On the numerical integration of the neutron transport equation by Herbert Bishop Keller

Cover of: On the numerical integration of the neutron transport equation | Herbert Bishop Keller

Published by Courant Institute of Mathematical Sciences, New York University in New York .

Written in English

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Edition Notes

Book details

Statementby Herbert B. Keller and Jack Heller.
ContributionsHeller, Jack, Keller, Herbert
The Physical Object
Pagination39 p.
Number of Pages39
ID Numbers
Open LibraryOL20424743M

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Abstract A procedure for the direct numerical integration of the steady-state, elastic scattering neutron transport equation is presented. Conditions for the existence, uniqueness and non-negativeness of the numerical solution are obtained.

Under these conditions it is shown that the scheme is stable and the numerical solution converges, with Author: Herbert B. Keller. A difference form of the Boltzmann equation is derived as the final expression for machine computation.

Comparisons are given of the numerical solutions with an analytical solution for a constant source distribution, and with NIOBE calculations and experimental spectra for neutron transport in water, with good agreement obtained between by: 7.

For any given azimuth‐independent scattering law, it is shown that the neutron transport equation with external source is rigorously equivalent to a coupled system of Fredholm integral equations. These are derived both for vacuum and periodic boundary : Herbert S.

Wilf. Neutron Transport Equation Boundary Conditions. Scalar Flux and Current. Partial Currents. Integral Transport Theory Isotropic Point Source. Isotropic Plane Source. Anisotropic Plane Source. Transmission and Absorption Probabilities. Escape Probability. First‐Collision Source for Diffusion Theory.

For any given azimuth-independent scattering law, it is shown that the neutron transport equation with external source is rigorously equivalent to a coupled system of Fredholm integral equations.

These are derived both for vacuum and periodic boundary conditions. A numerical integration scheme is given for solving these integral equations with no angular truncation error, thereby permitting.

The integral equation is solved numerically and the critical radius of the sphere is obtained. The results are compared with those from the P N approximation as reported by Yildiz and Alcan (Yildiz, C., Alcan, On the numerical integration of the neutron transport equation book, The effect of strong anisotropic scattering on the critical sphere problem in neutron transport theory using a synthetic kernel.

The subject of this work is computational modeling of neutron trans-port relevant to economical and safe operation of nuclear facilities. The general mathematical model of neutron transport is provided by the linear Boltzmann’s transport equation and the thesis begins with its precise mathematical formulation and presentation of known con.

Numerical methods for solving the integrodifferential, integral, and surface-integral forms of the neutron transport equation are reviewed. The solution methods are shown to evolve from only a few. Boltzman Transport Equation.

The Boltzman transport equation is a balance statement that conserves neutrons. Each term represents a gain or a loss of a neutron, and the balance, in essence, claims that neutrons gained equals neutrons lost.

It must be added it is much easier to derive the Boltzmann transport equation than it is to solve it. book, as. stated. anisotropic multi-velocity kinetic equation of neutron transport and the balance equation of delayed neutrons.

use standard methods for numerical integration. The governing integral equations may be solved with several analytical and/or numerical techniques, which will not be discussed in this text in any detail.

An example of analytical techniques is the use of Chandrasekhar's X- and ϒ- functions, based on the principle of invariance, which is described in some detail in Chandrasekhar's book [ 16 ].

On the Numerical Integration of the Neutron Transport Equation. | | 0. CiteSeerX — Citation Query On the numerical integration of. Neutron transport (also known as neutronics) is the study of the motions and interactions of neutrons with materials.

Nuclear scientists and engineers often need to know where neutrons are in an apparatus, what direction they are going, and how quickly they are moving. It is commonly used to determine the behavior of nuclear reactor cores and experimental or industrial neutron beams. The Multi-P N (MP N) approximation to the first-order neutron transport equation in one- and two-dimensional geometries is presented using a variational directional variations of the angular flux have been modeled by separate spherical harmonics expansions in M subdomains, while finite elements have been utilized to represent the spatial dependence.

average integral derivative (IAD method) to the solution of transport equation neutrons to describe the physical processes taking place in the nuclear reactor core. The result after the problems of nuclear reactors given matrix equation calculated neutron flux for each energy group.

Diffusion Equation for Neutron and the IAD Method. ferential equation to a system of ordinary differential equations. We can use ODE theory to solve the characteristic equations, then piece together these characteristic curves to form a surface. Such a surface will provide us with a solution to our PDE.

Example 1. Find a solution to the transport equation, ut +aux = 0: (). The steady-state neutron transport equation is derived based on the neutron balance principle. Before giving a brief introduction to the numerical methods of the neutron transport equation, some other useful forms of transport equations are also introduced.

Finally, the adjoint transport equation is formulated from the foregoing equation. ment of the neutron transport equation for finite cylindrical geometry followed by a discussion on its applicability to neutron transport calculations in shields.

The method is based on the use of quasi-cartesian coordi-nates for the phase space and the application of numerical integration to the scattering. In this chapter, we start with a brief introduction to numerical simulation of transport phenomena. We consider mathematical models that express certain conservation principles and consist of convection-diffusion-reactionequations written in integral, differential, or weak form.

In particular, we discuss the qualitative properties of. @article{osti_, title = {Computational methods of neutron transport}, author = {Lewis, E E and Miller, W F}, abstractNote = {This books presents a balanced overview of the major methods currently available for obtaining numerical solutions in neutron and gamma ray transport.

It focuses on methods particularly suited to the complex problems encountered in the analysis of reactors, fusion. In the context of neutron transport, the polynomial collocation coupled with the DE quadrature is a very attractive prescription for the numerical solution of the singular integral equations.

The results of the test problems confirm this supposition. A NUMERICAL SOLUTION OF THE TIME-DEPENDENT NEUTRON TRANSPORT EQUATION USING THE CHARACTERISTIC METHOD. presented in the Clark-Hansen book) 2. The time-dependent neutron transport equation in a moving system in lagrangian the time derivative can be displaced into the integral.

But for a moving medium (as the fissile mass in a nuclear. A numerical method for solving the steady-state one-velocity neutron transport equation in x-y geometry is presented.

It is based on the concept of combining the spherical harmonics theory with the discrete-ordinate method. Neutron Diffusion and Migration Lengths Numerical Solution of Diffusion Equation Finite Difference Equations in One Dimension.

Forward Elimination/Backward Substitution Spatial Solution Procedure. Power Iteration on Fission Source Integral Transport Formulation. Collision Probability Method. The neutron transport equation exists in two forms: integro-differential form and integral form.

Analytical solutions of the neutron transport equation are limited to simple cases with very simple geometries. Practical cases of interest necessitate the use of numerical methods which rely on discretization in space, angle, and energy. For the.

Integral Neutron Transport and New Computational Methods: A Review They address a broad range of topics, from problems of existence and uniqueness for singular integral equations on domain boundaries to numerical integration via finite and boundary elements, conservation laws, hybrid methods, and other quadrature-related approaches.

() Discontinuous Galerkin finite element method applied to the 1-D spherical neutron transport equation. Journal of Computational Physics() Multigrid algorithms for solving multigroup neutron transport equations. The transport equation for the neutron flux is transformed into the inte- gral equation form and with a second equation for the currents, that leave the considered re- gion, one gets a linear system of equations, the collision probability equations.

For instance, we include (i) a discussion of the lack of smoothness of the angular flux in multidimensional geometries (this has a negative impact on numerical simulations); (ii) derivations of the transport equation in specialized 1-D, 2-D, and 3-D geometries; (iii) a derivation of the time-dependent integral transport equation; (iv) an.

SOLVING THE TRANSPORT EQUATION IN SPHERICAL GEOMETRY by WILLIAM ERVIN ABBOTT, B.S. A THESIS IN PRESENTATION OF NUMERICAL METHOD 3 III. NUMERICAL RESULTS 8 IV. ANALYSIS OF NUMERICAL METHOD 23 a positive numerical method for solving the neutron transport equation in spherical geometry is presented.

This method is analyzed for consis­. Integral Methods in Science and Engineering: Progress in Numerical and Analytic Techniques will be a valuable resource for researchers in applied mathematics, physics, and mechanical and electrical engineering, for graduate students in these disciplines, and for various other professionals who use integration as an essential tool in their work.

The Boltzmann equation or Boltzmann transport equation (BTE) describes the statistical behaviour of a thermodynamic system not in a state of equilibrium, devised by Ludwig Boltzmann in The classic example of such a system is a fluid with temperature gradients in space causing heat to flow from hotter regions to colder ones, by the random but biased transport of the particles making up.

Thus, myriad numerical methods have been set up for its resolution and will be discussed briefly in this chapter.

Beginning inwith the Manhattan Project for the design of the atomic bomb in the United States, numerical methods specific to the transport equations have been schemed for. @article{osti_, title = {A PROGRAM FOR THE NUMERICAL INTEGRATION OF THE BOLTZMANN TRANSPORT EQUATION- NIOBE. Project No. }, author = {Preiser, S and Rabinowitz, G and deDufour, E}, abstractNote = {A code (NIOBE) for numerically integrating the timeindependent neutron or gamma-ray Boltzmann transport equation was written for the IBM the three-dirnensional neutron transport equation by using time­ saving differenceThe discrete angular flux is computed in 14 directions.

This paper shows how to use a set of 24 angular directions identical with those proposed by B.G. Carlson and K.D. Lathrop for S4-calculations. Zusammenfassung M.R.

Wagner. The roots of Discontinuous Galerkin (DG) methods is usually attributed to Reed and Hills in a paper published in on the numerical approximation of the neutron transport equation [18].

In fact, the adventure really started with a rather thoroughfull series of five papers by Cockburn and Shu in the late 80's [7, 5, 9, 6, 8]. Then, the fame of the method, which could be seen as a compromise. neutron transport case, Keller and Wendroff [15] and Wendroff [16] were able to establish sufficient conditions for the convergence of a finite-difference scheme to the truncated system.

For the numerical integration of the quasi- linear type of equations which arise from the half-range truncated system. The roots of Discontinuous Galerkin (DG) methods is usually attributed to Reed and Hills in a paper published in on the numerical approximation of the neutron transport equation [18].

In fact, the adventure really started with a rather thoroughfull series of five papers by Cockburn and Shu in the late 80's [7, 5, 9, 6, 8]. Then, the fame of the method, which could be seen as a compromise. The neutron transport equation for a slab geometry with the extremely anisotropic scattering kernel is considered.

The albedo and transmission factors are calculated using the variation method. The numerical technique is applied to three-dimensional space-time neutron diffusion equations with average one group of delayed neutrons in the different nuclear reactors. A solution of the neutron transport equation is obtained by expanding the flux Phi (r Omega) at position r in direction Omega as a series of the form: Phi (r, Omega)= Sigma l=0 N (2l+1) Sigma m=0 l P l m (cos theta)(psi lm (r)cos(m phi)+ gamma lm (r)sin(m phi)) where P l m (cos theta) is the associated Legendre polynomial of order l, m with theta and phi the axial and azimuthal angles.The two forms in turn lead to contrasting approaches to reducing the differential-integral equations to sets of linear equations and in the formulation of iterative methods suitable for the numerical solution of large engineering design problems.

Second-order neutron transport methods. In Nuclear Computational Science: A Century in Review.

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