Betz Limit Not an Exact Optimum

Abstract

The Betz high speed ratio limit has not been derived in the complete generality that would prove it absolute for an unshrouded actuator. The assumed uniform wake or thrust loading is hard to prove optimal. Many authors also assume uniform flow at the rotor or equivalently the differential form of the momentum equation but neither is necessary in the derivation or in fact true. Calculations of the expanding vortex ring wake have indicated lower velocity at the rotor's edge and slightly higher in the center. It is shown that if the uniform wake is perturbed by a fraction of the same variation function, the Betz power coefficient is raised. The increase is very small for a Hawt but might be bigger for a 2D actuator with a larger variation function. Thus the Betz value is very close to a local optimum, but not exact or absolute.