Diffusion of Heat Through a Rectangular Bar and the Cooling and Insulating Effect of Fins. I. The Steady State

Exact solutions for temperature distribution and heat flow through a rectangular fin are rederived in more general and simple form than previously given. These are studied numerically and graphically and shown to yield new results not contained in the approximate formulas hitherto used. Except for the factor of thermal conductivity of the fin material, the cooling effect is a function only of the height-width ratio β of the fin, and of a dimensionless ``relative boundary resistivity'' μ governing the heat loss to the environment. It is shown that, for any value of this boundary resistivity, there is a fairly sharp optimum value of the height-width ratio beyond which further increase in height will not lead to much improvement in cooling effect. This optimum β decreases with decreasing μ until, at a certain critical value μ* in the neighborhood of 1, it becomes zero. At the critical μ*, the cooling effect is altogether independent of β, and the presence of the fin is a matter of indifference. Below μ*, the cooling effect decreases with increasing β, so that the presence of the fin serves merely to insulate the hot plate. This analysis of a single fin is the basis upon which a study of the conditions for optimal cooling by an array of fins will be carried out in a subsequent publication.