Starting and Low Wind Speed Performance - Small Wind Turbines: Analysis, Design, and Application - page 106

sin 2h p ð 1 a Þ WU ¼ 1

2 rU T

ð 6 : 4a Þ

When the blade is stationary, there is no need to distinguish between U 0 and U 1
so U is used for the wind speed. (This equivalence of wind speeds will also be
assumed for the starting analysis.) Similarly, W is the constant (with downstream
distance) circumferential velocity behind the blades. The axial momentum equa-
tion gives

a ð 1 a Þ U 2 ¼ 1

2 rU T cos 2 h p

ð 6 : 4b Þ

it follows from ( 6.4a ) and ( 6.4b ) that

a 1

2 r cos 2 h p

ð 6 : 5a Þ

and

a 0 ¼ a tan h p ð 6 : 5b Þ

Typically, r \ 0.2 for modern blades, and h P lies in the range 0-30, so that
a 0 \ a as well as a, a 0 1 so the assumption made about the wind speed after
( 6.4a ) is justified. Further, U t & (1 - a)U & U. Alternatively, U t /U & 1 and
a and a 0 can (and will) be neglected in the analysis of starting torque. By nor-
malising all lengths by R, and all velocities by U, Q s (in Nm), is determined by
Q s ¼ 1

2 NqU 2 R 3 I cp

ð 6 : 6 Þ

where the ''chord-pitch integral'' I cp , and its integrand, i cp , are defined by

I cp ¼ Z

1

i cp dr ¼ Z

1

cr sin 2h p dr

ð 6 : 7 Þ

r h r h

The lower limit on the integral, the ''hub'' radius r h , is assumed to be the
beginning of the aerodynamic section of the blade. It is straightforward to integrate
( 6.7 ) for an optimal power-producing blade because cr is constant by Eq. 5.12a :

16p
9Nk p C l ; max

cr ¼ ð 6 : 8 Þ

where k p is the design tip speed ratio for rated power. Note that the constancy of cr
means that the integral in ( 6.7 ) receives its largest contributions where h P is
largest, that is, in the hub region. Thus most starting torque is generated near the
hub.
With a max again defining the angle for maximum lift:drag, it follows from
( 5.13a )that