Analytical and Numerical Solutions to the Three-Phase Stefan Problem With Simultaneous Occurrences of Melting, Solidification, Boiling, and Condensation Phenomena

Abstract

Abstract. The one-dimensional (1D) Stefan problem is a prototypical heat and mass transfer problem that analyzes the temperature distribution in a material undergoing phase change. In addition, it describes the evolution of the phase change front within the phase change material (PCM). Analytical solutions to the two-phase Stefan problem which describes the melting of a solid or boiling of a liquid have been extensively discussed in the literature. Density change effects and associated fluid flow phenomena during phase change are typically ignored to simplify the analysis. As the PCM boils or condenses, it undergoes a density change of 1000 or more. The effects of density changes and convection cannot be ignored when dealing with such problems. In our recent work (Thirumalaisamy and Bhalla, 2023, “A Low Mach Enthalpy Method to Model Non-Isothermal Gas–Liquid–Solid Flows With Melting and Solidification,” Int. J. Multiphase Flow, 169, p. 104605), we found analytical solutions to the two-phase Stefan problem that account for a jump in the thermophysical properties of the two phases, including density. In this work, we extend our prior analyses to obtain analytical solutions to the three-phase Stefan problem in which an initially solid PCM melts and boils under imposed temperature conditions. This scenario is typical of metal additive manufacturing and welding processes, wherein a high-power laser melts and boils the metal powder or substrate. Each phase of the PCM (solid, liquid, and vapor) has distinct thermophysical properties, including density, thermal conductivity, and heat capacity. While deriving the analytical solution, all relevant jump conditions, including density and kinetic energy, are accounted for. It is shown that the three-phase Stefan problem admits similarity transformations and similarity solutions. Similarity solutions to the three-phase Stefan problem require solving two coupled transcendental equations. To our knowledge, this is the first work that presents an analytical solution to the three-phase Stefan problem with simultaneous melting, solidification, boiling, and condensation (MSNBC). Furthermore, we describe a numerical method for solving the three-phase Stefan problem with second-order accuracy. Numerical solutions for vapor–liquid and liquid–solid interface positions and temperature distributions are compared to analytical solutions. It is demonstrated that the proposed sharp–interface technique can simulate phase change phenomena in moving domains with second-order spatiotemporal accuracy.