## 5 Numerical Models Used in Studies Regarding PCMs
### Performance Evaluation of Computers and Numerical Models  
The performance evaluation method shown in the present work can be applied to compare numerical models or different computers. Regarding geometrical characteristics, the analysis can be performed in different geometrical configurations, since they have the same number of dimensions (i.e., two-dimensional domains). It is worth mentioning that several characteristics of the computers are not included in the method, such as frequency, number of cores, or memory bandwidth. Such a method consists of two simple normalization steps based on the number of mesh elements and on the number of time steps evaluated. The first step consists of the mean number of time steps solved per time unit, through Equation (10), i.e.,  
\[dt=\frac{n_{dt}}{t_{e}} \tag{10}\]  
where dt is the rate of time steps solved, n\({}_{\text{dt}}\) is the number of time steps, and t\({}_{\text{e}}\) is the computational time elapsed to solve n\({}_{\text{dt}}\). The second normalization step is based on the grid size, aimed to obtain the rate of units solved (\(\eta\)), using Equation (11), i.e.,  
\[\eta=n_{e}dt \tag{11}\]  
where n\({}_{\text{e}}\) is the number of elements of the grid.  
As a brief comparison of the evolution of processing capabilities of computers, we choose two studies from the 1980s, namely [PERSON] [133] and [PERSON] et al. [103], and two other studies dated after year 2000, namely [PERSON] et al. [48] and [PERSON] et al. [109]. Despite the greater complexity in the most recent studies, all are characterized by two-dimensional geometries and consider natural convection. The obtained outcomes are shown in Table 4.  
Table 4 shows that the performance of modern computers to solve phase change problems is 1-2 orders of magnitude higher than that of supercomputers in the late 1980s and at least three orders of magnitude higher than that of regular computers in the late 1980s. However, it is worth mentioning that the considered parameter is not precisely a processing capacity indicator, as many relevant factors in the studies are different, such as geometries, numerical configurations, solution methods, and the number of iterations per time step.  
The implementation of the numerical methods for solving melting and solidification problems in latent heat storage, as well as details about the accuracy, grid independence and validation codes  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Reference** & **n\({}_{\text{e}}\)** & **dt [s\({}^{-1}\)]** & **\(\eta\) [s\({}^{-1}\)]** \\ \hline Bennon and Incropera [133]b ** & 1344 & 90–180 & 44.1–88.2 \\ [PERSON] et al. [103]a  & 1764 & 4 & 1.4 \\ [PERSON] et al. [103]b  & 1764 & 773 & 288.6 \\ [PERSON] et al. [48]a  & 3520 & 4500–6000 & 4400–5866 \\ [PERSON] et al. [109]a  & 13565 & 2000 & 7536 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Performance indicators of numerical studies of phase change.  
using commercial software, as well as self-developed codes can be found in works [135; 136]. It is worth emphasizing that the in-lab-scale or full-scale heat exchangers have also been discussed in the literature (see [PERSON] et al., 2016 [137]).