CALCULATION of the mean or centroidal value of discrete

Transcription

1 636 IEEE TRANSACTIONS ON COMPONENTS AND PACKAGING TECHNOLOGIES, VOL 29, NO 3, SEPTEMBER 2006 Influence Coefficient Method for Calculating Discrete Heat Source Temperature on Finite Convectively Cooled Substrates Yuri S Muzychka 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 heat sources, there exists effects of source strength and position on any one heat source This includes a self effect (source of interest) and 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 Index Terms Compound systems, conduction, electronics cooling, heat sinks, heat spreaders, orthotropic properties, spreading resistance NOMENCLATURE Linear dimensions, Fourier coefficients Modified Fourier coefficients Influence coefficient, K/W Contact conductance or film coefficient, W/m K Thermal conductivities, W/m K Indices for summations Number of heat sources Heat flow rate, W Thermal resistance, K/W Total and layer thicknesses, Surface temperature, K Sink temperature, K Heat source centroid, m Greek Symbols Eigenvalues, Eigenvalues, Manuscript received November 26, 2004 revised October 31, 2005 This work was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) This work was recommended for publication by Associate Editor Y Joshi upon evaluation of the reviewers comments The author is with the Faculty of Engineering and Applied Science, Memorial University of Newfoundland, St John s, NF A1B 3X5, Canada ( yuri@engrmunca) Digital Object Identifier /TCAPT Transform variable Temperature excess,, Mean temperature excess,, Centroidal temperature excess,,k Surface temperature excess,,k Relative conductivity, Eigenvalues, Spreading function Dummy variable, Subscripts Effective value Denotes the and sources -plane -plane Superscripts Mean value Centroid value I INTRODUCTION CALCULATION of the mean or centroidal value of discrete heat sources on a rectangular substrate 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 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 [2], 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 multi-layered systems /$ IEEE

2 MUZYCHKA: INFLUENCE COEFFICIENT METHOD 637 Fig 1 Multisource system [2] 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, as 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 manageable 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 [2] also allow for convective cooling in the die plane in addition to the sink plane In this work, the solution of Muzychka et al [3] is recast in terms of influence coefficients, which allow the effects of neighboring 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 II 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 integrated circuit (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 multilayer substrate containing discrete heat sources The substrate base was assumed to be attached to a heat sink of fixed temperature Discrete heat sources were dealt with using the die Fig 2 Single eccentric heat source [3] plane boundary condition Ellison [1] developed a method referred to as TAMS This method is similar to that of Hein and Lenzi [4], but considers multiple layers More recently, Culham et al [2] 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 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 III 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 2 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 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, ie, thermal wake effects are neglected The addition of convection in the source plane is dealt with in a separate section

3 638 IEEE TRANSACTIONS ON COMPONENTS AND PACKAGING TECHNOLOGIES, VOL 29, NO 3, SEPTEMBER 2006 A 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: and (5) B Centroidal Source Temperature The maximum or centroidal heat source temperature may be determined from (1) when,, and 0 This gives (6),, and 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 uniformly distributed over the entire source plane, 0 The general 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 : (1) C Mean Source Temperature The mean heat source temperature is obtained by integrating the local source temperature over the source area, ie, This leads to the following result for the mean temperature excess of a single eccentric heat source: (7) (2) is replaced by,, or, accordingly The spreading parameter accounts for the effects of conductivity, thickness, and convection cooling The final Fourier coefficients,, and were obtained by taking Fourier series expansions of the boundary condition in the source plane, 0 This yielded the following expressions for the Fourier coefficients: and 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 2 Finally, values for the coefficients in the uniform flow solution are given by (3) (4) The results given by (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 IV 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 A Surface Temperature Distribution For discrete heat sources the maximum temperatures occur in the source plane The surface temperature distribution is obtained from (8) (9)

4 MUZYCHKA: INFLUENCE COEFFICIENT METHOD 639 which may be written as (14) Using (10) results in the following expression for the mean temperature excess contribution of the heat source in the region of the heat source Fig 3 Multiple heat sources [3] is the temperature excess for each heat source by itself The temperature excess of each heat source may be computed using (1) evaluated at the surface The Fourier coefficients are now evaluated at each of the source characteristics, ie,,,, and (10) heat B Centroidal Source Temperature The maximum or centroidal temperature is now the sum of all heat source contributions at the point of interest Thus, using (10) evaluated at the centroid of the heat source, we may write (11) (15) Equation (14) represents the sum of the effects of all sources over an arbitrary region Equation (15) is evaluated over the region of interest located at, The coefficients,, and are then evaluated at each of the source parameters V 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 [12] 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 neighboring heat sources on thermal resistance, and hence the mean or centroidal temperature excess of each discrete heat source We begin by examining (12) and (15), for the centroidal and mean temperature excess Beginning first with (11) we may write (12) The present notation, denotes the effect of the heat source in the region of the heat source C Mean Source Temperature The mean heat source temperature of an arbitrary rectangular patch of dimensions and, ie, the heat source, located at and, may be computed by integrating (7) over the region, ie, which may be written as or (16) (17) (18) (13) (19)

5 640 IEEE TRANSACTIONS ON COMPONENTS AND PACKAGING TECHNOLOGIES, VOL 29, NO 3, SEPTEMBER 2006 Finally, we may write the temperature excess in the following matrix form: (25) or (20) are modified Fourier coefficients, since has now been factored out Once again it is noted that the coefficients are evaluated at each of the heat source characteristics, ie,,,, an 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 heat source using (14) which may be written as or (21) (22) (23) (26) 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 (18) or (23) reveals that an influence effect arises by virtue of proximity and strength In other words, 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 (27) This property significantly reduces computation for systems more than five sources are present In general, for a system of sources, a symmetric matrix results for the influence coefficients As a result of this symmetry only 2 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 (9) The reciprocity is a result of the property of Greens functions [15], ie, the potential at, due to a unit heat input at, is the same as the potential at, due to a unit heat input at, 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 A Thermal Resistance Finally, if we consider defining a thermal resistance, for each heat source, it can be shown that (28) (24) When, the contribution is a self effect, ie, the effect of the source acting alone When the contribution to the temperature excess is an influence effect The self effect, is merely the single source thermal resistance The influence effects, are affected by two factors: source strength and the location and size of neighboring sources, ie, a geometry effect The influence coefficients are clearly functions only of the location of the neighboring sources or (29) 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, ie,, and the relative strength ratio, ie Changing location or

6 MUZYCHKA: INFLUENCE COEFFICIENT METHOD 641 strength of any source leads to a new value of thermal resistance Later, a comparison of the self effect and the influence effect on total thermal resistance is presented for a simple case using (29) VI 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 becomes Fig 4 Compound system [3] TABLE I CASE STUDIES (30) denotes the film coefficient in the source plane and denotes the film coefficient in the sink plane Further, the spreading function becomes (31) TABLE II SOURCE LAYOUT Both (30) and (31) reduce to (2) and (20), when adiabatic source plane 0, ie, VII 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 isotropic model to effectively model a resistive or conductive layer placed on a rectangular substrate Each modification is discussed below A Orthotropic Systems If the rectangular flux channel is orthotropic such that the in plane and through plane conductivities are different, ie, then the following transformations may be made to apply the present method to such systems (Muzychka et al [9]): eff (32) and represent the in-plane and through-plane thermal conductivity, and eff (33) 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 B 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 and (34) with, and is replaced by,, or, accordingly Further, the coefficient is now given by (35) This modification can only be applied to the case when there is no convection in the source plane VIII APPLICATION OF RESULTS The results may now be applied to a simple system Three cases will be examined: isotropic, compound, and orthotropic Since the fundamental solution has been previously verified [2], [3], the studies are presented for illustrative purposes only The thermal property and component thicknesses are given in Table I In all three cases, the heat source layout summarized in Table II is used along with the following substrate properties: 200 mm, 100 mm, and 100 W/m K

7 642 IEEE TRANSACTIONS ON COMPONENTS AND PACKAGING TECHNOLOGIES, VOL 29, NO 3, SEPTEMBER 2006 TABLE III ISOTROPIC SUBSTRATE RESULTS TABLE IV COMPOUND SUBSTRATE RESULTS 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 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 are on the order of seconds depending on whether an isotropic or compound system was considered and whether the centroid or mean value of temperature was computed TABLE V ORTHOTROPIC SUBSTRATE RESULTS TABLE VI CONTRIBUTION TO TOTAL RESISTANCE FOR ISOTROPIC SUBSTRATE IX CONCLUSION 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 heat is dissipated in the source plane was also discussed Reasonable accuracy is also obtained when only a small number of terms ( 20 25) are used in the series APPENDIX Simple Maple Release 8 code for Case A results Define Influence Coefficient In the first case, an isotropic substrate which is cooled in the 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 eff mm and eff W/mK using the properties in Table I 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 III V, which report the centroidal and mean temperature excess for each case Table VI reports the relative contributions of the self effect and influence effect to the total resistance on the basis of the mean temperature excess, for the isotropic substrate 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 > 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 a b t/k 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 cos lambda X i sin lambda c i cos delta Y i sin delta d i a b c i d i k lambda delta beta phi beta > f i value B add Bm i cos lambda X j sin lambda c j lambda c j m add Bn i cos delta Y j sin delta d j delta d j n

8 MUZYCHKA: INFLUENCE COEFFICIENT METHOD 643 add add Bmn i cos lambda X j sin lambda c j cos delta Y j sin delta d j lambda c j delta d j m n : Input System Parameters > baseparameters a b k h t > sourceparameters c d X Y Q c d X Y Q c d X Y Q c d X Y Q Calculate Influence Coefficients > f s seq evalf subs j i n baseparameters sourceparameters f i > f s seq evalf subs j i n baseparameters sourceparameters f i > f s seq evalf subs j i n baseparameters sourceparameters f i > f s seq evalf subs j i n baseparameters sourceparameters f i Calculate Source Temperature Excesses > Source Theta subs j sourceparameters add Q i f s i i > Source Theta subs j sourceparameters add Q i f s i i > Source Theta subs j sourceparameters add Q i f s i i > Source Theta subs j sourceparameters add Q i f s i i ACKNOWLEDGMENT The author would like to thank Dr M M Yovanovich for comments given during manuscript preparation n n n n REFERENCES [1] G Ellison, Thermal Computations for Electronic Equipment Malabar, FL: Krieger, 1984 [2] J R Culham, M M Yovanovich, and T F Lemczyk, Thermal characterization of electronic packages using a three-dimensional Fourier series solution, J Electron Packag, vol 122, pp , 2000 [3] Y S Muzychka, M M Yovanovich, and J R Culham, Thermal spreading resistance of eccentric heat sources on rectangular flux channels, J Electron Packag, vol 125, pp , 2003 [4] V L Hein and V D Lenzi, Thermal analysis of substrates and integrated circuits, in Proc Electron Comp Conf, 1969, pp [5] A Kokkas, Thermal analysis of multiple-layer structures, IEEE Trans Electron Dev, vol ED-21, no 11, pp , Nov 1974 [6] V Kadambi and N Abuaf, An analysis of thermal response of power chip packages, IEEE Trans Electron Dev, vol ED-32, no 6, pp , Jun 1985 [7] M J H Krane, Constriction resistance in rectangular bodies, J Electron Packag, vol 113, pp , 1991 [8] M M Yovanovich, Y S Muzychka, and J R Culham, Spreading resistance of isoflux rectangles and strips on compound flux channels, J Thermophys Heat Transf, vol 13, pp , 1999 [9] Y S Muzychka, M M Yovanovich, and J R Culham, Thermal spreading resistances in compound and orthotropic systems, J Thermophys Heat Transf, vol 18, no 1, pp 45 51, Jan/Mar 2004 [10] Y S Muzychka, J R Culham, and M M Yovanovich, Thermal spreading resistances of rectangular flux channels: part II edge cooling, in Proc 36th AIAA Thermophys Conf, Orlando, FL, Jun 2003, pp 1 9 [11] K J Negus and M M Yovanovich, Thermal resistance of arbitrarily shaped contacts, in Proc 3rd Int Conf Numer Methods Thermal Probl, Seattle, WA, 1983, pp [12] K J Negus, M M Yovanovich, and J W DeVaal, Development of thermal constriction resistance for anisotropic rough surfaces by the method of infinite images, in Proc Nat Heat Transf Conf, Denver, CO, 1985, pp 1 11 [13] K J Negus and M M Yovanovich, Transient temperature rise at surface due to arbitrary contacts on half spaces, Trans CSME, vol 13, pp 1 9, 1987 [14], Thermal computations in a semiconductor die using surface elements and infinite images, in Proc Int Symp Cooling Technol Electron Equip, Honolulu, HI, 1987, pp [15] P M Morse and H Feshbach, Methods of Theoretical Physics, Part I New York: McGraw-Hill, 1953 [16] Waterloo Maple, Inc, Maple Release 8, Tech Rep, Waterloo, ON, Canada, 2002 Yuri S Muzychka is an Associate Professor of mechanical engineering at Memorial University of Newfoundland, St John s, NF, Canada His research focus is on the development of robust models for characterizing transport phenomena using fundamental theory These models are validated using experimental and/or numerical results He has published approximately 35 papers in refereed journals and conference proceedings in these areas Presently, his research is focused on the modeling of complex fluid dynamics and heat transfer problems in internal flows These include transport in porous media, compact heat exchangers, two phase flow in oil and gas operations, fluid film lubrication models, noncontinuum flow, micro-channel flows, and non-newtonian flows He also undertakes research in electronics packaging, contact heat transfer, and thermal design/optimization of energy systems Dr Muzychka is a member of the American Institute for Aeronautics and Astronautics (AIAA) and the American Society of Mechanical Engineers (ASME) He is also a registered professional engineer