Heat transfer has direction as well as magnitude. The rate of heat conduction

Transcription

1 cen58933_ch2.qd 9/1/22 8:46 AM Page 61 HEAT CONDUCTION EQUATION CHAPTER 2 Heat tansfe has diection as well as magnitude. The ate of heat conduction in a specified diection is popotional to the tempeatue gadient, which is the change in tempeatue pe unit length in that diection. Heat conduction in a medium, in geneal, is thee-dimensional and time dependent. That is, T T(, y, z, t) and the tempeatue in a medium vaies with position as well as time. Heat conduction in a medium is said to be steady when the tempeatue does not vay with time, and unsteady o tansient when it does. Heat conduction in a medium is said to be one-dimensional when conduction is significant in one dimension only and negligible in the othe two dimensions, two-dimensional when conduction in the thid dimension is negligible, and thee-dimensional when conduction in all dimensions is significant. We stat this chapte with a desciption of steady, unsteady, and multidimensional heat conduction. Then we deive the diffeential equation that govens heat conduction in a lage plane wall, a long cylinde, and a sphee, and genealize the esults to thee-dimensional cases in ectangula, cylindical, and spheical coodinates. Following a discussion of the bounday conditions, we pesent the fomulation of heat conduction poblems and thei solutions. Finally, we conside heat conduction poblems with vaiable themal conductivity. This chapte deals with the theoetical and mathematical aspects of heat conduction, and it can be coveed selectively, if desied, without causing a significant loss in continuity. The moe pactical aspects of heat conduction ae coveed in the following two chaptes. CONTENTS 2 1 Intoduction One-Dimensional Heat Conduction Equation Geneal Heat Conduction Equation Bounday and Initial Conditions Solution of Steady One-Dimensional Heat Conduction Poblems Heat Geneation in a Solid Vaiable Themal Conductivity k (T ) 14 Topic of Special Inteest: A Bief Review of Diffeential Equations 17 61

2 cen58933_ch2.qd 9/1/22 8:46 AM Page HEAT TRANSFER Hot baked potato Magnitude of tempeatue at a point A (no diection) 5 C 8 W/m 2 A Magnitude and diection of heat flu at the same point FIGURE 2 1 Heat tansfe has diection as well as magnitude, and thus it is a vecto quantity. Hot Cold medium medium Cold Hot medium medium Q = 5 W Q = 5 W FIGURE 2 2 Indicating diection fo heat tansfe (positive in the positive diection; negative in the negative diection). 2 1 INTRODUCTION In Chapte 1 heat conduction was defined as the tansfe of themal enegy fom the moe enegetic paticles of a medium to the adjacent less enegetic ones. It was stated that conduction can take place in liquids and gases as well as solids povided that thee is no bulk motion involved. Although heat tansfe and tempeatue ae closely elated, they ae of a diffeent natue. Unlike tempeatue, heat tansfe has diection as well as magnitude, and thus it is a vecto quantity (Fig. 2 1). Theefoe, we must specify both diection and magnitude in ode to descibe heat tansfe completely at a point. Fo eample, saying that the tempeatue on the inne suface of a wall is 18 C descibes the tempeatue at that location fully. But saying that the heat flu on that suface is 5 W/m 2 immediately pompts the question in what diection? We can answe this question by saying that heat conduction is towad the inside (indicating heat gain) o towad the outside (indicating heat loss). To avoid such questions, we can wok with a coodinate system and indicate diection with plus o minus signs. The geneally accepted convention is that heat tansfe in the positive diection of a coodinate ais is positive and in the opposite diection it is negative. Theefoe, a positive quantity indicates heat tansfe in the positive diection and a negative quantity indicates heat tansfe in the negative diection (Fig. 2 2). The diving foce fo any fom of heat tansfe is the tempeatue diffeence, and the lage the tempeatue diffeence, the lage the ate of heat tansfe. Some heat tansfe poblems in engineeing equie the detemination of the tempeatue distibution (the vaiation of tempeatue) thoughout the medium in ode to calculate some quantities of inteest such as the local heat tansfe ate, themal epansion, and themal stess at some citical locations at specified times. The specification of the tempeatue at a point in a medium fist equies the specification of the location of that point. This can be done by choosing a suitable coodinate system such as the ectangula, cylindical, o spheical coodinates, depending on the geomety involved, and a convenient efeence point (the oigin). The location of a point is specified as (, y, z) in ectangula coodinates, as (,, z) in cylindical coodinates, and as (,, ) in spheical coodinates, whee the distances, y, z, and and the angles and ae as shown in Figue 2 3. Then the tempeatue at a point (, y, z) at time t in ectangula coodinates is epessed as T(, y, z, t). The best coodinate system fo a given geomety is the one that descibes the sufaces of the geomety best. Fo eample, a paallelepiped is best descibed in ectangula coodinates since each suface can be descibed by a constant value of the -, y-, o z-coodinates. A cylinde is best suited fo cylindical coodinates since its lateal suface can be descibed by a constant value of the adius. Similaly, the entie oute suface of a spheical body can best be descibed by a constant value of the adius in spheical coodinates. Fo an abitaily shaped body, we nomally use ectangula coodinates since it is easie to deal with distances than with angles. The notation just descibed is also used to identify the vaiables involved in a heat tansfe poblem. Fo eample, the notation T(, y, z, t) implies that the tempeatue vaies with the space vaiables, y, and z as well as time. The

3 cen58933_ch2.qd 9/1/22 8:46 AM Page 63 z z 63 CHAPTER 2 z y P(, y, z) z y φ (a) Rectangula coodinates (b) Cylindical coodinates (c) Spheical coodinates P(, φ, z) z y φ θ P(, φθ, ) y FIGURE 2 3 The vaious distances and angles involved when descibing the location of a point in diffeent coodinate systems. notation T(), on the othe hand, indicates that the tempeatue vaies in the -diection only and thee is no vaiation with the othe two space coodinates o time. Steady vesus Tansient Heat Tansfe Heat tansfe poblems ae often classified as being steady (also called steadystate) o tansient (also called unsteady). The tem steady implies no change with time at any point within the medium, while tansient implies vaiation with time o time dependence. Theefoe, the tempeatue o heat flu emains unchanged with time duing steady heat tansfe though a medium at any location, although both quantities may vay fom one location to anothe (Fig. 2 4). Fo eample, heat tansfe though the walls of a house will be steady when the conditions inside the house and the outdoos emain constant fo seveal hous. But even in this case, the tempeatues on the inne and oute sufaces of the wall will be diffeent unless the tempeatues inside and outside the house ae the same. The cooling of an apple in a efigeato, on the othe hand, is a tansient heat tansfe pocess since the tempeatue at any fied point within the apple will change with time duing cooling. Duing tansient heat tansfe, the tempeatue nomally vaies with time as well as position. In the special case of vaiation with time but not with position, the tempeatue of the medium changes unifomly with time. Such heat tansfe systems ae called lumped systems. A small metal object such as a themocouple junction o a thin coppe wie, fo eample, can be analyzed as a lumped system duing a heating o cooling pocess. Most heat tansfe poblems encounteed in pactice ae tansient in natue, but they ae usually analyzed unde some pesumed steady conditions since steady pocesses ae easie to analyze, and they povide the answes to ou questions. Fo eample, heat tansfe though the walls and ceiling of a typical house is neve steady since the outdoo conditions such as the tempeatue, the speed and diection of the wind, the location of the sun, and so on, change constantly. The conditions in a typical house ae not so steady eithe. Theefoe, it is almost impossible to pefom a heat tansfe analysis of a house accuately. But then, do we eally need an in-depth heat tansfe analysis? If the Time = 2 PM 15 C 7 C 12 C (a) Tansient Q 1 15 C 7 C 15 C Q 1 Time = 5 PM 5 C 7 C Q 2 Q 1 Q 2 = Q 1 (b) Steady-state FIGURE 2 4 Steady and tansient heat conduction in a plane wall.

4 cen58933_ch2.qd 9/1/22 8:46 AM Page HEAT TRANSFER pupose of a heat tansfe analysis of a house is to detemine the pope size of a heate, which is usually the case, we need to know the maimum ate of heat loss fom the house, which is detemined by consideing the heat loss fom the house unde wost conditions fo an etended peiod of time, that is, duing steady opeation unde wost conditions. Theefoe, we can get the answe to ou question by doing a heat tansfe analysis unde steady conditions. If the heate is lage enough to keep the house wam unde the pesumed wost conditions, it is lage enough fo all conditions. The appoach descibed above is a common pactice in engineeing. 8 C 8 C 8 C z T(, y) y 7 C 7 C 7 C 65 C 65 C 65 C FIGURE 2 5 Two-dimensional heat tansfe in a long ectangula ba. Negligible heat tansfe Q y Q Pimay diection of heat tansfe FIGURE 2 6 Heat tansfe though the window of a house can be taken to be one-dimensional. Q Multidimensional Heat Tansfe Heat tansfe poblems ae also classified as being one-dimensional, twodimensional, o thee-dimensional, depending on the elative magnitudes of heat tansfe ates in diffeent diections and the level of accuacy desied. In the most geneal case, heat tansfe though a medium is thee-dimensional. That is, the tempeatue vaies along all thee pimay diections within the medium duing the heat tansfe pocess. The tempeatue distibution thoughout the medium at a specified time as well as the heat tansfe ate at any location in this geneal case can be descibed by a set of thee coodinates such as the, y, and z in the ectangula (o Catesian) coodinate system; the,, and z in the cylindical coodinate system; and the,, and in the spheical (o pola) coodinate system. The tempeatue distibution in this case is epessed as T(, y, z, t), T(,, z, t), and T(,,, t) in the espective coodinate systems. The tempeatue in a medium, in some cases, vaies mainly in two pimay diections, and the vaiation of tempeatue in the thid diection (and thus heat tansfe in that diection) is negligible. A heat tansfe poblem in that case is said to be two-dimensional. Fo eample, the steady tempeatue distibution in a long ba of ectangula coss section can be epessed as T(, y) if the tempeatue vaiation in the z-diection (along the ba) is negligible and thee is no change with time (Fig. 2 5). A heat tansfe poblem is said to be one-dimensional if the tempeatue in the medium vaies in one diection only and thus heat is tansfeed in one diection, and the vaiation of tempeatue and thus heat tansfe in othe diections ae negligible o zeo. Fo eample, heat tansfe though the glass of a window can be consideed to be one-dimensional since heat tansfe though the glass will occu pedominantly in one diection (the diection nomal to the suface of the glass) and heat tansfe in othe diections (fom one side edge to the othe and fom the top edge to the bottom) is negligible (Fig. 2 6). ikewise, heat tansfe though a hot wate pipe can be consideed to be onedimensional since heat tansfe though the pipe occus pedominantly in the adial diection fom the hot wate to the ambient, and heat tansfe along the pipe and along the cicumfeence of a coss section (z- and -diections) is typically negligible. Heat tansfe to an egg dopped into boiling wate is also nealy one-dimensional because of symmety. Heat will be tansfeed to the egg in this case in the adial diection, that is, along staight lines passing though the midpoint of the egg. We also mentioned in Chapte 1 that the ate of heat conduction though a medium in a specified diection (say, in the -diection) is popotional to the tempeatue diffeence acoss the medium and the aea nomal to the diection

5 cen58933_ch2.qd 9/1/22 8:46 AM Page 65 of heat tansfe, but is invesely popotional to the distance in that diection. This was epessed in the diffeential fom by Fouie s law of heat conduction fo one-dimensional heat conduction as T 65 CHAPTER 2 dt slope d < dt Q cond ka (W) (2-1) d whee k is the themal conductivity of the mateial, which is a measue of the ability of a mateial to conduct heat, and dt/d is the tempeatue gadient, which is the slope of the tempeatue cuve on a T- diagam (Fig. 2 7). The themal conductivity of a mateial, in geneal, vaies with tempeatue. But sufficiently accuate esults can be obtained by using a constant value fo themal conductivity at the aveage tempeatue. Heat is conducted in the diection of deceasing tempeatue, and thus the tempeatue gadient is negative when heat is conducted in the positive -diection. The negative sign in Eq. 2 1 ensues that heat tansfe in the positive -diection is a positive quantity. To obtain a geneal elation fo Fouie s law of heat conduction, conside a medium in which the tempeatue distibution is thee-dimensional. Figue 2 8 shows an isothemal suface in that medium. The heat flu vecto at a point P on this suface must be pependicula to the suface, and it must point in the diection of deceasing tempeatue. If n is the nomal of the isothemal suface at point P, the ate of heat conduction at that point can be epessed by Fouie s law as T Q n ka (W) (2-2) n T() Q > Heat flow FIGURE 2 7 The tempeatue gadient dt/d is simply the slope of the tempeatue cuve on a T- diagam. In ectangula coodinates, the heat conduction vecto can be epessed in tems of its components as Q n Q i Q y j Q z k (2-3) whee i, j, and k ae the unit vectos, and Q, Q y, and Q z ae the magnitudes of the heat tansfe ates in the -, y-, and z-diections, which again can be detemined fom Fouie s law as T T T Q ka, Q y ka y, and Q z ka z (2-4) y z Hee A, A y and A z ae heat conduction aeas nomal to the -, y-, and z-diections, espectively (Fig. 2 8). Most engineeing mateials ae isotopic in natue, and thus they have the same popeties in all diections. Fo such mateials we do not need to be concened about the vaiation of popeties with diection. But in anisotopic mateials such as the fibous o composite mateials, the popeties may change with diection. Fo eample, some of the popeties of wood along the gain ae diffeent than those in the diection nomal to the gain. In such cases the themal conductivity may need to be epessed as a tenso quantity to account fo the vaiation with diection. The teatment of such advanced topics is beyond the scope of this tet, and we will assume the themal conductivity of a mateial to be independent of diection. z A y y Q z P Q y Q Q n An isothem A FIGURE 2 8 The heat tansfe vecto is always nomal to an isothemal suface and can be esolved into its components like any othe vecto. A z n

6 cen58933_ch2.qd 9/1/22 8:46 AM Page HEAT TRANSFER FIGURE 2 9 Heat is geneated in the heating coils of an electic ange as a esult of the convesion of electical enegy to heat. Wate Sola adiation q s Sun Sola enegy absobed by wate g() = q s, absobed () FIGURE 2 1 The absoption of sola adiation by wate can be teated as heat geneation. Heat Geneation A medium though which heat is conducted may involve the convesion of electical, nuclea, o chemical enegy into heat (o themal) enegy. In heat conduction analysis, such convesion pocesses ae chaacteized as heat geneation. Fo eample, the tempeatue of a esistance wie ises apidly when electic cuent passes though it as a esult of the electical enegy being conveted to heat at a ate of I 2 R, whee I is the cuent and R is the electical esistance of the wie (Fig. 2 9). The safe and effective emoval of this heat away fom the sites of heat geneation (the electonic cicuits) is the subject of electonics cooling, which is one of the moden application aeas of heat tansfe. ikewise, a lage amount of heat is geneated in the fuel elements of nuclea eactos as a esult of nuclea fission that seves as the heat souce fo the nuclea powe plants. The natual disintegation of adioactive elements in nuclea waste o othe adioactive mateial also esults in the geneation of heat thoughout the body. The heat geneated in the sun as a esult of the fusion of hydogen into helium makes the sun a lage nuclea eacto that supplies heat to the eath. Anothe souce of heat geneation in a medium is eothemic chemical eactions that may occu thoughout the medium. The chemical eaction in this case seves as a heat souce fo the medium. In the case of endothemic eactions, howeve, heat is absobed instead of being eleased duing eaction, and thus the chemical eaction seves as a heat sink. The heat geneation tem becomes a negative quantity in this case. Often it is also convenient to model the absoption of adiation such as sola enegy o gamma ays as heat geneation when these ays penetate deep into the body while being absobed gadually. Fo eample, the absoption of sola enegy in lage bodies of wate can be teated as heat geneation thoughout the wate at a ate equal to the ate of absoption, which vaies with depth (Fig. 2 1). But the absoption of sola enegy by an opaque body occus within a few micons of the suface, and the sola enegy that penetates into the medium in this case can be teated as specified heat flu on the suface. Note that heat geneation is a volumetic phenomenon. That is, it occus thoughout the body of a medium. Theefoe, the ate of heat geneation in a medium is usually specified pe unit volume and is denoted by g, whose unit is W/m 3 o Btu/h ft 3. The ate of heat geneation in a medium may vay with time as well as position within the medium. When the vaiation of heat geneation with position is known, the total ate of heat geneation in a medium of volume V can be detemined fom V G g dv (W) (2-5) In the special case of unifom heat geneation, as in the case of electic esistance heating thoughout a homogeneous mateial, the elation in Eq. 2 5 educes to G g V, whee g is the constant ate of heat geneation pe unit volume.

7 cen58933_ch2.qd 9/1/22 8:46 AM Page 67 EXAMPE 2 1 Heat Gain by a Refigeato 67 CHAPTER 2 Heat tansfe In ode to size the compesso of a new efigeato, it is desied to detemine the ate of heat tansfe fom the kitchen ai into the efigeated space though the walls, doo, and the top and bottom section of the efigeato (Fig. 2 11). In you analysis, would you teat this as a tansient o steady-state heat tansfe poblem? Also, would you conside the heat tansfe to be one-dimensional o multidimensional? Eplain. SOUTION The heat tansfe pocess fom the kitchen ai to the efigeated space is tansient in natue since the themal conditions in the kitchen and the efigeato, in geneal, change with time. Howeve, we would analyze this poblem as a steady heat tansfe poblem unde the wost anticipated conditions such as the lowest themostat setting fo the efigeated space, and the anticipated highest tempeatue in the kitchen (the so-called design conditions). If the compesso is lage enough to keep the efigeated space at the desied tempeatue setting unde the pesumed wost conditions, then it is lage enough to do so unde all conditions by cycling on and off. Heat tansfe into the efigeated space is thee-dimensional in natue since heat will be enteing though all si sides of the efigeato. Howeve, heat tansfe though any wall o floo takes place in the diection nomal to the suface, and thus it can be analyzed as being one-dimensional. Theefoe, this poblem can be simplified geatly by consideing the heat tansfe to be onedimensional at each of the fou sides as well as the top and bottom sections, and then by adding the calculated values of heat tansfe at each suface. FIGURE 2 11 Schematic fo Eample 2 1. EXAMPE 2 2 Heat Geneation in a Hai Dye The esistance wie of a 12-W hai dye is 8 cm long and has a diamete of D.3 cm (Fig. 2 12). Detemine the ate of heat geneation in the wie pe unit volume, in W/cm 3, and the heat flu on the oute suface of the wie as a esult of this heat geneation. SOUTION The powe consumed by the esistance wie of a hai dye is given. The heat geneation and the heat flu ae to be detemined. Assumptions Heat is geneated unifomly in the esistance wie. Analysis A 12-W hai dye will convet electical enegy into heat in the wie at a ate of 12 W. Theefoe, the ate of heat geneation in a esistance wie is equal to the powe consumption of a esistance heate. Then the ate of heat geneation in the wie pe unit volume is detemined by dividing the total ate of heat geneation by the volume of the wie, G G 12 W g 212 W/cm 3 (D 2 /4) [(.3 cm) 2 /4](8 cm) V wie Similaly, heat flu on the oute suface of the wie as a esult of this heat geneation is detemined by dividing the total ate of heat geneation by the suface aea of the wie, G G 12 W q 15.9 W/cm D (.3 cm)(8 cm) 2 A wie Hai dye 12 W FIGURE 2 12 Schematic fo Eample 2 2.

8 cen58933_ch2.qd 9/1/22 8:46 AM Page HEAT TRANSFER Discussion Note that heat geneation is epessed pe unit volume in W/cm 3 o Btu/h ft 3, wheeas heat flu is epessed pe unit suface aea in W/cm 2 o Btu/h ft ONE-DIMENSIONA HEAT CONDUCTION EQUATION G Volume element A Conside heat conduction though a lage plane wall such as the wall of a house, the glass of a single pane window, the metal plate at the bottom of a pessing ion, a cast ion steam pipe, a cylindical nuclea fuel element, an electical esistance wie, the wall of a spheical containe, o a spheical metal ball that is being quenched o tempeed. Heat conduction in these and many othe geometies can be appoimated as being one-dimensional since heat conduction though these geometies will be dominant in one diection and negligible in othe diections. Below we will develop the onedimensional heat conduction equation in ectangula, cylindical, and spheical coodinates. Q Q + + A = A + = A FIGURE 2 13 One-dimensional heat conduction though a volume element in a lage plane wall. Heat Conduction Equation in a age Plane Wall Conside a thin element of thickness in a lage plane wall, as shown in Figue Assume the density of the wall is, the specific heat is C, and the aea of the wall nomal to the diection of heat tansfe is A. An enegy balance on this thin element duing a small time inteval t can be epessed as o Rate of heat conduction at Rate of heat Rate of heat geneation conduction inside the at element E element Rate of change of the enegy content of the element Q Q G element (2-6) t But the change in the enegy content of the element and the ate of heat geneation within the element can be epessed as E element E t t E t mc(t t t T t ) CA(T t t T t ) (2-7) G element g V element g A (2-8) Substituting into Equation 2 6, we get Dividing by A gives T tt T t Q Q g A CA (2-9) t 1 Q Q T tt T t g C (2-1) A t

9 cen58933_ch2.qd 9/1/22 8:46 AM Page CHAPTER 2 Taking the limit as and t yields since, fom the definition of the deivative and Fouie s law of heat conduction, Q lim 1 A T T ka g C (2-11) t Q Q kat (2-12) Noting that the aea A is constant fo a plane wall, the one-dimensional tansient heat conduction equation in a plane wall becomes T Vaiable conductivity: k T g C (2-13) t The themal conductivity k of a mateial, in geneal, depends on the tempeatue T (and theefoe ), and thus it cannot be taken out of the deivative. Howeve, the themal conductivity in most pactical applications can be assumed to emain constant at some aveage value. The equation above in that case educes to Constant conductivity: 2 T g 1 T (2-14) 2 k t whee the popety k/c is the themal diffusivity of the mateial and epesents how fast heat popagates though a mateial. It educes to the following foms unde specified conditions (Fig. 2 14): (1) Steady-state: (/t ) (2) Tansient, no heat geneation: (g ) (3) Steady-state, no heat geneation: (/t and g ) d 2 T g (2-15) d 2 k 2 T 1 T (2-16) 2 t (2-17) Note that we eplaced the patial deivatives by odinay deivatives in the one-dimensional steady heat conduction case since the patial and odinay deivatives of a function ae identical when the function depends on a single vaiable only [T T() in this case]. Heat Conduction Equation in a ong Cylinde Now conside a thin cylindical shell element of thickness in a long cylinde, as shown in Figue Assume the density of the cylinde is, the specific heat is C, and the length is. The aea of the cylinde nomal to the diection of heat tansfe at any location is A 2 whee is the value of the adius at that location. Note that the heat tansfe aea A depends on in this case, and thus it vaies with location. An enegy balance on this thin cylindical shell element duing a small time inteval t can be epessed as d 2 T d 2 Geneal, one dimensional: No Steadygeneation state 2 T g 1 T 2 k t Steady, one-dimensional: d 2 T d 2 FIGURE 2 14 The simplification of the onedimensional heat conduction equation in a plane wall fo the case of constant conductivity fo steady conduction with no heat geneation. Q Q + G + Volume element FIGURE 2 15 One-dimensional heat conduction though a volume element in a long cylinde.

10 cen58933_ch2.qd 9/1/22 8:46 AM Page 7 7 HEAT TRANSFER o Rate of heat conduction at Rate of heat Rate of heat geneation conduction inside the at element E element Rate of change of the enegy content of the element Q Q G element (2-18) t The change in the enegy content of the element and the ate of heat geneation within the element can be epessed as E element E t t E t mc(t t t T t ) CA(T t t T t ) (2-19) G element g V element g A (2-2) Substituting into Eq. 2 18, we get T tt T t Q Q g A CA (2-21) t whee A 2. You may be tempted to epess the aea at the middle of the element using the aveage adius as A 2( /2). But thee is nothing we can gain fom this complication since late in the analysis we will take the limit as and thus the tem /2 will dop out. Now dividing the equation above by A gives 1 Q Q T tt T t g C (2-22) A t Taking the limit as and t yields 1 T T ka g C (2-23) A t since, fom the definition of the deivative and Fouie s law of heat conduction, Q lim Q Q T ka (2-24) Noting that the heat tansfe aea in this case is A 2, the onedimensional tansient heat conduction equation in a cylinde becomes 1 T T Vaiable conductivity: k g C (2-25) t Fo the case of constant themal conductivity, the equation above educes to 1 g 1 T Constant conductivity: T (2-26) k t

11 cen58933_ch2.qd 9/1/22 8:46 AM Page CHAPTER 2 whee again the popety k/c is the themal diffusivity of the mateial. Equation 2 26 educes to the following foms unde specified conditions (Fig. 2 16): g (1) Steady-state: (/t ) (2) Tansient, no heat geneation: (g ) (3) Steady-state, no heat geneation: (/t and g ) dt d k (2-27) 1 T T t (2-28) dt d (2-29) Note that we again eplaced the patial deivatives by odinay deivatives in the one-dimensional steady heat conduction case since the patial and odinay deivatives of a function ae identical when the function depends on a single vaiable only [T T() in this case]. Heat Conduction Equation in a Sphee Now conside a sphee with density, specific heat C, and oute adius R. The aea of the sphee nomal to the diection of heat tansfe at any location is A 4 2, whee is the value of the adius at that location. Note that the heat tansfe aea A depends on in this case also, and thus it vaies with location. By consideing a thin spheical shell element of thickness and epeating the appoach descibed above fo the cylinde by using A 4 2 instead of A 2, the one-dimensional tansient heat conduction equation fo a sphee is detemined to be (Fig. 2 17) 1 T Vaiable conductivity: 2 k T g C (2-3) t 2 1 d d 1 d d (a) The fom that is eady to integate d dt d d (b) The equivalent altenative fom d dt 2 T d 2 d FIGURE 2 16 Two equivalent foms of the diffeential equation fo the onedimensional steady heat conduction in a cylinde with no heat geneation. G Q + Q + R Volume element FIGURE 2 17 One-dimensional heat conduction though a volume element in a sphee. which, in the case of constant themal conductivity, educes to 1 g 1 T Constant conductivity: 2 T (2-31) k t whee again the popety k/c is the themal diffusivity of the mateial. It educes to the following foms unde specified conditions: (1) Steady-state: (/t ) (2) Tansient, no heat geneation: (g ) (3) Steady-state, no heat geneation: (/t and g ) 1 d g 2 dt (2-32) d d k T 2 T (2-33) 2 t d d 2 2 dt d o 2 T dt 2 (2-34) d d 2 d whee again we eplaced the patial deivatives by odinay deivatives in the one-dimensional steady heat conduction case.

12 cen58933_ch2.qd 9/1/22 8:46 AM Page HEAT TRANSFER Combined One-Dimensional Heat Conduction Equation An eamination of the one-dimensional tansient heat conduction equations fo the plane wall, cylinde, and sphee eveals that all thee equations can be epessed in a compact fom as 1 T n k T g C (2-35) n t whee n fo a plane wall, n 1 fo a cylinde, and n 2 fo a sphee. In the case of a plane wall, it is customay to eplace the vaiable by. This equation can be simplified fo steady-state o no heat geneation cases as descibed befoe. EXAMPE 2 3 Heat Conduction though the Bottom of a Pan Conside a steel pan placed on top of an electic ange to cook spaghetti (Fig. 2 18). The bottom section of the pan is.4 cm thick and has a diamete of D 18 cm. The electic heating unit on the ange top consumes 8 W of powe duing cooking, and 8 pecent of the heat geneated in the heating element is tansfeed unifomly to the pan. Assuming constant themal conductivity, obtain the diffeential equation that descibes the vaiation of the tempeatue in the bottom section of the pan duing steady opeation. 8 W FIGURE 2 18 Schematic fo Eample 2 3. SOUTION The bottom section of the pan has a lage suface aea elative to its thickness and can be appoimated as a lage plane wall. Heat flu is applied to the bottom suface of the pan unifomly, and the conditions on the inne suface ae also unifom. Theefoe, we epect the heat tansfe though the bottom section of the pan to be fom the bottom suface towad the top, and heat tansfe in this case can easonably be appoimated as being onedimensional. Taking the diection nomal to the bottom suface of the pan to be the -ais, we will have T T () duing steady opeation since the tempeatue in this case will depend on only. The themal conductivity is given to be constant, and thee is no heat geneation in the medium (within the bottom section of the pan). Theefoe, the diffeential equation govening the vaiation of tempeatue in the bottom section of the pan in this case is simply Eq. 2 17, d 2 T d 2 which is the steady one-dimensional heat conduction equation in ectangula coodinates unde the conditions of constant themal conductivity and no heat geneation. Note that the conditions at the suface of the medium have no effect on the diffeential equation. EXAMPE 2 4 Heat Conduction in a Resistance Heate A 2-kW esistance heate wie with themal conductivity k 15 W/m C, diamete D.4 cm, and length 5 cm is used to boil wate by immesing

13 cen58933_ch2.qd 9/1/22 8:46 AM Page CHAPTER 2 it in wate (Fig. 2 19). Assuming the vaiation of the themal conductivity of the wie with tempeatue to be negligible, obtain the diffeential equation that descibes the vaiation of the tempeatue in the wie duing steady opeation. SOUTION The esistance wie can be consideed to be a vey long cylinde since its length is moe than 1 times its diamete. Also, heat is geneated unifomly in the wie and the conditions on the oute suface of the wie ae unifom. Theefoe, it is easonable to epect the tempeatue in the wie to vay in the adial diection only and thus the heat tansfe to be one-dimensional. Then we will have T T ( ) duing steady opeation since the tempeatue in this case will depend on only. The ate of heat geneation in the wie pe unit volume can be detemined fom G G 2 W g W/m 3 (D 2 /4) [(.4 m) 2 /4](.5 cm) V wie Noting that the themal conductivity is given to be constant, the diffeential equation that govens the vaiation of tempeatue in the wie is simply Eq. 2 27, 1 d d dt d g k Wate Resistance heate FIGURE 2 19 Schematic fo Eample 2 4. which is the steady one-dimensional heat conduction equation in cylindical coodinates fo the case of constant themal conductivity. Note again that the conditions at the suface of the wie have no effect on the diffeential equation. EXAMPE 2 5 Cooling of a Hot Metal Ball in Ai A spheical metal ball of adius R is heated in an oven to a tempeatue of 6 F thoughout and is then taken out of the oven and allowed to cool in ambient ai at T 75 F by convection and adiation (Fig. 2 2). The themal conductivity of the ball mateial is known to vay linealy with tempeatue. Assuming the ball is cooled unifomly fom the entie oute suface, obtain the diffeential equation that descibes the vaiation of the tempeatue in the ball duing cooling. SOUTION The ball is initially at a unifom tempeatue and is cooled unifomly fom the entie oute suface. Also, the tempeatue at any point in the ball will change with time duing cooling. Theefoe, this is a one-dimensional tansient heat conduction poblem since the tempeatue within the ball will change with the adial distance and the time t. That is, T T (, t ). The themal conductivity is given to be vaiable, and thee is no heat geneation in the ball. Theefoe, the diffeential equation that govens the vaiation of tempeatue in the ball in this case is obtained fom Eq. 2 3 by setting the heat geneation tem equal to zeo. We obtain 1 2 k T 2 C T t Metal ball 6 F 75 F FIGURE 2 2 Schematic fo Eample 2 5. Q

14 cen58933_ch2.qd 9/1/22 8:46 AM Page HEAT TRANSFER which is the one-dimensional tansient heat conduction equation in spheical coodinates unde the conditions of vaiable themal conductivity and no heat geneation. Note again that the conditions at the oute suface of the ball have no effect on the diffeential equation. 2 3 GENERA HEAT CONDUCTION EQUATION In the last section we consideed one-dimensional heat conduction and assumed heat conduction in othe diections to be negligible. Most heat tansfe poblems encounteed in pactice can be appoimated as being onedimensional, and we will mostly deal with such poblems in this tet. Howeve, this is not always the case, and sometimes we need to conside heat tansfe in othe diections as well. In such cases heat conduction is said to be multidimensional, and in this section we will develop the govening diffeential equation in such systems in ectangula, cylindical, and spheical coodinate systems. Volume element z Q g y z Q y y Q z + z Q z y z Q y + y Q + FIGURE 2 21 Thee-dimensional heat conduction though a ectangula volume element. Rectangula Coodinates Conside a small ectangula element of length, width y, and height z, as shown in Figue Assume the density of the body is and the specific heat is C. An enegy balance on this element duing a small time inteval t can be epessed as Rate of heat Rate of heat Rate of heat conduction geneation conduction at inside the at,, y, and z y y, and z z element o Rate of change of the enegy content of the element E element Q Q y Q z Q Q y y Q z z G element (2-36) t Noting that the volume of the element is V element yz, the change in the enegy content of the element and the ate of heat geneation within the element can be epessed as E element E t t E t mc(t t t T t ) Cyz(T t t T t ) G element g V element g yz Substituting into Eq. 2 36, we get Q Q y Q z Q Q y y Q z z g yz Cyz Dividing by yz gives T tt T t t 1 Q Q yy Q Q 1 y 1 Q zz Q z T tt T t g C yz z y y z t (2-37)

15 cen58933_ch2.qd 9/1/22 8:46 AM Page 75 Noting that the heat tansfe aeas of the element fo heat conduction in the, y, and z diections ae A yz, A y z, and A z y, espectively, and taking the limit as, y, z and t yields 75 CHAPTER 2 T k T k T k T g C (2-38) y y z z t since, fom the definition of the deivative and Fouie s law of heat conduction, lim lim y lim z 1 yz 1 z 1 y Q Q Q yy Q y y Q zz Q z z 1 Q 1 T kyz yz yz 1 Q y 1 T kz z y z y y y 1 Q z 1 T ky y z y z z z k T k T y k T z Equation 2 38 is the geneal heat conduction equation in ectangula coodinates. In the case of constant themal conductivity, it educes to whee the popety k/c is again the themal diffusivity of the mateial. Equation 2 39 is known as the Fouie-Biot equation, and it educes to these foms unde specified conditions: (1) Steady-state: (called the Poisson equation) (2) Tansient, no heat geneation: (called the diffusion equation) (3) Steady-state, no heat geneation: (called the aplace equation) 2 T T 2 T g 1 2 T (2-39) 2 y 2 z 2 k t 2 T g 2 T 2 T (2-4) 2 y 2 z 2 k 2 T 1 T 2 T 2 T (2-41) 2 y 2 z 2 t 2 T 2 T 2 T (2-42) 2 y 2 z 2 2 T g 2 T 2 T 2 y 2 z 2 k 2 T 1 T 2 T 2 T 2 y 2 z 2 t 2 T 2 T 2 T 2 y 2 z 2 FIGURE 2 22 The thee-dimensional heat conduction equations educe to the one-dimensional ones when the tempeatue vaies in one dimension only. Note that in the special case of one-dimensional heat tansfe in the -diection, the deivatives with espect to y and z dop out and the equations above educe to the ones developed in the pevious section fo a plane wall (Fig. 2 22). dz z d Cylindical Coodinates The geneal heat conduction equation in cylindical coodinates can be obtained fom an enegy balance on a volume element in cylindical coodinates, shown in Figue 2 23, by following the steps just outlined. It can also be obtained diectly fom Eq by coodinate tansfomation using the following elations between the coodinates of a point in ectangula and cylindical coodinate systems: cos, y sin, and z z φ z dφ FIGURE 2 23 A diffeential volume element in cylindical coodinates. y

16 cen58933_ch2.qd 9/1/22 8:46 AM Page 76 z 76 HEAT TRANSFER φ θ d dθ dφ FIGURE 2 24 A diffeential volume element in spheical coodinates. y Afte lengthy manipulations, we obtain 1 T k 1 T T k k T g C (2-43) z z t Spheical Coodinates The geneal heat conduction equations in spheical coodinates can be obtained fom an enegy balance on a volume element in spheical coodinates, shown in Figue 2 24, by following the steps outlined above. It can also be obtained diectly fom Eq by coodinate tansfomation using the following elations between the coodinates of a point in ectangula and spheical coodinate systems: cos sin, y sin sin, and z cos Again afte lengthy manipulations, we obtain T 1 1 T T k k T k sin g C 2 2 sin 2 2 sin t (2-44) Obtaining analytical solutions to these diffeential equations equies a knowledge of the solution techniques of patial diffeential equations, which is beyond the scope of this intoductoy tet. Hee we limit ou consideation to one-dimensional steady-state cases o lumped systems, since they esult in odinay diffeential equations. Metal billet z R 6 F T = 65 F φ Heat loss FIGURE 2 25 Schematic fo Eample 2 6. EXAMPE 2 6 Heat Conduction in a Shot Cylinde A shot cylindical metal billet of adius R and height h is heated in an oven to a tempeatue of 6 F thoughout and is then taken out of the oven and allowed to cool in ambient ai at T 65 F by convection and adiation. Assuming the billet is cooled unifomly fom all oute sufaces and the vaiation of the themal conductivity of the mateial with tempeatue is negligible, obtain the diffeential equation that descibes the vaiation of the tempeatue in the billet duing this cooling pocess. SOUTION The billet shown in Figue 2 25 is initially at a unifom tempeatue and is cooled unifomly fom the top and bottom sufaces in the z-diection as well as the lateal suface in the adial -diection. Also, the tempeatue at any point in the ball will change with time duing cooling. Theefoe, this is a two-dimensional tansient heat conduction poblem since the tempeatue within the billet will change with the adial and aial distances and z and with time t. That is, T T (, z, t ). The themal conductivity is given to be constant, and thee is no heat geneation in the billet. Theefoe, the diffeential equation that govens the vaiation of tempeatue in the billet in this case is obtained fom Eq by setting the heat geneation tem and the deivatives with espect to equal to zeo. We obtain 1 T k k T z z C T t

17 cen58933_ch2.qd 9/1/22 8:46 AM Page CHAPTER 2 In the case of constant themal conductivity, it educes to 1 which is the desied equation. T 2 T z 2 1 T t 2 4 BOUNDARY AND INITIA CONDITIONS The heat conduction equations above wee developed using an enegy balance on a diffeential element inside the medium, and they emain the same egadless of the themal conditions on the sufaces of the medium. That is, the diffeential equations do not incopoate any infomation elated to the conditions on the sufaces such as the suface tempeatue o a specified heat flu. Yet we know that the heat flu and the tempeatue distibution in a medium depend on the conditions at the sufaces, and the desciption of a heat tansfe poblem in a medium is not complete without a full desciption of the themal conditions at the bounding sufaces of the medium. The mathematical epessions of the themal conditions at the boundaies ae called the bounday conditions. Fom a mathematical point of view, solving a diffeential equation is essentially a pocess of emoving deivatives, o an integation pocess, and thus the solution of a diffeential equation typically involves abitay constants (Fig. 2 26). It follows that to obtain a unique solution to a poblem, we need to specify moe than just the govening diffeential equation. We need to specify some conditions (such as the value of the function o its deivatives at some value of the independent vaiable) so that focing the solution to satisfy these conditions at specified points will esult in unique values fo the abitay constants and thus a unique solution. But since the diffeential equation has no place fo the additional infomation o conditions, we need to supply them sepaately in the fom of bounday o initial conditions. Conside the vaiation of tempeatue along the wall of a bick house in winte. The tempeatue at any point in the wall depends on, among othe things, the conditions at the two sufaces of the wall such as the ai tempeatue of the house, the velocity and diection of the winds, and the sola enegy incident on the oute suface. That is, the tempeatue distibution in a medium depends on the conditions at the boundaies of the medium as well as the heat tansfe mechanism inside the medium. To descibe a heat tansfe poblem completely, two bounday conditions must be given fo each diection of the coodinate system along which heat tansfe is significant (Fig. 2 27). Theefoe, we need to specify two bounday conditions fo one-dimensional poblems, fou bounday conditions fo two-dimensional poblems, and si bounday conditions fo thee-dimensional poblems. In the case of the wall of a house, fo eample, we need to specify the conditions at two locations (the inne and the oute sufaces) of the wall since heat tansfe in this case is one-dimensional. But in the case of a paallelepiped, we need to specify si bounday conditions (one at each face) when heat tansfe in all thee dimensions is significant. The diffeential equation: d 2 T d 2 Geneal solution: T() C 1 C 2 Abitay constants Some specific solutions: T() 2 5 T() 12 T() 3 T() 6.2 M T 5 C FIGURE 2 26 The geneal solution of a typical diffeential equation involves abitay constants, and thus an infinite numbe of solutions. Some solutions of d 2 T = d 2 15 C The only solution that satisfies the conditions T() = 5 C and T() = 15 C. FIGURE 2 27 To descibe a heat tansfe poblem completely, two bounday conditions must be given fo each diection along which heat tansfe is significant.

18 cen58933_ch2.qd 9/1/22 8:46 AM Page HEAT TRANSFER The physical agument pesented above is consistent with the mathematical natue of the poblem since the heat conduction equation is second ode (i.e., involves second deivatives with espect to the space vaiables) in all diections along which heat conduction is significant, and the geneal solution of a second-ode linea diffeential equation involves two abitay constants fo each diection. That is, the numbe of bounday conditions that needs to be specified in a diection is equal to the ode of the diffeential equation in that diection. Reconside the bick wall aleady discussed. The tempeatue at any point on the wall at a specified time also depends on the condition of the wall at the beginning of the heat conduction pocess. Such a condition, which is usually specified at time t, is called the initial condition, which is a mathematical epession fo the tempeatue distibution of the medium initially. Note that we need only one initial condition fo a heat conduction poblem egadless of the dimension since the conduction equation is fist ode in time (it involves the fist deivative of tempeatue with espect to time). In ectangula coodinates, the initial condition can be specified in the geneal fom as T(, y, z, ) f(, y, z) (2-45) whee the function f(, y, z) epesents the tempeatue distibution thoughout the medium at time t. When the medium is initially at a unifom tempeatue of T i, the initial condition of Eq can be epessed as T(, y, z,) T i. Note that unde steady conditions, the heat conduction equation does not involve any time deivatives, and thus we do not need to specify an initial condition. The heat conduction equation is fist ode in time, and thus the initial condition cannot involve any deivatives (it is limited to a specified tempeatue). Howeve, the heat conduction equation is second ode in space coodinates, and thus a bounday condition may involve fist deivatives at the boundaies as well as specified values of tempeatue. Bounday conditions most commonly encounteed in pactice ae the specified tempeatue, specified heat flu, convection, and adiation bounday conditions. 15 C T(, t) 7 C T(, t) = 15 C T(, t) = 7 C FIGURE 2 28 Specified tempeatue bounday conditions on both sufaces of a plane wall. 1 Specified Tempeatue Bounday Condition The tempeatue of an eposed suface can usually be measued diectly and easily. Theefoe, one of the easiest ways to specify the themal conditions on a suface is to specify the tempeatue. Fo one-dimensional heat tansfe though a plane wall of thickness, fo eample, the specified tempeatue bounday conditions can be epessed as (Fig. 2 28) T(, t) T 1 T(, t) T 2 (2-46) whee T 1 and T 2 ae the specified tempeatues at sufaces at and, espectively. The specified tempeatues can be constant, which is the case fo steady heat conduction, o may vay with time.

19 cen58933_ch2.qd 9/1/22 8:46 AM Page 79 2 Specified Heat Flu Bounday Condition When thee is sufficient infomation about enegy inteactions at a suface, it may be possible to detemine the ate of heat tansfe and thus the heat flu q (heat tansfe ate pe unit suface aea, W/m 2 ) on that suface, and this infomation can be used as one of the bounday conditions. The heat flu in the positive -diection anywhee in the medium, including the boundaies, can be epessed by Fouie s law of heat conduction as T Heat flu in the q k (W/m 2 ) (2-47) positive -diection Then the bounday condition at a bounday is obtained by setting the specified heat flu equal to k(t/) at that bounday. The sign of the specified heat flu is detemined by inspection: positive if the heat flu is in the positive diection of the coodinate ais, and negative if it is in the opposite diection. Note that it is etemely impotant to have the coect sign fo the specified heat flu since the wong sign will invet the diection of heat tansfe and cause the heat gain to be intepeted as heat loss (Fig. 2 29). Fo a plate of thickness subjected to heat flu of 5 W/m 2 into the medium fom both sides, fo eample, the specified heat flu bounday conditions can be epessed as T(, t) T(, t) k 5 and k 5 (2-48) 79 CHAPTER 2 Heat flu Conduction T(, t) q = k Conduction T(, k t) = q Heat flu FIGURE 2 29 Specified heat flu bounday conditions on both sufaces of a plane wall. Note that the heat flu at the suface at is in the negative -diection, and thus it is 5 W/m 2. Special Case: Insulated Bounday Some sufaces ae commonly insulated in pactice in ode to minimize heat loss (o heat gain) though them. Insulation educes heat tansfe but does not totally eliminate it unless its thickness is infinity. Howeve, heat tansfe though a popely insulated suface can be taken to be zeo since adequate insulation educes heat tansfe though a suface to negligible levels. Theefoe, a well-insulated suface can be modeled as a suface with a specified heat flu of zeo. Then the bounday condition on a pefectly insulated suface (at, fo eample) can be epessed as (Fig. 2 3) T(, t) T(, t) k o (2-49) That is, on an insulated suface, the fist deivative of tempeatue with espect to the space vaiable (the tempeatue gadient) in the diection nomal to the insulated suface is zeo. This also means that the tempeatue function must be pependicula to an insulated suface since the slope of tempeatue at the suface must be zeo. Insulation T(, t) 6 C T(, t) = T(, t) = 6 C FIGURE 2 3 A plane wall with insulation and specified tempeatue bounday conditions. Anothe Special Case: Themal Symmety Some heat tansfe poblems possess themal symmety as a esult of the symmety in imposed themal conditions. Fo eample, the two sufaces of a lage hot plate of thickness suspended vetically in ai will be subjected to

20 cen58933_ch2.qd 9/1/22 8:46 AM Page 8 8 HEAT TRANSFER Zeo slope Cente plane Tempeatue distibution (symmetic about cente plane) the same themal conditions, and thus the tempeatue distibution in one half of the plate will be the same as that in the othe half. That is, the heat tansfe poblem in this plate will possess themal symmety about the cente plane at /2. Also, the diection of heat flow at any point in the plate will be towad the suface close to the point, and thee will be no heat flow acoss the cente plane. Theefoe, the cente plane can be viewed as an insulated suface, and the themal condition at this plane of symmety can be epessed as (Fig. 2 31) 2 T(/2, t) (2-5) T(/2, t) = FIGURE 2 31 Themal symmety bounday condition at the cente plane of a plane wall. which esembles the insulation o zeo heat flu bounday condition. This esult can also be deduced fom a plot of tempeatue distibution with a maimum, and thus zeo slope, at the cente plane. In the case of cylindical (o spheical) bodies having themal symmety about the cente line (o midpoint), the themal symmety bounday condition equies that the fist deivative of tempeatue with espect to (the adial vaiable) be zeo at the centeline (o the midpoint). EXAMPE 2 7 Heat Flu Bounday Condition q Wate 11 C FIGURE 2 32 Schematic fo Eample 2 7. Conside an aluminum pan used to cook beef stew on top of an electic ange. The bottom section of the pan is.3 cm thick and has a diamete of D 2 cm. The electic heating unit on the ange top consumes 8 W of powe duing cooking, and 9 pecent of the heat geneated in the heating element is tansfeed to the pan. Duing steady opeation, the tempeatue of the inne suface of the pan is measued to be 11 C. Epess the bounday conditions fo the bottom section of the pan duing this cooking pocess. SOUTION The heat tansfe though the bottom section of the pan is fom the bottom suface towad the top and can easonably be appoimated as being one-dimensional. We take the diection nomal to the bottom sufaces of the pan as the ais with the oigin at the oute suface, as shown in Figue Then the inne and oute sufaces of the bottom section of the pan can be epesented by and, espectively. Duing steady opeation, the tempeatue will depend on only and thus T T (). The bounday condition on the oute suface of the bottom of the pan at can be appoimated as being specified heat flu since it is stated that 9 pecent of the 8 W (i.e., 72 W) is tansfeed to the pan at that suface. Theefoe, dt() k q d whee Heat tansfe ate.72 kw q 22.9 kw/m Bottom suface aea 2 (.1 m) 2

21 cen58933_ch2.qd 9/1/22 8:46 AM Page CHAPTER 2 The tempeatue at the inne suface of the bottom of the pan is specified to be 11 C. Then the bounday condition on this suface can be epessed as T() 11 C whee.3 m. Note that the detemination of the bounday conditions may equie some easoning and appoimations. 3 Convection Bounday Condition Convection is pobably the most common bounday condition encounteed in pactice since most heat tansfe sufaces ae eposed to an envionment at a specified tempeatue. The convection bounday condition is based on a suface enegy balance epessed as Heat conduction Heat convection at the suface in a at the suface in selected diection the same diection Fo one-dimensional heat tansfe in the -diection in a plate of thickness, the convection bounday conditions on both sufaces can be epessed as and T(, t) k h 1 [T 1 T(, t)] (2-51a) T(, t) k h 2 [T(, t) T 2 ] (2-51b) whee h 1 and h 2 ae the convection heat tansfe coefficients and T 1 and T 2 ae the tempeatues of the suounding mediums on the two sides of the plate, as shown in Figue In witing Eqs fo convection bounday conditions, we have selected the diection of heat tansfe to be the positive -diection at both sufaces. But those epessions ae equally applicable when heat tansfe is in the opposite diection at one o both sufaces since evesing the diection of heat tansfe at a suface simply eveses the signs of both conduction and convection tems at that suface. This is equivalent to multiplying an equation by 1, which has no effect on the equality (Fig. 2 34). Being able to select eithe diection as the diection of heat tansfe is cetainly a elief since often we do not know the suface tempeatue and thus the diection of heat tansfe at a suface in advance. This agument is also valid fo othe bounday conditions such as the adiation and combined bounday conditions discussed shotly. Note that a suface has zeo thickness and thus no mass, and it cannot stoe any enegy. Theefoe, the entie net heat enteing the suface fom one side must leave the suface fom the othe side. The convection bounday condition simply states that heat continues to flow fom a body to the suounding medium at the same ate, and it just changes vehicles at the suface fom conduction to convection (o vice vesa in the othe diection). This is analogous to people taveling on buses on land and tansfeing to the ships at the shoe. Convection Conduction T(, t) h 1 [T 1 T(, t)] = k h 1 T 1 Conduction Convection T(, k t) = h 2 [T(, t) T 2 ] h 2 T 2 FIGURE 2 33 Convection bounday conditions on the two sufaces of a plane wall. Convection Convection Conduction T(, t) h 1 [T 1 T(, t)] = k h 1, T 1 Conduction T(, t) h 1 [T(, t) T 1 ] = k FIGURE 2 34 The assumed diection of heat tansfe at a bounday has no effect on the bounday condition epession.

22 cen58933_ch2.qd 9/1/22 8:46 AM Page HEAT TRANSFER If the passenges ae not allowed to wande aound at the shoe, then the ate at which the people ae unloaded at the shoe fom the buses must equal the ate at which they boad the ships. We may call this the consevation of people pinciple. Also note that the suface tempeatues T(, t) and T(, t) ae not known (if they wee known, we would simply use them as the specified tempeatue bounday condition and not bothe with convection). But a suface tempeatue can be detemined once the solution T(, t) is obtained by substituting the value of at that suface into the solution. EXAMPE 2 8 Convection and Insulation Bounday Conditions Insulation h1 T 2 1 FIGURE 2 35 Schematic fo Eample 2 8. Steam flows though a pipe shown in Figue 2 35 at an aveage tempeatue of T 2 C. The inne and oute adii of the pipe ae 1 8 cm and cm, espectively, and the oute suface of the pipe is heavily insulated. If the convection heat tansfe coefficient on the inne suface of the pipe is h 65 W/m 2 C, epess the bounday conditions on the inne and oute sufaces of the pipe duing tansient peiods. SOUTION Duing initial tansient peiods, heat tansfe though the pipe mateial will pedominantly be in the adial diection, and thus can be appoimated as being one-dimensional. Then the tempeatue within the pipe mateial will change with the adial distance and the time t. That is, T T (, t ). It is stated that heat tansfe between the steam and the pipe at the inne suface is by convection. Then taking the diection of heat tansfe to be the positive diection, the bounday condition on that suface can be epessed as T( 1, t) k h[t T( 1 )] The pipe is said to be well insulated on the outside, and thus heat loss though the oute suface of the pipe can be assumed to be negligible. Then the bounday condition at the oute suface can be epessed as T( 2, t) That is, the tempeatue gadient must be zeo on the oute suface of the pipe at all times. 4 Radiation Bounday Condition In some cases, such as those encounteed in space and cyogenic applications, a heat tansfe suface is suounded by an evacuated space and thus thee is no convection heat tansfe between a suface and the suounding medium. In such cases, adiation becomes the only mechanism of heat tansfe between the suface unde consideation and the suoundings. Using an enegy balance, the adiation bounday condition on a suface can be epessed as Heat conduction at the suface in a selected diection Radiation echange at the suface in the same diection

23 cen58933_ch2.qd 9/1/22 8:46 AM Page 83 Fo one-dimensional heat tansfe in the -diection in a plate of thickness, the adiation bounday conditions on both sufaces can be epessed as (Fig. 2 36) and T(, t) k 1 [T su, 4 1 T(, t) 4 ] (2-52a) T(, t) k 2 [T(, t) 4 Tsu, 4 2] (2-52b) whee 1 and 2 ae the emissivities of the bounday sufaces, W/m 2 K 4 is the Stefan Boltzmann constant, and T su, 1 and T su, 2 ae the aveage tempeatues of the sufaces suounding the two sides of the plate, espectively. Note that the tempeatues in adiation calculations must be epessed in K o R (not in C o F). The adiation bounday condition involves the fouth powe of tempeatue, and thus it is a nonlinea condition. As a esult, the application of this bounday condition esults in powes of the unknown coefficients, which makes it difficult to detemine them. Theefoe, it is tempting to ignoe adiation echange at a suface duing a heat tansfe analysis in ode to avoid the complications associated with nonlineaity. This is especially the case when heat tansfe at the suface is dominated by convection, and the ole of adiation is mino. Radiation 83 CHAPTER 2 Conduction 1 [T su, 1 T(, T(, t) ε σ 4 t)4 ] = k ε 1 ε 2 T su, 1 T su, 2 Conduction Radiation T(, k t) = ε 2 σ [T(, t)4 T 4 su, 2 ] FIGURE 2 36 Radiation bounday conditions on both sufaces of a plane wall. Inteface 5 Inteface Bounday Conditions Some bodies ae made up of layes of diffeent mateials, and the solution of a heat tansfe poblem in such a medium equies the solution of the heat tansfe poblem in each laye. This, in tun, equies the specification of the bounday conditions at each inteface. The bounday conditions at an inteface ae based on the equiements that (1) two bodies in contact must have the same tempeatue at the aea of contact and (2) an inteface (which is a suface) cannot stoe any enegy, and thus the heat flu on the two sides of an inteface must be the same. The bounday conditions at the inteface of two bodies A and B in pefect contact at can be epessed as (Fig. 2 37) and T A (, t) T B (, t) (2-53) T A (, t) T B (, t) k A k B (2-54) Mateial A Conduction Mateial B T A (, t) = T B (, t) T A (, t) T B (, t) Conduction T k A (, t) T A = k B (, t) B FIGURE 2 37 Bounday conditions at the inteface of two bodies in pefect contact. whee k A and k B ae the themal conductivities of the layes A and B, espectively. The case of impefect contact esults in themal contact esistance, which is consideed in the net chapte.

24 cen58933_ch2.qd 9/1/22 8:46 AM Page HEAT TRANSFER 6 Genealized Bounday Conditions So fa we have consideed sufaces subjected to single mode heat tansfe, such as the specified heat flu, convection, o adiation fo simplicity. In geneal, howeve, a suface may involve convection, adiation, and specified heat flu simultaneously. The bounday condition in such cases is again obtained fom a suface enegy balance, epessed as Heat tansfe Heat tansfe to the suface fom the suface (2-55) in all modes in all modes This is illustated in Eamples 2 9 and 2 1. EXAMPE 2 9 Combined Convection and Radiation Condition Metal ball T su = 525 R Conduction T i = 6 F Radiation Convection FIGURE 2 38 Schematic fo Eample 2 9. T = 78 F A spheical metal ball of adius is heated in an oven to a tempeatue of 6 F thoughout and is then taken out of the oven and allowed to cool in ambient ai at T 78 F, as shown in Figue The themal conductivity of the ball mateial is k 8.3 Btu/h ft F, and the aveage convection heat tansfe coefficient on the oute suface of the ball is evaluated to be h 4.5 Btu/h ft 2 F. The emissivity of the oute suface of the ball is.6, and the aveage tempeatue of the suounding sufaces is T su 525 R. Assuming the ball is cooled unifomly fom the entie oute suface, epess the initial and bounday conditions fo the cooling pocess of the ball. SOUTION The ball is initially at a unifom tempeatue and is cooled unifomly fom the entie oute suface. Theefoe, this is a one-dimensional tansient heat tansfe poblem since the tempeatue within the ball will change with the adial distance and the time t. That is, T T (, t ). Taking the moment the ball is emoved fom the oven to be t, the initial condition can be epessed as T(, ) T i 6 F The poblem possesses symmety about the midpoint ( ) since the isothems in this case will be concentic sphees, and thus no heat will be cossing the midpoint of the ball. Then the bounday condition at the midpoint can be epessed as The heat conducted to the oute suface of the ball is lost to the envionment by convection and adiation. Then taking the diection of heat tansfe to be the positive diection, the bounday condition on the oute suface can be epessed as k T(, t) T(, t) h[t( ) T ] [T( ) 4 T 4 su]

25 cen58933_ch2.qd 9/1/22 8:46 AM Page CHAPTER 2 All the quantities in the above elations ae known ecept the tempeatues and thei deivatives at and. Also, the adiation pat of the bounday condition is often ignoed fo simplicity by modifying the convection heat tansfe coefficient to account fo the contibution of adiation. The convection coefficient h in that case becomes the combined heat tansfe coefficient. EXAMPE 2 1 Combined Convection, Radiation, and Heat Flu Conside the south wall of a house that is.2 m thick. The oute suface of the wall is eposed to sola adiation and has an absoptivity of.5 fo sola enegy. The inteio of the house is maintained at T 1 2 C, while the ambient ai tempeatue outside emains at T 2 5 C. The sky, the gound, and the sufaces of the suounding stuctues at this location can be modeled as a suface at an effective tempeatue of T sky 255 K fo adiation echange on the oute suface. The adiation echange between the inne suface of the wall and the sufaces of the walls, floo, and ceiling it faces is negligible. The convection heat tansfe coefficients on the inne and the oute sufaces of the wall ae h 1 6 W/m 2 C and h 2 25 W/m 2 C, espectively. The themal conductivity of the wall mateial is k.7 W/m C, and the emissivity of the oute suface is 2.9. Assuming the heat tansfe though the wall to be steady and one-dimensional, epess the bounday conditions on the inne and the oute sufaces of the wall. SOUTION We take the diection nomal to the wall sufaces as the -ais with the oigin at the inne suface of the wall, as shown in Figue The heat tansfe though the wall is given to be steady and one-dimensional, and thus the tempeatue depends on only and not on time. That is, T T (). The bounday condition on the inne suface of the wall at is a typical convection condition since it does not involve any adiation o specified heat flu. Taking the diection of heat tansfe to be the positive -diection, the bounday condition on the inne suface can be epessed as k h 1 [T 1 T()] The bounday condition on the oute suface at is quite geneal as it involves conduction, convection, adiation, and specified heat flu. Again taking the diection of heat tansfe to be the positive -diection, the bounday condition on the oute suface can be epessed as k dt() d dt() d h 2 [T() T 2 ] 2 [T() 4 T 4 sky] q sola whee q sola is the incident sola heat flu. Assuming the opposite diection fo heat tansfe would give the same esult multiplied by 1, which is equivalent to the elation hee. All the quantities in these elations ae known ecept the tempeatues and thei deivatives at the two boundaies. Inne suface h 1 T 1 Convection South wall Conduction Conduction T sky Radiation Sola Convection h 2 T 2 Oute suface Sun FIGURE 2 39 Schematic fo Eample 2 1.

26 cen58933_ch2.qd 9/1/22 8:46 AM Page HEAT TRANSFER Heat tansfe poblem Mathematical fomulation (Diffeential equation and bounday conditions) Geneal solution of diffeential equation Application of bounday conditions Solution of the poblem FIGURE 2 4 Basic steps involved in the solution of heat tansfe poblems. Note that a heat tansfe poblem may involve diffeent kinds of bounday conditions on diffeent sufaces. Fo eample, a plate may be subject to heat flu on one suface while losing o gaining heat by convection fom the othe suface. Also, the two bounday conditions in a diection may be specified at the same bounday, while no condition is imposed on the othe bounday. Fo eample, specifying the tempeatue and heat flu at of a plate of thickness will esult in a unique solution fo the one-dimensional steady tempeatue distibution in the plate, including the value of tempeatue at the suface. Although not necessay, thee is nothing wong with specifying moe than two bounday conditions in a specified diection, povided that thee is no contadiction. The eta conditions in this case can be used to veify the esults. 2 5 SOUTION OF STEADY ONE-DIMENSIONA HEAT CONDUCTION PROBEMS So fa we have deived the diffeential equations fo heat conduction in vaious coodinate systems and discussed the possible bounday conditions. A heat conduction poblem can be fomulated by specifying the applicable diffeential equation and a set of pope bounday conditions. In this section we will solve a wide ange of heat conduction poblems in ectangula, cylindical, and spheical geometies. We will limit ou attention to poblems that esult in odinay diffeential equations such as the steady one-dimensional heat conduction poblems. We will also assume constant themal conductivity, but will conside vaiable conductivity late in this chapte. If you feel usty on diffeential equations o haven t taken diffeential equations yet, no need to panic. Simple integation is all you need to solve the steady one-dimensional heat conduction poblems. The solution pocedue fo solving heat conduction poblems can be summaized as (1) fomulate the poblem by obtaining the applicable diffeential equation in its simplest fom and specifying the bounday conditions, (2) obtain the geneal solution of the diffeential equation, and (3) apply the bounday conditions and detemine the abitay constants in the geneal solution (Fig. 2 4). This is demonstated below with eamples. EXAMPE 2 11 Heat Conduction in a Plane Wall T 1 Plane wall T 2 Conside a lage plane wall of thickness.2 m, themal conductivity k 1.2 W/m C, and suface aea A 15 m 2. The two sides of the wall ae maintained at constant tempeatues of T 1 12 C and T 2 5 C, espectively, as shown in Figue Detemine (a) the vaiation of tempeatue within the wall and the value of tempeatue at.1 m and (b) the ate of heat conduction though the wall unde steady conditions. FIGURE 2 41 Schematic fo Eample SOUTION A plane wall with specified suface tempeatues is given. The vaiation of tempeatue and the ate of heat tansfe ae to be detemined. Assumptions 1 Heat conduction is steady. 2 Heat conduction is onedimensional since the wall is lage elative to its thickness and the themal

27 cen58933_ch2.qd 9/1/22 8:46 AM Page CHAPTER 2 conditions on both sides ae unifom. 3 Themal conductivity is constant. 4 Thee is no heat geneation. Popeties The themal conductivity is given to be k 1.2 W/m C. Analysis (a) Taking the diection nomal to the suface of the wall to be the -diection, the diffeential equation fo this poblem can be epessed as with bounday conditions d 2 T d 2 T() T 1 12 C T() T 2 5 C Diffeential equation: Integate: d 2 T d 2 The diffeential equation is linea and second ode, and a quick inspection of it eveals that it has a single tem involving deivatives and no tems involving the unknown function T as a facto. Thus, it can be solved by diect integation. Noting that an integation educes the ode of a deivative by one, the geneal solution of the diffeential equation above can be obtained by two simple successive integations, each of which intoduces an integation constant. Integating the diffeential equation once with espect to yields dt d C 1 whee C 1 is an abitay constant. Notice that the ode of the deivative went down by one as a esult of integation. As a check, if we take the deivative of this equation, we will obtain the oiginal diffeential equation. This equation is not the solution yet since it involves a deivative. Integating one moe time, we obtain T() C 1 C 2 which is the geneal solution of the diffeential equation (Fig. 2 42). The geneal solution in this case esembles the geneal fomula of a staight line whose slope is C 1 and whose value at is C 2. This is not supising since the second deivative epesents the change in the slope of a function, and a zeo second deivative indicates that the slope of the function emains constant. Theefoe, any staight line is a solution of this diffeential equation. The geneal solution contains two unknown constants C 1 and C 2, and thus we need two equations to detemine them uniquely and obtain the specific solution. These equations ae obtained by focing the geneal solution to satisfy the specified bounday conditions. The application of each condition yields one equation, and thus we need to specify two conditions to detemine the constants C 1 and C 2. When applying a bounday condition to an equation, all occuences of the dependent and independent vaiables and any deivatives ae eplaced by the specified values. Thus the only unknowns in the esulting equations ae the abitay constants. The fist bounday condition can be intepeted as in the geneal solution, eplace all the s by zeo and T ( ) by T 1. That is (Fig. 2 43), T() C 1 C 2 C 2 T 1 dt C d 1 Integate again: T() C 1 C 2 Geneal solution Abitay constants FIGURE 2 42 Obtaining the geneal solution of a simple second ode diffeential equation by integation. Bounday condition: T() T 1 Geneal solution: T() C 1 C 2 Applying the bounday condition: T() C 1 C 2 { T 1 Substituting: T 1 C 1 C 2 C 2 T 1 It cannot involve o T() afte the bounday condition is applied. FIGURE 2 43 When applying a bounday condition to the geneal solution at a specified point, all occuences of the dependent and independent vaiables should be eplaced by thei specified values at that point.

28 cen58933_ch2.qd 9/1/22 8:46 AM Page HEAT TRANSFER The second bounday condition can be intepeted as in the geneal solution, eplace all the s by and T ( ) by T 2. That is, T() C 1 C 2 T 2 C 1 T 1 C 1 T 2 T 1 Substituting the C 1 and C 2 epessions into the geneal solution, we obtain T 2 T 1 T() T 1 (2-56) which is the desied solution since it satisfies not only the diffeential equation but also the two specified bounday conditions. That is, diffeentiating Eq with espect to twice will give d 2 T /d 2, which is the given diffeential equation, and substituting and into Eq gives T () T 1 and T () T 2, espectively, which ae the specified conditions at the boundaies. Substituting the given infomation, the value of the tempeatue at.1 m is detemined to be (5 12) C T(.1 m) (.1 m) 12 C 85 C.2 m (b) The ate of heat conduction anywhee in the wall is detemined fom Fouie s law to be dt T 2 T 1 T 1 T 2 Q wall ka kac 1 ka ka (2-57) d The numeical value of the ate of heat conduction though the wall is detemined by substituting the given values to be T 1 T 2 (12 5) C Q ka (1.2 W/m C)(15 m 2 ) 63 W.2 m Discussion Note that unde steady conditions, the ate of heat conduction though a plane wall is constant. EXAMPE 2 12 A Wall with Vaious Sets of Bounday Conditions Conside steady one-dimensional heat conduction in a lage plane wall of thickness and constant themal conductivity k with no heat geneation. Obtain epessions fo the vaiation of tempeatue within the wall fo the following pais of bounday conditions (Fig. 2 44): dt() (a) k q 4 W/cm 2 and T() T 15 C d dt() dt() (b) k q 4 W/cm 2 and k q 25 W/cm d d 2 dt() dt() (c) k q 4 W/cm 2 and k q 4 W/cm d d 2

29 cen58933_ch2.qd 9/1/22 8:46 AM Page CHAPTER 2 15 C 4 W/cm 2 Plane wall T() 4 W/cm 2 Plane wall T() 4 W/cm 2 Plane wall T() 25 W/cm 2 4 W/cm 2 (a) (b) (c) FIGURE 2 44 Schematic fo Eample SOUTION This is a steady one-dimensional heat conduction poblem with constant themal conductivity and no heat geneation in the medium, and the heat conduction equation in this case can be epessed as (Eq. 2 17) whose geneal solution was detemined in the pevious eample by diect integation to be T() C 1 C 2 whee C 1 and C 2 ae two abitay integation constants. The specific solutions coesponding to each specified pai of bounday conditions ae detemined as follows. (a) In this case, both bounday conditions ae specified at the same bounday at, and no bounday condition is specified at the othe bounday at. Noting that C 1 the application of the bounday conditions gives and d 2 T d 2 dt() q k q kc 1 q C 1 d k T() T T C 1 C 2 C 2 T Substituting, the specific solution in this case is detemined to be dt d q T() T k Theefoe, the two bounday conditions can be specified at the same bounday, and it is not necessay to specify them at diffeent locations. In fact, the fundamental theoem of linea odinay diffeential equations guaantees that a

30 cen58933_ch2.qd 9/1/22 8:46 AM Page 9 9 HEAT TRANSFER Diffeential equation: T () Geneal solution: T() C 1 C 2 (a) Unique solution: kt() q q T() T T() T k (b) No solution: kt() q T() None kt() q (c) Multiple solutions: kt() q q T() C kt() k 2 q Abitay FIGURE 2 45 A bounday-value poblem may have a unique solution, infinitely many solutions, o no solutions at all. unique solution eists when both conditions ae specified at the same location. But no such guaantee eists when the two conditions ae specified at diffeent boundaies, as you will see below. (b) In this case diffeent heat flues ae specified at the two boundaies. The application of the bounday conditions gives and dt() k q kc 1 q C 1 d dt() k q kc 1 q C 1 d Since q q and the constant C 1 cannot be equal to two diffeent things at the same time, thee is no solution in this case. This is not supising since this case coesponds to supplying heat to the plane wall fom both sides and epecting the tempeatue of the wall to emain steady (not to change with time). This is impossible. (c) In this case, the same values fo heat flu ae specified at the two boundaies. The application of the bounday conditions gives dt() q k q kc 1 q C 1 d k q k q k and dt() q k q kc 1 q C 1 d k Resistance heate 12 W Insulation Base plate 3 cm 2 T = 2 C Thus, both conditions esult in the same value fo the constant C 1, but no value fo C 2. Substituting, the specific solution in this case is detemined to be q T() C k 2 which is not a unique solution since C 2 is abitay. This solution epesents a family of staight lines whose slope is q /k. Physically, this poblem coesponds to equiing the ate of heat supplied to the wall at be equal to the ate of heat emoval fom the othe side of the wall at. But this is a consequence of the heat conduction though the wall being steady, and thus the second bounday condition does not povide any new infomation. So it is not supising that the solution of this poblem is not unique. The thee cases discussed above ae summaized in Figue h EXAMPE 2 13 Heat Conduction in the Base Plate of an Ion FIGURE 2 46 Schematic fo Eample Conside the base plate of a 12-W household ion that has a thickness of.5 cm, base aea of A 3 cm 2, and themal conductivity of k 15 W/m C. The inne suface of the base plate is subjected to unifom heat flu geneated by the esistance heates inside, and the oute suface loses heat to the suoundings at T 2 C by convection, as shown in Figue 2 46.

31 cen58933_ch2.qd 9/1/22 8:46 AM Page CHAPTER 2 Taking the convection heat tansfe coefficient to be h 8 W/m 2 C and disegading heat loss by adiation, obtain an epession fo the vaiation of tempeatue in the base plate, and evaluate the tempeatues at the inne and the oute sufaces. SOUTION The base plate of an ion is consideed. The vaiation of tempeatue in the plate and the suface tempeatues ae to be detemined. Assumptions 1 Heat tansfe is steady since thee is no change with time. 2 Heat tansfe is one-dimensional since the suface aea of the base plate is lage elative to its thickness, and the themal conditions on both sides ae unifom. 3 Themal conductivity is constant. 4 Thee is no heat geneation in the medium. 5 Heat tansfe by adiation is negligible. 6 The uppe pat of the ion is well insulated so that the entie heat geneated in the esistance wies is tansfeed to the base plate though its inne suface. Popeties The themal conductivity is given to be k 15 W/m C. Analysis The inne suface of the base plate is subjected to unifom heat flu at a ate of Q 12 W q 4, W/m 2.3 m 2 A base The oute side of the plate is subjected to the convection condition. Taking the diection nomal to the suface of the wall as the -diection with its oigin on the inne suface, the diffeential equation fo this poblem can be epessed as (Fig. 2 47) with the bounday conditions d 2 T d 2 dt() k q 4, W/m d 2 Heat flu Base plate Conduction q = k dt() d Conduction h T Convection k dt() = h[t() T ] d FIGURE 2 47 The bounday conditions on the base plate of the ion discussed in Eample dt() k h[t() T ] d The geneal solution of the diffeential equation is again obtained by two successive integations to be and dt d C 1 T() C 1 C 2 (a) whee C 1 and C 2 ae abitay constants. Applying the fist bounday condition, dt() q k q kc 1 q C 1 d k Noting that dt /d C 1 and T () C 1 C 2, the application of the second bounday condition gives

32 cen58933_ch2.qd 9/1/22 8:46 AM Page HEAT TRANSFER dt() k h[t() T ] kc 1 h[(c 1 C 2 ) T ] d Substituting C 1 q /k and solving fo C 2, we obtain q q C 2 T h k Now substituting C 1 and C 2 into the geneal solution (a) gives 1 T() T q (b) k which is the solution fo the vaiation of the tempeatue in the plate. The tempeatues at the inne and oute sufaces of the plate ae detemined by substituting and, espectively, into the elation (b): and 1 T() T q k h 2 C (4, W/m 2.5 m 1 ) 533 C 15 W/m C 8 W/m 2 C 1 4, W/m 2 T() T q 2 C 52 C 8 W/m 2 C h Discussion Note that the tempeatue of the inne suface of the base plate will be 13 C highe than the tempeatue of the oute suface when steady opeating conditions ae eached. Also note that this heat tansfe analysis enables us to calculate the tempeatues of sufaces that we cannot even each. This eample demonstates how the heat flu and convection bounday conditions ae applied to heat tansfe poblems. h EXAMPE 2 14 Heat Conduction in a Sola Heated Wall T 1 Plane wall Conduction ε α Sola Radiation Sun Space FIGURE 2 48 Schematic fo Eample Conside a lage plane wall of thickness.6 m and themal conductivity k 1.2 W/m C in space. The wall is coveed with white pocelain tiles that have an emissivity of.85 and a sola absoptivity of.26, as shown in Figue The inne suface of the wall is maintained at T 1 3 K at all times, while the oute suface is eposed to sola adiation that is incident at a ate of q sola 8 W/m 2. The oute suface is also losing heat by adiation to deep space at K. Detemine the tempeatue of the oute suface of the wall and the ate of heat tansfe though the wall when steady opeating conditions ae eached. What would you esponse be if no sola adiation was incident on the suface? SOUTION A plane wall in space is subjected to specified tempeatue on one side and sola adiation on the othe side. The oute suface tempeatue and the ate of heat tansfe ae to be detemined.

33 cen58933_ch2.qd 9/1/22 8:46 AM Page CHAPTER 2 Assumptions 1 Heat tansfe is steady since thee is no change with time. 2 Heat tansfe is one-dimensional since the wall is lage elative to its thickness, and the themal conditions on both sides ae unifom. 3 Themal conductivity is constant. 4 Thee is no heat geneation. Popeties The themal conductivity is given to be k 1.2 W/m C. Analysis Taking the diection nomal to the suface of the wall as the -diection with its oigin on the inne suface, the diffeential equation fo this poblem can be epessed as with bounday conditions k dt() d d 2 T d 2 T() T 1 3 K [T() 4 T 4 space] q sola whee T space. The geneal solution of the diffeential equation is again obtained by two successive integations to be T() C 1 C 2 (a) whee C 1 and C 2 ae abitay constants. Applying the fist bounday condition yields T() C 1 C 2 C 2 T 1 Noting that dt /d C 1 and T () C 1 C 2 C 1 T 1, the application of the second bounday conditions gives dt() k T() 4 q sola kc 1 (C 1 T 1 ) 4 q sola d Although C 1 is the only unknown in this equation, we cannot get an eplicit epession fo it because the equation is nonlinea, and thus we cannot get a closed-fom epession fo the tempeatue distibution. This should eplain why we do ou best to avoid nonlineaities in the analysis, such as those associated with adiation. et us back up a little and denote the oute suface tempeatue by T () T instead of T () C 1 T 1. The application of the second bounday condition in this case gives dt() k T() 4 q sola kc 1 T 4 q sola d Solving fo C 1 gives C 1 q sola T 4 k (b) Now substituting C 1 and C 2 into the geneal solution (a), we obtain q sola T 4 T() T 1 (c) k

34 cen58933_ch2.qd 9/1/22 8:46 AM Page HEAT TRANSFER (1) Reaange the equation to be solved: T T 1 4 The equation is in the pope fom since the left side consists of T only. (2) Guess the value of T, say 3 K, and substitute into the ight side of the equation. It gives T 29.2 K (3) Now substitute this value of T into the ight side of the equation and get T K (4) Repeat step (3) until convegence to desied accuacy is achieved. The subsequent iteations give T K T K T K Theefoe, the solution is T K. The esult is independent of the initial guess. FIGURE 2 49 A simple method of solving a nonlinea equation is to aange the equation such that the unknown is alone on the left side while eveything else is on the ight side, and to iteate afte an initial guess until convegence. which is the solution fo the vaiation of the tempeatue in the wall in tems of the unknown oute suface tempeatue T. At it becomes T T 1 (d ) k which is an implicit elation fo the oute suface tempeatue T. Substituting the given values, we get T which simplifies to T T 1 4 (.6 m) 3 K This equation can be solved by one of the seveal nonlinea equation solves available (o by the old fashioned tial-and-eo method) to give (Fig. 2 49) T K Knowing the oute suface tempeatue and knowing that it must emain constant unde steady conditions, the tempeatue distibution in the wall can be detemined by substituting the T value above into Eq. (c):.26 (8 W/m 2 ).85 ( W/m 2 K 4 )(292.7 K) 4 T() 3 K 1.2 W/m K which simplifies to q sola T 4.26 (8 W/m 2 ).85 ( W/m 2 K 4 ) T W/m K T() (121.5 K/m) 3 K Note that the oute suface tempeatue tuned out to be lowe than the inne suface tempeatue. Theefoe, the heat tansfe though the wall will be towad the outside despite the absoption of sola adiation by the oute suface. Knowing both the inne and oute suface tempeatues of the wall, the steady ate of heat conduction though the wall can be detemined fom T T ( ) K q k (1.2 W/m K) 146 W/m.6 m 2 Discussion In the case of no incident sola adiation, the oute suface tempeatue, detemined fom Eq. (d ) by setting q sola, will be T K. It is inteesting to note that the sola enegy incident on the suface causes the suface tempeatue to incease by about 8 K only when the inne suface tempeatue of the wall is maintained at 3 K. T 2 T1 1 2 FIGURE 2 5 Schematic fo Eample EXAMPE 2 15 Heat oss though a Steam Pipe Conside a steam pipe of length 2 m, inne adius 1 6 cm, oute adius 2 8 cm, and themal conductivity k 2 W/m C, as shown in Figue 2 5. The inne and oute sufaces of the pipe ae maintained at aveage tempeatues of T 1 15 C and T 2 6 C, espectively. Obtain a geneal elation

35 cen58933_ch2.qd 9/1/22 8:46 AM Page CHAPTER 2 fo the tempeatue distibution inside the pipe unde steady conditions, and detemine the ate of heat loss fom the steam though the pipe. SOUTION A steam pipe is subjected to specified tempeatues on its sufaces. The vaiation of tempeatue and the ate of heat tansfe ae to be detemined. Assumptions 1 Heat tansfe is steady since thee is no change with time. 2 Heat tansfe is one-dimensional since thee is themal symmety about the centeline and no vaiation in the aial diection, and thus T T ( ). 3 Themal conductivity is constant. 4 Thee is no heat geneation. Popeties The themal conductivity is given to be k 2 W/m C. Analysis The mathematical fomulation of this poblem can be epessed as with bounday conditions d dt d d T( 1 ) T 1 15 C T( 2 ) T 2 6 C Integating the diffeential equation once with espect to gives dt C d 1 whee C 1 is an abitay constant. We now divide both sides of this equation by to bing it to a eadily integable fom, dt d C 1 Again integating with espect to gives (Fig. 2 51) T() C 1 ln C 2 We now apply both bounday conditions by eplacing all occuences of and T ( ) in Eq. (a) with the specified values at the boundaies. We get T( 1 ) T 1 C 1 ln 1 C 2 T 1 T( 2 ) T 2 C 1 ln 2 C 2 T 2 which ae two equations in two unknowns, C 1 and C 2. Solving them simultaneously gives T 2 T 1 T 2 T 1 C 1 and C 2 T 1 ln ln( ln( 1 2 / 1 ) 2 / 1 ) Substituting them into Eq. (a) and eaanging, the vaiation of tempeatue within the pipe is detemined to be T() ln(/ 1) (T 2 T 1 ) T 1 (2-58) ln( 2 / 1 The ate of heat loss fom the steam is simply the total ate of heat conduction though the pipe, and is detemined fom Fouie s law to be (a) Diffeential equation: d dt d d Integate: dt C d 1 Divide by ( ): dt d C 1 Integate again: T() C 1 ln C 2 which is the geneal solution. FIGURE 2 51 Basic steps involved in the solution of the steady one-dimensional heat conduction equation in cylindical coodinates.

36 cen58933_ch2.qd 9/1/22 8:46 AM Page HEAT TRANSFER C 1 dt T 1 T 2 Q cylinde ka k(2) 2kC 1 2k (2-59) d ln( 2 / 1 ) The numeical value of the ate of heat conduction though the pipe is detemined by substituting the given values (15 6) C Q 2(2 W/m C)(2 m) 786 kw ln(.8/.6) DISCUSSION Note that the total ate of heat tansfe though a pipe is constant, but the heat flu is not since it deceases in the diection of heat tansfe with inceasing adius since q Q /(2). T 2 EXAMPE 2 16 Heat Conduction though a Spheical Shell T 1 1 FIGURE 2 52 Schematic fo Eample Conside a spheical containe of inne adius 1 8 cm, oute adius 2 1 cm, and themal conductivity k 45 W/m C, as shown in Figue The inne and oute sufaces of the containe ae maintained at constant tempeatues of T 1 2 C and T 2 8 C, espectively, as a esult of some chemical eactions occuing inside. Obtain a geneal elation fo the tempeatue distibution inside the shell unde steady conditions, and detemine the ate of heat loss fom the containe. SOUTION A spheical containe is subjected to specified tempeatues on its sufaces. The vaiation of tempeatue and the ate of heat tansfe ae to be detemined. Assumptions 1 Heat tansfe is steady since thee is no change with time. 2 Heat tansfe is one-dimensional since thee is themal symmety about the midpoint, and thus T T ( ). 3 Themal conductivity is constant. 4 Thee is no heat geneation. Popeties The themal conductivity is given to be k 45 W/m C. Analysis The mathematical fomulation of this poblem can be epessed as d 2 dt d d with bounday conditions T( 1 ) T 1 2 C T( 2 ) T 2 8 C Integating the diffeential equation once with espect to yields 2 dt C d 1 whee C 1 is an abitay constant. We now divide both sides of this equation by 2 to bing it to a eadily integable fom, dt C 1 d 2

37 cen58933_ch2.qd 9/1/22 8:46 AM Page CHAPTER 2 Again integating with espect to gives C 1 T() C 2 (a) We now apply both bounday conditions by eplacing all occuences of and T ( ) in the elation above by the specified values at the boundaies. We get C 1 1 T( 1 ) T 1 C 2 T 1 C 1 2 T( 2 ) T 2 C 2 T 2 which ae two equations in two unknowns, C 1 and C 2. Solving them simultaneously gives 1 2 C 1 (T 1 T 2 ) and C Substituting into Eq. (a), the vaiation of tempeatue within the spheical shell is detemined to be T 2 1 T 1 T() (T 1 T 2 ) (2-6) ( 2 1 ) 2 1 The ate of heat loss fom the containe is simply the total ate of heat conduction though the containe wall and is detemined fom Fouie s law C T 2 1 T dt T Q sphee ka k(4 2 1 T 2 ) 4kC 1 4k 1 2 (2-61) d 2 1 The numeical value of the ate of heat conduction though the wall is detemined by substituting the given values to be (2 8) C Q 4(45 W/m C)(.8 m)(.1 m) 27,14 W (.1.8) m Discussion Note that the total ate of heat tansfe though a spheical shell is constant, but the heat flu, q Q /4 2, is not since it deceases in the diection of heat tansfe with inceasing adius as shown in Figue Q 1 Q 2 = Q q 1 q 2 < q 1 q Q kw = = kw/m 2 1 = 1 A1 4 π (.8 m) 2 q Q kw = = 216. kw/m 2 2 = 2 A2 4 π (.1 m) 2 FIGURE 2 53 Duing steady one-dimensional heat conduction in a spheical (o cylindical) containe, the total ate of heat tansfe emains constant, but the heat flu deceases with inceasing adius. Chemical eactions 2 6 HEAT GENERATION IN A SOID Many pactical heat tansfe applications involve the convesion of some fom of enegy into themal enegy in the medium. Such mediums ae said to involve intenal heat geneation, which manifests itself as a ise in tempeatue thoughout the medium. Some eamples of heat geneation ae esistance heating in wies, eothemic chemical eactions in a solid, and nuclea eactions in nuclea fuel ods whee electical, chemical, and nuclea enegies ae conveted to heat, espectively (Fig. 2 54). The absoption of adiation thoughout the volume of a semitanspaent medium such as wate can also be consideed as heat geneation within the medium, as eplained ealie. Nuclea fuel ods Electic esistance wies FIGURE 2 54 Heat geneation in solids is commonly encounteed in pactice.

38 cen58933_ch2.qd 9/1/22 8:46 AM Page HEAT TRANSFER Heat geneation is usually epessed pe unit volume of the medium, and is denoted by g, whose unit is W/m 3. Fo eample, heat geneation in an electical wie of oute adius and length can be epessed as E g.electic V wie I g 2 R e (W/m 3 ) (2-62) o 2 h, T V k Heat geneation E gen = gv Q = E gen FIGURE 2 55 At steady conditions, the entie heat geneated in a solid must leave the solid though its oute suface. T s whee I is the electic cuent and R e is the electical esistance of the wie. The tempeatue of a medium ises duing heat geneation as a esult of the absoption of the geneated heat by the medium duing tansient stat-up peiod. As the tempeatue of the medium inceases, so does the heat tansfe fom the medium to its suoundings. This continues until steady opeating conditions ae eached and the ate of heat geneation equals the ate of heat tansfe to the suoundings. Once steady opeation has been established, the tempeatue of the medium at any point no longe changes. The maimum tempeatue T ma in a solid that involves unifom heat geneation will occu at a location fathest away fom the oute suface when the oute suface of the solid is maintained at a constant tempeatue T s. Fo eample, the maimum tempeatue occus at the midplane in a plane wall, at the centeline in a long cylinde, and at the midpoint in a sphee. The tempeatue distibution within the solid in these cases will be symmetical about the cente of symmety. The quantities of majo inteest in a medium with heat geneation ae the suface tempeatue T s and the maimum tempeatue T ma that occus in the medium in steady opeation. Below we develop epessions fo these two quantities fo common geometies fo the case of unifom heat geneation (g constant) within the medium. Conside a solid medium of suface aea A s, volume V, and constant themal conductivity k, whee heat is geneated at a constant ate of g pe unit volume. Heat is tansfeed fom the solid to the suounding medium at T, with a constant heat tansfe coefficient of h. All the sufaces of the solid ae maintained at a common tempeatue T s. Unde steady conditions, the enegy balance fo this solid can be epessed as (Fig. 2 55) Rate of Rate of heat tansfe enegy geneation (2-63) fom the solid within the solid o Q g V (W) (2-64) Disegading adiation (o incopoating it in the heat tansfe coefficient h), the heat tansfe ate can also be epessed fom Newton s law of cooling as Q ha s (T s T ) (W) (2-65) Combining Eqs and 2 65 and solving fo the suface tempeatue T s gives g V T s T (2-66) ha s

39 cen58933_ch2.qd 9/1/22 8:46 AM Page CHAPTER 2 Fo a lage plane wall of thickness 2 (A s 2A wall and V 2A wall ), a long solid cylinde of adius o (A s 2 o and V o 2 ), and a solid sphee of adius (A s 4o 2 and V 4 o), 3 Eq educes to 3 g T s, plane wall T h (2-67) g o T s, cylinde T 2h (2-68) g o T s, sphee T 3h (2-69) A Q = E gen Note that the ise in suface tempeatue T s is due to heat geneation in the solid. Reconside heat tansfe fom a long solid cylinde with heat geneation. We mentioned above that, unde steady conditions, the entie heat geneated within the medium is conducted though the oute suface of the cylinde. Now conside an imaginay inne cylinde of adius within the cylinde (Fig. 2 56). Again the heat geneated within this inne cylinde must be equal to the heat conducted though the oute suface of this inne cylinde. That is, fom Fouie s law of heat conduction, dt ka g V (2-7) d whee A 2 and V 2 at any location. Substituting these epessions into Eq. 2 7 and sepaating the vaiables, we get dt g k(2) g ( 2 ) dt d d 2k Integating fom whee T() T to o whee T( o ) T s yields g o 2 T ma, cylinde T o T s (2-71) 4k whee T o is the centeline tempeatue of the cylinde, which is the maimum tempeatue, and T ma is the diffeence between the centeline and the suface tempeatues of the cylinde, which is the maimum tempeatue ise in the cylinde above the suface tempeatue. Once T ma is available, the centeline tempeatue can easily be detemined fom (Fig. 2 57) T cente T o T s T ma (2-72) The appoach outlined above can also be used to detemine the maimum tempeatue ise in a plane wall of thickness 2 and a solid sphee of adius, with these esults: V E gen = gv FIGURE 2 56 Heat conducted though a cylindical shell of adius is equal to the heat geneated within a shell. T ma T o = T ma T T s T s Heat geneation Symmety line FIGURE 2 57 The maimum tempeatue in a symmetical solid with unifom heat geneation occus at its cente. o T g 2 T ma, plane wall 2k (2-73) g o 2 T ma, sphee 6k (2-74)

40 cen58933_ch2.qd 9/1/22 8:46 AM Page 1 1 HEAT TRANSFER Again the maimum tempeatue at the cente can be detemined fom Eq by adding the maimum tempeatue ise to the suface tempeatue of the solid. Wate T o D = 4 mm T s = 15 C FIGURE 2 58 Schematic fo Eample Q q EXAMPE 2 17 Centeline Tempeatue of a Resistance Heate A 2-kW esistance heate wie whose themal conductivity is k 15 W/m C has a diamete of D 4 mm and a length of.5 m, and is used to boil wate (Fig. 2 58). If the oute suface tempeatue of the esistance wie is T s 15 C, detemine the tempeatue at the cente of the wie. SOUTION The suface tempeatue of a esistance heate submeged in wate is to be detemined. Assumptions 1 Heat tansfe is steady since thee is no change with time. 2 Heat tansfe is one-dimensional since thee is themal symmety about the centeline and no change in the aial diection. 3 Themal conductivity is constant. 4 Heat geneation in the heate is unifom. Popeties The themal conductivity is given to be k 15 W/m C. Analysis The 2-kW esistance heate convets electic enegy into heat at a ate of 2 kw. The heat geneation pe unit volume of the wie is Q gen Q gen 2 W g W/m 3 o 2 (.2 m) 2 (.5 m) V wie Then the cente tempeatue of the wie is detemined fom Eq to be g o 2 ( W/m 3 )(.2 m) 2 T o T s 15 C 126 C 4k 4 (15 W/m C) Discussion Note that the tempeatue diffeence between the cente and the suface of the wie is 21 C. Wate 226 F We have developed these elations using the intuitive enegy balance appoach. Howeve, we could have obtained the same elations by setting up the appopiate diffeential equations and solving them, as illustated in Eamples 2 18 and g FIGURE 2 59 Schematic fo Eample EXAMPE 2 18 Vaiation of Tempeatue in a Resistance Heate A long homogeneous esistance wie of adius.2 in. and themal conductivity k 7.8 Btu/h ft F is being used to boil wate at atmospheic pessue by the passage of electic cuent, as shown in Figue Heat is geneated in the wie unifomly as a esult of esistance heating at a ate of g 24 Btu/h in 3. If the oute suface tempeatue of the wie is measued to be T s 226 F, obtain a elation fo the tempeatue distibution, and detemine the tempeatue at the centeline of the wie when steady opeating conditions ae eached.

41 cen58933_ch2.qd 9/1/22 8:46 AM Page CHAPTER 2 SOUTION This heat tansfe poblem is simila to the poblem in Eample 2 17, ecept that we need to obtain a elation fo the vaiation of tempeatue within the wie with. Diffeential equations ae well suited fo this pupose. Assumptions 1 Heat tansfe is steady since thee is no change with time. 2 Heat tansfe is one-dimensional since thee is no themal symmety about the centeline and no change in the aial diection. 3 Themal conductivity is constant. 4 Heat geneation in the wie is unifom. Popeties The themal conductivity is given to be k 7.8 Btu/h ft F. Analysis The diffeential equation which govens the vaiation of tempeatue in the wie is simply Eq. 2 27, 1 d d dt d g k This is a second-ode linea odinay diffeential equation, and thus its geneal solution will contain two abitay constants. The detemination of these constants equies the specification of two bounday conditions, which can be taken to be T( ) T s 226 F dt() d = and T T() dt() d The fist bounday condition simply states that the tempeatue of the oute suface of the wie is 226 F. The second bounday condition is the symmety condition at the centeline, and states that the maimum tempeatue in the wie will occu at the centeline, and thus the slope of the tempeatue at must be zeo (Fig. 2 6). This completes the mathematical fomulation of the poblem. Although not immediately obvious, the diffeential equation is in a fom that can be solved by diect integation. Multiplying both sides of the equation by and eaanging, we obtain d g dt d d k FIGURE 2 6 The themal symmety condition at the centeline of a wie in which heat is geneated unifomly. g Integating with espect to gives dt g 2 C 1 (a) d k 2 since the heat geneation is constant, and the integal of a deivative of a function is the function itself. That is, integation emoves a deivative. It is convenient at this point to apply the second bounday condition, since it is elated to the fist deivative of the tempeatue, by eplacing all occuences of and dt /d in Eq. (a) by zeo. It yields dt() g C 1 C 1 d 2k

42 cen58933_ch2.qd 9/1/22 8:47 AM Page HEAT TRANSFER Thus C 1 cancels fom the solution. We now divide Eq. (a) by to bing it to a eadily integable fom, dt g d 2k Again integating with espect to gives g T() 2 C 2 (b) 4k We now apply the fist bounday condition by eplacing all occuences of by and all occuences of T by T s. We get g g T s 2 C 2 C 2 T s 4k 4k Substituting this C 2 elation into Eq. (b) and eaanging give g T() T s ( 2 2 ) (c) 4k which is the desied solution fo the tempeatue distibution in the wie as a function of. The tempeatue at the centeline ( ) is obtained by eplacing in Eq. (c) by zeo and substituting the known quantities, g 24 Btu/h in 12 in. T() T s 226 F 3 (.2 in.) F 4k 4 (7.8 Btu/h ft F) 1 ft 2 Discussion The tempeatue of the centeline will be 37 F above the tempeatue of the oute suface of the wie. Note that the epession above fo the centeline tempeatue is identical to Eq. 2 71, which was obtained using an enegy balance on a contol volume. 2 EXAMPE 2 19 Heat Conduction in a Two-aye Medium Inteface Wie 2 Ts = 45 C 1 Ceamic laye FIGURE 2 61 Schematic fo Eample Conside a long esistance wie of adius 1.2 cm and themal conductivity k wie 15 W/m C in which heat is geneated unifomly as a esult of esistance heating at a constant ate of g 5 W/cm 3 (Fig. 2 61). The wie is embedded in a.5-cm-thick laye of ceamic whose themal conductivity is k ceamic 1.2 W/m C. If the oute suface tempeatue of the ceamic laye is measued to be T s 45 C, detemine the tempeatues at the cente of the esistance wie and the inteface of the wie and the ceamic laye unde steady conditions. SOUTION The suface and inteface tempeatues of a esistance wie coveed with a ceamic laye ae to be detemined. Assumptions 1 Heat tansfe is steady since thee is no change with time. 2 Heat tansfe is one-dimensional since this two-laye heat tansfe poblem possesses symmety about the centeline and involves no change in the aial diection, and thus T T ( ). 3 Themal conductivities ae constant. 4 Heat geneation in the wie is unifom. Popeties It is given that k wie 15 W/m C and k ceamic 1.2 W/m C.

43 cen58933_ch2.qd 9/1/22 8:47 AM Page CHAPTER 2 Analysis etting T I denote the unknown inteface tempeatue, the heat tansfe poblem in the wie can be fomulated as with dt wie d T wie ( 1 ) T I dt wie () d This poblem was solved in Eample 2 18, and its solution was detemined to be g T wie () T I ( 2 ) (a) Noting that the ceamic laye does not involve any heat geneation and its oute suface tempeatue is specified, the heat conduction poblem in that laye can be epessed as with T ceamic ( 1 ) T I T ceamic ( 2 ) T s 45 C This poblem was solved in Eample 2 15, and its solution was detemined to be ln(/ 1 ) T ceamic () (T s T I ) T I (b) ln( 2 / 1 ) We have aleady utilized the fist inteface condition by setting the wie and ceamic laye tempeatues equal to T I at the inteface 1. The inteface tempeatue T I is detemined fom the second inteface condition that the heat flu in the wie and the ceamic laye at 1 must be the same: dt wie ( 1 ) dt ceamic ( 1 ) g 1 T s T I k wie k ceamic k d d 2 ceamic ln( 2 / 1 ) Solving fo T I and substituting the given values, the inteface tempeatue is detemined to be g T I ln T s 2k ceamic (5 1 6 W/m 3 )(.2 m) 2.7 m ln 45 C C 2(1.2 W/m C).2 m Knowing the inteface tempeatue, the tempeatue at the centeline ( ) is obtained by substituting the known quantities into Eq. (a), g 1 2 (5 1 6 W/m 3 )(.2 m) 2 T wie () T I C C 4 (15 W/m C) 4k wie 1 1 d d d d 4k wie dt ceamic d g k

44 cen58933_ch2.qd 9/1/22 8:47 AM Page HEAT TRANSFER Thus the tempeatue of the centeline will be slightly above the inteface tempeatue. Discussion This eample demonstates how steady one-dimensional heat conduction poblems in composite media can be solved. We could also solve this poblem by detemining the heat flu at the inteface by dividing the total heat geneated in the wie by the suface aea of the wie, and then using this value as the specifed heat flu bounday condition fo both the wie and the ceamic laye. This way the two poblems ae decoupled and can be solved sepaately. Themal conductivity (W/m K) Tungsten Aluminum oide Silve Coppe Gold Aluminum Platinum Ion Stainless steel, AISI VARIABE THERMA CONDUCTIVITY, k (T ) You will ecall fom Chapte 1 that the themal conductivity of a mateial, in geneal, vaies with tempeatue (Fig. 2 62). Howeve, this vaiation is mild fo many mateials in the ange of pactical inteest and can be disegaded. In such cases, we can use an aveage value fo the themal conductivity and teat it as a constant, as we have been doing so fa. This is also common pactice fo othe tempeatue-dependent popeties such as the density and specific heat. When the vaiation of themal conductivity with tempeatue in a specified tempeatue inteval is lage, howeve, it may be necessay to account fo this vaiation to minimize the eo. Accounting fo the vaiation of the themal conductivity with tempeatue, in geneal, complicates the analysis. But in the case of simple one-dimensional cases, we can obtain heat tansfe elations in a staightfowad manne. When the vaiation of themal conductivity with tempeatue k(t) is known, the aveage value of the themal conductivity in the tempeatue ange between T 1 and T 2 can be detemined fom Pyoceam 2 Fused quatz Tempeatue (K) FIGURE 2 62 Vaiation of the themal conductivity of some solids with tempeatue. T 2 T 1 k(t)dt k ave (2-75) T 2 T 1 This elation is based on the equiement that the ate of heat tansfe though a medium with constant aveage themal conductivity k ave equals the ate of heat tansfe though the same medium with vaiable conductivity k(t). Note that in the case of constant themal conductivity k(t) k, Eq educes to k ave k, as epected. Then the ate of steady heat tansfe though a plane wall, cylindical laye, o spheical laye fo the case of vaiable themal conductivity can be detemined by eplacing the constant themal conductivity k in Eqs. 2 57, 2 59, and 2 61 by the k ave epession (o value) fom Eq. 2 75: T1 T 2 T 1 T 2 A Q plane wall k ave A k(t)dt (2-76) T 1 T 2 2 Q cylinde 2k ave k(t)dt (2-77) ln( 2 / 1 ) T1 ln( 2 / 1 ) T 1 T Q sphee 4k ave 1 2 k(t)dt (2-78) 2 1 T T 2 T 2

45 cen58933_ch2.qd 9/1/22 8:47 AM Page 15 The vaiation in themal conductivity of a mateial with tempeatue in the tempeatue ange of inteest can often be appoimated as a linea function and epessed as k(t) k (1 T) (2-79) whee is called the tempeatue coefficient of themal conductivity. The aveage value of themal conductivity in the tempeatue ange T 1 to T 2 in this case can be detemined fom T T 1 15 CHAPTER 2 Plane wall k(t) = k (1 + β T) β > β = T 2 k (1 T)dT T 1 T 2 T 1 k ave k 1 k(t ave ) (2-8) T 2 T 1 2 Note that the aveage themal conductivity in this case is equal to the themal conductivity value at the aveage tempeatue. We have mentioned ealie that in a plane wall the tempeatue vaies linealy duing steady one-dimensional heat conduction when the themal conductivity is constant. But this is no longe the case when the themal conductivity changes with tempeatue, even linealy, as shown in Figue β < FIGURE 2 63 The vaiation of tempeatue in a plane wall duing steady one-dimensional heat conduction fo the cases of constant and vaiable themal conductivity. T 2 EXAMPE 2 2 Vaiation of Tempeatue in a Wall with k(t ) Conside a plane wall of thickness whose themal conductivity vaies linealy in a specified tempeatue ange as k (T ) k (1 T ) whee k and ae constants. The wall suface at is maintained at a constant tempeatue of T 1 while the suface at is maintained at T 2, as shown in Figue Assuming steady one-dimensional heat tansfe, obtain a elation fo (a) the heat tansfe ate though the wall and (b) the tempeatue distibution T ( ) in the wall. SOUTION A plate with vaiable conductivity is subjected to specified tempeatues on both sides. The vaiation of tempeatue and the ate of heat tansfe ae to be detemined. Assumptions 1 Heat tansfe is given to be steady and one-dimensional. 2 Themal conductivity vaies linealy. 3 Thee is no heat geneation. Popeties The themal conductivity is given to be k (T ) k (1 T ). Analysis (a) The ate of heat tansfe though the wall can be detemined fom Q k ave A T 1 T 2 T 1 Plane wall k(t) = k (1 + β T) T 2 FIGURE 2 64 Schematic fo Eample 2 2. whee A is the heat conduction aea of the wall and k ave k(t ave ) k 1 is the aveage themal conductivity (Eq. 2 8). (b) To detemine the tempeatue distibution in the wall, we begin with Fouie s law of heat conduction, epessed as Q k(t) A dt d T 2 T 1 2

46 cen58933_ch2.qd 9/1/22 8:47 AM Page HEAT TRANSFER whee the ate of conduction heat tansfe Q and the aea A ae constant. Sepaating vaiables and integating fom whee T () T 1 to any whee T ( ) T, we get Q d A k(t)dt Substituting k (T ) k (1 T ) and pefoming the integations we obtain Q Ak [(T T 1 ) (T 2 )/2] Substituting the Q epession fom pat (a) and eaanging give 2k ave T T (T 1 T 2 ) T1 2 T 1 k which is a quadatic equation in the unknown tempeatue T. Using the quadatic fomula, the tempeatue distibution T ( ) in the wall is detemined to be 1 T() 1 2k ave 2 k (T 1 T 2 ) T1 2 2 T 1 The pope sign of the squae oot tem ( o ) is detemined fom the equiement that the tempeatue at any point within the medium must emain between T 1 and T 2. This esult eplains why the tempeatue distibution in a plane wall is no longe a staight line when the themal conductivity vaies with tempeatue. T T 1 T 2 1 EXAMPE 2 21 Heat Conduction though a Wall with k(t ) T 1 = 6 K Bonze plate k(t) = k (1 + β T) T 2 = 4 K FIGURE 2 65 Schematic fo Eample Q Conside a 2-m-high and.7-m-wide bonze plate whose thickness is.1 m. One side of the plate is maintained at a constant tempeatue of 6 K while the othe side is maintained at 4 K, as shown in Figue The themal conductivity of the bonze plate can be assumed to vay linealy in that tempeatue ange as k (T ) k (1 T ) whee k 38 W/m K and K 1. Disegading the edge effects and assuming steady one-dimensional heat tansfe, detemine the ate of heat conduction though the plate. SOUTION A plate with vaiable conductivity is subjected to specified tempeatues on both sides. The ate of heat tansfe is to be detemined. Assumptions 1 Heat tansfe is given to be steady and one-dimensional. 2 Themal conductivity vaies linealy. 3 Thee is no heat geneation. Popeties The themal conductivity is given to be k (T ) k (1 T ). Analysis The aveage themal conductivity of the medium in this case is simply the value at the aveage tempeatue and is detemined fom k ave k(t ave ) k 1 (38 W/m K) 1 ( K 1 ) 55.5 W/m K T 2 T 1 2 (6 4) K 2

47 cen58933_ch2.qd 9/1/22 8:47 AM Page CHAPTER 2 Then the ate of heat conduction though the plate can be detemined fom Eq to be Q k ave A T 1 T 2 (55.5 W/m K)(2 m.7 m) (6 4)K.1 m 155,4 W Discussion We would have obtained the same esult by substituting the given k (T ) elation into the second pat of Eq and pefoming the indicated integation. TOPIC OF SPECIA INTEREST A Bief Review of Diffeential Equations* As we mentioned in Chapte 1, the desciption of most scientific poblems involves elations that involve changes in some key vaiables with espect to each othe. Usually the smalle the incement chosen in the changing vaiables, the moe geneal and accuate the desciption. In the limiting case of infinitesimal o diffeential changes in vaiables, we obtain diffeential equations, which povide pecise mathematical fomulations fo the physical pinciples and laws by epesenting the ates of change as deivatives. Theefoe, diffeential equations ae used to investigate a wide vaiety of poblems in science and engineeing, including heat tansfe. Diffeential equations aise when elevant physical laws and pinciples ae applied to a poblem by consideing infinitesimal changes in the vaiables of inteest. Theefoe, obtaining the govening diffeential equation fo a specific poblem equies an adequate knowledge of the natue of the poblem, the vaiables involved, appopiate simplifying assumptions, and the applicable physical laws and pinciples involved, as well as a caeful analysis (Fig. 2 66). An equation, in geneal, may involve one o moe vaiables. As the name implies, a vaiable is a quantity that may assume vaious values duing a study. A quantity whose value is fied duing a study is called a constant. Constants ae usually denoted by the ealie lettes of the alphabet such as a, b, c, and d, wheeas vaiables ae usually denoted by the late ones such as t,, y, and z. A vaiable whose value can be changed abitaily is called an independent vaiable (o agument). A vaiable whose value depends on the value of othe vaiables and thus cannot be vaied independently is called a dependent vaiable (o a function). A dependent vaiable y that depends on a vaiable is usually denoted as y() fo claity. Howeve, this notation becomes vey inconvenient and cumbesome when y is epeated seveal times in an epession. In such cases it is desiable to denote y() simply as y when it is clea that y is a function of. This shotcut in notation impoves the appeaance and the Identify impotant vaiables Apply elevant physical laws Apply applicable solution technique Physical poblem A diffeential equation Solution of the poblem Make easonable assumptions and appoimations Bounday and initial conditions FIGURE 2 66 Mathematical modeling of physical poblems. *This section can be skipped if desied without a loss in continuity.

48 cen58933_ch2.qd 9/1/22 8:47 AM Page HEAT TRANSFER y y( + ) y() y() y eadability of the equations. The value of y at a fied numbe a is denoted by y(a). The deivative of a function y() at a point is equivalent to the slope of the tangent line to the gaph of the function at that point and is defined as (Fig. 2 67) dy() y y( ) y() y() lim lim (2-81) d Tangent line + FIGURE 2 67 The deivative of a function at a point epesents the slope of the tangent line of the function at that point. z z ( ) y FIGURE 2 68 Gaphical epesentation of patial deivative z /. y Hee epesents a (small) change in the independent vaiable and is called an incement of. The coesponding change in the function y is called an incement of y and is denoted by y. Theefoe, the deivative of a function can be viewed as the atio of the incement y of the function to the incement of the independent vaiable fo vey small. Note that y and thus y() will be zeo if the function y does not change with. Most poblems encounteed in pactice involve quantities that change with time t, and thei fist deivatives with espect to time epesent the ate of change of those quantities with time. Fo eample, if N(t) denotes the population of a bacteia colony at time t, then the fist deivative N dn/dt epesents the ate of change of the population, which is the amount the population inceases o deceases pe unit time. The deivative of the fist deivative of a function y is called the second deivative of y, and is denoted by y o d 2 y/d 2. In geneal, the deivative of the (n 1)st deivative of y is called the nth deivative of y and is denoted by y (n) o d n y/d n. Hee, n is a positive intege and is called the ode of the deivative. The ode n should not be confused with the degee of a deivative. Fo eample, y is the thid-ode deivative of y, but (y) 3 is the thid degee of the fist deivative of y. Note that the fist deivative of a function epesents the slope o the ate of change of the function with the independent vaiable, and the second deivative epesents the ate of change of the slope of the function with the independent vaiable. When a function y depends on two o moe independent vaiables such as and t, it is sometimes of inteest to eamine the dependence of the function on one of the vaiables only. This is done by taking the deivative of the function with espect to that vaiable while holding the othe vaiables constant. Such deivatives ae called patial deivatives. The fist patial deivatives of the function y(, t) with espect to and t ae defined as (Fig. 2 68) y y(, t) y(, t) lim (2-82) y y(, t t) y(, t) lim (2-83) t t t Note that when finding y/ we teat t as a constant and diffeentiate y with espect to. ikewise, when finding y/t we teat as a constant and diffeentiate y with espect to t. Integation can be viewed as the invese pocess of diffeentiation. Integation is commonly used in solving diffeential equations since solving a diffeential equation is essentially a pocess of emoving the deivatives

49 cen58933_ch2.qd 9/1/22 8:47 AM Page CHAPTER 2 fom the equation. Diffeentiation is the pocess of finding y() when a function y() is given, wheeas integation is the pocess of finding the function y() when its deivative y() is given. The integal of this deivative is epessed as y()d dy y() C (2-84) since y()d dy and the integal of the diffeential of a function is the function itself (plus a constant, of couse). In Eq. 2 84, is the integation vaiable and C is an abitay constant called the integation constant. The deivative of y() C is y() no matte what the value of the constant C is. Theefoe, two functions that diffe by a constant have the same deivative, and we always add a constant C duing integation to ecove this constant that is lost duing diffeentiation. The integal in Eq is called an indefinite integal since the value of the abitay constant C is indefinite. The descibed pocedue can be etended to highe-ode deivatives (Fig. 2 69). Fo eample, y()d y() C (2-85) dy y C y d y C y d yc y d yc y (n) d y (n 1) C FIGURE 2 69 Some indefinite integals that involve deivatives. This can be poved by defining a new vaiable u() y(), diffeentiating it to obtain u() y(), and then applying Eq Theefoe, the ode of a deivative deceases by one each time it is integated. Classification of Diffeential Equations A diffeential equation that involves only odinay deivatives is called an odinay diffeential equation, and a diffeential equation that involves patial deivatives is called a patial diffeential equation. Then it follows that poblems that involve a single independent vaiable esult in odinay diffeential equations, and poblems that involve two o moe independent vaiables esult in patial diffeential equations. A diffeential equation may involve seveal deivatives of vaious odes of an unknown function. The ode of the highest deivative in a diffeential equation is the ode of the equation. Fo eample, the ode of y (y) is 3 since it contains no fouth o highe ode deivatives. You will emembe fom algeba that the equation 3 5 is much easie to solve than the equation because the fist equation is linea wheeas the second one is nonlinea. This is also tue fo diffeential equations. Theefoe, befoe we stat solving a diffeential equation, we usually check fo lineaity. A diffeential equation is said to be linea if the dependent vaiable and all of its deivatives ae of the fist degee and thei coefficients depend on the independent vaiable only. In othe wods, a diffeential equation is linea if it can be witten in a fom that does not involve (1) any powes of the dependent vaiable o its deivatives such as y 3 o (y) 2, (2) any poducts of the dependent vaiable o its deivatives such as yy o yy, and (3) any othe nonlinea functions of the dependent vaiable such as sin y o e y. If any of these conditions apply, it is nonlinea (Fig. 2 7). (a) A nonlinea equation: 3(y ) 2 4yy + e 2y = 6 2 Powe Poduct (b) A linea equation: 3 2 y 4y + e 2 y = 6 2 Othe nonlinea functions FIGURE 2 7 A diffeential equation that is (a) nonlinea and (b) linea. When checking fo lineaity, we eamine the dependent vaiable only.

50 cen58933_ch2.qd 9/1/22 8:47 AM Page HEAT TRANSFER A linea diffeential equation, howeve, may contain (1) powes o nonlinea functions of the independent vaiable, such as 2 and cos and (2) poducts of the dependent vaiable (o its deivatives) and functions of the independent vaiable, such as 3 y, 2 y, and e 2 y. A linea diffeential equation of ode n can be epessed in the most geneal fom as y (n) f 1 ()y (n 1) f n 1 ()yf n ()y R() (2-86) (a) With constant coefficients: y6y 2y e 2 Constant (b) With vaiable coefficients: y y y e 1 2 Vaiable FIGURE 2 71 A diffeential equation with (a) constant coefficients and (b) vaiable coefficients. (a) An algebaic equation: y 2 7y 1 Solution: y 2 and y 5 (b) A diffeential equation: y 7y Solution: y e 7 FIGURE 2 72 Unlike those of algebaic equations, the solutions of diffeential equations ae typically functions instead of discete values. A diffeential equation that cannot be put into this fom is nonlinea. A linea diffeential equation in y is said to be homogeneous as well if R(). Othewise, it is nonhomogeneous. That is, each tem in a linea homogeneous equation contains the dependent vaiable o one of its deivatives afte the equation is cleaed of any common factos. The tem R() is called the nonhomogeneous tem. Diffeential equations ae also classified by the natue of the coefficients of the dependent vaiable and its deivatives. A diffeential equation is said to have constant coefficients if the coefficients of all the tems that involve the dependent vaiable o its deivatives ae constants. If, afte cleaing any common factos, any of the tems with the dependent vaiable o its deivatives involve the independent vaiable as a coefficient, that equation is said to have vaiable coefficients (Fig. 2 71). Diffeential equations with constant coefficients ae usually much easie to solve than those with vaiable coefficients. Solutions of Diffeential Equations Solving a diffeential equation can be as easy as pefoming one o moe integations; but such simple diffeential equations ae usually the eception athe than the ule. Thee is no single geneal solution method applicable to all diffeential equations. Thee ae diffeent solution techniques, each being applicable to diffeent classes of diffeential equations. Sometimes solving a diffeential equation equies the use of two o moe techniques as well as ingenuity and mastey of solution methods. Some diffeential equations can be solved only by using some vey cleve ticks. Some cannot be solved analytically at all. In algeba, we usually seek discete values that satisfy an algebaic equation such as When dealing with diffeential equations, howeve, we seek functions that satisfy the equation in a specified inteval. Fo eample, the algebaic equation is satisfied by two numbes only: 2 and 5. But the diffeential equation y 7y is satisfied by the function e 7 fo any value of (Fig. 2 72). Conside the algebaic equation Obviously, 1 satisfies this equation, and thus it is a solution. Howeve, it is not the only solution of this equation. We can easily show by diect substitution that 2 and 3 also satisfy this equation, and thus they ae solutions as well. But thee ae no othe solutions to this equation. Theefoe, we say that the set 1, 2, and 3 foms the complete solution to this algebaic equation. The same line of easoning also applies to diffeential equations. Typically, diffeential equations have multiple solutions that contain at least one abitay constant. Any function that satisfies the diffeential equation on an

51 cen58933_ch2.qd 9/1/22 8:47 AM Page 111 inteval is called a solution of that diffeential equation in that inteval. A solution that involves one o moe abitay constants epesents a family of functions that satisfy the diffeential equation and is called a geneal solution of that equation. Not supisingly, a diffeential equation may have moe than one geneal solution. A geneal solution is usually efeed to as the geneal solution o the complete solution if evey solution of the equation can be obtained fom it as a special case. A solution that can be obtained fom a geneal solution by assigning paticula values to the abitay constants is called a specific solution. You will ecall fom algeba that a numbe is a solution of an algebaic equation if it satisfies the equation. Fo eample, 2 is a solution of the equation 3 8 because the substitution of 2 fo yields identically zeo. ikewise, a function is a solution of a diffeential equation if that function satisfies the diffeential equation. In othe wods, a solution function yields identity when substituted into the diffeential equation. Fo eample, it can be shown by diect substitution that the function 3e 2 is a solution of y4y (Fig. 2 73). Function: f 3e CHAPTER 2 Diffeential equation: y 4y Deivatives of f: f 6e 2 f 12e 2 Substituting into y 4y : f 4f 12e 2 4 3e 2 Theefoe, the function 3e 2 is a solution of the diffeential equation y 4y. FIGURE 2 73 Veifying that a given function is a solution of a diffeential equation. SUMMARY In this chapte we have studied the heat conduction equation and its solutions. Heat conduction in a medium is said to be steady when the tempeatue does not vay with time and unsteady o tansient when it does. Heat conduction in a medium is said to be one-dimensional when conduction is significant in one dimension only and negligible in the othe two dimensions. It is said to be two-dimensional when conduction in the thid dimension is negligible and thee-dimensional when conduction in all dimensions is significant. In heat tansfe analysis, the convesion of electical, chemical, o nuclea enegy into heat (o themal) enegy is chaacteized as heat geneation. The heat conduction equation can be deived by pefoming an enegy balance on a diffeential volume element. The onedimensional heat conduction equation in ectangula, cylindical, and spheical coodinate systems fo the case of constant themal conductivities ae epessed as 2 T g 2 k 1 g T k 1 g 2 T k 2 1 T t 1 T t 1 T t whee the popety k/c is the themal diffusivity of the mateial. The solution of a heat conduction poblem depends on the conditions at the sufaces, and the mathematical epessions fo the themal conditions at the boundaies ae called the bounday conditions. The solution of tansient heat conduction poblems also depends on the condition of the medium at the beginning of the heat conduction pocess. Such a condition, which is usually specified at time t, is called the initial condition, which is a mathematical epession fo the tempeatue distibution of the medium initially. Complete mathematical desciption of a heat conduction poblem equies the specification of two bounday conditions fo each dimension along which heat conduction is significant, and an initial condition when the poblem is tansient. The most common bounday conditions ae the specified tempeatue, specified heat flu, convection, and adiation bounday conditions. A bounday suface, in geneal, may involve specified heat flu, convection, and adiation at the same time. Fo steady one-dimensional heat tansfe though a plate of thickness, the vaious types of bounday conditions at the sufaces at and can be epessed as Specified tempeatue: T() T 1 and T() T 2 whee T 1 and T 2 ae the specified tempeatues at sufaces at and. Specified heat flu: dt() dt() k q and k q d d whee q and q ae the specified heat flues at sufaces at and.

52 cen58933_ch2.qd 9/1/22 8:47 AM Page HEAT TRANSFER Insulation o themal symmety: dt() dt() and d d Convection: dt() dt() k h 1 [T 1 T()] and k h 2 [T() T 2 ] d d whee h 1 and h 2 ae the convection heat tansfe coefficients and T 1 and T 2 ae the tempeatues of the suounding mediums on the two sides of the plate. Radiation: dt() k 1 [T 4 su, 1 T() 4 ] and d dt() k 2 [T() 4 T 4 su, d 2] whee 1 and 2 ae the emissivities of the bounday sufaces, W/m 2 K 4 is the Stefan Boltzmann constant, and T su, 1 and T su, 2 ae the aveage tempeatues of the sufaces suounding the two sides of the plate. In adiation calculations, the tempeatues must be in K o R. Inteface of two bodies A and B in pefect contact at : dt A ( ) dt B ( ) T A ( ) T B ( ) and k A k d B d whee k A and k B ae the themal conductivities of the layes A and B. Heat geneation is usually epessed pe unit volume of the medium and is denoted by g, whose unit is W/m 3. Unde steady conditions, the suface tempeatue T s of a plane wall of thickness 2, a cylinde of oute adius o, and a sphee of adius o in which heat is geneated at a constant ate of g pe unit volume in a suounding medium at T can be epessed as whee h is the convection heat tansfe coefficient. The maimum tempeatue ise between the suface and the midsection of a medium is given by T ma, plane wall T ma, cylinde T ma, sphee g 2 2k g 2 o 4k g 2 o 6k When the vaiation of themal conductivity with tempeatue k(t) is known, the aveage value of the themal conductivity in the tempeatue ange between T 1 and T 2 can be detemined fom T 2 k(t)dt T 1 k ave T 2 T 1 Then the ate of steady heat tansfe though a plane wall, cylindical laye, o spheical laye can be epessed as T 1 T 2 A Q plane wall k ave A k(t)dt T1 T 2 T 1 T 2 2 Q cylinde 2k ave k(t)dt ln( 2 / 1 ) T1 ln( 2 / 1 ) T 2 T 1 T Q sphee 4k ave 1 2 k(t)dt 2 1 T The vaiation of themal conductivity of a mateial with tempeatue can often be appoimated as a linea function and epessed as k(t) k (1 T) whee is called the tempeatue coefficient of themal conductivity. T 2 T s, plane wall T T s, cylinde T T s, sphee T g h g o 2h g o 3h REFERENCES AND SUGGESTED READING 1. W. E. Boyce and R. C. Dipima. Elementay Diffeential Equations and Bounday Value Poblems. 4th ed. New Yok: John Wiley & Sons, J. P. Holman. Heat Tansfe. 9th ed. New Yok: McGaw-Hill, 22.

53 cen58933_ch2.qd 9/1/22 8:47 AM Page F. P. Incopea and D. P. DeWitt. Intoduction to Heat Tansfe. 4th ed. New Yok: John Wiley & Sons, S. S. Kutateladze. Fundamentals of Heat Tansfe. New Yok: Academic Pess, CHAPTER 1 5. M. N. Ozisik. Heat Tansfe A Basic Appoach. New Yok: McGaw-Hill, F. M. White. Heat and Mass Tansfe. Reading, MA: Addison-Wesley, PROBEMS* Intoduction 2 1C Is heat tansfe a scala o vecto quantity? Eplain. Answe the same question fo tempeatue. 2 2C How does tansient heat tansfe diffe fom steady heat tansfe? How does one-dimensional heat tansfe diffe fom two-dimensional heat tansfe? 2 3C Conside a cold canned dink left on a dinne table. Would you model the heat tansfe to the dink as one-, two-, o thee-dimensional? Would the heat tansfe be steady o tansient? Also, which coodinate system would you use to analyze this heat tansfe poblem, and whee would you place the oigin? Eplain. 2 4C Conside a ound potato being baked in an oven. Would you model the heat tansfe to the potato as one-, two-, o thee-dimensional? Would the heat tansfe be steady o tansient? Also, which coodinate system would you use to solve this poblem, and whee would you place the oigin? Eplain. FIGURE P2 4 *Poblems designated by a C ae concept questions, and students ae encouaged to answe them all. Poblems designated by an E ae in English units, and the SI uses can ignoe them. Poblems with an EES-CD icon ae solved using EES, and complete solutions togethe with paametic studies ae included on the enclosed CD. Poblems with a compute-ees icon ae compehensive in natue, and ae intended to be solved with a compute, pefeably using the EES softwae that accompanies this tet. 2 5C Conside an egg being cooked in boiling wate in a pan. Would you model the heat tansfe to the egg as one-, two-, o thee-dimensional? Would the heat tansfe be steady o tansient? Also, which coodinate system would you use to solve this poblem, and whee would you place the oigin? Eplain. 2 6C Conside a hot dog being cooked in boiling wate in a pan. Would you model the heat tansfe to the hot dog as one-, two-, o thee-dimensional? Would the heat tansfe be steady o tansient? Also, which coodinate system would you use to solve this poblem, and whee would you place the oigin? Eplain. Boiling wate Hot dog FIGURE P C Conside the cooking pocess of a oast beef in an oven. Would you conside this to be a steady o tansient heat tansfe poblem? Also, would you conside this to be one-, two-, o thee-dimensional? Eplain. 2 8C Conside heat loss fom a 2- cylindical hot wate tank in a house to the suounding medium. Would you conside this to be a steady o tansient heat tansfe poblem? Also, would you conside this heat tansfe poblem to be one-, two-, o thee-dimensional? Eplain. 2 9C Does a heat flu vecto at a point P on an isothemal suface of a medium have to be pependicula to the suface at that point? Eplain. 2 1C Fom a heat tansfe point of view, what is the diffeence between isotopic and unisotopic mateials? 2 11C What is heat geneation in a solid? Give eamples. 2 12C Heat geneation is also efeed to as enegy geneation o themal enegy geneation. What do you think of these phases? 2 13C In ode to detemine the size of the heating element of a new oven, it is desied to detemine the ate of heat tansfe though the walls, doo, and the top and bottom section of the oven. In you analysis, would you conside this to be a

54 cen58933_ch2.qd 9/1/22 8:47 AM Page HEAT TRANSFER steady o tansient heat tansfe poblem? Also, would you conside the heat tansfe to be one-dimensional o multidimensional? Eplain. 2 14E The esistance wie of a 1-W ion is 15 in. long and has a diamete of D.8 in. Detemine the ate of heat geneation in the wie pe unit volume, in Btu/h ft 3, and the heat flu on the oute suface of the wie, in Btu/h ft 2, as a esult of this heat geneation. D FIGURE P2 14E q g 2 19 Wite down the one-dimensional tansient heat conduction equation fo a plane wall with constant themal conductivity and heat geneation in its simplest fom, and indicate what each vaiable epesents. 2 2 Wite down the one-dimensional tansient heat conduction equation fo a long cylinde with constant themal conductivity and heat geneation, and indicate what each vaiable epesents Stating with an enegy balance on a ectangula volume element, deive the one-dimensional tansient heat conduction equation fo a plane wall with constant themal conductivity and no heat geneation Stating with an enegy balance on a cylindical shell volume element, deive the steady one-dimensional heat conduction equation fo a long cylinde with constant themal conductivity in which heat is geneated at a ate of g. 2 15E Reconside Poblem 2 14E. Using EES (o othe) softwae, evaluate and plot the suface heat flu as a function of wie diamete as the diamete vaies fom.2 to.2 in. Discuss the esults In a nuclea eacto, heat is geneated unifomly in the 5-cm-diamete cylindical uanium ods at a ate of W/m 3. If the length of the ods is 1 m, detemine the ate of heat geneation in each od. Answe: kw 2 17 In a sola pond, the absoption of sola enegy can be modeled as heat geneation and can be appoimated by g g e b, whee g is the ate of heat absoption at the top suface pe unit volume and b is a constant. Obtain a elation fo the total ate of heat geneation in a wate laye of suface aea A and thickness at the top of the pond. Radiation beam being absobed Sola enegy FIGURE P Stating with an enegy balance on a spheical shell volume element, deive the one-dimensional tansient heat conduction equation fo a sphee with constant themal conductivity and no heat geneation. Sola pond FIGURE P R 2 18 Conside a lage 3-cm-thick stainless steel plate in which heat is geneated unifomly at a ate of W/m 3. Assuming the plate is losing heat fom both sides, detemine the heat flu on the suface of the plate duing steady opeation. Answe: 75, W/m 2 Heat Conduction Equation FIGURE P Conside a medium in which the heat conduction equation is given in its simplest fom as 2 T 2 1 T t

55 cen58933_ch2.qd 9/1/22 8:47 AM Page 115 (a) Is heat tansfe steady o tansient? (b) Is heat tansfe one-, two-, o thee-dimensional? (c) Is thee heat geneation in the medium? (d) Is the themal conductivity of the medium constant o vaiable? 2 25 Conside a medium in which the heat conduction equation is given in its simplest fom as 1 d d g d (a) Is heat tansfe steady o tansient? (b) Is heat tansfe one-, two-, o thee-dimensional? (c) Is thee heat geneation in the medium? (d) Is the themal conductivity of the medium constant o vaiable? 2 26 Conside a medium in which the heat conduction equation is given in its simplest fom as 1 1 T 2 T t 2 dt k (a) Is heat tansfe steady o tansient? (b) Is heat tansfe one-, two-, o thee-dimensional? (c) Is thee heat geneation in the medium? (d) Is the themal conductivity of the medium constant o vaiable? 2 27 Conside a medium in which the heat conduction equation is given in its simplest fom as d 2 T dt d 2 d (a) Is heat tansfe steady o tansient? (b) Is heat tansfe one-, two-, o thee-dimensional? (c) Is thee heat geneation in the medium? (d) Is the themal conductivity of the medium constant o vaiable? 2 28 Stating with an enegy balance on a volume element, deive the two-dimensional tansient heat conduction equation in ectangula coodinates fo T(, y, t) fo the case of constant themal conductivity and no heat geneation Stating with an enegy balance on a ing-shaped volume element, deive the two-dimensional steady heat conduction equation in cylindical coodinates fo T(, z) fo the case of constant themal conductivity and no heat geneation. 2 3 Stating with an enegy balance on a disk volume element, deive the one-dimensional tansient heat conduction equation fo T(z, t) in a cylinde of diamete D with an insulated side suface fo the case of constant themal conductivity with heat geneation Conside a medium in which the heat conduction equation is given in its simplest fom as 115 CHAPTER 1 (a) Is heat tansfe steady o tansient? (b) Is heat tansfe one-, two-, o thee-dimensional? (c) Is thee heat geneation in the medium? (d) Is the themal conductivity of the medium constant o vaiable? 2 32 Conside a medium in which the heat conduction equation is given in its simplest fom as 1 z 2 T 2 T k FIGURE P2 29 Disk z FIGURE P2 3 z + z 2 T y 2 + Insulation 1 T t k T g z z (a) Is heat tansfe steady o tansient? (b) Is heat tansfe one-, two-, o thee-dimensional? (c) Is thee heat geneation in the medium? (d) Is the themal conductivity of the medium constant o vaiable? 2 33 Conside a medium in which the heat conduction equation is given in its simplest fom as T 2 T 2 T t 2 2 sin 2 2 t (a) Is heat tansfe steady o tansient? (b) Is heat tansfe one-, two-, o thee-dimensional? (c) Is thee heat geneation in the medium? (d) Is the themal conductivity of the medium constant o vaiable? g A = constant z

56 cen58933_ch2.qd 9/1/22 8:47 AM Page HEAT TRANSFER Bounday and Initial Conditions; Fomulation of Heat Conduction Poblems 2 34C What is a bounday condition? How many bounday conditions do we need to specify fo a two-dimensional heat tansfe poblem? 2 35C What is an initial condition? How many initial conditions do we need to specify fo a two-dimensional heat tansfe poblem? 2 36C What is a themal symmety bounday condition? How is it epessed mathematically? 2 37C How is the bounday condition on an insulated suface epessed mathematically? 2 38C It is claimed that the tempeatue pofile in a medium must be pependicula to an insulated suface. Is this a valid claim? Eplain. 2 39C Why do we ty to avoid the adiation bounday conditions in heat tansfe analysis? 2 4 Conside a spheical containe of inne adius 1, oute adius 2, and themal conductivity k. Epess the bounday condition on the inne suface of the containe fo steady onedimensional conduction fo the following cases: (a) specified tempeatue of 5 C, (b) specified heat flu of 3 W/m 2 towad the cente, (c) convection to a medium at T with a heat tansfe coefficient of h. FIGURE P2 4 Spheical containe 2 41 Heat is geneated in a long wie of adius at a constant ate of g pe unit volume. The wie is coveed with a plastic insulation laye. Epess the heat flu bounday condition at the inteface in tems of the heat geneated Conside a long pipe of inne adius 1, oute adius 2, and themal conductivity k. The oute suface of the pipe is subjected to convection to a medium at T with a heat tansfe coefficient of h, but the diection of heat tansfe is not known. Epess the convection bounday condition on the oute suface of the pipe Conside a spheical shell of inne adius 1, oute adius 2, themal conductivity k, and emissivity. The oute suface of the shell is subjected to adiation to suounding sufaces at T su, but the diection of heat tansfe is not known. 1 2 Epess the adiation bounday condition on the oute suface of the shell A containe consists of two spheical layes, A and B, that ae in pefect contact. If the adius of the inteface is, epess the bounday conditions at the inteface Conside a steel pan used to boil wate on top of an electic ange. The bottom section of the pan is.5 cm thick and has a diamete of D 2 cm. The electic heating unit on the ange top consumes 1 W of powe duing cooking, and 85 pecent of the heat geneated in the heating element is tansfeed unifomly to the pan. Heat tansfe fom the top suface of the bottom section to the wate is by convection with a heat tansfe coefficient of h. Assuming constant themal conductivity and one-dimensional heat tansfe, epess the mathematical fomulation (the diffeential equation and the bounday conditions) of this heat conduction poblem duing steady opeation. Do not solve. Steel pan Wate FIGURE P E A 2-kW esistance heate wie whose themal conductivity is k 1.4 Btu/h ft F has a adius of.6 in. and a length of 15 in., and is used fo space heating. Assuming constant themal conductivity and one-dimensional heat tansfe, epess the mathematical fomulation (the diffeential equation and the bounday conditions) of this heat conduction poblem duing steady opeation. Do not solve Conside an aluminum pan used to cook stew on top of an electic ange. The bottom section of the pan is.25 cm thick and has a diamete of D 18 cm. The electic heating unit on the ange top consumes 9 W of powe duing cooking, and 9 pecent of the heat geneated in the heating element Aluminum pan Stew 18 C FIGURE P2 47

57 cen58933_ch2.qd 9/1/22 8:47 AM Page 117 is tansfeed to the pan. Duing steady opeation, the tempeatue of the inne suface of the pan is measued to be 18 C. Assuming tempeatue-dependent themal conductivity and one-dimensional heat tansfe, epess the mathematical fomulation (the diffeential equation and the bounday conditions) of this heat conduction poblem duing steady opeation. Do not solve Wate flows though a pipe at an aveage tempeatue of T 5 C. The inne and oute adii of the pipe ae 1 6 cm and cm, espectively. The oute suface of the pipe is wapped with a thin electic heate that consumes 3 W pe m length of the pipe. The eposed suface of the heate is heavily insulated so that the entie heat geneated in the heate is tansfeed to the pipe. Heat is tansfeed fom the inne suface of the pipe to the wate by convection with a heat tansfe coefficient of h 55 W/m 2 C. Assuming constant themal conductivity and one-dimensional heat tansfe, epess the mathematical fomulation (the diffeential equation and the bounday conditions) of the heat conduction in the pipe duing steady opeation. Do not solve. Insulation h T 1 Wate FIGURE P Electic heate 2 49 A spheical metal ball of adius is heated in an oven to a tempeatue of T i thoughout and is then taken out of the oven and dopped into a lage body of wate at T whee it is cooled by convection with an aveage convection heat tansfe coefficient of h. Assuming constant themal conductivity and tansient one-dimensional heat tansfe, epess the mathematical fomulation (the diffeential equation and the bounday and initial conditions) of this heat conduction poblem. Do not solve. 2 5 A spheical metal ball of adius is heated in an oven to a tempeatue of T i thoughout and is then taken out of the oven and allowed to cool in ambient ai at T by convection and adiation. The emissivity of the oute suface of the cylinde is, and the tempeatue of the suounding sufaces is T su. The aveage convection heat tansfe coefficient is estimated to be h. Assuming vaiable themal conductivity and tansient one-dimensional heat tansfe, epess the mathematical fomulation (the diffeential equation and the bounday Metal ball T i T su FIGURE P2 5 Radiation 117 CHAPTER 1 Convection and initial conditions) of this heat conduction poblem. Do not solve Conside the noth wall of a house of thickness. The oute suface of the wall echanges heat by both convection and adiation. The inteio of the house is maintained at T 1, while the ambient ai tempeatue outside emains at T 2. The sky, the gound, and the sufaces of the suounding stuctues at this location can be modeled as a suface at an effective tempeatue of T sky fo adiation echange on the oute suface. The adiation echange between the inne suface of the wall and the sufaces of the walls, floo, and ceiling it faces is negligible. The convection heat tansfe coefficients on the inne and oute sufaces of the wall ae h 1 and h 2, espectively. The themal conductivity of the wall mateial is k and the emissivity of the oute suface is 2. Assuming the heat tansfe though the wall to be steady and one-dimensional, epess the mathematical fomulation (the diffeential equation and the bounday and initial conditions) of this heat conduction poblem. Do not solve. h 1 T 1 h 2 T 2 Wall FIGURE P2 51 T sky T h

58 cen58933_ch2.qd 9/1/22 8:47 AM Page HEAT TRANSFER Solution of Steady One-Dimensional Heat Conduction Poblems 2 52C Conside one-dimensional heat conduction though a lage plane wall with no heat geneation that is pefectly insulated on one side and is subjected to convection and adiation on the othe side. It is claimed that unde steady conditions, the tempeatue in a plane wall must be unifom (the same eveywhee). Do you agee with this claim? Why? 2 53C It is stated that the tempeatue in a plane wall with constant themal conductivity and no heat geneation vaies linealy duing steady one-dimensional heat conduction. Will this still be the case when the wall loses heat by adiation fom its sufaces? 2 54C Conside a solid cylindical od whose ends ae maintained at constant but diffeent tempeatues while the side suface is pefectly insulated. Thee is no heat geneation. It is claimed that the tempeatue along the ais of the od vaies linealy duing steady heat conduction. Do you agee with this claim? Why? 2 55C Conside a solid cylindical od whose side suface is maintained at a constant tempeatue while the end sufaces ae pefectly insulated. The themal conductivity of the od mateial is constant and thee is no heat geneation. It is claimed that the tempeatue in the adial diection within the od will not vay duing steady heat conduction. Do you agee with this claim? Why? 2 56 Conside a lage plane wall of thickness.4 m, themal conductivity k 2.3 W/m C, and suface aea A 2 m 2. The left side of the wall is maintained at a constant tempeatue of T 1 8 C while the ight side loses heat by convection to the suounding ai at T 15 C with a heat tansfe coefficient of h 24 W/m 2 C. Assuming constant themal conductivity and no heat geneation in the wall, (a) epess the diffeential equation and the bounday conditions fo steady one-dimensional heat conduction though the wall, (b) obtain a elation fo the vaiation of tempeatue in the wall by solving the diffeential equation, and (c) evaluate the ate of heat tansfe though the wall. Answe: (c) 63 W Base plate 2 57 Conside a solid cylindical od of length.15 m and diamete.5 m. The top and bottom sufaces of the od ae maintained at constant tempeatues of 2 C and 95 C, espectively, while the side suface is pefectly insulated. Detemine the ate of heat tansfe though the od if it is made of (a) coppe, k 38 W/m C, (b) steel, k 18 W/m C, and (c) ganite, k 1.2 W/m C Reconside Poblem Using EES (o othe) softwae, plot the ate of heat tansfe as a function of the themal conductivity of the od in the ange of 1 W/m C to 4 W/m C. Discuss the esults Conside the base plate of a 8-W household ion with a thickness of.6 cm, base aea of A 16 cm 2, and themal conductivity of k 2 W/m C. The inne suface of the base plate is subjected to unifom heat flu geneated by the esistance heates inside. When steady opeating conditions ae eached, the oute suface tempeatue of the plate is measued to be 85 C. Disegading any heat loss though the uppe pat of the ion, (a) epess the diffeential equation and the bounday conditions fo steady one-dimensional heat conduction though the plate, (b) obtain a elation fo the vaiation of tempeatue in the base plate by solving the diffeential equation, and (c) evaluate the inne suface tempeatue. Answe: (c) 1 C 2 6 Repeat Poblem 2 59 fo a 12-W ion Reconside Poblem Using the elation obtained fo the vaiation of tempeatue in the base plate, plot the tempeatue as a function of the distance in the ange of to, and discuss the esults. Use the EES (o othe) softwae. 2 62E Conside a steam pipe of length 15 ft, inne adius 1 2 in., oute adius in., and themal conductivity k 7.2 Btu/h ft F. Steam is flowing though the pipe at an aveage tempeatue of 25 F, and the aveage convection heat tansfe coefficient on the inne suface is given to be h 1.25 Btu/h ft 2 F. If the aveage tempeatue on the oute Steam 25 F h FIGURE P2 59 h FIGURE P2 62E T 2 = 16 F C

59 cen58933_ch2.qd 9/1/22 8:47 AM Page CHAPTER 1 sufaces of the pipe is T 2 16 F, (a) epess the diffeential equation and the bounday conditions fo steady onedimensional heat conduction though the pipe, (b) obtain a elation fo the vaiation of tempeatue in the pipe by solving the diffeential equation, and (c) evaluate the ate of heat loss fom the steam though the pipe. Answe: (c) 16,8 Btu/h 2 63 A spheical containe of inne adius 1 2 m, oute adius m, and themal conductivity k 3 W/m C is filled with iced wate at C. The containe is gaining heat by convection fom the suounding ai at T 25 C with a heat tansfe coefficient of h 18 W/m 2 C. Assuming the inne suface tempeatue of the containe to be C, (a) epess the diffeential equation and the bounday conditions fo steady one-dimensional heat conduction though the containe, (b) obtain a elation fo the vaiation of tempeatue in the containe by solving the diffeential equation, and (c) evaluate the ate of heat gain to the iced wate Conside a lage plane wall of thickness.3 m, themal conductivity k 2.5 W/m C, and suface aea A 12 m 2. The left side of the wall at is subjected to a net heat flu of q 7 W/m 2 while the tempeatue at that suface is measued to be T 1 8 C. Assuming constant themal conductivity and no heat geneation in the wall, (a) epess the diffeential equation and the bounday conditions fo steady one-dimensional heat conduction though the wall, (b) obtain a elation fo the vaiation of tempeatue in the wall by solving the diffeential equation, and (c) evaluate the tempeatue of the ight suface of the wall at. Answe: (c) 4 C T 1 q FIGURE P Repeat Poblem 2 64 fo a heat flu of 95 W/m 2 and a suface tempeatue of 85 C at the left suface at. 2 66E A lage steel plate having a thickness of 4 in., themal conductivity of k 7.2 Btu/h ft F, and an emissivity of.6 is lying on the gound. The eposed suface of the plate at is known to echange heat by convection with the ambient ai at T 9 F with an aveage heat tansfe coefficient of h 12 Btu/h ft 2 F as well as by adiation with the open sky with an equivalent sky tempeatue of T sky 51 R. Also, the tempeatue of the uppe suface of the plate is measued to be 75 F. Assuming steady one-dimensional heat tansfe, (a) epess the diffeential equation and the bounday conditions fo heat conduction though the plate, (b) obtain a elation fo the vaiation of tempeatue in the plate by solving 75 F T sky Radiation Gound FIGURE P2 66E ε h, T Convection Plate the diffeential equation, and (c) detemine the value of the lowe suface tempeatue of the plate at. 2 67E Repeat Poblem 2 66E by disegading adiation heat tansfe When a long section of a compessed ai line passes though the outdoos, it is obseved that the moistue in the compessed ai feezes in cold weathe, disupting and even completely blocking the ai flow in the pipe. To avoid this poblem, the oute suface of the pipe is wapped with electic stip heates and then insulated. Conside a compessed ai pipe of length 6 m, inne adius cm, oute adius 2 4. cm, and themal conductivity k 14 W/m C equipped with a 3-W stip heate. Ai is flowing though the pipe at an aveage tempeatue of 1 C, and the aveage convection heat tansfe coefficient on the inne suface is h 3 W/m 2 C. Assuming 15 pecent of the heat geneated in the stip heate is lost though the insulation, (a) epess the diffeential equation and the bounday conditions fo steady one-dimensional heat conduction though the pipe, (b) obtain a elation fo the vaiation of tempeatue in the pipe mateial by solving the diffeential equation, and (c) evaluate the inne and oute suface tempeatues of the pipe. Answes: (c) 3.91 C, 3.87 C Compessed ai Insulation FIGURE P Electic heate 1 C 2 69 Reconside Poblem Using the elation obtained fo the vaiation of tempeatue in the pipe mateial, plot the tempeatue as a function of the adius in

60 cen58933_ch2.qd 9/1/22 8:47 AM Page HEAT TRANSFER the ange of 1 to 2, and discuss the esults. Use the EES (o othe) softwae. 2 7 In a food pocessing facility, a spheical containe of inne adius 1 4 cm, oute adius 2 41 cm, and themal conductivity k 1.5 W/m C is used to stoe hot wate and to keep it at 1 C at all times. To accomplish this, the oute suface of the containe is wapped with a 5-W electic stip heate and then insulated. The tempeatue of the inne suface of the containe is obseved to be nealy 1 C at all times. Assuming 1 pecent of the heat geneated in the heate is lost though the insulation, (a) epess the diffeential equation and the bounday conditions fo steady one-dimensional heat conduction though the containe, (b) obtain a elation fo the vaiation of tempeatue in the containe mateial by solving the diffeential equation, and (c) evaluate the oute suface tempeatue of the containe. Also detemine how much wate at 1 C this tank can supply steadily if the cold wate entes at 2 C. Spheical containe FIGURE P2 7 Hot wate 1 C Insulation 2 71 Reconside Poblem 2 7. Using the elation obtained fo the vaiation of tempeatue in the containe mateial, plot the tempeatue as a function of the adius in the ange of 1 to 2, and discuss the esults. Use the EES (o othe) softwae. Heat Geneation in a Solid 2 72C Does heat geneation in a solid violate the fist law of themodynamics, which states that enegy cannot be ceated o destoyed? Eplain. 2 73C What is heat geneation? Give some eamples. 2 74C An ion is left unattended and its base tempeatue ises as a esult of esistance heating inside. When will the ate of heat geneation inside the ion be equal to the ate of heat loss fom the ion? 2 75C Conside the unifom heating of a plate in an envionment at a constant tempeatue. Is it possible fo pat of the heat geneated in the left half of the plate to leave the plate though the ight suface? Eplain. 1 2 Electic heate 2 76C Conside unifom heat geneation in a cylinde and a sphee of equal adius made of the same mateial in the same envionment. Which geomety will have a highe tempeatue at its cente? Why? 2 77 A 2-kW esistance heate wie with themal conductivity of k 2 W/m C, a diamete of D 5 mm, and a length of.7 m is used to boil wate. If the oute suface tempeatue of the esistance wie is T s 11 C, detemine the tempeatue at the cente of the wie. D 11 C Resistance heate 2 78 Conside a long solid cylinde of adius 4 cm and themal conductivity k 25 W/m C. Heat is geneated in the cylinde unifomly at a ate of g 35 W/cm 3. The side suface of the cylinde is maintained at a constant tempeatue of T s 8 C. The vaiation of tempeatue in the cylinde is given by g 2 T() 1 T s 2 k Based on this elation, detemine (a) if the heat conduction is steady o tansient, (b) if it is one-, two-, o thee-dimensional, and (c) the value of heat flu on the side suface of the cylinde at Reconside Poblem Using the elation obtained fo the vaiation of tempeatue in the cylinde, plot the tempeatue as a function of the adius in the ange of to, and discuss the esults. Use the EES (o othe) softwae. 2 8E A long homogeneous esistance wie of adius.25 in. and themal conductivity k 8.6 Btu/h ft F is being used to boil wate at atmospheic pessue by the passage of Wate FIGURE P2 77 Resistance heate FIGURE P2 8E T h