Blade Element Calculations - Small Wind Turbines: Analysis, Design, and Application - page 90

Fig. 5.6 Predicted
distribution of blade thrust for
the measurements of
Anderson et al. [ 5 ]

0.12

0.1

λ = 8
λ = 10
λ = 12

0.08

0.06

0.04

0.02

0

0 0.2 0.4 0.6 0.8 1

radius, r/R

Having shown the reasonable accuracy of the blade element calculations for the
rotor properties, it is necessary to consider the radial dependence. At the maximum
efficiency point, k = 10, a and hence U 1 is roughly constant with a value not much
greater than the Betz-Joukowsky value of 2/3, Fig. 5.3 . Note that the figure shows
U 1 = 1 - a, not the value of the wind speed at the blades which would be 1 - a b .
The decrease in U 1 near the tips is a consequence of the tip loss correction; without
this correction the distribution for k = 10 would be almost flat for r/R [ 0.4.
Measurements of the velocity immediately behind the blades by Anderson et al.
[ 5 ] for k = 10 (only) show considerable scatter, but are in general agreement with
the calculated values of U 1 . Similarly, C is approximately constant over the blade,
k = 10 suggesting that the region where the bound vorticity decreases and the
trailing vorticity is formed, is narrow. C is distributed quite differently on an
efficient wind turbine than on a helicopter rotor in hover, but there are strong
similarities with the behaviour of propeller blades. Finally, there is remarkably
little variation in the Reynolds number along the blade for all three operating
conditions. This is partly due to the reduction in chord as radius and effective
velocity increase.
Figure 5.4 shows C is most uniform along the blade when k is at its optimum
value, and tends to decrease with increasing k. The decrease in circulation near the
tips is due to the tip loss correction. We will see later that, without this correction,
Ck is approximately constant for a wind turbine.
Figures 5.5 and 5.6 show that most of the power and thrust is produced near the
tip, simply because of the rapidly increasing contribution of Xr to U T . It follows
that the blade design near the hub is not critical for power extraction, so that
modifications to accommodate the attachment to the hub and structural consid-
erations, such as increasing the thickness of the blade to withstand the centrifugal
loads, can be made without compromising power performance. We will see in
Chap. 6 that starting performance introduces important aerodynamic consider-
ations to the hub region which may work against these modifications.
Comparison of the power_calc.m output for blade element a with Fig. 5.1
suggests that, as expected, optimum performance occurs when l/d is maximised.