How does blade solidity ratio relate to thrust/power/torque of a propeller?

Question

for quite some time I was using ABBOTT formula for static thrust estimation on a two-bladed propellers:

$$ T=6.8\times10^{-5}\times D^{3}\times p\times RPM^{2} $$

$T$ is Static Thrust (N); $D$ is propeller diameter (m), $p$ is propeller pitch (m); RPM (1/s)

I was, however, unable to find any formula which would include number of propeller blades.

Standard thrust estimation methods all seem to rely upon the "propeller disk area" which is always assumed for a two-bladed propellers.

The closest I was able to get is something called "blade solidity ratio", or "rotor solidity":

$$ \sigma = (B c)/2\pi R $$

where $B$ is number of propeller blades, $c$ is chord of each blade, $R$ is radius of the rotor.

The question is - how does blade solidity relate to thrust/power/torque?

Answer 1

Each propeller blade is a wing in itself, and like a wing carries the weight of the plane, the propeller blade carries its fraction of the total thrust of the propeller. The more blades, the lower the fraction of each blade.

Low disc loadings are associated with two- or three-bladed propellers. Those can be found on GA aircraft and older, slow designs like pre-WW II aircraft. With turboprops and near-transsonic designs, more blades are needed to distribute the aerodynamic loads and to reduce the lift coefficient especially at the tips.

There is no strict formula, but in general a higher disk loading is associated with a higher solidity ratio.

Answer 2

You'll find this info in helicopter performance design literature. The power to drive the rotor (and a propellor) can be sub-divided into three parts: useful power, induced power, and profile power. The solidity ratio shows up in the profile power part.

Like in fixed wing aerodynamics, the power/thrust equations are often made dimensionless and evaluated in coefficients. From Helicopter Test and Evaluation by Cooke and Fitzpatrick, section 2.4:

$$C_P = C_T \left( \frac{V_C + v_i}{V_T} \right) + \frac{s \cdot C_D}{8} $$

with $s$ = solidity ratio, $V_C$ = climb speed, $v_i$ = induced velocity, $V_T$ = blade tip speed. So solidity ratio shows up in the profile drag power portion, which makes sense.

Answer 3

Let's calculate the thrust given by a blade of airfoil NACA 0012, length r, constant chord c, and pitch p (constant along the blade). Prop turns at Omega rad/s. Derivation taken from https://aviation.stackexchange.com/a/80626/16042 All units SI. The pitch angular value p is in radians.

The differential expression for the thrust given by element area ds=dr·c of one blade is: $$dL=2,86\cdot c\cdot \rho \cdot (p\cdot \Omega ^{2}\cdot r^{2}-w\cdot \Omega r+p\cdot w^{2}-w^{3}/\Omega r)\cdot dr$$

This expression accounts indirectly for the solidity of the propeller, since the chord is one of the variables. And, after all, as solidity s= c/π·r, the differential expression above may be re-written inserting s·π·r in place of c…

Going back to the original differential expression above, and for constant values of chord, air density, inflow, prop angular velocity and blade pitch, we may integrate only the first term, since in the static case inflow is zero or close to zero. So we have that, for one blade:

$$L=2,86\cdot c\cdot \rho \cdot p\cdot \Omega ^{2} \int_{0}^{r}r^{2} dr$$

Inserting values for an example two-blade propeller with blade length 0,86m, Omega = 2124 rpm = 222 rad/s, rho = 1,23 kg/m3, chord = 0,12 m, blade pitch = 14º = 0,244 rad

$$L=2,86\cdot 0,12\cdot 1,23 \cdot 0,244\cdot \ 222^{2} \int_{0}^{0,86}r^{2} dr$$

Integrating…

$$L=2,86\cdot 0,12\cdot 1,23 \cdot 0,244\cdot 222^{2}\cdot 0,86^{3}/3$$

We end up with a thrust of 1076 newton per blade… That’s 2152 N for the two-blade prop. Almost the same value as with Abbott’s formula… (Values for pitch & chord duly tweaked so that the result agrees with Abbott's…)