Flow Question 1

[Mysterrose]

Ok, have a tank of Argon which I am using to posotively pressurize a small container. Positive pressure is only to keep outside air out of container ensuring my small container is pretty much all argon in it.

What I'm trying to figure out is how long my tank will last. I know I need to calculate the volumetric flow rate passing through my small container in a given amount of time, but all I know is the following

1) Have 20L tank of Argon
2) Small container is approx. 1 cubic ft.
3) Pressure regulator at argon tank will basically be set as low as possible just to keep posotive pressure going to my small container (assuming it's going to be somewhere in 1 to 5 psi depending on losses through the system which shouldn't be all that much)
4) Inlet pipe and outlet pipe to small container are 1/4". I did plumb in some valves if I want to restrict the flow at all.


What I'm stuck on is calculating flow as I only know the inlet pressure and inlet and outlet area.

Even if I measure the outlet pressure which would basically tell me my pressure drop through the system I still don't know my flow rate other than I can use the Bernoulli equation to calculate ratio of flow at inlet compared to flow at outlet.

Mass convservation of mass just told me velocity in equals velocity out since density is same and inlet and outlet oriface area are the same.

There's got to be some equation I am forgetting here or not applying right.

 

[TD2K]

The length of time that your tank of argon will last depends on the leakage rate from the small container. You won't be able to estimate it based on the system between the argon tank and the regulator because the regulator area is constantly changing to maintain the downstream pressure.

Can you pressure the small container and then measure how long it takes the pressure to drop? From the volume for gas in the small container and the change in pressure with time you can guesstimate the leakage rate of argon. The argon tank should have how much argon it holds, you may need to adjust that depending how low you can draw the pressure down in the tank.

 

[MintJulep]

Can't get there from here.

Need to know the size of the leak - which could then be converted to an equivalent orifice for use in a calculation.

But since the only way to know the size of the leak is to test, you might as well just test to find the flow rate instead.

So install a flow meter in a line and call it a day.

And by-the-way, density is not constant throughout the system you describe.

 

[rb1957]

leakage will be proportional to pressure.
you have a 20L reservior of gas, at some pressure.
you're filling a 1cf tank = 28L at something like 1.1 atm (16psi); so the reservior pressure should be quite a bit more than this, maybe 10 atm ?
so your regulator is taking your reservior pressure and limiting it to 1.1 atm and supplying this to the tank, and keeping up with some unknown leakage.

you can see that the regulator will keep the pressure in the tank at 1.1 atm until the pressure in the reservior falls to 1.1 atm.

if leakage is negligible, (maybe you have an outflow valve in the tank to allow the tank volume to be replaced with gas) then reservior pressure needs to be at least 1.1*(28+20)/20 = 2.64 atm to pressurise both volumes.

 

[zekeman]

"leakage will be proportional to pressure."

By what law?

 

[zdas04]

D'Arcy? It is a bit messy, but if the downstream pressure is constant (like a discharge to atmosphere) then changing upstream pressure in an incompressible fluid changes the flow rate in the same direction. I agree that you are not going to find a constant of proportionality for that relationship, but in non-tech speak when two numbers change in the same direction then they are often said to be "proportional". I don't know how language gets so lazy, but it does.

David Simpson, PE
MuleShoe Engineering

"Belief" is the acceptance of an hypotheses in the absence of data.
"Prejudice" is having an opinion not supported by the preponderance of the data.
"Knowledge" is only found through the accumulation and analysis of data.

 

[zekeman]

Mass flow leakage is proportional to the gauge pressure times the square root of the absolute pressure in the container.

So, if you do it experimentally, you pressurize the small container, to say 5psig, if possible and valve it off as IDF2 suggests; you get

1) -dM/dt=k*sqrt(P)*(P+Pa)
which is the fundamental mass loss equation

p= gauge pressure
Pa= atmospheric pressure

M= instantaneous mass in the small container= (P+Pa)/RT*volume

Substitute
dM/dt=dp/dt/RT*Volume
-dp/dt=k*sqrt(p)*(p+pa)*RT/volume
solve ( a closed form solution might be challenging) to get
p=p(k,t) and
Now use the end points
p0=p0(k,0)
pf=pf(k,tf)
to get k
Now go back to eq 1 to find loss rate=dM/dt where p is the controlled constant pressure in the container.
You now have the mass loss and since the initial mass in the tank is M1=(pi+Pa)/RT * volume tank
approximate time for depletion=pi/RT* vol tank/loss rate

 

[rb1957]

i didn't say "directly proportional" ... i meant the higher the pressure, the higher the leakage.

 

[MiketheEngineer]

Weigh an empty tank and a full one. Continue weighing while in use. Might give you an approx rate??

 

[btrueblood]

Well, if the pressure is low enough, and the leak path small enough, the flow will be laminar, and the mass flow directly proportional to pressure...

 

[dgallup]

I agree with btrueblood. Most small leaks behave more like laminar flow unless the materials are soft enough that the area of the leak can change with pressure, then all bets are off.

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The Help for this program was created in Windows Help format, which depends on a feature that isn't included in this version of Windows.

 

[rb1957]

as i understand the problem, you'll have to test the set-up, to determine how the tank leaks. i thought it was worth pointing out that your "small" container is bigger than your reservior, so obviously you've got high pressure Ar gas regulated for something higher than 1 atm. simply filling the container will cause a significant drop in reservior pressure.

 

[Mysterrose]

I thought about closing it off and then doing a leak down test to see how much time it took to drop X number of psi.

I like the assumption that if the leak is small enough you can approximate it as laminar flow. So think I'm gonna do the following:

1) Make sure to set the valve I plumbed in on the inlet pipe open more than the valve I plumbed in on the outlet pipe to ensure I can have a greater inflow than outflow when the regulator tries to maintain pressure inside the tank

2) Calculate flow through my pipe using Q = (Pi (R^4) (P - Po))/(8 N L). P inside tank is known, P at other end is atmosphere.


I realize that this doesn't take into account loses for the valve or the valve restricting flow, but I may actually just make inlet pipe larger than outlet pipe which would also work and just take valve out of equation.


Thanks guys, this will get me plenty close enough for this. I learned a lot, and also glad to confirm I'm not crazy in thinking my available data wouldn't get me there.

 

[Cockroach]

RB1957 is close, Zekeman. Actually average leak rate is constant and proportional to the amount present of that leaking medium. This is simply because the average rate of change in volume is constant. So you work out delta V over V equals a constant and solve it as a boundary value problem where V (t=0) = initial volume.

For example, the leak rate at high pressures and therefore high volumes is higher than what can be expected at lower pressures, hence lower volumes. I base this on experimental datum collected on a car tire which I submerged in a tub of water and pounded a nail through the wall. What I was after was the leak rates across Seats in Ball and Gate Valves. My observations found that the average rate of change in a leaking, gaseous fluid is constant. I counted bubbles emerging from the orifice and using high speed photography, photographed the bubble sphere to be measured in order to estimate change in volumes. Good times in the garage with a case of beer over a long weekend.

The reason why the equation falls apart for pressure is that density varies with pressure and volume, so you need to account for compressability of the working medium. I could never get it to work quite right, my suspicion is that humidity played a role with air quality.

Regards,
Cockroach

 

[zekeman]

"RB1957 is close, Zekeman. Actually average leak rate is constant and proportional to the amount present of that leaking medium. This is simply because the average rate of change in volume is constant. So you work out delta V over V equals a constant and solve it as a boundary value problem where V (t=0) = initial volume "

Cockroach,
You couldn't possibly mean this.
dV/dt/V= constant ??

I think something is missing .
The problem at hand has 2 volumes, the small container and the large holding tank, which are constant.