Y. S. Muzychka Assistant Professor, Faculty of Engineering and Applied Science, Thermal Spreading Resistance of Memorial University of Newfoundland, St. John’s, Newfoundland, Canada A1B 3X5 Eccentric Heat Sources on J. R. Culham Rectangular Flux Channels Associate Professor, Mem. ASME A general solution, based on the separation of variables method for thermal spreading resistances of eccentric heat sources on a rectangular flux channel is presented. M. M. Yovanovich Solutions are obtained for both isotropic and compound flux channels. The general solu- Distinguished Professor Emeritus, tion can also be used to model any number of discrete heat sources on a compound or Fellow ASME isotropic flux channel using superposition. Several special cases involving single and multiple heat sources are presented. ͓DOI: 10.1115/1.1568125͔ Microelectronics Heat Transfer Laboratory, Department of Mechanical Engineering, Keywords: Conduction, Electronics Cooling, Heat Spreaders, Heat Sinks, Spreading University of Waterloo, Resistance Waterloo, Ontario, Canada N2L 3G1
ϭ Introduction where Ab ab, and h is a heat transfer coefficient which may be a contact conductance or effective fin conductance. Thermal spreading resistance results from discrete heat sources The total thermal resistance is defined as in many engineering systems. Typical applications include cooling of electronic devices both at the package and system level, and ¯T ϪT ϭ s f cooling of power semi-conductors using heat sinks. In these ap- RT (3) plications, heat dissipated by electronic devices is conducted Q through electronic packages into printed circuit boards or heat where ¯T is the average source temperature given by sink baseplates which are convectively cooled. In heat sink appli- s cations the convective film resistance is replaced by an effective 1 ¯T ϭ ͵͵ T͑x,y,0͒dA (4) extended surface film coefficient. s A s Analytical results for an eccentric heat source on a finite rect- s As angular flux channel are obtained for isotropic and compound sys- The thermal resistance for the configuration shown in Fig. 1 is tems. These solutions may be used to model single or multi-source a function of systems by means of superposition. The present approach differs ϭ ͑ ͒ from other methods ͓1–3͔, in that the heat source specification is R f a,b,c,d,Xc ,Y c ,t1 ,h,k1 (5) incorporated in the definition of the thermal boundary conditions rather than in the governing partial differential equation. This re- Problem Statement sults in analytical expressions which can be easily manipulated in The governing equation for the system shown in Fig. 1 is ͓ ͔ most advanced mathematical software packages 4–7 . Laplace’s equation The general solution for the spreading resistance of a single 2Tץ 2Tץ 2Tץ constant flux eccentric heat source with convective or conductive (2Tϭ ϩ ϩ ϭ0 (6ٌ z2ץ y 2ץ x2ץ cooling at one boundary will be presented, see Fig. 1. A review of the literature reveals that this particular configuration has not yet been analyzed ͓8͔. The two-dimensional eccentric strip heat which is subjected to a uniform flux distribution ͒ ץ source for a semi-infinite flux channel was obtained by Veziroglu T ͑Q/As and Chandra ͓9͔. While the solution for a centrally located heat ͯ ϭϪ (7) z kץ source on an isotropic rectangular plate was obtained by Krane zϭ0 1 ͓ ͔ ͓ ͔ 10 . Recently, Yovanovich et al. 11 obtained a solution for a ϭ centrally located heat source on a compound rectangular flux within the heat source area, As cd, and Tץ .channel The general solution will depend on several dimensionless geo- ͯ ϭ0 (8) zץ metric and thermal parameters. In general, the total resistance is zϭ0 given by outside the heat source area, and a convective or mixed boundary condition on the bottom surface ϭ ϩ RT R1D Rs (1) T hץ ͯ ϭϪ ͓ ͑ ͒Ϫ ͔ (T x,y,t1 T f (9 ץ -where R is the spreading resistance and R is the one dimen s 1D z ϭ k1 sional resistance of the system given by z t1 In extended surface applications such as heat sinks, the value of t 1 h is replaced by an effective value which accounts for both the ϭ 1 ϩ R1D (2) heat transfer coefficient on the fin surface and the increased sur- k1Ab hAb face area Contributed by the Electronic and Photonic Packaging Division for publication in 1 ϭ the JOURNAL OF ELECTRONIC PACKAGING. Manuscript received by the EPPD Di- heff (10) vision February 26, 2002. Guest Editors: Y. Muzychka and R. Culham. AbRfins
178 Õ Vol. 125, JUNE 2003 Copyright © 2003 by ASME Transactions of the ASME ͒ϭ ͒ T1͑x,y,t1 T2͑x,y,t1 (13) and Tץ Tץ 1ͯ ϭ 2ͯ (14) ץ k2 ץ k1 z ϭ z ϭ z t1 z t1 ϭ ϩ while along the bottom surface z t1 t2 , the boundary condition to be satisfied becomes T hץ 2ͯ ϭϪ ͓ ͑ ϩ ͒Ϫ ͔ (T2 x,y,t1 t2 T f (15 ץ z ϭ ϩ k2 z t1 t2 The full solution is obtained for the isotropic case, however, it may easily be applied to the compound system with only slight modification, using the results of Yovanovich et al. ͓11͔.
General Solution The solution for the isotropic plate may be obtained by means of separation of variables ͓12–15͔. The solution is assumed to have the form (x,y,z)ϭX(x)*Y(y)*Z(z), where (x,y,z) ϭ Ϫ T(x,y,z) T f . Applying the method of separation of variables yields the following general solution for the temperature excess in the plate which satisfy the thermal boundary conditions along (x ϭ0, xϭa) and (yϭ0, yϭb) ϱ ͒ϭ ϩ ϩ ͒ ͒ϩ ͒ ͑x,y,z A0 B0z ͚ cos͑ x ͓A1 cosh͑ z B1 sinh͑ z ͔ mϭ1 ϱ ϩ ␦ ͒ ␦ ͒ϩ ␦ ͒ ͚ cos͑ y ͓A2 cosh͑ z B2 sinh͑ z ͔ nϭ1 ϱ ϱ ϩ ͑ ͒ ͑␦ ͓͒ ͑ ͒ Fig. 1 Isotropic plate with eccentric heat source ͚ ͚ cos x cos y A3 cosh z mϭ1 nϭ1 ϩ  ͒ B3 sinh͑ z ͔ (16) Along the edges of the plate, the following conditions are also ϭ ␦ϭ ϭͱ2ϩ␦2 required: where m /a, n /b, and . The solution contains four components, a uniform flow solution T and three spreading ͑or constriction͒ solutions which vanish whenץ ͯ ϭ0 (11) x the heat flux is distributed uniformily over the entire surface zץ xϭ0,a ϭ0. Since the solution is a linear superposition of each compo- and nent, they may be dealt with separately. Application of the bound- ary conditions in the z direction will yield solutions for the un- Tץ ͯ ϭ0 (12) known constants. y ϭץ yϭ0,b Application of the thermal boundary condition at z t1 for an isotropic rectangular plate yields the following result for the Fou- The general solution for the total thermal resistance and tem- rier coefficients: perature distribution will be obtained for the system shown in Fig. ϭϪ ͒ ϭ 1. In a later section, the solution will be extended to compound Bi ͑ Ai i 1,2,3 (17) systems as shown in Fig. 2. where In a compound system Laplace’s equation must be solved in ͒ϩ ͒ each layer. In addition to the boundary conditions specified for the sinh͑ t1 h/k1 cosh͑ t1 ͑͒ϭ (18) isotropic system, the following conditions with perfect contact at cosh͑t ͒ϩh/k sinh͑t ͒ the interface are required: 1 1 1 and is replaced by , ␦,or, accordingly. The final Fourier coefficients Am , An , and Amn are obtained by taking Fourier series expansions of the boundary condition at the surface zϭ0. This results in ϩ ͐Xc c/2 ͒ Ϫ cos͑ mx dx Q Xc c/2 A ϭ (19) 1 ͑ ͒ ͐a 2͑ ͒ bck1 m m 0 cos mx dx or ϩ ͒ Ϫ ͒ ͑2Xc c ͑2Xc c 2Qͫsinͩ ͪ Ϫsinͩ ͪͬ 2 m 2 m ϭ Am 2 ͑ ͒ (20) abck1 m m Fig. 2 Compound plate with eccentric heat source and
Journal of Electronic Packaging JUNE 2003, Vol. 125 Õ 179 ϩ ͐Y c d/2 ␦ ͒ Ϫ cos͑ ny dy Q Y c d/2 A ϭ (21) 2 ␦ ͑␦ ͒ ͐b 2͑␦ ͒ adk1 n n 0 cos ny dy or ϩ ͒ Ϫ ͒ ͑2Y c d ͑2Y c d 2Qͫsinͩ ␦ ͪ Ϫsinͩ ␦ ͪͬ 2 n 2 n ϭ An ␦2͑␦ ͒ (22) abdk1 n n and ϩ ϩ ͐Y c d/2͐Xc c/2 ͒ ␦ ͒ Ϫ Ϫ cos͑ mx cos͑ ny dxdy Q Y c d/2 Xc c/2 A ϭ 3  ͑ ͒ • ͐b͐a 2͑ ͒ 2͑␦ ͒ cdk1 m,n m,n 0 0 cos mx cos ny dxdy (23) or 1 1 16Q cos͑ X ͒sinͩ cͪ cos͑␦ Y ͒sinͩ ␦ dͪ m c 2 m n c 2 n ϭ Amn  ␦  ͒ abcdk1 m,n m n ͑ m,n Fig. 3 Special cases of a plate with eccentric heat source (24) Finally, values for the coefficients in the uniform flow solution are given by Compound Systems. In many applications an interface ma- terial may be added to reduce thermal contact conductance and/or Q t 1 ϭ ͩ 1 ϩ ͪ promote thermal spreading. The solution obtained for the isotropic A0 (25) ab k1 h rectangular flux channel may be used for a compound flux channel ͓ ͔ and with only minor modifications. In Yovanovich et al. 11 the au- thors obtained a solution for a compound rectangular flux channel. Q It is not difficult to show that in Eqs. ͑20͒, ͑22͒, ͑24͒ can be ϭϪ B0 (26) replaced by k1ab ͑ ϩ ͒ ͑ ϩ ͒ ͑␣ 4 t1Ϫ 2 t1͒ϩ ͑ 2 2t1 t2 Ϫ␣ 2 t1 t2 ͒ Mean Temperature Excess. The general solution for the e e % e e ͑͒ϭ ͑ ϩ ͒ ͑ ϩ ͒ (30) mean temperature excess of a single heat source may be obtained ͑␣e4 t1ϩe2 t1͒ϩ%͑e2 2t1 t2 ϩ␣e2 t1 t2 ͒ ͑ ͒ by integrating Eq. 16 over the heat source area. Carrying out the where necessary integrations leads to the following expression for the mean source temperature: ϩh/k 1Ϫ ϭ 2 ␣ϭ % Ϫ and ϩ 1 h/k2 1 ϱ cos͑ X ͒sinͩ cͪ m c 2 m with ϭk /k , and is replaced by , ␦,or, accordingly. This ¯ϭ¯ ϩ 2 1 1D 2 ͚ Am simple extension is possible, since the effect of the additional ϭ c m 1 m layer results from solving for the unknown coefficients by appli- 1 cation of the boundary conditions in the z direction. The general ϱ cos͑␦ Y ͒sinͩ ␦ dͪ solution for all but one of the Fourier coefficients is identical to n c 2 n ϩ the case for a central source. Since the spreading resistance is 2 ͚ An ␦ nϭ1 nd based upon the mean source temperature at the surface of the flux channel, it is not necessary to resolve the complete system of 1 1 equations. However, complete solution for all coefficients is re- ϱ ϱ cos͑␦ Y ͒sinͩ ␦ dͪ cos͑ X ͒sinͩ cͪ n c 2 n m c 2 m quired for calculating the temperature within the solid. ϩ 4 ͚ ͚ Amn ␦ Additionally, mϭ1 nϭ1 mc nd Q t1 t2 1 (27) ¯ ϭ ͩ ϩ ϩ ͪ (31) 1D ab k k h where 1 2 for a compound system. Application of the above results is only Q t 1 ¯ ϭ ͩ 1 ϩ ͪ valid for computing the temperature distribution at the surface of 1D (28) ab k1 h the baseplate and to compute the spreading resistance. Equation ͑30͒ is merely the reciprocal of a similar expression reported in for an isotropic system. Yovanovich et al. ͓11͔. Thermal Spreading Resistance. The thermal spreading re- sistance may now be computed using the mean temperature ex- Special Cases cess. In general, the total resistance as defined by Eq. ͑3͒, results in Several special cases of an eccentric heat source may be ob- tained from the general solution. These are shown in Fig. 3. So- ¯ lution for a central heat source on an isotropic plate was obtained R ϭ ϭR ϩR (29) by Krane ͓10͔. However, the results are only presented for the T Q 1D s thermal resistance based upon the maximum or centroidal tem- ͓ ͔ where R1D is the one-dimensional thermal resistance and Rs is perature difference. Recently, Yovanovich et al. 11 obtained the the thermal spreading resistance. The thermal spreading resis- solution for the spreading resistance of a centrally located heat tance component is defined by the three series solution terms in source on a compound plate. A special case of the solution of Eq. ͑27͒. Yovanovich et al. ͓11͔ is that for an isotropic plate; see Fig. 4.
180 Õ Vol. 125, JUNE 2003 Transactions of the ASME Fig. 5 Compound plate with central heat source
ϭ ␦ with k2 /k1 . The eigenvalues for these solutions are: m ϭ ϭ  ϭͱ␦2 ϩ2 m /c, n n /d and m,n m n. Edge and Corner Heat Sources. The solution obtained by Fig. 4 Isotropic plate with central heat source Yovanovich et al. ͓11͔ may also be used to model the three addi- tional special cases shown in Fig. 3. By means of symmetry, the solution for the spreading resistance may be obtained by consid- ering that each of the special cases represents an element of the Additionally, the remaining cases in Fig. 3 may also be obtained system with a centrally located heat source. For an edge source, ϭ from the solution for a central heat source using the method of the resistance is given by Rs 2R, and for a corner heat source ϭ images. the total resistance is given by Rs 4R, where R is the resistance of the system composed by mirroring the image͑s͒ of the edge or Central Heat Source. The spreading resistance of Yovanov- corner heat sources to obtain a system with a central heat source. ich et al. ͓11͔ is obtained from the following general expression which shows the explicit relationships with the geometric and Semi-Infinite Isotropic Flux Channel. Additional results thermal parameters of the system according to the notation in may be obtained for semi-infinite flux channels for the case where →ϱ Figs. 4 and 5: t1 and the effect of the conductance h is no longer a factor. ϱ ϱ The solution for a semi-infinite flux channel is obtained when the 1 sin2͑a␦ ͒ 1 sin2͑b ͒ ϭ m ͑␦ ͒ϩ n parameter Rs 2 ͚ 3 m 2 ͚ 3 2a cdk ϭ ␦ • 2b cdk ϭ 1 m 1 m 1 n 1 n ͑͒ϭ1 (35) ϱ ϱ 1 sin2͑a␦ ͒sin2͑b ͒ Semi-Infinite Compound Flux Channel. The general ex- ͑ ͒ϩ m n ͑ ͒ n 2 2 ͚ ͚ 2 2 m,n pression for ͑͒ reduces to a simpler expression when t →ϱ, • a b cdk1 mϭ1 nϭ1 ␦  • 2 m n m,n ͑see Fig. 2͒. The solution for this particular case arises from Eq. (32) ͑30͒ with where ͑e2 t1ϩ1͒ϩ͑e2 t1Ϫ1͒ ͑͒ϭ (36) ͑ 2 t1Ϫ ͒ϩ͑ 2 t1ϩ ͒ ͑e2 tϩ1͒Ϫ͑1Ϫe2 t͒h/k e 1 e 1 ͑͒ϭ 1 2tϪ ͒ϩ ϩ 2t͒ (33) ͑e 1 ͑1 e h/k1 where the influence of the convective conductance has vanished, but the influence of the substrate remains. If the system is composed of two layers, then Eccentric Strip Solutions. Finally, solutions for both isotro- ͑ ϩ ͒ ͑ ϩ ͒ ͑␣e4 t1ϩe2 t1͒ϩ%͑e2 2t1 t2 ϩ␣e2 t1 t2 ͒ pic and compound eccentric strips may be obtained from the gen- ͑͒ϭ ͑ ϩ ͒ ͑ ϩ ͒ (34) ͑␣e4 t1Ϫe2 t1͒ϩ%͑e2 2t1 t2 Ϫ␣e2 t1 t2 ͒ eral solution. If the dimensions of the heat source extend to two of the boundaries to form a continuous strip, the general solution where simplifies considerably. In Eq. ͑27͒, the general solution consists of four terms, a uniform flow component, two strip solutions ϩh/k 1Ϫ ϭ 2 ␣ϭ ͑single summations͒ and a rectangular source solution ͑double % Ϫ and ϩ ͒ h/k2 1 summation . For the case of an eccentric strip, one need only
Journal of Electronic Packaging JUNE 2003, Vol. 125 Õ 181 Table 1 Results for case 1„a… ˆ ¯ ˆ ¯ ¯ T1 T1 T2 T2 Tb Present 84.97 80.56 108.43 101.81 53.06 Culham ͓16͔ 85.95 80.20 109.92 101.26 53.06
which may be written N N 1 ¯ ϭ ͵͵ ͑ ͒ ϭ ¯ j ͚ i x,y,0 dAj ͚ i (40) ϭ A ϭ i 1 j A j i 1 Using Eqs. ͑27͒ and ͑40͒ results in the following expression for the mean temperature of the jth heat source: N ¯ Ϫ ϭ ¯ T j T f ͚ i (41) iϭ1 where 1 ϱ cos͑ X ͒sinͩ c ͪ m c, j 2 m j ¯ ϭ i ϩ i i Ao 2 ͚ Am mϭ1 mc j 1 ϱ cos͑␦ Y ͒sinͩ ␦ d ͪ ϱ ϱ n c, j 2 n j ϩ i ϩ i 2 ͚ An ␦ 4 ͚ ͚ Amn nϭ1 nd j mϭ1 nϭ1 1 1 cos͑␦ Y ͒sinͩ ␦ d ͪ cos͑ X ͒sinͩ c ͪ n c, j 2 n j m c, j 2 m j ϫ ␦ mc j nd j Fig. 6 Plate with multiple heat sources Equation ͑41͒ represents the sum of the effects of all sources over an arbitrary location. Equation ͑41͒ is evaluated over the region of interest c j , d j located at Xc, j , Y c, j , with the coefficients i i i i consider the appropriate strip solution and the uniform flow solu- A0 , Am , An and Amn evaluated at each of the ith source param- tion. The remaining summations fall out of the solution when b eters. ϭd or cϭa. In the case of semi-infinite eccentric strip solutions, The coordinate system is based on the origin placed at the Eqs. ͑35͒ and ͑36͒ also hold. lower left corner of the plate, and each source is located using the coordinates of the centroid. Multiple Heat Sources ͑ ͒ If more than one heat source is present see Fig. 6 , the solution Application of Results for the temperature distribution on the surface of the circuit board or heat sink may be obtained using superposition. For N discrete To demonstrate the usefulness of the present approach, several heat sources, the surface temperature distribution is given by examples of systems containing multiple sources are presented. First, an example is given which shows the effect of a heat N spreader on a low conductivity substrate. Next, the method is ͒Ϫ ϭ ͒ T͑x,y,0 T f ͚ i͑x,y,0 (37) ϭ applied to model a heat sink containing a number of discrete heat i 1 sources uniformly located on the baseplate. Finally, a uniform flux where i is the temperature excess for each heat source by itself. heat source of complex shape is analyzed. The temperature excess of each heat source may be computed using Eq. ͑16͒ evaluated at the surface Case 1. In the first case, the dimensions of a plate or circuit board are: aϭ300 mm, bϭ300 mm, t ϭ10 mm, hϭ10 W/m2K ϱ ϱ 1 and kϭ10 W/mK, with T ϭ25°C. Two heat sources having di- ͑ ͒ϭ i ϩ i ͑ ͒ϩ i ͑␦ ͒ f i x,y,0 A0 ͚ Am cos x ͚ An cos y mensions cϭ25 mm and dϭ25 mm each. The first source with a mϭ1 nϭ1 ϭ ϭ ϭ power of Q 10 W is located at Xc Y c 90 mm and the second ϱ ϱ ϭ ϭ ϭ having a power of Q 15 W at Xc Y c 210 mm. The basic equa- ϩ i ͑ ͒ ͑␦ ͒ tions may be programmed into any symbolic or numerical math- ͚ ͚ Amn cos x cos y (38) mϭ1 nϭ1 ematics software package. For the present calculations, the sym- bolic mathematics program Maple V ͓4͔ was employed. A total of ͑ ͒ ͑ ͒ ϭ¯ ͑ ͒ with defined by Eq. 18 or 30 and A0 1D given by Eq. 28 50 terms were used in each of the single and double summations. or ͑31͒. The results for the mean ¯T and centroidal Tˆ temperatures of the The mean temperature of an arbitrary rectangular patch of di- first case are presented in Table 1. In this example the centroidal mensions c and d , located at X and Y may be computed by j j c, j c, j temperatures of each heat source were found to be 84.97°C and integrating Eq. ͑37͒ over the region A ϭc d j j j 108.43°C. A three-dimensional plot of the surface temperature N 1 1 profile is given in Fig. 7. Comparisons with a generalized Fourier ¯ ϭ ͵͵ ϭ ͵͵ ͑ ͒ ͓ ͔ j dAj ͚ i x,y,0 dAj (39) series approach of Culham and Yovanovich 16 yields 85.95°C A A iϭ1 j A j j A j and 109.92°C. The primary difference between the present ap-
182 Õ Vol. 125, JUNE 2003 Transactions of the ASME Table 2 Results for case 1„b… spreader has reduced the maximum source temperatures consider- ably and equalized the temperature. ˆ ¯ ˆ ¯ ¯ T1 T1 T2 T2 Tb Case 2. In this example, the baseplate of a heat sink is to be Present 56.55 56.05 59.94 59.19 53.06 ϭ Culham ͓16͔ 56.63 56.03 60.08 59.16 53.06 analyzed. The dimensions of the baseplate are a 50 mm, b ϭ50 mm, kϭ150 W/mK, tϭ10 mm, and an effective extended surface heat transfer coefficient hϭ400 W/m2K. Four sources having the characteristics summarized in Table 3, are attached to proach and that of Culham and Yovanovich ͓16͔ is that the present the baseplate assuming negligible contact or interface resistance. work yields simplified expressions which may be easily pro- The temperature results are summarized in Table 4 and in Fig. 9. grammed in any Mathematics or Spreadsheet software, whereas the latter uses a numerical least squares approximation to solve for Case 3. In the final example, a heat source with a complex a mixed boundary value problem. shape is analyzed by the present approach. The source is com- For the same configuration, a thin, tϭ2 mm, highly conductive posed of five square heat sources each having dimensions c ϭ ϭ layer kϭ350 W/mK, is added to the original substrate and the 20 mm by d 20 mm and dissipating 5 W. The heat sources are problem reanalyzed. The results are summarized in Table 2. In this arranged in the form of a cross in the center of a plate having example the centroidal temperatures of each heat source were dimensions aϭ100 mm by bϭ100 mm, thickness of tϭ10 mm. found to be 56.55 and 59.94°C. Comparisons with a generalized This results in an irregular shaped isoflux heat source which can- Fourier series approach of Culham and Yovanovich ͓16͔ yields not be solved using conventional approaches. The thermal con- 56.63 and 60.08°C. A three-dimensional plot of the surface tem- ductivity of the plate is kϭ50 W/mK, while hϭ50 W/m2K, and ϭ perature profile is given in Fig. 8. It is clearly seen that adding a T f 25°C. The value of the maximum temperature at the center of
Fig. 7 Surface temperature for an isotropic plate with two heat sources
Fig. 8 Surface temperature for a compound plate with two heat sources
Journal of Electronic Packaging JUNE 2003, Vol. 125 Õ 183 Table 3 Source characteristics for case 2 Table 4 Source temperatures for case 2
Q c d Xc Yc Tˆ ¯T W mm mm mm mm Source 1 97.40 96.85 Source 1 10 10 10 12.5 12.5 Source 2 103.82 102.33 Source 2 25 10 10 37.5 12.5 Source 3 101.28 100.10 Source 3 20 10 10 12.5 37.5 Source 4 99.94 99.07 Source 4 15 10 10 37.5 37.5
source is present. The location and strength of each additional heat the plate is found to be 80.61°C. The area weighted mean source source will affect the value of thermal resistance for a particular temperature is found to be 78.87°C. The thermal contour plot is source of interest. given in Fig. 10. Finally, if the ambient temperature increases as a result of heat transfer due to the film coefficient h, a wake effect may be ap- Discussion proximated in the final solution by defining an ambient tempera- The general solution obtained may easily be coded in a number ture which is a function of flow position of ways. The simplest approach is through the use of mathemati- Q x ͓ ͔ ϭ ϩ cal programs 4–7 . These packages allow for symbolic and nu- T f Ti (42) merical computation of mathematical expressions. They also pro- m˙ C p L vide graphical functions for generating three-dimensional plots where Ti is the inlet temperature, m˙ is the mass flow through the such as those presented earlier. One advantage of these packages system, Q is the sum of all heat sources, and x/L the local position is the minimal effort required to enter the basic equations. Com- in the flow direction. putation time varies among packages and is also dependent upon the number of sources specified. The present results were obtained using Maple V6 ͓4͔ with 50 terms in each of the summations. A Summary and Conclusions single source calculation typically required a few seconds. Rea- General expressions for determining the spreading resistance of sonable convergence with 50 terms is obtained for most problems. an eccentric isoflux rectangular source on the surface of finite Another method of computation which was assessed is the use isotropic and compound rectangular flux channels were presented. of computer languages such as C or Basic. Computation time is The solution for the temperature at the surface of a rectangular much faster with a compiled code, however, a considerable flux channel was also presented. It was shown that this solution amount of code is required to achieve the same results as those may be used to predict the centroidal temperatures for any number produced using mathematical software. The method is also ame- of heat sources using superposition. Additional special cases of nable to spreadsheet calculations with or without the use of mac- spreading resistance from single eccentric heat sources on isotro- ros. However, the use of macros allows for easier development. pic, compound, finite, and semi-infinite flux channels were also In addition to providing details of the surface temperature dis- presented. Finally, it was shown that the solution for a central heat tribution and centroidal and mean source temperatures, the effec- source may be used to compute the spreading resistance for corner tive thermal resistance may also be computed for each source. and edge heat sources using the method of images. Several appli- This approach was not taken in the present work, since no unique cations of the general solution to systems with multiple heat value of thermal resistance is definable when more than one sources were also given.
Fig. 9 Surface temperature for a heat sink with four heat sources
184 Õ Vol. 125, JUNE 2003 Transactions of the ASME Fig. 10 Surface temperature for a complex heat source
ϭ ϵ Ϫ Acknowledgments temperature excess, T T f ,K ¯ ϭ ϵ¯ Ϫ The authors acknowledge the financial support of the Natural mean temperature excess, T T f ,K ˆ Sciences and Engineering Research Council of Canada. The au- ϭ centroidal temperature excess, ϵTˆ ϪT ,K thors would also like to thank Dave Fast of PowerSink Technolo- f ¯ ϭ ϵ¯ Ϫ gies Inc., for demonstrating that the present approach can be b mean base temperature excess, Tb T f , coded in a Microsoft Excel spreadsheet using Visual Basic Mac- K ϭ ros. relative conductivity, k2 /k1 ϭ m eigenvalues, (m /a) Nomenclature ϭ spreading function ϭ relative thickness, ϵt/L ϭ Ϫ a, b, c, d linear dimensions, m ϭ dummy variable, m 1 ϭ 2 Ab baseplate area, m ϭ 2 As heat source area, m ϭ References A0 , Am , An , Amn Fourier coefficients Bi ϭ Biot no., hL/k ͓1͔ Ellison, G., 1984, Thermal Computations for Electronic Equipment, Krieger ϭ Publishing, Malabar, FL. C p heat capacity, J/Kg K ϭ • ͓2͔ Kokkas, A., 1974, ‘‘Thermal Analysis of Multiple-Layer Structures,’’ IEEE h contact conductance or film coefficient, Trans. Electron Devices, Ed-21͑14͒, pp. 674–680. 2 W/m •K ͓3͔ Hein, V. L., and Lenzi, V. D., ‘‘Thermal Analysis of Substrates and Integrated i ϭ index denoting layers 1 and 2 Circuits,’’ pp. 166–177, Bell Telephone Laboratories, unpublished report. ϭ ͓4͔ Maple V Release 6, 2000, Waterloo Maple Software, Waterloo, ON, Canada. k, k1 , k2 thermal conductivities, W/m K L ϭ • ͓5͔ Mathcad Release 6, 1995, Mathsoft Inc., Cambridge, MA. arbitrary length scale, m ͓6͔ Mathematica, 1996, Release 3, Wolfram Research, Champaign-Urbana, IL. m˙ ϭ mass flow rate, kg/s ͓7͔ MatLab Release 4, 1997, Mathworks Inc., Natick, MA. m, n ϭ indices for summations ͓8͔ Yovanovich, M. M., 1998, ‘‘Chapter 3: Conduction and Thermal Contact Re- N ϭ no. of heat sources sistance ͑Conductance͒,’’ Handbook of Heat Transfer, eds., W. M. Rohsenow, ϭ J. P. Hartnett, and Y. L. Cho, McGraw-Hill, New York, NY. Q heat flow rate, W ͓9͔ Verziroglu, T. N., and Chandra, S., 1969, ‘‘Thermal Conductance of Two Di- 2 q ϭ heat flux, W/m mensional Constrictions,’’ Prog. Astronaut. Aeronaut., Vol. 21 Thermal Design R ϭ thermal resistance, K/W Principles of Spacecraft and Entry Bodies, ed., G. T. Bevins, pp. 617–620. ͓ ͔ R* ϭ dimensionless thermal resistance, ϵkRL 10 Krane, M. J. H., 1991, ‘‘Constriction Resistance in Rectangular Bodies,’’ ϭ ASME J. Electron. Packag., 113, pp. 392–396. R1D one-dimensional resistance, K/W ͓11͔ Yovanovich, M. M., Muzychka, Y. S., and Culham, J. R., 1999, ‘‘Spreading ϭ Rs spreading resistance, K/W Resistance of Isoflux Rectangles and Strips on Compound Flux Channels,’’ J. ϭ Thermophys. Heat Transfer, 13, pp. 495–500. RT total resistance, K/W ϭ ͓12͔ Arpaci, V., 1966, Conduction Heat Transfer, Addison-Wesley, New York, NY. t, t1 , t2 total and layer thicknesses, m ͓ ͔ ϭ 13 Carslaw, H. S., and Jaeger, J. C., 1959, Conduction of Heat in Solids, Oxford T1 , T2 layer temperatures, K University Press, Oxford, UK. ¯ ϭ ͓14͔ Ozisik, N. A., 1980, Heat Conduction, John Wiley and Sons, Inc., New York, Ts mean source temperature, K ϭ NY. T f sink temperature, K ͓ ͔ ϭ 15 Luikov, A. V., 1968, Analytical Heat Diffusion Theory, Academic Press, New Ti inlet or initial temperature, K York, NY. Xc, Yc ϭ heat source centroid, m ͓16͔ Culham, J. R., and Yovanovich, M. M., 1997, ‘‘Thermal Characterization of  ϭ ϵͱ2 ϩ␦2 Electronic Packages Using a Three-Dimensional Fourier Series Solution,’’ The m,n eigenvalues, m n ␦ ϭ Pacific Rim/ASME International, Intersociety Electronic and Photonic n eigenvalues, (n /b) Packaging Conference, INTERpack ’97, Kohala Coast, Island of Hawaii, ⑀ ϭ relative contact size, ϵb/a June 15–19.
Journal of Electronic Packaging JUNE 2003, Vol. 125 Õ 185