## 4 The First-Round Optimal Design of Aerostatic Thrust Bearings
### FEM Modeling of Aerostatic Bearings  
The flow status of an orifice restrictor is simplified as the compressible flow through an ideal nozzle, and the orifice is simplified as a point. When the air flows through the orifice, the air pressure drops from \(P_{s}\) to \(P_{d}\) immediately. Considering the difference between the actual mass flow rate of the orifice restrictor and the ideal theoretical mass flow rate, a constant called discharge coefficient \(C_{d}=0.8\) is introduced to correct the mass flow rate. The mathematical models for calculating the theoretical mass flow rate of the orifice restrictor are Equations (1) and (2) [20]. \[m=C_{d}Ap_{s}\sqrt{\frac{2\rho_{a}}{p_{a}}}y_{s}, \tag{1}\]  
\[\psi_{s}=\left\{ \begin{array}{c}\left[\frac{k}{2}\left(\frac{2}{k+1}\right)^ {(k+1)/(k-1)}\right]^{1/2};\left(\frac{p_{d}}{p_{s}}\leq\beta_{k}\right)\\ \left\{\frac{k}{k-1}\left[\left(\frac{p_{d}}{p_{s}}\right)^{2/k}-\left(\frac{ p_{d}}{p_{s}}\right)^{(k+1)/k}\right]\right\};\left(\frac{p_{d}}{p_{s}}>\beta_{k} \right)\end{array} \right.\left(\beta_{k}=\left(2/(k+1)\right)^{(k+1)/k}\right) \tag{2}\]  
where \(A=\pi d^{2}/4\) is the cross-section area of orifice, \(P_{a}\) is the atmosphere pressure, \(\rho_{a}\) is the density of air under standard state, \(\psi_{s}\) is flow rate function, and \(k\) is the specific heat ratio of air. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \(D_{1}\) (mm) & \(D_{2}\) (mm) & \(D_{3}\) (mm) & \(d\) (mm) & \(h\) (\(\mu\)m) & \(d_{1}\) (mm) & \(h_{1}\) (mm) & \(n\) & \(L\) (mm) \\ \hline
60 & 88.3 & 130 & 0.2 & 16 & 6 & 0.05 & 6 & 30 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The design parameters of the aerostatic thrust bearing. Figure 2: The configuration of an aerostatic spindle and the design parameters of its thrust bearing. (**a**) The configuration of the aerostatic spindle and (**b**) the design parameters of the thrust bearing. The fluid domain of the aerostatic thrust bearing is described using the governing equation in the cylindrical coordinate system [20] below:  
\[\frac{\partial}{\partial\xi}\left(\overline{h}^{3}\frac{\partial\overline{p}^{2} }{\partial\xi}\right)+\frac{\partial}{\partial\theta}\left(\overline{h}^{3} \frac{\partial\overline{p}^{2}}{\partial\theta}\right)+\overline{r^{2}Q} \delta_{i}=\Lambda\overline{r}\frac{\partial}{\partial\theta}\left(\overline{ h}\overline{p}\right), \tag{3}\]  
where \(\overline{h}\) is the dimensionless thickness of air film, \(\overline{p}\) is the dimensionless pressure, \(\overline{\xi}\) is the dimensionless coordinate in the radial direction, \(r\) is the coordinate in the radial direction, and \(\theta\) is the coordinate in the circumferential direction. \(\overline{Q}\) is the mass flow rate factor of the orifice restrictor. \(\delta_{i}\) is a constant coefficient with a value of 1 or 0 (in the computational domain where there is an orifice, it equals 1; otherwise, it equals 0.) \(\Lambda\) is the dimensionless bearing number. \[\overline{h}=\frac{h}{h_{m}},\overline{r}=\frac{r}{r_{0}},\overline{\xi}= \ln\overline{r},\overline{p}=\frac{p}{p_{s}},\overline{Q}=\frac{24\eta r_{0} ^{3}p_{s}}{h_{m}^{3}p_{s}^{2}p_{s}}\rho\tilde{v},\Lambda=\frac{12\eta v_{ \theta 2}r_{0}}{h_{m}^{2}p_{s}}, \tag{4}\]  
where \(h\) is the thickness of air film. \(r_{0}\) is the characteristic length, \(h_{m}\) is the reference air film thickness, \(p\) is the gas pressure distribution function, \(\rho\) is the air density, \(\eta\) is air dynamic viscosity, \(\tilde{v}\) is the velocity of orifice gas flow, and \(v_{\theta 2}\) is the velocity components in the circumferential direction. In this paper, the performance of aerostatic thrust bearing is analyzed under motionless conditions, and therefore, \(\Lambda\) = 0.