Approximate solutions to the Stefan problem with internal heat generation

Abstract

Using a quasi-static approach valid for Stefan numbers less than one, we derive approximate equations governing the movement of a phase change front for materials which generate internal heat. These models are applied for both constant surface temperature and constant surface heat flux boundary conditions, in cylindrical, spherical, plane wall and semi-infinite geometries. Exact solutions with the constant surface temperature condition are obtained for the steady-state solidification thickness using the cylinder, sphere, and plane wall geometries which show that the thickness depends on the inverse square root of the internal heat generation. Under constant surface heat flux conditions, closed form equations can be obtained for the three geometries. In the case of the semi-infinite wall, we show that for constant temperature and constant heat flux out of the wall conditions, the solidification layer grows then remelts.