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Assuming a simple rocket engine, with a combustion chamber and de laval nozzle, fueled by liquid methane and liquid oxygen, what temperature must the combustion chamber, choke and nozzle withstand?

From a university article, the adiabatic temperature of a perfect mix would be about 5000C:

adiabatic temperature

However, when combustion products expand they cool, so this question could be rephrased as what pressures exist in the three main parts of the engine.

Really I am asking "how tough does the inside of a rocket engine need to be"? What is the max temperature and pressure the materials must be able to withstand?

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    $\begingroup$ "It's hot. Damn hot! Real hot! Hottest things is my shorts. I could cook things in it. A little crotch pot cooking." (Robin Williams, Good Morning Vietnam) $\endgroup$ – geoffc Jul 1 '15 at 19:52
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I found a paper by 4 authors that compares Kerosene engines to Methane engines. It states the chamber temperature of a Methane rocket is 3533K, comparable to the RP-1 3676K chamber temperature. The optimal temperature in a vacuum is 3563K, compared to 3701K. So it seems the temperature constraints are very similar to RP-1 rockets. See the paper for other comparisons.

This isn't the highest temperature, but results in the highest ISP, as determined parametrically.

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