One year of high school physics and several attempts at Googling have failed me. Chief Delphi? I need your help.

There are two closed containers, each holding different amounts of the same gas at different pressures. The two systems are opened to each other. I am looking for an/the equation that models the rate of flow (change in mass/time) from one container to the other.

The solution I’m seeking is to calculate how long it will take for one container to reach a certain psi. After finding the rate of flow, it’s pretty simple to calculate the (change in pressure/time) which can then be integrated for change in pressure. Then, it’s just a matter of solving.

I assume that the closer the pressures, the slower the rate will be. Is there already an equation for (change in pressure/time)? Does the area of the opening matter? Am I wrong anywhere in the above post?

For starters, you should look up conductance equations and pumping speeds.

There’s a simple “flow control” device (an orifice) which will regulate gas flow rates based on pressure differential. Lenox Laser has a wiki providing gas flow control equations. With these equations, you can integrate flow equations (keeping in mind gas laws for pressure, temperature, and volume ) to solve your problem.

Each of these appear to be exactly what I’m looking for, but with different equations. Does the second provide less accurate estimates, or something like that? It looks like the primary difference is the first’s inclusion of molecular weight (they mention it’s to accommodate gasses other than air; not necessary in this case) and temperature (Is that a major factor? There may be variations of 5-10 Kelvin from room temperature, only about a 2-4% difference).

Pretty darn close is more than close enough (within 10% would be good). It certainly isn’t high school physics. Honestly, I could just experiment, but I felt learning the calculations would be significantly more beneficial.

Fluid dynamics is generally a 2nd or 3rd year ME course which many find difficult. The real difficulty in what you are asking is that the volumes, transitions, and connections are extremely important to get an accurate answer. While your desire for 10% accuracy may seem relatively low, many fluid dynamics problems using values from test data will only claim to get you within 30% or so.

Check out introduction to fluid dynamics by Fox and MacDonald. They have a ton of great examples in that book and you may find one that will work well for you. If the pressure differential is relatively low, you could probably use something like bernoulli equations of “incompressible” fluid theory (even though it is clearly compressed). If it is relatively high pressure differential, then you would need to use a completely different set of equation.

This depends on the runner length (connection between volumes) and the velocities involved and the precision you are looking for with regards to timing. With air, a similar effect to “water hammering” can occur with relatively fast flows. That is how tuned intake systems for cars can actual increase torque at certain rpms by adjusting runner lengths (extra air can be pumped in due to the pulsing nature).

Basel,
Eric gave you a really good link for constant pressure systems. You could either try to work out the math for the changing pressures, or you could use excel (or some other program) to do a time based integration technique.
Time based is similar to the “Numerical Integration](Numerical integration - Wikipedia)” technique you likely lerned in calculus.

Figure out the initial flow rate. Assume it to be nearly constant over a given small time-frame. Assuming ideal gas PV=NRT, you could then get a change in pressure of both containers after that discreet amount of flow and thus have two new pressures. Put this into an expanding table until the pressures eventually equalize…
Or,
You can make a table that has various balanced transition states. for example assuming equal volume containers, and one is at 100 psi, and the other is at 0, they should balance at 50 psi. Then take various static slices between those balance points. 100:0, 90:10, 80:20…60:40, 55:45 (I think it would be linear, but please check using the ideal gas law). By calculating flow at those various slices, you can use the average of the two ends to figure out an approximate time it would take to get between the states. While not perfect, this will give you a pretty good approximation, especially with more and more slices. A neat way of looking at this would be to compare the “answer” with various resolutions. The coarsest would be using just the two end conditions 100:0 and 50:50. Then use 3 states (2 regions), then 5 states (4 regions)…
Essentially this is the foundation of many Finite Element Analysis type tools.

I use this technique a lot for buidling simple models for dynamic phsyics problems. With a little bit practice, you can build eerily accurrate tools. Also, this will give you some insight into the “dangers” of “bad FEA”. For a problem like this, you should see that your initial answer of the 2 state 1 region differs drastically from your “final” which will likely be on the order of 20+ regions. Resolution in critical errors often leads to poor decisions from bad FEA.

Also play around with using different discharge coefficients. As you can see in the table, they have a huge effect on the flow velocities, and thus the “answer”. This brings up the other important thing about FEA which is boundary constraints and assumptions.