Betz limit and power available in the wind

I agree, though the effect is probably marginal.

Also reel-out speed in the paper states reel-out in the direction of the wind. This is not a realistic scenario as elevation angle should be at least in the range of 20-30 degrees. The harvested power is made by multiplying the reel out speed and the tension of the tether, in the direction of the tether. I think this means that in theory you could produce a bit more than the lift mode Betz limit

I am not sure if these two statements both represent the same effect.

Im not sure if the effect is entirely positive though. We are saying that one wingtip will have fresh wind at the top of the spiral/fig8 while otherwise in the wake of previous wing. This will create additional stress on the wing.

BTW: I received the paper for free and it was a good read.

Recall: traction 6000 N X 1/3 wind speed = 20000.
Now assuming the elevation angle is 30°, leading to a cosine of 0.8660.
The reel-out speed in the horizontal becomes 2.955 m/s.
And the apparent wind speed becomes 10 - 2.955 = 7.045 m/s instead of 6.66 m/s.
7.045² = 49.63. Then 49.63/100 = 0.4963 instead of 4/9, so a bit more (10%).

IMHO the 4/27 of a flat plane is also the 4/27 of a same area swept by a lift (yoyo) crosswind kite as both go downwind.
Indeed an interesting Dave Lang’s message on mentions:

So (at least by my derivation) for a device like a “parachute” or a “Ship on a down-wind run under sail”,

1. The Max power extraction fraction = “(4/27)* Cd”

2. Peak power extraction occurs at a “downwind device speed” of “(1/3) Vwind”.

OF course the devil is in the details concerning the value of Cd, since it is itself a quantified measurement of a device’s propensity to catch air. For instance, a flat plate has a Cd frequently quoted as about 1.1, Thus, a flat plate operating as a drag device to capture wind power, will exhibit the ability to extract only about 15% of the incident power (compared to the classical turbine limit of 59%).

Thanks for posting this talk in the Yahoo group @PierreB.

Moritz Diehl at AWEC 2015: Multiple Wing Systems - an Alternative to upscaling?

All of the talks from the conference are here:

And AWEC 2017:

Hi would like to add a possibly new observation, that maybe should lead to a reevaluation of the 16/27 limit for hovering [drag] mode designs.

Because an AWE plant in general (especially if self-erect) will be rotating with the blades in a plane that is rotated according to the plant elevation angle, presumably in the range 20 - 40 degrees.

This means blades will be travelling slightly downwind sometimes and upwind sometimes. Slightly is maybe an understatement, as the speed may be to the order of 13% of blade ground speed. Though the effect is not linearly symmetric. For instance, blades travelling upwind will have wakes with larger spacing than those travelling downwind.

In short the actuator disc is not uniformly loaded at all. I believe this would lead to a further reduction of the Betz limit after 16/27, if one did thorough calculations

This affects bounding [lift] mode plants less, because these will presumably have active winch speed control to compensate for elevation angle. Still, the end result would be quite difficult to analyze, and probably no better than the 4/27 Betz limit already stated

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Isn’t it the cosine loss that we already know?

No. The cosine loss only deals with the geometric alignment of the kite vs power generation. This latest post regarded what happens when you approach the Betz limit. This would be a very interesting analysis to be done…

Also interesting question is whether cosine loss affects the Betz limit. I dont think it does, in general.

What I am talking about is uneven distribution of power over the actuator disk, and that some of the disk is moving upwind or downwind.

To see someone way ahead of me in terms of forward thinking about Betz limit in awe look to

An engineering model for the induction of crosswind kite power systems by Gaunaa et al

Abstract. The present paper introduces a new, physically consistent definition of effective induction that should be used in engineering models for power kite performance that use aerodynamic coefficients for the wing. It is argued that in such cases it is physically inconsistent to use disc-based induction models – like momentum models – and thus a new, physically consistent induction model using vortex theory methodology is derived. Simulation results using the new induction model are compared to the previously often used momentum method and Actuator Line (AL) CFD simulations. The comparison shows that the new vortex based model is in much better agreement with the AL results than the momentum method. The new model is as computationally light as the momentum induction method.


… , where the tip speed ratios are not too low (usually TSR>6) and where the aerodynamically effective part of the blades extend almost to the rotation axis. For the kite case, there is (usually) only one “blade” circling in the “rotor-plane”, and the effective “root” (inner kite tip) of the blade is often at 80%-90% of the radius of the “tip” (outer kite tip). In this case the Goldstein function would be immensely different from Prandtl’s tip correction, and it would be an extremely large correction to the raw momentum results. This case is very far from the comfort zone of the methods used in the wind turbine community.

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Does that mean I’ve just wasted a day doing this?

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AWE lift power systems = yo-yo AWES as we know. We go from 16/27 to 4/27 because of the 2/3 apparent wind speed due to reel-out speed of 1/3 wind speed. At 16/27 (Betz limit) we have the complete wind speed cubed x 16/27. And at 4/27 we have 2/3 wind speed cubed which is 1/4 of 16/27 (Betz limit).

The area can be swept by a crosswind kite (SkySails, KitePower), or by a tethered-aligned parachute kite. For this last eventuality, the parachute kite (or any static and tethered-aligned kite) constitutes the swept area (not counting the additional area due to downwind operation in yo-yo mode) which moves downwind.

If we assume that the parachute (being not yet a kite) has only drag, we just need to apply M. Loyd’s formula (but by eliminating the lift coefficient (Cl) and (L/D)² since there is no lift and no crosswind flight), and we see that we obtain 4/27 with a drag coefficient (Cd) of 1.

We will take as an example a swept area of 1 m², for an air density of 1.2, and a wind speed of 10 m/s.
At Betz limit (16/27): 1/2 x 1 x 1.2 x 10³ x 16/27 = about 355.55.
In yo-yo mode (4/27) by using the usual formula of kite power (but without Cl and (L/D)² as stated above: 1/2 x 2/27 x 10³ = about 88.88.

That said some parachutes have High drag coefficient, and a publication indicates that “the CD value of the parasail was found to be 2.727” and “As there were additional benefits of using a parasail in the arrest system such as the higher drag coefficient and lift force present”.

If the Cd is 2.727, we perhaps could state that this AWE power system can harvest up to 4/27 x (2.727) = about 0.4 of the power available in the wind. So the power coefficient (Cp) is 0.4. Is this wrong or right?

And it is said that this parasail has also a high lift force, so a high Cl. The thrust or pull can be a basis to calculate the power (?) as it combines the force of lift and drag.
By some calculations with Cd of 2.727 and with reasonable elevation angles, being comparable to average elevation angles of crosswind AWES, we can have:

  • For an elevation angle of 35°, the tangent is 0.7002. We obtain a Cl of 1.9094454, for a cosine of 0.8191, leading to a thrust (pull) coefficient of 3.3292638.

  • For an elevation angle of 30°, the tangent is 0.5774. We obtain a Cl of 1.5745698, for a cosine of 0.8660, leading to a thrust (pull) coefficient of 3.1489607.

  • For an elevation angle of 25°, the tangent is 0.4663. We obtain a Cl of 1.2716001, for a cosine of 0.9063, leading to a thrust (pull) coefficient of 3.0089374.

There may be errors, and this may be wrong, so thanks for any corrections. The seminal publication can be sufficient in order to correct, above all by the equation (4) about a simple kite:

I think there is benefit to being aware of these figures, but with variability wind and the array of kites available it may be hard to find data for anything other than a ballpark estimate.

I’m sure where I heard this but apparently the approximate power range is up to 30x minimum power?

If this is the case in Yo Yo mode or with any rotating AWES we could measure power over time. At X RPM the AWES produces an average of X amount of torque.

This is something more straightforward for “torque” AWES like Daisys or Superturbines, but for “pulling” AWES, there needs to be some standard of a R for Rotation.

For instance should the loop be a standard size or duration, that is difficult because each subsequent loop gets wider? Looking for suggestions

This would only be good for a standardised comparison of AWE designs. Changing the size and shape of the kite path would be a low hanging fruit for optimization. So in a future where AWE was in continuous operation at scale, one would certainly spend time optimizing these things.

Torque AWES like Daisy or SuperTurbine ™ can be assimilated to “drag” devices (flygens like Makani wings) or even conventional wind turbines, because the swept area does not move. So they are subject to the Betz limit (16/27).

In contrast pulling (yo-yo mode) AWES are subject to a limit of 4/27 of the swept area because it is going downwind at theoretically 1/3 wind speed, whether the use of crosswind or static (tethered-aligned) AWES.

In my last comment I wanted to show that a tethered-aligned AWES could go beyond the limit of 4/27 if the drag coefficient is higher than 1. I exposed a different conception where the swept area is the reference, in contrast to the usual reference of the kite area, at least for yo-yo mode. Parachutes with High drag coefficient and Parasails could lead to a better optimization of the swept area, so a better Power to space use ratio.

And we should keep in mind that the generation component of any AWES (as for a conventional wind turbine) always goes horizontal, being opposed to the lift component which goes vertical. It is the reason why crosswind kites flight at an average elevation angle of only about 30-35 degrees, so not too far from the horinzontal in order to mitigate cosine loss.

This is possible for well-sharpened rigid wings like the rigid blades of a gyroplane rotor. But as such a rotor scales up, the mass will quickly become much higher (squared-cubed scaling law for rigid things) than that of the parachute which “fills” the swept area. So, in yo-yo use, 30x power is not useful if we reason by swept area (here where the power available in the wind is limited to 4/27) as we should in my opinion.

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