The term fuel management is used to describe the strategies according to which the fuel is loaded, unloaded or reshuffled in the reactor. Sometimes this term is also used to describe what happens to the fuel outside the reactor (fabrication, reprocessing, etc.).

10.2. Continuous and discontinuous refuelling

During reactor operation fissile material is being burnt and the resulting change in the reactor composition should be compensated, either by continuously adding new fuel or by removing control rods or other neutron absorbers. This reactivity compensation can partly take place automatically in the reactor through conversion of fertile into fissile material (238U into 239Pu, 232Th into 233U, 240Pu into 241Pu) or through burn-up of absorbers (burnable poisons) which have been added to the core.

Various types of fuel-management schemes are used or are under study for high-temperature reactors. As two extreme limits one can consider a continuous reloading and a batch loading. In the first case, after a running-in phase the core composition does not change with time because the spent fuel is continuously replaced. This type of operation is possible in the pebble-bed reactors because of the very high number of small fuel elements. The other extreme (batch loading) consists of loading a reactor with fresh fuel and unloading it completely after the fuel has been burnt. In this case the reactivity loss due to fuel consumption and fission product build-up is compensated by control-rod movement and burn-up of burnable poisons. This type of operation is usually uneconomical. A very high excess reactivity should normally be built in the fresh loading. The excess neutron production is then absorbed in control rods or other poisons instead of being used for breeding purposes, increasing in this way the fuel-cycle costs. It is theoretically possible to have a flat reactivity versus time curve, even without using poisons, with fertile isotopes like 240Pu in which case a batch loading would be possible without economical losses<4"~6) but a fuel with the proper Pu composition and self-shielding would not be easy to obtain.

Between the two extreme cases of continuous reloading and batch loading a great number of partial core reloading schemes are possible. For reactors with prismatic fuel one should distinguish the case in which refuelling is possible while the reactor is in operation (on-load refuelling) from the cases where the reactor has to be shut down in order to be recharged (off-load refuelling). In the first case these operations can be so frequent that a continuous reloading can be simulated. In the second case the reloading intervals are less frequent because it is economically impossible to shut down the reactor too frequently. This means that a certain neutron loss occurs, but its economical impact is rather small if four to six reloadings are possible during the fuel lifetime. Fuel-management schemes should also consider the effects of the reloading operations on the flux, power and temperature distribution, as well as on the control-rod requirement and reactor safety.

In order to flatten the power distribution fuel reshuffling can be considered. For example, it is possible to load fresh fuel in the outer core regions and to shift it toward the centre after a certain burn-up period. This is done in pebble-bed reactors. Reshuffling operations are much more difficult with prismatic fuel. An axial reshuffling might be used in order to obtain an axial power distribution as near as possible to the one given by eqn. (10.8), loading fresh fuel at the gas inlet side and shifting it towards the gas exit side as the burn-up increases.

After a running-in time the curve of reactivity as a function of time becomes periodic, so that it may be in the first approximation sufficient to study the period between two subsequent recharges. In pebble-bed reactors the fluctuations due to reloading are negligible and the reactor composition is constant in the equilibrium phase.

In a spherical harmonics expansion it is difficult to fulfil a vacuum boundary condition, e. g. a boundary at which neutrons are only going out but not entering the system. Even with high-order P, approximations such a condition can only be approximatively satisfied. The same thing applies to boundaries between media with very different cross-sections. A possible solution consists of expanding the angular — dependent flux density separately in the two half spaces. The procedure first proposed by Yvon is called double Pi and fulfils exactly the above-mentioned boundary conditions, but may lead to worse approximations in case of big homogeneous layers. Furthermore, it can only be applied to simple geometries (plane or spherical).

8.1. The spectrum calculations

It is now routine for thermal reactor calculations to perform detailed spectrum calculations in a high number of energy groups with only a rough approximation of the spatial dependence and then to perform few-group space-dependent calculations with constants calculated by averaging the fine group cross-sections on this spectrum.

Various types of methods and codes are being used to perform these multi-group spectrum calculations. All of these codes contain a library in a fine energy structure (40 to 200 groups) including cross-sections for all interesting reactions, transfer matrices (the £so.,-.k. £sl, i_k or higher-order components of the scattering cross-section described in Chapter 4), resonance parameters, fission spectra, etc.

In the thermal-energy range the transfer matrices of the moderators are stored for each material for various temperatures in order to take into account the effect of thermal motion of the moderator. Transfer matrices are sometimes stored for the moderators only, while the more sophisticated codes include transfer matrices for heavy materials in order to enable treatment of inelastic scattering in the fast energy range.

Normally the interval between fission energy and thermal energy (at which the neutron energy spectrum is in equilibrium with the thermal energy of the moderator) is subdivided into a certain number of energy ranges.

First one can distinguish between a fast and a thermal range. In the fast-energy range neutrons are slowed down by every collision. When the neutron energy has been decreased to the range of the thermal energy of the moderator it becomes possible for the neutrons to gain energy in a collision. As the Maxwell-Boltzmann distribution of thermal energy gives probabilities different from zero for the whole energy range, there is no clear cut between fast and thermal energy. The boundary between these two energy ranges is normally set at a level at which the probability for a neutron of gaining energy in a collision becomes negligibly small. This boundary depends, of course, on the moderator temperature so that it has to be higher in HTRs. Once fixed, this boundary is kept constant, even for calculations of a reactor in cold conditions.

The normal value for HTR calculations lies between 2 and 4 eV. This value is very high if compared with the traditionally used Cd cut-off energy of ~0.68 eV (below which energy a Cd foil sharply becomes opaque to neutrons).

Within the fast-energy range further subdivisions are possible, especially according to the representation of the cross-section of the heavy metals. For these nuclides

neutron absorption occurs mainly in resonances whose energies are well separated and whose parameters are well known in a lower energy range (up to 3 8 keV) while the

individual resonance levels are not resolved experimentally at higher energies. Above ~ 100 keV the resonance structure of the heavy metal cross-sections becomes unimpor­tant for reactor calculations. According to those three fast-energy ranges different calculational methods are used.

The methods used for the treatment of resonances in the resolved and unresolved energy ranges have been described in Chapter 7.

The temperature coefficient depends on burn-up, temperature, cell geometry (lumping) and reactor geometry (leakage). The burn-up dependence is due to the build-up of 233U, Pu and fission products.

This depends upon the type of fuel cycle (U-Th or low enriched U) and upon the fuel-management scheme.

The insertion of control rods gives a more negative temperature coefficient because the absorption of a black absorber tends to remain constant with temperature while the fuel absorption decreases. Calculations made without rods are then conservative. The temperature coefficient has to be calculated as function of time in the case of batch loading. In case of continuous reloading there will be no time-dependence after the equilibrium composition is achieved, but the running-in period again requires a time-dependent investigation.

The worst possible temperature coefficient (calculated considering burn-up condition and Xe concentration) must be assumed for accident analysis.

The temperature coefficient is temperature-dependent. The Doppler coefficient decreases with increasing temperature. The moderator coefficient is often negative at room temperature and tends to become positive (or less negative) at operating temperature, depending on the reactor composition. If Pu and Xe have a strong influence, the temperature coefficient tends to become negative again at very high temperature. An example of the breakdown of the various components of the temperature coefficient is given in Table 11.1<2) for an HTR with Th fuel cycle. See also Fig. 11.1.<2)

As already mentioned the temperature coefficient can be strongly influenced by the fuel and reactor geometry. The fuel geometry and its cooling system influences not only the temperature coefficient, but also the delay with which the moderator temperature coefficient is responding to a transient. The part of the moderator which is intimately mixed with the fuel responds promptly, while the remaining moderator responds with various delays. If the moderator is separately cooled, this delay can be very large (a re-entrant cooling of the AGR type would keep the moderator at a practically constant temperature and eliminate a great part of the moderator temperature coefficient). Unfortunately these factors are usually fixed by other considerations. The fuel cycle optimization, temperature limitations and total power output largely determine the fuel and reactor geometry.

Any strong modification of the temperature coefficient will tend again to shift the reactor away from the economically optimum case. Such modifications are not usually necessary because if the moderator temperature coefficient is positive, the overall coefficient remains usually negative in HTRs. Furthermore, even a positive overall temperature coefficient would not exclude the possibility of a safe reactor operation because of the great time delay associated with the positive moderator coefficient (the prompt coefficient is always strongly negative).

If a more stabilizing temperature coefficient is anyhow required, beside geometrical changes, composition changes are possible. The moderator temperature coefficient can be modified in given temperature intervals introducing absorbing isotopes whose absorption cross-section increases with energy.

The positive contribution of low-lying fission resonances can be counteracted by the addition of isotopes having absorption resonances in about the same energy range (e. g. 103Rh to compensate 233U resonances and Er to compensate 239Pu resonance).

Those absorbers are, of course, adversely affecting the neutron economy. In a more economical way modifications of the temperature coefficient can sometimes be obtained changing the neutron spectrum with a modification of the fuel to moderator ratio or changing the Doppler effect with a modification of the fertile loading. This would practically mean a reactor optimization with a constraint on the value of the temperature coefficient.

References

1. P. U. Fischer and N. F. Wikner, An interim report on the temperature coefficient of the 40 Mw(e) HTGR, GA-2307; 1961.

2. R. C. Dahlberg, Physics of gas cooled reactors, ANS CONF-720901.

Especially in two — or three-dimensional codes the limitation on the total number of mesh points due to the capacity of the computer and to its speed can be very restrictive. It is then necessary to reduce the total number of mesh points to a minimum.

As it is obvious from the derivation of the difference equations the mesh should be small where the flux is expected to vary rapidly, while it can be larger in regions of flat flux. Numerical difficulties can also be encountered if

Xm+t-Xm xm — Xm-,

is either too large or too small (where X represents any of the chosen space coordinates). This means that the mesh size has to vary smoothly. Very often the only way of knowing if a mesh is sufficiently fine is to test whether appreciable changes in the results occur when a finer mesh is used. This is not always possible on the full size geometry and has to be tried in simplified cases. There are codes like CRAM(7) where it is possible to give an instruction “double”, in which case after convergence is reached with the given mesh the calculation is continued with a doubled number of mesh points, using as a flux guess values interpolated between the results of the first calculation. This is theoretically very useful, but its use is limited in practice by the fact that eventually doubling the number of mesh points leads to the computer-storage capacity being exceeded.

(a) General definitions

The upper part of the neutron spectrum is particularly important because of the damage induced in the reactor materials by fast neutrons.

The methods we have seen so far are normally used to calculate ф(Е) also in this upper energy range. Monte Carlo calculations have also been used139’ and they show a reasonable agreement with other transport methods. As the damage is a function of the neutron spectrum, it is unfortunately difficult to express it in one quantity.

One usually speaks of fast neutron dose (or fluence)

or if the flux is constant with time

d = фТ = nvT

dimensionally d is expressed as

Sometimes the area is measured in kilobarn (1 barn = КГ24 cm2) so that the dose can be also measured in neutrons per kilobarn. The real problem consists in defining the fast flux ф (see ref. 40).

A simple definition is

ф = [ ф(Е) dE (8.42)

J E0

where usually E0= 1 MeV, but also lower limits (e. g. 0.18 MeV) have been used. Another definition is related to threshold detectors which are often used to measure the fast flux.

If o-(E) is the detector cross-section, one defines a fission flux ф1,

where o-Ni = 107 mb, calculated according to eqn. (8.44).

(b) Graphite damage

Of primary importance in HTRs is the graphite damage. This can be related to the number of displaced atoms per unit volume.

The carbon displacement rate Cd is given by139’401

Cd=§=f <t>(E)crs(E)p(E)dE

where x = number of displaced atoms per unit volume, o-s(E) = carbon scattering cross-section,

p(E) = number of carbon atom displacements caused by a collision with a neutron of incident energy E.

p(E) is known,<40’41) so that expression (8.46) can be easily calculated. An equivalent graphite fission flux can be defined as

x(E) dE

As many irradiations have been performed in the DIDO reactor at Harwell, this has been used as a standard to define the Equivalent Dido Nickel Flux (EDNF)

Calculated results give1′

= 1260 x КГ24.

V ФNi/ DIDO

As the number of displaced carbon atoms per unit volume is (at a given temperature) related to changes in electrical resistivity, the relative calibration of various irradiation facilities is usually based on measurements of resistivity changes in standardized reference graphite samples.<43)

Although the equations written up to now are valid in general for most reactor types, the approximations made for their solution should take into account the characteristics of HTRs. As has already been mentioned, more than one fuel element type will have to be considered in order to represent the different burn-up stages. Space-dependent heat-transfer calculations are necessary, while zero-dimensional neutron kinetics calculations are, in many cases, sufficient. Besides, the coefficients appearing in the kinetics equation in one or more energy groups will have to be calculated by codes able to treat properly the neutron spectrum of HTRs (see Chapter 8). The approximation often used in the past, which consisted in performing one thermal group calculations, representing all epithermal events by a resonance escape probability, is not valid for HTRs (more than 10% of all fissions occur above thermal energy).

In comparison to reactors having metal of oxide fuel, HTRs have a very high thermal capacity. This is due to the very high specific heat of graphite, and to the fact that a great amount of graphite is mixed with the fuel. The high thermal capacity gives more time to the control system to intervene. On the other hand, if the temperature coefficient is negative, a higher thermal capacity will eventually cause higher tempera­tures if no scram occurs.

Figure 12.2 shows the time dependence of the total power Q as well as the maximum fuel and gas outlet temperatures for a 3% Ak step. The full line shows a calculation performed with the normal heat capacity of an HTR. In the case of the broken line the heat capacity has been reduced by a factor of 4. In reactors with a very low thermal capacity the maximum temperatures occur so soon that the control system has no possibility of limiting them. By the time the scram comes the temperature coefficient has already stabilized the reactor to a lower power.

In HTRs the temperature increase is relatively slow. As a consequence the temperature coefficient feedback comes also relatively late, so that higher power level, and eventually higher temperatures can be reached, if a scram does not occur.

3.1. Potential scattering and resonance reactions

All reactions between neutrons and atomic nuclei can be divided into two main categories.

1. Potential scattering. The neutron is deflected by the potential field of the nucleus. If the neutron wavelength is much greater than the radius of the nucleus, the potential scattering cross-section is approximately energy independent.

2. Resonance reactions. A compound nucleus is formed. This compound nucleus is in an excited state the energy of which consists of the binding energy between nucleus and neutron and of the neutron kinetic energy. In heavy nuclides there is a high density of possible excited states. Compound nuclei are easily formed when the sum of binding and collision energy corresponds to quantum states of the compound nucleus so that the cross-section presents sharp peaks around these energies (resonances). The compound nucleus decays in different ways (“channels”): neutron emission (scatter­ing), у emission (capture), fission, etc., each mode having a different probability of occurrence.

In the case of resonance scattering we can distinguish the case in which after neutron emission the nucleus returns to normal unexcited state (elastic scattering) and the case in which part of the neutron energy is retained as excitation energy of the nucleus (inelastic scattering). In light nuclides excited states have too high an energy level to be of interest in nuclear reactors, while in heavy nuclides these energies are much lower. If A; is the decay constant of the compound nucleus along the ith channel, the probability of decay along this channel is

where Г, = ЙА, partial width,

Г = X Гі total level width.

These widths correspond to the energy uncertainty of each compound state. We have then Г„, neutron width (probability of neutron emission), I, radiation width (capture with у radiation emission), etc.

Let us suppose an infinite homogeneous mixture of moderator and resonance absorber. Above the resonances the flux will be ф(Е) = ЦЕ (see § 6.1). The reaction rate within a resonance is

/ = J (Та(Е)ф(Е) dE resonance integral,

a resonance integral I, can be defined for each resonance i.

In the resonance region, because of the neutron absorption the flux will not in general follow the ЦЕ behaviour, except if the concentration of resonance absorber is

negligible (infinite dilution). In that case we have

infinite dilution resonance integral

where и is the lethargy.

For multi-group calculations the cross-section for group g is given by

If the flux over the group interval can still be considered to have 1 IE behaviour, considering that

f <t>(E)dE= f Щ=А щ

J A Eg J AEK LL

eqn. (7.1) becomes

If more resonances are included in the energy range of group g their contributions to Is have to be added.

The flux in the resonance region in an homogeneous infinite mixture of moderator and resonance absorber with a homogeneously distributed source of fission neutrons of energy well above the resonance range is given by the slowing-down eqn. (6.7) which in this case takes the formt

where 1 indicates moderator and 0 absorber.

For numerical reasons this equation is usually rewritten in terms of the collision

de"Si, V F(E)-*(£)*,№) and instead of E the lethargy is used as a variable.

It is always possible to solve numerically this equation obtaining the resonance integral and effective group cross sections as in eqn. (7.1) assuming а ЦЕ flux above the resonance region.

Both 238U and 232Th can be used as fertile materials for high temperature reactors. In case of 238U the ratio of fissile to fertile material in the fresh fuel corresponds to an enrichment of 5 to 8%. In the case of 232Th this fertile material has to be mixed with highly enriched (usually 93%) 235U. The Th cycle has thus the disadvantage of needing highly enriched uranium. On the other hand, the 233U produced in this cycle is a very good nuclear fuel with a very high tj value (number of fission neutrons produced per neutron captured in the fissile material).

Typical tj values, averaged over an HTR spectrum are:

233U tj = 2.29 235U tj = 2.05 239Pu tj = 1.80

These are only indicative values and can change from design to design according to the core composition. The high tj value of 233U allows a higher conversion factor and a value higher than 1 is theoretically possible. This means that it would be actually possible to breed fuel in an HTR. To achieve this it would be necessary to limit very greatly the fuel burn-up in order to reduce neutron losses in fission products, and to use at least in part Be as moderator in order to take advantage of its good moderating properties and of its n, 2n reactions. This turns out to be too expensive in practice so that breeding would be economically nonsensical in HTRs, nevertheless the Th cycle retains the advantage of a higher conversion factor and hence a better use of the natural fuel resources. In general it is possible to state that from the point of view of neutron economy the Th cycle is best suited to exploit the spectral characteristics of HTRs, but also other factors (enrichment, reprocessing, refabrication, existing industrial equip­ment, symbiosis with the fuel cycle of other reactor types) have an important weight in the choice of the fuel cycle.

As one can see from Tables 9.1 and 9.2 the breed fuel 233U or 239Pu is produced by the /З-decay of 233Pa or of 239Np. This means not only a certain delay in the production of the fissile isotopes, but also losses of neutrons and of fissile material because of parasitic absorption in 233Pa or 239Np. This fact is not very important in the case of 239Np because of its short half-life (2.33 d), while it is much more important in the case of 233Pa (half-life 27.4 d).

The losses in 233Pa are proportional to the thermal flux level, i. e. to the fuel rating (power per unit weight of fissile material). The losses due to decay of M1Pu in 241 Am are negligible during reactor operation because of the long half-life of “‘Pu (13.2 y), but can become important if plutonium fuel is stored for several years.