*4.2. Exact Analytical Solution for Changing the Air Temperature at the Inlet to the Heat Accumulator in the Form of a Ramp*

Modelling of the heat accumulator was carried out using the explicit finite-difference method and the Crank–Nicolson method to test their suitability for determining the air and filling temperature distribution with time-dependent air temperature at the inlet to the accumulator. For this reason, the heat transfer in the accumulator was modelled under the particularly challenging boundary condition of a step change in temperature. Exact analytical solutions made it possible to assess the accuracy of the heat storage unit modelling using numerical methods for different time-dependent boundary conditions, as so far, there are no methods to assess the accuracy of numerical methods. There was no guarantee that both numerical methods used in the paper would give correct results.

The time–space temperature distribution during the flow of hot medium through the accumulator was determined. The initial temperature of the air and packing was constant and was equal to *T*0. At the beginning of the heating process, the air temperature increased in steps by Δ*Tg* (Figure 7). The air temperature then rose at a constant rate *vT* for the time *tcn* until the temperature of the medium reached *Tnom*. For time *tcn* ≤ *t,* the air temperature was constant and equal to *Tnom*.

**Figure 7.** Changes in gas temperature over time at the accumulator inlet (**a**), which were decomposed into three components according to the superposition method: *f* I(*t*) (**b**), *f* II(*t*) (**c**), and *f* III(*t* − *tcn*) (**d**).

The initial air and packing temperatures were assumed to be zero to simplify considerations. This assumption did not reduce the generality of the solutions obtained because the initial temperature *Tg*,0 had to be added to the packing and air temperatures obtained at the zero initial temperature.

The temperature distribution in the air and the pipe wall at the boundary condition shown in Figure 7a was determined using the superposition method [35]. According to this method, the air and packing temperatures were defined as the following:

$$T\_{\mathcal{S}}(\mathbf{x},t) = T\_1(\mathbf{x},t) + T\_2(\mathbf{x},t),\ 0 \le t \le t\_{cn} \tag{27}$$

$$T\_w(\mathbf{x}, t) = T\_{w1}(\mathbf{x}, t) + T\_{w2}(\mathbf{x}, t), \text{ 0 } \le t \le t\_{cm} \tag{28}$$

$$T\_{\mathcal{S}}(\mathbf{x},t) = T\_1(\mathbf{x},t) + T\_2(\mathbf{x},t) + T\_3(\mathbf{x},t - t\_{\text{cn}}), \ t\_{\text{cn}} \le t \tag{29}$$

$$T\_{\rm w}(\mathbf{x},t) = T\_{\rm w1}(\mathbf{x},t) + T\_{\rm w2}(\mathbf{x},t) + T\_{\rm w3}(\mathbf{x},t - t\_{\rm cm}), \ t\_{\rm cm} \le t \tag{30}$$

The boundary conditions for solutions *T*1, *T*2, and *T*3; i.e., the change in air temperature over the time at the accumulator inlet, are shown in Figure 7b–d, respectively.
