# Analysis of an Elasto-Hydrodynamic Seal by Using the Reynolds Equation
## 3 Results and Discussions
### Proof-of-Concept Studies
The mass flow rate (also referred as the leakage rate) is increasing with increasing inlet pressures and peaks at around 5.5 MPa, before decaying to lower values at higher inlet pressures. This quadratic behavior of the mass flow rate can become advantageous for sCO\({}_{2}\) power generation applications, where the inlet pressures can  
Figure 6: (**a**) Line plot of the pressure variation along the clearance for various inlet pressure conditions between 1 MPa and 11 MPa in intervals of 1 MPa. The inlet pressure value for each curve is the value on the corresponding curve at \(x=0\). (**b**) Surface plot of the pressure variation in the thin film between the rotor and sleeve at \(P_{0}=5\) MPa. Figure 7: (**a**) Variation of radial displacement of the sleeve and the film thickness along the sleeve length at different inlet pressure conditions between 0.2 MPa and 11 MPa in increments of 1 MPa. (**b**) von Mises stress on the inner surface of the sleeve for \(P_{0}=5\) MPa. reach and exceed 11 MPa. Similar leakage behavior was observed in [46; 47; 48; 49], where the researchers investigated the Morrison and Parry seals. Although the Morrison and Parry seals differ in seal configuration and application, the sealing mechanism and the underlying physics are considered to be the similar for both. Therefore, the findings of the current study are supported by the data reported in [46; 47; 48; 49]. It should be noted that the thermal effects in the solid domain are neglected for simplicity. Nevertheless, the goal of this paper was to prove the working mechanism of the proposed seal concept by using fundamental physics and modern engineering tools to lay a foundation for future model developments (see Section 4 for further discussion). As a general rule, a mesh independency study must be conducted to prove that the simulation results are not changing with decreasing mesh sizes or increasing number of mesh elements. Figure 9 shows the results of the mesh independency study using quadratic elements for FEM discretization and using 1, 0.9, 0.8, and 0.7 mm mesh sizes along the seal length. The pressure variation and the film thickness results did not change beyond the mesh element size of 1 mm. The rotor and the sleeve were assumed to be concentric in this study. Thus, the effects of rotational motion were assumed to be negligible since the majority of the flow would be in the axial direction of the rotor. Nonetheless, we carried out a comparison study to reveal the effects of the rotations on seal behavior. The comparisons between the simulation results of 3D with rotational effects, 3D without rotational effects, and 2D without rotational effects are presented in Figure 10. Apparently, the results are overlapped. Thus, it was concluded that it is safe to carry out the simulations in 2D by ignoring the rotational effects, which helped save significant computational time and resources. In fact, this is one of the apparent advantages of COMSOL's Thin-Film Flow module, specifically, that simulations can be conducted with minimal time and effort without loss of accuracy. In this study, a single simulation took less than 5 seconds to converge. Figure 8: Variation of the mass flow rate for different inlet pressure conditions, between 0.1 MPa and 11 MPa, obtained from the fully coupled multiphysics simulations at the design conditions listed in Table 1.  
Figure 10: Comparisons of the simulations results for \(P_{0}\) = 5.5 MPa: (**a**) 3D with rotational effects versus 2D without rotational effects and (**b**) 3D without rotational effects versus 2D without rotational effects. Figure 9: Mesh independency study using linear and quadratic shape functions for FEM discretization, and using 1, 0.9, 0.8, and 0.7 mm mesh sizes at a film thickness of 0.0127 mm and an inlet pressure condition of 5.5 MPa. The solution changes with decreasing mesh size with a linear shape function, while the results with the quadratic element show that the solution is mesh-independent.