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Sample records for monoenergetic transport equation

  1. Green's function method for the monoenergetic transport equation in heterogeneous plane geometry

    International Nuclear Information System (INIS)

    Ganapol, B.D.

    1995-01-01

    For the past several years, a series of papers by the transport group at the University of Arizona dealing with benchmark solutions of the monoenergetic transport equation has appeared. The approach has been to take advantage of highly successful numerical Laplace Fourier transform inversions to provide benchmark quality solutions in infinite media, half-space in one and two dimensions and in homogeneous slabs. This paper extends the set of solutions to include heterogeneous slab geometry by using the recently established Green's Function Method (GFM). Analytical benchmark solutions are an essential part of the quality control of computational algorithms developed for particle transport. In addition, benchmarking methods have applications in the classroom by providing examples of how computational mathematics is used to solve physical problems to obtain meaningful answers. In a structural context, monoenergetic solutions are directly applicable to the investigation of the microlight environment within a leaf. The leaf is considered to be a composition of alternating layers of highly absorbing pigments and water superimposed on a refractively scattering background

  2. Numerical solutions of the monoenergetic neutron transport equation with anisotropic scattering

    International Nuclear Information System (INIS)

    Dahl, B.

    1985-01-01

    The Boltzmann equation for monoenergetic neutrons has been solved numerically with high accuracy for homogeneous slabs and spheres with various degree of linear anisotropy. Vacuum boundary conditions are used. The numerical method is based on previous work by Carlvik. Benchmark values of the criticality factor and higher order eigenvalues are given for multiplying systems of thickness or diameter from 10 -5 to 20 mean free paths and with anisotropy coefficients from 0.0 to 0.3. For slab geometry, both even and odd mode eigenvalues are treated. With increasing anisotropy, an increasing number of complex eigenvalues is observer. The total flux is calculated from the eigenvector and tables of the fundamental mode flux are given. Accurate extrapolation distances are derived for various dimensions and anisotropy coefficients from our eigenvalue results on slabs and spheres and from the work by Sanchez on infinite cylinders.The time eigenvalue spectrum in subcritical systems has also been studied. First, the connection between the eigenvalues arising from the time dependent and stationary transport equation is established. Based on this, the spectrum of real time eigenvalues in slabs and spheres is calculated. For spheres, the existence of complex time eigenvalues in the region beyond the value corresponding to the Corngold limit is numerically established. The presence of such eigenvalues has earlier not been proved. It is further shown that the Boltzmann equation for a sphere is significantly simplified when the decay constant is at the Corngold limit. The spectrum of sphere diameters corresponding to this decay constant is calculated for various linear anisotropies, and detailed numerical results are given. (Author)

  3. Solution of the linear transport equation, monoenergetic in multiregions with anisotopic scattering by the method F sub(N)

    International Nuclear Information System (INIS)

    Pontedeiro, E.M.B.D.; Maiorino, J.R.

    1982-01-01

    The linear equation transport, monoenergetic, with anysotropic scattering, in multiregions, by F sub(N) method, is resolved. The mathematical analysis used for this method consists in to use parcially the expansion method in singular autofunctions, or Case's method, aiming to derive a set of integral equations coupled to the angular distribution in the boundaries and interfaces, and then to approximate these distributions by polynomics of N order, aiming to derive, with the use of these boundary and continuity conditions in the interfaces, a set of algebric equations for the coef. of polynomical approximation. With the goal to obtain numerical results, a computer code (FNAM-1) with options for the number of regions, boundary conditions, F sub(N) approx order, were developed. Numerical results were then obtained for various sample problems and compared with the results published in the literature with the objective to demonstrate the precision and applicability of the F sub(N) method. (E.G.) [pt

  4. Presentation of some methods for the solution of the monoenergetic neutrons transport equation

    International Nuclear Information System (INIS)

    Valle G, E. del.

    1978-01-01

    The neutrons transport theory problems whose solution has been reached were collected in order to show that the transport equation is so complicated that different techniques were developed so as to give approximative numerical solutions to problems concerning the practical application. Such a technique, which had not been investigated in the literature dealing with these problems, is described here. The results which were obtained through this technique in undimensional problems of criticity are satisfactory and speaking in a conceptual way this method is extremely simple because it times. There is no limitation to deal with problems related neutrons sources with an arbitrary distribution and in principle the application of this technique can be extended to unhomogeneous environments. (author)

  5. Solution to the monoenergetic time-dependent neutron transport equation with a time-varying source

    International Nuclear Information System (INIS)

    Ganapol, B.D.

    1986-01-01

    Even though fundamental time-dependent neutron transport problems have existed since the inception of neutron transport theory, it has only been recently that a reliable numerical solution to one of the basic problems has been obtained. Experience in generating numerical solutions to time-dependent transport equations has indicated that the multiple collision formulation is the most versatile numerical technique for model problems. The formulation coupled with a moment reconstruction of each collided flux component has led to benchmark-quality (four- to five-digit accuracy) numerical evaluation of the neutron flux in plane infinite geometry for any degree of scattering anisotropy and for both pulsed isotropic and beam sources. As will be shown in this presentation, this solution can serve as a Green's function, thus extending the previous results to more complicated source situations. Here we will be concerned with a time-varying source at the center of an infinite medium. If accurate, such solutions have both pedagogical and practical uses as benchmarks against which other more approximate solutions designed for a wider class of problems can be compared

  6. A solution of the monoenergetic neutral particle transport equation for adjacent half-spaces with anisotropic scattering

    Energy Technology Data Exchange (ETDEWEB)

    Ganapol, B.D., E-mail: ganapol@cowboy.ame.arizona.edu [Department of Aerospace and Mechanical Engineering, University of Arizona, Tucson, AZ (United States); Mostacci, D.; Previti, A. [Montecuccolino Laboratory, University of Bologna, Via dei Colli, 16, I-40136 Bologna (Italy)

    2016-07-01

    We present highly accurate solutions to the neutral particle transport equation in a half-space. While our initial motivation was in response to a recently published solution based on Chandrasekhar's H-function, the presentation to follow has taken on a more comprehensive tone. The solution by H-functions certainly did achieved high accuracy but was limited to isotropic scattering and emission from spatially uniform and linear sources. Moreover, the overly complicated nature of the H-function approach strongly suggests that its extension to anisotropic scattering and general sources is not at all practical. For this reason, an all encompassing theory for the determination of highly precise benchmarks, including anisotropic scattering for a variety of spatial source distributions, is presented for particle transport in a half-space. We illustrate the approach via a collection of cases including tables of 7-place flux benchmarks to guide transport methods developers. The solution presented can be applied to a considerable number of one and two half-space transport problems with variable sources and represents a state-of-the-art benchmark solution.

  7. Monoenergetic time-dependent neutron transport in an infinite medium with time-varying cross sections

    International Nuclear Information System (INIS)

    Ganapol, B.D.

    1987-01-01

    For almost 20 yr, the main thrust of the author's research has been the generation of as many benchmark solutions to the time-dependent monoenergetic neutron transport equation as possible. The major motivation behind this effort has been to provide code developers with highly accurate numerical solutions to serve as standards in the assessment of numerical transport algorithms. In addition, these solutions provide excellent educational tools since the important physical features of neutron transport are still present even though the problems solved are idealized. A secondary motivation, though of equal importance, is the intellectual stimulation and understanding provided by the combination of the analytical, numerical, and computational techniques required to obtain these solutions. Therefore, to further the benchmark development, the added complication of time-dependent cross sections in the one-group transport equation is considered here

  8. Solutions of Boltzmann`s Equation for Mono-energetic Neutrons in an Infinite Homogeneous Medium

    Science.gov (United States)

    Wigner, E. P.

    1943-11-30

    Boltzman's equation is solved for the case of monoenergetic neutrons created by a plane or point source in an infinite medium which has spherically symmetric scattering. The customary solution of the diffusion equation appears to be multiplied by a constant factor which is smaller than 1. In addition to this term the total neutron density contains another term which is important in the neighborhood of the source. It varies as 1/r{sup 2} in the neighborhood of a point source. (auth)

  9. Complex eigenvalues for neutron transport equation with quadratically anisotropic scattering

    International Nuclear Information System (INIS)

    Sjoestrand, N.G.

    1981-01-01

    Complex eigenvalues for the monoenergetic neutron transport equation in the buckling approximation have been calculated for various combinations of linearly and quadratically anisotropic scattering. The results are discussed in terms of the time-dependent case. Tables are given of complex bucklings for real decay constants and of complex decay constants for real bucklings. The results fit nicely into the pattern of real and purely imaginary eigenvalues obtained earlier. (author)

  10. Transport equation solving methods

    International Nuclear Information System (INIS)

    Granjean, P.M.

    1984-06-01

    This work is mainly devoted to Csub(N) and Fsub(N) methods. CN method: starting from a lemma stated by Placzek, an equivalence is established between two problems: the first one is defined in a finite medium bounded by a surface S, the second one is defined in the whole space. In the first problem the angular flux on the surface S is shown to be the solution of an integral equation. This equation is solved by Galerkin's method. The Csub(N) method is applied here to one-velocity problems: in plane geometry, slab albedo and transmission with Rayleigh scattering, calculation of the extrapolation length; in cylindrical geometry, albedo and extrapolation length calculation with linear scattering. Fsub(N) method: the basic integral transport equation of the Csub(N) method is integrated on Case's elementary distributions; another integral transport equation is obtained: this equation is solved by a collocation method. The plane problems solved by the Csub(N) method are also solved by the Fsub(N) method. The Fsub(N) method is extended to any polynomial scattering law. Some simple spherical problems are also studied. Chandrasekhar's method, collision probability method, Case's method are presented for comparison with Csub(N) and Fsub(N) methods. This comparison shows the respective advantages of the two methods: a) fast convergence and possible extension to various geometries for Csub(N) method; b) easy calculations and easy extension to polynomial scattering for Fsub(N) method [fr

  11. On an analytical formulation for the mono-energetic neutron space-kinetic equation in full cylinder symmetry

    International Nuclear Information System (INIS)

    Oliveira, F.R.; Bodmann, B.E.J.; Vilhena, M.T.; Carvalho, F.

    2017-01-01

    Highlights: • The present work presents an exact solution to neutron spatial kinetic equation. • It is an exact solution in a heterogeneous cylinder with temporal dependence. • The solution was constructed through the separation of variables method. - Abstract: In the present work we discuss a system of partial differential equations that model neutron space-kinetics in cylindrical geometry and are defined by two sectionally homogeneous cylinder cells, mono-energetic neutrons and one group of delayed neutron precursors. The solution is determined using the technique of variable separation. The associated complete spectra with respect to each variable separation are analysed and truncated such as to allow a parameterized global solution. For the obtained solution we present some numerical results for the scalar neutron flux and its time dependence and projection on the cylinder axis z and the radial and cylinder axis projection. As a case study we consider an insertion of an absorbing medium in the upper cylinder cell. Continuity of the scalar flux at the interface between the two cylinder elements and conserved current density is explained and related to scale invariance of the partial differential equation system together with the initial and boundary conditions. Some numerical results for the scalar angular neutron flux and associated current densities are shown.

  12. Computational programs for shielding calculation with transport of one dimensional and monoenergetic SN

    International Nuclear Information System (INIS)

    Nunes, Carlos Eduardo A.; Barros, Ricardo C.

    2009-01-01

    This paper describes a computational program for result simulation of neutron transport problems at one velocity with isotropic scattering in Cartesian onedimensional geometry. Describing the physical modelling, the next phase is a mathematical modelling of the physical problem for simulation of the neutron distribution. The mathematical modelling uses the linearized Boltzmann equation which represents a balance among the production and loss of particles. The formulation of the discrete ordinates S N consists of discretization of angular variables at N directions (discrete ordinates), and using a set of angular quadratures for the approximation of integral terms of scattering sources. The S N equations are numerically solved. This work describes three numerical methods: diamond difference, step and characteristic step. The paper also presents numerical results for illustration of the efficiency of the developed program

  13. Saturation and linear transport equation

    International Nuclear Information System (INIS)

    Kutak, K.

    2009-03-01

    We show that the GBW saturation model provides an exact solution to the one dimensional linear transport equation. We also show that it is motivated by the BK equation considered in the saturated regime when the diffusion and the splitting term in the diffusive approximation are balanced by the nonlinear term. (orig.)

  14. Monoenergetic neutral particle transport in an anisotropically scattering half-space

    International Nuclear Information System (INIS)

    Ganapol, B.D.; Garth, J.C.; Woolf, S.

    1995-01-01

    During the past several years, a considerable effort has been underway to develop reliable analytical benchmark solutions to the one-speed transport equation in various geometries. The reader may well ask open-quotes whyclose quotes such a task has been undertaken, given the recent rapid advances in numerical transport theory. The simple answer is that reliable numerical solutions do not yet exist, and their development still represents a mathematical challenge. However, regardless of how mathematically challenging the development is, there are more compelling reasons for this effort which are rooted in the very fundamentals of science and technology. In particular, these solutions, which are highly accurate numerical evaluations of analytical solution representations, serve as open-quotes industry standardsclose quotes to which other more approximate methods or approximations can be compared. Thus analytical benchmarks are part of the process control and continuous improvement of numerical transport methods and are therefore integral components in TQM (Total Quality Management) as applied to transport methods development. With the above reasoning in mind, the authors begin the development and application of a new analytical solution and evaluation for a half-space featuring anisotropic scattering. This work is an extension of previous efforts in which isotropically scattering half-spaces were treated. The previous benchmarks were obtained most conveniently via a numerical Laplace transform inversion which could be applied in a straightforward manner to the case of isotropic scattering. The application of the Laplace transform method is problematical for anisotropic scattering and does not admit to the direct identification of the scalar flux from integral transport theory

  15. Transport equation and shock waves

    International Nuclear Information System (INIS)

    Besnard, D.

    1981-04-01

    A multi-group method is derived from a one dimensional transport equation for the slowing down and spatial transport of energetic positive ions in a plasma. This method is used to calculate the behaviour of energetic charged particles in non homogeneous and non stationary plasma, and the effect of energy deposition of the particles on the heating of the plasma. In that purpose, an equation for the density of fast ions is obtained from the Fokker-Planck equation, and a closure condition for the second moment of this equation is deduced from phenomenological considerations. This method leads to a numerical method, simple and very efficient, which doesn't require much computer storage. Two types of numerical results are obtained. First, results on the slowing down of 3.5 MeV alpha particles in a 50 keV plasma plublished by Corman and al and Moses are compared with the results obtained with both our method and a Monte Carlo type method. Good agreement was obtained, even for energy deposition on the ions of the plasma. Secondly, we have calculated propagation of alpha particles heating a cold plasma. These results are in very good agreement with those given by an accurate Monte Carlo method, for both the thermal velocity, and the energy deposition in the plasma

  16. Monoenergetic particle transport in a semi-infinite medium with reflection

    International Nuclear Information System (INIS)

    Ganapol, B.D.

    1993-01-01

    Next to neutron or photon transport in infinite geometry, particle transport in semi-infinite geometry is probably the most investigated transport problem. When the mean free path for particle interaction is small compared to the physical dimension of the scattering medium, the infinite or semi-infinite geometry assumption is reasonable for a variety of applications. These include nondestructive testing, photon transport in plant canopies, and inverse problems associated with well logging. Another important application of the transport solution in a semi-infinite medium is as a benchmark to which other more approximate methods can be compared. In this paper, the transport solution in a semi-infinite medium with both diffuse and specular reflection at the free surface is solved analytically and numerically evaluated. The approach is based on a little-known solution obtained by Sobelev for the problem with specular reflection, which itself originates from the classical albedo problem solution without reflection. Using Sobelev's solution as a partial Green's function, the exiting flux for diffuse reflection can be obtained. In this way, the exiting flux for a half-space with both constant diffuse and specular reflection coefficients is obtained for the first time. This expression can then be extended to the complex plane to obtain the interior flux as an inverse Laplace transform, which is numerically evaluated

  17. LOCFES-B: A program for solving the one-dimensional particle transport equation with user-selected CLOF methods

    International Nuclear Information System (INIS)

    Jarvis, R.D.; Nelson, P.

    1995-01-01

    LOCFES-B solves the steady-state, monoenergetic and azimuthally symmetric neutral-particle transport equation in one-dimensional plane-parallel geometry. LOCFES-B is designed to facilitate testing and comparison of different spatial approximations in neutron transport. Accordingly, it permits performance of user-provided CLOF spatial approximations to be compared directly on successively refined mesh sizes and user-input physical problems with automatic comparison of results. if desired, to user-supplied benchmark results

  18. Normal scheme for solving the transport equation independently of spatial discretization

    International Nuclear Information System (INIS)

    Zamonsky, O.M.

    1993-01-01

    To solve the discrete ordinates neutron transport equation, a general order nodal scheme is used, where nodes are allowed to have different orders of approximation and the whole system reaches a final order distribution. Independence in the election of system discretization and order of approximation is obtained without loss of accuracy. The final equations and the iterative method to reach a converged order solution were implemented in a two-dimensional computer code to solve monoenergetic, isotropic scattering, external source problems. Two benchmark problems were solved using different automatic selection order methods. Results show accurate solutions without spatial discretization, regardless of the initial selection of distribution order. (author)

  19. The Laplace transformation of adjoint transport equations

    International Nuclear Information System (INIS)

    Hoogenboom, J.E.

    1985-01-01

    A clarification is given of the difference between the equation adjoint to the Laplace-transformed time-dependent transport equation and the Laplace-transformed time-dependent adjoint transport equation. Proper procedures are derived to obtain the Laplace transform of the instantaneous detector response. (author)

  20. Comparison of two Ssub(infinity) methods for solving the neutron transport equation

    International Nuclear Information System (INIS)

    Mennig, J.; Brandt, D.; Haelg, W.

    1978-01-01

    A semianalytic method (S 0 sub(infinity)) is presented for solving the monoenergetic multi-region transport equation. This method is compared with results from S 1 sub(infinity)-theory given in the literature. Application of S 1 sub(infinity)-theory to reactor shields may lead to negative neutron fluxes and to flux oscillations. These unphysical effects are completely avoided by the new method. Numerical results demonstrate the limitations of S 1 sub(infinity) and confirm the numerical stability of (S 0 sub(infinity)). (Auth.)

  1. The 'generalized Balescu-Lenard' transport equations

    International Nuclear Information System (INIS)

    Mynick, H.E.

    1990-01-01

    The transport equations arising from the 'generalized Balescu-Lenard' collision operator are obtained and some of their properties examined. The equations contain neoclassical and turbulent transport as two special cases having the same structure. The resultant theory offers a possible explanation for a number of results not well understood, including the anomalous pinch, observed ratios of Q/ΓT on TFTR, and numerical reproduction of ASDEX profiles by a model for turbulent transport invoked without derivation, but by analogy with neoclassical theory. The general equations are specialized to consideration of a number of particular transport mechanisms of interest. (author). Letter-to-the-editor. 10 refs

  2. The gBL transport equations

    International Nuclear Information System (INIS)

    Mynick, H.E.

    1989-05-01

    The transport equations arising from the ''generalized Balescu- Lenard'' (gBL) collision operator are obtained, and some of their properties examined. The equations contain neoclassical and turbulent transport as two special cases, having the same structure. The resultant theory offers potential explanation for a number of results not well understood, including the anomalous pinch, observed ratios of Q/ΓT on TFTR, and numerical reproduction of ASDEX profiles by a model for turbulent transport invoked without derivation, but by analogy to neoclassical theory. The general equations are specialized to consideration of a number of particular transport mechanisms of interest. 10 refs

  3. On discontinuous Galerkin and discrete ordinates approximations for neutron transport equation and the critical eigenvalue

    International Nuclear Information System (INIS)

    Asadzadeh, M.; Thevenot, L.

    2010-01-01

    The objective of this paper is to give a mathematical framework for a fully discrete numerical approach for the study of the neutron transport equation in a cylindrical domain (container model,). More specifically, we consider the discontinuous Galerkin (D G) finite element method for spatial approximation of the mono-energetic, critical neutron transport equation in an infinite cylindrical domain ??in R3 with a polygonal convex cross-section ? The velocity discretization relies on a special quadrature rule developed to give optimal estimates in discrete ordinate parameters compatible with the quasi-uniform spatial mesh. We use interpolation spaces and derive optimal error estimates, up to maximal available regularity, for the fully discrete scalar flux. Finally we employ a duality argument and prove superconvergence estimates for the critical eigenvalue.

  4. Spherical harmonics solutions of multi-dimensional neutron transport equation by finite Fourier transformation

    International Nuclear Information System (INIS)

    Kobayashi, Keisuke

    1977-01-01

    A method of solution of a monoenergetic neutron transport equation in P sub(L) approximation is presented for x-y and x-y-z geometries using the finite Fourier transformation. A reactor system is assumed to consist of multiregions in each of which the nuclear cross sections are spatially constant. Since the unknown functions of this method are the spherical harmonics components of the neutron angular flux at the material boundaries alone, the three- and two-dimensional equations are reduced to two- and one-dimensional equations, respectively. The present approach therefore gives fewer unknowns than in the usual series expansion method or in the finite difference method. Some numerical examples are shown for the criticality problem. (auth.)

  5. Development of interfacial area transport equation

    International Nuclear Information System (INIS)

    Kim, Seung Jin; Ishii, Mamoru; Kelly, Joseph

    2005-01-01

    The interfacial area transport equation dynamically models the changes in interfacial structures along the flow field by mechanistically modeling the creation and destruction of dispersed phase. Hence, when employed in the numerical thermal-hydraulic system analysis codes, it eliminates artificial bifurcations stemming from the use of the static flow regime transition criteria. Accounting for the substantial differences in the transport mechanism for various sizes of bubbles, the transport equation is formulated for two characteristic groups of bubbles. The group 1 equation describes the transport of small-dispersed bubbles, whereas the group 2 equation describes the transport of large cap, slug or churn-turbulent bubbles. To evaluate the feasibility and reliability of interfacial area transport equation available at present, it is benchmarked by an extensive database established in various two-phase flow configurations spanning from bubbly to churn-turbulent flow regimes. The geometrical effect in interfacial area transport is examined by the data acquired in vertical air-water two-phase flow through round pipes of various sizes and a confined flow duct, and by those acquired in vertical co-current downward air-water two-phase flow through round pipes of two different sizes

  6. Simulation of transport equations with Monte Carlo

    International Nuclear Information System (INIS)

    Matthes, W.

    1975-09-01

    The main purpose of the report is to explain the relation between the transport equation and the Monte Carlo game used for its solution. The introduction of artificial particles carrying a weight provides one with high flexibility in constructing many different games for the solution of the same equation. This flexibility opens a way to construct a Monte Carlo game for the solution of the adjoint transport equation. Emphasis is laid mostly on giving a clear understanding of what to do and not on the details of how to do a specific game

  7. Swarm analysis by using transport equations

    International Nuclear Information System (INIS)

    Dote, Toshihiko.

    1985-01-01

    As the basis of weak ionization plasma phenomena, the motion, i.e. swarm, of charged particles in the gas is analyzed by use of the transport equations, from which basic nature of the swarm is discussed. The present report is an overview of the studies made in the past several years. Described are principally the most basic aspects concerning behaviors of the electrons and positive ions, that is, the basic equations and their significance, characteristics of the behaviors of the electron and positive ion swarms as revealed by solving the equations, and various characteristics of the swarm parameters. Contents are: Maxwell-Boltzmann's transport equations, behavior of the electron swarm, energy loss of the electrons, and behavior of the positive ion swarm. (Mori, K.)

  8. Adaptive integral equation methods in transport theory

    International Nuclear Information System (INIS)

    Kelley, C.T.

    1992-01-01

    In this paper, an adaptive multilevel algorithm for integral equations is described that has been developed with the Chandrasekhar H equation and its generalizations in mind. The algorithm maintains good performance when the Frechet derivative of the nonlinear map is singular at the solution, as happens in radiative transfer with conservative scattering and in critical neutron transport. Numerical examples that demonstrate the algorithm's effectiveness are presented

  9. Quadrature with arbitrary weight for the numerical solution of the critical slab Neutron Transport Equation

    International Nuclear Information System (INIS)

    Sanchez G, J.

    2007-01-01

    A standard procedure for the solution of singular integral equations is applied to the one-dimensional transport equation for monoenergetic neutrons. The results obtained with two versions of the procedure, differing only in the extent of the basic region to which they are applied, are compared with analytically derived results available for benchmarking. The procedures considered yield consistent results for the calculated neutron densities and eigenvalues. Several approximate expressions of the neutron density are used to render closed-form formulas for the densities which can then be analytically operated on to obtain expressions for extrapolation distances or angular densities or serve other purposes that require an analytical expression of the neutron density. (Author)

  10. General particle transport equation. Final report

    International Nuclear Information System (INIS)

    Lafi, A.Y.; Reyes, J.N. Jr.

    1994-12-01

    The general objectives of this research are as follows: (1) To develop fundamental models for fluid particle coalescence and breakage rates for incorporation into statistically based (Population Balance Approach or Monte Carlo Approach) two-phase thermal hydraulics codes. (2) To develop fundamental models for flow structure transitions based on stability theory and fluid particle interaction rates. This report details the derivation of the mass, momentum and energy conservation equations for a distribution of spherical, chemically non-reacting fluid particles of variable size and velocity. To study the effects of fluid particle interactions on interfacial transfer and flow structure requires detailed particulate flow conservation equations. The equations are derived using a particle continuity equation analogous to Boltzmann's transport equation. When coupled with the appropriate closure equations, the conservation equations can be used to model nonequilibrium, two-phase, dispersed, fluid flow behavior. Unlike the Eulerian volume and time averaged conservation equations, the statistically averaged conservation equations contain additional terms that take into account the change due to fluid particle interfacial acceleration and fluid particle dynamics. Two types of particle dynamics are considered; coalescence and breakage. Therefore, the rate of change due to particle dynamics will consider the gain and loss involved in these processes and implement phenomenological models for fluid particle breakage and coalescence

  11. Range of validity of transport equations

    International Nuclear Information System (INIS)

    Berges, Juergen; Borsanyi, Szabolcs

    2006-01-01

    Transport equations can be derived from quantum field theory assuming a loss of information about the details of the initial state and a gradient expansion. While the latter can be systematically improved, the assumption about a memory loss is not known to be controlled by a small expansion parameter. We determine the range of validity of transport equations for the example of a scalar g 2 Φ 4 theory. We solve the nonequilibrium time evolution using the three-loop 2PI effective action. The approximation includes off-shell and memory effects and assumes no gradient expansion. This is compared to transport equations to lowest order (LO) and beyond (NLO). We find that the earliest time for the validity of transport equations is set by the characteristic relaxation time scale t damp =-2ω/Σ ρ (eq) , where -Σ ρ (eq) /2 denotes the on-shell imaginary-part of the self-energy. This time scale agrees with the characteristic time for partial memory loss, but is much shorter than thermal equilibration times. For times larger than about t damp the gradient expansion to NLO is found to describe the full results rather well for g 2 (less-or-similar sign)1

  12. A method for solving neutron transport equation

    International Nuclear Information System (INIS)

    Dimitrijevic, Z.

    1993-01-01

    The procedure for solving the transport equation by directly integrating for case one-dimensional uniform multigroup medium is shown. The solution is expressed in terms of linear combination of function H n (x,μ), and the coefficient is determined from given conditions. The solution is applied for homogeneous slab of critical thickness. (author)

  13. Alternative formulation of the monokinetic transport equation

    International Nuclear Information System (INIS)

    Coppa, G.; Ravetto, P.; Sumini, M.

    1985-01-01

    After recalling a technique already exploited in stationary neutron transport, the dynamic linear monokinetic equation for general geometry is cast into an integro-differential form where a second order space Laplace operator and both a second and first time derivatives appear. The introduced unknowns are given a physical interpretation for plane geometry and their relations with the total flux and current are derived

  14. The transport equation in general geometry

    International Nuclear Information System (INIS)

    Pomraning, G.C.

    1990-01-01

    As stated in the introduction to the paper, the motivation for this work was to obtain an explicit form for the streaming operator in the transport equation, which could be used to compute curvature effects in an asymptotic analysis leading to diffusion theory. This sign error was discovered while performing this analysis

  15. Aspheric surface testing by irradiance transport equation

    Science.gov (United States)

    Shomali, Ramin; Darudi, Ahmad; Nasiri, Sadollah; Asgharsharghi Bonab, Armir

    2010-10-01

    In this paper a method for aspheric surface testing is presented. The method is based on solving the Irradiance Transport Equation (ITE).The accuracy of ITE normally depends on the amount of the pick to valley of the phase distribution. This subject is investigated by a simulation procedure.

  16. Swarm analysis by using transport equations, 1

    International Nuclear Information System (INIS)

    Dote, Toshihiko; Shimada, Masatoshi

    1980-01-01

    By evolving Maxwell-Boltzmann transport equations, various quantities on swarm of charged particles have been analyzed. Although this treatment is properly general, and common transport equations for charged particles ought to be given, in particular, equations only for electrons were presented here. The relation between the random energy and the drift energy was first derived and the general expression of the electron velocity was deduced too. For a simple example, one dimensional steady-state electron swarm in a uniform medium was treated. Electron swarm characteristics numerically calculated in He, Ne or Ar exhibited some interesting properties, which were physically clearly elucidated. These results were also compared with several data already published. Agreements between them were qualitatively rather well in detailed structures. (author)

  17. Diffusive limits for linear transport equations

    International Nuclear Information System (INIS)

    Pomraning, G.C.

    1992-01-01

    The authors show that the Hibert and Chapman-Enskog asymptotic treatments that reduce the nonlinear Boltzmann equation to the Euler and Navier-Stokes fluid equations have analogs in linear transport theory. In this linear setting, these fluid limits are described by diffusion equations, involving familiar and less familiar diffusion coefficients. Because of the linearity extant, one can carry out explicitly the initial and boundary layer analyses required to obtain asymptotically consistent initial and boundary conditions for the diffusion equations. In particular, the effects of boundary curvature and boundary condition variation along the surface can be included in the boundary layer analysis. A brief review of heuristic (nonasymptotic) diffusion description derivations is also included in our discussion

  18. Decay constants for pulsed monoenergetic neutron systems with quadratically anisotropic scattering

    International Nuclear Information System (INIS)

    Sjoestrand, N.G.

    1977-06-01

    The eigenvalues of the time-dependent transport equation for monoenergetic neutrons have been studied numerically for various combinations of linearly and quadratically anisotropic scattering assuming a space dependence of e β . The results, presented in the form of tables and graphs, show that quadratic anisotropy leads to a more complicated eigenvalue spectrum. However, no drastic changes occur in comparison to purely linear anistropy.(author)

  19. Moment equation approach to neoclassical transport theory

    International Nuclear Information System (INIS)

    Hirshman, S.P.

    1978-01-01

    The neoclassical cross-field fluxes for a toroidally confined, axisymmetric plasma are calculated in terms of the thermodynamic forces from the fluid continuity and momentum balance equations. This macroscopic formulation of neoclassical transport theory unifies the numerous complex expressions for the transport coefficients, previously obtained by solving the Fokker--Planck equation, and elucidates their physical basis. In the large aspect ratio limit, the continuous transition in the scaling of the diffusion coefficient throughout various collisionality regimes is shown to depend on the ratio of parallel viscosity coefficients of the plasma species. Comparison of the present results with the kinetic theory expressions for the neoclassical fluxes determines the parallel viscosity coefficients for a multispecies plasma in the long-mean-free-path regime

  20. Exact solution of the neutron transport equation in spherical geometry

    Energy Technology Data Exchange (ETDEWEB)

    Anli, Fikret; Akkurt, Abdullah; Yildirim, Hueseyin; Ates, Kemal [Kahramanmaras Suetcue Imam Univ. (Turkey). Faculty of Sciences and Letters

    2017-03-15

    Solution of the neutron transport equation in one dimensional slab geometry construct a basis for the solution of neutron transport equation in a curvilinear geometry. Therefore, in this work, we attempt to derive an exact analytical benchmark solution for both neutron transport equations in slab and spherical medium by using P{sub N} approximation which is widely used in neutron transport theory.

  1. A Green function of neutron transport equation

    International Nuclear Information System (INIS)

    Simovic, R.

    1993-01-01

    In this paper the angularly dependent Green function of the neutron transport equation is derived analytically and approximately. By applying the analytical FDPN approximation up to eighth order, numerical values of the Green functions are obtained with the accuracy of six significant figures in the whole range of parameter c, angle cosine μ and distances x up to the ten optical lengths from the neutron source. (author)

  2. Solving the equation of neutron transport

    International Nuclear Information System (INIS)

    Nasfi, Rim

    2009-01-01

    This work is devoted to the study of some numerical methods of resolution of the problem of transport of the neutrons. We started by introducing the equation integro-differential transport of the neutrons. Then we applied the finite element method traditional for stationary and nonstationary linear problems in 2D. A great part is reserved for the presentation of the mixed numerical diagram and mixed hybrid with two types of uniform grids: triangular and rectangular. Thereafter we treated some numerical examples by implementations in Matlab in order to test the convergence of each method. To finish, we had results of simulation by the Monte Carlo method on a problem of two-dimensional transport with an aim of comparing them with the results resulting from the finite element method mixed hybrids. Some remarks and prospects conclude this work.

  3. Generation of monoenergetic positrons

    International Nuclear Information System (INIS)

    Hulett, L.D. Jr.; Dale, J.M.; Miller, P.D. Jr.; Moak, C.D.; Pendyala, S.; Triftshaeuser, W.; Howell, R.H.; Alvarez, R.A.

    1983-01-01

    Many experiments have been performed in the generation and application of monoenergetic positron beams using annealed tungsten moderators and fast sources of 58 Co, 22 Na, 11 C, and LINAC bremstrahlung. This paper will compare the degrees of success from our various approaches. Moderators made from both single crystal and polycrystal tungsten have been tried. Efforts to grow thin films of tungsten to be used as transmission moderators and brightness enhancement devices are in progress

  4. Neutron transport equation - indications on homogenization and neutron diffusion

    International Nuclear Information System (INIS)

    Argaud, J.P.

    1992-06-01

    In PWR nuclear reactor, the practical study of the neutrons in the core uses diffusion equation to describe the problem. On the other hand, the most correct method to describe these neutrons is to use the Boltzmann equation, or neutron transport equation. In this paper, we give some theoretical indications to obtain a diffusion equation from the general transport equation, with some simplifying hypothesis. The work is organised as follows: (a) the most general formulations of the transport equation are presented: integro-differential equation and integral equation; (b) the theoretical approximation of this Boltzmann equation by a diffusion equation is introduced, by the way of asymptotic developments; (c) practical homogenization methods of transport equation is then presented. In particular, the relationships with some general and useful methods in neutronic are shown, and some homogenization methods in energy and space are indicated. A lot of other points of view or complements are detailed in the text or the remarks

  5. Maximal stochastic transport in the Lorenz equations

    Energy Technology Data Exchange (ETDEWEB)

    Agarwal, Sahil, E-mail: sahil.agarwal@yale.edu [Program in Applied Mathematics, Yale University, New Haven (United States); Wettlaufer, J.S., E-mail: john.wettlaufer@yale.edu [Program in Applied Mathematics, Yale University, New Haven (United States); Departments of Geology & Geophysics, Mathematics and Physics, Yale University, New Haven (United States); Mathematical Institute, University of Oxford, Oxford (United Kingdom); Nordita, Royal Institute of Technology and Stockholm University, Stockholm (Sweden)

    2016-01-08

    We calculate the stochastic upper bounds for the Lorenz equations using an extension of the background method. In analogy with Rayleigh–Bénard convection the upper bounds are for heat transport versus Rayleigh number. As might be expected, the stochastic upper bounds are larger than the deterministic counterpart of Souza and Doering [1], but their variation with noise amplitude exhibits interesting behavior. Below the transition to chaotic dynamics the upper bounds increase monotonically with noise amplitude. However, in the chaotic regime this monotonicity depends on the number of realizations in the ensemble; at a particular Rayleigh number the bound may increase or decrease with noise amplitude. The origin of this behavior is the coupling between the noise and unstable periodic orbits, the degree of which depends on the degree to which the ensemble represents the ergodic set. This is confirmed by examining the close returns plots of the full solutions to the stochastic equations and the numerical convergence of the noise correlations. The numerical convergence of both the ensemble and time averages of the noise correlations is sufficiently slow that it is the limiting aspect of the realization of these bounds. Finally, we note that the full solutions of the stochastic equations demonstrate that the effect of noise is equivalent to the effect of chaos.

  6. Numerical solution of the radionuclide transport equation

    International Nuclear Information System (INIS)

    Hadermann, J.; Roesel, F.

    1983-11-01

    A numerical solution of the one-dimensional geospheric radionuclide chain transport equation based on the pseudospectral method is developed. The advantages of this approach are flexibility in incorporating space and time dependent migration parameters, arbitrary boundary conditions and solute rock interactions as well as efficiency and reliability. As an application the authors investigate the impact of non-linear sorption isotherms on migration in crystalline rock. It is shown that non-linear sorption, in the present case a Freundlich isotherm, may reduce concentration at the geosphere outlet by orders of magnitude provided the migration time is comparable or larger than the half-life of the nuclide in question. The importance of fixing dispersivity within the continuum approach is stressed. (Auth.)

  7. Diffusion equation and spin drag in spin-polarized transport

    DEFF Research Database (Denmark)

    Flensberg, Karsten; Jensen, Thomas Stibius; Mortensen, Asger

    2001-01-01

    We study the role of electron-electron interactions for spin-polarized transport using the Boltzmann equation, and derive a set of coupled transport equations. For spin-polarized transport the electron-electron interactions are important, because they tend to equilibrate the momentum of the two-s...

  8. Fundaments of transport equation splitting and the eigenvalue problem

    International Nuclear Information System (INIS)

    Stancic, V.

    2000-01-01

    In order to remove some singularities concerning the boundary conditions of one dimensional transport equation, a split form of transport equation describing the forward i.e. μ≥0, and a backward μ<0 directed neutrons is being proposed here. The eigenvalue problem has also been considered here (author)

  9. Hot electrons in superlattices: quantum transport versus Boltzmann equation

    DEFF Research Database (Denmark)

    Wacker, Andreas; Jauho, Antti-Pekka; Rott, S.

    1999-01-01

    A self-consistent solution of the transport equation is presented for semiconductor superlattices within different approaches: (i) a full quantum transport model based on nonequilibrium Green functions, (ii) the semiclassical Boltzmann equation for electrons in a miniband, and (iii) Boltzmann...

  10. Symmetrized neutron transport equation and the fast Fourier transform method

    International Nuclear Information System (INIS)

    Sinh, N.Q.; Kisynski, J.; Mika, J.

    1978-01-01

    The differential equation obtained from the neutron transport equation by the application of the source iteration method in two-dimensional rectangular geometry is transformed into a symmetrized form with respect to one of the angular variables. The discretization of the symmetrized equation leads to finite difference equations based on the five-point scheme and solved by use of the fast Fourier transform method. Possible advantages of the approach are shown on test calculations

  11. An integral equation arising in two group neutron transport theory

    International Nuclear Information System (INIS)

    Cassell, J S; Williams, M M R

    2003-01-01

    An integral equation describing the fuel distribution necessary to maintain a flat flux in a nuclear reactor in two group transport theory is reduced to the solution of a singular integral equation. The formalism developed enables the physical aspects of the problem to be better understood and its relationship with the corresponding diffusion theory model is highlighted. The integral equation is solved by reducing it to a non-singular Fredholm equation which is then evaluated numerically

  12. Exact analytical solution of time-independent neutron transport equation, and its applications to systems with a point source

    International Nuclear Information System (INIS)

    Mikata, Y.

    2014-01-01

    Highlights: • An exact solution for the one-speed neutron transport equation is obtained. • This solution as well as its derivation are believed to be new. • Neutron flux for a purely absorbing material with a point neutron source off the origin is obtained. • Spherically as well as cylindrically piecewise constant cross sections are studied. • Neutron flux expressions for a point neutron source off the origin are believed to be new. - Abstract: An exact analytical solution of the time-independent monoenergetic neutron transport equation is obtained in this paper. The solution is applied to systems with a point source. Systematic analysis of the solution of the time-independent neutron transport equation, and its applications represent the primary goal of this paper. To the best of the author’s knowledge, certain key results on the scalar neutron flux as well as their derivations are new. As an application of these results, a scalar neutron flux for a purely absorbing medium with a spherically piecewise constant cross section and an isotropic point neutron source off the origin as well as that for a cylindrically piecewise constant cross section with a point neutron source off the origin are obtained. Both of these results are believed to be new

  13. Discontinuous Galerkin for the Radiative Transport Equation

    KAUST Repository

    Guermond, Jean-Luc; Kanschat, Guido; Ragusa, Jean C.

    2013-01-01

    This note presents some recent results regarding the approximation of the linear radiative transfer equation using discontinuous Galerkin methods. The locking effect occurring in the diffusion limit with the upwind numerical flux is investigated and a correction technique is proposed.

  14. Discontinuous Galerkin for the Radiative Transport Equation

    KAUST Repository

    Guermond, Jean-Luc

    2013-10-11

    This note presents some recent results regarding the approximation of the linear radiative transfer equation using discontinuous Galerkin methods. The locking effect occurring in the diffusion limit with the upwind numerical flux is investigated and a correction technique is proposed.

  15. Dynamic modeling of interfacial structures via interfacial area transport equation

    International Nuclear Information System (INIS)

    Seungjin, Kim; Mamoru, Ishii

    2005-01-01

    The interfacial area transport equation dynamically models the two-phase flow regime transitions and predicts continuous change of the interfacial area concentration along the flow field. Hence, when employed in the numerical thermal-hydraulic system analysis codes, it eliminates artificial bifurcations stemming from the use of the static flow regime transition criteria. Accounting for the substantial differences in the transport phenomena of various sizes of bubbles, the two-group interfacial area transport equations have been developed. The group 1 equation describes the transport of small-dispersed bubbles that are either distorted or spherical in shapes, and the group 2 equation describes the transport of large cap, slug or churn-turbulent bubbles. The source and sink terms in the right-hand-side of the transport equations have been established by mechanistically modeling the creation and destruction of bubbles due to major bubble interaction mechanisms. In the present paper, the interfacial area transport equations currently available are reviewed to address the feasibility and reliability of the model along with extensive experimental results. These include the data from adiabatic upward air-water two-phase flow in round tubes of various sizes, from a rectangular duct, and from adiabatic co-current downward air-water two-phase flow in round pipes of two sizes. (authors)

  16. A modular spherical harmonics approach to the neutron transport equation

    International Nuclear Information System (INIS)

    Inanc, F.; Rohach, A.F.

    1989-01-01

    A modular nodal method was developed for solving the neutron transport equation in 2-D xy coordinates. The spherical harmonic expansion was used for approximating the second-order even-parity form of the neutron transport equation. The boundary conditions of the spherical harmonics approximation were derived in a form to have forms analogous to the partial currents in the neutron diffusion equation. Relations were developed for generating both the second-order spherical harmonic equations and the boundary conditions in an automated computational algorithm. Nodes using different orders of the spherical harmonics approximation to the transport equation were interfaced through mixed-type boundary conditions. The determination of spherical harmonic orders implemented in the nodes were determined by the scheme in an automated manner. Results of the method compared favorably to benchmark problems. (author)

  17. Relativistic transport equation for a discontinuity wave of multiplicity one

    Energy Technology Data Exchange (ETDEWEB)

    Giambo, S; Palumbo, A [Istituto di Matematica, Universita degli Studi, Messina (Italy)

    1980-04-14

    In the framework of the theory of the singular hypersurfaces, the transport equation for the amplitude of a discontinuity wave, corresponding to a simple characteristic of a quasi-linear hyperbolic system, is established in the context of special relativity.

  18. The accuracy of time dependent transport equation ergodic approximation

    International Nuclear Information System (INIS)

    Stancic, V.

    1995-01-01

    In order to predict the accuracy of the ergodic approximation for solving the time dependent transport equation, a comparison with respect to multiple collision and time finite difference methods, has been considered. (author)

  19. Analytical solution to the hybrid diffusion-transport equation

    International Nuclear Information System (INIS)

    Nanneh, M.M.; Williams, M.M.R.

    1986-01-01

    A special integral equation was derived in previous work using a hybrid diffusion-transport theory method for calculating the flux distribution in slab lattices. In this paper an analytical solution of this equation has been carried out on a finite reactor lattice. The analytical results of disadvantage factors are shown to be accurate in comparison with the numerical results and accurate transport theory calculations. (author)

  20. Unconditionally stable diffusion-acceleration of the transport equation

    International Nuclear Information System (INIS)

    Larson, E.W.

    1982-01-01

    The standard iterative procedure for solving fixed-source discrete-ordinates problems converges very slowly for problems in optically thick regions with scattering ratios c near unity. The diffusion-synthetic acceleration method has been proposed to make use of the fact that for this class of problems, the diffusion equation is often an accurate approximation to the transport equation. However, stability difficulties have historically hampered the implementation of this method for general transport differencing schemes. In this article we discuss a recently developed procedure for obtaining unconditionally stable diffusion-synthetic acceleration methods for various transport differencing schemes. We motivate the analysis by first discussing the exact transport equation; then we illustrate the procedure by deriving a new stable acceleration method for the linear discontinuous transport differencing scheme. We also provide some numerical results

  1. Unconditionally stable diffusion-acceleration of the transport equation

    International Nuclear Information System (INIS)

    Larsen, E.W.

    1982-01-01

    The standard iterative procedure for solving fixed-source discrete-ordinates problems converges very slowly for problems in optically large regions with scattering ratios c near unity. The diffusion-synthetic acceleration method has been proposed to make use of the fact that for this class of problems the diffusion equation is often an accurate approximation to the transport equation. However, stability difficulties have historically hampered the implementation of this method for general transport differencing schemes. In this article we discuss a recently developed procedure for obtaining unconditionally stable diffusion-synthetic acceleration methods for various transport differencing schemes. We motivate the analysis by first discussing the exact transport equation; then we illustrate the procedure by deriving a new stable acceleration method for the linear discontinuous transport differencing scheme. We also provide some numerical results

  2. Deferred correction approach on generic transport equation

    International Nuclear Information System (INIS)

    Shah, I.A.; Ali, M.

    2004-01-01

    In this study, a two dimensional Steady Convection-Diffusion was solved, using Deferred correction approach, and results were compared with standard spatial discretization schemes. Numerical investigations were carried out based on the velocity and flow direction, for various diffusivity coefficients covering a range from diffusive to convective flows. The results show that the Deferred Ted Correction Approach gives more accurate and stable results in relation to UDS and CDs discretization of convective terms. Deferred Correction Approach caters for the wiggles for convective flows in case of central difference discretization of the equation and also caters for the dissipative error generated by the first order upwind discretization of convective fluxes. (author)

  3. Dynamic modeling of interfacial structures via interfacial area transport equation

    International Nuclear Information System (INIS)

    Seungjin, Kim; Mamoru, Ishii

    2004-01-01

    Full text of publication follows:In the current thermal-hydraulic system analysis codes using the two-fluid model, the empirical correlations that are based on the two-phase flow regimes and regime transition criteria are being employed as closure relations for the interfacial transfer terms. Due to its inherent shortcomings, however, such static correlations are inaccurate and present serious problems in the numerical analysis. In view of this, a new dynamic approach employing the interfacial area transport equation has been studied. The interfacial area transport equation dynamically models the two-phase flow regime transitions and predicts continuous change of the interfacial area concentration along the flow field. Hence, when employed in the thermal-hydraulic system analysis codes, it eliminates artificial bifurcations stemming from the use of the static flow regime transition criteria. Therefore, the interfacial area transport equation can make a leapfrog improvement in the current capability of the two-fluid model from both scientific and practical point of view. Accounting for the substantial differences in the transport phenomena of various sizes of bubbles, the two-group interfacial area transport equations have been developed. The group 1 equation describes the transport of small-dispersed bubbles that are either distorted or spherical in shapes, and the group 2 equation describes the transport of large cap, slug or churn-turbulent bubbles. The source and sink terms in the right hand-side of the transport equations have been established by mechanistically modeling the creation and destruction of bubbles due to major bubble interaction mechanisms. The coalescence mechanisms include the random collision driven by turbulence, and the entrainment of trailing bubbles in the wake region of the preceding bubble. The disintegration mechanisms include the break-up by turbulence impact, shearing-off at the rim of large cap bubbles and the break-up of large cap

  4. Generalized heat-transport equations: parabolic and hyperbolic models

    Science.gov (United States)

    Rogolino, Patrizia; Kovács, Robert; Ván, Peter; Cimmelli, Vito Antonio

    2018-03-01

    We derive two different generalized heat-transport equations: the most general one, of the first order in time and second order in space, encompasses some well-known heat equations and describes the hyperbolic regime in the absence of nonlocal effects. Another, less general, of the second order in time and fourth order in space, is able to describe hyperbolic heat conduction also in the presence of nonlocal effects. We investigate the thermodynamic compatibility of both models by applying some generalizations of the classical Liu and Coleman-Noll procedures. In both cases, constitutive equations for the entropy and for the entropy flux are obtained. For the second model, we consider a heat-transport equation which includes nonlocal terms and study the resulting set of balance laws, proving that the corresponding thermal perturbations propagate with finite speed.

  5. Homogenization of the critically spectral equation in neutron transport

    Energy Technology Data Exchange (ETDEWEB)

    Allaire, G. [CEA Saclay, 91 - Gif-sur-Yvette (France). Dept. de Mecanique et de Technologie]|[Paris-6 Univ., 75 (France). Lab. d' Analyse Numerique; Bal, G. [Electricite de France (EDF), 92 - Clamart (France). Direction des Etudes et Recherches

    1998-07-01

    We address the homogenization of an eigenvalue problem for the neutron transport equation in a periodic heterogeneous domain, modeling the criticality study of nuclear reactor cores. We prove that the neutron flux, corresponding to the first and unique positive eigenvector, can be factorized in the product of two terms, up to a remainder which goes strongly to zero with the period. On terms is the first eigenvector of the transport equation in the periodicity cell. The other term is the first eigenvector of a diffusion equation in the homogenized domain. Furthermore, the corresponding eigenvalue gives a second order corrector for the eigenvalue of the heterogeneous transport problem. This result justifies and improves the engineering procedure used in practice for nuclear reactor cores computations. (author)

  6. Homogenization of the critically spectral equation in neutron transport

    International Nuclear Information System (INIS)

    Allaire, G.; Paris-6 Univ., 75; Bal, G.

    1998-01-01

    We address the homogenization of an eigenvalue problem for the neutron transport equation in a periodic heterogeneous domain, modeling the criticality study of nuclear reactor cores. We prove that the neutron flux, corresponding to the first and unique positive eigenvector, can be factorized in the product of two terms, up to a remainder which goes strongly to zero with the period. On terms is the first eigenvector of the transport equation in the periodicity cell. The other term is the first eigenvector of a diffusion equation in the homogenized domain. Furthermore, the corresponding eigenvalue gives a second order corrector for the eigenvalue of the heterogeneous transport problem. This result justifies and improves the engineering procedure used in practice for nuclear reactor cores computations. (author)

  7. Stable solutions of nonlocal electron heat transport equations

    International Nuclear Information System (INIS)

    Prasad, M.K.; Kershaw, D.S.

    1991-01-01

    Electron heat transport equations with a nonlocal heat flux are in general ill-posed and intrinsically unstable, as proved by the present authors [Phys. Fluids B 1, 2430 (1989)]. A straightforward numerical solution of these equations will therefore lead to absurd results. It is shown here that by imposing a minimal set of constraints on the problem it is possible to arrive at a globally stable, consistent, and energy conserving numerical solution

  8. Solution of the transport equation with account for inelastic collisions

    International Nuclear Information System (INIS)

    Kalashnikov, N.P.; Remizovich, V.S.; Ryazanov, M.I.

    1980-01-01

    The theory of charged particle scattering in a matter with account for inelastic collisions is considered. In ''directly-forward'' approximation the transport equation at the absence of elastic collisions is obtained. The solution of the transport equation is made without and with account for fluctuation of energy losses. Formulas for path-energy relation are given. Energy spectrum and distribution of fast charged particles with respect to paths are studied. The problem of quantum mechanical approach to the theory of multiple scattering of fast charged particles in a matter is discussed briefly

  9. Quantum Non-Markovian Langevin Equations and Transport Coefficients

    International Nuclear Information System (INIS)

    Sargsyan, V.V.; Antonenko, N.V.; Kanokov, Z.; Adamian, G.G.

    2005-01-01

    Quantum diffusion equations featuring explicitly time-dependent transport coefficients are derived from generalized non-Markovian Langevin equations. Generalized fluctuation-dissipation relations and analytic expressions for calculating the friction and diffusion coefficients in nuclear processes are obtained. The asymptotic behavior of the transport coefficients and correlation functions for a damped harmonic oscillator that is linearly coupled in momentum to a heat bath is studied. The coupling to a heat bath in momentum is responsible for the appearance of the diffusion coefficient in coordinate. The problem of regression of correlations in quantum dissipative systems is analyzed

  10. Two-scale approach to oscillatory singularly perturbed transport equations

    CERN Document Server

    Frénod, Emmanuel

    2017-01-01

    This book presents the classical results of the two-scale convergence theory and explains – using several figures – why it works. It then shows how to use this theory to homogenize ordinary differential equations with oscillating coefficients as well as oscillatory singularly perturbed ordinary differential equations. In addition, it explores the homogenization of hyperbolic partial differential equations with oscillating coefficients and linear oscillatory singularly perturbed hyperbolic partial differential equations. Further, it introduces readers to the two-scale numerical methods that can be built from the previous approaches to solve oscillatory singularly perturbed transport equations (ODE and hyperbolic PDE) and demonstrates how they can be used efficiently. This book appeals to master’s and PhD students interested in homogenization and numerics, as well as to the Iter community.

  11. Exactly averaged equations for flow and transport in random media

    International Nuclear Information System (INIS)

    Shvidler, Mark; Karasaki, Kenzi

    2001-01-01

    It is well known that exact averaging of the equations of flow and transport in random porous media can be realized only for a small number of special, occasionally exotic, fields. On the other hand, the properties of approximate averaging methods are not yet fully understood. For example, the convergence behavior and the accuracy of truncated perturbation series. Furthermore, the calculation of the high-order perturbations is very complicated. These problems for a long time have stimulated attempts to find the answer for the question: Are there in existence some exact general and sufficiently universal forms of averaged equations? If the answer is positive, there arises the problem of the construction of these equations and analyzing them. There exist many publications related to these problems and oriented on different applications: hydrodynamics, flow and transport in porous media, theory of elasticity, acoustic and electromagnetic waves in random fields, etc. We present a method of finding the general form of exactly averaged equations for flow and transport in random fields by using (1) an assumption of the existence of Green's functions for appropriate stochastic problems, (2) some general properties of the Green's functions, and (3) the some basic information about the random fields of the conductivity, porosity and flow velocity. We present a general form of the exactly averaged non-local equations for the following cases. 1. Steady-state flow with sources in porous media with random conductivity. 2. Transient flow with sources in compressible media with random conductivity and porosity. 3. Non-reactive solute transport in random porous media. We discuss the problem of uniqueness and the properties of the non-local averaged equations, for the cases with some types of symmetry (isotropic, transversal isotropic, orthotropic) and we analyze the hypothesis of the structure non-local equations in general case of stochastically homogeneous fields. (author)

  12. An Implementation of Interfacial Transport Equation into the CUPID code

    Energy Technology Data Exchange (ETDEWEB)

    Park, Ik Kyu; Cho, Heong Kyu; Yoon, Han Young; Jeong, Jae Jun

    2009-11-15

    A component scale thermal hydraulic analysis code, CUPID (Component Unstructured Program for Interfacial Dynamics), is being developed for the analysis of components for a nuclear reactor, such as reactor vessel, steam generator, containment, etc. It adopted a three-dimensional, transient, two phase and three-field model. In order to develop the numerical schemes for the three-field model, various numerical schemes have been examined including the SMAS, semi-implicit ICE, SIMPLE. The governing equations for a 2-phase flow are composed of mass, momentum, and energy conservation equations for each phase. These equation sets are closed by the interfacial transfer rate of mass, momentum, and energy. The interfacial transfer of mass, momentum, and energy occurs through the interfacial area, and this area plays an important role in the transfer rate. The flow regime based correlations are used for calculating the interracial area in the traditional style 2-phase flow model. This is dependent upon the flow regime and is limited to the fully developed 2-phase flow region. Its application to the multi-dimensional 2-phase flow has some limitation because it adopts the measured results of 2-phase flow in the 1-dimensional tube. The interfacial area concentration transport equation had been suggested in order to calculate the interfacial area without the interfacial area correlations. The source terms to close the interfacial area transport equation should be further developed for a wide ranger usage of it. In this study, the one group interfacial area concentration transport equation has been implemented into the CUPID code. This interfacial area concentration transport equation can be used instead of the interfacial area concentration correlations for the bubbly flow region.

  13. An Implementation of Interfacial Transport Equation into the CUPID code

    International Nuclear Information System (INIS)

    Park, Ik Kyu; Cho, Heong Kyu; Yoon, Han Young; Jeong, Jae Jun

    2009-11-01

    A component scale thermal hydraulic analysis code, CUPID (Component Unstructured Program for Interfacial Dynamics), is being developed for the analysis of components for a nuclear reactor, such as reactor vessel, steam generator, containment, etc. It adopted a three-dimensional, transient, two phase and three-field model. In order to develop the numerical schemes for the three-field model, various numerical schemes have been examined including the SMAS, semi-implicit ICE, SIMPLE. The governing equations for a 2-phase flow are composed of mass, momentum, and energy conservation equations for each phase. These equation sets are closed by the interfacial transfer rate of mass, momentum, and energy. The interfacial transfer of mass, momentum, and energy occurs through the interfacial area, and this area plays an important role in the transfer rate. The flow regime based correlations are used for calculating the interracial area in the traditional style 2-phase flow model. This is dependent upon the flow regime and is limited to the fully developed 2-phase flow region. Its application to the multi-dimensional 2-phase flow has some limitation because it adopts the measured results of 2-phase flow in the 1-dimensional tube. The interfacial area concentration transport equation had been suggested in order to calculate the interfacial area without the interfacial area correlations. The source terms to close the interfacial area transport equation should be further developed for a wide ranger usage of it. In this study, the one group interfacial area concentration transport equation has been implemented into the CUPID code. This interfacial area concentration transport equation can be used instead of the interfacial area concentration correlations for the bubbly flow region

  14. Analysis of discrete reaction-diffusion equations for autocatalysis and continuum diffusion equations for transport

    Energy Technology Data Exchange (ETDEWEB)

    Wang, Chi-Jen [Iowa State Univ., Ames, IA (United States)

    2013-01-01

    In this thesis, we analyze both the spatiotemporal behavior of: (A) non-linear “reaction” models utilizing (discrete) reaction-diffusion equations; and (B) spatial transport problems on surfaces and in nanopores utilizing the relevant (continuum) diffusion or Fokker-Planck equations. Thus, there are some common themes in these studies, as they all involve partial differential equations or their discrete analogues which incorporate a description of diffusion-type processes. However, there are also some qualitative differences, as shall be discussed below.

  15. Correction of the wavefront using the irradiance transport equation

    Science.gov (United States)

    García, M.; Granados, F.; Cornejo, A.

    2008-07-01

    The correction of the wavefront in optical systems implies the use of wavefront sensors, software, and auxiliary optical systems. We propose evaluated the wavefront using the fact that the wavefront and its intensity are related in the mathematical expression the irradiance transport equation (ITE)

  16. From statistic mechanic outside equilibrium to transport equations

    International Nuclear Information System (INIS)

    Balian, R.

    1995-01-01

    This lecture notes give a synthetic view on the foundations of non-equilibrium statistical mechanics. The purpose is to establish the transport equations satisfied by the relevant variables, starting from the microscopic dynamics. The Liouville representation is introduced, and a projection associates with any density operator , for given choice of relevant observables, a reduced density operator. An exact integral-differential equation for the relevant variables is thereby derived. A short-memory approximation then yields the transport equations. A relevant entropy which characterizes the coarseness of the description is associated with each level of description. As an illustration, the classical gas, with its three levels of description and with the Chapman-Enskog method, is discussed. (author). 3 figs., 5 refs

  17. A stochastic solution of the advective transport equation with uncertainty

    International Nuclear Information System (INIS)

    Williams, M.M.R.

    1991-01-01

    A model has been developed for calculating the transport of water-borne radionuclides through layers of porous materials, such as rock or clay. The model is based upon a purely advective transport equation, in which the fluid velocity is a random variable, thereby simulating dispersion in a more realistic manner than the ad hoc introduction of a dispersivity. In addition to a random velocity field, which is an observable physical phenomenon, allowance is made for uncertainty in our knowledge of the parameters which enter the equation, e.g. the retardation coefficient. This too, is assumed to be a random variable and contributes to the stochasticity of the resulting partial differential equation of transport. The stochastic differential equation can be solved analytically and then ensemble averages taken over the associated probability distribution of velocity and retardation coefficient. A method based upon a novel form of the central limit theorem of statistics is employed to obtain tractable solutions of a system consisting of many serial legs of varying properties. One interesting conclusion is that the total flux out of a medium is significantly underestimated by using the deterministic solution with an average transit time compared with that from the stochastically averaged solution. The theory is illustrated numerically for a number of physically relevant cases. (author) 8 figs., 4 tabs., 7 refs

  18. Recent advances in the spectral green's function method for monoenergetic slab-geometry fixed-source adjoint transport problems in S{sub N} formulation

    Energy Technology Data Exchange (ETDEWEB)

    Curbelo, Jesus P.; Alves Filho, Hermes; Barros, Ricardo C., E-mail: jperez@iprj.uerj.br, E-mail: halves@iprj.uerj.br, E-mail: rcbarros@pq.cnpq.br [Universidade do Estado do Rio de Janeiro (UERJ), Nova Friburgo, RJ (Brazil). Instituto Politecnico. Programa de Pos-Graduacao em Modelagem Computacional; Hernandez, Carlos R.G., E-mail: cgh@instec.cu [Instituto Superior de Tecnologias y Ciencias Aplicadas (InSTEC), La Habana (Cuba)

    2015-07-01

    The spectral Green's function (SGF) method is a numerical method that is free of spatial truncation errors for slab-geometry fixed-source discrete ordinates (S{sub N}) adjoint problems. The method is based on the standard spatially discretized adjoint S{sub N} balance equations and a nonstandard adjoint auxiliary equation expressing the node-average adjoint angular flux, in each discretization node, as a weighted combination of the node-edge outgoing adjoint fluxes. The auxiliary equation contains parameters which act as Green's functions for the cell-average adjoint angular flux. These parameters are determined by means of a spectral analysis which yields the local general solution of the S{sub N} equations within each node of the discretization grid. In this work a number of advances in the SGF adjoint method are presented: the method is extended to adjoint S{sub N} problems considering linearly anisotropic scattering and non-zero prescribed boundary conditions for the forward source-detector problem. Numerical results to typical model problems are considered to illustrate the efficiency and accuracy of the o offered method. (author)

  19. 1D equation for toroidal momentum transport in a tokamak

    International Nuclear Information System (INIS)

    Rozhansky, V A; Senichenkov, I Yu

    2010-01-01

    A 1D equation for toroidal momentum transport is derived for a given set of turbulent transport coefficients. The averaging is performed taking account of the poloidal variation of the toroidal fluxes and is based on the ambipolar condition of the zero net radial current through the flux surface. It is demonstrated that taking account of the Pfirsch-Schlueter fluxes leads to a torque in the toroidal direction which is proportional to the gradient of the ion temperature. This effect is new and has not been discussed before. The boundary condition at the separatrix, which is based on the results of the 2D simulations of the edge plasma, is formulated.

  20. Quality factors for monoenergetic neutrons

    International Nuclear Information System (INIS)

    Cross, W.G.; Ing, H.

    1984-01-01

    Mean quality factors anti Q(E/sub n/), for the dose resulting from first interactons of monoenergetic neutrons in tissue, have been calculated at energies from thermal to 14.7 MeV. Knowledge of these values, along with kerma factors, allows the calculation of the heavy-particle dose equivalent for any known neutron spectrum in tissue. The partial quality factors for the dose delivered by C, N, and O elastic and inelastic recoils are found to be virtually independent of the scattering angular distributions and are given by simple expressions

  1. Sn approach applied to the solution of transport equation

    International Nuclear Information System (INIS)

    Lopes, J.P.

    1973-09-01

    In this work the origin of the Transport Theory is considered and the Transport Equation for the movement of the neutron in a system is established in its more general form, using the laws of nuclear physics. This equation is used as the starting point for development, under adequate assumptions, of simpler models that render the problem suitable for numerical solution. Representation of this model in different geometries is presented. The different processes of nuclear physics are introduced briefly and discussed. In addition, the boundary conditions for the different cases and a general procedure for the application of the Conservation Law are stated. The last chapter deals specifically with the S n method, its development, definitions and generalities. Computational schemes for obtaining the S n solution in spherical and cylindrical geometry, and convergence acceleration methods are also developed. (author)

  2. Transport equations in an enzymatic glucose fuel cell

    Science.gov (United States)

    Jariwala, Soham; Krishnamurthy, Balaji

    2018-01-01

    A mathematical model is developed to study the effects of convective flux and operating temperature on the performance of an enzymatic glucose fuel cell with a membrane. The model assumes isothermal operating conditions and constant feed rate of glucose. The glucose fuel cell domain is divided into five sections, with governing equations describing transport characteristics in each region, namely - anode diffusion layer, anode catalyst layer (enzyme layer), membrane, cathode catalyst layer and cathode diffusion layer. The mass transport is assumed to be one-dimensional and the governing equations are solved numerically. The effects flow rate of glucose feed on the performance of the fuel cell are studied as it contributes significantly to the convective flux. The effects of operating temperature on the performance of a glucose fuel cell are also modeled. The cell performances are compared using cell polarization curves, which were found compliant with experimental observations.

  3. Finite element approximation to the even-parity transport equation

    International Nuclear Information System (INIS)

    Lewis, E.E.

    1981-01-01

    This paper studies the finite element method, a procedure for reducing partial differential equations to sets of algebraic equations suitable for solution on a digital computer. The differential equation is cast into the form of a variational principle, the resulting domain then subdivided into finite elements. The dependent variable is then approximated by a simple polynomial, and these are linked across inter-element boundaries by continuity conditions. The finite element method is tailored to a variety of transport problems. Angular approximations are formulated, and the extent of ray effect mitigation is examined. Complex trial functions are introduced to enable the inclusion of buckling approximations. The ubiquitous curved interfaces of cell calculations, and coarse mesh methods are also treated. A concluding section discusses limitations of the work to date and suggests possible future directions

  4. Numerical Integration of the Transport Equation For Infinite Homogeneous Media

    Energy Technology Data Exchange (ETDEWEB)

    Haakansson, Rune

    1962-01-15

    The transport equation for neutrons in infinite homogeneous media is solved by direct numerical integration. Accounts are taken to the anisotropy and the inelastic scattering. The integration has been performed by means of the trapezoidal rule and the length of the energy intervals are constant in lethargy scale. The machine used is a Ferranti Mercury computer. Results are given for water, heavy water, aluminium water mixture and iron-aluminium-water mixture.

  5. New numerical method for solving the solute transport equation

    International Nuclear Information System (INIS)

    Ross, B.; Koplik, C.M.

    1978-01-01

    The solute transport equation can be solved numerically by approximating the water flow field by a network of stream tubes and using a Green's function solution within each stream tube. Compared to previous methods, this approach permits greater computational efficiency and easier representation of small discontinuities, and the results are easier to interpret physically. The method has been used to study hypothetical sites for disposal of high-level radioactive waste

  6. Deterministic methods to solve the integral transport equation in neutronic

    International Nuclear Information System (INIS)

    Warin, X.

    1993-11-01

    We present a synthesis of the methods used to solve the integral transport equation in neutronic. This formulation is above all used to compute solutions in 2D in heterogeneous assemblies. Three kinds of methods are described: - the collision probability method; - the interface current method; - the current coupling collision probability method. These methods don't seem to be the most effective in 3D. (author). 9 figs

  7. Comparison of neutronic transport equation resolution nodal methods

    International Nuclear Information System (INIS)

    Zamonsky, O.M.; Gho, C.J.

    1990-01-01

    In this work, some transport equation resolution nodal methods are comparatively studied: the constant-constant (CC), linear-nodal (LN) and the constant-quadratic (CQ). A nodal scheme equivalent to finite differences has been used for its programming, permitting its inclusion in existing codes. Some bidimensional problems have been solved, showing that linear-nodal (LN) are, in general, obtained with accuracy in CPU shorter times. (Author) [es

  8. Numerical solution of the transport equation describing the radon transport from subsurface soil to buildings

    International Nuclear Information System (INIS)

    Savovic, S.; Djordjevich, A.; Ristic, G.

    2012-01-01

    A theoretical evaluation of the properties and processes affecting the radon transport from subsurface soil into buildings is presented in this work. The solution of the relevant transport equation is obtained using the explicit finite difference method (EFDM). Results are compared with analytical steady-state solution reported in the literature. Good agreement is found. It is shown that EFDM is effective and accurate for solving the equation that describes radon diffusion, advection and decay during its transport from subsurface to buildings, which is especially important when arbitrary initial and boundary conditions are required. (authors)

  9. On an analytical representation of the solution of the one-dimensional transport equation for a multi-group model in planar geometry

    Energy Technology Data Exchange (ETDEWEB)

    Fernandes, Julio C.L.; Vilhena, Marco T.; Bodmann, Bardo E.J., E-mail: julio.lombaldo@ufrgs.br, E-mail: mtmbvilhena@gmail.com, E-mail: bardo.bodmann@ufrgs.br [Universidade Federal do Rio Grande do Sul (UFRGS), Porto Alegre, RS (Brazil). Dept. de Matematica Pura e Aplicada; Dulla, Sandra; Ravetto, Piero, E-mail: sandra.dulla@polito.it, E-mail: piero.ravetto@polito.it [Dipartimento di Energia, Politecnico di Torino, Piemonte (Italy)

    2015-07-01

    In this work we generalize the solution of the one-dimensional neutron transport equation to a multi- group approach in planar geometry. The basic idea of this work consists in consider the hierarchical construction of a solution for a generic number G of energy groups, starting from a mono-energetic solution. The hierarchical method follows the reasoning of the decomposition method. More specifically, the additional terms from adding energy groups is incorporated into the recursive scheme as source terms. This procedure leads to an analytical representation for the solution with G energy groups. The recursion depth is related to the accuracy of the solution, that may be evaluated after each recursion step. The authors present a heuristic analysis of stability for the results. Numerical simulations for a specific example with four energy groups and a localized pulsed source. (author)

  10. Transport parameter estimation from lymph measurements and the Patlak equation.

    Science.gov (United States)

    Watson, P D; Wolf, M B

    1992-01-01

    Two methods of estimating protein transport parameters for plasma-to-lymph transport data are presented. Both use IBM-compatible computers to obtain least-squares parameters for the solvent drag reflection coefficient and the permeability-surface area product using the Patlak equation. A matrix search approach is described, and the speed and convenience of this are compared with a commercially available gradient method. The results from both of these methods were different from those of a method reported by Reed, Townsley, and Taylor [Am. J. Physiol. 257 (Heart Circ. Physiol. 26): H1037-H1041, 1989]. It is shown that the Reed et al. method contains a systematic error. It is also shown that diffusion always plays an important role for transmembrane transport at the exit end of a membrane channel under all conditions of lymph flow rate and that the statement that diffusion becomes zero at high lymph flow rate depends on a mathematical definition of diffusion.

  11. Quantum-mechanical transport equation for atomic systems.

    Science.gov (United States)

    Berman, P. R.

    1972-01-01

    A quantum-mechanical transport equation (QMTE) is derived which should be applicable to a wide range of problems involving the interaction of radiation with atoms or molecules which are also subject to collisions with perturber atoms. The equation follows the time evolution of the macroscopic atomic density matrix elements of atoms located at classical position R and moving with classical velocity v. It is quantum mechanical in the sense that all collision kernels or rates which appear have been obtained from a quantum-mechanical theory and, as such, properly take into account the energy-level variations and velocity changes of the active (emitting or absorbing) atom produced in collisions with perturber atoms. The present formulation is better suited to problems involving high-intensity external fields, such as those encountered in laser physics.

  12. Approximate solution to neutron transport equation with linear anisotropic scattering

    International Nuclear Information System (INIS)

    Coppa, G.; Ravetto, P.; Sumini, M.

    1983-01-01

    A method to obtain an approximate solution to the transport equation, when both sources and collisions show a linearly anisotropic behavior, is outlined and the possible implications for numerical calculations in applied neutronics as well as shielding evaluations are investigated. The form of the differential system of equations taken by the method is quite handy and looks simpler and more manageable than any other today available technique. To go deeper into the efficiency of the method, some typical calculations concerning critical dimension of multiplying systems are then performed and the results are compared with the ones coming from the classical Ssub(N) approximations. The outcome of such calculations leads us to think of interesting developments of the method which could be quite useful in alternative to other today widespread approximate procedures, for any geometry, but especially for curved ones. (author)

  13. Delbrueck scattering of monoenergetic photons

    International Nuclear Information System (INIS)

    Kahane, S.

    1978-05-01

    The Delbrueck effect was experimentally investigated in high Z nuclei with monoenergetic photons in the range 6.8-11.4 MeV. Two different methods were used for measurements of the differential scattering cross-section, in the 25-140 deg range and in the forward direction (theta = 1.5 deg), respectively. The known Compton scattering cross-section was used in a new and unique way for the determination of the elastic scattering cross-section. Isolation of the contribution of the real Delbrueck amplitudes to the cross-section was crried out successfully. Experimental confirmation of the theoretical calculations of Papatzacos and Mork and measurement, for the first time, of the Rayleigh scattering in the 10 MeV region are also reported. One of the most interesting findings is the presence of Coulomb corrections in Delbrueck scattering at these energies. More theoretical effort is needed in this last direction. (author)

  14. Discontinuous nodal schemes applied to the bidimensional neutron transport equation

    International Nuclear Information System (INIS)

    Delfin L, A.; Valle G, E. Del; Hennart B, J.P.

    1996-01-01

    In this paper several strong discontinuous nodal schemes are described, starting from the one that has only two interpolation parameters per cell to the one having ten. Their application to the spatial discretization of the neutron transport equation in X-Y geometry is also described, giving, for each one of the nodal schemes, the approximation for the angular neutron flux that includes the set of interpolation parameters and the corresponding polynomial space. Numerical results were obtained for several test problems presenting here the problem with the highest degree of difficulty and their comparison with published results 1,2 . (Author)

  15. Coupling of neutron transport equations. First results; Couplage d`equations en transport neutronique. premiere approche 1D monocinetique

    Energy Technology Data Exchange (ETDEWEB)

    Bal, G.

    1995-07-01

    To achieve whole core calculations of the neutron transport equation, we have to follow this 2 step method: space and energy homogenization of the assemblies; resolution of the homogenized equation on the whole core. However, this is no more valid when accidents occur (for instance depressurization causing locally strong heterogeneous media). One solution consists then in coupling two kinds of resolutions: a fine computation on the damaged cell (fine mesh, high number of energy groups) coupled with a coarse one everywhere else. We only deal here with steady state solutions (which already live in 6D spaces). We present here two such methods: The coupling by transmission of homogenized sections and the coupling by transmission of boundary conditions. To understand what this coupling is, we first restrict ourselves to 1D with respect to space in one energy group. The first two chapters deal with a recall of basic properties of the neutron transport equation. We give at chapter 3 some indications of the behaviour of the flux with respect to the cross sections. We present at chapter 4 some couplings and give some properties. Chapter 5 is devoted to a presentation of some numerical applications. (author). 9 refs., 7 figs.

  16. Transport methods: general. 8. Formulation of Transport Equation in a Split Form

    International Nuclear Information System (INIS)

    Stancic, V.

    2001-01-01

    The singular eigenfunction expansion method has enabled the application of functional analysis methods in transport theory. However, when applying it, the users were discouraged, since in most problems, including slab problems, an extra problem has occurred. It appears necessary to solve the Fredholm integral equation in order to determine the expansion coefficients. There are several reasons for this difficulty. One reason might be the use of the full-range expansion techniques even in the regions where the function is singular. Such an example is the free boundary condition that requires the distribution to be equal to zero. Moreover, at μ = 0, the transport equation becomes an integral one. Both reasons motivated us to redefine the transport equation in a more natural way. Similar to scattering theory, here we define the flux distribution as a direct sum of forward- and backward-directed neutrons, e.g., μ ≥ 0 and μ < 0, respectively. As a result, the plane geometry transport equation is being split into coupled-pair equations. Further, using an appropriate transformation, this pair of equations reduces to a self-adjoint one having the same form as the known full-range single flux. It is interesting that all the methods of full-range theory are applicable here provided the flux as well as the transformed transport operator are two-dimensional matrices. Applying this to the slab problem, we find explicit expressions for reflected and transmitted particles caused by an arbitrary plane source. That is the news in this paper. Because of space constraints, only fundamentals of this approach will be presented here. We assume that the reader is familiar with this field; therefore, the applications are noted only at the end. (author)

  17. Modeling Blazar Spectra by Solving an Electron Transport Equation

    Science.gov (United States)

    Lewis, Tiffany; Finke, Justin; Becker, Peter A.

    2018-01-01

    Blazars are luminous active galaxies across the entire electromagnetic spectrum, but the spectral formation mechanisms, especially the particle acceleration, in these sources are not well understood. We develop a new theoretical model for simulating blazar spectra using a self-consistent electron number distribution. Specifically, we solve the particle transport equation considering shock acceleration, adiabatic expansion, stochastic acceleration due to MHD waves, Bohm diffusive particle escape, synchrotron radiation, and Compton radiation, where we implement the full Compton cross-section for seed photons from the accretion disk, the dust torus, and 26 individual broad lines. We used a modified Runge-Kutta method to solve the 2nd order equation, including development of a new mathematical method for normalizing stiff steady-state ordinary differential equations. We show that our self-consistent, transport-based blazar model can qualitatively fit the IR through Fermi g-ray data for 3C 279, with a single-zone, leptonic configuration. We use the solution for the electron distribution to calculate multi-wavelength SED spectra for 3C 279. We calculate the particle and magnetic field energy densities, which suggest that the emitting region is not always in equipartition (a common assumption), but sometimes matter dominated. The stratified broad line region (based on ratios in quasar reverberation mapping, and thus adding no free parameters) improves our estimate of the location of the emitting region, increasing it by ~5x. Our model provides a novel view into the physics at play in blazar jets, especially the relative strength of the shock and stochastic acceleration, where our model is well suited to distinguish between these processes, and we find that the latter tends to dominate.

  18. Nodal collocation approximation for the multidimensional PL equations applied to transport source problems

    International Nuclear Information System (INIS)

    Verdu, G.; Capilla, M.; Talavera, C. F.; Ginestar, D.

    2012-01-01

    PL equations are classical high order approximations to the transport equations which are based on the expansion of the angular dependence of the angular neutron flux and the nuclear cross sections in terms of spherical harmonics. A nodal collocation method is used to discretize the PL equations associated with a neutron source transport problem. The performance of the method is tested solving two 1D problems with analytical solution for the transport equation and a classical 2D problem. (authors)

  19. Nodal collocation approximation for the multidimensional PL equations applied to transport source problems

    Energy Technology Data Exchange (ETDEWEB)

    Verdu, G. [Departamento de Ingenieria Quimica Y Nuclear, Universitat Politecnica de Valencia, Cami de Vera, 14, 46022. Valencia (Spain); Capilla, M.; Talavera, C. F.; Ginestar, D. [Dept. of Nuclear Engineering, Departamento de Matematica Aplicada, Universitat Politecnica de Valencia, Cami de Vera, 14, 46022. Valencia (Spain)

    2012-07-01

    PL equations are classical high order approximations to the transport equations which are based on the expansion of the angular dependence of the angular neutron flux and the nuclear cross sections in terms of spherical harmonics. A nodal collocation method is used to discretize the PL equations associated with a neutron source transport problem. The performance of the method is tested solving two 1D problems with analytical solution for the transport equation and a classical 2D problem. (authors)

  20. Mixed-hybrid finite element method for the transport equation and diffusion approximation of transport problems

    International Nuclear Information System (INIS)

    Cartier, J.

    2006-04-01

    This thesis focuses on mathematical analysis, numerical resolution and modelling of the transport equations. First of all, we deal with numerical approximation of the solution of the transport equations by using a mixed-hybrid scheme. We derive and study a mixed formulation of the transport equation, then we analyse the related variational problem and present the discretization and the main properties of the scheme. We particularly pay attention to the behavior of the scheme and we show its efficiency in the diffusion limit (when the mean free path is small in comparison with the characteristic length of the physical domain). We present academical benchmarks in order to compare our scheme with other methods in many physical configurations and validate our method on analytical test cases. Unstructured and very distorted meshes are used to validate our scheme. The second part of this thesis deals with two transport problems. The first one is devoted to the study of diffusion due to boundary conditions in a transport problem between two plane plates. The second one consists in modelling and simulating radiative transfer phenomenon in case of the industrial context of inertial confinement fusion. (author)

  1. On standard forms for transport equations and fluxes: Part 2

    International Nuclear Information System (INIS)

    Ross, D.W.

    1990-03-01

    Quasilinear expressions for anomalous particle and energy fluxes arising from electrostatic plasma turbulence in a tokamak are reviewed yet again. Further clarifications are made, and the position taken in a previous report is modified. There, the total energy flux, Q j , and the conductive heat flux, q j , were correctly defined, and the anomalous Q j was correctly calculated. It was shown that the anomalous energy transport can be correctly described by ∇·Q* j , where Q* j = 3/5 Q j , with all remaining source terms such as left-angle p j ∇·Vj} cancelling. Here, a revised discussion is given of the identification of the anomalous conductive flux, q j , in which the distinction between Q j and Q* j is reconsidered. It is shown that there is more than one consistent way to define q j . Transport calculations involving only theoretical electrostatic turbulent fluxes are unaffected by these distinctions since Q j or Q* j , rather than q j , is the quantity naturally calculated in the theory. However, an ambiguity remains in experimental transport analysis if the measured particle flux Γ j = n j V j is to be used in the energy equation. This is because we cannot be sure how properly to treat the source terms p j ∇·V j or { p j ∇·V j }. 17 refs

  2. Numerical method for solving integral equations of neutron transport. II

    International Nuclear Information System (INIS)

    Loyalka, S.K.; Tsai, R.W.

    1975-01-01

    In a recent paper it was pointed out that the weakly singular integral equations of neutron transport can be quite conveniently solved by a method based on subtraction of singularity. This previous paper was devoted entirely to the consideration of simple one-dimensional isotropic-scattering and one-group problems. The present paper constitutes interesting extensions of the previous work in that in addition to a typical two-group anisotropic-scattering albedo problem in the slab geometry, the method is also applied to an isotropic-scattering problem in the x-y geometry. These results are compared with discrete S/sub N/ (ANISN or TWOTRAN-II) results, and for the problems considered here, the proposed method is found to be quite effective. Thus, the method appears to hold considerable potential for future applications. (auth)

  3. Heat conduction in multifunctional nanotrusses studied using Boltzmann transport equation

    International Nuclear Information System (INIS)

    Dou, Nicholas G.; Minnich, Austin J.

    2016-01-01

    Materials that possess low density, low thermal conductivity, and high stiffness are desirable for engineering applications, but most materials cannot realize these properties simultaneously due to the coupling between them. Nanotrusses, which consist of hollow nanoscale beams architected into a periodic truss structure, can potentially break these couplings due to their lattice architecture and nanoscale features. In this work, we study heat conduction in the exact nanotruss geometry by solving the frequency-dependent Boltzmann transport equation using a variance-reduced Monte Carlo algorithm. We show that their thermal conductivity can be described with only two parameters, solid fraction and wall thickness. Our simulations predict that nanotrusses can realize unique combinations of mechanical and thermal properties that are challenging to achieve in typical materials

  4. On the Solution of the Neutron Transport Equation

    Energy Technology Data Exchange (ETDEWEB)

    Depken, S

    1962-12-15

    The neutron transport equation has occupied the attention of many authors since Placzek, Wick and others made their first attempts to solve it, Even in the simple case of energy independent cross-sections, and disregarding the motion of the scattering nucleons, it is difficult to find a solution in an analytical form which is easily surveyable and fitted for numerical calculations. In Part I of this paper some new viewpoints will be introduced which enable the solution to be presented in its simplest possible form. Part II is devoted to an investigation of some functions introduced in Part I. In Part III the results are applied to the case of large energy lethargy, and the validity of derived formulas is discussed.

  5. A simple Boltzmann transport equation for ballistic to diffusive transient heat transport

    International Nuclear Information System (INIS)

    Maassen, Jesse; Lundstrom, Mark

    2015-01-01

    Developing simplified, but accurate, theoretical approaches to treat heat transport on all length and time scales is needed to further enable scientific insight and technology innovation. Using a simplified form of the Boltzmann transport equation (BTE), originally developed for electron transport, we demonstrate how ballistic phonon effects and finite-velocity propagation are easily and naturally captured. We show how this approach compares well to the phonon BTE, and readily handles a full phonon dispersion and energy-dependent mean-free-path. This study of transient heat transport shows (i) how fundamental temperature jumps at the contacts depend simply on the ballistic thermal resistance, (ii) that phonon transport at early times approach the ballistic limit in samples of any length, and (iii) perceived reductions in heat conduction, when ballistic effects are present, originate from reductions in temperature gradient. Importantly, this framework can be recast exactly as the Cattaneo and hyperbolic heat equations, and we discuss how the key to capturing ballistic heat effects is to use the correct physical boundary conditions

  6. Generalized fluid equations for parallel transport in collisional to weakly collisional plasmas

    International Nuclear Information System (INIS)

    Zawaideh, E.S.

    1985-01-01

    A new set of two-fluid equations which are valid from collisional to weakly collisional limits are derived. Starting from gyrokinetic equations in flux coordinates with no zeroth order drifts, a set of moment equations describing plasma transport along the field lines of a space and time dependent magnetic field are derived. No restriction on the anisotropy of the ion distribution function is imposed. In the highly collisional limit, these equations reduce to those of Braginskii while in the weakly collisional limit, they are similar to the double adiabatic or Chew, Goldberger, and Low (CGL) equations. The new transport equations are used to study the effects of collisionality, magnetic field structure, and plasma anisotropy on plasma parallel transport. Numerical examples comparing these equations with conventional transport equations show that the conventional equations may contain large errors near the sound speed (M approx. = 1). It is also found that plasma anisotropy, which is not included in the conventional equations, is a critical parameter in determining plasma transport in varying magnetic field. The new transport equations are also used to study axial confinement in multiple mirror devices from the strongly to weakly collisional regime. A new ion conduction model was worked out to extend the regime of validity of the transport equations to the low density multiple mirror regime

  7. A Photon Free Method to Solve Radiation Transport Equations

    International Nuclear Information System (INIS)

    Chang, B

    2006-01-01

    The multi-group discrete-ordinate equations of radiation transfer is solved for the first time by Newton's method. It is a photon free method because the photon variables are eliminated from the radiation equations to yield a N group XN direction smaller but equivalent system of equations. The smaller set of equations can be solved more efficiently than the original set of equations. Newton's method is more stable than the Semi-implicit Linear method currently used by conventional radiation codes

  8. Basic equations of interfacial area transport in gas-liquid two-phase flow

    International Nuclear Information System (INIS)

    Kataoka, I.; Yoshida, K.; Naitoh, M.; Okada, H.; Morii, T.

    2011-01-01

    The rigorous and consistent formulations of basic equations of interfacial area transport were derived using correlation functions of characteristic function of each phase and velocities of each phase. Turbulent transport term of interfacial area concentration was consistently derived and related to the difference between interfacial velocity and averaged velocity of each phase. Constitutive equations of turbulent transport terms of interfacial area concentration were proposed for bubbly flow. New transport model and constitutive equations were developed for churn flow. These models and constitutive equations are validated by experimental data of radial distributions of interfacial area concentration in bubbly and churn flow. (author)

  9. Approximate solution of the transport equation by methods of Galerkin type

    International Nuclear Information System (INIS)

    Pitkaranta, J.

    1977-01-01

    Questions of the existence, uniqueness, and convergence of approximate solutions of transport equations by methods of the Galerkin type (where trial and weighting functions are the same) are discussed. The results presented do not exclude the infinite-dimensional case. Two strategies can be followed in the variational approximation of the transport operator: one proceeds from the original form of the transport equation, while the other is based on the partially symmetrized equation. Both principles are discussed in this paper. The transport equation is assumed in a discretized multigroup form

  10. Normal and adjoint integral and integrodifferential neutron transport equations. Pt. 2

    International Nuclear Information System (INIS)

    Velarde, G.

    1976-01-01

    Using the simplifying hypotheses of the integrodifferential Boltzmann equations of neutron transport, given in JEN 334 report, several integral equations, and theirs adjoint ones, are obtained. Relations between the different normal and adjoint eigenfunctions are established and, in particular, proceeding from the integrodifferential Boltzmann equation it's found out the relation between the solutions of the adjoint equation of its integral one, and the solutions of the integral equation of its adjoint one (author)

  11. Parallel computing for homogeneous diffusion and transport equations in neutronics

    International Nuclear Information System (INIS)

    Pinchedez, K.

    1999-06-01

    Parallel computing meets the ever-increasing requirements for neutronic computer code speed and accuracy. In this work, two different approaches have been considered. We first parallelized the sequential algorithm used by the neutronics code CRONOS developed at the French Atomic Energy Commission. The algorithm computes the dominant eigenvalue associated with PN simplified transport equations by a mixed finite element method. Several parallel algorithms have been developed on distributed memory machines. The performances of the parallel algorithms have been studied experimentally by implementation on a T3D Cray and theoretically by complexity models. A comparison of various parallel algorithms has confirmed the chosen implementations. We next applied a domain sub-division technique to the two-group diffusion Eigen problem. In the modal synthesis-based method, the global spectrum is determined from the partial spectra associated with sub-domains. Then the Eigen problem is expanded on a family composed, on the one hand, from eigenfunctions associated with the sub-domains and, on the other hand, from functions corresponding to the contribution from the interface between the sub-domains. For a 2-D homogeneous core, this modal method has been validated and its accuracy has been measured. (author)

  12. Effect of magnetic axis shift on neoclassical transport in helical torus

    International Nuclear Information System (INIS)

    Sanuki, Heiji; Todoroki, Jiro

    2004-01-01

    Neoclassical transport for large helical device (LHD) configurations is studied by solving the bounce-averaged Fokker-Planck equation. Numerical code employed in the present paper (CHD1) is much faster and more efficient than existing transport codes. Effects of the magnetic axis shift on the mono-energetic transport coefficients are studied in detail for the LHD configurations, revealing that a strong inward shift of the magnetic axis can reduce remarkably the neoclassical ripple transport. (author)

  13. Variational formulation and projectional methods for the second order transport equation

    International Nuclear Information System (INIS)

    Borysiewicz, M.; Stankiewicz, R.

    1979-01-01

    Herein the variational problem for a second-order boundary value problem for the neutron transport equation is formulated. The projectional methods solving the problem are examined. The approach is compared with that based on the original untransformed form of the neutron transport equation

  14. Parallel computing solution of Boltzmann neutron transport equation

    International Nuclear Information System (INIS)

    Ansah-Narh, T.

    2010-01-01

    The focus of the research was on developing parallel computing algorithm for solving Eigen-values of the Boltzmam Neutron Transport Equation (BNTE) in a slab geometry using multi-grid approach. In response to the problem of slow execution of serial computing when solving large problems, such as BNTE, the study was focused on the design of parallel computing systems which was an evolution of serial computing that used multiple processing elements simultaneously to solve complex physical and mathematical problems. Finite element method (FEM) was used for the spatial discretization scheme, while angular discretization was accomplished by expanding the angular dependence in terms of Legendre polynomials. The eigenvalues representing the multiplication factors in the BNTE were determined by the power method. MATLAB Compiler Version 4.1 (R2009a) was used to compile the MATLAB codes of BNTE. The implemented parallel algorithms were enabled with matlabpool, a Parallel Computing Toolbox function. The option UseParallel was set to 'always' and the default value of the option was 'never'. When those conditions held, the solvers computed estimated gradients in parallel. The parallel computing system was used to handle all the bottlenecks in the matrix generated from the finite element scheme and each domain of the power method generated. The parallel algorithm was implemented on a Symmetric Multi Processor (SMP) cluster machine, which had Intel 32 bit quad-core x 86 processors. Convergence rates and timings for the algorithm on the SMP cluster machine were obtained. Numerical experiments indicated the designed parallel algorithm could reach perfect speedup and had good stability and scalability. (au)

  15. On the Boltzmann Equation of Thermal Transport for Interacting Phonons and Electrons

    Directory of Open Access Journals (Sweden)

    Amelia Carolina Sparavigna

    2016-05-01

    Full Text Available The thermal transport in a solid can be determined by means of the Boltzmann equations regarding its distributions of phonons and electrons, when the solid is subjected to a thermal gradient. After solving the coupled equations, the related thermal conductivities can be obtained. Here we show how to determine the coupled equations for phonons and electrons.

  16. Cable Connected Spinning Spacecraft, 1. the Canonical Equations, 2. Urban Mass Transportation, 3

    Science.gov (United States)

    Sitchin, A.

    1972-01-01

    Work on the dynamics of cable-connected spinning spacecraft was completed by formulating the equations of motion by both the canonical equations and Lagrange's equations and programming them for numerical solution on a digital computer. These energy-based formulations will permit future addition of the effect of cable mass. Comparative runs indicate that the canonical formulation requires less computer time. Available literature on urban mass transportation was surveyed. Areas of the private rapid transit concept of urban transportation are also studied.

  17. Transport methods: general. 7. Formulation of a Fourier-Boltzmann Transformation to Solve the Three-Dimensional Transport Equation

    International Nuclear Information System (INIS)

    Stancic, V.

    2001-01-01

    This paper presents some elements of a new approach to solve analytically the linearized three-dimensional (3-D) transport equation of neutral particles. Since this task is of such special importance, we present some results of a paper that is still in progress. The most important is that using this transformation, an integro-differential equation with an analytical solution is obtained. For this purpose, a simplest 3-D equation is being considered which describes the transport process in an infinite medium. Until now, this equation has been analytically considered either using the Laplace transform with respect to time parameter t or applying the Fourier transform over the space coordinate. Both of them reduce the number of differential terms in the equation; however, evaluation of the inverse transformation is complicated. In this paper, we introduce for the first time a Fourier transform induced by the Boltzmann operator. For this, we use a complete set of 3-D eigenfunctions of the Boltzmann transport operator defined in a similar way as those that have been already used in 3-D transport theory as a basic set to transform the transport equation. This set consists of a continuous part and a discrete one with spectral measure. The density distribution equation shows the known form asymptotic behavior. Several applications are to be performed using this equation and compared to the benchmark one. Such an analysis certainly would be out of the available space

  18. On the derivation of vector radiative transfer equation for polarized radiative transport in graded index media

    International Nuclear Information System (INIS)

    Zhao, J.M.; Tan, J.Y.; Liu, L.H.

    2012-01-01

    Light transport in graded index media follows a curved trajectory determined by Fermat's principle. Besides the effect of variation of the refractive index on the transport of radiative intensity, the curved ray trajectory will induce geometrical effects on the transport of polarization ellipse. This paper presents a complete derivation of vector radiative transfer equation for polarized radiation transport in absorption, emission and scattering graded index media. The derivation is based on the analysis of the conserved quantities for polarized light transport along curved trajectory and a novel approach. The obtained transfer equation can be considered as a generalization of the classic vector radiative transfer equation that is only valid for uniform refractive index media. Several variant forms of the transport equation are also presented, which include the form for Stokes parameters defined with a fixed reference and the Eulerian forms in the ray coordinate and in several common orthogonal coordinate systems.

  19. Solution and study of nodal neutron transport equation applying the LTSN-DiagExp method

    International Nuclear Information System (INIS)

    Hauser, Eliete Biasotto; Pazos, Ruben Panta; Vilhena, Marco Tullio de; Barros, Ricardo Carvalho de

    2003-01-01

    In this paper we report advances about the three-dimensional nodal discrete-ordinates approximations of neutron transport equation for Cartesian geometry. We use the combined collocation method of the angular variables and nodal approach for the spatial variables. By nodal approach we mean the iterated transverse integration of the S N equations. This procedure leads to the set of one-dimensional averages angular fluxes in each spatial variable. The resulting system of equations is solved with the LTS N method, first applying the Laplace transform to the set of the nodal S N equations and then obtained the solution by symbolic computation. We include the LTS N method by diagonalization to solve the nodal neutron transport equation and then we outline the convergence of these nodal-LTS N approximations with the help of a norm associated to the quadrature formula used to approximate the integral term of the neutron transport equation. (author)

  20. The plasma transport equations derived by multiple time-scale expansions and turbulent transport. I. General theory

    International Nuclear Information System (INIS)

    Edenstrasser, J.W.

    1995-01-01

    A multiple time-scale derivative expansion scheme is applied to the dimensionless Fokker--Planck equation and to Maxwell's equations, where the parameter range of a typical fusion plasma was assumed. Within kinetic theory, the four time scales considered are those of Larmor gyration, particle transit, collisions, and classical transport. The corresponding magnetohydrodynamic (MHD) time scales are those of ion Larmor gyration, Alfven, MHD collision, and resistive diffusion. The solution of the zeroth-order equations results in the force-free equilibria and ideal Ohm's law. The solution of the first-order equations leads under the assumption of a weak collisional plasma to the ideal MHD equations. On the MHD-collision time scale, not only the full set of the MHD transport equations is obtained, but also turbulent terms, where the related transport quantities are one order in the expansion parameter larger than those of classical transport. Finally, at the resistive diffusion time scale the known transport equations are arrived at including, however, also turbulent contributions. copyright 1995 American Institute of Physics

  1. An Equation-Type Approach for the Numerical Solution of the Partial Differential Equations Governing Transport Phenomena in Porous Media

    KAUST Repository

    Sun, Shuyu; Salama, Amgad; El-Amin, Mohamed

    2012-01-01

    A new technique for the numerical solution of the partial differential equations governing transport phenomena in porous media is introduced. In this technique, the governing equations as depicted from the physics of the problem are used without extra manipulations. In other words, there is no need to reduce the number of governing equations by some sort of mathematical manipulations. This technique enables the separation of the physics part of the problem and the solver part, which makes coding more robust and could be used in several other applications with little or no modifications (e.g., multi-phase flow in porous media). In this method, one abandons the need to construct the coefficient matrix for the pressure equation. Alternatively, the coefficients are automatically generated within the solver routine. We show examples of using this technique to solving several flow problems in porous media.

  2. An Equation-Type Approach for the Numerical Solution of the Partial Differential Equations Governing Transport Phenomena in Porous Media

    KAUST Repository

    Sun, Shuyu

    2012-06-02

    A new technique for the numerical solution of the partial differential equations governing transport phenomena in porous media is introduced. In this technique, the governing equations as depicted from the physics of the problem are used without extra manipulations. In other words, there is no need to reduce the number of governing equations by some sort of mathematical manipulations. This technique enables the separation of the physics part of the problem and the solver part, which makes coding more robust and could be used in several other applications with little or no modifications (e.g., multi-phase flow in porous media). In this method, one abandons the need to construct the coefficient matrix for the pressure equation. Alternatively, the coefficients are automatically generated within the solver routine. We show examples of using this technique to solving several flow problems in porous media.

  3. Approximate solutions for the two-dimensional integral transport equation. The critically mixed methods of resolution

    International Nuclear Information System (INIS)

    Sanchez, Richard.

    1980-11-01

    This work is divided into two part the first part (note CEA-N-2165) deals with the solution of complex two-dimensional transport problems, the second one treats the critically mixed methods of resolution. These methods are applied for one-dimensional geometries with highly anisotropic scattering. In order to simplify the set of integral equation provided by the integral transport equation, the integro-differential equation is used to obtain relations that allow to lower the number of integral equation to solve; a general mathematical and numerical study is presented [fr

  4. Source of monoenergetic electrons for beta dosimetry

    International Nuclear Information System (INIS)

    Graham, C.L.; Elliott, J.H.

    1983-01-01

    We have developed an electron spectrometer which can produce a continuous beam of monoenergetic electrons. The spectrometer uses 20 millicuries of Cs-137 as a source of electrons which can be magnetically focused at the exit port. Various electron energies can be selected by changing the magnetic field. The maximum electron energy and dose rate for the present design are approximately 630 keV and 1.5 rads per hour, respectively

  5. Arbitrary quadratures determination of the monoenergetic neutron density in an homogeneous finite sphere with isotropic scattering

    International Nuclear Information System (INIS)

    Sanchez G, J.

    2015-09-01

    The solution of the so-called Canonical problems of neutron transport theory has been given by Case, who developed a method akin to the classical eigenfunction expansion procedure, extended to admit singular eigenfunctions. The solution is given as a set consisting of a Fredholm integral equation coupled with a transcendental equation, which has to be solved for the expansion coefficients by iteration. CASE's method make extensive use of the results of the theory of functions of a complex variable and many successful approaches to solve in an approximate form the above mentioned set have been reported in the literature. We present here an entirely different approach which deals with the canonical problems in a more direct and elementary manner. As far as we know, the original idea for the latter method is due to Carlvik who devised the escape probability approximation to the solution of the neutron transport equation in its integral form. In essence, the procedure consists in assuming a sectionally constant form of the neutron density that in turn yields a set of linear algebraic equations obeyed by the assumed constant values of the density. Very well established techniques of numerical analysis for the solution of integral equations consist in independent approaches that generalize the sectionally constant approach by assuming a sectionally low degree polynomial for the unknown function. This procedure also known as the arbitrary quadratures method is especially suited to deal with cases where the kernel of the integral equation is singular. The author wishes to present the results obtained with the arbitrary quadratures method for the numerical calculation of the monoenergetic neutron density in a critical, homogeneous sphere of finite radius with isotropic scattering. The singular integral equation obeyed by the neutron density in the critical sphere is introduced, an outline of the method's main features is given, and tables and graphs of the density

  6. Arbitrary quadratures determination of the monoenergetic neutron density in an homogeneous finite sphere with isotropic scattering

    Energy Technology Data Exchange (ETDEWEB)

    Sanchez G, J., E-mail: julian.sanchez@inin.gob.mx [ININ, Carretera Mexico-Toluca s/n, 52750 Ocoyoacac, Estado de Mexico (Mexico)

    2015-09-15

    The solution of the so-called Canonical problems of neutron transport theory has been given by Case, who developed a method akin to the classical eigenfunction expansion procedure, extended to admit singular eigenfunctions. The solution is given as a set consisting of a Fredholm integral equation coupled with a transcendental equation, which has to be solved for the expansion coefficients by iteration. CASE's method make extensive use of the results of the theory of functions of a complex variable and many successful approaches to solve in an approximate form the above mentioned set have been reported in the literature. We present here an entirely different approach which deals with the canonical problems in a more direct and elementary manner. As far as we know, the original idea for the latter method is due to Carlvik who devised the escape probability approximation to the solution of the neutron transport equation in its integral form. In essence, the procedure consists in assuming a sectionally constant form of the neutron density that in turn yields a set of linear algebraic equations obeyed by the assumed constant values of the density. Very well established techniques of numerical analysis for the solution of integral equations consist in independent approaches that generalize the sectionally constant approach by assuming a sectionally low degree polynomial for the unknown function. This procedure also known as the arbitrary quadratures method is especially suited to deal with cases where the kernel of the integral equation is singular. The author wishes to present the results obtained with the arbitrary quadratures method for the numerical calculation of the monoenergetic neutron density in a critical, homogeneous sphere of finite radius with isotropic scattering. The singular integral equation obeyed by the neutron density in the critical sphere is introduced, an outline of the method's main features is given, and tables and graphs of the density

  7. Differential equation of exospheric lateral transport and its application to terrestrial hydrogen

    Science.gov (United States)

    Hodges, R. R., Jr.

    1973-01-01

    The differential equation description of exospheric lateral transport of Hodges and Johnson is reformulated to extend its utility to light gases. Accuracy of the revised equation is established by applying it to terrestrial hydrogen. The resulting global distributions for several static exobase models are shown to be essentially the same as those that have been computed by Quessette using an integral equation approach. The present theory is subsequently used to elucidate the effects of nonzero lateral flow, exobase rotation, and diurnal tidal winds on the hydrogen distribution. Finally it is shown that the differential equation of exospheric transport is analogous to a diffusion equation. Hence it is practical to consider exospheric transport as a continuation of thermospheric diffusion, a concept that alleviates the need for an artificial exobase dividing thermosphere and exosphere.

  8. A variational solution of transport equation based on spherical geometry

    International Nuclear Information System (INIS)

    Liu Hui; Zhang Ben'ai

    2002-01-01

    A variational method with differential forms gives better precision for numerical solution of transport critical problem based on spherical geometry, and its computation seems simple than other approximate methods

  9. Application of Trotter approximation for solving time dependent neutron transport equation

    International Nuclear Information System (INIS)

    Stancic, V.

    1987-01-01

    A method is proposed to solve multigroup time dependent neutron transport equation with arbitrary scattering anisotropy. The recurrence relation thus obtained is simple, numerically stable and especially suitable for treatment of complicated geometries. (author)

  10. Solution of linear transport equation using Chebyshev polynomials and Laplace transform

    International Nuclear Information System (INIS)

    Cardona, A.V.; Vilhena, M.T.M.B. de

    1994-01-01

    The Chebyshev polynomials and the Laplace transform are combined to solve, analytically, the linear transport equation in planar geometry, considering isotropic scattering and the one-group model. Numerical simulation is presented. (author)

  11. An introduction to the Boltzmann equation and transport processes in gases

    CERN Document Server

    Kremer, Gilberto M; Colton, David

    2010-01-01

    This book covers classical kinetic theory of gases, presenting basic principles in a self-contained framework and from a more rigorous approach based on the Boltzmann equation. Uses methods in kinetic theory for determining the transport coefficients of gases.

  12. TLC scheme for numerical solution of the transport equation on equilateral triangular meshes

    International Nuclear Information System (INIS)

    Walters, W.F.

    1983-01-01

    A new triangular linear characteristic TLC scheme for numerically solving the transport equation on equilateral triangular meshes has been developed. This scheme uses the analytic solution of the transport equation in the triangle as its basis. The data on edges of the triangle are assumed linear as is the source representation. A characteristic approach or nodal approach is used to obtain the analytic solution. Test problems indicate that the new TLC is superior to the widely used DITRI scheme for accuracy

  13. Evaluation of finite difference and FFT-based solutions of the transport of intensity equation.

    Science.gov (United States)

    Zhang, Hongbo; Zhou, Wen-Jing; Liu, Ying; Leber, Donald; Banerjee, Partha; Basunia, Mahmudunnabi; Poon, Ting-Chung

    2018-01-01

    A finite difference method is proposed for solving the transport of intensity equation. Simulation results show that although slower than fast Fourier transform (FFT)-based methods, finite difference methods are able to reconstruct the phase with better accuracy due to relaxed assumptions for solving the transport of intensity equation relative to FFT methods. Finite difference methods are also more flexible than FFT methods in dealing with different boundary conditions.

  14. New diffusion-like solutions of one-speed transport equations in spherical geometry

    International Nuclear Information System (INIS)

    Sahni, D.C.

    1988-01-01

    Stationary, one-speed, spherically symmetric transport equations are considered in a conservative medium. Closed-form expressions are obtained for the angular flux ψ(r, μ) that yield a total flux varying as 1/r by using Sonine transforms. Properties of this solution are studied and it is shown that the solution can not be identified as a diffusion mode solution of the transport equation. Limitations of the Sonine transform technique are noted. (author)

  15. Discrete Ordinates Approximations to the First- and Second-Order Radiation Transport Equations

    International Nuclear Information System (INIS)

    FAN, WESLEY C.; DRUMM, CLIFTON R.; POWELL, JENNIFER L. email wcfan@sandia.gov

    2002-01-01

    The conventional discrete ordinates approximation to the Boltzmann transport equation can be described in a matrix form. Specifically, the within-group scattering integral can be represented by three components: a moment-to-discrete matrix, a scattering cross-section matrix and a discrete-to-moment matrix. Using and extending these entities, we derive and summarize the matrix representations of the second-order transport equations

  16. Discrete Ordinates Approximations to the First- and Second-Order Radiation Transport Equations

    CERN Document Server

    Fan, W C; Powell, J L

    2002-01-01

    The conventional discrete ordinates approximation to the Boltzmann transport equation can be described in a matrix form. Specifically, the within-group scattering integral can be represented by three components: a moment-to-discrete matrix, a scattering cross-section matrix and a discrete-to-moment matrix. Using and extending these entities, we derive and summarize the matrix representations of the second-order transport equations.

  17. Relativistic neoclassical transport coefficients with momentum correction

    International Nuclear Information System (INIS)

    Marushchenko, I.; Azarenkov, N.A.

    2016-01-01

    The parallel momentum correction technique is generalized for relativistic approach. It is required for proper calculation of the parallel neoclassical flows and, in particular, for the bootstrap current at fusion temperatures. It is shown that the obtained system of linear algebraic equations for parallel fluxes can be solved directly without calculation of the distribution function if the relativistic mono-energetic transport coefficients are already known. The first relativistic correction terms for Braginskii matrix coefficients are calculated.

  18. Adomian decomposition method for solving the telegraph equation in charged particle transport

    International Nuclear Information System (INIS)

    Abdou, M.A.

    2005-01-01

    In this paper, the analysis for the telegraph equation in case of isotropic small angle scattering from the Boltzmann transport equation for charged particle is presented. The Adomian decomposition is used to solve the telegraph equation. By means of MAPLE the Adomian polynomials of obtained series (ADM) solution have been calculated. The behaviour of the distribution function are shown graphically. The results reported in this article provide further evidence of the usefulness of Adomain decomposition for obtaining solution of linear and nonlinear problems

  19. Numerical evaluation of the intensity transport equation for well-known wavefronts and intensity distributions

    Science.gov (United States)

    Campos-García, Manuel; Granados-Agustín, Fermín.; Cornejo-Rodríguez, Alejandro; Estrada-Molina, Amilcar; Avendaño-Alejo, Maximino; Moreno-Oliva, Víctor Iván.

    2013-11-01

    In order to obtain a clearer interpretation of the Intensity Transport Equation (ITE), in this work, we propose an algorithm to solve it for some particular wavefronts and its corresponding intensity distributions. By simulating intensity distributions in some planes, the ITE is turns into a Poisson equation with Neumann boundary conditions. The Poisson equation is solved by means of the iterative algorithm SOR (Simultaneous Over-Relaxation).

  20. Determination of a closed-form solution for the multidimensional transport equation using a fractional derivative

    International Nuclear Information System (INIS)

    Zabadal, J.; Vilhena, M.T.; Segatto, C.F.; Pazos, R.P.Ruben Panta.

    2002-01-01

    In this work we construct a closed-form solution for the multidimensional transport equation rewritten in integral form which is expressed in terms of a fractional derivative of the angular flux. We determine the unknown order of the fractional derivative comparing the kernel of the integral equation with the one of the Riemann-Liouville definition of fractional derivative. We report numerical simulations

  1. Determination of a closed-form solution for the multidimensional transport equation using a fractional derivative

    Energy Technology Data Exchange (ETDEWEB)

    Zabadal, J. E-mail: jorge.zabadal@ufrgs.br; Vilhena, M.T. E-mail: vilhena@mat.ufrgs.br; Segatto, C.F. E-mail: cynthia@mat.ufrgs.br; Pazos, R.P.Ruben Panta. E-mail: rpp@mat.pucrgs.br

    2002-07-01

    In this work we construct a closed-form solution for the multidimensional transport equation rewritten in integral form which is expressed in terms of a fractional derivative of the angular flux. We determine the unknown order of the fractional derivative comparing the kernel of the integral equation with the one of the Riemann-Liouville definition of fractional derivative. We report numerical simulations.

  2. The discontinuous finite element method for solving Eigenvalue problems of transport equations

    International Nuclear Information System (INIS)

    Yang, Shulin; Wang, Ruihong

    2011-01-01

    In this paper, the multigroup transport equations for solving the eigenvalues λ and K_e_f_f under two dimensional cylindrical coordinate are discussed. Aimed at the equations, the discretizing way combining discontinuous finite element method (DFE) with discrete ordinate method (SN) is developed, and the iterative algorithms and steps are studied. The numerical results show that the algorithms are efficient. (author)

  3. The H-N method for solving linear transport equation: theory and application

    International Nuclear Information System (INIS)

    Kaskas, A.; Gulecyuz, M.C.; Tezcan, C.

    2002-01-01

    The system of singular integral equation which is obtained from the integro-differential form of the linear transport equation as a result of Placzec lemma is solved. Application are given using the exit distributions and the infinite medium Green's function. The same theoretical results are also obtained with the use of the singular eigenfunction of the method of elementary solutions

  4. Two analytic transport equation solutions for particular cases of particle history

    International Nuclear Information System (INIS)

    Simovic, R.

    1997-01-01

    For anisotropic scattering and plane geometry, the linear transport equation of particles generated by a monodirectional unit source A(x,μ) = δ(x-0)δ(μ - μ 0 ) > 0, can be stated in the form of an integral equation

  5. Optimal partial mass transportation and obstacle Monge-Kantorovich equation

    Science.gov (United States)

    Igbida, Noureddine; Nguyen, Van Thanh

    2018-05-01

    Optimal partial mass transport, which is a variant of the optimal transport problem, consists in transporting effectively a prescribed amount of mass from a source to a target. The problem was first studied by Caffarelli and McCann (2010) [6] and Figalli (2010) [12] with a particular attention to the quadratic cost. Our aim here is to study the optimal partial mass transport problem with Finsler distance costs including the Monge cost given by the Euclidian distance. Our approach is different and our results do not follow from previous works. Among our results, we introduce a PDE of Monge-Kantorovich type with a double obstacle to characterize active submeasures, Kantorovich potential and optimal flow for the optimal partial transport problem. This new PDE enables us to study the uniqueness and monotonicity results for the active submeasures. Another interesting issue of our approach is its convenience for numerical analysis and computations that we develop in a separate paper [14] (Igbida and Nguyen, 2018).

  6. Some results on the neutron transport and the coupling of equations; Quelques resultats sur le transport neutronique et le couplage d`equations

    Energy Technology Data Exchange (ETDEWEB)

    Bal, G. [Electricite de France (EDF), Direction des Etudes et Recherches, 92 - Clamart (France)

    1997-12-31

    Neutron transport in nuclear reactors is well modeled by the linear Boltzmann transport equation. Its resolution is relatively easy but very expensive. To achieve whole core calculations, one has to consider simpler models, such as diffusion or homogeneous transport equations. However, the solutions may become inaccurate in particular situations (as accidents for instance). That is the reason why we wish to solve the equations on small area accurately and more coarsely on the remaining part of the core. It is than necessary to introduce some links between different discretizations or modelizations. In this note, we give some results on the coupling of different discretizations of all degrees of freedom of the integral-differential neutron transport equation (two degrees for the angular variable, on for the energy component, and two or three degrees for spatial position respectively in 2D (cylindrical symmetry) and 3D). Two chapters are devoted to the coupling of discrete ordinates methods (for angular discretization). The first one is theoretical and shows the well posing of the coupled problem, whereas the second one deals with numerical applications of practical interest (the results have been obtained from the neutron transport code developed at the R and D, which has been modified for introducing the coupling). Next, we present the nodal scheme RTN0, used for the spatial discretization. We show well posing results for the non-coupled and the coupled problems. At the end, we deal with the coupling of energy discretizations for the multigroup equations obtained by homogenization. Some theoretical results of the discretization of the velocity variable (well-posing of problems), which do not deal directly with the purposes of coupling, are presented in the annexes. (author). 34 refs.

  7. Role of Dielectric Constant on Ion Transport: Reformulated Arrhenius Equation

    Directory of Open Access Journals (Sweden)

    Shujahadeen B. Aziz

    2016-01-01

    Full Text Available Solid and nanocomposite polymer electrolytes based on chitosan have been prepared by solution cast technique. The XRD results reveal the occurrence of complexation between chitosan (CS and the LiTf salt. The deconvolution of the diffractogram of nanocomposite solid polymer electrolytes demonstrates the increase of amorphous domain with increasing alumina content up to 4 wt.%. Further incorporation of alumina nanoparticles (6 to 10 wt.% Al2O3 results in crystallinity increase (large crystallite size. The morphological (SEM and EDX analysis well supported the XRD results. Similar trends of DC conductivity and dielectric constant with Al2O3 concentration were explained. The TEM images were used to explain the phenomena of space charge and blocking effects. The reformulated Arrhenius equation (σ(ε′,T=σoexp(-Ea/KBε′T was proposed from the smooth exponential behavior of DC conductivity versus dielectric constant at different temperatures. The more linear behavior of DC conductivity versus 1000/(ɛ′×T reveals the crucial role of dielectric constant in Arrhenius equation. The drawbacks of Arrhenius equation can be understood from the less linear behavior of DC conductivity versus 1000/T. The relaxation processes have been interpreted in terms of Argand plots.

  8. dispersion equation parameters of solute transport in agricultural

    African Journals Online (AJOL)

    Jane

    2011-08-31

    Aug 31, 2011 ... fields for predicting soil quality property. Key words: ... The classical approach of modeling solute transport in porous media uses the deterministic ... concentration of the solution in the liquid phase, u0 is the mean velocity and ...

  9. Elliptic random-walk equation for suspension and tracer transport in porous media

    DEFF Research Database (Denmark)

    Shapiro, Alexander; Bedrikovetsky, P. G.

    2008-01-01

    . The new theory predicts delay of the maximum of the tracer, compared to the velocity of the flow, while its forward "tail" contains much more particles than in the solution of the classical parabolic (advection-dispersion) equation. This is in agreement with the experimental observations and predictions......We propose a new approach to transport of the suspensions and tracers in porous media. The approach is based on a modified version of the continuous time random walk (CTRW) theory. In the framework of this theory we derive an elliptic transport equation. The new equation contains the time...... of the CTRW theory. (C) 2008 Elsevier B.V. All rights reserved....

  10. Resolution of the neutron transport equation by massively parallel computer in the Cronos code

    International Nuclear Information System (INIS)

    Zardini, D.M.

    1996-01-01

    The feasibility of neutron transport problems parallel resolution by CRONOS code's SN module is here studied. In this report we give the first data about the parallel resolution by angular variable decomposition of the transport equation. Problems about parallel resolution by spatial variable decomposition and memory stage limits are also explained here. (author)

  11. Modified two-fluid model for the two-group interfacial area transport equation

    International Nuclear Information System (INIS)

    Sun Xiaodong; Ishii, Mamoru; Kelly, Joseph M.

    2003-01-01

    This paper presents a modified two-fluid model that is ready to be applied in the approach of the two-group interfacial area transport equation. The two-group interfacial area transport equation was developed to provide a mechanistic constitutive relation for the interfacial area concentration in the two-fluid model. In the two-group transport equation, bubbles are categorized into two groups: spherical/distorted bubbles as Group 1 while cap/slug/churn-turbulent bubbles as Group 2. Therefore, this transport equation can be employed in the flow regimes spanning from bubbly, cap bubbly, slug to churn-turbulent flows. However, the introduction of the two groups of bubbles requires two gas velocity fields. Yet it is not practical to solve two momentum equations for the gas phase alone. In the current modified two-fluid model, a simplified approach is proposed. The momentum equation for the averaged velocity of both Group-1 and Group-2 bubbles is retained. By doing so, the velocity difference between Group-1 and Group-2 bubbles needs to be determined. This may be made either based on simplified momentum equations for both Group-1 and Group-2 bubbles or by a modified drift-flux model

  12. Numerical simulation for fractional order stationary neutron transport equation using Haar wavelet collocation method

    Energy Technology Data Exchange (ETDEWEB)

    Saha Ray, S., E-mail: santanusaharay@yahoo.com; Patra, A.

    2014-10-15

    Highlights: • A stationary transport equation has been solved using the technique of Haar wavelet collocation method. • This paper intends to provide the great utility of Haar wavelets to nuclear science problem. • In the present paper, two-dimensional Haar wavelets are applied. • The proposed method is mathematically very simple, easy and fast. - Abstract: In this paper the numerical solution for the fractional order stationary neutron transport equation is presented using Haar wavelet Collocation Method (HWCM). Haar wavelet collocation method is efficient and powerful in solving wide class of linear and nonlinear differential equations. This paper intends to provide an application of Haar wavelets to nuclear science problems. This paper describes the application of Haar wavelets for the numerical solution of fractional order stationary neutron transport equation in homogeneous medium with isotropic scattering. The proposed method is mathematically very simple, easy and fast. To demonstrate about the efficiency and applicability of the method, two test problems are discussed.

  13. Least-squares finite element discretizations of neutron transport equations in 3 dimensions

    Energy Technology Data Exchange (ETDEWEB)

    Manteuffel, T.A [Univ. of Colorado, Boulder, CO (United States); Ressel, K.J. [Interdisciplinary Project Center for Supercomputing, Zurich (Switzerland); Starkes, G. [Universtaet Karlsruhe (Germany)

    1996-12-31

    The least-squares finite element framework to the neutron transport equation introduced in is based on the minimization of a least-squares functional applied to the properly scaled neutron transport equation. Here we report on some practical aspects of this approach for neutron transport calculations in three space dimensions. The systems of partial differential equations resulting from a P{sub 1} and P{sub 2} approximation of the angular dependence are derived. In the diffusive limit, the system is essentially a Poisson equation for zeroth moment and has a divergence structure for the set of moments of order 1. One of the key features of the least-squares approach is that it produces a posteriori error bounds. We report on the numerical results obtained for the minimum of the least-squares functional augmented by an additional boundary term using trilinear finite elements on a uniform tesselation into cubes.

  14. Stability result for Navier-Stokes equations with entropy transport

    Czech Academy of Sciences Publication Activity Database

    Michálek, Martin

    2015-01-01

    Roč. 17, č. 2 (2015), s. 279-285 ISSN 1422-6928 R&D Projects: GA ČR GA13-00522S Institutional support: RVO:67985840 Keywords : compressible Navier-Stokes system * entropy transport * effective viscous flux Subject RIV: BA - General Mathematics Impact factor: 1.023, year: 2015 http://link.springer.com/article/10.1007%2Fs00021-015-0205-x

  15. Generalized fluid equations for parallel transport in collisional to weakly collisional plasmas

    International Nuclear Information System (INIS)

    Zawaideh, E.; Najmabadi, F.; Conn, R.W.

    1986-01-01

    A new set of two-fluid equations that are valid from collisional to weakly collisional limits is derived. Starting from gyrokinetic equations in flux coordinates with no zero-order drifts, a set of moment equations describing plasma transport along the field lines of a space- and time-dependent magnetic field is derived. No restriction on the anisotropy of the ion distribution function is imposed. In the highly collisional limit, these equations reduce to those of Braginskii, while in the weakly collisional limit they are similar to the double adiabatic or Chew, Goldberger, and Low (CGL) equations [Proc. R. Soc. London, Ser. A 236, 112 (1956)]. The new set of equations also exhibits a physical singularity at the sound speed. This singularity is used to derive and compute the sound speed. Numerical examples comparing these equations with conventional transport equations show that in the limit where the ratio of the mean free path lambda to the scale length of the magnetic field gradient L/sub B/ approaches zero, there is no significant difference between the solution of the new and conventional transport equations. However, conventional fluid equations, ordinarily expected to be correct to the order (lambda/L/sub B/) 2 , are found to have errors of order (lambda/L/sub u/) 2 = (lambda/L/sub B/) 2 /(1-M 2 ) 2 , where L/sub u/ is the scale length of the flow velocity gradient and M is the Mach number. As such, the conventional equations may contain large errors near the sound speed (Mroughly-equal1)

  16. Molecular representation of molar domain (volume), evolution equations, and linear constitutive relations for volume transport.

    Science.gov (United States)

    Eu, Byung Chan

    2008-09-07

    In the traditional theories of irreversible thermodynamics and fluid mechanics, the specific volume and molar volume have been interchangeably used for pure fluids, but in this work we show that they should be distinguished from each other and given distinctive statistical mechanical representations. In this paper, we present a general formula for the statistical mechanical representation of molecular domain (volume or space) by using the Voronoi volume and its mean value that may be regarded as molar domain (volume) and also the statistical mechanical representation of volume flux. By using their statistical mechanical formulas, the evolution equations of volume transport are derived from the generalized Boltzmann equation of fluids. Approximate solutions of the evolution equations of volume transport provides kinetic theory formulas for the molecular domain, the constitutive equations for molar domain (volume) and volume flux, and the dissipation of energy associated with volume transport. Together with the constitutive equation for the mean velocity of the fluid obtained in a previous paper, the evolution equations for volume transport not only shed a fresh light on, and insight into, irreversible phenomena in fluids but also can be applied to study fluid flow problems in a manner hitherto unavailable in fluid dynamics and irreversible thermodynamics. Their roles in the generalized hydrodynamics will be considered in the sequel.

  17. Transition flow ion transport via integral Boltzmann equation

    International Nuclear Information System (INIS)

    Darcie, T.E.

    1983-10-01

    A new approach is developed to solve the Integral Boltzmann Equation for the evolving velocity distribution of a source of ions, undergoing electrostatic acceleration through a neutral gas target. The theory is applicable to arbitrarily strong electric fields, any ion/neutral mass ratio greater than unity, and is not limited to spatially isotropic gas targets. A hard sphere collision model is used, with a provision for inelasticity. Both axial and radial velocity distributions are calculated for applications where precollision radial velocities are negligible, as is the case for ion beam extractions from high pressure sources. Theoretical predictions are tested through an experiment in which an atmospheric pressure ion source is coupled to a high vacuum energy analyser. Excellent agreement results for configurations in which the radial velocity remains small. Velocity distributions are applied to predicting the efficiency of coupling an atmospheric pressure ion source to a quadrupole mass spectrometer and results clearly indicate the most desirable extracting configuration. A method is devised to calculate ion-molecule hard sphere collision cross sections for easily fragmented organic ions

  18. Solution and Study of the Two-Dimensional Nodal Neutron Transport Equation

    International Nuclear Information System (INIS)

    Panta Pazos, Ruben; Biasotto Hauser, Eliete; Tullio de Vilhena, Marco

    2002-01-01

    In the last decade Vilhena and coworkers reported an analytical solution to the two-dimensional nodal discrete-ordinates approximations of the neutron transport equation in a convex domain. The key feature of these works was the application of the combined collocation method of the angular variable and nodal approach in the spatial variables. By nodal approach we mean the transverse integration of the SN equations. This procedure leads to a set of one-dimensional S N equations for the average angular fluxes in the variables x and y. These equations were solved by the old version of the LTS N method, which consists in the application of the Laplace transform to the set of nodal S N equations and solution of the resulting linear system by symbolic computation. It is important to recall that this procedure allow us to increase N the order of S N up to 16. To overcome this drawback we step forward performing a spectral painstaking analysis of the nodal S N equations for N up to 16 and we begin the convergence of the S N nodal equations defining an error for the angular flux and estimating the error in terms of the truncation error of the quadrature approximations of the integral term. Furthermore, we compare numerical results of this approach with those of other techniques used to solve the two-dimensional discrete approximations of the neutron transport equation. (authors)

  19. Nature of complex time eigenvalues of the one speed transport equation in a homogeneous sphere

    International Nuclear Information System (INIS)

    Dahl, E.B.; Sahni, D.C.

    1990-01-01

    The complex time eigenvalues of the transport equation have been studied for one speed neutrons, scattered isotropically in a homogeneous sphere with vacuum boundary conditions. It is shown that the complex decay constants vary continuously with the radius of the sphere. Our earlier conjecture (Dahl and Sahni (1983-84)) regarding disjoint arcs is thus shown to be true. We also indicate that complex decay constants exist even for large assemblies, though with rapid oscillations in the corresponding eigenvectors. These modes cannot be predicted by the diffusion equation as this behaviour of the eigenvectors contradicts the assumption of 'slowly varying flux' needed to derive the diffusion approximation from the transport equation. For an infinite system, the existence of complex modes is related to the solution of a homogeneous equation. (author)

  20. A time dependent mixing model to close PDF equations for transport in heterogeneous aquifers

    Science.gov (United States)

    Schüler, L.; Suciu, N.; Knabner, P.; Attinger, S.

    2016-10-01

    Probability density function (PDF) methods are a promising alternative to predicting the transport of solutes in groundwater under uncertainty. They make it possible to derive the evolution equations of the mean concentration and the concentration variance, used in moment methods. The mixing model, describing the transport of the PDF in concentration space, is essential for both methods. Finding a satisfactory mixing model is still an open question and due to the rather elaborate PDF methods, a difficult undertaking. Both the PDF equation and the concentration variance equation depend on the same mixing model. This connection is used to find and test an improved mixing model for the much easier to handle concentration variance. Subsequently, this mixing model is transferred to the PDF equation and tested. The newly proposed mixing model yields significantly improved results for both variance modelling and PDF modelling.

  1. Second order time evolution of the multigroup diffusion and P1 equations for radiation transport

    International Nuclear Information System (INIS)

    Olson, Gordon L.

    2011-01-01

    Highlights: → An existing multigroup transport algorithm is extended to be second-order in time. → A new algorithm is presented that does not require a grey acceleration solution. → The two algorithms are tested with 2D, multi-material problems. → The two algorithms have comparable computational requirements. - Abstract: An existing solution method for solving the multigroup radiation equations, linear multifrequency-grey acceleration, is here extended to be second order in time. This method works for simple diffusion and for flux-limited diffusion, with or without material conduction. A new method is developed that does not require the solution of an averaged grey transport equation. It is effective solving both the diffusion and P 1 forms of the transport equation. Two dimensional, multi-material test problems are used to compare the solution methods.

  2. Transport equations, Level Set and Eulerian mechanics. Application to fluid-structure coupling

    International Nuclear Information System (INIS)

    Maitre, E.

    2008-11-01

    My works were devoted to numerical analysis of non-linear elliptic-parabolic equations, to neutron transport equation and to the simulation of fabrics draping. More recently I developed an Eulerian method based on a level set formulation of the immersed boundary method to deal with fluid-structure coupling problems arising in bio-mechanics. Some of the more efficient algorithms to solve the neutron transport equation make use of the splitting of the transport operator taking into account its characteristics. In the present work we introduced a new algorithm based on this splitting and an adaptation of minimal residual methods to infinite dimensional case. We present the case where the velocity space is of dimension 1 (slab geometry) and 2 (plane geometry) because the splitting is simpler in the former

  3. Comparison of preconditioned generalized conjugate gradient methods to two-dimensional neutron and photon transport equation

    International Nuclear Information System (INIS)

    Chen, G.S.

    1997-01-01

    We apply and compare the preconditioned generalized conjugate gradient methods to solve the linear system equation that arises in the two-dimensional neutron and photon transport equation in this paper. Several subroutines are developed on the basis of preconditioned generalized conjugate gradient methods for time-independent, two-dimensional neutron and photon transport equation in the transport theory. These generalized conjugate gradient methods are used. TFQMR (transpose free quasi-minimal residual algorithm), CGS (conjuage gradient square algorithm), Bi-CGSTAB (bi-conjugate gradient stabilized algorithm) and QMRCGSTAB (quasi-minimal residual variant of bi-conjugate gradient stabilized algorithm). These sub-routines are connected to computer program DORT. Several problems are tested on a personal computer with Intel Pentium CPU. (author)

  4. A general analytical approach to the one-group, one-dimensional transport equation

    International Nuclear Information System (INIS)

    Barichello, L.B.; Vilhena, M.T.

    1993-01-01

    The main feature of the presented approach to solve the neutron transport equation consists in the application of the Laplace transform to the discrete ordinates equations, which yields a linear system of order N to be solved (LTS N method). In this paper this system is solved analytically and the inversion is performed using the Heaviside expansion technique. The general formulation achieved by this procedure is then applied to homogeneous and heterogeneous one-group slab-geometry problems. (orig.) [de

  5. Generalized multivariate Fokker-Planck equations derived from kinetic transport theory and linear nonequilibrium thermodynamics

    International Nuclear Information System (INIS)

    Frank, T.D.

    2002-01-01

    We study many particle systems in the context of mean field forces, concentration-dependent diffusion coefficients, generalized equilibrium distributions, and quantum statistics. Using kinetic transport theory and linear nonequilibrium thermodynamics we derive for these systems a generalized multivariate Fokker-Planck equation. It is shown that this Fokker-Planck equation describes relaxation processes, has stationary maximum entropy distributions, can have multiple stationary solutions and stationary solutions that differ from Boltzmann distributions

  6. Variance estimates for transport in stochastic media by means of the master equation

    International Nuclear Information System (INIS)

    Pautz, S. D.; Franke, B. C.; Prinja, A. K.

    2013-01-01

    The master equation has been used to examine properties of transport in stochastic media. It has been shown previously that not only may the Levermore-Pomraning (LP) model be derived from the master equation for a description of ensemble-averaged transport quantities, but also that equations describing higher-order statistical moments may be obtained. We examine in greater detail the equations governing the second moments of the distribution of the angular fluxes, from which variances may be computed. We introduce a simple closure for these equations, as well as several models for estimating the variances of derived transport quantities. We revisit previous benchmarks for transport in stochastic media in order to examine the error of these new variance models. We find, not surprisingly, that the errors in these variance estimates are at least as large as the corresponding estimates of the average, and sometimes much larger. We also identify patterns in these variance estimates that may help guide the construction of more accurate models. (authors)

  7. Advances in the solution of three-dimensional nodal neutron transport equation

    International Nuclear Information System (INIS)

    Pazos, Ruben Panta; Hauser, Eliete Biasotto; Vilhena, Marco Tullio de

    2003-01-01

    In this paper we study the three-dimensional nodal discrete-ordinates approximations of neutron transport equation in a convex domain with piecewise smooth boundaries. We use the combined collocation method of the angular variables and nodal approach for the spatial variables. By nodal approach we mean the iterated transverse integration of the S N equations. This procedure leads to the set of one-dimensional averages angular fluxes in each spatial variable. The resulting system of equations is solved with the LTS N method, first applying the Laplace transform to the set of the nodal S N equations and then obtaining the solution by symbolic computation. We include the LTS N method by diagonalization to solve the nodal neutron transport equation and then we outline the convergence of these nodal-LTS N approximations with the help of a norm associated to the quadrature formula used to approximate the integral term of the neutron transport equation. We give numerical results obtained with an algebraic computer system (for N up to 8) and with a code for higher values of N. We compare our results for the geometry of a box with a source in a vertex and a leakage zone in the opposite with others techniques used in this problem. (author)

  8. Solving the transport equation with quadratic finite elements: Theory and applications

    International Nuclear Information System (INIS)

    Ferguson, J.M.

    1997-01-01

    At the 4th Joint Conference on Computational Mathematics, the author presented a paper introducing a new quadratic finite element scheme (QFEM) for solving the transport equation. In the ensuing year the author has obtained considerable experience in the application of this method, including solution of eigenvalue problems, transmission problems, and solution of the adjoint form of the equation as well as the usual forward solution. He will present detailed results, and will also discuss other refinements of his transport codes, particularly for 3-dimensional problems on rectilinear and non-rectilinear grids

  9. The Transport Equation in Optically Thick Media: Discussion of IMC and its Diffusion Limit

    Energy Technology Data Exchange (ETDEWEB)

    Szoke, A. [Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States); Brooks, E. D. [Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)

    2016-07-12

    We discuss the limits of validity of the Implicit Monte Carlo (IMC) method for the transport of thermally emitted radiation. The weakened coupling between the radiation and material energy of the IMC method causes defects in handling problems with strong transients. We introduce an approach to asymptotic analysis for the transport equation that emphasizes the fact that the radiation and material temperatures are always different in time-dependent problems, and we use it to show that IMC does not produce the correct diffusion limit. As this is a defect of IMC in the continuous equations, no improvement to its discretization can remedy it.

  10. Development of a polynomial nodal model to the multigroup transport equation in one dimension

    International Nuclear Information System (INIS)

    Feiz, M.

    1986-01-01

    A polynomial nodal model that uses Legendre polynomial expansions was developed for the multigroup transport equation in one dimension. The development depends upon the least-squares minimization of the residuals using the approximate functions over the node. Analytical expressions were developed for the polynomial coefficients. The odd moments of the angular neutron flux over the half ranges were used at the internal interfaces, and the Marshak boundary condition was used at the external boundaries. Sample problems with fine-mesh finite-difference solutions of the diffusion and transport equations were used for comparison with the model

  11. Coupled force-balance and particle-occupation rate equations for high-field electron transport

    International Nuclear Information System (INIS)

    Lei, X. L.

    2008-01-01

    It is pointed out that in the framework of balance-equation approach, the coupled force-balance and particle-occupation rate equations can be used as a complete set of equations to determine the high-field transport of semiconductors in both strong and weak electron-electron interaction limits. We call to attention that the occupation rate equation conserves the total particle number and maintains the energy balance of the relative electron system, and there is no need to introduce any other term in it. The addition of an energy-drift term in the particle-occupation rate equation [Phys. Rev. B 71, 195205 (2005)] is physically inadequate for the violation of the total particle-number conservation and the energy balance. It may lead to a substantial unphysical increase of the total particle number by the application of a dc electric field

  12. Simulation of neutron transport equation using parallel Monte Carlo for deep penetration problems

    International Nuclear Information System (INIS)

    Bekar, K. K.; Tombakoglu, M.; Soekmen, C. N.

    2001-01-01

    Neutron transport equation is simulated using parallel Monte Carlo method for deep penetration neutron transport problem. Monte Carlo simulation is parallelized by using three different techniques; direct parallelization, domain decomposition and domain decomposition with load balancing, which are used with PVM (Parallel Virtual Machine) software on LAN (Local Area Network). The results of parallel simulation are given for various model problems. The performances of the parallelization techniques are compared with each other. Moreover, the effects of variance reduction techniques on parallelization are discussed

  13. Neutron filters for producing monoenergetic neutron beams

    International Nuclear Information System (INIS)

    Harvey, J.A.; Hill, N.W.; Harvey, J.R.

    1982-01-01

    Neutron transmission measurements have been made on high-purity, highly-enriched samples of 58 Ni (99.9%), 60 Ni (99.7%), 64 Zn (97.9%) and 184 W (94.5%) to measure their neutron windows and to assess their potential usefulness for producing monoenergetic beams of intermediate energies from a reactor. Transmission measurements on the Los Alamos Sc filter (44.26 cm Sc and 1.0 cm Ti) have been made to determine the characteristics of the transmitted neutron beam and to measure the total cross section of Sc at the 2.0 keV minimum. When corrected for the Ti and impurities, a value of 0.35 +- 0.03 b was obtained for this minimum

  14. Unified implicit kinetic scheme for steady multiscale heat transfer based on the phonon Boltzmann transport equation

    Science.gov (United States)

    Zhang, Chuang; Guo, Zhaoli; Chen, Songze

    2017-12-01

    An implicit kinetic scheme is proposed to solve the stationary phonon Boltzmann transport equation (BTE) for multiscale heat transfer problem. Compared to the conventional discrete ordinate method, the present method employs a macroscopic equation to accelerate the convergence in the diffusive regime. The macroscopic equation can be taken as a moment equation for phonon BTE. The heat flux in the macroscopic equation is evaluated from the nonequilibrium distribution function in the BTE, while the equilibrium state in BTE is determined by the macroscopic equation. These two processes exchange information from different scales, such that the method is applicable to the problems with a wide range of Knudsen numbers. Implicit discretization is implemented to solve both the macroscopic equation and the BTE. In addition, a memory reduction technique, which is originally developed for the stationary kinetic equation, is also extended to phonon BTE. Numerical comparisons show that the present scheme can predict reasonable results both in ballistic and diffusive regimes with high efficiency, while the memory requirement is on the same order as solving the Fourier law of heat conduction. The excellent agreement with benchmark and the rapid converging history prove that the proposed macro-micro coupling is a feasible solution to multiscale heat transfer problems.

  15. Inelastic Quantum Transport in Superlattices: Success and Failure of the Boltzmann Equation

    DEFF Research Database (Denmark)

    Wacker, Andreas; Jauho, Antti-Pekka; Rott, Stephan

    1999-01-01

    the whole held range from linear response to negative differential conductivity. The quantum results are compared with the respective results obtained from a Monte Carlo solution of the Boltzmann equation. Our analysis thus sets the limits of validity for the semiclassical theory in a nonlinear transport...

  16. Parallel algorithms for 2-D cylindrical transport equations of Eigenvalue problem

    International Nuclear Information System (INIS)

    Wei, J.; Yang, S.

    2013-01-01

    In this paper, aimed at the neutron transport equations of eigenvalue problem under 2-D cylindrical geometry on unstructured grid, the discrete scheme of Sn discrete ordinate and discontinuous finite is built, and the parallel computation for the scheme is realized on MPI systems. Numerical experiments indicate that the designed parallel algorithm can reach perfect speedup, it has good practicality and scalability. (authors)

  17. On the relativistic transport equation for a discontinuity wave of multiplicity one

    International Nuclear Information System (INIS)

    Giambo, Sebastiano; Palumbo, Annunziata

    1980-01-01

    In the framework of the theory of the singular hypersurfaces, the transport equation for the amplitude of a discontinuity wave, corresponding to a simple characteristic of a quasi-linear hyperbolic system, is established in the context of special relativity [fr

  18. Measuring the contour of a wavefront using the Irradiance Transport Equation (ITE)

    Science.gov (United States)

    Castillo-Rodríguez, Luis; Granados-Agustín, Fermín; Fernández-Guasti, Manuel; Cornejo-Rodríguez, Alejandro

    2006-01-01

    The Irradiance Transport Equation (ITE), found by Teague, had been used in optics with different applications. One of the field where had been used is in optical testing, for example, with the method developed by Takeda. In this paper following the idea of using different optical and mathematical analysis method, theorical and experimental results are presented.

  19. On the history of a stochastic ansatz for solving the transport equation

    International Nuclear Information System (INIS)

    Williams, M.M.R.

    2010-01-01

    A very useful approximate tool for understanding the role of random material properties on solutions of the transport equation is described and its historical derivation given. The development of this stochastic tool, from its introduction by Randall, to its use in describing current problems involving dichotomic or pseudo-dichotomic Markov processes is discussed.

  20. FMCEIR: a Monte Carlo program for solving the stationary neutron and gamma transport equation

    International Nuclear Information System (INIS)

    Taormina, A.

    1978-05-01

    FMCEIR is a three-dimensional Monte Carlo program for solving the stationary neutron and gamma transport equation. It is used to study the problem of neutron and gamma streaming in the GCFR and HHT reactor channels. (G.T.H.)

  1. Solving the two-dimensional stationary transport equation with the aid of the nodal method

    International Nuclear Information System (INIS)

    Mesina, M.

    1976-07-01

    In this document the two-dimensional stationary transport equation for the geometry of a fuel assembly or for a system of square boxes has been formulated as an algebraic eigenvalue problem, and the solution was achieved with the computer code NODE 2 which was developed for this purpose. (orig.) [de

  2. Analytical solution for the transport equation for neutral particles in cylindrical and Cartesian geometry

    International Nuclear Information System (INIS)

    Goncalves, Glenio Aguiar

    2003-01-01

    In this work, we are reported analytical solutions for the transport equation for neutral particles in cylindrical and cartesian geometry. For the cylindrical geometry, it is applied the Hankel transform of order zero in the S N approximation of the one-dimensional cylindrical transport equation, assuming azimuthal symmetry and isotropic scattering. This procedure is coined HTSN method. The anisotropic problem is handled using the decomposition method, generating a recursive approach, which the HTSN solution is used as initial condition. For cartesian geometry, the one and two dimensional transport equation is derived in the angular variable as many time as the degree of the anisotropic scattering. This procedure leads to set of integro-differential plus one differential equation that can be really solved by the variable separation method. Following this procedure, it was possible to come out with the Case solution for the one-dimensional problem. Numerical simulations are reported for the cylindrical transport problem both isotropic and anisotropic case of quadratic degree. (author)

  3. Description of deeply inelastic collisions in terms of a transport equation

    International Nuclear Information System (INIS)

    Weidenmueller, H.A.

    1977-01-01

    A transport equation for deeply inelastic collisions is derived from a random-matrix model for the form factors for inelastic scattering and transfer reactions. The parametrization of these form factors is discussed. Results in one dimension indicate the importance of quantum fluctuations, and limitations of other approaches to the same problem. Results of three dimensions are compared with the data

  4. Application of finite element method in the solution of transport equation

    International Nuclear Information System (INIS)

    Maiorino, J.R.; Vieira, W.J.

    1985-01-01

    It is presented the application of finite element method in the solution of second order transport equation (self-adjoint) for the even parity flux. The angular component is treated by expansion in Legendre polinomials uncoupled of the spatial component, which is approached by an expansion in base functions, interpolated in each spatial element. (M.C.K.) [pt

  5. Steady-state transport equation resolution by particle methods, and numerical results

    International Nuclear Information System (INIS)

    Mercier, B.

    1985-10-01

    A method to solve steady-state transport equation has been given. Principles of the method are given. The method is studied in two different cases; estimations given by the theory are compared to numerical results. Results got in 1-D (spherical geometry) and in 2-D (axisymmetric geometry) are given [fr

  6. Resolution of the neutron transport equation by a three-dimensional least square method

    International Nuclear Information System (INIS)

    Varin, Elisabeth

    2001-01-01

    The knowledge of space and time distribution of neutrons with a certain energy or speed allows the exploitation and control of a nuclear reactor and the assessment of the irradiation dose about an irradiated nuclear fuel storage site. The neutron density is described by a transport equation. The objective of this research thesis is to develop a software for the resolution of this stationary equation in a three-dimensional Cartesian domain by means of a deterministic method. After a presentation of the transport equation, the author gives an overview of the different deterministic resolution approaches, identifies their benefits and drawbacks, and discusses the choice of the Ressel method. The least square method is precisely described and then applied. Numerical benchmarks are reported for validation purposes

  7. A numerical solution of the coupled proton-H atom transport equations for the proton aurora

    International Nuclear Information System (INIS)

    Basu, B.; Jasperse, J.R.; Grossbard, N.J.

    1990-01-01

    A numerical code has been developed to solve the coupled proton-H atom linear transport equations for the proton aurora. The transport equations have been simplified by using plane-parallel geometry and the forward-scattering approximations only. Otherwise, the equations and their numerical solutions are exact. Results are presented for the particle fluxes and the energy deposition rates, and they are compared with the previous analytical results that were obtained by using additional simplifying approximations. It is found that although the analytical solutions for the particle fluxes differ somewhat from the numerical solutions, the energy deposition rates calculated by the two methods agree to within a few percent. The accurate particle fluxes given by the numerical code are useful for accurate calculation of the characteristic quantities of the proton aurora, such as the ionization rates and the emission rates

  8. Radiative transport equation for the Mittag-Leffler path length distribution

    Science.gov (United States)

    Liemert, André; Kienle, Alwin

    2017-05-01

    In this paper, we consider the radiative transport equation for infinitely extended scattering media that are characterized by the Mittag-Leffler path length distribution p (ℓ ) =-∂ℓEα(-σtℓα ) , which is a generalization of the usually assumed Lambert-Beer law p (ℓ ) =σtexp(-σtℓ ) . In this context, we derive the infinite-space Green's function of the underlying fractional transport equation for the spherically symmetric medium as well as for the one-dimensional string. Moreover, simple analytical solutions are presented for the prediction of the radiation field in the single-scattering approximation. The resulting equations are compared with Monte Carlo simulations in the steady-state and time domain showing, within the stochastic nature of the simulations, an excellent agreement.

  9. Application of the finite element method to the neutron transport equation

    International Nuclear Information System (INIS)

    Martin, W.R.

    1976-01-01

    This paper examines the theoretical and practical application of the finite element method to the neutron transport equation. It is shown that in principle the system of equations obtained by application of the finite element method can be solved with certain physical restrictions concerning the criticality of the medium. The convergence of this approximate solution to the exact solution with mesh refinement is examined, and a non-optical estimate of the convergence rate is obtained analytically. It is noted that the numerical results indicate a faster convergence rate and several approaches to obtain this result analytically are outlined. The practical application of the finite element method involved the development of a computer code capable of solving the neutron transport equation in 1-D plane geometry. Vacuum, reflecting, or specified incoming boundary conditions may be analyzed, and all are treated as natural boundary conditions. The time-dependent transport equation is also examined and it is shown that the application of the finite element method in conjunction with the Crank-Nicholson time discretization method results in a system of algebraic equations which is readily solved. Numerical results are given for several critical slab eigenvalue problems, including anisotropic scattering, and the results compare extremely well with benchmark results. It is seen that the finite element code is more efficient than a standard discrete ordinates code for certain problems. A problem with severe heterogeneities is considered and it is shown that the use of discontinuous spatial and angular elements results in a marked improvement in the results. Finally, time-dependent problems are examined and it is seen that the phenomenon of angular mode separation makes the numerical treatment of the transport equation in slab geometry a considerable challenge, with the result that the angular mesh has a dominant effect on obtaining acceptable solutions

  10. Some results on the neutron transport and the coupling of equations

    International Nuclear Information System (INIS)

    Bal, G.

    1997-01-01

    Neutron transport in nuclear reactors is well modeled by the linear Boltzmann transport equation. Its resolution is relatively easy but very expensive. To achieve whole core calculations, one has to consider simpler models, such as diffusion or homogeneous transport equations. However, the solutions may become inaccurate in particular situations (as accidents for instance). That is the reason why we wish to solve the equations on small area accurately and more coarsely on the remaining part of the core. It is than necessary to introduce some links between different discretizations or modelizations. In this note, we give some results on the coupling of different discretizations of all degrees of freedom of the integral-differential neutron transport equation (two degrees for the angular variable, on for the energy component, and two or three degrees for spatial position respectively in 2D (cylindrical symmetry) and 3D). Two chapters are devoted to the coupling of discrete ordinates methods (for angular discretization). The first one is theoretical and shows the well posing of the coupled problem, whereas the second one deals with numerical applications of practical interest (the results have been obtained from the neutron transport code developed at the R and D, which has been modified for introducing the coupling). Next, we present the nodal scheme RTN0, used for the spatial discretization. We show well posing results for the non-coupled and the coupled problems. At the end, we deal with the coupling of energy discretizations for the multigroup equations obtained by homogenization. Some theoretical results of the discretization of the velocity variable (well-posing of problems), which do not deal directly with the purposes of coupling, are presented in the annexes. (author)

  11. Spatially adaptive hp refinement approach for PN neutron transport equation using spectral element method

    International Nuclear Information System (INIS)

    Nahavandi, N.; Minuchehr, A.; Zolfaghari, A.; Abbasi, M.

    2015-01-01

    Highlights: • Powerful hp-SEM refinement approach for P N neutron transport equation has been presented. • The method provides great geometrical flexibility and lower computational cost. • There is a capability of using arbitrary high order and non uniform meshes. • Both posteriori and priori local error estimation approaches have been employed. • High accurate results are compared against other common adaptive and uniform grids. - Abstract: In this work we presented the adaptive hp-SEM approach which is obtained from the incorporation of Spectral Element Method (SEM) and adaptive hp refinement. The SEM nodal discretization and hp adaptive grid-refinement for even-parity Boltzmann neutron transport equation creates powerful grid refinement approach with high accuracy solutions. In this regard a computer code has been developed to solve multi-group neutron transport equation in one-dimensional geometry using even-parity transport theory. The spatial dependence of flux has been developed via SEM method with Lobatto orthogonal polynomial. Two commonly error estimation approaches, the posteriori and the priori has been implemented. The incorporation of SEM nodal discretization method and adaptive hp grid refinement leads to high accurate solutions. Coarser meshes efficiency and significant reduction of computer program runtime in comparison with other common refining methods and uniform meshing approaches is tested along several well-known transport benchmarks

  12. Transport methods: general. 3. An Additive Angular-Dependent Re-balance Acceleration Method for Neutron Transport Equations

    International Nuclear Information System (INIS)

    Cho, Nam Zin; Park, Chang Je

    2001-01-01

    An additive angular-dependent re-balance (AADR) factor acceleration method is described to accelerate the source iteration of discrete ordinates transport calculation. The formulation of the AADR method follows that of the angular-dependent re-balance (ADR) method in that the re-balance factor is defined only on the cell interface and in that the low-order equation is derived by integrating the transport equation (high-order equation) over angular subspaces. But, the re-balance factor is applied additively. While the AADR method is similar to the boundary projection acceleration and the alpha-weighted linear acceleration, it is more general and does have distinct features. The method is easily extendible to DP N and low-order S N re-balancing, and it does not require consistent discretizations between the high- and low-order equations as in diffusion synthetic acceleration. We find by Fourier analysis and numerical results that the AADR method with a chosen form of weighting functions is unconditionally stable and very effective. There also exists an optimal weighting parameter that leads to the smallest spectral radius. The AADR acceleration method described in this paper is simple to implement, unconditionally stable, and very effective. It uses a physically based weighting function with an optimal parameter, leading to the best spectral radius of ρ<0.1865, compared to ρ<0.2247 of DSA. The application of the AADR acceleration method with the LMB scheme on a test problem shows encouraging results

  13. Self-Adjoint Angular Flux Equation for Coupled Electron-Photon Transport

    International Nuclear Information System (INIS)

    Liscum-Powell, J.L.; Lorence, L.J. Jr.; Morel, J.E.; Prinja, A.K.

    1999-01-01

    Recently, Morel and McGhee described an alternate second-order form of the transport equation called the self adjoint angular flux (SAAF) equation that has the angular flux as its unknown. The SAAF formulation has all the advantages of the traditional even- and odd-parity self-adjoint equations, with the added advantages that it yields the full angular flux when it is numerically solved, it is significantly easier to implement reflective and reflective-like boundary conditions, and in the appropriate form it can be solved in void regions. The SAAF equation has the disadvantage that the angular domain is the full unit sphere and, like the even- and odd- parity form, S n source iteration cannot be implemented using the standard sweeping algorithm. Also, problems arise in pure scattering media. Morel and McGhee demonstrated the efficacy of the SAAF formulation for neutral particle transport. Here we apply the SAAF formulation to coupled electron-photon transport problems using multigroup cross-sections from the CEPXS code and S n discretization

  14. Self-adjoint angular flux equation for coupled electron-photon transport

    International Nuclear Information System (INIS)

    Liscum-Powell, J.L.; Prinja, A.K.; Morel, J.E.; Lorence, L.J. Jr.

    1999-01-01

    Recently, Morel and McGhee described an alternate second-order form of the transport equation called the self-adjoint angular flux (SAAF) equation that has the angular flux as its unknown. The SAAF formulation has all the advantages of the traditional even- and odd-parity self-adjoint equations, with the added advantages that it yields the full angular flux when it is numerically solved, it is significantly easier to implement reflective and reflective-like boundary conditions, and in the appropriate form it can be solved in void regions. The SAAF equation has the disadvantage that the angular domain is the full unit sphere, and, like the even- and odd-parity form, S n source iteration cannot be implemented using the standard sweeping algorithm. Also, problems arise in pure scattering media. Morel and McGhee demonstrated the efficacy of the SAAF formulation for neutral particle transport. Here, the authors apply the SAAF formulation to coupled electron-photon transport problems using multigroup cross sections from the CEPXS code and S n discretization

  15. Analysis of an upstream weighted collocation approximation to the transport equation

    International Nuclear Information System (INIS)

    Shapiro, A.; Pinder, G.F.

    1981-01-01

    The numerical behavior of a modified orthogonal collocation method, as applied to the transport equations, can be examined through the use of a Fourier series analysis. The necessity of such a study becomes apparent in the analysis of several techniques which emulate classical upstream weighting schemes. These techniques are employed in orthogonal collocation and other numerical methods as a means of handling parabolic partial differential equations with significant first-order terms. Divergent behavior can be shown to exist in one upstream weighting method applied to orthogonal collocation

  16. Resolution of the steady state transport equation for Lagrangian geometry with cylindrical symmetry

    International Nuclear Information System (INIS)

    Samba, G.

    1983-05-01

    The purpose of this work is to solve the steady state transport equation for (r, z) geometries given by hydrodynamics calculations. The discontinuous finite element method for the space variables (r, z) provides a stable scheme which satisfies the particle balance equation. We are able to sweep cells for each direction over the mesh to have an explicit scheme. The graph theory provides a very efficient algorithm to compute this ordering array. Previously, we must divide all the quadrilaterals into two triangles to get only convex cells. Thus, we get a fast, vectorized calculation which gives a good accuracy on very distorted meshes [fr

  17. Electron and ion transport equations in computational weakly-ionized plasmadynamics

    International Nuclear Information System (INIS)

    Parent, Bernard; Macheret, Sergey O.; Shneider, Mikhail N.

    2014-01-01

    A new set of ion and electron transport equations is proposed to simulate steady or unsteady quasi-neutral or non-neutral multicomponent weakly-ionized plasmas through the drift–diffusion approximation. The proposed set of equations is advantaged over the conventional one by being considerably less stiff in quasi-neutral regions because it can be integrated in conjunction with a potential equation based on Ohm's law rather than Gauss's law. The present approach is advantaged over previous attempts at recasting the system by being applicable to plasmas with several types of positive ions and negative ions and by not requiring changes to the boundary conditions. Several test cases of plasmas enclosed by dielectrics and of glow discharges between electrodes show that the proposed equations yield the same solution as the standard equations but require 10 to 100 times fewer iterations to reach convergence whenever a quasi-neutral region forms. Further, several grid convergence studies indicate that the present approach exhibits a higher resolution (and hence requires fewer nodes to reach a given level of accuracy) when ambipolar diffusion is present. Because the proposed equations are not intrinsically linked to specific discretization or integration schemes and exhibit substantial advantages with no apparent disadvantage, they are generally recommended as a substitute to the fluid models in which the electric field is obtained from Gauss's law as long as the plasma remains weakly-ionized and unmagnetized

  18. Exponentially-convergent Monte Carlo for the 1-D transport equation

    International Nuclear Information System (INIS)

    Peterson, J. R.; Morel, J. E.; Ragusa, J. C.

    2013-01-01

    We define a new exponentially-convergent Monte Carlo method for solving the one-speed 1-D slab-geometry transport equation. This method is based upon the use of a linear discontinuous finite-element trial space in space and direction to represent the transport solution. A space-direction h-adaptive algorithm is employed to restore exponential convergence after stagnation occurs due to inadequate trial-space resolution. This methods uses jumps in the solution at cell interfaces as an error indicator. Computational results are presented demonstrating the efficacy of the new approach. (authors)

  19. The use of non-dimensional representation of the solute transport equations

    International Nuclear Information System (INIS)

    Laurens, J.-M.

    1988-07-01

    This report presents the results obtained in a pilot investigation into the use of non-dimensional representations of the solute transport equations, so as to improve the efficiency of the PRA codes used by the DoE and its contractors. A reduced set of parameters was obtained for a single layer transport model. As expected, the response was shown to be highly sensitive on the new parameters. A faster convergence of the system was observed when the sampling technique used was changed to take into account the properties of the new parameters, such that uniform coverage of the reduced parameter hyperspace was achieved. (author)

  20. Tokamak fluidlike equations, with applications to turbulence and transport in H mode discharges

    International Nuclear Information System (INIS)

    Kim, Y.B.; Biglari, H.; Carreras, B.A.; Diamond, P.H.; Groebner, R.J.; Kwon, O.J.; Spong, D.A.; Callen, J.D.; Chang, Z.; Hollenberg, J.B.; Sundaram, A.K.; Terry, P.W.; Wang, J.F.

    1990-01-01

    Significant progress has been made in developing tokamak fluidlike equations which are valid in all collisionality regimes in toroidal devices, and their applications to turbulence and transport in tokamaks. The areas highlighted in this paper include: the rigorous derivation of tokamak fluidlike equations via a generalized Chapman-Enskog procedure in various collisionality regimes and on various time scales; their application to collisionless and collisional drift wave models in a sheared slab geometry; applications to neoclassical drift wave turbulence; i.e. neoclassical ion-temperature-gradient-driven turbulence and neoclassical electron-drift-wave turbulence; applications to neoclassical bootstrap-current-driven turbulence; numerical simulation of nonlinear bootstrap-current-driven turbulence and tearing mode turbulence; transport in Hot-Ion H mode discharges. 20 refs., 3 figs

  1. CACTUS, a characteristics solution to the neutron transport equations in complicated geometries

    International Nuclear Information System (INIS)

    Halsall, M.J.

    1980-04-01

    CACTUS has been written to solve the multigroup neutron transport equation in a general two-dimensional geometry. The method is based upon a characteristics formulation for the problem in which the transport equation is integrated explicitly along straight line tracks that are suitably distributed throughout the problem. Source distributions and scattering are assumed to be isotropic, but the only restriction on geometry is that the outer boundary should be rectangular. Within this rectangular boundary the user is free to build his problem geometry using any combination of intersecting straight lines and circular arcs. The theory of the method is described, followed by some details of a coding, a sensitivity study on the number of tracks required to integrate fluxes in a particular problem, a user's guide, and a few test cases. (author)

  2. Numerical solution of the time dependent neutron transport equation by the method of the characteristics

    International Nuclear Information System (INIS)

    Talamo, Alberto

    2013-01-01

    This study presents three numerical algorithms to solve the time dependent neutron transport equation by the method of the characteristics. The algorithms have been developed taking into account delayed neutrons and they have been implemented into the novel MCART code, which solves the neutron transport equation for two-dimensional geometry and an arbitrary number of energy groups. The MCART code uses regular mesh for the representation of the spatial domain, it models up-scattering, and takes advantage of OPENMP and OPENGL algorithms for parallel computing and plotting, respectively. The code has been benchmarked with the multiplication factor results of a Boiling Water Reactor, with the analytical results for a prompt jump transient in an infinite medium, and with PARTISN and TDTORT results for cross section and source transients. The numerical simulations have shown that only two numerical algorithms are stable for small time steps

  3. Numerical solution of the time dependent neutron transport equation by the method of the characteristics

    Energy Technology Data Exchange (ETDEWEB)

    Talamo, Alberto, E-mail: alby@anl.gov [Nuclear Engineering Division, Argonne National Laboratory, 9700 South Cass Avenue, Lemont, IL 60439 (United States)

    2013-05-01

    This study presents three numerical algorithms to solve the time dependent neutron transport equation by the method of the characteristics. The algorithms have been developed taking into account delayed neutrons and they have been implemented into the novel MCART code, which solves the neutron transport equation for two-dimensional geometry and an arbitrary number of energy groups. The MCART code uses regular mesh for the representation of the spatial domain, it models up-scattering, and takes advantage of OPENMP and OPENGL algorithms for parallel computing and plotting, respectively. The code has been benchmarked with the multiplication factor results of a Boiling Water Reactor, with the analytical results for a prompt jump transient in an infinite medium, and with PARTISN and TDTORT results for cross section and source transients. The numerical simulations have shown that only two numerical algorithms are stable for small time steps.

  4. RTk/SN Solutions of the Two-Dimensional Multigroup Transport Equations in Hexagonal Geometry

    International Nuclear Information System (INIS)

    Valle, Edmundo del; Mund, Ernest H.

    2004-01-01

    This paper describes an extension to the hexagonal geometry of some weakly discontinuous nodal finite element schemes developed by Hennart and del Valle for the two-dimensional discrete ordinates transport equation in quadrangular geometry. The extension is carried out in a way similar to the extension to the hexagonal geometry of nodal element schemes for the diffusion equation using a composite mapping technique suggested by Hennart, Mund, and del Valle. The combination of the weakly discontinuous nodal transport scheme and the composite mapping is new and is detailed in the main section of the paper. The algorithm efficiency is shown numerically through some benchmark calculations on classical problems widely referred to in the literature

  5. Modeling of amorphous pocket formation in silicon by numerical solution of the heat transport equation

    International Nuclear Information System (INIS)

    Kovac, D.; Otto, G.; Hobler, G.

    2005-01-01

    In this paper we present a model of amorphous pocket formation that is based on binary collision simulations to generate the distribution of deposited energy, and on numerical solution of the heat transport equation to describe the quenching process. The heat transport equation is modified to consider the heat of melting when the melting temperature is crossed at any point in space. It is discretized with finite differences on grid points that coincide with the crystallographic lattice sites, which allows easy determination of molten atoms. Atoms are considered molten if the average of their energy and the energy of their neighbors meets the melting criterion. The results obtained with this model are in good overall agreement with published experimental data on P, As, Te and Tl implantations in Si and with data on the polyatomic effect at cryogenic temperature

  6. A multi scale approximation solution for the time dependent Boltzmann-transport equation

    International Nuclear Information System (INIS)

    Merk, B.

    2004-03-01

    The basis of all transient simulations for nuclear reactor cores is the reliable calculation of the power production. The local power distribution is generally calculated by solving the space, time, energy and angle dependent neutron transport equation known as Boltzmann equation. The computation of exact solutions of the Boltzmann equation is very time consuming. For practical numerical simulations approximated solutions are usually unavoidable. The objective of this work is development of an effective multi scale approximation solution for the Boltzmann equation. Most of the existing methods are based on separation of space and time. The new suggested method is performed without space-time separation. This effective approximation solution is developed on the basis of an expansion for the time derivative of different approximations to the Boltzmann equation. The method of multiple scale expansion is used for the expansion of the time derivative, because the problem of the stiff time behaviour can't be expressed by standard expansion methods. This multiple scale expansion is used in this work to develop approximation solutions for different approximations of the Boltzmann equation, starting from the expansion of the point kinetics equations. The resulting analytic functions are used for testing the applicability and accuracy of the multiple scale expansion method for an approximation solution with 2 delayed neutron groups. The results are tested versus the exact analytical results for the point kinetics equations. Very good agreement between both solutions is obtained. The validity of the solution with 2 delayed neutron groups to approximate the behaviour of the system with 6 delayed neutron groups is demonstrated in an additional analysis. A strategy for a solution with 4 delayed neutron groups is described. A multiple scale expansion is performed for the space-time dependent diffusion equation for one homogenized cell with 2 delayed neutron groups. The result is

  7. Exact solutions of the population balance equation including particle transport, using group analysis

    Science.gov (United States)

    Lin, Fubiao; Meleshko, Sergey V.; Flood, Adrian E.

    2018-06-01

    The population balance equation (PBE) has received an unprecedented amount of attention in recent years from both academics and industrial practitioners because of its long history, widespread use in engineering, and applicability to a wide variety of particulate and discrete-phase processes. However it is typically impossible to obtain analytical solutions, although in almost every case a numerical solution of the PBEs can be obtained. In this article, the symmetries of PBEs with homogeneous coagulation kernels involving aggregation, breakage and growth processes and particle transport in one dimension are found by direct solving the determining equations. Using the optimal system of one and two-dimensional subalgebras, all invariant solutions and reduced equations are obtained. In particular, an explicit analytical physical solution is also presented.

  8. A scalar flux - oriented method for the transport equation in slab geometry

    International Nuclear Information System (INIS)

    Budd, C.

    1981-01-01

    A new method for solving the neutron transport equation is described. An unusual feature of this method is that it deals principally with scalar fluxes rather than angular fluxes. An alternative approach in slab geometry promises to be cheaper to run and does not suffer from many of the problems of the discrete ordinates method. It also appears possible to extend the method to several dimensions and this is discussed. (U.K.)

  9. A parallel version of a multigrid algorithm for isotropic transport equations

    International Nuclear Information System (INIS)

    Manteuffel, T.; McCormick, S.; Yang, G.; Morel, J.; Oliveira, S.

    1994-01-01

    The focus of this paper is on a parallel algorithm for solving the transport equations in a slab geometry using multigrid. The spatial discretization scheme used is a finite element method called the modified linear discontinuous (MLD) scheme. The MLD scheme represents a lumped version of the standard linear discontinuous (LD) scheme. The parallel algorithm was implemented on the Connection Machine 2 (CM2). Convergence rates and timings for this algorithm on the CM2 and Cray-YMP are shown

  10. Numerical solution of neutron transport equations in discrete ordinates and slab geometry

    International Nuclear Information System (INIS)

    Serrano Pedraza, F.

    1985-01-01

    An unified formalism to solve numerically, between other equation, the neutron transport in discrete ordinates, slab geometry, several energy groups and independents of time, has been developed recently. Such a formalism cover some of the conventional schemes as diamond difference, (WDD) characteristic step (SC) lineal characteristic (LC), quadratic characteristic (QC) and lineal discontinuous. Unified formation gives before hand the convergence order of the previously selected scheme. In fact it allows besides to generate a big amount of numerical schemes, with which is also possible to solve numerical equations as soon as neutron transport. The essential purpose of this work was to solve the neutron transport equations in slab geometry and discrete ordinates considering several energy groups without to take under advisement time dependence based in the above mentioned unified formalism. To reach this purpose it was necesary to design a computer code with the name TNOD1 (Neutron transport in discrete ordinates and 1 dimension) which includes each one of the schemes already pointed out. there exist two numerical schemes, also recently developed, quadratic continuous (QC) and cubic continuous (CN), although covered by unified formalism, it has been possible to include them inside this computer code without make substantial changes in its structure. In chapter I, derivative of neutron transport equation independent of time is taken, for angular flux, including boundary conditions and discontinuity. In chapter II the neutron transport equations are obtained in multigroups, independents of time, for approximation of discrete ordinates. Description of theory related with unified formalism and its relationship with mentioned discretization schemes is presented in chapter III. Chapter IV describes the computer code developed and finally, in chapter V different numerical results obtained with TNOD1 program are shown. In Appendix A theorems and mathematical arguments used

  11. Finite-difference solution of the space-angle-lethargy-dependent slowing-down transport equation

    Energy Technology Data Exchange (ETDEWEB)

    Matausek, M V [Boris Kidric Vinca Institute of Nuclear Sciences, Vinca, Belgrade (Yugoslavia)

    1972-07-01

    A procedure has been developed for solving the slowing-down transport equation for a cylindrically symmetric reactor system. The anisotropy of the resonance neutron flux is treated by the spherical harmonics formalism, which reduces the space-angle-Iethargy-dependent transport equation to a matrix integro-differential equation in space and lethargy. Replacing further the lethargy transfer integral by a finite-difference form, a set of matrix ordinary differential equations is obtained, with lethargy-and space dependent coefficients. If the lethargy pivotal points are chosen dense enough so that the difference correction term can be ignored, this set assumes a lower block triangular form and can be solved directly by forward block substitution. As in each step of the finite-difference procedure a boundary value problem has to be solved for a non-homogeneous system of ordinary differential equations with space-dependent coefficients, application of any standard numerical procedure, for example, the finite-difference method or the method of adjoint equations, is too cumbersome and would make the whole procedure practically inapplicable. A simple and efficient approximation is proposed here, allowing analytical solution for the space dependence of the spherical-harmonics flux moments, and hence the derivation of the recurrence relations between the flux moments at successive lethargy pivotal points. According to the procedure indicated above a computer code has been developed for the CDC -3600 computer, which uses the KEDAK nuclear data file. The space and lethargy distribution of the resonance neutrons can be computed in such a detailed fashion as the neutron cross-sections are known for the reactor materials considered. The computing time is relatively short so that the code can be efficiently used, either autonomously, or as part of some complex modular scheme. Typical results will be presented and discussed in order to prove and illustrate the applicability of the

  12. Approximate method for solving the velocity dependent transport equation in a slab lattice

    International Nuclear Information System (INIS)

    Ferrari, A.

    1966-01-01

    A method is described that is intended to provide an approximate solution of the transport equation in a medium simulating a water-moderated plate filled reactor core. This medium is constituted by a periodic array of water channels and absorbing plates. The velocity dependent transport equation in slab geometry is included. The computation is performed in a water channel: the absorbing plates are accounted for by the boundary conditions. The scattering of neutrons in water is assumed isotropic, which allows the use of a double Pn approximation to deal with the angular dependence. This method is able to represent the discontinuity of the angular distribution at the channel boundary. The set of equations thus obtained is dependent only on x and v and the coefficients are independent on x. This solution suggests to try solutions involving Legendre polynomials. This scheme leads to a set of equations v dependent only. To obtain an explicit solution, a thermalization model must now be chosen. Using the secondary model of Cadilhac a solution of this set is easy to get. The numerical computations were performed with a particular secondary model, the well-known model of Wigner and Wilkins. (author) [fr

  13. From analytical solutions of solute transport equations to multidimensional time-domain random walk (TDRW) algorithms

    Science.gov (United States)

    Bodin, Jacques

    2015-03-01

    In this study, new multi-dimensional time-domain random walk (TDRW) algorithms are derived from approximate one-dimensional (1-D), two-dimensional (2-D), and three-dimensional (3-D) analytical solutions of the advection-dispersion equation and from exact 1-D, 2-D, and 3-D analytical solutions of the pure-diffusion equation. These algorithms enable the calculation of both the time required for a particle to travel a specified distance in a homogeneous medium and the mass recovery at the observation point, which may be incomplete due to 2-D or 3-D transverse dispersion or diffusion. The method is extended to heterogeneous media, represented as a piecewise collection of homogeneous media. The particle motion is then decomposed along a series of intermediate checkpoints located on the medium interface boundaries. The accuracy of the multi-dimensional TDRW method is verified against (i) exact analytical solutions of solute transport in homogeneous media and (ii) finite-difference simulations in a synthetic 2-D heterogeneous medium of simple geometry. The results demonstrate that the method is ideally suited to purely diffusive transport and to advection-dispersion transport problems dominated by advection. Conversely, the method is not recommended for highly dispersive transport problems because the accuracy of the advection-dispersion TDRW algorithms degrades rapidly for a low Péclet number, consistent with the accuracy limit of the approximate analytical solutions. The proposed approach provides a unified methodology for deriving multi-dimensional time-domain particle equations and may be applicable to other mathematical transport models, provided that appropriate analytical solutions are available.

  14. On the equation of transport for cosmic-ray particles in the interplanetary region

    International Nuclear Information System (INIS)

    Webb, G.M.; Gleeson, L.J.

    1979-01-01

    Two new alternative derivations of the equation of transport for cosmic-ray particles in the interplanetary region are provided. Both derivations are carried out by using particle position r and time t in a frame of reference fixed in the solar system, and the particle momentum p' is specified relative to a local frame of reference moving with the solar wind. The first derivation is carried out by writing down a continuity equation for the cosmic rays, taking into account particle streaming and energy changes, and subsequently deriving the streaming and energy change terms in this equation. The momentum change term in the continuity equation, previously considered to be due to the adiabatic deceleration of particles in the expanding magnetic fields carried by the solar wing, appears in the present analysis as a dynamic effect in which the Lorentz force on the particle does not appear explicitly. An alternative derivation based on the ensemble averaged Liouville equation for charged particles in the stochastic interplanetary magnetic field using (r,p',t) as independent coordinates is also given. The latter derivation confirms the momentum change interpretation of the first derivation. A new derivation of the adiabatic rate as a combination of inverse-Fermi and betatron deceleration processes is also provided. (Auth.)

  15. Revisiting Wiedemann-Franz law through Boltzmann transport equations and ab-initio density functional theory

    Science.gov (United States)

    Nag, Abhinav; Kumari, Anuja; Kumar, Jagdish

    2018-05-01

    We have investigated structural, electronic and transport properties of the alkali metals using ab-initio density functional theory. The electron energy dispersions are found parabolic free electron like which is expected for alkali metals. The lattice constants for all the studied metals are also in good agreement within 98% with experiments. We have further computed their transport properties using semi-classical Boltzmann transport equations with special focus on electrical and thermal conductivity. Our objective was to obtain Wiedemann-Franz law and hence Lorenz number. The motivation to do these calculations is to see that how the incorporation of different interactions such as electron-lattice, electron-electron interaction affect the Wiedeman-Franz law. By solving Boltzmann transport equations, we have obtained electrical conductivity (σ/τ) and thermal conductivity (κ0 /τ) at different temperatures and then calculated Lorenz number using L = κ0 /(σT). The obtained value of Lorenz number has been found to match with value derived for free electron Fermi gas 2.44× 10-8 WΩK-2. Our results prove that the Wiedemann-Franz law as derived for free electron gas does not change much for alkali metals, even when one incorporates interaction of electrons with atomic nuclei and other electrons. However, at lower temperatures, the Lorenz number, was found to be deviating from its theoretical value.

  16. Modelling uncertainties in the diffusion-advection equation for radon transport in soil using interval arithmetic.

    Science.gov (United States)

    Chakraverty, S; Sahoo, B K; Rao, T D; Karunakar, P; Sapra, B K

    2018-02-01

    Modelling radon transport in the earth crust is a useful tool to investigate the changes in the geo-physical processes prior to earthquake event. Radon transport is modeled generally through the deterministic advection-diffusion equation. However, in order to determine the magnitudes of parameters governing these processes from experimental measurements, it is necessary to investigate the role of uncertainties in these parameters. Present paper investigates this aspect by combining the concept of interval uncertainties in transport parameters such as soil diffusivity, advection velocity etc, occurring in the radon transport equation as applied to soil matrix. The predictions made with interval arithmetic have been compared and discussed with the results of classical deterministic model. The practical applicability of the model is demonstrated through a case study involving radon flux measurements at the soil surface with an accumulator deployed in steady-state mode. It is possible to detect the presence of very low levels of advection processes by applying uncertainty bounds on the variations in the observed concentration data in the accumulator. The results are further discussed. Copyright © 2017 Elsevier Ltd. All rights reserved.

  17. Analytical solutions of a fractional diffusion-advection equation for solar cosmic-ray transport

    International Nuclear Information System (INIS)

    Litvinenko, Yuri E.; Effenberger, Frederic

    2014-01-01

    Motivated by recent applications of superdiffusive transport models to shock-accelerated particle distributions in the heliosphere, we analytically solve a one-dimensional fractional diffusion-advection equation for the particle density. We derive an exact Fourier transform solution, simplify it in a weak diffusion approximation, and compare the new solution with previously available analytical results and with a semi-numerical solution based on a Fourier series expansion. We apply the results to the problem of describing the transport of energetic particles, accelerated at a traveling heliospheric shock. Our analysis shows that significant errors may result from assuming an infinite initial distance between the shock and the observer. We argue that the shock travel time should be a parameter of a realistic superdiffusive transport model.

  18. Molecular dynamics studies of transport properties and equation of state of supercritical fluids

    Science.gov (United States)

    Nwobi, Obika C.

    Many chemical propulsion systems operate with one or more of the reactants above the critical point in order to enhance their performance. Most of the computational fluid dynamics (CFD) methods used to predict these flows require accurate information on the transport properties and equation of state at these supercritical conditions. This work involves the determination of transport coefficients and equation of state of supercritical fluids by equilibrium molecular dynamics (MD) simulations on parallel computers using the Green-Kubo formulae and the virial equation of state, respectively. MD involves the solution of equations of motion of a system of molecules that interact with each other through an intermolecular potential. Provided that an accurate potential can be found for the system of interest, MD can be used regardless of the phase and thermodynamic conditions of the substances involved. The MD program uses the effective Lennard-Jones potential, with system sizes of 1000-1200 molecules and, simulations of 2,000,000 time-steps for computing transport coefficients and 200,000 time-steps for pressures. The computer code also uses linked cell lists for efficient sorting of molecules, periodic boundary conditions, and a modified velocity Verlet algorithm for particle displacement. Particle decomposition is used for distributing the molecules to different processors of a parallel computer. Simulations have been carried out on pure argon, nitrogen, oxygen and ethylene at various supercritical conditions, with self-diffusion coefficients, shear viscosity coefficients, thermal conductivity coefficients and pressures computed for most of the conditions. Results compare well with experimental and the National Institute of Standards and Technology (NIST) values. The results show that the number of molecules and the potential cut-off radius have no significant effect on the computed coefficients, while long-time integration is necessary for accurate determination of the

  19. Solution of the Boltzmann-Fokker-Planck transport equation using exponential nodal schemes

    International Nuclear Information System (INIS)

    Ortega J, R.; Valle G, E. del

    2003-01-01

    There are carried out charge and energy calculations deposited due to the interaction of electrons with a plate of a certain material, solving numerically the electron transport equation for the Boltzmann-Fokker-Planck approach of first order in plate geometry with a computer program denominated TEOD-NodExp (Transport of Electrons in Discreet Ordinates, Nodal Exponentials), using the proposed method by the Dr. J. E. Morel to carry out the discretization of the variable energy and several spatial discretization schemes, denominated exponentials nodal. It is used the Fokker-Planck equation since it represents an approach of the Boltzmann transport equation that is been worth whenever it is predominant the dispersion of small angles, that is to say, resulting dispersion in small dispersion angles and small losses of energy in the transport of charged particles. Such electrons could be those that they face with a braking plate in a device of thermonuclear fusion. In the present work its are considered electrons of 1 MeV that impact isotropically on an aluminum plate. They were considered three different thickness of plate that its were designated as problems 1, 2 and 3. In the calculations it was used the discrete ordinate method S 4 with expansions of the dispersion cross sections until P 3 order. They were considered 25 energy groups of uniform size between the minimum energy of 0.1 MeV and the maximum of 1.0 MeV; the one spatial intervals number it was considered variable and it was assigned the values of 10, 20 and 30. (Author)

  20. A new approach to obtaining the roots of the dispersion equation for slab geometry multiplying media

    International Nuclear Information System (INIS)

    Silva, Davi J.M.; Barros, Ricardo C.; Alves Filho, Hermes

    2013-01-01

    In this work we describe an alternative approach for obtaining the roots of the dispersion equation. For the mathematical model, we used the slab-geometry neutron transport equation in the discrete ordinates (S N ), formulation, considering isotropic scattering and monoenergetic model. The basic idea is to find a basis for the kernel of the S N differential operator, whose elements are exponential eigenfunctions corresponding to distinct eigenvalues which are the roots of the dispersion equation. That strategy yields a gain in programming computational codes, including the strategy used to obtain the purely imaginary eigenvalues and their associated complex eigenfunctions, that appear in the spectral analysis of the S N equations in multiplying media. These eigenvalues and corresponding eigenfunctions are used to obtain the parameters of the auxiliary equations of the spectral nodal methods, e.g., the spectral diamond (SD) auxiliary equation. (author)

  1. Development of two-group interfacial area transport equation for confined flow-2. Model evaluation

    International Nuclear Information System (INIS)

    Sun, Xiaodong; Kim, Seungjin; Ishii, Mamoru; Beus, Stephen G.

    2003-01-01

    The bubble interaction mechanisms have been analytically modeled in the first paper of this series to provide mechanistic constitutive relations for the two-group interfacial area transport equation (IATE), which was proposed to dynamically solve the interfacial area concentration in the two-fluid model. This paper presents the evaluation approach and results of the two-group IATE based on available experimental data obtained in confined flow, namely, 11 data sets in or near bubbly flow and 13 sets in cap-turbulent and churn-turbulent flow. The two-group IATE is evaluated in steady state, one-dimensional form. Also, since the experiments were performed under adiabatic, air-water two-phase flow conditions, the phase change effect is omitted in the evaluation. To account for the inter-group bubble transport, the void fraction transport equation for Group-2 bubbles is also used to predict the void fraction for Group-2 bubbles. Agreement between the data and the model predictions is reasonably good and the average relative difference for the total interfacial area concentration between the 24 data sets and predictions is within 7%. The model evaluation demonstrates the capability of the two-group IATE focused on the current confined flow to predict the interfacial area concentration over a wide range of flow regimes. (author)

  2. Quadratic inner element subgrid scale discretisation of the Boltzmann transport equation

    International Nuclear Information System (INIS)

    Baker, C.M.J.; Buchan, A.G.; Pain, C.C.; Tollit, B.; Eaton, M.D.; Warner, P.

    2012-01-01

    This paper explores the application of the inner element subgrid scale method to the Boltzmann transport equation using quadratic basis functions. Previously, only linear basis functions for both the coarse scale and the fine scale were considered. This paper, therefore, analyses the advantages of using different coarse and subgrid basis functions for increasing the accuracy of the subgrid scale method. The transport of neutral particle radiation may be described by the Boltzmann transport equation (BTE) which, due to its 7 dimensional phase space, is computationally expensive to resolve. Multi-scale methods offer an approach to efficiently resolve the spatial dimensions of the BTE by separating the solution into its coarse and fine scales and formulating a solution whereby only the computationally efficient coarse scales need to be solved. In previous work an inner element subgrid scale method was developed that applied a linear continuous and discontinuous finite element method to represent the solution’s coarse and fine scale components. This approach was shown to generate efficient and stable solutions, and so this article continues its development by formulating higher order quadratic finite element expansions over the continuous and discontinuous scales. Here it is shown that a solution’s convergence can be improved significantly using higher order basis functions. Furthermore, by using linear finite elements to represent coarse scales in combination with quadratic fine scales, convergence can also be improved with only a modest increase in computational expense.

  3. Thermophoresis of a spherical particle: Modeling through moment-based, macroscopic transport equations

    Science.gov (United States)

    Padrino, Juan C.; Sprittles, James; Lockerby, Duncan

    2017-11-01

    Thermophoresis refers to the forces on and motions of objects caused by temperature gradients when these objects are exposed to rarefied gases. This phenomenon can occur when the ratio of the gas mean free path to the characteristic physical length scale (Knudsen number) is not negligible. In this work, we obtain the thermophoretic force on a rigid, heat-conducting spherical particle immersed in a rarefied gas resulting from a uniform temperature gradient imposed far from the sphere. To this end, we model the gas dynamics using the steady, linearized version of the so-called regularized 13-moment equations (R13). This set of equations, derived from the Boltzmann equation using the moment method, provides closures to the mass, momentum, and energy conservation laws in the form of constitutive, transport equations for the stress and heat flux that extends the Navier-Stokes-Fourier model to include rarefaction effects. Integration of the pressure and stress on the surface of the sphere leads to the net force as a function of the Knudsen number, dimensionless temperature gradient, and particle-to-gas thermal conductivity ratio. Results from this expression are compared with predictions from other moment-based models as well as from kinetic models. Supported in the UK by the Engineering and Physical Sciences Research Council (EP/N016602/1).

  4. From the Dyson-Schwinger to the Transport Equation in the Background Field Gauge of QCD

    CERN Document Server

    Wang, Q; Stöcker, H; Greiner, W

    2003-01-01

    The non-equilibrium quantum field dynamics is usually described in the closed-time-path formalism. The initial state correlations are introduced into the generating functional by non-local source terms. We propose a functional approach to the Dyson-Schwinger equation, which treats the non-local and local source terms in the same way. In this approach, the generating functional is formulated for the connected Green functions and one-particle-irreducible vertices. The great advantages of our approach over the widely used two-particle-irreducible method are that it is much simpler and that it is easy to implement the procedure in a computer program to automatically generate the Feynman diagrams for a given process. The method is then applied to a pure gluon plasma to derive the gauge-covariant transport equation from the Dyson-Schwinger equation in the background covariant gauge. We discuss the structure of the kinetic equation and show its relationship with the classical one. We derive the gauge-covariant colli...

  5. Exact solutions of Fisher and Burgers equations with finite transport memory

    International Nuclear Information System (INIS)

    Kar, Sandip; Banik, Suman Kumar; Ray, Deb Shankar

    2003-01-01

    The Fisher and Burgers equations with finite memory transport, describing reaction-diffusion and convection-diffusion processes, respectively have recently attracted a lot of attention in the context of chemical kinetics, mathematical biology and turbulence. We show here that they admit exact solutions. While the speed of the travelling wavefront is dependent on the relaxation time in the Fisher equation, memory effects significantly smoothen out the shock wave nature of the Burgers solution, without any influence on the corresponding wave speed. We numerically analyse the ansatz for the exact solution and show that for the reaction-diffusion system the strength of the reaction term must be moderate enough not to exceed a critical limit to allow a travelling wave solution to exist for appreciable finite memory effect

  6. Exact solutions of Fisher and Burgers equations with finite transport memory

    CERN Document Server

    Kar, S; Ray, D S

    2003-01-01

    The Fisher and Burgers equations with finite memory transport, describing reaction-diffusion and convection-diffusion processes, respectively have recently attracted a lot of attention in the context of chemical kinetics, mathematical biology and turbulence. We show here that they admit exact solutions. While the speed of the travelling wavefront is dependent on the relaxation time in the Fisher equation, memory effects significantly smoothen out the shock wave nature of the Burgers solution, without any influence on the corresponding wave speed. We numerically analyse the ansatz for the exact solution and show that for the reaction-diffusion system the strength of the reaction term must be moderate enough not to exceed a critical limit to allow a travelling wave solution to exist for appreciable finite memory effect.

  7. Solution of neutron transport equation using Daubechies' wavelet expansion in the angular discretization

    International Nuclear Information System (INIS)

    Cao Liangzhi; Wu Hongchun; Zheng Youqi

    2008-01-01

    Daubechies' wavelet expansion is introduced to discretize the angular variables of the neutron transport equation when the neutron angular flux varies very acutely with the angular directions. An improvement is made by coupling one-dimensional wavelet expansion and discrete ordinate method to make two-dimensional angular discretization efficient and stable. The angular domain is divided into several subdomains for treating the vacuum boundary condition exactly in the unstructured geometry. A set of wavelet equations coupled with each other is obtained in each subdomain. An iterative method is utilized to decouple the wavelet moments. The numerical results of several benchmark problems demonstrate that the wavelet expansion method can provide more accurate results by lower-order expansion than other angular discretization methods

  8. Finite element analysis of the neutron transport equation in spherical geometry

    International Nuclear Information System (INIS)

    Kim, Yong Ill; Kim, Jong Kyung; Suk, Soo Dong

    1992-01-01

    The Galerkin formulation of the finite element method is applied to the integral law of the first-order form of the one-group neutron transport equation in one-dimensional spherical geometry. Piecewise linear or quadratic Lagrange polynomials are utilized in the integral law for the angular flux to establish a set of linear algebraic equations. Numerical analyses are performed for the scalar flux distribution in a heterogeneous sphere as well as for the criticality problem in a uniform sphere. For the criticality problems in the uniform sphere, the results of the finite element method, with the use of continuous finite elements in space and angle, are compared with the exact solutions. In the heterogeneous problem, the scalar flux distribution obtained by using discontinuous angular and spatical finite elements is in good agreement with that from the ANISN code calculation. (Author)

  9. The solution of the multigroup neutron transport equation using spherical harmonics

    International Nuclear Information System (INIS)

    Fletcher, K.

    1981-01-01

    A solution of the multi-group neutron transport equation in up to three space dimensions is presented. The flux is expanded in a series of unnormalised spherical harmonics. Using the various recurrence formulae a linked set of first order differential equations is obtained for the moments psisup(g)sub(lm)(r), γsup(g)sub(lm)(r). Terms with odd l are eliminated resulting in a second order system which is solved by two methods. The first is a finite difference formulation using an iterative procedure, secondly, in XYZ and XY geometry a finite element solution is given. Results for a test problem using both methods are exhibited and compared. (orig./RW) [de

  10. Matrix-oriented implementation for the numerical solution of the partial differential equations governing flows and transport in porous media

    KAUST Repository

    Sun, Shuyu; Salama, Amgad; El-Amin, Mohamed

    2012-01-01

    In this paper we introduce a new technique for the numerical solution of the various partial differential equations governing flow and transport phenomena in porous media. This method is proposed to be used in high level programming languages like

  11. Calculation of radiation effects in solids by direct numerical solution of the adjoint transport equation

    International Nuclear Information System (INIS)

    Matthes, W.K.

    1998-01-01

    The 'adjoint transport equation in its integro-differential form' is derived for the radiation damage produced by atoms injected into solids. We reduce it to the one-dimensional form and prepare it for a numerical solution by: --discretizing the continuous variables energy, space and direction, --replacing the partial differential quotients by finite differences and --evaluating the collision integral by a double sum. By a proper manipulation of this double sum the adjoint transport equation turns into a (very large) set of linear equations with tridiagonal matrix which can be solved by a special (simple and fast) algorithm. The solution of this set of linear equations contains complete information on a specified damage type (e.g. the energy deposited in a volume V) in terms of the function D(i,E,c,x) which gives the damage produced by all particles generated in a cascade initiated by a particle of type i starting at x with energy E in direction c. It is essential to remark that one calculation gives the damage function D for the complete ranges of the variables {i,E,c and x} (for numerical reasons of course on grid-points in the {E,c,x}-space). This is most useful to applications where a general source-distribution S(i,E,c,x) of particles is given by the experimental setup (e.g. beam-window and and target in proton accelerator work. The beam-protons along their path through the window--or target material generate recoil atoms by elastic collisions or nuclear reactions. These recoil atoms form the particle source S). The total damage produced then is eventually given by: D = (Σ)i ∫ ∫ ∫ S(i, E, c, x)*D(i, E, c, x)*dE*dc*dx A Fortran-77 program running on a PC-486 was written for the overall procedure and applied to some problems

  12. Comparison of preconditioned generalized conjugate gradient methods to two-dimensional neutron and photon transport equation

    International Nuclear Information System (INIS)

    Chen, G.S.; Yang, D.Y.

    1998-01-01

    We apply and compare the preconditioned generalized conjugate gradient methods to solve the linear system equation that arises in the two-dimensional neutron and photon transport equation in this paper. Several subroutines are developed on the basis of preconditioned generalized conjugate gradient methods for time-independent, two-dimensional neutron and photon transport equation in the transport theory. These generalized conjugate gradient methods are used: TFQMR (transpose free quasi-minimal residual algorithm) CGS (conjugate gradient square algorithm), Bi-CGSTAB (bi-conjugate gradient stabilized algorithm) and QMRCGSTAB (quasi-minimal residual variant of bi-conjugate gradient stabilized algorithm). These subroutines are connected to computer program DORT. Several problems are tested on a personal computer with Intel Pentium CPU. The reasons to choose the generalized conjugate gradient methods are that the methods have better residual (equivalent to error) control procedures in the computation and have better convergent rate. The pointwise incomplete LU factorization ILU, modified pointwise incomplete LU factorization MILU, block incomplete factorization BILU and modified blockwise incomplete LU factorization MBILU are the preconditioning techniques used in the several testing problems. In Bi-CGSTAB, CGS, TFQMR and QMRCGSTAB method, we find that either CGS or Bi-CGSTAB method combined with preconditioner MBILU is the most efficient algorithm in these methods in the several testing problems. The numerical solution of flux by preconditioned CGS and Bi-CGSTAB methods has the same result as those from Cray computer, obtained by either the point successive relaxation method or the line successive relaxation method combined with Gaussian elimination

  13. Method for the determination of the dominant eigenvalue of the neutron transport equation in a slab using fractional derivative

    International Nuclear Information System (INIS)

    Sperotto, Fabiola Aiub; Segatto, Cynthia Feijo; Zabadal, Jorge

    2002-01-01

    In this work, we determine the dominant eigenvalue of the one-dimensional neutron transport equation in a slab constructing an integral form for the neutron transport equation which is the expressed in terms of fractional derivative of the angular flux. Equating the fractional derivative of the angular flux to the integrate equation, we determine the unknown order of the fractional derivative comparing the kernel of the integral equation with the one of Riemann-Liouville definition of fractional derivative. Once known the angular flux the dominant eigenvalue is calculated solving a transcendental equation resulting from the application of the boundary conditions. We report the methodology applied, for comparison with available results in literature. (author)

  14. Finite-element discretization of 3D energy-transport equations for semiconductors

    Energy Technology Data Exchange (ETDEWEB)

    Gadau, Stephan

    2007-07-01

    In this thesis a mathematical model was derived that describes the charge and energy transport in semiconductor devices like transistors. Moreover, numerical simulations of these physical processes are performed. In order to accomplish this, methods of theoretical physics, functional analysis, numerical mathematics and computer programming are applied. After an introduction to the status quo of semiconductor device simulation methods and a brief review of historical facts up to now, the attention is shifted to the construction of a model, which serves as the basis of the subsequent derivations in the thesis. Thereby the starting point is an important equation of the theory of dilute gases. From this equation the model equations are derived and specified by means of a series expansion method. This is done in a multi-stage derivation process, which is mainly taken from a scientific paper and which does not constitute the focus of this thesis. In the following phase we specify the mathematical setting and make precise the model assumptions. Thereby we make use of methods of functional analysis. Since the equations we deal with are coupled, we are concerned with a nonstandard problem. In contrary, the theory of scalar elliptic equations is established meanwhile. Subsequently, we are preoccupied with the numerical discretization of the equations. A special finite-element method is used for the discretization. This special approach has to be done in order to make the numerical results appropriate for practical application. By a series of transformations from the discrete model we derive a system of algebraic equations that are eligible for numerical evaluation. Using self-made computer programs we solve the equations to get approximate solutions. These programs are based on new and specialized iteration procedures that are developed and thoroughly tested within the frame of this research work. Due to their importance and their novel status, they are explained and

  15. Stochastic interpretation of the advection-diffusion equation and its relevance to bed load transport

    Science.gov (United States)

    Ancey, C.; Bohorquez, P.; Heyman, J.

    2015-12-01

    The advection-diffusion equation is one of the most widespread equations in physics. It arises quite often in the context of sediment transport, e.g., for describing time and space variations in the particle activity (the solid volume of particles in motion per unit streambed area). Phenomenological laws are usually sufficient to derive this equation and interpret its terms. Stochastic models can also be used to derive it, with the significant advantage that they provide information on the statistical properties of particle activity. These models are quite useful when sediment transport exhibits large fluctuations (typically at low transport rates), making the measurement of mean values difficult. Among these stochastic models, the most common approach consists of random walk models. For instance, they have been used to model the random displacement of tracers in rivers. Here we explore an alternative approach, which involves monitoring the evolution of the number of particles moving within an array of cells of finite length. Birth-death Markov processes are well suited to this objective. While the topic has been explored in detail for diffusion-reaction systems, the treatment of advection has received no attention. We therefore look into the possibility of deriving the advection-diffusion equation (with a source term) within the framework of birth-death Markov processes. We show that in the continuum limit (when the cell size becomes vanishingly small), we can derive an advection-diffusion equation for particle activity. Yet while this derivation is formally valid in the continuum limit, it runs into difficulty in practical applications involving cells or meshes of finite length. Indeed, within our stochastic framework, particle advection produces nonlocal effects, which are more or less significant depending on the cell size and particle velocity. Albeit nonlocal, these effects look like (local) diffusion and add to the intrinsic particle diffusion (dispersal due

  16. Global existence of weak solutions to dissipative transport equations with nonlocal velocity

    Science.gov (United States)

    Bae, Hantaek; Granero-Belinchón, Rafael; Lazar, Omar

    2018-04-01

    We consider 1D dissipative transport equations with nonlocal velocity field: where is a nonlocal operator given by a Fourier multiplier. We especially consider two types of nonlocal operators: (1) , the Hilbert transform, (2) . In this paper, we show several global existence of weak solutions depending on the range of γ, δ and α. When , we take initial data having finite energy, while we take initial data in weighted function spaces (in the real variables or in the Fourier variables), which have infinite energy, when .

  17. Gluon transport equation with effective mass and dynamical onset of Bose–Einstein condensation

    International Nuclear Information System (INIS)

    Blaizot, Jean-Paul; Jiang, Yin; Liao, Jinfeng

    2016-01-01

    We study the transport equation describing a dense system of gluons, in the small scattering angle approximation, taking into account medium-generated effective masses of the gluons. We focus on the case of overpopulated systems that are driven to Bose–Einstein condensation on their way to thermalization. The presence of a mass modifies the dispersion relation of the gluon, as compared to the massless case, but it is shown that this does not change qualitatively the scaling behavior in the vicinity of the onset.

  18. Solution of the neutron transport equation by means of Hermite-Ssub(infinity)-theory

    International Nuclear Information System (INIS)

    Brandt, D.; Haelg, W.; Mennig, J.

    1979-01-01

    A stable numerical approximation Hsub(α)-Ssub(infinity) is obtained through the use of Hermite's method of order α(Hsub(α)) in the spatial integration of the ID neutron transport equation. The theory for α = 1 is applied to a one-group shielding problem. Numerical calculations show the new method to converge much faster than earlier versions of Ssub(infinity)-theory. Comparison of H 1 - Ssub(infinity) with the well-known Ssub(N)-code ANISN indicates a large gain in computing time for the former. (Auth.)

  19. Asymptotic formulae for solutions of the two-group integral neutron-transport equation

    International Nuclear Information System (INIS)

    Duracz, T.

    1976-01-01

    The steady-state, two-group integral neutron-transport equation is considered for two cases. First, for plane geometry, formulae for the asymptotic flux are obtained, under assumptions of homogeneous medium with isotropic scattering, extended to infinity (whole space and half-space), with sources vanishing at infinity as 0(esup(-IXI)). Next, for spherical geometry, the Milne problem is considered and formulae for the asymptotic flux are obtained. These formulae have the form of asymptotic expansions for small and large radii of the black sphere. (orig.) [de

  20. Solution of charged particle transport equation by Monte-Carlo method in the BRANDZ code system

    International Nuclear Information System (INIS)

    Artamonov, S.N.; Androsenko, P.A.; Androsenko, A.A.

    1992-01-01

    Consideration is given to the issues of Monte-Carlo employment for the solution of charged particle transport equation and its implementation in the BRANDZ code system under the conditions of real 3D geometry and all the data available on radiation-to-matter interaction in multicomponent and multilayer targets. For the solution of implantation problem the results of BRANDZ data comparison with the experiments and calculations by other codes in complexes systems are presented. The results of direct nuclear pumping process simulation for laser-active media by a proton beam are also included. 4 refs.; 7 figs

  1. Transport equation theory of electron backscattering and x-ray production

    International Nuclear Information System (INIS)

    Fathers, D.J.; Rez, P.

    1978-02-01

    A transport equation theory of electron backscattering and x ray production is derived and applied to energy dissipation of 30-KeV electrons for copper as a function of depth and to the energy distribution of backscattered electrons for copper, aluminum, and gold. These results are plotted and compared with experiment. Plots for variations of backscattering with atomic number and with angle of incidence, and polar plots of backscattering for 30-keV electrons at normal incidence are also presented. 10 references, seven figures

  2. Numerical Solution of the Electron Transport Equation in the Upper Atmosphere

    Energy Technology Data Exchange (ETDEWEB)

    Woods, Mark Christopher [Sandia National Lab. (SNL-NM), Albuquerque, NM (United States); Holmes, Mark [Rensselaer Polytechnic Inst., Troy, NY (United States); Sailor, William C [Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)

    2017-07-01

    A new approach for solving the electron transport equation in the upper atmosphere is derived. The problem is a very stiff boundary value problem, and to obtain an accurate numerical solution, matrix factorizations are used to decouple the fast and slow modes. A stable finite difference method is applied to each mode. This solver is applied to a simplifieed problem for which an exact solution exists using various versions of the boundary conditions that might arise in a natural auroral display. The numerical and exact solutions are found to agree with each other to at least two significant digits.

  3. A piecewise bi-linear discontinuous finite element spatial discretization of the Sn transport equation

    International Nuclear Information System (INIS)

    Bailey, Teresa S.; Warsa, James S.; Chang, Jae H.; Adams, Marvin L.

    2011-01-01

    We present a new spatial discretization of the discrete-ordinates transport equation in two dimensional Cartesian (X-Y) geometry for arbitrary polygonal meshes. The discretization is a discontinuous finite element method (DFEM) that utilizes piecewise bi-linear (PWBL) basis functions, which are formally introduced in this paper. We also present a series of numerical results on quadrilateral and polygonal grids and compare these results to a variety of other spatial discretization that have been shown to be successful on these grid types. Finally, we note that the properties of the PWBL basis functions are such that the leading-order piecewise bi-linear discontinuous finite element (PWBLD) solution will satisfy a reasonably accurate diffusion discretization in the thick diffusion limit, making the PWBLD method a viable candidate for many different classes of transport problems. (author)

  4. A Piecewise Bi-Linear Discontinuous Finite Element Spatial Discretization of the Sn Transport Equation

    International Nuclear Information System (INIS)

    Bailey, T.S.; Chang, J.H.; Warsa, J.S.; Adams, M.L.

    2010-01-01

    We present a new spatial discretization of the discrete-ordinates transport equation in two-dimensional Cartesian (X-Y) geometry for arbitrary polygonal meshes. The discretization is a discontinuous finite element method (DFEM) that utilizes piecewise bi-linear (PWBL) basis functions, which are formally introduced in this paper. We also present a series of numerical results on quadrilateral and polygonal grids and compare these results to a variety of other spatial discretizations that have been shown to be successful on these grid types. Finally, we note that the properties of the PWBL basis functions are such that the leading-order piecewise bi-linear discontinuous finite element (PWBLD) solution will satisfy a reasonably accurate diffusion discretization in the thick diffusion limit, making the PWBLD method a viable candidate for many different classes of transport problems.

  5. Theoretical analysis of integral neutron transport equation using collision probability method with quadratic flux approach

    International Nuclear Information System (INIS)

    Shafii, Mohammad Ali; Meidianti, Rahma; Wildian,; Fitriyani, Dian; Tongkukut, Seni H. J.; Arkundato, Artoto

    2014-01-01

    Theoretical analysis of integral neutron transport equation using collision probability (CP) method with quadratic flux approach has been carried out. In general, the solution of the neutron transport using the CP method is performed with the flat flux approach. In this research, the CP method is implemented in the cylindrical nuclear fuel cell with the spatial of mesh being conducted into non flat flux approach. It means that the neutron flux at any point in the nuclear fuel cell are considered different each other followed the distribution pattern of quadratic flux. The result is presented here in the form of quadratic flux that is better understanding of the real condition in the cell calculation and as a starting point to be applied in computational calculation

  6. Theoretical analysis of integral neutron transport equation using collision probability method with quadratic flux approach

    Energy Technology Data Exchange (ETDEWEB)

    Shafii, Mohammad Ali, E-mail: mashafii@fmipa.unand.ac.id; Meidianti, Rahma, E-mail: mashafii@fmipa.unand.ac.id; Wildian,, E-mail: mashafii@fmipa.unand.ac.id; Fitriyani, Dian, E-mail: mashafii@fmipa.unand.ac.id [Department of Physics, Andalas University Padang West Sumatera Indonesia (Indonesia); Tongkukut, Seni H. J. [Department of Physics, Sam Ratulangi University Manado North Sulawesi Indonesia (Indonesia); Arkundato, Artoto [Department of Physics, Jember University Jember East Java Indonesia (Indonesia)

    2014-09-30

    Theoretical analysis of integral neutron transport equation using collision probability (CP) method with quadratic flux approach has been carried out. In general, the solution of the neutron transport using the CP method is performed with the flat flux approach. In this research, the CP method is implemented in the cylindrical nuclear fuel cell with the spatial of mesh being conducted into non flat flux approach. It means that the neutron flux at any point in the nuclear fuel cell are considered different each other followed the distribution pattern of quadratic flux. The result is presented here in the form of quadratic flux that is better understanding of the real condition in the cell calculation and as a starting point to be applied in computational calculation.

  7. Discontinuous Galerkin discretization and hp-refinement for the resolution of the neutron transport equation

    International Nuclear Information System (INIS)

    Fournier, Damien; Le-Tellier, Romain; Herbin, Raphaele

    2013-01-01

    This paper presents an hp-refinement method for a first order scalar transport reaction equation discretized by a discontinuous Galerkin method. First, the theoretical rates of convergence of h- and p-refinement are recalled and numerically tested. Then, in order to design some meshes, we propose two different estimators of the local error on the spatial domain. These quantities are analyzed and compared depending on the regularity of the solution so as to find the best way to lead the refinement process and the best strategy to choose between h- and p-refinement. Finally, the different possible refinement strategies are compared first on analytical examples and then on realistic applications for neutron transport in a nuclear reactor core. (authors)

  8. A Piecewise Bi-Linear Discontinuous Finite Element Spatial Discretization of the Sn Transport Equation

    Energy Technology Data Exchange (ETDEWEB)

    Bailey, T S; Chang, J H; Warsa, J S; Adams, M L

    2010-12-22

    We present a new spatial discretization of the discrete-ordinates transport equation in two-dimensional Cartesian (X-Y) geometry for arbitrary polygonal meshes. The discretization is a discontinuous finite element method (DFEM) that utilizes piecewise bi-linear (PWBL) basis functions, which are formally introduced in this paper. We also present a series of numerical results on quadrilateral and polygonal grids and compare these results to a variety of other spatial discretizations that have been shown to be successful on these grid types. Finally, we note that the properties of the PWBL basis functions are such that the leading-order piecewise bi-linear discontinuous finite element (PWBLD) solution will satisfy a reasonably accurate diffusion discretization in the thick diffusion limit, making the PWBLD method a viable candidate for many different classes of transport problems.

  9. Parallel computing for homogeneous diffusion and transport equations in neutronics; Calcul parallele pour les equations de diffusion et de transport homogenes en neutronique

    Energy Technology Data Exchange (ETDEWEB)

    Pinchedez, K

    1999-06-01

    Parallel computing meets the ever-increasing requirements for neutronic computer code speed and accuracy. In this work, two different approaches have been considered. We first parallelized the sequential algorithm used by the neutronics code CRONOS developed at the French Atomic Energy Commission. The algorithm computes the dominant eigenvalue associated with PN simplified transport equations by a mixed finite element method. Several parallel algorithms have been developed on distributed memory machines. The performances of the parallel algorithms have been studied experimentally by implementation on a T3D Cray and theoretically by complexity models. A comparison of various parallel algorithms has confirmed the chosen implementations. We next applied a domain sub-division technique to the two-group diffusion Eigen problem. In the modal synthesis-based method, the global spectrum is determined from the partial spectra associated with sub-domains. Then the Eigen problem is expanded on a family composed, on the one hand, from eigenfunctions associated with the sub-domains and, on the other hand, from functions corresponding to the contribution from the interface between the sub-domains. For a 2-D homogeneous core, this modal method has been validated and its accuracy has been measured. (author)

  10. Solution of the Boltzmann equation for primary light ions and the transport of their fragments

    Directory of Open Access Journals (Sweden)

    J. Kempe

    2010-10-01

    Full Text Available The Boltzmann equation for the transport of pencil beams of light ions in semi-infinite uniform media has been calculated. The equation is solved for the practically important generalized 3D case of Gaussian incident primary light ion beams of arbitrary mean square radius, mean square angular spread, and covariance. The transport of the associated fragments in three dimensions is derived based on the known transport of the primary particles, taking the mean square angular spread of their production processes, as well as their energy loss and multiple scattering, into account. The analytical pencil and broad beam depth fluence and absorbed dose distributions are accurately expressed using recently derived analytical energy and range formulas. The contributions from low and high linear energy transfer (LET dose components were separately identified using analytical expressions. The analytical results are compared with SHIELD-HIT Monte Carlo (MC calculations and found to be in very good agreement. The pencil beam fluence and absorbed dose distributions of the primary particles are mainly influenced by an exponential loss of the primary ions combined with an increasing lateral spread due to multiple scattering and energy loss with increasing penetration depth. The associated fluence of heavy fragments is concentrated at small radii and so is the LET and absorbed dose distribution. Their transport is also characterized by the buildup of a slowing down spectrum which is quite similar to that of the primaries but with a wider energy and angular spread at increasing penetration depths. The range of the fragments is shorter or longer depending on their nuclear mass to charge ratio relative to that of the primary ions. The absorbed dose of the heavier fragments is fairly similar to that of the primary ions and also influenced by a rapidly increasing energy loss towards the end of their ranges. The present analytical solution of the Boltzmann equation

  11. Fourier analysis of a new P1 synthetic acceleration for Sn transport equations

    International Nuclear Information System (INIS)

    Turcksin, B.; Ragusa, J. C.

    2010-10-01

    In this work, is derived a new P1 synthetic acceleration scheme (P1SA) for the S N transport equation and analyze its convergence properties through the means of a Fourier analysis. The Fourier analysis is carried out for both continuous (i.e., not spatially discretized) S N equations and linear discontinuous Fem discretization. We show, thanks to the continuous analysis, that the scheme is unstable when the anisotropy is important (μ - >0.5). However, the discrete analysis shows that when cells are large in comparison to the mean free path, the spectral radius decreases and the acceleration scheme becomes effective, even for highly anisotropic scattering. In charged particles transport, scattering is highly anisotropic and mean free paths are very small and, thus, this scheme could be of interest. To use the P1SA when cells are small and anisotropy is important, the scheme is modified by altering the update of the accelerated flux or by using either K transport sweeps before the application of P1SA. The update scheme performs well as long as μ - - ≥0.9, the modified update scheme is unstable. The multiple transport sweeps scheme is convergent with an arbitrary μ - but the spectral radius increases when scattering is isotropic. When anisotropic increases, the frequency of use of the acceleration scheme needs to be decreased. Even if the P1SA is used less often, the spectral radius is significantly smaller when compared with a method that does not use it for high anisotropy (μ - ≥0.5). It is interesting to notice that using P1SA every two iterations gives the same spectral radius than the update method when μ - ≥0.5 but it is much less efficient when μ - <0.5. (Author)

  12. Application of preconditioned GMRES to the numerical solution of the neutron transport equation

    International Nuclear Information System (INIS)

    Patton, B.W.; Holloway, J.P.

    2002-01-01

    The generalized minimal residual (GMRES) method with right preconditioning is examined as an alternative to both standard and accelerated transport sweeps for the iterative solution of the diamond differenced discrete ordinates neutron transport equation. Incomplete factorization (ILU) type preconditioners are used to determine their effectiveness in accelerating GMRES for this application. ILU(τ), which requires the specification of a dropping criteria τ, proves to be a good choice for the types of problems examined in this paper. The combination of ILU(τ) and GMRES is compared with both DSA and unaccelerated transport sweeps for several model problems. It is found that the computational workload of the ILU(τ)-GMRES combination scales nonlinearly with the number of energy groups and quadrature order, making this technique most effective for problems with a small number of groups and discrete ordinates. However, the cost of preconditioner construction can be amortized over several calculations with different source and/or boundary values. Preconditioners built upon standard transport sweep algorithms are also evaluated as to their effectiveness in accelerating the convergence of GMRES. These preconditioners show better scaling with such problem parameters as the scattering ratio, the number of discrete ordinates, and the number of spatial meshes. These sweeps based preconditioners can also be cast in a matrix free form that greatly reduces storage requirements

  13. LOCFES-B: Solving the one-dimensional transport equation with user-selected spatial approximations

    International Nuclear Information System (INIS)

    Jarvis, R.D.; Nelson, P.

    1993-01-01

    Closed linear one-cell functional (CLOF) methods constitute an abstractly defined class of spatial approximations to the one-dimensional discrete ordinates equations of linear particle transport that encompass, as specific instances, the vast majority of the spatial approximations that have been either used or suggested in the computational solution of these equations. A specific instance of the class of CLOF methods is defined by a (typically small) number of functions of the cell width, total cross section, and direction cosine of particle motion. The LOCFES code takes advantage of the latter observation by permitting the use, within a more-or-less standard source iteration solution process, of an arbitrary CLOF method as defined by a user-supplied subroutine. The design objective of LOCFES was to provide automated determination of the order of accuracy (i.e., order of the discretization error) in the fine-mesh limit for an arbitrary user-selected CLOF method. This asymptotic order of accuracy is one widely used measure of the merit of a spatial approximation. This paper discusses LOCFES-B, which is a code that uses methods developed in LOCFES to solve one-dimensional linear particle transport problems with any user-selected CLOF method. LOCFES-B provides automatic solution of a given problem to within an accuracy specified by user input and provides comparison of the computational results against results from externally provided benchmark results

  14. Angular finite volume method for solving the multigroup transport equation with piecewise average scattering cross sections

    Energy Technology Data Exchange (ETDEWEB)

    Calloo, A.; Vidal, J.F.; Le Tellier, R.; Rimpault, G., E-mail: ansar.calloo@cea.fr, E-mail: jean-francois.vidal@cea.fr, E-mail: romain.le-tellier@cea.fr, E-mail: gerald.rimpault@cea.fr [CEA, DEN, DER/SPRC/LEPh, Saint-Paul-lez-Durance (France)

    2011-07-01

    This paper deals with the solving of the multigroup integro-differential form of the transport equation for fine energy group structure. In that case, multigroup transfer cross sections display strongly peaked shape for light scatterers and the current Legendre polynomial expansion is not well-suited to represent them. Furthermore, even if considering an exact scattering cross sections representation, the scattering source in the discrete ordinates method (also known as the Sn method) being calculated by sampling the angular flux at given directions, may be wrongly computed due to lack of angular support for the angular flux. Hence, following the work of Gerts and Matthews, an angular finite volume solver has been developed for 2D Cartesian geometries. It integrates the multigroup transport equation over discrete volume elements obtained by meshing the unit sphere with a product grid over the polar and azimuthal coordinates and by considering the integrated flux per solid angle element. The convergence of this method has been compared to the S{sub n} method for a highly anisotropic benchmark. Besides, piecewise-average scattering cross sections have been produced for non-bound Hydrogen atoms using a free gas model for thermal neutrons. LWR lattice calculations comparing Legendre representations of the Hydrogen scattering multigroup cross section at various orders and piecewise-average cross sections for this same atom are carried out (while keeping a Legendre representation for all other isotopes). (author)

  15. Application of the multigrid amplitude function method for time-dependent transport equation using MOC

    International Nuclear Information System (INIS)

    Tsujita, K.; Endo, T.; Yamamoto, A.

    2013-01-01

    An efficient numerical method for time-dependent transport equation, the mutigrid amplitude function (MAF) method, is proposed. The method of characteristics (MOC) is being widely used for reactor analysis thanks to the advances of numerical algorithms and computer hardware. However, efficient kinetic calculation method for MOC is still desirable since it requires significant computation time. Various efficient numerical methods for solving the space-dependent kinetic equation, e.g., the improved quasi-static (IQS) and the frequency transform methods, have been developed so far mainly for diffusion calculation. These calculation methods are known as effective numerical methods and they offer a way for faster computation. However, they have not been applied to the kinetic calculation method using MOC as the authors' knowledge. Thus, the MAF method is applied to the kinetic calculation using MOC aiming to reduce computation time. The MAF method is a unified numerical framework of conventional kinetic calculation methods, e.g., the IQS, the frequency transform, and the theta methods. Although the MAF method is originally developed for the space-dependent kinetic calculation based on the diffusion theory, it is extended to transport theory in the present study. The accuracy and computational time are evaluated though the TWIGL benchmark problem. The calculation results show the effectiveness of the MAF method. (authors)

  16. Estimating the solute transport parameters of the spatial fractional advection-dispersion equation using Bees Algorithm.

    Science.gov (United States)

    Mehdinejadiani, Behrouz

    2017-08-01

    This study represents the first attempt to estimate the solute transport parameters of the spatial fractional advection-dispersion equation using Bees Algorithm. The numerical studies as well as the experimental studies were performed to certify the integrity of Bees Algorithm. The experimental ones were conducted in a sandbox for homogeneous and heterogeneous soils. A detailed comparative study was carried out between the results obtained from Bees Algorithm and those from Genetic Algorithm and LSQNONLIN routines in FracFit toolbox. The results indicated that, in general, the Bees Algorithm much more accurately appraised the sFADE parameters in comparison with Genetic Algorithm and LSQNONLIN, especially in the heterogeneous soil and for α values near to 1 in the numerical study. Also, the results obtained from Bees Algorithm were more reliable than those from Genetic Algorithm. The Bees Algorithm showed the relative similar performances for all cases, while the Genetic Algorithm and the LSQNONLIN yielded different performances for various cases. The performance of LSQNONLIN strongly depends on the initial guess values so that, compared to the Genetic Algorithm, it can more accurately estimate the sFADE parameters by taking into consideration the suitable initial guess values. To sum up, the Bees Algorithm was found to be very simple, robust and accurate approach to estimate the transport parameters of the spatial fractional advection-dispersion equation. Copyright © 2017 Elsevier B.V. All rights reserved.

  17. A numerical spectral approach to solve the dislocation density transport equation

    International Nuclear Information System (INIS)

    Djaka, K S; Taupin, V; Berbenni, S; Fressengeas, C

    2015-01-01

    A numerical spectral approach is developed to solve in a fast, stable and accurate fashion, the quasi-linear hyperbolic transport equation governing the spatio-temporal evolution of the dislocation density tensor in the mechanics of dislocation fields. The approach relies on using the Fast Fourier Transform algorithm. Low-pass spectral filters are employed to control both the high frequency Gibbs oscillations inherent to the Fourier method and the fast-growing numerical instabilities resulting from the hyperbolic nature of the transport equation. The numerical scheme is validated by comparison with an exact solution in the 1D case corresponding to dislocation dipole annihilation. The expansion and annihilation of dislocation loops in 2D and 3D settings are also produced and compared with finite element approximations. The spectral solutions are shown to be stable, more accurate for low Courant numbers and much less computation time-consuming than the finite element technique based on an explicit Galerkin-least squares scheme. (paper)

  18. Solution of the Neutron transport equation in hexagonal geometry using strongly discontinuous nodal schemes

    International Nuclear Information System (INIS)

    Mugica R, C.A.; Valle G, E. del

    2005-01-01

    In 2002, E. del Valle and Ernest H. Mund developed a technique to solve numerically the Neutron transport equations in discrete ordinates and hexagonal geometry using two nodal schemes type finite element weakly discontinuous denominated WD 5,3 and WD 12,8 (of their initials in english Weakly Discontinuous). The technique consists on representing each hexagon in the union of three rhombuses each one of which it is transformed in a square in the one that the methods WD 5,3 and WD 12,8 were applied. In this work they are solved the mentioned equations of transport using the same discretization technique by hexagon but using two nodal schemes type finite element strongly discontinuous denominated SD 3 and SD 8 (of their initials in english Strongly Discontinuous). The application in each case as well as a reference problem for those that results are provided for the effective multiplication factor is described. It is carried out a comparison with the obtained results by del Valle and Mund for different discretization meshes so much angular as spatial. (Author)

  19. Angular finite volume method for solving the multigroup transport equation with piecewise average scattering cross sections

    International Nuclear Information System (INIS)

    Calloo, A.; Vidal, J.F.; Le Tellier, R.; Rimpault, G.

    2011-01-01

    This paper deals with the solving of the multigroup integro-differential form of the transport equation for fine energy group structure. In that case, multigroup transfer cross sections display strongly peaked shape for light scatterers and the current Legendre polynomial expansion is not well-suited to represent them. Furthermore, even if considering an exact scattering cross sections representation, the scattering source in the discrete ordinates method (also known as the Sn method) being calculated by sampling the angular flux at given directions, may be wrongly computed due to lack of angular support for the angular flux. Hence, following the work of Gerts and Matthews, an angular finite volume solver has been developed for 2D Cartesian geometries. It integrates the multigroup transport equation over discrete volume elements obtained by meshing the unit sphere with a product grid over the polar and azimuthal coordinates and by considering the integrated flux per solid angle element. The convergence of this method has been compared to the S_n method for a highly anisotropic benchmark. Besides, piecewise-average scattering cross sections have been produced for non-bound Hydrogen atoms using a free gas model for thermal neutrons. LWR lattice calculations comparing Legendre representations of the Hydrogen scattering multigroup cross section at various orders and piecewise-average cross sections for this same atom are carried out (while keeping a Legendre representation for all other isotopes). (author)

  20. Numerical solution of the Neutron Transport Equation using discontinuous nodal methods at X-Y geometry

    International Nuclear Information System (INIS)

    Delfin L, A.

    1996-01-01

    The purpose of this work is to solve the neutron transport equation in discrete-ordinates and X-Y geometry by developing and using the strong discontinuous and strong modified discontinuous nodal finite element schemes. The strong discontinuous and modified strong discontinuous nodal finite element schemes go from two to ten interpolation parameters per cell. They are describing giving a set D c and polynomial space S c corresponding for each scheme BDMO, RTO, BL, BDM1, HdV, BDFM1, RT1, BQ and BDM2. The solution is obtained solving the neutron transport equation moments for each nodal scheme by developing the basis functions defined by Pascal triangle and the Legendre moments giving in the polynomial space S c and, finally, looking for the non singularity of the resulting linear system. The linear system is numerically solved using a computer program for each scheme mentioned . It uses the LU method and forward and backward substitution and makes a partition of the domain in cells. The source terms and angular flux are calculated, using the directions and weights associated to the S N approximation and solving the angular flux moments to find the effective multiplication constant. The programs are written in Fortran language, using the dynamic allocation of memory to increase efficiently the available memory of the computing equipment. (Author)

  1. An extended step characteristic method for solving the transport equation in general geometries

    International Nuclear Information System (INIS)

    DeHart, M.D.; Pevey, R.E.; Parish, T.A.

    1994-01-01

    A method for applying the discrete ordinates method to solve the Boltzmann transport equation on arbitrary two-dimensional meshes has been developed. The finite difference approach normally used to approximate spatial derivatives in extrapolating angular fluxes across a cell is replaced by direct solution of the characteristic form of the transport equation for each discrete direction. Thus, computational cells are not restricted to the geometrical shape of a mesh element characteristic of a given coordinate system. However, in terms of the treatment of energy and angular dependencies, this method resembles traditional discrete ordinates techniques. By using the method developed here, a general two-dimensional space can be approximated by an irregular mesh comprised of arbitrary polygons. Results for a number of test problems have been compared with solutions obtained from traditional methods, with good agreement. Comparisons include benchmarks against analytical results for problems with simple geometry, as well as numerical results obtained from traditional discrete ordinates methods by applying the ANISN and TWOTRAN-II computer programs

  2. Tourism sector, Travel agencies, and Transport Suppliers: Comparison of Different Estimators in the Structural Equation Modeling

    Directory of Open Access Journals (Sweden)

    Kovačić Nataša

    2015-11-01

    Full Text Available The paper addresses the effect of external integration (EI with transport suppliers on the efficiency of travel agencies in the tourism sector supply chains. The main aim is the comparison of different estimation methods used in the structural equation modeling (SEM, applied to discover possible relationships between EIs and efficiencies. The latter are calculated by the means of data envelopment analysis (DEA. While designing the structural equation model, the exploratory and confirmatory factor analyses are also used as preliminary statistical procedures. For the estimation of parameters of SEM model, three different methods are explained, analyzed and compared: maximum likelihood (ML method, Bayesian Markov Chain Monte Carlo (BMCMC method, and unweighted least squares (ULS method. The study reveals that all estimation methods calculate comparable estimated parameters. The results also give an evidence of good model fit performance. Besides, the research confirms that the amplified external integration with transport providers leads to increased efficiency of travel agencies, which might be a very interesting finding for the operational management.

  3. Estimating the solute transport parameters of the spatial fractional advection-dispersion equation using Bees Algorithm

    Science.gov (United States)

    Mehdinejadiani, Behrouz

    2017-08-01

    This study represents the first attempt to estimate the solute transport parameters of the spatial fractional advection-dispersion equation using Bees Algorithm. The numerical studies as well as the experimental studies were performed to certify the integrity of Bees Algorithm. The experimental ones were conducted in a sandbox for homogeneous and heterogeneous soils. A detailed comparative study was carried out between the results obtained from Bees Algorithm and those from Genetic Algorithm and LSQNONLIN routines in FracFit toolbox. The results indicated that, in general, the Bees Algorithm much more accurately appraised the sFADE parameters in comparison with Genetic Algorithm and LSQNONLIN, especially in the heterogeneous soil and for α values near to 1 in the numerical study. Also, the results obtained from Bees Algorithm were more reliable than those from Genetic Algorithm. The Bees Algorithm showed the relative similar performances for all cases, while the Genetic Algorithm and the LSQNONLIN yielded different performances for various cases. The performance of LSQNONLIN strongly depends on the initial guess values so that, compared to the Genetic Algorithm, it can more accurately estimate the sFADE parameters by taking into consideration the suitable initial guess values. To sum up, the Bees Algorithm was found to be very simple, robust and accurate approach to estimate the transport parameters of the spatial fractional advection-dispersion equation.

  4. A linear multiple balance method for discrete ordinates neutron transport equations

    International Nuclear Information System (INIS)

    Park, Chang Je; Cho, Nam Zin

    2000-01-01

    A linear multiple balance method (LMB) is developed to provide more accurate and positive solutions for the discrete ordinates neutron transport equations. In this multiple balance approach, one mesh cell is divided into two subcells with quadratic approximation of angular flux distribution. Four multiple balance equations are used to relate center angular flux with average angular flux by Simpson's rule. From the analysis of spatial truncation error, the accuracy of the linear multiple balance scheme is ο(Δ 4 ) whereas that of diamond differencing is ο(Δ 2 ). To accelerate the linear multiple balance method, we also describe a simplified additive angular dependent rebalance factor scheme which combines a modified boundary projection acceleration scheme and the angular dependent rebalance factor acceleration schme. It is demonstrated, via fourier analysis of a simple model problem as well as numerical calculations, that the additive angular dependent rebalance factor acceleration scheme is unconditionally stable with spectral radius < 0.2069c (c being the scattering ration). The numerical results tested so far on slab-geometry discrete ordinates transport problems show that the solution method of linear multiple balance is effective and sufficiently efficient

  5. Assessment of intermittency transport equations for modeling transition in boundary layers subjected to freestream turbulence

    International Nuclear Information System (INIS)

    Suluksna, Keerati; Juntasaro, Ekachai

    2008-01-01

    The γ-Re θ transition model of Menter et al. [Menter, F.R., Langtry, R.B., Volker, S., Huang, P.G., 2005. Transition modelling for general purpose CFD codes. ERCOFTAC International Symposium Engineering Turbulence Modelling and Measurements] is a highly generalized transport equation model in which it has been developed based on the concept of local variables compatible with modern CFD methods where the unstructured grid and the parallel computing technique are usually integrated in. To perform the prediction with this model, two essential parameters, F length which is used to control the length of the transition region and Re θc which is used to control the onset of the transition location, must be specified to close the model. At present, both parameters are proprietary and their formulations are unpublished. For the first time here, the relations for both parameters are formulated by means of numerical experiments and analysis under the assumption of Re θc = Re θt corresponding with the bypass transition behavior. Based on this analysis, the optimized values of the parameters are found and their relations can be constructed as follows: Re θc = 803.73(Tu ∞ , le + 0.6067) -1.027 and F length = 163 ln(Tu ∞ , le ) + 3.625. The performance of this transition model is assessed by testing with the experimental cases of T3AM, T3A, and T3B. Detailed comparisons with the predicted results by the transition models of Suzen and Huang [Suzen, Y.B., Huang, P.G., 2000. Modeling of flow transition using an intermittency transport equation. J. Fluids Eng. 122, 273-284] and Lodefier et al. [Lodefier, K., Merci, B., De Langhe, C., Dick, E., 2003. Transition modelling with the SST turbulence model and intermittency transport equation. ASME Turbo Expo, Atlanta, GA, USA, June 16-19], and also with the predicted results by the k-ε model of Launder and Sharma [Launder, B.E., Sharma, B., 1974. Application of the energy dissipation model of turbulence to the calculation of

  6. Finite moments approach to the time-dependent neutron transport equation

    International Nuclear Information System (INIS)

    Kim, Sang Hyun

    1994-02-01

    Currently, nodal techniques are widely used in solving the multidimensional diffusion equation because of savings in computing time and storage. Thanks to the development of computer technology, one can now solve the transport equation instead of the diffusion equation to obtain more accurate solution. The finite moments method, one of the nodal methods, attempts to represent the fluxes in the cell and on cell surfaces more rigorously by retaining additional spatial moments. Generally, there are two finite moments schemes to solve the time-dependent transport equation. In one, the time variable is treated implicitly with finite moments method in space variable (implicit finite moments method), the other method uses finite moments method in both space and time (space-time finite moments method). In this study, these two schemes are applied to two types of time-dependent neutron transport problems. One is a fixed source problem, the other a heterogeneous fast reactor problem with delayed neutrons. From the results, it is observed that the two finite moments methods give almost the same solutions in both benchmark problems. However, the space-time finite moments method requires a little longer computing time than that of the implicit finite moments method. In order to reduce the longer computing time in the space-time finite moments method, a new iteration strategy is exploited, where a few time-stepwise calculation, in which original time steps are grouped into several coarse time divisions, is performed sequentially instead of performing iterations over the entire time steps. This strategy results in significant reduction of the computing time and we observe that 2-or 3-stepwise calculation is preferable. In addition, we propose a new finite moments method which is called mixed finite moments method in this thesis. Asymptotic analysis for the finite moments method shows that accuracy of the solution in a heterogeneous problem mainly depends on the accuracy of the

  7. Flux-probability distributions from the master equation for radiation transport in stochastic media

    International Nuclear Information System (INIS)

    Franke, Brian C.; Prinja, Anil K.

    2011-01-01

    We present numerical investigations into the accuracy of approximations in the master equation for radiation transport in discrete binary random media. Our solutions of the master equation yield probability distributions of particle flux at each element of phase space. We employ the Levermore-Pomraning interface closure and evaluate the effectiveness of closures for the joint conditional flux distribution for estimating scattering integrals. We propose a parameterized model for this joint-pdf closure, varying between correlation neglect and a full-correlation model. The closure is evaluated for a variety of parameter settings. Comparisons are made with benchmark results obtained through suites of fixed-geometry realizations of random media in rod problems. All calculations are performed using Monte Carlo techniques. Accuracy of the approximations in the master equation is assessed by examining the probability distributions for reflection and transmission and by evaluating the moments of the pdfs. The results suggest the correlation-neglect setting in our model performs best and shows improved agreement in the atomic-mix limit. (author)

  8. A modified two-fluid model for the application of two-group interfacial area transport equation

    International Nuclear Information System (INIS)

    Sun, X.; Ishii, M.; Kelly, J.

    2003-01-01

    This paper presents the modified two-fluid model that is ready to be applied in the approach of the two-group interfacial area transport equation. The two-group interfacial area transport equation was developed to provide a mechanistic constitutive relation for the interfacial area concentration in the two-fluid model. In the two-group transport equation, bubbles are categorized into two groups: spherical/distorted bubbles as Group 1 while cap/slug/churn-turbulent bubbles as Group 2. Therefore, this transport equation can be employed in the flow regimes spanning from bubbly, cap bubbly, slug to churn-turbulent flows. However, the introduction of the two groups of bubbles requires two gas velocity fields. Yet it is not desirable to solve two momentum equations for the gas phase alone. In the current modified two-fluid model, a simplified approach is proposed. The momentum equation for the averaged velocity of both Group-1 and Group-2 bubbles is retained. By doing so, the velocity difference between Group-1 and Group-2 bubbles needs to be determined. This may be made either based on simplified momentum equations for both Group-1 and Group-2 bubbles or by a modified drift-flux model

  9. Predicting fractional bed load transport rates: Application of the Wilcock‐Crowe equations to a regulated gravel bed river

    Science.gov (United States)

    Gaeuman, David; Andrews, E.D.; Krause, Andreas; Smith, Wes

    2009-01-01

    Bed load samples from four locations in the Trinity River of northern California are analyzed to evaluate the performance of the Wilcock‐Crowe bed load transport equations for predicting fractional bed load transport rates. Bed surface particles become smaller and the fraction of sand on the bed increases with distance downstream from Lewiston Dam. The dimensionless reference shear stress for the mean bed particle size (τ*rm) is largest near the dam, but varies relatively little between the more downstream locations. The relation between τ*rm and the reference shear stresses for other size fractions is constant across all locations. Total bed load transport rates predicted with the Wilcock‐Crowe equations are within a factor of 2 of sampled transport rates for 68% of all samples. The Wilcock‐Crowe equations nonetheless consistently under‐predict the transport of particles larger than 128 mm, frequently by more than an order of magnitude. Accurate prediction of the transport rates of the largest particles is important for models in which the evolution of the surface grain size distribution determines subsequent bed load transport rates. Values of τ*rm estimated from bed load samples are up to 50% larger than those predicted with the Wilcock‐Crowe equations, and sampled bed load transport approximates equal mobility across a wider range of grain sizes than is implied by the equations. Modifications to the Wilcock‐Crowe equation for determining τ*rm and the hiding function used to scale τ*rm to other grain size fractions are proposed to achieve the best fit to observed bed load transport in the Trinity River.

  10. Diffusion Coefficient Calculations With Low Order Legendre Polynomial and Chebyshev Polynomial Approximation for the Transport Equation in Spherical Geometry

    International Nuclear Information System (INIS)

    Yasa, F.; Anli, F.; Guengoer, S.

    2007-01-01

    We present analytical calculations of spherically symmetric radioactive transfer and neutron transport using a hypothesis of P1 and T1 low order polynomial approximation for diffusion coefficient D. Transport equation in spherical geometry is considered as the pseudo slab equation. The validity of polynomial expansionion in transport theory is investigated through a comparison with classic diffusion theory. It is found that for causes when the fluctuation of the scattering cross section dominates, the quantitative difference between the polynomial approximation and diffusion results was physically acceptable in general

  11. Transport Equations Resolution By N-BEE Anti-Dissipative Scheme In 2D Model Of Low Pressure Glow Discharge

    International Nuclear Information System (INIS)

    Kraloua, B.; Hennad, A.

    2008-01-01

    The aim of this paper is to determine electric and physical properties by 2D modelling of glow discharge low pressure in continuous regime maintained by term constant source. This electric discharge is confined in reactor plan-parallel geometry. This reactor is filled by Argon monatomic gas. Our continuum model the order two is composed the first three moments the Boltzmann's equations coupled with Poisson's equation by self consistent method. These transport equations are discretized by the finite volumes method. The equations system is resolved by a new technique, it is about the N-BEE explicit scheme using the time splitting method.

  12. A new treatment of the heat transport equation with a transport barrier and applications to ECRH experiments in Tore Supra

    Energy Technology Data Exchange (ETDEWEB)

    Zou, X.L.; Giruzzi, A.G.; Bouquey, F.; Clary, J.; Darbos, C.; Lennholm, M.; Magne, R.; Segui, J.L. [CEA Cadarache, Dept. de Recherches sur la Fusion Controlee, 13 - Saint-Paul-lez-Durance (France); Clemencon, A. [MIT, Electrochemical Energy Laboratory, Cambridge, MA (United States); Guivarch, C. [Ecole Nationale des Ponts et Chaussees, 77 - Marne-la-Vallee (France)

    2004-07-01

    An exact analytical solution of the electron heat diffusion equation in a cylinder has been found with a step-like diffusion coefficient, plus a monomial increase in the radial direction and a constant damping term. This model is sufficiently general to describe heat diffusion in the presence of a critical gradient threshold or a transport barrier, superimposed to the usual trend of increasing heat diffusivity from the plasma core to the edge. This type of representation allows us to see some well-known properties of heat transport phenomena in a different light. For instance, it has been shown that the contributions of the Eigenmodes to the time dependent solution grow at speeds that depend on the Eigenmode order i.e. at the beginning of the heating phase all the Eigenmodes are equally involved, whereas at the end only the lower order ones are left. This implies, e.g., that high frequency modulation experiments provide a characterization of transport phenomena that is intrinsically different with respect to power balance analysis of a stationary phase. It is particularly useful to analyse power switch on/off events and whenever high frequency modulations are not technically feasible. Low-frequency (1-2 Hz) ECRH modulation experiments have been performed on Tore Supra. A large jump (a factor of 8) in the heat diffusivity has been clearly identified at the ECRH power deposition layer. The amplitude and phase of several harmonics of the Fourier transform of the modulated temperature, as well as the time evolution of the modulated temperature have been reproduced by the analytical solution. The jump is found to be much weaker at lower ECRH power (one gyrotron)

  13. Generation of monoenergetic ion beams with a laser accelerator

    International Nuclear Information System (INIS)

    Pfotenhauer, Sebastian M.

    2009-01-01

    A method for the generation of monoenergetic proton and ion beams from a laser-based particle accelerator is presented. This method utilizes the unique space-charge effects occurring during relativistic laser-plasma interactions on solid targets in combination with a dot-like particle source. Due to this unique interaction geometry, MeV proton beams with an intrinsically narrow energy spectrum were obtained, for the first time, from a micrometer-scale laser accelerator. Over the past three years, the acceleration scheme has been consistently improved to enhance both the maximum particle energy and the reliability of the setup. The achieved degree of reliability allowed to derive the first scaling laws specifically for monoenergetic proton beams. Furthermore, the acceleration scheme was expanded on other target materials, enabling the generation of monoenergetic carbon beams. The experimental work was strongly supported by the parallel development of a complex theoretical model, which fully accounts for the observations and is in excellent agreement with numerical simulations. The presented results have an extraordinarily broad scope way beyond the current thesis: The availability of monoenergetic ion beams from a compact laser-plasma beam source - in conjunction with the unique properties of laser-produced particle beams - addresses a number of outstanding applications in fundamental research, material science and medical physics, and will help to shape a new generation of accelerators. (orig.)

  14. A monoenergetic electron source generated by nuclear stimulated desorption

    International Nuclear Information System (INIS)

    Kelson, I.; Levy, Y.; Nir, D.; Haustein, P.E.

    1994-01-01

    A series of measurements of nuclear stimulated desorption was performed for 103 Ru, using thin ruthenium films irradiated by thermal neutrons. The magnitude, time dependence and electric charge state of the outgoing 103m Rh flux was investigated. The utilization of monoenergetic electrons accompanying the 103 Rh decay for thin film thickness measurement is considered. (Author)

  15. Generation of monoenergetic ion beams with a laser accelerator

    Energy Technology Data Exchange (ETDEWEB)

    Pfotenhauer, Sebastian M.

    2009-01-29

    A method for the generation of monoenergetic proton and ion beams from a laser-based particle accelerator is presented. This method utilizes the unique space-charge effects occurring during relativistic laser-plasma interactions on solid targets in combination with a dot-like particle source. Due to this unique interaction geometry, MeV proton beams with an intrinsically narrow energy spectrum were obtained, for the first time, from a micrometer-scale laser accelerator. Over the past three years, the acceleration scheme has been consistently improved to enhance both the maximum particle energy and the reliability of the setup. The achieved degree of reliability allowed to derive the first scaling laws specifically for monoenergetic proton beams. Furthermore, the acceleration scheme was expanded on other target materials, enabling the generation of monoenergetic carbon beams. The experimental work was strongly supported by the parallel development of a complex theoretical model, which fully accounts for the observations and is in excellent agreement with numerical simulations. The presented results have an extraordinarily broad scope way beyond the current thesis: The availability of monoenergetic ion beams from a compact laser-plasma beam source - in conjunction with the unique properties of laser-produced particle beams - addresses a number of outstanding applications in fundamental research, material science and medical physics, and will help to shape a new generation of accelerators. (orig.)

  16. Discrete energy formulation of neutron transport theory applied to solving the discrete ordinates equations

    International Nuclear Information System (INIS)

    Ching, J.; Oblow, E.M.; Goldstein, H.

    1976-01-01

    An algebraic equivalence between the point-energy and multigroup forms of the Boltzmann transport equation is demonstrated that allows the development of a discrete energy, discrete ordinates method for the solution of radiation transport problems. In the discrete energy method, the group averaging required in the cross-section processing for multigroup calculations is replaced by a faster numerical quadrature scheme capable of generating transfer cross sections describing all the physical processes of interest on a fine point-energy grid. Test calculations in which the discrete energy method is compared with the multigroup method show that, for the same energy grid, the discrete energy method is much faster, although somewhat less accurate, than the multigroup method. However, the accuracy of the discrete energy method increases rapidly as the spacing between energy grid points is decreased, approaching that of multigroup calculations. For problems requiring great detail in the energy spectrum, the discrete energy method is therefore expected to be far more economical than the multigroup technique for equivalent accuracy solutions. This advantage of the point method is demonstrated by application to the study of neutron transport in a thick iron slab

  17. The Factors Influencing Satisfaction with Public City Transport: A Structural Equation Modelling Approach

    Directory of Open Access Journals (Sweden)

    Pawlasova Pavlina

    2015-12-01

    Full Text Available Satisfaction is one of the key factors which influences customer loyalty. We assume that the satisfied customer will be willing to use the ssame service provider again. The overall passengers´ satisfaction with public city transport may be affected by the overall service quality. Frequency, punctuality, cleanliness in the vehicle, proximity, speed, fare, accessibility and safety of transport, information and other factors can influence passengers´ satisfaction. The aim of this paper is to quantify factors and identify the most important factors influencing customer satisfaction with public city transport within conditions of the Czech Republic. Two methods of analysis are applied in order to fulfil the aim. The method of factor analysis and the method Varimax were used in order to categorize variables according to their mutual relations. The method of structural equation modelling was used to evaluate the factors and validate the model. Then, the optimal model was found. The logistic parameters, including service continuity and frequency, and service, including information rate, station proximity and vehicle cleanliness, are the factors influencing passengers´ satisfaction on a large scale.

  18. Variable order spherical harmonic expansion scheme for the radiative transport equation using finite elements

    International Nuclear Information System (INIS)

    Surya Mohan, P.; Tarvainen, Tanja; Schweiger, Martin; Pulkkinen, Aki; Arridge, Simon R.

    2011-01-01

    Highlights: → We developed a variable order global basis scheme to solve light transport in 3D. → Based on finite elements, the method can be applied to a wide class of geometries. → It is computationally cheap when compared to the fixed order scheme. → Comparisons with local basis method and other models demonstrate its accuracy. → Addresses problems encountered n modeling of light transport in human brain. - Abstract: We propose the P N approximation based on a finite element framework for solving the radiative transport equation with optical tomography as the primary application area. The key idea is to employ a variable order spherical harmonic expansion for angular discretization based on the proximity to the source and the local scattering coefficient. The proposed scheme is shown to be computationally efficient compared to employing homogeneously high orders of expansion everywhere in the domain. In addition the numerical method is shown to accurately describe the void regions encountered in the forward modeling of real-life specimens such as infant brains. The accuracy of the method is demonstrated over three model problems where the P N approximation is compared against Monte Carlo simulations and other state-of-the-art methods.

  19. Measurement of Membrane Characteristics Using the Phenomenological Equation and the Overall Mass Transport Equation in Ion-Exchange Membrane Electrodialysis of Saline Water

    Directory of Open Access Journals (Sweden)

    Yoshinobu Tanaka

    2012-01-01

    Full Text Available The overall membrane pair characteristics included in the overall mass transport equation are understandable using the phenomenological equations expressed in the irreversible thermodynamics. In this investigation, the overall membrane pair characteristics (overall transport number , overall solute permeability , overall electro-osmotic permeability and overall hydraulic permeability were measured by seawater electrodialysis changing current density, temperature and salt concentration, and it was found that occasionally takes minus value. For understanding the above phenomenon, new concept of the overall concentration reflection coefficient ∗ is introduced from the phenomenological equation. This is the aim of this investigation. ∗ is defined for describing the permselectivity between solutes and water molecules in the electrodialysis system just after an electric current interruption. ∗ is expressed by the function of and . ∗ is generally larger than 1 and is positive, but occasionally ∗ becomes less than 1 and becomes negative. Negative means that ions are transferred with water molecules (solvent from desalting cells toward concentrating cells just after an electric current interruption, indicating up-hill transport or coupled transport between water molecules and solutes.

  20. libmpdata++ 1.0: a library of parallel MPDATA solvers for systems of generalised transport equations

    Science.gov (United States)

    Jaruga, A.; Arabas, S.; Jarecka, D.; Pawlowska, H.; Smolarkiewicz, P. K.; Waruszewski, M.

    2015-04-01

    This paper accompanies the first release of libmpdata++, a C++ library implementing the multi-dimensional positive-definite advection transport algorithm (MPDATA) on regular structured grid. The library offers basic numerical solvers for systems of generalised transport equations. The solvers are forward-in-time, conservative and non-linearly stable. The libmpdata++ library covers the basic second-order-accurate formulation of MPDATA, its third-order variant, the infinite-gauge option for variable-sign fields and a flux-corrected transport extension to guarantee non-oscillatory solutions. The library is equipped with a non-symmetric variational elliptic solver for implicit evaluation of pressure gradient terms. All solvers offer parallelisation through domain decomposition using shared-memory parallelisation. The paper describes the library programming interface, and serves as a user guide. Supported options are illustrated with benchmarks discussed in the MPDATA literature. Benchmark descriptions include code snippets as well as quantitative representations of simulation results. Examples of applications include homogeneous transport in one, two and three dimensions in Cartesian and spherical domains; a shallow-water system compared with analytical solution (originally derived for a 2-D case); and a buoyant convection problem in an incompressible Boussinesq fluid with interfacial instability. All the examples are implemented out of the library tree. Regardless of the differences in the problem dimensionality, right-hand-side terms, boundary conditions and parallelisation approach, all the examples use the same unmodified library, which is a key goal of libmpdata++ design. The design, based on the principle of separation of concerns, prioritises the user and developer productivity. The libmpdata++ library is implemented in C++, making use of the Blitz++ multi-dimensional array containers, and is released as free/libre and open-source software.

  1. libmpdata++ 0.1: a library of parallel MPDATA solvers for systems of generalised transport equations

    Science.gov (United States)

    Jaruga, A.; Arabas, S.; Jarecka, D.; Pawlowska, H.; Smolarkiewicz, P. K.; Waruszewski, M.

    2014-11-01

    This paper accompanies first release of libmpdata++, a C++ library implementing the Multidimensional Positive-Definite Advection Transport Algorithm (MPDATA). The library offers basic numerical solvers for systems of generalised transport equations. The solvers are forward-in-time, conservative and non-linearly stable. The libmpdata++ library covers the basic second-order-accurate formulation of MPDATA, its third-order variant, the infinite-gauge option for variable-sign fields and a flux-corrected transport extension to guarantee non-oscillatory solutions. The library is equipped with a non-symmetric variational elliptic solver for implicit evaluation of pressure gradient terms. All solvers offer parallelisation through domain decomposition using shared-memory parallelisation. The paper describes the library programming interface, and serves as a user guide. Supported options are illustrated with benchmarks discussed in the MPDATA literature. Benchmark descriptions include code snippets as well as quantitative representations of simulation results. Examples of applications include: homogeneous transport in one, two and three dimensions in Cartesian and spherical domains; shallow-water system compared with analytical solution (originally derived for a 2-D case); and a buoyant convection problem in an incompressible Boussinesq fluid with interfacial instability. All the examples are implemented out of the library tree. Regardless of the differences in the problem dimensionality, right-hand-side terms, boundary conditions and parallelisation approach, all the examples use the same unmodified library, which is a key goal of libmpdata++ design. The design, based on the principle of separation of concerns, prioritises the user and developer productivity. The libmpdata++ library is implemented in C++, making use of the Blitz++ multi-dimensional array containers, and is released as free/libre and open-source software.

  2. Finite element approximation of the radiative transport equation in a medium with piece-wise constant refractive index

    International Nuclear Information System (INIS)

    Lehtikangas, O.; Tarvainen, T.; Kim, A.D.; Arridge, S.R.

    2015-01-01

    The radiative transport equation can be used as a light transport model in a medium with scattering particles, such as biological tissues. In the radiative transport equation, the refractive index is assumed to be constant within the medium. However, in biomedical media, changes in the refractive index can occur between different tissue types. In this work, light propagation in a medium with piece-wise constant refractive index is considered. Light propagation in each sub-domain with a constant refractive index is modeled using the radiative transport equation and the equations are coupled using boundary conditions describing Fresnel reflection and refraction phenomena on the interfaces between the sub-domains. The resulting coupled system of radiative transport equations is numerically solved using a finite element method. The approach is tested with simulations. The results show that this coupled system describes light propagation accurately through comparison with the Monte Carlo method. It is also shown that neglecting the internal changes of the refractive index can lead to erroneous boundary measurements of scattered light

  3. Quantitative phase microscopy for cellular dynamics based on transport of intensity equation.

    Science.gov (United States)

    Li, Ying; Di, Jianglei; Ma, Chaojie; Zhang, Jiwei; Zhong, Jinzhan; Wang, Kaiqiang; Xi, Teli; Zhao, Jianlin

    2018-01-08

    We demonstrate a simple method for quantitative phase imaging of tiny transparent objects such as living cells based on the transport of intensity equation. The experiments are performed using an inverted bright field microscope upgraded with a flipping imaging module, which enables to simultaneously create two laterally separated images with unequal defocus distances. This add-on module does not include any lenses or gratings and is cost-effective and easy-to-alignment. The validity of this method is confirmed by the measurement of microlens array and human osteoblastic cells in culture, indicating its potential in the applications of dynamically measuring living cells and other transparent specimens in a quantitative, non-invasive and label-free manner.

  4. Azimuthal anisotropy in heavy-ion collisions using non-extensive statistics in Boltzmann transport equation

    International Nuclear Information System (INIS)

    Tripathy, S.; Tiwari, S.K.; Younus, M.; Sahoo, R.

    2017-01-01

    One of the major goals in heavy-ion physics is to understand the properties of Quark Gluon Plasma (QGP), a deconfined hot and dense state of quarks and gluons existed shortly after the Big Bang. In the present scenario, the high-energy particle accelerators are able to reach energies where this extremely dense nuclear matter can be probed for a short time. Here, we follow our earlier works which use non-extensive statistics in Boltzmann Transport Equation (BTE). We represent the initial distribution of particles with the help of Tsallis power law distribution parameterized by the nonextensive parameter q and the Tsallis temperature T, remembering the fact that their origin is due to hard scatterings. We use the initial distribution (f in ) with Relaxation Time Approximation (RTA) of the BTE and calculate the final distribution (f fin ). Then we calculate ν 2 of the system using the final distribution in the definition of ν2

  5. On the spectral analysis of iterative solutions of the discretized one-group transport equation

    International Nuclear Information System (INIS)

    Sanchez, Richard

    2004-01-01

    We analyze the Fourier-mode technique used for the spectral analysis of iterative solutions of the one-group discretized transport equation. We introduce a direct spectral analysis for the iterative solution of finite difference approximations for finite slabs composed of identical layers, providing thus a complementary analysis that is more appropriate for reactor applications. Numerical calculations for the method of characteristics and with the diamond difference approximation show the appearance of antisymmetric modes generated by the iteration on boundary data. We have also utilized the discrete Fourier transform to compute the spectrum for a periodic slab containing N identical layers and shown that at the limit N → ∞ one obtains the familiar Fourier-mode solution

  6. Solving the multigroup adjoint transport equations using the method of cyclic characteristics

    Energy Technology Data Exchange (ETDEWEB)

    Assawaroongruengchot, M.; Marleau, G. [Ecole Polytechnique de Montreal, Inst. de genie nucleaire, Montreal, Quebec (Canada)]. E-mail: monchai.assawar@polymtl.ca

    2005-07-01

    The adjoint transport solution algorithm based on the method of cyclic characteristics (MOCC) is developed for the heterogeneous 2D geometries. The adjoint characteristics equation associated with a cyclic tracking line is formulated, then a closed form for adjoint angular flux can be determined. The acceleration techniques are implemented using the group-reduction and group-splitting techniques. To demonstrate the efficacy of the algorithm, the calculations are performed on the 37 pin CANDU cell and on the Watanabe-Maynard benchmark problem. Comparisons of adjoint flux and k{sub eff} results obtained by MOCC and collision probability (CP) methods are performed. The mathematical relationship between pseudo-adjoint flux obtained by CP method and adjoint flux by MOCC method is presented. (author)

  7. Solving the multigroup adjoint transport equations using the method of cyclic characteristics

    International Nuclear Information System (INIS)

    Assawaroongruengchot, M.; Marleau, G.

    2005-01-01

    The adjoint transport solution algorithm based on the method of cyclic characteristics (MOCC) is developed for the heterogeneous 2D geometries. The adjoint characteristics equation associated with a cyclic tracking line is formulated, then a closed form for adjoint angular flux can be determined. The acceleration techniques are implemented using the group-reduction and group-splitting techniques. To demonstrate the efficacy of the algorithm, the calculations are performed on the 37 pin CANDU cell and on the Watanabe-Maynard benchmark problem. Comparisons of adjoint flux and k eff results obtained by MOCC and collision probability (CP) methods are performed. The mathematical relationship between pseudo-adjoint flux obtained by CP method and adjoint flux by MOCC method is presented. (author)

  8. Program to solve the multigroup discrete ordinates transport equation in (x,y,z) geometry

    International Nuclear Information System (INIS)

    Lathrop, K.D.

    1976-04-01

    Numerical formulations and programming algorithms are given for the THREETRAN computer program which solves the discrete ordinates, multigroup transport equation in (x,y,z) geometry. An efficient, flexible, and general data-handling strategy is derived to make use of three hierarchies of storage: small core memory, large core memory, and disk file. Data management, input instructions, and sample problem output are described. A six-group, S 4 , 18 502 mesh point, 2 800 zone, k/sub eff/ calculation of the ZPPR-4 critical assembly required 144 min of CDC-7600 time to execute to a convergence tolerance of 5 x 10 -4 and gave results in good qualitative agreement with experiment and other calculations. 6 references

  9. A non overlapping parallel domain decomposition method applied to the simplified transport equations

    International Nuclear Information System (INIS)

    Lathuiliere, B.; Barrault, M.; Ramet, P.; Roman, J.

    2009-01-01

    A reactivity computation requires to compute the highest eigenvalue of a generalized eigenvalue problem. An inverse power algorithm is used commonly. Very fine modelizations are difficult to tackle for our sequential solver, based on the simplified transport equations, in terms of memory consumption and computational time. So, we propose a non-overlapping domain decomposition method for the approximate resolution of the linear system to solve at each inverse power iteration. Our method brings to a low development effort as the inner multigroup solver can be re-use without modification, and allows us to adapt locally the numerical resolution (mesh, finite element order). Numerical results are obtained by a parallel implementation of the method on two different cases with a pin by pin discretization. This results are analyzed in terms of memory consumption and parallel efficiency. (authors)

  10. Numerical tilting compensation in microscopy based on wavefront sensing using transport of intensity equation method

    Science.gov (United States)

    Hu, Junbao; Meng, Xin; Wei, Qi; Kong, Yan; Jiang, Zhilong; Xue, Liang; Liu, Fei; Liu, Cheng; Wang, Shouyu

    2018-03-01

    Wide-field microscopy is commonly used for sample observations in biological research and medical diagnosis. However, the tilting error induced by the oblique location of the image recorder or the sample, as well as the inclination of the optical path often deteriorates the imaging quality. In order to eliminate the tilting in microscopy, a numerical tilting compensation technique based on wavefront sensing using transport of intensity equation method is proposed in this paper. Both the provided numerical simulations and practical experiments prove that the proposed technique not only accurately determines the tilting angle with simple setup and procedures, but also compensates the tilting error for imaging quality improvement even in the large tilting cases. Considering its simple systems and operations, as well as image quality improvement capability, it is believed the proposed method can be applied for tilting compensation in the optical microscopy.

  11. Transmission probability method for solving neutron transport equation in three-dimensional triangular-z geometry

    Energy Technology Data Exchange (ETDEWEB)

    Liu Guoming [Department of Nuclear Engineering, Xi' an Jiaotong University, Xi' an, Shaanxi 710049 (China)], E-mail: gmliusy@gmail.com; Wu Hongchun; Cao Liangzhi [Department of Nuclear Engineering, Xi' an Jiaotong University, Xi' an, Shaanxi 710049 (China)

    2008-09-15

    This paper presents a transmission probability method (TPM) to solve the neutron transport equation in three-dimensional triangular-z geometry. The source within the mesh is assumed to be spatially uniform and isotropic. At the mesh surface, the constant and the simplified P{sub 1} approximation are invoked for the anisotropic angular flux distribution. Based on this model, a code TPMTDT is encoded. It was verified by three 3D Takeda benchmark problems, in which the first two problems are in XYZ geometry and the last one is in hexagonal-z geometry, and an unstructured geometry problem. The results of the present method agree well with those of Monte-Carlo calculation method and Spherical Harmonics (P{sub N}) method.

  12. Green's theorem and Green's functions for the steady-state cosmic-ray equation of transport

    International Nuclear Information System (INIS)

    Webb, G.M.; Gleeson, L.J.

    1977-01-01

    Green's Theorem is developed for the spherically-symmetric steady-state cosmic-ray equation of transport in interplanetary space. By means of it the momentum distribution function F 0 (r,p), (r=heliocentric distance, p=momentum) can be determined in a region rsub(a) 0 . Examples of Green's functions are given for the case rsub(a)=0, rsub(b)=infinity and derived for the cases of finite rsub(a) and rsub(b). The diffusion coefficient kappa is assumed of the form kappa=kappa 0 (p)rsup(b). The treatment systematizes the development of all analytic solutions for steady-state solar and galactic cosmic-ray propagation and previous solutions form a subset of the present solutions. (Auth.)

  13. Gluon transport equation in the small angle approximation and the onset of Bose–Einstein condensation

    International Nuclear Information System (INIS)

    Blaizot, Jean-Paul; Liao, Jinfeng; McLerran, Larry

    2014-01-01

    To understand the evolution of a dense system of gluons, such as those produced in the early stages of ultra-relativistic heavy ion collisions, is an important and challenging problem. We describe the approach to thermal equilibrium using the small angle approximation for gluon scattering in a Boltzmann equation that includes the effects of Bose statistics. The role of Bose statistical factors in amplifying the rapid growth of the population of the soft modes is essential. With these factors properly taken into account, one finds that elastic scattering alone provides an efficient mechanism for populating soft modes, and in fact leads to rapid infrared local thermalization. Furthermore, recent developments suggest that high initial overpopulation plays a key role and may lead to dynamical Bose–Einstein condensation. The kinetics of condensation is an interesting problem in itself. By solving the transport equation for initial conditions with a large enough initial phase-space density the equilibrium state contains a Bose condensate, and we present numerical evidence that such over-occupied systems reach the onset of Bose–Einstein condensation in a finite time. It is also found that the approach to condensation is characterized by a scaling behavior. Finally we discuss a number of extensions of the present study

  14. Gluon transport equation in the small angle approximation and the onset of Bose–Einstein condensation

    Energy Technology Data Exchange (ETDEWEB)

    Blaizot, Jean-Paul [Institut de Physique Théorique, CNRS/URA 2306, CEA Saclay, F-91191 Gif-sur-Yvette (France); Liao, Jinfeng [Physics Dept. and CEEM, Indiana University, 2401 N Milo B. Sampson Lane, Bloomington, IN 47408 (United States); RIKEN BNL Research Center, Bldg. 510A, Brookhaven National Laboratory, Upton, NY 11973 (United States); McLerran, Larry [Physics Dept., Bldg. 510A, Brookhaven National Laboratory, Upton, NY 11973 (United States); RIKEN BNL Research Center, Bldg. 510A, Brookhaven National Laboratory, Upton, NY 11973 (United States); Physics Department, China Central Normal University, Wuhan (China)

    2014-11-15

    To understand the evolution of a dense system of gluons, such as those produced in the early stages of ultra-relativistic heavy ion collisions, is an important and challenging problem. We describe the approach to thermal equilibrium using the small angle approximation for gluon scattering in a Boltzmann equation that includes the effects of Bose statistics. The role of Bose statistical factors in amplifying the rapid growth of the population of the soft modes is essential. With these factors properly taken into account, one finds that elastic scattering alone provides an efficient mechanism for populating soft modes, and in fact leads to rapid infrared local thermalization. Furthermore, recent developments suggest that high initial overpopulation plays a key role and may lead to dynamical Bose–Einstein condensation. The kinetics of condensation is an interesting problem in itself. By solving the transport equation for initial conditions with a large enough initial phase-space density the equilibrium state contains a Bose condensate, and we present numerical evidence that such over-occupied systems reach the onset of Bose–Einstein condensation in a finite time. It is also found that the approach to condensation is characterized by a scaling behavior. Finally we discuss a number of extensions of the present study.

  15. Multi-dimensional upwinding-based implicit LES for the vorticity transport equations

    Science.gov (United States)

    Foti, Daniel; Duraisamy, Karthik

    2017-11-01

    Complex turbulent flows such as rotorcraft and wind turbine wakes are characterized by the presence of strong coherent structures that can be compactly described by vorticity variables. The vorticity-velocity formulation of the incompressible Navier-Stokes equations is employed to increase numerical efficiency. Compared to the traditional velocity-pressure formulation, high order numerical methods and sub-grid scale models for the vorticity transport equation (VTE) have not been fully investigated. Consistent treatment of the convection and stretching terms also needs to be addressed. Our belief is that, by carefully designing sharp gradient-capturing numerical schemes, coherent structures can be more efficiently captured using the vorticity-velocity formulation. In this work, a multidimensional upwind approach for the VTE is developed using the generalized Riemann problem-based scheme devised by Parish et al. (Computers & Fluids, 2016). The algorithm obtains high resolution by augmenting the upwind fluxes with transverse and normal direction corrections. The approach is investigated with several canonical vortex-dominated flows including isolated and interacting vortices and turbulent flows. The capability of the technique to represent sub-grid scale effects is also assessed. Navy contract titled ``Turbulence Modelling Across Disparate Length Scales for Naval Computational Fluid Dynamics Applications,'' through Continuum Dynamics, Inc.

  16. Discontinuous finite element and characteristics methods for neutrons transport equation solution in heterogeneous grids

    International Nuclear Information System (INIS)

    Masiello, E.

    2006-01-01

    The principal goal of this manuscript is devoted to the investigation of a new type of heterogeneous mesh adapted to the shape of the fuel pins (fuel-clad-moderator). The new heterogeneous mesh guarantees the spatial modelling of the pin-cell with a minimum of regions. Two methods are investigated for the spatial discretization of the transport equation: the discontinuous finite element method and the method of characteristics for structured cells. These methods together with the new representation of the pin-cell result in an appreciable reduction of calculation points. They allow an exact modelling of the fuel pin-cell without spatial homogenization. A new synthetic acceleration technique based on an angular multigrid is also presented for the speed up of the inner iterations. These methods are good candidates for transport calculations for a nuclear reactor core. A second objective of this work is the application of method of characteristics for non-structured geometries to the study of double heterogeneity problem. The letters is characterized by fuel material with a stochastic dispersion of heterogeneous grains, and until now was solved with a model based on collision probabilities. We propose a new statistical model based on renewal-Markovian theory, which makes possible to take into account the stochastic nature of the problem and to avoid the approximations of the collision probability model. The numerical solution of this model is guaranteed by the method of characteristics. (author)

  17. A Novel Algorithm for Solving the Multidimensional Neutron Transport Equation on Massively Parallel Architectures

    Energy Technology Data Exchange (ETDEWEB)

    Azmy, Yousry

    2014-06-10

    We employ the Integral Transport Matrix Method (ITMM) as the kernel of new parallel solution methods for the discrete ordinates approximation of the within-group neutron transport equation. The ITMM abandons the repetitive mesh sweeps of the traditional source iterations (SI) scheme in favor of constructing stored operators that account for the direct coupling factors among all the cells' fluxes and between the cells' and boundary surfaces' fluxes. The main goals of this work are to develop the algorithms that construct these operators and employ them in the solution process, determine the most suitable way to parallelize the entire procedure, and evaluate the behavior and parallel performance of the developed methods with increasing number of processes, P. The fastest observed parallel solution method, Parallel Gauss-Seidel (PGS), was used in a weak scaling comparison with the PARTISN transport code, which uses the source iteration (SI) scheme parallelized with the Koch-baker-Alcouffe (KBA) method. Compared to the state-of-the-art SI-KBA with diffusion synthetic acceleration (DSA), this new method- even without acceleration/preconditioning-is completitive for optically thick problems as P is increased to the tens of thousands range. For the most optically thick cells tested, PGS reduced execution time by an approximate factor of three for problems with more than 130 million computational cells on P = 32,768. Moreover, the SI-DSA execution times's trend rises generally more steeply with increasing P than the PGS trend. Furthermore, the PGS method outperforms SI for the periodic heterogeneous layers (PHL) configuration problems. The PGS method outperforms SI and SI-DSA on as few as P = 16 for PHL problems and reduces execution time by a factor of ten or more for all problems considered with more than 2 million computational cells on P = 4.096.

  18. Solute transport in aquifers: The comeback of the advection dispersion equation and the First Order Approximation

    Science.gov (United States)

    Fiori, A.; Zarlenga, A.; Jankovic, I.; Dagan, G.

    2017-12-01

    Natural gradient steady flow of mean velocity U takes place in heterogeneous aquifers of random logconductivity Y = lnK , characterized by the normal univariate PDF f(Y) and autocorrelation ρY, of variance σY2 and horizontal integral scale I. Solute transport is quantified by the Breakthrough Curve (BTC) M at planes at distance x from the injection plane. The study builds on the extensive 3D numerical simulations of flow and transport of Jankovic et al. (2017) for different conductivity structures. The present study further explores the predictive capabilities of the Advection Dispersion Equation (ADE), with macrodispersivity αL given by the First Order Approximation (FOA), by checking in a quantitative manner its applicability. After a discussion on the suitable boundary conditions for ADE, we find that the ADE-FOA solution is a sufficiently accurate predictor for applications, the many other sources of uncertainty prevailing in practice notwithstanding. We checked by least squares and by comparison of travel time of quantiles of M that indeed the analytical Inverse Gaussian M with αL =σY2 I , is able to fit well the bulk of the simulated BTCs. It tends to underestimate the late arrival time of the thin and persistent tail. The tail is better reproduced by the semi-analytical MIMSCA model, which also allows for a physical explanation of the success of the Inverse Gaussian solution. Examination of the pertinent longitudinal mass distribution shows that it is different from the commonly used Gaussian one in the analysis of field experiments, and it captures the main features of the plume measurements of the MADE experiment. The results strengthen the confidence in the applicability of the ADE and the FOA to predicting longitudinal spreading in solute transport through heterogeneous aquifers of stationary random structure.

  19. Solution and study of nodal neutron transport equation applying the LTS{sub N}-DiagExp method

    Energy Technology Data Exchange (ETDEWEB)

    Hauser, Eliete Biasotto; Pazos, Ruben Panta [Pontificia Univ. Catolica do Rio Grande do Sul, Porto Alegre, RS (Brazil). Faculdade de Matematica]. E-mail: eliete@pucrs.br; rpp@mat.pucrs.br; Vilhena, Marco Tullio de [Pontificia Univ. Catolica do Rio Grande do Sul, Porto Alegre, RS (Brazil). Instituto de Matematica]. E-mail: vilhena@mat.ufrgs.br; Barros, Ricardo Carvalho de [Universidade do Estado, Nova Friburgo, RJ (Brazil). Instituto Politecnico]. E-mail: ricardo@iprj.uerj.br

    2003-07-01

    In this paper we report advances about the three-dimensional nodal discrete-ordinates approximations of neutron transport equation for Cartesian geometry. We use the combined collocation method of the angular variables and nodal approach for the spatial variables. By nodal approach we mean the iterated transverse integration of the S{sub N} equations. This procedure leads to the set of one-dimensional averages angular fluxes in each spatial variable. The resulting system of equations is solved with the LTS{sub N} method, first applying the Laplace transform to the set of the nodal S{sub N} equations and then obtained the solution by symbolic computation. We include the LTS{sub N} method by diagonalization to solve the nodal neutron transport equation and then we outline the convergence of these nodal-LTS{sub N} approximations with the help of a norm associated to the quadrature formula used to approximate the integral term of the neutron transport equation. (author)

  20. Real time quantitative phase microscopy based on single-shot transport of intensity equation (ssTIE) method

    Science.gov (United States)

    Yu, Wei; Tian, Xiaolin; He, Xiaoliang; Song, Xiaojun; Xue, Liang; Liu, Cheng; Wang, Shouyu

    2016-08-01

    Microscopy based on transport of intensity equation provides quantitative phase distributions which opens another perspective for cellular observations. However, it requires multi-focal image capturing while mechanical and electrical scanning limits its real time capacity in sample detections. Here, in order to break through this restriction, real time quantitative phase microscopy based on single-shot transport of the intensity equation method is proposed. A programmed phase mask is designed to realize simultaneous multi-focal image recording without any scanning; thus, phase distributions can be quantitatively retrieved in real time. It is believed the proposed method can be potentially applied in various biological and medical applications, especially for live cell imaging.

  1. Note: A monoenergetic proton backlighter for the National Ignition Facility

    Energy Technology Data Exchange (ETDEWEB)

    Rygg, J. R.; LePape, S.; Bachmann, B.; Khan, S. F.; Sayre, D. B. [Lawrence Livermore National Laboratory, Livermore, California 94551 (United States); Zylstra, A. B.; Séguin, F. H.; Gatu-Johnson, M.; Lahmann, B. J.; Petrasso, R. D.; Sio, H. W. [Plasma Science and Fusion Center, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 (United States); Craxton, R. S.; Garcia, E. M.; Kong, Y. Z.; McKenty, P. W. [Laboratory for Laser Energetics, University of Rochester, Rochester, New York 14623 (United States); Rinderknecht, H. G. [Lawrence Livermore National Laboratory, Livermore, California 94551 (United States); Plasma Science and Fusion Center, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 (United States); Rosenberg, M. J. [Plasma Science and Fusion Center, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 (United States); Laboratory for Laser Energetics, University of Rochester, Rochester, New York 14623 (United States)

    2015-11-15

    A monoenergetic, isotropic proton source suitable for proton radiography applications has been demonstrated at the National Ignition Facility (NIF). A deuterium and helium-3 gas-filled glass capsule was imploded with 39 kJ of laser energy from 24 of NIF’s 192 beams. Spectral, spatial, and temporal measurements of the 15-MeV proton product of the {sup 3}He(d,p){sup 4}He nuclear reaction reveal a bright (10{sup 10} protons/sphere), monoenergetic (ΔE/E = 4%) spectrum with a compact size (80 μm) and isotropic emission (∼13% proton fluence variation and <0.4% mean energy variation). Simultaneous measurements of products produced by the D(d,p)T and D(d,n){sup 3}He reactions also show 2 × 10{sup 10} isotropically distributed 3-MeV protons.

  2. Thermalization of monoenergetic neutrons in a concrete room

    Energy Technology Data Exchange (ETDEWEB)

    Vega C, H.R.; Manzanares A, E.; Hernandez D, V.M.; Mercado, G.A. [UAZ, A.P. 336, 98000 Zacatecas (Mexico); Iniguez, M.P.; Martin M, A. [Universidad de Valladolid, (Spain)

    2006-07-01

    The thermalization of neutrons from monoenergetic neutron sources in a concrete room has been studied. During calibration of neutron detectors it is mandatory to make corrections due to neutron scattering produced by the room walls, therefore this factor must be known in advance. The scattered neutrons are thermalized and produce a neutron field that is directly proportional to source strength and inversely proportional to room total wall-surfaces, the proportional coefficient has been calculated for neutrons whose energy goes from 1 eV to 20 MeV. This coefficient was calculated using Monte Carlo methods for 150, 200 and 300 cm-radius spherical cavity, where monoenergetic neutrons were located at the center, along the spherical cavity radius neutron spectra were calculated at several source-to-detector distances inside the cavity. The obtained coefficient is almost three times larger than the factor normally utilized. (Author)

  3. Fission product yield measurements using monoenergetic photon beams

    Science.gov (United States)

    Krishichayan; Bhike, M.; Tonchev, A. P.; Tornow, W.

    2017-09-01

    Measurements of fission products yields (FPYs) are an important source of information on the fission process. During the past couple of years, a TUNL-LANL-LLNL collaboration has provided data on the FPYs from quasi monoenergetic neutron-induced fission on 235U, 238U, and 239Pu and has revealed an unexpected energy dependence of both asymmetric fission fragments at energies below 4 MeV. This peculiar FPY energy dependence was more pronounced in neutron-induced fission of 239Pu. In an effort to understand and compare the effect of the incoming probe on the FPY distribution, we have carried out monoenergetic photon-induced fission experiments on the same 235U, 238U, and 239Pu targets. Monoenergetic photon beams of Eγ = 13.0 MeV were provided by the HIγS facility, the world's most intense γ-ray source. In order to determine the total number of fission events, a dual-fission chamber was used during the irradiation. These irradiated samples were counted at the TUNL's low-background γ-ray counting facility using high efficient HPGe detectors over a period of 10 weeks. Here we report on our first ever photofission product yield measurements obtained with monoenegetic photon beams. These results are compared with neutron-induced FPY data.

  4. Fission product yield measurements using monoenergetic photon beams

    Directory of Open Access Journals (Sweden)

    Krishichayan

    2017-01-01

    Full Text Available Measurements of fission products yields (FPYs are an important source of information on the fission process. During the past couple of years, a TUNL-LANL-LLNL collaboration has provided data on the FPYs from quasi monoenergetic neutron-induced fission on 235U, 238U, and 239Pu and has revealed an unexpected energy dependence of both asymmetric fission fragments at energies below 4 MeV. This peculiar FPY energy dependence was more pronounced in neutron-induced fission of 239Pu. In an effort to understand and compare the effect of the incoming probe on the FPY distribution, we have carried out monoenergetic photon-induced fission experiments on the same 235U, 238U, and 239Pu targets. Monoenergetic photon beams of Eγ = 13.0 MeV were provided by the HIγS facility, the world's most intense γ-ray source. In order to determine the total number of fission events, a dual-fission chamber was used during the irradiation. These irradiated samples were counted at the TUNL's low-background γ-ray counting facility using high efficient HPGe detectors over a period of 10 weeks. Here we report on our first ever photofission product yield measurements obtained with monoenegetic photon beams. These results are compared with neutron-induced FPY data.

  5. Spectral finite element methods for solving fractional differential equations with applications in anomalous transport

    Energy Technology Data Exchange (ETDEWEB)

    Carella, Alfredo Raul

    2012-09-15

    Quantifying species transport rates is a main concern in chemical and petrochemical industries. In particular, the design and operation of many large-scale industrial chemical processes is as much dependent on diffusion as it is on reaction rates. However, the existing diffusion models sometimes fail to predict experimentally observed behaviors and their accuracy is usually insufficient for process optimization purposes. Fractional diffusion models offer multiple possibilities for generalizing Flick's law in a consistent manner in order to account for history dependence and nonlocal effects. These models have not been extensively applied to the study of real systems, mainly due to their computational cost and mathematical complexity. A least squares spectral formulation was developed for solving fractional differential equations. The proposed method was proven particularly well-suited for dealing with the numerical difficulties inherent to fractional differential operators. The practical implementation was explained in detail in order to enhance reproducibility, and directions were specified for extending it to multiple dimensions and arbitrarily shaped domains. A numerical framework based on the least-squares spectral element method was developed for studying and comparing anomalous diffusion models in pellets. This simulation tool is capable of solving arbitrary integro-differential equations and can be effortlessly adapted to various problems in any number of dimensions. Simulations of the flow around a cylindrical particle were achieved by extending the functionality of the developed framework. A test case was analyzed by coupling the boundary condition yielded by the fluid model with two families of anomalous diffusion models: hyperbolic diffusion and fractional diffusion. Qualitative guidelines for determining the suitability of diffusion models can be formulated by complementing experimental data with the results obtained from this approach.(Author)

  6. A fast, parallel algorithm to solve the basic fluvial erosion/transport equations

    Science.gov (United States)

    Braun, J.

    2012-04-01

    Quantitative models of landform evolution are commonly based on the solution of a set of equations representing the processes of fluvial erosion, transport and deposition, which leads to predict the geometry of a river channel network and its evolution through time. The river network is often regarded as the backbone of any surface processes model (SPM) that might include other physical processes acting at a range of spatial and temporal scales along hill slopes. The basic laws of fluvial erosion requires the computation of local (slope) and non-local (drainage area) quantities at every point of a given landscape, a computationally expensive operation which limits the resolution of most SPMs. I present here an algorithm to compute the various components required in the parameterization of fluvial erosion (and transport) and thus solve the basic fluvial geomorphic equation, that is very efficient because it is O(n) (the number of required arithmetic operations is linearly proportional to the number of nodes defining the landscape), and is fully parallelizable (the computation cost decreases in a direct inverse proportion to the number of processors used to solve the problem). The algorithm is ideally suited for use on latest multi-core processors. Using this new technique, geomorphic problems can be solved at an unprecedented resolution (typically of the order of 10,000 X 10,000 nodes) while keeping the computational cost reasonable (order 1 sec per time step). Furthermore, I will show that the algorithm is applicable to any regular or irregular representation of the landform, and is such that the temporal evolution of the landform can be discretized by a fully implicit time-marching algorithm, making it unconditionally stable. I will demonstrate that such an efficient algorithm is ideally suited to produce a fully predictive SPM that links observationally based parameterizations of small-scale processes to the evolution of large-scale features of the landscapes on

  7. Analysis and development of spatial hp-refinement methods for solving the neutron transport equation

    International Nuclear Information System (INIS)

    Fournier, D.

    2011-01-01

    The different neutronic parameters have to be calculated with a higher accuracy in order to design the 4. generation reactor cores. As memory storage and computation time are limited, adaptive methods are a solution to solve the neutron transport equation. The neutronic flux, solution of this equation, depends on the energy, angle and space. The different variables are successively discretized. The energy with a multigroup approach, considering the different quantities to be constant on each group, the angle by a collocation method called SN approximation. Once the energy and angle variable are discretized, a system of spatially-dependent hyperbolic equations has to be solved. Discontinuous finite elements are used to make possible the development of hp-refinement methods. Thus, the accuracy of the solution can be improved by spatial refinement (h-refinement), consisting into subdividing a cell into sub-cells, or by order refinement (p-refinement), by increasing the order of the polynomial basis. In this thesis, the properties of this methods are analyzed showing the importance of the regularity of the solution to choose the type of refinement. Thus, two error estimators are used to lead the refinement process. Whereas the first one requires high regularity hypothesis (analytical solution), the second one supposes only the minimal hypothesis required for the solution to exist. The comparison of both estimators is done on benchmarks where the analytic solution is known by the method of manufactured solutions. Thus, the behaviour of the solution as a regard of the regularity can be studied. It leads to a hp-refinement method using the two estimators. Then, a comparison is done with other existing methods on simplified but also realistic benchmarks coming from nuclear cores. These adaptive methods considerably reduces the computational cost and memory footprint. To further improve these two points, an approach with energy-dependent meshes is proposed. Actually, as the

  8. Central moments of ion implantation distributions derived by the backward Boltzmann transport equation compared with Monte Carlo simulations

    International Nuclear Information System (INIS)

    Bowyer, M.D.J.; Ashworth, D.G.; Oven, R.

    1992-01-01

    In this paper we study solutions to the backward Boltzmann transport equation (BBTE) specialized to equations governing moments of the distribution of ions implanted into amorphous targets. A central moment integral equation set has been derived starting from the classical plane source BBTE for non-central moments. A full generator equation is provided to allow construction of equation sets of an arbitrary size, thus allowing computation of moments of arbitrary order. A BBTE solver program has been written that uses the residual correction technique proposed by Winterbon. A simple means is presented to allow direct incorporation of Biersack's two-parameter ''magic formula'' into a BBTE solver program. Results for non-central and central moment integral equation sets are compared with Monte Carlo simulations, using three different formulae for the mean free flight path between collisions. Comparisons are performed for the ions B and As, implanted into the target a-Si, over the energy range 1 keV-1 MeV. The central moment integral equation set is found to have superior convergence properties to the non-central moment equation set. For As ions implanted into a-Si, at energies below ∼ 30 keV, significant differences are observed, for third- and fourth-order moments, when using alternative versions for the mean free flight path. Third- and fourth-order moments derived using one- and two-parameter scattering mechanisms also show significant differences over the same energy range. (Author)

  9. Solution algorithms for a PN-1 - Equivalent SN angular discretization of the transport equation in one-dimensional spherical coordinates

    International Nuclear Information System (INIS)

    Warsa, J. S.; Morel, J. E.

    2007-01-01

    Angular discretizations of the S N transport equation in curvilinear coordinate systems may result in a streaming-plus-removal operator that is dense in the angular variable or that is not lower-triangular. We investigate numerical solution algorithms for such angular discretizations using relationships given by Chandrasekhar to compute the angular derivatives in the one-dimensional S N transport equation in spherical coordinates with Gauss quadrature. This discretization makes the S N transport equation P N-1 - equivalent, but it also makes the sweep operator dense at every spatial point because the N angular derivatives are expressed in terms of the N angular fluxes. To avoid having to invert the sweep operator directly, we must work with the angular fluxes to solve the equations iteratively. We show how we can use approximations to the sweep operator to precondition the full P N-1 equivalent S N equations. We show that these pre-conditioners affect the operator enough such that convergence of a Krylov iterative method improves. (authors)

  10. A fully coupled Monte Carlo/discrete ordinates solution to the neutron transport equation. Final report

    Energy Technology Data Exchange (ETDEWEB)

    Baker, Randal Scott [Univ. of Arizona, Tucson, AZ (United States)

    1990-01-01

    The neutron transport equation is solved by a hybrid method that iteratively couples regions where deterministic (SN) and stochastic (Monte Carlo) methods are applied. Unlike previous hybrid methods, the Monte Carlo and SN regions are fully coupled in the sense that no assumption is made about geometrical separation or decoupling. The hybrid method provides a new means of solving problems involving both optically thick and optically thin regions that neither Monte Carlo nor SN is well suited for by themselves. The fully coupled Monte Carlo/SN technique consists of defining spatial and/or energy regions of a problem in which either a Monte Carlo calculation or an SN calculation is to be performed. The Monte Carlo region may comprise the entire spatial region for selected energy groups, or may consist of a rectangular area that is either completely or partially embedded in an arbitrary SN region. The Monte Carlo and SN regions are then connected through the common angular boundary fluxes, which are determined iteratively using the response matrix technique, and volumetric sources. The hybrid method has been implemented in the SN code TWODANT by adding special-purpose Monte Carlo subroutines to calculate the response matrices and volumetric sources, and linkage subrountines to carry out the interface flux iterations. The common angular boundary fluxes are included in the SN code as interior boundary sources, leaving the logic for the solution of the transport flux unchanged, while, with minor modifications, the diffusion synthetic accelerator remains effective in accelerating SN calculations. The special-purpose Monte Carlo routines used are essentially analog, with few variance reduction techniques employed. However, the routines have been successfully vectorized, with approximately a factor of five increase in speed over the non-vectorized version.

  11. Correspondence Between One- and Two-Equation Models for Solute Transport in Two-Region Heterogeneous Porous Media

    KAUST Repository

    Davit, Y.; Wood, B. D.; Debenest, G.; Quintard, M.

    2012-01-01

    In this work, we study the transient behavior of homogenized models for solute transport in two-region porous media. We focus on the following three models: (1) a time non-local, two-equation model (2eq-nlt). This model does not rely on time

  12. LUCKY-TD code for solving the time-dependent transport equation with the use of parallel computations

    Energy Technology Data Exchange (ETDEWEB)

    Moryakov, A. V., E-mail: sailor@orc.ru [National Research Centre Kurchatov Institute (Russian Federation)

    2016-12-15

    An algorithm for solving the time-dependent transport equation in the P{sub m}S{sub n} group approximation with the use of parallel computations is presented. The algorithm is implemented in the LUCKY-TD code for supercomputers employing the MPI standard for the data exchange between parallel processes.

  13. Cellular neural network to the spherical harmonics approximation of neutron transport equation in x–y geometry

    International Nuclear Information System (INIS)

    Pirouzmand, Ahmad; Hadad, Kamal

    2012-01-01

    Highlights: ► This paper describes the solution of time-dependent neutron transport equation. ► We use a novel method based on cellular neural networks (CNNs) coupled with the spherical harmonics method. ► We apply the CNN model to simulate step and ramp perturbation transients in a core. ► The accuracy and capabilities of the CNN model are examined for x–y geometry. - Abstract: In an earlier paper we utilized a novel method using cellular neural networks (CNNs) coupled with spherical harmonics method to solve the steady state neutron transport equation in x–y geometry. Here, the previous work is extended to the study of time-dependent neutron transport equation. To achieve this goal, an equivalent electrical circuit based on a second-order form of time-dependent neutron transport equation and one equivalent group of neutron precursor density is obtained by the CNN method. The CNN model is used to simulate step and ramp perturbation transients in a typical 2D core.

  14. Application of synthetic diffusion method in the numerical solution of the equations of neutron transport in slab geometry

    International Nuclear Information System (INIS)

    Valdes Parra, J.J.

    1986-01-01

    One of the main problems in reactor physics is to determine the neutron distribution in reactor core, since knowing that, it is possible to calculate the rapidity of occurrence of different nuclear reaction inside the reactor core. Within different theories existing in nuclear reactor physics, is neutron transport the one in which equation who govern the exact behavior of neutronic distribution are developed even inside the proper neutron transport theory, there exist different methods of solution which are approximations to exact solution; still more, with the purpose to reach a more precise solution, the majority of methods have been approached to the obtention of solutions in numerical form with the aim of take the advantages of modern computers, and for this reason a great deal of effort is dedicated to numerical solution of the equations of neutron transport. In agreement with the above mentioned, in this work has been developed a computer program which uses a relatively new techniques known as 'acceleration of synthetic diffusion' which has been applied to solve the neutron transport equation with 'classical schemes of spatial integration' obtaining results with a smaller quantity of interactions, if they compare to done without using such equation (Author)

  15. An inexact Newton method for fully-coupled solution of the Navier-Stokes equations with heat and mass transport

    Energy Technology Data Exchange (ETDEWEB)

    Shadid, J.N.; Tuminaro, R.S. [Sandia National Labs., Albuquerque, NM (United States); Walker, H.F. [Utah State Univ., Logan, UT (United States). Dept. of Mathematics and Statistics

    1997-02-01

    The solution of the governing steady transport equations for momentum, heat and mass transfer in flowing fluids can be very difficult. These difficulties arise from the nonlinear, coupled, nonsymmetric nature of the system of algebraic equations that results from spatial discretization of the PDEs. In this manuscript the authors focus on evaluating a proposed nonlinear solution method based on an inexact Newton method with backtracking. In this context they use a particular spatial discretization based on a pressure stabilized Petrov-Galerkin finite element formulation of the low Mach number Navier-Stokes equations with heat and mass transport. The discussion considers computational efficiency, robustness and some implementation issues related to the proposed nonlinear solution scheme. Computational results are presented for several challenging CFD benchmark problems as well as two large scale 3D flow simulations.

  16. Modeling of isothermal bubbly flow with interfacial area transport equation and bubble number density approach

    Energy Technology Data Exchange (ETDEWEB)

    Sari, Salih [Hacettepe University, Department of Nuclear Engineering, Beytepe, 06800 Ankara (Turkey); Erguen, Sule [Hacettepe University, Department of Nuclear Engineering, Beytepe, 06800 Ankara (Turkey); Barik, Muhammet; Kocar, Cemil; Soekmen, Cemal Niyazi [Hacettepe University, Department of Nuclear Engineering, Beytepe, 06800 Ankara (Turkey)

    2009-03-15

    In this study, isothermal turbulent bubbly flow is mechanistically modeled. For the modeling, Fluent version 6.3.26 is used as the computational fluid dynamics solver. First, the mechanistic models that simulate the interphase momentum transfer between the gas (bubbles) and liquid (continuous) phases are investigated, and proper models for the known flow conditions are selected. Second, an interfacial area transport equation (IATE) solution is added to Fluent's solution scheme in order to model the interphase momentum transfer mechanisms. In addition to solving IATE, bubble number density (BND) approach is also added to Fluent and this approach is also used in the simulations. Different source/sink models derived for the IATE and BND models are also investigated. The simulations of experiments based on the available data in literature are performed by using IATE and BND models in two and three-dimensions. The results show that the simulations performed by using IATE and BND models agree with each other and with the experimental data. The simulations performed in three-dimensions give better agreement with the experimental data.

  17. Modeling of isothermal bubbly flow with interfacial area transport equation and bubble number density approach

    International Nuclear Information System (INIS)

    Sari, Salih; Erguen, Sule; Barik, Muhammet; Kocar, Cemil; Soekmen, Cemal Niyazi

    2009-01-01

    In this study, isothermal turbulent bubbly flow is mechanistically modeled. For the modeling, Fluent version 6.3.26 is used as the computational fluid dynamics solver. First, the mechanistic models that simulate the interphase momentum transfer between the gas (bubbles) and liquid (continuous) phases are investigated, and proper models for the known flow conditions are selected. Second, an interfacial area transport equation (IATE) solution is added to Fluent's solution scheme in order to model the interphase momentum transfer mechanisms. In addition to solving IATE, bubble number density (BND) approach is also added to Fluent and this approach is also used in the simulations. Different source/sink models derived for the IATE and BND models are also investigated. The simulations of experiments based on the available data in literature are performed by using IATE and BND models in two and three-dimensions. The results show that the simulations performed by using IATE and BND models agree with each other and with the experimental data. The simulations performed in three-dimensions give better agreement with the experimental data

  18. Prediction of adiabatic bubbly flows in TRACE using the interfacial area transport equation

    International Nuclear Information System (INIS)

    Talley, J.; Worosz, T.; Kim, S.; Mahaffy, J.; Bajorek, S.; Tien, K.

    2011-01-01

    The conventional thermal-hydraulic reactor system analysis codes utilize a two-field, two-fluid formulation to model two-phase flows. To close this model, static flow regime transition criteria and algebraic relations are utilized to estimate the interfacial area concentration (a i ). To better reflect the continuous evolution of two-phase flow, an experimental version of TRACE is being developed which implements the interfacial area transport equation (IATE) to replace the flow regime based approach. Dynamic estimation of a i is provided through the use of mechanistic models for bubble coalescence and disintegration. To account for the differences in bubble interactions and drag forces, two-group bubble transport is sought. As such, Group 1 accounts for the transport of spherical and distorted bubbles, while Group 2 accounts for the cap, slug, and churn-turbulent bubbles. Based on this categorization, a two-group IATE applicable to the range of dispersed two-phase flows has been previously developed. Recently, a one-group, one-dimensional, adiabatic IATE has been implemented into the TRACE code with mechanistic models accounting for: (1) bubble breakup due to turbulent impact of an eddy on a bubble, (2) bubble coalescence due to random collision driven by turbulent eddies, and (3) bubble coalescence due to the acceleration of a bubble in the wake region of a preceding bubble. To demonstrate the enhancement of the code's capability using the IATE, experimental data for a i , void fraction, and bubble velocity measured by a multi-sensor conductivity probe are compared to both the IATE and flow regime based predictions. In total, 50 air-water vertical co-current upward and downward bubbly flow conditions in pipes with diameters ranging from 2.54 to 20.32 cm are evaluated. It is found that TRACE, using the conventional flow regime relation, always underestimates a i . Moreover, the axial trend of the a i prediction is always quasi-linear because a i in the

  19. A high-order Petrov-Galerkin method for the Boltzmann transport equation

    International Nuclear Information System (INIS)

    Pain, C.C.; Candy, A.S.; Piggott, M.D.; Buchan, A.; Eaton, M.D.; Goddard, A.J.H.; Oliveira, C.R.E. de

    2005-01-01

    We describe a new Petrov-Galerkin method using high-order terms to introduce dissipation in a residual-free formulation. The method is developed following both a Taylor series analysis and a variational principle, and the result has much in common with traditional Petrov-Galerkin, Self Adjoint Angular Flux (SAAF) and Even Parity forms of the Boltzmann transport equation. In addition, we consider the subtleties in constructing appropriate boundary conditions. In sub-grid scale (SGS) modelling of fluids the advantages of high-order dissipation are well known. Fourth-order terms, for example, are commonly used as a turbulence model with uniform dissipation. They have been shown to have superior properties to SGS models based upon second-order dissipation or viscosity. Even higher-order forms of dissipation (e.g. 16.-order) can offer further advantages, but are only easily realised by spectral methods because of the solution continuity requirements that these higher-order operators demand. Higher-order operators are more effective, bringing a higher degree of representation to the solution locally. Second-order operators, for example, tend to relax the solution to a linear variation locally, whereas a high-order operator will tend to relax the solution to a second-order polynomial locally. The form of the dissipation is also important. For example, the dissipation may only be applied (as it is in this work) in the streamline direction. While for many problems, for example Large Eddy Simulation (LES), simply adding a second or fourth-order dissipation term is a perfectly satisfactory SGS model, it is well known that a consistent residual-free formulation is required for radiation transport problems. This motivated the consideration of a new Petrov-Galerkin method that is residual-free, but also benefits from the advantageous features that SGS modelling introduces. We close with a demonstration of the advantages of this new discretization method over standard Petrov

  20. High energy ion range and deposited energy calculation using the Boltzmann-Fokker-Planck splitting of the Boltzmann transport equation

    International Nuclear Information System (INIS)

    Mozolevski, I.E.

    2001-01-01

    We consider the splitting of the straight-ahead Boltzmann transport equation in the Boltzmann-Fokker-Planck equation, decomposing the differential cross-section into a singular part, corresponding to small energy transfer events, and in a regular one, which corresponds to large energy transfer. The convergence of implantation profile, nuclear and electronic energy depositions, calculated from the Boltzmann-Fokker-Planck equation, to the respective exact distributions, calculated from Monte-Carlo method, was exanimate in a large-energy interval for various values of splitting parameter and for different ion-target mass relations. It is shown that for the universal potential there exists an optimal value of splitting parameter, for which range and deposited energy distributions, calculated from the Boltzmann-Fokker-Planck equation, accurately approximate the exact distributions and which minimizes the computational expenses

  1. Transport Equations for CAD Modeling of Al(x)Ga(1-x)N/GaN HEMTs

    Science.gov (United States)

    Freeman, Jon C.

    2003-01-01

    BEMTs formed from Al(x)Ga(1-x)N/GaN heterostructures are being investigated for high RF power and efficiency around the world by many groups, both academic and industrial. In these devices, the 2DEG formation is dominated by both spontaneous and piezoelectric polarization fields, with each component having nearly the same order of magnitude. The piezoelectric portion is induced by the mechanical strain in the structure, and to analyze these devices, one must incorporate the stress/strain relationships, along with the standard semiconductor transport equations. These equations for Wurtzite GaN are not easily found in the open literature, hence this paper summarizes them, along with the constitutive equations for piezoelectric materials. The equations are cast into the format for the Wurtzite crystal class, which is the most common way GaN is grown epitaxially.

  2. Coupling of discrete ordinates methods by transmission of boundary conditions in solving the neutron transport equation in slab geometry; Couplage de discretisations aux ordonnees discretes d`equations de transport 1D par passage de conditions frontieres

    Energy Technology Data Exchange (ETDEWEB)

    Bal, G. [Departement MMN, Service IMA, Direction des Etudes et Recherches, Electricite de France (EDF), 92 - Clamart (France)

    1995-10-01

    Neutron transport in nuclear reactors is quite well modelled by the linear Boltzmann transport equation. Its solution is relatively easy, but unfortunately too expensive to achieve whole core computations. Thus, we have to simplify it, for example by homogenizing some physical characteristics. However, the solution may then be inaccurate. Moreover, in strongly homogeneous areas, the error may be too big. Then we would like to deal with such an inconvenient by solving the equation accurately on this area, but more coarsely away from it, so that the computation is not too expensive. This problem is the subject of a thesis. We present here some results obtained for slab geometry. The couplings between the fine and coarse discretization regions could be conceived in a number of approaches. Here, we only deal with the coupling at crossing the interface between two sub-domains. In the first section, we present the coupling of discrete ordinate methods for solving the homogeneous, isotropic and mono-kinetic equation. Coupling operators are defined and shown to be optimal. The second and the third sections are devoted to an extension of the previous results when the equation is non-homogeneous, anisotropic and multigroup (under some restrictive assumptions). Some numerical results are given in the case of isotropic and mono-kinetic equations. (author) 15 refs.

  3. Application of longshore transport equations to Andhra coast, East coast of India

    Digital Repository Service at National Institute of Oceanography (India)

    Chandramohan, P.; Nayak, B.U.; RamaRaju, V.S.

    during November to February. The longshore transport rate is high during the southwest monsoon period from June to September. A higher sediment transport rate is observed for the coastline oriented at 80 degrees east of north. The annual net transport...

  4. Evaluation of an analytic linear Boltzmann transport equation solver for high-density inhomogeneities

    Energy Technology Data Exchange (ETDEWEB)

    Lloyd, S. A. M.; Ansbacher, W. [Department of Physics and Astronomy, University of Victoria, Victoria, British Columbia V8W 3P6 (Canada); Department of Physics and Astronomy, University of Victoria, Victoria, British Columbia V8W 3P6 (Canada) and Department of Medical Physics, British Columbia Cancer Agency-Vancouver Island Centre, Victoria, British Columbia V8R 6V5 (Canada)

    2013-01-15

    Purpose: Acuros external beam (Acuros XB) is a novel dose calculation algorithm implemented through the ECLIPSE treatment planning system. The algorithm finds a deterministic solution to the linear Boltzmann transport equation, the same equation commonly solved stochastically by Monte Carlo methods. This work is an evaluation of Acuros XB, by comparison with Monte Carlo, for dose calculation applications involving high-density materials. Existing non-Monte Carlo clinical dose calculation algorithms, such as the analytic anisotropic algorithm (AAA), do not accurately model dose perturbations due to increased electron scatter within high-density volumes. Methods: Acuros XB, AAA, and EGSnrc based Monte Carlo are used to calculate dose distributions from 18 MV and 6 MV photon beams delivered to a cubic water phantom containing a rectangular high density (4.0-8.0 g/cm{sup 3}) volume at its center. The algorithms are also used to recalculate a clinical prostate treatment plan involving a unilateral hip prosthesis, originally evaluated using AAA. These results are compared graphically and numerically using gamma-index analysis. Radio-chromic film measurements are presented to augment Monte Carlo and Acuros XB dose perturbation data. Results: Using a 2% and 1 mm gamma-analysis, between 91.3% and 96.8% of Acuros XB dose voxels containing greater than 50% the normalized dose were in agreement with Monte Carlo data for virtual phantoms involving 18 MV and 6 MV photons, stainless steel and titanium alloy implants and for on-axis and oblique field delivery. A similar gamma-analysis of AAA against Monte Carlo data showed between 80.8% and 87.3% agreement. Comparing Acuros XB and AAA evaluations of a clinical prostate patient plan involving a unilateral hip prosthesis, Acuros XB showed good overall agreement with Monte Carlo while AAA underestimated dose on the upstream medial surface of the prosthesis due to electron scatter from the high-density material. Film measurements

  5. Modeling strategy of the source and sink terms in the two-group interfacial area transport equation

    International Nuclear Information System (INIS)

    Ishii, Mamoru; Sun Xiaodong; Kim, Seungjin

    2003-01-01

    This paper presents the general strategy for modeling the source and sink terms in the two-group interfacial area transport equation. The two-group transport equation is applicable in bubbly, cap bubbly, slug, and churn-turbulent flow regimes to predict the change of the interfacial area concentration. This dynamic approach has an advantage of flow regime-independence over the conventional empirical correlation approach for the interfacial area concentration in the applications with the two-fluid model. In the two-group interfacial area transport equation, bubbles are categorized into two groups: spherical/distorted bubbles as Group 1 and cap/slug/churn-turbulent bubbles as Group 2. Thus, two sets of equations are used to describe the generation and destruction rates of bubble number density, void fraction, and interfacial area concentration for the two groups of bubbles due to bubble expansion and compression, coalescence and disintegration, and phase change. Based upon a detailed literature review of the research on the bubble interactions, five major bubble interaction mechanisms are identified for the gas-liquid two-phase flow of interest. A systematic integral approach, in which the significant variations of bubble volume and shape are accounted for, is suggested for the modeling of two-group bubble interactions. To obtain analytical forms for the various bubble interactions, a simplification is made for the bubble number density distribution function

  6. Development of two-group interfacial area transport equation for confined flow-1. Modeling of bubble interactions

    International Nuclear Information System (INIS)

    Sun, Xiaodong; Kim, Seungjin; Ishii, Mamoru; Beus, Stephen G.

    2003-01-01

    This paper presents the modeling of bubble interaction mechanisms in the two-group interfacial area transport equation (IATE) for confined gas-liquid two-phase flow. The transport equation is applicable to bubbly, cap-turbulent, and churn-turbulent flow regimes. In the two-group IATE, bubbles are categorized into two groups: spherical/distorted bubbles as Group 1 and cap/slug/churn-turbulent bubbles as Group 2. Thus, two sets of equations are used to describe the generation and destruction rates of bubble number density, void fraction, and interfacial area concentration for the two groups of bubbles due to bubble expansion and compression, coalescence and disintegration, and phase change. Five major bubble interaction mechanisms are identified for the gas-liquid two-phase flow of interest, and are analytically modeled as the source/sink terms for the transport equations based on certain assumptions for the confined flow. These models include both intra-group (within a certain group) and inter-group (between two groups) bubble interactions. The comparisons of the prediction by the one-dimensional two-group IATE with experimental data are presented in the second paper of this series. (author)

  7. Two-dimensional Haar wavelet Collocation Method for the solution of Stationary Neutron Transport Equation in a homogeneous isotropic medium

    International Nuclear Information System (INIS)

    Patra, A.; Saha Ray, S.

    2014-01-01

    Highlights: • A stationary transport equation has been solved using the technique of Haar wavelet Collocation Method. • This paper intends to provide the great utility of Haar wavelets to nuclear science problem. • In the present paper, two-dimensional Haar wavelets are applied. • The proposed method is mathematically very simple, easy and fast. - Abstract: This paper emphasizes on finding the solution for a stationary transport equation using the technique of Haar wavelet Collocation Method (HWCM). Haar wavelet Collocation Method is efficient and powerful in solving wide class of linear and nonlinear differential equations. Recently Haar wavelet transform has gained the reputation of being a very effective tool for many practical applications. This paper intends to provide the great utility of Haar wavelets to nuclear science problem. In the present paper, two-dimensional Haar wavelets are applied for solution of the stationary Neutron Transport Equation in homogeneous isotropic medium. The proposed method is mathematically very simple, easy and fast. To demonstrate about the efficiency of the method, one test problem is discussed. It can be observed from the computational simulation that the numerical approximate solution is much closer to the exact solution

  8. Spatial discretizations for self-adjoint forms of the radiative transfer equations

    International Nuclear Information System (INIS)

    Morel, Jim E.; Adams, B. Todd; Noh, Taewan; McGhee, John M.; Evans, Thomas M.; Urbatsch, Todd J.

    2006-01-01

    There are three commonly recognized second-order self-adjoint forms of the neutron transport equation: the even-parity equations, the odd-parity equations, and the self-adjoint angular flux equations. Because all of these equations contain second-order spatial derivatives and are self-adjoint for the mono-energetic case, standard continuous finite-element discretization techniques have proved quite effective when applied to the spatial variables. We first derive analogs of these equations for the case of time-dependent radiative transfer. The primary unknowns for these equations are functions of the angular intensity rather than the angular flux, hence the analog of the self-adjoint angular flux equation is referred to as the self-adjoint angular intensity equation. Then we describe a general, arbitrary-order, continuous spatial finite-element approach that is applied to each of the three equations in conjunction with backward-Euler differencing in time. We refer to it as the 'standard' technique. We also introduce an alternative spatial discretization scheme for the self-adjoint angular intensity equation that requires far fewer unknowns than the standard method, but appears to give comparable accuracy. Computational results are given that demonstrate the validity of both of these discretization schemes

  9. Solution of unidimensional problems from monoenergetics neutrons diffusion through finite differences

    International Nuclear Information System (INIS)

    Filio Lopez, Carlos.

    1979-01-01

    A calculation program (URA 6.F4) was elaborated on FORTRAN IV language, that through finite differences solves the unidimensional scalar Helmholtz equation, assuming only one energy group, in spherical cylindrical or plane geometry. The purpose is the determination of the flow distribution in a reactor of spherical cylindrical or plane geometry and the critical dimensions. Feeding as entrance datas to the program the geometry, diffusion coefficients and macroscopic transversals cross sections of absorption and fission for each region. The differential diffusion equation is converted with its boundary conditions, to one system of homogeneous algebraic linear equations using the box integration technique. The investigation on criticality is converted then in a succession of eigenvalue problems for the critical eigenvalue. In general, only is necessary to solve the first eigenvalue and its corresponding eigenvector, employing the power method. The obtained results by the program for the critical dimensions of the clean reactors are admissible, the existing error as respect to the analytic is less of 0.5%; by the analysed reactors of three regions, the relative error with respect to the semianalytic result is less of 0.2%. With this program is possible to obtain one quantitative description of one reactor if the transversal sections that appears in the monoenergetic model are adequatedly averaged by the energy group used. (author)

  10. Optical diffraction tomography microscopy with transport of intensity equation using a light-emitting diode array

    Science.gov (United States)

    Li, Jiaji; Chen, Qian; Zhang, Jialin; Zhang, Zhao; Zhang, Yan; Zuo, Chao

    2017-08-01

    Optical diffraction tomography (ODT) is an effective label-free technique for quantitatively refractive index imaging, which enables long-term monitoring of the internal three-dimensional (3D) structures and molecular composition of biological cells with minimal perturbation. However, existing optical tomographic methods generally rely on interferometric configuration for phase measurement and sophisticated mechanical systems for sample rotation or beam scanning. Thereby, the measurement is suspect to phase error coming from the coherent speckle, environmental vibrations, and mechanical error during data acquisition process. To overcome these limitations, we present a new ODT technique based on non-interferometric phase retrieval and programmable illumination emitting from a light-emitting diode (LED) array. The experimental system is built based on a traditional bright field microscope, with the light source replaced by a programmable LED array, which provides angle-variable quasi-monochromatic illumination with an angular coverage of ±37 degrees in both x and y directions (corresponding to an illumination numerical aperture of ∼0.6). Transport of intensity equation (TIE) is utilized to recover the phase at different illumination angles, and the refractive index distribution is reconstructed based on the ODT framework under first Rytov approximation. The missing-cone problem in ODT is addressed by using the iterative non-negative constraint algorithm, and the misalignment of the LED array is further numerically corrected to improve the accuracy of refractive index quantification. Experiments on polystyrene beads and thick biological specimens show that the proposed approach allows accurate refractive index reconstruction while greatly reduced the system complexity and environmental sensitivity compared to conventional interferometric ODT approaches.

  11. Computational diffraction tomographic microscopy with transport of intensity equation using a light-emitting diode array

    Science.gov (United States)

    Li, Jiaji; Chen, Qian; Zhang, Jialin; Zuo, Chao

    2017-10-01

    Optical diffraction tomography (ODT) is an effective label-free technique for quantitatively refractive index imaging, which enables long-term monitoring of the internal three-dimensional (3D) structures and molecular composition of biological cells with minimal perturbation. However, existing optical tomographic methods generally rely on interferometric configuration for phase measurement and sophisticated mechanical systems for sample rotation or beam scanning. Thereby, the measurement is suspect to phase error coming from the coherent speckle, environmental vibrations, and mechanical error during data acquisition process. To overcome these limitations, we present a new ODT technique based on non-interferometric phase retrieval and programmable illumination emitting from a light-emitting diode (LED) array. The experimental system is built based on a traditional bright field microscope, with the light source replaced by a programmable LED array, which provides angle-variable quasi-monochromatic illumination with an angular coverage of +/-37 degrees in both x and y directions (corresponding to an illumination numerical aperture of ˜ 0.6). Transport of intensity equation (TIE) is utilized to recover the phase at different illumination angles, and the refractive index distribution is reconstructed based on the ODT framework under first Rytov approximation. The missing-cone problem in ODT is addressed by using the iterative non-negative constraint algorithm, and the misalignment of the LED array is further numerically corrected to improve the accuracy of refractive index quantification. Experiments on polystyrene beads and thick biological specimens show that the proposed approach allows accurate refractive index reconstruction while greatly reduced the system complexity and environmental sensitivity compared to conventional interferometric ODT approaches.

  12. Electronic structure and transport in the low-temperature thermoelectric CsBi4Te6: Semiclassical transport equations

    DEFF Research Database (Denmark)

    Lykke, Lars; Iversen, Bo Brummerstedt; Madsen, Georg

    2006-01-01

    The band structure of the low-temperature thermoelectric material, CsBi4Te6, is calculated and analyzed using the semiclassic transport equations. It is shown that to obtain a quantitative agreement with measured transport properties, a band gap of 0.08 eV must be enforced. A gap in reasonable...... agreement with experiment was obtained using the generalized gradient functional of Engel and Vosko [E. Engel and S. H. Vosko, Phys. Rev. B 47, 13164 (1993)]. We found that the experimental p-type sample has a carrier concentration close to optimal. Furthermore, the conduction bands have a form equally well...

  13. Improvement of the symbolic Monte-Carlo method for the transport equation: P1 extension and coupling with diffusion

    International Nuclear Information System (INIS)

    Clouet, J.F.; Samba, G.

    2005-01-01

    We use asymptotic analysis to study the diffusion limit of the Symbolic Implicit Monte-Carlo (SIMC) method for the transport equation. For standard SIMC with piecewise constant basis functions, we demonstrate mathematically that the solution converges to the solution of a wrong diffusion equation. Nevertheless a simple extension to piecewise linear basis functions enables to obtain the correct solution. This improvement allows the calculation in opaque medium on a mesh resolving the diffusion scale much larger than the transport scale. Anyway, the huge number of particles which is necessary to get a correct answer makes this computation time consuming. Thus, we have derived from this asymptotic study an hybrid method coupling deterministic calculation in the opaque medium and Monte-Carlo calculation in the transparent medium. This method gives exactly the same results as the previous one but at a much lower price. We present numerical examples which illustrate the analysis. (authors)

  14. Coupled energy-drift and force-balance equations for high-field hot-carrier transport

    International Nuclear Information System (INIS)

    Huang, Danhong; Alsing, P.M.; Apostolova, T.; Cardimona, D.A.

    2005-01-01

    Coupled energy-drift and force-balance equations that contain a frictional force for the center-of-mass motion of electrons are derived for hot-electron transport under a strong dc electric field. The frictional force is found to be related to the net rate of phonon emission, which takes away the momentum of a phonon from an electron during each phonon-emission event. The net rate of phonon emission is determined by the Boltzmann scattering equation, which depends on the distribution of electrons interacting with phonons. The work done by the frictional force is included into the energy-drift equation for the electron-relative scattering motion and is found to increase the thermal energy of the electrons. The importance of the hot-electron effect in the energy-drift term under a strong dc field is demonstrated in reducing the field-dependent drift velocity and mobility. The Doppler shift in the energy conservation of scattering electrons interacting with impurities and phonons is found to lead to an anisotropic distribution of electrons in the momentum space along the field direction. The importance of this anisotropic distribution is demonstrated through a comparison with the isotropic energy-balance equation, from which we find that defining a state-independent electron temperature becomes impossible. To the leading order, the energy-drift equation is linearized with a distribution function by expanding it into a Fokker-Planck-type equation, along with the expansions of both the force-balance equation and the Boltzmann scattering equation for hot phonons

  15. Solution of multigroup transport equation in x-y-z geometry by the spherical harmonics method using finite Fourier transformation

    International Nuclear Information System (INIS)

    Kobayashi, Keisuke; Kikuchi, Hirohiko; Tsutsuguchi, Ken

    1993-01-01

    A neutron multigroup transport equation in x-y-z geometry is solved by the spherical harmonics method using finite Fourier transformation. Using the first term of the Fourier series for the space variables of spherical harmonics moments, three-point finite difference like equations are derived for x-, y- and z-axis directions, which are more consistent and accurate than those derived using the usual finite difference approximation, and these equations are solved by the iteration method in each axis direction alternatively. A method to find an optimum acceleration factor for this inner iteration is described. It is shown in the numerical examples that the present method gives higher accuracy with less mesh points that the usual finite difference method. (author)

  16. A quasi moment description of the evolution of an electron gas towards a state dominated by a reduced transport equation

    International Nuclear Information System (INIS)

    Oeien, A.H.

    1980-09-01

    For electrons in electric and magnetic fields which collide elastically with neutral atoms or molecules a minute evolution study is made using the multiple time scale method. In this study a set of quasi moment equations is used which is derived from the Boltzmann equation by taking appropriate quasi moments, i.e. velocity moments where the integration is performed only over velocity angles. In a systematic way the evolution in a transient regime is revealed where processes take place on time scales related to the electron-atom collision frequency and electron cyclotron frequency and how the evolution enters a regime where it is governed by a reduced transport equation is shown. This work has relevance to the theory of evolution of gases of charged particles in general and to non-neutral plasmas and partially ionized gases in particular. (Auth.)

  17. Elliptic equation for random walks. Application to transport in microporous media

    DEFF Research Database (Denmark)

    Shapiro, Alexander

    2007-01-01

    We consider a process of random walks with arbitrary residence time distribution. We show that in many cases this process may not be described by the classical (Fick) parabolic diffusion equation, but an elliptic equation. An additional term proportional to the second time derivative takes into a...

  18. Analysis of a block Gauss-Seidel iterative method for a finite element discretization of the neutron transport equation

    International Nuclear Information System (INIS)

    Lorence, L.J. Jr.; Martin, W.R.; Luskin, M.

    1985-01-01

    We prove the convergence of a finite element discretization of the neutron transport equation. The iterative solution of the resulting linear system by a block Gauss-Seidel method is also analyzed. This procedure is shown to require less storage than the direct solution by Gaussian elimination, and an estimate for the rate of convergence is used to show that fewer arithmetic operations are required

  19. Calculation of the critical buckling of a lattice based on the integral form of the transport equation

    International Nuclear Information System (INIS)

    Benoist, P.

    1990-06-01

    The migration area, which relates the buckling to the multiplication factor, can be calculated by means of the Deniz formula. This formula involves the direct and adjoint angular fluxes. It is shown in this note that it is possible, using the integral form of the transport equation, to establish an equivalent formula in which only angle-integrated quantities appear. This formulation is more suitable for the calculation by the collision probably method [fr

  20. Solution of two-dimensional equations of neutron transport in 4P0-approximation of spherical harmonics method

    International Nuclear Information System (INIS)

    Polivanskij, V.P.

    1989-01-01

    The method to solve two-dimensional equations of neutron transport using 4P 0 -approximation is presented. Previously such approach was efficiently used for the solution of one-dimensional problems. New an attempt is made to apply the approach to solution of two-dimensional problems. Algorithm of the solution is given, as well as results of test neutron-physical calculations. A considerable as compared with diffusion approximation is shown. 11 refs

  1. Solution of the within-group multidimensional discrete ordinates transport equations on massively parallel architectures

    Science.gov (United States)

    Zerr, Robert Joseph

    2011-12-01

    The integral transport matrix method (ITMM) has been used as the kernel of new parallel solution methods for the discrete ordinates approximation of the within-group neutron transport equation. The ITMM abandons the repetitive mesh sweeps of the traditional source iterations (SI) scheme in favor of constructing stored operators that account for the direct coupling factors among all the cells and between the cells and boundary surfaces. The main goals of this work were to develop the algorithms that construct these operators and employ them in the solution process, determine the most suitable way to parallelize the entire procedure, and evaluate the behavior and performance of the developed methods for increasing number of processes. This project compares the effectiveness of the ITMM with the SI scheme parallelized with the Koch-Baker-Alcouffe (KBA) method. The primary parallel solution method involves a decomposition of the domain into smaller spatial sub-domains, each with their own transport matrices, and coupled together via interface boundary angular fluxes. Each sub-domain has its own set of ITMM operators and represents an independent transport problem. Multiple iterative parallel solution methods have investigated, including parallel block Jacobi (PBJ), parallel red/black Gauss-Seidel (PGS), and parallel GMRES (PGMRES). The fastest observed parallel solution method, PGS, was used in a weak scaling comparison with the PARTISN code. Compared to the state-of-the-art SI-KBA with diffusion synthetic acceleration (DSA), this new method without acceleration/preconditioning is not competitive for any problem parameters considered. The best comparisons occur for problems that are difficult for SI DSA, namely highly scattering and optically thick. SI DSA execution time curves are generally steeper than the PGS ones. However, until further testing is performed it cannot be concluded that SI DSA does not outperform the ITMM with PGS even on several thousand or tens of

  2. Generation of monoenergetic ion beams via ionization dynamics (Conference Presentation)

    Science.gov (United States)

    Lin, Chen; Kim, I. Jong; Yu, Jinqing; Choi, Il Woo; Ma, Wenjun; Yan, Xueqing; Nam, Chang Hee

    2017-05-01

    The research on ion acceleration driven by high intensity laser pulse has attracted significant interests in recent decades due to the developments of laser technology. The intensive study of energetic ion bunches is particularly stimulated by wide applications in nuclear fusion, medical treatment, warm dense matter production and high energy density physics. However, to implement such compact accelerators, challenges are still existing in terms of beam quality and stability, especially in applications that require higher energy and narrow bandwidth spectra ion beams. We report on the acceleration of quasi-mono-energetic ion beams via ionization dynamics in the interaction of an intense laser pulse with a solid target. Using ionization dynamics model in 2D particle-in-cell (PIC) simulations, we found that high charge state contamination ions can only be ionized in the central spot area where the intensity of sheath field surpasses their ionization threshold. These ions automatically form a microstructure target with a width of few micron scale, which is conducive to generate mono-energetic beams. In the experiment of ultraintense (< 10^21 W/cm^2) laser pulses irradiating ultrathin targets each attracted with a contamination layer of nm-thickness, high quality < 100 MeV mono-energetic ion bunches are generated. The peak energy of the self-generated micro-structured target ions with respect to different contamination layer thickness is also examined This is relatively newfound respect, which is confirmed by the consistence between experiment data and the simulation results.

  3. Target shape effects on monoenergetic GeV proton acceleration

    Energy Technology Data Exchange (ETDEWEB)

    Chen Min; Yu Tongpu; Pukhov, Alexander [Institut fuer Theoretische Physik I, Heinrich-Heine-Universitaet Duesseldorf, 40225 Duesseldorf (Germany); Sheng Zhengming, E-mail: pukhov@tp1.uni-duesseldorf.d [Department of Physics, Shanghai Jiao Tong University, Shanghai 200240 (China)

    2010-04-15

    When a circularly polarized laser pulse interacts with a foil target, there are three stages: pre-hole-boring, hole-boring and light sail acceleration. We study the electron and ion dynamics in the first stage and find the minimum foil thickness requirement for a given laser intensity. Based on this analysis, we propose using a shaped foil for ion acceleration, whose thickness varies transversely to match the laser intensity. Then, the target evolves into three regions: the acceleration, transparency and deformation regions. In the acceleration region, the target can be uniformly accelerated producing a mono-energetic and spatially collimated ion beam. Detailed numerical simulations are performed to check the feasibility and robustness of this scheme, such as the influence of shape factors and surface roughness. A GeV mono-energetic proton beam is observed in three-dimensional particle-in-cell simulations when a laser pulse with a focus intensity of 10{sup 22} W cm{sup -2} is used. The energy conversion efficiency of the laser pulse to the accelerated proton beam with the simulation parameters is more than 23%.

  4. Target shape effects on monoenergetic GeV proton acceleration

    International Nuclear Information System (INIS)

    Chen Min; Yu Tongpu; Pukhov, Alexander; Sheng Zhengming

    2010-01-01

    When a circularly polarized laser pulse interacts with a foil target, there are three stages: pre-hole-boring, hole-boring and light sail acceleration. We study the electron and ion dynamics in the first stage and find the minimum foil thickness requirement for a given laser intensity. Based on this analysis, we propose using a shaped foil for ion acceleration, whose thickness varies transversely to match the laser intensity. Then, the target evolves into three regions: the acceleration, transparency and deformation regions. In the acceleration region, the target can be uniformly accelerated producing a mono-energetic and spatially collimated ion beam. Detailed numerical simulations are performed to check the feasibility and robustness of this scheme, such as the influence of shape factors and surface roughness. A GeV mono-energetic proton beam is observed in three-dimensional particle-in-cell simulations when a laser pulse with a focus intensity of 10 22 W cm -2 is used. The energy conversion efficiency of the laser pulse to the accelerated proton beam with the simulation parameters is more than 23%.

  5. Method of solution of the neutron transport equation in multidimensional cartesian geometries using spherical harmonics and spatially orthogonal polynomials

    International Nuclear Information System (INIS)

    Fenstermacher, T.E.

    1981-01-01

    The solution of the neutron transport equation has long been a subject of intense interest to nuclear engineers. Present computer codes for the solution of this equation, however, are expensive to run for large, multidimensional problems, and also suffer from computational problems such as the ray effect. A method has been developed which eliminates many of these problems. It consists of transforming the transport equation into a set of linear partial differential equations by the use of spherical harmonics. The problem volume is divided into mesh boxes, and the flux components are approximated within each mesh box by spatially orthogonal quadratic polynomials, which need not be continuous at mesh box interfaces. A variational principle is developed, and used to solve for the unknown coefficients of these polynomials. Both one dimensional and two dimensional computer codes using this method have been written. The codes have each been tested on several test cases, and the solutions checked against solutions obtained by other methods. While the codes have some difficulty in modeling sharp transients, they produce excellent results on problems where the characteristic lengths are many mean free paths. On one test case, the two dimensional code, SHOP/2D, required only one-fourth the computer time required by the finite difference, discrete ordinates code TWOTRAN to produce a solution. In addition, SHOP/2D converged much better than TWOTRAN and produced more physical-appearing results

  6. Cellular neural networks (CNN) simulation for the TN approximation of the time dependent neutron transport equation in slab geometry

    International Nuclear Information System (INIS)

    Hadad, Kamal; Pirouzmand, Ahmad; Ayoobian, Navid

    2008-01-01

    This paper describes the application of a multilayer cellular neural network (CNN) to model and solve the time dependent one-speed neutron transport equation in slab geometry. We use a neutron angular flux in terms of the Chebyshev polynomials (T N ) of the first kind and then we attempt to implement the equations in an equivalent electrical circuit. We apply this equivalent circuit to analyze the T N moments equation in a uniform finite slab using Marshak type vacuum boundary condition. The validity of the CNN results is evaluated with numerical solution of the steady state T N moments equations by MATLAB. Steady state, as well as transient simulations, shows a very good comparison between the two methods. We used our CNN model to simulate space-time response of total flux and its moments for various c (where c is the mean number of secondary neutrons per collision). The complete algorithm could be implemented using very large-scale integrated circuit (VLSI) circuitry. The efficiency of the calculation method makes it useful for neutron transport calculations

  7. Macroscopic transport equations in many-body systems from microscopic exclusion processes in disordered media: a review

    Directory of Open Access Journals (Sweden)

    Marta Galanti

    2016-08-01

    Full Text Available Describing particle transport at the macroscopic or mesoscopic level in non-ideal environments poses fundamental theoretical challenges in domains ranging from inter and intra-cellular transport in biology to diffusion in porous media. Yet, often the nature of the constraints coming from many-body interactions or reflecting a complex and confining environment are better understood and modeled at the microscopic level.In this paper we review the subtle link between microscopic exclusion processes and the mean-field equations that ensue from them in the continuum limit. We show that in an inhomogeneous medium, i.e. when jumps are controlled by site-dependent hopping rates, one can obtain three different nonlinear advection-diffusion equations in the continuum limit, suitable for describing transport in the presence of quenched disorder and external fields, depending on the particular rule embodying site inequivalence at the microscopic level. In a situation that might be termed point-like scenario, when particles are treated as point-like objects, the effect of crowding as imposed at the microscopic level manifests in the mean-field equations only if some degree of inhomogeneity is enforced into the model. Conversely, when interacting agents are assigned a finite size, under the more realistic extended crowding framework, exclusion constraints persist in the unbiased macroscopic representation.

  8. A Spatial Discretization Scheme for Solving the Transport Equation on Unstructured Grids of Polyhedra

    International Nuclear Information System (INIS)

    Thompson, K.G.

    2000-01-01

    In this work, we develop a new spatial discretization scheme that may be used to numerically solve the neutron transport equation. This new discretization extends the family of corner balance spatial discretizations to include spatial grids of arbitrary polyhedra. This scheme enforces balance on subcell volumes called corners. It produces a lower triangular matrix for sweeping, is algebraically linear, is non-negative in a source-free absorber, and produces a robust and accurate solution in thick diffusive regions. Using an asymptotic analysis, we design the scheme so that in thick diffusive regions it will attain the same solution as an accurate polyhedral diffusion discretization. We then refine the approximations in the scheme to reduce numerical diffusion in vacuums, and we attempt to capture a second order truncation error. After we develop this Upstream Corner Balance Linear (UCBL) discretization we analyze its characteristics in several limits. We complete a full diffusion limit analysis showing that we capture the desired diffusion discretization in optically thick and highly scattering media. We review the upstream and linear properties of our discretization and then demonstrate that our scheme captures strictly non-negative solutions in source-free purely absorbing media. We then demonstrate the minimization of numerical diffusion of a beam and then demonstrate that the scheme is, in general, first order accurate. We also note that for slab-like problems our method actually behaves like a second-order method over a range of cell thicknesses that are of practical interest. We also discuss why our scheme is first order accurate for truly 3D problems and suggest changes in the algorithm that should make it a second-order accurate scheme. Finally, we demonstrate 3D UCBL's performance on several very different test problems. We show good performance in diffusive and streaming problems. We analyze truncation error in a 3D problem and demonstrate robustness in a

  9. Investigating transport capacity equations in sediment yield modelling for the Cariri semi-arid region of Paraiba-PB/Brazil

    Directory of Open Access Journals (Sweden)

    E. E. De Figueiredo

    2015-03-01

    Full Text Available In the semi arid Cariri region of the state of Paraiba, Brazil, runoff is of the Hortonian type generated by excess of rainfall over infiltration capacity, and soil erosion is governed by rainfall intensity and sediment size. However, the governing sediment transport mechanism is not well understood. Sediment transport generally depends on the load of sediment provided by soil erosion and on the transport capacity of the flow. The latter is mainly governed by mechanisms such as water shear stress, or stream power. Accordingly, the load of sediment transported by the flow may vary depending on the mechanism involved in the equation of estimation. Investigation of the sediment transport capacity of the flow via a distributed physically-based model is an important and necessary task, but quite rare in semi-arid climates, and particularly in the Cariri region of the state of Paraíba/Brazil. In this study, the equations of Yalin, Engelund & Hansen, Laursen, DuBoys and Bagnold have been coupled with the MOSEE distributed physically based model aiming at identifying the mechanisms leading to the best model simulations when compared with data observed at various basin scales and land uses in the study region. The results obtained with the investigated methods were quite similar and satisfactory suggesting the feasibility of the mechanisms involved, but the observed values were better represented with Bagnold’s equation, which is physically grounded on the stream power, and we recommend it for simulations of similar climate, runoff generation mechanisms and sediment characteristics as in the study region.

  10. A stochastic model of multiple scattering of charged particles: process, transport equation and solutions

    International Nuclear Information System (INIS)

    Papiez, L.; Moskvin, V.; Tulovsky, V.

    2001-01-01

    The process of angular-spatial evolution of multiple scattering of charged particles can be described by a special case of Boltzmann integro-differential equation called Lewis equation. The underlying stochastic process for this evolution is the compound Poisson process on the surface of the unit sphere. The significant portion of events that constitute compound Poisson process that describes multiple scattering have diffusional character. This property allows to analyze the process of angular-spatial evolution of multiple scattering of charged particles as combination of soft and hard collision processes and compute appropriately its transition densities. These computations provide a method of the approximate solution to the Lewis equation. (orig.)

  11. On an integral equation arising in the transport of radiation through a slab involving internal reflection

    International Nuclear Information System (INIS)

    Williams, M.M.R.

    2005-01-01

    The integral equation derived by Nieuwenhuizen and Luck for transmission of radiation through an optically thick diffusive medium is reconsidered in the light of radiative transfer theory and extended to slabs of arbitrary thickness. (author)

  12. Normal and adjoint integral and integrodifferential neutron transport equations. Pt. 1

    International Nuclear Information System (INIS)

    Velarde, G.

    1976-01-01

    Using some simplifying hypotheses, different expressions of the Boltzmann integrodifferential equation are obtained. Posteriorly, they are applied to some particular cases: slowing down, thermalization, multigroups, critical reactors and virtual critical reactors with k, α and lambda. (author)

  13. Transport and relaxation properties of superfluid 3He. I. Kinetic equation and Bogoliubov quasiparticle relaxation rate

    International Nuclear Information System (INIS)

    Einzel, D.; Woelfle, P.

    1978-01-01

    The kinetic equation for Bogoliubov quasiparticles for both the A and B phases of superfluid 3 He is derived from the general matrix kinetic equation. A condensed expression for the exact spin-symmetric collision integral is given. The quasiparticle relaxation rate is calculated for the BW state using the s--p approximation for the quasiparticle scattering amplitude. By using the results for the quasiparticle relaxation rate, the mean free path of Bogoliubov quasiparticles is calculated for all temperatures

  14. The LTSN method used in transport equation, applied in nuclear engineering problems

    International Nuclear Information System (INIS)

    Borges, Volnei; Vilhena, Marco Tulio de

    2002-01-01

    The LTS N method solves analytically the S N equations, applying the Laplace transform in the spatial variable. This methodology is used in determination of scalar flux for neutrons and photons, absorbed dose rate, buildup factors and power for a heterogeneous planar slab. This procedure leads to the solution of a transcendental equations for effective multiplication, critical thickness and the atomic density. In this work numerical results are reported, considering multigroup problem in heterogeneous slab. (author)

  15. Solution of the Boltzmann-Fokker-Planck transport equation using exponential nodal schemes; Solucion de la ecuacion de transporte de Boltzmann-Fokker-Planck usando esquemas nodales exponenciales

    Energy Technology Data Exchange (ETDEWEB)

    Ortega J, R.; Valle G, E. del [IPN-ESFM, 07738 Mexico D.F. (Mexico)]. e-mail: roj@correo.azc.uam.mx

    2003-07-01

    There are carried out charge and energy calculations deposited due to the interaction of electrons with a plate of a certain material, solving numerically the electron transport equation for the Boltzmann-Fokker-Planck approach of first order in plate geometry with a computer program denominated TEOD-NodExp (Transport of Electrons in Discreet Ordinates, Nodal Exponentials), using the proposed method by the Dr. J. E. Morel to carry out the discretization of the variable energy and several spatial discretization schemes, denominated exponentials nodal. It is used the Fokker-Planck equation since it represents an approach of the Boltzmann transport equation that is been worth whenever it is predominant the dispersion of small angles, that is to say, resulting dispersion in small dispersion angles and small losses of energy in the transport of charged particles. Such electrons could be those that they face with a braking plate in a device of thermonuclear fusion. In the present work its are considered electrons of 1 MeV that impact isotropically on an aluminum plate. They were considered three different thickness of plate that its were designated as problems 1, 2 and 3. In the calculations it was used the discrete ordinate method S{sub 4} with expansions of the dispersion cross sections until P{sub 3} order. They were considered 25 energy groups of uniform size between the minimum energy of 0.1 MeV and the maximum of 1.0 MeV; the one spatial intervals number it was considered variable and it was assigned the values of 10, 20 and 30. (Author)

  16. Study of a method to solve the one speed, three dimensional transport equation using the finite element method and the associated Legendre function

    International Nuclear Information System (INIS)

    Fernandes, A.

    1991-01-01

    A method to solve three dimensional neutron transport equation and it is based on the original work suggested by J.K. Fletcher (42, 43). The angular dependence of the flux is approximated by associated Legendre functions and the finite element method is applied to the space components is presented. When the angular flux, the scattering cross section and the neutrons source are expanded in associated Legendre functions, the first order neutron transport equation is reduced to a coupled set of second order diffusion like equations. These equations are solved in an iterative way by the finite element method to the moments. (author)

  17. Mixed-hybrid finite element method for the transport equation and diffusion approximation of transport problems; Resolution de l'equation du transport par une methode d'elements finis mixtes-hybrides et approximation par la diffusion de problemes de transport

    Energy Technology Data Exchange (ETDEWEB)

    Cartier, J

    2006-04-15

    This thesis focuses on mathematical analysis, numerical resolution and modelling of the transport equations. First of all, we deal with numerical approximation of the solution of the transport equations by using a mixed-hybrid scheme. We derive and study a mixed formulation of the transport equation, then we analyse the related variational problem and present the discretization and the main properties of the scheme. We particularly pay attention to the behavior of the scheme and we show its efficiency in the diffusion limit (when the mean free path is small in comparison with the characteristic length of the physical domain). We present academical benchmarks in order to compare our scheme with other methods in many physical configurations and validate our method on analytical test cases. Unstructured and very distorted meshes are used to validate our scheme. The second part of this thesis deals with two transport problems. The first one is devoted to the study of diffusion due to boundary conditions in a transport problem between two plane plates. The second one consists in modelling and simulating radiative transfer phenomenon in case of the industrial context of inertial confinement fusion. (author)

  18. Mixed-hybrid finite element method for the transport equation and diffusion approximation of transport problems; Resolution de l'equation du transport par une methode d'elements finis mixtes-hybrides et approximation par la diffusion de problemes de transport

    Energy Technology Data Exchange (ETDEWEB)

    Cartier, J

    2006-04-15

    This thesis focuses on mathematical analysis, numerical resolution and modelling of the transport equations. First of all, we deal with numerical approximation of the solution of the transport equations by using a mixed-hybrid scheme. We derive and study a mixed formulation of the transport equation, then we analyse the related variational problem and present the discretization and the main properties of the scheme. We particularly pay attention to the behavior of the scheme and we show its efficiency in the diffusion limit (when the mean free path is small in comparison with the characteristic length of the physical domain). We present academical benchmarks in order to compare our scheme with other methods in many physical configurations and validate our method on analytical test cases. Unstructured and very distorted meshes are used to validate our scheme. The second part of this thesis deals with two transport problems. The first one is devoted to the study of diffusion due to boundary conditions in a transport problem between two plane plates. The second one consists in modelling and simulating radiative transfer phenomenon in case of the industrial context of inertial confinement fusion. (author)

  19. Applicability of angular flux discontinuity factor preserving region-wise leakage for integro-differential transport equation

    International Nuclear Information System (INIS)

    Sakamoto, Tatsuya; Endo, Tomohiro; Yamamoto, Akio

    2014-01-01

    In the current core analysis, spatial homogenization is utilized to reduce the computational time. The discontinuity factor (DF) is one of the effective correction factors to reduce spatial homogenization error. The DF in diffusion equation is widely used; on the other hand the DF in transport equation has not been put to practical use although several efforts have been carried out. In this paper, the angular flux discontinuity factor (AFDF) as the DF for the integro-differential transport equation (e.g., the discrete-ordinate method, the method of characteristics) is theoretically described and its applicability is discussed. The AFDF is used to preserve the region-wise neutron leakage at each spatial mesh and defined as a ratio of heterogeneous and homogeneous angular fluxes at the homogenized region surface. In a homogeneous calculation with the AFDF, the angular flux is discontinuous at the region surface. In this paper the applicability of the AFDF to fuel pin cell homogenization is verified for one-dimensional slab geometry. As a result of this verification, it is confirmed that the AFDF has the capability to reduce the spatial homogenization error of fuel pin cell homogenization. (author)

  20. Heat transport in two-dimensional materials by directly solving the phonon Boltzmann equation under Callaway's dual relaxation model

    Science.gov (United States)

    Guo, Yangyu; Wang, Moran

    2017-10-01

    The single mode relaxation time approximation has been demonstrated to greatly underestimate the lattice thermal conductivity of two-dimensional materials due to the collective effect of phonon normal scattering. Callaway's dual relaxation model represents a good approximation to the otherwise ab initio solution of the phonon Boltzmann equation. In this work we develop a discrete-ordinate-method (DOM) scheme for the numerical solution of the phonon Boltzmann equation under Callaway's model. Heat transport in a graphene ribbon with different geometries is modeled by our scheme, which produces results quite consistent with the available molecular dynamics, Monte Carlo simulations, and experimental measurements. Callaway's lattice thermal conductivity model with empirical boundary scattering rates is examined and shown to overestimate or underestimate the direct DOM solution. The length convergence of the lattice thermal conductivity of a rectangular graphene ribbon is explored and found to depend appreciably on the ribbon width, with a semiquantitative correlation provided between the convergence length and the width. Finally, we predict the existence of a phonon Knudsen minimum in a graphene ribbon only at a low system temperature and isotope concentration so that the average normal scattering rate is two orders of magnitude stronger than the intrinsic resistive one. The present work will promote not only the methodology for the solution of the phonon Boltzmann equation but also the theoretical modeling and experimental detection of hydrodynamic phonon transport in two-dimensional materials.

  1. Atlas of photoneutron cross sections obtained with monoenergetic photons

    International Nuclear Information System (INIS)

    Dietrich, S.S.; Berman, B.L.

    1988-01-01

    Photoneutron cross-section and integrated cross-section data obtained with monoenergetic photons are presented in a uniform format. All of the measured partial photoneutron cross sections, the total photoneutron cross section, and the photoneutron yield cross section are plotted as functions of the incident photon energy, as are the integrated photoneutron cross sections and their first and second moments. The values of the integrated cross sections and the moments of the integrated total cross section up to the highest photon energy for which they were measured are tabulated, as are the parameters of Lorentz curves fitted to the total photoneutron cross-section data for medium and heavy nuclei (A>50). This compilation is current as of June 1987. copyright 1988 Academic Press, Inc

  2. Monoenergetic electron parameters in a spheroid bubble model

    Science.gov (United States)

    Sattarian, H.; Sh., Rahmatallahpur; Tohidi, T.

    2013-02-01

    A reliable analytical expression for the potential of plasma waves with phase velocities near the speed of light is derived. The presented spheroid cavity model is more consistent than the previous spherical and ellipsoidal models and it explains the mono-energetic electron trajectory more accurately, especially at the relativistic region. The maximum energy of electrons is calculated and it is shown that the maximum energy of the spheroid model is less than that of the spherical model. The electron energy spectrum is also calculated and it is found that the energy distribution ratio of electrons ΔE/E for the spheroid model under the conditions reported here is half that of the spherical model and it is in good agreement with the experimental value in the same conditions. As a result, the quasi-mono-energetic electron output beam interacting with the laser plasma can be more appropriately described with this model.

  3. Monoenergetic electron parameters in a spheroid bubble model

    International Nuclear Information System (INIS)

    Sattarian, H.; Rahmatallahpur, Sh.; Tohidi, T.

    2013-01-01

    A reliable analytical expression for the potential of plasma waves with phase velocities near the speed of light is derived. The presented spheroid cavity model is more consistent than the previous spherical and ellipsoidal models and it explains the mono-energetic electron trajectory more accurately, especially at the relativistic region. The maximum energy of electrons is calculated and it is shown that the maximum energy of the spheroid model is less than that of the spherical model. The electron energy spectrum is also calculated and it is found that the energy distribution ratio of electrons ΔE/E for the spheroid model under the conditions reported here is half that of the spherical model and it is in good agreement with the experimental value in the same conditions. As a result, the quasi-mono-energetic electron output beam interacting with the laser plasma can be more appropriately described with this model. (physics of gases, plasmas, and electric discharges)

  4. A generalized master equation approach to modelling anomalous transport in animal movement

    International Nuclear Information System (INIS)

    Giuggioli, Luca; Sevilla, Francisco J; Kenkre, V M

    2009-01-01

    We present some models of random walks with internal degrees of freedom that have the potential to find application in the context of animal movement and stochastic search. The formalism we use is based on the generalized master equation which is particularly convenient here because of its inherent coarse-graining procedure whereby a random walker position is averaged over the internal degrees of freedom. We show some instances in which non-local jump probabilities emerge from the coupling of the motion to the internal degrees of freedom, and how the tuning of one parameter can give rise to sub-, super- and normal diffusion at long times. Remarks on the relation between the generalized master equation, continuous time random walks and fractional diffusion equations are also presented.

  5. Magnetoresistance in organic semiconductors: Including pair correlations in the kinetic equations for hopping transport

    Science.gov (United States)

    Shumilin, A. V.; Kabanov, V. V.; Dediu, V. I.

    2018-03-01

    We derive kinetic equations for polaron hopping in organic materials that explicitly take into account the double occupation possibility and pair intersite correlations. The equations include simplified phenomenological spin dynamics and provide a self-consistent framework for the description of the bipolaron mechanism of the organic magnetoresistance. At low applied voltages, the equations can be reduced to those for an effective resistor network that generalizes the Miller-Abrahams network and includes the effect of spin relaxation on the system resistivity. Our theory discloses the close relationship between the organic magnetoresistance and the intersite correlations. Moreover, in the absence of correlations, as in an ordered system with zero Hubbard energy, the magnetoresistance vanishes.

  6. Assessing links between energy consumption, freight transport, and economic growth: evidence from dynamic simultaneous equation models.

    Science.gov (United States)

    Nasreen, Samia; Saidi, Samir; Ozturk, Ilhan

    2018-06-01

    We investigate this study to examine the relationship between economic growth, freight transport, and energy consumption for 63 developing countries over the period of 1990-2016. In order to make the panel data analysis more homogeneous, we apply the income level of countries to divide the global panel into three sub-panels, namely, lower-middle income countries (LMIC), upper-middle income countries (UMIC), and high-income countries (HIC). Using the generalized method of moments (GMM), the results prove evidence of bidirectional causal relationship between economic growth and freight transport for all selected panels and between economic growth and energy consumption for the high- and upper-middle income panels. For the lower-middle income panel, the causality is unidirectional running from energy consumption to economic growth. Also, the results indicate that the relationship between freight transport and energy use is bidirectional for the high-income countries and unidirectional from freight transport to energy consumption for the upper-middle and lower-middle income countries. Empirical evidence demonstrates the importance of energy for economic activity and rejects the neo-classical assumption that energy is neutral for growth. An important policy recommendation is that there is need of advancements in vehicle technology which can reduce energy intensity from transport sector and improve the energy efficiency in transport activity which in turn allows a greater positive role of transport in global economic activity.

  7. Dynamical equations and transport coefficients for the metals at high pulse electromagnetic fields

    International Nuclear Information System (INIS)

    Volkov, N B; Chingina, E A; Yalovets, A P

    2016-01-01

    We offer a metal model suitable for the description of fast electrophysical processes in conductors under influence of powerful electronic and laser radiation of femto- and picosecond duration, and also high-voltage electromagnetic pulses with picosecond front and duration less than 1 ns. The obtained dynamic equations for metal in approximation of one quasineutral liquid are in agreement with the equations received by other authors formerly. New wide-range expressions for the electronic conduction in strong electromagnetic fields are obtained and analyzed. (paper)

  8. The nature and role of advection in advection-diffusion equations used for modelling bed load transport

    Science.gov (United States)

    Ancey, Christophe; Bohorquez, Patricio; Heyman, Joris

    2016-04-01

    The advection-diffusion equation arises quite often in the context of sediment transport, e.g., for describing time and space variations in the particle activity (the solid volume of particles in motion per unit streambed area). Stochastic models can also be used to derive this equation, with the significant advantage that they provide information on the statistical properties of particle activity. Stochastic models are quite useful when sediment transport exhibits large fluctuations (typically at low transport rates), making the measurement of mean values difficult. We develop an approach based on birth-death Markov processes, which involves monitoring the evolution of the number of particles moving within an array of cells of finite length. While the topic has been explored in detail for diffusion-reaction systems, the treatment of advection has received little attention. We show that particle advection produces nonlocal effects, which are more or less significant depending on the cell size and particle velocity. Albeit nonlocal, these effects look like (local) diffusion and add to the intrinsic particle diffusion (dispersal due to velocity fluctuations), with the important consequence that local measurements depend on both the intrinsic properties of particle displacement and the dimensions of the measurement system.

  9. A mass conservative numerical solution of vertical water flow and mass transport equations in unsaturated porous media

    International Nuclear Information System (INIS)

    Lim, S.C.; Lee, K.J.

    1993-01-01

    The Galerkin finite element method is used to solve the problem of one-dimensional, vertical flow of water and mass transport of conservative-nonconservative solutes in unsaturated porous media. Numerical approximations based on different forms of the governing equation, although they are equivalent in continuous forms, can result in remarkably different solutions in an unsaturated flow problem. Solutions given by a simple Galerkin method based on the h-based Richards equation yield a large mass balance error and an underestimation of the infiltration depth. With the employment of the ROMV (restoration of main variable) concept in the discretization step, the mass conservative numerical solution algorithm for water flow has been derived. The resulting computational schemes for water flow and mass transport are applied to sandy soil. The ROMV method shows good mass conservation in water flow analysis, whereas it seems to have a minor effect on mass transport. However, it may relax the time-step size restriction and so ensure an improved calculation output. (author)

  10. A spatially adaptive grid-refinement approach for the finite element solution of the even-parity Boltzmann transport equation

    International Nuclear Information System (INIS)

    Mirza, Anwar M.; Iqbal, Shaukat; Rahman, Faizur

    2007-01-01

    A spatially adaptive grid-refinement approach has been investigated to solve the even-parity Boltzmann transport equation. A residual based a posteriori error estimation scheme has been utilized for checking the approximate solutions for various finite element grids. The local particle balance has been considered as an error assessment criterion. To implement the adaptive approach, a computer program ADAFENT (adaptive finite elements for neutron transport) has been developed to solve the second order even-parity Boltzmann transport equation using K + variational principle for slab geometry. The program has a core K + module which employs Lagrange polynomials as spatial basis functions for the finite element formulation and Legendre polynomials for the directional dependence of the solution. The core module is called in by the adaptive grid generator to determine local gradients and residuals to explore the possibility of grid refinements in appropriate regions of the problem. The a posteriori error estimation scheme has been implemented in the outer grid refining iteration module. Numerical experiments indicate that local errors are large in regions where the flux gradients are large. A comparison of the spatially adaptive grid-refinement approach with that of uniform meshing approach for various benchmark cases confirms its superiority in greatly enhancing the accuracy of the solution without increasing the number of unknown coefficients. A reduction in the local errors of the order of 10 2 has been achieved using the new approach in some cases

  11. A spatially adaptive grid-refinement approach for the finite element solution of the even-parity Boltzmann transport equation

    Energy Technology Data Exchange (ETDEWEB)

    Mirza, Anwar M. [Department of Computer Science, National University of Computer and Emerging Sciences, NUCES-FAST, A.K. Brohi Road, H-11, Islamabad (Pakistan)], E-mail: anwar.m.mirza@gmail.com; Iqbal, Shaukat [Faculty of Computer Science and Engineering, Ghulam Ishaq Khan (GIK) Institute of Engineering Science and Technology, Topi-23460, Swabi (Pakistan)], E-mail: shaukat@giki.edu.pk; Rahman, Faizur [Department of Physics, Allama Iqbal Open University, H-8 Islamabad (Pakistan)

    2007-07-15

    A spatially adaptive grid-refinement approach has been investigated to solve the even-parity Boltzmann transport equation. A residual based a posteriori error estimation scheme has been utilized for checking the approximate solutions for various finite element grids. The local particle balance has been considered as an error assessment criterion. To implement the adaptive approach, a computer program ADAFENT (adaptive finite elements for neutron transport) has been developed to solve the second order even-parity Boltzmann transport equation using K{sup +} variational principle for slab geometry. The program has a core K{sup +} module which employs Lagrange polynomials as spatial basis functions for the finite element formulation and Legendre polynomials for the directional dependence of the solution. The core module is called in by the adaptive grid generator to determine local gradients and residuals to explore the possibility of grid refinements in appropriate regions of the problem. The a posteriori error estimation scheme has been implemented in the outer grid refining iteration module. Numerical experiments indicate that local errors are large in regions where the flux gradients are large. A comparison of the spatially adaptive grid-refinement approach with that of uniform meshing approach for various benchmark cases confirms its superiority in greatly enhancing the accuracy of the solution without increasing the number of unknown coefficients. A reduction in the local errors of the order of 10{sup 2} has been achieved using the new approach in some cases.

  12. almaBTE : A solver of the space-time dependent Boltzmann transport equation for phonons in structured materials

    Science.gov (United States)

    Carrete, Jesús; Vermeersch, Bjorn; Katre, Ankita; van Roekeghem, Ambroise; Wang, Tao; Madsen, Georg K. H.; Mingo, Natalio

    2017-11-01

    almaBTE is a software package that solves the space- and time-dependent Boltzmann transport equation for phonons, using only ab-initio calculated quantities as inputs. The program can predictively tackle phonon transport in bulk crystals and alloys, thin films, superlattices, and multiscale structures with size features in the nm- μm range. Among many other quantities, the program can output thermal conductances and effective thermal conductivities, space-resolved average temperature profiles, and heat-current distributions resolved in frequency and space. Its first-principles character makes almaBTE especially well suited to investigate novel materials and structures. This article gives an overview of the program structure and presents illustrative examples for some of its uses. PROGRAM SUMMARY Program Title:almaBTE Program Files doi:http://dx.doi.org/10.17632/8tfzwgtp73.1 Licensing provisions: Apache License, version 2.0 Programming language: C++ External routines/libraries: BOOST, MPI, Eigen, HDF5, spglib Nature of problem: Calculation of temperature profiles, thermal flux distributions and effective thermal conductivities in structured systems where heat is carried by phonons Solution method: Solution of linearized phonon Boltzmann transport equation, Variance-reduced Monte Carlo

  13. Label-free nanoscale characterization of red blood cell structure and dynamics using single-shot transport of intensity equation

    Science.gov (United States)

    Poola, Praveen Kumar; John, Renu

    2017-10-01

    We report the results of characterization of red blood cell (RBC) structure and its dynamics with nanometric sensitivity using transport of intensity equation microscopy (TIEM). Conventional transport of intensity technique requires three intensity images and hence is not suitable for studying real-time dynamics of live biological samples. However, assuming the sample to be homogeneous, phase retrieval using transport of intensity equation has been demonstrated with single defocused measurement with x-rays. We adopt this technique for quantitative phase light microscopy of homogenous cells like RBCs. The main merits of this technique are its simplicity, cost-effectiveness, and ease of implementation on a conventional microscope. The phase information can be easily merged with regular bright-field and fluorescence images to provide multidimensional (three-dimensional spatial and temporal) information without any extra complexity in the setup. The phase measurement from the TIEM has been characterized using polymeric microbeads and the noise stability of the system has been analyzed. We explore the structure and real-time dynamics of RBCs and the subdomain membrane fluctuations using this technique.

  14. Flow and transport simulation of Madeira River using three depth-averaged two-equation turbulence closure models

    Directory of Open Access Journals (Sweden)

    Li-ren Yu

    2012-03-01

    Full Text Available This paper describes a numerical simulation in the Amazon water system, aiming to develop a quasi-three-dimensional numerical tool for refined modeling of turbulent flow and passive transport of mass in natural waters. Three depth-averaged two-equation turbulence closure models, k˜−ε˜,k˜−w˜, and k˜−ω˜ , were used to close the non-simplified quasi-three dimensional hydrodynamic fundamental governing equations. The discretized equations were solved with the advanced multi-grid iterative method using non-orthogonal body-fitted coarse and fine grids with collocated variable arrangement. Except for steady flow computation, the processes of contaminant inpouring and plume development at the beginning of discharge, caused by a side-discharge of a tributary, have also been numerically investigated. The three depth-averaged two-equation closure models are all suitable for modeling strong mixing turbulence. The newly established turbulence models such as the k˜−ω˜ model, with a higher order of magnitude of the turbulence parameter, provide a possibility for improving computational precision.

  15. Calculations of reactivity based in the solution of the Neutron transport equation in X Y geometry and Lineal perturbation theory

    International Nuclear Information System (INIS)

    Valle G, E. del; Mugica R, C.A.

    2005-01-01

    In our country, in last congresses, Gomez et al carried out reactivity calculations based on the solution of the diffusion equation for an energy group using nodal methods in one dimension and the TPL approach (Lineal Perturbation Theory). Later on, Mugica extended the application to the case of multigroup so much so much in one as in two dimensions (X Y geometry) with excellent results. Presently work is carried out similar calculations but this time based on the solution of the neutron transport equation in X Y geometry using nodal methods and again the TPL approximation. The idea is to provide a calculation method that allows to obtain in quick form the reactivity solving the direct problem as well as the enclosed problem of the not perturbed problem. A test problem for the one that results are provided for the effective multiplication factor is described and its are offered some conclusions. (Author)

  16. Application of Trotter approximation for solving time dependent neutron transport equation; Primena Trotterove aproksimacije za resavanje vremenski zavisne transportne jednacine neutrona

    Energy Technology Data Exchange (ETDEWEB)

    Stancic, V [Institut za nuklearne nauke Boris Kidric, Vinca, Beograd (Yugoslavia)

    1987-07-01

    A method is proposed to solve multigroup time dependent neutron transport equation with arbitrary scattering anisotropy. The recurrence relation thus obtained is simple, numerically stable and especially suitable for treatment of complicated geometries. (author)

  17. Existence of weak solutions for compressible Navier-Stokes equations with entropy transport

    Czech Academy of Sciences Publication Activity Database

    Maltese, D.; Michálek, Martin; Mucha, P.; Novotný, A.; Pokorný, M.; Zatorska, E.

    2016-01-01

    Roč. 261, č. 8 (2016), s. 4448-4485 ISSN 0022-0396 R&D Projects: GA ČR GA13-00522S Institutional support: RVO:67985840 Keywords : Navier-Stokes equations Subject RIV: BA - General Mathematics Impact factor: 1.988, year: 2016 http://www.sciencedirect.com/science/article/pii/S0022039616301656

  18. Two split cell numerical methods for solving 2-D non-equilibrium radiation transport equations

    International Nuclear Information System (INIS)

    Feng Tinggui

    2004-11-01

    Two numerically positive methods, the step characteristic integral method and subcell balance method, for solving radiative transfer equations on quadrilateral grids are presented. Numerical examples shows that the schemes presented are feasible on non-rectangle grid computation, and that the computing results by the schemes presented are comparative to that by the discrete ordinate diamond scheme on rectangle grid. (author)

  19. Modelling of thermal transport using Fokker-Planck equations in laser produced plasma

    International Nuclear Information System (INIS)

    Nakarmi, J.J.; Jha, L.N.

    1996-12-01

    The kinetic equation with Fokker-Planck collision term has been presented to obtain the distribution function in the corona of inertial confinement fusion, in the presence of the self generated magnetic field. The resulting distribution has non-local form with the convolution in Maxwellian. An expression for thermal flux with self generated magnetic field is obtained. (author). 22 refs

  20. The Factors Influencing Satisfaction with Public City Transport: A Structural Equation Modelling Approach

    OpenAIRE

    Pawlasova Pavlina

    2015-01-01

    Satisfaction is one of the key factors which influences customer loyalty. We assume that the satisfied customer will be willing to use the ssame service provider again. The overall passengers´ satisfaction with public city transport may be affected by the overall service quality. Frequency, punctuality, cleanliness in the vehicle, proximity, speed, fare, accessibility and safety of transport, information and other factors can influence passengers´ satisfaction. The aim of this pap...

  1. Development and validation of bubble breakup and coalescence constitutive models for the one-group interfacial area transport equation

    International Nuclear Information System (INIS)

    Pellacani, Filippo

    2012-01-01

    A local mechanistic model for bubble coalescence and breakup for the one-group interfacial area transport equation has been developed, in agreement and within the limits of the current understanding, based on an exhaustive survey of the theory and of the state of the art models for bubble dynamics simulation. The new model has been tested using the commercial 3D CFD code ANSYS CFX. Upward adiabatic turbulent air-water bubbly flow has been simulated and the results have been compared with the data obtained in the experimental facility PUMA. The range of the experimental data available spans between 0.5 to 2 m/s liquid velocity and 5 to 15 % volume fraction. For the implementation of the models, both the monodispersed and the interfacial area transport equation approaches have been used. The first one to perform a detailed analysis of the forces and models to reproduce the dynamic of the dispersed phase adequately and to be used in the next phases of the work. Also two different bubble induced turbulence models have been tested to consider the effect of the presence of the gas phase on the turbulence of the liquid phase. The interfacial area transport equation has been successfully implemented into the CFD code and the state of the art breakup and coalescence models have been used for simulation. The limitations of the actual theory have been shown and a new bubble interactions model has been developed. The simulations showed that a considerable improvement is achieved if compared to the state of the art closure models. Limits in the implementation derive from the actual understanding and formulation of the bubbly dynamics. A strong dependency on the interfacial non-drag force models and coefficients have been shown. More experimental and theory work needs to be done in this field to increase the prediction capability of the simulation tools regarding the distribution of the phases along the pipe radius.

  2. SHADOK-3-6, Transport Equation with Anisotropic Diffusion in P1 Approximation for Spherical and Cylindrical Geometry

    International Nuclear Information System (INIS)

    Ligou, J.; Thomi, P.A.

    1973-01-01

    1 - Nature of physical problem solved: Integral transport equation, anisotropy of diffusion in P1 approximation. SHADOK3 - cylindrical geometry; direct solution of the linear system. SHADOK4 - cylindrical geometry; Thermalization iteration; solution of the linear system with inverse matrix calculation. SHADOK5 - like SHADOK3 for spherical geometry. SHADOK6 - like SHADOK4 for spherical geometry. 2 - Method of solution: Analysis in terms of annuli for each of which polynomial approximation is applied. Dynamic allocation (for formulas see report TM(10)). 3 - Restrictions on the complexity of the problem: Relative accuracy of the Bickley functions about 1.0E-13

  3. Analysis of an a posteriori error estimator for the transport equation with SN and discontinuous Galerkin discretizations

    International Nuclear Information System (INIS)

    Fournier, D.; Le Tellier, R.; Suteau, C.

    2011-01-01

    We present an error estimator for the S N neutron transport equation discretized with an arbitrary high-order discontinuous Galerkin method. As a starting point, the estimator is obtained for conforming Cartesian meshes with a uniform polynomial order for the trial space then adapted to deal with non-conforming meshes and a variable polynomial order. Some numerical tests illustrate the properties of the estimator and its limitations. Finally, a simple shielding benchmark is analyzed in order to show the relevance of the estimator in an adaptive process.

  4. Neoclassical transport simulations for stellarators

    International Nuclear Information System (INIS)

    Turkin, Y.; Beidler, C. D.; Maassberg, H.; Murakami, S.; Wakasa, A.; Tribaldos, V.

    2011-01-01

    The benchmarking of the thermal neoclassical transport coefficients is described using examples of the Large Helical Device (LHD) and TJ-II stellarators. The thermal coefficients are evaluated by energy convolution of the monoenergetic coefficients obtained by direct interpolation or neural network techniques from the databases precalculated by different codes. The temperature profiles are calculated by a predictive transport code from the energy balance equations with the ambipolar radial electric field estimated from a diffusion equation to guarantee a unique and smooth solution, although several solutions of the ambipolarity condition may exist when root-finding is invoked; the density profiles are fixed. The thermal transport coefficients as well as the ambipolar radial electric field are compared and very reasonable agreement is found for both configurations. Together with an additional W7-X case, these configurations represent very different degrees of neoclassical confinement at low collisionalities. The impact of the neoclassical optimization on the energy confinement time is evaluated and the confinement times for different devices predicted by transport modeling are compared with the standard scaling for stellarators. Finally, all configurations are scaled to the same volume for a direct comparison of the volume-averaged pressure and the neoclassical degree of optimization.

  5. Methodology for solving the equation of transport ordered discrete TORT code in the reactor IPEN/MB-01; Metodologia para resolver la ecuacion del transporte con el codigo de Ordenadas Discretas TORT en el reactor IPEN/MB-01

    Energy Technology Data Exchange (ETDEWEB)

    Bernal, A.; Abarca, A.; Barrachina, T.; Miro, R.; Verdu, G.

    2013-07-01

    The resolution of the neutron transport equation in steady state in pool-type nuclear reactors, is normally achieved through 2 different numerical methods: Monte Carlo (stochastic) and discrete ordinates (deterministic). The discrete ordinates method solves the neutron transport equation for a set of specific addresses, obtaining a set of equations and solutions for each direction, where the solution for each direction is the angular flux. With the aim of treating energy dependence, used energy multigroup approximation, thus obtaining a set of equations that depends on the number of energy groups considered.

  6. Laser Acceleration of Quasi-Monoenergetic Protons via Radiation Pressure Driven Thin Foil

    International Nuclear Information System (INIS)

    Liu, Chuan S.; Shao Xi; Liu, T. C.; Dudnikova, Galina; Sagdeev, Roald Z.; Eliasson, Bengt

    2011-01-01

    We present a theoretical and simulation study of laser acceleration of quasi-monoenergetic protons in a thin foil irradiated by high intensity laser light. The underlying physics of radiation pressure acceleration (RPA) is discussed, including the importance of optimal thickness and circularly polarized light for efficient acceleration of ions to quasi-monoenergetic beams. Preliminary two-dimensional simulation studies show that certain parameter regimes allow for stabilization of the Rayleigh-Taylor instability and possibility of acceleration of monoenergetic ions to an excess of 200 MeV, making them suitable for important applications such as medical cancer therapy and fast ignition.

  7. Calculation of X-ray emission produced by a quasi-monoenergetic electron distribution

    International Nuclear Information System (INIS)

    Fanaei, M.; Sadighi-Bonabi, R.

    2010-01-01

    Complete text of publication follows. By using an intense ultrafast laser interaction with plasma, generation of accelerated relativistic electrons with quasi monoenergetic spectrum has been possible. Analytic expressions for spectra and emission efficiencies of x-rays bremsstrahlung and characteristic line emission produced by a quasi-monoenergetic electron distribution from several targets are investigated. In this work, a Gaussian profile is assumed for the quasi-monoenergetic electron spectrum. The produced x-ray radiations are compared with the previous achieved results for a Maxwellian electron profile. These results and achievements are discussed in detail. Also, the outcomes can be evaluated with the experimental and simulated results.

  8. Approximate solutions for the two-dimensional integral transport equation. Solution of complex two-dimensional transport problems

    International Nuclear Information System (INIS)

    Sanchez, Richard.

    1980-11-01

    This work is divided into two parts: the first part deals with the solution of complex two-dimensional transport problems, the second one (note CEA-N-2166) treats the critically mixed methods of resolution. A set of approximate solutions for the isotropic two-dimensional neutron transport problem has been developed using the interface current formalism. The method has been applied to regular lattices of rectangular cells containing a fuel pin, cladding, and water, or homogenized structural material. The cells are divided into zones that are homogeneous. A zone-wise flux expansion is used to formulate a direct collision probability problem within a cell. The coupling of the cells is effected by making extra assumptions on the currents entering and leaving the interfaces. Two codes have been written: CALLIOPE uses a cylindrical cell model and one or three terms for the flux expansion, and NAUSICAA uses a two-dimensional flux representation and does a truly two-dimensional calculation inside each cell. In both codes, one or three terms can be used to make a space-independent expansion of the angular fluxes entering and leaving each side of the cell. The accuracies and computing times achieved with the different approximations are illustrated by numerical studies on two benchmark problems and by calculations performed in the APOLLO multigroup code [fr

  9. Nystro¨m method applied to integral formulation of the neutron transport equation in X-Y geometry

    Energy Technology Data Exchange (ETDEWEB)

    Azevedo, Fabio S.; Sauter, Esequia; Konzen, Pedro H.A.; Barichello, Liliane B., E-mail: fabio.azevedo@ufrgs.br, E-mail: esequia.sauter@ufrgs.br, E-mail: pedro.konzen@ufrgs.br, E-mail: lbaric@mat.ufrgs.br [Universidade Federal do Rio Grande do Sul (UFRGS), Porto Alegre, RS (Brazil). Departamento de Matem´atica Pura e Aplicada

    2017-07-01

    Neutron transport problems in X-Y geometry have been solved with several techniques in last decades but it is still a challenge to produce a good balance between computational efficiency and accuracy. In this work, we address this problem by efficiently applying the Nystr¨om method to the integral formulation of the transport equation. Analytical techniques, modern numerical packages and optimized implementation were applied to reduce the computational time. This method presented results free of ray effects leading to high accurate numerical results for two-dimensional scalar flux. Our implementation simulates homogeneous problems with vacuum and reflective boundary conditions. Results were validated with up to seven significant digits and compared with those available in the literature. (author)

  10. A method for solving the spherical harmonics equations applied for space-energy transport of fast and resonance neutrons

    International Nuclear Information System (INIS)

    Matausek, M.

    1972-01-01

    A new proposed method for solving the space-energy dependent spherical harmonics equations represents a methodological contribution to neutron transport theory. The proposed method was applied for solving the problem of spec-energy transport of fast and resonance neutrons in multi-zone, cylindrical y symmetric infinite reactor cell and is related to previously developed procedure for treating the thermal energy region. The advantages of this method are as follows: a unique algorithm was obtained for detailed determination of spatial and energy distribution of neutrons (from thermal to fast) in the reactor cell; these detailed distributions enable more precise calculations of criticality conditions, obtaining adequate multigroup data and better interpretation of experimental data; computing time is rather short

  11. Neutron wave reflexions in interface media with transport equation P1 approximation

    International Nuclear Information System (INIS)

    Oliveira Vellozo, S. de.

    1977-01-01

    The propagation of neutron waves in non multiplying media is investigated employing the Telegrapher's equation obtained from the P 1 approximation of the time, space and energy dependent Boltzmann equation. Solution of the problem of propagation of sinusoidally modulated source incident on one face of the medium is obtained by analysing the Fourier component of a pulsed source introduced, for the corresponding frequency. The amplitude and the phase of the flux are computed as a function of frequency in media consisting of one, two and three regions in order to study the effects of reflection at the interfaces. The results are compared with those from the Diffusion approximation obtained by neglecting the term involving the second order time derivative. (author)

  12. Multigroup neutron transport equation in the diffusion and P{sub 1} approximation

    Energy Technology Data Exchange (ETDEWEB)

    Obradovic, D [Boris Kidric Institute of nuclear sciences Vinca, Belgrade (Yugoslavia)

    1970-07-01

    Investigations of the properties of the multigroup transport operator, width and without delayed neutrons in the diffusion and P{sub 1} approximation, is performed using Keldis's theory of operator families as well as a technique . recently used for investigations into the properties of the general linearized Boltzmann operator. It is shown that in the case without delayed neutrons, multigroup transport operator in the diffusion and P{sub 1} approximation possesses a complete set of generalized eigenvectors. A formal solution to the initial value problem is also given. (author)

  13. Equation of state and transport properties of uranium and plutonium carbides in the liquid region

    International Nuclear Information System (INIS)

    Sheth, A.; Leibowitz, L.

    1975-09-01

    By the use of available low-temperature data for various thermophysical and transport properties for uranium and plutonium carbides, values above the melting point were estimated. Sets of recommended values have been prepared for the compounds UC, PuC, and (U,Pu)C. The properties that have been evaluated are density, heat capacity, enthalpy, vapor pressure, thermal conductivity, viscosity, and emissivity

  14. Unified solution of the Boltzmann equation for electron and ion velocity distribution functions and transport coefficients in weakly ionized plasmas

    Science.gov (United States)

    Konovalov, Dmitry A.; Cocks, Daniel G.; White, Ronald D.

    2017-10-01

    The velocity distribution function and transport coefficients for charged particles in weakly ionized plasmas are calculated via a multi-term solution of Boltzmann's equation and benchmarked using a Monte-Carlo simulation. A unified framework for the solution of the original full Boltzmann's equation is presented which is valid for ions and electrons, avoiding any recourse to approximate forms of the collision operator in various limiting mass ratio cases. This direct method using Lebedev quadratures over the velocity and scattering angles avoids the need to represent the ion mass dependence in the collision operator through an expansion in terms of the charged particle to neutral mass ratio. For the two-temperature Burnett function method considered in this study, this amounts to avoiding the need for the complex Talmi-transformation methods and associated mass-ratio expansions. More generally, we highlight the deficiencies in the two-temperature Burnett function method for heavy ions at high electric fields to calculate the ion velocity distribution function, even though the transport coefficients have converged. Contribution to the Topical Issue "Physics of Ionized Gases (SPIG 2016)", edited by Goran Poparic, Bratislav Obradovic, Dragana Maric and Aleksandar Milosavljevic.

  15. Numerical solution of the equation of neutrons transport on plane geometry by analytical schemes using acceleration by synthetic diffusion

    International Nuclear Information System (INIS)

    Alonso-Vargas, G.

    1991-01-01

    A computer program has been developed which uses a technique of synthetic acceleration by diffusion by analytical schemes. Both in the diffusion equation as in that of transport, analytical schemes were used which allowed a substantial time saving in the number of iterations required by source iteration method to obtain the K e ff. The program developed ASD (Synthetic Diffusion Acceleration) by diffusion was written in FORTRAN and can be executed on a personal computer with a hard disc and mathematical O-processor. The program is unlimited as to the number of regions and energy groups. The results obtained by the ASD program for K e ff is nearly completely concordant with those of obtained utilizing the ANISN-PC code for different analytical type problems in this work. The ASD program allowed obtention of an approximate solution of the neutron transport equation with a relatively low number of internal reiterations with good precision. One of its applications would be in the direct determinations of axial distribution neutronic flow in a fuel assembly as well as in the obtention of the effective multiplication factor. (Author)

  16. Sediment transport modeling in deposited bed sewers: unified form of May's equations using the particle swarm optimization algorithm.

    Science.gov (United States)

    Safari, Mir Jafar Sadegh; Shirzad, Akbar; Mohammadi, Mirali

    2017-08-01

    May proposed two dimensionless parameters of transport (η) and mobility (F s ) for self-cleansing design of sewers with deposited bed condition. The relationships between those two parameters were introduced in conditional form for specific ranges of F s , which makes it difficult to use as a practical tool for sewer design. In this study, using the same experimental data used by May and employing the particle swarm optimization algorithm, a unified equation is recommended based on η and F s . The developed model is compared with original May relationships as well as corresponding models available in the literature. A large amount of data taken from the literature is used for the models' evaluation. The results demonstrate that the developed model in this study is superior to May and other existing models in the literature. Due to the fact that in May's dimensionless parameters more effective variables in the sediment transport process in sewers with deposited bed condition are considered, it is concluded that the revised May equation proposed in this study is a reliable model for sewer design.

  17. Analysis of a HP-refinement method for solving the neutron transport equation using two error estimators

    International Nuclear Information System (INIS)

    Fournier, D.; Le Tellier, R.; Suteau, C.; Herbin, R.

    2011-01-01

    The solution of the time-independent neutron transport equation in a deterministic way invariably consists in the successive discretization of the three variables: energy, angle and space. In the SNATCH solver used in this study, the energy and the angle are respectively discretized with a multigroup approach and the discrete ordinate method. A set of spatial coupled transport equations is obtained and solved using the Discontinuous Galerkin Finite Element Method (DGFEM). Within this method, the spatial domain is decomposed into elements and the solution is approximated by a hierarchical polynomial basis in each one. This approach is time and memory consuming when the mesh becomes fine or the basis order high. To improve the computational time and the memory footprint, adaptive algorithms are proposed. These algorithms are based on an error estimation in each cell. If the error is important in a given region, the mesh has to be refined (h−refinement) or the polynomial basis order increased (p−refinement). This paper is related to the choice between the two types of refinement. Two ways to estimate the error are compared on different benchmarks. Analyzing the differences, a hp−refinement method is proposed and tested. (author)

  18. Exact harmonic solutions to Guyer-Krumhansl-type equation and application to heat transport in thin films

    Science.gov (United States)

    Zhukovsky, K.; Oskolkov, D.

    2018-03-01

    A system of hyperbolic-type inhomogeneous differential equations (DE) is considered for non-Fourier heat transfer in thin films. Exact harmonic solutions to Guyer-Krumhansl-type heat equation and to the system of inhomogeneous DE are obtained in Cauchy- and Dirichlet-type conditions. The contribution of the ballistic-type heat transport, of the Cattaneo heat waves and of the Fourier heat diffusion is discussed and compared with each other in various conditions. The application of the study to the ballistic heat transport in thin films is performed. Rapid evolution of the ballistic quasi-temperature component in low-dimensional systems is elucidated and compared with slow evolution of its diffusive counterpart. The effect of the ballistic quasi-temperature component on the evolution of the complete quasi-temperature is explored. In this context, the influence of the Knudsen number and of Cauchy- and Dirichlet-type conditions on the evolution of the temperature distribution is explored. The comparative analysis of the obtained solutions is performed.

  19. Matrix-oriented implementation for the numerical solution of the partial differential equations governing flows and transport in porous media

    KAUST Repository

    Sun, Shuyu

    2012-09-01

    In this paper we introduce a new technique for the numerical solution of the various partial differential equations governing flow and transport phenomena in porous media. This method is proposed to be used in high level programming languages like MATLAB, Python, etc., which show to be more efficient for certain mathematical operations than for others. The proposed technique utilizes those operations in which these programming languages are efficient the most and keeps away as much as possible from those inefficient, time-consuming operations. In particular, this technique is based on the minimization of using multiple indices looping operations by reshaping the unknown variables into one-dimensional column vectors and performing the numerical operations using shifting matrices. The cell-centered information as well as the face-centered information are shifted to the adjacent face-center and cell-center, respectively. This enables the difference equations to be done for all the cells at once using matrix operations rather than within loops. Furthermore, for results post-processing, the face-center information can further be mapped to the physical grid nodes for contour plotting and stream lines constructions. In this work we apply this technique to flow and transport phenomena in porous media. © 2012 Elsevier Ltd.

  20. Numerical Calculation of Transport Based on the Drift Kinetic Equation for plasmas in General Toroidal Magnetic Geometry

    International Nuclear Information System (INIS)

    Reynolds, J. M.; Lopez-Bruna, D.

    2009-01-01

    This report is the first of a series dedicated to the numerical calculation of the evolution of fusion plasmas in general toroidal geometry, including TJ-II plasmas. A kinetic treatment has been chosen: the evolution equation of the distribution function of one or several plasma species is solved in guiding center coordinates. The distribution function is written as a Maxwellian one modulated by polynomial series in the kinetic coordinates with no other approximations than those of the guiding center itself and the computation capabilities. The code allows also for the inclusion of the three-dimensional electrostatic potential in a self-consistent manner, but the initial objective has been set to solving only the neoclassical transport. A high order conservative method (Spectral Difference Method) has been chosen in order to discretized the equation for its numerical solution. In this first report, in addition to justifying the work, the evolution equation and its approximations are described, as well as the baseline of the numerical procedures. (Author) 28 refs

  1. An upscaled two-equation model of transport in porous media through unsteady-state closure of volume averaged formulations

    Science.gov (United States)

    Chaynikov, S.; Porta, G.; Riva, M.; Guadagnini, A.

    2012-04-01

    We focus on a theoretical analysis of nonreactive solute transport in porous media through the volume averaging technique. Darcy-scale transport models based on continuum formulations typically include large scale dispersive processes which are embedded in a pore-scale advection diffusion equation through a Fickian analogy. This formulation has been extensively questioned in the literature due to its inability to depict observed solute breakthrough curves in diverse settings, ranging from the laboratory to the field scales. The heterogeneity of the pore-scale velocity field is one of the key sources of uncertainties giving rise to anomalous (non-Fickian) dispersion in macro-scale porous systems. Some of the models which are employed to interpret observed non-Fickian solute behavior make use of a continuum formulation of the porous system which assumes a two-region description and includes a bimodal velocity distribution. A first class of these models comprises the so-called ''mobile-immobile'' conceptualization, where convective and dispersive transport mechanisms are considered to dominate within a high velocity region (mobile zone), while convective effects are neglected in a low velocity region (immobile zone). The mass exchange between these two regions is assumed to be controlled by a diffusive process and is macroscopically described by a first-order kinetic. An extension of these ideas is the two equation ''mobile-mobile'' model, where both transport mechanisms are taken into account in each region and a first-order mass exchange between regions is employed. Here, we provide an analytical derivation of two region "mobile-mobile" meso-scale models through a rigorous upscaling of the pore-scale advection diffusion equation. Among the available upscaling methodologies, we employ the Volume Averaging technique. In this approach, the heterogeneous porous medium is supposed to be pseudo-periodic, and can be represented through a (spatially) periodic unit cell

  2. Production of a monoenergetic electron bunch in a self-injected laser-wakefield accelerator

    International Nuclear Information System (INIS)

    Chang, C.-L.; Hsieh, C.-T.; Ho, Y.-C.; Chen, Y.-S.; Lin, J.-Y.; Wang, J.; Chen, S.-Y.

    2007-01-01

    Production of a monoenergetic electron bunch in a self-injected laser-wakefield accelerator is investigated with a tomographic method which resolves the electron injection and acceleration processes. It is found that all the electrons in the monoenergetic electron bunch are injected at the same location in the plasma column and then accelerated with an acceleration gradient exceeding 2 GeV/cm. The injection position shifts with the position of pump-pulse focus, and no significant deceleration is observed for the monoenergetic electron bunch after it reaches the maximum energy. The results are consistent with the model of transverse wave breaking and beam loading for the injection of monoenergetic electrons. The tomographic method adds a crucial dimension to the whole array of existing diagnostics for laser beams, plasma waves, and electron beams. With this method the details of the underlying physical processes in laser-plasma interactions can be resolved and compared directly to particle-in-cell simulations

  3. Transport equation for the time scale of a turbulent scalar field

    International Nuclear Information System (INIS)

    Kurbatskij, A.F.

    1999-01-01

    The two-parametric turbulence models cause serious difficulties by modeling the near-wall flows due to absence of the natural boundary condition on the wall for dissipation of the ε turbulence energy and the ε θ scalar field destruction. This difficulty may be overcome, if instead of the ε and ε θ , as the second parameter of the model, to apply the time scales of the turbulent dynamic and scalar fields. The equation of the scalar field is derived and numerical coefficients included therein, are determined from the simplest problems on the turbulent heat transfer [ru

  4. Effect of trapping on transport coherence III: Dissipation in the stochastic Liouville equation approach

    Energy Technology Data Exchange (ETDEWEB)

    Barvik, I [International Centre for Theoretical Physics, Trieste (Italy); Polasek, M [Charles Univ., Prague (Czech Republic). Inst. of Physics; Herman, P [Pedagogical Univ., Hradec Kralove (Czech Republic)

    1995-08-01

    We used the formal stochastic Liouville equations within Haken-Strobl-Reineker parametrization for the description of the influence of the bath on the memory functions entering the GME for a dimer and a linear trimer with a trap (here modeled as a sink). The often used inclusion of the noncoherent regime in the MF by an exponentially damped prefactor (after Kenkre`s prescription) does not hold for finite systems. The analytical form of the MF is changed more pronouncely and the influence of the sink in the center of the trimer runs parallel with the influence of the bath in destroying the coherence. (author). 60 refs.

  5. Analytical synthetic methods of solution of neutron transport equation with diffusion theory approaches energy multigroup

    International Nuclear Information System (INIS)

    Moraes, Pedro Gabriel B.; Leite, Michel C.A.; Barros, Ricardo C.

    2013-01-01

    In this work we developed a software to model and generate results in tables and graphs of one-dimensional neutron transport problems in multi-group formulation of energy. The numerical method we use to solve the problem of neutron diffusion is analytic, thus eliminating the truncation errors that appear in classical numerical methods, e.g., the method of finite differences. This numerical analytical method increases the computational efficiency, since they are not refined spatial discretization necessary because for any spatial discretization grids used, the numerical result generated for the same point of the domain remains unchanged unless the rounding errors of computational finite arithmetic. We chose to develop a computational application in MatLab platform for numerical computation and program interface is simple and easy with knobs. We consider important to model this neutron transport problem with a fixed source in the context of shielding calculations of radiation that protects the biosphere, and could be sensitive to ionizing radiation

  6. Cusping, transport and variance of solutions to generalized Fokker-Planck equations

    Science.gov (United States)

    Carnaffan, Sean; Kawai, Reiichiro

    2017-06-01

    We study properties of solutions to generalized Fokker-Planck equations through the lens of the probability density functions of anomalous diffusion processes. In particular, we examine solutions in terms of their cusping, travelling wave behaviours, and variance, within the framework of stochastic representations of generalized Fokker-Planck equations. We give our analysis in the cases of anomalous diffusion driven by the inverses of the stable, tempered stable and gamma subordinators, demonstrating the impact of changing the distribution of waiting times in the underlying anomalous diffusion model. We also analyse the cases where the underlying anomalous diffusion contains a Lévy jump component in the parent process, and when a diffusion process is time changed by an uninverted Lévy subordinator. On the whole, we present a combination of four criteria which serve as a theoretical basis for model selection, statistical inference and predictions for physical experiments on anomalously diffusing systems. We discuss possible applications in physical experiments, including, with reference to specific examples, the potential for model misclassification and how combinations of our four criteria may be used to overcome this issue.

  7. Bounds and maximum principles for the solution of the linear transport equation

    International Nuclear Information System (INIS)

    Larsen, E.W.

    1981-01-01

    Pointwise bounds are derived for the solution of time-independent linear transport problems with surface sources in convex spatial domains. Under specified conditions, upper bounds are derived which, as a function of position, decrease with distance from the boundary. Also, sufficient conditions are obtained for the existence of maximum and minimum principles, and a counterexample is given which shows that such principles do not always exist

  8. Long and short time quantum dynamics: I. Between Green's functions and transport equations

    Czech Academy of Sciences Publication Activity Database

    Špička, Václav; Velický, Bedřich; Kalvová, Anděla

    2005-01-01

    Roč. 29, - (2005), s. 154-174 ISSN 1386-9477 R&D Projects: GA ČR(CZ) GA202/04/0585; GA AV ČR(CZ) IAA1010404 Institutional research plan: CEZ:AV0Z10100521; CEZ:AV0Z10100520 Keywords : non-equilibrium * Green functions * quantum transport * density functional the ory Subject RIV: BE - The oretical Physics Impact factor: 0.946, year: 2005

  9. Study of response of 3He detectors to monoenergetic neutrons

    International Nuclear Information System (INIS)

    Abanades, A.; Andriamonje, S.; Arnould, H.; Barreau, G.; Bercion, M.; Casagrande, F.; Cennini, P.; Del Moral, R.; Gonzales, E.; Lacoste, V.; Pdemay, G.; Pravikoff, M.S.

    1997-01-01

    In the search of a hybrid system (the coupling of the particle accelerator to an under-critical reactor) for radioactive waste transmutation the TARC (Transmutation by Adiabatic Resonance Crossing) program has been developed. Due to experimental limitations, the time-energy relation at higher neutron energies, particularly, around 2 MeV, which is an important domain for TARC, cannot be applied. Consequently the responses of the 3 He ionization neutron detector developed for TARC experiment have been studied using a fast monoenergetic neutron source. The neutrons were produced by the interaction of the proton delivered by Van de Graaff accelerator of CENBG. The originality of the detector consists in its structure of three series of electric conductors which are mounted around the anode: a grid ensuring the detector proportionality, a cylindrical suit of alternating positive voltage and grounded wires aiming at eliminating the radial end effects, serving as veto and two cylinders serving as end plugs to eliminate the perpendicular end effects. Examples of anode spectra conditioned (in anticoincidence) by the mentioned vetoes are given. One can see the contribution of the elastic scattering from H and 3 He. By collimating the neutron beam through a borated polyethylene system it was possible to obtain a mapping of the detector allowing the study of its response as a function of the irradiated zones (anode and grid)

  10. LTSN solution of the adjoint neutron transport equation with arbitrary source for high order of quadrature in a homogeneous slab

    International Nuclear Information System (INIS)

    Goncalves, Glenio A.; Orengo, Gilberto; Vilhena, Marco Tullio M.B. de; Graca, Claudio O.

    2002-01-01

    In this work we present the LTS N solution of the adjoint transport equation for an arbitrary source, testing the aptness of this analytical solution for high order of quadrature in transport problems and comparing some preliminary results with the ANISN computations in a homogeneneous slab geometry. In order to do that we apply the new formulation for the LTS N method based on the invariance projection property, becoming possible to handle problems with arbitrary sources and demanding high order of quadrature or deep penetration. This new approach for the LTS N method is important both for direct and adjoint transport calculations and its development was inspired by the necessity of using generalized adjoint sources for important calculations. Although the mathematical convergence has been proved for an arbitrary source, when the quadrature order or deep penetration is required the LTS N method presents computational overflow even for simple sources (sin, cos, exp, polynomial). With the new formulation we eliminate this drawback and in this work we report the numerical simulations testing the new approach

  11. A non-linear optimal Discontinuous Petrov-Galerkin method for stabilising the solution of the transport equation

    International Nuclear Information System (INIS)

    Merton, S. R.; Smedley-Stevenson, R. P.; Pain, C. C.; Buchan, A. G.; Eaton, M. D.

    2009-01-01

    This paper describes a new Non-Linear Discontinuous Petrov-Galerkin (NDPG) method and application to the one-speed Boltzmann Transport Equation (BTE) for space-time problems. The purpose of the method is to remove unwanted oscillations in the transport solution which occur in the vicinity of sharp flux gradients, while improving computational efficiency and numerical accuracy. This is achieved by applying artificial dissipation in the solution gradient direction, internal to an element using a novel finite element (FE) Riemann approach. The amount of dissipation added acts internal to each element. This is done using a gradient-informed scaling of the advection velocities in the stabilisation term. This makes the method in its most general form non-linear. The method is designed to be independent of angular expansion framework. This is demonstrated for the both discrete ordinates (S N ) and spherical harmonics (P N ) descriptions of the angular variable. Results show the scheme performs consistently well in demanding time dependent and multi-dimensional radiation transport problems. (authors)

  12. A device adaptive inflow boundary condition for Wigner equations of quantum transport

    International Nuclear Information System (INIS)

    Jiang, Haiyan; Lu, Tiao; Cai, Wei

    2014-01-01

    In this paper, an improved inflow boundary condition is proposed for Wigner equations in simulating a resonant tunneling diode (RTD), which takes into consideration the band structure of the device. The original Frensley inflow boundary condition prescribes the Wigner distribution function at the device boundary to be the semi-classical Fermi–Dirac distribution for free electrons in the device contacts without considering the effect of the quantum interaction inside the quantum device. The proposed device adaptive inflow boundary condition includes this effect by assigning the Wigner distribution to the value obtained from the Wigner transform of wave functions inside the device at zero external bias voltage, thus including the dominant effect on the electron distribution in the contacts due to the device internal band energy profile. Numerical results on computing the electron density inside the RTD under various incident waves and non-zero bias conditions show much improvement by the new boundary condition over the traditional Frensley inflow boundary condition

  13. Various types of numerical schema for the one-dimensional spherical geometry transport equation

    International Nuclear Information System (INIS)

    Jaber, Abdelouhab.

    1981-07-01

    Mathematical and numerical studies of new schemas possessing high accuracy spatial variable properties are described and the corresponding studies presented. In order to do this, the [0,R] x [-1,+1] rectangle is decomposad into Ksub(ij) = [rsub(i),rsub(i+1)] x [μsub(j),μsub(j+1) ] rectangles. Continuous finite element methods employing polynominals of degree 1 in μ and degree 2 in r are defined for each elements. In chapter I, different ways of rendering the particular equation (for μ = -1) discrete are studied. In chapter II, numerical schemas are described and their stability investigated. In chapter III, error estimation theories are exposed and numerical results for different second members, S, given [fr

  14. Multi-domain/multi-method numerical approach for neutron transport equation; Couplage de methodes et decomposition de domaine pour la resolution de l'equation du transport des neutrons

    Energy Technology Data Exchange (ETDEWEB)

    Girardi, E

    2004-12-15

    A new methodology for the solution of the neutron transport equation, based on domain decomposition has been developed. This approach allows us to employ different numerical methods together for a whole core calculation: a variational nodal method, a discrete ordinate nodal method and a method of characteristics. These new developments authorize the use of independent spatial and angular expansion, non-conformal Cartesian and unstructured meshes for each sub-domain, introducing a flexibility of modeling which is not allowed in today available codes. The effectiveness of our multi-domain/multi-method approach has been tested on several configurations. Among them, one particular application: the benchmark model of the Phebus experimental facility at Cea-Cadarache, shows why this new methodology is relevant to problems with strong local heterogeneities. This comparison has showed that the decomposition method brings more accuracy all along with an important reduction of the computer time.

  15. High-order discontinuous Galerkin nonlocal transport and energy equations scheme for radiation hydrodynamics

    Czech Academy of Sciences Publication Activity Database

    Holec, M.; Limpouch, J.; Liska, R.; Weber, Stefan A.

    2017-01-01

    Roč. 83, č. 10 (2017), s. 779-797 ISSN 0271-2091 R&D Projects: GA MŠk EF15_008/0000162; GA MŠk LQ1606 Grant - others:ELI Beamlines(XE) CZ.02.1.01/0.0/0.0/15_008/0000162 Institutional support: RVO:68378271 Keywords : radiation hydrodynamics * nonlocal transport * Knudsen number * multigroup diffusion * radiation coupling Subject RIV: BG - Nuclear, Atomic and Molecular Physics, Colliders OBOR OECD: Nuclear physics Impact factor: 1.652, year: 2016

  16. Discontinuous finite element and characteristics methods for neutrons transport equation solution in heterogeneous grids; Resolution de l'equation du transport des neutrons par les methodes des elements finis discontinus et des caracteristiques structurees appliquees a des maillages heterogenes

    Energy Technology Data Exchange (ETDEWEB)

    Masiello, E

    2006-07-01

    The principal goal of this manuscript is devoted to the investigation of a new type of heterogeneous mesh adapted to the shape of the fuel pins (fuel-clad-moderator). The new heterogeneous mesh guarantees the spatial modelling of the pin-cell with a minimum of regions. Two methods are investigated for the spatial discretization of the transport equation: the discontinuous finite element method and the method of characteristics for structured cells. These methods together with the new representation of the pin-cell result in an appreciable reduction of calculation points. They allow an exact modelling of the fuel pin-cell without spatial homogenization. A new synthetic acceleration technique based on an angular multigrid is also presented for the speed up of the inner iterations. These methods are good candidates for transport calculations for a nuclear reactor core. A second objective of this work is the application of method of characteristics for non-structured geometries to the study of double heterogeneity problem. The letters is characterized by fuel material with a stochastic dispersion of heterogeneous grains, and until now was solved with a model based on collision probabilities. We propose a new statistical model based on renewal-Markovian theory, which makes possible to take into account the stochastic nature of the problem and to avoid the approximations of the collision probability model. The numerical solution of this model is guaranteed by the method of characteristics. (author)

  17. Discontinuous finite element and characteristics methods for neutrons transport equation solution in heterogeneous grids; Resolution de l'equation du transport des neutrons par les methodes des elements finis discontinus et des caracteristiques structurees appliquees a des maillages heterogenes

    Energy Technology Data Exchange (ETDEWEB)

    Masiello, E

    2006-07-01

    The principal goal of this manuscript is devoted to the investigation of a new type of heterogeneous mesh adapted to the shape of the fuel pins (fuel-clad-moderator). The new heterogeneous mesh guarantees the spatial modelling of the pin-cell with a minimum of regions. Two methods are investigated for the spatial discretization of the transport equation: the discontinuous finite element method and the method of characteristics for structured cells. These methods together with the new representation of the pin-cell result in an appreciable reduction of calculation points. They allow an exact modelling of the fuel pin-cell without spatial homogenization. A new synthetic acceleration technique based on an angular multigrid is also presented for the speed up of the inner iterations. These methods are good candidates for transport calculations for a nuclear reactor core. A second objective of this work is the application of method of characteristics for non-structured geometries to the study of double heterogeneity problem. The letters is characterized by fuel material with a stochastic dispersion of heterogeneous grains, and until now was solved with a model based on collision probabilities. We propose a new statistical model based on renewal-Markovian theory, which makes possible to take into account the stochastic nature of the problem and to avoid the approximations of the collision probability model. The numerical solution of this model is guaranteed by the method of characteristics. (author)

  18. The discrete ordinates method for solving the azimuthally dependent transport equation in plane geometry

    International Nuclear Information System (INIS)

    Chalhoub, Ezzat Selim

    1997-01-01

    The method of discrete ordinates is applied to the solution of the slab albedo problem with azimuthal dependence in transport theory. A new set of quadratures appropriate to the problem is introduced. In addition to the ANISN code, modified to include the proposed formalism, two new programs, PEESNC and PEESNA, which were created on the basis of the discrete ordinates formalism, using the direct integration method and the analytic solution method respectively, are used in the generation of results for a few sample problems. Program PEESNC was created to validate the results obtained with the discrete ordinates method and the finite difference approximation (ANISN), while program PEESNA was developed in order to implement an analytical discrete ordinates formalism, which provides more accurate results. The obtained results for selected sample problems are compared with highly accurate numerical results published in the literature. Compared to ANISN and PEESNC, program PEESNA presents a greater efficiency in execution time and much more precise numerical results. (author)

  19. The recovery of a time-dependent point source in a linear transport equation: application to surface water pollution

    International Nuclear Information System (INIS)

    Hamdi, Adel

    2009-01-01

    The aim of this paper is to localize the position of a point source and recover the history of its time-dependent intensity function that is both unknown and constitutes the right-hand side of a 1D linear transport equation. Assuming that the source intensity function vanishes before reaching the final control time, we prove that recording the state with respect to the time at two observation points framing the source region leads to the identification of the source position and the recovery of its intensity function in a unique manner. Note that at least one of the two observation points should be strategic. We establish an identification method that determines quasi-explicitly the source position and transforms the task of recovering its intensity function into solving directly a well-conditioned linear system. Some numerical experiments done on a variant of the water pollution BOD model are presented

  20. Influence of Superconducting Leads Energy Gap on Electron Transport Through Double Quantum Dot by Markovian Quantum Master Equation Approach

    International Nuclear Information System (INIS)

    Afsaneh, E.; Yavari, H.

    2014-01-01

    The superconducting reservoir effect on the current carrying transport of a double quantum dot in Markovian regime is investigated. For this purpose, a quantum master equation at finite temperature is derived for the many-body density matrix of an open quantum system. The dynamics and the steady-state properties of the double quantum dot system for arbitrary bias are studied. We will show that how the populations and coherencies of the system states are affected by superconducting leads. The energy parameter of system contains essentially four contributions due to dots system-electrodes coupling, intra dot coupling, two quantum dots inter coupling and superconducting gap. The coupling effect of each energy contribution is applied to currents and coherencies results. In addition, the effect of energy gap is studied by considering the amplitude and lifetime of coherencies to get more current through the system. (author)

  1. Least-Squares PN Formulation of the Transport Equation Using Self-Adjoint-Angular-Flux Consistent Boundary Conditions

    Energy Technology Data Exchange (ETDEWEB)

    Laboure, Vincent M.; Wang, Yaqi; DeHart, Mark D.

    2016-05-01

    In this paper, we study the Least-Squares (LS) PN form of the transport equation compatible with voids [1] in the context of Continuous Finite Element Methods (CFEM).We first deriveweakly imposed boundary conditions which make the LS weak formulation equivalent to the Self-Adjoint Angular Flux (SAAF) variational formulation with a void treatment [2], in the particular case of constant cross-sections and a uniform mesh. We then implement this method in Rattlesnake with the Multiphysics Object Oriented Simulation Environment (MOOSE) framework [3] using a spherical harmonics (PN) expansion to discretize in angle. We test our implementation using the Method of Manufactured Solutions (MMS) and find the expected convergence behavior both in angle and space. Lastly, we investigate the impact of the global non-conservation of LS by comparing the method with SAAF on a heterogeneous test problem.

  2. Least-Squares PN Formulation of the Transport Equation Using Self-Adjoint-Angular-Flux Consistent Boundary Conditions.

    Energy Technology Data Exchange (ETDEWEB)

    Vincent M. Laboure; Yaqi Wang; Mark D. DeHart

    2016-05-01

    In this paper, we study the Least-Squares (LS) PN form of the transport equation compatible with voids in the context of Continuous Finite Element Methods (CFEM).We first deriveweakly imposed boundary conditions which make the LS weak formulation equivalent to the Self-Adjoint Angular Flux (SAAF) variational formulation with a void treatment, in the particular case of constant cross-sections and a uniform mesh. We then implement this method in Rattlesnake with the Multiphysics Object Oriented Simulation Environment (MOOSE) framework using a spherical harmonics (PN) expansion to discretize in angle. We test our implementation using the Method of Manufactured Solutions (MMS) and find the expected convergence behavior both in angle and space. Lastly, we investigate the impact of the global non-conservation of LS by comparing the method with SAAF on a heterogeneous test problem.

  3. Approximated transport-of-intensity equation for coded-aperture x-ray phase-contrast imaging.

    Science.gov (United States)

    Das, Mini; Liang, Zhihua

    2014-09-15

    Transport-of-intensity equations (TIEs) allow better understanding of image formation and assist in simplifying the "phase problem" associated with phase-sensitive x-ray measurements. In this Letter, we present for the first time to our knowledge a simplified form of TIE that models x-ray differential phase-contrast (DPC) imaging with coded-aperture (CA) geometry. The validity of our approximation is demonstrated through comparison with an exact TIE in numerical simulations. The relative contributions of absorption, phase, and differential phase to the acquired phase-sensitive intensity images are made readily apparent with the approximate TIE, which may prove useful for solving the inverse phase-retrieval problem associated with these CA geometry based DPC.

  4. Optimization of the spatial mesh for numerical solution of the neutron transport equation in a cluster-type lattice cell

    International Nuclear Information System (INIS)

    Davis, R.S.

    2012-01-01

    For programs that solve the neutron transport equation with an approximation that the neutron flux is constant in each space in a user-defined mesh, optimization of that mesh yields benefits in computing time and attainable precision. The previous best practice does not optimize the mesh thoroughly, because a large number of test runs of the solving software would be necessary. The method presented here optimizes the mesh for a flux that is based on conventional approximations but is more informative, so that a minimal number of parameters, one per type of material, must be adjusted by test runs to achieve thorough optimization. For a 37 element, natural-uranium, CANDU lattice cell, the present optimization yields 7 to 12 times (depending on the criterion) better precision than the previous best practice in 37% less computing time. (author)

  5. Non-interferometric determination of optical anisotropy in highly-oriented fibres using transport intensity equation technique

    Science.gov (United States)

    Sokkar, T. Z. N.; El-Farahaty, K. A.; El-Bakary, M. A.; Raslan, M. I.; Omar, E. Z.; Hamza, A. A.

    2018-03-01

    The optical setup of the transport intensity equation (TIE) technique is developed to be valid for measuring the optical properties of the highly-oriented anisotropic fibres. This development is based on the microstructure models of the highly-oriented anisotropic fibres and the principle of anisotropy. We provide the setup of TIE technique with polarizer which is controlled via stepper motor. This developed technique is used to investigate the refractive indices in the parallel and perpendicular polarization directions of light for the highly-oriented poly (ethylene terephthalate) (PET) fibres and hence its birefringence. The obtained results through the developed TIE technique for PET fibre are compared with that determined experimentally using the Mach-Zehnder interferometer under the same conditions. The comparison shows a good agreement between the obtained results from the developed technique and that obtained from the Mach-Zehnder interferometer technique.

  6. Fast Solutions of Maxwell's Equation for High Resolution Electromagnetic Imaging of Transport Pathways; TOPICAL

    International Nuclear Information System (INIS)

    DAY, DAVID M.; NEWMAN, GREGORY A.

    1999-01-01

    A fast precondition technique has been developed which accelerates the finite difference solutions of the 3D Maxwell's equations for geophysical modeling. The technique splits the electric field into its curl free and divergence free projections, and allows for the construction of an inverse operator. Test examples show an order of magnitude speed up compared with a simple Jacobi preconditioner. Using this preconditioner a low frequency Neumann series expansion is developed and used to compute responses at multiple frequencies very efficiently. Simulations requiring responses at multiple frequencies, show that the Neumann series is faster than the preconditioned solution, which must compute solutions at each discrete frequency. A Neumann series expansion has also been developed in the high frequency limit along with spectral Lanczos methods in both the high and low frequency cases for simulating multiple frequency responses with maximum efficiency. The research described in this report was to have been carried out over a two-year period. Because of communication difficulties, the project was funded for first year only. Thus the contents of this report are incomplete with respect to the original project objectives

  7. A transport equation for the evolution of shock amplitudes along rays

    Directory of Open Access Journals (Sweden)

    Giovanni Russo

    1991-05-01

    Full Text Available A new asymptotic method is derived for the study of the evolution of weak shocks in several dimension. The method is based on the Generalized Wavefront Expansion derived in [1]. In that paper the propagation of a shock into a known background was studied under the assumption that shock is weak, i.e. Mach Number =1+O(ε, ε ≪ 1, and that the perturbation of the field varies over a length scale O(ε. To the lowest order, the shock surface evolves along the rays associated with the unperturbed state. An infinite system of compatibility relations was derived for the jump in the field and its normal derivatives along the shock, but no valid criterion was found for a truncation of the system. Here we show that the infinite hierarchy is equivalent to a single equation that describes the evolution of the shock along the rays. We show that this method gives equivalent results to those obtained by Weakly Nonlinear Geometrical Optics [2].

  8. Equation of state and transport properties of warm dense helium via quantum molecular dynamics simulations

    Energy Technology Data Exchange (ETDEWEB)

    Li, Zhi-Guo [College of Physical Science and Technology, Sichuan University, Chengdu 610064 (China); National Key Laboratory for Shock Wave and Detonation Physics Research, Institute of Fluid Physics, Chinese Academy of Engineering Physics, Mianyang 621900 (China); Key Laboratory of High Energy Density Physics and Technology of Ministry of Education, Sichuan University, Chengdu 610064 (China); Cheng, Yan [College of Physical Science and Technology, Sichuan University, Chengdu 610064 (China); Key Laboratory of High Energy Density Physics and Technology of Ministry of Education, Sichuan University, Chengdu 610064 (China); Chen, Qi-Feng, E-mail: chenqf01@gmail.com, E-mail: xrchen@scu.edu.cn [National Key Laboratory for Shock Wave and Detonation Physics Research, Institute of Fluid Physics, Chinese Academy of Engineering Physics, Mianyang 621900 (China); Chen, Xiang-Rong, E-mail: chenqf01@gmail.com, E-mail: xrchen@scu.edu.cn [College of Physical Science and Technology, Sichuan University, Chengdu 610064 (China)

    2016-05-15

    The equation of state, self-diffusion, and viscosity coefficients of helium have been investigated by quantum molecular dynamics (QMD) simulations in the warm dense matter regime. Our simulations are validated through the comparison with the reliable experimental data. The calculated principal and reshock Hugoniots of liquid helium are in good agreement with the gas-gun data. On this basis, we revisit the issue for helium, i.e., the possibility of the instabilities predicted by chemical models at around 2000 GPa and 10 g/cm{sup 3} along the pressure isotherms of 6309, 15 849, and 31 623 K. Our calculations show no indications of instability in this pressure-temperature region, which reconfirm the predictions of previous QMD simulations. The self-diffusion and viscosity coefficients of warm dense helium have been systematically investigated by the QMD simulations. We carefully test the finite-size effects and convergences of statistics, and obtain numerically converged self-diffusion and viscosity coefficients by using the Kubo-Green formulas. The present results have been used to evaluate the existing one component plasma models. Finally, the validation of the Stokes-Einstein relationship for helium in the warm dense regime is discussed.

  9. Use of simple transport equations to estimate waste package performance requirements

    International Nuclear Information System (INIS)

    Wood, B.J.

    1982-01-01

    A method of developing waste package performance requirements for specific nuclides is described. The method is based on: Federal regulations concerning permissible concentrations in solution at the point of discharge to the accessible environment; a simple and conservative transport model; baseline and potential worst-case release scenarios. Use of the transport model enables calculation of maximum permissible release rates within a repository in basalt for each of the scenarios. The maximum permissible release rates correspond to performance requirements for the engineered barrier system. The repository was assumed to be constructed in a basalt layer. For the cases considered, including a well drilled into an aquifer 1750 m from the repository center, little significant advantage is obtained from a 1000-yr as opposed to a 100-yr waste package. A 1000-yr waste package is of importance only for nuclides with half-lives much less than 100 yr which travel to the accessible environment in much less than 1000 yr. Such short travel times are extremely unlikely for a mined repository. Among the actinides, the most stringent maximum permissible release rates are for 236 U and 234 U. A simple solubility calculation suggests, however, that these performance requirements can be readily met by the engineered barrier system. Under the reducing conditions likely to occur in a repository located in basalt, uranium would be sufficiently insoluble that no solution could contain more than about 0.01% of the maximum permissible concentration at saturation. The performance requirements derived from the one-dimensional modeling approach are conservative by at least one to two orders of magnitude. More quantitative three-dimensional modeling at specific sites should enable relaxation of the performance criteria derived in this study. 12 references, 8 figures, 8 tables

  10. Numerical simulations of air–water cap-bubbly flows using two-group interfacial area transport equation

    International Nuclear Information System (INIS)

    Wang, Xia; Sun, Xiaodong

    2014-01-01

    Highlights: • Two-group interfacial area transport equation was implemented into a three-field two-fluid model in Fluent. • Numerical model was developed for cap-bubbly flows in a narrow rectangular flow channel. • Numerical simulations were performed for cap-bubbly flows with uniform void inlets and with central peaked void inlets. • Code simulations showed a significant improve over the conventional two-fluid model. - Abstract: Knowledge of cap-bubbly flows is of great interest due to its role in understanding of the flow regime transition from bubbly to slug or churn-turbulent flows. One of the key characteristics of such flows is the existence of bubbles in different sizes and shapes associated with their distinctive dynamic natures. This important feature is, however, generally not well captured by many available two-phase flow modeling approaches. In this study, a modified two-fluid model, namely a three-field, two-fluid model, is proposed. In this model, bubbles are categorized into two groups, i.e., spherical/distorted bubbles as Group-1 while cap/churn-turbulent bubbles as Group-2. A two-group interfacial area transport equation (IATE) is implemented to describe dynamic changes of interfacial structure in each bubble group, resulting from intra- and inter-group interactions and phase changes due to evaporation and condensation. Attention is also paid to appropriate constitutive relations of the interfacial transfers due to mechanical and thermal non-equilibrium between the different fields. The proposed three-field, two-fluid model is used to predict the phase distributions of adiabatic air–water flows in a confined rectangular duct. Good agreement between the simulation results from the proposed model and relevant experimental data indicates that the proposed model is promising as an improved computational tool for two-phase cap-bubbly flow simulations in rectangular flow ducts

  11. Correspondence Between One- and Two-Equation Models for Solute Transport in Two-Region Heterogeneous Porous Media

    KAUST Repository

    Davit, Y.

    2012-07-26

    In this work, we study the transient behavior of homogenized models for solute transport in two-region porous media. We focus on the following three models: (1) a time non-local, two-equation model (2eq-nlt). This model does not rely on time constraints and, therefore, is particularly useful in the short-time regime, when the timescale of interest (t) is smaller than the characteristic time (τ 1) for the relaxation of the effective macroscale parameters (i. e., when t ≤ τ 1); (2) a time local, two-equation model (2eq). This model can be adopted when (t) is significantly larger than (τ 1) (i.e., when t≫τ 1); and (3) a one-equation, time-asymptotic formulation (1eq ∞). This model can be adopted when (t) is significantly larger than the timescale (τ 2) associated with exchange processes between the two regions (i. e., when t≫τ 2). In order to obtain insight into this transient behavior, we combine a theoretical approach based on the analysis of spatial moments with numerical and analytical results in several simple cases. The main result of this paper is to show that there is only a weak asymptotic convergence of the solution of (2eq) towards the solution of (1eq ∞) in terms of standardized moments but, interestingly, not in terms of centered moments. The physical interpretation of this result is that deviations from the Fickian situation persist in the limit of long times but that the spreading of the solute is eventually dominating these higher order effects. © 2012 Springer Science+Business Media B.V.

  12. Time-dependent quantum transport through an interacting quantum dot beyond sequential tunneling: second-order quantum rate equations

    International Nuclear Information System (INIS)

    Dong, B; Ding, G H; Lei, X L

    2015-01-01

    A general theoretical formulation for the effect of a strong on-site Coulomb interaction on the time-dependent electron transport through a quantum dot under the influence of arbitrary time-varying bias voltages and/or external fields is presented, based on slave bosons and the Keldysh nonequilibrium Green's function (GF) techniques. To avoid the difficulties of computing double-time GFs, we generalize the propagation scheme recently developed by Croy and Saalmann to combine the auxiliary-mode expansion with the celebrated Lacroix's decoupling approximation in dealing with the second-order correlated GFs and then establish a closed set of coupled equations of motion, called second-order quantum rate equations (SOQREs), for an exact description of transient dynamics of electron correlated tunneling. We verify that the stationary solution of our SOQREs is able to correctly describe the Kondo effect on a qualitative level. Moreover, a comparison with other methods, such as the second-order von Neumann approach and Hubbard-I approximation, is performed. As illustrations, we investigate the transient current behaviors in response to a step voltage pulse and a harmonic driving voltage, and linear admittance as well, in the cotunneling regime. (paper)

  13. Time-dependent integral transport equation kernels, leakage rates and collision rates for plane and spherical geometry

    International Nuclear Information System (INIS)

    Henderson, D.L.

    1987-01-01

    Time-dependent integral transport equation flux and current kernels for plane and spherical geometry are derived for homogeneous media. Using the multiple collision formalism, isotropic sources that are delta distributions in time are considered for four different problems. The plane geometry flux kernel is applied to a uniformly distributed source within an infinite medium and to a surface source in a semi-infinite medium. The spherical flux kernel is applied to a point source in an infinite medium and to a point source at the origin of a finite sphere. The time-dependent first-flight leakage rates corresponding to the existing steady state first-flight escape probabilities are computed by the Laplace transform technique assuming a delta distribution source in time. The case of a constant source emitting neutrons over a time interval, Δt, for a spatially uniform source is obtained for a slab and a sphere. Time-dependent first-flight leakage rates are also determined for the general two region spherical medium problem for isotropic sources with a delta distribution in time uniformly distributed throughout both the inner and outer regions. The time-dependent collision rates due to the uncollided neutrons are computed for a slab and a sphere using the time-dependent first-flight leakage rates and the time-dependent continuity equation. The case of a constant source emitting neutrons over a time interval, Δt, is also considered

  14. Fast linear solver for radiative transport equation with multiple right hand sides in diffuse optical tomography

    International Nuclear Information System (INIS)

    Jia, Jingfei; Kim, Hyun K.; Hielscher, Andreas H.

    2015-01-01

    It is well known that radiative transfer equation (RTE) provides more accurate tomographic results than its diffusion approximation (DA). However, RTE-based tomographic reconstruction codes have limited applicability in practice due to their high computational cost. In this article, we propose a new efficient method for solving the RTE forward problem with multiple light sources in an all-at-once manner instead of solving it for each source separately. To this end, we introduce here a novel linear solver called block biconjugate gradient stabilized method (block BiCGStab) that makes full use of the shared information between different right hand sides to accelerate solution convergence. Two parallelized block BiCGStab methods are proposed for additional acceleration under limited threads situation. We evaluate the performance of this algorithm with numerical simulation studies involving the Delta–Eddington approximation to the scattering phase function. The results show that the single threading block RTE solver proposed here reduces computation time by a factor of 1.5–3 as compared to the traditional sequential solution method and the parallel block solver by a factor of 1.5 as compared to the traditional parallel sequential method. This block linear solver is, moreover, independent of discretization schemes and preconditioners used; thus further acceleration and higher accuracy can be expected when combined with other existing discretization schemes or preconditioners. - Highlights: • We solve the multiple-right-hand-side problem in DOT with a block BiCGStab method. • We examine the CPU times of the block solver and the traditional sequential solver. • The block solver is faster than the sequential solver by a factor of 1.5–3.0. • Multi-threading block solvers give additional speedup under limited threads situation.

  15. Coaxial Mono-Energetic Gamma Generator for Active Interrogation

    International Nuclear Information System (INIS)

    Ludewigt, Bernhard A.; Antolak, A.J.; Henestroza, E.; Leitner, M.; Leung, K.-N.; Waldron, W.; Wilde, S.; Kwan, J.W.

    2008-01-01

    Compact mono-energetic photon sources are sought for active interrogation systems to detect shielded special nuclear materials in, for example, cargo containers, trucks and other vehicles. A prototype gamma interrogation source has been designed and built that utilizes the 11B(p,gamma)12C reaction to produce 12 MeV gamma-rays which are near the peak of the photofission cross section. In particular, the 11B(p,gamma)12C resonance at 163 kV allows the production of gammas at low proton acceleration voltages, thus keeping the design of a gamma generator comparatively small and simple. A coaxial design has been adopted with a toroidal-shaped plasma chamber surrounding a cylindrical gamma production target. The plasma discharge is driven by a 2 MHz rf-power supply (capable up to 50 kW) using a circular rf-antenna. Permanent magnets embedded in the walls of the plasma chamber generate a multi-cusp field that confines the plasma and allows higher plasma densities and lower gas pressures. About 100 proton beamlets are extracted through a slotted plasma electrode towards the target at the center of the device that is at a negative 180 kV. The target consists of LaB6 tiles that are brazed to a water-cooled cylindrical structure. The generator is designed to operate at 500 Hz with 20 mu s long pulses, and a 1percent duty factor by pulsing the ion source rf-power. A first-generation coaxial gamma source has been built for low duty factor experiments and testing.

  16. Impact of Monoenergetic Photon Sources on Nonproliferation Applications Final Report

    Energy Technology Data Exchange (ETDEWEB)

    Geddes, Cameron [Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States); Ludewigt, Bernhard [Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States); Valentine, John [Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States); Quiter, Brian [Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States); Descalle, Marie-Anne [Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States); Warren, Glen [Pacific Northwest National Lab. (PNNL), Richland, WA (United States); Kinlaw, Matt [Idaho National Lab. (INL), Idaho Falls, ID (United States); Thompson, Scott [Idaho National Lab. (INL), Idaho Falls, ID (United States); Chichester, David [Idaho National Lab. (INL), Idaho Falls, ID (United States); Miller, Cameron [Univ. of Michigan, Ann Arbor, MI (United States); Pozzi, Sara [Univ. of Michigan, Ann Arbor, MI (United States)

    2017-03-01

    Near-monoenergetic photon sources (MPSs) have the potential to improve sensitivity at greatly reduced dose in existing applications and enable new capabilities in other applications, particularly where passive signatures do not penetrate or are insufficiently accurate. MPS advantages include the ability to select energy, energy spread, flux, and pulse structures to deliver only the photons needed for the application, while suppressing extraneous dose and background. Some MPSs also offer narrow angular divergence photon beams which can target dose and/or mitigate scattering contributions to image contrast degradation. Current bremsstrahlung photon sources (e.g., linacs and betatrons) produce photons over a broad range of energies, thus delivering unnecessary dose that in some cases also interferes with the signature to be detected and/or restricts operations. Current sources must be collimated (reducing flux) to generate narrow divergence beams. While MPSs can in principle resolve these issues, they remain at relatively low TRL status. Candidate MPS technologies for nonproliferation applications are now being developed, each of which has different properties (e.g. broad vs. narrow angular divergence). Within each technology, source parameters trade off against one another (e.g. flux vs. energy spread), representing a large operation space. This report describes a broad survey of potential applications, identification of high priority applications, and detailed simulations addressing those priority applications. Requirements were derived for each application, and analysis and simulations were conducted to define MPS parameters that deliver benefit. The results can inform targeting of MPS development to deliver strong impact relative to current systems.

  17. Approximate solutions of the two-dimensional integral transport equation by collision probability methods

    International Nuclear Information System (INIS)

    Sanchez, Richard

    1977-01-01

    A set of approximate solutions for the isotropic two-dimensional neutron transport problem has been developed using the Interface Current formalism. The method has been applied to regular lattices of rectangular cells containing a fuel pin, cladding and water, or homogenized structural material. The cells are divided into zones which are homogeneous. A zone-wise flux expansion is used to formulate a direct collision probability problem within a cell. The coupling of the cells is made by making extra assumptions on the currents entering and leaving the interfaces. Two codes have been written: the first uses a cylindrical cell model and one or three terms for the flux expansion; the second uses a two-dimensional flux representation and does a truly two-dimensional calculation inside each cell. In both codes one or three terms can be used to make a space-independent expansion of the angular fluxes entering and leaving each side of the cell. The accuracies and computing times achieved with the different approximations are illustrated by numerical studies on two benchmark pr

  18. A verification regime for the spatial discretization of the SN transport equations

    Energy Technology Data Exchange (ETDEWEB)

    Schunert, S.; Azmy, Y. [North Carolina State Univ., Dept. of Nuclear Engineering, 2500 Stinson Drive, Raleigh, NC 27695 (United States)

    2012-07-01

    The order-of-accuracy test in conjunction with the method of manufactured solutions is the current state of the art in computer code verification. In this work we investigate the application of a verification procedure including the order-of-accuracy test on a generic SN transport solver that implements the AHOTN spatial discretization. Different types of semantic errors, e.g. removal of a line of code or changing a single character, are introduced randomly into the previously verified S{sub N} code and the proposed verification procedure is used to identify the coding mistakes (if possible) and classify them. Itemized by error type we record the stage of the verification procedure where the error is detected and report the frequency with which the errors are correctly identified at various stages of the verification. Errors that remain undetected by the verification procedure are further scrutinized to determine the reason why the introduced coding mistake eluded the verification procedure. The result of this work is that the verification procedure based on an order-of-accuracy test finds almost all detectable coding mistakes but rarely, 1.44% of the time, and under certain circumstances can fail. (authors)

  19. Tracking techniques for the characteristics method applied to the resolution of the neutrons transport equation in multi scale domains

    International Nuclear Information System (INIS)

    Fevotte, F.

    2008-01-01

    At the various stages of a nuclear reactor's life, numerous studies are needed to guaranty the safety and efficiency of the design, analyse the fuel cycle, prepare the dismantlement, and so on. Due to the extreme difficulty to take extensive and accurate measurements in the reactor core, most of these studies are numerical simulations. The complete numerical simulation of a nuclear reactor involves many types of physics: neutronics, thermal hydraulics, materials, control engineering, Among these, the neutron transport simulation is one of the fundamental steps, since it allows computation - among other things - of various fundamental values such as the power density (used in thermal hydraulics computations) or fuel burn-up. The neutron transport simulation is based on the Boltzmann equation, which models the neutron population inside the reactor core. Among the various methods allowing its numerical solution, much interest has been devoted in the past few years to the Method of Characteristics in unstructured meshes (MOC), since it offers a good accuracy and operates in complicated geometries. The aim of this work is to propose improvements of the calculation scheme bound on the two dimensions MOC, in order to decrease the needed resources number. (A.L.B.)

  20. Conductivity equations of protons transporting through 2D crystals obtained with the rate process theory and free volume concept

    Science.gov (United States)

    Hao, Tian; Xu, Yuanze; Hao, Ting

    2018-04-01

    The Eyring's rate process theory and free volume concept are employed to treat protons (or other particles) transporting through a 2D (two dimensional) crystal like graphene and hexagonal boron nitride. The protons are assumed to be activated first in order to participate conduction and the conduction rate is dependent on how much free volume available in the system. The obtained proton conductivity equations show that only the number of conduction protons, proton size and packing structure, and the energy barrier associated with 2D crystals are critical; the quantization conductance is unexpectedly predicted with a simple Arrhenius type temperature dependence. The predictions agree well with experimental observations and clear out many puzzles like much smaller energy barrier determined from experiments than from the density function calculations and isotope separation rate independent of the energy barrier of 2D crystals, etc. Our work may deepen our understandings on how protons transport through a membrane and has direct implications on hydrogen related technology and proton involved bioprocesses.

  1. Solution to the transport equation with anisotropic dispersion in a BWR type assembly using the AZTRAN code

    International Nuclear Information System (INIS)

    Chepe P, M.; Xolocostli M, J. V.; Gomez T, A. M.; Del Valle G, E.

    2016-09-01

    Due to the current computing power, the deterministic codes for analyzing nuclear reactors that have been used for several years are becoming more relevant, since much more precise solution techniques can be used; the last century would have been very difficult, since memory and processor capacities were very limited or had high prices on the components. In this work we analyze the effect of the anisotropic dispersion of the effective dispersion section, compared to the isotropic dispersion. The anisotropy implementation was carried out in the AZTRAN transport code, which is part of the AZTLAN platform for nuclear reactors analysis (in development). The AZTRAN code solves the Boltzmann transport equation in one, two and three dimensions at steady state, using the multi-group technique for energy discretization, the RTN-0 nodal method in spatial discretization and for angular discretization the discrete ordinates without considering anisotropy originally. The effect of the anisotropy dispersion on the effective multiplication factor and the axial and radial power on a fuel assembly BWR type are analyzed. (Author)

  2. Multigroup discrete ordinates solution of Boltzmann-Fokker-Planck equations and cross section library development of ion transport

    International Nuclear Information System (INIS)

    Prinja, A.K.

    1995-08-01

    We have developed and successfully implemented a two-dimensional bilinear discontinuous in space and time, used in conjunction with the S N angular approximation, to numerically solve the time dependent, one-dimensional, one-speed, slab geometry, (ion) transport equation. Numerical results and comparison with analytical solutions have shown that the bilinear-discontinuous (BLD) scheme is third-order accurate in the space ad time dimensions independently. Comparison of the BLD results with diamond-difference methods indicate that the BLD method is both quantitavely and qualitatively superior to the DD scheme. We note that the form of the transport operator is such that these conclusions carry over to energy dependent problems that include the constant-slowing-down-approximation term, and to multiple space dimensions or combinations thereof. An optimized marching or inversion scheme or a parallel algorithm should be investigated to determine if the increased accuracy can compensate for the extra overhead required for a BLD solution, and then could be compared to other discretization methods such as nodal or characteristic schemes

  3. Finite difference numerical method for the superlattice Boltzmann transport equation and case comparison of CPU(C) and GPU(CUDA) implementations

    International Nuclear Information System (INIS)

    Priimak, Dmitri

    2014-01-01

    We present a finite difference numerical algorithm for solving two dimensional spatially homogeneous Boltzmann transport equation which describes electron transport in a semiconductor superlattice subject to crossed time dependent electric and constant magnetic fields. The algorithm is implemented both in C language targeted to CPU and in CUDA C language targeted to commodity NVidia GPU. We compare performances and merits of one implementation versus another and discuss various software optimisation techniques

  4. Finite difference numerical method for the superlattice Boltzmann transport equation and case comparison of CPU(C) and GPU(CUDA) implementations

    Energy Technology Data Exchange (ETDEWEB)

    Priimak, Dmitri

    2014-12-01

    We present a finite difference numerical algorithm for solving two dimensional spatially homogeneous Boltzmann transport equation which describes electron transport in a semiconductor superlattice subject to crossed time dependent electric and constant magnetic fields. The algorithm is implemented both in C language targeted to CPU and in CUDA C language targeted to commodity NVidia GPU. We compare performances and merits of one implementation versus another and discuss various software optimisation techniques.

  5. Efficient method for the solution of the energy dependent integral Boltzmann transport equation in the resolved resonance energy region

    International Nuclear Information System (INIS)

    Schwenk, G.A. Jr.

    1980-01-01

    The calculation of neutron-nuclei reaction rates in the lower resolved resonance region (167 eV - 1.855 eV) is considered in this dissertation. Particular emphasis is placed on the calculation of these reaction rates for tight lattices where their accuracy is most important. The results of the continuous energy Monte Carlo code, VIM, are chosen as reference values for this study. The primary objective of this work is to develop a method for calculating resonance reaction rates which agree well with the reference solution, yet is efficient enough to be used by nuclear reactor fuel cycle designers on a production basis. A very efficient multigroup solution of the two spatial region energy dependent integral transport equation is developed. This solution, denoted the Broad Group Integral Method (BGIM), uses escape probabilities to obtain the spatial coupling between regions and uses an analytical flux shape within a multigroup to obtain weighted cross sections which account for the rapidly varying resonance cross sections. The multigroup lethargy widths chosen for the numerical integration of the two region energy-dependent neutron continuity equations can be chosen much wider (a factor of 30 larger) than in the direct numerical integration methods since the analytical flux shape is used to account for fine structure effects. The BGIM solution is made highly efficient through the use of these broad groups. It is estimated that for a 10 step unit cell fuel cycle depletion calculation, the computer running time for a production code such as EPRI-LEOPARD would be increased by only 6% through the use of the more accurate and intricate BGIM method in the lower resonance energy region

  6. Solution of the neutron transport equation by the collision probability for 3D geometries; Resolution de l`equation du transport pour les neutrons par la methode des probabilites de collision dans le geometries 3D

    Energy Technology Data Exchange (ETDEWEB)

    Oujidi, B.

    1996-09-19

    The TDT code solves the multigroup transport equation by the interface current method for unstructured 2D geometries. This works presents the extension of TDT to the treatment of 3D geometries obtained by axial displacement of unstructured 2D geometries. Three-dimensional trajectories are obtained by lifting the 2D trajectories. The code allows for the definition of macro-domains in the axial direction to be used in the interface-current method. Specular and isotropic reflection or translations boundary conditions can be applied to the horizontal boundaries of the domain. Numerical studies have shown the need for longer trajectory cutoffs for trajectories intersecting horizontal boundaries. Numerical applications to the calculation of local power peaks are given in a second part for: the local destruction of a Pyrex absorbent and inter-assembly (UO{sub 2}-MOX) power distortion due to pellet collapsing at the top of the core. Calculations with 16 groups were performed by coupling TDT to the spectral code APOLLO2. One-group comparisons with the Monte Carlo code TRIMARAN2 are also given. (author). 30 refs.

  7. Solution of the neutron transport equation by the collision probability method for 3D geometries; Resolution de l`equation du transport par les neutrons par la methode des probabilites de collision dans les geometries 3D

    Energy Technology Data Exchange (ETDEWEB)

    Oujidi, B

    1996-09-19

    The TDT code solves the multigroup transport equation by the interface-current method for unstructured 2D geometries. This works presents the extension of TDT to the treatment of 3D geometries obtained by axial displacement of unstructured 2D geometries. Three-dimensional trajectories are obtained by lifting the 2D trajectories. The code allows for the definition of macro-domains in the axial direction to be used in interface-current method. Specular and isotropic reflection or translations boundary conditions can be applied to the horizontal boundaries of the domain. Numerical studies have shown the need for longer trajectory cutoffs for trajectories intersecting horizontal boundaries. Numerical applications to the calculation of local power peaks are given in a second part for: the local destruction of a Pyrex absorbent, inter-assembly (U02-MOX) power distortion due to pellet collapsing at the top of the core. Calculations with 16 groups were performed by coupling TDT to the spectral code APOLLO2. One-group comparisons with the Monte Carlo code TRIMARAN2 are also given. (author) 30 refs.

  8. Analysis and development of spatial hp-refinement methods for solving the neutron transport equation; Analyse et developpement de methodes de raffinement hp en espace pour l'equation de transport des neutrons

    Energy Technology Data Exchange (ETDEWEB)

    Fournier, D.

    2011-10-10

    The different neutronic parameters have to be calculated with a higher accuracy in order to design the 4. generation reactor cores. As memory storage and computation time are limited, adaptive methods are a solution to solve the neutron transport equation. The neutronic flux, solution of this equation, depends on the energy, angle and space. The different variables are successively discretized. The energy with a multigroup approach, considering the different quantities to be constant on each group, the angle by a collocation method called SN approximation. Once the energy and angle variable are discretized, a system of spatially-dependent hyperbolic equations has to be solved. Discontinuous finite elements are used to make possible the development of hp-refinement methods. Thus, the accuracy of the solution can be improved by spatial refinement (h-refinement), consisting into subdividing a cell into sub-cells, or by order refinement (p-refinement), by increasing the order of the polynomial basis. In this thesis, the properties of this methods are analyzed showing the importance of the regularity of the solution to choose the type of refinement. Thus, two error estimators are used to lead the refinement process. Whereas the first one requires high regularity hypothesis (analytical solution), the second one supposes only the minimal hypothesis required for the solution to exist. The comparison of both estimators is done on benchmarks where the analytic solution is known by the method of manufactured solutions. Thus, the behaviour of the solution as a regard of the regularity can be studied. It leads to a hp-refinement method using the two estimators. Then, a comparison is done with other existing methods on simplified but also realistic benchmarks coming from nuclear cores. These adaptive methods considerably reduces the computational cost and memory footprint. To further improve these two points, an approach with energy-dependent meshes is proposed. Actually, as the

  9. Uncertainty analysis of an interfacial area reconstruction algorithm and its application to two group interfacial area transport equation validation

    International Nuclear Information System (INIS)

    Dave, A.J.; Manera, A.; Beyer, M.; Lucas, D.; Prasser, H.-M.

    2016-01-01

    Wire mesh sensors (WMS) are state of the art devices that allow high resolution (in space and time) measurement of 2D void fraction distribution over a wide range of two-phase flow regimes, from bubbly to annular. Data using WMS have been recorded at the Helmholtz-Zentrum Dresden-Rossendorf (HZDR) (Lucas et al., 2010; Beyer et al., 2008; Prasser et al., 2003) for a wide combination of superficial gas and liquid velocities, providing an excellent database for advances in two-phase flow modeling. In two-phase flow, the interfacial area plays an integral role in coupling the mass, momentum and energy transport equations of the liquid and gas phase. While current models used in best-estimate thermal-hydraulic codes (e.g. RELAP5, TRACE, TRACG, etc.) are still based on algebraic correlations for the estimation of the interfacial area in different flow regimes, interfacial area transport equations (IATE) have been proposed to predict the dynamic propagation in space and time of interfacial area (Ishii and Hibiki, 2010). IATE models are still under development and the HZDR WMS experimental data provide an excellent basis for the validation and further advance of these models. The current paper is focused on the uncertainty analysis of algorithms used to reconstruct interfacial area densities from the void-fraction voxel data measured using WMS and their application towards validation efforts of two-group IATE models. In previous research efforts, a surface triangularization algorithm has been developed in order to estimate the surface area of individual bubbles recorded with the WMS, and estimate the interfacial area in the given flow condition. In the present paper, synthetically generated bubbles are used to assess the algorithm’s accuracy. As the interfacial area of the synthetic bubbles are defined by user inputs, the error introduced by the algorithm can be quantitatively obtained. The accuracy of interfacial area measurements is characterized for different bubbles

  10. Uncertainty analysis of an interfacial area reconstruction algorithm and its application to two group interfacial area transport equation validation

    Energy Technology Data Exchange (ETDEWEB)

    Dave, A.J., E-mail: akshayjd@umich.edu [Department of Nuclear Engineering and Rad. Sciences, University of Michigan, Ann Arbor, MI 48105 (United States); Manera, A. [Department of Nuclear Engineering and Rad. Sciences, University of Michigan, Ann Arbor, MI 48105 (United States); Beyer, M.; Lucas, D. [Helmholtz-Zentrum Dresden-Rossendorf, Institute of Fluid Dynamics, 01314 Dresden (Germany); Prasser, H.-M. [Department of Mechanical and Process Engineering, ETH Zurich, 8092 Zurich (Switzerland)

    2016-12-15

    Wire mesh sensors (WMS) are state of the art devices that allow high resolution (in space and time) measurement of 2D void fraction distribution over a wide range of two-phase flow regimes, from bubbly to annular. Data using WMS have been recorded at the Helmholtz-Zentrum Dresden-Rossendorf (HZDR) (Lucas et al., 2010; Beyer et al., 2008; Prasser et al., 2003) for a wide combination of superficial gas and liquid velocities, providing an excellent database for advances in two-phase flow modeling. In two-phase flow, the interfacial area plays an integral role in coupling the mass, momentum and energy transport equations of the liquid and gas phase. While current models used in best-estimate thermal-hydraulic codes (e.g. RELAP5, TRACE, TRACG, etc.) are still based on algebraic correlations for the estimation of the interfacial area in different flow regimes, interfacial area transport equations (IATE) have been proposed to predict the dynamic propagation in space and time of interfacial area (Ishii and Hibiki, 2010). IATE models are still under development and the HZDR WMS experimental data provide an excellent basis for the validation and further advance of these models. The current paper is focused on the uncertainty analysis of algorithms used to reconstruct interfacial area densities from the void-fraction voxel data measured using WMS and their application towards validation efforts of two-group IATE models. In previous research efforts, a surface triangularization algorithm has been developed in order to estimate the surface area of individual bubbles recorded with the WMS, and estimate the interfacial area in the given flow condition. In the present paper, synthetically generated bubbles are used to assess the algorithm’s accuracy. As the interfacial area of the synthetic bubbles are defined by user inputs, the error introduced by the algorithm can be quantitatively obtained. The accuracy of interfacial area measurements is characterized for different bubbles

  11. Models for microtubule cargo transport coupling the Langevin equation to stochastic stepping motor dynamics: Caring about fluctuations.

    Science.gov (United States)

    Bouzat, Sebastián

    2016-01-01

    One-dimensional models coupling a Langevin equation for the cargo position to stochastic stepping dynamics for the motors constitute a relevant framework for analyzing multiple-motor microtubule transport. In this work we explore the consistence of these models focusing on the effects of the thermal noise. We study how to define consistent stepping and detachment rates for the motors as functions of the local forces acting on them in such a way that the cargo velocity and run-time match previously specified functions of the external load, which are set on the base of experimental results. We show that due to the influence of the thermal fluctuations this is not a trivial problem, even for the single-motor case. As a solution, we propose a motor stepping dynamics which considers memory on the motor force. This model leads to better results for single-motor transport than the approaches previously considered in the literature. Moreover, it gives a much better prediction for the stall force of the two-motor case, highly compatible with the experimental findings. We also analyze the fast fluctuations of the cargo position and the influence of the viscosity, comparing the proposed model to the standard one, and we show how the differences on the single-motor dynamics propagate to the multiple motor situations. Finally, we find that the one-dimensional character of the models impede an appropriate description of the fast fluctuations of the cargo position at small loads. We show how this problem can be solved by considering two-dimensional models.

  12. Development of a discrete-ordinate approximation of the neutron transport equation for coupled xy-R-geometry

    International Nuclear Information System (INIS)

    Maertens, H.D.

    1982-01-01

    The inhomogenious structure of modern heavy water reactor fuel elements result in a strong spacial dependence of the neutron flux. The flux distribution can be calculated in detail by numerical methods, which describe exactly the geometrical heterogeniety and take into account the neutron flux anisotropy by higher transport theoretical approximations. Starting from the discrete ordinate method an approximation of the neutron transport equation has been developed, allowing for a cylindrical representation of the fuel-elements in a rectangular array of rods. The discretisation of the space variables, is based on the finite-difference approximation, defining a rectangular lattice in a two-dimensional cartesian coordinate system, which can be cut and replaced by circular mesh elements of a partially one-dimensional cylindrical coordinate system at arbitrary space points. To couple the two spacial regions the outer circle line of a cylindrical geometry is approximated in the cartesian system by a polygon with n >= 8. A cylindrical geometry is approximated in the cartesian system by a polygon with n>=8. A cylindrical geometry is thus enclosed by a system of two-dimensional rectangular, triangular and trapezoid mesh elements. The directional distribution of the neutron flux is conserved when switching from the xy-system to the cylindrical coordinate system. The angle discretisation by balanced sets of squares (EQsub(n)) allows a simple definition of transfer-coefficients for the redistribution of the neutron flux due to coordinate transformations. The procedure is verified and tested by selected problems. Possible applications and limits are discussed. (orig.) [de

  13. Solution of the transport equation in stationary state and X Y geometry, using continuous and discontinuous hybrid nodal schemes

    International Nuclear Information System (INIS)

    Xolocostli M, V.; Valle G, E. del; Alonso V, G.

    2003-01-01

    In this work it is described the development and the application of the NH-FEM schemes, Hybrid Nodal schemes using the Finite Element method in the solution of the neutron transport equation in stationary state and X Y geometry, of which two families of schemes were developed, one of which corresponds to the continuous and the other to the discontinuous ones, inside those first its are had the Bi-Quadratic Bi Q, and to the Bi-cubic BiC, while for the seconds the Discontinuous Bi-lineal DBiL and the Discontinuous Bi-quadratic DBiQ. These schemes were implemented in a program to which was denominated TNHXY, Transport of neutrons with Hybrid Nodal schemes in X Y geometry. One of the immediate applications of the schemes NH-FEM it will be in the analysis of assemblies of nuclear fuel, particularly of the BWR type. The validation of the TNHXY program was made with two test problems or benchmark, already solved by other authors with numerical techniques and to compare results. The first of them consists in an it BWR fuel assemble in an arrangement 7x7 without rod and with control rod providing numerical results. The second is a fuel assemble of mixed oxides (MOX) in an arrangement 10x10. This last problem it is known as the Benchmark problem WPPR of the NEA Data Bank and the results are compared with those of other commercial codes as HELIOS, MCNP-4B and CPM-3. (Author)

  14. Laser Radiation Pressure Accelerator for Quasi-Monoenergetic Proton Generation and Its Medical Implications

    Science.gov (United States)

    Liu, C. S.; Shao, X.; Liu, T. C.; Su, J. J.; He, M. Q.; Eliasson, B.; Tripathi, V. K.; Dudnikova, G.; Sagdeev, R. Z.; Wilks, S.; Chen, C. D.; Sheng, Z. M.

    Laser radiation pressure acceleration (RPA) of ultrathin foils of subwavelength thickness provides an efficient means of quasi-monoenergetic proton generation. With an optimal foil thickness, the ponderomotive force of the intense short-pulse laser beam pushes the electrons to the edge of the foil, while balancing the electric field due to charge separation. The electron and proton layers form a self-organized plasma double layer and are accelerated by the radiation pressure of the laser, the so-called light sail. However, the Rayleigh-Taylor instability can limit the acceleration and broaden the energy of the proton beam. Two-dimensional particle-in-cell (PIC) simulations have shown that the formation of finger-like structures due to the nonlinear evolution of the Rayleigh-Taylor instability limits the acceleration and leads to a leakage of radiation through the target by self-induced transparency. We here review the physics of quasi-monoenergetic proton generation by RPA and recent advances in the studies of energy scaling of RPA, and discuss the RPA of multi-ion and gas targets. The scheme for generating quasi-monoenergetic protons with RPA has the potential of leading to table-top accelerators as sources for producing monoenergetic 50-250 MeV protons. We also discuss potential medical implications, such as particle therapy for cancer treatment, using quasi-monoenergetic proton beams generated from RPA. Compact monoenergetic ion sources also have applications in many other areas such as high-energy particle physics, space electronics radiation testing, and fast ignition in laser fusion.

  15. Dose point kernel simulation for monoenergetic electrons and radionuclides using Monte Carlo techniques.

    Science.gov (United States)

    Wu, J; Liu, Y L; Chang, S J; Chao, M M; Tsai, S Y; Huang, D E

    2012-11-01

    Monte Carlo (MC) simulation has been commonly used in the dose evaluation of radiation accidents and for medical purposes. The accuracy of simulated results is affected by the particle-tracking algorithm, cross-sectional database, random number generator and statistical error. The differences among MC simulation software packages must be validated. This study simulated the dose point kernel (DPK) and the cellular S-values of monoenergetic electrons ranging from 0.01 to 2 MeV and the radionuclides of (90)Y, (177)Lu and (103 m)Rh, using Fluktuierende Kaskade (FLUKA) and MC N-Particle Transport Code Version 5 (MCNP5). A 6-μm-radius cell model consisting of the cell surface, cytoplasm and cell nucleus was constructed for cellular S-value calculation. The mean absolute percentage errors (MAPEs) of the scaled DPKs, simulated using FLUKA and MCNP5, were 7.92, 9.64, 4.62, 3.71 and 3.84 % for 0.01, 0.1, 0.5, 1 and 2 MeV, respectively. For the three radionuclides, the MAPEs of the scaled DPKs were within 5 %. The maximum deviations of S(N←N), S(N←Cy) and S(N←CS) for the electron energy larger than 10 keV were 6.63, 6.77 and 5.24 %, respectively. The deviations for the self-absorbed S-values and cross-dose S-values of the three radionuclides were within 4 %. On the basis of the results of this study, it was concluded that the simulation results are consistent between FLUKA and MCNP5. However, there is a minor inconsistency for low energy range. The DPK and the cellular S-value should be used as the quality assurance tools before the MC simulation results are adopted as the gold standard.

  16. Solutions of transport equation in (X-Y-Z) three-dimensional geometry by finite element method and spherical harmonic expansion

    International Nuclear Information System (INIS)

    Fernandes, A.; Maiorino, J.R.

    1989-01-01

    This work presents a method to solve the neutron transport equation in thre space dimensions. The angular flux is aproximated by spherical harmonics and the finite element method is applied to the space component. The program originated by the analytical development is being tested and some results are presented. (author) [pt

  17. Construction of the Cauchy problem for solution of the integral equation of transport in a sphere with a central absorbing region

    International Nuclear Information System (INIS)

    Ezhov, A.A.

    1978-01-01

    On the basis of the integral equation for neutron transport in a homogeneous isotropically-scattering sphere with an absolutely black central part an initial value problem has been formulated which permits the construction of a numerical scheme to find the neutron flux density

  18. A non-conforming generalization of Raviart-Thomas elements to the spherical harmonic form of the even-parity neutron transport equation

    Energy Technology Data Exchange (ETDEWEB)

    Van Criekingen, S. [Commissariat a l' energie atomique (CEA-Saclay), DEN/DM2S/SERMA/LENR (Bat 470), 91191 Gif-sur-Yvette Cedex (France)]. E-mail: serge.van-criekingen@cea.fr

    2006-05-15

    The Raviart-Thomas finite elements provide an appropriate spatial discretization of the mixed-dual form of the diffusion equation. This discretization can then be coupled to an efficient solution method. The high performances achieved by such an approach triggered research on its possible generalization to the transport equation using a spherical harmonic (or P {sub N}) angular approximation. In view of the difficulty of developing a straightforward generalization within the mixed-dual framework, we here consider 2D non-conforming (i.e., allowing interface discontinuities) finite elements coupled to the second-order form of the transport equation. This non-conforming approach keeps the mixed-dual property of the relaxation of the flux interface continuity constraint. We investigate different non-conforming elements and compare them to the well-known Lagrangian conforming elements.

  19. A non-conforming generalization of Raviart-Thomas elements to the spherical harmonic form of the even-parity neutron transport equation

    International Nuclear Information System (INIS)

    Van Criekingen, S.

    2006-01-01

    The Raviart-Thomas finite elements provide an appropriate spatial discretization of the mixed-dual form of the diffusion equation. This discretization can then be coupled to an efficient solution method. The high performances achieved by such an approach triggered research on its possible generalization to the transport equation using a spherical harmonic (or P N ) angular approximation. In view of the difficulty of developing a straightforward generalization within the mixed-dual framework, we here consider 2D non-conforming (i.e., allowing interface discontinuities) finite elements coupled to the second-order form of the transport equation. This non-conforming approach keeps the mixed-dual property of the relaxation of the flux interface continuity constraint. We investigate different non-conforming elements and compare them to the well-known Lagrangian conforming elements

  20. SEE cross section calibration and application to quasi-monoenergetic and spallation facilities

    Directory of Open Access Journals (Sweden)

    Alía Rubén García

    2017-01-01

    Full Text Available We describe an approach to calibrate SEE-based detectors in monoenergetic fields and apply the resulting semi-empiric responses to more general mixed-field cases in which a broad variety of particle species and energy spectra are involved. The calibration of the response functions is based both on experimental proton and neutron data and considerations derived from Monte Carlo simulations using the FLUKA code. The application environments include the quasi-monoenergetic neutrons at RCNP, the atmospheric-like VESUVIO spallation spectrum and the CHARM high-energy accelerator test facility.

  1. Comparison of virtual monoenergetic and polyenergetic images reconstructed from dual-layer detector CT angiography of the head and neck.

    Science.gov (United States)

    Neuhaus, Victor; Große Hokamp, Nils; Abdullayev, Nuran; Maus, Volker; Kabbasch, Christoph; Mpotsaris, Anastasios; Maintz, David; Borggrefe, Jan

    2018-03-01

    To compare the image quality of virtual monoenergetic images and polyenergetic images reconstructed from dual-layer detector CT angiography (DLCTA). Thirty patients who underwent DLCTA of the head and neck were retrospectively identified and polyenergetic as well as virtual monoenergetic images (40 to 120 keV) were reconstructed. Signals (± SD) of the cervical and cerebral vessels as well as lateral pterygoid muscle and the air surrounding the head were measured to calculate the CNR and SNR. In addition, subjective image quality was assessed using a 5-point Likert scale. Student's t-test and Wilcoxon test were used to determine statistical significance. Compared to polyenergetic images, although noise increased with lower keV, CNR (p 0.05) of the cervical, petrous and intracranial vessels were improved in virtual monoenergetic images at 40 keV and virtual monoenergetic images at 45 keV were also rated superior regarding vascular contrast, assessment of arteries close to the skull base and small arterial branches (p virtual monoenergetic images reconstructed from DLCTA at low keV ranging from 40 to 45 keV improve the objective and subjective image quality of extra- and intracranial vessels and facilitate assessment of vessels close to the skull base and of small arterial branches. • Virtual monoenergetic images greatly improve attenuation, while noise only slightly increases. • Virtual monoenergetic images show superior contrast-to-noise ratios compared to polyenergetic images. • Virtual monoenergetic images significantly improve image quality at low keV.

  2. Boundary-artifact-free phase retrieval with the transport of intensity equation II: applications to microlens characterization.

    Science.gov (United States)

    Zuo, Chao; Chen, Qian; Li, Hongru; Qu, Weijuan; Asundi, Anand

    2014-07-28

    Boundary conditions play a crucial role in the solution of the transport of intensity equation (TIE). If not appropriately handled, they can create significant boundary artifacts across the reconstruction result. In a previous paper [Opt. Express 22, 9220 (2014)], we presented a new boundary-artifact-free TIE phase retrieval method with use of discrete cosine transform (DCT). Here we report its experimental investigations with applications to the micro-optics characterization. The experimental setup is based on a tunable lens based 4f system attached to a non-modified inverted bright-field microscope. We establish inhomogeneous Neumann boundary values by placing a rectangular aperture in the intermediate image plane of the microscope. Then the boundary values are applied to solve the TIE with our DCT-based TIE solver. Experimental results on microlenses highlight the importance of boundary conditions that often overlooked in simplified models, and confirm that our approach effectively avoid the boundary error even when objects are located at the image borders. It is further demonstrated that our technique is non-interferometric, accurate, fast, full-field, and flexible, rendering it a promising metrological tool for the micro-optics inspection.

  3. Asymmetric modes and complex time eigenvalues of the one-speed neutron transport equation in a homogeneous sphere

    International Nuclear Information System (INIS)

    Paranjape, S.D.; Kumar, V.; Sahni, D.C.

    1993-01-01

    The one-speed, time-dependent, isotropically scattering, integral transport equation in a homogeneous sphere has been converted into a criticality-like problem by considering exponential time behaviour of the scalar flux. This criticality problem has been converted into a matrix eigenvalue problem using the Fourier transform technique. The time eigenvalues λ, which are complex in general, have been determined for spherically symmetric as well as asymmetric modes. For the former case, the real decay constants and the real parts of complex decay constants decrease monotonically with increasing system size and form two distinct families of single-valued functions. For the spherically asymmetric modes, certain new features emerge. The real decay constants are found to be multi-valued functions of system size and they do not always decrease monotonically with increasing system size. As the system size increases from zero onwards, the decay constants alternate between complex and real values and the real and complex decay constant curves interlace. (Author)

  4. Out of equilibrium transport through an Anderson impurity: probing scaling laws within the equation of motion approach.

    Science.gov (United States)

    Balseiro, C A; Usaj, G; Sánchez, M J

    2010-10-27

    We study non-equilibrium electron transport through a quantum impurity coupled to metallic leads using the equation of motion technique at finite temperature T. Assuming that the interactions are taking place solely in the impurity and focusing on the infinite Hubbard limit, we compute the out of equilibrium density of states and the differential conductance G(2)(T, V) in order to test several scaling laws. We find that G(2)(T, V)/G(2)(T, 0) is a universal function of both eV/T(K) and T/T(K), T(K) being the Kondo temperature. The effect of an in-plane magnetic field on the splitting of the zero bias anomaly in the differential conductance is also analyzed. For a Zeeman splitting Δ, the computed differential conductance peak splitting depends only on Δ/T(K), and for large fields approaches the value of 2Δ. Besides studying the traditional two leads setup, we also consider other configurations that mimic recent experiments, namely, an impurity embedded in a mesoscopic wire and the presence of a third weakly coupled lead. In these cases, a double peak structure of the Kondo resonance is clearly obtained in the differential conductance while the amplitude of the highest peak is shown to decrease as ln(eV/T(K)). Several features of these results are in qualitative agreement with recent experimental observations reported on quantum dots.

  5. MPN-1 : A computation module for the solution of transport equation in multiregions by P sub(N) method

    International Nuclear Information System (INIS)

    Yamaguchi, M.; Maiorino, J.R.

    1981-11-01

    The MPN-1, which is a computer program made in FORTRAN IV is described. The MPN-1 uses P sub(N) method to solve the one-speed, anisotropic linear steady-state transport equation in non-multiplying multiregions, slab geometry. The MPN-1 allows a maximum number of 12 regions, and maximum order of approximation, N=15. Besides, the anisotropic scattering is treated by an expansion in Legendre polynomial up to maximum degree of L=15. The boundary conditions used are: I) free surface; II) isotropic incidence; III) cosine incidence (aμ sup(α1 + bμ sup(α2) + cμ sup(α3)); IV) monodirectional incidence (delta function); V) total reflection. The required input are: order of approximation P sub(N'), average number of secondary particles in each region, external sources, thickness of each region, coefficients in a Legendre expansion of the transfer function in each region, and type of boundary conditions at the boundaries. The output are: albedo, transmission factor, disadvantage factor, total and angular fluxes and current. (Author) [pt

  6. Analysis and development of a numerical discretization scheme of the neutron transport equation in three-dimensional geometries

    International Nuclear Information System (INIS)

    Naymeh, L.

    2013-01-01

    The method of characteristics is a flexible and efficient method solving the transport equation. It has been largely used in two dimension calculations because it enables to study complex geometries and it has a good time/precision ratio. However, despite a great improvement in storage capacities and computing power, a direct three dimension calculation is still unreachable. In the following work, we introduce and analyze several modifications of the method of characteristics (MOC) in order to reduce the memory usage as well as calculation burden. This document aims at studying a higher order spatial approximation for the flux. It steps away from the classical method (constant MOC) by introducing an increase of details of the representation of the flux, which may enable to reduce the size of the grid while keeping a good precision. Numerical results tested on benchmarks show an improvement of time/precision ratio. Regarding the memory storage, the number of trajectories has an influence on the amount of data to be stored. Hence, we study a tracking method based on local tracks defined for all sub-domains having the same geometry. Redundancies happening in a reactor core suggest an important reduction of required memory. Two tracking methods have been studied, the first one being a non-uniform tracking method including sub-domain discontinuities and the other being a method based on periodic and continuous trajectories for a sub-domain to another. (author) [fr

  7. Time-dependent integral equations of neutron transport for calculating the kinetics of nuclear reactors by the Monte Carlo method

    Energy Technology Data Exchange (ETDEWEB)

    Davidenko, V. D., E-mail: Davidenko-VD@nrcki.ru; Zinchenko, A. S., E-mail: zin-sn@mail.ru; Harchenko, I. K. [National Research Centre Kurchatov Institute (Russian Federation)

    2016-12-15

    Integral equations for the shape functions in the adiabatic, quasi-static, and improved quasi-static approximations are presented. The approach to solving these equations by the Monte Carlo method is described.

  8. Rotina computacional e equação simplificada para modelar o transporte de sedimentos num Latossolo Vermelho Distrófico Computational routine and simplified equation for modeling sediment transport capacity in a Dystrophic Hapludox

    Directory of Open Access Journals (Sweden)

    Gilmar E. Cerquetani

    2006-08-01

    Full Text Available Os objetivos do presente trabalho foram desenvolver rotina computacional para a solução da equação de Yalin e do diagrama de Shields e avaliar uma equação simplificada para modelar a capacidade de transporte de sedimento num Latossolo Vermelho Distrófico que possa ser utilizada no Water Erosion Prediction Project - WEPP, assim como em outros modelos de predição da erosão do solo. A capacidade de transporte de sedimento para o fluxo superficial foi representada como função-potência da tensão cisalhante, a qual revelou ser aproximação da equação de Yalin. Essa equação simplificada pôde ser aplicada em resultados experimentais oriundos de topografia complexa. A equação simplificada demonstrou acuracidade em relação à equação de Yalin, quando calibrada utilizando-se da tensão média cisalhante. Testes de validação com dados independentes demonstraram que a equação simplificada foi eficiente para estimar a capacidade de transporte de sedimento.The objectives of the present work were to develop a computational routine to solve Yalin equation and Shield diagram and to evaluate a simplified equation for modeling sediment transport capacity in a Dystrophic Hapludox that could be used in the Water Erosion Prediction Project - WEPP, as well as other soil erosion models. Sediment transport capacity for shallow overland flow was represented as a power function of the hydraulic shear stress and which showed to be an approximation to the Yalin equation for sediment transport capacity. The simplified equation for sediment transport could be applied to experimental data from a complex topography. The simplified equation accurately approximated the Yalin equation when calibrated using the mean hydraulic shear stress. Validation tests using independent data showed that the simplified equation had a good performance in predicting sediment transport capacity.

  9. A comparison of numerical solutions of partial differential equations with probabilistic and possibilistic parameters for the quantification of uncertainty in subsurface solute transport.

    Science.gov (United States)

    Zhang, Kejiang; Achari, Gopal; Li, Hua

    2009-11-03

    Traditionally, uncertainty in parameters are represented as probabilistic distributions and incorporated into groundwater flow and contaminant transport models. With the advent of newer uncertainty theories, it is now understood that stochastic methods cannot properly represent non random uncertainties. In the groundwater flow and contaminant transport equations, uncertainty in some parameters may be random, whereas those of others may be non random. The objective of this paper is to develop a fuzzy-stochastic partial differential equation (FSPDE) model to simulate conditions where both random and non random uncertainties are involved in groundwater flow and solute transport. Three potential solution techniques namely, (a) transforming a probability distribution to a possibility distribution (Method I) then a FSPDE becomes a fuzzy partial differential equation (FPDE), (b) transforming a possibility distribution to a probability distribution (Method II) and then a FSPDE becomes a stochastic partial differential equation (SPDE), and (c) the combination of Monte Carlo methods and FPDE solution techniques (Method III) are proposed and compared. The effects of these three methods on the predictive results are investigated by using two case studies. The results show that the predictions obtained from Method II is a specific case of that got from Method I. When an exact probabilistic result is needed, Method II is suggested. As the loss or gain of information during a probability-possibility (or vice versa) transformation cannot be quantified, their influences on the predictive results is not known. Thus, Method III should probably be preferred for risk assessments.

  10. Development of a monoenergetic neutron beam (Theoretical aspects, experimental developments and applications)

    International Nuclear Information System (INIS)

    Varela G, A.

    2003-01-01

    By the use of a neutron time of flight system at the Tandem Accelerator of the National Nuclear Research Institute; with neutrons provided by means of the 2 H(d, n) 3 He we intend to use the associated particle technique in order to have monoenergetic neutrons. This neutron beam will be used both in basic and applied research. (Author)

  11. Monoenergetic neutron fields for the calibration of neutron dosemeters at the accelerator facility of the PTB

    International Nuclear Information System (INIS)

    Lesiecki, H.; Cosack, M.; Schoelermann, H.

    1987-01-01

    The present state in the realization of monoenergetic standard neutron fields and the possibility of calibrating neutron dose- and doserate meters at the accelerator facility of the PTB are described. There are excellent conditions for the performance of irradiations in the neutron energy range of 1 keV to 14.8 MeV. (orig.) [de

  12. 8-group relative delayed neutron yields for monoenergetic neutron induced fission of 239Pu

    International Nuclear Information System (INIS)

    Piksaikin, V.M.; Kazakov, L.E.; Isaev, S.G.; Korolev, G.G.; Roshchenko, V.A.; Tertychnyj, R.G

    2002-01-01

    The energy dependence of the relative yield of delayed neutrons in an 8-group model representation was obtained for monoenergetic neutron induced fission of 239 Pu. A comparison of this data with the available experimental data by other authors was made in terms of the mean half-life of the delayed neutron precursors. (author)

  13. Advanced virtual monoenergetic images: improving the contrast of dual-energy CT pulmonary angiography

    International Nuclear Information System (INIS)

    Meier, A.; Wurnig, M.; Desbiolles, L.; Leschka, S.; Frauenfelder, T.; Alkadhi, H.

    2015-01-01

    Aim: To investigate the value of advanced virtual monoenergetic image reconstruction (mono-plus) from dual-energy computed tomography (CT) for improving the contrast of CT pulmonary angiography (CTPA). Materials and methods: Forty consecutive patients (25 women, mean 62.5 years, range 28–87 years) underwent 192-section dual-source CTPA with dual-energy CT (90/150 SnkVp) after the administration of 60 ml contrast media (300 mg iodine/ml). Conventional virtual monochromatic images at 60 keV and 17 mono-plus image datasets from 40–190 keV (in 10 keV steps) were reconstructed. Subjective image quality (artefacts, subjective noise) was rated. Attenuation was measured in the pulmonary trunk and in the right lower lobe pulmonary artery; noise was measured in the periscapular musculature. The signal-to-noise (SNR) and contrast-to-noise ratios (CNR) were calculated for each patient and dataset. Comparisons between monochromatic images and mono-plus images were performed by repeated measures analysis of variance (ANOVA) with post-hoc Bonferroni correction. Results: Interreader agreement was good to excellent for subjective image quality (ICC: 0.616–0.889). As compared to conventional 60 keV images, artefacts occurred less (p=0.001) and subjective noise was rated lower (p<0.001) in mono-plus 40 keV images. Noise was lower (p<0.001), and the SNR and CNR in the pulmonary trunk and right lower lobe pulmonary artery were higher (both, p<0.001) in mono-plus 40 keV images compared to conventional monoenergetic 60 keV images. Transient interruption of contrast (TIC) was found in 14/40 (35%) of patients, with subjective contrast being similar 8/40 (20%) or higher 32/40 (80%) in mono-plus 40 keV as compared to conventional monoenergetic 60 keV images. Conclusions: Compared to conventional virtual monoenergetic imaging, mono-plus images at 40 keV improve the contrast of dual-energy CTPA. - Highlights: • Advanced monoenergetic image reconstruction from dual-energy CT

  14. TIMEX: a time-dependent explicit discrete ordinates program for the solution of multigroup transport equations with delayed neutrons

    International Nuclear Information System (INIS)

    Hill, T.R.; Reed, W.H.

    1976-01-01

    TIMEX solves the time-dependent, one-dimensional multigroup transport equation with delayed neutrons in plane, cylindrical, spherical, and two-angle plane geometries. Both regular and adjoint, inhomogeneous and homogeneous problems subject to vacuum, reflective, periodic, white, albedo or inhomogeneous boundary flux conditions are solved. General anisotropic scattering is allowed and anisotropic inhomogeneous sources are permitted. The discrete ordinates approximation for the angular variable is used with the diamond (central) difference approximation for the angular extrapolation in curved geometries. A linear discontinuous finite element representation for the angular flux in each spatial mesh cell is used. The time variable is differenced by an explicit technique that is unconditionally stable so that arbitrarily large time steps can be taken. Because no iteration is performed the method is exceptionally fast in terms of computing time per time step. Two acceleration methods, exponential extrapolation and rebalance, are utilized to improve the accuracy of the time differencing scheme. Variable dimensioning is used so that any combination of problem parameters leading to a container array less than MAXCOR can be accommodated. The running time for TIMEX is highly problem-dependent, but varies almost linearly with the total number of unknowns and time steps. Provision is made for creation of standard interface output files for angular fluxes and angle-integrated fluxes. Five interface units (use of interface units is optional), five output units, and two system input/output units are required. A large bulk memory is desirable, but may be replaced by disk, drum, or tape storage. 13 tables, 9 figures

  15. Experimental evaluation of the ‘transport-of-intensity’ equation for magnetic phase reconstruction in Lorentz transmission electron microscopy

    Energy Technology Data Exchange (ETDEWEB)

    Kohn, Amit, E-mail: akohn@post.tau.ac.il [Department of Materials Science and Engineering, Faculty of Engineering, Tel Aviv University, 69978 Tel Aviv (Israel); Habibi, Avihay; Mayo, Martin [Department of Materials Engineering, Ben-Gurion University of the Negev, 84105 Beer Sheva (Israel)

    2016-01-15

    The ‘transport-of-intensity’ equation (TIE) is a general phase reconstruction methodology that can be applied to Lorentz transmission electron microscopy (TEM) through the use of Fresnel-contrast (defocused) images. We present an experimental study to test the application of the TIE for quantitative magnetic mapping in Lorentz TEM without aberration correction by examining sub-micrometer sized Ni{sub 80}Fe{sub 20} (Permalloy) elements. For a JEOL JEM 2100F adapted for Lorentz microscopy, we find that quantitative magnetic phase reconstructions are possible for defoci distances ranging between approximately 200 μm and 800 μm. The lower limit originates from competing sources of image intensity variations in Fresnel-contrast images, namely structural defects and diffraction contrast. The upper defocus limit is due to a numerical error in the estimation of the intensity derivative based on three images. For magnetic domains, we show quantitative reconstructions of the product of the magnetic induction vector and thickness in element sizes down to approximately 100 nm in lateral size and 5 nm thick resulting in a minimal detection of 5 T nm. Three types of magnetic structures are tested in terms of phase reconstruction: vortex cores, domain walls, and element edges. We quantify vortex core structures at a diameter of 12 nm while the structures of domain walls and element edges are characterized qualitatively. Finally, we show by image simulations that the conclusions of this experimental study are relevant to other Lorentz TEM in which spherical aberration and defocus are dominant aberrations. - Highlights: • Testing TIE for quantitative magnetic phase reconstruction in Lorentz TEM. • Quantitative magnetic phase reconstructions for defoci distances in 200–800 μm range. • Minimal detection of the product of the magnetic induction and thickness is 5 T nm. • Quantitative phase reconstruction for vortex core structures at 12 nm diameter. • Observations

  16. A closed-form solution for the two-dimensional Fokker-Planck equation for electron transport in the range of Compton Effect

    International Nuclear Information System (INIS)

    Rodriguez, B.D.A.; Vilhena, M.T.; Borges, V.; Hoff, G.

    2008-01-01

    In this paper we solve the Fokker-Planck (FP) equation, an alternative approach for the Boltzmann transport equation for charged particles in a rectangular domain. To construct the solution we begin applying the P N approximation in the angular variable and the Laplace Transform in the x-variable, thus obtaining a first order linear differential equation in y-variable, which the solution is straightforward. The angular flux of electrons and the parameters of the medium are used for the calculation of the energy deposited by the secondary electrons generated by Compton Effect. The remaining effects will not be taken into account. The results will be presented under absorbed energy form in several points of interested. We present numerical simulations and comparisons with results obtained by using Geant4 (version 8) program which applies the Monte Carlo's technique to low energy libraries for a two-dimensional problem assuming the screened Rutherford differential scattering cross-section

  17. A closed-form solution for the two-dimensional transport equation by the LTS{sub N} nodal method in the energy range of Compton effect

    Energy Technology Data Exchange (ETDEWEB)

    Rodriguez, B.D.A., E-mail: barbararodriguez@furg.b [Universidade Federal do Rio Grande, Instituto de Matematica, Estatistica e Fisica, Rio Grande, RS (Brazil); Vilhena, M.T., E-mail: vilhena@mat.ufrgs.b [Universidade Federal do Rio Grande do Sul, Departamento de Matematica Pura e Aplicada, Porto Alegre, RS (Brazil); Hoff, G., E-mail: hoff@pucrs.b [Pontificia Universidade Catolica do Rio Grande do Sul, Faculdade de Fisica, Porto Alegre, RS (Brazil); Bodmann, B.E.J., E-mail: bardo.bodmann@ufrgs.b [Universidade Federal do Rio Grande do Sul, Departamento de Matematica Pura e Aplicada, Porto Alegre, RS (Brazil)

    2011-01-15

    In the present work we report on a closed-form solution for the two-dimensional Compton transport equation by the LTS{sub N} nodal method in the energy range of Compton effect. The solution is determined using the LTS{sub N} nodal approach for homogeneous and heterogeneous rectangular domains, assuming the Klein-Nishina scattering kernel and a multi-group model. The solution is obtained by two one-dimensional S{sub N} equation systems resulting from integrating out one of the orthogonal variables of the S{sub N} equations in the rectangular domain. The leakage angular fluxes are approximated by exponential forms, which allows to determine a closed-form solution for the photons transport equation. The angular flux and the parameters of the medium are used for the calculation of the absorbed energy in rectangular domains with different dimensions and compositions. In this study, only the absorbed energy by Compton effect is considered. We present numerical simulations and comparisons with results obtained by using the simulation platform GEANT4 (version 9.1) with its low energy libraries.

  18. Solution of the two dimensional diffusion and transport equations in a rectangular lattice with an elliptical fuel element using Fourier transform methods: One and two group cases

    International Nuclear Information System (INIS)

    Williams, M.M.R.; Hall, S.K.; Eaton, M.D.

    2014-01-01

    Highlights: • A rectangular reactor cell with an elliptical fuel element. • Solution of transport and diffusion equations by Fourier expansion. • Numerical examples showing convergence. • Two group cell problems. - Abstract: A method for solving the diffusion and transport equations in a rectangular lattice cell with an elliptical fuel element has been developed using a Fourier expansion of the neutron flux. The method is applied to a one group model with a source in the moderator. The cell flux is obtained and also the associated disadvantage factor. In addition to the one speed case, we also consider the two group equations in the cell which now become an eigenvalue problem for the lattice multiplication factor. The method of solution relies upon an efficient procedure to solve a large set of simultaneous linear equations and for this we use the IMSL library routines. Our method is compared with the results from a finite element code. The main drawback of the problem arises from the very large number of terms required in the Fourier series which taxes the storage and speed of the computer. Nevertheless, useful solutions are obtained in geometries that would normally require the use of finite element or analogous methods, for this reason the Fourier method is useful for comparison with that type of numerical approach. Extension of the method to more intricate fuel shapes, such as stars and cruciforms as well as superpositions of these, is possible

  19. A non-local-thermodynamic equilibrium formulation of the transport equation for polarized light in the presence of weak magnetic fields. Doctoral thesis

    International Nuclear Information System (INIS)

    McNamara, D.J.

    1977-01-01

    The present work is motivated by the desire to better understand solar magnetism. Just as stellar astrophysics and radiative transfer have been coupled in the history of research in physics, so too has the study of radiative transfer of polarized light in magnetic fields and solar magnetism been a history of mutual growth. The Stokes parameters characterize the state of polarization of a beam of radiation. The author considers the changes in polarization, and therefore in the Stokes parameters, due to the transport of a beam through an optically thick medium in a weak magnetic field. The transport equation is derived from a general density matrix equation of motion. This allows the possibility of interference effects arising from the mixing of atomic sublevels in a weak magnetic field to be taken into account. The statistical equilibrium equations are similarly derived. Finally, the coupled system of equations is presented, and the order of magnitude of the interference effects, shown. Collisional effects are not considered. The magnitude of the interference effects in magnetic field measurements of the sun may be evaluated

  20. Cellular neural network to the spherical harmonics approximation of neutron transport equation in x-y geometry. Part I: Modeling and verification for time-independent solution

    International Nuclear Information System (INIS)

    Pirouzmand, Ahmad; Hadad, Kamal

    2011-01-01

    Highlights: → This paper describes the solution of time-independent neutron transport equation. → Using a novel method based on cellular neural networks (CNNs) coupled with P N method. → Utilize the CNN model to simulate spatial scalar flux distribution in steady state. → The accuracy, stability, and capabilities of CNN model are examined in x-y geometry. - Abstract: This paper describes a novel method based on using cellular neural networks (CNN) coupled with spherical harmonics method (P N ) to solve the time-independent neutron transport equation in x-y geometry. To achieve this, an equivalent electrical circuit based on second-order form of neutron transport equation and relevant boundary conditions is obtained using CNN method. We use the CNN model to simulate spatial response of scalar flux distribution in the steady state condition for different order of spherical harmonics approximations. The accuracy, stability, and capabilities of CNN model are examined in 2D Cartesian geometry for fixed source and criticality problems.

  1. Laser radiation in active amplifying media treated as a transport problem - Transfer equation derived and exactly solved

    Science.gov (United States)

    Gupta, S. R. D.; Gupta, Santanu D.

    1991-10-01

    The flow of laser radiation in a plane-parallel cylindrical slab of active amplifying medium with axial symmetry is treated as a problem in radiative transfer. The appropriate one-dimensional transfer equation describing the transfer of laser radiation has been derived by an appeal to Einstein's A, B coefficients (describing the processes of stimulated line absorption, spontaneous line emission, and stimulated line emission sustained by population inversion in the medium) and considering the 'rate equations' to completely establish the rational of the transfer equation obtained. The equation is then exactly solved and the angular distribution of the emergent laser beam intensity is obtained; its numerically computed values are given in tables and plotted in graphs showing the nature of peaks of the emerging laser beam intensity about the axis of the laser cylinder.

  2. Quasi-monoenergetic proton acceleration from cryogenic hydrogen microjet by ultrashort ultraintense laser pulses

    Science.gov (United States)

    Sharma, A.; Tibai, Z.; Hebling, J.; Fülöp, J. A.

    2018-03-01

    Laser-driven proton acceleration from a micron-sized cryogenic hydrogen microjet target is investigated using multi-dimensional particle-in-cell simulations. With few-cycle (20-fs) ultraintense (2-PW) laser pulses, high-energy quasi-monoenergetic proton acceleration is predicted in a new regime. A collisionless shock-wave acceleration mechanism influenced by Weibel instability results in a maximum proton energy as high as 160 MeV and a quasi-monoenergetic peak at 80 MeV for 1022 W/cm2 laser intensity with controlled prepulses. A self-generated strong quasi-static magnetic field is also observed in the plasma, which modifies the spatial distribution of the proton beam.

  3. Filtered epithermal quasi-monoenergetic neutron beams at research reactor facilities

    International Nuclear Information System (INIS)

    Mansy, M.S.; Bashter, I.I.; El-Mesiry, M.S.; Habib, N.; Adib, M.

    2015-01-01

    Filtered neutron techniques were applied to produce quasi-monoenergetic neutron beams in the energy range of 1.5–133 keV at research reactors. A simulation study was performed to characterize the filter components and transmitted beam lines. The filtered beams were characterized in terms of the optimal thickness of the main and additive components. The filtered neutron beams had high purity and intensity, with low contamination from the accompanying thermal emission, fast neutrons and γ-rays. A computer code named “QMNB” was developed in the “MATLAB” programming language to perform the required calculations. - Highlights: • Quasi-monoenergetic neutron beams in energy range from (1.5–133) keV. • Interference between the resonance and potential scattering amplitudes. • Epithermal neutron beams used in BNCT

  4. Multiple quasi-monoenergetic electron beams from laser-wakefield acceleration with spatially structured laser pulse

    Energy Technology Data Exchange (ETDEWEB)

    Ma, Y.; Li, M. H.; Li, Y. F.; Wang, J. G.; Tao, M. Z.; Han, Y. J.; Zhao, J. R.; Huang, K.; Yan, W. C.; Ma, J. L.; Li, Y. T. [Beijing National Laboratory of Condensed Matter Physics, Institute of Physics, CAS, Beijing 100080 (China); Chen, L. M., E-mail: lmchen@iphy.ac.cn [Beijing National Laboratory of Condensed Matter Physics, Institute of Physics, CAS, Beijing 100080 (China); Department of Physics and Astronomy and IFSA Collaborative Innovation Center, Shanghai Jiao Tong University, Shanghai 200240 (China); Li, D. Z. [Institute of High Energy Physics, CAS, Beijing 100049 (China); Chen, Z. Y. [Institute of Fluid Physics, China Academy of Engineering Physics, Mianyang, Sichuan 621999 (China); Sheng, Z. M. [Department of Physics and Astronomy and IFSA Collaborative Innovation Center, Shanghai Jiao Tong University, Shanghai 200240 (China); Department of Physics, Scottish Universities Physics Alliance, University of Strathclyde, Glasgow G4 0NG (United Kingdom); Zhang, J. [Department of Physics and Astronomy and IFSA Collaborative Innovation Center, Shanghai Jiao Tong University, Shanghai 200240 (China)

    2015-08-15

    By adjusting the focus geometry of a spatially structured laser pulse, single, double, and treble quasi-monoenergetic electron beams were generated, respectively, in laser-wakefield acceleration. Single electron beam was produced as focusing the laser pulse to a single spot. While focusing the laser pulse to two spots that are approximately equal in energy and size and intense enough to form their own filaments, two electron beams were produced. Moreover, with a proper distance between those two focal spots, three electron beams emerged with a certain probability owing to the superposition of the diffractions of those two spots. The energy spectra of the multiple electron beams are quasi-monoenergetic, which are different from that of the large energy spread beams produced due to the longitudinal multiple-injection in the single bubble.

  5. Accurate monoenergetic electron parameters of laser wakefield in a bubble model

    Science.gov (United States)

    Raheli, A.; Rahmatallahpur, S. H.

    2012-11-01

    A reliable analytical expression for the potential of plasma waves with phase velocities near the speed of light is derived. The presented spheroid cavity model is more consistent than the previous spherical and ellipsoidal model and it explains the mono-energetic electron trajectory more accurately, especially at the relativistic region. As a result, the quasi-mono-energetic electrons output beam interacting with the laser plasma can be more appropriately described with this model.

  6. Efficiency determination of resistive plate chambers for fast quasi-monoenergetic neutrons

    Energy Technology Data Exchange (ETDEWEB)

    Roeder, M.; Cowan, T.E.; Kempe, M.; Yakorev, D. [Helmholtz-Zentrum Dresden-Rossendorf, Dresden (Germany); Technische Universitaet Dresden, Dresden (Germany); Elekes, Z. [MTA ATOMKI, Debrecen (Hungary); Helmholtz-Zentrum Dresden-Rossendorf, Dresden (Germany); Aumann, T.; Caesar, C. [GSI Helmholtzzentrum fuer Schwerionenforschung, Darmstadt (Germany); Technische Universitaet Darmstadt, Darmstadt (Germany); Bemmerer, D.; Sobiella, M.; Stach, D.; Wagner, A. [Helmholtz-Zentrum Dresden-Rossendorf, Dresden (Germany); Boretzky, K.; Hehner, J.; Heil, M. [GSI Helmholtzzentrum fuer Schwerionenforschung, Darmstadt (Germany); Maroussov, V. [Universitaet zu Koeln, Koeln (Germany); GSI Helmholtzzentrum fuer Schwerionenforschung, Darmstadt (Germany); Nusair, O. [GSI Helmholtzzentrum fuer Schwerionenforschung, Darmstadt (Germany); Al-Balqa Applied University, Salt (Jordan); Prokofiev, A.V. [Uppsala University, The Svedberg Laboratory, Uppsala (Sweden); Reifarth, R. [Johann Wolfgang Goethe - Universitaet, Frankfurt am Main (Germany); Zilges, A. [Universitaet zu Koeln, Koeln (Germany); Zuber, K. [Technische Universitaet Dresden, Dresden (Germany); Collaboration: R3B Collaboration

    2014-07-15

    Composite detectors made of stainless-steel converters and multigap resistive plate chambers have been irradiated with quasi-monoenergetic neutrons with a peak energy of 175 MeV. The neutron detection efficiency has been determined using two different methods. The data are in agreement with the output of Monte Carlo simulations. The simulations are then extended to study the response of a hypothetical array made of these detectors to energetic neutrons from a radioactive ion beam experiment. (orig.)

  7. Calculation of quality factor for monoenergetic neutrons - in accordance with ICRU Report 40

    International Nuclear Information System (INIS)

    Bregadze, Y.I.; Maslyaev, P.F.; Nurlibaev, K.N.

    1988-01-01

    The quality factors for heavy particle dose in tissue from first interactions of monoenergetic neutrons calculated directly from the new quality factor determination given in ICRU Report 40 are presented as a function of neutron energy. The results of the calculation are compared with previously published calculations based on old concepts. For neutron energies from 100 keV up to 1 MeV these differences are about a factor of two. (author)

  8. Development of the in vivo measurement system of bone mineral content using monoenergetic gamma rays

    International Nuclear Information System (INIS)

    Nardocci, A.C.

    1990-08-01

    A system, developed for in vivo measurement of bone mineral content (BMC) using monoenergetic gamma-rays of 241 Am, is described. It presents a discussion of the theoretical and practical aspects of the technique, with details of acquisition and data processing and also discusses the calibration procedure used. The results obtained with in vivo measurements are presented and BMC values of clinically normal subjects and chronic renal patients are compared. (author)

  9. On the quasi-monoenergetic electron beam generation in the laser wakefield acceleration

    International Nuclear Information System (INIS)

    Bulanov, S.V.; Tajima, Toshiki

    2005-01-01

    A new phase of laser acceleration research has entered, as signified by the recent reports in Nature 9/30/05 of the generation of quasi-monoenergetic electron beam by laser wakefield acceleration in three experiments. We survey the current status of experiments and offer their theoretical interpretation. We understand why the choice of parameters is of such importance and why the earlier experiments showed energy spectra far from monoenergy. (author)

  10. Calibration of Cherenkov detectors for monoenergetic photon imaging in active interrogation applications

    Energy Technology Data Exchange (ETDEWEB)

    Rose, P.B., E-mail: prose6@gatech.edu; Erickson, A.S., E-mail: anna.erickson@me.gatech.edu

    2015-11-01

    Active interrogation of cargo containers using monoenergetic photons offers a rapid and low-dose approach to search for shielded special nuclear materials. Cherenkov detectors can be used for imaging of the cargo provided that gamma ray energies used in interrogation are well resolved, as the case in {sup 11}B(d,n-γ){sup 12}C reaction resulting in 4.4 MeV and 15.1 MeV photons. While an array of Cherenkov threshold detectors reduces low energy background from scatter while providing the ability of high contrast transmission imaging, thus confirming the presence of high-Z materials, these detectors require a special approach to energy calibration due to the lack of resolution. In this paper, we discuss the utility of Cherenkov detectors for active interrogation with monoenergetic photons as well as the results of computational and experimental studies of their energy calibration. The results of the studies with sources emitting monoenergetic photons as well as complex gamma ray spectrum sources, for example {sup 232}Th, show that calibration is possible as long as the energies of photons of interest are distinct.

  11. Value of monoenergetic dual-energy CT (DECT) for artefact reduction from metallic orthopedic implants in post-mortem studies.

    Science.gov (United States)

    Filograna, Laura; Magarelli, Nicola; Leone, Antonio; Guggenberger, Roman; Winklhofer, Sebastian; Thali, Michael John; Bonomo, Lorenzo

    2015-09-01

    The aim of this ex vivo study was to assess the performance of monoenergetic dual-energy CT (DECT) reconstructions to reduce metal artefacts in bodies with orthopedic devices in comparison with standard single-energy CT (SECT) examinations in forensic imaging. Forensic and clinical impacts of this study are also discussed. Thirty metallic implants in 20 consecutive cadavers with metallic implants underwent both SECT and DECT with a clinically suitable scanning protocol. Extrapolated monoenergetic DECT images at 64, 69, 88, 105, 120, and 130 keV and individually adjusted monoenergy for optimized image quality (OPTkeV) were generated. Image quality of the seven monoenergetic images and of the corresponding SECT image was assessed qualitatively and quantitatively by visual rating and measurements of attenuation changes induced by streak artefact. Qualitative and quantitative analyses showed statistically significant differences between monoenergetic DECT extrapolated images and SECT, with improvements in diagnostic assessment in monoenergetic DECT at higher monoenergies. The mean value of OPTkeV was 137.6 ± 4.9 with a range of 130 to 148 keV. This study demonstrates that monoenergetic DECT images extrapolated at high energy levels significantly reduce metallic artefacts from orthopedic implants and improve image quality compared to SECT examination in forensic imaging.

  12. Value of monoenergetic dual-energy CT (DECT) for artefact reduction from metallic orthopedic implants in post-mortem studies

    International Nuclear Information System (INIS)

    Filograna, Laura; Magarelli, Nicola; Leone, Antonio; Bonomo, Lorenzo; Guggenberger, Roman; Winklhofer, Sebastian; Thali, Michael John

    2015-01-01

    The aim of this ex vivo study was to assess the performance of monoenergetic dual-energy CT (DECT) reconstructions to reduce metal artefacts in bodies with orthopedic devices in comparison with standard single-energy CT (SECT) examinations in forensic imaging. Forensic and clinical impacts of this study are also discussed. Thirty metallic implants in 20 consecutive cadavers with metallic implants underwent both SECT and DECT with a clinically suitable scanning protocol. Extrapolated monoenergetic DECT images at 64, 69, 88, 105, 120, and 130 keV and individually adjusted monoenergy for optimized image quality (OPTkeV) were generated. Image quality of the seven monoenergetic images and of the corresponding SECT image was assessed qualitatively and quantitatively by visual rating and measurements of attenuation changes induced by streak artefact. Qualitative and quantitative analyses showed statistically significant differences between monoenergetic DECT extrapolated images and SECT, with improvements in diagnostic assessment in monoenergetic DECT at higher monoenergies. The mean value of OPTkeV was 137.6 ± 4.9 with a range of 130 to 148 keV. This study demonstrates that monoenergetic DECT images extrapolated at high energy levels significantly reduce metallic artefacts from orthopedic implants and improve image quality compared to SECT examination in forensic imaging. (orig.)

  13. Value of monoenergetic dual-energy CT (DECT) for artefact reduction from metallic orthopedic implants in post-mortem studies

    Energy Technology Data Exchange (ETDEWEB)

    Filograna, Laura [University of Zurich, Department of Forensic Medicine and Imaging, Institute of Forensic Medicine, Zurich (Switzerland); Catholic University of Rome, School of Medicine, University Hospital ' ' A. Gemelli' ' , Department of Radiological Sciences, Rome (Italy); Magarelli, Nicola; Leone, Antonio; Bonomo, Lorenzo [Catholic University of Rome, School of Medicine, University Hospital ' ' A. Gemelli' ' , Department of Radiological Sciences, Rome (Italy); Guggenberger, Roman; Winklhofer, Sebastian [University Hospital Zurich, Institute of Diagnostic and Interventional Radiology, Zurich (Switzerland); Thali, Michael John [University of Zurich, Department of Forensic Medicine and Imaging, Institute of Forensic Medicine, Zurich (Switzerland)

    2015-09-15

    The aim of this ex vivo study was to assess the performance of monoenergetic dual-energy CT (DECT) reconstructions to reduce metal artefacts in bodies with orthopedic devices in comparison with standard single-energy CT (SECT) examinations in forensic imaging. Forensic and clinical impacts of this study are also discussed. Thirty metallic implants in 20 consecutive cadavers with metallic implants underwent both SECT and DECT with a clinically suitable scanning protocol. Extrapolated monoenergetic DECT images at 64, 69, 88, 105, 120, and 130 keV and individually adjusted monoenergy for optimized image quality (OPTkeV) were generated. Image quality of the seven monoenergetic images and of the corresponding SECT image was assessed qualitatively and quantitatively by visual rating and measurements of attenuation changes induced by streak artefact. Qualitative and quantitative analyses showed statistically significant differences between monoenergetic DECT extrapolated images and SECT, with improvements in diagnostic assessment in monoenergetic DECT at higher monoenergies. The mean value of OPTkeV was 137.6 ± 4.9 with a range of 130 to 148 keV. This study demonstrates that monoenergetic DECT images extrapolated at high energy levels significantly reduce metallic artefacts from orthopedic implants and improve image quality compared to SECT examination in forensic imaging. (orig.)

  14. Thermal transport property of Ge34 and d-Ge investigated by molecular dynamics and the Slack's equation

    International Nuclear Information System (INIS)

    Han-Fu, Wang; Wei-Guo, Chu; Yan-Jun, Guo; Hao, Jin

    2010-01-01

    In this study, we evaluate the values of lattice thermal conductivity κ L of type II Ge clathrate (Ge 34 ) and diamond phase Ge crystal (d-Ge) with the equilibrium molecular dynamics (EMD) method and the Slack's equation. The key parameters of the Slack's equation are derived from the thermodynamic properties obtained from the lattice dynamics (LD) calculations. The empirical Tersoff's potential is used in both EMD and LD simulations. The thermal conductivities of d-Ge calculated by both methods are in accordance with the experimental values. The predictions of the Slack's equation are consistent with the EMD results above 250 K for both Ge 34 and d-Ge. In a temperature range of 200–1000 K, the κ L value of d-Ge is about several times larger than that of Ge 34 . (condensed matter: structure, thermal and mechanical properties)

  15. Numerical Calculation of Transport Based on the Drift-Kinetic Equation for Plasmas in General Toroidal Magnetic Geometry: Numerical Methods

    International Nuclear Information System (INIS)

    Reynolds, J. M.; Lopez-Bruna, D.

    2009-01-01

    In this report we continue with the description of a newly developed numerical method to solve the drift kinetic equation for ions and electrons in toroidal plasmas. Several numerical aspects, already outlined in a previous report [Informes Tecnicos Ciemat 1165, mayo 2009], will be treated now in more detail. Aside from discussing the method in the context of other existing codes, various aspects will be now explained from the viewpoint of numerical methods: the way to solve convection equations, the adopted boundary conditions, the real-space meshing procedures along with a new software developed to build them, and some additional questions related with the parallelization and the numerical integration. (Author) 16 refs

  16. High-field transport of electrons and radiative effects using coupled force-balance and Fokker-Planck equations beyond the relaxation-time approximation

    International Nuclear Information System (INIS)

    Huang, Danhong; Apostolova, T.; Alsing, P.M.; Cardimona, D.A.

    2004-01-01

    The dynamics of a many-electron system under both dc and infrared fields is separated into a center-of-mass and a relative motion. The first-order force-balance equation is employed for the slow center-of-mass motion of electrons, and the Fokker-Planck equation is used for the ultrafast relative scattering motion of degenerate electrons. This approach allows us to include the anisotropic energy-relaxation process which has been neglected in the energy-balance equation in the past. It also leads us to include the anisotropic coupling to the incident infrared field with different polarizations. Based on this model, the transport of electrons is explored under strong dc and infrared fields by going beyond the relaxation-time approximation. The anisotropic dependence of the electron distribution function on the parallel and perpendicular kinetic energies of electrons is displayed with respect to the dc field direction, and the effect of anisotropic coupling to an incident infrared field with polarizations parallel and perpendicular to the applied dc electric field is shown. The heating of electrons is more accurately described beyond the energy-balance equation with the inclusion of an anisotropic coupling to the infrared field. The drift velocity of electrons is found to increase with the amplitude of the infrared field due to a suppressed momentum-relaxation process (or frictional force) under parallel polarization but decreases with the amplitude due to an enhanced momentum-relaxation process under perpendicular polarization

  17. GANTRAS - a system of codes for the solution of the multigroup transport equation with a rigorous treatment of anisotropic neutron scattering

    International Nuclear Information System (INIS)

    Schwenk-Ferrero, A.

    1986-11-01

    GANTRAS is a system of codes for neutron transport calculations in which the anisotropy of elastic and inelastic (including (n,n'x)-reactions) scattering is fully taken into account. This is achieved by employing a rigorous method, so-called I * -method, to represent the scattering term of the transport equation and with the use of double-differential cross-sections for the description of the emission of secondary neutrons. The I * -method was incorporated into the conventional transport code ONETRAN. The ONETRAN subroutines were modified for the new purpose. An implementation of the updated version ANTRA1 was accomplished for plane and spherical geometry. ANTRA1 was included in GANTRAS and linked to another modules which prepare angle-dependent transfer matrices. The GANTRAS code consists of three modules: 1. The CROMIX code which calculates the macroscopic transfer matrices for mixtures on the base of microscopic nuclide-dependent data. 2. The ATP code which generates discretized angular transfer probabilities (i.e. discretizes the I * -function). 3. The ANTRA1 code to perform S N transport calculations in one-dimensional plane and spherical geometries. This structure of GANTRAS allows to accommodate the system to various transport problems. (orig.) [de

  18. Weak second-order splitting schemes for Lagrangian Monte Carlo particle methods for the composition PDF/FDF transport equations

    International Nuclear Information System (INIS)

    Wang Haifeng; Popov, Pavel P.; Pope, Stephen B.

    2010-01-01

    We study a class of methods for the numerical solution of the system of stochastic differential equations (SDEs) that arises in the modeling of turbulent combustion, specifically in the Monte Carlo particle method for the solution of the model equations for the composition probability density function (PDF) and the filtered density function (FDF). This system consists of an SDE for particle position and a random differential equation for particle composition. The numerical methods considered advance the solution in time with (weak) second-order accuracy with respect to the time step size. The four primary contributions of the paper are: (i) establishing that the coefficients in the particle equations can be frozen at the mid-time (while preserving second-order accuracy), (ii) examining the performance of three existing schemes for integrating the SDEs, (iii) developing and evaluating different splitting schemes (which treat particle motion, reaction and mixing on different sub-steps), and (iv) developing the method of manufactured solutions (MMS) to assess the convergence of Monte Carlo particle methods. Tests using MMS confirm the second-order accuracy of the schemes. In general, the use of frozen coefficients reduces the numerical errors. Otherwise no significant differences are observed in the performance of the different SDE schemes and splitting schemes.

  19. Stochastic uncertainty analysis for solute transport in randomly heterogeneous media using a Karhunen‐Loève‐based moment equation approach

    Science.gov (United States)

    Liu, Gaisheng; Lu, Zhiming; Zhang, Dongxiao

    2007-01-01

    A new approach has been developed for solving solute transport problems in randomly heterogeneous media using the Karhunen‐Loève‐based moment equation (KLME) technique proposed by Zhang and Lu (2004). The KLME approach combines the Karhunen‐Loève decomposition of the underlying random conductivity field and the perturbative and polynomial expansions of dependent variables including the hydraulic head, flow velocity, dispersion coefficient, and solute concentration. The equations obtained in this approach are sequential, and their structure is formulated in the same form as the original governing equations such that any existing simulator, such as Modular Three‐Dimensional Multispecies Transport Model for Simulation of Advection, Dispersion, and Chemical Reactions of Contaminants in Groundwater Systems (MT3DMS), can be directly applied as the solver. Through a series of two‐dimensional examples, the validity of the KLME approach is evaluated against the classical Monte Carlo simulations. Results indicate that under the flow and transport conditions examined in this work, the KLME approach provides an accurate representation of the mean concentration. For the concentration variance, the accuracy of the KLME approach is good when the conductivity variance is 0.5. As the conductivity variance increases up to 1.0, the mismatch on the concentration variance becomes large, although the mean concentration can still be accurately reproduced by the KLME approach. Our results also indicate that when the conductivity variance is relatively large, neglecting the effects of the cross terms between velocity fluctuations and local dispersivities, as done in some previous studies, can produce noticeable errors, and a rigorous treatment of the dispersion terms becomes more appropriate.

  20. Novel design concepts for generating intense accelerator based beams of mono-energetic fast neutrons

    International Nuclear Information System (INIS)

    Franklyn, C.B.; Govender, K.; Guzek, J.; Beer, A. de; Tapper, U.A.S.

    2001-01-01

    Full text: Successful application of neutron techniques in research, medicine and industry depends on the availability of suitable neutron sources. This is particularly important for techniques that require mono-energetic fast neutrons with well defined energy spread. There are a limited number of nuclear reactions available for neutron production and often the reaction yield is low, particularly for thin targets required for the production of mono-energetic neutron beams. Moreover, desired target materials are often in a gaseous form, such as the reactions D(d,n) 3 He and T(d,n) 3 He, requiring innovative design of targets, with sufficient target pressure and particle beam handling capability. Additional requirements, particularly important in industrial applications, and for research institutions with limited funds, are the cost effectiveness as well as small size, coupled with reliable and continuous operation of the system. Neutron sources based on high-power, compact radio-frequency quadrupole (RFQ) linacs can satisfy these criteria, if used with a suitable target system. This paper discusses the characteristics of a deuteron RFQ linear accelerator system coupled to a high pressure differentially pumped deuterium target. Such a source, provides in excess of 10 10 mono- energetic neutrons per second with minimal slow neutron and gamma-ray contamination, and is utilised for a variety of applications in the field of mineral identification and materials diagnostics. There is also the possibility of utilising a proposed enhanced system for isotope production. The RFQ linear accelerator consists of: 1) Deuterium 25 keV ion source injector; 2) Two close-coupled RFQ resonators, each powered by an rf amplifier supplying up to 300 kW of peak power at 425 MHz; 3) High energy beam transport system consisting of a beam line, a toroid for beam current monitoring, two steering magnets and a quadrupole triplet for beam focusing. Basic technical specifications of the RFQ linac