A variational problem relating to cooling fins with heat generation.

Abstract

Optimum fin geometry for maximum heat dissipation per unit weight is derived. The calculus of variations is used. The heat generation function is assumed to be a function of the coordinate along the fin. Euler equations are obtained, and sufficient conditions are derived. The solution is also provided for the case in which the heat generation function is linearly dependent on temperature only. The temperature is found to be a hyperbolic sine function. Explicit expressions for the fin shape and for the maximum heat dissipation are derived for rectangular and circular fins. (L.N.N.)