## 2 Theoretical Analysis of Air Refrigeration Cycles  
Air refrigeration cycles can be configured with three types of reverse Brayton cycle, including the closed cycle, the open-loop low-pressure cycle (TC cycle) and the open-loop high-pressure cycle (CT cycle), as shown in Figure 1. In the open-loop low-pressure cycle, air of ambient temperature can directly expand to low-temperature air of sub-ambient pressure, and the hot-side heat exchanger is not used. After the cooling power is consumed in a cold-side heat exchanger, the air is compressed and discharged into the outside. In the open-loop high-pressure cycle, the air after expansion is directly supplied to the room, and the cold-side heat exchanger is not used. The condensed water can be removed in the hot-side heat exchanger or after the expander in the case of high-humidity ambient air, and low-temperature, dry air is supplied to the room. Figure 1: Schematic of air refrigeration cycles. T: turbo-expander; C: compressor; H: heat exchanger; I: indoor; O: outdoor. The coefficient of performance (COP) of the cycle is defined as the ratio of the cooling capacity to the power consumption of the system. In order to simplify the calculation, the pressure drop in the pipes and the heat exchanger is ignored. The compression ratio and expansion ratio (\(Er\)) in the cycle are the same, and the working fluid is air which is considered an ideal gas. In the closed cycle, the output shaft power \(W_{1}\) and input shaft power of the cycle \(W_{2}\) are calculated as Equations (1) and (2), respectively. \[W_{T}=q_{m}\frac{k}{k-1}RT_{\text{AC}}\Big{[}1-(Er)^{\frac{1-k}{k}}\Big{]} \times\eta_{T} \tag{1}\]  
\[W_{\text{C}}=q_{m}\frac{k}{k-1}RT_{\text{CI}}\Big{[}(Er)^{\frac{k-1}{k}}-1 \Big{]}/\eta_{\text{C}} \tag{2}\]  
The cooling capacity of cycle is determined by Equation (3):  
\[Q_{\text{c}}=q_{m}(h_{\text{CI}}-h_{EO})=q_{m}(h_{\text{CI}}-h_{\text{AC}}+h_{ AC}-h_{\text{EO}})=q_{m}\bigg{[}h_{\text{CI}}-h_{\text{AC}}+\frac{k}{k-1}RT_{ \text{AC}}\Big{(}1-Er^{\frac{1-k}{k}}\Big{)}\times\eta_{T}\bigg{]} \tag{3}\]  
The COP of the cycle is calculated by Equation (4):  
\[\text{COP}_{\text{Closedcycle}}=\eta_{\text{C}}\frac{(T_{\text{CI}}-T_{\text{ AC}})+\eta_{T}T_{\text{AC}}\Big{(}1-Er^{\frac{1-k}{k}}\Big{)}}{\Big{[}T_{\text{CI}}Er^{ \frac{k-1}{k}}-T_{\text{CI}}-\eta_{\text{C}}\eta_{T}\eta_{M}T_{\text{AC}} \Big{(}1-Er^{\frac{1-k}{k}}\Big{)}\Big{]}}=\eta_{\text{C}}\frac{\eta_{T}T_{ \text{AC}}\Big{(}Er^{\frac{k-1}{k}}-1\Big{)}-(T_{\text{AC}}-T_{\text{CI}})Er^ {\frac{k-1}{k}}}{\Big{(}T_{\text{CI}}Er^{\frac{k-1}{k}}-\eta_{\text{C}}\eta_{ M}T_{\text{AC}}\Big{)}\Big{(}Er^{\frac{k-1}{k}}-1\Big{)}} \tag{4}\]  
Similarly, the COP of the open-loop low-pressure cycle and high-pressure cycle is determined by Equations (5) and (6):  
\[\text{COP}_{\text{TCycle}}=\eta_{\text{C}}\frac{(T_{\text{CI}}-T_{\text{O}})+ \eta_{T}T_{\text{O}}\Big{(}1-Er^{\frac{1-k}{k}}\Big{)}}{\Big{[}T_{\text{CI}} Er^{\frac{k-1}{k}}-T_{\text{CI}}-\eta_{\text{C}}\eta_{T}\eta_{M}T_{\text{O}} \Big{(}1-Er^{\frac{1-k}{k}}\Big{)}\Big{]}}=\eta_{\text{C}}\frac{\eta_{T}T_{ \text{O}}\Big{(}Er^{\frac{k-1}{k}}-1\Big{)}-Er^{\frac{k-1}{k}}(T_{\text{O}}- T_{\text{CI}})}{\Big{[}T_{\text{CI}}Er^{\frac{k-1}{k}}-T_{\text{CI}}\eta_{T}\eta_{M}T_{ \text{O}}\Big{)}\Big{(}Er^{\frac{k-1}{k}}-1\Big{)}} \tag{6}\]  
where \(T_{\text{AC}}\) is the air temperature after the hot-side heat exchanger, \(T_{\text{CI}}\) is the air temperature after the cold-side heat exchanger. \(T_{\text{I}}\) is the indoor temperature, and \(T_{\text{O}}\) is the outdoor temperature. It is clear that the mechanical efficiency \(\eta_{\text{M}}\), compressor isentropic efficiency \(\eta_{\text{C}}\) and turbo-expander isentropic efficiency \(\eta_{\text{T}}\) determine the system performance. Moreover, the expander efficiency has a relatively greater impact, which not only affects the output work of the expander, but also determines the enthalpy drop in the expander and hence the cooling capacity of the system.