I'm doing some simulations of a 1D system with diffusion. One boundary has a no-flux boundary condition, while the other boundary has a prescribed-flux boundary condition with a specified mass transfer coefficient. I'm writing some tests to make sure my simulation code works as expected, and I have a number of natural test cases for the diffusion part, but I can't think of any good ones for the mass transfer part.
My thinking so far is that the mass transfer coefficient, $k_w$, has units of speed, so dividing the width of the system, $L$, by the mass transfer coefficient ought to give me a time scale that I can use for something. If the mass-transfer Biot number ($Bi = k_w L / D$, where $D$ is the diffusivity) is much smaller than one, then the diffusion is fast enough to keep the concentration approximately constant within my domain, so in that case, I would expect the remaining concentration to decay exponentially with the time scale given by $\tau = L/k_w$. Does that make sense?
Are there any other simple cases where I can know the results from simple, analytical considerations, that I can use to test my simulation code?
Just thought I could add some plots from my simulations to help explain the case I'm describing. This first plot shows the situation when $Bi \ll 1$ (or actually 0.14 in this case), in which case concentration remains almost constant inside the domain, and the exponential decay I describe works well.
In the next plot, I show the same results, but this time the diffusion is slower, and the mass transfer is higher, such that $Bi \gg 1$ (or actually 14 in this case). Here, we see the concentration near the surface is depleted, and the decay is not well described by the same exponential decay (as expected).