Because of rather homogeneous core geometry with little self-shielding effect on 238U absorptions, enriched uranium has to be used in these reactors. It is then possible to choose between a low enriched uranium cycle (3-10% enrichment) and highly enriched uranium (93% 233U) with 232Th as fertile material. It is also possible to burn Pu in HTRs in conjunction either with 232Th or with natural uranium.

Unfuel led

In the case of low enriched uranium fuel cycle, more heterogeneous geometries require a lower enrichment (e. g. pin-block compared with multi-hole block), but this does not necessarily mean lower fuel-cycle costs.

Reactors with prismatic fuel can be refuelled either on-load or off-load; this choice determines the frequency of these operations because off-load refuelling cannot take place too frequently.

In pebble-bed reactors the refuelling is continuous. The fuel elements can either flow many times through the reactor before being discarded (6 times in the 300 MWe THTR) or only once in the so-called OTTO cycle (Once Through Then Out).

It is normally not necessary to calculate the full matrix as(E’ -> E) for all considered energy points, but it is sufficient to calculate half of this matrix because a,(E’ -* E) is related to as(E -»E’) by the so-called principle of detailed balance.

In an infinite non-absorbing medium a thermodynamic equilibrium is reached in which the neutron spectrum has the form of a Maxwellian distribution M(E, T):

M(E, T) = e~ElkT.

In this equilibrium state the number of neutrons scattered from energy E’ to E must be equal to the number scattered from E to E’

M(E’, T)as (E’ -» E) = M(E, T)as(E -* E’). (6.15)

Equation (6.15) expresses the principle of detailed balance, well known from statistical mechanics. It is valid for any arbitrary scatterer, independently from the existence of chemical bindings.

A systematic optimization with the help of a computer requires a mathematical model of the reactor, capable of expressing the power-generating cost and the given constraints as a function of the independent variables which have been chosen.

The optimization procedure consists then in searching for the set of independent variables giving the minimum power-generating cost, subject to the constraints.

Mathematically one can express this problem as the minimization of the power unit cost function

C = /(xi, x2, x3,…, xN)

which is a function of N variables x„, subject to m restrictions of the type

glixi, x2,…, xN) «а, і = 1,…, m,, gi(x,-, x2,…, xN) = bj 7 = 1,…, m2.

Equation (10.1) is called the objective function and the restrictions (10.2) and (10.3) are called the constraints.

The function (10.1) is usually non-linear and often discontinuous. It is beyond the scope of this book to describe the various methods used to solve this problem.

The number of constraints is usually high in nuclear systems and the optimization method must be able to deal efficiently with them and with a high number of variables. If the function (10.1) is continuous with its first and second derivatives, gradient methods can be used. These methods attempt to change simultaneously many variables along the direction of the gradient of the function /.

Stochastic Monte Carlo procedures can be also used but present difficulties in handling constraints. Drastic simplifications are often needed in order to express analytically the power generating cost as a function of the chosen independent variables. In the case of fuel-cycle cost the function (10.1) should represent the complexity of series of burn-up and fuel management studies. The independent variables are typically

P = power density

T = fuel residence time (by constant power level), e= initial fuel enrichment (in case of low enriched U),

Th

— = thorium over carbon ratio of atomic concentrations (in Th cycle),

5= carbon over fissile atomic ratio.

Other variables like cell geometry (e. g. lattice pitch), coolant mass flow, inlet and outlet temperatures can be considered as independent parameters to be optimized.

A series of burn-up calculations and subsequent fuel-cycle-cost calculations gives the fuel cycle costs for each set of the above listed parameters. Usually very fast zero-dimensional equilibrium burn-up codes (e. g. BASS, GAFFEE) are used for these survey calculations.

The total power-generating cost must be calculated taking into account the effect of
the above parameters on the cost of the out-of-core reactor components (e. g. pressure-vessel size and cost as a function of power density P) and on the plant efficiency (e. g. effect of the blower power). The objective function (10.1) is then often obtained as a polynomial least-square fit of the results of the above calculations.

Each burn-up and cost calculation must be accompanied by an analysis of the constraints. Considering power density, burn-up, age factors, coolant inlet temperature, mass flow, etc., simplified temperature calculations are performed, while the maximum burn-up and fast neutron dose can be easily obtained from the result of burn-up calculations.

These optimization procedures can be pushed to various degrees of sophistication, but can only with difficulty treat a number of detailed problems which can still considerably influence the costs of power generation. These are the obvious limits to optimizations in any sort of engineering problems, although with modern computers we observe a continuous improvement of optimization procedures.

The process of designing a reactor is an iterative one, as with any other engineering design. First the designer has to make a certain number of hypothesis and perform simplified calculations, then as a result of these calculations the hypothesis are refined, the range of variables restricted and more detailed calculations are performed. The optimization is usually first performed zero dimensionally over the equilibrium reactor condition. Some suppositions are then made over the running-in period in order to assess the fuel investment in the core. These suppositions must be checked at a later stage against a detailed calculation. The same is done with various other parameters like the leakage which in a zero-dimensional calculation is often represented by a polyno­mial fitting of the Buckling as function of the reactor size and of its height to diameter ratio.

After the first optimization, some more detailed calculations are performed on the chosen optimum case and, if the previous hypothesis appear to have been too inaccurate, an optimization calculation might have to be repeated, often limiting the variation of the variables to a more restricted range. Usually the problem of optimal reactor design is simplified by the fact that the zone of minimum costs is a rather broad region in which the costs are not strongly dependent on the independent parameters. In this case an optimization calculation is performed in order to find this region. Within this optimum region the final choice of the parameters is then made on purely technical reasons without a consistent optimization.

This approach, which is the most consistent with the discretization typical of digital computer methods, was first developed by Carlson and is called the S„ method.

In its earlier version this technique consisted in dividing the solid angle into n segments approximating the neutron density within each segment by a linear combina­tion of its values at the end of the segment. The abbreviation S indicated straight lines rather than P used for polynomials in the spherical harmonics method. In the modern versions of the S„ method the difference procedure is not applied directly to the analytical form of the Boltzmann equation but to averages of the neutron density over discrete intervals in space, energy and solid angle.

In this way the neutron balance is more easily respected. The angular dependence is discretized in M discrete directions defined by their direction cosines p. m, t)„, fm. Each direction can be thought of as a point on the surface of a unit sphere with which a surface area (weight) wm is associated. If angular areas are measured in units of 4-7Г,

M

X vvm = 1.

m = 1

The definition of these weights results from the definition of the angular meshes.

Usually the meshes must be defined in such a way as to preserve computational invariance under geometric transformation (i. e. the solution should not depend on the orientation of the coordinates). If we choose to use n meshes for each of the three variables /x, 17 and f and impose symmetry conditions, the total number of discrete directions can be shown to be

M = n{n + 2) with n = 2,4, 6,…, or, if the geometry is not three-dimensional,

M = 2dn (n + 2)/8

where d is the number of geometric dimensions (see ref. 3). The value n defines the order of the S„ approximation. The angular dependence of the scattering term of the Boltzmann equation is usually dealt with by expanding the scattering cross-section in spherical harmonics.

The energy and space dependence of the flux is discretized in energy groups and spatial mesh points, so that all independent variables of the Boltzmann equation are treated as discrete. This method, which has been programmed in various computer codes, is one of the most accurate ways to solve the Boltzmann equation. As for all numerical methods, the order n of the approximation has to be determined by experience. In general S4 calculations with Pt treatment of the scattering cross-sections is sufficient for most core design problems, with a mesh spacing of the order of the mean free paths. Higher-order approximations usually require finer spatial meshes, otherwise negative fluxes can be obtained (this may be unimportant in some cases, e. g. in the centre of large control rods).04 (For a detailed treatment of this approximation see refs. 2 and 3.)

A characteristic of HTRs is the use of coated particle fuel. This generates a double heterogeneity: a microscopic one at the level of the coated particles and a macroscopic one at the level of the fuel compacts. Sometimes two types of particles are present: feed particles containing only fissile material and breed particles containing a mixture of fissile and fertile material.

One can distinguish a fuel region, i. e. the particle kernels and two moderator regions, one within the compacts, including coating and compacting graphite, and the second one outside the compacts, i. e. the structural graphite of the HTR core.

The treatment of the double heterogeneity consists normally in homogenizing the microscopic structure using either the equivalence relation or energy-dependent self-shielding in the case in which a multi-group calculation is performed over the resonance (this is particularly the case for low-energy resonances).

If a micro-cell can be defined including the grain and the surrounding moderator, a two-region transport calculation can be performed using either the collision probability equations (7.17) or other transport codes. The DSN method, has, for example, been used in the Ispra WRETCH code122’ for the low-energy Pu resonances.

The boundary conditions to be assumed at the boundary of this micro-cell involve assumptions on the statistical distributions of the grains which are not easily justified. Monte Carlo calculations’23’ have shown that the result is rather independent of the lattice structure and packing of the grains.

As the above-mentioned homogenization methods involve rather expensive compu­tations, simplified methods have been developed.

Let us first consider an infinite system of grains embedded in a moderator. If P0 is the escape probability from a single grain, the escape probability РІ from the system of grains is given by (7.24). The Dancoff correction C appearing in this expression has been derived by Lane et a/.’24’ for a random distribution of grains. Let be the mean free path of the moderator and

L = -4r — nag

be the mean free path between grains if the moderator were not present (we remind the reader that for grains we intend only the kernels of the coated particles, the coating belongs to the moderator zone).

The symbols are: Nc = atomic density of moderator,

cr0 = moderator scattering cross-section, n = number of grains per unit volume,

<rg = average geometrical cross-section of the grain.

The probability of hitting a grain in the path between r and r + dr is and the probability of travelling this distance without colliding with the moderator is

C is the probability of hitting a grain before making a moderator collision

This statistical treatment is valid only if the volume fraction Va of the absorber grains in the fuel compacts is small, Va ^ L

_ V.

П Vo

with Vo = grain volume.

It can be proved that for any convex body

So

4

lo is the mean chord length of the grain,

where No = atomic density of the absorber in the grain

NcCTc

VaNc

we have then

with а и total cross-section in the grain.

The equivalence relation can be used to homogenize the micro-structure within the fuel lumps.

trt = (r.(l-C) = ^f. (7.36)

i>0»0

For the double heterogeneous geometry Lane et al.(24> have derived the following expression:

P’o=P’oM~P’om) + PoP’om (7.37)

where Po = escape probability from an isolated grain,

Pom = escape probability from an infinite system of microscopic grains,

Ром = escape probability from the macroscopic cell structure.

This equation expresses the two possible ways of escape from the double heteroge­neous structure. The first term gives the probability of collision with the moderator of the micro-cell, while the second term gives the escape from the macro-cell into the outside moderator.

Pom is calculated by Lane et al.(24> using the rational approximation as seen above (Pl„ is actually the Po of eqn. (7.35). Here the suffix m has been added in order to distinguish it from Pom). Making use of this expression Journet’25’ derived an equivalence relation for double heterogeneous geometry.

For pebble-bed reactors Teuchert<26> calculated the escape probabilities of (7.37) following the path of a neutron in coated particles and macroscopic spherical fuel elements. A different approach has been used by Walti who calculates self-shielding factors in a fine group structure for a microscopic cell consisting of kernel and moderator’2*’ with a method used by Sauer’29’ for fuel lattice calculations.

The following balance are used for the kernel:

%rO<fio Vo = J Go + Qo Vo(l — Po) = Qo Vo + / — J +

and for the moderator

£„</>, V, = Г G, + Q, V,(l — P,) = Q, V, + Г — Г (7.38)

where, for к = 0 and 1,

Gk = probability that a neutron entering region к is absorbed or scattered outside the considered energy group after any number of within-group collisions,

Pk = probability that a neutron born in region к escapes after any number of within-group collisions,

Vo = volume of kernel,

Vi = volume of moderator (coating plus binder),

So = surface of kernel,

J~(E) = current through So into the kernel,

J+(E) = current through So into the moderator,

Q0(E), Q,(E)= average source in kernel and moderator, фо(Е), ф,(Е)= average flux in kernel and moderator,

Sro, Sri = removal cross-section in kernel and moderator,
lo, h = mean chord length in kernel and moderator,

Г = ф0 т = 12r = optical thickness,

from (7.38) one can obtain

ті pQjGo + Gі — GoGi — PoGi) + PiGo To (G0+ Gi — GoG, — PG0) + pQPoGi

_ lo _ Vo P 1, V,’

Assuming flat flux at the outer micro-cell boundary, isotropic and spatially constant neutron emission in each region and cosine distributed currents J+ and J~ one has (see ref. 29):

Gk = TkPk, (7.40)

Pk is given by Sauer by the following expression:

1 + Tk (1 + Hk)

which has the form of the Wigner rational approximation (7.19), the deviation from which is given by the function Hk, the so-called “augment” introduced by Sauer. Inserting (7.40) and (7.41) in (7.38) we have

Pk are multiple collision escape probabilities. The first collision escape probabilities are easily obtained. Defining with P and Я the first collision values we have

1 + Tk (1 + Hk)

so that knowing Pk it is possible to have Hk. Walti proves that for realistic grain dimensions

Як — Як

The other unknown parameter in (7.42) is Q, but since Г does not depend strongly on Q, a reasonable guess is sufficient.

As already mentioned the fuel-temperature coefficient is determined by the Doppler — broadening of the resonances of the heavy isotopes contained in the fuel and appears in the term (1 lp)(dp IdT) of eqn. (11.2).

The important resonance absorbers are the fertile materials 232Th, 238U, 240Pu (Doppler-broadening in the 1.054 eV resonance of 240Pu is a second-order effect compared with its effect on the moderator temperature coefficient). Doppler — broadening of fission resonances is of little importance in thermal reactors.

The Doppler effect decreases with increasing temperature and is strongly dependent upon concentration and lumping. In reactors with very diluted fuel the resonance integral is not temperature dependent. Considerable increases of the Doppler effects are obtained by lumping or by increasing the absorber concentration in a homogeneous design. Also the grain size of the coated particle has a significant effect.

Beside (1 lp)(dpldT), also (1 /P/)(3P//dT) is influenced by the Doppler effect: the increase in resonance absorption decreases the fast diffusion coefficient and then the fast leakage. This means that the contribution of (1 /P/)(3P//3T) is usually a positive one although rather small. On the other hand, the contribution of (1 lp)(dpIdT) is always negative and prompt, being bound to the fuel temperature. This is a very important inherent safety factor. Even reactors with an overall positive temperature coefficient have usually a prompt negative coefficient. In HTRs the Doppler effect is more pronounced than in other reactor types with more heterogeneous fuel. A heterogeneous fuel design increases the resonance self-shielding more than the thermal self-shielding. This means that in a homogeneous fuel the ratio of breeding in the resonance energy to breeding in thermal energy is greater than in more heterogeneous reactors.

The enrichment has, of course, to be higher. For the same conversion factor HTRs have a greater Doppler coefficient. This is an important safety advantage.

5.1. Analytical solutions of the diffusion equation

The diffusion equation is mathematically relatively simple and analytical solutions can be easily found if geometry and boundary conditions are not too complicated.

These classical solutions are very instructive and have been largely used in the past before the generalized utilization of high-speed computers. We refer the reader for these solutions to existing literature (e. g. ref. 1, §6-2, paras. 6 to 8).

In general the analytical solution of the diffusion equation can be based on a series expansion of the flux of the type

4>(r, E) = ‘2<pAE)fAr) (5.1)

n

for each homogeneous region / of the reactor, where the fm(r) are the eigenfunctions of the wave equation

(V2 + Bli)fm(r) = 0

with the boundary conditions fni(r) = 0 at the extrapolated reactor boundary and of continuity of flux and current at the region boundaries (see §§ 5.5 and 5.6). In this case using the multi-group formulation with N groups it can be demonstrated that in each region the summation (5.1) is limited to N eigenfunctions (see ref. 2). Another way of solving analytically the diffusion equation consists of using expansion (5.1) this time not for each region, but over the whole reactor. This is possible because the wave equation

(V2 + B„2)f„(r) = 0

together with the boundary conditions /„(r) = 0 at the extrapolated reactor boundary defines a complete system of eigenfunctions f„(r).

As usual the equations for the coefficients of each eigenfunction /, are obtained by multiplying the diffusion equation by ft, integrating over the whole volume and using the orthogonality properties of the /„ functions (see ref. 3). In this case the expansion

(5.1) is not limited to as many terms as groups, but extends in principle to infinity and the accuracy depends on the number of terms taken into consideration.

Nowadays finite difference methods have proved to be best suited to modern high-speed computers, so that these analytical methods have lost much of their importance. This is mainly due to the difficulty of treating complicated geometries with analytical methods (and because of engineering problems reactor geometries are

usually rather complicated). Besides it is much easier in a computer programme to adjust the number of meshpoints of finite difference treatments to achieve the required accuracy than to change the number of terms considered in expansions of the type (5.1). This is the same reason as that for which S„ methods are now generally preferred to Pi methods.

Some simplified analytical treatment is sometimes used in order to obtain a first guess of the flux distribution for a numerical code. Analytical methods are also sometimes used in conjunction with finite difference calculations in which in order to save computer time very coarse meshes are chosen. In these cases analytical solutions can be used in order to take into account the within-mesh flux distribution (see § 5.4).

All gas-cooled reactors contain a high number of coolant holes, which, because of the low density of the gas, can be considered as empty for neutron calculations. Besides coolant channels, control-rod holes also contribute to the empty space in the reactor cells when these rods are extracted. Holes which are small compared to the neutron mean free path in the surrounding material can be simply spread over the cell which will then have a lower homogeneized density. In most cases, however, this condition is not satisfied, at least not for the axial direction where the holes usually extend to the full reactor length. It is then clear that the effect of neutron streaming through those holes will influence the neutron leakage.

In diffusion calculations the leakage is expressed by the term

F = DB2 (8.9)

or more generally for anisotropic systems

F = ^DkBk2 (8.10)

where к is the geometrical variable (k = x, y, z or к = r, z, etc.) and Dk is a diffusion coefficient valid for the direction к (it is evident that the diffusion coefficient can have different values in the directions parallel or orthogonal to the coolant channels).

Diffusion theory is in general not valid in a heterogeneous reactor cell, particularly in the presence of holes, so that we are confronted with the problem of averaging D over the various regions of a heterogeneous cell.

The first treatment of the neutron streaming effect has been made by Behrens.<34) He calculated the ratio r2/r02 of the migration areas (r is the distance, as the crow flies, travelled by a neutron in the course of the N collisions it has before being absorbed or leaving the energy group under consideration, the suffix 0 referring to the case without holes). Behrens does not consider cell heterogeneities except for the holes. Without holes we have

r2=f r2^e"lkdr = 2Nk2 (8.11)

Jo A

where A is the mean free path and (1/Л)е-г/л is the probability for a free path of having length r. We have then r0 as a root-mean-square average of each free path, which is obvious for a random scattering process with no correlation between the directions of successive free paths. Let ф be the volume ratio hole/material, let £(1, ft)Sl be the cross-section of passages through the hole in the direction ft having lengths between l and l + 81, and p(r, ft, l) be the probability that during the course of a free path of length r in the direction ft a neutron emerges into a hole of chord l.

In the case in which the holes are so far apart that passages through more than one hole during a free path are very unlikely we have instead of (8.11) the equation

P’=4 Jg [Jo (r + l)2p(r, n,l)dl + r2{l-Jg p(r, n,l)dl]]j-e-r/xdrj£- (8Л2>

where ш is the angle between the direction under consideration and the direction of ft. For cylindrical holes the effect is twice as great in the axial as in the transverse directions:

where Q is still the average value for all directions.

Many diffusion codes allow the use of different diffusion coefficients for the different directions.

For the input of diffusion codes one normally needs instead of L2 the diffusion coefficient D :

where Do and 2o0 are the values corresponding to the full material. The absorption cross-section is corrected for the voidage

= po2„o

At the limit of very small holes (no streaming) eqn. (8.15) gives

D = —Do.

Po

The diffusion coefficient obtained from the codes for spectrum calculations is already corrected for voidage (if the atomic densities given in input were corrected for voidage). Before being used in diffusion codes this diffusion coefficient must therefore be multiplied by the factor

which tends obviously to unity in the case of very small holes.

A more accurate treatment of this problem is due to Benoist.<35~37) He noted that in the Behrens theory the diffusion area is calculated not from the mean square distance until absorption, but from the sum of the mean squares of the elementary paths, using in a lattice a property true only in a homogeneous medium.

The only function of the diffusion coefficient consists in reproducing the exact leakage F in eqns. (8.9) or (8.10). The leakage cut of a zone V of the rector is given by

(8.18)

Benoist starts from the integral form of the Boltzmann equation and considers the cell of volume V subdivided in regions of volume V,, transport cross-section 2и,„ and average flux <£,. Using (8.18) and (8.10) he obtains the expression (see ref. 36)

2 2 и, Ф,-2р-,к

Dk = ————— ^———— (8.19)

з 2 v,4>,

і

where Dk, ф, Pl. t are in general energy dependent so that eqn. (8.19) can be considered as valid for a given energy group. The quantities Р‘,,к are called by Benoist transport probabilities.

where ф(г’, Cl’) = angular flux,

Ok, ftL = components of the unit vectors ft and ft’ in the direction k,

G(r, ft; r’, ft’) = Green function: kernel giving the angular flux at point r, direction ft produced by a neutron born at point r’ along the direction ft’.

It is usually possible to replace ф(г’, ft’) by its average value фі/4 = for the region i. In this case

Р’»* = Т^Г f dr* f dCl [ dr’3 [ dfl’G(r, ft; r’, ft’)OkO( (8.21)

47Г Vj Jj J 4W J і J 4W

from which follows

itr. ivip;i. k = itr, ivlpik,

2рі‘= і­/

In an earlier derivation of eqn. (8.19) Benoist sought a solution of the Boltzmann equation in its integral form using a Liouville-Neumann expansion/25’ In that case one can see that

(8.22)

If the lattice is uniform along the direction к and the scattering isotropic the collision probabilities corresponding to / s* 1 (usually called angular correlation or secondary terms) vanish (this is often true for the axial direction).

Expression (8.23) is similar to an ordinary collision probability apart from the term ftk2.

The average over all directions к gives

D = 1X Dk

к

and the probability Рц, к reduces to an ordinary collision probability

з 2 JV = Ри-

The terms Py.’k are related to the angular correlations between neutron paths separated by / collisions. The series (8.22) converges very slowly if the channels are big compared with the mean free path in the moderator. The Behrens method corresponds to the P„,k terms only, where the diffusion area (which is proportional to the mean square of the distance as the crow flies, travelled by the neutron from source to absorption) is resolved into a sum of the mean squares of the elementary paths. To these terms one

has to add the sum of the mean scalar products of two elementary paths separated by l collisions. These terms do not vanish in an heterogeneous medium, even in case of isotropic scattering. Benoist illustrates simply this effect saying that a neutron which has crossed a cavity has a greater probability of travelling a longer distance during its next path if it is scattered backward than if it is scattered in the forward direction. In the Behrens theory the neutron is only aware of the direction of its path and forgets the direction of its emission. In an heterogeneous medium a great number of collisions are necessary for the neutron to forget the direction of its emission.

Besides, in case of annular holes, Behrens does not consider that a neutron can cross twice the same cavity without suffering a collision. In the practical case of HTR calculations the reactor cells are rather homogeneous, and the holes usually sufficiently small, so that the simpler Behrens theory can be often used.

If the terms for 1 ss 1 can be neglected, the calculation of eqn. (8.23) can be rather easily programmed (e. g. the ARIADNE routine of the WIMS code).’111

For simple geometries (e. g. a three-region problem: fuel region, coolant channel and moderator) expressions for Р-,,к have been given by Benoist’371 (these expressions are also programmed in the WIMS code). In the case of very big holes, like empty columns during reactor refuelling or the empty space on the top of a pebble-bed reactor, diffusion theory fails completely and different transport or Monte Carlo methods have to be used.

As already mentioned, the reactor dynamic behaviour is in most cases determined by the delayed neutrons. The fractional yield of delayed neutrons per fission neutron is a function of the fissionable isotope.

Defining /Зі as the yield of group і of delayed neutrons

m

2 A = A

і = 1

j8i and j8 for various fissionable isotopes are given in Table 12.1’7’ together with the decay constant of the delayed neutron precursors.

Table 12.1 (cent.)
Delayed-Neutron Fractions p

Fast fission Thermal fission

Another form of delayed neutrons are the photoneutrons, resulting from interaction of y-rays with moderator atoms. The only moderators whose neutron binding energy lies within the spectrum of fission product y-rays are deuterium and beryllium. Delayed photoneutron parameters for Be are given in Table 12.2.(7>

The delayed neutron fraction can be burn-up dependent. The replacement of 235U by 233U reduces /3, while the replacement of 239Pu by M1Pu increases it.

The energy of the delayed neutrons is lower than the average fission neutron energy. This fact results in a higher importance of the delayed neutrons. Neutrons born with a lower energy have a higher probability of reaching thermal energy without leaking out of the core or being absorbed.

The absorption cross-section in the energy range between the fission energy and the delayed neutron energy is usually very small, but the leakage probability can be high in small cores. The higher importance is taken into account using in the calculations an effective delayed neutron fraction /3eff higher than /3.

This fact is taken into account in expression (12.12) where the /3, are weighted over the delayed neutron spectra and their importance. As in the case of Л, the effective delayed neutron fraction can be calculated once the energy and space distribution of the flux and adjoint flux are known.

A way of calculating /3eff/y3 consists in performing two multi-group space-dependent reactivity calculations, once using the prompt fission spectrum and once the spectrum of the delayed neutrons. The number of groups should be high enough to describe the
fission and delayed neutron spectra. This would require an impractically high number of groups for multi-dimensional calculations.

The problem can be solved by performing these two calculations with zero­dimensional codes (e. g. MUPO) in which the leakage is represented by means of energy-dependent bucklings obtained from previous few group space-dependent calcu­lations. The ratio of the effective multiplication factors calculated once with the fission spectrum and once with the delayed neutron spectrum gives the ratio /3ctt 1/3. If the fast leakage is small the difference between and /8 can be within the experimental uncertainties of the measurements (3-5%).

A somewhat similar method is also described by Henry.141 It must be noted that if the dynamics calculation is performed with a sufficiently high number of energy groups, and space-dependence accounts properly for fast leakage, a separate calculation of /8eff is not necessary. But this is seldom the case in practice. As in most reactor physics experiments the core reactivity is obtained from period measurements, i. e. from transients which are dominated by the delayed neutrons, it is convenient to measure the reactivity in terms of /8eff.

The reactivity which makes the reactor prompt critical (p = /8eff) is called dollar:

1 dollar = )3eff.

This unit is mostly used by experimental reactor physicists, while in the theoretical calculations the reactivity is usually measured in percent or niles (1 nile reactivity means (keff-l)/keff = 0.01). Submultiples of these units are the cent (10 2 dollars) and the millinile (10“3 niles). In the French literature the p. c.m. is sometimes used (1 p. c.m. = 1 millinile).

Many possibilities are available for the choice of the geometrical location of the control rods. In the block design the control rods can be located in some of the hexagonal blocks (see Fig. 1.2).<4> The choice of the control-rod location is strongly related to the design of the number of penetrations in the upper part of the pressure vessel (which in modern HTRs is of pre-stressed concrete). The control rods can either be located in the same penetrations used for refuelling, or in separate penetrations.

In the case of pebble-bed reactors the control rods are directly inserted among the fuel elements. This requires higher forces and allows a smaller insertion velocity than in other reactors. In order to ease this problem it is possible to use also control rods located in the reflector (e. g. THTR). These rods alone are not sufficient to control the reactor, but may be used either for the fine regulation where frequent rod movements are needed, or for a fast insertion in case of an emergency shut-down.

1.2. Limitations to the performance of the HTR core

Whereas in other reactor types the melting point of some reactor components gives well-defined limits to the core performance, the limitations of the fully ceramic HTR core are much more difficult to define. Safe and reliable reactor operation requires mechanical stability of the core components and a primary circuit activity kept below limits established by maintenance and safety requirements. Damage to the core material is induced by the combined effect of temperature and irradiation. Collisions with neutrons above a certain energy (0.1 MeV) displace atoms from the original crystal structure, thus altering the properties of the reactor materials with resulting changes in dimensions, mechanical and heat transfer properties. The effect of neutron irradiation is temperature dependent. The thermal movement of the atoms influences the further consequences of the displacements due to neutron collision and can also cause the return of the displaced atoms to their original positions.

Gradients in temperature and fast neutron dose cause stresses, particularly in large fuel or reflector blocks.

Furthermore, fissions can cause damage to the coated particles because of fission fragment recoil, and cause build-up of pressure due to gaseous fission products. The latter effect is also influenced by temperature.

If coated particles operate under a significant temperature gradient an internal corrosive attack and transport of carbon takes place (amoeba effect). By proper choice of particle design and operating conditions the amount of attack on the inner PyC layer can be held within safe limits.

Sectional View of THTR Pressure Vessel

Another effect of temperature is to determine the rate of diffusion and the rate of reaction with graphite of H20 and C02 impurities present in the coolant. The temperature at the surface of the coolant channels is particularly important for this effect, whereas the central fuel temperature is responsible for coated particle damage.

The primary circuit coolant activity is determined by the balance between the emission of gaseous fission products from the fuel elements, and their elimination by the coolant clean-up system (this is a purification plant to which a part of the coolant flow is continuously by-passed).

Virtually all non-gaseous fission products emitted by the coated particles are either retained in the graphite structure or are evaporated in the coolant and subsequently deposited on the cooler parts of the circuit (e. g. heat exchangers, circulator blades, etc.).

The release of fission products from the fuel elements is due to migration through the coating of the particle, the possible presence of broken particles, and the presence of U impurities outside the coated particles in the fresh fuel. Before reaching the coolant the fission products have also to migrate through the carbon matrix and the outer graphite layers of the fuel elements. The migration of fission products through pyrocarbon coating and through graphite is strongly influenced by temperature. An improvement of the retention of metallic fission products at high temperature is obtained with a SiC layer in the coating of the particles.

From the above mentioned facts it results that temperature and dose limitations cannot be easily expressed by fixed data in HTRs.’791

Higher temperature transients usually give rise to only a temporary increase in activity which can be often accepted. This means that some temperature rises can be easily tolerated during accidents.

Also during normal operation the need of power and temperature flattening is less stringent than in other reactors: the same release rate can be theoretically obtained with all particles at a given temperature or with a few at a higher temperature and the rest at a lower level. This fact is present to an even greater degree in pebble-bed reactors where, because of the continuous movement, it is not always the same element which experiences the highest temperatures.’10’ Also the rupture of the coating can be tolerated if it is limited to a very small fuel fraction.

Besides higher temperatures usually occur in fresh fuel where, because of the low burn-up and dose, they are better tolerated.

References

1. G. E. Lockett and R. A. U. Huddle, Development of the design of the high temperature gas cooled reactor experiment. Dragon Project Report 1, Jan. 1960.

2. R. A. U. Huddle et al., Coated particle fuel for the Dragon reactor experiment. Dragon Project Report 116, Oct. 1962.

3. A. L. Bickerdike, H. C. Ranson, C. Vivante and G. Hughes. Studies on coated particle fuel involving coating, consolidation and evaluation. Dragon Project Report 139, Jan. 1963.

4. D. A. Nehrig, A. J. Neylan and E. O. Winkler, Design features of the core and support structures for the Fort St. Vrain Nuclear Generating Station. Conference on Graphite Structure for Nuclear Reactors, London, 7-9 March 1972.

5. M. V. Quick, D. G. Richardson and W. R. Hough, High temperature reactor core design and behaviour. Conference on Graphite Structure for Nuclear Reactors, London, 7-9 March 1972.

6. E. Cramer, G. Hagstotz and A. Eiermann: Components of the THTR300 heat transfer system. Conference on Component Design in High Temperature Reactors Using Helium as a Coolant, London, 3-4 May 1972.

7. L. W. Graham and H. Hick, Performance limits of coated particle fuel. Dragon Project Report 850, Sept. 1973.

8. H. Nickel, E. Balthesen, L. W. Graham and H. Hick, HTR fuel development for advanced applications. BNES Conference on the High Temperature Reactor and Process Heat Applications, London, Nov. 1974.

9. M. R. Everett, D. F. Leushacke and W. Delle, Graphite and matrix materials for very high temperature reactors. BNES Conference on the High Temperature Reactor and Process Heat Applica­tions, London, Nov. 1974.

10. L. Massimo and U. A. Schmid, Statistical temperature distribution calculation in pebble bed reactors. Nucl. Engng and Design, 10, 367-372 (1969).