In two-dimensional lattice calculation, the subgroup method with ultrafine groups was implemented in the two-dimensional resonance self-shielding by incorporating MOC to get the accurate self-shielded cross-sections. In FSAR, the number of the ultrafine energy group is 2164 and that of subgroups is determined according to the change amplitude of the cross section (maximum is three groups). For the heterogeneous system, the neutron source term is assumed to be isotropic and the steady Boltzmann transport equation shows as follows: Ω → ⋅ ∇ ϕ r → , E + Σ t r → , E ϕ r → , E = Q r → , E (1) where Q is the neutron source. Based on the treatments of the neutron source, the subgroup method can be divided into the statistical model and the fixed source model (Takeda et al., 1991).

In the statistical model, the probability that the neutron source in group g lies in the subgroup sg is equal to the subgroup probability of sg. The corresponding subgroup transport equation is simplified as follow: Ω → ⋅ ∇ ϕ s g , g r → + Σ t , s g , g r → ϕ s g , g r → = Q s g , g r → = p s g , g Q g r → (2) where p _sg,g is the subgroup probability of sg. The neutron source term Q includes scattering source and fission source and the detail flux spectrum is required for the calculation of the scattering source and the fission source. In this way, the resonance calculation and the transport calculation are coupled together in the statistical model and completed in one time.

In the fixed source model, the narrow resonance approximation is implemented, the corresponding subgroup transport is simplified as below: Ω → ⋅ ∇ ϕ s g , g r → + Σ t , s g , g r → ϕ s g , g r → = Σ p r → 4 π (3) where Σ_p is the macroscopic potential cross section.

Compared to the statistical model, the fixed source model has better parallel performance. The subgroup flux of different subgroups in the entire energy range is coupled with each other, and can be calculated together. The self-shielding cross section is collapsed as: σ x , g r → = ∑ s g ∈ g p s g , g σ x , s g , g ϕ s g , g r → ∑ s g ∈ g p s g , g ϕ s g , g r → (4)

Comparing the above two models, it can be noticed that the statistical model needs iterations, while the fixed source model does not. It means that the latter requires fewer fixed source iterations. In the lattice calculation, the bondarenko iteration (Stamm’ler and Abbate, 1983) is used to deal with the resonance interference effect. When one isotope is treated as a resonance isotope, the other isotopes are treated as non-resonant isotopes, and the subgroup total cross section is written as follows: Σ t , s g , g r → = N r r → σ t , s g , g + ∑ k ≠ r N k r → σ t , g , k r → (5)

The subgroup flux is then calculated by Eq. 3. The new self-shielding cross section of isotope r is updated by Eq. 4. The flow chart of the resonance calculation is shown in Figure 1.

The single assembly problems with reflective boundary condition are used to determine the homogenized cross-sections of fuel material. For better consideration of the strong space coupling in different geometric sizes due to different mean free path of neutrons, the super homogenization method (Zhang et al., 2016) is optional to be used as the homogenization technique. For the structural materials and other non-fuel materials, the super cell or assembly models are used to determine the SPH factors and the cross-sections. The cross-sections are modified as follows: Σ ∼ i , g hom = μ i , g Σ i , g hom (6)

Based on the multi-group cross-sections generated by the lattice calculation, the three-dimensional whole core calculation is carried out by the core simulator. The S _N method with triangular grid is applied as the solver of the transport equations. The three-dimensional multi-group neutron transport equation within the triangular prism grid can be written as follows, which assumes that the fission source is isotropic and scattering sources is anisotropic in the derivation. μ m ∂ Ψ g m x , y , z ∂ x + η m ∂ Ψ g m x , y , z ∂ y + ξ m h z ∂ Ψ g m x , y , z ∂ z + Σ t g Ψ g m x , y , z = Q ^ g x , y , z (7) where m represents a certain angular direction, μ m , η m , ξ m is the component of the angular direction m on the coordinate axis (x, y, z), Ψ g m x , y , z is the angular flux of the g-th group.

For the reactor system without outer neutron source, the source Q ^ g includes fission source and scattering source. The fission source is usually assumed to be isotropic, while the scattering source is anisotropic in reactor system. The assumption that the scattering source is isotropic limits the application of the triangular discrete ordinate nodal method in practical problems. The isotropic scattering source written as follows is considered in FSAR: Q s r , E , Ω = ∫ 0 ∞ ∫ 4 π Σ s r , E ′ f r , E ′ → E , Ω ′ → Ω ψ r , E ′ , Ω ′ d Ω ′ d E ′ (8) where f represents the scattering function.

The micro-depletion scheme is applied to simulate the core burn up and the Chebyshev Rational Approximation Method is used to solve the depletion equation. In FSAR, the depletion chains containing 21 heavy isotopes and 49 fission products is provided for the advanced reactor core system.

UO₂ and MOX pin cell problems from JAEA benchmark (Yamamoto et al., 2002) are utilized for the thermal spectrum and the geometry is shown in Figure 2. The two problems have the same mesh division. In the radial direction, the fuel, clad and moderator are divided into three and one and two equal-volume rings respectively. Each ring is divided into 8 equal-volume sectors. The MOC parameters are set to be three polar angles and 16 angles per octant, and 0.01 cm ray spacing. The multigroup transport equation with P₀ scattering was solved with the pure MOC calculation without any accelerations. In the calculations, the inter-molecular thermal scattering effect is not considered.

For the thermal spectrum problems, the results of eigenvalue are presented in Table 1, and the self-shielding cross section errors are shown in Figures 3, 4, 5. In the table, the errors of eigenvalue are calculated as follows: E r r k = k M O C − k r e f ∗ 10 5 (9)

In the figures, the errors of cross section are calculated as follows: E r r x s = σ g , M O C − σ g , r e f ∗ ϕ g , r e f ∑ g ∈ G σ g , r e f ϕ g , r e f * 10 5 (7b)

It can be found that the cross-sections have high accuracy for most groups. However, in the UO₂ problem, the absorption cross section of ²³⁸U has a large deviation at 20 eV energy. While the absorption cross section error of ²³⁸U at 20 eV energy of the MOX problem is smaller. This is because the MOX fuel including serval Pu isotopes hardens the neutron flux spectrum, as shown in Figure 6. Furthermore, affected by the harder flux spectrum, there is no significant increase in the errors of eigenvalue for the MOX fuel problem, even the obvious differences of the cross-sections appear in the energy range between 10 and 100 eV.

In order to determine the error source of the eigenvalue, the cross-sections tallied from OpenMC are used to get the macroscopic cross-sections. The multi-group model in OpenMC is used to perform the transport calculation. The results are shown in Table 2. It can be found that the result of FSAR and multi-group OpenMC is close and the anisotropic scattering effect would be the main error source.

The hexagonal UO₂ cell with high enrichment (Lee, 2014) is utilized as the intermediate spectrum problem. The graphite is used as the moderator. The geometry is shown in Figure 7. Both fuel and moderator are divided into one ring. Each ring is divided into 6 equal-volume sectors.

For the intermediate spectrum problem, the reference eigenvalue obtained by OpenMC is 1.28829 ± 8 pcm and the value of FSAR is 1.28465. The results of cross-sections are shown in Figure 8. The error results are similar to that of the UO₂ problem. There is also a significant deviation at 20 eV energy and small deviations at other positions. It can be explained by the invalidation of narrow approximation around 20 eV.

In order to test the performance of FSAR in the fast reactor, the calculations of UO₂ cell with high enrichment and MOX cell are carried out. The same hexagonal geometry of the two problems is shown in Figure 9. The clad material is SS 316 alloy, and two conventional coolants in fast reactors, sodium and potassium, are selected in this section. The compositions of the materials are obtained from the reference (Li et al., 2017). Finally, two fast spectrum problems are formed, expressed as MOX-SS316-Na and UO₂-SS316-K respectively.

The eigenvalue results are shown in Table 3. The errors of cross-sections are shown in Figures 10, 11, 12. The numerical results indicate that the calculation in fast spectrum problems has higher accuracy than that of other problems. This is because the resonance peak is narrower in the high energy range and the resonance effect treatment is more applicable.