# Hydrodynamic lubrication of scroll compressor thrust bearing with grooves and circular pockets.

INTRODUCTION

A flat un-profiled thrust bearing is commonly used in scroll compressors. The bearing consists of two parallel plates where one is fixed while the other plate undergoes an orbital motion. These bearings operate under high speed and high load conditions. In order to increase the compressor efficiency reduction in friction and wear at the thrust interface is one of the major factors.

Over the past years it has been shown that lubrication performance in parallel surface thrust bearings can be improved by two predominant phenomena, i.e. thermal wedge effects and geometric wedge effects. Thermal wedge effects depend on changes occurring in the properties of the lubricant and thermal distortion of thrust surface during operation of the thrust bearing. These changes cannot be precisely controlled and this results in unstable lubrication and load support. On the other hand the geometric wedge effect provides consistent and stable lubrication performance.

In the quest for understanding the orbiting thrust bearing Kulkarni (1990) proposed an approach to design the inner and outer radii of thrust bearing to take into account axial load as well as any twisting moment. Tatsuya et al. (2004) in their theoretical study provided an explanation for the existing lubrication condition at the thrust surface by taking into account wedge formation between the thrust surfaces caused due to the elastic deformation of thrust plate under large loads. This elastic deformation cannot be controlled and hence cannot be used to improve the existing performance. Noriaki et al. (2004) in their experimental study showed improvement in the performance of thrust slide bearing by using the pressure difference between the interior and exterior of the thrust bearing. Wang et al. (2002) studied the effect of micro pores formed by laser texturing on a SIC surface. Furthermore, Etsion and Halperin (2002) showed that optimally textured micro pockets improved the lubrication performance in mechanical seals. Yu and Sadeghi (2001) showed that excellent lubrication and load support is achieved by imparting uniformly distributed radial grooves on the fixed thrust surface of a thrust bearing, in which the top plate undergoes rotational motion. In the case of rotational thrust bearings couette velocity component is present only in the tangential direction. However for a thrust bearing in which the top plate undergoes orbital motion couette velocity components in both the radial and tangential direction are present. These velocity components are a function of the position of the eccentric crankshaft that drives the plate undergoing orbital motion. In order to deal with orbital situation it was proposed to use pockets instead of grooves on the fixed thrust surface so that lubrication performance of the thrust slide bearing can be enhanced.

In this study an analytical model was developed to study the effect of circular pockets and grooves on the hydrodynamic lubrication of the thrust slide bearing undergoing an orbital motion. The primary purpose of using circular pockets was to utilize the geometric wedge effect in the radial as well as in the tangential direction and thus increase the load support. An isothermal, transient, polar coordinate Reynolds equation was discretized using control volume finite difference approach and solved using Newton's method. Reynolds boundary condition was used to take into consideration cavitation effects. The analytical model thus developed was used to predict the lubrication performance of the thrust slide bearing for the entire range of operating conditions (i.e. number of pockets, size, depth and position of pockets on the thrust surface). The performance of circular pockets was compared with that of the grooves and results indicate that for the orbiting thrust bearing circular pockets surpass the performance of the uniformly distributed radial grooves in terms of frictional characteristics, minimum film thickness and load support.

Analytical Model

Figure 1 illustrates the principle on which the hydrody-namic lubrication model for orbital motion is based. The model consists of a top plate undergoing orbital motion that is separated from the stationary annular thrust surface at the bottom by a lubricating film of height ([h.sub.0]) when subjected to an axial load [F.sub.z]. In order to achieve orbital motion a crankshaft (not shown) with eccentricity (e) rotating at a constant angular velocity ([omega]) is used to drive the top plate. In this investigation, we assume that the bearings are completely submerged in an oil bath. The velocities of any control volume centroid P on the top plate in polar coordinates as shown in Figure 1 is given by;

[FIGURE 1 OMITTED]

[V.sub.r] = e[omega]sin ([theta]-[[theta].sub.2]) (1)

[V.sub.[theta]] = e[omega]cos ([theta]-[[theta].sub.2]) (2)

Figure 3d illustrates the fixed surface of the thrust slide bearing that contains circular pockets. The annular surface has inner and outer radii [R.sub.1] and [R.sub.2] respectively. Circular pockets are located at pitch radius [R.sub.p] and have width [G.sub.w] and depth [G.sub.d]

Mathematical Formulation

The thrust bearing surface is an annular ring hence it is more convenient to use the polar coordinate system. For orbiting motion, velocities in both the radial and the tangential direction exist; these velocities are a function of crank position that drives the orbiting plate. Solution for the entire annular domain is necessary because of the asymmetric nature of the problem. The following assumptions were made while deriving the governing Reynolds equation:

1. Lubricant is Newtonian and the flow is laminar

2. Inertia effects are negligible

3. Film thickness is small as compared to other dimensions hence properties such as pressure, density and viscosity are constant across the film thickness

4. Body forces neglected

5. Thermal distortion of thrust surface neglected

6. Edge effects due to surface features neglected. During modeling of grooves/pockets the edge shape is like chamfered edge because of linear discretization of the geometry of pockets however theoretically at limit of mesh refinement the shape of edge will tend to be sharp.

Using the above assumptions, the Navier stokes equation in polar coordinates can be simplified to

r-Direction:

[partial derivative]P/[partial derivative]r = [eta][[partial derivative].sup.2][V.sub.r]/[partial derivative][z.sup.2] (3)

[theta]-Direction:

1[partial derivative]p/r[partial derivative][theta] = [eta][[partial derivative].sup.2][V.sub.[theta]]/[partial derivative][z.sup.2] (4)

z-Direction

[[partial derivative].sup.2][V.sub.z]/[partial derivative][z.sup.2] = 0 (5)

Integrating Equation 3 twice with respect to z yields the following:

[V.sub.r] = [z.sup.2][partial derivative]p/2[eta][partial derivative]r + [A.sub.1]z + [B.sub.1] (6)

Here [A.sub.1] and [B.sub.1] are integration constants. Applying boundary conditions [V.sub.r] (r, [theta], 0) = [V.sub.ra] and [V.sub.r] (r, [theta], h) = 0 results in the radial velocity component as follows:

[V.sub.r] = z(z-h)[partial derivative]p/2[eta][partial derivative]r + [V.sub.ra](h-z)/h (7)

Similarly integrating Equation 4 twice with respect to z yields the following:

[V.sub.[theta]] = [z.sup.2][partial derivative]p/2r[eta][partial derivative][theta] + [A.sub.2]z + [B.sub.2] (8)

Here [A.sub.2] and [B.sub.2] are integration constants. Applying boundary conditions [V.sub.[theta]] (r, [theta], 0) = [V.sub.[theta]a]] and [V.sub.[theta]] (r, [theta], h) = 0 results in the circumferential velocity component as follows:

[V.sub.[theta]] = z(z-h)[partial derivative]p/2r[eta][partial derivative][theta] + [V.sub.[theta]a](h-z)/h (9)

Similarly integrating Equation 5 twice with respect to z yields the following:

[V.sub.z] = [A.sub.3]z + [B.sub.3] (10)

Here [A.sub.3] and [B.sub.3] are integration constants. Applying boundary conditions [V.sub.z] (r, [theta], 0) = 0 and [V.sub.z] (r, [theta], h) = [[partial derivative]h/[partial derivative]t results in the squeeze velocity component as follows:

[V.sub.2] = z[partial derivative]h/h[partial derivative]t (11)

Integral form of continuity equation in polar coordinates is given by Equation 12:

[h.[integral] 0] ([partial derivative]p/[partial derivative]t + 1[partial derivative](r[rho][V.sub.r])/[partial derivative]r + 1[partial derivative]*[rho][V.sub.[theta])/[partial derivative][theta] + [partial derivative]([rho](rho][V.sub.z])/[partial derivative]z) dz = 0 (12)

Recall Leibniz rule for integration is given by the following:

[b(z) [integral] a(z)] ([partial derivative]f/[partial derivative]z) dx = [partial derivative]f/[partial derivative]z [b(z) [integral] a(z)] f(x, z)dx - f[b(z), z] [partial derivative]b/[partial derivative]z + f[a((z), z)] [partial derivative]a/[partial derivative]z (13)

Using Leibniz rule for integration the volume flow rate per unit length in r and 6 direction can be determined.

[q.sub.[theta]] = [h.[integral] 0] [V.sub.[theta]dz] = -[h.sup.3]/12r[eta]([partial derivative]p/[partial derivative][theta]) + [V.sub.[theta][a.sup.h]/2 (14)

[q.sub.r] = [h [integral] 0] [V.sub.r]dz = -[h.sup.3]/12[eta] ([partial derivative]p/[partial derivative]r) + [V.sub.ra]h/2 (15)

Substituting Equations 14 and 15 into Equation 12, and then expanding the derivatives, we obtain isothermal, time dependent, constant density and viscosity, polar coordinate Reynolds equation.

[partial derivative]/[partial derivative]r (r[h.sup.e]([partial derivative]p/[partial derivative]r)) + 1[partial derivative]/r[partial derivative][theta](h.sup.3]([partial derivative]p/[partial derivative][theta])) = 6[eta][partial derivative]/[partial derivative]r (rh[V.sub.ra]) + 6[eta][partial derivative]/[partial derivative][theta](h[V.sub.[theta]a]) + 12r[eta][partial derivative]h/[partial derivative]t (16)

In order to solve Equations 16 we need the following boundary conditions:

1. Pressure at inner and outer radii are ambient, at r = [R.sub.i] and at r = [R.sub.0], p = [P.sub.atm]

2. There is a periodic boundary at [theta] = [0.sup.0] and [theta] = [360.sup.0]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Introducing dimensionless variables:

[bar.[eta]] = [eta]/[[eta].sub.0], [bar.H] = h/[h.sub.0], [[bar.V].sub.ra] = [V.sub.ra]/[R.sub.0][omega], [bar.P] = p/[P.sub.atm], t - t/[t.sub.0]

[gamma] = 6[[eta].sub.0][omega]/[P.sub.atm] [([R.sub.0]/[h.sub.0]).sup.2] [sigma] = 12[eta].sub.0][t.sub.0][P.sub.atm] [([R.sub.0]/[h.sub.0]).sup.2] (17)

and substitution of dimensionless parameters from Equation 17 into Equation 16 we obtain nondimensional Reynolds equation:

[partial derivative]/[partial derivative][bar.R]([bar.H][[bar.H].sup.3] ([partial derivative][bar.P]/[partial derivative]R)) + 1[partial derivative]/R[partial derivative][theta] ([[bar.H].sup.3] ([partial derivative][bar.P]/[partial derivative][theta])) = [gamma][bar.[eta]] [partial derivative]/[partial derivative]r([bar.R][bar.H][[bar.V].sub.r]) + [gamma][bar[eta]] [partial derivative]/[partial derivative][theta]([bar.H][[bar.V].sub.[theta]a) + [sigma][bar.[eta]][bar.R][partial derivative][bar.H]/[partial derivative]t (18)

With dimensionless boundary conditions:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Solution Procedure

Nondimensional Reynolds Equation 18 can be expressed in vector form as follows:

[nabla].J = S + T (19)

Here,

J = [GAMMA]([nabla][bar.P]) (19a)

[GAMMA] = [bar.R][[bar.H].sup.3] (19b)

S = [gamma][bar.[eta]] [partial derivative]/[partial derivative]r ([bar.R][bar.H][[bar.V].sub.ra]) + [gamma][bar.[eta]] [partial derivative]/[partial derivative][theta] ([H.sub.H][[bar.V].sub.[theta]a]) (19c)

T = [sigma][bar.[eta]][bar.R] [partial derivative][bar.H]/[partial derivative]t (19d)(19d)

Equation 19 is the diffusion equation with a time dependent source term. Reynolds equation is solved using control volume finite difference approach of Patankar (1980). The arrangement of the cells is shown in Figure 2. Our focus is on cell P and its neighbors cells E, W, N and S. Discrete value of P and r are stored at cell centroid. The faces e, w, n and s are associated with area vectors [A.sub.e], [A.sub.w], [A.sub.n], and [A.sub.s]. The vectors are positive pointing outward from the cell P. The volume of cell P is [partial derivative]V = [R.sub.p] [partial derivative][theta][partial derivative][bar.R].

[FIGURE 2 OMITTED]

Multiplying Equation 19 by [partial derivative]t[partial derivative]V and integrating over control volume we obtain the following:

[[integral] [DELTA]t][[integral] [DELTA]v] ([[nabla]*J) [partial derivative]V[partial derivative]t = [[integral] [DELTA]t][[integral] [DELTA]v] (S + T) [partial derivative]V[partial derivative]t (20)

The divergence theorem is used to evaluate the volume integral on the LHS of Equation 20:

[[integral] [DELTA]t][[integral] [DELTA]A] J*d A[partial derivative]t = [[integral] [DELTA]t] S[partial derivative]V[partial derivative]t + [[integral] [DELTA]t]T[partial derivative]V[partial derivative]t (21)

The first integral in LHS of Equation 21 is an integral over the control surface. It is assumed that J varies linearly over each face of cell P so that it may be represented by its value at the face centroid. It is also assumed that the mean value of the source term over the control volume is [bar.S] and [bar.T] respectively.

[[integral] [DELTA]t] [[(J*A).sub.e] + [(J*A).sub.w] + [(J*A).sub.n] + [(J*A).sub.s] [DELTA]t = [[integral] [DELTA]t [bar.S][partial derivative]V[partial derivative]t + [[integral] [DELTA]t] [bar.T][partial derivative]V[partial derivative]t (22)

Consider the left hand side of Equation 22 it is at this point we need an assumption about how J varies over time on a typical control volume.

[[integral] [DELTA]t] [summation over (f = e, w, n, s)] [J.sub.i].[A.sub.][partial derivative]t = (f [summation over (f = e, w, n, s)] [J.sub.i.sup.1].[A.sub.i]) + (1 - f) [summation over (f = e, w, n, s)] [J.sub.i.sup.1].[A.sub.i.sup.0] (23)

Here f is a weighting factor between 0 and 1. Where f = 0 corresponds to fully explicit scheme, f = 1 corresponds to fully implicit scheme and f = 0.5 corresponds to Crank-Nicholson scheme. A fully implicit scheme (f = 1) is chosen as it is unconditionally stable and thus Equation 23 reduces to the following:

[[integral] [DELTA]t] [summation over (f = e, w, n, s)] [J.sub.i].[A.sub.i][partial derivative]t = ([summation over (f = e, w, n, s)] [J.sub.i.sup.1].[A.sub.i]) (24)

The face areas [A.sub.e] and [A.sub.n] are given by the following:

[A.sub.e] = [partial derivative] [vector.R] [e.sub.[theta]] and [A.sub.n] = [[vector.R].sub.n] [partial derivative] [theta] [e.sub.r]

area vectors can be written analogously for other face areas. Furthermore, we make linear profile assumption of [bar.P] between cell centroids, thus

[J.sub.e]*[A.sub.e] = [[GAMMA].sub.e]/[[bar.R].sub.e]([[bar.P].sub.E] - [[bar.P].sub.p]/[partial derivative][theta])[partial derivative][bar.R]

[J.sub.n]*[A.sub.n] = [[bar.R].sub.n][[GAMMA].sub.n] ([[bar.P].sub.n] - [P.sub.p]/[partial derivative][[bar.R]]) [partial derivative][theta] (25)

The transport equations in other directions can be written analogously. Similarly using a fully implicit scheme for the first term on right hand side of Equation 22 and expanding the gradients using finite difference scheme in Equation 21we obtain the following:

[[integral] [DELTA]t] [bar.S][partial derivative]V[partial derivative]t = [bar.[S.sup.1]][partial derivative]v[partial derivative]t (26)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (27)

Integrating second term on right hand side of Equation 22 directly we obtain the following:

[[integral] [DELTA]t] [bar.T][partial derivative]V[partial derivative]t = [sigma][bar.R]([H.sub.1] - [H.sup.0]) [partial derivative]V (28)

Using Equations 22 through 28 the discretized form of Equation 18 can be written as follows:

[[[[bar.R].sub.n][[GAMMA].sub.n] ([P.sub.n] - [P.sub.p]/[partial derivative][bar.R]) [partial derivative][theta] - [[GAMMA].sub.S][R.sub.S] ([P.sub.p] - [P.sub.s]/[partial derivative][bar.R]) [partial derivative][theta] + [[GAMMA].sub.e]/[R.sub.e] ([P.sub.E] - [P.sub.p]/[partial derivative][theta]) [partial derivative][bar.R] - [[GAMMA].sub.w]/[R.sub.w]([P.sub.p] - [P.sub.2]/[partial derivative][theta]) [partial derivative][bar.R]].sup.1] = [[[gamma][bar.[eta]][partial derivative]/[partial derivative]r([bar.R][bar.H][[bar.V].sub.ra]) + [gamma][bar.[eta]][partial derivative]/[partial derivative][theta] ([bar.H][[bar.V].sub.[theta]a]).sup.1] [partial derivative]V + [sigma][bar.[eta]]R([H.sup.1] - [H.sup.0]) [partial derivative]V[partial derivative]t (29)

Equation 29 represents a complete discretized form for [[bar.P].sub.P] and can be written in algebraic form as follows:

[a.sub.P][P.sub.P] = [a.sub.E][P.sub.E] + [a.sub.E][P.sub.E] + [a.sub.E][P.sub.N] + [a.sub.E][P.sub.S] + b (30)

The linear system of Equation 30 was solved using Newton's method and Stone's implicit procedure. The discrete equation expresses a balance of discrete flux J and source term. Thus conservation over the individual control volume is guaranteed. After the convergence for pressure and load is achieved the following parameters were evaluated at various operating conditions.

Nondimensional Load Capacity

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (31)

Nondimensional Tangential Velocity

[[bar.V].sub.[theta]] = 3/[sigma][bar.z]([bar.z] - [bar.H])[partial derivative][bar.P]/[partial derivative][theta] + [[bar.V].sub.0a] ([bar.H] - [bar.z]/[bar.H] (32)

Nondimensional Radial Velocity

[[bar.V].sub.r] = 3/[sigma][bar.z]([bar.z] - [bar.H]) [partial derivative][bar.P]/[partial derivative]R + [[bar.V].sub.ra] ([bar.H] - [bar.z]/[bar.H] (33)

Friction Coefficient

Viscous forces experienced by the fixed plate in radial and tangential direction are as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (34)

Coefficient of friction is then calculated by the following:

[mu] = [square root of (([[bar.F].sub.r.sup.2] + [[bar.F].sub.[theta].sup.2])/[[bar.F].sub.z]] (35)

Results and Discussion

In order to examine the details, the current numerical model was evaluated for several cases with an objective of verifying the performance of orbiting thrust bearing with radial grooves and circular surface pockets. Table 1 contains the various operating parameters used in this investigation. These are typical dimensions for a scroll compressor thrust surface. Figure 3a-3c illustrates modeling of various cross sections of grooves as given in Razzaque (1999), whereas Figure 3d shows a circular surface pocket with Np = 3 and Ng = 20. Modeling of the circular pockets is done by superimposing spherical geometry of pockets on the flat surface in cylindrical coordinates.

[FIGURE 3 OMITTED]

Time-Dependent Solution

Figure 4 depicts the pressure distribution at various time steps. The model involves two disks one stationary (thrust surface that has pockets/grooves) at the bottom and one on top which orbits. The disks are brought in contact under the action of external load. In this case, it should be noted that velocity components of the orbiting plate are a function of crankshaft position that is used to drive the orbit plate. Therefore, as time progresses the position of the crankshaft changes and velocities need to be updated at each time step. Since the velocities are updated after each time step the solution matrix becomes stiff and under relaxation is required for gauss - siedel scheme to solve the stiff PDE. The results indicate that initially the load is supported by the squeezing action thus building up pressure on entire thrust surface. However as the time progresses the fluid squeeze film effect diminishes and hydro-dynamic pressure starts to build in the converging sections of the surface pockets and fluid cavitates [Dowson (1979), Elrod (1981) and Brewe (1986)] in diverging regions. In order to understand the pressure build development in the case of orbital motion, it should be noted that kinematic (Please refer Appendix A for the kinematic analysis of the orbiting thrust bearings) position analysis shows that the locus of an arbitrary control volume centroid P on the orbiting disk shown in Figure 1 is a circle of radius, equal to eccentricity (e) of the crankshaft driving the plate, hence during a part of rotation of crankshaft point P moves out of pocket experiencing a converging wedge and giving rise to pressure spike at the edge of the pocket and during remaining part it experiences a diverging wedge causing the fluid to cavitate. The above phenomenon when extended to entire orbit plate causes the pressure spikes to oscillate about the centerline and along the edges of the pocket as the crank rotates. Figure 5 illustrates variation of minimum film thickness with advance in time. Minimum film thickness decreases very fast initially depicting the presence of squeeze film effect but later (approximately after 0.35 seconds) becomes almost constant indicating that steady state operation has been achieved. In general we are only concerned with the steady state performance of the thrust bearing hence the rest of the discussion will be restricted to steady state conditions.

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

Effect of Number of Grooves/Pockets, Ng

Figure 6 illustrates the effect of number of grooves on the load carrying capacity for different surface profiles. In the case of grooves (circular, triangular and trapezoidal) the load carrying capacity for triangular grooves is highest for any number of grooves. This is due to the fact that with the same groove depth ratio and groove width ratio the area of cross section for triangular grooves is the least which causes average film thickness for triangular grooves to be minimum consequently giving greater load support. Maximum load carrying capacity for grooves occurs when the number of grooves is 16. As the number of grooves on the bearing increases, geometric wedge effect increases resulting in greater load support. On the other hand leakage through grooves increases with more number of grooves. The net load carrying capacity is function aforementioned mechanisms. The load carrying capacity of circular pockets is significantly larger as compared to that of the grooves. This can be attributed to the fact that in case of circular pockets the curvature effects are present in both radial and tangential direction, (i.e. geometric wedge effect occurs in both directions) this causes a large magnitude of pressure build up at the edge of pocket. Figure 7 depicts the effect of number of grooves on coefficient of friction, [mu] for different surface profiles. Friction coefficient depends on pressure gradients and couette velocity of the plate. For fixed speed of crankshaft the friction depends only on pressure gradients. In the case of circular pockets, pressure builds up almost symmetric about centerline of the pocket which causes significant reduction in friction coefficient as that compared to that of the grooves.

[FIGURE 6 OMITTED]

[FIGURE 7 OMITTED]

Effect of Groove Depth Ratio, [DELTA]

Figure 8 depicts the effect of groove depth on the load carrying capacity of the thrust bearing for various surface features. It is observed that maximum load carrying capacity occurs at [DELTA] = 1 for grooved surfaces and is independent of groove cross section. For a circular pocket surface the maximum load carrying capacity occurs at [DELTA] = 2. It should be noted that the load carrying capacity is a function of average film thickness and geometric wedge effects. As [DELTA] increases the geometric wedge effect increases thus enhancing the load carrying capacity, however with increase in [DELTA] average film thickness also increases thus reducing the load support. The net load carrying capacity is a sum of both the effects. Reynolds equation is based on the assumption that film thickness is of the order of groove depth and at small groove depths (order 10 to 50 microns) inertia effects of the fluid are negligible, hence the value of [DELTA] is restricted to less than 5 in order to ensure the validity of the Reynolds equation. Apart from the above two effects with increase in [DELTA] the groove/pocket acts as a lubricant sump thus ensuring stable fluid film between surfaces. Figure 8 can also be used to ascertain the sensitivity of the load carrying capacity due to error in manufacturing the specified groove depth. Figure 9 shows the effect of groove depth ratio [DELTA] on coefficient of friction ([mu]) for different surface profiles. At shallow groove depths ([DELTA] < 1) grooves perform better than pockets, however as the groove depth increases frictional performance of circular pocket surfaces is significantly better as compared to that of grooved surfaces.

[FIGURE 8 OMITTED]

[FIGURE 9 OMITTED]

Effect of Groove Width Ratio, [lambda]

Figure 10 illustrates the effect of groove width on load carrying capacity for various surface features. The results indicate that for [lambda] < 500 the load carrying capacity of grooves is higher as compared to that of circular pockets and as [lambda] becomes greater than 500, a bearing with pockets supports higher load than a bearing with grooves. At lower groove width ratios the side leakage is less for grooves but as the width of groove increases, side leakage through grooves becomes large. On the other hand because circular pockets are a locked geometry, side leakage is not a function of pocket width. With increase in value of [lambda] the load carrying capacity for grooves/ pocket increases and there is no optimum value of [lambda]; however, the upper limit on [lambda] is imposed by the size of the thrust surface. Figure 11 shows the effect of groove width on the coefficient of friction ([mu]) for various surface features. In most cases, frictional performance is inversely proportional to load carrying capacity, consequently the friction coefficient for pockets is higher than that for grooves for [lambda] < 500, but the friction coefficient ([mu]) significantly decreases as the value of [lambda] increases beyond 500.

[FIGURE 10 OMITTED]

[FIGURE 11 OMITTED]

Effect of Number of Pockets in Radial Direction, Np

Figure 12 illustrates the effect of number of pockets in the radial direction (Np) has on load carrying capacity and coefficient of friction ([mu]) for a constant area density (area density is the ratio of total area occupied by pockets to total available area on the thrust surface). It should be noted that in order to keep the area density constant with increase in Np, width of the pockets is required to be reduced. The results indicate that smaller number of pockets with a large groove width gives the best performance however the width of the pockets is limited by the size of the thrust surface. This can be attributed to the fact that as Np increases, interaction between the pressure spikes of pockets increases causing steep pressure gradients. The friction coefficient is a function of pressure gradients hence its value increases with increase in Np.

[FIGURE 12 OMITTED]

CONCLUSION

A Numerical model for an orbiting thrust bearing was developed. This model can be used to predict the frictional performance of grooves and circular pockets on the thrust surfaces. The results obtained from the model confirm that circular pocket thrust surfaces provide significantly better performance in orbiting situations as compared to that of uniformly distributed radial grooves. This can be attributed to the fact that in the orbiting case couette flow velocities in both radial and tangential direction are present. Circular pockets having curvature symmetric about their centerline tend to create geometric wedge effects in both the directions thus increasing the load support. Uniformly distributed radial grooves, on the other hand, have no geometric wedge in the radial direction also due to the presence of couette flow radial velocity component side leakage through the grooves is more. The results also indicate that pockets with large width, minimum Np and shallow depths (order of film thickness) are the best configuration for maximizing load carrying and minimizing friction. It should be noted that this model is based on the assumption of full film lubrication and therefore it should be ensured that roughness of the thrust surface is sufficiently below the minimum film thickness.

Reynolds boundary condition does not take into account reformation of full film from in the cavitation zone. However, the numerical method suggested by Elrod (1981) or Payvar and Salant (1992) to remove this restriction resulted in instability of the numerical solution for orbital motion. Insight about temperature distribution at the thrust interface can be obtained by removing the isothermal assumption and coupling the Reynolds equation with the energy equation. In order to evaluate performance of very deep pockets the inclusion of inertia effects Pinkus et al. (1981) would be necessary, but due to presence of radial velocity components a complete solution to the Navier Stokes equation would be required.

ACKNOWLEDGMENT

The authors would like to express their deepest appreciations to the Tecumseh Company for their support of this research study.

NOMENCLATURE

[F.sub.z] = normal load, N

[G.sub.d] = depth of a groove or pocket, m

[G.sub.w] = width of a groove or pocket, m

[h.sub.0] = reference film thickness, m

[h.sub.min] = minimum film thickness, m

h = film thickness, m

[[bar.h].sub.min] = dimensionless minimum film thickness, [h.sub.min]/[h.sub.0]

[bar.h] = dimensionless film thickness, h/[h.sub.0]

[W.sub.z] = nondimensional load, [W.sub.z] = [F.sub.z]/([P.sub.atm][R.sub.0.sup.2])

[N.sub.p] = number of uniformly distributed grooves or pocke along circumference

[N.sub.p] = number of uniformly distributed pockets along radius

P = pressure, MPa

[P.sub.a] = ambient pressure, MPa

[bar.P] = dimensionless pressure P/[P.sub.a]

[D.sub.1] = inner diameter of the annular thrust surface, m

[D.sub.2] = outer diameter of the annular thrust surface, m

[R.sub.1] = inner radius of the annular thrust surface, m

[R.sub.2] = outer radius of the annular thrust surface, m

r = radial coordinate, m

[bar.R] = dimensional radial coordinate r/[R.sub.0]

t = time, s

[t.sub.0] = reference time 1 /[omega]

t = dimensionless time, t/[t.sub.0]

[V.sub.r] = radial velocity component, m/s

[V.sub.[theta]] = tangential velocity component, m/s

[[bar.V].sub.r] = dimensionless radial velocity component [V.sub.[theta]]/[R.sub.0][omega]

[[bar.V].sub.[theta]] = dimensionless radial velocity component [V.sub.r]/[R.sub.0][omega]

z = axial coordinate, m

[bar.Z] = dimensionless axial coordinate z/[h.sub.0]

[[theta].sub.2] = input angle of crank shaft in degrees

[gamma] = characteristics number, 6[[eta].sub.0][omega][R.sub.0.sup.2]/([P.sub.atm][h.sub.0.sup.2])

[sigma] = squeeze number, 12[[eta].sub.0][R.sub.0.sup.2]/([P.sub.atm][h.sub.0.sup.2][t.sub.0])

[DELTA] = groove depth ratio, [G.sub.d]/[h.sub.0]

[lambda] = groove width ratio, [G.sub.w]/[h.sub.0]

[eta] = absolute viscosity, N-sec/[m.sup.2]

[[eta].sub.0] = absolute viscosity of the lubricant at ambient temperature, Pa * s

[bar.[eta]] = dimensionless absolute viscosity, [eta]/[[eta].sub.0]

[omega] = angular velocity of crankshaft, rad/s

[mu] = coefficient of friction

Subscripts

e = east boundary surface of control volume

E = east grid point

n = north boundary surface of control volume

N = north grid point

P = grid point of interest

s = south boundary surface of the control

S = south grid point

w = west boundary surface of control volume

W = west grid point

Superscripts

0 = value from previous time step

1 = value from current time step

REFERENCES

Salama, M.E., 1950. The Effect of Macro-roughness on the performance of Parallel Thrust Bearings. Proc. Inst. Mech. Eng. 163: pp. 149.

Kulkarni, S. S. 1990. Scroll compressor.- Thrust bearing design under laminar conditions. Proc. Int. Compressor. Eng. Conf. pp. 327-332.

Wang Xiaolei, Kato Koji, Adachi Koshi. 2002. The lubrication effect of Micro-Pits on Parallel Sliding Faces of SiC in water. Lubrication Engineers. vol 58(8):27-34.

Yu, T.H., and Sadeghi, F. 2001. Groove effects on thrust washer lubrication. J. Tribol. 123(2):295-304.

Tatsuya Oku, Keiko Anami, Noriaki Ishii, and Kiyoshi Sano. 2004. Lubrication Mechanism at Thrust Slide-Bearing of Scroll compressors (Theoretical Study). Proc. Int. Compressor. Eng. Conf. C104:1-10.

Noriaki Ishii, Tatsuya Oku, Keiko Anami, and Akinori Fakuda. 2004. Lubrication Mechanism at Thrust Slide-Bearing of Scroll compressors (Experimental Study). Proc. Int. Compressor. Eng. Conf. C104:1-10.

Elrod, H.G. 1981. A Cavitation Algorithm. ASME J. Lubr. Technol. 103:350-354.

Dowson, D., and Taylor, C. M. 1979. Cavitation in Bearing. Annu. Rev. Fluid Mech. 190:1-116.

Brewe, D. E. 1986. Theoretical Modeling of Vapor Cavita-tion in Dynamically Loaded Journal Bearings. ASME J. Tribol. 108:628-638.

Pinkus, O., and Lund, J. W. 1981. Centrifugal Effects in Thrust Bearings and Seals under Laminar Conditions. ASME J. Tribol. 103:126-136.

Razzaque, M., M., Kato, T. 1999. Effects of groove orientation on hydrodynamic behavior of wet clutch coolant films. Journal of Tribology, Transactions of the ASME 121(1):56-61.

Payvar, P. and Salant, R. F. Jan 1992. Computational method for cavitation in a wavy mechanical seal. Journal of Tri-bology, Transactions of the ASME 114(1):199-204.

Etsion, I., and Halperin, G. July 2002. A laser surface textured hydrostatic mechanical seal.Tribology Transactions 45(3):430-434.

Patankar, Suhas V. 1980. Numerical Heat Transfer and Fluid Flow. Washington, D.C: Hemisphere Publishing Corp.

Hamrock, Bernard J. 1994. Fundamentals of Fluid Film Lubrication. New York: McGraw-Hill, Inc.

Ferziger, J. H., and Peric, M. 1999. Computational Methods for Fluid Dynamics. Germany: Springer.

APPENDIX A KINEMATIC ANALYSIS

Figure 1 illustrates the vector loop for determining the radial and tangential velocities of an arbitrary control volume centroid P on the orbit plate. Vector equation for control volume centroid P can be written as follows:

[[vector].R] = [bar.[r.sub.2]] + [bar.[r.sub.7]] (A1)

The x andj components of Equation A1 are as follows:

Rcos[theta] = [r.sub.2]cos[[theta].sub.2] + [r.sub.7]cos[alpha] (A2)

Rsin[theta] = [r.sub.2]sin[[theta].sub.2] + [r.sub.7]sin[alpha] (A3)

Differentiating Equations A2 and A3 with respect to time we obtain the following

Rcos[theta] - Rsin[theta][theta] = [r.sub.2][omega]sin[[theta].sub.2] (A4)

Rsin[theta] + Rcos[theta][theta] = [r.sub.2][omega]cos[[theta].sub.2] (A5)

Equations A4 and A5 in a matrix form are given by the following:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A6)

Solution to Equation A6 can be obtained inverting the matrix as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A7)

Equation A7 can be expanded to obtain the following:

R = -[r.sub.2][omega]sin[[theta].sub.2]cos[theta] + [r.sub.2][omega]sin[theta]cos[[theta].sub.2] (A8)

[theta] = [r.sub.2][omega]sin[theta]sin[[theta].sub.2] + [r.sub.2][omega]cos[theta]cos[[theta].sub.2] (A9)

The equations for radial velocity ([V.sub.r]) and tangential velocity ([V.sub.[theta]]) can be obtained form Equations A8 and A9 as follows:

[V.sub.r] = R = e [omega] sin ([theta] - [[theta].sub.2] (A10)

[V.sub.[theta]] = R[theta] = e [omega] cos ([theta] - [[theta].sub.2](A11)

Here e = [r.sub.2] = |[r.sub.s.sup.[tau]]} and [omega] = [[theta].sub.2] = [partial derivative] [[theta].sub.2]/[partial derivative]t.

Amit Vaidya

Farshid Sadeghi, PhD

Amit Vaidya is a mechanical engineer, Farshid Sadeghi is a professor in the School of Mechinical Engineering, Purdue University, West Lafayette, IN

A flat un-profiled thrust bearing is commonly used in scroll compressors. The bearing consists of two parallel plates where one is fixed while the other plate undergoes an orbital motion. These bearings operate under high speed and high load conditions. In order to increase the compressor efficiency reduction in friction and wear at the thrust interface is one of the major factors.

Over the past years it has been shown that lubrication performance in parallel surface thrust bearings can be improved by two predominant phenomena, i.e. thermal wedge effects and geometric wedge effects. Thermal wedge effects depend on changes occurring in the properties of the lubricant and thermal distortion of thrust surface during operation of the thrust bearing. These changes cannot be precisely controlled and this results in unstable lubrication and load support. On the other hand the geometric wedge effect provides consistent and stable lubrication performance.

In the quest for understanding the orbiting thrust bearing Kulkarni (1990) proposed an approach to design the inner and outer radii of thrust bearing to take into account axial load as well as any twisting moment. Tatsuya et al. (2004) in their theoretical study provided an explanation for the existing lubrication condition at the thrust surface by taking into account wedge formation between the thrust surfaces caused due to the elastic deformation of thrust plate under large loads. This elastic deformation cannot be controlled and hence cannot be used to improve the existing performance. Noriaki et al. (2004) in their experimental study showed improvement in the performance of thrust slide bearing by using the pressure difference between the interior and exterior of the thrust bearing. Wang et al. (2002) studied the effect of micro pores formed by laser texturing on a SIC surface. Furthermore, Etsion and Halperin (2002) showed that optimally textured micro pockets improved the lubrication performance in mechanical seals. Yu and Sadeghi (2001) showed that excellent lubrication and load support is achieved by imparting uniformly distributed radial grooves on the fixed thrust surface of a thrust bearing, in which the top plate undergoes rotational motion. In the case of rotational thrust bearings couette velocity component is present only in the tangential direction. However for a thrust bearing in which the top plate undergoes orbital motion couette velocity components in both the radial and tangential direction are present. These velocity components are a function of the position of the eccentric crankshaft that drives the plate undergoing orbital motion. In order to deal with orbital situation it was proposed to use pockets instead of grooves on the fixed thrust surface so that lubrication performance of the thrust slide bearing can be enhanced.

In this study an analytical model was developed to study the effect of circular pockets and grooves on the hydrodynamic lubrication of the thrust slide bearing undergoing an orbital motion. The primary purpose of using circular pockets was to utilize the geometric wedge effect in the radial as well as in the tangential direction and thus increase the load support. An isothermal, transient, polar coordinate Reynolds equation was discretized using control volume finite difference approach and solved using Newton's method. Reynolds boundary condition was used to take into consideration cavitation effects. The analytical model thus developed was used to predict the lubrication performance of the thrust slide bearing for the entire range of operating conditions (i.e. number of pockets, size, depth and position of pockets on the thrust surface). The performance of circular pockets was compared with that of the grooves and results indicate that for the orbiting thrust bearing circular pockets surpass the performance of the uniformly distributed radial grooves in terms of frictional characteristics, minimum film thickness and load support.

Analytical Model

Figure 1 illustrates the principle on which the hydrody-namic lubrication model for orbital motion is based. The model consists of a top plate undergoing orbital motion that is separated from the stationary annular thrust surface at the bottom by a lubricating film of height ([h.sub.0]) when subjected to an axial load [F.sub.z]. In order to achieve orbital motion a crankshaft (not shown) with eccentricity (e) rotating at a constant angular velocity ([omega]) is used to drive the top plate. In this investigation, we assume that the bearings are completely submerged in an oil bath. The velocities of any control volume centroid P on the top plate in polar coordinates as shown in Figure 1 is given by;

[FIGURE 1 OMITTED]

[V.sub.r] = e[omega]sin ([theta]-[[theta].sub.2]) (1)

[V.sub.[theta]] = e[omega]cos ([theta]-[[theta].sub.2]) (2)

Figure 3d illustrates the fixed surface of the thrust slide bearing that contains circular pockets. The annular surface has inner and outer radii [R.sub.1] and [R.sub.2] respectively. Circular pockets are located at pitch radius [R.sub.p] and have width [G.sub.w] and depth [G.sub.d]

Mathematical Formulation

The thrust bearing surface is an annular ring hence it is more convenient to use the polar coordinate system. For orbiting motion, velocities in both the radial and the tangential direction exist; these velocities are a function of crank position that drives the orbiting plate. Solution for the entire annular domain is necessary because of the asymmetric nature of the problem. The following assumptions were made while deriving the governing Reynolds equation:

1. Lubricant is Newtonian and the flow is laminar

2. Inertia effects are negligible

3. Film thickness is small as compared to other dimensions hence properties such as pressure, density and viscosity are constant across the film thickness

4. Body forces neglected

5. Thermal distortion of thrust surface neglected

6. Edge effects due to surface features neglected. During modeling of grooves/pockets the edge shape is like chamfered edge because of linear discretization of the geometry of pockets however theoretically at limit of mesh refinement the shape of edge will tend to be sharp.

Using the above assumptions, the Navier stokes equation in polar coordinates can be simplified to

r-Direction:

[partial derivative]P/[partial derivative]r = [eta][[partial derivative].sup.2][V.sub.r]/[partial derivative][z.sup.2] (3)

[theta]-Direction:

1[partial derivative]p/r[partial derivative][theta] = [eta][[partial derivative].sup.2][V.sub.[theta]]/[partial derivative][z.sup.2] (4)

z-Direction

[[partial derivative].sup.2][V.sub.z]/[partial derivative][z.sup.2] = 0 (5)

Integrating Equation 3 twice with respect to z yields the following:

[V.sub.r] = [z.sup.2][partial derivative]p/2[eta][partial derivative]r + [A.sub.1]z + [B.sub.1] (6)

Here [A.sub.1] and [B.sub.1] are integration constants. Applying boundary conditions [V.sub.r] (r, [theta], 0) = [V.sub.ra] and [V.sub.r] (r, [theta], h) = 0 results in the radial velocity component as follows:

[V.sub.r] = z(z-h)[partial derivative]p/2[eta][partial derivative]r + [V.sub.ra](h-z)/h (7)

Similarly integrating Equation 4 twice with respect to z yields the following:

[V.sub.[theta]] = [z.sup.2][partial derivative]p/2r[eta][partial derivative][theta] + [A.sub.2]z + [B.sub.2] (8)

Here [A.sub.2] and [B.sub.2] are integration constants. Applying boundary conditions [V.sub.[theta]] (r, [theta], 0) = [V.sub.[theta]a]] and [V.sub.[theta]] (r, [theta], h) = 0 results in the circumferential velocity component as follows:

[V.sub.[theta]] = z(z-h)[partial derivative]p/2r[eta][partial derivative][theta] + [V.sub.[theta]a](h-z)/h (9)

Similarly integrating Equation 5 twice with respect to z yields the following:

[V.sub.z] = [A.sub.3]z + [B.sub.3] (10)

Here [A.sub.3] and [B.sub.3] are integration constants. Applying boundary conditions [V.sub.z] (r, [theta], 0) = 0 and [V.sub.z] (r, [theta], h) = [[partial derivative]h/[partial derivative]t results in the squeeze velocity component as follows:

[V.sub.2] = z[partial derivative]h/h[partial derivative]t (11)

Integral form of continuity equation in polar coordinates is given by Equation 12:

[h.[integral] 0] ([partial derivative]p/[partial derivative]t + 1[partial derivative](r[rho][V.sub.r])/[partial derivative]r + 1[partial derivative]*[rho][V.sub.[theta])/[partial derivative][theta] + [partial derivative]([rho](rho][V.sub.z])/[partial derivative]z) dz = 0 (12)

Recall Leibniz rule for integration is given by the following:

[b(z) [integral] a(z)] ([partial derivative]f/[partial derivative]z) dx = [partial derivative]f/[partial derivative]z [b(z) [integral] a(z)] f(x, z)dx - f[b(z), z] [partial derivative]b/[partial derivative]z + f[a((z), z)] [partial derivative]a/[partial derivative]z (13)

Using Leibniz rule for integration the volume flow rate per unit length in r and 6 direction can be determined.

[q.sub.[theta]] = [h.[integral] 0] [V.sub.[theta]dz] = -[h.sup.3]/12r[eta]([partial derivative]p/[partial derivative][theta]) + [V.sub.[theta][a.sup.h]/2 (14)

[q.sub.r] = [h [integral] 0] [V.sub.r]dz = -[h.sup.3]/12[eta] ([partial derivative]p/[partial derivative]r) + [V.sub.ra]h/2 (15)

Substituting Equations 14 and 15 into Equation 12, and then expanding the derivatives, we obtain isothermal, time dependent, constant density and viscosity, polar coordinate Reynolds equation.

[partial derivative]/[partial derivative]r (r[h.sup.e]([partial derivative]p/[partial derivative]r)) + 1[partial derivative]/r[partial derivative][theta](h.sup.3]([partial derivative]p/[partial derivative][theta])) = 6[eta][partial derivative]/[partial derivative]r (rh[V.sub.ra]) + 6[eta][partial derivative]/[partial derivative][theta](h[V.sub.[theta]a]) + 12r[eta][partial derivative]h/[partial derivative]t (16)

In order to solve Equations 16 we need the following boundary conditions:

1. Pressure at inner and outer radii are ambient, at r = [R.sub.i] and at r = [R.sub.0], p = [P.sub.atm]

2. There is a periodic boundary at [theta] = [0.sup.0] and [theta] = [360.sup.0]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Introducing dimensionless variables:

[bar.[eta]] = [eta]/[[eta].sub.0], [bar.H] = h/[h.sub.0], [[bar.V].sub.ra] = [V.sub.ra]/[R.sub.0][omega], [bar.P] = p/[P.sub.atm], t - t/[t.sub.0]

[gamma] = 6[[eta].sub.0][omega]/[P.sub.atm] [([R.sub.0]/[h.sub.0]).sup.2] [sigma] = 12[eta].sub.0][t.sub.0][P.sub.atm] [([R.sub.0]/[h.sub.0]).sup.2] (17)

and substitution of dimensionless parameters from Equation 17 into Equation 16 we obtain nondimensional Reynolds equation:

[partial derivative]/[partial derivative][bar.R]([bar.H][[bar.H].sup.3] ([partial derivative][bar.P]/[partial derivative]R)) + 1[partial derivative]/R[partial derivative][theta] ([[bar.H].sup.3] ([partial derivative][bar.P]/[partial derivative][theta])) = [gamma][bar.[eta]] [partial derivative]/[partial derivative]r([bar.R][bar.H][[bar.V].sub.r]) + [gamma][bar[eta]] [partial derivative]/[partial derivative][theta]([bar.H][[bar.V].sub.[theta]a) + [sigma][bar.[eta]][bar.R][partial derivative][bar.H]/[partial derivative]t (18)

With dimensionless boundary conditions:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Solution Procedure

Nondimensional Reynolds Equation 18 can be expressed in vector form as follows:

[nabla].J = S + T (19)

Here,

J = [GAMMA]([nabla][bar.P]) (19a)

[GAMMA] = [bar.R][[bar.H].sup.3] (19b)

S = [gamma][bar.[eta]] [partial derivative]/[partial derivative]r ([bar.R][bar.H][[bar.V].sub.ra]) + [gamma][bar.[eta]] [partial derivative]/[partial derivative][theta] ([H.sub.H][[bar.V].sub.[theta]a]) (19c)

T = [sigma][bar.[eta]][bar.R] [partial derivative][bar.H]/[partial derivative]t (19d)(19d)

Equation 19 is the diffusion equation with a time dependent source term. Reynolds equation is solved using control volume finite difference approach of Patankar (1980). The arrangement of the cells is shown in Figure 2. Our focus is on cell P and its neighbors cells E, W, N and S. Discrete value of P and r are stored at cell centroid. The faces e, w, n and s are associated with area vectors [A.sub.e], [A.sub.w], [A.sub.n], and [A.sub.s]. The vectors are positive pointing outward from the cell P. The volume of cell P is [partial derivative]V = [R.sub.p] [partial derivative][theta][partial derivative][bar.R].

[FIGURE 2 OMITTED]

Multiplying Equation 19 by [partial derivative]t[partial derivative]V and integrating over control volume we obtain the following:

[[integral] [DELTA]t][[integral] [DELTA]v] ([[nabla]*J) [partial derivative]V[partial derivative]t = [[integral] [DELTA]t][[integral] [DELTA]v] (S + T) [partial derivative]V[partial derivative]t (20)

The divergence theorem is used to evaluate the volume integral on the LHS of Equation 20:

[[integral] [DELTA]t][[integral] [DELTA]A] J*d A[partial derivative]t = [[integral] [DELTA]t] S[partial derivative]V[partial derivative]t + [[integral] [DELTA]t]T[partial derivative]V[partial derivative]t (21)

The first integral in LHS of Equation 21 is an integral over the control surface. It is assumed that J varies linearly over each face of cell P so that it may be represented by its value at the face centroid. It is also assumed that the mean value of the source term over the control volume is [bar.S] and [bar.T] respectively.

[[integral] [DELTA]t] [[(J*A).sub.e] + [(J*A).sub.w] + [(J*A).sub.n] + [(J*A).sub.s] [DELTA]t = [[integral] [DELTA]t [bar.S][partial derivative]V[partial derivative]t + [[integral] [DELTA]t] [bar.T][partial derivative]V[partial derivative]t (22)

Consider the left hand side of Equation 22 it is at this point we need an assumption about how J varies over time on a typical control volume.

[[integral] [DELTA]t] [summation over (f = e, w, n, s)] [J.sub.i].[A.sub.][partial derivative]t = (f [summation over (f = e, w, n, s)] [J.sub.i.sup.1].[A.sub.i]) + (1 - f) [summation over (f = e, w, n, s)] [J.sub.i.sup.1].[A.sub.i.sup.0] (23)

Here f is a weighting factor between 0 and 1. Where f = 0 corresponds to fully explicit scheme, f = 1 corresponds to fully implicit scheme and f = 0.5 corresponds to Crank-Nicholson scheme. A fully implicit scheme (f = 1) is chosen as it is unconditionally stable and thus Equation 23 reduces to the following:

[[integral] [DELTA]t] [summation over (f = e, w, n, s)] [J.sub.i].[A.sub.i][partial derivative]t = ([summation over (f = e, w, n, s)] [J.sub.i.sup.1].[A.sub.i]) (24)

The face areas [A.sub.e] and [A.sub.n] are given by the following:

[A.sub.e] = [partial derivative] [vector.R] [e.sub.[theta]] and [A.sub.n] = [[vector.R].sub.n] [partial derivative] [theta] [e.sub.r]

area vectors can be written analogously for other face areas. Furthermore, we make linear profile assumption of [bar.P] between cell centroids, thus

[J.sub.e]*[A.sub.e] = [[GAMMA].sub.e]/[[bar.R].sub.e]([[bar.P].sub.E] - [[bar.P].sub.p]/[partial derivative][theta])[partial derivative][bar.R]

[J.sub.n]*[A.sub.n] = [[bar.R].sub.n][[GAMMA].sub.n] ([[bar.P].sub.n] - [P.sub.p]/[partial derivative][[bar.R]]) [partial derivative][theta] (25)

The transport equations in other directions can be written analogously. Similarly using a fully implicit scheme for the first term on right hand side of Equation 22 and expanding the gradients using finite difference scheme in Equation 21we obtain the following:

[[integral] [DELTA]t] [bar.S][partial derivative]V[partial derivative]t = [bar.[S.sup.1]][partial derivative]v[partial derivative]t (26)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (27)

Integrating second term on right hand side of Equation 22 directly we obtain the following:

[[integral] [DELTA]t] [bar.T][partial derivative]V[partial derivative]t = [sigma][bar.R]([H.sub.1] - [H.sup.0]) [partial derivative]V (28)

Using Equations 22 through 28 the discretized form of Equation 18 can be written as follows:

[[[[bar.R].sub.n][[GAMMA].sub.n] ([P.sub.n] - [P.sub.p]/[partial derivative][bar.R]) [partial derivative][theta] - [[GAMMA].sub.S][R.sub.S] ([P.sub.p] - [P.sub.s]/[partial derivative][bar.R]) [partial derivative][theta] + [[GAMMA].sub.e]/[R.sub.e] ([P.sub.E] - [P.sub.p]/[partial derivative][theta]) [partial derivative][bar.R] - [[GAMMA].sub.w]/[R.sub.w]([P.sub.p] - [P.sub.2]/[partial derivative][theta]) [partial derivative][bar.R]].sup.1] = [[[gamma][bar.[eta]][partial derivative]/[partial derivative]r([bar.R][bar.H][[bar.V].sub.ra]) + [gamma][bar.[eta]][partial derivative]/[partial derivative][theta] ([bar.H][[bar.V].sub.[theta]a]).sup.1] [partial derivative]V + [sigma][bar.[eta]]R([H.sup.1] - [H.sup.0]) [partial derivative]V[partial derivative]t (29)

Equation 29 represents a complete discretized form for [[bar.P].sub.P] and can be written in algebraic form as follows:

[a.sub.P][P.sub.P] = [a.sub.E][P.sub.E] + [a.sub.E][P.sub.E] + [a.sub.E][P.sub.N] + [a.sub.E][P.sub.S] + b (30)

The linear system of Equation 30 was solved using Newton's method and Stone's implicit procedure. The discrete equation expresses a balance of discrete flux J and source term. Thus conservation over the individual control volume is guaranteed. After the convergence for pressure and load is achieved the following parameters were evaluated at various operating conditions.

Nondimensional Load Capacity

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (31)

Nondimensional Tangential Velocity

[[bar.V].sub.[theta]] = 3/[sigma][bar.z]([bar.z] - [bar.H])[partial derivative][bar.P]/[partial derivative][theta] + [[bar.V].sub.0a] ([bar.H] - [bar.z]/[bar.H] (32)

Nondimensional Radial Velocity

[[bar.V].sub.r] = 3/[sigma][bar.z]([bar.z] - [bar.H]) [partial derivative][bar.P]/[partial derivative]R + [[bar.V].sub.ra] ([bar.H] - [bar.z]/[bar.H] (33)

Friction Coefficient

Viscous forces experienced by the fixed plate in radial and tangential direction are as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (34)

Coefficient of friction is then calculated by the following:

[mu] = [square root of (([[bar.F].sub.r.sup.2] + [[bar.F].sub.[theta].sup.2])/[[bar.F].sub.z]] (35)

Results and Discussion

In order to examine the details, the current numerical model was evaluated for several cases with an objective of verifying the performance of orbiting thrust bearing with radial grooves and circular surface pockets. Table 1 contains the various operating parameters used in this investigation. These are typical dimensions for a scroll compressor thrust surface. Figure 3a-3c illustrates modeling of various cross sections of grooves as given in Razzaque (1999), whereas Figure 3d shows a circular surface pocket with Np = 3 and Ng = 20. Modeling of the circular pockets is done by superimposing spherical geometry of pockets on the flat surface in cylindrical coordinates.

[FIGURE 3 OMITTED]

Table 1. Parameters Used in Analytical Model Inter radius, [R.sub.1] 31.75 mm Outer radius, [R.sub.2] 46.54 mm Eccentricity of crank shaft, e 2.97 mm Groove/pockets width, [G.sub.w] 8.00 mm Groove/pocket depth, [G.sub.d] 0.020 mm Number of grooves in tangential direction, [N.sub.g] 8.00 mm Number of grooves in tangential direction, [N.sub.p] 1.00 mm Lubricant viscosity, [[eta].sub.0] 0.018 N.s/ [m.sup.2] Rotational speed, [omega] 3000 rpm Minimum film thickness, [h.sub.min] 5e-3 mm Ambient pressure, [P.sub.atm] 0.101325 MPa Cavitation pressure, [P.sub.C] 0.1 MPa Applied normal load 3500 N

Time-Dependent Solution

Figure 4 depicts the pressure distribution at various time steps. The model involves two disks one stationary (thrust surface that has pockets/grooves) at the bottom and one on top which orbits. The disks are brought in contact under the action of external load. In this case, it should be noted that velocity components of the orbiting plate are a function of crankshaft position that is used to drive the orbit plate. Therefore, as time progresses the position of the crankshaft changes and velocities need to be updated at each time step. Since the velocities are updated after each time step the solution matrix becomes stiff and under relaxation is required for gauss - siedel scheme to solve the stiff PDE. The results indicate that initially the load is supported by the squeezing action thus building up pressure on entire thrust surface. However as the time progresses the fluid squeeze film effect diminishes and hydro-dynamic pressure starts to build in the converging sections of the surface pockets and fluid cavitates [Dowson (1979), Elrod (1981) and Brewe (1986)] in diverging regions. In order to understand the pressure build development in the case of orbital motion, it should be noted that kinematic (Please refer Appendix A for the kinematic analysis of the orbiting thrust bearings) position analysis shows that the locus of an arbitrary control volume centroid P on the orbiting disk shown in Figure 1 is a circle of radius, equal to eccentricity (e) of the crankshaft driving the plate, hence during a part of rotation of crankshaft point P moves out of pocket experiencing a converging wedge and giving rise to pressure spike at the edge of the pocket and during remaining part it experiences a diverging wedge causing the fluid to cavitate. The above phenomenon when extended to entire orbit plate causes the pressure spikes to oscillate about the centerline and along the edges of the pocket as the crank rotates. Figure 5 illustrates variation of minimum film thickness with advance in time. Minimum film thickness decreases very fast initially depicting the presence of squeeze film effect but later (approximately after 0.35 seconds) becomes almost constant indicating that steady state operation has been achieved. In general we are only concerned with the steady state performance of the thrust bearing hence the rest of the discussion will be restricted to steady state conditions.

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

Effect of Number of Grooves/Pockets, Ng

Figure 6 illustrates the effect of number of grooves on the load carrying capacity for different surface profiles. In the case of grooves (circular, triangular and trapezoidal) the load carrying capacity for triangular grooves is highest for any number of grooves. This is due to the fact that with the same groove depth ratio and groove width ratio the area of cross section for triangular grooves is the least which causes average film thickness for triangular grooves to be minimum consequently giving greater load support. Maximum load carrying capacity for grooves occurs when the number of grooves is 16. As the number of grooves on the bearing increases, geometric wedge effect increases resulting in greater load support. On the other hand leakage through grooves increases with more number of grooves. The net load carrying capacity is function aforementioned mechanisms. The load carrying capacity of circular pockets is significantly larger as compared to that of the grooves. This can be attributed to the fact that in case of circular pockets the curvature effects are present in both radial and tangential direction, (i.e. geometric wedge effect occurs in both directions) this causes a large magnitude of pressure build up at the edge of pocket. Figure 7 depicts the effect of number of grooves on coefficient of friction, [mu] for different surface profiles. Friction coefficient depends on pressure gradients and couette velocity of the plate. For fixed speed of crankshaft the friction depends only on pressure gradients. In the case of circular pockets, pressure builds up almost symmetric about centerline of the pocket which causes significant reduction in friction coefficient as that compared to that of the grooves.

[FIGURE 6 OMITTED]

[FIGURE 7 OMITTED]

Effect of Groove Depth Ratio, [DELTA]

Figure 8 depicts the effect of groove depth on the load carrying capacity of the thrust bearing for various surface features. It is observed that maximum load carrying capacity occurs at [DELTA] = 1 for grooved surfaces and is independent of groove cross section. For a circular pocket surface the maximum load carrying capacity occurs at [DELTA] = 2. It should be noted that the load carrying capacity is a function of average film thickness and geometric wedge effects. As [DELTA] increases the geometric wedge effect increases thus enhancing the load carrying capacity, however with increase in [DELTA] average film thickness also increases thus reducing the load support. The net load carrying capacity is a sum of both the effects. Reynolds equation is based on the assumption that film thickness is of the order of groove depth and at small groove depths (order 10 to 50 microns) inertia effects of the fluid are negligible, hence the value of [DELTA] is restricted to less than 5 in order to ensure the validity of the Reynolds equation. Apart from the above two effects with increase in [DELTA] the groove/pocket acts as a lubricant sump thus ensuring stable fluid film between surfaces. Figure 8 can also be used to ascertain the sensitivity of the load carrying capacity due to error in manufacturing the specified groove depth. Figure 9 shows the effect of groove depth ratio [DELTA] on coefficient of friction ([mu]) for different surface profiles. At shallow groove depths ([DELTA] < 1) grooves perform better than pockets, however as the groove depth increases frictional performance of circular pocket surfaces is significantly better as compared to that of grooved surfaces.

[FIGURE 8 OMITTED]

[FIGURE 9 OMITTED]

Effect of Groove Width Ratio, [lambda]

Figure 10 illustrates the effect of groove width on load carrying capacity for various surface features. The results indicate that for [lambda] < 500 the load carrying capacity of grooves is higher as compared to that of circular pockets and as [lambda] becomes greater than 500, a bearing with pockets supports higher load than a bearing with grooves. At lower groove width ratios the side leakage is less for grooves but as the width of groove increases, side leakage through grooves becomes large. On the other hand because circular pockets are a locked geometry, side leakage is not a function of pocket width. With increase in value of [lambda] the load carrying capacity for grooves/ pocket increases and there is no optimum value of [lambda]; however, the upper limit on [lambda] is imposed by the size of the thrust surface. Figure 11 shows the effect of groove width on the coefficient of friction ([mu]) for various surface features. In most cases, frictional performance is inversely proportional to load carrying capacity, consequently the friction coefficient for pockets is higher than that for grooves for [lambda] < 500, but the friction coefficient ([mu]) significantly decreases as the value of [lambda] increases beyond 500.

[FIGURE 10 OMITTED]

[FIGURE 11 OMITTED]

Effect of Number of Pockets in Radial Direction, Np

Figure 12 illustrates the effect of number of pockets in the radial direction (Np) has on load carrying capacity and coefficient of friction ([mu]) for a constant area density (area density is the ratio of total area occupied by pockets to total available area on the thrust surface). It should be noted that in order to keep the area density constant with increase in Np, width of the pockets is required to be reduced. The results indicate that smaller number of pockets with a large groove width gives the best performance however the width of the pockets is limited by the size of the thrust surface. This can be attributed to the fact that as Np increases, interaction between the pressure spikes of pockets increases causing steep pressure gradients. The friction coefficient is a function of pressure gradients hence its value increases with increase in Np.

[FIGURE 12 OMITTED]

CONCLUSION

A Numerical model for an orbiting thrust bearing was developed. This model can be used to predict the frictional performance of grooves and circular pockets on the thrust surfaces. The results obtained from the model confirm that circular pocket thrust surfaces provide significantly better performance in orbiting situations as compared to that of uniformly distributed radial grooves. This can be attributed to the fact that in the orbiting case couette flow velocities in both radial and tangential direction are present. Circular pockets having curvature symmetric about their centerline tend to create geometric wedge effects in both the directions thus increasing the load support. Uniformly distributed radial grooves, on the other hand, have no geometric wedge in the radial direction also due to the presence of couette flow radial velocity component side leakage through the grooves is more. The results also indicate that pockets with large width, minimum Np and shallow depths (order of film thickness) are the best configuration for maximizing load carrying and minimizing friction. It should be noted that this model is based on the assumption of full film lubrication and therefore it should be ensured that roughness of the thrust surface is sufficiently below the minimum film thickness.

Reynolds boundary condition does not take into account reformation of full film from in the cavitation zone. However, the numerical method suggested by Elrod (1981) or Payvar and Salant (1992) to remove this restriction resulted in instability of the numerical solution for orbital motion. Insight about temperature distribution at the thrust interface can be obtained by removing the isothermal assumption and coupling the Reynolds equation with the energy equation. In order to evaluate performance of very deep pockets the inclusion of inertia effects Pinkus et al. (1981) would be necessary, but due to presence of radial velocity components a complete solution to the Navier Stokes equation would be required.

ACKNOWLEDGMENT

The authors would like to express their deepest appreciations to the Tecumseh Company for their support of this research study.

NOMENCLATURE

[F.sub.z] = normal load, N

[G.sub.d] = depth of a groove or pocket, m

[G.sub.w] = width of a groove or pocket, m

[h.sub.0] = reference film thickness, m

[h.sub.min] = minimum film thickness, m

h = film thickness, m

[[bar.h].sub.min] = dimensionless minimum film thickness, [h.sub.min]/[h.sub.0]

[bar.h] = dimensionless film thickness, h/[h.sub.0]

[W.sub.z] = nondimensional load, [W.sub.z] = [F.sub.z]/([P.sub.atm][R.sub.0.sup.2])

[N.sub.p] = number of uniformly distributed grooves or pocke along circumference

[N.sub.p] = number of uniformly distributed pockets along radius

P = pressure, MPa

[P.sub.a] = ambient pressure, MPa

[bar.P] = dimensionless pressure P/[P.sub.a]

[D.sub.1] = inner diameter of the annular thrust surface, m

[D.sub.2] = outer diameter of the annular thrust surface, m

[R.sub.1] = inner radius of the annular thrust surface, m

[R.sub.2] = outer radius of the annular thrust surface, m

r = radial coordinate, m

[bar.R] = dimensional radial coordinate r/[R.sub.0]

t = time, s

[t.sub.0] = reference time 1 /[omega]

t = dimensionless time, t/[t.sub.0]

[V.sub.r] = radial velocity component, m/s

[V.sub.[theta]] = tangential velocity component, m/s

[[bar.V].sub.r] = dimensionless radial velocity component [V.sub.[theta]]/[R.sub.0][omega]

[[bar.V].sub.[theta]] = dimensionless radial velocity component [V.sub.r]/[R.sub.0][omega]

z = axial coordinate, m

[bar.Z] = dimensionless axial coordinate z/[h.sub.0]

[[theta].sub.2] = input angle of crank shaft in degrees

[gamma] = characteristics number, 6[[eta].sub.0][omega][R.sub.0.sup.2]/([P.sub.atm][h.sub.0.sup.2])

[sigma] = squeeze number, 12[[eta].sub.0][R.sub.0.sup.2]/([P.sub.atm][h.sub.0.sup.2][t.sub.0])

[DELTA] = groove depth ratio, [G.sub.d]/[h.sub.0]

[lambda] = groove width ratio, [G.sub.w]/[h.sub.0]

[eta] = absolute viscosity, N-sec/[m.sup.2]

[[eta].sub.0] = absolute viscosity of the lubricant at ambient temperature, Pa * s

[bar.[eta]] = dimensionless absolute viscosity, [eta]/[[eta].sub.0]

[omega] = angular velocity of crankshaft, rad/s

[mu] = coefficient of friction

Subscripts

e = east boundary surface of control volume

E = east grid point

n = north boundary surface of control volume

N = north grid point

P = grid point of interest

s = south boundary surface of the control

S = south grid point

w = west boundary surface of control volume

W = west grid point

Superscripts

0 = value from previous time step

1 = value from current time step

REFERENCES

Salama, M.E., 1950. The Effect of Macro-roughness on the performance of Parallel Thrust Bearings. Proc. Inst. Mech. Eng. 163: pp. 149.

Kulkarni, S. S. 1990. Scroll compressor.- Thrust bearing design under laminar conditions. Proc. Int. Compressor. Eng. Conf. pp. 327-332.

Wang Xiaolei, Kato Koji, Adachi Koshi. 2002. The lubrication effect of Micro-Pits on Parallel Sliding Faces of SiC in water. Lubrication Engineers. vol 58(8):27-34.

Yu, T.H., and Sadeghi, F. 2001. Groove effects on thrust washer lubrication. J. Tribol. 123(2):295-304.

Tatsuya Oku, Keiko Anami, Noriaki Ishii, and Kiyoshi Sano. 2004. Lubrication Mechanism at Thrust Slide-Bearing of Scroll compressors (Theoretical Study). Proc. Int. Compressor. Eng. Conf. C104:1-10.

Noriaki Ishii, Tatsuya Oku, Keiko Anami, and Akinori Fakuda. 2004. Lubrication Mechanism at Thrust Slide-Bearing of Scroll compressors (Experimental Study). Proc. Int. Compressor. Eng. Conf. C104:1-10.

Elrod, H.G. 1981. A Cavitation Algorithm. ASME J. Lubr. Technol. 103:350-354.

Dowson, D., and Taylor, C. M. 1979. Cavitation in Bearing. Annu. Rev. Fluid Mech. 190:1-116.

Brewe, D. E. 1986. Theoretical Modeling of Vapor Cavita-tion in Dynamically Loaded Journal Bearings. ASME J. Tribol. 108:628-638.

Pinkus, O., and Lund, J. W. 1981. Centrifugal Effects in Thrust Bearings and Seals under Laminar Conditions. ASME J. Tribol. 103:126-136.

Razzaque, M., M., Kato, T. 1999. Effects of groove orientation on hydrodynamic behavior of wet clutch coolant films. Journal of Tribology, Transactions of the ASME 121(1):56-61.

Payvar, P. and Salant, R. F. Jan 1992. Computational method for cavitation in a wavy mechanical seal. Journal of Tri-bology, Transactions of the ASME 114(1):199-204.

Etsion, I., and Halperin, G. July 2002. A laser surface textured hydrostatic mechanical seal.Tribology Transactions 45(3):430-434.

Patankar, Suhas V. 1980. Numerical Heat Transfer and Fluid Flow. Washington, D.C: Hemisphere Publishing Corp.

Hamrock, Bernard J. 1994. Fundamentals of Fluid Film Lubrication. New York: McGraw-Hill, Inc.

Ferziger, J. H., and Peric, M. 1999. Computational Methods for Fluid Dynamics. Germany: Springer.

APPENDIX A KINEMATIC ANALYSIS

Figure 1 illustrates the vector loop for determining the radial and tangential velocities of an arbitrary control volume centroid P on the orbit plate. Vector equation for control volume centroid P can be written as follows:

[[vector].R] = [bar.[r.sub.2]] + [bar.[r.sub.7]] (A1)

The x andj components of Equation A1 are as follows:

Rcos[theta] = [r.sub.2]cos[[theta].sub.2] + [r.sub.7]cos[alpha] (A2)

Rsin[theta] = [r.sub.2]sin[[theta].sub.2] + [r.sub.7]sin[alpha] (A3)

Differentiating Equations A2 and A3 with respect to time we obtain the following

Rcos[theta] - Rsin[theta][theta] = [r.sub.2][omega]sin[[theta].sub.2] (A4)

Rsin[theta] + Rcos[theta][theta] = [r.sub.2][omega]cos[[theta].sub.2] (A5)

Equations A4 and A5 in a matrix form are given by the following:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A6)

Solution to Equation A6 can be obtained inverting the matrix as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A7)

Equation A7 can be expanded to obtain the following:

R = -[r.sub.2][omega]sin[[theta].sub.2]cos[theta] + [r.sub.2][omega]sin[theta]cos[[theta].sub.2] (A8)

[theta] = [r.sub.2][omega]sin[theta]sin[[theta].sub.2] + [r.sub.2][omega]cos[theta]cos[[theta].sub.2] (A9)

The equations for radial velocity ([V.sub.r]) and tangential velocity ([V.sub.[theta]]) can be obtained form Equations A8 and A9 as follows:

[V.sub.r] = R = e [omega] sin ([theta] - [[theta].sub.2] (A10)

[V.sub.[theta]] = R[theta] = e [omega] cos ([theta] - [[theta].sub.2](A11)

Here e = [r.sub.2] = |[r.sub.s.sup.[tau]]} and [omega] = [[theta].sub.2] = [partial derivative] [[theta].sub.2]/[partial derivative]t.

Amit Vaidya

Farshid Sadeghi, PhD

Amit Vaidya is a mechanical engineer, Farshid Sadeghi is a professor in the School of Mechinical Engineering, Purdue University, West Lafayette, IN

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Author: | Vaidya, Amit; Sadeghi, Farshid |
---|---|

Publication: | ASHRAE Transactions |

Date: | Jul 1, 2008 |

Words: | 6018 |

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