How can jet engines be more efficient than propellers? [duplicate]

Question

If jet engines are smaller in diameter than propellers, is it fair to say that propellers accelerate a lot of air a little while a jet engine accelerates a little air a lot? Basically, a jet engine accelerates the air flowing through it by more than a propeller engine of the same thrust, right?

If so, propellers should be way more efficient because, by accelerating a larger mass of air, they're able to generate more force (thrust) while performing less work (change in kinetic energy of the accelerated air) due to the fact that force is proportional to $\Delta V$ but energy is proportional to $\Delta V^2$, but both are proportional to mass.

So, if this is true, shouldn't propulsion systems try to maximize the mass of air that it accelerates in order to minimize the $\Delta V$ required? This is, after all, the whole concept behind using large wings to stay aloft (they can grab onto a large mass of air).

So how can jet engines be more efficient than propeller engines? What other factors contribute to the efficiency of the propulsion system besides just the mass and velocity of accelerated air? And similarly, why aren't propellers way larger than they typically are? Because by this logic airplanes should have propellers the size of helicopter rotors.

Answer 1

Why aren't propellers way larger than they typically are?

There are some "propellers" these days with a 400 foot disk diameter, known as windmills. But, they are doing their best to extract energy from the airstream and here lies the issue.

"Relative wind" is conveniently used to combine the net direction and force of two aerodynamic effects, but let's look at an old question of: can a sail boat ever sail faster than than wind? This is very easy to explain by separating the "wind" components, rather than combining them. Ice boats can go several times faster than the wind by using wind across the beam as "thrust", and using "wind" from the motion of the boat to create lift.

This is exactly how powered aircraft fly, the ice boat is simply doing it on its side (hence the tiller is the "elevator").

Which brings us to propellers, which can be more accurately described as generating lift than "moving air". But, as the graph shows, net thrust of props decreases as speed increases because disk drag increases with speed.

Turbojets become more efficient at higher speeds than props because they have much less disk drag subtracting from their thrust. Even though they are less efficient at producing thrust, their net thrust out put is much more linear over speed as "ram effect" (increasing amount of air available to produce more thrust) and drag cancel each other out, while the propeller falls victim to its own disk drag.