Tube fracture - volume of gas lost

[stuckoutthere]

Good Afternoon,

I am looking to work out the volume of gas lost in a fractured tube line.

If I have a tube that measures 12mm outer diameter with a 1.5mm wall thickness giving a 9mm internal diameter that has a gas going through it at 150psi which develops a 1mm hole how would I calculate the volume of gas lost in 1 hour?

Any help will be gratefully received.

 

[zdas04]

Take a look at faq378-1864 for the calculations.

The first thing to do is assume that the 9 mm tube is big enough to allow pressure at the hole to stay above the minimum for critical flow. This is a three step process. First calculate the flow rate out the hole using the critical flow equation in the FAQ. Then calculate the velocity of that flow rate down the tube (if it is less than 0.6 M then you can assume a constant mass flow rate out the hole, if it is more than 0.6 M then there is no good way to solve this problem since you can't know the actual pressure at the hole, if it is less than 0.6 M go to step 3). Finally, use the Fully Turbulent Gas Flow equation to calculate the dP from the header to the hole (using the flow rate through the hole at choked flow) and assume that the header pressure will not drop. Once you have that dP, you can re-calculate the flow out the hole (lower pressure at the hole results in a lower mass flow rate at the same sonic velocity). Recalculate the dP and then recalculate the flow out the hole. I find that within 3 iterations I get an adequate match between assumed pressure and calculated pressure at the hole.

David Simpson, PE
MuleShoe Engineering

In questions of science, the authority of a thousand is not worth the humble reasoning of a single individual. Galileo Galilei, Italian Physicist

 

[racookpe1978]

1. You are assuming a 0.6 M limit as 0.6 Mach for the pressure, temperature and flow at the inlet of the tube, middle (approximately) of the 9 mm ID tube, or the location right at the little hole in the tube wall?

(Not 0.6 Mach for the gas after it leaves the tube wall hole, right? (Cooling only happens in the expansion region outside of the tube, as i understand the OP's problem and assumptions.)

2. The OP has two contradictions: If the gas is flowing through such a little tube at a (assumed ??) gas pressure of 150 psi (assumed GAGE pressure - and why is he switching units ???) , then it MUST be constantly resupplied at the tube inlet from a gas main or large chamber. If so, then the flow will rapidly stabilize so the inlet pressure remains 150 psig, and the outlet pressure after an UNKNOWN length of tube = 150 psig - flow losses through tube to the halfway (???) point at 100% flow - PVnRT mass losses through the side diversion of the 1 mm hole - flow losses of the remaining gasses through the remainder of the tube - exit losses when the remaining gasses leave the 9mm ID tube at whatever condition they originally were facing.

It appears you're using a simple variety of this constant flow assumption: The flow is from a near-constant 150 psig (at centerline of the tube at radius 0.0) through the inlet flow losses at the wall at 4.5 mm radius, through the 1.5 mm thick wall hole 1.0 mm dia, through the exit losses at the outside tube wall, then against atmospheric pressure.

None of which are stated in this apparent homework problem. 8<(

3. Now, IF BOTH end of the tube were isolated immediately when the leak occurred, then the end psig = 0.0 because the 9 mm ID tube will be "empty" of gas after 1 hour. The volume of gas lost = initial volume of the tube (@ PVnRT mass conditions at 164.7 psia) minus final volume of the gas (@ PVnRT conditions at 14.7 psia)

4. Assume the tube is isolated at the inlet. Then the final pressure will approximate the outlet pressure of the reciever or reservoir or tank if it is above atmospheric pressure, again minus flow losses from the final pressure in the unknown length of tubing from the end to the leak point.

 

[stuckoutthere]

Thanks for the above replies.

I will try and re-word part of the question.

If I had the above tube with the same defect being constantly fed (failure has gone unnoticed,compressor keeps running) with the normal gauge pressure being 150psi how long would it take for it to be classified as a major release. A major release being defined as below

'major if greater than 300kg of gas or two phase fluid and 9000kg of liquid is released'

 

[zdas04]

stuckoutthere,
Your restatement is the same as your original question. The answer to both is in my post. If you are just trying to get a time-to-major-release you can just assume that there is no pressure drop from the header to the hole (your answer will be high, but probably not horribly so) and use the choked flow equation in the FAQ to get the flow rate under standard conditions. Convert that volume flow rate to a mass flow rate (by multiplying standard volume flow rate times density at STP) and then divide the answer into your limits.

racookpe1978,
1. Yes, I was taking the 0.6 Mach as the transition from being able to ignore dynamic pressure (i.e., the incompressible flow assumption) and not being able to ignore it. If I can't assume incompressible flow then I can't calculate friction drop down the tube to calculate the mass flow rate out the hole.
2. These really aren't contridictions. This is a common calculation in real systems. Shell and tube heat exchangers have a multitude of small tubes between two large headers. A leak in one tube may or may not cause the header pressure to change. That is why I said you have to do the evaluation in steps and then iterate the steps.
3. Not sure what you mean here, in this kind of situation there is never an isolation valve for the tubes.
4. see 3.

David Simpson, PE
MuleShoe Engineering

In questions of science, the authority of a thousand is not worth the humble reasoning of a single individual. Galileo Galilei, Italian Physicist

 

[racookpe1978]

OK, I see what you mean.

I was thinking of a single long tube (more like am isolable instrument tube or flex hose) that leaks somewhere between the inlet at Point A to the outlet at point B, and the leak occurring somewhere in the middle. Definitely NOT a single tube leaking in the middle of a tube bundle.