A benchmark for neutron transport simulations of fast reactors with lead coolant and mixed-nitride uranium-plutonium fuel is presented. An approach for developing adapted test problems is proposed for extending an open database of neutron transport simulation test problems with a simple and full description to widen a potential audience. The formulation of the benchmark is based on a conceptual computational model with no excessive detailing of design elements, which reduces the number of potential errors in the preparation of initial data and the model description in the tool. In addition, a methodology for developing a test problem with an equilibrium composition of nuclear fuel with a minimal reactivity margin for burn-up based on existing fast reactor design concepts is described to expand the cross-verification base of fast reactors. The proposed core contains fuel in three zones (regions) with fuel rods having different fuel column heights. This ensures a simplified core design, a negative void effect of reactivity, and effective flattening of the energy deposition distribution along the core radius. The benchmark may also be of use in an initial verification of neutron transport simulation tools for finding the errors related to the methods implemented in the tool. Comparative results of the benchmark calculations with the ENDF/B-VII.1-based data of SCALE 6.2 and the BNAB-RF data are presented, providing a reference point for the potential audience of the benchmark problem. Besides, the work highlights an educational purpose of the problem and prospects of creating a system of computational benchmarks for innovative fast nuclear reactors included in the Generation IV list.

Benchmark, reactor core, stepped fuel distribution, fast reactor, lead coolant

Within the framework of the new technological platform for nuclear energy, based on closed nuclear fuel cycle technologies with fast neutron reactors, the primary goal of nuclear safety is to prevent severe accidents at nuclear power plants (Adamov et al. 2022). At the stage of neutron transport simulations, it is necessary to exclude severe reactivity-induced accidents with a prompt neutron runaway to achieve this goal.

Reactivity-induced accidents can be prevented by (Slesarev et al. 2021):

• using dense fuel, which allows for the formation of a core with an equilibrium composition of nuclear fuel and a minimal reactivity margin for burn-up;
• ensuring the overall negative temperature coefficient of reactivity.

The results of the development of inherent safety technologies include the technical design of the BREST-OD-300 reactor installation and the corresponding power unit for construction and experimental operation. Innovative technical solutions of the BREST-OD-300 reactor unit will be used to develop and construct a high-power reactor with a lead coolant, the BR-1200, which will be part of the industrial power complex. The development of the BR-1200 is already underway, and significant scientific and technical results have been obtained to date — the feasibility of maintaining a reactivity margin of less than the effective delayed neutron fraction β_eff has been substantiated based on neutron transport calculations in the preliminary design stage.

Currently, domestic programs for the development of Generation IV fast reactors lack a sufficient base of neutron-physics test problems, and the existing base is usually based on problems for oxide fuel (Transient and accident analysis of a BN-800 type LMFR with near zero void effect, IAEA-TECDOC-1139 2000; BN-600 Hybrid Core Benchmark Analysis, IAEA-TECDOC-1623 2010; BN-600 MOX Core Benchmark Analysis, IAEA-TECDOC-1700 2013) or materials for which a complete description of the problem is not available (Belov et al. 2014; Bereznev et al. 2023). This can be explained by the need to protect commercial solutions for proposed projects and the requirements for non-proliferation of nuclear materials, especially relevant for closed nuclear fuel cycle technologies with fast neutron reactors.

In the 1990s and 2000s, much attention was paid to the development of international benchmarks. The Kurchatov Institute (Obninsk, Russian Federation) proposed a design in which the upper blanket of the core was replaced with a sodium chamber consisting of shells filled with sodium. In this case, increased axial neutron leakage led to a strong negative reactivity effect in the event of voids in the chamber, which compensated for most of the positive void reactivity coefficient (void coefficient) of sodium in the core. This method of reducing the void coefficient was adopted in the BN-800 reactor project. In 1994, the IAEA proposed a comparative study of a severe accident for a BN-800 type reactor with an innovative core with a near-zero void coefficient (Transient and accident analysis of a BN-800 type LMFR with near zero void effect, IAEA-TECDOC-1139 2000). The initial data were prepared by the Russian side, the study was divided into six stages, and specialists from Germany, France, Great Britain, Italy, Japan, and India participated. The main results of the study showed that all codes predicted the onset of boiling, which meant that the negative feedback from the sodium chamber was not enough to prevent the onset of boiling. The forecast of the time to the onset of boiling for different codes coincided within 14%. The feedback affects the delay to the onset of boiling. The total reactivity is negative, which is beneficial for reactor safety. The results are not very sensitive to uncertainties in the basic physical parameters.

In 1999, under the auspices of the IAEA, a comparative study of the core of the BN-600 reactor with a hybrid UOX/MOX fuel was launched to study various nuclear data and levels of approximation in calculating reactivity-related safety effects and their influence on uncertainties in forecasts of transient processes (IAEA 2010). The studies consisted of a homogeneous reference test RZ (stage 1), a homogeneous reference test Hex-Z (stage 2), and a heterogeneous reference test Hex-Z and burn-up (stage 3). In stage 4 of the reference analysis, conducted in 2002–2003, the core of the BN-600 reactor, fully loaded with MOX fuel containing weapons-grade plutonium, was studied. The initial data were prepared by the Russian side, and specialists from different countries participated in the study. In stages 1–4, particular attention was paid to calculating reactivity coefficients related to safety and the influence of calculation uncertainties on the modeling of transient processes. In 2005, stage 5 confirmed the calculation uncertainty of reactivity coefficients and, in particular, the reactivity of sodium voids during calculations of BFS-62-3A experiments. The main results of the study showed that to obtain accurate results, it is necessary to use codes for solving the neutron transport equation with no diffusion approximation. In this case, for most participants, the results of criticality and reactivity effects, including the reactivity of voids in the sodium chamber, obtained using their own initial data and codes, were consistent with equivalent measured values within the range of uncertainties. At the same time, significant deviations were observed among the participants for some reactivity coefficients and power distributions.

The research was continued (stage 6) with a model of the BN-600 reactor with MOX fuel containing plutonium and minor actinides (MA) coming from spent light water reactor fuel (IAEA 2013). The results were considered for criticality, reactivity loss during burn-up, and reactivity coefficients, as well as the results of transient ULOF analyses. The main results of the study showed that the values of criticality, reactivity coefficients, reactivity loss during burn-up, and changes in fuel composition are qualitatively consistent. The absolute value of the Doppler constant of the fuel decreases by about 50% or more, the coefficient of sodium density increases significantly (in magnitude), the effective fraction of delayed neutrons and the neutron lifetime decrease. Significant discrepancies were also present among the results presented by different participants, mainly due to different nuclear data used, especially for the hybrid core case. Uncertainties are more significant in the case of a core with fuel containing MA.

This work discusses the creation of an open database of neutron-physics test problems for Generation IV reactors on the OECD/NEA platform. The test problems listed in (Transient and accident analysis of a BN-800 type LMFR with near zero void effect, IAEA-TECDOC-1139 2000; BN-600 Hybrid Core Benchmark Analysis, IAEA-TECDOC-1623 2010; BN-600 MOX Core Benchmark Analysis, IAEA-TECDOC-1700 2013) can serve as examples for the database being created. The formation of the database should be based on the following principles:

• test problems are based on original ideas that are not considered in other open test problems;
• conceptually assume a solution to the problem of preventing prompt-neutron reactivity-induced accidents;
• use only publicly available information, including patents;
• have an adapted formulation.

An adapted formulation implies that test problems should be based on a conceptual computational model without excessive detailing of design elements that do not affect neutron physics. Reducing the amount of data in the formulation of test problems should reduce the number of errors in the development of calculation models for codes and contribute to increasing the number of participants in testing.

A test problem based on the idea (Leonov et al. 2015a, 2015b) is considered as a pilot. The core of the reactor contains three parts — central, middle, and outer regions with identical fuel assemblies in each, but with fuel assemblies in different regions containing fuel rods with different fuel column heights. The radial distribution of fuel in the cores has a stepped shape in the axial section. The proposed stepped distribution has a similar effect on the void coefficient to that of flattening the central part of the core, which increases the neutron leakage and allows achieving a negative void coefficient for this part. This, together with the influence of the lateral and end reflectors, as well as the neutron absorber placed at the top of the fuel rods, ensures that a negative void coefficient is achieved for the entire reactor. The pilot test problem was developed in two stages:

• adaptation of the calculation model;
• selection of the initial fuel loading.

The adapted calculation model was created based on one of the test problems that participated in the independent testing of new generation codes of the “Proryv” project (Ternovykh and Bogdanova 2020; Tikhomirov et al. 2021). The goal of adaptation is to reduce the amount of initial data for the model. Such a test problem can be useful for the initial verification of neutron transport simulation codes and for finding programming errors. The visual representation of the prototype and the test calculation model is shown in Fig. 1.

Figure 1. 

a. Prototype model; b. Test model.

In the proposed calculation model, the key simplifications were the elimination of detailed descriptions of control elements that are completely withdrawn from the core and the unification of assemblies within zones. The criteria for compliance of the test calculation model with the prototype were primarily the preservation of the reactor multiplication factor and the mass of fissile materials in the reactor for the initial loading, respectively:

$k\approx k_{\text{prot}}$

$M\approx M_{\text{prot}}$

In addition, the requirements for the adaptation are the minimal influence on the values of reactivity coefficients and changes during burn-up. The characteristics of the calculation models are compared in Table 1. The number of fresh fuel compositions in the model is given for the problem of calculating the energy release in the zones of the core with different heights of the fuel column. Note that expanding the range of calculated functionals to the reload mode sharply increases this number.

The second stage in the formation of the pilot test problem was the selection of the initial fuel loading of the reactor operating in a self-sustaining fuel mode. The loading must provide the formation of a core with a balanced composition of nuclear fuel and a minimal burn-up reactivity margin. In the work (Ternovykh et al. 2017), it is shown for the BN-1200 reactor model that the composition of plutonium changes relatively quickly during the first campaign, and then reaches a complete equilibrium for a sufficiently long period. The most important parameters in terms of the required reactivity margin are the first few cycles of irradiation. Therefore, the concepts of “equilibrium core” and “equilibrium fuel” are proposed to be separated. An equilibrium core is a core in which reactivity changes during the campaign do not exceed β_eff. Equilibrium fuel is the composition of the fuel, primarily the isotopic composition of plutonium, which is formed during the repeated reprocessing of fuel operating in an equilibrium core for several decades. The test problem requires the following conditions:

• maintaining the reactor criticality during depletion, i.e. k(t) > 1;
• an equilibrium core, i.e. | k(0) – k(t)| < β;
• a self-sustaining fuel mode, Δ m _Pu ≥ 0;
• equilibrium fuel, ρ ^In _Pu ≈ ρ ^Out _Pu.

The scenarios for changes in the reactor reactivity between reloads for the test problems are shown in Fig. 2.

Figure 2. 

“Ideal” scenarios of multiplication factor change with equilibrium core.

The initial fuel loading composition should match the equilibrium fuel. The system of test problems being developed provides the ability to analyze various reactor operating scenarios and the formation of equilibrium fuel. The characteristics of the equilibrium fuel are included in the calculation: fuel density, fraction of fuel in the fuel assembly, fraction of plutonium in the fuel, fraction of minor actinides in the fuel (not discussed in the pilot version). The composition obtained by iterative modeling of fuel depletion without taking into account reloads was used in the test problem, and it satisfies the conditions of the equilibrium fuel and self-sustaining mode. The results of modeling the reactor criticality during burn-up without taking into account reloads, depending on the relative change in the fuel fraction in the fuel assembly, are shown in Fig. 3, and the results for the change in the fraction of plutonium in the fuel are shown in Fig. 4.

Figure 3. 

Change in k behavior during depletion with fuel fraction change with no consideration of refueling: 1 – 0.95; 2 – 0.975; 3 – 1.0; 4 – 1.025; 5 – 1.05.

Figure 4. 

Change in k behavior during depletion with plutonium fraction change with no consideration of refueling: 1 – 13.0%; 2 – 13.2%; 3 – 13.4%; 4 – 13.6%; 5 – 13.8%.

Based on the change in k during burn-up without taking into account reloads, it is possible to estimate the change in reactivity during the micro-campaign, taking into account the presence of fuel assemblies that have been irradiated for a different number of micro-campaigns, using the ratio:

$k_{\text{real}}\left(t\right)=\frac{1}{M}\sum _{m=1}^{M}k(t+(m-1\left)T\right)$

where M is the number of fuel micro-campaigns; T is the duration of the micro-campaign, eff. days.

The results of such an estimate are shown in Fig. 5. The similar behavior of the curves in Fig. 3 suggests that, in a real simulation of the core, it is possible to ensure the criticality of the reactor at the beginning of the micro-campaign by changing the fraction of fuel in the fuel assembly without changing the shape of the curve in Fig. 5.

Figure 5. 

Assumed core k change during cycle.

The results obtained correspond to the formulated requirements for the equilibrium core, and the proposed test problem is offered for calculations using various codes and libraries of constants. The test problem presented in this work can be further supplemented with a set of end-of-equilibrium-cycle fuel functionals and the results of their calculations.

In the pilot test problem, the core of the reactor model consists of fuel assemblies of three types, located in the central, middle, and outer regions, with identical fuel assemblies in each, but with fuel assemblies in different zones containing fuel rods with different fuel column heights. The core model consists of assemblies of several types, which are divided into homogeneous physical zones in the axial direction. The core map of the test problem is shown in Fig. 6. The core is surrounded by eleven layers of fuel assemblies 4 (1156 assemblies of the reflector).

Figure 6. 

Core configuration for test.

The central core contains 72 fuel assemblies of type 1 and 13 assemblies of type 5 for control elements. The middle core contains 186 fuel assemblies of type 2 and 54 assemblies of type 6 for control elements. The outer core contains 222 fuel assemblies of type 3. All assemblies have an overall side-to-side size of 19.9 cm and a height of 400 cm. The material compositions and heights of the vertical sections of fuel assemblies of types 1, 2, 3, 4, 5, and 6 are shown in Fig. 7. The nuclide compositions for these material compositions are given in Table 2. When performing neutron transport calculations for compositions containing fuel, the temperature is assumed to be 1200 K, and for compositions not containing fuel, the temperature is assumed to be 900 K.

Figure 7. 

Materials and their location of fuel assemblies, cm: a. Inner core; b. Middle core; c. Outer core; d. Reflector.

Fig. 7 shows the material compositions and heights of the vertical sections of fuel assemblies for (a) the central region; (b) the middle region; (c) the outer region of the core; and (d) the reflector.

To obtain the characteristics of irradiated fuel assemblies, the following characteristics of the reactor operation in a mode without reloads are considered: the thermal power of the reactor is 3440 MW_th (an average of 55.55 kW per 1 kgHM), the reactor operation time is 1800 effective days.

For a comparison, the following functionals are suggested:

• the reactor multiplication factor with the initial fuel composition, k;

• the reactivity coefficient with respect to fuel density (when the density of the fuel is reduced by 1% in all zones),

$W_{\text{fuel}}=\frac{dk}{d{\rho }_{\text{fuel}}}\frac{{\rho }_{\text{fuel}}}{k^2},\Delta {\rho }_{\text{fuel}}=-0.01{\rho }_{\text{fuel}}$

• the void reactivity coefficient (when the density of the coolant is reduced by 99% in the corresponding zone),

$W_{\mathrm{Na}}^{empty}=\frac{dk}{d{\rho }_{\mathrm{Na}}}\frac{{\rho }_{\mathrm{Na}}}{k^2},\Delta {\rho }_{\mathrm{Na}}=-0.99{\rho }_{\mathrm{Na}}$

• the Doppler constant on the fuel (T_fuel1 = 1200 K),

$K_D=\frac{\Delta K}{\ln \left(\frac{T_{\text{fuel}2}}{T_{\text{fuel}1}}\right)},\Delta K=\frac{1}{k\left(T_{\text{fuel}1}\right)}-\frac{1}{k\left(T_{\text{fuel}2}\right)}$

The functionals of the calculation model with depletion are:

• the reactor multiplication factor k with the initial fuel composition and at an average fuel burn-up of 50 and 100 MWd/kgHM;
• the relative average power release in the fuel assemblies by region;
• the change in the nuclide composition in the irradiated fuel assemblies by zone, in g per 1 kg of fuel, for an average reactor burn-up of 100 MWd/kgHM.

The results of the calculations using the SCALE code system with the KENO-VI code for the following sets of constants have been obtained for a comparative analysis:

• ENDF/B-VII.1-based continuous (pointwise) cross-sections;
• 252-group cross-sections based on ENDF/B-VII.1;
• 299-group cross-sections of the BNAB-RF, prepared by CONSYST-RF;
• 28-group cross-sections of the BNAB-RF.

The multiplication factors for various values of the average fuel burn-up are given in Table 3, the reactivity coefficients with respect to fuel are given in Table 4, and the void reactivity coefficients are given in Table 5. The relative deviations of the results with respect to the ENDF/B-VII.1 (B7) are indicated.

Table 5 shows the void reactivity coefficient for the different zones. Fig. 5 shows the estimated change in reactivity during a cycle, taking into account the presence of fuel assemblies that have been irradiated for a different number of cycles.

The results of the calculations for the Doppler constant (K_D) for various temperature ranges are presented in Table 6. The statistical uncertainty in the calculation of k is 1∙10^-5. The maximum difference between K_D values for the ENDF/B-VII.1 and BNAB-RF libraries is 8%. At the same time, the difference between k values for all temperature ranges does not exceed 0.1%.

Table 7 shows the relative average power release in the fuel assemblies by regions of the core for various values of the average fuel burn-up in the reactor, obtained using calculations with 252-group cross-sections from ENDF/B-VII.1. It should be noted that, firstly, the average power release in the fuel volume of the fuel assemblies differs by more than 20%. Secondly, there is a redistribution of power release during irradiation; therefore, the proposed test model offers opportunities for modification to take into account a real reload scenario.

For an average reactor burn-up of 100 MWd/kgHM, the change in the nuclide composition in the irradiated fuel assemblies with respect to the fresh fuel is shown in Table 8.

The proposed core is a basis for a system of test problems. The importance of computational test problems (along with experimental ones) for the development of Generation IV fast reactors is well recognized by specialists. In the absence of a working reactor prototype, significant efforts are needed to develop test problems and obtain accurate results. The proposed system of test problems is a realistic model for the expected behavior of the reactors under consideration, although it uses some simplified models and data. The problem is representative for assessing the methodological accuracy of the calculation codes, the influence of nuclear data (Chereshkov et al. 2023; Ryzhkov et al. 2025), technological and operational uncertainties (Ryzhkov et al. 2023) on the safety of operation of innovative nuclear reactor installations.

In addition, test problems have an important educational function. Their use in training allows for a deeper understanding of the concepts of modern computational methods, such as the error of approximation and convergence, and mastering them in practice.