LUBRICATION

Purpose of Lubrication

• The primary purpose of lubrication is to reduce wear and heat between contacting surfaces in relative motion.

• Because heat and wear are associated with friction, both effects can be minimized by reducing the coefficient of friction between the contacting surfaces.

• Lubrication is also used to reduce oxidation and prevent rust; to provide insulation in transformer applications; to transmit mechanical power in hydraulic fluid power applications; and to seal against dust, dirt, and water.

Lubricants

• Reduced wear and heat are achieved by inserting a lower-viscosity

(shear strength) material between wearing surfaces that have a

relatively high coefficient of friction. Any material used to reduce

friction in this way is a lubricant. Lubricants are available in liquid,

solid, and gaseous forms.

• Industrial machinery ordinarily uses oil or grease.

• Solid lubricants such as molybdenum disulfide or graphite are used

when the loading at contact points is heavy. In some applications

the wearing surfaces of a material are plated with a different metal to

reduce friction.

Viscosity

• Viscosity is a measure of a fluid's resistance to motion. It

represents the fluid internal friction.

• Newton’s viscous law:

where:

is the shear stress;

is a proportionality constant and defines the dynamic or absolute viscosity as the

tangential force per unit area required to move one horizontal plane with respect to

the other at unit velocity when maintained a unit distance apart by the fluid;

is the shear rate or velocity gradient.

F

A du

dy

du

dy

Viscosity

• Kinematic viscosity is the ratio between absolute viscosity and mass

density.

- It is measured by viscosimeters (viscometers) in which the

fluid drives the oil through a calibrated capillary.

where:

is the kinematic viscosity;

is the absolute viscosity;

ρ is the mass density.

Example of a viscosimeter:

Saybolt Universal Viscosimeter

Units of Dynamic Viscosity

• Dynamic viscosity:

• SI system:

- 1 Pa.s = 1 N.s/m2 = 1 kg/m.s

• Metric CGS (centimeter-gram-second) system:

- 1 Poise (P) = 1 dyne.s/cm2 = 1 g/cm.s = 0.1 Pa.s

• It's usual to divide the Poise by 100 into the smaller unit:

- 1 centiPoise (cP) = 0.01 P

• Water at 20.2oC has an absolute viscosity of 1 cP.

• British IPS (inch-pound-second) system:

- 1 reyn = 1 lbf.s/in2 = 6890 Pa.s

Units of Kinematic Viscosity

• Kinematic viscosity:

• SI system:

- 1 m2/s

• Metric CGS (centimeter-gram-second) system:

- 1 Stoke (St) = 1 cm2/s = 10-4 m2/s

• It's usual to divide the Stoke by 100 into the smaller unit:

- 1 centistoke (cSt) = 0.01 St = 1 mm2/s = 10-6 m2/s

• Water at 20.2oC has a kinematic viscosity of 1 cSt.

• British IPS (inch-pound-second) system:

- 1 in2/s = 645.2 cSt = 6.452E-04 m2/s

• Conversion from Saybolt Universal seconds (SUS):

- 1 cSt = 0.22 (SUS) – 180/(SUS)

Viscosity Classifications

• ISO (from ASTM D2422): Based only on viscosity ranges at 40ºC.

• Each viscosity grade is designated by the nearest whole number to its midpoint

kinematic viscosity in mm2/s (cST) at 40ºC, and a range of +/- 10 percent of this value

is permitted.

Viscosity Classifications

• SAE: Based primarily on viscosity at 100ºC, with an additional low-

temperature requirement for winter grades (ASTM D5293-Cranking

and ASTM D455-Pumping).

SAE Crankcase oil grades

Comparative Viscosity Classifications

Viscosity-Temperature Charts

• Oil viscosity decreases rapidly with increasing temperature.

Viscosity-Temperature Relations

• Some of the most commonly used viscosity-temperature equations:

• Walther equation:

where:

is the kinematic viscosity in cSt;

T is the absolute temperature in ºK;

A and B are constants for any given oil.

• This relation forms the basis of the ASTM viscosity-temperature

chart and holds well for most mineral and synthetic oils, at high

pressures, and at a variety of given shear rates for polymer solutions

in multigrade automotive oils.

TlogBA)7.0log(log

Viscosity-Temperature Relations

• Some of the most commonly used viscosity-temperature equations:

• Vogel equation:

where:

ref is the absolute viscosity at reference temperature Tref;

β is a viscosity-temperature coefficient that can be estimated from the viscosities at

two temperatures from the relation:

• The Vogel equation is a more accurate representation of viscosity

over an extended temperature range.

)TT(

refrefe

12

12

TT

)/ln(

Viscosity Index

• The Viscosity Index (VI) is an empirical indicator of the relative

decrease in viscosity with increasing temperature.

Viscosity Index

• Originally a VI of 100 represented the least temperature-sensitive

Pennsylvania petroleum oil, and a VI of 0 represented the most

temperature-sensitive Gulf Coast oil.

• For an oil with VI ≤ 100:

where, for the same 100ºC kinematic viscosity of the oil in question:

H is the 40ºC viscosity of the 100 VI oil;

L is the 40ºC viscosity of the 0 VI oil;

U is the 40ºC viscosity of the unknown VI oil.

HL

ULVI

Viscosity Index

• For an oil with VI > 100:

where:

and

y is the viscosity in cSt at 100ºC for the fluid of interest.

• This method is now the ASTM procedure when VI exceeds 100,

because the former gave confusing results and even some double

values in this VI range.

10000715.0

1)Nloganti(VI

Ylog

)UlogH(logN

Viscosity Index

• Data for the evaluation

of viscosity index:

Viscosity-Temperature Coefficient

• Viscosity-Temperature Coefficient (VTC) is useful for some synthetic

oils and multigrade mineral oils not well defined by the usual VI

system.

• VTC gives a measure of the fractional change in kinematic viscosity

in going from 40ºC to 100ºC.

• VTC = 0 for an oil viscosity unaffected by temperature;

• VTC = 1 for an 100ºC oil viscosity = 0.

40

100

40

10040 1)(

VTC

100

Viscosity-Pressure Charts

• Oil viscosity increases steeply with increasing pressure.

Viscosity-Pressure Relations

• Some of the most commonly used viscosity-pressure equations:

• Barus equation:

where:

μ0 is the absolute viscosity in cP at atmospheric (nearly zero) pressure;

μ is the viscosity at pressure P at the same temperature;

α is the pressure-viscosity coefficient.

• This relation applies for many oils at pressures up to about 70 Mpa

such as those encountered in conventional fluid-film bearings and in

hydraulic systems.

P

0e

Viscosity-Pressure Relations

• Some of the most commonly used viscosity-pressure equations:

• Roelands equation:

where:

μ0 is the absolute viscosity in cP at atmospheric (nearly zero) pressure;

Z is a constant which is characteristic of the lubricant and is relatively independent of

temperature;

C is a constant with a value that depends on the pressure units used (196.1 MPa or

28440 psi), as necessary to make P/C dimensionless.

• This isothermal relation applies over a broad range of gauge

pressures up to quite high values.

Z

01010 )C/P1).(1200(log1200log

Viscosity-Pressure Index Z

• Representative values

of viscosity-pressure index Z:

Viscosity-Pressure Index Z

• Values of Z for most mineral oils, synthetic hydrocarbons, diesters,

and polyesters, can be estimated by the relations:

• where:

)F()HH(81.7Z 40

5.1

10040

)1200)log(log(H 4040

)1200)log(log(H 100100

)H864.0885.0(F 4040

Petroff’s Equation

• Bearing friction was first analyzed by Petroff in 1883.

• The shaft is vertical and considered concentric. It is assumed that

the bearing load is small and that oil completely fills the clearance

space without side leakage.

Petroff’s Equation

• Shearing stress in the lubricant:

with: N in rev/s – Shaft velocity;

U = 2πrN – Surface velocity.

• Torque required to shear the oil film:

• Frictional torque:

with: P = W/2rl – Bearing pressure.

• Solving for the coefficient of friction, we find the:

Petroff’s equation:

c

Nr2

h

U

c

Nlr4rrl2

c

Nr2r)A(T

32

flPr2r)rlP2(ffWrT 2

c

r

P

N2f 2

Sommerfeld Number

• The bearing characteristic number, or the Sommerfeld number is:

with: μN/P – Dimensionless parameter;

r/c – Dimensionless radial clearance ratio;

S – Dimensionless Sommerfeld number.

• Sommerfeld number and coefficient of friction relation:

P

N

c

rS

2

S2c

r

P

N2

c

rf 2

2

2

Petroff’s Law Viscous Friction

Model • In the thick film lubrication regime, Petroff’s law gives a good

approximation to friction losses.

Stable Lubrication

• Stribeck-Gumbel or McKee-Petroff curves:

• In stable thick film lubrication variations are self-correcting. For example an increase

in lubricant temperature will decrease both viscosity, μN/P and coefficient of friction

resulting in less heat generated and then in a drop of lubricant temperature.

• In unstable thin film lubrication variations have compounded results.

Beauchamp Tower's Experiments

• Beauchamp Tower’s experiments (1883-1885) to determine the

friction of the journal of a railway-car wheel with a lubricator hole, in

the partial bearing, drilled in the course of the experiments:

Beauchamp Tower's Experiments

• The experiments showed some surprising results. It became clear

that the lubricant between the journal and bearing was under

pressure. At sufficient rotational speed the pressure distribution

within the oil film was such that the film actually generated the load-

carrying forces.

Reynolds Equation - Assumptions -

• Beauchamp Tower’s results were explained by Osbourne Reynolds using a

differential equation first published in 1886.

• Reynolds equation can be derived from the Navier-Stokes and continuity

equations or directly from laws of viscous flow and principle of mass

conservation, with the following required simplifying assumptions: • (1) The lubricant fluid is Newtonian (shear stress is proportional to the shear rate);

• (2) Flow of fluid is laminar (no vortex flow or turbulence);

• (3) The weight of the fluid is negligible.

• (4) The forces due to the inertia of the lubricant are negligible.

• (5) The lubricant fluid is assumed to be incompressible;

• (6) The pressure remains constant across the thickness of the relatively thin lubricant fluid;

• (7) The viscosity of the fluid is assumed to be constant throughout the film;

• (8) There is no slip between the lubricant and the bounding surfaces;

• (9) There is no end leakage (infinite length bearing);

• (10) The curvature of the bearing can be ignored when compared with the very small

thickness of the fluid film (translational instead of rotational velocities).

Reynolds Equation - Control Volume -

• The Reynolds equation can be derived directly by considering a

control volume fixed in space and extended across the lubricant film.

Reynolds Equation - Velocity Distribution-

• Summing the forces in the x direction gives

• reducing to

• Using the Newton’s viscous law

• we obtain

• or

0dxdzdyy

dxdzdydzdxdx

dpppdydzFx

ydx

dp

dy

du

2

2

y

u

dx

dp

dx

dp1

y

u2

2

Reynolds Equation - Velocity Distribution -

• Integrating twice with respect to y

• Applying the lubricant velocity boundary conditions

• gives the velocity distribution of the lubricant in the film

1Cydx

dp1

y

u

21

2 CyCydx

dp

2

1u

0C0u0y 2

dx

dp

2

h

h

UCUuhy 1

yh

Uhyy

dx

dp

2

1u 2

Reynolds Equation - Velocity Distribution -

• The velocity distribution is obtained by superposing a parabolic

(Poiseuille) distribution onto a linear (Couette) distribution.

(a) Couette flow induced by

viscous drag force

Resulting velocity profiles

(b) Poiseuille flow induced by

pressure gradient

When the pressure is

maximum, dp/dx = 0

and u = Uy/h

Reynolds Equation - Mass Conservation -

• The volumetric flow rate in the x direction per unit of width in the z

direction is

• Substituting the value of u and integrating gives

• Since the lubricant is assumed as incompressible, the flow must be

the same for any cross section

h

0

z,h

0xx udy

z

udydz

z

Qq

2

Uh

dx

dp

12

hq

3

x

dx

dhU6

dx

dph

dx

d0

2

Uh

dx

dp

12

h

dx

d0

dx

dq 33

x

Reynolds Equation

• Reynolds equation for one-dimensional flow in a journal bearing

• Reynolds equation for two-dimensional flow when side leakage is

not neglected

• There is no general analytical solution to this equation. An

approximate solution due to Sommerfeld can be expressed as

• where ɸ indicates a functional relationship.

dx

dhU6

dx

dph

dx

d 3

x

hU6

z

ph

zx

ph

x

33

SP

N

c

rf

c

r2