Top-level rotor optimisations based on actuator disc theory

Abstract

Ahead of the elaborate rotor optimisation modelling that would support detailed design, it is shown that significant insight and new design directions can be indicated with simple, high-level analyses based on actuator disc theory. The basic equations derived from actuator disc theory for rotor power, axial thrust and out-of-plane bending moment in any given wind condition involve essentially only the rotor radius, R, and the axial induction factor, a. Radius, bending moment or thrust may be constrained or fixed, with quite different rotor optimisations resulting in each case. The case of fixed radius or rotor diameter leads to conventional rotor design and the long-established result that power is maximised with an axial induction factor, a = 1/3. When the out-of-plane bending moment is constrained to a fixed value with axial induction variable in value (but constant radially) and when rotor radius is also variable, an optimum axial induction of 1/5 is determined. This leads to a rotor that is expanded in diameter 11.6 %, gaining 7.6 % in power and with thrust reduced by 10 %. This is the low-induction rotor which has been investigated by Chaviaropoulos and Voutsinas (2013). However, with an optimum radially varying distribution of axial induction, the same 7.6 % power gain can be obtained with only 6.7 % expansion in rotor diameter. When without constraint on bending moment, the thrust is constrained to a fixed value, and the power is maximised as a→ 0, which for finite power extraction would require R→∞. This result is relevant when secondary rotors are used for power extraction from a primary rotor. To avoid too much loss of the source power available from the primary rotor, the secondary rotors must operate at very low induction factors whilst avoiding too high a tip speed or an excessive rotor diameter. Some general design issues of secondary rotors are explored. It is suggested that they may have the most practical potential for large vertical axis turbines avoiding the severe penalties on drivetrain cost and weight implicit in the usual method of power extraction from a central shaft.