The problem of calculating the long time reactor behaviour is complicated by the spatial dependence of neutron flux and core composition. Even in a homogeneous reactor the composition becomes space dependent after burn-up because the fuel burns more rapidly in the high flux regions. In general the effect of burn-up is to flatten the power distribution because the zones of high power are more rapidly depleted. Furthermore, spatial variations in flux distribution and composition are produced by reloadings, control-rod movements, etc. Also the space-dependence of temperature, which has very little effect on the feett of static calculations, results in local perturbations of reactions rates which can have a considerable effect on burn-up. This means that one-, two — or even three-dimensional calculations may be necessary in order to describe the reactor burn-up.

Usually the system of eqns. (9.1) is programmed in a programme sub-routine which is part of a more general code performing diffusion calculations (in order to have power and flux distribution), neutron spectrum calculations, cross-section averaging, simula­tion of control rod movement and reloading operations.<7)

Given the space-dependent composition of the fresh reactor (or of the reactor after a given reloading operation) it is first necessary to perform a spectrum calculation in order to obtain the constants for the diffusion and burn-up calculations. These spectrum calculations are usually performed in rather broad regions for which an average composition is calculated.

Then a diffusion calculation is performed on the reactor system in one-, two — or three-space dimensions and flux distributions are obtained. As a diffusion calculation gives unnormalized fluxes these fluxes must be normalized to the required power level. The number of energy groups considered in the diffusion calculation varies according to the problem, in general between two and ten groups are used.

In order to perform the actual depletion calculation the reactor must be divided in a certain number of burn-up regions for each of which eqns. (9.1) are solved. Theoreti­cally this should be done for every mesh point of the diffusion calculation, but in order to reduce the computational effort these burn-up regions include many mesh points. These regions must be chosen in such a way as to have a reasonably flat flux because for each of them only one flux level is being considered in the solution of the burn-up equations. An average one-group flux and average one-group cross-sections are calculated for every burn-up region and eqns. (9.1) are solved for a certain number of time steps. At each time step the level of the neutron flux is usually readjusted in order to keep the total reactor power constant, taking into account the variation in fissile concentration. This flux readjustment is usually performed without always repeating the diffusion calculation, which is only made after a certain number of time steps. This renormalization creates some problem because in reality only the thermal flux level changes with burn-up, while the fast flux does not change if the power density remains constant. In this way one can define three different time steps:

Дtb: for the solution of the burn-up equations and flux renormalization,

Atd: at each of these steps a diffusion calculation is performed,

Ats at each of these time steps a spectrum calculation is performed.

These time steps are multiples of each other

In a similar way the space is discretized in three different levels: the meshes of the diffusion calculation, the burn-up regions, and the zones for spectrum calculations.

This scheme can be varied according to the code used for the burn-up calculations, and not all operations are necessarily performed within the same code, so that in some cases it may be necessary to stop the burn-up code in order to perform spectrum calculations which produce new constants with which to restart the burn-up code.

The latter procedure is advisable for expensive three-dimensional calculations where it is better to check for possible errors before proceeding to new time steps.

Spectral changes during burn-up can in some cases be very important. A rather extreme example is given in Fig. 9.5 for the case of a Pu-fuelled HTR with batch loading/8’

In the case of zero-, one — or two-dimensional calculations in which a transverse leakage is expressed by means of energy-dependent bucklings, the bucklings determined for the fresh core are not valid after burn-up if the neutron spectrum has changed. Because of the return of fast neutrons thermalized in the reflector, the thermal leakage is usually negative. If this is expressed as DB2<£, an increase of thermal flux would artificially increase this inward leakage, while in reality it remains almost constant because the fast flux does not change with burn-up. Although this problem can only be completely avoided by the use of a full space dependent calculation, a way of improving the accuracy consists in keeping constant with burn-up instead of D, B,2, the leakage per source neutron’9’ defined as

D, V 2ф,
j, 2 ^пФі

eff j

In the case of zero-dimensional calculations the diffusion equation instead of the usual form

No lumping of fuel

850 days correspond to a burn up

of 1-125 fissions per initial fissile

atom

Beginning of life

2

takes then the form

N __ ту — N

-2w$i + X *і-‘Фі + 1X ч^пФі = 0-

j=1 Keflf j=1

This problem does not exist in the case of equilibrium calculations of reactors with continuous refuelling because in that case neither flux level nor spectrum change with time. Various types of spectrum calculations can be performed in burn-up codes. In some cases a few group library (e. g. 10-12 groups) is used throughout the calculation and the spectrum calculation simply consists in condensing from this library the one-group constants of eqn. (9.1). This method is used in many General Atomic codes. In other cases (e. g. the WIMS code) more detailed cell calculations are used. The codes developed by the Dragon Project (e. g. BASS) base their spectrum calculation on the forty-three-group MUPO code while other codes (e. g. MAFIA and VSOP) are based on more sophisticated GGC and Thermos spectrum calculations.

Another important item is the criticality search.

As the reactor must be kept critical it is usually necessary after each diffusion calculation to readjust the position of the control rods. This can be done automatically in many codes after having specified in input the sequence of movement of these rods, which are represented through extrapolation lengths or poisoned regions. At specified intervals recharge or reshuffle operations occur, in which the composition of given regions is either substituted by that of the fresh fuel or exchanged with that of another region. In many cases the spent fuel is being reprocessed and the fissile material present in it is being totally or partially reinserted in the fresh fuel. These operations can also be easily treated in the burn-up calculations. If the fuel self-shieldings change appreciably during life it is necessary to recalculate them periodically during the execution of the burn-up calculation. This would mean that from time to time a neutron transport calculation should be performed over the fuel cell. This can be very much time-consuming and often approximations are used. In some codes simplified calcula­tions are made using the collision probability methods. In other cases a set of transport calculations is performed in advance and a fitting is made of the self-shielding as a function of the concentration of the interesting isotope (e. g. self-shielding of 240Pu as a function of its concentration). These fitted curves are then used by the computer code during the burn-up calculation.’91

For fast survey calculations zero dimensional burn-up codes are normally used. Even in zero-dimensional codes it is possible to simulate recharge or reshuffle operations. In this case the composition of a certain number of fuel batches must be stored and burnt independently of each other. The core reactivity is calculated from an average composition resulting from the mixture of all batches, but for all other operations these batches must be treated independently. The diffusion calculation is substituted in these zero-dimensional codes by a neutron balance where the leakage is expressed as DB2<f>. As examples of such codes we quote the HELIOS’101 and the GARGOYLE’1’1 codes.