I wonder which of these has higher or lower rotational speed:
Does anybody know the typical speeds for these kind of engines?
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First of all: There is not one single speed on many aircraft engines. Big propeller engines have a reduction gearbox so the propeller runs at a lower speed than the core engine, and multi-spool turbines have an inner, high-pressure section with higher rotational speeds than the outer, low-pressure parts of both the compressor and the turbine. Take the Progress D-27 turboprop for example: It has three spools and a reduction gear, so its components run at four different speeds.
Generally: The bigger the diameter, the lower the typical speed. Fan and propeller blades should run at a high dynamic pressure, but still at subsonic speeds to avoid the higher losses involved with supersonic flow. Since the tip will be the part with the highest speed, and propellers have bigger diameters than fans, propellers run at lower speeds. Also, their bigger diameter makes propellers more efficient. The more air mass is involved in thrust creation, the higher the propulsive efficiency will be. Propellers on slow-flying electric or human-powered airplanes run at only hundreds of RPM, well below speeds where their tips would approach Mach 1, in order to convert as much of the limited power to thrust as possible.
If flow speed allows, core engine parts run at the highest RPM possible, because the faster a turbine or a piston engine runs, the more power it will produce for a given size. Therefore, core engines of the same size run at the same speed, regardless of their type, their bypass ratios or their use. Only stationary gas turbines for power generation might run a bit slower than their aircraft-mounted siblings to increase their life.
The high pressure compressor and turbines of an airliner engine will run at above ten thousand RPM, while the propeller of a large turboprop engine wants to run at about 1800 to 2200 RPM. Both the fan and the propeller are powered by separate low pressure turbines. The fan and low-pressure compressor of a big airliner engine run at between 2500 and 4000 RPM, and the turbine of the PT-6, a classical turboprop engine, at 30.000 RPM. Since power is torque times RPM, a faster running shaft will be lighter than one running at low speed, so the speed is only reduced directly at the propeller mount.
A fan's optimum speed is between that of the compressor and the propeller and would ideally also need a gearbox, but here the transmitted power is so high that a compact gearbox will be hard to design. Even if it runs at 98% efficiency, it will produce waste heat in the order of several hundred kilowatts in case of a big airliner engine. Some small engines with just a single turbine use gearboxes already (e.g. the Lycoming ALF507), and now the next generation of efficient fan engines are also introducing gearboxes. But so far, most engine designs have preferred to run the fan faster than what would be ideal. Note that the tip speed of a modern, ungeared fan is already well above Mach 1.