## 3 Mathematical Model  
### Modeling of Cycle  
Based on the first thermodynamic laws, a mathematical model is established to evaluate the performance of two systems from the perspective of energy efficiency. In or  
Figure 1: Schematic of CHPVC.  
Figure 2: Schematic of HPTVC.  
der to simplify the theoretical calculation, the following assumptions are made as follows [13; 14; 24; 25; 26; 27]:  
1. All components operate under steady conditions.
2. The temperature loss and pressure loss of the working fluids are ignored.
3. The superheating degree is set to 5 \({}^{\circ}\)C, and the sub-cooling degree is set to 3 \({}^{\circ}\)C.
4. The heat transfer efficiency of the heat exchanger is assumed to be 100%.
5. The minimum heat transfer temperature difference in all heat exchangers is assumed to be 5 \({}^{\circ}\)C.
6. The efficiency of the pump is assumed to be 80%.
7. The vapor at the outlet of the compressor is saturated.  
For the R134a heat pump cycle, the power consumption and the heating capacity can be calculated as follows:  
\[W_{\text{c,R134a}}=m_{1}(h_{2}-h_{1}) \tag{1}\]  
\[Q_{\text{H,R134a}}=m_{1}(h_{2}-h_{3}) \tag{2}\]  
\[h_{2}=\frac{h_{2s}-h_{1}}{\eta_{\text{is,R134a}}}+h_{1} \tag{3}\]  
For the R245 fa heat pump cycle, the power consumption and the heating capacity can be calculated as follows:  
\[W_{\text{c,R245 fa}}=m_{7}(h_{8}-h_{7}) \tag{4}\]  
\[Q_{\text{C,R245 fa}}=m_{7}(h_{7}-h_{10}) \tag{5}\]  
\[Q_{\text{H,R245 fa}}=m_{7}(h_{8}-h_{9}) \tag{6}\]  
\[h_{8}=\frac{h_{8s}-h_{7}}{\eta_{\text{is,R245 fa}}}+h_{7} \tag{7}\]  
For the single-stage water vapor compression, the injected water pressure is equal to the discharge pressure of the vapor compressor, which means that the injected water is used to reduce the discharge temperature of the vapor compressor and does not consume the compression work. Hence, the power consumption and the energy increment (\(\Delta E\)) of the working fluid can be calculated as follows:  
\[W_{\text{P,I}}=m_{11}(h_{11}-h_{14}) \tag{8}\]  
\[W_{\text{P,II}}=m_{16}(h_{16}-h_{13}) \tag{9}\]  
\[h_{11}=\frac{h_{11s}-h_{14}}{\eta_{\text{P}}}+h_{14} \tag{10}\]  
\[h_{16}=\frac{h_{16s}-h_{13}}{\eta_{\text{P}}}+h_{13} \tag{11}\]  
\[W_{\text{c,vapor}}=m_{15}\frac{(1+x)h_{17}-(h_{15}+xh_{16})}{\eta_{\text{c, vapor}}} \tag{12}\]  
\[\Delta E_{\text{vapor}}=m_{5}(h_{17}-h_{5}) \tag{13}\]  
where \(x\) is the water-injection ratio, which can refer to [PERSON]'s work [18]. The calculation process of the pump can refer to [PERSON]'s work [28].  
For the two-stage water vapor compression, the power consumption and the energy increment (\(\Delta E\)) of the working fluid can be calculated as follows:  
\[W_{\text{P,I}}=m_{9}(h_{11}-h_{9}) \tag{14}\]  
\[W_{\text{P,II}}=m_{13}(h_{13}-h_{9}) \tag{15}\]  
\[W_{\text{P,III}}=m_{16}(h_{16}-h_{14}) \tag{16}\]\[W_{\text{cl,vapor}}=m_{10}\frac{n}{n-1}ZR_{g}T_{10}[(\frac{p_{12}}{p_{10}})^{\frac{ n-1}{n}}-1] \tag{17}\]  
\[W_{\text{cl,vapor}}=m_{15}\frac{(1+x)h_{17}-(h_{15}+xh_{16})}{\eta_{\text{c,vapor}}} \tag{18}\]  
\[\Delta E_{\text{vapor}}=m_{5}(h_{17}-h_{5}) \tag{19}\]  
\[\eta_{\text{pol,vapor}}=\frac{n/(n-1)}{k/(k-1)} \tag{20}\]  
where \(n\), \(k\), and \(Z\) are the polytropic index, isentropic index, and compressibility factor, respectively. The calculation process of the centrifugal vapor compressor can refer to [PERSON]'s work [18].