Application of Standard and Refined Heat Balance Integral Methods to One-Dimensional Stefan Problems

Abstract

Abstract. The work in this paper concerns the study of conventional and refined heat balance integral methods for a number of phase change problems. These include standard test problems, both with one and two phase changes, which have exact solutions to enable us to test the accuracy of the approximate solutions. We also consider situations where no analytical solution is available and compare these to numerical solutions. It is popular to use a quadratic profile as an approximation of the temperature, but we show that a cubic profile, seldom considered in the literature, is far more accurate in most circumstances. In addition, the refined integral method can give greater improvement still and we develop a variation on this method which turns out to be optimal in some cases. We assess which integral method is better for various problems, showing that it is largely dependent on the specified boundary conditions.

Optimal exponent heat balance and refined integral methods applied to Stefan problems

“…The Heat Balance Integral Method (HBIM) was developed by Goodman, see [7,12], to find approximate solutions to the heat equation and Stefan problems. In fact it is an adaptation of the earlier method of von Kárman and Pohlhausen for analysing boundary layer flow [24].…”

“…Wood [26] showed that even for the quadratic there are six distinct formulations and the one employed by Goodman is generally only the third most accurate. In fact there is a seventh formulation, see [12]. Antic & Hill [2] employ a cubic profile in the study of diffusion in grain silos.…”

One-dimensional solidification of supercooled melts

“…In the original HBIM the value n = 2 was employed, although other values have been used in later studies (often motivated by numerical solutions), see [19]. For now we leave it unspecified.…”

“…The popularity of the HBIM is mainly due to its simplicity. However, in its original form there are a number of drawbacks, primarily a lack of accuracy for certain problems but also the rather arbitrary choice of approximating function, see [19] for a more detailed description of the method and problems. Recently a number of variants of the HBIM have been developed which address the issues and have led to simple solution methods that, over physically realistic parameter ranges, have proved more accurate than second order perturbation solutions [20,21,22,23].…”

“…with s(0) = 0, and primes denote differentiation with respect to η. Denoting ν = −G (0)/β ≥ 0 we use the Stefan condition to eliminate t α s t from (19) and so the model becomes…”