## 5 Numerical Models Used in Studies Regarding PCMs  
As solidification and melting processes have latent heat and velocity transitions between the phases, a proper numerical approach uses a combination of at least one of each group of methods described in the previous sections. In numerical models, the use of simplifying conditions is common to speed up convergence and allow the solution in a reasonable time. Such simplifying conditions can be assumed regarding geometry, material properties, physical phenomena associated with the problem, and boundary conditions. Among the geometrical simplifications, it is worth mentioning the use of a reduced domain (one- or two-dimensional) and the application of symmetry conditions: the former needs fewer equations to be solved and the latter reduces the size of the domain and consequently the number of cells. Therefore, the computational cost is dramatically reduced by adopting the simplifications mentioned above. Regarding material properties, simplifications include the assumption of continuous properties in each phase, independently of temperature, or even continuous properties for both phases, i.e., no property difference between solid and liquid. Another useful simplifying condition is the well-known Boussinesq approach, which considers the density constant for all equations, except for the body forces term in the momentum equation [51]. In such an assumption, convective motions are included in the model with low additional processing. Simplifications regarding physical phenomena contemplate mainly the conductive heat transfer, which \begin{table}
\begin{tabular}{c c c} \hline \hline
**Reference** & **Viscosity Function** & **Viscosity Ratio (\(\mu_{\text{s}}/\mu_{\text{l}}\))** \\ \hline [PERSON] and [PERSON] [78] & \(\mu=\frac{\mu}{B}\) & \(\infty\) \\ [PERSON] et al. [92] & \(\mu=\mu_{\text{l}}+B(L-h_{\text{d}})\) & \(10^{3}\) \\ [PERSON] and [PERSON] [82] & \(\mu=\mu_{\text{l}}+\left(T+\frac{T_{\text{s}}-T_{\text{s}}}{T_{\text{s}}-T_{ \text{s}}}\right)\frac{\mu_{\text{l}}-B}{T_{\text{s}}-T_{\text{s}}}\) & \(10^{10}\) \\ [PERSON] et al. [95] & Not specified & \(\infty\) \\ [PERSON] and [PERSON] [32] & Not specified & \(10^{30}\) \\ [PERSON] et al. [96] & Not specified & \(5.5\times 10^{8}\)–\(5.5\times 10^{10}\) \\ [PERSON] et al. [94] & Not specified & \(10^{8}\) \\ [PERSON] et al. [97] & Not specified & \(10^{4}\) \\ [PERSON] et al. [98] & Linear & \(2.2\times 10^{7}\) \\ [PERSON] et al. [76] & Linear & \(2.2\times 10^{5}\)–\(2.2\times 10^{7}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Viscosity functions adopted in VVM studies. represents a massive reduction in computation, as suppresses all the velocity terms, resuming the model to heat transfer equation with only conduction and phase change. However, such simplification limits the accuracy of the results, as primarily shown in the analysis of the phase morphology (see Section 3). Other physical simplifications may include the suppression of surface tension between solid and liquid or to consider the material isotropic. Boundary conditions are normally applied in the heat transfer surfaces of the PCM container. Such conditions include: (i) neglecting the effects of the container wall thickness, (ii) application of constant temperature or heat flux over time, and (iii) ignoring the impact of fluid flow outside of the container, which generates different values of heat flux in different positions. On a general basis, it is evident that fewer simplifications in the model imply more difficulties in the resolution and additional processing time. Computer and processor evolution that has been developed so far has allowed researchers to increase the complexity of the models by including more features as exhaustive properties, boundary conditions, physical conditions, and more detailed or different geometries.