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
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ci represents the concentration of the ion i (SI unit: mol/m3),
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zi its valence,
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Di the diffusion coefficient (SI unit: m2/s),
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um,i its mobility (SI unit: s·mol/kg),
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F denotes the Faraday constant (SI unit: C/mol),
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 ϕl the electrolyte potential,
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u is, the velocity vector (SI unit: m/s), and
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Ji denotes the molar flux relative to the convective transport.
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.
The Nernst-Planck Equations can be written in nonconservative or conservative form. The nonconservative formulations are applicable to systems that contain incompressible electrolytes. For these electrolytes, the following continuity equation is satisfied:
The velocity vector in the convective term can therefore be moved out of the divergence operator which gives the nonconservative form of the Nernst-Planck equations:
On the other hand, for compressible electrolytes, the convective term cannot be simplified and the use of the following conservative form of the Nernst-Planck equations is applicable:
For mass transport in porous systems, the accumulation term is deviating between the conservative or nonconservative formulations. The electrolyte volume fraction (porosity), εi, is only differentiated in the conservative form giving the following formulation:
While it is not for the nonconservative case:
Considering a system with no producing or consuming species reactions, these formulations ensure constant species concentration (mass conservation) over the model domains together with accumulated mass across in/outflow boundaries over time.