## 2 Exergy Analysis of Cascaded Phase Change Heat Storage
### Analysis of Exergy Efficiency of Thermal Fluid  
In the ideal case, the exergy completely converted by the heat flow of the thermal fluid in the cascaded phase change heat storage device can be expressed as [28]:  
\[Ex_{\max}=\int_{T_{0}}^{T_{\text{in}}}\dot{inc}_{P}\bigg{(}1-\frac{T_{0}}{T} \bigg{)}\text{d}T=\dot{mic}_{P}(T_{\text{in}}-T_{0}-T)[T_{\text{in}}-T_{0}-T_{0 }\text{Ln}(T_{\text{in}}/T_{0})] \tag{3}\]  
The ratio of the exergy \(Ex\) of the thermal fluid after heat exchange by the device to the exergy \(Ex_{\max}\) of the complete conversion of the heat flow is defined as the exergy efficiency, which is expressed as [28]:  
\[\begin{split}&\eta_{n}=\frac{Ex}{Ex_{\max}}=\frac{\big{(}T_{ \text{in}}-T_{\text{opt},n,1}-\Delta T_{1}\big{)}\big{(}1-\frac{T_{0}}{T_{ \text{opt},n,1}}\big{)}}{T_{\text{in}}-T_{0}-T_{0}\text{Ln}(T_{\text{in}}/T_{ 0})}+\\ &\frac{\sum_{i=2}^{n}\big{[}\big{(}T_{\text{opt},n,i-1}-\Delta T _{i-1}\big{)}-\big{(}T_{\text{opt},n,i}+\Delta T_{i}\big{)}\big{]}\big{(}1- \frac{T_{0}}{T_{\text{opt},n,i}}\big{)}}{T_{\text{in}}-T_{0}-T_{0}\text{Ln}( T_{\text{in}}/T_{0})}\end{split} \tag{4}\]  
When \(T_{0}\) = 298.15 K and \(T_{\text{in}}\) = 328.15 K, the difference between the air fluid temperature at the outlet of the dehumidifier and the phase change temperature of the PCMs in each stage \(\Delta T\) is 0, 5 and 10 K, respectively, and the stage number of PCMs is taken as \(n\) = 1, 2, 3 and 4 separately; the exergy efficiency of the thermal fluid \(\eta_{n}\) is calculated and listed in Figure 2.  
It can be seen from Figure 2 that, with the number of PCM stages increasing, the exergy efficiency of the thermal fluid increases significantly; however, the relationship between them is not linear. When \(\Delta T\) is 5 K, and \(n\) is equal to 1 and 4, respectively, the exergy efficiency of the phase change heat storage \(\eta_{n}\) is 35.6% and 56.4%, respectively. This means the exergy efficiency can be significantly improved when PCMs are arranged in cascades, and the \(\eta_{n}\) has nothing to do with the physical properties of PCMs.  
When \(n\) is constant, it can be found that, with the increase of \(\Delta T\), \(\eta_{n}\) becomes lower. Considering that the heat transfer process needs a certain temperature difference as the heat transfer driving force, it is recommended to use \(\Delta T\) = (3-5) K in the actual device design.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{\(\Delta\)_T/K_} & \multirow{2}{*}{**Stage Number \(i\)**} & \multicolumn{4}{c}{\(T_{\text{opt},n,i}\)/K} \\ \cline{3-6}  & & \(n\) = 1 & \(n\) = 2 & \(n\) = 3 & \(n\) = 4 \\ \hline \multirow{4}{*}{0} & 1 & 312.79 & 317.83 & 320.38 & 321.92 \\  & 2 & — & 307.80 & 312.79 & 315.80 \\  & 3 & — & — & 305.38 & 309.81 \\  & 4 & — & — & — & 303.90 \\ \hline \multirow{4}{*}{5} & 1 & 310.40 & 314.59 & 316.71 & 317.99 \\  & 2 & — & 306.26 & 310.40 & 312.91 \\  & 3 & — & — & 304.21 & 307.91 \\  & 4 & — & — & — & 302.99 \\ \hline \multirow{4}{*}{10} & 1 & 307.99 & 311.34 & 313.03 & 314.05 \\  & 2 & — & 304.67 & 307.99 & 309.99 \\ \cline{1-1}  & 3 & — & — & 303.03 & 305.99 \\ \cline{1-1}  & 4 & — & — & — & 302.05 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Optimum phase change temperature in each stage of the cascaded phase change heat storage device.