Deriving Equations for the Atmospheric Pressure Profile in a rotating Cylinder

Deriving the Differential Equation

We regard a sector of a cylinder of height b and radius R with an angle θ. We can derive the differential equation for the change in pressure dP in a small volume of the section dV at a distance r from the cylinder center when we change the distance to r + dr.


The increase in force and pressure in the direction of r is caused by the mass of the air in the volume element dV due to the centrifugal acceleration aC:


So the pressure differential is then:


The density ρ(r) can be expressed from pressure P(r) using the ideal gas law:


So we can insert this into equation (4):


Now bringing the pressure term to the left hand side gives the follong DE:


Solving the Differential Equation

We have to integrate both sides: 1/P dP = ln(P):


We assume a constant temperature T and rotation rate ω. Solving the integrals yields:


Applying the limits:


To get rid of the logarithms we raise both sides to the power of e:


Bringing Po to the right gand side we get:

More Page Infos / Sitemap
Created Wednesday, November 9, 2022
Scroll to Top of Page
Changed Wednesday, November 9, 2022