Derivation of "heat" equation w/ diffusion and convection

[Haydo]

Homework Statement

The goal is to derive the heat equation with convection,
u_t=α²u_xx-vu_x
but for the case where u(x,t) instead models concentration changes by diffusion and convection. The idea is to use conservation of mass to do this.

Homework Equations

We are given:
Change of mass inside [x,x+Δx] = Change due to diffusion + Change due to material being carried across boundary

The Attempt at a Solution

I can solve this problem for the case where we are using the actual heat equation, as it becomes a flux problem with Fourier's Law and some calc tricks, but I can't figure out how to set up this problem for material flow.

I think that u(x,t) should be in units of mass/vol of some sort, making u_t have units of mass/(vol*time), and for the heat equation, LHS = d/dt(∫cρAu(s,t)ds), so I'd expect it to look something like that. Perhaps without the thermal capacity constant c in the equation. For the RHS, I'm pretty lost, as I can't use Fourier's law for a concentration problem, or at least I don't think I can.

 

[Orodruin]

Change of mass inside [x,x+Δx] = Change due to diffusion + Change due to material being carried across boundary

Diffusion is also material (of what you are computing the concentration for) being carried across the boundary, with the net transport going in the direction of lower density. This is described by Fick's law, which is nothing else than the material equivalent of Fourier's law. Add a convection current and the mathematics are equivalent to the heat transfer situation.

Edit: Typo removed.

 

[Brian T]

I would set up an equation relating the change in concentration over time to the flux. After, relate the flux to the gradient (in one dimension just w.r.t x).

 

[Haydo]

Thanks guys! Fick's law was what I was looking for. Couldn't figure out what it was called. After that, the derivation is essentially that same as for heat.