Modeling Current Distributions in a Molten Salt Electro-Refiner

Today, we welcome a new guest blogger, Alexandre Oury of SIMTEC. He discusses the analysis of current distributions in a molten salt electro-refiner.

In a webinar highlighting electrochemical recycling processes, SIMTEC presented a computational approach for predicting current distributions in a molten salt electro-refiner. The three main types of current distribution (primary, secondary, and tertiary) were treated, with a particular emphasis on the first two types. Using COMSOL Multiphysics, we implemented primary and secondary current distributions in an electrolysis cell.

Molten Salt Electro-Refining

A common pyrometallurgical route for the recovery of many metals and rare earths is high-temperature molten salt electrolysis. This process involves a molten electrolyte made of a molten salt in which the metal to be recovered, most commonly present in its oxide form, is dissolved. When a current is applied between the cathode and anode, the metal is deposited as a solid or a liquid at the cathode, while some gas (generally a carbon oxide, such as CO or CO₂) evolves at the carbon-based anode, as depicted in the figure below.

Principle of a molten salt electrolysis cell for the refining of rare earths.

Numerical simulation can be advantageously employed to predict the main cell features (e.g., the reaction rate distribution on the electrodes, the cell voltage, or the electrolyte temperature) in order to optimize the design and operational conditions of the process. Even though high-temperature electrolysis involves many physical phenomena that are complex and strongly coupled to one another, the reaction rate distribution on the electrodes can be approximated with a rather simple electrostatic approach that involves the calculation of the current densities associated with the reactions.

Simulating the Current Distribution

The current flow in an electrochemical device can be described by three different types of distributions: primary, secondary, and tertiary. The type of distribution depends on the reaction kinetics and whether the cell is limited by the transport of active species.

The primary current distribution assumes that the reaction kinetics are so fast that no reaction overpotential is associated with the activation of the reactions. The cell can be considered a simple resistive charge conductor. The only equation to solve is the conservation of the current density in all domains. An example of the equation and possible boundary conditions to implement in order to compute the primary current are represented schematically in the following image. In this example, the cathode potential is arbitrarily set to zero, while a current is prescribed at the anode.

An example of equations and boundary conditions of a primary current model of an electrolysis cell.

The secondary current distribution accounts for the activation overpotentials of the reaction(s) at the electrode/electrolyte interfaces, which can be required when the electrochemical reactions are significantly slow. A kinetic relation is implemented as a boundary condition, which relates the electrode overpotential to the Faradaic current density. An ordinary kinetics expression is the so-called Butler-Volmer equation, which includes exponential activation terms. A schematic of the secondary current approach is illustrated below, with an example of a classical Butler-Volmer relation expressed with the exchange current density i₀.

An example of equations and boundary conditions of a secondary current model of an electrolysis cell.

The design of the molten salt electro-refiner considered here is represented in the following figure. It is made up of several anode rods mounted in a circle around a central cathode rod. The electrodes are immersed in a molten salt-based electrolyte containing the metal (a rare earth, such as neodymium) to be recovered at the cathode. Symmetrical considerations allow the implementation of 1/32^nd of the geometry without the loss of information, which significantly diminishes the degrees of freedom within the system.

The geometry of the molten salt electro-refiner considered for simulation and an elementary unit obtained taking symmetry planes into consideration.

The primary and secondary current models are implemented in COMSOL Multiphysics using the Electric Currents physics interface of the AC/DC Module. Predefined physics interfaces are available in the electrochemistry-based COMSOL Multiphysics modules, such as the Electrodeposition Module, that provide easy definitions of terms such as the Butler-Volmer equations and enable the simulation of current distributions in electrochemical devices. However, we chose to use the Electric Currents physics interface found in the core COMSOL Multiphysics product and introduce the electrochemical reaction kinetics as expressions in the source term. Three electric currents physics are used, one for each domain (anode, cathode, and electrolyte). For both models, a Ground condition (V = 0) and a Normal Current Density condition are set on top of the cathode and on top of the anode, respectively.

In the case of the primary current model, the Electric Potential condition is set at both electrode surfaces, with the value defined in the primary current model example above. For the secondary current model, the Normal Current Density condition is used on these boundaries, selecting “Inward Current Density” with values defined by the Butler-Volmer relations in the previous secondary current model example. The conductivity of the electrodes and that of the electrolyte are set as constant. The kinetics parameters associated with the secondary current description are i₀ = 1 A.cm^-2 for the cathodic reaction and i₀ = 0.001 A.cm^-2 for the anodic reaction, corresponding respectively to fast (metal deposition) and slow (gas evolution) kinetics.

The domains are meshed using tetrahedral elements. There is no need for a special level of refinement. However, it is important to ensure that there are enough elements on top of the anode, where the current is injected, as well as on the electrode edges, where current peaks are expected (edge effects).

The current densities simulated at the electrode surfaces for a total prescribed current of 1000 Amps are shown for both models in the next image.

Current densities simulated at the electrode surface for the primary and secondary current models.

In both current approaches, it is observed that the gas evolution reactions on the anodes are mainly concentrated on the inner parts facing the cathode. The deposition rate is quite uniform along the cathode height, with a significant current peak taking place around the bottom edge. The main difference between the primary and secondary descriptions is the uniformity of the current associated with the anodic reaction.

In the case of a primary distribution, the current density ranges from 0 to nearly 8 A/cm². In the secondary distribution, the anodic current density range is twice as small, with a maximum value of 3.63 A/cm². This is due to the homogenization effect induced by the activation overpotentials, which are accounted for in the secondary model. The difference between the primary and secondary currents is much more attenuated at the cathode, where a similar level of heterogeneity is observed due to the high value of i₀ considered in the secondary model for the cathodic reaction.

The simulated cell voltages can be decomposed into several contributions. In the primary current model, the cell voltage (8 V) is the sum of the thermodynamic voltage \DeltaE⁰ (1.7 V) and the Ohmic drops occurring across the cell (6.3 V). In the secondary model, there is an additional activation term of 2.1 V due to the electrode kinetics, which brings the cell voltage up to 10 V.

The secondary current model is then expected to be more accurate than the primary model in terms of current and voltage, especially for such processes involving kinetically slow gas evolution reactions at the anode.

To bring the current model further, a tertiary distribution can be implemented to take into account the effect of the active species concentration in the electrode vicinity. However, such a model is more complicated as it requires an appropriate description of the species transport as well as the electrolyte convective motions in the cell. While an electrochemistry-based module, such as the Electrodeposition Module, has specific physics interfaces for defining and simulating tertiary current distributions, other physics sometimes need to be considered within the simulation. In our case, gas evolution at the anode as well as natural convection occurring through thermal gradients can affect the electrolyte concentration near the anode.

An example of bubbly flow simulated in the multi-anode cell with the Laminar Bubbly Flow interface is given in the following figure. It is shown that the main electrolyte movement occurs around the anodes where gas bubbles are released and rise, while slower zones are also apparent, for example, at the bottom of the cathode.

An example of a bubbly flow simulation in the electro-refiner.

Conclusion

In this short presentation, we reported on two current models developed with COMSOL Multiphysics for simulating the electrochemical reaction rate distribution in a typical molten salt electrolysis cell. The primary current model considers the reactions to be infinitely fast, while the secondary current model includes kinetics relations at the electrode/electrolyte interface. The results differ in terms of current uniformity (more uniform distributions for the secondary distribution) as well as cell voltage (significant contribution of the activation overpotentials in the secondary model). For such cells, a secondary approach is clearly better. While a tertiary current distribution is not presented, an example of the convective flow due to gas bubble formation at the anode is provided and can be used to solve for the tertiary currents.

About the Guest Author

Alexandre Oury received a master’s degree in materials science and engineering from Joseph Fourier University (Grenoble). He later earned his PhD in electrochemistry and process engineering from the National Polytechnic Institute of Grenoble, with a research topic related to battery storage systems. He currently works at SIMTEC as a modeling engineer, developing numerical models that are mostly applied to electrochemical systems and electrolysis cells.