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Determining group constants in heterogeneous reactors

Perhaps it would be appropriate to motivate the need for homogenized cross sections. The vast majority of modern power generating nuclear reactors are heterogeneous, ie they consist of a large number of small diameter clad fuel rods in a tight lattice surrounded by coolant and moderator (which commonly are the same material). First of all, diffusion theory is not valid in such a media which obviously consists of a very non-homogeneous composition with "slabs" of different materials which have an extension much less than 2-3 scattering mean free paths required by the diffusion method. Secondly, describing this kind of geometry demands a very high number of points and we are in most cases interested only in the overall behavior of the fluxes. These two circumstances make is plausible to obtain some mean values of the cross sections over the lattice which, multiplied by the flux solution, give the right reaction rates. Many different methods have been applied in the past for the (more or less approximate) evaluation of homogenized cross sections. One of the best methods is obtained by solving the Boltzmann transport equation for the smallest entity of the core; in the case of a boiling water reactor commonly a fuel element and possibly control rod. A large thermal power reactor has a small outgoing net leakage^1.10 of neutrons which makes it possible to apply a symmetry condition on the boundary of the entity, ie a boundary condition of zero net leakage. At RISØ^1.11 the program cccmo builds upon this strategy and solves the Boltzmann transport equation in 76 energy groups. With this solution at hand the program calculates the different homogenized cross sections having access to microscopic cross sections in 76 groups of the different elements and corresponding number densities^1.12 of the elements present in the given setup. At this point we have to state that apart from the geometry and the material composition of the fuel assembly which are fixed, the cross sections depend on both the vapor contents, $\mbox{$<\!{\alpha}\!>$}$, and to a lesser extent the mean fuel temperature, $\mbox{T}_{\mbox{f}}$. In order to account for this dependence we generate the homogenized cross sections for a few different $(\mbox{$<\!{\alpha}\!>$}, \mbox{T}_{\mbox{f}})$ values, which cover every possible situation. In regard to the multi-group diffusion equations we are now able to calculate the cross sections for any pair of $(\mbox{$<\!{\alpha}\!>$}, \mbox{T}_{\mbox{f}})$ by means of, for instance, 2-D linear interpolation.

Using cccmo, however, presents the following difficulties

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The assumption of cylinder geometry.

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Restrictions in regard to the enrichment pattern within the fuel assembly.

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No direct way of calculating cross sections for a reflector region.

We return to these problems in the sections concerning the natural circulation boiling water reactor from General Electric Co. which is chosen as a "test case".

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