Beating Betz's Law: A larger fundamental upper bound for wind energy harvesting

Abstract

Betz's law, purportedly, says an ideal wind harvester cannot extract more than 16/27 ($\sim$59\%) of the wind energy. As the law's derivation relies on momentum and energy conservation with incompressible flow and not the physical mechanism coupling the wind-field to the extraction of work it is ubiquitously regarded as a "universal" upper bound on efficiency, as inclusion of mechanics, aerodynamics and thermodynamics are presumed to worsen this upper bound. Here we show that when unneeded assumptions in the Betz's law derivation are relaxed a higher bound of 2/3 ($\sim$67\%) can be achieved. A concrete example, strictly obeying the identical energy and momentum conservation used to derive the Betz's law, is given that violates Betz's law by achieving our higher 2/3 bound. Thus Betz's law is not a universal limit on wind energy harvesting efficiency. More surprisingly, we show Betz law is not simply the limit case of a vanishingly-thin turbine either. In 2-D models specific for turbines, radial flow is known to occur to occur as a consequence of angular momentum,\cite{Sharpe2004} but here we show in a 1-D modelthat allowing any radial flux out of the harvester cross-section can increase the efficiency without any need to consider angular momentum or explicit 2-D models. A key design insight we glean is that for high-efficiency harvesters it is better to strive for the least pressure build up (to increase flux) -- the exact opposite of the Betz model's sole operational principle of high pressure differentials. Additionally, we derive an alternative metric of harvester efficiency which takes into account the downstream wake expansion ignored by the conventional definition of power conversion factors, and the resulting upper bound this places on power extraction from dense grids of harvesters.