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P
iir
1 +∞ 2π
= ∫ dρ
∫ dϕ{
2λi
− 3K
i5
(0) + 3K
i5
( λi
)} (7.1-72)
4πΣ
V −∞ 0
i
i
Thus we have the double integration of the linear combination of K in function as a final form of
the collision probabilities for the two-dimensional cylindrical system.
7.1.6 Ray-Trace Method for Integration of Collision Probabilities
The integration by ρ and ϕ in Eqs.(7.1-66) and (7.1-67) is carried out by the trapezoidal
integration formula with equal width and weight in a general two-dimensional geometry. As seen, a
pair of ρ and ϕ determines a neutron line across the cell.
In one-dimensional geometries such as cylinder and sphere, no integration over the azimuthal
angle ϕ is needed as the geometry is invariant about the azimuthal angle ϕ. In this case, the range of
the integration over ρ ; (0, r N ) is sub-divided into N regions by r i where r i is the outer boundary of the
i-th annular region, in order to perform an efficient Gaussian quadrature in each sub-division, so that
we can avoid the singularity in the integrand. That is, the argument λ i vanishes as ρ approaches r i and
this causes the integrand to have an undefined derivative at this point. The efficiency of the Gaussian
quadrature is shown by an example that the 10-point Gaussian integration for (r 2 - ρ 2 ) 1/2 gives the area
of a circle by an accuracy of 0.1%.
For the two-dimensional cell of complex geometry which includes a number of pin rods where
the same situation occurs as in a one-dimensional cell, we, however, have no means than to apply the
trapezoidal rule. We know that the finer interval of Δρ and Δϕ gives the more accurate results.
Implementation of this integration scheme requires the development of a tracing routine to
calculate the intersections traversed by the line of integration. To maximize the computing efficiency,
specialized routines are prepared for a variety of geometries which have been shown in Sect.2.4.
The calculation of collision probabilities is performed in two steps. First, the tracing routine is
used to get the geometrical information called “trace table” by each line and accumulate on a large
scale storage, say, disc. In the second step, these data, together with the cross sections, are used to
perform the integration of collision probabilities. The second step is repeated for every energy group.
(1) How to Compose Trace Table
We shall describe how to compose the "trace table" using a sample geometry which is a
hexagonal cell including six fuel rods equally spaced on a circular ring. Shown in Fig.7.1-6a are the
purely geometrical region (S-Region) numbers. And the corresponding physical region (T-Region)
numbers are indicated in Fig.7.1-6b. The latters are treated as the spatial variables after considering
any symmetric condition. The rods consist of two concentric layers and they, together with the coolant,
are divided further by the circle through the centers of the rods into inner and outer regions. Hence a
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