Asymptotic analysis of a two-phase Stefan problem in annulus: Application to outward solidification in phase change materials

Abstract

Stefan problems provide one of the most fundamental frameworks to capture phase change processes. The problem in cylindrical coordinates can model outward solidification, which ensures the thermal design and operation associated with phase change materials (PCMs). However, this moving boundary problem is highly nonlinear in most circumstances. Exact solutions are restricted to certain domains and boundary conditions. It is therefore vital to develop approximate analytical solutions based on physically tangible assumptions, e.g., a small Stefan number. A great amount of work has been done in one-phase Stefan problems, where the initial state is at its fusion temperature, yet very few literature has considered two-phase problems particularly in cylindrical coordinates. This paper conducts an asymptotic analysis for a two-phase Stefan problem for outward solidification in a hollow cylinder, consisting of three temporal and four spatial scales. The results are compared with the enthalpy method that simulates a mushy region between two phases by numerical iterations, rather than a sharp interface in Stefan problems. After studying both mathematical models, the role of mushy-zone thickness in the enthalpy method is also unveiled. Moreover, a wide range of geometric ratios, thermophysical properties and Stefan numbers are selected from the literature to explore their effects on the developed model with regards to interface motion and temperature profile. It can be concluded that the asymptotic solution is capable of tracking the moving interface and evaluating the transient temperature for various geometric ratios and thermophysical properties in PCMs. The accuracy of this solution is found to be affected by Stefan number only, and the computational cost is much less compared with the enthalpy method and other numerical schemes.