Steady symmetrical temperature field in a hollow spherical particle with temperature-dependent thermal conductivity

Transcription

1 Arch. Mech., 64, 4, pp , Warszawa 212 Steady symmetrical temperature field i a hollow spherical particle with temperature-depedet thermal coductivity A. MOOSAIE Departmet of Mechaical Egieerig Yasouj Uiversity Yasouj, Ira amimoosaie@gmail.com I this work, a exact aalytical solutio to the axisymmetric heat coductio equatio for hollow spherical objects with temperature-depedet thermal coductivity is preseted. The oliear differetial equatio is first trasformed ito a liear oe by meas of a itegral trasform method. The, the separatio of variables method is employed to solve the trasformed liear equatio. Ultimately, we use the iverse trasform to obtai the physical temperature field. Furthermore, two examples are worked out, i.e., the oe-dimesioal heat coductio i the radial directio ad the two-dimesioal case with axial symmetry. The solutio is preseted as a ifiite series i terms of Legedre fuctios. The problem with spherical symmetry is also solved by usig perturbatio methods up to the third-order approximatio, ad the results are compared with the exact solutio. Key words: heat coductio, steady-state, aalytical solutio, temperature-depedet thermal coductivity, oliear equatio, hollow sphere. Copyright c 212 by IPPT PAN 1. Itroductio Heat coductio i spherical objects is a importat problem i egieerig practice. It is also a iterestig problem from a fudametal/mathematical poit of view. Aalytical methods are ofte limited to liear problems, i.e., problems with liear differetial equatios ad boudary coditios. I heat coductio cotext, this implies a costat or at most a space-depedet (but ot temperature-depedet) thermal coductivity. However, the assumptio of a costat thermal coductivity is valid whe the rage of temperatures ivolved is ot wide. Whe we ecouter a wide rage of temperatures i a problem, the the temperature depedece of the thermal coductivity is usually to be take ito accout. Aalytical solutio of liear heat coductio problem i a spherical object is a rather classical problem, see for example [1]. Also, some recet aalytical works ca be foud o the o-fourier heat coductio i a hollow sphere [2, 3]. However, these are liear cases. Aalytical solutios for oliear cases

2 46 A. Moosaie with temperature-depedet thermal coductivity are rare. Trostel [4, 5] has proposed a method to treat this kid of problems aalytically usig the Kirchhoff s trasform. He has applied his method to the oe-dimesioal problem of oliear heat coductio i a hollow cylider (i the radial directio). A perturbatio method is utilized i [6] to solve the oliear heat coductio problem i a fi with temperature-depedet thermal coductivity. The homotopy aalysis method (HAM) is used i [7] to aalytically ivestigate the thermal performace of a straight fi of trapezoidal profile with temperature-depedet thermal coductivity. Hybrid aalytical-umerical methods are becomig more attractive amog researchers. Ofte i these approaches, the oliear goverig equatio is reduced i dimesios by usig some symmetry argumets, e.g., by usig the Lie group theory, ad the the reduced problem (usually a ordiary differetial equatio) is solved umerically, e.g., see [8]. I this work, we aalytically solve the problems of spherically-symmetric ad axisymmetric heat coductio i a hollow sphere with temperature-depedet thermal coductivity. The solutio to this oliear problem is obtaied as a ifiite series i terms of Legedre fuctios. We make use of the Kirchhoff s itegral trasform to solve the problem. The remaider of this paper is orgaized as follows. The goverig differetial equatios are preseted i Sec. 2. Sectio 3 cotais the solutio method i geeral ad the applicatio of the geeral solutio to hollow spherical objects with worked-out examples. The paper is cocluded i Sec Goverig equatios I this sectio, we preset the goverig equatios of steady heat coductio i a hollow sphere with temperature-depedet thermal coductivity. For this purpose, we start with the steady eergy coservatio equatio without heat geeratio: (2.1) q =, where ad q are the abla operator ad heat flux vector, respectively. This is a scalar equatio (2.2) q i x i = q 1 x 1 + q 2 x 2 + q 3 x 3 =, for three compoets of the heat flux vector q, i.e., q i s. I order to close the system, we require a costitutive equatio. I this work, we use the Fourier heat coductio law: (2.3) q = λ ϑ,

3 Steady symmetrical temperature field i which the scalar quatity ϑ is the temperature, ad λ is the thermal coductivity tesor. For isotropic materials, the thermal coductivity tesor λ reduces to a spherical tesor, i.e., (2.4) λ = λ1, with λ ad 1 beig the thermal coductivity ad the idetity tesor, respectively. Substitutig Eq. (2.4) ito Eq. (2.3) yields (2.5) q = λ ϑ. The Fourier costitutive equatio (2.5) alog with the eergy coservatio equatio (2.1) gives the followig field equatio for the temperature: (2.6) (λ ϑ) = λ ϑ + λ ϑ =, where is the Laplacia operator. The thermal coductivity λ ca deped o the spatial coordiates (e.g., due to material ihomogeeities) ad/or temperature. The former case leads to a liear partial differetial equatio (PDE) with variable coefficiets whereas the latter case results i a oliear PDE. I the simplest case, λ is assumed to be a costat (with λ = ), ad Eq. (2.6) reduces to the Laplace equatio: (2.7) ϑ =. Experimetal observatios show that, i geeral, λ does deped o temperature, i.e., λ = λ(ϑ) [9]. The assumptio of a costat thermal coductivity is a good approximatio whe the rage of temperatures ivolved is small. This assumptio is ofte made because it offers a great simplificatio i the mathematical aalysis of heat coductio problems. However, i problems which ivolve a broad rage of temperatures, this assumptio becomes less accurate ad oe eeds to take ito accout the depedece of λ o the temperature. By doig so, we have λ = dλ dϑ ϑ, ad Eq. (2.6) reads (2.8) dλ ϑ ϑ + λ(ϑ) ϑ =, dϑ which is obviously oliear. I the ext sectio, we aalytically solve the oliear problem of heat coductio i hollow spherical objects with temperature-depedet thermal coductivity.

4 48 A. Moosaie 3. Aalytical solutio I this sectio, we preset aalytical solutios to the oliear PDE (2.6) (or (2.8)) i hollow spherical objects with ier ad outer radii r i ad r o, respectively. First, we preset the solutio strategy for this type of problems i Subsec The, we proceed to solve the oliear PDE (2.6) i the spherical coordiate system show i Fig. 1. I geeral, for a steady three-dimesioal case we have ϑ = ϑ (r,ψ,ϕ). However, we cosider two reduced cases i this paper. The first case, preseted i Subsec. 3.2, cosiders the temperature field with spherical symmetry (oe-dimesioal i the radial directio), that is (3.1) ϑ ψ = ϑ =, ϑ = ϑ (r). ϕ The secod case, preseted i Subsec. 3.3, is the axisymmetric case (two-dimesioal) which takes place whe (3.2) ϑ =, ϑ = ϑ (r,ψ). ϕ Fig. 1. Spherical coordiate system Solutio strategy Trostel [4] has developed a methodology to deal with the oliear equatio (2.6). It is based o the followig itegral trasform of the temperature field: (3.3) Θ (ϑ) = 1 λ ϑ ϑ= λ( ϑ) d ϑ, λ = λ(ϑ = ). Takig the gradiet of Eq. (3.3) we have (3.4) Θ = 1 λ λ(ϑ) ϑ,

5 Steady symmetrical temperature field i which the Leibiz itegral theorem is used. Substitutio of Eq. (3.4) ito the oliear heat equatio (2.6) yields the Laplace partial differetial equatio for the trasformed temperature Θ: (3.5) Θ =. It is see that we obtai a liear PDE which ca be solved aalytically. The oliearity ow lies i the itegral trasform (3.3), that is, Θ oliearly depeds o ϑ ad vice versa. For a wide rage of egieerig materials, oe ca assume a liear depedece of the thermal coductivity λ o the temperature ϑ, that is (3.6) λ(ϑ) = λ λ 1 ϑ, with λ ad λ 1 beig material costats. Isertig Eq. (3.6) ito the itegral trasform (3.3) results i (3.7) Θ (ϑ) = ϑ ε 2 ϑ2, ε = λ 1 λ. The oliear algebraic equatio (3.7) describes the trasformed temperature Θ as a fuctio of the physical temperature ϑ. I tur, oe ca derive a equatio for ϑ i terms of Θ by ivertig (3.7): (3.8) ϑ 1,2 (Θ) = 1 ε ( 1 ± ) 1 2εΘ. I order to decide which sig reveals physically acceptable temperatures, we look at the itegral trasform (3.3) i the limitig case λ 1 (i.e., ε ) which represets the case of a costat thermal coductivity. Equatios (3.3) ad (3.7) show that ϑ = Θ i this case. Now, we take the limit of expressio (3.9) as ε. For the plus sig, we have For the mius sig, oe writes lim ϑ 1 ( 1 = lim 1 + ) 1 2εΘ = 2 ε ε ε =. Usig the l Hopital s rule, we have lim ϑ 1 1 2εΘ 2 = lim ε ε ε lim ϑ 1 ( 2 = lim 1 ) 1 2εΘ = ε ε ε. ( = lim 1 ) ( 2Θ) ε εΘ = Θ.

6 41 A. Moosaie Therefore, we choose the mius sig which yields (3.9) ϑ (Θ) = 1 ( 1 ) 1 2εΘ. ε It shall be oted here that Eq. (3.9) yields physical temperatures whe 2εΘ < 1 or Θ < 1/2ε. This might seem too restrictive at the first glace. However, such iequality typically holds for egieerig materials over a cosiderable rage of temperatures. For example, for mild steel we have ε = ad thus Θ < 857 which traslates to ϑ < 1714 C Case with spherical symmetry I this case, the temperature oly depeds o the radial coordiate ad we have ϑ = ϑ (r). Thus, the goverig Eq. (2.6) reduces to (3.1) [λ(ϑ) ϑ] = 1 [ d r 2 r 2 λ(ϑ) dϑ ] =. dr dr The Dirichlet s boudary coditios are (3.11) ϑ (r = r i ) = ϑ i, ϑ (r = r o ) = ϑ o. Usig the liear depedece of λ o temperature (3.6) ad the trasform (3.7), the goverig Eq. (3.1) i terms of the trasformed temperature Θ reads (3.12) Θ = 1 ( d r 2 r 2dΘ ) =. dr dr This is a Cauchy Euler differetial equatio with the geeral solutio: (3.13) Θ (r) = C 1 + C 2 r. The itegratio costats C 1 ad C 2 are to be determied from the boudary coditios. To this aim, we first eed to trasform the boudary coditios (3.11): (3.14) Θ (r = r i ) = ϑ i ε 2 ϑ2 i = Θ i, Θ (r = r o ) = ϑ o ε 2 ϑ2 o = Θ o. Applyig Eq. (3.14) to Eq. (3.13) we have (3.15) C 1 = Θ i r o r i r o (Θ o Θ i ) = r oθ o r i Θ i r o r i, C 2 = r i r o r i r o (Θ o Θ i ).

7 Steady symmetrical temperature field As a example, here we cosider a hollow sphere with r i =.6 cm ad r o = 1. cm. The boudary temperatures are assumed to be ϑ i =. C ad ϑ = 1. C. Three cases are cosidered. First, we cosider a costat thermal coductivity which leads to the liear differetial equatio (2.7) for the temperature field. I this case, the temperature field is idepedet of the value of the thermal coductivity. As for the secod case, we cosider a sphere made of mild steel for which we have [4] λ =.12 cal cm 1 sec 1 C 1, λ 1 = cal cm 1 sec 1 C 2. A positive λ 1 meas that the thermal coductivity decreases with icreasig the temperature. Also, we preset aother fictitious material with λ =.12 cal cm 1 sec 1 C 1, λ 1 = cal cm 1 sec 1 C 2, whose thermal coductivity icreases with icreasig the temperature. The temperature profiles for the above-metioed cases are show i Fig. 2. For the first case with costat λ we have ϑ = Θ ad we get the classical solutio. For the secod case (mild steel) the temperature profile deviates from the first case. Except from the boudaries r i ad r o, the temperature is lower across the sphere thickess. ϑ/ r of the oliear temperature is smaller tha that of the liear oe i the viciity of the ier surface ad gets larger by approachig the outer surface. For the third case with λ 1 = 7 1 5, we have the opposite behavior. The temperature is greater across the sphere thickess. The temperature gradiet is greater tha that of the liear temperature adjacet to the ier surface ad it gets smaller by approachig the outer surface. 1 8 ϑ [ C] 6 4 λ 1 =. 2 λ 1 = +7.x1-5 λ 1 = -7.x r [cm] Fig. 2. Spherically symmetric temperature profiles i a hollow sphere with r i =.6 cm, r o = 1.cm, ϑ i = C, ϑ o = 1 C ad λ = λ λ 1 ϑ with λ =.12. Three cases are show: costat λ, λ decreasig with temperature ad λ icreasig with temperature. Values of λ 1 are show i the figure.

8 412 A. Moosaie It should be oted here that the oliear effects are proouced whe the temperature rage is wide. For example, if we reduce ϑ o from 1 C to 4 C, we observe that the two temperature profiles with o-zero λ 1 approach the oe with zero λ 1, as show i Fig. 3. If we decrease the temperature differece eve more, say ϑ o = 1 C, the the three profiles fall o top of each other, see Fig. 4. This shows that for applicatios ivolvig a wide rage of temperatures, oe has to take the oliearity ito accout. However, whe the temperature rage is arrow, the liear model with costat λ is sufficietly accurate ϑ [ C] λ 1 =. 8 λ 1 = +7.x1-5 λ 1 = -7.x r [cm] Fig. 3. Spherically symmetric temperature profiles i a hollow sphere with r i =.6 cm, r o = 1. cm, ϑ i = C, ϑ o = 4 C ad λ = λ λ 1 ϑ with λ =.12. Three cases are show: costat λ, λ decreasig with temperature ad λ icreasig with temperature. Values of λ 1 are show i the figure ϑ [ C].4 λ 1 =..2 λ 1 = +7.x1-5 λ 1 = -7.x r [cm] Fig. 4. Spherically symmetric temperature profiles i a hollow sphere with r i =.6 cm, r o = 1. cm, ϑ i = C, ϑ o = 1 C ad λ = λ λ 1 ϑ with λ =.12. Three cases are show: costat λ, λ decreasig with temperature ad λ icreasig with temperature. Values of λ 1 are show i the figure. All three cases lie o top of each other.

9 3.3. Case with axial symmetry Steady symmetrical temperature field For the axisymmetric case with λ = λ λ 1 ϑ, the goverig Eq. (2.8) reduces to the followig oliear PDE: [( ) ϑ 2 (3.16) λ r r 2 + (λ λ 1 ϑ) ( ) ϑ 2 ] ψ The Dirichlet s boudary coditios read [ 2 ϑ r ϑ r r + 1 r 2 (3.17) ϑ (r = r i,ψ) = ϑ i (ψ), ϑ (r = r o,ψ) = ϑ o (ψ). ( 2 )] ϑ ϑ + cot ψ =. ψ2 ψ Usig the temperature trasform (3.7), Eq. (3.16) reduces to the Laplace equatio for the trasformed temperature Θ: (3.18) Θ = 2 Θ r Θ r r + 1 ( 2 ) Θ r 2 ψ 2 + cotψ Θ, ψ subjected to the trasformed boudary coditios (3.19a) (3.19b) Θ (r = r i,ψ) = ϑ i (ψ) ε 2 (ϑ i (ψ)) 2 = Θ i (ψ), Θ (r = r o,ψ) = ϑ o (ψ) ε 2 (ϑ o (ψ)) 2 = Θ o (ψ). Now, we solve this problem by the use of a separatio asatz as Θ (r,ψ) = R (r) Ψ (ψ). Substitutig this asatz ito Eq. (3.18) yields (3.2) r 2d2 R dr 2 + 2r dr dr ( + 1) R (r) =, (3.21) d 2 Ψ dψ 2 + cot ψdψ dψ + ( + 1) Ψ (ψ) =. Equatio (3.2) is the Cauchy Euler differetial equatio ad its geeral solutio reads (3.22) R (r) = A 1 r + B 1 r +1. Equatio (3.21) ca be rewritte as ( 1 d si ψ dψ ) + ( + 1) Ψ (ψ) =. si ψ dψ dψ

10 414 A. Moosaie We ow utilize the followig trasform of agle ψ: (3.23) Ψ (ψ) = Ψ (ξ (ψ)) with ξ (ψ) = cos ψ, 1 ξ 2 = si 2 ψ ad dξ dψ = si ψ. Applyig this trasform to Eq. (3.21), we obtai the Legedre differetial equatio i terms of ξ: [ d (1 (3.24) ξ 2 ) ] dψ + ( + 1) Ψ (ξ) =, dξ dξ which has the followig geeral solutio. (3.25) Ψ (ξ) = Ψ (cos ψ) = A 2 P (ξ) + B 2 Q (ξ), where P (ξ) ad Q (ξ) are the spherical fuctios, i.e., Legedre fuctios, of first ad secod kid, respectively. Sice we have ξ = cos ψ 1 ad the spherical fuctio of secod kid is ot defied o this iterval, the geeral solutio (3.25) reduces to (3.26) Ψ (ξ) = Ψ (cos ψ) = A 2 P (ξ). With the help of abbreviatios A = A 1 A 2 ad B = B 1 A 2 we have ( R (r) Ψ (ψ) = A r + B ) r +1 P (ξ). Therefore, the trasformed temperature field becomes (3.27) Θ (r,cosψ) = Θ (r,ξ) = = = R (r) Ψ (ψ) = ( A r + B ) r +1 P (ξ). Now, we have to determie the coefficiets A ad B ( =,1,2,...) by eforcig the boudary coditios (3.19). The trasformed boudary temperatures are fuctios of ψ whereas the trasformed temperature field is a fuctio of ψ through ξ = cos ψ. Therefore, we write the fuctios Θ o (ψ) ad Θ i (ψ) as Θ o = Θ o (ξ) = Θ o (cos ψ) ad Θ i = Θ i (ξ) = Θ i (cosψ). By doig so, we have (3.28) A ro + B r (+1) A ri + B r (+1) o i = c (o), = c (i),

11 Steady symmetrical temperature field i which the fuctios Θ o (ξ) ad Θ i (ξ) are expaded as (3.29a) (3.29b) Θ o (ξ) = Θ i (ξ) = = = c (o) P (ξ), c (i) P (ξ). The coefficiets c (o) ad c (i) are determied from the orthogoality of spherical fuctios P (ξ) o the iterval ξ [ 1,+1], amely (3.3) This yields +1 1, m, P (ξ) P m (ξ) dξ = , m =. (3.31a) (3.31b) c (o) = c (i) = Θ o (ξ) P (ξ) dξ, Θ i (ξ) P (ξ) dξ. Now, the costats A ad B ( =,1,2,...) are obtaied by solvig the liear equatio system (3.28): (3.32a) (3.32b) A = α (i) c (i) + α (o) c (o), B = β (i) c (i) + β (o) c (o), where (3.33a) α (o) = r (+1) i, (3.33b) (3.33c) (3.33d) = r (+1) o, α (i) β (o) β (i) = r i, = r o,

12 416 A. Moosaie i which (3.33e) = r o r (+1) i r (+1) o r i. Thus, the trasformed temperature field Θ ca be writte as (3.34) Θ (r,ξ) = where (3.35) η (r) = ( α (i) c (i) + α (o) η (r) P (ξ), = c (o) ) r + ( β (i) c (i) + β (o) c (o) ) r (+1). Fially, the temperature field ϑ (r,ψ) = ϑ (r,ξ) ca be obtaied by utilizig the iverse trasform (3.9): ϑ (r,ξ) = 1 ( 1 ) (3.36) 1 2εΘ (r,ξ) ε = 1 [ ( ) 1/2 ] 1 1 2ε η (r) P (ξ). ε First, we show that for ϑ i (ψ) = ϑ i ad ϑ o (ψ) = ϑ o, the solutio (3.34) reduces to the oe-dimesioal solutio (3.13) with itegratio costats give by (3.15). For this purpose, we start with costats (3.31): (3.37a) (3.37b) c (o) = Θ o c (i) = Θ i = P (ξ) dξ = P (ξ) dξ = { Θ o, =,, 1. { Θ i, =,, 1. This meas that oly the term with = is o-zero i the series (3.34) ad all other terms with 1 vaish. From Eq. (3.33e) we have = (r o r i ) /r i r o. The solutio (3.34) reduces to (3.38) Θ (r) = A + B r, with (3.39a) (3.39b) A = r oθ o r i Θ i r o r i, B = r i r o r i r o (Θ o Θ i ). This is exactly the solutio we obtaied i Subsec. 3.2.

13 Steady symmetrical temperature field As a example here, we solve the case with ϑi (ψ) = C ad ϑo (ψ) = ϑo si ψ with ϑo = 1 C. Agai, we have ri =.6 cm ad ro = 1. cm. The depedece of the boudary temperature Θo o ψ leads to a two-dimesioal temperature field, i.e., Θ = Θ (r, ψ). I this case, we have c(o) (3.4) Z+1 = Θo (ξ) P (ξ) dξ. 1 (i) The itegrals i (3.4) are evaluated umerically. Moreover, we have c = for all. The other coefficiets are computed usig the give formulae. Fially, the trasformed temperature Θ (r, ψ) ad cosequetly the physical temperature ϑ (r, ψ) are obtaied. λ1=-7.e-5 λ1=. 1.5 λ1=+7.e Fig. 5. Axisymmetric temperature fields i a hollow sphere with ri =.6 cm, ro = 1. cm, ϑi (ψ) = C, ϑo (ψ) = 1 si ψ C ad λ = λ λ1 ϑ with λ =.12. Three cases are show: costat λ, λ decreasig with temperature ad λ icreasig with temperature. Values of λ1 are show i the figure. Horizotal axis is x = r si ψ ad vertical axis is z = r cos ψ. Colors map the temperature. The temperature fields for three differet cases are show i Fig. 5. These three cases have the same material properties as of the example i Subsec We, agai, observe that the temperature of the case with costat λ lies i betwee the other two cases with positive ad egative λ1. We also observe that the high-temperature regio is more exteded i the case with egative λ1 compared to other two cases. Moreover, oe-dimesioal temperature profiles i the radial directio r at ψ = π/2 ad i the zeithal directio ψ at r = (ri + ro ) /2 are plotted i Figs. 6

14 418 A. Moosaie 1 8 ϑ [ C] 6 4 λ 1 =. 2 λ 1 = -7.x1-5 λ 1 = +7.x r [cm] Fig. 6. Axisymmetric temperature profiles i a hollow sphere with r i =.6 cm, r o = 1. cm, ϑ i (ψ) = C, ϑ o (ψ) = 1 si ψ C ad λ = λ λ 1 ϑ with λ =.12. The profiles are alog the radial directio r at ψ = π/2. Three cases are show: costat λ, λ decreasig with temperature ad λ icreasig with temperature. Values of λ 1 are show i the figure. 8 6 ϑ [ C] 4 2 λ 1 =. λ 1 = -7.x1-5 λ 1 = +7.x ψ Fig. 7. Axisymmetric temperature profiles i a hollow sphere with r i =.6 cm, r o = 1. cm, ϑ i (ψ) = C, ϑ o (ψ) = 1 si ψ C ad λ = λ λ 1 ϑ with λ =.12. The profiles are alog the zeithal directio ψ at r =.8 cm. Three cases are show: costat λ, λ decreasig with temperature ad λ icreasig with temperature. Values of λ 1 are show i the figure. ad 7, respectively. Agai, we observe that the temperature is lower for the case with λ 1 > ad greater for the case with λ 1 < as compared to the liear case with λ 1 =. Figure 6 shows that ot oly the values, but also the shape of the temperature profile is chaged for differet cases. However, Fig. 7 reveals that, i the zeithal directio ψ, the shape of the temperature profile is preserved ad oly the values are chaged.

15 Steady symmetrical temperature field Compariso with perturbatio solutio Aother approach to solve the oliear heat equatio (2.8) is the use of perturbatio methods. This gives a approximate solutio to the problem. Here, we compare such a approximate solutio with our exact solutio ad examie the covergece of the perturbatio series. I order to perform a perturbatio solutio, we rewrite Eq. (2.8) i the followig form: (3.41) (1 εϑ) ϑ = ε ϑ ϑ. Note that we have assumed a liear variatio of the thermal coductivity with temperature, i.e., λ = λ λ 1 ϑ = λ (1 εϑ) with ε = λ 1 /λ. Now, we assume that the temperature field ca be expressed as a power series i the small parameter ε: (3.42) ϑ = ε ϑ = ϑ + εϑ 1 + ε 2 ϑ 2 +. = Substitutig this asatz i Eq. (3.41), we have (3.43) ( 1 ) ε +1 ϑ ε ϑ = = = ε +1 ϑ ε ϑ. Groupig terms with similar power of ε yields the followig series of differetial equatios: = = (3.44a) (3.44b) (3.44c) (3.44d) ε : ϑ =, ε 1 : ϑ 1 = ϑ ϑ, ε 2 : ϑ 2 = ϑ ϑ ϑ ϑ 1, ε 3 : ϑ 3 = ϑ ϑ 2 + ϑ 1 ϑ ϑ ϑ 2 + ϑ 1 ϑ 1, which are called zeroth-, first-, secod- ad third-order approximatios, respectively. Also, appropriate boudary coditios are to be derived. To this aim, we isert the asymptotic expasio (3.42) ito Eq. (3.11). Groupig the terms with similar power of ε, we get (3.45a) (3.45b) (3.45c) (3.45d) ε : ϑ (r = r i ) = ϑ i, ϑ (r = r o ) = ϑ o, ε 1 : ϑ 1 (r = r i ) =, ϑ 1 (r = r o ) =, ε 2 : ϑ 2 (r = r i ) =, ϑ 2 (r = r o ) =, ε 3 : ϑ 3 (r = r i ) =, ϑ 3 (r = r o ) =.

16 42 A. Moosaie This approach ca be exteded to th-order approximatio. However, it demads a cosiderable amout of calculatio effort whe is large. Here, we calculate up to the third-order approximatio ad compare the results of the perturbatio approximatios of differet orders ( 3) with the exact solutio. The results are plotted i Fig. 8 for the parameters of mild steel (ε = ). We observe that although the first-order solutio gives cosiderable improvemet to the liear solutio, it is ot eough to get the temperature profile accurately. The secod- ad third-order solutios are almost 1 8 ϑ [ C] 6 4 zeroth-order first-order 2 secod-order third-order exact r [cm] Fig. 8. Spherically symmetric temperature profiles i a hollow sphere with r i =.6 cm, r o = 1. cm, ϑ i = C, ϑ o = 1 C ad λ = λ λ 1 ϑ with λ =.12 ad λ 1 = Also, perturbatio solutios of differet orders are show ϑ [ C] 4 zeroth-order first-order 2 secod-order third-order exact r [cm] Fig. 9. Spherically symmetric temperature profiles i a hollow sphere with r i =.6 cm, r o = 1. cm, ϑ i = C, ϑ o = 1 C ad λ = λ λ 1 ϑ with λ =.12 ad λ 1 = Also, perturbatio solutios of differet orders are show.

17 Steady symmetrical temperature field idistiguishable from the exact solutio. The first-order perturbatio solutio is ofte used i approximatios of oliear equatios. Here, we see that oly a first-order approximatio does ot produce accurate results. If we icrease the perturbatio parameter, say ε = , the eve the secod- ad third-order solutios deviate from the exact oe, as show i Fig Coclusios I this paper, we have developed a exact aalytical solutio for steady oliear heat coductio equatio with temperature-depedet thermal coductivity i hollow spherical objects. For this purpose, we have employed a itegral trasform which trasforms the oliear equatio ito a liear oe (the Laplace equatio). Oce the Laplace equatio is solved for the trasformed temperature subjected to trasformed boudary coditios, oe ca compute the physical temperature usig the iverse trasform. Two problems are solved for demostratio of the proposed solutio. First, the temperature field i a hollow sphere with spherical symmetry is ivestigated. This is a oe-dimesioal problem i the radial directio. Secod, we solve for the axisymmetric temperature field i a hollow sphere which is a two-dimesioal problem. Fially, we ivestigated perturbatio solutios of the oe-dimesioal problem ad compared them with the exact solutio. With this, we are able to examie the covergece of the perturbatio solutios. Refereces 1. R. Trostel, Istatioäre Wärmespauge i eier Hohlkugel, Igeieur-Archiv, 24, , R. Shirmohammadi, A. Moosaie, No-Fourier heat coductio i a hollow sphere with periodic surface heat flux, It. Commu. Heat Mass Trasf., 36, , A. Moosaie, Axisymmetric o-fourier temperature field i a hollow sphere, Arch. Appl. Mech., 79, , R. Trostel, Statioäre Wärmespauge mit temperaturabhägige Stoffwerte, Igeieur-Archiv, 26, , R. Trostel, Wärmespauge i Hohlzylider mit temperaturabhägige Stoffwerte, Igeieur-Archiv, 26, , A. Aziz, T.Y. Na, Periodic heat trasfer i fis with variable thermal parameters, It. J. Heat Mass Trasf., 24, , F. Khai, A. Aziz, Thermal aalysis of a logitudial trapezoidal fi with temperaturedepedet thermal coductivity ad heat trasfer coefficiet, Commu. Noliear Sci. Numer. Simul., 15, 59 61, 21.

18 422 A. Moosaie 8. O.D. Makide, R.J. Moitsheki, O solutios of oliear heat diffusio model for thermal eergy storage problem, It. J. Physical Scieces, 5, , VDI-Gesellschaft Verfahrestechik ud Chemieigeieurwese, Wärmeatlas, Spriger, Berli 26. Received March 11, 212; revised versio May 29, 212.