Page 86 - Wind Energy Handbook

60 AERODYNAMICS OF HORIZONTAL-AXIS WIND TURBINES

3.5.2 Blade element theory

It is assumed that the forces on a blade element can be calculated by means of two-
dimensional aerofoil characteristics using an angle of attack determined from the
incident resultant velocity in the cross-sectional plane of the element; the velocity
component in the span-wise direction is ignored. Three-dimensional effects are also
ignored.
The velocity components at a radial position on the blade expressed in terms of
the wind speed, the flow factors and the rotational speed of the rotor will determine
the angle of attack. Having information about how the aerofoil characteristic coeffi-
cients C d and C d vary with the angle of attack the forces on the blades for given
values of a and a9 can be determined.
Consider a turbine with N blades of tip radius R each with chord c and set pitch
angle â measured between the aerofoil zero lift line and the plane of the disc. Both the
chord length and the pitch angle may vary along the blade span. Let the blades be
rotating at angular velocity Ù and let the wind speed be U 1 . The tangential velocity
Ùr of the blade element shown in Figure 3.13 combined with the tangential velocity
of the wake a9Ùr means that the net tangential flow velocity experienced by the
blade element is (1 þ a9)Ùr. Figure 3.14 shows all the velocities and forces relative
to the blade chord line at radius r.
From Figure 3.14 the resultant relative velocity at the blade is
q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
W ¼ U (1 a) þ Ù r (1 þ a9) 2 (3:41)
2
2 2
1
which acts at an angle ö to the plane of rotation, such that

Ωra'
δr
r

U (1-a)

Ωr
r

Ω

Figure 3.13 A Blade Element Sweeps Out an Annular Ring