Power Coefficient for Large Wind Turbines Considering Wind Gradient Along Height

The Betz constant is the well-known aerodynamic limit of the maximum power which can be extracted from wind using wind turbine technologies, under the assumption that the wind speed is uniform across a blade disk. However, this condition may not hold for large wind turbines, since the wind speed may not be constant along their height; rather, it may vary with the location due to surface friction from tall buildings and trees, the topography of the Earth’s surface, and radiative heating and cooling in a 24 h cycle. This paper derives a new power coefficient for large wind turbines based on the power law exponent model of the wind gradient and height. The proposed power coefficient is a function of the size of the rotor disk and the Hellmann exponent, which describes the wind gradient based on wind stability at various locations, and it approaches the same value as the Betz limit for wind turbines with small rotor disks. It is shown that for large offshore wind turbines, the power coefficient was about 1.27% smaller than that predicted by the Betz limit, whereas for onshore turbines in human-inhabited areas with stable air, the power coefficient was about 8.7% larger. Our results are significant in two ways. First, we achieve generalization of the well-known Betz limit through elimination of the assumption of a constant wind speed across the blade disk, which does not hold for large wind turbines. Second, since the power coefficient depends on the location and air stability, this study offers guidelines for wind power companies regarding site selection for the installation of new wind turbines, potentially achieving greater energy efficiency than that predicted by the Betz limit.