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.     

Transport calculations of neutron energy spectra in a graphite cylinder with a D-T source

Neutron energy spectra resulting from the transport of 14.7 MeV neutrons from a collimated D-T source through a graphite cylinder, have been calculated with the discrete-ordinates 2-D transport dot 4.2 code, with multigroup cross-sections generated using the njoy code from the ENDF/B (IV & V) libraries. The results confirm the conclusion of Goldfeld et al. (1985), that energy spectra at mesh points close to the axis of the system, in front of the collimated beam, consist mainly of one-collision contributions of elastically or inelastically scattered neutrons. Investigation of the dependence of the calculated spectra on the order of truncation of the Legendre polynomials expansion of the flux and of the cross-sections (i.e. the order of scattering) leads to the following observations:

• 1.(a) the P₆ or P₇ approximations seem to be adequate enough for flux calculations, with less than 3% error, in spite of the high degree of the source and the cross-section's anisotropy;
• 2.(b) the calculation error is reduced significantly by increasing the order of scattering from P₄ to P₇, mainly in mesh points close to the axis and of those energies in which the anisotropy of the elastic and discrete level inelastic scattering processes is most pronounced.

Finally, the dot 4.2 calculations are compared with Monte Carlo mcnp calculations; both calculated spectra are in a good agreement.     

Recent progress in the application of the finite element method to the neutron transport equation

The finite element technique is applied to the even-parity form of the neutron transport equation by means of a variational functional. A computer code, FELICIT, has been developed which solves criticality and source problems in R–Z and X–Y geometries assuming isotropic scattering in the lab-system. The code is compared with a number of exact analytic solutions in the P₁ approximation and in exact transport theory thus enabling an estimate of its accuracy to be made. Many tables and graphs are given which illustrate the usefulness of the method for a variety of source and criticality problems in complicated geometries.     

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.     

The integral form of the neutron transport equation for backward–forward scattering in bare spheres

An integral form of the transport equation for bare spheres is developed which incorporates fission plus an arbitrary proportion of isotropic, backward and forward scattering. 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, E., 1995. The effect of strong anisotropic scattering on the critical sphere problem in neutron transport theory using a synthetic kernel. Annals of Nuclear Energy, 22, 671). and some interesting anomalies are found and discussed.     

A multigroup finite-element solution of the neutron transport equation—III: Anisotropy of scattering

The extension of the theory of a multigroup finite-element method of solving the neutron transport equation, to include general anisotropy of scattering and an anisotropic spatially-dependent source, is described. The method, based on a variational principle applied to the even-parity transport equation, employs spherical harmonics for the angular basis functions. To illustrate the development in the associated computer code, three test problems, all including energy-dependent anisotropy, are solved, and the results presented in the form of tables and graphs.     

An extended variational principle for an albedo boundary condition

The K⁺(ø⁺) variational principle for neutron transport maintained by volume and surface sources is extended so that it is applicable to an albedo boundary condition. This condition arises from the use of a forward-backward-isotropic scattering kernel and the subsequent transformation of the transport equation to an equivalent isotropic form. Thus a finite element code for neutron transport which has been established for isotropic scattering can be used to provide solutions for the forward-backward-isotropic case of anisotropic scattering. In this way benchmarks can be found to check finite element codes written specifically for anisotropic scattering.     

Discontinuous 1-D Spherical Transport Solvers using Diffusion Preconditioner and Iterative Methods

Using discontinuous Galerkin finite-element methods to solve the hyperbolic components of the transport operator in the 1-D spherically symmetric case, the convergence behavior of various iterative methods are compared and evaluated. Diffusion synthetic accelerator, a preconditioner based on the diffusion approximation to the transport equation, is presented formally for this geometry for the first time. Compared with classical, finite-difference like methods (diamond difference methods), it is found that DG diffusion based preconditioners performed extremely well in resolving problems with strong scattering effects and material discontinuities.     

The energy-dependent backward-forward-isotropic scattering model with some applications to the neutron transport equation

A multigroup formalism is developed for the backward-forward-isotropic scattering model of neutron transport. Some exact solutions are obtained in two-group theory for slab and spherical geometry. The results are useful for benchmark problems involving multigroup anisotropic scattering.     

Solution of the neutron transport problem with anisotropic scattering in cylindrical geometry by the decomposition method

An analytical solution has been obtained for the one-speed stationary neutron transport problem, in an infinitely long cylinder with anisotropic scattering by the decomposition method. Series expansions of the angular flux distribution are proposed in terms of suitably constructed functions, recursively obtainable from the isotropic solution, to take into account anisotropy. As for the isotropic problem, an accurate closed-form solution was chosen for the problem with internal source and constant incident radiation, obtained from an integral transformation technique and the F_N method.     

Finite element spherical harmonics (PN) solutions of the three-dimensional Takeda benchmark problems

A set of multi-group eigenvalue (K_eff) benchmark problems in three-dimensional homogenised reactor core configurations have been solved using the deterministic finite element transport theory code EVENT and the Monte Carlo code MCNP4C. The principal aim of this work is to qualify numerical methods and algorithms implemented in EVENT. The benchmark problems were compiled and published by the Nuclear Data Agency (OECD/NEACRP) and represent three-dimensional realistic reactor cores which provide a framework in which computer codes employing different numerical methods can be tested. This is an important step that ought to be taken (in our view) before any code system can be confidently applied to sensitive problems in nuclear criticality and reactor core calculations. This paper presents EVENT diffusion theory (P₁) approximation to the neutron transport equation and spherical harmonics transport theory solutions (P₃–P₉) to three benchmark problems with comparison against the widely used and accepted Monte Carlo code MCNP4C. In most cases, discrete ordinates transport theory (S_N) solutions which are already available and published have also been presented. The effective multiplication factors (K_eff) obtained from transport theory EVENT calculations using an adequate spatial mesh and spherical harmonics approximation to represent the angular flux for all benchmark problems have been estimated within 0.1% (100 pcm) of the MCNP4C predictions. All EVENT predictions were within the three standard deviation uncertainty of the MCNP4C predictions. Regionwise and pointwise multi-group neutron scalar fluxes have also been calculated using the EVENT code and compared against MCNP4C predictions with satisfactory agreements. As a result of this study, it is shown that multi-group reactor core/criticality problems can be accurately solved using the three-dimensional deterministic finite element spherical harmonics code EVENT.     

Application of the TN method to critical slab problem for one-speed neutrons with forward and backward scattering

The critical slab problem which includes isotropic forward and backward scattering has been studied in one-speed neutron transport equation using first kind of Chebyshev polynomials. The critical half-thicknesses are computed for different degrees of c and forward and backward scattering with Mark and Marshak boundary conditions in the uniform finite slab. It is shown that T_N method gives accurate results in one-dimensional geometry and the results are agreement P_N approximation.     

Effect of anisotropic scattering in neutronics analysis of BWR assembly

The anisotropic scattering effect to keff is studied for UO₂ and MOX fueled BWR assemblies. The anisotropic scattering effect increases the assembly k_∞ by 0.44% Δk for the UO₂ assembly with 0% void fraction, and by 0.21% Δk for the MOX assembly with 0% void fraction. This is because the anisotropic scattering effect flattens the intra-assembly thermal flux, and the absorption rate in the surrounding water gap is decreased, but the absorption rates in the MOX fuel rods are increased compared to the UO₂ rods. Therefore, the total decrease in absorption rates in the UO₂ assembly is relatively large, and the k_∞ is increased in the UO₂ assembly. The dependence of the anisotropic scattering effect on the void fraction is investigated, and the significant difference of 0.62% Δk/k is found for the 0% and the 80% void fractions. The BWR assemblies with Gd rods are also considered. Furthermore, the usefulness of the transport cross section is investigated, and it is found that the transport cross section gives reasonable anisotropic scattering effect, though not satisfactory.     

Numerical methods for the generation of the spectrum of the multigroup slab-geometry discrete ordinates operator in neutron transport theory

We describe two numerical methods applied to the first-order form of the multigroup slab-geometry discrete ordinates equations modelling fixed-source neutron transport problems with anisotropic scattering. The numerical methods described in this article generate the spectrum and a vector basis for the null space of the multigroup slab-geometry discrete ordinates operator defined in a homogeneous domain. The first method is a more general approach of a numerical method described in a recent work by others. We then come to consider numerical and computational aspects of the first method and we propose a second method. The second method is a multigroup extension of a numerical method described in a more recent work by the present author. In order to provide those interested in implementing either method with a reference set, we present numerical results for some multigroup slab-geometry model problems with anisotropic scattering. We conclude this article with a discussion and directions for future work.     

Calculation of Azimuthally Dependent Thermal-Neutron Albedos for Slab by Method of Invariant Embedding

The method of invariant embedding has been applied to the calculation of differential thermal-neutron albedos for a semi-infinite ordinary concrete slab. The calculations have been performed in both cases of isotropic and anisotropic scattering in the laboratory system.

The calculated albedo data are compared with those obtained by the experiments and the semi-empirical formula fitted the detailed data obtained by Monte Carlo method. The calculated results assuming isotropic scattering are in good overall agreement with the values obtained by Monte Carlo and S_N methods, but there are some errors for azimuthally anisotropic scattering when azimuthal angle becomes large.

In this method, much less computing times within given accuracy are required for azimuthally isotropic scattering, but it is pronounced that the necessary computing times are heavily dependent on N in DP _(N/2)-1 (ξ)T_N(μ) quadrature sets when the azimuthally anisotropic scattering is considered.

It is found that, except for large N for the case of azimuthally anisotropic scattering, the calculation of differential albedo data by using invariant embedding method is much faster than those by using the Monte Carlo and the discrete ordinates methods.     

Analytical solutions to the moment transport equations-II: Multiregion,multigroup 1-D slab,cylinder and sphere criticality and source problems

Analytical multigroup multiregion P_N criticality solutions are obtained for 1-D slabs, cylinders and spheres. A two-group two-/three-region slab with reflective boundary conditions, a one-group linear anisotropic cylinder with Marshak boundary condition and a six-group GODIVA benchmark k_eff problem with Mark, Marshak and importance-weighted boundary conditions are compared with literature results. Importance-weighted boundary conditions yield more accurate GODIVA k_effs than do Marshak or Mark boundary conditions. The analytic P₉ GODIVA k_effs are as accurate as S₆₄ with 160 nodes. A five-region one-group heterogeneous slab-source problem is solved analytically in P_N and double-P_N approximation and compared with F_N. The P_N scalar flux agrees well with F_N, but some angular fluxes are negative. The double-P_NN angular flux solutions are positive in the regions where the distribution is highly decoupled into forward and backward components.     

P1各向异性散射特征线方法研究及应用

基于各向同性散射的中子输运方程特征线方法,计算实际组件能谱时经输运修正后,散射矩阵中P0自散射截面可能出现一定量的负值,影响数值稳定性。本文开发了P1各向异性散射特征线方法,并研制了计算程序PEACH-A。压水堆栅元基准问题的验证结果表明,PEACH-A程序具有较高的计算精度。对典型富集度的UO_2、MOX燃料栅元及其组合问题进行了敏感性分析,结果表明,针对MOX燃料及富集度差异较大的UO_2燃料栅元组合问题有必要采用P1各向异性散射。     

Transport Calculations of MOX and UO2 Pin Cells by the Method of Characteristics

A new transport theory code for two-dimensional calculations of both square and hexagonal fuel lattices by the method of characteristics has been developed. The ray tracing procedure is based on the macroband method, which permits more accurate spatial integration in comparison to the equidistant method of tracing. The neutron source within each region is approximated by a linear function and linearly anisotropic scattering can be optionally accounted for. Efficient new techniques for both azimuthal and polar integration are presented. The spatial discretization problem in case of P ₁-scattering has been studied. Detailed analyses show that the P ¹-scattering in case of regular infinite array of fuel cells is significant, especially for MOX fuel, while the transport correction is inadequate in case of real geometry multi-group calculations. Finally, the complicated nature of the angular flux in MOX and UO₂ fuel cells is demonstrated.     

A new spectral nodal method based on Larsen’s extended diamond approach

In this article, we describe a new spectral nodal method for solving discrete ordinates (S_N) neutron transport problems with anisotropic scattering for arbitrary order N of angular quadrature. The key to our new spectral nodal method is a consistent derivation of nonstandard auxiliary equations that relate angular neutron fluxes only in the upwind directions. These nonstandard equations are angularly coupled extensions of very basic auxiliary equations proposed by Edward W. Larsen in his extended diamond scheme of solving S₂ problems in the presence of scattering and free from spatial truncation error. The resulting method here is also free from spatial truncation error and, in contrast to previously developed spectral nodal methods, it is compatible with an efficient use of iteration on the scattering source and is free from the storage of cell-edge angular fluxes.     

Solution of the radiative heat transfer equation with internal energy sources in a slab by the Green's Functions Decomposition Method for anisotropic scattering

In this work we apply the Green's Function Decomposition Method to solve numerically the radiative transport equation in a slab. The method consists of converting the radiative transport equation into an integral equation and projecting the integral operators involved into a finite dimensional space. This methodology does not involve any a priori discretization on the angular variable μ, requiring only the numerical integration of the kernel on μ. Numerical results are provided for isotropic, linearly anisotropic, and Rayleigh scattering.