Optimization of main and branched electronic expansion valves for twostage compression onestep throttling cycle
Twostage compression onestep throttling interstage incomplete cooling cycle (referred to Figure 2.7) and twostage compression twostep throttling interstage incomplete cooling cycle' (referred to Figure 2.9) are widely applied to engineering practice. In the two heat pump cycle systems under ideal operating conditions(no heat transfer temperature difference, no pressure drop in the heat exchanger and complete separation of gas and liquid in the flash tank), the suction, discharge and intermediate pressures of the onestep throttling cycle shown in Figure 2.7 are the same as those of the twostep throttling cycle shown in Figure 2.9. The mass flow rate of the refrigerant flowing through the branched EEV and the main EEV in the onestep throttling cycle obey the Equation (2.34) and the Equation (2.36), respectively, where the entrainment ratio E = 0. Because the mass flow rate of refrigerant flowing through the condenser in twostep throttling cycle is the same as that flowing through the firststep EEV, the mass flow rate of the refrigerant flowing through the secondstep EEV also satisfies Equation (2.36) where the entrainment ratio E = 0. Therefore, when the heating capacity is the same, the ideal onestep throttling cycle and the twostep throttling cycle have the following similarities:
 1) The state of the inlet and outlet refrigerants of the branched and main EEVs in onestep throttling cycle are the same as those of the firststep and secondstep EEVs in twostep throttling cycle, respectively, except for the inlet pressure of the main and the secondstep EEVs. According to Equation (5.2), the flow coefficient and the inlet refrigerant density in Equation (5.1) are correspondingly identical between the above two systems;
 2) The throttling pressure drop of the branched EEV in the onestep throttling cycle corresponding to Equation (5.1) is the same as the throttling pressure drop of the firststep EEV in twostep throttling cycle;
 3) The refrigerant mass flow rate of the main EEV in the onestep throttling cycle is the same as that of the secondstep EEV in the twostep throttling cycle.
However, the following differences exist:
 1) The throttling pressure drop of the main EEV in onestep throttling cycle is the sum of the throttling pressure drops of the firststep and the secondstep EEVs in the twostep throttling cycle;
 2) The refrigerant mass flow rate of the branched EEV in the onestep throttling cycle is equal to that of the firststep EEV in the twostep throttling cycle multiplied by the firststep throttling quality (flash quality).
From the above analysis, combined with Equation (5.1) and Equation (2.34), it can be deduced that when the heating capacity is constant, under ideal operating conditions, the ratio of the flow cross section area of branched EEV in the onestep throttling cycle to the flow cross section area of the firststep EEV in the twostep throttling cycle is equal to the firststep throttling quality (flash quality). In addition, the ratio of the flow cross section area of the main EEV in the onestep throttling cycle to the that of the secondstep EEV in the twostep throttling cycle is equal to the square root of the ratio of the secondstep throttling pressure drop to the total throttling pressure drop.
Taking the variable volume ratio twostage compression onestep throttling interstage incomplete cooling heat pump cycle as an example, the flow cross section areas of the branched and main EE Vs of heat pump system are calculated as described above. The results are listed in Table 5.6. The rated heating capacity of heat pump is 12kW and other calculation conditions are the same as Table 5.1. When calculating, firstly the flow cross section areas of the branched and main EEVs in onestep throttling cycle with the rated heating capacity of 4kW are calculated referred to Table 5.1; then, according to the Equation (5.1), the flow cross section area of the EEV of onestep throttling cycle of 12kW is converted.
TABLE 5.6
Flow cross section area of EEV for variable volume ratio twostage compression onestep throttling heat pump system with rated heating capacity of 12 kW
R410A 
R410A 
R134a 
R134a 
R32 
R32 

Evaporation temperature (°C) 
0 
30 
0 
30 
0 
30 
Volume ratio of compressor 
0.8 
0.4 
0.8 
0.4 
0.8 
0.4 
Mass flow rate of branched EEV (kg/h) 
43.8 
46.2 
46.9 
57.4 
22.3 
22.0 
Mass flow rate of main EEV (kg/h) 
171.9 
140.8 
190.2 
159.9 
117.7 
95.6 
Pressure drop of branched EEV (kPa) 
1397.7 
1635.8 
684.9 
829 
1456.1 
1652.6 
Pressure drop of main EEV (kPa) 
1932.8 
2463.2 
867.1 
1075.5 
1981.7 
2521.4 
Flow cross section area of branched EEV (mm^{2}) 
0.320 
0.311 
0.417 
0.460 
0.174 
0.161 
Flow cross section area of main EEV (nun^{2}) 
0.956 
0.663 
1.386 
0.984 
0.712 
0.492 
Similarly, the selection design of EEV can be processed after calculating the above mentioned flow cross section area.
For comparison, referring to Table 5.5, the calculation results of the flow cross section area of EEV in the singlestage heat pump system of 12 kW are listed in Table 5.7. According to Tables 5.5 and 5.7, when the heating capacity is the same, the flow cross section area of the branched EEV of the variable volume ratio twostage compression onestep throttling interstage incomplete cooling heat pump system is much smaller than that of the single stage heat pump system and that of the twostep throttling heat pump system. In addition, the flow cross section area of the main EEV of the variable volume ratio twostage compression onestep throttling interstage incomplete cooling heat pump system is smaller than that of the singlestage heat pump system and that of the twostep throttling heat pump system.
TABLE 5.7
Flow cross section area of EEV for singlestage heat pump system with rated heating capacity of 12 kW
Refrigerant 

R410A 
R410A 
R134a 
R134a 
R32 
R32 

Condensation temperature (°C) 
45 
45 
45 
45 
45 
45 
Outlet subcooling of condenser (°C) 
5 
5 
5 
5 
5 
5 
Evaporation temperature (°C) 
0 
30 
0 
30 
0 
30 
Mass flow rate (kg/h) 
212.9 
180.0 
236.2 
213.5 
138.6 
114.2 
Flow cross section area of singlestage EEV (mm^{2}) 
1.316 
0.955 
1.847 
1.399 
0.924 
0.652 
Flow cross section area ratio of branched EEV to singlestage EEV 
0.243 
0.326 
0.226 
0.329 
0.189 
0.248 
Flow cross section area ratio of main EEV to singlestage EEV 
0.727 
0.694 
0.750 
0.703 
0.771 
0.755 
In the the singlestage compression heat pump, the outlet subcooling of condenser is generally greater than that of the twostage compression interstage incomplete cooling heat pump, resulting in a decrease in the inlet temperature of EEV, and increases in the inlet density and flow coefficient. In addition, the outlet specific enthalpy of the condenser decreases while the inlet specific enthalpy of the condenser increases (as the discharge temperature of singlestage compression is higher), so the specific heating capacity increases and actual mass flow rate is reduced compared with the theoretical calculation result. It can be known from Equation (5.1) that the actual flow cross section area of EEV of the singlestage compression heat pump will be smaller than the theoretical calculation result. Therefore, the actual ratio of the flow cross section areas of the firststep EEV and the secondstep EEV of twostage compression twostep throttling heat pump to that of the EEV of singlestage compression heat pump, or the actual ratio of the flow cross section areas of the branched EEV and the main EEV of the twostage compression onestep throttling heat pump to that of the EEV of singlestage compression heat pump, will be larger than the theoretical calculation result.