17. Maximum wind turbine efficiency - the Betz limit.

The simplest model of a wind turbine is the so-called actuator disc model where the turbine is replaced by a circular disc through which the airstream flows with a velocity U_t and across which there is a pressure drop from P₁ to P₂ as shown in the sketch. At the outset, it is important to stress that the actuator disc theory is useful (as will be shown) in discussing overall efficiencies of turbines but it does not help at all with how to design the turbine blades to achieve a desired performance.

The power developed by the wind turbine is

where A_t is the turbine disc area. Volume flow continuity gives

From momentum conservation, the force exerted on the turbine is equal to the momentum change between the flow far upstream of the disc to the flow far downstream of the disc. Thus

The final basic equations are Bernoulli's equation applied upstream and downstream of the actuator disc

where P_∞ is the ambient pressure in the flow both far upstream and far downstream of the actuator disc.
From equations (4a),(4b), (3) and (2)

whence

i.e. the velocity through the actuator disc is the mean of the upstream and downstream velocities in the stream tube.
Finally, from equations (1), (5) and (3), the efficiency is given by

The figure below shows the variation of efficiency (often referred to as the power coefficient, cp) with the ratio of downstream to upstream velocity. By differentiating equation (7), it is easy to show that the maximum efficiency oocurs when U_d/U_u=1/3 (i.e. when A_d/A_u=3). The efficiency is then η=16/27 ≈ 59%. This is the maximum achievable efficiency of a wind turbine and is known as the Betz limit - after Albert Betz who published this result in 1920.

The point to note here is that as you reduce the downstream velocity in the expectation of increasing the power extracted from the wind, the area of the upstream stream tube that passes through the turbine reduces in size. In the limit as the downstream velocity is reduced to zero, the area of the upstream stream tube that passes through the turbine is just half the turbine area and the efficiency is thus 50%.