Читать книгу Wind Energy Handbook - Michael Barton Graham - Страница 90

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 = Ωr/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.