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When we consider the doubly heterogeneous case as seen in the HTTR block, each region must
be specified as l in I to denote the microscopic region l (= f or m) in the macroscopic region I.
Hence, the collision probability P ij above defined takes the form,
( l,
k)
, where the index l
indicates fuel grain f or diluent m in the macroscopic region I and the index k does fuel grain f or
diluent m in the macroscopic region J. According to the derivation by Leslie & Jonsson 70) , the
off-diagonal element of the collision probability is given by
P
WI
l,
k)
= α l Iα k J P ,
(7.3.5-3)
W
IJ ( IJ
lI
and the diagonal element by
P
WI
l,
k)
= QI
( l,
k)
− (1 − PII
) αlIαk
,
(7.3.5-4)
W
II ( I
lI
where WlI
= ΣlIVlI
, WI
= ΣIVI
and ∑αlI
= 1 ,
lI
lI
lI
l
α , Σ and V are the fraction of collision, the total cross-section and the volume of the region ( l
in I ), respectively, Σ I and V I are the corresponding quantities for the macroscopic region I. The
quantity
Q I
( l,
k)
is the probability defined to an imaginary infinite lattice consisting of the
microscopic cell of the region I, which can be evaluated numerically by the formalism presented in
Appendix B of the reference 36) assuming a spherical cell with the white boundary condition. The
quantity P IJ denotes the collision probability between macroscopic regions I and J, which is assumed
to be obtained using a homogenized cross-section which substitutes for the heterogeneous fuel region.
lI
If we can determine, independently of the macroscopic configuration of I, both of the quantity
α and the homogeneous-equivalent cross-section Σ I of the medium I, we can obtain ( l,
k)
by
the following procedure:
Now, we define the self-shielding factor f for the fuel grain so as to give an equivalent collision
cross-section to the macroscopic fuel region under the assumption of a uniform flux distribution
through the microscopic cell. That is to say, the fraction of collision rate α l
in the region
assumed to be given by the effective collision cross-section Σ F and the self-shielding factor f, i.e.,
V f f Σ f
α f = ,
(7.3.5-5)
V Σ
F
F
P IJ
P IJ
l
is
Vm
Σm
α m = ,
(7.3.5-6)
V Σ
F
F
α α =1 ,
(7.3.5-7)
f
+ m
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