Fluid mechanics - Additional liquid capacity due to compression

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SUMMARY

The discussion focuses on the additional liquid capacity in a cylindrical tube due to compression effects when pressurized by a pump. It highlights the application of Hooke's law (σ = ε.E) for the solid tube's deformation and the modulus of elasticity (k) for the liquid's compressibility. The analysis indicates that the hoop stress in the tube wall significantly influences the volume change, while the compression of the fluid is less impactful. Understanding these principles is essential for calculating the percentage increase in liquid capacity due to compressibility.

PREREQUISITES
  • Understanding of Hooke's law and its application in material deformation.
  • Knowledge of fluid mechanics, specifically the behavior of compressible fluids.
  • Familiarity with the concept of hoop stress in cylindrical pressure vessels.
  • Basic principles of elasticity and compressibility in materials.
NEXT STEPS
  • Study the derivation of the relationship between pressure and volume in compressible fluids.
  • Learn about the calculations involved in hoop stress for cylindrical structures.
  • Research the modulus of elasticity and its significance in material science.
  • Explore practical applications of pressure vessels in engineering contexts.
USEFUL FOR

Engineers, students in mechanical and civil engineering, and professionals involved in the design and analysis of pressure vessels and fluid systems will benefit from this discussion.

amora
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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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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 ?
 
Yes, it is
 
please help
 
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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