The Nernst-Planck Equations
The general mass balance for the diluted species in an electrolyte is described by the following equations for each species i:
where Ni is the total flux of species i (SI unit: mol/(m2·s)). The flux in an electrolyte is described by the Nernst-Planck equations and accounts for the flux of charged solute species (ions) by diffusion, migration, and convection; these are respectively the first, second, and third term on right side in the equation below.
where
ci represents the concentration of the ion i (SI unit: mol/m3),
zi its valence,
Di the diffusion coefficient (SI unit: m2/s),
um,i its mobility (SI unit: s·mol/kg),
F denotes the Faraday constant (SI unit: C/mol),
u is, the velocity vector (SI unit: m/s), and
Ji denotes the molar flux relative to the convective transport.
(3-2)
The net current density can be described using the sum of all species fluxes:
where il denotes the current density vector (SI unit: A/m2) in the electrolyte.