Analysis of Thin Pressure Vessels

In this tutorial, you will be introduced to Pressure Vessels and learn what factors govern the design of Thin Pressure Vessels. You will learn the stresses associated with such a vessel in application and other processes used to make a thin vessel functional.

Contents:

  1. What are Pressure Vessels?
  2. What are Thin Cylinders?
  3. Tangential Stresses in Thin Cylinders
  4. Longitudinal Stresses in Thin Cylinders
  5. Thin Spherical Vessels
  6. What are Gaskets?
  7. What is Autofrettage?
  8. Safety and Maintenance of Pressure Vessels

What are Pressure Vessels?

A Pressure Vessel is a term reserved for containers that are designed to handle liquids or gases at elevated temperatures during transit or in storage. Pressure Vessels can be dangerous if not designed accurately, and history holds many examples of events resulting from vessels that failed under load.

  • A vessel is said to be pressurized if there exists a pressure difference separated by a boundary. Oxygen Cylinders, Boilers, Autoclaves, and even balloons are examples of pressure vessels.
  • Pressure vessels can be designed to satisfy any shape imaginable, but the shapes are usually limited to spheres, cylinders, and in some cases, cones. A common design is a cylinder capped at its ends with hemispheres.
  • Historically, pressure vessels have been made from various grades of steel, depending on the application. However, research has indicated that composites can be a viable replacement. Composite Materials can be wound around the container, reinforcing it and providing durability and isolation.
  • Pressure vessels find application in industrial and private sectors like boilers and storage tanks and can be also used to protect their occupants in high-pressure environments, like the passenger cabin of an airplane or deep diving cylinder and submarines.
  • From a designer’s aspect, the vessels can be categorized as thin or thick pressure cylinders depending on the thickness of the boundary separating the pressure regions.

What are Thin Pressure Vessels?

As discussed earlier, pressure vessels can be divided into categories of thick or thin depending on their wall thickness. A vessel is considered thin when the ratio of its inner diameter to wall thickness is greater than 15.

  • Thin vessels are subjected to failure most commonly when their internal stresses exceed the limits of the material of which the vessel is fabricated.
  • Thin vessels are placed under circumferential tangential stresses and longitudinal stresses when in application.
  • Thin Pressure vessels are usually used as pipes for commercial or domestic uses, boilers for domestic application, and storage tanks for handling fluids or gases.

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Tangential Stresses in Thin Cylinders

In a pressure vessel shaped like a cylinder, failure will occur along the length if the circumferential or tangential stresses exceed their limits. These stresses occur along the circumference of the vessel and can be visualized to radiate outwards from the center of the cross-section of the vessel.
The diagram demonstrates the application of Tangential Stress on the cylinder wall due to the pressurized fluid and the resulting restoring forces in the wall.

Tangential Stresses in Thin Cylinders

Balancing the equations we get
DiPi=2σtt
Rearranging, we get
σt=\(\frac{P_iD_i}{2t}\)
Here,
Pi = Internal Pressure of the Pressurized Fluid
Di = Internal Diameter of the Cylinder
t = Thickness of the wall
t = Tangential Stress

Longitudinal Stresses in Thin Cylinders

In a pressure vessel shaped like a cylinder, failure will occur along the transverse direction if the longitudinal stresses exceed their limits. These stresses occur along the ends of the vessel and can be visualized to push outwards along the caps of the cylinder.
The diagram demonstrates the application of Longitudinal Stress on the cylinder wall due to the pressurized fluid and the resulting restoring forces in the wall.

Longitudinal Stresses in Thin Cylinders

Balancing the equations we get
\(P_i (\frac{π}{4} D_i^2)\) = l (πDit)
Rearranging, we get
Pi = \(\frac{P_i D_i}{4 t}\)
Here,
Pi = Internal Pressure of the Pressurized Fluid
Di = Internal Diameter of the Cylinder
t = Thickness of the wall
l = Longitudinal Stress

Thin Spherical Vessels

In a pressure vessel shaped like a sphere, the sphere is inherently symmetrical and shows only tangential or circumferential stresses when in application. As such pressure vessels can be analyzed as two hemispherical halves stressed by the fluid pressure.
The diagram demonstrates the application of Tangential Stress on the Spherical vessel’s wall due to the pressurized fluid and the resulting restoring forces generated in the wall.

Tangential Stresses in Thin Spherical Vessels

Balancing the equations we get
\(P_i (\frac{π}{4} D_i^2)\) = l (πDit)
Rearranging, we get
Pi = \(\frac{P_i D_i}{4 t}\)
Here,
Pi = Internal Pressure of the Pressurized Fluid
Di = Internal Diameter of the Sphere
t = Thickness of the wall
t = Tangential Stress

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What are Gaskets?

A gasket is any device that is used to create and maintain a boundary between regions to prevent the transfer of fluids. It is a mechanical seal that is used to fill in the space between mating surfaces of assemblies and prevents leakage when under a compressive environment.
The following diagram shows some common shapes of gaskets used in the industry.

Common Types of Gaskets
  • There are two types of gaskets available, metallic, or non-metallic types.
  • Metallic gaskets are made of sheets of copper, aluminum, or in some cases, even lead. They are used in high-temperature and pressure environments. They take on a permanent shape when compressed and have minimal recovery characteristics to compensate for any changes. They are also susceptible when used in a chemical or corrosive environment.
  • Non-metallic gaskets are made of materials like cork, rubber, or plastic. Asbestos gaskets have resistance to crushing loads. Rubber seals are cheap and used in steam lines.
  • A compression ring is often added to allow for higher flange compression while preventing gasket failure. Often, a guiding ring is added to the gasket to allow for easier installation and to serve as a compression inhibitor.
  • A gasket can fail when it experiences an unevenly distributed pressing force. This can be due to asymmetric bolt preloading, causing a discontinuity in the gasket. The surface deforms and fluid leakage may occur.
  • An uneven surface provides for paths through which the fluid may leak out. As such the surfaces of the gasket must be polished or machined flat. A surface with a 32RMS finish is often a good choice.

What is Autofrettage?

Autofrettage is a process that involves prestressing the cylinder before it is put in application. It is often used in high-pressure applications like gun barrels. When the cylinder is placed under high internal pressure, the circumferential stresses limit the pressure capacity of the cylinder. In the pre-stressing process, residual compressive stresses are developed at the inner surface, which begins to decrease as the pressure is gradually increased in the cylinder. There are three methods for prestressing.

  • A compound cylinder consists of two concentric cylinders with the outer cylinder shrunk on the inner one. This results in induced compressive stresses in the inner cylinder.
  • Overloading a cylinder involves stressing the inner cylinder in the plastic region while the outer surface is maintained at elastic regions. When the pressure is released, the outer region contracts, exerting pressure on the inner surface.
  • Another method is to wind a wire under tension around the cylinder, which results in residual compressive stresses.
  • The Autofrettage process increases the pressure capacity of the cylinder and also serves to close any cracks within the cylinder resulting in enhanced endurance strength.

Safety and Maintenance of Pressure Vessels

Fatal accidents have happened in the history of pressure vessel development and operation. The Bhopal Gas Tragedy is one of the world’s worst disasters and occurred due to the backflow of water in a MIC tank. The pipelines and valves were in poor condition and safety systems were not functioning. This disaster highlighted the importance of proper maintenance and safety precautions in high-pressure vessel systems.
Some Safety and maintenance features used include

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  • The concept of “leak before burst” describes a pressure vessel that is constructed so that a crack in the wall will spread through the wall, allowing the contained fluid to escape, and decreasing the pressure, before becoming large enough to cause fracture at the operating pressure.
  • Because the pressure vessel is constructed to a specific pressure, there is usually a safety valve or relief valve to guarantee that this pressure is not exceeded while in use.
  • Pressure vessel closures are pressure-retaining constructions that provide quick access to pipelines, pressure vessels, pig traps, filters, and filtration systems. Pressure vessel closures usually give maintenance staff access. Internal pressure prevents it from being opened unexpectedly when under load.

Key Points to Remember

Here is the list of key points we need to remember about “Analysis of Thin Pressure Vessels”.

  • Pressure Vessels are classified as thick or thin depending on the value of the ratio of the inner diameter to the wall thickness.
  • Thin Pressure Vessels have a value greater than 15 for this ratio and see the widespread application.
  • Thin Cylinders face Longitudinal and Tangential Stresses from the Pressure of the fluid contained within them.
  • Thin Spherical vessels are symmetric about their origin and only see the development of tangential stresses in their walls.
  • Gaskets are used to create mechanical seals between two mating surfaces to prevent leaks by sealing the joints when in compression. They can be made of metals, plastic, rubber, etc.
  • Autofrettage is a process of creating compressive stress in the inner surface of the vessel to increase the pressure handling capacity of the vessel.
  • Safety and maintenance are paramount when it comes to pressure vessel systems, as system failure can lead to catastrophic losses, as is evident in our history.

If you find any mistake above, kindly email to [email protected]

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Manish Bhojasia - Founder & CTO at Sanfoundry
Manish Bhojasia, a technology veteran with 20+ years @ Cisco & Wipro, is Founder and CTO at Sanfoundry. He lives in Bangalore, and focuses on development of Linux Kernel, SAN Technologies, Advanced C, Data Structures & Alogrithms. Stay connected with him at LinkedIn.

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