4.2.3. Solution T3(x, t) for Air Temperature

The differential equations describing the changes in air temperature and packing have the following forms:

$$
\pi\_{\mathcal{S}} \frac{\partial T\_3}{\partial t} + \frac{1}{N\_{\mathcal{S}}} \frac{\partial T\_3}{\partial \mathbf{x}^+} = -(T\_3 - T\_w) \tag{46}
$$

$$
\tau\_w \frac{\partial T\_{w3}}{\partial t} = T\_3 - T\_{w3} \tag{47}
$$

The air temperature *T*<sup>3</sup> at the inlet to the accumulator increases linearly with time and the initial air and packing temperatures are equal to zero. The boundary condition and initial conditions take the form:

$$\left.T\_3\right|\_{x=0} = f\_\Pi(t - t\_{cn}) = -\nu\_T(t - t\_{cn}), \; t \ge t\_{cn} \tag{48}$$

$$|T\_3|\_{t=0} = 0\tag{49}$$

$$T\_{\mathfrak{w}\mathfrak{Z}}|\_{t=0} = 0 \tag{50}$$

where *fII*(*t*) = *vT*(*t* − *tcn*), *tcn* ≤ *t*. The symbol *vT* is the rate of change of the air temperature at the inlet to the accumulator (Figure 7d).

$$T\_3 = -\upsilon\_T \tau\_{\rm tr} \left\{ e^{-\left(\frac{\tau}{\xi} + \eta\_1\right)} \left[ (\eta\_1 - \xi)lI + \tilde{\xi}l\_0 \left( 2\sqrt{\xi\eta\_1} \right) + \sqrt{\xi\eta\_1}l\_1 \left( 2\sqrt{\xi\eta\_1} \right) \right] \right\}, \ t \ge t\_{\rm cr} \tag{51}$$

$$T\_{\rm n3} = T\_3 + \upsilon\_T \pi\_{\rm tr} e^{-(\xi + \eta\_1)} \left[ \mathcal{U} - I\_0 \left( 2\sqrt{\xi \eta\_1} \right) \right], \ t \ge t\_{\rm cn} \tag{52}$$

where *ξ* = *xNg Lx* , *<sup>η</sup>* <sup>=</sup> *<sup>t</sup>*−*tpr <sup>τ</sup><sup>w</sup>* , *<sup>η</sup>*<sup>1</sup> <sup>=</sup> *<sup>t</sup>*−*tcn*−*tpr <sup>τ</sup><sup>w</sup>* , and *tpr* <sup>=</sup> *<sup>x</sup>*+*Ngτg*.

The number of heat transfer units *Ng*, the time constant *τg*, and the wall time constant *τ<sup>w</sup>* are defined by Equations (6) and (9), respectively.

#### **5. Numerical Modelling of a Heat Accumulator**

The differential equations describing the air and packing temperature changes were solved using the finite-difference method. Two finite difference schemes were used: the explicit Euler method and the Crank–Nicolson method.

The accuracy of the explicit Euler method is of the first order [36], and that of the Crank–Nicolson method [36] is the second order; i.e., the latter method is more accurate with the same number of nodes in the difference mesh. The advantage of these methods was that they could consider that the thermophysical properties of the packing and air were temperature dependent. The air temperature at the inlet to the accumulator could change over time. With the numerical methods, the accumulator air and packing temperature distribution could be determined using any temporal variation of the air temperature at its inlet. Unlike the exact analytical method, the finite difference procedure did not require the boundary condition to be a step or linear change in the air temperature.

Cooling of the accumulator filling, during which heat was extracted by the flowing air and then transferred to the finned air–water heat exchanger, was modelled using two methods: the explicit Euler method and the implicit Crank–Nicolson method. To assess the accuracy of these methods, an exact analytical solution was also found for selected temporal variations in the air temperature at the accumulator inlet. The mutual comparison of the two different numerical methods and their comparison with the exact analytical solution was necessary because, as shown in [14], the accuracy of the solutions obtained by the finite-difference (finite volume) method depends very strongly on the number of divisions of the accumulator length into calculation cells. The air temperature over the length of one cell was approximated using the arithmetic mean of the inlet and outlet temperatures for the cell. However, it turned out that the temperature distribution inside a given cell was exponential, and at low air velocities, the air temperature dropped quickly over the short length from the air inlet to the cell. In this case, the arithmetic mean did not give the true average temperature over the length of a single cell. Computational tests of air heating in an accumulator [14] have shown that when the length of the accumulator was divided into 10 control volumes, inaccurate—one might say absurd—results were obtained. For example, at an accumulator filling temperature of 300 ◦C, the air temperature at the heat accumulator outlet reached 500 ◦C, which was obviously not possible. For this reason, the assessment of the accuracy of the two numerical methods used was given more attention.
