A reduced model for phase-change problems with radiation using simplified PN approximations

Radiative heat transfer in phase-change media is of great interest in many thermal applications in sciences and engineering involving internal melting or solidification. In these problems at high temperature, a mathematical model used to describe the heat transfer and phase change should also include equations accounting for thermal radiation. Using the integro-differential equation for the radiative intensity in these models results in a system of coupled equations for which its numerical solution is computationally very demanding. In the present study, we develop a class of efficient reduced models for phase-change problems accounting for grey thermal radiation. The novelty in these models lies in the fact that effects of thermal radiation are well captured in phase-change materials without solving the computationally demanding radiative transfer equation. The model is derived from the enthalpy formulation and the simplified $P_N$ approximations of spherical harmonics. The integro-differential equation for the full radiative transfer is replaced by a set of differential equations which are independent of the angle variable and easy to solve using conventional computational methods. To solve the coupled equations, we implement a second-order implicit scheme for the time integration and a mixed finite element method for the space discretization. A Newton-based algorithm is also adopted for solving the nonlinear systems resulting from the considered monolithic approach. The performance of the proposed reduced models is analyzed on several test examples for coupled radiative heat transfer and phase-change problems in two and three space dimensions. The results presented in this study demonstrate that the proposed models can accurately predict the temperature distributions and capture the phase-change interfaces in melting and solidification examples, all while maintaining a very low computational cost.