A multiphysics simulation of a fluorine electrolysis cell

We modelled a laboratory-scale fluorine reactor which employed fully coupled, fundamental electron, heat, mass and momentum transfer (two-phase) equations to deliver a transient simulation. Hydrodynamic quasisteady-state results were produced for the current density, electric field, temperature, reactive species concentration, gas and liquid velocity profiles as well as gas fraction distribution within the reactor. Simulation results were verified by modelling and comparing models from published works on similar reactors, as the laboratory-scale reactor is still in construction phase. Comparisons were favourable.


Introduction
Industrial manufacture of fluorine requires the extraction of hydrogen fluoride from fluorspar, the electrolysis of hydrogen fluoride to form fluorine gas, and, finally, purification by a separation step. 1 Moissan was the first to produce fluorine gas via electrolysis. 2His original cell has been refined over the years, but the fundamental operating principles have not changed much. 3ry little is currently known, at least in the open literature, about the hydrodynamic behaviour of fluorine electrolysers.Experimental studies are difficult and dangerous, because they involve corrosive chemicals, at elevated temperatures, and high electric currents.A theoretical study was therefore commissioned to better understand typical cell operation.The hydrodynamic behaviour inside a typical two-electrode reactor was mathematically modelled using applicable computational fluid dynamics simulation software (COMSOL Multiphysics ® ).The reactor will be built and studied using recommendations found during the simulation procedure at a later date.
Fluorine electrolysis typically operates by subjecting molten potassium-acid-fluoride (KF•xHF, 1.8 ≤ x 2.2) to an electric field.The potassium-fluoride matrix is required because of the low electrical conductivity of anhydrous hydrogen fluoride.Hydrogen forms at the cathode and fluorine at the carbon anode.A separator skirt prevents explosive recombination of the gaseous components. 4bble formation severely complicates cell operation and is the major source of electrolyte flow caused by gaseous convection, which in turn causes mixing of the diluted reactant species.Furthermore the high electrical resistivity of the gaseous bubbles compared to the electrolyte results in several phenomena on the electrode. 5The anode is particularly susceptible to bubble phenomena as a consequence of the tendency of fluorine bubbles to stick to the electrode surface and move up slowly along the anode as a result of buoyancy forces. 3ermodynamic hydrogen fluoride decomposition requires a potential of 2.9 V, but an anode-cathode voltage of 8-10 V is required to maintain a current density of 10-12 A/dm 2 in industrial cells. 6The reversible cell voltage (Φ RV ) or thermodynamic decomposition voltage is the minimum potential required for product formation during electrolysis.Any voltage supplied that surpasses the reversible voltage (done to achieve the desired current density) produces heat through Ohmic heating. 3,7 a consequence of the unavailability of experimental data at simulation completion, the modelling technique was evaluated through a comparison with published fluorine cell simulations.Published models 8,9 were compared with self-produced simulacrums that we recreated under the same conditions.[12]

Modelling procedure
Four coupled transfer processes were identified as critical to ensure an accurate model: electron, heat, mass and two-phase momentum transfer.Roustan and co-workers 9 modelled uncoupled electron, heat and momentum transfer (single phase), using Flux-Expert ® .They investigated the two-phase momentum transfer using Flux-Expert ® and Ested Astrid code using predetermined experimental values for electron and heat transfer.We attempted to encompass all of the above-mentioned transfer processes into a single coupled model.Standard fundamental transfer equations were used where possible.Widely used empirical correlations were employed where fundamental equations were not available.A detailed description of the modelling procedure as well as the equations and variables used during modelling can be found in the Online Supplementary Material.
Because of the complex coupling (complete interdependence) between the featured transfer processes, a four-step solution method was employed.In the first step, the cell potential was slowly ramped up to its final value in a timeindependent calculation.This value was then used as a starting value to do a time-dependent calculation to solve heat and momentum transfer.A third calculation was done in which transient mass transfer was calculated using the values calculated in the previous calculation as a starting value.Finally, all of the values calculated above were used to calculate the transient fully coupled electron, heat, mass and momentum transfer processes up to a point at which a hydrodynamic steady state was reached.
Two meshes were used to ensure a mesh-independent solution.The first was a very fine mesh used to calculate the first three iterations.The second mesh was as fine but utilised rectangular mesh elements in which complex variable interaction occurred.These rectangular mesh elements are extremely useful in situations in which a lot of change occurs in one direction, but very little in the perpendicular direction.The meshes were employed on the http://www.sajs.co.za July/August 2015 electrodes and separator skirt.Mesh-dependent solutions were further deterred by ensuring that the solutions converged to a common solution with mesh refinement.

Results and discussion
For the results shown, we assume a static homogeneous molten electrolyte at simulation initiation.Time-dependent results of interest and importance are shown at 100 s after simulation (and electrochemical reaction) commenced; this point in time was identified as a hydrodynamic steady state within the reactor.
In general, arrows represented in the results indicate direction and are proportional to the norm of the vector quantity represented, at the arrow starting point.Colours indicate values as given by the legend to the right of the image.

Published result comparison
Simulation results were not justified with experimental data; instead it was decided to simulate published fluorine electrolysers while construction and commissioning of the lab-scale electrolyser was taking place.Roustan et al.'s 9 publication was used, from which electron, heat and single-phase momentum transfer could be compared.All comparisons were favourable; one such comparison is shown in Figure 1.
We used a two-phase momentum transfer equation; a suitable model comparison was therefore required and Roustan et al.'s 9 was chosen.
The two-phase momentum transfer results as found by Espinasse and co-workers 8 is shown in Figure 2a.We found similar plume shapes (as shown in Figure 2b).The predicted gas fraction was, however, significantly lower, but supported by photographic evidence.

Simulations
Our simulations were based on a simple lab-scale fluorine electrolyser comprised of one anode and a corresponding cathode.A cross section of the middle of the reactor was modelled and is shown in subsequent figures.

Momentum transfer
Figure 3 shows a well-developed hydrogen plume, and detachment from the cathode occurs as expected.Very little hydrogen migration into the fluorine section is observed; therefore the chance of explosive recombination of product gases is very low.This finding is also good from a productivity standpoint, as fewer product gases are lost and less purification of product streams will be required.
The electrolyte movement (Figure 4) induced by gaseous (hydrogen) movement is evidenced by the swirling liquid phase eddy between the separator skirt and the cathode at the top right of the reactor.This same eddy has the effect of causing convective stirring throughout the reactor, increasing heat and mass transfer (which in turn increases current density and reaction rate).These observations align well with what is observed in industrial and other lab-scale reactors. Fraction_vol.

Electron transfer
Figure 5 shows the electric potential and electric potential contour lines within the electrolyte.Electric potential drops from the anode to the cathode, from 9.1 V to 0 V, as expected.This result corresponds to the potential change expected from the literature.The bending of the electric potential contour lines along the separator skirt corresponds to the electric current density field lines that bend around the skirt.Boundary effects such as bubbles cause a large electrical conductivity decrease, as a result of the low conductivity of the gaseous phase, resulting in an exponential potential drop over these bubbles on the boundaries.

Heat transfer
From the internal temperature distribution results it was found that heat flux followed the same path as the electrolyte convection path shown in Figure 4.This shows that heat convection is the dominant heat flux component that removes heat from the electrode tips and separation skirt and transfers the heat to the cooled reactor wall.A parametric study was done on the value of electrolyte thermal conductivity as it is not widely available.It was found that a thermal conductivity value of 1.25 W/(m•K) (the thermal conductivity of potassium fluoride) was more than sufficient because heat conduction is completely overshadowed by heat convection during operation.In the interests of brevity and space, heat transfer images are not shown here.

Mass transfer
HF is produced at the anode and consumed at the cathode.The concentration gradient as a result of consumption at the cathode is a contributor to flux in the form of diffusion.From the scale bar on the right, it is clear that more HF is consumed than produced, as was predicted by the electrode half reactions (Equations 10 and 11 in the Online Supplementary Material).From Figure 8, it is evident that the secondary contributors to flux are convection and migration due to electrical field.
The ion flux and concentration of HF 2 -is not shown, but is a mirror image of Figure 8.The HF 2 -ion is produced at the cathode and consumed at the anode.The concentration gradient indicates ion flux from the cathode to the anode, as expected.

Conclusions and recommendations
Results obtained from the simulations, under the quasi-steady-state assumption, are reasonable and within expectations.The simulated results show a strong correlation between the gaseous phase movement (induced by buoyancy forces) and that of the liquid phase.The gasphase flux seen in Figure 3 shows that little or no hydrogen gas transfers to the fluorine compartment.All comparative simulations also deliver satisfactory results when compared with the published works.Current density and electric potential field line predictions correspond to expectations and match up satisfactorily with those found by Espinasse et al. 8 The shape of the gaseous plume of hydrogen that forms at the anode is the same as that reported in the literature when compared to the results from Mandin et al. 5 and Roustan et al. 9 There is, however, a difference in the qualitative gaseous fraction of the published and simulated reactors.
It is therefore recommended that the results found be used by the construction and experimental team of the physical reactor as an indication of what to expect and what to investigate.We also recommend that the physical parameters (mostly estimates) and empirical correlations, specifically the kinetics, be investigated to deliver more accurate results in future simulations.

Electron transfer
The chemical reaction (electrolytic decomposition of the electrolyte into F2 and H2) is induced by electric potential as prescribed by Equation 1 (Laplace's equation) which models the primary current distribution and adheres to the assumption of Equation 28 : −∇(∇Φ) = 0 Equation 1 Current density distribution is modelled using Equation 3 (the Butler-Volmer equation) and Equation 49 : Electrical conductivity was in turn modelled using the empirical relation shown in Equation 5.This equation assumes a linear relationship between electrical conductivity and gas fraction.Low gas fractions result in maximum conductivity, and vice versa.

Heat transfer
Heat transfer as a result of convection and conduction inside the reactor is modelled using Equation 63 : Heat generation in turn is modelled using Equation 73 :

Mass transfer
Reactant and product species movement is modelled using Equation 8 11 .This correlation incorporates convection, conduction, ion migration due to electric field and chemical reaction: Table 1 shows the three species assumed to comprise the system.The dissociation reactions are given by Equation 9: Relevant electrode half reactions are given by Equation 10(anode) and Equation 11(cathode) as supplied by Groult et al. 2 : Dilute species flux at the electrodes was further modified to include the effect of bubbles on the electrode surface.This was implemented by coupling the calculated dilute species flux and liquid fraction (through multiplication) at the electrode boundary.

Momentum transfer
Flow induced inside the reactor was modelled by Equation 12, Equation 13 and Equation 14, representing the momentum transport, continuity and laminar bubbly flow equations, respectively.Subscripts 'l' and 'g' denote the liquid and gas phases, respectively 4,8 : ρ   ⁄ + ∇(  ρ   ⃑⃑  ) = 0 Equation 14 The following assumptions allow for a simplified modelling procedure:  The gas density is negligible compared to the liquid density. The motion of the gas bubbles relative to the liquid is determined by a balance between viscous drag and pressure forces.
 The two phases share the same pressure field. Gas volume fraction is less than 0.1.

Starting and boundary conditions
The starting conditions for the reactor are given in Table 2.  Empirical equations used in the modelling procedure are given in Table 4.  , = −Φ−Φ 0,   =  0  (•(−25℃)) ⁄ A list of constants used during modelling is given in Table 5.

7 k_electrolyteFigure 1 :
Figure 1: (a) Temperature profile within the electrolyser as simulated 1 and redrawn for this publication.(b) Temperature profiles (K) inside the reactor as simulated in this study.

Figure 2 :
Figure 2: (a) Mean hydrogen gas distribution for two different current densities, low on the left and high on the right, reproduced from Roustan et al. 9 (b) Our simulation of the published electrolyser of Roustan et al. 9 , showing relative gas fraction.

Figure 3 :Figure 4 :
Figure 3: Hydrogen gas phase fraction in the reactor.

Figure 7 :
Figure 7: Current density variation (in A/m 2 ) along the perimeter length (in m) of the anode from the outside at electrolyte level to the inside of the anode at electrolyte level.

Table 1 :
Chemical species assumed to be present during the electrolytic process

Table 2 :
Starting conditions used in the modelBoundary conditions used in the model are given in Table3; representing electron, heat, mass and momentum transfer boundary conditions, respectively.

Table 3 :
List of parameters and expressions used during the simulation Boundary Condition Anode surface Gas flux specified.Thermal insulation.Specified current density.Cathode surface No slip for liquid flow.Gas flux specified.Thermal insulation.Specified current density.Cooling walls No slip for liquid.Temperature specified as TW.Electrical insulation.Electrolyte level No electron or heat flow permitted (insulation).Slip condition for liquid flow.Other boundaries Thermal and electrical insulation.Liquid no slip condition.