Flux Definitions and Material Balances
The total flux Ni (mol/m2/s) of each individual species is defined as
(4-12)
where the first term ji represents the diffusive (and migrative) flux and the second term the convective flux, where u (m/s) is the mass-averaged velocity of the electrolyte.
The electrolyte current density Il (A/m2) is defined as
(4-13)
The material balances for the individual species are defined as
(4-14)
where εl (dimensionless) is the electrolyte volume fraction and Ri the molar source term.
Summing all individual material balances and multiplying by the corresponding species charges results in the charge balance equation for the electrolyte
(4-15)
When formulating the material balances in the interface, it is assumed that the continuity equation for the fluid (which is not solved for by the interface) is fulfilled
(4-16)
This allows for reformulating the individual material balances as
(4-17)
which is the actual (nonconservative) form of the material balance equation used internally by the interface. This equation form avoids accuracy issues related to evaluating derivatives of the velocity variable.