1. 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 e. Putting back the dimensional quantities we get

2. Sommerfeld number Which gives Where W = total load L = axial length
U = surface speed
h = viscosity
C = radial clearance
R = bearing radius
The variable on the left hand side is known as the Sommerfeld number and is often designated by S or D. It is more usual to work with the reciprocal

3. 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.

4. Friction coefficient Sommerfeld number
Ref:

5. Reynold’s condition p = 0, dp/dq = 0, at some value of q > p
The start of the curve is assumed at the point of maximum oil thickness q = 0
The pressure equation obtained earlier is
If the pressure starts at q =0 (g = 0), then C = 0
If p*= 0 at any other value of q, C will not be zero

6. We know that h = c(1 + ecosq) and therefore ho = c(1 + ecosqo) where ho is the film thickness when dp/dq = 0 at q = qo
This equation is symmetrical about q = 180o, hence qo can have 2 values, one when pressure p is maximum and the other when p is minimum
Therefore we can write
On the g scale we can write
Where b on the g scale corresponds to a on the q scale
p* = 0 at g = p + b, sin(p+b) = -sinb and cos(p + b) = -cosb

7. Reynold’s condition, p = 0, dp/dq = 0, at some value of q > p
Start of pressure curve
Bearing
Rdq q Wy Wx
= p+a
(min. pressure = 0) W Y
Shaft
Pressure curve
= p-a
(max. pressure)

8. e in terms of b
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 b

9. Values of b for different e (obtained by Cameron and Wood)
b (rad.)
0.8871
0.9
1.0
1.1
1.2
1.3
1.352 e
0.9727
0.7574
0.5383
0.3204
0.1073
e = e/c = 1 when e = c (eccentricity = radial clearance)
= 0 when eccentricity = 0, i.e. the shaft and bearing are coaxial
The values of b can be inserted into the pressure equation and integrated to give the loads Wx and Wy

10. Now The first term is zero as p = 0 at q = 0 and (p+a).
Using Sommerfeld substitution the required integral in terms of g comes out to be

11. P = 0 at q = 0 and p + a, therefore the 1st. Term disappears
P = 0 at q = 0 and p + a, therefore the 1st. Term disappears. Using Sommerfeld’s substitution we get

12. Therefore
Once y has been found we can find W from Wsiny or Wcosy

13. As e  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

14. The minimum film thickness ho = c(1-e)
Hence the expression for load carried can be written as
Where D is the Sommerfeld number

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
Oxygen reacts preferentially with additive molecules
Oxygen
Oil
Oil
Additive
Preferential oxidation: Additive is more susceptable to oxidation than oil
Oil particles are therefore prevented from oxidising

22. Oxidation prevention additives
Additive reacts with metal particles
Metal catalyst
Oil
Oil
Additives
Additive covers metal particles by forming a coating
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

23. Oxidation prevention additives- peroxide decomposers
Hydrocarbon + oxygen
Hydroxyperoxides
Materials susceptible to oxidation by decomposed peroxides
Hydroxyperoxides
Decompose
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
Acids formed by oxidation
Harmless products
Shield from air/water
Polar additive layer or chemically reacted layer
Metal
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