expression for load we have seen that under sommerfeld’s condition w = w y in non-dimensional...
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Expression for load
• We have seen that under Sommerfeld’s condition W = Wy
• In non-dimensional terms, W* = Wy* =
• This is a function solely of the eccentricity ratio . Putting back the dimensional quantities we get
)2/1()1( 22/12
)2/1()1(c
LRU6
c
WLRU6W
22/122
2
2
*2
Sommerfeld number•Which gives
WhereW = total loadL = axial lengthU = surface speed = viscosityC = radial clearanceR = bearing radiusThe variable on the left hand side is known as the Sommerfeld number and is often designated by S or . It is more usual to work with the reciprocal
)29...()2/1()1(
6
R
c.
U
L/W22/122
2
• Sommerfeld’s number is used as abcissa for a number of design curves.
• The ordinate can be selected to allow the friction value, film thickness, oil leakage, temperature rise etc to be determined.
• Design curves have been produced of various variables against the Sommerfeld number using computer techniques by A.A Raimondi and J.Boyd of Westinghouse Research Labs(ASLE Transactions Vol 1 No 1 April 1958).
• These graphs include compensation for end leakage and eccentricity.
Ref: http://www.roymech.co.uk/Useful_Tables/Tribology/Liquid_Lubrication.htm
Sommerfeld number
Fric
tion
coeffi
cien
t
5
Reynold’s condition
• p = 0, dp/d = 0, at some value of > • The start of the curve is assumed at the point of
maximum oil thickness = 0• The pressure equation obtained earlier is
• If the pressure starts at =0 (= 0), then C = 0• If p*= 0 at any other value of , C will not be zero
C)1(
4/)2sin(2/sin2
cos1
1
)1(
sinp
2/52
22
o
2
2/32*
6
• We know that h = c(1 + cos) and therefore ho = c(1 + coso) where ho is the film thickness when dp/d = 0 at = o
• This equation is symmetrical about = 180o, hence o can have 2 values, one when pressure p is maximum and the other when p is minimum
• Therefore we can write
• On the scale we can write
• Where on the scale corresponds to on the scale
• p* = 0 at = + , sin(+) = -sin and cos( + ) = -cos
o
o
7
Reynold’s condition, p = 0, dp/d = 0, at some value of >
Pressure curve
W
Bearing
Shaft
= -
(max. pressure)
=+
(min. pressure = 0)
Rd
WyWx
=
Start of pressure curve
8
in terms of
• Substituting the above in the pressure equation we get p* = 0
• Expanding and multiplying out we find that
• Or
• This equation can be used to determine
cos1
4/)2sin(
2/)(sin2)(
sin)()1(
1 2
2
2/32
sin2)(cos)(2cossin0
)30...(
)(cossin
cos)(sin2c/e
9
Values of for different (obtained by Cameron and Wood)
(rad.)
0.8871 0.9 1.0 1.1 1.2 1.3 1.352
1.0 0.9727 0.7574 0.5383 0.3204 0.1073 0
= e/c = 1 when e = c (eccentricity = radial clearance)
= 0 when eccentricity = 0, i.e. the shaft and bearing are coaxial
The values of can be inserted into the pressure equation and integrated to give the loads Wx and Wy
10
Now
0
0
x
y
dcospRL
dsinpRL
W
W
cosW
sinWtan
0 00dcos
d
dpcospdsinp
The first term is zero as p = 0 at = 0 and (+).
Using Sommerfeld substitution the required integral in terms of comes out to be
0 0 2/32
d)1(
)(cos
cos1
cos11dcos
d
dp
dcosp
dsinp
0
0
11
)cos1()1(2
sincos)(2)(cossin2/32
2
)cos1()1(
sincos)(2/32
000
dcosd
dpsinpdcosp
P = 0 at = 0 and + , therefore the 1st. Term disappears. Using Sommerfeld’s substitution we get
020 )cos1(
)cos1(1
1
1dcosp
)cos1()1(
sincos)(sin*W
2/32
12
)cos1)(1(2
)cos1(2
2
2
2/12
)cos1(
cos)(sin)1(2tan
Therefore
Once has been found we can find W from Wsin or Wcos
)cos1)(1(2
)cos1(*Wcos*W
2
2
x
13
• As 1, The eccentricity radial clearance, therefore the infinite journal bearing approaches the value for 2 discs with internal contact.
• The expression is
• Where
• Now R1 - R2 = c, the radial clearance, so 1/Rred = c/R2, where R is taken as the radius of the shaft
ored h/UR447.2L
W
21
21
12red RR
RR
R
1
R
1
R
1
14
• The minimum film thickness ho = c(1-)
• Hence the expression for load carried can be written as
• Where is the Sommerfeld number
)1(R
c)1(
U
L/W
UR
h)L/W(447.2
2
2
red
o
15
Dielectric strength• A measure of the electrical insulating strength
• Measured as the maximum voltage it can withstand without conducting (expressed as volts/thickness)
• Less moisture- better insulators
• Dehydrating techniques are used to improve the dielectric strength
16
Carbon residue• Carbon residue is formed by evaporation and oxidation of
lubricant
• The test of the tendency of a lubricant to form carbon residue is called the “Conradson” test
• The test sample is heated until it is completely evaporated (cannot ignite)
• The residue is cooled and weighed
• Result interpreted as weight ratio of residue to oil sample
17
Lubricant additives
18
Lubricant additives or agents
• Added to preserve, improve and/or provide additional useful properties to a lubricant
• Protect the surface forming a film• Keep surfaces and lubricant passageways
clean• Inhibitors prevent the formation of harmful
products• Some are consumed (sacrificial), others are
not (non-sacrificial)
19
Lubricant additives- classification
• Oxidation inhibitors• Viscosity index improvers• Boundary and extreme pressure additives• Rust inhibitors• Detergents• Dispersants• Pour point depressants• Anti-foaming agents• Friction modifiers
20
Oxidation products
• Sludge: Black tar-like substance consisting of water, carbon, engine oil, organic residue and dirt
• Engine gum: Acts as a binder causing residue to stick to machinery components
• Varnish: Petroleum gum exposed to high temperature and ironed out on surfaces
• Laquer: Thin layer of reacted varnish• Carbon deposits: Combination of soot from fuel
burning and oxidation of lubricating oil
21
Oxidation prevention additives
• Preferential oxidation: Additive is more susceptable to oxidation than oil
• Oil particles are therefore prevented from oxidising
Oxygen
Additive
Oil
Oxygen reacts preferentially with additive molecules
Oil
22
Oxidation prevention additives
Metal deactivators-• Metal particles in the oil act as catalysts for the
oxidation reactions• The additives either react with the metal particles
or form a coating over them
Metal catalyst
Additives
Oil
Additive reacts with metal particles
Additive covers metal particles by forming a coating
Oil
23
Oxidation prevention additives- peroxide decomposers
Hydrocarbon + oxygen
Hydroxyperoxides
Hydroxyperoxides DecomposeMaterials susceptible to
oxidation by decomposed peroxides
Additives + Hydroxyperoxides
Non-oxidizing product
24
Rust Inhibitors- effect of water
• Below boiling point, water is present in a lubricating system.
• Water contaminant can lead to formation of rust
• Water enters by condensation and/or leakage from coolers or steam heating coils
• Some oils are hygroscopic and therefore physically absorb moisture
25
Rust prevention additives
• Rust inhibitors neutralize acids formed by oxidation• Polar additives form a protective layer on the metal surface
due to attraction by the surface• Chemically react with the metal surfaces to form a protective
film• E.g. metal sulphonates, fatty acids, phosphates
Rust prevention additive
Acids formed by oxidation
Harmless products
Polar additive layer or chemically reacted layer
Metal
Shield from air/water