# Analysis of an Elasto-Hydrodynamic Seal by Using the Reynolds Equation
## 3 Results and Discussions
### Parametric Studies  
To see the effects of significant design parameters on seal performance, we performed parametric sweep analyses. The parameters that we considered included seal material, seal clearance, seal length, seal radius, and seal thickness. These parameters along with their ranges are listed in Table 2. Each range included 10 data points. It should be noted that when a parameter of interest was varied, all else remained at their design values listed in Table 1, except for the inlet pressure, \(P_{0}\). The effects of the seal material, seal clearance, seal length, seal radius, and seal thickness on the mass flow rate are given in Figure 11 in a mass flow rate vs. parameter range format. It can be seen that the mass flow rate is increasing with increasing values of clearance (Figure 11a,b), which makes sense since the larger clearances reduce the flow resistance, increasing the mass flow rate for the same inlet pressure. In other words, less energy is spent to overcome the frictional effects in the clearance. When comparing the steel and aluminum material cases, it can be seen that the mass flow rate values are higher for the steel material, which agrees with our intuitions because the elastic modulus of steel is higher than that of aluminum, making the steel seal stiffer to bend. As for the effect of seal length, the mass flow rate is decreasing with increasing seal length (Figure 11c,d), which is expected. This is because the longer the seal becomes, the larger the frictional losses are, reducing the mass flow rate. Again, the mass flow rate values were higher for the steel seal due to the same reason discussed above. Regarding the effect of rotor radius on the mass flow rate, as the rotor radius increases, the mass flow rate decreases (Figure 11e,f). This also agrees with our expectations since the thin-walled cylinder effect becomes stronger with increasing rotor diameters, reducing the effective stiffness of the seal. This, in turn, increases the bending of the seal, resulting in lower mass flow rates. Compared to the aluminum case, steel material resulted in larger mass flow rates because it has a higher elastic modulus and thus stiffness compared to aluminum. Lastly, the mass flow rate increases with increasing seal thicknesses (Figure 11g,h). This is reasonable because the larger seal thicknesses reduce the thin-walled cylinder effect, reducing the deformations on the seal, and thus increasing the mass flow rate. Again, compared to aluminum, the steel material resulted in larger mass flow rates because of its greater elastic modulus compared to aluminum. It should be noted that due to convergence issues that occurred during the simulations, some of the mass flow rate plots do not have data points at high inlet pressure values. Nonetheless, this did not obscure the trends observed and discussed above. Such trends can clearly be seen in Figure 11a-h.  
To validate the assumptions listed in Section 2.2, the Reynolds numbers for all simulations are listed on the right side in Figure 11a-h. For the flow through the annular ring, the Reynolds number is given by  
\[\text{Re}=\frac{\rho Vh}{\mu} \tag{13}\]  
where \(\rho\) is the density of the fluid, \(V\) is the velocity of the fluid, \(h\) is the height of the clearance, and \(\mu\) is the dynamic viscosity of the fluid. As can be seen, the Reynolds number was below 2000 in all cases, where the Reynolds equation can safely be used. However, it should be note that despite the accuracy, simplicity, and very fast convergence time the \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Range** \\ \hline
**EHD seal initial clearance, \(h_{0}\)** & 0.00635–0.0254 mm \\
**EHD seal length, \(L_{seal}\)** & 12.7–38 mm \\
**Rotor radius, \(r_{rotar}\)** & 25.4–50.8 mm \\
**EHD seal thickness, \(t_{seal}\)** & 0.25–0.75 mm \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters with their respective ranges. present model is valid only under the restrictive assumptions listed in Section 2.1.