3.3.1. Settings

Unlike the previous cases, the tank in question is divided into two regions by an insulated intermediate floor. A vertical open-ended pipe serves as a hydraulic connection between the upper and the lower zone: it compensates the water density changes in the lower region and it prevents the whole tank from under-pressure or over-pressure. Furthermore, the intermediate floor—acting like an obstacle to natural convection—allows hot water in the lower zone to be stored at higher pressure if compared to an ordinary TES tank.

Since the simulation of a two-zone heat storage tank falls outside the scope of this work, only the upper region of the tank has been modeled: according to [38], it is a cylindrical tank with a height of 30 m and an internal diameter of 20 m (resulting in a volume of 9420 m3). A constant volume flow of 50 m<sup>3</sup>/<sup>h</sup> at 95 ◦C was introduced throughout the day by means of a radial di ffuser placed at the top of the tank. The same inverse mass flow exited the storage tank from the bottom outlet. In order to evaluate the thermal stratification, the tank was equipped with a distributed temperature sensing (DTS) measurement system (resolution of about 0.1 ◦C) and 28 vertically aligned PT100 sensors.

#### 3.3.2. Simulation Results

The tank model is made up of 25 evenly spaced nodes. Initial conditions were set on the basis of the information given in [38]. For ease of comparison, the water temperature daily evolution was recorded for seven specific tank heights, at four-hour intervals. The simulation was performed by means of the Matlab ® ode45 variable-step solver, which took 0.71 s to simulate the daily operation of the storage tank, with a standard laptop.

The simulation result and absolute error are shown in Table 4, together with the experimental temperatures mentioned in the reference article.

The simulated temperature profile is in good accordance with the experimental data both for the top and the bottom of the considered region; the average absolute error is equal to 1 ◦C. It should be observed that a slight increase in the absolute error occurs over time at the half-height of the tank region; it is probably due to the mixing of chilled water—entering the upper zone at the temperature of about 62 ◦C—from the compensation pipe, as reported in [38]. Since the interaction between the two zones of the storage tank is not considered in the proposed model, the simulated temperature gradients seem to be smaller than the experimental one.

Even considering this issue, the model has proved to be performant in the representation of the real operation of a large heat storage tank: the maximum absolute error does not exceed 6 ◦C.


Note: Exp. = experimental; Sim. = simulated; Abs. = absolute error.

**Table 4.** Comparative table of experimental and simulated daily temperature evolution inside the TES tank.

## **4. Discussion**

In this paper, a new Matlab ®/Simulink ® model for the simulation of stratified sensible heat storage systems was presented. The 1D model was built using the multi-node approach, solving the volume and energy balance equations for each node. Because of its innovative structure, the model is highly customizable in node number and dimension, enabling detailed investigation of the thermal stratification phenomenon.

Three experimental datasets from the literature were used as references for the model analysis and validation, by considering the charge phase, the discharge phase, and the nominal operation of heat storage tanks with di fferent sizes.

In the case of the charge phase, the tank—initially at a low temperature—was heated up with the injection of hot-water at the top. This simulation was performed three times, by varying the number of nodes while maintaining all the other parameter values (such as tank dimension, inlet and outlet flow rates, and initial temperature of the stored water).

In the case of the discharge phase, the tank—initially at a warm temperature—was cooled down with the injection of cold-water at the bottom. In this case, both the number of nodes and the inlet flow rates were varied throughout the simulations, while maintaining the other parameter values. For a better comparison of the results, an appropriate point was chosen as reference; it corresponds to the location of the fifth thermocouple in the experimental system, it is far enough from the edges of the tank and it was matched with a specific node in every simulation.

In the real operation case, the tank was fed with hot water from the top radial di ffuser. The daily temperature evolution was simulated by means of a model made up of 25 evenly spaced nodes. The simulated and experimental data were compared in order to evaluate the model performance when dealing with large storage tanks, commonly used in district heating applications.

The comparison between simulated and experimental data confirms that the choice of the number of nodes plays a significant role in the representation accuracy of thermal stratification inside the storage tank: an increase in the number of nodes—which, for a given tank capacity, corresponds to a decrease in node volume—improves the simulation results (i.e., temperature variations in the tank) allowing more accurate temperature gradients.

The model proposed in this paper proved to be able to give an accurate physical representation of stratification and heat exchange phenomena in sensible heat storage systems and can be a useful tool to reliably simulate temperature changes in stratified storage tanks. Its innovative features are flexibility and adaptability, which make it possible to choose the number and dimensions of each node in the model, allowing the user to focus the simulation on a specific zone of interest.

However, the model shows limitations for some specific storage configurations. If the ratio between the inlet flow rate and the node volume is too small, the thermal stratification dynamics slows down and the simulation becomes inaccurate; the larger the node volume, the greater the mass of water contained and the higher the mixing e ffect that dampens the incoming flow temperature. Furthermore, the simplified representation of convection can lead to discrepancies between the real and the simulated temperature evolutions, as is the case for the models reported in the literature. Another limitation was detected in small storage tank simulations; for high flow rates and large temperature di fferences between the incoming and the stored fluid, a fraction of the incoming fluid may flow directly to the exit duct without exchanging heat with the stored water. In other words, part of the fluid is bypassed, and its thermal energy is not stored. The developed model is not able to reproduce this phenomenon as—due to the 1D approach followed—all the incoming flows pass through all the nodes.

It should be recalled that the stratified storage model was developed to become part of a library of physics-based components for the dynamic simulation of district heating networks. In these applications the involved storage tanks are large and the temperature di fferences between the incoming and stored water temperature are usually fairly low, and therefore the above-mentioned drawbacks of the proposed 1D model are acceptable, compared to the advantage of keeping low calculation times.

**Author Contributions:** Conceptualization, A.G., M.M. and M.R.; funding acquisition and supervision, A.G. and M.M.; investigation, validation and writing, N.C. and A.D.L.

**Funding:** This work was supported by the "Efficity—Efficient Energy Systems for Smart Urban Districts" project (CUP E38I16000130007) co-funded by the Emilia-Romagna Region through the European Regional Development fund 2014–2020.

**Conflicts of Interest:** The authors declare no conflict of interest.
