## 6 Experimental Estimation of Porous Material Permeability Coefficient (\(\kappa\))  
Figure 2 displays a photograph of a cylindrical PGB with physical dimensions listed in Table 1. The table also details the conditions for measurement of the supplied flow rate and estimation of \(\kappa\). A product specification sheet is available in [27]. Without a shaft installed in the bearing, measurements of the air supply pressure (\(p_{S}\)) and ensuing mass flow rate (\(\dot{M}_{\text{max}}\)) deliver an estimation of the permeability coefficient (\(\kappa\)) [20]. From Equation (11),  
\[\kappa=\frac{\dot{M}_{\text{max}}}{\pi\;D\;L}\frac{2\;t_{p}\;\mu\;R_{S}T}{(p_{S} ^{2}-p_{a}{}^{2})}. \tag{33}\]  
In the measurements, the supply pressure (\(p_{S}\)) increases from 2.4 bar to 6.5 bar (absolute), and an accurate turbine type meter records the flow. Figure 3 shows \(\dot{M}_{\text{max}}\) vs. (\(p_{S}{}^{2}-p_{a}{}^{2}\)), and the line fitting the data has a correlation factor \(R^{2}=1.0\). From the experimental data, \(\kappa=8.2\times 10^{-16}\) m\({}^{2}\), which is typical for the product. \begin{table}
\begin{tabular}{c c} \hline \hline Bearing Length, \(L=1.17\;D\) & 88.8 mm (3.5 inch) \\ \hline inner diameter, \(D\) & 76.2 mm (3.0 inch) \\ outer diameter, \(D_{out}\) & 99.5 mm (3.9 inch) \\ \hline Carbon-graphite permeability, \(\kappa\) & 8.2 \(\times\) 10\({}^{-16}\) m\({}^{2}\) \\ Porous layer radial thickness, \(t_{p}\) & 2.71 mm (0.11 inch) \\ Equivalent clearance for porous layer, \(c_{\kappa}\) & 3 \(\mu\)m \\ \hline Supply pressure, \(p_{S}\) & 2–8 bar \\ Exit pressure, \(p_{a}\) & 1 bar \\ Temperature, \(T\) & 294 K \\ \hline Air density at (\(p_{a}\), \(T\)), \(\rho_{a}\) & 1.2 kg/m\({}^{3}\) \\ viscosity at (\(p_{S}\), \(T\)), \(\mu\) & 18.3 \(\times\) 10\({}^{-6}\) Pa-s \\ gas constant, \(R_{g}\) & 287.05 J/(kg\(\cdot\)K) \\ \hline Parameters & For \(c=0.010\) mm \\ \hline Feed flow parameter & \(\Lambda_{\kappa}\)= \(\frac{3\;\kappa}{t_{p}c}\left(\frac{D}{c}\right)^{2}\) = 5.3 \\ \hline \multicolumn{3}{c}{At \(p_{S}=6\) bar, and \(\Omega=\omega=22\),618 rad/s (25 krm)} \\ Speed and frequency numbers & \(\Lambda_{\Omega}\)= 6 \(\frac{\mu\;\Omega}{p_{S}}\left(\frac{R}{c}\right)^{2}\) = 7, \(\Lambda_{\omega}\)= 12 \(\frac{\mu\;\omega}{p_{S}}\left(\frac{R}{c}\right)^{2}\) = 14 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Dimensions of porous journal bearing and operating conditions. Figure 2: Photograph of a porous material cylindrical journal bearing. **7. PGB Mass Flow Rate, Peak Pressure, and Aerostatic Stiffness vs. Clearance**  
From Equation (9), the pressure at the bearing mid-plane (\(z=0\)) is \(p_{(z=0)}=\sqrt{p_{S}^{2}+\left(p_{a}^{2}-p_{S}^{2}\right)\left[\cosh(\gamma \frac{L}{D})\right]^{-1}}\). For the PGB with geometry listed in Table 1 and \(e=0\), Figure 4 depicts the mass flow rate (\(\dot{M}\)),\(p_{(z=0)}\)and the aerostatic stiffness \(K_{S}\) (\(=K_{XXs}=K_{YYs}\)) vs. clearance for the PGB supplied with air at \(p_{S}=6\) bar. Note the equivalent clearance for the porous layer \(c_{\kappa}=(12\kappa\ t_{p})^{1/3}\)-\(3\) um. The graphs include results from the exact solution and the FE computational model (1872 elements = 72 circumferential \(\times\) 26 axial). Note that shaft speed has no effect on both parameters; see Reynolds Equation (7). The FE model flowrate is almost identical to the exact solution (difference less than 1%). For small clearances (\(c<0.040\) mm), \(\dot{M}\) decreases quickly, \(p_{(z=0)}\to p_{S}\), \(K_{S}\) reaches a maximum (184 MN/m) at \(c=0.010\) mm, and then sharply decreases as \(c\to 0\). On the other hand, for \(c>0.06\) mm, \(\dot{M}\rightarrow\dot{M}_{\max c\to 0}\)= 0.73 g/s, \(p_{(z=0)}\to p_{a}\), and \(K_{S}\to 0\), i.e., a complete loss of load carrying capacity. Note \(c=0.010\) mm (D/\(c=7610\)) \(\sim\) 3.3 \(c_{\kappa}\), though appearing to be tight (too small bearing clearances increase cost and make installation difficult), is quite appropriate for a gas bearing.