Corresponding author: Olga N. Andrianova ( oandrianova@ippe.ru ) Academic editor: Georgy Tikhomirov
© 2020 Olga N. Andrianova, Gennady M. Zherdev, Gleb B. Lomakov, Yevgeniya S. Teplukhina.
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:
Andrianova ON, Zherdev GM, Lomakov GB, Teplukhina YS (2020) Consideration of heterogeneous effects in preparing neutron multigroup constants in the CONSYST/BNABRF system. Nuclear Energy and Technology 6(1): 6370. https://doi.org/10.3897/nucet.6.52112

The need for building mutually selfagreed computational models for highprecision and engineering neutronic codes is defined by requirements to certification and verification of software products and nuclear data in accordance with the Regulations for Verification and Expert Examination of Neutronic Calculation Software Tools (RB06111). The key requirement in RB06111 is that there shall be a methodically transparent and reproducible procedure to estimate the methodological and nuclear data component of the neutronic parameter uncertainties to be implemented only if there are mutually selfagreed computational neutronic models. Using an example of a series of measurements carried out on three critical BFS61 configurations, factors are discussed which need to be taken into account when building such types of neutronic models and the peculiarities of their application for calculating the neutronic parameters of BFS61 assemblies. Improved functional capabilities of updated software tools and nuclear data for the computational and experimental analysis of integral BFS experiments (ROSFOND/BNABRF, CONSYST and MMKRF) have been demonstrated, allowing a much shorter time and the smallest risks of errors in preparing mutually selfagreed computational models for various neutronic codes, as well as correct estimation of the methodological and nuclear data components of the uncertainties in neutronic parameters in accordance with RB06111. The results of estimating the uncertainty in neutronic parameters with respect to the group approximation approach, are presented. It has been shown based on an analysis of the obtained results that the discrepancies in the calculations of the BFS61 configurations in the transition from the ROSFOND evaluated neutron data library to its group version, BNABRF, does not exceed 0.3% in criticality (heterogeneous effects uncertainty of 0.2 to 0.8 %). The estimated spectral index data biases lie in the limits of the Monte Carlo statistical error. Based on the results of a computational and experimental analysis for the entire set of measurements performed on a series of BFS61 assemblies, the ROSFOND library is the optimal nuclear data library.
Integral experiments, BFS, effective neutron multiplication factor, nuclear data uncertainty, Monte Carlo, ROSFOND, BNABRF, spectral indices
The purpose of the paper is to demonstrate the improved functional capabilities of updated software products and nuclear data to support the computational and experimental analysis of integral experiments based on BFS assemblies (using BFS61 critical assemblies as the example). These capabilities make it possible to reduce considerably the time and to minimize the risks of errors in preparation of models, to build mutually selfagreed models for precision and engineering codes, and to estimate the methodological and nuclear data components of uncertainties in neutronic parameters required for verification and qualification of codes in accordance with the Regulations RB06111 (RB06111). The uncertainties in engineering calculations, as specified in RB06111 (par. 13–15), are determined by comparing the calculation results against the respective results obtained based on precision (reference) neutronic codes (par. 22). One of the world’s most commonly used codes of the kind is MCNP, a US code (MCNP 2008). The MCU code is an extensively used Russian analog (
The MMKK code developed at IPPE (
The constant component of an error should be estimated based on comparing the results of the benchmark experiment calculations, performed with the use of the codes implementing the Monte Carlo method, in detailed and group representations of neutron crosssections. A computational description of experiments performed on the BFS facilities in a group representation of neutron crosssections is associated with the problem of taking into account, in a correct manner, the effects of the crosssection resonance selfshielding due to the heterogeneous arrangement of materials (
Critical assemblies of the BFS61 series were set up in 1990 at the BFS1 facility to investigate the characteristics of lead cooled nuclear reactors. In 2009, the experiments conducted based on the BFS61 critical assemblies were evaluated for international handbooks of evaluated benchmark experiments (
The BFS61 assemblies (Fig.
The program of experimental studies based on the BFS61 assemblies included measuring the ratios of the neutron interaction crosssections (〈σ_{f}^{i}/〈σ_{f}^{j}) averaged by the spectrum at a certain point of the critical assembly (spectral indices) containing ample information on the properties of the fast neutron reactor composition under investigation (
$\left.\right)"\; close=">">{\sigma}_{x}^{i}$
where φ(E) is the spectral density of the neutron flux; 〈 is the symbol of averaging by energy; i, j are the isotope type; and x is the reaction type (e.g., f is fission). The ratio of the average fission crosssection for the isotopes ^{232}Th, ^{233, 238}U, and ^{239}Pu to the average fission crosssection for ^{235}U, and the ratio of the average fission crosssection for the minor actinides ^{238, 240, 241, 242}Pu, ^{241}Am, and ^{237}Np to the average fission crosssection for ^{239}Pu were measured on the BFS61 assemblies using a smallsize fission chamber (SFC).
The SFC design is described in (
To measure the spectral indices, the chamber is installed such that the middle of the fissile material layer inside the chamber coincided with the median plane of the assembly core (with the middle of the fuel composition in the central tube). Fig.
The experiment evaluation procedure suggests that the description of a mathematical model in the calculated value shall include estimated contributions from all of the adopted assumptions and approximations of the methods in use. A practicable way to estimate these assumptions and approximations is to do this successively while relying on the results of the calculations based on a highprecision neutronic models. Schematically, the calculation procedure for the uncertainty evaluation is shown in Fig.
A computational neutronic model is prepared at the initial stage for the precision Monte Carlo code that allows performing calculations with a pointwise representation of the neutron crosssections without introducing any approximations and simplifications associated with the description of the actual geometry and the material composition of the neutron multiplying system.
Based on the above computational model, neutron multigroup calculations are proceeded to at the second stage while estimating the nuclear data component of the uncertainty by comparing the results of two calculations. The methodological component of the uncertainty, associated with the implementation of the neutron group approximation in another particular code using the Monte Carlo method, can be also estimated at this stage (
One can proceed from the precision specification of the assembly configuration and composition to homogenized compositions at stage 3 while making changes to the geometry description in the computation neutronic model.
Comparing the results of the Monte Carlo calculations with the detailed representation of the neutron crosssections for the homogeneous model and for the heterogeneous model makes it possible to estimate the computational adjustment factor for heterogeneity. The following uncertainty components can be determined using precision codes: a nuclear data component, a methodological component (associated with the applied neutron group approximation) and the uncertainty caused by the transition to the homogeneous neutronic model.
When comparing the results of the calculations for homogeneous models, based on various engineering neutronic codes, with the results of the calculation for the same model, using a highprecision neutronic code with a single nuclear data basis, it is possible to estimate the uncertainties of the approximations the engineering neutronic codes.
The CONSYST processing system implements various methods to take into account heterogeneous effects when preparing neutron multigroup macroscopic crosssections based on the principle of equivalence of homogeneous and heterogeneous media (theorem of equivalence):
– “manual” introduction of corrections – the user can calculate the corrections for the dilution crosssection manually: a) by introducing the fictitious isotope of the δscattering, b) by identifying additional material blocking zones;
– automatic calculation of corrections – the system is capable to compute automatically the values of the corrections if the user has described the actual heterogeneous structure using the GETER module.
The concentrations of fictitious materials need to be calculated in the “manual” mode of introducing the corrections to take into account the spatial heterogeneity.
First, if the dimensions of the uniform region are comparable with the length of the free path and it includes isotopes with resonances, a fictitious isotope named “DSC”, the socalled δscattering that does not change the neutron transport process but influences the way the resonance selfshielding of neutron crosssections is taken into account, can be introduced in this region. The full crosssection of the δscattering is equal to 1 barn, and the average cosine of the scattering angle is equal to unity. The concentration of the δ scattering is selected as equal to 1/l, where l = 4V/S is the mean chord or the average value of the neutron path length in a zone without regard for collisions; V is the volume; and S is the surface area.
Second, the CONSYST system comprises two classes of material zones having different nuclide compositions – material blocking zones and physical zones. The blocking zones serve to calculate the resonance selfshielding factors, and the neutron fields are calculated in a system consisting of physical zones each of which is assigned a blocking zone. A zone with a homogeneous composition, including a set of physical zones (a tube, pellets of various materials, etc.), can be chosen as the blocking zone.
Heterogeneous effects can be taken into account by delivering information on the actual cell structure of the medium into the CONSYST. In this case, the program will recalculate the dilution crosssections of the resonance isotopes with regard for the particular geometry of the heterogeneous cell with the following corrections computed in the dilution crosssection:
– the Dancoff correction that takes into account the interactions of blocks in an infinite periodic array;
– the Bell factor depending on the block thickness (
The concentration of the δ scattering for all of the materials will be computed automatically.
Another important change in the CONSYST system is that the system determines independently for which zones and how the anisotropy of the full crosssection needs to be taken into account. To do this, it is enough for the user to set the respective indicator.
The neutronic parameters were calculated for the three BFS61 configurations using the MMKK, MMKRF, and MCNP codes. Table
Results of comparing the criticality calculations for various codes and nuclear data libraries.
BFS61 configuration  MMKRF  MCNP  MMKK  MMKK  

Mutually selfagreed computation model  Model ( 

ROSFOND  BNABRF  ROSFOND  BNABRF GETER  BNABRF GETER  BNABRF  
BFS610  0.99661(20)  0.99131(21)  0.99663(21)  1.00008(20)  0.99969(21)  0.99932(21) 
BFS611  0.99463(19)  0.99094(21)  0.99475(20)  0.99974(20)  0.99671(21)  0.99674(21) 
BFS612  0.99398(19)  0.99068(20)  0.99377(20)  0.99733(20)  0.99507(21)  0.99499(20) 
Time, min  56  50  262  156  48 
The values shown in columns 5 and 6 of Table
The results of the calculations shown in the table formed the basis for estimating the value of the nuclear data component for the uncertainty in neutronic parameters caused by the transition from the pointwise representation of neutron crosssections to the group approximation (this makes ~0.3% for the considered systems). Taking into account the spatial heterogeneity when preparing the macroscopic constants leads to a 0.8% finer criticality calculation.
The spectral characteristics measured by the SFC in the tube gap to the right of the BFS61 assembly central tube in the core median plane were calculated (see Fig.
Apart from the criticality calculation results, the base version of the MMKK code is capable to produce a flux in the preset zone with the same energy structure with which the CONSYST system prepared the macroscopic crosssections. Therefore, unlike the MCNP code, the MMKK code does not include capabilities for computing integrals of the ∫φ(E)dE and ∫φ(E)σ_{x} (E)dE form. Calculating such integrals requires extra programs for postprocessing of calculation results. Apart from the capability to compute any types of convolution with the flux, MMKRF, an improved version of the MMKK, (
Fig.
Biases in the calculations of the BFS610 assembly neutron spectrum, calculated using the BNABRF neutron group library and the MCNP (1) and MMKRF (2) codes, relative to the spectrum of neutrons calculated based on the MCNP code with the ROSFOND nuclear data library; 3 – statistical error of the spectrum calculation (3σ).
As can be seen from the diagram, the results of the neutron spectra calculations in the neutron crosssection group and detailed representations agree well in the limits of the statistical calculation error. Spectral indices can be calculated based on these energy distributions of neutron fluxes.
Table
The data presented in the table demonstrate a good agreement of the spectral indices calculation results obtained based on the MCNP and MMKRF codes using the ROSFOND library.
Calculated values of the fission rates measured at the BFS610 assembly.
Index  Experiment  Calculation  

MMKRF  MCNP  
σ_{f}^{Th232}/σ_{f}^{U235}  0.00947 ± 0.0003  0.00730 ± 0.00003  0.00724 ± 0.00004 
σ_{f}^{U233}/σ_{f}^{U235}  1.513 ± 0.03  1.4666 ± 0.0035  1.4673 ± 0.006 
σ_{f}^{U238}/σ_{f}^{U235}  0.0320 ± 0.0008  0.0307 ± 0.0001  0.0305 ± 0.0002 
σ_{f}^{Pu239}/σ_{f}^{U235}  1.057 ± 0.015  1.0337 ± 0.003  1.046 ± 0.005 
σ_{f}^{Pu240}/σ_{f}^{Pu239}  0.2575 ± 0.007  0.2674 ± 0.0008  0.2646 ± 0.002 
σ_{f}^{Pu241}/σ_{f}^{Pu239}  1.259 ± 0.03  1.2936 ± 0.003  1.2885 ± 0.006 
σ_{f}^{Pu242}/σ_{f}^{Pu239}  0.1883 ± 0.005  0.1908 ± 0.0007  0.1890 ± 0.0009 
σ_{f}^{Am241}/σ_{f}^{Pu239}  0.1963 ± 0.005  0.1935 ± 0.0006  0.1965 ± 0.0009 
The quality of the criticality calculation results obtained using different libraries (ROSFOND, BNABRF, and BNAB93) and the spectral indices measurements can be judged from the data presented in Table
Calculation/experiment discrepancies for measurements at the BFS61 assemblies.
BFS  Change  Value  δ_{e},%  (C/E – 1), %  δ_{c},%  

ROSFOND  BNABRF  BNAB93  
610  k_{eff}  1.0003  0.3  –0.3  –0.02  –0.06  0.01 
611  1.0004  0.3  –0.5  –0.07  –0.1  0.01  
612  1.0004  0.3  –0.6  –0.3  –0.1  0.01  
610  σ_{f}^{U238}/σ_{f}^{U235}  0.0320  3.0  –4.7  –9.9  –1.1  1.9 
σ_{f}^{Pu240}/σ_{f}^{Pu239}  0.2575  3.0  2.8  1.2  4.1  1.1  
σ_{f}^{Pu242}/σ_{f}^{Pu239}  0.1883  3.0  0.4  –2.0  4.6  1.2  
σ_{f}^{Th232}/σ_{f}^{U235}  0.0095  3.0  –23.6  –28.9  –20.5  2.0  
σ_{f}^{Pu239}/σ_{f}^{U235}  1.0570  1.5  –1.0  –2.2  –0.7  0.8  
σ_{f}^{Pu241}/σ_{f}^{Pu239}  1.2590  3.0  2.3  2.7  1.0  0.8  
σ_{f}^{U233}/σ_{f}^{U235}  1.5130  3.0  –3.1  –3.1  –1.2  0.8  
σ_{f}^{Am241}/σ_{f}^{Pu239}  0.1963  3.0  0.1  –3.2  14.8  1.4  
611  σ_{f}^{Pu238}/σ_{f}^{Pu239}  0.5412  3.0  12.1  14.6  17.6  0.9 
612  σ_{f}^{Pu239}/σ_{f}^{U235}  1.0520  1.5  –1.4  –2.7  –1.0  0.8 
σ_{f}^{Np237}/σ_{f}^{Pu239}  0.2610  3.0  0.2  –1.2  2.3  0.8  
µ_{k}^{2} for k_{eff}  3.09  0.39  0.11  
µ_{k}^{2} (all experiments)  1.13  1.30  2.28 
As can be seen from the table, the calculation/experiment discrepancies for most of the spectral indices lie within the experimental error limits of 3σ for all of the libraries. The exception is the calculation/experiment discrepancies for the ratio of the ^{232}Th fission crosssection to the ^{235}U fission crosssection and the ratio of the ^{238}Pu fission crosssection to the ^{239}Pu fission crosssection, the values of which are several times as great as the experimental error value. Such large discrepancy values are explained by technical aspects of the fission crosssection measurements for these isotopes. The experimental data for these spectral indices were ignored for the subsequent computational and experimental analysis.
The data shown in Table
Table
${\mu}_{k}^{2}=\frac{1}{N}\sum _{n=1}^{N}\left[{\left(\left({C}_{n}^{k}{\mathrm{E}}_{n}\right)\xb7100/{\mathrm{C}}_{n}^{k}\right)}^{2}/\left({\delta}_{\mathrm{C}n}^{2}+{\delta}_{\mathrm{E}n}^{2}\right)\right]$,
where N is the number of the experiments; n is the experiment number; and k is the identifier of the nuclear data library.
L.N. Usachev defines an optimized system of nuclear data that does not need to be adjusted to describe a set of N experimental data as the system of nuclear data, for which the value of the integral efficiency indicator μ_{k}^{2} lies in the interval of zero to unity.
As shown by the results of the calculation and experimental analysis (see the closing line in Table
Peculiarities have been considered of preparing mutually selfagreed computational models for precision and engineering neutronic codes and calculating, on the basis of these, the neutronic parameters of the three critical BFS61 assembly configurations.
The improved functional capabilities of updated software products and nuclear data have been demonstrated to support the computational and experimental analysis of integral experiments based on the BFS assemblies which make it possible to reduce considerably the time for and to minimize the probability of errors in preparing models and building mutually selfagreed neutronic models for highprecision and engineering neutronic codes. The new capabilities of the CONSYSTRF system allow simplifying to a great extent and automating the preparation of computational tasks for taking into account the heterogeneous selfshielding effects, while avoiding the need for additional codes to be involved. The updated version of the MMKRF code includes a superimposed grid function which permits computations of spectral characteristics and spatial distributions of the BFSmeasured reaction rates. Besides, the MMKRF code can be used for calculations in a neutron multigroup approximation with a pointwise representation of nuclear crosssections and using a combined technique. Since the MMKRF code has a better calculation rate than the MCNP code thanks to a specialized geometrical module, it can be chosen as the Russian highprecision neutronic code for the computational support of experiments on the BFS critical facilities.
The procedures have been tested for preparing neutron multigroup constants with regard for the effects of the spatial heterogeneity implemented in the CONSYSTRF code. It has been shown that the difference between the group approach and the pointwise representation of neutron crosssections in the criticality calculations does not exceed 0.3% in criticality for the BFS61 assemblies. The biases in the results for the spectral indices are in the limits of the statistical calculation error. Based on the results of the computational and experimental analysis for the entire set of measurements performed based on a series of the BFS61 assemblies, the ROSFOND library is the optimal nuclear data library.