Multigroup Diffusion Approximation Research Articles

Multiscale model reduction for neutron diffusion equation

Modelling of dynamic processes in nuclear reactors is carried out, mainly, on the basis of the multigroup diffusion approximation for the neutron flux. The neutron diffusion approximation is widely used for reactor analysis and applied in most engineering calculation codes. In this paper, we attempt to employ a model reduction technique based on the multiscale method for neutron diffusion equation. The proposed method is based on the use of a generalized multiscale finite element method. The main idea is to create multiscale basis functions that can be used to effectively solve on a coarse grid. From calculation results, we obtain that multiscale basis functions can properly take into account the small-scale characteristics of the medium and provide accurate solutions. The results calculated with the GMsFEM are compared with the reference fine-grid calculation results.

Preliminary investigation on the sodium fast reactor concave cores with near‐zero or negative sodium void reactivity

A preliminary investigation on the sodium fast reactor (SFR) concave cores was conducted with the aim of obtaining near-zero or negative sodium void reactivity (SVR). The geometry of the MOX fueled 750 MWe Japanese Sodium Fast Reactor (JSFR) core is transformed into concave cores by maintaining the same power level and heavy metal fuel inventory. The inner and outer core (with axial blanket in the lower part) configuration is maintained as in the original JSFR but the height of the inner core was set lower than the outer core to improve the SVR. Also a short inner blanket (20 cm thick) is inserted into the inner core for the same purpose. Radial blanket with the same height is also adopted as in the original JSFR to maintain the breeding ratio. In this investigation, the MOX fuel composition for the fresh fuel does not contain minor actinides. JOINT-FR code system with JENDL-4.0 library is used for neutronics, burnup, kinetics parameters as well as temperature and SVR calculations in 2-D R-Z geometry multigroup diffusion approximation. Several candidates of concave cores with favorable near-zero or negative SVR have been found whose burnup performance is comparable with the original JSFR core.

Small nuclear power plants for power supply in arctic regions: assessment of spent nuclear fuel radioactivity

The purpose of the present study is the investigation of mass composition of long-lived radionuclides accumulated in the fuel cycle of small nuclear power plants (SNPP) as well as long-lived radioactivity of spent fuel of such reactors. Analysis was performed of the published data on the projects of SNPP with pressurized water-cooled reactors (LWR) and reactors cooled with Pb-Bi eutectics (SVBR). Information was obtained on the parameters of fuel cycle, design and materials of reactor cores, thermodynamic characteristics of coolants of the primary cooling circuit for reactor facilities of different types. Mathematical models of fuel cycles of the cores of reactors of ABV, KLT-40S, RITM-200M, UNITERM, SVBR-10 and SVBR-100 types were developed. The KRATER software was applied for mathematical modeling of the fuel cycles where spatial-energy distribution of neutron flux density is determined within multi-group diffusion approximation and heterogeneity of reactor cores is taken into account using albedo method within the reactor cell model. Calculation studies of kinetics of burnup of isotopes in the initial fuel load (235U, 238U) and accumulation of long-lived fission products (85Kr, 90Sr, 137Cs, 151Sm) and actinoids (238,239,240,241,242Pu, 236U, 237Np, 241Am, 244Cm) in the cores of the examined SNPP reactor facilities were performed. The obtained information allowed estimating radiation characteristics of irradiated nuclear fuel and implementing comparison of long-lived radioactivity of spent reactor fuel of the SNPPs under study and of their prototypes (nuclear propulsion reactors). The comparison performed allowed formulating the conclusion on the possibility in principle (from the viewpoint of radiation safety) of application of SNF handling technology used in prototype reactors in the transportation and technological process layouts of handling SNF of SNPP reactors.

Solution of neutron-transport multigroup equations system in subcritical systems

An iteration method has been implemented to solve a neutron transport equation in a multigroup diffusion approximation. A thermoelectric generator containing plutonium dioxide, used as a source of thermal and electric power in spacecraft, was studied. Neutron yield and multigroup diffusion approximation data was used to obtain a continuous and group distribution of neutron flux density spectra in a subcritical multiplying system. Numerical multigroup approaches were employed using BNAB-78, a system of group constants, and other available evaluated nuclear data libraries (ROSFOND, BROND, BNAB, EXFOR and ENDSF). The functions of neutron distribution in the zero iteration for the system of multigroup equations were obtained by approximating an extensive list of calculated and experimental data offered by the EXFOR and ENDSF nuclear data libraries. The required neutronic functionals were obtained by solving a neutron transport equation in a 28-group diffusion approximation. The calculated data was verified. The approach used is more efficient in terms of computational efforts (the values of the neutron flux density fractions converge in the third iteration). The implemented technique can be used in nuclear and radiation safety problems.

State change modal method for numerical simulation of dynamic processes in a nuclear reactor

Modeling of dynamic processes in nuclear reactors is carried out, mainly, on the basis of the multigroup diffusion approximation for the neutron flux. The basic model includes a multidimensional set of coupled parabolic equations and ordinary differential equations. Dynamic processes are modelled by a successive change of the reactor states, which are characterized by given coefficients of the equations. In the modal method, the approximate solution is represented as an expansion on the first eigenfunctions of some spectral problem. The numerical-analytical method is based on the use of the dominant time-eigenvalues of a multigroup diffusion model taking into account delayed neutrons. In this work, the application of the modal methodology based on calculation of the dominant eigenvalues and eigenfunctions of α-eigenvalue problem has been tested for the VVER-1000 reactor test model. The last is characterized by the fact that some eigenvalues are complex. Reactor dynamics behavior is simulated for symmetrical and non-symmetrical control rods insertion/withdrawal. The power calculation results obtained with the modal method were compared with the numerical solution of the dynamics problem. A rather good agreement was shown for the problem with single delayed neutron precursor group.

Neutronic calculation of fast reactors by the EUCLID/V1 integrated code

This article considers neutronic calculation of a fast-neutron lead-cooled reactor BREST-OD-300 by the EUCLID/V1 integrated code. The main goal of development and application of integrated codes is a nuclear power plant safety justification. EUCLID/V1 is integrated code designed for coupled neutronics, thermomechanical and thermohydraulic fast reactor calculations under normal and abnormal operating conditions. EUCLID/V1 code is being developed in the Nuclear Safety Institute of the Russian Academy of Sciences. The integrated code has a modular structure and consists of three main modules: thermohydraulic module HYDRA-IBRAE/LM/V1, thermomechanical module BERKUT and neutronic module DN3D. In addition, the integrated code includes databases with fuel, coolant and structural materials properties. Neutronic module DN3D provides full-scale simulation of neutronic processes in fast reactors. Heat sources distribution, control rods movement, reactivity level changes and other processes can be simulated. Neutron transport equation in multigroup diffusion approximation is solved. This paper contains some calculations implemented as a part of EUCLID/V1 code validation. A fast-neutron lead-cooled reactor BREST-OD-300 transient simulation (fuel assembly floating, decompression of passive feedback system channel) and cross-validation with MCU-FR code results are presented in this paper. The calculations demonstrate EUCLID/V1 code application for BREST-OD-300 simulating and safety justification.

Spectral properties of dynamic processes in a nuclear reactor

As a rule, mathematical modeling of dynamic processes in nuclear reactors is conducted using an approach that treats a neutron flux in the multigroup diffusion approximation. In this approach, the basic model involves a multidimensional system of coupled parabolic-type equations. Similarly to common thermal phenomena, it is possible here to separate a regular mode of nuclear reactor operation that is associated with a selfsimilar development of a neutron field at large times. In this case, the main feature of dynamic processes is a minimal eigenvalue of the corresponding spectral problem. In the present paper, calculations of various eigenvalues are performed via the two-group model and discussed for the VVER-1000 reactor without a reflector and HWR reactor.

Calculation of the Covariation Matrix of the Neutron Flux Density in the Multigroup Diffusion Model of a Subcritical Reactor

The reconstruction of fields and adjustment of the parameters of reactor models are increasingly prioritized for reactor physics because the efficiency and safety of the control of a nuclear plant depend on solving them. The results of reconstruction and parameter adjustments are largely determined by the probability distributions of the computed and measured data. The probability distribution of the computed data in turn depends on the properties of the neutron-physical model and probabilistic characteristics of the parameters of this model. The present work is devoted to the search for and investigations of efficient algorithms for calculating the covariation functions of the neutron flux density. A model of a stationary subcritical reactor with an internal neutron source is studied. Linear perturbation theory, the theorem on the spectral decomposition of operators, and the theory of stochastic processes are the tools used for performing the analysis. The uncertainty of the nuclear data, the technological parameters of the core, and the inaccuracies in the model approximations give rise to a significant error in neutron-physical calculations. For this reason, reactor models must be verified and adjusted on the basis of measurements in order to make the control and design of reactor setups more reliable and efficient. The measurements and calculations can be compared with one another correctly only if their errors are taken into account. The problem of quantitative assessment of the computational uncertainty in the model parameters is solved by means of the theory of sensitivity and uncertainty analysis [1‐3]. Methods for evaluating the uncertainty of linear and nonlinear functions of the neutron field are now adequately developed [4, 5]. Less attention is given to the evaluation of the uncertainty of distributed characteristics: the neutron and temperature fields, distribution of the nuclear concentration, energy release fields, and so on. The probabilistic characteristics of the distributed parameters are used in the algorithms for reconstruction, adjustment, and optimal filtration of computational models [6‐9]. Of special value for these algorithms are the second moments of the function of state of a model ‐ the covariation matrices, to whose study the work is devoted. The aim of the present work is to study the structure of and search for efficient algorithms for calculating the covariation matrices of the neutron field on the basis of the multigroup diffusion approximation. The analysis is performed for a stationary subcritical reactor with a distributed internal neutron source.

靶丸变形实验的多群扩散整体数值模拟

靶丸变形实验的多群扩散整体数值模拟

Radiative gas-dynamic model of a continuous optical discharge in a gravitational field: quasi-optical approximation

We consider a continuous optical discharge (COD) sustained by a vertically directed weakly focused CO2 laser beam, in a gravitational field. We used a full two-dimensional radiative gas-dynamic numerical model for the COD, which uses realistic quasi optics and takes refraction of the laser radiation in the plasma properly into account in the description of the laser beam propagation. The model is applied to calculate the parameters of the COD in a converging CO2 laser beam in free air atmosphere as a function of the laser power. We also demonstrate the effect of the selection of spectral groups, used in the multi-group diffusion approximation of the thermal radiation transport, on the model solutions.

Modeling of radiation transfer and emission spectra in laser-induced plasma of Al vapor

The influence of radiation transfer on dynamics of laser-induced plasma of Al vapor is analyzed numerically. Mathematical description of the plasma is performed in the framework of the transient 2D radiation gas dynamics (RGD); the multi-group diffusion approximation with 7–1000 spectral intervals is used to describe plasma radiation transfer. The performed research allows us to conclude that radiative transfer substantially affects the expansion of laser plasma for the nanosecond action intensity of 10 9– 2×10 10 W/ cm 2 and the pulse energies of 0.02– 0.4 J . The radiative energy losses can reach 60% of the laser pulse energy absorbed by the plasma. The spectral composition of escaping radiation is non-equilibrium and qualitatively corresponds to the optically thin approximation. The application of the Planck averaging technique make it possible to predict the correct values of total quantities (integrated over the entire frequency spectrum) using several tens of groups.

Simulation of Laser Plasma Dynamics: Influence of Ambient Pressure and Intensity of Laser Radiation

A mathematical model, describing the dynamics of laser plasma (gas dynamic stage) in air medium with the pressure variation in the range of 1-100 Bar, is presented. The simulation is based on the transient axially symmetric system of radiative gas dynamic equations. The application on the multi-group diffusion approximation for the simulation of radiation transfer is discussed. The solution of gas dynamic equations is made by the explicit finite-difference FLIC method. For the energy equation solving jointly with the averaged equation of radiation diffusion, the implicit difference scheme is used. The later is realized by the proposed nonlinear iterative procedure. The evolution of laser plasma in the near threshold values (for plasma formation) of radiation intensity G = 5 × 107-5 × 108 W/cm2 (λ = 1.06 μm) is analyzed. It is shown that the mechanism of the plasma expansion strongly depends on the pressure of the gaseous medium: it varies from the light detonation regime of fast combustion at low pressures; to the slow combustion in the form of the subsonic radiative wave at high pressures.

Near-infinite-medium solutions of the equation of transfer

We consider certain aspects of the asymptotic derivation of the diffusion (differential) approximation to the equation of transfer. We argue that it is not necessary to assume small absorption as has been traditional in the classic asymptotic derivation of the diffusion approximation. As part of these considerations, we consider a family of asymptotic scalings in the equation of transfer, each yielding a diffusion equation but with different diffusion coefficients if the scattering phase function is anisotropic. A preferred scaling is obtained by appealing to the local infinite medium solution of the equation of transfer. In a related development, consideration of an entirely different scaling, which corresponds to a perturbation about the local infinite medium solution, gives a slightly different diffusion equation for an anisotropic source. For an isotropic source such as is extant in radiative transfer, this new diffusion equation reduces to that obtained from the preferred scaling in the earlier treatment. In either treatment, the relaxation of the requirement of small absorption leads to a boundary condition different from that predicted by classic (small absorption) asymptotic analysis. Applying these considerations to the multigroup equation of transfer establishes, for linearly anisotropic scattering, the multigroup diffusion approximation as an asymptotic limit of transport theory, a result heretofore not available.

An implicit scheme for calculating time- and frequency-dependent flux limited radiation diffusion in one dimension

A flexible and accurate method for solving nonlinear time- and frequency-dependent flux-limited radiation diffusion and radiation-matter coupling problems in one dimension is developed. The method is based on converting the set of partial differential equations for the flux-limited multigroup diffusion approximation into a system of ordinary differential equations which are integrated by a generalized stiff equation solver. Unlike previous operator-splitting techniques which can introduce instabilities into the solution or which produce smooth but highly inaccurate solutions, this scheme provides a reliable accurate solution with reduced overall computational effort. The solutions describe both the streaming and diffusion limits of transport theory and are stable with time steps comparable to the changes in physical variables.

The solution of two-dimensional non-stationary problems of radiation gas dynamics

METHODS of solving the two-dimensional equations of the dynamics of a radiating gas are presented. Radiative transport is considered in the multigroup diffusion approximation. The equations of gas-dynamics are solved in the Lagrangian coordinate system.

Iterative synthesis of solutions of the neutron transport equations

Iterative methods for the synthesis of neutron transport equation solutions in multidimensional problems are discussed. These methods are based on the assumption that the calculated function describing neutron space-energy distribution in a nuclear reactor may be represented by a sum of functions with separated variables. Algorithms of methods of synthesis in sets of continuous and discontinuous functions are presented. A number of examples illustrate methods of obtaining iterative synthesis equations. Calculations indicate that it is possible to achieve results of high accuracy by means of iterative synthesis. A brief description of program, RADAR, a program for two-dimensional reactor calculation in the multigroup diffusion approximation, is presented. (auth)

Interpretation of Pulsed-Neutron-Source Experiments in a Multiregion Reactor

By expanding the static flux into kinetic fluxes containing the fundamental and higher modes, we derive a kind of inhour equation which can express kinetic distortion, and can relate the static reactivity to prompt decay constants and to kinetic and adjoint fluxes. The equation is developed using the time-dependent, multigroup diffusion approximation, and is applied to the interpretation of pulsed-neutron-source experiments in a multiregion reactor. It is shown from this application that the Simmons-King formula can express the dynamic reactivity only under conditions of constant generation time and no kinetic distortion, and that the conventional inhour equation derived by the perturbation method cannot distinguish between the static and dynamic reactivities differing from each other on account of kinetic distortion.

Interpretation of Pulsed-Neutron-Source Experiments in a Multiregion Reactor

By expanding the static flux into kinetic fluxes containing the fundamental and higher modes, we derive a kind of inhour equation which can express kinetic distortion, and can relate the static reactivity to prompt decay constants and to kinetic and adjoint fluxes. The equation is developed using the time-dependent, multigroup diffusion approximation, and is applied to the interpretation of pulsed-neutron-source experiments in a multiregion reactor. It is shown from this application that the Simmons-King formula can express the dynamic reactivity only under conditions of constant generation time and no kinetic distortion, and that the conventional inhour equation derived by the perturbation method cannot distinguish between the static and dynamic reactivities differing from each other on account of kinetic distortion.