THERMAL spreading resistance occurs whenever heat leaves a

Size: px

Start display at page:

Download "THERMAL spreading resistance occurs whenever heat leaves a"

William Mathews
5 years ago
Views:

1 JOURNAL OF THERMOPHYSICS AND HEAT TRANSFER Vol. 3, No. 4, October December 999 Spreading Resistance of Iso ux Rectangles and Strips on Compound Flux Channels M. M. Yovanovich, Y. S. Muzychka, and J. R. Culham University of Waterloo, Waterloo, Ontario NL 3G, Canada The general expression for the spreading resistance of an iso ux, rectangular heat source on a two-layer rectangular ux channel with convective or conductive cooling at one boundary is presented. The general expression depends on several dimensionless geometric and thermal parameters. Expressions are given for some two- and three-dimensional spreading resistances for two-layer and isotropic nite and semi-in nite systems. The effect of heat ux distribution over strip sources on two-dimensional spreading resistances is discussed. Tabulated values are presented for three ux distributions, the true isothermal strip, and a related noniso ux, nonisothermal problem. For narrow strips, the effect of the ux distribution becomes relatively small. The dimensionless spreading resistance for an iso ux square source on an isotropic square ux tube is discussed, and a correlation equation is reported. The closed-form expression for the dimensionless spreading resistance for an iso ux rectangular source on an isotropic half-space is given. Nomenclature A = channel conduction area, m p A s = heat source area, m A = characteristic length of contact area, m a; b = half-lengths of source area, m Bi = Biot number, hl=k c; d = half-lengths of ux channel, m h = contact conductanceor lm coef cient, W/m K i = index denoting layer and layer J º.x/ = Bessel function of rst kind, order º k; k ; k = thermal conductivities,w/m K L = arbitrary length scale, m m; n = indices for summations Q = heat ow rate, W q = heat ux, W/m R = thermal resistance, K/W R s = spreading resistance, K/W R total = total resistance, K/W R D = one-dimensionalresistance, K/W NT sink = mean sink temperature, K NT source = mean source temperature, K T ; T = layer temperatures, K t; t ; t = total and layer thicknesses, m u = relative local position in strip, x=c x; y; z = Cartesian coordinates, m = conductivity parameter,. /=. C / = eigenvalues, p.± C / 0 = gamma function ± = eigenvalues,.m¼=c/ ²; ² ; ² = relative contact size; ² a=c and ² b=d = eigenvalue = relative conductivity, k =k = eigenvalues,.n¼=d/ ¹ % = heat ux shape parameter, ; 0; = aspect ratio of rectangular source area, a=b Presented as Paper at the 36th Aerospace Sciences Meeting, Reno, NV, 5 January 998; received 9 May 998; revision received 7 June 999; accepted for publication 5 June 999. Copyright c 999 by the authors. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission. Professor and Director, Microelectronics Heat Transfer Laboratory, Department of Mechanical Engineering. Fellow AIAA. PostdoctoralFellow, Microelectronics Heat Transfer Laboratory,Department of Mechanical Engineering. Research Associate Professor, Department of Mechanical Engineering. 495 ; ; = relative layer thickness, t=l, t =L, and t =L, respectively Á m ; Á n = two-dimensional spreading functions Á m;n = three-dimensionalspreading function Ã = dimensionless spreading resistance, R s k L Introduction THERMAL spreading resistance occurs whenever heat leaves a heat source of nite dimensionsand enters into a larger region, as shown in Fig.. Figure shows a planar rectangularheat source situated on one end of a compound heat ux channelthat consistsof two layers having thicknesses t and t and thermal conductivities k and k, respectively. The heat ux channel is cooled along the bottom surface through a uniform lm coef cient or a uniform contact conductance h. The heat source area can be rectangular having dimensionsa by b or it may be a strip of width a, when b D d. The dimensions of the heat ux channel are c by d, as shown in Fig.. The lateral boundaries of the heat ux channel are adiabatic. The heat ow rate through the heat ux channel Q is related to the mean temperature of the heat source NT source, the mean heat sink temperature NT sink, and the total system thermal resistance R total through the relationship Q R total D NT source NT sink () The totalthermalresistanceof the systemis de nedby the relation R total D R s C R D () where R s is the thermal spreading resistance of the system and R D is the one-dimensionalthermal resistance de ned as R D D t =k A C t =k A C =h A (3) The conduction area in Eq. (3) is A D 4cd. For an iso ux source area, the heat ow rate through the system is Q D q A s, where q is the uniform heat ux and A s D 4ab is the heat source area. For the general case of a rectangularsource area on a rectangular heat ux channel, as shown in Fig., the spreading resistance will depend on several geometric and thermal parameters such as R s D f.a; b; c; d; t ; t ; k ; k ; h/ (4) This paper has three objectives.one is to obtain a generalsolution for the system shown in Fig.. Second is to report several two- and three-dimensionalcases that arise from the general solution. Third is to report several previously unpublished results that examine the

2 496 YOVANOVICH, MUZYCHKA, AND CULHAM X i.x/ Y i.y/ Z i.z/. The computer algebra system MAPLE V (Ref. 9) was used to accomplish all of the required algebraic manipulations to obtain the two temperature distributions. The spreading resistance was obtained by means of the de nition proposed by Mikic and Rohsenow : R s D. NT source NT contact plane /=Q () The mean temperature of the heat source area is obtained from NT source D 4ab a a b b T.x; y; 0/ dx dy () and the mean temperatureof the contactplane z D 0 is obtainedfrom Fig. Finite compound channel with rectangular heat source. NT contact plane D 4cd c c d d T.x; y; 0/ dx dy (3) effect of heat ux distribution for the two-dimensional strip heat source. Several investigators 7 have examined spreading resistance in rectangularisotropictwo- and three-dimensionalsystems. Recently, Yovanovich 8 reviewed and summarized past work in the area of spreading resistance for circular and rectangular systems. As a result, several solutions of practical interest in engineering systems are now available. The general solution presented in the next section reduces to several cases that were previously unavailable. 8 Problem Statement The temperature distributions T and T within the two layers must satisfy the Laplace equation r T i D 0; i D ; (5) where for the rectangularheat source/rectangular ux channel system, the three-dimensionallaplacian operator is Along the common interface z D t, the two temperatures must satisfy the perfect contact T D T ; Along the lateral boundaries x D c and y D d, the two temperatures must satisfy the adiabatic D (6) D 0; i D ; (7) Along the bottom surface z D t C t, the Robin boundary condition must be D h k.t NT sink / (8) The parameter h can representa uniform lm coef cient or a uniform contact conductance.over the top surface z D 0, the boundary conditions are ) the iso D q k ; a < x < a; b < y < b (9) General Spreading Resistance Expression The methodology just described was used to obtain the solution for the general problem de ned earlier. The spreading resistance is obtained from the following general expression that shows the explicit and implicit relationships with the geometric and thermal parameters of the system: R s D a cdk C b cdk m D C a b cdk n D sin.a± m / ± 3 m Á.± m / sin.b n/ Á. n/ 3n m D n D sin.a± m / sin.b n/ ± m n m;n Á. m;n / (4) The general expression for the spreading resistance consists of three terms. The single summations account for two-dimensional spreading in the x and y directions, respectively, and the double summation term accounts for three-dimensionalspreadingfrom the rectangularheat source. Figure shows the superpositionof the two strip solutions and the rectangular solution that yield the general expression. p The eigenvalues are ± m D m¼=c, n D n¼=d, and m;n D.± m C n /. The eigenvalues ± and, corresponding to the two strip solutions, depend on the ux channel dimensions and the indices m and n, respectively. The eigenvalues for the rectangular solution are functions of the other two eigenvalues. The contributionsof the layer thicknessest and t, the layer conductivitiesk and k, andthe uniform conductanceh to the spreading resistance are determined by means of the general expression where Á. / D. e4 t C e t / C e.t C t/.t C C t/ e (5). e 4 t e t / C e.t C t/ e.t C t/ D C Bi= L Bi= L D. /=. C / over the heat source area and ) the D 0 (0) for all points that lie outside the heat source area. The separationof variablesmethodwas employedto nd the solutions for T and T, by assuming solutions of the form T i.x; y; z/ D Fig. Superposition of solutions.

YOVANOVICH, MUZYCHKA, AND CULHAM 497 Table Summary of solutions for iso ux source Figure Con guration Limiting values Rectangular heat source: three-dimensional solutions Finite compound rectangular

3 YOVANOVICH, MUZYCHKA, AND CULHAM 497 Table Summary of solutions for iso ux source Figure Con guration Limiting values Rectangular heat source: three-dimensional solutions Finite compound rectangular ux channel a; b; c; d; t ; t ; k ; k ; h 3 Semi-in nite compound rectangular ux channel t! 4 Finite isotropic rectangular ux channel k D k 5 Semi-in nite isotropic rectangular ux channel t! 6 Isotropic half-space c! ; d! ; t! 7 Compound half-space c! ; d! ; t! Strip heat source: two-dimensional solutions 8 Finite compound rectangular ux channel a; c; b D d; t ; t ; k ; k ; h 9 Semi-in nite compound rectangular ux channel t! 0 Finite isotropic rectangular ux channel k D k Semi-in nite isotropic rectangular ux channel t! Fig. 3 Semi-in nite coated channel with rectangular heat source. Fig. 4 Finite isotropic channel with rectangular heat source. with D k =k and Bi D hl=k, where L is an arbitrarylength scale employed to de ne the dimensionless spreading resistance: Ã D R s k L (6) that is based on the thermal conductivityof the layer adjacent to the heat source.varioussystem lengthsmay be used, and the appropriate choice depends on the system of interest. In all summations Á. / is evaluated in each series using D ± m, n, and m;n. Spreading Resistance for Three-Dimensional Systems The dimensionless spreading resistance Ã depends on six independentdimensionlessparameterssuch as ) the relativesizes of the rectangularsourcearea.² D a=c and² D b=d/, )thelayerconductivity ratio. D k =k /, 3) the relative layer thicknesses. D t =L and D t =L/, and 4) the Biot number.bi D hl=k /. Thus, correlation of the general solution or graphical representation of the resistance is not possible. However, the general solution reduces to many special cases, such as those shown in Figs. 3. This section examines all of the three-dimensional solutions that may be obtained from the general solution given by Eqs. (4) and (5). All of the special three-dimensionalcases are summarized in Table. Semi-In nite Compound Rectangular Flux Channel The general expression for Á. / reduces to a simpler expression when t! (Fig. 3). The solution for this particular case arises from Eq. (4) with Á. / D.e t / C.e t C /.e t C / C.e t / where the in uence of the contact conductance has vanished. (7) Fig. 5 Semi-in nite isotropic channel with rectangular heat source. Finite Isotropic Rectangular Flux Channel The general expression for Á. / reduces to a simpler expression when D (Fig. 4). The solution for this particularcase arises from Eq. (4) with Á. / D.e t C /. e t /Bi=L.e t / C. C e t /Bi=L where the in uence of has vanished. (8) Semi-In nite Isotropic Rectangular Flux Channel When the relative thickness is suf ciently large, Á!, for the three basic solutions of Eq. (4), then Ã D Ã.² ; ² / is independent of and Bi. This correspondsto the case of a rectangularheat source on a semi-in nite rectangular ux channel (Fig. 5). This solution was rst reported by Mikic and Rohsenow.

498 YOVANOVICH, MUZYCHKA, AND CULHAM Fig.

6 Isotropic half-space with rectangular heat source. Fig.

7 Compound half-space with rectangular heat source. Fig.

The general solution may also be used to obtain the solution for an iso ux square area on the

5 For the special case of a square heat source on a semi-in nite square, isotropic ux tube, the

4 498 YOVANOVICH, MUZYCHKA, AND CULHAM Fig. 9 Semi-in nite compound channel with strip heat source. Fig. 6 Isotropic half-space with rectangular heat source. Fig. 0 Finite isotropic channel with strip heat source. Fig. 7 Compound half-space with rectangular heat source. Fig. Semi-in nite isotropic channel with strip heat source. The general solution may also be used to obtain the solution for an iso ux square area on the end of a square semi-in nite ux tube. 5 For the special case of a square heat source on a semi-in nite square, isotropic ux tube, the general solution reduces to a simpler expression that depends on one parameter only. The solution was recast into the form 5 k A s R s D ¼ 3 ² m D sin.m¼ ²/ m 3 Fig. 8 Finite compound channel with strip heat source. C ¼ ² m D n D sin.m¼²/ sin.n¼²/ m n p m C n (9) where the characteristic length was selected as L D p A s. The relative size of the heat source was de ned as ² D p.a s =A c /, where

5 YOVANOVICH, MUZYCHKA, AND CULHAM 499 A c is the ux tube area. A correlation equation was reported for Eq. (9): k A s R s D 0:4730 0:6075² C 0:98² 3 (0) in the range 0 ² 0:5, with a maximum relative error of approximately 0:3%. The constant on the right-hand side of the correlation equation is the value of the dimensionless spreading resistance of an iso ux square source on an isotropic half-space when the square root of the source area is chosen as the characteristic length. Iso ux Rectangular Heat Sources on a Half-Space The spreading resistance for an iso ux rectangular source of dimensions a b on an isotropic half-space(fig. 6) whose thermal conductivity is k has a closed-form solution 6 : k p % A s R s D ¼ sinh % C % 3 C % 3 C % C % sinh % 3 () where% D a=b is the aspectratio of the rectangle.when the scale length is L D p A s, the dimensionlessspreadingresistancebecomes a weak functionof %. For a squareheat source,the numericalvalueof the dimensionlessspreadingresistanceis k p A s R s D 0:473, which is very close to the numerical value for the iso ux circular source on an isotropic half-space and other singly connected heat source geometries, such as an equilateral triangle and a semicircular heat source. The solution for the rectangular heat source on a compound half space (Fig. 7) can be obtained from the general solution for the nite compound ux channel, provided that t!, c!, and d!. No closed-form solution such as that given by Eq. () exists. The size of the computational domain may be determined by comparing the series solution with the closed-form solution [Eq. ()] using the dimensions of the source and the conductivity of the more conductive material to determine the approximate outer dimensions,which have little in uence on the isotropicresult. Spreading Resistance for Two-Dimensional Systems Severaltwo-dimensionalsolutionsmay be obtainedfrom the general solution presented earlier. Four special cases that are summarized in Table are discussed next. Finite Compound Rectangular Flux Channel When the rectangular heat source on the system shown in Fig. has dimensionssuch that b D d, the systemshown in Fig. 8 results. The solution may be written as R s D a ck where Á is given by Eq. (5). m D sin.a± m / ± 3 m Á.± m / () Semi-In nite Compound Rectangular Flux Channel The solution for R s when t! (Fig. 9) is obtained from Eq. () with Á de ned by Eq. (7). Finite Isotropic Rectangular Flux Channel The solutionfor R s when D (Fig. 0) is obtainedfrom Eq. () with Á de ned by Eq. (8). For this system the appropriate scale lengthmay be chosento be L D c, the half-widthof the ux channel. The general solution may then be written in an alternative form: k R s D ¼ 3 ² n D with ² D a=c; D t=c, and Bi D hc=k. sin.n¼ ²/ n¼ C Bi tanh.n¼ / n 3 n¼ tanh.n¼ / C Bi (3) Semi-In nite Isotropic Rectangular Flux Channel When the relativethicknessexceedsthe criticalvalue > :65=¼, the earlier result reduces to the result for the case shown in Fig., an iso ux strip on an isotropic, semi-in nite ux channel for which the spreading resistance is obtained from the expression k R s D ¼ 3 ² n D that depends on the relative strip size only. sin.n¼²/ n 3 (4) Effect of Heat Source Flux Distribution The effect of the heat ux distributionon a semi-in nite isotropic strip source was examined by Yovanovich. 7 Flux distributions are of the form f.u/ D. u / ¹, where u D x=a is the arbitrary relative position in the strip source and the ux shape parameter is ¹. Yovanovich 7 reported the general result k R s D ¼ 0 ¹ C 3 ² n D sin.n¼²/ n n¼² ¹ C J ¹ C.n¼ ²/ (5) where J ¹ C = is the Bessel function of the rst kind of order ¹ C. By means of the general expression, Yovanovich 7 obtained results for three ux distributions:)equivalentisothermal ux distribution, when ¹ D, ) iso ux strip, when ¹ D 0, and 3) parabolic ux distribution, when ¹ D. The general expression with ¹ D for the equivalent isothermal ux distributionreduces to the previously reported result k R s D ¼ ² n D n sin.n¼ ²/J 0.n¼²/ (6) This expression can be compared against the true isothermal closed-form expression ;3 k R s D.=¼/.fsin[.¼=/²]g / (7) For ² < 0:, the earlier result approaches the asymptote k R s D ¼.=¼²/. The parabolic ux distribution result 7 was obtained by setting ¹ D : k R s D ¼ 3 ² n D n 3 sin.n¼²/j.n¼²/ (8) Numerical values of Ã D k R s are given in Table for the ux distributionsde ned by the ux distributionparameter¹ D ; 0; Table Numerical values of for =, 0, and Parameter ² ¹ D Eq. (6) ¹ D 0 Eq. (4) ¹ D Eq. (8) T D const Eq. (7)

6 500 YOVANOVICH, MUZYCHKA, AND CULHAM Table 3 Typical numerical values of Eq. (9) and the average of Eqs. (4) and (6) ² Eq. (9) [Eq:.4/ C Eq:.6/]= % Difference Fig. In nite channel with abrupt change in channel width. and the true isothermal result for a range of the relative strip size parameter ². For completeness, the analytical, closed-form result for the ux channel shown in Fig. is reported. In this case the ux channel is isotropic, the cross section changes abruptly from a width of a to a width of b. The boundary condition over the interface between the upper and lower parts is not known. For the general case, ² D a=b <, the boundary condition is neither isothermal nor iso ux. The true condition is an unknown variable temperature distributionand an unknownvariable ux distribution.when ² D, the temperature and ux distributions are known; however, the spreading resistance is not present. The spreading resistance can be obtained by means of the closed-form result 4 kr s D ¼ ² C ² C ² ² C ² 4² (9) We observe that the numerical values for the equivalent isothermal ux distribution [Eq. (6)] and the true isothermal [Eq. (7)] approach each other as ²! 0; however, there are large differences in the numerical values for ² > 0:6. The numerical values for the parabolic distribution are greater than the iso ux values, which are greater than the values for the isothermal strip. For very narrow strips, ² < 0:0, the maximum difference between the highest values corresponding to ¹ D and the lowest values corresponding to ¹ D differ by less than 5%. This implies that the spreading resistance for very narrow strips depends weakly on the heat ux distribution. In Table 3 the numerical values obtained from Eq. (9) are compared against the mean values of Eqs. (4) and (6) for a range of the relative strip sizes. The differencesare less than % for ² 0:0, and the differences become negligible for ²! 0. Conclusion A general expression for the spreading resistance of an iso ux rectangular source on the surface of a nite compound rectangular ux channel is presented. The series solution consists of three summations that correspond to two strip solutions and a rectangle solution.in general, the dimensionlessspreadingresistancedepends on several dimensionless geometric and thermal parameters. Results are presented for isotropic nite and semi-in nite rectangular ux channelsfor the stripsource.resultsare also presentedfor the iso ux rectangularand square source areas on an isotropic halfspace. A correlation equation is reported for the three-dimensional spreading resistance for an iso ux square source on an isotropic semi-in nite square ux tube. Expressionsthat show the effect of heat ux distributionover the strip source area are presented. Tabulated values of the dimensionlessspreadingresistancefor various ux distributionsare also given. Acknowledgment The rst author acknowledges the continued nancial support of the Natural Sciences and Engineering Research Council of Canada. References Mikic, B. B., and Rohsenow, W. M., Thermal Contact Resistance, Heat Transfer Lab., Rept ,Massachusetts Inst. of Technology,Cambridge, MA, Sept Sexl, R. U., and Burkhard, D. G., An Exact Solution for Thermal Conduction Through a Two-Dimensional Eccentric Constriction, Thermal Design Principles of Spacecraft and Entry Bodies, edited by G. T. Bevins, Vol., Progress in Astronautics and Aeronautics, AIAA, New York, 969, pp Verziroglu, T. N., and Chandra, S., Thermal Conductance of Two Dimensional Constrictions, Thermal Design Principles of Spacecraft and Entry Bodies, edited by G. T. Bevins, Vol., Progress in Astronautics and Aeronautics, AIAA, New York, 969, pp Smythe, W. R., Static and Dynamic Electricity, 3rd ed., McGraw Hill, New York, 968, pp Negus, K. J., Yovanovich, M. M., and Beck, J. V., On the Nondimensionalization of Constriction Resistance for Semi-In nite Heat Flux Tubes, Journal of Heat Transfer, Vol., No. 3, 989, pp Carslaw, H. S., and Jaeger, J. C., Conduction of Heat in Solids, Oxford Univ. Press, London, 959, p Yovanovich, M. M., Two-Dimensional Constriction Resistance with Arbitrary Flux Distribution, Microelectronics Heat Transfer Lab., Univ. of Waterloo, Rept. UW/MHTL-9909, Waterloo, ON, Canada, Yovanovich,M. M., Conductionand Thermal Contact Resistance (Conductance), Handbook of Heat Transfer, edited by W. M. Rohsenow, J. P. Hartnett, and Y. L. Cho, McGraw Hill, New York, 998, Chap MAPLE V, Ver. 4, Waterloo Maple Software, Waterloo, ON, Canada, 998.

THE generalsolutionfor the spreadingresistanceof a ux speci-

THE generalsolutionfor the spreadingresistanceof a ux speci- JOURNAL OF THERMOPHYSICS AND HEAT TRANSFER Vol. 5, No. 3, July September 00 Thermal Spreading Resistances in Compound Annular Sectors Y. S. Muzychka Memorial University of Newfoundland, St. John s, Newfoundland