Page 364 - Wind Energy Handbook

338 CONCEPTUAL DESIGN OF HORIZONTAL-AXIS TURBINES

blade designs optimized for a number of different rotational speeds but the same
rated power produce substantially the same energy yield, so the choice of rotational
speed is based on machine cost rather than energy yield.
One of the key cost drivers is the rotor torque at rated power, as this is the main
determinant of the drive train cost. For a given tip radius and machine rating, the
rotor torque is inversely proportional to rotational speed, which argues for the
adoption of a high rotational speed. However increasing the rotational speed has
adverse effects on the rotor design, which are explored in the following sections.

6.4.1 Ideal relationship between rotational speed and solidity

Equation (3.67a) in Section 3.7.2 gives the chord distribution of a blade optimized to
give maximum power at a particular tip speed ratio in terms of the lift coefficient,
ignoring drag and tip loss:

8=9
ó r ºC l ¼ s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (3:67a)
2
2
1 2
2 2
(1 ) þ º ì 1 þ
2 2
3 9º ì
where º is the tip speed ratio, ó r is the solidity and ì ¼ r=R. Over the outboard half
of the blade, which produces the bulk of the power, the local speed ratio, ºì, will
normally be large enough to enable the denominator to be approximated as ºì,
giving:

Nc(ì) 8
ó r ºC l ¼ ºC l ¼ (6:7)
2ðR 9ºì

where N is the number of blades. After rearrangement, this gives

2
ÙR 16ðR 1
c(ì) ¼ (6:8)
U 1 9C l N ì

Hence it can be seen that, for a family of designs optimized for different rotational
speeds at the same wind speed, the blade chord at a particular radius is inversely
proportional to the square of the rotational speed, assuming that N and R are fixed
and the lift coefficient is maintained at a constant value by altering the local blade
pitch to maintain a constant angle of attack.
Note that Equation (6.8) does not apply if energy yield is optimized over the full
range of operating wind speeds for a pitch-regulated machine. In this case, it has
been demonstrated that the blade chord at a particular radius is approximately
inversely proportional to rotational speed rather than to the square of it (Jamieson
and Brown, 1992).