Energy2green Wind And Solar Power System

Fig. B.1 indicates these five rotor flow states in the Cdax- a diagram.

Wind turbines normally operate in the windmill state, with 0 < a < 0.5. Glauerts well-known, and well established blade element momentum theory is generally recommended for use in this flow state, resulting in the following relationship between the thrust coefficient and the axial induction factor:

This relationship can also be applied for negative axial induction factors (i.e. a < 0). In this so-called propellor state energy is added to the wake thereby producing an upwind force (i.e. Cdax < 0).

Vortex ring

Vortex ring

Figure B.1: Thrust coefficient as function of the axial induction factor and corresponding rotor flow states. Solid curve: momentum theory, dashed curve: momentum theory not valid, dashed-dotted curve: Glauert empirical relation, and ♦, •: measured data [282].

In the turbulent wake state the relationship between the axial induction factor and the thrust coefficient according to the actuator disk/momentum theory has to be replaced by an empirical relation. The explanation for this is that the momentum theory predicts a decreasing thrust coefficient with an increasing axial induction factor, while data obtained from wind turbines show an increasing thrust coefficient [279] (see Figure B.1). Thus, the momentum theory is considered to be invalid for axial induction factors larger than 0.5. A number of empirical relations have been derived in order to improve agreement between theory and experiment. These empirical relations are compared in Fig. 2.1 on page 25 for perpendicular flow. Here Glauert's empirical model is depicted in order to illustrate that the Cdax- a curve from actuator disk/momentum theory fails to describe to measured results for 0.5 < a < 1.0. This is consistent with the fact that the (flow streamline) assumptions on which this theory is based are violated in this region.

Wilson and Lissaman extended Eq. (B.1) (knowing that the assumptions are violated) to the region a > 0.5 in order to understand the observed differences between momentum theory and experimental results. The resulting equation for the thrust coefficient as function of the axial induction factor is as follows:

When the induction factor is somewhat over unity, the rotor flow state is called the vortex ring state. For axial induction factors greater than unity, where the rotor reverses the direction of the flow, the state is termed the propellor brake state. The interested reader is referred to Stoddard [282] for a more comprehensive survey.

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