Page 97 - Wind Energy Handbook

BLADE GEOMETRY 71

substituting for sin ö gives
W c 2
N C l (1 a) ¼ 8ðºì a9(1 a) (3:66)
U 1 R
From which is derived

2
N c 4ºì a9
¼
2ð R W
C l
U 1
The only unknown on the right-hand side of the above equation is the value of the
lift coefficient C l and so it is common to include it on the left-side of the equation
with the chord solidity as a blade geometry parameter. The lift coefficient can be
chosen as that value which corresponds to the maximum lift/drag ratio C l =C d as
this will minimize drag losses; even though drag has been ignored in the determi-
nation of the optimum flow induction factors and blade geometry it cannot be
ignored in the calculation of torque and power. Blade geometry also depends upon
the tip speed ratio º so it is also included in the blade geometry parameter. Hence

2 2
N c 4º ì a9
ó r ºC l ¼ ºC l ¼ p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (3:67)
2ð R (1 a) þ (ºì(1 þ a9)) 2
2
Introducing the optimum conditions of Equations (3.63)

8
9
ó r ºC l ¼ s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (3:67a)

1 2 2 2
2 2
1 þº ì 1 þ
2 2
3 9(º ì )
The parameter ºì is called the local speed ratio and is equal to the tip speed ratio
where ì ¼ 1.
If, for a given design, C l is held constant then Figure 3.17 shows the blade plan-
form for increasing tip speed ratio. A high design tip speed ratio would require a
long, slender blade (high aspect ratio) whilst a low design tip speed ratio would
need a short, fat blade. The design tip speed ratio is that at which optimum
performance is achieved. Operating a rotor at other than the design tip speed ratio
gives a less than optimum performance even in ideal drag free conditions.
1
In off-optimum operation the axial inflow factor is not uniformly equal to ,in
3
fact it is not uniform at all.
The local inflow angle ö at each blade station also varies along the blade span as
shown in Equation (3.68) and Figure 3.18
1 a
tan ö ¼ (3:68)
ºì(1 þ a9)