Wind Energy Handbook. Michael Barton Graham
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Название: Wind Energy Handbook
Автор: Michael Barton Graham
Издательство: John Wiley & Sons Limited
Жанр: Физика
isbn: 9781119451167
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Figure 3.19 Optimum blade design for three blades and λ = 6: (a) blade twist distribution, and (b) blade plan‐form.
Figure 3.20 Uniform taper blade design for optimal operation.
The expression for this chord distribution approximation to the optimum plan‐form (Figure 3.20) is
The 0.8 in Eq. (3.75) refers to the 80% point, approximating in this case the solid line between target points 0.7 and 0.9 by the tangent at 0.8, which is very close to it.
Equation (3.75) can then be combined with Eq. (3.72) to give the modified spanwise variation of Cl for optimal operation of the uniformly tapered blade (Figure 3.21):
Close to the blade root the lift coefficient approaches the stalled condition and drag is high, but the penalty is small because the adverse torque is small in that region.
Assuming that stall does not occur, for the aerofoil in question, which has a 4% camber (this approximates to a zero lift angle of attack of −4o), the lift coefficient is given approximately by
where α is in degrees and 0.1 is a good approximation to the gradient of the Cl vs αo for most aerofoils, so
The blade twist distribution can now be determined from Eqs. (3.74) and (3.45) and is shown in Figure 3.22.
Figure 3.21 Spanwise distribution of the lift coefficient required for the linear taper blade.
Figure 3.22 Spanwise distribution of the twist in degrees required for the linear taper blade.
The twist angle close to the root is still high but lower than for the constant Cl blade.
3.8.4 Effects of drag on optimal blade design
If, despite the views of Wilson et al. (1974) – see Section 3.5.3, the effects of drag are included in the determination of the flow induction factors, we must return to Eq. (3.48) and follow the same procedure as described for the drag‐free case.
In the current context, the effects of drag are dependent upon the magnitude of the lift/drag ratio, which, in turn, depends on the aerofoil profile but largely on Reynolds number and on the surface roughness of the blade. A high value of lift/drag ratio would be about 150, whereas a low value would be about 40.
Unfortunately, with the inclusion of drag, the algebra of the analysis is complex. Polynomial equations have to be solved for both a and a′. The details of the analysis are left for the reader to discover.
In the presence of drag, the axial flow induction factor for optimal operation is not uniform over the disc because it is in the hypothetical drag‐free situation. However, the departure of the axial flow distribution from uniformity is not great, even when the lift/drag ratio is low, provided the flow around a blade remains attached.
The radial variation of the axial and tangential flow induction factors is shown in Figure 3.23 for zero drag and for a lift/drag ratio of 40. The tangential flow induction factor is lower in the presence of drag than without because the blade drags the fluid around in the direction of rotation, opposing the general rotational reaction to the shaft torque.
From the torque/angular momentum Eq. (3.52), the blade geometry parameter becomes
(3.76)
Figure 3.24 compares the blade geometry parameter distributions for zero drag and a lift/drag ratio of 40, and, as СКАЧАТЬ