# Efficiency estimation for small-scale (~15cm) propellers

First up, apologies if this isn't quite the place for this; Aerospace is still in Area 51 so there doesn't seem to be a better place for it. If it helps, feel free to give theoretical answers and I'll go crunch the numbers myself.

I'm working on a model aircraft which will act as a testbed for various configurations of wing. The overall length is about a metre; the wingspan will be between one and two metres depending on the configuration being tested. We're looking at two types of propulsion, both driven by electric motor: free propeller, and ducted fan. Either way, it'll be rear-mounted to keep airflow over the wings clean.

My knowledge of aerodynamics is not what it once was, and "what it once was" was a little shaky. Here's what I know:

• All else being equal (airspeed, RPM, thrust) a free propeller is less efficient than a ducted fan of the same diameter.
• Ducted fan efficiency increases with RPM, so they're built to run fast, which gives more thrust per unit disc area, so ducted fans tend to be smaller and spin faster.
• Efficiency increases as the momentum change per unit mass of air decreases, yielding best efficiency at zero-thrust cruise (sadly).
• Ducted fans become less attractive at high speeds because the duct itself adds drag.

Given a rough estimate of required thrust and cruising airspeed, how do I determine which of the two systems is more efficient?

Some details:

• Aircraft:
• wingspan around 1.5m (give or take 30-50cm)
• fuselage diameter roughly 15-20cm
• roughly 5-7kg operating mass
• airspeed 30m/s (very roughly)
• Propeller would be rear-mounted ("pusher"), anything up to about 30cm in diameter, 2-4 blades. Probably folding if that matters.
• Ducted fan would be any size that fits inside the fuselage (including duct clearance), so 15-18cm(ish). Duct would be integrated in a manner similar to jet fighter intakes; overall from intake to nozzle around 30-50cm. Fuselage shape is affected as little as possible.

I've tried to work out some of the finer details, but things like Reynolds number calculations have given me very varying results so I don't really trust them. Any help here would also be appreciated.

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Due to the complex nature of the flow and the variance in the properties of each option (variable pitch, propeller shape et al.) working this out with a pen and paper will involve a vast number of estimations and the use of correlations. The accuracy of this is likely to be very poor. From reading about ducted fans on Wikipedia, it is my contention that these provide a much better option for your model aircraft. In fact it seems to me that ducted fans out do propellers in many aspects, probably apart from cost. I hope this helps. – Killercam Aug 19 '13 at 17:17
I figured as much. Always worth asking, though. Thanks! – anaximander Aug 19 '13 at 19:23
As you might have concluded from your points, ducted propellers are better when you need these two conditions: low speed, high load (think of quadcopters carrying heavy-ish stuff). At your speeds of 30m/s, it's a pretty darn safe bet that the free propeller is the better option. Even if there is a ducted propeller that's somehow more efficient at those speeds, I doubt it'll be by any significant amount. – Esteban Aug 28 '13 at 21:30
But you might get more acceleration from ducted fans... The way to make this assumption free, is to get the efficiency and thrust curves. If buying from manufacturer, they should have something. If not, most accurate way to do it would be to test it yourself. – Esteban Aug 28 '13 at 21:44