Can I model propellers used in electric aircraft the same way as I model RC quadcopter drones?

Question

I am writing an article about mathematically modelling a flying car, using D'Alembert's principle. I need some information about the propellers ... When I worked with the mathematical model of a small RC quadcopter I found that many researchers consider the thrust of the propeller proportional with the square of its angular velocity and the same for the torque of the propeller. I am interested in something similar for the bigger propellers ... Is the assumption still holding? Can someone give some numerical values for the constants? The flying car (VTOL vehicle) I am modelling has six propellers, therefore I need less then 1000 N of thrust for each propeller. Therefore my first question is: Is it ok to say

1. $F = a \cdot \omega^2$
2. $T = b \cdot \omega^2$ for big propellers? $F$ is the thrust of the propeller along its axis, and $T$ is the torques developed by the motor to rotate the propeller. Please give some examples of commercial products ... if possible. Thank you!

Answer 1

The thrust T of a rotor can be given as: $$ T = C_T \cdot \rho \cdot A \cdot (\omega \cdot R)^2$$

With:

• $\rho$ = air density [1.225kg/m$^3$] at sea level
• $A$ = area of the rotor disc [m$^2$]
• $\omega$ = angular speed of the rotor [rad/s]
• $C_T$ = thrust coefficient.

$C_T$ depends on the configuration of the rotor: blade pitch, number of blades, blade chord etc. It can range from 0 to 0.01, without knowing more about the config of your rotor we can take the average at 0.005 the value 0.001, see update below.

The torque Q follows a similar format:

$$ Q = C_Q \cdot \rho \cdot A \cdot (\omega \cdot R)^2 \cdot R $$

For a given $C_T$, we can look up $C_q$ in the following figure, valid for a propeller solidity of 0.1. For a $C_T$ of 0.005 we find a $C_Q$ (= $C_P$!) of 0.0004, for the updated $C_T$ of 0.001 the $C_Q$ = 0.00015 Source:

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Update

The above $C_T$ is valid for helicopter rotors. For quadcopters with their much smaller rotors, induced power losses, tip losses etc are much higher. This document gives a detailed discussion on $C_T$ of small rotors, although a direct comparison cannot be made since they use a different definition of $C_T$. With this in mind, and using the definition of $C_T$ as above, a better choice for $C_T$ would be 0.001, and the corresponding $C_Q$ is 0.00015.