For weakly soluble components like noble metals, it may be efficient to precipitate them in a low-temperature section of the molten salt loop. This will help prevent clogging of the pipes.

9.3.2 Electrolysis

Before examining the application of electrolysis to the separation of metals in a fluoride bath, we think it useful to give a short derivation of the main rela­tions used in electrochemistry.

Much closer to the configuration of what an ADSR could be, the MUSE (Multiplication Source Externe) project is an experimental programme dedicated to the study of neutronic multiplication in a subcritical reactor. The first experiment was carried out in 1995 at CEA/Cadarache.

The principle of the MUSE experiment consists of coupling a fast neutron subcritical reactor (MASURCA) to an external neutron source. MASURCA is a very low power (maximum 5 kW) experimental reactor. The core is rather small (60 cm in height and about 100 cm in diameter). The fuel is a MOx fuel (UO2-PuO2) enriched with 25% plutonium. Sodium or lead rodlets are put in the fuel elements to simulate the coolant. It is also possible to remove these rodlets to simulate a gas cooled reactor.

The first tests have been done with a strong californium source, placed in the centre of the core. Today, a pulsed neutron generator is used (GENEPI: Generateur de Neutrons Pulse; Intense), in order to study the time response of the subcritical core to a burst of source neutrons. The neutrons are produced by a deuteron beam, impinging on a deuterium target (yielding around

Figure 13.1. Geometry of the MUSE experiment with the deuteron accelerator GENEPI feeding into the (sub)critical fast-neutron pile MASURCA.

2 MeV neutrons) or a tritium target (14 MeV neutrons) placed at the centre of the reactor. A schematic view of the coupling is shown in figure 13.1. A lead buffer zone is placed around the source to simulate a spallation target. Different subcriticality levels can be investigated (k = 0.95 up to criticality) by playing with the configuration of the fuel elements. The pulse width is less than 1 ms, with a shape very close to that of a square gate. These charac­teristics allow very precise measurements of the evolution of the neutron flux in the core over a few tens of milliseconds. The main characteristics of the neutron generator are listed in table 13.1.

Different types of measurement are performed. On one hand, the spectrum shape is studied, at different positions in the core, target and reflector, using different types of detector (fission chambers, helium detector

for example). On the other hand, the time response of the system to a neutron pulse is measured, in order to devise a method able to determine the subcriticality level without starting the reactor in a critical configuration, as is done at present. The subcriticality level is determined by following the calculation described in chapter 7. The results of such an experimental pro­gramme are very important in the context of the definition of an experimental ADSR at significant power. In particular, the MUSE project has to provide a precise method to control the subcriticality level of the system in operating conditions.

Alvarez was the first to propose to consider the containment tank and the drift tubes together as a high-frequency resonating cavity [180]. This gave birth to the modern concept of proton and nucleus Linacs. In order to under­stand the principle of the Alvarez concept, it is useful to consider the simple case of a cylindrical cavity.

One possible way to allow the production of energy essential to the develop­ment of developing countries, which account for the majority of humankind, without catastrophically increasing greenhouse gas emissions might be to rely increasingly on renewable energies. In order to assess their possibilities, we briefly review the main ones.

In nuclear reactors the fission of a nucleus results from the absorption of a neutron. This fission is accompanied by the emission of v neutrons, with v between 2.2 and 3, depending on the fissioning species.

These neutrons, in turn, may induce fissions, and thus produce new neutrons. However, each neutron does not produce a fission. It may be absorbed either in a non-fissile or in a fissile nucleus without fission of the said (fission probability after neutron absorption by a fissile nucleus is never 100%). A neutron created in a medium (which we first consider infinite) containing fissile nuclei will give birth to ko second-generation neutrons. The number of neutrons of the third generation will be k1 and that of generation n, kO—1. Each neutron generation is the result of a neutron-producing nuclear reaction which can be a fission or, more rarely, an (n, xn) reaction. The total number of neutrons following the apparition of a neutron in the multiplying medium will be[16]

1

nchain = 1 + ki + koo + ‘"+k1o + ‘" = 1 _ k. (3.71)

1 ko

The total number of neutrons created in the medium per source neutron is simply konchain. One defines a neutronic ‘gain’ as the ratio of the total number of neutrons (source + created) to the number of source neutrons. This gain is then 1/(1 — ko). Since all neutrons are ultimately absorbed, the number of absorption reactions is thus nreac = nchain. For finite media one has to replace ko by an effective value of kef which is less than ko due to neutrons escaping from the system. One should also consider local values, ks, dependent on the specific location of the apparition of the initial neutron. If keff is larger than unity the reaction diverges, i. e. from

one initial neutron the final number of neutrons goes to infinity. A controlled divergence allows one to start a reactor. When uncontrolled it leads to a criticality accident as in Chernobyl. Of course, in the case of nuclear weapons, the divergence is sought. When keff is kept equal to unity one obtains a critical reactor. The possibility to keep precisely the condition keff = 1 is due to the presence of a small fraction of delayed neutrons[17] which allow time to compensate for deviation of the criticality coefficient keff from unity. If keff is less than unity an incident neutron gives birth to a finite number of secondary neutrons. The medium is said to be multiplying. The multiplication factor is 1/(1 — keff).

The simulation of the neutronic behaviour (safety features, fuel evolution, radioprotection, etc.) of a nuclear reactor requires precise knowledge of the reaction rates, mainly elastic and inelastic scattering, fission, radiative capture, (n, a) and (n, xn) reactions, fission product yields, angular and energy distribution of the produced neutrons, and, of course, the decay time of the radioactive nuclei. The neutron energies in a reactor range from 0.01 eV to a few MeV.

The ADSR concept extends the neutron energy up to a few hundred MeV, and requires a precise knowledge of the interactions of charged particles with the nuclei of the spallation target (neutron emission, angular distribution, production of residues, etc.). These specific codes and data are presented in chapter 6.

At present, most of the codes used to simulate the neutronics of a reactor core use data libraries describing neutron-induced reactions from 10 peV to 20 MeV. The deterministic method and the Monte Carlo approach do not use the data bases in the same way. Both methods require, however, a huge number of data in order to perform precise calculations. The neutronic libraries provide evaluated values of the cross-sections at a given temperature.

There is at present no nuclear model able to predict exactly the neutron cross-sections in the energy range covered by reactor physics. Thus, most of the data available in the neutronic libraries are evaluated from experimental measurements. Different effects have to be taken into account, for instance background, impurities in the sample, etc. In general, the evaluation of neutronic data requires different types of measurement, which do not necessarily correspond to the same energy range. Thus, a compilation step is required, to combine different types of experimental data together, and produce a list of evaluated parameters of the cross-sections. This very impor­tant step consists of fitting the data with a nuclear model. One of the approaches that can be considered is the Bayesian one, which allows a compilation of different data sets. This statistical method is based on the equation

p(AB)^p(BA)p(A), (5.1)

where A is the quantity to be determined (a resonance parameter for example) and B is the set of data. p(A/B) gives the probability distribution of the evaluated parameter. p(B/A) is the probability that quantity B will be observed if the parameter is A (likelihood function). This quantity is deter­mined by the nuclear model chosen. Finally the last term p(A) represents what is known about A before the experimental measurement. This equation shows how the information contained in a new measurement can be added to the initial data base corresponding to p(A), in order to update the knowledge of the parameter A. A simplified example of this approach is detailed below.

Depending on the neutron energy, different models (p(B/A)) are used to extract the cross-section parameters from the measurements. The R-matrix theory [67, 68] is generally applied in the energy domain where the resonances can be resolved experimentally. This range of energy depends on the nucleus studied and on the quality of the measurements. It corresponds in general to thermal energies, up to a few tens of keV. This formalism considers only binary collisions: an ingoing wave function describes the two incident particles (the neutron and the nucleus); an outgoing wave function describes the emerging reaction products (the neutron and the excited nucleus, or fission fragments, etc.). The space is divided into, on one hand, the external area, where nuclear forces are negligible, and, on the other hand, the internal region, where nuclear forces dominate. The external region is handled by pure Coulombic wave functions of the particles. Different approximations can be used to describe the excited nucleus, the Breit and Wigner approach being preferred for an isolated resonance. For the different absorption reactions i, the cross-section is a Lorentzian:

where Гп is the neutronic width and Г = ^1 rt the total width.

In these equations,

• A = ЙД/2^ECM where ^ is the reduced mass and ECM the energy of the particle in the centre of mass system.

• gJ = (2J + 1) /2(2/ + 1) where I is the spin of the target nucleus, and J the spin of the compound nucleus level.

• E0 is the energy of the resonance being studied.

More precise models calculate the resonance profile in more complex cases. However, the different parameters (width, mean energy) have to be fitted on experimental measurements. We give here a simplified example of Bayes’ method, using the least-squares approximation. Let us consider a resonance of a given nucleus. We will assume that the partial widths of this resonance are perfectly known, and that we are trying to determine only its position in energy.

We consider that a first experimental measurement has been performed, and that we have prior knowledge of the resonance position, in the form of an estimated value e1 and an associated error oy. The prior distribution of E0, given e1 and o1, is then

p(E01"1,О) dE0 « exp ^- (E°2o.2"1 ) ) dE0. (5.4)

We suppose now that we perform a new experiment concerning the position of this resonance (for example a fission experiment), and that we obtain a data set fyig. We choose the Breit-Wigner description which leads to a theoretical observable fyth « oBW(Ei)g. Thus, the likelihood of obtaining the values f yi g, given the theoretical f ythg, is

P(yil/h df yig ~ exp (- X (Уі 2fly’ ) ) df Уіg (5.5)

where o2 = ((yi — yth)2) and dfyg = dyj dy2 • —dyN. Finally we obtain a new distribution of the energy position, given by the product of the like­lihood function by the prior distribution:

p(E„)iE„ = Ц-ДХ — X) dE«. (5.6)

This evaluation contains the knowledge taken from the two successive experimental data sets. The correlation between prior and new data has been neglected here, but could be taken into account.

This approach can be generalized to a vector of parameters X = {X1, X2,XN g (for example position, total and partial widths of a resonance). Prior knowledge of this vector is in the form of an estimated vector E and an associated covariance matrix A (with A = ((E — X)^(E — X))), which describes the uncertainties and correlations of these parameters. The probability distribution of X can be written as

p(XE, A) ~ exp(— 1 (E — X)fA—1 (E — X)) dX (5.7)

where dX is the volume element of the N-dimensional parameter space.

This evaluation step requires sophisticated computer programs. For example, the SAMMY [69] code has been developed at the Oak Ridge National Laboratory (ORNL), and allows one to parametrize the experi­mental data (essentially neutron time-of-flight data), using Bayes’ method, together with the multilevel R-matrix description. This code gives the best fit of the experimental data and the associated covariance matrix. Physics effects, such as the Doppler effect, are taken into account.

For heavy nuclei, the resonances can be separated for energies below a few tens of keV. Above this energy, the resonances are too close to be experi­mentally separated. In this domain, the evaluated cross-sections are built with resonance parameters that are chosen randomly. Individual partial widths are chosen following the Porter-Thomas [70][29] distribution:

n nx 2" 1

2Г(ЙІ T) e with x = г(а)/(Г(а)).^ The average values (T(q)) are extrapolated from the region of well-separated resonances or from nuclear model estimates. In the Porter-Thomas distribution, n is the number of degrees of freedom. For neutron elastic scattering widths, there is only one final state, so that n = 1. For gamma rays, there are many available levels for the primary gamma decays, n ~ 30-40. For fission, the relevant degrees of freedom are the Bohr and Wheeler transition states, and one finds typi­cally n ~ 3-4. Note that large values of n correspond to small fluc­tuations around the average. Resonance energies are chosen according to the Wigner interval distribution [71] between next-neighbour levels with same spin and parity which reads P(S) =2 kS e—4KS with S = D/(D) and D the distance between two nearest neighbours.*

Families of resonances with different spins and/or parities are handled independently.

At higher energies, the resonance widths are larger than the spacing between each of them, and the cross-section varies smoothly with energy. In this energy domain, the optical model is generally used to evaluate the experimental measurements.

The evaluated cross-sections are usually found in nuclear data evaluated files like ENDF-B6, JEF 2.2, JENDL or BROND. These files contain the parameters of each resonance and sometimes the correlation matrix of these parameters, which allow sensitivity calculations. These files, as well as the experimental files ([30].EXFOR files in the CSIRS library), can be found on the National Nuclear Data Center (NNDC) site* at Brookhaven National Laboratory.

In general, an additional step is required and consists of decoding the data files, and producing a new one at the chosen temperature, which will be consistent with the code and the system studied. For deterministic codes, the reaction cross-section must be calculated and averaged on different energy groups, while Monte Carlo codes require, in general, continuous cross-section values obtained by a list of points and an interpola­tion procedure. For example, the program NJOY reads an ENDF format file, and writes a specific file for a Monte Carlo code such as MCNP, taking into account the Doppler broadening of the resonances.

The spallation source of the ADSR implies neutron energies of a few hundred MeV. Present nuclear data libraries contain neutronic cross-sections up to 20 MeV. Different experimental programmes aim to widen the neutronic cross-section available in the libraries up to 200 MeV.

One has to decide which are the relevant parameters for controlling the evolution (fixed ks=eff, fixed power, …). For example, in the study by Brandan et al. [77], the average reactor power density (per fuel volume unit) was fixed at 330W/cm3, which permits local values of the order of 600W/cm3 at the reactor centre, an acceptable value from a thermo­dynamical point of view. The proton beam intensity (in this study a true spallation target was generated by FLUKA) is adjusted during the reactor evolution to compensate the reactivity fluctuations from one cycle to the next. This restriction imposes a constant number of fissions per unit time, so that it is possible to associate a fixed burn-up to all cycles (defined at fixed operation periods). In turn, this allows the direct comparison among inventories after different cycles for all types of reactor fuel.

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

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.

To describe the development capability of a fuel cycle, it can be useful to cal­culate the doubling time of a given system. It corresponds to the time needed to breed enough fissile element to start a new reactor. It can be defined by

Td(years)=-F— (П.3)

-FNa

0.99 0.98 0.97 0.96

where I is the inventory of the fissile element, 1 — ц the fraction of the elec­tricity produced which has to be used for the accelerator (ц = 1 for a critical system) and F the fission rate per year (around 1000kg/GWe depending on the thermal efficiency of the system).

An alternative definition of the doubling time can be the time which is needed by Nr reactors to double the installed power, with Nr very large. We denote this doubling time T“. In this condition, we can write that the fissile breeding during dt is NrFNadt, and the number of new reactors which can be built during dt is

which leads to

Nr = Nr(t = 0) exp ^FNa t

and

For a solid fuel cycle, the cooling time (around 5 years) has to be taken into account: one has to alternate two cores.[55] The fissile inventory is thus twice as large, and is around 15 tons in a fast lead cooled reactor of 2500 thermal GW. Considering ц = 0.9 and Na = 0.3, we find a doubling time of 55 years (T“ = 38 years). This is a typical doubling time for a fast- spectrum core.

The breeding rates can be improved using gas as coolant, which leads to a faster spectrum, as compared with lead cooled systems, and makes the fission of 232Th non-negligible (around 8% of the total fissions). The core can be surrounded by a large thorium blanket, in order to minimize neutron leakage and to optimize the captures on 232Th. The elements of this blanket have to be changed every 2 months, in order to avoid modifying the evolution of the reactivity. The 233U produced is then accumulated to start a new reactor. The 233U of the core is reprocessed, with all other actinides. Different subcriticality levels are obtained by playing on the proportion of fuel in the core. Figure 11.5 shows the annual growth factor of such a system, as a function of the multiplication factor, for two different blanket configurations. A doubling time TU of 40 years is obtained for k = 0.97 and 60 years for k = 1 (by extrapolation). With this the advantages of hybrid systems for development of fast thorium reactors can be quantified. Subcriticality can help for a deployment of the nuclear power on a few tens of

years at the scale of a single country, but appears not to be sufficient to consider a fast increase of the share of nuclear power in the world in 20 or 30 years.

At the end of this section we mention a scenario which associates fast — neutron reactors and molten salt thorium reactors, allowing a very fast growth of world nuclear energy.

The solution to equation (I.5) in a semi-infinite environment defined by an interface at x = L, and for N nuclei, gives the current at the interface:

For stable nuclei (A = 0), the flow has a maximum at time:

L2

Ts = 6D

which can be considered as a delay to the egression of the radioelements. The width of the egression time distribution is about

It appears that the clay layer plays a twofold role: it delays biosphere contamination (thus leaving time for many isotopes to decay); it spreads the contamination over a duration that is proportional to the delay. The
maximum value determines the amplitude of the risk for the critical popula­tion. The risk, then, is proportional to the diffusion constant and inversely proportion to the square of the layer thickness. The quadratic dependence on the clay layer thickness emphasizes its importance and, as a consequence, that of a judicious selection of the geological formation used for disposal.

These formulas are easily modified so as to take the decay constant of the radioelements into consideration. Similarly, the time spread of radio­activity release from the fuel elements can be taken into consideration.