Engineering Tribology Episode 1 Part 6 pptx

100 ENGINEERING TRIBOLOGY TEAM LRN HYDRODYNAMIC 4 LUBRICATION 4.1 INTRODUCTION In the previous chapters the basic physical properties of lubricants, their composition and applications have been discussed. The fundamental question to be answered is: what causes a lubricant to lubricate? If some specific concepts of lubrication are formulated then the following questions become pertinent. What conditions must be fulfilled to fully separate two loaded surfaces in relative motion? How can we manipulate these conditions in order to minimize friction and wear? Is there only one or are there several mechanisms of lubrication? In what specific applications do they operate? What factors determine the classification of a load-carrying mechanism to a specific category? How do thrust and journal bearings operate? How can the design parameters for such bearings be estimated? How does cavitation or oil whirl affect bearing performance? Engineers are usually expected to know the answers to all of these questions. In this chapter the basic principles of hydrodynamic lubrication will be discussed. The mechanisms of hydrodynamic film generation and the effects of operating variables such as velocity, temperature, load, design parameters, etc., on the performance of such films are outlined. This will be explained using bearings commonly found in many engineering applications as examples. Secondary effects in hydrodynamic lubrication such as viscous heating, compressible and non-Newtonian lubricants, bearing vibration and deformation, will be described and their influence on bearing performance assessed. 4.2 REYNOLDS EQUATION The serious appreciation of hydrodynamics in lubrication started at the end of the 19th century when Beauchamp Tower, an engineer, noticed that the oil in a journal bearing always leaked out of a hole located beneath the load. The leakage of oil was a nuisance so the hole was plugged first with a cork, which still allowed oil to ooze out, and then with a hard wooden bung. The hole was originally placed to allow oil to be supplied into the bearing to provide ‘lubrication’. When the wooden bung was slowly forced out of the hole by the oil, Tower realized that the oil was pressurized by some as yet unknown mechanism. Tower then measured the oil pressure and found that it could separate the sliding surfaces by a hydraulic force [1]. At the time of Beauchamp Tower's discovery Osborne Reynolds and other theoreticians were working on a hydrodynamic theory of lubrication. By a most fortunate TEAM LRN 102 ENGINEERING TRIBOLOGY coincidence, Tower's detailed data was available to provide experimental confirmation of hydrodynamic lubrication almost at the exact time when Reynolds needed it. The result of this was a theory of hydrodynamic lubrication published in the Proceedings of the Royal Society by Reynolds in 1886 [2]. Reynolds provided the first analytical proof that a viscous liquid can physically separate two sliding surfaces by hydrodynamic pressure resulting in low friction and theoretically zero wear. At the beginning of the 20th century the theory of hydrodynamic lubrication was successfully applied to thrust bearings by Michell and Kingsbury and the pivoted pad bearing was developed as a result. The bearing was a major breakthrough in supporting the thrust of a ship propeller shaft and the load from a hydroelectric rotor. At the present level of technology, loads of several thousand tons are carried, at sliding speeds of 10 to 50 [m/s], in hydroelectric power stations. The operating surfaces of such bearings are fully separated by a lubricating film, so the friction coefficient is maintained at a very low level of about 0.005 and the failure of such bearings rarely occurs, usually only after faulty operation. Reynolds' theory explains the mechanism of lubrication through the generation of a viscous liquid film between the moving surfaces. The condition is that the surfaces must move, relatively to each other, with sufficient velocity to generate such a film. It was found by Reynolds and many later researchers that most of the lubricating effect of oil could be explained in terms of its relatively high viscosity. There are, however, some lubricating functions of an oil as opposed to other liquids which cannot be explained in terms of viscosity and these are described in more detail in Chapter 8 on ‘Boundary and Extreme Pressure Lubrication’. All hydrodynamic lubrication can be expressed mathematically in the form of an equation which was originally derived by Reynolds and is commonly known throughout the literature as the ‘Reynolds equation’. There are several ways of deriving this equation. Since it is a simplification of the Navier-Stokes momentum and continuity equation it can be derived from this basis. It is, however, more often derived by considering the equilibrium of an element of liquid subjected to viscous shear and applying the continuity of flow principle. There are two conditions for the occurrence of hydrodynamic lubrication: · two surfaces must move relatively to each other with sufficient velocity for a load- carrying lubricating film to be generated and, · surfaces must be inclined at some angle to each other, i.e. if the surfaces are parallel a pressure field will not form in the lubricating film to support the required load. There are two exceptions to this last rule: hydrodynamic pressure can be generated between parallel stepped surfaces or the surfaces can move towards each other (these are special cases and are discussed later). The principle of hydrodynamic pressure generation between moving non-parallel surfaces is schematically illustrated in Figure 4.1. It can be assumed that the bottom surface, sometimes called the ‘runner’, is covered with lubricant and moves with a certain velocity. The top surface is inclined at a certain angle to the bottom surface. As the bottom surface moves it drags the lubricant along it into the converging wedge. A pressure field is generated as otherwise there would be more lubricant entering the wedge than leaving it. Thus at the beginning of the wedge the increasing pressure restricts the entry flow and at the exit there is a decrease in pressure boosting the exit flow. The pressure gradient therefore causes the fluid velocity profile to bend inwards at the entrance to the wedge and bend outwards at the exit, as shown in Figure 4.1. The generated pressure separates the two surfaces and is also able to support a certain load. It is also possible for the wedge to be curved or wrapped around a shaft to form a journal bearing. If the wedge remains planar then a pad bearing is obtained. The entire process of hydrodynamic pressure generation can be described mathematically to enable accurate prediction of bearing characteristics. TEAM LRN HYDRODYNAMIC LUBRICATION 103 Pressure profile p max p z x h 0 h ¯ h 1 h Oil U 0 FIGURE 4.1 Principle of hydrodynamic pressure generation between non-parallel surfaces. Simplifying Assumptions In most engineering applications the controlling processes are too complicated to be easily described by exact mathematical equations. There are many interacting factors and variables in the real processes which make such a description extremely difficult, if not impossible. For example, with fluid mechanics in the early days of modelling, the terms of internal fluid friction were ignored. The mathematician John Newman observed sarcastically that these approximations have nothing to do with real fluids. It was like trying to study the flow of ‘dry-water’. The situation dramatically changed with the introduction of computers so that mechanical systems could be studied in a more detailed fashion. Similarly in hydrodynamics, several simplifying approximations have to be made before a mathematical description of the fundamental underlying mechanisms can be derived. All the simplifying assumptions necessary for the derivation of the Reynolds equation are summarized in Table 4.1 [3]. The Reynolds equation can now be conveniently derived by considering the equilibrium of an element (from which the expressions for fluid velocities can be obtained) and continuity of flow in a column. Equilibrium of an Element The equilibrium of an element of fluid is considered. This approach is frequently used in engineering to derive formulae in stress analysis, fluid mechanics, etc. Consider a small element of fluid from a hydrodynamic film shown in Figure 4.2. For simplicity, assume that the forces on the element are acting initially in the ‘x’ direction only. Since the element is in equilibrium, forces acting to the left must balance the forces acting to the right, so τ dxdy xx ∂τ ∂z (τ + dz)dxdy = x pdydz + ∂p ∂x (p + dx)dydz + (4.1) which after simplifying gives: ∂τ ∂z dxdydz = x ∂p ∂x dxdydz (4.2) TEAM LRN 104 ENGINEERING TRIBOLOGY TABLE 4.1 Summary of simplifying assumptions in hydrodynamics. Assumption Comments Always valid, since there are no extra outside fields of forces acting on the fluids with an exception of magnetohydrodynamic fluids and their applications. Body forces are neglected Always valid, since the thickness of hydrodynamic films is in the range of several micrometers. There might be some exceptions, however, with elastic films. Pressure is constant through the film Always valid, since the velocity of the oil layer adjacent to the boundary is the same as that of the boundary. No slip at the boundaries Usually valid with certain exceptions, e.g. polymeric oils.Lubricant behaves as a Newtonian fluid Usually valid, except large bearings, e.g. turbines.Flow is laminar Valid for low bearing speeds or high loads. Inertia effects are included in more exact analyses. Fluid inertia is neglected Usually valid for fluids when there is not much thermal expansion. Definitely not valid for gases. Fluid density is constant Crude assumption but necessary to simplify the calculations, although it is not true. Viscosity is not constant throughout the generated film. Viscosity is constant throughout the generated fluid film 8 7 6 5 4 3 2 1 z x ∂p ∂x y dy dx dz (p + dx)dydz τ dxdy x x ∂τ ∂z (τ+ dz)dxdy x pdydz FIGURE 4.2 Equilibrium of an element of fluid from a hydrodynamic film; p is the pressure, τ x is the shear stress acting in the ‘x’ direction. Assuming that dxdydz ≠ 0 (i.e. non zero volume), both sides of equation (4.2) can be divided by this value and then the equilibrium condition for forces acting in the ‘x’ direction is obtained, ∂τ ∂z = x ∂p ∂x (4.3) TEAM LRN HYDRODYNAMIC LUBRICATION 105 A similar exercise can be performed for the forces acting in the ‘y’ (out of the page) direction, yielding the second equilibrium condition, ∂τ ∂z = y ∂p ∂y (4.4) In the ‘z’ direction since the pressure is constant through the film (Assumption 2) the pressure gradient is equal to zero: = 0 ∂p ∂z (4.5) It should be noted that the shear stress in expression (4.3) is acting in the ‘x’ direction while in expression (4.4) it is acting in the ‘y’ direction, thus the values of the shear stress in these expressions are different. Remembering the formula for dynamic viscosity discussed in Chapter 2, the shear stress ‘τ’ can be expressed in terms of dynamic viscosity and shear rates: ∂u ∂z τ = η x u h = η (4.6) where: τ x is the shear stress acting in the ‘x’ direction [Pa]. Since ‘u’ is the velocity along the ‘x’ axis, the shear stress ‘τ’ is also acting along this direction. Along the ‘y’ (out of the page) direction, however, the velocity is different and consequently the shear stress is different: ∂v ∂z τ = η y v h = η (4.7) where: τ y is the shear stress acting in the ‘y’ direction [Pa]; v is the sliding velocity in the ‘y’ direction [m/s]. Substituting (4.6) into (4.3) and (4.7) into (4.4), the equilibrium conditions for the forces acting in the ‘x’ and ‘y’ directions are obtained: = ∂p ∂x ∂ ∂z ∂u ∂z ) η ( (4.8) = ∂p ∂y ∂ ∂z ∂v ∂z ) η ( (4.9) TEAM LRN 106 ENGINEERING TRIBOLOGY The above equations can now be integrated. Since the viscosity of the fluid is constant throughout the film (Assumption 8) and it is not a function of ‘z’ (i.e. η ≠ f(z)), the process of integration is simple. For example, separating the variables in (4.8), ∂z = ∂p ∂x ∂ ∂u ∂z ) η ( and integrating gives: z + C 1 = ∂p ∂x ∂u ∂z η Separating variables again, z + C 1 ∂p ∂x ∂z = η∂u ( ( and integrating again yields: + C 1 z + C 2 = ηu ∂p ∂x z 2 2 (4.10) Since there is no slip or velocity discontinuity between liquid and solid at the boundaries of the wedge (Assumption 3), the boundary conditions are: u = U 2 at z = 0 u = U 1 at z = h In the general case, there are two velocities corresponding to each of the surfaces ‘U 1 ’ and ‘U 2 ’. By substituting these boundary conditions into (4.10) the constants ‘C 1 ’ and ‘C 2 ’ are calculated: C 2 =ηU 2 η h C 1 = (U 1 − U 2 ) − ∂p ∂x h 2 Substituting these into (4.10) yields: + ∂p ∂x z 2 2 (U − U) 12 ηz h − ∂p ∂x hz 2 + ηU = ηu 2 Dividing and simplifying gives the expression for velocity in the ‘x’ direction: u = ∂p ∂x − zh 2η ( ( z 2 (U − U) 12 z h ++U 2 (4.11) In a similar manner a formula for velocity in the ‘y’ direction is obtained. TEAM LRN HYDRODYNAMIC LUBRICATION 107 v = ∂p ∂y − zh 2η ( ( z 2 (V − V) 12 z h ++V 2 (4.12) The three separate terms in any of the velocity equations (4.11) and (4.12) represent the velocity profiles across the fluid film and they are schematically shown in Figure 4.3. U 2 velocity U 2 U 1 = 0 ∂p ∂x z 2 − zh 2η Parabolic distribution due to pressure gradient ∂p ∂x () ++ (U 1 − U 2 ) z h Linear distribution due to the parallel velocity of surfaces ‘Couette velocity’ z x FIGURE 4.3 Velocity profiles at the entry of the hydrodynamic film. Continuity of Flow in a Column Consider a column of lubricant as shown in Figure 4.4. The lubricant flows into the column horizontally at rates of ‘q x ’ and ‘q y ’ and out of the column at rates of ( q x + ∂q x ∂ x dx ) and ( q y + ∂q y ∂ y dy ) per unit length and width respectively. In the vertical direction the lubricant flows into the column at the rate of ‘w 0 dxdy’ and out of the column at the rate of ‘w h dxdy’, where ‘w 0 ’ is the velocity at which the bottom of the column moves up and ‘w h ’ is the velocity at which the top of the column moves up. The principle of continuity of flow requires that the influx of a liquid must equal its efflux from a control volume under steady conditions. If the density of the lubricant is constant (Assumption 7) then the following relation applies: q x dy + q y dx + w 0 dxdy = ∂q x ∂x ( ( q x + dx dy + ∂q y ∂y ( ( q y + dy dx + w h dxdy (4.13) TEAM LRN 108 ENGINEERING TRIBOLOGY z x y dy dx dz ∂q y ∂y (q y + dy)dx q x dy ∂q x ∂x (q x + dx)dy w 0 dxdy q y dx w h dxdy h FIGURE 4.4 Continuity of flow in a column. simplifying: ∂q y ∂y ∂q x ∂x dxdy + dxdy + (w h − w 0 )dxdy = 0 (4.14) Since ‘dxdy ≠ 0’ equation (4.14) can be rewritten as: ∂q x ∂x + ∂q y ∂y + (w h − w 0 ) = 0 (4.15) which is the equation of continuity of flow in a column. Flow rates per unit length, ‘q x ’ and ‘q y ’, can be found from integrating the lubricant velocity profile over the film thickness, i.e.: q x = ⌠ ⌡ 0 h udz and (4.16) q y = ⌠ ⌡ 0 h vdz (4.17) substituting for ‘u’ from equation (4.11) yields: ∂p 2η∂x z 2 h 2 ( ( z 3 + (U 1 − U 2 ) z 2 2h + U 2 z q x = 3 − 0 h which after simplifying gives the flow rate in the ‘x’ direction, q x = − h 3 12η ∂p ∂x + (U 1 + U 2 ) h 2 (4.18) Similarly the flow rate in the ‘y’ direction is found by substituting for ‘v’ from equation (4.12): TEAM LRN HYDRODYNAMIC LUBRICATION 109 q y = − h 3 12η ∂p ∂y + (V 1 + V 2 ) h 2 (4.19) Substituting now for flow rates into the continuity of flow equation (4.15): ∂ ∂x + (U 1 + U 2 ) h 2 h 3 12η ∂p ∂x − [] + ∂ ∂y + (V 1 + V 2 ) h 2 h 3 12η ∂p ∂y − [] + (w h − w 0 ) = 0 (4.20) Defining U = U 1 + U 2 and V = V 1 + V 2 and assuming that there is no local variation in surface velocity in the ‘x’ and ‘y’ directions (i.e. U ≠ f(x) and V ≠ f(y)) gives: ∂ ∂x h 3 12η ∂p ∂x − () + U 2 dh dx ∂ ∂y h 3 12η ∂p ∂y − () + V 2 dh dy + (w h − w 0 ) = 0 Further rearranging and simplifying yields the full Reynolds equation in three dimensions. ∂ ∂x h 3 η ∂p ∂x () + dh dx + 12(w h − w 0 ) ∂ ∂y h 3 η ∂p ∂y () = 6 ( U + dh dy V ) (4.21) Simplifications to the Reynolds Equation It can be seen that the Reynolds equation in its full form is far too complex for practical engineering applications and some simplifications are required before it can conveniently be used. The following simplifications are commonly adopted in most studies: · Unidirectional Velocity Approximation It is always possible to choose axes in such a way that one of the velocities is equal to zero, i.e. V = 0. There are very few engineering systems, in which, for example, a journal bearing slides along a rotating shaft. V U Assuming that V = 0 equation (4.21) can be rewritten in a more simplified form: ∂ ∂x h 3 η ∂p ∂x () + dh dx + 12(w h − w 0 ) ∂ ∂y h 3 η ∂p ∂y () = 6U (4.22) · Steady Film Thickness Approximation It is also possible to assume that there is no vertical flow across the film, i.e. w h - w 0 = 0. This assumption requires that the distance between the two surfaces remains constant during the TEAM LRN [...]... h1 (4.42) Substituting into (4. 41) the constants ‘h’ and ‘C’ are: h = 2h 0h 1 h1 + h0 C= (4.43) 1 h1 + h0 The maximum film thickness ‘h1’, can also be expressed in terms of the convergence ratio ‘K’: K = h1 - h0 h0 Thus: h1 = h0(K + 1) (4.44) Substituting into (4.43) the constants ‘h’ and ‘C’ in terms of ‘K’ are: (K + 1) ¯ = 2 h0 (K + 2) h (4.45) 1 C= h0 (K + 2) Substituting into (4. 41) gives: Kh0 1. .. and substituting for ‘p’, equation (4. 46) , yields: 6 Uη B W = L Kh0 ⌠ ⌡ 0 B ( − ) 1 1 h (K + 1) + 0 + dx h h2 (K + 2) h0 (K + 2) (4.49) Again there are two variables in (4.49), ‘x’ and ‘h’, and one has to be replaced by the other before the integration can be performed Substituting (4.39) for ‘dx’, TEAM LRN 12 0 ENGINEERING TRIBOLOGY ( ) 6 Uη B B ⌠ h1 1 1 W h (K + 1) = + dh − + 0 L Kh0 K h0 ⌡ h2 (K +... h0 (K + 2) h (4.45) 1 C= h0 (K + 2) Substituting into (4. 41) gives: Kh0 1 1 h (K + 1) p=− + 0 + 6Uη B h h2 (K + 2) h0 (K + 2) or: TEAM LRN HYDRODYNAMIC LUBRICATION p= ( 6 Uη B 1 1 h (K + 1) − + 0 + K h0 h h2 (K + 2) h0 (K + 2) ) 11 9 (4. 46) Note that the velocity ‘U’, in the convention assumed, is negative, as shown in Figure 4 .1 It is useful to find the pressure distribution in the bearing expressed... (4.24) may be neglected and the equation becomes: ( ) ∂ 3 ∂p dh h = 6Uη ∂y ∂y dx (4.28) Also, since h ≠ f(y) then (4.28) can be further simplified, d2p 6 Uη dh = dy2 h3 dx (4.29) Integrating once, dp 6 Uη dh = y + C1 dy h3 dx (4.30) TEAM LRN HYDRODYNAMIC LUBRICATION 11 3 and again gives: p= 6 Uη dh y2 + C1 y + C2 h3 dx 2 (4. 31) From Figure 4 .6 the boundary conditions are: p = 0 at y = ± L/2 i.e at the edges... ] 3 K − 2 (K + 2) ln(K + 1) 6 K − 3 (K + 2) ln(K + 1) (4 .64 ) where: µ* = B µ h0 (4 .65 ) The optimum bearing geometry which gives a minimum value of coefficient of friction can now be calculated Differentiating (4 .64 ) with respect to ‘K’ and equating to zero gives: K = 1. 55 which is the optimum convergence ratio for a minimum coefficient of friction TEAM LRN 12 4 ENGINEERING TRIBOLOGY As stated previously,... dh Kh0 (4.39) Substituting into (4.27) yields: TEAM LRN 11 8 ENGINEERING TRIBOLOGY dp h−¯ h = 6 Uη 3 B h dh Kh0 and after simplifying and separating variables: Kh0 h−¯ h dp = dh 6UηB h3 (4.40) which is the differential formula for pressure distribution in this bearing Equation (4.40) can be integrated to give: Kh0 1 ¯ +C h p=− + 6UηB h 2h2 (4. 41) The boundary conditions, taken from the bearing's inlet... 10 8 0 .1 6 4 2 0 0 1 2 3 4 5 6 7 8 0 K FIGURE 4 .10 Variation of load capacity and coefficient of friction with a convergence ratio in a linear pad bearing It is quite easy to see what coefficient of friction can be anticipated in a linear pad bearing For example, for a 0 .1 [m] bearing width, a film thickness of 0 .1 [mm] is typical The minimum value of ‘µ*’ is approximately 5 and the ratio B/h0 is 10 00... which is very convenient for quick engineering analysis Since ∂p/∂y = 0, the second term of the Reynolds equation (4.24) is also zero and equation (4.24) simplifies to: ( ) ∂ ∂p dh h3 = 6 Uη ∂x ∂x dx (4.25) which can easily be integrated, i.e.: h3 dp = 6U η h + C dx (4. 26) TEAM LRN HYDRODYNAMIC LUBRICATION 11 1 pmax ; ; ;;;;;; ;;;;;; ;;;;;;;; ;;;;; ; ;;;;;;;; ; ; y z h1 h0 x FIGURE 4.5 U Pressure distribution... load capacity occurs at K = 1. 2 but the minimum coefficient of friction is obtained when K = 1. 55 In bearing design there must consequently be a compromise and ‘K’ is chosen between these two values, i.e 1. 2 < K < 1. 55 to give the optimum performance This is evident when plotting ‘µ*’ and ‘6W*’ (known as the load coefficient) against ‘K’ as shown in Figure 4 .10 − 6 W* 0.2 12 Dimensionless load Normalized... ⌡ ⌡ h 0 0 (4.55) The first part of the above equation must be integrated by parts According to the theorems of integration, the general mathematical formula to integrate by parts is: ⌠ adb = a b −⌠ bda ⌡ ⌡ So if, a= h 2 and db = dp dx dx TEAM LRN 12 2 ENGINEERING TRIBOLOGY then: da = 1 dh 2 b =⌠ dp dx = p ⌡ dx and substituting: −⌠ B ⌡ 0 h dp dx = − 2 dx ( h p 2 B 0 −⌠ ⌡ 0 B 1 pdh 2 ) B Since p = 0 at . 1) h ¯ (K + 2) C = 1 h 0 (K + 2) (4.45) Substituting into (4. 41) gives: p = − Kh 0 1 h6UηB h 0 h 2 + (K + 1) (K + 2) + 1 h 0 (K + 2) or: TEAM LRN HYDRODYNAMIC LUBRICATION 11 9 p = Kh 0 1 h h 0 h 2 + (K. 2h 0 h 1 h 1 + h 0 C = 1 h 1 + h 0 (4.43) The maximum film thickness ‘h 1 ’, can also be expressed in terms of the convergence ratio ‘K’: K = h 1 - h 0 h 0 Thus: h 1 = h 0 (K + 1) (4.44) Substituting. − h 3 12 η ∂p ∂y + (V 1 + V 2 ) h 2 (4 .19 ) Substituting now for flow rates into the continuity of flow equation (4 .15 ): ∂ ∂x + (U 1 + U 2 ) h 2 h 3 12 η ∂p ∂x − [] + ∂ ∂y + (V 1 + V 2 ) h 2 h 3 12 η ∂p ∂y − [] +