Pressure vessels are essential systems commonly used to store fluids. However, these vessels are often subjected to large internal pressure due to the fluid. A small rupture or failure of these vessels can cause an explosion. Therefore, engineers need to understand the different stresses that pressure vessels experience before designing (or choosing) one for an application.

Figure 1: Engineers need to understand the different stresses that pressure vessels experience before designing (or choosing) one for an application. Source: gooutsight/Adobe Stock

Understanding the classification of pressure vessels

Pressure vessels are categorized according to their:

• Dimension
• End construction

Classification of pressure vessels according to their dimension

Engineers classify pressure vessels as thin-wall or thick-wall vessels based on the ratio of the vessel’s inside radius to the wall thickness.

Equation 1: Pressure vessel inside radius to wall thickness equation.

Where:

R_i = inside radius

t = wall thickness

As a rule, thin wall vessels have their radius to wall thickness ratio greater than 10, while vessels with a ratio less than 10 are considered thick wall vessels. So, for instance, engineers will find thin wall vessels commonly used in boilers, pipes, and regular industrial tank applications. On the other hand, thick wall vessels are commonly used in high-pressure cylinders and tanks.

Classification of pressure vessels according to their end construction

Pressure vessels can be categorized according to their end construction as open-ended or close-ended vessels. As its name implies, the open-ended vessel has at least one of its ends open. Examples include pipes, the cylinder of a press and the piston cylinder. In contrast, close-ended vessels have both ends closed, such as tanks.

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Stresses in a thin cylindrical shell under internal pressure

When a thin cylindrical shell or vessel is under internal pressure, it experiences three fundamental types of stress:

• Radial stress
• Circumferential (or hoop) stress
• Longitudinal stress

In the design of pressure vessels, engineers usually assume that the magnitude of the radial stress is small and can be ignored. Therefore, these vessels are likely to fail circumferentially or longitudinally.

Circumferential stress and failure

To understand how circumferential stress affects a cylindrical shell, consider Figure 2, which shows the failure of a cylindrical shell due to circumferential stress. It Is represented by σ_y.

Figure 2. Stress directions of a cylindrical shell, with directions of stress. Source: Cdang/CC BY-SA 3.0

The circumferential stress (also known as hoop stress) is simply tensile stress acting tangential to the vessel's circumference. And the effect of this may split the pipe into two halves. It can be calculated using:

Equation 2: Pressure vessel circumferential stress equation.

Where:

P = internal pressure (N/m² or Pa)

d = inner diameter of the cylindrical shell (m)

t = thickness of the shell (m)

This stress is assumed to be constant across the wall thickness. Also, this equation can be used to determine the maximum tangential stress in pressure vessels based on the assumption that the end enclosures do not provide any support, as is the case with cylinders and pipes.

So consider a scenario where a thin cylindrical pressure vessel of 1.3 m diameter is required to deliver steam at 2 MPa. If the vessel is to be designed to have a thickness of 25 mm, then the circumferential stress experienced by the vessel will be obtained as 52 MPa, as shown below:

Equation 3: Example equation of pressure vessel circumferential (hoop) stress.

Longitudinal stress and failure

Figure 2 also shows the failure of a thin cylindrical shell due to longitudinal stress. It Is represented by σ_x.

Notice how the longitudinal stress acts along the transverse section of the shell. Longitudinal stress is the tensile stress that tends to cause failure by splitting the cylinder into two smaller cylindrical shells. It can be calculated using:

Equation 4: Pressure vessel longitudinal stress equation.

Where:

P = intensity of internal pressure (N/m² or Pa)

d = internal diameter of the cylindrical shell (m)

t = thickness of the cylindrical shell (m)

From this equation, it can be observed that the longitudinal stress is half of the hoop stress. Therefore, when designing pressure vessels, engineers should ensure their design calculations are based on the possible maximum circumferential stress in the vessel.

Conclusion

While it is important to consider circumferential and longitudinal stresses, there are several other things that engineers must consider when designing pressure vessels for a particular application. For instance, engineers must also consider the thermal conditions that the vessels will be subjected to and choose the ideal engineering material to meet these requirements.

Therefore, engineers are advised to reach out to pressure vessel suppliers to discuss their application needs.