# Amount of material required for a pressure tank

I read the answer for the question Why is a hot air balloon “stiff”? and thought something sounded ridiculous. My engineering requirement is that the walls be strong enough. Here $T$ will be the tension (for a surface, not sure about those units) and $R$ is radius of curvature of the wall. Requirement is:

$$\Delta p < 2 \frac{T}{R}$$

Let $d$ be the thickness of the wall and $\sigma$ the material tensile strength.

$$T \propto d \sigma$$

This would indicate that thickness increases linearly with scale. That sounds ridiculous.

Why it sounds so silly

• Volume scales as $R^3$ and surface area as $R^2$. SA x (thickness) = material volume = constant, so that implies there are no economies of scale for pressure tanks in terms of pressurized volume divided by structural materials. That sounds nonsensical. That means a chemical plant wouldn't save any materials by buying a large tank as opposed to 1,000 tiny tanks.
• Say that I have a tank shape in mind. If I build a small tank and a large tank, they will be geometrically congruent. That is, if thickness if 5% of the diameter of the small one, it will be 5% of the diameter of the large one.

Please prove me wrong. And if you can't prove me wrong, please establish a physical intuition as to why this should be the case.

I think there was another question that I commented on like this, but the gist of both of them is that weight scales with volume, $L^3$ while the yield strength of the supporting structures scales with cross sectional area $L^2$. This is why elephants and dinosaurs have proportionally thick legs. Similar logic applies to buildings. In other words, the diameter of a leg must grow compared to a measurement of the body as $L$. In the case of pressure vessels, the factor is $1$. Note, my problem with this conclusion was that I wanted it to be less than $1$. For both of these we should ask why.