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3.8.2 Optimal design for variable‐speed operation
ОглавлениеA turbine operating at variable speed can maintain the constant tip speed ratio required for the maximum power coefficient to be developed regardless of wind speed. To develop the maximum possible power coefficient requires a suitable blade geometry, the conditions for which will now be derived.
For a chosen tip speed ratio λ the torque developed at each blade station is given by Eq. (3.49) and is maximised if
giving
From Eqs. (3.51) and (3.52) a relationship between the flow induction factors can be obtained. Dividing Eq. (3.52) by the modified Eq. (3.51), modified to include the additional loss of axial momentum from the pressure drop term Δpd2 in the far wake due to the centrifugal swirl generated radial pressure gradient, leads to:
The flow angle ϕ is given by
Substituting Eq. (3.62) into Eq. (3.61) gives
Simplifying:
At this stage the process is made easier to follow if drag is ignored; Eq. (3.63) then reduces to
Differentiating Eq. (3.64) with respect to a′ gives
and substituting Eq. (3.60) into (3.65)
Equations (3.64, 3.66), together, give the flow induction factors for optimised operation:
These are consistent at the rotor tip (where μ = 1) with Eq. (3.2) provided a′ is sufficiently small compared with unity for terms in a′2 to be neglected. This is normally true at the rotor tip, and these results agree exactly with the momentum theory prediction, because no losses such as aerodynamic drag have been included, and the number of blades is assumed to be large. This last assumption means that every fluid particle that passes through the rotor disc interacts strongly with a blade, resulting in the axial velocity being more uniform over the area of the disc. If the same analysis is followed excluding the swirl pressure drop term, then a = 1/3 – a small term ∼2/(9λμ)2, which is negligible except very close to the axis (blade root) or when the rotor tip speed ratio is very low.
To achieve the optimum conditions, the blade design has to be specific and can be determined from either of the fundamental Eqs. (3.48) and (3.49). Choosing Eq. (3.49), because it is the simpler, ignoring the drag, and assuming a′ ≪ 1, the torque developed in optimised operation is
The component of the lift per unit span in the tangential direction is therefore
By the Kutta–Joukowski theorem the lift per unit span is
where Γ is the sum of the individual blade circulations and W is the component of incident velocity mutually perpendicular to both Γ and L.
It is important to note that where the incident velocity varies spatially, as here, W takes the value that would exist at the effective position of the bound vortex representing the local blade circulation excluding its own induced velocity.
Consequently,
so
If, therefore, a is to take everywhere the optimum value (1/3), the circulation must be uniform along the blade span, and this is a condition for optimised operation.
To determine the blade geometry, that is, how should the chord size vary along the blade and what pitch angle β distribution is necessary, neglecting the effect of drag, we must return to Eq. (3.52) with CD set to zero:
substituting for sinϕ gives
The value of the lift coefficient Cl in the above equation is an input, and it is commonly included as above on the left side of Eq. (3.70) with a ‘chord solidity’ parameter representing blade geometry. The lift coefficient can be chosen as that value that corresponds to the maximum lift/drag ratio , as this will minimise drag losses: even though drag has been ignored in the determination of the optimum flow induction factors and blade geometry, it cannot be ignored in the calculation of torque and power. Blade geometry also depends upon the tip speed ratio λ, which is also an input. From Eq. (3.70) the blade geometry parameter can be expressed as
Hence
Introducing the optimum conditions of Eq. (3.67),
The parameter λμ is the local speed ratio λr and is equal to the tip speed ratio where μ = 1.
If, for a given design, Cl is held constant, then Figure 3.17 shows the blade plan‐form for increasing tip speed ratio. A high design tip speed ratio would require a long, slender blade (high aspect ratio) whilst a low design tip speed ratio would need a short, fat blade. The design tip speed ratio is that at which optimum performance is achieved. Operating a rotor at other than the design tip speed ratio gives a less than optimum performance even in ideal drag‐free conditions.
In off‐optimum operation, the axial inflow factor is not uniformly equal to 1/3; in fact, it is not uniform at all.
The local inflow angle ϕ at each blade station also varies along the blade span, as shown in Eq. (3.73) and Figure 3.18:
Figure 3.17 Variation of blade geometry parameter with local speed ratio.
Figure 3.18 Variation of inflow angle with local speed ratio.
which, for optimum operation, is
Close to the blade root the inflow angle is large, which could cause the blade to stall in that region. If the lift coefficient is to be held constant such that drag is minimised everywhere, then the angle of attack α also needs to be uniform at the appropriate value. For a prescribed angle of attack variation, the design pitch angle β = ϕ − α of the blade must vary accordingly.
As an example, suppose that the blade aerofoil is National Advisory Committee for Aeronautics (NACA) 4412, popular for hand‐built wind turbines because the bottom (high‐pressure) side of the profile is almost flat, which facilitates manufacture. At a Reynolds number of about 5 ⋅ 105, the maximum lift/drag ratio occurs at a lift coefficient of about 0.7 and an angle of attack of about 3°. Assuming that both Cl and α are to be held constant along each blade and there are to be three blades operating at a tip speed ratio of 6, then the blade design in pitch (twist) and plan‐form variation are shown in Figures 3.19a and b, respectively. This blade solidity becomes very large at the root but can be accommodated to around r/R = 0.1 depending on the location of the blade axis.
