## 2 Theoretical Analysis of Air Refrigeration Cycles
Another factor affecting system performance is the expansion ratio \(Er\), or the compression ratio. As the expansion ratio increases, the enthalpy drop in the expander increases, leading to an increase in the cooling capacity of the system. However, the raised expansion ratio also increases the power consumption, and the value should be optimized to obtain the maximum COP of the cycle. By taking the partial derivative of COP with respect to \(Er^{(k-1)/k}\), the optimal \(Er\) can be obtained by making the derivative equal to zero. It can be found that the optimal \(Er\) satisfies Equation (7):  
\[Er^{\frac{k-1}{k}}\text{Closedcycle, opt}=\frac{\eta_{T}\frac{T_{\text{AC}}}{T_{ \text{CI}}}+\sqrt{\Big{(}\eta_{T}\frac{T_{\text{AC}}}{T_{\text{CI}}}\Big{)}^{2 }-\Big{(}1-\frac{T_{\text{AC}}}{T_{\text{CI}}}+\eta_{T}\frac{T_{\text{AC}}}{T_ {\text{CI}}}\Big{)}\Big{[}\eta_{T}\frac{T_{\text{AC}}}{T_{\text{CI}}}+\eta_{ C}\eta_{T}\eta_{M}\frac{T_{\text{AC}}}{T_{\text{CI}}}\Big{(}\frac{T_{\text{AC}}}{T_ {\text{CI}}}-1\Big{)}\Big{]}}}{1-\frac{T_{\text{AC}}}{T_{\text{CI}}}+\eta_{T} \frac{T_{\text{AC}}}{T_{\text{CI}}}} \tag{7}\]  
Similarly, the optimal expansion ratios of the open-loop low-pressure and high-pressure cycles can be obtained by Equations (8) and (9):\[Er_{\text{TCcycle, opt}}=\left[\frac{\eta_{T}\frac{T_{\text{O}}}{T_{\text{CI}}}+ \sqrt{\left(\eta_{T}\frac{T_{\text{O}}}{T_{\text{CI}}}\right)^{2}-\left(1-\frac{ T_{\text{O}}}{T_{\text{CI}}}+\eta_{T}\frac{T_{\text{O}}}{T_{\text{CI}}}\right) \left[\eta_{T}\frac{T_{\text{O}}}{T_{\text{CI}}}+\eta_{T}\eta_{M}\frac{T_{ \text{O}}}{T_{\text{CI}}}\left(\frac{T_{\text{O}}}{T_{\text{CI}}}-1\right) \right]}}{1-\frac{T_{\text{O}}}{T_{\text{CI}}}+\eta_{T}\frac{T_{\text{O}}}{T_{ \text{CI}}}}\right]^{\frac{k}{k-1}} \tag{8}\]  
\[Er_{\text{TCcycle, opt}}=\left[\frac{\eta_{T}\frac{T_{\text{AC}}}{T_{\text{I}}}+ \sqrt{\left(\eta_{T}\frac{T_{\text{AC}}}{T_{\text{I}}}\right)^{2}-\left(1- \frac{T_{\text{AC}}}{T_{\text{I}}}+\eta_{T}\frac{T_{\text{AC}}}{T_{\text{I}} }\right)\left[\eta_{T}\frac{T_{\text{AC}}}{T_{\text{I}}}+\eta_{C}\eta_{T}\eta _{M}\frac{T_{\text{AC}}}{T_{\text{I}}}\left(\frac{T_{\text{AC}}}{T_{\text{I}} }-1\right)\right]}}{1-\frac{T_{\text{AC}}}{T_{\text{I}}}+\eta_{T}\frac{T_{ \text{AC}}}{T_{\text{I}}}}\right]^{\frac{k}{k-1}} \tag{9}\]  
When \(T_{\text{I}}=300.15\) K, \(T_{\text{O}}=308.15\) K, \(T_{\text{AC}}=313.15\) K, \(T_{\text{CI}}=295.15\) K, \(\eta_{\text{T}}=0.8\), \(\eta_{\text{C}}=0.75\), \(\eta_{\text{M}}=0.99\), it can be obtained that \(Er_{\text{Closed cycle, opt}}=2.27\), \(\text{COP}_{\text{Closed cycle, max}}=0.659\). For the open-loop low-pressure cycle, it can be obtained that \(Er_{\text{TC cycle, opt}}=1.96\), \(\text{COP}_{\text{TC cycle, max}}=0.739\). For the open-loop high-pressure cycle it can be obtained that \(Er_{\text{TC cycle, opt}}=1.94\), \(\text{COP}_{\text{CT cycle, max}}=0.743\). Due to the temperature difference in the heat exchanger, \(T_{\text{CI}}\) is lower than the indoor temperature \(T_{\text{I}}\), and \(T_{\text{AC}}\) is higher than the outdoor temperature \(T_{\text{O}}\), leading to a lower COP of the closed cycle than that of the open-loop cycles. As the discharge pressure and temperature for one-stage compression is relatively high, a two-stage compression with intercooling can be employed to reduce the compression power. By integrating the expander with one of the compressors, the matching characteristics of the aerodynamic performance can be improved while the expansion work is recovered. Therefore, the TC cycle can be modified to the TC-C cycle and C-TC cycle, and the CT cycle can be modified to the C-CT cycle and CT-C cycle, as shown in Figure 1:0  
The COP of the two open-loop low-pressure refrigeration cycles (TC-C cycle and C-TC cycle) can be determined by Equations (10) and (11), respectively:  
Figure 1:1  
Figure 1:2 Schematic of open-loop air refrigeration cycles with two-stage compressor.