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@Uwe's excellent answer to the question Why would sub-cooled LOX tanks need to “topped-off” until the last minute or so? explains that the LOX is sub-cooled in situ within the rocket's tank using evaporative cooling/refrigeration by passing helium bubbles through the tank so that the oxygen evaporates into the bubbles. Read that answer for further details.

How easily could this be adapted to methane or hydrogen cryogenic propellants?

From this answer:

Naively I would guess that it should work for methane in a straightforward way (boils at 112 K versus 90 K for oxygen), but I don't know how easily this could be applied to liquid hydrogen with a boiling point of only 20 K.

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Hydrogen is liquid below 21.15 K and goes solid at 14.01 K.

Helium is liquid below 4.15 K and goes solid at 0.95 K under huge pressure of 2.5 MPa.

So it should be possible to use gaseous helium to sub cool liquid hydrogen. It is gaseous even below the temperature of solid Hydrogen.

Methane boils at 111.65 K and solidifies at 90.7 K. Sub cooling it with gaseous helium should be possible too. According to this NASA paper the solubility of helium in subcooled liquid methane is small, about 0.0001 mole fraction at 25° F (13.9 K), but these temperature values seems to be wrong. In a graph a value of 98 K is found on page 20.

Helium does not react with hydrogen or methane. I don't know if helium is soluble in liquid hydrogen or methane. But helium is still gaseous in a liquid air separation plant without using an extra cooling stage for Helium. Helium is used to pressurize liquid hydrogen in rockets, so its solvability should be at least small if not zero.

According to this page there might be a hot metallic alloy of hydrogen and helium within the core of Jupiter and Saturn. But this alloy exists only under enormous pressure.

Here is some information about solubility:

The solubility of helium in liquid hydrogen was measured as a function of temperature and pressure. It was found that the solubility varied from 0.59 mole% for T = 26.8°K and 1.98 atm partial pressure of helium to 11.1 mole% for T = 19.8°K and 7.0 atm partial pressure of helium.

But the partial pressure of helium in a LH2 rocket tank would be lower than 7 bar.

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    $\begingroup$ +1 Thanks for the interesting and well thought-out answer! $\endgroup$ – uhoh Jun 7 at 1:02

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