On the Maximum of Wind Power Efficiency

Gerhard Kramm^1*, Gary Sellhorst², Hannah K. Ross³, John Cooney⁴, Ralph Dlugi⁵, Nicole Mölders⁶

¹Engineering Meteorology Consulting, Fairbanks, USA.
²Geophysical Institute, University of Alaska Fairbanks, Fairbanks, USA.
³Department of Mechanical Engineering, University of Washington, Seattle, USA.
⁴Department of Atmospheric Sciences, Texas A & M University, College Station, USA.
⁵Arbeitsgruppe Atmosphärische Prozesse (AGAP), Munich, Germany.
⁶College of Natural Science and Mathematics and Geophysical Institute, University of Alaska Fairbanks, Fairbanks, USA.

DOI: 10.4236/jpee.2016.41001                

Abstract

In our paper we demonstrate that the filtration equation used by Gorban’ et al. for determining the maximum efficiency of plane propellers of about 30 percent for free fluids plays no role in describing the flows in the atmospheric boundary layer (ABL) because the ABL is mainly governed by turbulent motions. We also demonstrate that the stream tube model customarily applied to derive the Rankine-Froude theorem must be corrected in the sense of Glauert to provide an appropriate value for the axial velocity at the rotor area. Including this correction leads to the Betz-Joukowsky limit, the maximum efficiency of 59.3 percent. Thus, Gorban’ et al.’s 30% value may be valid in water, but it has to be discarded for the atmosphere. We also show that Joukowsky’s constant circulation model leads to values of the maximum efficiency which are higher than the Betz-Jow-kowsky limit if the tip speed ratio is very low. Some of these values, however, have to be rejected for physical reasons. Based on Glauert’s optimum actuator disk, and the results of the blade-element analysis by Okulov and Sørensen we also illustrate that the maximum efficiency of propeller-type wind turbines depends on tip-speed ratio and the number of blades.

Keywords

Wind Power, Power Efficiency, General Momentum Theory, Axial Momentum Theory, Blade Element Analysis, Betz-Joukowsky Limit, Joukowsky’s Constant Circulation Model, Glauert’s Optimum Actuator Disk, Balance Equation for Momentum, Equation of Continuity, Balance Equation for Kinetic Energy, Reynolds’ Average, Hesselberg’s Average, Favre’s Average, Bernoulli’s Equation, Integral Equations

Conflicts of Interest

The authors declare no conflicts of interest.

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