Hello,

there was a discussion on how battery capacity should be chosen in another posting today. I therefore sat down and gave the problem some thoughts. After some time I had a solution and wrote the results down in a short report (see attachment battery_capacity.pdf ).

For those of you, not interested in reading the whole report, I found a very short and simple way to choose the battery capacity to optimize flight time. The calculations show that the battery should be twice as heavy as the airframe without batteries (but all other things installed).

I guess this rule gets as simple as it can :D

Regards,

Andrés

p.d. I added a new version of the report (just use link above)

## Comments

further on the topic. This includes a calculation for helicopters and also a

calculation of the best compromise between flight time and total airframe

weight.

@I.S.: I made a calculation for hovering helicopters and it turns out that the

optimality conditions are the same (just the efficiencies change). So the

results are also valid for helicopters ;)

@Fabien: I checked for the best compromise between flight time and total

weight. It seems to be at a mass ratio of about 0.8, i.e. the battery should

weight about 80% of the battery-free airframe. If I remember right, the

EasyStar weight about 750 g without battery, so taking a battery of about 600g

should be a good compromise.

@Darren & Ramón: Yeah, there are a lot of effects, which influence flight

times. However, at the end of the day, when you design an airframe, you cannot

rely on e.g. thermals being on the airplanes way. It would be great to have an

autopilot, which thermals by itself, though ... One thing I really have to

consider, is the energy needed to climb to the level flight height. Next

version ;)

This is exactly the sort of calculation I was suggesting in the other topic, your report looks very professional.

So it would seem a 1.5 kg battery is the best to put in an Easyglider? It sounds kind of heavy, but it would be cool to trey and see if it still flies! (or with 1kg worth of batteries, considering your remark about the plane stall)

@Andrés

All your calculus and the "simplified formula" is only for planes (fixed wing), or it can be also applicable to rotatory wings as multicopters?

Some things to consider... have you taken into account some real world loiter time extending events? for example utilizing thermal lifting during missions, or assisted assent, perhaps powered and unpowerd (rockets vs balloon). So begin usage of battery only at altitude; or how about regenerative braking, such as during power off events like a desent to the next waypoint... probably more but should stop there... lol

edit: major typos and a revision or two... ;)

@Niklas: Thanks. Imho, there's nothing better for equations than LaTex :D

@Abey: Sure, C_Lis bounded and lift is produced then by increasing flight speed. This ends up easily in problems at take-off, etc. (see comments at the end of the short report). However, the optimal value of twice battery weight compared to airframe weight might be still be applicable (in particular for EPP airplanes). Nonetheless, I think that less battery is still fine and more affordable. Thank's for the reference, I'll have a look at it. Optimizing a solar airplane is a special situation, because you have also to optimize for span/wing area and keep also constrains on stiffness and solidity. I did the calculation some months ago and ended up with an airplane with an impressive span of above 40m span (remember NASA's Helios?).

@Marko: The linear relation is only an approximation, but for example, I tend to increase capacity or voltage by using mutliples of a battery. So, iguess assuming a linear relation is quiet realistic ;)

Very nice analysis! If I read it correctly, you make the assumption that the energy content of the battery is strictly proportional to its mass. This is probably not too far from the truth, but I would imagine that there is at least some "fixed overhead" weight for a battery, i.e. m_{bat} = m_{0,bat} + \alpha * E. I don't think a 4400mAh battery is exactly twice as heavy as a 2200mAh one (but then again, maybe I am wrong and it is...).

Again, I really like your approach, and I think the result in general is very useful!

Marko

For a given airframe, C_L is bounded, i.e. there is no way to increase lift beyond some maximum value (unless you increase significantly the speed, which requires a significant scaling of the propulsion system, which will impact m_0, etc.). Therefore, Equ. 1 cannot be true for any value of m. You can demonstrate this problem experimentally by taking some airframe and progressively increasing its weight by adding payload (camera, air quality sensor, rocks,...) and increasing battery size accordingly (twice the total weight). Soon enough, your aircraft won't even take-off.

I suggest you take a look at André Noth's PhD thesis (linked on this page: http://www.sky-sailor.ethz.ch/publications.htm). He has modelled a large number of UAV subsystem (airframe, propulsion systems, energy storage systems, etc.) to optimally design a solar-powered glider. His approach can very well be applied to pretty much any airframe.

Very well written report, and LaTeX ftw!

I have not had the time to confirm your results, but they seem reasonable at first glance.

/Niklas