Conjugate Problems in Convective Heat Transfer

400 Pages·2012·6.43 MB·English

by  Abram S. Dorfman

Conjugate Problems
in Convective
Heat Transfer
Abram S. Dorfman
Boca Raton London New York
CRC Press is an imprint of the
Taylor & Francis Group, an informa business
© 2010 by Taylor & Francis Group, LLC
8237X_C000.indd 3 7/24/09 6:33:43 PM
CRC Press
Taylor & Francis Group
6000 Broken Sound Parkway NW, Suite 300
Boca Raton, FL 33487-2742
© 2010 by Taylor and Francis Group, LLC
CRC Press is an imprint of Taylor & Francis Group, an Informa business
No claim to original U.S. Government works
Printed in the United States of America on acid-free paper
10 9 8 7 6 5 4 3 2 1
International Standard Book Number: 978-1-4200-8237-1 (Hardback)
This book contains information obtained from authentic and highly regarded sources. Reasonable efforts
have been made to publish reliable data and information, but the author and publisher cannot assume
responsibility for the validity of all materials or the consequences of their use. The authors and publishers
have attempted to trace the copyright holders of all material reproduced in this publication and apologize to
copyright holders if permission to publish in this form has not been obtained. If any copyright material has
not been acknowledged please write and let us know so we may rectify in any future reprint.
Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmit-
ted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented,
including photocopying, microfilming, and recording, or in any information storage or retrieval system,
without written permission from the publishers.
For permission to photocopy or use material electronically from this work, please access www.copyright.
com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood
Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and
registration for a variety of users. For organizations that have been granted a photocopy license by the CCC,
a separate system of payment has been arranged.
Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used
only for identification and explanation without intent to infringe.
Library of Congress Cataloging-in-Publication Data
Dorfman, A. Sh. (Abram Shlemovich)
Conjugate problems in convective heat transfer / Abram S. Dorfman.
p. cm. -- (Heat transfer)
Includes bibliographical references and index.
ISBN 978-1-4200-8237-1 (hardcover : alk. paper)
1. Heat--Transmission. 2. Heat--Convection. 3. Fluid dynamics. I. Title.
TJ260.D64 2009
621.402’25--dc22 2009018850
Visit the Taylor & Francis Web site at
http://www.taylorandfrancis.com
and the CRC Press Web site at
http://www.crcpress.com
© 2010 by Taylor & Francis Group, LLC
8237X_C000.indd 4 7/24/09 6:33:43 PM
In Memory
In memory of our dearest, lovely,
loving daughter, Ella Fridman.
© 2010 by Taylor & Francis Group, LLC
8237X_C000.indd 5 7/24/09 6:33:43 PM
Contents
List of Examples ....................................................................................................xi
Preface .................................................................................................................xvii
Acknowledgments .............................................................................................xxi
The Author ........................................................................................................xxiii
Introduction .......................................................................................................xxv
Nomenclature ..................................................................................................xxvii
Part I: Approximate Solutions
Chapter 1
Analytical Methods for the Estimation of Heat Transfer from
Nonisothermal Walls ............................................................................................3
1.1 Basic Equations...............................................................................................3
1.2 Self-Similar Solutions of the Boundary Layer Equations ........................4
1.3 Solutions of the Boundary Layer Equations in the Power Series ...........5
1.4 Integral Methods ............................................................................................7
1.5 Method of Superposition ............................................................................12
1.6 Solutions of the Boundary Layer Equations in the Series
of Shape Parameters ....................................................................................16
References ...............................................................................................................18
Chapter 2
Approximate Solutions of Conjugate Problems in Convective
Heat Transfer ........................................................................................................21
2.1 Formulation of a Conjugate Problem of Convective
Heat Transfer ................................................................................................21
2.2 The Case of Linear Velocity Distribution across the Thermal
Boundary Layer ............................................................................................22
2.3 The Case of Uniform Velocity Distribution across the Thermal
Boundary Layer (Slug Flow) .......................................................................28
2.4 Solutions of the Conjugate Convective Heat Transfer Problems
in the Power Series .......................................................................................35
2.5 Solutions of the Conjugate Heat Transfer Problems by Integral
Methods .........................................................................................................46
References ...............................................................................................................50
vii
© 2010 by Taylor & Francis Group, LLC
8237X_C000.indd 7 7/24/09 6:33:43 PM
viii Contents
Part II: Theory and Methods
Chapter 3
Heat Transfer from Arbitrary Nonisothermal
Surfaces in a Laminar Flow ...............................................................................55
3.1 The Exact Solution of the Thermal Boundary Layer Equation
for an Arbitrary Surface Temperature Distribution ...............................55
3.2 Generalization for an Arbitrary Velocity Gradient
in a Free Stream Flow ..................................................................................62
3.3 General Form of the Influence Function of the
Unheated Zone: Convergence of the Series..............................................64
3.4 The Exact Solution of the Thermal Boundary Layer
Equation for an Arbitrary Surface Heat Flux Distribution ....................69
3.5 Temperature Distribution on an Adiabatic Surface in an
Impingent Flow ............................................................................................73
3.6 The Exact Solution of an Unsteady Thermal Boundary Layer
Equation for Arbitrary Surface Temperature Distribution ....................75
3.7 The Exact Solution of a Thermal Boundary Layer Equation for
a Surface with Arbitrary Temperature in a Compressible Flow ...........78
3.8 The Exact Solution of a Thermal Boundary Layer
Equation for a Moving Continuous Surface with Arbitrary
Temperature Distribution ...........................................................................79
3.9 The Other Solution of a Thermal Boundary Layer Equation
for an Arbitrary Surface Temperature Distribution ...............................83
3.9.1 Non-Newtonian Fluid with a Power Law Rheology ..................83
3.9.2 The Effect of Mechanical Energy Dissipation .............................86
3.9.3 Axisymmetric Streamlined and Rotating Bodies .......................87
3.9.4 Thin Cylindrical Bodies ..................................................................89
References ...............................................................................................................91
Chapter 4
Heat Transfer from Arbitrary Nonisothermal Surfaces
in Turbulent Flow ................................................................................................95
4.1 Basis Relations for the Equilibrium Boundary Layer .............................95
4.2 Solution of the Thermal Turbulent Boundary Layer Equation
for an Arbitrary Surface Temperature Distribution ...............................99
4.3 Intensity of Heat Transfer from an Isothermal Surface:
Comparison with Experimental Data .....................................................104
4.3.1 Reynolds Analogy .........................................................................104
4.3.2 Relations for Heat Transfer from an Isothermal
Surface in Gradientless Flow ........................................................105
4.3.3 Relations for Heat Transfer from an Isothermal
Surface in Gradient Flows.............................................................109
4.4 The Effect of the Turbulent Prandtl Number on Heat Transfer
on Flat Plates ...............................................................................................111
© 2010 by Taylor & Francis Group, LLC
8237X_C000.indd 8 7/24/09 6:33:43 PM
Contents ix
4.5 Coefficients g of Heat Flux Series for Nonisothermal Surfaces .........112
k
4.6 Approximate Relations for Heat Flux in a Transition Regime ............116
References .............................................................................................................118
Chapter 5
General Properties of Nonisothermal and Conjugate Heat Transfer ........121
5.1 The Effect of Temperature Head Distribution on Heat Transfer
Intensity .......................................................................................................121
5.1.1 The Effect of the Temperature Head Gradient ..........................122
5.1.2 The Effect of Flow Regime ............................................................132
5.1.3 The Effect of Pressure Gradient ...................................................137
5.2 Gradient Analogy and Reynolds Analogy ............................................137
5.3 Heat Flux Inversion ...................................................................................141
5.4 Zero Heat Transfer Surfaces .....................................................................148
5.5 Examples of Optimizing Heat Transfer in Flow over Bodies..............150
References .............................................................................................................158
Chapter 6
Analytical Methods for Solving Conjugate Convective Heat
Transfer Problems ..............................................................................................161
6.1 A Biot Number as a Criterion of the Conjugate Heat
Transfer Rate ...............................................................................................161
6.2 General Boundary Condition for Convective Heat Transfer
Problems: Errors Caused by Boundary Condition of the Third Kind ....162
6.3 Reduction of a Conjugate Convective Heat Transfer Problem
to an Equivalent Heat Conduction Problem ..........................................166
6.4 Temperature Singularities on the Solid–Fluid Interface ......................179
6.4.1 Basic Equations...............................................................................179
6.4.2 Examples of Singularities for Different Flow Regimes
and Conditions ...............................................................................181
6.4.3 Estimation of Accuracy of the Assumption of a
Thermally Thin Body ....................................................................183
6.5 Universal Functions for Solving Conjugate Heat Transfer
Problems — Solution Examples ...............................................................183
6.6 Reducing the Unsteady Conjugate Convective Heat Transfer
Problem to an Equivalent Heat Conduction Problem ..........................198
6.6.1 Validity of Quasi-Steady Approximation — The Luikov
Number as a Conjugate Criterion in Unsteady Heat Transfer .....199
6.6.2 Universal Eigenfunctions for Unsteady Conjugate Heat
Transfer Problems ..........................................................................201
6.7 Integral Transforms and Similar Methods ............................................206
6.8 Solutions in Asymptotic Series in Eigenfunctions ...............................210
6.9 Superposition and Other Methods ..........................................................220
6.10 Green’s Function and the Method of Perturbation ...............................224
References .............................................................................................................227
© 2010 by Taylor & Francis Group, LLC
8237X_C000.indd 9 7/24/09 6:33:43 PM
x Contents
Chapter 7
Numerical Methods for Solving Conjugate Convective
Heat Transfer Problems ....................................................................................231
7.1 Analytical and Numerical Methods .......................................................231
7.2 Approximate Analytical and Numerical Methods for Solving
Differential Equations ...............................................................................234
7.3 Difficulties in Computing Convection-Diffusion and Flow ................240
7.3.1 The Control-Volume Finite-Difference Method ........................240
7.3.1.1 Computing Pressure and Velocity ................................240
7.3.1.2 Computing Convection–Diffusion ...............................242
7.3.1.3 False Diffusion .................................................................245
7.3.1.4 A Case of a Moderate Mach Number ...........................246
7.3.2 The Control-Volume Finite-Element Method ............................246
7.4 Numerial Methods of Conjugation .........................................................249
7.5 Examples of Numerical Studies of the Conjugate Convective
Heat Transfer in Pipes and Channels .....................................................252
7.6 Examples of Numerical Studies of the Conjugate Convective
Heat Transfer in Flows around and inside Bodies ................................264
References .............................................................................................................283
Part III: Applications
Chapter 8
Thermal Treatment of Materials ....................................................................289
8.1 Moving Materials Undergoing Thermal Processing ............................289
8.2 Simulation of Industrial Processes ..........................................................296
8.3 Drying of Continuous Moving Materials...............................................307
References .............................................................................................................318
Chapter 9
Technological Processes ...................................................................................321
9.1 Multiphase and Phase-Change Processes ..............................................321
9.2 Drying and Food Processing ...................................................................333
References .............................................................................................................341
Chapter 10
Manufacturing Equipment Operation ..........................................................343
10.1 Heat Exchangers and Finned Surfaces ...................................................343
10.2 Cooling Systems .........................................................................................358
10.2.1 Electronic Packages .......................................................................358
10.2.2 Turbine Blades and Rockets .........................................................362
10.2.3 Cooling by Rewetting Surfaces ....................................................368
References .............................................................................................................377
Conclusion ...........................................................................................................379
© 2010 by Taylor & Francis Group, LLC
8237X_C000.indd 10 7/24/09 6:33:44 PM
List of Examples
Chapter 1
Example 1.1: The Effect of Boundary Conditions...............................................5
Example 1.2: Friction and Heat Transfer from an Arbitrary
Nonisothermal Plate ......................................................................................8
Example 1.3: Calculating the Influence Function .............................................13
Example 1.4: Accuracy of Linear Velocity Distribution ...................................14
Chapter 2
Example 2.1: An Incompressible Flow with a Large Prandtl Number
Past Plate with a Heat Source .....................................................................24
Example 2.2: Transient Heat Transfer in the Thermal Entrance Region
of a Parallel Plate Channel with the Fully Developed Flow ..................25
Example 2.3: Transient Heat Transfer from a Plate of Appreciable
Thermal Capacity with Heat Generation of q to Turbulent Flow ........28
0
Example 2.4: Unsteady Heat Transfer between a Fluid with
Time-Varying Temperature and a Plate ....................................................31
Example 2.5: Heat Transfer between Two Fluids Separated
by a Thin Plate ..............................................................................................34
Example 2.6: A Thermally Thin Plate with Heat Sources
in a Compressible Fluid ..............................................................................36
Example 2.7: Temperature of a Radiating Thin Plate Placed
in Supersonic Gas Flow ...............................................................................39
Example 2.8: Temperature of a Thin Plate after an Oblique Shock................44
Example 2.9: Heat Transfer between a Thin Plate and
an Incompressible Flow Past It ..................................................................46
Example 2.10: Heat Transfer between a Thin Radiating Wedge
and an Incompressible Fluid Flow Past It ................................................47
Chapter 5
Example 5.1: Linear Temperature Head along the Plate ...............................123
Example 5.2: A Plate Heated from One End (Qualitative Analysis) ............124
Example 5.3: Power Law Temperature Head along the Plate ......................124
Example 5.4: Sinusoidal Temperature Head Variation along the Plate .......125
Example 5.5: Linear Temperature Head near Stagnation Point ...................126
Example 5.6: Transverse Flow Past Nonisothermal Cylinder — Linear
Temperature Head .....................................................................................127
Example 5.7: Transverse Flow Past Nonisothermal Cylinder
with q = Constant ......................................................................................129
w
xi
© 2010 by Taylor & Francis Group, LLC
8237X_C000.indd 11 7/24/09 6:33:44 PM
xii List of Examples
Example 5.8: A Jump of Heat Flux on a Transversally Streamlined
Cylinder .......................................................................................................130
Example 5.9: Transverse Flow Past Nonisothermal Cylinder — Effect
of Dissipation ..............................................................................................131
Example 5.10: Comparison between the Effects of Nonisothermicity
in Turbulent and Laminar Flows — Linear Temperature Head .........132
Example 5.11: Different Temperature Head and Heat Flux Variations
for Turbulent Flow — Comparison with Experimental Data ..............133
Example 5.12: Stepwise Temperature Head — Comparison with
Experimental Data .....................................................................................136
Example 5.13: Heat Transfer Inversion in Unsteady Flow — Linear
Temperature Head .....................................................................................144
Example 5.14: Heat Flux Inversion in Gradientless Flow for
a Parabolic and Some Other Temperature Heads .................................146
Example 5.15: Distributing Heat Sources for Optimal Temperature ...........150
Example 5.16: Mode of Change of Temperature Head ..................................153
Example 5.17: Heat Flux Pattern .......................................................................156
Chapter 6
Example 6.1: Heat Transfer from Fluid to Fluid in a Flow Past Two
Sides of a Plate ............................................................................................163
Example 6.2: Heat Transfer from Thermally Thin Plate Heated
from One End .............................................................................................164
Example 6.3: Heat Transfer from Elliptic Cylinder with Semiaxis
Ratio a/b=4 and Uniformly Distributed Source of Power q ...............165
v
Example 6.4: A Continuous Plate (Strip) of Polymer at
Temperature T Extruded from a Die and Passed at Velocity U
0 w
through a Bath with Water (Pr=6.1) at Temperature T ........................165
∞
Example 6.5: Heat Transfer from Liquid to Liquid in a Flow Past
a Thin Plate .................................................................................................167
Example 6.6: Heat Transfer between Two Fluids Separated by a
Plate — An Exact Solution ........................................................................170
Example 6.7: Heat Transfer between an Elliptical Cylinder and
Flowing Laminar Flow ..............................................................................175
Example 6.8: A Plate Heated from One End in a Symmetrical Flow ...........187
Example 6.9: A Plate Streamlined on One Side and Is Isolated
from Another and a Plate in a Flow Past Two Sides .............................189
Example 6.10: A Plate with Inner Heat Sources ..............................................191
Example 6.11: A Plate in a High-Speed Compressible Flow .........................192
Example 6.12: A Plate in the Flow with a Pressure Gradient........................194
Example 6.13: A High Heated Radiating Plate ...............................................194
Example 6.14: Two Countercurrent Flowing Fluids Separated
by a Thin Wall .............................................................................................196
© 2010 by Taylor & Francis Group, LLC
8237X_C000.indd 12 7/24/09 6:33:45 PM
List of Examples xiii
Example 6.15: A Parallel Plate Duct with Isolated Outer
Surfaces and a Periodically Varying Inlet Temperature
of Laminar Flow .........................................................................................206
Example 6.16: Alternative Quasi-Steady Approach .......................................208
Example 6.17: A Parallel Plate Duct with an Outer Wall Subjected
to Convection with Environment and Turbulent Flows
with a Periodically Varying Inlet Temperature .....................................209
Example 6.18: Heat Transfer between a Flush-Mounted Source
on an Infinite Slab and an Incompressible Fluid Flow .........................211
Example 6.19: Heat Transfer between Two Fluids Separated by a
Thin Wall Flowing Concurrently or Countercurrently in the
Double Pipe — Conjugate Graetz Problem............................................213
Example 6.20: Temperature Field in the Entrance of a Plane Duct
with a Fully Developed Velocity Profile .................................................215
Example 6.21: The Pipe with Hydrodynamically and Thermally
Fully Developed Flow and a Thick Wall with Sufficient Heat
Conduction and Capacity .........................................................................217
Example 6.22: Heat Transfer between Two Laminar or
Turbulent Concurrent or Countercurrent Flows Separated
by a Thin Plate ............................................................................................220
Example 6.23: Two Quiescent Fluids Separated by a Vertical Thin Plate ......221
Example 6.24: Heat Transfer between Two Fluids Separated
by a Vertical Thin Plate Local Similarity Approach ..............................222
Example 6.25: A Radiating Plate with an Internal Source
and an Insulated Bottom in Laminar or Turbulent Flows ...................223
Example 6.26: Heat Transfer from a Small Heated Strip of
Length 2a Embedded in an Infinite Plate Streamlined
by an Incompressible Laminar Flow .......................................................224
Example 6.27: Heat Transfer between a Translating Liquid Drop
and Another Immiscible Liquid of Infinite Extent under
the Influence of an Electric Field at a Low Peclet Number ..................226
Chapter 7
Example 7.1: One-Dimensional Conduction Problem for a Plane Wall ......236
Example 7.2: A Horizontal Channel Heated from below
by q = Constant Fully Developed Laminar Flow .................................252
w
Example 7.3: A Pipe with a Fully Developed Laminar Flow Heated
Symmetrically at the Outer Surface by Uniform Heat Flux ................254
Example 7.4: A Thick-Walled Parallel-Plate Channel with
Poiseuille-Couette Incompressible Laminar Flow and
a Moving Upper Wall ................................................................................255
Example 7.5: The Thick-Walled Pipe of an Arbitrary Shape
of Cross-Section with Fully Developed Velocity ...................................257
© 2010 by Taylor & Francis Group, LLC
8237X_C000.indd 13 7/24/09 6:33:45 PM