**1. Introduction**

In recent years, there has been increasing interest in both heat and electricity storage. This is mainly due to the time-varying power generated by renewable energy sources such as wind and photovoltaic farms. Ceramic-filled heat accumulators or electrically heated hot water storage tanks can utilise surplus electrical energy from photovoltaics and wind farms, which will improve the controllability of the entire energy system.

Water, solid materials, and phase-change materials (PCMs) are most commonly used as thermal storage materials [1]. Large water heat storage tanks with hot and cold water stratification are widely used in urban central heat supply systems. The paper [2] presents CFD modelling of a water heat accumulator with water temperature stratification used in a large municipal thermal power plant comprising coal-fired grate boilers and natural gas-fired internal combustion engines. Pressurised hot water storage tanks are also used to improve the flexibility of thermal power units. One way of managing excess electricity at night; i.e., during times of reduced electricity demand, is through the accumulation of hot water in pressurised storage tanks. The heat flow rate supplied to the power plant with the fuel is partly used during the night period to heat up the hot water in the accumulators, resulting in a reduction in the electrical power generated. The boiler in this situation can operate above the technical minimum without burning heavy fuel oil in the chamber. In the case of boilers fired with pulverised coal, when the boiler load is below the technical minimum, pulverised coal and fuel oil are burnt in the boiler simultaneously to avoid the pulverised coal burners being extinguished. Thanks to these tanks, it is

**Citation:** Taler, D.; Taler, J.; Sobota, T.; Tokarczyk, J. Cooling Modelling of an Electrically Heated Ceramic Heat Accumulator. *Energies* **2022**, *15*, 6085. https://doi.org/10.3390/en15166085

Academic Editors: Artur Bartosik and Dariusz Asendrych

Received: 26 July 2022 Accepted: 19 August 2022 Published: 22 August 2022

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**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

possible to increase the maximum load of the power unit by a few percent and decrease the value of the minimum acceptable load of the power unit by dozens of percent [3]. A concept of a nuclear plant with improved flexibility was described in [4]. Water tanks and borehole thermal energy-storage systems (BTES) can be used for thermal energy storage for district heating systems utilising waste heat from data centres [5]. The application of water-to-water heat pumps in a solar system for space heating and domestic hot water preparation in a cold climate was presented by Pinamonti et al. [6]. A seasonal water storage tank for a single-family house was used to store the excess heat. The long-term storage tank was completely buried under the ground near the house. Electric water heaters affected the operation of the entire energy system. The flexibility potential of electric water heaters was strongly influenced by the daily hot water demand [7]. Hot water storage tanks can be used to capture the excess electricity generated by photovoltaic cells [8]. A system consisting of photovoltaic cells and an electric water heater can essentially meet a household's electricity and heat requirements for heating a building [9]. A novel hot water storage unit with low thermal inertia was proposed for floor heating and domestic hot water preparation using solar collectors. The proposed solution ensured stable system operation even after sunset [10].

The use of a proportional water flow controller in a hot water preparation system with a hot water reservoir was presented in [11]. The effect of thermometer inertia on temperature control quality in an electrically heated hot water reservoir was investigated in [12]. The correct operation of the temperature control in the hot water storage tank was strongly influenced by random temperature measurement errors. The accidental errors in the thermometer indications could result in the faulty operation of the PID controller, since the water temperature in the tank could rise steadily despite exceeding the set temperature [13].

Solid-fill accumulators are often used for heating buildings and domestic hot water preparation. The filling of the accumulator can be either structured or random. A ceramic filling resistant to high temperatures in the order of 800 ◦C is used in the designed battery. As a result, a large amount of heat can be stored in the lightweight heat storage unit, which is used to heat the building. The use of fireproof ceramic cylinders in the proposed accumulator is an inexpensive and safe solution. The accumulator filling can be made of other materials; e.g., magnetite, as it has a high specific gravity and high heat capacity. In the general case, high-density, high-heat-capacity, high-temperature-resistant materials are the materials from which the filling can be made. Due to the high temperature of the accumulator packing when charged, materials such as concrete or stones are unsuitable due to their low resistance to thermal stress cracking. The ceramic elements are arranged in an orderly manner so that the porosity of the filling can be reduced and the pressure loss when air flows through the accumulator is reduced. This results in a reduction in the energy consumed by a fan pushing air through the accumulator.

A numerical model of a channel heat accumulator heated by hot air was presented in [14]. Computer simulations showed that the number of finite elements along the duct length had to be large to ensure satisfactory accuracy of the air temperature calculations at a low air velocity.

The standard approach in heat accumulator modelling, in which the average temperature of the flowing gas along the length of one cell is approximated by the arithmetic mean of the inlet and outlet temperatures of the cell, can lead to substantial errors, especially for small mass flow rates of the gas [14]. It is necessary to divide the accumulator model into a considerable number of cells; e.g., into 50 or more, to achieve a numerical solution of the system of differential equations for an accumulator that does not change with the increasing number of cells and agrees well with the experiment. This requirement applies to the finite-difference method, including both the explicit Euler method and the Crank–Nicolson method. Furthermore, in contrast to heat conduction issues, the stability and accuracy of modelling heat accumulators with different numerical methods have not been sufficiently investigated in the literature.

A hybrid finite volume method–finite element method was used to simulate the operation of the accumulator with an organised structure [15]. The results of the calculations carried out in [15] were compared with the measurements. A heat storage unit made of steel or concrete cylindrical elements was modelled using the COMSOL Multiphysics 4.3a software [16]. The operation of a honeycomb ceramic heat accumulator was simulated using a one-dimensional model in the object-oriented modeling language Modelica [17]. A simple transient floor heating model was presented in [18]. The solution of the onedimensional transient heat conduction was solved using a Laplace transform. The storage capacity of the floor and the energy transferred from the floor to the heated space were estimated. A resistance-heated energy storage unit made of firebricks was studied in [19]. The thermal performance of the industrial heat accumulator was studied numerically using the finite volume method in conjunction with the Crank–Nicolson method. The paper [20] presented the concept of a graphite heat accumulator. The heat was accumulated in a graphite block in which steel water tubes were installed. The advantage of such an accumulator over this type of concrete accumulator was the higher operating temperature of the graphite block and lower thermal stresses in the graphite block due to the high value of the graphite thermal conductivity of about 88 W/(m·K). A combined heat and compressed air energy-storage system using a packed bed accumulator and an electrical heater was proposed in [21]. The thermal oil was resistively heated to a high temperature, stored in a reservoir, and then directed to an interstage air superheater located between the air turbine's high- and low-pressure parts. An air-based high-temperature heat and power storage system that cogenerated heat and electricity with high efficiency was presented in [22]. During the charging process, the packed bed of stones was heated up to 685 ◦C. The effect of the thermal conductivity of the solid particles that made up the filling of a heat accumulator was analysed in [23].

In the last few decades, there has been a lot of research on the application of phasechange materials (PCMs) for heat storage in technical systems. Numerical modelling of the combined sensible–latent heat-storage unit with a periodic flow used in concentrated solar power plants was carried out in [24]. Solid (GF/Sn-Bi composite) and PCM (GF/MgCl2 composite) were used in numerical simulations of the proposed solution. A significant problem in the broader use of PCM-based heat storage is the high thermal inertia. Usually, a small area close to the components to be heated or cooled; e.g., pipes or flat surfaces, is used. A large part of the tank does not undergo a phase change in either the heating or cooling phase. The appropriate tank structure and fin arrangement [25] or metal foam inside the storage tank [26] can improve the heat performance of thermal storage systems. A heat accumulator filled entirely with rocks and an accumulator filled in the upper part with PCM capsules and in the lower part with rocks was simulated numerically and studied experimentally in [27]. The performance of the heat accumulator during charging and discharging processes, as well as its cyclic operation, were investigated. Sodium nitrate (NaNO3), which has a melting point of 306 ◦C, was used as the PCM [28]. The PCM was placed between two steel plates. To improve the thermal performance of the multi-slab phase-change thermal energy storage block, 0.14 mm-thick fins were used inside the PCM. The effect of the position of the PCM layer on the temperature distribution in the wall and on heat loss to the surroundings was investigated in [29]. The PCM layer was placed under a layer of plaster near the interior of the building or on the exterior surface of the brick wall and covered with a layer of extruded polystyrene. The variability of electricity production from wind farms causes major disruption to electricity and heat networks. Distributed electric heating storage units were used to capture excess electricity generated at wind farms [30]. To improve solar energy utilisation and the stability of solar heating systems, an air-type solar collector with heat accumulation was designed [31]. Phase-change material was placed inside the solar vacuum tube to reduce the impact of solar radiation fluctuations on indoor heating.

This paper presents a mathematical model for the cooling of a ceramic electric accumulator with an ordered structure. Conventional heat accumulators are inconvenient to

use. The disadvantages of traditional electric heat accumulators for space heating are the high casing temperature and the significant decrease in the heat flow rate given off by the accumulator with time. Dynamic discharge heat storage units that are placed inside rooms have the disadvantages of fan noise and dust deposits on the room walls. The high-temperature accumulator analysed in this paper can be used in replacement of a conventional solid-fuel boiler or oil- or gas-fired boilers that are located in the boiler room. By locating the accumulator and the finned air–water heat exchanger in a separate room; e.g., the previous boiler room, they are not troublesome for the room occupants in the heated building. The central heating system does not have to be replaced. The coupling element between the heat accumulator and the central heating system is a finned air–water heat exchanger. The accumulator filling is heated by low-cost electricity at low loads in the power system or excess electricity from photovoltaic cells. The term "low-cost electricity" refers to the price of 1 kWh of electricity at night, when electricity demand is low, which is much lower than the price of 1 kWh during peak load times, which typically occur in the evening.

The new idea of replacing traditional coal-, oil-, or gas-fired central heating boilers with heat storage units is presented in this paper. Accumulators heated using inexpensive electricity will help to reduce the danger of carbon monoxide poisoning or explosions present in traditional heating systems. In addition, carbon dioxide emissions and lowparticulate-matter emissions will be reduced compared to traditional home heating systems.

Solid-fill heat accumulators are typically used for direct space heating. The electrically heated storage unit analysed in this paper can be used as a heat source in a hydronic central heating system for a building. In this way, solid-fuel-, oil-, or gas-fired boilers can be eliminated from existing building heating systems. The electrically heated accumulators can be used to capture excess electricity from wind farms. In summer, the heat stored in the accumulator can be used to heat domestic hot water. An example of using a heat accumulator to heat a building is shown in Figure 1.

**Figure 1.** Use a heat accumulator heated with low-cost electricity to heat a building or provide domestic hot water. 1—thermal power plant, 2—wind farm, 3—power network, 4—solid heat accumulator, 5—electric resistance heater, 6—fan, 7—building, 8—finned air–water heat exchanger, 9—hydronic central heating system, 10—circulation pump.

During the period of overproduction of electricity in the system, which usually occurs during the night, the heat storage unit is heated using electricity. During the night period, the demand for electricity is low. Thermal power plants operate with the minimum allowable load, as it is not worthwhile to shut them down due to the very high restart costs. In addition, the wind velocity during the night period is usually higher than during the day, resulting in increased output from wind farms. Due to the excess of electricity during the night period when electricity demand is low, the electricity supply is greater than the electricity demand. This results in an extreme reduction in the electricity price per kWh at night. One way to utilise this excess electricity is through heat accumulators with a fixed packing, which can be heated to a high temperature; e.g., up to 600 ◦C. When electricity prices rise due to increased demand during the daytime period, the heat accumulator is not supplied with electricity. During the day, the accumulator is discharged using the air flowing through it. The accumulator can be located in a heated room, or it can replace a

conventional fossil fuel-fired boiler situated in the boiler room. The latter solution is very beneficial for historic city centres, where there are many old buildings without access to a central heating network. A favourable alternative to replacing a masonry heater (tiled stove) or other heat sources is a designed accumulator that does not emit harmful substances. By heating the accumulator using inexpensive electricity in times of low demand, the heating costs are competitive with other techniques to heat buildings. A central heating system in which an accumulator has replaced a conventional gas-fired boiler is shown in Figure 1. The accumulator, which is heated up during the night, is cooled down by flowing air that is then cooled down in a finned air–water heat exchanger. This heat exchanger works as a central heating boiler. The proposed solution is simple and inexpensive.

This paper developed a new simple numerical model of the heat storage unit. A new method for determining the correlation for the heat transfer coefficient from the packing surface to the flowing air was proposed using the least-squares method. The accuracy of the numerical accumulator models was assessed by comparing the air and packing temperatures at the accumulator outlet with those obtained using an analytical exact solution that was obtained using the superposition method. A numerical mathematical model of the accumulator based on the explicit Euler method and the Crank–Nicolson method was verified by exact solutions for a step change in air temperature at the inlet to the accumulator and a ramp change in inlet air temperature. The verification of numerical solutions is of great importance in the case of heat accumulators. If the number of finite volumes is too low, the accuracy of the numerical solutions can frequently be unsatisfactory [14]. Compared to heat conduction, the number of finite volumes along the length of the accumulator needs to be greater or several times higher to achieve satisfactory accuracy of the accumulator modelling. The computer calculation time using the developed model, especially using the Crank–Nicolson method, was very short. This made it possible to apply the developed numerical model in an automatic air temperature control system. In the predictive air temperature control, the proposed mathematical model of the heat storage unit could be used to determine the fan rotational speed so that the air temperature at its outlet was equal to the set temperature. In this way, despite the discharging of the accumulator during the day; i.e., a decrease in the packing temperature, it was possible to maintain the set temperature in the heated room by changing the airflow velocity.
