The design of hydrodynamic water-lubricated step thrust bearings using CFD method

– Water-lubricated bearings are expected to be widely used because of convenience, green, safe and energy saving. The purpose of this study is to provide references for designing hydrodynamic water-lubricated step thrust bearings. The numerical analysis is undertaken under the condition of diﬀerent pad dimensions, step heights, step positions, water ﬁlm thicknesses and rotational speeds of thrust rings based on computational ﬂuid dynamics (CFD). The results including pressure distribution, load carrying capacity, friction torque and friction coeﬃcient are gained and compared for optimizing geometry parameters. A reference to determine water-lubricated step thrust bearing dimensions and a formula to check the minimum water ﬁlm thickness are proposed.


Introduction
Step thrust bearings are widely used in pumps, highspeed turbomachineries to support axial loads because of its easy manufacturing and low cost [1,2].Presently the conventional lubricant is oil, but it has many disadvantages such as serious waste of resources, environmental pollution caused by oil leakage, explosive hazard, large bulk and complex structure, poor flexibility and maneuverability [3].Moreover, oil-lubrication is infeasible in some special condition such as nuclear pumps.Water resources are rich and water has advantages of convenience, green, safe and energy saving, therefore, more and more attention has been paid to water-lubrication [4].However, water has a lower viscosity than that of oil, and the load carrying capacity of water-lubricated bearings is much smaller than that of oil-lubricated bearings.Works on water-lubricated bearings [5][6][7][8] showed that the bearings usually operate in the regime of mixed lubrication and the lubrication mode might convert to hydrodynamic when load decreases.Wang et al. [5,6] studied the critical load, considering the sudden increase in the friction coefficient as the transition of lubrication mode.The present work focuses on the hydrodynamic water-lubrication, considers the minimum water film thickness as a criterion of lubrication mode and aims at giving a reference for designing hydrodynamic water-lubricated step thrust bearings.
a Corresponding author: yinzw@sjtu.edu.cn Step thrust bearings have traditionally been analyzed using Reynolds equation since the early twentieth century.In 1918 Rayleigh [9] analyzed several kinds of slider bearing profiles and found that a step convergence showed in Figure 1a would provide the greatest load per unit width neglecting side leakage.Archibald [10] considered side leakage when studying the load carrying capacity of a square step slider.Kettleborough investigated step thrust bearings with an electrolytic tank in 1953 [11] and by experiments in 1955 [12].Then Rohde [13] used the finite element method to solve the problem.The optimum variables they gained for designing a straight step bearing are summarized in Table 1.It indicates that the optimum step height δ is about 0.7 times of the minimum film thickness h 2 , and the optimum L 1 /L 2 changes with pad width-to-length ratio B/L.However, the sector pad was simplified to rectangle slider when they did the theoretical derivation or numerical simulation, and oil grooves were ignored, thereby reducing computation.There are also many research works in which infinite width step slider bearing hydrodynamic problems were solved by solving Reynolds Equation using numerical schemes or analytical method [1,14].
In recent years, with the development of computer technology, many researchers have used commercial CFD programs which are based on the full Navier-Stokes equations to solve lubrication problems.Chen et al. [15] demonstrated the validity of using CFD software to handle hydrodynamic lubrication problems pertaining to slider bearings, step bearings, journal bearings and  The present work investigates the step thrust bearings with sector pads and straight radial grooves under hydrodynamic lubrication by water.Three-dimensional CFD models with different bearing dimensions, film thicknesses, step heights and positions were computed to study the effects of them on load carrying capacities, friction torques and friction coefficients.

Step thrust bearing model
Figure 1b shows the three-dimensional flow model of a sector pad and its groove.The thrust bearing is submersed in water.The hydrodynamic action generates dynamic pressure in water, primarily in the convergent part of the thrust pad, to counteract the load thereby separating the ring surface from the bearing surface with a thin lubricant film.Geometry parameters -inner radius R 1 , outer radius R 2 , number of pads n, pad ratio k, step height δ and step position L 1 /L 2 -all influence the load carrying capacity of step thrust bearings.According to the design of oil-lubricated thrust bearings [20], R 2 is usually 1.5−3 times of R 1 .The number of pads n is generally 6 to 12.The pad ratio k, the percentage of pad area in the whole thrust surface, is typically 0.7−0.85.Pad width-to-length ratio B/L is a pad parameter determined by R 1 /R 2 , n and k, and the relation is It has an effect on the optimum value of the step height δ and step position L 1 /L 2 .Rotational speed N , fluid viscosity μ and the minimum film thickness h 2 determine the bearing carrying performance as well.Rotational speed and fluid viscosity are determined by work conditions.The minimum film thickness should not be less than a safety value depending on surface roughness and system vibration, or the lubrication mode might change from full dynamic lubrication to boundary or mixed lubrication.3 Numerical analysis

Governing equations
The flow is considered laminar isothermal, steady and incompressible, with zero gravitational and other external body forces.The bearing is fully submerged into water.To solve such flow, the following governing equations must be solved: Mass conservation equation Momentum conservation equations When flow enters the groove, pressure might fall below the saturation water vapor pressure, and the liquid would rupture and cavitation occurs.Thus cavitation is taken into account.In the present work the CFD code AN-SYS FLUENT is used.There are three available cavitation models in ANSYS FLUENT: Singhal et al. model, Zwart-Gerber-Belamri model and Schnerr and Sauer model.The Singhal et al. model is numerically less stable and more difficult to use.The Zwart-Gerber-Belamri and the Schnerr and Sauer models are robust and converge quickly [21].In this case CFD models with the latter two cavitation models are calculated when studying the mesh refinement.Table 6 shows that the Zwart-Gerber-Belamri model is less sensitive to mesh density.Thus the Zwart-Gerber-Belamri model is employed.
In cavitation, the liquid-vapor mass transfer (evaporation and condensation) is governed by the vapor transport equation [21]: where R g and R c account for the mass transfer source terms connected to the growth and collapse of the vapor bubbles, α v is vapor volume fraction and ρ v is vapor density.In Zwart-Gerber-Belamri model, R g and R c are defined as follows [22]: where F evap = evaporation coefficient = 50, F cond = condensation coefficient = 0.01, R b = bubble radius = 10 −6 m, α nuc = nucleation site volume fraction = 5 × 10 −4 , ρ l = liquid density, p v = pressure of vapor.Supposed that there's no installation error and the load of a thrust bearing is distributed uniformly over all pads, the load and friction torque of a thrust bearing can be calculated by integrating the pressure and shear stress over the rotating wall as follows: The friction coefficient is To compare with results in the literature, nondimensional pressure, non-dimensional load carrying capacity and non-dimensional friction torque are defined as

Boundary conditions
The boundary condition is set as shown in Figure 1b.Rotational periodic boundary condition is used to simplify the flow model and to reduce the computational cost.The operating pressure is set to 101 325 Pa.Since the bearing is fully submerged, the pressure at the inlet and outlet boundaries is taken as zero (gauge pressure).A no-slip condition is imposed on the solid walls.The bottom wall is stationary and the upper one is assumed to be rotating at a constant rotational speed N .

CFD models
In the present study, water properties at 20 • C listed in Table 2 are employed.In all cases, the groove depth h g = 2 mm and pad ratio k = 0.75.For the purpose of finding relations between B/L, δ and L 1 /L 2 , the inner radius R 1 is set to 30 mm, number of pads n is 12, rotational speed N is 3000 r.min −1 and R 2 , δ * , L 1 /L 2 vary as shown in Table 3.The non-dimensional step height δ * and the step position L 1 /L 2 vary to seek out the optimum ones.The outer radius R 2 changes to find the effect of B/L on the non-dimensional load W * , the optimum δ * and the optimum L 1 /L 2 .Then models with different sizes and rotational speeds are analyzed to get design reference for bearing dimensions.Inner radius R 1 ranges from 5 mm to 40 mm and outer radius R 2 is 1.5 times, twice and 3 times R 1 , respectively.The number of pads varies as shown in Table 4. Pad width-to-length ratio B/L can be calculated from equation (1).The step height δ and the step position L 1 /L 2 are the optimum values gained from the study above.The minimum film thicknesses increase as the bearings get larger to ensure that the bearings operate under hydrodynamic lubrication.The values are shown in Table 5.The rotational speeds range from 100 r.min −1 to 5000 r.min −1 .
Due to the existence of the thin film in hydrodynamic bearings, the dominating feature for bearing CFD models is the large aspect ratio of the grid which is 330 ∼ 4500 in this study.However, the suggested value of aspect ratio is less than 100 or 200 for normal analysis.Thus double precision calculations are employed to avoid the negative influence of the large aspect ratio according to ANSYS FLUENT user manual [21].The CFD models are meshed using hexahedron grids in Gambit 2.3.In order to obtain accurate solutions, a mesh refinement study is carried out.Since the groove area contributes little to the load carrying capacity, the groove length L g is divided into 15 cells and the groove height h g is divided into 20 cells for all cases.An expansion ratio of 1.4 is employed in h g to avoid sudden change of mesh density near the thin film thickness.A uniform mesh is employed in other edges of the model.The influence of the mesh density of h 2 , δ, L 1 , L 2 , B on load for the bearing model with R 1 = 30 mm, R 2 = 45 mm, n = 12, h 2 = 10 μm, δ = 7 μm, L 1 /L 2 = 1, N = 3000 r.min −1 is studied and the results are summarized in Table 6.When the Zwart-Gerber-Belamri model is used, the relative difference in load between case 6 and case 9 is only 0.2%.Based on this, case 6 is employed in this study.Appropriate mesh sizes are also found for other sizes of bearing models.Non-dimensional radii of the maximum pressure R * pmax , for pads with B/L = 1.02, 1.70, 4.00 are 0.537, 0.62 and 0.78 respectively.Figure 3 presents the distribution of p * at R pmax in Figures 2a-2d.Comparing the three solid lines, it is found that the maximum p * on the rotating wall rises from 0.244 to 0.305 as B/L increases from 1.02 to 1.70, but only increases by 0.007 as B/L increases from 1.70 to 4.00.It indicates that as the pressure center moves outwards, the pressure buildup is still affected by side leakage for pads with large B/L values.The line "B/L = 1.02 sw" shows that at the step (θ = 20 • ) and at the entrance to the thin film from the groove (θ = 7.5 • ) pressure on the stationary wall has a sudden increase, which is known as the "ram effect" [23].When the fluid flows into the groove (θ = 0 • , θ = 30 • ), pressure on the stationary wall drops rapidly, which might cause cavitation if it is less than 2 340 Pa, i.e. gauge pressure -98 985 Pa.The vapor fraction distribution of the model in Figure 2a is shown in Figure 4.As the fluid velocity near the outer radius is larger than that near the inner radius, the pressure drop is more significant and cavitation occurs more easily.From the CFD analyses, it is found that the cavitation zone will get larger when the minimum film thickness decreases or the fluid velocity increases.

Effect of step height
Figure 5 presents the non-dimensional load carrying capacity W * versus the non-dimensional pad height δ * for a range of h 2 from 5 μm to 10 μm with B/L = 1.02 and L 1 /L 2 = 1.25.Results illustrate that although the optimum step height is different for different minimum film thicknesses, the optimum ratio of them, the nondimensional pad height δ * , is always around 0.67, similar     is less than 0.4 or more than 1.1, but less than 1 percent when δ * is between 0.6 and 0.8.Figures 7a-7f present the influence of non-dimensional pad height δ * on non-dimensional friction torque M * fr and friction coefficient f for pads with B/L = 0.39, 1.02 and 3.12 respectively.As the step height increases, M * fr decreases monotonically, and f first decreases and then increases with a minimum at δ * = 0.8.The results follow the same trend for different B/L values.Figures 7b, 7d, 7f also show that f decreases by almost 25% from δ * = 0.3 to δ * = 0.8, but it varies slightly when δ * is in the range [0.6, 1.0].Thus considering synthetically the load carrying capacity and friction performance, the step height can be designed as follows:

Effect of step position
Figure 8 presents the dependence of non-dimensional load carrying capacity on step position, i.e.L 1 /L 2 , for pads with different widths and δ * = 0.7.Similar to the relation of W * with δ * , there is an optimum L 1 /L 2 maximizing W * .It can be seen that the values of the optimum L 1 /L 2 vary with B/L. Figure 9 illustrates that the optimum L 1 /L 2 resembles a sigmoid curve with the pad width-to-length ratio B/L.It is about 1.1 for small width pads (B/L < 1) and 1.85 for large width pads (B/L > 3).For a range of B/L [1,2] which is commonly used in design, the optimum L 1 /L 2 increases rapidly from 1.2 to 1.75.For large width pads, the optimum L 1 /L 2 is smaller than Rohde's solution (see Tab. 1), which can also be explained by the centrifugal effect in the rotating sector film.
Figures 10a-10b present non-dimensional friction torque M * fr and friction coefficient f versus L 1 /L 2 with B/L = 1.02 respectively.Compared with Figure 7, the curves have the same trend.Nevertheless, the step position has a larger effect on friction torque, which decreases by about 14% with L 1 /L 2 increasing from 0.75 to 3, as shown in Figure 10a.The friction coefficient gets the minimum value at L 1 /L 2 = 1.75-2, about 0.55−0.8larger than the value of L 1 /L 2 at the maximum load, but f changes very small from L 1 /L 2 at the maximum load to L 1 /L 2 at the minimum friction torque.The same rule is found when B/L is equal to other values.So the nondimensional load carrying capacity can be taken as the principle objective function and L 1 /L 2 can be designed according to Figure 9.  show a design reference obtained by calculating numerical models with different dimensions and rotational speeds.The number on the curve is the load carrying capacity.The outer diameter is 1.5 times, twice and 3 times the inner diameter, respectively.When the load and rotational speed are defined, the bearing inner diameter can be selected from the graphs.For example, as shown in Figures 11-13 (the dash-dotted line), when the rotational speed is 3500 r.min −1 and 200 N is required for the load carrying capacity, the inner diameter should be over 72.5 mm when D 2 = 1.5D 1 , 40 mm when D 2 = 2D 1 or 22 mm when D 2 = 3D 1 .Designers could determine the bearing diameters according to the mounting dimensions.can also be used to check if a specified bearing can meet the load requirements.This is a rough design of bearing diameters.Designers should check the minimum film thickness for further determination.

Effect of pad width-to-length ratio and check computation of the minimum film thickness
Figures 14a-14c present the obtained non-dimensional load carrying capacity W * , non-dimensional friction torque M * fr and friction coefficient f , versus pad widthto-length ratio B/L for δ * = 0.7 and L 1 /L 2 = 1.25, respectively.As B/L increases from 0.39 to 4, W * grows more than 10-fold, from 0.018 to 0.214, while M * fr only increases by 70%.The friction coefficient sharply decreases from 0.037 to 0.008 when B/L increases from 0.39 to 1, and then gradual decreases to 0.001.The slope of the load curve is 0.1 for small values of B/L (<1), while reduces to 0.034 for large values of B/L(>2.5).This is because for small B/Lvalues, the pressure buildup is more affected by side leakage.W * for B/L = 0.5, 1, 2, 4 is larger than that obtained by Rohde, and W * for B/L = 4 is even larger than Rayleigh's solution for infinite width sliders, W * = 0.206.It indicates that sector pads provide greater load carrying capacity than rectangular ones.This can be explained by the pressure distributions for pads with different widths, which is presented in Figure 2. As B/L increases, pressure center in a sector pad moves towards the outside radius, where the fluid velocity is much larger than that at the middle radius.Thus larger pressure and greater load are gained in a sector pad than those in a rectangular pad.
The minimum film thickness h 2 can be calculated from W * .As illustrated in Figure 14a, B/L is the main influence factor on the non-dimensional load carrying capacity W * for a constant non-dimensional step height δ * and step position L 1 /L 2 .This means for a certain B/L value, W * is definite.Then the minimum film thickness In order to ensure that the bearings operate under hydrodynamic lubrication the surface roughness values of the thrust pad and the thrust ring should satisfy the following formula [25]: where R q, p = rms (root mean square) surface of the pad surface, R q, r = rms surface roughness of the ring surface.If the surface roughness values could not meet the design requirements, the bearing parameters should be corrected according to equation (10) and equation (11).

Conclusions
In this research, the hydrodynamic lubrication model of the water-lubricated step thrust bearing with sector pads and straight radial grooves has been fully studied using CFD method.Zwart-Gerber-Belamri model is used to simulate cavitation in the bearings.The effects of step height, step position and pad width-to-length ratio on lubrication properties are discussed.A reference to   5. The minimum film thickness should be checked in order to ensure that the bearings operate under hydrodynamic lubrication.

Fig. 1 .
Fig. 1.(a) Three-dimensional flow model of a rectangular pad (not to scale) which is studied in the literature.(b) Threedimensional flow model of a sector pad and its groove (not to scale).(c) Geometry parameters of model in (b).(Stationary wall is the thrust pad and moving wall is the thrust ring.)

4. 1
Figures 2a-2d present the gauge pressure distribution on the pad surface (the stationary wall) for pads with δ * = 0.7, L 1 /L 2 = 1.25, B/L = 1.02 and pressure distribution on the ring surface (the rotating wall) for pads with different B/L values.Pressure distribution for pads with B/L = 4.00 is presented here for comparison.Figures2a−2bshow that the maximum pressure of the whole film is located at the step of the pad.Figures 2b−2d illustrate that as B/L increases, the pressure center moves towards the outer radius, where the fluid velocity is much larger than that at the middle radius.Non-dimensional radii of the maximum pressure R * pmax , for pads with B/L = 1.02, 1.70, 4.00 are 0.537, 0.62 and 0.78 respectively.Figure3presents the distribution of p * at R pmax in Figures2a-2d.Comparing the three solid lines, it is found that the maximum p * on the rotating wall rises from 0.244 to 0.305 as B/L increases from 1.02 to 1.70, but only increases by 0.007 as B/L increases from 1.70 to 4.00.It indicates that as the pressure center moves outwards, the pressure buildup is still affected by side leakage for pads with large B/L values.The line "B/L = 1.02 sw" shows that at the step (θ = 20 • ) and at the entrance to the thin film from the groove (θ = 7.5 • ) pressure on the stationary wall has a sudden increase, which is known as the "ram effect"[23].When the fluid flows into the groove (θ = 0 • , θ = 30 • ), pressure on the stationary wall drops rapidly, which might cause cavitation if it is less than 2 340 Pa, i.e. gauge pressure -98 985 Pa.The vapor fraction distribution of the model in Figure2ais shown in Figure4.As the fluid velocity near the outer radius is larger than that near the inner radius, the pressure drop is more significant and cavitation occurs more easily.From the CFD analyses, it is found that the cavitation zone will get larger when the minimum film thickness decreases or the fluid velocity increases.

Fig. 3 .
Fig.3.Non-dimensional pressure distribution at the radius of the maximum pressure in Figures2a-2d("sw" refers to the stationary wall and "rw" refers to the rotating wall).

h 2 ,
an important criterion of lubrication mode, can be calculated from equation (10):

Table 1 .
The optimum design of a straight step bearing in the literature.

Table 3 .
Values of variable parameters.

Table 4 .
Number of pads for different bearing dimensions.

Table 5 .
The minimum film thicknesses.

Table 7 .
Side mass flow rate of a sector flow model compared with that of a rectangular flow model for pads with different B/L values (Unit: kg.s −1 ).