Page 117 - Wind Energy Handbook

THE EFFECTS OF A DISCRETE NUMBER OF BLADES 91

Again, the effects of tip-loss are confined to the blade tip.
The power coefficient for an optimized rotor, operating at the design tip speed
ratio, without drag and tip-losses is equal to the Betz limit 0.593 but with tip-loss
there is obviously a reduced optimum power coefficient. Equation (3.20) determines
the power coefficient distribution along the blade, see Figure 3.39.
The power coefficient
ð
P 2 1 3
C P ¼ ¼ 8º a9(1 a)ì dì (3:87)
1 3 2 0
rU ðR
2 1
for which a9 and a are obtained from Equations (3.83) and (3.84).
The maximum power coefficient that can be achieved in the presence of both
drag and tip-loss is significantly less than the Betz limit at all tip speed ratios. As is
shown in Figure 3.42 drag reduces the power coefficient at high tip speed ratios but
the effect of tip-loss is most significant at low tip speed ratios because the separation
of the helicoidal vortex sheets is large.

3.8.6 Incorporation of tip-loss for non-optimal operation

The blade element–momentum Equations (3.51), (3.51a) and (3.52) are used to
determine the flow induction factors for non-optimal operation. With tip-loss
included the BEM equations have to be modified. The necessary modification

1

R dCp/dr 0.5

0
0 0.2 0.4 0.6 0.8 1
r/R
With tip-loss uniform circulation
With tip-loss optimized
No tip-loss
Figure 3.39 Span-wise Variation of Power Extraction in the Presence of Tip-loss for Three
Blades with Uniform Circulation and of Optimized Design for a Tip Speed Ratio of 6