# Numerical Simulation Study on the Volumetric Efficiency Loss of CO2 Swash Plate Axial Piston Pumps
## 4 Numerical Simulation Method and Simulation Method Validation  
In a numerical simulation of the piston pump fluid domain, the flow process of fluid CO\({}_{2}\) needs to follow the continuity equation, momentum conservation equation, and energy conservation equation in order to make the control equation closed with the introduction of a turbulence model.  
(1) Continuity equation  
\[\frac{\partial\rho}{\partial t}+\frac{\partial(\rho u)}{\partial x}+\frac{ \partial(\rho v)}{\partial y}+\frac{\partial(\rho w)}{\partial z}=0 \tag{1}\]  
where \(\rho\) is the fluid density; \(t\) is time; \(u\) is the velocity component in the x-direction; \(v\) is the velocity component in the y-direction; and \(w\) is the velocity component in the z-direction.  
(2) Momentum equation  
\[\frac{\partial\rho u}{\partial t}+\
abla\cdot\rho uu=-\
abla P+\
abla\tau+f_{v} \tag{2}\]  
where \(P\) is pressure; \(\tau\) is the viscous stress tensor; and \(f_{v}\) is the volumetric force acting on the fluid.  
(3) Energy equation  
\[\frac{\partial(\rho E)}{\partial t}+\
abla(\rho Eu)=-\
abla(Pu)+\
abla(\tau u )+\rho fu+\
abla(k\
abla T)+\tilde{q} \tag{3}\]  
where \(u\) is the velocity vector; \(k\) is the heat transfer coefficient; \(T\) is the temperature; \(\tilde{q}\) is the heat source term per unit volume; and \(E\) is the total energy per unit volume of fluid, calculated as  
\[E=e+\frac{1}{2}\left|\overset{\rightarrow}{u}\right|^{2} \tag{4}\]  
where \(e\) is the thermodynamic internal energy per unit mass of fluid.  
The physical properties of CO\({}_{2}\) are extremely complex and sensitive to changes in temperature and pressure. The phase state is variable, which causes difficulties in numerical simulations. Refprop9.1 is a software product developed by the National Institute of Standards and Technology (NIST) to reference the physical parameters of fluids. This software is widely used in scientific research, the energy and chemical industries, aerospace, and other fields, and its physical property library includes most fluids. Based on a wide range of experimental data and theoretical models, the software can accurately provide the physical properties of CO\({}_{2}\), including its density, viscosity, and heat conduction, which were used in this study. Due to the wide range of pressure changes during the working process of the plunger pump, in order to accurately simulate CO\({}_{2}\) physical properties, the real gas NIST model was selected in the numerical simulation. This model was based on the Refprop  
Figure 4: Grid-independent validation of liquid film leakage volume loss.  
database, and the physical properties were determined according to the experimental data and the equation of state. Using this model for CO\({}_{2}\) ensured the accuracy of its physical properties and more accurately reflected its real flow state. By querying the Refprop software for the pressure range from 2.5 MPa to 12.5 MPa and the temperature range from 240K to 260K, physical parameters such as density, viscosity, thermal conductivity, and specific heat could be obtained. The values derived from Refprop were consistent with those from the real gas NIST model in the fluent software. Taking density and viscosity as an example, changes in the physical properties of CO\({}_{2}\) with temperature and pressure are shown in Figure 5.