Until now we have considered spectrum calculations based on B„ or Pi approxima­tion to transport theory. Subject to the assumptions discussed in § 4.10 it is possible to calculate neutron spectra using the multi-group diffusion formulation. This would be rather inaccurate in case of scattering by light nuclides like hydrogen, but it is usually sufficient for graphite. The computer codes based on this method are very fast and often used for HTR calculations.

These codes are based on a multi-group formulation of eqn. (4.63)

(8.7)

This is a homogeneous system of algebraic linear equations and ксЯ is its greatest eigenvalue. First ксЯ has to be calculated and then the can be obtained for all groups i.

If В2 is not specified as a function of energy it is also possible to set ксЯ = 1 and B2 is then the eigenvalue of the problem. The programme can then calculate the smallest eigenvalue. This is equivalent to calculate the dimensions of the critical bare reactor whose material properties are defined by the parameters of eqn. (8.7).

Both approaches are possible in the MUPO code/7’8’ which solves eqns. (8.7) in forty-three groups. This code which is largely used in Europe for HTR calculations has the advantage of being extremely fast in comparison with most other spectrum calculations. Being based on multi-group diffusion theory it cannot be used for light moderators like water, but is sufficiently accurate for graphite or beryllium. Besides in this code only slowing down due to collisions with the moderator is considered, so that inelastic scattering with heavy atoms is neglected. This is also quite an acceptable approximation for most thermal reactors.

The equivalence relation is used in MUPO for resonance calculations.

Reactor kinetics is represented by the time-dependent Boltzmann equation (4.2) associated to a set of equations expressing the balance of the delayed neutron precursors.

For our practical applications a diffusion theory approximation is usually sufficient. We obtain in this way(1)

The symbols are those of the diffusion equation (4.39) with m = number of delayed neutron groups,

Ci = concentration of delayed neutron precursors of group /,

A, = decay constant for delayed neutron precursors of group i, f3i = fraction of the total number of fission neutrons emitted as delayed neu­trons of group i,

p=’Zpu

і = I

X(E) = fission spectrum of prompt neutrons,

Xi(E) = spectrum of delayed neutrons of group і (often assumed to be the same for all groups),

v = neutron velocity corresponding to energy E.

This equation is suitable for computer calculations after discretizing the space, time and energy coordinates. Energy is discretized in groups, space in mesh points, and time in time steps (see § 12.12).

The problem of the boundary conditions in diffusion theory calculation is rather difficult because diffusion theory is not valid in the vicinity of a boundary.

As has been seen previously these boundaries represent either an external empty space or a “non-diffusion region” defining a control rod. In the first case for a convex body the vacuum is equivalent to a “black” absorber from which no neutron can return to the reactor. In the case of a region representing a control rod it is possible to define its “blackness” as the probability that a neutron entering this region will be absorbed in it (after any number of collisions). The blackness is in general an energy-dependent quantity, most control rods being “black” at thermal energy and “grey” for neutrons of higher energy. In the diffusion calculations the boundary condition is usually given in the form of an extrapolation length

d = —- calculated at the boundary, (5.9)

grad ф

d is in general an energy dependent quantity.

In order to obtain exact boundary conditions one has to perform transport calcula­tions on the region surrounding the boundary where diffusion theory is not valid. The extrapolation length of a control rod is then obtained performing a transport theory calculation over a control-rod cell. This cell is usually defined as the part of the reactor core corresponding to this control rod.

What complicates considerably the problem is that eqn. (5.9) cannot be calculated using the exact fluxes obtained at the boundary with a transport calculation. From the diffusion calculation performed over the complete reactor it is usually required to obtain an accurate overall power distribution and reactivity. The boundary conditions used should satisfy this condition and should not try to reproduce the exact flux distribution in the immediate vicinity of a boundary where diffusion theory is not valid.

If we consider the exact solution of the transport equation we see that it is possible to split it into two parts: an asymptotic neutron distribution valid far from the sources (or from the localized absorbers) and a transient part which is only important near the source (or absorber)

Ф(Г) = 4>As(r) + <(>trans(r);

this transient part vanishes within a few mean free paths from the boundary.

The asymptotic solution of the transport equation is practically coincident with the diffusion theory result in a weakly absorbing medium. As an example we can consider the monoenergetic case of a point source in an infinite homogeneous medium. The result can be extended to absorbers which can be represented by an appropriate distribution of negative point sources.

The diffusion equation takes the form

Ф(г) = S

with S = source strength

The transport theory solution is (see ref. 36, p. 236) ф(г) = <Mr) + фtrans(r) = -^-r є "

with

к = V32,2«(l-£) (1-22./52,),

2′ = 12, r = 1 2«=2,-2,

The asymptotic solution is smaller by a factor у than the diffusion theory solution. This is due to the fact that the diffusion theory flux is too low near the source, and must therefore be higher far from the source, in order to give the same number of absorbed neutrons

2 0(/>(r)dr = S.

J r = 0

In reality for usual cases у ~ 1 because 2„ <^2(.

Applying these considerations to the opposite case of an absorber rod we have the situation shown in Fig. 5.2.

The extrapolation length to be used in diffusion calculations should not be obtained from the exact transport theory solution, but from the asymptotic solution

d =—

grad ф As

In practice if the transport calculation has been performed with numerical methods it is impossible to separate the asymptotic from the transient solution. This is only possible when analytical methods are used. For example, in the analytical solution of the Pi equations in multi-region cylindrical cells the total flux and current are obtained as a sum of Bessel functions of different arguments out of which it is possible to separate the asymptotic terms (see ref. 37).

In practice for the external boundary of the reactor it is possible to use the extrapolated length

0.

71 0.71

2tr 2Д1-Д)

which is obtained from the asymptotic solution of the so-called Milne problem’38’ (vacuum boundary in slab geometry with sources at — °°). In the case of control rods it is possible to use expressions derived by Kusheriuk and McKay relating the blackness with the extrapolated length.’39’40’ These expressions are based on the following considerations. Under the assumption of validity of the diffusion theory on the boundary it is possible to relate the blackness to the extrapolation length. The Pt expansion of the angular flux is [see eqn. (4.19)]

ф(г, ІІ) = ^ф(г) + ^Ш(г)

which in one-dimensional cylindrical geometry takes the form

The neutron current is [see eqn. (4.13)]

J(r)=f ф(г, fl) cos 0 dfl = 27Г f ф(г, fl) cos в sin в d6. (5.12)

J4ir Jo

At a boundary the current can be resolved into two components (remembering that the positive J is in the direction of increasing r, i. e. the outward direction),

Substituting (5.11) in (5.14) and (5.15) we have

J = — + —

•/out ^ ‘ 2>

Ф

4

Defining the blackness /3 as the probability that a neutron entering the rod is absorbed in it

(5.18)

Substituting (5.16) and (5.17) in (5.18) under consideration of Fick’s law (4.31) and of (5.9) we obtain for the extrapolation length the expression

d=ikИ] <5,9)

this expression assumes the validity of diffusion theory on the boundary. Considering the asymptotic solution of the transport equation Kushneriuk and McKay found that the value I is more accurately replaced by a function of the radius R measured in moderator mean free paths. The expression takes then the form

£[£-*<*-4 <5’20)

or in presence of an air gap between rod and moderator

sb ‘*’■)] <5’21)

where Ra is the external radius of the gap, R the rod radius and 2,r refers to the surrounding medium. This formula is very often used to calculate control rods of HTRs. The blackness /3(R) is usually calculated with collision probability codes based on the numerical solution of the integral form of the Boltzmann equation (e. g. the MINOTAUR code, ref. 41).

The problem of calculating the long time reactor behaviour is complicated by the spatial dependence of neutron flux and core composition. Even in a homogeneous reactor the composition becomes space dependent after burn-up because the fuel burns more rapidly in the high flux regions. In general the effect of burn-up is to flatten the power distribution because the zones of high power are more rapidly depleted. Furthermore, spatial variations in flux distribution and composition are produced by reloadings, control-rod movements, etc. Also the space-dependence of temperature, which has very little effect on the feett of static calculations, results in local perturbations of reactions rates which can have a considerable effect on burn-up. This means that one-, two — or even three-dimensional calculations may be necessary in order to describe the reactor burn-up.

Usually the system of eqns. (9.1) is programmed in a programme sub-routine which is part of a more general code performing diffusion calculations (in order to have power and flux distribution), neutron spectrum calculations, cross-section averaging, simula­tion of control rod movement and reloading operations.<7)

Given the space-dependent composition of the fresh reactor (or of the reactor after a given reloading operation) it is first necessary to perform a spectrum calculation in order to obtain the constants for the diffusion and burn-up calculations. These spectrum calculations are usually performed in rather broad regions for which an average composition is calculated.

Then a diffusion calculation is performed on the reactor system in one-, two — or three-space dimensions and flux distributions are obtained. As a diffusion calculation gives unnormalized fluxes these fluxes must be normalized to the required power level. The number of energy groups considered in the diffusion calculation varies according to the problem, in general between two and ten groups are used.

In order to perform the actual depletion calculation the reactor must be divided in a certain number of burn-up regions for each of which eqns. (9.1) are solved. Theoreti­cally this should be done for every mesh point of the diffusion calculation, but in order to reduce the computational effort these burn-up regions include many mesh points. These regions must be chosen in such a way as to have a reasonably flat flux because for each of them only one flux level is being considered in the solution of the burn-up equations. An average one-group flux and average one-group cross-sections are calculated for every burn-up region and eqns. (9.1) are solved for a certain number of time steps. At each time step the level of the neutron flux is usually readjusted in order to keep the total reactor power constant, taking into account the variation in fissile concentration. This flux readjustment is usually performed without always repeating the diffusion calculation, which is only made after a certain number of time steps. This renormalization creates some problem because in reality only the thermal flux level changes with burn-up, while the fast flux does not change if the power density remains constant. In this way one can define three different time steps:

Дtb: for the solution of the burn-up equations and flux renormalization,

Atd: at each of these steps a diffusion calculation is performed,

Ats at each of these time steps a spectrum calculation is performed.

These time steps are multiples of each other

In a similar way the space is discretized in three different levels: the meshes of the diffusion calculation, the burn-up regions, and the zones for spectrum calculations.

This scheme can be varied according to the code used for the burn-up calculations, and not all operations are necessarily performed within the same code, so that in some cases it may be necessary to stop the burn-up code in order to perform spectrum calculations which produce new constants with which to restart the burn-up code.

The latter procedure is advisable for expensive three-dimensional calculations where it is better to check for possible errors before proceeding to new time steps.

Spectral changes during burn-up can in some cases be very important. A rather extreme example is given in Fig. 9.5 for the case of a Pu-fuelled HTR with batch loading/8’

In the case of zero-, one — or two-dimensional calculations in which a transverse leakage is expressed by means of energy-dependent bucklings, the bucklings determined for the fresh core are not valid after burn-up if the neutron spectrum has changed. Because of the return of fast neutrons thermalized in the reflector, the thermal leakage is usually negative. If this is expressed as DB2<£, an increase of thermal flux would artificially increase this inward leakage, while in reality it remains almost constant because the fast flux does not change with burn-up. Although this problem can only be completely avoided by the use of a full space dependent calculation, a way of improving the accuracy consists in keeping constant with burn-up instead of D, B,2, the leakage per source neutron’9’ defined as

D, V 2ф,
j, 2 ^пФі

eff j

In the case of zero-dimensional calculations the diffusion equation instead of the usual form

No lumping of fuel

850 days correspond to a burn up

of 1-125 fissions per initial fissile

atom

Beginning of life

2

takes then the form

N __ ту — N

-2w$i + X *і-‘Фі + 1X ч^пФі = 0-

j=1 Keflf j=1

This problem does not exist in the case of equilibrium calculations of reactors with continuous refuelling because in that case neither flux level nor spectrum change with time. Various types of spectrum calculations can be performed in burn-up codes. In some cases a few group library (e. g. 10-12 groups) is used throughout the calculation and the spectrum calculation simply consists in condensing from this library the one-group constants of eqn. (9.1). This method is used in many General Atomic codes. In other cases (e. g. the WIMS code) more detailed cell calculations are used. The codes developed by the Dragon Project (e. g. BASS) base their spectrum calculation on the forty-three-group MUPO code while other codes (e. g. MAFIA and VSOP) are based on more sophisticated GGC and Thermos spectrum calculations.

Another important item is the criticality search.

As the reactor must be kept critical it is usually necessary after each diffusion calculation to readjust the position of the control rods. This can be done automatically in many codes after having specified in input the sequence of movement of these rods, which are represented through extrapolation lengths or poisoned regions. At specified intervals recharge or reshuffle operations occur, in which the composition of given regions is either substituted by that of the fresh fuel or exchanged with that of another region. In many cases the spent fuel is being reprocessed and the fissile material present in it is being totally or partially reinserted in the fresh fuel. These operations can also be easily treated in the burn-up calculations. If the fuel self-shieldings change appreciably during life it is necessary to recalculate them periodically during the execution of the burn-up calculation. This would mean that from time to time a neutron transport calculation should be performed over the fuel cell. This can be very much time-consuming and often approximations are used. In some codes simplified calcula­tions are made using the collision probability methods. In other cases a set of transport calculations is performed in advance and a fitting is made of the self-shielding as a function of the concentration of the interesting isotope (e. g. self-shielding of 240Pu as a function of its concentration). These fitted curves are then used by the computer code during the burn-up calculation.’91

For fast survey calculations zero dimensional burn-up codes are normally used. Even in zero-dimensional codes it is possible to simulate recharge or reshuffle operations. In this case the composition of a certain number of fuel batches must be stored and burnt independently of each other. The core reactivity is calculated from an average composition resulting from the mixture of all batches, but for all other operations these batches must be treated independently. The diffusion calculation is substituted in these zero-dimensional codes by a neutron balance where the leakage is expressed as DB2<f>. As examples of such codes we quote the HELIOS’101 and the GARGOYLE’1’1 codes.

While at low energies resonances are well separated, they become broader and closer together at higher energies, so that closer together single resonance parameters can no longer be determined experimentally. The lower energy parameters are extrapolated for these higher energies and statistical distributions are used for the width and energy separation of these unresolved resonances.

The NR approximation is not valid for some of the most important low-energy resonances of the fertile materials. In this case it is possible to assume the validity of NR for moderator collisions, while the energy loss due to collisions with the absorber atoms is neglected (infinite mass of the absorber). The first integral in eqn. (7.3) becomes then

lim t—^— І*’"0 <t>(E’)%s0^= Ф(Е)ЇАЕ) (7.6)

while the second integral is treated as in the NR case so that we have

ф(Е)%!{Е) = ф(E)Xso + — g 2,i

2i (E) = Xso(E) + 2,i + Xa(E)

(7.7)

As we have seen in both cases of continuous or discontinuous reloading, after some years of reactor operation an equilibrium condition is reached in which all burn-up stages of the fuel are present at the same time in the reactor. The average reactor composition remains then constant (in the case of continuous refuelling) or has periodical variations (in the case of discontinuous refuelling, in which case the period is given by the refuelling interval). As only fresh fuel is present in the first core, a transition period (running-in) is necessary before equilibrium is reached. This period is economically very important because its fuel-cycle costs, usually higher than in equilibrium, can strongly influence the mean costs averaged over the reactor lifetime. Besides the maximum temperatures and control-rod requirements during running-in can pose limiting values for the reactor layout. The optimum running-in strategy is the one incurring the least economic penalty for operating off the optimum point during the first periods of the reactor life/73 All the constraints about temperature, dose, gradients, etc., are obviously valid for this phase as well as for the equilibrium condition.

For economical reasons it is necessary to limit the fissile inventory of the first core, but a lower limit is given by the power peaking in the fresh elements of the following loadings, which gets too high if the average fission cross-section in the core is low. The value of these power peaks should not be higher than in the equilibrium condition.

An ideal running-in strategy could consist in simulating the equilibrium core in the first loading. This would require a very high number of fuel element types with different fuel loadings, simulating the various burn-up stages of the equilibrium core. Fission products could be simulated by means of poisons (B, Gd, etc.). Normal refuelling operations would then immediately start. This would mean that some fuel would be discharged at a very low irradiation level, with the consequence of a high economic penalty. Besides the fabrication of so many different elements is hardly conceivable.

The opposite extreme would consist in loading entirely the reactor with fresh fuel elements of the type used in equilibrium. In this case the fuel loading and reactivity would be too high. A very high quantity of poison (or a high number of control rods) would be required, and the high fuel loading would also mean a high initial investment.

In pebble-bed reactors it is possible to use for the first loading the same type of fresh fuel elements used for the equilibrium condition.’81 The in-core fissile inventory is limited by mixing these elements with dummy graphite elements. In this case it is not possible to have the same ratio of fertile to fissile material as in equilibrium because the fertile material burns very slowly and in equilibrium is present in a considerable amount in the depleted elements, which are not present in the first core. The missing fertile material and the missing fission products are compensated with burnable poisons concentrated in absorber balls. Considering the passage time through the core, the rate of burn-up of the poisons should be chosen in such a way as not to disturb the axial flux distribution. Besides this poison burn-up should locally compensate the fuel burn-up as nearly as possible. From the beginning of the power operation dummy elements start being discharged and substituted with fuel elements.

The use of dummy elements has been also analysed in the case of reactors with prismatic fuel, but engineering problems are posed by non-power-producing channels and large flux tilts can occur with high local power peaks, so that this solution does not appear to be attractive. The best solution in this case appears to be the use of a certain number of fuel element types (four or more) with different fissile loading (or enrichment) in the first core.’7’ The excess reactivity is controlled by means of burnable poisons.

There is usually an initial non-refuelling period. The loading of equilibrium fresh fuel directly after this period may still give rise to too high power peaks, so that often fuel elements with intermediate fissile loading are needed for this intermediate phase.

The safety characteristics and control-rod requirements of the running-in phase must be carefully analysed. In particular the use of burnable poisons can have a strong effect on the temperature coefficient. Also the load-following capability (Xe overriding) during running-in can considerably differ from the equilibrium case.

In the energy dependent case the diffusion equation cannot be simply obtained from the Pі eqns. (4.22) since the energy dependent equivalent of Fick’s law is

5grad <Mr, J5) + 2,(J5)J(r, J5)= f 2„(E’ -> E)J(r, E’) dE’

in which the flux ф(г, E) does not only depend on the current at energy E, but through the Sji terms is related to the current at all other energies E’ from which scattering into energy E is possible.

This fact can also be easily seen using the multi-group formulation. The multi-group form of the P, equations can be obtained integrating (4.22) on the energy range of each group і (see § 4.7).

div Ji(r) + 2Ііфі(г) = ^ ‘2*о. к~іфк(г)+-г — X‘ 2 п2іифк(г),

к = 1 Keff к = 1

[grad <fr(r) + 2!.,JKr) = X 2si. k~Jk(r).

Here 21 і is weighted by the flux while 2!.,- is weighted by the current. It is convenient to write these equations in matrix form.<9) We then obtain

div J(r) + 2,ф(г) = у- р2іФ(г) + 2s0Ф(г),

Keff

J(r) = — Dgrad ф(г),

where

The second equation of (4.38a) gives a matrix form of Fick’s law. In order to have the ordinary diffusion equation D should be a diagonal matrix.

Off-diagonal terms obviously disappear for isotropic scattering, in which case 2,i = 0. In general we have in analogy to (4.16) and (4.17)

f X.,(E-*E’) dE’

&o{E) = ————————

I 2,о(E-*E’)dE’

so that

J 2,i(E -*E’) dE’ = jio(E) I Xs0(E->E’) dE’ = Д„(Е)2,(Е).

Inelastic scattering tends to be isotropic because it involves compound nucleus formation and one can assume that the neutron emitted has forgotten the direction of the incident neutron. In the case of elastic scattering its degree of anisotropy depends only on the mass of the scattering nucleus (see §6.1).

COS So — fJLо — — . .

ЗА

Without the need of going as far as assuming isotropic scattering, the energy dependent diffusion equation can be obtained supposing that

f X„(E’ -* E)J(r, E’) dE’ = f X„(E -» E’)J(r, E) dE’

in that case the term

can be written in the form

J(r, E) I Z„(E->E’) dE’ = J(r, E)Zs(E)jlo(E)

and Fick’s law in its usual form can be obtained

J(r, E) = — D(E) grad ф(г, E)

with

1

3[2.(E) + 2«,(E)]’

2«,(Е) = 2.(Я)(1-До).

The energy-dependent diffusion equation takes then the form — D(E)VV(r, E) + 2,(E)0(r, E)

= 12s0(Е’->Е)ф(г, Е’)<ІЕ’ + у-х(Е) J v(E’)Sf(E^(r, E’)dE’ + S(r, E).

(4.39)

In the same way for the multi-group formulation it is possible to diagonalize the matrix D thus obtaining the ordinary multi-group diffusion equations

— DiV20,(r) + 2«0i(r)

2 2s0л~іфк(г) + — г-Хі2,

k=1 Keff к = 1

Let us summarize the conditions under which the diffusion equation is valid:

(a) Using the Pi approximation we have neglected in the spherical harmonics expansion the terms corresponding to / > 1. This implies that one must be sufficiently far from surfaces where the angular flux distribution is greatly anisotropic (e. g. external boundaries and strong absorbers).

(b) In order to obtain the multi-group Fick’s law we have assumed

J 2.,(E’->E)J(r, E’)dE’ = Js.,(E->E’)J(r, E)dE’.

This means that the scattering collision densities are slowly varying functions of energy over the maximum allowable energy change per collision. This cannot be easily fulfilled by light moderators like hydrogen where the energy loss per collision can be very big, but is reasonable for heavier moderators like graphite or beryllium.

Until now in the treatment of spectrum calculations we have only considered an homogeneous reactor. In all practical cases we are confronted with reactor systems having various levels of heterogeneity. All power reactors consist of at least two
regions, a core and a reflector. For reasons of power shaping the core is normally radially and axially subdivided in regions of different composition. Usually a spectrum calculation has to be performed independently for each of the reactor regions. The coupling between these regions is taken into account through the use of energy — dependent bucklings, obtained for each region from a few group two — or three­dimensional calculation over the complete reactor.

As has already been mentioned in the proceding sections an iteration is then necessary. With a guess on the bucklings first spectrum calculations are performed producing condensed few-group constants for two — or three-dimensional calculations, out of which new energy-dependent bucklings are obtained for each region. This iterative procedure is continued until the desired accuracy is achieved. A second level of heterogeneity is given by the fine structure of the reactor cells.

Even if HTRs are rather homogeneous compared with many other reactor types, heterogeneity in the fuel structure cannot be neglected. All reactors consist of rather regular lattices of cells containing fuel pins, coolant channels, moderator and absorbers.

In spectrum calculations one has the choice between a multi-group space-dependent transport treatment, or a calculation on a homogenized cell. The codes we have examined so far follow this second procedure. The homogenization of the lattice cell implies a previous few-group transport calculation on the cell. Once the fine structure of the flux within the cell is known, the homogenization is performed multiplying the cross-sections by energy-dependent self-shieldings (sometimes called disadvantage factors). These self-shieldings Sk(E) are defined for each material к in such a way as to give the same reaction rate in the homogenized calculation as in the real cell. The homogenized calculation deals only with a reference flux фя(Е) while in reality the flux ф(г, E) depends on the position within the cell. As a reference flux one can take the flux at any point of the cell, e. g. at its centre, or at the outer boundary, or an average value

фн{Е) —

where V is the cell volume and the integral is extended over the whole cell. The spectrum which is obtained from the flux calculation depends, of course, on the reference which has been chosen, although reaction rates and reactivity are indepen­dent of this choice.

In order to obtain the proper reaction rates the self-shielding Sk (here we omit, for simplicity, the indication of the energy-dependence, or of the group index) must satisfy the equation

where Nk(r) space-dependent concentration of isotope k,

— average concentration of isotope k, crk cross-section of isotope k.

The self-shieldings must then be defined as

(8.8)

The cross-section of each material must be multiplied by its self-shielding. Very often the average cell flux is used as фя, but this might give rise to some difficulty in imposing boundary conditions, so that sometimes the flux at the outer cell boundary is used as reference. This is anyhow not very important in HTR calculations, because self- shieldings are usually very near to unity. Cell transport calculations are performed with one of the methods described in Chapter 4.

In general simplifications are made in order to avoid three-dimensional cell calcula­tions. The cell is usually considered as infinitely long and the cell outer boundary, which has usually a polygonal shape, is replaced by a cylindrical boundary giving the same cell volume (Wigner-Seitz approximation). In this way one-dimensional transport calcula­tions are often sufficient. Reflective conditions are required at the external cell boundary since this procedure assumes an infinite lattice of identical cells. In the case of the cell cylindrization these reflective conditions tend to give too high a flux at the outer boundary, while a diffuse reflection (white boundary condition) gives better results. If these conditions are not available in the transport code being used, it is often possible to simulate them by surrounding the cell with a pure scatterer.

In order to save computer time transport cell calculations are performed in a limited number of energy groups, otherwise this procedure would not differ from a space — dependent spectrum calculation in a heterogeneous cell. The few-group self-shieldings obtained in this way must then be adapted to the higher number of groups of the spectrum calculation. It is usually sufficiently accurate to use the same self-shielding for all the fine groups included within each broad group of the transport calculation. For the few-group transport cell calculation few-group constants must be obtained with a code for spectrum calculations for which a guess on the self-shieldings is needed. Here again we have an iterative procedure. Actually these iterative procedures are simplified by the fact that in most practical cases good guesses are available. The reactor design is a slow evolution from one version to the next and very seldom is a calculation performed without having previous experience of similar cases. This, of course, applies also to the buckling iteration.

A third level of heterogeneity is given in HTRs by the presence of coated particles. A homogenization by means of self-shieldings as in the case of cell calculations is possible. In the case of resonance absorption this grain structure can become very important (see §7.13).

In the most frequently used form of the kinetics equation the energy and space — dependence is eliminated averaging over all energies (one-group approximation) and assuming that the flux, source and delayed neutron precursor concentrations are separable in space and time:

Ф(г, t) = fair)<f(t),

C(r, t) = fo(r)C,(t),

S(r, t) = /„(r)S(t),

where /u(r) is the fundamental mode of the flux, first eigenfunction of the equation

v2/ + b2/ = o.

This means that the flux shape during the transient is supposed to remain the same as in the steady state. This approximation is sufficient in many cases. Higher modes of the flux distribution are excited during the transient, but they decay very rapidly. As the heat capacity of HTR fuel is very high, these short-lived higher modes do not last long enough to produce appreciable temperature increases and can therefore be neglected. The assumption of space-time separability is then valid in all cases in which the fundamental mode of the flux does not appreciably change during the transient.

After introducing the above expressions for ф(г, t), Ci(r, t) and S(r, t) into eqn.

(12.1) and substituting V2/o by ~B2fa the following equations are obtained:

-^~ = C-p)i’^it) + fJ ,C(t) + S(t) — ОВ2ф«) — Хаф(1),

V dt (12.2)

= Р, сЪ! ф(1) — ЛіСі(0 (f = l,…m)

In most cases S(t) = 0 and then, neglecting the delayed neutrons eqn. (12.2) reads:

(12.6)

If we eliminate all sources (including fission) from the first eqn. (12.2) we have

— фП is then the decay rate of the neutron flux in absence of all sources, whence the name “neutron lifetime”.

Equation (12.5) is the general equation describing a population growth where kctt is the gain between two generations. Considering also the delayed neutrons the zero­dimensional one-group equation takes the form

Equation (12.7), which can be easily obtained from (12.2), (12.3) and (12.4), is used for most space-independent dynamic calculations and is a simple equation if A к is independent of feedback effects. As A к is usually a function of temperature, the problem is complicated by the equations describing the temperature feedback, and eqn. (12.7) is no longer linear. In the most general case A к can represent also the whole control system feedback. Defining

kcff~ 1 А к. . л оч

p=— ——- = -— reactivity, (12.8)

kctt kctt

A = generation time. (12.9)

kctt vvZf

Equation (12.7) can take the form

Фф p ~ (3 , , — Vі, c,

-cr = — 7- Ф + V 2j AiCi + vS,

ut а і = і

dCt /3,

—77 = 77- ф — A, C, (i = 1,. . . m ).

dt Av

In a critical reactor l = Л, and in most cases the difference between l and Л is negligible compared to the uncertainty with which l is known. Equations (12.7a) can be still further simplified defining

Д = p — /3 prompt reactivity.

If ДзїО the reactor is critical on the prompt neutrons alone (prompt critical).

The method we have used here to obtain the one-group space-independent kinetics eqns. (12.7), while giving a clear picture of the approximations to be made, does not show explicitly the way of obtaining the energy — and space-independent constants appearing in these equations, starting from the energy — and space-dependent reactor parameters.

Because of this reason we show here also a different method of obtaining eqn. (12.7), based on perturbation theory (see for details ref. 4 or ref. 1, p. 179), starting from the time-dependent Boltzmann equation [or simply from eqn. (12.1)] and from the time-independent adjoint equation of a just-critical reference system. The flux is expressed as the product of an amplitude <f>(f) multiplied by a shape function/(r, E, t)

ф(г, Е,П = 4>(t)f(r, E,t). (12.10)

In order to obtain the space-independent equations, the time dependence of / is ignored.

Substituting (12.10) in (12.1), multiplying (12.1) by the time-independent adjoint flux Фо(г, E), multiplying the adjoint equation by ф(г, E, t), subtracting the resulting equations and integrating over volume, angle and energy it is possible to obtain eqns. (12.7a).

In this case the coefficients of (12.7a) have the following definitions:

Л(0 = 4: f —f(r, E, f)<^o(r, E) dV dE, (12.11)

r J V

0.(0 = -^ f x<(E)Pii’Zf(r, E’)f(r, E’,tWo(r, E)dVdEdE’ (12.12)

with

E = J x(E)vi,(r, E’)f(r, E’, t )фо{г, E) dV dE dE’.

It can easily be seen that eqn. (12.11) represents an energy and space weighted version of eqn. (12.9).