Flow rate and pressure drops

[Josh2008]

How do you calculate the time for a container (fixed volume) filled with air, to drop from 100psi to atmospheric 14.7psi? Say the container had a ball valve that was closed and then opened once it reached 100psi.

I assume that the larger the container, the longer it would take.

Case 1: 1000 gal container
Case 2: 5000 gal container

I understand that it has everything to do with flow rate, I just don't know how to get it. It must be a differential equation, because once the pressure lowers, so does the flow rate.

Josh

 

[mjpetrag]

I think you are going to have to look into your college gas dynamics book and use the tables in the back regarding pressure ratios to find your velocity, then mass flow. I never liked using differential equations for this type of problem. The best way I think would be to set up an excel spreadsheet with some of the following parameters

P2, P2/P1, Mach Number, Velocity, mass flow, time (probably want to use 0.1 second intervals), mass loss (found by multiplying flow rate by your time column and subtracting from initial air mass in tank [initial mass = PV/RT]). P2 would change at every 0.1 second interval. You will have to calculate the new pressure in the tank using the new air mass at each interval and solving P = mRT/V. Just keep dragging and dropping the the parameters in Excel until your flow rate is zero and your pressure inside the tank is atmospheric.

One thing I see happening is choked flow occurring for a while depending on the throat area of the valve. You will have a constant flow rate out of the valve until the pressure inside the tank is small enough for flow to become subsonic.

If you graph the flow rate over time, I think you will see a flat line for a while and then a negative parabolic drop hitting an inflection point and leveling out.

-Mike

 

[zdas04]

I've done this a bunch of times and there are two distinct regions of flow. Until pressure reaches the critical pressure (depends on the gas, but it is somewhere around 10-15 psig), the velocity of the flow is sonic. Mass flow rate changes continously, but velocity is constant. In the spreadsheet that mjpetrag is talking about, I calculate the mass that left during 1 second and recalculate pressure and mass flow rate for the next second.

You can get very close to the actual time required to get to critical this way (I once calculated it to within 10 seconds out of 90 minutes on a big pipeline blowdown). Then it gets REALLY hard. Determining non-choked flow rate depends on using an empirical equation, most are not very precise at the limits of their applicability. In the blowdown referenced above I said it would take 87.5 minutes to get to non-critical flow and 35 minutes to blow down to zero psig, I nailed the first part, but the second part actually took just over 3 hours. I've done this sort of thing a few dozen times (it is useful for scheduling the start of the blowdown so that work can begin at daylight) and never been very close on the last section.


David Simpson, PE
MuleShoe Engineering

http://www.muleshoe-eng.com

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[mjpetrag]

right...my mistake the flow rate will still change at sonic velocity with varying upstream pressure.

-Mike