Page 111 - Dynamics and Control of Nuclear Reactors

CHAPTER

9
Space-time kinetics

9.1 Introduction
Point kinetics models have proven their value in reactor dynamic simulation, but are
extreme simplifications of what goes on in a nuclear reactor. Changes in the local
neutron flux during a transient are often important. Consequently, space-time neu-
tronic models have been developed and implemented.
The most complete description of the spatial distribution of neutrons in a reactor
is given by neutron transport theory. Transport theory defines a reactor in terms of
seven independent variables: three position coordinates, two direction vectors,
energy and time. The transport theory equation is called the Boltzmann equation.
Computer codes have been developed for neutron transport, but they suffer from
complexity and long computing time. Diffusion theory provides a simpler, yet often
satisfactory, approach.

9.2 Diffusion theory
Most reactor studies treat neutron motion as a diffusion process—that is, neutrons
tend to diffuse from regions of high neutron density to regions of low neutron den-
sity. Diffusion theory ignores the direction dependence of the neutrons.
Other processes besides neutron diffusion have diffusion theory models. See
Chapter 10 for heat conduction theory based on heat diffusion theory. Diffusion the-
ory models use partial differential equations. Such models are called distributed
parameter models. Models involving ordinary differential equations are called
lumped parameter models.
Exact solutions are available for some distributed parameter models. For exam-
ple, exact solutions are available for heat conduction in a homogeneous solid in slab,
cylindrical, or spherical geometry. Solutions are even available for a layered solid,
but they are complex. Exact solutions for highly inhomogeneous media (like a reac-
tor core) are intractable.
A typical approach for inhomogeneous media simulations is to treat the space as
an array of segments with internally averaged properties and coupling terms to treat
segment-to-segment transfers. This approach results in a set of ordinary differential
equations that can be solved by standard ordinary differential equation solvers.
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