Constant mass model for the liquid-solid phase transition on a one-dimensional Stefan problem: Transient and steady state regimes

Abstract

One way to study the dynamics of first order phase transitions is through solving the Stefan problem. The goal of this work is to develop a framework that allows researchers to predict the behaviour of a one-dimensional liquid-solid phase transition in the asymptotic time regime by taking into account volumetric effects through mass conservation. In most works on classical Stefan problems, only thermal diffusion in each phase and the variation of the heat flux through the interface by means of the Stefan condition are considered. In this paper, the model is generalized by considering mass conservation as well, which has not been reported before in the literature to the authors knowledge. By using a higher order finite difference scheme and the heat balance integral method for two different types of boundary conditions (isothermal and adiabatic), the improved model is solved.As a consequence of mass conservation, a subtle change in the thermal balance at the interface (Stefan's condition), is needed. The numerical and semi-analytical solutions for the improved model and the asymptotic values for the interface position, system length, temperature profile and liquid and solid masses, are compared. In the case of isothermal boundary conditions, due to mass conservation, more general asymptotic values are found. A good agreement between the numerical and semi-analytical solutions and the asymptotic time values is observed. In the case of adiabatic boundary conditions, the interface position at thermodynamic equilibrium can be overestimated by the numerical and semi-analytical solutions if only conservation of energy is considered. Finally, by applying mass conservation, a significant error decrease between the numerical and semi-analytical solutions and the predicted values at thermodynamic equilibrium is observed.