# Characteristics of Air Resistance in Aerostatic Bearings
## 2 Materials and Methods
### Orifice Throttling Air Resistance Theory
The velocity distribution of the air in the vertical section of the tube is nonuniform. The air close to the boundary layer has a low velocity. The main features of air resistance in a pneumatic system are pressure loss, system heating, and system efficiency reduction. Equation (5) shows that the ratio of pressure loss \(\Delta p\) is related to the heat ratio \(\gamma\), resistance coefficient along the path \(\lambda\), air density \(\rho\), pipeline length \(l\), equivalent diameter of the pipeline \(d\) (as the cross-sectional area \(A\) and equivalent diameter \(d\) can be converted to each other, they are regarded as the same variable), and flow \(q\). For air, the heat ratio \(\gamma\) is 1.4; the air density represents the compressibility. In addition, the air volume is restricted by the shape of the inner flow channel. For horizontal, short, and straight pipes, the cross-sectional area \(A\) of the internal flow and the length of the inner flow channel \(l\) are also important parameters that affect the air resistance and are mostly constant. The only uncertain and variable parameter is the resistance coefficient along the path \(\lambda\), which is easily affected by the roughness of the inner wall of the pipeline. Therefore, when the air flow state does not change, the air resistance is determined by the air density \(\rho\), length of the pipeline \(l\), and equivalent diameter \(d\) of the pipeline. However, when the air flows at a high speed, it is turbulent, and the increase in air pressure not only changes the density \(\rho\) but also causes a change in the resistance coefficient along the path \(\lambda\). Here, the capillary tube is considered an example. To address the inaccurate calculation of the resistance coefficient along the path, a new parameter is defined, and the symbol \(R_{g}\) is used to indicate air resistance. The function of air resistance is expressed as  
\[R_{g}=f(k,\rho,l,d) \tag{8}\]  
where \(R_{g}\) is the air resistance and \(k\) is a parameter that changes with the air flow state and includes the resistance coefficient along the path \(\lambda\). Based on Equation (5), when the air flow remains unchanged, \(\Delta p\) is proportional to \(q^{2}\); thus, Equation (5) can be rewritten as  
\[\Delta p=R_{g}q^{2} \tag{9}\]  
From Equation (9), the air resistance is dimensional, and the unit of air resistance can be determined as kg/m\({}^{7}\). In summary, a definition of air resistance applicable to capillary tubes is proposed, whose dimensional formula is ML\({}^{-7}\). Air resistance represents the transport capacity of air in the pipeline. The greater the air resistance, the greater the pressure loss after the air flows through the pipeline. The formula of air flowing at high speeds in the capillary tube is  
\[\frac{\Delta p}{\rho g}=\frac{\gamma}{\gamma-1}\frac{p_{1}-p_{2}}{\rho_{m}g}= \left(\frac{{V_{2}}^{2}}{2g}-\frac{{V_{1}}^{2}}{2g}\right)+(z_{2}-z_{1})+h_{f} \tag{2}\]0  
The characteristics of the parameter \(k\) will be further analyzed  
flow density and average flow rate of the measured pipeline. The mathematical model is as follows:  
\[\frac{\Delta p}{\rho g}=\frac{\gamma}{\gamma-1}\frac{p_{1}-p_{2}}{\rho_{m}g}= \left(\frac{{V_{2}}^{2}}{2g}-\frac{{V_{1}}^{2}}{2g}\right)+(z_{2}-z_{1})+h_{f} \tag{2}\]1  
where \(d_{1}\) is the hydraulic diameter of the measured pipeline and \(U_{s}\) is the local speed of sound. After obtaining the average flow rate in the pipeline under test \(q_{m}\) from the average flow rate \(U_{1}\), the air resistance model of the pipeline under test can be obtained:  
\[\frac{\Delta p}{\rho g}=\frac{\gamma}{\gamma-1}\frac{p_{1}-p_{2}}{\rho_{m}g}= \left(\frac{{V_{2}}^{2}}{2g}-\frac{{V_{1}}^{2}}{2g}\right)+(z_{2}-z_{1})+h_{f} \tag{2}\]2  
Using COMSOL simulation software, stainless-steel pipes with inner diameters of 2, 3, and 4 mm were selected as the simulation objects. For each diameter, lengths of 0.5 and 1.0 m were selected.