- Sommerfeld number
In the design of fluid bearings, the Sommerfeld number, or bearing characteristic number, is a
dimensionless quantity used extensively inhydrodynamic lubrication analysis. The Sommerfeld number is very important in lubrication analysis because it contains all the variables normally specified by the designer.Definition
The Sommerfeld Number is typically defined by the following equation [Shigley 1989, p.484.] .
:S = left( frac{r}{c} ight)^2 frac {mu N}{P}
Where:: S is the Sommerfeld Number or bearing characteristic number: r is the journal radius: c is the radial clearance: μ is the absolute viscosity of the lubricant: N is the speed of the rotating shaft in rev/s: P is the load per unit of projected bearing area
Derivation
Petroff's Law
Petroff's method of lubrication analysis, which assumes a
concentric shaft and bearing, was the first to explain the phenomenon of bearingfriction . This method, which ultimately produces the equation known as Petroff's Law, is useful because it defines groups of relevant dimensionless parameters, and predicts a fairly accurate coefficient of friction, even when the shaft is not concentric [Shigley 1989, p.483.] .Considering a vertical shaft rotating inside a bearing, it can be assumed that the bearing is subjected to a negligible load, the radial clearance space is completely filled with lubricant, and that leakage is negligible. The surface velocity of the shaft is: U = 2 pi r N, where N is the rotational speed of the shaft in rev/s.
The
shear stress in the lubricant can be represented as follows::au = mu left.frac{partial u}{partial y} ight|_{y = 0}Assuming a constant rate of shear,:au = mu frac{U}{h} = frac{2 pi r mu N}{c}
The
torque required to shear the film is:T = left( au A ight) left( r ight) = left( frac {2 pi r mu N}{c} ight) left( 2 pi r l ight) left( r ight) = frac {4 pi r^3 l mu N}{c}If a small force W acts on the bearing, the torque can also be represented as:T = f mathrm{Wr} = 2 mathrm{r}^2 f mathrm{lP}
Where: W is the force acting on the bearing: P is the pressure on the bearing: f is the coefficient of friction
Setting the two expressions for torque equal to one another and solving for the coefficient of friction yields
f = 2 pi^2 frac{mu N}{P} frac{r}{c}
Which is known as Petroff's Law.
Sommerfeld number
Multiplying both sides of Petroff's Law by the "clearance ratio" r/c,
:f frac{r}{c} = 2 pi^2 frac{mu N}{P} frac{r}{c}^2 = 2 pi^2 S:S = left( frac{r}{c} ight)^2 frac {mu N}{P}
Notes
References
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