Monte Carlo criticality analysis of random media under bounded fluctuation driven by normal noise

In Monte Carlo criticality analysis under material distribution uncertainty, it is necessary to evaluate the response of neutron effective multiplication factor (k_eff) to the space-dependent random fluctuation of volume fractions within a prescribed bounded range. Normal random variables, however, cannot be used in a straightforward manner since the normal distribution has infinite tails. To overcome this issue, a methodology has been developed via forward–backward-superposed reflection Brownian motion (FBSRBM). Here, the forward–backward superposition makes the variance of fluctuation spatially constant and the reflection Brownian motion confines the fluctuation driven by normal noise in a bounded range. In addition, the power spectrum of FBSRBM remains the same as that of Brownian motion. FBSRBM was implemented using Karhunen–Loève expansion (KLE) and applied to the fluctuation of volume fractions in a model of UO₂–concrete media with stainless steel. Numerical results indicate that the non-negligible and significant fluctuation of k_eff arises due to the uncertainty of media formation and just a few number of terms in KLE are enough to ensure the reliability of criticality calculation.     

Neutronics Calculations of HTTR with Several Nuclear Data Libraries

Benchmark calculations for several HTTR core states were performed with four cross-section sets which were generated from JENDL-3.3, JENDL-3.2, ENDF/B-VI.8 and JEFF-3.0 using a continuous energy Monte Carlo code MVP. The core states were a critical approach in which an annular core was formed at room temperature and solid cores at room temperature and at full power operation. Study of k_eff discrepancies caused by difference of the nuclear data libraries and identification of nuclides which have large effects on the k_eff discrepancies were carried out. Comparison of the respective k_eff from calculations and experiments was also carried out. As the results, for each of the HTTR core states, JENDL-3.3 yields a k_eff agreeing with the experiments within 1.5%Δk, JENDL-3.2 yields k_eff agreement within 1.7%Δk, and ENDF/B-VI.8 and JEFF-3.0 yield k_eff agreement within 1.8%Δk. There is little k_eff discrepancy between ENDF/B-VI.8 and JEFF-3.0. The k_eff between JENDL-3.3 and JENDL-3.2 is caused by difference of ²³⁵U data and has temperature dependency. The k_eff discrepancy between JENDL-3.3 and ENDF/B-VI.8 or JEFF-3.0 is mainly caused by difference in graphite data.     

International Symposium on Isotope Separation and Chemical Exchange Uranium Enrichment

From the viewpoint of nuclear criticality safety, it is important to comprehend the reactivity of fuel solutions induced by oscillatory movements such as earthquakes. This paper intends to figure out the reactivity of a fuel solution system with a free surface formed by oscillation by evaluating the fluctuation of the neutron multiplication factor (k _eff ) obtained from a static calculation. To fulfill this intension, criticality calculations with reflecting fluid calculation results have been carried out. In the fluid calculations, the finite volume method and the volume of fluid (VOF) method have been applied in tracking the free surface formed by oscillation. The continuous energy Monte Carlo calculation method has been applied in the criticality calculations. As a result, it has been found that the variation patterns of the k _eff and those of the shape of fuel solutions are classified according to oscillation frequency and the ratio of solution height to the width of the tank (H/L). If a sloshing motion is generated, the k _eff fluctuates widely and has a threshold, with which we can classify the fluctuation type of the k _eff , despite the kind of reflector. If H/L is above the threshold, i.e., H/L =0.4, the k _eff fluctuates to a value below that obtained in the resting state. On the contrary, if H/L is below the threshold, the k _eff fluctuates to a value above that obtained in the resting state. This result implies the criticality calculation for a fuel solution with a free surface using the Monte Carlo method may give a slightly smaller threshold than using other approaches.     

Effects of Fission Products on Physical Quantities Accompanying Burn-Up of Large Pu-U Fast Reactors

The effects brought by the presence of fission products (F.P.) on the effective multiplication factor k _eff, the Na-void reactivity, the breeding ratio, the fuel composition and kinetics parameters have been calculated as functions of burn-up for Pu-U fast reactor with 3,000l core volume.

The F.P. sharply reduce k _eff and increase the positive values of the Na-void reactivity. Moreover, at a given burn-up, this effect of F.P. on k _eff and Na-void reactivity is governed largely by the total amount of the F.P. found accumulated at the time of observation, and is independent of the history of the material.

The F.P. hardly influence the transformation accompanying burn-up undergone by the ratio of Pu to ²³⁸U atoms and by the isotopic composition of Pu. Similarly, the effect on the internal breeding ratio also is very small. The total breeding ratio increases gradually with accumulation of the F.P. The effect on the effective delayed neutron fraction β_eff is only slight, while that on the prompt-neutron lifetime l _p is appreciably larger.     

Investigation on Criticality and Infinite Multiplication Factor of High Conversion Light Water Reactor Using Zone-Type FCA-HCLWR Core Fueled with Enriched Uranium

—A series of reactor physics experiments have been carried out at the FCA to examine the availability of the nuclear data and computational method currently employed to evaluate the nuclear characteristics of the High Conversion Light Water Reactor. Experimental results of the effective and infinite multiplication factors k_eff and k_∞ are compared with the calculated ones for three zone-type FCA-HCLWR cores fueled with enriched uranium. The calculated k_eff and k_∞ values with use of the SRAC system and the cross section set based on the JENDL-2 data file show a good agreement with the measured ones. The calculated-to-experimental (C/E) values for k_eff and k_∞ do not depend on the cell parameters such as the fuel enrichment, the moderator voidage state and the moderator-to-fuel volume ratio, and these values are similar with each other There is also no inconsistency between the C/E values for k_eff and k_∞ : The average C/E values are 989 and 0 988 for k_eff and k_∞ respectively     

Measurement of effective atomic numbers using energy-resolved computed tomography

For ion radiation therapy, the measurement of effective atomic numbers, Z_eff, is necessary to know the material distribution in a human body; the range of ions entering the human body is influenced by the material distribution along their paths. Z_eff, however, cannot be measured at hospitals because monochromatic X-rays with different energies are necessary and are used only at synchrotron facilities. To make Z_eff measurements at hand, we propose energy-resolved computed tomography (CT) using a “transXend detector”. By assigning two narrow energy ranges in the unfolding process of the data obtained by the transXend detector, Z_eff for acrylic and aluminum can be estimated by energy-resolved CT. The estimated Z_eff are compared with those obtained by dual-energy and monochromatic X-ray CT.     

Whole Core Calculations of Power Reactors by Use of Monte Carlo Method

Abstract

Whole core calculations have been performed for a commercial size PWR and a prototype LMFBR by using vectorized Monte Carlo codes. Geometries of cores were precisely represented in a pin by pin model. The calculated parameters were k _eff, control rod worth, power distribution and so on. Both multigroup and continuous energy models were used and the accuracy of multigroup approximation was evaluated through the comparison of both results. One million neutron histories were tracked to considerably reduce variances. It was demonstrated that the high speed vectorized codes could calculate k _eff, assembly power and some reactivity worths within practical computation time. For pin power and small reactivity worth calculations, the order of 10 million histories would be necessary. It would be difficult for the conventional scalar code to solve such large scale problems while the present codes consumed computation time less than 30 min for a PWR and 1 hour for an LMFBR. Required number of histories to achieve target design accuracy were estimated for those neutronic parameters.     

The comparison of MCNP perturbation technique with MCNP difference method in critical calculation

For a nuclear fission system, we calculated Δk_eff, which arise from system material composition changes, by two different approaches, the MCNP perturbation technique and the MCNP difference method. For every material composition change, we made four different runs, each run with different cycles or each cycle generating different neutrons, then we compared the two Δk_eff that are obtained by two different approaches. As a material composition change in any particular cell of the nuclear fission system is small compared to the material compositions in the whole nuclear fission system, in other words, this composition change can be treated as a small perturbation, the Δk_eff results obtained from the MCNP perturbation technique are much quicker, much more efficient and reliable than the results from the MCNP difference method.When a material composition change in any particular cell of the nuclear fission system is significant compared to the material compositions in the whole nuclear fission system, both the MCNP perturbation technique and the MCNP difference method can give satisfactory results. But for the run with the same cycles and each cycle generating the same neutrons, the results obtained from the MCNP perturbation technique are systemically less than the results obtained from the MCNP difference method. To further confirm our calculation results from the MCNP4C, we run the exact same MCNP4C input file in MCNP5, the calculation results from MCNP5 are the same as the calculation results from MCNP4C.We need caution when using the MCNP perturbation technique to calculate the Δk_eff as the material composition change is large compared to the material compositions in the whole nuclear fission system, even though the material composition changes of any particular cell of the fission system still meet the criteria of MCNP perturbation technique.     

Uncertainty analysis in Monte Carlo criticality computations

Uncertainty analysis is imperative for nuclear criticality risk assessments when using Monte Carlo neutron transport methods to predict the effective neutron multiplication factor (k_eff) for fissionable material systems. For the validation of Monte Carlo codes for criticality computations against benchmark experiments, code accuracy and precision are measured by both the computational bias and uncertainty in the bias. The uncertainty in the bias accounts for known or quantified experimental, computational and model uncertainties. For the application of Monte Carlo codes for criticality analysis of fissionable material systems, an administrative margin of subcriticality must be imposed to provide additional assurance of subcriticality for any unknown or unquantified uncertainties. Because of a substantial impact of the administrative margin of subcriticality on economics and safety of nuclear fuel cycle operations, recently increasing interests in reducing the administrative margin of subcriticality make the uncertainty analysis in criticality safety computations more risk-significant. This paper provides an overview of two most popular k_eff uncertainty analysis methods for Monte Carlo criticality computations: (1) sampling-based methods, and (2) analytical methods. Examples are given to demonstrate their usage in the k_eff uncertainty analysis due to uncertainties in both neutronic and non-neutronic parameters of fissionable material systems.     

Tally efficiency analysis for Monte Carlo Wielandt method

The Monte Carlo Wielandt method has the potential to eliminate most of a variance bias because it can reduce the dominance ratio by properly controlling the estimated eigenvalue (k_e). However, it requires increasingly more computation time to simulate additional fission neutrons as the estimated eigenvalue becomes closer to the effective multiplication factor (k_eff). Therefore, its advantage over the conventional Monte Carlo (MC) power method in the calculation efficiency may not always be ensured. Its efficiency of the tally estimation needs to be assessed in terms of a figure of merit based on a real variance as a function of k_e. In this paper, the real variance is estimated by using an inter-cycle correlation of the fission source distribution for the MC Wielandt calculations. Then, the tally efficiency of the MC Wielandt method is analyzed for a 2 × 2 fission matrix system and weakly coupled fissile array problems with different dominance ratios (DRs). It is shown that the tally efficiency of the MC Wielandt method depends strongly on k_e, there is a k_e value resulting in the best efficiency for a problem with a large DR, and the efficiency curve as a function of L, the average number of fission neutrons per history, follows a long tail after the best efficiency.     

The Sandwich Method for Determining Source Convergence in Monte Carlo Calculation

In slow source convergence problems, it is often difficult to ascertain whether the source iteration has converged or not. In order to solve this problem, a new “sandwich method” has been proposed. The essence of this method is that a finally converged eigenvalue k_eff is approached starting from two kinds of initial source guesses which give higher and lower neutron multiplication factors. It is especially important for evaluating nuclear criticality safety to know how to choose a biasing source to obtain an upper limit for k_eff . In this paper, (1) an example is shown to explain the difficulties in ascertaining the source convergence, (2) a method is proposed to obtain the upper and lower limit curves for k_eff by biasing the initial source distribution, (3) the sandwich method is applied to four benchmark problems proposed by the source convergence group of the OECD/NEA Working Party on Nuclear Criticality Safety.

Our calculation results show that the sandwich method is an effective means to confirm source convergence in such slow convergence problems. Appendix is prepared to support the method theoretically.     

Comparison of Cell Models for Analysis of Heterogeneous Fast Critical Assembly

We have investigated cell calculation models to be used in the analysis of neutronic characteristics of a heterogeneous fast critical assembly. As cell models we have considered a single drawer model with a critical buckling, a single drawer model with group dependent bucklings and a multidrawer model which consists of some fuel and blanket drawers. We have compared the cell averaged cross sections obtained from these cell models with the results of a reference transport calculation and estimated the effects of the cell models on k _eff, reaction rate ratios, reaction rate distributions and sodium void worths in the heterogeneous fast critical assembly ZPPR-13A. The multidrawer model and the single drawer model with group dependent buckling give reasonable cell averaged cross sections and have large effects on k _eff and ²³⁸U fission rate distribution.     

Neutron cross section sensitivity analysis of UO2 and MOX core physics experiments on light water reactors

The perturbation theory based on the transport calculation has been applied to study sensitivity of neutron multiplication factors (k_eff's) to neutron cross sections used for the reactivity analysis of UO₂ and MOX core physics experiments on light water reactors. The studied cross sections were neutron capture, fission and elastic scattering cross sections, and a number of fission neutrons, ν. The obtained sensitivities were multiplied to relative differences in the cross sections between JENDL-4.0 and JENDL-3.3 in order to estimate the reactivity effects. The results show that the increase in k_eff, 0.3%Δk/kk′, from JENDL-3.3 to JENDL-4.0 for the UO₂ core is mainly attributed to the decreases in the capture cross sections of ²³⁸U. On the other hand, there are various contributions from the differences in the cross sections of U, Pu, and Am isotopes for the MOX cores. The major contributions to increase in k_eff are decreases in the capture cross sections of ²³⁸U,²³⁸Pu, ²³⁹Pu, and those to decrease in k_eff are decreases in ν of ²³⁹Pu and increases in the capture cross sections of²⁴¹Am. They compensate each other, and the difference in k_eff between JENDL-3.3 and JENDL-4.0 is less than 0.1%Δk/kk′ and relatively small.     

Problems and Experience in Guaranteeing the Nuclear Safety of the Storage of Spent Fuel from Nuclear Power Plants

Nuclear-safety problems are examined and the results of investigations of nuclear safety of storage sites are presented for spent nuclear fuel from nuclear power plants. The initial events of anticipated and unanticipated accidents, methods and errors in the calculation of k _eff taking account of burnup to ensure nuclear safety, the possibility of measuring k _eff of storage sites experimentally, and new forms of fuel with a consummable absorber are calculated.     

Investigation of thermal neutron scattering data for BeF2 and LiF crystals

The influences of thermal neutron scattering data for BeF₂ and LiF crystals on molten salt reactor physics are investigated in this work. Based on the structure parameters of BeF₂ and LiF, the coherent scattering for both crystals is added to NJOY source code. The ENDF6 format thermal neutron scattering sub-libraries for both crystals are evaluated with their phonon spectra using LEAPR; the ACE format data are produced by NJOY subsequently. Finally, the effect of thermal neutron scattering of BeF₂ and LiF crystals on k _eff and spectrum are investigated. The result shows that thermal neutron scattering for bound state of BeF₂ and LiF influence k _eff and spectrum obviously. The elastic scattering cross section for bound state of crystals is smaller than free atom; it makes k _eff decrease (1%–2%) and spectra be hardened. The higher temperature the bound state has, the smaller coherent elastic scattering cross section it gets; therefore, k _eff decreases with temperature. It is suggested that the thermal neutron data of LiF and BeF₂ should be taken into account for molten salt reactor.     

Fission matrix based Monte Carlo criticality calculations

We have described a fission matrix based method that allows to cancel the inactive cycles in Monte Carlo criticality calculations. The fission matrix must be sampled in the course of the Monte Carlo calculation using a space mesh with sufficiently small zones as it causes the fission matrix be insensitive to errors in the initial fission source. The k_eff and other quantities can be derived by means of the final fission matrix. The confidence interval for the k_eff estimate can be conservatively determined via the variance in the fission matrix.     

Monte Carlo uncertainty quantification of the effective delayed neutron fraction

The applicability of Monte Carlo techniques, namely the Monte Carlo sensitivity method and the random-sampling method, for uncertainty quantification of the effective delayed neutron fraction β_eff is investigated using the continuous-energy Monte Carlo transport code, MCNP, from the perspective of statistical convergence issues. This study focuses on the nuclear data as one of the major sources of β_eff uncertainty. For validation of the calculated β_eff, a critical configuration of the VENUS-F zero-power reactor was used. It is demonstrated that Chiba's modified k-ratio method is superior to Bretscher's prompt k-ratio method in terms of reducing the statistical uncertainty in calculating not only β_eff but also its sensitivities and the uncertainty due to nuclear data. From this result and a comparison of uncertainties obtained by the Monte Carlo sensitivity method and the random-sampling method, it is shown that the Monte Carlo sensitivity method using Chiba's modified k-ratio method is the most practical for uncertainty quantification of β_eff. Finally, total β_eff uncertainty due to nuclear data for the VENUS-F critical configuration is determined to be approximately 2.7% with JENDL-4.0u, which is dominated by the delayed neutron yield of ²³⁵U.     

Estimation of time-dependent neutron transport from point source based on Monte Carlo power iteration

This study proposes a method for calculating time-dependent neutron transport from a point source with a continuous-energy Monte Carlo code. To deal with neutron multiplication and attenuation in orders of magnitude, the power iteration method conventionally used to estimate the effective multiplication factor k_eff was utilized. The time of a neutron flying in a cycle from emission of its ancestor at the point source was estimated. In the estimation, the decay time of the delayed neutron precursor was considered. The neutron flux was tallied in time bins in each cycle. The source strength in the cycle was considered as the product of k_eff estimators from the first to the previous cycle. By summing up the tallied flux multiplied by the strength, the neutron flux variation with time was obtained. This method was verified against a UO₂ fuel lattice moderated and reflected by light water.     

Criticality Safety Benchmark Experiment on 10% Enriched Uranyl Nitrate Solution Using a 28-cm-Thickness Slab Core

The second series of critical experiments with 10% enriched uranyl nitrate solution using a 28-cm-thick slab core have been performed with the Static Experiment Critical Facility of the Japan Atomic Energy Research Institute. Systematic critical data were obtained by changing the uranium concentration of the fuel solution from 464 to 300 gU/l under various reflector conditions. In this paper, the thirteen critical configurations for water-reflected cores and un reflected cores are identified and evaluated. The effects of uncertainties in the experimental data on k ^eff are quantified by sensitivity studies. Benchmark model specifications that are necessary to construct a calculational model are given. The uncertainties of k _eff's included in the benchmark model specifications are approximately 0.1%δk _eff. The thirteen critical configurations are judged to be acceptable benchmark data. Using the benchmark model specifications, sample calculation results are provided with several sets of standard codes and cross section data.     

On Effective Thermal Conductivity and Structural Change of UO2 Powder Compacted Fuel in Thermal Simulation Experiment

The studies described in this paper relate to the effective thermal conductivity K^eff of powder compacted UO₂ fuel material under simulated thermal conditions of a reactor, and to structural changes brought about by large thermal gradient at elevated temperatures. From the results obtained, it is concluded that: (1) Structural change in the material occurring under such thermal conditions greatly affects K_eff , which is thereby increased as a result of columnar grain growth; (2) Columnar grains developed the path of heat flow are single crystals growing in the direction (111); (3) K_eff is higher when measured during cooling than during heating, and the relation between K_eff and temperature is similar to the case of a UO₂ single crystal; (4) K_eff measured in 90% He- 10% H₂ at atmospheric pressure is much higher than in vacuum, from which it can lie inferred that the gas contained in the fuel pin has marked influence on K_eff .