Wind Energy Handbook. Michael Barton Graham

Wind Energy Handbook

2 Baseline r Over one half rho upper U Subscript infinity Baseline Superscript 3 Baseline pi upper R squared EndFraction equals StartFraction 8 left-parenthesis 1 minus a right-parenthesis a prime lamda Subscript r Baseline Superscript 2 Baseline r Over upper R squared EndFraction"/>

(3.20) StartFraction italic d upper C Subscript upper P Baseline Over d mu EndFraction equals 8 left-parenthesis 1 minus a right-parenthesis a prime lamda squared mu cubed

where mu equals StartFraction r Over upper R EndFraction _.

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

StartLayout 1st Row one half rho upper U Subscript infinity Baseline Superscript 2 Baseline left-parenthesis 1 minus a right-parenthesis squared plus one half rho normal upper Omega squared r squared plus one half rho w squared plus p Subscript upper D Superscript plus 2nd Row equals one half rho upper U Subscript infinity Baseline Superscript 2 Baseline left-parenthesis 1 minus a right-parenthesis squared plus one half rho normal upper Omega squared left-parenthesis 1 plus 2 a prime right-parenthesis squared r squared plus one half rho w squared plus p Subscript upper D Superscript minus EndLayout

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

Consequently,

normal upper Delta p Subscript upper D Baseline equals 2 rho normal upper Omega squared left-parenthesis 1 plus a prime right-parenthesis a prime r squared

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

(3.21) normal upper Delta p Subscript upper D Baseline 1 Baseline equals 2 rho normal upper Omega squared a prime r squared

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) normal upper Delta p Subscript upper D Baseline 2 Baseline equals 2 rho normal upper Omega squared a prime squared r squared

Δp_D2 can be shown to provide a radial, static pressure gradient

StartFraction italic d p Over italic d r EndFraction equals rho left-parenthesis 2 normal upper Omega a prime right-parenthesis squared r

in the rotating wake that balances the centrifugal force on the rotating fluid, because [see Eq. (3.33) a^′(r) = a′(R)R²/r². This pressure causes a small discontinuity in the pressure at the wake boundary equal to 2ρ(a′(R)ΩR)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.

3.3.3 Maximum power

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) StartFraction italic d a Over italic d a prime EndFraction equals StartFraction 1 minus a Over a prime EndFraction

From Eq. (3.18)

StartFraction italic d a Over italic d a prime EndFraction equals StartFraction lamda Subscript r Baseline Superscript 2 Baseline Over 1 minus 2 a EndFraction

giving

(3.24) a prime lamda Subscript r Baseline Superscript 2 Baseline equals left-parenthesis 1 minus a right-parenthesis left-parenthesis 1 minus 2 a right-parenthesis

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

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