Starting and Low Wind Speed Performance - Small Wind Turbines: Analysis, Design, and Application - page 111

have been inconsistently formulated in blade element theories. The situation is
complicated by the implication from Fig. 4.6 b that what appear to be aspect ratio
effects may, in fact, be due to Re differences between the 2.5 and 0.97 m blades.
More fundamental experimental information is needed. In the meantime Eq. 6.16
will be used in the expectation that the dual optimisation to be pursued in the next
chapter requires only the relative starting performance of the competing blades.

6.4 Estimating the Rotor Inertia

Equation 6.20 shows that the rotor inertia is needed to calculate starting. For
simplicity only blades with a uniform density, q b , are considered and the blade
attachment is ignored. The moment of inertia, J, of a rotor of N blades about the
x-axis (the turbine axis) is given by
J ¼ Nq b Z

dxdydz

y 2 þ z 2 ð 6 : 21 Þ

where the z-axis is in the radial direction and y is in the direction of rotation of the
blade. The integration is over the blade volume. (The normalisation by blade
radius is delayed until Eq. 6.25 .) The blade-fixed Cartesian co-ordinate system
used in ( 6.21 ) has its origin on the axis of rotation with y in the direction of the
wind, and z along the blade. Because BET does not constrain the position of the
blade elements along their chord, the determination of J at the design stage always
carries some uncertainty. It is assumed that the centroids of the elements lie along
the z-axis, and the z-position of a blade element is its radius, r. Equation 6.21 can
be rewritten as

Þ¼ Z r 2 dxdydr þ Z y 2 dxdydr ¼ J 1 þ J 2 J = Nq b ð ð 6 : 22 Þ

where J 1 should dominate as c/r 1 for most wind turbines. The first integral is
just

J 1 ¼ A Z c ð 2 dr ð 6 : 23 Þ

where A is the dimensionless result of dividing the area of the aerofoil section by
the square of the chord. As determined by trapezoidal integration, the values of
A for several aerofoils are listed in Table 6.2 .
An approximate expression for J 2 can be found by assuming each blade section is
rectangular with thickness t 0 such that A = t 0 /c, and the centroid is along the z-axis.
Now change to x 0 ,y 0 co-ordinates such that the latter is along the chord line. Thus

J 2 Z dr Z

c = 2 t 0 = 2

R

dy 0 Z

dx 0

cos 2 h p y 0 2 þ sin 2 h p x 0 2 þ 2 cos h p sin h p x 0 y 0

0 c = 2 t 0 = 2

ð 6 : 24a Þ