Fluid mechanics - Additional liquid capacity due to compression

In summary: The wall must be thick enough to resist the hoop stress. Now use Hooke's law to calculate the deformation at the cylinder wall. The deformation will be small, so use a small angle approximation. The change in volume is related to the change in diameter. Use the percentage change.In summary, a cylindrical tube filled with a liquid and pressurized with a pump will experience changes in its dimensions due to hoop stress and compression of the fluid. The most significant factor contributing to the change in volume is the hoop stress, followed by the increased length of the cylinder. To calculate the percentage change in volume, one must consider the wall thickness and use Hooke's law to approximate the deformation at the cylinder wall.
  • #1
amora
3
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A cylindrical tube (diameter = D, width = L) is completely filled with a liquid (density = ρ). A pump pressurizes the system with a pressure P. Consequently, 1) the solid tube is compressed and deformed according to Hooke's law (σ = ε.E), and 2) the liquid is compressed and deformed, following the modulus of elasticity (k).
The question is: what additional percentage of liquid will the tube accommodate as a result of the compressibility effect?

Tips:
Start by:
m = ρV
dm = ρdV + Vdρ
dm/m = ρdV/m + Vdρ/m
Apply Hooke's law to the first term and the modulus of compressibility to the second term.

Mentor note: Moved to homework forum from technical forum so no template.
 
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  • #2
Welcome to PF.

The cylinder is a pressure vessel. The pressure inside the cylinder places the wall material in tension, to a greater degree than the internal pressure thins the wall. The hoop stress in the wall causes the most significant change, increasing the volume by the greatest amount. The next significant factor will be the increased length of the cylinder. Compression of the fluid will probably be least important.

Is this a homework assignment ?
 
  • #3
Yes, it is
 
  • #4
please help
 
  • #5
We do not do your work for you.
You must read about the subject and ask questions when you get stuck.

The cylinder needs a wall thickness.
Now you need to understand hoop stress.

Assume the ends of the cylinder are flat and do not bend.
The ends are pushed apart by the internal pressure, which stretches the cylinder.
 

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