Heat conduction in concentric cylinders: an application to electronic packages

A numerical analysis on the heat conduction characteristics of concentric cylinders is presented. The study is relevant to heat transfer in electronic equipment wherein cylindrical electrical conductors or thermal vias (typically characterized by high thermal conductivities) are surrounded by electrical insulators having a different thermal conductivity. The heat conduction equation is solved using a control-volume finite-difference technique. Results are presented in terms of the thermal spreading resistance, the heat removal contribution of the inner cylinder (pin) and the heat spreading contribution of the surrounding material for a wide range of governing parameters. In particular, the effects of the thermal conductivities of the cylinders and the geometric dimensions on the heat spreading characteristics are investigated. It is shown that there is no spreading when the ratio of the conductivity of the pin to the conductivity of the surrounding material (k/K) is larger than about 1000; as found in copper pins surrounded by Teflon or Fr-4 type of a dielectric material. Substantial reduction in the overall thermal resistance is obtained with k/K less than 10, which could be practically achieved by using silicon carbide or aluminum nitride as the surrounding material.