Misled by Betz and unsteady flow

Abstract

A turbine array is an adjustable flow resistance R placed in a tidal channel. Ideally, it is designed and operated to maximise energy yield. Garrett & Cummins (2005), using optimal control theory applied to the RCelement channel (R) and basin (C), showed: the energy extraction from the flow PT + PD is maximised when the flow rate is slowed down by a factor of 1/√3. This result is independent of the ratio of the extracted mechanical power PT to the total power extraction including the power loss PD due to the mixing of the bypass flows within the turbine field. The optimisation task for turbine arrays is maximising PT. This objective raises two questions: ”What is the maximum power PT that can be extracted, and what is the optimal design (size, topology) and operation to achieve this output?” When addressing them, the literature still uses the Betz ‘limit’ as a reference. The work presented highlights two major problems. First, the Betz ’limit’ is not a constant upper bound for open channel flow. This problem has been discussed and solved by the first author (2011, 2020). Second and more importantly, the presented paper points out the misconception under which several research studies referred to array topologies as ‘optimal’ with regard to design and operation. Hereby, the presented paper contributes to the advancement of tidal power on an axiomatic basis. The misleading by Betz and overvaluing of transient effects is made transparent in a scientific discourse.

Keywords

• Betz limit
• blockage
• coefficient of performance
• optimisation
• quasi-steady flow
• tidal array