A SHOOTING METHOD SOLUTION FOR A CONVECTIVE-RADIATIVE STRAIGHT LONGITUDINAL FIN WITH VARIABLE THERMAL PROPERTIES AND NONLINEAR BOUNDARY CONDITIONS

Abstract

We develop in this work a simple and highly efficient shooting approach for solving the fin energy equation with multiple nonlinearities. The present fin problem is characterized by temperature-dependent thermal conductivity, heat transfer coefficient, and surface radiation emissivity, where the fin base is imposed to a constant temperature and the fin tip is subjected to a combination of convective and radiative heat losses. The governing fin boundary value problem is first reduced into an equivalent initial value problem and then integrated using the fourth-order Runge-Kutta method. The temperature gradient at the tip is approximated by a five-point backward finite difference formula, and computed iteratively using the secant method on the base, which is decisive for numerical integration. The fin problem is solved and compared for two cases of tip boundary condition: an adiabatic fin tip and a convective-radiative fin tip. A thermal analysis is performed using Biot number, Stark number, and the geometrical number that stands for the ratio of fin surface area to its cross-sectional area. Solutions are computed and compared to those obtained by the boundary value problem method, Galerkin method, and the Adomian decomposition method under the assumption of adiabatic fin tip. The accuracy of the nonlinear shooting method is checked by evaluating the absolute errors. Comparative results show that the fin temperature distribution and the fin efficiency are significantly impacted not only by Biot number, Stark number, and geometrical number but also by the type of the tip boundary condition, which could dramatically degrade the fin efficiency mainly for small values of geometrical number.