Multi-Rotor Propeller Thrust Formulas

During the construction of my unmanned aerial vehicle (AUV), I needed to know how much thrust my motors would produce, or more, how much mass they could support. The objective is to optimize the mass to thrust ratio using differentiation. After about two hours of research and assistance, I found the formula for thrust, the relation to how much mass the given thrust can support. Next, a friend of mine demonstrated how to solve for air velocity before knowing the math by using the second method of solving a system of equations.

Thrust:

According to AirWolf II, "A commonly used rule is that velocity of the air at the propeller is v=½Δv of the total change in air velocity."

Where,

T	Thrust which is a kind of force (F).
D	The diameter of the propeller (m)
ρ	Air density - approximately 1.225 kg / m3
v	The velocity of the air pulled into the propeller (m/s)
Δv	The velocity of the air pushed away from the propeller (m/s)

Next, we can relate mass to the thrust by substituting T with the function for force (Force = mass (m) x acceleration (a)) and solve for m, giving us:

In order to find mass, we need to first know Δv - this is what slowed me down for a little while. So, in order to solve for Δv without first knowing the mass, we'll need to create a second equation (power = force x velocity):

Where,

η	Efficiency of motor (approximately .75)
W	The max wattage of the motor (W = IV or W = I2r  or  W = V2/r)
C	The scaling factor from 0 to 1 (one being maximum throttle)
m	Mass (g)
a	Acceleration
Δv	The velocity of the air pushed away from the propeller

Substitute and solve for Δv:

Then to solve for mass given thrust, simply plug Δv into one of the above formulas and vuala, mass is related to thrust and can be adjusted by either the power of the motors or the size of the propellers:

A real-world example using this formula:
I have brushless motors that run with a max current of 10.5A at 11.1V with a prop radius of 4.5" (0.1143m). Given the following:

C: 100%	η: 80%
ρ: 1.225 kg/m3	D: 0.2286 m
a: 9.8 m/s2	W: 116.55

...each motor with it's mounted prop running at 80% efficiency and the throttle maxed, can carry a maximum approximate weight of 615 grams (or 0.614646... kg). Since I have 6 mounted to my chassis, my AUV will be able to support a maximum [hover] load of approximately 3.69kg. (since the motors are arranged in pairs, this number may be further increased via an increase in efficiency by re-evaluating for v given the previous solved equation)