Maximum wind turbine performance at low tip speed ratio

Abstract

Wind turbines can approach the Betz-Joukowsky limit on maximum power only at sufficiently high tip speed ratio: in practice, for ratios in excess of about seven. This paper analyses the performance of a turbine with an infinite number of blades as the tip speed ratio decreases to zero, beginning with the two traditional ways of determining the maximum power. The first is the “Glauert” optimization of the power extracted at every radius and the second is the “Betz-Goldstein” optimization of the whole rotor with the wake represented as a rigid helicoidal sheet of constant pitch. At high tip speed ratio, the two methods give very similar power and both asymptote to the Betz-Joukowsky limit. As the ratio approaches zero, the differences become significant. It is shown that Glauert's analysis does not account for effect of the varying pitch on the axial velocity. In addition, there is a large and unphysical energy extraction by a stationary rotor, and an unphysical constant circumferential velocity. Glauert's analysis, however, gives positive torque on a stationary rotor, which is necessary to start a wind turbine, but the torque at very low tip speed ratio seems too high. In contrast, the Betz-Goldstein analysis implies no torque on a stationary rotor but the circumferential velocity is zero on the axis of rotation. A modification is proposed to the Betz-Goldstein analysis to yield positive torque on a stationary rotor. The modified Betz-Goldstein torque coefficient never exceeds 1/2 and the power is less than the Glauert optimum. Finally, the analysis for varying pitch is corrected and the power maximized numerically. The modification applied to the Betz-Goldstein rotor is required to produce torque on the stationary rotor. The maximum torque coefficient was 0.55, and the maximum power was again less than the Glauert maximum. As tip speed ratio increases, the power from all methods asymptotes to the Betz-Joukowsky limit.