Power to space use ratio

Engineering Crosswind Kites / Planes
101

Well, it is true they didn’t mention how a parachutal drag Cd (aka Ct in HAWT) of 2.0 was measured, I tend to believe it is correct, because predicted Ct for axial induction factors greater than 0.5 will not fit the theory of nice laminar flowing wind turbines - as in figure 5 in the overly mentioned paper.pdf When the so-called axial induction factor exceeds values of 0.5 empirical data confirms that Ct gradually increases towards 2 instead of dropping to 0.

Here-s how actual thrust goes vs. charted from Betz numerical derivation:

chart
chart514×406 23.3 KB

Image taken from the longer explanation here

The reason Betz computed thrust coefficient failes above 0.5 value of axial induction factor is that leads to negative wind speed downwind, which is impossible

Betz limit and power available in the wind
102

On the referenced website (below) there is no “empirical data”. There is “empirical correction” with a formula.

I asked a question about this statement, without any convincing answer on the topic in question:

Let @blimpyway prove his “theories” supposedly refuting the document entitled The Betz limit applied to Airborne Wind Energy and Betz’s law here and there and elsewhere. For that a peer-reviewed publication would be better than unnecessarily cluttering (to remain polite) the two topics in question: Betz limit and power available in the wind and the current topic entitled Power to space use ratio.

Now back to the topic.

103

As I indicated, the document intiled The Betz limit applied to Airborne Wind Energy brilliantly demonstrated the 4/27 limit for the reeling mode (also called yo-yo, or “lift” to use their terminology) in crosswind flight, compared to the 16/27 (Betz limit) applied to wind turbines, and also AWES in “drag” mode (which also implies crosswind flight but in fly-gen mode). The latter would then be the winner in terms of density (power-to-space ratio). But I asked some questions about crosswind flight that could apply to both “lift” and “drag” modes.

Now, let’s transpose this towards two non-crosswind systems, such as those presented on AWE in China: a parachute system in yo-yo mode, and a blimp carrying wind turbines. The 16/27 Betz limit could apply to the latter, while the 4/27 limit would apply to the parachute system. In a real world, the two systems will not reach their respective limits. See, for example this evaluation.

But if we look at the photos of the blimp carrying the wind turbines, with a total diameter of 40 meters, a quick calculation shows that the total wind area captured by the turbines represents only a small fraction of the total diametrical surface area, because the two sections of the airship occupy a considerably larger space. The two functions (aerostatic lift and turbine support) take up a significant amount of room, while the entire parachute-like flying system is dedicated to power generation.

Therefore, it’s not so easy to make a definitive judgment, to take this example.

104

Well guess what, empirical correction formulae is what engineers try to come up with when some theory doesn’t fit empirical data.

Point is the brilliant Brazilian paper assumes a max. Ct of 1.0 (as predicted by Betz math) while empirical data shows up a Ct up to 2.0.

I’m done with this useless discussion, thanks and sorry for the prolonged rant.

105

As I repeat, there is no “empirical data” on the publication in question. It’s “empirical correction,” which is not at all the same thing.
Empirical Correction of a Dynamical Model. Part I: Fundamental Issues in: Monthly Weather Review Volume 127 Issue 11 (1999)

I’ve already corrected this error: there’s no point in repeating it; it won’t become a truth, regardless of the argument used.

Slow Chat III