Engineering Tribology Episode 1 Part 9 pps

HYDRODYNAMIC LUBRICATION 175 · Conducted/Convected Heat Ratio From the above equations the ratio of conducted to convected heat can be calculated to determine which of the two mechanisms of heat removal from the bearing is the more significant. Combining (4.124) and (4.125) the ratio is given by: = K ρσ 2B Uh 2 H cond H conv () (4.126) where: K/ρσ = χ is the thermal diffusivity of the fluid [m 2 /s]. Typical values of density, specific heat, thermal conductivity and thermal diffusivity are shown in Table 2.7, Chapter 2. EXAMPLE Find the ratio of conducted to convected heat in a journal bearing of diameter D = 0.1 [m] and length L = 0.157 [m] which operates at 3000 [rpm]. The hydrodynamic film thickness is h = 0.0001 [m]. The bearing is lubricated by mineral oil of thermal diffusivity χ = 0.084 × 10 -6 [m 2 /s]. Since D = 0.1 [m], B = πD/2 = 0.157 [m] and = 0.084 × 10 −6 2 × 0.157 3000(π × 0.1/60)0.0001 2 H cond H conv = 0.168 It is clear from the above example that for journal bearings the amount of convected heat is quite significant. When the film thickness of lubricant is much smaller or a highly conductive fluid such as mercury is used as a lubricant then the ratio of conducted to convected heat will be different. Hydrodynamic lubrication at much thinner film thicknesses is discussed in the chapter on ‘Elastohydrodynamic Lubrication’. The significance of the ratio of conducted to convected heat for hydrodynamic bearings is that convection must be included in the equations of heat transfer in a hydrodynamic film. This condition renders the numerical analysis of the heat transfer equations much more complicated than would otherwise be the case, as is demonstrated in the next chapter. Another ramification of the above result is that conductive heat transfer is still significant although it is often the smaller component of overall heat transfer. Most hydrodynamic bearings operate under a condition between adiabatic and isothermal heat transfer. Adiabatic heat transfer can be modelled by a perfectly insulating shaft and bush. In this case, all the heat is transferred by the lubricant as convection. Isothermal heat transfer represents a bearing made of perfectly conductive material which maximizes heat transfer by conduction in the lubricant. The adiabatic model gives the lowest load capacity since the highest possible lubricant temperatures are predicted. The isothermal model conversely predicts the maximum attainable load capacity. Combined solution of the two models provides valid upper and lower limits of load capacity. If a more accurate estimate of load capacity is required then it is necessary to estimate heat transfer coefficients to the surrounding bearing structure. This is a very complex task and is still under investigation [35]. This topic is discussed further in the chapter on ‘Computational Hydrodynamics’. As is probably very clear here, exact analysis of thermal effects in bearings is a demanding task and most designers of bearings have used the ‘effective viscosity’ methods even though they are at least partly based on supposition. TEAM LRN 176 ENGINEERING TRIBOLOGY Isoviscous Thermal Analysis of Bearings A simple method of estimating the loss in load capacity due to frictional heat dissipation is to adopt an isoviscous model. It is assumed that the lubricant viscosity is lowered by frictional heating to a uniform value over the whole film. Viscosity may vary with time during operation of the bearing but its value remains uniform throughout the lubricating film. An ‘effective temperature’ is introduced with a corresponding ‘effective viscosity’ which is used to calculate load capacity. Two methods are available to find the effective temperature and viscosity, an ‘iterative method’ which requires computation and a ‘constant flow method’ which can be executed on a pocket calculator. These are discussed below. · Iterative Method The iterative method is effective and accurate in finding the value of effective viscosity. The standard procedure is conducted in the following stages: · An effective bearing temperature is initially assumed for the purposes of iteration. The assumed value must lie between the inlet temperature ‘T inlet ’ and the maximum temperature ‘T max ’ of the bearing material, i.e.: T inlet < T eff,s1 < T max where: T eff,s1 is the effective temperature at the start ‘s’ of iteration, first cycle [°C]. The maximum temperature is usually set by the manufacturer and for most bearing materials ‘T max ’ is about 120 [°C]. For computing purposes, the initial value for ‘T eff,s1 ’ is usually assumed as equal to the inlet temperature. · the corresponding effective viscosity ‘η eff,s1 ’ is found from ‘T eff,s1 ’ using the ASTM viscosity chart or applying the appropriate viscosity-temperature law, e.g. Vogel equation. · for a given film geometry and effective viscosity, bearing parameters such as friction force ‘F’ and lubricant flow rate ‘Q’ can now be calculated. · the values of ‘F’ and ‘Q’ are used to calculate the new effective temperature. This will be different from the previous value unless they happen to coincide. The effective temperature is calculated from [4]: T eff = T inlet + k∆T (4.127) where: k is an empirical constant with a value of 0.8 giving good agreement between theory and experiment [4]; ∆T is the frictionally induced temperature rise dependent on ‘F’ and ‘Q’ [°C]. The frictionally induced temperature rise is found from the following argument. The heat generated in the bearing is: H = FU TEAM LRN HYDRODYNAMIC LUBRICATION 177 At equilibrium, the heat generated by friction balances the heat removed by convection assuming an adiabatic bearing, thus: FU = Qρσ ∆T 2 Rearranging this gives: ∆T = 2FU Qρσ Substituting into (4.127) yields: T eff = T inlet + 1.6FU Qρσ (4.128) A new effective temperature is then calculated from (4.128). · for the new effective temperature ‘T eff,n1 ’ (‘n’ denotes new) a corresponding effective viscosity is then found from, for example, the ASTM chart or Vogel equation. · if the difference between the new effective viscosity and the former effective viscosity is less than a prescribed limit then the iteration is terminated. If the difference in viscosities is still too large the new viscosity value ‘η eff,s2 ’ is assumed and the procedure is continued until the required convergence is achieved. A relaxation factor is usually incorporated at this stage (see program listed in Appendix) to prevent unstable iteration. The iteration procedure is summarized in a flow-chart shown in Figure 4.52 and a computer program ‘SIMPLE’ written in Matlab to perform this analysis for narrow journal bearings is listed in the Appendix. End Find new effective viscosity η eff,n1 No Has viscosity converged? η eff,s1 = η eff,n1 Yes Calculate new effective temperature T eff,n1 Calculate friction force F and oil flow Q Define bearing geometry h = f(x) Find effective starting viscosity η eff,s1 Start Assume effective starting temperature T eff,s1 Assume new effective starting viscosity η eff,sn (n = 1, 2, 3 ) FIGURE 4.52 Flow chart for the iterative method in isoviscous analysis. TEAM LRN 178 ENGINEERING TRIBOLOGY · Constant Flow Method The constant flow method is simpler than the formal iterative method and does not require a computer. In journal bearings, lubricant flow remains approximately constant between eccentricity ratios 0.6 < ε < 0.95 as can be seen in Figure 4.36. It was found experimentally that in this range the lubricant flow can be approximated by the formula [3,4]: Q = 0.3 L D () 2 − UcL The friction is approximated by the ‘Petroff friction’ (eq. 4.117): F = 2πηULR c Substituting into (4.128) yields: T eff = T inlet + 10.67 πηUR c 2 ρσ(2 − L/D) (4.129) The solution can now be obtained by applying only one equation which is easily programmed into a pocket calculator. As with the iterative method it is necessary to perform some iteration since an initial value of effective temperature ‘T eff,s1 ’ has to be assumed. The corresponding viscosity ‘η eff,s1 ’ is found from the ASTM chart or Vogel equation as described previously. It is important to choose a sensible value of guessed initial effective temperature, i.e. between the inlet temperature and bearing material limit. This value is then substituted into (4.129) which in turn produces a new value of the effective temperature ‘T eff,n1 ’. A new corresponding viscosity ‘η eff,n1 ’ is then found and (4.129) re-applied. This procedure is repeated until satisfactory convergence is obtained. Non-Isoviscous Thermal Analysis of Bearings With Locally Varying Viscosity The assumption that lubricant viscosity is uniform across the film is in fact erroneous and inevitably causes inaccuracy. This error can be quite large especially when the lubricant viscosity has been severely reduced by frictional heating. Lubricant viscosity varies with position in the film, both in the plane of sliding and normal to the direction of sliding. An example of computed film temperatures inside a journal bearing operating at an eccentricity ratio of 0.8 and L/D = 1 is shown in Figure 4.53. The figure shows a temperature field through a radial section parallel to the load line and an axial section through the midplane of the bearing. The temperature distribution was calculated for a bearing speed of 2,000 [rpm], lubricant inlet temperature of 33°C and ambient temperature of 23°C [36]. It can be seen from Figure 4.53 that the hottest part of the lubricant film is at the centre of the bearing close to the minimum film thickness. The temperature at this location is 50°C which is 27°C higher than ambient temperature and causes a large viscosity loss in most lubricating oils. On the other hand, the shaft tends to a uniform temperature with angular position because it is rotating at a high speed. Bearing temperatures, even at the coolest point, are higher than the lubricant inlet temperature. Frictional heat accumulates in the bearing and there is a large temperature rise from the initial level to ensure sufficient dissipation of heat by convection or conduction through the external bearing structure. The uniformity of shaft temperature ensures a temperature difference across the lubricant film since the temperature TEAM LRN HYDRODYNAMIC LUBRICATION 179 of the bush varies with angular position. Both the temperatures of the shaft and bush vary in the axial direction. The temperature characteristic of a journal bearing is clearly non-uniform and all ‘effective viscosity’ methods are, at best, approximations to a complex problem. The solution to the problem of calculating the pressure field and load capacity with variable viscosity involves the simultaneous solution of a variable viscosity form of the Reynolds equation and a heat transfer equation for the lubricant film. This is clearly beyond the scope of analytical solution and numerical methods are almost exclusively employed. The solution method which is generally referred to as ‘thermohydrodynamics’ is discussed in the next chapter. Thermal effects render invalid many of the predictions of classical ‘isoviscous hydrodynamics’, e.g. that the load capacity is proportional to surface speed, and constitute the prime reason why the viscosity index of an oil is such an important property for the maintenance of viscosity at high localized operating temperatures, as was discussed in Chapter 2. y z 41° 42° 43° 45° 46° 48° 47° 49°50° 45° 44° 42° 40° 37° 35° 30° 27° Shaft Oil film Bush 36.5° 37° 41° 43° 45° 48° 49° 50° 39° 41° 43° 37.5° Oil supply groove Shaft Oil film Bush Attitude angle β W 45° ω All temperatures are in °C 0L/2 Bearing midline FIGURE 4.53 Example of computed temperature distribution in a hydrodynamic bearing (adapted from [36]). Multiple Regression in Bearing Analysis The multiple regression method is very useful in finding the correlation between variables, and also in expressing one variable, selected as a dependent variable, in terms of all the other variables which are independent variables. Any variable present in a particular process can be selected as a dependent variable and expressed in terms of the other variables. Some form of approximating function is assumed and polynomials or exponential functions are the most frequently used. Correlation coefficients between the variables and coefficients of TEAM LRN 180 ENGINEERING TRIBOLOGY approximation are usually computed. The technique is used for analysing experimental data and has also been applied to bearing analysis [39]. Theoretical data from several hundred bearings was analysed by multiple regression resulting in a set of equations which directly provide information about the design and performance parameters of journal bearings. The equations are given in an exponential form, i.e.: z = Cv 1 a v 2 b v 3 c v n m where: z is the dependent variable; C is the calculated regression constant; v 1 ,v 2 , ,v n are the independent variables, n =1,2,3, ; a,b, ,m are the calculated exponents. The dependent and independent variables together with calculated constants and exponents are shown in Table 4.5 [39]. For example, if the load capacity is required to be calculated for a specific journal bearing, then the following equation from Table 4.5, row 1, can be used: W = 2.7861 × 10 1 υ 37.8°C −1.1 υ 93.3°C 2.46 L 2.515 D 0.563 N 0.528 c −1.09 T S −0.383 () ε 1 − ε 2 1.385 or row 7: W = 1.7575 × 10 4 υ 37.8°C −0.793 υ 93.3°C 2.033 L 2.596 D 1.042 N 0.884 c −1.51 T mean −2.02 () ε 1 − ε 2 1.108 TABLE 4.5 Multiple regression relationships between journal bearing design and performance parameters [39]. (Note that the temperature is in degrees Fahrenheit [°F], i.e. T°F = 1.8 × T°C + 32). Nº Dependent parameter Curve constant υ 37.8°C [cS] υ 93.3°C [cS] L [m] D [m] N [rps] c [m] T s [°F] T mean [°F] 1 + lnW* W [N] ε 1 − ε 1 W [N] 2.7861× 10 1 -1.100 2.460 2.512 0.563 0.528 -1.090 -0.383 - - - 1.385 - 2 H [W] 3.9307× 10 3 -0.706 1.577 0.477 2.240 1.287 0.249 -0.204 - 1 .324 - - 3 ε 1.2666× 10 -2 0.536 1.120 -1.050 -0.578 -0.217 0.476 0.214 - - - 4 Q s [kg/s] 1.4791× 10 3 0.524 -1.070 0.212 1.381 0.821 1.457 0.276 - - 5 T max [°F] 3.8608 0.137 -0.063 0.024 0.387 0.272 -0.311 0.081 - - - 6 T max −T s [°F] 2.0528 × 10 -2 -0.783 1.730 -0.367 0.881 0.496 -0.690 -0.348 - - - 7 W [N] 1.7575×10 4 -0.793 2.033 2.596 1.042 0.884 -1.510 - - 1.108 - 8 H [W] 2.7915× 10 3 -0.579 1.530 0.873 2.500 1.642 -0.225 - - 9 ε 1.0516× 10 2 0.399 -1.040 1.372 -0.539 -0.458 0.765 - - 10 T max [°F] 2.1918× 10 -1 0.097 -0.055 -0.064 -0.314 0.170 -0.176 - - 11 1.7580× 10 -1 0.005 0.134 0.010 -0.302 0.190 -0.244 - - - 0.422 - 1.699 - - -0.011 - 0.162 - - -2.020 -1.470 - 0.962 - 0.426 - 0.396 - 0.659 - 0.535 - 0.194 - T max [°F] - -0.142 1 − ε 2 × 10 0 TEAM LRN HYDRODYNAMIC LUBRICATION 181 One of the equations is expressed in terms of lubricant supply temperature and the other in terms of mean lubricant temperature. Both, however, should give a similar result. The results predicted by these equations are acceptable and can be used in quick engineering analysis. Bearing Inlet Temperature and Thermal Interaction Between Pads of a Michell Bearing Frictional heat not only affects the load capacity of a bearing by directly influencing the viscosity but also leads to thermal interaction between adjacent bearings. For example, Michell pads are used in a combination of several pads distributed around a circle to form a larger thrust bearing. According to isoviscous theory a minimum of space should be left between the pads to maximize load bearing area. However, experimental measurements have revealed an optimum pad coverage fraction for maximum load capacity [37]. A certain amount of space is required between the pads for the hot lubricant discharged from one pad to be replaced by cool lubricant before entrainment in the following pad. In practice, the replacement of lubricant is never perfect and a phenomenon known as ‘hot oil carry over’ is almost inevitable. This phenomenon is illustrated schematically in Figure 4.54. It was found from boundary layer theory that the lubricant inlet temperature can be calculated from [38]: T inlet = T s (1 − m) / (1 − 0.5m) + 0.5T outlet m/(1 − 0.5m) (4.130) where: T inlet is the lubricant inlet temperature [°C]; T s is the lubricant supply temperature [°C]; T outlet is the lubricant outlet temperature [°C]; m is the hot oil carry over coefficient. Cool oil Pad Pad Hot oil Sliding T out T in FIGURE 4.54 ‘Hot oil carry over’ in a multiple pad bearing. The hot oil carry over coefficient is a function of sliding speed and space between adjacent pads. For a small gap width of 5 [mm] between pads, ‘m’ has a value of 0.8 at 20 [m/s] and 0.7 at 40 [m/s]. For a large gap width of 50 [mm] between pads, ‘m’ has a value of 0.55 at 20 [m/s] and 0.5 at 40 [m/s]. The minimum value of hot oil carry over coefficient occurs at approximately 40 [m/s] with a sharp rise in ‘m’ beyond this speed [38]. The effect of ‘hot oil carry over’ in a multipad thrust bearing was found to be sufficiently strong to ensure that individual pad temperatures were reduced when the number of pads was reduced from 8 to 3 [37]. This reduction in temperature occurred in spite of the greater TEAM LRN 182 ENGINEERING TRIBOLOGY concentration of frictional power dissipation per pad at constant load and speed. Even when the ratio of combined pad area to area swept by the pads was lowered to less than 35% the bearing still functioned efficiently. It is therefore unnecessary to fit a large number of closely spaced pads in a high-speed thrust bearing because this merely allows hot lubricant to recirculate almost indefinitely. The number of pads can be reduced for the same load capacity thus achieving considerable economies in the manufacture of the bearing. This is illustrated in Figure 4.55. It can be seen that bearing temperature is reduced especially at high loads when four pads are used instead of eight. The reduction in bearing temperature coincides with improved load capacity. Removal of pads can raise load capacity under certain conditions, as shown in Figure 4.55, and this is the result of a diminished loss of lubricant operating viscosity. 50 60 70 80 90 01020 a) 0 12 0 b) Maximum pad temperature T [°C] Power loss H [kW] Thrust load W [kN] Thrust load W [kN] 2 4 6 8 10 10 20 30 40 6050 3 000 [rpm] 5 000 [rpm] FIGURE 4.55 Effect of pad number on the performance of a thrust bearing, a) bearing temperature, b) power loss (adapted from [37] and [64]). 4.7 LIMITS OF HYDRODYNAMIC LUBRICATION As has been implied throughout this chapter, hydrodynamic lubrication is only effective when an appreciable sliding velocity exists. A sliding velocity of 1 [m/s] is typical of many bearings. As the sliding velocity is reduced the film thickness also declines to maintain the pressure field. This process is very effective as pressure magnitudes are proportional to the square of the reciprocal of film thickness. Eventually though the film thickness will have diminished to such a level that the small high points or asperities on each surface will come into contact. Contact between asperities causes wear and elevated friction. This condition, where the hydrodynamic film still supports most of the load but cannot prevent some contact between the opposing surfaces, is known as ‘partial hydrodynamic lubrication’. When TEAM LRN HYDRODYNAMIC LUBRICATION 183 the sliding speed is reduced still further the hydrodynamic lubrication fails completely and solid contact occurs. A lubricant may still, however, influence the coefficient of friction and wear rate to some degree, as is discussed in subsequent chapters. Original research into the limits of hydrodynamic lubrication was performed early in the 20th century by Stribeck [40] and Gumbel [41]. The limits of hydrodynamic lubrication are summarized in a graph shown in Figure 4.56. When the friction measurements from a journal bearing were plotted on a graph against a controlling parameter defined as ‘ηU/W’ it was found that, for all but very small sliding speeds, friction ‘µ’ was proportional to the above parameter which is known as the ‘Stribeck number’. When a critically low value of this parameter was reached, the friction rose from values of about 0.01 to much higher levels of 0.1 or more. The rapid change in the coefficient of friction represents the termination of hydrodynamic lubrication. Later work revealed that hydrodynamic lubrication persists until the largest asperities are separated by only a few nanometres of fluid. It was found that a minimum film thickness of more than twice the combined roughness of the opposing surfaces ensures full hydrodynamic lubrication of perfectly flat surfaces [3]. With the level of surface roughness attainable today on machined surfaces, a minimum film thickness of a few micrometres could thus be acceptable. In fact for small bearings, e.g. a journal bearing of 80 [mm] diameter, it is possible to use twice the combined roughness as a minimum limit for film thickness. On the other hand for large hydrodynamic bearings, larger clearances are usually selected because of the great difficulty in ensuring that such a small minimum film thickness is maintained over the entire bearing surface. Even if the bearing surfaces are machined accurately, elastic or thermal deflection would almost certainly cause contact between the bearing surfaces. If contact between sliding surfaces occurs then, particularly at high speeds of e.g. 10 [m/s], the dramatic increase in frictional power dissipation can cause overheating of the lubricant and possibly seizure of the bearing. Most hydrodynamic bearings, particularly the larger bearings, are designed to operate at film thicknesses well above the estimated transition point between fully hydrodynamic lubrication and wearing contact because: · the transition loads and speeds are difficult to specify accurately, · bearing failure is almost inevitable if hydrodynamic lubrication is allowed to fail even momentarily. 0 0.1 0.2 0.3 Friction coefficient dependent on non-hydrodynamic characteristics Increasing coefficient of friction caused by partial contact between shaft and bush Rise in friction due to high eccentricity at limit of hydrodynamic lubrication Friction coefficient determined by hydrodynamic theory ηU W µ Zero friction coefficient at zero sliding speed according to Petroff theory FIGURE 4.56 Schematic diagram of changes in friction coefficient at the limits of hydrodynamic lubrication. TEAM LRN 184 ENGINEERING TRIBOLOGY 4.8 HYDRODYNAMIC LUBRICATION WITH NON-NEWTONIAN FLUIDS Most fluids in use as lubricants either have a rheology that cannot be described as Newtonian or are modified by additives to cause deviations from Newtonian behaviour. All fluids have a non-zero density and therefore the hydrodynamic equations should ideally include the effect of inertia and at high bearing speeds, turbulent flow can also occur. Some of these flow effects are deleterious to bearing performance but others can be beneficial. The Reynolds equation presented so far does not include the characteristics of complex fluids and the refinement of hydrodynamic lubrication theory to model complex fluids is a subject of many current research projects. Some of the problems associated with complex fluids in hydrodynamics are outlined in this section. Turbulence and Hydrodynamic Lubrication In most bearings operating at moderate speeds, laminar flow of lubricant prevails but at high speed, the lubricant flow becomes turbulent and this affects the load capacity and particularly the friction coefficient of the bearing. Turbulent flow in a hydrodynamic bearing can be modelled by introducing ‘turbulence coefficients’ into the Reynolds equation. An example of a modified Reynolds equation used in the analysis of Michell pad bearings is in the form [24]: + ∂ ∂x ∂p ∂x () ∂h ∂x ∂ ∂y ∂p ∂y () = G x h 3 η 1 2 R 2 L 2 G y h 3 η (4.131) where: G x and G y are the turbulence coefficients defined as: G x = 1 12(1 + 0.00116Re h 0.916 ) G y = 1 12(1 + 0.00120Re h 0.854 ) Re h is the local Reynolds number defined as: Re h = hUρ / η R is the radius of a bearing [m]; h is the film thickness [m]; U is the surface velocity [m/s]; L is the length of the bearing [m]; ρ is the oil density [kg/m 3 ]; η is the oil viscosity [Pas]. The solution to this equation is usually obtained by computation since the ‘turbulence coefficients’ are functions of film thickness. At the high speeds where turbulence occurs, heating effects are quite significant and must be incorporated in the solution. It was found that the onset of turbulence in the bearing results in a slight increase in load capacity and marginal effect on stiffness coefficients (i.e. the variation in hydrodynamic load with film thickness change), which can slightly alter the limiting speed before bearing vibration occurs (i.e. vibration stability threshold). 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Bearings, ASLE Transactions, Vol 23, 19 80, pp 4 31- 4 41 29 J.T Burwell, The Calculated Performance of Dynamically Loaded Sleeve Bearings, Journal Appl Mech., Vol 69, 19 47, pp 2 31- 245 30 A.J Smalley and H McCallion, The Effect of Journal Misalignment on the Performance of a Journal Bearing under Steady Running Conditions, Proc Inst Mech Engrs., Vol 18 1, Pt 3B, 19 66 - 19 67, pp 45-54 31 J.R Stokley and R.R Donaldson, . -0.5 39 -0.458 0.765 - - 10 T max [°F] 2 . 19 18× 10 -1 0. 097 -0.055 -0.064 -0. 314 0 .17 0 -0 .17 6 - - 11 1. 7580× 10 -1 0.005 0 .13 4 0. 010 -0.302 0 . 19 0 -0.244 - - - 0.422 - 1. 699 - - -0. 011 - 0 .16 2. 0. 496 -0. 690 -0.348 - - - 7 W [N] 1. 7575 10 4 -0. 793 2.033 2. 596 1. 042 0.884 -1. 510 - - 1. 108 - 8 H [W] 2.7 91 5 × 10 3 -0.5 79 1. 530 0.873 2.500 1. 642 -0.225 - - 9 ε 1. 0 516 × 10 2 0. 399 -1. 040 1. 372. 2.460 2. 512 0.563 0.528 -1. 090 -0.383 - - - 1. 385 - 2 H [W] 3 .93 07× 10 3 -0.706 1. 577 0.477 2.240 1. 287 0.2 49 -0.204 - 1 .324 - - 3 ε 1. 2666× 10 -2 0.536 1. 120 -1. 050 -0.578 -0. 217 0.476 0. 214 -