EFFICIENT HYDRAULIC AND THERMAL ANALYSIS OF HEAT SINKS USING VOLUME AVERAGING THEORY AND GALERKIN METHODS

Abstract

Air- and water-cooled heat sinks are still the most common heat rejection devices in electronics, making their geometric optimization a key issue in thermal management. Because of the complex geometry, the use of finite-difference, finite-volume, or finite-element methods for the solution of the governing equations becomes computationally expensive. In this work, volume averaging theory is applied to a general heat sink with periodic geometry to obtain a physically accurate, but geometrically simplified, system model. The governing energy and momentum equations are averaged over a representative elementary volume, and the result is a set of integro-partial differential equations. Closure coefficients are introduced, and their values are obtained from data available in the literature. The result of this process is a system of closed partial differential equations, defined on a simple geometry, which can be solved to obtain average velocities and temperatures in the system. The intrinsic smoothness of the solution and the simplified geometry allow the use of a modified Fourier‐Galerkin Method for efficient solutions to the set of differential equations. Modified Fourier series are chosen as the basis functions because they satisfy the boundary conditions a priori and lead to a sparse system of linear equations for the coefficients. The validity of the method is tested by applying it to model the hydraulic and thermal behavior of an air-cooled pin-fin and a water-cooled micro-channel heat sink. The convergence was found to be O(N i3:443 ), while the runtime was »0.25 s for N = 56. The numerical results were validated against the experimental results, and the agreement was excellent with an average error of »4% and a maximum error of »5%.