Blade Design, Manufacture, and Testing - Small Wind Turbines: Analysis, Design, and Application - page 142

e ¼ d 1 d 2

j j

ð 7 : 4 Þ

M

where M is the sum of the blade masses and d is the product of the blade mass and
centre of mass. Equation 7.4 holds only if the blade masses lie in the same plane
and are exactly 180 apart. For N = 3, the formula is

h i

1
2M 2

Þ 2 þ d 1 d 3 Þ 2 þ d 2 d 3 Þ 2

e 2 ¼

ð d 1 d 2 ð ð ð 7 : 5 Þ

provided the blades are planar and exactly 120 apart.
There is very little publicly-available information on actual and allowable
values of e. Ramlau and Niebsch [ 16 ] describe a large rotor with
e & 0.003R suggesting the value of 0.005R is a reasonable upper limit. The
author's experience supports this.
If the blades are made in a batch larger than N the question then arises of how to
optimally form the rotors in such a way as to minimise the sum of e 2 over all
rotors. This problem is addressed by Hitz and Wood [ 17 ] to which the interested
reader is referred for more details. They found that when the standard deviation in
mass and centre of mass was 1% or less, optimal matching is nearly equivalent to
ordering the blades in terms of d. One fascinating aspect of the problem is that
( 7.4 ) and ( 7.5 ) are unchanged if the blade indices are permuted. However, this is
not the case for N C 4 and the optimisation of blade matching for large N is far
from simple, if not a major problem for mainstream turbines.

7.7.1 Exercises

1. The discussion at the start of Sect. 7.4 of the advantages and disadvantages of
three rather than two blades was brief and ignored several important consid-
erations. What are the missing considerations?
2. Interpret the optimal power trajectory in Fig. 7.1 in terms of the performance
curve such as Fig. 5.2 a by ignoring Reynolds number effects.
3. The generator used for the blade design example apparently shows a significant
decrease in efficiency as power decreases. How would this alter the trajectory in
Fig. 7.1 ?
4. The blade section manufacturing errors in Figs. 7.9 and 7.12 may have an effect
on power extraction but should have little effect on starting performance. Is this
statement correct?
5. Use the internet and other sources to find out what information you can on
locally-available timbers that could be used for small blades. What information
is required and can you find sufficient for each candidate species?
6. Derive Eqs. 7.4 and 7.5 .
7. For an N-bladed rotor in which the blades are co-planar and equi-spaced, show
the eccentricity of the centre of mass e is given by