Читать книгу Wind Energy Handbook - Michael Barton Graham - Страница 73

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 A_D = 2πrδr

(3.48)

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 Δp_d2 when considered previously in Eq. (3.22).

Equating the torque on the elements, given by Eq. (3.47), with the rate of change of angular momentum of the air passing through the swept annulus, given by Eq. (3.34),

(3.49)

If drag is eliminated from the above two equations, to make a comparison with the results of the vortex theory of Section 3.4, the flow angle ϕ can be determined:

However, from the velocity triangle at a blade element given by Eq. (3.44), the flow angle is also

Equating the two above expressions for tanϕ

(3.50a)

At the outer edge of the rotor μ = 1 and a^′ = a^′_t, so

(3.50b)

Equation (3.2) is consistent with the earlier Eqs. (3.32) and (3.33).

With drag included the thrust Eq. (3.48) can be reduced to

(3.51)

where the parameter _.

If the pressure drop term Δp_d2 is not ignored, the right hand side of Eq. (3.51) becomes 8πμ{a(1 − a) + (a^′λμ)²}. The additional term (a^′λμ)² is small and usually negligible except very close to the rotor axis or at low tip speed ratios.

The torque Eq. (3.49) simplifies to

(3.52)

It is convenient to put

(3.53a)

and

(3.53b)

Solving Eqs. (3.51) and (3.52) to obtain values for the flow induction factors a and a' using 2‐D aerofoil characteristics requires an iterative process for which the following equations, derived from (3.51), (3.52), and (3.53a and b), are convenient. The right hand sides are evaluated using existing values of the flow induction factors, yielding simple equations for the next iteration of the flow induction factors:

(3.54a)

(3.55)

If the additional pressure‐drop term Δp_d2 at the rotor due to wake rotation is included in the analysis, following from Eq. (3.48), Eq. (3.54a) becomes

(3.54b)

Blade solidity σ is defined as total blade area divided by the rotor disc area and is a primary parameter in determining rotor performance. Chord solidity σ_r is defined as the total blade chord length at a given radius divided by the circumferential length around the annulus at that radius:

(3.56)

It is argued by Wilson et al. (1974) that the drag coefficient should not be included in Eqs. (3.54a or b) and (3.55) because the velocity deficit caused by drag is confined to the narrow wake that flows from the trailing edge of the aerofoil. Furthermore, Wilson and Lissaman reason, the drag based velocity deficit is only a feature of the wake and does not contribute to the velocity deficit upstream of the rotor disc. The basis of the argument for excluding drag in the determination of the flow induction factors is that, for attached flow, drag is caused only by skin friction and does not affect the pressure drop across the rotor. Clearly, in stalled flow the drag is overwhelmingly caused by pressure. In attached flow – see, e.g. Young and Squire (1938) – the modification to the inviscid pressure distribution around an aerofoil caused by the boundary layer has a small effect both on lift and drag. The ratio of pressure drag to total drag at zero angle of attack is approximately the same as the thickness to chord ratio of the aerofoil and increases as the angle of attack increases.

One last point about the BEM theory: the theory neglects the axial components of the pressure forces at curved boundaries between streamtubes. It is more accurate if the blades have uniform circulation, i.e. if a is uniform. For non‐uniform circulation there is increased radial interaction and exchange of momentum as a result of normal pressure and viscous shear forces between flows through adjacent elemental annular streamtubes. However, in practice, it appears that the error involved is small for tip speed ratios greater than three.