Michael Graham - Wind Energy Handbook

that acts at an angle ϕ to the plane of rotation such that

(3.44)

The angle of attack α is then given by

(3.45)

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

(3.46)

The torque on an annular ring is

(3.47)

where B is the number of blades.

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= 2 πrδr

(3.48)

It should be noted here that the right hand side of Eq. (3.48)ignores the effect of the swirl velocity (2 a 'Ω R ) 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 Δ p d2when 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 tan ϕ

(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 Δ p d2is not ignored, the right hand side of Eq. (3.51)becomes 8 πμ { a (1 − a ) + ( a ′ λμ ) 2}. The additional term ( a ′ λμ ) 2is 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 ( 3.53aand 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 Δ p d2at the rotor due to wake rotation is included in the analysis, following from Eq. (3.48), Eq. (3.54a)becomes

(3.54b)