2.3.1. Thermal Convection

Based on the initial temperature *T*ini and final temperature *T*fin of the thermal storage, the thermal convection from bricks into the isolation layer is given by Equation (3). The thermal convection equation is conveniently expressed by Newton's law of cooling [23].

$$
\phi\_{\rm conv} = h\_{\rm conv} \left( T\_{\rm fin} - T\_{\rm ini} \right) A\_{\rm cha} = N\_u \frac{\lambda}{D} \left( T\_{\rm fin} - T\_{\rm ini} \right) A\_{\rm cha} \tag{3}
$$

where *h*conv is the surface heat transfer coefficient of the bricks, *A*cha is the surface area of the round holes in the bricks as shown in Figure 3. Many bricks are superimposed to form the core frame of the thermal storage. Two heat bricks are stacked together to form a circular hole in the center of the bricks as shown in the zoom figure. Electric heating wires are placed in the middle of the hole. *L*, *W*, *H*, and *D* are the length, width, height, and diameter of the brick, respectively. *A*cha is the area of hot airflow channel which is stacked by two bricks together, *A*cha = *L* × *D* × *N*, where *N* is the number of bricks in a row. Based on the Nusselt–Number *Nu*, the heat transfer coefficient *h*conv can be calculated by the Zukauskas formula [24]. *λ* is the thermal conductivity.

**Figure 3.** Schematic diagram of bricks placement.

In heat convection, multiple round holes can be approximately regarded as the tube bundle. The average heat transfer performance of the tube bundle is related to the flow Reynolds-number (*Re*) [24].

$$R\_{\mathfrak{C}} = \mathfrak{u}\_{\mathfrak{A}} \times \mathrm{D} / \vtharpoonright\_{T\_{\text{uvv}}} \tag{4}$$

where *u*<sup>a</sup> is the hot air flow rate, *D* is bricks round hole diameter, and *νT*ave is the aerodynamic viscosity at the temperature of *T*ave = *T*fin/2. According to the range of *Re*, the Nusselt-number (*Nu*) can be calculated by Zukauskas formula in Equation (5).

$$\begin{cases} \begin{aligned} \mathcal{N}\_{\boldsymbol{u}} &= 0.52 R\_{\varepsilon}^{0.5} P\_{r\_{\boldsymbol{u}}}^{0.36} P\_{r\_{\boldsymbol{c}}} & \text{if } 10^{2} < R\_{\varepsilon} \le 10^{3} \\\ N\_{\boldsymbol{u}} &= 0.27 R\_{\varepsilon}^{0.63} P\_{r\_{\boldsymbol{c}}}^{0.36} P\_{r\_{\boldsymbol{c}}} & \text{if } 10^{3} < R\_{\varepsilon} \le 20^{5} \\\ N\_{\boldsymbol{u}} &= 0.033 R\_{\varepsilon}^{0.8} P\_{r\_{\boldsymbol{u}}}^{0.36} P\_{r\_{\boldsymbol{c}}} & \text{if } 20^{5} < R\_{\varepsilon} \le 20^{6} \end{aligned} \end{cases} \tag{5}$$

where *Pr*<sup>a</sup> is the Prandtl-number (*PN*) of the fluid average temperature, *PN* reflects the contrast between momentum diffusion and thermal diffusion in fluid. *Pr*<sup>c</sup> is a constant. *Pr*<sup>c</sup> = *Pr*a/*Pr*bri0.25, *Pr*bri is the *PN* of the bricks average surface temperature. According to the temperature *T*int and *T*fin, the *Pr*<sup>a</sup> and *Pr*bri can be queried.

When 10<sup>2</sup> < *Re* ≤ 206, the corresponding *Nu* can be selected. In most cases, the calculation results satisfy the condition that 10<sup>3</sup> < *Re* ≤ 205. This paper uses *Nu* = 0.27*R*0.63 *<sup>e</sup> P*0.36 *<sup>r</sup>*<sup>g</sup> *Pr*<sup>c</sup> as an example. Substituting Equation (5) into Equation (3), the thermal energy *φ*conv is obtained in Equation (6).

$$
\phi\_{\rm conv} = 0.27 R\_c^{0.63} P\_{r\_\rm g}^{0.36} P\_{r\_\rm c} \frac{\lambda}{D} \left( T\_{\rm fin} - T\_{\rm lin} \right) A\_{\rm cha}.\tag{6}
$$

#### 2.3.2. Thermal Radiation

The thermal energy *φ*rad released from the bricks to the flow channel hot air, is expressed by Equation (7), which is based on the Stefan–Boltzmann law [23].

$$
\phi\_{\rm rad} = \varepsilon \sigma \left( (T\_{\rm fin} + 273)^4 - (T\_{\rm fini} + 273)^4 \right) A\_{\rm bri} \tag{7}
$$

where *ε* is the emissivity of the bricks' surface, *σ* is the blackbody radiation constant (Stefan–Boltzmann constant is 5.67 <sup>×</sup> <sup>10</sup>−<sup>8</sup> W/m<sup>2</sup> K4), and *A*bri is the heat transfer surface area of the bricks.
