Doctoral Dissertations

Date of Award

8-1995

Degree Type

Dissertation

Degree Name

Doctor of Philosophy

Major

Nuclear Engineering

Major Professor

Paul N. Stevens

Committee Members

H. Lee Dodds, Raphael Perez, Peter Groer, William Lawkins

Abstract

The moments method is a semi-analytical technique for solving the Boltzmann transport equation. This method was also the first deterministic technique to be successfully applied to the Boltzmann transport equation for solutions useful to reactor shielding. The moments method was used for many years (since about 1950) as the benchmark for calculating the total flux (particles/cm²-sec). However, to this day, the ability to calculate the angular flux (particles/cm²-sec by discrete angles) has not been successfully accomplished.

A knowledge of all the moments would enable, in principle, both the total and the angular fluxes to be reconstructed exactly. However, because the series expansions are truncated, there are an infinite number of functions with moments that correspond to the finite number of moments calculated. Therefore, a method to accelerate convergence with the amount of information available was deemed beneficial, Because Padė approximants are used in numerical analyses to accelerate the convergence of sequences and iterative processes, and because there is a close connection between Padé approximants and orthogonal polynomials (the moments method uses orthogonal polynomials), there appeared to be a good basis for applying Padė approximants to the moments method. By using Padė approximants, the linear type of approximation used in the moments method, was changed to a rational type of approximation.

Seven-and eight-moment problems were chosen because it had been observed that fewer moments gave a better flux curve (ie., less oscillations about the true flux). The source is a plane monodirectional source at an initial energy of 1.0 MeV.The medium is lead. In other words, this problem was chosen because it provided difficulties in trying to reconstruct the angular energy flux.

The results of this work show that at low scattering angles, the Pade approximant solution and the solution using the current methodology provide results similar to those obtained from MCNP 4A (MCNP 4A provided the benchmark for these calculations). The Pade approximant solution estimated the flux to be slightly less than two times the flux calculated from MCNP at large scattering angles. At these same scattering angles, the current method estimated the fluxes to be over ten times the calculated fluxes from MCNP. Additional improvement should be possible by first smoothing the angular distributions, as described by L. V. Spencer, prior to reconstructing the angular flux.

This research has not only provided closure to the moments method, but has also demonstrated the expanded application of the Padé approximant procedure to practical problems. Through the use of Pade approximants, not only can the angular flux be determined but the solution, as it should be, is invariant. That is, the solution should be unaffected by the group of mathematical operations under consideration.The same results were obtained when the polynomials used to expand the spatial variables were changed from Laguerre polynomials to the U & V polynomials. The Same results were also obtained when the number of moments was decreased from eight to seven, and when the angular quadrature was changed from Δω = 0.1 to 0.05.In addition, the total flux can now be determined at deeper penetrations than before.

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