Approximate solutions to the Stefan problem with internal heat generation

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 semiinfinite wall, we show that for constant temperature and constant heat flux out of the wall conditions, the solidification layer grows then remelts. List of symbols a, b, c temperature profile constants cp specific heat Dhf latent heat of fusion k thermal conductivity L half-width of plane wall _ q internal heat generation Q nondimensional internal heat generation (constant temperature) Qf nondimensional heat flux (semi-infinite wall) _ Q nondimensional internal heat generation (constant heat flux) Q¢¢ nondimensional heat flux r radius variable r0 radius of cylinder or sphere s distance to the phase change front (for cylinder, spherical and plane wall geometries) ~s nondimensional distance to phase change front (semi-infinite wall) St Stefan number t time ~t nondimensional time (semi-infinite wall) T temperature T0 surface temperature Tm melting or fusion temperature x distance variable a thermal diffusivity q density s Fourier number n nondimensional distance to the phase change front Subscripts liq liquid phase sol solid phase