Keywords

1 Introduction

Neutron diffusion equation is a great simplification of Boltzmann transport equation, while it can obtain sufficient accuracy with less requirements for computer resources, so neutron diffusion theory is still a main method to solve the neutron flux at full-core scale. Nodal method is widely used in most neutron diffusion codes like DIF3D [1], DYN3D [2], and SIMULATE-4 [3], while which is different from the numerical methods of common Thermal-Hydraulic code, so coupling between neutronics and thermal-hydraulics is usually in manner of “code to code”, which often requires a lot of modifications to the original codes, and such kind of coupling is usually “loosely”. An alternative coupling method, called multi-physics modeling approach, can solve variables of each system simultaneously, which promotes the research on the solution of neutron diffusion problem based on multi-physics platform. Neutronic modeling based on multi-physics platform like COMSOL [4, 5], OpenFOAM [6, 7], NURESIM [8, 9] has been widely researched. In this work, a full-core neutron diffusion application is developed based on MOOSE, a powerful Finite Element Method (FEM) multi-physics coupling framework. 3D-IAEA Benchmark with given few-group constants is applied for code verification.

In addition, solutions of neutron diffusion equation in conventional deterministic neutronic codes are generally based on fixed assembly shape and arrangement (such as rectangular or hexagonal assemblies). While with the trend of modularization and miniaturization of Generation IV nuclear reactors, the core structure is becoming more and more complex. Therefore, deterministic neutronics needs a more convenient modeling method. The MOOSE-based neutron diffusion application in this paper can achieve a free manner of reactor core modeling.

Full-core diffusion calculation need few-group constants as input. Traditionally, complicated procedure of self-shielding and lattice transport calculation was adopted to generate few-group constants, while problem of accuracy exists because of simplifications made in this procedure. Recent years, generation of few-group constants using Monte Carlo method has been widely researched, aim to improve accuracy. Monte Carlo code like Serpent [10], McCARD [11], cosRMC [12] already has the ability to generate homogenized few-group constants. In order to realize the practicability of the developed MOOSE-based diffusion program, the problem of the giving of few-group constants must be solved. Therefore, this work research on few-group constants generation using MC code OpenMC. The homogenized few-group constants of a Sodium-cooled Fast Reactor (SFR) benchmark is generated in this work, and the MOOSE-based diffusion application is applied on this benchmark.

Rest of this paper is organized as follows. Section 2 gives a brief introduction of codes and computational methods used in this paper. Section 3 & 4 presents the diffusion calculation of 3D-IAEA (with given few-group constants) and ABR-MOX (with few-group constants generated by OpenMC) respectively. Section 5 summarizes the conclusions.

2 Code and Method Introduction

2.1 MOOSE Introduction

MOOSE [13] (Multiphysics Object-Oriented Simulation Environment) is an open-source FEM computational framework developed by Idaho National Laboratory (INL), aims at solution of coupled physics systems. JFNK algorithm and physics-based preconditioning of MOOSE framework enable it to solve multi-physics systems in a “fully coupled” manner. The layered coupling structure makes the development of MOOSE-based application relatively easy, developers only need to pay attention to the definition of the top-level physical field or “kernels”, rather than the solution of algebraic equations and data transmission. Well-known nuclear fuel performance code BISON [14], general purpose reactor physics code MAMMOTH [15] are all developed based on MOOSE.

Solution of steady-state neutron diffusion equation is an eigenvalue problem, and the equation is written as:

$$ \begin{gathered} - \nabla \cdot D_{g} \nabla \phi_{g} + {\Sigma }_{t,g} \phi_{g} = \mathop \sum \limits_{{g^{\prime} = 1}}^{G} {\Sigma }_{{g^{\prime} \to g}} \phi_{{g^{\prime}}} + \frac{{\chi_{g} }}{{k_{eff} }}\mathop \sum \limits_{{g^{\prime} = 1}}^{G} \left( {\nu {\Sigma }_{f} } \right)_{{g^{\prime}}} \phi_{{g^{\prime}}} \hfill \\ \begin{array}{*{20}c} {g = 1,2, \cdots ,G} \\ \end{array} \hfill \\ \end{gathered} $$
(1)

Where \({\Sigma }_{t,g}\) is macroscopic total cross section, \({D}_{g}\) is neutron diffusion coefficient, \({\phi }_{g}\) is scalar neutron flux, \({\chi }_{g}\) is normalized fission spectrum, and \({k}_{eff}\) is effective multiplication factor. The first step of using FEM to solve a partial difference equation (PDE) is forming it into Weak form, the weak form of diffusion equation can be written as:

$$ \begin{aligned} {\varvec{F}}_{{\phi_{g} }} \left( {\varvec{U}} \right) = & \left( {\nabla {\varvec{B}},D_{g} \nabla \phi_{g} } \right) - \langle{\varvec{B}},D_{g} \nabla \phi_{g} \cdot {\varvec{n}}\rangle + \left( {{\varvec{B}},{\Sigma }_{{{\text{r}},g}} \phi_{g} } \right) \\ & - \left( {{\varvec{B}},\mathop \sum \limits_{{g^{\prime} = 1}}^{G} {\Sigma }_{{g^{\prime} \to g}} \phi_{{g^{\prime}}} } \right) - \frac{1}{{k_{eff} }}(B,\chi_{g} \mathop \sum \limits_{{g^{\prime} = 1}}^{G} \left( {\nu {\Sigma }_{f} } \right)_{{g^{\prime}}} \phi_{{g^{\prime}}} ) \\ \end{aligned} $$
(2)

Five terms at the right end of the equation are leakage term, boundary condition, removal term, scattering source term and fission source term, the parentheses term represent the volume integral term, and the angle brackets term represent the surface integral term (boundary condition) of the calculation domain. Because neutronics is export controlled physics, MOOSE itself has no modules related to neutronic calculation. Therefore, for the MOOSE-based diffusion application of this work, the kernels corresponding to each term of Eq. (2) are developed. As mentioned before, steady-state neutron diffusion equation is an eigenvalue problem, and the built-in Eigenvalue Executioner of MOOSE can be used to solve such kind of problem.

2.2 MC-Based Few-Group Constants Generation

OpenMC [16] is a Monte Carlo neutron and photon transport simulation code, developed by Massachusetts Institute of Technology (MIT). It is capable of performing fixed source, k-eigenvalue, and subcritical multiplication calculations like usual MC code, and has many unique features like its rich, extensible Python and C/C++ programming interfaces, depletion calculation ability and so on. Energy condensation and spatial homogenization in a few-group constants generation process can be accomplished in OpenMC by its tally system, which process is based on the principle of reaction rate conservation, as:

$$ \begin{array}{*{20}c} {{\Sigma }_{x,g,k} = \frac{{\mathop \smallint \nolimits_{{E_{g} }}^{{E_{g - 1} }} dE^{\prime}\int_{{V_{k} }} {d{\varvec{r}}{\Sigma }_{{\varvec{x}}} \left( {{\varvec{r}},E^{\prime}} \right)\phi \left( {{\varvec{r}},E^{\prime}} \right)} }}{{\mathop \smallint \nolimits_{{E_{g} }}^{{E_{g - 1} }} dE^{\prime}\int_{{V_{k} }} {d{\varvec{r}}\phi \left( {{\varvec{r}},E^{\prime}} \right)} }}} \\ \end{array} $$
(3)

Where \({\Sigma }_{x,g,k}\) is the homogenized few-group cross section of reaction \(x\) in energy group \(g\) and region \(k\), which can be used directly in neutron diffusion code. The generated group constants for diffusion calculation include total cross section, scattering cross section, diffusion coefficient and fission cross section (multiplied by normalized fission spectrum and average fission neutrons produced per fission). Considering the prominent anisotropic scattering of fast reactor, transport correction is used in the process of scattering cross section generation, as:

$$ \begin{array}{*{20}c} {{\Sigma }_{{s,g^{\prime} \to g}} = \frac{{\left\langle {{\Sigma }_{{s,0,g^{\prime} \to g}} \phi } \right\rangle - \delta_{{gg^{\prime}}} \mathop \sum \nolimits_{{g^{\prime\prime}}} \left\langle {{\Sigma }_{{s,1,g^{\prime\prime} \to g}} \phi } \right\rangle }}{\langle \phi \rangle }} \\ \end{array} $$
(4)

Where \({\Sigma }_{s,{g}^{\mathrm{^{\prime}}}\to g}\) is transport-correction cross section. In addition, (n, xn) reactions are negligible for the hard neutron spectrum of fast reactor [17]. The effect of neutron multiplication from (n, xn) reactions is incorporated into the transport-correction cross section.

Most MC-based few-group constants generation methods are somewhat similar to the traditional 2-step method, i.e., cell or assembly level transport calculation with leakage-corrected model is applied in few-group constants generation [18]. While in this work, considering the long neutron free path and strong global neutron spectrum coupling of LMFR, an approach of using a full-core homogeneous MC model to generate the homogenized few-group constants is researched, which can directly model the neutronic coupling between fuel assemblies and reflectors.

The overview Technique Route of the OpenMC-based few-group constants generation and full-core neutronic diffusion calculation by MOOSE-based diffusion application is depicted in Fig. 1. In which process, an in-house tool to automatically write the input file of MOOSE is developed, which can read the output of OpenMC and automatically write the few-group constants into the format of MOOSE input file; And an in-house post-processing tool, which can process the output of MOOSE and output the normalized assembly power factor.

Fig. 1.
figure 1

Overview of Technique Route

Full size image

3 Code Verification

3.1 Benchmark Modeling Description

A 3D PWR full-core benchmark 3D-IAEA [19] is calculated by the diffusion application for code verification. 3D-IAEA is a full-core steady-state two group neutron diffusion problem, there are 177 fuel assemblies and 64 reflector assemblies, and there are 9 fully inserted and 4 partially inserted control rods, the total height of the core is 380 cm. The core structure is shown in Fig. 2. And the 2-group constants set is given as Table 1. Considering the symmetry structure of the core, a 1/4 geometry is constructed for calculation, the geometry and unstructured mesh used in the calculation is shown in Fig. 3. The number of elements and DOF is 4.94 million and 1.71 million respectively. The high-performance computer used in the calculation procedure has dual Intel Core i9-7900X ten-core CPU (3.30 GHz) and contains 64 GB of RAM. The total run time of this problem was 93.3 min using 16 processors with MPI.

Fig. 2.
figure 2

Core structure of 3D-IAEA

Full size image
Fig. 3.
figure 3

(a) Quarter-Core geometry and (b) slice of unstructured mesh of 3D-IAEA

Full size image
Table 1. 2-group constants of 3D-IAEA
Full size table

3.2 Computation and Results

The calculation results of the MOOSE-based diffusion application are listed below. Table 2 shows the comparison of effective multiplication factor (keff) between calculation and reference. The calculated keff is 1.02900, and the relative error with the reference value of 1.02903 is only 3 pcm. The calculated peak-to-average power density is 2.429, which is 3.2% different from the reference value of 2.354. Figure 4 shows the comparison between the calculated normalized assembly power factor distribution and the reference value. The maximum relative error is 1.91%, and the average relative error is 0.52%, which is in good agreement with the reference value, which proves the accuracy of the diffusion application. The distributions of fast and thermal neutron flux are depicted in Fig. 5.

Table 2. Comparison of calculated results and reference
Full size table
Fig. 4.
figure 4

Comparison of calculated assembly power distributions with reference of 3D-IAEA

Full size image
Fig. 5.
figure 5

(a) (c) Fast flux and (b) (d) Thermal flux distribution of 3D-IAEA problem

Full size image

It could be concluded from above calculation that the MOOSE-based neutron diffusion application is valid and accurate, the solution of neutron diffusion problem based on unstructured mesh is realized, which can handle complex geometry. The benchmark verified in this section is a given few-group constants problem, while as mentioned before, most practical problems can not directly obtain few-group constants. Therefore, the content of the next section is to research the generation of few-group constants based on MC, to solve the input-given problem of the developed diffusion application.

4 LMFR Application

4.1 Homogenized Few-Group Constants Generation

The developed MOOSE-based neutron diffusion application is used to calculate the benchmark problem of a sodium cooled fast reactor [20]. The benchmark set gives two large size core designs and two medium size core designs with different fuel types. In this work, the design of 1000 MWth medium size core loaded with MOX fuel is selected for verification, and the layout is shown in Fig. 6.

Fig. 6.
figure 6

Radial layout of ABR-MOX-1000

Full size image

According to the core layout and material density given by the benchmark problem, a full core assembly-wise homogenization model is constructed using OpenMC to generate the group constants, as shown in Fig. 7. A widely-used equal-lethargy 33-group cross sections structure [21] is used for fast reactor analysis, as presented in Table 3. A total number of 1 billion particles (500000 particles per batch, with 100 inactive batches and 1900 active batches) was used in the criticality MC problem to reduce the variance of the output.

Fig. 7.
figure 7

(a) Aixal and (b) Radial view of OpenMC calculation model

Full size image
Table 3. 33-group structure for fast reactor analysis.
Full size table

A 1/6 full-core model with unstructured mesh is constructed for deterministic diffusion calculation, as shown in Fig. 8. The number of elements and DOF is 1.49 million and 8.5 million respectively. The fuel assemblies, reflector assemblies, coolant and other structural components are distinguished as different blocks, and the 33-group few-group constants of each block are generated by OpenMC. The ACE formatted continuous energy cross section library used in OpenMC calculation is generated by ENDF/B-VIII.0 evaluated nuclear data library.

Fig. 8.
figure 8

(a) Geometry and (b) Axial & (c) Radial view of unstructured mesh of ABR-MOX-1000

Full size image

4.2 Computation and Results

The calculation results of the diffusion application are compared with those calculated directly by OpenMC. The effective multiplication factor calculated by MOOSE is 1.01951, which is only 12 pcm different from the OpenMC calculated value of 1.01963. The comparison of assembly power factors is shown in Fig. 9. The maximum relative error is 1.98%, and the average relative error is 0.56%. Figure 10 shows the results of neutron spectrum calculated by MOOSE-based diffusion application and OpenMC respectively, and they are almost identical. It can be seen that the peak of neutron spectrum appears in the tenth group, i.e. neutron energy between 111 keV and 183 keV, which indicating the hard neutron spectrum of LMFR (Table 4).

Table 4. Comparison of calculated results between MOOSE-based diffusion application and OpenMC
Full size table
Fig. 9.
figure 9

Comparison of calculated assembly power distributions with reference of ABR-MOX-1000

Full size image
Fig. 10.
figure 10

Comparison of calculated neutron spectrum of ABR-MOX-1000

Full size image

The calculated neutron flux distribution is shown in Fig. 11. It can be seen that fast neutrons are generated in the fuel assemblies with higher energy. Since the fission and absorption cross sections are relatively low at higher energy, the neutron flux of the fuel assembly is relatively high at this energy, as shown in Fig. 11(a); And also because the fission and absorption cross sections are relatively low at higher energy, neutrons gradually diffuse and slow down, and the neutron flux distribution in the core is almost uniform at the order of hundreds keV, as shown in Fig. 11(b). As the moderation continues, the resonance absorption effect of the fuel assemblies appears, the fission cross section increases, the neutron flux inside the assembly decreases, and the coolant neutron flux is higher than that inside the fuel assembly, as shown in Fig. 11(c). However, it can also be seen that the neutron moderation of fast reactors is relatively weak, and the overall neutron flux is very low at the order of several keV. This is also the reason why fast reactor needs higher fissile enrichment than light water reactor to maintain criticality.

Fig. 11.
figure 11

Relative distribution of neutron flux of (a) group-1 (b) group-10 (c) group-20 (d) group-30

Full size image

The calculation results above show that the approach of MC-based few-group constants generating by full-core modeling is feasible for LMFR analysis, the accuracy of the neutron MOOSE-based diffusion application is also further verified. It can be seen that the neutron diffusion theory could achieve the acceptable accuracy compared with MC method in the calculation of homogenized full-core scale, while some calculations can be carried out better through deterministic codes than MC method, such as neutron diffusion kinetics analysis, sensitivity and uncertainty analysis, etc., which is the significance of develo** deterministic diffusion code. At the same time should also see, in the more detailed pin-by-pin neutronic calculations, some assumptions of the diffusion theory itself do not hold, and more accurate neutron transport codes are required, which is also our next research direction.

5 Conclusion

In this paper, a neutron diffusion application is developed based on the multi-physics framework MOOSE, which realizes the FEM solution of neutron diffusion problems and a free manner of core modeling. A LWR benchmark 3D-IAEA with given few-group constants is calculated using the diffusion application, the calculation result shows the accuracy of it. In addition, the research on the few-group constants generation based on MC code OpenMC is carried out, using a full-core homogeneous model, and is applied to a LMFR full-core benchmark diffusion calculation, the result shows the feasibility of this approach, which solves the problem of the few-group constants given of the developed application. The above verification shows that the developed diffusion application is universal and accurate, which can be further applied to the coupling between neutronic and thermal-hydraulic calculation in our future work.

It can be seen from this work that the neutron diffusion theory has good accuracy in the full-core scale calculation with homogenized assemblies. While at the same time, it should be noted that with the development of computing power, the requirements of neutron calculations are more refined, and a more precise neutron flux distribution with the resolution of pin level is pursued. At this scale, the neutron diffusion theory itself has the problem of accuracy, so a more accurate neutron transport theory is needed. And it is generally deemed that diffusion approximation is inadequate for fast reactor calculations due to the strong neutron leakage and heterogeneity. Therefore, the development of a neutron transport program based on MOOSE and the corresponding multi-physics coupling research is our further research directions.