Introduction to Tribology Viscosity, Lubrication and Friction Theories
: Sec: 1
Introduction The study of friction, wear, and lubrication has long been of enormous practical importance, since the functioning of many mechanical, electromechanical and biological systems depends on the appropriate friction and wear values. In recent decades, this field, termed "tribology," has received increasing attention as it has become evident that the wastage of resources resulting from high friction and wear is greater than 6% of the Gross National Product. The potential savings offered by improved tribological knowledge, too, are great. The background of most engineers in this important technological area, however, is seriously deficient. For example, an undergraduate engineering student receives less than an hour of instruction in tribology. Moreover, most reference works of tribology provide little guidance to solving real-world problems. Tribology is the science and technology of interacting surfaces in relative motion. It includes the study and application of the principles of friction, lubrication and wear.
1. Viscosity Definition:
Viscosity is a measure of the resistance of a fluid which is being deformed by either shear stress or extensional stress. In everyday terms (and for fluids only), viscosity is "thickness". Thus, water is "thin", having a lower viscosity, while honey is "thick" having a higher viscosity. Viscosity describes a fluid's internal resistance to flow and may be thought of as a measure of fluid friction. All real fluids (except superfluids) have some resistance to stress, but a fluid which has no resistance to shear stress is known as an ideal fluid or inviscid fluid. The study of viscosity is known as rheology. 1.1. Viscosity coefficients
When looking at a value for viscosity, the number that one most often sees is the coefficient of viscosity. There are several different viscosity coefficients depending on the nature of applied stress and nature of the fluid. They are introduced in the main books on hydrodynamics and rheology. •
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Dynamic viscosity (or absolute viscosity) determines the dynamics of an incompressible Newtonian fluid; Kinematic viscosity is the dynamic viscosity divided by the density for a Newtonian fluid; Volume viscosity (or bulk viscosity) determines the dynamics of a compressible Newtonian fluid; Shear viscosity is the viscosity coefficient when the applied stress is a shear stress (valid for nonNewtonian fluids); Extensional viscosity is the viscosity coefficient when the applied stress is an extensional stress (valid for non-Newtonian fluids).
Shear viscosity and dynamic viscosity are much better known than the others. That is why they are often referred to as simply viscosity. Simply put, this quantity is the ratio between the pressures exerted on the surface of a fluid, in the lateral or horizontal direction, to the change in velocity of the fluid as you move down in the fluid (this is what is referred to as a velocity gradient). For example, at room temperature, water −3 −3 has a nominal viscosity of 1.0 × 10 Pa·s and motor oil has a nominal apparent viscosity of 250 × 10 Pa·s. Extensional viscosity is widely used for characterizing polymers. Volume viscosity is essential for Acoustics in fluids, see Stokes' law (sound attenuation).
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1.2. Newton's theory
In general, in any flow, layers mov at different velocities and the fluid's visc sity arises from the shear stress between the layers that ultimat ly oppose any applied force. Newton postulated that, for straight, parallel and uniform flow, the shear st ess, τ, between layers is proportional to the velocity gradient, ∂u/ ∂y, in the direction perpendicular to the l yers.
Here, the constant µ is known as the coefficient of viscosity, the viscosity, the dynamic viscosity, or the Newtonian viscosity. Many fluids, s ch as water and most gases, satisfy Newton's criterion and are known as Newtonian fluids. Non-Newtonia fluids exhibit a more complicated relatio ship between shear stress and velocity gradient than simple linearity.
1.3. Viscosity Measurements
Dynamic viscosity is measured with various types of rheometer. Close tempera ture control of the fluid is essential to accurate measurements, articularly in materials like lubricants, whos viscosity can double with a change of only 5 °C. For some fluids, it is a constant over a wide range of shear rates. These are Newtonian fluids. The fluids without a constant viscosi y are called non-Newtonian fluids. Their viscosity cannot be described by a single number. Non-Newtonian fluids exhibit a variety of different correlations between shear stress and shear rate. Units of measure
Dynamic viscosity The cgs physical unit for dynamic vi scosity is the poise (P), named after Jean Louis Marie Poiseuille. It is more commonly expressed, particularly in ASTM standards, as centipoise (c ). Water at 20 °C has a viscosity of 1.0020 cP.
1 P = 1 g·cm−1·s−1. The relation between poise and pasca l-seconds is: 10 P = 1 kg·m−1·s−1 = 1 Pa·s, 1 cP = 0.001 Pa·s = 1 mPa·s. 2
Kinematic viscosity
The cgs physical unit for kinematic viscosity is the stokes (St), named after George Gabriel Stokes. It is sometimes expressed in terms of centistokes (cSt or ctsk). In U.S. usage, stoke is sometimes used as the singular form. 1 stokes = 100 centistokes = 1 cm2·s−1 = 0.0001 m2·s−1. 1 centistokes = 1 mm2·s−1 = 10−6m2·s−1. 1.4. Viscosity Index
Viscosity index is a term. It is a lubricating oil quality indicator, an arbitrary measure for the change of kinematic viscosity with temperature. The viscosity of liquids decreases as temperature increases of a lubricant is closely related to its ability to reduce friction. Generally, you want the thinnest liquid/oil which still forces the two moving surfaces apart. If the lubricant is too thick, it will require a lot of energy to move the surfaces (such as in honey); if it is too thin, the surfaces will rub and friction will increase. 1.5. Heat Effects Liquids: As the temperature of the liquid fluid increases its viscosity decreases. In the liquids the cohesive forces between the molecules predominates the molecular momentum transfer between the molecules, mainly because the molecules are closely packed (it is this reason that liquids have lesser volume than gases. The cohesive forces are maximum in solids so the molecules are even more closely packed in them). When the liquid is heated the cohesive forces between the molecules reduce thus the forces of attraction between them reduce, which eventually reduces the viscosity of the liquids. 2
µ = µ o / (1 + αt + βt ) Where: µ - Viscosity of the liquid at t degree Celsius n poise o µ o – Viscosity of the fluid at 0 Celsius in poise α, β – are the constants 1.6. Viscosity in Gases
Suppose now we repeat Newton’s suggested experiment, the two parallel plates with one at rest the other moving at steady speed, but with gas rather than liquid between the plates. It is found experimentally that the equation F/A= η v/ d still describes the force necessary to maintain steady motion, but, not surprisingly, for gases anywhere near atmospheric pressure the coefficient of viscosity is far lower than that for liquids (not counting liquid helium—a special case):
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Note first that, in contrast to the liquid case, gas viscosity increases with temperature. Even more surprising, it is found experimentally that over a very wide range of densities, gas viscosity is independent of the density of the gas! Returning to the two plates, and picturing the gas between as made up of layers moving at different speeds as before:
The previous picture of crowded molecules jostling each other is completely irrelevant! As we shall discuss in more detail later, the molecules of air at room temperature fly around at about 500 meters per second, the molecules have diameter around 0.35 nm, are around 3 or 4 nm apart on average, and travel of order 70 nm between collisions with other molecules .
2. Reynolds number Definition
Reynolds type equations are widely used in the field of Tribology. Tribology is a multidisciplinary field, which deals with the science, practice and technology of lubrication, wear prevention and friction control in machines. This enables lubrication engineers to minimize cost of moving parts. In this way machinery can be made more efficient, more reliable and more cost effective. In the field of hydrodynamic lubrication, the flow of fluid through machine elements such as bearings, gearboxes and hydraulic systems may be governed by the Reynolds equation. The Reynolds type equations are often used in analyzing the influence of surface roughness on the hydrodynamic performance of different machine elements when a lubricant is flowing through it. The two surfaces through which a lubricant flows, may have any of the following characteristics: • both surfaces are rough and moving, • one surface is rough and stationary while the other is smooth and moving. 2.1.
Flow in Pipe
For flow in a pipe or tube, the Reynolds number is generally defined as:
where: • • • • • • •
is the mean fluid velocity (SI units: m/s) D is the diameter (m) µ is the dynamic viscosity of the fluid (Pa·s or N·s/m²) ν is the kinematic viscosity ( ν = µ / ρ) (m²/s) ρ is the density of the fluid (kg/m³) Q is the volumetric flow rate (m³/s) A is the pipe cross-sectional area (m²)
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Definitions in non-SI units usually ave a number (coefficient) in front, for ex mple 124, where the pipe diameter is in inches, the velocity i feet per second, the fluid density in pounds per cubic foot, and the viscosity in centipoise, which are tra itional USA and UK units. 2.2.
Assumptions
Some assumptions on which the Lub ication Approximation (Reynolds Equation) is based are given below: 1. Laminar Flow: i. ii. iii.
This is flow which is like she ring a deck of cards. Flow is in one direction only, x. Flow is a continuous function of y (i.e. no discontinuities, i.e. no slip)
2. Steady State i. ii. iii.
In = Out Isothermal: Heat in = heat out Incompressible: Mass in=Mass out
3. Newtonian (Viscosity is constant in dg/dt): t = h dg/dt. (For non-Newtonian h = f(dg/dt).) 4. Viscosity dominates over inertia (Creeping Flow), i.e. a couette viscometer is low of two flat plates. 5. Taking Navier-Stokes, r Dv/Dt = -ÑP + h Ñ2v + rg, for these conditions, ∇P = η ∇2v and for velocity only in the x direction, δP/ δx = η δ2vx / δy2 which n integration yields: Vx(y) = Vx (1 - y/H) + H/2η (δP δx) y(y/H -1)
where v = V x at y = 0 and v = 0 at y = H. The pressure driven velocity is m ximum in the middle. The first term in this form of the Reynolds Equation is linear and the second is arabolic. The sum gives a skewed parabolic profile. 6. For Reynolds flow between 2 pla es, integration over y yields: qx = VxH/2 +H3 /12η (-δP/ δx) 2.3.
Reynolds Equation Derivati n
Substitute velocities into continuity e uation, and then integrate across film thickness
First and third terms require Leibnitz s rule:
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Reynolds Equation
• Describes flow through convergent channel • Left side: tangential & out of plane flows • Film thickness h = h( x, z) • Pressure p = p( x, z) • Boundary velocities on surfaces:
(U 1, V 1, W 1), (U 2, V 2, W 2)
3. Hydrodynamic Lubrication heory 3.1. Lubrication Zones Boundary lubrication Contact between Jour al and Bearing Mixed-film lubrication Intermittent contact Hydrodynamic lubrication Journal rides on a flui film. Film is created by the Motion of the j urnal.
3.2. Newtonian Fluid
Newtonian fluid is any fluid whose s ear stress and transverse rate of deformation are related through the equation.
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3.3. Pumping Action
When dry, friction will cause the journal to try to climb bearing inner wall. When lubricant is introduced, the climbing action and the viscosity of he fluid will cause lubricant to be drawn ar und the journal creating a film between the journals and bearin . The lubricant pressure will push the journal to the side.
3.4. Journal Bearing Nomencla ure
β is equal to 2π for a full bearing.
If β is less than 2π, it is known as a partial bearing. We will only be considering the full beari g case.
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3.5. Analysis Assumptions
1) Lubricant is a Newtonian flui . 2) Inertia forces of the lubricant are negligible. 3) Incompressible. 4) Constant viscosity. 5) Zero pressure gradient along the length of the bearing. 6) The radius of the journal is la ge compared to the film thickness. 3.6. Analysis Geometry
Actual Geometry
Unrolled Geometry
From boundary layer theory, the pres sure gradient in the y direction is constant.
X-Momentum Equation
Boundary Con itions
y=0, u=0 y=h(x), u=-U
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Note that h(x) and dp/dx are not known at this point.
Mass Flow Rate
Conversion of mass
Reynolds Equation
h(x) Relationship
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3.7. Sommerfeld Solution
A. Sommerfeld solved these equations in 1904 to find the pressure distribution around the bearing. It is known as a long bearing solution because there is no flow in the axial direction.
3.8. Short-Bearing Pressure Distributions
3.9. Short & Long BearingCom parisons
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4. Elasto-Hydrodynamic Lubrication theory Hydrodynamic lubrication is the best described regime in literature and is based on the so called Reynolds equation (1886). For detailed information regarding hydrodynamic lubrication the reader is referred to Moes (1997). This equation describes the relation between the pressure and film shape as a function of the viscosity and the velocity. The Reynolds equation can be written as:
With: + v+: the sum velocity of the moving surfaces ( v = v1 + v2 + ) x, y : spatial Cartezian coordinates t: time p: pressure h: film thickness ρ: density η : viscosity
The right-hand-side of Eq. 2.1 denotes three possible effects that generate pressure in the gap between the opposing moving surfaces. The first term is referred to as the wedge term, the second as the stretch term and the last as the squeeze term. The stretch term is omitted in this thesis, i.e. the sum velocity is constant in the direction of motion. The chosen moving direction is the x-direction. This thesis deals with line-contacts, the pressure is constant along the y-direction and therefore the second term in the left-hand side of Eq. 2.1 can be omitted. In this thesis the steady-flow case is considered. The squeeze term in the Reynolds equation can thus be left out as well. The remaining Reynolds equation reads:
In order to solve the Reynolds equation boundary conditions are needed. The pressure is defined as zero at the edges of the gap, thus:
By solving the Reynolds equation, pressure distribution is obtained. The integral over pressure distribution in the gap results into the load applied to the contact:
With B the length of the cylinder. Reynolds’ equation includes parameters like the geometry or shape of the surfaces (i.e. the film thickness), the viscosity h and the density r. These three parameters are pressure dependent. The next three sections present a literature review on the relationships between these three parameters and pressure. Film shape Due to the pressure generated in the contact the surface may deform. If this is the case elasto-hydrodynamic lubrication takes place. The distribution of the pressure in the gap is solved by using the Reynolds equation when the geometry or shape of the surface is known. The equation which describes the film shape between two deformed cylinders can be written by using a parabolic approximation: 11
with: h¥ : constant R: reduced radius of the undeformed cylinders w: deformation. The reduced radius is defined by: with R1 and R2 the radii of cylinder 1 and 2 respectively. The deformation of a cylinder by a pressure distribution is calculated according to Timosenko and Goodier (1982) as:
Where E’ is reduced elasticity modulus. The reduced elastic modulus is given by:
With E1and E2 the elasticity modulus and ν1 and ν2 the Poisson ratios of surfaces ν1 and ν2 respectively
5. Tribology and friction Friction is essential in our daily life. Friction makes it possible to walk, cycle, skate etc. On one hand, there are cases where a high friction is demanded, as for instance in brakes, traction drives and clutches. But on the other hand, in many industrial applications low friction between the contacting surfaces is required, such as gears, bearings and cam & tappet system. One of the developments in design is to reduce the size of the components in constructions while transmitting the same or even higher loads resulting in severe operational contact conditions. Therefore, a higher quality of the materials is required and the need of tribological knowledge increases as well. Furthermore the tribo-systems are optimized with respect to low friction and wear. This is often realised by lubrication. When the lubricant is able to separate the surfaces, then friction is considerably less, compared to the situation when the surfaces are in direct contact. If the surfaces are separated by a fluid film due to motion, the lubrication mechanism is called hydrodynamic lubrication (HL) and when the contacting bodies deform elastically due to the contact pressure the lubrication mechanism refers to as elasto-hydrodynamic lubrication (EHL).
Schematic representation of two surfaces in contact. I local contact separated by a boundary layer and II direct contact between the opposing surfaces Triboelectric effect Rubbing dissimilar materials against one another can cause a build-up of electrostatic charge, which can be hazardous if flammable gases or vapours are present. When the static build-up discharges, explosions can be caused by ignition of the flammable mixture. 12
