Approximate, analytical procedure for rectangular annular fins by accommodating the Cauchy–Euler equation

Abstract

An approximate, analytical treatment is presented for the rectangular annular fin by transforming the complicated modified Bessel equation of zero order into a rudimentary Cauchy-Euler equation. The essential step in the computational procedure revolves around a simple manipulation of the radial coordinate that sets up a variable coefficient in the third term of the modified Bessel equation of zero order. In the third term, the radial variable will be replaced by the mean radius of the inner and outer radius, whereas the radial variable prevails in the first and second terms. This action paves the way to the easier Cauchy-Euler equation. For a collection of rectangular annular fins of interest in engineering applications, approximate, analytical temperature distributions and heat transfer rates (via the fin efficiency) written in terms of two binomials demonstrate excellent quality levels in all cases. Additionally, relative error distributions are presented in detailed manner using as the baseline cases the classical exact, analytical temperature distributions and heat transfer rates expressed in terms of the complicated modified Bessel functions of first and second kind.