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How would one figure out the maximum PSI (air only) a particular container may hold knowing only its dimensions?

I understand the water formula. P=.434 x height

I would like a formula for air pressure. Thank you.

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Although this is really more an engineering question, I will give a few pointers.

First - the pressure equation you give relates pressure in a liquid to depth - I think. But I can't quite make out the units you use.

Air pressure is "essentially constant" over most volumes - although air pressure does change with height, the density is so low that you have to go a long way up to feel a difference. To calibrate you: when you go down 10 m in water, the pressure increases by 1 atmosphere. When you go up 1 km in air, the pressure decreases by about 1/10th of that...

So when you need to compute the pressure an air container can hold, you need to look at the area of a cross section - the force due to the air pressure is given by the pressure times the area of the container, $P\times \pi r^2$, and this force is supported by the walls of the container, shown here with thickness $t$ and so with approximate area $A=2\pi r t$ (when $t<<r$).

enter image description here

Now you can compute the stress on the wall based on the pressure (for a spherical container - but similar calculations will work for other shapes) from

$$\sigma = \frac{\pi r^2}{ 2\pi r t} P= \frac{r}{2t} P$$

So the stress in the walls will be greater as the ratio of vessel size to wall thickness increases. The exact calculation for the maximum stress will depend on the shape of the vessel - you should repeat the above calculation for the section with the largest area of gas for the smallest area of wall.

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