Michael Graham - Wind Energy Handbook

Hence

(3.71)

Introducing the optimum conditions of Eq. (3.67),

(3.72)

The parameter λμ is the local speed ratio λr and is equal to the tip speed ratio where μ = 1.

If, for a given design, C lis held constant, then Figure 3.17shows the blade plan‐form for increasing tip speed ratio. A high design tip speed ratio would require a long, slender blade (high aspect ratio) whilst a low design tip speed ratio would need a short, fat blade. The design tip speed ratio is that at which optimum performance is achieved. Operating a rotor at other than the design tip speed ratio gives a less than optimum performance even in ideal drag‐free conditions.

In off‐optimum operation, the axial inflow factor is not uniformly equal to 1/3; in fact, it is not uniform at all.

The local inflow angle ϕ at each blade station also varies along the blade span, as shown in Eq. (3.73)and Figure 3.18:

(3.73)

Figure 3.17 Variation of blade geometry parameter with local speed ratio.

Figure 3.18 Variation of inflow angle with local speed ratio.

which, for optimum operation, is

(3.74)

Close to the blade root the inflow angle is large, which could cause the blade to stall in that region. If the lift coefficient is to be held constant such that drag is 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 ⋅ 10 5 ,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 C land 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 3.19aand 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.

The blade design of Figure 3.19is 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 3.20has 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

(3.75)

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 C lfor 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 −4 o), the lift coefficient is given approximately by

where α is in degrees and 0.1 is a good approximation to the gradient of the C lvs α ofor 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 C lblade.

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.