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
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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.
 

1. How does compression affect the additional liquid capacity in a fluid system?

Compression can increase the additional liquid capacity in a fluid system by decreasing the volume of the container and forcing the liquid to occupy a smaller space. This allows for more liquid to be added to the system without overflowing.

2. What is the formula for calculating the additional liquid capacity due to compression?

The formula for calculating the additional liquid capacity due to compression is V = P x ΔV, where V is the additional liquid capacity, P is the pressure applied to the fluid, and ΔV is the change in volume of the container.

3. Can the additional liquid capacity due to compression be negative?

Yes, the additional liquid capacity due to compression can be negative if the pressure applied to the fluid causes the container to expand, resulting in a decrease in the overall volume of the system.

4. How does the compressibility of the liquid affect the additional liquid capacity in a fluid system?

The compressibility of the liquid plays a significant role in the additional liquid capacity in a fluid system. Liquids with higher compressibility will experience a larger change in volume when subjected to pressure, resulting in a greater increase in additional liquid capacity.

5. What are some real-world applications of understanding the additional liquid capacity due to compression in fluid mechanics?

Understanding the additional liquid capacity due to compression is crucial in various industries, such as oil and gas, chemical processing, and hydraulic systems. It is also essential in designing and maintaining pipelines, storage tanks, and other fluid systems to prevent overfilling and potential hazards.

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