One-speed Neutron Transport Equation Research Articles

Case’s Method for Anlı-Güngör Scattering Formula

Case method is a powerful method in solving one-speed neutron transport equation. The method can be applied to one-speed neutron transport problems and pseudo- geometry problems. The method basis on the usage of Case’s eigenfunctions and the orthogonality relations with the certain boundary conditions according to the interested problem. The scattering effects can be investigated via Mika scattering formula and also İnönü’s scattering formula. In this study Case method’s formalism is derived by using Anlı-Güngör scattering formula as an analogue of Mika’s scattering function. This study is about the Case’s eigenfunctions, normalization relation and the orthogonality properties among these eigenfunctions and, moreover; Case’s eigenfunctions and the orthogonality properties must be rewritten according to the studied scattering order as the number of scattering order increase in Anlı-Güngör scattering formula.

A numerically stable spherical harmonics solution for the neutron transport equation in a spherical shell

A numerically stable version of the spherical harmonics (PN) method for solving the one-speed neutron transport equation with Lth order anisotropic scattering in a spherical shell is developed. Implementing a stable PN solution for this problem is a challenging task for which no satisfactory answer has been given in the literature. The approach used in this work follows and generalizes a previous work on a problem whose domain is defined by the exterior of a sphere. First, a transformation is used to reduce the original transport equation in spherical geometry to a plane-geometry-like transport equation, where the angular redistribution term in spherical geometry is treated as a source. Then, a PN solution in plane geometry given by a combination of the solution of the associated homogeneous equation and a particular solution is developed. This is followed by a post-processing step which is very effective in improving the PN solution. An additional amount of work with respect to that required for solving problems in plane geometry occurs in the form of a system of N+1 Volterra integral equations of the second kind that has to be solved for the coefficients of the particular solution. The proposed approach has, in any case, the merit of avoiding the ill-conditioning caused by the presence of modified spherical Bessel functions in the standard PN solution, as demonstrated by numerical results tabulated for some test cases.

Solution of the one-speed neutron transport equation in anisotropic scattering medium with a Pomraning-Eddington approximation approach

In this work, a combined modified K-integral approach and Pomraning-Eddington approximation to solve the neutron transport equation for general anisotropic scattering law is presented. This combined approach leads to the establishment of general expressions of the even e(x, μ) and odd o(x, μ) functions of the angular variable μ. Both functions are slowly varying in x. Furthermore, these expressions can be reduced to those reported in literature. In addition, the parameters such as and with p = 1, 2, … are obtained without any integral calculus and the variables we used α1, α2, r1 and r2 verify the relation . As application, diffusion lengths are calculated for linear scattering and the neutron heat fluxes are evaluated graphically for four-parameter scattering phase function.

Modified FN solution of the neutron transport equation for the Milne problem with FBIS kernel

The solution of the Milne problem is studied by one-speed neutron transport equation in plane geometry with Inonu’s scattering kernel, which is known as a linear combination of the forward, backward and isotropic scattering kernel (FBIS kernel). The solution of the neutron transport equation with Inonu’s scattering kernel can be written in terms of the solution of the neutron transport equation for isotropic scattering case. The extrapolation distance is calculated with modified FN (or $$H_{\mathrm{N}}$$ ) method. The numerical values of the extrapolation distance are obtained depending on the secondary neutron numbers and anisotropy coefficients and compared with the available data in the literature values.

On eigenvalue problems of the one-speed neutron transport equation for isotropic scattering

In this paper, we consider eigenvalue problems of the one-speed neutron transport equation with isotropic scattering in a steady state and prove the Rayleigh–Faber–Krahn type inequality for the first eigenvalue.

A new analytic solution of the one-speed neutron transport equation for adjacent half-spaces with isotropic scattering

This paper derives a new analytic solution of the one-speed neutron transport equation for two isotropically scattering half-spaces, where each half-space has its own isotropic internal source. The constant source problem has been studied previously by Auerbach (1961), and this analysis has been generalised to consider a source whose strength varies linearly with position. This problem arises in connection with assessing the accuracy of linear treatments of the thermal emission source in Monte Carlo thermal radiation transport algorithms. Simple expressions for the angular flux at the interface as well as its zeroth and first moments are provided as a function of the scattering ratios in the two half-spaces.

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.

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

Abstract The one-speed neutron transport equation which includes isotropic forward and backward scattering has been studied using the TN approximation. Critical thicknesses for different values of the reflection coefficient R and c values have been calculated. The study shows that the TN method gives nearly the same results as the PN approximation in one-dimensional geometry. The results obtained in this study are presented in tables and are compared with PN approximation results.

The albedo problem for pure-quadratic anisotropic scattering with İnönü scattering

Abstract In this study the half-space albedo problem has been investigated for highly anisotropic scattering. For this highly anisotropic scattering the İnönü scattering has been added to the pure-quadratic anisotropic scattering. The general solution of the one-speed neutron transport equation can be written as linear combination of the solutions which involve these scatterings. Hence, Case's eigenfunctions and the related semi-analytical methods can be used to get the albedo calculations. Some selected numerical data have been tabulated. To compare these results with the İnönü scattering values the linear interpolation technique has been used. Successful results have been achieved.

Efficiency of reflection coefficients in the isotropic neutron transport equation and computation of the critical-half thicknesses

Abstract The critical slab problem of the reflected isotropic one-speed neutron transport equation is solved with the Chebyshev polynomial approximation. The efficiency of the reflection coefficient, R, in the neutron transport equation is obtained for different c values. For the solution, the Mark boundary condition is used. The values obtained from this approximation are compared with results obtained by the spherical harmonics method.

Calculating near-singular eigenvalues of the neutron transport operator with arbitrary order anisotropic scattering

The homogeneous one-speed neutron transport equation in plane geometry can be solved using a separation of variables technique. In the case of isotropic scattering and a sub-critical system, it is well known [15,5] that there are exactly two discrete eigenvalues with opposite sign associated to this separation procedure. In the case of arbitrary order anisotropic scattering, there can be many more of those plus–minus pairs of discrete eigenvalues. When in a subsequent step, the corresponding eigenfunctions are used as a basis set for the expansion of a general solution to the neutron transport equation, it is of utmost importance to be able to find all discrete eigenvalues in order to have a complete set. In this paper we briefly describe the three step procedure we have developed to calculate all discrete eigenvalues. During numerical tests with this procedure, we found that there exist cases where there is a discrete eigenvalue located extremely close to the singular point at unity. The main part of this paper is devoted to the description of how we needed to modify our routines in order to calculate these so-called near-singular eigenvalues. Our most important boundary condition was that we did not wish to resort to high-precision fixed point arithmetic but would solely rely on IEEE 754 double precision arithmetic.

A polynomial analytical solution to the one-speed neutron transport equation in slab geometry under inherently verified Mark-Marshak boundary conditions

A polynomial analytical solution to the one-speed neutron transport equation in slab geometry under inherently verified Mark-Marshak boundary conditions

Unified treatment of the P (λ) n approximation to solve the reflected slab criticality problem with strong anisotropy

Abstract Ultraspherical polynomial P (λ) n approximations are used in reflected slab criticality calculations for strongly anisotropic scattering. A unique formulation is developed for reflected slab criticality conditions whose special cases are spherical harmonics approximations, Chebyshev polynomial approximations of first and second kind. It is shown in the calculations that all ultraspherical polynomial approximations are equivalent regarding the convergence of the solution of one-speed neutron transport equation for various degrees of anisotropy and cross section parameters. Although some ultraspherical polynomials approximations converge in the mean as the order of approximation is increased and some converge uniformly, they all converge to the same critical thickness which implies their equi-convergence. It is also demonstrated that the first order approximation P (λ) 1 would give the exact critical thickness for a certain value of the parameter λ.

Cellular neural networks (CNN) simulation for the T N 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 complete algorithm could be implemented using very large-scale integrated circuit (VLSI) circuitry. The efficiency of the calculation method makes it useful for neutron transport calculations.

Chebyshev polynomial (TN ) approximation to neutron transport theory and application to the critical slab problem

Abstract The critical slab problem has been studied in the one-speed neutron transport equation with isotropic scattering by using the first kind of Chebyshev Polynomials. The moment criticality solutions were obtained for the uniform finite slab using Mark and Marshak type vacuum boundary conditions. The results obtained by this approximation are presented in tables which also include the results obtained by the PN method for comparison.

Neutron transport through semi-infinite continuous stochastic media using Gaussian statistics

The stationary solution of the one-speed neutron transport equation in a semi-infinite stochastic medium with linear anisotropic scattering is considered. The cross-section function of the medium is assumed to be a continuous random function of position with fluctuations about the mean taken as Gaussian distributed. The joint probability distribution function of these Gaussian random variables is used to calculate the ensemble-averaged quantities, such as radiant neutron energy and net neutron flux, for an arbitrary correlation function. The problem is solved at first in the deterministic case, then the solution is averaged using Gaussian joint probability distribution function. A modified Gaussian probability distribution function is also used to average the solution. Numerical results are given for the sake of comparison.

Jacobi polynomials approximation to the one-speed neutron transport equation

Jacobi polynomials, the largest family of classical orthogonal polynomial sequence, are used to obtain eigenspectrum of one-speed neutron transport equation for strongly anisotropic scattering. Previously detailed Ultraspherical or Gegenbauer polynomials approximation including spherical harmonics, Chebyshev polynomials approximation of first and second kinds is a special case of Jacobi polynomials approximation in which the two variables of the Jacobi polynomials are equal. Eigenspectrum calculations using the so-called Jacobi polynomials approximation have demonstrated that plane symmetrical systems prefer Ultraspherical polynomials approximation since their well-known peculiarity or symmetry property meets the requirement of symmetry in angular flux with respect to spatial coordinates for every order of approximation. In the more general case when two variables of Jacobi polynomial are unequal, symmetry in eigenvalues disappear in the low-order approximations. However, it has been shown that symmetrical eigenvalue pairs are reached asymptotically as the order of approximation is increased for unequal variables too. Jacobi polynomials approximation is further applied to homogeneous slab criticality problem with strongly anisotropic scattering and reflected boundaries. Additionally, very useful analytical recursive relations for the calculation of various types of integrals involving Jacobi polynomials are derived which are needed in the eigenspectrum and criticality calculations.

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 P 1 method to calculate the extrapolation distances z 0. Moreover, we prefer new solution of transport equation in one-dimensional slab geometry.

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.

Spectral [formula omitted] approximation(s) to the transport problems in spherical media

In this study, the applicability of ultraspherical-polynomials ( P N ( λ ) ) approximation as a spectral approach to spherical media transport/transfer problems is investigated for strongly anisotropic scattering media in which the scattering kernel is an admixture of linearly anisotropic and strongly backward/forward anisotropic parts. It is shown that the P N method (corresponding to the sub-case λ→1/2 in the P N ( λ ) approximation), among all ultraspherical-polynomials methods, is the unique method converting the one-speed neutron transport equation into an algebraic eigenvalue problem in spectral functions G n ( ν ) . It is also demonstrated that the T N method (corresponding to the sub-case λ→0 in the P N ( λ ) approximation) is another algebraic spectral approach suitable for spherical transport/transfer problems with restriction of strongly peaking anisotropy in scattering. The P N ( λ ) approximation as λ→−1/2 is found to be a prospective candidate, which can be converted into an algebraic spectral case by forcing some spectral functions into pre-selected proper forms. Even though the P N ( λ ) approximations applicable as a spectral approach to the spherical media transfer problems are shown to be limited to the sub-cases aforementioned, the unifying P N ( λ ) formalism makes it possible to study spherical media transfer problems in general. As an example, the P N ( λ ) method with any variable λ is demonstrated to be applicable to the bare homogeneous sphere criticality problems by using pseudo-slab approach. Critical radii calculated are in very close agreement with literature values and equiconvergent.