Research Article 
Corresponding author: Abdus Sattar Mollah ( mosattar54@gmail.com ) Academic editor: Georgy Tikhomirov
© 2022 Md. Imtiaj Hossain, Yasmin Akter, Mehraz Zaman Fardin , Abdus Sattar Mollah.
This is an open access article distributed under the terms of the Creative Commons Attribution License (CC BY 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Citation:
Hossain MdI, Akter Y, Fardin MZ, Mollah AS (2022) Neutronics and burnup analysis of VVER1000 LEU and MOX assembly computational benchmark using OpenMC Code. Nuclear Energy and Technology 8(1): 111. https://doi.org/10.3897/nucet.8.78447

A handful of computational benchmarks that incorporate VVER1000 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 VVER1000 LEU & MOX assembly benchmarks in different states were considered. In this work, the VVER1000 LEU and MOX Assembly computationalbenchmark 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/BVII.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 State5 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/BVII.1 nuclear data library was successfully implemented. It is planned to use OpenMC code for calculation of neutronics and burnup of the VVER1200 reactor to be commissioned in Bangladesh by 2023/2024.
VVER1000, OpenMC, multiplication factor, burnup, isotope concentrations, Light Enriched Uranium (LEU) Assembly, Mixed Oxide (MOX) fuel Assembly, Benchmark
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 VVER1200 (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 VVER1000 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
The cell type geometry specifications are given in Table
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 zaxis with reflecting boundary conditions bounded the geometry, which is equivalent to the geometry being infinite in the zaxis. 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
State  State name  Fuel temperature in K  Nonfuel 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 nonpoisoned 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 
In OpenMC code, there are three ways to calculate keigenvalue, including tracklength estimator, collision estimator, and absorption estimator, respectively. They are expressed in the following equations:
${k}_{\text{tracklength}}=\frac{{\Sigma}_{\text{all flights}}{w}_{j}{d}_{j}v{\Sigma}_{f}}{W}$
${k}_{\text{Collision}}=\frac{{\Sigma}_{\text{all collisions}}{w}_{j}\left(\frac{v{\Sigma}_{f}}{{\Sigma}_{t}}\right)}{W}$
${k}_{\text{tracklength}}=\frac{{\Sigma}_{\text{all absorption}}{w}_{j}\left(\frac{v{\Sigma}_{f}}{{\Sigma}_{a}}\right)}{W}$
where W is the total weight starting each generation (or batch), w_{j} is the precollision weight of the particle as it enters event j, d_{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 (
The OpenMC results of the present study were compared to the results of SERPENT code as well as the MCNP results (
Table
LEU  SERPENT(SE) (ENDF/BVII.0)  OpenMC (OP) (ENDF/BVII.1)  MCNP (JEFF 2.2)  ∆K (OPSE)*10^{5} 

State 1  1.13997 ±8.8E05  1.13923 ± 2E04    74 
State 2  1.17587 ± 8.8E05  1.17520± 2E04  1.1800 ± 6E05  67 
State 3  1.18996 ± 8.6E05  1.18849± 2E04  1.1925 ± 6E05  147 
State 4  1.24993 ± 8.7E05  1.24896± 2.5E04  1.2531 ± 7E05  97 
State 5  1.32305 ±7.7E05  1.32210± 2E04  1.3235 ±6E05  95 
MOX  SERPENT(SE) (ENDF/BVII.0)  OpenMC (OP) (ENDF/BVII.1  MCNP (JEFF 2.2)  ∆K (OPSE)*10^{5} 

State 1  1.17382 ± 8.4E05  1.17131 ±1.8E04    51 
State 2  1.19762 ± 8.6E05  1.19740 ±1.8E04  1.1922 ± 7E05  22 
State 3  1.21429 ± 8.4E05  1.21378 ± 1.8E04  1.2091 ± 6E05  51 
State 4  1.24923 ±8.4E05  1.24822 ± 1.8E04  1.2430 ± 6E05  99 
State 5  1.33013 ± 7.6E05  1.33033 ± 1.8E04  1.3256 ± 6E05  20 
The variation of k_{inf} with respect to burnup (MWd/kgHM) is shown in Fig.
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
Initial state  final state  Effect  (K_{init.}– K_{fin}) / (Kinit. * K_{fin})*1000 (mk)  

OpenMC  Benchmark mean  SERPENT  (OPSE)  
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 
Initial state  final state  Effect  (K_{init.}– K_{fin}) / (Kinit. * K_{fin})*1000 (mk)  

OpenMC  Benchmark mean  SERPENT  (OPSE)  
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 cell1 and cell24 (as shown in Figs
The isotopic concentrations for cell1 are shown in Figs
The isotopic concentrations for cell 24 are shown in Figs
Assembly average concentration changes with burnup is shown in Figs
Figs
The k_{inf} values were calculated for VVER1000 LEU & MOX assemblies that are typically of the advanced Russian designs in different reactor operating states using OpenMC code with nuclear data library ENDF/BVII.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 State1. It is concluded that the OpenMC code along with the nuclear data library ENDF/BVII.1 was successfully implemented at the Department of Nuclear Science and Engineering Department, MIST. In Bangladesh, two Russian design VVER1200 (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 VVER1200 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.