Multicopter propulsion theory

I have worked a little on the physics of an electric multicopter (true for helicopters as well) to prove one of the ways to optimize for longer flight time.

Original post: http://www.rcgroups.com/forums/showthread.php?t=1733300#post22754326

Why the maximum flight time for a given frame is obtained with a battery of equal weight (=1/2 of total weight) ?

Firstly, for a propeller we have:
thrust F = kt * p * d^3 * w^2
input power P = kp * p * d^4 * w^3
where p, d – propeller pitch and diameter, kt and kp – specific constants and w – angular velocity.

Playing with the 2 formulas we get: F / P = kt * sqrt(kt) / kp * sqrt(p * d / F)

But we have a given frame(p, d – constant) therefore P = k * F * sqrt(F), where k is a constant that shows proportionality (from now on I’ll use k to show that, but it’ll not be the same k, meaning it will not have the same value), and its value doesn’t matter.

Note that we have worked out P as mechanical power, we need to find out P(el) – that is electrical power drained by motors from the battery.
Now I’m cheating a bit, and assume that P(el) is not propotional to P(mechanical output) for an electric motor, but with P * sqrt(F). Meaning that the more we “push” a motor, the lower its efficiency (I know it’s only half true). The term sqrt(F) might not seem to make much sense at this point, but we’ll see later on that it drastically simplifies the demonstration.

So we write:
P(el) = k * F * sqrt(F) * sqrt(F) = k * F^2

Then F must be equal to copter weight (hover), which is (m+M) * g. M is frame mass(constant), m is battery mass and g – gravitational acceleration.

Therefore:
P(el) = k * (m+M)^2

Now, battery energy is proportional to its mass, E = k * m

Flight time is t = E / P(el)

Substituting P(el) we get t = k * m / (m+M)^2

After playing with inequalities, we find out that the expression m / (m+M)^2 has a maximum value of 1 / (4*M) (remember that M is constant and m is variable), reached if and only if m=M. This means t is maximum when for the given frame, we choose a battery with m=M.

I don’t have enough batteries at the moment to experiment and check the validity of the theory, however I will attach a real data graph I found on the mikrokopter site, which seems to be in accordance.
The maximum flight time is achieved for 14 000mAh capacity which on 4s weighs around 1kg – very close to the empty frame weight (970g) :