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.

 (1) (2)

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:

 (3)

So the pressure differential is then:

 (4)

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

 (5)

So we can insert this into equation (4):

 (6)

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

(7)

Solving the Differential Equation

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

 (8)

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

 (9)

Applying the limits:

 (10) (11)

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

 (12)

Bringing Po to the right gand side we get:

(13)
with
and

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