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If we assume the cylindricalized cell with the perfect reflective outer boundary, more terms like
those in Eq.(7.1-46) are required as follows;
K
i3
+ K
( λ ) − K
i3
( λ ) − K
+ ............................................................................................
+ K
i3
3
ij
4
ij
n
ij
i3
( λ ) − K
( λ + λ ) − K
i3
i3
3
ij
4
ij
( λ + λ ) − K
n
ij
( λ + λ ) − K
( λ + λ ) + K
( λ + λ ) + K
( λ + λ ) + K
( λ + λ + λ )
( λ + λ + λ )
( λ + λ + λ )
+ ...........................................................................................
where
3
λij
= λij
+ λ j + 2
4 2
λ = λ + λ + 2
ij
n
ij
1
ij
λ = λ
n−2
ij
j
∑
∑
+ λ + 2
j
N
k
k = i+
1
N
λ
λ
k
k = i+
1
....................................... ,
N
∑
i
i
i
λ ,
k
k = i+
1
,
,
i3
i3
i3
3
ij
4
ij
n
ij
j
j
j
i3
i3
i3
3
ij
4
ij
n
ij
i
i
i
j
j
j
These terms to be used in the integrand of Eq.(7.1-46) are the generalized form of those appearing in
the expression of P given by Takahashi 35) 1→2
, while the integration variable has not yet been
transformed to ρ.
As regards the directional probabilities in the cylindrical coordinates, we know for the axial
2 2
direction 3 = 3cos θ and for the radial direction 3 2 = (3/ 2) sin 2 θ . For the latter, P ijr is
Ω z
obtained by multiplying the integrand in Eq.(7.1-40) by
here the whole expressions for each condition. It is enough for us to know only the fact that all the
terms expressed by K in (x) must be replaced by (3/2)K i (n+2) (x). Similarly to the slab system, the
following relation holds:
Ω r
(3/ 2)sin
2 θ . It is not worth while to repeat
P
ij
1 2
= Pijz
+ Pijr
(7.1-53)
3 3
We know that the isotropic boundary condition brings more accurate result and is less
time-consuming than the perfect reflective boundary condition to evaluate the flux distribution in the
real cell by the cylindricalized model. In this case the probability, P is that a neutron emitted in the shell
i escapes from the outer boundary without suffering any collision is required. It is easily obtained as
P
is
2
=
Σ V
i
i
r i
1
1
2
2
{ K ( λ ) − K ( λ + λ ) + K ( λ ) − K ( λ λ }
∫ dρ i3 is i3
is i i3
is i3
is + i ) , (7.1-54)
0
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