Propagation of damage from an event in a single subassembly is only one way in which an accident affecting the whole core might be ini­tiated. There are other ways involving malfunction of the control or cooling systems, the frequency of occurrence of which can be estim­ated using event trees in a similar way to that described in section 5.4.2.

In all cases the frequency is very low because they can take place only if the protective system, which is designed to be very reliable, fails to operate.

To determine the risks posed by this range of hypothetical acci­dents it is usual to analyse the consequences of a small number of well — defined initiating events, chosen so that they cover the range. Damage would be caused by overheating, overheating would be caused either by excessive reactor power or ineffective cooling, and the extent of damage would depend on the rate at which these happened. By con­vention a set of standard accident scenarios are analysed. These are described in Table 5.2 and illustrated in Figure 5.11.

(In the event that the single subassembly accident described in section 5.4.2 propagates to the whole core its consequences would be included in the range covered by these standard scenarios. For example if it were to give rise to a vapour explosion the resulting pressure wave might cause widespread cladding failures in other subassemblies, which in turn might cause extensive blockage of the coolant flow. This would not be worse than the Fast LOF standard scenario. Thus the risk posed by single subassembly accidents of this type can be

included in the overall assessment of risk by including the frequency of its propagation (10-8 per year in Figure 5.10) in the frequency of initiation of Fast TOPs.)

In reality any of these accidents would be terminated safely by the reactor protective system. A trip would be activated by high neutron flux, high core outlet temperature or low coolant flow rate. The trip would act to insert the absorber rods (both control rods and shut­off rods). Redundancy is ensured because only a fraction of the total number of rods would have to be inserted to shut the reactor down and reduce the power to zero. Diversity can be provided by making the absorber rods of different designs, to guard against common-mode failure.

Each of these accident scenarios can in principle be analysed in terms of an event tree, in which the low probability of failure of the various components of the protective system is taken into account at the various branch points. In general the most serious consequences, posing the most severe risks, arise if the protective trip system fails entirely, allowing the reactivity to increase towards prompt critical. This might be the direct result of a Fast TOP, or in the other scenarios it might be caused by ejection of boiling sodium from the centre of the core or compaction of fuel after widespread cladding failure or fuel melting. Accidents involving an increase of reactivity towards prompt critical are usually known as “core-disruptive accidents” or CDAs.

5.4.2 Core-Disruptive Accidents — the Initiation Phase

The increase in reactivity to or close to prompt critical is sometimes, confusingly, called the “initiation phase” of a CDA, in spite of the fact that the accident sequence must have started earlier with an initiating event and the failure of many components of the protective system.

This initiation phase has received close attention since the earliest days of fast reactor development because it addresses the most obvious risk — that because there is excess reactivity available in a fast reactor core it might become prompt critical and suffer a violent power excur­sion liberating large quantities of energy in a very short time. The fear has been that a fast reactor could “explode”, and the object has been to prove that this is impossible and that any actual energy release could be safely contained.

The response time in the prompt-critical regime is related to the prompt neutron lifetime, which in a fast reactor core is of the order of 0.5 p, s. Since this is short compared with the time for a sound wave to cross the core there is a possibility that the pressure might rise high enough to generate a shock wave and cause mechanical damage to the containment. This would be analogous to a chemical explosion.

(The mechanism involved differs from that of a nuclear weapon in that the prompt neutron lifetime in the latter is shorter by an order of magnitude or more, so that much larger amounts of energy can be liberated before the core is dispersed. For this reason it is incorrect to characterise fast reactors as “potential bombs”. Even with very pessimistic assumptions calculated energy yields for large fast reactor cores do not exceed a few GJ at most, many orders of magnitude less than the yields of the most modest nuclear weapons, which are in the range of 10 TJ or more.)

The first attempts to predict what would happen if a fast reactor were subject to a severe reactivity accident used a drastically simplified model in which feedback reactivity was assumed to be due to thermal expansion of the fuel and therefore to depend linearly on the average
fuel temperature (Fuchs, 1946). This approximation was appropriate to very small metal-fuelled cores. It was later realised that in larger cores the fuel could melt and possibly vaporise, and that dispersion of the fuel would be a more important source of feedback (Bethe and Tait, 1956). These early calculations indicated that, for a small core, the energy released by such reactivity transients as could be envisaged was small enough to be contained, but when the same methods were used to assess the effects in larger cores the energy release appeared to be unacceptably large. However as the size of the core increases the neut­ron energy spectrum becomes softer and the Doppler effect becomes a more important source of feedback, reducing the energy release considerably (Hicks and Menzies 1965). For large power reactors the Doppler coefficient is the parameter that has the greatest effect on the response of the core in the initiation phase of a core-disruptive accident.

Applied to a modern reactor conclusions drawn from the results of these early calculations are not accurate and would not be convincing in a safety case. Nevertheless, because the calculation model is so simple it exposes the essence of the initiation phase and shows clearly if not accurately what would happen. The following illustrations make use of the simple approach introduced by Fuchs.

The main simplifying approximations are to assume that everything happens on the timescale of the prompt neutron lifetime so that it is possible to ignore the delayed neutrons; to represent the state of the reactor core by a single parameter, the average fuel temperature; and to assume that the reactivity feedback depends in a simple way on this temperature. With these assumptions the point-kinetics dynamic equation becomes

dP (p — в)Р

dt Л

and the response of the core is given by an energy equation in the form

Here t is time, P is the reactor power and P0 is the initial power when t = 0. p is the reactivity, в is the delayed neutron fraction and Л is the prompt neutron lifetime. T is the average fuel temperature and C is the reciprocal of the heat capacity of the fuel. (It is assumed that in the short time of the transient all the heat is retained in the fuel.)

The reactivity has two components, the input disturbance pi that causes the transient, and p/,the feedback from the fuel temperature. Thus

p = Pi + p /. (5.3)

Figure 5.12 shows, according to this idealised model, the response of a large oxide-fuelled power reactor core when reactivity is added at a steady rate of $50 per second, assuming (for the purpose of illus­tration) that it remains intact and that the only source of negative reactivity feedback is the Doppler effect. Specifically in equation 5.3 we have pi = rt, where t is time and r = 0.15 s-1 is the ramp-rate; and Pf = — Dln(T/T0), where D = 0.008 is the Doppler coefficient and T0 is the initial fuel temperature.

For the first few milliseconds after prompt critical little happens but after 5 ms when the ramp reactivity has reached 0.075% above prompt critical the power doubling time falls below 0.5 ms. The power rises very steeply to nearly 2 TW and with it the fuel temperature. This causes strong negative feedback from the Doppler effect which reduces the power as quickly as it has risen and shuts the reactor down after a power spike lasting for about 2 ms, which deposits just over 4 GJ of energy into the fuel. In this unrealistic model the input reactivity ramp continues and after a further 15 ms there is a second power spike, higher this time because the fuel temperature is higher so the Doppler feedback is smaller.

The Doppler effect is not the only source of negative feedback. In addition reactivity would be reduced by dispersion of the fuel. This would be driven by expansion as the fuel temperature rises, especially when it reaches the melting point. After melting the dispersion would be accelerated by the increasing vapour pressure of the molten fuel. Possibly more important would be release of fission-product gas from the fuel crystals (see section 2.3.5). Gas retained in the fuel matrix is in effect stored at very high pressure, and as the crystal structure breaks down this pressure would act on the fuel to disperse it.

Figure 5.13 shows the effect of dispersion of the fuel. The power rises in a sharp spike as before, but in this case subsequent spikes are suppressed by the strong negative feedback as the fuel is displaced outwards.

The energy release of 4.4 GJ shown in Figure 5.13 is typical for a large fast reactor with a Doppler coefficient in the range 0.005-0.010. The implication of calculations of this type is that it is possible to design the reactor containment so that, however severe the pressure transient during the nuclear excursion might be, it can be contained

without allowing core material to be dispersed to the environment. It is nevertheless necessary to continue the analysis of the accident to ascertain what happens next, because at the conclusion of the initiation phase the core is left, probably largely molten, containing a large amount of energy and in an unstable state.

1.5.1 Materials

The reactor is controlled and shut down by moving control rods incor­porating neutron-absorbing material into or out of the core. Other methods of control such as moving fuel or parts of a neutron reflector around the core were used in early experimental reactors but are not feasible in a large power reactor.

The absorbing material is usually boron, possibly enriched in 10B, in the form of the carbide B4C. Alternatives are metallic tantalum or oxides of europium or gadolinium. 10B has a very high capture cross-section that varies with energy E as 1/VE up to 0.1 MeV and thus captures neutrons predominantly at the low-energy end of the spectrum.

A disadvantage of boron is that it captures neutrons by an (n, a) reaction so that while it is in an operating reactor helium atoms accu­mulate within the crystals of the boron carbide. These tend to form little bubbles of gas that disrupt the structure of the crystals and dam­age the material. This, together with loss of the l0B, limits the life of a rod used for controlling the reactor while it is operating. Shut-off rods, used only to shut the reactor down and hold it subcritical during refuelling, are not subject to this limitation.

Uranium metal adopts different crystalline forms at different tem­peratures. The stable phases are an а-phase consisting of anisotropic orthorhombic crystals below 668 °C, a tetragonal p — phase, from 668 °C to 776 °C, and a у-phase consisting of isotropic cubic crystals from 776 °C up to the melting point. Plutonium is soluble in concentrations up to 16% in the а-phase and completely soluble in the y-phase.

The а-phase is dimensionally unstable under irradiation. The high — energy fission fragments from each fission event displace some of the metal atoms from their positions in the crystal lattice as they slow down, in effect melting a small volume of the crystal. As this cools and resolidifies the shape of the crystal changes and it grows aniso- tropically. Individual crystals growing in different directions tend to

push each other apart by opening large irregular empty voids on the boundaries between them. This process is called “tearing” or “cavit­ation”, and causes the fuel to swell. Uranium fuel at temperatures in the range 400 to 500 °C swells particularly severely, but the presence of plutonium reduces the effect.

Cavitation swelling can be avoided by alloying with a metal such as molybdenum or zirconium that stabilises the y-phase crystal structure. Figure 2.18 is a part of the uranium-zirconium phase diagram showing the effect of the alloying metal in increasing the temperature range in which the у-phase is stable. Typical fuel alloys are ternary alloys of U and Pu with 10% Mo or Zr.

Unfortunately the elimination of cavitation only allows another swelling mechanism to operate. Fission-product gas tends to diffuse out of the individual metal crystals and accumulate in voids on the grain boundaries. The voids are small with dimensions of the order of 10 gm and the gas in them, being confined by a form of surface tension, is at high pressure and therefore high density, but even so the swelling is substantial, amounting to an increase in volume of around 50% at 2% burnup of U-10%Zr, and slightly less for U-Pu-Zr fuel. This implies that the pressure in the gas-filled voids, which is determined by the shear strength of the metal, is around 5 MPa. In a cylindrical element the fuel swells much more rapidly radially than axially. This anisotropy is thought to be an effect of the radial variation of temperature. The voids in metal fuel do not migrate in the way that pores in oxide fuel do (see section 2.4.1) because the temperatures are much lower.

As long as the gas is retained in the inter-grain voids the swelling is unavoidable and eventually causes the cladding to fail. As a result, in the early stages of development, fuel elements with an initial smear density (see section 2.3.1) of 85% had to be limited to a maximum burnup of around 3%. It was this limitation that in the 1960s caused most of the worldwide development effort to switch from metal to oxide fuel.

The development in the United States that made it possible to overcome this barrier was to reduce the smear density substantially, to around 75%. This allows a volume increase in excess of 30% before the fuel makes contact with the cladding. At this point the voids on the grain boundaries have grown large enough to join together and release the gas from the fuel matrix into a plenum. Figure 2.19 shows the form of these voids and Figure 2.20 shows the dependence of gas release on the smear density.

Once gas has escaped from the fuel matrix swelling is dramatically reduced but not stopped. The solid fission-product atoms together with some 20% of the gaseous atoms are retained within the metal crystals, resulting in inexorable solid-phase swelling at a rate about half the burnup rate (0.5% increase in volume for each 1% increase in burnup).

2.5.2 Mechanical Behaviour during Irradiation

A typical metal fuel element consists of a steel tube 6.0 mm OD, 5.4 mm ID, containing cast U-Pu-Zr pellets initially 4.7 mm in diameter, to give a smear density of 76%. The initial gap between fuel and cladding

is filled with the sodium in which the whole stack of fuel pellets is immersed. Above and below the core fuel pellets there may be U-Zr breeder pellets, and above the upper breeder there is a gas plenum initially filled with helium and with a volume approximately equal to that of the core fuel. (The option of placing the plenum below the core, which is available for oxide fuel, is precluded by the presence of the sodium.)

If the element is irradiated at a peak linear power of ~50 kWm-1 to start with the fuel swells rapidly and makes contact with the cladding when the burnup reaches about 2%. The smaller swelling in the axial direction tends to reduce the reactivity. At about the same time as the fuel contacts the cladding its volume has increased by about 30% and its porosity has reached the point at which the majority of the fission-product gas is released to the plenum.

Solid fission products and the small fraction of the fission-product gas that is retained within the crystals continue to accumulate so that the metal continues to expand. The porous fuel is however now weak compared with the cladding with which it is in contact, so further over­all swelling, both radial and axial, is restrained. The cladding itself may be swelling (see section 3.3.2) allowing the overall fuel volume to increase a little, but otherwise the solid-state swelling is taken up by reduction in the porosity. In spite of this irradiation experience indic­ates that, at up to 20% burnup or so, the pores remain interconnected and the fission-product gas continues to be released to the plenum. The cladding increases in diameter due to its own swelling and to creep strain caused by the plenum gas pressure. There is no evidence that the fuel itself imposes any significant stress on the cladding.

Radial cracks appear in the early stages of irradiation but are com­pletely healed at 10% burnup. There is evidence that the sodium even­tually permeates the porous structure, increasing the effective thermal conductivity and reducing the central temperature. At high burnup some cladding materials exhibit an increased “break-away” swelling rate and start to swell faster than the fuel so that the gap between fuel

and cladding increases. Figure 2.21 shows recommended values of the swelling of the fuel and the fraction of the fission-product gas that is released as functions of burnup for U-Pu-Zr fuel alloy with a wide range of compositions, based on experimental results.

The main impurities found in the sodium are oxygen and hydrogen. Oxygen enters the primary circuit as an impurity in the argon cover gas and as moisture on the surface of new fuel. Hydrogen, liberated by corrosion of the steam generator tubes, enters the secondary circuits by diffusing through the tube walls, and from there diffuses through the intermediated heat exchangers into the primary circuit.

As pointed out in section 3.3.4 oxygen dissolved in the sodium causes corrosion of stainless steel, and if the corrosion rate is to be limited the concentration of oxygen must not be too high. On the other hand there is some evidence that a little oxide in the sodium acts as a lubricant preventing self-welding between steel surfaces in contact and facilitating the operation of immersed mechanisms. It is usually controlled at a level of about 10 parts per million by weight or less.

The solubility of sodium monoxide Na2O in sodium as a function of temperature is shown in Figure 4.6. If the oxygen concentration is high oxide tends to be precipitated in cool parts of the circuit. This is another reason for controlling the oxygen concentration because precipitation of oxide could block narrow passages such as coolant monitoring pipes.

Fortunately the low solubility of oxide at low temperature also affords means of controlling and monitoring the oxygen concentration. It can be controlled by passing the sodium through a “cold trap”. This is a device in which the sodium is cooled to a temperature below 160 °C or so and then passed through a bed of stainless steel mesh or rings in which oxide is precipitated, so that the oxygen concentration is reduced to about 10 parts per million. To obtain lower oxygen concentrations a “hot trap” may be needed. In this the sodium is heated to 600 or 700 °C and passed over zirconium, which has a greater affinity than sodium for oxygen. A hot trap can reduce the oxygen concentration to 1-2 parts per million. Hot traps and cold traps cannot be used together because oxygen would be transported from the cold traps to the hot traps.

Hydrogen forms sodium hydride, NaH. The solubility of sodium hydride in sodium is also shown in Figure 4.6. It behaves similarly to oxide and is precipitated in cold traps in the same way. Because there is a continuous source of hydrogen cold traps in the secondary circuits have to be operated continually to prevent precipitation of hydride in the coldest parts of the steam generators. If it is not trapped in the secondary circuits hydride appears in the primary cold traps.

Tritium, 3H, is radioactive with a half-life of 12.4 years, emitting a weak p. It is generated in the reactor core in three ways. It is formed directly in fission by rare ternary fission events, by the neutron capture reaction 6Li (n, a) 3H in lithium present as an impurity in the sodium, and by the capture reaction 10B (n,2a) 3H in boron in the control rods. The total production rate is of the order of 4 TBq per day (3 x l016 atoms per second) in a 2500 MW (heat) reactor. Tritium diffuses read­ily through the tubes of the intermediate heat exchangers and tends to be precipitated in the secondary cold traps because these are usually in continuous operation. This may cause difficulty in disposing of the cold trap packing when it is full of hydride.

Flowmeter

Figure 4.7 A plugging meter.

The decline in Europe and America did not affect Asia. A long­standing programme in India had continued, somewhat separated from work in the rest of the world. In Japan activity had continued in spite of the Monju fire, and there was continuing interest in the Republic of Korea. Most significantly work had started in China. Nuclear power is seen to be necessary for the future economic health of all of these countries and breeding will be an essential component by the second half of the century. Because there is so much more experience with it than with any alternative the main emphasis is on sodium as a coolant.

In the West as well the revival of nuclear power has been accompan­ied by increased interest in fast reactors, but here it is not so clear that the future lies with the sodium-cooled oxide-fuelled critical breeder that had been the norm in the 1970s. Fast reactors may have diverse applications (consumption as well as breeding) in the years to come, and there may be several design variations (lead or gas coolants as well as sodium, accelerator-driven subcritical reactors as well as critical, and metal fuel as well as oxide).

Because there is so much more experience of sodium-cooled breed­ers most of the content of the following chapters is about systems of this type, but the intention is also to provide an introduction to the alternative coolants, fuel materials and design styles that may become important as fast reactors are deployed for a range of purposes in a growing and diversifying nuclear industry across the world.

GENERAL REFERENCES

Chang, Y. I. and C. E. Till (2011) Plentiful Energy: The Story of the Integral Fast Reactor, CreateSpace online publishers Forrest, J. S. (Ed.) (1977) The Breeder Reactor, Scottish Academic Press, Edinburgh

IAEA (2013) Status of Fast Reactor Research and Technology Development Technical Report TECDOC-1691, International Atomic Energy Agency, Vienna

IWGFR (1985) Status of Liquid Metal Cooled Fast Breeder Reactors Technical Report 246, International Atomic Energy Agency, Vienna IWGFR (1999) Status of Liquid Metal Cooled Fast Reactor Technology Tech­nical Report TECDOC-1083, International Atomic Energy Agency, Vienna IWGFR (2007) Liquid Metal Cooled Reactors: Experience in Design and Operation Technical Report TECDOC-1569, International Atomic Energy Agency, Vienna

Judd, A. M. (1981) Fast Breeder Reactors: An Engineering Introduction, Pergamon, Oxford

Judd, A. M. (1983) Fast Reactors, pp 297-333 in W. Marshall (ed.) Nuclear Power Technology, Volume 1: Reactor Technology, Clarendon, Oxford

Sweet, C. (Ed.) (1980) The Fast Breeder Reactor: Need? Cost? Risk?, Macmil­lan, London

Waltar, A. E. and A. B. Reynolds (1981) Fast Breeder Reactors, Pergamon, New York

Waltar, A. E., D. R. Todd and P. V. Tsvetkov (2012) Fast Spectrum Reactors, Springer, New York

There are two ways in which an ADR can be considered as an amplifier, augmenting the output of the accelerator. Considered as a source
of neutrons, its gain Gn is (n + S)/S (see Figure 1.29), and since ke = n/(n + S),

Gn = 1/(1 — ke). (1.51)

Considered as a source of power, however, the gain is greater because each proton interaction in the spallation target produces many more neutrons than a fission. For proton energies up to about 1 GeV impinging on a lead target nn, the number of neutrons produced, is roughly proportional to the proton energy. A1 GeV proton produces about 18 neutrons, so that nn и Ep/E0 where E0, the energy associated with each neutron, is about 55 MeV.

In the subcritical assembly for each source neutron 1/S neutrons are absorbed in fissile material (see Figure 1.29). Of these events 1/S(1 + a) are fissions, each giving Ef и 200 MeV of energy out­put. Thus the energy gain Ge is approximately Ef /E0S(1 + a). Since S = n(1 — ke)/ke and n(1 + a) = v,

Since ke in the numerator can be taken to be и1, and with Ef/E0 200/55 = 3.6,

The numerator in this expression depends on the nature of the spalla­tion target and v depends on the fissile material.

While the reactor is operating the heat transferred from the fuel arises almost entirely from fission. Q is given by

Q « AEfJ2 ,

g

where AEf is the difference between the internal energy of the react­ants and products of a fission event, assuming there is no temperature change and that the products are at rest. AEf is about 200 MeV, or

3.2 x 10-11 J, for 239Pu and is only slightly different for other isotopes. ^fgf1 is the group fission cross-section for the fuel material. It is not the same as £fg in equation 1.9, which is an average cross-section for a region of the reactor including structure and coolant.

Equation 3.2 is not exact. Some of the energy is transferred by neutrons and radiation and appears in the structure and coolant and even in the shielding. Some is transferred when radioactive fis­sion products decay and so appears after the fission has taken place. While the reactor is operating y-heating in the structure, the outer parts of the breeder and the shield is important, and when it is shut down radioactive decay heating in the fuel is important. Nevertheless during operation some 97% of the energy appears promptly in the fuel.

The neutron spectrum is nearly the same throughout the core so Q is roughly proportional to the total flux, the distribution of which is shown in Figure 1.13. The maximum value, Qmax, occurs at the centre of the core or at the inside of the outer enrichment zones. The average Q along the most highly rated fuel element is lower than Qmax and the average Q for the whole core is lower still.

To take account of the variation of power density across the core “form factors” are defined. An axial form factor fz is given by

where Q(z) is the value of Q at a distance z from the bottom of the most highly rated fuel element and H is the height of the core. The radial form factor, fr, is then defined by

frfz = / Qdv/VQmax, (3.4)

Jcore

where the integral runs over the whole core and V is its volume. Both fr and fz are usually about 0.8 and the average power density over the whole core is about 0.64 of the maximum. The power density at the ends of the most highly rated element is about 0.4 Qmax, and at the extremities of the core it is as low as 0.2 Qmax.

If the peak linear rating is 50 kWm-1 therefore the average for the most highly rated element is about 40 kWm-‘ and the average over the whole core is about 32 kWm_l, whereas the least highly rated element has an average linear rating of about 18 kWm-’. If a reactor is to produce 2500 MW of heat it requires a minimum of about 78 km of fuel elements in total, whatever their radius. If the fuel radius is 2.5 mm the total volume of fuel is at least 1.53 m3.

A major advantage of a liquid-metal-cooled reactor from the point of view of safety is that the coolant pressure is low, so that the primary coolant containment is only modestly stressed and is unlikely to fail, and even if it should fail the coolant does not vaporise. This is in complete contrast to gas- and water-cooled reactors where the coolant pressure is high and extensive protection has to be provided against loss-of-coolant accidents.

It is possible to design a lead-cooled or sodium-cooled reactor so that in the event of a primary circuit rupture the core can be cooled without the provision of emergency supplies of coolant. This can be done by surrounding the vessels and pipework by a leak jacket. Further protection from the consequences of failure of the leak jacket can be provided by surrounding it with a strong concrete structure or by siting the reactor vessel underground. This protects a pool reactor against overheating because the primary coolant cannot fall below the level of the core and the intermediate heat exchangers. Decay heat can be removed indefinitely by the secondary coolant provided the intermediate heat exchangers are intact, or by means of an emergency heat rejection system. In a loop system the coolant pipes have to be connected to the reactor vessel above the level of the core if the core is to remain covered in the event of a pipe break. Decay heat can be rejected if at least one of the primary coolant circuits remains intact.

It is normal to provide auxiliary pump motors to guarantee circu­lation of the primary coolant at a sufficient rate to remove the decay heat even if all the main pump motors or the electrical supply to them should fail. This may not actually be necessary, however, because nat­ural circulation may be quite adequate to remove decay heat from the core without excessive overheating.

An additional safety feature, particularly in pool reactors, is provided by the large mass of the primary coolant, which may exceed 2000 tonnes. Provided it circulates, by means of pumps or natural con­vection, its temperature rises only slowly even if there is no secondary cooling at all. This is illustrated by Figure 5.1, which shows the rise in the mean primary coolant temperature for a sodium-cooled reactor, assuming the reactor shuts down and simultaneously all secondary cooling, including the decay-heat rejection system, is lost. There are several hours in which to make secondary cooling available before the temperature rises enough to cause widespread fuel failure (which might happen in the range of 800-1000 °C, depending on the details of the fuel design and the burnup). At atmospheric pressure sodium boils at 892 °C.

To determine the group cross-section for a neutron group containing a resonance we have to know how the flux varies in the resonance region. A simple approximation, which is fairly accurate in most cases, is the “narrow resonance approximation”. Suppose we have a region made up of two materials, one of which has a constant scattering cross­section while the other has only a single capture resonance. Suppose

also that the resonance is narrow so that its width Y < Z E0, where Z is the mean change in ln E in neutron scattering events. It can be shown that the flux ф (E) varies as 1/EEt where £t is the total cross-section — i. e. £t = £s + Nac where N is the number of atoms of the capturing

material per unit volume. The capture cross-section for a group that contains this resonance is then

and the denominator of this expression is 0g, the total flux in the group. Substituting for £t and making various approximations we obtain

and

Thus the group capture cross-section and the total capture rate depend both on the temperature (via Z) and on the scattering cross-section. The variation of J with в for various values of Z is shown in Figure 1.3. It can be seen that when p is large J is independent of Z, but when в is smaller J increases as Z increases. This means that at “infinite dilution” when there is very little of the capturing material the capture rate is independent of temperature, but when more is present the cap­ture rate increases as the temperature increases. This is known as the Doppler effect on the capture rate. It happens because as the temper­ature increases the effective resonance becomes lower and broader, as Figure 1.2 indicates. The flux at the peak of the resonance is less depressed, but this does not quite compensate for the lower cross­section so the reaction rate per unit energy (фас) at the resonance peak decreases. At the sides of the resonance, however, ac increases more than ф decreases and фас increases. The increase at the sides out­weighs the decrease at the peak and the total reaction rate (/ фасйЕ) increases.

This applies to all resonance reactions, and of particular importance in a uranium-cycle fast reactor is the fact that both the fission rate in 235U or 239Pu and the capture rate, predominantly in 238U, increase with temperature. The former tends to increase reactivity and the latter to decrease it. In a breeder reactor containing a large amount of 238U the capture effect is greater and the resulting temperature coefficient of reactivity, known as the Doppler coefficient, is negative, making for stability. If the reactor is not designed to breed, however, and contains less 238 U or none at all this stabilising property is reduced and may be lost.

This discussion of resonance absorption and the Doppler effect is a simple version of the whole story. Other effects that have to be taken into account are the variation of the flux integral (the denominator in equation 1.37, which is not constant), the fact that resonances in one material are not in general isolated but overlap both with each other and with those of other materials, the fact that the narrow resonance approximation is not accurate, and the existence of unresolved reson­ances. The way these problems can be treated is described by Hummel and Okrent (1970).

Various isotopes of krypton and xenon are produced by fission. The cumulative yields of the stable and long-lived isotopes are shown in Table 2.1. Most of them are formed not immediately but as the end — products of beta-decay chains. The half-lives of most of the decay processes are of the order of a few minutes, and in only two cases (131Xe and 132Xe) of a few days. The total yield is about 0.04 atoms of krypton and 0.22 atoms of xenon per fission.

This is a very large quantity. If fuel of density 104 kg m-3 undergoes 10% burnup and the krypton and xenon generated exist as gas at 400 °C and atmospheric pressure they occupy 53 times the volume of the fuel. Alternatively if they are confined in a volume equal to that of the fuel, they exert a pressure of 5.3 MPa at 400 °C or 6.8 MPa at 600 °C.

The krypton and xenon are not formed as gases, however, but as single atoms in solution in the fuel. The solubility is very low so the solution is soon supersaturated and the atoms are precipitated to form bubbles. They enter the bubbles by thermal diffusion (and therefore at a rate that depends on temperature), and they may also leave the bubbles and return to the fuel by the process known as fission-induced resolution. This comes about as follows. If a fission occurs close to a gas bubble one of the fission-fragments may enter it. It may have 100 MeV or more of kinetic energy, and if it collides with a gas atom it can easily impart enough energy to it to make it reenter the surrounding fuel crystals. In effect the gas atom is knocked out of the bubble and forced back into solution.

The size and rate of growth of a bubble depend on the rates at which it gains and loses gas atoms. Bubbles within a grain gain by diffusion and also by sweeping up atoms or other bubbles as they move through the grain (see section 2.4.1 for the mechanism of this process) and lose by resolution. They are typically spherical with a diameter of the order of 50 nm. Bubbles at the grain boundaries, however, grow as the grain boundaries move. The hotter part of the fuel is continually being recrystallised, and as one grain grows at the expense of a neighbour the grain boundary sweeps through the fuel collecting gas atoms and bubbles as it goes. This is a very effective growth mechanism and as a result grain-boundary bubbles are larger than bubbles within the grains. Grain-boundary bubbles are sometimes flat or lenticular in shape and up to 20 nm across. Figure 2.5 shows typical gas bubbles. As the grain-boundary bubbles grow they link up and eventually form

channels by which the gas escapes from the fuel. The fraction of the gas that is released in this way depends in a complex manner on burnup and temperature. Because the process is so complicated it is usual to assume when designing the fuel elements that all the krypton and xenon generated will be released as gas.

In reality, however, some 10 or 20% is retained in the fuel, and this may become very important in the course of an accident. If the fuel becomes overheated gas retained in the colder parts may suddenly be released and affect the subsequent course of events (see section 5.4.5). An approximate estimate of the amount available can be made by assuming that 2% of the gas generated in fuel at temperatures above 1800 °C is retained, 50% in fuel between 1800 °C and 1400 °C, and 70% in fuel below 1400 °C.

2.3.3 Sealed or Vented Fuel

Given that allowance has to be made for this large quantity of gas to be given off by the fuel during irradiation the next question is whether it should be retained within the fuel element or released from it. If it is to be retained space has to be provided for it. If a volume equal to that of the fuel is provided, after 10% burnup the pressure inside the fuel element may be some 5-7 MPa as we have seen. A smaller volume would necessitate thicker cladding; a larger volume would make the fuel element longer so that the reactor vessel and the fuel handling equipment would have to be larger and therefore more expensive and the pressure drop in the coolant circuit would be greater. If the gas could be released from the fuel elements there would be considerable advantages in a reduction of the height of the reactor vessel and possibly of the cladding thickness. There would also, however, be important disadvantages, the worst of which is the difficulty of keeping the coolant out of the fuel elements.

If sodium comes into contact with the fuel under some circum­stances a mixture of sodium uranate and plutonate (Na3UO4 and Na3PuO4) is formed. These compounds have low densities causing the fuel to swell and possibly bursting the cladding. If the gas is to be released, therefore, each fuel element has to be fitted with a vent to allow the gas to leave while preventing coolant from entering even when the reactor is shut down and cooled and the pressure inside the fuel elements falls. It has not proved possible to design a vent that is both sufficiently reliable and sufficiently cheap to manufacture in large numbers.

A second disadvantage of vented fuel is that it inevitably allows the release of radioactive materials into the coolant. In additionto the isotopes listed in Table 2.1 various short-lived isotopes of krypton and xenon such as 90Kr (half-life 33 s), metastable 83Kr (114 min) and 133Xe (5.3 d) appear as fission products. To prevent or minimise release of these isotopes a device would have to be incorporated to delay the release of the gases until they have decayed, because the presence in the coolant of their decay products, such as 90Sr of which 90Kr is a precursor, is very undesirable. Of course nothing could be done to prevent the release of 85Kr from vented fuel and it would be very hard to guarantee that other volatile fission products such as 131I (8 d) and 137Cs (29 yr), of which the yields are high, would not be released with the inert gases.