Basdevant J.-L., Rich J., Spiro M. Fundamentals in Nuclear Physics: From Nuclear Structure to Cosmology

Подождите немного. Документ загружается.

6.8 Nuclear reactors 313

5.8 3.1 4.7 3.0

12.7 5.3 5.3 3.6 2.9 2.5 2.0 2.1

6.6 9.4 8.7 4.3 4.6 6.0 5.3

3.8 4.6 5.3 10.1 4.8 4.6 3.9 3.4 3.1

4.0 6.4 6.5 7.1 9.0 8.8 3.6 3.0 5.2

2.5 2.6 4.2 4.2 5.6 5.2 8.5 7.4 4.1 3.5

2.4 2.3 2.8 3.1 3.4 5.1 4.0 7.5 8.6 7.9 6.2 6.7 3.7 2.9

1.6 1.3 1.6 1.8 2.7 2.4 3.2 3.8 4.5 5.1 7.6 6.2 7.4 6.5 3.2

0.6 0.0 1.5 1.6 2.2 3.3 4.3 5.0 5.4 4.1 4.9 4.0 6.9

258

235

238

n−capture

−3.4

16.3 14.9 17.1

13.8

11.9 11.3 13.1

11.4

15.4

11.4 11.2 14.7 13.0 11.5

10.6

10.0 10.5

−0.7 −2.5

Fig. 6.12. The production of trans-uranium elements by neutron capture on

235

and

238

U in a nuclear reactor. The production proceeds by a series of neutron

captures and β

−

-decays. The ﬁgures in each box give the logarithm of the half-

life in seconds (7.5 ⇒ t

1/2

= 1 yr) and the shape of the box gives the dominant

decay mode. Short-lived nuclei (log t

1/2

< 6) generally decay while long-lived nuclei

capture neutrons. The sequence ends at

258

Fm that decays by spontaneous ﬁssion

with t

1/2

= 370 µ s. Elements beyond

258

Fm cannot be fabricated via neutron

capture.

238

U have been burnt by neutron absorption leading to the production of

239

Pu. Altogether, 826 kg of uranium have been consumed.

Thisbalanceshowsthattheeﬃciencyispoor.Oneconsumes92tonsof

natural uranium in order to recover the ﬁssion energy of 826 kg of nuclear

matter, i.e. 0.9%. This observation explains the interest for breeders, which

can increase the energetic potential by a factor of 100.

Heavy water thermal-neutron reactors. Despite its high price, heavy

water has been chosen as a moderator in the Canadian system CANDU. In

such reactors, one can use natural uranium as a fuel (Table 6.2), which is an

obvious advantage and compensates for the price of the moderator, since no

enrichment is necessary.

In the CANDU system, the pellets are placed in pressurized tubes where

heavy water circulates at a temperature of 200 C (pressure of 90 atm). Heavy

water is both the moderator and the coolant.

Graphite-gas thermal-neutron reactors. In these systems, the fuel is

natural metallic uranium. The moderator is graphite, and the coolant is car-

bon dioxide CO

Among the drawbacks, there are :

314 6. Fission

• The use of metallic uranium whose low melting point limits the core tem-

perature and therefore the thermal eﬃciency.

• The smaller moderation power which leads to larger sizes than water reac-

tors for the same power.

The RMBK reactors in the former Soviet Union are slightly diﬀerent.

They use weakly enriched uranium (1.8%). Graphite is the moderator and

boiling water is the coolant. The major drawback is that k can increase

when the temperature increases (see below), and therefore increases when

the density decreases due, for example, to loss of the coolant. Therefore, such

reactors are not self-regulated, as illustrated in the Chernobyl accident.

Void coeﬃcient. A very important characteristic of the various reactors is

the variation of k with the density of the coolant. This is characterized by the

void coeﬃcient. This determines the behavior of the reactor if, for instance,

there is a leak in the coolant system, and the temperature increases. This

results from the combination of two eﬀects. The ﬁrst is that if the amount of

moderator diminishes, the neutrons have higher energies and deposit more en-

ergy. However, if the neutrons are more energetic, the cross-sections decreases

as can be seen on Fig. 6.7, and this reduces the neutron ﬂux. Altogether, in

PWR reactors, the void coeﬃcient is negative, and therefore such reactors

are self regulated against ﬂuctuations or leak of the coolant.

In RMBK reactors, the net eﬀect of these two opposing characteristics

varies with the power level. At the high power level of normal operation, the

temperature eﬀect predominates, so the global void coeﬃcient is negative.

However, at a lower power output of less than 20% the maximum, the positive

void coeﬃcient eﬀect is dominant and the reactor becomes unstable and prone

to sudden power surges. This was a major factor in the development of the

Chernobyl accident (which was above all the product of a lack of “safety

culture”).

Reactor control . The number density of neutrons obeys the equation

= λ

ﬁs

(k − 1)n (6.65)

where k is the number of neutrons produced per ﬁssion, taking into account

absorption and geometrical losses. The ﬁssion rate per neutron is given by

ﬁs

= n

fuel

σ

ﬁs

v (6.66)

where n

fuel

is the number density of ﬁssionable nuclei and σv is the mean

ﬁssion cross-section times velocity. The solution is

n = n

exp

(k − 1)t

(6.67)

where τ =1/λ

ﬁs

is of the order of 0.1 ms in a thermal reactor. If k>1, we

observe a rapid exponential increase of the ﬂux and the reactor will become

6.8 Nuclear reactors 315

impossible to control. For instance, if k=1.1, in one tenth of a second the

power of the reactor is multiplied by 20000.

This would have been a terribly diﬃcult problem to solve technically,

if it had not been for the existence of a small number of delayed neutrons

produced in ﬁssion reactions. Most neutrons are emitted immediately in the

ﬁssion reaction. However, a small number of neutrons (7.5 per thousand for

235

U) are emitted by highly-excited daughters of β-unstable ﬁssion fragments

(Sect. 2.7). An example of a delayed neutron, due to the decay of

Br, is

shown on Fig. 6.13.

E (MeV)

Kr + n

55.6 s

76 m

stable

2.3%

48 Gyr

Fig. 6.13. Decay diagram of

Br. 2.3% of the β-decays are to excited states of

Kr that have excitation energies greater than the neutron separation energy of

Kr, S

=5.518 MeV. These states decay by neutron emission.

Before the reactor is ignited, control rods are in the core of the reactor.

The rods are made of steel impregnated with strong neutron absorbers such

as boron or cadmium. If the control rods are deep inside the core, the chain

reaction cannot occur. The rods are then slowly withdrawn to allow for a

steady regime. In case the neutron ﬂux increases too much the rods are low-

ered in the core. These operations take several seconds to complete. In order

for this process to eﬀectively prevent an supercritical reactor from diverging,

we must have

1 − β

, (6.68)

316 6. Fission

where β is the fraction of ﬁssion neutrons that are delayed neutrons. If this

condition is not satisﬁed, the divergence would be due to prompt neutrons

and it would not be possible to control it by mechanically lowering the control

rods.

Another characteristic that is important for reactor control is that k

should decrease if the temperature increases by error or ﬂuctuation or if

the cooling ﬂuid is lost. This is the case for PWRs:

• Temperature increase. In PWRs, k decreases with increasing temperature

because of the resonances in the neutron capture of

238

U (Fig. 6.7). The

increased thermal agitation eﬀectively widens the resonances, resulting in

increased neutron absorption before thermalization.

• An unexpected emptying of the coolant in the core. Here, k decreases be-

cause the loss of ﬂuid increases the neutron mean-free-path thus increasing

geometric losses of neutrons. This is not necessarily the case in reactors

where the cooling and moderating functions are separated, as in graphite-

moderated reactors.

Present projects to improve reactor performance are concentrating on

increasing security. For example the European Pressurized Reactor (EPR)

project, a French–German collaboration should design with more secure re-

actors than the present PWRs.

There are several ingredients. One is to use more sophisticated control

systems. In particular enriched boron carbide (B4C) control rods, sealed in-

side hafnium tubes will replace soluble boron. Another improvement will be

in the mechanical quality of the fuel tubes. This can lead to a 50% increase

in the power delivered.

Finally, several research programs are made concerning severe accidents,

in particular the risk that the “corium,” i.e. the melted core which is a magma

at 3000 K, can traverse the steel vessel.

6.8.2 Fast neutron reactors

Fast neutron reactors can be breeders that produce more nuclear fuel than

they consume, by using an intermediate fertile nucleus such as

238

Uor

232

Th.

This is possible because more than two neutrons per ﬁssion are produced in

fast-neutron

239

Pu reactors (Table 6.2). One of these neutrons can be used

to maintain the chain reaction and the others can create further

239

Pu via

neutron absorption on

238

U. If the probability for this to happen is suﬃciently

close to unity, the

239

Pu destroyed by ﬁssion can be replaced by a

239

created by radiative capture on

238

U. The ﬁnal result is that

238

Uisthe

eﬀective fuel of the reactor.

The following remarks are in order. Consider, for deﬁniteness, a breeder

with the fertile nucleus

238

U. The ﬁssile nucleus is

239

Pu for two reasons.

Firstly, it is produced in neutron absorption by

238

U, which leads to a closed

6.8 Nuclear reactors 317

cycle, secondly it is produced abundantly in nuclear technologies, whereas one

can only rely on the natural resources of

235

U. In order for a fertile capture

of a neutron to produce an appreciable amount of the ﬁssile

239

Pu inside

the fuel, the probability for this capture must not be too small compared to

the probability that the various nuclei in the medium undergo ﬁssion. This

probability depends both on the amounts of

239

Pu,

235

Uand

238

U, and on

the physical design of the fuel elements. It can be calculated in terms of the

amount of various nuclides and of the capture and ﬁssion cross-sections of,

respectively,

238

U and the pair

239

Pu −

235

U. We can read oﬀ from table

(6.2) that for thermal neutrons one has σ

(238)/σ

(239) ∼ 3.610

−3

whereas

for fast neutrons, on the contrary, the same ration is of order 1, i.e. 300 times

larger. The same feature appears for the

232

Th −

233

U pair. This is why fast

neutron reactors are used in breeders.

Furthermore, this also explains why in

238

U-

239

Pu breeders, the design

of fuel elements consists in a central core of

239

Pu surrounded by a mantle of

238

U depleted in

235

U in order to lower the amount of ﬁssion in the external

fertile region.

Consequently fast neutron reactors are used as breeders, and two fertile-

ﬁssile pairs are a priori possible,

238

U −

239

Pu and

232

Th −

233

U. Present

nuclear industry is oriented toward the ﬁrst, because the thermal reactors

produce plutonium which is separated in the fuel processing operation.

The

232

Th −

233

U couple is under study at present. It has many advan-

tages, among which that it does not lead to appreciable amounts of danger-

ous trans-uranium elements such as americium and curium. These “minor

actinides” are dangerous, because they are produced in appreciable amounts

and they have half-lives, and therefore activities, lying in the dangerous re-

gion, neither small enough to decay suﬃciently rapidly on a human scale nor

long enough to be ignored, such as natural uranium and thorium. The half

lives of some of these isotopes are 432 years for

241

Am, 7400 years for

243

and 8500 years for

247

Cm.

The consequences of using fast neutrons. The use of fast neutrons has

several important consequences:

• One must avoid the slowing down of neutrons through the presence of light

nuclei. In particular, water cannot be used a coolant.

• Since ﬁssion cross-sections are much smaller than with thermal neutrons,

so it is more diﬃcult to reach the critical regime.

• Great care must be taken concerning the mechanical damage caused by

the fast neutrons to the structures and construction materials.

In order to obtain the divergence of the reactor, one must use a fuel

containing a high proportion of

239

Pu (of the order of 15 %). The mass of

ﬁssile material inside a breeder is therefore larger than for a thermal neutron

reactor. In the core of the Superphenix breeder in France, there was the

equivalent of 4.8 tons of

239

Pu for an electric power of 1200 MW, whereas

318 6. Fission

the core of a usual thermal reactor contains roughly 3.2 tons of

235

U for an

electric power of 1300 MW. Therefore, the initial investment in ﬁssile material

is larger for a fast neutron reactor. This is an economic drawback, but it has

the advantage of having a more compact core and a larger neutron ﬂux.

Materials. In the core of a fast-neutron reactor one must minimize the neu-

tron leakage in order to ensure breeding. Therefore the core has a particular

composition. There are three concentric shells:

• An internal shell composed of ﬁssile fuel. In the case of Superphenix, this

was a mixture of 15 % of plutonium oxide and of 85 % of uranium oxide.

• An intermediate shell of fertile material called the “mantle ” in which the

238

U is converted into

239

Pu.

• An external shell contains steel elements which protect the vessel of the

reactor and backscatter neutrons into the core.

A breeder produces more plutonium than it burns. It can therefore self-

feed itself in ﬁssile plutonium, provided that it is associated with a processing

plant. This plant extracts from the irradiated fuel the useful plutonium nec-

essary for further use.

Under these conditions, everything occurs as if the reactor consumes only

238

U. As an example, the yearly consumption of uranium by the reactor

Superphenix (1200 MW electric power) was one ton of

238

U (an initial 4.8 ton

of plutonium was provided). This is to be compared with the consumption of

a PWR of 900 MW electric power which uses 92 ton of natural uranium per

year.

This explains the interest of breeders in view of the preservation of ura-

nium resources.

The use of sodium. Fast neutron reactors must use a coolant ﬂuid with

only heavy nuclei so as to avoid neutron-energy losses. For this reason, water

cannot be used, and the general choice is to use liquid sodium.

Melted sodium has very good thermal exchange properties. Furthermore,

it melts at 98 C and boils at 882 C. It is used at a maximal temperature of

550 C, and it does not necessitate any pressurizing, which is a favorable fea-

ture for the mechanical conception and safety of the installations. Another

feature is that it has hydraulic properties similar to water at room tempera-

ture, which is a useful coincidence for testing materials.

Another favorable feature is that sodium can be used at higher tempera-

tures than water, which improves the thermal/electric conversion factor.

On the other hand, sodium has the big drawback that it burns sponta-

neously in the air and in water. This is a major disadvantage for security

considerations. Furthermore, sodium becomes radioactive by neutron activa-

tion. Therefore, two circuits are necessary. In order to prevent the primary

circuit from leaking into the secondary circuit, the latter is maintained at a

higher pressure than the former.

6.8 Nuclear reactors 319

6.8.3 Accelerator-coupled sub-critical reactors

We have made much of the fact that for a chain reaction to be self-sustaining,

the number of neutrons per ﬁssion available to produce further ﬁssions must

be greater than unity, k>1. Sub-critical system (k<1) still generate energy

(200 MeV per ﬁssion) but the nuclear “ﬁre” must be continuously “re-lit”

by the injection of neutrons from an external source. Such systems have

the advantage that they cannot burn out of control and can be stopped by

removing the neutron source.

The most convenient neutron source is a particle accelerator that directs a

beam of charged particles on a target. This produces neutrons through (spal-

lation) reactions that breakup nuclei in the target. The target is surrounded

by a sub-critical mantle containing ﬁssile material. A schematic example is

shown in Fig. 6.14. Such accelerator-coupled detectors are often called hybrid

reactors.

The spallation target. The interaction of protons with energies E

100 MeV in a target suﬃciently thick to stop the beam, gives rise to a copious

emission of neutrons. They are produced both in the primary proton-nucleus

reaction and by further interactions of secondary particles (p,n,π...) inside the

thick target. This results in a cascade contained in the target cylinder, whose

length is roughly the stopping distance of the incident beam. The diameter

of the beam is optimized so that a maximum number of neutrons leave the

target and interact with the outside material.

The mean number ν

of spallation neutrons emitted per incident proton

is observed to be

∼ 30 × E

(6.69)

where E

is the incident beam energy in GeV. The high value of this number

determines, as we shall see, the energetic feasibility of the system. The energy

spectrum of the neutrons emitted goes from a few keV to the beam energy

according to their production mechanism (spallation, ﬁssion, evaporation).

The low energy part (1–2 MeV) is favored by a thick target, compared to the

high energy part which corresponds to direct spallation.

The sub-critical system. The thick target is surrounded by a sub-critical

system characterized by

ν and k, the numbers of neutrons per ﬁssion before

and after correction for absorption and geometrical losses.

The ν

neutrons injected into the sub-critical system by the proton are

therefore successively multiplied by k. The total number of neutrons per

incident proton is equal to:

= ν

(1 + k + k

+ ...)=

(1 − k)

(6.70)

Among these N

neutrons, N

− ν

are produced by ﬁssion. In this respect,

we notice that the higher the value of k<1, the larger the proportion of

ﬁssion neutrons in the sub-critical medium. Therefore, the neutron energy

320 6. Fission

proton beam tub

fuel

heat exchanger

cold lead falling

hotlead rising

Fig. 6.14. A possible design of a hybrid reactor [66]. The proton beam is directed

downward entering a region containing molten lead and surrounded by thorium fuel.

The neutrons created in the lead induce ﬁssion in the thorium. The heat generated

causes the lead to rise toward heat exchangers.

spectrum will be determined more by the sub-critical medium than by the

origin of the primary neutrons.

Since each ﬁssion produces

ν neutrons, the total number of ﬁssions per

incident proton is equal to :

ﬁs

(N − ν

)

1 − k

. (6.71)

6.8 Nuclear reactors 321

The gain of the system, i.e. the energy released in ﬁssion divided by the

incident proton energy is

G =

ﬁs

1 − k

, (6.72)

where E

ﬁs

∼ 200 MeV is the energy release per ﬁssion. The thermal power is:

P (MW) = E

ﬁs

(MeV) I(A)

1 − k

(6.73)

where I(A) is the beam current. We notice that the beam intensity necessary

to reach a given power of the reactor decreases as criticality (k =1)is

approached.

This system can achieve an energetic self-suﬃciency condition provided

the power delivered by the reactor is greater than the power which is necessary

in order to make the accelerator run (neglecting all other energetic needs).

In fact, given the powers under consideration, nearly all the power given

to the accelerator is absorbed by the beam. The self-suﬃciency condition is

expressed by:

G



> 1 (6.74)

where 

and 

represent respectively the thermal conversion eﬃciency of the

reactor, and the electric eﬃciency of the accelerator. This condition, which

is called the “break-even ” provides a deﬁnition of a minimum value k

min

for

k, below which there is no longer energetic self-suﬃciency :

min

1+



ﬁs

)(ν

/ν)

. (6.75)

The value of k

min

lies between 0.61 and 0.72, depending on the values of 

and 

.Thefactor

depends mainly on the temperature of the reactor; it

lies between 0.3 and 0.4. The other factor is more diﬃcult to estimate.

The high intensity accelerator. The intensity of a high energy beam

for a system whose power is equivalent to a standard reactor of 3000 MW

(thermal) lies between a few mA to 360 mA. State-of-the-art accelerators

have somewhat less power. The most powerful are either linear accelerators

(linacs) working in a continuous mode, (e.g. the Los Alamos 800 MeV, 1 mA

linac) or cyclotrons, (e.g., the PSI 600 MeV, 0.8 mA cyclotron).

The technological feasibility of reaching the necessary power requires

progress in several areas. Beam losses must be limited to very small val-

ues (of the order of 10

−8

−1

for a linac) in order to avoid activating the

structures of the accelerator up to a level where manual interventions be-

come impossible. The eﬃciency of the RF system (radio frequency) must be

increased in order to obtain the best possible eﬃciency 

. This eﬃciency 

depends on the losses by Joule eﬀect in the surface of the RF cavities. The

power loss in the cavities is of the order of several times that of the beam for

a hot linac and it becomes negligible for a superconducting linac.

322 6. Fission

The proposal shown in Fig. 6.14 is based on the thorium cycle with fast

neutrons. The fertile material is

232

Th. The ﬁssile material is

233

U. The pro-

posal is called an energy ampliﬁer because the gain which is aimed at is

very large (k =0.98). Molten lead is used both as spallation source and as

coolant, using its natural convection. The advantages of such a system are

the following :

• The thorium cycle produces little plutonium and heavier elements. It is

therefore “cleaner ” than the Uranium cycle.

• Thorium is twice as abundant as uranium and does not require isotopic

separation.

• If one masters the technology of molten lead, one avoids the drawbacks of

liquid sodium.

• The accelerator introduces a better control of the system thanks to the

quickness of reactions (of the order of a microsecond) compared to the

mechanical handling of control bars (of the order of one second).

6.8.4 Treatment and re-treatment of nuclear fuel

Nuclear fuel requires considerable treatment before burning and careful

“management” after burning because of its high radioactivity. The global

processing of the nuclear fuel starts with the extraction of the ore. This is

followed by the concentration, conversion and enrichment of the uranium, the

manufacturing of the fuel elements, and, ﬁnally the processing of the used

fuel, and the disposal and storage of the waste.

For thermal-neutron reactors, the production of 75 ton of fuel requires

156 ton of natural uranium. The unburned natural uranium contains 1.1 ton

235

U out of which only 775 kg is retained in the enriching procedure.

The most critical step in the preparation of uranium-based fuels is isotopic

separation. The possibilities for this are

• Diﬀusive enrichment using the higher drift speed of gaseous

235

com-

pared to

238

when traversing a capillary system. This is the most com-

monly used system.

• Centrifugal enrichment using higher centrifugal force on

238

compared

to that on

235

in high-speed rotors.

• Mass-spectrometer enrichment using the diﬀering values of Q/M of the

two isotopes. This method is only possible for small quantities.

• Laser ionization methods, using the slight isotopic shifts in atomic spectra

of the two isotopes. Isotopes are selectively ionized by photon absorption

and then collected in an electric ﬁeld.

The nuclear fuel stays between three and four years in the core of the

reactor. Gradually, it loses its ﬁssile matter and is enriched in plutonium

and other trans-uranium elements, and ﬁssion fragments many of which are

highly radioactive.