In recent years a significant effort has been invested in the development of computer programmes for space-dependent reactor dynamics.

These codes are all based on the time-dependent multi-group diffusion equation. — дфЛг’П = а Ч2ф<(г, t) — 2«ф, + І £, (г, о + (1 — jS )*1 2*фк(г, t)

Vi dt /Г=і *Г|

+ 2х«С,(г. П [і = 1————— л]. (12.34)

. J = l

dCii/t~ = £ P, vk2^Hr, t)-,Q(r, t) [j = 1,… m]

where the symbols are the same as in eqns. (12.1) and (4.40), with Xi = fraction of fission neutrons emitted in energy group i,

Хіі = fraction of delayed neutrons of group j emitted in energy group і (often assumed to be the same for all j), n = total number of energy groups, m = total number of delayed neutron groups.

Usually two energy groups are sufficient for HTR dynamics calculations.

Numerical solutions are obtained subdividing the time and space coordinates in an appropriate mesh.

In a representation of this type ke„ does not appear, so that it is not possible to introduct temperature and control feedbacks in the way used in eqn. (12.7). In this case all the parameters of the diffusion equation have to be given as a function of temperature. This is usually done by means of a polynomial fitting. Control rods can be simulated as a diffused poison added to certain regions. The cause of the transients is represented by changes in the absorption cross-section, or of any other parameter of the diffusion equation, over given regions.

In the case of slow transients, the equations for Xe and Sm must be taken into account beside eqns. (12.34), but the delayed neutrons, as well as the time derivative of the flux can be neglected. In practice, however, some codes (e. g. COSTANZA*2 8 9I) calculate the time derivative of the flux even if this term must vanish. The code iterates on the absorption cross-section of given regions (simulation of control-rod insertion) until this derivative vanishes, and this iteration substitutes the outer iterations of conventional diffusion codes. These methods are used for one or more dimensional codes.

Various methods have been considered in order to improve the efficiency of this finite difference approach."01 A classical mathematical approach is given by the modal expansion in which the flux is expanded in the normal modes of the system

Ф(г, П = ^фІ(ПМг). (12.35)

The f(r) are the eigenfunctions of the time-independent problems corresponding to the eigenvalues ki. While only the eigenfunction corresponding to the highest eigenvalue k0 is interesting for the static case, higher modes become important during the transients. If we suppose that the flux shape remains constant during the transient, we have

ф(г,1) = ф(ПМг) (12.36)

which corresponds to the point kinetics.

An improved approximation is given by the so-called adiabatic approximation (this

name should not generate confusion with the meaning it has in the case of ther­modynamic problems). In this approximation f0(r) of eqn. (12.36) is changed with time, but it is supposed to be at every instant coincident with the static flux shape corresponding to the reactor conditions of that instant. In this way the changes in flux distribution during the transient (e. g. control-rod movement, composition changes, etc.) can be taken into account, but only the fundamental mode is considered, the higher modes being neglected.

In the quasi-static method the spatial shape function f0(r) of eqn. (12.36) is also changed with time, and can be obtained solving the full space-time equation using very large time steps. In other words, one can say that both the adiabatic and the quasi-static methods use an assumption of the type (12.10),

ф(г, t) = f0(r, t)4>(t) (12.37)

where /o(r, t) has only a weak time dependence. In the adiabatic method f0(r, t) is calculated using static methods, while in the quasi-static method /0(r, t) is calculated solving numerically the space-time reactor equations using very large time steps. Another promising method is the nodal method in which the reactor is partitioned in a number of sub-regions (nodes) for which average power or flux is calculated. The method can be compared to a finite difference method with very coarse meshes. The essential problem consists in obtaining coupling constants between the nodes, maintain­ing neutron balance within each node and between nodes.