We have seen above that a large increase of the share of nuclear power in meeting energy needs will most probably require the deployment of a large breeder park. How quickly such a park could be developed will depend on the amount of plutonium available from standard reactor fuel reprocessing, the initial inventory of the breeders, as well as on the doubling time of the breeder park. A relevant study has been made in [30] where the deployment of 3000 PWR reactors in 2030 and an additional 6000 breeders by 2050 was considered. Such a reactor park could stabilize the global temperature, while preserving the possibility of a strong increase of world energy consumption. The question addressed was whether such a deployment is compatible with uranium reserves and doubling times of the breeders.

Two possible breeding cycles were considered: the U-Pu cycle using fast neutron reactors and the Th-U cycle using thermal neutron reactors. In both cases the initial loads are assumed to be mixtures of the fertile element (U or Th) with plutonium taken from the spent fuels of PWR and BWR reactors. It is important to make sure that the amounts of plutonium available would be sufficient to supply all the breeding reactors by 2050.

Finally, reactors are designed so that a power increase leads to a reactivity decrease. This has the interesting consequence that the reactor is auto­stable. Placing the control rods in a predetermined position ensures that the reactor power will converge to a given value.

Slow reactivity insertion

As an example we consider the case of a PWR. The average normal tempera­ture of the coolant is around 300 °C. For fresh fuel the reactivity change
between zero and nominal power is close to —0.016 [48]. The nominal power is taken to be 3 GWth. We make the simplifying assumptions that the tem­perature is proportional to the reactor’s power and that the power rise is slow enough for the equilibrium temperature to be reached at each power level. We neglect the contribution of radioactive processes to the power. The initial power is assumed to be 1 MWth.[20] The initial value of the reactivity is taken to be p = 0.016.^ The evolution of the power [see equation (3.101)] is given by the set of equations

— = p(T (W)) W/rD

W — W(0)

T(W) = T(W(0)) + (T(WnQm) T(W(0))) Wnom —

where W(0) is the initial power, which we chose to be W(0) = 1 MW. Wnom is the nominal thermal power of the reactor, which we chose to be 3000 MW. The temperature at nominal power is T(Wnom) which we take to be 300 °C while the initial temperature is 30 °C. This set of equations reduces to

dW = p(T(W(0))) ґ Ґ W(0) W2

dt td I I (WnQm — W (0))J WnQm — W (0)J

whose solution is found to be

a

b + e—at(a — b)

with

and

The result obtained with this approximate treatment for a typical PWR reactor is shown in figure 3.8. We see in the figure that power stabilization occurs within around 50 s.

Monte Carlo results represent an average of contributions from many histories sampled during the run. A statistical error (or uncertainty) is associated with the result. This number is of course very important but it cannot be taken into account alone. It is also very important to know how this uncertainty evolves with the number of histories. Indeed, this behaviour can reflect whether the result is statistically well behaved; if this is not the case, the uncertainty will not reflect the true confidence interval of the result which thus could be completely erroneous. MCNP is certainly one of the best codes that provide very detailed and efficient methods to deter­mine the quality of the confidence interval, as well as methods to improve the precision. Here, we just want to give an insight as an introduction of all the methods that can be used.

Evaporation calculations are based on the Weisskopf [100] approach, with a few using the Hauser-Feshbach [81] one. Most actual calculations are based on the code developed by Dresner [84]. The most important ingredients of the codes are the level densities. It is important to account for the influence of shell effects on the level density parameters and of their washing out with nuclear temperature. In this respect important improvements have been made with respect to the original Dresner code. They often resort to the level density formula derived by Ignatyuk et al. [101]. Shell effects also have to be treated carefully in their influence on fission barriers. Most INC calculations use the fission model of Atchison [102]. Atchison gives a differ­ent treatment for the fission of heavy nuclei with Z > 88 and for that of nuclei with 70 < Z < 89. In the first case a fixed fission barrier, Bf = 6 MeV, is used and Tn/Tf is assumed to depend only on the charge of the fissioning nucleus, not on its energy. These approximations are based on the experimental data reviewed by Vandenbosch and Huizenga [103]. For lighter nuclei a statistical model calculation is carried out, using fissility dependent parametrizations of Bf and of af/an. Above a few MeV the results of the calculations are in reasonable agreement with experiments. However, it might be timely to improve the present treatment of fission by including such recently discovered features as the time delay to fission [104], which decreases the fission probability at high excitation energies, the temperature dependence of the fission barriers and symmetry and surface dependence of the level density parameters.

Figure 9.2 shows that the valence state of neptunium has a strong influence on the distribution coefficient. This is a general behaviour and elements with different valence states behave like different elements. In this context, it is useful to know the valence states of actinides in acid solutions.

Thorium. Thorium nitrate Th(NO3)4 is very soluble in water, thorium being in a tetravalent state. It can, rather easily, form complexes with TBP.

Uranium. In aqueous solution uranium can be found in trivalent U3+, quadrivalent U4+, pentavalent UVO+, and hexavalent UVIO2+ states. However the trivalent and pentavalent states are unstable so that, practically,
only the tetravalent and hexavalent states are important. The hexavalent state in the form of uranyl nitrate is highly soluble in TBP.

Neptunium. In acid aqueous solutions neptunium can be found in trivalent Np3+, quadrivalent Np4+, pentavalent NpVO+, and hexavalent NpVIO2+ states. Np4+ and NpVIO2+ are the more soluble states in TBP.

Plutonium. In aqueous solution plutonium can be found in trivalent Pu3+, quadrivalent Pu4+, pentavalent PuVO+, hexavalent PuVIO2+ and heptavalent PuVIIO2+ states. The heptavalent and pentavalent states are unstable. The tetravalent and hexavalent states can form complexes with TBP, while the trivalent state cannot and is not soluble in organic solvents.

Americium. In aqueous solution americium can be found in trivalent Am3+, pentavalent AmVO+ and hexavalent AmVIO2+ states. The tetravalent state is highly unstable. In the presence of oxidizable species, only Am3+ is of practical importance. It is moderately soluble in TBP. However, in the Purex process Am is not coextracted with uranium and plutonium and follows the fission products.

Curium. Curium is only stable in the trivalent state and less soluble in TBP than americium.

The renewed interest in ADSRs originates essentially in the proposals made by Bowman [2] and Rubbia [3]. The first advocated a molten salt thermal — neutron subcritical reactor, while the second advocated a solid fuel, lead cooled fast-neutron subcritical reactor. Almost all more recent work on ADSRs elaborates on these two ‘primordial’ proposals. This is why we devote this chapter essentially to them.

12.1 Solid fuel reactors

We shall concentrate on the system proposed by Rubbia. Similar proposals have been made or are being considered in the US [136] and Japan [137].

We consider a one-dimensional heat source with a heat production function w(x). The temperature distribution is given by the heat equation,

w(x)+a d = 0- (1.14)

We consider the simplified situation where the heat source is at x = 0 and w(x) = w0S(x), where S(x) is the usual Dirac distribution and a the thermal conductivity. It follows that

d2T (x) dx2

and by integration the derivative of the temperature follows as

dT (x) w0

=—— 0 H(x)+ constant (1.16)

dx a

where H(x) is the Heaviside step function. A discontinuity of dT(x)/dx is observed at x = 0.

Let us consider a confining layer thickness L = LB + LH where LB;H are the distances of the storage site to the lower (higher) borders of the layer. The temperature decreases from its maximum value Td at the storage site to T0 at

both bounding limits. It is then easy to obtain Td:

w0 Lb lh

TD — T0 + ■

a Lb + Lh

From equation (I.17) one sees that Td is maximum for the symmetric configuration Lb — Lh, which is unfavourable. In contrast, as shown by the quadratic dependence of the diffusion time [equation (I.7)] this con­figuration maximizes the diffusion times, which is optimal in that respect. If the storage temperature is one of the design constraints of the storage and should not exceed Tdesign, equation (1.17) shows that the average heat production density should be proportional to Tdesign — T0, and, con­sequently, that the surface of the storage (and its cost) should be inversely proportional to Tdesign — T0. It might be necessary to find a trade-off between the diffusion time and the temperature of the storage.

It is outside the scope of this book to give a detailed account of neutron propagation theory. The interested reader should refer to standard books on nuclear reactor theory like those of Weinberg and Wigner [54], Bussac and Reuss [48], Lamarsh [55], etc. However, we think it is useful to review the basics of neutron propagation, so that schematic calculations of hybrid reactors can be understood and carried out. The most general theoretical approach is the Boltzmann equation.

Nuclear energy production is accompanied by the production of various radioactive wastes:

• fission products

• activation products due to neutron captures by nuclei belonging to the structure of the reactor, such as, for example, cobalt 60

• transuranic nuclei due to neutron captures by the nuclear fuel.

Nuclear wastes are characterized by their radiotoxicity and their half­life. Only wastes with half-lives exceeding ten years are associated with significant storage problems. These are essentially some fission products (LLFP) and transuranic elements. Their noxiousness is, traditionally, measured by their ingestion radiotoxicity.

The previous examples show integrated flux over a cell, independent of energy and time. It is, however, useful to know the energy and/or time dependence of such quantities. Below are three examples related to the three previous tallies:

E4 0.01 0.1 1. 10.

T14 0.1 0.2 0.3 0.4 E24 0.1 1. 10.

T24 10. 99I 1000.

The first line defines, for tally 4, an energy binning where the upper bounds of each bin are 0.01, 0.1, 1 and 10MeV; tally 14 give the flux in cell 1 and in cell 2 as a function of time (from 0.1 to 0.4 shakes). Tally 24 will give the mean flux of cells 1 and 2 as a function of energy (three bins) and of time (101 linear bins from 10 to 1000 shakes).

• The first bin is always from 0 to the first specified upper bound.

• MCNP gives also the sum of all bins (1D binning), or for 2D binning the sum of all energy bins as a function of time and the sum of all time bins as a function of energy.[32]