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will be seen in the reference 65) . Moreover, there might be a problem whether or not we should treat a
region with small amount of resonance absorbers as the R-Region; this problem would not be essential,
since the effective cross-sections in such a region should be nearly infinite dilution cross-sections.
Anyway, a complex heterogeneity can be treated consistently without introducing any simplification
of geometry.
7.3.4 Direct Method for Calculating Neutron Flux Distribution (the Method in ‘PEACO’)
We assume that heterogeneous systems are built up of an infinite number of ‘unit cells’ and the
neutron balance in a heterogeneous system can be described by using the first-flight collision
probabilities. To reduce the numerical errors caused by the flat-flux assumption, each region of the
system may be divided into sub-regions as many as necessary or possible. Then, assuming the
isotropic elastic scattering, the neutron balance in a cell may be written by the neutron slowing down
equation
V Σ ( u)
Ψ ( u)
=
i
i
i
J
∑
P ( u)
V
K
∑
ji j
j= 1 k = 1
S
jk
( u)
(7.3.4-1)
S
jk
1 u
( u)
= ∫ exp
−
{ − ( u − u'
)}
Σ s jk ( u'
) Ψ j ( u'
) du'
(7.3.4-2)
1 − α u ε k
k
with
α
k
⎛ A
⎜
⎝
− 1 ⎞
2 ⎟ ε −
⎠
α
k
=
⎜
and k = ln k
Ak
+ 1⎟
(7.3.4-3)
Here, the subscripts i and j stands for the sub-region number and the k corresponds to nuclear species.
The quantity P ji is the effective probability in a unit cell that a neutron scattered isotropically in region
j into lethargy u will have its first collision in region i and other notation has the customary meanings.
By letting
V Ψ ( u) exp( u)
= Ψ ( u)
, we have
i
i
i
Σ ( u ) Ψ ( u)
P ( u)
S jk ( u)
i
i
= ∑∑
j
k
ji
0
(7.3.4-4)
with
S
0
jk
1
( u)
=
1 − α
k
u
∫u
− ε k
F
jk
( u'
) du'
(7.3.4-5)
F
jk
( u)
= Σ ( u)
Ψ ( u)
s jk
j
(7.3.4-6)
Here, note that the equations (7.3.4-4) and (7.3.4-5) for Ψ i (u) is more simple than Eqs.(7.3.4-1) and
(7.3.4-2).
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