Radiation transport in a light duct using a one-dimensional model

Abstract

A study is made of radiative transfer along a cylindrical duct using the one-dimensional approximation of Larsen. This method enables three-dimensional effects to be included in a one-dimensional equation. We have generalised Larsen's approach so that the internal reflection from the cylinder surface is governed by the Fresnel law. Two problems are considered. In the first, we assume an infinite cylinder with a plane source at the origin and, in the second, we have a semi-infinite cylinder with an incident beam on the surface. A Fourier transform is employed for the infinite cylinder and the Wiener-Hopf technique for the semi-infinite one. We are able to calculate the radiation intensity as a function of position along the cylinder and the mean square distance of travel of a photon. For the semi-infinite case, we obtain the albedo, the surface intensity and the mean distance of travel. Several features arise which demonstrate that the one-dimensional approximation preserves important features of the three-dimensional problem, thereby making the present approach a valuable tool in dealing with more complex problems.

Particle Transport in a 3D Duct by Adding and Doubling

“…Assuming classical Lambertian duct wall scattering, neutral particles, such as photons and neutrons move through a duct filled with a nonparticipating medium. Through a clever moments formulation, several investigators, including Ed Larsen, Roberto Garcia and MMR Williams [2][3][4][5][6][7][8][9][10][11][12], investigated solutions to the resulting moments equations via basis functions. In several very informative articles, Garcia [6,11] and co-authors studied the solution for up to three basis functions with conventional numerical discrete ordinates (NDO) and the analytical discrete ordinates (ADO) methods.…”

Improvements in the ADO Method for a One-Dimensional Model of Neutral Particle Transport in Ducts

“…More recently, transport problems in other fields of research (Williams, 2007; have also been investigated using similar ideas.…”

Uma solução particular baseada em funções de Green para formulações nodais derivadas da equação de transporte bidimensional

“…Neste contexto, daremos atenção aos métodos nodais [2,3,7], já que estes são comumente utilizados na resolução de problemas multidimensionais onde, através da integração em uma das variáveis espaciais, reduz-se o sistema de EDP's (originados da discretização da integral angular) em sistemas de EDO's. O uso de esquemas nodais reduz a complexidade do modelo e possibilita a utilização de vários instrumentos de análise espacial [5,6,8,11].…”

“…Neste contexto, daremos atenção aos métodos nodais [2,3,7], já que estes são comumente utilizados na resolução de problemas multidimensionais onde, através da integração em uma das variáveis espaciais, reduz-se o sistema de EDP's (originados da discretização da integral angular) em sistemas de EDO's. O uso de esquemas nodais reduz a complexidade do modelo e possibilita a utilização de vários instrumentos de análise espacial [5,6,8,11].O método ADO [4] tem sido intensivamente utilizado com sucesso para encontrar a solução das equações nodais em ordenadas discretas, derivadas do problema de transporte de nêutrons bidimensional, de uma maneira concisa e exata, a baixo custo computacional [5,6,10]. O problema teste apresentado aquié diferente daquele tratado em [5,6].…”