Page 41 - Fluid Power Engineering

Basics of W ind Energy and Power 19

Pressure difference was computed in Eq. (2-19), so power is:

1 1
0 2 2 2
P = (p − p )A r v r = ρA r v r v − v = ρA r v r (v 0 − v 2 )(v 0 + v 2 )
r r 0 2
2 2
(2-22)
Note that
1 2 2 1 2 2
P = ρA r v r v − v 2 = ˙ m v − v 2
0
0
2 2
which is change in kinetic energy applied to the flow of mass per unit
time through the rotor. That is, the work done by force due to pressure
difference is equal to the change in kinetic energy.
Combining Eqs. (2-20) and (2-22):
2
2
P = ρA r v (v 0 − v 2 ) = 2ρA r v (v 0 − v r ) (2-23)
r r
Maximum power is realized when:
∂ P 2
= 0 = 2v r v 0 − 3v r (2-24)
∂v r
2
v r = v 0 (2-25)
3
This implies:

1
v 2 = v 0 (2-26)
3
8
2
P = 2ρA r v (v 0 − v r ) = ρA r v 3
r 0
27
Max power extracted 1 3 16
= P/ ρA r v = = 0.593 = C p (2-27)
0
Power available 2 27
C p is called the power coefficient. A related concept is the thrust co-
efficient, C T , which is
F 8
= = C T (2-28)
1 ρA r v 2 9
2 0
C p is referred to as the Betz limit and states that the maximum power
anidealrotorcanextractfromwindis59.3%.SeeFig.2-8foragraphical
representation of the above function.
An ideal rotor of the type described above is called an “actuator
disk.” The actuator disk induces a reduction of the free-stream wind