A Step-by-Step Approach to Pressure Vessel Design Calculations

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Designing pressure vessels involves many complicated computations ranging from the selection of thickness of the vessel wall to evaluation of stress gradients. These calculations are also important to understand that the vessel will not fail under the conditions of its operation. This blog will also present practical guidance on the key computations performed in pressure vessel design, so that these structures meet appropriate safety, performance, and code requirements. If you are a beginner in pressure vessel design or if you simply want to improve your knowledge of this topic, the information outlined in this guide will help you with that.

1. Understanding Design Requirements

The things which should be known in the case of pressure vessel design is the requirement and specific detail of the pressure vessel since they are involved in the calculation process. Each vessel’s design will depend on:

  • Operating Pressure and Temperature: Depending on the surreal pressure and temperature inside a vessel; the kind of materials used and thickness of the wall does vary.
  • Contents: Different gases or liquids stored in the vessel affect the choice of material to be used to construct the vessel because, for example, some are corrosive or react with specific materials.
  • Safety Factor: The following typically involves a safety factor that considers disturbing factors from operating conditions as well as imperfection of the material to ensure that the vessel endures pressure over the limit operating pressure.
  • Code Requirements: ASME BPVC or other codes and standards that spell out minimum requirements concerning safety are widely used.

With a clear understanding of the vessel’s specifications, let’s proceed to the critical calculations.

2. Calculating Wall Thickness

This is important for determining the thickness of the wall in order to resist the internal pressure threat of the vessel. Depending on the shape of the vessel there are two wall thickness computations possible.: cylindrical and spherical.

2.1 Cylindrical Pressure Vessels

For cylindrical pressure vessels, the wall thickness, t, is given by the formula:

       

Where:

  • P = Internal pressure (in psi or Pa)
  • R = Inside radius of the cylinder (in inches or mm)
  • S = Allowable stress of the material (in psi or Pa)
  • E = Joint efficiency (typically between 0.7 and 1, depending on welding quality)

The joint efficiency factor (E) includes the impact of weld quality that reaches a first-class value of 1.00 and decreases progressively to below a first-class value for low-quality welds.

2.2 Spherical Pressure Vessels

For spherical vessels, the wall thickness calculation is slightly different due to their geometry:

This calculation takes into account the fact that, by applying uniform stress on the shell, pressure vessels in the form of a sphere can contain higher pressure more effectively than cylindrical pressure vessels.

3. Stress Analysis

Stress analysis makes sure that the material chosen for the vessel’s construction will endure the force experienced internally and externally as the vessel operates. It will be necessary to differentiate three major categories of stress in pressure vessels: circumferential (hoop) stress, longitudinal stress, and radial stress.

3.1 Circumferential (Hoop) Stress

For a cylindrical vessel, hoop stress is given by:

     

Where:

  • σh​ = Hoop stress (in psi or Pa)

Being the maximum stress in a pressure vessel, hoop stress is normally considered the governing stress and consequently influences material selection most.

3.2 Longitudinal Stress

The longitudinal stress, experienced along the length of the vessel, is half of the hoop stress:

   

This stress is not as extreme as hoop stress, but cannot and should not be ignored when calculating the confines of the pressure vessel in question.

3.3 Radial Stress

Radial stress is the stress that act along a radial direction that passes from the outer surface of the vessel to the inner surface of the vessel. The pressure increases from the value at the inner wall up to zero at the outer wall. For thick walled pressure vessels, radial stress is more significant than longitudinal stress, while for thin-walled pressure vessels, it is relatively insignificant in many cases.

4. Material Selection and Allowable Stress

The choice of an appropriate material for construction of a pressure vessel is crucial in order to supply the necessary stress conditions and withstand corrosion. Allowable stress (S) is calculated from the material yield strength and the ultimate tensile strength, less another factor of safety.

  1. Yield Strength: This is the stress at which point the material starts to take a new permanent shape. In the case of the pressure vessels, the material is required to be resistant to the applied forces with resulting deformation.
  2. Tensile Strength: This is the ultimate strength of the product, the maximum stress which it can withstand before it fails. It is the maximum that a vessel can suffer and get back into operation.
  3. Corrosion Allowance: Corrosion allowance is normally incorporated into the design thickness value since walls will degrade with service periods. This additional thickness is determined with respect to corrosion rate and life of the slab in years.

5. Thermal Stress and Expansion

Thermal stresses result from temperature changes within the vessel, especially in applications with high operating temperatures.

5.1 Thermal Stress Calculation

Thermal stress σt can be calculated as:

  

Where:

  • E = Young’s modulus of the material (in psi or Pa)
  • α = Coefficient of thermal expansion (per °F or °C)
  • ΔT = Temperature difference (in °F or °C)

Thermal stress analysis helps prevent cracking and failure in high-temperature applications.

5.2 Thermal Expansion

This effect can alter the dimensions of the vessel and must be accounted for in uses where the temperatures change appreciably. Expansion joints and allowances must be provided to let the building move as much as the structure permits due to the effect of thermal expansion.

6. Fatigue Analysis

Cyclic loading is a common load that acts on pressure vessels and therefore pressure vessels may experience fatigue failure. Fatigue can be considered as another central parameter for the characterization of cyclic behavior of pressure vessels and for evaluation of residual life at different Pressure cycles.

6.1 S-N Curve

Fatigue behaviour is commonly represented by an S-N curve, which shows the relationship between stress amplitude (S) and the number of cycles to failure (N). For a given stress amplitude, the S-N curve provides an estimate of the number of cycles the vessel can endure before failing.

6.2 Fatigue Life Calculation

Fatigue life is obtainable through calculation formulas or determined from the S-N curve of the engineering material. Subsequently, engineers operate at parameters or design and control the stress that the vessel endures to visible optimize its time of use.

7. Safety Factor and Design Margins

A safety factor is applied in pressure vessel design to account for uncertainties and variations in material properties and operating conditions. A typical safety factor ranges from 1.5 to 4, depending on the application, regulatory requirements, and potential hazards.

7.1 Determining Safety Factor

The safety factor F is determined based on:

  • The consequences of failure (e.g., human safety and environmental risks).
  • The accuracy of load predictions.
  • Material consistency and properties.

The formula for allowable stress with a safety factor is:

     

8. Testing and Validation

In order to check calculations, as well as to ensure the integrity of the structure, prior to operation of the pressure vessel, it has to be tested.

8.1 Hydrostatic Testing

Hydrostatic testing is conducted by inundating the vessel with water which is then pumped in at pressures to check on the vessel’s capacity to handle the design pressure. This test is normally carried out at 1.5 multiples of the operating pressure of the vessel without signs of leakage or structural failure.

8.2 Pneumatic Testing

Pneumatic testing applies compressed gas normally in the form of air or nitrogen to place pressure on the vessel. It is more effective than hydrostatic testing but much more dangerous in case the vessel blows up; thus, it is used only in cases where it is impossible to perform hydrostatic testing.

9. Design Documentation and Compliance

After doing calculations and testing, remaining documentation is equally crucial in terms of legal requirements for construction projects or other inspections. Documentation should include:

  • Material Certifications: For Some applications, ASME or other code requirements should be met to take the materials utilized.
  • Calculation Reports: Original documented drawings of the engineering design process and the required calculations.
  • Inspection and Testing Records: Record of hydrostatic test, pneumatic test, non destructive test, and other evaluation tests.

These are requirements that are standard for most pressure vessels in industries and anyone willing to design pressure vessels must observe the ASME Section VIII to the letter.

10. Advanced Considerations: Finite Element Analysis (FEA)

For more complicated pressure vessel designs, there is also an analytic tool called finite element analysis, or FEA, that gives a much finer picture of stress weight and possible points of failure. Together with multidisciplinary design spaces, FEA will compute the pressure temperature distribution of the vessel and predict regions of high stress within the design requiring modification.

Conclusion

The pressure vessel design calculations are a series of steps and a strong emphasis should be laid on principles of engineering. Each of them is important starting from the understanding of requirements for each wall thickness calculation down to stress, fatigue, and thermal expansion calculations for the vessel. It is about following industry codes, performing tests and calculating every move to demonstrate the pressure vessels are secure and can last as intended.

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