Michael Graham - Wind Energy Handbook

The power coefficient for an optimised rotor, operating at the design tip speed ratio, without drag and tip‐losses is equal to the Lanchester–Betz limit 0.593, but with tip‐loss there is obviously a reduced optimum power coefficient. Equation (3.20)determines the power coefficient; see Figure 3.39:

(3.93)

for which a ′and a are obtained from Eqs. (3.89)and (3.90). This differs from the result given by Eq. (3.20)by the term a′ / f in the denominator, which is very small except close to the root at low tip speed ratio.

Figure 3.39 Spanwise variation of power extraction in the presence of tip‐loss for three blades with uniform circulation and of optimised design for a tip speed ratio of 6.

Figure 3.40 The variation of maximum C Pwith design λ for various lift/drag ratios and including tip‐losses for a three bladed rotor.

The maximum power coefficient that can be achieved in the presence of both drag and tip‐loss is significantly less than the Betz limit at all tip speed ratios. As is shown in Figure 3.40, drag reduces the power coefficient at high tip speed ratios, but the effect of tip‐loss is most significant at low tip speed ratios because the pitch of the helicoidal vortex sheets is larger.

An alternative formulation for incorporating tip‐loss effect is to assume that tip‐loss effect may be applied to correct the blade section forces directly ensuring that they fall to zero at the blade root and tip. Thus the tip‐loss only appears as a factor f multiplying the right hand sides of Eqs. (3.48)and (3.49), which predict δ T and δ Q in terms of the momentum losses in the wake; see, for example, Wilson et al. (1974) and Jamieson (2018). This formulation if used simplifies the foregoing analysis of power coefficient because f only appears as a factor multiplying the expression for C P.

But in the following analysis, we will continue to follow the method of applying the tip‐loss factor derived in Section 3.9.2.

The BEM Eqs. (3.54a)and (3.55)are used to determine the flow induction factors for non‐optimal operation. With tip‐loss included the BEM equations have to be modified. The necessary modification depends upon whether the azimuthally averaged values of the flow factors are to be the determined or the maximum (local to a blade element) values. If the former alternative is chosen, then, in the momentum terms, the averaged flow factors a and a ′remain unmodified, but in the blade element terms, the flow factors must appear as the average values divided by the tip‐loss factor. Choosing to determine the maximum values of the flow factors, i.e. a band a b ′, means that they are not modified in the blade element terms but are multiplied by the tip‐loss factor in the momentum terms. The former choice allows the simpler modification of Eqs. (3.54a)and (3.55):

(3.54c)

(3.55a)

where the flow factor values determined are the averaged values a and a ′.

There remains the problem of the breakdown of the momentum theory when wake mixing occurs. The helicoidal vortex structure may not exist, and so Prandtl's approximation is less physically appropriate. Nevertheless, due to the finite length of the blades and radius of the vortex wake, the application of a tip‐loss factor is necessary. Prandtl's approximation is the only practical method available and so is commonly used. In view of the manner in which the experimental results of Figure 3.16were gathered, it is the average value of a that should determine at which stage the momentum theory breaks down.

The flow approaching the rotor is expanding because it is slowing down and so is not axial, that is, it is not parallel to the rotation axis or the undisturbed flow direction. Consequently, there is a radial flow velocity component at the upwind side of the rotor that arises because there is a radial pressure gradient with lower pressure in the tip region than in the inner region. The change of radial momentum at a point on the rotor disc is approximately balanced by the equal and opposite radial momentum at the diametrically opposite point. The magnitude of the radial velocity increases with radius, and so its effects will be greatest at the tip region. The kinetic energy associated with the radial flow does not directly affect the energy capture because it does not influence the aerodynamic force on the blade.

At the blade tip the blade chord length becomes zero (usually but not always in a gradual fashion) and so must also the axial force exerted on the air flow beyond the blade tip that bypasses the rotor. The idealised actuator disc theory predicts a logarithmically singular radial velocity at the tip. This is not possible, and the pressure difference across the disc must fall continuously radially over a small tip region to zero at the tip.

Both a and ψ , which is the angle of the resultant flow to the axial direction at the rotor plane, will vary radially and will change according to how the circulation on the disc varies radially. Disc circulation, or the bound vorticity on the disc, must also rise and fall from blade root to blade tip, as shown in Figure 3.41.

Figure 3.41 The variation of circulation along the length of a blade.

Using just the momentum theory, it is not possible to determine the manner of the variation of a and ψ , but it is clear that the integration with respect to radius r of Eq. (3.93)with (3.89)would result in a value for the optimised power coefficient that would be less than the Betz limit.

Throughout the BEM analysis, it is assumed implicitly that the swirl component generated in the wake of the rotor is sufficiently small that its influence on the pressure field may be ignored and specifically that the pressure far downstream in the wake where the momentum balance is calculated is uniform and ambient. However, as discussed earlier at the end of Section 3.3.2, under lower tip speed ratio conditions, typically within streamtubes that pass close to the blade roots so that the local speed ratio λ = Ω r/U ∞< 2, this is increasingly untrue. However, there is not as yet any fully agreed analysis for this effect except that it may offer the possibility of achieving local power coefficients in excess of the Betz limit. In practical terms the possible increase in total rotor power is unlikely to be very significant.