298 7 Wind Power Generation The above equation reveals that wind capacity depends on the wind speed ’s third power (PWi ~ vWi3); 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 (S1) and far behind the rotor (S2) (Fig. 7.1 and 7.2), wind velocity vWi is supposed to decrease steadily according to Equation (7.2). On the other hand wind pressure must increase accordingly. At the actual rotor plane SRot the theoretical rotor capacity is almost abruptly extracted from the wind flow. However, at this point wind velocity cannot change discontinuously; hence, power extraction requires a sudden pressure change ǻpWi (Fig. 7.2). Regardless of these circumstances the wind pressure pWi,0 far before and behind the wind turbine must be taken into account. Wind pressure is subject to weather changes. vWi
x pWi
'pWi
pWi,0
x
S1
SRot
S2
Fig. 7.2 Pressure and speed curve long before the wind energy converter, at the wind energy converter level (rotor plane), and far behind the wind energy converter (for an explanation 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 (FWi,WEC) must be equal to the force given by the wind energy converter, which slows down the wind flow (FWi,slow) (Equation (7.5)).
FWi ,WEC
FWi ,slow
m Wi (vWi ,1 vWi , 2 )
(7.5)
Within the rotor plain SRot, the wind force FWi,WEC together with the wind velocity at rotor level vWi,Rot must be equal to the theoretical rotor power PRot,th or the power extracted by the rotor PWi,ext (Equation (7.6)); whereby P = F v and power P, force F and velocity v.
PWi,ext
PRot,th
FWi,WEC vWi,Rot
m Wi (vWi,1 vWi, 2 ) vWi,Rot
(7.6)
By equating the relations of (7.3) and (7.6), wind velocity within the rotor level is derived as an arithmetical mean from vWi,1 and vWi,2 (Froude Rankin ’s theorem;