Propeller efficiency and propulsive efficiency

Propeller efficiency, or the propulsive efficiency of a propulsive propeller is the ratio of the useful power for forward motion to the kinetic power it gives the fluid: Rp = 2 / (1 + V2/V0). The less the propeller accelerates the fluid, the better this efficiency, which is why a large propeller moving a lot of fluid slowly is more economical than a small one.

This notion deserves a double explanation: first in equations, then in words. It is essential to propeller design, but a good propeller is first of all one suited to its purpose. A helicopter propeller hovering has zero propulsive efficiency (the vehicle does not advance) and is still a good propeller; the notion is mostly useful to measure the cost of travel as a function of distance covered.

In this page:

1: Propeller propulsive efficiency: equations and formula

The efficiency of a propeller compares the power it produces with the power supplied to it. For this efficiency to mean anything, we must specify how each power is determined.

The Froude's theory for propulsive propellers gave us the thrust (or traction) of the propeller:

T = m° × (V2 − V0)

Stream tube around a propeller: speed V0 far upstream, V1 at the propeller plane, V2 in the downstream jet, thrust T, jet radius r and control surface radius R

The stream tube crossing the propeller contracts downstream: the fluid goes from V0 far upstream to V1 at the disc, then to V2 in the jet of radius r. The thrust T is the reaction to this acceleration.

Creating thrust therefore means increasing the speed of a mass flow of fluid. Now, a power is a force times a speed. With the forward speed V0 and the thrust, we obtain the useful power Pu:

Pu = T × V0 = m° × (V2 − V0) × V0

This is the power that matters for transport, because it introduces useful displacement. If the vehicle is stopped, moored or with its brakes locked on the runway, V0 is 0 and so is Pu. Yet the propeller turns, an engine feeds it, and it moves fluid: its power is not zero. We must therefore measure it independently of the forward speed.

To do so, we compute the kinetic power of the fluid downstream, minus that of the fluid upstream. Recall that kinetic energy is 1/2 × mass × speed²; the available power Pd is therefore:

Pd = 1/2 × m° × V2² − 1/2 × m° × V0²

Even when V0 is zero, the propeller does give power to the fluid. We now have two powers, and their ratio is the propulsive efficiency Rp:

Rp = Pu / Pd = [m° × (V2 − V0) × V0] / [1/2 × m° × (V2² − V0²)]

Rp = 2 × (V2 − V0) × V0 / [(V2 − V0) × (V2 + V0)] = 2 × V0 / (V2 + V0) = 2 / (1 + V2/V0)

This expression comes from Froude's ideal theory: it counts neither profile drag nor the rotation given to the fluid. It already shows the essentials. The further V2 is from V0, the lower the efficiency, and the more energy we waste setting fluid in motion for nothing. When the propeller turns with the vehicle stopped, Pu is zero and so is Rp: propulsive efficiency varies with vehicle speed. Whatever that speed, we look for a minimal gap between V0 and V2.

The thrust T = m° × (V2 − V0) can increase in two ways: by increasing the mass flow rate m°, or the gap (V2 − V0). We have just seen that it is better to keep the gap small. For an economical thrust, we therefore act on the mass flow rate, and the only lever left, at constant speeds, is the propeller diameter. When building our propeller, we will watch the propulsive efficiency and, as far as possible, make it as large as possible.

2: Propeller propulsive efficiency: explanation in words

Let us restate the propulsion problem without equations.

To measure the quality of the propeller, we distinguish two powers at play:

To understand why these two powers matter, imagine a boat moored to the quay, or an airplane with brakes on before takeoff, whose propeller turns and produces thrust.

Photo of a mooring knot on a braided rope

A moored boat: the propeller can push without the vehicle moving.

The vehicle is fixed, so the speed is zero and so is the useful power. Yet the thrust and the power of the propeller are very real: it turns, driven by an engine that consumes energy. To quantify it, we use the change in kinetic energy that the propeller applies to the fluid. Note in passing that the energy consumed with the vehicle stopped is lost. A good propeller, one that carries a load with a minimum of energy, gives the maximum useful power for the minimum kinetic power.

The efficiency of the propeller is therefore the ratio useful power / kinetic power. Breaking the acceleration down to show the role of the speeds:

Note that the useful power is proportional to the speed gap (V2 − V0), while the kinetic power also contains the factor (V2 + V0). The speed gap therefore increases the kinetic power more than the useful power. We need a propeller that creates thrust while changing the fluid speed as little as possible. Since thrust is mass flow rate × acceleration, it is better to increase the flow rate without increasing the speed. A flow rate is a speed × a section: to increase it at constant speed, we enlarge the propeller diameter.

The larger the propeller, the less it has to change the fluid speed to create thrust, and the better its propulsive efficiency.

One may thus be led to place propellers as high as possible, to enlarge their diameter while still allowing takeoff and landing.

Artist's view of a long-winged solar airplane flying at night above the clouds, propellers carried high

Artist's view of a solar airplane at night, propellers carried high.

The Solar Impulse project aims to fly an aircraft powered exclusively by solar energy, autonomously by day and night, all the way around the world without fuel or pollution.

To go further, we can look at the induced speeds of Froude's theory, or follow a complete design with the airplane propeller design tutorial.