In section 3.5 we discussed some aspects of the control and safety of critical reactors. We discuss in what respect these are modified by subcriticality.

As mentioned in section 3.5.3 a reactivity insertion of more than 1 $ in a critical reactor leads to a fast exponential excursion, equation (3.101):

W(0=W„exp( P(pr°mpt)t) • (7.1)

For lead cooled fast reactors rn = 3 x 10—8 s [7]. It follows that the power would be multiplied by 100 after 14 x 10—8/p(prompt) seconds. Even for P(prompt) = 0.001, i. e. a total reactivity insertion of 0.004 (for a 233U fuelled reactor) the power is multiplied by 100 after 0.14 ms! Consider now the case of a hybrid system with k = 0.98, and a reactivity increase of 0.4% equal to that just considered for the critical system. The energy gain is proportional to 1/(1 — k) and increases by 25% only!

The preceding considerations are very schematic. An example of a realistic calculation [76] is displayed in figure 7.1, where the behaviour of a critical system is compared with that of a subcritical system. In the figure, the total reactivity inserted is as much as 2.55 $. Temperature reactivity dependence is taken into account. The advantage of hybrid reactors, even with a moderate amount of subcriticality, is quite striking.

A critical system is designed so as to make a prompt criticality almost impossible. In a subcritical system the reactivity margins can be increased and new types of fuel can be considered, which would be very difficult to use in a critical system. Note that the power increase due to a reactivity insertion in a subcritical core does not depend on the proportion of delayed neutrons. Thus, this safety parameter is no longer restrictive.

The comparison becomes much more complex in the case of an incident leading to an automatic stopping of the critical system, for instance a temperature increase or the loss of heat extraction. A critical reactor is dimensioned so that such an incident leads immediately to a decrease of the reactivity. Let us take the example of a diminution of the coolant mass

Time (ms) Figure 7.1. Comparison of the power increase of a critical reactor and of different subcritical systems after insertion of an additional reactivity. The additional reactivity amounts to an increase rate of 170 $/s for 15 ms, after which the reactivity remains constant. Note that hybrid reactors are not supposed to be less than 6 $ subcritical. Figure from Rubbia et al. [76].

in the core of a PWR. Several aspects play an important role: on one hand, the capture rates on hydrogen and boron decrease in the core. This effect has a positive impact on the reactivity. On the other hand, the neutron slowing down decreases and the neutronic spectrum becomes harder, leading to a decrease of the reactivity, as shown in figure 7.2, which represents the infinite multiplication factor as a function of the energy of the incident neutron: the neutronic captures become much more frequent (compared with fission) when the energy of the neutrons increases. This modification of the spectrum shape has, thus, a negative effect on the reactivity. Finally, a diminution of the heat extraction leads to an increase of the fuel temperature, which has two main consequences: a shift of the thermal spectrum towards higher energies, and the Doppler effect, which increases the reaction rates in the resonances. As already mentioned, a hardening of the spectrum leads to a decrease of the reactivity. In a PWR, the Doppler effect leads to an increase of neutron captures in the resonance of 238U at 4.6 eV, and makes the reactivity decrease. The characteristics (geometry, fuel composition, coolant proportion, boron concentration, etc.) are determined so that the global effect leads to an immediate decrease of reactivity, without any external intervention. The power is reduced to the residual heat, essentially
due to the radioactivity of fission products, which represents, a few seconds after the end of the chain reaction, a few per cent of the nominal power.

If the same incident occurs in a subcritical core, the reactivity loss (—Sp) due to passive effects explained before (void, temperature, etc.) makes the power decrease in a few seconds and converge towards

P0I1 + ^

0 p0

For a typical Sp of the order of a few tens of ppm (part per million), and p of the order of 300 ppm, the power decrease is of the order of a few per cent only. An additional step is thus required, which consists of stopping the external neutron source. This task is not difficult to carry out in itself, but requires a quick and precise detection of the problem, which could be not so crucial for a critical system. Some designs propose passive systems to manage this kind of incident, as the TIER concept proposed by Bowman [133] where fuse-wire put inside the fuel would immediately stop the beam after a temperature increase. In the Energy Amplifier design proposed by Rubbia et al. [3], the safety is provided by an overflow of the molten lead into the beam pipe, thus stopping the beam outside the subcritical assembly. In the case of an incident which could not be managed by the self-responding behaviour of the fuel and which would require detection and human or automatic intervention, the fall of the control rods of a critical system can be compared with beam stopping. Note that this can be immediately
achieved, a few fractions of a second after incident detection, whatever the core geometry, while the fall of control rods requires a few seconds and can be prevented by core deformation in the case of a major accident.

In any case, an accelerator driven subcritical reactor requires precise measurement of the reactivity, during operation as well as during rest periods. Different methods are considered to measure the reactivity. They are the object of present experimental campaigns, for example the MUSE experiment [134]. One of the aims of this experimental program is to determine a method to calculate the reactivity of a subcritical system, based on the measurement of the time evolution of the neutronic flux in response to a pulsed neutron source. The theoretical time response of a subcritical system characterized by its prompt effective multiplication factor kpff to a delta excitation is, in the approximation of the one-group point kinetics:

N (і) = Щ(і) e-Qt

where Q is the decrease rate defined by

П

where і is the average time between a fission and the previous one in the chain reaction. In the one-group point kinetics approach, і is constant and equal to the mean lifetime of a neutron. і could be calculated by simulation and kpff could thus be determined by the measurement of the time evolution of the subcritical core, following a neutron source pulse.

A recent simulation development [135] shows that this simplified approach is not sufficient to determine the subcriticality level with enough accuracy and proposes a more detailed formulation which takes into account, on the one hand, the differences of the neutron source and the subsequent fission source (energy and spatial distributions), and, on the other hand, the reflector effect, which strongly modifies the time response of the subcritical core.

Monte Carlo simulations, performed on a simplified system, show that the decrease rate Q is clearly not constant with time, as is shown in figure 7.3. This strongly modifies the shape of the population decrease and forbids its description as an exponential decrease. The decrease rate Q(t) converges towards an asymptotic value іїж, which is different from the decrease rate defined by the one-group point kinetics approach. The time required to converge towards corresponds to several hundreds of generations. This very slow evolution is due to the presence of the reflector where the absorp­tion cross-sections are low, and where the neutrons may spend a lot of time before coming back to the fissile zone. Furthermore, the importance of these long-lived neutrons increases with the level of subcriticality, for the same reasons that a population with a small birth rate is mainly old.

The method proposed by Perdu and coworkers [135] is to calculate, by detailed Monte Carlo simulation, the intergeneration time distribution, denoted P(t), which satisfies

P(t) dT = kpff.

It can be shown that this distribution is essentially determined by the characteristics of the reflector and does not depend strongly on the composi­tion of the fuel, nor on the value of kpff. Once the function P(t) is calculated, the fission rate at time t is a solution of

1

Nf(t — t)P(t) dT

0 which leads, for a Dirac pulse, to the expression of Nf(t)

Nf = P + P * P + P * P * P + •••

where the star denotes convolution. Since the distribution P(t) does not depend on the value of kpff, we can write

Nf = f + (kpff )2PI * P +(kpff)3P’ * P* P*■■■

where P is the normalized intergeneration time distribution

P(t )

1

P(u) du

0

Then, the decrease rate can be calculated by

1 dNf(t)

Nf(t) dt

This means that we are able to calculate theoretically the decrease rate Ukp (t) for any given value of kpff.

An experimental measurement of the response of a subcritical core to a neutron pulse provides an experimental decrease rate ^exp(t). It is possible to determine the value of kpff which gives the best agreement between ^exp(t) and Qkf(t). This method could provide a determination of the subcriticality level, which takes into account complex effects, like the energy spectrum of the source neutrons, or the effects of the reflector, where some neutrons spend a lot of time before coming back to the core.

As far as residual heat extraction is concerned, hybrid reactors have essentially the same properties as a critical reactor using the same technology as the subcritical reactor: hence, following the considerations of section 3.4.4, the potential interest of lead cooled and high-temperature gas reactors.