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`

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`
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`\
`
`FUNDAMENTALS OF
`
`I
`
`THIRD EDITION
`
`School of Mechanical
`Purdue Uni\'er51*}"
`
`
`
`Intel Corp. e_’;_a1_.___T_iE>ghib_11
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`
`
`
`.
`.
`Dedicated to those wonderful women in our lives,
`
`_
`Amy, Andrea, Dabble, Donna, Jody,
`
`Karen, Shalmna, and Terri
`who, through the years, have blessed us with
`their love, patience, and understanding.
`
`E
`5-
`
`-
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`C
`
`' N"1981l9851
`
`JhnW
`
`AII:ightsxescrvcd;PublishedsimultancouslyinCanada
`Reproduction or translation of any pain of
`thiswotkbeyenddaarpermittedbyseetions
`107 and we of the 1975 United States Copyright
`Actwithauttthepernlissionoidlecopjlfight
`owner is unlawful. Requests for p-mni.u5on
`or further infommian should be addressed to
`:hereun:ssaomnepaxmen;,Johhw;ieygsm_
`
`L&wyaj7Canpan¢Iafl$%iPn&|niinn-flats:
`Inempen. Frank P.
`Finnclamultalfis of heat and mass. transfer/Frank P. Incropera, David
`P. DeW‘m.~——?mi=ed.
`P-
`cn_=!-
`
`1.fleasi.—Trannni:nIvinn. Lhiasuansier.
`1934».
`II. 1311-;
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`

`
`PREFACE
`
`With the passage of approximately nine years since publication of the first
`edition, this text has been transformed from the status of a newcomer to a
`mature representative of heat transfer pedagogy. Despite this maturation,
`however, we like to think that, while remaining true to certain basic tenets, our
`treatment of the subject is constantly evolving.
`Preparation of the first edition was strongly motivated by the belief that,
`above all, a first course in heat transfer should do two things. First. it should
`instill within the student a genuine appreciation for the physical origins of the
`subject. It should then establish the relationship of these origins to the
`behavior of thermal systems. In so doing, it should develop methodologies
`which facilitate application of the subject
`to a broad range of practical
`problems, and it should cultivate the facility to perform the kind of engineer-
`ing analysis which, if not exact, still provides useful information concerning
`'the design and/or performance of a particular systan or process. Require»
`mentsofsuchananalysisindudetheahilitytodiscemrelevanttranspon
`processesandsimphfymgassumpfionnidmfifyimpmtantdepmdmtand
`indqaendent variables, develop appropriate expressions from first prineiplim
`andintroducerequisitematuialfiomtheheatuansferknowledgebaselnthc
`fifrst
`achievement of-this objective was fostered by coaching many of
`the examples and end-of-chapter prohlms in terms of actual aigineerirrg
`
`-
`systems.
`1heseccndeclifionwasalsodrivenbytbeforegoingobjectives,asweilJm
`by input derived from it sent tooverllwcollugnes who used, or
`with-,thefirstedit:ion.Ama_;or' consequenceofthis
`inputwaspuihlicationoftwoversionsofthe-shook, Faardmnenraboffleatmrd
`Mass Transfer a:nd=In1rnductian.ro Heat Transfer. As in the first edition, the
`Fundamemals version included mass transfer,
`an: integrated treat-
`mento£hatt,massandmomenni1nu-ansferbyconvectionmdsepante
`tre:mnentsofheatandmasstransferbydi&‘usicn.TheIun'adueticn versinnof
`thebookwas-inlendedfcrnsuswhounbracedtheueatmenttofheatttmsfer
`hndidnotwish;tocnvumassnamfu=eEeets.lnbothvem’ens,_signifieailé
`inrprovunmtswueamdeinthetr%entscf"inmraica!nIethnds_andliene
`transferwithphaseclnnge.
`_
`_
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`pnnens.canuauoutnatuuunuzmmgsar:aemaeayiamnnnetpun»-
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`
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`
`Intel Corp. et al. Exhibit 1014
`
`

`
`
`
`vi
`
`Preface
`
`compact heat exchangers, as well as by the addition of new material on
`submerged jets (Chapter 7) and free convection in open, parallel plate chan-
`nels (Chapter 9). Submerged jets are widely used for industrial cooling and
`drying operations, while free convection in parallel plate channels is pertinent
`to passive cooling and heating systems. Expanded discussions of physical
`principles are concentrated in the chapters on single-phase convection
`(Chapters 7 to 9) and relate, for example, to forced convection in tube banks
`and to free convection on plates and in cavities. Other improvements relate to
`the methodology of performing a first law analysis, a more generalized lumped
`capacitance analysis, transient conduction in semi-infinite media, and finite-
`difference solutions.
`
`the old Chapter 14, which dealt with multimode heat
`In this edition,
`transfer problems, has been deleted and many of the problems have been
`transferred to earlier chapters. This change was motivated by recognition of
`the importance of multimode effects and the desirability of impacting student
`consciousness with this importance at
`the earliest possible time. Hence,
`problems involving more than just a superficial consideration of multimode
`effects begin in Chapter 7 and increase in number through Chapter 13.
`in this
`The last. but certainly not
`the least
`important,
`improvement
`edition is the inclusion of nearly 300 new problems. In the spirit of our past
`efforts, we have attempted to address contemporary issues in many of the
`problems. Hence, as well as relating to engineering applications such as energy
`conversion and conservation, space heating and cooling, and thermal protec-
`tion, the problems deal with recent interests in electronic cooling, manufactur-
`ing, and material processing. Many of the problems are drawn from our
`accumulated research and consulting experiences;
`the solutions, which fre-
`quently are not obvious, require thoughtful implementation of the tools of heat
`transfer. It is our hope that in addition to reinforcing the student’s understand-
`ing of principles and applications, the problems serve a motivational role by
`relating the subject to real engineering needs.
`Over the past nine years, we have been fortunate to have received
`constructive suggestions from many colleagues throughout the United States
`and Canada. It is with pleasure that we express our gratitude for this input.
`
`W6! Lafayette. Indiana
`
`FRANK P. INCROPERA
`DAVID P_
`
`-.r.-..I
`
`5.
`
`3.3
`
`ii'3'
`_fit‘1'.
`
`
`
`Intel Corp. et al.
`
`Intel Corp. et al. Exhibit 1014
`
`

`
`CONTENTS
`
`Symbols
`
`Chapter 1
`
`INTRODUCTION
`1.1 What and How?
`
`1.2 Physical Origins and Rate Equations
`1.2.1 Conduction
`1.2.2 Convection
`1.2.3 Radiation
`
`1.2.4 Relationship to Thermodynamics
`1.3 The Conservation of Energy Requirement
`1.3.1 Conservation of Energy for a Control Volume
`1.3.2 The Surface Energy Balance
`1.3.3 Application of the Conservation Laws:
`Methodology
`1.4 Analysis of Heat Transfer Problems: Methodology
`1.5 Relevance of Heat Transfer
`1.6 Units and Dimensions
`
`1.7 Summary
`Problems
`
`Chapter 2
`
`INTRODUCTION TO CONDUCTION
`2.1 The Conduction Rate Equation
`2.2 The Thermal Properties of Matter
`2.2.1 Thermal Conductivity
`2.2.2 Other Relevant Properties
`2.3 The Heat Difiusion Equation
`2.4 Boundary and Initial Conditions
`2.5 Summary
`References
`Problems
`
`Clnpter 3 ONE-DIMENSIONAL, STEADY-STATE CONDUCTION
`3.1 The Plane Wall
`
`3.1.1 Temperature Distribution
`3.1.2 Thermal Resistance
`
`3.1.3 The Composite Wall
`3.1.4 Contact Resistance
`
`3.2 An Alternative Conduction Analysis
`3.3 Radial Systems
`3.3.1 The Cylinder
`3.3.2 The Sphere
`
`E.
`
`*0"--llhl-I-lhéh-I\p;«.L;JL:-J\DO'\|-.4-l|aJl'-3|-'
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`
`Intel Corp. et al. Exhibit 1014
`
`

`
`Viii
`
`Contents
`
`Chapter 4
`
`3.4 Summary of One-Dimensional Conduction Results
`3.5 Conduction with Thermal Energy Generation
`3.5.1 The Plane Wall
`3.5.2 Radial Systems
`3.5.3 Application of Resistance Concepts
`3.6 Heat Transfer from Extended Surfaces
`3.6.1 A General Conduction Analysis
`3.6.2 Fins of Uniform Cross-Sectional Area
`3.6.3 Fin Performance
`3.6.4 Overall Surface Efficiency
`3.6.5 Fin Contact Resistance
`3.7 Summary
`References
`Problems
`
`TWO-DIMENSIONAL, STEADY-STATE CONDUCTION
`4.1 Alternative Approaches
`4.2 The Method of Separation of Variables
`4.3 The Graphical Method
`4.3.1 Methodology of Constructing a Flux Plot
`4.3.2 Determination of the Heat Transfer Rate
`4.3.3 The Conduction Shape Factor
`4.4 Finite-Difierenoe Equations
`4.4.1 The Nodal Network
`4.4.2 Finite-Dilferenoe Form of the Heat Equation
`4-4-3 The Energy Balance Method
`Finite—Difl'crence Solutions
`4.5.1 The Matrix Inversion Method
`4.5.2 Gauss-Seidel Iteration
`4.5.3 Some Precautions
`4.6 Snnlmaljr
`References
`Problems
`
`4.5
`
`Charter 5
`
`TRANSIENT‘ CONDUCTION
`5-1
`“=€1_-mnped Capacitance Method
`5-2 Vahdny of the I-limped Capacitance Method
`5.3 General Lttmped
`‘mace
`5 4 Spatial Effects
`5-5 The Plane Wall with Convection
`5.5.1 Exact Solution
`551 APP1’0ximate Solution
`5.5.3 Total Energy Transfer
`5.5.4 Graphical Bepzuenmeom
`
`107
`108
`103
`114
`119
`119
`122
`123
`130
`134
`138
`141
`142
`142
`
`171
`172
`173
`177
`173
`179
`180
`184
`185
`135
`137
`194
`194
`200
`203
`203
`294
`204
`
`225
`225
`229
`234
`237
`239
`239
`24:;
`240
`242
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`

`
`
`
`Conlents
`
`ix
`
`4
`
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`v
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`
`101
`103
`103
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`122
`123
`130
`134
`138
`
`141
`142
`142
`
`171
`"172
`173
`177
`
`179
`180
`184
`185
`185
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`203
`2233
`204
`
`225
`235
`229
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`237
`
`m.
`
`239
`240
`240
`
`Zfl
`245‘
`
`245'
`246
`247
`249
`
`5.7 The Semi-infinite Solid
`5.8 Multidimensional Efi’ects
`5.9 Finite-Difference Methods
`5.9.1 Discretization of the Heat Equation:
`The Explicit Method
`5.9.2 Discretization of the Heat Equation:
`The Implicit Method
`5.10 Summary
`References
`Problems
`
`Chapter 6
`
`INTRODUCTION TO CONVECTION
`6.1 The Convection Transfer Problem
`6.2 The Convection Boundary Layers
`6.2.1 The Velocity Boundary Layer
`6.2.2 The Thermal Boundary Layer
`6.2.3 The Concentration Boundary Layer
`6.2.4 Significance of the Boundary Layers
`
`6.4 The Convection Transfer Equations
`6.4.1 The Velocity Boundary Layer
`6.4.2 The Thermal Boundary LayerL
`6.4.3 The Coneentratronllotmdary. ayer
`_
`6.5 Approximations and Conditions .
`6.6 Boundary Layer Slmxlanty: The Normalized Convection
`Transfer Equations
`_
`_
`_
`6.6.1
`Layer Smnlanty Parameters
`6.6.2 Functional Form of the Solutions
`6.7 Physical Signifieancealof the Dimensionless Parameters
`6.8 Boundary Layer An ogies
`6.8.1 The Heat and Mass Transfer Analogy
`6.8.2 Evaporative Cooling
`6.8.3 The Reynolds Analogy
`6.9 The Elfects of Turbulence
`5.10 The Convection Coeflicients
`6.11 Summary
`References
`Problems
`
`(liqueur 7 EXTERNAL FLOW
`7.1 The Empirical Method
`7.2 The Flat Plate in Parallel Flow
`
`7.2.1 Laminar Flow: A Similarity Solution
`7.2.2 Turbulent Flow
`
`7.2.3 Mixed Boundary Layer Conditions
`7.2.4 Special Cases
`7.3 Methodology for a- Convection Caleulafion
`
`259
`263
`270
`
`271
`
`279
`237
`237
`233
`
`312
`312
`318
`318
`319
`320
`323
`
`326
`326
`331
`335
`341
`343
`344
`346
`
`355
`359
`363
`364
`357
`363
`368
`359
`
`335
`387
`339
`
`389-
`396
`
`397
`399
`401
`
`Intel Corp. et a1.’ Exhibit 1014
`
`Intel Corp. et al. Exhibit 1014
`
`

`
`
`
`1'.
`
`Contents
`
`7.4
`
`7.5
`7.6
`7.7
`
`7.8
`7.9
`
`The Cylinder in Cross Flow
`7.4.1 Flow Considerations
`
`7.4.2 Convection Heat and Mass Transfer
`
`The Sphere
`Flow Across Banks of Tubes
`
`impinging Jets
`7.7.1 Hydrodynamic and Geometric Considerations
`7.7.2 Convection Heat and Mass Transfer
`Packed Beds
`
`Summary
`References
`Problems
`
`Chapter 8
`
`INTERNAL FLOW
`8.1
`
`Hydrodynamic Considerations
`8.1.1 Flow Conditions
`8.1.2 The Mean Velocity
`
`8.2
`
`8.3
`
`8.5
`
`8.6
`8.‘?
`8.8
`8.9
`
`Thermal Considerations
`8.2.1 The Mean Temperature
`8.2.2 Newton's Law of Cooling
`8.2.3 Fully Developed Conditions
`The Energy Balance
`8.3.1 General Considerations
`
`%£:::Section Correlations: Turbulent Flow in Circular
`Convection Correlations: Noncircular Tubes
`The Concentric Tube Ammlns
`Heat Transfer Enhancement
`
`onaVertina.l Surface
`
`411
`41?
`420
`
`431
`431
`433
`438
`
`441
`442
`
`46?
`468
`468
`469
`470
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`472
`474
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`475
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`
`489
`
`494
`
`495
`
`502
`
`505
`
`507
`
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`
`529
`530
`533
`535
`536
`539
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`
`
`Intel Corp. et al.
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`
`Intel Corp. et al. Exhibit 1014
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`

`
` 9.6 Empirical Correlations: External Free Convection Flows
`
`Contents
`
`9.6.1 The Vertical Plate
`9.6.2 Inclined and Horizontal Plates
`
`9.6.3 The Long Horizontal Cylinder
`9.6.4 Spheres
`9.7 Free Convection within Parallel Plate Channels
`9.7.1 Vertical Channels
`
`9.7.2 Inclined Channels
`
`433
`440
`441
`442
`
`46-’!
`463
`468
`469
`470
`
`472
`474
`475
`476
`476
`430
`430
`432
`435
`
`439
`439
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`9.3 Empirical Correlations: Enclosures
`9.8.1 Rectangular Cavities
`9.8.2 Concentric Cylinders
`9.8.3 Concentric Spheres
`9.9 Combined Free and Forced Convection
`9.10 Convection Mass Transfer
`9.11 Summary
`References
`Problems
`
`Chapter to BOILING AND CONDENSATION
`10.1 Dimensionless Parameters in Boiling and Condensation
`10.2 Boiling Modes
`10.3 Pool Boiling
`10.3.1 The Boiling Curve
`10.3.2 Modes of Pool Boiling
`10.4 Pool Boiling Correlations
`10.4.1 Nucleate Pool
`10.4.2 Critical I-{eat Flux for Nueleate Pool Boiling
`10.4.3 Minimum Heat Flux
`10.4.4 Film Pool Boiling
`10.4.5 Parametric Efiects on Pool Boiling
`10.5 Forced-Convection Boiling
`10.5.1 External Forced-Convection Boiling
`10.5.2 Two-Phase Flow
`
`10.6 Condensation: Physical Mechanisms
`10.7 Laminar Film Condensation on a Vertiul Plate
`10.8 Turbulent Film Condensation
`
`10.9 Film Condensation on Radial Systems
`10.10 Film Condensation in Horizontal Tubes
`
`10.11 Dmpwise Condensation
`1012 Summary
`References
`Problems
`
`(lunar It
`
`I-[PAT EXCHANGERS
`11.1 Heat Exchanger Types
`11.2 The Overall Heat Transfer Coefiicient
`
`113 HeatB:tchangerAnal’ysis:UseoftheLogMean
`Temperature Difference
`11.3.1 The Parana!-Flow Heat Exchanger
`
`xi
`
`541
`542
`546
`
`550
`553
`555
`555
`
`558
`
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`563
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`610
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`
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`622
`
`623
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`
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`
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`646
`
`Intel Corp. et al. Exhibit 1014
`
`Intel Corp. et al. Exhibit 1014
`
`

`
`
`
`11.3.2 The Counterflow Heat Exchanger
`11.3.3 Special Operating Conditions
`11.3.4 Multipass and Cross-Flow Heat Exchangers
`11.4 Heat Exchanger Anaiysis: The Efi"ectiveness— NTU
`Method
`11.4.1 Definitions
`11.4.2 Efi'ectiveness—NTU Relations
`11.5 Methodology of :1 Heat Exchanger Calculation
`11.6 Compact Heat Exchangers
`11.7 Summary
`References
`Problems
`
`RADIATION PROCESSES AND PROPERTIES
`12.1 Fundamental Concepts
`12.2 Radiation Intensity
`12.2.1 Definitions
`12.2.2 Relation to Emission
`12.2.3 Relation to Irradiation
`12.2.4 Relation to Radiosity
`12.3 Blackbody Radiation
`12.3.1 The Planck Distribution
`12.3.2 Wien’s Displacement Law
`12.3.3 The Stefan-Boltzmann Law
`12.3.4 Band Emission
`12.4 Surface Enttission
`12.5 Surface Absorption, Reflection, and Transmission
`115.1 Absoxptivity
`12.5.2 Reflectivity
`12.5.3 Transmissivity
`12.5.4 Special Considerations
`17.6 Kirchhofi"s Law
`117 The Gray Surface
`12.8 Environmental Radiation
`12.9 Summary
`Referenea
`Problems
`
`649
`650
`650
`
`658
`658
`660
`666
`672
`678
`679
`680
`
`695
`696
`699
`699
`702
`706
`708
`709
`710
`712
`712
`713
`719
`729
`731
`732
`734
`734
`749
`742
`749
`756
`753
`759
`
`RADIATION EXCHANGE BETWEEN SURFACES
`13.1 The View Factor
`13.1.1 The View Factor In
`13 1 2 View Factor Reiations
`13.2
`Iilaekbody Radiation Exchange
`13.3 Radiation Exchange Between Dnrug, Guy Sm-faces
`:;:E;;=.*°*m=.....
`3-
`-
`'
`'m1 Exehmge at a Surface
`13.3.2 Radiation Exchange Between surfaces
`13.3.3 The Two-Surface Enclosure
`13 3.4 Radiation Shields
`13 3.5 The Rcradiating Surface
`
`791
`79;;
`792
`794
`303
`aw
`806
`303
`314:
`315
`319
`Intel Corp. et al.
`
`
`
`
`
`-ps>..I._'—'-'5-«LJ-
`
`Eytitibit 1014
`
`
`
`
`Intel Corp. et al. Exhibit 1014
`
`

`
`
`
`
`
`649
`650
`650
`
`658
`658
`550
`666
`
`572
`573
`679
`530
`
`695
`595
`699
`699
`702
`705
`708
`709
`-no
`712
`712
`713
`719
`729
`731
`732
`734
`734
`740
`33
`
`742
`
`753
`759
`
`791
`
`792
`792
`794
`503
`
`806
`sod
`808
`314
`316
`819
`
`
`
`'
`
`_.
`_’
`'.
`'
`
`.
`
`-I
`
`V
`1
`3
`ll
`L_'
`""
`-3
`1
`31
`
`"
`*1
`
`:-A
`
`.1"
`
`Con tents
`
`13.4 Multimode Heat Transfer
`13.5 Additional Etrects
`13.5.1 Volumetric Absorption
`13.5.2 Gaseous Emission and Absorption
`13.6 Summary
`References
`Problems
`
`Chapter 14 DIFFUSION MASS TRANSFER
`14.1
`Physical Origins and Rate Equations
`14.1.1 Physical Origins
`14.1.2 Mixture Composition
`14.1.3 Fick’s Law or Diffusion
`14.1.4 Restrictive Conditions
`14.1.5 Mass Dimision Coeflicient
`14.2 Conservation of Species
`14.2.1 Conservation of Species for a Control Volume
`14.2.2 The Mass Diffusion Equation
`14.3 Boundary and Initial Conditions
`14.4 Mass Dilftision Without Homogeneous Chemical
`Reactions
`14.4.1 Stationary Media with Specified Surface
`Concentrations
`14.4.2 Stationary Media with Catalytic Surface Reactions
`14.4.3 Eqnimolar Counterdifiusion
`14.4.4 Evaporation in a Column
`14.5 Mass Difiusion with Homogeneous Chemical Reactions
`14.6 Transient Diffusion
`References
`Problems
`
`Appendix A 'I'I-IERMOPHYSICAL PROPERTIES OF MATTER
`Appendix 3 MATHEMATICAL RELATIONS AND FUNCTIONS
`
`Appendix (2 AN INTEGRAL LAMINAR BOUNDARY LAYER
`SOLUTION FOR PARALLEL FLOW OVER A FLAT
`PLATE
`
`Index
`
`324
`327
`323
`829
`833
`833
`834
`
`371
`372
`372
`373
`375
`875
`330
`880
`881
`881
`884
`
`888
`
`889
`893
`896
`900
`902
`906
`910
`91 1
`
`A1
`31
`
`C1
`
`11
`
`
`
`Intel Corp. et al. Exhibit 1014
`
`Intel Corp. et al. Exhibit 1014
`
`

`
`
`
`
`
`CHAPTER 5
`
`.K}
`isp
`Lltsina
`
`and COIJIZIEE
`assed. Under
`uniform heat
`
`water flow pr vi
`f h = 5000 W
`
`'
`
`anirc
`.'ady-state
`msideratio . W may
`
`‘
`
`the prec
`dated. use
`.[Ll1'€S.
`
`page
`
`'e-dL‘- !
`
`
`
`
`
`’%‘~§‘£*:’€%A§fi.‘€(1.0?Lumam;s._.;t';_H,az~41sw.e2_x;
`
`TRANSIENT
`
`CONDUCTION
`
`
`
`Intel Corp. et al. Exhibit 1014
`
`

`
`226
`
`Chapter 5 Transient Conduction
`
`'
`
`In our treatment of conduction we have gradually considered more compli
`cated conditions. We began with the simple case of one-dimensional, steady-
`state conduction with no internal generation, and we subsequently considered
`complications due to multidimensional and generation effects. However, we
`have not yet considered situations for which conditions change with time.
`We now recognize that many heat transfer problems are time dependent
`Such unsteady, or transient, problems typically arise when the boundary
`conditions of a system are changed. For example, if the surface temperature of
`a system is altered, the temperature at each point in the system will also begin
`to change. The changes will continue to occur until a sread_v—srare temperaturr
`distribution is reached. Consider a hot metal billet that is removed from
`furnace and exposed to a cool airstream. Energy is transferred by conveciiflfl
`and radiation from its surface to the surroundings. Energy transfer by oondlw '
`non also occurs from the interior of the metal
`to the surface, and tilt
`f¢'11P¢1’3i|1r€ at each point in the billet decreases until a steady-state condition
`IS reached. Such time-dependent effects occur in many industrial heating“
`Cooling processes.
`To determine the time dependence of the temperature distribution Willlifi
`3 ‘°‘*d “Wins 3 transient process. we could begin by solving the approvm
`f°m]_°f ‘ht heal °q113'i0n, for example, Equation 2.13. Some cases for
`'
`solutions have been obtained are discussed in Sections 5.4 to 5.3- H0"““'
`“ch 5011-ltions are often diflicult to obtain, and where possible a '
`appmach is P1"~‘fel‘1'6d. One such approach may be used under conditi0I15 m‘
`which. temperaturc gradients within the solid are small. It is termed the WM
`Wpflfflance method.
`
`Intel Corp. et al. Exhibi It 14
`
`Intel Corp. et al. Exhibit 1014
`
`

`
`
`
`5.1 The Lumped Capacitance Method
`
`22'?
`
`MPAlG.?~5
`
`decrease for time t > 0, until it eventually reaches Tm. This reduction is due to
`convection heat
`transfer at
`the soIid—liquid interface. The essence of the
`lumped capacitance method is the assumption that _the__t_e_rI_1pera_ture of the
`solid is sparfal'l}_*_ uniform at any instant during the transient process. This
`assfitioifruiplies that temperature gradients within the solid are negligible.
`From Fourier’s law. hea_t conductigtgjrl fl.'l§_§b_§€;1_1C€
`of a temperature
`
`grgdient implies the{§_:3i_s_ten7c§ __c>i__iritii_1"ite thermal conductivity. Such a condi-
`tion"ts—clEar1y impossible. However. although the condition is never satisfied
`exactly. it is closely approximated if the resistance to conduction within the
`solid is small compared with the resistance to heat transfer between the solid
`and its surroundings. For now we assume that this is, in fact, the case.
`In neglecting temperature gradients within the solid, we can no longer
`consider the problem from within the framework of the heat equation. Instead,
`the transient temperature response is determined by formulating an overall
`Energy balance on the solid. This balance must relate the rate of heat loss at
`the surface to the rate of change of the internal energy. Applying Equation
`1.11:: to the control volume of Figure 5.1. this requirement takes the form
`.»_E-om = E“
`/L), [
`QUUJ
`
`or
`
`'
`
`'
`
`
`
`more oompli
`sional. steady-
`
`itly considered
`However. it
`with time.
`
`me dependent
`the boundary
`temperatureol
`will also begin
`
`re temperature
`moved from
`
`by convection
`.fer by conduc-
`iace, and lb?
`state condition
`.al heating 4”
`
`ribution Viilllifi
`he appI0P1'l3“
`Lases for which
`5.8. Howe?!-
`
`dT
`"!A,(T- Tm) = pVc—
`at:
`
`Introducing the temperature difference
`
`.
`
`r — T,
`
`and recognizing that (d9/dr) = (dT/dr), it follows that
`
`(5-2)
`
`(5.3)
`
`We J3
`he‘!
`dr _ —a
`
`Separating Variables and integrating from the initial condition. for which I = 0
`and 7(0) = 7",. We then obtain
`
`PVC
`ad0
`,
`hi!‘ 61?: ‘lid!
`
`"..:l.E§'5'%ARY_U0;.t.Ltt-‘ta.mA.c
`ible a siJIJPl"
`
`
`Intel Corp. et al. Exhibit 1014
`
`

`
`Intel Corp. et al. Exhibit 1014
`
`

`
`Intel Corp. et al. Exhibit 1014
`
`

`
`3. Chaplerfi
`
`'I'ranm'cnIConducti0n
`
`T-= "
`
`Figure 5.4 Effect of Bic-t nmnber an
`steady—state temperature disuibutian in!
`
`plane wall with surface convecfim.
`
`““°“.“.°‘““‘° Value» 12,2. for which Tm < 1; 1 < 1; ,. Hence under sleadY'5““
`wndmons the smfice 511313)’ balance, Equation 1:12. reduces to
`
`char. It gumuty (Md/R)
`
`Intel Corp. et al. Exhi_b_ '14
`
`Intel Corp. et al. Exhibit 1014
`
`

`
`
`
`
`5.2 Validity of the Lumped Capacitance Method
`
`231
`
`
`
`B'<<l
`'
`T:.-Tu)
`
`3'21
`‘
`T= Ttr. it
`
`Bi >>l
`T=Tl.r.£)
`
`Figure 5.5 Transient temperature distribution for diflerent Biot numbers in a plane
`wall symmetrically cooled by convection
`
`
`
`number on
`stribution in 1
`ivection
`
`:r steady-stilt
`
`
`
`the
`number, and three conditions are shown in Figure 5.5. For 3:‘ 4: 1
`temperature gradient in the solid is small and T(x, 1) == TU). Virtually all
`the temperature difference is between the solid and the fluid. and the solid
`temperature remains nearly uniform as it decreases to Tm. For moderate to
`large values of the Biot number, however, the temperature gradients within the
`solid are significant. Hence T = T(x, 1). Note that for Bi‘ :3» 1, the tempera-
`ture diflerence across the solid is now much larger than that between the
`surface and the fluid.
`We conclude this section by emphasizing the importance of the lumped
`capacitance method. Its inherent simplicity renders it the preferred method for
`
`
`the error associated with using the tumped capacitance method is small. For
`°°11Vettience, it is customary to define the characteristic length of Equation 5.10
`as the ratio of the solid’s volume to surface area. L; 5 V/Ar Such 3 d°fi‘1i‘i°“
`facilitates calculation of LC for solids of complicated shape and reduces *0 I116
`hfilflthickness L for a plane wall of thickness 2L (Figure 5.5), to r,,/2 for a
`
`33316 corresponding to the maximum spatial temperature difference. Accord-
`“ISIY. for a symmetrically heated (or cooled) plane wall of thickness 2L, LL.
`would remain equal to the half-thickness L. However, f0? 3 109% °Y1i1~‘Id€T 0'
`Slihere, L1. would equal the actual radius re, rather than ro/2 or ra/3.
`
`
`
`._VyH'n).‘...,.
`
`
`
`q.-.«.t__.._,—...._
`
`Intel Corp. et al.
`
`I
`
`I
`
`Intel Corp. et al. Exhibit 1014
`
`

`
`
`
`Finally. we note that, with LE 2 V/A5, the exponent of Equation 5.6 may
`
`Accordin
`
`excellem
`
`Intel Corp. et al.
`
`B2
`
`Chapter 5 Transient Conduction
`
`M5:
`
`in
`
`M. k t
`
`M. at
`
`A mtmprions:
`
`1. Temperau
`
`2. Radiation
`
`3. Losses by
`4. Constant
`
`Analysis:
`
`1. Because I
`the solutit
`
`capacitan
`approach
`determin:
`fact that
`
`Intel Corp. et al. Exhibit 1014
`
`

`
`
`
`5.2 Validity of the Lumped Capacitance Method B3
`
`
`
`
`
`
`1;: 200's
`1; ~_- JJEXJW/m K
`D
`
`
`
`Thermocouple k = 20 Wim - K
`junction
`c = 400 Jjkg- K
`7,. = 25 “C
`,. = 3500 kgfrn‘
`
`Assumptions:
`
`Equation 5.6 my
`
`(511)
`
`
`
`
`
`
`1- Because the junction diameter is unknown, it is not possible to begin
`The solution by determining whether the tuiterion for using the lumped
`
`
`C3133‘-‘itance method, Equation 5.10, is satisfied. However, a reasonable
`3PProach is to use the method to find the diameter and to then
`
`
`determine whether the criterion is satisfied. From Equation 5.7 and the
`fact that .4, = rd)’ and V = «D3/6 for a sphere, it follows am
`*"’.“”’?"i~’“fl“':5'.'?-'
`
`
`'
`mm
`
`
`1' ==
`1
`x pmnac
`*m-K.
`1 f
`’"’D2
`5
`
`
`fm,_-wig-=+ -
`€‘-anangmg and subsutuung numerical values.
`am,
`5x_4ooW/m2-texts’ =_m6Xm_,m 4
`
`in‘:T
`pc
`8500kg/m3X4(I)J/kg-K.
`
`
`
`
`
`
`1. Temperature of junction is uniform at any instant.
`2. Radiation exchange with the surroundings is negligible.
`3- Losses by conduction through the leads are negligible.
`
`4. Constant properties.
`
`Analy_n'5_-
`
`
`
`Intel Corp. et al. Exhibit 1014
`
`

`
`Chapter 5 Transient Conduction
`
`234
`
`
`
`2. From Equation 5.5 the time required for the junction to reach T=
`199°C is
`
`_ P(‘-‘T93/5)C
`h(-:rD2)
`
`In
`
`W
`T, - T
`I-1;
`
`I
`09
`T — T
`pDc
`——1
`6hnT~Tm
`
`= ssuukg/m3 x 7.06 x104 m x 400]/kg - K.
`25 -200
`I
`6x4o0w/m1~1(
`199-200
`
`:=5.2s=sr,
`
`4
`
`Comments: Heat losses due to radiation exchange between the junction
`and the surrouridtnga and conduction through the leads would necessitate
`
`“S1113 3 51113116? Junction diameter to achieve the desired time response.
`
`
`
`Intel Corp. et al.
`
`.014
`
`Intel Corp. et al. Exhibit 1014
`
`

`
`
`
`'
`
`i
`
`1
`
`-
`
`¥
`
`'
`= '
`
`
`
`..E-__
`
`.-
`-
`
`_5:’
`
`.,
`ll. 7'
`~
`_
`-
`
`
`
`'
`
`
`
`5.3 General Lumped Capacitance Analysis B5
`
`heat flux, and intemal energy generation. It is presumed that, initially (t = 0),
`the temperature of the solid (T,—) dilfers from that of the fluid, Tan, and the
`surroundings, 1;“, and that both surface and volumetric heating (q;’ and 4)
`are initiated. The imposed heat flux qj’ and the convection—radiation heat
`transfer occur at mutually exclusive portions of the surface, Am” and A,“ ,,,
`rmpectively, and convection—~radiation transfer is presumed to be from the
`surface. Applying conservation of energy at any instant I, it follows from
`Equation 1.11a that
`
` q.:'As.h + £5 _ (qeimv + qi';d)A.t(c,r) = PVC d:
`
`E
`
`or, from Equations 1.3a and 1.7,
`dT
`4
`-6
`H’
`'
`q,A,_,, + as — [h(T— 1;,)+ ea(T — T,,,,)]A,,,,,, = pVc:£-
`
`(S .14)
`
`(5.15)
`
`lby oonveclill
`E5
`; mad nuyht
`._g_»mpa-agmmi
`
`
`
`first-onder, nonhomoge-
`Unfortunately, Equation 5.15 is a nonlinear,
`neous, ordinary differential equation which cannot be integrated to obtain an
`exact solution.‘ However, exact solutions may be obtained for simplified
`versions of the equation. For example,
`if there is no imposed heat flux or
`generation and convection is either nonexistent (a Vacuum) or negligible
`relative to radiation, Equation 5.15 reduces to
`PVC; = -eA.v..a(T‘ - 1;.)
`
`(5-16)
`
`41*
`
`
`
`
`
`temperature T becomes
`
`V
`I = ———._..
`4EA....nT.?.
`
`Tsar - T _,,,|,
`
`“ {ml-r
`
`r,,,,+r
`
`_‘An applzoximate,
`"““a=iv=tseczions.9}an¢».mntug.nneso:unonou:incinu:- '
`
`t5=6m=
`
`1;.
`
`Intel Co .wa1..f. E
`
`Intel Corp. et al. Exhibit 1014
`
`

`
`236
`
`Chapter 5 Transient Conduction
`
`deep space). Returning to Equation 5.17, it is readily shown that, for T” =0,
`
`(5.19:
`
`An exact solution to Equation 5.15 may also be obtained if radiationmzp
`be neglected and h is independent of time. Introducing a reduced temperature.
`9 '=' T " Tee’ Where d9/df = 537/41. Equation 5.15 reduces to a linear. firs!-
`Ofdef, nonhornogeneous differential equation of the fonn
`
`'
`
`d9
`E:-5-(I0~b.—_-0
`
`;"h°’° 0 5 WI.../r=Vc) and b 2 [(q;’A3_ ,, + rig)/pvc]. Although Equation
`'20 may be 5°l"°d by Summing its homogeneous and particular solutions. all
`ah°ma‘i""- 3PP1'0&Ch is to eliminate the nonhomogeneity by introducing 113
`transformation
`
`,
`b
`9 E9‘ "
`G
`
`(5.21)
`
`Recognizing that d9’ 4 =
`(5.20) ‘O yield
`/ 1‘
`‘. __,._
`
`-
`-
`-
`d€/a't, Equation 5.21 may be substituted IIIW
`
`d0’
`‘ET 4' G0’ = 0
`
`Separating Variables and integrating from 0 to I (9,-' to 6'), it follows that
`
`(533)
`
`Intel Corp. et al. Exhibits" 14
`
`Intel Corp. et al. Exhibit 1014
`
`

`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`t, for T“ =0,
`
`5.4 SPATIAL EFFECTS
`
`5.4 Spatial Effects
`
`237
`
`(5.19)
`
`radiation may
`
`alinear,fittt-
`
`(5-20}
`
`igh Equatinl
`' solution, I
`
`Situations frequently arise for which the lumped capacitance method is inap-
`propriate, and alternative methods must be used. Regardless of the particular
`form of the method, we must now cope with the fact that gradients within the
`medium are no longer negligible.
`In their most general form, transient conduction problems are described
`by the heat equation, Equation 2.13 for rectangular coordinates or Equations
`2.20 and 2.23. respectively, for cylindrical and spherical coordinates. The
`solution to these partial differential equations provides the variation of tem-
`perature with both time and the spatial coordinates. However,
`in many
`problems, such as the plane wall of Figure 5.5, only one spatial coordinate is
`needed to describe the internal temperature distribution. With no internal
`generation and the assumption of constant thermal conductivity, Equation
`113 then reduces to
`
`
`
`1—_ ~ -133air
`33:2 - O.’ 3:
`
`(5.25)
`
`(.511)
`
`Jstituted 1|!”
`
`is
`it
`To solve Equation 5.26 for the temperature distribution T(x, I),
`I106?-Ssary to specify an initial condition and two boundruy conditions. For the
`typical transient conduction problem of Figure 5.5, the
`condition is
`
`T(x,o) = T,
`
`(521)
`
`and the boundary conditions are
`
`32'
`
`
`
`x x-0
`
`(51231
`
`and
`
`(IT
`
`"*3"
`
`I -x_L
`
`= h[T(L, I) - Tm]
`
`(5.27)
`
`(5-29)
`
`
`
`
`
`
`
`
`
`Efillalion 5.27 presumes a uniform temperature distribution at time I = 0;
`Equation 5.23 reflects the symmemr reqtérenlenl for the fl1idP15-'-113° of the W311:
`“*1 Equation 5.29 describes the surface condition experienced f°I time ‘ > 0-
`Fmm Equations 5.26 to 529, it is evident that. in addition to dvependins on x
`"*“‘“'~ temperamios in thewal1alsodepet1donanuu11ber0§P11Y5l'°31P31'im3¢'
`‘"3 In Particular
`(530)
`Tr: 115:, r, r,.,1;,, L, k,a, 1:)
`The foregdngprohlmnmaybeso[vedanaIy1iCfllll“°‘“f“'F‘_l°a5Y-These
`
`“metheadvantages that maybeobtained.-by
`
`Inte1'Corp_. et..a1.
`
`Intel Corp. et al. Exhibit 1014
`
`

`
`I38 Chapteri Transient Conduction
`
`ing equations. This may be done by arranging the relevant variables into
`suitable groups. Consider the dependent variable T. If the temperature differ-
`ence 9 E T - Ta is divided by the maximum possible temperature dtfirente
`8,. E T, - Tan, a dimensionless form of the dependent variable may be defined
`as
`
`a
`3- ‘DB:
`
`1'-1-"
`I "
`
`($339 T
`
`Accordingly, 9* must lie in the range 0 g 0* 5 1. A dimensionless spatial
`coordinate may be defined as
`
`'
`
`'
`
`-
`
`I
`..
`
`.
`x’?
`
`x
`e—
`
`_
`p
`-
`(33
`where L is the half-thickness of the plane wall, and a dimensionless time E1133’
`be defined as
`
`-T
`
`,
`
`‘|
`
`fit;
`
`Intel Corp. et al. Exhibilt, 14
`
`Intel Corp. et al. Exhibit 1014
`
`

`
`
`
`5.5 The Plane Wall with Convection
`
`239
`
`
`
`..
`
`‘_
`I
`
`_'
`
`"
`
`,
`'
`
`.
`
`=
`
`.:
`'
`
`I
`
`'
`
`,
`
`_
`
`
`
`;_
`
`'
`
`~;
`
`, '
`I
`
`:
`=_
`A
`.
`_
`'
`
`,,
`
`y
`
`I
`
`
`
`qtiatinnill
`Equati<ms5J6 '
`
`'
`
`534)
`
`535]
`l
`'
`
`a
`
`A
`
`A
`
`
`
`Equation 5.38 implies that for a prescribed geometry, the transient temperature
`distribution is a universal function of x*, Fe, and Bi. That is. the dimensionless
`solution assumes a prescribed form that does not depend on the particular
`value of 1'], Too, L, k, 0!, or 1:. Since this generalization greatly simplifies the
`presentation and utilization of transient solutions, the dimensionless variables
`are used extensively in subsequent sections.
`
`5.5 THE PLANE WALL WITH -CONVECTION
`
`Exact, analytical solutions to transient conduction problems have been ob-
`tained for many simplified geometries and boundary conditions and are well
`documented in the literature [I-4]. Several mathematical techniques, including
`the "method