Transient Heat Conduction in a Planar Slab With Convection and Radiation Effects

This article presents new semi-analytical solutions for transient heat conduction in a slab, which exchanges heat at the surface with its surroundings via convection, radiation, or simultaneous convection–radiation mechanisms. The concept of thermal penetration depth together with the integral method of heat balances form the basis of the formulation. Explicit expressions are derived that eliminate the need for numerical integration of the ordinary differential equations (ODEs) obtained from space integration of the heat equation. Verification of the temperature relations with a convection boundary condition is performed using an exact solution. In the case of radiation and combined convection–radiation boundary conditions, the accuracy of the explicit solutions is assessed against a numerical model. In all three cases, the new relations predict the surface temperature with high precision. For the radiation boundary condition, the computed midpoint temperature is also in excellent agreement with the numerical solution. In the case of convection heat exchange at the surface, the temperature at internal locations has close agreement with the exact solution for Biot numbers up to B_i = 1. The model prediction deviates from the exact solution in the interior positions for higher values of B_i with the highest deviation occurring at the center. For example, at B_i = 3, the highest relative error is 7.5%. Likewise, for the case of combined convection–radiation boundary conditions, the predicted midpoint temperature is found to closely follow the numerical solution at early stages of the process only. The predictability of the explicit solution of the convection-only boundary condition is more accurate than other approximate solutions at a Fourier number less than 0.2. Additional relations are presented for determination of the total heat transfer as a function of the Fourier and Biot numbers.