## 5 Numerical Models Used in Studies Regarding PCMs
### Formulations  
The full phase change models are defined by the combination of one numerical method for latent heat in the energy conservation equation and one numerical method for velocity transition in the momentum equation. Among the abundant possible combinations of numerical methods, the most common are shown in Table 3, in which the latent heat and velocity transition formulations are combined.  
Several studies are based on the combination of E-EM with Darcy STM, while no literature was found that deals with the combinations of E-STM and STM, E-TTM and STM or E-EM and SOM. Besides the combinations of two single numerical formulations used for phase-change simulations, some authors have tried to adopt \"mixed methods,\" i.e., the combinations of various formulations. [PERSON] and [PERSON] [82] used E-TTM-AHC, as well as E-STM combined with VVM, for solidification and melting with different mushy zone ranges, reporting an increase in convergence difficulty as the mushy zone range reduces. [PERSON] et al. [132] used E-TTM-AHC and E-STM for solidification outside a horizontal finned tube in which only thermal conduction was contemplated. Regarding models for metallic alloys, [PERSON] and [PERSON] [78; 133] utilized E-EM with STM coupled with a specific formulation for the alloy composition and solubility.  
Several authors have compared different combinations of numerical formulations referred to phase change processes. Most of the tests have been performed on rectangular cavities with adiabatic top and bottom surfaces, and heat transfer in one or both side surfaces. Such domains were tested in the following evaluations. Regarding latent heat formulations, [PERSON] and [PERSON] [90] compared different results of latent heat for solidification: E-TTM-HI, E-TTM-AHC, E-TTM-EHC, E-EM, and a combination of E-EM and E-TTM-AHC. It was demonstrated that the suitability of each method depends on the type of phase change (isothermal or with continuous properties), as well as on the relationship between sensible and latent heat. [PERSON] et al. [134] compared E-EM and E-TTM-EHC for solidification and melting, reporting slight nonconformities in the results for different temperature  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & & \multicolumn{4}{c}{**Latent Heat Formulations**} \\ \cline{3-5}  & & **E-STM** & **E-TTM** & **E-EM** \\ \hline \multirow{2}{*}{
\begin{tabular}{c} **Velocity** \\ **Transition** \\ **Formulations** \\ \end{tabular} } & **SOM/RSOM** & [101] & [89] & - \\  & **STM/RSTM** & - & - & [102] \\  & **Darcy STM** & [51; 60; 103] & [104] & [42; 48; 49; 52; 56; 57; 105; 106; 107; 108; 109; 110; 111; 112; 113; 114; 115; 116; 117; 118; 119; 120; 121; 122; 123; 124; 125; 126; 127; 128] \\  & **VVM** & [75; 76; 94; 98] & [96; 97; 129; 130] & [73; 95; 131] \\ \hline \hline \end{tabular}
\end{table}
Table 3: Numerical formulations used in studies of PCMs.  
ranges used in E-TTM-EHC. Regarding the velocity transition formulation, [PERSON] et al. [92] compared SOM, VVM, and Darcy STM, all combined with E-EM for solidification in a rectangular cavity: Implementation was considered more accessible with Darcy STM when compared to the other formulations. [PERSON] and [PERSON] [32] performed a comparison of RSOM, RSTM, and VVM, all using the E-TTM latent heat formulation in a rectangular cavity. Ramped methods proved to improve the convergence if compared with non-ramped (SOM and STM) formulations, and the results of RSOM and RSTM were more accurate than VVM. Finally, [PERSON] et al. [96] compared VVM and a mixed method of VVM and Darcy STM, both with E-TTM-AHC in a rectangular cavity: the mixed method had faster convergence than VVM.