One dimensional Stefan problems for a semi-infinite material with temperature dependent thermal coefficients are considered. Existence and uniqueness of solution are obtained imposing a Dirichlet, a Neumann or a Robin type condition at fixed face x = 0 . Moreover, it is proved that the solution of the problem with the Robin type condition converges to the solution of the problem with the Dirichlet condition at the fixed face. Computational examples are provided.

Existence and uniqueness of solution for two one-phase Stefan problems with variable thermal coefficients

A one-dimensional two-phase Stefan problem for the melting of a semi-infinite material with a power-type temperature-dependent thermal conductivity was considered. The assumption of taking thermal parameters as functions of temperature found its basis in physical and industries applications, allowing for a more precise and realistic description of phase change processes. By imposing a Dirichlet condition at the fixed face, a theoretical and approximate study was developed. Through a similarity transformation, an equivalent ordinary differential problem was obtained from which an integral problem was deduced. The existence of at least one analytical solution was guaranteed by using the Banach fixed point theorem. Due the unavailability of an analytical solution, a heat balance integral method was applied, assuming a quadratic temperature profile in space, to simulate temperature variations and the location of the interface during the melting process. For constant thermal conductivity, results can be compared with the exact solution available in the literature to check the accuracy of the approximate method. 

A two-phase Stefan problem with power-type temperature-dependent thermal conductivity. Existence of a solution by two fixed points and numerical results

In this paper we consider two different Stefan problems for a semi-infinite material for the non classical heat equation with a source which depends on the heat flux at the fixed face x = 0. One of them (with constant temperature on x = 0) was studied in [4] where it was found a unique exact solution of similarity type 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 obtain some kind of approximate solution. For the above reason, the exact solution of each problem is compared with approximate solutions obtained by applying the heat balance integral method and the refined heat balance integral method, assuming a quadratic temperature profile in space. In all cases, a dimensionless analysis is carried out by using the parameters: Stefan number (Ste) and the generalized Biot number (Bi). In addition it is studied the case when Bi 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

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Integral balance methods applied to non-classical Stefan problems

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Integral balance methods applied to non-classical Stefan problems