Research Article 
Corresponding author: Md Imtiaj Hossain ( imtiajhossain854@gmail.com ) Academic editor: Maria Shchurovskaya
© 2023 Md Imtiaj Hossain, Abdus Sattar Mollah, Yasmin Akter, Mehraz Zaman Fardin.
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 MI, Mollah AS, Akter Y, Fardin MZ (2023) Neutronic calculations for the VVER1000 MOX core computational benchmark using the OpenMC code. Nuclear Energy and Technology 9(4): 215225. https://doi.org/10.3897/nucet.9.91090

The goal of this study is to perform neutronic calculations of the VVER1000 MOX core computational benchmarks with an OpenMC code along with ENDF/BVII.1 nuclear data library. The results of neutronic analysis using the OpenMC Monte Carlo code for the VVER1000 MOX core, containing 30% mixed oxide fuel with low enriched uranium fuel, are presented in this study. As per the benchmark report, all six states are considered in the present study. The k_{eff} values, assembly average fission reaction rates, and pinbypin fission rates were calculated as per benchmark criteria. In addition, 2D thermal and fast neutronflux distribution were also generated. The reactivity results and neutron flux distribution were compared with other results in which benchmark analysis was performed using the same core geometry and it showed great similarity with slight deviation. This shows that the modeling of the VVER1000 MOX core was done successfully using OpenMC. Because OpenMC was successfully used for neutronics calculation of the VVER1000 whole core, it may be mentioned here that OpenMC code can also be utilized for neutronics and other reactor core physics analyses of the VVER1200 reactor which is to be commissioned in Bangladesh in the upcoming year.
VVER1000 MOX core, OpenMC, Low Enriched Uranium (LEU) Assembly, Mixed Oxide (MOX) fuel Assembly, Neutron Flux, Burnup
The nuclear reactor, which is the center of a nuclear power plant, generates thermal power that is then converted to electric power for use in the economy by a variety of means. To avoid any unfortunate situation occurring, it is necessary to execute several core parameter calculations continuously. Calculations of multiplication factors, reactivity coefficients, fuel temperature (Doppler) and poison effect on reactivity, burnup, reactivity and isotopic concentration changes with burnup, fast and thermal neutron flux density, axial and radial power peaking of the core, fission rates distribution, power distribution of the core, etc. are among the crucial calculations (Lamarash 1988). To guarantee the integrity of the nuclear reactor core during operation, these calculations are carried out and evaluated regularly. The neutronic behavior of fuel assemblies and the core of a nuclear reactor with various combinations of fuel with different enrichments, moderator materials, and nonfuel structural components has been studied by using a suitable neutronic simulation code.
An OECDNEA paper contains a comprehensive list of benchmarks that can be used to carry out this type of verification (
Various benchmark problems may be used to extensively assess the core of a VVER reactor. For this investigation, a VVER1000 full core containing 30% MOX fuel was used as a benchmark problem which was obtained from a benchmark analysis (
The designed model is a VVER1000 reactor full core which contains 30% mixed oxide fuel alongside lowenriched uranium fuels. The modeling was done in OpenMC in a jupyter notebook with Python 3.9. The core includes both fresh and burned fuel from various burnups, which are arranged in a peripherytocenter pattern inside the core. Because fresh fuel can achieve a higher burnup compared to once or twiceburned fuel and produces a lot of power compared to other burned fuel, the neutron flux associated with this assembly is likewise a lot higher. The core has seven different types of fuel assemblies as mentioned in the benchmark problem, which are as follows:
Each assembly contains 331 elementary cells of various types such as different enriched fuel, gadolinium pins, guide tubes, and central tubes and for state6, some control rods are inserted in some specific assemblies, as mentioned later.
Various Assumptions were taken during the modeling process, they are:
Several steps must be followed to model a full core. At the very beginning, each sort of elementary cell was designed. These elementary cells include fuel cells, fuel cells with gadolinium absorbers, guide tube cells, central tube cells, and absorber rod cells. Fig.
Cells Name  Cell Radius (cm) 

Fuel cell  R_{1} = 0.386 
R_{2} = 0.455  
Central tube cell  R_{1} = 0.55 
R_{2} = 0.63  
Guide tube cell  R_{1} = 0.55 
R_{2} = 0.63  
Guide tube with absorber rod  R_{1} = 0.35 
R_{2} = 0.41  
R_{3} = 0.55  
R_{4} = 0.63 
Following the design of the elementary cells, seven different types of fuel assemblies were designed, and for state six, an additional five fuel assemblies containing absorber rod cells were created. Figs
After the successful modeling of the assemblies necessary for modeling the whole core, the core description from the benchmark report was followed and the full VVER1000 core consisting of 163 fuel assemblies was designed. The 1/6^{th} portion of the geometry description and the full core which was modeled using OpenMC is shown in Figs
After successful modeling of the whole core, various parameters were calculated for analysis purposes using OpenMC. There are six states described in the benchmark report in which the calculation was performed. The operational states’ description is given in Table
States  State name  Fuel temperature (K)  Nonfuel temperature (K)  Reflector temperature (K)  Moderator in the fuel assembly  Water hole, water gap, and downcomer material  Absorber rod 

State 1  Working state  1027  575  560  M575B1.3  M60B1.3   
State 2  State with constant temperature  575  575  560  M575B1.3  M560B1.3   
State 3  Cold state with high boron content  300  300  300  M300B2.8  M300B2.8   
State 4  Working state without boron  1027  575  560  M600B0  M560B0   
State 5  State with constant temperature without boron  575  565  560  M560B0  M560B0   
State 6  State with control rods inserted  565  565  560  M553B0  M553B0  Inserted 
Here, in Table
The models were represented in the OpenMC using python (python 3.7) code in jupyter notebook in the latest version of OpenMC (OpenMC 0.13.0). OpenMC has the feature to model hexagonal geometry which was used to design each type of assembly separately. Different materials in different regions inside the assemblies were defined using Boolean operation for modeling which is also known as constructive solid geometry. A hexagonal prism with an assembly pitch of 23.6 cm was used to bind the geometry giving it a hexagonal shape. For each pin cell, 1.275 cm of cell pitch was used. Two separate planes on the zaxis with reflecting boundary conditions, which is equivalent to the geometry being infinite on the zaxis, were defined. After completing the design of the seven types of assemblies, they were placed inside another hexagonal prism to produce the core. The core consists of a total of 163 fuel assemblies. To account for the thermal scattering at lower energies, S(α,β) table was provided. A total of six states were considered for the calculation of different parameters which are given in Table
Computing a value known as the Shannon entropy of the fission source distribution, H_{src}, has been done in research work to evaluate the convergence of the fission source distribution for the Monte Carlo method (
${H}_{src}=\sum _{J=1}^{N}{P}_{J}\xb7{\mathrm{ln}}_{2}\left({P}_{J}\right)$
where N_{s} is the number of grid boxes in the superimposed mesh, and P_{J} = (number of source sites in Jth grid box)/(total number of source sites). H_{src} varies between 0 for a point distribution to ln_{2}(N_{s}) for a uniform distribution.
The Shannon entropy curve is shown in Fig.
The VVER1000 whole core benchmark was first introduced and the effective multiplication factor was calculated for six different states from state1 to state6. The result obtained from OpenMC was compared with other results from benchmark reports such as MCNP4c, MCU, which used the MCUDAT2.1 data library as the basic data library, and MCNP5 (
Table
State  OpenMC (OP)  MCNP5  MCNP4C  MCU  BM*  ∆K 

(ENDF/BVII.1)  (ENDF/BVI.6)  (JEF2.2)  (MCUDAT 2.1)  (OP−BM)/(OP) × 100%  
1  1.0337 ±0.006  1.03614 ±0.007  1.03770 ±0.007  1.03341 ±0.013  1.03769  0.386 
2  1.0465 ±0.006  1.04339 ±0.010  1.05132 ±0.010  1.04719 ±0.012  1.04989  0.315 
3  0.9294 ±0.009  0.93397 ±0.011  0.93416 ±0.011  0.93237 ±0.01  0.93286  0.367 
4  1.1310 ±0.004  1.13511 ±0.010  1.13871 ±0.010  1.1339 ± 0.012  1.13781  0.432 
5  1.1472 ±0.004  1.14333 ±0.010  1.15400 ±0.010  1.14932 ±0.012  1.15302  0.507 
6  1.0506 ±0.002  1.03914 ±0.010  1.04729 ±0.011  1.04267 ±0.009  1.04498  +0.535 
The thermal output of the VVER1000 core is roughly 3000 MW. The overall power is distributed across the 163 assemblies that constitute the core. Each assembly or pin within an assembly does not produce the same amount of power, and the power that it produces also changes depending on its enrichment and composition. The reactor power is proportional to fission reaction rates. Assembly average fission reaction rates for assemblies 1 through 28 were determined, along with their standard deviation, and compared to the findings from data from the literature review’s MCNP4C, MCU, and Radar (
The neutron flux density spectrum was obtained from the flux tally via OpenMC. The flux spectrum is a 2D slice plot. Since the current version of OpenMC can’t generate an isometric plot of the neutron flux density, a 2D plot was generated for State1, and State6 only. The four slice plots of thermal fastflux spectrum plots are shown in Figs
Here, the symmetric behavior of the core is seen. State6 is a very special state, where control rods are inserted in some specific places inside the guide tubes in some selected assemblies. Hence, the thermal and the fast neutron flux density are less in the middle of the core due to control rod insertion, as illustrated by Fig.
The benchmark report lacks a neutron energy spectrum for comparison. In their 2009 article,
The OpenMC code was used in this investigation to calculate the effective multiplication factor for states one through six, assembly average fission reaction rates, and pinbypin fission reaction rates. In addition, 2D thermal and fast neutron flux density distributions were calculated. Following that, the obtained results were contrasted with those from MCU and MCNP as well as other findings from the literature values. It was evident from the comparisons of k_{eff} values that OpenMC had been successfully implemented for the model mentioned in the OECD benchmark problem. The assembly average fission reaction rates also showed slight deviation from other assemblies, as shown in the result sections. The absence of the three mm water layer right outside the core could be one of the causes. A very substantial discrepancy was seen at interior assemblies compared to periphery assemblies for state six as well. As can be seen from the obtained neutron flux spectrum in various states as well as assembly average fission reaction rates, OpenMC demonstrated a very good capability in performing neutronic calculations for VVER geometrybased nuclear reactors, with the exception of the minor deviations caused by modeling errors and the use of a new data library.
Md. Imtiaj Hossain: Methodology, Data collection, Formal analysis, Writing – original draft. A. S. Mollah: Supervision, Conceptualization, Results interpretation, Writing – review & editing. Yasmin Akter: Resources, Data analysis, Writing. Mehraz Zaman Fardin: Resources, Literature review, Writing. All authors reviewed the results and approved the final version of the manuscript.
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 anonymous referees for their critical reading of the paper and for the improvements they suggested.