A control volume method based interface movement equation for one-dimensional Stefan problem achieving mass conservation

Abstract

For heat-transfer-controlled or diffusion-controlled phase transformation simulations, the movement of the interface between different phases needs to be determined, which is commonly called the Stefan problem. Various numerical methods have been developed for this and the movement of the interface is conventionally calculated by a flux balance equation, which is usually called the Stefan condition equation. However, mass conservation errors have been reported in the literature for some of the numerical methods. In this research, a control volume method based interface movement equation was proposed to achieve mass conservation during calculation. The obtained maximum relative mass conservation error for the new interface movement equation is less than 10−13 and the total mass is constant to the machine precision. This is much smaller than that of the Stefan condition discrete equation, which is in the order of 10−2. It was also found that the mass conservation accuracy of the numerical methods partly depends on the choice of the interface movement equation. The mass change term in the Stefan condition discrete equation is only a part of the whole mass change during interface movement, leading to the mass non-conservation found in the literature. In contrast, the piecewise integration treatment in the proposed interface movement equation in this research gives a more accurate mass change description during the interface movement. The correctness of the proposed interface movement equation was also proved by a comparison with the analytical solutions.