Pressure variation in a rotating tube

[boileroo]

Homework Statement

An enclosed vertical tube rotates about its vertical axis at w=3000rpm. At the axis, r=0, P=1.5bar and T=293K.

What is the pressure distribution as a function of r? And hence calculate the pressure at r=2m.

The Attempt at a Solution

I have seen this type of questions before with a horizontal tube, where the pressure difference across an element is balanced out by the centripetal force (ie [P-(P+dP)]A = rho*A*w²*r dr).

I would appreciate some help adapting this to work for the tube aligned vertically as stated above. Obviously the mass of the element must also be taken into account, but I am struggling to figure out how! Thank you

 

[hikaru1221]

Consider a volume dV at position $$(r, \phi, z)$$. Then:
_ The pressure on the 2 vertical sides<---> Centripetal force (that is, considering P(r,z) and P(r+dr,z))
_ The pressure on the 2 horizontal sides <---> Weight (that is, considering P(r,z) and P(r,z+dz))
In general, pressure is a function of not only r but also z: P = P(r,z).