As already mentioned the feedback of the control system on the reactor core acts through two quantities: the control-rod position and the coolant mass flow. For accident analysis (short-time dynamics) it is not usually necessary to introduce a description of the control system in the already complicated system of equations describing the core dynamic behaviour. It is then sufficient to suppose that when a given quantity (e. g. neutron flux, coolant outlet temperature, etc.) exceeds a previously determined threshold value, a scram occurs. This means that a negative Дkc and a reduction of the coolant mass flow are introduced as given functions of time.

In the reactor operation and stability studies (long-time dynamics) the control system has to be simulated in the dynamics equations. It is usually required that the power Qr(t) released by the reactor follows a given power diagram Q(t) while a given gas-outlet temperature Tg must be maintained. Various methods of regulation can be used. A regulation of the thermal power through the coolant mass flow would be in general represented by

(12.22)

where KP, Ki and KD are the coefficients of the proportional, integral and differential member respectively, and

Qr = cPm(T, 2- T,,).

A regulation of the gas outlet temperature through the control rod reactivity Дkc would
be represented by

Дкс = KP(Tg — Tc2) + KiJ (Ts-Tc2)dt+KD-^(Tg-Tc2). (12.23)

See also § 12.14 for details on spatial instability and sector controllers.

1.1. Basic concepts

The temperature limitations of conventional reactors impose a low thermal effi­ciency, use of special turbines and consequently cause higher fuel consumption and thermal pollution of the environment. Furthermore, the low outlet temperature limits the possibilities of using nuclear power as an industrial source of heat.

High-temperature reactors have been developed in order to avoid these limitations using a core composed exclusively of ceramic materials and employing an inert coolant.10 This limits the choice of the moderator to graphite or beryllium.

Beryllium is a very good moderator, but its use is limited by technical and economic difficulties, so that nowadays graphite appears to be the only possible moderator for HTRs.

The coolant must be chemically inert, not undergo any phase change, have good heat-exchange properties and not be activated by neutrons. Helium is the only coolant able to satisfy these conditions and that is naturally available in sufficient quantity. The coolant outlet temperature of the HTRs now in operation ranges between 750° and 950°C with an inlet temperature around 300°C. As the high temperature excludes the use of metallic fuel sleeves, the fission product retention takes place initially at a microscopical level through the adoption of coated fuel particles.<2,3> These particles consist of kernels of oxides or carbides (of U, Th or Pu), whose diameters range between 200 and 800 /x m, coated with various layers of pyrolytic carbon (PyC). Layers of silicon carbide (SiC) are sometimes used in the coating in order to improve the retention of metallic fission products.® The coating thickness is of the order of 150-200 fi m.

If no SiC is present the coating consists of an inner low density buffer layer and an outer high density layer (BISO or duplex particles). If SiC is used the coating consists of an inner low density buffer layer, an inner high density pyrolytic carbon layer, a silicon carbide layer and an outer high density pyrolytic carbon layer (TRISO or triplex particles, Fig. 1.1).

These particles are dispersed in a graphite matrix.

In order to avoid corrosion due to impurities of the coolant and to further improve the fission product retention the part of the fuel element which contains the coated particles is separated from the coolant by means of graphite.

This basic concept leaves the possibility of various types of fuel geometries. In the first experimental reactors the fuel element had the form of hexagonal prismatic

(Dragon) or cylindrical (Peach Bottom) rods, with the coolant flowing between these rods.

Since then a more robust geometry has been devised for power reactors. The fuel consists of big hexagonal blocks (40 to 60 cm across flats) containing a regular pattern of fuel and coolant holes. The coolant holes can either contain the fuel in the form of pins or the fuel can be located in geometrically separated holes. In the second case (General Atomic version) the heat has to flow across the bulk moderator before reaching the coolant (Fig. 1.2).<4) In the first case (pin-block design) the fuel pins consist of a carbon matrix containing the coated particles, isolated from the coolant by a layer of unfuelled carbon (Fig. 1.3).<5) These pins can have different geometries, some of which are represented in Fig. 1.4.

The tubular design is shown on the left: the coolant flows inside and outside the pin. The dotted area represents the fuel containing matrix. On the right side is shown the teledial geometry, so called because of its resemblance to a telephone dial (Fig. 1.4).<5)

The core arrangement of a reactor with multi-hole block fuel is shown in Figs. 1.5 and

1.6.<4)

In parallel to the HTR with prismatic fuel a version with spherical fuel elements has been developed in Germany (pebble-bed reactor). The fuel elements have the form of graphite spheres of 6 cm diameter with a fuelled inner zone (see Fig. 1.7). It is possible to vary the fuel loading by mixing dummy graphite elements with the fuel spheres. These spheres which are contained in a reflector of graphite blocks are loaded pneumatically by means of tubes from the upper reflector and extracted from one or more tubes located in the bottom reflector (see Fig. 1.8).<6)

Apart from the described geometrical differences the HTR fuel elements differ also according to fabrication methods which influence the material structure and therefore the behaviour under heat flow and irradiation. Carbon can be in the form of machined graphite, or of a compacted matrix. The first pebble-bed fuel elements, for example,

Graphite

sleeves

consisted of a machined graphite shell which contained in the inner part a fuel compact of coated particles dispersed in a carbon matrix. The present THTR elements have a similar geometry, but the outer shell consists also of a pressed carbon matrix. The multi-hole blocks used by General Atomic for the Fort St. Vrain reactor are made of machined graphite, but by analogy with the THTR fuel, integral blocks made of a pressed carbon matrix are being developed in Germany (monolithic blocks). An ultimate development of this integral block technique may eventually consist in dispersing the fuel throughout the block instead of concentrating it in the fuel holes, only leaving small unfuelled layers around the coolant channels.

The principal differences between the various fabrication methods concern the behaviour under irradiation and the heat conduction properties, while for the reactor physicist the fuel is sufficiently characterized by geometry and atomic concentration of each region.

The calculation of scattering kernels with the full treatment of the crystal binding is a very lengthy procedure. In practice it is not possible to compute a kernel for each calculation, but these data are stored on computers for various discrete values of temperature. For high moderator temperatures this full treatment is no longer neces­sary. Table 6.2 shows that whereas at 300 К chemical binding effects can reduce ксЯ by up to 2.4% at 1200 К the reduction is only up to 0.4%; the latter is small and shows that the free gas treatment is adequate above 1200 K. For this reason at high temperatures even solid moderators can be treated as ideal monoatomic gases (free gas model).

In this case the calculation of the scattering kernel becomes very simple. Supposing that the gas atoms have a Maxwellian velocity distribution, the following expression based on simple classical hard sphere collision has been derived by Wigner and Wilkins (ref. 9; see also ref. 1, § 10.1, and ref. 2, p. 26):

ME’- E> = І f erf [, ~ P V#]

+erf [, v|- p VS] — (•-“«* NS+о V5]

-e“”ertNS+WS])if^fi) <6’14)

where crsf is the free atom-scattering cross-section here supposed to be energy independent, T the temperature and

A + 1 A — 1

V 2VA P 2VA ’

This expression can be quickly calculated by a computer so that in some cases the free gas kernels are not even stored, but directly calculated for each case. One must notice that when developing a set of scattering kernels at various temperatures it is important to have a smooth variation of kernel with temperature in order that calculated reactor quantities vary smoothly with temperature. This means that chemical binding should be considered to temperatures somewhat above what might otherwise be thought neces­sary.

10.1. Objectives

The objective of the core designer consists in obtaining with the minimum costs a reliable and safe reactor capable of satisfying given requirements.

In the case in which the reactor is used for electricity production these requirements are given by the maximum power-level and load-following capabilities, this second requirement becoming less important in the case of heat production for industrial purposes.

Safety and reliability can only be obtained if a certain number of constraints are satisfied concerning core materials, reactivity, etc.

Among all possible choices, the core designer has to find the solution of minimum cost under the given constraints. Such a process is called optimization. The possible choices are represented by a certain number of independent parameters which can be varied by the designer; their choice can be rather arbitrary, at the limit only the total reactor power and its load-following performances being fixed from the beginning. The optimization would then be extended from general decisions like the choice of the reactor type and its siting, down to the last technical details.

As this is obviously too complex a task, it is necessary to separate the general problem into a number of partial optimizations. In a general plant optimization one will, for example, limit the representation of the pressure vessel to an expression giving its cost as function of the core size, leaving any detailed design to a separate optimization. Whenever possible, different plant components are optimized separately. In this way the core optimization in a plant with steam turbine can often be performed without considering the secondary circuit, which is indepently optimized.

Previous experience is normally used to fix a high number of variables in the form of a reference design, in order to restrict the optimization to a limited number of parameters. Also the representation of the constraints depends on the amount of computational work one can invest in the optimization. On the fuel element side one can, for example, simply define a limit on the age factor (ratio between the maximum and the life average value of the macroscopic fission cross-section of the element) supposing that this parameter sufficiently defines the difference between the maximum fuel temperatures of various fuel cycles.

In a more detailed optimization simplified temperature calculations can be introduced in the procedure in order to limit temperatures instead of age factors.

An even better representation would set a limit on the probability of coated particle rupture and fission product release, calculating this probability as a function of the exposures to temperature, burn-up and fast flux the fuel has had during its life.

As the designer has to remain on the conservative side, every excessive simplification in the constraint representation tends to exclude as impossible some low cost cases which might be actually acceptable.

We have seen how the treatment of the energy dependence in the transport equation leads to the multi-group formulation. Let us now tackle the problem of the angular dependence of the neutron flux. The approximations used for all practical reactor calculations are characterized by the methods and assumptions with which the angular dependence is treated.

These methods substitute the angular-dependent Boltzmann equation with a system of differential equations in which the angular dependence has been eliminated.

We can distinguish the following approaches:

1. Numerical discretization of the angular dependence (S„ method).

2. Expansion in eigenfunctions (spherical harmonics): Pi or B„ method.

3. Pi expansion truncated to / = 1 and assumption of slowly varying collision density (diffusion approximation).

A rather different approach consists of writing the Boltzmann equation in an integral form in which angular dependence does not explicitly appear (collision probability methods).

The methods we have seen up to now to treat resonance absorption require a considerable computational effort, so that it is often desirable to use more rapid approximate methods. This is usually done by establishing a library of resonance integrals in homogeneous medium, and for all practical cases by looking for the homogeneous mixture which is equivalent to the actual heterogeneous case.

Such an equivalence can be theoretically proved under the following assumptions:

1. Narrow resonance approximation for moderator collisions.

2. Rational approximation for the collision probabilities.

Under assumption 1, eqn. (7.3) for the homogeneous case becomes:

(7.30)

and eqn. (7.14) for the heterogeneous medium:

Assumption 2 implies with = 2,(1 — С) = (1 — С)Цо (the results are also valid if a Bell factor a is used in this formula) which substituted in (7.31) gives

(7.32)

A comparison between (7.32) and (7.30) shows that the heterogeneous system is equivalent to a homogeneous system in which the term has been added both to the scattering and the total cross-section. This term can be interpreted as a pseudo-cross­section representing escape from the lump (escape cross-section). The equivalence relation can also be explained in the following way. The absorber region is limited geometrically to the fuel region and energetically to the resonance region (which in this approximation is supposed to be narrow). A neutron can escape resonance absorption leaving this region either geometrically or energetically (scattering collision in the narrow resonance approximation). The equivalence relation consists in simulating the geometrical escape by the addition of a fictitious scattering cross-section (energetical escape). This result is, of course, also valid if NR or IM approximations are made for the collision with the absorber atoms.

Very often the macroscopic cross-sections 2to, 2sm, etc., are transformed into microscopic cross-sections dividing them by the atomic density N0 of the absorber in the fuel region. In this way one can speak of a scattering cross-section per absorber atom.

In the NR or IM approximations where for the resonance absorber only the potential scattering appears one speaks of a aP, potential scattering cross-section per absorber atom, and of a apctt, effective potential scattering cross-section per absorber atom

0-p eft = crP + cr e (7.33)

and

if the equivalence relation is used.

Using the equivalence relation it is possible to save a considerable amount of computer time in codes for spectrum calculations. Instead of performing a detailed calculation for every case, many computer codes have resonance integrals stored for different homogeneous mixtures and different temperatures, i. e. for each resonance absorber two-dimensional tables of resonance integrals are stored as function of crpPtt and temperature. If different moderators are considered a second type of equivalence can be established between any given moderator and hydrogen. This procedure is used in the UKAEA WIMS code<l8) where resonance integrals are stored only for homoge­neous mixtures of resonance absorbers and hydrogen. For each moderator it is possible to establish the equivalent amount of hydrogen to be used in WIMS.<,9,20)

11.1. Definitions

The temperature coefficient of a reactor may be defined as the rate of change of reactivity per unit temperature change, and is in general a function of temperature.

A negative temperature coefficient leads to a self-stabilizing reactor. On the contrary, if the temperature coefficient is positive, the reactor is unstable and any flux distur­bance tends to amplify itself. In this case the reactor can only be operated if the control system counteractions stabilize it. A positive temperature coefficient imposes much more stringent requirements on the control system. On the other hand, a negative coefficient of very high absolute value, while improving the reactor safety, requires a large number of control rods to shut down the reactor in cold conditions. The temperature coefficient is the result of various effects and can be then subdivided in different ways.

It is customary to separate the fuel from the moderator contribution,

(11.1)

defining in this way a fuel and a moderator temperature coefficient.

Since following a transient the various contributions to the temperature coefficient may respond with different delays, it is also customary to speak of a prompt and a delayed temperature coefficient.

The two definitions are not necessarily the same, especially in high-temperature reactors where part of the moderator is intimately mixed with the fuel. The distinction between prompt and delayed coefficient depends on the fuel and moderator geometry and may be rather arbitrary. The distinction between fuel and moderator temperature coefficient is based on better-defined physical reasons.

The fuel temperature is determining the Doppler-broadening of resonances, while the moderator temperature defines the thermal spectrum. An interaction of these two effects can be given by low-lying resonances which can interact with the thermal spectrum (e. g. 240Pu). The effect of the thermal expansion and density change in the reactor is important in some reactor types, but negligible in high temperature graphite moderated reactors. In terms of the “four-factor formula” the temperature coefficient can be separated in the following way"’

where Р/ and P, h are respectively the fast and thermal non-leakage probability. This

expression is not used for calculating the temperature coefficient, for which multi-group computer codes are used, but this splitting provides a better physical insight.

Formula (11.2) can also be written

1 dke« 1 dkx 1 6Pnl,..

ійіг "Laf u ’

PNL being the total (fast and thermal) non-leakage probability. In this way it is possible to see the effect of temperature on к^ and on the leakage. The total leakage contribution is usually negative. The thermal leakage provides always a negative contribution because the thermal diffusion coefficient increases with temperature. The fast leakage has usually a small positive contribution because Doppler broadening may decrease the fast diffusion coefficient with temperature. The result is usually negative, so that a high leakage has a stabilizing effect on the reactor.

Sometimes a “power coefficient” is defined. This coefficient has only a meaning if the temperatures are in the equilibrium condition corresponding to a given power and if each power level corresponds to a given coolant mass flow. In this case, knowing the temperature coefficient, it is possible to calculate the rate of change in reactivity per unit power change.

During the reactor operation temperatures are space-dependent. Suitable average temperatures have to be defined in order to be able to define a temperature coefficient. Usually arithmetic averages are sufficient. Average fuel and moderator temperatures have to be defined either for the whole reactor (for zero-dimensional dynamics calculations) or for various reactor regions (for space-dependent dynamics calcula­tions). HTR temperature coefficients are of the magnitude KT5/°C, so that they can be calculated with sufficient accuracy as a difference of two multi-group static calcula­tions, performed at temperatures differing a few hundred degrees.

For a reactor with no external sources the diffusion eqn. (4.39) takes the form

(4.59)

Let us now consider the case of a homogeneous medium with energy independent boundary conditions of the type

(4.60)

where дфідп is the derivative in the direction perpendicular to the boundary, and d the so-called extrapolation distance. This condition is equivalent to ф = 0 at the extrapo­lated boundary (except in the case d ->» in which the condition is rather equivalent to дфідп = 0 at the boundary).

In this case the wave equation

V2f(r) + Bffi (r) = 0

determines a complete system of eigenfunctions on the considered domain so that one can in general expand the flux

(4.62)

One can substitute this expansion in the diffusion equation (4.59). If the equation so obtained is multiplied by f (r) and integrated over the whole volume, remembering that

these eigenfunctions are orthogonal, i. e. and

V2f,=-B2ft

we have

DB2<Pi(E) + X,(Pi(E) = J 2s0(E’^ £)<?,(£’) d£’ +

This equation can only be satisfied by one value of В from eqn. (4.62)

<Hr, E) = <p(E)f(r).

As ф(г, Е) must be positive throughout the reactor, f(r) must be the fundamental solution of the wave equation (4.61) corresponding to the smallest eigenvalue B2.

This is only valid if the coefficients D, X,, Xf, etc., are space independent and can be extracted from the integral over the reactor volume.

Under the above-mentioned restrictions (homogeneous medium with energy inde­pendent boundary conditions) is valid what is usually called

First Fundamental Theorem of Reactor Theory (see ref. 1, p. 382):

The stationary neutron distribution in a critical bare reactor is separable in space and energy

ф (r, E) = <p (E)f(r) (4.63)

where f(r) is the fundamental solution of the wave equation

V2/(r) + B2/(r) = 0 (4.64)

that is that solution which is positive throughout the reactor and vanishes at the extrapolated boundary.

The equation for cp(E) is then

DB2<p(E) + X,<p{E) = jXs0(E’^E)<p(E’)dE’+ f v(E’)X,(E’)<p(E’) dE’ (4.65)

and the leakage term is expressed as function of В2 which is called geometrical buckling and is the smallest eigenvalue of (4.64).

DV2</> (r, E) = D<p(E)V2f(r) = DB2<p(E)f(r) = — DB2<p(rE) (4.66)

it is also possible to obtain this theorem starting from the Boltzmann equation, for the flux integrated over the angle.<23)

This first fundamental theorem is of limited practical use because it applies only to a homogeneous bare reactor. In a reflected reactor, or in any region of a multi-region reactor, the boundary conditions (4.60) are energy dependent, for example there is a leakage of fast neutrons towards the reflector and a return of thermal neutrons to the core. In practice the problem is solved in multi-group formalism introducing group dependent bucklings B2 for each group i.<24)

Within each group і eqns. (4.63) and (4.64) are valid if the boundary conditions are supposed to be energy independent in each of these energy ranges. In this way it is possible in eqn. (4.40) to substitute

-ДУ2фі(г) = ПВі2фі(г). (4.67)

The leakage term takes then the form of an absorption term and the total losses are

(QB,2 + 2„)«fr.

This substitution can be used to reduce a space-dependent to a space-independent problem, e. g. in the case of spectrum calculations (see § 8.2) provided the В2 are known. In order to obtain the B2 it is theoretically necessary to solve the multi-group space-dependent problem, but a rough approximation of the Bf is sufficient for many applications. Energy-dependent bucklings can also be used to reduce the number of space dimensions. This implies the assumption of separability of the space dependence of the flux along the different coordinates. It is then possible to solve a three­dimensional problem treating explicitly the space dependence in one or two dimen­sions, representing the leakage in the other directions by means of energy-dependent bucklings.

For example,

-DV20(x, У, z) = — DV2[0(x) • ф(у) • ф(2)] = — D~ ф + DBІуф.

This separation of the x, у and z dependence constitutes, of course, an approximation which has to be justified from case to case.

References

1. A. M. Weinberg and E. P. Wigner, The Physical Theory of Neutron Chain Reactors, University of Chicago Press, 1958.

2. G. E. Bell and S. GlaSSTOne, Nuclear Reactor Theory, Van Nostrand Reinhold Co., 1970.

3. B. G. Carlson and K. D. Lathrop, Transport theory—the method of discrete ordinates. In Computing Methods in Reactor Physics ; edited by H. Greenspan, C. Kelber and D. Okrent, Gordon & Breach, 1968.

4. M. M. Merrill, Nuclear design methods and experimental data in use at Gulf General Atomic, Gulf-Ga-A 12652 (GA-LTR-2), July 1973.

5. K. D. Lathrop, DTF-IV, a Fortran-IV program for solving the multigroup transport equation with anisotropic scattering, LA-3373, 12 Nov. 1965.

6. K. D. Lathrop and F. W. Brinkley, Theory and use of the general geometry TWOTRAN program, LA-4432, May 1970.

7. S. FranceSCON, The Winfrith DSN programme, AEEW-R273.

8. L. Massimo, A programme for the solution of the monoenergetical transport equation in the P 5 approximation for a multiregion cylindrical geometry, EUR-2584e, 1965.

9. G. C. PomraninG, Reduction of transport theory to multigroup diffusion theory. Journal of Nuclear Energy, Parts A/B, 18, 497-512 (1964).

10. G. D. JoanOU and J. S. Dudek, GAM-II, a B3 code for the calculation of fast neutron spectra and associated multigroup constants, GA-4265, Sept. 1963.

11. M. M. R. Williams, The Slowing Down and Thermalization of Neutrons, North-Holland Publishing Co., 1966.

12. H. C. HonECK, THERMOS—a thermalization transport theory code for reactor lattice calculations, BNL-5826 (1961).

13. J. E. BeardwOOD, The solution of the transport equation by collision probability methods, ANL-7050, p. 93, 1965.

14. J. R. Askew, Review of the status of collision probability methods, IAEA-SM-154/69-A.

15. В. K. Chesterton, “MINOTAUR” TNPG/PDI 378, unpublished.

16. R. Bonalumi, Neutron first collision probabilities in reactor physics. Energia Nucleare, 8, 326 (1961).

17. J. R. Askew, Some boundary condition problems arising in the application of collision probability methods, IAEA-SM-154/69-B.

18. J. R. Askew, A coarse mesh correction for collision probabilities, AEEW-M889, June 1969.

19. J. R, Askew, Mesh requirements for neutron transport calculations, AEEW-M760.

20. R. H. W. Stace, A review of the use of collision probabilities in lattice calculations. Bournemouth Symposium on the Physics of Graphite Moderated Reactors, 4 to 7 April, 1962.

21. A. Jansson, THESEUS—a one group collision probability routine for annular systems, AEEW-R253, 1963.

22. M. H. Kalos, F. R. Nakache and J. Celnik, Monte Carlo Methods in Reactor Computations. In Computing Methods in Reactor Physics, edited by H. Greenspan, C. Kelber and D. Okrent, Gordon & Breach, 1968.

23. S. Corno, Report to be published.

24. T. D. Beynon, On the concept of energy dependent buckling in multi-group diffusion theory. Journal of Nuclear Energy, 25, 503-511 (1971).

In order to limit the computer time, the number of groups used in diffusion and transport calculations should be kept as low as possible, as long as a sufficient accuracy can be guaranteed. This may lead to different group structures for the treatment of different problems.

A higher number of groups can be easily used in fast running one-dimensional diffusion calculations, while in the treatment of three-dimensional cases practical considerations limit the number of groups. The effect of the reduction of groups on accuracy can be tested in the one-dimensional case. While ксЯ can be calculated with sufficient accuracy with relatively few groups, an accurate power distribution in regions with strong space-dependence of the neutron spectrum (e. g. core-reflector boundary) requires a high number of groups particularly in the thermal range. An example of the effect of the number of thermal groups on power distribution is given in Fig. 8.3<20) (the example of this figure is rather extreme because of the very high fuel loading of this case). Usually group structures are defined by trying to separate, if possible, different

types of nuclear reactions or other significant phenomena. This separation, which is sometimes necessary for accuracy reasons, provides better information on the effect of different reactions. It is then possible to distinguish a range of the fission source spectrum, a range of unresolved and of resolved resonances, a thermal energy range, etc. Particular energy partitions are used in the interpretation of experiments in which threshold detectors are used. In the case of high Pu loading particular groups may be chosen to include the low-lying Pu resonances.

A high number of fast groups may be required if the core calculation has to provide a neutron source to be used in subsequent shielding calculations. Considerable detail in the fast spectrum may also be required for the calculation of neutron damage in reactor materials. In the case of calculations with a very limited number of groups care must be taken to avoid the artificial transfer of neutrons to energy ranges they cannot physically reach. A typical example is given by neutrons up-scattered above the boundary between thermal and fast range. As this boundary is usually assumed to be around 2 eV for HTR calculations, a certain up-scatter exists above it. If the fast group above this boundary is broad enough to reach the resonance range, the up-scattered neutrons can be artificially absorbed in these resonances while in reality the probability of being scattered up to that energy is negligibly small. This means that there should be a group between the thermal and the resonance range, otherwise it would be more accurate to neglect up-scattering altogether. As an example Tables 8.2 and 8.3 give the group structures used for the calculations of the Fort St. Vrain and of the THTR reactors. The

boundary at 17.6 eV has been chosen to keep the up-scattered neutrons outside the resonance range.

The prompt neutron lifetime is important in determining the time dependence of excursions in which the reactivity excess is so great that the reactor is critical or near critical on the prompt neutrons alone (Akctt /2) without the delayed neutron contribu­tion. Otherwise the transient is governed by the delayed neutrons and the value of / does not affect strongly the results. In multi-group space-dependent calculations the lifetime / does not appear explicitly, but the coefficients of (12.4) must be given for each energy group and region. This means that beside the parameters for static calculations, the velocity v has to be calculated (usually with spectrum codes) for each group and region. As it happens with /, those velocities do not need to be very accurate in transients governed by delayed neutrons. In the one group approximation the mean neutron lifetime / has been introduced in eqn. (12.4) but the use of this expression with one group averaged cross-sections is not very accurate so that eqn. (12.11) is often used.

The energy and space distribution of the adjoint flux can be calculated with many codes for multi-group diffusion calculations, and routines for the calculation of Л and are sometimes included in these codes.

Equation (12.11) suggests also the calculation of Л as the reactivity worth of an absorber with a 1/r cross-section.<4’5> This method allows the calculation of Л with static diffusion or transport theory codes.

A simple and accurate procedure’6’ consists in calculating the stable reactor period T corresponding to a given Ak with a space-dependent code in the absence of delayed neutrons, and to obtain / as

/ = T Ak.

Very important in the spatial averaging is the effect of the reflector, which is returning neutrons to the core with a certain delay and causes therefore a considerable increase in the prompt neutron lifetime of the reactor.