Let us suppose that we want to compute the secondary current distribution of an electrolytic cell composed by anode, electrolyte and cathode. The governing equation in each subdomain isgiven by Laplace equation on the potential, coming from conservation of current and Ohm's law, and supposing that my experiment is potentiostatic, I impose potentials on the some part of the boundary of the electrodes for boundary conditions. Any other surface is assumed isolated (homogenous Neumann BC)
Now, on the electrode/electrolyte interfaces I would need to impose some kinetic relation between the current crossing the interface and the overpotential, this would imply that I will have some discontinuities for the potential along these interfaces. I was wondering how to solve such a problem using a FEM approach. Solving each subdomain at the time with some initial guesses of the overpotentials until convergence? or maybe some discontinous Galerkin approach? Any suggestion?
Dear Alvaro,
For the boundary condition, it is common to assume some form of kinetic relation, and for this case, for example a Tafel relation is suitable (E = A ln(j/j0)), or the full Butler-Volmer equation. The easiest way to calculate this is honestly to buy commercial software, for example COMSOL multiphysics has all this built-in in their Electrochemistry package(s). It is also often easier to approach the problem first by solving a constant current boundary condition (at the electrodes) because it is easier to know realistic values, and you can avoid some discontinuities for example due to the electrode edge (electrode BC -> isolated wall BC).
Hope this helps!
Thank you Thomas Holm. I'm attempting to write my own code, since I would like to couple the electrochemical problem with other effects (fluid dynamics, thermal effects, etc.) Do you know a good reference I could check on the numerical treatment of these type of problems. I tried the COMSOL reference manual and the Electrochemical Interface manual, but it's mostly oriented to COMSOL user interface guidance. Appreciate any help.
Exactly the coupling of various physics is perhaps Comsol's and similar software's biggest strengths. I would suggest starting with the simplest problem, 1D geometry and a single set of equations (only electrochemistry). For reference and especially regarding the nature of the boundary conditions, introductory books on computational fluid dynamics are often in-depth on the topic. The boundary conditions you will apply for all other physics (than fluid dynamics) will likely be of similar nature, for example, in electrochemistry, constant current is a Neumann boundary condition (BC), and constant potential condition is either a Dirichlet BC (during reversible reaction or mass transport control), a Neumann BC (because the surface current is dependent on the potential in a kinetic relation, for example the Butler-Volmer equation or the Tafel equation) or a combination of these (concentration dependent Butler-Volmer equation as an example). The challenge in numerical modeling is as a rule to pose a well-posed problem ( https://en.wikipedia.org/wiki/Well-posed_problem), which is a problem of more fundamental nature and which compounds as you add physics to your system.
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