# Analysis of an Elasto-Hydrodynamic Seal by Using the Reynolds Equation
## 3 Results and Discussions
### Proof-of-Concept Studies  
For proof-of-concept purposes, the seal material was selected to be structural steel. However, as discussed in Section 3.3, the aluminum seal material was also investigated. The working fluid was chosen to be CO\({}_{2}\). As known, the density and viscosity of CO\({}_{2}\) change with pressure and temperature. Considering the pressure range is taken to be 0.1 to 11 MPa for proof-of-concept purposes, the viscosity and density data, as a function of pressure and temperature, was extracted from the NIST's lookup tables and was imported into COMSOL to include the effect of the compressibility factor [43]. The outlet boundary condition for the fluid flow is zero-gauge pressure, and the inlet boundary condition is the working pressure. The pressure distribution and deformations of the sleeve design were solved for varying working (inlet) pressures. As for the meshing, a swept mesh scheme was used. First, a mapped mesh was used to mesh the outer surface of the cylinder with a 1 mm element size, with the surface mesh swept along the thickness, generating four layers of elements. A total of 20,160 hexahedra mesh elements were generated with 26,100 vertices, maintaining the minimum mesh quality of 0.99 and the element volume ratio of 0.99. Quadratic elements were used for finite element discretization. The fully coupled analyses are extremely difficult with analytical approaches. Therefore, we conveniently achieved the fully coupled solutions with the help of the Thin-Film Flow and Solid Mechanics modules of COMSOL. Figure 6 shows the pressure variation of the fluid along the clearance between the rotor and the seal for the inlet working pressure of 5 MPa obtained from the fully coupled simulation. The left and right sides are the inlet and outlet, respectively. As shown in Figure 6, the pressure is decreasing along the flow direction with the reduction rate being much prominent closer to the exit. This throttling effect of the seal is analogous to a typical throttling valve, where a similar pressure variation is observed across the valve. Depending on the strength of the throttling effect, the level of the recovered pressure varies. The deformation of the seal and the film thickness for varying inlet pressures and the von Mises stress plot at \(P_{0}\) = 5 MPa are shown in Figure 7. The film thickness is the difference between the initial clearance and the radial deformation of the inner surface of the sleeve. The film thickness is 0.0127 mm at the left end, and it decreases to a minimum value before increasing to the initial clearance at the fixed end on the right. The location of the minimum film thickness shifts towards the fixed end of the sleeve and increases nonlinearly with the working pressure. The sleeve deformation results proved our hypothesis, which was discussed in Section 2.1. The throttling effect is clearly visualized in Figure 7a. It is also interesting to note that the throttling effect becomes stronger with increasing inlet pressures. The deformation of the seal is also reflected on the von [PERSON] stress plot, clearly depicting the throttling effect in 3D view (Figure 7b). The inner surface of the sleeve was \begin{table}
\begin{tabular}{c c} \hline \hline
**Property/Parameter** & **Value** \\ \hline
**Young’s modulus of the aluminum seal (\(E\))** & \(70\times 10^{9}\) N/m\({}^{2}\) \\
**Density of CO\({}_{2}\) (\(\rho\))** & \(f(P,T)\) \\
**Working pressure (\(P_{0}\))** & Varying \\
**Ambient pressure (\(P_{e}\))** & 0 Gauge \\
**Gas constant of CO\({}_{2}\)** & 0.1889 kJ/(kg\cdot\)K) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Cont. subjected to varying fluid pressures, and the outer surface was under constant working pressure. As expected, the stress is maximized at the fixed end. **Pressure (Pa) variation along the clearance**  
The mass flow rate data, obtained from the fully coupled multiphysics simulation, for the design conditions listed in Table 1 are given in Figure 8. The inlet pressure ranged from 0.1 MPa to 11 MPa.