# Mathematical Models [10]

In the first stage of the study, the models were drawn at ParasolidĀ® software, later imported to COMSOLĀ® to determine the hydrodynamic properties of the fluid, selecting a laminar flow. In the next step the electrolyte distribution in the reactor and its effect on the current distribution was studied. Therefore, once the geometry of the reactor established, a study on the distribution of electrolyte in the reactor was performed to ensure current distribution as uniform as possible. Subsequently, was used the module tertiary electrodeposition Nernst-Planck in order to know the amount of silver that was deposited on the steel cathode 304, by simulating the reactor operation for 2 h at room temperature [11]. In Fig. 1, are shown the geometries of the two types of electrochemical reactors simulated, smooth (RMS) and striated (RST).

## Hydrodynamic Properties

The momentum transport of fluid flow is described by the Navier-Stokes (Eq. 7) at steady state:

where, n is the dynamic viscosity (N s m^{-2}), **u **is the velocity gradient, p is the pressure, p the fluid density (kg m^{-3}) and F is the volumetric force (N m^{-3}). At the entrance one must specify a velocity vector (**u**) normal to the contour (**n**):

**Fig. 1 ****Models of reactor filter-press used in this research**

In the output contour, the pressure (p) is equal to the initial pressure (p_{0}) according to the Eq. (3):

Finally, in the reactor walls, applies a boundary condition of no slip:

The mass transport in the reactor is given by equations convection and diffusion as shown in Eq. (11):

where, D_{i} is the diffusion coefficient (m^{2} s^{-1}), and R_{i} denotes the end rate of reaction (molm^{-3}s^{-1}). For the boundary conditions, must be specified the input concentrations (c_{i}):

In the output, the mass flow through the contour, is dominated by the diffusion process. This assumes that any mass flow which crosses the contour due to diffusion is zero.

Considering the flow of species i (mol m^{2} s ^{1}), Eq. (8) yields:

Finally, at the reactor walls no mass transport across the contours: