Thispageintentionallyleftblank Principles of Turbomachinery Thispageintentionallyleftblank Principles of Turbomachinery Seppo A. Korpela The Ohio State University WILEY A JOHN WILEY & SONS, INC., PUBLICATION Copyright © 2011 by John Wiley & Sons, Inc. All rights reserved. Published by John Wiley & Sons, Inc., Hoboken, New Jersey. Published simultaneously in Canada. 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, scanning or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4470, or on the web at www.copyright.com. 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TJ267.K57 2011 621.406—dc23 2011026170 Printed in the United States of America. 10 987654321 To my wife Terttu, to our daughter Liisa, and to the memory of our daughter Katja Thispageintentionallyleftblank CONTENTS Foreword Ml Acknowledgments 1 Introduction 1.1 Energy and fluid machines 1 1.1.1 Energy conversion of fossil fuels 1 1.1.2 Steam turbines 2 1.1.3 Gas turbines 3 1.1.4 Hydraulic turbines 4 1.1.5 Wind turbines 5 1.1.6 Compressors 5 1.1.7 Pumps and blowers 5 1.1.8 Other uses and issues 6 1.2 Historical survey 7 1.2.1 Water power 7 1.2.2 Wind turbines 8 1.2.3 Steam turbines 9 1.2.4 Jet propulsion 10 1.2.5 Industrial turbines 11 1.2.6 Note on units 12 Viii CONTENTS Principles of Thermodynamics and Fluid Flow 15 2.1 Mass conservation principle 15 2.2 First law of thermodynamics 17 2.3 Second law of thermodynamics 19 2.3.1 Tds equations 19 2.4 Equations of state 20 2.4.1 Properties of steam 21 2.4.2 Ideal gases 27 2.4.3 Air tables and isentropic relations 29 2.4.4 Ideal gas mixtures 31 2.4.5 Incompressibility 35 2.4.6 Stagnation state 35 2.5 Efficiency 36 2.5.1 Efficiency measures 36 2.5.2 Thermodynamic losses 42 2.5.3 Incompressible fluid 43 2.5.4 Compressible flows 44 2.6 Momentum balance 47 Exercises 54 Compressible Flow through Nozzles 57 3.1 Mach number and the speed of sound 57 3.1.1 Mach number relations 59 3.2 Isentropic flow with area change 61 3.2.1 Converging nozzle 65 3.2.2 Converging-diverging nozzle 67 3.3 Normal shocks 69 3.3.1 Rankine-Hugoniot relations 73 3.4 Influence of friction in flow through straight nozzles 75 3.4.1 Polytropic efficiency 75 3.4.2 Loss coefficients 79 3.4.3 Nozzle efficiency 82 3.4:4 Combined Fanno flow and area change 84 3.5 Supersaturation 90 3.6 Prandtl-Meyer expansion 92 3.6.1 Mach waves 92 3.6.2 Prandtl-Meyer theory 93 3.7 Flow leaving a turbine nozzle 100 Exercises 103 4 Principles of Turbomachine Analysis 105 CONTENTS ix 4.1 Velocity triangles 106 4.2 Moment of momentum balance 108 4.3 Energy transfer in turbomachines 109 4.3.1 Trothalpy and specific work in terms of velocities 113 4.3.2 Degree of reaction 116 4.4 Utilization 117 4.5 Scaling and similitude 124 4.5.1 Similitude 124 4.5.2 Incompressible flow 125 4.5.3 Shape parameter or specific speed 128 4.5.4 Compressible flow analysis 128 4.6 Performance characteristics 130 4.6.1 Compressor performance map 131 4.6.2 Turbine performance map 131 Exercises 132 Steam Turbines 135 5.1 Introduction 135 5.2 Impulse turbines 137 5.2.1 Single-stage impulse turbine 137 5.2.2 Pressure compounding 146 5.2.3 Blade shapes 150 5.2.4 Velocity compounding 152 5.3 Stage with zero reaction 158 5.4 Loss coefficients 160 Exercises 162 Axial Turbines 165 6.1 Introduction 165 6.2 Turbine stage analysis 167 6.3 Flow and loading coefficients and reaction ratio 171 6.3.1 Fifty percent (50%) stage 176 6.3.2 Zero percent (0%) reaction stage 178 6.3.3 Off-design operation 179 6.4 Three-dimensional flow 181 6.5 Radial equilibrium 181 6.5.1 Free vortex flow 183 6.5.2 Fixed blade angle 186 6.6 Constant mass flux 187 6.7 Turbine efficiency and losses 190 6.7.1 Soderberg loss coefficients 190 X CONTENTS 6.7.2 Stage efficiency 191 6.7.3 Stagnation pressure losses 192 6.7.4 Performance charts 198 6.7.5 Zweifel correlation 203 6.7.6 Further discussion of losses 204 6.7.7 Ainley-Mathieson correlation 205 6.7.8 Secondary loss 209 6.8 Multistage turbine 214 6.8.1 Reheat factor in a multistage turbine 214 6.8.2 Polytropic or small-stage efficiency 216 Exercises 217 Axial Compressors 221 7.1 Compressor stage analysis 222 7.1.1 Stage temperature and pressure rise 223 7.1.2 Analysis of a repeating stage 225 7.2 Design deflection 230 7.2.1 Compressor performance map 234 7.3 Radial equilibrium 235 7.3.1 Modified free vortex velocity distribution 236 7.3.2 Velocity distribution with zero-power exponent 239 7.3.3 Velocity distribution with first-power exponent 240 7.4 Diffusion factor 242 7.4.1 Momentum thickness of a boundary layer 244 7.5 Efficiency and losses 247 7.5.1 Efficiency* 247 7.5.2 Parametric calculations 250 7.6 Cascade aerodynamics 252 7.6.1 Blade shapes and terms 252 7.6.2 Blade forces 253 7.6.3 Other losses 256 7.6.4 Diffuser performance 257 7.6.5 Flow deviation and incidence 257 7.6.6 Multistage compressor 259 7.6.7 Compressibility effects 261 Exercises 262 Centrifugal Compressors and Pumps 265 8.1 Compressor analysis 266 8.1.1 Slip factor 267 8.1.2 Pressure ratio 269 CONTENTS xi 8.2 Inlet design 274 8.2.1 Choking of the inducer 278 8.3 Exit design 281 8.3.1 Performance characteristics 281 8.3.2 Diffusion ratio 283 8.3.3 Blade height 284 8.4 Vaneless diffuser 285 8.5 Centrifugal pumps 290 8.5.1 Specific speed and specific diameter 294 8.6 Fans 302 8.7 Cavitation 302 8.8 Diffuser and volute design 305 8.8.1 Vaneless diffuser 305 8.8.2 Volute design 306 Exercises 309 9 Radial Inflow Turbines 313 9.1 Turbine analysis 314 9.2 Efficiency 319 9.3 Specific speed and specific diameter 323 9.4 Stator flow 329 9.4.1 Loss coefficients for stator flow 333 9.5 Design of the inlet of a radial inflow turbine 337 9.5.1 Minimum inlet Mach number 338 9.5.2 Blade stagnation Mach number 343 9.5.3 Inlet relative* Mach number 345 9.6 Design of the Exit 346 9.6.1 Minimum exit Mach number 346 9.6.2 Radius ratio r%s/r2 348 9.6.3 Blade height-to-radius ratio 62/^2 350 9.6.4 Optimum incidence angle and the number of blades 351 Exercises 356 10 Hydraulic Turbines 359 10.1 Hydroelectric Power Plants 359 10.2 Hydraulic turbines and their specific speed 361 10.3 Pelton wheel 363 10.4 Francis turbine 370 10.5 Kaplan turbine 377 10.6 Cavitation 380 Exercises 382 XII CONTENTS 11 Hydraulic Transmission of Power 385 11.1 Fluid couplings 385 11.1.1 Fundamental relations 386 11.1.2 Flow rate and hydrodynamic losses 388 11.1.3 Partially filled coupling 390 11.2 Torque converters 391 11.2.1 Fundamental relations 392 11.2.2 Performance 394 Exercises 398 12 Wind turbines 401 12.1 Horizontal-axis wind turbine 402 12.2 Momentum and blade element theory of wind turbines 403 12.2.1 Momentum Theory 403 12.2.2 Ducted wind turbine 407 12.2.3 Blade element theory and wake rotation 409 12.2.4 Irrotational wake 412 12.3 Blade Forces 415 12.3.1 Nonrotating wake 415 12.3.2 Wake with rotation 419 12.3.3 Ideal wind turbine 424 12.3.4 Prandtl's tip correction 425 12.4 Turbomachinery and future prospects for energy 429 Exercises 430 Appendix A: Streamline curvature and radial equilibrium 431 A.l Streamline curvature method 431 A.