Integral balance methods applied to non-classical Stefan problems

Abstract

We consider two different Stefan problems for a semi-infinite material for the non-classical heat equation with a source that depends on the heat flux at the fixed face. One of them, with constant temperature at the fixed face, was already studied in literature and the other, with a convective boundary condition at the fixed face, is presented in this work. Due to the complexity of the exact solution it is of interest to compare with approximate solutions obtained by applying heat balance integral methods, assuming a quadratic temperature profile in space. A dimensionless analysis is carried out by using the parameters: Stefan number and the generalized Biot number. In addition it is studied the case when Biot number goes to infinity, recovering the approximate solutions when a Dirichlet condition is imposed at the fixed face. Some numerical simulations are provided in order to verify the accuracy of the approximate methods.