Deriving Laplace's law for spherical pressure vessels

[gkiverm123]

I'm reading a biology paper which uses LaPlace's law in the analysis. Basically the tension within a spherical pressure vessel is half the product of the radius and pressure. I'm trying to understand how this equation is derived but don't have a strong background in fluids or physics so I'm not really understanding the general process. Could anyone give a more general overview description?

In my undergrad we learned about thin-walled pressure vessels, and how the tension within these is a function of radius, pressure, and thickness. How does this fit into LaPlace's law?

 

[jbriggs444]

I'm reading a biology paper which uses LaPlace's law in the analysis. Basically the tension within a spherical pressure vessel is half the product of the radius and pressure. I'm trying to understand how this equation is derived but don't have a strong background in fluids or physics so I'm not really understanding the general process. Could anyone give a more general overview description?

In my undergrad we learned about thin-walled pressure vessels, and how the tension within these is a function of radius, pressure, and thickness. How does this fit into LaPlace's law?

Consider a slice that divides the pressure vessel into two hemispheres. The force that holds the one hemisphere to the other hemisphere is tension across the slice.

The net force due to pressure on a hemisphere is the internal pressure times the cross-sectional area of the hemisphere in the plane of the cut. [This is a general result which holds for any shape, not just a hemisphere] $$F=P \pi r^2$$
The tension per unit length along the cut is the total force divided by the length of the cut.$$t=\frac{F}{2 \pi r}$$
Put the two equations together and you get $$t=\frac{Pr}{2}$$