Study on Lubrication Theory of Orificed Throttle Based on Molecule Collision

Orifice restriction is a traditional form of throttle used in aerostatic guide way. Practically, self-excited vibration or even air hammer phenomenon will appear in gas film interior when supply gas enlarges or the film thickness decreases to micron scale. In order to analyze the process from the pressure inlet to the entrance of completely developed laminar flow, Reynolds Equation coupled with the first order velocity slip is used. Since it has rarefied property, turbulence model and molecular dynamics theory are adopted based on molecular collision theory. This paper reveals the internal cyclone law in gas chamber and the relationship between essence of the shock formation and the gas molecule velocity excitation. Further study shows that, supply pressure, film parameters and chamber parameters have important influences on cyclone phenomenon and shock wave.


Introduction
Orifice restriction is a traditional type of throttle in aerostatic guide way.Lots of research reports on orifice compensated air bearing at home and abroad are available.R.R.Willis published an articles on gas pressure in orifice plate in 1828, which is the earliest literature of gas lubrication [1] .G.Belfforte derived gas flow coefficient of orifice compensated thrust bearing by experiments.And he also derived empirical formula for calculation of gas consumption and pressure distribution with a view to Reynolds number analysis and the two throttling action in 2006 [2] .One important phenomenon that orifice throttling is different from the inherent throttling is that the gas chamber structure will form a cyclone.In that case, there may be a shock phenomenon near gas supply hole.
Carfagno S P found that through experiment: in area between the gas supply hole and the pressure gas film, the radial pressure distribution curve had a sudden drop according to laminar assumption [3] .The phenomenon of gas film fluctuation in the air static pressure guide rail was studied by AOYAMA.In order to restrain the fluctuation of gas film [4][5] , the structure of the throttle was reformed.
Through numerical simulation; CHEN proposed that orifice gas bearing had a large vortex in the groove, so the bearing is not stable enough [6] .According to Reynolds equation, Giving appropriate initial values and boundary conditions, Meng Xian and Liu Fan used turbulence model under isothermal conditions coming to the conclusion that the gas has spiral movement along the circumferential direction of gas chamber before the gas is discharged from the gas chamber [7] .In the view of the problem of air film fluctuation in air static pressure guide, Chen Dongju found that there was a close connection between the cyclone phenomenon and the film vibration [8] .The traditional explanation of the shock in the throttle is due to the improper parameter selection, and then it produces supersonic zone near the throttle hole.Gas velocity is changed to subsonic after shock wave.These documents had no complete description of the orifice throttling process.In this paper, Reynolds Equation where: ' l is the fluid slip length, ' a molecular tangential momentum regulation coefficient, which indicates the proportion of molecules that diffuse reflection at the surface of the object.O is the average free path of gas molecules.U is the relative velocity of the slide plate and the guide rail.u , v , Z is the velocity component of x, y, z direction.u , v , Z is calculated by the momentum equation of x, y, z direction and formula (1),formula (2).U is eliminated through gas state equation under isothermal condition.At last get the Reynolds equation of micro scale effect by the gas continuity equation: where: ' 2 k is Knudsen number.
Dimensionless mathematical model was shown as Formula (3), and the Reynolds equation was obtained under micro scale: where:

Theoretical research
The external high pressure gas flows through the gas supply hole.Partial gas velocity vector is not change, but the others enter the gas chamber directly in form of diffusion.The molecular without speed vector changes has an impact on the bottom of the gas chamber.
Molecular kinetic energy is converted into gas pressure energy, elastic deformation energy of support surface and Heat energy, etc. Hypothesis from the beginning of impact, all kinetic energy converted into pressure energy.
The speed of the molecules that impact the bottom of the gas chamber is reduced.Temperature at the bottom of the gas chamber can be reduced, which makes the Heat flux in this part has instantaneous maximum.This induces fluctuation of gas film.

02029-p.2
Assuming that the v V portion of the flow molecule is completely diffuse reflection, the rest (1 ) v V is completely specular reflection.In the case of specular reflection, normal velocity component is changed only at the impact surface.In the case of diffuse reflection, the velocity is in line with Maxwell distribution.Both specular reflection and diffuse reflection can change the movement direction of the gas molecules, which makes the angle between the two sides of the gas molecules that reflected less than 180 degrees.Through the molecular interaction and the collapse of the surface of the gas film, a small fraction of molecules get into gas film directly.
Given the thickness of gas film is h, most of the molecules collide with each other and collide with the surface of the gas chamber.The molecules rise to the upper surface of gas chamber with spiral movement.Then molecules form vortex.
When h is taken from the micron level, the area of 22 A must be less than the area of 21 A , the area of 22 A less than the area of 12 A .It is clear that at least two cross section flow occurs when the external high pressure gas flows from the inlet to the gas chamber and then to the rail gap.According to the flow continuity equation, the inlet cross section area is 10 times of the area of the outlet cross section.Therefore, the external molecular velocity out v of the gas chamber is 10 times of the molecular velocity in v in the gas chamber.The presence of a high pressure gas chamber makes a large number of high-speed molecules near the exit of the gas chamber.

Shock phenomenon
The system working pressure is about (0.5~0.6) MPa , generally no more than 1 MPa .The thickness of gas film is generally a few microns to a dozen microns.As shown in Figure4,Figure 5, Pressure of gas film in radial direction and radial velocity with different supply pressures that are 0.2 MPa ,0.5 MPa ,0.6 MPa ,0.7 MPa ,0.8 MPa ,1 MPa respectively and the gas film thickness is 10μm .When supply pressure suddenly increases to more than 0.8 MPa , molecular velocity in the gas chamber increase suddenly.Molecules through the gas film gap get out of the gas chamber.Velocity increases rapidly until the supersonic.The volume of a large number of high speed movement molecules is rapidly compressed and the collisions between molecules are intense.Then shock wave phenomenon occurs.The molecular kinetic energy is converted into internal energy after the impact of the collision and gas flow along the radius direction, which is equivalent to the one dimensional flow of the tube whose cross-section area continues to become large.Molecular motion velocity decreases rapidly to the subsonic speed.Phenomenon of shock is more obvious with the increasing of supply pressure, which will bring huge fluctuations affecting the stability and rigidity of the air static pressure guide.We should try our best to avoid this phenomenon.Under the premise of ensuring the bearing capacity, the working pressure should be controlled below 0.8 MPa .As shown in Figure 5, the frontier of the velocity curve is basically coincident under different air supply pressure.As shown in Figure 6 and Figure 7, pressure of gas film in radial direction and radial velocity with film thickness that are 6μm,7μm,8μm,9μm,10μm respectively and the supply pressure is 0.8 MPa .Figure 7 shows that when the gas film thickness is less than 10μm , the variation tendencies of radial velocity are almost the same.
When the film thickness is 10μm , the speed change is obvious.Shock wave phenomenon occurs.Figure 6 shows that, when the film thickness is 10μm , pressure in chamber change rapidly.The support ability of the high pressure gas chamber is reduced.Shock wave causes the pressure drop that causes a larger gas film wave fluctuation and the decrease of static stiffness of rail guide.
Gas molecules in the gas chamber have the same speed variation trend on the condition that film thickness is not sufficient to cause a shock.There are plenty of molecules to flow out through the gas film at the same time when the film thickness increases appropriately causing shock 02029-p.4 waves.Reducing gas film thickness appropriately can improve the bearing capacity of high pressure gas chamber.(2)The factors that influence the stability of air static pressure guide rail are revealed.The direction of collision between the incident gas molecules and the bottom surface of the gas film change, and interaction between the gas molecules and the interaction between molecules and the wall surface causing the phenomenon of cyclone.
The shock phenomenon caused by the pressure of the gas supply to the velocity of the excited molecules to the supersonic.The special structure of the orifice throttle caused the existence of the cyclone.

Fig 1
Fig 1 Diagram of orifice throttling in Descartes coordinate system As shown in Fig.1, a single orifice compensated throttle was located in the center of a disc aerostatic thrust bearing.Definition of effective flow sections:

Fig 2 5 ) 3 . 1 Fig 3
Fig 2 Diagram of supersonic zone around throttle As shown in Figure 2: Gas flow along the radius direction, which is equivalent to the one dimensional flow of the tube whose cross-section area continues to become large.The speed of molecules in the place 0 r r reaches sound speed.Obviously, it is a critical state.With speed increasing, the molecule is rapidly compressed when velocity reached supersonic.Meanwhile, shock wave generates at s r r .With the collision between each other

Fig 4 Fig 5
Fig 4 Pressure of gas film in radial direction with different supply pressures

Fig 6 Fig 7
Fig 6 Pressure of gas film in radial direction with different film thickness

Fig 8 4 Conclusions( 1 )
Fig 8 Pressure of gas film in radial direction with different chamber height