On eigensolutions of the one-speed neutron transport equation in plane geometry (original) (raw)
A family of transport equations in neutron transport theory
Annals of Nuclear Energy, 1991
The backward formulation of problems in neutron transport is considered. Boundary conditions are developed from the adjoint system which in turn is rigorously obtained from a variational principle. The variational principle is derived by constraining the flux with the forward transport equation and aff'diated boundary/initial conditions, avoiding physically motivated Importance arguments. Extremising the constrained functional yields both the forward and adjoint equations as well as boundary/'mitial conditions as the Euler Lagrange system. The backward formulation provides a means of developing exact equations for the scalar flux and emergent current as well as internal distributions such as the slowing down density.
2020
A new approach for the development of a numerical method of spectral nodal class for the solution of multigroup, anisotropic slab geometry, discrete ordinates transport problems with fixed-source is analyzed in this paper. The method, denominated Spectral Deterministic Method (SDM), is based on the spectral analysis of the neutron transport equations in the formulation of discrete ordinates (SN). The unknowns in the methodology are the cell-edge, and cell average angular fluxes, the numerical values computed for these quantities concur with the analytic solution of the discrete ordinate’s equation. Numerical results are given and compared with the traditional finemesh DD method, the spectral nodal method, spectral Green’s function (SGF) and the FN method to illustrate the method’s numerical accuracy.
Annals of Nuclear Energy, 2007
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.
Journal of Quantitative Spectroscopy and Radiative Transfer, 2002
The one-speed time-dependent and stationary neutron transport equation in spherical geometry with forward scattering is considered. A formal equivalence between the transport equations for a critical and for a decaying system is established. By considering the pseudo-slab problem the scaled transport equation is solved using the F N method. Numerical values of radii for a critical and time-dependent systems are tabulated as a function of the scattering parameters and the fundamental decay constant. Some of the results are discussed and compared with others obtained using various methods. The results agree for four or ÿve signiÿcant ÿgures with the published results. It is shown that the F N method yields good numerical results for the problem considered. Finally, a few remarks about the e ect of the forward anisotropy on the radius is also given.
Annals of Nuclear Energy, 1991
The integral transform (IT) method has been generalized to obtain spherically symmetric as well as asymmetric eigenmodes of the one-speed, integral neutron transport equation in a homogeneous sphere with isotropic scattering. The IT method as developed originally is applicable only to symmetric modes with real eigenvalues below the Corngold limit. The method presented in this paper can be used to investigate the discrete time eigenvalue spectrum in the full complex domain. The total neutron flux is expanded in spherical harmonics of the spatial unit vector r/r and decoupled integral equations are obtained for the spherical harmonic moments. The kernel in each of these equations is then decomposed in a bilinear form. This decomposition reduces the integral equation to an infinite system of algebraic equations, which is then truncated for numerical solution. The matrix elements for this sytem of equations are shown to be identical to those encountered in the case of spherically symmetric modes. These are evaluated by generalizing Hembd's procedure to the complex domain. The time eigenvalues corresponding to the first few eigenmodes are presented as a function of system size.
Progress in Nuclear Energy, 1996
The paper reviews some recent adva,nces pertaining to the criticality and time eigenvalues of the one-speed tra,nsport equation. We discuss the nature, real or complex of the k,~f, c and X eigenvalues and their variation with size and other parameters. Our empha.sis is on accurate numerical results that display some novel features. In some cases these are explained theoretically. We also discuss the recent findings about continuous spectrum of time eigenvalue problem as it consists of a set of discrete lines rather than a half plane.
Boundary and interface function method for one dimensional neutron transport
1979
A new numerical method, the Boundary and Interface Function (BIF). Method, for solving the one-dimensional neutron transport equation is formulated. The method derives its mathematical foundation from a two-sided Laplace transform in the space variable with a Fourier transform in the angular variable. The mathematical solution requires that the scalar flux at the boundaries and interfaces (the boundary and interface functions) satis fy systems of singular integral equations. The theory shows that a rather simple numerical method for the solution of the transport equation is feasible which uses only edge-and interface function approximations and no interior space points in the main program. This drastically reduces the order of the associated matrix problem.as well as improving the computational execution time on the computer. A computer code has been implemented and numerical results are presented and compared with a reference solution and solutions obtained by conventional methods for one and two region problems.
Nuclear Science and Engineering, 1977
The singular-eigenfunction-expansion method and the principle of invariance are combined to reduce the two-half-space Milne problem to a regular computational form in the two-group isotropic scattering model. The method used here consists in considering a problem of two contiguous half-spaces with surface sources at the interface. The problem is equivalent to the Milne problem in the sense that the expansion coefficients are to be determined from the same equation. The emergent distributions are obtained from coupled regular integral equations. The expansion coefficients can then be obtained using the halfrange orthogonality relation of the eigenfunctions. Numerical results are reported for light-water media.
Study of the Eigenvalue Spectra of the Neutron Transport Problem in PN Approximation
EPJ Web of Conferences, 2021
The study of the steady-state solutions of neutron transport equation requires the introduction of appropriate eigenvalues: this can be done in various different ways by changing each of the operators in the transport equation; such modifications can be physically viewed as a variation of the corresponding macroscopic cross sections only, so making the different (generalized) eigenvalue problems non-equivalent. In this paper the eigenvalue problem associated to the time-dependent problem (α eigenvalue), also in the presence of delayed emissions is evaluated. The properties of associated spectra can give different insight into the physics of the problem.