Reactivity estimation for source-driven systems using first-order perturbation theory. Page: 2 of 11
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PHYSOR 2002, Seoul, Korea, October 7-10, 2002
dynamic response of a source-driven system is governed by both the associated dynamic reactivity
and the external source. The neutronic behavior of a subcritical core is usually rather sensitive to the
source characteristics such as energy spectrum, spatial distribution, etc. Previously, concerning the
reactivity of source-driven systems, Gandini proposed a special concept of reactivity, the so-called
generalized reactivity, which incorporates both the conventional dynamic reactivity and the external
source. However, the suitability of a dynamic reactivity definition generally depends on its
consistency. Thus, the conventional definition of the dynamic reactivity is utilized in this paper.
The objective of this paper is to address the validity of the FOP theory for predicting the change of the
dynamic reactivity in the source-driven system. In Section II, a consistent reactivity expression is
derived for the exact point kinetics equations and the perturbation theory expressions for the reactivity
are presented in Section III. Section IV contains the numerical test results for benchmark problems.
Lastly, conclusions and future works are provided in Section V.
II. POINT KINETICS EQUATIONS FOR SOURCE-DRIVEN SYSTEMS
For a source-driven system with an independent source S, the space-energy-dependent dynamics
equations for the neutron flux (F and delayed neutron precursors Ck can be written in an operator
form  as
1 (F -M)+ +S, (la)
v at -(p- 4Zd~~
a =-AC + JF dE, (lb)
where v is the neutron velocity, F is the prompt fission source operator, Fdk are the delayed fission
source operators, Al is the neutron migration and loss operator, Ak is the decay constant for the k-th
group delayed neutron precursors, and %dk is the emission spectrum of the k-th delayed neutron
group. The flux can be factorized without any approximation into an amplitude function p(t) and a
space-, energy-, and time-dependent shape function P(r, E, t) such that
F(r, E, t)= p(t) - I(r, E, t), (2)
with a constraint on the shape function
Jfiw(r, E)P(r, E, t)dE dV = K0, (3)
where w is a weighting function and Ko is a constant.
The formally exact point kinetics equations can consistently be derived by inserting the flux
factorization into Eq. (1) and integrating over space and energy with a weighting function. In this
model, a dynamic reactivity is defined in terms of the weighting function and time-dependent shape
function. The weighting function should be chosen such that the reactivity is insensitive to errors in
the shape function. For an initially critical reactor, the initial adjoint flux fulfils this requirement.
However, for a subcritical reactor with an independent source, the adjoint function is not uniquely
defined, and thus the reactivity can be defined in various ways, depending on the weighting function.
Typically, the X-mode adjoint function of the initial state has been used as the weighting function,
since it leads to a formulation that eliminates the first-order flux errors. The X-mode adjoint function
of the initial state Q* can be obtained by solving the following adjoint equation
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Kim, Y.; Yang, W. S.; Taiwo, T. A. & Hill, R. N. Reactivity estimation for source-driven systems using first-order perturbation theory., article, July 2, 2002; Illinois. (digital.library.unt.edu/ark:/67531/metadc734030/m1/2/: accessed December 12, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.