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.