Analytical solutions to the Stefan problem with internal heat generation

A first-order, ordinary differential equation modeling the Stefan problem (solid-liquid phase change) with internal heat generation in a plane wall is derived and the solutions are compared to the results of a computational fluid dynamics analysis. The internal heat generation term makes the governing equations non-homogeneous so the principle of superposition is used to separate the transient from steady-state portions of the heat equation, which are then solved separately. There is excellent agreement between the solutions to the differential equation and the CFD results for the movement of both the solidification and melting fronts. The solid and liquid temperature profiles show a distinct difference in slope along the interface early in the phase change process. As time increases, the changes in slope decrease and the temperature profiles become parabolic. The system reaches steady-state faster for larger Stefan numbers and inversely, the time to steady-state increases as the Stefan number decreases.