Convective Heat Transfer Solved Problems Michel Favre-Marinet Sedat Tardu
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Convective Heat Transfer
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Convective Heat Transfer Solved Problems Michel Favre-Marinet Sedat Tardu
First published in France in 2008 by Hermes Science/Lavoisier entitled: Écoulements avec échanges de chaleur volumes 1 et 2 LAVOISIER, 2008 First published in Great Britain and the United States in 2009 by ISTE Ltd and John Wiley & Sons, Inc. Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address: ISTE Ltd John Wiley & Sons, Inc. 27-37 St George s Road 111 River Street London SW19 4EU Hoboken, NJ 07030 UK USA www.iste.co.uk ISTE Ltd, 2009 www.wiley.com The rights of Michel Favre-Marinet and Sedat Tardu to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988. Library of Congress Cataloging-in-Publication Data Favre-Marinet, Michel, 1947- [Ecoulements avec échanges de chaleur. English] Convective heat transfer : solved problems / Michel Favre-Marinet, Sedat Tardu. p. cm. Includes bibliographical references and index. ISBN 978-1-84821-119-3 1. Heat--Convection. 2. Heat--Transmission. I. Tardu, Sedat, 1959- II. Title. TJ260.F3413 2009 621.402'25--dc22 2009016463 British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN: 978-1-84821-119-3 Printed and bound in Great Britain by CPI Antony Rowe, Chippenham and Eastbourne.
Table of Contents Foreword.... xiii Preface... xv Chapter 1. Fundamental Equations, Dimensionless Numbers... 1 1.1. Fundamental equations... 1 1.1.1. Local equations... 1 1.1.2. Integral conservation equations... 4 1.1.3. Boundary conditions... 7 1.1.4. Heat-transfer coefficient... 7 1.2. Dimensionless numbers... 8 1.3. Flows with variable physical properties: heat transfer in a laminar Couette flow... 9 1.3.1. Description of the problem... 9 1.3.2. Guidelines... 10 1.3.3. Solution... 10 1.4. Flows with dissipation... 14 1.4.1. Description of the problem... 14 1.4.2. Guidelines... 14 1.4.3. Solution... 15 1.5. Cooling of a sphere by a gas flow... 20 1.5.1. Description of the problem... 20 1.5.2. Guidelines... 21 1.5.3. Solution... 21
vi Convective heat Transfer Chapter 2. Laminar Fully Developed Forced Convection in Ducts... 31 2.1. Hydrodynamics... 31 2.1.1. Characteristic parameters... 31 2.1.2. Flow regions... 32 2.2. Heat transfer... 33 2.2.1. Thermal boundary conditions... 33 2.2.2. Bulk temperature... 34 2.2.3. Heat-transfer coefficient... 34 2.2.4. Fully developed thermal region... 34 2.3. Heat transfer in a parallel-plate channel with uniform wall heat flux.. 35 2.3.1. Description of the problem... 35 2.3.2. Guidelines... 36 2.3.3. Solution... 37 2.4. Flow in a plane channel insulated on one side and heated at uniform temperature on the opposite side... 46 2.4.1. Description of the problem... 46 2.4.2. Guidelines... 47 2.4.3. Solution... 47 Chapter 3. Forced Convection in Boundary Layer Flows... 53 3.1. Hydrodynamics... 53 3.1.1. Prandtl equations... 53 3.1.2. Classic results... 55 3.2. Heat transfer... 58 3.2.1. Equations of the thermal boundary layer... 58 3.2.2. Scale analysis... 58 3.2.3. Similarity temperature profiles... 59 3.3. Integral method... 62 3.3.1. Integral equations... 62 3.3.2. Principle of resolution using the integral method... 64 3.4. Heated jet nozzle... 65 3.4.1. Description of the problem... 65 3.4.2. Solution... 67 3.5. Asymptotic behavior of thermal boundary layers... 68 3.5.1. Description of the problem... 68 3.5.2. Guidelines... 69 3.5.3. Solution... 69 3.6. Protection of a wall by a film of insulating material... 74 3.6.1. Description of the problem... 74 3.6.2. Guidelines... 76 3.6.3. Solution... 77
Table of Contents vii 3.7. Cooling of a moving sheet... 83 3.7.1. Description of the problem... 83 3.7.2. Guidelines... 84 3.7.3. Solution... 86 3.8. Heat transfer near a rotating disk... 93 3.8.1. Description of the problem... 93 3.8.2. Guidelines... 94 3.8.3. Solution... 96 3.9. Thermal loss in a duct... 106 3.9.1. Description of the problem... 106 3.9.2. Guidelines... 107 3.9.3. Solution... 108 3.10. Temperature profile for heat transfer with blowing... 117 3.10.1. Description of the problem... 117 3.10.2. Solution... 118 Chapter 4. Forced Convection Around Obstacles... 119 4.1. Description of the flow... 119 4.2. Local heat-transfer coefficient for a circular cylinder... 121 4.3. Average heat-transfer coefficient for a circular cylinder... 123 4.4. Other obstacles... 125 4.5. Heat transfer for a rectangular plate in cross-flow... 126 4.5.1. Description of the problem... 126 4.5.2. Solution... 126 4.6. Heat transfer in a stagnation plane flow. Uniform temperature heating... 128 4.6.1. Description of the problem... 128 4.6.2. Guidelines... 129 4.6.3. Solution... 130 4.7. Heat transfer in a stagnation plane flow. Step-wise heating at uniform flux... 131 4.7.1. Description of the problem... 131 4.7.2. Guidelines... 132 4.7.3. Solution... 133 4.8. Temperature measurements by cold-wire... 135 4.8.1. Description of the problem... 135 4.8.2. Guidelines... 136 4.8.3. Solution... 137
viii Convective heat Transfer Chapter 5. External Natural Convection.... 141 5.1. Introduction... 141 5.2. Boussinesq model... 142 5.3. Dimensionless numbers. Scale analysis... 142 5.4. Natural convection near a vertical wall... 145 5.4.1. Equations... 145 5.4.2. Similarity solutions... 146 5.5. Integral method for natural convection... 149 5.5.1. Integral equations... 149 5.5.2. Solution... 150 5.6. Correlations for external natural convection... 152 5.7. Mixed convection... 152 5.8. Natural convection around a sphere... 155 5.8.1. Description of the problem... 155 5.8.2. Solution... 155 5.9. Heated jet nozzle... 157 5.9.1. Description of the problem... 157 5.9.2. Solution... 158 5.10. Shear stress on a vertical wall heated at uniform temperature... 161 5.10.1. Description of the problem... 161 5.10.2. Solution... 162 5.11. Unsteady natural convection... 164 5.11.1. Description of the problem... 164 5.11.2. Guidelines... 166 5.11.3. Solution... 167 5.12. Axisymmetric laminar plume... 176 5.12.1. Description of the problem... 176 5.12.2. Solution... 177 5.13. Heat transfer through a glass pane... 183 5.13.1. Description of the problem... 183 5.13.2. Guidelines... 183 5.13.3. Solution... 184 5.14. Mixed convection near a vertical wall with suction... 189 5.14.1. Description of the problem... 189 5.14.2. Guidelines... 190 5.14.3. Solution... 190 Chapter 6. Internal Natural Convection... 195 6.1. Introduction... 195 6.2. Scale analysis... 195
Table of Contents ix 6.3. Fully developed regime in a vertical duct heated at constant temperature... 197 6.4. Enclosure with vertical walls heated at constant temperature... 198 6.4.1. Fully developed laminar regime... 198 6.4.2. Regime of boundary layers... 199 6.5. Thermal insulation by a double-pane window... 199 6.5.1. Description of the problem... 199 6.5.2. Solution... 200 6.6. Natural convection in an enclosure filled with a heat generating fluid. 201 6.6.1. Description of the problem... 201 6.6.2. Solution... 203 6.7. One-dimensional mixed convection in a cavity... 206 6.7.1. Description of the problem... 206 6.7.2. Guidelines... 207 6.7.3. Solution... 208 Chapter 7. Turbulent Convection in Internal Wall Flows... 211 7.1. Introduction... 211 7.2. Hydrodynamic stability and origin of the turbulence... 211 7.3. Reynolds averaged Navier-Stokes equations... 213 7.4. Wall turbulence scaling... 215 7.5. Eddy viscosity-based one point closures... 216 7.6. Some illustrations through direct numerical simulations... 227 7.7. Empirical correlations... 231 7.8. Exact relations for a fully developed turbulent channel flow... 233 7.8.1. Reynolds shear stress... 233 7.8.2. Heat transfer in a fully developed turbulent channel flow with constant wall temperature... 238 7.8.3. Heat transfer in a fully developed turbulent channel flow with uniform wall heat flux... 240 7.9. Mixing length closures and the temperature distribution in the inner and outer layers... 243 7.9.1. Description of the problem... 245 7.9.2. Guidelines... 245 7.9.3. Solution... 246 7.10. Temperature distribution in the outer layer... 252 7.10.1. Description of the problem... 252 7.10.2. Guidelines... 252 7.10.3. Solution... 252
x Convective heat Transfer 7.11. Transport equations and reformulation of the logarithmic layer... 255 7.11.1. Description of the problem... 257 7.11.2. Guidelines... 257 7.11.3. Solution... 258 7.12. Near-wall asymptotic behavior of the temperature and turbulent fluxes 261 7.12.1. Description of the problem... 261 7.12.2. Guidelines... 261 7.12.3. Solution... 261 7.13. Asymmetric heating of a turbulent channel flow... 264 7.13.1. Description of the problem... 264 7.13.2. Guidelines... 265 7.13.3. Solution... 266 7.14. Natural convection in a vertical channel in turbulent regime... 270 7.14.1. Description of the problem... 270 7.14.2. Guidelines... 271 7.14.3. Solution... 274 Chapter 8. Turbulent Convection in External Wall Flows.... 281 8.1. Introduction... 281 8.2. Transition to turbulence in a flat plate boundary layer... 281 8.3. Equations governing turbulent boundary layers... 282 8.4. Scales in a turbulent boundary layer... 284 8.5. Velocity and temperature distributions... 284 8.6. Integral equations... 285 8.7. Analogies... 286 8.8. Temperature measurements in a turbulent boundary layer... 289 8.8.1. Description of the problem... 289 8.8.2. Solution... 290 8.9. Integral formulation of boundary layers over an isothermal flat plate with zero pressure gradient... 292 8.9.1. Description of the problem... 292 8.9.2. Guidelines... 293 8.9.3. Solution... 294 8.10. Prandtl-Taylor analogy... 297 8.10.1. Description of the problem... 297 8.10.2. Guidelines... 297 8.10.3. Solution... 298 8.11. Turbulent boundary layer with uniform suction at the wall... 301 8.11.1. Description of the problem... 301 8.11.2. Guidelines... 301 8.11.3. Solution... 302
Table of Contents xi 8.12. Turbulent boundary layers with pressure gradient. Turbulent Falkner-Skan flows... 306 8.12.1. Description of the problem... 306 8.12.2. Guidelines... 306 8.12.3. Solution... 307 8.13. Internal sublayer in turbulent boundary layers subject to adverse pressure gradient... 312 8.13.1. Description of the problem... 312 8.13.2. Guidelines... 312 8.13.3. Solution... 313 8.14. Roughness... 319 8.14.1. Description of the problem... 319 8.14.2. Guidelines... 320 8.14.3. Solution... 320 Chapter 9. Turbulent Convection in Free Shear Flows... 323 9.1. Introduction... 323 9.2. General approach of free turbulent shear layers... 323 9.3. Plumes... 326 9.4. Two-dimensional turbulent jet... 328 9.4.1. Description of the problem... 328 9.4.2. Guidelines... 329 9.4.3. Solution... 330 9.5. Mixing layer... 335 9.5.1. Description of the problem... 335 9.5.2. Guidelines... 335 9.5.3. Solution... 336 9.6. Determination of the turbulent Prandtl number in a plane wake... 340 9.6.1. Description of the problem... 340 9.6.2. Guidelines... 341 9.6.3. Solution... 342 9.7. Regulation of temperature... 348 9.7.1. Description of the problem... 348 9.7.2. Guidelines... 350 9.7.3. Solution... 351 List of symbols... 363 References... 367 Index... 371
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Foreword It is a real surprise and pleasure to read this brainy book about convective heat transfer. It is a surprise because there are several books already on this subject, and because the book title is deceiving: here solved problems means the structure of the field and the method of teaching the discipline, not a random collection of homework problems. It is a pleasure because it is no-nonsense and clear, with the ideas placed naked on the table, as in elementary geometry. The field of convection has evolved as a sequence of solved problems. The first were the most fundamental and the simplest, and they bear the names of Prandtl, Nusselt, Reynolds and their contemporaries. As the field grew, the problems became more applied (i.e. good for this, but not for that), more complicated, and much more numerous and forgettable. Hidden in this stream, there are still a few fundamental problems that emerge, yet they are obscured by the large volume. It is here that this book makes its greatest contribution: the principles and the most fundamental problems come first. They are identified, stated and solved. The book teaches not only structure but also technique. The structure of the field is drawn with very sharp lines: external versus internal convection, forced versus natural convection, rotation, combined convection and conduction, etc. The best technique is to start with the simplest problem solving method (scale analysis) and to teach progressively more laborious and exact methods (integral method, selfsimilarity, asymptotic behavior). Scale analysis is offered the front seat in the discussion with the student. This is a powerful feature of the book because it teaches the student how to determine (usually on the back of an envelope) the proper orders of magnitude of all the physical features (temperature, fluid velocity, boundary layer thickness, heat flux) and the correct dimensionless groups, which are the fewest such numbers. With