Betz's law

The Betz law comes from the German physicist Albert Betz ( 1885-1968 ). He formulated it for the first time in 1919. Seven years later it appeared in his book, wind energy and its utilization by windmills. The British engineer Frederick W. Lanchester (1868-1946) published in 1915, similar considerations.

The law states that a wind turbine maximum 16/27 ( almost 60 percent ) that can convert mechanical power that would carry through the projection of the wind without the brake rotor, into useful power.

The reason is that the energy release is accompanied by a reduction in flow velocity and an air storage, which can avoid some of the oncoming air, the rotor area.

Designation diversity

Because of this theoretical limit, which has not otherwise defined for machine efficiency, Betz led to the ratio of unused to incoming wind power, cP = PNutz/P0, the term quality factor a. In 1926, he called him coefficient of performance, is now in the literature, the term power coefficient customary harvest level and efficiency are also common.

The term " Betz'scher power coefficient " is common for both the considered of Betz and executed in the following chapter theoretical dependence of this ratio from the deceleration of the flow, for which the maximum value, as well as generally, imposed by Betz for wind turbines, quality factor.

The term crop factor is used for both cP and for the ratio of plant- emitted with a ( wind or solar) during their lifetime amount of energy to that which was spent on its construction.

Derivation

The conditions to which Betz is out, are:

• The wind turbine has no extension in the flow direction, and is therefore an area.
• You created in this area A a negative jump in the pressure curve and
• Uses lossless thereby the flow of power drawn from P.
• The density of the medium is constant, ie incompressible flow and no heat flow between flow and wind turbine.

The pressure before and far away behind the contact is equal to the negative pressure jump is compensated for by a gradual increase in pressure, both before and after the effective area. Associated with the increase in pressure is a decrease of the flow velocity V1 on the output of the v = Vavg far in the effective area V2 in behind it. Inversely proportional to the flow velocity, the cross- sectional area of ​​the stream tube from A1 to A2 to about A, see figure, since all their cross sections are traversed by the mass flow ( continuity equation).

If according to the energy conservation law, the extracted power equates with the difference of the kinetic energy associated with v1 and v2, one obtains

At the same time ( for the same v) must also apply conservation of momentum, ie

This is possible because only

The extracted power in dependence on the relative residual rate is therefore

Based on the rotor surface A and the power density of the wind, there is the power coefficient to

The course of this function is shown in the diagram to the right of x = 0.1 to 1. x = 1 means v2 = v1, so no deceleration and hence no power output. x = 0.1 means A2: A = v: v2 = 0,55:0,1 = 5.5. The outlet cross section A2 of the stream tube would have to be 5.5 times larger than the rotor surface - an operating condition that is not only from the result unfavorable, but at all difficult to achieve.

Reaches its maximum function at x = 1/3, see the calculation of minima and maxima. The means A1: A = v: V1 = 2/3. The remaining third of the incoming flow deviates from the active surface. This ' loss ' corresponds to the first term in. The second term means that the flow is removed by the active surface 8/9 of its energy.

Ways to overcome

It is frequently claimed to have received a cP > cP ( 1/3). Within the conditions of the model but that is as impossible as a violation of energy and momentum conservation. Complicated currents in the effective area of ​​about merely reduce the power coefficient, if one breaks down the active surface into small sub-areas: For each corresponding stream tube in each case above derivation applies, so 3 v1 must apply over the entire active surface v = 2 / to reach the global optimum.

Often jacketed for wind turbines "cross - Betz value " claimed. Although the disc model Betz does not cover these constructions, but it provides an explanation: The acting as a diffuser casing runs parallel to the narrower part of Betz 's electricity tube and allows a displacement of the rotor forward so that it can be smaller. Since the shell the 'right' reference surface (those with ) surrounds what is essentially won.

So far, only a theoretically possible solution is known, has already been reported by Betz himself: If the as a single, infinitely thin " active wheel " (English actuator disk) modeled rotor awarded a finite thickness, so could transversely to the main flow existing turbulent fluctuations extra energy upstream into the volume between the two and bring downstream to be discriminated active disks.

This idea was elaborated in detail by Loth and McCoy in 1983 for a Darrieus rotor with a vertical axis of rotation. They received cP ≈ 0.62.

Executed rotors

Since the rotor losses are by far the greatest losses of a wind turbine, all manufacturers are working to achieve the highest possible power coefficients. Modern rotors achieve power coefficients of cp = 0.4 to 0.5, which are thus about 70% to 85 % of the theoretically possible. Real Darrieus rotors reached cP = 0.3.