## 2 Establishing the Model  
As shown in Figure 1, \(R_{o}\), \(R_{i}\), and \(R_{g}\) respectively represent the outer radius, the inner radius, and the groove root radius of the end face of the moving ring. Parameters \(\theta_{1}\) and \(\theta_{2}\) respectively represent the angle of the spiral groove area and the weir area corresponding to the center of the circle. Parameter \(\theta\) represents the helix angle, which is defined as the angle between the tangent of any point on the helix and the tangent on the circle where it is located. The following definitions are introduced for simplicity: groove-diameter ratio \(\beta=\left(R_{g}-R_{i}\right)/\left(R_{o}-R_{i}\right)\), groove-weir ratio \(\gamma=\theta_{1}/\left(\theta_{1}+\theta_{1}\right)\). The phase transition mechanism of the spiral groove liquid film seal is relatively complicated. For simplicity purposes, the following assumptions are made [31]:  
1. The fluid medium is a Newtonian fluid. 2. The fluid flow between the sealing interfaces is continuous medium laminar, and the fluid temperature and viscosity do not change with time. 3. The sealing surface is smooth, i.e., the effect of its roughness on the fluid flow is assumed to be negligible. 4. The film thickness is very thin, and the pressure and density remain unchanged in the thickness direction. 5. The temperature of the sealing ring and the mechanical properties of the material do not change with time. 6. There is no relative slip between the fluid medium and the seal face. 7. Disturbance and vibration of the system during operation are not accounted for. [PERSON] et al. [14] established independent conservation equations for the liquid and vapor phases. Evaporation and condensation terms of the mass transfer equation are shown respectively in Equations (1) and (2):  
\[\dot{m}_{l\to\
u}=\lambda_{c}\alpha_{l}\rho_{l}\frac{T_{l}-T_{\text{sat}}}{T _{\text{sat}}}(T_{l}>T_{\text{sat}}) \tag{1}\]  
\[\dot{m}_{\
u\to l}=\lambda_{c}\alpha_{\
u}\rho_{\
u}\frac{T_{\
u}-T_{\text{sat }}}{T_{\text{sat}}}(T_{\
u}<T_{\text{sat}}) \tag{2}\]  
In Equations (1) and (2), \(T_{\text{sat}}\) represents the evaporation temperature, and \(\lambda_{c}\) is the phase transition mass transfer coefficient, which can be experimentally obtained. The subscripts l and v indicate liquid phase and vapor phase, respectively. Finally, \(\alpha\) and \(\rho\) represent the volume fraction and density, respectively. The phase transition mass transfer coefficient \(\lambda_{c}\) can be expressed as [32]:  
\[\lambda_{c}=\frac{6}{d}\beta\sqrt{\frac{M}{2\pi RT_{s}}}h_{\text{fg}}\bigg{(} \frac{\rho_{l}}{\rho_{l}-\rho_{g}}\bigg{)} \tag{3}\]  
where \(d\) is the bubble diameter, \(\beta\) is the adjustment coefficient, \(M\) is the molar mass, \(R\) is the general gas constant, \(h_{\text{fg}}\) is the latent heat of vaporization, \(T_{s}\) is the fluid saturation temperature, and \(\rho_{l}\) and \(\rho_{\
u}\) are density of liquid and vapor phase. After the phase transition of the liquid film, the entire fluid film transforms into a homogeneous mixture made up of liquid and vapor. According to [PERSON] et al. [33], Equation (4) can accurately explain the relationship between the density \(\rho\) and viscosity \(\mu\) of the mixture:  
\[\left\{ \begin{array}{l}\frac{1}{\rho}=\phi\frac{1}{\rho_{\
u}}+(1-\phi) \frac{1}{\rho_{l}}\\ \frac{1}{\mu}=\phi\frac{1}{\mu_{\
u}}+(1-\phi)\frac{1}{\mu_{l}}\end{array} \right. \tag{4}\]  
where \(\mu_{\
u}\) and \(\mu_{l}\) are the dynamic viscosities of the vapor and liquid phases, respectively, and \(\varnothing\) is the mass fraction of the vapor phase in the mixed phase. Figure 1: Structure of end face of rotating seal ring. Temperature not only affects the degree of liquid film phase transition, but also affects the viscosity of the liquid phase in the liquid film and the saturated vapor pressure of the sealing medium. Because the viscosity of the vapor phase is much smaller than that of the liquid phase and is less affected by the temperature, its effect can be assumed to be negligible.