Two-dimensional fin with convective base condition

When solving a two-dimensional model of an isolated fin, researchers have mainly concentrated on either a constant or a periodic fin base temperature. It is possible to obtain a numerical solution by a convective boundary condition on the fin base. However, in an analytical solution, one cannot calculate an arbitrary constant because of the convective boundary condition of the separation of variables. Therefore a heat balance is applied here to resolve this difficulty. In addition, a modified solution is presented which does not involve any additional mathematics with respect to the classical approach of solving a one-dimensional model. For different values of the Biot number $B_{22}$, a comparison of one- and two-dimensional solutions is given. Relative errors of the heat flow rates predicted by the classical and modified one-dimensional solutions, and the respective exact two-dimensional solution with respect to an, are computed. It is found that, for large values of $B_{22}$ (say 50.0) modified solution, by using a convective condition at the fin base gives significant accuracy improvements in comparison to the classical one-dimensional technique.