In the density distribution of tertiary current, mass transport by diffusion, convection and migration are considered. The net charge ion transport is the electroneutrality and the electrolyte supporting with concentration negligible gradients, which means that the potential distribution in the electrolyte can be described by Ohm’s Law to introduce a dependence on mass transport in this model, the species oxidized at the anode is mass transport limited and its concentration is, c (mol/m^{3}) therefore, affect the kinetics of the electrodes. The anodic branch of the Butler-Volmer expression (Eq. 15) at the anode has a concentration dependence.

where, I is the electrode current, i_{0} is the density of exchange current, n is the number of electrons involved in the electrode reaction, A is the active surface area of the electrode, T is temperature, R is the constant of gases (J K^{-1} mol^{-1}), a is the coefficient of charge transfer, E and E_{eq} are the cell potential and the equilibrium respectively. The Butler-Volmer equation is based on pure metals electrodes. The expression, now, for the distribution of tertiary current is as shown in Eq. (22).

Here c_{0} (mol m^{-3}) denotes a reference concentration and is equal to the inlet concentration and P is the symmetry factor. The above equation is applied to the anode, for the cathode is maintained the model of density distribution secondary current.

The overpotential, p, is the difference between the electrode potential (E_{electrode}) and the equilibrium potential (E_{eq}) for the reaction in electrode, and is defined as follows:

This results in the following expressions for overvoltage at the cathode (p_{c}) and the anode (n^) with respect to the potential of the electrochemical cell (E_{cell}), the ionic potential of the liquid in the cathode (ф[, _{c}), in the anode (ф[, _{a}) and the equilibrium potential at the anode (E_{eq, a)} and cathode (E_{eq, c}):

In addition, a balance on momentum is introduced to describe convection. In this case, the flow is assumed laminar incompressible stationary, using the Navier-Stokes equations (Eq. 7), the equation for mass transport of the reacting species is:

where c is the concentration (mol m^{-3}), z the valence, D the diffusivity (m^{2} s^{-1}), m the mobility (mol m^{2} (s VA)), F is the Faraday constant (A s mol^{-1}), ф the ionic potential and u the velocity vector (m s ^{1}). For all boundary limits, conditions non-flow except for the input, output and anode are applied. At the entrance is fixed a concentration. Faraday’s law is used to specify the net molar flow at the anode where the species is consumed:

Electrochemical Silver Recovery

The electrodeposition simulates the evolution of the accumulation of the silver deposits with time on the electrode surface. The ion transport in the electrolyte occurs by convection and diffusion. The electrode kinetics is described by the expression of Butler-Volmer, dependent concentration [4]. The reaction at the electrode will move the limit in the normal direction with a speed v_{dep} (m s^{-1}) according to:

where M_{AG} is the molecular weight, and p_{Ag} the density of silver, respectively. The problem is time-dependent to simulate the electrode deformation for a time t in seconds.