Misled by Betz and unsteady flow

Review on turbine arrays falsely deemed ‘optimal’

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.