On the Maximum of Wind Power Efficiency

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-Jowkowsky 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 propellertype wind turbines depends on tip-speed ratio and the number of blades.

Sustainability of Wind Energy under Changing Wind Regimes—A Case Study

“…It is expressed by the Weibull two-parameter distribution Table 1. Specifications of the wind turbines considered in this study (adopted from [48]).…”

Wind Power Potential in Interior Alaska from a Micrometeorological Perspective

“…The integration of Equations (2.18) to (2.20) over a time-independent control volume, V c , encompassing the rotor of the wind turbine, leads to [19,45] …”

“…Based on the local balance equation of momentum and the equation of continuity we derive the local balance equation of kinetic energy for macroscopic and turbulent systems, and in a further step, Bernoulli's equation and integral equations that customarily serve as the key equations in momentum theory and blade-element analysis. It mainly follows that of Kramm et al [19] and serves to understand the generation of electricity by extracting kinetic energy from the wind field by using a propeller-type wind turbine and its inherent limitation characterized by power efficiency. In accord with Storm et al [15], Storm and Basu [16], Mölders and Kramm [10], Mölders et al [18], and Hinzman and Kramm [14], we also consider low-level wind field characteristics predicted by WRF/Chem [20] to assess the wind power potential over Interior Alaska.…”

“…Here, x x T = ⋅ e T is the axial force acting on the rotor of a propeller-type wind turbine (e.g., [19,45]). If we assume that the axial direction coincides with any horizontal direction, the term x ϕ ⋅∇ e will be nearly equal to zero.…”

“…Apparently, the integral relation (2.34) underlines the importance of Bernoulli's equation in wind power studies. As documented by Sørensen [45] and Kramm et al [19], Equations (2.32) to (2.34) are key equations for momentum theory and blade-element analysis. Figure 6 illustrates the so-called Lanchester-Betz-Joukowsky (LBJ) limit (e.g., [45,48,49]) for the maximum of the (wind) power efficiency (e.g., [50] P is the power extracted from the wind field by the turbine, and, in accord with Equation (2.13),…”