Correlation functions of non-linear nuclear reactor models

The steady-state correlation function of a non-linear point nuclear reactor model with instantaneous temperature feedback, reactivity fluctuations and neglection of delayed neutrons is calculated by means of a continued fraction expansion method. Near criticality, where other methods fail, this method gives good results for early times. A first-order truncation gives a spectral density with two corner frequencies. Results are given for a non-vanishing independent neutron source. In this case no exact results are known. For a vanishing source our results are compared with exact ones and with those of a linear analysis and a system size expansion. The necessity of a proper treatment of non-Markovian effects for non-white reactivity fluctuations is pointed out. The existence, in the spectral density, of a corner frequency associated with the correlation time of the noise is shown.     

A stochastically perturbed nonlinear point reactor model

We present an exact analysis of the fluctuations of neutron density in a stochastically perturbed nonlinear point reactor model in the absence of delayed neutrons. The reactivity and feedback coefficients are assumed to have white noise Gaussian component. The time development of probability distribution is obtained from the Fokker-Planck equation. It is found that for small density of neutron population the stationary probability distribution is more sensitive to the random change in the reactivity than in the feedback coefficient, whereas for large neutron density the probability distribution is more sensitive to the random change in the feedback coefficient than in the reactivity.     

Non-linear response of point-reactors to stochastic inputs

Statistical properties of the non-linear response of a point reactor to a white Gaussian reactivity insertion and an external source are investigated through the general Fokker-Planck theory for linear systems with random coefficients. The autocorrelation function and power spectral density of the reactor power are obtained, and the effect of non-linearities on the corner frequencies is discussed. The response to a Gaussian (not necessarily white) reactivity insertion and an arbitrary neutron source is also considered in the absence of delayed neutrons.     

A Method for Measuring Absolute Reactor Power through Neutron Fluctuation

A new principle is presented for obtaining absolute reactor power by processing the random fluctuation of neutron flux based on the stochastic nature of nuclear reactions. The required combination of instruments to carry out experiments is described, and experimental results obtained in a swimming pool reactor are reported. The power spectral density of the output current of an ion chamber located near the reactor core is determined by reactor kinetic parameters such as delayed neutron yield, life time, ν (mean number of neutrons generated per fission) and counter efficiency as well as by the total number of neutrons in the core, which is a measure of absolute power.

Using either logarithmic amplifier or reactivity meter, absolute reactor power can be measured without any information about detector efficiency. This method has such merits as easiness and simplicity in operation, ability to measure absolute power in the range 0.01～100 W where other methods are inapplicable, and negligible effect of changes in core configuration or in detector position.

The results of actual reactor experiments with this method proved to agree fairly well with those of absolute measurement by gold foil activation.     

‘Quantization’ of stochastic variables: Description and effects on the input noise sources in a BWR

A set of macrostochastic and discrete variables, with Markovian properties, is used to characterize the state of a BWR, whose input noise sources are of interest.The ratio between the auto-power spectral density (APSD) of the neutron noise fluctuations and the square modulus of the transfer function (SMTF) defines ‘the total input noise source’ (TINS), the components of which are the different noise sources corresponding to the relevant variables.A white contribution to TINS arises from the birth and death processes of neutrons in the reactor and corresponds to a ‘shot noise’ (SN). Non-white contributions arise from fluctuations of the neutron cross-sections caused by fuel temperature and steam content variations. These terms, called ‘flicker noises’ (FN) are characterized by cut-off frequencies related to time constants of reactivity feedback effects.The respective magnitudes of the shot and flicker noises depend not only on the frequency, the feedback reactivity coefficients or the power of the reactor, but also on the ‘quantization’ of the continuous variables introduced, such as fuel temperature and steam content.The effects of this last ‘quantization’ on the shapes of the noise sources and their sum are presented in this paper.     

An analytical solution for the point reactor kinetics equations with one group of delayed neutrons and the adiabatic feedback model

The point reactor kinetics equations with one group of delayed neutrons and the adiabatic feedback model are solved analytically. The analytical solution is based on an expansion of the neutrons density in powers of the small parameter, the prompt neutrons generation time, into the second order differential equation in the neutron density. The relation between the time and the reactivity for reactor excursions near prompt critical is derived. Also, the neutron density and the average density of delayed neutron precursors as functions of reactivity are presented. The relations of reactivity, neutron density and temperature with time are calculated, drawn, and compared with other analytic method.     

Application of the Wiener-Hermite functional method to point reactor kinetics driven by random reactivity fluctuations

The Wiener-Hermite Functional (WHF) method has been applied to the point reactor kinetic equation excited by Gaussian random reactivity noise under stationary conditions. Delayed neutrons and any feedback effects are disregarded. The neutron steady-state value and the power spectral density (PSD) of the neutron flux have been calculated in a second-order (WHF-2) approximation. Two cases are considered: in the first case, the noise source is assumed to be white, while in the second case the source is low-pass white noise. In both cases the WHF-2 approximation of the neutron PSDs leads to relatively simple analytical expressions. The accuracy of the approach is determined by comparison with exact solutions of the problem. The investigations show that the WHF method is a powerful approximative tool for studying the nonlinear effects in the stochastic differential equation.     

Determination of neutron generation time in miniature neutron source reactor by measurement of neutronics transfer function

The prompt neutron generation time Λ and the total effective fraction of delayed neutrons (including the effect of photoneutrons) β have been experimentally determined for the miniature neutron source reactor (MNSR) of Syria. The neutron generation time was found by taking measurements of the reactor open-loop transfer function using newly devised reactivity-step- ejection method by the reactor pneumatic rabbit system. Small reactivity perturbations i.e. step changes of reactivity starting from steady state, were introduced into the reactor during operation at low power level i.e. zero-power. Relative neutron flux and reactivity versus time were obtained. Using transfer function analysis as well as least square fitting techniques and measuring the delayed neutrons fraction, the neutron generation time was determined to be 74.6±1.57 μs. Using the prompt jump approximation of neutron flux, the total effective fraction of delayed neutrons was measured and found to be 0.00783±0.00017. Measured values of Λ and β were found to be very consistent with calculated ones reported in the Safety Analysis Report.     

The effect of delayed neutrons on the time of establishing a stable fission chain

The stochastic process of delayed neutron multiplication is considered in limited fission chains in a reactor which is supercritical with respect to prompt neutrons. Equations are obtained for the instants of distribution of the output of delayed neutron precursors and for the mean square deviation. The effect of delayed neutrons on the time of establishment of a stable fission chain is investigated. Equations are compiled and their approximate solutions are given. It is shown that for a small reactivity and with a weak source, the average time for establishing the first stable fission chain can be reduced by the delayed neutrons by a factor of 10 or more.Translated from Atomnaya Énergiya, Vol. 18, No. 6, pp. 578–583, June, 1965     

Reactivity Measurements by Reactor Noise Analysis using Two-Detector Correlation Method

For a subcritical reactor, power spectral density measurement of neutron fluctuations can be used to evaluate the quantity α=(1 — ρ/β)/β/l, where ρ/β, i.e.the reactivity, can be determined using the quantity β/l measured at critical state.

In an actual experiment however, chamber noise is a hindrance, particularly at the highly sub-critical state. This makes large reactivity determination by reactor noise analysis difficult.

The correlation technique was used to eliminate the chamber noise component, thereby considerably extending the possible range of reactivity determination by reactor noise analysis.     

On some matters concerning the backward and forward equations of probability balance for neutrons in a medium with randomly time-varying properties

The recent contributions to combined zero-power and at-power neutron noise analysis raises a number of interesting questions which deserve further discussion [Kitamura, Y., Pal, L., Pazsit, I., Yamamoto, A., Yamane, Y., Some properties of zero power neutron noise in a time-varying medium with delayed neutrons. Annals of Nuclear Energy, in press.]. It is shown, for example, by direct calculation, that the forward equation of probability balance is exact when combined with a dichotomic Markov process to describe the random physical behaviour of a medium, whereas the backward equation contains errors. This confirms and corroborates the assertions of Pal L. and Pazsit I. [2006. Neuron fluctuations in a multiplying medium randomly varying in time. Physica Scripta 74, 62.] who first pointed out this anomaly. Extensions to spatially random media are discussed, together with an application to the calculation of the extinction probability. Other methods of solution such as that of polynomial chaos are briefly touched upon. In order to carry out the calculations described we use the stochastic ansatz concept [Williams, M.M.R., 2008a. A stochastic ansatz and its relationship with the dichotomic Markov process. Physica A 387, 4997.].     

Concerning the stability parameter in point reactor kinetics driven by random reactivity noise

The paper is concerned with the stability criterion in the point reactor kinetics model when it is excited by Gaussian reactivity white noise in the presence of delayed neutrons. This criterion has been established by Behringer (Ann. Nucl. Energy 18, 397, 1991) where the Wiener-Hermite Functional method, and in particular with respect to the stability problem, the Fokker-Planck method, have been applied. The stability parameter introduced there has appeared in two mathematically different forms. The present work shows numerically, by a six-group treatment of the delayed neutrons, that these two forms are equivalent. The one-group form of the stability parameter, when the delayed neutron data are a priori condensed into one group, exhibits a very good approximation in comparison with the exact results.     

液态熔盐堆运行安全特性初步研究

液态燃料反应堆与固态燃料反应堆相比,原理上有较大不同。液态熔盐堆中由于燃料流动带走缓发中子先驱核在堆外衰变导致堆芯反应性降低,且裂变产物在堆外回路中衰变也会引起一回路发热。本文使用熔盐堆中子动力学程序Cinsf1D探讨2 MW熔盐堆的临界动力学特性和安全特性,研究零功率临界下不同熔盐流速启泵和停泵导致的缓发中子先驱核流失所需改变的控制棒棒位。同时还计算了2 MW恒定功率情况下稳态运行及降低流速时一回路温度分布,并模拟了2 MW额定功率下停泵事件。停泵后由于缓发中子损失减少反应堆功率先缓慢增加,然后迅速降低到接近余热水平。停泵后堆芯温度缓慢增加后稳定在安全值以内,说明熔盐堆具有本征安全性。     

Analytical solution of point kinetics equations for linear reactivity variation during the start-up of a nuclear reactor

The analytical solution of point kinetics equations with a group of delayed neutrons is useful in predicting the variation of neutron density during the start-up of a nuclear reactor. In the practical case of an increase of nuclear reactor power resulting from the linear insertion of reactivity, the exact analytical solution cannot be obtained. Approximate solutions have been obtained in previous articles, based on considerations that need to be verifiable in practice. In the present article, an alternative analytic solution is presented for point kinetics equations in which the only approximation consists of disregarding the term of the second derivative for neutron density in relation to time. The results proved satisfactory when applied to practical situations in the start-up of a nuclear reactor through the control rods withdraw.     

Noise analysis performed at the sodium cooled power reactor KNK I for malfunction diagnosis and plant surveillance

In order to use neutron noise analysis as an effective tool for early malfunction detection it is necessary to identify the driving forces and to calculate their contributions to the power fluctuations. In this paper the influence of a considerable number of measured noise sources on neutron noise within a large frequency range (10⁻³ Hz to 10³ Hz) is investigated for the sodium cooled power reactor KNK I (thermal core, 58 MW_th).

The experimental basis for the analysis is numerous records of the following signals at various power levels: neutron noise which has been measured with an in-core fission chamber and 3 ex-core ionisation chambers; the sodium inlet temperature and the coolant flow in both primary coolant loops and the movement of the control rods. In addition signals from acoustic-, seismic- and pressure transducers and the coolant outlet temperature were collected.

The influence of the thermohydraulic- and of the control system on neutron noise has also been calculated by means of the relations for linear and multiple-input systems. Important for this analysis is the reactivity-power transfer function. Calculations of this function could be confirmed by measurements using a pseudo-random binary signal as reactivity input.

The following results were obtained from the analysis of the auto-power spectral densities of the neutron flux: Fluctuations of the coolant inlet temperature and the coolant flow are relatively small sources for neutron noise. However, reactivity adjustments resulting from the automatic control system because of the inherent instability of the reactor turned out to be an important driving force.

The influence of still unknown driving forces increased considerably with the reactor power. Since the coolant flow was proportional to the reactor power in order to keep the coolant temperature constant, this result indicates that turbulent flow must have induced stochastical movements of core components. These movements are considered to have mainly caused the unknown reactivity driving forces. Their magnitude could be determined reliably only in the frequency range, in which external feedback mechanisms through the primary coolant system were negligible. For 30 to 50 % reactor power the contribution was about 30 % (for f > 5·10⁻³ Hz) and for full power it increased to about 80 % (for f > 5·10⁻² Hz) of the measured neutron noise. For frequencies > 5 Hz the white detection noise prevails. Single peaks in this frequency region could be explained by coherence function investigations between in-core and ex-core neutron detector signals and by correlation of these signals with displacement- and pressure fluctuations.

Though the measured neutron noise could not be unambiguously related to driving forces, the combination of analytical and empirical methods makes the results also applicable for the design of surveillance techniques for other sodium cooled reactors (e.g. LMFBRs). Examples for possible applications are given.     

时空中子倍增公式研究

以中子倍增理论作为出发点,考虑到中子输运过程的空间连续性,将空间概念引入中子倍增公式中,建立了时空中子倍增公式。该公式可严格描述中子密度(中子注量率)随时间的变化,相对于考虑缓发中子在内的中子倍增公式,时空中子倍增公式具有更大的普适性。     

Theory of stochastic space-dependent neutron kinetics with a Gaussian parametric excitation

Neutron kinetics and statics in a multiplying medium with a statistically fluctuating reactivity are unified and systematically studied by applying the Novikov-Furutsu formula. The parametric or multiplicative noise is spatially distributed and of Gaussian nature with an arbitrary spectral profile.It is found that the noise introduces a new definite production term into the conventional balance equation for the mean neutron number. The term is characterized by the magnitude and the correlation function of the random excitation. Its relaxation phenomena bring forth a non-Markoffian or a memory effect, which is conceptualized by introducing ‘pseudo-precursors’ or ‘pseudo-delayed neutrons’.By using the concept, some typical reactor physical problems are solved; they are (1) reactivity and flux perturbation originating from the random dispersal of core materials and (2) analysis of neutron decay mode and its relaxation constant, and derivation of the corresponding new inhour equation.     

252Cf源驱动噪声分析法测量α初步研究

在一球形浓缩铀临界装置上,采用²⁵²Cf源驱动噪声分析方法对其次缓发临界状态下的α进行测量,在频域内分析数据得到α。对比频域内的两种数据处理方式（功率谱实虚部相除方法和功率谱取模直接拟合方法）,用互功率谱密度函数实虚部相除方法得到在-0.1$情况下,脉冲堆的α为0.58μs^-1,与Rossi-α方法的结果相吻合。     

Analytical Study of Operating Reactor Noise on a Single Channel and a Single Phase Flow Model, (I)

Basic formulas to describe the propagation of a periodically modulated driving source are derived on the distributed model of at-power reactors.

Disturbances in the neutron field are regarded as being promptly spread over the whole core in the frequency domain of usual interest for noise and frequency response analyses. On the basis of this approximation, a coupled set of kinetic equations for neutrons, fuel and coolant temperature is derived and solved for the case where a driving source of unit strength is imparted. The relations between the solutions obtained and the formulas for analyzing experiments are established. Comparisons are made qualitatively between the results and those of the lumped model. The existence of a sink and a resonance frequency is examined. The power spectral density of neutron fluctuations in a water reactor under forced circulated cooling shows a sharply varying configuration when the inlet coolant temperature fluctuations are the major noise sources. When the reactor is under natural convective cooling or under liquid metal cooling, no such characteristic features appear.