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I was trying to imagine an ideal cylindrical vessel under internal pressure. Assuming it is a thin walled pressure vessel.

I.e. For storing a given Volume of contained fluid ( ie volume of cylinder) at a known pressure what would be the optimal aspect ratio that minimizes the amount of steel ( or any other material that the cylinder is built from)

If I use the usual formulae for thickness of a vessel under internal pressure that's proportional to both P and diameter. If I do the math then the volume of steel comes out as independent of the aspect ratio entirely.

In that case why are pressure vessels mostly long and narrow diameter? Of course the thickness is indeed proportional to dia but the d^2 term also appears in the Volume contained equation.

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    $\begingroup$ That would be a sphere: greatest volume cf external surface area. $\endgroup$
    – Solar Mike
    Feb 19 at 8:02
  • $\begingroup$ The primary structural issues are the deflection and stress risers in the are where the end caps join. There are three typical end cap designs, and those have been studied to death by NASA and others. There are some closed form solutions and a lot of heuristics for local reinforcements at the connection of the ends. $\endgroup$
    – Phil Sweet
    Feb 19 at 12:14
  • $\begingroup$ Yep, for given internal pressure and material stress limit, you'll get thickness/diam ratio independent of length. The ratio of sidewall_volume / internal_volume will also be a function of this t/d ratio, and again independent of length. That leaves the endcaps as Phil says , and they should get lighter with smaller d. $\endgroup$
    – Pete W
    Feb 19 at 14:26
  • $\begingroup$ @SolarMike but the required thickness is a function of curvature. The only reason a pressure cylinder would weigh more than a pressure sphere is that the axial stress in the cylinder shell is less than the hoop stress, so the sphere ends up being a bit more efficient, at least as far thin-wall approximations are concerned. $\endgroup$
    – Phil Sweet
    Feb 20 at 0:14

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What about the enclosures at the ends? These are more expensive than the same area of steel plate from which the cylindrical part is manufactured. Flat ends require greater thickness and domed ends require forming. So the more cylindrical part in between the ends, the cheaper it gets.

There are also some additional reasons:

  • Thicker plates are more prone to internal flaws and may be more difficult to weld. This is why they tend to have lower allowable stress compared to thinner plates from the same material.
  • Transportation maybe impose limits on diameter, while the length can be much greater.
  • Aerodynamics is also an issue in transportation, especially when you transport things on a rocket, which is basically a flying pressure vessel. Slender design also simplifies staging and it may also be of some comfort that the precious cargo is some distance from the hot engine nozzle.
  • There may be technological reasons. For example shell and tube heat exchangers are most efficient when long with small diameter. Distillation columns also need to be quite slender.

This being said, one use of short large diameter cylindrical vessel is for vertical storage tanks. These are not strictly classified as pressure vessels, because there is just a hydrostatic pressure. The shorter this tank is, the lesser the maximum hydrostatic pressure at the bottom, although you need bigger area to accommodate the tank. These tanks save a lot on the end enclosures, because the bottom one sits on the ground and does not need to be especially thick and the upper one (roof) does not need to withstand much pressure (sometimes it is even designed not to withstand certain pressure, so it can blow of in case of accident (frangible roof), which is preferable to tearing at the bottom). Another type is a floating roof, which is just a pontoon floating on the surface of the stored liquid.

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