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

`THIRD EDITION
`icca
`
`FUNDAMENTALS OF
`HEAT AND
`MASS TRANSFER
`
`New York * Chichester + Brisbane « Toronto = Stagapors
`
`FRANKP. INCROPERA
`DAVID P. DeWITT
`School of Mechanical Engineering
`Purdue University
`
`JOHN WILEY & SONS
`
`
`
`Page 2 of 98
`
`

`

`Dedicated to those wonderful womenin our lives,
`
`Amy, Andrea, Debbie, Donna, Jody,
`Karen, Shaunna, and Terri
`
`who, through the years, have blessed us with
`their love, patience, and understanding.
`
`With th
`edition,
`mature
`
`howeve
`treatme
`
`9
`
`P.
`above a
`
`subject.
`behavio
`which f
`problem
`ing anz
`the desi
`ments ©
`
`pro CESse
`indepen
`and int
`first edi
`
`systems
`
`Reproduction or translation of any part of
`this work beyond that permitted by Sections
`107 and 108 of the 1976 United States Copyright
`Act without the permission of the copyright
`owner is unlawful. Requests for permission
`or further information should be addressed to
`the Permissions Department, John Wiley & Sons.
`
`Copyright © 1981, 1985, 1990, by John Wiley & Sons, Inc.
`
`All rights reserved. Published simultaneously in Canada.
`
`Library of Congress Cataloging in Publication Data:
`Incropera, Frank P-
`Fundamentals of heat and mass. transfer/Frank P. Incropera, David
`P. DeWitt.—3rd ed.
`
`cm.
`p-
`Includes bibliographical references.
`ISBN 0-471-61246-4
`1. Heat—Transmission.
`1934—.
`IL. Title.
`
`2, Mass transfer.
`‘
`
`1990
`QC320.145
`621.402°2—de20
`
`Printed in the United States of America
`
`1987654321
`
`L DeWitt, David P.,
`
`89-38319
`CIP
`
`
`
`Page 3 of 98
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`

`

`Eva nee
`/
`
`PREFACE
`
`Se
`
`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 system or process. Require-
`ments of such an analysis include the ability to discern relevant transport
`processes and simplifying assumptions, identify important dependent and
`independent variables, develop appropriate expressions from first principles,
`and introduce requisite material from the heat transfer knowledge base. In the
`first edition, achievement of this objective was fostered by couching many of
`the examples and end-of-chapter. problems m terms of actual ‘engineering
`systems.
`The second edition was also driven by the foregoing objectives, as well as
`by input derived from a questionnaire sent to over 100 colleagues who used, or
`were otherwise familiar with, the first edition. A major consequence of this
`input was publication of two versions of the book, Fundamentals of Heat and
`Mass Transfer and Introduction to Heat Transfer. As in the first edition, the
`Fundamentals version included mass transfer, providing an integrated treat-
`ment of heat, mass and momentum transfer by convection and separate
`treatments of heat and mass transfer by diffusion. The Introduction version of
`the book was intended for users who embraced the treatment of heat transfer
`but did not wishto cover mass transfer effects. In both versions, significant
`improvements were made in the treatments of numerical methods and heat
`transfer with phase change.
`In this latest edition, changes have been motivated by the desire to
`expand the scope of applications and to enhance the exposition of physical
`principles. Consideration of a broader range of technically important prob-
`lems is facilitated by increased coverage of existing material on thermal
`-~ oo)
`fesistance, fin performance, convective heat transfer enhancement, and
`
`
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`Page 4 of 98
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`

`

`vi
`
`Preface
`
`Davip P. DeWrrr
`
`compact heat exchangers, as well as by the addition of new material on
`submerged jets (Chapter 7) and free convectionin 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 andin 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 multimodeeffects 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 recentinterests 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.
`
`West Lafayette, Indiana
`
`FRANK P. INCROPERA
`
`
`
`Page 5 of 98
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`

`

`CONTENTS
`
`Chapter 1
`
`Symbols
`
`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 Diffusion Equation
`2.4 Boundary and Initial Conditions
`2.5 Summary
`References
`Problems
`
`Chapter 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
`
`
`
`vu
`
`PSs
`
`een!ES
`aiaiaPec
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`of 98
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`viii
`
`Contents
`
`Chapter 4
`
`5.6.4 Graphical Representation
`
`Chapter 5|TRANSIENT CONDUCTION
`‘aco Capacitance Method
`5.1
`5.2
`ity of
`the Lumped Capacitance Method
`3.3
`General Lumped
`acitance Analysis
`Spatial Effects hs
`5.4
`a
`The Plane Wall with Convection
`5.5.1 Exact Solution
`5.5.2 Approximate Solution
`5.5.3 Total Energy Transfer
`5.5.4 Graphical Representations
`Radial Systems with Convection
`5.6.1 Exact Solutions
`5.6.2 Approximate Solutions
`5.6.3 Total Energy Transfer
`
`Summary of One-Dimensional Conduction Results
`Conduction with Thermal Energy Generation
`3.5.1 The Plane Wall
`3.5.2 Radial Systems
`3.5.3 Application of Resistance Concepts
`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
`Summary
`References
`Problems
`
`TWO-DIMENSIONAL, STEADY-STATE CONDUCTION
`4.1
`Alternative Approaches
`42
`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
`Finite-Difference Equations
`4.4.1 The Nodal Network
`4.4.2 Finite-Difference Form ofthe Heat Equation
`4.4.3 The Energy Balance Method
`Finite-Difference Solutions
`4.5.1 The Matrix Inversion Method
`4.5.2 Gauss-Seidel Iteration
`4.5.3 Some Precautions
`
`References
`Problems
`
`
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`5.7 The Semi-infinite Solid
`5.8 Multidimensional Effects
`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
`
`Contents
`
`ix
`
`259
`263
`270
`
`271
`
`279
`287
`
`tibetdiclateallNee
`
`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
`Laminar and Turbulent Flow
`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
`Boundary 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 Effects of Turbulence
`6.10 The Convection Coefficients
`6.11 Summary
`References
`Problems
`
`Chapter 6
`
`baddpeatST.
`
`Chapter 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 Calculation
`
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`x
`
`Contents
`
`Chapter 8
`
`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
`i
`7.7.1 Hydrodynamic and Geometric Considerations
`7.7.2 Convection Heat and Mass Transfer
`Packed Beds
`Summary
`References
`Problems
`
`539
`
`INTERNAL FLOW
`8.1 Hydrodynamic Considerations
`8.1.1 Flow Conditions
`8.1.2 The Mean Velocity
`8.1.3 Velocity Profile in the Fully Developed Region
`8.1.4 Pressure Gradient and Friction Factor in Fully
`Developed Flow
`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
`8.3.2 Constant Surface Heat Flux
`8.3.3 Constant Surface Temperature
`Laminar Flow in Circular Tubes: Thermal Analysis and
`Convection Correlations
`8.4.1 The Fully Developed Region
`8.4.2 The Entry Region
`oe Correlations: Turbulent Flow in Circular
`Convection Correlations: Noncircular Tubes
`The Concentric Tube Annulus
`Heat Transfer Enhancement
`
`Chapter 9
`
`FREE CONVECTION
`pysical Considerations
`91
`92
`93
`Similarity Considerations
`9.4
`Laminar Free Convection on a Vertical Surface
`9.5
`The Effects of Turbulence
`
`530
`533
`535
`536
`
`
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`Contents
`
`xi
`
`11.3.1 The Parallel-Flow Heat Exchanger
`
`Chapter 10 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 Boiling
`10.4.2 Critical Heat Flux for Nucleate Pool Boiling
`10.4.3 Minimum 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 Boiling
`10.5.2 Two-Phase Flow
`10.6 Condensation: Physical Mechanisms
`10.7 Laminar Film Condensation on a Vertical Plate
`10.8 Turbulent Film Condensation
`10.9 Film Condensation on Radial Systems
`10.10 Film Condensation in Horizontal Tubes
`10.11 Dropwise Condensation
`10.12 Summary
`References
`Problems
`
`541
`
`9.6 Empirical Correlations: External Free Convection Flows
`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
`9.8 Empirical Correlations: Enclosures
`9.8.1 RectangularCavities
`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
`
`HEAT EXCHANGERS
`11.1 Heat Exchanger Types
`11.2 The Overall Heat Transfer Coefficient
`11.3 Heat Exchanger Analysis: Use of the Log Mean
`Temperature Difference
`
`
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`
`
`649
`650
`650
`
`658
`
`11.3.2 The Counterfiow Heat Exchanger
`11.3.3 Special Operating Conditions
`11.3.4 Multipass and Cross-Flow Heat Exchangers
`11.4 Heat Exchanger Anaiysis: The Effectiveness-NTU
`Method
`11.4.1 Definitions
`11.4.2 Effectiveness-NTU Relations
`11.5 Methodology of a 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 Emission
`12.5 Surface Absorption, Reflection, and Transmission
`125.1 Absorptivity
`12.5.2 Reflectivity
`12.5.3 Transmissivity
`12.5.4 Special Considerations
`12.6 Kirchhoff’s Law
`12.7 The Gray Surface
`12.8 Environmental Radiation
`12.9 Summary
`References
`Problems
`
`RADIATION EXCHANGE BETWEEN SURFACES
`13.1 The View Factor
`13.1.1 The View Factor Integral
`13.1.2 View Factor Relations
`13.2 BlackbodyRadiation
`13.3 Radiation Exchange Between Diffuse, Gray Surfaces
`in anEnclosure
`
`13.3.5 The Reradiating Surface
`
`iii
`
`etekon
`
`(Dglaneal
`
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`

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`Contents
`
`13.4 Multimode Heat Transfer
`13.5 Additional Effects
`13.5.1 Volumetric Absorption
`13.5.2 Gaseous Emission and Absorption
`
`References
`Problems
`
`PLATE
`
`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 of Diffusion
`14.1.4 Restrictive Conditions
`14.1.5 Mass Diffusion Coefficient
`Conservation of Species
`14.2.1 Conservation of Species for a Control Volume
`14.2.2 The Mass Diffusion Equation
`Boundary and Initial Conditions
`Mass Diffusion 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 Equimolar Counterdiffusion
`14.4.4 Evaporation in a Column
`Mass Diffusion with Homogeneous Chemical Reactions
`Transient Diffusion
`References
`Problems
`
`THERMOPHYSICAL PROPERTIES OF MATTER
`
`MATHEMATICAL RELATIONS AND FUNCTIONS
`
`AN INTEGRAL LAMINAR BOUNDARY LAYER
`SOLUTION FOR PARALLEL FLOW OVER A FLAT
`
`
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`Page 12 of 98
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`
`
`
`/m - K) and contain
`rater is passed. Unde
`its in a uniform heat
`water flow providest
`fh = 5000 W/m -K
`‘ady-state temperature
`ynsiderations, we mij
`the preceding pag
`lated, use a finite-dt
`tures.
`
`Page 13 of 98
`
` Va _LIBRARYU.OFLURBANA-CHAMPAIGN
`
`
`TRANSIENT
`CONDUCTION
`
` CHAPTER?)
`
`
`
`
`
`
`
`Page 13 of 98
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`

`

`cated conditions. We began with thesimple ca
`state conduction with no internal generation
`ge
`complications due to multidimensional and
`chang
`have not yet considered situations for whict conditions
`Wenow recognize that many heat transfer problems aretime dependent
`Such unsteady, or transient, problemstypically arise when the bounday
`conditions of a system are changed. For example,if the surface temperature
`a system is altered, the temperature at each point in thesystem will also begin
`to change. The changes will continue to occuruntil a steady-state temperatutt
`distribution is reached. Consider a hot metal billet that is removed from:
`furnace and exposed to a cool airstream. Energy is transferred by convection
`andradiation from its surface to the surroundings. Energytransfer by condit
`tion also occurs from the interior of the metal
`to the surface, and the
`temperature at each point in the billet decreases until a steady-state conditiot
`is reached. Such time-dependenteffects occur in many industrial heating an!
`cooling processes.
`.
`To determine the time dependence of the temperaturedistribution withis
`a solid during a transient process, we could begin by solving the appropmi
`formof the heat equation, for example, Equation 2.13. Some cases for wh
`solutions have been obtained are discussed in Sections 5.4 to 5.8. Howett
`such solutions are often difficult to obtain, and where possible a 5
`approachis preferred. One such approach may be used under conditions for
`which temperature gradients within the solid are small. It is termed the
`capacitance method.
`
`Cooling of a hot Metal forging.
`
`t<0
`T=T
`
`5.1 THE LUMPED CAPACITANCE METHOD
`* simple, yet common, transient conduction problem is one in which a sold
`experiences a sudden
`Coram aes# sudden change inits thermal environment. Consider a bt
`immersing
`;
`‘S mitially at a uniform temperature 7, and is quem
`quenchine n:2 liuid of lower temperature T,, < 7, (Figure 51). ¥
`hing is said to begin at time 1 = 0, the temperature of the solid
`
`‘
`
`
`
`Page 14 of 98
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`

`

`
`
`5.1 The Lumped Capacitance Method
`
`227
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`decrease for time ¢ > 0, until it eventually reaches T,,. This reduction is due to
`convection heat
`transfer at
`the solid—liquid interface. The essence of the
`lumped capacitance method is the assumptionthat thetemperature of the
`solid is spatiallyuniform at any instant during the transient process. This
`assumptionimplies that temperature gradients within the solid are negligible.
`From Fourier’s law, heatconductionin theabsence of a temperature
`gradient implies the existenceof infinite thermal conductivity. Such a condi-
`tion is clearlyimpossible. 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 assumethatthis 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 mustrelate the rate of heatloss at
`the surface to the rate of change of the internal energy. Applying Equation
`L.1la to the control volume of Figure 5.1, this requirement takes the form
`= p,t a solid
`(5.1)
`
`or
`
`|
`
`aaeARYU.OF|.URBANA-CHAMPAIGN
`
`«|
`
`te
`Si
`ae
`
`dT
`~hA,(T — T,,) = pVe—-
`dt
`Introducing the temperature difference
` @=T-T
`See ises
`fe
`
`G2)
`
`G3
`
`;
`
`and recognizing that (d@/dt) = (dT/dt), it follows that
`eVe dé
`os.”
`Separating variables and integrating from the initial condition, forwhich i= 0
`and T(0) = T., we then obtain
`pVc
`-9dé
`;
`hA,
`J, @ ~ fa
`
`where
`
`
`
`more compl.
`sional, steady-
`itly considered
`. However, we
`with time.
`me dependent.
`the boundary
`temperature of
`will also begin
`te temperature
`moved from 4
`by convection
`fer by conduc
`‘face, and the
`state condition
`al heating and
`
`ribution withit
`he appropriate
`‘ases for
`5.8. Hower,
`ible a simple
`conditions for
`ned the hompel
`
`
`
`
`
`Wenham
`5 Valuating the integrals it follows that
`
`
`
`
`
`
`aye ae pk
`
`ls
`
`lie
`
`‘
`
`Page 15 of 98
`
`
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`Page 15 of 98
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`

`

`228
`
`Chapter 5 Transient Conduction
`
`iaad oe slowly to changes in its thermal environment andwill
`
`Equation 5.5 may be used to determine the time roigaired for the solid to read
`some temperature T, or, conversely, Equation 5.6 may be used to computetht
`temperature reached by the solid at some time 1.
`:
`_, he foregoing results indicate that the difference between the sold 2
`fluid temperatures must decay exponentially to zero as t approaches inlini
`This
`behavioris shown in Figure 5.2. From Equation 5.6 it is also evidenttt!
`
`Te1 7,2 3 Tea
`Figure 5.2 Transient temperature response of
`lumped capacitance solids corresponding to
`different thermal time constants 7:
`or
`
`where R, recta ane 10 convection heat transfer and C, is the /wm
`“4pacitance of
`the solid. Any increase in R, or C, wileae
`
`
`
`Page 16 of 98
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`

`

`|
`
`aetdehee
`
`ietT
`an|
`H74
`e)
`
`ai
`
`increases.
`
`S2 VALIDITY OF THE LUMPED CAPACITANCE METHOD
`From the foregoing results it is easy to see whythere is a strong preference for
`using the lumped capacitance method. It is certainly the simplest and most
`convenient method that can be used to solve transient conduction problems.
`Hence it is important to determine under what conditions it may be usedwith
`
`.
`
`and the othersurface is exposed to
`Surface is maintained at a temperature T,
`4 fluid of temperature T,, < T. ,. The temperature of this surface will be some
`
`Page 17 of 98
`
`5.2 Validity of the Lumped Capacitance Method
`
`229
`
`
`
`Fie
`
`Figure 53 Equivalent thermal circuit for a
`lumped capacitance solid.
`To determine the total energy transfer Q occurring up to some time f, we
`simply write
`
`o- [ade = ha,f'0dt
`Substituting for @ from Equation 5.6 and integrating, we obtain
`
`
`
`
`
`The quantityQis, of course, related to the change in the internal energy of the
`solid, and from Equation 1.11b
`
`
`For quenching Q is
`positi
`‘ences
`a decrease in
`:
`positive and the solid experiences
`>
`in energy
`Equations 5.5, 5.6, and 5.8a also apply to situations where the solid is heated
`(8 <0), in which case Q is negative and the internal energy of the solid
`
`
`
`Page 17 of 98
`
`

`

`230
`
`)=6Chapter 5 Transient Conduction
`
`A
`
`or
`onGuantity (hL/k) appearing in Equation 5.9 is a dimensionlesspars
`JS termed the Biot number, and it plays a fundamental role #
`aa co that involve surface convection effects. According ©
`oo as illustrated in Figure 5.4, the Biot number provide +
`arie
`temperature drop in the solid relative to the temperall#
`erence
`between the surface and the fluid. Note especially the
`
`Figure5.4 Effect of Biot number on
`steady-state temperature distribution ini
`plane wall with surface convection.
`
`intermediate value, T,., for which T,, < T,, <T, ,. Hence under steady-at
`conditions the surface energy balance, Equation 1.12, reduces to
`kA
`FZ (Tia - T2) = hA(T,, - T,)
`
`= T, and experiences convection
`reais
`« < T. The problem may be
`Position sithaca We are interested in the temperature varia?
`(x, 1). This variation is a strong function of per
`
`
`
`Page 18 of 98
`
`

`

`
`
`5.2 Validity of the Lumped Capacitance Method
`
`231
`
` number on
`stributionins
`wvection.
`
`Bi>1
`Biz1
`Bi<1
`T= Tix, 4)
`T= Tix, ¢)
`T=Tit)
`Figure 5.5 Transient temperature distribution for different Biot numbers in a plane
`wall symmetrically cooled by convection.
`number, and three conditions are shown in Figure 5.5. For Bi <1 the
`temperature gradient in the solid is small and T(x, t) = T(t). Virtually all
`the temperature difference is between the solid and the fluid, and the solid
`temperature remains nearly uniform as it decreases to T,,. For moderate to
`large values of the Biot number, however, the temperature gradients within the
`solid are significant. Hence T = T(x, t). Note that for Bi > 1, the tempera-
`ture difference 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 inherentsimplicity renders it the preferred method for
`solving transient conduction problems. Hence, when confronted with such a
`problem, the very first thing that one should do is calculate the Biot number. It
`the following condition is satisfied
`
`
`
`
`the error associated with using the lumped capacitance method is small. For
`convenience, it is customary to define the characteristic length of Equation 5.10
`as the ratio of the solid’s volume to surface area, L. = V/A,. Such a definition
`facilitates calculation of L, for solids of complicated
`shape and reduces to the
`half-thickness L for a plane wall of thickness 2L (Figure 5.5), to r,/2 for a
`long cylinder, and to r,/3 for a sphere. However, if one wishes to implement
`the criterion in a conservative fashion, L, should be associated with the length
`scale corresponding to the maximum spatial temperature difference. Accord-
`ingly, for a symmetrically heated (or cooled) plane wall of thickness 21, L-
`Would remain equal to the half-thickness L. However, for a long cylinder or
`sphere, L. would equal the actual radius r,, rather than r,/2 or r,/3.
`
`,
`
` oe
`
`Page 19 of 98
`
`
`
`Page 19 of 98
`
`

`

`232>=Chapter5 Transient Conduction
`
`Finally, we note that, with L. = V/A,, the exponent of Equation 5.6 my
`be expressed as
`
`Schematic:
`
`
`ht
`hA,t
`pVc Y' pcL,
`
`
`¢t
`AL. k
`k pe L?
`
`AL, at
`kai?
`
`7, = 200°C
`h = 400 W/m K
`
`Gas stream
`
`Assumptions:
`
`Radiation
`
`Pwn Temperati
`
`Losses by
`Constant
`
`
`
`
`
`
`
`
`
`
`
`
`
`a, fsaPe NagsAe =
`rs
`Eee
`
`
`is termed the Fourier number.It is a dimensionless time, which, with the Bit
`number, characterizes transient conduction problems. Substituting Equatiot
`5.11 into 5.6, we obtain
`
`Gh:
`Gee 7.
`7 Faz 7 TP (-B- Fo)
`(5.8)
`
`
`
`
`ae junction, which may be approximated as a sphere, is 10
`Or temperature measurementin a gas stream. The convection
`
`
`jag
`ths
`stingsn Surface and the gas is known to be h = 400 W/ati
`kg: Keanduermophysical properties are k = 20 W/m~K, «= #0
`the
`Bed kg/m.
`ine the junction diameter geol
`is placed in a
`ve a time constantof1 s. If the junctionis at 27°C
`
`
`to reach 199°
`am that is at 200°C, how longwill it take for thejuni
`
`
`sure temperature of aiProperties of thermocouplejunction used tomea
`
`
`
`Analysis:
`
`1, Because t
`the soluti
`capacitan
`approach
`determin
`fact that
`
`
`
`EXAMPLE 5.1
`
`SOLUTION
`
`Known: Thermophys;
`
`;
`
`equired
`2. Ti
`at
`
`"
`
`1 199°C in gas stream at 200°C.
`
`Accord
`L=ra
`excellen|
`
`Page 20 of 98
`
`
`
`Page 20 of 98
`
`

`

`
`
`
`
`
`
`
`
`
`
`
`
`6 X 400W/m? -K X18 oagese
`6he,
`
`pe 8500 kg/m x 400 J/kg -K
`
`With L. = r,/3 it thenfollows fromEquation5.10that
`aha 400W/m?-K x3.53X107m=235.0
`
`3 x20W/m-kK
`sopranaatbeientte0 Wag
`-— Accordingly, igs Sueiguhe&
`
`
`5.2 Validity of the Lumped Capacitance Method
`
`233
`
`
`
`mk \\ /]
`Thermocouple, & = 20 W/m-K
`quae:
`junction
`c = 400 J/kg K
`
`T, = 25°C|» = 8500 kg/m’
`
`Assumptions:
`
`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.
`
`Analysis:
`
`1. Because the junction diameter is unknown, it is not possible to begin
`the solution by determining whether the criterion for using the lumped
`approach is to use the method to find the diameter and to then
`determine whether thecriterionis satisfied. FromEquation5.7 andthe
`fact that A, = 7D? and V = 7D°/6 for a sphere, it follows that
`1
`a paD?
`,i
`‘
` hrD?
`Gay
`
`Rearranging and substituting numericalvalues,
`
`
`
`Bi =
`
`Page 21 of 98
`
`
`
`
`
`Page 21 of 98
`
`

`

`ee
`
`Chapter 5 Transient Conduction
`
`ae
`
`2. From Equation 5.5 the time required
`199°C is
`
`‘aneously influenced by convection, radiation, an applied
`
`53 GENERAL LUMPED CAPACITANCE ANALYSIS
`Although transient conduction in a solid iscommonly initiated by convection
`heat ‘transfer to or from an adjoining fluid, other processes mayi
`‘ransient thermal conditions within the solid. For example, a solid may ®
`ake — large surroundings by a gas or vacuum.If the temperatures0
`cs and
`surroundings differ, radiation exchange could cause the internal
`“nergy, and hence the temperature, of the solid to change. Tempe
`4
`j
`of the eee also be induced by applying a heat fiux at a portion, ofu
`
`ee6 FT DeRar
`Sine je f=?) 6h TT,
`
`8500 kg/m? x 7.06 x 10-* m x400J/kg - K
`J
`ai
`
`f=35.2s = 51,
`Comments: Heatlosses due to radiation exchange between the junction
`and the surroundings and conduction throughtheleads would necessitate
`using a smaller junction diameter to achieve the desired time response.
`
`ture ch
`
`mayin opt . Situation for which thermal conditions within a solid
`
`
`
`Page 22 of 98
`
`

`

`assailed daaT.
`om
`eu!
`
`5.3. General Lumped Capacitance Analysis
`
`235
`
`
`
`heat flux, and internal energy generation.Itis presumed that, initially (¢ = 0),
`the temperature of the solid (7;) differs from that of the fluid, T,,, and the
`surroundings, T,,,, and that both surface and volumetric heating (q;’ and 4)
`are initiated. The imposed heat flux q/ and the convection—radiation heat
`transfer occur at mutually exclusive portions of the surface, Agyy and Age, ,)>
`respectively, and convection-radiation transfer is presumed to be from the
`surface. Applying conservation of energy at any instant 1, it follows from
`Equation 1.11a that
`aT
`r
`<.
`q; A, + E, i oer t Gra)Aste,r) at ree.
`
`(5.14)
`
`or, from Equations 1.3a and 1.7,
`dT
`2
`avA, , + E,— [h(T - T,.) + e0(T* - Te,)|Anen = VEG (5-15)
`Unfortunately, Equation 5.15 is a nonlinear, first-order, nonhomoge-
`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
`
`5.17)
`
`heh
`
`solution maybeobtainedby dicrefizing whetime
`
`
`
`
`
`
`
`
`
`(5.16)
`aTpo = ~eA,,(Tt ~ Tex)
`
`Separating variables and integrating from the initial condition to any time t, it
`follows that
`
`;
`I, Teena:
`pVe I,
`x
`See
`aT
`eA, 7 pt
`
`Evaluating both integrals and rearranging, the time required toreach the
`temperature T becomes
`
`
`Tyo + Ti
`: ESF
`pve
`fg
`
`
`
`WA,af \"\Fe =F] Boe
`
`+2|tan~* apa S
`
`| Thisexpression cannotbeusedtoevaluate Texplicitlyinterms of,7), and
`
`
`
`
`Tr
`
`niue-diference
`
`«Am
`
`amprosimete
`
`Page 23 of 98
`
`
`
`
`
`Page 23 of 98
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`

`

`236)
`
`«Chapter 5 Transient Conduction
`
`eee
`f
`ra
`
`“a
`
`(5.13)
`
`An exact solution to Equation 5.15 mayalsobe obtainedif radiation my
`be neglected and h is independent of time. Introducing areduced temperatur
`@=T ~ T,,, where d6/dt = dT/dt, Equation
`5.15
`reduces to a linear, fint
`order, nonhomogeneous differential equationof theform
`d6
`42
`—+a0-b=0
`dt
`
`(5.2)
`
`where a =(hA,./pVc) and b = ((q'A, , + E,)/pVc]. Although Equation
`5.20 may be solved by summingits homogeneous and particular solutions
`alternative approach is to eliminate the nonhomogeneityby introducing the
`transformation
`
`a
`
`(5.2)
`reese?
`Recognizing that 0’/dt = d0/dt, Equation 5.21 may be substituted ilo
`(5.20) to yield
`
`—
`
`“State conditions.
`
`T~T
`
`akSe sn b/a 5.25)
`a exp ( @) + FzIl - exp (-at)]
`(
`AS it must, Equation 5.25 Teduces to
`(5.6) when b = 0 andyields T=!
`Equation 5.25 reduces to (T — T,,) = (6/2). which
`ing an energy balance on the control surface
`
`dé’
`7 +26’
`(
`=0
`oe
`521
`Separating variables and integrating from 0 to f (8 to 6’), it follows that
`4’
`7 7% (—ar)
`ae
`
`Or substituting for @” and 6,
`Em Ei i (b/a)
`
`TT,=(yay7° (-a)
`Hence.
`
`
`
`~
`
`2
`
`ee
`
`
`
`Page 24 of 98
`
`

`

`
`
`54 SPATIAL EFFECTS
`
`5.4 Spatial Effects
`
`
`
`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 thevariation 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
`generatio