## 2 The FSI Model of Reciprocating Compressors
### The Fluid Domain  
The fluid domain model is established based on the actual flow field structure of the compressor, as shown in Figure 3. The fluid domain of the compressor can be divided into three main parts: the suction part, the discharge part, and the cylinder part. The discharge part includes the discharge hole, discharge chamber, and discharge pipe. The suction part consists of the suction chamber and suction hole.  
The governing equations for the fluid domain in the FSI model are shown in Equations (1)-(3):  
\[\frac{\partial\rho}{\partial t}+\
abla\cdot(\rho v)=0 \tag{1}\]  
\[\frac{\partial\rho v}{\partial t}+\
abla\cdot(\rho vv-\tau)=f^{B} \tag{2}\]  
\[\frac{\partial\rho E}{\partial t}+\
abla\cdot(\rho vE-\tau\cdot v+q)=f\cdot v +q^{B} \tag{3}\]  
where \(\rho\) is the density, \(t\) is the time, \(q^{B}\) is the body force vector of the fluid medium, \(v\) is the velocity vector, \(E\) is the specific total energy, \(\tau\) is the stress tensor, \(q^{B}\) is the specific rate of heat generation, and \(q\) is the heat flux.  
Figure 3: The fluid domain of the compressor in the FSI model.  
Figure 2: The schematic diagram of the compressor with a reed valve.  
In this paper, the physical properties of the working fluid are calculated by the Redlich-Kwong (R-K) model. The R-K equation can be successfully used in the calculation of the gas phase P-V-T, and the accuracy of the R-K equation in the calculation of the gas phase volume is greatly improved compared with the Van [PERSON] equation [26]. It needs to be pointed out that the calculation accuracy of the gas phase volume is very important in compressor work. The model not only has high accuracy, but also is widely used as a two-parameter model in engineering calculations.  
The general form of the R-K model is expressed in Equation (4):  
\[p=\frac{R_{sp}T}{v-b+c}-\frac{a}{v(v+b)} \tag{4}\]  
The expressions of \(a\), \(b\), and \(c\) are  
\[a=a_{0}\left(\frac{T}{T_{c}}\right)^{-\frac{1}{2}},\;b=0.08664\frac{R_{sp}T_{c }}{p_{c}},\;a_{0}=0.42747\frac{R_{sp}^{2}T_{c}^{2}}{p_{c}},\;c=0\]  
where \(T_{c}\) is the critical temperature, \(R_{sp}\) is the gas constant, and \(p_{c}\) is the critical pressure.  
The working medium is air, and the physical parameters of the air are shown in Table 1.  
The gap boundary conditions are applied between these parts to separate the cylinder from the discharge part and the suction part [24]. Different initial temperature and pressure conditions are set for these three parts, respectively.  
In the fluid domain, the model adopts a low-speed compressible flow. The pressure at the suction part is set to 0.103 MPa, and the temperature at the suction part is set to 293.15 K. The initial temperature and pressure in the discharge and cylinder part are set to 338.15 K and 0.4 MPa, respectively.  
The RNG k-\(\epsilon\) model has been used to simulate the turbulent flow in the cylinder of the reciprocating compressors--where the standard pressure interpolation scheme was adopted--while the SIMPLE method was used to solve the pressure-velocity coupling problem. The eight-node 3D element is used to discretize the fluid domain.