COMSOL 6.4 - Two-Phase Flow Modeling of Copper Electrowinning Using Bubbly Flow

Two-Phase Flow Modeling of Copper Electrowinning Using Bubbly Flow

Copper electrowinning is the process of copper extraction from an electrolyte solution, and its subsequent deposition at the cathode surface, by passing an external current through the electrolytic cell. In the cell, an insoluble anode is typically used. During the process, oxygen bubbles are generated at the anode surface leading to a large recirculation zone between the anode and cathode surfaces.

In the model example presented here, charge and mass transports of ionic species are modeled using the Tertiary Current Distribution, Nernst–Planck interface and the two-phase flow generated due to oxygen gas evolution at the anode surface is modeled using the Bubbly Flow, Laminar Flow interface.

The model example is based on a scientific paper (Ref. 1).

Model Definition

The electrowinning cell model geometry is shown in Figure 1. The two vertical boundaries represent the anode and the cathode surfaces, respectively. Since the oxygen gas evolves at the anode surface, this boundary also acts a gas inlet. The top boundary serves as an inlet for the liquid phase and an outlet for the gas phase. The outlet boundary for both the liquid and gas phases is as shown Figure 1.

Figure 1: Model domain with boundaries corresponding to the anode, cathode, and vertical symmetry walls.

Mass transport by convection, diffusion and migration for three ionic species (copper, Cu2+, hydrogen, H+and bisulfate, HSO4-) is solved using the Tertiary Current Distribution, Nernst-Planck physics interface:

where εl is the electrolyte volume fraction. εl is defined in terms of the liquid phase fraction which is solved for in the Bubbly Flow, Laminar Flow interface, as discussed later in this section.

The flux for each of the ions in the electrolyte is given by the Nernst–Planck equation:

where Ni denotes the transport vector (mol/(m2·s)), ci the concentration in the electrolyte (mol/m3), zi the charge for the ionic species, ui,eff the mobility of the charged species (m2/(s·J·mole)), F Faraday’s constant (As/mole), and ϕl the potential in the electrolyte (V).

The transport properties such as diffusion coefficients are defined using the electrolyte volume fraction:

The velocity field used in the mass transport equation comes from the Bubbly Flow, Laminar Flow interface and is corrected for the electrolyte volume fraction.

The electroneutrality condition is given by the following expression:

Electrochemical reactions

The copper reduction and the oxygen evolution due to water decomposition are the two electrochemical reactions considered in the copper electrowinning model presented here.

The copper reduction at the cathode surface proceeds as:

The oxygen evolution at the anode surface proceeds as:

Concentration dependent kinetics is used to model the copper reduction and oxygen evolution oxidation reactions, which will set the local current density according to:

where i0,m is the exchange current density, CR,m is the reduced species expression, CO,m is the oxidized species expression, αa,m is the anodic transfer coefficient, αc,m is the cathodic transfer coefficient, and ηm is the overpotential for species m (Cu and H, respectively).

The overpotential ηm (V) is calculated from:

The total current of 5.14 A is applied at the anode surface which is used to calculate the external electric potential,

The equilibrium potentials for the copper reduction and oxygen evolution oxidation reactions are calculated using the Nernst equation:

where

and

are the standard equilibrium potentials for the copper reduction and oxygen evolution oxidation reactions and are considered to be 0.34 V and 1.23 V, respectively. Moreover, cCu,ref and cH,ref are the reference concentrations for copper and hydrogen ions, respectively.

At the anode and cathode surface boundaries, fluxes of ionic species are defined in terms of the electrochemical reactions as:

where νm is the stoichiometric coefficient, iloc,m is the local current density, nm is the number of electrons, and F is Faraday’s constant (96,485 C/mol). This will set the flux to be proportional to the electrode current density according to Faraday’s law.

The electrode kinetics parameters: i0,Cu=100 A/m2 and i0,H= 3 × 107 A/m2 are taken from Ref. 1.

Bubbly Flow interface

The Bubbly Flow interface sets up a two-phase flow model for oxygen gas bubbles in a liquid electrolyte. The physics interface tracks the averaged gas-phase concentration rather than each bubble in detail. The physics interface solves for the liquid velocity, the pressure, and the volume fraction of the gas phase. Details of the governing equations are presented in the theory section for the Bubbly Flow interfaces in the CFD Module User’s Guide.

The Physical Model settings for the Bubbly Flow interface provides a low gas concentration option which is active per default. This option is applicable if the gas concentration is about 1%, in which case the transport equations can be simplified compared to cases with higher gas concentrations. Considering the possibility of higher gas-phase fraction, particularly near the anode surface, the low gas concentration option is disabled in this model.

For demonstration purpose, density and viscosity of the liquid phase are assumed to be uniform and are considered to be 1200 kg/m3 and 0.835 × 10−3 kg/m/s, respectively. However, they can be set to be dependent upon the copper concentration.

For laminar flow the gas velocity ug is calculated from:

where ul stands for the liquid-phase velocity, and uslip stands for the relative velocity between gas and liquid, the so-called slip velocity.

The slip velocity is calculated from a slip model. The Bubbly Flow interface provides several slip models. The most appropriate slip model for this electrowinning cell is a pressure-drag balance slip model with a drag coefficient tuned for small spherical bubbles using Hadamard–Rybczynski model. The gas bubbles diameter is considered to be 50 μm.

The top boundary of the electrowinning cell acts as an inlet for the liquid phase and as an outlet for the gas phase. The Inlet boundary feature is used here to set the normal inflow velocity to 0.0001 m/s and to set the gas outlet boundary condition.

The Outlet boundary feature is used to set zero pressure for the liquid phase and to set gas outlet boundary condition for the gas phase for the boundary shown in Figure 1.

The Wall boundary feature is used to set no slip boundary condition for the liquid phase and to set the gas mass flux at the anode surface as

where

is the gas mass flux for oxygen bubbles;

is the molar mass of oxygen; iloc,H is the local current density of oxygen evolution reaction evaluated at the Tertiary Current Distribution, Nernst–Planck physics interface; and F is Faraday’s constant.

Results and Discussion

Figure 2 shows surface plot of the liquid phase velocity magnitude along with arrow plot of the liquid phase velocity field after 60 s of deposition operation. A large vortex can be seen at the top of the electrowinning cell in Figure 2. The higher magnitude of liquid phase velocity at the anode surface indicates the generation of oxygen bubbles due to water decomposition electrochemical reaction and their rise to the top of the cell due to buoyancy effects.

Figure 2: Surface plot of the liquid phase velocity magnitude along with arrow plot of the liquid phase velocity field after 60 seconds of deposition operation.

Figure 3 shows surface plot of the gas phase volume fraction along with streamline plot of the liquid phase velocity field after 60 s of deposition operation. The gas phase volume fraction is seen to be higher near anode surface in Figure 3, similar to the liquid phase velocity shown in Figure 2. The buoyancy force of oxygen bubbles generated at the anode surface is a strong driving force for the fluid flow in the electrowinning cell. The highest magnitude of the gas phase volume fraction is found to be just over 0.02.

Figure 3: Surface plot of the gas phase volume fraction along with streamline plot of the liquid phase velocity field after 60 seconds of deposition operation.

Figure 4 shows surface plot of the copper concentration along with streamline plot of the total flux after 60 s of deposition operation. It can be seen that the copper concentration is quite uniform throughout the electrowinning cell due to a strong stirring effect of oxygen bubbles. The lowest copper concentration is observed at the cathode surface (which is not clearly visible in Figure 4) since copper ions are consumed here during deposition.

Figure 4: Surface plot of the copper concentration along with streamline plot of the total flux after 60 seconds of deposition operation.

To bring out the depletion of copper ions at the cathode surface, several line plots are plotted next for the copper concentration.

Figure 5 shows line plot of the copper concentration along the cathode surface after 60 s of deposition operation. It can be seen that the copper concentration varies considerably with distance from the bottom of the cathode surface. The peak in the copper concentration at the top of the cathode surface corresponds to a vortex in the liquid phase velocity magnitude observed in Figure 2.

Figure 5: Line plot of the copper concentration along the cathode surface after 60 seconds of deposition operation.

Figure 6 shows line plot of the change in copper concentration with time for 60 s of deposition operation at three different locations on the cathode surface. It can be seen that the copper concentration attained a steady state at the top and middle of the cathode surface whereas it decreased continuously at the bottom of the cathode surface.

Figure 6: Line plot of the change in copper concentration with time at different locations on the cathode surface.

Figure 7 shows line plot of the change in copper concentration across the boundary layer thickness at a specific position on the cathode surface after 60 s of deposition operation. It can be seen that the copper concentration varies considerably across boundary layer thickness and it remains uniform toward the bulk of the electrolyte solution. This result confirms that the mesh used in the model is fine enough to resolve the change in copper concentration across the boundary layer at the cathode surface.

Figure 7: Line plot of the change in copper concentration within the boundary layer at a specific position on the cathode surface after 60 seconds of deposition operation.

Figure 8 shows line plot of the change in copper deposition thickness along the cathode surface after 60 s of deposition operation. It can be seen that the copper deposition is considerably higher at the top half of the cathode surface when compared to the bottom half. The peak in the copper deposition thickness at the top of the cathode surface corresponds to a vortex in the liquid phase velocity magnitude observed in Figure 2.