Research on Equivalent One-Dimensional Cylindrical Modeling Method for Lead–Bismuth Fast Reactor Fuel Assemblies

Abstract

The lead-cooled fast reactor (LFR), a Generation IV nuclear system candidate, presents unique neutronic characteristics distinct from pressurized water reactors. Its neutron spectrum spans wider energy ranges with fast neutron dominance, exhibiting resonance phenomena across energy regions. These features require a fine energy group structure for fuel lattice calculations, significantly increasing computational demands. To balance local heterogeneity modeling with computational efficiency, researchers across the world adopt fuel assembly equivalence methods using 1D cylindrical models through volume equivalence principles. This approach enables detailed energy group calculations in simplified geometries, followed by lattice homogenization for few-group parameter generation, effectively reducing whole-core computational loads. However, limitations emerge when handling strongly heterogeneous components like structural/control rods. This study investigates the 1D equivalence method’s accuracy in lead–bismuth fast reactors under various fuel assembly configurations. Through comprehensive analysis of material distributions and their equivalence impacts, the applicability of the one-dimensional equivalence approach to fuel assemblies of different geometries and material types is analyzed in this paper. The research further proposes corrective solutions for low-accuracy scenarios, enhancing computational method reliability. This paper is significant in its optimization of the physical calculation and analysis process of a new type of fast reactor component and has important engineering application value.

1. Introduction

LFR, as one of the six advanced reactor types selected by the Generation IV International Forum (GIF), has been ranked as the most promising option for commercialization due to its closed fuel cycle capability, high nuclear waste transmutation efficiency, and inherent safety. In the engineering design of lead–bismuth fast reactors, core physics calculation, as the core link of reactor physics analysis, plays a decisive role in reactor optimization and safety assessment. This involves key technologies such as complex neutron transport theory modeling, burnup analysis, and multi-physics field coupling simulation. The simplification of the computational process and the effective release of computational resources have become a hot research topic as of late.

1.1. Current Status of LFR Development

• China

The lead-based reactor (CLEAR) developed by the FDS team at the Institute of Nuclear Energy Safety and Technology (INEST) of the Chinese Academy of Sciences (CAS), which completed criticality testing in 2017 and sub-criticality validation with an integrated gas pedal in 2019, will also be commercialized in the future for transmutation of nuclear waste and energy production [1]. The KYLIN-II Thermal Hydraulic natural circulation test loop system developed by the Institute investigates the flow characteristics of natural circulation in lead–bismuth eutectic (LBE) loops, providing key data to support the design and safety assessment of advanced nuclear systems such as accelerator-driven subcritical systems ADS and lead-cooled fast reactors [2]. In 2020, Zhao et al. from Nanhua University designed a type of ultra-long-life small-sized natural-cycle lead–bismuth fast reactor, SPALLER-100. The existing small-sized lead-bismuth fast reactor design with problems such as large fuel loading, low burnup, and positive temperature coefficient was optimized in this project, which is suitable for scenarios such as offshore platforms, remote areas, and other scenarios [3].

• European Union

The ELSY project, funded by the European Union’s Sixth Framework Program FP6 in 2011, aims to design a 600 MWe pooled lead-cooled fast reactor (LFR) with pure lead coolant, a pooled design that reduces the risk of coolant leakage, simplifies the system architecture, improves natural circulation, and reduces cost and maintenance difficulties through removable components [4]. ALFRED, a fourth-generation reactor demonstration project led by the European FALCON consortium, adopts MOX fuel and hexagonal fuel assemblies, which can realize a simplified structure and efficient use of fuel, and has already verified key technologies such as thermo-hydraulics, material corrosion, and oxygen control [5]. ENEA has established Europe’s largest experimental platform for lead-based coolants—HLM, covering thermo-hydraulics, safety analysis, and material corrosion research—and has made significant progress in key LFR technology areas through multi-scale experiments and interdisciplinary cooperation [6].

• United States of America

SSTAR, the U.S. Generation IV lead-cooled fast reactor program, utilizes a “box design” for a single seal, eliminating the need for on-site material changes and reducing the risk of nuclear proliferation. It integrates the advantages of long life, high safety, proliferation resistance, and modularity, which are in line with the goals of GNEP and Generation IV nuclear energy systems [7]. Westinghouse has proposed a pre-design for a demonstration lead-cooled fast reactor, the DLFR, utilizing a compact pool primary system with a main vessel containing all primary elements submerged in liquid lead. It is rated at 500 MWt (210 MWe), with a design capability to facilitate power ramp-up (up to 700 MWt) once the initial demonstration mission is completed [8].

• Russia

The Russian lead-cooled fast reactor (LFR) SVBR-75/100, developed on the basis of nuclear submarine reactor technology, is designed to enhance the sustainability of nuclear energy and meet the needs of energy security and low-carbon requirements through a modular design, a closed fuel cycle, and intrinsic safety, with an outstanding performance in terms of safety, economy, and sustainability [9]. The Russian-developed BREST-300 lead-cooled fast neutron reactor adopts a dual-loop design, with the first loop using high-purity lead as the coolant without the risk of chemical reaction, which ensures that there will be no fire or explosion in case of an accident; replacing the uranium regeneration zone with a lead reflector layer, which is capable of eliminating the production of weapons-grade plutonium; and adopting a nitride material ((U + Pu + MA)N), which is capable of utilizing plutonium/secondary actinide elements in natural uranium and spent fuel, enabling fuel self-sustainability [10].

All projects focus on the inherent safety, fuel flexibility, and waste management potential of LFR, but the technical pathways vary according to national strategic needs, forming a complementary development landscape. China is using nuclear waste transmutation as a breakthrough point, simultaneously advancing applications in small reactor scenarios; the EU is leveraging its experimental platform advantages to focus on system integration and safety technology standardization; the US is emphasizing non-proliferation characteristics, developing plug-and-play modular reactors; and Russia is utilizing its military technology heritage to achieve a closed fuel cycle and deep integration with non-proliferation measures.

1.2. Fast Reactor Core Physics Calculation

Fast reactor core physics calculation is an important foundation for the design, development, and operation of fast neutron reactors. Its purpose is to obtain the core’s effective multiplication factor, neutron flux distribution, burnup characteristics, and other key physical parameters through accurate simulation and calculation of neutron transport, fission, capture, and other physical processes in the core, so as to ensure the core’s criticality and safety and, at the same time, provide a scientific basis for the optimization of the core design, operation control, and fuel management.

In 2023, Xia-Nan Du et al. from Xi’an Jiaotong University (XJTU) used SARAX to perform core physics calculations and the DAKOTA framework for multi-objective optimization and uncertainty analysis. The experimental data show that the SARAX-DAKOTA framework developed by them can effectively deal with the multi-objective optimization and uncertainty analysis problems in fast reactor design, which significantly improves the efficiency of fast reactor design and analysis [11]. Based on the two-step method of component homogenization-core transport calculation, Xiao Peng et al. from China Nuclear Power Design Institute (CNPDI) carried out research on the method of producing homogenized few-group components for core transport calculation based on the Monte Carlo procedure in terms of anisotropy, energy group structure, and leakage correction. It is found that the homogenized few-group model based on the Monte Carlo program can well accommodate the resonance effects of medium-mass nuclides [12].

In 2024, Wu Hongchun of Xi’an Jiaotong University and Yang Hongyi of China Academy of Atomic Energy Sciences (CAEAS) summarized the current status of research on the physical analysis methods of fast neutron reactor cores and put forward suggestions for their development. In their paper, it was mentioned that although each country has its own characteristics in the algorithmic model of the fast reactor program, it has consistent basic features from the perspective of the basic theory of physical analysis, such as the two-step core physical analysis process and the fine multi-group database energy group division. The paper also suggested that the Monte Carlo method is becoming more and more widely used in fast reactors and that the multi-physics coupling analysis method will guide the direction of development [13].

Youqi Zheng et al. from Xi’an Jiaotong University proposed an energy group structure that can be adaptively determined according to the neutron energy spectrum properties of different reactors, thus improving the accuracy of reactor core calculations. For thermal neutron reactors, the proposed 31-group structure can meet the accuracy requirements of k_eff and neutron flux calculations. For reactors with neutron spectra comprising mixed energy between hot and fast reactors, the adaptive energy group structure based on the neutron production rate improves the calculation accuracy more than the simple refinement of the energy group division. The new method is able to provide highly accurate core analysis for full neutron energy spectrum applications without significantly increasing the cost of energy discretization. Its method of capturing the main neutron energy spectral properties of the core through a one-dimensional equivalent model is informative for the fuel assembly analysis calculations in this paper [14].

In order to solve the limitations of resonance self-shielding calculations under broad-spectrum conditions, Qiang Zhao et al. from Harbin Engineering University proposed a new generalized framework. The framework designs a new energy group structure and uses a subgroup method based on the narrow resonance approximation to reconstruct the resonance self-shielding cross-section. It also proposes a fission spectrum calculation method for complex fuel compositions with high accuracy for problems of different energy spectrum types. However, it lacks the consideration of thermal or neutral energy neutron contributions from non-fuel regions in the thermal spectrum problem [15].

Using the Monte Carlo volumetric flux homogenization method and the super homogenization equivalent correction method for the control rods, Guo Hui et al. from Shanghai Jiaotong University achieved a reduction in the overestimation of the value of the control rods in the core diffusion calculation from 13.5% to 0.35%. The proposed Monte Carlo flux homogenization method greatly reduces the core transport calculation error of MET-1000. The optimization scheme of core physics calculation for advanced non-uniformly arranged fast reactors mentioned in the paper has been informative in the development of our research [16].

Abdullah O. Albugami et al. from King Saud University used the OpenMC program to model and simulate the nuclear reactor core physics of a commercial pressurized water reactor, and by comparing it with the VERA core physics benchmarking methodology, the results showed that OpenMC has high accuracy in the calculation of the radial power distributions of the fuel rods for the full-core three-dimensional simulation, etc., which verified the accuracy and performance of OpenMC in simulating key nuclear reactor characteristics and fuel rod power distribution [17].

Federico Ledda et al. from Politecnico di Torino, Italy, carried out a comparative study on two Monte Carlo codes, PHITS and OpenMC, for three-dimensional neutronics analysis of compact fusion reactors, and found that both PHITS and OpenMC show good performance in dealing with neutronics analysis of compact fusion reactors and that the choice of different nuclear databases has a significant effect on the neutron spectra, and the selection of different nuclear databases has a significant effect on the neutron spectrum, which requires the selection of appropriate libraries according to specific applications [18].

Sungtaek Hong et al. of the Korea Institute of Technology (KIST) applied a simplified version of generalized equivalence theory (GET) to improve the accuracy of neutron diffusion equation (NDE)-based calculations in a cylindrical liquid molten salt fast reactor (MSFR). A simplified method is proposed to enhance the accuracy of neutronics analysis of MSFR by applying flux-volume-weighted homogenized cross-sections and representative discontinuity factors (DFs) [19].

Fast reactor core physics calculations are evolving toward forward multi-scale coupling, intelligence, and high precision. The integration of Monte Carlo methods with deterministic methods, inline multi-physics coupling, and AI-driven optimization frameworks has become a key breakthrough area. The general analytical model of deterministic methods employs a two-step calculation method involving “uniformization of few-body cross-section generation followed by core neutron transport calculations.” The one-dimensional equivalent model method in this study can be used to generate uniformization parameters for components, enabling the integration of Monte Carlo methods with deterministic methods.

1.3. One-Dimensional Equivalence Method

The two-step method used in the core design and analysis of traditional pressurized water reactors (PWRs) consists of the following steps: First, a detailed two-dimensional analysis of fuel assemblies or control rod assemblies is performed to obtain homogenized few-group cross-section parameters at the assembly level. Second, these homogenized few-group cross-section parameters are used to conduct a three-dimensional computational analysis of the whole core. Unlike PWRs, neutrons in lead–bismuth fast reactors (LFR) exhibit complex resonance phenomena across a wide energy range and significant inelastic scattering effects [13]. The continuous energy Monte Carlo method is well-suited for two-dimensional analysis of LFR assemblies to meet cross-section generation requirements. However, the Monte Carlo method suffers from low computational efficiency and high resource consumption, leading to prolonged computation times and demanding high-performance computing resources. Deterministic methods offer a potential solution to improve computational efficiency, but the complex resonance phenomena in fast reactors often require energy group structures with thousands of groups, making direct two-dimensional calculations at the assembly level challenging. Therefore, the one-dimensional equivalence method is commonly adopted during two-dimensional analysis to enhance computational efficiency while maintaining accuracy, which is currently a widely used approach for fast reactor calculations.

Numerous studies have been conducted by researchers worldwide on one-dimensional equivalence models for fast reactors. Michael G. Jarret from the University of Michigan used the MC2-3 code to generate homogenized and partially homogenized (explicit coolant channels) geometric structures [20], followed by solving them using the finite element-based transport code PROTEUS. The k_eff values were compared with those from continuous-energy Monte Carlo models of the same geometry. The results showed that the agreement between PROTEUS and Monte Carlo solutions fell within a range of 80–180 pcm, with larger errors observed in control rod withdrawal scenarios compared to insertion cases.

In 2024, Qiang Zhao et al. from Harbin Engineering University proposed a new generalized framework to address the limitations of resonance self-shielding calculations under broad-spectrum conditions. The framework designs a new energy group structure and uses a subgroup method based on the narrow resonance approximation to reconstruct the resonance self-shielding cross-section. It also proposes a fission spectrum calculation method for complex fuel compositions with high accuracy for problems of different energy spectrum types. However, it lacks the consideration of thermal or neutral energy neutron contributions from non-fuel regions in the thermal spectrum problem [15].

This one-dimensional equivalent method can reduce the generation time of homogenization parameters and release computational resources. Despite these advancements, existing studies on the accuracy of two-dimensional calculations and one-dimensional equivalence methods in fast reactor assembly homogenization remain insufficient. This paper analyzes conventional and complex fuel assemblies in lead–bismuth fast reactors using continuous-energy Monte Carlo models, investigating the accuracy of one-dimensional equivalence algorithms in two-dimensional analysis for different fuel assemblies. Through comparative validation, the applicability of the one-dimensional equivalence algorithm is evaluated, and cases where it is unsuitable are identified and addressed.

2. Research Objects and Computational Methods

2.1. Research Objects

The lead–bismuth eutectic fast reactor (LFR) is one of the six candidate reactor types for Generation IV nuclear energy systems. The core is typically composed of hexagonal fuel assemblies, which are closely packed to enhance neutron utilization efficiency while reducing coolant flow resistance [21]. Designed with a fast neutron spectrum, the reactor requires high-enrichment nuclear fuel in its assemblies due to the relatively low fission cross-section of fast neutrons to sustain a chain reaction. The fuel cladding and structural materials must be resistant to corrosion from the lead–bismuth coolant. Control rods are arranged either by encapsulating neutron-absorbing materials within the structural materials or by distributing them within the fuel assemblies.

The calculation object in this study employs UO₂ as the fuel, Pb-Bi alloy as the coolant, and HT-9 stainless steel as the structural material [22]. Neutron moderator rods and control rods are appropriately configured, with BeO as the moderator material and B₄C as the control rod material. Some assemblies utilize a composite material of BeO-B₄C as structural rods. Detailed material properties are presented in

Table 1. Material composition.

Materials	Major Constituent Elements and Atomic Percentage	Density
UO2	5% 235U, 28.3% 238U, 66.7% O	10.94 g/cm3
HT-9	70% Fe, 11.5% Cr, 1% Mo	7.8 g/cm3
BeO	36% Be, 64% O	3.0 g/cm3
B4C	80% B, 20% C	2.52 g/cm3
BeO-B4C	38.96% Be, 39% O, 17.64% B, 4.4% C	3.0 g/cm3
Pb-Bi	44.3% Pb, 55.7% Bi	10.29 g/cm3

[23]. The geometric dimensions of the research objects are shown in

Table 2. Geometric parameters.

Calculation Scheme	Geometric Scale
Fuel rod inner/outer diameter	1.651/2.210 mm
Helium gap thickness	0.254 mm
Shell thickness	0.316 mm
Lattice diameter ratio	1.7144
Component box thickness	1.016 mm
Component box side length	56.134 mm

.

In the design of fuel assemblies for advanced fast reactors, the structural model incorporating beryllium oxide material within the assembly warrants discussion. Beryllium oxide, as an excellent moderator material, can reduce the number of fast neutrons and increase the number of neutrons in the medium-energy region. More neutrons can be resonantly absorbed by the fuel. Its larger (n,2n)/(γ,n) reaction cross-section in the high-energy region can generate more neutrons within the reactor, thereby improving the neutron economy and breeding capability of the reactor core.

We conducted a comparative analysis of the impact of different fuel assembly configurations on the accuracy of one-dimensional equivalent calculations. This study focuses on three primary configurations of fuel assemblies and their influence on the accuracy of one-dimensional equivalent calculations, as shown in Figure 1: full fuel rod arrangement, regular full-circle arrangement of structural or control rods, and irregular arrangement of structural or control rods.

2.2. Calculation Code

OpenMC is a Monte Carlo neutron transport simulation code developed by the Massachusetts Institute of Technology (MIT), capable of performing calculations such as effective multiplication factor (k_eff), burnup, and homogenized group constants [24]. In this study, the open-source code OpenMC is used to conduct the k_eff calculations. Each calculation is set to run 500 batches, with 20,000 particles per batch, and the results from the last 400 batches are taken as the final computational outcomes.

2.3. Calculation Procedure

The calculation procedure of this study is shown in Figure 2. First, an accurate two-dimensional assembly geometry is constructed in OpenMC, and the k_eff of this model is calculated using the continuous-energy Monte Carlo code as a reference value. In the second step, the previously constructed geometry is equivalently transformed into a cylinder based on the principle of equal volume. The equivalent cylinder is then used as the new geometric input for the second calculation, and the results are compared with the reference value to verify the applicability of the one-dimensional equivalence algorithm. The third step involves improving the parameters of the cylindrical equivalents with larger errors based on the results obtained in the previous step and exploring methods to reduce the errors.

The equivalence process is divided into the following steps: First, the hexagonal fuel assembly is decomposed into one hexagon and six hollowed-out hexagons, in order from the inside to the outside. Second, the areas of fuel, helium gas gap, cladding, structural material or absorber material, and coolant in each hexagon are calculated. Third, the seven hexagons are sequentially equivalently transformed into circular or annular shapes from the inside to the outside. After equivalence, each hexagon is radially divided into fuel-helium gas gap-cladding-structural material coolant, while the outermost hexagon is radially divided into coolant-structural material after equivalence, as shown in Figure 3.

Taking the third layer as an example, let the area of the hollowed-out hexagon be S, and the areas of fuel, helium gas gap, cladding, and structural material be S₁, S₂, S₃, and S₄, respectively. Let the outer radius of the coolant in the second layer be R1. The equivalence process is as follows:

Coolant Area:

$$S_5=S-S_1-S_2-S_3-S_4$$

(1)

Fuel Outer Diameter:

$$R_2=\sqrt{R_1^2+S_1/\pi }$$

(2)

Helium Gas Gap Outer Diameter:

$$R_3=\sqrt{R_2^2+S_2/\pi }$$

(3)

Cladding Outer Diameter:

$$R_4=\sqrt{R_3^2+S_3/\pi }$$

(4)

Structural Material Outer Diameter:

$$R_5=\sqrt{R_4^2+S_4/\pi }$$

(5)

Coolant Outer Diameter:

$$R_6=\sqrt{R_5^2+S_5/\pi }$$

(6)

3. Calculation Results and Analysis

In this paper, for the different types of fuel assembly structural materials and absorption materials, the calculation of five models is carried out, which are full fuel rod arrangement, HT-9 structural material, BeO structural material, B₄C absorption material, and BeO-B₄C composite structural material. Among them, the non-full fuel rod arrangement is categorized into regular and irregular arrangements of structural or absorbing materials, and the irregular arrangement is categorized into two cases, 1 and 2 (as shown in Figure 1c,d).

3.1. Full Fuel Rod Arrangement

3.1.1. Effective Multiplication Factor

Table 3. k_eff before and after equivalence at different enrichment levels.

Enrichments of 235U	Before Equivalence keff	After Equivalence keff	Δkeff
17.81%	1.40993 ± 0.00023	1.41118 ± 0.00023	+15
14.83%	1.31181 ± 0.00023	1.31116 ± 0.00023	−65
12.36%	1.21061 ± 0.00023	1.21105 ± 0.00023	+44
9.89%	1.09578 ± 0.00023	1.09549 ± 0.00023	−29
7.91%	0.98595 ± 0.00023	0.98670 ± 0.00023	+75
4.94%	0.78269 ± 0.00023	0.78321 ± 0.00023	+52

shows the results of the calculations before and after the one-dimensional equivalence at different ²³⁵U enrichments for the all-fuel rod arrangement, with error values all within 100 pcm. The results show that the accuracy of the 1D equivalent model is very high under the full fuel rod arrangement, and the change in enrichment will not affect its calculation accuracy.

3.1.2. Homogenized Cross-Section

The numerical analysis results of this study are shown in Figure 4, Figure 5, Figure 6, Figure 7, Figure 8 and Figure 9, which demonstrate the comparative analysis results before and after the equivalence based on the structural division of the 33 energy groups, including the distribution characteristics of the key parameters such as the total reactivity, neutron flux density, and total macroscopic cross-section, respectively.

The analysis results show that the distributions of the total reactivity, neutron flux density, and total macroscopic cross-section before and after the equivalence treatment remain highly consistent but show significant differences (with the maximum relative deviation as high as 95%) between some of the energy groups in the 40–300 eV and 10–20 Mev energy regions. It is noteworthy that the neutron flux density amplitude tends to be close to zero in the energy region of these differences, a phenomenon that is attributed to the spectral properties of the neutron transport process. Therefore, the total reactivity calculated before and after the equivalence shows good conservation properties in all energy groups, which verifies the physical reasonableness of the one-dimensional equivalence method and provides an important theoretical basis for the neutron transport calculations in all-fuel rod assemblies.

3.1.3. Comparison of Different Methods

In order to compare the computational accuracy of the one-dimensional equivalent model with advanced Monte Carlo methods with flux homogenization techniques, we output the homogenization parameters of the fuel assembly and perform criticality calculations. The results show that the accuracy of the flux homogenization method is significantly lower than that of the one-dimensional equivalent method. The homogenization process is shown in Figure 10. A comparison of the accuracy of the two methods is shown in

Table 4. Comparison of One-dimensional equivalent model and Flux homogenization.

Method	Before Equivalence	After Equivalence	Δkeff
One-dimensional equivalent model	1.31181 ± 0.00023	1.31116 ± 0.00023	−65
Flux homogenization	1.31181 ± 0.00023	1.30924 ± 0.00023	−257

.

3.2. HT-9 Structural Material

3.2.1. Regular Arrangement of HT-9 Structural Material

When the structural rods composed of HT-9 are regularly distributed in different rings in a full-circle pattern, the results before and after one-dimensional equivalence are shown in

Table 5. Structural materials distributed at different locations before and after k_eff.

At the Respective Ring	Before Equivalence	After Equivalence	Δkeff
1	1.33833 ± 0.00023	1.33776 ± 0.00023	+57
2	1.33690 ± 0.00023	1.33761 ± 0.00023	+71
3	1.33593 ± 0.00023	1.33606 ± 0.00023	+13
4	1.33468 ± 0.00023	1.33471 ± 0.00023	+3
5	1.33248 ± 0.00023	1.33300 ± 0.00023	+52
6	1.33178 ± 0.00023	1.33190 ± 0.00023	+22

. The trend of error variation is illustrated in Figure 11. It can be seen that when the structural material rods are regularly distributed in different rings in a full-circle pattern, the error after equivalence shows a periodic pattern of first increasing and then decreasing in the radial direction. In the first period, the error is the largest when the second ring is entirely composed of structural material, reaching +71 pcm; in the second period, the error is the largest when the fifth ring is entirely composed of structural material, reaching +52 pcm.

We selected the case with the largest deviation, i.e., the second ring consisting entirely of structural material, for correction. In order to solve the problem of high structural bias in the calculation of k_eff values after equivalence, an attempt was made to reduce the outer radius of the fuel region of the second ring. This approach inevitably affects the radius values of all outer ring regions (as shown in Figure 12). According to the calculation results, it can be seen that the error shows a tendency of decreasing and then increasing, and it shows an unstable phenomenon as the reduction increases (as shown in Figure 13).

It can be concluded that when there is an error after equivalence, whether increasing or decreasing the radius, selecting an appropriate equivalent value can reduce the error after equivalence.

3.2.2. Irregular Arrangement of HT-9 Structural Material

When the structural rods composed of HT-9 are irregularly distributed within the fuel assembly, the results before and after one-dimensional equivalence are shown in

Table 6. Equivalent calculation results before and after for Irregular Arrangement of HT-9 Structural Material.

Forms of Arrangement	Before Equivalence	After Equivalence	Δkeff
Irregular Arrangement 1	1.31863 ± 0.00023	1.31807 ± 0.00023	−56
Irregular Arrangement 2	1.33020 ± 0.00023	1.32913 ± 0.00023	−107

. In the case of irregular arrangement, the maximum error is only at the order of 100 pcm, and the calculation accuracy can be ensured.

3.3. BeO Structural Material

As a typical core structure/reflector material, beryllium oxide (BeO) has attracted a lot of attention due to its outstanding comprehensive performance: its high thermal conductivity (up to 225 W/m-K at room temperature) ensures the core heat transfer efficiency, and its excellent neutron economy and high temperature stability significantly enhance the operational safety margin of the reactor. In terms of nuclear physics, beryllium exhibits a unique neutron modulation capability through the (n,2n) reaction and elastic scattering process, and its large inelastic scattering cross-section enables it to have the dual functions of neutron reflection and slowing [25].

In this study, beryllium oxide is innovatively used as a multifunctional structural component for the optimization of core layout. In order to systematically evaluate the effects of different structural configurations on the neutron energy spectrum and effective multiplication factor (k_eff), a one-dimensional equivalent physical model is developed using the OpenMC program, and the accuracy of the equivalent model is investigated using the control variable method.

When the fuel assembly is equipped with structural rods made of BeO, the calculation results before and after equivalence are shown in

Table 7. Calculation results before and after equivalent for arrangement of BeO materials in different layers.

At the Respective Ring	Before Equivalence	After Equivalence	Δkeff
1	1.33162 ± 0.00023	1.33147 ± 0.00023	−15
2	1.33017 ± 0.00023	1.33057 ± 0.00023	+40
3	1.32958 ± 0.00023	1.32912 ± 0.00023	−46
4	1.32777 ± 0.00023	1.32766 ± 0.00023	−11
5	1.32569 ± 0.00023	1.32607 ± 0.00023	+38
6	1.32511 ± 0.00023	1.32611 ± 0.00023	+100

and

Table 8. Calculation results before and after equivalent for Irregularly distributed BeO materials.

Forms of Arrangement	Before Equivalence	After Equivalence	Δkeff
Irregular Arrangement 1	1.31685 ± 0.00023	1.31589 ± 0.00023	−96
Irregular Arrangement 2	1.32852 ± 0.00023	1.32878 ± 0.00023	+26

. Regardless of whether it is a regular or irregular distribution, the error is within 100 pcm.

3.4. B₄C Absorber Material

Boron carbide (B₄C), as a key component of neutron absorption control materials and radiation shielding materials for nuclear reactors, has an irreplaceable role in reactor safety control systems. Its excellent neutron absorption performance mainly originates from the high neutron absorption cross-section of the B10 isotope, supplemented by the outstanding high-temperature stability and anti-irradiation properties of the material itself, which makes it the preferred material for the control rods of high-temperature reactors such as fast neutron reactors. Existing sodium-cooled fast reactor designs in the international nuclear energy field (e.g., Russian BN-600, French Phénix, etc.) have adopted B₄C as the core functional material for control rods. This study addresses the differences in the spatial layout of B₄C control rod assemblies in the reactor core and systematically investigates the mechanism of their spatial configurations on the computational accuracy of the one-dimensional equivalent model of the reactor [26].

When control rods made of B₄C are arranged in the fuel assembly, the results before and after equivalence are shown in

Table 9. Δk_eff of different layers.

(a)
At the Respective Ring	Before Equivalence	After Equivalence	Δkeff
1	1.30796 ± 0.00023	1.30850 ± 0.00023	+56
2	1.29648 ± 0.00023	1.29736 ± 0.00023	+88
3	1.28154 ± 0.00023	1.28258 ± 0.00023	+104
4	1.26482 ± 0.00023	1.26537 ± 0.00023	+55
5	1.24572 ± 0.00023	1.25053 ± 0.00023	+481
6	1.23284 ± 0.00023	1.24327 ± 0.00023	+1043
(b)
At the 6th Ring	Before Equivalence	After Equivalence	Δkeff
Fully absorbent material rod	1.23284 ± 0.00023	1.24327 ± 0.00023	+1043
Half absorbent material rod	1.29014 ± 0.00023	1.29226 ± 0.00023	+212

a and Table 10. It is found that, in the case of regular distribution, the equivalence accuracy is high when they are distributed in the first to fourth rings. However, the error is larger in the fifth and sixth rings. The accuracy in the case of irregular distribution is higher than that in the case of regular distribution. Thus, it can be concluded that when there is absorber material in the fuel assembly, the more uniform the material distribution is, the higher the accuracy of the one-dimensional equivalence calculation method. When the absorber material is concentrated in distribution, the accuracy is reduced, and the accuracy error is the largest when it is concentrated in the outer periphery (as shown in Figure 14).

To address this, we reduced the amount of beryllium oxide distributed around the periphery. For example, in the sixth layer of the component, 15 rods were fuel rods, and 15 rods were absorber rods, with the two types of rods arranged in a crisscross pattern. We then performed criticality calculations before and after the equivalence adjustment and found that the error was greatly reduced, from 1043 pcm to 212 pcm (as shown in Table 9b). It can thus be seen that adding too much absorbent material will greatly increase the error.

3.4.1. Regular Arrangement of B₄C Absorber Material

• Effective Multiplication Factor

• Homogenized Cross-Section

The neutron properties of the B₄O material when the control rods are centrally arranged in the fifth circle, and the distribution characteristics of the total macroscopic cross-section and the neutron flux density are obtained by numerical calculations (as shown in Figure 15, Figure 16, Figure 17 and Figure 18).

The results show the following: before the equivalent treatment, the neutron injection rate of each energy group is higher than the value after the equivalent treatment, and the relative error is as high as 20%; the variation in the total macroscopic cross-section is mainly reflected in the energy region from 14 eV to 300 eV, and the magnitude of the difference in other energy groups is less than 10%. It is worth noting that the neutron flux density in this characteristic energy region (14–300 eV) is close to 0. Based on the sensitivity analysis, it can be concluded that the systematic underestimation of the post-equivalent neutron flux density is the main reason for the deviation of the effective multiplication factor by 0.34%.

3.4.2. Irregular Arrangement of B₄C Absorber Material

The calculation results are shown in Table 10.

Table 10. Calculation results before and after equivalent for Irregularly distributed B₄C Absorber Material.

Forms of Arrangement	Before Equivalence	After Equivalence	Δkeff
Irregular Arrangement 1	1.15292 ± 0.00023	1.15290 ± 0.00023	−2
Irregular Arrangement 2	1.27606 ± 0.00023	1.27592 ± 0.00023	−14

Table 10. Calculation results before and after equivalent for Irregularly distributed B₄C Absorber Material.

Forms of Arrangement	Before Equivalence	After Equivalence	Δkeff
Irregular Arrangement 1	1.15292 ± 0.00023	1.15290 ± 0.00023	−2
Irregular Arrangement 2	1.27606 ± 0.00023	1.27592 ± 0.00023	−14

3.4.3. Research on the Applicability of Optimization Methods

In this study, the problem of the concentration of B₄C absorbers in the peripheral region leads to a significant reduction in the calculation accuracy of the one-dimensional equivalent model and further verifies the applicability of the optimization method described in Section 3.2.1. The typical working condition where B₄C is centrally arranged in the fifth layer fuel assembly is selected as the research object, and its influence on the calculation error of the 1D model is systematically analyzed by adjusting the outer radius parameter of the B₄C absorber in the annular region of the fifth layer after the equivalence is made. The calculation results are shown in

Table 11. Comparison of Δk_eff for correction methods.

Real Value	Equivalence Radius	Value After Equivalence	Δkeff
1.24572	2.07	1.25185	+613
	2.08	1.25039	+467
	2.0804	1.25053	+481
	2.081	1.25082	+510
	2.09	1.21321	−3248
	2.1	1.21094	−3478

.

The experimental results show that when parameter optimization is carried out for the equivalent outer diameter of the annular region of the fifth layer, the calculated values show significant fluctuation characteristics, and there is a gap of more than 400 pcm between the numerical solution and the theoretical value, and it does not show a tendency to converge to the ideal solution. This phenomenon indicates that the optimization method established in Section 3.2.1 has obvious limitations under the working condition of non-uniform distribution of the radial direction of the absorbing material. Specifically, when the B₄C absorber is centrally distributed in the peripheral region of the core, the original equivalent radius optimization criterion is unable to effectively correct the computational deviation caused by the heterogeneity of the spatial distribution of the material, and a new equivalent modeling method needs to be developed for this special arrangement.

3.5. BeO-B₄C Composite Structural Material

In view of the large effect of beryllium oxide and boron carbide on the neutron energy spectrum, beryllium oxide and boron carbide composites as structural rods are also considered in this study.

When the fuel assembly is equipped with structural rods composed of BeO-B₄C composite material, the results before and after equivalence are shown in

Table 12. Calculation results before and after equivalent for arrangement of BeO-B₄C Composite Structural Material in different layers.

At the Respective Ring	Before Equivalence	After Equivalence	Δkeff
1	1.32545 ± 0.00023	1.32552 ± 0.00023	+7
2	1.32162 ± 0.00023	1.32189 ± 0.00023	+27
3	1.31605 ± 0.00023	1.31602 ± 0.00023	−3
4	1.30990 ± 0.00023	1.31016 ± 0.00023	+26
5	1.30385 ± 0.00023	1.30395 ± 0.00023	+10
6	1.29494 ± 0.00023	1.29487 ± 0.00023	−7

and

Table 13. Calculation results before and after equivalent for Irregularly distributed BeO-B₄C Composite Structural Material.

Forms of Arrangement	Before Equivalence	After Equivalence	Δkeff
Irregular Arrangement 1	1.26545 ± 0.00023	1.26530 ± 0.00023	−15
Irregular Arrangement 2	1.31448 ± 0.00023	1.31469 ± 0.00023	+21

. The error before and after equivalence does not exceed 30 pcm, indicating that the accuracy of the one-dimensional equivalence calculation is very high.

4. Conclusions

In this study, systematic research is carried out to address the applicability of the one-dimensional equivalent homogenization model in the physical calculations of fuel assemblies for lead–bismuth fast reactors. Based on the Monte Carlo program OpenMC [27], different configurations of fuel assemblies including full fuel rod arrangement, HT-9 structural material, beryllium oxide (BeO), boron carbide (B₄C), and their hybrid materials were characterized in neutronics, and the main research results are as follows:

• The Critical Characterization

The typical configurations of all-fuel rod arrangement, HT-9 uniform/non-uniform arrangement, BeO uniform/non-uniform arrangement, BeO-B₄C uniform/non-uniform arrangement, and B₄C hybrid arrangement show good applicability of the one-dimensional equivalent model, and the deviation of the effective multiplication factor (k_eff) calculations is controlled to be within ±100 pcm. However, it is worth noting that when the B₄C absorber is arranged in a peripheral concentration, the equivalent model shows a significant deviation, with the k_eff error on the order of thousands of pcm.

2.

The Evaluation of Calculation Accuracy

The analysis of cross-section conservativeness shows that the difference in energy group distribution of the total macroscopic cross-section and neutron flux density before and after the equivalence is mainly concentrated in the energy regions of 40–300 eV and 10–20 Mev, but the relative deviation of the total reactivity is <6% due to the influence of flux magnitude in this energy region.

Under the abnormal working condition (B₄C peripheral arrangement), the flux distribution distortion in the 40–300 eV and 10–20 Mev energy regions leads to obvious deviation in k_eff calculation.

3.

The Method in Model Correction

An adaptive correction method based on equivalent radius optimization is proposed to reduce the k_eff deviation to the order of 10 pcm in the HT-9 structural material arrangement scenario. However, for strong gradient cases such as the B₄C peripheral arrangement, the corrected calculation results show oscillatory characteristics of hundreds or even thousands of pcm, indicating that the applicability of the method is limited in the strongly nonuniform system.

5. Discussion

In this study, the applicability of the one-dimensional equivalent model to the fuel assemblies of lead–bismuth fast reactors is investigated, and it is found that the calculation accuracy of the model is high in general cases such as the full fuel rod arrangement, whereas in the case where the absorbing material is centrally distributed in the periphery, the calculation results obtained by the method have a large discrepancy with the actual values. This may be due to the fact that the free range of neutrons in the absorbing material is much smaller than that in the fuel, and the one-dimensional cylindrical model equivalent to the assembly with a large number of absorbing materials concentrated in the periphery cannot simulate the neutronics characteristics of the hexagonal assembly well, which leads to a large calculation error.

Through the study of various homogenization methods at home and abroad [28], an adaptive correction method based on the optimization of the equivalence radius is proposed to be able to reduce the computational error in the general arrangement case, but it cannot solve the strong gradient situation in the periphery of the beryllium oxide concentrated arrangement. More importantly, the one-dimensional equivalent model method can only achieve the generation of homogenization parameters at the component level, which cannot reflect the non-uniformity of the complex arrangement of local materials in the full core scope, and thus, further optimization of the model may yield limited results. We are currently working on a 3D full-core physical computation method based on 2D full-core component homogenization, aiming at solving the problem of excessive computational resources at the full-core scale, so as to facilitate the subsequent coupling with the thermal engineering direction.

Sun proposed a CFD modeling method applicable to fuel assemblies with 19 fuel rod bundles [29], using a grid deformation-based method for preprocessing, which can greatly reduce the consumption of computing resources. This method can be combined with subsequent three-dimensional coupling with thermal engineering.