Before describing the physics specific to hybrid reactors, it is appropriate to review the basics of nuclear reactor theory. We first recall some elements of neutron physics which apply to both critical and subcritical systems.

 

3.1 Interaction of neutrons with nuclei

 

A — 1 A + 1

 

A — 1 A + 1

 

cos(e)

 

1

 

(3.1)

 

1

 

‘ The absorption cross-section <ra = ac + <rF.

If the scattering in the centre of mass is isotropic it follows that all final energies between E0 and E0{[{A — 1)/(A + l)]2} are equiprobable. Since, in this latter case, the neutron energy loss is proportional to its initial energy, it is convenient to measure energies in terms of lethargy u = ln(E0/E) where E0 is some arbitrary initial energy (usually the average energy of fission neutrons), and E is the actual neutron energy. We define

It is convenient to define the average lethargy gain, or, equivalently, average logarithmic energy loss per collision

f$E° E0 dE, $

C = ln = 1 +

e0 E E0(1 — $) 1 — $

which expressed as function of the mass A yields

For large A, C — 2/(A +1).

In general, the evolution of the nuclear fuel is followed by solving the Bateman equations, which read

dn (t) dt

where ni is the number of nuclei of type i per unit volume, Ai j is the decay constant of nucleus i to nucleus j, acj is the capture cross-section of nucleus i giving nucleus j, ctt, the total cross-section of nucleus i, is the sum of the capture and fission cross-sections, ‘ is the neutron flux. These equations are summarized in the vector-matrix form.

dn

dt

In order to describe a cell, one can use intersections, unions or exclusions of surfaces (and also exclusion of other cells). Thus, a cell will be defined with

• a cell number, identifying the cell

• either a material number with its density or a ‘0’ if the cell is void,

• a list of surfaces (or cells) with operators (intersection: a blank; union: a colon ‘:’; exclusion ‘#’).

First, let us see how cells are built from surfaces; for the sake of simpli­city we will deal with a two-dimensional empty cell (see figure 5.3).

10-123-4

©

Figure 5.3. A simple geometry (a) and a somewhat more difficult one (b). Circled num­bered are the cell numbers, plain numbers are surface numbers.

In this example, the ‘-’ sign before surface 1 means that one considers the region below surface 1 (i. e. y — d1 < 0, if surface 1 is given by y — dj = 0). Similarly, the region on the left of surface 4 is considered (x — d4 < 0) whereas the ‘positive’ surfaces 2 and 3 mean that the region being considered is to the right and above them respectively. Cell 1 is thus the intersection of these four ‘signed’ surfaces. The ‘exterior’, i. e. cell 2, can be defined as

2 0 1:-2:-3:4

that is, as the union of the outside region defined by the surfaces; the exclusion operator can also be used:

2 0 #1

which means that cell 2 is the whole space except cell 1.

Consider now the more difficult case of figure 5.3(b). First, forget cell 3 and surface 6. Because cell 1 has a concave corner the cell has to be defined also with unions, namely:

10-123 (-4:-5) (Fig. 5.3(b) without surface 6)

Now, taking into account surface 6 and cell 3,

10-123 (-4:-5) 6 (Fig. 5.3(b))

3 0-6

where ‘-6’ means the inside of circular surface 6.

The outside of cell 1 could be defined as

2 0 1:-2:-3:(4 5)

or, with the exclusion operator,

2 0 #1 #3

Don’t forget to exclude cell 3 also! Indeed, #1 means ‘the whole space except cell 1′ and cell 3, although it is inside cell 1, does not belong to that cell.

To illustrate a real input geometry with a short example, let us consider the geometry of figure 5.4.

First simple geometry c

c Cell cards c

10-1 $ the inner sphere

20-23-41 $ the cylinder without the sphere

3 0 #2 #1 $ exterior c

c Surface cards

c

1 SO 5 $ centred sphere with R=5 cm

2 CZ 20 $ infinite cylinder with R=20cm

3 PZ -20 $ bottom plane intersecting the cyl.

4 PZ 20 $ top plane intersecting the cyl.

c end of the file

Suppose that this geometry is written in a file named ‘mygeom’. In order to see this geometry, simply type on the command line:

mcnp ip i=mygeom

A prompt ‘plot>’ allows you to view a section perpendicular to the z axis at the value z0 by the command

pz z0

Other views can be seen with ‘px’ and ‘py’. This shows a 2D view of the geometry. If there are errors in the geometry the ‘bad’ surfaces are drawn with a dotted red line.

An alternative way to measure neutron multiplicities has been used by the TARC collaboration [57]. The goal of the TARC experiment was to evaluate the possibility to transmute long-lived fission products like 99Tc by adiabatic resonance crossing, as discussed in section 3.2.5. To this aim a large 330 ton

block of pure lead was used as a target for the CERN PS protons at various energies. A number of holes were drilled through the block, allowing the placement of different types of neutron detector inside it. In particular, 3He gas proportional counters and silicon detectors viewing 6Li or 233U targets allowed the measurement of the neutron fluxes. In the range between 1 eV and 1 keV, neutron energies were determined by their slowing down time as discussed in section 3.2.5. A detailed mapping of the neutron flux could, therefore, be obtained. This mapping was reproduced by a Monte Carlo simulation, as shown in figure 6.11.

The figure shows excellent agreement between the different types of measurement and the calculated values. The simulation found a multiplicity of 31/p/GeV for neutrons falling below the threshold of 20MeV, in agree­ment with the HERMES calculation of Hilscher et al. [110].

Organic solvents suffer significantly from radiolysis. Therefore, before reprocessing they require sizeable cooling times of the spent fuels. Typical cooling times, such as those used at the COGEMA The Hague plant, are of the order of 5 years although cooling times as short as 1 year have been considered [138]. Shorter cooling times lead to a shorter lifetime of the expensive solvent, higher costs and larger amounts of wastes. When it is very important to minimize the cooling time, the fragility of organic solvents becomes a very serious drawback. This is especially the case in two occurrences: fuel reprocessing of breeders and molten salt reactors. In the first case the cooling time reflects directly on the doubling time. In the second case, online reprocessing involves treatment of an extremely active fuel. Dry inorganic processes appear to be much less sensitive to radiation effects and have been proposed and developed for the two aforementioned cases. In the breeder case, reprocessing of fast reactor fuels by chlorination was proposed, for example in the frame of the ‘integral fast reactor’ by the Argonne National Laboratory [151]. For molten salt reactors the reference is the fuel which was proposed in the MSBR project [50], which consisted of a mixture of fluorides.[49] Because of the subject of this book we shall restrict our considerations to the processing of fluoride salts rather than chloride salts. However, with the exception of vaporization of fluorides the possibili­ties offered by fluorination and chlorination are rather similar.

Elemental separation processes from a mixture of fluoride salts use one of the following techniques:

• vaporization of volatile fluorides

• gas purge

• liquid-liquid exchange

• selective precipitation

• electrolysis.

Efficiencies of these processes depend strongly on the relative amounts of fluor ions in the salt mixture. Large amounts of fluor ions lead to oxidizing conditions and shift the valence states of metals to high values. Oxidizing conditions often accelerate reaction rates and may be sought. However, the composition of the salts is often determined by other considerations such as fusion temperatures or corrosion reduction. Optimization with respect to such properties led the proponents of the molten salt breeder reactor to choose a mixture of the 7LiF, BeF2, ThF4 and UF4 fluorides in proportions (71.7: 16:12:0.3mol%) [154]. This mixture serves as a reference for our discussion. During irradiation 232Th is transmuted into 233Pa which can capture neutrons or decay to 233U. It is efficient to minimize the sterile captures in 233Pa, and therefore to extract it from the salt, and let it decay into U outside the neutron flux. Subsequently, the U can be re-injected into the salt or stored for future use. Similarly the quantity of fission products in the salt should be limited as much as possible. This is especially true for rare earths which have large neutron capture cross-sections. Reprocessing of the MSBR fuel, thus, consisted of extraction of 233Pa and of as many fission products as possible, using the different techniques mentioned above.

Not many cost estimates of hybrid reactors are available. It is, however, interesting to check whether the adjunction of a high-intensity accelerator to a reactor would not lead to an unacceptable cost increase. In this respect, although there are obviously large uncertainties, the CERN group cost estimate of its proposed Energy Amplifier [162] is interesting. The IEPE of Grenoble [53] has evaluated the estimate of the CERN group and has examined several cost scenarios differing by the intercalary costs, the standardization effects, the size and prototype cost incidence, as well as the fuel cycle reprocessing cost. Before giving the global results of these cost calculations[58] we review some of the elements given by the CERN

group, especially those corresponding to non-classical parts of the Energy Amplifier:

• The accelerator system involves four cyclotrons:

1. Two injector cyclotrons yielding 6 mA of 10 MeV protons with a cost of 4Mc each, and a total of 8Mc.

2. One intermediate energy separated sector cyclotron yielding 120 MeV protons for a total cost of 32 Mc.

3. The main separated sector cyclotron yielding 1 GeV proton for a cost of 80 Mc.

The total cost of the accelerator system, including the beam handling, power supplies and biological shieldings, amounts to 160 Mc.

• The cost of the subcritical system is evaluated at 50 Mc including:

1. 6Mc for the 10000 tons of lead.

2. 12 Mc for the main vessel with a total weight of 400 tons.

Table 12.1 is taken from the work of the IEPE. It compares the invest­ment costs given by the CERN group with those revised by the IEPE and with those of a PWR (with French conditions).

The investment costs would be of the same order for the PWR and the Energy Amplifier. Note that the cost of the non-classical part amounts only to a relatively small fraction of the total cost, so that the rather large uncertainties on this part have a limited influence on the total cost.

Table 12.2 compares the kWh production cost according to the CERN estimate and to the three scenarios considered by the IEPE to that of the PWR.

One sees that the cost of the energy produced by the Energy Amplifier would be, at worst, comparable to that of present PWRs. It is hoped that

the additional cost of the accelerator complex would be balanced by the simplifications on the reactor, due to its inherent subcriticality. Reductions on the fuel cost stem from the suppression of the uranium enrichment step and from the fewer reprocessings (larger burn-up). Note that an uncertainty exists concerning the best reprocessing technology (Thorex or pyro-electrolysis). Furthermore, the costs presented above also have large uncertainties and should be considered as preliminary approaches. Indeed, an alternative approach was made by Bacher [172] based on the experience of sodium cooled fast breeders. This analysis found a kWh cost twice as expensive as that obtained with PWRs.

The price comparison of electricity produced by different techniques and in different countries is instructive and is given in table 12.3, taken from the IEPE study. The table[59] shows that the conditions under which nuclear plants are built have a very large influence on their cost.

The accident occurred at 01:23 hr on Saturday, 26 April 1986, when the two explosions destroyed the core of Unit 4 and the roof of the reactor building. The two explosions sent a shower of hot and highly radioactive debris and graphite into the air and exposed the destroyed core to the atmosphere. The plume of smoke, radioactive fission products and debris from the core and the building rose up to about 1 km into the air. The heavier debris in the plume was deposited close to the site, but lighter components, including fission products and virtually all of the noble gas inventory, were blown by the prevailing wind to the north-west of the plant.

Fires started in what remained of the Unit 4 building, giving rise to clouds of steam and dust, and fires also broke out on the adjacent turbine hall roof and in various stores of diesel fuel and inflammable materials. Over 100 fire-fighters from the site and called in from Pripyat were needed, and it was this group that received the highest radiation exposures and suf­fered the greatest losses in personnel. These fires were put out by 05:00 hr of the same day, but by then the graphite fire had started. Many firemen added to their considerable radiation doses by staying on call on site. The intense graphite fire was responsible for the dispersion of radionuclides and fission fragments high into the atmosphere. The emissions continued for about 20 days, but were much lower after the tenth day when the graphite fire was finally extinguished.

Anthropic CO2 emission amounted, in 2000, to 24 Gigatons.[3] The regional emissions of polluting agents and greenhouse gases are summarized in table 2.4 [27].

The comparison between emissions from different countries, apparent in table 2.5, is instructive and gives a measure of the effort which these countries will have to achieve in order to improve the situation [28].

CO2 is essentially a greenhouse gas. Although sulfur and nitrogen oxides are much more effective greenhouse gases than CO2 at the molecular level, their much smaller concentrations and shorter lifetimes in the atmosphere make them contribute relatively little to the overall greenhouse effect. They are the main cause of acid rains, as well as of atmospheric ozone. The three main different fossil fuels (coal, gas and oil) have different greenhouse gas emission rates, as shown in table 2.6 [28].

Table 2.6 also shows that, whenever possible, the co-utilization of elec­tricity and heat allows significant gains on the emission rates.[4] Aside from CO2, methane also has a strong greenhouse effect, about one third of that of CO2. Apart from leaks in the gas transportation system and releases from coal mining, methane is essentially produced in the agriculture sector and will not be considered further here. The same applies to CFC and other stable and complex gaseous molecules.

The solution of the diffusion equation (3.24) requires that the one-group cross-sections be known. Some of the most important fission and capture cross-sections for heavy nuclei are given in tables 3.1 and 3.2. Table 3.1 gives the cross-sections for a PWR spectrum while table 3.2 gives the cross-sections for a fast reactor of the Superphenix type.

Aside from energy production, it is important to evaluate the potential of hybrid reactors for transmutation, i. e. to what extent they produce excess neutrons. A standard reactor can be viewed as a device producing energy and neutrons. Both energy and neutrons are primarily produced by fission. Fission releases about 200 MeV and 2.5 neutrons. It follows that one may

say that 80 MeV are needed to produce one neutron. The spallation process requires only 30 MeV to produce one neutron. Should the fission 200 MeV be available for proton acceleration one would, then, get more than nine neutrons per fission (2.5 ± 6.6)! True enough, no usable energy would be produced. In fact, assuming a thermodynamic efficiency of 40% and an accelerating efficiency of 40%, one finds that about 6GeV are needed to accelerate a proton to 1 GeV. It would then be possible to obtain about 3.5 neutrons per fission, still without producing usable energy. For more realistic scenarios one sees that an accelerator allows an increase of the number of neutrons available for transmutation at the expense of usable energy. It is interesting to see if, as far as neutron availability is concerned, hybrid reactors are more or less efficient than the association of a critical reactor and an accelerator. The number of neutrons produced in the hybrid reactor is

N0

N = ^; (44)

while the number of fissions is

On average a fission is produced by (<rF ± <rc)/aF neutrons. The total number of neutrons needed to produce NF fissions is

where ] is the number of neutrons produced following the capture of an initial neutron by a fissile nucleus. The total number of neutrons available for transmutation is therefore

NDhyb = N — Nnf = 1 — k). (4.7)

We now consider a critical reactor coupled to an accelerator. NDr is the number of neutrons available when using a reactor producing NF fissions, in addition to the N0 spallation neutrons. The number of neutrons necessary per fission is

= 1 + a (4.8)

°F

while the number of neutrons produced per fission is v. It follows that the number of neutrons available per fission is v — 1 — a. The total number of neutrons available in the reactor is then

and the total number of neutrons available for the reactor + accelerator system is

N°- = N»(>+*rh)(v -1 — a)) = (’ — k)’ (4Л0)

Thus

NDhyb = NDr — (411)

It follows that the choice of a specific value of k is irrelevant as far as the transmutation capabilities are concerned. Whatever the method of coupling between the fission reactor and the accelerator, the number of available neutrons is

Nd = N0 + Nf(v — 1 — a). (4.12)

From the preceding, it is seen that using 10% of the available energy allows us to obtain about 0.1 additional neutrons per fission. Although small, this number has to be compared with the number of neutrons which are effectively available in reactors. We know that the maximum number of available neu­trons per fission amounts to v — 1 — a. In practice the real number is smaller than this value due to captures in structural materials and to transmutations of fertile nuclei. Let the number of such capture neutrons be vc. The number of available neutrons is then v — 1 — a — vc. Captures in structural materials cannot be much less than 0.2 neutron per fission, particularly as reactivity changes are counterbalanced by the presence of consumable neutronic poisons. For each fissioning nucleus a fissile nuclei suffer neutron capture leading, in general, to a fertile nucleus. If one requires regeneration of the nuclear fuel, one sees that vc = 0.2 + 1 + a at least. The number of available neutrons amounts to v — 2(1 + a) — 0.2. We consider four cases.

1. The thermal 238U-239Pu system. Then, v = 2.871, a = 0.36. The number of available neutrons is 2.871 -(2 x 1.36)- 0.2 = —0.05. Regeneration is not possible and no neutrons are available for transmutation. The 0.1 additional neutrons made available by the use of an accelerator would allow regeneration.

2. The thermal 232Th-233U system. In this case v = 2.492, a = 0.09. The number of available neutrons is 2.492 -(2 x 1.09)- 0.2 = 0.11. Regenera­tion is possible and 0.1 neutrons are available for transmutation. The addi­tional number of neutrons that an accelerator would bring is significant.

3. The fast 238U-239Pu system. In this case v = 2.98, a = 0.14. The number of available neutrons is 2.98 -(2 x 1.14)- 0.2 = 0.5. Regeneration is easy. The advantage of an accelerator is not compelling.

4. The fast 232Th-233U system. In this case v = 2.492, a = 0.093. The number of available neutrons becomes 2.492 -(2 x 1.093)- 0.2 = 0.10. Regen­eration is possible. The additional number of neutrons that an accelerator would bring is significant.