How does the nozzle diameter affect the thrust of a ducted propeller?

I currently do some experiments with ducted propellers in which I try to figure out which effect a nozzle has on the thrust produced. My theory goes like this: If I reduce the exit diameter of the duct the pressure is going to decrease and the air velocity and thus the thrust is going to increase. Consequently, you would want a small exit diameter.

However, in my experiments I measured the thrust of a 12x12 inch propeller at around 5500 rpm and got 20 N without a nozzle (just a constant diameter duct) and only 4 N with a nozzle reducing the exit diameter to 50 percent of the prop diameter.

These results are contrary to my theory! Does anybody have an explanation for this ? And what should I change to actually increase the thrust compared to the prop without a nozzle ?

Here are some pictures:

• I would encourage you to get more results with various exhaust diameters, and, lengthening your duct so there is more distance between the fan and the exit. Turbulence within the duct may be affecting the performance of your rotor. Also, you may try a wide range of power settings. Good data from this one! Commented May 6, 2019 at 0:13

Welcome. I'm afraid your theory wasn't actually working. Reducing the duct's exit diameter led to an increase in its internal pressure, increasing load on the propeller, and likely even causing some reverse flow.

Nozzle design is a complex subject. I can't think of a way to condense it, even limited to one specific case, into a suitable answer; maybe I don't get it enough myself. Keep that in mind; the below is just one small shard of the whole and by no means the full picture.

In general, the job of a nozzle is to match the pressure at jet engine exit to that outside it. If the pressure is different, it gets matched outside the engine, where it doesn't produce thrust.

When the engine is a rocket, which creates high pressure, the nozzle needs to expand the gas, converting pressure to thrust through acting on the nozzle. When the engine is a cold fan, which accelerates air, it's the opposite - the nozzle needs to compensate for the loss of pressure with a bit of compression, so that the air stream can exit without fighting the pressure of outside air at the back.

It's important for a convergent nozzle not to compress the exhaust to a higher pressure than the outside air, else it will destroy thrust. That was your case, the nozzle was too narrow, so it compressed the air to above ambient - which caused it to try and blow back through the fan.

To give a practical answer, an optimal nozzle at these velocities would be very similar to a simple duct, narrowing just a percent or two at the end, with a smooth exit shape.

• Thanks for your answer! So do I get this right that after the propeller the air has a higher pressure than the air outside and the nozzle should narrow the duct just as much so that the pressure is equal to that outside the duct ? And because my nozzle decreased the diameter too much the pressure got too low and the jet had to "fight against the outside pressure" ? Commented May 26, 2018 at 12:22
• Almost, but in reverse. The accelerated air has a lower pressure, but that pressure increases if there is an obstruction in the duct - like a convergent nozzle. In your case, the nozzle was far too narrow, so it pressurized the air to above ambient - and then it started pushing back against the air entering the propeller. Commented May 26, 2018 at 12:44
• Ok, but doesn't the pressure decrease and the air accelerate when the diameter decreases ? Commented May 26, 2018 at 12:53
• The pressure decrease happens in the obstruction and after it - it's increased by the obstruction, between the fan and the nozzle exit. Commented May 26, 2018 at 13:10
• Ok so to sum it up: after the propeller the air has a higher pressure and velocity. The job of the nozzle is to convert some of that pressure into more velocity until the pressure matches the ambient pressure. If it continues narrowing though, the pressure increases again so the air eventually pushes back against the propeller decreasing the thrust ? Commented May 28, 2018 at 8:47

To add to @Therac's answer, you will probably add some drag on the outside of the nozzle by contracting it. The air flowing around it will separate if the contraction angle is too steep.

A bit of contraction makes sense, as the accelerated flow aft of the propeller will need less cross section for the given mass flow. You will also wish to make the capture area a bit larger than the cross section in the propeller plane. Just calculate the speed increase through the propeller disc and assume that half of that is reached in the propeller plane. This will ensure that pressure is about constant along the whole duct and losses are minimized.

This subject is one that the helicopter people have given a lot of thought. From this figure from Leishman we can see that the wake contracts by itself already.

The contracting shroud in your test setup has higher pressure just behind the propellor than at the shroud exhaust. This static pressure gradient exerts a force on the shroud area, resulting in negative thrust. Plus friction forces from the airstream in the duct.

The same book has a bit of a treatise on tail rotor fan-in-fin design based on momentum theory, which actually depicts a widening shape. More details in the masters thesis report, mentioned in this answer.

The thrust equation gives us Thrust = Mass x Acceleration

You increased the airspeed but reduced the airflow.

• Ok thanks for your answer! But doesnt the airspeed increase so that the same mass of air can get through the pipe ? Commented May 28, 2018 at 8:07

This is great work, and you are off to a good start. You may wish to review jet engine design. What you are building seems to be the compressor half. Narrowing the "exhaust end" will increase pressure in the duct, which is what you want a compressor to do. Forward motion created when applied to an aircraft will add to this affect. This could be an air "scoop" for a piston engine!

The best way to test your designs might be full throttle in level flight, as this would also give you nacelle drag data. Top speed comparisons will probably show by far a properly pitched prop with no duct will win the power to thrust efficiency test, but not without much being learned.

Ducted fans look great on scale model designs, but generally drain the batteries much faster than props. They do have advantages at very low or hovering speeds, but props take over from around 50 to around 400 mph.

I would definitely continue this work for application of boosting power in piston engines. Sports cars have air scoops on their hoods, this one might be better.

• At subsonic speeds, velocity increases with decreasing cross-sectional area and static pressure decreases. Commented May 5, 2019 at 14:35