Crosswind kite power formula: lift or thrust coefficient?

[PierreB]

A video has been linked. I put it (15:00) to illustrate this topic, as well as a quote from Diehl2013.pdf (page 5) below said video:

GAMM 2015 - 11) Plenary Lecture - Prof. Moritz Diehl

@PierreB

The power P that can be generated with a tethered airfoil operated either in drag
or in lift mode had under idealized assumptions been estimated by Loyd [8] to be
approximately given by P = 2/27ρAvw³CL (CL/CD)²
where A is the area of the wing, CL the lift and CD the drag coefficients, and vw the
wind speed.

We see that CL is counted twice. This is also underlined in Drag power kite with very high lift coefficient, page 291:

Because the power increases cubically with CL and decreases only quadratically with CD,eq,[…]

My question: when CL occurs for the first time in the formula, could it not be a thrust coefficient CT that incorporates the lift and drag coefficients? Intuitively, it could make sense.

For the examples above, considering a usual high L/D ratio and a low CD, the CT would be very close to the CL.

But for crosswind kites with a low L/D ratio, including Magnus effect-based balloons (very high CL and fairly high CD), there would be a difference, certainly not huge, but notable. It may partially be an explain for:

@PierreB

Sixth, they have a massive pull relative to their size.

The “resultant force coefficient CR” could be assimilated to the thrust coefficient CT mentioned above. We can see on the curve that for the single-keel NPW wing, this coefficient is 1.3 for an L/D ratio of 2, leading to a CL of about 1.16, and a CD of about 0.58. So there is a small but notable difference between 1.16 and 1.3.

If the use of CT proves to be correct, it could give rise to crosswind kites with a relatively low L/D ratio, and a very high CT (resultant force (thrust) coefficient from very high lift and fairly high drag coefficients), which can look NPW kites, or crosswind Magnus effect-based balloons.