How to calculate lift/thrust effect of adding blade to a propeller?

Question

Source:

Say I have blades of a propeller of an aircraft (airplane or helicopter). The blades have exactly the same profile, size, dimension, and weight, and will be coupled to a shaft with the same shaft power. I will test the propeller if I have only install two blades. Then I will try 3 blades, then 4 blades (of course there will be engineering work to make them work). My question is, what is the formula to calculate the thrust/lift blades number is in variable (say in n-blades, which by adding more blades will add weight too to the total load).

Answer 1

If the geometry of the blades is kept constant, then thrust and torque is simply proportional to the number of blades $N_b$.

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What you need is called Blade Element Momentum Theory (BEMT) and it is something more advanced than the simple momentum theory (where there's no sign of actual blade geometry) but still simple enough to be solvable by hand.

It is based on the hypothesis that any blade is composed by a continuous sequence of adjacent airfoils, each of them generating therefore lift and drag according to the simple and well know $l=½ \rho V²S C_l$ and $d=½ \rho V²S C_d$.

$V$ is the local airflow's speed seen by each slice of blade and is given by the sum of the following 3 terms:

1. the speed due to the rotation of the blade around the shaft; this speed varies linearly along the bladespan from 0 at the root to $\omega R$ at the tip, where $\omega$ is the rotating speed and $R$ the bladespan;
2. the speed at which the propeller is flying with the aircraft;
3. during its rotation, the blade bumps into the wake shed by the previous blade; and after one complete rotation, it bumps into its own wake; these wakes have also to be taken into account.

Terms 1. and 2. are know, since $\omega$, $R$ and the aircraft's speed are known; the speed 3. due to the wake depends itself on the lift and gives rise to a vicious circle: the lift depends on the wake and the wake depends on the lift! This vicious circle is broken by either using simplified models of the wake and/or wind tunnel measurements and/or CFD simulations.

Once lift and drag for each slice of blade have been calculated, they are summed up (integrated) along the bladespan, from root to tip, to get total lift and total drag per blade; these lift and drag are finally decomposed in 1) a force parallel to the propeller shaft, which is the thrust generated by the propeller; and 2) a force perpendicular to it, which gives the torque needed to make the propeller spin.

Let's make a simple example built upon the following simplifications:

• the aircraft is at rest, i.e. term 2. is null;
• the wake in/upon the propeller is constant everywhere, i.e. term 3. has a constant value;
• the blades have a hyperbolic twist, which is a good approximation of real propellers;
• and each airfoil is operating at its $\alpha$ of maximum efficiency i.e. maximum $C_l/C_d$; this is also a good approximation for variable-pitch propellers.

Then we get that:

$C_T=¼ \frac{N_b c_{tip}}{\pi R} C_{l_{\alpha}} \alpha_{@maxC_l/C_d}$

where:

• $N_b$ is the number of blades;
• $c_{tip}$ is the chord at the tip of the blade;
• $C_{l_{\alpha}}$ is the slope of the airfoil's lift coefficient;
• and $\alpha_{@maxC_l/C_d}$ is the airfoil's $\alpha$ for maximum efficiency.

Also for cases more complicated than the one in this example, BEMT can be used to get $C_T$. For a complete overview I'd suggest you to have a look at some standard books about helicopter aerodynamics: as seen, the theory behind isn't that complicated but for sure is a bit lengthy.

Note that in the propeller (or helicopter) world a coefficient of thrust $C_T$ exist and not of lift! From $C_T$, the thrust is calculated with an equation similar to the one for lift or drag but based on the speed of the blade's tip and the surface of the propeller disk:

$T = ½ \rho (\omega R)^2 (\pi R^2) C_T$

P.s.: if you are interested, I gave a very similar answer but applied to Ingenuity.

Answer 2

Propeller thrust ratings are (unfortunately) avoided by many people because of a variety of factors that affect prop performance.

Matching props to engines can be done by trial and error, refined by previous experience matching engine horsepower to prop pitch and diameter.

However, prop thrust output is strongly dependent on airspeed.

Let's go from a 2 blade to a 3 blade. The engine must work harder to achieve the same RPM due to prop drag. Assuming it can reach the same RPM (this would usually take a larger engine or turbocharging), the new found thrust improvement can be tested as follows:

Try flying Vy and see what your pitch angle is to the horizon. Assuming no changes in aircraft drag, your 3 blade should be giving a greater angle of climb.

Now you can calculate the thrust improvement by comparing 3 blade sine climb angle × weight and comparing it to the 2 blade sine climb angle × weight.

What will probably be found is that the increase in rate of fuel consumption will be greater than the performance gain because of prop blade interference. If fuel consumption (horsepower) is held constant, the 2 blade will outperform the 3 blade.

Fuel consumption flying Vy in level flight provides some idea of aircraft drag at that speed (for total thrust estimation), but will be somewhat skewed by prop and engine efficiency issues at lower operating RPM. Airframe drag can also be calculated from glide ratio as sine angle of descent × aircraft weight.