The Great Copter’s Theorem

Upd: I made online tool to see hovering time vs battery capacitance or fuel tank.

Theoretically possible maximal thrust of a propeller engine is determined by engine power, size of the propeller and density of air (e.g. http://www.dept.aoe.vt.edu/~lutze/AOE3104/thrustmodels.pdf ).

(1)

Where P is engine power, D is diameter of propeller and r is air density. If copter has N propellers (for example N=4 for quadcopter)

(2)

At hovering thrust is equal to weight of the copter (T_total=M_total*g), where g is gravity acceleration. The power needed to keep copter in air can be determined from (2):

(3)

Hovering time depends on energy stored in a battery or in a fuel tank: Time=E_bat/P_total (where E is an energy in W*h). Mass of fuel or battery is determined by energy density M_bат=E_bat/w, where w is energy density (W*h/ kg). And hovering time becomes:

(4)

Where M_craft is mass of copter and a load, excluding battery (or fuel).

Maximum possible flight time vs battery (fuel) energy looks like (from (3)):

First, with increasing of battery (fuel) weight, hovering time gets longer (almost linearly), after maximum it decreases because of battery weight. Maximum possible flight time is reached at point A in the above plot.

At maximum (point A) the derivative must be equal to zero:

(5)

Using (4) and (5) we get:

(6)

or

(7)

Maximal hovering time is reached, when fuel (or battery) weight is equal to double weight of the copter.

This is exact equation and does not depend on type of the fuel. Fuel may be gasoline, electrical battery or other type, even human power (with some abstraction)).

This is so universal and simple. I call it “The great copter’s theorem” 🙂 (CGT). (actually this equation is not specific to type of engine etc, this is general equation from dependence of energy and weights).

Using (6) and (4) we get maximal flight time:

(8)

Where N_rotor is number of craft’s propellers, M_craft is mass of the craft without battery (fuel), w is energy density and D is diameter of the propeller. Using constants, we get incredibly simple approximate equation (), I consider it as a part of GCT:

(9)

T_max is expressed in minutes w is energy density (W*h/kg), D (m), M_craft (kg). Here is some mix of SI units and units used in common life (minutes). Also do not expect proper units, this is a price of simplicity (all coefficients are gone).