Optimal performance of actuator disc models for horizontal-axis turbines

Abstract

This study analyzes actuator disc (AD) models of horizontal-axis turbines to determine optimal performance, defined as the maximum power extracted at any tip speed ratio. We use the calculus of variations to maximize rotor torque relative to the thrust without making any assumptions about the rotor loading. The torque was obtained from the angular momentum equation and the thrust from the Kutta-Joukowsky equation which depends on the circumferential velocity and tip speed ratio. The optimality requirement is that the pitch of the vorticity exiting the rotor must be constant across the wake and equal to the ratio of torque to thrust. This result generalizes the classical finding of Betz and Goldstein that optimal lightly-loaded ADs have constant pitch. Optimizing the torque in the far-wake, well downstream of the rotor, leads to the same requirement of constant pitch. This implies that the pitch of an optimal rotor is constant everywhere in the wake at all tip speed ratios. We show that it is not possible for the pitch to reach its optimal value because of the vorticity distribution in the wake, and propose modifications to the pitch at the rotor and in the far-wake. The axial and circumferential velocities in the far-wake, which are easily determined, were used to find those at the rotor from the “disc loading equation” for the angular momentum which is also the normalized bound circulation at the rotor. For the simplest case of a lightly-loaded rotor at zero tip speed ratio, the induced circumferential velocity is linear in radius and the axial component is quadratic, As the tip speed ratio increases, the optimal power and thrust asymptote to the familiar Betz-Joukowsky values, and the induced axial velocity and rotor bound circulation become constant. At low tip speed ratios, the optimal wakes are constrained by the need to avoid breakdown of the flow at high swirl, and the conventional thrust equation, involving the axial velocity only, is inaccurate. As found in previous studies, the power coefficient increases monotonically with tip speed ratio, but the thrust coefficient reaches a maximum value slightly above the Betz-Joukowsky limit at a tip speed ratio of two, before decreasing towards the limit.

A revision of blade element/momentum theory for wind turbines in their high-thrust region

“…Eq. 12 gives the vortex pitch as the ratio of the blade element torque to reduced thrust, which also holds for the ideal actuator disc model of Wood and Hammam (2022), for which dT = dT*. Thus, it is not surprising that the runaway is associated with p ↓ 0.…”

“…Their relation to the usual thrust equation is easily derived by ignoring circumferential variations and the expansion term. Wood and Hammam (2022) showed that the high−λ circumferentially uniform wakes of optimal rotors satisfy "helical symmetry:" pu = wr where p = (1 − u)/(w/r + λ) is the pitch of the helical vorticity. Applying these equations turns the right side of Eq.…”

“…The classical one-dimensional analysis that leads to the Betz-Joukowsky limit ignores drag and so gives no hint of a runaway. Using an actuator disc model of an ideal rotor, Wood and Hammam (2022) found that the Betz-Joukowsky values of C P = 16/27 and C T = 8/9 were approached as λ → ∞: ideal rotors do not run away. Thus, blade element drag is sufficient to cause a high-thrust region but it may not be the only mechanism to cause a runaway.…”

Calculation of the velocities induced by the trailing vorticity in the rotor plane of a horizontal-axis turbine or propeller

“…and radius as this is the only case with closed-form expressions for the induced velocities. The restriction of constant pitch is not significant as Wood and Hammam (2022) showed that the pitch is constant throughout the wake of optimal rotors as it is the ratio of torque to thrust. The restriction of constant radius is possibly more significant, but at least, the constant radius analysis captures the important features of circumferential periodicity for identical, equispaced blades.…”

“…Γ is easily determined from the element lift and drag, but this determination is outside the scope of the present work. All other parameters for the induced velocities are set by the rotor geometry, except for p. It can be calculated from the blade element thrust and torque, or equivalently, from u and w immediately behind the blades (Wood and Hammam, 2022). Wood et al (2021) analyzed methods to find u and w at x = θ = 0 due to a vortex also starting at x = θ = 0.…”