Approximate Temperature Profiles and Companion Heat Transfer Rates of Uniform Annular Fins Using Finite-Differences Instead of Bessel Functions

Abstract

The temperature variation along annular fins of uniform thickness and constant thermal conductivity is governed by a differential equation of second order with variable coefficients which is called the modified Bessel equation of zero order. This educational paper addresses a simplistic finite-difference procedure for solving this kind of Bessel equation employing a reduced system of algebraic equations. Approximate temperature distributions and companion heat transfer rates have been computed with the elimination of unknowns by hand and also with the Gauss elimination method using the symbolic algebra software Maple V (Char et al., 1991) on a personal computer. Instructors and students of heat transfer courses may benefit from this alternative computational procedure that seeks to avoid the use and operations with Bessel functions and still produce numerical results of good quality. Rudimentary knowledge of numerical techniques is the only mathematical background that students need to possess in order to implement the computational scheme explained here.