Thesis (Ph. D.)--University of Texas at Austin, 2006. / Vita. Includes bibliographical references.

Thesis (Ph. D.)--University of Texas at Austin, 2008. / Vita. Includes bibliographical references.

Thesis (M.S.)--University of Michigan, 1984.

Thesis (Ph. D.)--University of Wisconsin--Madison, 1969. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references.

Thesis (Ph.D.)--University of California, Berkeley, 1960. / "Physics & Mathematics, UC-34" -t.p. "TID-4500 (15th Ed.)" -t.p. Includes bibliographical references.

This is Part B. / <p> Another approach is adopted for deriving the moments equations in spherical geometry using a spherical harmonics expansion of the neutron transport equation over a variable range of the direction cosine. Because of complications and uncertainties in establishing boundary conditions for the equations, only the zero'th order equations are solved, in an idealized situation, in order that a feel for equations and boundary conditions may be obtained.</p> <p> The equations are compared to equations given in a paper 'Directionally Discontinuous Harmonic Solutions of the Neutron Transport Equation in Spherical Geometry', by A. A. Harms and E. A. Attia. Analytical solutions for the zero'th order equations are given for equations developed there and to the equations developed in this paper. Numerical values are presented to give an idea of what accuracies might be expected. It is hoped that similar techniques can be used to solve the higher order equations analytically, and that appropriate boundary conditions can be found.</p> / Thesis / Master of Engineering (MEngr)

The purpose of this thesis is to examine the accuracy of the RAPID (Real-Time Analysis for Particle Transport and In-situ Detection) eigenvalue algorithm based on a few problems from the ICSBEP (International Criticality Safety Benchmark Evaluation Project) Handbook. RAPID is developed based on the MRT (Multi-Stage Response-Function Transport) methodology and it uses the fission matrix (FM) method for performing eigenvalue calculations. RAPID has already been benchmarked based on several real-world problems including spent fuel pools and casks, and reactor cores. This thesis examines the accuracy of the RAPID eigenvalue algorithm for modeling the physics of problems with unique geometric configurations. Four problems were selected from the ICSBEP Handbook; these problems differ by their unique configurations which can effectively examine the capability of the RAPID code system. For each problem, a reference Serpent Monte Carlo calculation has been performed. Using the same Serpent model in the pRAPID (pre- and post-processing for RAPID) utility code, a series of fixed-source Serpent calculations are performed to determine spatially-dependent FM coefficients. RAPID calculations are performed using these FM coefficients to obtain the axially-dependent, pin-wise fission density distribution and system eigenvalue for each problem. It is demonstrated that the eigenvalues calculated by RAPID and Serpent agree with the experimental data within the given experimental uncertainty. Further, the detailed 3-D pin-wise fission density distribution obtained by RAPID agrees with the reference prediction by Serpent which itself has converged to less than 1% weighted uncertainty. While achieving accurate results, RAPID calculations are significantly faster than the reference Serpent calculations, with a calculation time speed-up of between 4x and 34x demonstrated in this thesis. In addition to examining the accuracy of the RAPID algorithm, this thesis provides useful information on the use of the FM method for simulation of nuclear systems. / Master of Science / In the modeling and simulation of nuclear systems, two parameters are of key importance: the system eigenvalue and the fission distribution. The system eigenvalue, known as kef f , is the ratio of neutron production from fission in the current neutron generation compared with the absorption and leakage of neutrons from the system in the previous neutron generation. When this ratio is equal to one, the system is critical and is a self-sustaining chain reaction. Knowledge of the fission distribution is important in the nuclear power industry, as it enables engineers to determine the best reactor core assembly configuration to maintain an even power distribution. Several methods have been developed over the years to effectively solve for a nuclear systems fission distribution and system eigenvalue. Aspects of both Monte Carlo and deterministic transport methods have been combined into RAPID’s MRT methodology. It is capable of accurately determining the system eigenvalue and fission distribution in real time. This thesis examines the accuracy of the RAPID algorithm using four unique problems from the ICSBEP handbook. These problems help us to test the limits of the FM method in RAPID through the modeling of small, unique geometric configurations not seen in large, uniformly configured power reactor cores and spent fuel pools. For comparison, each problem is modeled using the Serpent Monte Carlo code, an accurate code meant to serve as the industry standard for determination of the fission distribution of each problem. This model is then used to generate a set of FM coefficients for use in RAPID calculations. It is demonstrated that the eigenvalues calculated by RAPID and Serpent agree with the experimental data within the given experimental uncertainty. The fission distribution obtained by RAPID is also in agreement with the Serpent reference model. Finally, the RAPID eigenvalue calculation is significantly faster than the corresponding Serpent reference model, with speed-ups ranging from 4x to 34x demonstrated.

A novel collision source method has been developed to solve the Linear Boltzmann Equation (LBE) more efficiently by adaptation of the angular quadrature order. The angular adaptation method is unique in that the flux from each scattering source iteration is obtained, with potentially a different quadrature order used for each. Traditionally, the flux from every iteration is combined, with the same quadrature applied to the combined flux. Since the scattering process tends to distribute the radiation more evenly over angles (i.e., make it more isotropic), the quadrature requirements generally decrease with each iteration. This method allows for an optimal use of processing power, by using a high order quadrature for the first few iterations that need it, before shifting to lower order quadratures for the remaining iterations. This is essentially an extension of the first collision source method, and is referred to as the adaptive collision source (ACS) method. The ACS methodology has been implemented in the 3-D, parallel, multigroup discrete ordinates code TITAN. This code was tested on a variety of test problems including fixed-source and eigenvalue problems. The ACS implementation in TITAN has shown a reduction in computation time by a factor of 1.5-4 on the fixed-source test problems, for the same desired level of accuracy, as compared to the standard TITAN code. / Ph. D.

Thesis: Sc. D., Massachusetts Institute of Technology, Department of Nuclear Engineering, 1977 / Vita. / Includes bibliographical references. / Sc. D. / Sc. D. Massachusetts Institute of Technology, Department of Nuclear Engineering

This thesis is a blend of neutron transport theory and numerical analysis. We start with the study of the problem of the Mika/Case eigenexpansion used in the solution process of the homogeneous one-speed Boltzmann neutron transport equation with anisotropic scattering for plane symmetry. The anisotropic scattering is expressed as a finite Legendre series in which the coefficients are the ``scattering coefficients'. This eigenexpansion consists of a discrete spectrum of eigenvalues with its corresponding eigenfunctions and the continuous spectrum [-1,+1] with its corresponding eigendistributions. In the general case where the anisotropic scattering can be of any (finite) order, multiple discrete eigenvalues exist and these have to be located to have the complete spectrum. We have devised a stable and robust method that locates all these discrete eigenvalues. The method is a two-step process: first the number of discrete eigenvalues is calculated and this is followed by the calculation of the discrete eigenvalues themselves, now being able to count them down and make sure none are forgotten. During our numerical experiments, we came across what we called near-singular eigenvalues: discrete eigenvalues that are located extremely close to the continuum and hence lead to near-singular behaviour in the eigenfunction. Our solution method has been adapted and allows for the automatic detection of such a near-singular eigenvalue. For the elements of the continuous spectrum [-1,+1], there is no non-zero function satisfying the associated eigenequation but there is a non-zero distribution that does satisfy it. It is not feasible to compute a distribution as such but one can evaluate integrals in which this distribution appears. The continuum part of the eigenexpansion can hence only be characterised by its (angular) moments. Accurate and fast numerical quadrature is needed to evaluate these integrals. Several quadrature methods have been evaluated on a representative test function. The eigenexpansion was proved to be orthogonal and complete and hence can be used to represent the infinite medium Green's function. The latter is the building block of the Boundary Sources Method, an integral solution method for the neutron transport equation. Using angular and angular/spatial moments of the Green's function, it is possible to solve with high accuracy slab problems. We have written a one-dimensional slab code implementing this Boundary Sources Method allowing for media with arbitrary order anisotropic scattering. Our results are very good and the code can be considered as a benchmark code for others. As a final application, we have used our code to study the discrete spectrum of a well-known scattering kernel in radiative transfer, the Henyey-Greenstein kernel. This kernel has one free parameter which is used to fit the kernel to experimental data. Since the kernel is a continuous function, a finite Legendre approximation needs to be adopted. Depending on the free parameter, the approximation order and the number of secondaries per collision, the number of discrete eigenvalues ranges from two to thirty and even more. Bounds for the minimum approximation order are derived for different requirements on the approximation: non-negativity, an absolute and relative error tolerance.