Wind Energy Handbook. Michael Barton Graham
minimised everywhere, then the angle of attack α also needs to be uniform at the appropriate value. For a prescribed angle of attack variation, the design pitch angle β = ϕ − α of the blade must vary accordingly.
As an example, suppose that the blade aerofoil is National Advisory Committee for Aeronautics (NACA) 4412, popular for hand‐built wind turbines because the bottom (high‐pressure) side of the profile is almost flat, which facilitates manufacture. At a Reynolds number of about 5 ⋅ 105, the maximum lift/drag ratio occurs at a lift coefficient of about 0.7 and an angle of attack of about 3°. Assuming that both Cl and α are to be held constant along each blade and there are to be three blades operating at a tip speed ratio of 6, then the blade design in pitch (twist) and plan‐form variation are shown in Figures 3.19a and b, respectively. This blade solidity becomes very large at the root but can be accommodated to around r/R = 0.1 depending on the location of the blade axis.
3.8.3 A simple blade design
The blade design of Figure 3.19 is efficient but complex to build and therefore costly. Suppose the plan‐form was prescribed to have a uniform taper such that the outer part of the blade corresponds closely to Figure 3.19b. The straight line given by Eq. (3.75) and shown as the solid line in Figure 3.20 has been derived to minimise the departure from the true curve [Eq. (3.72))] in the outer region 0.7 < r/R < 0.9. This linear taper not only simplifies the plan‐form but removes a lot of material close to the root.
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