9.1. The depletion equations and the methods to solve them

One of the most important problems in reactor design is the calculation of the time dependence of the reactor behaviour. If the time involved is short (from fractions of seconds to hours) one has to consider the time-dependent form of the Boltzmann equation: these problems will be dealt with in Chapter 12. The burn-up calculations deal with the time evolution of reactor parameters over long periods involving the complete lifetime of the reactor.

For these calculations the time derivative of the neutron flux can be neglected and the static form of the Boltzmann equation (or of the diffusion equation) can be used.

For each isotope it is possible to write a balance equation relating the loss and production contributions to its concentration (depletion equations)

ИМ m q p

~ = Ni(rflyik + ф 2 Nscrasysk + 2 N, Aja№ — AkNk — фЫкСтак (9.1)

(It і = l s = r І =n

where Nk = atomic concentration of isotope k, ф = flux,

CTp = isotope і fission cross-section,

(Tai = isotope і absorption cross-section,

Ai = isotope і decay constant,

ylk = yield of isotope к due to a fission in isotope i,

ysk = probability that a neutron absorption in isotope s produces isotope k, alk = probability that the decay of isotope j produces isotope k.

The first summation (index i) extends to all fissionable isotopes; the second summation (index s) extends to those isotopes who can produce isotope к after a neutron absorption; the third summation (index j) extends to those isotopes whose decay product can be isotope k.

All cross-sections are here one-group values, averaged over the whole energy spectrum.

This equation applies to all isotopes present in the fresh fuel or produced by fission and those which originate from neutron absorption or decay of other nuclides. Since the nuclei of the primary fission products generally have a considerable excess of neutrons, they are unstable and decay often via complicated chains.

The number of isotopes to be treated in a burn-up calculation is of the order of 100.

They include the heavy nuclides, the fission products and the isotopes derived from them by neutron absorption and decay.

The types of heavy nuclides present in the reactor fuel depend on the fuel cycle which has been chosen.

In all power reactors the fission process produces more neutrons than the number strictly necessary to sustain the chain reaction. These excess neutrons are used to convert fertile isotopes into fissile ones. Both 238U and 232Th can be used as fertile materials for high temperature reactors.

The ratio

_ No. of atoms fissile material produced No. of atoms of burnt fissile material

is called conversion ratio. The most important reactions involving fertile and fissile material are shown in Tables 9.1 and 9.2 for the 232Th and the 238U chains respectively.

Table 9.1. Thorium Cycle

Th

Table 9.2. Low Enriched U Cycle

In these tables fission is only shown for those isotopes which are fissionable by thermal neutrons. Fast neutron-induced fissions can occur also in other isotopes like 232Th and 238U and these processes are easily taken into account in the calculations, where all heavy metals are usually treated as fissile isotopes. The most important fission product chains are given in Tables 9.3 to 9.6. In these tables a vertical line means neutron absorption while a horizontal line means decay.

The fission yields yik defined as the probability that a fission in isotope і produces

— l54Gd

♦

-l55Gd

l56Gd

І

l57Gd

l58Gd

isotope к are obtained experimentally and stored in the nuclear libraries of the codes for burn-up calculations. Each fission produces two fragments (ternary fissions exist but are negligible) so that

? — 2-

Fission yield data can be found in refs. 1 and 2. Not all fission products need to be treated in a burn-up calculation, but only those having either a non-negligible absorp­tion cross-section, or decaying into isotopes with non-negligible absorption cross­section. For this second case the parent nucli, de needs only to be treated if its decay

time is sufficiently long. Otherwise it is sufficient to treat the decay product, attributing to it its cumulative yield (which is the sum of the direct yield of the nuclide considered and of the yields of the isotopes of the decay chain leading to it). As a typical example Figs. 9.1 and 9.2 give the concentration of the fission products in an HTR as a function of time.

In many cases the system of differential equations (9.1) has been simplified by neglecting the interconnections between some of the different equations for the fission products, i. e. neglecting some links of the chains of Tables 9.3 to 9.6. This can be reasonable for low burn-up, but leads to intolerable errors if the burn-up is very high.® When the chains are neglected it is not necessary to treat independently all fission products, but they can be grouped according to their absorption cross-section in a certain number of pseudo fission products.14’ The yield of a pseudo-element is the sum of the yields of all the fission products which are represented by it, and its absorption cross-section is an average of the single cross-sections weighted over the yields

Fig. 9.1. Atomic density of fission products. Fig. 9.2. Atomic density of fission products.

In this way the pseudo elements should give the same overall absorption as the real elements. It is unfortunately impossible to reproduce in this simple form the proper time behaviour of this absorption. In high temperature reactors the burn-up is usually very high so that these simplifications lead to intolerable errors, and it is essential to treat the system of eqns. (9.1) in its full form for all important chains grouping together only the less important low absorption isotopes in the so called “non-saturating fission product aggregates”.

The importance of neutron absorptions in fission products is shown in Fig. 9.3<5> where the average number of neutrons lost in fission products per neutron absorbed in fuel is plotted as a function of burn-up measured in fissions per initial fissile atom (fifa).

The effect of neglecting the treatment of fission product chains on reactivity is shown in Fig. 9.4 (Ak is the difference between calculated without chains and a reference calculation with all chains).

If the coefficients of eqn. (9.1) are constant, analytical solutions are possible. Unfortunately this is not true in many cases. The cross-sections appearing in (9.1) are averaged over the whole energy range, so that they are very sensitive to any spectral change in the reactor. Furthermore, the flux level in each zone may change during burn-up.

If, in a reactor, the power is kept constant, and the fuel reloading is not continuous, between two reloadings the fissile concentration decreases and the thermal neutron flux must increase in order to maintain a constant fission rate. The fast flux remains constant

because the power and consequently the fission neutron source is unchanged. This means that the neutron spectrum changes with time consequentially changing the average absorption and fission cross-sections. Furthermore, changes in composition also modify the self-shieldings of the fuel elements and this is another cause of variation of the cross-sections.

Because of these reasons analytical solutions cannot always be used for eqn. (9.1). On the other hand, numerical solutions can be relatively simply obtained subdividing the time variable in a discrete number of time steps. A combination of analytical and numerical methods is often used so that within a time step the coefficients of eqn. (9.1) are supposed to be constant and analytical solutions are used.<6) These calculations are very fast on modern computers.