Pressure puts a hoop stress of P × d ÷ (2 × t) into a cylindrical wall, so a 30 cm vessel with a 4 mm wall carries about 38 MPa at 10 atmospheres and about 570 MPa at 150 atmospheres, far beyond what ordinary steel can take. Temperature raises pressure in three ways: gases expand in proportion to absolute temperature, liquids boil into vapour whose pressure rises steeply, and a liquid-full sealed vessel gains roughly 4.6 atmospheres per degree Celsius. No allowable-stress figure is given here; take that from tested material, and keep relief valves at 90% of the vessel maximum.
Part of Industrial Chemistry
The arithmetic of overpressure and thermal runaway in closed vessels: gas law, vapour pressure, liquid expansion and wall stress.
Four Ways A Closed Vessel Reaches Overpressure
- Heating a gas. At fixed volume, pressure is proportional to absolute temperature (kelvin, °C + 273).
- Boiling a liquid. Vapour pressure climbs far faster than the gas law predicts.
- Heating a liquid with no gas space. The liquid expands and the walls do not.
- Chemical generation of gas, such as decomposition or a runaway reaction (Heat Management).
The carrier’s rules for the response are: never exceed the rated pressure, install pressure relief valves, test at 1.5 times operating pressure with water, and set relief at 90% of the maximum (see Reaction Vessels).
What You Need
- Absolute temperatures and pressures in every calculation.
- A tested allowable stress for the actual wall material. Nothing in this article supplies one.
- A calibrated gauge, a relief valve and a vent to a safe place.
- Never a sealed, liquid-full vessel.
Method: Checking A Closed Vessel
Step 1: Gas law check. P₂ = P₁ × T₂ ÷ T₁ (absolute).
Step 2: Vapour pressure check for any liquid that can boil at the maximum temperature.
Step 3: Liquid-expansion check: is there ever a moment with no gas space?
Step 4: Wall stress check with σ = P × d ÷ (2 × t) against the tested allowable stress.
Step 5: Relief and vent: confirm the valve opens below the vessel maximum and discharges to a safe place.
Worked Example 1: Heating A Sealed Vessel Of Gas
Air sealed at 1 atmosphere and 20 °C (293 K), heated to 200 °C (473 K): P₂ = 1 × 473 ÷ 293 = 1.6 atmospheres absolute. A modest rise; the gas law alone is not the main danger.
Worked Example 2: Steam Is The Danger
The corpus states that 121 °C is reached by sealing water under about 15 psi (Pressure Sterilization); 15 psi is 15 ÷ 14.7 = 1.02 atmospheres above the surroundings, so about 2 atmospheres absolute. Water at 200 °C has a vapour pressure of about 15.5 atmospheres (a recalled steam-table value). From 121 °C to 200 °C, 79 °C of heating, the pressure rises about 15.5 ÷ 2 = 7.8 times. A sealed pot of water is a bomb that is only a fire away; see also Boiler Construction and Pressure Vessel Theory.
Worked Example 3: A Liquid-Full Sealed Vessel
The pressure rise per degree for a liquid in a rigid container is the volumetric expansion coefficient α times the bulk modulus K. For water near 20 °C, α ≈ 2.1 × 10⁻⁴ per °C and K ≈ 2.2 GPa (recalled values).
- dP/dT = 2.1 × 10⁻⁴ × 2.2 × 10⁹ = 4.6 × 10⁵ Pa per °C = 0.46 MPa per °C.
- In atmospheres: 0.46 ÷ 0.1013 = 4.6 atmospheres per °C.
- A rise of 10 °C in a perfectly rigid, liquid-full vessel is about 46 atmospheres. Real walls stretch and real vessels have small gas pockets, so the rise is lower, but the order of magnitude explains why a full sealed drum left in the sun can burst.
Never fill a vessel completely, and keep a vent or relief path to a gas space.
Worked Example 4: Wall Stress
For a thin-walled cylinder, hoop stress σ = P × d ÷ (2 × t), with diameter d and wall t. This is valid for walls thin relative to the diameter, t below about d ÷ 10.
- 10 atm = 10 × 0.1013 = 1.013 MPa. For a vessel 30 cm across (d = 0.30 m) with a 4 mm wall (t = 0.004 m): σ = 1.013 × 0.30 ÷ (2 × 0.004) = 0.304 ÷ 0.008 = 38 MPa.
- 150 atm = 15.2 MPa: σ = 15.2 × 0.30 ÷ 0.008 = 570 MPa.
- Ordinary mild steel yields at a few hundred MPa (about 250 MPa is a recalled typical figure, varying by grade), so 570 MPa is more than twice yield even before allowing for a safety factor, and welds are weaker than the plate.
- With a placeholder allowable stress of 100 MPa (not a tested value), the wall for 150 atm would need t = P × d ÷ (2 × σ) = 15.2 × 0.30 ÷ 200 = 0.0228 m = 23 mm, nearly six times the 4 mm drum.
The calculation shows the shape of the problem: stress scales linearly with pressure and diameter and inversely with wall. It does not certify any vessel. The high-pressure ammonia case is in Haber Process.
Common Mistakes
- Using gauge pressure in the gas law. Use absolute.
- Ignoring vapour pressure because the gas law gave a gentle answer.
- Filling a vessel solid.
- Relying on a safety valve that can freeze, corrode shut or be blocked by product.
- Treating temperature rating and pressure rating as independent. Strength falls as steel gets hot.
Safety
A closed vessel stores energy and fails without a second chance
Heating a sealed container of water or solvent, or any liquid-full container, is how many accidents begin. Fit a relief valve discharging to a safe place, set at 90% of the maximum, never plug it, test vessels with water not gas, stand behind a barrier during tests, and never use a pressure cooker, drum or cylinder as a chemical reactor. See Safety and Waste.
What’s Next
- Learn next: Safety and Waste — the wider hazard picture
- Combine with: Equipment Design — turning the numbers into hardware