Proof of Betz law

**Introduction**
In this page the proof of the Betz law is given, as it is given in his book Wind-Energie in 1926. In the figure there the formulation of the Betz's law in German can be sawn.
**Proof of Betz's law**
Let us make the reasonable assumption that the average wind speed through the rotor area is the average of the undisturbed wind speed before the wind turbine, V_{1}, and the wind speed after the passage through the rotor plane, V_{2}. Actually, Betz offers a proof of this. This way, the average wind speed V through the rotor is given by:

V=(V_{1}+V_{2})/2

The mass of the air streaming through the rotor during one second is:

m =d F (V_{1}+V_{2})/2

where * m: *mass per second,

* d: * density of air

F: rotor swept area

According to Newton's 2^{nd} law the power extracted from the wind by the rotor is equal to the mass times the drop in the wind speed squared:

P = (1/2) m (V_{1}^{2} - V_{2}^{2})

Substituting *m* into this expression from the first equation, the following expression for the power extracted from the wind comes:

P = (d/4) (V_{1}^{2} - v_{2}^{2}) (V_{1}+V_{2}) F

The total power in the undisturbed wind streaming through exactly the same area with no rotor blocking the wind is:

P_{0} = (d/2) V _{1}^{3} F

The ratio between the power extracted from the wind and the power in the undisturbed wind is then:

P/P_{0} = (1/2) [1 - (V_{2} / V_{1})^{2}] [1 + (V_{2} / V_{1})]

Ploting the ratio *P/P*_{0} as a function of *V*_{2}/V_{1} the following graph comes:
It can be seen that the ratio* P/P*_{0} reaches its maximum for *V*_{2}/V_{1} = 1/3 and that the maximum value for the power extracted from the wind is or 16/27=0,59 of the total power in the wind.