Variational reactivity estimates: new analyses and new results Page: 2 of 16
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International Conference on Mathematics, Computational Methods & Reactor Physics (M&C 2009)
Saratoga Springs, New York, May 3-7, 2009, on CD-ROM, American Nuclear Society, LaGrange Park, IL (2009)
VARIATIONAL REACTIVITY ESTIMATES:
NEW ANALYSES AND NEW RESULTS
Jeffrey A. Favorite
Applied Physics Division, Los Alamos National Laboratory
P.O. Box 1663, MS P365
A modified form of the variational estimate of the reactivity worth of a perturbation was
previously developed to extend the range of applicability of variational perturbation theory for
perturbations leading to negative reactivity worths. Recent numerical results challenged the
assumptions behind the modified form. In this paper, more results are obtained, leading to the
conclusion that sometimes the modified form extends the range of applicability of variational
perturbation theory for positive reactivity worths as well, and sometimes the standard variational
form is more accurate for negative-reactivity perturbations. In addition, this paper proves that
using the exact generalized adjoint function would lead to an inaccurate variational reactivity
estimate when the error in the first-order estimate is large; the standard generalized adjoint
function, an approximation to the exact one, leads to r.ore accurate results. This conclusion is
also demonstrated numerically. Transport calculations use the PARTISN multigroup discrete
Key Words: Variational perturbation theory, transport theory
In previous work [1-3], it was found that the variational estimate of a quantity of interest
associated with a perturbation leading to a negative change was less accurate than the variational
estimate of the quantity of interest associated with a similarly sized perturbation leading to a
positive change. In other words, the range of applicability of the variational estimate was more
limited on the negative side than the positive side. This phenomenon was observed in a sodium
slab inhomogeneous source problem with transport theory (using an unknown number of energy
groups), shown in Fig. 1 ; a 24-group fast spherical reactor with diffusion theory ; and a
two-group thermal slab reactor with diffusion theory, shown in Fig. 2 . In Ref. 1, the quantity
of interest was the flux exiting the slab; in Refs. 2 and 3, the quantity of interest was the
reactivity worth of the perturbation. Figures 1 and 2 show the characteristic quadratic shape of
the variational (second-order) estimate.
Over a decade ago, an attempt to extend the range of applicability of the variational reactivity
estimate resulted in the development of a modified form . Positive reactivities were to be
estimated using the standard form, but negative reactivities were to be estimated using the
modified form. This development resulted in more accurate negative-reactivity estimates  for
the 24-group fast spherical reactor with diffusion theory and the two-group thermal slab reactor
with diffusion theory.
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Favorite, Jeffrey A. Variational reactivity estimates: new analyses and new results, article, January 1, 2009; [New Mexico]. (digital.library.unt.edu/ark:/67531/metadc934023/m1/2/: accessed December 17, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.