In case of off-load refuelling the equilibrium composition and reactivity are not constant, but periodical, the period being the interval between two subsequent reloadings. In reality the period can be longer than this interval because the refuelling of one region is not equivalent to the reloading of another one, but usually this is only considered in detailed calculations and not in survey studies. In a zero-dimensional case the average reactor concentration Nk{t) of isotope к as a function of time during the refuelling period of length Tp is given by

і M-1

M(D = tt 2 Nk(t + nTP)

1*1 n =0

Fig. 9.6. Concentration of isotope к as a function of time.

The calculation is performed with methods similar to those used for the continuous refuelling. The reactivity at the end of each period must have the values specified as the minimum acceptable for control and operation reasons. In order to reach this an iteration on the feed-fuel composition can be performed.

A typical zero-dimensional code of this type is the General Atomic GAFFEE code which uses analytical solutions/14’ A modified version (MOGA) developed in Ispra can deal with concentration-dependent self-shieldings.

For analytical solutions a further approximation is needed because in the case of discontinuous reloading the flux does not remain constant during the period between two subsequent reloading. An analytical solution of eqn. (9.1) requires constant flux. It is a reasonable approximation to use the flux corresponding to the time average of the concentrations during the period.

As for the case of continuous refuelling, also for off-load refuelling the methods used for zero-dimensional equilibrium calculations can be extended to deal with space dependence. Using one group diffusion calculations based on nodal methods the GAFFEE type of calculation has been extended to one and two-dimensional calcula­tions (codes RACE and TRACE’15’).

9.2. Burn-up calculations in pebble-bed reactors

A special problem is posed by the burn-up of pebble-bed reactors. In this case the spherical fuel elements are loaded on the top of the reactor, flow across the core and are discharged from the bottom.

In the pebble-bed reactors now in operation or under construction (AVR and THTR) the fuel elements flow many times through the reactor before being finally discharged. Reactors in which the fuel elements flow only once through the core are being also considered for future applications (the so-called OTTO fuel cycle). In the normal case with many passages of the fuel elements, pebble-bed reactors are the perfect example of continuous reloading.

In any volume element of the core all burn-up stages of the fuel are represented and, because of the small size of the fuel elements, the neutron spectrum is determined by the average composition. The burn-up of the elements flowing out of the bottom of the core is measured (usually by means of a small reactor,(16>) the elements whose burn-up is higher than a given limit are discarded, the others are reloaded. A two-region core for radial power flattening can be obtained loading preferably fresh elements in the outer core and older elements in the inner parts.

The problem is complicated by the radial dependence of the axial fuel velocity, which is usually higher at the core centre and lower near the reflector.

First assessments are usually made with zero-dimensional burn-up codes for continu­ous refuelling, in which the effect of the axial fuel element movement does not need to be considered. Later more detailed space-dependent calculations are needed in which the peculiarities of this reactor type are taken into consideration.

Space-dependent burn-up calculations for pebble-bed reactors can be performed with standard burn-up codes treating a very high number of regions and considering each region as composed of a high number of fuel element classes representing different burn-up stages (and possibly different fuel-element types). The burn-up of each class must be followed independently of the others. The axial fuel element movement can then be simulated with a very high number of reloading-reshuffling operations in which the composition of each axial zone is shifted into the next one. If necessary different spectral regions can be considered. For each of these regions the cross-sections for the burn-up equations are calculated with spectrum calculations.1171 This method can give very accurate results but is not suited for fast calculations.

The simplified methods which have been developed for this reactor type have proved in practice quite fast and accurate. Supposing that the fuel moves only in the axial direction it is possible to write the general burn-up equation’181

v _ ф 2 ]sjj(Tfiy. k + Ф 2 Nscrasysk + 2 NAjCXjk — ЛkNk — 4>Nk(Tak (9.4)

Ol OZ i=l s=r

where Nk = atomic concentration of isotope k, v = axial ball velocity v = f(r, t), ф = flux, ф = f(r, z, f),

<Jfi = isotope і fission cross-section,

(Tai = isotope і absorption cross-section,

Лі = isotope і decay constant,

yik = yield of isotope к due to a fission in isotope i, ysk = probability that a neutron absorption in isotope s produces isotope k, a, k = probability that the decay of isotope і produces isotope k, r = radius,

z = axial coordinate—the balls are loaded in the point z = 0 and move in the direction of increasing z,

H = depth of the pebble bed.

This equation is identical to (9.1) with the addition on the left-hand side of the term

dNk

~z— v dz

taking into account the axial movement of the fuel. In equilibrium condition

and v = f(r), ф = f(r, z) are supposed to be known. In this case it is possible to solve numerically eqn. (9.4), if the boundary condition Nk(r)z.0 is given for all k. The difficulty consists in relating Nk(r)z,0 with Nk(r)z. H because the composition of the discharged balls is not known.

A simple way of solving this problem is to assume that the composition of a fuel element is fully defined by its irradiation. This method is used by the KUGEL code.’19’ This programme calculates the statistical space distribution of the fuel elements in a pebble-bed reactor taking into account the radial dependence of the passage time through the core. A fuel element is followed during all its passages through the core until it is discharged and the probability of finding it in all points of the core is calculated as a function of its irradiation.

The composition of the fuel element as a function of irradiation (here always measured in full power days) is read in by KUGEL in form of a table. In this way the burn-up equations do not appear in the code. This table is obtained from a zero­dimensional burn-up code which calculates numerically the burn-up of a fuel element in a constant flux taking into account the most important fission product chains. The flux level is the one obtained from a zero-dimensional burn-up calculation for continuous refuelling (e. g. the BASS code"2’) which must have been performed in a previous assessment stage. The KUGEL code divides radially the reactor in a certain number of axial channels.

In each channel the pebble velocity is supposed to be constant and it is assumed that the movement is restricted to the vertical direction. The total irradiation time of a fuel element is subdivided in a discrete number of age groups of width т (this age is an irradiation (ф x t) and not a time, but it is convenient to measure it in full power days). The fresh elements belong to group 1. Usually between 100 and 200 age groups are considered. An age spectrum A(f) is defined as the probability of finding a fuel element as function of its age t. If the age is discretized in age groups, the age spectrum is a vector A with as many components ak as age groups k. t At each passage through the core the irradiation of a fuel element increases by a fixed amount h which is

tin the following treatment we define with a small letter the components of a vector defined by the corresponding capital letter, so that ak are the components of vector А, Ь’л the components of vector Bl, etc.

characteristic of each channel і (proportional to neutron flux and inversely proportional to the axial velocity). This means that an element belonging to age group к if loaded in channel і will, at its exit from the core, belong to age group к — t — Sr, where 8, is a “delay” characteristic of this particular channel

(9-5)

t ф

where і = channel index,

tt = passage time in channel i, фі = average flux in channel i, ф = average core flux

(the discretization of the age groups means also that Si will have to be rounded off to the nearest integer).

At the beginning of each passage j it is possible to define a “loading spectrum” L’ whose components /*’ give the probability that a fuel element, at the beginning of its jth passage through the core, belongs to age group k. At the first passage

In each passage the fuel elements are distributed among the various channels according to their age, following a given distribution law. This distribution law should insure a constant level of the pebbles in all channels and a flat radial power distribution.

Let us define W‘ as the probability that a fuel element in passage j is loaded in channel i, and

(9.5)

where j = passage index, J = total number of passages.

In order to have the same level of pebbles in each channel the probability p, should be proportional to channel cross-section and inversely proportional to channel passage time

St It,

2 s, it,

where Si = cross-section of channel /, N = total number of channels.

In order to obtain a radially flat power distribution it is usually necessary to define two core zones. The inner zone includes channels 1 to R and the outer zone channels R + 1 to N. In this case it is possible to define

wi,

і = 1 І = 1

Pt and P2 are the probabilities that a fuel element (in the total of its passages) comes in the inner or the outer zone respectively. We then have:

One could assume that all elements whose age is smaller than T* are loaded into the outer region, those whose age is greater than T* are loaded into the inner region and those with age greater than T are discarded.

Defining

J К*

a? _

<=£ — j к

2 2 ь’

= 1 k = 1

where X* and К are the age groups corresponding to T* and T respectively. It would then result

P, = 1 — 5£, P2 = $.

К’ or T* would be defined by condition (9.8) and no parameter would remain free to allow for power flattening.

In order to avoid that, one can introduce two corrections.

ai Probability for an element of age < T* to be loaded into the inner zone.

a2 Probability for an element of age > T* to be loaded into the outer zone.

It follows:

P, = a,^ + (l-a2)(l-^),

(9.9)

P2 = (l-a,)^ + a2(l-^).

Pi and P2 must still satisfy the conditions (9.8), but now two parameters (ai and a2) are free for adjustments, is only known if the problem of the fuel element distribution as function of age has been solved, i. e. it can be only obtained through an iterative process. As first guess for the iteration it is assumed

(This formula would be valid for infinite axial velocity.) The loading spectrum L’ is subdivided into two parts Lx and L2‘ belonging to the inner and outer core zones respectively. For an age group k, corresponding to an age smaller than T we have:

and for an age group k corresponding to an age greater than T we have:

Within each core zone the distribution is then proportional to the channel cross-section and inversely proportional to the passage time. The age spectrum B’ at the entrance of each channel і for each passage j can be obtained from the loading spectrum L1 according to the distribution law.

The elements of the age spectrum В/ are obtained in the following way:

ik t 1 к R

2 Silt,
and

■ S It-

Ь[к= 12k—n——1— for the outer zone. (9.12)

2 S, lt,

i=R + 1

Let us define Ql as the age spectrum at the end of channel і for the passage j. Its elements q’ik can be obtained from the elements of the age spectrum В/,

qik+s,= b’,k. (9.13)

Out of the age spectrum Ql the part belonging to an age < T is used to produce the loading spectrum for the passage j + 1,

llk+’ = ‘Zq[k for k = K — (9.14)

1 = 1

The part corresponding to an age greater than T forms the discharge spectrum D,

dk=’Z’Zq[k for all k>K (9.15)

j-i i=i

defined as summation of the discharged elements over all channels N and all passages J.

The integrated loading spectrum

(9.16)

к

gives for each passage j the probability of still finding a fuel element in the reactor at the jth passage. The calculation is continued until yJ+’ =0.

J is then the greatest number of passages which a fuel element can have through the reactor.

Having started with a loading spectrum normalized to 1, also the discharge spectrum must give:

2 dk = 1.0.

к

The average age of a discarded element is

Td = t 2 kdk. (9.17)

к

Until now we have assumed that it is possible to measure exactly the age of a fuel element.

In reality one has a Gaussian distribution of the measured values so that eqns. (9.8) to 9.11) have to be modified in order to take into account this effect.

The probability that a fuel element of age T — e is classified as having an age greater than T is

-5= [ e~h2x2dx, (9.18)

V 7Г J.

where h is the precision index. This same expression gives the probability that a fuel element of age T + e is classified as having an age lesser than T. (It is assumed that h is the same for all age groups.)

Let us define

gn=~y=( e 1,1×1 dx, (9.19)

V7T J„T-T,2

g„ is the probability that a fuel element which is classified as belonging to an age group equal or greater than k, belongs in reality to group к — n (or the probability that a fuel element which is classified as belonging to an age group equal or smaller than k, belongs in reality to group к + n). We have then a vector G whose components are

go, g 1, gl ■ ■ ■ gs

assuming that all g„ with n > s are negligible.

Equations (9.14) and (9.15) are then substituted by

N

/i+, = 2qU for k^K-s, (9.20)

і = I

a = 2 (9.21)

І=I i=l

N

and (i+‘ = 2 q[kG~gK-k) for K-s<k^K,

і — 1

A =2 2 41*0 “ft-*). (9.22)

І~• i=1

N

and Tk+’ = 2 qlkgk-к for K<kSK + s,

і = 1

dk= 2 2^U for к > К + s. (9.23)

j-ii-i

In a similar way eqns. (9.10) and (9.11) are changed in order to take into account the errors in the measurements of T*.

After having calculated the age spectra B! for each channel і and each passage j, it is possible to calculate the total age spectrum A, at the entry of each channel,

J

A = 2 B,‘. (9.24)

i-i

These age spectra are then normalized so that

2 L

к

From the total age spectrum at the entry of each channel it is then possible to obtain the total age spectrum in every other point z,

alk+,= a,.k. (9.25)

In order to calculate s one needs an assumption on the flux distribution along the channel

<p(x) dx

———— 5,.

<p (x) dx

The discretization of the age in groups means that 5 will have to be rounded off to the nearest integer. Following this procedure the KUGEL code calculates a two­dimensional matrix of total age spectra. It is then easy to convert these total age spectra in isotopic compositions.

Let A* be the total age spectrum in a given point x. A previously performed burn-up calculation relates each age group к with a concentration vector Сь (This vector has as many components as isotopes and gives for each isotope the concentration correspond­ing to age group k.)

The composition in every point x is then given by

r=^CA’. (9.27)

к

As already pointed out, this treatment assumes that the composition of a fuel element is fully defined by its irradiation, independently of the actual form of the time dependence of the flux during burn-up. In reality, as the flux is not constant in the axial direction, the fuel elements are not burnt in a constant flux.

The error introduced by this simplification is very small except for those isotopes (like,35Xe and 233Pa) with short half-lives whose concentration is the result of a competition between flux dependent processes (production and absorption) and flux independent processes (decay). The concentration of 135Xe can be calculated at a later stage independently of the burn-up calculations (see Chapter 12).

Assuming that the axial fuel element velocity is radially constant, it is possible to perform an equilibrium burn-up calculation without the need of the above assumption.

In this case one can check the error introduced by this assumption, which is very small, even in the case of a greatly distorted axial flux distribution. Figure 9.7 gives the discharge spectrum for the 300 MWe THTR reactor calculated once without error in the burn-up measurement, and once with an error of ± 125 days.

A low axial fuel element velocity has the effect of shifting the fissile concentration, and hence the power, towards the top of the core. This can be seen in Fig. 9.8 where the THTR axial power distribution is compared with a hypothetical case with infinite velocity.

This fact can be used in order to approximate an exponential axial power. This same principle of separation of the burn-up calculation from the calculation of the fuel element distribution can be used for running-in calculations.<20> The fuel is again classified in age groups. The core is subdivided radially in channels and axially in zones. After each time step a zone is shifted into the next one. At defined time intervals a two-dimensional diffusion calculation is performed. The burn-up measurement and the

Full power days [t] Fig. 9.7. Discharge spectrum for the 300 MWe THTR.

Depth in the core [m] Fig. 9.8. Influence of the passage time of the fuel elements on the axial power distribution of pebble-bed reactor.

radial distribution of the recirculated elements is taken into account in the calculation with the same methods used for the equilibrium case. In the first core the power flatten­ing is obtained with a proper radial distribution of dummy elements and burnable poison.

These simplified burn-up methods cannot be applied in all cases. A very strong axial dependence of the power distribution influences the axial dependence of the neutron spectrum and of the cross-sections. An extreme case is given by the fuel cycles with only one passage through the core (OTTO) where the very strong axial dependence of burn-up is used to approximate an exponential axial power distribution with the aim of temperature flattening. In this case the simplification of performing a burn-up calcula­tion with constant flux and cross-sections, separate from the calculation of the fuel element distributions is no longer acceptable. On the other hand, the fact of having only one passage through the core simplifies the space-dependent burn-up calculations performed with standard methods. A high number of core regions can be used and the axial movement can be discretized in many axial shifts of one region into the next. An iteration is necessary since in order to obtain the axial flux distribution, which is needed for the burn-up calculation, one needs the axial dependence of the composition which is obtained from the burn-up calculation.