## 4 Simulation Results  
The solution method employs an iterative numerical approach implemented in FORTRAN, which refines the results through successive approximations until convergence is achieved. These results are then made accessible in Excel via the \"Get Data From Text/CSV\" function, which read the data from a.csv file, thus allowing the construction of a thermodynamic model used to evaluate the energy efficiency of the TES system. The whole algorithm is graphically described by the flowchart depicted in Figure 5. The model offers the possibility of carrying out a comparative study between various working pairs used in different working conditions. The working conditions considered refer to three different possible applications of the TES system: buildings, transport and industry. Each application is characterized by different boundary conditions expressed in terms of different temperature levels, as reported in Table 1.  
The boundary conditions were selected based on typical operating temperatures in buildings, transport, and industrial applications, ensuring practical relevance and compatibility with real-world thermal energy storage systems. These values align with established literature and industry standards while considering the feasibility of adsorption-based TES integration. The selected values provide a representative yet flexible framework that captures the performance of the TES system across multiple sectors while remaining within experimentally validated temperature ranges. The study aimed to balance generalizability with sector-specific relevance, ensuring that the findings could be applied to practical scenarios. Further refinement of these boundary conditions could be considered in future work based on additional experimental data and emerging industry trends.  
Several working pairs were selected, consisting of different adsorbent materials and three refrigerants: methanol, ethanol and water. Not all working pairs can be used for the three applications considered. In fact, for some working pairs the boundary conditions identify uptake values outside the valid range. This happens especially for industrial applications, where the working conditions are extreme for many solutions. Table 2 shows the working pairs considered acceptable for each application (for practical reasons, each pair is associated with an identification code).  
The next paragraphs summarize the main results obtained for the three selected applications. For each of these, the thermodynamic model is used to simulate two operating conditions:  
* Short-term storage (t = 0);
* Long-term storage (t = \(\infty\));  
As described by the formulas presented in the previous section, the amount of energy stored and made available by the TES system also depends on the amount of metal mass  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Buildings** & **Transport** & **Industry** \\ \hline \(T_{cup}\) (\({}^{\circ}\)C) & 15 & 10 & 50 \\ \(T_{cond}\) (\({}^{\circ}\)C) & 40 & 40 & 60 \\ \(T_{1}\) (\({}^{\circ}\)C) & 40 & 40 & 60 \\ \(T_{3}\) (\({}^{\circ}\)C) & 90 & 90 & 190 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Boundary conditions of each application.  
Figure 5: Block diagram of the computational program.  
present in the adsorbent bed. To take into account this dependence, the following parameter, the metal/adsorbent mass ratio, is introduced:  
\[\phi=\frac{m_{met}}{m_{als}} \tag{19}\]  
For each case considered, the storage efficiency is calculated for different values assumed by \(\phi\) (i.e., \(\eta\)0 for \(\phi\) = 0, \(\eta\)1 for \(\phi\) = 0.5, \(\eta\)2 for \(\phi\) = 1, \(\eta\)3 for \(\phi\) = 2 and \(\eta\)4 for \(\phi\) = 3).