Oxygen Flow Through a Concentration Gradient

TheChannigan

Hi guys

First things first, I'll lay out the problem. I have a box of volume V containing a constant sink of oxygen (e.g. a candle or an animal); this box is sealed except for a smallish aperture of area, A and depth, L (the L meaning the walls of the box have finite thickness).

After a significant time has passed from the introduction of the oxygen sink I would expect a dynamic equilibrium to have formed giving a constant, but lower, concentration of oxygen inside the box with oxygen 'flowing' through the hole to sustain this equilibrium - with the outside atmosphere being equivalent to a well mixed infinite reservoir of oxygen at a constant concentration.

What I am looking for is to be able to find the rate of flow of oxygen through the hole if all of the necessary parameters are known. I have essentially come up with a debauched version of Fick's first law of diffusion to fulfil this and wanted opinions on whether I'm barking up the right tree or if there are any better methods.

I started with Fick's law, which is: [itex]J=-D\frac{∂\phi}{∂x}[/itex], where J is diffusive flux, D is the diffusion constant, [itex]\phi[/itex] is the concentration and x is position.

I then reasoned that for small values of x and small differences in concentration that:
[itex]\frac{∂\phi}{∂x}=\frac{1000(C_{outside}-C_{inside})}{24.5L}[/itex]
Where C-outside and C-inside are the fractional components of Oxygen in the air outside and inside, L is the length of the hole and 1000/24.5 is the approximate number of moles/m³ of gas at room temp and pressure, thus giving a gradient in moles/m⁴, which are odd units that cancel down to being just moles/s when multiplied by the diffusion constant (m²/s) and the area of the hole.

I'm trying to get a practical estimation of what this flow will be, so any help, advice or corrections would be appreciated immensely.

Thanks,
Chan

mfb

The approach looks fine, so you get numbers if you add the diffusion constant and some arbitrary concentration difference.
Diffusion is very slow, however - for a realistic setup, I would expect larger contributions from turbulence and other air movements.

If L is not much larger than the diameter (or other length scale) of the hole, concentration differences in the box / outside close to the hole could be relevant, too.

Velikovsky

Are the products of combustion expected to vent through the same hole that is feeding the system oxygen?