Michael Graham - Wind Energy Handbook

3.8.1 Introduction

The purpose of most wind turbines is to extract as much energy from the wind as possible, and each component of the turbine has to be optimised for that goal. Optimal blade design is influenced by the mode of operation of the turbine, that is, fixed rotational speed or variable rotational speed and, ideally, the wind distribution at the intended site. In practice engineering compromises are made, but it is still necessary to know what would be the best design.

Optimising a blade design means maximising the power output, and so a suitable solution to BEM Eqs. (3.54or (3.59)and (3.55)) is necessary.

A 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

(3.60)

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 Δ p d2in the far wake due to the centrifugal swirl generated radial pressure gradient, leads to:

(3.61)

The flow angle ϕ is given by

(3.62)

Substituting Eq. (3.62)into Eq. (3.61)gives

Simplifying:

(3.63)

At this stage the process is made easier to follow if drag is ignored; Eq. (3.63)then reduces to

(3.64)

Differentiating Eq. (3.64)with respect to a ′gives

(3.65)

and substituting Eq. (3.60)into (3.65)

(3.66)

Equations (3.64, 3.66), together, give the flow induction factors for optimised operation:

(3.67)

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 ′ 2to 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/(9 λμ ) 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,

(3.68)

so

(3.69)

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 C Dset to zero:

substituting for sin ϕ gives

(3.70)

The value of the lift coefficient C lin 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