(Bejan's) early vs. late regimes method applied to entropy generation in one-dimensional conduction

In this paper, we treat the unsteady entropy generation rate due to an instantaneous internal heat generation in a solid slab. Following the basic ideas developed by Bejan [Heat Transfer, Wiley, 1993], we conduct a multiple-scale analysis identifying the "early" and "late" regimes to derive, in a very simple way, the nondimensional unsteady temperature profile for small values of the Biot number, Bi. In consequence, the nondimensional spatial average entropy generation rate per unit volume, Φ and the corresponding average steady-state entropy generation rate, Ψ, were evaluated for different values of the nondimensional heat generation parameter β. This parameter represents physically the ratio of the temperature of the solid slab (due to the internal heat generation) to the fluid temperature. We show that for the assumed values of this parameter β, the nondimensional temperature and entropy generation rate variables present a very sensible dependence between both parameters, indicating a direct relationship between the basic heat transfer mechanics: heat conduction, heat convection and internal heat generation.