Corresponding author: Alexander N. Pisarev ( a.n.pisarev93@gmail.com ) Academic editor: Yury Kazansky
© 2020 Alexander N. Pisarev, Valerii 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 (2020) A study into the propagation of the uncertainties in nuclear data to the nuclear concentrations of nuclides in burnup calculations. Nuclear Energy and Technology 6(3): 161166. https://doi.org/10.3897/nucet.6.57802

The key papers on estimating the uncertainties in nuclear data deal with the influence of these uncertainties on the effective multiplication factor by introducing the socalled sensitivity factors and only some of these are concerned with the influence of such uncertainties on the life calculation results. On the other hand, the uncertainties in reaction rates, the neutron flux, and other quantities may lead to major distortions in findings, this making it important to be able to determine the influence of uncertainties on the nuclear concentrations of nuclides in their burnup process. The possibility for the neutron flux and reaction rate uncertainties to propagate to the nuclear concentrations of nuclides obtained as part of burnup calculations are considered using an example of a MOXfuel PWR reactor cell. To this end, three burnup calculation cycles were performed, and the propagation of uncertainties was analyzed. The advantages of the uncertainty estimation method implemented in the VisualBurnOut code consists in that all rootmeansquare deviations are obtained as part of one calculation as the statistical method, e.g. GRS (Generation Random Sampled), requires multiple calculations. The VisualBurnOut calculation results for the rootmeansquare deviations in nuclear concentrations were verified using a simple model problem. It is shown that there is a complex dependence of the propagation of the rootmeansquare deviations in the nuclear concentrations of nuclides in the process of fuel burnup, and, therefore, further studies need to aim at investigating the influence of uncertainties in nuclear data on the nuclear concentrations of nuclides.
Reactor plant, burnup calculations, uncertainties in nuclear data, uncertainties in nuclear concentrations of nuclides, Monte Carlo method
At the present time, a great deal of attention is given to estimating the influence of uncertainties in nuclear data on the parameters of different reactor plants. Largely, however, the influence of uncertainties in nuclear data on the key parameters (k_{eff}, reaction rates, and others), are studied using sensitivity factors. Thus, estimated uncertainties for different thermal reactors with a burnup of 40 to 100 MW∙day/kg equal 0.5 to 2% for k_{eff} (
Nuclear data for high energies are less accurate, and estimated uncertainties for fast reactors have, accordingly, higher values than those for thermal reactors, e.g., 1 to 4% for k_{eff} (
Only several works deal with the effects of uncertainties in nuclear data on the nuclear concentrations of nuclides obtained in the process of calculating the fuel burnup in the reactor (
Quantitatively estimating the uncertainties in concentrations as part of burnup calculations will allow a better prediction of the fuel’s isotopic composition for optimization tasks of transmutation, recycling, and waste removal. Transmutation indices depend greatly on nuclear data, and it is important to quantitatively estimate the resulting uncertainties in integral parameters of fast reactor systems, since their fuel may be heavily saturated with minor actinides, such as americium and curium, the crosssections (reaction rates) of which are poorly known. Knowing the isotopic composition during the reactor operation is essential for estimating the longterm radiotoxicity and decay heat of spent fuel, changes in the reactivity margin, and the growth in the pressure of gas and in the concentration of volatile fission products (which form the source term in emergencies) (
Let N(t)= [N_{1}(t), N_{2}(t), N_{M} (t)]^{T} be the vector of nuclear concentrations consisting of M different nuclides at the time t. The timedependent change in the nuclear concentrations in the neutron field is described by a system of differential equations
d N/dt = AN = λN + σ^{eff}ΦN, (1)
where A is the matrix of the transitions of certain nuclides to others due to nuclear reactions and decays; λ is the matrix of the values for the decay constants of the size M×M; σ^{eff} is the matrix of singlegroup neutron crosssections; and Φ is the averaged integral neutron flux. Taking into account that N_{0} = N(0) is the initial vector of nuclear concentrations, we shall write down the solution as
N(t) = exp(At)N_{0} (2)
with the constant neutron spectrum (therefore, with constant singlegroup microscopic crosssections and the constant neutron flux) throughout the time step [0, t].
Nuclear systems, in which changes in the isotopic composition of fuel affect the neutron flux distribution, require a series of combined neutronic and fuel burnup calculations. Such combined calculations have the entire burnup cycle divided into several successive time intervals. The ionizing radiation transport is calculated for each time interval, and the reaction rates and the neutron flux are also calculated which are used to solve the burnup equation and to obtain the isotopic composition of fuel at the respective time interval end.
The purpose of the work is not only to calculate the vector N of the timedependent nuclear isotope concentration but also to estimate the effects of different uncertainty sources arising as the result of a combined burnup calculation of nuclear concentrations.
Let there be no uncertainty in the initial nuclear concentrations of nuclides though it is possible to take these into account as well. Uncertainties can be found in all the parameters a burnup equation includes, i.e., in decay constants λ, in singlegroup effective microscopic crosssections σ^{eff}, and in the integral neutron flux Φ: N = N(λ, σ^{eff}, Φ, t] whence it follows that ∆N depends on ∆λ, ∆σ^{eff}, ∆Φ, where the symbol ∆ means the uncertainty or the relative error of the respective value.
1. Uncertainties in decay constants ∆λ are normally given in evaluated nuclear data libraries.
2. Uncertainties in singlegroup effective microscopic crosssections ∆σ^{eff} depend both on uncertainties in evaluated nuclear data of microscopic crosssections ∆σ^{g} and on uncertainties in the neutron flux ∆Φ^{g} (E) (the index g here means the energy group number). Knowing the covariance matrices which can be obtained from respective files of evaluated neutron data, one can obtain uncertainties in singlegroup effective microscopic crosssections ∆σ^{eff}, e.g. using the ERRORJ module of the NJOY software package.
An analysis of uncertainties in the nuclear data available in the latest international nuclear data libraries has shown that:
In this case, the results obtained using this data should be viewed as a sort of a "proof of principle"; calculations should be repeated up to obtaining more accurate data. On the other hand, uncertainties in the neutron flux are explained by uncertainties in microscopic crosssections and nuclear concentrations of nuclides (generally, in all input data required for the transport equation) and by the statistical nature of the neutron transport Monte Carlo calculation as such.
3. Uncertainties in the integral neutron flux ∆Φ. A normalizing coefficient is required to obtain the neutron flux value. As a rule, constant power is assumed to be such a factor, that is, there is a control mechanism which will change (compensate for) the neutron flux value to keep the required constant power level. If we designate full power as P, then
P = K N σ_{f} Φ V, (3)
where V is the core volume; N σ_{f} Φ is the fission rate; and K is the conversion factor. It can be seen from the equation that the uncertainty in the integral neutron flux will depend on the uncertainties in the nuclear concentrations of nuclides and uncertainties in the singlegroup microscopic crosssections of the fissile material.
Therefore, the sources of uncertainties in burnup calculations can be classified as follows:
1) uncertainties in initial nuclear data;
2) uncertainties caused by the statistical nature of the neutron transport Monte Carlo calculation;
3) uncertainties caused by the normalizing factor.
N = N (λ, σ^{eff}, Φ, t) = N (λ, σ^{eff}, ∆Φ, t). (4)
The paper investigates the effects of uncertainties in the neutron flux and in the reaction rates for different nuclides on burnup calculations.
The most intuitively understood method to propagate the uncertainties in nuclear data to nuclear concentrations in the process of burnup consists in selecting, for each calculation, the reaction rates and/or the neutron flux in a statistical (e.g., lognormal) manner with the preset rootmeansquare deviation. The calculation is performed repeatedly. The method is simple and versatile but is extremely costly since it often requires a calculation to be repeated hundreds of times. Several patterns were developed to estimate uncertainties with only one set of solutions (
It was shown in (
The computational model is a squareshaped PWR cell containing MOX fuel. The geometry and the composition of the fresh fuel, the cladding and the moderator are shown in Tables
Singlegroup reaction rates and fluxes calculated by Monte Carlo method were used to calculate the isotope kinetics using the VisualBurnOut code. The calculations considered 254 nuclides for which singlegroup crosssections, obtained based on the JENDL3.2 and JEF2.2 libraries, were calculated. Each calculation had 1000 neutron generations traced with 1000 neutron histories per generation.
The following model problem is set to verify the calculations of uncertainties in nuclear concentrations obtained using the VisualBurnOut code: only the fission and radiation capture of two nuclides, ^{240}Pu and ^{241}Pu, are considered. Then the system of differential equations is written as
$\left\{\begin{array}{c}\frac{d{N}_{40}\left(t\right)}{dt}={\sigma}_{c}^{40}\Phi {N}_{40}\left(t\right){\sigma}_{f}^{40}\Phi {N}_{40}\left(t\right)\\ \frac{d{N}_{41}\left(t\right)}{dt}={\sigma}_{c}^{41}\Phi {N}_{41}\left(t\right){\sigma}_{f}^{41}\Phi {N}_{41}\left(t\right)+{\sigma}_{c}^{40}\Phi {N}_{40}\left(t\right)\end{array}\right.$ (5)
The analytical solution is written as
$\left\{\begin{array}{c}{N}_{40}\left(t\right)={N}_{40}^{0}\mathrm{exp}\left(\left({\sigma}_{c}^{40}+{\sigma}_{f}^{40}\right)\Phi t\right)\\ {N}_{41}\left(t\right)={N}_{41}^{0}\mathrm{exp}\left(\left({\sigma}_{c}^{41}+{\sigma}_{f}^{41}\right)\Phi t\right)+\frac{{\sigma}_{c}^{40}{N}_{40}^{0}}{{\sigma}_{c}^{41}+{\sigma}_{f}^{41}{\sigma}_{c}^{40}{\sigma}_{f}^{40}}\\ \times \left[\mathrm{exp}\left(\left({\sigma}_{c}^{40}+{\sigma}_{f}^{40}\right)\Phi t\right)\mathrm{exp}\left(\left({\sigma}_{c}^{41}+{\sigma}_{f}^{41}\right)\Phi t\right)\right]\end{array}\right.$ (6)
The sensitivity factors with respect to the neutron flux are as follows:
$\left\{\begin{array}{c}\frac{\partial {N}_{40}\left(t\right)}{\partial \Phi}={N}_{40}^{0}\left({\sigma}_{c}^{40}+{\sigma}_{f}^{40}\right)t\xb7\mathrm{exp}\left(\left({\sigma}_{c}^{40}+{\sigma}_{f}^{40}\right)\Phi t\right)\\ \left.\frac{\partial {N}_{41}\left(t\right)}{\partial \Phi}={N}_{41}^{0}\left({\sigma}_{c}^{41}+{\sigma}_{f}^{41}\right)t\xb7\mathrm{exp}\left(\left({\sigma}_{c}^{41}+{\sigma}_{f}^{41}\right)\Phi t\right)+\frac{{\sigma}_{c}^{40}{N}_{40}^{0}t}{{\sigma}_{c}^{41}+{\sigma}_{f}^{41}{\sigma}_{c}^{40}{\sigma}_{f}^{40}}\times \left({\sigma}_{c}^{41}+{\sigma}_{f}^{41}\right)\xb7\mathrm{exp}\left(\left({\sigma}_{c}^{41}+{\sigma}_{f}^{41}\right)\Phi t\right)\right]\\ \times \left[\left({\sigma}_{c}^{40}+{\sigma}_{f}^{40}\right)\mathrm{exp}\left(\left({\sigma}_{c}^{40}+{\sigma}_{f}^{40}\right)\Phi t\right)+\left({\sigma}_{0}^{4}\right)\xb7\right.\end{array}\right.$ (7)
Uncertainties in the nuclear concentrations are further found for the nuclides ^{240}Pu and ^{241}Pu: ∂N_{40}(t)∆Φ×100%/∂ΦN_{40}(t) и ∂N_{41}(t)∆Φ×100%/∂ΦN_{41}(t).
Figs
It can be seen from the figures that the rootmeansquare deviations behave in a complicated manner depending on the burnup time; secondly, the analytically obtained results agree well with the VisualBurnOut numerical calculation results.
Three burnup cycles were conducted further using the VisualBurnOut code: up to 16, 32 and 48 GW×day/t for a PWR with MOX fuel as described above. The neutron flux for keeping the linear power was recalculated with a step of 30 days which approximately corresponds to 1.1 GW×day/t. Singlegroup constants were recalculated as part of each burnup step. We calculated the burnup defining the rootmeansquare deviations for the neutron flux and the reaction rates, and further analyzed the behavior of the nuclear concentration uncertainties depending on time while taking into account both the individual contribution of the uncertainty for each quantity and the joint contribution. The rootmeansquare deviations of the burnup calculation input data amounted to 10%.
The results of calculating the rootmeansquare deviations in the nuclear concentrations of nuclides, depending on uncertainties in various nuclear data, are presented below. It is assumed for all cases that the rootmeansquare deviations of the reaction rates and the neutron flux amount to 10%.
Fig.
Figs
The uncertainty in the ^{239}Pu nuclear concentration is formed largely by uncertainties in the ^{239}Pu fission reaction and ^{238}U radiation capture rates. This can be explained by the fact that ^{239}Pu is obtained from ^{238}U in the chain of the following transformations:
^{238}U (n, γ) → ^{239}U → ^{239}Np → ^{239}Pu. (8)
The major contributors to the uncertainty in the nuclear concentration of ^{241}Pu are uncertainties in the ^{241}Pu fission reaction and ^{240}Pu radiation capture rates for the same reason as for ^{239}Pu. The uncertainty in the nuclear concentration of ^{235}U is formed largely by uncertainties in the ^{235}U fission reaction and radiation capture rates.
An increase or a decrease in the rootmeansquare deviation of the nuclear concentrations stems from the change in the behavior of the nuclear concentrations as such. Figs
Pitch, cm  1.3127 
External radius, cm  0.475 
Cladding thickness, cm  0.065 
Fuel pellet radius, cm  0.410 
Isotope  Isotopic composition, % of Pu_{total} 

^{238}Pu  0.05 
^{239}Pu  93.6 
^{240}Pu  6.0 
^{241}Pu  0.3 
^{242}Pu  0.05 
Isotope  Isotopic composition, % of U_{total} 

^{234}U  0.00119 
^{235}U  0.25000 
^{238}U  99.74881 
Content of plutonium in MOX fuel, % of Pu_{total}/[U+Pu]  MOX fuel enrichment, % of Pu_{fissile}/[U+Pu] 

4.377  4.110 
Isotope  Nuclear concentrations, 10^{24} nuclei/cm^{3} 
Zircaloy2 (5.8736 g/cm^{3} – reduced density)  
Zr (natural)  3.8657E–2 
Fe (natural)  1.3345E–4 
Cr (natural)  6.8254E–5 
Coolant/moderator (600 ppm of boron; 0.7245 g/cm^{3})  
H  4.8414E–2 
O  2.4213E–2 
^{10}B  4.7896E–6 
^{11}B  1.9424E–5 
Material  Temperature, K 

Fuel  900 
Cladding  620 
Coolant/moderator  575 
Isotope  Nuclear concentrations, 10^{24} nuclei/cm^{3} 

^{234}U  2.7043E–7 
^{235}U  5.6570E–5 
^{238}U  2.2286E–2 
^{238}Pu  4.5941E–7 
^{239}Pu  8.5640E–4 
^{240}Pu  5.4669E–5 
^{241}Pu  2.7221E–6 
^{242}Pu  4.5180E–7 
^{16}O  4.6515E–2 
The results of the studies have shown that there is a complex dependence of the rootmeansquare deviations in the nuclear concentrations of nuclides caused by uncertainties in the reaction rates and in the neutron flux on burnup time. As fuel burns up, the behavior of the rootmeansquare deviations is not at all times monotonous and depends, for the considered isotope, on the uncertainty source, the reaction rate type, and the precursor nuclei.