Refined Betz Limit For Rotors With A Finite Number Of Blades

摘要

The criterion of Betz for optimum rotors with a finite number of blades is revisited and used to determine the performance of wind turbines. The Betz criterion states that ideal efficiency is obtained when the distribution of circulation along the blade produces a rigidly helicoid wake that moves in the direction of its axis with a constant velocity. Theories based on this criterion were more than 60 years ago developed by Goldstein and Theodorsen. Although these theories made it possible to determine optimum circulation distributions on rotors, they fail to determine correctly the associated optimum power coefficient. Especially, they fail to reproduce the results of the general momentum theory when the number of blades goes to infinity. The present theory is a modification to the original model of Goldstein using a new analytical solution to the wake vortex problem. In contrast to earlier models, the new model is consistent with the general momentum theory and enables for the first time to determine the theoretical maximum efficiency of rotors with an arbitrary number of blades.