Commentary to the paper A Betz-Inspired Principle for Kite-Power Generation…(S. Costello et al., preprint 2014)

by Giancarlo Abbate

First of all, it should be noted that this paper has been submitted in 2014 to the Elsevier Journal Renewable Energy but it was never published. In fact, a search in the journal site gives a “no results were found” answer. The paper however is available at the main author’s academic site at the following link (https://infoscience.epfl.ch/record/181236?ln=en).

The sole scope of the present commentary is to apply the authors’ findings at a particular practical case, namely at kite developed for the KiteGen (KG) project. Thus, I won’t enter in any details of the demonstrations all along the paper. If the case, I’ll just comment shortly some assumptions made.

The KG project kite was designed, realized and tested (preliminary tests) at the KiteGen facility in Turin with the aim to build up a first industrial prototype for energy harnessing from high altitude winds, so having a theoretical estimation of its power limits in different flying conditions is certainly relevant. However, I must warn the reader about the limited validity of the paper results, because it cannot be considered as public knowledge, accepted by the scientific community.

The KG kite is a semi-flexible airfoil made of composite materials, having a hybrid Eppler-NACA profile. It is composed by 9 rigid sectors linked together by flexible joints. Its main measured parameters are:

1. Total mass: m=250 kg

2. Surface area: A=120 m²

3. Wing chord: l= 4m

4. Lift-to-drag coefficients ratio: C_l/d=28

For sake of simplicity, it is assumed a lift coefficient C_l=1. However, this is a quite conservative assumption because for airfoils of similar profile and the high Reynolds number, which characterizes the KG kite flight conditions, C_l is always greater than 1 (and may arrive to values around 2).

Now, after their maximization procedure to find the limit, i.e. the maximum power that a flying kite can extract from a wind of given velocity, the authors arrive to the following formula:

Where r is the air density, V_w is the wind velocity, and C_D the drag coefficient. The quite obvious conclusion was that the ‘lift power’ configuration is optimal. At this point, the authors introduce the right argument that this power must be transferred to the ground by means of a ‘force’ and that force is the rope tension, so it is parallel to the rope. As a consequence, of primary importance is the angle between the rope and the aerodynamic force, say angle z. After a series of demonstration, they arrive to an efficiency factor that is e=cos³(z₀), where z₀ is the angle z averaged over the whole flight path. Actually, the efficiency factor e can be slightly smaller than that, but the difference is in most cases negligible, as it is actually the case for the KG kite. The factor e for an angle z₀=48° is e=0.3. Of course, it increases with decreasing angle and vice versa. Therefore, the final formula for the maximum power that can be transferred to a ground generator is:

It is worth noting that in the paper there are some assumptions, which in general are limiting the maximum power, that are not necessary and, in one relevant case, also questionable. First, the wind direction is assumed horizontal and constant, and this will be not always the case. Second, more important, z₀ is assumed as mainly dependent on the tether-to-ground angle F_TG, and this is not correct, because the tether is a flexible body that can change its direction, and the direction of the force that transmits, for instance by means of a pulley, without any efficiency loss, apart from the rope/pulley friction. Moreover, the direction of the tether close to the kite is certainly quite different from the tether direction close to the ground. All these considerations mean that what Costello et al. call ‘Practical Upper Bound for a Generic Kite System’ are in fact a too severe constraint and we shouldn’t be surprised to find greater experimental power values.

Anyway, let’s continue and apply the paper final formula, with the small efficiency factor e=0.3 provided in the paper, to the KG kite parameters. We obtain

· for a wind speed, V_W=10 m/s,

· for a wind speed, V_W=15 m/s,

· for a wind speed, V_W=20 m/s,

If we, instead of using the measured Lift-to-drag coefficients ratio C_l/d=28, want to use in the formula a much lower value, say C_l/d=20, we obtain the following:

· for a wind speed, V_W=10 m/s,

· for a wind speed, V_W=15 m/s,

· for a wind speed, V_W=20 m/s,

Considering that the target maximum power, at which the KiteGen machine should work, is 3 MW, it is easily deduced that this target can be reached, according to the findings of the paper, at a wind speed of 13 m/s with a kite of C_l/d=20, and at wind speed of 10,4 m/s with the present KG kite. Both these speed values are reached by wind at the working altitude of the KG project (800-1500 m) for a large fraction of time, confirming its feasibility and correctness. Moreover, considering the excessive limitations imposed by some unnecessary, or incorrect, assumptions of the paper, a final conclusion of this commentary is that even a kite with a smaller area, or exhibiting a smaller aerodynamic efficiency, might be suitable for building a KG machine of nominal power in the MW range.