Expansion efficiency is the efficiency of converting static pressure to motion. This can be thought of as analogous to streamlining an aircraft - just asa streamilned aircraft will move through the air more efficiently than a big cube, so too will a well designed rocket engine more efficiently expand its propellant than a simple hole.
Another form of efficiency is thermodynamic. Rocket engines are a modification of the Brayton cycle which has high thermal efficiency compared to say an Otto cycle. They are still well below the Carnot efficiency, and the fact is most heat engines can closely approximate adiabatic expansion.
But for most applications of rocket engines, what we are really concerned about is thrust production. Rockets produce a lot of thrust for a given mass of propellant, and have excellent thrust to weight ratios - hence their use in mass constrained systems like spaceships. However, the energy contained within that mass of propellant could potentially produce substantially more thrust. Remember that thrust is proportional to the mass and the velocity of the exhaust, but the energy required to get it moving up to speed is proportional to the square of the velocity. To get the most bang for your buck, you want to accelerate as much working fluid as you can by a small amount. As an aside, this is why turbofans are more efficient than turbojets. Rockets have very poor performance in this regard, but if you're in space or need to reach high speeds there may be no viable alternatives.
As for the Aeopile the issue was never with the rocket-like jets. The Aeopile is a simple version of a reaction turbine - the working fluid is expelled through a nozzle and the turbine experiences an equal and opposite force. This is the same principle in use today with low pressure steam turbines. The blades of these turbines don't look like rocket engines, but they perform the same expansion upon the steam. The problem with the Aeopile was that the pressure was just far too low. The water being boiled was at atmospheric pressure and the steam generated was immediately vented so pressure never built up. If they had check valves which could form a good seal and survive high temperatures and pressures, they could have achieved much higher performance with essentially the same setup. However it would still literally run out of steam in a short period of time without a pump which could force additional water into the vessel at high pressure as the steam. Of course the ancients also didn't know how to make supersonic expansion nozzles either, but they could probably get by without those at the beginning.
Interesting discussion. Though it doesn't refute my issue with rocket efficiency when combined with a rotating lever shaft. A kg of propellant will could yield 4.5MJ of reactionary thrust. And outright burning a kg of propellant releases 14.5 MJ of thermal energy.
The supersonic expansion nozzles are important for any aeolpile I would figure though I've not seen it discussed yet till now.
You can't measure thrust in MJ. With perfect efficiency your rocket could produce 5385 N.s of impulse. There is no meaningful comparison between this and thermal energy.
You most certainly can use rockets on levers to turn a shaft and power a generator, it's just impractical and less efficient than the alternatives in most cases. Turbines work on the same principle but can have efficiencies in excess of 90%, far superior to a rocket, and with far simpler construction. That said, there have been proposals to power rotorcraft like helicopters with blade-tip rocket engines. By this point though you are getting very far away from the Aeolipile, which was an ancient steam turbine.
Expansion efficiency is the efficiency of converting static pressure to motion. This can be thought of as analogous to streamlining an aircraft - just asa streamilned aircraft will move through the air more efficiently than a big cube, so too will a well designed rocket engine more efficiently expand its propellant than a simple hole.
Another form of efficiency is thermodynamic. Rocket engines are a modification of the Brayton cycle which has high thermal efficiency compared to say an Otto cycle. They are still well below the Carnot efficiency, and the fact is most heat engines can closely approximate adiabatic expansion.
But for most applications of rocket engines, what we are really concerned about is thrust production. Rockets produce a lot of thrust for a given mass of propellant, and have excellent thrust to weight ratios - hence their use in mass constrained systems like spaceships. However, the energy contained within that mass of propellant could potentially produce substantially more thrust. Remember that thrust is proportional to the mass and the velocity of the exhaust, but the energy required to get it moving up to speed is proportional to the square of the velocity. To get the most bang for your buck, you want to accelerate as much working fluid as you can by a small amount. As an aside, this is why turbofans are more efficient than turbojets. Rockets have very poor performance in this regard, but if you're in space or need to reach high speeds there may be no viable alternatives.
As for the Aeopile the issue was never with the rocket-like jets. The Aeopile is a simple version of a reaction turbine - the working fluid is expelled through a nozzle and the turbine experiences an equal and opposite force. This is the same principle in use today with low pressure steam turbines. The blades of these turbines don't look like rocket engines, but they perform the same expansion upon the steam. The problem with the Aeopile was that the pressure was just far too low. The water being boiled was at atmospheric pressure and the steam generated was immediately vented so pressure never built up. If they had check valves which could form a good seal and survive high temperatures and pressures, they could have achieved much higher performance with essentially the same setup. However it would still literally run out of steam in a short period of time without a pump which could force additional water into the vessel at high pressure as the steam. Of course the ancients also didn't know how to make supersonic expansion nozzles either, but they could probably get by without those at the beginning.