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: $J=-D\frac{∂\phi}{∂x}$, where J is diffusive flux, D is the diffusion constant, $\phi$ is the concentration and x is position.

I then reasoned that for small values of x and small differences in concentration that:
$\frac{∂\phi}{∂x}=\frac{1000(C_{outside}-C_{inside})}{24.5L}$
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?