Ok for this question lets assume two gas turbines of the same power, one driving a propeller and one driving a fan (for this assume basically a normal turbofan but with duct removed). (The numbers I throw out here will be for theory purposes and probably will not reflect that in real life)
If we compare a King air to a similar sized Citation the propeller will be much larger in diameter compared to the fan and have less blades, whereas the smaller diameter fan will have more blades to absorb the engine power. Smaller fan rotates at a higher RPM giving lets assume a tip speed of 200kts and the big prop rotates slowly but due to big diameter lets assume tip speed of 200kts.
I have been told via sources online a fan basically accelerates a smaller mass of air quick and a prop accelerates a large mass slowly. Given that the prop and fan in this case have the same tip speed and same blade area to absorb the same power I just can't get my head around how this works?
I don't think I have worded my thoughts to well but hopefully someone out there can understand what I am trying to get at here, I think I need to take a fluid dynamics course after I finish my instructor course!