Performance evaluation of porous fin with prescribed tip temperature: An analytical and numerical approach

Abstract

The present investigation aims at estimating the efficiency and effectiveness of porous fins of various profiles in a natural convection environment with both the ends imposed at certain temperatures. Unlike the other boundary conditions (insulated tip, convective tip) the maximum heat transfer rate for the present case cannot be determined considering the whole fin being at the base temperature. The maximum heat transfer rate is calculated by first finding the temperature profile of the fin corresponds to a very high value of thermal conductivity (i.e. keff→∞). Using the local maximum temperature values the maximum heat transfer rate is calculated. The variation of the temperature as well as the performance parameters for the longitudinal porous fin as a function of fin parameter (SH), is presented for four different profiles (rectangular, trapezoidal, parabolic concave and cubic concave). The variation of the fin efficiency with SH for different values of fin tip to base temperature ratio (θL/θb), depicts that for certain value of θL/θbthe fin behaves differently in the two different zones SH<SH* andSH>SH*. The results indicate that for θL/θb ≥ 0.45 there exists a range of SH for which the fin efficiency is reduced and there exist local maxima and minima. This range of SH is undesirable to operate and it should be avoided for the proper design of the fin. The nonlinear governing differential equations are solved using the finite difference method followed by an iterative solver. To get approximate analytical expressions for the temperature as well as the fin efficiency and effectiveness semi- analytical solutions are also presented using Galerkin's method of weighted residual.