Page 1 of 92
`
`Samsung Exhibit 1007
`Samsung Electronics Co., Ltd. v. Daniel L. Flamm
`
`

`
`Page 2 of 92
`
`

`
`THIRD EDITION
`
`FUNDAMENTALS OF
`
`HEAT AND
`
`MASS TRANSFER
`
`FRANK P. INCROPERA
`
`DAVID P. DEWITT
`
`School of Mechanical Engineering
`Purdue University
`
`WILEY
`
`JOHN WILEY & SONS
`
`New York ° Chichester - Brisbane - Toronto - Singapore
`
`Page 3 of 92
`
`

`
`Dedicated to those wonderful women in our lives,
`
`Amy, Andrea, Debbie, Donna, Jody
`Karen, Shaunna, and Terri
`'
`
`who, through the years, have blessed us with
`their love, patience, and understanding.
`
`Copyright © 1981, 1985, 1990, by John Wiley & Sons, Inc.
`
`All rights reserved. Published simultaneously in Canada.
`
`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 & fions.
`
`_
`
`\ A
`
`‘A
`
`Library of Congress Cataloging in Publication‘: Data:
`Incropera, Frank P.
`M“
`Fundamentals of heat and mass transfer/Frank P. Incropera, David
`P. DeWitt.—3rd ed.
`
`n
`
`,
`
`cm.
`p.
`Includes bibliographical references.
`ISBN 0-471-61246-4
`1. I-Ieat—Tra11smission.
`1934-.
`II. Title.
`
`2. Mass transfer.
`
`1990
`QC320.I45
`6Z1.402’2—de2fl
`Printed in the United States of America
`
`10 9 8 7 6 5 4 3 2 1
`
`I. DeWitt, David P.,
`
`89-38319
`CIP
`
`Page 4 of 92
`
`

`
`K) and con '
`is passed. U
`1 a uniform heat
`,
`W provi
`=
`0 W/m -
`state temperature
`
`arations, we may
`preceding page,
`., use a finite-dif-
`
`TRANSIENT T
`
` CONDUCTION
`
`Page 5 of 92
`
`

`
`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 rrgiltidimensional 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 steady-state temperature
`distribution is reached. Consider a hot metal billet that is removed from a
`furnace and exposed to a cool airstream. Energy is transferred by convection
`and radiation from its surface to the surroundings. Energy transfer byaconduc-
`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 condition
`is reached. Such time-dependent effects occur in many industrial heating and
`cooling processes.
`To determine the time dependence of the temperature" distribution within
`a solid during a transient process, we could begin by solving the appropriate
`form of the heat equation, for example, Equation 2.13. Some cases for which
`solutions have been obtained are discussed in Sections 5.4 to 5.8. However,
`such solutions are often diflicult
`to obtain, and where possible a simpler
`approach is preferred. One such approach may be used under conditions for
`which temperature gradients within the solid are small. It is termed the lumped
`capacitance method.
`
`3,
`’
`
`.
`
`.
`
`,
`
`C
`
`decrease
`convecti<
`lumped
`solid is
`assumpt
`Fro
`gradient
`tion is c
`exactly,
`solid is
`and its
`
`‘
`:
`
`3
`
`In ,
`conside:
`the trar
`energy‘
`the sur:
`1.11a t(
`
`§
`3
`‘
`
`'
`
`1;
`‘- ~-
`
`II1t1‘QCll
`
`5.1 THE LUMPED CAPACITANCE METHOD
`
`A simple, yet common, transient conduction problem is one in which a solid
`experiences a sudden change in its thermal environment. Consider a hot metal
`forging that is initially at a uniform temperature T, and is quenched by
`immersing it in a liquid of lower temperature Too < T, (Figure 5.1). If the
`quenching is said to begin at time t = O, the temperature of the solid will
`
`Figure 5.1 Cooling of a hot rnetal forging.
`
`Page 6 of 92
`
`

`
`rnore compli-
`ional, steady-
`ly considered
`However, we
`vith time.
`
`ie dependent.
`he boundary
`zmperature of
`vill also begin
`2 temperature
`moved from a
`
`3y convection
`er by conduc-
`'ace, and the
`tate condition
`
`L1 heating and
`
`bution within
`
`.e appropriate
`Lses for which
`5.8. However,
`ble a simpler
`zonditions for
`
`ed the lumped
`
`which a solid
`er a hot metal '
`
`quenched by
`re 5.1). If the
`the solid will
`
`5.1 The Luniped Capacitance Method
`
`227
`
`decrease for time t > 0, until it eventually reaches Too. This reduction is due to
`convection heat
`transfer at
`the solid~liquid interface. The essence of the
`lumped capacitance method is the assumption that the temperature of the
`solid is spatially uniform at any instant during the transient process. This
`assumption implies that temperature gradients within the solid are negligible.
`From Fourier’s law, heat conduction in the absence of a temperature
`gradient implies the existence of infinite thermal conductivity. Such a condi-
`tion is clearly 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
`l.1la to the control volume of Figure 5.1, this requirement takes the form
`
`_Eout =
`
`dT
`~hAS(T— Too) = pVc—d-t-
`
`Introducing the temperature difference
`
`(5.2)
`
`and recognizing that (d0/dt) = (dT/dt), it follows that
`
`pVc d0
`hAs dt "
`
`Separating variables and integrating from the initial condition, for which t = O
`and T(0) = T,-, we then obtain
`
`V
`
`5-‘:
`hAs
`
`d0
`
`°— = — far
`1
`9. 0
`0
`
`where
`
`Page 7 of 92
`
`

`
`228
`
`Chapter 5 Transient Conduction
`
`7't,4
`7't,_ 3
`73,2
`Tt,1
`Figure 5.2 Transient temperature response of
`lumped capacitance ‘solids corresponding to
`difierent thermal time constants r,.
`
`01'
`
`ill: ;9I T 7" Tee; in HTl1A§
`".‘.€:XP.
`L PVC,
`
`M
`
`tip
`
`:
`
`pg
`
`f (56),
`:::.pA
`
`Equation 5.5 may be used to determine the time required for the solid to reach
`some temperature T, or, conversely, Equation 5.6 may be used to compute the
`temperature reached by the solid at some time t.
`The foregoing results indicate that the difierence between the solid and
`fluid temperatures must decay exponentially to zero as t approaches infinity.
`This behavior is shown in Figure 5.2. From Equation 5.6 it is also evident that
`the quantity (pVc/hAs) may be interpreted as a thermal time constant. This
`time constant may be expressed as
`
`t
`
`5,
`
`is the lumped
`is the resistance to convection heat transfer and C,
`where R ,
`thermal capacitance of the solid. Any increase in R , or C, will cause a solid to
`respond more slowly to changes in its thermal environment and will increase
`the timerequired to reach thermal equilibrium (0 = 0).
`It is useful to note that the foregoing behavior is analogous to the voltage
`decay that occurs when a capacitor is discharged through a resistor in an
`electrical RC circuit. Accordingly,
`the process may be represented by an
`equivalent thermal circuit, which is shown in Figure 5.3. With the switchclosed
`the solid is charged to the temperature 0,. When the switch is opened, the
`energy that is stored in the solid is discharged through the thermal resistance
`and the temperature of the solid decays with time. This analogy suggests that
`RC electrical circuits may be used to determine the transient behavior of
`thermal systems. In fact, before the advent of digital computers, RC circuits
`were widely used to simulate transient thermal behavior.
`
`
`
`
`
`
`
`
`
`py'_Imm$y,rW«;,Mg',?L,,,,."‘,;,9t4'¢‘t"“_"""W‘-“~'-h+m.'.~.«:,,......a..,x,,.t.I-sA.':t.V
`
`
`
`
`
`
`
`
`
`
`
`The qt
`solid,
`
`For iq
`Equat
`(0 <1 s
`increa
`
`5.2 VAL
`
`From
`
`using
`
`conve 3
`Henc:
`F6380]
`
`v
`
`F]
`
`,
`
`the pi
`condi
`surfat
`a flui
`
`Page 8 of 92
`
`

`
`5.2 Validity of the Lumped Capacitance Method
`
`229
`
`Figure 5.3 Equivalent thermal circuit for a
`lumped capacitance solid.
`
`To determine the total energy transfer Q occurring up to some time 2‘, we
`simply write
`
`Q = jgqdt = hAsf0t0dt
`
`Substituting for 0 from Equation 5.6 and integrating, we obtain
`
`The quantity Q is, of course, related to the change in the internal energy of the
`solid, and from Equation 1.11b
`
`For quenching Q is positive and the solid experiences a decrease in energy.
`Equations 5.5. 5.6, and 5.8a also apply to situations where the solid is heated
`(6 < O), in which case Q is negative and the internal energy of the solid
`increases.
`
`5.2 VALIDITY OF THE LUMPED CAPACITANCE NEETHOD
`
`From the foregoing results it is easy to see Why there is a strong preference for
`usingithe 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 used with
`reasonable accuracy.
`'
`'
`To develop a suitable criterion consider steady-state conduction through
`the plane wall of area A (Figure 5.4). Although We are assuming steady-state
`conditions,
`this criterion is readily extended to transient processes. One
`surface is maintained at a temperature T“ and the other surface is exposed to
`a fluid of temperature Tm < TS,1. The temperature of this surface will be some
`
`. .6) a
`
`solid to reach
`
`compute the
`
`;he solid and
`
`ches infinity.
`> evident that
`onstant. This
`
`s the lumped
`JSC a solid to
`will increase
`
`0 the voltage
`esistor in an
`
`ented by an
`switch‘ closed
`opened, the
`ial resistance
`
`suggests that
`behavior of
`RC circuits
`
`Page 9 of 92
`
`

`
`230
`
`Chapter 5 Transient Conduction
`
`Tm ’ h
`
`I’
`
`*
`
`Figure 5.4 Effect of Biot number on
`steady-state temperature distribution in a
`' plane wall with surface convection.
`
`intermediate value, Tsiz, for which Too < 3,2 < T“. Hence under steady-state
`conditions the surface energy balance, Equation 1.12, reduces to
`kA
`"Z‘(7l,1 “ 71,2) = hA(Ts,2 ‘ T)00
`
`where k is the thermal conductivity of the solid. Rearranging, we then obtain
`
`coud 5
`=<--—, 2-: Bi
`T..."(1/hA.)::l."R5 5 ink
`'
`: conv”
`
`M
`
`p
`
`:
`
`(5.9) "
`
`The quantity (hL/k) appearing in Equation 5.9 is a dimensionless param-
`eter. It
`is termed the Biot number, and it plays a fundamental role in
`conduction problems that involve surface convection effects. According to
`Equation 5.9 and’ as illustrated in Figure 5.4,
`the Biot number provides a
`measure of the temperature drop in the solid relative to the temperature
`difference between the surface and the fluid. Note especially the conditions
`corresponding to Bi << 1. The results suggest that, for these conditions, it is
`reasonable to assume a uniform temperature distribution across a solid at any
`time during a transient process. This result may also be associated with
`interpretation of the Biot numberas a ratio of thermal resistances, Equation
`5.9. If Bi << 1, the resistance to conduction within the solid is much less than the
`resistance to convection across the fluid boundary layer. Hence the assumption of
`a uniform temperature distribution is reasonable.
`We have introduced the Biot number because of its significance to
`transient conduction problems. Consider the plane wall of Figure 5.5, which is
`initially at a uniform temperature T, and experiences convection cooling when
`it is immersed in a fluid of Too < 1}. The problem may be treated as one
`dimensional in x, and we are interested in the temperature variation with
`position and time, T(x, t). This variation is a strong function of the Biot
`
`Page 10 of 92
`
`

`
`5.2 Validity of the Lumped Capacitance Method
`
`231
`
`umber on
`ibution in a
`ction.
`
`;teady-state
`
`Lhen obtain
`
`
`
`(5.9) :
`
`less param-
`:al role in
`
`cording to
`provides a
`smperature
`conditions
`
`itions, it is
`
`olid at any w
`iated with
`
`, Equation
`Iss than the
`
`imption of
`
`ificance to
`
`5, which is
`
`Jljng when
`:ed as one
`ation with
`
`f the Biot
`
`T: T(t)
`
`T = The, t)
`
`T = T(x, t)
`
`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, z) z 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 Tea. 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 inherent simplicity 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. If
`the following condition is satisfied
`'
`‘
`
` i hL
`‘ ,‘;<_o.1 ‘
`Bi_-‘-7
`
`,
`
`l
`
`V
`
`T
`
`.
`
`l -
`
`{(5.10),
`
`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, LC —=— V/As. Such a definition
`facilitates calculation of LC for solids of complicated shape and reduces to the
`half—thickness L for a plane Wall of thickness 2L (Figure 5.5), to r0/2 for a
`long cylinder, and to r,,/3 for a sphere. However, if one wishes to implement
`the criterion in a conservative fashion, LC 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 2L, LC
`would remain equal to the half-thickness L. However, for a long cylinder or
`sphere, LC would equal the actual radius ro, rather than ro/2 or r,,/3.
`
`Page 11 of 92
`
`

`
`232
`
`Chapter 5 Transient Conduction
`
`Finally, we note that, with LC 2 V/As, the exponent of Equation 5.6 may
`be expressed as
`
`hASt
`
`pVc
`
`ht
`
`p CL‘,
`
`MrS
`PVC = '
`
`- F0,
`
`Where
`
`'
`
`(5.11)
`
`‘ j p
`
`H
`
`cg
`
`if L’
`
`[M
`
`M
`
`‘p
`
`is termed the Fourier number. It is a dimensionless time, which, with the Biot
`number, characterizes transient conduction problems. Substituting Equation
`5.11 into 5.6, we obtain
`1
`
`“
`r—2;°
`0
`-0-’: T_ T =exp(—Bz-F0)
`
`(5.13)
`
`EXAMPLE 5.1
`
`A thermocouple junction, which may be approximated as a sphere, is to be
`used for temperature measurement in a gas stream. The convection coefficient
`between the junction surface and the gas is known to be h = 4004 W/m2 ‘ K,
`and the junction thermophysical properties are k = 20 W/m - K, c = 400 A
`J/kg - K, and p = 8500 kg/m3. Determine the junction diameter needed for
`the thermocouple to have a time constant of 1 s. If the junction is at 25°C and
`is placed in a gas stream that is at 200°C, how long will it take for the junction
`to reach 199°C?
`
`SOLUTION
`
`Known: Thermophysical properties of thermocouple junction used to mea-
`sure temperature of a gas stream.
`
`Find:
`
`1.
`Junction diameter needed for a time constant of 1 s.
`2. Time required to reach 199°C in gas stream at 200°C.
`
`Page 12 of 92
`
`

`
`tion 5.6 may
`
`Schematic:
`
`5.2 Validity of the Lumped Capacitance Method
`
`233
`
`Thermocouple k = 20 W/m - K
`junction
`c = 400 J/kg- K
`T; = 25 °C
`p = 8500 kg/m3
`
`(5.11)
`
`5
`
`5
`Am the mot
`1g Equation
`
`(5.13)
`
`Assumptions:
`
`L
`
`_
`
`i
`
`1
`
`it
`
`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:
`
`are’ is ‘to be
`n coefficient
`W/ID2 ' K,
`K, c = 400
`' needed for
`at 25°C and
`the junction,
`
`_
`
`yj
`5
`
`5
`
`Because the junction diameter is unknown, it is not possible to begin
`the solution by determining whether the criterion for using the lumped
`capacitance method, Equation 5.10, is satisfied. However, a reasonable
`approach 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 A3 = 7rD2 and V = 77D 3/6 for a sphere, it follows that
`
`T! = -
`
`1
`
`IWD
`
`D3
`
`2 X P”
`6
`
`C
`
`Rearrangmg and substituting numerical values,
`
`6hr,
`pc
`
`6><400.W/m2-K><1s
`= ———~——-—— = 7.06
`ssookg/m3><400 J/kg-K
`
`10-4
`_
`
`m
`
`X
`
`<1
`
`With L6 = r0/3 it then follows from Equation 5.10 that
`
`B‘ —
`’
`
`h(ro/3)
`k
`
`400 W/ml - K x 3.53 x 104 m
`>
`3 x 20 W/m - K
`
`= 2.35
`
`10‘4
`
`X
`
`Accordingly, Equation 5.10’ is satisfied (for L0 = r0, as well as for
`LC = r0/3) and the lumped capacitance method may be used to an
`excellent approximation.
`
`Page 13 of 92
`
`

`
`234
`
`Chapter 5 Transient Conduction
`
`2. From Equation 5.5 the time required for the junction to reach T =
`199°C is
`
`P(7TD3/(5)6
`t = —-——————ln
`h(7rD2)
`
`Ti ‘
`T—~ Tm
`
`PDC
`= ——ln
`6h
`
`Ti ” Too
`T— Too
`
`3500 kg/m3 x 7.06 x 10*‘ m x 400 J/kg - K
`
`t_—_'
`
`6 x 400 W/m2 - K
`
`t=5.2s==5r,
`
`1
`
`25 — 200
`n __.___
`199 - 200
`
`<1
`
`5
`
`‘
`
`°
`
`it
`,
`
`A
`
`L
`
`Comments: Heat losses due to radiation exchange between the -junction
`and the surroundings and conduction through the leads would necessitate
`using a smaller junction diameter to achieve the desired time response.
`
`5.3 GENERAL LUMPED CAPACITANCE ANALYSIS
`Although transient conduction in a solid is commonly initiated by convection
`heat
`transfer to or from an adjoining fluid, other processes may induce
`transient thermal conditions within the solid. For example, a solid may be
`separated from large surroundings by a gas or vacuum. If the temperatures of
`the solid and surroundings differ, radiation exchange could cause the internal
`thermal energy, and hence the temperature, of the solid to change. Tempera-
`ture changes could also be induced by applying a heat flux at a portion, or all,
`of» the surface and/or by initiating thermal energy generation within the solid.
`Surface heating could, for example, be applied by attaching a film or sheet
`electrical heater to the surface, while thermal energy could be generated by
`passing an electrical current through the solid.
`Figure 5.6 depicts a situation for which thermal conditions within a solid
`may be simultaneously influenced by convection, radiation, an applied surface
`
`‘
`
`‘
`
`i
`
`Surroundings
`
`1
`heat fir
`;
`the tel
`surrou =
`are in}
`T1
`transfc
`respec
`.
`surfacc
`Equal]
`
`11
`
`eous ;_
`exact
`Iv
`Vet510
`a
`gener:
`Elam _
`5
`
`Figure 5.6 Contral surface for general
`lumped capacitance analysis.
`
`Page 14 of 92
`
`

`
`53 General Lumped Capacitance Analysis
`
`235
`
`heat flux, and internal energy generation. It is presumed that, initially (t = O),
`the temperature of the solid (T,) differs from that of the fluid, T00, and the
`surroundings, TM, and that both surface and volumetric heating (q;’ and q)
`are initiated. The imposed heat flux q;’ and the convection—radiation heat
`transfer occur at mutually exclusive portions of the surface, As“) and AS“, r),
`respectively, 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
`
`qi‘/As, h + E-g — (qt/:bnv + q;lad)As(c, r) =
`
`dT
`
`or, from Equations 1.3a and 1.7,
`
`q;/A,,,, + E", — [h(T — Too) + ea(T4 4 T3,
`
`first—order, 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
`
`a'T
`pl/C71‘ = —eAS,,cr(T4 -— 7:3,)
`
`(5.16)
`
`Separating variables and integrating from the initial condition to any time 2‘, it
`follows that
`
`5
`
`A ,
`dT
`S‘ U]-1-dt = ‘I-7:7-j-Z
`I
`SUI‘
`pVc
`0
`T. T - T
`
`(5 .17)
`
`Evaluating both integrals and rearranging,
`temperature T becomes
`
`the time required to reach the
`
`<1
`
`junction
`:cessitate
`)11S6.
`
`' convection
`
`nay induce
`lid may be
`teratures of
`:he internal
`
`. Tempera-
`tion, or all,
`n the solid.
`In or sheet
`
`nerated by
`
`thin a solid
`ied surface
`
`1‘
`
`pVc
`= 43A 0T3
`V
`Slll‘
`
`{
`
`1”
`
`T—TS111‘
`
`TSur+T
`+2[tan'1
`
`Tsur+Ti
`_1n_____
`S11!‘
`1
`T -1".
`
`- tan”
`
`(5.18)
`
`This expression cannot be used to evaluate T explicitly in terms ‘of t, T,., and
`SUI’ ’
`T
`nor does it readily reduce to the limiting result for Tm = 0 (radiation to
`
`1 An approximate,‘finite-difference solution may be obtained by discretizing the time
`derivative (Section 5.9) and marching the solution out in time.
`-
`
`Page 15 of 92
`
`

`
`236
`
`Chapter 5 Transient Conduction
`
`t
`
`pVc
`
`1
`
`deep space). Returning to Equation 5.17, it is readily shown that, for Tm = O 7
`= 3.2 a(§~? ‘ F)
`(5-19)
`
`1
`
`5.4 SPA'l
`
`/Situati
`propri
`form(
`
`3
`
`*
`
`mediu 1
`
`I1
`by tl1€
`2.20 2
`solutii
`
`peratt
`proble (
`neede
`
`.
`
`gener:
`2.13 t
`
`An exact solution to Equation 5.15 may also be obtained if radiation may
`be neglected and h is independent of time. Introducing a reduced temperature,
`0 E T —— Tm, where d6/dt = dT/dz‘, Equation 5.15 reduces to a linear, first-
`order, nonhomogeneous differential equation of the form
`
`d0
`—C—l?+a0—b=0
`
`(5.20)
`
`where a 2 (hAM/pVc) and b 2 [(q;'AS, ,, + E3)/pVc]. Although Equation
`5.20 may be solved by summing its homogeneous and particular solutions, an
`alternative approach is to eliminate the nonhomogeneity by introducing the
`transformation
`
`0’ E 0 — —
`U
`
`(5.21)
`
`Recognizing that d0’/dt = d0/dt, Equation 5.21 may be substituted into
`(5.20) to yield
`
`am,
`—— +
`dt
`
`0
`
`‘'
`
`'=
`
`0
`
`5 22
`.
`
`)
`
`(
`
`Separating variables and integrating from 0 to I (6,? to 6’), it follows that
`0/
`E7 = exp (—at)
`
`(5.23)
`
`or substituting for 0’ and 9,
`
` _ (_ t)
`T.— Too — (b/a) “exp
`“
`
`Hence,
`
`T— Tm
`b/a
`Ti __ Tm = exp(—at) + Ti __ Too [1 - exp(-iat)]
`
`(5.25)
`
`As it must, Equation 5.25 reduces to (5.6) when b = O and yields T = Ti at
`t = 0. As t —-> oo, Equation 5.25 reduces to (T — TOO) = (b/a), which could
`also be obtained by performing an energy balance on the control surface of
`Figure 5.6 for steady—state conditions.
`
`Page 16 of 92
`
`

`
`S111’
`for T = 0,
`
`(5.19)
`
`idiation may
`temperature,
`linear, first-
`
`(5.20)
`
`;h Equation
`solutions, an
`
`roducing the
`
`(5.21)
`
`stituted into
`
`(5.22)
`
`ms ‘ that
`
`(5.23)
`
`(5.25)
`
`I
`.sT=T-at
`vhich could
`1 surface of
`
`5.4 SPATIAL EFFECTS
`
`5.4 Spatial Effects
`
`237
`
`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
`
`1 ar
`air
`9x2 " 01 3t
`
`(5 26)
`5'
`
`To solve Equation 5.26 for the temperature distribution T(x, t), it is
`necessary to specify an initial condition and two boundary conditions. For the
`typical transient conduction problem of Figure 5.5, the initial condition is
`
`T(x,0) = T,.
`
`.
`
`and the boundary conditions are
`
`6T
`——
`Bx
`
`(5.27)
`
`(5.28)
`
`(5.29)
`
`Equation 5.27 presumes a uniform temperature distribution at time t = 0;
`Equation 5.28 reflects the symmetry requirement for the midplane of the wall;
`and Equation 5.29 describes the surface condition experienced for time t > 0.
`From Equations 5.26 to 5.29, it is evident that, in addition to depending on x
`and t, temperatures in the wall also depend on a number of physical parame-
`ters. In particular
`
`vi:
`507
`:r= T(x,t T T L,k,a,h)
`
`'
`
`(5.30)
`
`The foregoing problem may be solved analytically or numerically. These
`methods will be considered in subsequent sections, but first it is important to
`note the advantages that may be obtained by nondimensionalizing the govern-
`
`Page 17 of 92
`
`

`
`238
`
`Chapter 5 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 0 E T ~ T00 is divided by the maximum possible temperature di erence
`01. E T’. — Tm, a dimensionless form of the dependent variable may be defined
`as
`
`k0*fT [*~f.=fZ;j
`
`:*=!°.°(
`
`y
`
`5
`
`3
`
`i
`
`j
`
`—
`
`’
`
`_l
`
`if
`
`5
`
`5.31
`
`Accordingly, 0* must lie in the range 0 s 30* s 1. A dimensionless spatial
`coordinate may be defined as
`
`where L is the half-thickness of the planewall, and a dimensionless time may
`be defined as
`
`r (5.33)
`
`where t* is equivalent to the dimensionless Fourier number, Equation 5.12.
`Substituting the definitions of Equations 5.31 to 5.33 into Equations 5.26
`to 5.29, the heat equation becomes
`
`6 20 *
`80*
`8x*2 = 6Fo
`
`3
`
`and the initial and boundary conditions become
`
`0*(x*,0) = 1
`60*
`
`(M
`
`x*=0
`
`J
`
`(5-34)
`
`(5.35)
`
`p
`
`p
`
`(5.36)
`
`and
`
`86*
`6 *
`X
`
`x*=1
`
`—Bi0*(1,t*)
`
`(5.37)
`
`where the Biot number is Bi 2 hL/k. In dimensionless form the functional
`dependence may now be expressed as
`
`0* =f(x*, F0, Bi)
`
`(5.38)
`
`Recall that this functional dependence, without the x* variation, was obtained
`for the lumped capacitance method, as shown in Equation 5.13.
`Comparing Equations 5.30 and 5.38, the considerable advantage associ-
`ated with casting the problem in dimensionless form becomes apparent.
`
`Page 18 of 92
`
`

`
`
`
`~yr-.«,.»ao.«.e.'jwmzs’%s~¢aéaJ;«a(w4a«e..;~M..s..sam..r..,.,.;'
`
`
`
`
`
`riables into
`ature differ-
`re di erence
`
`y be defined
`
`(5.31)
`
`1l6SS spatial
`
`5 (5.32)
`
`ss time may
`
`(5.33)
`
`tion 5.12.
`uations 5.26
`
`(5.34)
`
`(5.35)
`
`(5.36)
`
`(5.37)
`
`: functional
`
`mg)
`
`as obtained
`
`tage associ-
`3 apparent.
`
`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*, F0, and Bi. That is, the dimensionless
`solution assumes a prescribed form that does not depend on the particular
`value of T, Too, L, k, oz, or it. Since this generalization greatly simplifies the
`presentation and utilization of transient solutions, the dimensionless variables
`are used extensively in subsequent sections.
`
`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 [1—4]. Several mathematical techniques, including
`the method of separation of variables (Section 4.2), may be used for this
`purpose, and typically the solution for the dimensionless temperature distribu-
`tion, Equation 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
`single term and the results may be represented in a convenient graphical form.
`
`5.5.1 Exact Solution
`
`\\n‘
`<‘T:v<\‘~-.
`-5
`
`Consider the plane wall of thickness 2L (Figure 5.7a). If thethickness is small
`relative to the width and height of the wall, it is reasonable to assume that
`conduction occurs exclusively in the x direction. If the wall is initially at a
`uniform temperature, T(x,O) = T,., and is suddenly immersed in a fluid of
`Too #= T,., the resulting temperatures may be obtained by solving Equation 5.34
`subject
`to the conditions of Equations 5.35 to 5.37. Since the convection
`conditions for the surfaces at x* = i 1 are the same, the temperature distribu-
`tion at any instant must be symmetrical about the midplane (x* = 0). An
`
`T(r,O) = Ti
`
`ro
`
`iii
`
`Figure 5.7 One-dimensional systems with an initial uniform
`temperature subjected to sudden convection conditions. (a) Plane
`wall. (b) Infinite cylinder or sphere.
`
`(b)
`
`Page 19 of 92
`
`

`
`240
`
`Chapter 5 Transient Conduction
`
`exact solution to this problem has been obtained and is of the form [2]
`
`0* = 2 C” exp (— §n2Fo) cos (§,,x*)
`n=1
`I
`
`(5.39a)
`
`where the coefficient C" is
`
`C
`
`4 sin §,,
`
`and the discrete values (eigenvalues) of §,_ are positive roots of the transcen-
`dental equation
`
`fn tan {M = Bi
`
`(5.39c)
`
`The first four roots of this equation are given in Appendix B.3.
`
`5.5.2 Approximate Solution
`
`It can be shown (Problem 5.24) that for values of F0 2 0.2, the infinite series
`solution, Equation 5.39a, can be approximated by the first term of the series.
`Invoking this approximation,
`the dimensionless form of the temperature
`distribution becomes
`
`0* = C1exp(—§'12F0)cos(§1x*)
`
`0* = 0: cos (§1x*)
`
`where 0;‘ represents the midplane (x* = 0) temperature
`
`0; = C1exp(—§12Fo)
`
`(5.40a)
`
`(5.4%)
`
`(5.41)
`
`An important implication of Equation 5.40b is that the time dependence of the
`temperature at any location within the wall is the same as that of the midplane
`temperature. The coeflicients C1 and {I are evaluated from Equations 5.39b
`and 5.390, respectively, and are given in Table 5.1 for a range of Biot numbers.
`
`5.5.3 Total Energy Transfer
`
`In many situations it is useful to know the total energy that has left the wall
`up to any time t
`in the transient process. The conservation of energy
`requirement, Equation 1.11b, may be applied for the time interval bounded by
`the initial condition (if = O) and time t > 0 V
`
`Ein _ Eout = AEst
`
`Page 20 of 92
`
`

`
`Table 5.1 Coefiicients used in the one—term approximation
`to the series solutions for transient one-dimensional conduction
`
`PLANE WALL
`
`INFINITE
`CYLINDER
`
`SPHERE
`
`Bi"
`
`0.01
`0.02
`
`0.03
`0.04
`0.05
`0.06
`0.07
`0.08
`
`0.09
`
`0.10
`
`0.15
`
`Q
`(rad)
`
`0.0998
`0.1410
`
`0.1732
`0.1987
`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
`
`Q
`(rad)
`
`0.1412
`0.1995
`
`0.2439
`0.2814
`0.3142
`0.3438
`0.3708
`0.3960
`
`0.4195
`
`0.4417
`
`C1
`
`1.0025
`1.0050
`
`1.0075
`1.0099
`1.0124
`1.0148
`1.0173
`1.0197
`
`1.0222
`
`1.0246
`
`5
`(rad)
`
`0.1730
`0.2445
`
`0.2989
`0.3450
`0.3852
`0.4217
`0.4550
`0.4860
`
`0.5150
`
`0.5423
`
`0.3779
`
`1.0237
`
`0.5376
`
`1.0365‘
`
`0.6608
`
`Cl
`
`1.0030
`1.0060
`
`1.0090
`1.0120
`1.0149
`1.0179
`1.0209
`1.0239
`
`1.0268
`
`1.0298
`
`1.0445
`
`0.6
`
`0.7051
`
`1.0814
`
`1.0185
`
`1.1346
`
`1.2644
`
`1.1713
`
`(5.3%)
`
`1e transcen—
`
`(5390)
`
`lfinite Series
`.f the semi
`temperature
`
`(5.4021)
`
`6 1
`;]f{[V
`7 I
`I
`I
`
`2
`
`L
`
`jf._
`
`:7
`
`2
`
`0.20
`0.25
`0.30
`0.4
`0.5
`0.7
`0.8
`
`0.9
`
`1.0
`
`2.0
`3.0
`4.0
`
`5.0
`6.0
`7.0
`8.0
`9.0
`
`10.0
`20.0
`30.0
`40.0
`50.0
`
`0.4328
`0.4801
`0.5218
`0.5932
`0.6533
`0.7506
`0.7910
`
`0.8274
`
`0.8603
`
`1.0769
`1.1925
`1.2646
`
`1.3138
`1.3496
`1.3766
`1.3978
`1.4149
`
`1.4289
`1.4961
`1.5202
`1.5325
`1.5400
`
`1.0311
`1.0382
`1.0450
`1.0580
`1.0701
`1.0919
`1.1016
`
`1.1107
`
`1.1191
`
`1.1795
`1.2102
`1.2287
`
`1.2402
`1.2479
`1.2532
`» 1.2570
`1.2598
`
`1.2620
`1.2699
`1.2717
`1.2723
`1.2727
`
`0.6170
`0.6856
`0.7465
`0.8516
`0.9408
`1.0873
`1.1490
`
`1.2048
`
`1.2558
`
`1.5995
`1.7887 .
`1.9081
`
`1.9898
`2.0490
`2.0937
`2.1286
`2.1566
`
`2.1795
`2.2881
`2.3261
`2.3455
`2.3572
`
`1.0483
`1.0598
`1.0712
`1.0932
`1.1143
`1.1539
`1.1725
`
`1.1902
`
`1.2071
`
`1.3384
`1.4191
`1.4698
`
`1.5029
`1.5253
`1.5411
`1.5526
`1.5611
`
`1.5677
`1.5919
`1.5973
`1.5993
`1.6002
`
`0.7593
`0.8448
`0.9208
`1.0528 .
`1.1656
`1.3525
`14320
`
`1.5044
`
`1.5708
`
`2.0288
`2.2889
`_ 2.4556
`
`2.5704
`2.6537
`2.7165
`2.7654
`2.8044
`
`2.8363
`2.9857
`3.0372
`3.0632
`3.0788
`
`1.0592
`1.0737
`1.0880
`1.1164
`1.1441
`1.1978
`1.2236
`
`1.2488
`
`1.2732
`
`1.4793
`1.6227
`1.7201
`
`1.7870
`1.8338
`’ 1.8674
`1.8921
`1.9106
`
`1.9249
`1.9781
`1.9898
`1.9942
`1.9962
`
`100.0
`
`1.5552
`
`1.2731
`
`2.3809
`
`_
`
`1.6015
`
`3.1102
`
`1.9990
`
`“Bi = hL/k for the plane wall and hr‘,/k for the infinite cylinder and sphere. See Figure 5.7.
`
`~
`
`_
`2
` ”if}
`
`:
`
`.
`
`_
`
`.f;:
`
`'
`
`(5.4%)
`
`15 -41)
`
`..
`zdence of the
`he midplane
`mom 53%
`Qt numberS_
`
`1
`
`left the Wall
`L of emrgy
`bounded by.
`
`(5 .42)
`
`Page 21 of 92
`
`

`
`242
`
`Chapter 5 Transient Conduction
`
`Equating the energy transferred from the wall
`and AES‘ = E (t) — E (0), it follows that
`
`to
`
`out and setting Em = O
`
`Q = - [E(t) ~ E(0)l
`
`.
`
`Q = —]pc[T(r,z) — 1.] dV
`
`A
`
`(5~43a)
`
`(5.43b)
`
`where the integration is performed over the volume of the wall. It is conve-
`nient to nondimensionalize this result by introducing the quantity
`
`Q. = pcV(T.- - T...)
`
`(5-44)
`
`which may be interpreted as the initial internal energy of the wall relative to
`the fluid temperature. It is also the maximum amount of energy transfer which
`could occur if the process were continued to time t = oo. Hence, assuming
`constant properties, the ratio of the tot