Analytical and numerical solutions to the three-phase Stefan problem with simultaneous occurrences of melting, solidification, boiling, and condensation phenomena

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 that describe 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 the present work, we 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 (AM) and welding processes, wherein a high-power laser melts and boils the metal powder or substrate. Each phase of the PCM has distinct thermophysical properties, including density, thermal conductivity, and heat capacity. It is shown that the three-phase Stefan problem admits similarity transformations and similarity solutions. 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.