3.2.1. Settings

According to [18], a cylindrical storage tank with a height of 800 mm and an internal diameter of 400 mm (resulting in a volume of 0.1 m3) was modeled. Three different flow rates (5 dm<sup>3</sup>/min, 10 dm<sup>3</sup>/min, and 15 dm<sup>3</sup>/min) at 15 ◦C were introduced—by means of a slotting-type inlet—at the bottom of the tank initially at 60 ◦C. The whole incoming flow rate exited the storage tank from the top. Li et al. [18] divided the water tank into eight layers of the same dimension fitted with one thermocouple each with a measurement time interval of 5 s. The first and the last thermocouples were located at 50 mm from the top and the bottom of the tank, respectively, whereas the intermediate ones are placed 100 mm apart (Figure 15). A summary of the measurement instrumentation is given in Table 3.


**Table 3.** Measurement devices used for the discharge experimental trial.

As the real tank was equipped with eight thermocouples, the model was set with a basic number of eight nodes. The simulation time was set equal to the unit replacement time, i.e., 1536 s, 768 s, and 512 s for a volume flow rate of 5 dm<sup>3</sup>/min, 10 dm<sup>3</sup>/min, and 15 dm<sup>3</sup>/min, respectively.

**Figure 15.** Schematic representation of the experimental tank for the discharge phase [10]. The cold-water inlet (blue arrow) and the hot water outlet (red arrow) are placed at the bottom and the top of the tank, respectively.

#### 3.2.2. Sensitivity Analysis

Similarly to what was done for charging (Section 3.1), here again simulations were conducted in the three more significant cases: for the reference number of nodes (N = 8), for twice the reference number of nodes (2·N) and for half the reference number of nodes (N/2). In addition, for each specific number of nodes, the three different flow rates mentioned in the settings paragraph (Section 3.2.1) were introduced alternatively.

Finally, the fifth node—representative of the height at which the fifth thermocouple is located—was chosen as the reference node and the 100 mm above and below were split into ten nodes each (Figure 16). That choice is based on the fact that the first and the last nodes—where the first and the last thermocouples are located—are not far enough from the edges of the tank to allow the corresponding node to be split.

At the beginning, the temperature evolution trend of the first, middle, and last node was plotted and compared to the experimental time-temperature evolution [18] during discharging simulation for 5 dm<sup>3</sup>/min (Figure 17), 10 dm<sup>3</sup>/min (Figure 18), and 15 dm<sup>3</sup>/min (Figure 19) volume flow rates.

**Figure 16.** Reference node (#5) splitting for the investigation of the influence that node volume plays on model accuracy (heights in millimeters).

**Figure 17.** Temperature evolution during the discharge phase—experimental vs. simulated—5 dm<sup>3</sup>/min volume flow rate.

**Figure 18.** Temperature evolution during the discharge phase—experimental vs. simulated—10 dm<sup>3</sup>/min volume flow rate.

**Figure 19.** Temperature evolution during the discharge phase—experimental vs. simulated—15 dm<sup>3</sup>/min volume flow rate.

Then, similarly to what has been done for charging, during discharging the calculated temperature evolution of the reference node after splitting has been plotted for the three simulations with different volume flow rates (see Section 3.2) and then compared to experimental data [18] and to the simulation with four, eight, and sixteen nodes (Figures 20–22).

A first analysis shows that unit replacement times obtained from all the simulations remain comparable to the experimental results.

It can be observed that an increase in the number of nodes brings the temperature evolution closer to the experimental data, especially for small volume flow rates. It should be noted that in this case the simulated storage is small (800 mm high and 400 mm in diameter) and node splitting does significantly improve the sixteen-node model, as the node number is already limited. This feature can be better appreciated by observing the green and the orange curves in Figures 20–22, which become closer and closer to each other as the incoming flow rate decreases.

To better investigate the effects of flow rate variations on temperature evolution, three different simulations were performed, each one keeping the number of nodes equal to that considered in the experimental model (N = 8).

**Figure 20.** Experimental temperature evolution at node #5 vs. simulation results—5 dm<sup>3</sup>/min volume flow rate—discharge phase.

**Figure 21.** Experimental temperature evolution at node #5 vs. simulation results—10 dm<sup>3</sup>/min volume flow rate—discharge phase.

**Figure 22.** Experimental temperature evolution at node #5 vs. simulation results—15 dm<sup>3</sup>/min volume flow rate—discharge phase.

When observing the temperature evolution of the tank for a flow rate of 5 dm<sup>3</sup>/min (Figure 17), the time temperature evolution for node #8 is comparable to the experimental data. The mismatch slightly increases as it continues toward the upper nodes. It is clear that node #1, being the furthest from the fluid inlet, has a slower response to heat exchange because it is the last one to interact with the cold inlet (then its dynamics is affected by all the other nodes with which the heat exchange takes place beforehand). That trend is reversed with an increase in the inlet flow rate (Figures 18 and 19) in the sense that the higher the flow rate the better the curve representing node #1 fits the experimental data and vice versa for those representing the lower nodes (nodes #5 and #8). As a matter of fact, by observing Figures 18 and 19 compared to the above-mentioned Figure 17, it can be noted that the dynamic behavior becomes faster and faster (as a greater mass of fluid exchanges heat with the water mass in the tank). The temperature at node #8 takes around 300 s for a 15 dm<sup>3</sup>/min volume flow rate, about 400 s and 800 s for a 10 dm<sup>3</sup>/min and 5 dm<sup>3</sup>/min volume flow rate, respectively, to reach the injected flow temperature (i.e., 15 ◦C).

Looking at the experimental curves, even after the unit replacement time (i.e., the time required to replace the whole water mass in the tank) has passed, the water temperature does not reach the temperature of the incoming flow (i.e., 15 ◦C). This behavior might be due to the fact that—given the experimental storage size—for high flow rates, part of the incoming fluid can be directed straight to the exit duct, and it has no time to exchange heat with the water in the tank. Thus, the water remains at a higher temperature (around 25 ◦C). In other words, a fraction of the inlet flow rate is bypassed, and its thermal energy is not stored in the tank.

The above-mentioned temperature behavior is not present in the results given by the simulations, as the approach followed to build up the model does not allow the flow bypass event to be considered (i.e., the entire incoming flow passes through all nodes without exchanging heat).

From these simulations it is apparent that the model results in terms of temperature changes inside the tank show deviations from the experimental data when the 3D effects become more and more significant (i.e., when the size of the tank becomes smaller and fluid velocity becomes higher). However, it should be noted that the proposed model can satisfactorily describe the physical behavior of the component within a complex energy grid, without unacceptable increases in the calculation burden.

This work points out further topics for the future development of the model. For high volume flow rates (compared to the size of the tank), the nodes furthest from the inlet section are less affected by the dynamics of the intermediate nodes. Moreover, the global heat exchange is faster, as a greater mass of fluid exchanges energy with the water in the tank.

Finally, referring to the number of nodes, it can be concluded that for small-size storage tanks the node-splitting technique does not significantly improve the accuracy of the temperature evolution inside the tank.

#### *3.3. Model Application*

In the last case study, the real daily operation of the upper region of a large atmospheric two-zone heat storage tank was investigated [38]. The aim of this last trial is to examine how the model behaves when representing the real operation of a large Thermal Energy Storage tank.
