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piston engines of ww2 planes had pressure ratio of at least 10, while jet engines were still vastly superior with a pressure ratio of only 3.

Certainly pressure ratio of 1 would bring 0% fuel efficiency, while you don't gain as much going from 3 to 6,so there must be asymptotic behaviour here.

Is there a formula or some empirical rule i could use to predict what will be the relationship between fuel efficiency and pressure ratio?

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    $\begingroup$ The steady pressure ratio in a jet engine with continuous combustion is not directly comparable with the discontinuous pressure in the cylinder of a piston engine. The biggest limitation on the design of early jet engines was the temperature limits of the available materials for the combustor and turbine. Modern jet engine designs have pressure ratios up to around 80, not "3 to 6", and compressor exit temperatures (i.e. before combustion) higher than the early jet engine temperatures after combustion. $\endgroup$ – alephzero Sep 17 at 0:30
  • $\begingroup$ Also, your assumptions, or perhaps your terminology, are wrong. The Merlin had a 6:1 mechanical compression ratio. It's effective ratio was about 5.5:1. But it had a two speed, two stage, intercooled supercharger providing up to 20 inches hg of boost (relative to sea level pressure) at altitudes of 17,000 feet and WOT. And it had an exhaust velocity of 1300 mph in the ejector pipes. Jet engines quickly bumped their overall compression ratios up to 15:1 post war once the material issues had been solved. The Jumo 004 protoypes were redesigned in steel for mass production. That hurt them a lot. $\endgroup$ – Phil Sweet Sep 17 at 5:56
  • $\begingroup$ Now, please explain exactly what these returns are. You build an engine to do a job over a wide range of conditions, using the cheapest materials possible, and using the cheapest labor possible. What is the job and what are the limiting conditions? The Jumo 004 was 1/3 the cost of a piston engine overall. It was faster to build, needed no skilled labor to assemble, and met performance goals. There were other jet engines out there, it was the cheapest. Production was slated to hit 100,000 units per year. $\endgroup$ – Phil Sweet Sep 17 at 6:08
  • $\begingroup$ Your last question does sort-of have an answer from a thermo 101 perspective. - upload.wikimedia.org/wikipedia/commons/thumb/f/ff/GFImg7.png/… - but it doesn't include any practical considerations. $\endgroup$ – Phil Sweet Sep 17 at 6:21

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