Small pipe break for an ideal gas

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Discussion Overview

The discussion revolves around a homework problem involving the flow rate of an ideal gas through a small break in a large tube, where the gas is at a higher pressure than the surrounding environment. The scope includes theoretical considerations of adiabatic and isentropic processes, as well as the application of relevant equations in fluid dynamics.

Discussion Character

  • Homework-related
  • Exploratory
  • Technical explanation

Main Points Raised

  • One participant states that the process is adiabatic and isentropic, leading to the equation Δh + Δc²/2 = 0, assuming c1 = 0 and h2 = 0 due to the lower external pressure.
  • Another participant questions the assumption that h2 can be taken as zero and prompts consideration of the compressible flow version of the Bernoulli equation.
  • There is a discussion about the need for the size of the break, with one participant noting that if the size is zero, the flow rate would also be zero.
  • One participant proposes assuming a small size A for the break and suggests that this would allow them to find the velocity and density using the total state of the system.
  • Another participant expresses uncertainty about how to approach the problem after considering the previous points raised.
  • A hint is provided regarding the relationship between changes in enthalpy and velocity as the gas approaches the exit hole.

Areas of Agreement / Disagreement

Participants express differing views on the assumptions made regarding the flow conditions and the treatment of h2. There is no consensus on the correct approach to solving the problem, and uncertainty remains regarding the assumptions about the nature of the gas expansion.

Contextual Notes

Participants have not resolved the assumptions regarding the conditions of the gas flow, particularly whether it is isothermal or adiabatic, and the implications of these conditions on the equations used.

eXorikos
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Homework Statement


A large tube filled with an ideal gas at pressure p1 and temperature T1 has a small break in it towards an envirronement at p2, with p1 much larger than p2. What is the flow rate through the hole to the outside of the tube.

Homework Equations


pv=rT
Δh+Δc2/2=δq-δl
h1 + c12 = h2 + c22/2

The Attempt at a Solution


The proces is adiabatic and isentropic so Δh+Δc2/2=0
Since it is a large tube it can be presumed that c1=0. Since p2 is much lower than p1 we can presume h2 = 0.

Am I on the right track here?
 
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Is this the exact wording of the problem statement?
 
Yes. Do you need size of the break?
 
eXorikos said:
Yes. Do you need size of the break?
Sure. If the size of the break is zero, then the flow rate is zero.
 
Since the whole exercise is symbolic, let's assume size A which is small. Than the equation I mentioned would give the velocity, we have A and I can find the density using the total state of the system.

Is this correct?
 
eXorikos said:
Since the whole exercise is symbolic, let's assume size A which is small. Than the equation I mentioned would give the velocity, we have A and I can find the density using the total state of the system.

Is this correct?
What makes you think that h2 can be taken as zero? Are you familiar with the compressible flow version of the Bernoulli equation? Do you think that the gas in the tank approaching the exit hole will be experiencing something close to (a) isothermal expansion or (b) adiabatic expansion? Do you think that the gas flow will be close to reversible expansion or no?
 
Good points. Than I have no idea on how to approach this problem.

Can you point me into a direction?
 
My leading questions were to get you pointed in the right direction. Here's another hint: for the flow approaching the exit hole in the tank,
$$dh=-vdv$$
 

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