Pressure variation in a rotating tube

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SUMMARY

The discussion focuses on calculating the pressure distribution in a rotating vertical tube, specifically at a rotation speed of 3000 RPM, with an initial pressure of 1.5 bar at the axis (r=0) and a temperature of 293K. The pressure distribution is derived from balancing the centripetal force and the weight of the fluid element, leading to the conclusion that pressure is a function of both radial (r) and vertical (z) coordinates, expressed as P = P(r,z). The participant seeks assistance in adapting the principles used for horizontal tubes to this vertical configuration.

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  • Understanding of fluid dynamics principles, particularly pressure distribution in rotating systems.
  • Familiarity with centripetal force calculations and their application in fluid mechanics.
  • Knowledge of the relationship between pressure, density, and gravitational force in vertical columns of fluid.
  • Basic proficiency in calculus for deriving pressure as a function of multiple variables.
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  • Research the effects of centrifugal force on fluid behavior in vertical tubes.
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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
 
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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).
 

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