Influence Coefficient Method for Calculating Discrete Heat Source Temperature on Finite Convectively Cooled Substrates. Y.S.

Transcription

1 Influence Coefficient Method for Calculating Discrete Heat Source Temperature on Finite Convectively Cooled Substrates Y.S. Muzychka Faculty of Engineering and Applied Science Memorial University of Newfoundland St. John s, NF, Canada, A1B 3X5 Phone: (709) Fax: (709) ABSTRACT A simple method is developed for predicting discrete heat source temperatures on a finite convectively cooled substrate. The method is based on the principle of superposition using a single source solution for the mean or maximum contact temperature of an eccentric uniform heat source on a rectangular substrate. By means of influence coefficients, the effect of neighboring source strength and location may be assessed. It is shown that the influence coefficients represent localized thermal resistances, which must be weighted according to source strength. For a system of N heat sources, there exists N effects of source strength and position on any one heat source. This includes a self effect (source of interest) and N-1 influence effects (neighboring sources). The method is developed for isotropic, orthotropic, and compound systems. Convection in the source plane is addressed for isotropic and orthotropic systems. Expressions are developed for both mean and centroidal temperature. Keywords: Conduction, Electronics Cooling, Heat Spreaders, Compound Systems, Heat Sinks, Spreading Resistance, Orthotropic Properties NOMENCLATURE a, b, c, d = linear dimensions, m A 0,A m,a n,a mn = Fourier coefficients B 0,B m,b n,b mn = Modified Fourier coefficients ˆf,f = influence coefficient, K/W h,h 1,h = contact conductance or film coefficient, W/m K k,k 1,k = thermal conductivities, W/m K m,n = indices for summations N = number of heat sources Q = heat flow rate, W R = thermal resistance, K/W t,t 1,t = total and layer thicknesses, m Assistant Professor T s = surface temperature, K T f = sink temperature, K X c,y c = heat source centroid, m Greek Symbols β mn = eigenvalues, λ m + δn δ n = eigenvalues, nπ/b γ = transform variable k z /k xy θ = temperature excess, T T f, K θ = mean temperature excess, T T f, K ˆθ = centroidal temperature excess, ˆT T f, K θ s = surface temperature excess, T s T f, K κ = relative conductivity, k /k 1 λ m = eigenvalues, mπ/a φ = spreading function ζ = dummy variable, m 1 Subscripts eff = effective value i,j = denotes the i th and j th sources xy = xy-plane z = z-plane Superscripts ( ) = mean value ( ) ˆ = centroid value INTRODUCTION Calculation of the mean or centroidal value of discrete heat sources on a rectangular subststrate is of interest in electronic packages, circuit boards, and heat sink systems. In the simplest level of analysis, the total heat dissipated from all sources may be lumped together and evenly distributed over the substrate giving rise to the lowest system temperature. Further refinements in analysis may be made by treating this lumped heat source as a single discrete heat source with an area equivalent to the total area of all heat sources. In this case, the

2 discrete lumped formulation will usually give rise to the highest system temperature, depending upon the distribution and size of individual heat sources. In most cases, these two approaches will yield useful information for preliminary sizing of cooling systems. A more refined discrete heat source analysis is often desired to minimize hot spots and evenly distribute heat flows. This paper presents a simplified method of analysis for systems with multiple discrete heat sources, which enables the effects of neighboring source location and strength to be determined. Presently, a number of methods are widely used for examining systems with multiple packages or heat sources, see Fig. 1. These include the TAMS method of Ellison [1], the GENPaK model of Culham et al. [], and full numerical solutions using Finite Element Methods (FEM), among others. The analytic methods are based on a Fourier series solutions to Laplace s equation in isotropic or multilayered systems. They differ primarily in how the local heat source is introduced into the analysis. In the TAMS method, the source is specified through the governing partial differential equation, whereas in the GENPaK method, the heat source is specified through the die plane boundary condition. The present method is based on the latter approach with a significant simplification of the remaining boundary conditions and a limitation on the number of layers. This simplified system was recently addressed by Muzychka et al. [3], who obtained a managable Fourier series based solution for a single eccentrically located heat source on an isotropic or compound substrate, which is convectively cooled with a uniform film coefficient or contact conductance. Using the principles of superposition, this solution may be used for multiple discrete sources. The present analysis considers both heat loss through the sink plane and die plane. The method of Ellison [1] and Culham et al. [] also allow for convective cooling in the die plane in addition to the sink plane. Heat Sink Heat Sources k n, t n k n-1, t n-1. k, t k 1, t 1 Fig. 1 - Multi-Source System []. In this work, the solution of Muzychka et al. [3] is re-cast in terms of influence coefficients, which allow the effects of neighbouring heat sources to be easily assessed. A temperature for each heat source may be computed in terms of these influence coefficients which shows that the total temperature excess of any given heat source is comprised of a self effect in addition to the sum of all induced effects due to neighboring sources. Using the influence coefficients it is shown that a unique thermal resistance for each heat source cannot be defined in the presence of other heat sources. The present approach does allow for more efficient computation of heat source temperatures. LITERATURE REVIEW A review of the literature reveals that several approaches for computing the thermal spreading resistance and/or heat source temperature have been developed for a rectangular substrate with single or multiple discrete sources. In the case of multiple heat sources several approaches are found in the open literature. Hein and Lenzi [4] obtained a solution for an IC package using Fourier transforms. In their development, the heat source is specified by means of a Poisson equation using a piecewise function to model discrete heat sources. Both the die plane and sink plane are convectively cooled using uniform heat transfer coefficients. Later, Kokkas [5] obtained a Fourier/Laplace transform solution for a multi-layer substrate containing discrete heat sources. The substrate base was assumed to be attached to a heat sink of fixed temperture. Discrete heat sources were dealt with using the die plane boundary condition. Ellison [1] developed a method refered to as TAMS. This method is similar to that of Hein and Lenzi [4], but considers multiple layers. More recently, Culham et al. [] developed a three dimensional Fourier series model for an electronic packaging system. There model is very general and allows for the specification of a mixed boundary condition in the die plane. Heat sources are specified through the boundary condition in the die plane. Due to the complex nature of the die plane boundary condition, numerical analysis is required to complete the solution. In all of the above methods significant effort is required to code the analysis. In the case of a single discrete heat source, several approaches are readily found in the open literature. Kadambi and Abuaf [6] obtained steady and transient solutions for a central heat source on an isotropic rectangular substrate which was convectively cooled in the sink plane. Later, Krane [7] obtained a steady solution for a similar system in which the sink plane is at a constant temperature. More recently, Yovanovich et al. [8] and Muzychka et al. [3] obtained solutions for a compound convectively cooled rectangular substrate, containing a central and eccentric heat source, respectively.

3 Finally, Muzychka et al. [9] extended these solutions to orthotropic systems, while Muzychka et al. [10] obtained a solution for a central heat source on an isotropic convectively cooled rectangular substrate with edge cooling. MATHEMATICAL MODELLING The system of interest in the present work is idealized as a rectangular substrate which may be either isotropic, orthotropic, or compound in nature. For the time being we will only consider an isotropic system, see Fig.. Later, the effects of adding a conductive layer to promote the spreading of heat, and system orthotropy will be examined. In the present system all of the edges are assumed be adiabatic, a reasonable assumption in many electronics applications where edge area is significantly less than the area of the source and sink planes. Finally, there is no heat loss through the source plane, such that all heat is dissipated through the sink plane by means of a uniform film coefficient, i.e., thermal wake effects are neglected. The addition of convection in the source plane is dealt with in a separate section. Single Source Solution The single source solution of Muzychka et al. [3] for a single eccentric uniform heat source on an isotropic substrate has the following form: θ(x,y,z) = A 0 + C 0 z+ A m cos(λ m x)[cosh(λ m z) φ m (λ m )sinh(λ m z)]+ A n cos(δ n y)[cosh(δ n z) φ n (δ n )sinh(δ n z)] + A mn cos(λ m x)cos(δ n y) [cosh(β mn z) φ mn (β mn )sinh(β mn z)] (1) where λ m = mπ/a, δ n = nπ/b, and β mn = λ m + δ n are the eigenvalues. The origin of the coordinate system is taken to be the lower left corner of the substrate. The general solution contains four components, a uniform flow solution and three spreading (or constriction) solutions which vanish when the heat flux is uniformily distributed over the entire source plane, z = 0. The genral solution is a linear superposition of each component. Application of the boundary conditions in the through plane direction yields solutions for one half of the unknown constants and gives rise to the following expression for the spreading parameter φ: φ(ζ) = ζ sinh(ζt 1) + h/k 1 cosh(ζt 1 ) ζ cosh(ζt 1 ) + h/k 1 sinh(ζt 1 ) () where ζ is replaced by λ m, δ n, or β mn, accordingly. The spreading parameter accounts for the effects of conductivity, thickness, and convection cooling. Fig. - Single Eccentric Heat Source [3]. The final Fourier coefficients A m, A n, and A mn were obtained by taking Fourier series expansions of the boundary condition in the source plane, z = 0. This yielded the following expressions for the Fourier coefficients: A m = A n = [ ( ) ( )] Q sin (Xc+c) λ m sin (Xc c) λ m abck 1 λ mφ(λ m ) [ ( ) ( )] Q sin (Yc+d) δ n sin (Yc d) δ n abdk 1 δ nφ(δ n ) A mn = 16Qcos(λ mx c )sin( 1 λ mc)cos(δ n Y c )sin( 1 δ nd) abcdk 1 β mn λ m δ n φ(β mn ) (3) where X c and Y c are the coordinates of the centroid of an arbitrarily placed heat source with respect to the lower left corner of the substrate as shown in Fig.. Finally, values for the coefficients in the uniform flow solution are given by and A 0 = Q ab ( t1 + 1 ) k 1 h C 0 = Q k 1 ab (4) (5) Centroidal Source Temperature The maximum or centroidal heat source temperature may be determined from Eq. (1) when x = X c, y = Y c,

4 and z = 0. This gives: ˆθ = A 0 + A m cos(λ m X c ) + A n cos(δ n Y c )+ A mn cos(λ m X c )cos(δ n Y c ) (6) Mean Source Temperature The mean heat source temperature is obtained by integrating the local source temperature over the source area, i.e., θ = 1 A θ(x, y, 0)dA (7) This leads to the following result for the mean temperature excess of a single eccentric heat source: may be computed using Eq. (1) evaluated at the surface θ i (x,y,0) = A i 0 + A i m cos(λ m x) + A i n cos(δ n y)+ A i mn cos(λ m x)cos(δ n y) (10) The Fourier coefficients are now evaluated at each of the i th heat source characteristics, i.e. c i,d i,x c,i,y c,i and Q i. Centroidal Source Temperature The maximum or centroidal temperature is now the sum of all heat source contributions at the point of interest. Thus, using Eq. (10) evaluated at the centroid of the j th heat source, we may write θ = A o + cos(λ m X c )sin( 1 A λ mc) m + λ m c cos(δ n Y c )sin( 1 A δ nd) n + δ n d where ˆθ j = ˆθ i (X c,j,y c,j,0) = ˆθ ij (11) A mn 4cos(δ n Y c )sin( 1 δ nd)cos(λ m X c )sin( 1 λ mc) λ m cδ n d (8) The results given by Eqs. (6) and (8) may now be used to analyze systems containing multiple heat sources. These expressions may also be used as a fundamental surface element for analyzing irregularly shaped heat sources, by discretizing the region into several rectangular strip sources. MULTIPLE DISCRETE HEAT SOURCES If more than one heat source is present (see Fig. 3), the solution for the temperature distribution on the surface of the circuit board, heat sink, or chip substrate may be obtained using superposition. Both the centroidal and mean heat source temperatures will be obtained for each heat source. Surface Temperature Distribution For N discrete heat sources the maximum temperatures occur in the source plane. The surface temperature distribution is obtained from T s (x,y,0) T f = θ s = θ i (x,y,0) (9) where θ i is the temperature excess for each heat source by itself. The temperature excess of each heat source ˆθ ij = A i 0 + A i m cos(λ m X c,j ) + A i n cos(δ n Y c,j )+ A i mn cos(λ m X c,j )cos(δ n Y c,j ) (1) The present notation θ ij, denotes the effect of the i th heat source in the region of the j th heat source. Mean Source Temperature The mean heat source temperature of an arbitrary rectangular patch of dimensions c j and d j, i.e. the j th heat source, located at X c,j and Y c,j, may be computed by integrating Eq. (7) over the region A j = c j d j, i.e., θ j = 1 A j A j θda j = 1 A j which may be written as θ j = 1 A j A j A j θ i (x,y,0)da j = θ i (x,y,0)da j (13) θ ij (14) Using Eqs. (10) results in the following expression for the mean temperature excess contribution of the i th heat

5 source in the region of the j th heat source: θ ij = A i o + A i cos(λ m X c,j )sin( 1 λ mc j ) m + λ m c j A i cos(δ n Y c,j )sin( 1 δ nd j ) n + 4 δ n d j A i mn cos(δ n Y c,j )sin( 1 δ nd j )cos(λ m X c,j )sin( 1 λ mc j ) or which may be written as: ˆθ j = Q 1 ˆf1j + Q ˆfj + + Q N ˆfNj (17) ˆθ j = Q i ˆfij (18) λ m c j δ n d j (15) Equation (14) represents the sum of the effects of all sources over an arbitrary region c j d j. Equation (15) is evaluated over the region of interest c j d j located at X c,j,y c,j. The coefficients A i 0,A i m,a i n and A i mn are then evaluated at each of the i th source parameters. Fig. 3 - Multiple Heat Sources [3]. INFLUENCE COEFFICIENT METHOD The present results may now be used to define an influence coefficient. Influence coefficients were first proposed by Negus and Yovanovich [11] for semi-infinite domains and later applied by Negus et al. [1] and Negus and Yovanovich [13,14] for multiple sources on a half space. The concept of an influence coefficient for a finite substrate was partially addressed by Hein and Lenzi [4]. Influence coefficients offer an insightful assessment of the effect of neighbouring heat sources on thermal resistance, and hence the mean or centroidal temperature excess of each discrete heat source. We begin by examining Eq. (1) and Eq. (15), for the centroidal and mean temperature excess θ. Beginning first with Eq. (11) we may write ˆθ j = ˆθ 1j + ˆθ j ˆθ Nj (16) where ˆf ij = B 0 + where Bm i cos(λ m X c,j ) + Bn i cos(δ n Y c,j )+ Bmn i cos(λ m X c,j )cos(δ n Y c,j ) B 0 = 1 ab B i m = B i n = ( t1 + 1 ) k 1 h [ ( ) ( )] sin (Xc+c) λ m sin (Xc c) λ m abck 1 λ mφ(λ m ) [ ( ) ( )] sin (Yc+d) δ n sin (Yc d) δ n abdk 1 δ nφ(δ n ) (19) B i mn = 16cos(λ mx c )sin( 1 λ mc)cos(δ n Y c )sin( 1 δ nd) abcdk 1 β mn λ m δ n φ(β mn ) (0) are modified Fourier coefficients, since Q i has now been factored out. Once again it is noted that the coefficients are evaluated at each of the i th heat source characteristics, i.e. c i,d i,x c,i, an Y c,i. Thus, the influence coefficients are only functions of the substrate properties and dimensions and of heat source geometry and location. Similarly, we may obtain an expression for the mean temperature excess of the j th heat source using Eq. (14): or which may be written as: θ j = θ 1j + θ j θ Nj (1) θ j = Q 1 f 1j + Q f j + + Q N f Nj () θ j = Q i f ij (3)

6 where f ij = B o + Bm i cos(λ m X c,j )sin( 1 λ mc j ) + λ m c j Bn i cos(δ n Y c,j )sin( 1 δ nd j ) + δ n d j B i mn 4cos(δ n Y c,j )sin( 1 δ nd j )cos(λ m X c,j )sin( 1 λ mc j ) λ m c j δ n d j (4) When i = j, the contribution is a self effect, i.e., the effect of the source acting alone. When i j the contribution to the temperature excess is an influence effect. The self effect f ii, is merely the single source thermal resistance. The influence effects f ij, are affected by two factors: source strength and the location and size of neighboring sources, i.e. a geometry effect. The influence coefficients f ij are clearly functions only of the location of the neighboring sources. Finally, we may write the temperature excess in the following matrix form: or θ 1 θ θ 3.. θ N f 11 f 1 f 1N f 1 f f N = f 31 f 3 f 3N.... f N1 f N f NN Q 1 Q Q 3.. Q N (5) {θ} = [F ij ][Q] (6) where F ij is the matrix of influence coefficients. The influence coefficient method offers a number of advantages. First, it becomes obvious what the effect a neighboring heat source has on the thermal resistance of a particular heat source. Examination of Eqs. (18) or (3) reveals that an influence effect arises by virtue of proximity and strength. In otherwords, a remote and/or weak heat source has little influence on another heat source. Second, it can be shown that the influence coefficients also possess reciprocity for the case when i j, f ji = f ij (7) This property significantly reduces computation for systems where more than five sources are present. In general, for a system of N sources, a symmetric N N matrix results for the influence coefficients. As a result of this symmetry only (N +N)/ coefficients need be computed. An upper triangular matrix is all that is needed to compute the temperature excesses. Thus the influence method offers a substantial savings in computation over the use of Eq. (9). The reciprocity is a result of the property of Greens functions [15], i.e. the potential at X c,i,y c,i due to a unit heat input at X c,j,y c,j is the same as the potential at X c,j,y c,j due to a unit heat input at X c,i,y c,i. This property also holds upon integration over a finite region. The reciprocity of the influence coefficients was also observed by Negus and Yovanovich [11] for semi-infinite regions. Thermal Resistance Finally, if we consider defining a thermal resistance R j = θ j /Q j, for each heat source, it can be shown that or ˆR j = R j = Q i Q j ˆfij (8) Q i Q j f ij (9) The above equations clearly demonstrate that the concept of thermal resistance is not strictly applicable in multiple source systems, since the total resistance of any given source depends on both proximity of the neighboring heat source, i.e. f ij, and the relative strength ratio, i.e. Q i /Q j. Changing location or strength of any source leads to a new value of thermal resistance. CONVECTION IN THE SOURCE PLANE Convection in the source plane may now be dealt with using results of Hein and Lenzi [4]. Comparison of the solution of Muzychka et al. [3] with that of Hein and Lenzi [4] shows that coefficient B o becomes: ( 1 t1 + 1 ) ab k B 0 = 1 h 1 + h 1 + h (30) 1t 1 h k 1 where h 1 denotes the film coefficient in the source plane and h denotes the film coefficient in the sink plane. Further, the spreading function φ becomes: φ(ζ) = ( h1 k 1 ζ + k 1ζ h ) ( sinh(ζt 1 ) h 1 h k 1 ζ h cosh(ζt 1 ) + sinh(ζt 1 ) ) cosh(ζt 1 ) (31) Both Eqs. (30) and (31) reduce to Eqs. () and (0), when h 1 = 0, i.e. adiabatic source plane. COMPOUND AND ORTHOTROPIC SYSTEMS The results developed earlier may be easily adapted to compound and orthotropic systems with little effort. In a recent paper, Muzychka et al. [9], applied the necessary transformations to show the relationship between isotropic and orthotropic systems. Further, using the results of Yovanovich et al. [8], one may modify the

7 isotropic model to effectively model a resistive or conductive layer placed on a rectangular substrate. Each modification is discussed below. Orthotropic Systems If the rectangular flux channel is orthotropic such that the in plane and through plane conductivities are different, i.e. k xy k z, then the following transformations may be made to apply the present method to such systems (Muzychka et al. [9]): k k eff = k xy k z (3) where, k xy and k z represent the in-plane and throughplane thermal conductivity, and t t eff = t γ (33) where γ = k z /k xy is the conductivity ratio of the orthotropic system. The orthotropic transformation is also valid for a substrate which is convectively cooled in the source plane. Compound Systems The effect of an additional layer was also examined by Muzychka et al. [3]. It was shown that the effect of an additional layer (see Fig. 4) may be handled by means of the modified spreading parameter given by: ( ) ( αe 4ζt 1 e ζt1 + ϱ e ζ(t 1+t ) αe ζ(t1+t)) φ(ζ) = (αe 4ζt1 + e ζt1 ) + ϱ ( e ζ(t1+t) + αe ζ(t1+t)) where ϱ = ζ + h/k ζ h/k and α = 1 κ 1 + κ (34) with κ = k /k 1, and ζ is replaced by λ m, δ n, or β mn, accordingly. Further, the coefficient B 0 is now given by: B 0 = 1 ( t1 + t + 1 ) (35) ab k 1 k h This modification can only be applied to the case when there is no convection in the source plane. APPLICATION OF RESULTS The results may now be applied to a simple system. Three cases will be examined: isotropic, compound, and orthotropic. The thermal property and component thicknesses are given in Table 1. In all three cases, the heat source layout summarized in Table is used along with the following substrate properties: a = 00 [mm], b = 100 [mm], and h = 100 [W/m K]. In the first case, an isotropic substrate which is cooled in teh sink plane is considered. Next, the effect of a heat spreader is examined through the addition of a conductive layer. Finally, the effect of orthotropic properties is examined. This gives rise to t = t eff = [mm] and k = k eff = [W/mK] using the properties in Table 1. Maple V Release 8 [16] was used to perform the necessary calculations. The simple code is given in the Appendix for the isotropic case. To ensure convergence, 100 terms were used in each of the single summations and 50 terms in the double summation. The results of each of the six runs are summarized in Tables 3-5, which report the centroidal and mean temperature excess for each case. In general, convergence is much slower for the centroidal temperature excess due to an alternating series. In the case of the mean temperature excess, convergence is much more rapid since all terms are positive. The results illustrate the effect that a conductive layer or layers have on the temperature. The addition of a thin conductive layer in Case B, reduces the overall temperature level in addition to flattening the temperature distribution. Similar results are also obtained for the orthotropic case where the in-plane conductivity is higher than the through plane. In both cases thermal spreading is promoted due to the presence of a higher conductivity material. The results illustrate the ease with which discrete heat source temperatures may be determined. Typical computation times ranged between 100 and 400 seconds depending on whether an isotropic or compound system was considered and whether the centroid or mean value of temperature was computed. Table 1 - Case Studies t 1 t k 1 k k xy k z [mm] [W/mK] Case A Case B Case C Fig. 4 - Compound System [3].

8 Table - Source Layout Q c d X c Y c [W] [mm] [mm] [mm] [mm] Source Source Source Source Table 3 - Isotropic Substrate Results [ C] ˆθ θ Source Source Source Source Table 4 - Compound Substrate Results [ C] ˆθ θ Source Source Source Source Table 5 - Orthotropic Substrate Results [ C] ˆθ θ Source Source Source Source SUMMARY AND CONCLUSIONS A simple method for predicting mean and centroidal heat source temperature was developed by means of an influence coefficient. It was shown that this coefficient is only a function of source location and size. It was also shown that discrete heat source thermal resistance is weighted according to the relative source strength ratios. Further, it was also shown that the influence coefficients lend themselves to more efficient computation due to the reciprocity property. Several examples were computed to demonstrate the ease of application. Finally, the method was developed for isotropic, compound, and orthotropic systems. Modification of the basic equations for the case where heat is dissipated in the source plane was also discussed. ACKNOWLEDGMENTS The author acknowledges the support of the Natural Sciences and Engineering Research Council of Canada (NSERC). The author also thanks Prof. M.M. Yovanovich for comments given during manuscript preparation. REFERENCES [1] Ellison, G., Thermal Computations for Electronic Equipment, Krieger Publishing, Malabar, FL, [] Culham, J.R., Yovanovich, M.M., and Lemczyk, T.F., Thermal Characterization of Electronic Packages Using a Three-Dimensional Fourier Series Solution, Journal of Electronic Packaging, 000, Vol. 1, pp [3] Muzychka, Y.S., Yovanovich, M.M., and Culham, J.R., Thermal Spreading Resistance of Eccentric Heat Sources on Rectangular Flux Channels, Journal of Electronic Packaging, Vol. 15, pp , 003. [4] Hein, V.L. and Lenzi, V.D., Thermal Analysis of Substrates and Integrated Circuits, pp , Electronics Components Conference, [5] Kokkas, A., Thermal Analysis of Multiple-Layer Structures, IEEE Transactions on Electron Devices, Vol. Ed-1, No. 14, pp , [6] Kadambi, V. and Abuaf, N., Analysis of Thermal Response for Power Chip Packages, IEEE Trans. Elec. Dev., Vol. ED-3, No. 6, [7] Krane, M.J.H., Constriction Resistance in Rectangular Bodies, Journal of Electronic Packaging, Vol. 113, 1991, pp [8] Yovanovich, M.M., Muzychka, Y.S., and Culham, J.R., Spreading Resistance of Isoflux Rectangles and Strips on Compound Flux Channels, Journal of Thermophysics and Heat Transfer, Vol. 13, 1999, pp [9] Muzychka, Y.S., Yovanovich, M.M., and Culham, J.R., Thermal Spreading Resistances in Compound and Orthotropic Systems, Journal of Thermophysics and Heat Transfer, In Press, 003. [10] Muzychka, Y.S., Culham, J.R., and Yovanovich, M.M., Thermal Spreading Resistances of Rectangular Flux Channels: Part II Edge Cooling, 36th AIAA Thermophysics Conference, Orlando, FL, 003. [11] Negus, K.J. and Yovanovich, M.M., Thermal Resistance of Arbitrarily Shaped Contacts, Numerical Methods in Thermal Problems, Proceedings of the 3rd International Conference, Seattle, WA, 1983, pp [1] Negus, K.J., Yovanovich, M.M., and DeVaal, J.W., Development of Thermal Constriction Resistance for

9 Anisotropic Rough Surfaces by the Method of Infinite Images, National Heat Trnasfer Conference, Denver, CO, [13] Negus, K.J. and Yovanovich, M.M., Transient Temperature Rise at Surface Due to Arbitrary Contacts on Half Spaces, Transactions of the CSME, 1987, Vol. 13, pp [14] Negus, K.J. and Yovanovich, M.M., Thermal Computations in a Semiconductor Die Using Surface Elements and Infinite Images, International Symposium on Cooling Technology in Electronic Equipment, Honolulu, HI, 1987, pp [15] Morse, P.M., and Feshbach, H., Methods of Theoretical Physics, Part I, McGraw-Hill, New York, [16] Maple TM Release 8, Waterloo Maple Inc., Waterloo, ON, 00. APPENDIX Simple Maple Release 8 code for Case A results. Define Influence Coefficient > restart; > lambda:=m*pi/a; > delta:=n*pi/b; > beta:=sqrt(lambda^+delta^); > phi:=zeta->(zeta*sinh(zeta*t)+h/k* cosh(zeta*t))/(zeta*cosh(zeta*t)+ h/k*sinh(zeta*t)); > B[0]:=1/(a*b)*(t/k+1/h); > Bm[i]:=*(sin((*X[i]+c[i])*lambda/)- sin((*x[i]-c[i])*lambda/)) /(a*b*c[i]*k*lambda^*phi(lambda)); > Bn[i]:=*(sin((*Y[i]+d[i])*delta/)- sin((*y[i]-d[i])*delta/)) /(a*b*d[i]*k*delta^*phi(delta)); > Bmn[i]:=16*(cos(lambda*X[i])*sin(1/* lambda*c[i])*cos(delta*y[i])*sin(1/*delta*d[i]) )/(a*b*c[i]*d[i]*k*lambda*delta*beta*phi(beta)); > f[i]:=value(b[0]+*add(bm[i]*cos(lambda*x[j]) *sin(1/*lambda*c[j])/(lambda*c[j]),..100) +*add(bn[i]*cos(delta*y[j])*sin(1/*delta*d[j]) /(delta*d[j]),..100)+4*add(add(bmn[i]*cos( lambda*x[j])*sin(1/*lambda*c[j])*cos(delta*y[j] )*sin(1/*delta*d[j])/(lambda*c[j]*delta*d[j]),..100),..100)): Input System Parameters > baseparameters:={ a=0.,b=0.1,k=10,h=100,t=0.01}; > sourceparameters:={ c[1]=0.0,d[1]=0.0,x[1]=0.04,y[1]=0.03,q[1]=10, c[]=0.03,d[]=0.04,x[]=0.095,y[]=0.03,q[]=15, c[3]=0.03,d[3]=0.07,x[3]=0.155,y[3]=0.045,q[3]=5, c[4]=0.05,d[4]=0.01,x[4]=0.055,y[4]=0.075,q[4]=10}; Calculate Influence Coefficients > f1s:=[seq(evalf(subs(j=1,i=n,baseparameters, sourceparameters,f[i])),..4)]; > fs:=[seq(evalf(subs(j=,i=n,baseparameters, sourceparameters,f[i])),..4)]; > f3s:=[seq(evalf(subs(j=3,i=n,baseparameters, sourceparameters,f[i])),..4)]; > f4s:=[seq(evalf(subs(j=4,i=n,baseparameters, sourceparameters,f[i])),..4)]; Calculate Source Temperature Excesses > Source1Theta:=subs(j=1,sourceparameters, add(q[i]*f1s[i],..4)); > SourceTheta:=subs(j=,sourceparameters, add(q[i]*fs[i],..4)); > Source3Theta:=subs(j=3,sourceparameters, add(q[i]*f3s[i],..4)); > Source4Theta:=subs(j=4,sourceparameters, add(q[i]*f4s[i],..4));