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wake, are analogous to those behind an aircraft wing. The effects of helical trailing
vorticity on the flow over the blades is, however, much more difficult to calculate
than the effects of the nearly straight wing tip vortices.
One way to achieve this wake structure, and, therefore, to have a turbine whose
performance can approach the Betz-Joukowsky limit, is with constant bound
circulation of the blades. Combining Eq. 4.9 with Eqs. 3.5 and 3.11 , and ignoring
the drag in the latter, leads to
C 1
2 U T c l c
ð 4 : 12a Þ
For sufficiently high values of k, U T & kr, and
C 1
2 krc l c
ð 4 : 12b Þ
so that if C l remains roughly constant, c must decrease with radius, as shown in
Table 3.1 . This decrease is a feature of all well-designed wind turbine blades.
Furthermore, the higher the value of k, the smaller c needs to be. It is also a
feature of efficient blades that solidity decreases with increasing optimal k.
Equation 4.12 is important also because it links the aerofoil properties of the
blade elements with the strength of the trailing vortices. In the next chapter this
link is developed further to include U ? , the velocity in the far-wake. Since
U ? = 1/3 for the Betz-Joukowsky limit, considering the circulation leads to
specific equations for the chord and twist of a maximum-power-producing blade,
the first major goal of the blade designer. These equations also specify the value
of the optimum circulation.
4.6 Further Discussion on Reynolds Number,
High Incidence, and Aspect Ratio
The aerofoil data presented so far in this chapter have demonstrated graphically
the way that lift and drag can alter with angle of attack and Reynolds numbers.
Typical values of Re for small wind turbines were documented in Chap. 1 , and the
significance of Re for power maximisation is discussed in Chap. 7 . Here is a good
place, however, to continue the process of familiarisation with Re effects. In
addition, it is important to introduce the major differences between aerofoils and
three-dimensional lifting bodies in terms of the aspect ratio.
Figure 4.9 shows an idealised operating trajectory of the 500 W 1.94 m diameter
three-bladed turbine described in detail in Chaps. 6 and 9 in terms of a with Re for the
tip and the hub; note that Re is plotted logarithmically. The curves labeled ''starting''
were calculated for the blades starting from rest and a wind speed of 3 m/s. Starting
is completed when the turbine reaches its operational k of 7 and power production
commences. Then a does not change as long as the controller keeps k nearly constant
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