Theoretical Maximum Efficiency

Because both the thrust and power coefficients are functions of the induction factor, a, it is useful to seek the condition that corresponds to the maximum power output from the wind turbine. Differentiating Eq. 13.11 with respect to a gives

I /П

—- = 4(1 — 4a + 3a2) = 0 for a maximum. (13.17)

da

By inspection it is apparent that this condition is met when the wake induction factor a = 1/3 and the corresponding values of Ct and Cp are 8/9 and 16/27, respectively (see Fig. 13.4).

In summary, this means that at the most efficient operating condition then 1 8 16

a = -, CT = – = 0.89 and CP = — = 0.59. (13.18)

3 9 27

This operating condition is known as the Betz-Lanchester limit – see Glauert (1935,1983) and Bergy (1979). This condition gives the upper theoretical limit to the aerodynamic power (or airpower) that can be extracted by a conventional wind turbine, assuming no viscous or other losses. In practice it is found that airpower values of Cp of between 0.4 and 0.5 are typical of a modern wind turbine when at its design operating state. This suggests that a wind turbine has a maximum possible aerodynamic efficiency of between 66% and 83% (when compared to the maximum possible airpower extraction based on the simple momentum theory) and is comparable to the aerodynamic efficiency of a helicopter rotor in producing thrust for a given shaft power input.