Page 76 - Wind Energy Handbook

50 AERODYNAMICS OF HORIZONTAL-AXIS WIND TURBINES

From Equation (3.18)

d º 2 r
a ¼
da9 1 2a
giving

2
a9º ¼ (1 a)(1 2a) (3:22)
r
The combination of Equations (3.18) and (3.21) gives the required values of a and a9
which maximize the incremental power coefficient:

1 a(1 a)
a ¼ and a9 ¼ (3:23)
3 º ì
2 2
The axial flow induction for maximum power extraction is the same as for the non-
1
rotating wake case, that is, a ¼ and is uniform over the entire disc. On the other
3
hand a9 varies with radial position.
From Equation (3.20) the maximum power is
ð 1
2 3
C P ¼ 8(1 a)a9º ì dì
0
Substituting for the expressions in Equations (3.23)
ð
1 a(1 a) 16
2 3
2
C P ¼ 8(1 a) 2 2 º ì dì ¼ 4a(1 a) ¼ (3:24)
0 º ì 27
Which is precisely the same as for the non-rotating wake case.

3.3.4 Wake structure

The angular momentum imparted to the wake increases the kinetic energy in the
wake but this energy is balanced by a loss of static pressure:
1 2
˜ p r ¼ r(2Ùa9r) (3:25)
2

Substituting the expression for a9 given by Equations (3.23)

1 2 a(1 a) 2
˜p r ¼ rU 1 2 (3:26)
2 ºì
The tangential velocity increases with decreasing radius (Equation (3.23)) and so
the pressure decreases creating a radial pressure gradient. The radial pressure
gradient balances the centrifugal force on the rotating fluid. The pressure drop