Introduction to Modern Wind Turbines - Wind Turbine Technology: Fundamental Concepts of Wind Turbine Engineering - page 73

The integrations in Equations (2-12) are normally performed numerically, applying
Simpson's Rule. The magnitude of the rotor shaft power can then be calculated from shaft
torque and speed, as follows:

P R = Q W

(2-13)

where
P R = rotor shaft power (N-m/s)

Rotor Power Coeficient

The conventional measure of the aerodynamic performance of a wind turbine rotor (re-
gardless of coniguration) is its rotor power coeficient , which is the ratio of the rotor power
density (mechanical power at the turbine shaft per unit of swept area) to the wind power
density , or

C P , R = P R / A
p W

P R
0.5r U 3 A

(2-14)

=

where

C P,R = rotor power coeficient

P R = mechanical power at the turbine rotor shaft (W)

Axial Induction (Retardation) of the Wind

The axial induction factor, a , in Equation (2-11b) is the fractional decrease in wind speed
between the upwind free-stream and the plane of the rotor, as power is extracted at the rotor
from the wind. According to the axial momentum theory [Rankine 1865, W. Froude 1878, R.
Froude 1889], the wind speed downstream of the rotor is reduced further by the same amount
as upstream. On this basis, the total retardation of the free-stream wind, from upstream of the
rotor to the far wake downstream, is 2 a . This would indicate that the maximum value of a
for a wind turbine is 0.5, when the wind speed in the far wake is zero. However, momentum
theory is considered to be invalid for induction factors larger than about 0.4 [Wilson 1994].
By applying the Rankine-Froude theory to the calculation of rotor power, it has been
shown that

C P = 4 a (1- a ) 2 , a £ 0.4

(2-15a)

from which

(2-15b)

C P , max = 0.593 at a = 1/3

Therefore, an axial induction factor of approximately 1/3 is a common goal for a wind
turbine operating at its design wind speed (see Figure 2-12).
As discussed in detail by Eggleston and Stoddard [1987], the axial induction factor can
be determined from the rotor thrust force, normalized as follows:

T
0.5 r U 3 A

C T = (2-16)