Abstract:
Two new concepts have been explored in solving the neutron
diffusion equation in one and two dimensions. At the present time,
the diffusion equation is solved using source iterations. These
iterations are performed in a mathematical form which has a great
deal of physical significance. Specifically, the neutron production
term is on the right-hand side, while the absorption and leakage
terms are on the left side. In performing a single source iteration,
a distribution for the neutron flux is assumed so that the production
term can be calculated. This provides a "known" right-hand side.
Solving the difference equation for the flux, which corresponds to
this assumed source distribution, gives the next estimate for the
flux distribution. This type of iteration has the physically
significant characteristic of finding directly, for each iteration, a
flux which corresponds to an assumed source distribution. In this
thesis it was found that by subtracting the absorption term from both
sides of the diffusion equation, and performing "source iterations" with both absorption and production terms on the right-hand side (and
only the leakage term on the left-hand side), improved convergence
rates were attained in many cases.
In one neutron energy group, this new idea of putting the
absorption term on the right-hand side worked best with only one
region, and where reactor dimensions were large compared to the
thermal neutron diffusion length (a>>L). In small reactors, where
a=L, convergence behavior was similar for both forms of iteration.
This new idea was also found to work quite well in one-group multiregion
problems. However, due to problems with numerics (inherent
asymmetric treatment of the scattering terms), the method does not
work at all in a multi-energy group formulation.
Secondly, in two dimensions, a closed-form solution to a single
source iteration has been found. At this time, the standard method
of solution for a two-dimensional source iteration is to perform
"inner iterations" to approximately solve for the flux that
corresponds to an assumed source. The alternative, up until now, was
to solve a giant matrix of the order (N² x N²). This is a sparse
matrix, but it has always been considered as highly undesirable to
work with a solution (even though it may be closed-form) where the
matrix to be solved increases in order roughly as the fourth power of
the number of mesh intervals. The new algebraic form for this
closed-form solution involves a matrix of order (N x N), not (N² x
N²). The matrix is, however, a full matrix. What is done, essentially, is to solve simultaneously for all
the flux values along the vertical centerline of the two-dimensional
problem, and then use a reflective boundary condition across the core
centerline, and then the difference equation itself (in vector form)
as a set of flux-vector generating equations to generate the entire
flux field, line by line. In solving for the first flux vector (at
the x = o, or z = o, core centerline), the right-hand side of the
matrix problem incorporates all of the source values in the entire
problem space. The initial inversion of the full (N x N) matrix
algebraically guarantees that the (M+1)th flux vector (on the problem
space boundary) will go to zero. This matrix method for two-dimensional
neutronic analysis was shown to work well in both
cartesian and cylindrical coordinates.