Modeling contamination diffusion in a draining container

Question

I asked this on the Physics site, but it fits much better here, and I now can ask a more specific question.

For the scope of the problem, I have a 250-mL bottle filled with pure water and a sampling line at the bottom slowly draining the bottle to analyze, say 2.5 mL/min. The bottle is on the order of 10 cm high.

Since the bottle is open, a tiny amount of analyte (trace metal) is constantly entering the bottle from the top, at a certain mass flux. I observe that the concentration of this analyte increases over time as expected, but I'm interested in modeling the dynamics, perhaps for different bottle sizes.

First I want to model this with the 1D convection-diffusion equation, but I don't know how to properly apply boundary conditions. If the equation is,

$$\frac{\partial C}{\partial t} = D\frac{\partial^2 C}{\partial x^2} - v\frac{\partial C}{\partial x}$$

then I choose $v$ as the velocity that the liquid is draining (with the numbers I gave, $0.1$ cm/min).

For the boundary conditions, I'm unsure (for now I'm ignoring that the top face is moving downwards as the bottle drains). I think the top face should be a Robin boundary; since it has a constant flux, I could do

$$vC - D\frac{\partial C}{\partial x} = \phi, \quad x = 0$$

For the bottom face, I'm wondering if the boundary should simply be

$$\frac{\partial C}{\partial x} = 0, \quad x = L$$

since nothing can diffuse out of the bottom of the bottle, but I allow whatever concentration flows out of the system through the sampling tube to pass by. Is this correct?

Edit: I found a coordinate transform to include the effect of the level dropping, essentially mapping $x\in[0,\delta(t)]\to y=x/\delta ,y\in [0,1]$ (now x,y = 0 is the bottom face and y=1 is the top). I adapted the procedure in this paper After this transform, the advection term becomes something like $v(1−y)$; because the top boundary is moving due to the coordinate transform, it effectively sees no velocity in the volume there. Now what should the top boundary condition be?

Answer 1

If you are negleting the fact the the level of the flow gets lower as the bottle drains, then you are implicitly assuming that new water comes into the domain at speed $v$. Since the analyte inflows at a prescribed rate, too, I would use the condition $vC=\phi$ at the top of the bottle, that you can further split into $v=0.1\mathrm{cm/min}$ and $C=\phi/v$ if you need. If you include the diffusive flux of concentration in the top boundary condition you will potentially allow the analyte to diffuse outwards, depending on the concentration gradinent, which does not make much sense considering your physical system.

The bottom condition you wrote is ok because it allows the analyte to exit the domain only because of the flow of water. Consider that with 1D advection you do not need a boundary condition when the prescribed flow velocity blows towards the outside of the domain, so you are ok setting only a homogeneous Neumann condition on the concentration.