SOLID heat spreaders are frequently used for hotspot thermal

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1 IEEE TRANSACTIONS ON COMPONENTS, PACKAGING AND MANUFACTURING TECNOOGY, VO. 7, NO. 9, SEPTEMBER otspot Size Effect on Conductive eat Spreading ongtao Alex Guo, Kris F. Wiedenheft, and Chuan-ua Chen Abstract Solid heat spreaders, particularly those made of copper or graphite, are often benchmar solutions for hotspot thermal management. In this paper, we present exact and approximate analytical solutions of steady-state hotspot cooling with a planar heat spreader, which is subjected to adiabatic conditions except for a hotspot centered at the top surface and a constant temperature at the bottom surface. The approximate solution bridges exact solutions at two limits of hotspot size: infinitesimal hotspot at the center and uniform heat flux across the spreader. The approximate solution accounts for variable hotspot size and anisotropic thermal conductivity in a compact form, which is useful for estimating thermal parameters such as conduction shape factor and effective thermal conductivity. Index Terms Anisotropy, conduction, effective conductivity, heat spreader, hotspot cooling, shape factor. Fig. 1. Problem setup for hotspot cooling on a solid heat spreader. Circular dis of height and radius is subjected to a heat flux of q 0 localized on a circular area of radius a on the top, and constant temperature T 0 at the bottom. Rest of the heat spreader is adiabatic. isotropic and orthotropic media. With straightforward physical interpretations, our approximation can be used to deduce thermal design parameters such as conduction shape factor and effective thermal conductivity. I. INTRODUCTION SOID heat spreaders are frequently used for hotspot thermal management in electronic pacaging [1] [3]. These solid spreaders are made of a variety of materials including copper and graphite [1], with either isotropic or anisotropic thermal conductivity [4] [6]. Due to their simplicity, solid spreaders are also useful models for more complex systems such as vapor chambers, for which effective thermal conductivities are frequently reported [7] [9]. Analytical solutions for conductive heat spreading have been reported for many geometrical and thermal configurations, usually in the form of Fourier series [3], [10]. To simplify calculations, approximate solutions have been proposed, typically in the form of a polynomial see [11]. Although the series solutions and the polynomial approximations are very accurate, their mathematical complexity often obscures the physics. In this paper, we present an approximate solution for steadystate hotspot cooling on a solid heat spreader. Unlie prior wor, our approximation is basically a composite solution joining two limits in terms of hotspot size: infinitesimal hotspot at the center and uniform heat flux across the spreader. The approximate solution bridges the exact solutions at these two limits to account for the hotspot size effect for both Manuscript received October 24, 2016; accepted May 4, Date of publication May 31, 2017; date of current version August 31, This wor was supported in part by the National Science Foundation under Grant CBET and in part by the Intel Corporation. The wor of K. F. Wiedenheft was supported in part by the NSF Gradate Research Fellowship under Grant DGF and in part by the NSF Research Triangle MRSEC under Grant DMR Recommended for publication by Associate Editor G. Refai-Ahmad upon evaluation of reviewers comments. Corresponding author: Chuan-ua Chen. The authors are with the Department of Mechanical Engineering and Materials Science, Due University, Durham, NC USA chuanhua.chen@due.edu. Color versions of one or more of the figures in this paper are available online at Digital Object Identifier /TCPMT II. PROBEM SETUP The solid heat spreader in Fig. 1 is a circular dis with a height and a radius. A hotspot with a radius a and a constant heat flux q 0 is located at the center of the top surface. The bottom surface is isothermal at T 0. The rest of the heat spreader surface is adiabatic. The origin of the cylindrical coordinate system is located at the center of the bottom surface. We will discuss the axisymmetric solutions for both isotropic and orthotropic spreaders, and derive exact and approximate solutions for the temperature rise T = T r, z T 0. We will mainly study the maximum temperature rise at the center of the top surface ˆT T 0, T 0 1 which is arguably the most important indicator of the cooling performance. Toward the end, we will extend the conclusions from the maximum to average temperature and from cylindrical to Cartesian system. III. COMPOSITE SOUTION To understand the effect of the hotspot size, it is helpful to first examine two limiting cases. In the limit of uniform heat flux across the spreader with a =, the largest possible hotspot radius, the temperature rise on the top surface is uniform and given by Fourier s law T a= = q 0. 2 In the limit of an infinitesimal hotspot with a 0, the hotspot radius is the only relevant length scale in an essentially semiinfinite medium, so the temperature rise scales with a instead T a 0 q 0a = a q 0 3 where denotes an order-of-magnitude scaling relation. For a high-aspect-ratio heat spreader with / 1, the following IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See for more information.

2 1460 IEEE TRANSACTIONS ON COMPONENTS, PACKAGING AND MANUFACTURING TECNOOGY, VO. 7, NO. 9, SEPTEMBER 2017 composite solution reduces to 2 and 3 in the respective limits of hotspot radius: T a a q 0 a q0 = Ɣ 4 where Ɣ is a geometrical parameter for heat spreading Ɣξ ξ. 5 The heat spreader size does not enter the approximate solution explicitly, except for the high-aspect-ratio restriction which ensures the recovery of the uniform heat flux limit as a. More generally, the composite solution can be constructed as follows: T a Ɣξ T a=, ξ = T a 0 6 T a= where the argument ξ is given by the ratio of the two limits. The function Ɣξ represents the hotspot size effect by absorbing the geometrical correction to q 0 /, the Fourier s law solution in the case of uniform heat flux across the spreader. This composite solution turns out to be reasonably accurate for any hotspot size, as long as the heat spreader has a high aspect ratio as specified below. The geometrical function Ɣξ is readily lined to the conduction shape factor [12] S πa2 q 0 T πa2 Ɣξ. 7 IV. ISOTROPIC MEDIUM For an isotropic medium with a constant thermal conductivity, the steady-state axisymmetric distribution of temperature T r, z is governed by the heat conduction equation 2 T r T r r + 2 T z 2 = 0 8 subjected to the following boundary conditions: T z T r T z=0 = T 0 { q 0 /, 0 r a = z= 0, a < r = 0. 9 r= A. Exact Solution Using separation of variables, 8 is solved as T r, z = T 0 + q 0 a 2 z 2 + q 0a A n J 0 λ n r sinhλ nz coshλ n 10 where J 0 and J 1 are the Bessel functions of the first ind of order 0 and 1, respectively, and A n = 2J 1 λ n a λ 2 n 2 J 2 0 λ n, λ n = z 1n Fig. 2. otspot size effect on a cylindrical heat spreader with an aspect ratio / = 10. Approximate solution 13 and exact solution 11 of the maximum temperature rise ˆT overlap with each other in the limits of a/ 0 and a/ 1. where z 1n represents the nth positive root of J 1. According to 10, the maximum temperature rise at the hotspot center is ˆT = q 0 a q 0a A n tanhλ n. 11 The exact solution 11 is plotted in Fig. 2. For all exact solutions plotted in this paper, the truncation error is ept below 0.1%. The large and infinitesimal hotspot limits are apparent, signified by T a 0 and T a 1, respectively. In fact, 11 is arranged such that the first and second terms correspond to the limits represented by 2 and 3, respectively. owever, the series solution in the second term is too complex for the hotspot size effect to be apparent. B. Approximate Solution A more convenient form of the infinitesimal limit is obtained by approximating the heat spreader as a semi-infinite medium, as discussed in [13, Ch. 8.2.III] ˆT a 0 q 0a 0 J 1 λa dλ = q 0a = a q 0 λ 12 where λ is the continuous eigenvalue in the semi-infinite domain. The scaling relation in 3 turns out to be exact for the maximum temperature rise in a cylindrical system. Following 6, the approximate solution for the maximum temperature rise is constructed as ˆT a q 0 = Ɣ a q0. 13 This approximation is much more compact compared to their exact counterparts in 11, but it is reasonably accurate as long as the aspect ratio satisfies / 3. In Fig. 2, the approximate solution is plotted against the exact solution for a given aspect ratio of / = 10. The approximate solution nearly overlaps with the exact one at the two limits of hotspot size. A small error is introduced that

3 GUO et al.: OTSPOT SIZE EFFECT ON CONDUCTIVE EAT SPREADING 1461 Fig. 4. otspot size effect on heat spreaders with different thermal conductivity ratios r / z at a given aspect ratio / = 10. The exact and approximate solutions follow 18 and 21, respectively. for the consideration of anisotropy in the imposed heat flux with { T q 0 / z, 0 r a z = 16 z= 0, a < r. Fig. 3. a otspot size effect on heat spreaders with different aspect ratios /. The exact and approximate solutions follow 11 and 13, respectively. b Relative error of the approximation ε peas around a/ = 1. The error is within 7% for high-aspect-ratio heat spreaders with / 3. peas at intermediate hotspot sizes. The approximation error ε is assessed by ε = T exact T approx. 14 T exact The approximate solution 13 slightly underestimates the exact ˆT, but the error is less than 7% as long as the aspect ratio / 3, which is the case for most heat spreaders. For different aspect ratios, the exact and approximate solutions for ˆT a are plotted in Fig. 3a, and their difference is plotted in Fig. 3b. The maximum error occurs around a =, an intermediate hotspot size between the two limits. If the aspect ratio is too small / < 3, the approximate solution 13 is still accurate at the infinitesimal limit, but the error introduced at the large-hotspot limit may exceed 7%. V. ORTOTROPIC MEDIUM For an orthotropic medium in the cylindrical coordinate system, the axisymmetric governing equation becomes 2 T r r T 2 T + z r r z 2 = 0 15 where r and z are the radial and axial thermal conductivities, respectively. The boundary condition is the same as 9, except A. Exact Solution Using separation of variables again, 15 is solved as T r, z = T 0 + q 0 a 2 z z 2 + q 0a z r A n J 0 λ n r sinh r / z λ n z cosh r / z λ n 17 giving rise to ˆT = q 0 a 2 z 2 + q 0a A n tanh r / z λ n. 18 z r The solution for orthotropic medium can also be obtained from the isotropic solution 10 with an isotropic conductivity = r z 19 using the following transformations [10]: 1 1 z 4 r 4 r = r z = z q0 = r z r z 1 4 q0. 20 B. Approximate Solution Applying the above transformation to 13, the approximate solution to the orthotropic heat spreader becomes ˆT z r a q 0 21 z which approximates the exact solution 18 with an error of less than 7%, as long as the modified aspect ratio z / r 1/2 / 3. The exact and approximate solutions to the orthotropic problem are plotted in Fig. 4. When Fig. 4

4 1462 IEEE TRANSACTIONS ON COMPONENTS, PACKAGING AND MANUFACTURING TECNOOGY, VO. 7, NO. 9, SEPTEMBER 2017 Fig. 5. All the curves in Figs. 3a and 4 with various aspect ratios / and conductivity ratios r / z collapse onto a single curve according to 21. is compared to Fig. 3a, it is apparent that the anisotropic conductivity is effectively modifying the aspect ratio of the heat spreader. In accordance with 21, an effective hotspot size z / r 1/4 a normalized by an effective spreader height r / z 1/4 can be used to collapse all the curves in Figs. 3a and 4 onto a single curve in Fig. 5. We emphasize that the master curve according to 21 is far from apparent by direct examination of the Fourier Bessel series solution in 18. C. Effective Isotropic Conductivity As an example application of the approximate solution, we use it to assess the effective isotropic conductivity of an orthotropic heat spreader. The goal is to find an isotropic heat spreader with eff that spreads the same amount of heat q 0 πa 2 with the same maximum temperature rise ˆT. Equating the approximations 13 and 21, the effective isotropic conductivity is approximately eff a z r z. 22 a Theexactvalueof eff can be similarly obtained by equating the exact solutions 11 and 18. Both the exact and approximate solutions of the effective thermal conductivity are plotted in Fig. 6. The approximate solution closely follows the exact solution, validating the compact approximation in 22. The good match holds as long as the high-aspect-ratio requirement for approximations 13 and 21 are both satisfied. The limit of uniform heat flux a = is represented by a horizontal line independent of the radial conductivity, since the heat conduction in this limit is only in the axial z direction. The limit of infinitesimal hotspot a 0 approaches a diagonal line with eff = = r z 1/2, which can be obtained as a limit of 22. The slight difference from the 1/2 power law results from the finite hotspot size. Fig. 6. Effective isotropic thermal conductivity eff of an orthotropic heat spreader with an aspect ratio / = 10. The approximation solution dotted lines follows 22, and the exact solution solid lines is obtained by equating 11 and 18. VI. AVERAGE TEMPERATURE The same idea for constructing the approximate solution can be applied to the average temperature rise over the hotspot area a 0 T r, 2πrdr T πa 2 T According to 10, the average temperature rise for an isotropic heat spreader is T = q 0 a q 0a A n tanhλ n 2J 1λ n a. 24 λ n a The infinitesimal limit can again be obtained from the semiinfinite solution [13] T a 0 q 0a 0 2J1 2λa aλ 2 dλ = 8 q 0 a = 8a q 0 3π 3π. 25 Following 6, the average temperature rise is approximately T 3π 8 a q 0 8a q0 = Ɣ 3π. 26 The approximate and exact solutions for the average temperature are plotted in Fig. 7. When / 3, the maximum approximation error for the average temperature T is 12%, which is larger than the 7% error bound for the maximum temperature ˆT. Unlie the exact solution for the maximum temperature that quicly approaches q 0 / in the largehotspot limit a, the average temperature is appreciably smaller than q 0 / until the hotspot radius is exactly equal to the spreader radius a =. VII. CARTESIAN SYSTEM The approximate solution can be easily extended to the Cartesian coordinate system, where a square heater is subjected to the same boundary condition as in Fig. 1, except for the following change in geometry: The square heat spreader has a height andanareaof2 2, and the square hotspot

5 GUO et al.: OTSPOT SIZE EFFECT ON CONDUCTIVE EAT SPREADING 1463 is still applicable with z / xy as the conductivity ratio and a e as the equivalent radius. In Fig. 8, the approximate solutions using the equivalent radius agree very well with the exact solutions. For highaspect-ratio heat spreaders with / 3, the error of approximation is within 7% for the maximum temperature and within 14% for the average temperature. At the same aspect ratio / and hotspot size a/, the temperature rise is slightly higher for the square heater in Fig. 8 compared to the circular heater in Figs. 2 and 7, mainly because of the larger heat load 4a 2 q 0 instead of πa 2 q 0. VIII. CONCUSION Fig. 7. Average temperature rise T of a circular hotspot on a cylindrical heat spreader with an aspect ratio / = 10. The approximate solution 26 closely follows the exact solution 24. For a solid heat spreader subjected to a localized heat flux on one side and a constant temperature on the other side Fig. 1, we have developed an approximate solution in a compact form 6 that accounts for the variable hotspot size for both isotropic and orthotropic media. The approximate solution is a composite solution that bridges the exact solutions at two limits: infinitesimal hotspot at the center 3 and uniform heat flux across the heat spreader 2. The simple approximation is particularly useful for quic estimation of parameters in thermal design and analysis, e.g., the heat spreading parameter Ɣ that accounts for the hotspot size effect and the effective isotropic conductivity eff that characterizes an orthotropic heat spreader. These approximate parameters are used in [14] for the interpretation of hotspot cooling experiments. APPENDIX EXACT SOUTION IN CARTESIAN SYSTEM Fig. 8. otspot size effect on a square heat spreader with an aspect ratio / = 10. The horizontal axis is the half-width a of a square hotspot normalized by the half-width of a square spreader. The approximate solution 28 for the maximum temperature rise ˆT closely follows the exact solution 31. The approximation solution 29 for the average temperature T closely follows the exact solution 32. at the top center has an area of 2a 2a. For an isotropic medium, the exact solution is detailed in the Appendix. The approximate solutions above, including 13 and 26, still apply with an equivalent circular radius [6] a e = 2 a 27 π which converts the square hotspot with a half-width a to a circular one of the same area. Accordingly, the approximate solutions for the maximum and average temperature rises tae the form of ˆT q 0 T q 0 ae 3π 8 ae q 0 = q 0 = 2 π a π a π. 29 For an orthotropic heat spreader that is transversely isotropic x = y = xy = z, the approximate solution 21 For the Cartesian heat spreading problem in Section VII, the origin of the coordinate system is located at the bottom center of the heat spreader similar to Fig. 1, and the exact solution taes the form of T x, y, z = T 0 + q 0 mπx cos a 2 z 2 + q 0a where 2 nπa nπγ mn sin 2 mπa C mn = mπγ mn sin 4 mπa mnπ 2 γ mn a sin and cos nπy m=0 n=0 m+n =0 C mn sinhγmn z coshγ mn 30, m = 0, n 1, m 1, n = 0 sin nπa γ mn = π m 2 + n 2., m 1, n 1

6 1464 IEEE TRANSACTIONS ON COMPONENTS, PACKAGING AND MANUFACTURING TECNOOGY, VO. 7, NO. 9, SEPTEMBER 2017 Accordingly, the maximum and average temperature rises are ˆT = q 0 a q 0a C mn tanhγ mn 31 a 2 m=0 n=0 m+n =0 T = q q 0a [ Cm0 2 tanhγ m0 γ m0 2 2a m=1 + C0n 2 tanhγ 0n γ 0n 2 2a + Cmn 2 tanhγ mn γ ] mn a m=1 The infinitesimal limit follows the semi-infinite solution in [3, Sec ] ˆT a 0 4sinh 1 1 q 0 a = q 0a e π T a 0 4sinh /4 4/3 π q 0 a 33 = q 0a e 3π 34 where a is the half-width of the square hotspot and a e is the effective radius defined in 27. REFERENCES [1] X. C. Tong, Advanced Materials for Thermal Management of Electronic Pacaging. New Yor, NY, USA: Springer, 2011, ch. 9. [2] R. Mahajan, C.-P. Chiu, and G. Chrysler, Cooling a microprocessor chip, Proc. IEEE, vol. 94, no. 8, pp , Aug [3] M. M. Yovanovich and E. E. Marotta, Thermal spreading and contact resistances, in eat Transfer andboo, A. Bejan and A. D. Kraus, Eds. New Yor, NY, USA: Wiley, 2003, pp [4] D. P. Kennedy, Spreading resistance in cylindrical semiconductor devices, J. Appl. Phys., vol. 31, no. 8, pp , [5] P. ui and. S. Tan, Temperature distributions in a heat dissipation system using a cylindrical diamond heat spreader on a copper heat sin, Jpn. J. Appl. Phys., vol. 75, no. 2, pp , [6] Y. S. Muzycha, M. M. Yovanovich, and J. R. Culham, Influence of geometry and edge cooling on thermal spreading resistance, J. Thermophys. eat Transf., vol. 20, pp , Apr [7] I. Sauciuc, G. Chrysler, R. Mahajan, and R. Prasher, Spreading in the heat sin base: Phase change systems or solid metals?? IEEE Trans. Compon. Pacag. Technol., vol. 25, no. 4, pp , Dec [8] Y.-S. Chen, K.-. Chien, C.-C. Wang, T.-C. ung, Y.-M. Ferng, and B.-S. Pei, Investigations of the thermal spreading effects of rectangular conduction plates and vapor chamber, J. Electron. Pacag., vol. 129, pp , Sep [9]. Shabgard, M. J. Allen, N. Sharifi, S. P. Benn, A. Faghri, and T.. Bergman, eat pipe heat exchangers and heat sins: Opportunities, challenges, applications, analysis, and state of the art, Int. J. eat Mass Transf., vol. 89, pp , Oct [10] D. W. ahn and M. N. Ozisi, eat Conduction, 3rd ed. New Yor, NY, USA: Wiley, 2012, ch. 15. [11] M. M. Yovanovich, J. R. Culham, and P. Teertstra, Analytical modeling of spreading resistance in flux tubes, half spaces, and compound diss, IEEE Trans. Compon., Pacag., Manuf. Technol., A, vol. 21, no. 1, pp , Mar [12] T.. Bergman, F. P. Incropera, D. P. DeWitt, and A. S. avine, Fundamentals of eat and Mass Transfer, 7thed.NewYor,NY,USA: Wiley, 2011, ch. 4. [13]. S. Carslaw and J. C. Jaeger, Conduction of eat in Solids, 2nd ed. Oxford, U.K.: Clarendon, 1959, ch. 8. [14] K. F. Wiedenheft et al., otspot cooling with jumping-drop vapor chambers, Appl. Phys. ett., vol. 110, no. 14, p , ongtao Alex Guo received the B.E. degree in thermal energy and power engineering from Beihang University, Beijing, China, in 2015, and the M.Eng. degree in mechanical engineering from Due University, Durham, NC, USA, in 2016, where he is currently pursuing the Ph.D. degree with the Department of Mechanical Engineering and Materials Science. e is currently with the Microscale Physicochemical ydrodynamics aboratory, Due University. is current research interests include phase-change heat transfer, interfacial electrohydrodynamics, and impulsive biomechanics. Kris F. Wiedenheft received the B.S. degree in mechanical engineering from North Carolina Agricultural and Technical State University NC A&T, Greensboro, NC, USA, in e is currently pursuing the Ph.D. degree with the Department of Mechanical Engineering and Materials Science, Due University, Durham, NC, USA. e was a Mechanical Technician with the Prestone Products Research and Development aboratory, Danbury, CT, USA, from 2007 to e was an Engineering Intern with BASF, Wyandotte, MI, USA, in 2013 and also with the Jet Propulsion aboratory, Pasadena, CA, USA, in is current research interests include superhydrophobic structures and phase-change heat transfer. Mr. Wiedenheft is a member of the National Society of Professional Engineers. e was a recipient of the Namasar Award for Engineering Excellence from NC A&T, the Dean s Graduate Fellowship from Due University, the B. M. Goldwater Scholarship, and the NSF Graduate Research Fellowship. Chuan-ua Chen received the B.S. degree in applied mechanics from Peing University, Beijing, China, in 1998, and the Ph.D. degree in mechanical engineering from Stanford University, Stanford, CA, USA, in e was a Post-Doctoral Associate with Princeton University, Princeton, NJ, USA, and a Research Scientist with Rocwell Scientific Company, Thousand Oas, CA, USA. Since 2007, he has been an Assistant Professor and a unt Faculty Scholar of Mechanical Engineering and Materials Science with Due University, Durham, NC, USA, where he directs the Microscale Physicochemical ydrodynamics aboratory. In 2014, he was promoted to Associate Professor. Dr. Chen was a recipient of the NSF CAREER Award and the DARPA Young Faculty Award for his research integrating physicochemical hydrodynamics and interfacial engineering.