Spherical Shell Formula in Process Engineering
A spherical shell is a three-dimensional shape that looks like a hollow ball. It is made up of two concentric spheres, which means they have the same center, but different sizes. The spherical shell is the region between the inner and outer spheres. You can think of it as a ball that has been carved out from a bigger ball.
One example of a spherical shell is a soccer ball. The soccer ball has an inner layer of rubber and an outer layer of leather. The spherical shell is the space between these two layers. Another example is the Earth. The Earth has an inner core, an outer core, a mantle, and a crust. The spherical shell is the space between any two of these layers.
A spherical shell is a hollow sphere, commonly used in various engineering applications due to its uniform stress distribution under internal pressure. The formula for calculating the wall thickness of a spherical shell subjected to internal pressure (
) is given as:
![\[ t = \frac{{P \cdot r}}{{2 \cdot S}} \]](http://www.industrialseparation.com/wp-content/ql-cache/quicklatex.com-07e1addf3f4751235cf27f737bfa50f2_l3.png "Rendered by QuickLaTeX.com")
Where:
= wall thickness of the spherical shell- = internal pressure
= radius of the spherical shell
= allowable stress of the material
This formula helps engineers determine the necessary thickness of the shell to ensure it can withstand the pressure without experiencing failure.
Example:
Let’s consider a scenario where a process engineer is tasked with designing a spherical pressure vessel for storing compressed air. The vessel is required to withstand an internal pressure of 500 psi () and is made of stainless steel, which has an allowable stress () of 20,000 psi. The design specifies a radius () of 36 inches.
Calculation:
Using the spherical shell formula, we can calculate the required wall thickness ():
![\[ t = \frac{{500 \text{ psi} \cdot 36 \text{ inches}}}{{2 \cdot 20,000 \text{ psi}}} \]](http://www.industrialseparation.com/wp-content/ql-cache/quicklatex.com-ff837f5aa0fd59e2a9824428e49f2fb9_l3.png "Rendered by QuickLaTeX.com")
![\[ t = \frac{{18,000 \text{ psi-inch}}}{{40,000}} \]](http://www.industrialseparation.com/wp-content/ql-cache/quicklatex.com-b84530c51c76c1b680b964a4ca9245a7_l3.png "Rendered by QuickLaTeX.com")
![\[ t = 0.45 \text{ inches} \]](http://www.industrialseparation.com/wp-content/ql-cache/quicklatex.com-607b1192dcf484842a9b64f7dd7235fb_l3.png "Rendered by QuickLaTeX.com")
Results:
Based on the calculation, the required wall thickness of the spherical pressure vessel is 0.45 inches. This thickness ensures that the vessel can safely contain the internal pressure of 500 psi without experiencing failure.
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