Introduction & Context

The design of pressure vessels for high-pressure extraction, such as supercritical CO₂ systems, is a critical task in process engineering. Ensuring the structural integrity of the cylindrical shell is paramount to preventing catastrophic failure under high-pressure conditions. This calculation follows the ASME Boiler and Pressure Vessel Code (BPVC), Section VIII, Division 1, which provides the standardized framework for determining the minimum wall thickness required to safely contain internal pressure.

This methodology is typically employed during the preliminary design phase of extraction skids, reactor vessels, and storage tanks. It serves as the primary check to ensure that the selected material and geometry can withstand the intended operating conditions while accounting for safety factors, weld quality, and material degradation over time.

Methodology & Formulas

The calculation determines the minimum required thickness by balancing the hoop stress induced by internal pressure against the allowable stress of the material. The process follows these logical steps:

1. Minimum Theoretical Thickness: The base thickness t is calculated using the ASME formula for cylindrical shells under internal pressure:

\[ t = \frac{P \cdot R}{S \cdot E - 0.6 \cdot P} \]

2. Corrosion Allowance: To account for material loss over the vessel's service life, the corrosion allowance CA is added to the theoretical thickness:

\[ t_{ca} = t + CA \]

3. Nominal Thickness Selection: The calculated t_ca is rounded up to the nearest standard plate thickness t_nominal, accounting for manufacturer under-tolerance MT.

4. MAWP Verification: The Maximum Allowable Working Pressure MAWP is back-calculated using the available thickness to ensure the final design meets or exceeds the design pressure P:

\[ MAWP = \frac{S \cdot E \cdot (t_{nominal} - MT - CA)}{R + 0.6 \cdot (t_{nominal} - MT - CA)} \]

Regime / Condition	Criteria
Thin-Wall Validity	P < 0.385 · S · E
Joint Efficiency	0.0 < E ≤ 1.0
Denominator Stability	(S · E) - (0.6 · P) > 0
Design Adequacy	MAWP ≥ P

To ensure operational safety and regulatory compliance, process engineers must prioritize the following:

Adherence to ASME Boiler and Pressure Vessel Code Section VIII standards.
Implementation of redundant pressure relief devices, such as rupture disks and safety valves.
Selection of materials compatible with the solvent, specifically focusing on corrosion resistance and fatigue strength.
Verification of the maximum allowable working pressure (MAWP) against the peak process pressure.

Determining wall thickness requires a rigorous calculation based on the vessel geometry and operating conditions. Key factors include:

The design pressure and design temperature of the system.
The allowable stress values of the chosen material at the operating temperature.
The inclusion of a corrosion allowance to account for material degradation over the vessel lifecycle.
Joint efficiency factors based on the type of welding and non-destructive examination (NDE) performed.

Process engineers must design for specific failure modes that occur under high-pressure cycling:

Fatigue failure resulting from repeated pressurization and depressurization cycles.
Stress corrosion cracking caused by the interaction between the solvent and the vessel material.
Brittle fracture, particularly if the extraction process involves cryogenic temperatures.
Creep deformation if the process operates at elevated temperatures for extended durations.

Worked Example: Pressure Vessel Wall Thickness

A cylindrical pressure vessel is required for the static containment of supercritical CO₂ in a high-pressure extraction process. The following example calculates the minimum required wall thickness for the cylindrical shell section according to ASME Boiler and Pressure Vessel Code, Section VIII, Division 1.

Known Parameters

Design Pressure, \( P \): 40.0 MPa
Inside Radius of Cylinder, \( R \): 250.0 mm
Allowable Stress for SA-516 Gr. 70 Steel, \( S \): 138.0 MPa
Joint Efficiency for Radiographed Butt Weld, \( E \): 0.85
Corrosion Allowance, \( CA \): 1.5 mm
Manufacturer's Under-Tolerance, \( MT \): 0.25 mm

Step-by-Step Calculation

Validate Applicability of Thin-Wall Formula:
Check that \( P < 0.385 \cdot S \cdot E \).
\( 0.385 \cdot 138.0 \cdot 0.85 = 45.1605 \) MPa.
Since \( 40.0 \) MPa < \( 45.1605 \) MPa, the standard formula is applicable.
Calculate Minimum Theoretical Thickness:
Use the formula \( t = \frac{P \cdot R}{S \cdot E - 0.6 \cdot P} \).
First, compute the denominator: \( (138.0 \cdot 0.85) - (0.6 \cdot 40.0) = 93.3 \).
Therefore, \( t_{calc} = \frac{40.0 \cdot 250.0}{93.3} = 107.181 \) mm.
Add Corrosion Allowance:
\( t_{ca} = t_{calc} + CA = 107.181 + 1.5 = 108.681 \) mm.
Select Nominal Plate Thickness:
The calculated required thickness (108.681 mm) is rounded up to the next available standard plate thickness. The selected nominal thickness is \( t_{nominal} = 109.0 \) mm.
Verify Design with Maximum Allowable Working Pressure (MAWP):
The available metal thickness is \( t_{available} = t_{nominal} - MT - CA = 109.0 - 0.25 - 1.5 = 107.25 \) mm.
Back-calculate the MAWP using \( P_{MAWP} = \frac{S \cdot E \cdot t_{available}}{R + 0.6 \cdot t_{available}} \).
This yields \( P_{MAWP} = 40.02 \) MPa.
Final Validation:
The calculated MAWP (40.02 MPa) is greater than the design pressure (40.0 MPa). Therefore, the selected nominal thickness is adequate.

Final Answer

The minimum required nominal plate thickness for the cylindrical shell is 109.0 mm. This provides a validated Maximum Allowable Working Pressure of 40.02 MPa, which meets the design requirement of 40.0 MPa.

"Un projet n'est jamais trop grand s'il est bien conçu." — André Citroën

"La difficulté attire l'homme de caractère, car c'est en l'étreignant qu'il se réalise." — Charles de Gaulle