Computational model for the Mark-IV electrorefiner

4.3.5.1. Motivation and approach

A pair of computational models for INL’s Mark-IV ER have been developed as a joint project between INL, KAERI, SNU, and UI [219]. The Mark-IV ER has been used at INL to separate uranium from spent EBR-II driver fuel. KAERI and SNU developed a rigorous 3D model [222], while INL and UI focused on a less computationally intensive 2D model [103-105, 223]. These two computational models were derived based on fundamental electrochemical, thermodynamic, and kinetic theories. The 2D model is based on the interfaces within the Mark-IV ER across which mass transfer and electrochemical reactions occur, with the primary focus on the anode/molten salt and molten salt/cathode interfaces. To further reduce complexity, only three species have been investigated: uranium, plutonium, and zirconium, which constitute greater than 90% by mass of the spent EBR-II driver fuel [99].

4.3.5.2. Model development

The developed 2D model is based on fundamental electrochemical theory for both the cathode and anodes, as shown by the equations in Fig. 55 [104, 222]. With a combination of the three constraints shown, it is possible to develop a system of three equations to solve the problem.

For N species, these three equations result in N, N, and 1 total equations, respectively, creating a system of 2N+1 equations and 2N+1 unknowns (Ii, ηi, s, and Et). This system can be solved using any number of optimization routines. This new method allows the user to alter any of the

‘known’ variables (e.g. exchange current density, initial salt composition, mass transfer coefficients, interfacial surface area.) without affecting the overall solving scheme. It also enables the investigation of different studies for each interface, while not altering the solving scheme of the other interfaces.

FIG. 55. Modelling approach organization, where subscript i represents the species and subscript j represents the interface. 𝐸 is the equilibrium potential, 𝑋_, is the mole fraction at the electrode surface, 𝐶_, is the concentration at the electrode surface, 𝐶 is the total bulk salt concentration, 𝑁 is the molar mass transfer, 𝐼 is the partial current, 𝜂_, is the overpotential at the electrode surface, 𝐸 is the total electrode potential and all other variables are defined in Table 14.

4.3.5.3. Computational procedure

The commercial software ‘Matlab’ has been used to execute the aforementioned numerical scheme. To limit computational complexity, currently only U(III), Pu(III) and Zr(IV) have been considered. However, more species may be included with sufficient thermodynamic data. The selection of these three specific species is not arbitrary. First, uranium is the principle element of interest, as the purpose of the electrorefining process is to separate the uranium from the SNF and because it is the most prevalent species [183]. Second, plutonium is also an important constituent in these calculations because its behaviour in previous experimental work involving U-Pu-Zr alloys has been difficult to determine [183]. Third, zirconium is the most active of the noble metals present in the SNF; therefore, the separation of uranium and zirconium is of great interest and is one of the primary goals of the anodic process in the ER. It should also be noted that zirconium is the second most prevalent species in the SNF [223]. As there are three species in the present study, N is three and both the anode and cathode systems to be solved consist of seven equations and seven unknowns.

4.3.5.4. Important parameters

Many parameters can be obtained directly from system thermodynamics. However, there are other parameters that must be determined from directly specified conditions. Parameters used in the model and equations illustrated in Fig. 55 are summarized in Table 14.

TABLE 14. PARAMETERS USED IN THE MODEL AND EQUATIONS [216].

U/UCl3 Zr/ZrCl4 Pu/PuCl3 LiCl/

KCl Total/

mixture Ai,a, Initial anodic surface area (m²) 0.476 0.257 0.013 - 0.746 Ci,salt, Initial bulk salt concentration

(mol/m³) 557 0 16.9 27974 28547

Ei0, Standard reduction potential

(V vs. Ag/AgCl) -1.274 -1.088 -1.555 - -

F, Faraday constant (C/eq) 96485

i0, Exchange current density (A/m²) 0.5

Icell, Applied cell current (A) 40–100 (see Fig. 58) ki,a0, Initial anodic mass transfer

coefficient (m/s) 4.70×10^-6 4.00×10^-6 3.89×10^-6 - -

R, Ideal gas constant(J/molK) 8.314

T, Operating temperature (K) 773

zi, Valence (eq/mol) 3 4 3 1 -

α, Anodic transfer coefficient 0.4

γi,s, Activity coefficient 0.00139 0.00448 0.0041 - -

Data from a set of electrorefining experiments run in the Mark-IV ER have been used to facilitate the modelling process. The experiment was run with two anodes and one cathode, which was exchanged for a fresh one approximately 39 hours into the run. The modelled experiments were run in a controlled current mode with an applied current, Icell, changing periodically as shown in Fig. 56.

Accurate modelling of the interfacial surface areas at both electrodes is incredibly important as the electrochemical reactions of most interest take place across these interfaces. The total anodic surface area is modelled as decreasing with each individual species’ available surface area as a fraction of the total. The cathode surface area increases as the electrorefining process progresses and material is deposited onto the cathode. The cathode has been modelled as a smooth cylinder growing as current is passed.

The mass transfer coefficients at both the anode and cathode, ki,a and ki,c, are a measure of how well species i can move through the boundary layer separating the electrode surface and the bulk electrolyte. The initial values are calculated for both electrodes using a Sherwood correlation for a rotating cylinder. Anodic mass transfer coefficients are assumed to decrease as the SNF is dissolved as discussed in previous work.

4.3.5.5. Results and discussion

The calculated anode and cathode potentials as compared with the experimental potentials in the Mark-IV ER can be seen in Fig. 56. The modelled total potential trends are seen to match well with the experimental trends throughout the electrorefining process. The values calculated for the anode match the experimental results with an RMS error of 1.83% [104], while those calculated for the cathode have an RMS error of 3.79%. The large oscillations in the cathodic experimental data are likely due to the rough and uneven growth of deposits on the cathode. As previously mentioned, the cathode is modelled as a smooth cylinder, which results in a relatively smooth potential plot.

FIG. 56. (a) Anode and (b) cathode potential comparisons [216].

The partial currents at both electrodes are another important parameter to analyse. The partial current can be used to calculate the rate at which each species is being either dissolved from the anode or deposited onto the cathode using Faraday’s Law. The calculated partial currents can be seen in Fig. 57.

FIG. 57. (a) Anode and (b) cathode partial currents [216].

Figure 57 shows that the majority of the current is involved in uranium dissolution at the anode.

However, as the electrorefining process proceeds, more zirconium is electrochemically dissolved. At the cathode, the majority of the deposited product is, as desired, uranium. As the amount of zirconium being dissolved from the anode into the electrolyte increases, more zirconium ions are available in the bulk electrolyte to be deposited along with uranium at the cathode.

4.3.5.6. Two-dimensional distributions

The 2D potential and current distributions within the Mark-IV ER are essential in determining the mass balance and location of the different species within the ER. A material balance can be performed within the Mark-IV ER and simplified to show that the Laplace equation, 𝛻 𝜑 = 0, can be used to describe the potential distribution within the molten salt electrolyte. This equation is solved with a Neumann boundary condition, 𝛻𝜑 = 0, at the ER wall and the anode/cathode potentials, Et,a/Et,c, from the model at the respective electrode surfaces to find

the potential distribution. From the solution and a version of Ohm’s law, 𝑖 = −𝜅𝛻𝜑, the current distribution can be solved.

Matlab is used to calculate the potential and current distributions within the electrolyte because it has a partial differential equation solver using the finite element method. In the modelled run, the ER was operated with two anodes and one solid steel cathode as shown in Fig. 58. For simplicity, the rotating cruciform anodes and dendritically growing cathode have been considered cylindrical. The radius of the cathode increases throughout the electrorefining process as a function of the amount of deposited material as described previously.

To calculate the current distribution, the electrical conductivity, κ, of the bulk electrolyte needs to be calculated. For the modelled system containing Li⁺, K⁺, U³⁺, Zr⁴⁺, Pu³⁺, and Cl^- ions, the total conductivity is found to be almost constant throughout the electrorefining process at 205 S/m [105].

The total potential and current distributions at ten hours into the experiment are plotted in Fig. 58(a) and 58(b) respectively. From both distributions, the majority of the current is seen to flow directly between the two anodes and the cathode. The distributions throughout the electrorefining process appear similar with the potential gradients tending to increase as the cathode grows and the process proceeds [105]. The potential gradient in the lower region of Fig. 58(a) is small with little current (Fig. 58(b), and therefore there are few moving ions in this area of the ER. Hence, a possible second cathode could be placed in this region to better use the available space within the ER.

FIG. 58. (a) Potential distribution and (b) current distribution at 10 hours into the experimental run [105].

4.3.5.7. Summary

A simplistic optimization routine has been developed and applied to the electrochemical reactions occurring at both the anode and cathode of an ER utilized for the electrochemical processing of SNF. This routine has been incorporated into a computational model of the Mark-IV ER, currently operational at INL. Results from the modelling process reveal that plutonium is rapidly exhausted from the anode fuel baskets. Throughout the remainder of the electrorefining process uranium is the main species being both dissolved from the anode and deposited on the cathode. However, as time progresses, zirconium dissolution and deposition start to become important processes. The total electrode potentials as calculated by the model were compared with potentials from a set of experiments run in the Mark-IV. The calculated and experimental potentials are shown to match with RMS errors of 1.83% and 3.79% for the anode and cathode, respectively.

The 2D potential and current distributions within the bulk electrolyte of the Mark-IV ER have also been examined. The total electrical conductivity of the molten salt electrolyte depends upon composition and is shown to be ~205 S/m. The potential distributions show the highest potential gradients exist directly between the anodes and cathode. The regions furthest from the operating electrodes have small potential gradients and little ion movement.

Future work on the ER model development will include increasing the number of species being examined. The model currently includes species making up greater than 90% of the mass of the SNF and the inclusion of additional species should increase the accuracy of modelling the process. The dendritic deposition of uranium at the solid cathode will be further investigated to include this uneven, somewhat random, growth pattern. These studies will involve more accurately determining the actual exchange current density within the Mark-IV ER from experiments.