Thermal analysis of a longitudinal trapezoidal fin with temperature-dependent thermal conductivity and heat transfer coefficient

A homotopy analysis method (HAM) is used to develop analytical solution for the thermal performance of a straight fin of trapezoidal profile when both the thermal conductivity and the heat transfer coefficient are temperature dependent. Results are presented for the temperature distribution, heat transfer rate, and fin efficiency for a range of values of parameters appearing in the mathematical model. Since the HAM algorithm contains a parameter that controls the convergence and accuracy of the solution, its results can be verified internally by calculating the residual error. The HAM results were also found to be accurate to at least three places of decimal compared with the direct numerical solution of the mathematical model generated using a fourth–fifth-order Runge–Kutta–Fehlberg method. The HAM solution appears in terms of algebraic expressions which are not only easy to compute but also give highly accurate results covering a wide range of values of the parameters rather than the small values dictated by the perturbation solution.