Analytical solution for transient flow of energy in a one-dimensional radiating fin.

Abstract

An analytical solution is presented for the transient flow of energy in a one-dimensional, semi-infinite fin in the presence of a zero irradiation environment with nonlinear radiation at its surface. This analytical solution is obtained by the application of a similarity analysis that reduces the partial differential equation for the temperature distribution in the fin to an ordinary differential equation. The ordinary differential equation is then solved numerically, and the results are applied to the calculation of the heat flux from the fin. The results are compared with quasi-steady-state calculations to demonstrate the validity of such calculations. NomenclatureC p = specific heat k = thermal conductivity T = temp, deg absolute Tb = fin base temperature Tbi = initial base temperature of fin t = time to, XQ = arbitrary constants x,y,z = distance Zj = dependent variable in numerical calculations evaluated at r = r Zj,i = dependent variable in numerical calculations evaluated at a value of the independent variable of f y + A£ thermal diff usivity = 6<7/pCp\ j = € 0 X p a emissivity nondimensional temperature thickness of fin density Stefan-Boltzmann constant nondimensional variable given by Eq. (11) nondimensional variable given by Eq. (18)