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I think som submarines like Kockums use liquid oxygen instead of compressed air for driving their combustions engines. Is it more energy efficient to oxygen enrich the air before burning it compared to using common air?

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Apart from the volumetric efficiency of storing pure oxygen rather than compressed air mentioned by other replies, another important consideration is the exhaust gases of combustion. For submarines, stealth is important and ideally you want to minimise both sound and gases escaping from your air-independent propulsion system.

With pure O2 + fuel under optimal conditions, you get water and CO2 as the combustion byproducts. The former is obviously no issue, and at higher pressures, CO2 can be dissolved in water as well. The end result is no trail of bubbles behind your submarine. Furthermore, carbonic acid is denser than water so it will sink.

With compressed air + fuel, the nitrogen going into the combustion chamber emerges mostly unchanged (with the exception of some NOx). The gaseous nitrogen exhaust will leave a trail of bubbles behind your submarine. Furthermore, as they surface, they will enlarge in size as the hydrostatic pressure falls, and coalesce which altogether means a visible trail.

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the Kockums submarine stirling engine runs on heat generated by burning diesel fuel. Normally this is done in air but there is no plentiful air available underwater. To make this scheme even remotely viable, the submarine has to carry tanks full of pure liquid oxygen inside it for burning the fuel.

Why do this in the first place? the combustion in a stirling engine is not explosive, so it runs far more quietly than a diesel of the same power rating, which is important in a submarine. But electric motors are quiet too and don't require tanks of liquid oxygen to run them because they use batteries instead. Those batteries can be recharged at sea, but once the stirling sub has used up its liquid oxygen supply it cannot dive and must return to port on the surface and refill its tanks.

the fuel/oxygen ratio for diesel and liquid oxygen is about .45 gallon of diesel to gallon of LOX, or about 2 gallons LOX per gallon of diesel. If the sub carries 10,000 gallons of diesel it has to carry 20,000 gallons of LOX in cryogenic storage, which is a big deal.

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    $\begingroup$ The Air Independent Power System is a supplementary system. Much of the diesel will be combusted by snorting air. So the lox tanks aren't stoichiometrically proportional to the diesel tanks. $\endgroup$
    – Phil Sweet
    Jun 28 '20 at 17:24
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You can store more oxygen in the same volume as a liquid than you can as a gas.

There are safety issues for both cases though.

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