2.1 Theory of the DBRC method

Standard CXS scattering kernel ignores the energy-dependent scattering cross section during sampling of the target velocity. But when the target nucleus is resonant, such as ²³⁸U, the CXS is not reasonable. The DBRC method^[12] has been adopted in the latest continuous energy Monte Carlo code MCNP6 to consider the energy-dependent elastic scattering cross section during the elastic scattering events^[13]. The improved target probability density function in the DBRC method can be described as:

where v_n is the speed of incident neutron, v_r is the relative speed between incident neutron and target nucleus, M_t is the mass of target, V is the speed of the target, v_ξ is the dimensionless speed, and ${\sigma }_{\text{S}}^{\text{MAX}}$ (v_ξ,0) is the maximum value of elastic scattering cross sections within a range of the dimensionless speed determined by Eq.(3). In Eq.(1), μ is uniformly sampled from [-1, 1], V is sampled from $f_1(V)$ with probability P₁ and from $f_2(V)$ with probability P₂, v_r/(V+v_n) and σ_S(v_r,0)/${\sigma }_{\text{S}}^{\text{MAX}}$ (v_ξ,0) are the first and second rejection test applied to the sampled velocity of the target. The energy-dependent scattering cross section kernel model based on the DBRC method is implemented in PSG2/Serpent to accurately calculate the resonant elastic scattering of the heavy nuclei target.

2.2 Comparison of the DBRC and CXS for scattering kernel

²³⁸U is the most important heavy nuclei target because it has pronounced resonances in the elastic cross section in the epithermal range^[9]. Therefore, the first resonance of the ²³⁸U is analyzed in this section. Assuming an ideal case, the target material is a single nucleus of ²³⁸U in an infinite medium of constant density. The infinite medium is fixed at 1200 K, and the nuclear data library is based on ENDF/B-VII.0. Grey dashed line in Fig.1(a) represents the monochromatic energy of the incident neutron, 6.52 eV, which is lower than the first elastic scattering resonance peak at 6.78 eV. The average emerging neutron energy distribution is also given in Fig.1(a). It can be found that both the CXS model and the energy-dependent scattering cross section model based on DBRC method permit up-scattering. However, the probability of up-scattering in the DBRC method is higher than that in the CXS model, which means that more neutrons will be captured in the resonance peak of the ²³⁸U.

Fig. 1Full-screen View

(Color online) Comparison of normalized elastic scattering kernels using CXS and DBRC on ²³⁸U with incident energy 6.52 eV (a) and 6.90 eV (b)

Grey dashed line in Fig.1(b) represents the monochromatic energy of the incident neutron, 6.90 eV, which is larger than the first elastic scattering resonance peak at 6.78 eV. The average emerging neutron energy distribution is also given in Fig.1(b). It can be found that the probability of down-scattering in the DBRC method is also higher than that in the CXS model, which means that more neutrons will be captured in the resonance peak of the ²³⁸U.

Based on the above phenomenon, we can predict that if the DBRC is used, the neutron capture will increase and the multiplication coefficient decrease in thermal reactors.

2.3 Validation of the DBRC method

Before the application of the DBRC method, the Mosteller benchmark problem^[18] for LWR pin-cell with UOX fuel and MOX fuel is adopted to validate the DBRC implemented in Monte Carlo code PSG2/Serpent ^[19]. The results calculated with the use of Monte Carlo code TRIPOLI-4 have been used for the comparison.

It was found that k_inf calculated by the TRIPOLI-4 and PSG2/Serpent agree well; k_inf decrease if the DBRC model is adopted. However, there are some minor deviations caused by different codes, different cycles, and other reasons. From the Tables 1, 2, 3 and 4, compared with the results obtained by TRIPOLI-4, the energy-dependent scattering cross section scattering model based on the DBRC method which is implemented in PSG2/Serpent can accurately calculate the resonance scattering. Therefore, the PSG2/Serpent code with DBRC method can be used to evaluate the resonant scattering effect of the PB-FHR.

3.1 PB-FHR unit cell

The PB-FHR pebble unit cell is composed of randomly distributed TRISO coated fuel particles. In this work, the PB-FHR pebble unit cell was accurately modeled by the Monte Carlo code PSG2/Serpent using the explicit random geometric model^[20,21].

The impact of the energy-dependent scattering cross section model based on DBRC method on the burnup calculation for the PB-FHR unit cell with different TRISO packing factor (TPF) was investigated by performing two simulations: one with the original CXS scattering and another with the improved DBRC scattering. In the burnup calculation of the PB-FHR unit cell, temperatures of the fuel, coolant, and moderator were assumed to be 1200 K. Comparison of CXS and DBRC on the infinite multiplication factor during burnup for TPF=7.03% can be observed in Fig.2. The difference between CXS and DBRC on the infinite multiplication factor during burnup for TPF=7.03% is also shown in Fig.2; it can be found that the infinite multiplication factor decreases by 140~320 pcm during burnup if the DBRC is adopted. In addition, if the DBRC method is applied, more neutrons will be captured by ²³⁸U; therefore, more ²³⁹Pu will be generated. The comparison of CXS and DBRC on the ²³⁹Pu concentration during burnup for TPF=7.03% can be observed in Fig.3. The difference between CXS and DBRC on the ²³⁹Pu concentration during burnup for TPF=7.03% is also shown in Fig.3; it can be found that the ²³⁹Pu concentration increases by 1.2%~1.6% during burnup if the DBRC is adopted, that is to say, the computational accuracy of the ²³⁹Pu concentration increases by 1.2%~1.6%.

Fig. 2Full-screen View

(Color online) Comparison of k_inf during burnup for the unit cell with TPF=7.03%

Fig. 3Full-screen View

(Color online) Comparison of ²³⁹Pu concentration during burnup for the unit cell with TPF=7.03%

Similarly, comparison of CXS and DBRC on the infinite multiplication factor during burnup for TPF=15% can be observed in Fig.4. The difference between CXS and DBRC on the infinite multiplication factor during burnup for TPF=15% is shown in Fig.4; it can be found that the infinite multiplication factor decreases by 400~570 pcm during burnup if the DBRC is adopted. In addition, comparison of CXS and DBRC on the ²³⁹Pu concentration during burnup for TPF=15% can be observed in Fig.5. The difference between CXS and DBRC on the ²³⁹Pu concentration during burnup for TPF=15% is shown in Fig.5 too; it can be found that the ²³⁹Pu concentration increases by 1.4%~2.2% during burnup if the DBRC is adopted.

Fig. 4Full-screen View

(Color online) Comparison of k_inf (a) during burnup for the unit cell with TPF=15%

Fig. 5Full-screen View

(Color online) Comparison of ²³⁹Pu concentration during burnup for the unit cell with TPF=15%

In addition, comparison of CXS and DBRC on the infinite multiplication factor during burnup for TPF=30% can be observed in Fig.6. The difference between CXS and DBRC on the infinite multiplication factor during burnup for TPF=30% is shown in Fig.6; it can be found that the infinite multiplication factor decreases by 360~730 pcm during burnup if the DBRC is adopted. In addition, comparison of CXS and DBRC on the ²³⁹Pu concentration during burnup for TPF=30% can be observed in Fig.7. The difference between CXS and DBRC on the ²³⁹Pu concentration during burnup for TPF=30% is also shown in Fig.7; it can be found that the ²³⁹Pu concentration increases by 1.9%~2.7% during burnup if the DBRC is adopted.

Fig. 6Full-screen View

(Color online) Comparison of k_inf during burnup for the unit cell with TPF=30%

Fig. 7Full-screen View

(Color online) Comparison of ²³⁹Pu concentration during burnup for the unit cell with TPF=30%

3.2 PB-FHR full core

The PB-FHR is composed of randomly distributed fuel pebbles which consist of fully randomly distributed TRISO coated fuel particles; the PB-FHR full core is accurately modeled by the Monte Carlo code PSG2/Serpent using the random geometric model.

The impact of the energy-dependent scattering cross section model based on the DBRC method on the burnup calculation for the PB-FHR full core with TPF=7.03% is investigated by performing two simulations: one with the original CXS scattering and another with the improved DBRC scattering. In the burnup calculation of the PB-FHR full core, temperatures of the fuel, coolant, and moderator are assumed to be 1200 K. Comparison of CXS and DBRC on the effective multiplication factor during burnup can be observed in Fig.8. The difference between CXS and DBRC on the effective multiplication factor during burnup for the PB-FHR full core is shown in Fig.8; it can be found that the effective multiplication factor decreases by 144~205 pcm during burnup if DBRC is adopted. In addition, comparison of CXS and DBRC on the ²³⁹Pu concentration during burnup for the PB-FHR full core can be observed in Fig.9. The difference between CXS and DBRC on the ²³⁹Pu concentration during burnup for the PB-FHR full core is also shown in Fig.9; it can be found that the ²³⁹Pu concentration increases by 1.1%~1.3% during burnup if DBRC is adopted.

Fig. 8Full-screen View

(Color online) Comparison of k_eff during burnup for the PB-FHR full core

Fig. 9Full-screen View

(Color online) Comparison of ²³⁹Pu concentration during burnup for the PB-FHR full core