Application of thermal spreading resistance in compound and orthotropic systems

Abstract

A review of thermal spreading resistance in compound and orthotropic systems is presented. Transformation of the governing equations and boundary conditions for orthotropic systems is discussed. Relationships between the solutions for isotropic and orthotropic systems are developed. Solutions for spreading resistance are presented in both cylindrical and cartesian systems. NOMENCLATURE a, 6 a, b, c, d Ab As Am,An,Am Bn Bi h k keff C m,n Q q R RID Rs = radial dimensions, m = linear dimensions, m = baseplate area, m = heat source area, m n — Fourier coefficients = Biot number, h£/k = contact conductance or film coefficient, W/m • K = thermal conductivity, W/m-K = effective conductivity, W/mK = arbitrary length scale, m = indices for summations — heat flow rate, W, = qAs = heat flux, W/m = thermal resistance, K/W = one-dimensional resistance, K/W = spreading resistance, K/W * Assistant Professor I Distinguished Professor Emeritus, Fellow AIAA * Associate Professor Copyright ©2001 by Y.S. Muzychka, M.M. Yovanovich, and J.R. Culham. Published by the American Institute of Aeronautics and Astronautics, Inc. with permission. RT = total resistance, K/W t?ti,t2 = total and layer thicknesses, m teff = effective thickness, m T — temperature, K Ts = mean source temperature, K Tf = sink temperature, K Xc, Yc = heat source centroid, m Greek Symbols a = equation parameter, = | £m,n = eigenvalues, = \/m + l Sn — eigenvalues, (mr/6, myr/c) e = relative contact size, = a/b 6 = temperature excess, = T T/, K 0 = mean temperature excess, = T T/, K K = relative conductivity, k^/ki Am — eigenvalues, (rmr/a, nir/d) $,(p = spreading resistance functions ^ = spreading parameter, 4kaRs Q = equation parameter, == 5^^ r = relative thickness, = t/C ( = dummy variable, m~ ____ f = transform variable, = z/^/ktp/kip Subscripts i = index denoting layers 1 and 2 ip = in plane tp — through plane r = r-plane x = x-plane xy = xy-plane y = y-plane z = z-plane INTRODUCTION Thermal spreading resistance theory finds widespread application in electronics cooling, both at the board and chip level and in heat sink applications. It also arises (c)2001 American Institute of Aeronautics & Astronautics or Published with Permission of Author(s) and/or Author(s)' Sponsoring Organization. in the thermal analysis of bolted joints and other mechanical connections resulting in discrete points of contact. Recently, a comprehensive review of the theory and application of thermal spreading resistances was undertaken by one of the authors. Since this review, a number of new solutions and applications of spreading resistance theory have been addressed. These include, but are not limited to, prediction of thermal resistance of electronic devices known as Ball Grid Arrays (EGA), the effect of heat source eccentricity, the effect of heat spreaders in compound and the effect of orthotropic properties'. This paper presents a general review of thermal spreading resistance theory in compound and orthotropic systems. Presently, only a few analyses have been undertaken for orthotropic systems'. These solutions have only been presented for the circular disk and rectangular strip. Solutions for the thermal spreading resistance in compound disks and rectangular flux channels will be reviewed. It will be shown that with the appropriate transformation, these solutions may be applied to orthotropic systems with little effort. These new solutions may then be applied to a number of orthotropic systems, such as printed circuit boards. THERMAL SPREADING RESISTANCE Thermal spreading resistance arises in multidimensional applications where heat enters a domain through a finite area. The total thermal resistance of the system may be defined as _T.-Tf _es ~ Q ~ Q (i) In applications involving spreading resistance, the total thermal resistance is composed of two terms: a uniform flow or one-dimensional resistance and a spreading or multi-dimensional resistance which vanishes as the source area approaches the substrate area. These two components are combined as follows: