Monoenergetic Transport Equation Research Articles

Convergence of the discrete-cone method in plane-parallel geometry

Abstract In this paper, we study a relatively novel numerical technique, the discrete-cone (DCN ) method, in transport theory. Specifically, we use previous results for the discrete-ordinate (SN ) method to establish, under mathematically mild and physically reasonable conditions, the convergence of the discrete-cone method in the numerical solution of the monoenergetic transport equation for slab geometry.

The one-dimensional monoenergetic time-dependent transport equation in infinite media (revisited)

Abstract Classical integral transform methods are used to solve the monoenergetic time-dependent transport equation for the angular distribution and density in an infinite plane geometry with isotropic scattering for both a mono-directional and an isotropic, infinite, planar pulsed source (in other words, the initial-value Green's functions). In contrast to earlier work on the same problem, the methods presented here are relatively simple, and the results are—at least for the isotropic source—intuitively appealing, exact closed form solutions. For the isotropic, pulsed source problem, the expression for the density is shown to agree with both the closed form solution obtained by Ganapol and Grossman using the multiple collision method and the solution found by Case using the singular eigenfunction method. I also point out how exact closed form solutions for the density due to localized, isotropic pulsed sources in both spherical and cylindrical geometries can be obtained from the plane geometry result. Then, using the spherical geometry result for the density, I construct an algorithm for finding an exact solution to the physically interesting multigroup equations describing superthermal particle transport (with isotropic scattering).

Legendre expansion of neutron flux over entire multilayer slab geometry

The monoenergetic integral transport equation for a multilayer slab geometry has been solved by Legendre expansion method. The method utilizes an expansion of the neutron flux over entire multilayer slab geometry in Legendre polynomials of the position co-ordinate (Single Expansion Method—SEM). This formulation is an extension of Carlvik's (1968) method for a homogeneous slab. Earlier (Raghav, 1984) the expansion of the neutron flux was done in each layer in Legendre polynomials of the position co-ordinate (Multi Expansion Method—MEM). The aim in this paper is to compare both the approaches of SEM and MEM. A few multilayer slab systems with vacuum boundary conditions have been selected for this purpose and K eff, the effective multiplication factor of the system, has been compared. SEM requires the evaluation of the integrals where the limits are not −1 to +1 (as they are in MEM and where analytical expressions can be derived), in these cases we have derived recurrence relations (which are described in the Appendix) to evaluate such integrals.

Stationary transport with partially reflecting boundary conditions. II

We consider the stationary monoenergetic transport equation (the Lorentz model) in a semiinfinite geometry with most general nonmultiplying boundary conditions at the wall x=0, accounting for absorption, specular, diffuse, and ‘‘selective’’ reflection, or combinations thereof. Performing the numerical study of a (partially) ‘‘thermalizing’’ wall, we investigate the density profile n(x), the Milne extrapolation length xM, and the space dependent diffusion coefficient D(x) as functions of the accomodation and ‘‘selection’’ coefficients. The inversed density profile (equivalent to a negative diffusion coefficient) is numerically computed. As a simple analytically solvable example, a discrete velocity model is discussed in the Appendix.

On the spectral decomposition of the transport operator with anisotropic scattering and periodic boundary conditions

Abstract The time-dependent linear monoenergetic transport equation with linearly anisotropic scattering is considered in slab geometry With periodic boundary conditions. Adapting a result of Friedrichs, it is shown explicitly that the spectral theorem holds for the corresponding operator, for all but a set of zero measure in the space of parameters.

Stationary transport with partially reflecting boundary conditions

The stationary monoenergetic transport equation (the Lorentz model) is considered in a semiinfinite geometry with partially reflecting boundary conditions at x=0. The extrapolation length, the density, and the space-dependent diffusion coefficient are studied as functions of the accommodation coefficient α, both for constant and renormalized current. The hydrodynamical description at the level of the evolution equation is practically unaffected by α and is still valid at distances of several mean free paths (mfp) from the boundary. Yet, when consistently applying the boundary condition, the boundary layer has to be considered in terms of the Milne extrapolation length, which varies like (1−α)−1. The case α=1 is a singular limit in the sense that the boundary layer solution disappears completely.

Legendre expansion in multilayer slab geometry

The monoenergetic integral transport equation for a multilayer slab geometry has been solved by the Legendre expansion method. The method utilizes an expansion of the flux density in each layer in Legendre polynomials of the position co-ordinate. The use of these polynomials makes it possible to calculate most of the resulting matrix by means of recurrence formulae. These formulae have been obtained by a procedure which is an extension of Carlvik's method for a homogeneous slab. A code (MULREG) has been written for this purpose. Using MULREG a series of calculations have been performed for a homogeneous slab with vacuum boundary conditions. The slab has been divided into NR number of regions and in each region flux is expanded into Legendre polynomials of order NSI. For a particular value of NR, the NSI varies from 0 to 4. The effective multiplication factor k eff of the slab is calculated. By comparing the computational time for all the cases, it is studied as how severe it is to consider the flat flux approximation (conventional collision-probability approach) as compared to a case when more terms in the flux expansion are considered per layer.

A Semianalytic Solution of the Monoenergetic Transport Equation in Cylindrical Geometry and the Derivation of Fluxes from Singular Sources

The neutron flux (r,..cap omega..) at position r in the direction of unit vector ..cap omega.. is expanded as a series of the form (r,..cap omega..) = /SUP N/ ..sigma../sub 0/ (21 + 1) /SUP l/ ..sigma../sub 0/ /PSI/lm(r)P /SUP m/ /SUB l/ (cos/PHI/)cos(m) with P /SUP m/ /SUB l/ (cos/PHI/), the associated Legendre polynomial of order l,m. Variables /PHI/ and are the angles defining ..cap omega.. and N, an arbitrary odd number. Equations obtained for /PSI/..mu..(r) have solutions involving linear combinations of Bessel functions of the first and second kind. Expressions for the fluxes from point, line, and plane sources are found using a Green's function technique.

Finite-Difference Approximations and Superconvergence for the Discrete-Ordinate Equations in Slab Geometry

A unified framework is developed for calculating the order of the error for a class of finite-difference approximations to the monoenergetic linear transport equation in slab geometry. In particular, the global discretization errors for the step characteristic, diamond, and linear discontinuous methods are shown to be of order two, while those for the linear moments and linear characteristic methods are of order three, and that for the quadratic method is of order four. A superconvergence result is obtained for the three linear methods, in the sense that the cell-averaged flux approximations are shown to converge at one order higher than the global errors.

A Semianalytic Method for the Solution of the Neutron Transport Equation in Plane and Spherical Geometries

The source-free monoenergetic transport equation can be written as where for plane and for spherical geometries, respectively; Ω is the direction of motion with θ as its angle to the axis, An analytic expression for ψ(Z, cosθ) of the form is considered where Pn(cosθ) is the Legendre polynomial of order n; are combinations of exponential functions for the plane case and Bessel functions in the spherical formulation. A simple algorithm is developed that enables solutions to be found for any N using a small computer program.

Time-Dependent Moments of the Monoenergetic Transport Equation in Spherical and Cylindrical Geometry

The time-dependent spatial-Legendre moments for the monoenergetic transport equation in spherical geometry are expressed in terms of the half- and logarithmically weighted-Legendre moments of the flux in plane geometry. In cylindrical geometry, the moments are shown to be directly proportional to those in spherical geometry.

Reduced Two-Dimensional Critical Problem of Finite Cylindrical Reactor

The critical problem is considered for finite cylindrical reactors. By applying the concept embodied in the asymptotic solution of the mono-energetic transport equation, it is possible to reduce the two-dimensional transport equation to an axially and radially reduced one-dimensional form. In essence, the idea is to approximate one of the two space dimensions by the assumption of the asymptotic distribution, and to extend the reactor medium and the asymptotic distribution toward infinity. The resulting reduced equation is solved by making use of the singular eigenfunction expansion method introduced by Case. It is found that the two-dimensional flux distribution can be obtained with high accuracy from the reduced equation, excluding exceptional cases of extremely poor conditions. Numerical results are presented for a wide range of reactor multiplication factor, and are compared with the results obtained by the variational method. It is found that there is only a small contribution of the boundary transient term brought to the axially and radially reduced problem. The error introduced in reducing the space dimension can be evaluated by the perturbation theory. Numerical results for various sizes of reactor show that the perturbation is not very large, and that it does not bring marked changes to the eigenvalues of the transport operators.

Reduced Two-Dimensional Critical Problem of Finite Cylindrical Reactor

The critical problem is considered for finite cylindrical reactors. By applying the concept embodied in the asymptotic solution of the mono-energetic transport equation, it is possible to reduce the two-dimensional transport equation to an axially and radially reduced one-dimensional form. In essence, the idea is to approximate one of the two space dimensions by the assumption of the asymptotic distribution, and to extend the reactor medium and the asymptotic distribution toward infinity. The resulting reduced equation is solved by making use of the singular eigenfunction expansion method introduced by Case. It is found that the two-dimensional flux distribution can be obtained with high accuracy from the reduced equation, excluding exceptional cases of extremely poor conditions. Numerical results are presented for a wide range of reactor multiplication factor, and are compared with the results obtained by the variational method. It is found that there is only a small contribution of the boundary transien...

Exact Time-Dependent Spatial and Legendre Moments for the Monoenergetic Transport Equation

Time-dependent mixed spatial and Legendre moments are defined for the mono-energetic transport equation in spherical and slab coordinates. For an infinite medium, a pulsed source at the origin gives rise to finite closed sets of equations which can be solved to give exact results. Examples are given, and the transformation connecting spherical and slab moments is derived.

Singular Eigenfunction Solution of the Monoenergetic Neutron Transport Equation for Finite Radially Reflected Critical Cylinders

The normal mode expansion technique is applied to the transformed mono-energetic integral transport equation to develop a solution for the rotationally invariant and axially infinite critical two-region cylinder with a finite outer reflector boundary. The model assumes isotropic scattering and identical neutron mean-free-paths in the core and reflector regions. The solution in terms of singular integral equations is obtained by applying a completeness theorem found for the singular eigenfunctions. Numerical results for a variety of core and reflector multiplying properties and reflector thicknesses are presented and compared with the results of other methods. The completeness inherent in this solution and the high precision in the numerical calculations provide results which may be used as analytic standards for this problem.112

A Simple Analytic Transient Flux Expression for Use With Asymptotic Diffusion Theory

The monoenergetic integral transport equation is solved in an approximate manner by separating the integral into an asymptotic component plus a transient boundary component. Solutions to the result...

Solutions of Monoenergetic Time Dependent Neutron Transport Equation in Slab Geometry

Approximate solutions to the one-velocity time dependent neutron transport equation are derived by means of the eigenfunction expansion method of Case for a slab geometry. A finite set of discrete time eigenvalues is determined by a simple formula. This formula, which is derived for the case of large slab thickness, is found to be valid also for fairly thin slabs. Each time eigenvalue generates a pair of spatial modes. The expansion coefficients to these modes are examined by numerical evaluation for various slab thicknesses. Higher spatial modes are found to play an important role in the transient time evolution of neutrons.

Asymptotic PN and double PN approximations

The P N and double P N approximations to the monoenergetic transport equation are modified to improve their asymptotic behaviour. Free parameters are introduced into the P N and double P N equations through the use of a pseudo-scattering kernel. The parameters are evaluated by stipulating that the resulting scalar flux approximations to the infinite-medium Green's function yield exact asymptotic solutions while conserving specified spatial moments. Asymptotic P 3 and double P 1 solutions are presented for the infinite-medium and half-space Green's functions and for the source-free and constant-source Milne problems.

On the existence of discrete eigenvalues in slab lattice exponential experiments

A theoretical study is made of the neutron flux distributions in exponential experiments with regular alternating slabs of non-fissile materials. It is shown that discrete relaxation lengths need not exist in configurations where the source plane is perpendicular to the planes of the lattice slabs. The existence or non-existence depends on the relative thicknesses and the total cross sections of the lattice materials. For illustration, eigenvalue solutions of the monoenergetic transport equation are sought for two semi-infinite aluminium-water lattices of which only the first is found to support a discrete relaxation length. The flux distribution produced by a chosen source condition in the second lattice is studied by means of a two-dimensional transport calculation. The flux fine structure and spatial rate of decay are found to vary continuously in the lattice and to depend on the nature of the source.

S∞ Neutron Transport Theory in Plane Geometry

A semi-analytical method for solving the monoenergetic transport equation with isotropic scattering in plane geometry is developed, in which the slab system is subdivided into a number of discrete space points in x, while the angular variable is treated analytically. This is equivalent to taking N to ∞ in SN theory and avoids the numerical instabilities inherent in the limiting process. General boundary conditions are introduced allowing finite multilayer slabs, cells, and shielding problems with specified incident angular distribution of neutrons to be handled by the same formalism. Analytical expressions are derived for the angular distributions, and fluxes are obtained by solving a matrix problem, where the matrix elements are integrals over rational functions of the angular variable. Computing times are comparable to low-order SN calculations.