A SPECTRAL STREAMLINE DIFFUSION FINITE ELEMENT COUPLING METHOD OF UNSTEADY TRANSPORT EQUATION IN THE FIELD OF NEUTRON LOGGING

In this paper, a new numerical method, the coupling method of spherical harmonic function spectral and streamline diffusion finite element for unsteady Boltzmann equation in the neutron logging field, is discussed. The convergence and error estimations of this scheme are proved. Its applications in the field of neutron logging show its effectiveness.

NUMERICAL SIMULATION OF SINGLE-GROUP, STEADY STATE AND ISOTROPIC NEUTRON TRANSPORT EQUATION IN DIFFUSIVE REGIMES

We present an algorithm for numerical solution of transport equation in diffusive regimes, in which the transport equation is nearly singular and its solution becomes a solution of a diffusion equation. This algorithm, which is based on the Least-squares FEM in combination with a scaling transformation, presents a good approximation of a diffusion operator in diffusive regimes and guarantees an accurate discrete solution. The numerical experiments in 2D and 3D case are given, and the numerical results show that this algorithm is correct and efficient.

A RANDOM TRANSPORT-DIFFUSION EQUATION

In this paper we study the integral curve in a random vector field perturbed by white noise. It is related to a stochastic transport-diffusion equation. Under some conditions on the covariance function of the vector field, the solution of this stochastic partial differential equation is proved to have moments. The exact p-th moment is represented through integrals with respect to Brownian motions. The basic tool is Girsanov formula.

A meshless local Petrov–Galerkin method for solving the neutron diffusion equation

The goal of this study is to solve the neutron diffusion equation by using a meshless method and evaluate its performance compared to traditional methods. This paper proposes a novel method based on coupling the meshless local Petrov–Galerkin approach and the moving least squares approximation. This computational procedure consists of two main steps. The first involved applying the moving least squares approximation to construct the shape function based on the problem domain. Then, the obtained shape function was used in the meshless local Petrov–Galerkin method to solve the neutron diffusion equation.Because the meshless method is based on eliminating the mesh-based topologies, the problem domain was represented by a set of arbitrarily distributed nodes. There is no need to use meshes or elements for field variable interpolation. The process of node generation is simply and fully automated, which can save time. As this method is a local weak form, it does not require any background integration cells and all integrations are performed locally over small quadrature domains. To evaluate the proposed method,several problems were considered. The results were compared with those obtained from the analytical solution and a Galerkin finite element method. In addition, the proposed method was used to solve neutronic calculations in thesmall modular reactor. The results were compared with those of the citation code and reference values. The accuracy and precision of the proposed method were acceptable. Additionally, adding the number of nodes and selecting an appropriate weight function improved the performance of the meshless local Petrov–Galerkin method. Therefore, the proposed method represents an accurate and alternative method for calculating core neutronic parameters.

Analysis of anomalous transport based on radial fractional diffusion equation

Anomalous transport in magnetically confined plasmas is investigated by radial fractional transport equations.It is shown that for fractional transport models,hollow density profiles are formed and uphill transports can be observed regardless of whether the fractional diffusion coefficients(FDCs)are radially dependent or not.When a radially dependent FDC<D_(α)(r)1 is imposed,compared with the case under=D_(α)(r)1.0,it is observed that the position of the peak of the density profile is closer to the core.Further,it is found that when FDCs at the positions of source injections increase,the peak values of density profiles decrease.The non-local effect becomes significant as the order of fractional derivative a 1 and causes the uphill transport.However,as a 2,the fractional diffusion model returns to the standard model governed by Fick’s law.

Spherical harmonics method for neutron transport equation based on unstructured-meshes

Based on a new second-order neutron transport equation, self-adjoint angular flux (SAAF) equation, the spherical harmonics (PN) method for neutron transport equation on unstructured-meshes is derived. The spherical harmonics function is used to expand the angular flux. A set of differential equations about the spatial variable, which are coupled with each other, can be obtained. They are solved iteratively by using the finite element method on un- structured-meshes. A two-dimension transport calculation program is coded according to the model. The numerical results of some benchmark problems demonstrate that this method can give high precision results and avoid the ray effect very well.

NONLINEAR SEMIGROUP APPROACH TOTRANSPORT EQUATIONS WITH DELAYED NEUTRONS

This paper deal with a nonlinear transport equation with delayed neutron andgeneral boundary conditions. We establish, via the nonlinear semigroups approach, the exis-tence and uniqueness of the mild solution, weak solution, strong solution and local solutionon LP-spaces （1 ≤ p 〈 ＋∞）. Local and non local evolution problems are discussed.

Alternative constitutive relation for momentum transport of extended Navier–Stokes equations

The classical Navier–Stokes equation(NSE)is the fundamental partial differential equation that describes the flow of fluids,but in certain cases,like high local density and temperature gradient,it is inconsistent with the experimental results.Some extended Navier–Stokes equations with diffusion terms taken into consideration have been proposed.However,a consensus conclusion on the specific expression of the additional diffusion term has not been reached in the academic circle.The models adopt the form of the generalized Newtonian constitutive relation by substituting the convection velocity with a new term,or by using some analogy.In this study,a new constitutive relation for momentum transport and a momentum balance equation are obtained based on the molecular kinetic theory.The new constitutive relation preserves the symmetry of the deviation stress,and the momentum balance equation satisfies Galilean invariance.The results show that for Poiseuille flow in a circular micro-tube,self-diffusion in micro-flow needs considering even if the local density gradient is very low.

Analysis of High Concentration Spikes Appearing in Mass Plume in Nearly Homogeneous Turbulent Flow Based on the PDF Transport Equation

To evaluate the pollutant dispersion in background turbulent flows, most researches focus on statistical variation of concentrations or its fluctuations. However, those time-averaged quantities may be insufficient for risk assessment, because there emerge many high-intensity pollutant areas in the instantaneous concentration field. In this study, we tried to estimate the frequency of appearance of the high concentration areas in a turbulent flow based on the Probability Density Function (PDF) of concentration. The high concentration area was recognized by two conditions based on the concentration and the concentration gradient values. We considered that the estimation equation for the frequency of appearance of the recognized areas consisted of two terms based on each condition. In order to represent the two terms with physical quantities of velocity and concentration fields, simultaneous PIV (Particle Image Velocimetry) and PLIF (Planar Laser-Induced Fluorescence) measurement and PLIF time-serial measurement were performed in a quasi-homogeneous turbulent flow. According to the experimental results, one of the terms, related to the condition of the concentration, was found to be represented by the concentration PDF, while the other term, by the streamwise mean velocity and the integral length scale of the turbulent flow. Based on the results, we developed an estimation equation including the concentration PDF and the flow features of mean velocity and integral scale of turbulence. In the area where the concentration PDF was a Gaussian one, the difference between the frequencies of appearance estimated by the equation and calculated from the experimental data was within 25%, which showed good accuracy of our proposed estimation equation. Therefore, our proposed equation is feasible for estimating the frequency of appearance of high concentration areas in a limited area in turbulent mass diffusion.

New Dynamic Equations of Aerosol in Air of Certain Type

A new dynamic equation of aerosol in air is derived, using a model-in-model, by equilibrium of buoyancy, gravity and pressure, together with conservation laws of mass, momentum and energy via Reynolds transport theorem and supplemented by corresponding scientific laws for related properties of air and aerosols. This new dynamic equation of aerosol in air is a set of non-linear partial differential equations involved six unknown functions of mass densities, pressure, air and aerosol speeds and temperature. It has features: 1, it belongs to certain type;2, it emphases the effect of buoyancy in equilibrium and potential energy, and the Archimedes principle of buoyancy is firstly extended to lateral directions based on logical deduction, the phenomenon of stirring a glass of oil-water mixture and the recorded of Hurricane Isabel (2003) from space station. The later shows the evidence of existence of lateral buoyancy;3, the mass densities of air and aerosol of a point in our model are varied in different directions due to traction and are treated as vectors, and they have been used in the calculation of lateral buoyancy.

Alternative scattering kernels for the first estimates of a reactor:diffusion length

Diffusion-length calculations of neutrons are performed using the Chebyshev polynomials of the second kind. The neutrons are assumed to move with constant energy in a uniform homogeneous slab. An alternative scattering kernel called an Anl?–Gngr phase function and a traditional Henyey–Greenstein phase function are used for the scattering function in the stationary neutron transport equation. First, analytic expressions and then numerical results are obtained for the diffusion length for various values of the scattering and cross-sectional parameters.Numerical results obtained from both scattering kernels for the diffusion length of the neutrons are given in tables side by side for comparison. The applicability of the method is easily demonstrated by these results.

Unified semiclassical approach to electronic transport from diffusive to ballistic regimes

We show that by integrating out the electric field and incorporating proper boundary conditions,a Boltzmann equation can describe electron transport properties,continuously from the diffusive to ballistic regimes.General analytical formulas of the conductance in D = 1,2,3 dimensions are obtained,which recover the Boltzmann–Drude formula and Landauer–B ¨uttiker formula in the diffusive and ballistic limits,respectively.This intuitive and efficient approach can be applied to investigate the interplay of system size and impurity scattering in various charge and spin transport phenomena,when the quantum interference effect is not important.

Transportation Cost Inequalities for Stochastic Reaction-Diffusion Equations with Lévy Noises and Non-Lipschitz Reaction Terms

For stochastic reaction-diffusion equations with Levy noises and non-Lipschitz reaction terms,we prove that W\H transportation cost inequalities hold for their invariant probability measures and for their process-level laws on the path space with respect to the L1-metrie.The proofs are based on the Galerkin approximations.

Lattice Boltzmann method for simulation of time-dependent neutral particle transport

In this paper, a novel model is proposed to investigate the neutron transport in scattering and absorbing medium. This solution to the linear Boltzmann equation is expanded from the idea of lattice Boltzmann method(LBM) with the collision and streaming process. The theoretical derivation of lattice Boltzmann model for transient neutron transport problem is proposed for the first time.The fully implicit backward difference scheme is used to ensure the numerical stability, and relaxation time and equilibrium particle distribution function are obtained. To validate the new lattice Boltzmann model, the LBM formulation is tested for a homogenous media with different sources, and both transient and steady-state LBM results get a good agreement with the benchmark solutions.

Calculation of thermal neutron albedo for mono-material and bi-material reflectors

Thermal neutron albedo has been investigated for different thicknesses of mono-material and bi-material reflectors. An equation has been obtained for a bi-material reflector by considering the neutron diffusion equation. The bi-material reflector consists of binary combinations of water, graphite, lead, and polyethylene. An experimental measurement of thermal neutron albedo has also been conducted for mono-material and bi-material reflectors by using a^(241) Am–Be(5.2 Ci) neutron source and a BF3 detector. The maximum value of thermal neutron albedo was obtained for a polyethylene–water combination(0.95 ± 0.02).

Talagrand Inequality on Free Path Space and Application to Stochastic Reaction Diffusion Equations

By using a split argument due to[1],the transportation cost inequality is established on the free path space of Markov processes.The general result is applied to stochastic reaction diffusion equations with random initial values.

NEUMANN SERIES SOLUTION TO A NEUTRON TRANSPORT EQUATION OF SLAB GEOMETRY

This paper deals with the solution to an energy, dependent stationary neutrontransport equation of slab geometry. In L^p space, the equation is converted into an equiva-lent integral equation. By the study of the corresponding integral operator and its spectralradius, results of Neumann series solution are obtained, and an easy-verified condition thatthe transport equation has a nonnegative solution is given.

THE RATIONALITY OF THE DISCRETE ORDINATES METHOD FOR TIME DEPENDENT NEUTRON TRANSPORT EQUATION

ＴＨＥＲＡＴＩＯＮＡＬＩＴＹＯＦＴＨＥＤＩＳＣＲＥＴＥＯＲＤＩＮＡＴＥＳＭＥＴＨＯＤＦＯＲＴＩＭＥＤＥＰＥＮＤＥＮＴＮＥＵＴＲＯＮＴＲＡＮＳＰＯＲＴＥＱＵＡＴＩＯＮＬＩＸｕｅｚｈｉ；ＺＨＵＧｕａｎｇｔｉａｎ（ＩｎｓｔｉｔｕｔｅｏｆＳｙｓｔｅｍｓＳｃｉ...

Reconstruction of the Neutron Flux in a Slab Reactor

In this electronic article we use the one-dimensional multigroup neutron diffusion equation to reconstruct the neutron flux in a slab reactor from the nuclear parameters of the reactor, boundary and symmetry condition, initial flux and?keff. The diffusion equation was solved analytically for one single homogeneous fuel region and for two regions considering fuel and reflector. To validate the method proposed, the results obtained in this article were compared using reference methods found in the literature.