Page 90 - Wind Energy Handbook

64 AERODYNAMICS OF HORIZONTAL-AXIS WIND TURBINES

on the flow through a given annular ring is that due to the pressure drop across the
disc. However, in practice, it appears that the error involved in relaxing the above
constraint is small for tip speed ratios greater than 3.

3.5.4 Determination of rotor torque and power

The calculation of torque and power developed by a rotor requires a knowledge of
the flow induction factors, which are obtained by solving Equations (3.51) and
(3.52). The solution is usually carried out iteratively because the two-dimensional
aerofoil characteristics are non-linear functions of the angle of attack.
To determine the complete performance characteristic of a rotor, i.e., the manner
in which the power coefficient varies over a wide range of tip speed ratio, requires
the iterative solution. The iterative procedure is to assume a and a9 to be zero
initially, determining ö, C p and C d on that basis, and then to calculate new values
of the flow factors using Equations (3.51) and (3.52). The iteration is repeated until
convergence is achieved.
From Equation (3.50) the torque developed by the blade elements of span-wise
length är is
2
äQ ¼ 4ðrU 1 (Ùr)a9(1 a)r är
If drag, or part of the drag, has been excluded from the determination of the flow
induction factors then its effect must be introduced when the torque caused by drag
is calculated from blade element forces, see Equation (3.49),

1 2
2
äQ ¼ 4ðrU 1 (Ùr)a9(1 a)r är rW NcC d cos(ö)rär
2
The complete rotor, therefore, develops a total torque Q:
2 2 3 3
ð R N c
1 4 4 W 5 5
2
3
Q ¼ rU ðR º ì 2 8a9(1 a)ì R C d (1 þ a9) dì (3:54)
1
2 0 U 1 ð
The power developed by the rotor is P ¼ QÙ. The power coefficient is
P
C P ¼
1 3 2
rU ðR
2 1
Solving the BEM Equations (3.51) and (3.52) for a given, suitable blade geometrical
and aerodynamic design yields a series of values for the power and torque coeffi-
cients which are functions of the tip speed ratio. A typical performance curve for a
modern, high-speed wind turbine is shown in Figure 3.15.
The maximum power coefficient occurs at a tip speed ratio for which the axial
flow induction factor a, which in general varies with radius, approximates most