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