Wind Energy Handbook. Michael Barton Graham

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СКАЧАТЬ 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)

      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)

      which is precisely the same as for the non‐rotating wake case.

3.4 Vortex cylinder model of the actuator disc

      3.4.1 Introduction

The momentum theory of Section 3.1 uses the concept of the actuator disc across which a pressure drop develops, constituting the energy extracted by the rotor. In the rotor disc theory of Section 3.3, the actuator disc is depicted as being swept out by a multiplicity of aerofoil blades, each represented by a radial vortex of constant strength ΔΓ that denotes the bound circulation around each blade section (the totality of spanwise vorticity in the blade surface sheets). Each of these vortex lines is usually considered to lie along the quarter‐chord line of the blade but cannot terminate in the flow field at the tip. Therefore, each vortex is shed at the tip of the blade and convects downstream with the local flow velocity, forming a wake vortex in the form of a helix with strength ΔΓ. If the number, B, of blades is assumed to be very large but the solidity of the total is finite and small, then the accumulation of helical tip vortices will form the surface of a tube. As the number of blades approaches infinity, the tube surface will become a continuous tubular vortex sheet; see Figure 3.6.

Figure 3.6 Helical vortex wake shed by rotor with three blades each with uniform circulation ΔΓ.

      From the root of each blade, assuming it reaches to the axis of rotation, a line vortex of strength ΔΓ will extend downstream along the axis of rotation, contributing to the total root vortex of strength Γ(=BΔΓ). The streamtube will expand in radius as the flow of the wake inside the tube slows down. Because the axial convection of the tip vortices is therefore slowing from the rotor to the far wake, their spacing decreases and hence the vorticity density on the tube sheet representing the tip vortices increases. The vorticity is confined to the surface of this tube, the root vortex, and to the bound vortex sheet swept by the multiplicity of blades to form the rotor disc; elsewhere in the wake and everywhere else in the entire flow field the flow is irrotational.

The nature of the tube's СКАЧАТЬ