Blade Element Calculations - Small Wind Turbines: Analysis, Design, and Application - page 98

1 in twist. If the blade twist is reduced by this amount, the C P increases slightly to
0.465 at k = 10.
Of course the combination of Eqs. 5.12 a and 5.13 a will not be adequate at small
r, but Fig. 5.5 shows that the hub region contributes little to the overall power
production. As explained by Burton et al. [ 13 ] this fact is exploited by the
designers of large blades to improve their manufacturability. Most have linearly
tapering chord which approximates the 1/r optimal shape for the outer blade but is
more manageable near the hub. However, Chap. 6 shows that the hub region is
crucial for good starting and low wind speed performance, implying that linear
taper should not be used for small blades. Chapter 7 shows that the hub region of
small blades can be designed largely for starting in the context of optimising both
starting and power extraction.
The discussion of optimum efficiency has not considered the second order
effects of tip losses. These can be included but the details are complex, [ 3 ], and so
will not be addressed here. The numerical optimisation described in Chap. 7
includes tip losses.
5.5.1 Further Reading
The NREL web site has comprehensive documentation on their wind turbine
activities: http://www.nrel.gov/wind/ . The so-called ''Unsteady Aerodynamics
Experiment'', during which the photo in Fig. 5.12 was taken, is described at:
http://wind2.nrel.gov/amestest/
5.5.2 Exercises
1. Derive Eq. 5.7 .
2. Show that Eq. 5.10 can be derived by combining Eqs. 2.9 , 2.15 , and 2.16 .
3. A common question from interested non-specialists in wind turbine theory, is
''Well, if you can get x kW power from a turbine with two blades, why cannot
you get 1.5x kW by adding another blade?'' How would you answer that
question?
4. If the contribution to the thrust from each blade element, DC T , is linear in r,
show that the radial point of action of the thrust occurs at r/R = 2/3. Comment
on the significance of this result for the root bending moment of the blade.
5. If the DC p is linear in r, what is the point of action of the torque?
6. For a turbine operating at the Betz-Joukowsky limit, determine the approxi-
mate magnitude of W ? as a function of radius, r.
7. The tip vortex in Fig. 5.12 does not appear to increase in diameter. What does
that say about the operating condition at which the photo was taken?