# 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
### Calculation Method  
Due to the high rotation speed, the refrigerant leakage and the heat exchange between the cylinder wall and the refrigerant can be ignored in the refrigerator compressor [12]. Therefore, in the expansion and compression phases, the valves are closed and the refrigerant mass remains constant [16]. Moreover, the temperature and velocity gradients in compressor cylinder are small enough to neglect the irreversibility [12].  
In the compression and expansion phases, the compressor cylinder is regarded as a closed thermodynamic system with masses \(\dot{m}_{\text{com}}\) and \(\dot{m}_{\text{exp}}\), respectively. During the compression phase, the refrigerant in the cylinder includes the fresh refrigerant suctioning from the suction line and the refrigerant remaining in the clearance volume. During the expansion phase, the refrigerant in the cylinder only includes the refrigerant remaining in the clearance volume. Thus, the refrigerant mass flow rate of the compressor in one cycle can be calculated:  
\[\dot{m}=\dot{m}_{\text{com}}-\dot{m}_{\text{exp}} \tag{1}\]  
where \(\dot{m}\) is the refrigerant mass in the chamber, which is assumed to remain constant during the expansion and compression phases.  
Following Section 3.1, the compression and expansion phases approximate adiabatic processes. Based on thermodynamics, the two phases follow the relationship:  
\[\dot{m}_{\text{exp},com}=V_{c}\rho\left(p,\frac{C_{p}}{C_{v}}\right) \tag{2}\]  
where \(V_{c}\) is the cylinder volume at the corresponding time, and \(p\) is the refrigerant pressure in the cylinder. The ratio \(C_{p}/C_{v}\) is usually considered equal to the adiabatic exponent. Moreover, the density \(\rho\) of refrigerant in the cylinder is a function of \(p\) and \(C_{p}/C_{v}\). The physical property data for R600a are derived from NIST PEFPROP [16].  
In Equation (2), in fact, \(C_{p}/C_{v}\) is determined by the corresponding temperature and \(p\). However, the measurement of the instantaneous refrigerant temperature in the cylinder is a difficult problem. In this paper, due to the expansion phase and compression phase being close to the adiabatic process, the adiabatic exponent is approximately equal to the polytropic exponent obtained from the \(\log(p)\)-\(\log(V/Vmax)\) diagram [12]. In addition, \(p\) and \(V_{c}\) could be measured directly by experiment.  
Based on the Equation (2), a method is proposed to obtain the refrigerant mass in the two phases. A flow chart for calculating the refrigerant mass in the compression phase is shown in Figure 6. The calculation is started by the assumed temperature and the adiabatic exponent obtained experimentally. The pressure, density, and mass in the cylinder can be calculated by the temperature and adiabatic exponent. With the volume variation of the cylinder and adiabatic process, the variation of pressure is obtained. In addition, the convergence condition of the calculation is whether the difference between the experimental and the calculation pressure variations is sufficiently small. Subsequently, the refrigerant mass in the expansion phases can be obtained in the same way.