Depletion Calculations for an Integral Small Molten Salt Reactor With Serpent

The molten salt reactor (MSR) concept is one of the six Generation IV designs recommended by the Generation IV International Forum [1] as feasible for providing clean, safe, sustainable, and economical energy supplies to the world's population.

A typical integral MSR features a completely sealed reactor vessel containing a reactor core, primary pumps, and heat exchanger. Driven by primary pumps, the molten salt fuel flows through the core channels, generating and, as primary coolant, transporting fission heat. The heat generated in the fuel is transferred to a secondary coolant through an intermediate heat exchanger, and then through another heat exchanger to the power conversion system. The homogenous liquid fuel allows for online addition of fuel with variable composition to compensate for fuel burnup and growth of fission products (FPs). However, the circulating fluid fuel makes it difficult to model an MSR for reactor physics analysis as the existing reactor physics methods have been almost exclusively designed for systems with immobile, solid fuels. The purpose of this study is to explore a fuel depletion calculation method for simulating circulating liquid fuel systems.

For reactor physics analysis, fuel depletion (or burnup) calculations are primarily required to update the fuel material composition in a core model as burnup advances. Fuel depletion calculations are also required to predict radionuclide inventories and radio activity in the fuel, not only for radiological assessment of an operating reactor but also for the postirradiation waste characterization of spent fuel.

Serpent [2,3] is a three-dimensional Monte Carlo transport code with burnup capability developed at the VTT Technical Research Center of Finland. Serpent has become a code of choice for the modeling of different types of nuclear systems and has been verified by comparison with the MCNP (Monte Carlo N-Particle) code [4] in neutronics physics modeling based on the same physical laws. As a result, for similar reactor models, Serpent is able to reproduce MCNP neutronics results within statistical accuracy [3]. Furthermore, a number of depletion calculation models [5] have been developed in Serpent to deal with mass flows (i.e., material addition or separation), which may be used in simulating fluid-fuel flow in the MSR.

In this paper, fuel depletion calculations are performed using Serpent version 2.1.32 for a small modular fluoride-based MSR (sm-FMSR) that features circulating fuel and online fuel addition. Both step fueling (SF) and continuous fueling (CF) schemes are simulated. SF involves explicitly incrementing the fuel volume in the model and manually modifying the fuel composition between Serpent calculation cycles. CF utilizes the Serpent mass flow features combined with manual arithmetic calculations for adjustment of the Serpent built-in settings. In addition, the option of removing self-separated volatile gases or noble metals is investigated.

The sm-FMSR modeled in this paper is an integral channel type of molten salt reactor [6,7] that has a nominal thermal power of 400 MW and requires regular fuel addition to compensate for fuel burnup and FP accumulation. The modeled reactor may or may not retain the fission, activation, transmutation, and decay products³ inside its closed vessel for the whole operation cycle (up to seven years).

The sm-FMSR model is illustrated in Fig. 1. The reactor core consists of graphite moderator blocks with flow channels for the molten salt fuel. Core graphite blocks with the fuel flow channels are explicitly modeled. Other structures including the fuel bodies beyond the channels are approximated. The fuel contains low-enriched uranium fluoride (UF₄) diluted with sodium and potassium fluorides (NaF–KF) as a homogeneous mixture, which constitutes both fuel and primary coolant. Driven by the integral primary pump at the top, the liquid fuel flows upward through the core (heating up by fission), then through the heat exchanger (transferring heat to the secondary coolant), and downward through the core peripheral down-comer to re-enter the core from the bottom.

The reactor is initially loaded with a minimal volume of the salt–fuel mixture having a sufficient fissile content (²³⁵U-enriched) to provide reactivity for reactor operation at a nominal power at the maximum allowable temperature of the primary coolant (fuel). Short-term reactivity is controlled by a negative temperature feedback of reactivity, which is inherent to the molten salt fuel and the graphite moderator. This is achieved by using secondary pumps to alter the circulation flowrate, which changes the rate of heat removal and, consequently, changes the temperature of the primary coolant (fuel) and the graphite moderator. Long-term reactivity loss due to fuel depletion and FP accumulation is compensated for by online fuel addition. The fuel addition must have a larger fissile content (e.g., higher enrichment) than the initial fuel.

The main design parameters of the sm-FMSR are given in Table 1. A mixture of NaF–KF–UF₄ with 2.3–2.6 wt.% ²³⁵U/U (further shortened to “%” when referring to enrichment) is used for initial loading, and a mixture of the same molten salt composition and density (see Table 1), but with a higher enrichment of 5.5%, is used for online fueling.

The depletion model uses a homogenous fuel salt of a fixed, average temperature in all fuel channels. k_eff is maintained around 1.0 for depletion calculations. As k_eff indicates the level of fuel loading in the reactor at a given moment, it is also used to control fueling of the reactor.

where $m$ is the incremental index of calculation step at time $tm$ (days);$n$ is the index of fueling cycle (i.e., fuel addition step), which is incremented when the calculation k_eff drops to some lower bound, i.e., k_eff → k_min at $tn=tm$⁠;$Csys$ is the array of nuclide concentrations (g/cm³) in the system fuel (total load) of volume $Vsys$ (cm³); $Ccor$ is the array of nuclide concentrations (g/cm³) in the burnup fuel in a core of constant volume $Vcor$ (cm³);$Cadd$ is the array of nuclide concentrations (g/cm³) in the addition fuel of volume $ΔV$ (cm³);$SCR$ is the system-to-core fuel volume ratio;$δ$ is the Kronecker's delta function: $δ(0)=1; δ(≠0)=0$⁠.

For each calculation step (⁠$m$⁠), Serpent depletes only the fuel-in-core of constant volume (⁠$Vcor$⁠) from the start composition (⁠$Csys$⁠) to result in the end composition (⁠$Ccor$⁠), and then Serpent stops. The start fuel composition for the next step is manually evaluated using Eq. (1) and is then “reloaded” into the model to restart the Serpent depletion calculation. As the fuel-in-core is over-depleted over each step, i.e., k_eff drops to some lower bound of its range, fuel addition is required. This simplified approach, referred to as “manual mixing,” requires rather small time steps Δt (e.g., of the order of minutes), which would be impractical in a simulation of the whole reactor service lifespan of many years.

where $C0|nsys$ is the composition of the fuel load (⁠$Vnsys$⁠) at the start of depletion cycle n; $Cend|n−1sys$ is the composition of the previous fuel load (⁠$Vn−1sys$⁠) at the end of depletion cycle n−1.

Index m is not explicitly present in Eq. (2), being used only at the start, or at step zero (0|n), and at the last step of the previous depletion cycle, (end|n − 1), at the same time: $tm=tend|n−1=t0|n$⁠.

It should be noted that Serpent (or any neutronics solver) uses only nuclide concentrations in each material for the solution of a yet-to-normalize neutron flux, and processes nuclide cross section for each burnable material based upon it. With the modification, Serpent would solve the Bateman equations [8] using the correct flux spectrum and nuclide cross section data, and then normalize the depletion calculation results to a power contributed by the whole fuel in the circulation system. This modification also allows for larger calculation steps to be used by Serpent. However, Serpent must still be stopped for manual mixing with fuel addition using Eq. (2) when k_eff drops close to the lower bound.

The main disadvantage of the SF method is that it requires too many fuel addition steps (⁠$n$⁠) (requiring Serpent to stop and restart each time) for a realistic range of reactivity change.

Serpent has flow features [5] that allow for addition and removal of fuel components to and from the fuel-in-core in a continuous manner.

where $λoutρsysVsys$ is the mass rate (g/s) of fuel removal. By setting $Vsrc=Vsys$ and using the same density ρ (g/cm³) for both initial and top-up fuels (⁠$ρsys=ρsrc=ρ)$⁠, both “addfuel” and “overflow” can use the same flow constant (⁠$λout=λadd=λ).$ Note that Serpent incorporates these flow constants into the Bateman equation of each nuclide (iso) included in the flow as an additional source term (⁠$+λaddNisosrc$⁠) or decay term (⁠$−λoutNiso$⁠), or both, for the solution. There is no change in volume of any material involved.

where $P0$ is the actual reactor power (W) contributed by the whole actual fuel load in the reactor (i.e., $Vmsys$ at time t_m);$Pm$ is the power (W) contributed by only the fuel available in the model (i.e., of fixed $Vsys$ at all times); and norm is the fraction of the power from the previous time step that can be specified for normalization of the Serpent depletion results.

The parameters from Eqs. (3)–(6) must be manually evaluated and are applicable only when k_eff remains within an acceptable range. As the consumption of ²³⁵U per unit of thermal output (g/MWd) gradually decreases due to buildup of fissile plutonium (^fisPu = ²³⁹Pu + ²⁴¹Pu), the fuel addition constant (⁠$λ$⁠) should also decrease accordingly. As Serpent does not have an option to change that constant, the Serpent calculation must be stopped to update to a new flow constant $λ$ (in order to keep k_eff in range) and corresponding power normalization factors for the next period; this resembles the fuel depletion cycle in step fueling. However, CF requires only a few of Serpent depletion cycles as opposed to the tens of cycles required by SF.

In addition, an optional mass flow “separate” can be used to remove any individual nuclides at individual rates from the fuel-in-core. These nuclides can be volatile gases or noble metals, which self-separate from the circulating fluid in reality.

The fuel and coolant mixture, NaF–KF–UF₄ (21–44–35 mol. %), with an enrichment of 2.6% is initially loaded in the reactor model to a total volume of V^sys = 7.31 m³ (Table 1). This is supposed to provide enough reactivity for an operation duration (or fueling cycle) between approximately 110 and 120 full power days (FPD, equivalent to 400 MWd). Both fuel and moderator temperatures are set to 925 K, an average of the nominal core inlet and outlet temperatures. The Xe poison is set to an equilibrium value to avoid large changes of k_eff at the start of each depletion cycle. Fifteen million neutron histories per calculation step are used for Serpent to produce values of k_eff within 0.2 mk (reactivity × 1000) of statistical uncertainty and core total power and flux within 0.1%.

Illustrated in Fig. 3 are changes in fuel load in the system, Serpent-calculated k_eff (Fig. 3, upper), and fissile concentrations (Fig. 3, lower) as functions of reactor thermal output (i.e., FPD × 400 MWd) over the reactor fuel lifetime, using SF scheme.

Serpent-calculated k_eff values are used to control the fueling cycles of the reactor. Each cycle of Serpent depletion calculation ends when k_eff drops to approximately −11±0.5 mk, which is within a pre-assumed loading reactivity range of 25 mk that is compensable by temperature feedback [9]. In each fueling, a top-up fuel with a volume of 10% of the initial load (i.e., ΔV = 0.731 m³) is added to the burnup fuel load. The top-up fuel has the same composition and density as the initial fuel but a higher enrichment of 5.5% (Table 1). The depleted fuel is manually mixed with the fresh top-up fuel using Eq. (2), and the mixture of the combined volume is reloaded to start the next depletion cycle.

As seen from Fig. 3, the concentration of ²³⁵U in fuel decreases during each depletion cycle, but increases to a higher level than at the start of the previous cycle after mixing with the top-up fuel. In contrast, the concentration of plutonium isotopes increases during depletion (due to generation from ²³⁸U captures) but decreases when mixed with the top-up fuel (due to dilution in the increasing volume).

In general, the fissile concentrations (as well as fissile contents) in the burnup fuel increase with the reactor thermal output, and it appears that only the amount of ²³⁵U equal to (C^add,U235 − C^sys,U235)ΔV in the top-up fuel is useful in further providing both fission energy production and compensation for FP poisons. The rest would be added to the spent fuel inventory unless it is to be reused.

Continuous fueling with fission product accumulation fueling scheme simulates continuous addition of top-up fuel to the system with all fission products stay in the system. Serpent flows “addfuel” and “overflow” are turned on and the flow “separate” is off (see Fig. 2).

The reactor is initially loaded with a salt mixture of the same molar composition and density as that of the step fueling, but the ²³⁵U enrichment is reduced from 2.6% to 2.5%. The top-up fuel (5.5% enrichment) is virtually stored in an implicit source reservoir of V^src, which is set to V^src = V^sys = 7.31 m³. The volume for the removal of fuel V^out is arbitrary as nothing going out is of interest in the simulation. Xe is set to vary as the result of depletion calculations.

The initial flow constant (λ) for both “addflow” and “overflow” is 8.5 × 10⁻⁹ s⁻¹ in order to keep k_eff within ±3 mk for the first Serpent depletion cycle of 700 days. It is reduced to 5.2 × 10⁻⁹ s⁻¹ for the second cycle of 1000 days and, finally, to 3.3 × 10⁻⁹ s⁻¹ for the third and last cycle. Burnup steps are of 25 days each, but a few smaller steps (1 to 3 days) are used for the first few steps of each cycle (for catching FP poisons). The actual time-dependent volume of the fuel load at each step is precalculated using Eq. (4) and the normalization power is manually adjusted accordingly using Eq. (5).

Figure 4 presents the changes in fuel loading, Serpent-calculated k_eff, and fissile concentrations versus the reactor thermal output (in FPD × 400 MWd) for the case of continuous fueling of sm-FMSR with FP accumulation.

With a given “addflow” constant (λ) during each depletion cycle, the actual fuel load increases exponentially according to Eq. (4). Therefore, for each Serpent depletion cycle, a constant λ is chosen such that k_eff decreases first to above −3 mk (approximately), and then changes its course to increase to below +3 mk (approx.), that is, k_eff changes within a range of ±3 mk. An exception occurs in the first Serpent depletion cycle (with the initial load of fresh, poison-free fuel) in which k_eff first drops to the minimum during the first four days due to FP poisons, mostly from Xe-135 buildup to equilibrium (as expected), but it then quickly increases to a peak depending on the fuel addition rate (due to the combined ²³⁵U and initially generated ²³⁹Pu exceeding the fissile depletion) before following the anticipated course. In total, λ value is required to be reset three times, or three cycles of Serpent depletion calculations are needed, for the seven-year life cycle of the modeled sm-FMSR.

The modeled sm-FMSR is not considered to have the capability of reprocessing the irradiated fuel on site; thus, all FP are accumulated with the irradiated fuel in the reactor vessel during the whole fuel cycle. However, a number of FP elements are by nature insoluble in the fuel salt mixture, and are expected to self-separate from the circulating fuel at different rates depending on their mechanical properties.

With CF maintained by flows “addfuel” and “overflow,” the third mass flow, “separate,” is turned on to release the insoluble FP elements at their removal cycle times as provided in Ref. [9]. The volatile gases (Xe, Kr, Ar, Ne, and He) and noble metals (Se, Nb, Mo, Tc, Ru, Rh, Pd, Ag, Sb, and Te) have a full removal time of 20 s (or λ = 0.05 s⁻¹), and seminoble metals (Zr, Cd, In, and Sn) have a full removal time of 200 d (λ = 5.8 × 10⁻⁸ s⁻¹).

Figure 5 presents the changes in the fuel load and the Serpent-calculated k_eff and fissile concentrations versus the reactor thermal output (in FPD × 400 MWd) for the case of CFS, with separation of volatile gases, noble metals, and seminoble metals from the circulating burnup fuel.

When Xe and some of the other strong neutron absorbers are released from the fuel, a considerable reactivity worth held by FP poisons is freed, thus, the enrichment of the initial fuel can be reduced from 2.5% in CFA to 2.3%. In addition, the fuel addition constant (i.e., λ used in both “addfuel” and “overflow”) is reduced from 8.5 × 10⁻⁹, 5.2 × 10⁻⁹, and 3.3 × 10⁻⁹ s⁻¹ in CFA to 6.5 × 10⁻⁹, 4.5 × 10⁻⁹, and 3.0 × 10⁻⁹ s⁻¹ for three depletion cycles of approximately the same duration, respectively.

For the three fueling schemes (SF, CFA, and CFS) modeled in Serpent depletion calculations, Fig. 6 provides a comparison of changes in fissile loads (kg of ²³⁵U and ^fisPu) during the seven-year life cycle of the modeled sm-FMSR core unit, and Table 2 summarizes their values at the end of life (EOL).

Fissile plutonium (^fisPu = ²³⁹Pu + ²⁴¹Pu) buildups (total generation less removal) are similar in all fueling schemes because plutonium generation largely depends on reactor power and operation time, which are the same for all schemes. However, the removal of ^fisPu at a given time slightly depends on the consumed ²³⁵U (i.e., the initial amount plus the added amount and minus the EOL amount) as all the fissile materials (both ²³⁵U and ^fisPu) contribute to the total reactor power or fuel burnup. The more ²³⁵U consumed, the less removal of ^fisPu, resulting in a larger buildup of ^fisPu. As shown in Table 2, the CFA has the largest amount of ^fisPu at EOL (301 kg), which is slightly larger than with SF (294 kg), and CFS has the smallest amount of ^fisPu at EOL (268 kg). The ^fisPu buildups at EOL are consistent with the consumed ²³⁵U in CFA, SF, and CFS of 905, 878, and 815 kg, respectively. The total amount of fuel addition is 2.36, 2.20, and 1.74 times the initial fuel load, V^sys, for CFA, SF, and CFS, respectively.

²³⁵U not only dominates the fission energy produced but also provides compensation for FP poisons (drop in k_eff). Although the total reactor power output is the same in all fueling schemes, the k_eff with SF in this study is allowed to reach k_eff ≈ ±11 mk (Fig. 3; cf. k_eff > ±3 mk in CFA or CFS). As a result, the ²³⁵U consumption as well as fuel addition are lower in SF than in CFA. Both CFA and SF would result in similar fissile contents if k_eff of SF were maintained in the same range (i.e., ±3 mk) of CF, and would result in similar FP inventory as well.

For CFS, the strong neutron absorbing FP elements such as Xe, Kr, or Cd are removed from the system, and the remaining FP elements do not require as much ²³⁵U for compensation as required in CFA, therefore, a significantly smaller amount of ²³⁵U is required in CFS than in other fueling schemes. Fuel saving due to self-separation of insoluble FPs is an advantageous feature for the modeled sm-FMSR.

However, fissile content in the modeled sm-FMSR increases with the reactor power output (Fig. 6) in all three fueling schemes, i.e., there is a larger amount of ²³⁵U and fissile plutonium in the spent fuel than that in the initial fuel loading. This offers a strong motivation to recover and reuse of the spent fuel in integral channel type of sm-FMSR.

In the sm-FMSR model, k_eff values are used for control of the fueling process and provide a measure of fuel loading. It is believed that the sm-FMSR is able to sustain criticality within a range of approximately 25 mk of the calculated k_eff by changing the fuel and moderator temperatures [9]. However, it is a common practice to maintain k_eff to within 3–4 mk to minimize dependency on the calculated k_eff. With Serpent, a difference of 10 mk between two similar cases would result in a ∼3% difference in the final quantities of ²³⁵U.

CFA or CFS requires significantly fewer depletion cycles (three cycles; Fig. 4 or Fig. 5) in Serpent calculations than SF (22 cycles; Fig. 3) to keep k_eff in range over the seven-year service life. However, with the actual fuel volume explicitly set, SF allows for the generation of all results within a Serpent run. For CFA or CFS, the burnup fuel volumes at all steps must be precalculated and kept outside the Serpent environment (e.g., in a spreadsheet) for postprocessing of the Serpent output. Furthermore, with partial removal of FPs that are unknown prior to their depletion solutions, it is impossible to predict the mass or density change in the burnup fuel system due to such removals, which introduces additional uncertainties in the calculated results.

Step fueling requires the use of predetermined rates of fuel addition at each fueling step, whereas Serpent fuel addition or removal using mass flow feature only is set as an exponential of the flow constant (λ) (see Eq. (4)). This requires the changing of the flow constant λ in “addflow” and “overflow” at the beginning of every Serpent depletion step, making the Serpent flow feature no longer advantageous compared to manual adjustment of Serpent settings in modeling SF scheme. Presetting flow constant in Serpent is a desirable feature for advantageous computational performance and to make the model more realistic. In addition, if the material volumes can also be preset in Serpent, then the flow “overflow” is no longer needed, and Eqs. (5) and (6) become irrelevant. Therefore, being able to preset both flow constants fuel material volumes would make Serpent more efficient in fuel depletion calculations of circulating MSR fuel.

In order to make FMSR fuel depletion more realistic with current Serpent flow features, the system volume (V^sys) may be split into the core of fixed volume (V^cor) and the outer fuel volume such as in the down-comer (V^dnc). In this case, the three existing flows remain attached to the outer fuel only, and two additional flows are used to move fuel between the core and the down-comer. With this new flow setup, the outer fuel volume is still needed to be evaluated using Eq. (5) but the normalization power is contributed only by the fuel in core of a fixed volume and it does not change, making Eq. (6) be no longer needed. However, as Serpent only updates material compositions at the end of each calculation step, the calculation time-step-sizes must be in the order of minutes comparable to the fuel residence time in the core or down-comer. This makes the current Serpent flow features impractical for use to investigate MSR burnup.

The Monte Carlo code Serpent has been used for depletion calculations of the molten salt fuel in an integral channel type of sm-FMSR. Fuel depletion in three fueling schemes over a seven-year reactor life cycle have been simulated: SF, CFA, and CFS.

Using Serpent flow features in modeling the circulating fuel in the sm-FMSR, CFA or CFS requires only a few depletion cycles, each with a stop for adjustment of the flow constants and a restart, which is sufficient to keep k_eff within an acceptable range of ±3 mk. However, current flow features in Serpent are not suitable for application to SF that allows k_eff to be maintained within a larger range such as ±11 mk. Without flow features, SF requires more than 20 Serpent depletion cycles, each with a stop to allow for manual mixing of fuel with the top-up fuel and a restart.

With separation of insoluble fission products from fuel in CFS, a large worth of poison reactivity is removed. This allows for a reduction in the initial fuel fissile content and top-up fuel rate, leading to significant savings of fuel. However, removal of individual nuclides from a depletion (burnup) material introduces uncertainties in the fuel volume, which, in turn, affects the combined neutronics and depletion solutions of Serpent.

It should be noted that for the modeled sm-FMSR concept, the individual fissile quantities (both concentration and mass) increase with reactor thermal output as a result of fuel addition and plutonium generation, and they end up in the spent fuel. Recovery and reuse of the spent fuel should be considered for improving fuel economy in the modeled sm-FMSR design.

Enhancement in Serpent burnup calculations with the capabilities of presetting material volumes and flow constants will further facilitate the simulation of the fuel depletion process and allow for more realistic simulations of fuel circulations.

This work was funded by Atomic Energy of Canada Limited, under the auspices of the Federal Science and Technology Program. All the calculations are performed on CNL's High Performance Computer cluster.

• Atomic Energy of Canada Limited, under the auspices of the Federal Nuclear Science and Technology Program. The research was conducted at the Canadian Nuclear Laboratories (Funder ID: 10.13039/501100004953).

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

C =

concentration, kg/m³

k =

multiplication factor

mk =

unit of reactivity, reactivity × 1000

P =

power, W

SCR =

system-to-core volume ratio

t =

time, s

V =

volume, m³

δ =

Kronecker's delta function

Δt =

time interval, s

λ =

flow constant, s⁻¹

ρ =

density, kg/m³

Sub/Superscripts

add =

addition (fuel added)

cor =

core (fuel in core channels)

dnc =

down-comer

eff =

effective

fis =

fission

iso =

isotope (nuclide or element)

m =

calculation step index

min =

minimum

n =

fueling step or calculation cycle index

out =

outside (fuel removed)

src =

source (top-up fuel)

sys =

system total (total fuel load)

Abbreviations

CF =

continuous fueling

CFA =

continuous fueling with fission product accumulation

CFS =

continuous fueling with fission product separation

CNL =

Canadian Nuclear Laboratories

EOL =

end-of-life

FP =

fission product

FPD =

full power days

MCNP =

Monte Carlo N-Particle (code)

MSR =

molten salt reactor

SF =

step (cycle) fueling

sm-FMSR =

small modular fluoride molten salt reactor