The Stefan Problem With Internal Heat Generation in Spherical Coordinates

Abstract

A weakly time-dependent equation for the evolution of the solid–liquid interface in spherical coordinates, driven by internal heat generation, is derived for constant surface temperature boundary conditions. The derivation comes by making an assumption that the interface moves slowly compared to the changes in the temperature so that the technique of separation of variables may be applied for Stefan numbers less than one. Under this approximation, we can separate the nonhomogeneous heat diffusion equation into transient and steady-state terms, and then integrate to get the temperature relations. With the temperature equations in hand, the derivatives are inserted into the interface equation giving a first-order differential equation for the location of the solid–liquid interface as a function of time. The results are compared to a previously derived quasi-static solution and a numerical simulation generated using the method of catching of the front. This method allows for direct tracking of a moving boundary via the calculation of the time it takes to move from node to node in a discretized grid characteristic of classical finite difference methods. Plots of the interface evolution show excellent agreement between the three methods, especially for lower Stefan numbers. The quality of the approximation decreases as the Stefan number increases, but the model is more accurate than the previously studied quasi-static model. For the Stefan numbers St = 1.0 and 10.0, the weakly time-dependent solutions are in better agreement with the numerical results than the quasi-static solutions.