As the fuel temperature increases the resonance self-shielding in the fuel isotopes decreases. The resulting change in reactivity is called the Doppler effect because it is due to the dependence of the effective energy of a neutron on the relative motion of the nucleus with which it interacts. The decrease in self-shielding results in an increase in all the cross-sections of the fuel isotopes at energies below about 20 keV.

Although all the cross-sections increase, including scattering, the most significant changes are in £c and £ f. In all breeder reactors the increase in capture in 238U dominates over the increase in fission in 239Pu and the Doppler effect is negative, but the same is not always true for consumer reactors.

The Doppler effect is not independent of temperature. The dependence of J (equation 1.39) on T is complex. If fi/f0, where f0 is the peak value of f, is large (corresponding to a resonance with a low peak at higher energy) then J a T-1/2 and dJ/dT a T-3/2, but if fi/f0 is small J a T and dJ/dT is constant. Thus the Doppler effect varies in a different way with temperature for different resonances, but the total coefficient, involving a weighted sum of dJ/dT for all the resonances, varies roughly as 1 / T. As a result it is convenient to define a “Doppler constant”, — T dp/dT.

Figure 1.27 shows the contribution to the Doppler reactivity effect from various isotopes in different neutron energy groups for a breeder reactor with low enrichment. The most important effects are around

1 keV at which energy the resonances are not resolved. The effect of the iron resonance at 1 keV can be seen. This is the only resonance in a non-fissile or fertile isotope that makes any significant contribution to the Doppler effect.

In an operating reactor the fuel is not all at the same temperature, and as the Doppler coefficient is a function of temperature there is a question of the correct average temperature to use. In a cylindrical fuel element with constant thermal conductivity the volume-weighted mean temperature T is halfway between the central and surface tem­peratures. Because the Doppler coefficient is higher at low temper­atures the effective temperature Teff for the Doppler effect is slightly lower than T, but the difference is small. If the temperatures in the fuel element are in equilibrium, it can be shown that

(1.47)

where T0 is the fuel surface temperature. Temperature variation across the core can be taken into account by weighting changes with fgf* as indicated in equation 1.25.

As explained in section 1.4.1 isotopes of americium and curium are pro­duced in all uranium-fuelled reactors, in greater quantities the higher the burnup of the fuel. They constitute hazardous waste products and there may be an incentive to eliminate them. Because they are fission­able this can be done in principle by incorporating them as fuel in a fast reactor. However if the objective is to reduce their quantity the reactor should not generate more of them, and therefore, as in the case of a plutonium-consumer reactor, the fuel should contain little or no uranium and would have to be diluted.

Higher-actinide ceramic fuel would probably be reprocessed in dedicated facilities, separate from the mainstream, using either the Purex process or a modification of it (see section 2.3.3) with lower separation efficiency and a product stream matched to the fuel-cycle requirements. For this reason the fuel and any diluent materials must be soluble in nitric acid.

In addition the higher actinide fuel would have to be compatible with the coolant in the sense that, in the event of a small cladding failure, in the sense thatgross swelling or any other reaction that would cause the release of significant quantities of fuel material or fission products to the coolant would have to be impossible. This limits the choice of fuel material for a sodium-cooled reactor, but not necessarily for reactors cooled with gas or lead. In principle the fuel material could be a pure compound, a solid solution of fuel and a diluent compound, or a two-phase mixture of fuel and a diluent compound. The two-phase mixture could be either a cermet or a cercer.

Information for most of the candidate materials is incomplete. Physical property data on thermal conductivity or melting point are sometimes known or can be inferred, but there is usually nothing on properties such as thermal creep. There is little irradiation experience but some aspects of the behaviour under irradiation can be deduced from the phase diagram. (If there are phase changes in the operating temperature range the structure is unlikely to be stable.) There is often no empirical information on the compatibility with cladding or coolant, although theoretical inferences can sometimes be made. Information on curium compounds is scarce so the selections are mostly based on americium data.

The most obvious choices for pure compound fuel materials are summarised in Table 2.5. The most important considerations are that

carbides are pyrophoric, AmO2 has a high oxygen potential, and Am2 Оз has several phase changes.

Table 2.6 summarises the main candidate diluent materials for cermets or cercers. Nitrides might require enrichment in 15N to avoid production of 14C. In addition to the materials listed in the table Si3N4, TiN, YN, and AlN are also candidates.

The steam cycle in Figure 4.9 is very similar to that of a conven­tional fossil-fuelled power plant, because the maximum steam temper­atures are similar. The similarity is to some extent fortuitous, however, because the maximum temperature is set by different considerations in the two cases. In a fossil-fuelled power plant the maximum steam temperature is limited to about 565 °C because anything higher would require the use of austenitic steel rather than ferritic, and the increased cost would not be justified by the gain in efficiency. In a sodium-cooled fast reactor plant, however, austenitic steel is widely used, as we have seen. The main temperature limits are the maximum permissible fuel cladding temperature, and the temperature of the structure above the core in a pool reactor or of the hot leg pipework in a loop reactor. The parts of the structure in contact with hot primary sodium are subject to substantial thermal shock if the reactor is shut down suddenly in an emergency, and the primary sodium maximum temperature is limited to a level at which this shock can be withstood.

The problems of withstanding thermal shocks has even led some designers to propose reduction of the primary sodium maximum temperature to below 500 °C (Anderson, 1978; Horst, 1978). Steam superheaters are omitted, and moisture separators and reheaters are incorporated between some of the turbine stages to avoid the irre­versibilities associated with high moisture content in the steam. The resulting cycle is very similar to that of a boiling water reactor.

If maximum steam temperatures of 500 °C or above are permiss­ible, a conventional steam cycle with superheat and reheat can be

Secondary Sodium

water/Steam

Heat Transferred
1 High Feed Temp., Low Saturation Temp

Low Efficiency
2 Low Feed Temp., High Saturation Temp

High Efficiency

Figure 4.15 The interrelation of plant efficiency with the feedwater temperature.

used, as shown in Figure 4.9. The details of the cycle, and in particular of the feed heating system, may be slightly different from those of a fossil-fuelled power plant because the effect of final feed temperat­ure on efficiency is rather different if heat is being transferred from a relatively low-temperature coolant than if it is transferred from high — temperature gas. The point is discussed in detail by Haywood (1975) and can be illustrated by reference to Figure 4.15.

The plant efficiency is greater the higher the mean temperature at which the working fluid receives heat. Because a large portion of the heat supplied to the fluid is taken up by evaporating it, the higher the saturation temperature, and therefore the pressure, the greater the effi­ciency. If the cycle efficiency were to be increasedby increasing the steam pressure without changing the secondary sodium temperatures, it would be necessary to decrease the feedwater temperature by means of a different feed heating system, as shown in Figure 4.15. This illustrates
the fact that the feed and saturation temperatures in a plant of this type are interdependent. In contrast in a fossil-fuelled plant, they are independent (Haywood, 1975).

In this example the increased saturation temperature could, of course, be accommodated by allowing the minimum secondary, and also primary, sodium temperatures to rise. This in turn would require an increase in the secondary and primary sodium flow-rates and would involve the disadvantages of higher coolant speeds, more likelihood of vibration of heat exchanger tubes and fuel elements, greater pres­sure differences and stresses in the core, bigger circulating pumps and so on.

For the purposes of designing or operating a reactor the direction in which the neutrons are travelling is usually of little interest and there is an incentive to adopt a simpler calculation method that eliminates the two independent scalar variables Й. This is the treatment known as “diffusion theory”.

The basis of diffusion theory is to assume that ф is either isotropic or at most linearly anisotropic. (This is equivalent to P1 transport theory, with all but the first two terms on the right-hand side of equation (1.6) being ignored.) The transport equation is integrated over all directions and various approximations are made to obtain an energy-dependent diffusion equation in terms of the neutron flux ф, independent of direction, defined by

ф(г, E) = ф(г, E, to)dto, (1.7)

4n

and the neutron current

J(r, E) = &ф(і, E, to)dto. (1.8)

4n

The assumption that the flux is at most linearly anisotropic is accurate except in a strongly absorbing medium or where the properties of the medium change substantially over distances comparable to the mean free path of the neutrons. For many fast reactor calculations these limitations are not particularly important. Fast neutron cross­sections are usually small and mean free paths are typically 0.1 m or longer. The nuclear properties of fuel, coolant and structure are very different but because the dimensions of individual fuel elements and structural members are usually only a few millimetres, over distances comparable with the mean free path large regions of the reactor can be treated as homogeneous and diffusion theory can be used. Even in control rods the capture cross-sections for fast neutrons are low enough for diffusion theory to be a good approximation for many purposes. The worst inaccuracies arise at the edge of the core, and deep in the neutron shielding that surrounds the reactor.

There are some types of reactors for which transport theory has to be used, such as the small experimental fast reactors of the 1950s and 1960s that had cores with dimensions comparable with the mean free path, and also experimental reactors assembled from various materials in the form of thin plates, typically 50 mm square, arranged in arrays comparable in size with the mean free paths.

The great advantage of diffusion theory over transport theory is that, because for a steady-state calculation it deals with only four independent scalar variables than six, it makes smaller demands on computing power. For this reason up to the first decade of the 21st century most reactor design and operational flux calculations made use of diffusion theory. However in recent years the capacity of computers has increased to the extent that transport theory calculations have become more feasible and cheaper and therefore more widely used.

The rest of this discussion of fast reactor physics is presented in terms of diffusion theory because its formulation is much simpler than that of transport theory. Nevertheless it should be borne in mind that the same considerations apply in the latter, but the algebra is heavier.

A major uncertainty in predicting the fuel temperature is the value of the conductance across the interface or gap between the fuel and the cladding. This is hf in equation 2.1.

If the fuel is manufactured in the form of pellets they are inser­ted into the cladding tubes with a radial clearance of some 50 цm or more when cold. This is reduced by the differential thermal expansion when the reactor is at power. The thermal expansion coefficient of the cladding (about 18 x 10-6 K-1) is greater than that of the fuel (about 13 x 10-6 K-1), but as the increase in the fuel temperature is greater the gap is reduced. The extent of the reduction is hard to predict, how­ever, as the expansion of the fuel is irregular because it cracks (see section 2.4.2).

To increase hf the fuel elements are usually filled with helium when they are manufactured because it has a high thermal con­ductivity of about 0.3 Wm-1 K-1 at 600 °C. If a helium-filled radial gap is 20 gm wide and symmetrical when at power, hf is about 15 kWm-2 K-1.

As irradiation proceeds the fuel swells and the gap becomes nar­rower until the fuel and the cladding are in contact, but because the surfaces are rough this does not mean that the effective width of the gap is zero. When two solid bodies touch each other contact is made only at a few places where extreme points on the two rough surfaces touch. The fraction of the apparent interface over which there is actual contact may be of the order of 1% and increases if there is a normal stress pushing the surfaces together. The effective thickness of the gas layer when the surfaces are in contact depends on their roughness and may be 0.5 gm if they are smooth, but cracks in the fuel increase it to an extent that is uncertain. Also the estimation of the area of solid-to-solid contact is uncertain.

A further large uncertainty is introduced by the changes in the composition of the gas in the gap. As irradiation proceeds inert fission- product gases, mainly xenon with some krypton, are released from the fuel (see section 2.3.5). Because of their high atomic weights they have much lower thermal conductivities than helium (.026 Wm-1 K-1 for krypton and.016 Wm-1 K-1 for xenon at 600 °C), and as they are released from the fuel they reduce the conductivity of the helium substantially by diluting it, introducing another uncertainty in the conductance.

In principle it is possible to estimate hf experimentally, either by examining fuel after irradiation and deducing the temperature distri­bution from the observed changes in the structure, or by measuring the fuel temperature during operation by inserting thermocouples into it. Either method involves large uncertainties however, mainly because the conductivity of the fuel itself is not known accurately as explained earlier.

As a result it is usually best to assume an approximate value of hf of about 5 or 8 kWm-2 K-1, and to recognise that it is very uncertain and that the actual value may differ by a factor of two either way. For Rf = 3 mm and q = 50 kWm-1, hf = 6 kWm-2 K-1 gives an interfacial temperature difference between fuel and cladding of 440 K, but the actual value may be anywhere between 200 and 800 K.

Sodium has the effect of removing nickel and chromium from the sur­face layers of austenitic stainless steel. The chromium forms sodium chromite and the nickel is removed by dissolution in the sodium. Figure 3.16 shows the nickel and chromium concentrations near the surface of stainless steel exposed to flowing sodium. In this region the nickel concentration may be reduced to about 1% and chromium to 5-8%, and as a result a surface layer of ferrite is formed some

because in all cases the controlling process is the dissolution of ferrite. Figure 3.17 indicates that, with an oxygen concentration of around 10 ppm, which can be attained in practice in a fast power reactor, cor­rosion rates up to 10 pm per year are to be expected in the hottest parts of the coolant circuit where the temperature is around 600 °C. Higher oxygen concentrations carry a risk of excessive corrosion.

Corrosion or, more accurately, wear can be made worse by the relative motion of surfaces in contact, usually called “fretting”. In a reducing environment such as sodium there is a tendency for surfaces in contact to become welded together at high spots. If the surfaces are forced to slide over each other, for example by some source of mechanical vibration, the welded high spots are sheared, and material may be transferred from one surface to the other. Fretting damage is a danger in parts of the core subject to vibration due to turbulent flow of the coolant. Self-welding is a problem for items of equipment such as fuel subassemblies that have to be removed from the reactor from time to time.

Depending on the activity of dissolved carbon, sodium can transfer carbon either to or from steels immersed in it. This process is known as carburization or decarburization depending on whether carbon is gained by the steel or lost from it. The rate of loss or gain of carbon depends on temperature because both the carbon activity in sodium and steel and the diffusion rates vary with temperature.

The carbon activity in austenitic steel is relatively low so it tends to be carburized even if the carbon concentration in the sodium is as low as 5 parts per million. Carbide is precipitated in the surface layers of the steel, reducing the ductility at low temperatures. Low-alloy ferritic steels, however, such as 2.25 Cr 1 Mo with about 0.1% carbon, tend to be decarburized and lose strength as a result.

Carbon activity is higher at low temperature, so there is a tend­ency for decarburization to be particularly important if a sodium cir­cuit incorporates ferritic steels in a low-temperature region such as a heat exchanger. If excessive carburization of austenitic steels in the

high-temperature regions is to be avoided, careful control of the car­bon activity in the sodium has to be maintained.

Steels are corroded more by water than by liquid metals. As a result the coolant circuits of water-cooled reactors become considerably more contaminated with 60Co and 56Mn than those of sodium — or lead-cooled reactors. This has a marked effect on the radiation doses experienced by the plant operating personnel, particularly the maintenance staff. The principal radiation source in the primary circuit of a sodium- cooled reactor is 24Na, but this decays quickly (T1/2 = 15 hours) so that a few days after shutdown, provided there have been no fuel failures, radiation levels are low and do not impede access for maintenance. If failed fuel is present the primary circuit cover gas may be contaminated with radioactive Kr and Xe.

Collective doses in the range 10-100 man-mSv per year are repor­ted for sodium-cooled power reactor plants, in contrast to doses in the range 1000-10000 man-mSv per year for BWRs and PWRs.

5.3.2 Sodium Fires

Sodium burns readily in air to produce dense white fumes of sodium peroxide. The reaction is

2 Na + O2 ^ Na2O2 + 11 MJ per kg of sodium.

The peroxide combines with atmospheric water vapour to form sodium hydroxide which is corrosive. The ignition temperature in air is ~250 °C for a quiescent pool but lower, ~125 °C, for a spray such as might emerge from a leaking pipe. Burning sodium can generate temperatures greater than 1000 °C and can cause severe damage to steel structures.

Because the primary sodium is radioactive it is necessary to enclose all the sodium-carrying vessels and pipes in a continuous leak jacket. The space inside the leak jacket is filled with inert gas (nitrogen or argon) and provided with sodium presence detectors. Major secondary circuit components have also to be provided with leak jackets, but they are not necessary for minor secondary components and pipework, which can be located in air-filled spaces with fire-suppression pans on the floors. These pans work by collecting any leaking sodium and directing it to a drain tank while restricting access of air to it so as to minimise the fire, as shown in Figure 5.4.

Sodium pool fires can be extinguished by smothering with common salt (NaCl) or, better, graphite powder which expands by exfoliation to form a blanket over the surface of the sodium.

The term “higher actinides” is used for the man-made nuclides of elements with atomic number of 93 or greater. They originate from neutron capture in uranium and are sometimes called “trans-uranium elements” or “trans-uranics”. In practice the term is also often used for the man-made nuclides formed by neutron capture in thorium.

The most important of the higher actinides, both for reactor oper­ation and commercially, are 239Pu and 233U produced by neutron cap­ture in 238U and 232Th respectively, as explained in the Introduction. They are both beginnings of long and complicated chains of reactions, mainly neutron captures and p decays, that produce many isotopes of several elements. Figures 1.14 and 1.15 are simplified diagrams of these chains. The horizontal arrows represent neutron captures and the vertical ones p — (upward) or p+ (downward) decays.

Figures 1.14 and 1.15 include the reactions that are of greatest prac­tical importance, but many details have been omitted in the interest of clarity. For example 243Am is p -active but this is not significant because its half-life is 7370 years. Almost all the nuclides in both diagrams that are not p-active are subject to а-decay with half-lives of greater than 1000 years so they are of little importance as far as reactor operation is concerned. (They are much more important in reprocessing and waste storage because the radiation is hazardous.) 241Am however has

a half-life of 433 years to produce 237Np, which in turn is а-active with a half-life of 2.1 x 106 years. This makes it one of the longest lived hazardous nuclear waste products.

Many of the nuclides decay by spontaneous fission but in most cases it is unimportant because the half-lives are so long. The excep­tions are 242Cm and 244Cm which have spontaneous fission half-lives of 7.2 x 106 and 13.2 x 106 years respectively. The neutrons generated are insignificant in normal operation of the reactor but not when it is shut down. As the quantity of 241Pu changes with burnup, so do the quantities of the curium isotopes and hence the neutron source strength. As a result the relationship between shutdown reactivity and subcritical power level varies both with burnup and also with time while the reactor is shut down due to 242Am and 244Am.

Almost all of the (n, y) reactions indicated in both diagrams are mirrored by (n,2n) reactions going in the opposite direction. Most of the latter can be ignored because the (n,2n) cross-sections are very small. However, there is one that is of some significance because it leads to the production of a particularly unpleasant waste product. Figure 1.16 show how the (n,2n) reaction in 233Pa leads eventually, after a series of a — and p-decays, to the production of 208Tl. This is intensely radioactive, emitting 2.6 MeV ys that pose a serious problem in a reprocessing plant.

It is found that, even if the plutonium is initially distributed uni­formly throughout the fuel, after only a short irradiation (less than 1% burnup) the relative concentrations of uranium and plutonium have changed, but the nature of the redistribution depends on x. Figure 2.14 shows experimental values of plutonium concentration at the centre of the fuel after irradiation, indicating that for x < -0.04 the plutonium moves outwards, whereas for x > -0.04 it moves inwards. In contrast the plutonium in the outer, cooler, part of the fuel is hardly affected. The distribution of plutonium in irradiated fuel with x = 0 is shown in Figure 2.15.

The mechanism for this redistribution seems to depend on the preferential evaporation and migration of different species. The stoi­chiometric ratio is important because if there is excess oxygen the mixture of vapours in equilibrium with the mixed oxide contains relat­ively more UO3, whereas if oxygen is deficient it contains more PuO, and for x < -0.03 or -0.04 PuO predominates. Thus for more positive values of x it is uranium in the form of UO3 that tends to evaporate from the hotter part of the fuel and condense in the cooler, whereas for more negative x plutonium in the form of PuO moves instead. This is not the whole story, however, because it does not explain why the segregation is so limited in extent. It may be that diffusion of vapour through the dense columnar grain region is in fact severely restricted.

If the segregation of plutonium and uranium were more severe it would have serious consequences. Because the plutonium generates most of the power its extensive migration to the centre would increase the fuel central temperature and possibly cause melting. It would also have the effect of delaying the Doppler feedback on reactivity (see section 2.3.2). For the limited segregation that is actually observed these effects are negligible.

The same mechanism also results in a tendency for the oxygen to migrate. For x > -0.04 the evaporation and recondensation of UO3 result in oxygen moving outwards so that the fuel at the outside is nearer stoichiometric while that at the centre has even less oxygen. The buildup of oxygen in the outer parts of the fuel tends to increase the partial pressure of UO3 there and so helps to oppose the outward movement of uranium.

The intermediate heat exchangers, in which heat is transferred from primary to secondary sodium coolant, are normally of a shell-and-tube design. Differential expansion of tubes and shell can be accommod­ated by means of expansion bellows or by bends in the tubes. To eliminate any possibility of radioactive primary coolant leaking into the secondary circuit the pressure of the secondary coolant in the heat exchanger has to be greater than that of the primary. The tubes are usually arranged in an annular bundle, with the secondary sodium flowing down through a central duct and then upwards through the tubes.

Because the coolants transfer heat so readily it is possible to keep the temperature difference between primary and secondary coolant small and yet keep the intermediate heat exchangers reasonably com­pact. For example with tubes 20 mm in diameter and coolant velocities of about 5 m s-1 a Nusselt number of about 10 on both shell and tube sides is possible (see section 3.2.4), giving surface heat transfer coeffi­cients of about 3 x 104 Wm-2 K-1. If the wall thickness is 1 mm this gives an overall heat transfer coefficient U of about 104 Wm-2 K-1. The heat transfer rate Q, heat transfer area A, and the logarithmic mean temperature difference ATm are related by

Q = UAATm, (4.1)

so that if Q = 3.6 GW and ATm = 30 K an area of 12000 m2 is needed for heat transfer. If the tubes are 8 m long and 20 mm in diameter some 24000 of them are needed, which could for example be arranged in six separate units each containing about 4000 tubes. The diameter of each of these tube bundles, allowing for the central secondary sodium inlet duct, would be about 2 m. The design of intermediate heat exchangers is discussed by Tang, Coffield, and Markley (1978), p. 319.

In a pool reactor the primary coolant is driven through the interme­diate heat exchangers by the pressure difference due to the difference in levels between the hot coolant within the inner vessel, and the cold coolant outside it.

In the event of an accident it might become impossible to reject heat from the secondary coolant or the steam plant, and an alternative means of removing the heat due to decay of fission products in the fuel would be needed (see section 5.2.4). For this reason an auxiliary secondary coolant system is provided. In a pool reactor this may take the form of separate auxiliary heat exchangers in the vessel, in which heat can be transferred to an emergency or “decay heat removal” cooling system. In a loop reactor there may be a separate auxiliary cooling loop in the primary circuit, or arrangements for emergency cooling of the secondary circuits.