# Analysis of an Elasto-Hydrodynamic Seal by Using the Reynolds Equation
## 2 Materials and Methods
### Design Methodology  
In this study, only the primary seal, which is illustrated in Figure 2, has been analyzed. The flow field inside the clearance region and the deformation in the sleeve was modeled by using the Reynolds equation and the elastostatics equations, respectively. The simulations were performed via both one-way and two-way coupled physics between the fluid and solid domains. For the one-way coupling, first, the flow field was solved. The pressure data was then transferred to the solid domain to calculate the deflection of the seal. For the fully coupled simulations, first, the fluid domain was solved. The pressure information was then imparted to the solid domain to determine the deformation on the seal, and vice versa. One might think that the analyses presented here might be easily achieved with analytical modeling. Although the flow field can be solved analytically and conveniently--when the flow is assumed to be in the axial direction only--the 2D nature of the bending makes it extremely difficult to solve the solid domain analytically. For proof-of-concept purposes, the fluid was chosen to be CO\({}_{2}\) rather than sCO\({}_{2}\) because the sCO\({}_{2}\) requires two-phase flow modeling, making it very challenging, and this was left as future work. The governing equations for the fluid flow and structural deformations are detailed in this section. Before elaborating on the  
Figure 1: Working mechanism of the proposed seal concept: (**a**) initial condition and (**b**) operating condition. modeling, the assumptions for both the fluid and solid domains are summarized as follows for convenience:  
* _The flow is steady._ There is no sudden impact assumed between the sleeve and the rotor as it is assumed that the pressure is increased gradually. * _The flow is laminar._ The laminar flow assumption was validated after the simulations (see Figure 11)
* _The flow is in 1D and along the shaft axis only._ Because of the high-pressure differentials along the seal clearance, the very high majority of the flow occurs in the x-direction. The viscous effects due to the rotation of the sleeve are strongly dominated by the pressure differentials in the \(x\)-direction. * _The rotational effects of the rotor are neglected since the majority of the flow is in the axial direction of the rotor and viscous effects are negligible as the working fluid is gas._ Thus, both the flow and solid domains are modeled as axisymmetric, and the inertial terms are omitted from the governing equations of motion of the seal. This assumption was validated in Figure 10.
* _The back ring, the stator, and the rotor are not modeled, and the deformations of these rigid structures are assumed to be zero._ This can be achieved by proper design of the seal. * _The rotor and the sleeve were assumed to be concentric in this study._ This can be achieved by proper design of the seal. The clearance between the seal and the rotor is relatively thin compared to the sleeve thickness of the seal. The narrow clearance can be modeled as a domain, and the Navier-Stokes equations can be solved for the prescribed domain. However, it requires very fine mesh in the clearance compared to the sleeve, which will incur issues in maintaining the mesh quality due to sudden transitions of in the mesh size at the fluid-solid interface. This will also greatly increase the computational time. Instead, the narrow clearance between the seal and the rotor can be modeled as a surface, and the Reynolds equation, a simplified version of the Navier-Stokes equations for thin flow regions, can be solved conveniently.