Neutron Transport Theory Research Articles

The Milne problem using the P$_{N}$ method for a nonabsorbing medium with linearly anisotropic scattering

The Milne problem, known as one of the classical problems of radiative transfer and neutron transport theory, is solved using the P$_{N}$ method for a nonabsorbing medium (c = 1) with a linearly anisotropic scattering kernel. Specular and diffuse reflection boundary conditions are taken into account. The numerical results are listed for different selected parameters. Some results are also compared with the literature.

Multiregion Annular Heterogeneous Core Concepts for Plutonium-Thorium Fuels in Pressure-Tube Heavy Water Reactors

Pressure-tube heavy water reactors (PT-HWRs) are highly advantageous for implementing plutonium-thorium (Pu-Th) fuels because of their high neutron economy and online refueling capability. The use of annular heterogeneous seed-blanket core concepts in a PT-HWR where higher-fissile-content seed fuel bundles are physically separate from lower-fissile-content blanket bundles allows more flexibility and control in fuel management. The lattice concept modeled was a 35-element bundle made with a homogeneous mixture of reactor-grade PuO2 (~67 wt% fissile) and ThO2, with a central zirconia rod to reduce coolant void reactivity. Eight annular heterogeneous seed-blanket core concepts with plutonium-thorium–based fuels in a 700-MW(electric)–class PT HWR were analyzed, using a once-through-thorium cycle. Blanket region(s) represented 50% to 75% of the total fuel volume. There were 1, 2, and 3 different blanket regions and 1, 2, and 3 different seed regions. The seed fuel tested was 3 wt% or 4 wt% PuO2, while the blanket fuel tested was 1 wt% PuO2, mixed with ThO2. The impact of different fuel combinations on the core-average burnup, fissile utilization (FU), power distributions, and other performance parameters were evaluated. WIMS-AECL 3.1 was used to perform lattice physics calculations using two-dimensional, 89-group integral neutron transport theory, while RFSP 3.5.1 was used to perform the core physics and fuel management calculations using three-dimensional two-group diffusion theory. Among the different core concepts investigated, there were cores where the FU was up to 25% higher than is achieved in a PT-HWR using natural uranium fuel bundles. There were cores where up to 60% of the Pu was consumed, cores where up to 41% of the energy was produced from 233U, and cores where up to 236 kg/yr of fissile uranium (mainly 233U) was produced in the discharged fuel. This study is an extension of previous work that involved the analysis of homogeneous cores, two-region (one seed, one blanket) and eight-region (four seeds, four blankets) annular, and checkerboard-type heterogeneous seed-blanket cores.

Galerkin partial least-square SN (GpLS–SN) method for fixed source problems in the neutron transport theory

The neutron transport equation in the integral form is of difficult solution and therefore, schemes for obtaining an equivalent differential formulation are used. In this paper is used the discrete ordinates method (SN method) in order to obtain the equivalent differential formulation. Inspired in the GLS method (Galerkin least square method) a variant of the GLS (Brooks and Hughes, 1982; Ern and Guermond, 2004; Johnson, 1990) method denoted GpLS (Galerkin partial least-square) is proposed for solving this equivalent differential formulation. The convergence rates as well as the robustness of GLS method are preserved by the GpLS method that possesses computational lower cost. Simulations comparing the convergence rates of GpLS method with other methods are presented, confirming the best performance of GpLS method. The numerical experiments were made using fixed source in multiplying medium, with the objective to test the robustness of GpLS method. All the numerical experiments presented confirm the good properties of approximation of GpLS method.

The dynamic adjoint as a Green’s function

The concept of the dynamic adjoint was introduced by Hugo van Dam for calculating the in-core neutron noise in boiling water reactors in the mid-70’s. This successful approach found numerous applications for calculating the neutron noise in both PWRs and BWRs since then. Although the advantages and disadvantages of using the direct (forward) or the adjoint (backward) approach for the calculation of the neutron noise were analysed in a number of publications, the direct relationship between the forward Green’s function and the dynamic adjoint has not been discussed. On the other hand, in particle transport theory the relationship between the direct and adjoint Green’s function has been discussed in detail, in which Mike Williams has had many seminal contributions. In this note we analyse the relationship between the direct Green’s function and the dynamic adjoint in the spirit of Mike’s work in neutron transport and radiation damage theory. The paper is closed with some personal remarks and reminiscences.

Development of a 2D-Multigroup Code (NFDE-2D) based on the neutron spatial-fractional diffusion equation

The first aim of this work is the numerical solution of the neutron spatial-fractional diffusion equation in two-dimension for energy multigroups using the shifted Grünwald–Letnikov approximation in the fractional derivatives. The second aim is related to illustrate a real application of fractional derivatives in the nuclear reactors. The novelty of this work is the discretization method of the neutron fractional diffusion equation in multigroup form for high heterogeneous multiplicative media where the effective multiplication factor (Keff) was rigorously calculated. When the classical Laplacian is converted to fractional Laplacian, the jump length statistics are unrestricted. For neutron population in the nuclear reactors there is a new version of the neutron diffusion equation which is established on the fractional space derivatives, called in this work as neutron fractional diffusion equation (NFDE). We have developed the NFDE-2D as a nuclear reactors core calculation code, which is validated with some numerical experiments where different orders of fractional derivative are considered. The results demonstrate that reactors exhibit the complex behavior against order of fractional derivative which is depended on the competition between neutron absorption and super-diffusion phenomena, which is comparable with the neutron transport theory.

On the Consistent Spatial Homogenization Method in Neutron Transport Theory

The recently developed consistent spatial homogenization method in neutron transport theory is based on basis functions in spatial and angular domains and spatial rehomogenization to correct for the core environment effects while reproducing the heterogeneous solution in full phase space resolution and accuracy. In this article, an efficient version of the method is developed that also simplifies implementation and memory requirements for data handling in multidimensional geometries. This is done by (1) replacing the Fourier series with B-spline basis functions for the spatial domain; (2) truncating the angular expansion to first order, and (3) developing a source iteration method to replace the original local fixed-source transport method for on-the-fly rehomogenization. The new method is tested in a 1D benchmark problem characteristic of pressurized water reactors with mixed oxide fuel. The method is found to be very accurate and more efficient than the original consistent spatial homogenization method.

Thermal Neutron Utilization Factor Calculation by Monte Carlo

The Neutron Transport Theory is accurate in reactor engineering analysis, but the calculation process is tedious and complicated. The objective of the present study obtains the thermal utilization factor f by Monte Carlo method. The study establishes the pressurized water reactor model by MCNP program firstly, and calculates the dioxide pellets, zirconium alloy cladding and moderator’s neutron flux density distribution. The thermal neutron disadvantage factor ζ will be gained according to the definition formula. Based on the functional relationship between the above thermal neutron disadvantage factor ζ and the thermal utilization factor f, this study finally obtains the thermal utilization factor f.

Heterogeneous Cores for Implementation of Thorium-Based Fuels in Heavy Water Reactors

New reactor concepts to implement thorium-based fuel cycles have been explored to achieve maximum resource utilization. Pressure tube heavy water reactors (PT-HWRs) are highly advantageous for implementing thorium-based fuels because of their high neutron economy and online refueling capability. The use of heterogeneous seed/blanket core concepts in a PT-HWR where higher-fissile-content seed fuel bundles are physically separate from lower-fissile-content blanket bundles allows more flexibility and control in fuel management to maximize fissile utilization (FU) and conversion of fertile fuel. The lattice concept chosen was a 35-element bundle made with a homogeneous mixture of reactor-grade PuO2 (~67 wt% fissile) and ThO2, with a central zirconia rod to reduce coolant void reactivity. Several annular and checkerboard-type heterogeneous seed/blanket core concepts with plutonium-thorium–based fuels in a 700-MW(electric)–class PT-HWR were analyzed, using a once-through thorium cycle. Different combinations of seed and blanket fuel were tested to determine the impact on core-average burnup, FU, power distributions, and other performance parameters. WIMS-AECL Version 3.1 was used to perform lattice physics calculations using two-dimensional, 89-group integral neutron transport theory, while RFSP Version 3.5.1 was used to perform the core physics and fuel management calculations using three-dimensional two-group diffusion theory. Among the different core concepts investigated, there were cores where the FU was up to 30% higher than that achieved in a PT-HWR using natural uranium fuel bundles. There were cores where up to 67% of the Pu was consumed, cores where up to 43% of the energy was produced from thorium, and cores where up to 363 kg/year of 233U was produced in the discharged fuel.

On Some Measure Convolution Operators in Neutron Transport Theory

We show how some fundamental spectral properties of neutron transport semigroups in L p spaces (1<p<+?), such as stability of essential spectra or critical spectra and related results, can be inferred from the study of two measure convolution operators on $\mathbb{R}^{n}$ .

Calculation of the critical thickness for one-speed neutrons in a reflected slab with backward and forward scattering using modified T N method

Abstract The reflected critical slab problem is investigated in one-speed neutron transport theory for reflecting boundary conditions. The neutron angular flux is expanded in a series of first kind Chebyshev polynomials and isotropic, backward and forward scattering kernel is used as the scattering function. The critical sizes for one-speed neutrons in a uniform finite slab surrounded by identical reflectors from both sides are calculated for various values of the collision parameter, backward and forward scattering and reflection coefficient. Numerical values calculated for the critical slab are presented in the tables and are found to be in good agreement with literature values.

The effect of strongly anisotropic scattering on the critical size of a slab in one-speed neutron transport theory: Modified UN method

The criticality problem for one-speed neutrons in a uniform finite slab is studied in the case of a combination of forward and backward scattering with linearly anisotropic scattering using UN method based on the Chebyshev polynomials of second kind. The effect of the linear anisotropy on the critical thickness of the slab is investigated. The critical slab thicknesses are calculated by using Marshak boundary condition for various values of the anisotropy parameters and they are presented in the tables. In comparison to the results obtained by other methods, the results of this study are in compatible with the former ones.

Kinetic diffusion equation in neutron transport theory

The neutron transport equation can be expressed in a form identical to that of the diffusion equation in the elementary theory of diffusion. The kinetic equation expressed in such a form is defined as the kinetic diffusion equation. The kinetic diffusion equation in a one-dimensional planar geometry is derived and physical validation is given for the parameters in the equation. The physical meaning of the parameters is illustrated for the numerical solution of Milne’s problem. It is shown that the diffusion equation in the elementary theory of diffusion is a particular case of the kinetic diffusion equation. The criteria of realizability of this particular case are determined. The kinetic diffusion equation can be extended to other geometries, including multidimensional.

Method of equivalent differences for the neutron transport equation

For boundary value problems in neutron transport theory, numerical methods are described that are based on the reduction of these problems to systems of nonlinear algebraic equations. Techniques based on the theory of nonlinear positive operators in partially ordered spaces are used to establish major results concerning the existence and uniqueness of solutions and solution-finding methods.

Effects on criticality of selected scattering phase functions in neutron transport equation using the Chebyshev approximation

Abstract Criticality and diffusion problems have been widely studied in neutron transport theory using polynomial expansion techniques in different geometries. In this study, instead of conventional scattering functions, two different phase function will be used in slab geometry transport equation by using the TN approximation and Marshak boundary conditions. Critical half thicknesses of the slab will be calculated. The numerical results obtained will be presented in tables.

Subgroup decomposition method in diffusion theory

A new method for solving the fine-group diffusion equation is developed by estimating the in-scattering and fission source terms with consistent coarse-group diffusion solutions iteratively. This is achieved by extending the subgroup decomposition (SGD) method to diffusion theory initially developed in neutron transport theory. Additionally, a new stabilizing scheme for on-the-fly cross section recondensation based on local fixed source calculations is developed in the subgroup decomposition framework. The method is derived in general geometry and tested in a 1D benchmark problem characteristic of boiling water reactors. It is shown that the SGD method reproduces the standard fine-group results with 3–4 times faster computational speed.

The criticality calculations for one-speed neutrons in a reflected slab with anisotropic scattering using the modified UN method

Abstract The reflected critical slab problem is studied using Chebyshev polynomials of second kind in one-speed neutron transport theory. The variation of the critical thickness of the medium is investigated both in the case of linearly anisotropic scattering and reflecting boundary conditions. The critical half-thicknesses of the slab are calculated for selected values of the reflection coefficient and the anisotropy parameter. Numerical results obtained for the critical half-thickness are identical with the ones obtained by the PN method and the ones available in literature.

About an analytical application of the spherical symmetric diffusion equation

A time-dependent solution of the radiative diffusion energy equation in a hot plutonium sphere close to radiative equilibrium is derived. Based on a general iteration scheme, the analytical solutions correct to first and second order are derived. The outer boundary condition has been adopted from neutron transport theory.

Sobolev Regularity in Neutron Transport Theory

The main purpose of this paper is to extend the W1,p regularity results in neutron transport theory, with respect to the Lebesgue measure due to Mokhtar-Kharroubi, (1991), and to abstract measures covering, in particular, the continuous models or multigroup models. The results are obtained for vacuum boundary conditions as well as periodic boundary conditions. H2 regularity results are derived when the velocity space is endowed with an appropriate class of measures (signed in the multidimensional case).

Quadratic vector equations

We study in a unified fashion several quadratic vector and matrix equations with nonnegativity hypotheses, by seeing them as special cases of the general problem Mx=a+b(x,x), where a and the unknown x are componentwise nonnegative vectors, M is a nonsingular M-matrix, and b is a bilinear map from pairs of nonnegative vectors to nonnegative vectors. Specific cases of this equation have been studied extensively in the past by several authors, and include unilateral matrix equations from queuing problems (Bini, Latouche, and Meini, 2005 [7]), nonsymmetric algebraic Riccati equations (Guo and Laub, 2000 [14]), and quadratic matrix equations encountered in neutron transport theory (Lu, 2005 [23,24]).We present a unified approach which treats the common aspects of their theoretical properties and basic iterative solution algorithms. This has interesting consequences: in some cases, we are able to derive in full generality theorems and proofs appeared in literature only for special cases of the problem; this broader view highlights the role of hypotheses such as the strict positivity of the minimal solution. In an example, we adapt an algorithm derived for one equation of the class to another, with computational advantage with respect to the existing methods. We discuss possible research lines, including the relationship among Newton-type methods and the cyclic reduction algorithm for unilateral quadratic equations.

On the quadratic integral equations and their applications

The class of quadratic integral equations contains, as a special case, numerous integral equations encountered in the theory of radiative transfer, the queuing theory, the kinetic theory of gases and the theory of neutron transport. As a pursuit of this, in the following pages, sufficient conditions are given for the existence of positive continuous solutions to some possibly singular quadratic integral equations. Meanwhile, we prove the existence of maximal and minimal solutions of our problems. The method used here depends on both Schauder and Schauder–Tychonoff fixed point principles. Unlike all previous contributions of the same type, no assumptions in terms of the measure of noncompactness were imposed on the nonlinearity of the given functions. As far as we know, the approach presented in this paper, in particular, the discussion of the existence of maximal and minimal solutions to the quadratic integral equations was never applied in the field of the quadratic integral equations and so is new.