A formulation of a reactivity variation for taking into account spatial and energy collapsing

This paper presents an original method to evaluate the reactivity worth induced by space or energy condensation procedures commonly used by reactor physicists. Exact and approximate formulas are derived, which allow assigning a reactivity worth to each computation region in order to determine its importance. Some examples are shown aiming at highlighting that results obtained using the proposed formulation have a physical meaning and could be used to define an optimised spatial mesh and to analyse in a new way the usual energy collapsing process. This formulation can be considered as complement to the classical perturbation theory definition of a reactivity variation.     

Modified explicit higher-order perturbation method with two energy groups

The Explicit High-Order Perturbation method is among various perturbation methods and its main feature is to expand neutron flux of a perturbed system in a series of the λ-mode eigenfunctions in an unperturbed system. Although various perturbation methods using λ-mode eigenfunctions have been proposed in the past, it was difficult to calculate accurately the perturbed flux and the reactivity with multi energy groups. To overcome this problem we derived a new perturbation formulation unlike the Flux Reconstructed Explicitly by Neutronic Calculated Harmonics method based on the Explicit Higher-Order Perturbation Method and applied it to a simplified core system. To show the validity of this new perturbation method numerical calculations with two energy groups in an x–y geometry were carried out and it was found that the perturbed flux and the reactivity could be accurately obtained by this new perturbation method.     

快堆三维瞬态反应性反馈计算研究

在处理快堆时空动力学计算的反应性反馈问题时，提出了一种反应性直接反馈的数学模型。结合快堆的反应性反馈机制，在快堆中子学软件NAS的基础上，给出一种在时空动力学计算中截面反馈与反应性直接反馈相结合的反馈模式。同时，将快堆并群系统加入到程序中，实现了在线并群。对中国实验快堆(CEFR)等温温升过程进行模拟，通过计算结果与CEFR温度反应性系数实验测量结果的对比，证明了本模型和程序的正确性。     

Measurements and Analyses of Reactivity Effect of Fission Product Nuclides in Epithermal Energy Range

Experimental data usable for evaluating cross sections of main fission product elements (Rh, Cs, Nd, Sm, Eu and Gd) in the epithermal energy range were measured. A cadmium-covered vessel containing a pure water or an aqueous solution of a fission product element was inserted at the center of TCA (Tank-type Critical Assembly) core. Reactivity effects were obtained by the difference in the critical water levels between a pure water and an aqueous solution in the vessel. The measured reactivity was more than 1 φ and it was greater than the experimental uncertainties. Since the adjoint thermal flux below the cadmium-cutoff energy are largely depressed in the vessel, the reactivity effects in epithermal energy range could be measured. The analyses for the experiments were performed using the SRAC code system and neutron transport calculation code TWOTRAN. The exact Perturbation theory was applied to calculate the reactivity effects of fission product elements. The calculated reactivity effects using JENDL-3.2 and ENDF/B-IV cross-section libraries were compared against the measured values. The analyses using JENDL-3.2 gave reasonable results for these measurements.     

Effect of Group Collapsing in Perturbation Theory for Sample Worth Analysis

The effect of group collapsing applied to the perturbation theory for sample worth analysis in fast reactor systems is theoretically and numerically examined assuming the validity of the thin sample approximation.

As a result, the calculated worths of scattering predominant materials placed at the center of core are found to be strongly influenced by the group collapsing. The effect on the sample worth when the sample is placed in positions off the core center decreases with increasing distance from the center. It is noted that the reactivity perturbation due to inelastic scattering is also affected significantly by group collapsing especially near the core-blanket interface.

Based on the above observations, it is concluded that the perturbation theory with about 70 energy groups appropriately arranged is necessary to reproduce the experimental values of Na, O and Fe sample reactivity worths with accuracy efficient for ordinary purposes.     

Acoustic emission events from sodium vapour bubble collapsing: A stochastic model

The forward Kolmogorov equation method has been applied to a zero-dimensional model which describes the time distribution of acoustic emissions from sodium vapour bubble collapsing.Processes taken into account as components for outlining the upstated phenomenon are: energy generation, energy dissipation, bubble creation, acoustic emission and energy release from bubble collapsing.Processes involve affect or are induced by a population of particles (bubbles, acoustic pulses) and pseudoparticles (energetic units).A formulation is obtained for the expected values of some stochastic indicators, i.e. factorial moments and cumulants, autocorrelation function, waiting time distribution between contiguous events, of the time series consisting of acoustic emission pulses as detected by a suitable sensor.Preliminary but promising validation of the model and a sound prelude to an effective boiling regime diagnosing is obtained by processing data from the out-of-pile CFNa loop in Grenoble, France. Data are collected from a piezoelectric accelerometer located nearby the circuit.     

Measurement and Analysis of Reactivity Worth of 237Np Sample in Cores of TCA and FCA

The reactivity worth of 22.87 grams of ²³⁷Np oxide sample was measured and analyzed in seven uranium cores in the Tank-Type Critical Assembly (TCA) and two uranium cores in the Fast Critical Assembly (FCA) at the Japan Atomic Energy Agency. The TCA cores provided a systematic variation in the neutron spectrum between the thermal and resonance energy regions. The FCA cores, XXI and XXV, provided a hard neutron spectrum of the fast reactor and a soft one of the resonance energy region, respectively. Analyses were carried out using the JENDL-3.3 nuclear data library with a Monte Carlo method for the TCA cores and a deterministic method for the FCA cores. The ratios of calculated to experimental (C/E) reactivity worth were between 0.97 and 0.91, and showed no apparent dependence on the neutron spectrum.     

Stopping power of Mylar for heavy ions up to copper

The stopping powers of Mylar for several heavy ions covering Z=11 to 29 in the energy range 0.3 to 2.3 MeV/n have been measured using the elastic recoil detection technique and twin detector system. The technique provided a unique method to generate a variety of variable energy ion species utilizing a fixed energy 140 MeV Ag¹³⁺ primary beam from the Pelletron accelerator facility at Nuclear Science Center, New Delhi, India. Most of these measurements are new. The experimentally measured stopping power values have been compared with those calculated using LSS theory, Ziegler et al. formulation and Northcliffe and Schilling tabulations. Merits and demerits of these formulations are highlighted. Stopping power calculations using the Hubert et al. formulation have been extended successfully beyond its recommended range of validity, i.e. 2.5–500 MeV/n down to energies as low as 0.5 MeV/n.     

Classification and uniformity optimization of mesh-plate DBD and its application in polypropylene modification

The classification of spatial characteristics and discharge modes of dielectric barrier discharge (DBD) are gaining increasing attention in industrial applications, especially in the field of surface treatment of materials. In this work, gray level histogram (GLH) and Fourier energy spectrum based on the digital image processing technology are applied to investigate the spatial structure and discharge mode of mesh-plate DBD. The coefficient of variation (CV) is calculated to describe the uniformity of the discharge. The results show that the discharge mode of mesh-plate DBD changes from periodic discharge to filamentary discharge when the applied voltage increases from 11–15 kV. Moreover, a more regular spatial structure is obtained under lower applied voltages during the discharge process. It is also found that the apertures of mesh electrodes which are below 1mm have smaller values of CV compared to plate electrodes, indicating more uniform discharge. Finally, polypropylene is treated by mesh-plate DBD for surface modification. The hydrophilicity is significantly improved as the water contact angle decreased by 64°, and the dyeing depth is also enhanced.     

Savitzky–Golay filter for reactivity calculation

A new filter method known as Savitzky–Golay allows the reduction of reactivity fluctuations. The filter reduces fluctuations that are found in the nuclear power signal does not attenuate to the reactivity value. The method can be applied with a time step of up to T = 0.01?s and a noise level of up to σ = 0.1. This formulation employs a Gram polynomial approximation of degree d = 2, to calculate the convolution coefficients by means of an analytic formula that is implemented computationally and avoids ill-conditioning issues caused by the inversion of a linear system. The results show better values in the maximum difference and the mean absolute errors of reactivity in comparison with the results reported in the literature.     

Analysis of a multigroup stylized CANDU half-core benchmark

An 8-group cross section library is provided to augment a previously published 2-group 3D stylized half-core Canadian deuterium uranium (CANDU) reactor benchmark problem. Reference eigenvalues and selected pin and bundle fission rates are also included. This benchmark is intended to provide computational reactor physicists and methods developers with a stylized model problem in more than two energy groups that is realistic with respect to the underlying physics. In addition to transport theory code verification, the 8-group energy structure provides reactor physicist with an ideal problem for examining cross section homogenization and collapsing effects in a full-core environment. To this end, additional 2-, 4- and 47-group full-core Monte Carlo benchmark solutions are compared to analyze homogenization-free transport approximations incurred as a result of energy group condensation.     

Global physical and numerical stability of a nuclear reactor core

Low order models are used to investigate the influence of integration methods on observed power oscillations of some nuclear reactor simulators. The zero-power point reactor kinetics with six-delayed neutron precursor groups are time discretized using explicit, implicit and Crank–Nicholson methods, and the stability limit of the time mesh spacing is exactly obtained by locating their characteristic poles in the z-transform plane. These poles are the s to z mappings of the inhour equation roots and, except for one of them, they show little or no dependence on the integration method. Conditions for stable power oscillations can be also obtained by tracking when steady state output signals resulting from reactivity oscillations in the s-Laplace plane cross the imaginary axis. The dynamics of a BWR core operating at power conditions is represented by a reduced order model obtained by adding three ordinary differential equations, which can model void and Doppler reactivity feedback effects on power, and collapsing all delayed neutron precursors in one group. Void dynamics are modeled as a second order system and fuel heat transfer as a first order system. This model shows rich characteristics in terms of indicating the relative importance of different core parameters and conditions on both numerical and physical oscillations observed by large computer code simulations. A brief discussion of the influence of actual core and coolant conditions on the reduced order model is presented.     

节块格林函数法的微扰计算

在反应堆物理设计和分析时,经常要进行微扰计算,以快速分析堆芯中子截面扰动下反应性的变化量。本文从微扰计算的普遍公式出发,给出了节块格林函数法（ＮＧＦＭ）下微扰计算的具体公式。通过对比验算,验证了ＮＧＦＭ下的微扰公式,并且证明微扰计算需要的是节块法的数学共轭解而不是物理共轭解。     

Tables of thermonuclear-reaction-rate data for neutron-induced reactions on heavy nuclei

The results of statistical model calculations of (n,γ), (n,p), and (n,α) cross sections and reaction-rate factors are presented in tabular form for over 500 target nuclei in the range 36 ≤ Z ≤ 83 (krypton to bismuth). Included in these tables is information on (i) the reaction cross section as a function of energy for the exoergic channel in the range 0.01 ≤ E(MeV) ≤ 3.0; (ii) the thermally averaged reaction-rate factor, N_A〈σv〉 and the nuclear partition function G(T) for temperatures in the range 10⁸ ≤ T(^oK) ≤ 3 × 10⁹; (iii) analytic fits to the reaction-rate factors and partition functions as functions of temperature; and (iv) nuclear level-density parameters and formulas for their extrapolation. Two types of reaction-rate factors have been computed. One, which may be called the “laboratory rate factor,” is based on the assumption that the target nuclei occupy only their ground states. The other, which shall be termed the “stellar rate factor,” is based on the more realistic assumption that the target nuclei occupy a thermal distribution of excited states at temperature T. A brief discussion of theory and instructions for usage of the tables are included. New fitting forms for statistical-model thermonuclear reaction rates are presented and justified.     

The ‘FRENCH’ (flux reconstructed explicitly by neutronic calculated harmonics) method in multidimensional and multigroup theory

The explicit high-order perturbation formulation which allows reconstruction of reactivity variations as well as the neutronic flux for very strong perturbations has been implemented for multidimensional calculations. A new method, the ‘FRENCH’ method, has been developed to overcome the difficulties found by the ‘Standard’ method for multigroup calculations. Practical applications on cases of interest have shown a significant improvement in the agreement with the results obtained by direct calculations.     

Electron Damage Coefficients in P-Type Silicon

Crucible grown p-type silicon, as used for the base of many commercial solar cells, exhibits a damage coefficient under electron bombardment that appears to involve a double defect in each recombination center. The damage coefficient has been fitted to an empirical formula that accounts for silicon resistivity and for energy of the bombarding particles. To apply this formulation to a practical evaluation of solar cell damage in orbit, two complications were considered. The first is the angular distribution of the space radiation incident on the cell. An experiment showed that electron flux, and not current, determines the extent of the damage. The second is the presence of a transparent coverslide over the cell. By the application of Monte Carlo data for the transmission loss and energy degradation of the electrons, modifications to the damage coefficient can be made to account for coverslide shielding. An exponential attenuation appears adequate to describe these shielding effects. The end product of this study is an analytic formula for the damage coefficient that allows a simple calculation of the damage caused by energetic electrons to n/p silicon solar cells with coverslides.     

Creep analysis of slightly oval cylindrical shells subjected to time-dependent loading, temperature and neutron flux

Accurate creep analysis of unsupported, in-reactor fuel cladding containing initial ovality requires an analytic model which includes the effects of time-dependent variations in external pressure, cladding temperature and neutron flux. The purpose of this paper is to accomplish this objective by correcting and extending an earlier study that utilized the concept of body forces analogy to derive a set of finite difference equations governing the creep deformation of a slightly oval cylindrical shell. The present formulation also enables the analyst to investigate the influence of geometric nonlinearities on the deformation via application of the pseudo-load technique to the equilibrium equations. Numerical results demonstrate the effect of variations in neutron flux, external pressure, cladding temperature, geometric parameters, mesh size and solution time step on collapse time.     

Modeling reactivity devices for advanced CANDU reactors using the code DRAGON

Full core analysis of typical power reactors generally performed uses few group diffusion theory, it is necessary to generate beforehand, using a lattice code, the required few group cross-sections and diffusion coefficients associated with each region in the core.

For the ACR™ (Advanced CANDU Reactor), the problem is more complex because these reactors contain vertical reactivity devices that are located between two horizontal fuel bundles. The usual calculation scheme relies in this case on a 2D fuel cell calculation to generate the few group fuel properties and on a 3D supercell calculation for the analysis of the reactivity devices present in the core. Because of its complexity, the supercell calculations have usually been performed using simplified fuel geometries. The development of new geometry features in DRAGON and the availability of faster computers have made it possible to improve the 2D cell and 3D supercell models by using explicitly 3D assemblies of clusters to simulate the reactivity devices in CANDU reactors, including the ACR. These studies will thus improve the fine reactor core results by generating more accurate and appropriate reactor databases.

In this paper, we will review the lattice-cell/supercell calculation procedure using the code DRAGON by introducing a new supercell model. The use of such an explicit 3D geometry implies a very fine spatial mesh discretization that can generate a large number of regions leading to problems that cannot be solved by the collision probability (CP) method. The method of characteristics (MoC) is then the only alternative for such cases. A comparison of results using these two methods will also be presented for 3D models with a coarse mesh discretization.     

Measurement of Reactivity Effect and Thermal Neutron Flux in Non-uniformly Distributed Fuel Assemblies

A reactivity effect due to spatial variation of nuclear fuel concentration is an important problem for nuclear criticality safety in a reprocessing plant. As a method theoretically estimating this reactivity effect, the Goertzel's necessary condition is well known. To investigate this theoretical prediction, we have performed systematically a series of measurements of reactivity effect due to nonuniformity in the fuel distribution using a solid moderated core of the Kyoto University Critical Assembly. Thermal neutron flux distributions have been also measured using the activation method with gold wire. A nonuniform assembly was found to have an excess reactivity of 0.2%Δk/k when it had the same uranium mass as the uniform assembly was exactly critical. The fuel concentration of this nonuniform assembly was 40% higher in the center region than the uniform one. Moreover, the spatial distribution of reaction rate of gold wire due to thermal neutrons was flatter in this nonuniform assembly than in the uniform one, as expected by the Goertzel's condition.     

Pulsed neutron source measurements in the subcritical ADS experiment YALINA-Booster

A subcritical zero-power source-driven coupled core, the YALINA-Booster, has been constructed for experimental investigations of neutron kinetics of source-driven systems. In this study, the reactivity of two subcritical configurations has been determined by the area ratio method. The prompt neutron decay constants have been evaluated through slope fitting of the prompt neutron decay as well as through the pulsed Rossi-α method. It is shown that the slope fitting method and the pulsed Rossi-α method give stable results whereas the area ratio method results show spatial dependence. The reasons for the spatial spread are addressed.