Page 75 - Wind Energy Handbook

ROTOR DISC THEORY 49

3
äP ¼ 2räA d U a(1 a) 2
1
Hence
3
2
2
2räA d U a(1 a) ¼ räA d U 1 (1 a)2Ù a9r 2
1
and
2
2 2
U a(1 a) ¼ Ù r a9
1
Ùr is the tangential velocity of the spinning annular ring and so º r ¼ Ùr=U 1 is
called the local speed ratio. At the edge of the disc r ¼ R and º ¼ ÙR=U 1 is known
at the tip speed ratio. Thus
2
a(1 a) ¼ º a9 (3:18)
r
The area of the ring is äA D ¼ 2ðrär therefore the incremental shaft power is, from
Equation (3.17),

1 3 2
äP ¼ dQÙ ¼ rU 2ðrär 4a9(1 a)º r
1
2
The term in brackets represents the power flux through the annulus, the term
outside the brackets, therefore, is the efficiency of the blade element in capturing
the power, or blade element efficiency:
ç r ¼ 4a9(1 a)º 2 (3:19)
r
In terms of power coefficient

2
2
3
d 4ðrU (1 a)a9º r 8(1 a)a9º r
1
r
r
C P ¼ ¼
dr 1 3 2 R 2
rU ðR
1
2
d 2 3
dì C P ¼ 8(1 a)a9º ì (3:20)
where ì ¼ r=R.
Knowing how a and a9 vary radially, Equation (3.20) can be integrated to
determine the overall power coefficient for the disc for a given tip speed ratio, º.
3.3.3 Maximum power

The values of a and a9 which will provide the maximum possible efficiency can be
determined by differentiating Equation (3.19) by either factor and putting the result
equal to zero. Whence

d 1 a
a ¼ (3:21)
da9 a9