Yilazer, A., et al.: Heat Cnductin in Cnvectively Cled Eccentric... THERMAL SCIENCE, Year 017, Vl. 1, N. 5, pp. 55-66 55 Intrductin HEAT CONDUCTION IN CONVECTIVELY COOLED ECCENTRIC SPHERICAL ANNULI A Bundary Integral Ment Methd by Ayhan YILMAZER * and Ceil KOCAR a Departent f Nuclear Engineering, Hacettepe University, Beytepe, Ankara, Turkey Original scientific paper1 https://di.rg/10.98/tsci1513118y In this paper heat cnductin equatin fr an eccentric spherical annulus with the inner surface kept at a cnstant teperature and the uter surface subjected t cnvectin is slved analytically. Eccentric prble dain is first transfred int a cncentric dain via frulating the prble in bispherical c-rdinate syste. Since an analytical Green s functin fr the heat cnductin equatin in bispherical c-rdinate fr an eccentric sphere subject t bundary cnditin f third type can nt be fund, an analytical Green's functin btained fr Dirichlet bundary cnditin is eplyed in the slutin. Utilizing this Green's functin yields a bundary integral equatin fr the unknwn nral derivative f the surface teperature distributin. The resulting bundary integral equatin is slved analytically using ethd f ents. The ethd has been applied t heat generating eccentric spherical annuli and results are cpared t the siulatin results f FLUENT CFD cde. A very gd agreeent was bserved in teperature distributin cputatins fr varius geetrical cnfiguratins and a wide range f Bit nuber. Variatin f heat dissipatin with radii and eccentricity ratis are studied and a very gd agreeent with FLUENT has been bserved. Keywrds: eccentric sphere, heat cnductin, bundary integral equatin, Green's functin Heat cnductin in a cncentric spherical annulus with r withut heat generatin is the subject f standard heat transfer textbks and well develped. Interested readers ay refer t the textbk by Ozisik [1] fr the analytical slutins f linear heat cnductin equatin fr cncentric spherical annuli expsed t unifr bundary cnditins f any kind and any functinal fr f space dependent heat generatin rate. Hwever, slving cnductin equatin analytically fr an eccentric spherical annulus has inherent difficulties assciated with eplying a bundary fitting c-rdinate syste and applicatin f bundary cnditins. Bispherical c-rdinate is a cnvenient rthgnal c-rdinate syste which fits bth bundaries f an eccentric spherical annulus and allws applicatin f first type bundary cnditins directly. Fr exaple, cnductin equatin fr tw adjacent spheres withut heat generatin lcated at any distance fr each ther was slved analytically by Alassar and Alinshawy [] in bispherical c-rdinate. They slved steady-state axisyetric heat cnductin equatin fr tw istheral spheres at different teperatures (first type bundary cnditins) with different radii. 1* Crrespnding authr, e-ail: yilazer@hacettepe.edu.tr
Yilazer, A., et al.: Heat Cnductin in Cnvectively Cled Eccentric... 56 THERMAL SCIENCE, Year 017, Vl. 1, N. 5, pp. 55-66 Alassar [3] slved the cnductin equatin analytically fr an eccentric spherical annulus with first type bundary cnditins and unifr heat generatin rate. The slutin was the superpsitin f slutins in bispherical c-rdinate f R-separable Laplace equatin and a particular slutin. It was liited t the aziuthally syetrical -D prbles and t the unifr heat generatin rate and culd nt be applied t a general spatially varying heat generated 3-D eccentric annuli. Since Helhltz differential equatin is nt separable r R- separable in bispherical c-rdinate syste [4] the cnductin equatin culd nt be slved fr an eccentric spherical annulus with heat generatin in bispherical c-rdinate using standard techniques such as eigenfunctin expansin r related ethds. Yilazer and Kcar [5] btained an exact slutin using Green s functin ethd t the 3-D cnductin equatin thrugh an eccentric spherical annulus with cnstant surface teperatures and with space dependent heat generatin. While there are several studies as previusly illustrated n heat cnductin thrugh cncentric and eccentric spheres, a literature survey reveals that there is nt an exact slutin t the cnductin equatin in a spherical annulus cled cnvectively (third type bundary cnditin) at ne r tw bundaries. In a recent paper, the authrs f this paper develped a new analytical apprach based n Green s functin ethd t arrive at a bundary integral equatin (BIE) fr -D steady heat cnductin equatin in an eccentric cylindrical annulus whse inner bundary was istheral and uter bundary was subjected t cnvectin [6]. Since an analytical Green s functin t the cnductin equatin in biplar c-rdinate fr an eccentric cylindrical annulus subject t bundary cnditin f third type culd nt be fund, the prble was treated as a secnd type bundary value prble. The ethd is based n develping a BIE in biplar c-rdinate fr the uter surface teperature distributin using the analytical Green s functin btained fr secnd type bundary cnditin. The resulting BIE was slved by ethd f ents, referred t as bundary integral ent ethd (BIMM), prducing very accurate results. In this study BIMM is applied in bispherical c-rdinate t slve heat cnductin equatin analytically in heat generating 3-D eccentric spherical annuli whse inner surface is kept at a cnstant teperature and uter surface is subjected t the cnvectin. Instead f treating the prble as a secnd type bundary value prble as in previus cylindrical annulus cnductin prble [6], the eccentric spherical annulus cnductin prble is handled as a first type bundary value prble yielding a BIE fr the unknwn nral derivative f uter bundary teperature which is slved by the ethd f ents. The BIMM slutin prpsed in this study which is based n Green's functin apprach intrduces an unknwn BIE fr the nral derivative f the surface teperature aking analytical slutin uch re invlved. The ethd is applied fr a wide range f heat generatin ranges and Bit nubers and fr varius geetrical cnfiguratins, i. e., eccentricity and radii ratis. The results f teperature distributin and heat transfer calculatins are cpared with the siulatin results btained fr CFD cde FLUENT [7]. Definitin f the prble Cnsider an eccentric spherical annulus with inner surface kept istheral at teperature T i and uter surface cled cnvectively by a clant at an abient teperature, T. Steady-state heat cnductin equatin fr the annulus is: q T 0 k (1a)
Yilazer, A., et al.: Heat Cnductin in Cnvectively Cled Eccentric... THERMAL SCIENCE, Year 017, Vl. 1, N. 5, pp. 55-66 57 T T i n the inner surface (1b) T k h( T T ) n the uter surface n where is the Laplacian, q [W 3 ] the vluetric heat generatin rate, k [W 1 K 1 ] the theral cnductivity, h [W K 1 ] the cnvectin cefficient, n the utward nral, and T [K] the abient clant teperature. Ang the knwn rthgnal c-rdinate systes it is the bispherical c-rdinate fr which it is pssible t express bth bundaries f the eccentric annulus with nly ne c-rdinate paraeter [4, 8, 9]. If radii f the inner and uter spheres are dented by r i and r, respectively, and center t center distance r eccentricity by e then a bispherical c-rdinate syste culd be specified by using eq. (3) crrespnding t tw eccentric cnstant, ξ, spheres lying alng the z-axis: where (1c) a a i sinh and sinh () r 1 1 ri ( eri r)( eri r)( eri r)( eri r) a (3) e The cnductin prble defined by eq. (1) fr the cnvectively cled eccentric annulus culd be expressed in bispherical c-rdinate syste: 1 1 h h h sin (,, ) Q(,, ) 0 3 h sin sin with the diensinless bundary cnditins: (4a) (,, ) 1 at i (4b) (,,) a Bi (,, ) at n r (4c) where Bit nuber is defined as Bi = hr / k and h 1 h sin h h, h a csh cs a csh cs (5) are nralized scale factrs f the bispherical c-rdinate syste. The BIMM slutin If the inner and uter surfaces f the annulus are dented by Si and S then the diensinless cnductin equatin defined by eq. (1) has the fllwing Green s functin slutin: G G (,, ) QG dv G ds G ds n' n' n' n' (6) V S i S
Yilazer, A., et al.: Heat Cnductin in Cnvectively Cled Eccentric... 58 THERMAL SCIENCE, Year 017, Vl. 1, N. 5, pp. 55-66 where G G(r / r ') G(,, /,, ) is the Green s functin and the prie shws the cputatinal dain. Green s functin satisfies the fllwing equatins: 1 1 h h h sin G(r / r ) (r r ) 0 3 h sin sin G,, /,, n (7a) G,, /,, 0 at i (7b) a Bi G,, /,, at r where Dirac s delta functin culd be expressed in bispherical c-rdinate: (7c) ( ) ( ) ( ) ( ) Y Y hhh hhh * (r r ) (, ) (, ) 0 (8) Since Helhltz equatin in bispherical c-rdinate is neither separable nr R-separable [4], it is nt pssible t find an analytical expressin in bispherical c-rdinate fr the Green s functin f an eccentric spherical annulus with bundary cnditin f third kind, i. e. cnvective bundary cnditin in ur prble. Hwever, the riginal prble described by eq. (7) with third type bundary cnditin culd be transfred int a prble with a first type bundary cnditin: 1 1 h h h sin (,, ) Q(,, ) 0 3 h sin sin with diensinless bundary cnditins: (9a) (,, ) 1 at = i (9b) r (,, ) r 1 (,, ) (,, ) at abi n abi h where the nral derivative f the teperature distributin h (,, )/ n the uter bundary is assued t be knwn as a priri. Siilarly, Green s functin satisfies: 1 1 h h h sin G(r / r ) (r r ) 0 3 h sin sin 1 (9c) (10a) G,, /,, 0 at i (10b) G,, /,, 0 at (10c) Green s functin described by eqs. (10) fr eccentric spheres in bispherical c-rdinate is derived in [5, 10 ]: * G,, /,, csh cs csh cs Y (, ) Y (, ) g (, ) 0 (11a)
Yilazer, A., et al.: Heat Cnductin in Cnvectively Cled Eccentric... THERMAL SCIENCE, Year 017, Vl. 1, N. 5, pp. 55-66 59 wherey * are spherical harnics functins and Y is the cplex cnjugate f Y and g (, ) is the radial part f the Green s functin given by: 1 1 sinh ( i) sinh ( ) 1 g (, ), 1 (1)sinh ( i) g (, ) 1 1 sinh ( i) sinh ( ) g (, ), 1 (1)sinh ( i) (11b) It shuld be nted the factr a in the deninatr f g 1 (, ) and g (, ) in [5] is disissed in eq. (11b). Because the scale factrs are nralized by factr a in the present discussin distinctly fr the derivatin in [5]. Intrducing Green's functin given by eqs. (11) and bundary cnditins (9b), (9c), (10b), and (10c) int eq. (6) we get: 1 sinh ( ) Ψ (,, ) S(,, ) csh cs P(cs ) e ( 0.5) i 0 1 sinh i) where we dented: 1 sinh ) r abi 1 i * csh cs, Y (, ) 0 sinh i) (1a) QG d V S(,, ) (1b) V and the ents, are defined: (,, ) csh cs sin (, )dd (1c) π π, Y 0 0 Taking ent f bth sides f eq. (1a) by perating: π π (.) csh cs sin Y (, )dd (13) 0 0 a linear syte f infinite nuber f unknwn ents, culd be btained. Truncating the series slutin as 0,1,,..., L and, 1,... 1,0,1,..., 1, fr the diensinless teperature given by eq. (1a), this linear syste beces:
Yilazer, A., et al.: Heat Cnductin in Cnvectively Cled Eccentric... 60 THERMAL SCIENCE, Year 017, Vl. 1, N. 5, pp. 55-66 L,,, 0 rsinh 1 abi 1 1 csh i) r 1 π, csh,, ( 1) I, abi 1 3 sinh ( i) π( 1) ( 1/) π i 0, S, e csh, 0, I, 0 1 3 (14) sinh ( i), 0, where 0,1,,..., and, 1,...,0,... 1,. Here, I, and I, are integrals invlving triple prducts f spherical harnics functins and they culd be calculated fr [8]: π π 1 3 Y (, ) Y (, ) (, )sin d d = 1 Y 3 0 0 ( 1)( 1)( 1) 1 3 1 3 1 3 4π 0 0 0 1 3 where 1 3 1 3 is Wigner 3-j sybl. Using eqs. (1b) and (13) the ents S, culd re explicitly be expressed fr a given diensinless heat generatin rate distributin Q(ξ,θ, ): (15) where S 1 h h I 1,, ( ),, ( ) csh,, ( 1), 0 3 π π i 0 0 (16a) Q(,, ) h ( ) sin Y (, ) g (, )ddd (16b) 5/ (csh cs ) Heat transfer rate Let us define as in [5] the fllwing diensinless heat transfer rate: ht ( T )ds π π S a Q Bi d d hh T 4πr i T (17) 0 0 0 k(4π r0 ) r0 Intrducing the expressin fr Ψ given by eq. (1a) int eq. (17), Q beces: 1 ( 1/) abi 0 Q 1 e,0 h ( ) (18) π 0 r0
Yilazer, A., et al.: Heat Cnductin in Cnvectively Cled Eccentric... THERMAL SCIENCE, Year 017, Vl. 1, N. 5, pp. 55-66 61 where the ents,0 are btained fr the slutin f the linear syste given by eq. (14) and h 0 ( ) is calculated fr the expressin given by eq. (16b). An applicatin fr unifr surce distributin In this sectin, the ethd develped is applied fr unifr surce distributin. Teperature distributins and diensinless heat transfer rates fr varius diensinless surce strengths and a wide range f Bit nuber are calculated. Cparisns f the analytical results are ade with the siulatin results f CFD cde FLUENT. The FLUENT was used t slve energy cnservatin equatin in 3-D dain by a secnd rder discretizatin schee. In the first sequence f cputatins, radii rati is fixed as r /r i = 5.0 and eccentricity rati is changed as e/r i =1.0,.0, and 3.0, respectively. The residual is set as 1 10 11 fr energy equatin. Cputatinal dain is discretized with the esh nubers 07000, 50000, and 90000 fr e/r i = 1.0,.0, 3.0, and 4.0, respectively. In the secnd set f cputatins, eccentricity rati is fixed as e/r i = 0.5 and radii rati is changed as, r /r i =.0, 3.0, 4.0, and 5.0, respectively. Cputatinal dain is discretized with the esh nubers 157000, 1000, 45000, 730000 fr r /r i =.0, 3.0, 4.0, and 5.0, respectively. The quadrilateral esh eleents were used fr all the cases. The inner bundary is set as cnstant teperature bundary while cnvective bundary cnditin is applied at uter bundary. The nuerical value f the heat transfer cefficient is derived fr the Bit nuber f the interested case. The CFD calculatins are dne n an AMD 3.0 GHz cputer having a 4 GB ery. Decrease in the Bit nuber and increase in the abslute value f the heat generatin resulted in increase f siulatin tie and nuber f iteratins. Teperature distributin The diensinless teperature distributin in the eccentric sphere fr a space dependent surce distributin is expressed by eq. (1a). Fr a cnstant diensinless heat generatin rate Q, the S(,, ) defined by eq. (1b) culd be calculated: * (,, ) csh cs (, ) ( ) S Y h 0 where the functin h ( ) defined by eq. (16b) is btained: (19) 4Q π( 1) h ( ),0 3 ( 1/) 1 ( 1/) 1 i e (cth cth )sinh ( i) e (cth cth i)sinh ( ) 1 sinh ( i) The unknwn ents, in eq. (1a) culd be fund by slving the linear syste f equatins given by eqs. (14)-(16) in which the functin h ( ) takes the fr specified by eq. (0) fr unifr heat generatin and this cpletes the slutin. Effect f surce strength n teperature distributin It is first aied at bserving the effect f surce strength n teperature distributin. Fr this purpse teperature distributin is calculated alng the centerline [AB] n yz-plane f uter sphere as shwn in fig. 1. Variatin f teperature alng the line [AB] is depicted in (0)
Yilazer, A., et al.: Heat Cnductin in Cnvectively Cled Eccentric... 6 THERMAL SCIENCE, Year 017, Vl. 1, N. 5, pp. 55-66 fig. fr varius cnstant heat generatin rates: Q = 50, 5, 0, 5, and 50 fr a geetrical cnfiguratin r /r i = 5.0 and e/r i =.0. The y c-rdinate alng the [AB] line which extends fr (0, r, a cth ξ ) t (0,r, a cth ξ ) is nn-diensinalized as y * = y / ( AB / ) where y * extends fr 1 t 1. It shuld be nted that r i, r, and e d nt define the geetrical cnfiguratin theselves. Instead the radii rati r /r i and eccentricity rati e / r i are the paraeters identifying geetrical cnfiguratin. Naely, irrespective f the values f r, r i and e, diensinless teperature is erely a functin f these diensinless ratis. Bit nuber is assigned as 5.0 in all cases. Perfect agreeent with CFD results are bserved in all cputatins. The dips in the teperature bserved at the iddle f [AB] fr every heat generatin rate is because f istheral bundary cnditin expsed n the inner surface f the annulus, i. e., unit diensinless teperature. Syetrical teperature distributins are bserved n bth sides f the iddle pint O 1 where heat generatin spans cparatively larger vlues than the lateral sides yielding lcal axia n bth sides f O 1. As nticed fr fig., siilar syetrical like distributin ccurs in ters f surce strengths with different parities, naely, Q = ±cnstant. Slight deviatin fr the syetry arises fr the additive cntributin f the inner bundary cnditin t the teperature distributin fr heat surce case Q > 0. On the cntrary, this bundary cnditin plays an decreasing rle n the abslute value f the diensinless teperature fr heat sink cases Q < 0. A(0, r, a cthξ ) O (0, 0, acthξ ) B(0 r, r, acthξ ) C(0 r, r ( rl e), acthξi + rl) O (0, 0, ) acthξ i l D(0, r ( r l e), a + r l ) i cthξ Diensinless teperature (Ψ) 14 1 10 8 6 4 0 4 6 8 Fluent CFD Q = 50 Q = 5 Q = 0 Q = 5 Q = 50 1.0 0.5 0.0 0.5 1.0 Diensinless y psitin (y*) Figure 1. The lines [AB] and [CD] thrugh which diensinless teperatures are pltted Figure. Variatin f the diensinless teperature alng the line [AB] by diensinless psitin y * fr varius diensinless heat generatin rates (r/ri = 5, e/ri =, Bi = 5.0) Effect f Bit nuber n teperature distributin Figure 3 shws hw teperature distributin varies alng the line [AB] fr varius values f Bit nuber as Bi = 5, 10, 15, and 50. Geetrical cnfiguratin fr all Bit nubers is kept as r /r i = 5.0, e/r i =.0 and diensinless heat generatin rate is chsen as Q = 50.0. It culd be nticed fr fig. 3 that fr the higher Bit nubers the re effective cling is bserved and lwer teperature values are btained thrughut the line [AB] as expected. It culd als be nticed fr fig. 3 that there exists a cplete cnsistency between BIMM and CFD results.
Yilazer, A., et al.: Heat Cnductin in Cnvectively Cled Eccentric... THERMAL SCIENCE, Year 017, Vl. 1, N. 5, pp. 55-66 63 Effect f eccentricity n teperature distributin T evaluate the effect f eccentricity n teperature distributin, radii rati is set at r /r i = 5.0 and varius eccentricity ratis as e/r i = 1.0,.0,.5, and 3.0 are used in cputatins. Diensinless heat generatin rate is chsen as Q = 50.0. Variatin f teperature distributin alng the line [AB] is depicted in fig. 4. As seen fr the fig. 4 BIMM results and CFD results are in a very gd agreeent. It culd be nticed fr the figure that as the eccentricity is increased the effect f inner bundary cnditin n teperature distributin alng the line [AB] diinishes since inner bundary f the annulus beces re and re separated fr the centerline [AB]. Hence, the gverning factr turns ut t be the diensinless heat generatin rate nt the inner bundary cnditin fr larger eccentricities. When eccentricity is increased high teperature sites bece re widespread since a relatively larger heat generating vlue in the upper part is fred where ttal heat generatin rate is cparably larger than re cncentric cases. It shuld als be stated that the crescent geetrical cnfiguratin (e = r r i ) des nt yield in cnvergent results, since the tw fci used t describe c-rdinate in bispherical c-rdinate syste verlap in this case. Effect f radii rati n teperature distributin In this part, eccentricity rati is fixed at e/r i = 0.5 t evaluate the effect f radii rati n teperature distributin. Fr this purpse, cputatins are dne fr the radii ratis r /r i =.0, 3.0, 4.0, and 5.0 with Q = 50.0 and Bi = 5.0. Teperature distributin alng the line [CD] is shwn in fig. 5 fr varius radii ratis selected. It culd be seen fr the fig. 5 that BIMM results and CFD results agree very well. Since the line [CD] is tangent t the tp f the inner bundary f the annulus Diensinless teperature (Ψ) 8 6 4 Fluent CFD Bi = 5 Bi = 10 Bi = 15 Bi = 50 0 1 0 1 Diensinless y psitin (y*) Figure 3. Variatin f the diensinless teperature alng the line [AB] by diensinless psitin y* fr varius Bit nubers (r/ri = 5, e/ri =, Q = 50.0) 50 Diensinless teperature (Ψ) 40 30 0 10 Fluent CFD e/r i = 1 e/r i = e/r i =.5 e/r i = 3 0 1 0 1 Diensinless y psitin (y*) Figure 4. Variatin f the diensinless teperature thrugh the line [AB] by diensinless psitin y* fr varius eccentricity ratis (r/ri = 5.0) Diensinless teperature (Ψ) 3.5 3.0.5.0 1.5 1.0 0.5 Fluent CFD r /r i = r /r i = 3 r /r i = 4 r /r i = 5 0.0 1 0 1 Diensinless y psitin (y*) Figure 5. Variatin f the diensinless teperature thrugh the line [CD] by diensinless psitin y* fr varius radii ratis (e/ri = 0.5, Q = 50.0, Bi = 5.0)
Yilazer, A., et al.: Heat Cnductin in Cnvectively Cled Eccentric... 64 THERMAL SCIENCE, Year 017, Vl. 1, N. 5, pp. 55-66 diensinless teperature takes the value f unity at the iddle f the [CD] line as seen fr the figure. Fr the fixed e/r i = 0.5 the diensinless eccentricities bece δ = (e/r i ) / (r /r i 1) = = 0.50, 0.5, 0.167, and 0.15, respectively, fr the radii ratis used in cputatins as r /r i =.0, 3.0, 4.0, and 5.0, respectively. Figure 5 shws that diensinless teperature beces larger fr larger radii ratis and cnsequently fr larger diensinless eccentricities. Table 1. Diensinless heat transfer rate Q fr varius eccentricities (r/ri = 5.0) (e/ri = 1.0) Q 30 0 30 Bi CFD BIMM % Diff. CFD BIMM % Diff. CFD BIMM % Diff. 0. 0.651 0.647 0.67 0.110 0.110 0.4 0.871 0.867 0.39 0.6 1.04 1.019 0.45 0.174 0.175 0.64 1.37 1.370 0.17.0 1.80 1.76 0.30 0.0 0. 0.79 1.71 1.71 0.0 6.0 1.379 1.375 0.5 0.39 0.41 0.85 1.857 1.858 0.04 10.0 1.400 1.397 0.3 0.44 0.46 0.86 1.888 1.888 0.05 15.0 1.411 1.408 0.3 0.46 0.48 0.87 1.903 1.904 0.06 (e/ri =.0) Q 30 0 30 Bi CFD BIMM % Diff. CFD BIMM % Diff. CFD BIMM % Diff. 0. 4.55 4.40 0.34 0.107 0.108 0.3 4.470 4.456 0.31 0.6 6.564 6.55 0.19 0.171 0.171 0.47 6.906 6.895 0.16.0 8.13 8.115 0.09 0.1 0. 0.58 8.564 8.559 0.06 6.0 8.75 8.71 0.06 0.45 0.46 0.63 9.14 9.13 0.0 10.0 8.858 8.854 0.05 0.51 0.5 0.65 9.360 9.359 0.01 15.0 8.97 8.93 0.04 0.54 0.55 0.65 9.435 9.434 0.01 (e/ri = 3.0) Q 30 0 30 Bi CFD BIMM % Diff. CFD BIMM % Diff. CFD BIMM % Diff. 0. 1.661 1.604 0.6 0.10 0.10 0.4 1.866 1.809 0.6 0.6 3.3 3.198 0.10 0.16 0.16 0.38 3.555 3.53 0.10.0 39.053 39.050 0.01 0.18 0.19 0.47 39.490 39.488 0.00 6.0 41.700 41.711 0.03 0.55 0.57 0.51 4.11 4.4 0.03 10.0 4.96 4.311 0.04 0.67 0.68 0.5 4.830 4.848 0.04 15.0 4.605 4.63 0.04 0.74 0.75 0.5 43.15 43.174 0.05 Heat transfer rate In this part, the diensinless heat transfer rates calculated using eq. (18) fr different cases are cpared with the Fluent CFD results. It culd be nticed fr eq. (0) that h l (ξ ) = 0 fr unifr heat generatin rate. Hence, the diensinless heat transfer rate given by eq. (18) takes the fr:
Yilazer, A., et al.: Heat Cnductin in Cnvectively Cled Eccentric... THERMAL SCIENCE, Year 017, Vl. 1, N. 5, pp. 55-66 65 Q 1 π 0 1e (1) ( 1/),0 where the unknwn ents l,0 are calculated slving (14)-(16) using eq. (0) Table 1 tabulates cputed diensinless heat transfer, Q, values fr eccentricity ratis f e/r i = 1.0,,0, and 3.0. In cputatins fr each eccentricity rati, radii rati is fixed at r /r i = 5.0, diensinless heat generatin rate takes values Q = 30, 15, 0, 15, 30, Bit nuber takes values Bi = 0., 0.6,.0, 6.0, 10, and 15. As seen fr the table, results f BIMM agree very well with FLUENT results fr all range f paraeters investigated. The axiu abslute relative errr is less than 1% and even uch less than 1% in the st f the calculatins. Eccentricity ratis used in BIMM cputatins as e/r i = 1.0,.0, and 3.0 crrespnd t the diensinless eccentricity values δ = (e/r i ) / (r /r i 1), = 0.5, 0.50, and 0.75, respectively. It culd be nticed fr the table that the diensinless heat transfer rate fr the sae diensinless heat generatin rate and Bit nuber increases as δ is increased. This is because f fratin f larger teperature gradients when is increased due t the reasns explained in Sectin Effect f Bit nuber n teperature distributin. A cnsiderably greater change f rate f heat transfer is bserved fr relatively saller Bit nuber values (Bi < 1) than fr the larger Bit nuber f nuber values (Bi > 1). When Bit nuber exceeds practical liits (Bi 10) rate f change in the heat transfer rate is indiscriinable. Cnclusins Heat cnductin equatin fr an eccentric spherical annulus with heat generatin whse inner surface is kept at a cnstant teperature and the uter surface is subjected t cnvectin can be slved analytically using BIMM. Stating the prble in a diensinless fr in bispherical c-rdinate transfrs it int a cncentric annulus prble. Since Helhltz equatin is nt separable r R-separable in bispherical c-rdinate an analytical Green s functin t the cnductin equatin in bispherical c-rdinate fr an eccentric annulus subject t bundary cnditin f third type (cnvectin) is nt pssible. Hwever, an analytical Green's functin fr the first type bundary cnditins (given teperatures) n inner and uter bundaries can be btained. Slutin f the cnductin equatin with third type bundary cnditin n the uter surface culd be expressed in ters f the unknwn nral derivative f the uter surface teperature using the analytical Green's functin btained fr the first type bundary cnditin. The resulting equatin is a Fredhl integral equatin f the secnd kind with separable kernel fr the unknwn nral derivative f the uter surface teperature. Taking ent f this integral equatin leads t a set f linear equatins which can readily be slved. It is als pssible t btain an analytical Green's functin fr the secnd type bundary cnditin n the uter surface f the eccentric spherical annulus whse inner surface is kept at a cnstant teperature r subjected t the cnstant heat flux. Hence, the cnductin equatin with third type bundary cnditin n the uter surface can als be slved analytically using BIMM. In this case, the analytical Green's functin btained fr the secnd type bundary cnditin is t be used in the ethd. The resulting equatin fr unknwn uter surface teperature will again be a Fredhl integral equatin f the sae kind which can be slved by ethds f ent. The ethd has been applied t the eccentric spheres with varius heat generatin rates and fr a wide range f Bit nuber and a perfect agreeent has been bserved when cpared t the results f the CFD cde FLUENT. This exeplifies the rbustness f the ethd and its applicability t the cnductin prbles where an analytical slutin fr the
Yilazer, A., et al.: Heat Cnductin in Cnvectively Cled Eccentric... 66 THERMAL SCIENCE, Year 017, Vl. 1, N. 5, pp. 55-66 third type bundary cnditin are nt accessible, but still Green's functins are btainable fr the first r secnd type bundary cnditins. References [1] Ozisik, M. N., Heat Cnductin, Jhn Wiley and Sns, New Yrk, USA, 1980 [] Alassar, R. S., Alinshawy, B. J., Heat Cnductin fr Tw Spheres, AIChE J., 56 (010), 9, pp. 48-56 [3] Alassar, R. S., Cnductin in Eccentric Spherical Annuli, Internatinal Jurnal f Heat and Mass Transfer, 54 (011), 15-16, pp. 3796-3800 [4] Mn, P., Spencer, D. E., Field Thery Handbk, Including Crdinate Systes, Differential Equatins, and Their Slutins, nd ed., Springer-Verlag, Berlin, Gerany, 1988 [5] Yilazer, A., Kcar, C., Exact Slutin f the Heat Cnductin Equatin in Eccentric Spherical Annuli, Internatinal Jurnal f Theral Sciences, 68 (013), June, pp.158-17 [6] Yılazer, A., Kcar, C., A Nvel Analytical Methd fr Heat Cnductin in Cnvectively Cled Eccentric Cylindrical Annuli, Internatinal Jurnal f Theral Sciences, 63 (014), Sept., pp. 1-15 [7] ***, FLUENT 6 User s Guide, Fluent Inc., 003. [8] Arfken, G., Matheatical Methds fr Physicists, nd ed., Acadeic Press, Orland, Fla., USA, 1970 [9] Mrse, P. M., Feshbach, H., Methds f Theretical Physics, McGraw-Hill Bk Cpany Inc, New Yrk, USA, 1953, pp. 198-1301 [10] Gngra-T, A., Ley-K, E., On the Evaluatin f the Capacitance f Bispherical Capacitrs, Revista Mexicana de Física, 4 (1996), 4, pp. 663-674 Paper subitted: Deceber 31, 015 Paper revised: May, 016 Paper accepted: May 3, 016 017 Sciety f Theral Engineers f Serbia. Published by the Vinča Institute f Nuclear Sciences, Belgrade, Serbia. This is an pen access article distributed under the CC BY-NC-ND 4.0 ters and cnditins.