Effect of Deterministic Asperity Geometry on Hydrodynamic Lubrication

Transcription

1 Ravinder B. Siripuram Research Assistant Lyndon S. Stephens Associate Professor (contact author) Bearings and Seals Laboratory, Department of Mechanical Engineering, University of Kentucky, 151 Ralph G. Anderson Building, Lexington, KY Effect of Deterministic Asperity Geometry on Hydrodynamic Lubrication This paper presents a numerical study of the effects of different shapes of deterministic microasperities in sliding surface lubrication when hydrodynamic films are found. Positive (protruding) and negative (recessed) asperities of constant height (depth) are considered with circular, square, diamond, hexagonal and triangular cross-sections. Of particular interest is the impact of asperity/cavity cross-sectional geometry on friction and leakage, which has importance in sealing applications. The results indicate that the friction coefficient is insensitive to asperity/cavity shape, but quite sensitive to the size of the cross-section. By contrast, leakage rates are found to be quite sensitive to both crosssectional shape and size, with triangular asperities giving the smallest leakage rate and square asperities giving a largest leakage rate. The minimum coefficient of friction for all shapes is found to occur at an asperity area fraction of 0. for positive asperities and 0.7 for negative asperities. Finally, the results indicate the existence of a critical asperity area fraction where the performance curves for positive and negative asperities cross over. These cross-over points are identified for friction coefficient and leakage rate. DOI: / Introduction Contributed by the Tribology Division for publication in the ASME JOURNAL OF TRIBOLOGY. Manuscript received by the Tribology Division March 1, 003 revised manuscript received July, 003. Associate Editor: M. Fillon. The use of applied surface textures to improve the hydrodynamic effect on lubricated surfaces through local cavitation is a widely accepted approach. Deterministic micro asperities are asperities of a prescribed shape, size, orientation, and distribution that comprise an engineered surface texture. Methods to fabricate such microasperities traditionally include photoetching 1 and laser ablation. Recently, MEMS based fabrication techniques have been developed to manufacture deterministic microasperities that may be either recessed into negative or protruding up from positive the surface. These asperities can be made from robust materials including metal, ceramics and plastic, making them ideal for a wide variety of bearing and sealing applications. The Bearings and Seals Laboratory at the University of Kentucky has established two such manufacturing processes 3,4. These are modified forms of the LIGA process 5,6 and the UV Photolithography process 7,8. LIGA is the more expensive of the two processes and is more ideal for fabricating positive asperities with a large height-to-width aspect ratio and ultra precision feature size down to 800 nm. UV Photolithography is comparatively less expensive and can be used to produce either positive or negative asperities, but with smaller aspect ratios and less accuracy feature size down to 1 microns. Of particular importance is that either of these techniques can be used to fabricate any prescribed asperity cross section. Figure 1 a shows a typical outcome of the LIGA based process, which is a field of hexagonal asperities positive, 15microns in height, 550 microns in average diameter and 165 microns edge-to-edge spacing. The asperities are arranged in a hexagonal pattern and are fabricated of nickel and used to texture the surface of a thrust ring 38.1 mm in outer diameter. Figure 1 b shows a second surface texture that resulted from the UV photolithography process, consisting of a square array of recessed hexagonal cavities 10 m deep, 600 m average diameter, and 50 m edge-toedge spacing. These manufacturing developments have spawned a renewed interest among the industrial and academic communities on the application of deterministic microasperities on thrust bearing surfaces to improve the tribological performance. In hydrodynamic lubrication, this depends upon the viscosity of the lubricant, the relative velocity of the moving surfaces, and the geometries of both the microasperities and the thrust ring. Generally, for a given application, the applied load, the viscosity ignoring the temperature effects, velocity and the dimensions of the thrust bearing are constant. The parameters then available for a surface texture designer are the microasperity shape, size, orientation and distribution. Early research on the effect of deterministic microasperities on lubrication was conducted in the late 1960 s and focused on the effect of asperity size and distribution 1,9,10. In these works the authors experimentally and theoretically investigated three samples consisting of circular or square shaped asperities distributed in a hexagonal and/or square array. For the case of the circular geometry, both positive and negative asperities were examined. From this work the authors concluded, among other things, that 1 hydrodynamic performance is largely insensitive to asperity shape; and negative asperity patterns result in less leakage. More recent work has focused on modeling and applying the micro-pore negative asperity patterns that result from the laser ablation process. These works, again, focus on the effect of the size and distribution of the microasperities. Etsion and Burstein 11 examined hemispherical pores on a square array for a fixed mechanical seal design. The authors found that asperity size significantly affects friction, leakage and film thickness over a wide range of asperity distributions. Etsion and Kligerman 1 further examined micro-pores distributed on a radial row around a seal ring circumference and improved the lubrication model to include interactions between adjacent asperities and the Reynolds cavitation condition. Ronen and Etsion 13 demonstrated that the friction in reciprocating automotive components is reduced by using the proper micro-pore surface texture. Finally, Arghir et al. 14 recently examined shear driven flows over rough surfaces and have concluded that convective inertia effects have a significant impact and are not well predicted using the traditional Reynolds or Stokes equations. This paper presents a numerical study of the effect of deterministic microasperity geometry on lubrication performance. Of particular interest is the effect of asperity shape and orientation in addition to asperity size and distribution on friction, leakage and Journal of Tribology Copyright 004 by ASME JULY 004, Vol. 16 Õ 57

2 Fig. 1 Hexagonal, nickel surface textures manufactured at the Bearings and Seals Laboratory 4 : a positive asperities fabricated using LIGA; and b negative asperities cavities fabricated using UV Photolithography. film thickness for a given set of operating parameters. Seven different asperity shapes/orientations hex, square, diamond, circle and triangle in both the positive and negative configuration are examined. In contrast to the earlier work, the authors of the present study have found that 1 the hydrodynamic performance is sensitive to deterministic asperity shape and orientation; and for certain conditions, positive asperities result in less leakage than negative asperities. Theory Figure shows the distribution of asperities on a typical thrust ring used in mechanical seal applications with an inner radius of r 0 and an outer radius of r 1 subjected to an external sealant pressure of P s and ambient pressure, P a 0. This application is the sample problem used for this study. The pressure at any asperity is due to the hydrodynamic component due to slider motion over the asperity and the hydrostatic component due to the pressure, P s. The focus of this paper is the effect of the asperities on the hydrodynamic component. Assuming a small face width (r 1 r 0 )as compared to the ring average radius, the hydrodynamic characteristics are analyzed in Cartesian coordinates by unrolling the thrust ring geometry into a strip, as shown. This results in a square array of asperities the asperity distribution in which the interaction between adjacent asperities must be considered. The approach taken is to consider a single unit cell with periodic boundary conditions that account for asperity interactions in the tangential x-direction. It is assumed that the effects of radial interactions are negligible by comparison. This assumption is more valid for small values of asperity area fraction,. Asperities that lie closer to the inner radius of the ring will have an average hydrostatic pressure close to ambient, and will more likely experience local cavitation 11. This study considers only these asperities, as they are responsible for the majority of the load capacity. Figure 3 gives the geometry of a single unit cell with either a positive Fig. 3 a or negative Fig. 3 b asperity within. The asperity height depth is denoted a, and the film thickness is denoted b. The slider velocity, U, isinthex-direction. For a given asperity cross-sectional shape, the asperity is centered within the unit cell. Table 1 summarizes the seven different asperity geometries considered in this study. These are as follows: 1 a circle; a square with the flat side perpendicular and parallel to the slider motion; 3 an oriented square diamond with the apex perpendicular and parallel to the slider motion; 4 a hexagon with the apex oriented perpendicular to the slider motion; 5 a hexagon with the apex oriented parallel to the slider motion; 6 a triangle with the apex oriented perpendicular to the slider motion; and 7 a triangle with the apex oriented parallel to the slider motion. In all cases, both positive and negative features are considered. The size of the asperity is measured using the asperity area fraction,, which is defined as the fraction of the asperity crosssectional area to that of the unit cell. Table 1 summarizes the equations used to compute and also gives the maximum possible value, max. Note that for the square unit cell examined in this paper, max is limited to values less than unity in all cases except for the square asperity. The film thickness for all geometries is given by h,y b above positive asperities (1) a b between positive aperities h,y a b b above negative asperities between negative asperities Assuming a thin, Newtonian lubricant film undergoing laminar, incompressible flow and neglecting temperature and inertial effects, the pressure, p(x,y), in the fluid film is governed by Reynolds equation h p 3 y h p 3 y 6 U h where is the fluid viscosity. Note that squeeze film effects have also been neglected. Further assumptions include no pressure variation across the film thickness and no slip at the film bound- Fig. Microasperities on thrust ring () (3) 58 Õ Vol. 16, JULY 004 Transactions of the ASME

3 aries. For the hydrodynamic and hydrostatic components of the pressure at the innermost radial row of asperities, the following boundary conditions are enforced on the unit cell boundary p p,l/ P cell (4) p, L/ 0 (5) p L/,y p L/,y (6) p L/,y L/,y (7) where P cell is the pressure at the outer boundary of the unit cell the sealed pressure, P s, divided by the number of asperities radially across the thrust face, and Eqs. 6 and 7 are periodic boundary conditions in the tangential direction that account for adjacent asperity interactions. The Reynolds cavitation condition is approximated using the well-known Swift-Steiber conditions at the vapor region in the film: p p cav 0 if p 0 dp dn 0 at the liquid-gas boundary (8) Fig. 3 a Side view of a positive asperity; b side view of a negative asperity; and c typical unit cell. Despite having the advantages of a regular geometry, a search for an analytical solution to Eq. 3 is set aside in favor of a finite difference based numerical solution, both for the ease of obtaining a solution and for implementing the cavitation boundary condition. The resulting finite difference equations are solved iteratively using successive over relaxation with a square, staggered grid. There is a likelihood of introducing error into the numerical calculations due to the irregularity of the asperity shapes as compared to the square unit cell. To minimize this error, the mesh density is increased. The approach taken in this paper is to study the effects of asperity cross-sectional geometry on hydrodynamic effects assuming a constant set of running conditions. Therefore, the bearing number U/WL is held constant and the film thickness is varied from one asperity geometry to the next. This requires an iterative solution to Eqs. 3 8, which is accomplished using an optimization routine that minimizes the difference between the desired and the computed. Using the pressure results, the average load per unit area for one unit cell is calculated from the expression: Journal of Tribology JULY 004, Vol. 16 Õ 59

4 L/ W 1 p,y dxdy (9) L L/ L/ The friction coefficient is computed by considering the total frictional force over the unit cell. The average shear stress on the fluid is given by L/ L/ F 1,y dxdy (10) L L/ L/ where the expression for shear stress is,y u,y (11) z The resulting expression for coefficient of friction is f F/W, which is approximated as L/ f U W b 1 a b for positive asperities (1) f U W 1 for negative asperities (13) b a b In case of positive asperities, for very large asperity area fractions, the area above the asperity dominates the friction coefficient, and for very low values the area between asperities dominates. It is quite the opposite for negative asperities. Leakage due to the sealed pressured, P s, is governed by Poiseuille s law and is computed directly from the resulting pressure distribution using Q 1 L/ h P 1 L/ 3,y dx (14) This leakage occurs through the channels in the radial direction film thickness and the gap between the positive asperities, and is dominated by the hydrostatic pressure difference across a single unit cell. For positive asperities, the leakage channel is formed by the film thickness and the gap between the asperities. For negative asperities, only the film thickness provides the leakage channel. Thus, the approximate expressions for leakage rate are: Q p r 0 r 1 1 r 1 r 0 b3 1 a b 3 for positive asperities (15) Q p r 0 r 1 1 r 1 r 0 b3 for negative asperities (16) Equation 15 for positive asperities includes the asperity area fraction, as it provides a blockage for the leakage path. Equation 16 for negative asperities is independent of asperity area fraction. Note that this study neglects flow effects due to rotation and surface tension 1. Parameter Symbol Table Values of Constants Units Parameter Value N Asp/mm 5.4 L m 43 U m/s.66 r 0 mm 11.1 r 1 mm a m 5 P s N/mm 0.07 N-s/m 0.04 W N/mm of friction if any factor other than asperity area fraction is considered for comparison. Therefore, is used for the basis of comparison in this study. Figure 4 shows the benchmark results for the two dimensional numerical model used in this paper. Comparison is made between a very long in the y-direction square asperity to that of the well known analytical solution of a one dimensional positive Rayleigh step bearing. The nondimensional peak pressure is plotted versus the asperity area ratio,. Results for both positive and negative asperities are in very close agreement with the analytical solution over the entire asperity area ratio range. Further, the positive and negative asperity results are symmetric as one would expect for the nondimensional case. Figures 5 and 6 show typical, nondimensional pressure distributions over the unit cell. The pressure values are converted into nondimensional form by dividing by the applied load per unit area, W. Figure 5 a shows the pressure distribution for a positive, square asperity and For such small asperity area fractions, there is negligible interaction between adjacent asperities, so the pressure gradient, dp/dx 0, at x L/ as if no additional asperity occurs before or after. By contrast, Fig. 5 b shows the pressure distribution for a positive, square asperity and For these larger asperity area fractions there is significant interaction between adjacent asperities as the pressure gradient, dp/dx 0 atx L/. In both cases, the Reynolds cavitation condition is apparent as the leading edge of the cavitation region is clearly beyond the centerline of the bearing (x 0) as one would expect for the square pad geometry which is similar to the Rayleigh step. Figure 5 c shows the pressure distribution for a negative, square asperity with Comparing this to Fig. 5 b, it is seen that the pressure is reduced at the edges of the square asperity for the positive case, making the square shape less pronounced as compared to the negative asperity case in Fig. 5 c. This is due to the presence of the leakage path adjoining the sides of the positive asperities. In the absence of such a factor for negative asperities, the pressure distribution follows the geometry of 3 Results and Discussion Table gives the numerical data for the mechanical seal ring considered in this study. In order to isolate the effect of asperity shape, the asperity distribution (N 5.4 asperities/mm on a square array, unit cell side length (L 43 m), and asperity height (a 5 m) are held constant. The bearing number is also held constant at U/WL 51 and the pressure being sealed is assumed to be P s 0.07 MPa 10 psi. These values are selected as they reflect typical values of surfaces that have been fabricated and tested by the authors in 3,4. The load support values are dependent on the asperity area fraction for a given pressure distribution. According to 1, due to the sensitivity of this dependence, there could be vast variations in the values of coefficients Fig. 4 Benchmarking of numerical solution 530 Õ Vol. 16, JULY 004 Transactions of the ASME

5 Fig. 6 a Pressure distribution for a positive triangular asperity, Ä0.10; and b pressure distribution for a negative triangular asperity, Ä0.10. asperity edges. This is due to the square staggered grid used in the finite difference calculation to approximate the triangular geometry, and is minimized by using a finer finite difference mesh. Figure 7 shows the resulting film thickness, b, versus asperity area fraction for both positive and negative asperity shapes. Recall that the film thickness, b, is the distance between the slider and the Fig. 5 a Pressure distribution for a positive square asperity, Ä0.05; b pressure distribution for a positive square asperity, Ä0.5; and c pressure distribution for a negative square asperity, Ä0.5. the square more closely. Finally, Fig. 6 compares the pressure distributions for a positive Fig. 6 a and a negative Fig. 6 b triangular asperity with Note, again, that the pressure distribution follows the shape of the asperity, however, the pressure distribution for the positive asperity case is jagged at the Fig. 7 Film thickness, b Journal of Tribology JULY 004, Vol. 16 Õ 531

6 Fig. 8 Comparison of coefficient of friction for a typical positive and negative asperity Fig. 10 Leakage due to P across ring first contact with the stationary surface, as shown in Fig. 3. Figure 7 indicates that negative asperities result in a consistently larger film thickness as compared to positive asperities. Further, there is a significant spread in film thickness versus asperity shape at certain values 0 percent between the square and the triangular at 0.3). This spread is less for positive asperities than it is for negative. A large film thickness is a good result from the standpoint of wear, as the chance of the two surfaces touching during operation is reduced. However, from a leakage standpoint a smaller film thickness is desirable as less area is open to flow between the two surfaces. Finally, it should be noted that the film thickness presented in Fig. 7 includes only the film above the asperities and does not include the open area between asperities. This difference is accounted for in the leakage calculation of Eqs Figure 8 compares the coefficients of friction for positive and negative square asperities for the entire range of. Both positive and negative asperities have a minimum value as one would expect from the film thickness curves that exhibited a maximum value over the range. For positive asperities, an asperity area fraction of 0. gives the minimum coefficient of friction and for a negative asperity 0.8 gives the approximate minimum. For most of the asperity area range, the coefficients of friction for a negative asperity are less than those of a positive asperity. But the graph indicates that there exists a critical value of cr 0., below which the coefficient of friction for a positive asperity is smaller. This indicates a selection for the surface texture designer. Figure 9 shows the comparative values of coefficient of friction for the positive and negative asperities of all the geometries. There is a close agreement of the results for a hexagon and a circle with that of a square. The values in the case of a triangle deviate from the above by not more than 5 percent. Figure 9 further confirms the result found in 1 that indicated the friction coefficient is largely independent of asperity shape but very sensitive to size. Figure 10 shows the leakage due to the pressure, P s, across the thrust ring, for each of the asperity shapes considered. In contrast to the friction coefficient, this graph indicates that the leakage rate is quite sensitive to both asperity size area fraction and shape cross-section. Here the oriented triangle has the smallest leakage rate for both the positive and the negative asperity, and the square has the largest leakage rate. These results indicate that there is a benefit to exploring different asperity shapes to enhance lubricant performance in the area of leakage rate. It is also clear from the graph that leakage follows the trend of film thickness in general. For a positive asperity, leakage is maximum when takes the value of 0.. This is also the value where the coefficient of friction is the minimum. In the case of negative asperities, leakage is a maximum when is equal to 0.5 or its limiting value in the case of a triangle, as compared to 0.8 for the minimum friction coefficient. If the film thickness is kept constant as in previous studies 1, it is found that the positive asperity generates more leakage than a negative asperity in the entire range of. But, for a constant load bearing number, there exists a critical asperity area fraction beyond which positive asperities have less leakage than the negative asperities. This critical value is between 0. and 0.4 for all asperities considered in this study. Figure 11 gives the same results as Fig. 10 except for the triangular geometries alone. This plot shows approximately a 0 percent difference between leakage rates based on the orientation of the triangular asperity shape. Two effects are noteworthy in this graph. First, for a positive asperity the simple triangle has a larger leak rate than the oriented triangle case #6. This is due to the fact that the oriented triangle presents more of its area to impede any flow in the radial direction see Table 1 for geometry. Second, for a negative asperity the simple triangle has a smaller leak Fig. 9 Friction coefficient Fig. 11 Comparison of leakage for different cases of a triangle 53 Õ Vol. 16, JULY 004 Transactions of the ASME

7 rate than the oriented triangle. This is due to the larger film thickness required to support the load for the oriented triangle. For the oriented triangle, the side of the triangle presented normal to the slider direction is greater than that for the simple triangle. This leads to smaller film thickness requirement for the simple triangle to generate the same average pressure as the oriented triangle. Table 3 summarizes the key results of this study for each asperity shape. The asperity area fraction that minimizes the friction coefficient, opt, is independent of asperity shape for positive asperities, ( opt 0.) but dependent on asperity shape for negative asperities (0.4 opt 0.8). This is because cannot exceed max as defined in Table 1. The critical asperity area fraction, cr, f, which defines the cross-over point between negative and positive asperities for friction coefficient is only slightly dependent on asperity shape The minimum friction coefficient, f min, is found to be largely independent of asperity shape and concavity. This confirms previous results found in 1. The maximum leakage rate, Q max, over all, however, is found to be quite dependent on asperity shape and concavity, varying by a factor of over the asperity shapes. The critical asperity area fraction, cr,q, which defines the cross-over point between negative and positive asperities for leakage is dependent on asperity shape and orientation Finally, the film thickness corresponding to Q max is found to be dependent on asperity shape and concavity, as one would expect. This result correlates with the leakage results as one would expect. 4 Conclusions This paper utilized numerical modeling techniques to explore the effect of basic asperity properties comprised of shape, size, concavity and orientation on lubrication characteristics for a simple thrust slider application with a fixed dimensionless bearing number of U/WL 51. Different regular shapes consisting of square, diamond, circular, hexagonal and triangular asperities, all distributed in a square array, were examined. From these results the authors conclude the following: 1 Friction coefficient is largely independent of asperity shape and orientation but very sensitive to asperity area fraction size. The asperity area fraction size that minimizes the friction coefficient is independent of asperity shape and orientation for positive features but dependent on asperity shape and orientation for negative features. 3 Leakage and film thickness are dependent on asperity shape, concavity, orientation and size. 4 For some asperity patterns there exists a critical asperity area fraction, cr, f, below which positive asperities have a lower friction coefficient than negative asperities, and above which the reverse is true. 5 The cross-over point defined by cr, f varies only slightly with asperity shape, orientation and concavity. 6 For some asperity patterns there exists a critical asperity area fraction, cr,q, below which negative asperities leak less than negative asperities, and above which the reverse is true. 7 The cross-over point defined by cr,q is dependent on asperity shape, orientation and concavity. While these conclusions are limited to the sample model in this paper, they do illustrate the potential benefits and limitations that each asperity geometry may bring to hydrodynamic lubrication of conformal surfaces. Acknowledgment The authors wish to thank National Science Foundation for their sponsorship of this work under the Surface Engineering and Materials Design Program Award No.: CMS Nomenclature F average Shear Stress Pa L period of the unit cell m N number of Asperities per unit area m P a atmospheric Pressure Pa P cell lubricant Pressure at unit cell outer boundary Pa P s lubricant Pressure at ring outer radius Pa Q leakage Rate m 3 /s R 0 equivalent Asperity Radius m U velocity m/s W average Load Support per unit area Pa a asperity Height m b film Thickness m f coefficient of Friction h film thickness m p pressure Solution Pa inner Radius of the Thrust Bearing m r 0 r 1 outer Radius of the Thrust Bearing m s side of a square/triangle/hexagon m p differential Pressure Pa dimensionless Bearing Number asperity Area Fraction viscosity Pa s shear Stress Pa Journal of Tribology JULY 004, Vol. 16 Õ 533

8 References 1 Anno, J. N., Walowit, J. A., and Allen, C. M., 1969, Load Support and Leakage From Microasperity-Lubricated Face Seals, ASME J. Lubr. Technol., pp Etsion, I., Halperin, G., and Ryk, G., 000, Improving Tribological Performance of Mechanical Components by Laser Surface Texturing, Journal of the Balkan Tribological Association, 6, pp Stephens, L. S., Siripuram, R., Hayden, M., and McCartt, B., 00, Deterministic Microasperities on Bearings and Seals Using a Modified LIGA Process, Proceedings of ASME Turbo Expo 00, Paper No. GT Kortikar, S. N., Stephens, L. S., Hadinata, P. C., and Siripuram, R. B., 003, Manufacturing of Microasperities on Thrust Surfaces Using Ultraviolet Photolithography, Proceedings of the ASPE 003 Winter Topical Meeting, 8, pp Becker, E. W., Ehrfeld, W., Hagmann, P., Maner, A., and Munchmeyer, D., 1986, Fabrication of Microstructures With Extreme Structural Heights by Synchrotron Radiation Lithography, Galvanoforming and Plastic Moulding LIGA Process, Microelectron. Eng., 4, pp Hagmann, P., Ehrfeld, W., and Vollmer, H., 1987, Fabrication of Microstructures With Extreme Structural Heights by Reaction Injection Molding, Makromolkulare Chemie-Macromolecular Symposia, 4, pp Madou, M., 1997, Fundamentals of Microfabrication, CRC Press LLC. 8 Senturia, S. D., 001, Microsystem Design, Kluwer Academic Publishers, Boston, MA. 9 Hamilton, D. B., Walowit, J. A., and Allen, C. M., 1966, A Theory of Lubrication by Microirregularities, ASME J. Basic Eng., pp Hamilton, D. B., Walowit, J. A., and Allen, C. M., 1968, Microasperity Lubrication, ASME J. Lubr. Technol., pp Etsion, I., and Burstein, L., 1996, A Model for Mechanical Seals With Regular Microsurface Structure, Tribol. Trans., 39 3, pp Etsion, I., Kligerman, Y., and Halperin, G., 1999, Analytical and Experimental Investigation of Laser-Textured Mechanical Seal Faces, Tribol. Trans., 4 3, pp Ronen, A., Etsion, I., and Kilgerman, Y., 001, Friction-Reducing Surface- Texturing in Reciprocating Automotive Components, Tribol. Trans., 44 3, pp Arghir, M., Roucou, N., Helene, M., and Frene, I., 003, Theoretical Analysis of the Incompressible Laminar Flow in a Macro-Roughness Cell, ASME J. Tribol., 15, pp Õ Vol. 16, JULY 004 Transactions of the ASME