Disc cooled with high-conductivity inserts that extend inward from the perimeter

This paper considers the fundamental problem of how to shape rectangular high-conductivity inserts (fins) that are mounted on the rim of and protrude into a disc-shaped body that generates heat. The objective is to minimize the global thermal resistance by optimizing geometrically the distribution of a fixed amount of high-conductivity material through the material of lower conductivity. In addition to the fin geometry, three other design parameters are considered: the ratio between the high conductivity and low conductivity k̃, the relative amount of high conductivity material φ, and the number of sectors of the disc-shaped body, N. It is shown analytically and numerically that the thermal resistance can be minimized with respect to the fin aspect ratio, λ. The optimized geometry and performance are reported graphically as functions of k̃, φ and N. Good agreement is found between the analytical solution and the numerical results.