Page 113 - Dynamics and Control of Nuclear Reactors

9.3 Multi-group diffusion theory 107

The factors in the equation are as follows:
n(r, E, t)¼neutron density at position, r, energy group, E, and time, t. It is also
equal to Φ/v, where v¼average neutron velocity in the group.
χ(E, E i )¼fraction of the neutrons that are born in energy group, E, because of
fissions in energy group, E i .
r¼the position vector.
E¼neutron energy group.
υ(E i )¼number of neutrons produced per fission caused by neutrons in energy
group, E i .
β¼total delayed neutron fraction.
β j ¼delayed neutron fraction for the j-th delayed neutron group.
Ʃ f (r, E i )¼macroscopic fission cross section at position, r, in energy group, E i .
Φ(r, E i ,t)¼neutron flux at position, r, for energy group, E i at time, t.
S(r, E, t)¼rate of neutrons released from an artificial source in energy group, E,
at position, r, at time, t.
Ʃ R (r, E i !E)¼macroscopic removal cross section for scattering from energy
group, E i , to energy group, E, at position, r.
Ʃ a (r, E i )¼macroscopic absorption cross section for energy group, E i at
position, r.
Ʃ R (r, E)¼macroscopic removal cross section for neutron scattering out of energy
group, E, at position, r.
L(r, E, t)¼leakage of neutrons from energy group, E, at position, r, at time, t.
g j (E)¼fraction of delayed neutrons from precursor group, j, that appear in energy
group, E.
λ j ¼decay constant for the j-th delayed neutron precursor group.
C j (r, t)¼concentration of the j-th delayed neutron precursor group at position, r,
and at time, t.
Two quantities that have not been encountered in previous discussions need expla-
nation. These are the removal cross section, Ʃ R , and the neutron leakage term, L(r,
E, t).
The removal cross section is the probability that a scattering event in energy
group, i, results in delivering the scattered neutron into energy group, j. Reactor
physics books [1] provide formulas for the removal cross section. The neutron leak-
age term, L, is given by the following:
Lr, E, tÞ ¼ rDr Φ r, E i ,tÞ (9.3)
ð
ð
D¼the diffusion coefficient¼1/(3Ʃ s ).
If the diffusion coefficient is constant, Eq. (9.3) becomes
2
ð
Lr, E, tÞ ¼ Dr Φ r, E i ,tÞ (9.4)
ð
2
r is the Laplacian operator, shown below for radial and axial dependence.
2
1 ∂ ∂ ∂
2
r ¼ r + (9.5)
r ∂r ∂r ∂z 2