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Research Papers

# Beating Betz: Energy Extraction Limits in a Constrained Flow Field

[+] Author and Article Information
Peter M. Jamieson

Garrad Hassan and Partners Ltd., 2064 Maryhill Road, Glasgow G20 0AB, UK

This result is most easily understood looking ahead to Eq. 5 with $Ct=89$ for $a=13$ and $a=am$.

Each point on the real diffuser characteristic may then be considered to represent some ideal diffuser with the particular value of $a0$ that corresponds to a characteristic passing through the particular point.

J. Sol. Energy Eng 131(3), 031008 (Jul 09, 2009) (6 pages) doi:10.1115/1.3139143 History: Received May 22, 2008; Revised December 05, 2008; Published July 09, 2009

## Abstract

Experiments with diffusers and other flow concentrating devices have shown that the power performance coefficient $Cp$ of an energy extraction device, defined in relation to the area of flow intercepted at the device, may exceed the Betz limit. “Beating Betz,” in that sense, has been long established but no theory has existed to define in a generalized way what ideal limit may apply to $Cp$ in such situations. Recent analysis has resolved this. This indicates that, irrespective of the presence of flow concentration systems or other influences that perturb the flow but do not in themselves extract energy, there is a universal ideal limit of energy extraction. This is found to be $89$ of the upstream kinetic energy in the streamtube associated with the energy extraction. Moreover, the familiar Betz equations for power and thrust coefficients can be generalized in a simple way to express this. Although this work has been developed in the context of wind energy, it will be apparent that the results are of general significance for any application of ducted rotors or propellers in a fluid stream.

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Copyright © 2009 by American Society of Mechanical Engineers
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## Figures

Figure 1

Flow fields regions

Figure 2

Energy extraction from a fixed upstream source region

Figure 3

Comparison of theory and CFD

Figure 4

Effect of diffuser inefficiency on power coefficient

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