Research Article 
Corresponding author: Aleksandr N. Pisarev ( a.n.pisarev93@gmail.com ) Academic editor: Yury Kazansky
© 2022 Aleksandr N. Pisarev, Valery V. Kolesov, Dmitry V. Kolesov.
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:
Pisarev AN, Kolesov VV, Kolesov DV (2022) The effect of errors in the neutron flux density on the uncertainties of nuclear concentrations of nuclides arising during the calculation of fuel burnup in cells with different neutron spectra. Nuclear Energy and Technology 8(4): 231235. https://doi.org/10.3897/nucet.8.96559

Computational studies have been carried out showing the complex time dependence of uncertainties in nuclear concentrations of various nuclides arising from the propagation of the neutron flux density errors in the burnup calculation process in cells with different neutron spectra on the above errors.
It is found that these uncertainties not only depend on the burnup time in a complex way, but also depend on the spectrum of the cell. The variants of the cell with thermal and fast neutron spectra were considered.
The calculations were performed using the VisualBurnOut program (
The influence of the number of calculated burnup points on the results of burnup calculations by the Monte Carlo method was investigated. Uncertainties arising in nuclear concentrations at intermediate calculation steps due to errors in nuclear concentrations appearing at the previous step were taken into account in the calculations.
Reactor plant, burnup calculations, uncertainties in nuclear data, uncertainties in nuclide nuclear concentrations, Monte Carlo method, neutron spectrum
Estimation of errors for different physical quantities obtained as the result of calculations due to errors (errors will universally hereafter mean to understand rootmeansquare deviations) in initial data, e.g., nuclear physical constants, is one of the most important problems in neutronic calculation.
In Monte Carlo burnup calculations, the burnup period is divided by a finite number of disjoint time steps. At the beginning of each burnup time step, the Monte Carlo method is used to calculate microscopic reaction rates. The calculated microscopic reaction rates are introduced and used for solving isotopic kinetics equations to determine the nuclear concentrations of all nuclides at the end of the burnup time step.
The reaction rates estimated by the Monte Carlo method at the beginning of each burnup time step require to contain errors arising due to errors in microscopic crosssections and initial nuclear concentrations of nuclides, as well as in the static error of the Monte Carlo calculations. These lead to errors in the nuclear concentrations obtained in solving burnup equations at the end of the burnup time step. Therefore, the errors in nuclear concentrations of nuclides and estimated reaction rates propagate to other burnup time steps. To estimate the propagation of errors during the burnup period, it is required to use many time steps and undertake multiple Monte Carlo calculations.
A great number of studies were undertaken for burnup calculations using Monte Carlo codes, such as MONTEBURNS (
In (
The proposed methodology for estimating the effect of errors in the burnup problem input parameters on the errors in nuclear concentrations obtained in the burnup process has been justified by the results obtained in (
To investigate the effect of errors in defining the neutron flux density on the nuclear concentrations of nuclides, a PWR reactor cell model was studied representing a threezone square cell (mesh spacing of 1.3127 cm) with MOX fuel (
N×10^{24} nuclei/cm^{3}  Fuel (PuO_{2}+UO_{2}), T=900K  Cladding (zircaloy), T=620K, external diameter 0.475 cm, thickness 0.065 cm  Moderator (H_{2}O), T=575K 

^{234}U  2.7043E7  
^{235}U  5.6570E5  
^{238}U  2.2286E2  
^{238}Pu  4.5941E7  
^{239}Pu  8.5640E4  
^{240}Pu  5.4669E5  
^{241}Pu  2.7221E6  
^{16}O  4.5180E7  
Zr (natural)  3.8657E2  
Fe (natural)  1.3345E4  
Cr (natural)  6.8254E5  
H  4.8414E2  
O  2.4213E2  
^{10}B  4.7896E6  
^{11}B  1.9424E5 
A fixed flux density error of 10% was used for the calculations. All of the results discussed below were obtained for the given flux density error. In reality, this error is much smaller and varies as a function of burnup. Therefore, this error can be assumed to be a certain model error. The actual (experimental) error can be determined, e.g., as the result of experiments to determine the neutron flux density in actual benchmarks.
The calculations were performed based on the VisualBurnOut code (
To avoid the effects of the initial fuel composition, the same PWR reactor cell was selected for investigating the fast spectrum effect but only with a hard spectrum. For a harder spectrum, the geometry and the concentration of hydrogen in the moderator were changed.
Fig.
The results obtained for the rootmeansquare deviations in the nuclear concentrations of nuclides for the fast spectrum were compared with the results obtained in (
Figs
Nuclear concentrations of nuclides and their errors as of the reactor life end with a 10% flux density error
Nuclide  Thermal spectrum  Fast spectrum  

Error, %  N×10^{24} nuclei/cm^{3}  Error, %  N×10^{24} nuclei/cm^{3}  
^{137}Cs  0.51  6.5526E5  0.51  6.7441E5 
^{155}Eu  0.74  6.2419E7  0.68  1.6952E6 
^{155}Gd  5.71  1.2394E8  0.18  3.4383E7 
^{157}Gd  1.92  2.3097E8  0.51  7.0382E7 
^{158}Gd  0.85  2.0824E6  0.58  4.5752E7 
^{234}U  1.36  1.7528E7  0.38  2.4967E7 
^{235}U  2.18  2.0311E5  1.71  2.6407E5 
^{236}U  1.76  6.6135E6  2.11  6.0805E6 
^{238}U  0.05  2.1544E2  0.20  2.0185E2 
^{237}Np  1.05  3.1880E6  1.59  7.0853E6 
^{238}Pu  0.93  3.8438E6  1.48  1.9074E6 
^{239}Pu  0.94  2.7440E4  0.68  1.7051E3 
^{240}Pu  0.43  1.8115E4  1.49  2.0610E4 
^{241}Pu  0.72  1.0621E4  1.54  1.5168E5 
^{242}Pu  1.85  3.6946E5  1.61  1.4526E6 
^{241}Am  1.96  5.3795E6  0.60  1.1407E6 
^{242m}Am  0.97  1.1430E7  2.06  1.5666E8 
^{243}Am  2.12  8.7295E6  1.95  9.6828E8 
^{242}Cm  3.80  1.5823E6  3.90  4.9779E8 
^{243}Cm  2.77  1.4596E8  2.88  1.9457E9 
^{244}Cm  3.03  3.7437E6  2.48  1.6731E8 
^{245}Cm  3.18  3.5184E7  2.65  8.7577E10 
A trend is observed for most nuclides towards smaller errors in the transition between the thermal neutron spectrum to the fast neutron spectrum. There are however nuclides for which an opposite trend is observed (^{236,238}U, ^{237}Np, ^{240,241}Pu,^{242m}Am,^{242,243}Cm). This will be investigated further.
The error becomes larger for isotopes with relatively small nuclear concentrations (^{242–245}Cm, ^{241}Am, ^{243}Am). The nuclear concentration error for ^{238}U is very small due to a large nuclear concentration of this isotope. Errors for fission products are largely below 1%. An exclusion is gadolinium isotopes which have a large crosssection in the thermal spectrum.
In Monte Carlo burnup calculations, the errors in the model’s input parameters propagate to the nuclear concentrations of nuclides through an exponential function. In these conditions, there is a probability that the errors in the nuclear concentrations of nuclides depend on the number of the calculated burnup points.
The errors in nuclear concentrations were estimated depending on the number of burnup steps. The rootmeansquare deviations in the nuclear concentrations of ^{235}U, ^{239}Pu and ^{241}Am obtained for different numbers of burnup steps (11, 22 and 44) are presented in Table
Errors in nuclear concentrations of nuclides depending on the number of calculated burnup points with a 10% flux density error
Burnup depth, GW∙day/t  Number of calculated burnup points  

44  22  11  
Errors in nuclear concentrations of nuclides, %  
^{235}U  ^{239}Pu  ^{241}Am  ^{235}U  ^{239}Pu  ^{241}Am  ^{235}U  ^{239}Pu  ^{241}Am  
16  0.85  0.84  0.94  1.21  1.18  1.34  1.69  1.66  1.78 
32  1.18  0.89  1.16  1.69  1.26  1.63  2.36  1.74  2.32 
48  1.49  0.79  1.37  2.11  1.08  1.96  2.97  1.49  2.66 
The cost of calculations in actual Monte Carlo calculations depends greatly on the costs concerned with the total number of neutron histories by the Monte Carlo method when preparing singlegroup crosssections at each burnup step. One cannot therefore increase infinitely the number of calculated burnup points but shall reasonably limit the calculation time. The obtained results of the error estimations depending on the number of steps make it possible to state that it is preferable to have a smaller number of calculated burnup points when it is not required to obtain highaccuracy results in estimation of errors and a larger number of calculated burnup points for cases, e.g., PIE (Postirradiation Experiments), which require highaccuracy results in nuclear concentrations of nuclides. In both cases, we obtain an upperbound estimate of the error.
Errors (rootmeansquare deviations) in nuclear concentrations of nuclides caused by the error in defining the neutron flux density have been estimated for the thermal and fast neutron spectra. Errors in the nuclear concentrations of various nuclides obtained based on the thermal and fast spectra have been compared.
These comparisons have shown that the effects of the input data error propagation (for the neutron flux density in the given case) in Monte Carlo burnup calculations for a PWR thermal cell depend in a complex way on the neutron spectrum. Issues have been identified which require an additional investigation. It will be reasonable to analyze further the effect of errors in reaction rates for the thermal and fast spectra and find out which reactions have the greatest effect on the errors in concentrations of particular nuclides.
We have also checked the effect of the number of calculated burnup points on errors in the nuclear concentrations of nuclides. The rootmeansquare deviation in the nuclear concentrations of nuclides in the considered investigation range for the effect of the number of steps has been found to diminish in 2^{1/2} increments. It therefore appears reasonable to undertake more studies to investigate the behavior of errors for a much larger number of steps.
It is planned to calculate the coefficients of sensitivity of nuclear concentrations to different nuclear data. This will make it possible to look into the issue of the behavior of nuclear concentration errors with different values of the input data errors.