Mathematical Validation for Maximum Heat Transfer in the Optimal Straight Fin with Circular Arc Profile Constrained by Material Volume

Abstract

The central objective of the theoretical study is to analyze mathematically the straight fin with circular arc profile proposed inquisitively by Isachenko et al. back in 1969. The importance of this kind of optimal straight fin with circular arc profile stems from the fact that it maximizes heat transfer for a fixed volume of material. The mathematical analysis will be pursued with a quasi one-dimensional heat equation, constant thermal conductivity and mean convection coefficient where the controlling thermo-geometric parameter is the Biot number. Owing to the irregular shape of the circular arc, the solution of the quasi one-dimensional heat conduction equation with variable coefficients cannot be carried out analytically, and forcibly has to be carried out numerically with the finite difference method. After post-processing the computed numerical temperatures for selected Biot numbers, the veracity of maximum heat transfer could be assessed or ignored. The main finding of the present work is to incorporate to the Isachenko’s theory the limiting factor of the Biot number constraint, which must be equal to one.