Propellers and momentum theory

The first lesson quoted a scaling law for thrust and promised a derivation. The derivation is one of the most useful pieces of physics in the whole subject, because it does not just give a formula, it explains why a quadcopter that flies for twenty minutes must be built around large propellers turning slowly, and why the racing habit of small propellers turning fast is paid for in flight time. The model is called momentum theory, and it replaces the messy reality of spinning blades with a single idealised disc.

The actuator disc

Picture the propeller as an infinitely thin disc of area $A = π R^{2}$ , where $R$ is the blade radius, that does one thing: it accelerates the air passing through it, pushing it downward. Far above the disc the air is still. As it approaches it speeds up, at the disc it is moving down with an induced velocity $v_{i}$ , and far below it has reached a final velocity $w$ . Thrust is the rate at which the disc adds downward momentum to the air, and by conservation of momentum that rate is the force pushing back up on the disc.

The mass of air passing through the disc each second is $\dot{m} = ρ A v_{i}$ , the density times the area times the speed at the disc. The air enters with no downward speed and leaves far below with speed $w$ , so the momentum added per second, which is the thrust, is

T = \dot{m} w = ρ A v_{i} w

A short argument from energy, which we take on trust here, shows that the air reaches half its final speed by the time it crosses the disc, so $w = 2 v_{i}$ . Substituting gives the central result of momentum theory:

T = 2 ρ A v_{i}^{2} \Rightarrow v_{i} = \sqrt{\frac{T}{2 ρ A}}

The induced velocity needed to make a given thrust falls as the disc area rises. A larger propeller makes the same thrust while throwing a larger mass of air more gently, and that gentleness is exactly what saves energy, as the next section shows.

Why hovering is expensive

The power the propeller must deliver to the air is the thrust times the speed of the air at the disc, $P = T v_{i}$ . Substituting the induced velocity gives the ideal hover power:

P = T v_{i} = T \sqrt{\frac{T}{2 ρ A}} = \frac{T^{3 / 2}}{\sqrt{2 ρ A}}

Two features of this expression govern flight time. First, power grows as thrust to the power of three halves, faster than thrust itself, so a craft that is fifty per cent heavier needs about eighty per cent more power to hover, and weight is punished more than linearly. Second, and this is the design lever, power falls as the square root of the disc area. Double the disc area, by making the propellers larger, and the hover power drops by a factor of $\sqrt{2}$ , about thirty per cent, for the same thrust. This is the physics behind the endurance rule: a long flight wants the largest propellers the frame can carry, turning as slowly as they can while still making the weight in thrust. A racer accepts terrible hover efficiency in exchange for the light, small, fast propellers that give instant response.

Example. A 500 g quadcopter hovers on four 5 inch propellers, radius about 0.064 m, so each disc has area $A = π (0.064)^{2} = 0.0129$ m². Each propeller carries a quarter of the 4.9 N weight, so $T = 1.23$ N. Taking $ρ = 1.225$ kg m⁻³, what is the induced velocity and the ideal power per motor?

$v_{i} = \sqrt{1.23 / (2 \times 1.225 \times 0.0129)} = \sqrt{38.9} = 6.24$ m s⁻¹. The ideal power is $P = T v_{i} = 1.23 \times 6.24 = 7.7$ W per motor, so about 31 W for the craft. Real propellers are far from ideal, so the true figure is two to three times this, but the scaling is exact and it is the scaling that matters.

Now you. The same craft is rebuilt on 7 inch propellers, radius about 0.089 m, disc area 0.0249 m², same thrust per motor. Find the new induced velocity and ideal power per motor.

Answer

$v_{i} = \sqrt{1.23 / (2 \times 1.225 \times 0.0249)} = \sqrt{20.2} = 4.49$ m s⁻¹, and $P = 1.23 \times 4.49 = 5.5$ W. Larger disc, gentler air, less power for the same lift.

From the disc back to the scaling law

Momentum theory as written involves the induced velocity, which is not something a builder chooses directly. What a builder chooses is the diameter $D = 2 R$ and, through the motor, the rotation rate $n$ . A blade element moves through the air at a speed proportional to $n D$ , and the thrust a blade makes rises with the square of its speed and with its area, which is proportional to $D^{2}$ . Collecting the factors gives the static thrust law of the first lesson:

T \propto ρ n^{2} D^{4}

and the same reasoning applied to power gives $P \propto ρ n^{3} D^{5}$ . These two are how propellers are actually specified in practice, folded into a single number each. Manufacturers publish a thrust coefficient and a power coefficient, and the constants that turn the proportionalities into equalities absorb the blade shape, the number of blades and the pitch. The lesson to carry forward is the exponents. Thrust goes as the square of speed and the fourth power of size; power goes as the cube of speed and the fifth power of size. Speed is cheap in thrust and expensive in power, which is the same statement as before that hovering fast is wasteful.

Pitch, the second dimension of a propeller

A propeller is sold by two numbers, its diameter and its pitch, written together as 5x4.3 or 7x4. The pitch is the distance the propeller would advance in one turn if it moved through the air like a screw through wood, and it sets how aggressively the blade bites. A high pitch propeller moves more air per turn and reaches a higher top speed, but it demands more torque and draws more current, and it stalls more readily when the craft is climbing hard. A low pitch propeller is gentle, efficient and forgiving. Pitch is the fine adjustment that diameter is the coarse one: pick the diameter for the class of craft, then the pitch to trade top speed against efficiency and load. A propeller too aggressive for its motor overheats the motor and the electronics, which is the mechanism behind a whole category of build failures, and the power law is the warning: pitch pushes on the same cubic power curve that size does.

What the model leaves out

Momentum theory is an idealisation and knows nothing of the blade itself. It assumes the air arrives smoothly, ignores the swirl the blade leaves behind, and treats the disc as loss-free, so its power figures are a floor that no real propeller reaches. The ratio of the ideal power to the real power is a propeller's figure of merit, typically around 0.7 for a good small propeller, and it is where blade design, tip losses and the air's viscosity all hide. None of that changes the exponents, which is why the model earns its place: it will not predict a propeller's thrust to the gram, but it explains, correctly and for good, why the endurance machine and the racer look so different, and why the single most effective thing a builder can do for flight time is fit the biggest propellers the frame will take.