# Pressure vessel wall dimensions

I am trying to design a small cooling setup which has two parts: the compressed air vessel and the turbines. My understanding is that the compressed air, upon being released, passes through the turbine and loses its energy to a useful process (the turbine could run another compressor for instance) and thus becomes cold. This cold air is then released where needed (a room, a fridge etc.)

Visualized:

Vessel material: stainless-steel 316-grade Design temperature is 150°C (Assume a value upto 200°C for easier calculations, if you want)

How thick should the walls at points A, B and C be to be safe with these three design pressures?

Design pressure 𝘼 =5516 kPa

Design pressure 𝑩 =4482 kPa

Design pressure 𝘾 =2758 kPa

I don't need exact answers, but a largest (for safety) approximate would be fine too. If you could give me the relevant formulae and laws (capsule shaped vessels) so I could calculate it myself, I would really appreciate it! Note that I am a newbie, so correct mistakes in the design itself.

Thank you!

• If there are quantities I haven't determined above, then ask if it's very user specific, otherwise assume (but mention) the quantity's value. – El Flea Mar 24 at 4:08
• – Solar Mike Mar 24 at 5:05
• @SolarMike I can't see anything in that link that can answer my question. If you're talking about the answer, then that's quite general and for a sphere. My vessel is very different. – El Flea Mar 24 at 8:38
• At a 101 theory level, see the Lamé equation etc for thick walled cylinders. There would be actual codes for pressure vessels in industrial use. – Pete W Mar 24 at 13:14
• I was going to say going back to basics, I believe there is a physics/chemistry formula for compressible gas. PV=nR*T. – Forward Ed Mar 24 at 18:39

First thing first, you need to get hand on the ASME Boiler and Pressure Vessel Code (BPVC Code). Starting division 8, section 1, for safety and quality of your pressure vessel.

After checking on all relevant provisions/standards, you can use the equations to derive the required wall thickness for your application.

Notation σ_H = hoop stress, psi or MPa D = outer diameter, in or mm E = modulus of elasticity, psi or MPa P = pressure under consideration, psi or MPa P_i = internal pressure, psi or MPa P_o = external pressure, psi or MPa r = radius to point of of interest, in or mm r_i = internal radius, in or mm r_o = external radius, in or mm t = wall thickness, in or mm ∆_P = change in pressure, psi or MPa

Note, the above formulas may be used with both imperial and metric units, just keep the units consistent.

• Thank you so much! Finally someone who satisfies my answer, not showing off about his "10 years experience working with GE". Thank you!!! – El Flea Mar 25 at 4:24

That is certainly the hard way to do it. You could go to a local welding supplier and buy or lease a standard steel pressure bottle / tank ; good for probably 3 X your maximum pressure (designed and built in conformance to ASME Sec. 8 , Div.1 ). Stainless steel is certainly not needed for air or most gasses. You don't need a turbine to get the cooling effect of expanding gas. By leasing a bottle with normal full pressure ( roughly 15000 kpa ) and using a standard pressure regulator , you could get several minutes of cooling.

• I don't need it the turbine, but I think I am extracting useful energy from the expansion, as that turbine could run another compressor, which will make my process more efficient. Am I misunderstanding something? By the way, what happens to that energy without the turbine? – El Flea Mar 24 at 17:58
• Sounds like the Holy Grail of thermodynamics -perpetual motion ! Turbine drives compressor to pressurize the next pressure vessel . – blacksmith37 Mar 24 at 19:06
• No, no no! Haha. Like I said, I might be misunderstanding something, but I don't believe in perpetual motion. All I'm saying is, some of the energy could definitely be extracted in the turbine? Obviously the next vessel won't be pressurized like the first so overall it won't be perpetual, duh. – El Flea Mar 25 at 4:22