Page 112 - Dynamics and Control of Nuclear Reactors

106 CHAPTER 9 Space-time kinetics

9.3 Multi-group diffusion theory
Neutron diffusion theory is more complicated than many other distributed parameter
processes because of the energy dependence of the neutron population and the
nuclear reaction rates. The usual approach for neutron diffusion is to invoke multiple
energy groups with average, energy-dependent properties in each group. Use of few
(two to four) energy groups is common for thermal reactor space-time analysis. Since
the nuclear reactions in fast reactors span a significant range of neutron energies,
space time analysis usually employs more energy groups than thermal reactors.
We begin the development of the neutron diffusion approximation by writing the
balance for an elemental volume in terms of the various reaction rates. The develop-
ment will be for group of neutrons with an average energy. In practice, multiplecoupled
energygroups are used to handlethe energydependence. The equations for the group of
neutrons with average energy, E, contains eleven terms in the neutron diffusion equa-
tion and the delayed neutron precursor equations. Note that the symbol, E, stands for an
energy group rather than a specific energy. Note also that the geometry and material
properties are constant. For clarity, each term is numbered and defined below.
1 ðÞ 2 ðÞ 3 ðÞ 4 ðÞ 5ðÞ
∂n X X X X
ð
ðÞ
ð
ð
ð
ð
ð r, E, tÞ ¼ χ E, E i Þ υ E i 1 βÞ ð r, E i Þ Φ r, E i ,tÞ + S r, E, tÞ + ð r, E i ! EÞ Φ r, E i ,tÞ
∂t i f i R
6 ðÞ 7 ðÞ 8 ðÞ
X X X
ð
ð r, EÞ Φ r, E i ,tÞ ð r, EÞ Φ r, E, tÞ Lr, E, tÞ + g EðÞ λ j C j r, tÞ (9.1)
ð
ð
ð
a R j j
The delayed neutron precursor equation contains three terms, numbered as follows:
10
9 ðÞ ðÞ ð 11Þ
∂Cj X X (9.2)
ðÞ
ð
ð
ð r, tÞ ¼ β j υ E i ð r, E i Þ Φ r, E i ,tÞ λ j C j r, tÞ
∂t i f
Following are the physical interpretations of each term:
(1) Rate of change of the neutron density at position, r, energy group, E, and time, t.
(2) Rate of production of fission neutrons at position, r, in all energy groups that
produce neutrons in energy group, E, at time, t.
(3) Rate of neutron production from an external source at position, r, energy group,
E, and time, t.
(4) Rate of scattering of neutrons from other energy groups at position, r, at time, t,
that appear in group, E.
(5) Rate of neutron absorptions at position, r, energy group, E, at time, t.
(6) Rate of scattering of neutrons at position, r, energy group, E, and time, t, that
fall into another energy group.
(7) Rate of neutron leakage at position, r, energy group, E, at time, t.
(8) Rate of release of delayed neutrons in all precursor groups at position, r, energy
group, E, and time, t.
(9) Rate of change of the j-th delayed neutron precursor.
(10) Rate of production of the the j-th delayed neutron precursor.
(11) Rate of decay of the j-th delayed neutron precursor.