*5.3. Computational Tests*

Two cases of a change in inlet air temperature were considered. In the first case, the air temperature decreased stepwise from an initial temperature *T*<sup>0</sup> = 600 ◦C to an ambient temperature of 20 ◦C (Figure 9). In the second case, the initial air and packing temperature increased stepwise from initial temperature *T*<sup>0</sup> = 0 ◦C to 100 ◦C, then rose at a constant rate of 0.25 K/s and then remained constant after reaching 600 ◦C (Figure 10).

The results shown in Figures 9 and 10 were obtained for the following data calculated for the accumulator discussed in Section 2: *Ng* = 1.275, *τ<sup>w</sup>* = 1013.627 s, *τ<sup>g</sup>* = 0.3577 s, *hg* = 11.55 W/(m2·K), *<sup>N</sup>* = 20, and <sup>Δ</sup>*<sup>t</sup>* = 0.005. Figure 10 shows the results of the calculations obtained for the same data except for the number of nodes, which was (*N* + 1) = 65.

The analysis of the results shown in Figure 9 demonstrated that even with a small number of nodes along the length of the regenerator, the agreement between the air and packing temperatures at the outlet of the regenerator calculated using the exact analytical method and the explicit finite-difference method was very good.

**Figure 9.** Air and packing temperature variations at the accumulator outlet determined for a step change in the air temperature from 600 ◦C to 20 ◦C of the air temperature at the accumulator inlet (**a**); and the relative difference between the temperature determined using the exact analytical formula and the finite-difference method (**b**).

**Figure 10.** Air and packing temperature changes at the outlet of the accumulator; at the beginning, the air temperature increased in steps from 0 to 100 ◦C, then increased at a constant rate of 0.25 K/s, then remained stable after reaching 600 ◦C (**a**); and the relative difference between temperature determined using the exact analytical formula, the explicit finite-difference method, and the Crank–Nicolson method (**b**).

The relative values of the difference between the air temperature determined by the exact analytical formula and the finite-difference method were calculated using the following formula:

$$c\_{T, \%} = 100 \frac{T\_{\%}^{exact} \Big|\_{x = L\_r} - T\_{\%}^{num} \Big|\_{x = L\_r}}{T\_{\%}^{exact} \Big|\_{x = L\_r}} \text{ \%} \tag{69}$$

The relative difference was calculated analogously for the packing temperatures:

$$\left.c\_{T,w} = 100 \frac{T\_w^{\text{exact}}\big|\_{x=L\_r} - \left.T\_w^{\text{num}}\right|\_{x=L\_r}}{T\_w^{\text{exact}}\big|\_{x=L\_r}},\text{ \textquotedblleft}\_{\text{\textquotedblleft}}\right. \tag{70}$$

where *Texact <sup>g</sup>* and *Texact <sup>w</sup>* —air and packing temperatures calculated using an exact analytical method, respectively; and *Tnum <sup>g</sup>* and *Tnum <sup>w</sup>* —air and packing temperatures calculated using a numerical method, respectively.

The maximum value of *eT*,*<sup>g</sup>* = 2.3% occurred at time *t* = 5400 s and the maximum value of *eT*,*<sup>w</sup>* = 5.43% at time *t* = 5700 s (Figure 9b).

The relative differences between the exact analytical solution and the numerical solution were smaller for the second case analysed due to the larger number of nodes equal to (*N* + 1) = 65 and the use of the Crank–Nicolson method to solve the system of equations, which had a second order of accuracy; i.e., it was more accurate than the explicit finite-difference method, the order of accuracy of which was one.

The absolute maximum relative difference values were *eT*,*<sup>g</sup>* = 0.425% and *eT*,w = 1.20% for the explicit finite-difference method and *eT*,*<sup>g</sup>* = 0.424% and *eT*,w = 1.20% for the Crank– Nicolson method (Figure 10b).

From a comparison of the results shown in Figures 9b and 10b, it can be seen that the accuracy of calculations using the explicit finite-difference method increased as the number *N* of finite volumes increased. When increasing *N* from 20 to 64, the relative error *eT*,*<sup>g</sup>* for air decreased by 5.4 times and for packing *eT*,w by 4.5 times. When dividing the pipe into *N* = 64 finite volumes (Figure 10b), the Crank–Nicolson method gave similar results to the explicit finite-difference method due to a large number of finite volumes. The computational tests carried out demonstrated that both the explicit finite-difference method and the Crank–Nicolson method could be used to simulate the operation of the accumulator under analysis. The accuracy of such a simulation was significantly affected by the number of finite volumes, especially at low air velocity. In order to achieve sufficient calculation accuracy, it is recommended that *N* ≥ 50.
