Betz' law reflects a theory for flow machines, developed by Albert Betz. It shows the maximum possible energy that may be derived by means of a rotor from a fluid flowing at a certain speed. Albert Betz (25 December 1885 - 16 April 1968) was a German Engineer and a pioneer of wind energy technology. ...

In order to calculate the maximum theoretical efficiency of a rotor (of, for example, a wind mill) one imagines it to be replaced by a disc that withdraws energy from the fluid passing through it. At a certain distance behind this disc, the fluid, that has passed through, flows with a reduced velocity. Pitstone Windmill, believed to be the oldest windmill in the British Isles A windmill is an engine powered by the energy of wind. ...

Let v₁ be the speed of the fluid in front of the rotor and v₂ that of the fluid downstream from it. The mean flow velocity through the disc representing the rotor is v_avg, where

With the area of the disc equal to S, and with ρ = fluid density, the mass flow rate (the mass of fluid flowing per unit time) is given by: Mass flow rate is the movement of mass per time. ...

The power delivered is the difference between the kinetic energies of the flows approaching and leaving the rotor in unit time: In physics, power (symbol: P) is the rate at which work is performed. ...

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The horizontal axis reflects the ratio , the vertical axis is the "coefficient of performance" C_p.

By differentiating with respect to for a given fluid speed v₁ and a given area S one finds the maximum or minimum value for . The result is that reaches maximum value when . Image File history File links Ratio. ... Image File history File links Ratio. ... In mathematics, a derivative is the rate of change of a quantity. ...

Substituting this value results in:

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The work rate obtainable from a cylinder of fluid with area S and velocity v₁ is:

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The "coefficient of performance" C_p (= ) has a maximum value of: C_p.max = = 0.593 (or 59.3 %; however, coefficients of performance are usually expressed as a decimal, not a percentage).

Rotor losses are the most significant energy losses in, for example, a wind mill. It is, therefore, important to reduce these as much as possible. Modern rotors achieve values for C_p in the range of 0.4 to 0.5, which is 70 to 80% of the theoretically possible.

Reference

Betz, A. (1966) Introduction to the Theory of Flow Machines. (D. G. Randall, Trans.) Oxford: Pergamon Press.