Effect of surface roughness on lubrication performance of water-lubricated bearing for energy recovery turbocharger

In this study, the lubrication performance of the water-lubricated bearing of the energy recovery turbocharger was studied considering the surface roughness. By simulating the rough surface of the bearing with the Weierstrass-Mandelbrot function and combining the water film thickness equation, Reynolds average equation, Greenwood-Tripp contact model and motion equation, the theoretical model of water-lubricated bearing is established and its solution is developed. The variational rules for the hydrodynamic pressure and force, asperity contact pressure and force, and other performance parameters of the water lubricated bearings during start-up and stable operation were obtained by solving this model, and the effect of different surface roughness on the start-up performance and steady operation lubrication performance is studied. The results show that the hydrodynamic force increases rapidly and the asperity contact force sharp decreases during the early start-up stage, and the smaller the surface roughness, the faster the hydrodynamic increase and asperity contact force decrease. The thickness distribution and pressure distribution of water film with different surface roughness have slight differences in stable operation. Therefore, the surface roughness has an important effect on the lubrication performance of the water lubricated bearing in the start-up stage, but has little effect on the stable operation.


Introduction
Oceans cover 70 percent of the Earth's surface, 97.5 percent of which is seawater, while fresh water accounts for only 0.26 percent of the global total.Therefore, the conversion of sea salt water into fresh water can solve the global freshwater shortage, and it is necessary to vigorously promote the development of the desalination technology industry [1].Energy recovery turbochargers are mainly used in large-scale reverse osmosis membrane desalination systems, its rotor components are waterlubricated bearings, with seawater used as a lubricant [2], which has the advantage of being green and pollution-free.However, due to the low viscosity of seawater, it is difficult to form lubricating water films, and there is a large gap in carrying capacity compared to oil films.When the machine starts and stops frequently and the running speed changes, the water-lubricated bearing will be in the boundary lubrication state and friction state, resulting in frictional contact between the bearing rough surfaces, reducing the stability of the bearing-rotor system and affecting the normal operation of the machine [3].
When studying the lubrication performance of water-lubricated bearings, the influence of rough bearing surface on them is usually ignored, but the bearing surface cannot be completely smooth, and the bearing surface wear state is also different under different working conditions.The purpose of this study is to provide a theoretical basis for the machining accuracy of water lubricated bearing surfaces in energy recovery turbocharger.

Weierstrass-Mandelbrot function
The Weierstrass-Mandelbrot function is widely used to describe fractal surfaces in three dimensions for engineered rough surfaces [4].The rough surfaces of shaft and bearing of water-lubricated bearing are characterized by applying the Weierstrass-Mandelbrot function, which can be given by ( ) cos cos(arctan( ) )

( )
where D is the fractal dimension of the rough surface, and its value range is 2 < D < 3. G is the fractal roughness.

Water film thickness equation
There is a certain gap between the main shaft and the bearing surface of the water-lubricated bearing, and thickness of the water film can be determined by their relative positions [5], denoted by ( ) where ux and uy are the displacements of shaft in the x and y directions.

Average Reynolds equation
The water-lubricated bearing of the turbine energy recovery integrated machine uses seawater as the lubricating medium, and assuming it is an incompressible fluid, and does not consider the dynamic viscosity μ, density ρ changes with the temperature and pressure during operation.The hydrodynamic pressure of water-lubricated bearing is obtained by solving the average Reynolds equation considering the surface roughness [6,7], which is expressed as where p is the hydrodynamic pressure.σ is the standard deviation of the roughness surface.U is the axial surface velocity.ϕx and ϕy are pressure flow factors in the x and y directions [8], ϕs and ϕc are shear flow factors and contact factors, respectively [9].

Asperity contact model
When the hydrodynamic force cannot support the shaft, the rough surface between the shaft and bearing will come into contact, but this is actually the contact between the rough bodies on the two surfaces.
The G-T contact model [10,11] can be applied to the case of contact between two rough surfaces, and the contact pressure of individual bodies can be expressed as ( ) where f (zs * ) is the asperity function of the Gaussian distribution.Dsum and β are the density and curvature radius of asperities, respectively.zs * and d are the height of asperity and the standard distance.ω * = zs *d is the interference distance of the asperities between the two rough surfaces.

Motion equation
Based on the above calculations, the hydrodynamic force and asperity contact force of the waterlubricated bearing can be obtained.Moreover, considering the external load and its own gravity, the analysis of force on the shaft is performed according to Newton's law of motion [12], and two equations in the x and y directions are obtained, as follows where x and y are accelerations of shaft moving in the x and y directions.The x and y denote the horizontal and vertical directions, respectively.

Verification of the model
Figure 2 shows the comparison between the shaft locus solved by this model and Cui's simulation results [13] under the applied load of 4.63kN and 1.51kN, respectively.

Effect on start-up performance
Since the lubrication state of the water-lubricated bearing is in a mixed lubrication and friction state when the energy recovery turbocharger is started, there will be local rough contacts between the bearing surfaces.Therefore, it is necessary to investigate the effect of surface roughness of the water-lubricated bearing on the start-up performance by simulating the fractal roughness G=30×10 -6 m, 60×10 -6 m, 90×10 -6 m, respectively.
The parameters of the water-lubricated bearing are shown in table 1, with a start-up time of 0.03s.Figure 3 shows the shaft locus, eccentricity ratio, and attitude angle of water-lubricated bearings with different surface roughness during start-up.It shows that the laws of motion of the shaft locus during start-up are mostly the same for the three different surface roughness, that the initial shaft position is different for the different surface roughness at the beginning of start-up, the initial shaft position is higher when the roughness of the surface is larger (figure 3 (a)).This is due to a larger number of roughness peaks and peak values, resulting in more rough contact between the shaft and the bearing surface to support the load of the bearing-rotor system.At a later stage of start-up, when the shaft is lifted from the bearing surface, the shaft locus with different surface roughness begin to become consistent (figure 3 (b)).Because shaft leaves the bearing surface, it is supported by the water film, and the rough surface of the bearing is negligible relative to thickness of the water film.4 shows the change in force during start-up of the water lubricated bearing with different surface roughness, as well as the lift-off time and speed.Figure 4 (a) shows the variation of hydrodynamic force and asperity contact force with rotational speed in the transient start-up process under different surface roughness.It shows that the hydrodynamic force gradually increases with the rotation speed during the initial start-up phase.When the surface roughness is small, the increase is faster and the corresponding asperity contact force decreases faster.Moreover, the smaller surface roughness helps the hydrodynamic force to reach its maximum quickly, and the asperity contact force decreases to zero.As a result, the smaller the surface roughness, the shorter the lift-off time for the shaft to leave the bearing surface and the smaller the lift-off speed, as shown in figure 4

Effect on operation performance
During stable operation, the energy recovery turbocharger mainly relies on the lubricating water film between the shaft and bearing to support and separate the two.However, the lubricating water film is extremely thin, the surface roughness of bearing has an important influence on the thickness and pressure distribution of the water film during stable operation.
Figure 5 shows the water film thickness distribution of the water-lubricated bearing under different surface roughness during stable operation.It shows that the water film thickness distribution is nearly the same for different roughness, but there is a slight difference in the value of water film thickness.When the surface roughness is small, the corresponding minimum water film thickness is also small.With the increase of surface roughness, the minimum water film thickness also increases.This is due to the fact that an increase in surface roughness leads to an increase in the amount and size of roughness on the bearing surface, and thus to an increase in the minimum film thickness required for the same load.

Conclusion
This study analysed the effects of surface roughness on start-up and stable operation performance by studying the evolutions of the hydrodynamic pressure and force, asperity contact force.During the initial start-up phase, the increase of hydrodynamic force and decrease of asperity contact force is faster when the surface roughness is smaller, and therefore the lift-off time and speed required for the shaft to leave the bearing surface are shorter.In stable operation, the thickness distribution and pressure distribution of the water film for different surface roughness are nearly the same, but slightly different.As the surface roughness increases, the minimum water film thickness slightly increases and the hydrodynamic pressure slightly increases.Therefore, in the rough contact region, the surface roughness of the bearing has a larger effect on its transient start-up, but a smaller effect on its operational performance after the shaft leaves the bearing surface.

Figure 1 .
Figure 1.Structure diagram of the energy recovery turbocharger.

Figure 2 .
Figure 2. Comparison calculation between this model and Cui's results.Based on the comparison results, it is shown that the shaft locus calculated by the numerical model is in excellent agreement with Cui's simulation results, which demonstrates the reliability of the model calculations.

Figure 3 .
Figure 3. (a) Locus of shaft center, (b) Eccentricity ratio and attitude angle.Figure 4 shows the change in force during start-up of the water lubricated bearing with different surface roughness, as well as the lift-off time and speed.Figure 4 (a) shows the variation of hydrodynamic force and asperity contact force with rotational speed in the transient start-up process

Figure
Figure 3. (a) Locus of shaft center, (b) Eccentricity ratio and attitude angle.Figure 4 shows the change in force during start-up of the water lubricated bearing with different surface roughness, as well as the lift-off time and speed.Figure 4 (a) shows the variation of hydrodynamic force and asperity contact force with rotational speed in the transient start-up process

Figure 4 .
Figure 4. (a) Hydrodynamic force and asperity contact force, (b) Lift-off time and lift-off speed.

Figure 5 .
Figure 5. Water film thickness distribution with different roughness.

Figure 6
Figure6shows the hydrodynamic pressure distribution of water-lubricated bearings with different surface roughness.It shows that under different roughness, the pressure value and pressure distribution of lubricating water film are almost the same, but there is a slight difference in the value of hydrodynamic pressure.As the surface roughness increases, the hydrodynamic pressure increases slightly in the axial and circumferential directions.