How can you calculate the pressure that a liquid develops when freezing in a pipe?

My current understanding is that in addition to the reasons pointed out in the answer to this question here, another reason for choosing ammonia in the space station external coolant loops is that the pipes in the radiators can withstand the burst pressure of freezing ammonia.

One aspect speaking against ammonia is its low boiling point at ambient pressures, so the radiator is run at pretty low temperatures, which decreases its efficiency. Water/Glycol would be better in that respect but water develops high pressures on freezing.

Given that, how can the pressure of an expanding liquid on freezing be calculated?

And how would the combination with a liquid that contracts on freezing factor in with that?

• Is there a reference for "one of the main reasons for choosing ammonia in the space station external coolant loops is that the pipes in the radiators can withstand the burst pressure of freezing ammonia"? If so, consider writing an answer to that question you linked to, that would be valuable info. Sep 15 at 12:28
• I do not have a reference, it was my impression after going through all the documents i could find on the matter.
– Joe
Sep 15 at 15:56

For calculating the pressure we could generate:

We can estimate the generated pressure, at constant volume, by multiplying the required contraction by the bulk modulus of the ice. The required contraction should be expressed logarithmically, so we have the formula

$$P=K\ln(\rho_{liquid}/\rho_{solid})$$

From Ref. 1 the bulk modulus $$K$$ of water ice at the melting point is $$8.4$$ GPa, and the logarithm of the density ratio factor, using density data from Wikipedia, is approximately $$0.08667$$. This gives about $$730$$ MPa, enough pressure to generate stresses well beyond the tensile strength of most materials in common pipes.

Mixing the water with another liquid such as glycerine which contracts upon freezing will not remove this pressure unless you use enough of the second liquid to keep the solutionliquid or have the second liquid freeze first. If water freezes out first, you may get a large fraction of the generated pressure given above, and there could be trouble.

References

1. Neumeier, J.. (2018). "Elastic Constants, Bulk Modulus, and Compressibility of H 2 O Ice I h for the Temperature Range 50 K–273 K". Journal of Physical and Chemical Reference Data. 47. 033101. https://doi.org/10.1063/1.5030640.
• Great answer thanks Oscar, can anyone elaborate on why water/glycol would still cause trouble at an example ratio of 50/50?
– Joe
Sep 17 at 13:47
• I have to look at the phase diagram, but except at a precise composition (the eutectic) one liquid freezes first in pure form. If it's water that still expands. You need more glycol than the wutectic to get the more benign glycol to freeze first. Sep 17 at 14:03