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How are cabin pressure and oxygen level maintained inside the airplane?

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    $\begingroup$ This is a pretty broad question with lots of information available here and elsewhere. (I'm not going to research for you and post links...) Have you already looked? Is there something specific you don't understand? (Short answer: compressor stage bleed air, valves, and pressure regulators. Oxygen ratio is same as ambient.) $\endgroup$ Jul 31, 2021 at 15:29

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Some of the air flowing through the aircraft's engine(s) is diverted into the cabin (its cooled and humidified first). As the air enters the fuselage (cabin, cockpit, etc.) It builds up pressure (like filling up a balloon) until the the resulting compressed air in the aircraft contains enough oxygen molecules (for a given volume of air) to be breathable.

To keep the aircraft from being over-pressurized (once the appropriate pressure level is reached) an outflow valve on the fuselage opens and allows pressure/air to be released to the outside. This outflow valve continually opens and closes (modulates) as necessary to keep the air pressure in the aircraft at a level equivalent to about a 7000 - 8000 foot altitude (for an air carrier/jet type aircraft) while cruising at a normal enroute altitude (e.g., 30,000 to 39,000 feet cruise)

The above is an extremely simplified explanation hopefully giving you a basic understanding in response to your question.

I'm confident that more detailed explanations and nuanced analysis of my response will be available in subsequent comments or answers by others.

Here is a good website that will aid in your understanding of airplane pressurization:

How aircraft pressurization works

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If the pressure in the cabin of a pressurised transport aeroplane is maintained to under 10000 ft, the partial pressure of Oxygen is enough not to cause any significant adverse physiological effects to humans. To achieve that, hot compressed air is taken from the engines and is passed in the cabin. There are also valves on the fuselage, which control the pressure and introduce fresh air into the cabin if needed.

In contrast most military fighter jets use a dedicated system to provide the pilot directly with oxygen. The two main types are OBOGS (onboard oxygen generating system) and pure oxygen supply through LOX bottles. They are usually supplied using ambient pressure, so above 40000 ft you need a oxygen supply under pressure, because the ambient pressure is not enough to sustain the diffusion of gases in the lung.

Some general aviation aircraft also possess a dedicated oxygen supply using LOX bottles for short flights above 10000 ft.

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