Pressure vessel

A pressure vessel is a closed, rigid container designed to hold gases or liquids at a pressure different from the ambient pressure.

Fine examples of pressure vessels are: diving cylinder, recompression chamber, distillation towers and many other vessels in oil refineries and petrochemical plants, nuclear reactor vessel, habitat of a space ship, habitat of a submarine, pneumatic reservoir, hydraulic reservoir under pressure, rail vehicle airbrake reservoir, road vehicle airbrake reservoir and storage vessels for liquified gases such as ammonia, chlorine, propane, butane and LPG.

In the industrial sector, pressure vessels are designed to operate safely at a specific pressure and temperature, technically referred to as the "Design Pressure" and "Design Temperature". A vessel that is inadequately designed to handle a high pressure constitutes a very significant safety hazard. Because of that, the design and certification of pressure vessels is governed by design codes such as the ASME Boiler and Pressure Vessel Code in North America, the Pressure Equipment Directive of the EU (PED), Japanese Industrial Standard (JIS), CSA B51 in Canada and other international standards like Lloyd's, Bureau Veritas, Germanische Lloyd, Det Norske Veritas, Stoomwezen etc.

Shape of a pressure vessel
In theory, indeed, a sphere gives the most economic design of a pressure vessel. The most used material for pressure vessels is steel. In case a sphere of steel is used, one has to weld the pressure vessel together from forged items. The quality of steel is increased by the forging, but for sure not by the welding. In case of welding, especially if the pressure vessel should be classified for use all over the world (on a ship,) very special steel with a high impact value must be chosen. Apart from that, in a spheric pressurevessel, inspection covers will have to be made (for checking the welds), which makes the vessel expensive, also by the high quality welding, testing and heat treatments that are needed. For this reason you nowadays see more and more pressure vessels, made from one piece of forged material (by the Pilgerschrittverfahren). At both ends sphere shaped covers are forged to the forged pipe (same sort of pipe that is used for hydraulic cylinders). Advantage is that in this case no welding is needed and very high tensile steel can be used, so that a certain pressure vessel can be made with lighter weight than ever before. Disadvantage of these vessels is the fact that larger diameters are relatively expensive so that the most economic shape of a 1000 litres, 250 bar pressure vessel might be a diameter of 450 mm and a length of 6500 mm.

Scaling
No matter what shape it takes, the minimum mass of a pressure vessel scales with the pressure and volume it contains. For a sphere, the mass of a pressure vessel is

$$M = {3 \over 2} p V {\rho \over \sigma}$$

Where $$M$$ is mass, $$p$$ is pressure, $$V$$ is volume, $$\rho$$ is the density of the pressure vessel material, and $$\sigma$$ is the maximum working stress that material can tolerate. Other shapes besides a sphere have constants larger than 3/2, although some tanks, such as non-spherical wound composite tanks can approach this.

As can be seen from the equation, there is no theoretical efficiency of scale to be had in a pressure vessel; and further, for storing gases, tankage efficiency can be easily shown to be independent of pressure.

So, for example, a typical design for a minimum mass tank to hold helium (as a pressurant gas) on a rocket would use a spherical chamber for a minimum shape constant, carbon fiber for best possible $$\rho / \sigma$$, and very cold helium for best possible $$M / {pV}$$.

A spherical tank has less surface area for a given volume than any other tank shape. Also, the hoop stress in the wall of a sphere is half that of a cylinder at the same pressure. Thus if the walls are made of the same material, the spherical tank can hold twice the pressure of the cylindrical tank, or at the same pressure, the spherical tank wall can be half the thickness.

Stress in thin-walled pressure vessels
The stress in a thin-walled pressure vessel in the shape of a sphere is:  $$\sigma_\theta = \frac{pr}{2t}$$  Where $$\sigma_\theta$$ is the hoop stress, or stress in the radial direction, p is the internal gage pressure, r is the radius of the sphere, and t is the thickness. A vessel can be considered "thin-walled" if the radius is at least 20 times larger than the wall thickness.

The stress in a thin-walled pressure vessel in the shape of a cylinder is:  $$\sigma_\theta = \frac{pr}{t}$$  $$\sigma_{long} = \frac{pr}{2t}$$  Where $$\sigma_\theta$$ is the hoop stress, or stress in the radial direction, $$\sigma_{long}$$ is the stress in the longitudinal direction, p is the internal gage pressure, r is the radius of the cylinder, and t is the wall thickness.