# Investigation on the Mass Flow Rate of a Refrigerator Compressor Based on the p-V Diagram
## 3 Method to Calculate the Refrigerant Mass Flow Rate  
### Validation of Adiabatic Process Hypothesis in the Expansion Phase and Compression Phase  
As shown in Figure 4, the experimental pressure variations in the expansion and compression phases are in good agreement with the theoretical adiabatic pressure variations. The error between the theoretical adiabatic pressure variations and experimental data are listed in Table 2. The maximum error in the compression and expansion phase is \(-2.50\%\) and \(4.64\%\), respectively. The approximate pressure variation implies that the leakage and heat transfer have little influence on the refrigerator  
Figure 3: Test facility for the refrigeration system.  
Figure 2: The installation of pressure sensor.  
compressor and can be ignored since the rotation speed of refrigerator compressor is high. Therefore, the compression and expansion phases can be approximately considered as an adiabatic process [14].  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \multicolumn{2}{c}{**Compression Phase**} & \multicolumn{4}{c}{**Expansion Phase**} \\ \hline
**Experimental** & **Adiabatic Line** & **Error (\%)** & **Experimental** & **Adiabatic Line** & **Error (\%)** \\
**Pressure (MPa)** & **(MPa)** & **Pressure (MPa)** & **(MPa)** & **Error (\%)** \\ \hline  
(-\)1.35 \\ \hline \hline \end{tabular}
\end{table}
Table 2: The error between theoretical adiabatic pressure variations and experimental data.  
Figure 4: Comparison of theoretical adiabatic pressure variations and experimental data in a p–V diagram.  
The polytropic exponent is an important indicator to evaluate the reciprocating compressor performance, since faultless and leaking valves show different gradients in the expansion and compression phase. The polytropic exponents in the expansion and compression phase are 1.0968 and 1.0935, respectively, as shown in Figure 5. The approximately equal exponents illustrate that that the modified compressor is faultless and that the heat transfer can be ignored at a higher rotation speed [15].  
Knowing if the compression and expansion phases are approximate to the theoretical adiabatic process is critical to the correctness of the method for calculating the mass flow rate of the compressor, which is proposed in Section 3.2.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multicolumn{3}{c}{**Compression Phase**} & \multicolumn{3}{c}{**Expansion Phase**} \\ \hline
**Experimental** & **Adiabatic Line** & **Error (\%)** & **Experimental** & **Adiabatic Line** & **Error (\%)** \\
**Pressure (MPa)** & **(MPa)** & **Error (\%)** & **Pressure (MPa)** & **(MPa)** & **Error (\%)** \\ \hline
0.190 & 0.194 & \(-\)2.01 & 0.151 & 0.153 & \(-\)1.41 \\
0.205 & 0.210 & \(-\)2.37 & 0.148 & 0.146 & 1.33 \\
0.223 & 0.228 & \(-\)1.92 & 0.145 & 0.139 & 4.70 \\
0.242 & 0.248 & \(-\)2.72 & 0.121 & 0.120 & 0.86 \\
0.268 & 0.272 & \(-\)1.31 & 0.111 & 0.111 & \(-\)0.16 \\
0.296 & 0.299 & \(-\)1.15 & 0.106 & 0.106 & 0.24 \\
0.325 & 0.331 & \(-\)1.74 & 0.094 & 0.091 & 3.97 \\
0.363 & 0.368 & \(-\)1.34 & 0.081 & 0.081 & \(-\)0.05 \\
0.412 & 0.412 & 0.09 & 0.075 & 0.078 & \(-\)3.75 \\
0.461 & 0.463 & \(-\)0.53 & 0.068 & 0.066 & 2.77 \\
0.531 & 0.525 & 1.17 & 0.061 & 0.063 & \(-\)2.97 \\ \hline \hline \end{tabular}
\end{table}
Table 2: _Cont._  
Figure 5: Comparison of experimental data and adiabatic calculation in a log(\(p\))–log(\(V\)/\(Vmax\)) diagram