The neutron transport equation in the lattice calculation is a steady-state equation without the time differential term in Eq. (2.1). Further, the neutron energy variable is discretized in the equation and therefore a multi-group form

is used in design codes as shown in Eq. (2.20). The neutron source of Eq. (2.21) is the multi-group form without the external neutron source of Eq. (2.2) at the critical condition.

-ІЇ-Уфя(г, ^)-^вСг)фя(г, Й)+^(т, Й)=0 (2.20)

The system to which the multi-group transport equation is applied is an infinite lattice system of a 2D fuel assembly (including assembly gap) with a reflective boundary condition. For a complicated geometry, two lattice calculations corresponding to a single fuel rod and a fuel assembly are often combined.

In practically solving Eq. (2.20) in the lattice model, the space variable (r) is also discretized in the equation and each material region is divided into several sub-regions where neutron flux is regarded to be flat. In liquid metal-cooled fast reactors (LMFRs), neutron flux in each energy group has an almost flat spatial distribution within the fuel assembly because the mean free path of the fast neutrons is long. A simple hexagonal lattice model covering a single fuel rod or its equivalent cylindrical model simplified to one dimension is used in the

Cladding

Moderator

Spatial Distribution of bast Neutron 1′ lux

Spatial Distribution of 1 hernial Neutron r lux

design calculation of LMFRs. The spatial division can also be simplified by assigning the macroscopic cross section by material.

On the other hand, thermal reactors have a highly non-uniform distribution (called the spatial self-shielding effect) of neutron flux in a fuel assembly as thermal neutron flux rises in the moderator region or steeply falls in the fuel and absorber as shown in Fig. 2.9. Moreover, control rod guide tubes or water rods are situated within fuel assemblies and differently enriched fuels or burnable poison (Gd2O3) fuels are loaded. In such a lattice calculation, therefore, it is necessary to make an appropriate spatial division in the input data predicting spatial distribution of thermal neutron flux and its changes with burnup.

Numerical methods of Eq. (2.20) include the collision probability method (CPM), the current coupling collision probability (CCCP) method, and the method of characteristics (MOC) [13]. The SRAC code adopts the collision probability method and can treat the geometrical models as shown in Fig. 2.10. The collision probability method has been widely used in the lattice calculation, but it has a disadvantage that a large number of spatial regions considerably raise the computing cost. The current coupling collision probability method applies the collision probability method to the inside of fuel rod lattices constituting a fuel assembly and combines neighboring fuel rod lattices by neutron currents entering and leaving the lattices. This approach can substan­tially reduce the assembly calculation cost. Since the method of characteristics solves the neutron transport equation along neutron tracks, it provides compu­tations at relatively low cost even for complicated geometrical shapes and it has become the mainstream in the recent assembly calculation [14].

ZD square plate fuel 2D square assembly assembly (KUCA) with pin rods (PWR)

Fig. 2.10 Lattice models of SRAC [7]

In lattice calculation codes, effective microscopic cross sections are first prepared from fine-group infinite dilution cross sections based on input data such as material compositions, dimensions, temperatures, and so on. The effec­tive cross sections are provided in solving Eqs. (2.20) and (2.21) by the use of the collision probability method, etc. and then multi-group neutron spectra are obtained in each divided region (neutron spectrum calculation).