Closed form correlations for thermal optimization of plate-fin heat sinks under natural convection

In the present paper, closed form correlations that allow for thermal optimization of vertical plate-fin heat sinks under natural convection in a fully-developed-flow regime are suggested. Analytical solutions for velocity and temperature distributions for high channel aspect ratios, high conductivity ratios, and low Rayleigh numbers are presented using the volume averaging approach. To validate the analytical solutions, both experimental and numerical studies are performed, and it is shown that the analytical solutions are applicable to the range of H/wc >> 1, ks/kf > 10, and Ra″<1000. From the analytical solutions, explicit correlations for optimal fin thickness and optimal channel width, which minimize thermal resistance for given height, width, and length of heat sink, are proposed. The correlations show that the optimal fin thickness depends on the height, the solid conductivity, and the fluid conductivity only and is independent of the Rayleigh number, the viscosity of the fluid, and the length of the heat sink.