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Fluid Mechanics,

Thermodynamics of

Turbomachinery

S.L. Dixon, B.Eng., PH.D.

Senior Fellow at the University of Liverpool

FOURTH EDITION in SI/METRIC UNITS

Fluid Mechanics,

Thermodynamics of

Turbomachinery

FOURTH EDITION in SI/METRIC UNITS

In memory of

Avril and baby Paul

Fluid Mechanics,

Thermodynamics of

Turbomachinery

S. L. Dixon, B.Eng., Ph.D.

Senior Fellow at the University of Liverpool

FOURTH EDITION in SI/METRIC UNITS

Butterworth-Heinemann

Linacre House, Jordan Hill, Oxford OX2 8DP

225 Wildwood Avenue, Woburn, MA 01801-2041

A division of Reed Educational and Professional Publishing Ltd

A member of the Reed Elsevier plc group

First published by Pergamon Press Ltd 1966

Second edition 1975

Third edition 1978

Reprinted 1979, 1982 (twice), 1984, 1986, 1989, 1992, 1995

Fourth edition 1998

 S.L. Dixon 1978, 1998

All rights reserved. No part of this publication

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Designs and Patents Act 1988 or under the terms of a

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Applications for the copyright holder’s written permission

to reproduce any part of this publication should be addressed

to the publishers

British Library Cataloguing in Publication Data

A catalogue record for this book is available from the British Library

ISBN 0 7506 7059 2

Library of Congress Cataloguing in Publication Data

A catalogue record for this book is available from the Library of Congress

Typeset by Laser Words, Madras, India

Printed and bound in

Contents

PREFACE TO FOURTH EDITION ix

PREFACE TO THIRD EDITION xi

ACKNOWLEDGEMENTS xiii

LIST OF SYMBOLS xv

1. Introduction: Dimensional Analysis: Similitude 1

Definition of a turbomachine 1

Units and dimensions 3

Dimensional analysis and performance laws 4

Incompressible fluid analysis 6

Performance characteristics 7

Variable geometry turbomachines 9

Specific speed 10

Cavitation 12

Compressible gas flow relations 15

Compressible fluid analysis 16

The inherent unsteadiness of the flow within turbomachines 20

References 21

Problems 22

2. Basic Thermodynamics, Fluid Mechanics: Definitions of Efficiency 23

Introduction 23

The equation of continuity 23

The first law of thermodynamics internal energy 24

The momentum equation Newton’s second law of motion 25

The second law of thermodynamics entropy 29

Definitions of efficiency 30

Small stage or polytropic efficiency 35

Nozzle efficiency 41

Diffusers 43

References 53

Problems 53

vi Contents

3. Two-dimensional Cascades 55

Introduction 55

Cascade nomenclature 56

Analysis of cascade forces 57

Energy losses 59

Lift and drag 59

Circulation and lift 61

Efficiency of a compressor cascade 62

Performance of two-dimensional cascades 63

The cascade wind tunnel 63

Cascade test results 65

Compressor cascade performance 68

Turbine cascade performance 70

Compressor cascade correlations 71

Fan blade design (McKenzie) 80

Turbine cascade correlation (Ainley) 81

Comparison of the profile loss in a cascade and in a turbine stage 86

Optimum space-chord ratio of turbine blades (Zweifel) 87

References 88

Problems 90

4. Axial-flow Turbines: Two-dimensional Theory 93

Introduction 93

Velocity diagrams of the axial turbine stage 93

Thermodynamics of the axial turbine stage 94

Stage losses and efficiency 96

Soderberg’s correlation 97

Types of axial turbine design 99

Stage reaction 101

Diffusion within blade rows 103

Choice of reaction and effect on efficiency 107

Design point efficiency of a turbine stage 108

Maximum total-to-static efficiency of a reversible turbine stage 112

Stresses in turbine rotor blades 114

Turbine flow characteristics 120

Flow characteristics of a multistage turbine 122

The Wells turbine 124

References 132

Problems 133

5. Axial-flow Compressors and Fans 137

Introduction 137

Two-dimensional analysis of the compressor stage 138

Velocity diagrams of the compressor stage 140

Thermodynamics of the compressor stage 141

Contents vii

Stage loss relationships and efficiency 142

Reaction ratio 143

Choice of reaction 143

Stage loading 144

Simplified off-design performance 145

Stage pressure rise 147

Pressure ratio of a multistage compressor 148

Estimation of compressor stage efficiency 149

Stall and surge phenomena in compressors 154

Control of flow instabilities 159

Axial-flow ducted fans 160

Blade element theory 162

Blade element efficiency 163

Lift coefficient of a fan aerofoil 164

References 165

Problems 166

6. Three-dimensional Flows in Axial Turbomachines 169

Introduction 169

Theory of radial equilibrium 169

The indirect problem 171

The direct problem 179

Compressible flow through a fixed blade row 180

Constant specific mass flow 181

Off-design performance of a stage 183

Free-vortex turbine stage 184

Actuator disc approach 186

Blade row interaction effects 190

Computer-aided methods of solving the through-flow problem 191

Secondary flows 193

References 195

Problems 196

7. Centrifugal Pumps, Fans and Compressors 199

Introduction 199

Some definitions 200

Theoretical analysis of a centrifugal compressor 202

Inlet casing 203

Impeller 203

Conservation of rothalpy 204

Diffuser 205

Inlet velocity limitations 205

Optimum design of a pump inlet 206

Optimum design of a centrifugal compressor inlet 208

Slip factor 213

Head increase of a centrifugal pump 218

viii Contents

Performance of centrifugal compressors 219

The diffuser system 227

Choking in a compressor stage 230

References 232

Problems 233

8. Radial Flow Gas Turbines 236

Introduction 236

Types of inward flow radial turbine 237

Thermodynamics of the 90 deg IFR turbine 239

Basic design of the rotor 241

Nominal design point efficiency 242

Mach number relations 246

Loss coefficients in 90 deg IFR turbines 247

Optimum efficiency considerations 248

Criterion for minimum number of blades 253

Design considerations for rotor exit 256

Incidence losses 260

Significance and application of specific speed 263

Optimum design selection of 90 deg IFR turbines 266

Clearance and windage losses 269

Pressure ratio limits of the 90 deg IFR turbine 269

Cooled 90 deg IFR turbines 271

References 272

Problems 273

9. Hydraulic Turbines 277

Introduction 277

Hydraulic turbines 278

The Pelton turbine 281

Reaction turbines 290

The Francis turbine 290

The Kaplan turbine 296

Effect of size on turbomachine efficiency 299

Cavitation 301

References 305

Problems 306

Bibliography 309

Appendix 1. Conversion of British and US Units to SI Units 310

Appendix 2. Answers to Problems 311

Index 315

Preface to the Fourth Edition

It is now twenty years since the third edition of this book was published and in

that period many advances have been made to the art and science of turboma￾chinery design. Knowledge of the flow processes within turbomachines has increased

dramatically resulting in the appearance of new and innovative designs. Some of

the long-standing, apparently intractable, problems such as surge and rotating stall

have begun to yield to new methods of control. New types of flow machine have

made their appearance (e.g. the Wells turbine and the axi-fuge compressor) and

some changes have been made to established design procedures. Much attention

is now being given to blade and flow passage design using computational fluid

dynamics (CFD) and this must eventually bring forth further design and flow effi￾ciency improvements. However, the fundamentals do not change and this book is

still concerned with the basics of the subject as well as looking at new ideas.

The book was originally perceived as a text for students taking an Honours degree

in engineering which included turbomachines as well as assisting those undertaking

more advanced postgraduate courses in the subject. The book was written for engi￾neers rather than mathematicians. Much stress is laid on physical concepts rather

than mathematics and the use of specialised mathematical techniques is mostly kept

to a minimum. The book should continue to be of use to engineers in industry

and technological establishments, especially as brief reviews are included on many

important aspects of turbomachinery giving pointers to more advanced sources of

information. For those looking towards the wider reaches of the subject area some

interesting reading is contained in the bibliography. It might be of interest to know

that the third edition was published in four languages.

A fairly large number of additions and extensions have been included in the

book from the new material mentioned as well as “tidying up” various sections

no longer to my liking. Additions include some details of a new method of fan

blade design, the determination of the design point efficiency of a turbine stage,

sections on centrifugal stresses in turbine blades and blade cooling, control of flow

instabilities in axial-flow compressors, design of the Wells turbine, consideration of

rothalpy conservation in impellers (and rotors), defining and calculating the optimum

efficiency of inward flow turbines and comparison with the nominal design. A

number of extensions of existing topics have been included such as updating and

extending the treatment and application of diffuser research, effect of prerotation

of the flow in centrifugal compressors and the use of backward swept vanes on

their performance, also changes in the design philosophy concerning the blading of

axial-flow compressors. The original chapter on radial flow turbines has been split

into two chapters; one dealing with radial gas turbines with some new extensions

and the other on hydraulic turbines. In a world striving for a ‘greener’ future it was

felt that there would now be more than just a little interest in hydraulic turbines. It

is a subject that is usually included in many mechanical engineering courses. This

chapter includes a few new ideas which could be of some interest.

x Preface to the Fourth Edition

A large number of illustrative examples have been included in the text and many

new problems have been added at the end of most chapters (answers are given at the

end of the book)! It is planned to publish a new supplementary text called Solutions

Manual, hopefully, shortly after this present text book is due to appear, giving the

complete and detailed solutions of the unsolved problems.

S. Lawrence Dixon

Preface to Third Edition

Several modifications have been incorporated into the text in the light of recent

advances in some aspects of the subject. Further information on the interesting

phenomenon of cavitation has been included and a new section on the optimum

design of a pump inlet together with a worked example have been added which

take into account recently published data on cavitation limitations. The chapter on

three-dimensional flows in axial turbomachines has been extended; in particular the

section concerning the constant specific mass flow design of a turbine nozzle has

been clarified and now includes the flow equations for a following rotor row. Some

minor alterations on the definition of blade shapes were needed so I have taken the

opportunity of including a simplified version of the parabolic arc camber line as

used for some low camber blading.

Despite careful proof reading a number of errors still managed to elude me in the

second edition. I am most grateful to those readers who have detected errors and

communicated with me about them.

In order to assist the reader I have (at last) added a list of symbols used in the

text.

S.L.D.

xi

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Acknowledgements

The author is indebted to a number of people and manufacturing organisations for

their help and support; in particular the following are thanked:

Professor W. A. Woods, formerly of Queen Mary College, University of London

and a former colleague at the University of Liverpool for his encouragement of the

idea of a fourth edition of this book as well as providing papers and suggestions for

some new items to be included. Professor F. A. Lyman of Syracuse University, New

York and Professor J. Moore of Virginia Polytechnic Institute and State University,

Virginia, for their helpful correspondence and ideas concerning the vexed question

of the conservation of rothalpy in turbomachines. Dr Y. R. Mayhew is thanked for

supplying me with generous amounts of material on units and dimensions and the

latest state of play on SI Units.

Thanks are also given to the following organisations for providing me with illus￾trative material for use in the book, product information and, in one case, useful

background historical information:

Sulzer Hydro of Zurich, Switzerland; Rolls-Royce of Derby, England; Voith

Hydro Inc., Pennsylvania; and Kvaerner Energy, Norway.

Last, but by no means least, to my wife Rose, whose quiet patience and support

enabled this new edition to be prepared.

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