Optimization of a soft wing with turbines aloft


It is an updated and completed report with some corrections and comprising a first study of a 350 kW range flexible flygen kite. With an elevation angle of 30 degrees the power is 245 kW due to cosine loss. After other losses it can be about 150 kW.

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Theoretical drag of a Tether

It would be good to see the derivations and workings explained, referenced and explored in a bit more depth


It is an old report that is updated and corrected. All the paper is about the appropriate swept area rA by the turbines aloft in regard to the kite area kA as the turbines aloft should add 50% drag of the kite alone. 221.47466 m² (kite area) x 0.3 (drag coefficient) = 66.442398, so two times 33.2212 (turbines area x drag or thrust coefficient = 1).

All is in the table about parameters.

In the utility-scale model 33 m² rA (swept area by the turbines, r = rotor, estimated drag coefficient being 1) go with a 221 m² soft wing.
33 m² = 8 turbines of 2.3 m diameter that are used for the 600 kW Makani M600.

All this is precised in the paper. But, as the conclusion indicates, a deeper analysis is required.

Hypothesis for a calculation of M600: 2/27 x 1.2 x 1.2 x 50 [supposed area] x 1000 x (10.6)² = roughly 600 kW before cosine loss.
L/D ratio wing alone = 10.6; optimized L/D ratio with turbines = 7.0666666; wing area estimated at 50 m². With a lift coefficient CL of 1.2, wing area x CL = 60; so wing area x CD should be 5.66, leading to a drag coefficient of 0.1132. The thrust (drag) of the turbines should add 50% of 5.66, so 2.83. The thrust coefficient of the turbines is 2.83/33 = 0.0857575.
Verification: 33 x 1.2/2 x 70.666666 x70.666666 x 70.666666 x 0.0857575 = roughly 600 kW.
It looks that the turbines are large due to E-VTOL requirement, then the thrust coefficient is lower during operation.

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Some precision: in the utility-scale example the kite area kA is 221.47466 m², and the drag coefficient of the kite alone kDC is 0.3, as specified in the document. 221.47466 m² x 0.3 = 66.4424, so two times the value of rD [rA (area swept by the turbine(s), in m²) x estimated drag coefficient of 1] for the turbines, so 33.2212 as specified, adding 50% drag to the kite alone.


Some precisions and corrections:
The drag coefficient rD of the turbines aloft is also called the thrust coefficient which is used for wind turbines or propellers for planes. The thrust coefficient depends of the propeller design https://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node86.html. On the report it is 1, that is a rather high value.

On http://s3.amazonaws.com/zanran_storage/www.gl-garradhassan.com/ContentPages/107547242.pdf,
Figure 1, the generic thrust coefficient curve goes from 1.5 to 0.3 as the wind speed increases from 4 m/s to 25 m/s, and some thust coefficients for some MW range wind turbines are still lower, about 0.1 for 1-2 MW-class, about 0.2 for 0.5-1 MW-class , at 25 m/s wind speed.

So if the thrust coefficient rD of the turbines aloft is 0.2 with 26.6 m/s apparent wind speed, the kite area could be 5 times smaller, so 44 m² and as always two times the drag of the turbines aloft in order to keep the optimization. The power becomes also 5 times lesser, 75596.68 W then 49137.842 W after cosine loss. The power coefficient of the turbines aloft decreases as their thrust coefficient also decreases. This can be due to the pitch or stall control in wind turbines http://drømstørre.dk/wp-content/wind/miller/windpower%20web/en/tour/wtrb/powerreg.htm .

So Optimization of a soft wing with turbines aloft is now corrected for the part about Makani M600 of which proportions between the turbines aloft and the wing are more appropriate than what I expected. See also some observations on Makani's presentation in AWEC2017.