Heat Balance Integral Method for One-Dimensional Finite Ablation

In this paper, the heat balance integral method is applied to a simple one-dimensional ablation problem. Previous authors have provided solutions that give good approximations over a short time scale; we attempt to provide a solution valid over a large time scale, both before and after ablation begins. We give motivation for the choice of a quartic approximating function n = 4 and compare our results with different polynomial approximations and numerical solutions. It is shown that lower values of polynomial degree n provide a better approximation to the temperature profile in the preablation phase, whereas higher values are more accurate during the ablation phase. The temperature gradient at the ablating surface, and consequently the ablation rate, shows a much weaker dependence on n. However, to ensure a positive ablation rate, it is not possible to switch to a higher value of n between preablation and ablation phases. The approximate solutions are compared with numerical and exact analytical solutions whenever possible. Finally, a simple analytical solution is presented that corresponds to the classical solution of Landau.