Small pipe break for an ideal gas

[eXorikos]

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?

 

[Chestermiller]

Is this the exact wording of the problem statement?

 

[eXorikos]

Yes. Do you need size of the break?

 

[Chestermiller]

Yes. Do you need size of the break?

Sure. If the size of the break is zero, then the flow rate is zero.

 

[eXorikos]

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?

 

[Chestermiller]

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?

 

[eXorikos]

Good points. Than I have no idea on how to approach this problem.

Can you point me into a direction?

 

[Chestermiller]

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,
$$dh=-vdv$$