`
`Page 1 of 98
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`Samsung Exhibit 1014
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`
`
`Page 1 of 98
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`Samsung Exhibit 1014
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`

`

`MYmtm'm'W-Tmm'fiw
`
`w F
`
`THIRD EDITION
`
`/ F
`
`UNDAMENTALS OF
`HEAT AND
`
`RANK P. INCROPERA
`
`DAVID P. DEWITT
`
`Sam; 01' Mechanical Engintcring
`Purdue University
`
`WILEY
`
`JOHN WILEY & SONS
`
`
`
`Page 2 of 98
`
`

`

`Dedicated to those wonderful women in our lives,
`
`Amy, Andrea, Debbie, Donna, Jody,
`Km Shawna, and Terri
`
`who, through the years, have blessed us with
`their love, patience, and understanding
`
`Copyright <13 1981. 1985. 1990, by John Wiley at Sons. Inc.
`
`All rights med. Published simultaneously in Canada.
`
`10987654321
`
`Reproduction at translation of any part o!
`[115stquth permittedbySections
`107 and 103 of the 1976 United States Copyright
`Act without the pamisa'ou o! the oopyn'ght
`misunlawfulflnqnestsfmpmnfimon
`orfurthu'infmxioushmldbeaddressedto
`thcPermissiomDepmnlanJohnWikyckSms.
`
`thcmmarmm:
`WkaP.
`Fundamentals of heat and mas: transits/Frank P. Inaupcra, David
`P. DeW'uL—er ed.
`
`p.cm.
`
`Includes bibliographical aim
`ISBN 0-471-6136-«4
`
`I. fiest-Tmnsmission. 1 Mass transfer.
`1%.
`II. Title.
`-
`
`L DeWin, David P..
`
`1990
`QC320.145
`521.402'2-«20
`
`PrimndinlheUnitedSmesofAmica
`
`39.38319
`(ZIP
`
`
`
`Page 3 of 98
`
`

`

`PREFACE
`
`.1
`
`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
`behafiorofthermalsystemslnsodoingitshoulddevelopmethodologifi
`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-
`inganalysiswhichifnotexact,sfiflprovidesusehflinformsuonconeernhig
`‘thedesignand/orpedormanceofsparfieularsystanorprocess.keqnire-
`mmtsofsuchananalysisindudetheabilitytodiscunrelevantuanspon
`processesandsimpfifyingassumpfimsidmfifyimponamdepmdemand
`independent variables, developapproprhtmnpressionsfromfirstprinciples,
`andinmdttcereqtusitematerialh'omtbeheat u'ansferknowledgebaselnthe
`firstedifiomachiemnentofthisobjecfivewasfostercdbywuchingmyof
`theexamplesandend—of-chapterproblansintflmsofacmaleny’neeting
`systems.
`'
`Thesecondedifionwesalsodlimbythefmegoingobjetzfimaswellas
`byinputderivedfrmnaquestionnaimsenttomlmcofleagueswhousedmr
`wereotbawisefanfihathhefimediMAmajorconseqmnceofthis
`input waspuhlicationoftwoversionsofthebook, Fundamrabafflmand
`Mass Transfer and Introdwrim to Heat Transfer. As in the first
`the
`Bahama}: vasion included mass transfer, providing an integrated treat—
`mentofheanmassandmomtumtransferbyeonvecfionandsepmte
`treatmentsofheatandmasstransferbydifi'usiommlumm versionof
`lhebookwssintendedforuserswhoembraeedthetreatmentofheat transfer
`butdidnotwishitocovumasstransfu' efi‘ects. Inbothversiongsignificant
`improvemeittswerenmdeinthetreauncmsofnnmericalmethodsandheat
`transfer with phasechange.
`hthishtestcdiuomchangeshnvebcenmofintedbythedesireto
`expandthescopeofapplieafimandtoenhancetheexposifionofphyfical
`prhdpMCmddunfimofabroadamngeotteehnicallyhupomtpmb-
`Jamisfacifimedhyhiereasedcoverageofefistingmatefialcuthermal
`u. resistaneefinpufmmeomectiveheatumsfermhamgmd
`
`
`
`Page 4 of 98
`
`

`

`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-
`dili‘erence solutions.
`
`DAVID P. DEWI'IT
`
`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 etfects 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.
`The last, but certainly not
`the least
`important,
`improvement
`in this
`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 probiems deal with recent interests in electronic cooling, manufactur-
`ing, and material processing. Many of the problems are drawn from our
`accumulated research and consulting expel-ices; the solutions, which fre-
`qutly 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.
`
`Wm Lafayette, Indiana
`
`FRANK P. INCROPERA
`
`
`
`Page 5 of 98
`
`

`

`CONTENTS
`
`Symbols
`
`Chapterl
`
`INTRODUCHON
`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
`Probluns
`
`INTRODUCTION TO CONDUCI'ION
`
`21 The Conduction Rate Equation
`22 The Thermal Properties of Matter
`2.2.1 Thermal Conductivity
`2.2.2 Other Relevant Properties
`2.3 The Heat Difl‘usion Equation
`2.4 Boundary and Initial Conditions
`2.5 Summary
`References
`Problems
`
`W3
`
`ONE-DIMENSIONAL, STEADY-STATE CONDUC'HON
`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
`
`
`
`xiv
`
`1
`2
`
`3
`3
`6
`9
`
`13
`13
`14
`19
`
`21
`22
`23
`24
`
`27
`29
`
`43
`
`44
`46
`47
`51
`53
`62
`65
`66
`66
`
`79
`80
`
`80
`82
`
`84
`36
`
`92
`96
`97
`103
`
`VI
`
`
`
`'15,.:—,2;.
`
`
`
`
`
`Whitiéfieo-IIA3.1-.“'uC-‘t.':'
`
`Page 6 of 98
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`
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`Page 6 of 98
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`

`

`viii
`
`Contents
`
`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 Surfaca
`3.6.1 A
`Conduction Analysis
`3.6.2 Fins of Unifmm Cress—Sectional Area
`3.6.3 Fin Performance
`3.6.4 OverallSnrfaoe Efficiency
`3.6.5 Fin Contact Resistance
`3.? Summary
`References
`Problems
`
`5.6.4 Graphical Representation
`
`TWO-DIMENSIONAL, SI'EADY-STATE CONDUCTION
`4.1 Alternative App
`4.2
`4.3
`
`5.6.3 Total Energy "Tl-amia-
`
`
`
`Page 7 of 98
`
`

`

`5.7 The Semi-infinite Solid
`5.8 Multidimensional Effects
`5.9 Finite-Difi'erenoe 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
`
`Contents
`
`ix
`
`259
`263
`270
`
`271
`
`279
`287
`287
`288
`
`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
`6.3 Laminar and Turbulent Flow
`
`-—.~
`
`6.4 The Convection Transfer Equations
`6.4.1 The Velocity Boundary Layer
`6.4.2 The Thermal Boundary Layer
`6.4.3 The Concentration Boundary Layer
`Approximations and Special Conditions
`Boumlary Layer Similarity: The Normalized Convection
`Transfer Equations
`6.6.1 Boundary Layer Similarity Parameters
`6.6.2 Functional Form of the Solutions
`
`Physical Significance of the Dimensionless Parameters
`Boundary Layer Analogies
`6.8.1 The Heat and Mass Transfer Analogy
`6.8.2 Evaporative Cooling
`6.8.3 The Reynolds Analogy
`6.9 The Efi’ects of Turbulence
`6.10 The Convection Coefficients
`
`6.11 Summary
`References
`Problems
`
`M 7 ETERNAL 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 Calculation
`
`396
`
`397
`399
`401
`
`.-._..rt
`
`$4.5...)..
`«Gianni!fl-'
`
`
`>ISilt-i933Join.
`
`
`
`Page 8 of 98
`
`

`

`The Cylinder in Cross Flow
`7.4.1 Flow Considerations
`7.4.2 Convection Heat and
`The Sphere
`Flow Across Banks of Tubes
`
`Transfer
`
`_.
`_
`Impinsins Jets
`7.7.1 Hydrodynamic and Geometric Considerations
`7.7.2 Convection Hem and Mass Transfer
`Packed Beds
`
`Summary
`References
`Problems
`
`The Concentric Tube Annulus
`
`8.1.2 The Mean Velocity
`8.1.3 Velocity Profile in the Fully DeveIOped Region
`8.1.4 Pressure Gradient and Friction Factor in Fully
`Deveioped Flow
`8.2 Thermal Considerations
`8.2.1 The Mean Temperature
`8.2.2 Newton’s Law of Cooling
`8.2.3 Fully Dcvelcped Conditions
`3.3 The Energy Balance
`8.3.1 (hieral Considerations
`8.3.2 Constant Surface Heat Flux
`8.3.3 Constant Sm'faoe Temperamre
`8.4 IannnarFlowinCircularTubeszThermalAnalysis and
`CODVDcthn Correlations
`
`Met 8
`
`INTERNAL FLOW
`
`8.1 Hydrodynamic Considerations
`8.1.1 Flow Conditions
`
`Convection Correlations: Turbulent Flow in Circular
`Tubes
`
`Convection Correlations: Noncircular Tubes
`
`
`
`Page 9 of 98
`
`

`

`Contents
`
`xi
`
`9.6
`
`Correlations: External Free Convection Flows
`9.6,1 The Vertical Plate
`9.6.2 Inclined and Horizontal Plates
`9.6.3 The LDng Horizontal "Cylinder
`9.15.4 Sphere-s
`I
`9.7 Free Convee'tion within Parallel Plate rChannels
`9.7.1 Vertical Channels
`9.7.2 Inclined Channels
`
`9.3 flintpirical Correlations: Enclosures
`9.8.1 Rectangular. Cavities
`9.3.2 Concentric Cylinders
`9.8.3 Coneentric‘Spheres
`9.9 Combined Free and Forced Convection
`9.10 Convection Mass Transfer
`
`541-
`542
`546
`550
`.553
`555
`555
`558
`
`558
`559
`562
`563
`566
`567
`567
`568
`570
`
`587
`588
`539
`590
`590
`592
`596
`596
`597
`598
`
`599
`
`11.3.1 The Parallel-Plow Heat Exchanger
`
`Chapter 10 BOILING AND CONDENSA'I‘ION
`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
`10.4 Pool Boiling Correlations
`10.4.1 Nucleate Pool Boiling
`10.4.2 Critical Heat Flux for Nucleate Pool Boiling
`10.4.3
`Heat Flux
`
`10.4.4 Film Pool Boiling
`10.4.5 Parametric Effects on Pool Boiling
`10.5 Forced-Convection Boiling
`10.5.1 External Forced-Convection
`10.5.2 Two-Phase Flow
`
`10.6 Condensation: Physical Mechanisms
`10.7 Laminar Film Condensation on a Vertical Plate
`
`108 Turbulent Film Condensation
`10.9 Film Condensation on Radial Systems
`10.10 Film Condensation in Horizontal Tubes
`
`10.11 Dropr Condensation
`
`11.2 The Overall Heat Transfer Coefi‘idenl
`
`11.3 Heat Exchanger Analysis: Use of the Log Mean
`Temperature Difierenoe
`
`
`
`Page 10 of 98
`
`

`

`
`
`11.4
`
`11.5
`11.6
`11.7
`
`11.3.2 The Counterflow Heat Exchanger
`11.3.3 Special Operating Conditions
`11.3.4 Multipass and Cross-Flow Heat Exchangers
`Heat Exchanger Analysis: The Efi‘ecfiveness—NTU
`Method
`11.4.1 Definitions
`
`11.4.2 Effectiveness—NW Relations
`Methodology of a Heat Exchanger Calculation
`Compact Heat Exchangers
`Summary
`References
`Problems
`
`RADIATION: PROCESSES AND PROPERTIES
`12.1
`Fundamental Concepts
`12.2
`Radiation Intensity
`12.2.1 Definitions
`
`12.2.2 Relation to EnnSSion
`12.2.3 Relation to Irradiation
`12.24 Relation to Radiosity
`Blackbody Radiation
`12.3.1 The Planck Distribution
`12.3.2 Wien’s Displacemem Law
`12.3.3 The Stefan-Boltzmann Law
`12.3.4 Band Emission
`Surface Emission
`Surface Absorption, Reflection, and Transmission
`12.5.1 Absorptivity
`115.2 Reflectivity
`12.5.3 Tranmlfissivity
`12.5.4 Special Considerations
`Kirchhofi's Law
`
`12.3
`
`12.4
`12.5
`
`12.6
`12.7
`12.8
`12.9
`
`649
`650
`650
`
`658
`658
`
`666
`672
`678
`679
`680
`
`695
`696
`699
`699
`702
`706
`708
`709
`710
`712
`712
`713
`719
`729
`731
`732
`734
`
`740
`742
`749
`756
`758
`759
`
`791
`792
`792
`
`803
`
`814
`816
`819
`
`,..;-'...
`
`a'v'
`
`iii
`
`
`
`Page 11 of 98
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`
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`Page 11 of 98
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`

`

`Con tents
`
`‘Mitltimofie Heat "Transfer
`1-3.4
`13.5 Additional‘Efi'ects
`135.71 'Volumeftii'o Absorption
`13.6.2 Gaseous Emission and Absorption
`
`1.3:.6
`
`Re'fere'nees
`Problems
`
`War 14 DIFFUSION MASS TRANSFER
`14.1
`Physical Origins and Rate Equations
`14.1.1 Physical Origins
`14.1.2
`Composition
`14.1.3 Fick’s Law of Diffusion
`14.1.4 Restrictive Conditions
`
`14.1.5 Mass Diflusion Coefficient
`Conservation of Species
`14.2.1 Conservation of Species for a Control Volume
`14.2.2 The M335 Diflusion Equation
`Boundary and Initial Conditions
`Mass DiflusiOn Without Homogeneous Chemical
`Reactions
`
`PLATE
`
`14.4.1 Stationary Media with Specified Surface
`Concentrations
`
`14.4.2 Stationary Media with Catalytic Surface Reactions
`14.4.3 Equimolar Counterdifi'usion
`14.4.4 Evaporation in a Column
`M355 Dilfusion with Homogeneous Chemical Reactions
`Transient Diffusion
`
`References
`Problems
`
`Appendix A THERMOPHYSICAL PROPERTIES OF MATTER
`
`Amer-Ilia: B MATHEMATICAL RELATIONS AND FUNCTIONS
`
`Appendix C AN INTEGRAL LAMNAR BOUNDARY LAYER
`SOLUTION FOR PARALLEL FLOW OVER A FLAT
`
`
`
`Page 12 of 98
`
`

`

`
`
`
`
`CHAPTER 5
`
`
`
`m.me
`
`sater is passed. Under
`tits in a uniform hm
`
`water flow provides;
`E h = 5000 W/uf-K
`:ady-state temperature
`ansiderations. we may
`
`the preceding pug:
`dated. use a finiEE-dif-
`.tures.
`
`
`
`
`
`
`99.39;; ‘Emmiii.OFLURBANASHAEJE
`
`TRANSIENT
`
`CONDUCTION
`
`Page 13 of 98
`
`
`
`
`
`Page 13 of 98
`
`

`

`with the-.iéllggf L“
`sated conditions. We
`gene:
`"
`'
`state conduction with no
`complications due to mulndirnensiénéi
`I._
`7.
`,_
`.7
`_
`j,
`have not yet considered situations for—whié _
`We now recognize that
`heat- transfer; pfobl'ensjsinrehfiine dependent
`Such mteady, or transient. problem's:
`when the bowler!
`conditions of a system are changed. For e‘xargpit'é;
`the :sur‘fpce temperatm of
`a system is altered, the temperature zit each
`in therfsystém
`also high
`to change. The changes will continue to
`until at rsgteezdyrsmre tempefam
`distribution is reached. Consider a hot metal enter- first: iswrernoved hon“
`furnace and exposed to arcool
`Energy is windowed by MW
`and radiation from its surface to the surroundingsgfinergyjtransfer by Wm’
`lion also occurs from the interior of the metal
`to jibe surface, and
`fempefamre at each point in the billet decreases until :1 steady-state condition
`ts reached. Such time-dependent effects
`marry Zindi-Istiial heating and
`cooling processes.
`I
`To determine the time dependence of the tomper-a'ture distribution
`3 so“ during a transient process, we .could begin by adlfifig the 3139‘”ng
`form-of the heat equation, for example, Equation 2.13. Some cases for which
`Soluuom have been obtained are discussed in Sections 5.4 10.5.8.1‘10W'
`such solutions are often diflimlt to obtain, and where P055“: 3_W
`*1th 15 Preferred. One such approach may be used under conditions I“
`“mic? tempflaml'e gradients within the solid are small; It is termed the W
`Capacitance method.
`
`l<0
`
`e
`
`is quenfi‘m
`5.1.
`gurfhc solid win
`
`
`
`Page 14 of 98
`
`

`

`
`5.1 The Lumped Capacitance Method
`22‘?
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`decrease for time I > 0, until it eventually reaches Tm. This reduction is due to
`convection heat
`transfer at
`the solid—liquid interface. The essence of the
`lumped capacitance method is the assumptiorLt‘hat _the__tc_mpe£atnre of the
`solid is spariallywumfarm at any instant during the transient process. This
`assumption—{fines that temperature gradients within the solid are negligible.
`From Fourier’s law, heaticondugignriq nghsence of a__ temperature
`gradient implies the ggigterfigpf fin—finite thermal conductivity. Such a condi-
`tion is clemsible. Howeyer, 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
`1.113 to the control volume of Figure 5.1. this requirement takes the form
`~55... = a.
`fl [ 4 win!
`(5-1)
`
`or
`
`'
`
`dT
`‘MAT - Ta.) = ch—
`d:
`
`Inthducing the temperature difl'erence
`
`5L'- {.7’ '
`
`74'- r
`u
`
`(5.2)
`
`(5.3)
`
`.
`
`and recogIIiZiEg that (dB/dr) = ((117111 ). it follows that
`PVC :10
`M, d: _- —0
`Separating Variables and integrating from the initial condition. for which I = 0
`
`and 7(0) = T,-. we then obtain
`
`i7
`
`
`
`-‘14—.-.
`
`Mg
`
`,ourt.URWA-CHAMPMEN
`
`m
`9?
`
`more compli-
`sional,
`itly considered
`. However. at
`with time.
`
`me dependent
`the boundary
`temperature of
`will also been
`re tempetatutt
`moved from a
`
`
`
`
`
`PVC
`ha,
`
`add
`I
`a, 7 = ‘Ld‘
`
`Where
`
`.4 ,1" .
`
`(5.4)
`
`I
`
`_.
`
`
`
`Eva]
`
`,
`Eryn-“.1
`
`
`“filing the integrals it follows that
`.
`._ a
`
`
`
`Page 15 of 98
`
`
`
`Page 15 of 98
`
`

`

`11.1
`
`“L2 11,3 Tu
`
`$05.2 Trmsimttnmperammponsegf
`hunpedcapacitame to
`
`difl'erenlthermalfimccomtams 1;.
`
`
`
`Page 16 of 98
`
`

`

`5.2 Validity of the Lumped Capacitance lehod
`
`229
`
`
`
`Fifi! 5.3 Equivalent thermal drum for a
`lumped capacitance solid
`
`jI:
`I
`
`|
`
`Humrfi.r
`
`~_41\—L'i-yl:u-n-r
`
`MW'mmM-flfle
`
`Todctermincthetmalenergytransferrocuninguptosomefimet,we
`simplywrite
`
`g= £qd:=u,fo'adx
`Substituting for 0 from Equation 5.6 and integrating, we obtain
`
`
`
`WWfiWQisormmmdmmmmmmmwotm
`s“Y'lld.am‘.lfromfilquatim1.1111!
`
`
`
`mkm.
`WM—stateomducfimthrmgh
`Todevdopasuinble aim-ion
`thfldeWryneSfi).
`
`kammamrfilmmm-muwm
`‘MOftmlpaammTu<1;rThemnpemmfc
`
`
`
`Page 17 of 98
`
`
`
`Page 17 of 98
`
`

`

`13. Chapters Transian Conduction
`
`01- Bio! umber on
`steadysiéte Ftemperatute
`1'“
`151.311: Wall with
`convection
`
`-< 2'7“. Hm: under steadym
`mm.' M3111: 11.2. for which :2; _<
`m3“ “1° “We energy balance, Equation 11.212; reduces to
`
`dlst“tuition is reasonable.
`
`“"1353! as a ratio .‘of thermal resistances, W":
`the
`W ‘0 Candacrion within the solid is m k”
`a;
`mum
`‘heflllid boundary layer. Hence the mm”
`
`
`
`Page 18 of 98
`
`

`

`5.2 Validity of the Lumped Capacitance Method
`
`231
`
`
`
`number, and three conditions are shown in Figure 5.5. For Bicl the
`temperature gradient in the solid is small and T(x,t) = m). Virmally all
`finmpmturc difl‘a'enoeisbetwem Ihesolid andthefluid, and the solid
`mperatum remains neafly uniform as it decreases to T”. For moderate to
`Iarzevalues ofthe Biol number, however, thetemperature gradients withinthe
`wfidmsignificanLHenoeT=T(x,r)-NotethntforBi>-1,thetempera-
`mmflmoeanrossthesofidisnowmummlhanthatbctwmthe
`maccnndthefluid.
`
`solving transient conduction problems. Hence, when confronted with such a
`Problem. :heneqfiqugmaxmshouIddonmlcuIaremmmnmmu
`the following condition is satisfied
`
`
`
`
`“Eherauoonhcsoud'svommumcem L,.-= V/A,.Snchadefinition
`immmmlculafionoch forsolidsofcomplicaledshapeandreducestothe
`half'thischms L foraplanewallofthicknaiSZL (figureiS), to'rflfota
`www.mmrj3 forasphanfiowcvctjfonewishcsmimplement
`“criminath L‘shouldbemodawdwiththelmgxh
`WWmmemximantmpcramdiflermmrd—
`my’m’asymflyhcawdmrooolemmamwafiohhimzh L,
`Wmmflwthehafl-ihickness L. However, foralongcylinderor
`9"“: Lcwolfldequalflieactaalrufinsrflmhetthanrflorrfl.
`
`
`
`
`
`Page 19 of 98
`
`
`
`Page 19 of 98
`
`

`

`
`
`m ChapterS Transient Conduction
`
`
`
`Finally, we note that. with L: ‘=' V/As, the exponent of Equation 5.6 may
`be expressed as
`
`
`
`Schematic:
`
`
`M1t_ ht _hLe_£_l‘=
`ch
`10ml.f
`R {De L3
`
`
`
`k L3
`
`
`
`
`
`
`
`is termed the Fourier number. It is a dimensionless time, which, with the
`number, characterizes transient conduction problems. Substituting 1'3qu
`5.1] into 5.6, we obtain
`
`
`
`
`
`T _ Tue
`
`3: 7,7. =€Xp(-B£-Fa)
`I
`r
`w
`
`(5J3)
`
`'{;=200"C
`h = fllm‘ 1K
`—+>
`—{>
`Gas stream
`
`Assumptions:
`
`1. Temperau
`2. Radiation
`
`3' Losses by
`4' Constant
`
`Analysis:
`
`1. Because t
`the soluti-
`
`capacitan
`approach
`determin:
`fact that
`
`TI =
`
`Remang
`
`
`
`
`
`
`.
`a m
`_' 85m kg/Fg- Determine the Juncuon diamefef
`an
`k to have a “me Constant of 1 s. If thcjunction Is at 25 C.
`I
`is 1“
`.
`Winch
`that ‘5 at 200°C, how long will it take for d1:me
`
`-
`
`
`
`
`Page 20 of 98
`
`
`
`Page 20 of 98
`
`

`

`5.2 Validity of the Lumped Capacitanc: Method
`
`233
`
`
`
`
`
`Thermocouple k = 20 W/m - K
`junction
`c = 400 J/kg- I”:1
`T,=25°C “=8500kgjm
`
`
`
`darn—4..A
`
`.‘u.
`‘1
`,1”
`
`
`
`,
`
`r
`
`_
`
`I?
`I'
`
`:h’ with
`lmfing
`
`L Tperature of juncfion is uniform ax any instant.
`2. Radiation exchange with the surroundings is negligible-
`3. Losses by conduction through the leads are negligible-
`4- Constant properties.
`
`Analysis:
`
`capacitance method, Equation 5.10, is satisfied. Em, a reassemble
`“PmachkwusethemethOdmfindthediambtfi.andwtbn.
`We whether the criterion is satisfied. FromEquat'lon- 5.7 and: thc
`5‘“ that A, = «D1 and v= «153/6 for asphere, it follows that
`
`
`
`
`6"": “Xm'w/mf'fix” =7.06x10"‘ni
`pc Whym’XMJ/kg-K
`
`<1
`
`With L: = rn/3 it than inflows from
`
`_
`
`
`
`
`
`x m“
`
`460W/fi- I1; xx‘sgsaixgmrl‘wgt
`'3
`u
`15—191 1:
`no}:
`
`145/3)
`
` k
`
`hI m
`
`.‘
`
`I
`
`.I.
`
`--
`
`r j
`
`.
`
`I
`
`I
`
`:
`
`III”
`
`I‘
`
`‘
`
`I
`
`I
`
`'I
`
`
`If.
`
`
`
`
`
`
`‘7
`
`n
`
`L;
`
`I
`
`..r
`
`"' Q4.“ C" =
`
`Page 21 of 98
`
`
`
`Page 21 of 98
`
`

`

`i
`
`— 'I‘Jfi'lbr
`
`Chapter 5 Transient Conduction
`
`'1 ,
`
`A
`.Y
`2. From Equation 5.5 the time required‘gfiinheyfi'tihfitifih'tn reach T:
`199°C is
`‘ " 3“"? -°‘
`'
`
`P(er3/6)cln-TJ.,_T® J '_
`11:
`kW.)
`-
`‘T--—
`
`=
`
`:heween the junction
`Comments: Hcat losses due to radiafiéir;
`and the surroundings and conduction through. {the treads would necessitate
`“51118 a smallerjuncnon diameter to achj'eye {he ticsired time response.
`
`5.3 GENERAL LUMPED
`
`ANALYSIS
`
`ermal energy, and hence-the temperature, of the solid to change-Temp“!-
`
`Although transient conduction in a '50de is'commgnlyi initiated by 00mm
`heat
`transfer [0 01‘ from an adjoining fluid,
`:other processes may W
`transient thermal conditions within the solid. For example, a solid may b‘
`separaltied from large Slfrroundings by a gas or vacu'flm. If the tetrlli'fimf'”esof
`[I]
`so (1 and surroundings difler, radiation exchange could cause it“ "new
`
`
`
`Page 22 of 98
`
`

`

`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`5.3 General Lumped Capacitance Analysis
`
`235
`
`heat flux, and internal energy generation. It is presumed that, initially (r = 0),
`the temperature of the solid (1}) differs from that of the fluid, Tan, and the
`surroundings, T
`and that both surface and volunetric heating (4: and d)
`5111"
`are initiated. The imposed heat flux qf and the convection—radiation heat
`surface, Am.) and Am“),
`transfer occur at mutually exclusive portions of the
`rapectively, and convection—radiation transfer is presiimed to be from the
`surface. Applying conservation of energy at any instant I, it follows from
`Pquafion 1.113. that
`r
`'
`q; ALI! + E3 .—
`
`' a.
`+ 4;)Aflar) = PVC—d7
`
`01', from Equations 1.3a and 1.7,
`dT
`.
`4:44” + ES — [h(T— T“) + eu(T‘ — 1,1,):]A,(w, = chE-
`
`(5.15)
`
`first—order, nonhomoge-
`Unfortunately, Equation 5.15 is a nonlinear,
`MODS, ordinary difl’erentiai 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
`generation and convection is either nonexistent
`fdative to radiation, Equation 5-15 reduces to
`
`GET
`PVC; = -8A,,,°(T‘ - 7:»)
`Separatingvariablesand hitegratingfromtheinitialcondiliontoanythner. it
`follows that
`
`(S .16)
`
`I
`
`T
`= __i__
`
` 8A1!“
`.
`f1; Tfiu‘r
`9"“ I’d!
`'
`Evatnanngbommmgrnsandmnginemmmfimd
`Want-ureTbecomcs
`
`(
`
`5.17
`
`)
`
`tomchrtlie
`
`I'm-FT
`
`
`
`-13..-
`
`
`
`
`
`
`Page 23 of 98
`
`
`
`
`
`
`
`Page 23 of 98
`
`

`

`236 Chapters Transient Conduction
`
`w
`x . 5n!
`deep space} Returning to Equation 1:;
`
`.
`
`,.
`that, tor Ts" =o_
`
`5-4
`
`VC
`
`" p
`
`7"
`
`-
`
`I-_
`'_
`
`4
`"
`A
`‘
`
`(5.1m
`
`— =
`b
`
`0
`
`a
`
`i
`
`10
`
`J
`
`(5
`
`Dilai'ned if radiation may
`A” “an 501mm to Equatibn
`a rcduced temperatum
`be neEviemtid and h is independent 76f 1mm. 111g
`9 E T ' Tac’ where dfl/dt = dT/Jdr. ‘_ r, magic-es.
`.19 a linear. tim-
`Ofdel’, nonhomogeneous differential figumiqfi {3f
`d9
`9
`_
`'_
`__' +
`d:
`
`or substituting for 9’ and 0
`
`transformation
`
`5
`9'50__
`0
`
`RmSniZing that “7"? = da/dl, Equation 5.21 may be substituted in”
`(5-20) to yield
`
`{5.13}
`
`Separating variables and integrating from o m I (at, to 9,), it follows that
`
`0;
`a? = CRIN-a!)
`
`(523]
`
`
`
`Page 24 of 98
`
`

`

`
`
`5.4 Spatial Effects
`
`157
`
`5.4 SPATIAL EFFECTS
`
`t, for Tut-l,
`
`(5'19)
`
`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
`2.13 then reduces to
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`321‘
`
`31:2
`
`1 3T
`
`a 3!
`
`(5.26)
`
`T(x, r), it is
`To solve Equation 5.26 for- the temperature
`W to specify an initial condition and two bomdary conditions. For the
`W transiem conduction problem of Figure 5.5, the
`condition is
`mm) = r.
`em)
`
`and the boundary conditions are
`
`ET
`
`'5;
`
`
`
`= 0
`
`
`
`
`Page 25 of 98
`
`
`
`Page 25 of 98
`
`

`

`33ft variables imn
`I'
`. @E-ral’gnipcrature difi'er-
`' pom-tare Hi creme
`9' maybe defined
`-
`
`‘
`11""
`
`'
`
`_
`
`I .1"- TC
`:0,
`22-1..
`
`m
`
`Accordingly, 9“ must lie in the ran'gc'D
`coordinate may be defined as
`
`g
`
`A.
`
`{anaemionless 5P3“
`
`V
`
`x
`
`where L is the half-thickness of the plane mu, m a dimensionless time may
`be defined as
`7
`'
`‘
`
`i? “human: to the dimensionless Fourier “W’s Equation 5'11
`Subsutulmg the definitions of Equamns 5.31 to 533 mm Equations 51"
`to 5.29. the heat equation becomes
`
`610*
`ax,2=
`
`as:
`a
`
`and the initial and boundary conditions become
`0’(x".0) = 1
`30*
`
`3;;
`
`x‘ ‘0
`
`80‘
`
`‘
`(5.34]
`
`(535}
`
`(5.36)
`
`238 Chaplch Transient Condqqfion
`
`.
`
`"
`...-.1
`
`ing equations. This may {doing
`suitable groups. Consider "611:: dcp’enfléfiifvfififiifiy _
`ence 0 a T — Ta is divided by
`I”
`0, E T, - To“, a dinnensionlcsjs. form rof in;
`as
`
`Bx‘
`
`
`
`Page 26 of 98
`
`

`

`l.
`at
`{3'
`“ETMT;i.._-,
`[1J
`
`'I
`
`i-
`
`" "‘
`
`“'wo
`‘
`L
`
`-
`
`5.5 The Plane Wall with Convection
`
`239
`
`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, a, or h. 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
`
`t man“ In
`flperam flu.
`!
`diam
`
`
`
`- ll-
`
`:sionless spatial.
`
`,
`
`Exact, analytical solutions to transient conduction problems have been ob-
`lained for many simplified geometries and boundary conditions and are well
`documented in the literature [1—4]. Several mathatical tecl'miques, including
`“18 method of separation of variables (Section 4.2), may be used for this
`unless‘ Purpose. and typically the solution for the dimensionless temperature distribu-
`tion, Eqmtion 5.38, is in the form of an infinite series. However, except for
`very small values of the Fourier number, this series may be approximated by a
`
`
`
`
`
`____——T.__—‘.____f_VW___T__
`
`5M,&mn&hfim
`
`\ V
`.
`@fimflnwt
`.34.].m'7 35151;}
`.
`‘4‘
`
` - Singletermand theresultsmayberepresentedinaconvenientgraphicalform.
`
`
`.
`
`Consider the plane wall of thickness 2L (Figure 5.711). It the-thickness is small
`Native to the width and height of the wall,- it is reasonable to assume that
`Won occum exclusively in the x direction. It the wall is initially at a
`uniform temperature, T(x,0) = 2;, and suddenly immersed in a fluid of
`'
`Tm * To the resulting temperann’es may lie-obtained by solvingme 5.34
`_ i '
`"
`We“ to the conditions of Equations 535. to 5.37. Since the mention
`[935:
`'-.'. Wmforthesurfaoesatfl= ilamthesamthetempuamredtstnbu-
`‘fmf’.
`mummyinstammmtbesymmotficalabonIthB-nndplaneex*=0).An
`
`Nell) =3 Ti
`
`
`
`Page 27 of 98
`
`h.
`
`u,
`lt
`"_:H
`.l
`n.
`
`.
`
`
`
`Page 27 of 98
`
`

`

`
`
`24» Chapters Transient Conduction
`
`Table 5.1 Coefficients us«
`to the series so
`
`PLANE WALL
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`em- - hL/k for the planewall a
`
`
`10.0
`
`20.0
`
`1.259?
`1.4149
`1.26X
`1.4289
`1.2691
`1.4961
`1.271".
`1.5202
`1.2722
`15325
`1.2727
`1.5400
`1.273]
`1.5552
`.________________
`
`exact solution to this problem has been obtained and is of the form [2]
`
`9* = E C.exp(-§3Fo)cos(§.x*)
`nil
`
`where the coaificient C1 is
`
`4 sin
`I"
`C
`n =m
`
`(5393}
`
`(5.3%)
`
`and the discrete values (eigenvalues) of Q'" are positive roots of the transmi-
`denial equation
`
`1,, tan Q, = Bi
`
`(539:)
`
`The first four roots of this equation are given in Appendix B.3.
`
`552 Approximate Solution
`
`It ca? be Show]? (Pmblem 524) that for values of F0 2 0.2, the infinite
`solution, Equation 5.3931 can be approximawd by me first term of the scout
`‘9‘”?“5 this approximation.
`the dimensionless form of the telnme
`distribution becomes
`
`
`
`
`
`
`
`
`
`
`
`
`9‘ = Ciexfl‘fizl’ohosmx‘)
`
`01'
`
`0‘ = 0: cos(§1x‘)
`
`where 9: represents the midplane (x- = 0) temperature
`v:=c.exp(-:.2Fo)
`
`‘5'“
`
`(Ev-41'
`
`{mameimplication of Equation 5.40!) is that me time dependede
`tempera“; afimy focation within the wall is the same as that of the
`.
`e coefficients Cl and {I are evaluated from Equafimss
`’
`I
`ively’ and 3‘3 given in Table 5.1 for a range of Biot “W
`
`
`
`5‘55 1"“ my Transfer
`
`
`
`Page 28 of 98
`
`§i
`(rad)
`
`0.0998
`0.1410
`0.1732
`
`0.1937
`0.2217
`0.2425
`0.2615
`0.2791
`0.2956
`0.3111
`
`C1
`
`1.0017
`1.0033
`1.0049
`
`1.0066
`1.0082
`1.0098
`1.0114
`1.0130
`1.0145
`1.0160
`
`0.3779
`0.4328
`0.4801
`0.5218
`
`0.5932
`
`0.6533
`0.7051
`0.7506
`0.7910
`
`0.8274
`0.8603
`1.0769
`
`1.1925
`1.2646
`1.3138
`1.3496
`1.3766
`1.3978
`
`1.0237
`1.0311
`1.0382
`1.0450
`
`1.0580
`
`1.0701
`1.0814
`1.0919
`1.1016
`
`1.1107
`1.1191
`1.1795
`
`1.2102
`1.2287
`1.2402
`1.2479
`1.2532
`1.2570
`
`Bi“
`
`0.01
`0.02
`
`0.03
`0.04
`0.05
`0.06
`0.07
`
`0.08
`0.09
`0.10
`0.15
`0.20
`
`0.25
`0.30
`0.4
`0.5
`0.6
`0.7
`
`0.8
`0.9
`
`1.0
`
`20
`3.0
`
`4.0
`5.0
`6.0
`
`7.0
`8.0
`9.0
`
`
`
`Page 28 of 98
`
`

`

`——I——_——
`
`Table 5.1 C'ocfi'lciems used in the one-term approximatiOH
`m the series solutions for transient one-dimensional conduction
`INFINITL‘.
`
`("(1.119 DERPLANE W A L 1. SP1 IERE
`
`
`
`‘11: =1r1.."1\ 11>r1hc Nun: wall and ham/k I'm I111: 1n11l111€ L-\-‘lin:1n-r;md sphum. Sm.- figure 5.7.
`
`.11
`1
`‘E C. trad; (“J 1 rad 1
`
`
`
`
`0 1412
`0.1730
`11.0991:
`1.0017
`1.0025
`1.0030
`[1.14111
`1.0033
`0.1995.
`1.0050
`0.2445
`1 0060
`17.17.12
`.0049
`0.24.19
`1 0075
`0.2939
`1.0090
`[1.1937
`01.166
`0.22114
`1 .0099
`0.3450
`1.01 20
`0.2217
`.0082.
`0.3142
`1.0124
`0.3052
`1.111411
`112425
`0093
`0.3.4311
`1.01411
`0.4217
`1.0179
`1.1.2615
`1.0114
`0.37014
`1.0173
`0.4550
`1.0209
`0.2791
`0130
`0.3960
`1.0197
`0.41160
`1.02.19
`0.2956
`1.0145
`0.4195
`1.0222
`0 5150
`1.02611
`0.3111
`1.0160
`0.4417
`1.0246
`0 5423
`1.0293
`113779
`.0237
`0.5376
`1.0365
`0.6603
`1.0445
`0.43211
`0311
`0.6170
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