Analysis of the 1-D heat conduction problem for a single fin with temperature dependent heat transfer coefficient: Part I - Extended inverse and direct solutions

A closed-form inverse solution of the 1-D heat conduction problem for a single fin or spine of constant cross section with an insulated tip is generalized to account for the effect of the tip heat loss. The heat transfer coefficient (HTC) is assumed to exhibit the power-law type dependence on the local excess temperature with arbitrary value of the exponent n in the range of −0.5 ⩽ n ⩽ 5. The form of the obtained inverse solution is the same as the one for a fin with an insulated tip. However, in addition to the dimensionless fin tip temperature Te and n, the fin parameter N also depends on the complex parameter ω2Bi. Using the inversion of this solution and a linearization procedure, the recurrent direct solution with a high convergence rate is derived. Based on the latter, the explicit direct closed-form solution for the accurate determination of the temperature distribution along a fin height at the given values of N, n, and ω2Bi is obtained. This allows one to determine the base thermal conductance G of the straight plate fin (SPF) and cylindrical pin fin (CPF). The relations between the fin parameters are systematized and collected in two tables for the SPF and CPF. They permit one to determine the arbitrary dimensionless geometrical or thermal fin parameter at given value of any other of its parameters and prescribed or calculated values of the main fin parameter(s) N or (and) G.