Rotation speed and operating point of a propeller Propeller design point and off-design Sizing VTOL propellers through AI prompts

see also: Tutorials how to design propeller

Optimising the rotation speed and the number of blades of a propeller means finding the speed at which its profiles work at their best lift-to-drag ratio, without the blades getting in each other's way. For a propulsion propeller, a wind turbine or a turbine, the rotation speed that gives the best efficiency depends on the number of blades, so the two must be searched together. Heliciel does this with an iterative search from the 3: Optimize tab, and the multiple analysis takes over for a more complete study.

On this page:

• 1: Which factors determine the optimum rotation speed of a propeller?
• 2: Propulsion propeller: turning fast to waste less fluid rotation
• 3: Wind turbine or tidal turbine propeller: harvesting energy with few blades
• 4: The limits of rotation speed: masking, cavitation, sound, stresses, drag
• 5: Speed imposed by the machine and choice of the number of blades
• 6: Optimising rotation speed and number of blades in Heliciel

The 3: Optimize tab for a two-blade propeller: the efficiency curve against rotation speed, the result of the search (maximum kinetic efficiency of 0.861 at 200 rpm), the number-of-blades settings, the cavitation detection and the Multiple analysis button.

A quick search of the optimum speed for a given geometry can be run from this tab: if you wish, Heliciel automatically calculates the rotation speed that gives the optimum efficiency of a propeller whose geometry you have defined.

For a more complete analysis, we use the multiple analysis, which offers many options for displaying and testing performance. When the propeller is coupled to an electric motor or a generator, its actual rotation speed is the one that balances the motor torque and the load torque. To know at what speed the system will turn, you simply superimpose the torque curves against rotation speed.

Propellers that push air or water and propellers that harvest energy (wind turbines, tidal turbines) will be treated alike, for now.

1: Which factors determine the optimum rotation speed of a propeller?

We first assume that we are free to choose the rotation speed of the propeller, and that we look for the one that gives the best efficiency. Here is a short summary of the situation.

The apparent wind received by the blade elements is made up of the rotation speed and the speed of the fluid. We do not discuss the induced velocity here, but of course it must be taken into account. The efficiency of a propeller is that of its elements, which can be compared to portions of a rotating wing. The best efficiency of an element is obtained when its lift-to-drag ratio, the glide ratio, is as high as possible. The best glide ratio is reached at one precise angle of attack, and this is what determines the twist of our blade.

The apparent wind vectors of each blade element, in Heliciel: the further from the axis, the longer the vector and the more it leans towards the plane of rotation.

Our blade is therefore twisted so that the angle between the profile chord and the direction of the actual speed matches the angle of attack that gives the profile its best glide ratio. The lift of the profiles is the useful force that we exploit.

The lift is perpendicular to the apparent wind, but the force that interests us is the thrust (for a propulsion propeller) or the torque (for a driven propeller such as a wind turbine). If the direction of the lift is aligned with that force, we can assume the efficiency will be optimum. By changing the rotation speed, we change the direction of the apparent wind, and therefore that of the lift. So there is, in principle, a rotation speed that orients the lift forces best.

On the left, lift and thrust on a propulsive propeller; on the right, lift and torque on a wind turbine propeller.

2: Propulsion propeller: turning fast to waste less fluid rotation

For a propeller designed for propulsion, lift and drag must give the maximum thrust for the minimum torque. The higher the rotation speed, the more the apparent speed seen by the elements points into the plane of rotation, and the more the lift points in the direction of the desired thrust. From the point of view of force orientation, the optimum rotation speed of a propulsion propeller would therefore be the maximum speed, infinite? No: there is a limit beyond which the efficiency of the propeller drops.

An engine with counter-rotating propellers (the CFM logo is visible on the nacelle). The second propeller partly straightens the rotation that the first gives to the air, which recovers energy; the page on the counter-rotating propeller details this principle.

Another approach, through the energy balance between the upstream and the downstream of the propeller, shows the benefit of a small number of blades and a high rotation speed. The propulsion effect we want is tied, by reaction (Newton's principle), to the acceleration the propeller gives to the fluid. But be careful: only the axial acceleration produces thrust. The tangential acceleration, which is the fluid being set in rotation, produces no thrust at all; it only produces torque on the shaft. The rotation of the fluid downstream of the propeller is lost energy. Two propellers of equal thrust are therefore distinguished by their ability to produce the least rotation of the fluid downstream.

The power delivered to the shaft equals the torque multiplied by the rotation speed: P (W) = torque (N·m) × rotation speed (rad/s). By reaction to the torque, the fluid is set in rotation downstream of the propeller (this is Heliciel's downstream tangential induction). For the same power, a high rotation speed therefore goes with a low torque. Since this torque is directly tied to the rotation of the fluid, we can say that, for equal shaft power, the propeller that turns faster generates less rotation of the fluid.

A propeller fitted with blades profiled for high speeds, with low drag, would therefore be optimum. But the rotation speed is limited by other undesirable effects (see below), such as the masking effect, which leads us to a theoretical optimum single-blade propeller. We say "theoretical" because the rotational balance must be handled mechanically.

A single-blade propulsion propeller, with its counterweight for balance.

The "best" propeller does not exist: everything is a matter of compromise, and each application must invent the propeller that achieves the best compromise.

3: Wind turbine or tidal turbine propeller: harvesting energy with few blades

For a wind turbine or tidal turbine propeller, it is a little trickier. At first, we might think that what we want is torque. The best orientation of the lift would then be perpendicular to the wind, to be entirely devoted to torque. The apparent speed should therefore be parallel to the propeller axis, that is, made up of the fluid speed alone. The optimum rotation speed would be zero: a stopped propeller.

To capture all the energy of the fluid crossing the disc swept by a "stopped" propeller, we would need blades over the whole disc, and there would still be gaps. Pumping wind turbines, which look for torque to drive slow machines in light wind, follow this approach. The most modern ones use the interaction between the blades, like the sails of a sailing boat, to improve efficiency. By our estimates, even the best pumping propellers of this type recover only 15% of the wind energy.

A multi-blade high-torque wind turbine, like pumping machines: many blades, a low rotation speed.

Another approach, through the energy balance between the upstream and the downstream of the energy-harvesting propeller, shows the benefit of a small number of blades and a high rotation speed:

• Let us think in terms of power: P (W) = torque (N·m) × rotation speed (rad/s).
• When we capture the kinetic energy of a fluid with a propeller, we turn that energy into torque on the shaft.
• By reaction to the torque generated (Newton's principle), the fluid is slowed down but also set in rotation (tangential induction).
• The energy balance between upstream and downstream gives the energy captured, and therefore the efficiency of the propeller. The harvesting propeller of optimum efficiency is thus the one that leaves the least energy downstream.
• The energy not captured downstream is made up of the axial speed left in the fluid (within the Betz limit) and the tangential speed induced by reaction to the torque. For two propellers whose downstream axial speed is equivalent, the one that captures more energy is the one that induces less tangential speed, hence the one that produces less torque. To get the same power with a low torque, the propeller must turn faster.
• To avoid losing energy through the masking effect between blades (see below) at high rotation speeds and low wind speed, the smallest possible number of blades is the most effective way to reach the highest possible rotation speed.

The efficiency of our wind turbine propeller is therefore optimum at high rotation speed. Single-blade propellers exploit this option, reckoning that the price of a single blade is a sure saving. This type of propeller is good in theory, but in practice it involves imbalances and rotation speeds that are tricky to manage.

A single-blade wind turbine: one blade only, so a counterweight and a balance to look after.

4: The limits of rotation speed: masking, cavitation, sound, stresses, drag

We could therefore say that the faster the propeller turns, the better its efficiency. But up to what limit? Here are the main causes that limit high rotation speeds:

• Interaction between blades (masking effect). At low rotation speed, the apparent wind is far from the plane of rotation; as the speed increases, it "lies down" in that plane. At a very high rotation speed, the apparent wind no longer "sees" a passage between the blades if they are too close together. The disturbance generated by the blades then makes the efficiency drop: the number of blades makes the efficiency fall at high rotation speeds.

The apparent wind according to rotation speed: at zero rotation it lies along the axis, at a suitable rotation it passes between the profiles, at too fast a rotation it lies against the plane of rotation and finds no passage.

• Cavitation, for a propeller working in a liquid, of course.

Cavitation in a tunnel: the bubbles form just after the leading edge of the blade.

• The speed of sound. The apparent speed at the blade tip must be watched. Near the speed of sound, the performance of airfoils changes and phenomena appear that mean this speed should not be exceeded. Tests of supersonic propellers suggest it might be possible to go beyond it, but the noise and supersonic shocks remain complex to manage.

The XF-88B, a test aircraft for supersonic propellers.

• Centrifugal force and thrust put mechanical stresses on the blades. Centrifugal force can be used to offset thrust, by giving propellers a coning angle; but the materials used sometimes force us to limit rotation speeds. Some manufacturers claim that the lifetime of a wind turbine is inversely proportional to its rotation speed (a claim we report without having verified it).

A wind turbine on fire: the mechanical limits are not just theory.

• The drag of the profiles also grows with the square of the apparent speed. At high speeds, the negative torque produced by the profile drag eats into the efficiency significantly.

Torque and drag on a small wind turbine, in the Heliciel 3D Prototype tab.

5: Speed imposed by the machine and choice of the number of blades

We had assumed that the rotation speed of our propeller was not imposed on us, but the factors that limit efficiency are quickly reached. One of the key factors being the interaction between blades, optimising the rotation speed means searching for the optimum number of blades.

Another constraint imposes the rotation speed: the machine connected to the shaft. An alternator, a pump, an aircraft engine or a ventilation motor has characteristics that the propeller must meet. The rotation speed and the torque may therefore be imposed by the machine. If the gap between the optimum speed of the propeller and that of the machine is too large, a reduction gear must be installed. That lowers the efficiency and raises the cost, weight and maintenance of the system. To match the speed and torque of a propeller to the machine, we can play on the geometric parameters of the blades, but the number of blades helps us change the optimum speed in a fairly simple way.

The number of blades increases the torque and lowers the optimum rotation speed. We can therefore tune the optimum speed by increasing or decreasing the number of blades. To preserve the efficiency, we must still change the geometry of the blades to reduce their interaction. As the number of blades increases, the load per blade decreases, and the blade thickness can be reduced since the blades no longer need to provide as much strength.

These indications are leads: there is no miracle recipe to optimise the rotation speed, and a compromise always has to be made. The short article on the number of blades of a propulsion propeller can also give you some leads.

6: Optimising rotation speed and number of blades in Heliciel

Each time you change the blade geometry, if you click rebuild, Heliciel rebuilds the optimum twist of the propeller and updates the performance at the imposed operating point. The chosen rotation speed determines the twist. But for the reasons given above, this speed may not be optimum. The parameters that govern the optimum rotation speed are numerous and interact in such a way that it is very complex to set up a law valid for all propellers.

To find the optimum rotation speed, Heliciel offers an iterative search. A range of rotation speeds is tested, the twists of the propeller are recalculated, and the performance at each speed is displayed in a graph. When the maximum efficiency is reached, the propeller is updated at the optimum rotation speed. Optimising for efficiency gives the most energy-economical propeller; it is a starting point for the study. Installation constraints may force us to depart from the optimum speed, but we will keep an eye on the quality of the propeller through its efficiency.

A second procedure optimises the number of blades. Heliciel iterates over rotation speed: it first looks for the maximum efficiency of the propeller with a minimum number of blades, then increases the number of blades and looks again for the maximum efficiency. As long as the maximum efficiency increases, the number of blades increases. When the maximum efficiency is found, the procedure stops and the propeller is rebuilt with the optimum number of blades, at the optimum rotation speed.

Propulsive propellers, fans, helicopters. The propulsive efficiency of a propeller is zero when the upstream fluid has zero speed, which is the case for a propeller at rest or hovering. To be able to optimise the rotation speeds of all propulsive propellers, the optimisation is therefore made in terms of kinetic efficiency: the procedure stops at the rotation speed that gives the best kinetic efficiency.

Driven propellers, wind engines, wind turbines, tidal turbines. The procedure stops at the rotation speed that gives the best power coefficient (Cp), which is the ratio between the propeller shaft power and the kinetic power of the fluid. The graph below shows how the power coefficient evolves with rotation speed, with a red line for the Betz limit (16/27, or 0.593). In this example, the optimum rotation speed gives a Cp of 0.557 for a 3-blade propeller at 1,050 rpm, close to the Betz limit.

The power coefficient of a 3-blade wind turbine propeller against rotation speed: the maximum (Cp = 0.557) is reached at 1,050 rpm, under the Betz limit.

Unfortunately, this performance must be reduced by bearing friction, losses in the generator, the rectifier, any gearbox, the batteries and the cable losses. If in the end we manage to extract 30 to 50% of the Betz limit to produce energy, that will already be good. Let us not forget that once the wind turbine or tidal turbine is built, the energy of the wind or of the currents costs nothing.

To go further, we can move on to the multiple operating point analysis, which offers many performance displays and test options. When the propeller is coupled to an electric motor or a generator, the actual rotation speed is the one that balances the motor torque and the load torque: superimposing the torque curves gives this speed. We can also read the page on the operating point of the propeller to see how the rotation speed and the fluid speed set the twist.

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