Page 81 - Wind Energy Handbook

VORTEX CYLINDER MODEL OF THE ACTUATOR DISC 55

where (W 3 ˆ) is a vector product.

d
Q ¼ rW 3 ˆr sin ö t ¼ rˆrU 1 (1 a) (3:35)
dr

Equating the two expressions gives

ˆ
a9 ¼
4ðr Ù
2
Hence

2
U a(1 a) a(1 a)
a9 t ¼ 1 ¼
2
2
2
Ù R (1 þ a9 t ) º (1 þ a9 t )
so
a(1 a)
a9 t (1 þ a9 t ) ¼ (3:36)
º 2

Equation (3.36) is not quite the same as Equation (3.23) of Section 3.3.3 and it can be
shown that this is a result of ignoring the wake expansion.

3.4.5 Torque and power

The torque on an annulus of radius r and radial width är is

3
d r4ðrU a(1 a) 2
1
Qär ¼ rWˆr sin ö t är ¼ är
dr Ù(1 þ a9 t )
1 3 2
d 2 rU 2ðr4a(1 a)
1
Q ¼ (3:37)
dr Ù(1 þ a9 t )
Power

1 3 2
d 2 rU 2ðr4a(1 a)
1
P ¼ Ù Q ¼ (3:38)
dr dr (1 þ a9 t )
1 3 2 2
rU ðR 4a(1 a)
1
P ¼ 2 (3:39)
(1 þ a9 t )