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.
(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.