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

Betz' law, purportedly, says an ideal wind harvester cannot extract more than 16/27 ($\sim$59%) of the wind energy. As the law's derivation relies on momentum and energy conservation with incompressible flow and not the mechanical mechanism coupling the wind-field to the extraction of work it is ubiquitously regarded as a universal upper bound on efficiency, as inclusion of mechanics, aerodynamics and thermodynamics are presumed to worsen this limit. Here we show that when unneeded assumptions in the Betz' law derivation are relaxed a higher limit 2/3 ($\sim$67%) can be achieved. A concrete example, implemented with the same energy and momentum conservation laws used to derive the Betz law is given, achieving the higher 2/3 limit. Thus Betz 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 thin turbine either. Radial flow has been considered previously as a consequence of including angular moment,[Sharpe,2004] but here we showed that allowing any lateral flux emission out of the harvester cross-section increases the efficiency without any need to consider angular momentum or explicit 2-D models. A key design insight for high load harvesters is 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.