1 Introduction

Breeder reactors are considered as a unique tool for fully exploiting natural nuclear resources. Thorium is 3–4 times more abundant than uranium, widely distributed in nature as an easily exploitable resource in many countries. As 232Th is the only isotope present in the thorium ore, enrichment of Th is not necessary for a thorium-fueled reactor [1, 2]. Ever since the inception of nuclear power program, it has been recognized that 232Th is of immense potential for efficiently breeding fissile isotope 233U in a thermal neutron reactor. MSR is the best candidate thermal reactor for thorium–uranium breeding, but researches show that a self-breeder MSRs requires a fast removal fission products (FP) and actinide metals (MA) online and the extraction of Pa and the re-introduction of the formed U [3]. Limited studies were carried out on the use of thorium in fast reactors (FRs), historically conceived as breeder reactors, due to the superiority of the uranium–plutonium fuel cycle. The thorium fuel cycle is known to offer a better neutron economy than the uranium fuel cycle at epithermal energies and does not have a large positive reactivity feedback upon spectrum hardening at high neutron energies because η(233U) does not vary with neutron energy as sharply as η(239Pu), and the fission cross section of 232Th has a higher threshold energy and smaller magnitude than those of 238U [1, 4, 5]. Also the use of thorium-based fuel endures FRs with enhanced inherent safety.

The attractive features of fluoride salts, as a coolant for new reactor types, include lower pressure, higher boiling temperature, optical transparency, higher volumetric heat capacities and lower chemical reactivity [6,7,8,9,10,11,12,13,14,15,16]. MSFR systems with favorable features (large negative temperature and void reactivity coefficients) are perceived as a candidate fast reactor for thorium–uranium breeding. However, since fissile material is dissolved in the molten fluoride salt in an MSFR system, the following problems have been reported [17, 18]:

  1. 1.

    Low fuel loading: poor breeding performance caused by limited fuel solubility in the molten salt;

  2. 2.

    Difficulties in separating Th, MA and FP from the liquid fuel with immature electrochemical separation technology;

  3. 3.

    Risk of radioactive leakage from the pump and heat exchanger due to lack of radioactive inclusion barrier; and

  4. 4.

    Necessary development of advanced neutronic and thermal–hydraulic coupling models in the system design.

Thorium breeder reactors using fluoride salt as a coolant based on solid fuel can avoid these problems. The concept of liquid-salt-cooled solid-fuel fast reactor (LSFR) was proposed by Forsberg in 2005 [19], who promoted the development of advanced high-temperature reactor (AHTR) in 2003 [20] and described the preliminary investigations of LSFRs which combined the AHTR plant design with traditional metal-clad-fuel fast reactor cores and the use of liquid salt coolants to improve fast reactor economics [6]. An LSFR uses solid fuel and a clean liquid salt as coolant. It has two implications [19, 21]:

  1. 1.

    Salt selection. Existing fluoride salt coolants are based on thermal spectrum, and they may not be suitable for fast spectrum. Studies show that 57%NaF–43%BeF2 (FNaBe) salt is a suitable coolant for LSFR core.

  2. 2.

    Material selection. It is necessary to evaluate the structure materials and fuel materials for LSFR core. Two coolants, helium and liquid fluoride salts, have been demonstrated in high-temperature nuclear systems. In the material selection of LSFR, the design of European GFR2400 [22] is used as a reference. The feed fuel is UC/ThC. Carbides have higher heavy metal density than oxides and nitrides, hence better breeding performance. SiC/SiC is used as a cladding material, due to the low neutron absorption rate, high temperature and irradiation resistance, and compatibility with liquid salt.

However, there is not enough research about comprehensive LSFR core neutronics characteristics, due to the lack of a concrete core layout and equilibrium physics analysis. Due to moderating ability of molten salt, Th-U-fuelled FRs possess breeding capability for a soft spectrum, hence the possibility of designing fluoride-salt-cooled fast breeder reactor.

In a typical low-leakage large fast breeder reactor, the breeding fissile material builds up faster near the axial center of the fuel assembly than the bottom and top of the fuel assembly. In researching Breed and Burn (B&B) reactors, Hou et al. at UCB [23] developed a three-dimensional in-core fuel management strategy and the engineering solution for the 3D shuffling by axially relocating the fuel. The 3D fuel shuffling reduced peak burn-up and increased uranium utilization of B&B reactors. In this paper, we present a large LSFR reference core with detailed information on the geometry and dimensions. Both 2D/3D shuffling methods are used for the reference LSFR core based on thorium–uranium fuel, to investigate the thorium breeding ability and analyze the equilibrium physics for important design parameters, such as fuel volume fraction, process parameters for removing fission gases and power density level.

Section 2 is a detailed description of the reference LSFR, including the 2D/3D fuel shuffling strategies and the modeling methodology used. In Sect. 3, equilibrium of core neutronics with 2D and 3D fuel shuffling is given, and the LSFR core parameters of equilibrium state for different influencing factors are presented. Section 4 is conclusion of the study.

2 LSFR core modeling

2.1 LSFR core description

The concept core is a large fluoride-salt-cooled fast reactor of low leakage rate. This LSFR core of Th-U cycle is designed in reference to the experience accumulated in the international development and design of fast reactors. As shown in Fig. 1, the cylindrical core is in similar geometry to the UCB B&B reactor [20], the core dimensions and composition of the regions modeled for the LSFR core are given in Table 1. The effective core height is 300 cm, with equivalent diameter of 504.6 cm, giving a total active core volume of 60.0 m3. The upper and radial reflectors are made of Zr3Si2. The lower and radial shields are made of B4C. The fuel compositions are given in Table 2. The fuel volume fraction of active zone is 0.45.

Fig. 1
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Schematic layout of cylindrical LSFR core

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Table 1 Dimensions and composition of the regions modeled for the LSFR core
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Table 2 Enriched fuel composition for the LSFR core studied
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2.2 Fuel management strategy

In order to breed 233U in low-reactivity assemblies and to decrease power peaking factor (ppf), a twelve-batch scheme is adopted to promote reactivity sharing between low-reactivity assemblies and high-reactivity assemblies. The 12-batch reload sequence is designed to optimize three competing goals. First, the scheme must promote reactivity sharing between partially burnt and fresh fuel assemblies. Second, the reduction of core leakage shall be minimized to improve the neutron economy. Third, a radial power shape shall be as flat as possible to minimize the ppf. After the comprehensive comparison for many different shuffling schemes, an optimal radial shuffling scheme for the reference LSFR core is shown in Fig. 2a. It minimizes the ppf and the burn-up reactivity swings at acceptable low core leakage. To maintain the low core leakage while minimizing the ppf, the 3D shuffling scheme optimization is the same with the principles of Ref. [23]. Figure 2b shows the optimized 3D shuffling scheme.

Fig. 2
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Schematics of an optimal 2D (a) and 3D (b) shuffling schemes for the reference LSFR core. The batch numbers are 1–12

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As the neutron mean free path in typical FRs is larger than the lattice pitch, it is common to represent each burn-up node for neutronic analysis as homogenized fuel, cladding, other structural materials, and the coolant are mixed by their volume fractions. A heterogeneous structure of a fuel assembly with the same fuel composition as the LSFR active zone is shown in Fig. 3, and its geometrical properties and materials are given in Table 3. Table 4 compares results from k inf, mean free path (MFP) and one-group cross section of 232Th and 233U for various residence time between the 0D mixture model and assembly model. The results obtained using a homogenized core model are usually in acceptable agreement with those obtained using a heterogeneous model, but the use of the homogenized model saves computational time significantly [24]. For further reducing the computational time, the numerical modeling adopts a 1/32 symmetry core model in θ direction. The model is divided into 12 equal volumes radially and 8 axially for a total of 96 burn-up zones, each sub-batch is axially divided into 2 burn-up zones. This is acceptable for burn-up calculation and effectively reflecting the axial discharge difference.

Fig. 3
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Cross-sectional view of the fuel assembly for the calculations

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Table 3 Geometrical properties of fuel element and assembly
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Table 4 Physical parameters of 0D model (0 M) and assembly model (AM) for various fuel residence time
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2.3 Calculation procedure

The calculation is based on MCNP5 and MOBAT [25]. MOBAT is a MCNP5-ORIGEN2 coupled program developed at Shanghai Institute of Applied Physics with ENDFB-VII database. Its role is to couple ORIGEN2 with MCNP5 for burn-up analysis for arbitrary core models, and the correctness and its effectiveness has been verified.

The design objectives for fast breeder reactors can be formulated as [1]: (1) high breeding ratio, (2) low fissile inventory and (3) high burn-up fuel. Previous research based on 0D model for the neutron balance equation unique solution could achieve these design objectives. The 233U atom percentage \( \epsilon \) in heavy metal (HM) and discharge burn-up for various leakage rates in the previous results [26] can initialize the parameters: \( \epsilon \) and cycle length T for the core fuel management. The initial concentration of nuclides for different batches in MCNP5 input card implies a hypothesis that the fuel burn-up intervals for different batches are uniform, and therefore, the concentration of nuclides at the same burn-up level in 0D calculation can initialize the core material composition.

Figure 4 shows the flowchart of the equilibrium calculation method for the reference LSFR core. First, the active zone leakage rate can be obtained from the estimate for MCNP5 calculation with the fuel shuffling. Second, according to estimated leakage rate for active zone, the initial \( \epsilon \) and T can be interpolated from the results of 0D model. Third, burn-up analysis is calculated by MOBAT. It is worth noting that using the 0D burn-up calculation to initialize the materials for different batches in MOBAT input card can save computing time for accelerating into equilibrium state. Fourth, the minimum effective multiple factor for equilibrium \( k_{\text{eq,min}} \) should be greater than unity to maintain criticality; 1.001 is assumed for the upper permissible \( k_{\text{eq,min}} \) at equilibrium to reduce \( \epsilon \) value. Or the calculation will modify \( \epsilon \) and T according to existing results. The whole calculations are terminated if the requirements are satisfied.

Fig. 4
figure 4

Flowchart of the equilibrium calculation method for the core

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An additional term FMR (fissile material ratio from t to t 0) is proposed for the 233U fissile material mass at any time t for multiple recycling of recovered thorium and uranium with FMR > 1.0 to a self-sustained core at least. FMR is defined as:

$$ {\text{FMR}}\left( t \right) = \frac{{^{233} {\text{U}}\;{\text{fissile}}\;{\text{material}}\left( t \right)}}{{^{233} {\text{U}}\;{\text{fissile}}\;{\text{material}}\left( {t_{0} } \right)}} $$
(1)

3 Results

3.1 2D and 3D fuel shuffling

LSFR core parameters with a volumetric power density of 50 MW/m3 for a total core power of 3000 MWth using 2D and 3D fuel shuffling are given in Table 5. The reactivity swing for LSFR core based on thorium–uranium fuel cycle is 0.607% with 2D fuel shuffling and 0.542% with 3D fuel shuffling, obviously less than the reactivity of FRs based on U–Pu fuel. With 3D fuel shuffling, the initial 233U ε was reduced to 7.664% due to 233U build up in the axial direction more uniformly.

Table 5 LSFR core parameters using 2D and 3D fuel shuffling at 50 MW/m3
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Figure 5 compares the fuel sub-zone average FMR at EOEC for different fuel batches of the two cores. The position of sub-zone represents batch number from Batch 1 to Batch 12 in horizontal direction and initial fresh fuel subassembly from the bottom to the top of core in vertical direction, which reflects the FMR evolution with fuel batch. Figure 6a shows the 233U content peaks at Batch 6 with FMR = 1.154, while Fig. 6b shows the peaks at Batch 8 with FMR = 1.182. This indicates 3D fuel shuffling improves the utilization of nuclear fuel. Moreover, the average FMR at discharges is 1.090 for 3D shuffling versus 1.013 for 2D shuffling.

Fig. 5
figure 5

FMR evolution of fuel sub-zone with fuel batch for 2D (a) and 3D (b) fuel shuffling

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Fig. 6
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Discharge burn-up of half-subassemblies using 2D and 3D fuel shuffling. A–B are of one subassembly, and so are the other pairs

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The discharge burn-up of eight half-subassemblies with 2D and 3D fuel shuffling is shown in Fig. 6. The 3D discharge burn-up peaking factor is only 1.09, allowing the discharge burn-up distributed quite uniformly over the axial fuel. The maximum–minimum discharge burn-up ratio is close to 2. The 3D fuel management strategy can reduce the peak burn-up and corresponding radiation damage of fuel cladding.

3D fuel shuffling requires maintaining the low core leakage, while minimizing the power peaking. The leakage rate at EOEC is 3.0% with 3D shuffling, while it is 3.7% with 2D shuffling. The ppfs of 2D and 3D fuel shuffling are close to each other. The axial flux distribution along the core at EOEC with 2D and 3D fuel shuffling is presented in Fig. 7. The flux peak occurs in the center position of the core for both the fuel shuffling, but the 3D shuffling has higher flux peak and lower flux at both ends, hence lower leakage rate.

Fig. 7
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Axial flux distribution along the core at EOEC with 2D and 3D fuel shuffling

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As 3D shuffling offers better breeding capability, the parameters using just 3D shuffling are given in following sections.

3.2 Equilibrium core for different design parameters

3.2.1 Fuel volume fraction

Many of the inter-connected parameters of the fast reactor core design can be attributed to the fuel volume fractions which affect the spectrum and parasitic absorption. LSFR core parameters for various fuel volume fractions are given in Table 6 with the total percentage of the fuel and active coolant volume percentage fixed as 65%.

Table 6 LSFR core parameters for various fuel volume fractions with 3D fuel shuffling
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From Table 6, the core of high fuel volume fractions requires low 233U fissile inventory and achieves good breeding performance. Taking the core of 0.60 fuel volume fraction for example, the initial 233U atoms percentage in HM is 5.0%, the breeding ratio is 1.153 at BOEC and 1.114 at EOEC, which leads to high average FMR at discharge of 1.433. The core of high fuel volume fraction reduces the parasitic absorption rate, hence low leakage rate and good neutron economy. At the fuel volume fractions of 0.45, 0.50 and 0.60, the leakage rates at EOEC of active zone are 3.00, 2.67 and 2.07%, respectively. However, at high fuel volume fraction, the reactivity swing, discharge burn-up peaking factor, peak radiation damage and ppf are greater. The axial discharge burn-up becomes more uneven as the fuel volume fraction increases.

Figure 8a shows the discharge burn-up of eight half-subassemblies for fuel volume fractions of 0.45–0.60. The discharge burn-up peaking factors are 1.09, 1.11 and 1.21 for fuel volume fractions of 0.45, 0.50 and 0.60, respectively. The ppf at BOEC reaches up to 5.67 at 0.60 fuel volume fraction, and this is obviously unacceptable. Moreover, at 0.60 fuel volume fraction, the peak cladding radiation damage is 285.23 dpa (displacements per atom), which exceeds the presently demonstrated clad radiation damage limit of 200 dpa. Figure 8b shows the normalized neutron flux for fuel volume fraction of 0.45–0.60. It can be seen that the core with high fuel volume fraction has hard energy spectrum.

Fig. 8
figure 8

Discharge burn-up of eight half-subassemblies (a) and normalized neutron flux (b) for fuel volume fractions of 0.45–0.60

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3.2.2 Power density level

At 50 MW/m3 power density and 0.45 fuel volume fraction, the cycle length is 1530 EFPD due to long retention time in the core for fuel assembly. Improving power density can reduce cycle length and increase the plant economy. Table 7 lists the LSFR core parameters at 100 MW/m3 power density level for fuel volume fraction of 0.45–0.60. The initial 233U atoms percentage in HM for the core at 100 MW/m3 power density is slightly larger due to smaller average FMR at discharges. The average discharge burn-up is 18.20% FIMA at 100 MW/m3, while it is 23.60% FIMA at 50 MW/m3. At 100 MW/m3 power density, the leakage rate is slightly larger, but the ppf and peak radiation damage are smaller. The difference of average discharge burn-up for the two power density levels diminishes as the fuel volume fraction increases. This indicates that the core for high fuel volume is conducive to the operation of the reactor at large power density level.

Table 7 LSFR core parameters at 100 MW/m3 power density level with 3D fuel shuffling
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Figure 9 shows the axial flux distribution along the core at EOEC for power densities of 50 and 100 MW/m3 at 0.45 fuel volume fraction. The flux at 50 MW/m3 is more flatting than at 100 MW/m3. The peak flux for 50 MW/m3 is 2.31 × 1014 at 135 cm from the bottom, while it is 4.26 × 1014 at 145 cm for 100 MW/m3.

Fig. 9
figure 9

Axial flux distribution along the core at EOEC for power density of 50 and 100 MW/cm3 at 0.45 fuel volume fraction

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3.2.3 Continuous removal of fission gases

Removal fractions for a variety of volatile fission products in FRs have been studied [27]. In this section, the continuous 100% (except for I, 75%) removal of fission product noble gases (Xe and Kr) and volatile gases at high temperatures (Cs, Cd, Rb and I) is assumed in every burn-up calculation through a nuclear process off-gases system, possibly helium bubbling [28]. Table 8 lists the LSFR core parameters using 3D fuel shuffling with continuous removal of fission gases at 0.45 fuel volume fraction. With continuous removing volatile fission products, LSFR core achieves a good breeding performance, with increased nuclear fuel utilization. There is little difference between the average FMR at discharges with and without removing fission gases, but clearly the average discharge burn-up and the peak FMR become greater with removing fission gases. At 50 MW/m3 power density, the initial 233U atoms percentages in HM are 30.54 and 23.6% FIMA, and the peak FMRs are 1.294 (at Batch 6) and 1.182, without removing fission gases, respectively. With the removal fractions for a variety of volatile fission gases, the leakage rate decreases slightly due to the reduction in the neutron parasitic absorption of fission products.

Table 8 LSFR core parameters using 3D fuel shuffling with continuous removal of fission gases
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3.3 Temperature reactivity coefficients

The temperature coefficients include the reactivity coefficient of fuel (or Doppler coefficient) T F, the coolant T C, the structure material in active zone T S and the reflector regions T R. The results for 0.45 fuel volume fraction at BOEC and EOEC with 3D shuffling are presented in Table 9. It can be seen that although the coolant displays a foremost positive reactivity coefficient part, the overall value is negative due to the large negative contribution of the fuel. The T F value decreases with burn-up, but T C has little relationship from BOEC to EOEC. The coolant and fuel reactivity coefficient as a function of fuel volume fraction at EOEC with 3D shuffling are shown in Fig. 10. The total temperature reactivity coefficient for all cases is negative which shows the enhanced inherent safety of the LSFR core with thorium–uranium fuel.

Table 9 LSFR core temperature reactivity coefficients for 0.45 fuel volume fraction at BOEC and EOEC with 3D shuffling
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Fig. 10
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Coolant (a) and fuel (b) coefficient at EOEC as a function of fuel volume fraction

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4 Conclusion

Both 2D and 3D in-core fuel management strategies are adopted for equilibrium analysis of the LSFR core. The 3D fuel shuffling method for the LSFR core equilibrium analyzes parameters such as fuel volume fractions, power density level and continuous removal of fission gases. The main conclusions are as follows:

Both 2D and 3D shuffling strategies can achieve a self-sustaining core, but 3D fuel shuffling can reduce the initial fissile inventory, flatten discharge burn-up and achieve a good breeding performance.

At 50 MW/m3, comparing to the core of 0.45 fuel volume fraction, the core of 0.60 fuel volume fraction offers about 35% reduction in 233U fissile inventory, 15.7% increase in the average discharge burn-up, 31% reduction in the leakage rate of active zone and 31.5% increase in average FMR at discharges, but brings larger discharge burn-up peaking factor, larger cladding peak radiation damage and the obvious ppf increase. The core of 0.60 fuel volume fraction at 50 MW/m3 power density has the best breeding performance of all cases, with an average breeding ratio of 1.134, while it is 1.062 for the core of 0.45 fuel volume fraction. It explains that the breeding capacity of the LSFR core based on thorium fuel is limited.

By doubling the power density level, the LSFR core has more than half reduction in cycle length and receives a 10–15% reduction in ppf and 21% reduction in cladding peak radiation damage for 0.45 fuel volume fraction. The peak cladding radiation damage for the core of 0.60 fuel volume fraction at 50 MW/m3 power density is 285.23 dpa, while at 100 MW/m3 it is 276.03 dpa, which exceeds the presently demonstrated clad radiation damage limit of 200 dpa. Moreover, low power density promotes the breeding behavior and is beneficial to smaller reactivity swing.

The employment of continuous removal of fission gases processing can reduce the 233U fissile inventory, increase discharge burn-up and obtain a good breeding performance significantly.

The design of a large fluoride-salt-cooled fast breeder reactor based on thorium–uranium fuel is feasible, but the breeding capacity is limited. It achieves: (1) a high discharge burn-up of 20–30% FIMA, (2) small reactivity swing over whole lifetime to improve the safety and simplify the reactivity control system, (3) the negative reactivity temperature coefficient for all cases with intrinsically safe and (4) the accepted cladding peak radiation damage. These characteristics of LSFR core provide a good alternative option for the deployment of a self-sustained thorium-based nuclear energy system in the future.

Future studies will give a simple thermal hydraulic analysis and the final reference design for LSFR core.