## 3 Model Validation  
The first TES analysis was performed in a single tank of chosen geometry, fill, and fluid with thermodynamic properties shown in Table 1 and described by Reference [36]. One of the advantages of having the governing equations reduced to their dimensionless form is that by giving the values of two dimensionless parameters, all that is necessary about the problem is known. In the simulation, an analysis of 10 charge and discharge cycles was required to stabilize the initial conditions so that the data comparison was made with the tenth cycle in the developed code. This is because in the first cycle, the behavior corresponds to a fully loaded tank, and as it progresses up to a certain number of cycles, the final and initial temperature distributions remain constant. Subsequently, Figure 1 plots the temperature curves in the tank axis for five discharge operation times. The dimensionless temperature gradient of the HTF is shown during 2.5 h of discharge with intervals of 0.5 h, according to the results of the numerical model solved in MATLAB R2018b based on the parameters in Table 1. Then, the experimental data from [36] were transformed into a dimensionless form in the same graph. This process references thermocline TES performance test results reported in the literature [36] to validate current modeling work. Experimental measurements used eutectic molten salts (NaNO\({}_{3}\)-KNO\({}_{3}\), 50% by 50%) as heat transfer fluid, quartzite rocks, and silica sands as filler material. Finally, it can be seen that the model consistently fits the experimental data. It should be noted that the global simulation of the process requires an analysis of 10 loads and unload cycles to stabilize the initial conditions in the developed code. Therefore, the data comparison was made with the tenth cycle. Due to this, in the first cycle, the behavior corresponds to a fully loaded tank, and as it progresses up to a certain number of cycles, the final and initial temperature distributions remain constant. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Value** & **Unit** \\ \hline Capacity & 2.3 & [MWh] \\ Working Fluid & Solar Molten Salt & - \\ Solid fill material & Quartzite rock and silica sand & - \\ Insulating material & Fiberglass & - \\ Insulating layer & 23 & [cm] \\ Discharge time & 2.5 & [h] \\ Tank height & 6.1 & [m] \\ Tank diameter & 3 & [m] \\ Volume & 42 & [m\({}^{3}\)] \\ Porosity & 0.22 & - \\ Particle diameter & 0.0191 & [m] \\ Mass flow & 5.46 & [kg/s] \\ \(T_{H}\) & 396 & [\({}^{\circ}\)C] \\ \(T_{C}\) & 289 & [\({}^{\circ}\)C] \\ Time Interval & 0.5 & [h] \\ Configuration & Porous bed & - \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Operating parameters of the experimental thermocline tank from Reference [36]. It should be noted that the differences found were explained by the considerations of the model that distance it from the actual operating conditions of the experiments. Furthermore, due to the uncertainty in experimental measurements and material properties, the agreement between the experimental data and the model prediction is believed to be quite good. It strongly validates the current modeling and its numerical solution method. However, to adopt a more conservative analysis of the comparison, six statistical parameters were calculated to validate the results obtained. Firstly, the Mean Absolute Error (\(MAE\)) is used to determine how similar the experimental data are to those calculated by the model. \[MAE=\frac{1}{n}\sum_{i=1}^{n}\abs{\Theta_{exp}-\Theta_{mn}}, \tag{14}\]  
where \(\Theta_{mn}\) is the dimensionless temperature of the numerical model, and \(\Theta_{exp}\) corresponds to the dimensionless experimental temperature. Subsequently, the Root-Mean-Square Error (\(RMSE\)) measurement was used. A smaller magnitude of \(RMSE\) shows a better predictive capacity of a model in terms of its absolute deviation.