Wind Energy Handbook. Michael Barton Graham
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Название: Wind Energy Handbook
Автор: Michael Barton Graham
Издательство: John Wiley & Sons Limited
Жанр: Физика
isbn: 9781119451167
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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
The torque on an annular ring is
(3.47)
where B is the number of blades.
3.5.3 The BEM theory
The basic assumption of the BEM theory is that the force of a blade element is solely responsible for the change of axial momentum of the air that passes through the annulus swept by the element. It is therefore to be assumed that there is no radial interaction between the flows through contiguous annuli: a condition that is, strictly, only true if pressure gradients acting axially on the curved streamlines can be neglected if the axial flow induction factor does not vary radially. In practice, the axial flow induction factor is seldom uniform, but experimental examination of flow through propeller discs by Lock (1924) shows that the assumption of radial independence is acceptable.
Equating the axial thrust on all blade elements, given by Eq. (3.46), with the rate of change of axial momentum of the air that passes through the annulus swept out by the elements, given by Eq. (3.9), with AD = 2πrδr
It should be noted here that the right hand side of Eq. (3.48) ignores the effect of the swirl velocity (2a'ΩR) on the axial momentum balance through generating a centrifugal pressure gradient in the far wake from the axis to the wake boundary. The resulting pressure reduction that generates an additional pressure drop across the disc was termed Δpd2 when considered previously in Eq. (3.22).
Equating the torque on the elements, given by СКАЧАТЬ