Force on air column in a solar updraft tower

[Anoonumos]

Homework Statement

For a solar tower compare the total force on the air column in the tower
(i) by calculating the pressure at its bottom and top
(ii) from Archimedes’ Law.
Hint: there is a temperature
difference between top and bottom of the tower.

I have to estimate height, temperature difference and such.

The Attempt at a Solution

According to Archimedes' law the force is just
F = g * A * h * (ρ2 - ρ1)
with g = 9.81 N/kg, A the area of the tower, h the height of the tower and (ρ2 - ρ1) the difference in air density between top and bottom. (ρ2 - ρ1) can be calculated with the temperature difference.

I don't know how I should calculate the pressure directly. I can only think of using the ideal gas law, but I'm not sure how that would work in this situation.

Any suggestions? Thanks.

 

[mfb]

According to Archimedes' law the force is just
F = g * A * h * (ρ2 - ρ1)
with g = 9.81 N/kg, A the area of the tower, h the height of the tower and (ρ2 - ρ1) the difference in air density between top and bottom. (ρ2 - ρ1) can be calculated with the temperature difference.

This cannot work. It would give a force which increases quadratically for small tower heights (where the total mass and therefore the total force just increases linearly), and it diverges for extremely large tower heights (where you just keep adding empty space to the volume).

Archimedes' law would give you the net force (relative to air outside), if ρ2 and ρ1 would be the average density inside/outside.

I can only think of using the ideal gas law, but I'm not sure how that would work in this situation.

Assume some values for air pressure and temperature at the bottom (or at the top), calculate the other values with the ideal gas law and momentum conservation (the air inside does not accelerate and I think you should neglect wind/friction).