## 4 Numerical Models for Phase Change Processes  
Isothermal phase change problems are also known as moving boundary problems or Stefan problems, due to the study of [PERSON] [77] of the ice cap formation in the North Pole region. The model was elaborated considering the interface between solid and liquid as a moving boundary with temperature conditions (T = T\({}_{\text{m}}\)). The solution consisted of solving the interface position over time. Such a model presented a solution about some specific cases, i.e., considering several simplifications in the properties of the material, boundary, and initial conditions, as pointed out in work [31]. It is worth mentioning that the temperature condition in the solid-liquid interface corresponds to an isothermal phase change process, with a sharp interface. However, when referring to phase changes with continuous properties, both solid and liquid coexist in the mushy zone, and the interface is not a well-defined boundary [78]. Therefore, numerical models are most suitable for continuous properties phase change as the mushy zone represents a steep gradient in the domain. On the other hand, for isothermal phase change, the discontinuity represented by phase change is the main difficulty in managing with numerical models [47]. Numerical phase change models found in the literature are commonly divided into two main groups: fixed grid or continuum models, and variable grid or two-phase models [31]. Variable grid models are based on the implementation of separate computational domains for solid and liquid, i.e., separate equation sets, which are coupled using transference terms between the phases [79]. Thus, such models are more suitable to isothermal phase change processes, as they make possible the definition of an interface whose thickness tends to zero [78]. Besides the higher amount of equations, these procedures need interface tracking methods and grid adjustment methods according to the interface position, consequently presenting more difficulties to solving the problem. The grid position adjustment following the interface can be made through the movement of the grid points (dynamic grid) or through the variation of the time steps (interface fitting). Both adjustments are performed to fit the interface position to the grid points [31]. Variable grid models were implemented by [PERSON] et al. [54; 80] with a dynamic grid and by [PERSON] and [PERSON] [81] with interface fitting. Fixed grid models use a single computational domain with a single set of equations applied to the entire domain: continuity, energy conservation, and momentum equations. Therefore, the implementation of fixed grid models is simpler than variable grid models with the results always being accurate [32]. As fixed grid models have no interface tracking, the use of such models is useful to describe continuous properties phase change processes, since in the produced results a gradient between solid and liquid appears [78]. Therefore, fixed grid models are present in a more significant amount in the literature. The regular equations applied in fixed grid models are usually modified to contemplate the physical phenomena associated with phase change, namely the latent heat and the velocity transition between solid and liquid. The velocity transition occurs as liquid behavior follows the laws of fluid mechanics (with shear stress causing deformation rate), while the solid keeps static (with shear stress producing deformation rate). Fixed grid models are classified according to latent heat modeling and velocity transition modeling. Latent heat models are applied in the energy equation and can be based on source term (E-STM), enthalpy (E-EM) or temperature; in this case, they are also known in the literature as temperature-transforming models (E-TTM) [82]. Temperature-based models are divided into three categories: apparent heat capacity (AHC), effective heat capacity (EHC), and heat integration (HI), as reported in work [31].