3.1.1. Settings

According to [17], a cylindrical storage tank with a height equal to 1800 mm, an internal diameter equal to 800 mm, and a resulting volume of 0.9 m<sup>3</sup> was considered. The side walls, the top, and the bottom of the tank were insulated with a 50-mm-thick layer of fiberglass, with thermal conductivity equal to 0.043 W/<sup>m</sup>·K. The initial temperature of the water was set to 20 ◦C for the whole tank and a constant volume flow rate of 16 dm<sup>3</sup>/min at 52 ◦C was introduced at the top of the tank, with a conventional inlet elbow. A summary of the measurement devices is given in Table 2.


**Table 2.** Measurement devices used for the charge experimental trial.

González-Altozano et al. [17] placed twelve thermocouples uniformly spaced along the vertical axis of the tank, located 150 mm apart and 75 mm from both the top and the bottom of the tank, as represented in Figure 5. Therefore, N = 12 was set as the reference number for the nodes in the model.

The simulation time has been fixed at 4073 s, equal to the time declared by González-Altozano et al. [17] to replace 120% of the total storage tank volume.

#### 3.1.2. Sensitivity Analysis

Simulations have been conducted in three significant cases: for the reference number of nodes (N = 12), for twice the reference number of nodes (2·N) and for half the reference number of nodes (N/2). The trend of the temperature evolution against time during charging simulation for a storage tank model with 6 (Figure 6), 12 (Figure 7), and 24 nodes (Figure 8) has been recorded and compared to the experimental [17] time-temperature evolution.

**Figure 5.** Schematic representation of the experimental tank for the charge phase [9]. The hot water inlet (red arrow) and the cold-water outlet (blue arrow) are placed at the top and the bottom of the tank, respectively.

**Figure 6.** Temperature evolution during the charge phase—experimental vs. simulated (6 nodes).

As a first remark, it may be noted that simulated and experimental values of replacement time (i.e., the times required to replace the total water mass in the tank) are comparable. The results show that when the number of nodes N is increased, the thermal dynamics speeds up and the replacement time decreases toward the experimental value. Specifically, the dynamics of the upper nodes becomes faster whereas the dynamics related to the middle and the lower nodes slows down slightly in the first part—until reaching approximately 40 ◦C—and then increases, giving a closer match to the overall time-temperature evolution of the experiment. This latter phenomenon may be due to the thermal inertia of the upper nodes. This feature could be better appreciated in Figure 9, where for the sake of readability only the comparison of the first, middle, and last node of the three different models is shown, based on the number of nodes.

**Figure 7.** Temperature evolution during the charge phase—experimental vs. simulated (12 nodes).

**Figure 8.** Temperature evolution during the charge phase—experimental vs. simulated (24 nodes).

Thus, on the one hand, an increase in the number of nodes leads to faster thermal dynamics and to an improvement in accuracy of the replacement time. However, on the other hand, this highlights a specific behavior of the model. In fact, the larger the volume of a node (i.e., the lower the total node number N), the larger the mass of water inside the node and the more significant the damping effect of the incoming flow temperature. This is the case of the simulation with 6 nodes where the temperature in the first node seems to better fit the experimental temperature. This may be due to the upper zone of the storage being turbulently mixed. In fact, the temperature measured by the three thermocouples at the top are almost superimposed. On the contrary, a small number of nodes would be less beneficial when large incoming flow rates (i.e., high speed of the incoming water) are involved and there are no mixing devices—i.e., diffusers—at the inlet. In this case the incoming fluid would flow very quickly through the storage directly to the outgoing zone and without exchanging thermal energy with the crossed nodes. This operating condition cannot be described by the model because of the 1D approach used.

Thus, since the number of nodes N plays a significant role, some nodes have been split to better point out the influence that node volume plays on model accuracy. Three nodes—representative of the heights where the corresponding thermocouples are located—were chosen as reference nodes (i.e., the third, sixth, and ninth) for this purpose. Each of them was split into ten nodes, as represented in Figure 10.

The temperature evolution of each reference node after splitting was recorded and compared to those related to the experimental data and simulated results from the models with six, twelve, and twenty-four nodes (Figures 11–13). The results show that by decreasing the volume of nodes and focusing the attention on a specific zone by refining the local 1D "mesh," the calculated temperature evolution matches the measurements better and better, especially for nodes belonging to the upper part of the storage tank. This is due to the fact that the lower nodes are negatively influenced by the dynamics of the upper ones.

**Figure 9.** Temperature evolution during the charge phase from models with a different number of nodes.

**Figure 10.** Reference node splitting for the investigation of the influence that node volume plays on model accuracy (heights in millimeters).

In conclusion, the choice of the number of nodes N determines the resolution with which the vertical temperature distribution can be modeled in the storage tank. In fact, an increase in the number of nodes will allow significant temperature gradients to be modeled more accurately (Figure 14). Thanks to versatility and accurate physical representation of stratification and heat exchanges, it seems that the model can be a useful simulation tool for the reliable prediction of temperature evolution in a stratified storage tank during charge operation.

**Figure 11.** Experimental temperature evolution during the charge phase at node #3 vs. simulation results.

**Figure 12.** Experimental temperature evolution during the charge phase at node #6 vs. simulation results.

**Figure 13.** Experimental temperature evolution during the charge phase at node #9 vs. simulation results.

**Figure 14.** Temperature stratification (during the charge phase) inside the storage tank at three different simulation stages.

#### *3.2. Model Analysis—Discharge Phase*

Thereafter, the discharge maneuver of a thermal energy storage tank was simulated. The tank, initially at high temperature, was cooled down with the introduction of a cold-water flow. The effects of the node number and flow rate variations on temperature evolution were investigated.
