Misled by Betz and unsteady flow

Abstract

A turbine array is an adjustable flow resistanceR placed in a tidal channel. Ideally, it is designedand operated to maximise energy yield. Garrett & Cummins(2005), using optimal control theory applied to the RCelementchannel (R) and basin (C), showed: the energyextraction from the flow PT + PD is maximised when theflow rate is slowed down by a factor of 1/√3. This resultis independent of the ratio of the extracted mechanicalpower PT to the total power extraction including the powerloss PD due to the mixing of the bypass flows within theturbine field. The optimisation task for turbine arrays ismaximising PT. This objective raises two questions: ”Whatis the maximum power PT that can be extracted, and what isthe optimal design (size, topology) and operation to achievethis output?” When addressing them, the literature stilluses the Betz ‘limit’ as a reference. The work presentedhighlights two major problems. First, the Betz ’limit’ isnot a constant upper bound for open channel flow. Thisproblem has been discussed and solved by the first author(2011, 2020). Second and more importantly, the presentedpaper points out the misconception under which severalresearch studies referred to array topologies as ‘optimal’with regard to design and operation. Hereby, the presentedpaper contributes to the advancement of tidal power on anaxiomatic basis. The misleading by Betz and overvaluingof transient effects is made transparent in a scientificdiscourse.