Analytic Solution of Thermal Spreading Resistance: Generalization to Arbitrary-shape Heat Sources on a Half-space

Thermal spreading resistance is an important phenomenon in contacting bodies and local heating e.g. laser heating. In contacting bodies, the real area of contact is typically less than 2% of the nominal contact area. In practice, due to random nature of contacting surfaces, the actual shape of microcontacts is unknown; therefore, it is advantageous to have a model applicable to any arbitrary-shape heat source. Starting from a half-space representation of the heat transfer problem, a compact model is proposed based on the generalization of the analytical solution of the spreading resistance of an elliptical source on a half-space. Using a "bottom-up" approach, uni ed relations are found that allow accurate calculation of spreading resistance over a wide variety of heat source shapes under both iso ux and isothermal conditions. NOMENCLATURE A = area, m2 a = major semi-axis, m B( ) = beta function b = minor semi-axis, m K( ) = complete elliptic integral of the rst kind , Eq.(4) k = thermal conductivity,W=mK L = characteristic length scale, m N = number of sides of a regular polygon n = geometric parameter for hyperellipse Q = heat ow rate,W q = heat ux,W=m2 R = thermal spreading resistance, K=W PhD Candidate. Corresponding author. E-mail: [email protected]. †Assistant Professor and Mem. ASME ‡Professor R = average temperature based thermal spreading resistance, K=W R0 = centroidal temperature based thermal spreading resistance, K=W R = non-dimensional spreading resistance RT = thermal spreading resistance, isothermal source, K=W r = radius, m T = average temperature, K T0 = centroidal temperature, K Greek α = angle, rad β = length ratio, b=a Γ(:) = gamma function ε = aspect ratio, [ ] η = length, xc=r ρ = distance in polar coordinate, m ω = angle, rad Subscripts c = geometrical center of area