Rotor Thrust Coefficient

To compare a light two-seater with a heavy-lift machine you have to divide the thrust by something, and for rotors that something is $\rho A (\Omega R)^2$: the air density, the disc area, and the square of the tip speed. So $C_T = T / \rho A (\Omega R)^2$. It is disc loading made dimensionless, and it is the currency rotor aerodynamics is quoted in. Main rotors run about 0.004 to 0.010.

THE CONVENTION TRAP, AND IT IS THE SECOND OF THIS SHARD'S CLASSIC MISTAKES. An aeroplane's lift coefficient divides by the dynamic pressure $q = \tfrac{1}{2}\rho V^2$, so the half is built into it. The rotor thrust coefficient, in the standard British and American convention and on this page, divides by $\rho A (\Omega R)^2$ — with no half at all. Some European texts and some rotor codes do include the half, and the result is that a $C_T$ quoted without its convention is off by exactly a factor of two. A rotor at 0.0072 in one convention is 0.0144 in the other, and both numbers look perfectly reasonable. This is the direct cousin of the reference-area trap the lift equation page teaches: an aeroplane's $C_D$ is divided by wing planform while a car's $C_d$ is divided by frontal area, and comparing them without saying so is a tidy way to be badly wrong in public. The general rule is the same in both cases. A coefficient is defined by the equation that produced it. Write the definition down beside the number, always, or the number is not data.

Why tip speed rather than some other reference velocity? Because the tip speed is what sets the dynamic pressure over the outer part of the blade, which is where nearly all the thrust is made — the local speed varies as $r$ and the pressure as $r^2$, so the outer third of the radius does most of the work. Tip speeds cluster tightly, around 200 to 220 m/s or 650 to 720 ft/s, for reasons the advancing-tip Mach page explains, and that clustering is what makes $C_T$ such a useful comparator: two rotors at the same $C_T$ really are working their air similarly.

Note what $C_T$ does not tell you. It says how hard the DISC is working, and the disc is mostly empty. It says nothing about the blades, which are the parts that stall. For that you divide by solidity and get the blade loading coefficient, which is the next page and the one to look at before deciding whether a rotor has any margin left.

This page takes the tip speed directly rather than asking separately for rotor speed and radius, so that a reader cannot enter a disc area and a radius that disagree with one another. If you have rotor speed in rpm, the tip speed is $V_{tip} = \Omega R$ and the linear-speed-from-rotation page will convert it, remembering that $\Omega$ must be in radians per second: 300 rpm is 31.4 rad/s, and the factor between them is $2\pi/60 \approx 0.1047$.