## 2 Thermochemical Energy Storage Design
### Experimental Procedure
The thermal power produced by the reactor R1, the heat recovered by the air-to-air heat exchanger HX2 and the heating rate of the water tank U1 are determined as follows:  
\[\dot{Q}_{[i]}(t)=\dot{m}_{[i]}C_{\mathrm{P},[i]}\Delta T_{[i]}(t). \tag{1}\]  
where \(\Delta T_{[i]}(t)\) represents the variation of the fluid temperature difference recorded at the \([i]\) component's boundaries; \([i]=\{r,a,w\}\) refers to the fluid that circulates through the reactor R1, the air-to-air heat exchanger HX2 or the water tank in the respective order. The heating rate of the water tank was determined from the thermal power on the cold side of the air-to-water heat exchanger HX1. The hydration thermal power \(\dot{Q}_{h}(t)\) is identical to \(\dot{Q}_{r}(t)\) in the stationary regime and was determined from the air humidity measurements:  
\[\dot{Q}_{h}(t)=\dot{m}_{r}\cdot\Delta w_{r}(t)\cdot\Delta H_{\mathrm{s}}. \tag{2}\]  
With \(\Delta w_{r}(t)\) is the variation of the air absolute humidity at the reactor boundaries. The amount of thermal energy generated by the reactor and charged to the water tank:  
\[E_{[i]}=\int\limits_{t_{0}}^{t_{max}}\dot{Q}_{[i]}(t)\mathrm{d}t,\;\;i=\{r,w\}. \tag{3}\]  
The dynamic effectiveness of the air-to-air or air-to-water heat exchanger was evaluated as given below:  
\[\epsilon_{[i]}(t)=\frac{\dot{Q}_{[i]}(t)}{\left\{\dot{m}_{[i]}C_{\mathrm{P},[ i]}\right\}_{min}\cdot\Delta T_{[i],max}(t)},\;\;i=\{a,w\}. \tag{4}\]  
where \(\left\{\dot{m}_{[i]}C_{\mathrm{P},[i]}\right\}_{min}\) determines the minimum product of the \([i]\) fluid mass flow rate and the associated heat capacity \(C_{\mathrm{P},[i]}\), and \(\Delta T_{[i],max}\) is the maximum possible temperature difference on the concerned heat exchanger. The energy storage density of the material can be referred to the energy prognosis either by the reactor, or by the water storage tank. It was calculated with the following formula:  
\[\Delta E_{[i]}=\rho_{\mathrm{s}}\frac{E_{[i]}}{m_{s,max}},\;\;i=\{r,w\}. \tag{5}\]  
where \(\rho_{\mathrm{s}}\) represents the packing density of anhydrous material and \(m_{s,max}\) is the total anhydrous mass processed by the reactor during the experimental time \([t_{0},t_{max}]\). For the fixed packed bed \(m_{s,max}\) represents only the anhydrous bed mass \(m_{s}\), while for the moving bed test it represents the quantity of anhydrous material spent on the time frame \([t_{0},t_{max}]\). Considering the energy prognoses by both reactor and the water storage tank, the discharging efficiency was also introduced as the installation performance indicator:  
\[\epsilon_{d}=\frac{\Delta E_{w}}{\Delta E_{r}}. \tag{6}\]  
It has to be noted, that the listed above performance indicators (1-6) depend on the heat losses along the ducts and pipework, the heat capacities of the used materials and the ratio between \(\dot{m}_{[i]}\)C\({}_{\text{P},[i]}\) products. The water mass uptake for the fixed packed bed was determined at each experimental time step \(t_{j}\in[t_{0},t_{max}]\) by the following formula:  
\[x(t)=\frac{1}{m_{s}}\int\limits_{t_{0}}^{t_{j}}\dot{x}_{s}(t)\text{d}t. \tag{7}\]  
With \(\dot{x}_{s}(t)\) is the sorption rate in kg/s determined from the air humidity measurements as \(\dot{x}_{s}(t)=\dot{m}_{r}\cdot\Delta w_{r}(t)\). The water mass uptake for the moving bed was estimated by solving the equation of the boundary problem:  
\[x(t)=x_{0},\ z=0; \tag{8}\] Figure 4:0  
With \(u_{s}\) is the estimated averaged solid velocity and \(m_{s}\) is the anhydrous bed mass. For simplicity reasons, this mass was taken as a constant.