Liquid Helium (4.2K) sealed, then raised to sub-LOX temperature (~70K) - what is the new pressure?

Question

In this question there is discussion of the large differences in temperature between liquid helium-4, subcooled liquid oxygen (sub-LOX) and cold kerosene (RP-1). Approximate temperatures are 4.2K, ~70K also here, and ~286K respectively.

The following question is related to this, but not necessarily based on SpaceX designs or plans.

The phase diagram for helium-4 (${}^4$He) is shown below. There is further helpful discussion here and here.

Above its critical point of 5.2K helium-4 becomes a supercritical fluid where the distinction between liquid and gas no longer applies, and so 'boiling until it reaches equilibrium' is not the right way to think above 5.2K. If you close it off at ambient pressure at 4.2K, it will reach 2.24 atmospheres at 5.19K.

Question: But if you continue to warm this sealed, very strong tank up to 90K (LOX at ambient pressure) or even ~70K (sub-cooled LOX), what would be the helium pressure now?

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note 1: that's a supercritical fluid above 5.2K, not to be confused with superfluid.

note 2: since putting tanks of helium-4 inside tanks of LOX has become a critical technology in space exploration, this particular question - what actually happens - should be still considered on topic here.

above x2: Phase diagrams of helium-4 from here.

Answer 1

We don't need to look into the phase diagram here, because it doesn't matter on which path we reach a certain end point. We can safely assume that we heat up the helium at constant pressure and then shrink its volume again.

Density of liquid helium boiling at 1 atm pressure is 0.125 $\rm g/cm^3$. Density of gaseous helium at 273 K is 0.18 $\rm mg/cm^3$. Hence, it will have the 700-fold volume, or, if volume is constraint, a pressure of 700 bar. At temperature of LOX, i.e. 70 K we can treat Helium as a perfect gas that behaves according to the ideal gas law as a first approximation. That is, $pV/T = const$. So, pressure will be a factor $T_{Lox}/273 \rm K = 0.26$ lower. A volume of liquid helium sealed and heated to 70 K will be at a pressure of 180 bar.

This is only true if we can assume that Helium still behaves like an ideal gas even at an pressure of 18 MPa. This deviation can be found in the compressibility factor and has been measured (See e.g. S.W. Van Sciver, Helium Cryogenics, Appendix 1). This factor is about 1.2 to 1.3 at our conditions, hence actual pressure will be about 230 bar.