*4.1. Exact Analytical Solution for a Step Change in the Air Temperature at the Heat Accumulator Inlet*

The temperature of the air flowing through the heat accumulator was a function of time and position. The operation of the heat accumulator was modelled when the air temperature at the inlet to the accumulator was stepped down (Figure 6).

**Figure 6.** Changes in air temperature at the accumulator inlet: (**a**) air temperature changes *Tg*; (**b**) changes in temperature difference *θg <sup>x</sup>*=<sup>0</sup> = *Tg <sup>x</sup>*=<sup>0</sup> − *T*0.

The differential equations describing the air and packing temperatures as a function of time and position were derived under the following simplifying assumptions:


Changes in the air temperature inside the accumulator were described by the conservation of energy Equation (5) and the packing temperature by Equation (8).

The system of Equations (5) and (8) was solved with the following boundary condition:

$$\left.T\_{\mathbb{S}}\right|\_{\mathbf{x}=\mathbf{0}} = T\_{\mathbb{S}^{\mathrm{in}}} \tag{13}$$

The initial temperature of the air and the accumulator packing was constant and equal to *T*0; i.e., the initial condition for air and wall has the following form:

$$\left.T\_{\mathcal{S}}\right|\_{t=0} = \left.T\_{\mathcal{S}}0\right|\tag{14}$$

$$\left.T\_w\right|\_{t=0} = T\_{\text{g.0}} \tag{15}$$

By introducing temperature differences:

$$
\theta\_{\mathcal{S}} = T\_{\mathcal{S}} - T\_{\mathcal{S},0} \tag{16}
$$

$$
\theta\_w = T\_w - T\_{\text{g.0}} \tag{17}
$$

Equation (5) becomes:

$$
\pi\_{\mathcal{S}} \frac{\partial \theta\_{\mathcal{S}}}{\partial t} + \frac{1}{N\_{\mathcal{S}}} \frac{\partial \theta\_{\mathcal{S}}}{\partial x^{+}} = - \left(\theta\_{\mathcal{S}} - \theta\_{\text{uv}}\right) \tag{18}
$$

The conservation of energy Equation (8) for accumulator packing can be written in the form

$$
\pi\_w \frac{\partial \theta\_w}{\partial t} = \theta\_\mathcal{g} - \theta\_w \tag{19}
$$

The boundary condition (13) and the initial conditions (14) and (15) can be written in the following manner (Figure 6):

$$\left.\theta\_{\mathcal{S}}\right|\_{x=0} = -\Delta T\_{\mathcal{S}} \tag{20}$$

$$\left.\theta\_{\S}\right|\_{t=0} = 0\tag{21}$$

$$\left.\theta\_w\right|\_{t=0} = 0\tag{22}$$

The analytical solution of the system of Equations (18) and (19) for a stepwise increase in the medium temperature by −Δ*Tg* and initial conditions (21) and (22) has the form [31]:

$$\frac{\theta\_{\mathcal{S}}}{\Delta T\_{\mathcal{S}}} = -\mathcal{U}(\xi,\eta)\exp[- (\xi+\eta)],\ t \ge t\_{pr} \tag{23}$$

$$\frac{\theta\_w}{\Delta T\_\circ} = -\left[\mathcal{U}(\tilde{\xi}, \eta) - \log\left(2\sqrt{\xi\eta}\right)\right] \exp[- (\tilde{\xi} + \eta)], \ t \ge t\_{pr} \tag{24}$$

where:

$$\xi = \frac{\mathbf{x} \mathbf{N}\_{\mathcal{S}}}{L\_{\mathbf{x}}} \ \eta = \frac{t - t\_{pr}}{\mathbf{r}\_{\mathcal{U}}} \ \mathbf{t}\_{pr} = \mathbf{x}^{+} \mathbf{N}\_{\mathcal{S}} \mathbf{r}\_{\mathcal{S}} \tag{25}$$

$$\mathcal{U}I = \mathcal{U}(\xi, \eta) = e^{\frac{\xi}{\xi} + \eta} - \sum\_{n=1}^{\infty} \left(\frac{\xi}{\eta}\right)^{\frac{\eta}{2}} I\_{\mathfrak{n}}\left(2\sqrt{\xi \eta}\right) \tag{26}$$

In Equations (23), (24) and (26), *I*0(*x*)—modified function of order zero *In*(*x*)—modified function of *n*-th order [34], and *tpr*—the time of the particle passage from the inlet (*x* = 0) to the point of *x* coordinate determined by the formula *tpr* = *x*/*wg*.
