Science and Technology of Nuclear Installations

Volume 2019, Article ID 3702014, 21 pages

https://doi.org/10.1155/2019/3702014

## Data-Driven and Precursor-Group Uncertainty Propagation of Lattice Kinetic Parameters in UAM Benchmark

^{1}Department of Nuclear, Plasma, and Radiological Engineering, University of Illinois at Urbana Champaign, Urbana, Illinois 61801, USA^{2}Reactor and Nuclear Systems Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37830, USA

Correspondence should be addressed to Tomasz Kozlowski; ude.sionilli@kxt

Received 31 January 2019; Accepted 19 March 2019; Published 2 May 2019

Academic Editor: Arkady Serikov

Copyright © 2019 Majdi I. Radaideh et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A new data-driven sampling-based framework was developed for uncertainty quantification (UQ) of the homogenized kinetic parameters calculated by lattice physics codes such as TRITON and Polaris. In this study, extension of the database for the delayed neutron data (DND) is performed by exploring more delayed neutron experiments and adding additional isotopes/actinides to the data libraries. Afterwards, the framework is utilized to obtain a deeper knowledge of the kinetic parameters’ sensitivity and uncertainty. The kinetic parameters include precursor-group-wise delayed neutron fraction (DNF) and decay constant. Input uncertainties include nuclear data (i.e., cross-sections) and DND (i.e., precursor group parameters and fractional delayed neutron yield). It is found that kinetic parameters, especially DNFs, have large uncertainties. The DNF uncertainty is driven by the cross-section uncertainties for LWR designs, while decay constant uncertainty is dominated by the DND uncertainties. The usage of correlated U-235 thermal DND in the UQ process significantly reduces the DND uncertainty contribution on the kinetic parameters. Large void fraction and presence of neutron absorber (e.g., control rod) increase the DNF uncertainty due to the hardening of neutron spectrum. High correlation between the DNF groups () is observed, while the decay constant groups () show weak correlation to each other and also to DNF groups. The DNF uncertainties of the dominant precursor group 4 for PWR, BWR, and VVER are about 7.5%, 9.4%, and 7.6%, respectively. The DNF uncertainty grows to larger values after fuel burnup. Kinetic parameters’ values and uncertainties provided here can be efficiently used in subsequent core calculations, point reactor kinetics, and other applications.

#### 1. Introduction

Uncertainty quantification (UQ) and sensitivity analysis (SA) are vital for assessing model’s input and output. Uncertainty propagation is a typical UQ task that involves propagating the uncertainty in the model parameters into the model output. SA is the process of quantifying the effect of changing each input factor on the model’s output. Both SA and UQ are used in tandem in various applications. SA can be classified in different categories such as local methods, regression-based methods, and variance-based methods [1, 2]. UQ can be implemented in different ways such as deterministic/sensitivity-based, stochastic/Monte Carlo-based, and reduced-order methods [3–5].

Nuclear reactor kinetic parameters describe the behavior of a specific type of neutrons inside the reactor called delayed neutrons. Delayed neutrons from their name are emitted later after the fission process by some fission product isotopes called delayed neutron precursors. Although delayed neutrons form small fraction (i.e., ~1%) compared to the prompt neutrons, they are very important for reactor control. Kinetic parameters have different classifications/definitions in different studies. However, most of the studies consider the delayed neutron fraction and the decay constant as the two main kinetic parameters to describe delayed neutrons. Delayed neutron fraction (DNF) expresses the fraction of delayed neutrons from the total number of neutrons emitted, while decay constant describes the timely decaying behavior of the isotope [6, 7]. These two quantities gained great importance in the previous decades in which many experiments have been conducted to analyze the delayed neutron behavior emitted from the fission of different actinides.

Delayed neutron experiments started back to 1940s [8, 9]. The idea behind these experiments is to irradiate a sample of an actinide (e.g., U-235, U-238, and Pu-239) with thermal or fast neutrons causing fission in that sample. The experiments are usually performed on tiny samples (few grams) using high intensity neutron sources to prevent neutron multiplication within the sample. In addition, the irradiation time should be instantaneous to capture the short-lived precursors. Indeed, the last group of precursors is usually not reported in the experiments, due to the difficulty in measuring this group as well as the large uncertainty associated with its measurement. The delayed neutron activity is then monitored and analyzed to fit the precursor groups. If an exponential decay is assumed to represent the delayed neutron activity after a burst fission, then the response (e.g., count rate) can be written with the independent time aswhere is the optimum number of periods/groups that minimizes the difference between the left and right hand sides. The parameters and are the fitting parameters which can be determined by least-squares. The study by [6] is among the first studies in this area, which concluded that is the optimum number of precursor groups. Following Keepin’s work, several studies have been conducted to report the values of the fitting parameters for different actinides using the suggested 6-group Keepin’s model. Examples of these experiments are provided in the next section. The 6-group model is not the only known model. Other models with 5, 7, and 8 groups have been investigated as well. The 8-group model suggested by Spriggs and Campbell [10, 11] gained more interest than the other two models (i.e., 5 and 7 groups). This model improves the time representation of the 6-group model by adding two additional groups and should reduce the uncertainty in the calculations. However, the 6-group model is still widely used due to the large number of experiments that have been performed to validate it compared to the recent 8-group model. Therefore, our focus will be on the 6-group Keepin’s model.

A framework to propagate the uncertainty of delayed neutron data (DND) and neutron cross-sections into the homogenized kinetic parameters was developed previously by Radaideh et al. [12] in the SCALE code system [13]. The framework methodology is described in detail in that paper, where a set of DND for 7 actinides was collected and used in the UQ process. The framework was applied on a simplified PWR pin-cell geometry for demonstration and a comparison of different DND sets for U-235 was conducted. The framework was then utilized in different applications such as reduced order modeling with Gaussian processes [14] and variance decomposition using Sobol indices [15]. In this study, a deep and comprehensive analysis of the uncertainty of the homogenized precursor-group-wise kinetic parameters is presented. First, a set of DND uncertainties for 20 actinides is developed based on a rigorous review of the delayed neutron experiments. The new set contains data from a wide range of experiments and review studies. The uncertainty of the kinetic parameters is analyzed based on the combination of DND and neutron cross-section uncertainties in different aspects such as the effects of data type (DND vs. neutron cross-sections) and the effects of operating conditions such as void fraction, control rod, and gadolinium absorber. The impact of DND correlation on the uncertainty is investigated, which was neglected in the previous studies. Fuel depletion and burnup effects on the kinetic parameters’ uncertainty are also explored. Finally, a set of kinetic parameters’ values and uncertainties are suggested for applications (e.g., core calculations and accident analysis) based upon the lattice models in UAM benchmark [16]. Since this framework is developed mainly to analyze kinetic parameters in precursor-group form, the parameters that are explored are the DNF and the decay constant for the six precursor groups.

The remaining sections of this study are organized as follows: Section 2 presents a survey and data collection of the DND for different actinides, followed by the presentation of the mathematical formulation of the kinetic parameters and their relation to the cross-sections and DND. Afterwards, the tools and implementation of the UQ approach are described, along with a brief description of the selected lattice models used for testing purposes. Section 3 presents the results of this study which include the following: the impact of DND correlation matrix, sensitivity to operating conditions, and the burnup effect on the kinetic parameters’ values and uncertainties, accompanied by a summary and discussion of the main findings. Finally, the conclusions of this study are presented in Section 4.

#### 2. Materials and Methods

##### 2.1. Delayed Neutron Data

In this subsection, DND parameters and their uncertainty are discussed. As mentioned in the introduction, the study performed by Keepin on the delayed neutrons of major actinides is considered one of the earliest and known studies in the area [6]. Most of the subsequent delayed neutron studies validated the results reported by Keepin. The reported delayed neutron parameters include group decay constant, group relative yield, and absolute delayed neutron yield from the fast fission of Th-232, U-233, U-235, U-238, Pu-239, and Pu-240 and thermal fission of U-233, U-235, and Pu-239. The experiment was conducted at Los Alamos National Laboratory using a bare U-235 metal assembly as the neutron source. Following Keepin’s experiment, a plethora of experiments have been conducted to determine the delayed neutron yield and group parameters for different actinides, with a focus on U-235 DND parameters. For example, Cox et al. [17] conducted a study on Cf-252, Cox [18] analyzed the delayed neutron emission from Pu-241, Gudkov et al. [19] performed a delayed neutron study based on irradiating various actinide samples in a fast reactor, Loaiza et al. [20] measured the group parameters of Np-237 and U-235 from fast fission, and Saleh et al. [21] used the Texas A & M research reactor to study the delayed neutron emission from thermal fission of U-235, Np-237, Am-241, and Am-243.

Tuttle [7] in his work reviewed the previous delayed neutron experiments that occurred before his study to report DND for the actinides with relevancy to reactor calculations. The study evaluated and revised a large number of previous experimental data and suggested a set of recommended values of DND. This makes Tuttle’s work [7] a valuable source of DND to this study. It is worth mentioning that Tuttle’s work relied extensively on Keepin’s data, and both of these studies are still widely used in reactor physics applications. Other studies are selected for isotopes which are not reported by Tuttle/Keepin or if the data by Tuttle is not accurate due to the limited experiments available at that time for a particular isotope. Waldo et al. [25] reported group parameters for some isotopes that have been rarely studied such as U-232, Pu-238, and Cm-245. This study is also used in our data library. For other actinides at which there is no experimental data for their group parameters, data from Wilson and England [27] is used. The data reported by [27] is computational based that comes from simulating the activity of the precursors, following a burst fission of each actinide, and then using least-squares fit to calculate the group parameters. It is worth mentioning that such computational data is used here for isotopes that are less relevant to LWR applications such as Th-227, U-234, and Pa-231. First, we need to define the DND parameters of interest in this study as follows:(1)Group fractions (): this parameter represents the relative abundance or fractional delayed neutron yield for the precursor group that results from fission in isotope/actinide . This parameter is expressed in normalized form (i.e., the sum of all precursor group fractions should be unity).(2)Group decay constant (): this parameter represents the effective decay constant of the precursor isotopes in the precursor group that results from fission in isotope/actinide . Group is the longest-lived group, while group contains the short-lived precursors.(3)Absolute delayed neutron yield (): this parameter expresses the average number of delayed neutrons emitted per fission (thermal or fast) in each actinide.(4)Fractional delayed neutron yield (): this parameter is the ratio of to the average number of neutrons (prompt and delayed) emitted per fission in each actinide (i.e., ).

The first three parameters, especially , are measured in delayed neutron experiments. The fourth parameter is not usually reported, since it requires , which is rarely reported in delayed neutron experiments. Consequently, a data source for is needed. In brief, the criteria followed for dataset construction are as follows:(i)In general, the group parameters (, ) and the absolute delayed neutron yield for most of the fast fission data are taken from Tuttle [7] due to their high accuracy.(ii)The group parameters and the absolute delayed neutron yield for thermal fission data of U-233, U-235, and Pu-239 are taken from [6]. Indeed, Tuttle [7] suggested using Keepin’s data for thermal fission on the basis of its quality.(iii)It is preferred to use the absolute delayed neutron yield from the same study as the group parameters, since the group fractions () are calculated (i.e., normalized) using the measured delayed neutron yield.(iv)If the isotope data is not available in either [7] or [6], a different source is used for the group parameters and the absolute delayed neutron yield.(v)If there is no experimental data available for the group parameters of a specific isotope, a computational-based data is selected from [27]. The computational data has no uncertainty and no effect on the UQ results.(vi)If there is no experimental data available for for a specific isotope in the delayed neutron experiments, the value and its uncertainty are taken from SCALE data and covariance libraries which are based on ENDF-B/VII.1.(vii)All and its uncertainty for all isotopes are taken from SCALE data and covariance libraries based on ENDF-B/VII.1.(viii)Exceptions to the previous points are minimal, and they are mentioned in the appropriate place in the text.

Based on the previous criteria, a total of 20 actinides and their DND are used in the framework. Table 1 lists the measured delayed neutron group parameters for the isotopes whose delayed neutrons are assigned to both thermal and fast sets. The data for the other isotopes whose either fast or thermal fission set is listed in Table 2 is in Appendix A. The reference used to obtain the data is reported for each isotope. Separating the data into thermal and fast fission sets is based on data availability and differences in between thermal and fast fissions. According to Tuttle [7], it is recommended to use two sets if there is a considerable difference in (see, e.g., U-235 in Table 3) emitted from a thermal and fast fission of an isotope. This approach is adopted only if there is experimental data that investigates the delayed neutron emission from fast and thermal fission of the same isotope. These two conditions are applicable only for the five isotopes in Table 1. The measured values of are given in Table 3. As mentioned previously, all values are taken from SCALE libraries for both fast and thermal ranges, except for Cf-252 spontaneous fission, which is obtained from [31]. In overall data tables shown in this study, “F” refers to fast fission, “T” to thermal fission, and “S” to spontaneous fission.