I am working on a mass diffusion problem within a porous medium with time- and space-varying porosity (epsilon) and apparent diffusion coefficient (D*).
That is, epsilon and D* are both fn(x, y, z, t)
For the general three-dimensional case, my best guess for the accurate diffusion equation is:
d (epsilon *C)/d t = div [ D* [ div (epsilon C)],
where the concentration C is defined in the porous space only.
But I also saw from some literature that wrote the mass diffusion equation as
d (epsilon *C)/d t = div [ epsilon* x D* (div C)].
I am pretty sure that porosity (epsilon) should be included within the time derivative at the left hand side. But I am not sure about the location of epsilon at the right hand side of the equation. It all depends on how you define the diffusion flux J:
J = epsilon x D* x div C
or
J = D* x div (epsilon C),
you might get different answers. This is important because different mass governing equation may lead to different simulation results, since porosity (epsilon) now is a function of space. I wonder if someone can help me and share some thoughts with me on this issue.
If nobody knows which flux is best physics a way to decide is to compare both theories with experiment. Solving such diffusion equations is considered in W.E.Schiesser, The numerical method of lines (Academic 1991). There are commercial program packets for the task.
A very useful book fro such problems is the 'Mathematics of Diffusion' by Crank.
http://www-eng.lbl.gov/~shuman/NEXT/MATERIALS&COMPONENTS/Xe_damage/Crank-The-Mathematics-of-Diffusion.pdf
The correct equation is your second guess:
d (epsilon *C)/d t = div [ epsilon* x D* grad (C)]. where epsilon* x D* = Effective diffusion coefficient (Deff)
This equation can be obtained using the homogenization technique, averaging the equation over a representative volume element. The mass conservation equation is
d (epsilon *C)/d t = div [ epsilon* x J], where J = D* x grad(C)
This has been done in e.g. Samson et al., 1999, Cement and Concrete Research, 29, 1341-1345. Check that paper for the details of the averaging
The answer largely depends on the problem formulation: is C the concentration of a component in a fully saturated by a liquid porous network or something else?
- Badges
- Science topic
- Similar topics
- Agricultural Science
- Agronomy