Solution of the Phase Change Stefan Problem With Time-Dependent Heat Flux Using Perturbation Method

Abstract

Theoretical understanding of phase change heat transfer problems is of much interest for multiple engineering applications. Exact solutions for phase change heat transfer problems are often not available, and approximate analytical methods are needed to be used. This paper presents a solution for a one-dimensional (1D) phase change problem with time-dependent heat flux boundary condition using the perturbation method. Two different expressions for propagation of the phase change front are derived. For the special case of constant heat flux, the present solution is shown to offer key advantages over past papers. Specifically, the present solution results in greater accuracy and does not diverge at large times unlike past results. The theoretical result is used for understanding the nature of phase change propagation for linear and periodic heat flux boundary conditions. In addition to improving the theoretical understanding of phase change heat transfer problems, these results may contribute toward design of phase change based thermal management for a variety of engineering applications, such as cooling of Li-ion batteries.

On a multiple timescales perturbation approach for a stefan problem with a time-dependent heat flux at the boundary

“…Most Stefan problems are solved approximately by using numerical methods (see for instance [4,10,12,13,16,18,19,24,25]). A smaller part of the Stefan prob-lems is approximately solved by using straightforward perturbation expansion (see for instance [3,6,9,12,13,15,17,26,28]). To expand the solution of the problem, a small parameter is usually introduced in the aforementioned papers, and this choice of the small parameter usually leads to a problem for an elliptic partial differential equation.…”

“…In this paper, a classical Stefan problem similar to the ones that have been studied in [3] and in [17] is considered and is formulated in Sect. 2 of this paper.…”

“…2 of this paper. In [3,17], the authors used the moving boundary variable as the time-like variable and assumed that the Stefan number is small. In this way, the problem was reformulated in a problem for a perturbed elliptic equation, which does not permit arbitrary initial conditions.…”

“…In this way, the problem was reformulated in a problem for a perturbed elliptic equation, which does not permit arbitrary initial conditions. In fact, the approach as is introduced in [3,17] starts off from the stationary solution. This also implies that the heat influx at the boundary cannot be an arbitrary, timedependent function.…”

“…In contrast to [3,17] where the authors used a straighforward, naive perturbation approach, a multiple timescales perturbation method is used and applied in Sect. 3 of this paper such that arbitrary initial conditions and an arbitrary time-dependent heat inflow at the boundary can be taken into account.…”