Let us set l = 1, and neglect the higher-order terms. Considering (4.12) and (4.13) we have then the expression:

ф(г, Е,П) = ^ф(г, Е) + ^Ш(г, Е) (4.19)

and for the scattering cross-sections

2S (£’->£, ft’ft) = SoiE’^E)+sdE’^E)p0

= -^lso(E’^E) + -^Zsl(E’^E)p0 (4.20)

where ju0 = ft ■ ft’.

The time-independent Boltzmann equation with isotropic sources is [see eqn. (4.4)]

ПЧф(г, E, ft) + 2,,ф(г, E, ft) = J Xs(E’ ->Е, П’-> И)ф(г, E’, ft’) dE’ dft’

+ _LxjE) Г jf(£^(r)£.i0.)y(£.)(j£. dft’ (4.21)

Keff 47Г J

+ jLs(r. E).

Substituting (4.19) and (4.20) in (4.21) and integrating over ft, once directly and once after multiplication by ft, we have the set of two P, equations

div J(r, E) + 2, (Е)ф(г, Е)= [ Xs0(E’ -»Е)ф(г, E’) dE’

Jo

+ -£-x(E)( V(E’)ME’№(r, E’)dE’+S(r, E), (4.22)

Keff JO

5grad ф(г, E) + 2,(E)J(r, E) = [ XS,(E’ ->E)J(r, E’) dE’.

Jo

A multi-group formulation of the Pi equations can be obtained starting from the multi-group form of the transport equation (4.6) with the group constants defined by

(4.7) and (4.8).

It is also possible to obtain the multi-group P, equations integrating directly eqn. (4.22) over the energy range of each group.

In this case it is not necessary to assume the separability

ф(г, E, ft) = f(r, Sl)(pi(E) within each group і

and it can be easily seen that some cross-sections (e. g. 2s0) will be weighted on the flux and some on the current (e. g. 2.,i). Furthermore, the 2,(1?) appearing in the first eqn. (4.22) will be weighted by the flux

while the 2,(i?) appearing in the second equation will be weighted by the current

In practice these equations are always used in codes with a very high number of groups so that this difference is unimportant.

A major problem in spectrum calculations is the treatment of the neutron slowing down in an heterogeneous reactor system in presence of leakage and anisotropic scattering.

The only rigorous treatment possible is when the flux distribution is of the form (see §4.10 and ref. 1, p. 436)

ф(г, Е,П)= Ф(в, E, £i)e, BZ

where В is the square root of the fundamental buckling. This implies that the space-dependence of the flux is of the type e’BZ for all energies. In the case the B„ approximation can be adopted using the Fourier transform to reduce the spatial dependence to a single buckling constant.

Because of the limited anisotropy of graphite scattering, the Bt approximation is sufficient for HTRs.

As pointed out in the paragraph on the B„ method, the components of the flux can be calculated exactly with this method, the only assumption being the truncation in the expansion of the scattering cross-section. No inverse Fourier transformation is needed as, in the approximation (8.1), the energy dependence of the angular flux is identical to the one of its Fourier transform.

In practice the validity of the B„ approach is limited because the single mode describes an asymptotic solution valid only if the system is homogeneous and large enough for such a distribution to be fully established. This is certainly not true for zones lying on the boundary between the reflector and an undermoderated core. In this case it is better to use the multigroup Pi approximation which allows the use of a group-dependent leakage, while this is not possible in the B„ approximation.

In the GAM-I code, for example,<2) the Pt eqns. (4.22) are integrated over a region of interest obtaining

J(E) + 1,(Е)ф(Е) = j Z,„(E’ -* Е)ф(Е’) dE’ + S(E), 2ф(Е) + 32,(E)J(E) = 3 j 2„(E’ -» E)J(E’) dE’,

where

Г V2cf> dV

EE = ————

)<t>dV

in a multi-group formulation we will then have group dependent leakage factors EE і for each group i. These factors may be obtained from few — or multi-group space-dependent calculations. It is then necessary to iterate between the multi-group spectrum calcula­tion and the space-dependent calculation in order to obtain the leakage factors EEt. An initial guess for the EE. factors is given by —B2, the buckling. If the number of groups used in the space-dependent calculation is sufficiently high, this process converges very rapidly because the multi-group constants are only weakly independent on the EE factors. In small reactors where the leakage has a strong influence on the neutron spectrum the few group cross-sections can depend greatly on the leakage and this iteration may present problems. In that case the number of groups of the space — dependent calculation has to be increased, thus reducing the leakage sensitivity of the cross-sections. Having performed the spectrum calculation one has to produce average constants for few group reactor calculations. These constants are then averaged according to eqns. (4.7) and (4.8).

where ф(Е) is obtained from (8.2). Transfer coefficients

—і———- 1———— !————- 1-

Е,-1 E, Ei-i Ei

For use in a space-dependent code it is usually necessary to form a macroscopic So- including elastic scattering, inelastic scattering and n, 2n reactors,

= 2 NkO-цсі + 2 inei + 2 2 МкО-ц(„,2n) (8.5)

к к к

where Nk is the atomic concentration of isotope к and the summation extends over all isotopes к present in the reactor.

Diffusion coefficient

In order to perform few group diffusion calculations it is necessary to calculate group diffusion coefficients using Fick’s law

HE)

grad ф(Е)

Other versions of the GAM code have been restricted to the B„ method, with all the disadvantages connected with the use of energy-independent bucklings.

The GAM code is only used to calculate the fast spectrum, while the thermal spectrum is calculated separately in the GATHER code.

The sources for the two energy ranges are fission and slowing down respectively and are treated as external sources, so that the problem of criticality is not posed and ксЯ is not calculated. The two codes have then been coupled in the various GGC versions.’2"6’

In the resonance range the 2, of eqn. (8.2) must be obtained from expressions of the type (7.1). See Chapter 7 for the description of the methods used in the various codes.

12.1. General considerations about reactor dynamics

Reactor dynamics is concerned with the analysis of the time-dependent behaviour of the reactor in normal operation and during accidents. Also the interpretation of most reactor physics experiments involving reactivity changes requires an analysis of core dynamics.

The kineticst of a nuclear reactor can be described by a set of equations connecting the variables describing the state of the system. The most important variables are the neutron flux and the temperatures of the various reactor components, together with the parameters which can influence these quantities (poison concentrations, cross-sections, coolant mass flow, etc.) and the data defining the causes of the transients. Properly called kinetics equation is the equation connecting the neutron flux to the other variables. As its coefficients are usually temperature-dependent, this equation cannot be solved without considering the reactor heat transfer equations, which in turn can involve the complete power plant system, down to the turbine. Simplifying assumptions will have to be made for each practical case. The feedback of the control system appears in the coefficients of the kinetics equation. Its detailed description would involve treating the control system with its electrical and mechanical parts, and here again simplifications are necessary.

The phenomenon which is in most cases characterizing the reactor dynamic behaviour, is the presence of the delayed neutrons. The /3-decay of a fission product leads in some cases to a highly excited state of the daughter product which can then emit a neutron. Those fission products are commonly referred to as delayed neutron precursors. The neutron emission follows almost immediately (~ КГ14 sec) the /3- radiation, and the delay of the neutron is determined by the /3-decay constants of the parent, which ranges from milliseconds to minutes. The delayed neutrons are usually grouped according to the /3-decay constant of the parent nuclides (the number of groups is often six).

The time variation of the reactor state is the result of various phenomena: fuel burn-up, fission product build-up and decay, temperature variations, reactivity changes due to movement of absorber rods or other geometrical and material changes within the reactor. Each of these phenomena is characterized by a different time constant. The results of reactivity changes are usually rapid transients whose time constant is determined by the lifetime of the prompt and delayed neutrons.

tThe term “kinetics" is generally used to indicate the time-dependence of the neutron population, while reactor "dynamics" includes also beside kinetics, the study of temperature and control feedbacks.

Temperature feedbacks have time constants determined by the heat capacity and conductivity of fuel and moderator. Fuel burn-up and fission product build-up and decay are usually characterized by very long time constants. Most short-lived fission products do not need to be treated explicitly, with the exception of 135Xe because of its enormously high thermal absorption cross-section. The stable nuclide 149Sm gives also rise to transients with short time constant because it has a high cross-section and a short-lived parent. For Xe and Sm transients the time constants vary between 9 and 48 hours.

Each of these phenomena may be mathematically expressed by a set of differential equations. As a general rule it is possible to treat independently transients with widely differing time constants. This means that in solving, for example, the differential equations representing the fuel burn-up and fission product build-up it is possible to assume that Хе, I, Pm, Sm, temperatures and delayed neutrons are in equilibrium condition and the time derivative of these quantities is zero. Thus we can distinguish three independent types of transients: burn-up, Xe and Sm, temperature and delayed neutron transients. The first type of transient does not belong to reactor dynamics and has been treated in Chapter 9. Xenon-135 and 149Sm transients are of the order of some hours and are particularly important for reactor shut-down, start-up, power following, and spatial Xe instability. For these phenomena (long time dynamics) the time derivatives of the neutron flux, delayed neutron precursor concentrations and tempera­tures can be neglected (an exception can be the cases in which the heat capacity of some core component is so big that its time constant approaches the order of magnitude of the 135Xe decay constant).

The other type of transients (short time dynamics) whose time constants vary from fractions of seconds to a few minutes, can be treated assuming constant Xe and Sm concentrations since they hardly change in the short time of the transient. They include all reactivity accidents, temperature stability analysis, and rapid power excursions.

Ordinary numerical methods can attain a high degree of accuracy, but especially for two- or three-dimensional cases, the time of computation required can be very high.

This problem becomes very important in cases where the diffusion calculations have to be repeated many times in the course of a time-dependent analysis (burn-up or kinetics codes).

Various time-saving methods have been developed especially for use in the diffusion theory routines of kinetics and burn-up codes. A technique sometimes used in these cases is the so-called “nodal method” in which the reactor is divided in a certain number of sub-regions or “nodes” for which average power or flux is calculated. This method can be compared with a finite difference method with very coarse meshes. The essential problem consists in obtaining coupling constants between the nodes, maintain­ing neutron balance within each node and between nodes. Analytical solutions of the diffusion equations with an appropriate choice of boundary conditions are sometimes employed in order to get the coupling between nodes/22’30’ A method using finite difference in coarse meshes with corrections taking into account the form of the within-mesh flux distribution is used in the UKAEA CRISP and APEX codes/31’32’ Sometimes the so-called “flux synthesis” methods are used in which approximate three-dimensional flux shapes are obtained combining the results of one — and two­dimensional calculations. The simplest way is to assume separability of the axial flux dependence. This assumption is sometimes made for obtaining rough approximations. A flux synthesis method based on a variational principle is described in ref. 33. A single-channel flux synthesis method is used by the General Atomic SC ANAL code135’ in which two to four two-dimensional GAUGE calculations, corresponding to horizontal slices through several axial elevations in the core, are coupled by means of a one-dimensional axial calculation. Each slice is homogenized and used as an axial zone in the one-dimensional FEVER code. In this code it is not actually the flux but the power distribution which is synthetized. This approach has been found to be more accurate than calculating the power distribution from synthetized fluxes.

9.1. The depletion equations and the methods to solve them

One of the most important problems in reactor design is the calculation of the time dependence of the reactor behaviour. If the time involved is short (from fractions of seconds to hours) one has to consider the time-dependent form of the Boltzmann equation: these problems will be dealt with in Chapter 12. The burn-up calculations deal with the time evolution of reactor parameters over long periods involving the complete lifetime of the reactor.

For these calculations the time derivative of the neutron flux can be neglected and the static form of the Boltzmann equation (or of the diffusion equation) can be used.

For each isotope it is possible to write a balance equation relating the loss and production contributions to its concentration (depletion equations)

ИМ m q p

~ = Ni(rflyik + ф 2 Nscrasysk + 2 N, Aja№ — AkNk — фЫкСтак (9.1)

(It і = l s = r І =n

where Nk = atomic concentration of isotope k, ф = flux,

CTp = isotope і fission cross-section,

(Tai = isotope і absorption cross-section,

Ai = isotope і decay constant,

ylk = yield of isotope к due to a fission in isotope i,

ysk = probability that a neutron absorption in isotope s produces isotope k, alk = probability that the decay of isotope j produces isotope k.

The first summation (index i) extends to all fissionable isotopes; the second summation (index s) extends to those isotopes who can produce isotope к after a neutron absorption; the third summation (index j) extends to those isotopes whose decay product can be isotope k.

All cross-sections are here one-group values, averaged over the whole energy spectrum.

This equation applies to all isotopes present in the fresh fuel or produced by fission and those which originate from neutron absorption or decay of other nuclides. Since the nuclei of the primary fission products generally have a considerable excess of neutrons, they are unstable and decay often via complicated chains.

The number of isotopes to be treated in a burn-up calculation is of the order of 100.

They include the heavy nuclides, the fission products and the isotopes derived from them by neutron absorption and decay.

The types of heavy nuclides present in the reactor fuel depend on the fuel cycle which has been chosen.

In all power reactors the fission process produces more neutrons than the number strictly necessary to sustain the chain reaction. These excess neutrons are used to convert fertile isotopes into fissile ones. Both 238U and 232Th can be used as fertile materials for high temperature reactors.

The ratio

_ No. of atoms fissile material produced No. of atoms of burnt fissile material

is called conversion ratio. The most important reactions involving fertile and fissile material are shown in Tables 9.1 and 9.2 for the 232Th and the 238U chains respectively.

Table 9.1. Thorium Cycle

Th

Table 9.2. Low Enriched U Cycle

In these tables fission is only shown for those isotopes which are fissionable by thermal neutrons. Fast neutron-induced fissions can occur also in other isotopes like 232Th and 238U and these processes are easily taken into account in the calculations, where all heavy metals are usually treated as fissile isotopes. The most important fission product chains are given in Tables 9.3 to 9.6. In these tables a vertical line means neutron absorption while a horizontal line means decay.

The fission yields yik defined as the probability that a fission in isotope і produces

— l54Gd

♦

-l55Gd

l56Gd

І

l57Gd

l58Gd

isotope к are obtained experimentally and stored in the nuclear libraries of the codes for burn-up calculations. Each fission produces two fragments (ternary fissions exist but are negligible) so that

? — 2-

Fission yield data can be found in refs. 1 and 2. Not all fission products need to be treated in a burn-up calculation, but only those having either a non-negligible absorp­tion cross-section, or decaying into isotopes with non-negligible absorption cross­section. For this second case the parent nucli, de needs only to be treated if its decay

time is sufficiently long. Otherwise it is sufficient to treat the decay product, attributing to it its cumulative yield (which is the sum of the direct yield of the nuclide considered and of the yields of the isotopes of the decay chain leading to it). As a typical example Figs. 9.1 and 9.2 give the concentration of the fission products in an HTR as a function of time.

In many cases the system of differential equations (9.1) has been simplified by neglecting the interconnections between some of the different equations for the fission products, i. e. neglecting some links of the chains of Tables 9.3 to 9.6. This can be reasonable for low burn-up, but leads to intolerable errors if the burn-up is very high.® When the chains are neglected it is not necessary to treat independently all fission products, but they can be grouped according to their absorption cross-section in a certain number of pseudo fission products.14’ The yield of a pseudo-element is the sum of the yields of all the fission products which are represented by it, and its absorption cross-section is an average of the single cross-sections weighted over the yields

Fig. 9.1. Atomic density of fission products. Fig. 9.2. Atomic density of fission products.

In this way the pseudo elements should give the same overall absorption as the real elements. It is unfortunately impossible to reproduce in this simple form the proper time behaviour of this absorption. In high temperature reactors the burn-up is usually very high so that these simplifications lead to intolerable errors, and it is essential to treat the system of eqns. (9.1) in its full form for all important chains grouping together only the less important low absorption isotopes in the so called “non-saturating fission product aggregates”.

The importance of neutron absorptions in fission products is shown in Fig. 9.3<5> where the average number of neutrons lost in fission products per neutron absorbed in fuel is plotted as a function of burn-up measured in fissions per initial fissile atom (fifa).

The effect of neglecting the treatment of fission product chains on reactivity is shown in Fig. 9.4 (Ak is the difference between calculated without chains and a reference calculation with all chains).

If the coefficients of eqn. (9.1) are constant, analytical solutions are possible. Unfortunately this is not true in many cases. The cross-sections appearing in (9.1) are averaged over the whole energy range, so that they are very sensitive to any spectral change in the reactor. Furthermore, the flux level in each zone may change during burn-up.

If, in a reactor, the power is kept constant, and the fuel reloading is not continuous, between two reloadings the fissile concentration decreases and the thermal neutron flux must increase in order to maintain a constant fission rate. The fast flux remains constant

because the power and consequently the fission neutron source is unchanged. This means that the neutron spectrum changes with time consequentially changing the average absorption and fission cross-sections. Furthermore, changes in composition also modify the self-shieldings of the fuel elements and this is another cause of variation of the cross-sections.

Because of these reasons analytical solutions cannot always be used for eqn. (9.1). On the other hand, numerical solutions can be relatively simply obtained subdividing the time variable in a discrete number of time steps. A combination of analytical and numerical methods is often used so that within a time step the coefficients of eqn. (9.1) are supposed to be constant and analytical solutions are used.<6) These calculations are very fast on modern computers.

In recent years a significant effort has been invested in the development of computer programmes for space-dependent reactor dynamics.

These codes are all based on the time-dependent multi-group diffusion equation. — дфЛг’П = а Ч2ф<(г, t) — 2«ф, + І £, (г, о + (1 — jS )*1 2*фк(г, t)

Vi dt /Г=і *Г|

+ 2х«С,(г. П [і = 1————— л]. (12.34)

. J = l

dCii/t~ = £ P, vk2^Hr, t)-,Q(r, t) [j = 1,… m]

where the symbols are the same as in eqns. (12.1) and (4.40), with Xi = fraction of fission neutrons emitted in energy group i,

Хіі = fraction of delayed neutrons of group j emitted in energy group і (often assumed to be the same for all j), n = total number of energy groups, m = total number of delayed neutron groups.

Usually two energy groups are sufficient for HTR dynamics calculations.

Numerical solutions are obtained subdividing the time and space coordinates in an appropriate mesh.

In a representation of this type ke„ does not appear, so that it is not possible to introduct temperature and control feedbacks in the way used in eqn. (12.7). In this case all the parameters of the diffusion equation have to be given as a function of temperature. This is usually done by means of a polynomial fitting. Control rods can be simulated as a diffused poison added to certain regions. The cause of the transients is represented by changes in the absorption cross-section, or of any other parameter of the diffusion equation, over given regions.

In the case of slow transients, the equations for Xe and Sm must be taken into account beside eqns. (12.34), but the delayed neutrons, as well as the time derivative of the flux can be neglected. In practice, however, some codes (e. g. COSTANZA*2 8 9I) calculate the time derivative of the flux even if this term must vanish. The code iterates on the absorption cross-section of given regions (simulation of control-rod insertion) until this derivative vanishes, and this iteration substitutes the outer iterations of conventional diffusion codes. These methods are used for one or more dimensional codes.

Various methods have been considered in order to improve the efficiency of this finite difference approach."01 A classical mathematical approach is given by the modal expansion in which the flux is expanded in the normal modes of the system

Ф(г, П = ^фІ(ПМг). (12.35)

The f(r) are the eigenfunctions of the time-independent problems corresponding to the eigenvalues ki. While only the eigenfunction corresponding to the highest eigenvalue k0 is interesting for the static case, higher modes become important during the transients. If we suppose that the flux shape remains constant during the transient, we have

ф(г,1) = ф(ПМг) (12.36)

which corresponds to the point kinetics.

An improved approximation is given by the so-called adiabatic approximation (this

name should not generate confusion with the meaning it has in the case of ther­modynamic problems). In this approximation f0(r) of eqn. (12.36) is changed with time, but it is supposed to be at every instant coincident with the static flux shape corresponding to the reactor conditions of that instant. In this way the changes in flux distribution during the transient (e. g. control-rod movement, composition changes, etc.) can be taken into account, but only the fundamental mode is considered, the higher modes being neglected.

In the quasi-static method the spatial shape function f0(r) of eqn. (12.36) is also changed with time, and can be obtained solving the full space-time equation using very large time steps. In other words, one can say that both the adiabatic and the quasi-static methods use an assumption of the type (12.10),

ф(г, t) = f0(r, t)4>(t) (12.37)

where /o(r, t) has only a weak time dependence. In the adiabatic method f0(r, t) is calculated using static methods, while in the quasi-static method /0(r, t) is calculated solving numerically the space-time reactor equations using very large time steps. Another promising method is the nodal method in which the reactor is partitioned in a number of sub-regions (nodes) for which average power or flux is calculated. The method can be compared to a finite difference method with very coarse meshes. The essential problem consists in obtaining coupling constants between the nodes, maintain­ing neutron balance within each node and between nodes.

If the resonances are well separated the cross-section for a reaction x is given by the Breit-Wigner single level formula

(Е-Еа)2+ЇГ2

J = spin of target nucleus,

I = spin of compound nucleus,

E0 = resonance energy,

К = reduced de Broglie neutron wavelength,

E = energy of incoming neutron, or relative energy between neutron and target nucleus if this is not at rest (see § 3.4).

As one can see from (3.1) the cross-section has a maximum for E = E0 and reaches half of this maximum value for E — E0 = Г12, so that Г is the width at half height of the resonance (Fig. 3.1). The Breit-Wigner formula cannot be extrapolated to E-» 0 because the parameter Г„ can be considered independent of energy only in small intervals around the resonance. At very low energy

r„ = t„0Ve

This formula substituted in the Breit-Wigner expression gives the l/v behaviour of the low-energy cross-section of most isotopes. This 1 lv part can often be calculated from the known resonance parameters, but for some isotopes (e. g. 232Th) negative energy resonances (bound levels) have to be considered in order to reproduce the experimental 1 lv cross-sections.

As we have seen the scattering cross-section is composed of a fairly constant potential scattering and a strongly energy-dependent resonance scattering. In the vicinity of resonances these two terms interfere so that the scattering cross-section can be smaller than the potential part alone. In general we have

0’s O’ s, pot "l — O’ s, res 3“ O’ s, int

where ors, inl is the interference part of the scattering cross-section. Considering as a reference frame a coordinate system centred on the centre of mass of the two particles (nucleus and neutron) involved in the collision and defining p0 the cosine of the scattering angle in this reference system, we can expand the scattering cross-section in Legendre polynomials

сгЛр-о) = 2 СГ, Р,(ро), (3.3)

/=0

/ = 0 gives the isotropic component (s-wave scattering) while the component / = 1 is called p-wave scattering.

The /th partial wave of the scattering cross-section is given by:

і—— resonance———— 1 і———— interference———- 1 |—————— —potential—— 1

= —p f ІГ2 [Г„2 — 2Äà sin2 S, + 2Г(2і — E0) sin 2&] + 4ttX2(21 + 1) sin2 Si

{Л — Ло) +4І

(3.4)

with Si = phase shift associated with the potential scattering.

If the resonance width is small compared to the maximum energy loss per collision with an absorber atom, one can assume that any collision, even with heavy absorber atoms, slows down the neutron outside the resonance. It is then possible to replace ф(Е’) on the right-hand side of (7.3) by its asymptotic ЦЕ’ solution. This equation can

tThe reader is reminded that in the past some authors (e. g. Wigner-Weinberg, Nordheim) used a different definition of a, so that instead of a in every formula appears 1-а.

then be easily integrated if we assume 2,о and 2,i to be independent of energy. This is generally true for 2,i while 2,о consists of a constant part Spot and a resonance part. In the NR approximation we can set 2,о = Spo. in (7.3) because the collisions in the resonance range do not contribute to the resonance flux. Considering that

1 fE/° 1 dE’ 2 1-а JE Е’г E’ E

<ME)S,(E) = —

і / n 2pot+ 2s,

Ф(Е)‘~ЁШ)

(7.5)

In order to make full use of the advantages of 233U it is necessary to extract the fissile material still present in the spent fuel and to reinsert it into the fresh fuel elements. This reprocessing operation implies of course further costs and complications.

A small part of the fuel will be lost during reprocessing and a reprocessing efficiency has to be taken into account in the fuel-cycle calculations. In the case of low enriched uranium cycle reprocessing is less interesting because the cross-sections of Pu being higher than the one of U, most of the bred Pu is burnt before the fuel element is discharged from the reactor. It must be noted, however, that because of increasing restrictions in disposing of irradiated fuel it might be necessary to reprocess also low enriched fuel. During reprocessing the uranium (or plutonium) present in the spent fuel is chemically separated from the fission products and other materials. It is not possible by these means to separate the fissile from the non-fissile isotopes of the same element. Continuing in this way to recycle the fuel one gets an accumulation of the parasitic absorbers like 236U and 237Np, as one can see from the chain of Table 9.1. This fact is very important in the case of the Th cycle because of the continuous introduction of 235U in the chain.

The problem is not important in the case of low enriched uranium cycle because, even in the case of reprocessing, only Pu would be reused and the parasitic absorber 242Pu, being the result of a long chain, has always a low concentration. In order to avoid this problem it is possible, in the Th cycle, to discharge selectively from the system those portions of the fuel that have experienced the largest burn-up. In order to achieve this it is necessary to separate physically the fertile from the fissile part of the fuel. This

can be done using different fertile and fissile particles (e. g. with different coating) so that it is possible to separate the bred 233U from the residual feed fuel. In particular the method chosen by General Atomic is to use a TRISO SiC coating on the fissile particles (containing the feed highly enriched uranium) using BISO coating for the fertile particles (containing 233U and Th). Another separation method would require the use of different diameters for the two types of coated particles.

The use of different elements containing the feed and the breed fuel has also been considered, in which case it might be possible to have a higher residence time in the reactor for the breed elements. This is very difficult in prismatic fuel because of resulting distortion in power distribution, but it has been seriously considered for pebble bed reactors.

If we arbitrarily assume that all neutrons have the same velocity we obtain the energy-independent form of the Boltzmann equation (for simplicity we have here included the fission neutrons in the source term):

The P, approximation consists then in the assumption [see eqns. (4.19) and 4.20)],

(4.24)

(4.25)

where po = ft’ • ft.

Proceeding as in the energy-dependent case we obtain the P, equations

div J(r) + 2гф(г) = 2sod>(r) + S(r),

I grad ф{г) + 2,J(r) = 2si J(r)

introducing the average cosine of the scattering angle [see eqn. (4.17)]

(4.27)

and defining the transport cross-section as

and 2„ = 2, — 2s0 we obtain

div J(r) + Хаф(г) = S(r), з grad ф (r) + J(r)(2a + 2,r) = 0. Defining the diffusion coefficient

D = 3(2,r + 2a) the second eqn. (4.29) takes the form

J(r) = — D grad ф(г),

eqn. (4.31) is called Fick’s law of diffusion.

Substituting (4.31) into the first eqn. (4.29) we have

DV2d>(r)- Хаф(г) + S(r) = 0

which is the neutron diffusion equation.

This equation which is a simplified form of the Boltzmann equation, but can also be directly derived, represents a neutron balance between

DV2d>(r) losses due to leakage,

-Хаф(г) losses due to absorptions,

S(r) source (external or fission source).

Defining the diffusion length

(4.33)

^Ф-1?Ф + о = 0-

The energy-independent diffusion equation is also valid if ф(г, E) is separable in space and energy ф(г, E) = ф(г)(р(Е) in which case the diffusion eqn. (4.32) is valid for ф(г) (one group theory).

The diffusion length is related to the average-distance as the crow flies travelled by a neutron from source to absorption. The solution of eqn. (4.34) in the case of a point source in an infinite homogeneous medium is

ф(г) = А^ (4.35)

where r is the distance from the source and A a coefficient related to the source strength. The number of neutrons absorbed between r and r + dr is 47rr2 drl, uф(r) so that the square of the average r is

[ r2 ■ 4тгг2Хаф(г) dr

4ттг2Хаф(г) dr

Substituting (4.35) in (4.36) and performing the integrations we obtain

r = 6L2, (4.37)

L2 is known as diffusion area.