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Thermodynamics fail.
The work to empty the cylinders increases.
There is no free lunch.
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The turbine engine is not the most thermodynamically efficient design out there, and right now a traditional piston reciprocating design is about as good as it gets. Some get close to 50% thermally efficient or higher. Turbines excel in situations where power to weight ratio is more important than fuel economy.
Without fully geeking out, it has to do with cycle efficiency...and in very simplest terms, the higher you can compress the intake charge, the more energy you can get out of it once you add the fuel.
You seem knowledgable. If most of the efficiency is lost due to heat, then why haven't people figured out a way to extract that heat and turn it into power? Electrical or otherwise.
Great question, and there is a lot of work in that space! The biggest reason you don't see a lot of other devices is that the added cost usually doesn't justify the additional gains.
Probably the most common way to recover waste heat is a turbocharger. While many folks see this as just a way to boost power (by adding in more air, and thus more fuel can be burned) it also typically increases thermal efficiency since the engine doesn't have to do extra work to intake that extra air. There are devices like turbo compounding devices that have a turbine that is geared to the crank shaft. There are all sorts of other turbo generator type devices that use exhaust to do useful work. The challenge is that a lot of heat goes into the engine coolant, and recovering that is a bit harder.
In some stationary genset applications that waste heat can be used to heat hot water and other things to at least use that heat rather than have to pay for another heat source to make hot water.
he engine doesn't have to do extra work to intake that extra air.
Thermodynamics fail.
The work to empty the cylinders increases.
There is no free lunch.
You are right, there is no free lunch...but using turbochargers you get back a lot of wasted energy and often SIGNIFICANTLY reduce pumping losses. In non-throttled applications (like diesels) with a good turbo match you actually get a slight amount of positive pumping work as you go higher in boost pressure. I.e. the positive pressure acting on the piston during the intake stroke is more than the pumping work to push the exhaust back out of the cylinder on the exhaust stroke. While it isn't "free" it is using the energy left in the exhaust for useful work. The reason for this is that a turbo isn't working on gas flow and pressure alone, it is actually using the expansion of that gas to extract the energy from it. If intake manifold pressure is > exhaust manifold pressure, you have positive pumping work.
Things change a lot when you add an intake throttle and waste-gate or VG turbos where you will still have some pumping losses, which is sometimes intentional to help with cylinder scavenging and EGR for emissions controls.
Let me know if we need to break out the P-v curves to diagram the cycle.