High-Order Homogenization Method in Diffusion Theory for Reactor Core Simulation and Design Calculation Page: 1 of 98
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FINAL REPORT - High-Order Homogenization Method in Diffusion Theory
for Reactor Core Simulation and Design Calculation
DOE Contract # DE-FG0-001D13960; B&R Code AF40
Principal Investigator: Farzad Rahnema, Georgia Institute of Technology
Report Period: July 1, 2000 - June 30, 2003
Most modern nodal methods in use by the reactor vendors and utilities are based on the
generalized equivalence theory (GET) that uses homogenized cross sections and flux
discontinuity factors. These homogenized parameters, referred to as infinite medium
parameters, are precomputed by performing single bundle fine-mesh calculations with
zero current boundary conditions. It is known that for configurations in which the node-
to-node leakage (e.g., surface current-to-flux ratio) is large the use of the infinite medium
parameters could lead to large errors in the nodal solution. This would be the case for
highly heterogeneous core configurations, typical of modern reactor core designs.
The main objective of this project was to develop a new high-order cross section
homogenization method based on the boundary condition perturbation theory to improve
the accuracy of nodal diffusion methods within the context of the GET. The new
homogenization method corrects the homogenized parameters and discontinuity factors
for the effect of the core environment (node-to-node leakage), to an arbitrary order of
accuracy, by expanding them in terms of the node surface current-to-flux ratios. The
method utilizes two adjoint functions to determine the expansion coefficients. Since
these adjoint functions are solutions to the infinite medium problem (zero current-to-flux
ratio), the expansion coefficients can be precomputed and included with the standard
homogenization parameters for use by a nodal code. As a result, the nodal method has
the capability of achieving an arbitrarily accurate solution by efficiently updating
(correcting) the homogenized parameters, including the discontinuity factor, as it
computes the node interface current-to-flux ratio. The level of accuracy for the high-
order corrected reactor flux solution is close to that of the fine-mesh calculation, which
would be computationally expensive and impractical to determine directly at the core
The numerical implementation of the homogenization method required the development
of a fine-mesh lattice code capable of providing, along with the standard homogenized
parameters, the two adj oint functions (the adj oint flux and an adj oint Green's function) as
additional homogenization parameters. When going from one-group to multigroup, the
forms of the equations to be solved and of the expressions to be evaluated become more
complex, due to energy coupling between groups. The method was first tested and
implemented for simple problems (one-group, 1-D geometry). The work was then
extended to more complex (two-group 1-D and then 2-D geometry) problems. The main
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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/m1/1/: accessed March 17, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.