3.4.11 Conclusions
Despite the exclusion of wake expansion, the vortex theory produces results in agreement with the momentum theory and enlightens understanding of the flow through an energy extracting actuator disc. However, the infinite radial velocity predicted at the outer edge of the disc is further evidence that the actuator disc is physically unrealisable.
3.5 Rotor blade theory (blade‐element/momentum theory)
3.5.1 Introduction
The aerodynamic lift (and drag) forces on the spanwise elements of radius r and length δr of the several blades of a wind turbine rotor are responsible for the rate of change of axial and angular momentum of all of the air that passes through the annulus swept by the blade elements. In addition, the force on the blade elements caused by the drop in pressure associated with the rotational velocity in the wake must also be provided by the aerodynamic lift and drag. As there is no rotation of the flow approaching the rotor, the reduced pressure on the downwind side of the rotor caused by wake rotation appears as a step pressure drop just as is that which causes the change in axial momentum. Because the wake is still rotating in the far wake, the pressure reduction associated with the rotation is still present and so does not contribute to the axial momentum change.
3.5.2 Blade element theory
It is assumed that the forces on a blade element can be calculated by means of two‐dimensional (2‐D) aerofoil characteristics using an angle of attack determined from the incident resultant velocity in the cross‐sectional plane of the element. Applying the independence principle (see Appendix A3.1), the velocity component in the spanwise direction is ignored. Three‐dimensional (3‐D) effects are also ignored.
The velocity components at a radial position on the blade expressed in terms of the wind speed, the flow factors, and the rotational speed of the rotor together with the blade pitch angle will determine the angle of attack. Having information about how the aerofoil characteristic coefficients Cl and Cd vary with the angle of attack, the forces on the blades for given values of a and a′ can be determined.
Consider a turbine with B blades of tip radius R each with chord c and set pitch angle β measured between the aerofoil chord‐line and the plane of the disc. (Note that in referencing the pitch to the blade chord line the zero incidence lift coefficient must be included). Both the chord length, section profile (thickness and camber), and the pitch angle may vary along the blade span. Let the blades be rotating at angular velocity Ω and let the wind speed be U∞. The tangential velocity experienced by the blade element shown in Figure 3.13 is (1 + a′)rΩ. The actuator disc is infinitesimally thin; the change in tangential velocity is abrupt, but it is only the component induced by the root vortex that contributes. This varies smoothly across the region of the actuator disc (Figure 3.10). The bound velocity induced by the vorticity on the disc does not contribute.
Figure 3.14 shows all the velocities and forces relative to the blade chord line at radius r.
From Figure 3.14, the resultant relative velocity at the blade is
(3.43)
Figure 3.13 A blade element sweeps out an annular ring.
Figure 3.14 Blade element velocities and forces: (a) velocities, and (b) forces.
that acts at an angle ϕ to the plane of rotation such that
The angle of attack α is then given by
The basic assumption of the blade element theory is that the aerodynamic lift and drag forces acting upon an element are the same as those acting on an isolated, identical element at the same angle of attack in 2‐D flow.
The lift force on a spanwise length δr of each blade, normal to the direction of W, is therefore
and the drag force parallel to W is
The axial thrust on an annular ring of the actuator disc is