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In pressure fed cycle a high pressure gas tank contents are exhausted into the fuel and oxidizer tanks to essentially push them into the combustion chamber. Could a portion of the fuel from an LNG tank flow through the heat exchanger to be used as the high pressure gas needed to inject fuel in to the combustion chamber, or would LNGs' high liquefication temperature or other phase transition properties make it unsuitable? I imagine helium is preffered for a good reason...

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For a pressure fed engine, the propellants entering the combustion chamber are at a significantly lower pressure than the propellant tank pressure - otherwise they wouldn't flow the right way. You'd have to do something to increase the pressure of the fluid that was headed back up to the tank ullage, otherwise your pressurization line would flow the wrong way. You might need a pump....

Here's a patent that describes such a pumped system (1 and 2 are the pumps)

enter image description here

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