In the case of continuous refuelling, after a running-in phase an equilibrium is reached, and the reactor composition does not change with time. In this equilibrium condition all burn-up stages of the fuel are present at the same time in the reactor (uniformly graded exposure). The mean reactor composition is then an average of the compositions of all burn-up stages of a fuel element. If Nk(t) is the concentration as a function of time of isotope к in a fuel element, the average concentration Nk of this isotope in the reactor is

where T is the residence time of the fuel elements in the reactor. This expression supposes a very high number of fuel elements in the reactor and continuous refuelling. This condition is exactly satisfied for pebble-bed reactors and still sufficiently well satisfied in the case of prismatic fuel with on-load refuelling.

The existence of an equilibrium condition supposes a constant power level of the reactor so that the time t of eqn. (9.2) actually corresponds to an irradiation tф. Periodic power variations due to load following do not disturb the equilibrium condition if the time period is short compared with the decay constant of the isotopes considered. In eqn. (9.2) the concentrations Nk(t) must then be calculated using an average power level. This approximation is valid for most isotopes. The most important exception is given by l35Xe. The concentration of this isotope is usually calculated separately considering the operational requirement of the reactor (see Chapter 12).

These equilibrium calculations are usually the first calculations performed when designing and optimizing a reactor. In this case one has to iterate in order to obtain a critical reactor. Usually a calculation is started with a guess on the initial composition and the average composition is obtained. This can be done for each reactor region separately (e. g. inner and outer core) or for the whole reactor (point model calcula­tions). Once the equilibrium composition is known a criticality calculation is performed. In the case of the point model

У. 2/fc

fee* =—- !——— — (9.3)

к

where the summation is extended over all the isotopes k. In the case of a multi-region reactor calculation, a diffusion calculation is performed.

If кея is different from the desired value (usually slightly above 1 for control reasons) one iterates on the concentration of the fertile material of the fresh fuel (or in some cases on the fuef residence time) until this value is reached.

The %ak and %fk appearing in (9.3) are obtained as

lk=jjk(t)dt

in analogy to (9.2). Here the 2k(f) are macroscopic cross-sections including self — shieldings. These 2k(f) must be calculated from the concentrations obtained by solving the depletion equation (9.1), taking into account the cross-section variations due to spectrum changes during the fuel lifetime.

Here one can distinguish between different cases. If the fuel elements are small compared to the neutron migration length, the neutron spectrum is determined by the average reactor composition and remains constant in the equilibrium phase, indepen­

dently of the burn-up of the single fuel element which is being studied. In this case the constants of eqn. (9.1) can only be calculated when the average composition given by

(9.2) is known.

A double iterative procedure is then necessary. First with a guess of the average composition the cross-sections are calculated and eqn. (9.1) is solved. Now it is possible to obtain the average composition from eqn. (9.2) from which new cross-sections for eqn. (9.1) are calculated. Once this iteration (inner iteration) has converged, an iteration (outer iteration) on the fertile concentration (or on the residence time) will start to obtain criticality, and the whole cycle of calculations is repeated.

In order to deal with reprocessing, at the end of each inner iteration the fresh composition for the next iteration consists of all the U (or Pu) isotopes contained in the discharged fuel (whose concentration is multiplied by a reprocessing efficiency) plus enough feed fuel to have either the same moderator to fuel ratio of the first given loading, or the same life average moderator-to-fuel ratio. This method can also be applied to treat two intimately mixed sorts of fuel (feed and breed) out of which, in some cases, only one type is reprocessed (fuel segregation). Those two types of fuel could possibly have two different residence times. If the fuel elements are so big that their neutron spectrum is determined only by their composition, the above described inner iterations are no longer necessary. This does not normally happen, but in big fuel blocks the neutron spectrum is usually influenced both by the composition of the block under study and of the surrounding blocks. An exact treatment of this would require a two-dimensional burn-up calculation. Usually first survey studies are performed with the simpler methods here described and later more detailed calculations are performed in which the effect of neighbouring fuel blocks, control rods, reflector blocks, etc., can be properly treated. An interesting way of taking into account the effect of neighbour­ing blocks is given by the use of collision probability methods in the multi-cell option of the WIMS code. These energy-dependent probabilities, which are used to couple the individual cell calculations, are usually obtained from diffusion theory calculations on the regions of interest (see also §4.11).

If the parameters of eqn. (9.1) can be considered as constant, analytical solutions are possible with a considerable saving in computing time in the calculation of equilibrium cycles. This supposition requires beside the Constance of the spectrum, also a Constance of the self-shieldings. As outside the resonances self-shieldings are very near to unity in HTR fuel, this assumption is often satisfied. Besides the concentration of fertile material, and hence its self-shielding, changes very little during burn-up.

A notable exception can be given by the Pu resonances in the low enriched-uranium fuel cycle where, as the Pu concentration changes with time, the approximation of constant self-shieldings can be rather crude.

Particularly interesting in the case of analytical solutions is a method developed by Blomstrand which, using Fourier transform of eqn. (9.1), relates the average composition to the feed composition, avoiding the inner iterations described above. In the BASS and BABS codes"21 written according to this method it is possible to specify directly the characteristics of the average equilibrium core composition (e. g. its moderator to fissile ratio S) and obtain the required feed composition.

For more detailed equilibrium calculations space dependence has to be considered. In space-dependent equilibrium burn-up codes it is possible to treat various core regions iterating on the feed fuel composition, or on the burn-up, of these regions in order to obtain a flat power distribution (e. g. same maximum power in all regions). The one-dimensional FLATTER code,’131 which uses the Blomstrand method combining the BASS code with a one-dimensional diffusion calculation, can divide the core in a number of burn-up sub-regions (independent BASS calculations). These sub-regions can be combined in two or three macro-regions whose initial composition or burn-up can be changed in order to achieve power flattening.