pressure distribution can then be plotted as in Fig. A6. Curve BCD
refers to the continuous expansion (or compression depending on the direction of flow),
while BCE represents the limiting pressure distribution above which the passage is acting
as a venturi.
A.4 Frictionless Bow in a constant area dud with heat transfer
We have seen that when A is constant and "[ is zero, heat transfer to the gas causes a
subsonic flow to accelerate towards M = I and a supersonic flow to decelerate towards
M = 1. This idealized flow is referred to as Rayleigh flow. One important effect, to which
reference was made in Chapter 6, is that heat transfer to a subsonic flow in a duct of
constant area must be accompanied by a fall in pressure. The pressure difference is
404
APPENDIX A SOME NOTES ON GAS DYNAMICS
necessary to provide the force required to accelerate the flow, i.e. to satisfy the momentum
equation.
The relevant equations for a Rayleigh flow from some arbitrary state M, T, etc., to the
state where M = 1 and quari.tities are denoted by asterisks, are as follows:
pjpT=p*/p*T*
pC = p*C*(= p*a*)
(p*C*2 - pC
2
) + (P* :.... p) = 0
Q = cp(T* - T) + !(C*2 - C
2
) = c/TIJ - To)
From the momentum equation, in conjunction with relations (7), we have
p 1+1'
p*=I+yM2 (16)
From this, together with the continuity equation and equation of state, we obtain
(17)
T* - 1 +yM2
From (16) and (17) and relations (8), by writing
= (E2.) ( (jJ:), and likewise for
Po P \P \Po To
we obtain
Finally, from the energy equation we have
1_ To
cpTt Til
(18)
(19)
(20)
Equations (16)-(19) are the Rayleigh functions, and values of p/p*, T/T*,
To/ are given in the tables referred to in the previous section. To obtain values of a
property at state 2, knowing the value in state I, we merely apply such equations as:
l!2 =P2/P* or T02 =
PI pdp* TOI TodT/)
For .example, given the inlet conditions Po]' Tal and M], we may obtain POl and M2 for any
given final temperature T02 (or temperature ratio T02/Tol ) as follows.
For given Ml read PoJ/P5 and ToJ/T/i
Evaluate from (T02/ToIXToJ/T/i)
Read Mz corresponding to T02/ T/i
Read corresponding to M2
Evaluate P02 from (P02/P'6)(P'S/POI)POI
Finally, we could read p,jPOJ corresponding to MJ from the isentropic flow table, and
evaluate the 'fundamental pressure loss factor' referred to in Chapter 6, namely
ADIABATIC FLOW IN A CONSTANT AREA DUCT WITH FRICTION
'"
'" .2"0
m
'" .c
.2
E
0. '" 21------

2
:§ 11---------
rn c __
r7l - 0 0.2 0.5
Inlet Mach number M,
FIG. A7 Stagnatioll pressure loss in Rayleigh flow
405
(POI - P02)/(P01 - PI)' If this were done for a series of subsonic values of MJ and
T02/Tol > 1, the results would appear as in Fig. A7. The chain dotted curve represents
limiting case where the heat transfer is sufficient to accelerate the flow to M2 = I: this
condition is often referred to as thermal choking. Note that as the inlet Mach number
approaches zero the fundamental pressure loss factor tends to [(T02/Tod - I]: c.f. the
incompressible flow result obtained in section 6.4.
A,S Adiabatic fllOw in a constant area duct with friction
The idealization considered here is referred to as Fanno flow. We have seen in section A.2
that friction accelerates subsonic flow towards M = 1 and decelerates supersonic flow
towards M = 1. The length of duct which is sufficient to change the Mach number from any
value M to the value unity will be referred to as L *. Friction data are normally given in
terms of relations between a friction factor f and Reynolds number Re. The definition off
used here (other definitions are also nsed) is
T =JpC
2
/2
The relevant equations for Fanno flow are
p/pT=p*/p*T*
pC = p*C*(= p*a*)
(p*C*2 _ pC
2
) + (P* - p) + Til = 0
cp(T* - T) + !(C*2 - C
2
) = c/T5 - To) = 0
From the energy equation the stagnation temperature is constant, and this together with the
To/T relation (8) yields
T 1'+1
T*=2+(y-I)M2
(21)
Using this in conjunction with the continuity equation and equation of state, we get
P 1[ )'+1 ]1
p* = M 2 + ()' - J)M2
(22)
406 APPENDIX A SOME NOTES ON GAS DYNAMICS
And using the Po/p relation (8) we obtain
.Po = ~ [2 + (}, - 1)M2](Y+1l!2(1'-I)
p ~ M 1'+ 1
(23)
The equation expressing L * for a given M and f is best found from the differential form
of the momentum equation (3), with" replaced by Y pC
2
. Dividing throughout by A, the
momentum equation becomes'
For circular ducts of diameter D,
dS rrDdL 4dL
A = rrD2/4 = D
And for non-circular ducts D can be replaced by the equivalent diameter (= 4 x flow
area/perimeter). Introducing the equation of state, continuity equation and relations (7) and
(8), after much algebra we arrive at a differential equation for 41 dLlD in terms of M. Inte-
grating between M and I we have finally
(24)
The Fanno functions (21) to (24) are given in the compressible flow tables. To find a
length L over which the flow changes from MI to M2 it is only necessary to subtract the two
values of 4fL * / D, namely
It should be appreciated that we have assumed f to be constant, i.e. that it does not change
along the duct. Since the Reynolds number is pCD / J.l = GD / J1, and the mass velocity G is
constant because the area A is constant, Re only changes due to the variation of Jl with T.
This is small for gases. Furthermore, in turbulent pipe flow f is a weak function of Re. Thus
little error is incurred by using an appropriate mean value off
In practical situations variation in area, heat addition, and friction may be present
simultaneously. Methods for combining the results of the preceding three sections can be
found in any text on gas dynamics.
A.6 Plane normal shock waves
When shock waves occur normal to the axis of:fiow, they are discontinuities which occupy
a finite but very short length of duct as depicted in Fig. A8(a). For this reason they can be
treated as adiabatic frictionless processes in a duct of constant cross-sectional area. In
general, shock waves are formed when the conditions are such that the three conservation
laws cannot be satisfied simultaneously with an assumption of reversible flow. What has to
be relinquished is the idealization of reversibility, even though the flow is being regarded as
frictionless. Then, if the process is adiabatic, all that the Second Law of Thermodynamics
requires is that there should be an increase in entropy in the direction of flow.
PLANE NORMAL SHOCK WAVES
(a)
FIG. All
(b)
The relevant equations relating properties on either side of a shock wave are
pd PI TI = P2/ PZT2
m = PICI = PZC2
( P z C ~ - PIeD + (P2 - pd = 0
ciTz - TI) + !(Ci - cD = Cp (T02 - TOI ) = 0
407
From the momentum and continuity equations, together with relations (7), we obtain the
pressure ratio across the shock wave as
pz I +yMr
PI 1 +yMl
Also, since T02 = !Ol from the energy equation, we have
T2 (TZ) (TOI) 2+(y-l)Ml
r;= T02 r; =2+(1'-l)Ml
(25)
(26)
At this stage the value of M2 is unknown but from the continuity equation, equation of
state, and relations (7) we can show that M2 is uniquely related to MI by the equation
Mz =12 1'!2
MI pz YTI
On substitution from (25) and (26) we get
Ml = (1' - I )Mf + 2
2yMl- (y - 1)
(27)
Finally, substituting for Mz in (25) and (26) we obtain the pressure and temperature ratio in
terms of MI given by
pz 2yMr- (1' - I)
PI y+ 1
(28)
'!2 = [2YM1- (y _ I)] [2 + (1' - I)Mf]
TI (]' + liMr
(29)
The pressure ratio POZ/PI can be found from (P02/P2)(P2/PI), using (28), (27) and the Po/p
relation (8), the result being
Poz = [y + 1 Mf])'!(Y-I) 1[2J'Ml- Cl' - l)]I/(Y-I)
PI 2 / l' + I
(30)
APPENDlX A SOME NOTES' ON GAS DYNAMICS
408 . . .
The normal shock. functions (27) to (30) .are ta1;lUlated in Ref. (l). If required, the stag- .
nation pressure can be found from .
'P02 (P02) (£1)
POI == \;;- \POI b fi d from the isentropic table for the given MI' The
remembering that pi/PO! can e oun as well as P02IPI' .
fuller tables of Ref. p021Pili f) r M > 1 M2 is less than 1; while for MI < 1,
Evaluation of equation . at 0 I .' analysis to suggest that both these
M2 is greater than 1. There IS nothing m fore/gOm
g
d T IT from (28) and (29), it is a
fl ·tuations are not possible. But lmowmg P2 PI an 2 frl
matter to calculate the change in entropy of the gas om
.tls == yln - (y - 1) In
CV TI PI 1 fTIT andp
For values of MI > 1 we find that 0 is The latter situation is physically
whereas when MI < 1 the S PYd Law of Thermodynamics. Thus the normal
impossible because it contradicts e econ f M 1
shock functions are only tabulated for 0 shock wave can only arise in a
From the foregoing we may deduce a duction in velocity to a subsonic value.
supersonic flow, and that it produces su en re d I > 1 for all MI > 1. Owing to
Thus the normal shock is a irocess there is a loss of stagnation
the irreversibility introduced by the s oc obvious from the tables
. I < 1 All these charactenstlcs are f(M ) that
pressure, I.e. P02 POI '. . F A9 We note also from the M2 == 1 curve,. .
or the graphical representanon mig... d'that M is close to this lower lumt
M2 is asymptotic to a definite value as denominator of equation
whenMI has to abodut 5infin: By is seen to equal [(1' - 1)/21']
(27) by M1 and MI ten to. I,
which is 0·38 forharT. ffi' t the nonual shock is as a compressor, because
It is clearly of mterest to see tc:r of turbojet and ramjet engines ,,:hen. these are
it can be made use of.at fron ds The sim Ie 'pitot' intake is illustrated m A8(l?).
operating at supersomc flight spee . b d t! d as the ratio of the ideal (lsentroP.lc)
The efficiency of the process can e e. efi a mven pressure ratio P2IPI' In which
tore rise to the actual temperature nse or ".
tempera ffi . where
case we have the isentropic e clency 11,
1.0
0.9
;f1 O.S
lD 0.7

§ 0.6
c
-€i 0.5
0.4
Q)
'§ 0.3
0 0.2
0.1
f' - TI [(P2Ipd
y
-
I
)/Y - 1]
2 _
11 == T2 - T
J
- T21Tj - 1
2
M
Inlet Mach number Ml
Inlet Mach number 1
FIG. A9 Changes in Mach number and pressure across a plane normal shock wave
PLANE NORMAL SHOCK WAVES
20
10
Inlet Mach number Ml
FIG. AIO Isentropic efficiency of plane shock wave
409
P21PJ and T21TI can be found from the nonnaI shock table for a range of Mh and hence 11
can be plotted versus MJ as in Fig. AlO. The efficiency falls off rapidly as MI is increased.
Nevertheless, when it is realized that Pz/PI = 4·5 for MI == 2 (from Fig. A9), and that this
pressure ratio has been achieved in a negligible length of intake, an efficiency of 78 per
cent (from Fig. AI0) is not too low to be useful. For this example, M2 is 0·58 (Fig. A9) and
after the shock wave a further pressure rise can be obtained by ordinary subsonic diffiIsion
in a divergent duct. Higher efficiencies can only be obtained by designing supersonic
intakes to operate with a system of oblique shocks. This type of shock fonns the subject of
the next section. .<
The main features of Rayleigh flow, Fanno flow and flow through nonnal shocks can be
summarized neatly by drawing the processes on a T-s diagram as in Fig. All. Such a
diagram is a useful mnemonic. The Fanno and nonnal shock processes are shown as
dashed lines because they are essentially irreversible processes. The three lines are all
drawn for the same value of mass flow per unit area (i.e. pC) which we have seen is
constant for all three types of flow. This is why states I and 2 on either side of the nonnal
shock coincide with the points of intersection of the Fanno and Rayleigh lines. There are
four features which perhaps are not emphasized sufficiently by notes on the figI1Ie. (i) A
Fanno process can occur only from state 2 towards state 3 or from 1 towards 3. It cannot
pass through 3 without a decrease in entropy which would contravene the Second Law of
Thermodynamics. (ii) A Rayleigh process can occur between I and 4 in either direction or
between 2 and 4 in either direction, but it cannot proceed through state 4 in either direction.
In practice, the picture is modified because friction is present simultaneously with heating
or cooling; but also it would be physically difficult to suddenly change from heating to
cooling at the point in the duct where the gas attains state 4 and, without doing so, passage
through state 4 would clearly contravene the Second Law. (iii) It is interesting to note that
when a subsonic flow is heated, the maximum temperature is reached at state 5 (where it
can be shown that M = 1I.h), although the stagiiation temperature must continue to rise
with energy input until the gas attains state 4. In other words, between 5 and 4 the density
decreases, and hence the velocity increases, at such a rate that the static temperature falls.
(iv) when point 3 on the Fanno line or point 4 on the Rayleigh line is reached the flow is
choked, and further friction (i.e. additional length of pipe), or heating in the case of
Rayleigh flow, causes the state to move on to another Farmo line or another Rayleigh line
respectively. M remains equal to unity. These would be lines appropriate to a reduced mass
flow per unit area, and would lie to the right of those shown because of the increase in
entropy due to additional friction in one case and heating in the other.
410
APPENDIX A SOME NOTESON GAS DYNAMICS
For shock
and Fanno
T01 = T02
1
I-
" i!! :;;;
:::l
.E
'"
,,"- Co
E


t)
\
/
?
IlPo in shock
P011 P02 P03
Ilpo in subsonic Fanno
t
v
:;;
5
P2
/,
f -;,.'
!
r-- \ P3 -<:":
?\ /.li cprJ
I r
1/1
I
I
I
I " Normal shock
I
! Fanno line
Entropy S
FIG. AU Millin featlllres of Rayleigh, Fanno and normal shock processes
A.7 Oblique shock waves
. . fi w shock waves are formed at an iociination to the
In certam types of. 0., . the anal sis of such phenomena involves
bde
obtained for plane normal shock waves can e use . . b
of plane oblique shock waves shows that for any gIVen mCldenl Mach num er,
. .. fI d fI f gle i5 which carmot be exceeded;
(a) there is a Inmtmg owe. ec IOnIan of M there are two physically possible plane
(b) the angle {3. The shock wa:e having the
smaller{3 is the weaker shock, i.e. it produces the smaller pressure ralio P21p! and,
FIG. A12 OblIque shock
OBLIQUE SHOCK WAVES
'Weal<' shock"
whene<om "
(a)
FIG. AB
411
(b)
although the flow velocity is reduced by the shock, the flow can still be supersonic
downstream of it.
Oblique shock waves are formed when a body is immersed in a supersonic stream. If
we consider the simple case of a wedge of semi-angle e, two types of flow are fOlmd to
occur as depicted in Fig. Al3.
(i) When e::: ilm a plane oblique shock wave is attached to the apex of the wedge. For
e < ilm it is the weak shock which is formed, while for e = om there is no distinction
between the two shocks and only one value of fJ is possible.
(li) When e > iJm a plane oblique shock wave is not possible, and it is replaced by a
detached curved shock wave as in Fig. A 13(b). (The same type of flow would be set
up by a round nosed body such as a compressor blade of aerofoil shape.) On the axis
of the wedge the shock is normal, 0 = 0 and M2 < I. For a certain distance from the
axis, 0 < om and the curved shock can be regarded as being made up of a number of
plane oblique shocks of elemental length and of successively increasing (j. These are
always the stronger of the two possible shocks, i.e. those having the larger fJ.
Downstream of these shocks M2 < I, and thus the flow in the region near the body is
subsonic. Further out in the mainstream, beyond the poiot where .5 has increased to
Om, fJ decreases until it is zero in the free stream (strictly speaking at an infinite
distance from the body). In this region the curved shock can be regarded as a number
of elemental plane shocks ofthe weale type (small p), and the flow remains supersonic
with M2 -+ MI as the distance from the body iocreases.
Reverting to Fig. A12, consider the components of the velocities C1 and C2 in the
tangential and normal directions relative to the plane shock. On either side of the shock
there is no change of state along the shock, and with no force acting in this direction there
can be no change in the tangential component of momentom or velocity across the shock.
Thus
C2 cos (f3 - 0) = C! cos fJ (31)
From the point of view of the normal component, the shock is a plane normal shock which
we have already analysed. Thus equation (27) can be used but with M2 sin (fJ - .5) sub-
stituted for M2 andM! sin fJ for M!. Similarly, equations (28) and (29) for hlp! and T2IT!
will apply if MI sin fJ replaces MI. To relate b, fJ and M) we apply the continuity equation
in the form
P2C2 sin (fJ - 0) = PI C! sin fJ
and substitute CdC2 from (31) to give
P2 tan fJ
PI tan (fJ - 0)
412
APPENDIX A SOME NOTES ON GAS DYNAMICS
But P2/ PI = (P2/PI)(TdT2) = f(Mbf3) fr?m the modified (28) and (29), so that we have an
equation for (j in terms of MJ and f3 which finally reduces to .
Mf sin
2
f3 - I
tan 0 = cot f3
- (Mf sin
2
P -1)
2
(32)
For given values of b( < om) this equation has been shovvn to have two real positive r?ots,
yielding the two values of P corresponding to the weak and .strong shocr:s. Afte: dIffer-
entiating, and some algebra, equation (32) also yields the deflection Om m terms
of MI' The results are often plotted as in Fig. A14. They are also gIVen m the obhque shock
table ofRefs.(I) and (2). Such a table gives the two possible of p.j, P2/Pj, T2/TI and
M
2
, for each value of MI and a series. of values of The curve dlVldes the weak and
strong shock regions in Fig. AI4 gives the maxnnmn deflection Om which the flow can
undergo with a shock wave attached to the apex of the wedge. From the cham-dotted curve
we see that although the Mach nmnber downstream of a strong shock is always less than
unity, it is not necessarily always greater than unity for a weak shock. . .
It is of interest to compare the efficiency of a compreSSIon by an oblique shock With
that of the normal shock considered io section A.6. We took MI = 2, and found that a
pressure ratio (P2/PI) of 4·5 was obtained with an of 78 per cent. From the
oblique shock table of Ref. (1), with MI = 2 and an arbltranly chosen value of 0 = 10
degrees, we obtain
Weak Strong
P
39·3° 83·7°
Mz 1·64 0·60
PZ/PI
1·71 4·45
T2/TI
1·17 1·68
1)
0·98 0·79
50
Shock wave angle f3 degrees
FIG. A14
ISENTROPIC TWO-DIMENSIONAL SUPERSONIC EXPANSION AND COMPRESSION 413

FIG. A15 Oswatitsch illtake

'Dump door' for excess air
FIG. A16 Intake with variable geometry
The last line is calculated from
with y = 1-403 as used in the tables.
M<1

'Dump door' used as scoop
to increase air flow
Sioce a pressure ratio of 1·71 is obtaioable from a weak plane oblique shock with an
efficiency of 98 per cent, it would appear that a supersonic intake designed to make use of
one or more such shocks might be more efficient than the simple type of 'pitot' intake
referred to in section A.6; and this has been found to be the case. In the Oswatitsch iotake,
which makes use of a conical centre body as shown in Fig. AIS, the flow is decelerated to a
low supersonic velocity by several successive oblique shocks (two are shown) with the
final transition to subsonic velocity takiog place via a normal shock. The main problems in
the design of such supersonic iotakes ,re (a) the establishment of a stable shock pattern
insensitive to minor changes in the flow downstream (e.g. in the engine), and (b) the
maintenance of a good performance under off-design conditions. For aircraft engine
intakes which have to operate over a complete speed range from MI = 0 to MI » I,
variable geometry is essential. This is accomplished by incorporating an adjustable centre
body and/or cowl, and bleed slots. Figure AI6 illustrates a variable geometry intake of the
'scoop' type, which is of rectangular cross-section: it may be slung under the wing or run
along the side of the fuselage.
A.8 Isentropic two-dimensional supersonic expansion and
compression
Perhaps the foregoing sections have left the reader with the impression that isentropic
expansion and compression of a supersonic stream is impossible in principle. A moment's
reflection on the existence of successful supersonic aeroplane wings, and efficient nozzles
and diffusers for supersonic wiod tunnels, should dispel this illusion. Processes which
would be isentropic apart from the effect of viscous mction are possible if the duct walls,
or innnersed body as the case may be, are correctly shaped. Certainly it is possible to avoid
the large loss of stagnation pressure due to breakdown of the flow which was illustrated in
Fig. A4.
Consider supersonic flow that is initially parallel to a surface but which encounters a
small change of direction of the surface; it may be a convex or concave deflection as in Fig.
414 APPENDIX A SOME NOTES ON GAS DYNAMICS
p p-dp
(a) Convex (b) Concave
FIG. AI7 Expansive allli cllmpressive Mach waves
AI7(a) and (b). If the change in direction is infinitesimal, the corner is the source of an
infinitesimal disturbance which is communicated to the main flow isentropically along a
Mach wave. From section A.l we know that this wave will make an angle.u = sin-I (ljM)
with the direction of flow. And from the velocity triangles of Fig. AI7 we see that the
convex corner produces an increase in velocity, dC, which must be accompanied by a
pressure drop dp; while the concave corner kads to a decrease in velocity implying a
compression.
Any rounded convex comer giving a finite deflection can be regarded as the source of a
series of infinitesimal deflections as illustrated in Fig. AIS(a). The Mach waves, or char-
acteristic lines as they are often called, diverge and do not interfere with each other, so
permitting the flow to accelerate smoothly and isentropically to the downstream pressure.
In the limit the same thing can occur at a sharp finite corner: in this case the flow expands
smoothly through a fan of Mach waves as shown in Fig. A18(b). This is Imown as Prandtl-
Meyer flow, and the evaluation of finite changes of pressure and Mach number associated
with a finite deflection is possible with the aid of tabulations ofPrandtl-Meyer 'expansion
angles'. We shall proceed no further with this topic, but say merely that the analysis of
supersonic flow patterns using a step-by-step method moving from one Mach line to the
next, is referred to as the method of characteristics. When it is realized that in most
situations the Mach waves will suffer reflections from neighbouring smfaces or jet
boundaries, or intersect aod interact with Mach waves from an opposing smface, it will be
appreciated that the analysis is complex and beyond the scope of this Appendix.
To conclude, let us consider briefly what happens in a finite concave comer. If it is
rounded as in Fig. A19(a), the Mach waves will appear as shown. They must converge
because the Mach number is decreasing aod hence the Mach aogle .u is increasing. For a
sharp concave corner there can be no compressive equivalent of an expansion fan, because
this would involve successive Mach lines marching upstream of one another. What
(a)
FIG. AI!! Isentropic expansion
ISENTROPIC TWO-DIMENSIONAL SUPERSONIC EXPANSION AND COMPRESSION 415
happens, since no comer is truly sharp, is that the Mach lines converge to a point in the
stream the and coalesce to fonn an oblique shock as in Fig. A19(b). The
process IS then no longer isentropic. In other words, an isentropic diffusion of
a supersomc stream can only occur in a passage bounded by gradual concave curves.
l1+dv 1L+2dv
p+dp p+2dp
M-dM M-2dM
\/
dv
(a)
Oblique shock

1//////////////////
(b)
FIG, A19 Compression (a) Isentropic (IJ) Non-isentropic
Appendix B
Problems
The following problems are numbered by chapters. Unless otherwise stated the following
values have been used:
For air:
For combustion gas:
For air and combustion gas:
Cp = 1·005 kJ/kg K, y == lAO
cp = 1·148 kJ/kg K, Y= 1·333
R=O·287 kJ/kg K
Instructors who adopt the text for class use may obtain a manual of solutions from the
publishers. . .' .'
Readers who wish to carry out more detaJled deslgn-pomt and off-deSign studies,
allowing for such factors as variable fluid properties, bleeds, will
find the commercially available program GASTURB useful. This can be obtamed from Dr
J. Kurzke, Max Feldbauer Weg 5, 855221 Dachau, Germany.
2.1 In an ideal gas turbine cycle with reheat, air at state (PI> T1) is compressed to pressure
rpl and heated to 1'3' The air is then expanded in two stages, .each turbine the same
pressure ratio, with reheat to T3 between the stages. Assummg working flUid be a
perfect gas with constant specific heats, and that the compreSSIOn and are
isentropic, show that the specific work output will be a maxunum when r IS gIVen by
r(I-1)/1 = (T3/Tli/3
2.2 A simple gas turbine with heat-exchanger has a compressor and turbine having
respective isentropic efficiencies rye and 1Jt. Show that the c?mbined effect of small pressure
drops !J.Phg (in gas-side of heat-exchanger) and /1p (total m combustIOn cham?er and aIr-
side of heat-exchanger) is to reduce the specific work output by an amount given by
y - 1 cp 1'
31Jt [ti + LlP]
-y- x r(y-I)/YPI Phg r
where T3 = turbine inlet temperature, PI = compressor inlet pressure and r = compressor
pressure ratio.
Assume that cp and y are constant throughout the cycle.
2.3 Consider the ideal cycle for a gas turbine with heat-exchanger and separate LP power
turbine. When the heat-exchanger is fitted in the normal way, using the exhaust from the
LP turbine to heat the air after compression, the ideal cycle efficiency is
1] = 1 - (c/t)
APPENDIX B PROBr.,EMS 417
where C'"' (P2/PI)(y-l)/1' and t= T3/h Another possibility is to pass the gas leaving the
HP turbine through the heat-exchanger before it enters the LP turbine, and to let the LP
turbine exhaust to atmosphere. Derive an expression for the ideal cycle efficiency of this
scheme in terms of c and t, and hence show J:.hat for all values of c > I the efficiency is
higher than that of the ordinary scheme.
Finally, by referring to sketches of the cycles on the T-s diagram, say in what respects
the nonnal scheme might be superior in spite of the lower ideal cycle efficiency.
2,4 In a gas turbine plant, air is compressed from state (Pj, T1) to a pressure rpI and
heated to temperature h The air is then expanded in two stages with reheat to T3 between
the turbines. The isentropic efficiencies of the compressor and each turbine are 1Jc and 1Jt. If
XPI is the intermediate pressure between the turbines, show that, for given values Ph T1, T3,
1Jc, 1Jt and Y, the specific work output is a maximum when X= Jr.
If this division of the expansion between the turbines is maintained, show that
(a) when r is varied, the specific work output is a maximum with r given by
r
3
/
2
= (Y{cY{t T3/TIY/(I-
I
)
(b) When a perfect heat-exchanger is added the cycle efficiency is given by
1'1 r(I- 1)/21'(r(I-1)/21 + lJ
1]=1
21Jc11t T3
Assume that the working fluid is a perfect gas with constant specific heats, and that
pressure losses in the heater, reheater, and heat-exchanger are negligible.
2.5 A gas turbine plant has a compressor in which air is compressed from atmospheric
pressure and delivered to two turbines arranged in parallel, the combustion gases
expanding to atmospheric pressure in each turbine. One of the turbines derives the
compressor, to which it is mechanically coupled, while the other develops the power output
of the plant. Each turbine has its own combustion chamber, the fuel supply to each being
capable of control independently of the other.
(a) The power output of the plant is reduced by varying the fuel supply to the combustion
chambers in such a way that the inlet temperature to the turbine driving the
compressor is kept constant, while the inlet temperature to the power turbine is
reduced. If the mass flow through a turbine is proportional to p/ JI, where P and l'
refer to the turbine inlet conditions, show that the mass flow through the compressor
is proportional to r1l1' and the power output of the plant is proportional to
r[r(]'-I)/1' - Il/[I - kr(l'-I)/1']
where r = pressure ratio
k= TJ/(Y{cY{t1'3)
T1 = compressor inlet temperature
T3 = temperature at inlet to the turbine driving the compressor
1'/c = isentropic efficiency of compressor
'1t = isentropic efficiency of turbine driving the compressor.
The isentropic efficiencies of the compressor and turbines may be assumed to remain
constant under all conditions of tlle problem. Pressure losses, mechanical losses and
the increase in the mass flow through the combustion chambers due to the mass of
fuel bumt, may be neglected. cp and y may be assumed to be constant.
(b) Derive corresponding expressions for the altemative method of control in which the
fuel supply to the combustion chambers is reduced in such a way that the inlet
temperatures to both turbines are always equal.
418
APPENDIX B PROBLEMS
(c) Calculate the percentage of full power developed under each of the foregoing
methods of control when the pressure ratio oflhe compressor has fallen to 3·0, on a
plant designed to give full power under the following conditions:
pressure ratio of the compressor
inlet temperature to the compressor
inlet temperature to both turbines
l/e = 0·85, 1/, = 0·88, y = 1·4
[0·521, 0·594]
4·0
288 K
llOO K
2.6 A compressor has an isentropic efficiency ofO·85 at a pressure ratio of 4·0. Calculate
the corresponding polytropic efficiency, and thence plot the variation of isentropic
efficiency over a range of pressure ratio from 2·0 to 10·0.
[0·876; 0·863 at 2·0 and 0·828 at 10·0]
2.7 A peak load generator is to be powered by a simple gas turbine with free power
turbine delivering 20 MW of shaft power. The following data are applicable:
Compressor pressure ratio
Compressor isentropic efficiency
Combustion pressure loss
Combustion efficiency
Turbine inlet temperature
Gas generator turbine isentropic efficiency
Power turbine isentropic efficiency
Mechanical efficiency (each shaft)
Ambient conditions Pa, Ta
Calculate the air mass flow required and the SFC.
[119-4 kg/s, 0·307 kg/kW h]
ll·O
0·82
0-4 bar
0·99
1150 K
0·87
0·89
0·98
1 bar, 288 K
2.8 A gas turbine is to consist of a compressor, combustion chamber, turbine and heat-
exchanger. It is proposed to examine the advantage of bleeding off a fraction (I'!.m/m) of
the air delivered by the compressor and using it to cool the turbine blades. By so doing the
maximum permissible cycle temperature may be increased from Tto (T + I'!.T). The gain in
efficiency due to the increase of temperature will be offset by a loss due to the decrease in
effective air flow through the turbine. Show that, on the following assumptions, there is no
net gain in efficiency when
I'!.m I'!.T/T
m (I +I'!.TjT)
and that this result is independent of the compressor and turbine efficiencies. Assumptions:
(1) No pressure loss in combustion chamber or heat-exchanger.
(2) The working fluid is air throughout and the specific heats are constant.
(3) The air bled for cooling purposes does no work in the turbine.
(4) The temperature of the air entering the combustion chamber is equal to that of the
turbine exhaust.
A plant of this kind operates with an inlet temperature of288 K, a pressure ratio of6·0,
a turbine isentropic efficiency of 90 per cent and a compressor isentropic efficiency of 87
per cent. Heat transfer calculations indicate that if 5 per cent ofthe compressor delivery is
bled off for cooling purposes, the maximum temperature of the cycle can be raised from
1000 to 1250 K. Find the percentage increase in (a) efficiency, and (b) specific work
output, which is achieved by the combined bleeding and cooling process. Make the same
assumptions as before and take y = 1·4 throughout.
APPENDIX B PROBLEMS 419
Whatresults you expect if these calculations were repeated for the plant with the
heat-exchanger OIrutted?
[25·1 per cent, 47·8 per cent]
2.? auxiliary gas torbine for use on a large airliner uses a single-shaft configuration
wIth all' bled the compressor dIscharge for aircraft services. The unit must provide
1·5 kg/s bleed all' and a shaft power 0[200 kW. Calculate Ca) the total compressor air mass
flow and (b), the power available with no bleed flow, assuming the following:
Compressor pressure ratio
Compressor isentropic efficiency
Combustion pressure loss
Turbine inlet temperature
Turbine isentropic efficiency
Mechanical efficiency (compressor rotor)
Mechanical efficiency (driven load)
Ambient conditions
[4·78 legis, 633 kW]
3·80
0·85
0·12 bar
1050 K
0·88
0·99
0·98
I bar, 288 K
2.10 A closed cycle gas turbine is to be used in conjunction with a gas cooled nuclear
reactor. The working fluid is helium (cp =5·19 kJ/kg K and y= 1·66).
The of the plant of two-stage compression with intercooling followed by
a heat-exchanger; after leavmg the cold side of the heat-exchanger the helium passes
throu.gh the react<Jr channels and on to the turbine; from the turbine it passes through the
?ot-slde of the and then a pre-cooler before returning to the compressor
inlet. The followmg data are applicable:
Compressor and turbine polytropic efficiencies 0·88
TemperahITe at LP compressor inlet 310 K
Pressure at LP compressor inlet 14·0 bar
Compressor pressure ratios (LP and HP) 2·0
Temperature at HP compressor inlet 300 K
Mass flow of helium 180 kg/ s
Reactor thennal output (heat input to gas turbine) 500 MW
Pressure loss in pre-cooler and intercooler (each) 0·34 bar
Pressure loss in heat-exchanger (each side) 0·27 bar
Pressure loss in reactor channels 1·03 bar
Helium temperature at entry to reactor channels 700 K
the power output and thennal efficiency, and the heat-exchanger effectiveness
Imphed by the data.
[214·5 MW, 0·429, 0·782]
3.1 A simple turbojet is operating with a compressor pressure ratio of 8·0, a turbine inlet
temperat:rre of 1200 K and a mass flow of 15 kg/s, when the aircraft is flying at 260 m/s
at .altItode of 7000 m. Assuming the following component efficiencies, and I.S.A.
condItIOns, calculate the propelling nozzle area required, the net thrust developed and the
SFC.
Polytropic efficiencies of compressor and turbine
Isentropic efficiency of intake
Isentropic efficiency of propelling nozzle
Mechanical efficiency
Combustion chamber pressure loss
Combustion efficiency
[0·0713 m
2
, 7896 N, 0·126 kg/h N]
0·87
0·95
0·95
0·99
6 per cent camp. de1h( press.
0-97
420
APPENDIX B PROBLEMS
. .
3.2 The gases in the pipe of the engine considered in 3.1· are reheated to
2000 K, ap.d the combustion pressure loss incurred is 3 per cent of the pressure at outlet
from the turbine. Calculate the percentage increase in nozzle area required if the mass flow
is to be unchanged, and also the percentage increaSe in net thrust.
[48.3 per cent, 64·5 per cent]
3.3 A naval aircraft is powered by a turbojet engine, with provision for flap blowing.
When landing at 55 mis, 15 per cent of the compressor delivery air is bled off for flap
blowing and it can be assumed to be discharged perpendicularly to the direction offlight.lf
a propelling nozzle area of 0·13 m
2
is used, calculate the net thrust during landing given
that the engine operating conditions are as follows.
Compressor pressure ratio
Compressor isentropic efficiency
Turbine inlet temperature
Turbine isentropic efficiency
Combustion pressure loss
Nozzle isentropic efficiency
Mechanical efficiency
Ambient conditions
9·0
0·82
1275 K
0·87
0·45 bar
0·95
0·98
1 bar, 288 K
The ram pressure and temperature rise can be regarded as negligible.
[18·77 kN]
3.4 Under take-off conditions when the ambient pressure and temperature are 1·01 bar
and 288 K, the stagnation pressure and temperature in the jet pipe of a turbojet engine are
2.4 bar and 1000 K, and the mass flow is 23 kg/so Assuming that the expansion in the
converging propelling nozzle is isentropic, calculate the exit area required and the thrust
produced.
For a new version of the engine the thrust is to be increased by the addition of an aft fan
which provides a separate cold exhaust stream. The fan has a bypass ratio of 2·0 and a
pressure ratio of 1.75, the isentropic efficiencies of the fan and fan-turbine sections being
0.88 and 0.90 respectively. Calculate the take-off thrust assuming that the expansion in the
cold nozzle is also isentropic, and that the hot nozzle area is adjusted so that the hot mass
flow remains at 23 kg/so
[0.0763 nl, 15·35 kN; 24·9 kN]
3.5 Extending the example on the turbofan in Chapter 3, with the additionai information
that the combustion efficiency is 0·99, determine the SFC. Also, calculate the thrust and
SFC when a combustion chamber is incorporated in the bypass duct and the 'cold'stream
is heated to 1000 K. The combustion efficiency and pressure loss for this process may be
assumed to be 0·97 and 0·05 bar respectively.
[0.0429 kgJh N; 55·95 kN, 0·128 kg/h N]
4.1 The following data refer to the eye of a single-sided impeller.
Inner radius
Outer radius
Mass flow
Ambient conditions
Speed
6·5 em
15·0 cm
8 kg/s
1·00 bar, 288 K
270 rev/s
Assuming no pre-whirl and no loses in the intake duct, calculate the blade inlet angle at
root and tip of the eye, and the Mach number at the tip of the eye.
[48.200 ,25.43
0
,0.843]
APPENDIX B PROBLEMS 421
4:2 aircraft engine is fitted with a: single-sided centrifugal compressor. The aircraft
flies WIth a speed of 230 mls at an altitude where the pressure is 0·23 bar and the
217 The duct ?fthe impeller eye contains fixed vanes which give the
au pre-.whirl of 25 . at all radll. The nmer and outer diameters of the eye are 18 and 33 em
respectively, the diameter of the impeller periphery is 54 em and the rotational speed
270 rev Is. Estimate the stagnation pressure at the compressor outlet when the mass flow is
3·60 kg/so
losses in the duct and fixed vanes, and assume that the isentropic
effiCIency of the compressor IS 0·80. Take the slip factor as 0·9 and the power input factor
as 1·04.
[1·75 bar]
4.3 The following results were obtained from a test on a small single-sided centrifugal
compressor:
Compressor delivery stagnation pressure
Compressor delivery stagnation temperature
Static pressure at impeller tip
2·97 bar
429K
1·92 bar
0·60 kg/s
766 revls
Mass flow
Rotational speed
Ambient conditious 0·99 bar and 288 K
Calculate. the overall isentropic efficiency of the compressor.
The diameter of the is 16·5 cm, the axial depth of the vaneless diffuser is
1 ..0 cm and the number ofunpellervanes (n) is 17. Making use of the Stanitz equation for
slip factor, namely (f = - (O·63nln), calculate the stagnation pressure at the impeller tip
and hence find the fraction of the overall loss which occurs in the impeller
[0·75; 3·35 bar, .
4.4 The following design data apply to a double-sided centrifugal compressor:
Outer diameter of impeller
Speed
Mass flow
Inlet temperature
Inlet pressure
Isentropic efficiency of impeller only
Radial gap of vaneless space
Axial depth of vaneless space
Slip factor
Power input factor
50 cm
270 rev/s
16·0 kg/s
288 K
1·01 bar
0·90
4·0 em
5·0 cm
0·9
1·04
(a) Calcul.ate the pressure and temperature at the outlet of the impeller
assummg no pre-whirl. '
(b) Show that the radial outlet.veloci!y at the impeller tip is about 96 mls and hence find
Mach n?IDber and au .leavmg .angle at the impeller tip. (In calculating the
cucumferential area at the. tip, the thickness of the impeller disc may be neglected)
(c) isentropic ,?ffusion in the vaneless space, find the correct angle of the
leading edges of the diffuser vanes, and also find the Mach number at this radius
[(a) 4·40 bar, 455 K (b) 1·01, 14·08° (c) 12.40°,0.842] .
4.5 A centrifugal compressor is to deliver 14 kgl s of air when operating at a
pressure ratio of4: 1 and a speed.of200.revls. The inlet stagnation conditions may be
taken as 28.8 K 1·0 Assummg a slip factor of 0·9, a power input factor of 1·04 and
an overall IsentropIc effiCIency of 0·80, estimate the overall diameter of the impeller.
422 APPENDIX B PROBLEMS
If the Mach number is not to exceed unity at the impeller tip, and 50 per cent ofthe
losses are assumed to occur in the impeller, find the minimum possible aXial depth of the
diffuser,
[68,9 cm, 5,26 cm]
5.1 An axial flow compressor stage has blade root, mean and tip velocities of 150, 200
and 250 mis, The stage is to be designed for a stagnation temperature rise of 20 K and an
axial velocity of 150 mis, both 'constant from root to tip, The work done factor is 0·93.
Assumiog 50 per cent reaction at mean radius calculate the stage air angles at root, mean
and tip and the degree of reaction at root and tip for a free vortex design,
[IXI==17·07° (==/32)' /31==45,73° (== 1X2) at mean radius; IXI==I3,77°, /31==54.88°,
/32 ==40·23°, 1X2 == 39·43° at tip; IXI = 22,25°, /31 == 30,60°, /32 = - 20.25°, 1X2 == 53·85° at
root; A == 11 ,2 per cent at root and 67 -4 per cent at tip]
5.2 Recalculate the stage air angles for the same data as in the previous question for a
stage with 50 per cent reaction at all radii, and compare the results with those for the free
vortex design,
[IXI==28,60° (==/32)' /31==48,27° (== 1X2) at tip; IXI==l,15° (==/32)' /31==44,42° (=1X2) at
root]
5.3 The first stage of an axial compressor is designed on free vortex principles, with no
inlet guide vanes, The rotational speed is 6000 rev/min and the stagnation temperature rise
is 20 K. The hub-tip ratio is 0,60, the work done factor is 0,93 and the isentropic efficiency
of the stage is 0,89, Assuming an inlet velocity of 140 m/s and ambient conditions of
1,01 bar and 288 K, calculate:
(a) The tip radius and corresponding rotor air angles /31 and /32, if the Mach number
relative to the tip is limited to 0,95,
(b) The mass flow entering the stage,
(e) The stage stagnation pressure ratio and power required,
(d) The rotor air angles at the root sectiou,
[(a) 0-456 m, 63,95° and 56-40°, (b) 65,S kg/s, (e) 1,233, 1317 kW, (d) 50,83° and
18,32°]
5.4 An axial flow compressor has an overall pressure ratio of 4,0 and mass flow of
3 kg/ s, If the polytropic efficiency is 88 per cent and the stagnation temperature rise per
stage must not exceed 25 K, calculate the number of stages required and the pressure ratio
of the first and last stages, Assume equal temperature rise in all stages, If the absolute
velocity approaching the last rotor is 165 m/ s at an angle of 20° from the axial direction,
the work done factor is 0,83, the velocity diagram is symmetrical, and the mean diameter
of the last stage rotor is 18 em, calculate the rotational speed and the length of the last
stage rotor blade at inlet to the stage, Ambient conditions are 1,01 bar and 288 K.
[7, 1,273, 1,178; 414 revis, 1,325 em]
5.5 A helicopter gas turbine requires an overall compressor pressure ratio of 10 : 1, This
is to be obtained using a two-spool layout consisting of a four-stage axial compressor
followed by a single-stage centrifugal compressor, The polytropic efficiency of the axial
compressor is 92 per cent and that of the centrifugal is 83 per cent
The axial compressor has a stage temperature rise of 30 K, using a 50 per cent reaction
design with a stator outlet angle of 20°, If the mean diameter of each stage is 25,0 cm and
each stage is identical, calculate the required rotational speed, Assume a work done factor
of 0,86 and a constant axial velocity of 150 mis,
Assumiog an axial velocity at the eye· of the impeller, an iropeller tip diameter of
33,0 em, a slip factor of 0,90 and a power input factor of 1 '04, calculate the rotational
speed required for the centrifugal compressor,
APPENDIX B PROBLEMS
423
Ambient conditions are 1,0 I bar and 288 K.
[Axial compressor 318 revis, centrifugal compressor 454 rev/s]
6.1 (Chapter 2 also refers) The reference hydrocarbon fuel for which the combustion
temperature rise curves of Fig, 2,15 have been drawn contains 13,92 per cent hydrogen and
86,08 per cent carbon, and the relevant value of the enthalpy of reaction, t:J!298K, is
-43 lOO kJ/kg of fueL An actual adiabatic steady flow combustion process employs this
fuel, with fuel and air entering at 298 K in the ratio 0,0150 by weight. A chemical analysis
of the products shows that 4,0 per cent of the carbon is burnt only to carbon monoxide, and
the combustion temperature rise iJ.T is found to be 583 K.
Using Fig, 2,15, calculate the combustion efficiency based on (a) ratio of actual to
theoretical f for the actual iJ.T, (b) ratio of actual to theoretical AT for the actual f Compare
these values with the efficiency based on the ratio of actual energy released to that
theoretically obtainable, given that t:J!298K for CO is -lO 110 kJ/kg,
[0,980,0,983,0,981]
6.2 (Chapter 2 also refers) A gas turbine combustion chamber is supplied with liquid fuel
at 325 K and air at 450 K. The fuel approximates to CIOHI2, and five tiroes the quantity of
air required for stoichiometric combustion is supplied, Calculate the fueljair ratio, and
estimate the fuel products temperature assuming the combustion to be adiabatic and
complete,
In addition to the following data, use appropriate values of ep from p, 17 of the abridged
tables of Ref. (8) in Chapter 2: the combustion temperature rise curves of Fig. 2,15 may be
used to obtain an initial approxiroate value of the products temperature,
Data:
CIOH12(1iq) + 1302 -+ lOC02 +6H20(liq); t:J!298K = - 42500 kJ/kg CIOH12
For water at 298 K, hfg == 2442 Ie] /kg,
For the liquid fuel, mean cp == 1,945 leJ /kg K.
Composition of air by volume: 0·79 N2, 0,21 O2,
[0,0148, 984 K]
6.3 The overall pressure loss factor of a combustion chamber may be assumed to vary
with the temperature ratio according to the law
Apo [T02]
2/2 A2 =Kj +K, --1
m PI m TOl
For a particular chamber having an inlet area of 0,0389 m
2
and a maxiroum cross-sectional
area Am 0[0,0975 m
2
, cold loss tests show that KI has the value 19,0, When tested under
design conditions the following readings were obtained,
Air mass flow m
Inlet stagnation temperature TO!
Outlet stagnation temperature T02
Inlet static pressure PI
Stagnation pressme loss /:!po
9,0 kg/s
475 K
1023 K
4·47 bar
0,27 bar
Estimate the pressure loss at a part load condition for which In is 7-40 legis, TOl is 439 K,
T02 is 900 K and PI is 3,52 bar,
Also, for these two operating conditions compare the values of (a) the velocity at inlet
to the chamber and (b) the pressure loss as a fraction of the inlet stagnation (i.e, com-
pressor delivery) pressure, and comment on the result
[0,213 bar; 70,2 mis, 67,7 m/s; 0,0593, 0,0597]
424
APPENDIX B PROBLEMS
7.1 A mean-diameter design of a turbine stage having equal inlet and outlet velocities
leads to the following data.
Mass flow m
20 kg/s
Inlet temperature TOI
1000 K
Inlet pressure POI
4·0 bar
Axial velocity (constant through stage). Ca 260 m/s
Blade speed U
360 m/s
Nozzle efflux angle 1X2
65 degrees
Stage exit swirl 1X3
10 degrees
Determine the rotor blade gas angles, degree of reaction, temperature drop coefficient
(2cpf1Tosl U
2
) and power output.
Assuming a nozzle loss coefficient liN of 0·05, calculate the nozzle throat area required
(ignoring the effect of friction on the critical conditions).
[37-15°,57.37°,0.29,3-35,4340 kW, 0·0398 n1]
7.2 The following particulars relate to a single-stage turbine of free vortex design.
Inlet temperature TOI
Inlet pressure POI
Pressure ratio POJ!P03
Outlet velocity C3
B lade speed' at root radius, Ur
Turbine isentropic efficiency 11t
1050 K
3·8 bar
2·0
275 m/s
300 m/s
0·88
The turbine is designed for zero reaction (A = 0) at the root radius, and the velocities at
inlet and outlet (el and C3) are both equal and axial. Calculate the nozzle efflux angle 1X2
and blade inlet gas angle f32 at the root radius.
If the tip/rootradius ratio of the annulus at exit from the nozzle blades is 1-4, determine
the nozzle efflux angle and degree of reaction at the tip radius.
Assuming a nozzle blade loss coefficient AN of 0·05 calculate the static pressure at inlet
and outlet of the rotor blades at the root radius and thereby show that even at the root there
is some expansion in the rotor blade passages UDder these conditions.
[61.15°,40.23°,52.25°,0-46; 1·72 and 1·64 bar]
7.3 The following data apply to a single-stage turbine designed on free-vortex theory.
Mass flow 36 kg/s
Inlet temperature ToI 1200 K
Inlet pressure POI 8·0 bar
Temperature drop 1l.TOI3 150 K
Isentropic efficiency IJt 0·90
Mean blade speed Urn 320 m/s
Rotational speed N 250 rev/s
Outlet velocity C3 400 mls
The outlet velocity is axial. Calculate the blade height and radius ratio of the annulus from
the outlet conditions.
The turbine is designed with a constant annulus area through the stage, i.e. with no
flare. Assuming a nozzle loss coefficient }'N of 0·07, show that continuity is satisfied when
the axial velocity at exit from the nozzles is 346 ml s. Thence calculate the inlet Mach
number relative to the rotor blade at the root radius.
[0·0591 m, ]·34; 0·81]
APPENDIX B PROBLEMS 425
7.4 The example of a free vortex turbine design considered in section 7.2 yielded the
following. results for the gas angles.
1X2 f32 1X3 f33
tip 54·93° 0° 8·52° 58·33°
mean 58·38° 20-49° 10° 54·96°
root 62·15° 39·32° 12·12° 51·]3°
The values of the radius ratios in plane 2 were (rm/frh= ]:164 and (rmlrth = 0·877.
Using the same mean diameter anglies, calculate f32 at tip and root for a constant nozzle
angle design in which 1X2 and Cw2r are constant over the annulus. Compare the two designs
by sketching the velocity diagrams and commenting qualitatively on such aspects as the
radial variation of degree of reaction and blade inlet Mach number.
To satisfy radial equilibrium with constant nozzle angle, the constant angular
momentum condition should strictly be replaced by constant Cwr'in2.. Show, by
recalculating f32 at tip and root, that this refinement has only a small effect on the
required blade angle.
[f32r 35·33°, f32t 0°; f32r 33·50°. f32t 3·20°]
7.5 In certain designs the maximum mass flow passed by a turbine may be determined by
choking in the turbine annulus at outlet instead of in the turbine nozzles. The maximum
mass flow will then depend not merely on the inlet conditions as in the case of expansion in
a nozzle, but also upon the work output per unit mass flow. Given fixed values of the inlet
conditions POI and TOb the temperature equivalent of the specific work output LlToI3, the
annulus area A at the turbine outlet and that there is no outlet swirl, show that for an
isentropic expansion to a varying outlet static temperature T3 the maximum flow can be
expressed by
m../Tol = {I [(_2 ) (1- 1l.To13)](Y+Il/(Y-
I
l}4
POI R y+ 1 TOI
Comment on the effect this will have on the turbine mass flow versus pressure ratio
characteristic.
S.l The following data refer to a single-shaft gas turbine running at its design speed.
Compressor characteristic Turbine characteristic
P21PI m../TJ!PI '1e P31P4 m../T31p3 111
5·0 32·9 0·80 5·0 14·2 0·845
4·7 33-8 0·79 4·5 14·2 0·850
4·5 34·3 0·77 4·0 14·2 0·842
The combustion pressure loss is 5 per cent of the compressor delivery pressure and the
ambient conditions are 1·01 bar and 288 K. Mechanical losses can be neglected. The 'non-
dimensional' flows are based on m in kg/s, P in bar and T in K, all pressures and
temperatures being stagnation values.
Calculate the power output when operating at a turbine inlet temperature of 1100 K.
Comment briefly on the variation in thermal efficiency as the load is reduced at constant
speed.
[264 kWj
426 APPENDIX B PROBLEMS
8.2 The following data refer to a gas turbine with a free pow()r turbine, operating at
design speed.
Compressor characteristic Gas generator turbine
characteristic
PZ/PI mJTIJpI l1e . P3/P4 mJT3/P3 1'/,
5·2 220 0·82 2·50 90·2 0·85
5·0 236 0·83 2·25 90·2 0·85
4·8 244 0·82 2·00 88·2 0·85
Assuming that the power turbine is choked, the value of mJT4/P4 being 188, detennine
the design values of compressor pressure ratio and turbine inlet temperature.
Neglect all pressure losses and assume the mechanical efficiency of the gas-generator
rotor to be 0·98 and take the ambient temperature as 288 K. The 'non-dimensional' flows
quoted are based on m in kg! s, P in bar and Tin K, all pressures and temperatures being
stagnation values. The suffixes 1, 2, 3 and 4 refer to the following locations:
l-<:ompressor inlet
2-<:ompressor delivery
3-gas-generator turbine inlet
4-power turbine inlet.
[5·10, 1170 K]
8.3 When running at a low power condition a gas turbine with free power turbine
operates at a compressor pressure ratio of2·60. The combustion chamber pressure loss is 4
per cent of the compressor delivery pressure and the exhaust pressure loss can be ignored.
The turbine characteristics are given below:
Gas-generator turbine Power turbine
P3/P4 mJT3/P3 111 P4/Pa mJT4!P4
1·3 20·0 0·85 1·4 60·0
1·5 44·0 0·85 1·6 85·0
1·8 62·0 0·85 1·8 95·0
The 'non-dimensional' flows are based on m in kg/ s, P in bar and Tin K, all pressures and
temperatures being stagnation values.
Calculate the gas-generator turbine pressure ratio at this condition. Assuming the
compressor characteristic to be knOWll, indicate briefly how you would calculate the
turbine inlet temperature.
[1·61]
8.4 A gas turbine with a free power turbine gives the following results when tested at
ambient conditions of 1·0 bar and 288 K.
NJTI (% design) mJTIJpI P2/PI l1e
100 454·5 4·60 0·859
95 420·0 4·00 0·863
90 370·0 3·60 0·858
APPENDIX B PROBLEMS
427
The 'non-dimensional' flows are based on m in kg! s, P in bar and Tin K, all pressures and
temperatures being stagnation values.
The power turbine remains choked for all of these conditions. The gas generator turbine
inlet temperature at 95 per cent design mechanical speed was found to be 1075 K.
Assuming single line turbine flow characteristics, constant turbine efficiency, and constant
mechanical efficiency of the gas-generator rotor, calculate
(a) the gas-generator turbine inlet temperature at design mechanical speed with the same
ambient conditions;
(b) the compressor power absorbed when running at 95 per cent design mechanical speed
with ambient conditions of 0·76 bar and 273 K.
Sketch the operating line on the compressor characteristic, and discuss the effect of
ambient temperature on net power output for a fixed gas-generator mechanical speed.
[1215 K, 3318 leW]
1l.5. A simple gas turbine is to be used as a source of compressed air. The turbine
produces just enough work to drive the compressor which delivers an airflow, me, greater
than that required for the turbine by an amount mb. The design point operating conditions
are:
meJTI = 22.8,12 = 4.0
PI PI
and at the design point we also have T3/TI = 3·3, l1e = 0·8 and 111 = 0·85.
A rough estimate of the equilibrium running line is required on the compressor
characteristic for the conditions where T3!TI is maintained constant and the amount of air
bled off, mb, is varied. For this purpose the efficiencies of the compressor and turbine can
be assumed constant, the mechanical transmission loss and the combustion pressure loss
can be neglected;" and the turbine 'non-dimensional' mass flow can be assumed
independent of speed and related to the pressure ratio r by the expression
where m = me - mb and k is a constant.
As an example of the calculations required, find the operating point on the compressor
characteristic for the condition where mbJTIJpl is reduced to three-quarters of its value at
the design point. Take l' = 1·40 throughout.
[P2/PI =2·1, meJTI/PI = 13·0]
Chapter 1
Appendix C
References
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American Society of Mechanical Engineers, paper 8 I -GT-I 71, 1981.
(2) MORRIS, R. E. The Pratt and Whitney PWI0O--evolution of the design concept,
Canadian Aeronautics and Space Journal, 28, 211-221, 1982.
(3) BRANDT, D. E. The design and development of an advanced heavy-duty gas turbine,
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1988.
(4) SCALZO, A. J., BANNISTER, R. L., de CORSO, M. and HOWARD, G. S.
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Chapter 2
(I) ROGERS, G. F. C. and MAYHEW, Y. R. Engineering Thermodynamics, Work and
Heat Transfer 4th edition (Longman, 1992).
(2) McDONALD, C. F. Heat-exchanger ubiquity in advanced gas turbine cycles,
American Society of Mechanical Engineers, Cogen Turbo Power, 94, 681-703,
1994.
(3) BANES, B., McINTYRE, R. W. and SIMS, J. A. Properiies of air and combustion
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(4) FIELDING, D. and TOPPS, J. E. C. Thermodynamic data for the calculation of gas
turbine performance, Aeronautical Research Council, R&M No. 3099 (HMSO,
1959).
(5) UTILE, D. A., BANNISTER, R. L. and WIANT, B. C. Development of advanced
gas turbine systems, American Society of Mechanical Engineers, Cogen Turbo Power,
93, 271-280, 1993.
(6) LUGAND, P. and PARlEm, C. Combined cycle plants with Frame 9F gas tj.Jrbines,
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REFERENCES 429
and 2 .with combined Cycle, Transactions of the American Society of Mechanical
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(8) ROGERS, G. F. C. and MAYHEW, Y. R. Thermodynamic and Transport Properties
of Fluids (Blackwell, 1995).
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systems, American Society of Mechanical Engineers, paper 94-GT-415, 1994.
Chapter 3
(1) HILL, P. G. and PETERSON, C. R. Mechanics and Thermodynamics of Propulsion,
2nd Edition (Addison-Wesley, 1992).
(2) Relationships between some common intake parameters, Royal Aeronautical Society,
data sheet 66028, 1966.
(3) SEDDON, J. and GOLDSMITH, E. L. Intake Aerodynamics (AlAA Education
Series, 1985).
(4) ASHWOOD, P. F. A review of the performance of exhaust systems for gas turbine
aero-engines. Proceedings of the Institution of Mechanical Engineers, 171, 1957,
129-58.
(5) YOUNG, P. H. Propulsion controls on the Concorde. Journal of the Royal
Aeronautical Society, 70, 1966, 863-81.
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(7) FROST, T. H. Practical bypass mixing systems for fan jet aero-engines. Aeronautical
Quarterly, 17, 1966, 141-60.
(8) SARAVANAMUTTOO, H. I. H. Modern turboprop engines. Progress in Aerospace
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Chapter 4
(I) WHITTLE, F. The early history of Whittle jet propulsion gas turbines, Proceedings of
the Institution of Mechanical Engineers, 152, 1945, 419-35.
(2) CHESHIRE, L. J. The design and development of centrifugal compressors for aircraft
gas turbines, Proceedings of the Institution of Mechanical Engineers, 153, 1945,
426-40.
(3) WRONG, C. B. An introduction to the IT 15D engine, Transactions of the American
Society of Mechanical Engineers, 69-GT-119, 1969.
(4) STANITZ, J. D. Some theoretical aerodynamic investigations of impellers in radial
and mixed-flow centrifugal compressors, Transactions of the American Society of
Mechanical Engineers, 74, 1952, 473-97.
(5) KENNY, D. P. A novel low-cost diffuser for high-performance centrifugal
compressors, Transactions of the American Society of Mechanical Engineers, Series
A, 91, 1969, 37-46.
(6) FERGUSON, T. B. The Centrifogal Compressor Stage (Butterworth, 1963).
(7) HANKINS, G. A. and COPE, W. F. Discussion on 'The flow of gases at sonic and
supersonic speeds', Proceedings of the Institution of Mechanical Engineers, 155,
1947, 401-16.
(8) ENGINEERING SCIENCES DATA UNIT: Fluid Mechanics-internal flow, 4-
Duct expansions and duct contractions, Data Sheets 73024, 74015, 76027.
(9) CAME, P. M. The development, application and experimental eva1uationof a design
procedure for centrifugal compressors, Proceedings of the Institution of Mechanical
Engineers, 192, No.5, 1978, 49-67.
430 REFERENCES
(10) HERBERT, M. V. A method of performance predi<;tion for centrifugal COriIpfessors, .
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Chapter 5
(1) CONSTANT, H. The early history of the axial type of gas turbine engine, .
Proceedings of the Institution of Mechanical Engineers, 153, W.E.P. No. 12, 1945.
(2) HOWELL, A. R. Fluid dynamics of axial compressors and Design of axial
compressors, Proceedings of the Institution of Mechanical Engineers, 153, W.E.P.
No. 12, 1945.
(3) HOWELL, A. R. The present basis of axial compressor design Part I-Cascade
Theory, Aeronautical Research Council, R&M No. 2095 (HMSO, 1942).
(4) JOHNSEN, I. A. and BULLOCK, R. O. Aerodynamic Design ofAxial-flow
Compressors, NASA SP-36, 1965.
(5) HORLOCK, 1. H. Axial Flow Compressors (Butterworth, 1958).
(6) LIEBLEIN, S. and JOHNSEN, I. A. Resume of transonic compressor research at
NACA Lewis Laboratory, Transactions of the American Society of Mechanical
Engineers, Journal of Engineering for Power, 83, 1961, 219--34.
(7) TODD, K. W. Practical aspects of cascade wind tunnel research, Proceedings of the
Institution of Mechanical Engineers, 157, W.E.P. No. 36, 1947.
(8) GOSTELOW, 1. P. Cascade Aerodynamics (pergamon Press, 1984).
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Proceedings of the Institution of Mechanical Engineers, 175, No. 16, 1961,775-88.
(10) MILLER, G. R., LEWIS, G. W. and HARTMAN, M. 1. Shock losses in transonic
blade rows, Transactions of the American Society of Mechanical Engineers, Journal
of Engineering for Power, 83, 1961,235-42.
(11) SCHWENK, F. C., LEWIS, G. W. and HARTMAN, M. 1. A preliminary analysis of
the magnitude of shock losses in transonic compressors, NACA RM E57A30, 1957.
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Aeronautics and Astronautics Journal, 19, 1981, 4-19.
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American Society of Mechanical Engineers, Journal of Fluids Engineering, 102,
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(14) STONE, A. Effects of stage characteristics and matching on axial-flow-compressor
performance, Transactions of the American Society of Mechanical Engineers, 80,
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(15) CARCHEDI, F. and WOOD, G. R. Design and development ofa 12: 1 pressure ratio
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of Mechanical Engineers, Journal of Engineering for Power, 104, 1982,823-31.
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(17) DENTON, 1. D. An improved time marching method for turbomachinery flow
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REFERENCES 431
Chapter 6
(1) SOTHERAN, A. The Rolls-Royce annular vaporizer combustor, American Society of
Mechanical Engineers, paper 83-GT-49, 1983.
(2) Technical Advances in Gas Turbine Design, Institution of Mechanical Engineers
Symposium, 1969.
(3) SPALDING, D. B. Some Fundamentals of Combustion (Butterworths Scientific
Publications, 1955).
(4) ROGERS, G. F. C. and MAYHEW, Y. R. Engineering Thermodynamics, Work and
Heat Transfer 4th edition (Longman, 1992).
(5) LIPPFERT, F. W. Correlation of gas turbine emissions data, American Society of
Mechanical Engineers, paper 72-GT-60, 1972.
(6) LEONARD, G. and STEGMAIER, 1. Development of an aero derivative gas turbine
dry low emissions combustion system, American Society of Mechanical Engineers,
paper 93-GT-288, 1993.
(7) DAVIS, 1. B. and WASHAM, R. M. Development of a dry low NOx combustor,
American Society of Mechanical Engineers, paper 89-GT-255, 1989.
(8) MAGHON, H., BERENBRlNK, P., TERMUEHLEN, H. and GARTNER, G.
Progress in NOx and CO emission reduction of gas turbines, American Society of
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(10) ETHERIDGE, C. 1. Mars SoLoNOx-lean pre-mix combustion technology in
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(13) SUMMERFIELD, A. R., PRITCHARD, D., TUSON, D. W. and OWEN, D. A.
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(14) CORBETT, N. C. and LINES, N. P. Control requirements for the RB211 low
emission combustion system, American Society of Mechanical Engineers, paper 93-
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(15) BAMMERT, K. Operating experiences and measurements on turbo sets of CCGT-
cogeneration plants in Germany, American Society of Mechanical Engineers, paper
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Chapter 7
(1) HAWTHORNE, w. R. ed. Aerodynamics of Turbines and Compressors (Oxford UP.,
1964).
(2) HORLOCK, 1. H. Axial Flow Turbines (Butterworth, 1966).
(3) AINLEY, D. G. and MATHIESON, G. C. R. An examination of the flow and pressure
losses in blade rows of axial flow turbines, Aeronautical Research Council, R&M
2891 (HMSO, 1955).
(4) JOHNSTON, I. H. and KNIGHT, 1. R. Tests on a single-stage turbine comparing the
performance of twisted with untwisted rotor blades, Aeronautical Research Council,
R&M 2927 (HMSO, 1953).
432 REFERENCES
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design, Aeronautical Research Council C.P. 868 (HMSO, 1966).
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axial-flow turbines, Aeronautical Research Council, R&M 2974 (HMSO, 1951).
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Aeronautical Research Council, R&M 3013 (HMSO, 1957).
(10) HAWTHORNE, W R. Thermodynamics of cooled turbines Parts I and n,
Transactions of the American Society of Mechanical Engineers, 78, 1956, 1765-81.
(ll) BARNES, 1. F. and FRAY, D. E. An experimental high-temperature turbine (No.
126), Aeronautical Research Council, R&M 3405 (HMSO, 1965).
(12) Technical Advances in Gas Turbine Design, Institution of Mechanical Engineers,
symposium (1969).
(13) DUNHAM, J. and CAME, P. M. Improvements to the Ainley-Mathieson method of
turbine performance prediction, Transactions of the American Society of Mechanical
Engineers, Journal of Engineering for Power, 92, 1970,252-256.
(14) KACKER, S. C. and OKAPUU, U. A mean line prediction method for axial flow
turbine efficiency, Transactions of the American Society of Mechanical Engineers,
Journal of Engineering for Power, 104, 1982, 111-119.
(15) MOWILL, J. and STROM, S. An advanced radial-component industrial gas turbine,
Transactions of the American Society of Mechanical Engineers, Journal of
Engineering for Power, 105, 1983, 947-52.
(16) DIXON, S. 1. Fluid Mechanics, Thermodynamics of Turbomachinery (Pergamon,
1975).
(17) Aerodynamic Performance of Radial Inflow Turbines. First (1963) and Second (1964)
Reports, Motor Industry Research Association. (Alternatively, the work is
summarized in: HIETT, G. F. and JOHNSTON, 1. H. Experiments concerning the
aerodynamic performance of inward radial flow turbines, Proceedings of the
Institution of Mechanical Engineers, 178, Part 3I(ii), 1964.
(18) BENSON, R. S. A review of methods for assessing loss coefficients in radial gas
turbines, International Journal of Mechanical Science, 12, 1970, 905-32.
(19) BRIDLE, E. A. and BOULTER, R. A. A simple theory for the prediction oflosses in
the rotors of inward radial flow turbines, Proceedings of the Institution of Mechanical
Engineers, 1112, Part 3H, 1968.
(20) BENSON, R. S. Prediction of performance of radial gas turbines in automotive
turbochargers, American Society of Mechanical Engineers, paper 71-GT-66, 1971.
Chapter 9
(l) MALLINSON, D. H. and LEWIS, W G. E. The part-load performance of various
gas-turbine engine schemes, Proceedings of the Institution of Mechanical Engineers,
159, 1948, 198-219.
(2) TREWBY, G. F. A. British naval gas turbines, Transactions of the Institution of
Marine Engineers, 66, 1954, 125-67.
(3) SWATMAN, I. M. and MALOHN, D. A. An advanced automotive gas turbine
concept, Transactions of the Society of Automotive Engineers, 69, 1961,219-27.
(4) BARBEAU, D. E. The performance of vehicle gas turbines, Transactions of the
Society of Automotive Engineers, 76Iv], 1967, 90.
REFERENCES 433
(5) KAYS, W M. and LONDON, A. 1. Compact Heat Exchangers (McGraw-Hill,
1964).
(6) RAHNKE, C. J. The variable-geometry power turbine, Transactions of the Society of
Automotive Engineers, 711[i], 1969, 213-23.
(7) YOUNG, P. H. Propulsion controls on the Concorde, Journal of the Royal
Aeronautical Society, 70, 1966, 863-81.
(8) SARAVANAMUTTOO, H. 1. H. and FAWKE, A. 1. Simulation of gas turbine
dynamic performance, American Society of Mechanical Engineers, paper 70-GT-23,
1970.
(9) FAWKE, A. 1. and SARAVANAMUTTOO, H. I. H. Experimental investigation of
methods for improving the dynamic response of a twin-spool turbojet engine,
Transactions of the American Society of Mechanical Engineers, 93, series A, 1971,
418-24.
(10) FAWKE, A. J. and SARAVANAMUTTOO, H. I. H. Digital computer methods for
prediction of gas turbine dynamic response, Transactions of the Society of Automotive
Engineers, 80 [iii], 1971, 1805-13.
(11) SARAVANAMUTTOO, H. I. H. and MacISAAC, B. D. An overview of engine
dynamic response and mathematical modelling concepts, AGARD Conference
Proceedings No. 324 'Engine Handling', 1982.
(12) SARAVANAMUTTOO, H.!. H. and MacISAAC, B. D. Thermodynamic models for
pipeline gas turbine diagnostics, Transactions of the American Society of Mechanical
Engineers, 105, Series A, 1983, 875-84.
Appendix A
(1) HOUGHTON, E. L. and BROCK, A. E. Tables for the Compressible Flow of Diy Air
(Arnold, I 970}.
(2) KEENAN, 1. H. and KAYE, 1. Gas Tables (Wiley, 1948).
(3) SHAPIRO, A. H. The Dynamics and Thermodynamics of Compressible Flow
(Ronald, 1954).
(4) LIEPMANN, H. W. and ROSHKO, A. Elements of Gas Dynamics (Wiley, 1957).
Supplementary texts
BATHIE, W W Fundamentals of Gas Turbines, 2nd Edition (John Wiley & Sons, 1995).
CUMPSTY, N. A. Compressor Aerodynamics (Longman, 1989).
DIXON, S. 1. Fluid Mechanics, Thermodynamics of TurbomachinelY, 3rd Edition
(pergamon, 1978).
HARMAN, R. T. C. Gas Turbine Engineering (Macmillan, 1981).
LEFEBVRE, A. H. Gas Turbine Combustion (McGraw-Hill, 1983).
MATTINGLEY, J. D., HEISER, W. H. and DALEY, D. H. Aircraft Engine Design (AIAA
Education Series, 1987).
OATES, G. C. Aerothermodynamics of Gas Turbine and Rocket Propulsion (AIAA
Education Series, 1984).
SAWYER, J. W. ed. Gas Turbine Handbook (Turbomachinery International Publications
3rd Edition, 1980).
SMITH, M. J. 1. Aircraft Noise (Cambridge University Press, 1989).
WHITFIELD, A. and BAINES, N. C. Design of Radial Turbomachines (Longman, 1990).
WILSON, D. G. Design of High Efficiency Turbomachinery and Gas Turbines (MIT Press,
1984).
Aerodynamic coupling, 229, 376, 380
Aft-fan, 116
Afterburning, 95, 120
pressure loss, 122
Air angles, 159, 183, 190, 199
Air cooling, 45, 251, 301, 320
Air/fuel ratio, see Fuel/air ratio
Air separation unit, 268
Aircraft
gas turbines, 12, 86
propulsion cycles, 86
Altitude, effect on performance, 103,
105, 234, 361, 364
Ambient conditions, effect of, 337, 363
Annular combustion chamber, 12,236,
244
Annulus, contraction, 140, 189
drag, 211
loss, 293, 314
radius ratio, 160, 170, 179, 285
Applications, industrial, 16
Aspect ratio, 205, 296
Atomization, 252
Axial compressor, 8, 154
blading, 155, 199, 206
characteristics, 225, 338, 366
stage, 155, 157
surging in, 222, 225, 228
vortex flow in, 169
Axial flow turbine, 271
blade profile, 294, 303
characteristics, 319, 339, 346
choking, 319, 346, 377
cooling, 320
Axial flow turbine, cant.
free power, 6, 65, 389
multi-stage, 319
nozzles, 271, 280, 321
stage, 271
Index
stage efficiency, 274, 279, 317
Backswept vanes, 128, 135
Biconvex blading, 156, 207, 218
Binary cycle: see Combined power
plant
Blade, aspect ratio, 205, 296
camber, 201, 206, 300
cascade, 199
c h o r ~ 163, 201, 296
efficiency, 213
fan, 207
loading coefficient, 274, 323
loss coefficient, 208, 211, 217, 278,
293,312,316
pitch, 163, 201, 296, 302
pressure distribution, 304
profile, 206, 218, 294, 303
relative temperature, 323
root, 302
stagger, 201, 206
stresses, 160, 298, 302, 310
taper, 160, 298
tip clearance, 211, 293, 314
velocity distribution, 163, 304
Blading design
constant nozzle angle, 291
constant reaction, 177, 194, 197
exponential, 176, 195, 197
436
Blading design, cont.
first power, 176
free vortex, 173, 190, 197, 288
Bleeds, cooling, 45, 61, 322
Blow-off, 229, 343, 366
Boundary layer, 200, 293
separation, 286, 304, 399
transition, 304, 326
Burner
double-cone, 265
dual-fuel, 253, 265
duplex, 253
hybrid, 264
simplex, 252
spill, 253
Bypass
engine: see Turbofan
ratio, 106, 112
Camber
angle, 201, 206, 300
line, 206
Carbon formation, 237
Cascade, notation, 202
of blades, 200
pressure loss, 203
test results, 203
tunnel, 200
Centrifugal bending stress, 299, 301
Centrifugal compressor, 12, 126
characteristics, 148
surging, 149
Centrifugal tensile stress, 160, 298,
310
Ceramic, rotor, 328
combustor lining, 30, 238
Characteristics, axial compressor, 225,
338, 368
centrifugal compressor, 148
load, 341
power turbine, 339, 346
propelling nozzle, 358, 367
torque, 354
turbine, 319, 339, 349
Chilling offiame, 239, 241
Choking, in axial compressor, 225,
228
in centrifugal compressor, 151
Choking, cant.
in duct, 402
INDEX
in propelling nozzle, 98, 358, 362
in turbine, 320, 346, 376
Chord, 163,201,296
Circular arc blading, 156, 207,218
Climb rating, 363
Closed cycle, 3, 9, 81, 267
Coal gasification, 31, 257, 267
Coefficient, annulus drag, 211
blade loading, 274, 323
blade loss, 208, 211, 217, 278, 293,
312,316
flow, 222, 275, 323
heat transfer, 82, 324
lift, 208,210
nozzle loss, 278, 313, 315, 330
overall drag, 212
profile drag, 208, 313
profile loss, 293, 313, 317
rotor loss, 331
secondary loss, 212, 314
temperature drop, 274, 323
Cogeneration plant (CHP), 26, 76, 79
Combined cycle plant, 4, 22, 31, 76
Combustion
efficiency, 58, 245
emissions, 258
fluidized bed, 30
intensity, 249
mixing in, 239, 242
pressure loss, 53, 241, 243
process, 239
stability, 239, 248
Combustor (combustion chamber),
annular, 12, 236, 244
can (or tubular), 12, 235, 244
cannular (or tubo-annular), 235,
244
dry low-NOx, 263
emissions, 258
flame tube, 239, 251
industrial, 237, 245, 264
reverse flow, 13, 235
silo, 237
Complex cycles, 7, 76, 372
Compressed air storage, 32
Compressibility effects, 396
in axial compressors, 161, 216
INDEX
Compressibility effects, cant.
. in centrifugal compressors, 141
in turbines, 280, 310
Compressor, axial, 8, 154
centrifugal, 12, 126
characteristics, 148, 225, 338, 368
supersonis; and transonic, 154, 156,
218
test rigs, 227
twin-spool, 8, 229
Conservation equations, 400
Constant mass flow design, 292
Constant nozzle angle design, 291
Constant pressure cycle, 3, 38
intercooling, 7, 44, 73
reheating, 7, 42, 67, 73
with heat exchange,S, 40, 63, 72
Constant reaction blading, 177, 194,
197
Constant volume cycle, 3
Control systems, 259, 392
Convective air cooling, 251, 301, 320
Convergent-divergent nozzle, 95, 280,
398
Cooled turbine, 320
Critical, pressure ratio, 95, 98, 278,
283, 358
Mach number in cascade, 217
Cruise rating, 363
Cycle, aircraft propulsion, 86, 91, 99,
102, 105
closed, 3, 9, 81
complex, 7, 76, 372
constant pressure, 3, 38
constant volume, 3
efficiency, 39, 45, 61, 70
heat-exchange, 40, 43, 63, 72
ideal, 37
intercooled, 44, 73
Joule, 38
open, 3, 38
reheat, 42, 67, 73
simple, 37, 70
shaft power, 37
steam and gas, 4, 22, 31, 76
turbofan, 12, 106, 112
turbojet, 12, 87, 99, 102
turboprop, 12, 117
turboshaft, 13, 120
437
Dampers, part span, 221
de Haller number, 163, 183-9, 208
Deflection, 163, 199,202, 204
nominal, 204
Degree of reaction: see Reaction
Design point perfOlmance, 63, 336
heat-exchange cycle, 63, 72
intercooled cycle, 73
reheat cycle, 67, 73
turbofan cycle, 106, 112
turbojet cycle, 99, 102
Deviation angle, 202, 205
Diffuser, 127, 134, 136, 140
vanes, 139, 150
volute, 141
Diffusion factor, 163, 164, 208
Dilution zone, 239
Dimensional analysis, 146
Drag, momentum, 87, 103, 105, 115,
122
pod, 114
Drag coefficient, 208, 212
annulus, 211
profile, 208
secondary loss, 212
Duplex burner, 253
Dynamic head, 54,242
temperature, 46, 247
Effectiveness of heat-exchanger, 54,
73, 83
Efficiency
blade, 213
combustion, 58, 245
compressor and turbine, 48, 49
compressor blade row, 213
cycle, 39, 45, 61, 70
Froude,88
intake, 92
isentropic, 48, 49
mechanical transmission, 55
nozzle, 97
overall, 89
part-load, 5, 337, 352, 371
polytropic, 50, 67, 216, 319
propelling nozzle, 97
propulsion, 87
stage, 214, 215, 274, 279, 317
438 INDEX INDEX 439
Efficiency, cant. Fuel, cant. Intake, efficiency, 92 Noise, 96, 115, 119, 123, 156
total-to-static, 274, 330 consumption, nOll"dimensionitl, 364 momentum drag, 87, 103, 105, 110, Nominal deflection, 204
274 injection, 252 122 Non-dimensional quantities, 146
Effusion cooling: see Transpiration staging, 263, 266 Oswatitsch, 413 fuel consumption, 364
cooling Fuels, 256 pitot, 408 pressure loss, 54, 242, 243
Electrical power generation, 20, 76, Fundamental pressure loss, 241.244, pressure recovery factor, 94 thrust, 361
337, 353, 357, 389 403 variable 413 Normal shock, 398, 406, 408
Emissions, 258 Intercooling, 7, 44, 73 Nozzle
End bend blading, 231 Intemational Standard Atmosphere, characteristics, 358, 367
Engine braking, 375 Gas 90, 125 convergent-divergent, 95, 280, 398
Engine health monitoring, 394 angles, 272 Isentropic efficiency, 97
Equilibrium running diagram, 336 bending stress, 299, 302, 310 efficiency, 48, 49 loss coefficient, 278, 313, 315, 330
gas generator, 345 dynamics, 396 flow, 402, 413 turbine, 271, 278, 281, 322
shaft power unit, 342, 348 generator, 6, 338, 345 (see also Propelling nozzle)
turbofan unit, 383 Gasification plant, 31, 267 Nusse1t number, 82, 326
turbojet unit, 361 Greenhouse gases, 28, 258
twin-spool unit, 375, 381 Jet pipe, 95, 99
Equilibrium running line, 336, 342, temperature, 248, 393
Oblique shock wave, 398, 410
348, 366, 381 Heat-exchanger, 5, 7, 13,43, 53, 63, Jou1e cycle, 38
Equivalent, flow, 151 72, 82
Off-design performance, 222, 336
power, 118 effectiveness, 54, 73, 83
free turbine engine, 346, 352
speed, 151 pressure loss, 53, 83
single-shaft engine, 340
Evaporative cooling, 357 Heat Lift coefficient, 208, 210
turbofan, 383
Exponential blading, 176, 195, 197 rate, 61 Load characteristic, 341
turbojet, 358
Eye of impeller, 129, 131, 142 release, 249 Low emission systems, 261
twin-spool engine, 375, 379, 390
transfer coefficient, 82, 324
Open cycle, 3, 38
Helium working fluid, 10, 81
Oswatitsch intake, 413
Hub-tip ratio, 160, 162, 169, 179,285
Overall efficiency, 89
Fan, blade, 221 Mach angle, 397
pressure ratio, 107, 112-4 Mach number, 90, 397, 401
Fanno flow, 405, 409 Ice harvesting, 357 before and after heat release, 404 Part-load
Fir tree root, 302 Ideal cycle, 37 change through shock wave, 407 efficiency, 5, 337, 352, 371
First power designs, 176 heat-exchange, 40 in axial compressors, 161, 181, 197, performance, 337, 350, 361, 371,
Flame, chilling of, 239, 241 intercooling, 44 217 374
stabilization, 239 reheat, 42 in diffuser, 145 Part-span dampers, 221
temperature, 260, 269 reheat and heat-exchange, 43 in impeller, 142 Peak-load generation, 5, 20, 32
tube, 235, 251 simple constant pressure, 38 in turbine, 280, 286, 310, 317 Pipelines, 20, 55, 234
Flow turbojet, 91 Mach wave, 397, 414 Pitch, 163, 201, 296, 302
coefficient, 222, 275, 323 Ignition, 253 Marine gas turbine, 23, 76, 372 Pitch/chord ratio, 204, 211, 296, 302
steady one-dimensional, 400 Impeller, 127 Matrix through flow method, 231 Pitot intake, 408
Fluidized bed combustor, 30 centrifugal stresses, 131 Mechanical losses, 55 Plane normal shock wave, 398, 406,
Foreign object damage, 205, 221 eye, 127, 142 Method of characteristics, 414 408
Free turbine, 6, 65, 338, 346, 352, 389 loss, 134 Mixing, in combustion, 239, 242 efficiency, 408
Free vortex blading, 173, 190, 197, vane inlet angle, 129, 132, 142 in nozzles, 111 pressure ratio, 407
288 vibration, 146, 149 Momentum Pod drag, 114
Froude efficiency, 88 Incidence, 201, 203, 217, 295 drag, 87, 103, 105, 110, 115, 122 Pollution, 28, 232, 258, 260
FueI/air ratio, 58, 237, 245, 248, 343 Industrial gas turbine, 17 thrust, 87, 96 Polytropic efficiency, 50, 67,216,319
Fuel, atomization, 252 Inlet guide vanes, 116, 142, 155, 162, Multi-spool, 8 Power input factor, 130
burner, 252, 264 206 Mu1ti-stage turbine, 319 Power turbine: see Free turbine
440
INDEX INDEX 441
Prandtl-Meyer flow, 414
Repowering;80 Stage, COrlt. Thrust, cont.
Pressure, stagnation, 47, 401 Residence tUne, 260 stackiIi:g, 224 power, 117
loss
Response 385, 389 turbine, 271 pressure, 87, 96
in cascade, 203
Reynolds number effect, 147,227, Stagger angle, 201, 206 specific, 90, 102, 105
in combustion system, 53, 241, 244 318, 326 Stagnation spoiler, 96
in cycle calculations, 53, 369 Rig testing, 227, 236
enthalpy, 46 Time marclllng method, 231
factor, 244
Rotating stall, 150, 227 pressure, 401 Tip clearance, compressor, 211
fundamental, 241, 404 Rotor
temperature, 46, 245, 401 turbine, 293, 314, 323
Pressure
blade loss coefficient, 279, 312, 316 thermocouple, 247 Tip speed
coefficient, 224
loss coefficient, 331 Stalling, 150, 155,203,224,229 axial compressor, 160
ratio, critical, 95, 98,278,283,358
Starting, 253, 389 centrifugal compressor, 129, 131
recovery factor, 94
Stator blades, 155, 271 Torch igniter, 256
thrust, 87, 96 Sauter mean diameter, 252
one-dimensional flow, 400 Torque characteristics, 354
Prewhirl, 142, 144 Secondary
Steam and gas cycle, 4, 22, 31, 76 Torque, net, 386, 389
Primary zone, 239 losses, 211, 293, 314
Steam cooling, 262 Total: see Stagnation
Profile, blade, 206, 218,294, 303 zone, 239
Steam injection, 28, 262 Total-energy: see Cogeneration plant
drag coefficient, 208 Selective catalytic reduction, 262
Streamline curvature method, 231 Total-to-static efficiency, 274, 330
loss coefficient, 293, 313, 317 Separation of boundary layer, 286,
Stresses, blade, 160,298,302,310 Total-to-total efficiency, 274
Propeller turbine engine, 12, 117,389 304,399
disc, 303 Transient performance, 385
Propelling nozzle, 95 Single-shaft engine, 5, 340, 352, 389
impeller, 131 Transient running line, 388, 390
characteristics, 358, 367 Shaft power cycles, J 7
Supersonic Transonic compressor, 154, 156, 218
choking, 98, 358, 362 Shock losses, 142, 145, 218
compressor, 154 Transpiration cooling, 251, 321
convergent V. conv.-div., 95 Shockwave, 141,398
diffusion, 92, 218, 402 Tubular combustion chamber, 12,235,
efficiency, 97 diffusion by, 408, 412
expansion, 280, 402 244
mixing in, 111 efficiency, 408, 412
Surface dischargs: igniter, 255 Turbine: see Axial flow turbine; Radial
trimmer, 99 in centrifugal compressor, 142, 145
Surge line, 151, 225, 336, 342 flow turbine
variable area, 96, 122, 366, 377, in turbine, 280, 286
Surging, 149, 225, 228, 336, 342, 390 Turbofan, 13, 106, 112, 116
380 oblique, 398,410
Swirl, angle, 272, 277 design point performance, 106, 112
Propulsion efficiency, 87 on aerofoi1, 399
in combustion, 239 off-design performance, 383
plane normal, 398, 406
Symmetrical blading, 168 mixing in nozzle, 111
Simplex burner, 252
Turbojet, 12, 87, 99, 102
Radial
Site rating, 80
design point performance, 99, 102
equilibrium 170, 172, 177, 288 Slip factor, 129
equilibrium running diagram, 361
flow compressor: see Centrifugal Solidity, 131, 165,218
Take-off rating, 363 off-design performance, 361, 364
compressor Sonic velocity, 91, 396
Temperature, dynamic, 46, 247 surging, 366, 380
flow turbine, 328 Specific
measurement, 245 Turboprop, 12, 117, 389
Radius ratio of annulus, 160, 169, 179, fuel consumption, 45, 70, 89, 102,
stagnation, 46, 245, 401 Turboshaft, 13
285 343, 351, 363, 372
static, 41 Twin-shaft engine, 6
Ram
heat, variation of, 56
weighted mean, 245 (see also Free turbine)
compression, 92, 105 thrust, 90, 102, 105
Temperature coefficient, 222 Twin-spool engine, 8, 229, 375, 379,
efficiency, 92 work output, 39
Temperature drop coefficient, 274, 323 390
Rayleigh flow, 403, 410 Spill burner, 253
Thermal choking, 405
Reaction, degree of Spray cooling, 321
Thermal-ratio: see Effectiveness
in compressor, 167, 173, 185 Stability of combustion, 239, 248
Thermocouple, 247, 393
Unducted fan, 119
in turbine, 274, 281, 288 Stage
Thickness/chord ratio, 294, 312
Regenerative cycle: see Cycle, heat- axial compressor, 155, 157
Thrust, augmentation, 120
exchange characteristics, 223
momentmn, 87, 96
Regenerator, 7, 32 efficiency, 214, 215, 274, 279, 317
net, 86, 105, 107, 361 Vanes, diffuser, 139, 150
Reheating, 7,42, 67, 73 repeating, 272
non-dimensional, 361 impeller, 127, 135, 138
442
Vanes, cont. .
inlet guide, 116, 142, 155, 162,206
VaIleless space, 137, 146
Vaporizer system, 240, 251
Variable
area propelling nozzle, 96, 122,
366, 377, 380
area power turbine stators, 373
compressor stators, 229
cycle engine, 104
geometry intake, 413
inlet guide vanes, 155
pitch fan, 119
Vehicular gas turbine, 25, 353, 372
Velocity diagram, axial compressor,
157, 163
turbine, 272, 273
Vibration, compressor blade, 205
Vibration; . cant;
fan piac:le; 221 .
impeller vane, 146, 149
. turbine blade, 286, 297
Volute, 141
Vortex 172
Vortex flow in compressor, 169
in turbine, 287
INDEX
Waste heat boiler, 22, 76, 79,262
Water injection, 28, 262
Weighted mean temperature, 245
Windage loss, 56, 130
Work done factor, 166
Yawmeter, 201

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