Equation 1.1 is far too complicated to be solved analytically without drastic simplification. Before it can be solved numerically it has to be recast with the continuous independent variables changed into discrete forms. This can be done in various ways, as follows. To make the explanation simpler we shall assume we are dealing with a reactor that is operating steadily so that we can ignore t.

Position. The position vector r is discretised in the form of a spa­tial mesh covering the reactor core and the surrounding breeder or consumer regions. It is usually convenient to make the geometry of the mesh coincide as far as possible with the actual structure of the core. As explained in Chapters 2 and 3, most fast reactor cores consist of hexagonal fuel subassemblies so a fully three-dimensional mesh is usually either hexagonal or triangular in the radial directions and linear in the axial. (This is in contrast to the square calculation meshes for thermal reactors, the cores of which are usually made up of square subassemblies.) For some purposes a cylindrically symmetric two-dimensional (r, z) approximation, linear in both directions, may be adequate.

Having defined the spatial mesh equation 1.2 can be recast in terms of the mesh intervals. There are two ways to do this. In the finite difference method ф is calculated at each mesh point, whereas in the nodal method the average value of ф over the volume of each cell of the mesh is calculated.

Energy. E is discretised by dividing the range of energy of interest (from about 10 MeV down to 0.1 eV or lower) into a series of intervals Eo — Ei, Ei — E2, etc., where E0 may be 10 MeV, and Ei < E0, E2 < Ei, etc. The neutrons with energies between Eg and Eg-1 are known as group-g neutrons. The flux фg in group g is then given by

Eg+1

ф^г, ft) = ф(г, E, ft)dE. (1.5)

Eg

If there are G groups £r, £f and v are vectors with G elements. £s becomes a G x G matrix the nature of which depends on how the directional variables ft are discretised.

Direction. The direction vector ft can be discretised in either of two ways. One is to specify a set of points on the surface of an imaginary unit sphere and then recast equation 1.1 in terms of the flux of neutrons moving in the directions of areas dft surrounding these points. This leads to “Sn ” methods, where n is the number of discrete directions. Alternatively ф can be expressed as the sum of an infinite series of spherical harmonic components, and equation 1.1 rewritten in terms of the coefficients of the series. In the case of a one-dimensional axially symmetric approximation

TO

ф = ^2 a«P« (cos в ), (1.6)

n=0

where the Pn are Legendre polynomials. This is called a “Pn” method. In either Sn or Pn calculations the greater the number of “mesh points” (i. e. the number of directions for Sn or terms of the series for Pn) the greater the accuracy. In practice n = 5 is usually found adequate.