# Numerical Simulation Study on the Volumetric Efficiency Loss of CO2 Swash Plate Axial Piston Pumps
## 4 Numerical Simulation Method and Simulation Method Validation
### Numerical Simulation Method for Fluid Domain of Friction Subfilm  
A transient numerical simulation of the mating pair, plunger pair, and sliding shoe pair was used to open the energy equation, and the outlet boundary was chosen to be a pressure outlet with a pressure of 3 MPa. The outlet pressure of the plunger pump fluctuated with the reciprocating motion of the plunger, as shown in Figure 6. It can be seen from the figure that the waist groove of the high-pressure area of the mating pair showed the characteristic of cyclic pressure pulsation, and the pressure amplitude change corresponded to the frequency of reciprocating motion of the plunger. The curve shown in Figure 6 was converted into a profile file as the inlet pressure value of the flow pair  
Figure 5: Schematic diagram of CO\({}_{2}\) physical property distribution. Considering the shear effect generated by cylinder rotation, a moving wall was set on the contact surface between the liquid film of the flow pair and the cylinder, and the rotational speed was 1000 rpm. The plunger sub-film at the lower dead point was the object of study. From the characteristics of plunger pump motion, it is known that the pressure in the plunger cavity is cyclic, and the phase characteristics of the transition from the low-pressure region to the high-pressure region are presented in the cycle. The pressure shown in Figure 7 was converted into a Profile file as the inlet pressure value of the plunger sub. The plunger sub rotates around the axis under actual working conditions. A slip grid was applied to the fluid domain of the liquid membrane of the plunger sub-film with a rotational speed of 1000 rpm, and a UDF was applied against the pressure inlet surface of the liquid film to describe the axial motion. The velocity expression is shown in Equation (5). The secondary liquid film of the slipper at the following dead point was the research object. The plunger is connected with the slipper through the ball head. The action of hydraulic pressure ensures that the slipper is close to the surface of the swashplate for rotary motion. The spherical hinge center of the slipper was projected onto the plane of the swashplate for circular motion, and sliding mesh motion was applied to the secondary liquid film of the slipper at 1000 rpm. For the plunger pump fluid domain selection of the RNG k-\(\epsilon\) model, the friction sub-fluid film flow pattern needs to be determined by the Reynolds number. In order to match the flow sub-fluid film as an example, the Reynolds number calculation formula was used, as shown in Equation (6) [28]. \[\text{Re}=\frac{\rho\omega Rh}{\mu} \tag{6}\]  
Figure 6: Liquid film inlet pressure value of distribution pair. Figure 7: Liquid film inlet pressure value of the plunger pair and slipper pair. where Re is the Reynolds number; \(\omega\) is the rotational speed; \(R\) is the equivalent radius of the sealing end face; \(h\) is the liquid film thickness; and \(\mu\) is the liquid carbon dioxide kinetic viscosity. After this calculation, Re = 385.46, and a laminar flow model was used. In order to verify the reliability of the numerical simulation method in the fluid domain of the piston pump in this paper, the specific working conditions of the swashplate axial piston pump in reference [29] were selected for comparison and verification, and the numerical model consistent with the experimental conditions in the literature was established. The flow at the pump outlet with different swashplate inclination and working speed was numerically simulated and calculated; the results are shown in Figure 8. It can be seen from the figure that the numerical simulation results of the pump outlet flow under different swashplate inclinations and working speeds were roughly the same as the experimental results in the literature. The maximum error was 7.95%, which verified the accuracy of the numerical simulation calculation model and the rationality of the boundary conditions, providing a reliable basis for the subsequent numerical simulation research.