Design of Pressure Vessels

Why it matters

Pressure vessels are critical components in industries such as chemical processing, power generation, and oil refining. They are designed to hold gases or liquids at a pressure substantially different from the ambient pressure, making their design crucial for safety and efficiency.

Key ideas

• Types of Pressure Vessels: Pressure vessels can be classified based on their geometry (cylindrical, spherical) and the type of pressure they contain (internal or external).
• Stress Analysis: The design involves analyzing stresses due to internal pressure, which includes hoop stress, longitudinal stress, and radial stress.
• Material Selection: Materials must be chosen based on their strength, toughness, and resistance to corrosion and temperature.
• Safety Factors: Design must incorporate safety factors to account for uncertainties in material properties and loading conditions.
• IS Codes: Indian Standards (IS) provide guidelines for the design and testing of pressure vessels.

Assumptions

The membrane formulas below apply to a thin-walled circular cylindrical shell away from ends, nozzles and other discontinuities. Use pressure difference across the wall; the longitudinal expression requires closed ends whose pressure thrust is carried by the shell. These are educational stress estimates, not a complete pressure-vessel design or code-compliance calculation. External pressure also introduces buckling concerns.

Formulas

• Hoop Stress: σ_h = (p·d) / (2·t)
  • σ_h: Hoop stress (Pa)
  • p: Internal gauge pressure relative to the exterior (Pa)
  • d: Internal diameter of the vessel (m)
  • t: Wall thickness (m)
• Longitudinal Stress: σ_l = (p·d) / (4·t)
  • σ_l: Longitudinal stress (Pa)
• Radial Stress at the inner surface: σ_r = -p; it varies through the wall to approximately zero at the outer surface when external gauge pressure is zero.
  • σ_r: Radial stress (Pa)

Worked example

Given: A cylindrical pressure vessel with an internal diameter of 1 m, wall thickness of 10 mm, and internal gauge pressure of 2 MPa, with closed ends.

1. Calculate Hoop Stress

   • Formula: σ_h = (p·d) / (2·t)
   • Calculation: σ_h = (2×10^6 Pa · 1 m) / (2 · 0.01 m)
   • Result: σ_h = 100×10^6 Pa
   • Hoop Stress = 100 MPa

2. Calculate Longitudinal Stress

   • Formula: σ_l = (p·d) / (4·t)
   • Calculation: σ_l = (2×10^6 Pa · 1 m) / (4 · 0.01 m)
   • Result: σ_l = 50×10^6 Pa
   • Longitudinal Stress = 50 MPa

Common mistakes

• Ignoring Radial Stress: Radial stress is normally neglected compared with membrane stresses in a thin-wall approximation; it is not uniformly -p through the wall. Thick walls need a through-thickness stress analysis.
• Incorrect Unit Conversion: Ensure all units are consistent, especially when converting mm to m or MPa to Pa.
• Neglecting Safety Factors: Always include safety factors in design calculations.

For GATE ME

Questions typically involve calculating stresses in thin-walled pressure vessels, understanding failure theories, and applying IS codes. Practice problems on stress distribution and safety factor calculations.

Quick check

1. What is the formula for hoop stress in a cylindrical vessel?
2. Why is material selection important in pressure vessel design?
3. What is the typical unit for measuring stress in pressure vessels?

Answers: 1. σ_h = (p·d) / (2·t) 2. To ensure strength and resistance to operating conditions 3. Pascal (Pa)