Corresponding author: Abdus Sattar Mollah (

Academic editor: Georgy Tikhomirov

A handful of computational benchmarks that incorporate VVER-1000 assemblies having low enriched uranium (LEU) and the mixed oxide (MOX) fuel have been put forward by many experts across the world from the Nuclear Energy Agency. To study & scrutinize the characteristics of one of the VVER-1000 LEU & MOX assembly benchmarks in different states were considered. In this work, the VVER-1000 LEU and MOX Assembly computational-benchmark exercises are performed using the OpenMC software. The work was intended to test the preciseness of the OpenMC Monte Carlo code using nuclear data library ENDF/B-VII.1, against a handful of previously obtained solutions with other computer codes. The k_{inf} value obtained was compared with the SERPENT and MCNP result, which presented a very good similarity with very few deviations. The k_{inf} variation with respect to burnup upto 40 MWd/kgHM was obtained for State-5 by using OpenMC code for both the LEU and MOX fuel assembly. The depletion curves of isotope concentrations against burnup upto 40 MWd/kg/HM were also generated for both the LEU and MOX fuel assembly. The OpenMC results are comparable with those of benchmark mean values. The neutron energy vs flux spectrum was also generated by using OpenMC code. Based on the OpenMC results such as k_{inf}, burnup, isotope concentrations and neutron energy spectrum, it is concluded that the OPenMC code with ENDF/B-VII.1 nuclear data library was successfully implemented. It is planned to use OpenMC code for calculation of neutronics and burnup of the VVER-1200 reactor to be commissioned in Bangladesh by 2023/2024.

The main purpose of nuclear reactor theory is calculation of distribution of neutrons in the reactor core. From knowledge of it, we can determine the rate of fission reaction occurring in a nuclear reactor and hence reactor power and operating point (sub critical, critical or supercritical) hence, stability of fission chain reaction can be inferred. Generally, neutronic analysis is performed based on ‘‘Deterministic” and ‘‘Stochastic” methods. In deterministic methods the transport equation is solved as a differential equation. In stochastic methods such as Monte Carlo, discrete particle histories are tracked and averaged in a random walk directed by interaction probabilities. In Bangladesh, two Russian design VVER-1200 (2400 MWth) type nuclear reactors are under construction and to be commissioned by 2023/2024. A program has been undertaken at the Department of Nuclear Science and Engineering of Military Institute of Science and Technology, Dhaka, Bangladesh to introduce some Monte Carlo computer codes such as MCNPX (

The benchmark model has two fuel assemblies (LEU & MOX) of the VVER-1000 reactor. Each of the assemblies consists of 331 elementary cells of four types for LEU assembly & six types for MOX assembly inside a hexagonal lattice. Twelve Gd_{2}O_{3} pins are located inside each assembly as a burnable absorber at completely different positions. The layout of both the assembly types obtained using OpenMC is shown in Figs ^{235}U except in Gd, bearing pins that contain 3.6% wt. of ^{235}U and 4% wt. of Gd_{2}O_{3}. The assembly lattice pitch & the hexagonal cell pitch are respectively 23.6 cm & 1.275 cm. All three types of unit cells along with their dimensions for each assembly are given in Figs

LEU fuel assembly.

MOX fuel assembly.

The cell type geometry specifications are given in Table

Fuel & non-fuel cells.

Cell type geometry specification

Type of the Cell | Cell Radius (in cm) |
---|---|

Fuel | Fuel pellet radius = 0.386 |

Cladding outer radius = 0.4582 | |

Guide tube cell | Cladding inner radius = 0.545 |

Cladding outer radius = 0.6323 | |

Central tube cell | Cladding inner radius = 0.48 |

Cladding outer radius = 0.5626 |

The models were represented in the OpenMC using python (python 3.7) code in Jupyter notebook. Initially, one of each of the different types of rods was created. To obtain the assemblies, they were placed in a hexagonal lattice with a lattice cell pitch of 1.275 cm and an assembly pitch of 23.6 cm. Modeling was done utilizing Boolean operations to define different zones within the cells. The hexagonal lattice and two different planes in the z-axis with reflecting boundary conditions bounded the geometry, which is equivalent to the geometry being infinite in the z-axis. For thermal scattering at low energies, S(α,β) table was used. The benchmark demands a solution for a variety of states, encompassing both hot and cold conditions, as shown in Table

Reactor States for both of the assembly

State | State name | Fuel temperature in K | Non-fuel temperature in K | Boron concentration (ppm) | ^{149}Sm ^{135}Xe |
Moderator in fuel & central/guide tube | Moderator density in g/cm^{3} |
---|---|---|---|---|---|---|---|

State 1 | Operating poisoned state | 1027 | 575 | 600 | Eq. | MOD1 | 0.7235 |

State 2 | Operating non-poisoned state | 1027 | 575 | 600 | 0 | MOD1 | 0.7235 |

State 3 | Hot state | 575 | 575 | 600 | 0 | MOD1 | 0.7235 |

State 4 | Hot state without boric acid | 575 | 575 | 0 | 0 | MOD2 | 0.7235 |

State 5 | Cold state | 300 | 300 | 0 | 0 | MOD3 | 1.0033 |

Pin-by-pin Model

Reflective boundary condition in all directions (x, y & z)

Finite boundary at z-axis with reflective boundary conditions.

Cross-section data library: ENDF/B-VII.1

1000 batches with 100 inactive batches & 11000 particles in each batch.

In OpenMC code, there are three ways to calculate

where W is the total weight starting each generation (or batch), _{j}_{j} is the length of the j^{th} trajectory, and υΣ_{f} and Σ_{a} are macroscopic neutron production cross section and absorption cross section.

The OpenMC Monte Carlo code was used to calculate the average k_{inf} based on the combined collision estimator, track length estimator, & absorption estimator (

The infinite multiplication factor was calculated in eigenvalue mode of the OpenMC (Version 0.12.2) monte carlo code. To obtain accurate results based on the initial guess value for the fission source distribution, analysis of the iteration method source convergence is necessary. A study into convergence of Monte Carlo criticality analysis has proved that the Shannon entropy of the fission source distribution, H_{src}, is an effective parameter for identifying the convergence of the fission source distribution (

Shannon entropy vs generation.

The OpenMC results of the present study were compared to the results of SERPENT code as well as the MCNP results (_{inf} values obtained by using OpenMC code are given in Tables

Table _{inf} for the State-5 only (Table _{inf} value for the same state is roughly 680 pcm for LEU and about 1695 pcm for MOX fuel, according to the table. All of the values for the various codes were collected from various sources that are cited in the references. We can conclude from this comparison that the OpenMC results are acceptable.

k_{inf} for Zero burnup (For LEU)

LEU | SERPENT(SE) (ENDF/B-VII.0) | OpenMC (OP) (ENDF/B-VII.1) | MCNP (JEFF 2.2) | ∆K (OP-SE)*10^{5} |
---|---|---|---|---|

State 1 | 1.13997 ±8.8E-05 | 1.13923 ± 2E-04 | - | -74 |

State 2 | 1.17587 ± 8.8E-05 | 1.17520± 2E-04 | 1.1800 ± 6E-05 | -67 |

State 3 | 1.18996 ± 8.6E05 | 1.18849± 2E-04 | 1.1925 ± 6E-05 | -147 |

State 4 | 1.24993 ± 8.7E05 | 1.24896± 2.5E-04 | 1.2531 ± 7E-05 | -97 |

State 5 | 1.32305 ±7.7E05 | 1.32210± 2E-04 | 1.3235 ±6E-05 | -95 |

k_{inf} for at Zero burnup (for MOX fuel)

MOX | SERPENT(SE) (ENDF/B-VII.0) | OpenMC (OP) (ENDF/B-VII.1 | MCNP (JEFF 2.2) | ∆K (OP-SE)*10^{5} |
---|---|---|---|---|

State 1 | 1.17382 ± 8.4E-05 | 1.17131 ±1.8E-04 | - | -51 |

State 2 | 1.19762 ± 8.6E-05 | 1.19740 ±1.8E-04 | 1.1922 ± 7E-05 | -22 |

State 3 | 1.21429 ± 8.4E-05 | 1.21378 ± 1.8E-04 | 1.2091 ± 6E-05 | -51 |

State 4 | 1.24923 ±8.4E-05 | 1.24822 ± 1.8E-04 | 1.2430 ± 6E-05 | -99 |

State 5 | 1.33013 ± 7.6E-05 | 1.33033 ± 1.8E-04 | 1.3256 ± 6E-05 | 20 |

Variation of k_{inf} vs burnup

Comparison of different results (State-5)

NAME (Data library) | LEU | MOX |
---|---|---|

SERPENT(JEFF3.1) | 1.32088 | 1.32908 |

OpenMC(JEFF3.3) (Present study) | 1.31670 | 1.3297 |

SCALE(ENDF/B-VII.0) | 1.31770 | 1.32652 |

MCNP4B(JEFF2.2) | 1.3235 | 1.3256 |

GETERA(BNAB-93) | 1.3175 | 1.31213 |

The variation of k_{inf} with respect to burnup (MWd/kgHM) is shown in Fig. ^{3} given in the benchmark problem. The benchmark values are also included in Fig. _{inf} is seen in Fig. _{inf} is observed at burnup of ~10 MWd/kgHM. The OpenMC results for the both LEU and MOX fuel assembly are comparable with those of benchmark mean values.

The reactivity effect was computed using the k_{inf} values obtained from various reactor operational states at zero burnup for LEU and MOX fuel assembly is given in Tables

Reactivity effects (at zero burnup for LEU fuel)

Initial state | final state | Effect | (K_{init.}– K_{fin}) / (Kinit. * K_{fin})*1000 (mk) |
|||
---|---|---|---|---|---|---|

OpenMC | Benchmark mean | SERPENT | (OP-SE) | |||

State 1 | State 2 effect on reactivity | ^{135}Xe &^{149}Sm |
-26.30 | -30.22 | -26.78 | -0.48 |

State 2 | State 3 | Fuel temperature (Doppler effect) | -9.43 | -09.86 | -10.07 | +0.64 |

State 3 | State 4 | Soluble boron effect | -40.18 | -40.23 | -40.31 | +0.13 |

State 4 | State 5 | Moderator temperature effect | -44.00 | -41.73 | -44.21 | +0.21 |

Reactivity effects (at zero Burnup for MOX fuel)

Initial state | final state | Effect | (K_{init.}– K_{fin}) / (Kinit. * K_{fin})*1000 (mk) |
|||
---|---|---|---|---|---|---|

OpenMC | Benchmark mean | SERPENT | (OP-SE) | |||

State 1 | State 2 | ^{135}Xe & ^{149}Sm effect on reactivity |
-23.31 | -24.15 | -23.89 | -0.58 |

State 2 | State 3 | Fuel temperature (Doppler) | -11.04 | -12.21 | -11.39 | +0.35 |

State 3 | State 4 | Soluble Boron | -23.43 | -23.19 | -23.10 | -0.33 |

State 4 | State 5 | Moderator Temperature | -49.44 | -47.95 | -48.69 | -0.75 |

Figs

Isotopic composition changes of nuclides ^{235}U, ^{236}U, ^{238}U, ^{239}Pu, ^{240}Pu, ^{241}Pu, ^{242}Pu, and ^{149}Sm in cell-1 and cell-24 (as shown in Figs ^{155}Gd and ^{157}Gd, are compared in cell-24. Because of its large thermal fission cross-section of approx. 583 barns, U-235 is used as the primary fuel in an LEU fuel assembly. As the fission process progresses and burnup increases, the concentration of U-235 in LEU assembly falls. Because Pu is the major fuel in MOX assembly, ^{239}PU shows similar behavior which can be seen from Fig.

The isotopic concentrations for cell-1 are shown in Figs

The isotopic concentrations for cell 24 are shown in Figs

Assembly average concentration changes with burnup is shown in Figs

Variation of ^{235}U concentration vs burnup.

Variation of ^{236}U concentration vs burnup.

Variation of ^{238}U concentration vs burnup.

Variation of ^{239}Pu concentration vs burnup.

Variation of ^{240}Pu concentration vs burnup.

Variation of ^{241}Pu concentration vs burnup.

Variation of ^{242}Pu concentration vs burnup.

Variation of ^{149}Sm concentration vs burnup.

Variation of ^{235}U concentration vs burnup.

Variation of ^{236}U concentration vs burnup.

Variation of ^{238}U concentration vs burnup.

Variation of ^{239}Pu concentration vs burnup.

Variation of ^{240}Pu concentration vs burnup.

Variation of ^{241}Pu concentration vs burnup.

Variation of ^{242}Pu concentration vs burnup.

Variation of ^{149}Sm concentration vs burnup.

Variation of ^{155}Gd concentration vs burnup.

Variation of ^{157}Gd concentration vs burnup.

Variation of ^{235}U average concentration.

Variation of ^{236}U average concentration.

Variation of ^{238}U average concentration.

Variation of ^{239}Pu average concentration.

Variation of ^{240}Pu average concentration.

Variation of ^{155}Gd average concentration.

Variation of ^{242}Pu average concentration.

Variation of ^{157}Gd average concentration.

Variation of ^{241}Pu average concentration.

Comparison of neutron flux spectrum at BOC, MOC and EOC for LEU fuel assembly.

Comparison of neutron flux spectrum at BOC, MOC and EOC for MOX fuel assembly.

A typical neutron energy spectrum of a Light Water Reactor used as a weighting function in a generation of ORIGEN library [

Figs ^{−2}.s^{−1}.eV^{−1}) = Φ(cm^{−2}.s^{−1})/ΔE(eV) by using OpenMC code. As may be seen from the Figs

The k_{inf} values were calculated for VVER-1000 LEU & MOX assemblies that are typically of the advanced Russian designs in different reactor operating states using OpenMC code with nuclear data library ENDF/B-VII.1. The k_{inf} values were also calculated against fuel burnup upto 40 MWd/kgHM. In addition, the isotope composition was also calculated for burnup upto 40 MWd/kgHM as per benchmark requirements. The calculated results were compared with the benchmark mean values along with the literature data. The OpenMC results showed very good agreement with the benchmark mean values alongwith other literature values. The neutron energy spectrum was successfully generated by using OpenMC code for both LEU and MOX fuel assembly for State-1. It is concluded that the OpenMC code along with the nuclear data library ENDF/B-VII.1 was successfully implemented at the Department of Nuclear Science and Engineering Department, MIST. In Bangladesh, two Russian design VVER-1200 (2400 MW_{th}) type nuclear reactors are under construction and to be commissioned by 2023/2024. Based on the experience achieved for implementation of OpenMC code in the field of neutronics and burnup calculations, it is planned to calculate k_{inf} or to perform burnup calculations for VVER-1200 and other PWR and BWR by using OpenMC along with SuperMC.

Md. Imtiaj Hossain: Methodology, Data collection, Formal analysis, Writing – original draft. Yasmin Akter: Resources, data analysis, writing. Mehraz Zaman Fardin: Resources, literature review, writing. A. S. Mollah: Supervision, Conceptualization, Results interpretation, Writing – review & editing.

The authors really acknowledge the efforts of the Department of Nuclear Science and Engineering, Military Institute of Science and Technology, Dhaka, Bangladesh for their academic support. The authors thank the referees for their critical reading of the paper and for the improvements they suggested.