Although some attempt of standardization of the fine group structure had been made,<21) there still remain considerable differences as we have seen in the preceding sections.

These differences have not only historical but also physical reasons. A code with a very fine group structure can be used for any sort of calculations, but requires a longer running time, while codes having a limited energy detail are very fast but their library is valid only for a more restricted range of calculations. In HTR survey analysis where many spectrum calculations have to be performed both approaches are possible.

The first approach,<22) used by General Atomic, consists in using a rather sophistica­ted code (MICROX) for a limited number of spectrum calculations in which the broad group cross-sections are obtained as a function of fuel temperature and C/Th ratio in the fast range, and as a function of moderator temperature and C/235U ratio in the thermal range. These results are stored and a subsequent interpolation is made for the actual composition of each single case.

The second approach consists of using a simplified code like MUPO, repeating the spectrum calculation for each case.

The fine energy structure for the spectrum codes most frequently used for HTR calculations is shown in Table 8.1.

The fission energy includes the fission fragment kinetic energy, the neutron kinetic energy, fission y-rays, /3- and y-rays from fission product decay. This energy distributes itself between fuel, moderator, reflector and shield, according to core size and composi­tion.

For dynamics calculations only the heat production in fuel and moderator is usually considered, and we can assume for HTRs that 92% of the energy is produced in the fuel and 8% in the moderator. Within the fuel and moderator blocks the time-dependent heat-conduction equation is used:

iI’=-^V2T + — (12.17)

dt Cp Cp

where T = temperature,

P = power per unit volume, c = specific heat, p = density,

к = thermal conductivity, t = time.

Integrating (12.17) over a given volume V and using Green’s theorem on the V2 term we obtain

3T=JL± f dT _P

dt cp V Js dn cp

where S is the surface corresponding to volume V and dT/dn the derivative in the direction perpendicular to this surface. Approximate solutions are obtained by sub­dividing the reactor components (fuel and moderator) into axial and radial zones in which the temperatures are supposed to be time-dependent but not space-dependent. The fuel is then subdivided in concentric radial zones. For each zone j eqn. (12.18) can be simplified to

The heat transfer equations for the coolant channels can be obtained considering the conservation of mass and of enthalpy together with the state equation of gases<2) applied to the axial zone Az of the coolant channel under consideration.

m, — nit-, = p, — p,

AAZi At

2 p ^ = cp(rihTc. i — conservation of enthalpy,

k = 1 Rk

p — = RTd state equation of gases

P

coolant specific heat,

number of different types of fuel elements giving power to the coolant, gas constant,

temperature of outer radial zone of type к fuel element, coolant temperature at axial level /,

heat resistance between outer radial zone of element к and gas, coolant mass flow at axial level і in the channel under consideration, coolant channel cross-section, height of axial zone, pressure.

In the second of these equations there is no accumulation term with time, i. e., no derivative A ■ Az ■ cp{d(pT)/dt}, because the helium coolant is supposed to behave as a perfect gas and therefore pT is constant, if we neglect pressure changes.

By substitution (and still neglecting pressure changes) we obtain the following heat transfer equation for the coolant:

The error introduced neglecting pressure changes is negligible even in case of rapid depressurizations, but it is in any case rather easy to treat these terms in computer codes.

Many coefficients of eqns. (12.19) and (12.20) are temperature-dependent. This difficulty is overtaken by iterative procedures correcting the coefficients at every iteration. The heat resistances Rk of eqn. (12.20) depend also on the mass flow.

These equations are also valid for pebble-bed reactors. In this case the coolant channel does not physically exist, but can be thought as an ideal vertical column in the pebble bed, containing a certain number of pebbles, with an area A free for the coolant passage. The fact of using heat resistances makes eqns. (12.19) and (12.20) independent of the fuel geometry (so that the same computer code can be used for different geometries), while this is not true for eqn. (12.17). The fuel geometry must obviously be considered in order to calculate the heat resistances.

In some cases the fuel elements are cooled inside and outside. In this case the distribution of the inner and outer mass flow must be determined iteratively imposing equal pressure drop along both ways of the channels. This same condition of equal pressure drop has to be applied when calculating the steady-state coolant flow partition between the different channels of the core. This calculation must take into account the
given gas outlet temperature profile in the core and the different pressure drops in the channels caused by acceleration losses, friction and discontinuity losses, gagging scheme. During transients caused either by reactivity excursions or mass flow variations, the split of the mass flow between different channels (and between the two semichannels of a tubular pin design) can vary. A mathematical solution of this problem can be found in ref. 2.

When only the zero-dimensional form of the neutron kinetics equation is considered, it is possible to make an approximation for the temperatures analogous to the one made for the fluxes in § 12.3, supposing that

T(r, t) = r(r)T(t)

where r(r) is the stationary temperature distribution.

This approximation can be sufficient in some cases but is not as good as the similar one for the fluxes. The axial temperature profile is usually distorted during the accident and the consideration of those distortions can result in corrections in the maximum temperature excursion ДТта* of the order of 10-15%. If a zero-dimensional heat — transfer calculation is performed the axial index і disappears from eqn. (12.19) (a certain number j of radial zones must be considered in any case).

Equation (12.20) is substituted by

І = cprh(T2 ~ T t) + c,,Ap~H — — T’ (12.21)

k = l Kk lit

with Tc = (Tc2- Tc,)/2,

Tci = coolant inlet temperature,

Tc2 = coolant outlet temperature,

H = core height.

In all these equations heat radiation has not been considered.

If a gap is present in the fuel it may be necessary to consider radiation. This implies the introduction of terms of the type

aiTf-TU)

to describe the heat radiation between region j and j + 1. In some cases it is sufficient to approximate this term with (T, — Tj+l)IRj choosing an appropriate resistance Rt, which will then be temperature dependent. In fuel element configurations where the mod­erator is physically separated from the fuel (e. g. pin and block design) the heat radiation becomes important especially when dealing with temperature excursions after a trip. In that case the exact treatment of heat radiation is necessary because after shut-down the coolant flow is strongly reduced.

The thermal power is usually considered to be proportional to the neutron flux. In reality some 8% of the power is due to fission product decay and is therefore released with various delays. As in the case of the delayed neutron precursors these fission products can be grouped in a certain number of groups (e. g. 2 in ref. 2). Also the repartition between fuel and moderator is different for the decay heat which, being due to absorption of /3- and y-rays, is more uniformly distributed (70% in fuel, 30% in moderator compared to 94% in fuel and 6% in moderator for the prompt heat).

A detailed treatment of decay heat, separate from prompt heat, is necessary in case of temperature calculations after shut-down, or accident during start-up (when no fission products are yet available).

In the heat-transfer equations appear two parameters: the mass flow m and the coolant inlet temperature Tc 1. The mass flow, together with the control-rod position, is one of the quantities through which the control system acts on the reactor core.

The gas-inlet temperature can be influenced by the feedback of the heat exchangers and secondary system. In most core dynamics studies this inlet temperature is supposed to be constant (or at most, a known function of the average plant power level). This is usually a good approximation which allows the separation of the core from the secondary system.

In eqns. (12.20) and (12.21) the left-hand term can receive contributions from different element types. In reactors with prismatic fuel usually each channel receives heat from only one type of fuel, or at most from the fuel pins and from the moderator. In the case of pebble-bed reactors various types of elements can give heat to the same channel.

In reactors with continuous or frequent reloading the different types of elements should represent the different burn-up stages. Both temperature distribution and temperature coefficient feedback are different in fresh and depleted elements. The grouping of all elements in one average class may lead to errors of the order of 20 to 30% in the maximum temperature increase following an accident.’3’ It is often sufficient to group the fuel elements in two classes, the fresh ones in one class and all the others in a second class. The treatment of different fuel element classes implies in the case of prismatic reactors the treatment of different channel types in each region.

The many years of HTR physics experience of the Dragon Project have seen great progress in the development of reactor physics to meet the special requirements of HTRs.

Because of the international character of the Project there has always been a very close collaboration by the Project’s physics team with the national groups. The policy has been to meet the requirements of the Dragon Reactor with its constantly changing layout of experiments as well as power reactor studies by getting the best methods and programmes where available, adapting and developing them, and originating within the Project programmes and programme systems only where suitable alternatives did not exist.

In this way a very close collaboration with the various organizations of the Dragon Signatory Countries was built up and very good contacts between the Project and centres in the U. S.A. established.

The Dragon Countries Physics Meetings (DCPM) were set up and have now been running for some years, involving research centres, industry, utilities and regulatory bodies in the Project’s Signatory Countries. The purpose of these meetings has been the exchange of information at the forefront of development and many bi- and tri-lateral collaborations on the theoretical as well as the experimental research and development side have been initiated.

It was, therefore, only natural that the Project should initiate and sponsor the writing of a book on the present state of HTR physics at a time when the HTR is entering the nuclear energy market and interest in it will widen on many fronts, at industry and university research levels, not only with respect to its use as a heat source for electricity generation, but, looking ahead, also as a source of very high-temperature heat for industrial processes.

The challenge of steadily increasing requirements, of high complexity, on the experimental side and demands for high accuracy has been met successfully over the years. The present state is characterized by the fact that the physics of the HTR is now so well understood and so well checked against experiments that present-day require­ments and demands for designing and commissioning HTRs can be adequately met.

The further development of the HTR to realize its potential as a source of heat at very high temperatures, and also the prospects which it offers to achieve high conversion ratios, requires continuing physics studies and progress in methods. In particular, higher accuracies in the predictions of temperatures and their gradients, as well as fast neutron doses and burn-ups over the lifetime of fuel elements will be wanted, since these are main features which have to be considered in the fuel and fuel

element endurance and performance, and determine to a great extent the release of fission products.

Present and future requirements seen together make clear that it is now the right time to survey what has been achieved and compile what is now fully available and will form the basis and starting point of any further progress.

Winfrith, Dorset July 1975 H. Gutmann

Head of Physics Branch, OECD High Temperature Reactor Project (DRAGON)

ACKNOWLEDGEMENTS

Thanks are due to the Chief Executive of the Dragon Project and to Dr. P. J. Marien for their continuous support, as well as to the Members of the Staff of the Dragon Project Physics Branch, to Dr. A. T. Butland and Mr. P. B. Kemshall of the United Kingdom Atomic Authority, and to all scientists who indirectly contributed to this work by discussing their methods and results at the Dragon Countries Physics Meetings.

The Author also wishes to thank the Commission of the European Communities for permission to publish this book

In the case of solid moderators it is necessary to consider the exchange of energy between the neutron and the crystal. Energy can only be transferred in discrete amounts corresponding to the vibrational quanta of the crystal. A quantum of vibrational energy of a crystal is called phonon and in the energy exchange with neutrons the crystal absorbs or emits phonons. In the case of graphite the carbon atoms
are arranged in a hexagonal pattern in planar sheets. The valence bond between atoms acts almost completely within the separate sheets. Binding between different sheets is weak. This gives rise to a very anisotropic structure and one must distinguish between the phonon frequency spectra parallel and perpendicular to the lattice planes, рц(ы) and р±(ш), where the probability of phonons having frequencies between ш and ш + da is ри(ы) du> and p±{u>) du>, respectively.

The scattering cross-section consists of two parts: coherent and incoherent. The first contains interference effects between the neutron waves scattered from the various nuclei, analogous to X-ray diffraction. In graphite these interference effects are noticeable even in material consisting of randomly oriented crystals, but since they largely occur in that part of the scattering cross-section referring to interactions without neutron energy change they need not be treated in neutron thermalization theory, which even for graphite may be quite accurately performed in the incoherent approximation.

Using the incoherent approximation one can obtain (see ref. 2)

a(E’ -> E, cos во) = ^ VlF J Є ‘ * ‘ dt (6.9)

where оь is the bound atom scattering cross-section

with ov free atom cross-section, A ratio of scatterer to neutron mass, hк is the neutron momentum change vector

йк = m( v’ — v)

h reduced Plank constant, m neutron mass, v neutron velocity.

The scattering function (k, t) relates the change in neutron momentum to the change in quantum state of the scatterer, averaged over all orientations of the scattering material with respect to the incident neutrons.

For graphite a model developed by Parks gives13,4’

*(",f) = exp{^/0 l —^[(h + lXe-^-D + hCe^’-DldldcoJ (6.11)

with

p(cO, l) = l2p±(co) + (1 — l2)pu((d) (6.12)

M mass of the scattering atom

where / is the cosine of the angle between к and the normal to the lattice planes, and h the average number of oscillator quanta excited at the existing temperature T,

1

ft — ^ htnIkT _ j —

Substitution of (6.11) in (6.9) gives the scattering cross-sections. The difficulty of performing the time integration is avoided by a series expansion of х(к, t) called phonon expansion. These equations are programmed in various codes (e. g. SUMMIT,<5> LEAP’6,7’). The scattering kernel to be used in spectrum calculations is then obtained as

cr(E’ -* E) = f a(E’ -> E, cos во) dCl = 2tt f a(E’ ->

The frequency spectra p±(ш) and рц(ш) used in these kernel calculations for graphite are usually those obtained by Yoshimori and Kitano.’8’ Their theory is based on the assumption of four types of forces between the carbon atoms. Two act wholly within the basal planes, the third is determined by the displacement perpendicular to this plane relative to the three nearest neighbours from the same plane. The fourth is the only coupling assumed to exist between different planes and is related to the axial component of the displacement relative to the nearest neighbours in the axial direction. Comparisons of experimental and theoretical results for neutron spectra in poisoned graphite at several temperatures’3’ have shown that this model is quite adequate for all reactor physics purposes.

In reactors with discontinuous refuelling, or in the running-in phase of reactors with continuous refuelling, it is sometimes necessary to compensate the reactivity changes due to burn-up using the so-called burnable poisons. These poisons should burn in such a way as to compensate as nearly as possible the kc« changes due to burn-up.

Usually the cross-section of burnable poisons (l0B, Gd, etc….) is rather high and in order to avoid a too rapid burn-up these materials are lumped in rods, so that the effective cross-section is reduced by geometrical self shielding effects. The burn-up of these rods is calculated with methods similar to those used for the burn-up of control rods (see § 13.6).

The rods are subdivided in a certain number of regions whose depletion is calculated as in conventional burn-up codes with the difference that in this case diffusion theory cannot normally be employed because of the high localized absorption, and the flux distribution must be obtained with transport theory. Collision probability methods are usually employed for this purpose/24’25’

The Boltzmann equation which is conceptually very simple can only be solved analytically in very few cases.

We will not deal here with these classical solutions although they are very instructive and important for the reactor physicist. The most important classical cases can be seen in many textbooks (e. g. refs. 1 and 2).

Numerical solutions are, of course, always possible, and these numerical methods are of fundamental interest for the reactor physicist. The problem is strongly complicated by the complex energy dependence of the parameters of the Boltzmann equation. The neutron cross-sections vary by many orders of magnitudes in the energy range of interest and most heavy isotopes exhibit sharp resonances where cross-sections have enormous variations in a very small energy interval.

During the fission reactions neutrons are emitted with an energy distribution which has an average value of approximatively 2 MeV and becomes negligibly small above 10-15 MeV so that the maximum energy of interest in nuclear reactors is of about 15 MeV. Neutrons are then slowed down by collisions with the reactor components approaching, in a thermal reactor, the energy distribution corresponding to the moderator temperature.

To describe the energy-dependent neutron spectrum in a reactor means to follow the neutron history in this energy range considering its probability of absorption, slowing down by elastic and inelastic collision, and of leakage out of the system.

The only accurate way of treating this complex problem consists of discretizing the energy dependence in a high number of energy ranges. Because of the very strong energy dependence of some neutron cross-sections this number of ranges should be extremely high. The problem has normally been solved confining the treatment to a rather limited number of energy ranges (groups). As the energy dependence of the cross-sections within these groups cannot be neglected, group average cross-sections have to be produced. The number of groups to be used varies according to the complexity of the calculation being performed.

Normally between 40 and 200 groups have been used for high-temperature neutron spectrum calculations, but in recent times codes with a much higher number of groups have been developed for calculations at resonance energies.

In an N group scheme, if the energy boundaries for group і are Е,-1 and Et, the angular flux of group і is defined as

(4.5)

A straightforward way of obtaining the multi-group equations would consist in integrating the Boltzmann equation over the energy range of each group. Unfortunately
this procedure would result in the definition of angle-dependent group cross-sections (ref. 2, p. 51).

[‘ Zx(r, E)il>(r, E,il)dE

2,. , (r, ft) = E[ with x = t, f, s, etc.

ф(г, E, ft) dE

A simple way of avoiding this problem is to assume a separation of the energy dependence from the space and angular dependence within each group and within the region of interest:

ф(г, E, SI) = f (r, ft)<p, ІЕ) within group i,

ф>(г, п)=fir, ft) /;

Integrating eqn. (4.4) over the group і we have, under the assumption of separability,

ft • Уфііг, ft) + 2u<Mr, ft) = X

k = 1

+ ТГЇ-2 *2/.* [ Фir, Sl)dSl’

#Ceff47T к = і J

where we have defined the average constants:

fj 2,ir, E)<p, iE) dE

S*,i(r) = E’~’ E|—————————- with x = t,/, s, etc.;

I <p, iE)dE

J E,_,

The fine energy dependence of <p,(.E) within the group і is not known. Approximations must be used with which we will deal later. In general, if the number of groups is sufficiently high, the cross-sections are fairly constant within the group, in which case the group cross-sections are almost independent of the assumption made for <p,(.E). This is not true in case of resonances unless the number of energy groups is extremely high (see Chapter 7).

Another possibility of obtaining a multi-group formulation while avoiding the definition of angle dependent cross-sections consists in first approximating the angular dependence by means of expansion in eigenfunctions (P, method) or numerical discretization (S„ method) and then integrate over the energy range of each group.

In the above treatment we have supposed that the groups define non-overlapping energy ranges. A treatment with overlapping groups has sometimes been used. For example, at a boundary between two regions (e. g. core and reflector) it is possible to suppose that the flux is a linear combination of the two spectra typical of the two adjacent regions. These two spectra can be considered as two overlapping groups. In order to calculate the fluxes it is necessary to have a physical model describing the transfer of neutrons from one group to another. The difficulty of finding an adequate model has until now strongly limited the utilization of this method.

The escape probability P0 can be easily calculated for isolated lumps in the two extreme cases of very large or very small mean free path, under the assumption of a flat source of neutrons in the region. If the body is large compared with the mean free path (black limit)

Vo volume, So surface (see ref. 3, p. 117). For very large mean free path obviously Po-»l.

For intermediate situations the following rational approximation has been proposed

or defining the escape cross-section 2e = l//0

The values given by this approximation are systematically too low, up to 20% in the neighborhood of /02,o = 1.

In general for isolated lumps (that is with a spacing of several mean free paths) P0 is given as a function of /02,o (see, for example, Case, de Hoffmann, Placzek<l3) where these functions are given for various simple geometries). If the fuel lumps are closely packed the so-called Dancoff correction has to be employed.

The Po to be used in (7.14) has in this case to be substituted by Pо which is given by the escape probability P0 of an isolated lump multiplied by the probability that the neutron escaped from the lump has its next collision in the moderator region.

This correction was first introduced by Dancoff and Ginsburg.<l4> Its exact calculation can only be performed with Monte Carlo methods. A useful approximation can be obtained as shown in Fig. 7.1.<l5>

Let us consider the neutron trajectory of Fig. 7.1. The probability of survival without collision on the leg Ri is

Г, =

The escape probability from one lump is given by

Ро = г=-(1-Г„) (7.22)

where l0 = mean chord length,

X, o = total cross-section,

Го = average of Г0 over all possible chords of the lump.

The probability of making the next collision in the moderator is

P’o = ї4-{(1 — Г„Ж1 — Г,) + Г, Г2(1 — Гз) + • • ]}• (7.23)

I oZlo

The averaging is over all possible trajectories from the surface of the first lump.

In eqn. (7.23) the terms linear in the Г, are the most important ones if the lumps are big compared with the mean free paths. For the higher terms the average of the products is then replaced by the product of the averages. This will also be a good approximation for small dimensions. We obtain then

p. 1 (1-ГоХі-Г,)

/оХіо 1 — FoFi

here we have assumed that in a lattice geometry the averages of Г, for fuel lumps (even i) are all equal to Г0 and those for the moderator (odd i) are all equal to Г,. Introducing the Dancoff factor

С = Г, = e

and using for P0 the expression (7.22) we obtain

p’ = P ________ * ~ Є______

0 °1 -(1-/oS, oPo)C-

Using the rational approximation for Po

, 1 2, 0 1 + lo2,o 2, + 2ю 2,(1-C) SI 2ю + 2,(1 — C) 2u> + 2,

we have

(7.25) (7.26)

where 2, is defined as

here the Dancoff correction appears as a shadowing of the lump surface due to the adjacent lumps. A rational approximation can also be found for C.

Since (1 — C) is the probability that a neutron entering the moderator collides in it, C is the moderator escape probability Pi. In the rational approximation we have [see eqn. (7.21)]

(7.27)

with

(7.28)

The accuracy of this last expression is not very high.

The Wigner rational approximation for P0 does not give the correct behaviour for grey fuel lumps.

An improvement has been proposed by Bell introducing the so-called Bell factor a:

a 2,

a 2, — H 2to

a is not constant but depends on the geometry and on the scattering cross-section of the moderator mixed with the fuel.<l6) The values of this factor for various geometries are given, for example, in ref. 17.

The cost of enriched uranium is based on the cost of U308, the conversion of U308 in UF6 and the enrichment costs. The enrichment costs are expressed in units of separative work which give a measure of the required isotopic separation effort."7’ The separation work is related to the quantities of material fed to and withdrawn from the process and their enrichment. An enrichment plant splits the fed uranium with enrichment e„ in two quantities, the product with enrichment eP and the tails (depleted U) with enrichment є,.

The following considerations are based on an ideal diffusion cascade. If we consider the balance of the mass of 235U and 238U in an enrichment plant we have

M„e„ = M„e„ + M, e, 235U balance,

238 (10.22) M„(l —e„) = Mp(l —eP) + M,(l-e,) 238U balance

where M, is the mass of natural uranium with enrichment e„, Mp is the mass of product uranium with enrichment eP, M, is the mass of tails with enrichment e„

out of (10.22) we obtain

M„ є„ — e, , M, eP — e„

TT =———— and TT =—————- •

Mp e„ ~ є, Mp e„ — Є,

The plant consists of a series of stages.

For each stage we can define

€г, feed stream enrichment,

ex, enrichment of the depleted stream leaving the stage, tv, enrichment of the enriched stream leaving the stage.

A stage separation factor is defined as

as a is very near to unity the quantity

ф = a — 1

is also often used. From (10.24) and (10.25) we have

fed + Ф)

Єу 1 + фех ‘

since ф < 1 it is possible to expand this equation in a series

Єу = Єх( + ф)( — фех + ф2ех2+ ■ ■ •)

and ignoring the terms in ф2 or higher powers

Єу — ex = фех(1 — єх). (10.28)

Having these preliminary definitions it is possible to look for a “value function” V(e) giving the value of a unit of uranium as a function of its enrichment є. This value function can then be used to define the increase in value effected by each stage and hence give a measure of the separative work.

The increase in value effected by a certain stage should be independent of the enrichment of that stage, because the work done by a stage is only dependent on the amount of material, but not on its enrichment. The value function we are looking for should satisfy this condition. Assuming the existence of this function, the increase in value effected by the stage is

A = MV(ex) + MV(ey) — 2MV(f:) (10.29)

and this A is defined to be the “separative work” done by the stage under consideration. We have here assumed that the mass 2M entering the stage is equally subdivided between enriched stream and depleted stream (it is possible to obtain the same definition of V(e) without this assumption).

2M, «

Substituting (10.30), (10.31) and (10.32) in (10.29) and neglecting the powers higher than two of the small difference (ey — є,) we have

A = M(ey — ex)2 V"(ex)l4. (10.33)

Substituting (єу — є») by expression (10.28) we obtain

A = ^[ex(l-ex)fV"(ex). (10.34)

A must be independent of enrichment. This means

^) = ІІЛТ^)Г (1035)

The solution of this differential equation is

V(ex)=C0+Ciex+(2ex — 1)In (737)- ПО-36)

If we set = 0 the two arbitrary constants Co and Ci we get a particularly convenient form of the “value function”,

Substituting eqn. (1037) in (1034) we have an expression for the “separative work" (or increase in value) effected by a stage, which satisfies the condition of being independent of enrichment. The total separative work done by the plant is the sum of the As corresponding to all individual stages.

Separative work = ^ Д for ideal plant.

All stages

For a real plant 2Д is greater than for an ideal plant, but the amount of separative work

All stages

delivered is the same because the plant does the same job.

Considering the value balance of the entire plant we have

Total separative work = MpV(ep) + M, V(et) — M„V(e„). (10.38)

As the value function is dimensionless, the separative work is expressed in kilograms of uranium. The number of units of separative work to obtain a unit mass of uranium with enrichment ep is

w = V(ep)’+W V(e,)~W V(en)- (l0-39)

The ratios Mp/M„ and M,/M„ can be obtained from (10.23) and the value function from (10.37), so that we have

Cp = Cf + WCS (10.41)

where Cf = unit cost of U in the form of UF6, Cs = unit cost of separative work, and from

(10.22)

Cp = 6p ~6> Cf + WCS. (10.42)

€n

The unit cost of separative work Cs is calculated in such a way as to cover all costs of the separation plant.

An example of the cost of enriched 235U as a function of enrichment is given in Fig. 10.4.<M) This cost is, of course, dependent upon the value of the parameters specified in the figure and is therefore subject to changements.

The price of other fissile materials like 233U or Pu can be derived comparing their neutronic worth with the one of 235U. If the bred fuel is reprocessed and burnt in the same reactor it is not necessary to put a price on it. If it is burnt in a different reactor its price can be defined as the one giving the same fuel-cycle costs to the two symbiotic reactors.

When the bred fuel is recycled in the same reactor it is also possible to define its price as the price of the amount of enriched uranium which would be required to substitute the unit weight of the fuel under consideration. (This value is sometimes called “indifference value” because it makes the fuel cycle cost indifferent to the decision of recycling or selling the bred fuel.)

The value of the bred fuel is dependent upon the percentage of undesirable isotopes.

For example the 235U discharged from HTRs contains considerable amounts of 236U which is a parasitic absorber, therefore the “indifference value” of this discharged fuel must be calculated considering the 236U concentration. Another example is given by the bred 233U which is accompanied by 232U and its daughters which are у emitters involving additional handling costs. Even without 232U the handling costs of 233U and of Pu are higher than for 235U because glove boxes are required for the manipulation of these isotopes.

10.13. Methods for fuel-cycle cost calculations

The easiest way to calculate fuel-cycle costs is the so-called “Discounted cash flow” method which consists of present-worthing the cash flow X and the energy E:"4’18’

F(t)X(t)

C =——————- (10.43)

2 F(t)E{t)

where

F(t) = (1 + /) (r ro>

is the present worth factor with the reference time t0 usually chosen as the reactor commercial start up.

The summations of eqn. (10.43) extend from the time of the first payments before the reactor start up to the first 10, 20 or 30 years of operation. The cash flow X must consider the expenditure for fissile material, enrichment, fertile material, shipping, fabrication and reprocessing: if bred fuel is being sold also these revenues must appear in the cash flow. At the end of the accounting period the value of the fuel present in the reactor must be assessed and credited to the cash flow.

This method has some disadvantages.

1. It gives only an average fuel-cycle cost and not a cost per reload interval or per segment.

2. The same interest rate must be used for the pre-irradiation, in-core, and post­irradiation time intervals.

3. It requires the evaluation of a final settlement at the end of the period.

These disadvantages are avoided by the “Discounted energy cost” method."4’ In this method each segment is divided into several accounting periods, usually corresponding to the reloading intervals. It is then possible to calculate the fuel-cycle cost of each segment s for a given reloading interval n. Let P,„ be the change in value of the segment s during the reloading interval n present-worthed to the midpoint (or to the beginning) of the interval.

The law according to which the fuel value is assumed to change with time during its residence in the core can only be defined in an arbitrary way because only the values before and after irradiation are known. It is possible to assume that the fuel value decreases linearly as function of the produced energy, or as function of burn-up. The Dragon KPD code"9’ assumes that the fabrication costs are completely assigned to the time interval at the beginning of which the segment is loaded, and that the reprocessing expenditures are assigned at the interval at the end of which the segment is discharged.

Let £sn be the energy produced by the segment s in the interval n present-worthed to the midpoint (or to the beginning) of the interval. Here again only the total energy (E„) produced by the core during the interval n is known, and the energy Es„ can be defined as the fraction of E„ corresponding to the volume fraction occupied by the segment s in the core.

En = 2 £„.

5

The fuel-cycle cost associated to the segment s in the interval n is

C„=!^. (10.44)

c,„

The fuel-cycle cost during the reloading interval n is

(10.45)

It is also possible to calculate C„ directly as

2 P

C„ = ——

En

without the need of defining the energies per segment £,„. This procedure is used by the KPD code."91

If the C, n have been calculated it is possible to define the average fuel cycle cost for segment s: 2 C, nEsnF„

2 E, nFn

where F„ is the present-worth factor for interval n.

The present-worthed levelized cost is given by

The differences between C, C„ and Cs are due to the fact that, especially during the running-in phase, the amount of fissile material consumed and the capital charges accumulated before producing energy can be different from one segment to another.

In all the above formulas we have not specified whether the energy E is an electrical or thermal energy.

If the product which is being sold by the plant is process heat E will be a thermal energy, while if the product is electricity E is an electrical energy. The thermal energy produced by the core in a given time interval can be easily calculated from the number of fissions which have taken place. The electrical energy is then calculated multiplying the thermal energy by the net plant efficiency.

10.14. Simplified calculations for the equilibrium cycle

With the above described method it is possible to obtain the fuel-cycle cost C„ for any reloading interval n. This cost will remain constant once equilibrium is reached and its value will give the cost of the equilibrium cycle.

For survey studies simplified methods are often used."8’ We can think that the fuel cycle cost C is composed of a running cost Cr (out-of-pocket cost without capital charges) and a cost C, due to interest on investment.

Considering a fuel segment

(10.50) where A = cost of the fuel charged in the reactor (including fabrication),

В = value of the discharged fuel (after deduction of shipping and reprocessing costs),

E = energy produced by the segment.

The calculation of the investment cost requires an evaluation of the value of the fuel present in the reactor.

A simplification which is often used consists in assuming that this value is<20)

A+B

2

so that

(10.52)

where Ta is an equivalent pre-irradiation time, Ть is an equivalent post-irradiation time, T is in-core residence time, і is interest rate.

While the running cost is independent of time but only depends on the fuel consump­tion, in eqn. (10.52) appears the residence time. A low load factor of the plant increases the residence time and hence the interest to be paid.

Perturbation theory deals with the calculation of the effect of small changes in the properties of a reactor. If the perturbation is sufficiently small it is not necessary to perform a complete new calculation of the reactor, but the adjoint function of the transport operator can be used to obtain the response to the small change (perturba­tion). Perturbation theory used to be a very important tool for reactor calculations before the general introduction of high-speed computers. It was used to calculate temperature coefficients, control rod worths, etc. In practice the so-called perturbations are seldom very small and therefore the accuracy of the method is rather limited. Because of this reason it is nowadays preferred to perform two numerical calculations one on the unperturbed and one on the perturbed system, obtaining as a difference the

effect of the perturbation (e. g. reactivity variation due to control rod insertion). It is very important that these two calculations are absolutely identical (e. g. same meshes) except for the perturbation. Only in cases where the perturbation is really small can perturbation theory be more accurate than numerical methods.

Because of the above considerations we will not discuss this method, for which we refer the reader to the existing literature (e. g. ref. 2, chap. 6).

We will simply here recall the definition of neutron importance. This quantity, which is seldom used for reactor calculation, is frequently used to give qualitative explana­tions and to get a “feeling” of what happens in a reactor.

The neutron importance ф*(г, SI, E) can be defined as the increase in the neutron population of a critical reactor due to the introduction of one neutron in point r, with energy E in the solid angle SI. It can be simply demonstrated (ref. 2, § 6. Id) that the neutron importance is the adjoint function of the transport operator. The definition of importance can be extended to the group diffusion approximation of transport theory. In one group theory the importance is coincident with the neutron flux. It is intuitively clear that the neutron importance is high in the centre of the core, while it is zero at the outer surface for the outward direction. Perturbation theory is often used in order to obtain the parameters for point model kinetics calculations (see § 12.3).

The fine group libraries with the energy structures listed in Table 8.1 must be produced and regularly updated starting from cross-section sets of the type of ENDF/B (see § 3.4).

Processing codes are used in order to generate from ENDF/B a library usable in codes for spectrum calculations.

The fine group cross-sections must be obtained from the point values of ENDF/B assuming a certain form of the flux within these fine groups (e. g. constant flux per unit lethargy in the fast range).

The thermal neutron scattering law data are represented in ENDF/B for each moderating molecule n, by a scattering law S„(a, /8, T) where /3 is proportional to the energy change, a is related to the momentum change, and T is the temperature in °K.

The differential scattering cross-section is given by:l23>

<т(Е -> E’, cos 0o, T) = 2 VI e"P/2S"(a’ & T)

where there are (NS + 1) types of atoms in the molecule (i. e. for H20, NS = 1), and

M„ = number of atoms of the nth type in the molecule,

P=(E’-E)lkT,

a = (E’ + E — 2 cos 0oVEE’)/Ao/cT,

An = mass of the nth type atom, A0 is the mass of the principal scattering atom in the molecule,

(Thn = bound atom scattering cross-section of the nth type atom,

Only the data for the principal scatterer of the molecule (e. g. H in H20), S0(a, /З, T) are tabulated in ENDF/B. The scattering properties for the other atom types (n = 1,2,…, NS) are represented by analytic functions.

Supplementary codes (e. g. FLANGE’24’) are used to produce the actual scattering kernels. This approach gives the possibility of obtaining scattering kernels for different energy group structures having stored the scattering law S„(a, /З, T). Various interpola­tion schemes are given to interpolate between the values of a, /3 and T. This scattering law is obtained starting from the phonon spectra of the scatterer under considera — tion.’2526’ (See also § 6.3.)

As the data stored in the transfer matrices have often only rather smooth variations it is possible in some processing codes to reduce the computer store requirement by means of polynomial fittings.

For nuclides like structural materials and fission products the resonance calculation is usually done by the processing codes assuming zero temperature and concentration.

For fuel and fertile materials this calculation is done later in the spectrum codes.

As an example of a processing code we will quote SUPERTOG<27) which averages the ENDF/B data over specified group widths. The flux per unit lethargy is assumed to be constant unless suitable weight functions are supplied by the user. When resonance data are available resonance contributions are calculated using Breit-Wigner expres­sions. The point cross-sections are integrated in order to obtain smooth cross-sections. Elastic scattering matrices are computed from Legendre coefficients of the scattering angular-distribution data. Inelastic scattering and (n, 2n) matrices are computed considering the excitation levels of the considered nuclides. This code can be used to produce libraries for codes for spectrum calculation or for other transport codes.

Other processing codes are described in refs. 28 to 32.