Synthetic Kernel Approximation Research Articles

Application of alternative synthetic kernel approximation to radiative transfer in regular and irregular two-dimensional media

Alternative synthetic kernel (ASKN) approximation, just as the standard SKN method, is derived from the radiative integral transfer equations in full 3D generality. The direct and diffuse terms of thermal radiation appear explicitly in the radiative integral transfer equations as surface and volume integrals, respectively. In standard SKN method, the approximation is employed to the diffuse terms while direct terms are evaluated analytically. The alternative formulation differs from the standard one in that the direct radiation wall contributions are also approximated with the same spirit of the synthetic kernel approximation. This alternative formulation also yields a set of coupled partial differential-the ASKN–equations which could be solved using finite volume methods. This approximation is applied to radiative transfer calculations in regular and irregular two-dimensional absorbing, emitting and isotropically scattering media. Four benchmark problems-one rectangular and three irregular media-are considered, and the net radiative flux and/or incident energy solutions along the boundaries are compared with available exact, standard discrete ordinates S4 and S12, modified discrete ordinates S4, Monte Carlo and collocation spectral method to assess the accuracy of the method. The ASKN approximation yields ray effect free incident energy and radiative flux distributions, and low order ASKN solutions are generally better than those of the high order standard discrete ordinates method.

Nodal synthetic kernel (N-SKN) method for solving neutron transport equation in one- and two-dimensional X–Y geometries

In this study, a nodal method based on the synthetic kernel (SKN) approximation is presented for solving the neutron transport equation in one- and two-dimensional cartesian geometries. The two-dimensional neutron transport equation for a node is transformed to one-dimensional transport equation based on the face-averaged scalar flux and the current. At the node interfaces, DP1 expansion is employed to the surface angular fluxes in conjunction with isotropic angular dependence of the transverse leakage term. The one-dimensional integral transport equation is obtained in terms of the node-face-averaged incoming/outgoing neutron flux and the currents. The synthetic kernel approximation is employed to the transport kernels and nodal-face contributions. The resulting SKN equations are solved analytically. One-dimensional interface-coupling nodal SK1 and SK2 equations (incoming/outgoing flux and current) are derived for the small nodal-mesh limit. These equations have simple recursive forms which do not pose burden on either the memory or the computational time. The method was applied to one- and two-dimensional benchmark problems and compared with the solutions obtained with nodal integral method.

Nodal synthetic kernel (N-SKN) method for solving radiative heat transfer problems in one- and two-dimensional participating medium with isotropic scattering

In this study, a nodal method based on the synthetic kernel (SKN) approximation is developed for solving the radiative transfer equation (RTE) in one- and two-dimensional cartesian geometries. The RTE for a two-dimensional node is transformed to one-dimensional RTE, based on face-averaged radiation intensity. At the node interfaces, double P1 expansion is employed to the surface angular intensities with the isotropic transverse leakage assumption. The one-dimensional radiative integral transfer equation (RITE) is obtained in terms of the node-face-averaged incoming/outgoing incident energy and partial heat fluxes. The synthetic kernel approximation is employed to the transfer kernels and nodal-face contributions. The resulting SKN equations are solved analytically. One-dimensional interface-coupling nodal SK1 and SK2 equations (incoming/outgoing incident energy and net partial heat flux) are derived for the small nodal-mesh limit. These equations have simple algebraic and recursive forms which impose burden on neither the memory nor the computational time. The method was applied to one- and two-dimensional benchmark problems including hot/cold medium with transparent/emitting walls. The 2D results are free of ray effect and the results, for geometries of a few mean-free-paths or more, are in excellent agreement with the exact solutions.

The SKN Approximation for Solving Radiative Transfer Problems in Absorbing, Emitting, and Linearly Anisotropically Scattering Plane-Parallel Medium: Part 2

The SKN (Synthetic Kernel) approximation is proposed for solving radiative transfer problems in linearly anisotropically scattering homogeneous and inhomogeneous participating plane-parallel medium. The radiative integral equations for the incident energy and the radiative heat flux using synthetic kernels are reduced to a set of coupled second-order differential equations for which proper boundary conditions are established. Performance of the three quadrature sets proposed for isotropic scattering medium are further tested for linearly anisotropically scattering medium. The method and its convergence with respect to the proposed quadrature sets are explored by comparing the results of benchmark problems using the exact, P11, and S128 solutions. The SKN method yields excellent results even for low orders using appropriate quadrature set.

Generalized Synthetic Kernel Approximation for Elastic Moderation of Fast Neutrons

Generalized Synthetic Kernel Approximation for Elastic Moderation of Fast Neutrons

Generalized Synthetic Kernel Approximation for Elastic Moderation of Fast Neutrons

A method of synthetic kernel approximation is examined in some detail with a view to simplifying the treatment of the elastic moderation of fast neutrons. A sequence of unified kernel {f N } is introduced, which is then divided into two subsequences {W n } and {Gn} according to whether N is odd (W n =f 2n-1, n=1, 2,…) or even (G n =f 2n , n=0,1, …). The W 1 and G 1 kernels correspond to the usual Wigner and GG kernels, respectively, and the W n and G n kernels for n ≧ represent generalizations thereof. It is shown that the W n kernel solution with a relatively small n (≧2) is superior on the whole to the Gn kernel solution for the same index n, while both converge to the exact values with increasing n. To evaluate the collision density numerically and rapidly, a simple recurrence formula is derived. In the asymptotic region (except near resonances), this recurrence formula allows calculation with a relatively coarse mesh width whenever ha ≧0.05 at least. For calculations in the transient lethargy region, ...

An approximate treatment of collision integrals for inelastic scattering and ( n, 2 n) reaction by synthetic kernel approximation

An approximate treatment of the collision integrals for inelastic scattering and ( n, 2 n) reaction is studied in order to take the energy distribution of neutrons emitted in these reactions into the slowing down calculations. In this paper the ( n, 2 n) reaction is treated by cluster model. An approximate synthetic kernel for inelastic scattering is proposed in order to calculate the inelastic collision integral by a simple recursion relation, which allows its rapid computation. The application of the present approximate method to calculations of fast neutron spectra and the fast effects proved that the present synthetic kernel was very useful for treatment of inelastic scattering and ( n, 2 n) reaction.

Generalized synthetic kernel for neutron slowing down

A generalized synthetic kernel for the elastic scattering is proposed for more rapid and precise computation of neutron slowing down problem. This generalized synthetic kernel will be named as ‘ G n -kernel’, in which higher rsslowing down moments are taken into account than in the ordinary Greuling Goertzel approximation. Two applications of the present synthetic kernel approximation are shown in this paper. One of them is the calculation of the Placzek function. The approximate Placzek functions calculated from the G n -kernel is shown to be composed of a non-oscillatory term and finite sums of exponential functions of lethargy. The other is the calculation of the fast neutron spectrum in the vicinity of the strong scattering resonance. These numerical results reveal that the generalized synthetic kernel proposed in this paper is very useful for the slowing down calculations.

Placzek Functions Calculated with Synthetic Kernel Approximation

Placzek Functions Calculated with Synthetic Kernel Approximation

Placzek Functions Calculated with Synthetic Kernel Approximation

Placzek Functions Calculated with Synthetic Kernel Approximation