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Sommerfeld number - Wikipedia, the free encyclopedia Page 1 of 3 Sommerfeld number From Wikipedia, the free encyclopedia 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. The Sommerfeld number is named after Arnold Sommerfeld (1868 –1951). Contents • 1 Definition • 2 Derivation • 2.1 Petroff's Law • 2.2 Sommerfeld number • 3 Notes • 4 References Definition The Sommerfeld Number is typically defined by the following equation[1]. Where: S is the Sommerfeld Number or bearing characteristic number r is the shaft radius c is the radial clearance • is the absolute viscosity of the lubricant N is the speed of the rotating shaft in revs/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[2]. http://en.wikipedia.org/wiki/Sommerfeld_number 5/30/2011 Sommerfeld number - Wikipedia, the free encyclopedia Page 2 of 3 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•rN, where N is the rotational speed of the shaft in rev/s. The shear stress in the lubricant can be represented as follows: Assuming a constant rate of shear, The torque required to shear the film is If a small radial load W acts on the shaft and hence the bearing, the frictional drag force can be considered equal to the product fW, with the friction torque represented as T = fWr = 2r2flP Where W is the force acting on the bearing P is the radial load per unit of project bearing area (Pressure) f is the coefficient of friction If the small radial load W is considered negligible, setting the two expressions for torque equal to one another and solving for the coefficient of friction yields Which is known as Petroff's Law or the Petroff equation. It provides a quick and simple means of obtaining reasonable estimates of coefficients of friction of lightly loaded bearings. Sommerfeld number Multiplying both sides of Petroff's Law by the clearance ratio r/c, http://en.wikipedia.org/wiki/Sommerfeld_number 5/30/2011 Sommerfeld number - Wikipedia, the free encyclopedia Page 3 of 3 Notes 1. ^ Shigley 1989, p.484. 2. ^ Shigley 1989, p.483. References Shigley, Joseph Edward; Mischke, Charles R. (1989). Mechanical Engineering Design. New York: McGraw-Hill. Retrieved from "http://en.wikipedia.org/wiki/Sommerfeld_number" Categories: Fluid dynamics | Bearings • This page was last modified on 3 April 2011 at 23:35. • Text is available under the Creative Commons Attribution-ShareAlike License; additional terms may apply. See Terms of Use for details. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc., a non-profit organization. http://en.wikipedia.org/wiki/Sommerfeld_number 5/30/2011