Fluid Mechanics and Thermodynamics of Turbomachinery Fourth Edition, in WMetric units S. L. Dixon, B.Eng., Ph.D Senior Fellow at the University of Liverpool 1 EINEMANN Boston Oxford Johannesburg Melbourne New Delhi Singapore Butterworth-Heinemann &c A member of the Reed Elsevier Group First published by Pergamon Press Ltd. 1966 Second edition 1975 Third edition 1978 Reprinted 1979, 1982 (twice), 1984, 1989, 1992, 1995 Foruth edition 1998 0 S. L. Dixon 1978, 1998 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photo- copying, recording, or otherwise, without the prior written permission of the publisher. Recognizing the importance of preserving what has been written, Butter- @ worth-Heinemann prints its books on acid-free paper whenever possible. 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For information, please contact: Manager of Special Sales Butterworth-Heinemann 225 Wildwood Avenue Woburn, MA 01801-2041 Tel: (781) 904-2500 Fa: (781) 904-2620 For information on all Butterworth-Heinemann publications available, contact our World Wide Web home page at: 10 9876543 Printed in the United States of America Typeset by Laser Words, Madras, India http://www.bh.com 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 Polytechc 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. List of Symbols area sonic velocity, position of maximum camber passage width, maximum camber tangential force coefficient lift and drag coefficients specific heat at constant pressure, pressure coefficient, pressure rise coefficient ideal pressure rise coefficient specific heat at constant volume axial and tangential force coefficients absolute velocity spouting velocity drag force, diameter equivalent diffusion ratio hydraulic mean diameter energy, specific energy centrifugal force in blade acceleration, friction factor gravitational acceleration head, blade height effective head head loss fue to friction gross head net positive suction head (NPSH) specific enthalpy rothalpy incidence angle constants nozzle velocity coefficient lift force, length of diffuser wall blade chord length, pipe length Mach number mass, molecular ‘weight’ rotational speed, axial length of diffuser specific speed (rev) power specific speed (rev) suction specific speed (rev) number of stages, polytropic index pressure List of Symbols xv Pa Pv 4 R Re RH Ro r S T t U v, v W AW X x, y. z Y Q S U W Yid yk YP YS z a! B r Y 6 < rl 0 A. CL P E e v 0 ab 0, T atmospheric pressure vapour pressure heat transfer, volume flow rate dryness fraction reaction, specific gas constant Reynolds number reheat factor universal gas constant radius entropy, power ratio blade pitch, specific entropy temperature time, thickness blade speed, internal energy specific internal energy volume, specific volume work transfer specific work transfer relative velocity axial force Cartesian coordinate directions tangential force, actual tangential blade load per unit span ideal tangential blade load per unit span tip clearance loss coefficient profile loss coefficient net secondary loss coefficient number of blades, Ainley blade loading parameter absolute flow angle relative flow angle circulation ratio of specific heats deviation angle fluid deflection angle, cooling effectiveness enthalpy loss coefficient, total pressure loss coefficient efficiency minimum opening at cascade exit blade camber angle, wake momentum thickness profile loss coefficient dynamic viscosity kinematic viscosity, blade stagger angle, velocity ratio density slip factor, solidity blade cavitation coefficient Thoma’s coefficient, centrifugal stress torque xvi Fluid Mechanics, Thermodynamics of Turbomachinery flow coefficient, velocity ratio stage loading factor speed of rotation (rads) specific speed (rad) power specific speed (rad) suction specific speed (rad) vorticity stagnation pressure loss coefficient Subscripts av D e h i id is m N n P R r re1 c 0 S ss t x, y, z 21 e average compressor, critical diffuser exit hydraulic, hub inlet, impeller ideal isentropic mean, meridional, mechanical, material nozzle normal component stagnation property, overall polytropic, constant pressure reversible process, rotor radial relative isentropic, stall condition stage isentropic turbine, tip, transverse velocity Cartesian coordinate components tangential Superscript time rate of change average blade angle (as distinct from flow angle) * nominal condition I 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 Jlows in axial turbomachines has been extended; in particular the section concerning the constant specijic mass Jlow 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 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 loolung 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 Contents PREFACE TO FOURTH EDITION ix PREFACE TO THIRD EDITION xi ACKNOWLEDGEMENTS xiii LIST OF SYMBOLS XiV 1. Introduction: Dimensional Analysis: Similitude 1 Definition of a turbomachine I Units and dimensions 3 Dimensional analysis and pegormance laws Incompressible jluid analysis 6 Performance characteristics 7 Variable geometry turbomachines 9 Specijc speed 10 Cavitation 12 Compressible gas flow relations 15 Compressible jluid analysis 16 The inherent unsteadiness of thejlow within turbomachines References 21 Problems 22 4 20 2. Basic Thermodynamics, Fluid Mechanics: Definitions of Efficiency 23 Introduction 23 The equation of continuity 23 The first law of thermodynamics - internal energy The momentum equation - Newton’s second law of motion The second law of thermodynamics - entropy Definitions of efficiency 30 Small stage or polytropic efficiency Nozzle efficiency 41 Dimers 43 References 53 Problems 53 24 25 29 35 [...]... ) and thermodynamic temperature (K) All the other units used in this book are derived from these basic units The unit offorce is the 4 Fluid Mechanics, Thermodynamics of Turbomachinery newton (N), defined as that force which, when applied to a mass of lkilogram, gives an acceleration to the mass of 1 m/s2 The recommended unit of pressure is the pascal (Pa) which is the pressure produced by a force of. .. Turbomachine considered as a control volume Fluid Mechanics, Thermodynamics of Turbomachinery 6 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, l , / D , 12/D, etc Incompressible fluid analysis The performance of a turbomachine can now be expressed in terms of the control variables,... fluid The subject fluid mechanics, thermodynamics of turbomachinery, therefore, is limited to machines enclosed by a closely fitting casing or shroud through which a readily measurable quantity of fluid passes in unit time The subject of open turbomachines is covered by the classic text of Glauert (1959) or by Duncan et al (1970), the elementary treatment of propellers by general fluid mechanics textbooks... This point is discussed more fully in the 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 axialflow turbomachine (e.g 1 2 Fluid Mechanics, Thermodynamics of Turbomachinery FIG... specific 12 Fluid Mechanics, Thermodynamics of Turbomachinery FIG 1.7 Range of pump impellers of equal inlet area speed implies that the machine design changes Broadly speaking, each different class of machine has its optimum efficiency within its own fairly narrow range of specific speed For a pump, eqn (1.8) indicates, for constant speed N , that N , is increased by an increase in Q and decreased by an... 22 Fluid Mechanics, Thermodynamics of Turbomachinery Reynolds, 0 (1882) On the internal cohesion of fluids Mem Proc Manchester Lit SOC., 3rd Series, 7, 1-19 Ryley, D J (1980) Hydrostatic stress in water Int J Mech Eng Educ., 8 (2) Shames, I H (1992) Mechanics o Fluids McGraw-Hill f Shepherd, D G (1956) Principles o Turbomachinery Macmillan f Sweeter, V L and Wylie, E B (1979) Fluid Mechanics (7th edn)... summarises the basic physical laws of fluid mechanics and thermodynamics, developing them into a form suitable for the study of turbomachines Following this, some of the more important and commonly used expressions for the efficiency of compression and expansion flow processes are given The laws discussed are: (1) the continuity offlow equation; (2) the first law o thermodynamics and the steady flow energy... design is being considered the parameters normally specified are the shaft power P, the head at turbine entry H and the rotational speed N A non-dimensional parameter called the spec$c speed, N,, referred to and conceptualised as the shape number, is often used to facilitate the choice of the most appropriate machne This new parameter is derived from the non-dimensional groups defined in eqn (1.3) in such... the rotational speed, N , is expressed in the units of revolutions per unit of time so that although N, is dimensionless, numerical values of specific speed need to be thought of as revs Alternative versions of eqns (1.8) and (1.9) in radians are also in common use and are written (1.8a) Q , = Qrn ~ ( g H ) 5 / 4’ There is a simple connection between N , and N,, (and between 52, and dividing eqn (1.9)... distributed over an area of 1 square metre Several other units of pressure are in widespread use, however, foremost of these being the bar Much basic data concerning properties of substances (steam and gas tables, charts, etc.) have been prepared in SI units with pressure given in bars and it is acknowledged that this alternative unit of pressure will continue to be used for some time as a matter of expediency . unit offorce is the 4 Fluid Mechanics, Thermodynamics of Turbomachinery newton (N), defined as that force which, when applied to a mass of lkilogram, gives an acceleration to the mass of. 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. torque xvi Fluid Mechanics, Thermodynamics of Turbomachinery flow coefficient, velocity ratio stage loading factor speed of rotation (rads) specific speed (rad) power specific speed (rad)