**2. Design of the Heat Accumulator**

A laboratory test stand was built to test an electric–water heating system for a building. The air heated in the accumulator gave up heat to the water flowing in the finned heat exchanger (car radiator), which in turn was the heat source for the central heating system.

Figure 2a shows a hybrid electric–water building heating system using a dynamic discharge heat accumulator, while a photograph of the heating system test rig is shown in Figure 2b. The central element of the heating system was a ceramic dynamic discharge heat accumulator with an ordered packing that was heated by low-cost electricity. The following systems can be specified in the test rig:


The hybrid electric–water heating system for a building analysed in the paper is presented in Figure 2. The Heat accumulator (Figure 2a) as a heat source in a central heating system was heated by resistance heaters during periods of low electricity demand; i.e., at night or during midday hours when electricity was inexpensive. This type of heating

system is particularly cost-effective in countries where electricity is inexpensive; e.g., in countries with a high proportion of nuclear, hydroelectric, or wind power stations in the energy system.

**Figure 2.** Test rig for a hybrid electric–water heating system for a building: (**a**) flow system of the test stand; (**b**) general view of the test stand. , Ceramic bed heat accumulator; , air–water heater (plate-finned and tube heat exchanger); , reversal duct; , forced draft fan; , water-heating radiators; FI, flow rate meter; TI, pre-calibrated thermocouples.

In the heat storage unit , a dynamic heat discharge was used. On the outer surface, the accumulator was well thermally insulated. Heat was extracted from the high-temperature filling by flowing air that, after heating, was cooled in the PFTHE , which was composed of oval tubes with continuous fins. The capacity of the PFTHE was about 20 kW when the air velocity before entering the heat exchanger was approximately 3 m/s. The area of continuous fins in the heat exchanger was several times larger than the area of the plain tubes, so the PFTHE dimensions and its cost were small. The water temperature in the central heating system supply had to be maintained with a controller to ensure a constant temperature in the room. A centrifugal pump forced the water flow in the central heating system. The water temperature at the PFTHE outlet could be changed by varying the rotational speed of the fan or by changing the number of pump impeller revolutions. The centrifugal fan forced the air through the PFTHE.

The PFTHE was located at the outlet of the heat accumulator in the left lower part of the test stand. The fan forced air through the accumulator. A water flow meter was situated in the centre of the stand above the accumulator.

The water circuit in the hybrid heating system consisted of a PFTHE, a central heating system with plate radiators, and a water circulation pump. The PFTHE was an automotive radiator used in spark-ignition engines with a capacity of 1600 cm3. The air heated in the accumulator was used to heat the water in the PFTHE, which performed the function of a classic boiler in a central heating system. The purpose of plate radiators in the installation was to dissipate the heat absorbed by the air as it flowed through the accumulator. The panel radiators were located in an adjacent room to avoid heating the air where the test rig was located. The water temperature in the central heating radiators decreased by the same number of degrees. This ensured a stable internal temperature during the tests.

The hot water at the outlet of the lamella heat exchanger was the feed water for the central heating system. The water temperature in the central heating radiators dropped by the same amount as it heated up in the PFTHE. This ensured that the entire hybrid system operated under steady-state conditions. The water flow in the central heating system was regulated by changing the rotational speed of the circulation pump via a current frequency converter. A centrifugal fan with a drive motor of 1.5 kW forced the air through the heat accumulator. The maximum volumetric capacity of the fan was 660 m3/h and the maximum air pressure was 4.2 kPa.

The heat accumulator with dynamic discharge was the main component in the hybrid heating system. The principle of the heat accumulator operation was based on the use of clean energy such as electricity, including electricity from renewable energy sources.

The heat storage unit consisted of two basic components: the outer casing and the core (packing). The outer casing was a steel tube made of heat-resistant steel. The cylindrical shape of the casing provided advantages that are not present in traditional accumulation heaters. The air flow through the circular cross-section of the battery was more uniform compared to the rectangular cross-section of the housing.

The filling of the accumulator consisted of cylindrical ceramic elements heated during the night using inexpensive electric energy. A view of the accumulator packing is illustrated in Figure 3a, and the dimensions of the outer shell and regenerator components are shown in Figure 3b.

Eight steel tubes (*4*) with an external diameter of 101.6 mm and a wall thickness of 3.6 mm were inside the outer casing (*3*) (Figure 3). Three electric heaters (*2*) were placed in each tube. Seven rows of cylinders (*1*) formed by cylinders of an equal diameter and height of 30 mm were inside each of the steel tubes along their entire lengths. The ceramic cylinders were made of heat-resistant corundum concrete. The packing of the accumulator then had an arranged structure. The air flowed longitudinally around the ceramic elements inside the steel tubes. An additional air stream flowed between the steel tubes on the shell side. Air enters the inter-tube space through 14 holes (*6*) that were 20 mm in diameter (Figure 3b). The front view of the accumulator illustrating the distribution of the ceramic elements in the steel tubes with visible connections for resistance heaters is presented in Figure 4c.

During the day, air flowed through the heat accumulator, which was pre-heated by the accumulator packing. Internal energy stored in the heated packing could be used directly for space heating or via an air–water heat exchanger to heat water in the central heating system.

The density and specific heat of the ceramic cylinders were high. As a result, the heat capacity of the accumulator also was high, and the heat stored in the accumulator was large, which in turn made it possible to heat a house with a large space. The thermal conductivity of cylinders was also significant. The thermal stresses in the cylindrical ceramic elements were therefore low, avoiding scratches and cracks.

**Figure 3.** Ceramic heat accumulator: (**a**) view of the accumulator packing; (**b**) outer shell and regenerator components. 1—ceramic cylinders, 2—resistance electric heaters, 3—accumulator outer casing, 4—steel tubes (inside of which there are seven rows of ceramic cylinders), 5—a bottom with openings through which air flowed into the inter-tube space, 6—the air intake opening to the inter-tube space.

**Figure 4.** Heat accumulator with dynamic discharge: (**a**) view of cylindrical filling elements; (**b**) external shell of the accumulator with visible steel tubes, inside of which the ceramic cylinders are located; (**c**) front view of the accumulator illustrating the distribution of the ceramic filling elements in the steel tubes with visible connections for resistance heaters.

#### **3. Mathematical Model of a Heat Accumulator**

A mathematical model of a heat accumulator was developed using the following assumptions:


The first assumption was commonly made in the literature on thermal modelling of heat accumulators [1]. The modelled heat storage unit met the requirements to consider the filling as an object with lumped thermal capacity. The filling element consisted of cylindrical elements and steel pipes. A body can be treated as an object of lumped heat capacity when the Biot number (Bi) for a given element is less than 0.1. A more detailed explanation of when a body can be treated as an object with a concentrated heat capacity can be found in the books by Kreith [32] and Taler and Duda [33]. In the case of the heat accumulator analysed in this paper, the maximum value of the Biot number (Bi*w* = *hgdc*/(2*kw*)) was less than 0.01, where *dc* is the outer diameter of the ceramic cylinder, *hg* is the heat transfer coefficient on the outer cylinder surface, and *kw* is the ceramic cylinder thermal conductivity. In addition, for steel tubes with a wall thickness *sw* equal to 3.6 mm located inside the accumulator, the Biot number (Bi), while taking into account that the tube was cooled on the external and internal surface, was Bi*<sup>w</sup>* = *hgsw*/(2*km*), where *km* is the thermal conductivity of the alloyed steel of which the tubes were made. The maximum Biot number (Bi*w*) shall not exceed 0.0005. Longitudinal heat conduction in exchanger walls is of secondary importance and is usually neglected [32]. This is due to the small temperature gradient of the air along its flow path. An additional factor that hinders heat conduction in cylindrical elements in the axial direction is the contact resistance at the interface between adjacent cylindrical elements. There were 67 ceramic cylinders arranged in a line along the length of the accumulator.

Figure 5 shows a circular heat storage unit, where *Dw* is the internal diameter of the accumulator and *Lr* is its length. The ordered accumulator packing consists of ceramic cylinders of diameter *dc* and height *Hc* (Figure 5).

**Figure 5.** Heat accumulator with a finite volume of a Δ*x* thickness.

The energy conservation equation for air (gas) for a control volume of thickness Δ*x* (Figure 5) has the following form:

$$\dot{m}\_{\mathcal{S}} \varepsilon\_{\rm pg} \left| \,^{\rm T\_{\mathcal{S}}}\_{0} T\_{\mathcal{S}} \right|\_{x} + hA\_{\rm puck} \frac{\Delta x}{L\_{r}} \left( T\_{\dot{w}} \big|\_{x + \frac{\Delta \tilde{w}}{2}} - T\_{\mathcal{S}} \big|\_{x + \frac{\Delta \tilde{w}}{2}} \right) = \dot{m}\_{\mathcal{S}} \varepsilon\_{\rm pg} \left| \,^{\rm T\_{\mathcal{S}}}\_{0} T\_{\mathcal{S}} \right|\_{x + \Delta x} + p\_{r} A\_{w} \Delta x \rho\_{\mathcal{S}} c\_{p\mathcal{S}} \frac{\partial T\_{\mathcal{S}}}{\partial t} \Big|\_{x + \frac{\Delta \tilde{w}}{2}} \tag{1}$$

where . *mg*—air mass flow rate, kg/s; *Aw* = *πD*<sup>2</sup> *<sup>w</sup>*/4—cross-section area of the packing, m2; *Dw*—inner diameter of the accumulator casing, m; *cpg Tg* <sup>0</sup> —the mean specific heat of air at constant pressure in the temperature range from 0 ◦C to *Tg*, J/(kg·K); *Tg*—air temperature, ◦C; *Tw*—packing temperature, ◦C; *<sup>h</sup>*—heat transfer coefficient, W/(m2·K); *Apack*—the surface area of ceramic and steel elements where heat exchange with air takes place, m2; *pr* = *Vg*/*Vc*—porosity; *Vg*—air volume in the accumulator, m3; *Vc*—the total capacity of the accumulator, m3; Δ*x*—the thickness of the control area, m; ρ*g*—air density, kg/m3; *Lr*—accumulator length, m; *cpg*(*Tg*)—specific heat of air at constant pressure at temperature *Tg*, J/(kg·K); and *t*—time, s.

Introducing the mean specific heat in the control volume over the temperature interval from *Tg <sup>x</sup>* to *Tg <sup>x</sup>*+Δ*<sup>x</sup>* is defined as:

$$c\_{p\%}(T) = \frac{c\_{p\%}|\_{0}^{T\_{\mathcal{S}}}T\_{\mathcal{S}}|\_{x+\Delta x} - c\_{p\%}|\_{0}^{T\_{\mathcal{S}}}T\_{\mathcal{S}}|\_{x}}{T\_{\mathcal{S}}|\_{x+\Delta x} - T\_{\mathcal{S}}|\_{x}}\tag{2}$$

which gives:

$$\dot{m}\_{\mathcal{S}}c\_{\mathcal{P}\mathcal{S}}(T)\left(\left.T\_{\mathcal{S}}\right|\_{x+\Delta x} - \left.T\_{\mathcal{S}}\right|\_{x}\right)\frac{1}{\Delta x} + p\_{r}A\_{w}\rho\_{\mathcal{S}}c\_{p\mathcal{S}}\frac{\partial T\_{\mathcal{S}}}{\partial t} = hA\_{pack}\frac{1}{L\_{r}}\left(\left.T\_{w}\right|\_{x+\frac{\Delta x}{2}} - \left.T\_{\mathcal{S}}\right|\_{x+\frac{\Delta x}{2}}\right) \tag{3}$$

where *cpg*(*T*) is the specific heat of the air at constant pressure at temperature *T*.

When Δ*x* → 0, then Equation (3) takes the form:

$$
\dot{m}\_{\mathcal{S}}c\_{p\mathcal{S}}(T)L\_r \frac{\partial T\_{\mathcal{S}}}{\partial \mathbf{x}} + p\_r A\_w L\_r \rho\_{\mathcal{S}} c\_{p\mathcal{S}}(T) \frac{\partial T\_{\mathcal{S}}}{\partial t} = hA\_{p\text{ack}} \left(T\_w - T\_{\mathcal{S}}\right) \tag{4}
$$

By introducing a dimensionless coordinate *x*<sup>+</sup> = *x*/*Lr*, Equation (4) can be written as:

$$\frac{1}{N\_{\mathcal{S}}} \frac{\partial T\_{\mathcal{S}}}{\partial x^{+}} + \tau\_{\mathcal{S}} \frac{\partial T\_{\mathcal{S}}}{\partial t} = T\_w - T\_{\mathcal{S}} \tag{5}$$

The number of units *Ng* and the time constant *τ<sup>g</sup>* are defined by the following expressions:

$$N\_{\mathcal{S}} = \frac{hA\_{pack}}{\dot{m}\_{\mathcal{K}}c\_{p\mathcal{K}}} \ \ \ \tau\_{\mathcal{S}} = \frac{m\_{\mathcal{K}}c\_{p\mathcal{S}}}{hA\_{pack}} \tag{6}$$

where *mg* = *prAwLrρg*(*Tg*)—mass of air in the accumulator, kg; *pr*—packing porosity; *ρg*(*Tg*) = *ρgn Tg* + 273.15 /273.15; *ρgn*—air density in standard temperature and pressure conditions, kg/m3; *<sup>ρ</sup>g*(*Tg*)—air density at mean air temperature *Tg*, kg/m3; and . *mg*—mass flow rate of the air flowing through the accumulator, kg/s.

Next, the differential equation for the packing was derived. The conservation energy equation for the packing is:

$$
\hbar A\_{\rm pack} \frac{\Delta \mathbf{x}}{L\_{\rm r}} \left( \left. T\_{\rm w} \right|\_{\mathbf{x} + \frac{\Delta \mathbf{x}}{2}} - \left. T\_{\mathcal{S}} \right|\_{\mathbf{x} + \frac{\Delta \mathbf{x}}{2}} \right) + (m\_{\rm m} c\_{\rm m} + m\_{\rm c} c\_{\rm c}) \frac{\Delta \mathbf{x}}{L\_{\rm r}} \frac{\partial T\_{\rm w}}{\partial t} = 0 \tag{7}
$$

where *Tw*—packing temperature, ◦C; *mm*—the mass of the steel structural components inside the accumulator including the mass of the accumulator casing, kg; *cm*—stainless steel specific heat, J/(kg·K); *mc*—the mass of the ceramic packing elements, kg; and *cc*—specific heat of packing elements, J/(kg·K).

Transformation of Equation (7) to dimensionless form gives:

$$
\pi\_w \frac{\partial T\_w}{\partial t} = -\left(T\_w - T\_\S\right) \tag{8}
$$

where *τ<sup>w</sup>* is given by the following relation:

$$
\pi\_w = \frac{m\_{\rm m}c\_{\rm m} + m\_{\rm c}c\_{\rm c}}{hA\_{\rm puck}} \tag{9}
$$

Equations (5) and (8) describe the changes in air and packing temperature as a function of time.

The boundary condition for air and the initial temperatures for air and packing have the following forms:

$$\left.T\_{\mathcal{S}}\right|\_{\mathbf{x}^+=\mathbf{0}} = \left.T'\_{\mathcal{S}}(t)\right|\tag{10}$$

$$\left.T\_{\mathcal{S}}\right|\_{t=0} = T\_0 \tag{11}$$

$$T\_w|\_{t=0} = T\_0 \tag{12}$$

where *T <sup>g</sup>*(*t*)—the air temperature at the accumulator inlet, ◦C; and *T*0—initial packing and air temperature, ◦C.

#### **4. Modelling the Operation of a Heat Accumulator Using Exact Analytical Methods**

Solutions of Equations (5) and (8) with the boundary condition (10) and initial conditions (11) and (12) can only be found for specific boundary and initial conditions when using exact analytical methods. In the general case, when the air temperature at the inlet to the accumulator is a function of time, it is not easy to find an exact analytical solution. In this paper, solutions to two problems are presented. In the first problem, the initial temperature of the packing and the air was uniformly equal to *Tg,*0. The air temperature at the inlet to the accumulator decreased stepwise by the value Δ*Tg*. In the second problem, the initial temperature of the packing and air was also equal to *T*0. The air temperature at the inlet to the accumulator increased first at a constant rate *vT* and then equalled the set nominal temperature *T*gnom. Exact analytical solutions were used to assess the accuracy of the numerical solutions obtained using the finite-difference method.
