## 2 Theoretical Model
### Governing Equations  
The thermo-elasto-hydrodynamic lubrication model was established by taking the thermal distortion, the inertial effect, the choked-flow effect, and the real gas effect of helium, into account. During the modeling, the thermal boundary conditions referred to the Refs. [41; 42]. The heat flux was set between the gas film and the friction interfaces of \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters** & **Units** & **Values** \\ \hline Outlet radius/\(r_{1}\) & mm & 44 \\ Inlet radius/\(r_{0}\) & mm & 50 \\ Spiral radius/\(r_{g}\) & mm & 48 \\ Spiral angle/\(\beta\) & - & 16 \\ Groove depth/\(h_{\mathrm{d}}\) & \(\upmu\)m & 5 \\ Groove numbers/\(N\) & - & 12 \\ Thickness of seal rings/\(h_{1}\), \(h_{2}\) & mm & 15 \\ Thickness of gas film/\(h\) & \(\upmu\)m & - \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Relevant geometrical parameters of the spiral-grooved face seal. Figure 1: The three-phase diagram of helium and the geometrical diagram of a spiral-grooved face seal. the seal rings, and the convective and adiabatic boundaries were imposed between the seal rings and their surroundings, which can be seen in Figure 1.  
The thermo-hydrodynamic characteristics of the supercritical helium spiral-grooved gas face seal were obtained by solving the governing equations jointly in the lubrication model, on the bases of the geometrical model and boundary conditions of the face seal. The governing equations mainly included the Reynolds equation, energy equation, heat conduction equation, and interface equation, which are described in detail as follows [43]. The steady-state Reynolds equation was used to describe the motion of the sealing medium, which is expressed in the polar coordinates as:  
\[\frac{\partial}{r\partial\theta}\left(\frac{h^{3}\rho}{\eta}\frac{\partial p}{ r\partial\theta}\right)+\frac{\partial}{r\partial r}\left(\frac{rh^{3} \rho}{\eta}\frac{\partial p}{\partial r}\right)=6\omega\frac{\partial(\rho h) }{\partial\theta}, \tag{1}\]  
where \(p\) is the pressure of the gas film, \(r\) and \(\theta\) are the radial and circumferential coordinates, respectively, and \(\eta\) and \(r\) are the viscosity and density of helium, respectively. The energy equation of the gas film was derived by analyzing the variation of the internal energy of the gas film, including the flow work, friction work, energy loss induced by the interfacial heat conduction, and heat extrusion, which is described in the polar coordinates as:  
\[ \begin{array}{c}\left(\frac{h^{3}}{12\eta}\frac{\partial p}{r\partial \theta}-\frac{w\eta h}{2}\right)\frac{\partial T}{r\partial\theta}+\left( \frac{h^{3}}{12\eta}\frac{\partial p}{\partial r}\right)\frac{\partial T}{ \partial r}\\ =-\frac{w\eta^{2}r^{2}}{h\rho c_{v}}+\frac{h^{3}}{12\eta\rho c_{v}}\left( \left(\frac{\partial p}{r\partial\theta}\right)^{2}+\left(\frac{\partial p}{ \partial r}\right)^{2}\right)-\frac{k_{g1}}{\rho c_{v}}\left(T_{s1}-T\right)- \frac{k_{g2}}{\rho c_{v}}\left(T_{s2}-T\right)\end{array}  \tag{2}\]  
where \(c_{v}\) is the specific heat capacity of helium gas, \(T_{s1}\) and \(T_{s2}\) are the temperature values of seal rings at their friction interfaces, and \(k_{g1}\) and \(k_{g2}\) are the convection heat transfer coefficients at the friction interfaces.