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Suppose a thermally insulated container is filled with atmospheric air until the pressure reaches 5000 psi. This could represent the filling of a diving cylinder, before thermal dissipation becomes significant.

Initially, then, some mass of air has atmospheric temperature $T_1$ and $p_1$. When forced in to the tank adiabatically, we expect its final temperature to be

\begin{align} T_2 &= T_1\left(\frac{p_2}{p_1}\right)^{1-1/\gamma} \\ &= (300\,\mathrm{K})\left(\frac{5000\,\mathrm{psi}}{14.5\,\mathrm{psi}}\right)^{1-1/1.4} \\ &\approx 1590\,\mathrm{K} \end{align}

This final temperature seems far too hot. A diving cylinder would likely melt at this temperature.

More likely, though, something is wrong with my reasoning. Is this not the correct use of the temperature-pressure relations, assuming an ideal gas? If not, what other information is needed to predict the final temperature of the container?

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up vote 4 down vote accepted

You're trying to ask a question about real life (would the tank melt) with a model that approximates away the very thing that you need (adiabatic). With the assumptions you have made, you are in fact using the correct equations.

The adiabatic assumption is only valid if it is thermally insulated which is not the case in real life. There will be losses of heat in the system while filling the tank unless the filling occurs virtually instantaneously.

A quick search indicates that tanks should be filled at around 500 psi/minute, so it would take about 10 minutes to fill up a tank from empty to 5000 psi. Over the course of those 10 minutes, the heat that is added to the air as it is compressed is absorbed into the steel and conducted away to the atmosphere. So the process is far from adiabatic, which is why the tank does not actually melt.

Lastly, I would point out that steels melt at much higher temperatures, over 2000K so even if adiabatic, a steel tank would not melt. Aluminum on the other hand melts around 1200K, so that one wouldn't work so well.

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An aside: Steel would yield well before this temperature though. Structural steel in highrise buildings is a severe fire hazard if not enclosed in concrete (heat from fire will swiftly make steel fail). Steels go through a weird sequence of "mushy" phases before actually melting. – WetSavannaAnimal aka Rod Vance Sep 24 '13 at 14:09
Some people immerse the tanks in water to cool them during pressurization. This practice is frowned upon for a couple of reasons, but the fact that it is used at all indicates that the tanks can get quite hot even during feasibly fast compression, never mind the instantaneous compression needed for the adiabatic case. – Beta Sep 24 '13 at 20:01

Your calculation looks correct to me (although 5000 psi is a little high for a typical diving cylinder). But this calculation gives the final temperature of the air inside the cylinder. If -- after rapid compression -- we wrap insulation around the cylinder and allow the cylinder and the gas to reach equilibrium, the final temperature will be...

...tricky to calculate without tables of specific heats at high temperatures and pressures...

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