Energy2green Wind And Solar Power System

For high induced velocities (exceeding approximately 40 % of the free-stream velocity), the momentum and vortex theory are no longer applicable because of the predicted reversal of flow in the turbine wake. The vortex structure disintegrates and the wake becomes turbulent and, in doing so, entrains energetic air from outside the wake by a mixing process. Thereby thus altering the mass flow rate from that flowing through the actuator disk. The turbine is now operating in the so-called "turbulent wake state", which is an intermediate state between windmill, and pro-pellor state (see Appendix B for an overview of the different flow states of a wind turbine rotor).

In the turbulent wake state the relationship between the axial induction factor and the thrust coefficient according to the momentum theory (i.e. Cdax = 4a(1 — a), with a the axial induction factor and Cdax the thrust coefficient) has to be replaced by an empirical relation (a = f (Cdax) for Cdax > Cdax. Note that the threshold value Cfax depends on the 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]. Thus, the momentum theory is considered to be invalid for axial induction factors larger than 0.5. This is consistent with the fact that when a = 0.5 the far wake velocity vanishes (i.e. a condition at which streamlines no longer exist), thereby violating the assumptions on which the momentum theory is based.

Most design codes include an empirical relation for induced velocities for these high disk loading conditions in order to improve agreement between theory and experiment. The following approximations are commonly used: Anderson [2], Garrad Hassan [29], Glauert [55, 82], Johnson [118], and Wilson [280, 308].

These five empirical relations are compared in Fig. 2.1 for perpendicular flow. The simple expression for the thrust coefficient, as derived from the momentum theory is added for comparison. Obviously, disagreement exists about how to model the flow field through a wind turbine under heavily loaded conditions, and the applied empirical approximations must thus be regarded as being only approximate at best.

With the recent developments towards wind turbines operating at variable speed, however, the importance of this phenomenon will become of lesser importance. After all, a wind turbine typically operates in turbulent wake state when the tip-speed ratio A exceeds 1.3 or 1.4 times the value for which Cp,max is achieved [267]. For a constant rotational speed wind turbine this implies that it occurs at wind velocities much lower than the rated wind velocity, while for a variable speed wind turbine it may not occur at all during normal operation.

Figure 2.1: Thrust coefficient Cdax as function of axial induction factor a. Solid curve: Johnson, dashed curve: Garrad Hassan,, dashed-dotted straight line: Anderson, *: transition point, dotted straight line: Wilson, o: junction point, dashed-dotted curve: Glauert. The equations are given on page 64-65.

Figure 2.1: Thrust coefficient Cdax as function of axial induction factor a. Solid curve: Johnson, dashed curve: Garrad Hassan,, dashed-dotted straight line: Anderson, *: transition point, dotted straight line: Wilson, o: junction point, dashed-dotted curve: Glauert. The equations are given on page 64-65.

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