# Cases for reasoning about mass transfer

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?

Update:

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).

1. $$k_w\rightarrow 0$$: you have two zero-flux boundary condities, i.e. constant concentration profiles.
2. $$k_w\rightarrow \infty$$: your fixed-flux BC becomes a fixed-concentration BC at the interface which gives a linear profile in concentration.
• I agree about setting $k_w=0$, that gives two no-flux boundaries, and if the concentration is initially constant, it will remain constant forever. That's a good test. But if I set $k_w$ to some large number, then the concentration will go to zero everywhere, but exactly how fast depends on the diffusion (in this case, I will have $Bi \gg 1$). So I'm not sure how I can use that as a test, other than observing that the concentration eventually becomes zero.