Page 74 - Wind Energy Handbook

48 AERODYNAMICS OF HORIZONTAL-AXIS WIND TURBINES

2a'Ωr

p D -_ 1 _ 2 ρ 2
ρ(2a'Ωr)
U (1-a)
∞
p D + a'Ωr

U (1-a)
U (1-a) ∞
∞

Ωr
Rotor motion

Figure 3.5 Tangential Velocity Grows Across the Disc Thickness

imparting the tangential velocity component to the air whereas the axial force acting
on the ring will be responsible for the reduction in axial velocity. The whole disc
comprises a multiplicity of annular rings and each ring is assumed to act indepen-
dently in imparting momentum only to the air which actually passes through the
ring.
The torque on the ring will be equal to the rate of change of angular momentum
of the air passing through the ring. Thus,

torque ¼ rate of change of angular momentum

¼ mass flow rate 3 change of tangential velocity 3 radius

äQ ¼ räA d U 1 (1 a)2Ùa9r 2 (3:17)

where äA d is taken as being the area of an annular ring.
The driving torque on the rotor shaft is also äQ and so the increment of rotor
shaft power output is
äP ¼ äQÙ

The total power extracted from the wind by slowing it down is therefore deter-
mined by the rate of change of axial momentum given by Equation (3.10) in Section
3.2.2