Multiplying Medium Research Articles

Effect of Radial Neutron Reflector on the Characteristics of Nuclear Fuel Burn-Up Wave in a Fast Neutron Energy Spectrum Multiplying Medium: A Consistent Parametric Approach

Abstract The influence of radial neutron reflector on the build-up and propagation of a nuclear fuel burn-up wave in a fast multiplying medium is investigated using a consistent parametric approach. Coupled multigroup neutron diffusion equations with a burn-up evolution model are simulated on the two-dimensional cylindrical reactor geometry with azimuthal symmetry. Uranium–plutonium transmutation model is considered, and the simulation is performed by using the finite element multiphysics software package comsol. Transient characteristics of the burn-up wave are represented by two new parameters, namely, transient time (TT) and transient length (TL). TT and TL are defined as the time and distance required for the burn-up wave to attain its steady-state nature. Steady-state phases are characterized in terms of wave velocity, full width half maximum (FWHM), and full width 10% of maximum (FW10M). A sensitivity study of steady-state and transient parameters is conducted for the different values of radial reflector thickness. The potential relevance of these characterization parameters on the development of optimal geometrical configuration of radial neutron reflector in breed and burn (B&B)-based reactor design is addressed based on the sensitivity study.

Build Up and Characterization of Ultraslow Nuclear Burn-Up Wave in Epithermal Neutron Multiplying Medium

Abstract A study is carried out on the build-up and characterization of ultraslow nuclear burn-up wave in epithermal neutron multiplying medium for slab geometry. Uranium-plutonium fissile medium is considered for the calculation. Transient part of the wave is characterized by transient time (TT), transient length (TL), and TT and TL are defined as the time and distance required to develop asymptotic neutron flux propagating through the media. Steady-state part of the wave is characterized with wave velocity and reaction zone width (full width half maximum (FWHM) and full width 10% of maximum (FW10M)). Parametric studies are carried out for different enrichment of 235U and different values of external source of neutron. It is observed that TT, TL, FWHM, and FW10M decrease with the increase in enrichment. The velocity of the wave increases with the enrichment of 235U. This study is beneficial for understanding the characteristics of nuclear burn-up wave in epithermal region as it will help in further researches in this area.

Numerical-Statistical and Analytical Study of Asymptotics for the Average Multiplication Particle Flow in a Random Medium

It is well known that, under rather general conditions, the particle flux density in a multiplying medium is asymptotically exponential in time t with a parameter lambda , i.e., with an exponent lambda t. If the medium is random, then lambda is a random variable, and the time asymptotics of the average number of particles (over medium realizations) can be estimated in some approximation by averaging the exponent with respect to the distribution of lambda . Assuming that this distribution is Gaussian, an asymptotic “superexponential” estimate for the average flux expressed by an exponential with the exponent t{text{E}}lambda + {{t}^{2}}{text{D}}lambda {text{/}}2 can be obtained in this way. To verify this estimate in a numerical experiment, a procedure is developed for computing the probabilistic moments of lambda based on randomizations of Fourier approximations of special nonlinear functionals. The derived new formula is used to study the COVID-19 pandemic.

Description of transient processes in multiplying medium by the physical birth-and-death model

The work presents an approach to describe neutron and neutron-breeding medium interaction process based on the physical birth-and-death model. This model is an extended form of the basic birth-and-death model and operates with breeding medium parameters. Analytical solutions and physical interpretation of the processes within reactor can be obtained by such method. Recently, thermal reactor theory with an internal neutron source in the form of delayed neutrons on the basis of birth-and-death model within linear growth approximation was proposed. Particles radioactive decay properties and equation analysis for describing breeding medium parameters using point thermal reactor framework were also studied within the model using Poisson and binomial distribution formalism. Experimental data of three nuclear subcritical facilities were used in this work, KUCA (Japan), MASURCA (France) and GUINEVERE (Belgium). Reactivity and the average number of particles in the system were estimated in this work within physical model framework for three subcritical systems.

Numerically statistical investigation of the partly super-exponential growth rate in the COVID-19 pandemic (throughout the world)

Abstract A number of particles in a multiplying medium under rather general conditions is asymptotically exponential with respect to time t with the parameter λ, i.e., with the index of power λ ⁢ t {\lambda t} . If the medium is random, then the parameter λ is the random variable. To estimate the temporal asymptotics of the mean particles number (via the medium realizations), it is possible to average the exponential function via the corresponding distribution. Assuming that this distribution is Gaussian, the super-exponential estimate of the mean particle number could be obtained and expressed by the exponent with the index of power t ⁢ E ⁢ λ + t 2 ⁢ D ⁢ λ 2 {t{\rm E}\lambda+t^{2}{\rm D}\frac{\lambda}{2}} . The application of this new formula to investigation of the COVID-19 pandemic is performed.

A Study of Transverse Buckling Effect on the Characteristics of Nuclides Burnup Wave in a Fast Neutron Multiplying Media

A study of transverse buckling effect on the characteristics of nuclides burnup wave in multiplying media (cylindrical geometry) has been carried out. The burnup wave is characterized in terms of velocity of propagation, transient length (TL), and transient time (TT) in establishing the burnup wave and full width at half maximum (FWHM) in the established region of the wave. The uranium–plutonium fuel cycle is considered. The sensitivity of the results is studied for different radial buckling led leakage of neutrons. It is discovered that the velocity of the wave increases with the increase in the radius of the cylinder (i.e., reduction in the transverse buckling and hence increase in radial neutron leakage). FWHM is relatively insensitive to radial neutron leakage. The transient time and transient length are very large for smaller radius; these decrease with the increase in radius. The study provides insight on the build-up of burnup wave in the neutron multiplying media and brings out the importance of transverse buckling led radial neutron leakage on the characteristics of fuel burnup wave in multiplying media.

Mid-Weighed Kinetic Parameters for Use in the Two-Group Diffusion Model of Reactor Dynamics with Fuel Based on a Mixture of Fission Isotopes

In the model of reactor kinetics based on the description of neutron transport in the two-group diffusion approximation, the number of equations describing the change in the concentration of delayed neutron precursors depends not only on the number of groups of delayed neutrons, but also on the number of fissile isotopes present in nuclear fuel. Since each isotope is characterized by six groups of delayed neutrons, the total number of differential equations describing concentrations of delayed neutron precursors is equal to the product of the number of fissile isotopes (M) and the number of groups of delayed neutrons for each isotope (i = 6). This is true provided that the decay constant of the concentrations of delayed neutron precursors that were formed from the division by fast or thermal neutrons can be taken in the same way. In fact, there is a difference, though small, in these values for the two energy groups. Therefore, the number of the corresponding equations is twice as high. In this paper, a mathematical expression is obtained for the weighted average decay constant of delayed neutron predecessors from fission by fast and thermal neutrons in a multiplying medium with several fissile isotopes. This, together with the conventional procedure of weighing the fraction of delayed neutrons from fission by fast or thermal neutrons in a similar multiplying medium, allows the two-group diffusion model of the reactor kinetics to be limited to only six equations for the concentrations of delayed neutron precursors and thus the kinetic model of the reactor to be simplified.

Application of an adjoint technique to one–speed [formula omitted]–geometry source–detector transport problems in the discrete ordinates formulation using a spectral nodal method

Described here is an application of an adjoint technique for solving source–detector discrete ordinates (SN) transport problems by using a spectral nodal method. For one–speed X,Y–geometry adjoint SN transport problems, considering linearly anisotropic scattering in non–multiplying media, the coarse–mesh adjoint spectral Green's function–constant nodal (SGF†–CN) method is presented. The discretized SGF†–CN equations are solved by using the adjoint partial one–node block inversion (NBI) iterative scheme. The partial adjoint NBI scheme performs SN transport sweeps from the four corners of the rectangular domain. One uses the most recent estimates for the adjoint node–edge average fluxes in the outgoing directions, to solve the resulting adjoint SN problem in that node for all the incoming adjoint node–edge average fluxes, which constitute the outgoing adjoint node–edge average fluxes for the adjacent nodes in the direction of the transport sweep. Numerical results are given to illustrate the method's accuracy in coarse–mesh adjoint SN calculations and some advantages of using the adjoint technique in source–detector problems.

Bioinoculants and AM fungus colonized nursery improved management of complex root disease of Coleus forskohlii Briq. under field conditions

Nursery treatment with efficient microbial inoculants, using vermicompost as rooting medium, is not popular technique and underutilized for the management of complex root diseases (involved phyto-pathogens: Fusarium, Ralstonia and Meloidogyne) of Coleus forskohlii. To find a probable profound solution of the above problem we designed an experiment to study the potential of efficient bioinoculants and AM fungus as nursery treatment, where the vermicompost was added as rooting and microbial inoculants multiplying medium, for effective management of complex root disease of C. forskohlii for higher root yield, forskolin content and also to sustain the health of soil. Nursery (cuttings of C. forskohlii) treated with efficient microbial inoculants Pseudomonas monteilii (PM) and Glomus intraradices (GI) showed maximum control efficacy for root-rot/wilt (complex disease) of C. forskohlii and improved the growth, yield attributes and also the content of forskolin in the plants. The efficient bioinoculant P. monteilii (strain CRC1) and AM fungus (G. intraradices) thus proved to be a potent/novel antagonist and can be better utilised for biological management of complex root disease of C. forskohlii under organic field conditions.

The build-up and characterization of nuclear burn-up wave in a fast neutron multiplying medium

A study has been carried out on the build-up and characterization of nuclear burn-up wave in fast neutron multiplying media. The focus of the study is more on the transient part of the burn-up wave though the steady state is also studied. The characteristics of the transient part are expressed in terms of new parameters, i.e., Transient Length (TL) and Transient Time (TT) elapsed in establishing the asymptotic burn-up wave. The TT and TL are defined, respectively, as the time and length needed to establish 95% and 99% of the asymptotic neutron flux propagating in the media. From these parameters, the transient wave velocity is determined. The characteristics of asymptotic part of the wave are determined in terms of wave velocity, Full Width at Half Maximum (FWHM) and Full Width at 10% Maximum (FW10M) of neutron flux distribution in the space. A parametric study is also carried out to investigate the sensitivity of these parameters to some of the physical parameters of the ignition zone and the breeder zone of the reactor. The characterization parameters would be very meaningful to understand the transient characteristics of the self-sustaining nuclear burn-up wave and in evaluating the quality of the wave by the researchers working in the field of nuclear burn-up wave build-up and propagation.

The extinction probability in systems randomly varying in time

The extinction probability of a branching process (a neutron chain in a multiplying medium) is calculated for a system randomly varying in time. The evolution of the first two moments of such a process was calculated previously by the authors in a system randomly shifting between two states of different multiplication properties. The same model is used here for the investigation of the extinction probability. It is seen that the determination of the extinction probability is significantly more complicated than that of the moments, and it can only be achieved by pure numerical methods. The numerical results indicate that for systems fluctuating between two subcritical or two supercritical states, the extinction probability behaves as expected, but for systems fluctuating between a supercritical and a subcritical state, there is a crucial and unexpected deviation from the predicted behaviour. The results bear some significance not only for neutron chains in a multiplying medium, but also for the evolution of biological populations in a time-varying environment.

Finite Element Method with Spectral Green's Function in Slab Geometry for Neutron Diffusion in Multiplying Media and One Energy Group

The physical phenomenon of neutrons transport associated with eigenvalue problems appears in the criticality calculations of nuclear reactors and can be treated as a diffusion process. This paper presents a new method to solve eigenvalue problems of neutron diffusion in slab geometry and one energy group. This formulation combines the Finite Element Method, considered an intermediate mesh method, with the Spectral Green's Function Method, which is free of truncation errors, and it is considered a coarse mesh method. The novelty of this formulation is to approach the spatial moments of the neutron flux distribution by the first-order polynomials obtained from the spectral analysis of diffusion equation. The approximations provided by the new formulation allow obtaining accurate results in coarse mesh calculations. To validate the method, we compare the results obtained with the methods described in the literature, specifically the Diamond Difference method. The accuracy and the computational performance of the proposed formulation were characterized by solving benchmarks problems with a high degree of heterogeneity.

Galerkin partial least-square SN (GpLS–SN) method for fixed source problems in the neutron transport theory

The neutron transport equation in the integral form is of difficult solution and therefore, schemes for obtaining an equivalent differential formulation are used. In this paper is used the discrete ordinates method (SN method) in order to obtain the equivalent differential formulation. Inspired in the GLS method (Galerkin least square method) a variant of the GLS (Brooks and Hughes, 1982; Ern and Guermond, 2004; Johnson, 1990) method denoted GpLS (Galerkin partial least-square) is proposed for solving this equivalent differential formulation. The convergence rates as well as the robustness of GLS method are preserved by the GpLS method that possesses computational lower cost. Simulations comparing the convergence rates of GpLS method with other methods are presented, confirming the best performance of GpLS method. The numerical experiments were made using fixed source in multiplying medium, with the objective to test the robustness of GpLS method. All the numerical experiments presented confirm the good properties of approximation of GpLS method.

Об использовании решения сопряженного неоднородного уравнения переноса при определении параметров размножающих сред

Об использовании решения сопряженного неоднородного уравнения переноса при определении параметров размножающих сред

Campbelling-type theory of fission chamber signals generated by neutron chains in a multiplying medium

The signals of fission chambers are usually evaluated with the help of the co-called Campbelling techniques. These are based on the Campbell theorem, which states that if the primary incoming events, generating the detector pulses, are independent, then relationships exist between the moments of various orders of the signal in the current mode. This gives the possibility to determine the mean value of the intensity of the detection events, which is proportional to the static flux, from the higher moments of the detector current, which has certain advantages.However, the main application area of fission chambers is measurements in power reactors where, as is well known, the individual detection events are not independent, due to the branching character of the neutron chains (neutron multiplication). Therefore it is of interest to extend the Campbelling-type theory for the case of correlated neutron events. Such a theory could address two questions: partly, to investigate the bias when the traditional Campbell techniques are used for correlated incoming events; and partly, to see whether the correlation properties of the detection events, which carry information on the multiplying medium, could be extracted from the measurements. This paper is devoted to the investigation of these questions. The results show that there is a potential possibility to extract the same information from fission chamber signals in the current mode as with the Rossi- or Feynman-alpha methods, or from coincidence and multiplicity measurements, which so far have required detectors working in the pulse mode. It is also shown that application of the standard Campbelling techniques to neutron detection in multiplying systems does not lead to an error for estimating the stationary flux as long as the detector is calibrated in in situ measurements.

Neutron spatial flux profile measurement in compact subcritical system using miniature neutron detectors

A zero power multiplying assembly in subcritical regime serves as a benchmark for validating subcritical reactor physics. The utilization of a subcritical assembly for the determination of nuclear parameters in a multiplying medium requires a well-defined neutron flux to carry out the experiments. For this it is necessary to know the neutron flux profile inside a subcritical system. A compact subcritical assembly BRAHMMA has been developed in India. The experimental channels in this assembly are typically less than 8mm diameter. This requires use of miniature detectors that can be mounted in these experimental channels. In this article we present the thermal neutron flux profile measurement in a compact subcritical system using indigenously developed miniature gas filled neutron detectors. These detectors were specially designed and fabricated considering the restrictive dimensional requirements of the subcritical core. Detectors of non-standard size with various sensitivities, from 0.4 to 0.001cps/nv were used for neutron flux of interest ranging from 103 to 107 n-cm−2s−1. A comparison of measured neutron flux using these detectors and simulated Monte Carlo calculations are also presented in this article.

Energy-higher order mode analyses in Feynman-α method

A subcriticality measurement by the Feynman-α method is generally affected by the spatial and energy-higher order modes. This paper focuses on the energy-higher order modes, which can never be eliminated unless the one-energy group approximation is adopted. An analytical formula of the Feynman Y-function that takes into account the energy-higher order modes has been derived for two-energy group systems based on the stochastic transport theory. A generalized formula for the space and energy-dependent Feynman-α method, which was originally derived by Endo et al. and Muñoz-Cobo et al., has been introduced to the study on the energy-higher order mode effects. The generalized formula used in this study is simplified for multi-group problems in a subcritical infinite multiplying medium. Numerical examples show that the energy-higher order mode effects are very minor in a thermal system with a commonly-used 1/υ detector. Thus, the conventional Feynman-α formula may be accurately applicable to subcriticality measurements. In a fast system, however, the energy-higher order mode effects are notable except when the system is near-critical.

Statistical Theory of Fission Chains and Generalized Poisson Neutron Counting Distributions

The neutron counting probability distribution for a multiplying medium was shown by Hage and Cifarelli to be a generalized Poisson distribution that depends on the fission chain number distribution. An analytic formula is obtained for this number distribution, the probability to produce a number of neutrons in a fission chain. The formula depends on the probability that a fission spectrum neutron induces a subsequent fission and depends on the probability distribution for a specific number of neutrons to be produced in an individual induced fission. The formula is an exact solution to a functional equation due to Böhnel for the probability generating function. The Böhnel equation is derived as the t →∞ limit of a rate equation for a neutron population generating function, related to a rate equation studied by Feynman. The Böhnel equation is also shown to be a fixed point of an iteration problem, related to one studied by Hawkins and Ulam, where the iteration generates the chain a generation at a time. The discrete iteration problem is shown to be connected to the continuous time evolution of the chain. An explicit solution for the time evolution of the chain is given in the simplified approximation where at most two neutrons are created by an induced fission. The t →∞ limit of this equation gives a simple analytic expression for the solution to the Böhnel equation in this approximation. A generalized Poisson counting distribution constructed from the theoretical fission chain probability number distribution is compared to experimental data for a multiplying Pu sample.

Neutron diffusion in a randomly inhomogeneous multiplying medium with random phase approximation

Neutron diffusion in a randomly inhomogeneous multiplying medium is studied. By making use of a random phase assumption we show that the average neutron density approximately satisfies an integral equation in Fourier space, which is solved using Kummer functions. We used multi-dimensional formulation. In the case of one dimension, we obtain the result of Rosenbluth and Tao for the mean total density for large t. In the three-dimensional case, a closed form of solution is derived for the mean total neutron density. Its asymptotic behavior is also investigated for large t.

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.].