Depletion-driven thermochemistry of molten salt reactors: review, method, and analysis

1 Introduction

Molten salt reactors (MSRs) utilize nuclear fuel dissolved in molten salt that can be pumped around a primary loop or kept roughly stagnant in pipes placed within the core. Due to the nature of their liquid fuel design, MSRs are considered to have the lowest technology readiness level (TRL) of all Generation IV nuclear reactors, as they are essentially a nuclear/chemical reactor hybrid (Gougar et al., 2015).

To increase the TRL of MSRs, private and public research focused on this cutting-edge reactor type has increased in recent years. Many MSR advocates strongly believe that investing in this unique advanced reactor type will reap distinct benefits in comparison to more traditional advanced reactor designs based on solid nuclear fuel. Thanks to the liquid nuclear fuel, MSRs open the possibility of designing ingenious and enhanced passive safety, online reprocessing schemes for fuel-cycle flexibility, and alternative economic uses for nuclear power including industrial heat and isotope production (Mignacca and Locatelli, 2020). For these reasons, MSR development is an attractive option for advanced nuclear reactor deployment. However, these reactor concepts are not without their drawbacks---mainly the low TRL that MSRs must overcome, and the utilization of an engineering methodology not traditionally seen in the nuclear engineering field for other reactors. Whereas solid-fueled reactors allow researchers and engineers to organize reactor analyses and design more easily into distinct physics (e.g., neutronics, thermal hydraulics, coolant chemistry, etc.), the physics of liquid-fueled reactors are inherently tightly coupled. This ultimately requires that MSRs are analyzed and designed in a manner that accounts for their multiphysics behavior (Abou-Jaoude et al., 2021).

One primary multiphysics phenomenon that drives the long-term behavior of MSRs during their operational lifetime is depletion-driven thermochemistry (Walker et al., 2023). Simply defined, depletion-driven thermochemistry is the process where the thermochemical equilibrium of the MSR fuel salt changes over time as the dissolved nuclear fuel is consumed and fission products (FPs) are generated via depletion. The new chemical species formed by FPs in the liquid salt may precipitate or volatilize into non-soluble species and hence, separate from the fuel salt and change the depletion evolution of the reactor. Additionally, this multiphysics process has several ramifications including corrosion, the behavior of volatile nuclide species, and the integral behavior of the reactor during normal operation and accident scenario transients (Walker and Ji, 2021a). Therefore, a multiphysics modeling and simulation approach, which couples fuel depletion and equilibrium thermochemistry, is an essential first step in modeling and designing MSR systems.

The work presented here represents a critical initial step toward building such modeling and simulation tools and identifying the experimental and thermochemical data limitations that remain. A new depletion-driven thermochemistry analysis capability was developed and implemented by coupling the U.S. Department of Energy’s (DOE) Nuclear Energy Advanced Modeling and Simulation (NEAMS) neutronics code Griffin (Lee et al., 2021) with the open-source Gibbs Energy Minimizer (GEM) Thermochimica (Poschmann et al., 2021). Here, Griffin performs a multi-region depletion calculation of the fuel salt with gaseous nuclide removal and Thermochimica performs a thermochemical equilibrium calculation of the fuel-salt composition using the Molten Salt Thermodynamic Database–Thermochemical (MSTDB-TC) (Ard et al., 2022).

The following sections delve into the specifics of this multiphysics modeling and simulation work elucidating depletion-driven thermochemistry in MSR systems. Section 2 further identifies, defines, and describes in detail the multiphysics effect named here as “depletion-driven thermochemistry”. Additionally, the state of the art of research on this topic is presented by way of a literature review, and the objective of this work in relation to the existing body of previous work is presented. The computational methodology and multiphysics coupling scheme of depletion and thermochemistry used in this work are detailed in Section 3.

In Section 4, the resulting multiphysics depletion-driven thermochemistry modeling capability is then applied to simulate the changing chemistry of both fluoride and chloride fuel-salt systems as depletion occurs. Additionally, chemical species and their associated vapor pressures in the fuel salt are identified, the availability of thermochemical data is discussed, and active chemistry control of fuel salts is simulated using this approach. Lastly, conclusions and key findings are reported and discussed in Section 5, along with a discussion of the future work necessary to more accurately model this multiphysics problem.

2 Review of depletion-driven thermochemistry

This section primarily introduces the multiphysics phenomenon that this work seeks to model via multiphysics simulations. The importance of depletion-driven thermochemistry in MSR systems is identified and its effect during the Molten Salt Reactor Experiment (MSRE) is described. The existing state of the art research capability is reviewed, and the objective of this work is discussed.

2.1 Defining depletion-driven thermochemistry

One of the largest remaining challenges in predicting MSR multiphysics behavior lies in understanding how the chemistry of fuel-bearing molten salts will change over the reactor lifetime as the MSR fuel salt undergoes depletion and how, in turn, chemical changes in the molten salt will affect the isotopic evolution of the reactor. This multiphysics coupling is coined by this work as “depletion-driven thermochemistry”. It is defined as the neutronic-driven changes in the fuel salt chemistry over time due to the process of depletion. Here, thermochemistry refers to the chemical compounds and chemical state of the molten salt ionic liquid, while depletion refers to the process of fission, decay, and neutron capture that depletes the fissionable material and creates an extensive inventory of fission and activation products in the fuel salt.

2.1.1 Definition of the redox potential in relation to the fluorine or chlorine potential

The chemical behavior of molten salts is largely dependent on the redox potential of the fuel salt. The baseline potential for measuring the redox potential is conveniently stated in reference to the chemical potential of the base ion of the ionic salt (i.e., fluorine or chlorine chemical potential) as seen in Eq 1 (Olander, 2002). Here, $R$ is the ideal gas constant in units of [J/(mol - °K)], $T$ is the temperature of salt in °K, and ${p_F}_2$ is the unitless partial pressure of fluorine gas (i.e., F_2(g)) in the molten salt, so the units of $\Delta {\overline{G}}_{F_2}$ are most commonly [J/mol] or [kcal/mol].

$\Delta {\overline{\mathit{G}}}_{{\mathit{F}}_{\mathbf{2}}}\equiv \mathit{R}\mathit{T}\mathit{ln}{{\mathit{p}}_{\mathit{F}}}_{\mathbf{2}}(1)$

Additionally, the fluorine chemical potential can be related to the electrochemical potential of the molten salt via Eq 2 (Kelleher, 2013).

$\Delta {\overline{\mathit{G}}}_{{\mathit{F}}_{\mathbf{2}}}\equiv \mathit{R}\mathit{T}\mathit{ln}{{\mathit{p}}_{\mathit{F}}}_{\mathbf{2}}=-\mathit{n}\mathit{F}\mathit{E}(2)$

Here, F = 96,485 [C/mol] is the Faraday constant representing the amount of charge per mol of electrons and $n$ is the number of electrons transferred in the reaction where the standard state reaction is given as Eq 3 (Baes, Jr, 1965). The reference potential for the standard state half-cell reaction is set at 0 [V] as shown in Eq 3.

${\mathit{F}}_{\mathbf{2}\left(\mathit{g}\right)}+\mathbf{2}{\mathit{e}}^{-}\leftrightarrow \mathbf{2}{\mathit{F}}^{-},{\mathit{E}}^{\mathit{O}}=\mathbf{0}\left[\mathit{V}\right](3)$

Essentially the fluorine chemical potential is a measurement of how strongly the fluorine in a system can oxidize/dissolve other elements in the system. In the remainder of this article, the electrochemical redox potential and the fluorine or chlorine chemical potential of the fuel salt will be used interchangeably with the understanding that the potential we refer to is linked to the base ion in the modeled salt system.

2.1.2 Importance of depletion-driven thermochemistry in molten salt reactors

The fluorine or chlorine potential of fluoride and chloride salts, respectively, is heavily influenced by the chemical composition of cations present which make up the ionic molten salt (Baes, Jr, 1965). During normal reactor operations, the chemical composition of the fuel salt will gradually change as the fuel salt undergoes nuclear fission, causing the fissionable heavy metals to be replaced with soluble or insoluble FPs. Additionally, non-fission transmutation of the fuel salt through capture reactions and radioactive decay will also occur altering the concentrations of actinides (e.g., neptunium, plutonium, etc.) as well as FPs in the fuel salt. The fuel salt chemical composition can also be altered by chemical species extracted from the fuel salt via off-gassing of volatile species or by additions of metallic nuclides to the fuel salt in the form of soluble species that are then dissolved in the fuel salt.

These changes in chemical composition due to depletion directly alter the fuel salt’s redox potential, which in turn determines how the fuel and FPs behave in solution, as well as how structural materials interact with the salt via corrosion mechanisms. Therefore, depletion-driven thermochemistry is a critical driving force in determining reactor behavior during normal operation and accident scenarios. This includes long-term corrosion effects on key reactor components and how radionuclide source terms (dictated by nuclide vapor pressures) in the molten salt evolve over reactor lifetimes. Ultimately, this multiphysics effect is an essential consideration for future MSR reactor design, licensing, and operation.

2.2 Depletion-driven thermochemistry in the Molten Salt Reactor Experiment

To better understand the problem of depletion-driven thermochemistry, it is helpful to review the first comprehensive operational example of an MSR---the Molten Salt Reactor Experiment (MSRE). The MSRE was a fluoride salt reactor that operated in Oak Ridge National Laboratory (ORNL) in the 1960s for over 11,500 equivalent full-power hours using both ²³⁵U- and ²³³U-based fuel dissolved in the molten salt (Haubenreich and Engel, 1970). The reactor was considered a successful proof of concept of liquid fuel technology, showcasing the benefits of such a design as well as identifying the remaining technical challenges that must be resolved by future research.

Of note for this present work, the implementation of chemistry control of the fuel-salt was a critical aspect of the reactor operation. It was anticipated that structural materials present in the Hastelloy N alloy used for the reactor vessel and piping would reduce UF₄ to UF₃ to some limited extent forming soluble corrosion products in the fuel salt as seen in Eq. 4, where the corrosion of chromium into the fuel salt is highlighted (Shaffer, 1971).

$\mathit{C}{\mathit{r}}_{\left(\mathit{s}\right)}^{\mathbf{0}}+\mathbf{2}\mathit{U}{\mathit{F}}_{\mathbf{4}}\leftrightarrow \mathbf{2}\mathit{U}{\mathit{F}}_{\mathbf{3}}+\mathit{C}\mathit{r}{\mathit{F}}_{\mathbf{2}}(4)$

Therefore, the molten salt system at thermodynamic equilibrium should be “poised” or “buffered” by the UF₄-UF₃ redox couple and a small amount of CrF₂ should be present. However, as UF₄ and UF₃ are consumed and transmuted during depletion, the total amounts and ratio of this redox couple change. If they change sigfnificantly enough, then the system could be increasingly poised or buffered by another redox couple.

In the case of the system shifting towards an oxidizing state where the UF₄ to UF₃ ratio is large or increasing, then the redox couple Cr_(s)-CrF₂ becomes dominant and poises/buffers the system. This system would then be in an active state of corrosion, where thermodynamic equilibrium is dependent on the amount of Cr_(s) that could be extracted from the structural surfaces in contact with the fuel salt. On the other hand, if the system shifts towards a reducing state where the UF₄ to UF₃ is small or decreasing, then the redox couple UF₃-U_(s) would become dominant and poise/buffer the system, and uranium would solidify and precipitate out of the fuel salt due to a lack of fluorine potential.

2.2.1 Understanding relative redox potentials of reactions

To illustrate this point more clearly, a standard half-cell redox potential table calculated from the MSRE is reproduced here as Table 1 (Baes, Jr, 1965). This table lists the standard half-cell electrochemical redox potentials calculated in FLiBe salt in relation to the fluorine chemical potential where the standard half-cell potential is defined by Eq 3. It is important to note that these redox potentials shown in Table 1 are standard at 1,000 [°K] whereas traditional standard cell potentials by definition mean that the voltage is measured or calculated at 298 [K], 1 [atm], and 1 [M] or 1.0E^-3 [mol/cm³] solutions. Since these ionic melts would not be liquid at 298 [°K], the standard potentials are calculated at 1,000 [°K] instead, and the other standard states (i.e., pressure and chemical activity) remain the same.

TABLE 1

TABLE 1. Calculated Electrode Potential in LiF-0.33BeF₂ (1,000 K)–Reformatted from (Baes, Jr, 1965).

The redox potential of a reaction not at standard states (i.e., redox potentials not calculated or measured at 1,000 [°K], 1 [atm], and 1.0E^-3 [mol/cm³] solutions) can be calculated by the Nernst equation shown in Eq 5 (Zhang et al., 2018). Here $E$ is the redox potential of the reaction at non-standard states, $E^O$ is the standard cell potential of the complete reaction (i.e., the addition of two half-cell reactions shown in Table 1), and $Q$ is the reaction quotient, which is a ratio of the relative chemical activities of the products to the reactants in the reaction.

$\mathit{E}={\mathit{E}}^{\mathit{O}}-\frac{\mathit{R}\mathit{T}}{\mathit{n}\mathit{F}}\mathit{ln}\left(\mathit{Q}\right)(5)$

Building upon this point then, the standard redox potential half-cell reactions in Table 1 show the relative likelihood of different reactions happening at different redox potentials with the implicit notion that the chemical activity of the elements being dissolved in the FLiBe salt are all equal to 1.0E^-3 [mol/cm³]. However, it should be stressed to the reader that the redox potential of reactions depend heavily on the chemical activities of the products and reactants of the reactions as seen in the $Q$ term of Eq 5. Therefore, the relative redox potentials of reactions in a real MSR system made up of base salt elements, fuel, fission products, and corrosion products is not as linear as the table may suggest and that a plethora of reactions shown (and more not shown) in Table 1 will be at various Q values for a given redox potential depending on the chemical activities of the elements in the fuel salt.

As can be at the top of Table 1, the most stable ions are the base salt ions Li⁺ and Be²⁺ in addition to a host of rare-earth fission product ions. The low redox potential values of these reactions means that it takes a very limited fluorine potential to sufficiently dissolve these elements in the molten salt, or that they readily form ion pairs with fluorine. Whereas for more positive redox potential values, a larger fluorine chemical potential is required to dissolve these elements in the molten salt.

As discussed earlier, if the redox potential is buffered or poised by the U⁴⁺-U³⁺ (UF₄-UF₃) pair in the salt as seen in Eq 4 (i.e., their chemical activities are thermochemically dominant in dictating the fluorine chemical potential), then the redox potential of the fuel salt at thermochemical equilibrium is held somewhere below −3.915 [V]. Therefore, more positive redox potential reactions further down Table 1 will not be significant (i.e., corrosion of Cr_(s), Fe_(s), and Ni_(s) is controlled).

As uranium fissions however, the relative chemical activities of the U⁴⁺-U³⁺ buffer pair is sufficiently altered, and the redox potential shifts towards the next thermochemically favorable buffer pair. The resulting two FPs generated in the fuel salt have an aggregated electrochemical potential that is more positive than the single uranium atom they replace and hence, the electrochemical potential of the system increases or becomes more oxidizing (Baes, Jr, 1974). Moving down Table 1 if the chemical activity of Cr_(s) in the structural material alloy in thermodynamic equilibrium with the salt is large enough, then the Cr_(s)-Cr²⁺ will become the dominant redox potential buffer pair and the redox potential of the fuel salt will be held somewhere below to −3.261 [V].

If fluorine has a less negative chemical potential, then UF₄ is overly dominant, and the formation of corrosion products will occur as seen in Eq 4 when the Cr_(s)-Cr²⁺ buffer becomes dominant. If the fluorine potential continues to increase, then in addition to significant corrosion of structural materials leaching Cr_(s), Fe_(s), and Ni_(s) into the fuel salt, UF_{5 (g)} and UF_{6 (g)} gas (not shown in Table 1) will eventually form as well.

2.2.2 Changing redox potential in the MSRE

The chemical activity ratio of UF₄ to UF₃ can be utilized to understand the current redox potential of the fuel salt and was therefore the primary method used to determine the chemical state of the fuel salt during the MSRE as seen in Figure 1 (Thoma, 1971). Ideally, the fluorine potential of the fuel salt would be actively controlled and held steady during reactor operation so that corrosion rates and FP behavior in the fuel salt would be predictable.

FIGURE 1

FIGURE 1. UF ₄ to UF₃ ratio during the ²³⁵U MSRE runs–reformatted from (Thoma, 1971).

However, as seen in Figure 1, the redox potential of the fuel-salt during the MSRE did not remain steady during the entire reactor operation (Thoma, 1971). During the first ∼2000 full power hours (or ∼2.5 MWd/Kg-U burnup) of the MSRE, the UF₄/UF₃ ratio rose without active chemistry control. This experimental data showcases how, as depletion occurs, the chemical environment of the fuel salt becomes more oxidizing where the fluorine chemical potential increases over time.

Another way to understand this process is that fuel salt develops a cationic charge deficiency due to depletion as dissolved uranium metal is consumed and the FPs that are generated are not as thermochemically stable in fluorides as the original uranium fuel (i.e., noble metal and noble gas FPs Mo, Tc, Xe, Kr) and do not form ion pairs with fluorine (i.e., their cationic charge = 0). To fill this cationic charge deficiency or to satisfy the increased fluorine chemical potential, the chemical reaction in Eq 4 comes into effect to form a new thermochemical equilibrium with a significant amount of corrosion products that have leached into the fuel salt from structural materials in direct contact with the fuel salt.

To limit corrosion, another possibility to reduce the redox potential of the fuel salt is to replace the cationic charge deficiency that develops due to depletion with a more thermochemically stable metal higher up Table 1 (e.g., more U or Be) as seen in Eq 6. In this case the chemical activity of the Be_(s)-Be²⁺ buffer ratio becomes thermochemically dominant and dictates the fluorine potential.

${\mathit{B}\mathit{e}}_{\left(\mathit{s}\right)}^{\mathbf{0}}+\mathbf{2}\mathit{U}{\mathit{F}}_{\mathbf{4}}\leftrightarrow \mathbf{2}\mathit{U}{\mathit{F}}_{\mathbf{3}}+\mathit{B}\mathit{e}{\mathit{F}}_{\mathbf{2}}(6)$

This is exactly what is occurring in Figure 1 when the UF₄/UF₃ ratio sharply decreases---the MSRE fuel salt is being actively reduced through the dissolution of Be into the fuel salt. This process was monitored by taking a fuel-salt sample at an interval of several days to determine the UF₄/UF₃ ratio, before the beryllium rod was be immersed into the fuel salt for a period of a few hours (Thoma, 1971).

2.3 Current state of the art and objective of work

As can be seen from the example of the MSRE, depletion-driven thermochemistry is an integral part of MSR behavior that must be modeled, measured, and controlled in future MSR designs. A survey of existing work in this area to identify the current state of the art will now be provided.

2.3.1 Literature review of similar works

The original research marking the onset of this topic is, of course, the experience of the MSRE identified in the previous subsection (Thoma, 1971). Additionally, in preparation for the proposed Molten Salt Breeding Reactor (MSBR), a simple study of the chemical consequences of fission was performed. The combination of assumed equilibrium oxidation states of FPs and their fission yields were calculated to determine an effective rate of cationic charge deficiency growth (Baes, Jr, 1974). Similarly, the same sort of cationic charge deficiency rate determination using equilibrium oxidation state assumptions of FPs in chloride salts was also performed (Chasanov, 1965). These studies aimed at estimating the amount of extra cationic charge needed to avoid significant corrosion during the operation of the MSRs at higher power.

Recent efforts build upon computational methodology advancements in thermochemistry such as the use of large thermochemical databases and the Calculation of Phase Diagrams (CALPHAD) method (McMuray et al., 2018). In support of these efforts, a new database for chemical analysis of MSRs (e.g., depletion-driven thermochemistry), the MSTDB-TC was developed (Ard et al., 2022). It uses the Modified Quasi-chemical Model in Quadruplet Approximation (MQMQA) for modeling molten salts that can be leveraged by the GEM solver Thermochimica (Poschmann et al., 2021).

These initial efforts culminated in the multiphysics coupling of Thermochimica using the MSTDB-TC in the Virtual Environment for Reactor Applications (VERA) tool suite (Graham et al., 2021). This multiphysics capability included thermal hydraulics in order to provide a temperature-informed spatial thermochemical solution of the MSR system at equilibrium. However, this work did not incorporate depletion and a time evolution of the fuel salt chemistry was not performed.

A parallel effort that utilized the Transient Simulation Framework of Reconfigurable Models (TRANSFORM) was also coupled with Thermochimica using the MSTDB-TC to perform dynamic mass accountancy during various reactor operational transients (Poschmann et al., 2022). This effort resulted in the transient analysis of volatile species formation and extraction into an off-gas system, as well as the freezing and precipitation of molten salt chemical constituents. However, this work did not focus on modeling depletion (i.e., chemical composition) driven thermochemistry changes, but rather focused on temperature effects.

Even more recently, the issue of depletion driven thermochemistry or chemistry-integrated fuel evolution was highlighted by the U.S. DOE’s Advanced Reactor Technology–Molten Salt Reactor Program (MSRP) (Shahbazi, 2022a). An excellent and thorough review of thermochemical modeling for radionuclide speciation in molten salts was completed and is highlighted here by the authors for the inquisitive reader (Shahbazi et al., 2021). Lastly, radionuclide species transport in the MSRE was conducted albeit without coupling depletion and thermochemistry in the analysis (Shahbazi et al., 2022b).

2.3.2 Objective of present work

The objective of this present work is to use the same methodology framework of MSR multiphysics analysis as identified in the literature review and apply this approach to model depletion-driven thermochemistry of MSRs using the DOE’s NEAMS toolkit built on the Multiphysics Object-Oriented Simulation Environment (MOOSE) framework (Martineau, 2021). This approach offers a greater flexibility of multiphysics and multiscale modeling capability, where other MOOSE based physics models for thermal-hydraulics, species mass transport, and corrosion can be easily incorporated in the future. Additionally, this work will mainly focus on modeling the changing redox potential of both fluoride and chloride salts as they undergo depletion and active chemistry control. Consequentially, this work will identify the remaining thermochemical data and modeling methodology challenges that are needed to comprehensively model depletion-driven thermochemistry in MSRs.

This present work is built upon smaller pieces that were previously published by the authors as key milestones in developing this depletion-driven thermochemistry analysis capability using NEAMS tools. Notably, the depletion solver in the neutronics code Griffin had to be altered to account for nuclide removal and multi-region depletion (Walker et al., 2022). Most recently, the coupling of Thermochimica using the MSTDB-TC with Griffin was performed to model material accountancy of gaseous species being extracted into the off-gas system (Walker et al., 2023). This work builds upon these initial steps to perform tightly coupled depletion-driven thermochemical analyses of MSRs focusing on the changing redox potential and the implementation of active redox control mechanisms.

3 Computational methodology and numerical implementation

This section details the computational methodology and numerical implementation of the multiphysics coupled code simulation capability developed for this analysis. Since much of this development work was done in previous reports documenting the incremental steps of this capability, readers are encouraged to review these for more details (Walker et al., 2022; Walker et al., 2023). A brief description for each individual code as well as the coupled system will be provided. Lastly, the current capabilities and limitations of the MSTDB-TC are identified for this work.

3.1 Griffin

Griffin is a NEAMS tool used for multi-fidelity neutronics and fuel depletion. Recent efforts incorporated a nuclide removal and multi region microscopic depletion capability for modeling off-gas and other waste streams in MSR systems during burnup (Walker et al., 2022).

The depletion equation solved by Griffin, also known as the Bateman equations, track the nuclide number density of the fuel and resulting FPs, and incorporates fission, neutron transmutation, and radioactive decay as seen in Eq 7 has additional terms not traditionally seen in the Bateman equations, which represent sources and sinks from removal waste streams and from chemical reactions at interfaces.

$\begin{array}{ll}\hfill \frac{\mathit{d}{\mathit{N}}_{\mathit{Z}}^{\mathit{A}}\left(\mathit{t}\right)}{\mathit{d}\mathit{t}}& ={\displaystyle \sum _{{\mathit{Z}}^{\prime },{\mathit{A}}^{\prime }=\mathbf{1}}^{\mathit{M}}}\left({\mathit{\gamma }}_{\mathit{Z}}^{\mathit{A}}{\displaystyle \sum _{\mathit{g}=\mathbf{1}}^{\mathit{G}}}{\mathbf{\sigma }}_{\mathit{f}\left({\mathit{Z}}^{\prime },,{\mathit{A}}^{\prime }\right),\mathit{g}}\left(\mathit{t}\right){\mathit{\varphi }}_{\mathit{g}}\left(\mathit{t}\right)+{\displaystyle \sum _{\mathit{g}=\mathbf{1}}^{\mathit{G}}}{\mathit{\sigma }}_{\left({\mathit{Z}}^{\prime },{\mathit{A}}^{\prime }\right)\to \left(\mathit{Z},\mathit{A}\right),\mathit{g}}\left(\mathit{t}\right){\mathit{\varphi }}_{\mathit{g}}\left(\mathit{t}\right)+{\mathit{r}}_{\left({\mathit{Z}}^{\prime },{\mathit{A}}^{\prime }\right)\to \left(\mathit{Z},\mathit{A}\right)}{\mathit{\lambda }}_{{\mathit{Z}}^{\prime }}^{{\mathit{A}}^{\prime }}\right){\mathit{N}}_{{\mathit{Z}}^{\prime }}^{{\mathit{A}}^{\prime }}\left(\mathit{t}\right)-{\mathit{\lambda }}_{\mathit{Z}}^{\mathit{A}}{\mathit{N}}_{\mathit{Z}}^{\mathit{A}}\left(\mathit{t}\right)\hfill \\ \hfill & -{\displaystyle \sum _{\mathit{g}=\mathbf{1}}^{\mathit{G}}}{\mathit{\sigma }}_{\mathit{a}\left(\mathit{Z},\mathit{A}\right),\mathit{g}}\left(\mathit{t}\right){\mathit{\varphi }}_{\mathit{g}}\left(\mathit{t}\right){\mathit{N}}_{\mathit{Z}}^{\mathit{A}}\left(\mathit{t}\right)-{\mathit{f}}_{\mathit{c}\mathit{h}\mathit{e}\mathit{m},\mathit{Z}}^{\mathit{A}}\left(\mathit{t}\right)-{\mathit{f}}_{\mathit{s}\mathit{t}\mathit{r}\mathit{e}\mathit{a}\mathit{m},\mathit{Z}}^{\mathit{A}}\left(\mathit{t}\right)\hfill \end{array}(7)$

Here $N_{Z^{\prime }}^{A^{\prime }}\left(t\right)$ is the number density of other nuclides with atomic number $Z^{\prime }$ and mass number $A^{\prime }$, which produce nuclide $N_Z^A\left(t\right)$, ${\gamma }_Z^A$ is the fission yield of nuclide $N_Z^A$ from $N_{Z^{\prime }}^{A^{\prime }}\left(t\right)$, ${\mathrm{\sigma }}_{f\left(Z^{\prime },A^{\prime }\right),g}\left(t\right)$ is the groupwise microscopic fission cross section of $N_{Z^{\prime }}^{A^{\prime }}\left(t\right)$, ${\varphi }_g\left(t\right)$ is the groupwise scalar neutron flux, ${\sigma }_{\left(Z^{\prime },A^{\prime }\right)\to \left(Z,A\right),g}\left(t\right)$ is the groupwise microscopic transmutation cross section of $N_{Z^{\prime }}^{A^{\prime }}\left(t\right)$ that produces $N_Z^A\left(t\right)$, $r_{\left(Z^{\prime },A^{\prime }\right)\to \left(Z,A\right)}$ is the branching ratio for radioactive decay, ${\lambda }_{Z^{\prime }}^{A^{\prime }}$ is the radioactive decay constant for $N_{Z^{\prime }}^{A^{\prime }}\left(t\right)$, ${\lambda }_Z^A$ is the radioactive decay constant for $N_Z^A\left(t\right)$, ${\sigma }_{a\left(Z,A\right),g}\left(t\right)$ is the groupwise absorption cross section for $N_Z^A\left(t\right)$, $f_{chem,Z}^A\left(t\right)$ is a chemical reaction source at interfaces, and $f_{stream,Z}^A\left(t\right)$ is a mass transfer removal source term for feed and waste streams.

The primary application of this work focuses on utilizing the $f_{stream,Z}^A\left(t\right)$, where nuclides will be withdrawn or added to the reactor system which alters the ongoing depletion of the fuel and nuclide inventory. By altering the $f_{stream,Z}^A\left(t\right)$ term, the chemical composition changes and this will drive any observed thermochemical equilibrium changes in the fuel salt. The form $f_{stream,Z}^A\left(t\right)$ takes is defined in Eq 8, a type of effective removal rate that depends on the existing amount of the nuclide in the fuel salt and a feed rate $L$, where $L$ can be positive or negative for material additions or subtractions from the system.

${\mathit{f}}_{\mathit{s}\mathit{t}\mathit{r}\mathit{e}\mathit{a}\mathit{m},\mathit{Z}}^{\mathit{A}}\left(\mathit{t}\right)=\mathit{L}{\mathit{N}}_{\mathit{Z}}^{\mathit{A}}\left(\mathit{t}\right)(8)$

3.2 Thermochimica

Thermochimica is the GEM that is currently being incorporated into the MOOSE framework for use in many NEAMS tools for thermochemistry analyses. The reader is encouraged to investigate the references in this section for more details involving the specific methodologies used to model the thermochemistry of ionic liquids. A high-level overview of this detailed subject is provided here.

Thermochimica uses the CALPHAD method and, for this case, the MSTDB-TC database to determine the equilibrium of an isothermal and isobaric system by minimizing the global Gibbs free energy of the chemical composition of the system. Eq 9 shows the general form of the Gibbs energy of the system where $G^{ref}$ is the reference Gibbs energy, $T$ is the temperature, $\Delta S$ is the change in configurational entropy, and $G^{XS}$ is the excess Gibbs energy (McMuray et al., 2018).

$\mathit{G}={\mathit{G}}^{\mathit{r}\mathit{e}\mathit{f}}-\mathit{T}\Delta \mathit{S}+{\mathit{G}}^{\mathit{X}\mathit{S}}(9)$

The $G^{XS}$ term accounts for the non-ideal mixing that arises from chemical species interactions occurring in the system. In an ideal gas, the $G^{XS}$ term is negligible, but this is not the case for ionic liquids like molten salts. This is where the MQMQA comes in handy to accurately characterize the chemical species interactions that occur in molten salts to determine the effect of non-ideal mixing (Poschmann et al., 2021).

Using the MQMQA and an extensive experimental thermochemical database, Thermochimica can proceed to solve Eq 9 and determine equilibrium phase values and chemical potentials of species in the molten salt. The MSTDB-TC database funded by the U.S. DOE contains one of the largest collections of self-consistent and vetted fluoride and chloride thermochemical data (Ard et al., 2022).

Utilizing the MSTDB-TC within Thermochimica allows for the thermochemical equilibrium of the fuel salt to be calculated at each depletion step. In addition to understanding how the chemical potentials of species change due to composition changes in the fuel salt, vapor pressures of chemical species in the fuel salt can also be calculated for volatilization and off-gas extraction analysis.

3.3 Multiphysics coupling schemes and numerical performance

This section examines the multiphysics coupling scheme used between Griffin-depletion and Thermochimica. Initially, a loosely coupled (or time-marching) scheme was implemented, with each code calculating one solution for each given time step. Although this served as a proof of concept for the depletion-driven thermochemistry capability, a more accurate coupling scheme was needed as oscillatory solutions were obtained for volatile isotopes with large extraction rates as seen in Figure 2. Figure 2 showcases the changing concentration of ¹³⁵Xe in the reactor core during depletion where volatile species and noble gases are being extracted to an off-gas system using a user specified system mass transfer coefficient $L$ as seen in Eq 8. The time marching multiphysics coupling scheme incorrectly overpredicts and underpredicts the solution marked by the orange sawtooth in Figure 2.

FIGURE 2

FIGURE 2. Comparison of ¹³⁵Xe core concentration using a time marching scheme versus a Picard iteration scheme.

The current tightly coupled implementation uses Picard iterations, with the two individual codes iterating their solutions until both solutions stop changing within a given tolerance as seen in Figure 3. Here the driving application is Griffin-depletion since it defines the fuel-salt composition needed by Thermochimica to determine the equilibrium thermochemistry of the system. Griffin calculates the nuclide inventory $A$ at time step $t^{n-1}$ and passes this to Thermochimica which calculates the thermochemical equilibrium of the fuel salt and determines the feed rate $f_{stream,Z}^A$ at time $t^{n-1}$ by accounting for the mass transport of volatile species. A time step is taken to time step $t^n$ where a new nuclide inventory $A$ and a new feed rate $f_{stream,Z}^A$ are calculated. The average of the two feed rates ${f_{stream,Z}^A}^{n-1}$ and ${f_{stream,Z}^A}^n$ is used to re-calculate the original time step $t^{n-1}$ until both solutions self-consistently converge to a set tolerance.

FIGURE 3

FIGURE 3. Griffin and Thermochimica multiphysics coupling scheme.

To illustrate how Thermochimica alters the nuclide inventory calculation done by Griffin, a truncated Thermochimica output example is provided in Table 2 which describes what sort of chemical species are present in the liquid fuel salt and the vapor pressures of these species if the salt is in contact with an off-gas system liquid-gas interface.

TABLE 2

TABLE 2. Truncated Thermochimica Output Example to Determine $f_{stream,Z}^A\left(t\right)$

At this point a user specified system mass transfer coefficient $L$ as seen in Eq. 8 can be used to extract fractions of the vapor pressure of gaseous species to an off-gas system which will determine the nuclide inventory calculated in Eq 9 as seen in Figure 3. The system mass transfer coefficient $L$ which details the mass transfer to the liquid-gas interface of an off-gas system should be determined by higher fidelity spatially resolved chemical species distributions in the reactor system (Tano, 2023a).

There are two primary assumptions that are made in this coupled multiphysics methodology that must be made evident. One is that the multiphysics solution is pointwise, or zero dimensional, where the reactor fuel-salt system is homogenized, and the spatial temperature or fuel-salt composition distributions are not accounted for in this analysis. Therefore, this proof-of-concept analysis provided here assumes a system mass transfer coefficient $L$ for the 0-D system rather than calculating it from a spatially resolved thermal-hydraulic solution. Additionally, this methodology assumes that the changes in the overall fuel salt composition due to depletion is a slow process and that the system is always in thermochemical equilibrium. For most nuclides which make up the elements in the molten salt this is a good assumption and the nuclides which are changing rapidly due to radioactive decay make up a negligible fraction of the element or total chemical composition.

For fast changes in the chemical composition, which occur during active redox potential control, a chemical redox kinetics model for corrosion and reduction at interfaces would be required to model the time and spatial effect of this process (Walker and Ji, 2021b; Tano, 2023b). The analysis of active redox potential control provided in this work will be stepwise accounting for the chemical potential before and after the fuel salt composition has been altered.

3.4 Discussion of current MSTDB-TC capability and limitations

Before diving into the results section, it is important to preface the results with a discussion of the database that was used, as well as the current limitations restricting the applicability and scope of this current analysis. The most up to date MSTDB-TC version at the time this work has been completed was MSTDB-TC V2.0, which was used in Thermochimica. It is important to note that the formal process of compiling these self-consistent thermochemical databases is a rigorous, time-consuming process which can become even more difficult as new species, elements, and their respective interactions are added (Ard et al., 2022).

As a result, the current version of the MSTDB only contains 16 elements for both fluoride and chloride salts as seen in Table 3. The respective fluoride and chloride databases differ in significant ways, the most obvious being the base elements of the ionic salts (i.e., Li, Be, F, Na, K, and Ca for fluoride salts, and Li, Na, Mg, Al, Cl, K, and Ca for chloride salts). Other differences are that the fluoride database includes more rare earth element FPs (i.e., La, Ce, and Nd), while the chloride database includes more corrosion product elements (i.e., Cr, Fe, and Ni). Lastly, both the databases include Cs and I. The chemical behavior of these elements is important due to their radiological effect on humans and are therefore important from a source term or safety perspective.

TABLE 3

TABLE 3. MSTDB-TC V2.0 Analysis.

While these databases may be well suited for thermochemistry calculations of fuel salt behavior at the beginning of life of the reactor, they are not yet able to completely capture the role of depletion in altering the redox potential, due to missing key FPs that are expected to interact with the base ion (F or Cl) and form ionic end members in the fuel salt. However, it should be noted that the importance of this effect will depend on the oxidation state and the relative chemical activity of these FPs (Chasanov, 1965; Baes, Jr, 1974). Thus, for low burnup, the current elements available in the MSTDB database are enough to demonstrate the general chemical evolution of MSRs due to depletion.

A few of these important corrosion and fission products currently missing from the MSTDB-TC are also listed in Table 3. Table 3 lists key FPs that buildup in non-negligible amounts after considerable depletion (Zr, Nd, Mo), which are important for determining the redox potential of the fuel salts towards the end of the reactor lifetime. Most important for fluoride salts is the inclusion of corrosion products Cr and Fe in the system since the chemical activity of Cr_(s)-CrF₂ will poise the redox potential after the UF₄/UF₃ ratio becomes too oxidizing as seen in Eq 4; Table 1 (Besmann and Schorne-Pinto, 2021).

It is important to note that, though the behavior of Mo in both salts is still being determined, it is likely insoluble and chemically neutral as a noble metal in fluoride salts during normal fluorine potentials poised by the UF₄/UF₃ couple, whereas it is believed to be soluble with a valence state of +3 for chloride salts during normal chlorine potentials (Thomas and Jerden, 2020). Here a “normal” fluorine potential is one poised by the UF₄-UF₃ couple as seen in Figure 1 where an ideal UF₄/UF₃ ratio is somewhere between 10 and 100 to limit corrosion (Guo et al., 2017) where the fluorine potential of the initial fuel-salt composition is maintained. A normal chlorine potential similarly would be one that is poised by the UCl₄-UCl₃ couple. For a corresponding electrochemical potential plot of Table 1 for chemical species in a molten chloride salt, the reader is advised to review (Guo et al., 2018). Lastly, note that elements involved in oxidation reactions from air or water (i.e., H and O) are currently excluded, and likely play an important role in altering the redox potential during initial or operational air and moisture ingress (Thoma, 1971).

Due to these thermochemical data shortcomings, it is anticipated that the multiphysics depletion-driven thermochemical analysis of MSRs in its current capability will not capture the evolving redox potential of the fuel salt in an absolute sense. Rather these calculations serve as a proof of concept of the multiphysics depletion-driven thermochemistry capability that will be validated against real fuel-salt systems as more vetted thermochemical data is added to the MSTDB-TC. A prime example of this would be to use the chemical data from the MSRE during the ²³⁵U runs (Thoma, 1971); this future work can be conducted once Zr (a primary fuel-salt component in the MSRE) is added to the MSTDB-TC.

4 Results: Depletion-driven thermochemical analysis of molten salt reactors

This section presents some key results obtained using the multiphysics-coupled, depletion-driven thermochemistry scheme detailed in the previous section. First, the specifications of the generic zero-dimensional fluoride and chloride MSRs are defined. The following subsections detail the changing fluorine and chlorine potential of the systems as they undergo depletion. Vapor pressures for chemical species in both systems are also calculated throughout burnup. Finally, active redox chemistry control is modeled, and the resulting changing thermochemical equilibrium of the fuel-salt is calculated.

4.1 Specification of fluoride and chloride zero-dimensional MSR systems

This subsection examines the specifications of two generic zero-dimensional fluoride and chloride salt reactors used in this analysis. These are hypothetical use cases that are leveraged to evaluate possible depletion-driven thermochemical effects during burnup.

Table 4 identifies the specifications used in the depletion-driven thermochemical analysis of these two generic reactor types. The fluoride salt composition was selected to be close to that of the MSRE fuel salt with the notable exception that the Zr component is missing as mentioned earlier (Thoma, 1971). Instead, it is very similar to pure FLiBe (2LiF-BeF₂) salt with the replacement of some beryllium with uranium fuel. Additionally, the ²³⁵U enrichment of the system was set to 33.3% to match the MSRE fuel salt.

TABLE 4

TABLE 4. Generic Reactor Specifications.

The chloride salt composition on the other hand is a chloride eutectic fuel salt originally proposed for fast chloride breeder reactors (Mourogov and Bokov, 2006). Simply put, this is molten table salt with some uranium fuel mixed in. Both the power densities and uranium enrichments are set equal so that the depletion histories can be compared between the two fuel-salt systems (i.e., the amount of uranium fuel consumed in both reactors is the same).

Due to the thermochemical data limitations discussed in the previous section, a low burnup of 2.87 [MWd/Kg-U] is chosen for both reactors. To simplify the analyses, a constant scalar neutron flux is chosen, and one condensed energy group is used to evaluate the mid-burnup micro cross sections of nuclides in the reactor as seen in (Walker et al., 2022; Walker et al., 2023) for both fluoride and chloride systems. Here the Monte Carlo code SERPENT-2 is used to generate the micro cross sections and to collapse the neutron flux into a single energy for both generic reactor systems (Aufiero et al., 2013; Walker et al., 2022).

4.2 Changes in redox potential due to depletion

This subsection investigates changes in the chemical fluorine and chlorine potential along with their associated electrochemical redox potentials due to depletion. It is important to note that these sets of calculations do not include any active redox control measures; so, the expected behavior of the redox potential should increase or become more oxidizing during depletion as observed in the MSRE (Thoma, 1971) and predicted for chloride systems (Chasanov, 1965).

4.2.1 Fluoride fuel salt analysis

The results of the multiphysics depletion-driven thermochemistry analysis of a fluoride fuel salt using the MSTDB-TC V2.0 during a 2.87 [MWd/Kg-U] burnup and 60 days of operation are presented below. Figure 4 showcases the change in fluorine potential and its associated redox potential where Eq 3 sets the standard half-cell potential.

FIGURE 4

FIGURE 4. Change in fluorine potential over 60 days or 2.87 [MWd/Kg-U] burnup.

Here, the fluorine potential is steadily increasing over the first 1.0 [MWd/Kg-U] of burnup. This indicates that the fuel salt is becoming more oxidizing which is the expected outcome of depletion as observed in Figure 1 for the MSRE, where the UF₄/UF₃ measurement of the redox potential is increasing. Although this analysis is not meant to be a direct comparison with the MSRE since the base MSRE fuel salt is not currently available in the MSTDB-TC V2.0, the redox potential is increasing more rapidly during the first 1 [MWd/Kg-U] of burnup compared with the MSRE in Figure 1. This is especially true when the fluorine potential rapidly increases between 1.0 - 1.5 [MWd/Kg-U] of burnup. Therefore, it is safe to say that the current MSTDB-TC V2.0 overestimates the depletion-driven evolving fluorine potential due to a lack of some important reducing fission products (e.g., Zr) for the fluoride fuel salt system.

The apparent step change in fluorine potential observed from 1.0–1.5 [MWd/Kg-U] of burnup is related to a significant shift in the chemical activity of redox buffers as discussed in Section 2. From 0.0–1.0 [MWd/Kg-U] of burnup, the fluorine potential is controlled by the UF₄/UF₃ redox buffer pair. However, the chemical activity of UF₃ is greatly decreasing as a cationic charge deficiency builds up due to depletion, and the UF₃ is forced to shift towards UF₄ as seen in Eq 4. Eventually UF₃ is depleted, and the redox buffer will shift to the chemical activity ratio of Cr_(s)-Cr²⁺. However, in this analysis using the MSTDB-TC V2.0, corrosion/structural products Cr and Fe are not currently available, so the support structure in thermodynamic equilibrium with fuel-salt available to be modeled in this analysis is pure Ni _(s) with a chemical activity of 1.0 since there is no alloy. Therefore, the redox potential shifts from being dictated by the chemical activity ratio of UF₄/UF₃ to the new redox buffer Ni_(s)-Ni²⁺ in this analysis as seen in Eq 10

${\mathit{N}\mathit{i}}_{\left(\mathit{s}\right)}^{\mathbf{0}}+\mathbf{2}\mathit{U}{\mathit{F}}_{\mathbf{4}}\leftrightarrow \mathbf{2}\mathit{U}{\mathit{F}}_{\mathbf{3}}+\mathit{N}\mathit{i}{\mathit{F}}_{\mathbf{2}}(10)$

This shift in redox potential buffers can occur so rapidly in the fluoride fuel salt composition chosen since the relative chemical activity of Uranium is so small in the base fuel salt as seen in Table 4 (i.e., 0.9 [mol%]). The missing chemical activities of key FPs may also be accentuated by the small amount of U contained in the fluoride fuel salt system when compared to the chloride fuel salt system analyzed in the following subsection. From 1.5 [MWd/Kg-U] of burnup onwards, the Ni_(s)-Ni²⁺ redox buffer poises the system and the fluoride fuel salt is in a state of active corrosion where NiF₂ attempts to replace the cationic charge deficiency generated through depletion (i.e., it tries to take up the excess fluorine potential generated through the generation of less thermochemically stable FPs).

Whereas this is a technically accurate description of the fluoride fuel salt system given the current thermochemical database, it is not necessarily representative of a real MSR system where the structural material is an alloy instead of pure Ni_(s). As seen in Table 1; Eq 4, a more physical scenario would be the oxidation of Cr_(s) from the alloy materials into the fuel salt (Besmann and Schorne-Pinto, 2021). Therefore, the resulting redox potential calculated here is similarly too high due to missing thermochemical data for Cr and Fe in fluoride salts and should be lower being controlled by the Cr_(s)-Cr²⁺ buffer instead.

Ongoing development of the MSTDB-TC will likely rectify this issue in the near term (McMurray et al., 2021). The addition of ZrF₄ (a base component of the MSRE fuel salt) and corrosion products CrF₂ and FeF₂ will enable the benchmarking of this multiphysics depletion-driven thermochemistry capability against the UF₄/UF₃ data observed during the MSRE (Thoma, 1971).

Finally, the changing chemical potentials of a select number of elements in the fluoride fuel salt due to depletion are plotted in Figure 5. As seen in red, the fluorine potential is the most negative chemical potential in the system as expected. As the fuel salt is depleted, the chemical potential of all cations (i.e., Li, Be, U, Pu, Cs) decrease to adjust for the increase of the chemical potential of the anions (F and I). The chemical potential of uranium decreases the most compared with other cations since it is being depleted by fission and is the initial redox buffer until it no longer poises the system, and rapidly decreases as fluorine looks for a more positive redox buffer to consume. Lastly, it is important to note that as the redox potential increases, so does the iodine potential which is an important consideration for vapor pressures and source term calculations as seen in the following subsections.

FIGURE 5

FIGURE 5. Change in chemical potentials in fluoride fuel salt over 60 days or 2.87 [MWd/Kg-U] burnup.

4.2.2 Chloride fuel salt analysis

The results of the multiphysics depletion-driven thermochemistry analysis of a chloride fuel salt using the MSTDB-TC V2.0 during a 2.87 [MWd/Kg-U] burnup and 60 days of operation are presented below. Figure 6 showcases the change in chlorine potential and its associated redox potential where the standard half-cell potential is defined by Eq 11

${\mathit{C}\mathit{l}}_{\mathbf{2}\left(\mathit{g}\right)}+\mathbf{2}{\mathit{e}}^{-}\leftrightarrow \mathbf{2}{\mathit{C}\mathit{l}}^{-},{\mathit{E}}^{\mathit{O}}=\mathbf{0}\left[\mathit{V}\right](11)$

FIGURE 6

FIGURE 6. Change in chlorine potential over 60 days or 2.87 [MWd/Kg-U] burnup.

Like the fluoride fuel salt, the redox potential gradually increases, meaning the fuel salt is becoming more oxidizing as anticipated. Unlike the fluoride fuel salt however, there is no acceleration of redox potential occurring meaning the redox buffer in place, likely UCl₄/UCl₃, is dictating the chlorine potential in the fuel salt. Here uranium is a much larger fuel salt constituent compared to the fluoride fuel salt as seen in Table 4 (i.e., 33.3 [mol%]) and therefore the chemical activities of the UCl₄/UCl₃ buffer ratio are more stable. Additionally, this analysis using the current thermochemical database would suggest that the cationic charge deficiency generated from the depletion of UCl₃ is less than the cationic charge deficiency that occurs from the depletion of UF₄.

Most of the key fission and activation products currently accounted for in the MSTDB-TC V2.0 for fluorides form trifluorides or MF₃ (e.g., CeF₃, NdF₃ PuF₃). The fission of UF₄ fuel therefore rapidly yields a significant cationic charge deficiency. Whereas for the chloride salt, the primary fuel is UCl_3, and the key fission and activation products currently accounted for in the MSTDB-TC V2.0 for chlorides also form trichlorides (e.g., CeCl₃ and PuCl₃). As a result, the rate of cationic charge deficiency is less pronounced for chlorides than fluorides given the current thermochemical database. In other words, based on the current available thermochemical data for fission and activation products included in MSTDB-TC V2.0, the missing charge from UCl₃ consumption (cationic charge of 3+) is less pronounced than the missing charge from UF₄ consumption (cationic charge of 4+).

Since the potential is increasing, there is not enough soluble material for the chlorine to dissolve. Like the oxidizing fluoride fuel salt example, a realistic chlorine fuel salt reactor in thermodynamic equilibrium with the structural material holding the salt will begin to dissolve the structural materials at the wall interfaces forming corrosion products CrCl₂ and FeCl₂. In this situation, Cr is available for chlorides in the MSTDB-TC V2.0. Like the fluoride fuel salt example where pure Ni_(s) was used as the structural material, here a pure Cr_(s) structural material with a chemical activity of 1.0 is used instead. Future work will refine this interaction by correctly accounting for the chemical activity of elements in the secondary phase model of the alloy (Besmann and Schorne-Pinto, 2021).

However, since the redox potential of CrCl₂ is similar to UCl₃ (i.e., it is the first corrosion product to form in significant amounts as seen in the electrochemical potential plot for chlorides in (Guo et al., 2018)) this is not a terrible approximation. Therefore, the primary and end time effect of corrosion on controlling the redox potential of the fuel salt can be modeled. That being said, the rate of this corrosion is too fast, since the chemical activity of pure Cr_(s) is far greater than it would be in an alloy, and the rate of chemical activity of Cr_(s) diffusing to the grain boundaries would require a redox kinetics model instead of a thermochemical equilibrium calculation as used in this analysis.

In this analysis, the initial chlorine potential of the chloride fuel salt readily corrodes the pure Cr_(s) structural material. This effect also contributes to the diminished change in the redox potential during burnup since the chemical activity of the Cr_(s)-Cr²⁺ couple entering the fuel salt is significant to nearly replace the cationic charge deficiency generated through less thermochemically stable FPs.

Lastly, the changing chemical potentials of a select number of elements in the chloride fuel salt due to depletion are plotted in Figure 7. As seen in green, the chlorine potential is the quite negative in the system as expected. Of particular interest here, the iodine chemical potential is more negative in the chloride fuel salt than it is in the fluoride fuel salt when comparing Figures 5, 7. Here the decrease in the uranium chemical potential is very minimal compared with the fluoride fuel salt, since the relative chemical activity of uranium in the chloride fuel salt is not changing as rapidly as in the fluoride fuel salt. Lastly, it appears that the chemical potential of cesium, which is important in determining vapor pressures covered in the next section, just so slightly increases over depletion. This is likely an effect of corrosion products entering the fuel salt, since the general behavior of the chemical potential of cations is to go down as the aggregate anion potential increases.

FIGURE 7

FIGURE 7. Change in chemical potentials in chloride fuel salt over 60 days or 2.87 [MWd/Kg-U] burnup.

4.3 Changes in vapor pressures of chemical species due to depletion

This subsection focuses on the consequences that a changing redox potential due to depletion will have on the vapor pressure of chemical species in the fuel salts. Both fluoride and chloride salts are analyzed based upon the changing redox potential observed in the previous section.

4.3.1 Fluoride fuel salt analysis

The results for the vapor pressures in the fluoride fuel salt without active redox control are shown in Figure 8. Here the highest vapor pressures are of the base salt constituents (i.e., F, Li, Be, except for U). This is because these elements have the highest partial pressures in the system. The general trend for vapor pressures of minor fission and transmutation product elements is that they increase as their chemical activity builds up in the fluoride fuel salt.

FIGURE 8

FIGURE 8. Changes in element vapor pressures in the fluoride fuel salt over 60 days or 2.87 [MWd/Kg-U] burnup.

One key item of note for accident source term analysis is that the vapor pressure of iodine in fluoride fuel salts significantly increases when the fluorine potential increases corresponding to Figures 4, 5. Additionally, the vapor pressure of Pu in the fluoride fuel salt shows how extremely stable this element is in fluoride fuel salts also evidenced by its chemical potential in Figure 5.

4.3.2 Chloride fuel salt analysis

The results for the vapor pressures in the chloride fuel salt without active redox control are shown below in Figure 9. Like the fluoride fuel salt analysis, the chloride salt remains quite stable throughout the burnup period as seen in Figure 7, although the redox potential is ever so slightly increasing over the reactor operation period.

FIGURE 9

FIGURE 9. Changes in element vapor pressures in the chloride fuel salt over 60 days or 2.87 [MWd/Kg-U] burnup.

As seen previously in Figure 8, the vapor pressures of elements are dictated by their relative concentration in the fuel salt, so base elements Cl, Na, and U have the highest vapor pressures in Figure 9. However, as the fuel salt becomes more oxidizing due to the rising redox potential, the rise in uranium volatility despite its declining chemical activity can be observed. In contrast to the fluoride salt, most of the elements are typically less stable in the chloride salt when comparing Figures 8, 9 due to the difference in the higher electronegativity of fluorine vs. chlorine. One last interesting item to note is that iodine is apparently more stable in chlorides than fluorides, whereas cesium has roughly the same stability in both; this is an important understanding when considering chemical speciation during accident scenarios.

4.4 Depletion-driven thermochemistry with active redox potential control

Lastly, the following analyses model scenarios where the chemistry in the MSR system is being actively controlled during operation. As mentioned previously, if the molten salt becomes more oxidizing over time as expected and seen during the MSRE, the redox potential is actively controlled by adding a reducing metal to the molten salt.

4.4.1 Fluoride fuel salt analysis

As seen in the MSRE and in the results of the previous analyses, the fluorine potential increases due to depletion. To maintain a well-behaved reactor and minimize the acceleration of corrosion, the MSRE would periodically reduce the redox potential of the salt by adding a reducing metal to the fuel-salt, typically Be as seen in Eq. 6 (Thoma, 1971; Kelleher et al., 2016). There are several methods for controlling the redox potential as identified here (Zhang et al., 2018). Actively controlling the redox potential through beryllium addition can be modeled using the new multiphysics depletion-driven thermochemistry capability and is illustrated in Figure 10.

FIGURE 10

FIGURE 10. Change in fluorine potential with active redox potential control.

Here the $L$ term in Eq 8 is utilized to feed beryllium into Eq 7 thereby controlling the redox potential of the system. As shown in the plot, the beryllium additions can control the redox potential as depletion proceeds. The redox potential looks much smoother when comparing against Figure 1 from the MSRE, because the simulation assumes a constant, continuous rate of beryllium addition whereas the MSRE gradationally added beryllium to reduce the redox potential.

It should be stressed that Figure 10 is not modeling a beryllium rod constantly being dissolved in the fuel salt. This type of analysis would require a redox kinetics model; subsequently if a large enough amount of beryllium was constantly available to dissolve into the salt, it would result in the fuel salt eventually being over-reduced and thereby precipitate solid uranium fuel as seen in Table 1 and can be inferred from the fast reduction that occurs in Figure 1. Instead, Figure 10 is modeling a discrete, constant rate of beryllium addition. A controlled specific mass of the material is being introduced to the system. In an ideal situation, a constant in situ redox potential measurement would be implemented, with a controller constantly monitoring the potential and raising or lowering a beryllium rod into the salt to balance the chemical state as depletion proceeds.

4.4.2 Chloride fuel salt analysis

Like the fluoride fuel salt analysis, the chloride fuel salt can also be controlled through reducing metal additions. The results for sodium additions to the chloride fuel salt can be seen in Figure 11. Here, sodium additions can serve the same purpose as the beryllium additions in the fluoride fuel salt. The reducing metal can help control or reduce the redox potential of the fuel salt during depletion. This reaction with a more thermochemically stable metal would then be dominant in absorbing excess chemical potential and would thereby limit the corrosion that the MSR system may experience. The discrete constant rates of sodium additions are shown in Figure 11 In both fluoride and chloride systems, the possibility of over-reducing the salt as seen in the blue circle marked lines in Figures 10, 11 can be a valid concern depending on the choice of reducing metal or redox control method that is used (Zhang et al., 2018; Newton et al., 2020).

FIGURE 11

FIGURE 11. Change in chlorine potential with active redox potential control.

4.5 Future improvements to the model

This research provides an initial starting point to build a complex multiphysics framework for species tracking simulation inside MSRs. This first step consisted of incorporating depletion-driven thermochemistry within the NEAMS framework. This effort will be steadily and continuously enhanced to incorporate higher levels of fidelity. Logical next steps to enhance this capability include coupling depletion-driven thermochemistry with thermal hydraulics so that the impact of advection can be captured as well as provide a multi-dimensional resolution of the system.

Parallel work has already laid the foundation for spatially informed depletion in MSR systems using a quasistatic approach to account for fuel circulation. This was performed using Griffin and the NEAMS coarse mesh thermal hydraulics code Pronghorn (Tano, 2023a). Preliminary work is underway to merge this new spatial depletion capability with the depletion-driven thermochemistry shown here. The end goal will be to fully couple Griffin, Thermochimica, and Pronghorn to leverage 3D temperatures/velocity fields to inform concentration gradients in the thermochemical equilibrium of MSR systems and more accurately determine the $f_{stream,Z}^A\left(t\right)$ term in Eq 7.

Additionally, redox corrosion kinetics plan to be incorporated in order to model short time frames where the molten salt is not in thermodynamic equilibrium represented by the $f_{chem,Z}^A\left(t\right)$ term in Eq 7. Uncertainty quantification and sensitivity studies can also be performed to see how uncertainties in nuclear data propagate changes in MSR thermochemistry throughout depletion. The comprehensive capability can also be applied towards performing material accountancy simulations that would be useful for safeguards and nonproliferation analyses.

The spatially informed depletion-driven thermochemistry model is expected be critical for understanding chemical species transport, deposition, dissolution, and corrosion during normal operation and accident scenarios. In this manner, multi-dimensional depletion-driven thermochemistry analyses of MSRs can be conducted to assess the space- and time-resolved chemistry of MSR systems.

5 Conclusion

A new multiphysics depletion-driven thermochemistry capability using NEAMS tools was developed and applied to analyze the role of depletion in altering the chemistry and multiphysics behavior of MSRs. The individual physics and coupled multiphysics methodology were introduced and discussed. While the capability in its current form is unable to provide absolute estimates for the redox potential in MSR systems (namely, due to database and code-coupling limitations), it can still generate informative results to model the chemical evaluation of both fluoride and chloride-based MSRs.

The phenomenon of depletion-driven thermochemistry is an essential aspect of MSR behavior and a critical starting point in analyzing these multiphysics systems. Specifically, this multiphysics phenomena dictates changes in corrosion, chemical speciation, and possible accident source terms. Consequentially, depletion-driven thermochemistry must be predicted for the reactor lifetime and subsequently measured and controlled by MSR instrumentation.

The result of this current work advances the state of the art of depletion-driven thermochemistry analysis by building a multiphysics framework using MOOSE-based NEAMS tools. The impact of this phenomena using the most up-to-date thermochemical data can be evaluated to a limited level of fidelity. This is expected to provide the foundation for more comprehensive models capturing more physics that impact species transport in MSRs.

More specifically, redox potential changes due to depletion can be modeled using this depletion-driven thermochemistry multiphysics capability over the reactor lifetime given the proper thermochemical data. This will allow upcoming MSR industry partners to accurately predict fuel salt behavior as depletion alters the chemistry of the system. Additionally, this work demonstrates how redox control mechanisms can be simulated over long periods of time. This is expected to help inform future instrumentation design for effective redox control of MSR systems. Lastly, this new capability can account for the change in element vapor pressures in the fuel salt as a function of burnup (and therefore redox potential). This is expected to be critical to performing source term calculations. Overall, this work is envisaged to lay the foundation for important design and licensing considerations of these multiphysics driven MSR systems.

Data availability statement

The original contributions presented in the study are included in the article/Supplementary material further inquiries can be directed to the corresponding author.

Author contributions

SW, MT, and AA-J contributed to conception and design of the study. OC contributed to developing new Griffin capabilities for multi-region microscopic depletion with nuclide feeds and SW contributed to testing and verifying these capabilities. SW, MT, and AA-J contributed to developing methodology approach for depletion-driven thermochemistry. SW and MT implemented the multiphysics coupling of Griffin and Thermochimica. MT enhanced coupling through Picard iterations. SW tested and verified capability of coupled multiphysics system. SW performed depletion-driven thermochemical analysis of MSR systems. SW wrote the first draft of manuscript. All authors contributed to the article and approved the submitted version.

Funding

This manuscript was authored at Idaho National Laboratory by Battelle Energy Alliance LLC under contract no. DE-AC07-05ID14517 with the U.S. Department of Energy (DOE). The funding agencies for this work are the DOE Office of Nuclear Energy programs: the Molten Salt Reactor campaign and the Nuclear Energy Advanced Modeling and Simulation (NEAMS) program.

Acknowledgments

This work was prepared for the U.S. DOE through a collaboration between the Molten Salt Reactor (MSR) campaign and the Nuclear Energy Advanced Modeling and Simulation (NEAMS) program. It made use of the resources of the High Performance Computing Center at Idaho National Laboratory, which is supported by the Office of Nuclear Energy of the U.S. Department of Energy and the Nuclear Science User Facilities under contract no. DE-AC07-05ID14517. The U.S. Government retains for itself, and others acting on its behalf, a paid-up nonexclusive, irrevocable worldwide license in said article to reproduce, prepare derivative works, distribute copies to the public, and perform publicly and display publicly, by or on behalf of the Government. The DOE will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan: energy. gov/downloads/doe-public-access-plan. The authors would also like to thank Yaqi Wang, and Javier Ortensi of Idaho National Laboratory for their support with implementing modifications to the CRAM solver in Griffin. As well as Jacob Yingling for help with integrating the new version of MSTDB with Thermochimica, and for providing some helpful insight in understanding how Thermochimica processes results of the database. Lastly, the authors would like to thank Max Poschmann and Markus Piro from the University of Toronto, for their help with setting up the Thermochimica evaluations.

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s note

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Nomenclature