Performance of Damaged Hydrostatic Bearings: Predictions Versus Experiments

Abstract

Contributed by the Tribology Division of the American Society of Mechanical Engineers for presentation at the ASME/STLE Tribology Conference, Cancun, Mexico, October 27–30, 2002. Manuscript received by the Tribology Division February 18, 2002 revised manuscript received May 7, 2002. Associate Editor: M. M. Khonsari.The technology of hybrid (hydrostatic/hydrodynamic) fluid film bearings for process fluid applications has improved in the last few years through extensive experimentation and analysis. The increasing number of applications encompasses cryogenic liquid turbopumps, A/C compressors, and canned motor pumps. Recently, a major firm in rotating machinery monitoring and protection introduced hydrostatic bearings as a revolutionary concept enabling rotor dynamically stable operation for industrial turbomachinery by providing adequate means of support stiffness and viscous damping dissipation 1. Hybrid bearings derive their load capacity from shear flow driven effects (hydrodynamic wedge) and from controlled pressure drop and flow resistance effects through feed restrictors and within the bearing film lands. Hydrostatic bearings can support large loads without journal rotation, and provide large (controllable) direct stiffnesses and good damping coefficients. However, hydrostatic bearings have the same limited stability characteristic (self-excited half frequency whirl) as plain cylindrical bearings, further aggravated by pneumatic-hammer instability in poorly designed configurations for highly compressible fluids and gases 2. In most process fluid applications, the lubricants are of low viscosity and the operating conditions demand large pressure differentials and high rotational speeds, thus bringing flow regimes characterized by turbulent flow conditions and dominated by fluid inertia effects. Comprehensive bulk-flow model for prediction of hybrid bearing static and dynamic forced performance are available 34. Several publications detail experimental results for hybrid bearings load capacity, flow rate and rotordynamic force coefficients. In particular, Childs et al. 567 has conducted a comprehensive test program with water-lubricated hybrid bearings emulating typical operating conditions found in cryogenic liquid turbopumps. Bulk-flow based predictions correlate well with the measurements, thus allowing the rapid design of hybrid bearings and increasing their reliability in field applications 8910. Recently, Laurant, and Childs 11 forward experimental results for the leakage and force coefficients from two damaged hydrostatic bearings. One five recess, water lubricated hydrostatic bearing has an injection orifice plugged, and the second has a locally enlarged clearance to simulate a worn condition (65 percent increase in local clearance). The first off-design represents failure of the feed and filtering system with debris contamination reaching a bearing supply orifice and preventing adequate lubrication of the film lands, thus potentially offsetting the bearing direct stiffness coefficients. The second off-design configuration includes an uneven bearing land surface simulating the inevitable wear attained during the frequent startup and shutdown processes in liquid rocket engine turbopump applications. In the experiments, the degrading operating features take place at the orifice and recess most sensitive to loaded operation, i.e. around the position of minimum film thickness. The authors in 11 indicate that analytical tools are not available to assess the bearing performance as compared to the test measurements, and call for their development. However, the computational flow models are readily available; and thus, the present technical note forwards comparisons of predictions to the experimental data. Details on the modeling and experimental procedure can be found in the appropriate references 34 and 567, respectively. Figures 1 show hybrid journal bearings along with the coordinate system for definition of journal displacements and recesses angular position. Figure 1(a) notes a plugged orifice at the bottom recess, and Fig. 1(b) depicts the worn surface profile also located at the bottom recess aligned with the X-axis. A liquid at high pressure Ps and inlet temperature Ts flows radially through orifice restrictors and enters into the bearing recesses with a mean pressure PR. The pressure field within the recesses is determined from flow continuity with the film lands, momentum exchange at the orifice plane and a viscous rise due to journal rotation. At the recess edges, an inertial pressure drop also occurs due to the sudden transition from the recess of depth HR into the film lands of thickness (H). Past the recesses, the liquid then flows through the film lands and the pressure drops to the discharge value Pa.The model considers the fully developed turbulent bulk-flow of a fluid whose material properties depend on its local thermophysical state of pressure and temperature. The general transport equations including these features are 34: (1)∂ρHψ∂t+∂ρHUxψ∂x+∂ρHUyψ∂y=SwhereVariableSource term, Sconservation of massequationtransport of circumferentialmomentum velocitytransport of axialmomentum velocityψ=10ψ=Ux−H ∂P∂x−μH κxUx−κJ ΩR2ψ=Uy−H ∂P∂y−μH κyUyThe film thickness (H) is a function of the nominal clearance (C) and journal center eccentricity eX,eY displacements, i.e., (2)H=Cθ+eX cos θ+eY sin θThe wall shear stress parameters κy=κx=1/2κJ+κB with κJ=fJRJ,κB=fBRB, and the friction factors fJ,B depend on the bearing and journal surface conditions and the flow Reynolds numbers relative to the rotating RJ and stationary RB surfaces 12. Small amplitude journal motions ΔX,ΔY with frequency (ω) about an equilibrium position eX0,eY0 allow expressing the film thickness and flow variables as the superposition of equilibrium and perturbed flow fields, i.e., H=H0+ΔX cos θ+ΔY sin θeiωt;(3)ψ=ψ0+ΔXψX+ΔYψYeiωt;ψ={P,Ux,Uy,κJ,κB}Substitution of the fields above into the transport equations in the film lands and recesses renders sets of zeroth and first-order flow equations for evaluation of the bearing static load capacity, drag torque, recess flow rates and dynamic force coefficients. Well known control-volume CFD schemes 1314 are implemented to solve the flow equations along with a Newton-Raphson scheme to satisfy the recess flow rate and pressure constraints. References 34 discuss the numerical method accuracy, its extension to compressible fluids and complex geometries, and parametric studies to determine the sensitivity of rotordynamic force predictions to variations in input (usually known) parameters and empirical coefficients. Integration of the equilibrium and first-order pressure fields on the journal surface render fluid film forces Fα=X,Y and rotordynamic force coefficients K,C,Mα,β=X,Y, i.e., (4)Fα=∫P0hαdxdy;αβ=X,YKαβ−ω2Mαβ+iωCαβ=∫Pβhαdxdy;with {hX=cos θ,hY=sin θ}. The whirl frequency ratio WFR determining the bearing rotordynamic stability is a complex function of the twelve force coefficients 3. For centered operation it reduces to WFR=KXY/ΩCXX in the absence of fluid inertia effects. Figures 1(a) and 1(b) show the bearings configurations with the plugged orifice and worn land surface, respectively. Table 1 details the geometry of the reference (undamaged) five-recess hydrostatic bearing lubricated with water. Reference 11 presents results for three operating speeds designated as (LS) low, (MS) middle and (HS) high, and corresponding to 10.2, 17.4, and 24.6 krpm, respectively. Supply pressure conditions are also noted as (LP) low, (MP) middle, and (HS) high for 40, 55 and 70 bar, respectively. To simulate the plugged-orifice in recess 1 (aligned with X-axis), the model considers a minute orifice diameter equal to 1 micron thus increasing greatly the flow resistance. In the simulated worn hydrostatic bearing, the film thickness increases locally to produce a maximum change of 65 percent (δ=0.066 mm) of the nominal radial clearance at the center of recess 1, as in the reported experiments. The shaft radius defines the arc of the worn surface, which is uniform along the axial direction. Note that in the reference bearing configuration, orifice 1 is not plugged and its surfaces are in pristine condition (no wear). Comparisons between the test results and the numerical predictions for the flow rate and the rotordynamic coefficients of the damaged bearings at the centered rotor position follow. In all cases, predictions and experimental results for the reference (undamaged) configuration are included to evidence the effects of the off-design characteristic. Note that the figures depicting the comparisons contain on their horizontal axis the operating conditions grouped in the left, middle and right as LS (low speed), MS (middle speed), and HS (high speed operation), respectively. In each sub group the supply pressure varies from low (LP), to middle (MP) to high (HP), respectively. In the numerical predictions, the orifice discharge coefficient Cd is found by matching as best as possible the flow rate for the medium MS/MP reference configuration. References 1516 list the recorded experimental clearances and actual feed pressures for each test condition. These parameters are used in the predictive process to determine the bearing static and dynamic performance characteristics. Laurant and Childs 11 also note that the uncertainties in the measurement are low for the flow rate, stiffness coefficients and direct damping coefficient (<2 percent), while relatively high for cross-coupled damping and inertia coefficients (∼16 percent). The limited extent of a technical note does not allow presenting comparisons for all experimentally recorded parameters; namely flow rate, twelve force coefficients and WFR, and including variations in the journal eccentricity. Thus, comparisons for selected parameters follow. Flow rate (Fig. 2) increases rapidly with feed pressure and decreases slightly with journal speed. The model predicts well an overall 15 percent decrease in measured flow rate (Q) from the reference to the plugged-orifice bearing. The reduction is clearly due to the loss of one feed orifice. Direct stiffness coefficients (Fig. 3) increase with supply pressure and are seldom affected by rotor speed. However, plugging one orifice produces an uneven pressure distribution that causes the sensitive drop in direct stiffness, in particular along the +X axis. The model predicts well the reduction (20 to 25 percent decrease) in direct stiffness KXX as shown. However, eccentric journal operation (loaded condition) causes a rapid increase in stiffness for the plugged orifice bearing as reported in 11. The model also predicts the rapid rise although results are not presented by brevity. Cross-coupled stiffness coefficients (Fig. 4) Cross-coupled stiffnesses increase rapidly with journal speed, and are affected by increasing supply pressure to the extent that the flow turbulence level increases (higher effective viscosity). The predictions show minimal differences between the reference and plugged orifice conditions, while the test results evidence a large increase in cross-coupled stiffness −KYX. Thus, the cross-stiffness coefficient is not just due to the hydrodynamic effect in the film lands but also related to the flow condition in the ineffective recess (plugged orifice). The model sensibly underpredicts the increase in cross-coupled stiffness KXY, except at high speeds for the off-design condition. Direct damping coefficients (Fig. 5) increase sensibly with supply pressure and the test plugged orifice condition renders larger coefficients. The model predicts well the slight variations of the direct damping coefficient CYY when plugging the injection orifice facing the +X axis. However, the model sensibly underpredicts the increase of the direct damping coefficient CXX from the reference to the plugged-orifice bearing configuration. Direct inertia coefficients (Fig. 6) of large magnitude are derived for the water lubricated bearing. At low speed operation, the test results for direct inertia MYY bracket predictions that show reduced magnitudes with increasing operating speeds. Note that the experimental inertia coefficients have a large uncertainty. Whirl frequency ratio (Fig. 7) The test results show a stability ratio ∼0.40 while the predictions show larger magnitudes, mainly due to over predicting the cross-coupled stiffness coefficients. At low speed, plugging one feed orifice leads to a noticeable decrease in the WFR, as shown by both experiments and predictions. Flow rate (Fig. 8) increases sensibly for the worn surface condition. The model predicts well the flow rate for all conditions, although differences of 5 percent to 14 percent are apparent at low and high speeds, respectively. Predicted direct stiffness coefficients (Fig. 9) correlate well with the test data and show a significant decrease for the worn condition, in particular at low speed operation. While agreement is fairly good for KXX (9 percent maximum difference), model predictions overestimate KYY by as much as 16 percent. Cross-coupled stiffness coefficients (Fig. 10) show little variation between the reference and worn surface configurations. This unexpected performance feature may be due to the local character of the simulated wear pattern affecting a small area of the bearing surface. The predictions for KXY, as for the case with one orifice plugged, do not correlate well with the experimental parameters. The discrepancy may be due to an over estimation of the inlet swirl fraction at the recess into land interface. Predicted direct damping coefficientsCXX (Fig. 11) show a slight decrease (8 percent) from the reference to the worn-land bearing configuration, whereas no firm variation trend is shown experimentally. The average difference between the predictions and test results is only 9 percent. Predictions for the inertia coefficients MXX,MYY show a nearly uniform 12 percent increase between the reference and worn surface condition. The experimental results are somewhat similar to the plugged orifice condition, and thus conform poorly to the present predictions. Figure 12 shows the whirl frequency ratio is not affected by the worn surface condition. The model, however, overpredicts the WFR indicating a more conservative value for stable rotordynamic performance.High speed, high pressure turbomachinery incorporates process fluid hybrid (hydrostatic/hydrodynamic) bearings to reduce number of parts, to eliminate expensive mineral lubricant storage and pumping hardware, and to satisfy ever stringent environmental constraints. Computational models and exhaustive experimental programs have steadily aided to advance the applications of hybrid bearings to modern machinery. Test results for water lubricated, off design (damaged) bearing configurations were released recently 11, but no correlation to predictions was available at that time. The damaged configurations include one bearing with a plugged-orifice denoting failure of the supply and filtering system, and a second bearing with a mechanically enlarged clearance representing the surface wear induced by frequent rotor rubs during machine shutdowns. Presently, a computational bulk-flow model forwards predictions for the two damaged bearings dynamic forced response characteristics. When compared to a reference (undamaged) bearing, the experimental results for the plugged-orifice bearing show increases in direct and cross-coupled stiffnesses and in direct damping coefficients, in particular at highly off-centered rotor conditions (large eccentricities). The bulk-flow model predictions for flow rate and direct stiffness and damping coefficients correlate well with the test data. However, significant discrepancies are evident for the cross-coupled stiffness coefficients and the whirl frequency ratio, which ultimately defines the limit of stable operation for the hybrid bearings. In sum, the experimental results and model predictions suggest that neither a single plugged orifice nor significant wear on the bearing lands will be likely causes for imminent failure of a well-designed hybrid bearing. The comparisons to the test data allow further validation of the flow model and extend its range of applicability to off-design conditions. The authors thank Professor Dara Childs for his encouragement to conduct the present comparisons and for releasing extensive information related to the experimental testing and results. C= reference radial clearance (m) Cdo= orifice discharge coefficient Cαγ= damping coefficients (Ns/m), α,γ=X,YD= journal diameter (m) do= orifice diameter (m) eX,eY= journal center displacements (m) fJ,B= Moody’s friction factor at rotor and stator =aM⌊1+104s˙rJ,B+bM/ReJ,BeM⌋H, HR= film thickness function, recess depth (m) Kαγ= stiffness coefficients (N/m), α,γ=X,YkJ,B= shear flow parameter at rotor and stator interfaces =fs˙ReJ,Bkx,ky= circumferential and axial shear flow parameters =kJ+kB/2L, l= bearing and recess axial lengths (m) Mαγ= inertia coefficients (kg), α,γ=X,YPa,Ps,PR= discharge, supply and recess pressures (Pa) Q= bearing mass flowrate (kg/s) RJ,B= Reynolds numbers relative to journal and bearing surfaces =ρ/μH[Ux−1/2DΩ2+Uy2]1/2,ρ/μH[Ux2+Uy2]1/2rJ,rB= mean surface relative roughness at journal and bearing Ts= feed temperature (K) t= time (s) Ux,Uy= circumferential and axial bulk-flow velocities (m/s) X, Y, Z= Cartesian coordinate system (m) x, y= circumferential and axial coordinates in plane of flow (m) ΔX,ΔY= journal center dynamic motions (m) μ = fluid viscosity (Pa⋅s) ξ = empirical film land entrance loss coefficient θ = angular coordinate =2x/D (rad) ΘR= recess arc length (rad) ρ = fluid density (kg/m3) Ω = rotor speed (rad/s) ω = excitation frequency (rad/s) Subscripts0 = zeroth-order fields α, β = first-order fields, X, Y directions