• An NPS card specifies the number of particle histories: NPS n means that the Monte Carlo calculation stops after n source particles.

• A CTME card specifies the computer time limit (in minutes): if a run duration is greater, the Monte Carlo calculation stops.

TotNu and NoNu cards

• NoNu means that fission is treated as a simple capture; this card may be useful to see primary fission neutrons.

• TotNu has two meanings, depending on the type of calculation: (1) In KCODE mode (critical calculation), if TOTNU is not present or is present with no argument, the total neutron number (including delayed neutrons) v is used; if TOTNU is followed by NO, the prompt v is used. (2) In non-KCODE mode, if TOTNU is absent or present with NO, the prompt v is used, whereas if TOTNU is present with no argument, the total v is used.

PHYS card

This card is used as

We show in chapter 8 that, if control rods are to be avoided, the multiplica­tion factor keff should be limited to about 0.98 for the standard fast-neutron hybrid reactors and 0.95 for the slow-neutron ones. This limitation on keff also limits the energy gain G0/(1 — keff) accordingly. An interesting sugges­tion was made as early as 1958 by Avery [131], in order to increase the energy gain of a hybrid system; it was reactivated by Abalin et al. [124] and Daniel and Petrov [132]. It consists of coupling two multiplying systems in such a way that neutrons produced in the first one can penetrate the second while those produced in the second cannot penetrate the first. We quantify the possible gain which can be obtained in this way.

Let one neutron be created in a multiplying medium. If absorbed it produces kx new neutrons. However, in a finite system it only produces keff neutrons.[43] Since keff = Pcapk1 the escape probability is

Pesc = 1 — k2 • (6.12)

k1

If we consider a system with N0 injected neutrons and multiplication keff, the number of escaping neutrons will be

N0 ki keff 1 — keff ki

Now, we consider two multiplying media which communicate. Let

k !21

K =

!12 k2 _

be the matrix of efficiencies: a neutron born in medium 1 gives k1 progeny in medium 1, and i12 in medium 2. Let и(1) and n(2) be the number of generation i neutrons in media 1 and 2. The number of neutrons of the next generation in each medium is

that is,

ni +1 = Knt

and the final number of neutrons as a function of the initial one:

which yields, for

while, if one neutron is created in medium 2, the final numbers are

If one could define a system where ^12 ^ 0 and ^21 = 0, one would get

Abalin et al. [124] propose that the first medium could be a fast-neutron multiplier but a strong thermal-neutron absorber while the second medium would be both a slowing down and thermal-neutron multiplier. The fast neutrons created in medium 1 could, eventually, reach medium 2 and be slowed down and multiplied. In contrast, slow neutrons from medium 2 could not reach medium 1 without being immediately absorbed in the strong thermal-neutron absorber (for example a gadolinium nucleus). As suggested by equation (6.12), an estimate of!12 is

which shows that it is interesting to maximize k1M, and thus to use as pure fissile material as possible. !12 will, in general, be of the order of unity. It is mandatory that (1 — k1)(1 — k2) > !21!12 ~ !21. In order for the system to be of interest as compared with standard ones, k1 and k2 should be close to 0.95. This means that the coupling term, !21, should be less than 2 x 10—3. A serious safety problem might arise from an unwanted decrease of the amount of absorber in medium 1.

Rather than using a difference between the neutronic properties of medium 1 and 2, it is possible to play on the relative geometrical arrangement of the two media. For example, consider that the first medium is a sphere with radius R1 surrounded by a spherical shell between R1 and R2 comprising medium 2. A neutron exiting medium 1 has, evidently, a unit probability of entering medium 2 so that!12 is given by equation (6.18). A neutron emitted from the inner surface of the shell has probability 1 — 1 — (R2/R2) of

entering medium 1. In the absence of medium 1, neutrons lost by medium 2 exit by the external surface of the shell. If the shell is not too thick, the number of neutrons crossing the inner surface of the shell should be approxi­mately equal to this last quantity. It follows that

The minimization of!21 implies a minimization of (k2l — k2)/k2l in agree­ment with other constraints. Typically, for breeder reactors, (k2l — k2)/k2l is close to 0.1. It follows that the condition on the product!12!21 implies that Rj/R2 < 0.2. With R1/R2 = 0.1, !12!21 — 5 x 10—4 and

n2F) = 500N0 (6.20)

for k1 = k2 = 0.95 and!12 = 1. In these conditions, if a neutron is created in medium 2, the final number of neutrons will be n1 = 0.25 and n2 = 25, so n1 + n2 = 25.25. The large amplification difference when neutrons are created in medium 1 and when they are created in medium 2 shows that very high neutron multiplication can be obtained in the case being discussed, the system remaining, however, safely far from criticality. This could give the possibility of reducing by almost one order of magnitude the power require­ment for the accelerator.

Figure 6.16 shows the result of a very simple Monte Carlo calculation which illustrates the preceding discussion, and shows how a very high multiplication can be obtained, while staying very far from criticality. The model reactor is made of a central plutonium sphere with a radius of 4.62 cm surrounded by a plutonium shell with an inner radius of 10 m and a thickness of 1.54 cm. Each single component is characterized by keff = 0.95. The very high values of k; for small i reflects the high value of p for pure plutonium. The sharp decrease is due to the large escape prob­ability of neutrons created in the inner sphere. After 20 generations, the multiplication process takes place essentially in the outer shell. The simulated value of ks = 0.997 is to be compared with the analytically calculated value of ks = 0.9964. Figure 6.16 also illustrates the consideration of section 4.1.3.

Finally Daniel and Petrov [132] have proposed the use of the difference in fissile concentration in zone 1 (booster) and zone 2 to obtain a high value of ks while keeping a reasonably small value of keff. They did a one-group

diffusion calculation for a two-zone fast reactor with a subcriticality p2 = 1 — keff2 = 0.03, for the external zone and kXl = 1.2, corresponding to keff1 = 0.98. For the spherical geometry, with a volume ratio between the two zones of 103, they obtained a booster gain of 3.6, allowing a corresponding decrease in the beam power.

This is not the place to provide a thorough and general discussion of electrolysis. We focus on the electrolysis of fluorides, and more specifically on the separation of lanthanide fluorides from thorium fluoride. The simplest method for separating various elements is to apply decreasing voltages to a batch mixture of the salts. Other, more elaborate, methods make use of the time variation of the electrolysis current when varying voltages are applied. However, all methods need a batch treatment.

The electrolysis process can be compared with the reduction process by Li which was described in section 9.3.3. Consider the metallic fluoride salt Mn+Fn. The reduction reaction reads:

MFn + nLi ^ M + nLiF [Qred] (9.62)

and the electrolysis reaction

Mn+ + ne— =) M [Qel]. (9.63)

Reaction (9.62) can be written as

Mn+ + ne— + nF— — ne— + nLi =)

M + nLi+ + ne—+ nF—— ne— [Qred]. (9.64)

Using equation (9.63), equation (9.64) can be written as

nLi ^ n(Li++ e—) [Qred — Qel]. (9.65)

It is seen that the difference between the reduction by Li and electrolysis reaction energies is expressed as a function of the Li ionization energy and of the ionization state of the metal:

Qred — Qel = nQ[Li =) Li+ + e ]. (9.66)

It follows that if some elements with different oxidation states are difficult to separate with the Li reduction technique, the addition of an additional electrolysis step may be helpful. An example is the separation of Th and La. The most stable oxidation state of Th is 4+ while that of La and other lanthanides is 3+. It follows that

Qel[Th]- Qel[La] = Qel[Th] — Qd[La]- Q[u)u++ e-]. (9.67)

An experimental programme, the MEGAPIE project, began in 2000 at the Paul Scherrer Institut (PSI) in Zurich (Switzerland) concerning the construc­tion of an intensive spallation neutron source at the existing SINQ facility. It appeared immediately that such a programme would show very interesting results for the ADSR problematics.

The objectives are to study all the steps of the operation of an industrial liquid metal spallation source: design, building, operating and decommission.

The feasibility studies were performed in 2000, and the conceptual design in 2001. The system integration could be done in 2003, and the first tests and commissioning in 2004.

The MEGAPIE specifications are shown in table 13.2, and a preliminary design in figure 13.2.

Attatchment for Lifting Gear

Connectors for Supplies and Diagnostics

Shie/ding Plug

Heat removal zone

Manifold for Enclosure Coolant

Neutron production zone

Thermal Shield

Main Beam Window Inner Enclosure Window Outer Enclosure Window

The charged ions injected into the accelerators are produced with ion sources which deliver beams with a finite area, energy and angular spread. Above, we briefly discussed the consequences of longitudinal energy spread and of the need for phase stabilization. Here we discuss the consequences of angular spread and the need for spatial focusing. The quality of the beam is expressed by its emittance.

It is usual to express the wind energy annual resource in terms of kWh/m2 of swept-through surface. The best sites are close to the sea coasts. As an example, we consider the case of France. There the best sites generate up to 5000 kWh/m2 per year. A 1000 m2 windmill, with a peak power of 1 MW, would yield about 5 GWh/year. Existing large windmills reach peak powers of a few MW. The average surface requirement is of the order of 8ha/MW, i. e. a production of 60 kWh/m2/year. This figure is noticeably less than that expected from photovoltaic facilities, which reach close to 300 kWh/m2/year.

For France, the production potential is estimated to be 66 TWh/year for ground-based facilities and 97 TWh/year for offshore facilities. This figure corresponds to approximately the production of 20 nuclear reactors (57 are in use at this time) and would require 100000 high power windmills with an average density of 20 windmills per km of coast. It is clear that the eco­nomically competitive potential is much less than the technically feasible potential and would depend upon the selling price as well as the environ­mental constraints. Note that since wind energy is intermittent, it can only become competitive when the facility is connected to a network. Under such conditions competitiveness is only marginal for good sites. However, the unpredictable intermittency of wind energy will make network control rather difficult, should the share of wind energy become significant. Further­more, the possibility of windless periods requires that backup electricity production systems be available. This means that wind energy is only able to save fuel but not investment. It is more adapted to fuel-intensive electricity production means, such as gas turbines, than to capital-intensive production means like nuclear reactors. It is also compatible with hydroelectricity.

We give a schematic determination of critical sizes and masses of two model homogeneous reactors: a lead cooled fast neutron reactor and a heavy-water moderated thermal neutron reactor.

Fast reactor

In the one-group diffusion formalism, the critical size of a spherical homo­geneous reactor is given by equation (3.38)

2T2 — L

ki = 1 + — дГ (3-78)

with

^

c _ xa _ 3sasT which leads to the minimum size of the sphere,

(3.80)

The physical characteristics of the medium components are given in table 3.3. The relative atomic concentrations are also given in table 3.3. The relative concentration of 232Th and 233U are in the proportion required for regeneration:

The number of lead atoms is taken as set to that of the fuel atoms. Table 3.4 gives the macroscopic cross-sections.

With a value of v = 2.53 one gets

ki= VP = 1.17

Ea

and

and

The mass of the fuel is around 7.2 metric tons.

Deterministic codes need an a priori knowledge of the solution in order to obtain the convergence of the iterative process described previously (for
example, the calculation of mean cross-sections is a very delicate step which is generally system dependent). They have been developed in the past years, when computers were too slow. The predictions of such codes are limited but they can be useful to study perturbations to a given system. The main drawback is that the discretization of space, time and energy implies approx­imations. Moreover, they are very memory and computer time consuming if a realistic system is to be described, and, in practice, it is not possible to have a reasonably good description of a real 3D system.

Monte Carlo codes are very well adapted to the description of complex 3D systems. There are no approximations due to discretization. These codes allow very detailed representations of all physical data. With increasing computer speed, very precise results can be obtained for a system within a few hours. The same codes could be used to describe experimental set-ups and give precise predictions to which experimental results can be compared, improving confidence in the code, in the case of good agreement. Code reliability lies in the validity of cross-sections (which are directly taken from evaluations); if they are not correct, the results will also be wrong. Of course, this is also true of deterministic codes.

The production of the metastable 242mAm (excitation energy 48.6 keV, half­life 141 years) has to be taken into account, assuming that 10% of the 241Am neutron capture produces this particular state [78].

піл 0^0

As we have seen in section 3.5, U is formed by neutron capture by Th followed by two beta decays:

The presence of protactinium imposes limits on the admissible neutron flux when using solid fuels. This limitation is due to two detrimental effects of protactinium:

1. Protactinium captures neutrons, and thus decreases the reactivity of the reactor.

nio ТОО

2. After a reactor stop, the Pa inventory decays to U, which leads to an increase of the reactivity and of k. This increase may lead to reactor criticality. The characteristic time for such an evolution is of the order of the half-life of 233Pa, i. e. about one month. Corrective actions could easily be taken by inserting a negative reactivity. However, the advantage that passive safety of hybrid systems represents would be lost. It is, thus, interesting to keep the system subcritical in all instances.

The evolution equations of the Th-Pa-U system read

Thus, at equilibrium,

and

«U = «Pa A = AdTh

«Th «Th aU’)’ a{j>(A + dPaa}’)

For thermal neutrons oPa = 43 barns and for fast neutrons = 1.12 barns.

The lifetime of Pa in the neutron flux is only significantly shortened if ‘ > A/oPa, i. e. ‘> 7 x 1015n/cm2/s for thermal reactors and

‘ > 2.7 x 1017 n/cm2/s for fast reactors. Except for the very high-flux molten salt reactor which has been proposed by Bowman [2], such fluxes are never reached, so that can be neglected with respect to A. Thus, at equilibrium,

It is seen that the amount of protactinium is a measure of the neutron flux. It is also proportional to the specific power of the reactor, itself proportional to the density of fission vz, with

Since the specific power is the limiting factor of reactor designs rather than the neutron flux we express the modifications to the reactivity due to Pa in terms of the number of captures in uranium. The multiplication coefficient reads

(a)

nU0U

(a) (a) (a)

nU0U + nTh°Th + nPa0Pa + P

and

For thermal reactors the ratio 0^ /oTh = 74 while /oTh = 24 for fast

reactors. It follows that, for a given decrease of kx, fast reactors allow a specific power 3 times larger than thermal reactors and, hence, three times more compact cores.

After a reactor stop, the protactinium will decay into 233U, leading to an increase of kx. Asymptotically, the final value of kx will be

Since the perturbation on the reactivity is small, we can write that the relative change with respect to the unperturbed value of kr is

In order to estimate the true reactivity excursion, the decrease of the reactivity during irradiation has to be taken into account so that the total, maximum excursion is

Akr «c(1 + a) (a) kr (a)

P^ = l0.5"-‘ + ~T"P-

For fast reactors we get

One sees that the limit on the specific power is ten times more stringent for thermal systems than for fast ones. For Akr/kr = 2 x 10~2 the correspond­ing capture densities are on the order of 3 x 1013 for fast systems and

2.5 x 1012 for thermal ones. The corresponding fluxes are then 4 x 1015 for fast reactors and 4 x 1013 for thermal reactors.

In conclusion, it appears that the protactinium effect greatly favours fast reactors if solid fuels and the Th-U cycle are to be used. This is not true for

■л-зп ahi

the U-Pu cycle where Np, which plays a role analogous to Pa, has a much shorter half-life of 2.35 days.