# Characteristics of Air Resistance in Aerostatic Bearings
## 2 Materials and Methods  
### Orifice Throttling Air Resistance Theory  
The Bernoulli equation is one of the basic theories used in fluid mechanics. Its essence is the conservation of fluid mechanical energy, where the sum of the kinetic energy, gravitational potential energy, and pressure potential energy is a constant [22]. Owing to the viscosity of air, the energy loss in the process should consider the actual pipe flow. A compressible fluid flowing at high speeds cannot exchange heat with air, so the flow can be regarded as adiabatic (i.e., isentropic flow) [23]. Assuming that the volume force is only gravity, the Bernoulli equation can be considered a one-dimensional flow with energy loss along the pipeline. Thus, the Bernoulli equation for an inviscid compressible fluid under adiabatic conditions can be expressed as follows:  
\[\frac{{{V_{1}}^{2}}}{2}+\frac{\gamma}{\gamma-1}\frac{p_{1}}{\rho_{1}}+gz_{1}= \frac{{{V_{2}}^{2}}^{2}}{2}+\frac{\gamma}{\gamma-1}\frac{p_{2}}{\rho_{2}}+gz_ {2}+gh_{f} \tag{1}\]where \(V_{1}\) and \(V_{2}\) are the flow velocity of a unit mass of fluid on cross sections 1 and 2, respectively; \(p_{1}\) and \(p_{2}\) are the pressure of a unit mass of fluid on cross sections 1 and 2, respectively; \(z_{1}\) and \(z_{2}\) are the position potential energy of a unit mass of fluid on cross sections 1 and 2, respectively; and \(h_{f}\) is the energy loss per unit mass of fluid from flow cross sections 1 to 2, also called the head loss, which usually includes the resistance loss and local resistance loss. Figure 1 presents a schematic diagram of the orifice restrictor. Based on the flow required by the restrictor, the inner diameter is set below 10 mm, forming a horizontal, short, and straight pipe. Equation (1) can be regarded as the Bernoulli equation when the viscous compressible fluid flows at a high speed under adiabatic conditions. This equation can also be used to derive the pressure loss calculation formula. Equation (1) can then be rewritten as  
\[\frac{\Delta p}{\rho g}=\frac{\gamma}{\gamma-1}\frac{p_{1}-p_{2}}{\rho_{m}g}= \left(\frac{{V_{2}}^{2}}{2g}-\frac{{V_{1}}^{2}}{2g}\right)+(z_{2}-z_{1})+h_{f} \tag{2}\]  
When the flow of the viscous compressible fluid is adiabatic, the change in the kinetic energy and potential energy for a fully developed laminar flow or turbulent flow is small and negligible. Therefore, Equation (2) can be rewritten as  
\[\Delta p=\frac{\gamma-1}{\gamma}\rho_{m}gh_{f} \tag{3}\]  
The energy loss of the viscous fluid flowing in the pipeline consists of pressure loss and local resistance loss. For the horizontal, short, and straight pipe, the local resistance loss can be ignored. Thus, the energy loss of the viscous fluid while moving in the pipe is equal to the pressure loss along the entire process. The pressure loss along the viscous fluid in the pipe is expressed as  
\[h_{f}=h_{l}=\frac{\Delta p}{\rho g}=\lambda\frac{l}{d}\frac{U^{2}}{2g} \tag{4}\]  
where \(\lambda\) is the resistance coefficient along the path; \(l\) is the tube length; \(d\) is the equivalent diameter of the pipeline, which is four times the ratio of the cross-sectional area to the circumference; and \(U\) is the average flow rate in the tube. By substituting Equation (4) into Equation (3), the pressure loss in the pipe can be calculated as follows:  
\[\Delta p=\frac{\gamma-1}{\gamma}\frac{\lambda\rho_{m}lU^{2}}{2d}=\frac{\gamma -1}{\gamma}\frac{\lambda\rho_{m}lq^{2}}{2 dA^{2}} \tag{5}\]  
where \(A\) is the cross-sectional area of the flow channel in the pipe. According to Hagen-Poiseuille low, the total flow cross a round tube can be expressed as:  
\[Q=\frac{\pi d^{4}}{128\mu l}\Delta p \tag{6}\]  
Figure 1: Schematic diagram of the orifice restrictor. So, the pressure loss can be expressed as:  
\[\Delta p=\frac{128\mu lQ}{\pi d^{4}} \tag{7}\]  
The Hagen-Poiseuille low can also provide a calculated method for pressure loss. When the air flows through the pipeline, the gas viscosity and roughness of the pipe wall influence the airflow [24].