Wind Energy Handbook. Michael Barton Graham

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      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^′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,

(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_D set to zero:

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