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 may be reproduced in any material form (including photocopying or storing in any medium by electronic means and whether or not transiently or incidentally to some other use of this publication) without the written permission of the copyright holder except in accordance with the provisions of the Copyright, Designs and Patents Act 1988 or under the terms of a license issued by the Copyright Licensing Agency Ltd, 90 Tottenham Court Road, London, England W1P 9HE. 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 P REFACE TO THIRD EDITION xi A CKNOWLEDGEMENTS xiii L IST 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. [...]... section of this chapter concerned with specific speed Turbomachines are further categorised according to the nature of the flow path through the passages of the rotor When the path of the through-flow is wholly or mainly parallel to the axis of rotation, the device is termed an axial flow turbomachine (e.g 1 2 Fluid Mechanics, Thermodynamics of Turbomachinery FIG 1.1 Diagrammatic form of various types of turbomachine... defined above, is at the point of maximum efficiency of a turbomachine, it becomes a parameter of great importance in selecting the type of machine required for a given duty The maximum efficiency condition replaces the condition of geometric similarity, so that any alteration in specific 12 Fluid Mechanics, Thermodynamics of Turbomachinery FIG 1.7 Range of pump impellers of equal inlet area speed implies... 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... mechanics and thermodynamics are the metre (m), kilogram (kg), second (s) and thermodynamic temperature (K) All the other units used in this book are derived from these basic units The unit of force is the 4 Fluid Mechanics, Thermodynamics of Turbomachinery newton (N), defined as that force which, when applied to a mass of 1 kilogram, gives an acceleration to the mass of 1 m/s2 The recommended unit of pressure... 1.1(f), is an example of an impulse turbine The main purpose of this book is to examine, through the laws of fluid mechanics and thermodynamics, the means by which the energy transfer is achieved in the chief types of turbomachine, together with the differing behaviour of individual types in operation Methods of analysing the flow processes differ depending upon the geometrical configuration of the machine,... form, a selection of the many different varieties of turbomachine encountered in practice The reason that so many different types of either pump (compressor) or turbine are in use is because of the almost infinite range of service requirements Generally speaking, for a given set of operating requirements there is one type of pump or turbine best suited to provide optimum conditions of operation This... examples In SI the units of all forms of energy are the same as for work The unit of energy is the joule (J) which is the work done when a force of 1 newton is displaced through 1 a distance of 1 metre in the direction of the force, e.g kinetic energy ( 2 mc2 ) has the 2 2 2 dimensions kg ð m /s ; however, 1 kg D 1 N s /m from the definition of the newton given above Hence, the units of kinetic energy must... unit of power; when 1 watt is applied for 1 second to a system the input of energy to that system is 1 joule (i.e 1 J) The hertz (Hz) is the number of repetitions of a regular occurrence in 1 second Instead of writing c/s for cycles/sec, Hz is used instead The unit of thermodynamic temperature is the kelvin (K), written without the ° sign, and is the fraction 1/273.16 of the thermodynamic temperature of. .. considered as a control volume 6 Fluid Mechanics, Thermodynamics of Turbomachinery variables on the performance must now be included The size of machine is characterised by the impeller diameter D, and the shape can be expressed by a number of length ratios, l1 /D, l2 /D, etc Incompressible fluid analysis The performance of a turbomachine can now be expressed in terms of the control variables, geometric... changes occurring simulataneously in the fluid The definition of a turbomachine as stated above, is rather too general for the purposes of this book as it embraces open turbomachines such as propellers, wind turbines and unshrouded fans, all of which influence the state of a not readily quantifiable flow of a fluid The subject fluid mechanics, thermodynamics of turbomachinery, therefore, is limited to machines enclosed . 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. specific enthalpy I rothalpy i incidence angle K, k constants K N nozzle velocity coefficient L lift force, length of diffuser wall l blade chord length, pipe length M Mach number m mass, molecular ‘weight’ N. 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