The maximum Cp may be determined by taking the derivative of a(1 — a)2 with respect to a and setting it equal to zero. This yields a = 1/3, and the corresponding value of Cp = CP;max is given by, CPjmax = 16/27 = 0.5926. This result is known as the Betz limit, and it indicates that the ideal rotor must be designed such that the wind speed at the rotor is 1 — a = 2/3 of the free stream wind speed, so it operates for maximum power extraction. This implies that the flow through the disc corresponds to an upstream cross-sectional area of 1 — a = 2/3 the disc area that expands to twice the area far downstream. It must be emphasized that the Betz limit is not a universal limit as it is based on the approximate actuator disc theory. Thus, in principle, it is possible to extract more power than that predicted by the Betz limit, but it is important to recognize that there is indeed a real physical limit to the maximum power extracted from the wind.
Although the maximum thrust is obtained for a = 1/2 corresponding to a Ct;max = 1, the thrust coefficient for maximum power transfer, based on the actuator disc model, is Ct = 8/9. There are a number of losses which detract the
wind turbine from achieving maximum power-transfer efficiency. These include the wake vortex rotation behind the disc which consumes energy, tip losses associated with blades and aerodynamic drag which is another form of energy dissipation. One way of including the effects of wake rotation is to consider the rotation of the flow which is addressed in the next section.
