Analytical Derivation of the Current Distribution

This topic is covered in significant detail in Newman s textbook. A summary, relevant to the ensuing discussion, is provided here. 

The current density is directly related to the ionic fiux, Nj, in the electrochemical cell. The fiux is typically described in terms of three major components diffusion of ions across a concentration gradient, migration of charged ions down the electric field, and transport of ions due to bulk electrolyte convection. Consequently, the flux of an ionic species j is given by 

The current density is determined by assigning the charge Fzj to the flux of each species j and summing over all ionic species  

We also recognize that the electrolyte conductivity, k, is given by 

Equation (8) indicates that the current density is determined by both the potential and the concentration gradients. The explicit velocity term is absent from (8) (owing to electroneutrahty) however, convection still affects the current density by controlling the concentration field. It should also be noted that the electrode kinetics which do not appear explicitly in (8) establish the boimdary conditions required for its solution. As subsequently shown, the electrode kinetics may infiuence quite significantly the current distribution. 

---