Betz limit and power available in the wind

[PierreB]

@blimpyway

while autogyro rotor data I linked somewhere else are 2.0 which already indicate a 8/27 power potential. And since an autogyro rotor in parachute mode is showing a counter example of an actual lift-based pumping system which contradicts the paper claiming a theoretical 4/27 limit

If you indicate a 8/27 power because the Cd is 2, you implicitly accept the base value of 4/27 which is half. How do you obtain the 4/27 value if the document is wrong, as you stated only one hour (!) after it was fully available? Perhaps you used my demonstration I put again:

@PierreB

Let us try to check this with an example, wind speed = 10 m/s, air density = 1.2.
The power of a HAWT is determined by the disk swept by the rotor. Supposing the disk area is 100 m².
Power at Betz limit: 1/2 x 100 x 1.2 x 1000 x 16/27 = 35555 W.
Now let us take a surface of 100 m². Traction (force): 1/2 x 100 x 1.2 x 100 = 6000 N. Let us reel-out this surface in the optimal speed of 1/3 wind speed, considering the wind speed on the surface is now only 2/3 wind speed, leading to the result: 20000 x 4/9 = 8888 W, so exactly 1/4 of 35555 W.
Checked!

In this demonstration, the 1/4 of power at Betz limit (first formula) is reached by using the formula of force multiplied by reel-out speed. This is not about Cd, only about what remains when the swept area goes downwind, the apparent wind becoming lower.

In the past, I produced a similar line of reasoning three times to arrive at the values ​​I cite below, in a form of question, but without explicitly contradicting the 4/27 limit mentioned in the document, having no sufficient knowledge to challenge the authors, and not having read the complete document at that time.

@PierreB

simplistic and probably incomplete view I suggested

@PierreB

For example if a tether-aligned (non-crosswind) parachute has a Cd of 3 (assuming it has no lift to simplify, or even a thrust coefficient (Tc) including a Cd of 3 and a lift coefficient (Cl) of about 1) “the power available in the wind” (in pumping mode, so with the quoted limit of 4/27) would be 3 times, so 12/27, is not it?

@PierreB

Apart from that, this would pose a theoretical problem for the yo-yo mode here in horizontal use. We know the limit of 4/27 (almost 15%, compared to 16/27 Betz limit) but with such drag coefficients (starting with 2.2) this limit would increase to a little more than 8/27

@PierreB

Conversely, subject to further investigation, it is not impossible that with very high drag coefficients Cd the Rogallo and other parachute kites could reach values ​​well above 4/27 (8/27 with Cd of 2 for example?) in reeling mode.

I would never have hastily asserted that the authors were wrong. I preferred to leave aside the elements that needed clarification, putting forward hypotheses here and there.

@blimpyway

I don’t know who you are addressing this to

To you, among others.