High-Order Homogenization Method in Diffusion Theory for Reactor Core Simulation and Design Calculation Page: 2 of 98
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difficulty requiring a substantial effort was the numerical implementation of the solution
method for precomputing the Green's function. A fine-mesh lattice code with the
capability mentioned above was developed for each of the three sets of problems: one-
speed 1-D, two-group 1-D and two-group 2-D.
In this project, it was shown that the perturbation expansion series for the homogenized
cross sections and discontinuity factors converge in a multigroup case. This is new in
that it has not been shown before in the literature. The benchmark configurations
consisted of two types of BWR assemblies in slab geometry for the one-dimensional case
and different types of assemblies in the HAFAS core for the two-dimensional case. The
benchmark configurations were analyzed for various magnitudes of the perturbation in
the boundary condition. It was shown that the perturbation method achieves an excellent
accuracy: the reference homogenized cross sections and discontinuity factors are almost
The new homogenization method was numerically implemented at the nodal level, in the
context of the GET, for one-speed 1-D, as well as for two-group 1-D configurations. For
each of these two types of configurations, a finite-difference coarse-mesh code with a
bilinear intra-nodal flux shape was developed. As compared to a standard nodal code for
solving the nodal diffusion equations, which has two levels of calculation (source
iteration and flux iteration), this code has an additional level (iteration) in which the the
homogenized parameters are corrected. Nodal equations were developed for
implementing the homogenization method at the nodal level for two-group 2-D problems,
and their numerical implementation is in progress. The code for solving the nodal
equations in this case is based on a transverse integrated method with a nodal expansion
used for solving the transverse-integrated equations. The associated system of equations
is solved by employing a non-linear iterative strategy. For the 2-D case some difficulty
might arise in determining a surface-dependent boundary condition (current-to-flux ratio)
from node-averaged quantities. Note that the Green's function is not constant at the node
interface. As a first approximation, the expansion parameter in the 2-D case would be
taken as an average over the node surface, which is consistent with the GET assumption.
The testing of the new homogenization method at the nodal level (for one- and two-group
one-dimensional problems) was performed on five benchmark configurations typical of a
BWR, from mildly to highly heterogeneous. Three of these five benchmarks, in which
each assembly is of the GE-9 bundle design, were newly developed because of the need
for more realistic benchmark configurations. It is anticipated that the technical
community in reactor physics and math and computations will benefit from the new
benchmarks developed in this study. It was shown that the homogenization method
provides excellent results. For all of the analyzed configurations, the node-integrated
flux is within 1.2% of the assembly reference (fine-mesh) flux in all nodes for each
group. There is a significant improvement from the zeroth order case (standard GET), in
which the node-averaged flux has a large error (e.g., up to 8% in group 1 and up to 14%
in group 2 for some of the analyzed configurations). It was also shown that the
reconstructed fine-mesh flux (or equivalently the power distribution) in the core
approximates the reference value very well. The reference flux distribution is almost
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Rahnema, Farzad. High-Order Homogenization Method in Diffusion Theory for Reactor Core Simulation and Design Calculation, report, September 30, 2003; United States. (digital.library.unt.edu/ark:/67531/metadc733610/m1/2/: accessed December 16, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.