minimum critical mass and flat flux in a 2-group model

Minimum critical mass and flat flux in a2-group model

Jeffery Lewins*

University of Cambridge, Engineering Department, Trumpington St. Cambridge CB2 1P2, UK

Received 31 May 2003; accepted 18 June 2003

Abstract

Solutions for a flat thermal flux in a two-group diffusion model are found for a range ofassumptions, since this is a requisite for both a minimum and a maximum fuel loading in the

model. It is shown that a maximum fuel loading then arises when such a region, at unityinfinite multiplication factor, is complemented by an outer core to bring the finite reactorcritical. Correspondingly, when a core with a flat flux is surrounded by a reflector, the corefuel may be so distributed as to flatten the flux using the ‘flux-trap’ phenomenon and provide

a minimum loading. Critical conditions and necessary fuel density are derived in finite andinfinitely thick reflectors. A curiosity is the possibility of observing ‘negative’ reflector savings.Fuel savings are estimated for a range of models for the minimum loading compared to a

correspondingly critical uniform core loading. In some circumstances a saving of up to 70% isindicated with economic and safety implications. It is shown how the reduction of the mini-mum loading solution from a two- to a one-group model retains the flat flux in the core but

fails to satisfy the thermal boundary condition unless a reflector, with the flux-trap, is used sothat without it, the two-group minimum fuel loading solution cannot fully transform to aminimum loading in one group. The full solution for a two-group model flat flux core with

finite thickness reflector is given. It is shown that as the reflector is reduced it becomes neces-sary to increase the fuel density at the centre to a point where this exceeds the capability of thechosen fuel, thus providing a secondary criticality equation. The increasing steepness (nega-tive slope) of the flux distribution is required to make the flux trap phenomenon possible in

the reducing reflector region. The range in which there may be two reflector thicknesses lead-ing to the same size core, but different fuel distributions observed by Williams, is determined.Constrained solutions that limit either the size of core or the maximum fuel density are con-

sidered, generalising the original work of Goertzel to a practicable core design.# 2003 Elsevier Ltd. All rights reserved.

Annals of Nuclear Energy 31 (2004) 541–576

www.elsevier.com/locate/anucene

0306-4549/$ - see front matter # 2003 Elsevier Ltd. All rights reserved.

doi:10.1016/j.anucene.2003.09.001

* Tel.: +44-1223-332100; fax: +44-1223-63637.

E-mail address: [email protected] (J. Lewins).

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1. Introduction

Studies of the problem of a minimum fuel loading in a reactor where the density offuel can be varied continuously have been performed. In a model due originally toGoertzel (1956) generalised by Shapiro (1961), that assumes the scattering propertiesare independent of the fuel loading, it can be shown (Lewins, 1963b) that the adjointor importance function for the movement of fuel is proportional to the thermal fluxand consequently that the minimum fuel loading is achieved when the thermal flux isconstant or flat. This is explicable on a perturbation basis, that if the square-fluxweighting were anywhere higher, fuel should be moved to that location from alocation of lower weighting. The model is good for say a heavy-water moderatedreactor with highly enriched fuel. A minimum fuel loading then has significant eco-nomic benefit as well as implications for safety. Further studies (van Dam and deLeege, 1987; Hirano et al., 1991; Williams, 2003) have extended the study to a widerclass of systems requiring numerical work outside the original quadratic variationalprinciple.

However, we have pointed out that a flat flux is also a condition for a maximumfuel loading, undesirable as this might be (Lewins, 1963a, Poon and Lewins, 1990).This is the only result available in one-group diffusion theory and the minimumproblem requires slowing down theory to allow a solution; one-group theory willnot serve. Williams, in studying a transport model for the minimum loading prob-lem rather than the original diffusion theory model, has pointed out an apparentanomaly (Cassel and Williams, 2003a,b). A multi-group energy model can bereduced to a one-group model by requiring fast scattering to become infinite andhence all fast neutrons enter the thermal group where they are born, without diffu-sion. But what happens if a solution to the minimum loading problem is found andsubjected to this limiting process that puts the scattering length to zero? Is the resultstill a minimum?

In this note we study solutions in two- and one-group models attempting to clarifythe matter. The fundamental physics can be explained in terms of the ‘thermal flux-trap’, as designed in certain research reactors (Fig. 1). If the fuel is removed from asmall region within the core, the thermal absorption cross-section is then muchdiminished although leakage of the fast flux into the region means the fast flux, thesource of thermal neutrons, remains high. Consequently the thermal flux is high.This effect is also seen close to a core-reflector interface. It is this flux-trap effect thatcan boost the thermal flux to make it flat towards the core edge despite fast leakage.We may think of the effect as allowing fast neutrons that would otherwise leak outof the system to leak back into the core, thus reducing the thermal leakage andproviding for a minimum fuel loading. We may anticipate that a significant para-meter of the problem will be the ratio L=Ls, the thermal diffusion length in themoderator to the slowing down length in the moderator (essentially the ratio of theirrespective mean-free paths). A small value of this parameter favours the ‘flux-trap’.

After reviewing the minimum/maximum theorem, solutions are given for both,starting with the simpler maximum solutions. An estimate of the fuel saving in theminimum configuration compared to a uniform loading of the same nature fuel is

542 J. Lewins / Annals of Nuclear Energy 31 (2004) 541–576

obtained. Then the minimum solution derived is subjected to a reduction from two-to one-group theory to assess this apparent anomaly. A finite reflector is finallystudied to observe the limit of reflector thickness required.

With finite reflector we can study the observation of Williams (2003) that twodifferent reflector thicknesses may lead to the same reflector saving and hence coresize, albeit with two different fuel loadings.

Finally, the effect of further constraints is studied: limited fuel density and limitedcore size. Only the latter was considered by Goertzel but treated as requiring aninfinitely dense loading of fuel at the core surface. This is evidently not a practicalsolution to the constraint and so the two constraints are considered with explicitsolutions.

2. Theorem

The flat thermal flux condition for either a minimum or a maximum fuel loading isestablished as follows. The Goertzel model assumes that the scattering properties,and hence the diffusion coefficients, are uniform throughout the system and areunperturbed by the presence of fuel with its large thermal absorption cross-section.A perturbation expression for the movement of a small amount of fuel would lead toa weighting of flux and importance that, on the one hand, is the thermal flux leadingto the production of neutrons and, on the other, the net importance of the fastneutrons produced by fission compared to the thermal neutrons removed by fissionand capture. Let the thermal flux be � rð Þ and the adjoint function or importance of afast neutron and a thermal neutron be �þ1 rð Þ; �þ rð Þ respectively. If the fission pro-duces � fast neutrons for every thermal neutron absorbed then the appropriateweighting for a perturbation in fuel cross-section that represents movement of fuel is

Fig. 1. The flux-trap; enhancement of the thermal flux by fast neutrons leaking into a region adjacent to

fuel.

J. Lewins / Annals of Nuclear Energy 31 (2004) 541–576 543

��þ1 � �þ� �

�. As pointed out by Goertzel (1956) in an integral equation formalismand by Lewins (1963b) for a differential formalism, the assumptions of the modellead to the proportionality that ��þ1 � �þ

� �� . Consequently,1 the perturbation

weighting for fuel movement is �2. This scattering model is certainly appropriate toa highly enriched reactor using heavy water as moderator. It will be approximatelytrue for a much wider range of reactor systems. Capture of fast neutrons in themoderator can be allowed for by a suitable reduction of the slowing down sourceterm in the thermal group, assumed to be a constant factor �. Fast fission and reso-nance capture in the fuel can be accounted for approximately by a suitable redefi-nition of �, the yield of neutrons in fissionable material.2 The differential proof 3 issummarised in an appendix that shows a generalisation allowing for variable mod-erator densities.

Data taken from Glasstone and Edlund (Glasstone and Sesonske, 1967) for ther-mal fission gives the following

U � 233 : � ¼ 2:27;U � 235 : � ¼ 2:07;Pu � 239 : � ¼ 2:09

All these would be reduced by any allowance for resonance escape and capture infuel cladding, etc.

We may now state the minimum theorem. If fuel can be moved to a region whereits weighting by �2 is higher, then reactivity will be gained and consequently the fuelloading can be reduced whilst still achieving criticality. In an unrestricted region,therefore, the thermal flux must be flat for a minimum loading. However, restric-tions or constraints are likely to arise. If the fuel is already at its maximum densityso that none can be added, then it is acceptable for the thermal flux there to riseabove the unrestricted constant value. If the extent of the core is constrained, it maybe necessary to load additional fuel, up to its physical limit, in such a region ofhigher thermal flux to achieve criticality, the overriding constraint.4 It is of partic-ular interest to see, in these circumstances, where the additional fuel shall be loaded:at the centre or at the edge. The answer is not intuitive and will be sought in whatfollows.

It is easily seen that a minimum condition calls for the best possible reflector, tosave fuel, and a maximum condition for no reflector, which would otherwise savefuel.

1 This is self-evident in self-adjoint one-group theory.2 This is not the conventional definition that would include allowance for moderator capture.3 This proof will show that the theorem holds more widely, for variable density moderator as well as

variable density fuel, assuming only that the ratio of properties for fuel and, separately, moderator

between fast and thermal flux are independent of density and hence of position. However, the variable

moderator density model is more difficult to solve and solutions given here are limited to uniform mod-

erators.4 Such an extreme addition, contemplated by Goertzel in his original study, would tend to violate the

condition that the scattering properties are unchanged. Actually we have proved Goertzel’s theorem under

a wider range that allows spatial variation of scattering properties as long as they are not linked directly to

the fuel properties.


3. Maximum loading

Equally, a flat thermal flux is the basic condition for a maximum loading. If theflux were anywhere lower than some constant value, fuel should be moved there as aregion of lower reactivity requiring additional fuel to be loaded to achieve criticality.This condition has an obvious interpretation, calling for an infinite multiplicationfactor of unity k1 ¼ 1. Such a core could be made infinitely big at this multi-plication factor and hence provide a maximum solution. If now the core size is lim-ited, additional fuel must be loaded at the edges to make up for the edge leakagebefore the reactor is critical. Since the thermal flux will now be lower in this region,such an outer core requires the fuel to be loaded at its maximum density or else thelower flux calls for additional fuel to be shifted there from within. In this model, thethermal flux will fall in the edge region towards the bare edge or in any imposedreflector (that will of course decrease the maximum fuel loading). However, althoughit would be advantageous in these circumstances to move fuel to this edge region, thisagain is at its physical upper limit. Thus the conditions for a maximum loading apply.

This maximum is predictable in one-group diffusion theory with or without areflector (Fig. 2). If slowing-down theory is included, however, there should prob-ably be no reflector or else the ‘flux-trap’ phenomenon may raise the thermal fluxabove the level in the inner core region (Fig. 3). If, however, there is a reflector andthe size constraint operates so that no fuel may be placed there, this restraint on thethermal flux in the reflector not exceeding the flat flux in the core no longer applies.

Analytical solutions for a two-group model are readily available in simple geo-metries. In slab geometry the inner region with k1 ¼ 1 has a maximum loadingregion either side in which the thermal flux falls off as a cosine (assuming a commonextrapolated boundary for both groups). A three-dimensional model has analogoussolutions available in spherical geometry although it is noted that the sphere pro-motes a minimum fuel loading for uniform fuel loading. More complex geometries

Fig. 2. The bare reactor in a maximum fuel-loading configuration.


are not fully separable for a simple analytical solution and recourse could be had tonumerical methods if a maximum solution were of any intrinsic interest.

4. Minimum loading

Analytical solutions of the minimum loading problem are of more interest. I studyfour successively more complicated sub-models to tease out the nature of the flatflux solution and how it changes on reducing from two diffusion groups to onegroup. In three cases it is assumed that the scattering and moderating properties areuniform everywhere and are not affected by the presence of the fuel. In the firstmodel, a bare core is studied with no capture in the moderator. In the second, cap-ture during slowing down and thermal capture in the moderator is added to the barecore. In Model 3, the original loss-free Model 1 is extended to infinity, offeringtherefore an infinite reflector, but with reflector thermal capture. And in the finalmodel, fast and thermal moderator capture in the core as well as reflector is added toModel 3. This model is explored in infinite and then finite reflector thickness.

As a preliminary, however, consider the minimum loading in a one-group model.This requires the fuel to be placed in the region of highest flux and importance, i.e.concentrated at the centre at its physical maximum density. The theorem is observedin the restraint; that is, there is no flat flux region. The minimum fuel loading is the

Fig. 3. The reflected reactor in a minimum fuel-loading configuration.

Table 1

Model assumptions

Model
1 2 3 4
Assumptions
Bare core Bare core Infinite capturing
reflector

Infinite capturing

reflector

Uniform scattering in

all models

No moderator

capture

Moderator

capture

No core moderator

capture

Moderator capture


maximum fuel loading without the central region of flat flux at unity infinite multi-plication factor, in a one-group model (Table 1).

5. Bare geometry

5.1. Model 1

It is instructive first to consider the simplest possible two-group diffusion modelfor the flat thermal flux problem as a bare core with no reflector. Amongst manysimplifications, suppose not only that it is a bare reactor (no reflector but nothomogenous in fuel loading) but that the only thermal absorption is in the fuel; thatthere are no absorptions in the first or fast group. Let the constant thermal flux be�o. The fast and thermal equations are then

D1r2 ��s

� ��1 þ ��f�o ¼ 0 and �s�1 ��f�o ¼ 0 ð1Þ

where �f xð Þ is the fuel thermal macroscopic absorption cross-section. The slowingdown or removal cross-section �s is constant and represents the scattering out of thefast group rather than the internal scattering that leads to a diffusion coefficient D1,although one is inversely proportional to the other.

Substitution gives

D1r2 þ �� 1ð Þ�s

� ��1 ¼ 0 ð2Þ

and writing B2s ¼

��1L 2

swhere L2

s ¼ D1

�sis the square of the fast scattering length, we

have an equation: r2 þ B2s

� ��1 ¼ 0. Then in slab geometry, assuming uniform flux

in the remaining orthogonal directions, �1 xð Þ ¼ �1 0ð ÞcosBsx, symmetry required.Putting the fast flux to zero at an extrapolated boundary in symmetric slab geometryat some half-size � gives the usual criticality condition Bsa ¼ �

2 and the core half-width for given properties.

This simplified model provides a criticality equation for constant thermal flux. It isnotable that this involves fast properties only save for the relative yield in fission that must of course be larger than one for a meaningful problem. This all arisesbecause of the assumption of no other thermal absorption than in fuel and no cap-ture in the fast group. The losses are only fast leakage.

The fuel cross-section �f has shape but arbitrary scaling. �f xð Þ ¼ �s�1 0ð Þ

�ocosBsx.

But because power is proportional to the product of thermal flux and fuel cross-section, we do see that the fast flux is proportional to power as one would expect:�1 0ð Þ ¼ 1

�s�o�f 0ð Þ. There will be a physical maximum �fm 0ð Þ and if this (or any

other specified cross-section) is employed, then the central fast flux is indeed pro-portional to the thermal flux.

The fundamental weakness of the initial model itself is the nature of the thermalflux at the bare core boundary. The physically derived edge-conditions for the ther-


mal flux are not satisfied by having a constant flux. It is essential to have a reflectorregion, the reflector providing the flux-trap phenomenon.

5.2. Model 2

The model is improved somewhat if we account for losses, both in the fast groupand thermal capture in the scattering material. A slightly more realistic model thenallows for thermal capture with cross-section � in the core and the capture in thefast region represented by a smaller production term in the thermal balance by afactor5 �4 1. This raises some further points of interest in the bare reactor model.The core equations become

D1r2 ��s

� ��1 þ ��f�o ¼ 0 and ��s�1 � �f þ�

� ��o ¼ 0 ð3Þ

Now we have

�f xð Þ ¼��s

�o�1 xð Þ �� and D1r

2 ��s�1

� �þ ��s�1 � ��0 ¼ 0 ð4Þ

Again for slab geometry with uniform distributions in the remaining directions, this has aparticular integral: ��o

��1ð Þ�sand symmetric complementary function: gcosBsx where now

B2s ¼

��1L 2

s. For a bare reactor we would, using extrapolated boundary condition, require

gcosBsa ¼ ��

�� 1ð Þ�s�o ð5Þ

at the extrapolated boundary.so that

�1 xð Þ ¼�

�� 1�o�

�s1 �

cosBsx

cosBsa

� �ð6Þ

Then

�f xð Þ ¼1

�� 1� 1 � ��

cosBsx

cosBsa

� �: ð7Þ

yielding a modified criticality condition involving a specified central fuel cross-section

cosBsa ¼ ��

�� 1ð Þ�fð0Þ ��ð8Þ

This reduces to the previous result as � ! 0 but note that the negative sign, forsufficiently small but finite �, implies that the critical half-size is given by Bsa >

�2.

5 This will not, however, allow for fast capture in the fuel, e.g. by U-238. If this is present, an

approximate reduction of the fission yield � could be made: � ! p�0 where p is the resonance escape

factor. Indeed one could include the fast fission term " of the conventional four-factor formula, as long as

it was understood that � ¼ "p�0, like � was independent of position. Such an approximation is of limited

use however.


The critical half-width now depends not only on the fast leakage but the ratio ofcentral fuel cross-section to moderator capture cross-section. However, the valuepredicted for the fuel cross-section at the extrapolated boundary is unphysical: �f ¼

�� at the extrapolated boundary. Since this is non-physical, the bare reactor cannotbe critical under the assumptions of Model 2.

The extrapolated boundary is not itself physical, of course, and we should reallybe looking at a physical boundary a distance of 2D1 inside. Even so, the more exactdiffusion boundary condition (zero return half-current) is similar in result that thebare reactor cannot be specified with flat thermal flux. Likewise a transport correc-tion to the extrapolated boundary condition would not suffice to make it feasible, sowe must consider a reflected core. Fundamentally, the bare core model cannotsatisfy the thermal flux boundary condition. It is necessary to have a reflector andmake use of the ‘flux-trap’ phenomenon.

6. Reflected geometry

6.1. Model 3

In view of the imperfection of the thermal boundary condition for the bare core, it isworth considering an infinite system where the outer region is purely a reflector. Togive a simplified result, however, we start by ignoring moderator capture in the corewhile imposing thermal capture � in the reflector, to ensure the vanishing of the fluxat infinity. No fuel is to be placed here so that the fundamental assumptions are stillmet over the adjoint weighting. The core equations are unchanged but in the reflec-tor, starting at x ¼ �a, we have

D1r2 ��s

� ��1 ¼ 0 and D2r

2 ��

�2 þ�s�1 ¼ 0: ð9Þ

For slab geometry, the first gives �1 xð Þ ¼ fe�x=Ls in the reflector (positive-x), allowingfor vanishing flux at infinity so that continuity of flux and current (we assume commondiffusion coefficient and slowing-down cross-section in core and reflector) yield

�1 0ð ÞcosBsa ¼ fe�x=Ls and �1 0ð ÞBssinBsa ¼fe�x=Ls

Lsð10Þ

Division yields the criticality condition that LsBstanBsa ¼ 1 so that for �� 1 <<1 then there is a reflector half-saving approximately Ls and the reduced half-corethickness is a � �

2Bs� Ls. Of course the exact core size is readily available from the

criticality condition.For the thermal reflector flux consider a particular integral and complementary

function leading (with zero gradient at the interface) to a reflector thermal flux

�2 x > að Þ ¼ �1 0ð Þ�s

�

cosBsa

1 � L2=L2s

� � e�x�aLs � L

Lse�

x�aL

h ið11Þ


e L2 ¼ D2=� defines a thermal diffusion length. Equating the gradient at the
wherinterface to zero then yields6
�o ¼ �1 0ð ÞcosBsa�

�s

1

1 þ L=Lsð12Þ

The fuel cross-section becomes (express cosine in tangents)

�f xð Þ ¼ �s�1 xð Þ

�o¼ � 1 þ L=Lsð Þ

cosBsx

cosBsa¼

ffiffiffi�

pffiffiffiffiffiffiffiffiffiffiffi�� 1

p � 1 þ L=Lsð ÞcosBsx ð13Þ

and is seen to be positive at the interface and hence a physically realisable solution:

�f að Þ ¼ � 1 þ L=Lsð Þ ð14Þ

At the centreline we have

�f 0ð Þ ¼ �1 þ L=Ls

cosBsa¼ �

1 þ L=Lsffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 � 1=�

p ð15Þ

The fast flux and fuel cross-section are no longer dependent on some arbitrarychoice for the centre-line fuel cross-section as in Model 1 because of the impositionof a thermal capture cross-section in the reflector.

6.2. Model 4

We finally include common moderator losses in the fast and thermal groups inboth core and reflector for the most general case studied. The reflector equations arenow (we assume common moderator properties in core and in reflector)

D1r2 ��s

� ��1 ¼ 0 and D2r

2 ��

�2 þ ��s�1 ¼ 0 ð16Þ

having solutions in slab geometry

�1 xð Þ ¼ �1 að Þe� x�að Þ=Ls and �2 xð Þ ¼��s

�

�1 að Þ

1 � L2=L2s

þ fe� x�að Þ=L: ð17Þ

Since the gradient of the thermal flux is to be zero at the interface we may find theconstant f ¼ �o 1 � Ls=Lð Þ and hence the interface fast flux in terms of the thermalflux: ��s�1 að Þ ¼ 1 þ L=Lsð Þ��o. If we now substitute in the core equations at theinterface we obtain a criticality equation

BsLstanBsa ¼1 þ L=Ls

L=Ls �1

��1

�1 þ L=Ls

Sð18Þ

6 If the scattering length and the thermal diffusion length happen to be equal (L ¼ Ls resonance) the

particular integral is to be modified in the usual way. Some particular case results are given in Appendix A.


us, given the properties, we may determine the core half width. Note that for
Thsmall enough L=Ls (large enough � ) or small enough �� then Bsa >
�2. The chan-

geover point is when S � LLs� 1

��1 ¼ 0 and S is negative for sufficiently large � orsmall ��. For the particular case that S ¼ 0 then cotBsa ¼ 0 and Bsa ¼ �

2.It is sometimes more convenient to express these results through the criticality

equation as

S

cosBsa¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL

Ls�

1

�� 1

�2

þ1 þ L=Lsð Þ

2

�� 1

�sand note !

1 þ L=Lsffiffiffiffiffiffiffiffiffiffiffiffiffi�� 1

p

¼1 þ L=Lsffiffiffiffiffiffiffiffiffiffiffi

Ls=Lp as S ! 0 ð19Þ

A conventional treatment of criticality would have an equation analogous to Eq.(18) but based on thermal as well as fast properties. The bare reactor critical equa-tion would have a form such as cotBa ¼ 0 so that the bare core size is a ¼ �

21B. When

reflected, the criticality equation might be cotBa ¼ MB where M is a characteristiclength (such as the migration length). The reduction in core size is called the (half)reflector saving7 and there is a well known approximation that for large cores andhence small buckling; we can write the saving as � MB.

In the present case we may loosely refer to the ‘reflector saving’ as being the dif-ference between the solution of Eq. (18) in the form cotBsa ¼ SBsLs and the formcotBsa ¼ 0, while noting that the flat flux solution is not available in a bare core sothat ‘reflector savings’ is somewhat of a misnomer. Never the less, the sameapproximation may be used, noting that the range of validity will be improved if S issmall: saving � SLsBs

The fast flux in the core is

�1 xð Þ ¼�

��s�o

��

�� 1þ S

cosBsx

cosBsa

� �ð20Þ

Note that the coefficient, S, changes sign with cosBsa. It follows that if thisreversal of sign has occurred, there is a region close to the core-reflector interface inwhich the contribution to the fast flux from the complementary function goesnegative. It would be unphysical if this brought the total fast flux below zero.However, it is readily shown that the total flux at the interface is positive, indepen-dent of the sign of S since ��

��1 þ S ¼ LLs

.While of course S

cosBsacan and has been transformed through the criticality con-

dition, Eq. (18), the explicit sign of S indicates more clearly perhaps a ‘negative’reflector saving, when S < 0.

The fuel cross-section becomes

�f xð Þ ¼ �1

�� 1þ S

cosBsx

cosBsa

� �ð21Þ

7 Hugo van Dam (indirect personal communication) has very properly corrected this term in general to

‘core savings’.


at at the core edge �f ¼ �L=Ls > 0. At the centre we have
so th
�f 0ð Þ � �fo ¼ �1

�� 1þ

S

cosB2a

� �> 0: ð22Þ

This again is physically satisfactory. It remains satisfactory as the moderatorthermal cross-section goes to zero since then there are no losses in the thermal groupin the reflector but the thermal flux does not decrease to infinity. For the reasonableapproximation that �� ¼ 2 and the particular case that L ¼ Ls then S ¼ 0 and�fo ¼ 3�. (The centreline density is required to be much greater, however, for athin, finite reflector.) We have a self-consistent diffusion-model solution satisfyinginterface and boundary conditions (Fig. 3).

It is also convenient to express the fast flux in terms of the centreline fuel cross-section so that

�1 xð Þ ¼�o�

��s

��

�� 1þ

�fo

��

1

�� 1

�cosBsx

� �ð20bÞ

7. Fuel saving

There will be interest of course in the fuel saving achieved in the two-group model.The amount of fuel loaded will be proportional to the fuel cross-section integratedover the reactor core region. We make comparisons on the somewhat arbitrary basisthat the uniformly loaded design uses the maximum fuel cross-section called for inan unconstrained optimum design, and that the reflector is infinitely thick. Exactcomparisons for more complex cases can be carried out numerically (Williams,2003).

7.1. Model 1

For the first minimum model we have a ¼

12�

Bsso that

mmin ¼ 2

ða

0

�f xð Þdx ¼ 2�f 0ð Þ

Bs

ð�2

0

cos d ¼2�f 0ð ÞLsffiffiffiffiffiffiffiffiffiffiffi�� 1

p : ð23Þ

Note that in this model there is no loss through thermal capture. Thus there is anunrealistic result that the centreline fuel cross-section and correspondingly its den-sity can be reduced to yield progressively smaller fuel mass. However, we may makea legitimate comparison with the two-group result for uniform fuel loading underthe same condition of common centreline fuel cross-section.

The uniform fuel loading in two groups, assuming a common extrapolatedboundary, yields a quadratic equation for the buckling:

�� 1 ¼ 1 þ L22 B2

� �ð1 þ L2

s B2� �

ð24Þ


e L2 ¼ D2=�f in the absence of further thermal absorption in this model. There
wher 2
will be two roots for the buckling and we require the positive value. If �� 1 << 1then the buckling is small and its square negligible so that this root is approximatedby the migration area approximation:

B2 �� 1

L2s þ L2

2

�� 1

M2c

: ð25Þ

In this case the fuel mass is proportional to

munif ¼ 2�f 0ð Þa0 ¼ �f�

B¼ �f

�Mcffiffiffiffiffiffiffiffiffiffiffiffiffiffi��0 � 1

p : ð26Þ

Then the ratio of the masses of fuel in the minimum and the uniform loading is

r ¼mmin

munif�

2Ls

�Mc<

2

�¼ 0:6366: ð27Þ

This result suggests that, under the somewhat arbitrary assumptions made, a well-distributed fuel can save perhaps one-third.

A better comparison would use spherical geometry for which

rspher ¼3

�2

Ls

M

�3

< 0:304 ð28Þ

with a potential 70% saving. This saving will presumably be reduced for a reflectedreactor, where the reflection saves more fuel for the uniform design than for theoptimum design, but is still an attractive result of the minimum fuel design. But thismodel is unrealistic so we consider instead Model 4.

7.2. Model 4

With a model including the reflector and all losses, we have to approximate theuniform loading solution even further to obtain an analytical result. We assume thatthe reflected core is reduced in size by a reflector half-saving of M noting that this isthe value in the reflector, not in the core where M2

c ¼ L2s þ D2= �f þ�

� �. The uni-

form fuel loading in the reflected case with losses is given by

munif ¼ 2�f 0ð Þa0 � 2�f 0ð ÞMcffiffiffiffiffiffiffiffiffiffiffiffiffiffi��0 � 1

p�

2� MrB

h ið29Þ

where a0 is the core half-width in the uniform loading model. The core buckling isnow defined using �0 ¼

��f

�fþ�and L2

2 ¼ D2

�fþ�with uniform properties.


The minimum fuel loading in slab geometry, however, becomes

mmin ¼2�S

BscosBsa

ðBsa

0

cos d þ2�a

�� 1

¼2� L þ Lsð Þ

�� 1þ

2�a

�� 1

�2�Ls

�� 11 þ

L

Ls

�þ

�

2ffiffiffiffiffiffiffiffiffiffiffiffiffi�� 1

p �S

1 þ L=Ls

� �ð30Þ

for small S. In the square brackets of the final approximation, the first term repre-sents the exact contribution of the cosine-distributed fuel, the second term the con-tribution from the additional uniform fuel loading imposed by the moderatorthermal cross-section over a range to Bsx

0 ¼ �=2 and the third term the approximatereduction in this uniformly distributed fuel due to the ‘reflector savings’. Note thepossibility, with the sign of S, that these reflector savings are in fact negative. Thenthe fuel saving ratio in this reflected and lossy slab geometry is

r ¼Ls

Mc

1 þL

Lsþ�

2

1ffiffiffiffiffiffiffiffiffiffiffiffiffi�� 1

p �S

1 þ L=Ls

1 þS �� 1ð Þ

cosBsa

� ��

2

1ffiffiffiffiffiffiffiffiffiffiffiffiffiffi��0 � 1

p �Mr

Mc

� � : ð31Þ

This may be evaluated exactly if we wish but this rather messy expression can besimplified under the assumption that S and hence the ‘reflector saving’ in the mini-mum loading problem is zero. Noting that as S ! 0

S �� 1ð Þ

cosBsa¼ S �� 1ð Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 þ

1 þ L=Ls

SBsLs

�2s

!ffiffiffiffiffiffiffiffiffiffiffiffiffi�� 1

p1 þ L=Lsð Þ ð32Þ

we may make the further approximation that both the moderator thermal cross-section and simultaneously the infinite multiplication factor excess over one aresmall (but retaining S ¼ 0) to arrive at a position where the prime values, etc, areindistinguishable from unprimed values and the following gives an estimate of theorder of magnitude of the potential fuel saving:

rlmi;slab ¼2

�

Ls

M

ffiffiffiffiffiffiffiffiffiffiffiffiffi�� 1

p<

2

�ð33Þ

It has to be said again that if the maximum fuel cross-section is appreciably higherthan that called for in the minimum loading, as used in this comparison, then thepotential savings are correspondingly reduced. This may well be the case forparticular fuels.


8. The reduction to a single neutron group

The reduction of a minimum fuel loading solution in a two-group model to a one-group model can now be considered. If we simply let�s ! 1 so that Ls ! 0;Bs !

1 then the criticality condition of the physically realistic Model 4 becomesffiffiffiffiffiffiffiffiffiffiffiffiffi�� 1

ptanBsa ¼ 1 and the half-width goes to zero: a �

cot�1ffiffiffiffiffiffiffiffi��1

p

Bs! 0. This would

imply a trivial result with a vanishing reactor core incapable of running at finite power.If the half-width is to remain finite, then the buckling must go to zero sufficientlyrapidly to overcome the increase of the scattering removal cross-section. This isachievable if simultaneously �� 1 ! 0 sufficiently rapidly. This condition is ofcourse the same as saying the infinite multiplication factor becomes unity, the flat fluxcondition in a maximum fuel loading. However, the consequence is that necessary fueldensity increases to a value not physically realisable. Note also that if indeed �� 1 !

0 and there is no capture in the moderator we have an infinite multiplication factor ofunity, sustaining a flat flux anyway.

Thus the reduction to one-group theory of the minimum fuel loading solution isconsistent in the inner core region with the flat flux of the maximum or minimumsolution. It is the further necessity of making the reactor critical that calls for anedge loading that means the solution will now model a maximum rather than aminimum loading.

9. Finite reflector

The analysis can be extended to a finite rather than an infinite reflector with theexpectation of larger fuel loadings as the reflector thickness T decreases. FollowingModel 4, with details in the first Appendix, we find now a criticality equation

cotBsa ¼BsLs

cothT=Ls1 �

�� 1 �L

Ls

cothT=Ls

cothT=L

�� 1ð Þ 1 � L2=L2

s

� �2664

3775

¼BsLsS Tð Þ

cothT=Ls

1 �L

Ls

cothT=Ls

cothT=L

1 � L2=L2s

ð34Þ

where

S Tð Þ �1 � L2=L2

s

1 �L

Ls

cothT=Ls

cothT=L

��

�� 1ð35Þ

is a generalisation from the infinite reflector case, and reduces to it when T issufficiently large. The equation may be solved exactly for core size or treated by the


linear expansion approximation to estimate the ‘reflector savings’. It is seen againthat S may be positive or negative, with positive or negative ‘reflector savings’8

�

2� Bsa �

1 �L

Ls

cothT=Ls

cothT=L

1 � L2=L2s

SBsLs ð36Þ

There then follows

�1 xð Þ ¼��o

��s

��

�� 1þ

S Tð Þ

cosBsacosBsx

� �ð37Þ

and

�f ¼ �1

�� 1þ S Tð Þ

cosBsx

cosBsa

� �ð38Þ

The first, constant term in this expression is engendered by the moderator thermalabsorption cross-section, as seen from the transition from Model 1 to Model 2. Thiswould disappear as we have seen in the absence of moderator capture in the core;the cosine-distributed component would then have arbitrary scaling since it isneeded to meet only fast losses.

The fuel loading as we have defined it can be determined over the constantcomponent and over the continuous cosine-distribution yielding

mmin ¼2�a

�� 1þ

2�Ls

�� 1

1 � L2=L2s

1 �L

Ls

cothT=Ls

cothT=L

26643775cothT=Ls ð39Þ

that reduces to the previous infinite reflector expression. Here a is to be the solutionof the criticality equation, Eq. (34). If the reflector thickness goes to zero we have anexpansion of the cosine-distributed term as

2�Ls

�� 1

1 � L2=L2s

1 �L

Ls

cothT=Ls

cothT=L

26643775cothT=Ls !

4�

�� 1

L2L2s

T3ð40Þ

The fuel loading to secure a flat flux thus becomes very large as the reflectorvanishes. In practical terms, therefore, there is a limit dictated by the maximumavailable fuel density. This constraint is considered below.

8 The linear expansion approximation is in powers of S but it is the totality of the right-hand side of

Eq. (34) that must be small for the cotangent to be small and approximated linearly.


Using the linear ‘reflector savings’ approximation for a then gives

mmin ��Ls

�� 1ð Þ32

�2�LsS Tð Þ

cothT=Ls

1 �L

Ls

cothT=L

cothT=Ls

1 � L2=L2s

� �� 1ð Þ

þ�Ls

�� 1

1 � L2=L2s

1

cothT=Ls�

L

Ls

1

cothT=L

ð41Þ

If S Tð Þ ¼ 0, a particular case, we may give an alternative form:

S Tð Þ

cosBsa!

��

�� 1ð Þ32

so that�1 xð Þ !��

�� 1

�

��s�o 1 þ

cosBsxffiffiffiffiffiffiffiffiffiffiffiffiffi�� 1

p

� �ð42Þ

and

�f xð Þ ¼�

�� 11 þ

��ffiffiffiffiffiffiffiffiffiffiffiffiffi�� 1

p cosBsx

� �with mmin

!��Ls

�� 1ð Þ32

1 þ��ffiffiffiffiffiffiffiffiffiffiffiffiffi�� 1

p

� �ð43Þ

As the reflector thickness vanishes, we may show that S Tð Þ becomes larger andhence is forced positive with positive ‘reflector savings’. However, the central fuelloading also gets larger with the slope or steepness at the core edge tying to com-pensate for the vanishing reflector thickness while still trying to employ the ‘flux-trap’ phenomenon to achieve the flat thermal flux. Thus we have

S T ! 0ð Þ !1 � L2=L2

s

1 �L

Ls

T=L

T=Ls

1 � 12 T=Lð Þ

2

1 � 12 T=Lsð Þ

2

" #��

�� 1¼ 2

L2

T2�

��

�� 1ð44Þ

that will be positive and larger as the reflector thickness is less than say a thermaldiffusion length

10. Dual-valued solutions

Williams (2003c) expects to see two solutions for the critical fuel loading at givencore thickness, at least in some cases. The dual result occurs when two differentreflector thicknesses can give the same ‘reflector savings’ and same critical size. Thelarger thickness corresponds to a smaller fuel density and smaller total fuel loading.

To examine this, consider the criticality equation in the form

cotBsa

BsLs¼ 1 � Gð Þtanh

T

Lsþ G

L

Lstanh

T

L� F Tð Þ ð45Þ


e
wher
G ¼��

�� 1ð Þ

1

1 � L2=L2s

ð46Þ

so G > 1 or G < 0 and F 1ð Þ ¼ 1 þ G � 1ð ÞL=Ls ¼ 1 � ��= �� 1ð Þ 1 þ L=Lsð Þ½ �. ThusF 1ð Þ can be positive or negative, above or below F 0ð Þ.

F Tð Þ is continuous and if it is to be double valued in this range, it must have aturning point in the range such that

1 � G

cosh2T=Ls¼ �

G

cosh2T=Lð47Þ

For L=Ls < 1;G > 1, putG

G � 1¼ H2 and take the positive root. We have H > 1

so that

coshT=L ¼ HcoshT=Ls ð48Þ

Then T=L > T=Ls and this is consistent with the assumption. For G ¼ 1 werequire �� 1 ¼ �L2

s =L2 ! 0 requiring � ! 0 as well as .

For L=Ls > 1;G < 0, put G1�G ¼ H2 and we have H < 1 requiring in Eq. (47)

that T=L < T=Ls consistent with the assumption. In the limit G ¼ 0 then 0 ¼

��1

1

1 � L2=L2s

� � again requiring � ! 0 at this limit.

At L ¼ Ls, G ! 1;H ! 1 thus accommodating the particular case. Now,however, employing L’Hopital’s rule,

F Tð Þ !cotBsa

BsLs! tanhT=Ls �

��

�� 1

LstanhT=Ls � LtanhT=L

Ls � L2=Ls

� �! tanhT=Ls þ

1

2

��

�� 1

T

Lscosk2T=Ls� tanh

T

Ls

� �; ð49Þ

starting at zero at the origin and with initial positive slope. Now the differentialbecomes

dF

dT!

1

Lscosh2T=Lsþ 1

2��1

�2tanhT=Ls

Lscosh2T=Ls

� �ð50Þ

and the turning point is the solution of TLstanhT=Ls ¼

��1�� < 1. Since the left-hand

side has a range between zero and infinity, there is indeed a solution.Eq. (48) does therefore have a unique solution in the range of 0;1½ Þ. Other than

at the turning point itself, there will then be two values of thickness T leading to thesame core size but different fuel densities over a certain part of the range. Naturallythe thicker reflector has the smaller fuel loading.


If F 1ð Þ < 0, the dual value range will be from T ¼ 0 to the second root (otherthan zero) of F Tð Þ ¼ 0. If F 1ð Þ > 0, the range for dual values will be from the finiteroot of F Tð Þ ¼ F 1ð Þ to infinity. Outside these ranges (and at the turning point itself)the function is single valued (Fig. 4). The range is complete, i.e. there are alwaysdual values (subject to the coalescence of the two roots at the turning point) of thereflector thickness for the same core size, if F 1ð Þ ¼ 0 or �� ¼ 1 þ Ls=L.

11. Constrained solutions

In his original paper, Goertzel addressed the constrained problem where the corewas prevented from being as large as the criticality equation calls for. He advancedthe solution to this constrained problem that additional fuel would be added at thecore-reflector interface to bring the system critical. His solution [and its correctlocation is confirmed by Williams (2003)] is given, however, as an infinitely denseaddition occupying no space. This is not physically realisable and we must in prac-tice propose that there is a finite region where the fuel cross-section is added at itsmaximum density (to meet the restraints of the variational principle). In the light ofthe Goertzel–Williams arguments, it can be expected that this addition is at the non-intuitive core edge rather than at the centre.

There is a further constrained problem to consider. The flat-flux solution calls fora certain maximum fuel density at the centre-line and this may be beyond the phy-sical limit. Again, where is the additional fuel to be located in a modified solutionthat achieves criticality with minimum fuel in the constrained space?

Fig. 4. The criticality function of reflector thickness F Tð Þ showing dual-value ranges.


A preliminary question then will be: can the reactor be made critical with the fuelavailable and in the constraint of core size. This will certainly be possible if the cri-tical size for the conventional uniformly loaded reactor is within the constraint.However, we can show that the minimum fuel solution is generally of a smaller size.We therefore address the question of the distributed fuel cross-section exceeding theavailable fuel density first because it has an explicit solution that bears on the otherconstrained problem. This leaves for further study the problem of minimum sizecore.

Thus we consider first the problem of a constraint on the fuel cross-section thatcalls for a central core region at the available maximum cross-section. Clearly wemust start with a region at the available maximum flux (for flux, read fuel density).It might be thought9 that this should then be followed by an outer core with thedistributed fuel starting at its maximum, i.e. continuity of fuel density, but thiscannot be valid. The flat-flux solution shows that both the fuel density and its deri-vative are continuous. Thus there has to be a drop at the core-core interface fromthe maximum density to some unknown fuel density.

The solutions for a conventional uniform fuel distribution in the two-group dif-fusion model based on the maximum available fuel cross-section �fm leads to afourth-order equation for the buckling given by10

L2s L2

2 B4 þ B2 L2s þ L2

2

� �þ 1 �

��fm

�fm þ�¼ 0 ð51Þ

2� �

where the thermal diffusion length is given by L2 ¼ D2= � þ�f . Assuming criti-cality is possible (sufficient �) there will a positive root B2

c and a negative root we canwrite as � C2

c . We have

B2c ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 þ 4 ��fm= �fm þ�

� �� 1

� �L2

s L22 = L2

s þ L22

� �2q� 1

�L2

s þ L22

� �2L2

s L22

ð52Þ

and for small buckling this gives approximately11

B2c �

��fm= �fm þ��

� 1

L2s þ L2

2

which is less than B2s ¼

�� 1

L2s

ð53Þ

Thus replacing the distributed fuel with the maximum density fuel at the centrelineis unlikely to save space and achieve any size constraint, even though the negativebuckling second solutions can be invoked to produce a flat thermal flux at theinterface between the central core and the distributed core region.

9 ‘‘It might be thought’’; i.e. I got it wrong originally.10 The notation used here will be different from the conventional notation because of our separate

consideration of fuel and moderator.11 A reflector would of course reduce the bare core size by approximately a migration length, M.


We assume that the unconstrained problem has been solved so that for theavailable moderator we know the critical half-size a and the maximum fuelcross-section required, �f 0ð Þ ¼ �fo say. Since the flat-flux solution exceeds thefuel density constraint we start with an inner core with fuel uniformly dis-tributed at its maximum physically available density, �fm < �fo followed by thecosine distribution of the distributed solution starting at ��fm where � is somepositive fraction less than 1. The core regions change at a value A say to bedetermined (Fig. 5).

Consider the solutions in the inner core. Using symmetry there are two unknownsin the solutions:

�2 ¼ cacosBcx þ cbcoshCcxso that�2 0ð Þ ¼ ca þ cb ð55Þ

and

�1 ¼ ca��fm

�s 1 þ L2s B2

c

� � cosBcx � cb��fm

�s L2s C2

c � 1� � coshCcx ð56Þ

noting that L2s C 2

c � 1 þ L2s =L

22 > 1 so that the second contribution to the fast flux

has the opposite sign to its contribution to the thermal flux.For the distributed solution we no longer have symmetry so that

�1 xð Þ ¼�þ�f xð Þ

��s�o ¼ dacosBs x � Að Þ þ dbsinBs x � Að Þ�1 Að Þ ¼ da ð57Þ

and

r�1 xð Þ ¼ �BsdasinBs x � Að Þ þ BsdbcosBs x � Að Þr�2 Að Þ ¼ �Bsdb ð58Þ

Fig. 5. Constrained fuel-density reactor in minimum fuel-loading configuration.


��

e A is to be the interface where the inner uniform core meets the outer fuel-
wherdistributed core. The thermal flux �o has to be flat and continuous, with its gradientzero, at the interface.
The distributed solution at the interface with uniformly loaded region has a fueldensity ��fm.

Matching the flux and gradient at this interface,

cacosBcA þ cbcoshCcA ¼ �o and 0 ¼ �caBcsinBcA þ cbCcsinhCcA ð59Þ

This determines the inner core constants together with the fast flux

cb ¼ caBcsinBcA

CcsinhBcAand �o ¼ cacosBcA 1 þ

BctanBcA

CctanhCcA

� �ð60Þ

in terms of the arbitrary thermal flux �o except that A is presently unknown. A isdetermined, however, by the requirement that the first core interface satisfies�1 Að Þ ¼ �o ��fm þ�

� ��s. This in turn requires

�1 Að Þ ¼�o��fm

�scosBcA

cosBcA

L2s B2

c þ 1�coshCcA

L2s C 2

c � 1

BcsinBcA

CcsinhCcA

1 þBctanBcA

CctanhCcA

¼ �o��fm þ�

��s

or, with the thermal flux cancelling, determines A from

��fm1

L2s B2

c þ 1�

Bc

Cc

tanBcA=tanhCcA

L2s C2

c � 1

� �¼ ��fm þ��

1 þBctanBcA

CctanhCcA

� �ð61Þ

which may be written as

��fm

L2s B2

c þ 1� ��fm þ��

¼BctanBcA

CctanhCcA

��fm

L2s C2

c � 1þ ��fm þ�

�ð62Þ

and with the solutions for the central region, as

fm þ��

L22 B2

c þ 1� �

� ��fm þ��

¼Bc

Cc

tanBcA

tanhCcA

�fm þ��

L22 C 2

c � 1� �

þ ��fm þ�� ð63Þ

The left-hand side is positive, as seen from the original uniform core Eq. (51) sothis equation is expected to have a solution in the first quadrant. For the particularcase that � ¼ 1 this equation may be further expressed as


tanhCcA

Cc¼tanBcA

Bc

so that an expansion in powers of A shows that A ¼ 0 is a solution.12

We may show that the equation has no other finite real solution by writing itas p xð Þ � BctanhCcx ¼ CctanBcx � q xð Þ with solution at x ¼ 0 or p ¼ 1 ¼ q.Then form f

0

xð Þ�1¼ cosh2Ccx=BcCc 5 g0 xð Þ

�1¼ cos2Bcx=BcCc to see there is no

further real root. This particular solution then meets the condition that the cen-tral fuel density of the original flat-flux solution is within the range of the maximumfuel density available and correspondingly calls for zero extent of an inner core atuniform density.

Thus in general � < 1 and we have in Eq. (63) an equation in two unknowns, � or" ¼ 1 � � say and A. To find these involves writing a further equation for the con-tinuity of the fast flux derivative at the core-core interface, in terms of the originalflat-flux solution and thus involving a further unknown. This is most convenientlyrepresented by using a prime to indicate a distance A

0

, initially the third unknown,in the co-ordinate system of the unconstrained problem. For simplicity we use aninfinite reflector and we have in terms of the (presumably ) calculated central fuelcross-section of the original design, the continuity of current expanded to second-order accuracy as

A0

A�� 1ð Þ�fm ��

�� 1ð Þ�fo ��¼ 1 � "

�� 1ð Þ�fo

�� 1ð Þ�fo ��ð65Þ

showing that for small constraint, at least, the critical size is made larger.A similar expansion for the continuity of fast flux gives

�o

��s�� 1ð Þ�fo þ� �� 1ð Þ�fo ��

� �B2s A

02

6

� �¼�o

��s�� 1ð Þ�m þ�þ �� 1ð Þ�fffm ��

� �� B2s A2

3ð66Þ

Substitution then yields, to first order in epsilon, and noting that we have shownthat O Að Þ ¼ O A

0� �as " ! 0

B2s A2 ¼

2" �� 1ð Þ�fo

�� 1ð Þ�fo ��! 3" for �� ¼ 2 and L ¼ Ls when �fo

¼ 3�: ð67Þ

Again for this particular case, A0

A ¼ 1 � 32 " and in general �f A

0� �¼ �fo 1 � 2"ð Þ that

demonstrates the discontinuity in the fuel density at the internal interface.

12 To first order the solution is indeterminate. To third order, only the zero root is obtained. To fifth

order a real root is obtained but this is lost at seventh order.


What we observe from this analysis is that the inner core region sustains a higherthermal flux in its constraint to maximum fuel loading but satisfies the restraint ofthe minimization theorem. But this uniform region will extend further than theunrealisable section of the distributed region it replaces.

Consequently if we now turn to the other constraint, limited core size, it isapparent that this cannot be overcome by adding uniform fuel in an inner core;indeed on examining Eq. (67) the solution is not available for " < 1. Instead theroles are reversed with a distributed inner core and a uniform outer core that makesuse of the flux-trap effect from the reflector to raise the flux above the constantthermal flux value �o, (Fig. 6). We make some general remarks before describing thesolution for a small constraint.

1. If the size constraint is active, the position of the core-reflector interface isknown. The criticality condition then determines the core-core interface location.

2. The central region is symmetric so the distributed solution has just the cosinecomponent. The outer uniform core has the anti-symmetric solutions added (e.g sinand sinh) with four coefficients. Two of these are determined at the core-reflectorinterface leaving two to be matched at the core-core interface where we have the sizeand slope of the fast flux that are themselves determined by the value of the thermalflux and zero gradient in the distributed solution. The resulting criticality equationdetermines the location of the core-core interface. For simplicity we assume aninfinite reflector.

The fast flux in the central region has the form

Fig. 6. Constrained core size reactor in minimum fuel-loading configuration.


�1 xð Þ ¼ gcosBsx þ �o�

��s

��

�� 1ð68Þ

and the thermal flux in the outer core has the form

�2 xð Þ ¼ bcosBc x � Að Þ þ csinBc x � Að Þ þ dcoshCc x � Að Þ þ esinhCc

� x � Að Þ ð69Þ

We write a0 ¼ A þ D; a � a0 ¼ " where a and a0 are both known and we seek D, thethickness of the outer core. The eight equations of the problem are as follows:

b þ d ¼ �o ð70Þ

cBc þ eCc ¼ 0 ð71Þ

gcosBsA þ �o�

��s

��

�� 1��fm þ�

��sL2

2 B2c þ 1

� �b þ

�fm þ�

��sL2

2 C � 1� �

d

¼ 0 ð72aÞ

�gBssinBsA ��fm þ�

��scBc L2

2 B2c þ 1

� �þ�fm þ�

��seCc L2

2 C2c � 1

� �¼ 0 ð73aÞ

��s

�

L2s

L2 � L2s

�1 a0ð Þ þ f � bcosBcD � csinBcD � dcoshCcD � esinhCcD ¼ 0 ð74Þ

��s

�Ls

L2s

L2 � L2s

�1 a0ð Þ �f

Lþ bBcsinBcD � cBccosBcD

� dCcsinhCcD � eCccoshCcD ¼ 0 ð75Þ

��s

�fm þ��1 a0ð Þ þ b L2

2 B2c þ 1

� �cosBcD þ c L2

2 B2c þ 1

� �sinBcD � d

� L22 C2

c � 1� �

coshCcD � e L22 C 2

c � 1ÞsinhCcD ¼ 0�

ð76aÞ

��s

�fm þ�

�1 a0ð Þ

Ls� bBc L2

2 B2c þ 1

� �sinBcD þ cBc L2

2 B2c þ 1

� �cosBcD � dCc

� L22 C2

c � 1� �

sinhCcD � eCc L22 C2

c � 1� �

coshCcD ¼ 0 ð77Þ


e assume that the constraint " ¼ a � a0 is small so that the necessary thickness of
Wouter core D is therefore small and expand four of these equations accordingly tofirst order to give:
gBs "þ Dð Þ þ�

��s

��

�� 1�0 �

�fm þ�

��s1 � L2

2 C2c

� ��0 þ L2

2 B2c þ C 2

c b� ��

¼ 0 ð72bÞ

�gBs ��fm þ�

��sL2

2 cBc B2c þ C 2

c

� �¼ 0 ð73bÞ

��s

�fm þ��1 a0ð Þ þ bL2

2 B2c þ C2

c

� �þ cBcL

22 B2

c þ C2c

� �D � �o L2

2 C2c � 1

� �¼ 0 ð76bÞ

and

��s

�fm þ�

�1 a0ð Þ

Lsþ b B2

c þ C2c

� �L2

2 C2c � B2

c

� �� 1

� �D þ cBcL

22 B2

c þ C2c

� �� 0C

2c

� L22 C2

c � 1� �

D ¼ 0

We have a system of eight equations with a vector solution of eight unknowns T ¼

�o; b; c; d; e; f; g; �1 a0ð Þ½ � of which again one, �o, can be arbitrarily assigned, togetherwith a matrix whose determinant must vanish, providing the equation for the furtherunknown D. The first two equations are simple and allow a reduction to sixth orderbut still leaves a determinant that should probably be evaluated numerically ingeneral cases. Whilst this can be done for any particular case, we may display ananalytical solution on noting that the unconstrained problem had a minimum fuelloading so that to first order the fuel loading is unchanged. Thus either directly or bynoting that in the unperturbed problem the perturbation weighting is a constant, �2

o,that cancels from the integrals, we have

�

�� 1L þ Ls þ að Þ ¼

��s

Bs�ogsinBsA þ

A�

�� 1þ D�fm ð78aÞ

Hence we may estimate the unknown g to first order. The algebra of what followsis significantly easier of we now make the particular assumption that S ¼ 0; so thatLs=L ¼ �� 1. This is a particular, not a special case, and we may reasonably expectthe results to be typical.

We can now find:

g��s

�0¼ BsLs

L

Ls� 1 þ

L

Lsþ"þ D

Ls

��fm

DLs

� �ð78bÞ


b ¼ �0L2

2 C2c � 1

L22 B2

c þ C2c

� �þ� 1 þL

Ls

�1 þ

"þ DLs

��fm þ��

L22 B2

c þ C2c

� �2664

3775

c ¼ ��0

�L

Ls1 þ

L

Lsþ"þ D

Ls

��fm

DLs

BcL �fm þ��

L22 B2

c þ C2c

� � ð79Þ

When Eqs. (76a,b) and (77) are used to eliminate �1 a0ð Þ we have

�o L22 C2

c � 1� �

1 þ C2c LsD

� �� ¼ b B2

c þ C2c

� �� L2

2 þ L2s L2

2 C 2c � B2

c

� �� 1

� �D=Ls

� �þ cL2

2 B2c þ C 2

c

� �Bc Ls þ Dð Þ ð80Þ

and the substitutions made, the zeroth-order terms cancel, as expected. With carefulsubstitution (no approximations for the buckling roots are needed) this gives thefirst-order expression as

L

Ls

"

D¼ 1 � 1 þ

L

Ls

�ð81Þ

�
where we have written ��fmþ�
. Now is less than unity and for large maximumfission cross-section therefore we haveD ¼ " L=Ls

recovering the Goertzel Dirac-type solution, that D ! 0 as �fm ! 1; ! 0.The amount of fuel in the outer core region is given by

mG ¼ �fmD ¼ �1 � ð ÞL=Ls

1 � 1 þ L=Lsð Þ" ! �

L

Ls" ð83Þ

where we recollect that at the assumed value S ¼ 0 the amount of fuel in the con-strained region width " is, to first order, "�. The discrepancy "� 1 � L=Lsð Þ is takenup in the change (to g) in the original distributed core region.

For the further particular value that �� ¼ 2 the minimum fission cross-section (atthe original reflector interface) is �fm ¼ � and the centreline value is �f 0ð Þ ¼ 3�.Using this for the smallest value of �fm gives ¼ 1

4 and D ¼ "=2 to illustrate thethickness of maximum fuel sheath for a small constraint on core size. At these fur-ther particular values, there is no discrepancy (to first order) between the Diracloading and the constraint saving but there is such a discrepancy more generally, notnoted in Goertzel’s original analysis where the continuous fuel distribution was leftunchanged.

One difference from the maximum fuel density constraint is that with the core sizeconstraint, the maximum fuel-loading region must indeed have a thermal flux higherthan the flux in the distributed region but the reflector is not placed under any


restraint, since by assumption no fuel is allowed there. This, and the location of themaximum fuel density region as an outer core, are the same in both the minimumand the maximum problem under size constraint.

It might seem strange at first sight13 that the extra fuel should not be loaded at thecentre of the reactor and thus minimise leakage. Of course adding fuel at the cen-treline does increase the reactivity but not necessarily in the optimum way. But theoptimum design has already minimised net thermal leakage so that this is illusory. Inretrospect at least, we can see several reasons for the maximum fuel density sheathbeing at the outer edge of the core. The previous constrained solution, Eq. (71) for" < 0, showed that fuel added at the centreline at a density higher than called for inthe unconstrained solution required a jump upwards at the inner interface; such ajump exceeds the assumed maximum fuel density. One might also say that by put-ting the additional fuel to the outside, it has the largest possible contrast with theunconstrained solution and hence the most effect. But fundamentally, the mini-misation turns on use of the ‘flux-trap’ and this will have its greatest effect close tothe reflector region.

We have provided a design through which either or both of the constraints onmaximum fuel loading and limited core size may be addressed. As both constraintsare imposed further, there will come a point where the distributed core is extin-guished and the only critical system within the constraints is a uniform maximumfuel loading. Beyond this, a critical system is not possible under the constraints.

12. Conclusion

The whole problem is intriguing and raises a number of issues. Without con-straints, the fuel saving in a distributed fuel reactor may be appreciable over itsuniform counterpart. One results suggests that the reduction of this two-groupproblem to a one-group problem needs careful consideration. The flat flux minimumsolution of the former is not physically realisable in the reduced model that never-theless can sustain a flat flux solution corresponding to a maximum fuel loadingwhen the edge-loading is considered.

The analysis suggests that the ‘flux-trap’ phenomenon is crucial to the minimumfuel-loading problem. Indeed the parameter L=Ls has played a significant role in theanalysis. It is this flux-trap that allows the fuel density to be decreased towards theedge of the core without reducing the thermal flux. Some of the fast flux that leaksfrom the core is returned to the core after slowing down rather than lost. Conse-quently in this condition, one may say that the fuel is expected to be a minimumbecause of the elimination of thermal neutron leakage from the core. As the reflectorthickness is reduced, the necessary steepness in fuel distribution at the core edge

13 But I did not get it wrong on this occasion.


increases, if the flux-trap phenomenon is to work, imposing an increasing fuel den-sity at the centre. This can be taken to the point where the fuel density reaches themaximum available physically from the fuel used and thus provides a secondarycriticality equation.

Thus we see a different pattern to the ‘criticality’ analysis

1. In the conventional uniform fuel case, it must first be shown that the infinite
multiplication factor based on all core thermal as well as fast propertiesexceeds unity. Then we prescribe the reflector and arrive at a criticalityequation to determine the size of the finite core.
2. In the present case, the ‘infinite multiplication factor’ must exceed unity but
this factor is determined by fast scattering properties and the yield in fissiononly, neglecting moderator thermal losses. A ‘partial’ criticality equation isderived to give the core size for a given reflector. This partial equation mustbe supplemented by the final step of assessing the maximum fuel density to seeit does not exceed what is physically feasible.
In a one-group model, however, this flux-trap phenomenon is not describable. Aflat flux can indeed be established in a central region with unity value of the infinitemultiplication factor. This solution in this central core region can indeed be descri-bed by collapsing the two-group solution. But the flux-trap mechanism to keep theflux flat in this core is lacking and a core edge region has to be imposed to bring thefinite system to criticality. Lacking the flux-trap mechanism, this finite region nowsupports a diminishing thermal flux and hence a maximum fuel loading.

An indication of the fuel savings relative to a uniformly distributed core where thefuel cross-section is that of the centre-line cross-section of the corresponding mini-mum fuel loading has been provided that suggests substantial economic and safetyissues.

We have found analytical justification for the observation by Williams that for afinite reflector there is a range in which a given distributed core region can be madecritical (with minimum conditions satisfied) by two different reflector thicknesses.

In the original work of Goertzel, studied further by Williams, consideration isgiven to a one of the two physical constraints. Our present study extends to boththe constraint on core size and the constraint on fuel loading and offers a physi-cal design rather than the infinite fuel densities of the earlier work. We concludethat:

1. for a fuel-density constraint, the inner core is provided at the physical limit
with a jump down to the fuel density of a distributed outer core;
2. for the size restraint, the reverse design is appropriate, replacing the outer
region with a maximum fuel-loading core. However, in general the con-tinuous fuel distribution is slightly modified, a new result.
A good additional exercise is to carry out the analysis in spherical geometry.Table 2 summarises the results of the four unconstrained models. It is an intriguing


aspect of the whole problem that in some circumstances, the presence of the reflectorleads to a negative ‘reflector saving’. I first met the flat-flux problem at my pre-doc-toral examination at MIT quizzed by Professor ‘Tommy’ Thompson in 1957 and Ican only acknowledge my dilatoriness in studying it more thoroughly.

Acknowledgements

My thanks to Mike Williams for making me study something I should have dealtwith 40 years ago and reading preliminary drafts and sharing his insight with me. Iam grateful to a reviewer for valuable corrections.

Appendix A. Flat flux with a finite reflector

In this appendix we give details of the full model solution for a finite reflector. Themoderating properties are uniform in the core and reflector, the latter of thickness T.Core equations

D1r2 ��s

� ��1ðxÞ þ ��f�o ¼ 0 and ��s�1 � �f þ�

� ��o ¼ 0 ðA1Þ

so �f ¼ �� þ��s�1

�o� �
D1r
2 þ �� 1�s �1ðxÞ � ��o ¼ 0 ðA2Þ

so that

�1ðxÞ ¼ AcosBsLs þ��

�� 1

�o

�sðA3Þ

D1 �� 1
where L2
s ¼�s

and B2s ¼

L2s

.

Reflector equations

D1r2 ��1

� ��1 x5 að Þ ¼ 0 D2r

2 ��

�2 x5 að Þ þ ��s�1 ¼ 0 ðA4Þ

and from the first,

�1 xð Þ ¼Gcosh x � að Þ=Ls þ Hsinh x � að Þ

LsðA5Þ

Taking an extrapolated boundary of zero flux at x ¼ a þ T gives

�1 a þ Tð Þ ¼ 0 ¼GcoshT=Ls þ HsinhT

Lsand H ¼ �GcothT=Ls ðA6Þ

so that


G ¼ �1 að Þ and r�1 að Þ ¼ ��1 að Þ

LscothT=Ls ðA7Þ

Next,

�2 xð Þ ¼ Ecoshx � a

Lþ Fsinh

x � a

L

h iþ��s

�

1

1 � L2=L2s

�1 xð Þ ðA8Þ

If this is to vanish at the common extrapolated boundary F ¼ �EcothT=L and forzero gradient at the interface

E ¼ ��1 að Þ��s

�

1

1 � L2=L2s

LcothT=Ls

LscothT=LðA9Þ

so that

�o ¼��s

�

�1 að Þ

1 � L2=L2s

1 �L

Ls

cothT=Ls

cothT=L

� �ðA10Þ

Then at the interface again

AcosB1a ¼ �1 að Þ 1 ��

�� 1

1

1 � L2=L2s

1 �LcothT=Ls

LscothT=L

�" #ðA11Þ

and

�AB1sinBsa ¼ ��1 að ÞcothT=Ls

LsðA12Þ

with primary criticality equation

cotBsa ¼BsLs

cothT=Ls1 �

��

�� 1

1 �LcothT=Ls

LscothT=L

1 � L2=L2s

26643775 ðA13Þ

or

cotBsa ¼BsLs

cothT=Ls

1 �LcothT=Ls

LscothT=L

�1 � L2=L2

s

1 � L2=L2s

1 �LcothT=Ls

LscothT=L

��

�� 1

26643775

�BsLs

cothT=Ls

1 �LcothT=Ls

LscothT=L

�1 � L2=L2

s

S Tð Þ ðA14Þ


where

S Tð Þ ¼1 � L2=L2

s

1 �LcothT=Ls

LscothT=L

��

�� 1ðA15Þ

Then S 1ð Þ reduces to the S of the infinite reflector. For small T we have

S T ! 0ð Þ !1 � L2=L2

s

1 �L

Ls

T=L 1 þ 16 T=Lð Þ

2� �

1 þ 12 T=Lsð Þ

2� �

T=Ls 1 þ 16 T=Lsð Þ

2� �

1 þ 12 T=Lð Þ

2� ��

��

�� 1

! 3L2

T2�

��

�� 1ðA16Þ

to second order. Thus for vanishing reflector, T << L, the function S Tð Þ is large andpositive.

In general we have

�1 x4 að Þ ¼��0

��s

��

�� 1þ S Tð Þ

cosBsx

cosBsa

� �ðA17Þ

Note that S Tð Þ changes sign with cosBsa so that the fast flux remains non-nega-tive, and

�f xð Þ ¼ ��þ ��1 xð Þ

�o¼

�

�� 1þ S Tð Þ�

cosBsx

cosBsaðA18Þ

also positive in the core despite the possible ‘negative reflector savings’.More explicitly,

�f 0ð Þ ¼�

�� 1

"1 þ

1 � L2=L2s

1 �L

Ls

cothT=Ls

cothT=L

0BB@1CCA

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�� 1 �

�� 1 �L

Ls

cothT=Ls

cothT=L

�1 � L2=L2

s

0BB@1CCA

2

þcoth2T=Lss

vuuuuuut# ðA19Þ


taking the positive root in view of the common sign for S Tð Þ and cosB1a.If S Tð Þ ¼ 0 then cosBsa ¼ 0 and we have the particular case that

S

cosBsa!

1 � L2=L2s

� �cothT=Ls

1 �L

Ls

cothT=Ls

cothT=L

! 1T ! 0 ðA20Þ

More generally, the fuel loading is given by

md ¼2�a

�� 1þ

S Tð Þ�

BscosBsa

ðBsa

0

cos d ¼2�a

�� 1þ

S Tð Þ�

BstanBsa

¼2�a

�� 1þ�Ls

�� 1

1 � L2=L2s

1

cothT=Ls�

L

Ls

1

cothT=L

ðA21Þ

The second, distributed term is positive for any reflector thickness and becomesincreasingly larger for vanishing thickness T: !

2�L 2s

��11

T 3.Here a is given by the inverse of the criticality equation a ¼ cot�1Bsa

Bs. If

BsLs=cothT=Ls is small (which will be the case if T is small and there are no reflectorsavings to speak of), a first-order approximation for the cotangent gives

Bsa ��

2�

BsLs

cothT=Ls1 �

��

�� 1

1 �LcothT=Ls

LscothT=L

1 � L2=L2s

26643775 ðA22Þ

so that

md ��Ls

�� 1ð Þ32

�2�LsS Tð Þ

cothT=Ls

1 �L

Ls

cothT=L

cothT=Ls

1 � L2=L2s

� �� 1ð Þ

þ�Ls

�� 1

1 � L2=L2s

1

cothT=Ls�

L

Ls

1

cothT=L

ðA23Þ

middle tem is the approximation, which becomes negligible with smaller reflec-
Thetor, while the third, distributed term, becomes larger and dominates.
For the particular case that L ¼ Ls the flux solutions require some modificationbut, using L’Hopital’s theorem, the final results become

S Tð Þ ¼2

1 �2T

Ls

1s

sinh2T=Ls

��

�� 1

and


cotBsa ¼BsLs

cothT=Ls1 �

��

�� 1

2T=Ls

2sinh2T=Ls

� �ðA24Þ

with

md ��Ls

�� 1ð Þ32

�� 2

�� 1

�Ls

cothT=Lsþ

2�Ls

�� 1cothT=Ls ðA25Þ

�f 0ð Þ !�

�� 11 þ cothT=Ls

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�� 2

cothT=Ls

�2

þ4ð�� 1

s24 35 ðA26Þ

and for T ! 1 then

�f 0ð Þ ! !�

�� 11 þ ��½ � ðA27Þ

Appendix B. Proof of flat thermal flux as adjoint weighting

In this annexure we give a differential proof based on Lewins (1963b), of the the-orem advanced by Goertzel to show that the perturbation weighting for fuel relo-cation in this model is the thermal flux. The proof allows a generalisation to variabledensities. It will be given for thermal fission only. The original reference extends theproof to fast fission.

The fuel is assumed to have negligible scattering properties in both groups and theresonance escape probability (from fast capture in the fuel) p is constant. Fuel per-turbation importance is then p��þ1 � �þ2 Then our equations are

r:D1 rð Þr ��s rð Þð Þ�1 rð Þ þ �p�f rð Þ�2 rð Þ ¼ 0

��s rð Þ�1 þ r:D2 rð Þr ��f rð Þ �� rð Þ� �

�2 rð Þ ¼ 0 ðB1Þ

We rewrite these equations in matrix form as

r:DD2r�� S�s�þ F�f where � ¼ 0� rð Þ ¼ f�1; �2gT ðB2Þ

and the three matrices are

D ¼

D1

D20

0 1

!;S ¼

�

�s0

��

�s1

0B@1CA;F ¼

0 p�0 �1

�ðB3Þ

Note that under the assumptions, all three matrices are constant. independent ofposition. We now seek a symmetrising matrix X that will simultaneously symmetries


all three matrices and hence make the modified matrix equation self-adjoint.The existence of such a matrix is due to Shapiro (1963). Such a matrix isgiven by

X ¼

1 p�

p�D2

D1�

p�

�

�

�s�

D2

D1

�0@ 1A ðB4Þ

It is seen also to be constant.We now have relations between the vector flux and the vector importance �þ ¼

f�þ1 ; �þ2 g

T of the form �þ ¼ XT� and � ¼ XT� ��1

�þ so that

�þ ¼

1 p�D2

D1

p� �p�

�

�

�s�

D2

D1

�0BB@

1CCA� and p��þ1 � �þ2

¼p�

�p�� 1ð Þ

D2

D1þ�

�s

� ��2e�2 ðB5Þ

noting the arbitrary scaling of importance functions here, a point not spelled out inmy original.

Appendix C. The constrained size solution to first order

We have the expansion equation, where we have assumed small quantities ";DS ¼ 0 and that

�o L22 C2

c � 1� �

1 þ C2c LsD

� �� ¼ b B2

c þ C2c

� �L2

2 þ L2s L2

2 C2c � B2

c

� �� 1

� �D=Ls

� �þ cL2

2 B2c þ C2

c

� �Bc Ls þ Dð Þ

ð80bÞ

On substituting for b; c the zeroth-order form is inherently satisfied:

�o L22 C2

c � 1� �

¼ �o L22 C 2

c � 1� �

þ �o ðÞÞ � �o ð1 þL

LsÞ ðC1Þ

To first order then

L

Ls

"

D¼ L2

2 C2c � 1

� �L2

s B2c þ

L2s

L22

�� 1 þ

L

Ls

�L2

s C2c � B2

c

� ��

L2s

L22

� �þ � 1 � ð Þ

Ls

LðC2Þ

Substituting the exact values for the buckling roots and noting that L22 ¼ L2


L22 C 2

c � 1� �

L2s B2

c þL2

s

L22

�¼ L2

2 L2s B4

c þ L2s þ L2

2

� �B2

c þ 1

¼ �� 1 � ð Þ and L2s C2

c � B2c

� ��

L2s

L22

¼ 1

so that

L

Ls

"

D¼ 1 � ð Þ 1 þ

Ls

L

�� 1 þ

L

Ls

�þ � 1 � ð Þ

Ls

L

¼ 1 � 1 þL

Ls

�ðC3Þ

At this value S ¼ 0 the ratio "=D would be negative when �� <�fmþ�

�fmor the system

fell below an infinite multiplication factor of unity and so may be ruled out.

References

Cassell, J.S., Williams, M.M.R., 2003a. A note on the behaviour of a two group minimum critical mass

problem as the slowing down length tends to zero. Annals Nuclear Energy 30, 865–871.

Cassell, J.S., Williams, M.M.R., 2003b. An integral equation arising in neutron transport theory. Annal

Nuclear Energy 30, 1009–1031.

Glasstone, S., Sesonske, A.C., 1967. Nuclear Reactor Engineering. Van Nostrand, New York.

Goertzel, G., 1956. Minimum critical mass and flat flux. J. Nuclear Energy 2, 193.

Hirano, Y., Yamane, Y., Nishina, K., Mitsuhashi, I., 1991. Reactivity effect of non-uniformly distributed

fuel in fuel solution systems. J. Nucl. Sci. Technol. 28, 595–607.

Lewins, J., 1963a. Minimum fuel loading in nuclear reactors. Nuclear Energy 103–109.

Lewins, J., 1963b. Two group importance functions without solving the adjoint equations. Nuclear

Energy 164–165.

Poon, P.W., Lewins, J.D., 1990. Minimum fuel loading. Annals Nuclear Energy 17, 245.

Shapiro, M.M., 1963. Minimum critical mass studies in variable density and epithermal reactors. Nucl.

Sci. Eng. 10, 159.

Van Dam, H., de Leege, P.F.A., 1987. Minimum critical mass systems. Ann. Nucl. Energy 14, 369–376.

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minimum critical mass and flat flux in a 2-group model

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