The passing of gas...

from one place to another, through a variable opening…
Inspired by this thread, which I thought deserved more attention than it got, I have two tangentially related questions about the movement of gases from one place to another.

Question the first: I have a sealed chamber of arbitrarily large size, divided down the middle by a wall with a shuttered opening in it. On one side of the wall is a mixture of two gases, A and B, at one atmosphere of pressure, such that the partial pressure of A is 0.5 atm and B is 0.5 atm. On the other side of the wall is a mix of gases B and C at the same total pressure, such that the partial pressure of B is 0.25 atm and C is 0.75 atm.

Now if I open the shutter, the gases will diffuse/effuse through the opening until they reach an equilibrium. I’d like to know how to figure out what the net movement of gas B is across the opening at the moment that the shutter is opened. I imagine that I would have to know something about the molecular weights of the gases involved, the total temperature, the size of the opening, and maybe some other things I haven’t thought about.

Second topic: while thinking about this, I started wondering how air moves through a doorway if the door is only partially open. Imagine a square door in the middle of a wall, such that there is wall space on all four sides of the door. As the door is opened, there is initially a tiny crack along three sides. As you swing the door open, it passes 90 degrees to the wall and eventually lies flat along the wall again, 180 degrees from its initial position. If there is a pressure difference across the doorway, air will move through the doorway at a rate that depends on the ‘effective cross-sectional area’ of the opening around the door.

So here are the questions - what sort of shape does the flow vs. angle-of-door graph make? Are there linear regions, or inflection points, or asymptotic curves, or what? Once the door passes 90 degrees, does it affect airflow any more? Is this something that can be easily described mathematically, or do you just have to build a wind tunnel and test things in the real world (especially if the shape of the door changes)?

In answer to the first part of your question, you’d have to take into account the diffusivities of the gasses and the geometry of the opening and the spaces around it in either chamber. For some simple shapes, like a rectangular box divided by a thin partition with its opening centered, you might find an analytic solution, but if I had to solve this particular problem I’d use finite element software to solve it.

Note, it will take forever for the gasses to mix completely. There will be various wavelengths of concentration difference implicit in the problem, and the wavelengths will each die down in a strictly exponential decay, with the time constant being proportional to the square of the wavelength and to the diffusivity of the gasses. Eventually whatever difference remains will be dominated by the longest wavelength in the problem.

I appreciate your response, Napier. What do you mean by a wavelength of concentration difference? Is that a quantum effect?

If I wanted to make it easier to calculate the initial flux of gas B across the opening, would it simplify things to say that the chambers on either side were of infinite size, the partition was of zero thickness, and the hole was circular?

Going back to the post that inspired my question, is it possible to predict the size of a circular hole that allows x moles of gas to diffuse across it per minute given a particular set of steady-state concentrations on either side, or is it something that must be modeled by trial and error?