## 6 Experimental Estimation of Porous Material Permeability Coefficient (\(\kappa\))
Hence, the selection of the PGB clearance is most important to both keep a low flow rate while also providing enough load capacity and centering ability. Having established the validity of the analytical solution versus a numerical solution, Figure 5 depicts \(p_{(z=0)}/p_{a}\) mass flow rate, and the (centering) aerostatic stiffness (\(K_{S}\)) vs. clearance (\(c\)) for operation with supply pressure (\(p_{S}\)) = 2, 4, 6, and 8 bar. The clearance ranges from a thin \(c=0.002\) mm (D/\(c=38\),105) to \(c=0.1\) mm (D/\(c=762\)). Clearly, the film peak pressure, mass flow, and aerostatic stiffness increase with \(p_{S}\). As noted earlier, too large clearances (\(c>0.040\) mm \(\rightarrow\) D/\(c<\) 1,900) produce no film pressure (\(p\to p_{a}\)); hence, the PGB leaks too much (\(\rightarrow\dot{M}_{\max c\rightarrow\infty}\)) and is devoid of load capacity (\(K_{S}\to 0\)). For a very tight clearance (\(c<0.005\) mm), the mid-plane pressure \(p_{z=0}\to p_{S}\), the flow (\(\dot{M}\)) is small, and the direct stiffness is along a decreasing path. Independent of the supply pressure, the peak aerostatic stiffness appears at the same (narrow) clearance, \(c\sim 0.010\) mm = 3.3 \(c_{\kappa}\). At this \(c\), \(p_{z=0}/p_{a}=0.92\) and \(\dot{M}/\dot{M}_{\max}=0.37\). Figure 3: PGB measured mass flow rate \(\dot{M}_{\max}\) vs. (\(p_{S}{}^{2}-p_{a}{}^{2}\)) and a line fit (no shaft installed). Figure 4: PGB mass flow, pressure at bearing mid-plane, and aerostatic stiffness vs. clearance (\(c\)); (**a**) Mass flow rate; (**b**) Pressure at mid-plane \(p_{(z=0)}/p_{a}\); (**c**) Static stiffness \(K_{S}\). Analytical and FE predictions. Air supply pressure \(p_{S}=6\) bar. Aerostatic operation at \(e=0\). Figure 5: FGB static pressure at mid-plane, mass flow rate, and aerostatic stiffness vs. clearance (\(c\)). Air supply pressure \(p_{s}\) = 2, 4, 6, and 8 bar; (**a**) Mass flow rate; (**b**) Pressure at mid-plane \(p_{(z=0)}/p_{a}\); (**c**) Static stiffness \(K_{S}\). Aerostatic operation (\(\Omega\) = 0) and null eccentricity (\(e\) = 0). Vertical lines denote clearance \(c\) = 0.01 mm for largest stiffness. Although not shown for brevity, a dimensionless aerostatic stiffness scaled as \(K=(K_{S}\ c/W_{*})\) clusters the data in Figure 5c. At \(c=0.010\) mm, \(K\sim 0.6\) for all \(p_{S}\). Being of order (1), the parametrization of \(K\) allows one to quickly estimate the (maximum) centering stiffness of any PGB having the same (\(\kappa/t_{p}\)). The finding is nearly the same as that reported by [PERSON] et al. [11; 12] in 1968, \(K\sim 0.7\). The PGB product specification sheet [27] recommends a PGB with clearance \(c=0.010\) mm and, for operation with pressure supply \(p_{S}=5.15\) bar [60 psig], quotes a static stiffness \(K_{S}=159\) MN/m and flow of 13.2 L per minute (LPM) at standard conditions. For the same conditions, the current analysis predicts \(K_{S}=155.5\) MN/m and a flow rate equaling to 9.86 LPM. The agreement in stiffness is remarkable, while the difference in flows indicates differing clearances. For example, for \(c=0.012\) mm (\(\sim\)20% larger), the analysis predicts 13.2 LPM flow and \(K_{S}=149\) MN/m. Since the porous layer is affixed to the bearing housing using an adhesive, the manufacturer [27] specifies a maximum operating supply pressure of 7.9 bar (100 psig).