We show that several diffusion-based approximations (classical diffusion or SP 1 , SP 2 , SP 3 ) to the linear Boltzmann equation can (for an infinite, homogeneous medium) be represented exactly by a non-classical transport equation. As a consequence, we indicate a method to solve diffusion-based approximations to the Boltzmann equation via Monte Carlo, with only statistical errors -no truncation errors.
This paper extends a recently introduced theory describing particle transport for random statistically homogeneous systems in which the distribution function p(s) for chord lengths between scattering centers is non-exponential. Here, we relax the previous assumption that p(s) does not depend on the direction of flight Ω; this leads to a new generalized linear Boltzmann equation that includes angular-dependent cross sections, and to a new generalized diffusion equation that accounts for anisotropic behavior resulting from the statistics of the system. This typically leads to the particle flux decreasing as an exponential function of the path-length (Beer-Lambert law).However, in an inhomogeneous random medium, particles will travel through different materials with randomly located interfaces. In atmospheric clouds, experimental studies have found evidence of a non-exponential attenuation law [1][2][3]. It has been suggested [4] that the locations of the scattering centers (in this case water droplets) are spatially correlated in ways that measurably affect radiative transfer within the cloud [5][6][7][8][9][10][11][12][13].An approach to this type of non-classical transport problem was recently introduced [14], with the assumption that the positions of the scattering centers are correlated but independent of direction Ω; that is, Σ t is independent of Ω but not s: Σ t = Σ t (x, E, s). A full derivation of this generalized linear Boltzmann equation (GLBE) and its asymptotic diffusion limit can be found in [15], along with numerical results for an application in 2-D pebble bed reactor (PBR) cores. Existence and uniqueness of solutions, as well as their convergence to the diffusion equation, are rigorously discussed in [16]. Furthermore, a similar kinetic equation with path-length as an independent variable has been derived for the periodic Lorentz gas [17].For specific random systems in which the locations of the scattering centers are correlated and dependent on the direction Ω, anisotropic particle transport arises [18,19]. This anisotropy is a direct result of the geometry of the random system -for instance, the packing of pebbles close to the boundaries of a pebble bed system lead to particles traveling longer distances in directions parallel to the boundary wall [20]. One may also expect that, due to the "gravitational" arrangement of pebbles in PBR cores, diffusion in the vertical and horizontal directions might differ. This behavior can only be captured if we allow the path-lengths of the particles to depend upon Ω; that is, Σ t = Σ t (x, Ω, E, s).The goal of this paper is to extend the GLBE formulation in [15] to include this angular dependence. For simplicity, we do not consider the most general problem here; similarly to [15], our analysis is based on five primary assumptions: i The physical system is infinite and statistically homogeneous.ii Particle transport is monoenergetic. (However, the inclusion of energy-or frequency-dependence is straightforward.)iii Particle transport is driven by a known interio...
We describe an analysis of neutron transport in the interior of model pebble bed reactor (PBR) cores, considering both crystal and random pebble arrangements. Monte Carlo codes were developed for (i) generating random realizations of the model PBR core, and (ii) performing neutron transport inside the crystal and random heterogeneous cores; numerical results are presented for two different choices of material parameters. These numerical results are used to investigate the anisotropic behavior of neutrons in each case and to assess the accuracy of estimates for the diffusion coefficients obtained with the diffusion approximations of different models: the atomic mix model, the Behrens correction, the Lieberoth correction, the generalized linear Boltzmann equation (GLBE), and the new GLBE with angular-dependent path-length distributions. This new theory utilizes a non-classical form of the Boltzmann equation in which the locations of the scattering centers in the system are correlated and the distance-to-collision is not exponentially distributed; this leads to an anisotropic diffusion equation. We show that the results predicted using the new GLBE theory are extremely accurate, correctly identifying the anisotropic diffusion in each case and greatly outperforming the other models for the case of random systems.
We show that, by correctly selecting the probability distribution function p(s) for a particle's distance-to-collision, the nonclassical diffusion equation can be represented exactly by the nonclassical linear Boltzmann equation for an infinite homogeneous medium. This choice of p(s) preserves the true mean-squared free path of the system, which sheds new light on the results obtained in previous work.
This paper introduces a mathematical approach that allows one to numerically solve the nonclassical transport equation in a deterministic fashion using classical numerical procedures. The nonclassical transport equation describes particle transport for random statistically homogeneous systems in which the distribution function for free-paths between scattering centers is nonexponential. We use a spectral method to represent the nonclassical flux as a series of Laguerre polynomials in the free-path variable s, resulting in a nonclassical equation that has the form of a classical transport equation. We present numerical results that validate the spectral approach, considering transport in slab geometry for both classical and nonclassical problems in the discrete ordinates formulation.
We investigate the accuracy of the recently proposed nonclassical transport equation. This equation contains an extra independent variable compared to the classical transport equation (the path-length s), and models particle transport taking place in homogenized random media in which a particle's distance-to-collision is not exponentially distributed. To solve the nonclassical equation one needs to know the s-dependent ensemble-averaged total cross section, Σ t (µ, s), or its corresponding path-length distribution function, p(µ, s). We consider a 1-D spatially periodic system consisting of alternating solid and void layers, randomly placed in the x-axis. We obtain an analytical expression for p(µ, s) and use this result to compute the corresponding Σ t (µ, s). Then, we proceed to numerically solve the nonclassical equation for different test problems in rod geometry; that is, particles can move only in the directions µ = ±1. To assess the accuracy of these solutions, we produce "benchmark" results obtained by (i) generating a large number of physical realizations of the system, (ii) numerically solving the transport equation in each realization, and (iii) ensemble-averaging the solutions over all physical realizations. We show that the numerical results validate the nonclassical model; the solutions obtained with the nonclassical equation accurately estimate the ensemble-averaged scalar flux in this 1-D random periodic system, greatly outperforming the widely-used atomic mix model in most problems.
We show that the recently introduced nonclassical simplified P N equations can be represented exactly by a nonclassical transport equation. Moreover, we validate the theory by showing that a Monte Carlo transport code sampling from the appropriate nonexponential free-path distribution function reproduces the solutions of the classical and nonclassical simplified P N equations. Numerical results are presented for four sets of problems in slab geometry.
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