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

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

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.

Friction coefficient Sommerfeld number Ref: http://www.roymech.co.uk/Useful_Tables/Tribology/Liquid_Lubrication.htm

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

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

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)

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

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

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

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

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

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

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

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

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

Lubricant additives

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)

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

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

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

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

Oxidation prevention additives- peroxide decomposers Hydrocarbon + oxygen Hydroxyperoxides Materials susceptible to oxidation by decomposed peroxides Hydroxyperoxides Decompose Additives + Hydroxyperoxides Non-oxidizing product

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

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