10
$\begingroup$

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!
$\endgroup$
  • $\begingroup$ Why the down vote ? $\endgroup$ – C Marius Sep 7 '17 at 22:03
10
$\begingroup$

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 Fig 2.7 Leishman Source:


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.

$\endgroup$
  • $\begingroup$ No worries mate. $\endgroup$ – Koyovis Sep 8 '17 at 7:44
  • $\begingroup$ To be honest I am interested in something between helicopter and small quadcopters ... I am interested in a flying car of mass $\approx 500 kg$ and having six propellers ... According to Fig. 11 page 20 in that document I think I will consider $C_T = 0.002$ and $C_P = 0.0002$ ... Is this OK ? $\endgroup$ – C Marius Sep 9 '17 at 19:41
  • $\begingroup$ Yes that would bring the rotors in the same size category as the tail rotor of the Robinson R22 helicopter, R = 0.533m, chord = 0.1m, 2 blades. Not easy to cross-reference all design data but your assumptions seem to be on the safe side. $\endgroup$ – Koyovis Sep 10 '17 at 4:57
  • $\begingroup$ Great! Thank you sir for sharing your knowledge! $\endgroup$ – C Marius Sep 10 '17 at 19:59

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.