This research investigation delves into the generation of diffusion coefficients and two-group constants for both 2D and 3D models representing fuel assemblies and detailed core configurations of the Chinese Small Modular Reactor (SMR) known as ACP-100. This study encompasses six distinct fuel types, employing various methods to produce diffusion coefficients available in SERPENT. The objective is to identify the most appropriate constants for solving the two-group diffusion equation. The ACP-100 (Access Control Point-100), characterized by an innovative Pressurized Water Reactor (PWR) design, incorporates a passive safety system and integrated reactor design technology, anticipating an electrical power output within the range of 100–125 MWe (Zhong 2014; Ishraq et al. 2024b). The ACP-100 reactor core consists of 57 partial-height fuel assemblies (FAs), each housing 264 fuel elements. Table 1 presents the principal design and technical parameters of the reactor core employed in this research endeavour. The configurations of the fuel element and fuel rod adhere to the standards of a typical Pressurized Water Reactor (PWR) assembly. The study encompasses six distinct fuels, comprising three variations of UOX with different enrichment, homogeneous Th, and both Weapon and Reactor Grade (WG, RG) MOX. A detailed description of these fuels is available in Table 2, while comprehensive details regarding the considered core models are provided in Table 3 (Ishraq et al. 2024a). Fig. 1 offers visual depictions of a standard fuel pin and radial view of an assembly. The radial geometries of the cores are presented in Fig. 2. Additionally, Fig. 3 illustrates the axial geometric views of both 2D. 3D mesh model of the core built in COMSOL Multiphysics is shown in Fig. 4.

Figure 1. 

Diagrams of a typical fuel element and a fuel assembly. (a) Representative fuel pin/element. (b) Fuel assembly.

Figure 2. 

Radial view of the considered core models. (a) Models 1, 2, 3 and 4. (b) Models 5 and 6.

Figure 3. 

Axial view of the 3D FA model.

Figure 4. 

3D mesh model of the core modeled in COMSOL.

The main design/technical parameters of the reactor core utilized in this research work are presented in Table 1.

In this study, 2D and 3D models of Fuel Assemblies (FAs) were constructed and detailed core models were developed for six distinct fuel configurations using SERPENT. To obtain essential two-group constants and diffusion coefficients, three distinct approaches were employed: the Out-Scattering Approximation (OSA), the Cumulative Migration Method (CMM), and the Hydrogen Transport Correction Method (TRC). The history of 50,000 neutrons were tracked for 3000 cycles/batches (the first 200 of which were skipped) with reflective boundary conditions for assembly models and vacuum boundary conditions for core models in this study. The constants derived from these approaches served as the input parameters for solving the two-group diffusion equation for the core model. The numerical simulations were then conducted within the framework of COMSOL Multiphysics, which allowed for the determination of the eigenvalue, specifically the effective multiplication factor (k_eff). To support the validation of using the two codes a comparative analysis of spatial (Radial) flux distribution was presented for the fuel with 4.45% enrichment. To get higher statistical accuracy in SERPENT and produce more accurate group constants 50,000 neutron population with 5,00,000 cycles (first 200 were skipped) was used.

The impacts of various fuels and soluble boric acid on diffusion coefficients and two-group constants were also investigated. The analysis involved the construction of a 315-energy group spectrum for diverse fueled cores, considering various zones within the core. Soluble boric acid (H₃BO₃) is a very important chemical compound to control excess reactivity. So, the effect of H₃BO₃ for six distinct fuels and their group constants in the reflector zones using TRC and OSA methods were observed. The sequential methodological approach utilized in this study to obtain the desired results is visually represented in Fig. 5 as a flowchart. Table 4 outlines diverse models employed for the generation of diffusion coefficients and group constants within the SERPENT framework. Subsequently, these parameters were utilized as input in COMSOL Multiphysics for the solution of Partial Differential Equations (PDEs). In this study assembly discontinuity factor (ADF) was not considered. The two-group diffusion equations, that were used in this study are shown bellow (Stacey 2018):

Figure 5. 

Flow chart describing the sequence of steps followed in the analysis.

$-\nabla ·{\mathrm{D}}^1\nabla {\Phi }_1\left(\overrightarrow{\mathbf{r}}\right)+\left({\Sigma }_a^1+\sum _s^{1\to 2}\right){\Phi }_1\left(\overrightarrow{\mathbf{r}}\right)=\frac{1}{k}\left(v_1{\Sigma }_f^1{\Phi }_1\left(\overrightarrow{\mathbf{r}}\right)+v_2{\Sigma }_f^2{\Phi }_2\left(\overrightarrow{\mathbf{r}}\right)\right)$ (1)

$-\nabla ·{\mathrm{D}}^2\nabla {\Phi }_2\left(\stackrel{\rightharpoonup }{\mathbf{r}}\right)+\sum _a^2{\Phi }_2\left(\overrightarrow{\mathbf{r}}\right)=\sum _s^1\to 2{\Phi }_1\left(\stackrel{\rightharpoonup }{\mathbf{r}}\right)$ (2)

where D¹ and D² are diffusion coefficients, ${\Sigma }_a^1$ and ${\Sigma }_a^2$ are absorption cross-sections, ${\Sigma }_f^1$ and ${\Sigma }_f^2$ are fission cross-sections, υ₁ and υ₂ are average neutron yield, ${\Phi }_1\left(\overrightarrow{\mathbf{r}}\right)$ and ${\Phi }_2\left(\overrightarrow{\mathbf{r}}\right)$ are neutron flux for energy group 1 and 2 respectively, ${\Sigma }_S^{1\to 2}$ is scattering cross-sections from energy group 1 to 2. Default two-energy-group structure of SERPENT was used in this study which is separated at 0.625eV (Leppänen et al. 2015).

Effective multiplication factor k_eff serves as a quantitative measure, representing the criticality state of the system. Within Table 5, a comparative analysis is presented, comparing the k_eff values derived from SERPENT and COMSOL Multiphysics using two-group constants obtained from diverse models of fuel assemblies, each representative of distinct fuel compositions generated from SERPENT. The statistical uncertainty associated with the k_eff values calculated using SERPENT was less than 3 pcm for all fuel types. For UOX-based fuel, models 2, 3, 4 show close results, while the deviations from the Monte Carlo calculation in these models are the smallest. For options with weapons-grade and reactor-grade plutonium, a higher deviation is observed, but still not exceeding 0.4%, which can be considered a good result. The higher deviation can be explained by the fact that a larger proportion of fissile nuclide is present in the fuel elements with Pu, which creates higher non-uniformity throughout the fuel assembly. This can also explain the better agreement between the results when using the CMM method to calculate the diffusion coefficient, this is especially evident for weapons-grade plutonium.e multiplication factor, denoted as k_eff, is precisely defined as the quantitative relationship between the number of neutrons generated from fission events within each succeeding generation and the corresponding amount of neutrons absorbed in the antecedent generation, within the context of a finite medium.

Two-group diffusion coefficients and macroscopic cross-sections for fuel and non-fuel materials were computed across various configurations of detailed cores containing different fuel materials. The diffusion coefficients were determined through the application of both the Out-scattering Approximation (OSA) and the Transport Correction Method (TRC) for non-fuel components. For fuel materials, only the default OSA method was used. For all the fuel types, the effective delayed neutron fraction (β_eff), which is an important safety parameter was calculated. The outcomes of these calculations are delineated in Tables 6–9. It is found that fuel materials do not affect the constants of radial water. So, for different fuels, it is not necessary to prepare constants for them. However, variations of fuels have noticeable effects on top and bottom reflectors and Zircaloy layers. Hence, these constants must be prepared in case of different fuels. Additionally, it is noteworthy that none of the constants exhibit significant variations across differently enriched UOX fuel compositions.

The radial distributions of thermal and fast neutron fluxes were analyzed and compared between the two programs. The results are presented in Fig. 6. Fig. 6(a) illustrates the thermal flux distribution across half of the active zone. A noticeable increase in neutron population can be observed at the fuel zone boundary, primarily attributed to the radial reflector’s neutron moderating properties. The maximum relative error in thermal flux between the two programs was approximately 10%. Fig. 6(b) depicts the radial flux distribution of fast neutrons, with a maximum relative error of around 3%.

Figure 6. 

Radial Flux distribution of thermal and fast neutrons across the half of the active zone.

The neutron flux spectrum within a nuclear reactor determines the distribution of neutron energies, exerting a direct influence on fission, capture, and scattering reactions, as well as critical reactor parameters such as reactivity and power density. Fig. 7 illustrates the normalized neutron flux per unit lethargy for designed core models at the inception of their operational life (BOL), featuring various fuels. To achieve this, the SERPENT code utilized a predefined group structure known as “Tripoli 315-group structure” (Zheng et al. 1998). It can be observed that no strong resonance is found in the thermal energy zones. So for thermal reactors, the two-group approximation can produce results with sufficient accuracy. 3% UOX fuel has the softest neutron energy spectrum whereas MOX-RG has the hardest spectrum. For thermal neutrons, the absorption cross-sections for ²³⁸U, ²³⁵U and ²³⁹Pu are 2.68, 99 and 270 barn respectively obtained from JANIS 4.0 (Lamarsh and Baratta 2001; Soppera et al. 2014). With the increase in enrichment, the fraction of ²³⁵U with higher absorption cross-section increases which hardens the neutron spectrum. In the case of MOX fuels, ²³⁹Pu has an even higher absorption cross-section. As a result, MOX-RG has the hardest spectrum due to the higher mass fraction of ²³⁹Pu (Table 3).

Figure 7. 

Spectrum of the core for different fuels.

Boric acid (H₃BO₃) is an important chemical compound to control excess reactivity. The presence of soluble boric acid in the moderator affects the group constants. As ¹⁰B has a higher absorption cross-sections over the entire energy range, it has effects on both thermal and fast constants. In this study, the impact of boric acid (enriched to 19.9% with ¹⁰B) on six distinct fuel types was examined across a concentration gradient of boric acid ranging from 900 ppm to 4700 ppm (900 ppm to 7500 ppm for MOX-RG) to show results in both supercritical and subcritical state. Comparison of the effectiveness of boric acid on different fuels is presented in Fig. 8. It can be observed that the effectiveness of boric acid to control reactivity is lower for MOX fuels, especially for MOX-RG. While for 4700 ppm of boric acid all the fuels show subcriticality (k_eff < 1), MOX-RG still remains in a supercritical state (k_eff = 1.067). For MOX-RG, the spectrum became more harder (Fig. 7), as Pu has comparatively higher absorption cross-sections than U and Th. Consequently, it reduces the effectiveness of boron regulation. So, additional analysis was undertaken with higher concentrations of boric acids in order to observe critical and subcritical states. For all of these concentrations, group constants were generated using TRC and OSA methods in SERPENT and those constants were used to determine k_eff from COMSOL Multiphysics. The comparisons of k_eff obtained from both programs are presented in Fig. 9. For all of the fuels, constants of the TRC method provide more accurate results except for MOX-RG. The maximum relative error of k_eff is less than 0.25% for the TRC method and around 0.45% for the OSA method. For MOX-RG, OSA method demonstrates higher accuracy. The maximum relative error is 0.86% for the OSA method and 1.06% for TRC method.

Figure 8. 

Dependence of k_eff on the concentration of boric acid for six distinct fuels.

Figure 9. 

Comparison of k_eff dependence on boric acid concentration from SERPENT and COMSOL for different fuels. (a) UOX 4.45%, (b) UOX 4%, (c) UOX 3%, (d) Th, (e) MOX-WG, (f) MOX-RG.