A “Boundary Layer” Method of Obtaining an Approximate Solution to the Infinite Fin Problem

Abstract

The classical infinite fin problem is considered in this study. First the exact solution is stated in which temperature, heat transfer rate, effectiveness and fin efficiency are all given. Then the boundary layer method is used to obtain alternative solutions in polynomial form. Boundary conditions are written for this method, and applied in various combinations to an assumed temperature profile. First, second, and third order approximate solutions are derived. Temperature profiles obtained from these solutions are compared to that calculated from the exact solution. It is shown that as more terms are included in the assumed profile, the resultant expression better fits the exact solution. Very good agreement between the third order and exact solution was obtained. Also derived from the approximate solutions was a distance along the fin beyond which the temperature difference between the fin and the surroundings is negligible. This arbitrary distance is analogous to the boundary layer thickness for boundary layer flow over a flat plate.