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 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.     

Neutron conservation and boundary conditions in the PN approximation for critical slabs—I

A global conservation equation is derived for the one-velocity P_N approximation for the critical slab with isotropic scattering. Its relationship to various sets of boundary conditions in the P₃ approximation (e.g. Marshak or Pomraning-Federighi) is examined, and its implications for the infinite slab (Milne problem) are discussed.     

The critical slab with the backward-forward-isotropic scattering model

The variation of the critical slab thickness from backward to forward scattering is studied for specified isotropic scattering. Numerical results are obtained from the zeroth and first-order approximate analytical expressions, derived using the method of elementary solutions. It is shown that the critical slab thickness increases monotonically with increasing forward anisotropy.     

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.     

On multi-media calculations in the theory of neutron diffusion

The F_N method is used to solve the critical problem for a three-region reactor and to compute the disadvantage factor required for thermal utilization calculations. Anisotropic scattering is allowed and numerical results are given.     

The criticality problems with the FN method for the FBIS model

In one-speed, time-independent, neutron transport theory, the F_N method is used for the FBIS (forward-backward-isotropic scattering) model to reinvestigate the behaviour of the critical size in plane and spherical geometries. For the FIS (forward-isotropic scattering) model the numerical results are compared with previously obtained variational results and it is shown that they are in agreement. For the BIS (backward-isotropic scattering) model exact results are obtained and compared with the first-order approximate results obtained using the method of elementary solutions.     

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.     

One-speed criticality problems for a bare slab and sphere: Some benchmark results

Benchmark results for the two canonical problems of critical slab and sphere are reported. Three different approaches, use of the X-function and Neumann iteration, F_N method, and discrete-ordinate method, are explored. It is found that to the orders of the approximations considered here, the critical half-thickness and critical radius values obtained through the three approaches are consistently accurate from 9 to 12 significant figures, and agree amongst themselves. The accuracy for the corresponding values for neutron density using X-function, and discrete-ordinate method (the only two explored for the density computations) is 9–10 figures. In several instances, the results constitute an improvement on the previously reported benchmark results in literature.     

The FN method for solving the critical problem for a slab with a finite reflector

The F_N method is used to solve the critical problem for a slab reactor with a finite reflector. Numerical results for the critical thickness are shown for different values of the mean number of secondary neutrons per collision and reflector thickness using various orders of the F_N approximation. The results show the capability of the method to provide accurate solutions.     

The discrete ordinates method compared to Carlvik's and Syros's methods for anisotropic neutron scattering in slabs

The discrete ordinates method has been used to determine the criticality factor of infinite slabs with monoenergetic neutrons scattering anisotropically. The slab thickness was 0.2, 1, 2 or 20 mean free paths and the average cosine of the scattering angle 0, 0.1 or 0.2. The calculations were extended up to the S₃₂ approximation for the thinnest slab. It was found that the S_n results extrapolate towards the values obtained by Dahl and Sjöstrand using Carlvik's method. The reason why Syros and Theocharopoulos obtain deviating results is not known.     

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.     

A closed-form solution for the two-dimensional transport equation by the LTSN nodal method in the energy range of Compton effect

In the present work we report on a closed-form solution for the two-dimensional Compton transport equation by the LTS_N nodal method in the energy range of Compton effect. The solution is determined using the LTS_N nodal approach for homogeneous and heterogeneous rectangular domains, assuming the Klein–Nishina scattering kernel and a multi-group model. The solution is obtained by two one-dimensional S_N equation systems resulting from integrating out one of the orthogonal variables of the S_N equations in the rectangular domain. The leakage angular fluxes are approximated by exponential forms, which allows to determine a closed-form solution for the photons transport equation. The angular flux and the parameters of the medium are used for the calculation of the absorbed energy in rectangular domains with different dimensions and compositions. In this study, only the absorbed energy by Compton effect is considered. We present numerical simulations and comparisons with results obtained by using the simulation platform GEANT4 (version 9.1) with its low energy libraries.     

Cellular neural networks (CNN) simulation for the TN approximation of the time dependent neutron transport equation in slab geometry

This paper describes the application of a multilayer cellular neural network (CNN) to model and solve the time dependent one-speed neutron transport equation in slab geometry. We use a neutron angular flux in terms of the Chebyshev polynomials (T_N) of the first kind and then we attempt to implement the equations in an equivalent electrical circuit. We apply this equivalent circuit to analyze the T_N moments equation in a uniform finite slab using Marshak type vacuum boundary condition. The validity of the CNN results is evaluated with numerical solution of the steady state T_N moments equations by MATLAB. Steady state, as well as transient simulations, shows a very good comparison between the two methods. We used our CNN model to simulate space–time response of total flux and its moments for various c (where c is the mean number of secondary neutrons per collision).     

The CN method applied to the stationary problem of particle reflection by a purely scattering or weakly absorbing half-space; extension to the asymptotic time-dependent case

The application of the classical C_N method fails for a pure scattering or weakly absorbing medium. The asymptotic C_N method, in stationary mode, is the way to solve the transport equation in this limiting case, when c is equal or very close to 1. The asymptotic method allows us to obtain the asymptotic time-dependent emergent angular distribution for a given impinging angular intensity at t = 0 whatever c may be. The numerical results for the classical and asymptotic methods are consistent in the overlapping range.     

An overview of the Boltzmann transport equation solution for neutrons, photons and electrons in Cartesian geometry

Questions regarding accuracy and efficiency of deterministic transport methods are still on our mind today, even with modern supercomputers. The most versatile and widely used deterministic methods are the P_N approximation, the S_N method (discrete ordinates method) and their variants. In the discrete ordinates (S_N) formulations of the transport equation, it is assumed that the linearised Boltzmann equation only holds for a set of distinct numerical values of the direction-of-motion variables. In this paper, looking forward to confirm the capabilities of deterministic methods in obtaining accurate results, we describe the recent advances in the class of deterministic methods applied to one and two dimensional transport problems for photons and electrons in Cartesian Geometry. First, we describe the Laplace transform technique applied to S_N two dimensional transport equation in a rectangular domain considering Compton scattering. Next, we solved the Fokker-Planck (FP) equation, an alternative approach for the Boltzmann transport equation, assuming a mono-energetic electron beam in a rectangular domain. The main idea relies on applying the P_N approximation, a recent advance in the class of deterministic methods, in the angular variable, to the two dimensional Fokker-Planck equation and then applying the Laplace Transform in the spatial x variable. Numerical results are given to illustrate the accuracy of deterministic methods presented.     

An overview of the Boltzmann transport equation solution for neutrons,photons and electrons in Cartesian geometry

Questions regarding accuracy and efficiency of deterministic transport methods are still on our mind today, even with modern supercomputers. The most versatile and widely used deterministic methods are the P_N approximation, the S_N method (discrete ordinates method) and their variants. In the discrete ordinates (S_N) formulations of the transport equation, it is assumed that the linearised Boltzmann equation only holds for a set of distinct numerical values of the direction-of-motion variables. In this paper, looking forward to confirm the capabilities of deterministic methods in obtaining accurate results, we describe the recent advances in the class of deterministic methods applied to one and two dimensional transport problems for photons and electrons in Cartesian Geometry. First, we describe the Laplace transform technique applied to S_N two dimensional transport equation in a rectangular domain considering Compton scattering. Next, we solved the Fokker–Planck (FP) equation, an alternative approach for the Boltzmann transport equation, assuming a mono-energetic electron beam in a rectangular domain. The main idea relies on applying the P_N approximation, a recent advance in the class of deterministic methods, in the angular variable, to the two dimensional Fokker–Planck equation and then applying the Laplace Transform in the spatial x variable. Numerical results are given to illustrate the accuracy of deterministic methods presented.     

Neutron conservation and boundary conditions in the PN approximation for critical slabs—II

The usual Marshak and Pomraning-Federighi (importance-weighted) free-surface boundary conditions for a critical slab are modified. For the infinite slab, neutron conservation and the exact extrapolation distance are imposed. A minimization criterion is used to complete the required set of equations. These boundary conditions are compared with the usual ones in the P₃ and P₅ approximations with respect to critical size and conservation for finite slabs.     

Calculation of extrapolation distances by low order Chebyshev polynomial approximation of transport equation in slab geometry

In this study, the problem of extrapolated end point has been studied in one-speed neutron transport equation with isotropic scattering by using the Chebyshev polynomial approximation which is called T_N method. Assuming neutrons of one speed, extrapolated end point are calculated for the uniform finite slab using Mark and Marshak type vacuum boundary conditions. It is shown that low order T_N method gives very good results of low order spherical harmonics approximation and diffusion theory for extrapolation of the flux of neutrons leaking from the medium. We present an alternative method which is similar to P₁ method to calculate the extrapolation distances z₀. Moreover, we prefer new solution of transport equation in one-dimensional slab geometry.