Calculation of reactivity using a finite impulse response filter

A new formulation is presented in this paper to solve the inverse kinetics equation. This method is based on the Laplace transform of the point kinetics equations, resulting in an expression equivalent to the inverse kinetics equation as a function of the power history. Reactivity can be written in terms of the summation of convolution with response to impulse, characteristic of a linear system. For its digital form the Z-transform is used, which is the discrete version of the Laplace transform. This new method of reactivity calculation has very special features, amongst which it can be pointed out that the linear part is characterized by a filter named finite impulse response (FIR). The FIR filter will always be, stable and non-varying in time, and, apart from this, it can be implemented in the non-recursive form. This type of implementation does not require feedback, allowing the calculation of reactivity in a continuous way.     

Novel fluctuation reduction procedure for nuclear reactivity calculations based on the discrete fourier transform method

A new method for calculating nuclear reactivity based on the Discrete Fourier Transform (DFT) – with two filters: a first-order delay low-pass filter and a Savitzky-Golay filter – is presented. The reactivity is calculated from an integrodifferential equation known as the inverse point kinetic equation, which contains the history of neutron population density. The new method can be understood as a convolution between the neutron population density signal and the response to the characteristic impulse of a linear system. The proposed method is based on the discrete Fourier transform (DFT) that performs a circular convolution. The fast Fourier transform algorithm (FFT) with the zero-padding technique is implemented to reduce the computational cost.     

Feasibility of Continuous Sub-criticality Monitoring using a Digital Reactivity Meter to Overcome the Criticality Accident

In the foregoing studies, it has been proved that the digital reactivity meter can be used for not only sub-criticality measurement but also continuous sub-criticality monitoring during criticality approach. Based on these studies, we investigated the applicability of a digital reactivity meter for continuous sub-criticality monitoring to intervene before a criticality accident occurs that is similar to the Tokai-mura accident in 1999. In our mock up numerical simulation, there are three calculation steps that are (1) reactivity transient calculation, (2) neutron transient calculation and (3) sub-criticality monitoring calculation. The reactivity transient was calculated using two-group diffusion nuclear constants and the neutron transient was calculated using the one-point reactor kinetics. In sub-criticality monitoring, three algorithms for reactivity evaluation were compared. We have investigated which algorithm is the most suitable to use in an actual system. In practice, we also advise use of some filtering algorithm to reduce the neutron transient fluctuation and a warning reactivity to know estimated sub-criticality as earlier as possible.     

Development of a potential based code for MHD analysis of LLCB TBM

A two dimensional solver is developed for MHD flows with low magnetic Reynolds’ number based on the electrostatic potential formulation for the Lorentz forces and current densities which will be used to calculate the MHD pressure drop inside the channels of liquid breeder based Test Blanket Modules (TBMs). The flow geometry is assumed to be rectangular and perpendicular to the flow direction, with flow and electrostatic potential variations along the flow direction neglected. A structured, non-uniform, collocated grid is used in the calculation, where the velocity (u), pressure (p) and electrostatic potential (?) are calculated at the cell centers, whereas the current densities are calculated at the cell faces. Special relaxation techniques are employed in calculating the electrostatic potential for ensuring the divergence-free condition for current density. The code is benchmarked over a square channel for various Hartmann numbers up to 25,000 with and without insulation coatings by (i) comparing the pressure drop with the approximate analytical results found in literature and (ii) comparing the pressure drop with the ones obtained in our previous calculations based on the induction formulation for the electromagnetic part. Finally the code is used to determine the MHD pressure drop in case of LLCB TBM. The code gives similar results as obtained by us in our previous calculations based on the induction formulation. However, the convergence is much faster in case of potential based code.     

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.     

Calculation of the void reactivity of CANDU lattices using the code system

The reactivity effect of coolant voiding in CANDU-type fuel lattices has been calculated with different methods using the code system. The known positive void reactivity coefficient of the original lattice was correctly obtained. A modified fuel bundle containing dysprosium and slightly enriched uranium to eliminate the positive reactivity effect was also calculated. Owing to the increased heterogeneity of this modified fuel the one-dimensional cylindrical calculation with XSDRN proved to be inadequate. Code options allowing bundle geometry were successfully used for the calculation of the strongly space dependent flux and spectrum changes which determine the void reactivity.     

Analysis of Heavy-Water-Moderated,Cluster-Type Fuel Lattices by Cluster Physics Code ‘MESSIAH’

This paper presents a comprehensive analysis performed by a new cluster analysis code ‘MESSIAH’ on reactor physics constants measured in the critical facility for a pressure-tube-type, heavy-water-moderated reactor. The MESSIAH code system utilizes the method of the collision probability to solve the neutron transport equation. The effective space dependent cross sections are calculated in the thermal and resonance energy range before the eigenvalue calculation for the whole energy range. With use of these cross sections, the multi-group, space dependent transport equation is solved, and the flux distribution, spectrum and k _eff are obtained to the input bucking. In the above three steps the method of the collision probability is used consistently and extensively. The treatment of leakage neutrons from lattices in MESSIAH is also confirmed by an independent method using a Monte Carlo calculation. The calculated reactor physics constants, especially the micro-parameters and the activation traverse of Dy, agreed fairly well with the experiment. The diffusion calculation with use of the group constants calculated by MESSIAH predicts the reactivity of 0% void core excellently (<0.12%). However, for a 100% void core, the calculated reactivity was slightly lower than the experiment (~0.74%), which was attributed to over prediction of the diffusion constants.     

A benchmark on the calculation of kinetic parameters based on reactivity effect experiments in the CROCUS reactor

Measurements in the CROCUS reactor at EPFL, Lausanne, are reported for the critical water level and the inverse reactor period for several different sets of delayed supercritical conditions. The experimental configurations were also calculated by four different calculation methods. For each of the supercritical configurations, the absolute reactivity value has been determined in two different ways, viz.: (i) through direct comparison of the multiplication factor obtained employing a given calculation method with the corresponding value for the critical case (calculated reactivity: ρ_calc); (ii) by application of the inhour equation using the kinetic parameters obtained for the critical configuration and the measured inverse reactor period (measured reactivity: ρ_meas). The calculated multiplication factors for the reference critical configuration, as well as ρ_calc for the supercritical cases, are found to be in good agreement. However, the values of ρ_meas produced by two of the applied calculation methods differ appreciably from the corresponding ρ_calc values, clearly indicating deficiencies in the kinetic parameters obtained from these methods.     

Accurate Estimation of Subcriticality Using Indirect Bias Estimation Method, (II)

Subcriticalities were estimated by applying the Indirect Bias Estimation Method to subcritical experiments on a light-water moderated/reflected low-enriched UO₂ lattice cores. Two measurable values, prompt neutron time-decay constant and spatial-decay constant were calculated using MCNP 4A and JENDL-3.2. With these calculation errors, the biases in calculated reactivity were derived from the Indirect Bias Estimation Method. The differences between the calculated and measured spatial-decay constants were more or less at the same extent of experimental errors. These results show that the accuracy of subcriticality estimation of MCNP 4A and JENDL-3.2 ranges within the uncertainty which can be achieved by the exponential experiment. The differences between calculated and measured prompt neutron decay constants derive significant biases in calculated reactivity. The subcriticalities were estimated by using the effective multiplication factors adjusted based on these biases in calculated reactivity.     

Reactivity Analysis by Spectral Corrected Garelis-Russel Method

A series of reactivity measurements has been performed in a subcritical assembly using pulsed neutron source. The study is aimed at gaining information on the differences in the neutron population decay curve caused by differences in energy response of the detector used in the measurement. A discussion is given on the effective difference brought upon neutron spectra and in turn upon detector response due to the fact that the prompt flux is observed in decaying state and the delayed flux in nearly steady state. A scheme is proposed for compensating this difference in observed neutron fluxes. Comparison with experimental results is given, showing that, by means of the proposed correction, the calculated reactivity can be made independent of both detector position and detector type. To obtain meaningful values of reactivity without making use of this spectral correction, detectors with only epi-thermal response should preferably be used, which should be installed in the reflector region.     

Modified Quasi-Steady-State Method to Evaluate the Mean Power Profiles of Nuclear Excursions in Fissile Solution

A modified quasi-steady-state method has been developed in order to evaluate the mean power during a nuclear excursion in fissile solution. The conventional method used the critical equation based on the one-group theory in order to calculate the reactivity. However, the one-group approximation reduces the calculation accuracy, and the geometrical buckling used in the critical equation is not applicable to complex geometries. Thus, we have modified the method to use the reactivity feedback coefficients, which are widely used in the calculation of one-point reactor kinetics. Although the modified method requires an external calculation to obtain the feedback coefficients, it is applicable to complex geometries and provides more accurate results than does the one-group approximation when the proper coefficients are given.

Moreover, a new method to calculate the boiling power has been developed. In this method, the power corresponding to the void fraction that compensated for the inserted reactivity along with the temperature feedback was calculated using the relationship, which was derived using the French SILENE experimental data.

Experimental analyses have been conducted to validate the new method for the French CRAC and Japanese TRACY experiments. The analytical results showed close agreement with the experimental results.     

High-accuracy fit of the poles of spectroscopy amplifiers designedfor mixed analog-digital filtering

In this paper, a method for the identification of the poles' and zeros' position of an analog amplifier for nuclear spectroscopy used as a prefilter for a subsequent digital filter setup is presented. The proposed technique is based upon a subspace-based system state-space identification (4SID) method, which is well suited to a data set constituted by a noisy measurement of the sampled impulse response of the circuit. The algorithm runs unassisted and does not require skills by the operator. The experiments confirm that by using the so-obtained pole values, the shape of the impulse response of the amplifier can be fit with much better than 1% accuracy. Consequently, the overall filtering (analog+digital) can have finite duration and a top with a flatness much better than 1%     

西安脉冲堆反应性测量空间效应修正方法研究

反应性是反应堆重要的物理参数,在西安脉冲堆上增加反应性实时监测功能,能够为操纵员提供反应性实时大小和变化趋势,有利于其安全运行。逆动态方法以其实时性好、能测量任意反应性引入而被广泛应用于反应性测量,但由于控制棒的移动会导致中子注量率空间分布前后不一致,而出现偏差。本文对逆动态方法和静态空间效应因子进行了理论分析,给出了相应的计算方法;以西安脉冲堆为研究对象,使用蒙特卡罗（MCNP）程序计算了探测器三维空间的响应函数,同时计算了归一化节块功率密度,由此得到静态空间效应因子;最后在脉冲堆上进行了不同棒速下插控制棒实验,处理实验数据得到控制棒积分价值曲线。结果表明,进行空间效应修正是必要的,经修正后计算得到的控制棒价值曲线更稳定,计算结果与真实值误差更小。      

Development of a fundamental crack tip strain rate equation and its application to quantitative prediction of stress corrosion cracking of stainless steels in high temperature oxygenated water

A formulation for the quantitative calculation of the stress corrosion cracking (SCC) growth rate was proposed based on a fundamental-based crack tip strain rate (CTSR) equation that was derived from the time-based mathematical derivation of a continuum mechanics equation. The CTSR equation includes an uncertain parameter r₀, the characteristic distance away from a growing crack tip, at which a representative strain rate should be defined. In this research, slow strain rate tensile tests on sensitized 304L stainless steel in oxygenated high temperature water were performed. By curve fitting the experimental results to the numerically calculated crack growth rate, the parameter r₀ was determined. Then, the theoretical formulation was used to predict the SCC growth rates. The results indicate that r₀ is on the order of several micrometers, and that the application of the theoretical equation in predicting the crack growth rate provides satisfactory agreement with the available data.     

Application of ‘natural coordinate system’ in the finite element solution of multigroup neutron diffusion equation

A simple method of generating stiffness matrices for the solution of multigroup diffusion equation by ‘natural coordinate system’ has been presented. A comparative study has been made using triangular elements with linear model, triangular elements with quadratic model and rectangular elements with bilinear model to demonstrate their relative efficiencies. The quadratic interpolation model has been shown to be superior to linear and bilinear models with respect to computing time, computer storage and relative error in K_eff for a two group diffusion example. The flexibility of the finite element treatment has been demonstrated by the calculation of the reactivity of a partially inserted control rod. Good agreement has been obtained with a perturbation calculation.     

Three-Dimensional First-Order Perturbation Calculation Method for Reactivity Changes Due to Core Deformations of Fast Breeder Reactors

A three-dimensional diffusion calculation method has been proposed to rapidly and accurately calculate reactivity changes of LMFBRs caused by assembly displacements in accidental events. The method requires shorter computation times and provides almost the same accuracy as a conventional direct eigenvalue calculation method. In this method, changes in macroscopic neutron cross-sections and diffusion coefficient are defined so that changes in both region volume and material composition can be treated in a mesh-centered finite-difference program under the same coarse mesh division as used for the normal, non-deformed core. Reactivity changes are calculated from the above-mentioned changes by the first-order perturbation method using normal and adjoint neutron fluxes calculated beforehand for the normal core.

The method was applied to deformations of a 1,000-MWe LMFBR core. Reactivity changes calculated by the method agreed within 0.4% with those by a conventional direct eigenvalue calculation method, while computation time was less than 1/35.     

An efficient analytical form for the period-reactivity relation of beryllium and heavy-water moderated reactors

A new mathematical form of the period-reactivity equation for heavy-water- and beryllium-moderated reactors has been developed. This form is represented in a polynomial form with a degree of G+1 for G-th group of delayed neutrons and photoneutrons. A general formula for the coefficients of such polynomial is derived. These coefficients have a linear dependence on the step reactivity insertion. The related constants of this linear dependence are calculated for both types of reactors. In addition, a comparison has been made for the stable reactor periods of an infinite U²³⁵-fueled, D₂O- or Be-moderated reactors, following step reactivity insertion, with and without delayed photoneutrons taken into consideration. Also, a comparison was made for the reactor response to reactivity changes when evaluated using fast and thermal fission delayed neutron group constants combined with and without D₂O- and Be-moderated reactors.     

Some Studies on the Effectiveness of Control Rods

The linear extraporation distances for epi-thermal nautrons on various nautron absorbers were obtained experimentally with the use of a natural UO₂-H₂O cyrindrical exponential column. The measured flux distributions or flux-depression distributions around the control rod were compared with calculated values. It was confirmed that, except in the region immediately around the absorber, the radial flux distributions or depressions can be calculated accurately by the two-group approximation using these linear extrapolation distances obtained experimentally for epi-thermal neutrons, and those calculated by the Kushneriuk-McKay method for thermal neutrons.

The flux depressions on the surface of the control rod were measured to be almost constant, and independent of the lattice position of the control rod inserted. This fact simplifies the calculation for evaluating the reactivity worth of the control rod by the perturbation theory.

The scattaring effect of the control rod on its effectiveness was studied experimentally with the use of SHE, a 20% enriched UO₂-graphite moderated critical assembly. Sometimes a scatterer (graphite slug) inserted inside a hollow conrol rod decreases the reactivity worth by more than 10%.     

基于超级电容储能的中性束注入系统弧电源设计

弧电源是中性束注入加热系统中最关键的设备之一,它的性能决定了弧放电的稳定性及束流引出的品质。为提高弧放电稳定性,降低电网容量,减少对电网的冲击,弧电源拓扑设计采用了基于超级电容储能和开关电源技术的DC/DC变换器结构。利用多个IGBT功率模块并联工作,可提高电源工作频率,实现更快的动态响应速度。在详细分析电源工作过程的基础上,设计了滤波电路和电流快速转移电路,根据电源的要求和具体参数,由一阶RL电路的电流响应特性,精确计算出滤波电感的最小值。最后,利用Matlab对电源性能指标进行了仿真验证,结果表明电源性能完全符合设计要求。     

A new digital Gaussian pulse shaping algorithm based on bilinear transformation

Nuclear pulse signal needs to be transformed to a suitable pulse shape to remove noise and improve energy resolution of a nuclear spectrometry system. In this paper, a new digital Gaussian shaping method is proposed.According to Sallen-Key analog Gaussian shaping filter circuits, the system function of Sallen-Key analog Gaussian shaping filter is deduced on the basis of Kirchhoff laws. The system function of the digital Gaussian shaping filter based on bilinear transformation is deduced too. The expression of unit impulse response of the digital Gaussian shaping filter is obtained by inverse z-transform. The response of digital Gaussian shaping filter is deduced from convolution sum of the unit impulse response and the digital nuclear pulse signal. The simulation and experimental results show that the digital nuclear pulse has been transformed to a pulse with a pseudo-Gaussian, which confirms the feasibility of the new digital Gaussian pulse shaping algorithm based on bilinear transformation.