Page 39 - Fluid Power Engineering

Basics of W ind Energy and Power 17

Equating (2-13) and (2-14):
0

F = A r p − p 2 = ρA r v r (v 0 − v 2 ) (2-15)
r r
Bernoulli’s law is next applied in two volumes: (a) Flow along stream-
lines from A 0 to the front face of the rotor; and (b) flow from the back
surface of rotor to A 2 .

1 2 0 1 2
p 0 + ρv = p + ρv r (2-16)
0
r
2 2
1 1
2
2
p + ρv = p 0 + ρv 2 (2-17)
r r 2
2 2
Subtracting (2-16) from (2-17):
0 2 1 2 2
p − p = ρ v − v (2-18)
r r 0 2
2
Equating (2-15) and (2-18):
F 0 2 ρ 2 2
= p − p = ρv r (v 0 − v 2 ) = v − v 2 (2-19)
r
r
0
A r 2
(v 0 + v 2 )
v r = (2-20)
2
Equation (2-20) implies that v r , the wind speed at the rotor, is aver-
age of the free-stream wind speed and the wind speed in the wake.
Note, the wind speed in wake (v 2 ) is where the pressure reaches free-
stream pressure (p 0 ). Equation (2-20) also implies that one-half the
wind speed loss occurs in front of the rotor and the other one-half
occurs downstream.
This is counterintuitive because in the rotor itself—between the
front and the back face of the rotor—there is no loss in wind speed,
and all the speed loss happens upstream and downstream. The power
is delivered (or work is done) by the force exerted because of pressure
difference across the rotor. Power is defined as force multiplied by
speed = Fv r .
Therefore, the mechanism that delivers power to the rotor is:
In the volume that is upstream of the rotor, some of the

free-stream kinetic energy is converted into static pressure
(Bernoulli’s equation). Kinetic energy is reduced and pre-
ssure is increased as wind approaches the face of rotor. Since
it is assumed that air is incompressible, that is, density is as-
sumed to be constant, the reduction in wind speed is accom-
panied by increase in the flow area.