Is Diffusion the Answer? Understanding Gas Density Change in an Open Box

[DonnerJack]

Hi all,

I'm abit confused about diffusion - I can't seem to understand how to translate the question at hand to equation (from there it is only math...).

I have a box with a gas at concentration n0 in it, which is divided from the rest of the box where you have vacuum (for simplicity, the left half part of the box is filled with a classical gas, and the right half is vaccum).

Now, the barrier is taken out of the box - I need to solve the density of the gas (dependent on time).

The main problem here, is why is it diffusion? The prof. said it's a diffusion problem, but won't the mean-free-path be smaller than the side of the box? therefore it's a simple flow?

furthermore, when I tried to solve it I got to the point where I can't seem to write the boundary/starting conditions!

1. it's not and impulse problem
2. it's not a constant/infinite source of particles/heat.

How can I treat something like a H function? (because the particles in the beginning end with the barrier)

Any help will be appreciated (Mind you - I don't want the whole solution! I need help stating the boundary condition in mathematical form).

Thanks again.

 

[Tide]

As you have stated the problem it is not diffusion. Have you left out some details?

 

[DonnerJack]

Nope

Didn't left anything out.

That's how the Q was stated.

IF I consider the gas to be highly dilute - would it be diffusion, or should I consider the other limit?

 

[Tide]

Perhaps there is gas in both parts of the box but you have a second component of some other gas in one part?

 

[Bystander]

(snip)IF I consider the gas to be highly dilute - would it be diffusion, or should I consider the other limit?

If you pick a concentration that puts you into the molecular flow regime, yeah, you can sort of call it diffusion --- no intermolecular collisions, so you'll be looking at concentration as a function of time and location in the box after you open the gate.

 

[DonnerJack]

But...

1. Nope. only vacuum in the other part of the box.
2. OK. if I consider what you said (molecular flow) - how do I state the boundary conditions? I can't seem to understand how to make that step. after I have the conditions it's either I know by heart how to solve the diff. eq. or I would go and look in the books...

I can think of a Theta function ( due to concentration in one part of the box) but I can't really work with that.

any suggestions?

 

[Bystander]

What do you know about the system at t₀ and at t = infinity?

 

[DonnerJack]

I know that n(t=0)=Tetha(-x)*n0 (the half of the box is chosen to be x=0) and n(t=inf)=Const. in the whole box.

 

[Bystander]

"... left half part of the box is filled with a classical gas, and the right half is vaccum..."

Density is low enough (assumed molecular flow regime) that you're looking at free expansion of an ideal gas, tells you all about P, T.

 

[Astronuc]

Treating this as a diffusion problem, I believe one applies Fick's law at the boundary - The current J = -D\frac{d\,n}{d\,x}

The diffusion equation then can be written as \frac{\partial{n}}{\partial{t}} = \frac{\partial^2{n}}{\partial{x}^2}

Intially the current out of the gas. i.e. from gas to vacuum is some initial value, but the current from vacuum to gas is zero.

At a fixed boundary \frac{\partial{n}}{\partial{x}}=0, because locally the density does not change spatially, i.e. there is not diffusion across a fixed boundary.

This is similar to neutron diffusion.

For a reference, try - http://www.timedomaincvd.com/CVD_Fundamentals/xprt/intro_diffusion.html