DESIGN
`AND ANALYSIS
`TUL
`NTTrds
`WTI)ICaI
`
`IPR2022-01299
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`eZ
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`Wiley Series in Thermal Managementof Microelectronic and Electronic Systems
`Allan D. Kraus and Avram Bar-Cohen, Series Editors
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`

`DESIGN AND ANALYSIS OF HEAT SINKS
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`WILEY SERIES IN THERMAL MANAGEMENT OF
`MICROELECTRONIC AND ELECTRONIC SYSTEMS
`Allan D. Kraus and Avram Bar-Cohen, Series Editors
`
`AnIntroduction to Heat Pipes: Modeling, Testing, and Applications
`G.P. Peterson
`
`Design and Analysis of Heat Sinks
`Allan D. Kraus
`Avram Bar-Cohen
`
`
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`

`PA
`—| /
`| &
` / J L / Ke
`DESIGN AND ANALYSIS =
`OF HEAT SINKS
`
`)
`
`Allan D. Kraus
`Allan D. Kraus Associates
`Pacific Grove, CA
`
`Avram Bar-Cohen
`Depariment of Mechanical Engineering
`University ofMinnesota
`Minneapolis, MN
`
`A Wiley-Interscience Publication
`JOHN WILEY & SONS,INC.
`New York / Chichester / Brisbane / Toronto / Singapore
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`
`
`This text is printed on acid-free paper.
`
`Copyright (©€)1995 by John Wiley & Sons, Inc.
`
`All rights reserved. Published simultancously in Canada.
`
`Reproductionor translation of any part of this work beyond
`that permitted by Section 107 or 108 of the 1976 United
`States Copyright Act without the permission of the capyright
`owneris unlawful. Requests for permission or further
`information should be addressed to the Permissions Department,
`John Wiley & Sons, Inc., 605 Third Avenue, New York, NY
`10158-0012
`
`This publication is designed to provide accurate and
`authoritative informationin regard to the subject
`matter covered. It is sold with the understanding that
`the publisher is not engaged in rendering legal, accounting,
`or other professional services. If legal advice or other
`expertassistanceis required, the services of a competent
`professional person should be sought.
`
`Library of Congress Cataloging in Publication Data:
`Kraus, Allan D.
`Design and analysis of heat sinks / Allan D. Kraus, Avram Bar-Cohen.
`p. cm. — (Wiley series in thermal managementof microelectronic
`and electronic systems)
`Includes index.
`ISBN 0-471-01755-8 (cloth: alk. paper)
`1. Heat sinks (Electronics) — Design and construction.
`I. Bar-Cohen, Avram. 1946 — II. Title. IIL Series.
`TK7872.H4K69
`1995
`821.381'046 —dc20
`Printed in the United States of America
`1098765432
`
`1 95-32934
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`

`

`To our teachers and mentors
`
`A. E. Bergles
`K, A. Gardner
`A, L. London
`and
`W. M. Rohsenow
`
`
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`
`
`CONTENTS
`
`PREFACE
`
`INTRODUCTION
`
`xiii
`
`1
`
`11
`1.2
`
`1.3
`
`1.4
`
`1.5
`
`/ 14
`
`Introduction / 1
`Heat-Transfer Fundamentals / 3
`Introduction / 3
`/ 3
`Conduction Heat Transfer
`/ 9
`Convective Heat Transfer
`/ 13
`Radiative Heat Transfer
`Thermal Resistance Network / 14
`Extended Surface and Heat Sinks
`Introduction / 14
`Surface Augmentation with Extended Surface / 15
`Types of Fins, Terminology, and Nomenclature / 16
`The Fin Efficiency / 18
`Modesof Heat Transfer Involving Fins and Surroundings / 19
`Limiting Assumptions / 19
`Chip Module Thermal Resistances / 20
`Definition / 20
`Internal Thermal Resistance / 22
`Substrate or PCB Conduction / 23
`External Resistance / 24
`Flow Resistance / 27
`Total Resistance —Single-Chip Packages
`The Heat Sink Design Procedure / 28
`
`/ 27
`
`vii
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`
`
`vu
`
`CONTENTS
`
`2
`
`LINEAR TRANSFORMATIONS
`
`35
`
`Introduction / 35
`2.1.
`The Longitudinal Fin of Rectangular Profile / 37
`2.2
`The Limitations of the Fin Efficiency / 42
`2.3.
`2.4 The LongitudinalFin of Rectangular Profile Revisited / 44
`2.5 ALinear Transformation / 46
`2.6 Other Linear Transformations / 46
`The Z- and Y-matrices
`/ 47
`The T- and I- matrices / 49
`The H- and G- matrices / 50
`Relationships between Parameters / 51
`Summary of All Conversions / 53
`2.7.
`Formal Developmentof the Linear Transformations
`2.8
`2.9 An Example of Finding the Parameters / 58
`2.10 The Input Admittance and the Thermal Transmission Ratio / 60
`The Input Admittance / 60
`The Thermal Transmission Ratio / 61
`
`/ 54
`
`3.
`
`ELEMENTS OF THE LINEAR TRANSFORMATIONS
`
`63
`
`Introduction / 63
`3.1
`3.2 The Longitudinal Fin of Rectangular Profile / 65
`The T-matrix / 65
`The T-matrix / 65
`The Z-matrix / 66
`The Y-matrix / 66
`The H-matrix / 66
`The G-matrix / 66
`3.3 The Longitudinal Fin of Rectangular Profile
`with One Face Insulated / 69
`3.4 The LongitudinalFin of Trapezoidal Profile / 70
`The T-matrix / 76
`The Z-matrix / 76
`The Y-matrix / 76
`The H-matrix / 77
`The G-matrix / 77
`3.5 The LongitudinalFin of Half TrapezoidalProfile / 79
`3.6 The Longitudinal Fin of Truncated Concave Parabolic Profile / 81
`The T-matrix / 84
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`
`
`
`CONTENTS
`
`ix
`
`The Z-matrix / 85
`The Y-matrix / 85
`The H-matrix / 85
`The G-matrix / 86
`The Radial Fin of Rectangular Profile / 86
`The T-matrix / 90
`The Z-matrix / 91
`The Y-matrix / 91
`The H-matrix / 91
`The G-matrix / 92
`A Generalized Differential Equation for Spines / 93
`The Cylindrical Spine / 95
`The Spine of Rectangular Cross Section / 96
`The Elliptical Spine / 97
`The Truncated Conical Spine / 99
`The T-matrix / 103
`The Z-matrix / 104
`The Y-matrix / 104
`The H-matrix / 104
`The G-matrix / 105
`The Truncated Concave Parabolic Spine / 105
`The T-matrix / 108
`The Z-matrix / 108
`The Y-matrix / 108
`The H-matrix / 109
`The G-matrix / 109
`Summary of Matrix Elements / 110
`Closure / 115
`
`w=ebw©Dbne)
`
`wy
`
`w
`
`3.13
`
`3.14
`3.15
`
`SINGULAR FINS AND SPINES AND SINGLE
`ELEMENTS
`
`117
`
`4.1
`4.2
`43
`4.4
`4.5
`4.6
`4.7
`
`Introduction / 117
`The Longitudinal Fin of Triangular Profile / 119
`Longitudinal Fin of Concave Parabolic Profile / 122
`The Conical Spine / 125
`The Concave Parabolic Spine / 128
`The Single Series Resistance / 129
`The Single Shunt Conductance / 131
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`
`
`CONTENTS
`
`48
`
`Summary and Closure / 133
`
`135
`
`157
`
`5
`
`6
`
`ALGORITHMS FOR FINNED ARRAY ASSEMBLY
`5.1
`5.2
`5.3
`5.4
`
`/ 137
`
`Introduction / 135
`Arrays with and without Loops
`The Cascade Algorithm / 137
`The Array Input Admittance / 140
`The Fin Efficiency / 145
`Bond Resistance / 145
`The Thermal Transmission Ratio / 148
`Fins in Cluster
`/ 148
`Fins in Parallel
`/ 151
`The Choking Phenomenon and Array Optimization / 154
`Closure / 155
`
`5.5
`5.6
`5.7
`5.8
`
`EXAMPLES OF FINNED ARRAY ANALYSIS
`6.1
`6.2
`6.3
`6.4
`6.5
`6.6
`6.7
`6.8
`
`Introduction / 157
`An Example of Heat Sink Analysis / 157
`Removal of the Choke / 162
`An Example of a Finned Heat Exchanger
`A Caution with Regard to Choking / 172
`The Exploitation of Symmetry / 173
`An Array Containing Radial Fins and Spines / 176
`Fins with Variable Heat-Transfer Coefficient
`/ 180
`The Han and Lefkowitz Solution / 181
`The Use of the Cascade Algorithm / 184
`The Numberof Subfins Required / 184
`
`/ 167
`
`7
`
`RECIPROCITY AND NODE ANALYSIS
`
`187
`
`7.1
`7.2
`7.3
`
`74
`
`Introduction / 187
`A Step Forward / 187
`Reciprocity of Regular Fins / 192
`Introduction / 192
`Reciprocity / 194
`Conditions for Reciprocity / 197
`The Equivalent Pi-Network / 197
`The Array Graph / 199
`The General Branch / 201
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`CONTENTS
`
`XL
`
`The Node Branch Incidence Matrix / 202
`Continuity / 203
`Node-to-Datum Analysis of Finned Arrays / 204
` TwoExamples / 205
`
`7.5
`
`A GENERAL ARRAY METHOD
`
`219
`
`Introduction / 219
`8.1
`The General Array Algorithm / 219
`8.2.
`A Technique for Node Reduction / 224
`8.3
`84 Three Examples / 225
`8.5 Closure / 236
`
`9
`
`CONVECTIVE OPTIMIZATIONS
`
`239
`
`9.3
`
`Introduction / 239
`9.1
`/ 240
`9.2 Longitudinal Fins
`The Longitudinal Fin of Rectangular Profile / 240
`The Longitudinal Fin of Triangular Profile / 242
`The Longitudinal Fin of Concave Parabolic Profile / 244
`Comparison of Optimum Fins
`/ 245
`Spines / 248
`The Cylindrical Spine / 249
`The Conical Spine / 250
`The Concave Parabolic Spine / 251
`9.4 Radial Fins
`/ 254
`The Radial Fin of Rectangular Profile / 254
`Radial Fins of Trapezoidal and Triangular Profiles / 259
`Radial Fins of Parabolic and Hyperbolic Profiles
`/ 265
`9.5 Closure / 271
`
`10
`
`HEAT TRANSFER-PARALLEL PLATE HEAT SINKS
`
`273
`
`10.1 Introduction / 273
`/ 274
`10.2 Forced Convection in a Parallel-Plate Channel
`/ 275
`Thermofluid Analysis of Laminar Flow ina Channel
`Turbulent, Fully Developed, Smooth Channel Flow / 283
`10.3. Natural Convection in a Parallel-Plate Channel
`/ 286
`The Elenbaas Correlation / 286
`Composite Relations / 287
`Symmetric, Isothermal Plates
`
`/ 289
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`xil
`
`CONTENTS
`
`/ 291
`
`10.4 Thermal Radiation from a Parallel-Plate Channel
`10.5 Arrays of Longitudinal, Rectangular Fins
`in Natural Convection / 302
`Introduction / 302
`Empirical Results
`/ 304
`Natural Convection Correlations / 310
`10.6 Natural Convection in Pin-Fin Arrays / 320
`10.7 Optimum (Least Material) Heat Sinks
`/ 322
`Introduction / 322
`Theoretical Background / 323
`Optimum Multiple Fin Array / 325
`/ 329
`Optimum Natural Convection Heat Sinks
`Optimum Forced Convection Arrays / 336
`25-kW Power Tube / 337
`10.8 Closure / 339
`
`'
`
`REFERENCES
`
`APPENDIX A MATRICES AND DETERMINANTS
`
`APPENDIX B NOMENCLATURE
`
`AUTHOR INDEX
`
`SUBJECT INDEX
`
`341
`
`447
`
`477
`
`489
`
`491
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`
`
`PREFACE
`
`Heat sinks are the most common thermal management hardware in use in
`the electronics industry. They are used to improve the thermal control of
`electronic components, assemblies, and modules by enhancingtheir exterior
`surface area through the useof fins or spines. The presence of the heat sink
`hasa positive impact on both thereliability and the functional performance
`of electronic, telecommunication, and power conversion systems.
`In the present day competitive environment, superior thermal perfor-
`mance must be attained simultaneously with lower manufacturing costs,
`minimum added mass, and shortened product developmentcycles. Thus, the
`successful design and/or selection of cost-effective heat sinks for electronic
`equipment requires a detailed understanding of the governing heat transfer
`phenomena, mastery of thermal modeling tools for geometrically complex
`fin structures, and an appreciation for thermal optimization opportunities.
`In this book, the authors have attempted to provide a comprehensivetreat-
`mentof the “Analysis and Design of Heat Sinks,” including developmentof
`the mathematical tools, discussion of the prevailing heat transport processes,
`articulation of the toolsfor a variety of fin geometries, and presentation of sim-
`ple “transfer function”relationships forsingle fin, cascadedfin, and fin array
`heat sinks. Indeed, manyofthe fin geometries examined in this book cannot
`be thermally characterized by conventional analytical techniques. Thus,this
`self contained book can serveas a text for the engineer interested in learning
`the subject of heat sink design and analysis in its entirety. And yet, it can be
`used as a source book for the accomplished thermal specialist seeking to hone
`his/her skills.
`The book opens with an extensive introduction, presenting the funda-
`mentals of heat transfer, thermal modeling of electronic packages, and the
`first-order thermal analysis of finned arrays. The next two chapters review
`the mathematicsof linear transformations and prepare the readerfor the ap-
`plication of this methodologyto fin structures, Chapters 4 through 9, the core
`
`xiii
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`xtv
`
`PREFACE
`
`of this heat sink book, systematically explore the useoflinear transformations
`in the thermal parameterizationof a wide variety of singlefin, cascaded fin,
`and fin array designs encountered in commercial heat sinks, as well as the
`least material optimization ofspecific fin shapes. Thelast chapterof the book
`is devotedto the thermalcharacterization and optimization of the ubiquitous
`plate-fin heat sink, including both convective and radiative cooling and pre-
`sentation of empirical results, as well as theoretical formulations. The reader
`will find a useful review of matrices and determinants, along withthelist of
`nomenclature and references in the appendices.
`Mastery ofthe contents of this book will equip the reader with a most
`powerful thermal analysis technique whichis uniquely suited to the analysis,
`design and optimization of convective heat sinks. This will provide a rich
`understanding of the thermal performance and optimizationpotential of the
`most commonly used plate-fin heat sinks and a deep appreciation of the
`thermal context of heat sink applications in the electronics industry.
`The authors are indebted to Abdul Aziz, Jim Welty, Bud Peterson, Dave
`Snider, and Mike Yovanovich for their continued encouragement, to Frank
`Cerra whose vision brought aboutthe series in thermal management, and to
`Howard and Jo Aksen who handled the production of the bookso ably.
`
`ALLAN D. KRAUS
`Pacific Grove, CA
`
`AVRAM BAR-COHEN
`Minneapolis, MN
`
`
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`Chapter
`1
`
`
`INTRODUCTION
`
`1.1
`
`INTRODUCTION
`
`This book is devoted to the heat sink which is used to enhance heattransfer
`from the surface of an electronic component or package to the surround-
`ing environment. Although the presence of the heat sink (and its bonding
`layer) produces a thermal resistance, when properly designed, the thermal
`resistance attributed to the heat sink will be substantially lower than the re-
`sistance associated with convection and radiation from the bare surface of
`the component. Thus, thermal design of electronic equipment often requires
`detailed consideration of the thermal resistance of the heat sink as part of the
`thermal path between the componentjunction and the “ground” temperature
`(usually the environment). The heat sink represents just one element along
`this thermal path, but, because it may be the controlling resistance, thermal
`analysis and design of heat sinks are of considerable importancein the devel-
`opmentof electronic products. Sometypical electronic equipmentheatsinks
`are shownin Fig.1.1.
`This chapter prepares the reader for the study of the methods used for
`expeditious design and analysis of heat sinks for electronic equipment and
`electronic components. Section 1.2 deals with the heat-transfer phenomenain
`terms of the thermalresistancesattributedto individual mechanisms. Section
`1.3 introduces the concept ofthe chip package resistance between the compo-
`nent junction and the surroundings and provides the context for the design
`and use of heat sinks. Section 1.4 provides a summaryof the considerations
`and proceduresinvolvedin the design and analysis of a typical array offins.
`This introductory chapter concludes with a general discussion of extended
`surfaces that will serve to introduce the detailed considerations that follow
`in Chapters 2 through 10.
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`
`
`2
`
`INTRODUCTION (e)
`
`(a) Radial fin cooler, (b) clamp-on pin fin cooler/heat sink, (c) plate fin
`Figure 1.1
`heat sink, (d) and (e) finned array heat sinks (photographs courtesy of Wakefield
`Engineering Co., Wakefield, MA).
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`
`
`HEAT-TRANSFER FUNDAMENTALS
`
`3
`
`1.2 HEAT-TRANSFER FUNDAMENTALS
`
`1.2.1
`
`Introduction
`
`To determine the temperature differences encountered in the flow of heat
`withinelectronic systems, it is necessary to recognize the relevant heat-transfer
`mechanismsandtheir governingrelations. In a typical system, heat removal
`from theactive regionsof the microcircuit(s} or chip(s) may require the use of
`several mechanisms, some operating in series and others in parallel, to trans-
`port the generated heatto the coolant or ultimate heat sink. Practitioners of
`the thermalarts and sciences generally deal with four basic thermal transport
`modes: conduction, convection, phase change, and radiation.
`The process by whichheatdiffuses through a solid or a stationary fluid is
`termed heat conduction. Situations in which heattransfer froma wetted surface
`is assisted by the motion of the fluid give rise to heat convection, and when
`the fluid undergoes a liquid / solid or liquid/vapor state transformation, at,
`or very near, the wetted surface, attention is focussed on this phase-change heat
`transfer. The exchangeof heat between surfaces, or between a surface and a
`surroundingfluid, by long-wavelength electromagnetic radiation is termed
`heat radiation.
`
`1.2.2 Conduction Heat Transfer
`
`One-Dimensional Conduction.
`Steady thermaltransport throughsolids is governed by the Fourier equation,
`which in one-dimensional form is expressible as
`dT
`(1.1)
`[W]
`= -kAT
`where q is the heatflow, & is the thermal conductivity of the medium,A is the
`cross-sectional area for the heat flow, and dT/dz is the temperature gradient
`in the direction of heat flow. In this book, heat flow produced by a negative
`temperature gradientwill be considered positive. This convention requires
`the insertion of the minussign in Eq. (1.1) to ensure a positive heat flow, ¢
`The temperature difference resulting from the steady-state diffusion of heat
`is thusrelated to the thermal conductivity of the material, the cross-sectional
`area, and the path length, L (Fig. 1.2), according to
`
`IK]
`
`1.2)
`
`(14 — T2)ea =975
`The form of Eq. (1.2) suggests that, by analogy to Ohm’s Law governing
`electrical current flow througha resistance,it is possible to define a thermal
`resistance for conduction, Reg, as
`(i-th) Lb
`Regs AB)
`
`IK/W]
`
`(1.3)
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`
`no)
`
` x
`
`4
`
`INTRODUCTION
`
`Figure 1.2 Heat transfer by conduction throughaslab.
`
`One-Dimensional Conduction with Internal Heat Generation.
`Situations in which a solid experiencesinternal heat generation, such as that
`produced by theflow ofan electric current, give rise to more complex govern-
`ing equations and require greater care in obtaining the appropriate tempera-
`ture differences. The axial temperature variation in a slim, internally heated,
`conductor whose edges(ends)are held at a temperature, T, (Fig. 1.3) is found
`to equal
`
`ano (3)
`
`Piro
`
`z\?2
`
`whenthe volumetic heat generationrate, g,,in W/m/?,is uniform throughout.
`The peak temperature is developed at the centerof the solid and is given by
`
`[2
`Tinos = T, + Ig ° 8k
`
`[K]
`
`(1.4)
`
`Alternatively, because q, is the volumetric heat generation g, = ¢/LW6,
`the center-to-edge temperature difference can be expressed as
`
`iP
`L
`Tinas — To = TeeraS~ {Ska
`(1.5)
`
`where the cross-sectional area, A, is the product of the width, W, and the
`thickness, 6. Comparison of Eq. (1.5) with Eq. (1.3), for a path length of L/2,
`reveals that the thermalresistance of a conductor witha distributed heat input
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`

`HEAT-TRANSFER FUNDAMENTALS
`
`2D
`
`xy
`
`Tmax
`Peak temperature
`
`Xo
`
`T,
`~—«— Edge temperature
`
`‘k_ _(——_—————“
`
`x
`
`Figure 1.3 Temperature variation in an internally heated conductor.
`
`is only one-quarter that of a structure in which all of the heat is generated at
`the center.
`
`Spreading Resistance
`In chip packagesthat provideforlateral spreading of the heat generated in
`the chip, the increasing cross-sectional area for heat flow at successive “Tay-
`ers” below the chip reduces the internal thermal resistance. Unfortunately,
`however, there is an additional resistance associated withthis lateral flow of
`heat. This, of course, must be taken into account in the determinationof the
`overall chip package temperature difference.
`For the circular and square geometries common in microelectronic ap-
`plications, Negus et al.
`(1989) provide an engineering approximation for
`the spreading resistance for a small heat source on a thick substrate or heat
`spreader. This can be expressed as
`0.475 — 0.626 + 0.13
`Rsp =
`kVAe
`,
`where« is the squarerootof the heat source area divided by the substrate area,
`k is the thermal conductivity of the substrate, and A,is the area of the heat
`source. It is to be noted that the use of Eq.(1.6) requires that the substrate be
`3 to 5 timesthicker than the square root of the heat sourcearea.
`Forrelatively thin layers on thicker substrates, such as encounteredin the
`use of thin lead frames, or heat spreaders interposed between the chip and
`substrate, Eq. (1.6) cannot provide an acceptable prediction of Rep. Instead,
`use can be made of the numerical results plotted in Fig. 1.4 to obtain the
`requisite value of the spreading resistance.
`
`[K/W]
`
`(1.6)
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`
`
`6
`
`INTRODUCTION
`
`
`“TTT Uniform flux
`|
`[os|)em | *
`
`lal
`
`0.30
`
`10°
`
`10°
`
`107!
`
`10°
`
`101
`
`102
`
`6/a
`
`Figure 1.4 The thermalresistancefor a circular heat source on a twolayer substrate
`(Yovanovich and Antonetti, 1988),
`
`Interface/Contact Resistance.
`Heat transfer across the interface between twosolids is generally accompa-
`nied by a measurable temperature difference, which can be ascribed to a
`contact or interface thermal resistance. For perfectly adhering solids, geo-
`metrical differences in the crystal structure (lattice mismatch) can impede
`the flow of phononsand electrons across the interface, but this resistance
`is generally negligible in engineering design. However, when dealing with
`real interfaces, the asperities present on each ofthe surfaces, as shownin an
`artist’s conceptionin Fig. 1.5, limit actual contact betweenthe twosolids to a
`very small fraction of the apparentinterface area. The flow of heat across the
`gap between twosolids in nominalcontactis, thus, seen to involvesolid con-
`duction in the areas of actual contact and fluid conduction across the “open”
`spaces. Radiation across the gap can be important in a vacuum environment
`or whenthe surface temperatures are high.
`The heat transferred across aninterface can be found by adding theeffects
`of the solid-to-solid conduction and the conduction through the fluid and
`recognizingthat the solid-to-solid conduction, in the contact zones, involves
`heat flowing sequentially through the twosolids. With the total contact con-
`ductance, feo, taken as the sum of the solid-to-solid conductance, h,, and the
`gap conductance,hg,
`
`heo = he + hy
`
`[W/m?-K]
`
`(1.74)
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`HEAT-TRANSFER FUNDAMENTALS
`
`7
`
`&
`
`Intimate contact
`
`Gap filled with fluid with
`thermal conductivity k¢
`
`I
`
`-- y lo
`
`Figure 1.5 Physical contact between two nonidealsurfaces.
`
`the contact resistance based on the apparent contact area, A,, may be defined
`as
`1
`IK/WI
`Reo =
`K/W
`——
`In Eq.(1.72), h, is given by Yovanovich and Antonetti (1988) as
`0.95
`
`m
`
`he = 125k, (=) (=)
`
`ie
`
`(1.70)
`1.7h
`
`(1.82)
`
`wherek, is the harmonic mean thermal conductivity for the two solids with
`thermal conductivities, k; and kp,
`
`2hy ko
`ks = i x ky
`
`[W/m-K]
`
`o is the effective rms surface roughness developed from the surface rough-
`nesses of the two materials, 7, and o»,
`
`o= ot + 03
`
`[y-m
`
`and mis the effective absolute surface slope composed ofthe individual slopes
`of the two materials, m, and m2,
`
`2
`Se ae
`m= y/my + ms
`
`and where P is the contact pressure, and H is the microhardnessof the softer
`material, both in N/m7. In the absence ofdetailed information, the m/a ratio
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`8
`
`INTRODUCTION
`
`can be taken equal to 0.111 to 0.200 microns“! forrelatively smooth surfaces
`(Yovanovich, 1990),
`In Eq.(1.72), h, is given by
`
`
`kg
`_
`hg = Y+M
`
`(1.88)
`
`where k, is the thermal conductivity of the gap fluid, Y is the distance between
`the mean planes(Fig. 1.5) given by
`
`Ye 1.185 |- In (s1325)]
`
`A
`
`o
`
`0.547
`
`.
`
`and Mis a gas parameter used to accountfor rarified gas effects:
`
`M=a6A
`
`In this relationship,a is an accomodation parameter (approximately equal to
`2.4 for air and clean metals), A is the mean free path of the molecules (equal
`to approximately 0.06 um for air at atmospheric pressure and 15°C), and 8
`is a fluid property parameter (equal to approximately 1.7 for air and other
`diatomic gases).
`Returning to Eq.(1.7a) and inserting the expressions from Eqs. (1.82) and
`(1.85), the contact resistance becomes
`
`=1
`
`
`Reo =§|1.25k, (™) (Fy +s
`
`
`
`
`(1.9)
`Ag
`
`
`ee kaJAE Y+M
`
`Lumped-Capacity Heating and Cooling.
`Aninternally heated solid of relatively high thermal conductivity, whichis
`experiencing no external cooling, will undergo a constant rise in temperature
`according to
`
`aT
`qd
`(1.10)
`ee [K/s]
`whereqis the rate of internal heat generation, m is the massofthe solid, and
`cis the specific heat of the solid. Equation (1.10) assumesthatall of the mass
`canbe represented by a single temperature. This relation is frequently termed
`the lumped-capacity solution for transient heating.
`Expanding upon the analogy between thermalandelectrical resistances
`suggested previously, the product of mass and specific heat can be viewed as
`analogousto electrical capacitance and to thus constitute the thermal capaci-
`tance,
`Whenthis samesolid is externally cooled, the temperature rises asymp-
`totically toward the steady-state temperature, whichis itself determined by
`the external resistance to heat flow, R.,. Consequently, the time variation of
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`HEAT-TRANSFER FUNDAMENTALS
`
`9
`
`the temperature of the solid, with theinitial temperature equal to the fluid
`temperature, is expressible as
`
`T(t) =T(t =0) + qRe.[1 — 6 '/ Ree|
`
`[K]
`
`(1.11)
`
`where the productof the external resistance, Re, and the thermal capacitance,
`me, is seen to constitute the thermal time constant of the system.
`
`1.2.3 Convective Heat Transfer
`
`The HeatTransfer Coefficient.
`Convective thermal transportfrom a surface to a fluid in motion can be related
`to the heat-transfercoefficient, i, the surface-to-fluid temperature difference,
`and the “wetted” surface area, S, in the form
`
`q = hS(Ts — Ty)
`
`(WI
`
`(1.12)
`
`The differences between convection to a rapidly moving fluid, a slowly
`flowing fluid, or a stagnantfluid, as well as variations in the convective heat-
`transfer rate among various fluids, are reflected in the values of h. For a
`particular geometry and flow regime, h may be found from available em-
`pirical correlations and/or theoretical relations. Moreover, use of Eq. (1.12)
`makesit possible to define the convective thermal resistance, as
`1
`Rey = hs
`
`[K/W]
`
`(1.13)
`
`Dimensionless Parameters.
`Common dimensionless parameters that are used in the correlation of heat-
`transfer data are the Nusselt number, Nu, which relates the convective heat-
`transfer coefficient to conduction in the fluid !
`
`aah hb
`
`the Prandtl number, Pr, which is a fluid property parameter relating the diffu-
`sion of momentum to the conduction of heat
`Colt
`P=
`
`the Grashof number, Gr, which accounts for the bouyancy effect produced by
`the volumetric expansion of the fluid
`
`_ p?BgLFAT
`2
`
`Gr
`
`lAt this point, it becomes necessary to use the subscript f! whenever there is a chance for
`confusion betweensolid and fluid properties.
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`
`
`10
`
`INTRODUCTION
`
`and the Reynolds number, Re, which relates the momentum in the flow to the
`viscous dissipation
`
`Re=
`
`ppVL
`Lb
`
`Natural Convection.
`Innatural convection,fluid motion is induced by density differences resulting
`from temperature gradients in the fluid. The heat-transfer coefficient for this
`regime can be related to the buoyancy and the thermal properties of the fluid
`through the Rayleigh number, whichis the productof the Grashof and Prandtl
`numbers,
`
`Ra = 29913ar
`jakyi
`wherethe fluid properties, p, @, cp, 4, and k, are evaluated at the fluid bulk
`temperature, and AT is the temperature difference between the surface and
`the fluid.
`Empirical correlations for the natural convection heat-transfer coefficient
`generally take the form
`
`h=C (4) (Ra)
`
`[W/m?-K]
`
`(1.14)
`
`where nis found to be approximately 0.25 for 10* < Ra < 10”, representing
`laminar flow, 0.33 for 10’ < Ra < 10", the region associated with the tran-
`sition to turbulent flow, and 0.4 for Ra > 10!*, when strong turbulent flow
`prevails. The precise value of the correlating coefficient, C, depends on the
`fluid, the geometry of the surface, and the Rayleigh number range. Neverthe-
`less, for commonplate, cylinder, and sphere configurationsin air, it is found
`to vary in the relatively narrow range of 0.45 to 0.65 for laminar flow and
`0.11 to 0.15 for turbulentflow past the heated surface (Kraus and Bar-Cchen,
`1983).
`Natural convection in vertical channels, such as those formed by arrays
`of longitudinal fins, is of major significance in the analysis and design of
`heat sinks. Elenbaas (1942) wasthe first to document a detailed study of
`this configuration. His experimentalresults for isothermal plates were later
`confirmed numerically by Bodoia and Osterle (1964) and have subsequently
`been extended to many other configurations.
`These many studies have revealed that the value of the Nusselt number
`lies between two extremesassociated with the separation betweenthe plates
`or the channel width. For wide spacing, the plates appear to havelittle influ-
`ence on one another, and the Nusselt number in this case achievesits isolated
`plate limit. On the other hand,for closely spaced plates or for relatively long
`channels, the fluid attainsitsfully developed velocity and temperature profiles,
`and the Nusselt numberreachesitsfully developed limit. Intermediate values of
`the Nusselt numbercan be obtained from a correlating expression following
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`HEAT-TRANSFER FUNDAMENTALS
`
`11
`
`the form suggested by Churchill and Usagi (1972) for smoothly varying pro-
`cesses, as has been verified by detailed experimental and numerical studies
`(Bar-Cohen and Rohsenow, 1984).
`Thus, the correlation for the average value of h along isothermal vertical
`plates separated by a spacing, z, is given by
`(1.15)
`kg [576
`2.873]?
`
`~ 2|(El? (BD?
`
`
`
`whereEl is the Elenbaas number,
`E= p*BgepzAT
`pkpl
`
`and AT = T, — Ty. The use ofthis correlation will be explored in some
`detail in an example in Sec. 1.4, and further details concerningits use will be
`provided in Chap.10.
`
`Forced Convection.
`For forced flow in long, or very narrow, parallel-plate channels, the heat-
`transfer coefficient attains an asymptotic value (a fully developed limit),
`which for symmetrically heated channel surfaces is equal approximately to
`
`_ Akp
`="7
`
`h
`
`[W/m?-K]
`
`(1.16)
`
`whered, is the hydraulic diameter defined in termsof the flow area, A, and the
`wetted perimeter of the channel, P..,,
`
`de
`
`4A
`Py
`
`In the inlet zones of such parallel-plate channels and alongisolated plates,
`the heat-transfer coefficient varies with the distance from the leading edge.
`The low velocity, or laminarflow, average convective heat-transfer cocfficient
`along a surface of length L, for Re < 3 x 10° is given for example, in Kraus
`and Bar-Cohen(1983)by:
`
`h = 0.664 (4) Repr?33
`
`[W/m?-K]
`
`(1.17)
`
`wherekf; is the fluid thermal conductivity, L is the characteristic dimension
`of the surface, and Reis the Reynolds number.
`A similar relation applies to a flow in tubes, pipes, annuli, or channels
`with the hydraulic diameter, d., serving as the characteristic dimension in
`both the Nusselt and Reynolds numbers. For laminar flow, Re < 2100,
`1/3
`0,14
`
`
`a = 1.86 xey(es) Gl
`
`(“)
`
`(1.18)
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`12
`
`INTRODUCTION
`
`whichis attributed to Sieder and Tate (1936), and wherej,, is the viscosity
`of the convective medium at the channel wall temperature. Observethat this
`relationship showstheheat-transfercoefficientto attain its maximum value at
`the inlet to the channel and decreases towards the asymptotic fully developed
`value given by Eq.(1.16) as d./L decreases.
`In higher-velocity turbulent flow along plates, the dependenceof the con-
`vective heat-transfer coefficient on the Reynolds numberincreases and, in the
`range Re > 3 x 10°,is typically given by (Kraus and Bar-Cohen 1983):
`
`h = 0.036 (“F) (Re)°8(Pr)°43
`
`[W/m?-K]
`
`(1.19)
`
`In pipes, tubes, annuli, and channels, fully turbulent flow occur