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`I-'|F'I'H|'-.['J|T|f."JI"-.1
`
`S. L. DIXON
`
`UTC-2006.001
`GE V. UTC
`
`Trial IPR2016-00952
`
`GE v. UTC
`Trial IPR2016-00952
`
`UTC-2006.001
`
`

`
`Fluid Mechanics,
`Thermodynamics of
`Turbomachinery
`
`Fifth Edition, in SI/Metric units
`
`S. L. Dixon, B.Eng., Ph.D.
`Senior Fellow at the University of Liverpool
`
`o
`
`I
`
`-
`
`,
`
`H.
`
`,5: ..
`.‘~.'
`'
`*
`F
`
`_
`.
`-
`,_.
`=
`EL5
`ER
`BUTTERWORTH
`HEINEMANN
`
`
`
`é »
`
`-
`
`AMSTERDAM - BOSTON - HEIDELBERG - LONDON
`NEW YORK - OXFORD - PARIS - SAN DIEGO
`
`SAN FRANCISCO - SINGAPORE - SYDNEY - TOKYO
`
`UTC-2006.002
`
`UTC-2006.002
`
`

`
`Acquisition Editor: Joel Stein
`Project Manager: Carl M. Soares
`Editorial Assistant: Shoshanna Grossman
`Marketing Manager: Tara Isaacs
`
`Elsevier Butterworth–Heinemann
`30 Corporate Drive, Suite 400, Burlington, MA 01803, USA
`Linacre House, Jordan Hill, Oxford OX2 8DP, UK
`
`First published by Pergamon Press Ltd. 1966
`Second edition 1975
`Third editon 1978
`Reprinted 1979, 1982 (twice), 1984, 1986 1989, 1992, 1995
`Fourth edition 1998
`
`© S.L. Dixon 1978, 1998
`
`No part of this publication may be reproduced, stored in a retrieval system, or transmitted in
`any form or by any means, electronic, mechanical, photocopying, recording, or otherwise,
`without the prior written permission of the publisher.
`
`Permissions may be sought directly from Elsevier’s Science & Technology Rights Department
`in Oxford, UK: phone: (+44) 1865 843830, fax: (+44) 1865 853333, e-mail:
`permissions@elsevier.com.uk. You may also complete your request on-line via the Elsevier
`homepage (http://elsevier.com), by selecting “Customer Support” and then “Obtaining
`Permissions.’
`
`Recognizing the importance of preserving what has been written, Elsevier prints its books on
`acid-free paper whenever possible.
`
`Library of Congress Cataloging-in-Publication Data
`Dixon, S. L. (Sydney Lawrence)
`Fluid mechanics and thermodynamics of turbomachinery.
`p.
`cm.
`Includes bibliographical references.
`1. Turbomachines—Fluid dynamics. I. Title.
`TJ267.D5 2005
`621.406—dc22
`
`2004022864
`
`British Library Cataloguing-in-Publication Data
`A catalogue record for this book is available from the British Library.
`
`ISBN: 0-7506-7870-4
`
`For information on all Elsevier Butterworth–Heinemann publications
`visit our Web site at www.books.elsevier.com
`
`05 06 07 08 09 10 10 9 8 7 6 5 4 3 2 1
`
`Printed in the United States of America
`
`UTC-2006.003
`
`

`
`Preface to the Fifth Edition
`
`In the earlier editions of this book, open turbomachines, categorised as wind
`turbines, propellers and unshrouded fans, were deliberately excluded because of
`the conceptual obstacle of precisely defining the mass flow that interacts with the
`blades. However, having studied and taught the topic of Wind Turbines for a number
`of years at the University of Liverpool, as part of a course on Renewable Energy, it
`became apparent this was really only a matter of approach. In this book a new chapter
`on wind turbines has been added, which deals with the basic aerodynamics of the wind
`turbine rotor. This chapter offers the student a short basic course dealing with the essen-
`tial fluid mechanics of the machine, together with numerous worked examples at
`various levels of difficulty. Important aspects concerning the criteria of blade selection
`and blade manufacture, control methods for regulating power output and rotor speed
`and performance testing are touched upon. Also included are some very brief notes
`concerning public and environmental issues which are becoming increasingly impor-
`tant as they, ultimately, can affect the development of wind turbines. It is a matter
`of some regret that many aspects of the nature of the wind, e.g. methodology of deter-
`mining the average wind speed, frequency distribution, power law and the effect of
`elevation (and location), cannot be included, as constraints on book length have to be
`considered.
`The world is becoming increasingly concerned with the very major issues sur-
`rounding the use of various forms of energy. The ever-growing demand for oil and the
`undeniably diminishing amount of oil available, global warming seemingly linked to
`increased levels of CO2 and the related threat of rising sea levels are just a few of these
`issues. Governments, scientific and engineering organisations as well as large (and
`small) businesses are now striving to change the profile of energy usage in many coun-
`tries throughout the world by helping to build or adopt renewable energy sources for
`their power or heating needs. Almost everywhere there is evidence of the large-scale
`construction of wind turbine farms and plans for even more. Many countries (the UK,
`Denmark, Holland, Germany, India, etc.) are aiming to have between 10 and 20% of
`their installed power generated from renewable energy sources by around 2010. The
`main burden for this shift is expected to come from wind power. It is hoped that this
`new chapter will instruct the students faced with the task of understanding the techni-
`calities and science of wind turbines.
`Renewable energy topics were added to the fourth edition of this book by way of the
`Wells turbine and a new chapter on hydraulic turbines. Some of the derivatives of the
`Wells turbine have now been added to the chapters on axial flow and radial flow tur-
`bines. It is likely that some of these new developments will flourish and become a major
`source of renewable energy once sufficient investment is given to the research.
`xi
`
`UTC-2006.004
`
`

`
`xii Preface to the Fifth Edition
`
`The opportunity has been taken to add some new information about the fluid
`mechanics of turbomachinery where appropriate as well as including various corrections
`to the fourth edition, in particular to the section on backswept vanes of centrifugal
`compressors.
`
`S.L.D.
`
`UTC-2006.005
`
`

`
`Preface to the Fourth Edition
`
`It is now 20 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 turbomachinery 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 efficiency improvements. However, the fundamen-
`tals 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 engineers
`rather than mathematicians. Much stress is laid on physical concepts rather than math-
`ematics 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 techno-
`logical 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 applica-
`tion 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 tur-
`bines with some new extensions and the other on hydraulic turbines. In a world striv-
`ing 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 mechan-
`xiii
`
`UTC-2006.006
`
`

`
`xiv Preface to the Fourth Edition
`
`ical engineering courses. This chapter includes a few new ideas which could be of some
`interest.
`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 com-
`plete and detailed solutions of the unsolved problems.
`
`S. Lawrence Dixon
`
`UTC-2006.007
`
`

`
`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 phe-
`nomenon 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 com-
`municated 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.
`
`xv
`
`UTC-2006.008
`
`

`
`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 conserva-
`tion 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 illustra-
`tive material for use in the book, product information and, in one case, useful back-
`ground 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.
`
`xvii
`
`UTC-2006.009
`
`

`
`List of Symbols
`
`A
`A2
`a
`a–
`a¢
`b
`Cc
`Cf
`CL, CD
`CP
`Cp
`
`Cpi
`Cv
`CX, CY
`c
`co
`D
`Deq
`Dh
`E, e
`F
`Fc
`f
`g
`H
`HE
`Hf
`HG
`HS
`h
`I
`i
`J
`j
`K, k
`KN
`L
`l
`
`area
`area of actuator disc
`sonic velocity, position of maximum camber
`axial-flow induction factor
`tangential flow coefficient
`passage width, maximum camber
`chordwise force coefficient
`tangential force coefficient
`lift and drag coefficients
`power coefficient
`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
`Prandtl correction factor
`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
`tip–speed ratio
`local blade–speed ratio
`constants
`nozzle velocity coefficient
`lift force, length of diffuser wall
`blade chord length, pipe length
`xix
`
`UTC-2006.010
`
`

`
`xx List of Symbols
`
`M
`m
`N
`NS
`NSP
`NSS
`n
`P
`p
`pa
`pv
`pw
`Q
`q
`R
`
`Re
`RH
`Ro
`r
`S
`s
`T
`t
`U
`u
`V, v
`W
`DW
`w
`X
`x, y, z
`Y
`Yid
`Yk
`Yp
`YS
`Z
`
`a
`b
`G
`g
`d
`e
`
`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
`power
`pressure
`atmospheric pressure
`vapour pressure
`rate of energy loss
`heat transfer, volume flow rate
`dryness fraction
`reaction, specific gas constant, tip radius of a blade, radius of
`slipstream
`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, pitch angle of blade
`circulation
`ratio of specific heats
`deviation angle
`fluid deflection angle, cooling effectiveness, drag–lift ratio
`
`UTC-2006.011
`
`

`
`z
`
`h
`Q
`q
`l
`m
`␯
`r
`s
`sb
`sc
`t
`f
`Y
`W
`WS
`WSP
`WSS
`w
`w–
`
`List of Symbols
`
`xxi
`
`enthalpy loss coefficient, total pressure loss coefficient, relative power
`coefficient
`efficiency
`minimum opening at cascade exit
`blade camber angle, wake momentum thickness
`profile loss coefficient, blade loading coefficient
`dynamic viscosity
`kinematic viscosity, blade stagger angle, velocity ratio
`density
`slip factor, solidity
`blade cavitation coefficient
`Thoma’s coefficient, centrifugal stress
`torque
`flow coefficient, velocity ratio, relative flow angle
`stage loading factor
`speed of rotation (rad/s)
`specific speed (rad)
`power specific speed (rad)
`suction specific speed (rad)
`vorticity
`stagnation pressure loss coefficient
`
`Subscripts
`av
`average
`compressor, critical
`c
`diffuser
`D
`exit
`e
`hydraulic, hub
`h
`inlet, impeller
`i
`id
`ideal
`is
`isentropic
`mean, meridional, mechanical, material
`m
`nozzle
`N
`normal component
`n
`stagnation property, overall
`o
`polytropic, constant pressure
`p
`reversible process, rotor
`R
`radial
`r
`rel
`relative
`isentropic, stall condition
`s
`stage isentropic
`ss
`turbine, tip, transverse
`t
`␯
`velocity
`
`UTC-2006.012
`
`

`
`xxii List of Symbols
`
`x, y, z
`q
`
`cartesian coordinate components
`tangential
`
`Superscript
`.
`time rate of change
`-
`average
`¢
`blade angle (as distinct from flow angle)
`*
`nominal condition
`
`UTC-2006.013
`
`

`
`Contents
`
`PREFACE TO THE FIFTH EDITION xi
`
`PREFACE TO THE FOURTH EDITION xiii
`
`PREFACE TO THE THIRD EDITION xv
`
`ACKNOWLEDGEMENTS xvii
`
`LIST OF SYMBOLS xix
`
`1. Introduction: Dimensional Analysis: Similitude
`
`1
`
`Definition of a turbomachine 1
`Units and dimensions 3
`Dimensional analysis and performance laws 5
`Incompressible fluid analysis 6
`Performance characteristics 7
`Variable geometry turbomachines 8
`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
`
`24
`
`Introduction 24
`The equation of continuity 24
`25
`The first law of thermodynamics—internal energy
`The momentum equation—Newton’s second law of motion 26
`The second law of thermodynamics—entropy 30
`Definitions of efficiency 31
`Small stage or polytropic efficiency 35
`Nozzle efficiency 42
`Diffusers 44
`References 54
`Problems 55
`
`v
`
`UTC-2006.014
`
`

`
`vi Contents
`3. Two-dimensional Cascades
`
`56
`
`Introduction 56
`Cascade nomenclature 57
`Analysis of cascade forces 58
`Energy losses 60
`Lift and drag 60
`Circulation and lift 62
`Efficiency of a compressor cascade 63
`Performance of two-dimensional cascades 64
`The cascade wind tunnel 64
`Cascade test results 66
`Compressor cascade performance 69
`Turbine cascade performance 72
`Compressor cascade correlations 72
`Fan blade design (McKenzie) 80
`Turbine cascade correlation (Ainley and Mathieson) 83
`Comparison of the profile loss in a cascade and in a turbine stage 88
`Optimum space–chord ratio of turbine blades (Zweifel) 89
`References 90
`Problems 92
`
`4. Axial-flow Turbines: Two-dimensional Theory
`
`94
`
`Introduction 94
`Velocity diagrams of the axial turbine stage 94
`Thermodynamics of the axial turbine stage 95
`Stage losses and efficiency 97
`Soderberg’s correlation 98
`Types of axial turbine design 100
`Stage reaction 102
`Diffusion within blade rows 104
`Choice of reaction and effect on efficiency 108
`Design point efficiency of a turbine stage 109
`Maximum total-to-static efficiency of a reversible turbine stage 113
`Stresses in turbine rotor blades 115
`Turbine flow characteristics 121
`Flow characteristics of a multistage turbine 123
`The Wells turbine 125
`Pitch-controlled blades 132
`References 139
`Problems 140
`
`5. Axial-flow Compressors and Fans
`
`145
`
`Introduction 145
`Two-dimensional analysis of the compressor stage 146
`
`UTC-2006.015
`
`

`
`Contents
`
`vii
`
`Velocity diagrams of the compressor stage 148
`Thermodynamics of the compressor stage 149
`Stage loss relationships and efficiency 150
`Reaction ratio 151
`Choice of reaction 151
`Stage loading 152
`Simplified off-design performance 153
`Stage pressure rise 155
`Pressure ratio of a multistage compressor 156
`Estimation of compressor stage efficiency 157
`Stall and surge phenomena in compressors 162
`Control of flow instabilities 167
`Axial-flow ducted fans 168
`Blade element theory 169
`Blade element efficiency 171
`Lift coefficient of a fan aerofoil 173
`References 173
`Problems 174
`
`6. Three-dimensional Flows in Axial Turbomachines
`
`177
`
`Introduction 177
`Theory of radial equilibrium 177
`The indirect problem 179
`The direct problem 187
`Compressible flow through a fixed blade row 188
`Constant specific mass flow 189
`Off-design performance of a stage 191
`Free-vortex turbine stage 192
`Actuator disc approach 194
`Blade row interaction effects 198
`Computer-aided methods of solving the through-flow problem 199
`Application of Computational Fluid Dynamics (CFD) to the design of axial turbomachines 201
`Secondary flows 202
`References 205
`Problems 205
`
`7. Centrifugal Pumps, Fans and Compressors
`
`208
`
`Introduction 208
`Some definitions 209
`Theoretical analysis of a centrifugal compressor 211
`Inlet casing 212
`Impeller 212
`Conservation of rothalpy 213
`Diffuser 214
`
`UTC-2006.016
`
`

`
`viii Contents
`
`Inlet velocity limitations 214
`Optimum design of a pump inlet 215
`Optimum design of a centrifugal compressor inlet 217
`Slip factor 222
`Head increase of a centrifugal pump 227
`Performance of centrifugal compressors 229
`The diffuser system 237
`Choking in a compressor stage
`References 242
`Problems 243
`
`240
`
`8. Radial Flow Gas Turbines
`
`246
`
`Introduction 246
`Types of inward-flow radial turbine 247
`Thermodynamics of the 90 deg IFR turbine 249
`Basic design of the rotor 251
`Nominal design point efficiency 252
`Mach number relations 256
`Loss coefficients in 90 deg IFR turbines 257
`Optimum efficiency considerations 258
`Criterion for minimum number of blades 263
`Design considerations for rotor exit 266
`Incidence losses 270
`Significance and application of specific speed 273
`Optimum design selection of 90 deg IFR turbines 276
`Clearance and windage losses 278
`Pressure ratio limits of the 90 deg IFR turbine 279
`Cooled 90 deg IFR turbines 280
`A radial turbine for wave energy conversion 282
`References 285
`Problems 287
`
`9. Hydraulic Turbines
`
`290
`
`Introduction 290
`Hydraulic turbines 291
`The Pelton turbine 294
`Reaction turbines 303
`The Francis turbine 304
`The Kaplan turbine 310
`Effect of size on turbomachine efficiency 313
`Cavitation 315
`Application of CFD to the design of hydraulic turbines 319
`References 320
`Problems 320
`
`UTC-2006.017
`
`

`
`Contents
`
`ix
`
`10. Wind Turbines
`
`323
`
`Introduction 323
`Types of wind turbine 325
`Growth of wind power capacity and cost 329
`Outline of the theory 330
`Actuator disc approach 330
`Estimating the power output 337
`Power output range 337
`Blade element theory 338
`The blade element momentum method 346
`Rotor configurations 353
`The power output at optimum conditions 360
`HAWT blade selection criteria 361
`Developments in blade manufacture 363
`Control methods (starting, modulating and stopping) 364
`Blade tip shapes 369
`Performance testing 370
`Performance prediction codes 370
`Comparison of theory with experimental data 371
`Peak and post-peak power predictions 371
`Environmental considerations 373
`References 374
`
`Bibliography
`
`377
`
`Appendix 1. Conversion of British and US Units to SI Units
`
`378
`
`Appendix 2. Answers to Problems
`
`379
`
`Index 383
`
`UTC-2006.018
`
`

`
`CHAPTER 4
`
`Axial-flow Turbines:
`Two-dimensional Theory
`
`Power is more certainly retained by wary measures than by daring counsels.
`(TACITUS, Annals.)
`
`Introduction
`The simplest approach to the study of axial-flow turbines (and also axial-flow com-
`pressors) is to assume that the flow conditions prevailing at the mean radius fully rep-
`resent the flow at all other radii. This two-dimensional analysis at the pitchline can
`provide a reasonable approximation to the actual flow, if the ratio of blade height to
`mean radius is small. When this ratio is large, however, as in the final stages of a steam
`turbine or in the first stages of an axial compressor, a three-dimensional analysis is
`required. Some important aspects of three-dimensional flows in axial turbomachines
`are discussed in Chapter 6. Two further assumptions are that radial velocities are zero
`and that the flow is invariant along the circumferential direction (i.e. there are no “blade-
`to-blade” flow variations).
`In this chapter the presentation of the analysis has been devised with compressible
`flow effects in mind. This approach is then applicable to both steam and gas turbines
`provided that, in the former case, the steam condition remains wholly within the vapour
`phase (i.e. superheat region). Much early work concerning flows in steam turbine
`nozzles and blade rows are reported in Stodola (1945), Kearton (1958) and Horlock
`(1960).
`
`Velocity diagrams of the axial turbine stage
`The axial turbine stage comprises a row of fixed guide vanes or nozzles (often called
`a stator row) and a row of moving blades or buckets (a rotor row). Fluid enters the
`stator with absolute velocity c1 at angle a1 and accelerates to an absolute velocity c2 at
`angle a2 (Figure 4.1). All angles are measured from the axial (x) direction. The sign
`convention is such that angles and velocities as drawn in Figure 4.1 will be taken as
`positive throughout this chapter. From the velocity diagram, the rotor inlet relative
`velocity w2, at an angle b2, is found by subtracting, vectorially, the blade speed U from
`the absolute velocity c2. The relative flow within the rotor accelerates to velocity w3 at
`an angle b3 at rotor outlet; the corresponding absolute flow (c3, a3) is obtained by
`adding, vectorially, the blade speed U to the relative velocity w3.
`94
`
`UTC-2006.019
`
`

`
`Axial-flow Turbines: Two-dimensional Theory
`
`95
`
`Nozzle row
`
`FIG. 4.1. Turbine stage velocity diagrams.
`
`The continuity equation for uniform, steady flow is
`
`IOlAlCx1 = lO2A2Cx2 = l03A3Cx3-
`
`In two-dimensional theory of turbomachines it is usually assumed, for simplicity, that
`
`the axial Velocity remains constant i.e. cxl = ,2 = ,3 = x.
`
`This must imply that
`
`p1A1 Z pg./‘lg Z /O3/l3 Z constant.
`
`Thermodynamics of the axial turbine stage
`
`The Work done on the rotor by unit mass of fluid, the specific Work, equals the
`
`stagnation enthalpy drop incurred by the fluid passing through the stage (assuming adi-
`
`abatic flow), or
`
`AW Z W/riz Z 110] — hog Z U(C’_,_-p_ + Cyg).
`
`(4,2)
`
`In eqn. (4.2) the absolute tangential Velocity components (Cy) are added, so as to
`adhere to the agreed sign convention of Figure 4.1. As no work is done in the nozzle
`
`row, the stagnation enthalpy across it remains constant and
`
`/101 Z 1102.
`
`(4.3)
`
`Writing ho = h + %(cf + 0;‘) and using eqn. (4.3) in eqn. (4.2) we obtain,
`
`1102 — hots = (I22 — I13) + §<cf:i2 — c§.3> = U(cy2 + cm.
`
`UTC-2006.020
`
`UTC-2006.020
`
`

`
`96 Fluid Mechanics, Thermodynamics of Turbomachinery
`
`hence,
`
`It is observed from the velocity triangles of Figure 4.1 that cy2 - U = wy2, cy3 +
`U = wy3 and cy2 + cy3 = wy2 + wy3. Thus,
`
`Add and subtract 1–2 cx
`2 to the above equation
`
`(4.4)
`Thus, we have proved that the relative stagnation enthalpy, h0rel = h + 1–2 w2, remains
`unchanged through the rotor of an axial turbomachine. It is implicitly assumed that no
`radial shift of the streamlines occurs in this flow. In a radial flow machine a more
`general analysis is necessary (see Chapter 7), which takes account of the blade speed
`change between rotor inlet and outlet.
`A Mollier diagram showing the change of state through a complete turbine stage,
`including the effects of irreversibility, is given in Figure 4.2.
`Through the nozzles, the state point moves from 1 to 2 and the static pressure
`decreases from p1 to p2. In the rotor row, the absolute static pressure reduces (in general)
`from p2 to p3. It is important to note that all the conditions contained in eqns. (4.2)–(4.4)
`are satisfied in the figure.
`
`FIG. 4.2. Mollier diagram for a turbine stage.
`
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`

`
`Axial-flow Turbines: Two-dimensional Theory 97
`Stage losses and efficiency
`In Chapter 2 various definitions of efficiency for complete turbomachines were given.
`For a turbine stage the total-to-total efficiency is
`
`At the entry and exit of a normal stage the flow conditions (absolute velocity and
`flow angle) are identical, i.e. c1 = c3 and a1 = a3. If it is assumed that c3ss = c3, which is
`a reasonable approximation, the total-to-total efficiency becomes
`
`(4.5)
`Now the slope of a constant pressure line on a Mollier diagram is (∂h/∂s)p = T,
`obtained from eqn. (2.18). Thus, for a finite change of enthaply in a constant pressure
`process, Dh ⯐ TDs and, therefore,
`
`(4.6a)
`
`(4.6b)
`Noting, from Figure 4.2, that s3s - s3ss = s2 - s2s, the last two equations can be com-
`bined to give
`
`(4.7)
`
`The effects of irreversibility through the stator and rotor are expressed by the dif-
`ferences in static enthalpies, (h2 - h2s) and (h3 - h3s) respectively. Non-dimensional
`enthalpy “loss” coefficients can be defined in terms of the exit kinetic energy from each
`blade row. Thus, for the nozzle row,
`
`For the rotor row,
`
`Combining eqns. (4.7) and (4.8) with eqn. (4.5) gives
`
`(4.8a)
`
`(4.8b)
`
`(4.9)
`
`When the exit velocity is not recovered (in Chapter 2, examples of such cases are
`quoted) a total-to-static efficiency for the stage is used.
`
`)
`(
`-
`h
`01
`z
`+
`c T T
`3
`2
`N
`)
`-(
`2
`h
`h
`1
`3
`where, as before, it is assumed that c1 = c3.
`
`(4.10)
`
`-
`1
`
`,
`
`˘˚˙
`
`12
`
`c
`
`+
`
`h
`3
`
`ss
`
`22
`
`h
`03
`
`)
`
`32
`
`w
`
`R
`
`-
`z
`
`h
`
`ts
`
`=
`
`(
`h
`01
`
`+
`
`ÈÎÍ
`
`1
`
`=
`
`UTC-2006.022
`
`

`
`98 Fluid Mechanics, Thermodynamics of Turbomachinery
`
`In initial calculations or, in cases where the static temperature drop through the rotor
`is not large, the temperature ratio T3/T2 is set equal to unity, resulting in the more con-
`venient approximations,
`
`(4.9a)
`
`(4.10a)
`
`So that estimates can be made of the efficiency of a proposed turbine stage as part
`of the preliminary design process, some means of determining the loss coefficients
`is required. Several methods for doing this are available with varying degrees of
`complexity. The blade row method proposed by Soderberg (1949) and reported
`by Horlock (1966), although old, is still remarkably valid despite its simplicity. Ainley
`and Mathieson (1951) correlated the profile loss coefficients for nozzle blades (which
`give 100% expansion) and impulse blades (which give 0% expansion) against flow
`deflection and pitch–chord ratio for stated values of Reynolds number and Mach
`number. Details of their method are given in Chapter 3. For blading between
`impulse and reaction the profile loss is derived from a combination of the impulse
`and reaction profile losses (see eqn. (3.42)). Horlock (1966) has given a detailed
`comparison between these two methods of loss prediction. A refinement of the
`Ainley and Mathieson prediction method was later reported by Dunham and Came
`(1970).
`Various other methods of predicting the efficiency of axial flow turbines have
`been devised such as those of Craig and Cox (1971), Kacker and Okapuu (1982)
`and Wilson (1987). It was Wilson who, tellingly, remarked that despite the emergence
`of “computer programs of great power and sophistication”, and “generally incor-
`porating computational fluid dynamics”, that these have not yet replaced the
`preliminary design methods mentioned above. It is, clearly, essential for a design to
`converge as closely as possible to an optimum configuration using preliminary design
`methods before carrying out the final design refinements using computational fluid
`dynamics.
`
`Soderberg’s correlation
`One method of obtaining design data on turbine blade losses is to assemble infor-
`mation on the overall efficiencies of a wide variety of turbines, and from this calculate
`the individual blade row losses. This system was developed by Soderberg (1949) from
`a large number of tests performed on steam turbines and on cascades, and extended to
`fit data obtained from small turbines with very low aspect ratio blading (small
`height–chord). Soderberg’s method was intended only for turbines conforming to the
`standards of “good design”, as discussed below. The method was used by Stenning
`(1953) to whom reference can be made.
`A paper by Horlock (1960) critically reviewed several different and widely used
`methods of obtaining design data for turbines. His paper confirms the claim made for
`
`UTC-2006.023
`
`

`
`Axial-flow Turbines: Two-dimensional Theory 99
`
`FIG. 4.3. Soderberg’s correlation of turbine blade loss coefficient with fluid deflection
`(adapted from Horlock 1960).
`
`Soderberg’s correlation that, although based on relatively few parameters, it is of com-
`parable accuracy with the best of the other methods.
`Soderberg found that for the optimum space–chord ratio, turbine blade losses (with
`“full admission” to the complete annulus) could be correlated with space–chord ratio,
`blade aspect ratio, blade thickness–chord ratio and Reynolds number. Soderberg used
`Zweifel’s criterion (see Chapter 3) to obtain the optimum space–chord ratio of turbine
`cascades based upon the cascade geometry. Zweifel suggested that the aerodynamic
`load coefficient yT should be approximately 0.8. Following the notation of Figure 4.1
`(4.11)
`
`The optimum space–chord ratio may be obtained from eqn. (4.11) for specified values
`of a1 and a2.
`For turbine blade rows operating at this load coefficient, with a Reynolds number of
`105 and aspect ratio H/b = blade height–axial chord of 3, the “nominal” loss coefficient
`z* is a simple function of the fluid deflection angle ⑀ = a1 + a2, for a given
`thickness–chord ratio (tmax/l). Values of z* are drawn in Figure 4.3 as a function of
`deflection ⑀, for several ratios of tmax/l. A frequently used analytical simplification of
`this correlation (for tmax/l = 0.2), which is useful in initial performance calculations, is
`
`(4.12)
`
`This expression fits Soderberg’s curve (for tmax/l = 0.2) quite well for e ⬉ 120° but
`is less accurate at higher deflections. For turbine rows operating at zero incidence,
`which is the basis of Soderberg’s correlation, the fluid deflection is little different from
`the blading deflection since, for turbine cascades, deviations are usually small. Thus,
`for a nozzle row, ⑀ = ⑀N = a¢2 + a¢1 and for a rotor row, ⑀ = ⑀R = b¢2 + b¢3 can be used
`(the prime referring to the actual blade angles).
`
`UTC-2006.024
`
`

`
`100 Fluid Mechanics, Thermodynamics of Turbomachinery
`If the aspect ratio H/b is other than 3, a correctio