Sommerfeld number

In the design of fluid bearings, the Sommerfeld number, or bearing characteristic number, is a dimensionless quantity used extensively in hydrodynamic 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 bearing friction. 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