Wind Power Generation - Renewable Energy: Technology, Economics and Environment - page 297

The above equation reveals that wind capacity depends on the wind speed's
third power ( P Wi ~ v Wi 3 ); which is of major importance when it comes to selecting
a particular site and for wind power plant technology.
When continuously observing the wind flow long before the rotor ( S 1 ) and far
behind the rotor ( S 2 ) (Fig. 7.1 and 7.2), wind velocity v Wi is supposed to decrease
steadily according to Equation (7.2). On the other hand wind pressure must in-
crease accordingly. At the actual rotor plane S Rot the theoretical rotor capacity is
almost abruptly extracted from the wind flow. However, at this point wind veloc-
ity cannot change discontinuously; hence, power extraction requires a sudden
pressure change ∆p Wi (Fig. 7.2). Regardless of these circumstances the wind pres-
sure p Wi, 0 far before and behind the wind turbine must be taken into account. Wind
pressure is subject to weather changes.

v Wi

x

p Wi

∆ p

Wi

p Wi, 0

x

S 1 S Rot S 2

Fig. 7.2 Pressure and speed curve long before the wind energy converter, at the wind en-
ergy converter level (rotor plane), and far behind the wind energy converter (for an expla-
nation of symbols see text)

Since according to Newton´s law action equals reaction, the dynamic effect
which the wind has on the wind energy converter ( F Wi,WEC ) must be equal to the
force given by the wind energy converter, which slows down the wind flow
( F Wi,slow ) (Equation (7.5)).

&

F = F = m ( v − v ) (7.5)

Wi , WEC Wi , slow Wi Wi , Wi , 2

Within the rotor plain S Rot, the wind force F Wi,WEC together with the wind veloc-
ity at rotor level v Wi,Rot must be equal to the theoretical rotor power P Rot,th or the
power extracted by the rotor P Wi,ext (Equation (7.6)); whereby P = F v and power
P , force F and velocity v .

P = P = F v = m & ( v − v ) v

(7.6)

Wi , ext Rot , th Wi , WEC Wi , Rot Wi Wi , Wi , 2 Wi , Rot

By equating the relations of (7.3) and (7.6), wind velocity within the rotor level
is derived as an arithmetical mean from v Wi, 1 and v Wi, 2 (Froude Rankin's theorem;