Integral transform method for solving multi-dimensional neutron transport problems with linearly anisotropic scattering

A method has been developed to solve the neutron transport equation in multi-dimensional convex and homogenous assemblies with linearly anisotropic scattering. The method consists of solving the Fourier transformed integral transport equations for flux and partial currents and is a generalization of the method developed by Sahni (1972) to treat the one group criticality problem for multi-dimensional geometries. The kernels of the transformed integral equations get factorized into components depending on only one of the dimensions of the assembly. These factorized kernels in each of the dimensions are then decomposed into their respective degenerate forms involving suitable spherical Bessel functions. The transformed flux and partial currents are expanded in a series of products of suitable spherical Bessel functions commensurate with the symmetry and dimensionality of the problem, which can be truncated after very few terms.The one group criticality problem is then converted into the eigenvalue problem of a matrix equation of finite order. The order of this matrix depends upon the truncation order of the transformed flux and partial currents; on the other hand the matrix elements themselves do not depend upon the order of truncation. A striking similarity between one group criticality problem of an infinite rectangular prism of dimensions 2a and 2b along x and y directions and a finite cylinder of diameter 2R and height 2H is brought out, as far as the structure of their matrix equations and calculational procedure of their general matrix elements is concerned. Some results of the criticality problems of the infinite rectangular prism and the finite cylinder are tabulated.     

Solution of one-speed neutron transport equation for strongly anisotropic scattering by TN approximation: Slab criticality problem

In this study, a recently proposed version of Chebyshev polynomial approximation which was used in spectrum and criticality calculations by one-speed neutron transport equation for slabs with isotropic scattering is further developed to slab criticality problems for strongly anisotropic scattering. Backward–forward-isotropic model is employed for the scattering kernel which is a combination of linearly anisotropic and strongly backward–forward kernels. Further to that, the common approaches of using the same functional form for scattering and fission kernels or embedding fission kernel into the scattering kernel even in strongly anisotropic scattering is questioned for T_N approximation via taking an isotropic fission kernel in the transport equation. As a starting point, eigenvalue spectrum of one-speed neutron transport equation for a multiplying slab with different degrees of anisotropy in scattering and for different cross-section parameters is obtained using Chebyshev method. Later on, the spectra obtained for different degree of anisotropies and cross-section parameters are made use of in criticality problem of bare homogeneous slab with strongly anisotropic scattering. Calculated critical thicknesses by Chebysev method are almost in complete agreement with literature data except for some limiting cases. More importantly, it is observed that using a different kernel (isotropic) for fission rather than assuming it equal to the scattering kernel which is a more realistic physical approach yields in deviations in critical sizes in comparison with the values presented in literature. This separate kernel approach also eliminates the slow convergency and/or non-convergent behavior of high-order approximations arising from unphysical eigenspectrum calculations.     

The integral-transform method for solving neutron transport problems with general anisotropic scattering in a cylinder of finite height

In this paper, we present a mathematical technique for solving the integral transport equation for the criticality of a homogeneous cylinder of finite height. The purpose of the present paper is two-fold : firstly, to show that our earlier formalism can be generalized to any order of anisotropy, and secondly to generate the numerical results, which could serve as benchmarks when scattering is linearly anisotropic. We expand the scattering function in spherical harmonics to retain the Lth order of anisotropy. Thereafter, we write the integral transport equations for the Fourier-transformed spherical harmonic moments of the angular flux. In conformity with the integral-transform method for multidimensional geometry, the kernels of these integral equations are represented in their respective factorized form, which consists of a series of products of suitable spherical Bessel functions. The Fourier-transformed spherical harmonic moments are also represented in their separable form by expanding them in a series of products of spherical Bessel functions, commensurate with the symmetry of finite cylindrical geometry. The criticality problem for the cylinder of finite height is then posed as a matrix eigen value problem whose eigen vector is composed of the expansion coefficients mentioned above. The general matrix element is expressed as a product of certain integrals of Bessel functions, which can be evaluated by recursion relations derived in this paper. Finally, a comparison between the present benchmark results and S_N results (twotran) in (r–z) goemetry is presented when scattering is linearly anisotropic.     

The finite-element method for multigroup neutron transport: Anisotropic scattering in 1-D slab geometry

Proof-tests on 1-D multigroup neutron transport problems are reported for strong anisotropic scattering. These tests have been undertaken as part of the validation of the 3-D multigroup finite-element transport code fel tran for ansisotropic scattering media. To illustrate the treatment of within-group and intergroup anisotropic scattering in the finite-element method the relevant theory is outlined. Ingroup scattering is checked using the backward-forward-isotropic (BFI) scattering law for source and eigenvalue problems. With this law anisotropic scattering problems can be transformed into equivalent isotropic scattering problems. In this way the well-validated isotropic scattering version of fel tran is used to validate the anisotropic version. Intergroup scattering effects are checked by solving few-group source problems for P₁ and P₃ scattering and the BFI scattering law. For P₁ and P₃ scattering checks are made with the discrete-ordinate finite-difference code anisn and the spherical harmonics finite-difference code marc/pn. For the BFI scattering law comparison is made with two-group exact solutions of Williams (1985) for 1-D systems.     

On the solution of the dispersion equation for monoenergetic neutron transport with linearly anisotropic scattering

The eigenvalues occurring in the stationary or time dependent, monoenergetic Boltzmann equation with linearly anisotropic scattering are investigated. A detailed analysis is made of the number, nature and behaviour of the eigenvalues for the stationary case. This is applied to an almost identical equation obtained by Williams in the theory of slowing down of particles. In the time dependent case, a semi-analytical proof is given for the existence of complex eigenvalues, which are not encountered for isotropic scattering.     

Transmission probability method based on triangle meshes for solving unstructured geometry neutron transport problem

In the advanced reactor, the fuel assembly or core with unstructured geometry is frequently used and for calculating its fuel assembly, the transmission probability method (TPM) has been used widely. However, the rectangle or hexagon meshes are mainly used in the TPM codes for the normal core structure. The triangle meshes are most useful for expressing the complicated unstructured geometry. Even though finite element method and Monte Carlo method is very good at solving unstructured geometry problem, they are very time consuming. So we developed the TPM code based on the triangle meshes. The TPM code based on the triangle meshes was applied to the hybrid fuel geometry, and compared with the results of the MCNP code and other codes. The results of comparison were consistent with each other. The TPM with triangle meshes would thus be expected to be able to apply to the two-dimensional arbitrary fuel assembly.     

Two-region Milne problem with anisotropic scattering

We consider the two-region Milne problem, defined as the steady-state monoenergetic linear transport problem for two adjoining homogeneous source-free half-spaces, with particle source coming from infinity in the positive half-space. The integral version of the transport equation is solved using trial functions based on Case’s eigen modes and exponential integral function. The solution of the Milne problem is formulated in terms of characteristic parameters such as the extrapolation length, the fractional scalar flux discontinuity and the fractional current discontinuity. Numerical results for the analytically evaluated parameters are then presented. Some of our numerical results are compared with the available published results.     

TN approximation on the critical size of time-dependent,one-speed and one-dimensional neutron transport problem with anisotropic scattering

The criticality problem is studied based on one-speed time-dependent neutron transport theory, for a uniform and finite slab, using the Marshak boundary condition. The time-dependent neutron transport equation is reduced to a stationary equation. The variation of the critical thickness of the time-dependent system is investigated by using the linear anisotropic scattering kernel together with the combination of forward and backward scattering. Numerical calculations for various combinations of the scattering parameters and selected values of the time decay constant and the reflection coefficient are performed by using the Chebyshev polynomials approximation method. The results are compared with those previously obtained by other methods which are available in the literature.     

Improved nodal expansion method for solving neutron diffusion equation in cylindrical geometry

The challenges encountered in the development of nodal expansion method (NEM) in cylindrical geometry and the method to circumvent these difficulties are introduced and discussed in this paper. Due to the fact that the azimuthal term contains a factor 1/r², the traditional transverse integration fails to produce a 1D transverse integrated equation in θ-direction; a simple but effective approach is employed to obtain the θ-directional transverse integration equation. When the traditional polynomials are used to solve the 1D transverse integral equation in r-direction, some additional approximations, which may undermine the precision of the method, are required in the derivation of the moment equations; in order to preserve the accuracy of calculations, the special polynomial approximation is used to solve the 1D transverse integrated equations in r-direction. Moreover, the Row-Column iterative scheme, which is considered to be the more efficient and convenient schemes in cylindrical geometry, is used to solve the partial currents equations. An improved NEM for solving the multidimensional diffusion equation in cylindrical geometry is implemented and tested. And its accuracy and efficiency are demonstrated through several benchmark problems.     

Multigroup multi-layer models of neutron reflection and transmission for reactor transport calculations with anisotropic scattering

In this article, we extend the one-speed multi-layer models to neutron reflection and transmission developed in our earlier work (de Abreu, M.P., 2005. Multi-layer models to neutron reflection and transmission for whole-core transport calculations, Annals of Nuclear Energy 32, 215) to multigroup transport theory. We begin by considering a two-layer boundary region, and we develop for such a region discrete ordinates models to the diffuse reflection and transmission of neutrons for multigroup nuclear reactor core problems with anisotropic scattering. We perform numerical experiments to show that our models to neutron reflection and transmission can be used to replace efficiently and accurately two nonactive boundary layers in whole-core transport calculations. We conclude this article with an inductive extension of our two-layer results to a boundary region with an arbitrary number of layers.     

The critical problem with high-order anisotropic scattering

A synthetic scattering kernel is used with one-speed transport theory to evaluate the effect of high-order anisotropic scattering on the critical half thickness for a multiplying, unreflected, plane-parallel medium.     

Spectral Green's function method for neutron transport: isotropic,forward, and backward scattering in 1-D slab geometry

Numerical solutions of one-group and one-dimensional neutron transport problems are reported for isotropic, forward, and backward scattering. Numerical solution is carried out by using two different methods, the SGF “ spectral Green's function ” method and the DD “ diamond-difference” scheme, to test the accuracy of the results. Results of cell-edge scalar fluxes obtained for both methods are presented in the tables.     

The analytic nodal method in cylindrical geometry

Nodal diffusion methods have been used extensively in nuclear reactor calculations, specifically for their performance advantage, but also for their superior accuracy. More specifically, the Analytic Nodal Method (ANM), utilising the transverse integration principle, has been applied to numerous reactor problems with much success. In this work, a nodal diffusion method is developed for cylindrical geometry. Application of this method to three-dimensional (3D) cylindrical geometry has never been satisfactorily addressed and we propose a solution which entails the use of conformal mapping. A set of 1D-equations with an adjusted, geometrically dependent, inhomogeneous source, is obtained. This work describes the development of the method and associated test code, as well as its application to realistic reactor problems. Numerical results are given for the PBMR-400 MW benchmark problem, as well as for a “cylindrisized” version of the well-known 3D LWR IAEA benchmark. Results highlight the improved accuracy and performance over finite-difference core solutions and investigate the applicability of nodal methods to 3D PBMR type problems. Results indicate that cylindrical nodal methods definitely have a place within PBMR applications, yielding performance advantage factors of 10 and 20 for 2D and 3D calculations, respectively, and advantage factors of the order of 1000 in the case of the LWR problem.     

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

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 P₁ 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_N) method.