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 Quote by Swedpat Pedalled drived airplanes have a 2-bladed propeller of around 3m diameter, so I understand that the most efficient is to have a large propeller which is rotating not much faster than the vehicle.
Bigest propeller have two advantages (good things):
1 good thing. Slow rotating, lees friction brearing.
2 good thing. It blow less to himself, like in this image:

And is other good thing. Then pitch is, for example, 10 inches and diameter propeller is 40 inches, then that propeller very not efficien. Best proportion is then pitch and diameter is about 1:1, for example best proportion is 10x10 inches, 1x1 meter, 2x2 meters. Midle propeller angle of atack must be about 45 degree.

Big propeller only not good things is that it must be wery thin profile (for pedal, propeller driven bicycle, who speed is about 20 km/h), so it easy breake.

And, maybe, difficult make transmision to rotate for exampla 0.5 meter propeller with pedal. Much easy, maybe, make for 3 meter diameter propeller transmision and beter be efficient transmision.

Small propeller( for example 0.5x0.5 meter), who efficienty max is on 20 km/h and we need that speed, for esample, not make enought thrus. So need make smaller pich, for example 0.5 diameter meter and pitch 0.1 meter and make biger rpm, but how I said that propeller proportion is not efficient. Or can use 0.5 meter diameter and 0.5 pitch meter propeller with highest rpm, but then it and now be not efficient, because max efficienty it be for example on 50 km/h, on 20 km/h it eficienty be, let say, about 30 precent (on 50 km/h be about 70 precent), so it makes enough thrust to ride on 20 km/h, but eficienty propeller be only 30 precent, so using bigest propeller, who diameter can be 3 meters and pitch can be 3 meters. And it make enough thrust to ride or fly on 20 km/h and it max efficienty is on 20 km/h.

For example in this video

http://www.youtube.com/watch?v=iGxWIbhTqwg

propeller efficienty max is on about 30-40 miles per hour. But it not make enought thrust to pull bicycle in that speed so it ride on about 10-20 km/h, but if human little bedal it ride on 35 mph how say author that video:

"It will run 35 mph - just need to pedal a little."