Michael Graham - Wind Energy Handbook

Thus

(3.18)

The area of the ring is δA D= 2 πrδr , therefore the incremental shaft power is, from Eq. (3.17),

The first term in brackets represents the power flux through the annulus in the absence of any rotor action; the term outside these brackets, therefore, is the efficiency of the blade element in capturing that power.

Blade element efficiency is

(3.19)

in terms of power coefficient

(3.20)

where .

Knowing how a and a ′vary radially [ Eq. (3.20)] can be integrated to determine the overall power coefficient for the disc for a given tip speed ratio λ .

It was argued by Glauert (1935b) that the rotation in the wake required energy that is taken from the flow and is unavailable for extraction, but this can be shown not to be the case. The residual rotation in the far wake is supplied by the rotation component a′ Ω induced at the rotor. The lift forces on the blades forming the rotor disc are normal to the resultant velocity relative to the blades, and so no work is done on or by the fluid. Therefore, Bernoulli's theorem can be applied to the flow across the disc, relative to the disc spinning at angular velocity Ω, to give for an annulus of radius r

where w is the radial component of velocity. which is assumed continuous across the disc.

Consequently,

The pressure drop across the disc clearly has two components. The first component

(3.21)

is shown to be, from Eq. (3.18), the same as that given by Eq. (3.9)in the simple momentum theory in which rotation plays no part. The second component is

(3.22)

Δ p D2can be shown to provide a radial, static pressure gradient

in the rotating wake that balances the centrifugal force on the rotating fluid, because [see Eq. (3.33) a ′( r ) = a′ ( R ) R 2/ r 2. This pressure causes a small discontinuity in the pressure at the wake boundary equal to 2 ρ ( a′ ( R )Ω R ) 2, which in reality, along with the other discontinuities there, is smeared out.

The kinetic energy per unit volume of the rotating fluid in the wake is also equal to the drop in static pressure of Eq. (3.22), and so the two are in balance and there is no loss of available kinetic energy.

However, the pressure drop of Eq. (3.22)balancing the centrifugal force on the rotating fluid does cause an additional thrust on the rotor disc. In principle, the low‐pressure region close to the axis caused by the centrifugal forces in the wake can increase the local power coefficient. This is because it sucks in additional fluid from the far upstream region that accelerates through the rotor plane. This effect would cause a slight reduction in the diverging of the inflow streamlines. However, the degree to which this effect might allow a useful increase in power to be achieved is still the subject of discussion; see, e.g. the analyses given by Sorensen and van Kuik (2011), Sharpe (2004), and Jamieson (2011). The ideal model with constant blade circulation right in to the axis is not consistent due to the effect on the blade angle of attack by the arbitrarily large rotation velocities induced there, and in reality, the circulation must drop off smoothly to zero at the axis, and the root vortex must be a vortex with a finite diameter. This is discussed later in Section 3.4, where the vortex model of the wake is analysed. Numerical simulations of optimum actuator discs by Madsen et al. (2007) have not found the optimum power coefficient ever to exceed the Betz limit. But the relevance of the issue is that it may be possible to extract more power than predicted by the Betz limit in cases of turbines running at very low tip speed ratios, even recognising that the rotor vortex has a finite sized core or is shed as a helix at a radius greater than zero, and taking account of the small amount of residual rotational energy lost in the far wake.

The values of a and a ′that will provide the maximum possible efficiency can be determined by differentiating Eq. (3.19)by either factor and putting the result equal to zero.

Hence

(3.23)

From Eq. (3.18)

giving

(3.24)

The combination of Eqs. (3.18)and (3.21)gives the required values of a and a ′that maximise the incremental power coefficient:

(3.25)

The axial flow induction for maximum power extraction is the same as for the non‐rotating wake case, that is, , and is therefore uniform over the entire disc. However, a ′varies with radial position.

From Eq. (3.20)the power coefficient for the whole rotor is

Substituting for the expression for a ′in Eq. (3.25)gives maximum power as

(3.26)