Residue minimization technique to analyze the efficiency of convective straight fins having temperature-dependent thermal conductivity

Abstract

In this article, we propose and implement a numerical technique based on residue minimization to solve the nonlinear differential equation, which governs the temperature distribution in straight convective fins having temperature-dependent thermal conductivity. The form of temperature distribution is approximated by a polynomial series, which exactly satisfies the boundary conditions of the problem. The unknown coefficients of the assumed series are optimized using the Nelder-Mead simplex algorithm such that the squared L₂ norm of the residue attains its minimum value within a specified tolerance limit. The near-exact solution thus obtained is further used to calculate the fin efficiency. For the case of constant thermal conductivity, the obtained results are validated with the analytical solutions, while for the case of variable thermal conductivity, the obtained results are corroborated with those previously published in the literature. An excellent agreement in each case consolidates the effectiveness of the proposed numerical technique.