Page 96 - Wind Energy Handbook

70 AERODYNAMICS OF HORIZONTAL-AXIS WIND TURBINES

Equations (3.60a) and (3.62), together, give the flow induction factors for optimized
operation

1 a(1 a)
a :¼ and a9 ¼ (3:63)
2 2
3 º ì
which agree exactly with the momentum theory prediction (Equation 3.23) because
no losses, such as aerodynamic drag, have been included and the number of blades
is assumed to be large; every fluid particle which passes through the rotor disc
interacts with a blade resulting in a uniform axial velocity over the area of the disc.
To achieve the optimum conditions the blade design has to be specific and can be
determined from either of the fundamental Equations (3.48) and (3.50). Choosing
Equation (3.50), because it is the simpler, and ignoring the drag, the torque de-
veloped in optimized operation is

U 3 1 2
2
äQ ¼ 4ðrU 1 (Ùr)a9(1 a)r är ¼ 4ðr a(1 a) rär
Ù
The component of the lift per unit span in the tangential direction is therefore

U 3 1 2
L sin ö ¼ 4ðr a(1 a)
Ù
By the Kutta–Joukowski theorem the lift per unit span is

L ¼ rWˆ

where ˆ is the sum of the individual blade circulations.
Consequently

U 3 1 2
rWˆ sin ö ¼ rˆU 1 (1 a) ¼ 4ðr a(1 a) (3:64)
Ù

so

U 2
ˆ ¼ 4ð 1 a(1 a) (3:65)
Ù
The circulation is therefore uniform along the blade span and this is a condition for
optimized operation.
To determine the blade geometry, that is, how should the chord size vary along
the blade and what pitch angle â distribution is necessary, we must return to
Equation (3.50a):

W 2 N c C l sin ö ¼ 8ðºì a9(1 a)
2
U 2 1 R