I'm looking for an answer that at least addresses the fundamental thermodynamics of a jet engine.
Jet engines use a Brayton cycle to get useful work out of a heat source. An ideal Brayton cycle engine has the input and output pressure be the same. If Exhaust pressure is greater than inlet pressure, then that's becoming closer to Diesel cycle further from a Brayton cycle (And to those who feel like correcting this point, a diesel cycle has many forms beyond what is typically thought of as a diesel engine but that's not want I want to talk about. Diesel cycle only cares about adiabatic compressions/expansion, constant pressure combustion, and V1 = V4, but I do not wish to dwell on this).
Also, when looking at nozzle design, we ideally try for the exhaust pressure to be equal to ambient pressure as opposed to an over expanded nozzle where the exhaust pressure is less than ambient or an under expanded nozzle where the exhaust pressure is more than ambient.
Both of these two things together seem to indicate that a jet engine would be designed to have the intake and exhaust pressure be the same (ie the EPR should be 1). Is there something that would make achieving this in the real world challenging? Or are the engine designers not even trying to have the exhaust pressure be equal to the ambient pressure? If the latter is the case, why would you not want to try to have the engine operate as closely to the ideal cycle and nozzle design as you could? If it's the former, what conditions would contribute to this becoming more challenging?
I get that if an engine has fixed geometry, if you change the amount of heat added in combustion, it will change the EPR, but there are plenty of jet engines have variable exit geometry. Therefore, I would think you could vary the geometry to get the ideal exhaust pressure. Even if you couldn't vary the engine geometry and only design for 1 power setting, I would expect that it is most efficient where the engine is expected to be used the most. For example, if I expect my engine to spend most of its time being used at 35,000 ft and generating a specific amount of thrust, I would design the nozzle so that the discharge pressure would be whatever ambient is in those conditions. That would translate in the real world to the engine operating at or very close to EPR of 1 during cruise.
It is my understanding that this is not the case, and if that's a correct understanding, I am missing a piece to the puzzle to make sense of why things are the way that they are.