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

Transcription

1 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 ate of change in tempeatue with distance in that diection. Heat conduction in a medium, in geneal, is thee-dimensional and time dependent, and the tempeatue in a medium vaies with position as well as time, that is, T T(, y, z, t). 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 pimay 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. OBJECTIVES When you finish studying this chapte, you should be able to: Undestand multidimensionality and time dependence of heat tansfe, and the conditions unde which a heat tansfe poblem can be appoimated as being one-dimensional, Obtain the diffeential equation of heat conduction in vaious coodinate systems, and simplify it fo steady one-dimensional case, Identify the themal conditions on sufaces, and epess them mathematically as bounday and initial conditions, Solve one-dimensional heat conduction poblems and obtain the tempeatue distibutions within a medium and the heat flu, Analyze one-dimensional heat conduction in solids that involve heat geneation, and Evaluate heat conduction in solids with tempeatuedependent themal conductivity. 63

2 64 HEAT CONDUCTION EQUATION 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. Q = 5 W Hot medium Cold medium Q = 5 W Cold Hot medium medium 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 (, f, z) in cylindical coodinates, and as (, f, u) in spheical coodinates, whee the distances, y, z, and and the angles f and u ae as shown in Fig 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.

3 65 CHAPTER 2 z z 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. The 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 is 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? Time = 2 PM 15 C 7 C 15 C Q 1 (a) Steady 15 C 7 C 12 C Q 1 (b) Tansient Time = 5 PM 7 C 5 C Q 2 = Q 1 Q 2 Q 1 FIGURE 2 4 Tansient and steady heat conduction in a plane wall.

4 66 HEAT CONDUCTION EQUATION 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 Pimay diection of heat tansfe FIGURE 2 6 Heat tansfe though the window of a house can be taken to be one-dimensional. Q y Q Q If the 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 most demanding conditions, it is lage enough fo all conditions. The appoach descibed above is a common pactice in engineeing. 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, f, and z in the cylindical coodinate system; and the, f, and u in the spheical (o pola) coodinate system. The tempeatue distibution in this case is epessed as T(, y, z, t), T(, f, z, t), and T(, f, u, 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 occus 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 is tansfeed to the egg in this case in the adial diection, that is, along staight lines passing though the midpoint of the egg. We 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 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 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. Fig. 2 8 shows an isothemal suface in that medium. The heat tansfe 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 67 CHAPTER 2 dt slope d < T() Q > Heat flow FIGURE 2 7 The tempeatue gadient dt/d is simply the slope of the tempeatue cuve on a T- diagam. T Q n ka (W) (2 2) n 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 A z 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. n

6 68 HEAT CONDUCTION EQUATION 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 e gen () = 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 mechanical, electical, nuclea, o chemical enegy into heat (o themal enegy). In heat conduction analysis, such convesion pocesses ae chaacteized as heat (o themal enegy) 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 e gen, 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 E gen e gendv (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 E gen e genv, whee e gen is the constant ate of heat geneation pe unit volume.

7 EXAMPE 2 1 Heat Geneation in a Hai Dye 69 CHAPTER 2 Hai dye 12 W The esistance wie of a 12-W hai dye is 8 cm long and has a diamete of D.3 cm (Fig. 2 11). 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 convets 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, FIGURE 2 11 Schematic fo Eample 2 1. E. gen Ė gen 12 W e gen 212 W/cm V 3 (pd 2 /4) [p(.3 cm) 2 wie /4](8 cm) 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, E. gen Ė gen 12 W Q s 15.9 W/cm pd p(.3 cm)(8 cm) 3 A wie 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 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 is dominant in one diection and negligible in othe diections. Net we develop the onedimensional heat conduction equation in ectangula, cylindical, and spheical coodinates. Heat Conduction Equation in a age Plane Wall Conside a thin element of thickness in a lage plane wall, as shown in Fig 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 Q Q + Δ E gen + Δ A = A + Δ = A Volume element FIGURE 2 12 One-dimensional heat conduction though a volume element in a lage plane wall. A

8 7 HEAT CONDUCTION EQUATION 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 E gen, 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) E gen, element e genv element e gen A (2 8) Substituting into Eq. 2 6, we get T t t T t Q Q e gen A ca (2 9) t Dividing by A gives Q. Q. 1 T t t T t e gen c (2 1) A t Taking the limit as and t yields since, fom the definition of the deivative and Fouie s law of heat conduction, lim 1 A T T ka e gen c (2 11) t Q Q Q kat (2 12) Geneal, one-dimensional: No geneation 2 T e gen 2 + = k Steady, one-dimensional: d 2 T d 2 Steadystate 1 T a t = FIGURE 2 13 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. 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 e gen 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 ė gen Constant conductivity: 2 T 1 T (2 14) 2 k a t whee the popety a 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 13):

9 71 CHAPTER 2 (1) Steady-state: (/t ) (2) Tansient, no heat geneation: (e gen ) (3) Steady-state, no heat geneation: (/t and e gen ) d 2 T (2 15) d 2 k d 2 T d 2 ė gen 2 T 1 T (2 16) 2 a 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 Fig 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 2p 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 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 E gen, 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) Q Q + Δ E gen + Δ Volume element FIGURE 2 14 One-dimensional heat conduction though a volume element in a long cylinde. E gen, element e genv element e gen A (2 2) Substituting into Eq. 2 18, we get T t t T t Q Q e gen A ca (2 21) t whee A 2p. You may be tempted to epess the aea at the middle of the element using the aveage adius as A 2p( /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 Q. Q. 1 T t t T t e gen c (2 22) A t Taking the limit as and t yields 1 A T ka T e gen c (2 23) t

10 72 HEAT CONDUCTION EQUATION (a) The fom that is eady to integate d d dt = d (b) The equivalent altenative fom d 2 T dt d 2 + = d FIGURE 2 15 Two equivalent foms of the diffeential equation fo the onedimensional steady heat conduction in a cylinde with no heat geneation. since, fom the definition of the deivative and Fouie s law of heat conduction, Q. Q. Q. T lim ka (2 24) Noting that the heat tansfe aea in this case is A = 2p, the one-dimensional tansient heat conduction equation in a cylinde becomes 1 T T Vaiable conductivity: k e gen c (2 25) t Fo the case of constant themal conductivity, the pevious equation educes to 1 1 T Constant conductivity: T (2 26) k a t whee again the popety a k/c is the themal diffusivity of the mateial. Eq educes to the following foms unde specified conditions (Fig. 2 15): (1) Steady-state: (/t ) (2) Tansient, no heat geneation: (e gen ) (3) Steady-state, no heat geneation: (/t and e gen ) ė gen 1 d ė gen dt (2 27) d d k 1 1 T T (2 28) t d dt (2 29) d d E gen Q Q + Δ + Δ R 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 4p 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 4p 2 instead of A 2p, the one-dimensional tansient heat conduction equation fo a sphee is detemined to be (Fig. 2 16) 1 T Vaiable conductivity: 2 k T e gen c (2 3) t 2 Volume element FIGURE 2 16 One-dimensional heat conduction though a volume element in a sphee. which, in the case of constant themal conductivity, educes to 1 ė 1 T Constant conductivity: 2 T gen (2 31) 2 k a t whee again the popety a k/c is the themal diffusivity of the mateial. It educes to the following foms unde specified conditions:

11 73 CHAPTER 2 (1) Steady-state: (/t ) (2) Tansient, no heat geneation: (e gen ) (3) Steady-state, no heat geneation: (/t and e gen ) 1 d 2 dt (2 32) d d k T 2 T (2 33) 2 a t d d 2 dt ė gen 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. 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 e gen 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 2 Heat Conduction though the Bottom of a Pan Conside a steel pan placed on top of an electic ange to cook spaghetti (Fig. 2 17). The bottom section of the pan is.4 cm thick and has a diamete of 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. SOUTION A steel pan placed on top of an electic ange is consideed. The diffeential equation fo the vaiation of tempeatue in the bottom of the pan is to be obtained. Analysis 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 one-dimensional. 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. 8 W FIGURE 2 17 Schematic fo Eample 2 2.

12 74 HEAT CONDUCTION EQUATION 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. Discussion Note that the conditions at the suface of the medium have no effect on the diffeential equation. Wate Resistance heate FIGURE 2 18 Schematic fo Eample 2 3. EXAMPE 2 3 Heat Conduction in a Resistance Heate A 2-kW esistance heate wie with themal conductivity k 15 W/m K, diamete D.4 cm, and length 5 cm is used to boil wate by immesing it in wate (Fig. 2 18). 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 of a wate heate is consideed. The diffeential equation fo the vaiation of tempeatue in the wie is to be obtained. Analysis 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 have T T() duing steady opeation since the tempeatue in this case depends on only. The ate of heat geneation in the wie pe unit volume can be detemined fom E. gen Ė gen 2 W e gen W/m V 3 (pd 2 /4) [p(.4 m) 2 wie /4](.5 m) 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 a dt d b which is the steady one-dimensional heat conduction equation in cylindical coodinates fo the case of constant themal conductivity. Discussion Note again that the conditions at the suface of the wie have no effect on the diffeential equation. ė gen k

13 75 CHAPTER 2 EXAMPE 2 4 Cooling of a Hot Metal Ball in Ai 75 F 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 19). 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 A hot metal ball is allowed to cool in ambient ai. The diffeential equation fo the vaiation of tempeatue within the ball is to be obtained. Analysis 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 changes with time duing cooling. Theefoe, this is a one-dimensional tansient heat conduction poblem since the tempeatue within the ball changes 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 T a 2 k T c b 2 t which is the one-dimensional tansient heat conduction equation in spheical coodinates unde the conditions of vaiable themal conductivity and no heat geneation. Discussion Note again that the conditions at the oute suface of the ball have no effect on the diffeential equation. Metal ball 6 F FIGURE 2 19 Schematic fo Eample 2 4. Q 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 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 develop the govening diffeential equation in such systems in ectangula, cylindical, and spheical coodinate systems. Rectangula Coodinates Conside a small ectangula element of length, width y, and height z, as shown in Fig 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 of change Rate of heat of the enegy Rate geneation conduction at conduction at, inside the content of, y, and z y y, and z z element the element Volume element Q e gen ΔΔyΔz z Q y y Δ Q z + Δz Q z Δy Δz Q y + Δy Q + Δ FIGURE 2 2 Thee-dimensional heat conduction though a ectangula volume element.

14 76 HEAT CONDUCTION EQUATION o E element Q Q y Q z Q Q y y Q z z E gen, 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 ) E gen, element e genv element e genyz Substituting into Eq. 2 36, we get Q Q y Q z Q Q y y Q z z e genyz cyz Dividing by yz gives T tt T t t Q. Q. Q. zz Q. yy Q. Q. 1 1 y 1 z e gen yz z y y z T tt T t c (2 37) t 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 T k T k T k T e gen 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. 1 Q 1 T kyz yz yz Q. yy Q. y 1 Q y 1 T kz y z y z y y y Q. zz Q. z z 1 Q z 1 T ky y z y z z z Eq is the geneal heat conduction equation in ectangula coodinates. In the case of constant themal conductivity, it educes to whee the popety a k/c is again the themal diffusivity of the mateial. Eq is known as the Fouie-Biot equation, and it educes to these foms unde specified conditions: ė gen k T k T y k T z 2 T T 2 T 1 2 T (2 39) 2 y 2 z 2 k a t

15 (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) ė gen 2 T 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 a t 2 T 2 T 2 T (2 42) 2 y 2 z 2 2 T 2 2 T 2 2 T 2 77 CHAPTER 2 2 T e gen + y T z 2 + k = 2 T 1 T + y T z 2 = a t 2 T + y T z 2 = 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 21). 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 Fig. 2 22, 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 f, y sin f, and z z Afte lengthy manipulations, we obtain FIGURE 2 21 The thee-dimensional heat conduction equations educe to the one-dimensional ones when the tempeatue vaies in one dimension only. dz z z d 1 T k 1 T T ak k T e gen c (2 43) T f f b 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 Fig. 2 23, 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 f sin u, y sin f sin u, and z cos u Again afte lengthy manipulations, we obtain T ak T ak sin u T 2 T 1 k e gen c f f b u u b 2 2 sin 2 u 2 sin u 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, since they esult in odinay diffeential equations. φ dφ FIGURE 2 22 A diffeential volume element in cylindical coodinates. z φ θ d dθ dφ FIGURE 2 23 A diffeential volume element in spheical coodinates. y y

16 78 HEAT CONDUCTION EQUATION Metal billet z R 6 F T = 65 F φ Heat loss FIGURE 2 24 Schematic fo Eample 2 5. EXAMPE 2 5 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 A shot cylindical billet is cooled in ambient ai. The diffeential equation fo the vaiation of tempeatue is to be obtained. Analysis The billet shown in Fig 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 changes with time duing cooling. Theefoe, this is a twodimensional tansient heat conduction poblem since the tempeatue within the billet changes 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 f equal to zeo. We obtain 1 ak T b z c In the case of constant themal conductivity, it educes to 1 T ak T z b 2 T z 2 1 T a t which is the desied equation. Discussion Note that the bounday and initial conditions have no effect on the diffeential equation. 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.

17 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 25). 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 26). 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 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 5 C 79 CHAPTER 2 The diffeential equation: d 2 T d 2 = Geneal solution: T() = C 1 + C 2 Abitay constants Some specific solutions: T() = T() = + 12 T() = 3 T() = 6.2 FIGURE 2 25 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 26 To descibe a heat tansfe poblem completely, two bounday conditions must be given fo each diection along which heat tansfe is significant. 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

18 8 HEAT CONDUCTION EQUATION tempeatue of T i, the initial condition in 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 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 27) T(, t) = 15 C T(, t) = 7 C FIGURE 2 27 Specified tempeatue bounday conditions on both sufaces of a plane wall. 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. Heat flu Conduction 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 a (W/m 2 ) (2 47) positive diection b. q = k T(, t) Conduction T(, t). k = q Heat flu FIGURE 2 28 Specified heat flu bounday conditions on both sufaces of a plane wall. 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 28). 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)

19 81 CHAPTER 2 Note that the heat flu at the suface at is in the negative -diection, and thus it is 5 W/m 2. The diection of heat flu aows at in Fig in this case would be evesed. 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 29) Insulation T(, t) 6 C 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. T(, t) = T(, t) = 6 C FIGURE 2 29 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 is subjected to the same themal conditions, and thus the tempeatue distibution in one half of the plate is the same as that in the othe half. That is, the heat tansfe poblem in this plate possesses themal symmety about the cente plane at /2. Also, the diection of heat flow at any point in the plate is towad the suface close to the point, and thee is 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 3) T(/2, t) (2 5) 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). Zeo slope 2 T(/2, t) Cente plane = Tempeatue distibution (symmetic about cente plane) FIGURE 2 3 Themal symmety bounday condition at the cente plane of a plane wall.

20 82 HEAT CONDUCTION EQUATION EXAMPE 2 6 Heat Flu Bounday Condition 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.. q Wate 11 C FIGURE 2 31 Schematic fo Eample 2 6. SOUTION An aluminum pan on an electic ange top is consideed. The bounday conditions fo the bottom of the pan ae to be obtained. Analysis 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 Fig 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 p(.1 m) 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. Discussion 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 at the suface in a selected diection Heat convection at the suface in the same diection