Here is a contra-rotating propeller. The front propeller rotates counterclockwise while the back propeller rotates clockwise. My questions are:
- Do they rotate at the same speed (RPM)?
- If not, which one is rotate faster, and why?
Most if not all contra-rotating propeller systems are geared together so that both sections turn at the same speed. This is done so that as well as allowing effective conversion of very high power into thrust, torque and P-factor are cancelled, making the aircraft easier to fly (especially in a single-engine or two-into-one installation).
This is why you see the same diameter propeller in front as in the rear -- if they were turning at different rates, the slower one could be larger diameter (a limitation on diameter is transonic conditions at the tips) -- but you always see the same diameter on front and rear.
It depends on the power source.
From efficiency point of view, counter-rotating propellers are worse than a single propeller. Exception to that is counter-rotating rotors on some helicopters where there is enough vertical separation so the upper rotor stream contracts and some fresh air is feed to the bottom rotor. As airplane propellers would work most of the time at high speed, the influx of fresh air is lost, the second propeller is operating in the stream of increased velocity, and so the efficiency is less. To answer your question I'll need more information, as what you want to achieve. For example if the goal is to eliminate torque, the rear propeller needs to rotate faster. If the goal is to reduce fuel burn then one propeller needs to be stopped and feathered.