Page 88 - Wind Energy Handbook

62 AERODYNAMICS OF HORIZONTAL-AXIS WIND TURBINES

The rate of change of axial momentum of the air passing through the swept annulus
is

2
rU 1 (1 a)2ðrär2aU 1 ¼ 4ðrU a(1 a)rär (3:47)
1
The drop in wake pressure caused by wake rotation is equal to the increase in
dynamic head, which is

1 2
r(2a9Ùr)
2
Therefore the additional axial force on the annulus is

1 2
r(2a9Ùr) 2ðrär
2
Thus

1 2 2 2
rW Nc(C l cos ö þ C d sin ö)är ¼ 4ðr[U a(1 a) þ (a9Ùr) ]rär
1
2
Simplifying,

W 2 c 2
N (C l cos ö þ C d sin ö) ¼ 8ð(a(1 a) þ (a9ºì) )ì (3:48)
U 2 1 R
The element of axial rotor torque caused by aerodynamic forces on the blade
elements is

1
2
(äL sin ö ä D cos ö)r ¼ rW Nc(C l sin ö C d cos ö)rär (3:49)
2
The rate of change of angular momentum of the air passing through the annulus is

2
rU 1 (1 a)Ùr2a9r2ðrär ¼ 4ðrU 1 (Ùr)a9(1 a)r är
Equating the two moments

1 2 2
rW Nc(C l sin ö C d cos ö)rär ¼ 4ðrU 1 (Ùr)a9(1 a)r är (3:50)
2
Simplifying,

W 2 c 2
N (C l sin ö C d cos ö) ¼ 8ðºì a9(1 a) (3:50a)
U 2 1 R
where the parameter ì ¼ r=R.