## 2 Research Model and Numerical Methods
### Numerical Method Validation  
In order to verify the effectiveness of the numerical methods, the performance test of the research model (EPC model) was carried out in the laboratory of turbo machinery of a northeastern university. In this test, the precision of temperature sensors is \(\pm\)0.1%. Pressure measurements are done by piezoresistive sensors with a precision of \(\pm\)0.3%. The rotor speed is measured by an external magnetic induction tachometer with a precision of \(\pm\)0.2%. To better simulate the actual HFC intake condition, a 90\({}^{\circ}\) bend was installed at the LPS inlet. In the experiment, the MFR, pressure and temperature data are collected by setting sensors at the inlet and outlet of the compressor, thus, the isentropic efficiency and the pressure ratio of the compressor can be calculated. The arrangement of compressor to be tested, sensors and interstage pipelines are shown in Figure 3(a), and the test principle is shown in the Figure 3(b).  
In order to eliminate the influence of the intake temperature on the performance of the compressor, the reduced parameters were used to characterize the compressor performance in which all the parameters are reduced to standard atmospheric pressure and 298 K. The formulas for calculating the reduced parameters are as follows:  
\[M_{\text{cor}}=M_{\text{c}}\frac{100000}{P_{1}}\sqrt{\frac{T_{1}}{298}} \tag{8}\]  
\[N_{cor}=n_{c}\sqrt{\frac{298}{T_{1}}} \tag{9}\]  
where \(M_{cor}\) is reduced mass flow rate; \(M_{c}\) is the actual mass flow rate; \(N_{cor}\) is reduced speed and \(n_{c}\) is the actual speed; \(P_{1}\) is the absolute total pressure of compressor inlet; \(T_{1}\) is the total temperature of compressor inlet.  
The pressure ratio and isentropic efficiency can be calculated by the following formulas:  
\[\pi=\frac{P_{2}}{P_{1}} \tag{10}\]  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Gird Number** & **m (kg/s)** & \(\pi\) & \(\eta\) \\ \hline
6048565 & 0.15 & 2.801 & 0.7631 \\
7426874 & 0.15 & 2.814 & 0.7665 \\
8864795 & 0.15 & 2.816 & 0.7669 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Gird independent verification of the research model.  
Figure 4: Compressor installation and test bench schematic diagram.  
\[\eta=\frac{T_{1}(\pi^{\frac{k-1}{K}}-1)}{T_{2}-T_{1}} \tag{11}\]  
where \(\pi\) is the pressure ratio; \(\eta\) is the isentropic efficiency; \(K\) is the isentropic exponent and its value is 1.4; \(P_{2}\) is the absolute total pressure of compressor outlet; \(T_{2}\) is the total temperature of compressor outlet.  
The compressor performance test and CFD calculations were both carried out at three speeds in this work to verify the effectiveness of the numerical methods. Figure 5 compares the performance curves between CFD and test results at the speed of 40 kr/min, 60 kr/min and 80 kr/min, respectively. It can be seen form Figure 5a,b that at low speed, the test and CFD results can match well with each other on both the pressure ratio and the efficiency curves; while with the speed increases, the deviation between the CFD and the test results is also growing. At 80 kr/min, the difference in pressure ratio and efficiency between CFD and the test were 1% and 2%, respectively, and it can be seen clearly that the test results are higher than that of the CFD results in both the pressure ratio and efficiency. The main reason for the difference is that the heat transfer of the solute is not considered during the CFD calculation, while in the tests, the volute dissipates heat outward, which will undoubtedly reduce the isentropic compression work and improve the test efficiency. In addition, in the actual work of the compressor, the rotor will expand due to the rise of temperature, thus reducing the distance between the impeller and the volute and mitigating the clearance leakage flow loss. In general, the results of the test and CFD are well matched, which verifies the effectiveness of the numerical method adopted in this work.