Proof of Betz law

The original formulation of Betz's law in German       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
Stream tube through the turbine        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, V1, and the wind speed after the passage through the rotor plane, V2. Actually, Betz offers a proof of this. This way, the average wind speed V through the rotor is given by:

The mass of the air streaming through the rotor during one second is:
                              m =d F (V1+V2)/2
      where   m: mass per second,
                    d: density of air
rotor swept area

According to Newton's 2nd 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 (V12 - V22)

Substituting m into this expression from the first equation, the following expression for the power extracted from the wind comes:
                     P = (d/4) (V12 - v22) (V1+V2) F

The total power in the undisturbed wind streaming through exactly the same area with no rotor blocking the wind is:
                            P0 = (d/2) V 13 F

The ratio between the power extracted from the wind and the power in the undisturbed wind is then:
             P/P0 = (1/2) [1 - (V2 / V1)2] [1 + (V2 / V1)]

Ploting the ratio P/P0 as a function of V2/V1 the following graph comes:

                                      Betz diagram

        It can be seen that the ratio P/P0 reaches its maximum for V2/V1 = 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.