GOVERNMENT OF INDIA ATOMIC ENERGY COMMISSION

B.A.R.C/M48

GOVERNMENT OF INDIA

ATOMIC ENERGY COMMISSION

PHYSICS DESIGN OF HEAVY WATER MODERATED REACTORS

Lectures given by K. R. SrinivasanNotes prepared by Kamala Balakrishnan

Reactor Engineering Division

BHABHA ATOMIC RESEARCH CENTRE

BQM3AY. INDIA

1 9 7 1

B. A. R. C. /I-148

GOVERNMENT OF INDIAATOMIC ENERGY COMMISSION'

PHYSICS DESIGN OF HEAVY WATER MODERATED REACTORS

Lectures given by K. R. SrinivasanNctes prepared by Kamals. Balakrishnan.

Reactor Engineering Division

BHABHA ATOMIC RESEARCH CENTREBOMBAY. INDIA

A B S T R A C T

This lecture course was arranged to make the members

of Theoretical Fhysics .Section fani l iar with the physics

aspects of DO moderated reactors . The main emphasis has

been on the methods for calculating l a t t i c e parameters

and long term reactivi ty change. The method described is

the conventional one, which lwes the four factor fomula.

Some aspects of fuel management have also teen covered. The

reactor theory \auderlying the calc-ulational methods has*.

been described in d e t a i l .

ACKNOWLEDGEMENT

I irish to thank Shri B.F. Bastogi for his constant

encouragement and for having made the suggestion \ t this

work should be carried out, and Miss H.A. Irani for doing

most of the typing work.

Contents

1. Introduction 12 . Optimisation 43 . ' Sffect of Varying the Parameters 34. The Heterogeneous Method 135.. The Homogeneous Methods 226. The Fast Fission Pastor 257. The Resonance 2scape Probability 348. The NDA - Model 449. HI for Clusters 4310. Thermal calculations 5211. The 'festcott Llodel 5612. The Neutron Tenperature 6713. Calculation of r 7014. The Theraal Uti l isat ion Factor 74

15. Calculation of L2 9316. Calculation of D 3417. D In the general case 9813. j'e-asurenent of Buckling 11415. The Substitution JIathad 11820. The Swedish "Method 13321. The Fast Fission Ratio 14622. I n i t i a l Conversion Hatio 15323. Spectrum Parameters r and T 155

24. The Fine Structure Hatio 16025. #he 3urn-up Equations 16126. The Saturating Fission Products 17527. Non-saturating Fission Products 17428. Pu-240 Cross-sections 18029. . Fuel Management Schemes I84.30. Continuous Fuelling 19231. Unidirectional Fuelling 20032. Hadial Fuelling 20133. The Roundelay Scheme 20234. Problems of In i t i a l Fuelling 202

Appendix 209References 213

The most important factor point: r.r to the i"s i r a b i l i t v of To(as amoderator in reac tors i s i t s ]c?r a r so rp t i cn c ross - sec t ion . Thffollowing table gives the values of or and SO" for three condor,moderators.

d— (barns) .5 <T (bams )OL22C0 *

H2O O.659 24.21"2O 0.00144 q . ?

-S <T represents the moderating a b i l i t y o r a moderator. Cbvious?v2?Q is net as s- cd a r r d e r a t o r as "-"C. - ' it 'he lo?; v^lue cf <T r?kssi t possible ;o build 7>C resc tors ^i IT. ?.~ tur^. 1 ursiiiu.T. 3.s f u e l . Thisleads to I c^ fuel l ing c o s t .

The comparatively I Q T value cf -5aifor 3p^ ref lected ir. the HP-?-:for a l a r r e r quantity of 22C to achieve thertr .al isation. 7ec-"uce "f t h i s ,the fuel eleT.en^? sr? ".^t =s ^r?"Iriei ^cp-^ther as they zv~ •i" a- '"o"r e a c t c r ; se t'-.= t fh?»ro is -"•;.ch .TOTC rcctr for r 'nysica] mar.ipulaticr. s r idiff"?T"?nt i("?'j?r<! l ike ""he prescure vessel and •cr-iosure tucs r f^c torshave evolved, panh h?.'"ir.r i t s CTT. advantages and disadvantages.

Pressure Y<?ssgl

This has the advantas-e cf a smaller quant i ty of s t r v ! viralmater ia l in the core , and c^risecuently l ess a^sorr^tion. rh^"^? arefewer j o i n t s leading: to les~ leakage cf I2C- The dipadvartare ii' thatthe moderator terrrsr^-t'jre i s r e l a t i ve ly his"h, r*"1;*!; t ir .r i". hij"h "C . t r r rtemperature and lo r ^ . Also, an innrease in s ize of the pre?cnre vesse!i r i l l r e a u i r e increased vess111''. '".'ill th ickr.o = £. Thi? is a l i t i :•" "icr. 0.".y*a^Q t o r S ' Z ^ . " " ^ o e o l a - ' T T ^ a c 77 - i l l n a ^ C n ^ 1 ' ! 1 ^ ^ ""-^ ' T? 1V TC t f' 1 p -"c •.- r> v i«p

tube kind cf reac to r , s ince mo? t o** c:r r eac to r s e.^"., ? / . " ar.d V.-'-?r =.rea l l cf d ia t tyos . The r.h-vsic" c&!c':lo-ricp_s ar.d T.odelp used will ho".'°v<?r,r o t be very different *"or the t " o .

The ?TPS s".rQ Tu'" 1 Peactcr

t h e d i s a d v a n - a r e s o-" t h e p r e s s u r e tube c o n c e p t ar.d vic^ ' ' c r " ; - . •'••'re

to be t h i c k . Cuts : ie -he ••r'3~s;r'? ":'rc i s thp -:s ] ar> ••ir: °. "-:bc ••'"."sctre i ' . i su la t i i? j a te r i? 1 . ne'v-esr tr.e f-l-o tc -r°ven- h pat '"TTT e;-" snir

::2'-, organic ccmpundr ^"i ""-T ny rases 'n3v>; beer s~u".:f'd.

Cost Caloulatloa

In all design calculations that are done r.a reactors, the aim iatwo fold. One is to find the reactor parameters for whioh the unit energycost will be least; and the other is to calculate various quantities whiohare needed for the day to day operation of the reactor, aa for instance thereactivity coefficients. For the f irst one, a large number of reactors haveto be evaluated and so the models used are comparatively crude. For the secondone, since values are needed for operational purposes, more accurate methodshave to be employed, and since only one reactor need be studied, 1his can bedons.

The unit energy cost in mili/Kwhr oan be expressed by the followingapproximate formula.

r» _ /ooo "P

C • unit capital eost In $/Kw(e)a - interest rate in # . This includes both the interest on tha capital and

the depreciation of the oapital arising from the fact that the l i f e ofthe reactor i s f inite .

L - load factor in fractionF - cost of fuel in Z/Zgm. U.B m burn-up In MWD/Te U.e - net efficiency

We oan make a calculation for a typieal example

C - $ 300/Kw(o)a - 7#L . 0.8P - $ 45/Xgm. 0B - 9000 MWD/Te U.e . 0.3The f irst term in the expression far energy oost in tha o ontrlbution of oapitalcoat. This is . 3JXLJCJL . , ^ / K , te.

876.6 z 0.8seeond term itiich la the fuelling oost - 100 x 45

24 x 9000 x 0.3

The operation and maintenance component i s of the order of 1 ail/KiteTotal oost - 4.7 mila/KwhrWe see tuat the most .Important component Is the oapital eta urge riiieh foraanearly 65^ of the whole.

Reaetor Parameters

Lee us now see the design of a particular reactor, say the HAPP. The totaleleotrle power is fixedt, say from oonslderatioaui of grid or doaand. Theoondenser temperature la also fixed beoause the lowest teaperatnre available

is conveniently the room temperature at the site. From metallur-gical considerations we decide that the fuel should be UOCladding for the fuel has to be selected on the basis of compat-ibility with the fuel, resistance to corrosion, low absorptioncross-section, etc. and this leads us to zircaloy sheath.

Now there are a number of other reactor parameters to bechosen, like temperatures and pressures, total mass flow ofcoolant, volume of fuel, number of rods in a cluster, and soon. The l i s t is lone and some of the parameters will be depet.d-ent on others. For instance, onoe the pressure tube thicknessis fixed the coolant pressure is automatically fixed. Howeveran attempt can be made to select one set of independent variables.One such set is given here. I t should be remembered that theset is not unique.

1. Coolant pressure2 A T — T Ta' ** l ~ outlet - inlet3. Height of the reactor4. *>ef lector thickness5. Total number of channels.6. Pin radius7. No. of rods in a cluster.8. Area of the coolant9. Pitch.

10. tadlal form factor.11. Moderator temperature12. Enrichment.

These are the variables in the reactor. Then there will betwo Independent variables connected with the heat exchanger; butin these lectures we will not be dealing with the heat exchanger.

The methods which are used to optimise these will bediscussed in the next lecture.

-: 4 :-

LECTURE 2

In addition to the 1'2 variables of the reactor andtwo of the heat exchanger that were mentioned in the lastlecture optimization of the reactor is subject to certainconstraints or limitations.

1. Temperature of ths fuel should not exceed a certainmaximum which may bo the melting point ofthe fuel.For U02 it is about ;H>OO°C.

2. Temperature of the sheath is constrained to remainbelow some value which is about 400°C for zircaloy.This is not very 3erious in pressurised water reactors;But in boiling water reactors the phenomenon of burnout,which is a limitation on the heat flux that can comeout through the clad, is very important.

3. The outlet temperature of the coolant has to be belowthe saturation temperature except in a boiling waterreactor.

4. There may be some restrictions arising from themethod of control. For example, suppose it is a controlrequirement that it should be possible to start thereactor within half an hour of shut down, and a certainamount of excess reactivity is provided for xenonover-ride, this imposes a limitation on the flux levelin the reactor. It is difficult to take such restrictionsinto account during optimization. But they are usuallychecked before a design is frozen.

The calculation of energy cost from the set of independentvariables we had selected can be done approximately along thefollowing lines.

All the 12 independent variables are given specificvalues, and the aim is to calculate the cor^sponding cost.The outlet coolant pressure is given. This fixes outlettemperature following the criterion that boiling should beavoided. AT being known we now get the inlet temperatureThese two temperatures along with the parameter ox the heatexchanger - turbine system gives the gross efficiency ofconversion of coolant heat to electrical power. An estimateis made of the electrical power needed for pumping and otherauxiliary servicos. This, added to the net electrical poweroutpui wanted from the reactor, and the whole divided by thegross efficiency gives the total thermal power that has to becarried by the coolant. Dividing by the no. of channels givesthe average channel power, and multiplication bv thp formfactor gives the maximum channel power, A heat transfer and

-: 5 :-

hydraulic calculation on the channel now skives the maximumsheath temoerature andthe total pressure drop. The pressuredrop with the assumed outlet pressure, gives the inlet pressure.If the inlet pressure and the maximum sheath temperature arenot within the acceptable limits, the coolant flow is variedand the entire sequence of calculations is repeated untilacceptable values are obtained.

Now we have the total power carried bv the coolant.The fission power of the reactor will be somewhat highersince some heat is ilwavs lost. For this reason the coolantpower is multiplied hv a factor of the order of l-*)5 to2ive fission power. Kith this knowledge now we .10 over tothe physics calculations. The aim of the calculation is toget the discharge t*-Tndlation and since the main part of thoselectures is to consist of the description of these calculations.Ve will not discuss it at the moment.

At the end of these calculations we know both the capitalcost and the fuel burn-un and so we are in a position to calculatethe unit energy cost. The calculation could be done for aiotherset of values of the 12 independent variables and thus thereactor could be optimised for minimum energy cost. An attemptto minimise cost with respect to 12 parameters could be verydifficult. Dynamic programming may be of help in this.

At this point it may be mentioned that the 5th narataet ersin our set, viz., the number of channels, may conveniently bereplaced by another useful parameter jkde . The significance ofthis quantity ia the following.

Consider a fuel pin in which the temperature distributionhas attained a steady state. The rate at which heat is produced

at any point will be a function of distance from the centreq(r). Take a unit length of the pin. The amount of heatproduced in a cylindrical shell of radius *.' and thicknessdr' is

. . the total quantity of heat produced in the pa*** °* thepin Inside the radius r is

- : 6 : -

The amount of beat conducted across the surface of radiusr is

where ® Is the temperature and K is the thermal conductivity.In the steady state all the heat produced within the radiusr has to be conducted across the cylindrical surface at r

Integrating from the fuel pin centre to the surface

ie COf ± L \ c*.') *».' » - f

Total ano<int of heat produced in the fuel Is•R

€1 =

-: 7 :-

factor depending upon the shape ofthe heat production profilewithin the rod. When the profile is flat, Its value la unity.In actual practical cases -L is very nearly unity.

Since f, IC and T . are more or less fixed, this gives a

relation between Q and T Obvlouslv if we do not wantcentre'

the fuel to melt in the centre. T + should remain below' centre

the melting point and this imposes a lower limit on Q, thetotal heat production In the rod.

The maximum specific power in the reactor should thusremain below.

/r i **

- : 8 : -

where P is the densitv of the fuel. In terras of heat flux,the maximum value imposed by the condition of centre-lineme 1 tins; is

Effect of Vary'^y the ParametersLet us now see how small changes in the different

independent variables reflect on the energy cost.

If the coolant pressure increases, the coolant temperaturecan increase. This increases the thernodynarnic efficiency andhence, for the same electrical power, thermal power can comedown, lending to smaller size of the reactor and consequentlyreduced capital cost. At the same time, thicker pressure tubeswill be needed to withstand the increased pressure. So neutronahsorntion will increase, decreasing the burn-up and raisingthe fuel 1 Ins: cost.

"".Tien &T decreases, the gross therraodynaraic efficiencyincreases. This is because T . . . is fixed by the coolant

pressure, andt'<e tetiporature of steam from the heat exchangeris a function of both T and T ( . . . Hut then the coolant

flow tias to increase. This needs increased pumping power andred'icrs the net station efficiency. Very high coolant flowcan also cause vibration of the pins and consequent difficulties.

A changp in reactor height a, affects the leakage. I t canbe said that for a bare homogeneous reactor, minimum leakageis obtained when U = 1,85, provided the volume is constant.

nVhen reflector thickness is the range 0 - 7 0 cms, the

reflector saving is equal to the reflector thickness. Asreflector thickness increases, leakage decreases and so burn-upincreases but at the same time capital cost increases due tothe cost of the reflector. Again form factor decreases so thatthe average power can increase and bring down this size and therebythe capital cost.

An increase in the number of channels decreases theleakage thus increasing burn-up, but simulatneously increases thecapital cost.

Mien the pin radius increases, the maximum allowablespecific power decreases increasing the capital cost. Physics-wisethere is an optimum pin radius which gives .the minimum leakage.This will be discussed more fully in the next lecture.

LECTUHB 3

Continuing the discussion on the consequences of thevariation of pin radius, we notice that the mailmumspecific power in the reactor is

so as Tt decreases, maximum specific power increases andthe reactor can be smaller. However, for thinner rods,fabrication is more costly, both because it is moreexpensive to manufacture the thinner rods, if we countthe cost per unit weight, and also because the rejectionrate Is also higher when the product has to conform tonarrower specifications. The net result is an increasein P, the cost of fabricated fue-l/kgm. The cost is roughLy

D S /M

Pin radius (cms) Cost $/kg.U Maximum burn-up for= i.O m-2

0.50.60.70.8

75594943

8450902092509220

-: 10 :-

For burn-up there is an optimum value of rod radius.

As the number of rods in the cluster increases, theflux depression in the cluster ii-oreases. This decreasesthe average specific power and fuel inventory increases.It should be noted that the number of rods in a clustercannot be increased in a random manner. The number shouldbe such that it should be possible to arrange the rodsin a way so as to have approximately equal amount ofcoolant available to tacts rod. Otherwise there may besharp temperature differences between parts of the coolant,It is found that suitable values of rod numbers are 7,12, 19, 22, 28, 37, etc. The total power in a channel is

whore II is the reactor height and Nc is the number of rodsin the cluster. So if Nc increases, the total number ofchannels can decrease. It may be necessary to have anincrease in pitch so thatthe reactor size does not changeappreciably. The main difference comes because now thereare fewer number of channels to be fabricated. The totalcost of channels is of the form

The fi^st term i s the cost of fixing the pressure tubeto the reactor. In the second term V . i s the weight ofthe pressure tube! When the pressure tube diameter increa-*96 9 the thickness has also to increase so that V . mayremain constant. The total cost wil l however decreasebecause of the f irst term. Burn-up decreases somewhat.

Another way i s to change the number of rods keepingvoltage" of fuel in the c luster constant, i e . by changingthe pin radius. If the number of rods increases, thepin radius decreases. This increases the cost but theincrease in maximum specific power accompanying thedecrease in radius can decrease the inventory. Physicswlae,a larger number of smaller rods means thatthe effectivesurface has increased. So the resonance absorptionincreases and reactivity comes down. There i s some gainin conversion ratio.

In connection with coolant area we note that thetotal channel power i s given by

where V is the mass flow and is equal to Av, A. being thecoolant area and V the coolant velocity, li A decreasesV increases giving rise to mechanical problems. Toosmall a coolant area might cause the rods to be so closetogether that slight mechanical vibrations may cause rodsto touQh. Ilot spots mi ht develop at these points ofcontac. since heat transfer there i s affected. Normallyin pressurised water reactors, coolant area i s fixed bysuch mechanical considerations.

In I1LW systems, the coolant area depends upon theenrichment. With natural uranium fuel, the area has tobe minimised because H O absorbs neutrons. With enriched

fuel, larger areas are permitted, and the consequence ofthis i s a very important safety consideration — a reducedpositive veld coefficient.

Pitch against burn-up Is some thing as shown

If pitch is decreased below the maximum of this curve,we los t some burn-up but save in capital cost because ofreduced D20 inventory. Meanwhile, mechanical considerations

demand that the pitch should not be too small. In theDouglas point reactor, the maximum burn-up was found fora pitch of 21.5 cms. Engineering considerations neededa larger pitch, 22.86 cms. Our calculations here placedthe optimum burn-up value at 24 cms.

In this connection i t may be pointed out that allthese burn-up calculations are made by certain computercodas which are extremely fallible things. The accuracyof our predictions i s dependent upon these codes and thephysical models on which the code i s based.

-: 12 -."

The moderator temperature in a CAN'DU type reactor isaround 70°F. If this increases the reactivity may decrease.So the moderator is circulated and cooled. The circulationwill need some auxiliary power. An alternative is to havebetter insulation of the coolant tubes but this will meanincroased absorption and will impose a burn-up penalty.

The form factor in a -eactor has three components,the radial, the axial, and the cluster form factors. Thespecific power is «;iven by

-*• z , « *

whore •*-.£ is the macroscopic fission cross sectlon/gm. ofthe fuel. E is the energy release per fission and C isa constant for converting Mev/sec. into watts. If thereactor is limited by the maximum flux, as it is in mostcases, a higher specific power can be had byreducing theform factor. For a bare homogeneous core, Faxial ss "/*and Fradial at 2.307. Fcluster is usually around 1.2 to 1.5.

The assault is normally made on Fradial because mostreactors are so designed that there is less flexibilityin the axial direction and within the cluster. A radialrefloctor helps in reducing the form factor since the fluxnow does not fall to zero at the core boundary. Assumingone group theory we have

Fradial = 1.2025 1

J» (2.405 Ttcore)

The effect of reflector saturates after a certain thickness.When n . n m

O T B ~ 0.8, the form factor is about 1.657.

To reduce the form factor still further, we have toresort to some other method. One comnonly employed la toflatten the flux in the central zone. A prefectly flatflux would mean that budding is zero. Assuming that theflux is separable in asial and radial directions, we cansay thai a radial variation in oore properties will changeonly the raJial buckling.

*• f w \

** - c —

Obviously Axoan be reduced by reducing Kn . One way ofdoing this is to introduce sorae poison into the central,region. Another is to change some lattice constant likepitch or enrichment to get a lower K.&, in the centraLzoao. A third is to irradiate the fuel in the centralzone to higher burn-ups by a i-elatively slower fuelmovement.

Vhen Kw is deceased in the central region, i t hasto be increased in the ontor zone. This Increases thebuckling and consequently the leakage also. So the burn-upin the outer zone decreases. This effect becomes morepronounced as the radius of the flattened zone increases.

Following one group theory, we now try to find anexpression for the buckling of the outer zone. The fluxprofile in the reactor locks something like to depictedin the figure.

R. is the radius of the central zone and ft is the extra-polated radius. The flux in the central zone is assumedto be perfectly flat. 1'hus the flux cin be written as

••[«• y»*o -, <

-: 14 '.-

s - •«•

The radial form factor is

In the central zone, the radial buckling is zero. Ifthe reactor height is made infinite the axial bucklingKill also be zero, ie. the total buckling will be zero.In other words the li <*, of the central zone will beunity, and since in a critical reactor K ._ =• 1, this

means that there is no leakage from the central zone.Sinco we have already assumed the separability of fluxinto axial and radial parts, we can conclude thatradial leakage from the central zone will be zero evenin a finite reactor. Thus the net current at the inter-face r = 1. should be zero.

: 15 :

3. C/» -R

.In the unflattenad reactor, y* s a " V o g . In the flattened case, we

P°* /•* « X i±2f» 30 that % directly giv«3 us an estinate of the

amount of flattening.

f TSlhen E, - 0, X - 1 . 1

If ne want, we can 3olve this, and get the valoa of X. for any

particular value of

In eqn. (1) we now subatitate

—•R

giving

: 16 J

Substituting for — from (2)A

a aat. x -t-

A y. c x") r_ z J.CxD -

3-.C~OY.CX:> - J.C*^Y.C**-) L

The figure shows that the form factor improve^ as the centralzone becomes larger, but the rate of improvement fall3 off after a while.

- : IT : -

The nlot of 3* against I./1* shows thatxincreases rapidly

alter a stage. The sharp increase in buckling leads toan equally sharp increase in leakage which reduces theaverage burn-up.

The last factor i s enrichment. As enrichment isincreased the burn-up will increase but will saturateafter a while because conversion ratio decreases. Theburn-un raay also be limited by metallurgical consider-ations. The cost of fuel/kqo. increases with enrich-ment. An optimum value should be found for the fuell-ing cost.

- : \ 2'. -

The Heterogeneous Method

The previous lectures must have made it clear theimportant quantities to be obtained by the physics calculationare the burn-up and the power distribution. From the physics pointof view, the limitation on burnup is that the Keff should be greaterthan unity. Our problem thus reduce to the calculation of Keff asa function of time, i.e. our oalculational methods should be able tooaloulate Keff under any given oondition.

Broadly, the oalculational methods can be divided into twocategories; the homogeneous methods *ioh are the more conventionalones, and the heterogeneous methods which are not very widely usedand in which not much progress has been made.

In the homogenous method, the reactor is divided intoseveral zones. Bach zone is assumed to consist of a number ofidentical lattice cells. The property of the zone is then equated tothe property of the cell. These properties are usually calculatedby considering an infinite array of cells or in other words, byusing the boundary oondition that the net current at the cell boundaryis zero. The microscopic flux distrubution in the cell is calculatedand the cell constants are obtained by averaging the aross.section withthis flux. These constants are now used to solve the reactor equations.As the reactor operates, the different regions in the reactor getdifferent irradiations and the assumption of a uniform lattice cellis no longer valid. It is however, still used to a limited extent.

In the heterogeneous mathod, the fuel rods are treated assources of fast noutrons and sinks of thermal neutrons. Certainsimplifying, assumptions are made.

1. The sources and -sinks are considered to be lines of infinitelength.

2. Elementary diffusfoa theory may be used for describing thebehaviour of neutrons in the moderator.

3* The moderator is assumed to extend up to infinity.

The first assumption is justified if the rod diameter issmall compared to the inter rod distance and the flux distribution withinthe rod is symmetric.

The intensity of the thermal sink in the kth rod is assumedto be proportional to the thermal flux at the surface of the rod.

-: 19:-

'- r T!•-• J

The fast neutron source in a rod should be propor-tional to the thermal sinic.

** = * 1***]. arraln Is a constant.

The fast source from a rod . ivea neutrons Into themoderator. The thermal source at a point x"' due tothe fast source in the rod at *?£ may written as.

j nelwith Fermi age theory we i»et

ij could be a Iternel - obtained usinn; any theory, sayth F

The thermal source at *• causes a thermal flux at _^ ->3? which could be expressed by a diffusion Kernal 4C l - «•'O

So the flux at *- due to the thermal source at- **•'caused by a fast source at x^ i s .

Flux at s». due to ail thermal sources from the fastsource at i j ,

The diffusion Kernal f can also bo used to findtho flux depression at *. due to thethermal sink atThe value is

the net flux

= ** c 12 -*k iThe flux at r-"dna to all the rods.

-: 20 :-

F is the Green's function and gives the thermal fluxat r due to the fast source at -n^

We now change *. C JI. - n^ the idea beingmerely to give us the value of the flux at all pointsin the vicinity of the n*'1 rod. Expression (i) can besplit up into two portions, one the contribution dueto the ntn rod itself, and that duo to all other rods.For the latter part the nxpression 3? — 3?^, which isthe distance of the spatial point from the K* rod,can be taken as nearly equal to the distance <i ,Abetween the rods in and IC since our spatial point isin the vicinity of n.

At the surface of the n rod, flux is

The two value of flux can be equated.

This can be written as

L --Vhere "*,»$. can be e a s i l y ohtalned from ( 2 ) . N i s theto ta l number of rods. There are N such equat ions .

If the system of homogeneous equations (3) isto have a solution the determinant of <*. should vanish

(4) can be solved to give N, which gives the number ofrods fo*- criticality.

Alternatively, we notice that ^ by definition, is thesource of fast neutrons in the rod for a thermal neutronsink of unit intensity. If we replace T A by T^ / *«uwe have decreased the multiplication factor by Kgff sothat the system now becomes critical and 1*1 shouldvanish. Solving it now -we can »et the value of IC __

As N becomes large it becomes increasinglydifficult to solve this.

The flux within the rod is taken as^ symmetric, ie.,the source is assumed to give out fast neutrons iao-tropically in all directions. A proposed correction isto replace the line source by two line sources whichwill take into account some assymmetry. The sjiathematicswill of course, become more involved.

The effect of afinite reflector can also be includedFor this purpose, the Green's function will have to beobtained by solving the diffusion equation with the pro-per boundary condition.

— • 00 •—

THE HOMOGENEOUS METHODS

In this again, there are broadly two approaches.

1) By following 1he neutron cycle ie., the four factor formulaand one or two group leakages. This is used In DUMLAC,LATREP etc.

2) The multi group treatment in which the diffusion or transportequations are solved in a number of groups. The number maybe of the order of 4 or 5, when it is called a few groupmethod, as in CAROL or METHUSELAH, or it may be of the orderof 40 or 50, when we call it a many group method, like

The neutron cycle method is to define "| , «, -p, -I ifor the lattice cells.

DUMLAC

LATREP

The difference between LATREP and BDMLAC ia that here theepithermal captures are given some special treatment whereas inUUMLAC they are included in ttie thermal captures.

- : 23 : -

pae is the capture escape probablltty excludingthe resonance. Here the neutron cycle has got twobranches. The capture escape is separated into twoparts, p due to U-238 and pcc due to all the rest. Itis also assumed that p takes place first and p Mafter that. If we break the cycle at the point 1>, wecan calculate the number of neutrons that come thereafter ona generation.

-*• L. a

If we take thu censured value of the buckling andcalculated values of all the Lattice parameters and

suhstltute them in the expression for K .. wo should

obtain unity. The departure of this value from unitygives a measure of the error in the code.

The Paat Fission FactorToday I shalL try to 2ive the definitions of the

various parameters and indicate the method of calcula-tion.

£ has been dcfiued in two ways; one is aa thenumber of fast neutrons escaping from the fuel for thefirst tirae for every fast neutron produced in thermalfission, the second is as the number of neutrons slow-ing down below the fast fission threshold for everyfast neutron produced in thermal fission. It doas notmatter which definition we take. But the definitionof p should be consistent with i t . In both UUMLAtand LAT'tfSP the first definition is used, though thereis a slight difference between the definitions in thetwo codes. Uumlac merely considers the total numberof neutrons loaving the rod for the f irst time; where-as Latrep considers the net number of neutrons abovethe fission threshold that escape from the rod andbelow the fission threshold merely counts those whichescape for* the first time. Thus DlfrlLAG neglects thoseneutrons which escape the rod above the fission thre-shold, return to the rod while s t i l l remaining abovethe threshold and then cause fission. The error causedby this nefiluct in DUMLA.L will be of the order of 3 ink.

For the calculation of C , two energy groups ofneutrons ar<* used, one with energies above i .4 mev.,the other below 1.4 nev. The method usually eranloycdis a col l is ion probability method. The rod ia dividedinto a number of concentric cylindrical regions.

-: 26 :-

it the n ' boundnry the current is J . The total current

the surface isthen

) i the n region, Q i s the source in the region. J

is thu nutwa'-ft current through the n surface, and J~

i 5 Via inward current. Then we oun write

-.i-hero W " i s the nrobabillty that a neutron born in the

vi)luine of tfie n region crosses i t s outor surface, etc

Tliose two equations are written for overy Rroup underconsideration, e^., in Duralac or Lat-cm there are twosuch sets , f) i s the source in the n region in a

•ift icu I ar »roun and ts producud by slowing down fromi.!u) hi't'ior groups of neutrons in the region. Fissionof neutr.wis from other groups will produce secondariesin this group too.

la short, a neutron which changes from one group toanother in a region, whether by slowing down or byfi-ision, ciMitributoH to the source of the lutter groupia that region. The probabilities ¥ are not merelythe f irst flight probabilities. It i s obvious fromthe way in which we have defined the source that aneutron whirti enters a region through say the outersur-f.'iRe is not included in the source even in the oase ,v/hen i t remains in the same group after col l i s ion.

ooTV yV

-:-27 :-

where t:^ is the mean number of secondaries per collie ion

L.Z.. is the scattering from the group to itself , and

^.y'A the mean number of neutrons born in the groupin me fission spectrum. The P are the first flightprobabilities. The infinite aeries (1) should converge,i e . , vre should have t^. "pj[v < i . The regions shouldtherefore not be very large.

For each surface «e can define a vector

where M-n, and x.^ ar-

- : 28 : -

(2) can be verified by directly substituting the values.

In the first zone there is no inner boundary. *»ofor n a 1 we can write

and we can have

which can also he verified by direct substitution. Letus take N as the number of tho outermost region. Thenwe write

- r - M J" •+• 61., x

»v rv-i

X •+• -V R Q X

- 29 -

Writing

we have

Here we have one vector equation, ie, two independ-ent equations. Unknowns are the three currents 3,~ ,

"t > °i* . Normally we start working from the first groupThe "purees in the first group in the different regionsare known. It comes directly from the fissions alone.

We use some boundary condition at the outer surface.It could be areflectlve boundary condition

JN * JN

or a zero incurrent boundary condition

J" = 0

The equation (4) can now be solved to give JT, JZ and

J^ Eqn. (3) now gives J* and thereby J . Repeated

application of the eqn (") can now give all the.J.

In the lHJMLuVU, only one region is taken In the fuel,and the zero incut-rent boundary condition Is applied atthe fuel boundary. . Thus N » i

- : 30 : -

The superscript represents the group. Two groups areused, the dividing point being 1.4 raev. e Is definedas the number of neutrons escaping from the rod for thef irst time.

Source In the first group is merely t^ where t^ la the

fraction of the fission neutrons born into the firstgroup. Source In 2nd group is

< o v'i> * t w c s * '•»There afe no Inner and outer currents since there Isonly one region. In LATTI2P, fuel i s divided into variousregions. The fast source in a region i s proportional tothe thermal f i s s ions in I t .

CO *V-n 4 '

•v- - V . z - t ~ , v - *—The normalisation i s to one neutron from primary f i s s ion .The fast source comes from thermal f i ss ions . The procedurei s to start by assuming a spat ia l ly f lat thermal flux.This can be used for the €. calculation. €. pomes asparameter in ^es tcot t ' s r factor, so that now we canobtain r. This gives us the thermal cross-sections andnow the diffusion equation for instance in the thermalgroup can be solved to ffive the thermal fluxes. '•I'hesource terra i s taken to be proportional to the S Z+ ofuach region for this purpose. These thermal fluxes cannow be used to gives the fast sources. This whole cyclecan be repeated a number of times. In LATRKP usuallytwo or three cycles are used.

The boundary condition used In LATTliP i s that therei s perfect ref lect ion at the ce l l boundary for the f irs tgroup and zero lncurrent for the second group at thefuel boundary.

. . 31

• • • ( 1 1 • •

JN * JN

6 I s def ined as the net number of neutrons thatenter the moderator in the f i r s t group plus the numberof second group heutrona that enter the moderator forthe f i r s t tlrau.

€ = JN2 ° JN2 * JN2

€ i s not measurable, but a re lated quantity which canbe experimental ly measured i s the f a s t f i s s i o n r a t i o .This i s defined as the r a t i o of the number of fastf i s s i o n s to the nunber of thermal f i s s i o n s .

where C, ' is the number of collisions in the firstnsroup. "5^ Is the fission source.

C ( D „ Q(i) ¥ ( D w j(i) * w(Div + j(l>- v(l)ovn Hn n n-i n "n "n

The total fast fission ratio is

COZ -J-n. CO—?r- c~

In DDMLAC, there la only one region, and the total fisssa source i sunity. So (5) gi*»s

and €•3 CO

is the arerage Z. in the cluster

I =cluster

In the calculation of € both in HJML4C and LATHEP, the clusteris homogenised. his leads to an error in the collision probabilities,beoause the fuel to fuel oollision probability in the actual ease ishigher than in the homogenised oase and so the £ la also oorrespon-ondingly higher, Pershagen applied a correction for this as follows.Take a unit pin oell as shorn.

• : 33 : -

€ i s calculated twice. €-« For the single rod takingT» C 2. *.«") as the c o l l i s i o n probability where 2. i8

the actual cross-section in the rod, and 6 ^ ^ for ahomogenised unit ce l l using "PC £*•»,} where 2. i s theaverage cross-section in the unit c e l l .

The difference C4 — 6 f L in used as correctionto the calculated value of C.

Since <£ i s the measurable quantity, i t may beuseful to find a relation connecting S and € inorder to s e t an estimate ofthe error in e . ^he totalnumber of fast neutrons produced p«x thermal f ission i s

v_ e . I t i s also equal to v •+• 4C""j "' - *«/.

- • -

e - , . V4 " -

This expression should not be used with the measured £to got a so called "experimental" value of € , becausethis re lat ion Is obtained from one group theory whereasthe 6 in LATTIEP and DUMLAC i s from two group theory.However, the error in i can be multiplied by O J - 1 "*to ?ive an estimate of the error in calculated by thecode.

LECTURE 6.

The Resonance Eaoara Probability :

The resonance escape probability i s defined byboth LATREP and DUMLAC in the same way. P i s the numberof neutrons that escape resonance capture in U-238 andbecome thermal for every fast neutron that ecupes fromthe rod. For calculational purposes we assume thatthese neutrons have the f ission spectrum, even thoughthis i s not true because there i s always some slowingdown In the fuel due to inelast ic co l l i s i ons . Theescaping neutrons are slowed down in the moderator andreturn to the fuel rod where they undergo resonancecapture.

The main d i f f icu l ty in tho theoretical calculationof p comes from the rapid variation of cross-section withthe energy. The exact cross-sections are obtained froma knowledge of the resonance energies and level widths.In earl ier days when this knowledge was lacking, integralmethods coupled with experimental measurements were deviced.The experimental part consists of the measurement of theeffect ive resonance integral .

The resonance Integral i s defined as follows. Ifthe flux at the surface of the fuel In an energy internaldE i s given by CdE/E, the total absorption of the fuelIs given.

v . <v. C <*O C

where No is the number of U-238 atoms. It is importantthat the flux at the surface of the rod should be i/E ornearly so if we want to use the resonance integral tocalculate the absorption. If there is a departure from1/E some correction has to be applied to the resonanceIntegral.

Til can be expressed In the form

Nowadays, with the knowledge of various resonanceparameters, it has been possible to calculate RI and theyare found to agree with experiments to within 5%.

- s 35 : -

To calculate p let us first assume that there area number of resonances, and they are all narrow. Considerthe first of these. The absorption in the resonance is

' - f . - "• v- C <*O, «•„where Qs i s the flux/unit lethargy at the surface of the rod.

i.e. ' ~ f • • "• v- C*O, ——" ' QaJO,

Now the slowing down density Just above the first resonanceIs taken to be one, and this will give

which is merely the conventional fornula tor the slowingdown density.

/ en \ C1)

' - f'

Sinilarly for the second resonance we oan write

where J»2 is probability of escaping resonance in both thefirst and second resonances. The slowing down densityimmediately above this resonance is p,*

.«. <p

Using Vbis In (2)

:- 36 :-

Xf there are N resonances, we will get for the totalresonance escape probability

' *^««fflL

Aa the number of resonances Increases and tho resonanceintegral for the Individual resonances decreases, this canbe approximated to

• • 3 T

» - 3T : -

and i f we assume tbatve get

i s Independent of 1,

where HIintegral.

L Is the total resonance

ofThe essent ial problem now reduce_s to the estimation

/cp.y.a . Crlasstone puts q>, % «pj , the average fluxIn the rod. The procedure adopted i s 0 take the cel l withfuel and moderator. The absorption cross-section of thefuel i s taken as ^ i . where U i s the lethargy width of theresonance region. In the moderator the removal cross-section,which i s for a l l practical purposes absorption as far as theresonance region i s concerned i s »iven b y - * i * / ^ . The scatter-ing cross-sections are known. The source Into the region Isthe slowing down source, and is taken as f l a t in the moderatorand zero in the fuel. The diffusion equation can now be solvedto give the flux from which we can get <T* .

The assumption of a spatial ly f la t source Is Incorrect.When we attempt to account properly for the spatial variationof the source, the d i f f icu l ty arises of knowing at what energythe slowing down source Is to be obtained. For this , Critophmade the simplifying assumption that a l l the resonance absorpt-

:- 38 :-

tlon takes place at one energy, say E«. If the spatialvariation of the flux is calculated ax this energy, wedirectly have the required <fc»/<pe»a.« Since flux Isproportional to the slowing down density, it is sufficientto oaloulate the slowing down density instead.

Crltoph uses the age theory for calculating the slowingdown density

where r Is the distance from the source of fast neutrons.In Crltoph1s theory, every cluster is replaced by a linesource of fast neutrons, and the slowing down density atEH at any spatial po-lnt will be obtained by summing up thecontributions « C».. «nO that point due to all thevarious line sources, t^is the age corresponding toAn Infinite lattice of clusters is assumed.

pitch s.Let uo take, as an example, a square l a t t i c e with

The slowing down density at a point (the l ine sources Is

*., -M ) due t o a l l

Using Poisaon's summation formula, we can write t h i s as

t LA. 71 ii

* . j . - «

: - 39 s -

The proof ia given in the Appendix.

Evaluating the slowing down density at the centre of a rod,and also at the corner point of the cell

-*= C ' - +«

Let us take, in a typical case for D20 reaotors •* «80n* , pr -* ~ AT cm . Eqn.(6) compares the flux at thecentre with that at the cel l boundary a*"* «rill give us anestimate of how flat the flux in the c«»ll i s .

Experimental measurement of the slowing down densityrevealed that i t is not possible to express i t in the fora.

A f i t using two «'-» gave be t te r agreement

A pliysioal reason for th i s could be that the source neutronswill have to be taken as having two energies. In a typicalcase

>. 40 :-

_ I * 8 C O- 677

=» O- 9S

Thus q at the centre is about 55$ higher than it is at theooll boundary. Obviously the assumption of a spatiallyflat flux would be wrong. For close pitches, ie., withsmall s, the error may not be large: but it is certainlyappreciable for D«0 reactors.

The above calculation was given merely to justifythe need for Crltoph's formulation.

In Crltoph's original formulation, he does i.ot make use ofthe expression (4) for p. His method Is to assume that allthe absorption takes place at one energy E_, and he calculatesp by calculating the number of neutrons that are absorbed inthe rods. If one neutron is emitted from a rod, 1 - p willbe the number of neutrons absorbed in this and all other rods,assuming that there are no sources in the other rods. We candivided the total absorption 1-p into two parts.

The first term gives the absorption in the rod in which theneutron was produced, and the summation gives the absorptionsin other rods.

For the first term we use eqa. (l)

iv. v. Ci —. o_ t=

/v. v%

: - 41 : -

The subscript L Is only to say that qu i s the slowingdown density in the appropriate lattice and not In thepure moderator.

An expression similar to (T) oan be written for the otherrods too

v. v .

The integral in the denominator is tne total slowing downdensity over all space. The above calculations were donefor one neutron emitted from one rod, assuming that thereis no emission from other rods. Since all absorption isat E_, and <yL is being calculated just above E_, it followsthat this integral should be unity.

/ - f m C ' -

rTo evaluate the expression in brackets, we note that theterms in the summation are <VL C"R , £-*) = slowing downdensity at U. due to a unit source ar r = 0. Since we areworking in aft infinite lattice, we can say, thett fromsymmetry considerations, this should be equal to the slowingdown density at r » 0 due to- a unit source at Tt.. So if wenow change the picture and imagine that, every rod is emittinga unit source, the total slowing down density at r = 0 willbe exactly equal to the expression we are seeking to evaluate.

For a square pitch., this quantity has already beencalculated. Putting x = y « 0 in eqn. (7) we get

: - 42 : -

V Co, Z. X

(Ve divide Q.by V cell to make i t dimension-less)

This corresponds to one value of c . Actually as wesaw earlier, the glowing down density consists of twocomponents given in (w) so Q will also have two components.

iv. v. C < * ^ C«O

for a square pitch. For a hexagonal pitch, $ comes out toho

If we follow the theory of p given a t the bcglnini? of t h i slec ture , we can soe how the expression ( i ) for a s ingleresonance, which is s imi lar to (9) went over to the form (4)when the nurabor of resonances increased very much. We mightexpect a similar change when Critoph' formula assuming a l lthe absorption at one energy is extended to a f in i t ed energyrange. Thus we set the expression used in DUMLAC

In LATRKP, more or less the sane theory is used, but weuow take 9 groups In the resonance region, the NBA -method, as it ia called. The highest ^rouu starts at1 mev, and the group boundaries are so selected aa tohave major resonances in the diddle of a group. Theslowing down is calculated with three components

Each of these

equation

is a solution of the age

with the boundary condition

at the cell boundary. •

For each group, a mean energy is taken and all theabsorption is assumed to take place there. The NDA - methodwill be discussed fully in the next lecture.

LECTURE T

The NIA - ModelThe NDA model assum. that resonance absorption

takes place at nine d i s t inc t t- ^ g i e s . The resonanceregion i s from 1 mev. to 5 ev. This i s divided into ninegroups. The II for each %roup has been calculated andsupplied in A ••• . 3 Ja/« form. The calculat ionwas done by a code which was calibrated using Uel lstrand'smeasurements for the tota l resonance integral . The expressionfor p in a number of groups wi l l have the form given in eqn(3),6th lecture .

the average value of flux in the fuel i s used in place of4 tlte surface value. In the NBA model, we have nine groups

and so nine values of **'4/<»att-we to be found. These arecalculated according to the a<*e theory. The slowing downdistribution in pure moderator due to a point source wasmeasured experimentally. It was f i t t e d to the expression

«y--7.

Three t -a were used because this was found to give a betterfit. Each of these c will follow the age equation.

Aso that

The thret ^ values for the pure moderator can beexperimentally determined at the indium resonance value oii . 4 ev. The corresponding c for the la t t i ce can be calculateddirect ly as

••45*•

The value corresponding to each of the nine energies Is

first thing to rto la to calculate the value ofI t is known that at any energy E

' ca l l .Ce)

where q is the solution of the ar;e equation.

If we assume that the total source In a cell is unity, wehave "Yej, = ' and the problem reduces to the calculationof the average q In the fuel fo- unit source.

In Latrep, this is done by solving a two region problem.Let us start with the calculation for the first resonance,

Here the total source is from the fission, q at E. is calculated aa+-he solution of the age equation. The total fast source in the fuelij unity, and the source ia assumed to be flat in the rod and zero inthe moderator. There are three Si. and correspondingly three ^and each of them ia a solution of the qge equation. A boundary conditionat the cell boundary is

The -a., are solved assuming unit source for each -v, i.e.,

The total q then will be given by

2ach q,. oan be averaged over the rod, and over the cell.,Ve call r

= «Vi C «-, O in this case.The solution yields the following expression for

*l C *> -

where «*.*, are constants given by

Thus /*C*O is the average «vC«O in the rod on theassumption of one nnut^on in the ce l l .

Assuming that the slowing down density is flat we can take

This was for the first group. For the eightsubsequent groups there is an additional complication. Theflux from the fission source is there of course, but thereis also a depression due to the absorption in the previousresonance. The procedure now is to assume that the-»-e is asource of one neutron at the fission and there are sinks atevery resonance of higher energy. The strength of a sink isput as equal to the nunbe- of absorptions in that resonance,

the average slowing down density in fuel is »iven by

• .C*-}33 the average slowing down density in fuel at the energyE i using unit neutron produced in fission.

« C-*O a same in the ce l l .

^ ^ ^ ^, in a situation where there are no absorptionsabove n. The summation is taken over all £-„. > £-«. • ft

at energy En when one neutron is injected at

A m. = total number of absorptions at E for one fissionneutron

In this troatnf»nt w<j assumed age theory. A fflulti-grouM treatment can also ho used to calculate «p / ^^n. .The diffusion equation is solved in a numbe** of groups takingthe absorption c-oss-section as T«i / -u. and the removalcrosa-acction as -S Z.. / «. . This latter p esents adifficulty because the variation of Ti with energy i s not

well known.

Even the nine group treatment does not give very?ood results. This nay be due to the limitations of agetheory.

ftl for- Clusters

It lias already been mentioned that the 711 of single>*od3 have been experimentally deterained and expressed in-the form

when isini? this formula to calculate HI for clusters, itshould bo renumbered that the inner surface in a clusteris shielded due to the proximity of other rods, so that ifwe take S to be tha total surface of all the rods in thecluster we will net wrong results. For this reason, itusual to define 3" Q^C for a cluster.

First let us consi 4ar an annular rod. The outersurface sees the moderator, ' 'he inner surface sees only thecoolant which is not as good a source of resonance neutronsas ttie moderator i s , since i ts volume is smaller. Lot thenumber of neutrons that hit the outer surface/unit area/sec.

— x.

- -49 -

The above quantity for the inner surface « s

x and x have to be calculated at the resonance ene-gy E_We assume that the flux i s >/E and f i r s t calculate x. T6do this we calculate the slowing down source into an e n c g ywidth dE^ at E^.

A. 4 C

The source is assumed to be spatially flat. To c»oe nowto be number of neutrons of energy E_ that cross unit areaof S . The number of neutrons that reach ds per aec is £ivenby

3C•"l Tt M.*

Replacing dV by its equivalent

this number is found to be

To get the total number of neutron crossing 4A from the

-gen -

right, we have to integrate this expression over the wholeof the apace on the *"ight of AS

I IE, AS f (* i 3 = - \ \

The source in the coolant is also the same

the total number of neutrons entering S^ is

where ^c C ^ i ) is the probability that a neutronborn in the coolant collides in the coolant.

Thus we oan write

S „„ = So -v 7 S i

. .at..

- 51, -

where 7 for an annular rod is given by the above expression.For a pair of inf ini te parallel plane plates separated by adistance d, y becomes

E_} C-*3 is a well known standard integral.

To extend the treatment to clusters is more difficult.DUMLAC identifies a number of coolant subchannels cf radiusA; to calculate y. So is put equal to the rubber band,surface.

£ S 4 } I , _ T>_ ( 2.+ *. • ) \ A 2.^ »• •

j- * L

HI of c lusters were experimentally measured and comparedwith the values calculated with these S*UL an(^ t n e

agreement i s sat isfactory. ®t

There i s sone reduction in surface because ofshielding by other rods in a closely packed lat t ice l ike thosefound in HgO reactors.

s o ( < -

where c i s the probability that a neutron entering themoderator with random orientation escapes co l l i s ion in i t .Be l l ' s rational approximation gives

where £ i s the mean chord length and the subscriptrefers to the moderator.

-v v, 2.,

Thermal Calculations

Having covered the high energy and resonance regions,we now corae to the therraa-1 range. The first problem here isto vet the values of the c^oss-sections. In multigi-ouptreatment when the group widths are araaLl, the calculation ofcroas-sections is a simple matter. But when the entire thermalg^oup is taken as one ?Toup, this becomes a problem, theessential difficulty being that of finding the spectrum overwhich to average the cross-section for the group.

. . 5 3 - .

In DIELAC and LATREP, a method suggested by V.eatcott i3 used.The thermal group is taken from 0 to 5 or 6 ev. say. The thermal3pectru3i in a reactor can never be pure Marwellian. Vestcottproposed that the spectrum can be taken as a Maxwellian with a 1/2component superposed on it at the high evergy side. The neutrondensity can be -Britten as

where ^.^^ V" J "" ji; *.T ^ *T '

Both the X ir^ nornalised to unity

AE a '

«C E 5 "

It is to be noted that X ,-t must have a low energy cut off.This ia the reason for having in it a function C which is ajoining function. The simplest kind of joining function will be

This was the first function used, with yu. « f . Laterthis was modified to avoid a sharp cut off

can be calculated using the condition that

It comes out that

/*., a 'i.b68

Experimental measurements showed that & should be roughlyof the form shown hern. The actual neutron density willnot have a hump. The Jiump. comes here only because theMaxwellian is subtracted' : from the actual spectrum. Anexpression was fitted to this experimental curve

This expression gives j*. as M^ = 3.681

The Vestcott model depends oa the fact that the thermal neutronspectrum can be replaced by a Maxvplilnn oo-»-<-esnondln!r to someterapeT-atu-re. Slight differences from a Maxwellian can beadjusted by changing the neutron temperature, but not largedifferences. Thu success of the method will depend upon whethertho raeasurenent of neutton'. temperature of the medium yields a

••55.

- 5 5 -

a unique value rega-dless of what the material used forthe measurement. This point wi l l become clear later whenthe measurement of neutron temperature i s described.

The assumption of a MaxwellIan spectrum in themoderator i s l ike ly to be correct. To what ext«nt i t i sjust i f ied in the fuel i s debatable.

10 •• -

IT » 4 T

i4S AT

LECTUBE 8

The 'fegtcott I!odel

Last t ine we saw t h a t the Weatcott model use3 neutron densi tyexpressed aa two parts

0•CO

We also saw the experimental curve for A , The bump in the curvecan be physically expressed as follows. In a medium without absorption,the slowing down density is constant. The flux j.s

•S A . *

This holds in the high energy region, ka energy decreases, then justabove the thermal region we have a situation with slight amounts ofupscattering. This decreases the effective value of 5 which nowbecomes a function of E. Thus <3? at these energies is higher thana '/c flux would be.

Whatever cross-sections we use and in *hich-ever way they maybe calculated, the final aim is always the same, ie, the reaction ratesshould be predicted correctly.

Reaction rate - Z.^n fmU

IShatever the flux spectrum, the reaction rate is always given by

(R. CR.

And the t rue . o x is given by

If we t r y to evaluate <y us ing eqn. (1) we ge t a log E a t nThis can be avoided by using some high energy cut off, but Westcoti:followed a d i f ferent course. He defined the so ca l led Westcott f lux

Then the flestcott effective cross-section

We pat

wb«»re o~» i s o~ at aaoo -m.^^«e. I t is easy to see toy Substitutingin (2) that for a •/•»• absorber a5- = cro . Thus if a l lmaterials had '/•» cross-sections we would have had ^ - ' and thecalculation of <r would have been straight forward. Putting

l,Ye can put ths inte,gral as I . This will be a kind of a resonanceintegral. Weatoott has calculated I for different materialstaking exaot & and <r . If a material does not haveresonances in the region of the hump & can be replaced by astep function. A = • above £ = yu. A.T .V/here /J. can becalculated as described last time. Thus

is the infinite dilution resonance integral from M * T upwards

-K T .

By substituting in (5) for a •/ «- cross-section we can verifythat in thi3 case, .« = o so that here <r is the same as cro

Now we can write

<r = - ^ ' - i J j x3 \

- 59 -

Substituting for the integrals from eqns. (3) and (4)

Since ?t and & are functions of T, both g and s will alsodepend on T. r is called the epithernal index, depending as i t doesmainly on f.

These Tfesteott cross-section are one group cross-sections takingthe the ent i re energy range into account. They should therefore containtwo parts, one coming from t he Maxvrellian, and the other from the epithermalpart. Since the total spectrum Should be normalised to unity, i t folloTrsthat the aaxwellian and epithersal spectra used here should be normalised toI- f and f respectively, where f is the fraction of epithermal neutrons.However, the division of the $~ into the g and s part 3 is not done onthe basis of the i'axvrellian and epithernal division. Rather, i t i s made7/ith a vien- to have one term independent of the strength of the epithernalflux. ?or th i s reason, the f i r s t term o". *K i s calculated using a

?!axwellian normalised to unity whereas the actual contribution of the•.{axvrellian to r would only be «" i^Ci - -^3 . This increase is of course,adjusted in the s term, which will therefore contain a '.'axwellian with negativesign and an epithermal spectrum with positive sign. Incidentally i t also turnsout that for a */xr -CMSS-section, these two contributions will cancel eachother.

In the case of U-238, the .'featcott r cannot be used in the fashiondescribed. This is because large quantities of U-253 are lumped together andso there is appreciable self shielding. However, we have seen that theepitherraal absorption of U-238 has been treated separately in calculating pusing measured <R1 . So the .vestcott cross-sect ion need take only theMaxwellian part .

. . 60 . .

In order to oonform with other materials, we take Waatoott orosa sectionwith -4 » o, and

The part 91 has been taken in the thermal oross-seotion, andso It should be subtracted somewhere.

{SIn the thermal range, far below the resonances we can assume that the cross,seotion is '/v or nearly so. To compensate for the slight departure from

11 / >•• we oan take 0" as equal to, not <r; w . / w but rather as *This change is made so that the value of V should satisfy (6).For a •/•* oroaa-seotion, we know that

X . C O

"re can make thi8 aubatitution An (6) and get the terra to be aubtrs.jtc*d as

• .61 • i

This is like a resonance integral term with a ' / •» cros3-section.(7)as i t stands should give the actual epithernal absorptions, and with«r - J*«>y«^ gives the extra tern TO have added. To subtract this, we

•v•write the resonance integral as

•w J t

inatead of

(8) is the quantity which is measured experimentally. The ' / v termcan be evaluated analytically aad subtracted. This is the fora in Tfaichthe HI is normally 3u?plisd.

.Vestcott has not tabulated the T± in his C •*• **••*)fora. In DCKLAC, quite often like in the calculation of f(thernalutilisation), i t is necessary to Imow 3t , which is basically the szaeas O"_ or O" . The one group cr^ would be

5 -These <r are available in literature for a Maxwellian distributionand are made use of in DUKLAC. So correspondingly the flux has to bechanged. Thu3 7/hat we use here is not the 7/estcott flux, but what wecall the "true flux" defined as

This fixes the 3> V <? term. But the L^ 9 term will be wrong ifwe use Vfestoott cross-section and true flux. To avoid this we should have

- 6 2 -

Where the cap denotes Weat-oott

These are the crosa-aectiona actually used in DOMLAC.

In LATREP we use a fixed energy thermal cut-off at 0.625 ev.The total absorption is considered in three parts. Starting fromhigh energies doma-wards, «e first have resonance capture in 11-238,then epithermal capture in all other materials, and then thermaloapture.

The thermal oross-section is

The thermal neutron density is

-tat J — - tor 1 \

The maxwellian becomes very weak beyond 0.625 ev.t 30 that we canreplace the upper limit in the first integral ty 00 . Now theintegral i3 just unity.

- sWe can define an average velocity for thermal neutrons

a- (17

-<R. 06 as

. - 4 x

I --t*-i

4 n -•where

•CK.

: —-2X

Nov/ the denominator of (10) can be written asand the numerator can be written using (1)

from(12),

Here again we have made uae of the fact that for Maximilian, theupper limit can be replaced by oo

. .65.

\ quantity a can be defined analogous to ga

x C e 3 f f - C e ) d eT

substituting (11), (15) and (15) into (14)

has been calculated and tabulated in the LATEEP library.

An exactly similar procedure is followed to calculate 'J"^It has already been msntioned that epithermal'captures (in allmaterials other than U-238)are treated separately, and we needcross-sections for these also.

Neglecting the Maxwellian above the cut-off, «e can 3ay that

Xa , C O AC

The flux is defined as

. . 6 5 5 . .

In LATRSP, for the calculation of epithermal reaction r a t e3 , we do notcalculate v«.hi t u t merely obtain the slowing down density fromage theory. I t is apparent from (16) that *r+*i. i s proportional to .the neutron density. The actual epithermal flux would be TV,^ ^ Wwhere ^Vbt is the average velocity of the epithermal neutrons.If the spectrum is assumed to be spat ia l ly constant ^v^ i s alsoconstant and so «$>„*;. is proportional to the actual flux and therebyto the slowing dowirdensity.

Osing (4) and (15)

- - c 1. - *}

. .57.

LECTtTRS - 9The TTeutron Temperature

I t ia obvious now that for the calculation of both g and 3,we need to know Tjjt the neutron temperature. Experimentally, thiscan be measured by means of certain reaction ra tes . This will bedescribed l a t e r .

In a medium without absorption when the neutrons are in thermalequilibrium with the atoms, the neutron temperature will be equal tothe physical temperature of the medium. In actual situations however,the neutron temperature increases owing1 to the preferential absorptionof low energy neutrons.

Since- fuel cross-sections are more important, i t ia the neutrontemperature in the fuel that we try to calculate more accurately. . TCeknow that there is l i t t l e thermalisatian in the fuel and the thermalspectrirum even in the fuel i s formed in ttie moderator and comes fromthere. So i t could have the same temperature *s the physical tempera-ture of the moderator. But the Tft changes due to two effects. Oneis the absorption in the fuel . The other is due to the fact that thefuel i s at a different temperature and the oxygen in i t has somethermalising effect. Thus, there is some tendency toward3 rethermali-sing the entering spectrum at moderator temperature to that correspon-ding to the fuel temperature. In DCMLA.C we make the assumption thatthe absorption and rethermalisation effects are separable. Thereforewe write

1- S ( T 6 - T_} + , ? C Tf - T O (1)

•p » physical temperature of the moderator. A T ^ is the effect dueto hardening in the moderator. This hardening1 is due to absorption inthe fuel. If the moderature was very large, th is effect would nothave been fe l t at points far from the fuel. But in the actual reactorth is never happens and the moderator spectrum is always distorted bythe presence of fuel nearby. For a system in thermal equilibrium, suchincrease in TJJ would have depended only on the absorption ra te , buthere there i s a competition between the absorption of thermal neutronsand the production of thermal neutrons by slowing down. kT,^ canthus be expressed as a function of lm.f3Z.M The practice however, is toexpress i t as a function of the epithennal index r , since this is alaoa quantity determined by the competing effect of thermal absorption andslowing- down.

3 o o *• ^ J | i r i..ft-,,

. . 6 8 , .

i , ?,., is L'ie increase of T,. due to absorption in the fuel. Here the

c.v.:'.::eti:v- roceases are absorption in the fuel and slowing down in the

r.ia r^nina. i - three terns are due to tie rethemalisation effect .:-9'ii.v5n9 fro..i i"wl, coolant and -reasure t'l'-e core to the moderator.";- -' in;? in T,. due to t'.2 retherralisatj /n effect of the neutrons

f r v tv; fu3l say, should be oroportiona3 to (Tr - T j . The . *should ni+tira].ly depend upon ths scnttorinu properties of fuel ( in thisca9o) Sirica t>.o tharnal spectrvmi in thz- i\iol will be different fron thatin the moclorator only to the exxent that there is scattering in the fuel.

-til the constants in these oxpressiona have been obtained experimentally.

In LATRiP the formula is written as

Th« las t two term3 '^^re are different from the corresponding terms of (1)This is because the « in the fourth term takes into account the effectof having the T7hole cluster at coolant temperature. A T . is calculated

by a different expression.

The « '* are calculated in LA'JJRSP using the nethod of tW) overlappingthermal groups. Let us say we are trying to calculate <x . Then

<* (Te - T^) should give the increase in *N of the cluster region over

the moderator tem-erature owing to the fact that the cluster region isat a higher temperature T . Vie consider the cluster region in an

infinite moderator. Let the equilibrium temperature in the euter region

. .69.

. .69 . .

be Tu and that in the inner region TH- The temperature in the inner

region is assumed to be perturbed due to the neutrons entering fromthe outer region. The addition of two ISaxwellians does not give athird Maxwellinn, but as a crude approximation we can say that theneutron temperature in the inner region is T

Where 0's are the fluxes in the inner region. We set about calcula-ting the fluxes with the normalisation that i neutron is removedfrom the Tu spectrum in the inner region. The flux corresponding tothis is 0 . I t is lost by two effects, one is absorption and theother is removal fror? the U spectrum_to the H spectrum.

zThe source into the hardened spectrum from the unhardened spectrumis

The number of absorptions in the hardened spectrum

•n a.a . M

. . T O .

- 7 0 -

Substituting for 3g from (2) we have

Sow the assumsd nmitron temperatars dme to the two llaxwalllans la T.t - TL i s th« lncteaae In Matron temperature das to the highar

tenparatore ?H. that la ,

TM - T ^

CALCULATIQH OP r

Vi start tor assailing that the slowing down flux i s 1/B. How

the rate at which thermal neutrons are being produced in the cel l is

The flux ia given by

O + 3* i^

We had assumed that

• t <»

So we can say that the epithermal flux Is given by

CO CMC C O -

We can substitute for X .(E) from Lecture 7, Eqn.(i) taking

We can equate this to Eqn,(3)

. J- <*d

This is r in the fuel. The calculation of r d will be straight -forward as we have assumed that the epithermal flux is spatially-f la t . So we can write an Eqn. similar to Eqn.(4). The r i g i t handside is the flux as calculated. The left hand side can be replacedby a similar expression for the moderator.

An iterat ion scheme will have to be followed. Firs t we assume" that=• Tm and calculate Tjif • Now ue can calculate 1he cross-sectionr f u e i . rm0(j ig needed only for the calculation of neutron tempe-

rature. We have started by putting rmod = 0 and getting Tjjf. So nowwe can make a thermal flux calculation which will also be used for thethermal ut i l isa t ion factor as explained in the next lecture. Thesefluxes are those defined by Eqn.(9)» lecture 8.

The ^ •• are known, the ^ 7 ^ ^ are known since TJJ in fuel andmoderator are known. So we can get ^ " ^ • With this value

•15'

we calculate rmQd and recalculate TNm, TSff r f u e l , cross-sectionaand re-do the flux calculation to get the thermal ut i l izat ionfactor. -Normally the process is terminated here, but any numberof i terat ions may be done if desired.

In EUMLAC, Tum is given as input and no iteration is doneon

KB/ppv4870

- '.74:-

cture 10

The Thermal Utilisation ?a~*:

Tho thermal u t i l i sa t ion factor f i s essentially a measureof the relative absorptions in different regions. Let i be the slowing

down source into the thermal group. Ir. OH.1AC TO taka in the r. region.

» is assumed to be spatially f l a t . In L^I32P —— ia

ciloul-rtei Just as i t was ione in the p calculation, taking the /*.•• e tc .Thin of co-arse has to be done at sose particular energy aid they haveta^-cn i t at 1 ..ii ev. Th normalisation used is «»*P\ » i . '.Ve can

torn mite «P^Lu«. m '*" F •

In the annular region outside the fuel but inside the cluster we write

so that ? can be obtained.

The thermal flux calculation is roughly similar to that whichve uied in the £ calculation. The to ta l incurrent and outcurrent atthe outer boundary of the n region are denoted by J~ and J T . 7/e can

then say that the absorption in the n region is

-ha total absorption in the cell mist equal the total source

so. Tie can define a thermal utilisation for each region.

L » -

"He now have to evaluate the J f s . This has already teen done ia

ocnneotion. with the calculation of €. . fte can write, Tlecture 5,eq.n. ( * )rf

1*1ma.

iSe take N as the las t region before the moderator. Moderator boundaryis at b .

.•a assj-ie perfect rsfl^otior. at tha moderator

This will also be equal to J^ since we have taken a reflective•boundary.

J,* - w i - vi

this into (2),

w — * W w .to

Iqn. (3} can also be 'written down directly from the fact that we'-.Ave as3u::.ei zero net current at the outer boundary. ".7e can say'-hat •,vha-.ever leaves the cell must be coming back. So J~ should

+• rrt v»« n

iipard only upon ^ nod acd J,j . The coefficients w .can be easilycalculated if the collision probability ^^^^ e t c . are known.Ho sever if the moderator is a l l treated as one region, theseprobabilities cannot be calculated very accurately. So a methodadopted in preference is to use diffusion theory in the ncderatorto calculate J~. This will give the coefficients W*.

The diffusion equation in the noderator region can be writteh.

»B ass'-iae that there i s no absorption in the moderator, and the totalsource in the noderator is unity. Then

where 7 is the moderator volume

At r = b, we have the boundary condition

aav *• *r*

At r - C, we eon write

iihars

- 7 9 -vi «

First, l e t us oalculate V ^ ^ the coefficient of 61 . 4 . Obviously,in thin oase «s are only interested in those source neutrons that v ;enter the fuel. If any of then come back tfaie does not affect w ^ JSo we can assume that the value of w ^ * * i l l be indapendent of thematerial within r < Q and BO we can replace It by a black boundary atC, and we take the boundary condition there as

Xwhere X i s the extrapolation distance at a blaok boundary.

3 c* \ i

Here we have replaced —1_— by B

With this we now have the exaot flux distribution In the moderator.So we can calculate the total number of absorptions in the moderator

oubstitutiag from (6) and carrying out the integration

- 8 0 -

B la than the total anaber of aboarptiona duo to xmlt aouroe. However,In the oalaolatlon of > we had neglected the abaarptlon. The H

abaorptloni way bo oonaldexed aa a negative aouro* giving — "Rabaorptiona. Wo will thua got an infinite eerie a, and we aaa say that thoeewhlah are not absorbed will oaeape into the Moderator.

Slaee thin la tt»e aoabex for oao aoana aaatroa, It la alaa aqoal to tao

oooffiaieat

I 1 1* la aalaolatad by (5). Bila la jwtlflad baaasaa, as alreriyand

tio»d, tiw aaln objootioo to oaloulatlng tba 1* fcoa eqaa.(4) and (5)la that the aodarator zogloa la too largo for a proper oaloulaUon of thaoolliaioa probabllitlaa. thla trill not b» rorj aarlooa far W>JJ alneo aaav

of tu* collision* vhioh return a Awl neutron back to the foal will oooitfIn th« vicinity of to* teams boundary. In the expression ve hav* derivedfor f r ^ , that only unknown quantity la X . In LaTBEP, an expression la

oaad for X a This was ess derived by Knihneriuk and given by

g ia a function which has been tabulated by Eushneriuk.

Th» problem has easenbially been that of calculating the flux. Wehave done this by applying1 the black boundary condition at e . In DUMLACalso the method i s same, but the boundary conditions applied are different.

We denote the current densities a t the interface by J ^ and J

••i-

The total absorption in the moderator is

Since there is perfect reflection at the outer boundary, the neutronconservation becomes

Vow we eaa follow pare diffusion theory *en the two ourrents aregiven by

Bqn. (3) oan then bt written a*

The diffusion equation In the moderator is

This oan be aolvsd In the straight forward manner using the boundarycondition

- 8 J -

«DCO of ooursa ia unknownt bat ?C>O i* s t i l l azpraaaad in tarma of

.(13) oan ba uaad to aHinlnato tha trnkoom <pC«3 • AltamatiTtly, «a oaauaa an axtaeapolatad diataooa booadaxy erudition.

At tha fael boundary «a us« tha aana boundary condition as bafara.

Ltna blank boundary hovavar, i s not wad . Batbar «a uaa a blaokataa faotordaf inad aa

fiiahnariuk haa darivad an. azpraaaioa for X

Frws aqna. (11), (12), (14) and (15) «• oan v i t a

SUaln*tin« & and > between (15), (16) and (17)

* a « { . CO

(9) and (10)

— J« — 2 \

{• - • " { • -

This gives the values of «, and «ca.

It is to be noted that the procedure given abov» i s the following.We have derived eqn. (10) which is exact. It however oontains the unknownquantity <pOO which has to be evaluated with the help of some boundarycondition at r • o. Here i t was done by Kushneriuk's expression (16) forthe extrapolated boundary. Kushneriuk derived this expression with theassumption that there are no sources in the moderator, but that all thesources are at infinity.

In terms of can also be obtainedAn expression for J N

in the following way. The total source at any point in the moderator Is-,he sum of the slowing down source and the soattered thermal flux.

Let us take the cylindrioal_polar co-ordinate system with origin at thecentre of the fuel.

Th« co-ordinatea of P are (r, © , z ) . We now calculate the number ofneutrons starting from P and falling on the rod surface at Q. The number

, .86a

- 86 -

of collisions in an elementary volume dV a' t>.

The solid angle which a unit area of the aurfaoe at Q will subtend at F is

So the munber of neutronB falling on unit area at Q from F

- * « -

To get the total inourrent at Q we have to integrate over all positions at

P. The moderator extends over e < x. < «o, - «> < x<«*. The moderator is

?f course, strictly not infinite, but this can be condoned on the groundshat the contribution from far off regions is small anyway. Only half themoderator oan contribute to jj^ at Q. This is that part which satisfies the

_ tcondition — JL >• _ >• •* .This condition i s the same as —

«• *• ^ < T

2 2 2let x = y cosh a. Then, since x - y + z , we h?ve z =• y sinh u

tjk c.«* o

The last integral i s one which has been evalxsated_numerioally and tabulated.

Some of its properties, which can be seen by direct integration, are

\ * . _ C«D««t«** 3

This ia now an area integral of the expression

over the whole of the (r, 6 ) plane lying to the right of the dotted line.

The expression (23) is defined at P'f the projection of P on the (r, 9 )plane through Q.

We can evaluate the integral in the (Y, «f> ) cor-ordinate syatem.

This has been calculated only for the thermal flux scattered from P. Weshould add a term for the slowing down source at P.

Expression (8) gives-the value of <f (r) . This however, i s a diffusiontheory calculation and may not be accurate near the fuel boundary. Kiishneriukmade calculations and deoided that there is some depression near the boundary.So he expressed the flux by

where Tail*. C*l i s 'the diffusion theory flux. «p*C^>is the transporttheory correction. Striotly speaking i t would depend upon the fuelmaterial. But Kushneriuk conoluded that this dependence will not bevery strong. He therefore oaloulated i t for a black body and used i t fora l l oases.

i u - Z*^C-*HT -

- 89 -

.Ve can substitute this into (24) and carry out the integration. Using (20),

we can see that the qp (c) term gives juat " _3 . The log term will

contribute

7 ^ (log) J.s merely a notation for the evaluated integral. Thecontribution of the »«?• - c* term oan be calculated using

• "/* a. a.

JJ*^~L © o

Using eqns. (21) and (22) for these two integrals we have

: c r i* •

c a n "S

.- C -h-. - TT >

The <y*(r) tern gives an expression which we denote by P ( Z o). The

actual contribution is

"a" ^t* *

Rushneriiik found that

where g has already been introduoed in oonnection with eqns.(7) and 0 6 )Their combined contribution thus becomes

a. Ti"«»«A «.--» ^ 2. c

Finally the last term due to V (r) juat gives J-^L sinoe we have

taken ^ ( r ) to be spatially flat.

Collecting everything, we end up .with

- 9 1 -

At the point oneJfeing may be north mentioning. The exact expressionfor the flux «p (r) has to come from

There should be a term- S(r')K(r,r') for the slowing down Sssa oourca in themoderator; but Kushneriuk omits it on the assumption that all the sources areat infinity. The integral equation could be solved if K(r,r') were known, but

the calculation of K(r,r') is no simple matter. It is easy to see that forpaths 1 and 2, the kernel oan be widely different. So Kushneriuk used therelation (25) in (27) and solved it by variational methods to get P( Z °)eto. *••*"*

Returning now to eqn. (26) we write it as

- 92 -

Z was used to calculate the scattering collisions in the moderator

aai 30 i t is nothing tat Z*. • l^llua TO c a n write the expression insquare brackets as

Let us compare this with eqn. (18) which we had derived earlier.The second term there contains J ^ - j . .

, (neglecting absorption, since g wascalculated by Kushneriuk assumingno absorption in the moderator)

Then (18) oan be written as

- 95 -

Comparing this with (28), we find that Q £ I — — ) in (29) has been

replaced by g*. I t has already been mentioned that there are no sourcesin the moderator. (28) was derived taking into account the souroes aswell, and BO the term

is the correction factor due to the sources. It gives the error in usingeqn.(i8) and i s usually negligible. DOJ1LAC uses (28).

Reading the text which immediately follows eqn.(4)» Lecture (5),we find that the set of equations for the J 'a can be solved if we canget one boundary condition at the outermost boundary. (19) gives us onesuch condition,. A more correct boundary condition can be obtained byeliminating cp (c) from eqn.(28) with the help of (9) and (10). Now wecan obtain a l l the J 's and then (1) gives us the thermal utilisationfactor.

The moderator absorption had been naglected in the calculationof the flux. This flux was then used to calculate the moderator absorption.

Now Z. «« /o and Z.*. « O • 3 > SO t h e neglect wasnot unjustified.

2Calculation of L

2L has been defined as

Now T *

The only unknowns here are the <y's. But the thermal utilisation factors

f. for the lifferent regions' are known and since

i t i s easy to get the <y e.

If there is a void region there will be some difficulty, because wewould then need 9 . . for the denominator of (30). However, a diffusion

theory calculation can then be used to obtain this . The currents at the two

boundaries r, and r, of the void region are known, ffe can write

Slnilarly

Salfltilation nf TI

The diffusion coefficient D is defined as

In the homogeneous case this is straight-forward. But when we have toaverage D over a heterogeneous cell, the question arises as to whether/if; isD which is to be averaged, or £ . which is to be averaged; or whether the

method to be followed is something quite different.

The b"sio point which is to be remembered is the following: D is used to get t>"the leakage term in the diffusion equation. This comes directly from the currentSo in any averaging process for D we should seek to conserve- the currents.

In calculating the current at any surface, the standard procedure is to

take the number of scattering collisions 2. «y and multiply it by the probability

e of reaching the surface without further collisions. The bulk of thecontribution will thus come from within a few mean free paths. Thus if regionsare small compared to the mean free path, all of them influence the current andso X. may be averaged.

L v-where

LIf the regions are small we can take o . to be nearly same for all i and then

ordinary volume averaging will suffice.

When the regions axe large, we fall back upon the diffusion equation todecide what should be done.

The first term gives the total leakage from unit volume. If we have a volume Vmade up of smaller volume-^ 7 , each being a region with diffusion coefficient

D., the total leakage from V should be conserved.

Taking into account the D for each region, the total leakage from V is

3 7 <y 4 W

and with the homogenised D i t i s

f a - ( •

The diffusion theory flux always sat isf ies an equation of the form

T CT> a

vHenoe

- * 5Thus we get that

< a.vV

» 3> »* = p t say. (30

- 97 -

We treat this more fully next tine.

- 1 8 -

LECTURE 11

5 in the General Case

Last time we saw how to calculate 5 when the regions are small.This time we give Bemoist's treatment for the general case. In tilecase of small regions also , i f there i s a void, i t i s preferable touse this method.

We f i r s t assume that the scattering i s i sotropic . Thisassumption i s not necessary in Bentjist'a treatment, but i t simplifiesthe equations« l e t

<p (r) - total f l u x at r

SL (r) - isotropic source at r

C (r) - number of secondaries per co l l i s ion at r , wi l l be

• -=— in the absence of regeneration

7 (r) • total cross-section at r • —*

The Integral Boltzmann equation can be written in the straightforward manner as

where VR - volume of the reactor

Here ZR i s a kind of an optical distance from "r to

H1 " r1 " r2

M - » R1 »H1

The vector^ current atr, is given by

- 99 -

Here X?. gives the direction of IL. J(r..) is a vector quantity, and its

component in any direction gives the number of neutrons crossing unit areaplaced at r1 and oriented perpendicular to this direction.

The number of neutrons leaving after a collision at r. is

where S is the operator of (3).

The source term i(r) at any point will consist of two parts; onewill be periodic because of the repeating cell structure in the reactor,and the other will be a macroscopic part arising from the finite size ofthe reactor. We assume that'this latter part will satisfy the buckling equation

V Y. + 3* T. a ° C °

that is, the flux in any group satisfies this equation and the source inany group is proportional to the flux in the previous groups.

Now let us write an equation similar to (4) for an infinite reactor

'. f %On the solution n of this we superpose the total reactor solution

and write

- 100 -

This is an approxinate solution of (4).

Mathematically, when an approximate solution is known, the exact solutioncan be written aa

[ C *C.i

Benoiat has solved the problem using this series* ait ss shall restrict ourselvesto the approximate solution

Following (2), the first order flux ia

_ "*• CO

Since n is the solution of the infinite reactor equation, (8) gives theoflux fine structure within the lattice oell. TOien this is multiplied bythe maoroacopio variation, we get the actual flux.

We nor prooeed to evaluate F defined by equation (31), Lecture 10. It is

expected to appear in the form DB . It may be recalled that F was definedfrom diffusion theory where flux was expanded in Taylor series and only one

term retained. So any higher powers of B that make their appearance in theevaluated F may be neglected without further loss of accuracy.

C dU» J W **

j(r) can be written as

J *' ~T.

J. j At

onusing (6), (7) , (8) and (9)

>; now, is a solution of Eqn.(5)« -e make the assumption that theintegration over V^ can be replaced by one over infinity. This isjustified because the bulk of the contribution to J comes from nearlypoints.

For ease of handling we now take the fourier transform of "f,

so that

substituting this into (5)f we get

(-«2 + B2) -r.(r) > 0

or c*» »

. .103. .

.; 102'

This we see that if f; (*") has to be a solution of (5)» its fourierintegral can contain only those values of the « vector for whichthe rector length is B. So we can replace £? by B^", where •?now is a unit vector. (11) can thus be written as

7<here the integral is over a surface which sat isf ies the conditionI £? | - 1. f ( *? ) is a complex number v.hioh '.rtll be decided

by the boundary com'dtions. TTere yje will not attempt to calculate i t .

Using 0 2 ) , (10) can now be written F •

, J l ^ —

Since R =» r - r 1 t we can write the numerator as

F numerator a

Using the identity

we can write

numeratorc f * a w . * . r _ ^ a

P T ..104..

..103..

,it(ve J \ 4«, i l _i «.

If we look a t the f i r s t term here, we can say that the integral overd'B, ia a function of r*, and since the integral is taken over a l lspace, and the only term which depends on the finite size of thereactor, i e , f(«o1, is not present in i t , we can conclude that thisintegral is periodic with -the same period as the la t t i ce- Thisbeing so, the average value of the divergence over the wb ,le spacei s zero.

Bencist has justified the separation of Q into tlie factorsY; and «y. on the asaumption that the variation of Y. over a

lat t ice cell i s small.' This means that , from (12), the variation ofe * » « . »v o v e r the cell is small. So in the integral over V in thef i rs t term of (13) we can replace the divergence by i ta average valuevhich is zero. Thus the f i r s t term vanishes and we are left with

numerat • \ 9 £ 3") <1^ \ A t^ J C^J 4« \

Using the relat ion

We can write (14) as

numera •»

..105.

I <*. C»O

Just as we explained in the case of the firat term of (13), herealso we can replace the integral over d t-t by its average value ina cell, or *hat amounts to the same thing, its average over allspace.

Thia should be divided by J^"4* but we omit it because we are goingto do a similar averaging in the denominator also. Vfith thisaveraging the denominator stands as

rr i»5.^ ABenoist has sho m that • \ e • *• turns out to be

independent of w in most practical cases. This can probably beroughly as follow. If the reactor is long enough axially and "y. *is slowly varying so that there is a kind of radial symmetry alsowithin every cell, vie may oonclude that there .are no preferreddirections within the volume V

The quantity J e At can now be cancelled from both thenumerator and denominator of F

..106..

(12) and (15) give us

X CO =

» - i _ ^ J * » —XL e ^ (18)n s

Using (17) and (18) in (16) _ £\

, „ (19)

. C«3 J

We can decompose Eqm(5) into i ts three components along the threeco-ordinate axes,

- Z T l

. co \..107..

»e can expand >*(-R) in a Taylor's series

• * . o

'.Ye have neglected the terms V A | V4 f c -y. etc. This ia becausefinally we are going to average i t over al l spape. The integrand inthe numerator of (21) contains H.A vA -y. (-1). Taking saycartesian co-ordinates, this should give for P , the three secondorder terms x

On integration over al l space, i . e . , J_ w for x, y and z i t i sobvious that ifae last two vanish. This can be seen by converting tocylindrical polar co-ordinates when the integration over 9 andcontain only the terms coming from i2 ( ^ e t c # A n o t h e r t e r n a dfependonly on r. For similar reasons, the integral over toe f irst termwill also vanish, and so far use in Eqn.(2i) we can write

^A -V. C- ^ 5 o * i •** x GO

And (21) now becomea

K CO J , .Wit

..108.

..107-.

Prom (19) and (21)

and since F - DB1 y B n2

2 2where B and B. are knowi, being the eigenvalues of Sqns.(5) and(20) with tiie Sight boundary conditions.

Consider now the integral

f -1£L IL ^ / « 3

The integral w.r.t. R can be integrated by-parts

4in w i J L

The lower limit is some value at which the integral is zero. In thefirst term here, the integral vanishes at H - «» because of the

..109.

presence of the exponential and at Rbecause of R. This leaves.

0 of conrse it vanishes

+•*(23)

-m* SI")(24)

+ •"

In (23) the indefinite integral has been replaced by the definiteintegral - J £ This ia allowable ainoe the limit of theprimitive at infinity will be aero. This can now be said to be theflux at r of neutrons moving in the direction iC and caused by thoseneutrons which have been scattered to r from all those points forwhich R is between R and .00 . How (24) comes about can be seen asfollows.

v (r 1 til) is the angular flux at IL, i.e., the flux of thoseneutrons moving in the direction XL in unit solid angle. '<?e can

see from the figure

\ *n — —-X**

..110..

This followa Btraight-away if we write down ttie expression similarto (1) for the total flux and substitute i t , , or what would herebe

Aci « V * Ail.. *

So (25) can be written as follows on changing the variable fromR" to H1

which is the substitution used in going from (25) to (24)

Putting (24) into

V A* ^ A*. A--— Jl f ci = 2 - * (26)

In a reactor without very strongly manifested anisotropy, we can saythat the D, are nearly equal, or if the reactor has some kind ofsymmetry so that the B§ are neatly equal, ire can say

With (26) this gives, since Z -ft-*.

Let us denote by P4; the probability that a neutron born in regionwith the space and j&galar distribution v (r^t A . ) makes itsfirst collision in the region ,1

..111.

.110 .

(27) can then te •nritten as

where Xj » ~ - total mean free path in region j1 hThis is the formula used in LATREP. If the regions are smallcompared to X , m have

substituting this into (28)

In the other limit nhen the regions are large

and D • -£••3

..112.

. 111..

Thus the problem of averaging D baa reduced essentially tothe calculation of oollision probabilities. These collisionprobabilities are different from those used In -(he calculation ofC and f • There we had taken only P , F . and F , which

are relatively simple to calculate. Here TO want P.., nSSutron bornin one region colliding in another. To see how complicated

this can be we look at the figure, P±j is the sum of collisions byneutrons moving In various paths 1, 2, 3, 4 etc. It is obvious howdifferent they axe and how difficult the Integration will be if anInfinite number of reflections at the outer boundary are considered.So Fji is usually calculated after taking may be one or two reflections*

We have already mentioned that the decomposition of Q intoft and ym Is only justified on the grounds -feat Tm is slowlyTarring over the cell. It should be added that the variation ofir any direction should also be small gyer the mean free path inthat direction. This is-because for J we sum up scattered neutronsand the major part of the contribution comes from within a few meanfree paths.

In a reactor with void,should have

will bo different from D f. So we

• * * -

Raving obtained D we can calculate

112...

2For calculating L<t we have the expression

where U is lethargy 'width of the fast group. Df can be found by asimilar process of averaging using Benoist's method. However, whenwe come to the calculation of the P^, there is some difficulty . mthe thermal group we had the ".estcott averaged cross-sections. Buthere i t is difficult to find some average cross-section for ttieentire fast group which.can be used to calculate ttje P^j. So l £is calculated by just volume weighting

2In the actual code, L 4 is calculated by a kind of correcting ofof the pure moderator which latter is known from experimentalmeasurements.

2It should be remembered that L4 would be markedly less for

a neutron which has undergone inelastic scattering In U-238; thedifference is around 30 cm. The difference AL? is known experi-mentally for the moderator. ^

Then we have

Where P is the probability of inelastic collision. The quantityis calculated during the 6 calculation. 6 in both LATBEP and

EUMLAC is calculated using two groups, divided at 1.4 MeV. Thispoint is also very nearly the inelastic collision threshold. Th»number of collisions in the fast group in the nth region is

ThiB quantity multiplied by -gr— of the n"i region gives theaumber of inelastic c r

straight forward way.

raumber of inelastic collisions.rFrom here we can go to P in a

That brings us to the end of the theory of LATHEP andexcept for the treatment of burn-up* Now we will describe

some of the experiments that are used to verify the theory.

Lecture 12

Measurement of Buckling

Tfo now describe aome of the experiments that are used for checkingour oodea.

One of the important integral measurements is that of materialbuckling:. In one-group theory.

2B oan be measured in various ways. It oan be oaloulatad from our oodea

using expression (1) and this value compared with the measured value.

There are three methods for the measurement of- JT1. Critioal experiment.2 a Exponential experiment.3. Substitution experiment.

We prooeed to desoribe briefly the first and r » third.

1. The Critioality Method - Here the experimental lattice is put into areaotor whioh is then made critical. The flux distribution is now measuredin the radial and axial directions. To do this, foils of Mn, In, Cu, Au,eto. are irradiated at different spatial points in the reaotor, and theiraotivities are measured. These are then correoted for variations in foilthiokness and irradiation time. The flux is proportional to this activation.

In a cylindrical reaotoxv the one group flux oan be representedas

<9 m A 3.

ZQ is the position at which the flux is oaxiinum, and < and X axe the

axial and radial buoklings. Sinoe we have assumed that there ia only onegroup of neutrons, the average v»looity of the neutrons will be the sameeverywhere. So the effective cross-aaotionp will also be the same every-where and the foil aotivation will be proportional to the flux. Then themeasured flux distribution can be used to find X and « bv a least squarefitting'.

Thia calculation will be valid in the oase where 'the one group theoryholds. We oan say that this so if V ^ / «pf io spatially constant. Tjis

• • • i '5 • • •

eondiyion is sore or leas satisfied in a bare reactor. However if the experinwnIs done ta a reflected reactor, the fast and thermal fluxea will have roughly,the shape shown in the figure. 7e can improve the situation by confining

our flux measurements to regions well removed from the reflector. Ehe extenof this region can be determined by measuring the cadmium ratio. At anypoint two irradiations are done, one on a plain indium foil, and the otheron a similar foil covered by cadmium. Cadmium ratio ia then defined aa

Uncovered activityCovered activity

Since the absorption crcae-seotion of cadmium is somewhat as shown in thefigure, most of the thermal neutrons are absorbed by the covering of cadmium,.

tOOO <r

o-o« 0.4, / •O

The cadmium cut-off energy is around 0.4 to 0.6 ev, the exact value whichshould be assumed will depend upon the thickness of the cadmium fo i l . The ieffective cut-off enersy is. defined as the cut-off of a perfect f i l t e r whichallows the same total absorption by the f i l tered sample of neutrons as doesthe sample actually produced by the given cadmium f i l te r .

. . . 116 . . .

Covered activity «t <$», i f

Uncovered act ivi ty « » 2.

Thus if the cadmium ratio is spatially constant over a certain region,we can say that ^ l / ^ is constant too. 3 0 we try to make our flux

measurements in a region over which the oadaium ratio is nearly constant.However, it may be pointed out that the smaller the region the more nearlyconstant the oadaium ratio will be; but this will be off-set by largerleast square fitting errors.

The situation can be bettered by using two group theory. Takingthe reactor to be a finite oylinder we cao write the two group flutes as

p, = I , '»J.

The S1 and S2 are the coupling coefficients. They axe the inverse of Glasstona'avalues.

The p is absent in Glasstone's book. But here we have tfl put it in becausein Glasstone's eqn.(8.39.i) it is more accurate to take the souroe term88

••• 117 •••

yu. and y correspond to the J. and I.. So, since the cylinder is

finite,

and * +

From (8.45.2) and (8.45.3) of Glasstone we know that

Combining (2) , (3) and (4)

The material backlinff

Let Z» and £ 4 , give the orass-sections of the foil * i c h i s

used for irradiation. The activity i s then given by

^ - •• V J - c ^

If the f o i l creas-sections are such that Z _ s> S. £ • , wo can

neglect the second term. In any case the activity can be written in the *'->form*

.- 118 -

Vfe can fit the measured values to this and try to get 9 ^ .With Is.foils the error in neglecting the second term is small.

The Substitution Method

The experiment here is to measure very accurately the changeinaritioal height of a reaotor when a certain known number of oentralrods are replaced by test rods. The experiment La repeated withdifferent numbers of rods, say 1,4,9, 16, and knowing the buckling ofthe reference lattice, that of the test lattice can be calculated-. Theadvantage of this method is that «a do not have to fabricate a largenumber of rods. The measurements are also faster.

Where only one replacement experiment has been done, the way ofoheoking the coda is something as follows. For the reference lattice aoomplate oritioality experiment, including flux mapping, is done, sothat not only B? but also the axial and radial extrapolation distancesare known, *hese latter are assumed to remain unchanged under the 2perturbation of substitution. Row the quantities like S.., S2, L^, L^,p are not known even for the reference lattice, we use our code tocalculate these quantities for both the reference and test zones, ji andcan also be obtained from these quantities. How we can use thedeterminant method given by Glasstone to calculate the critical height,the extrapolated radius baing assumed the sauna as inMhe zap unreplacedoase. ^his oalculated critical height is compared with the measuredvalue and gives an idea of the combined accuracy of the lattice code andthe determinant method. In making this.-statement however, we should notforget one more source of inaccuracy. The lattice parameters arecalculated by the oode for an infinite lattice and we are assuming thatthe same values hold for the finite one. To obviate this error, thePranoh have made the measurement with progressive substitution.

N m total number of rods

n • number cf reds replaced

RQ m Badtua of the replaced zone - t A

H, - *• JM

H 8 - extrapolated height for the ' reference lattice

H e + o*H - extrapolated height after substitution

..120..

B . buckling of the reference lattice

B'2 - buckling of the test lattice. a.y8» • radial buckling in the uneubstituted reactora.

y» m radial buckling in tb£ reference lattice afterreplacement.

A - radial buckling in the teat lattice.

B2 - ^ * C^O*' JL

The difference in buckling of the teat and reference lattices can bewritten aa

patting

lie get

The aim of thia method is to obtain thia quantity directly from the £Hmeaaureoents to give us an "experimental" ££ as in the critical expsrimeo

From the measurement, «e can get the value *ich we shall call SB

--120-

• /aj - / » * , ' ft*" (*) M * (7)V »

On putting A • yflL - * - < / » , *•

- * A ^ .GO

From (8) and (10)

is the quantity m want, 5 3 is the quantity *» lufwmeasured. So now we need to know A /a / 4y3 . To findthis* we have to take xeooune to some theoxetio&l treatment.

We oonaider the tare xeaotor and apply two group tiieory.

The funotiona X and 7 tax the zefexenoe core aa» given by

. . 122 . .

la, aa «• attain* in aqn. (5)

baza ac« tba i i a aa tnaaa glvan %gr Glaaatana, with tlw padded of eopcM.

fosr S^fiMa (4)

•"4 i

" tf - i

..123.

- 122 -

After replace^eat we have the test zone in the centre and we denote its• r t

properties by primed quantitiea S, , 8X , A . The X and Y functions in

the central zone are

* o y - »• cy».#

In the outer zone _

* « J. €/»»D + * Y.

, *O • *' K C ^ O

The fluxes and currents have to be continuous at the boundary betweenthe two regions. Let ua represent the X and Y functions in the testregion by the subscript 0, and in the reference regios by the unsubscript edquantities themselves. The derivative'w.r.t. r ia denoted by X',.. etc.e also make the assumption that D is the same in both regions. Then weget the following at r • R

x. t- •"*. Y. m x • -^.y

s, x % ••• T*.. *» y» • •. * "**•**•• y

Eliminating Yft between (*3) and (14), and Y« between (15) and (16).

define the quantities

- 123-

A « _o-o

PPOB (17) and (18)

Cs, - « l ~ C

C s, - s; )

further define

s. - saami

s«. - «;

• *. A » —

* -t- -«.*y

X ^ -m S y

er X + A

-124 -

Similarly, •liainatlag XQ b t t m n (13) and (14), andX^ b c t m n (15) «t&

K * A * ' ) * - » C y - A ' y ) « o

Ualnff (11) «al (12) ia (20) aad (21), aai xmlag tte

ia ordar to siaplify vrltiag

•liainata • ftraa thes« two tqiation

. . 125 . . .

c.os '

If we haul worked in one group, our equation correaponding to (22)would have been

Since we have assuoed that the disturbance due to the teat zoneia snail enough to be considered a perturbation, we must have ft rathersmall. So in the denoainator cf (23) we can replace € by i t s approximatevalue obtained from (24).

r i l S

In the practical case we can make some simplificationsperturbation beln? 3Jail, we can write

since £ ia approximately given by (24). Thus w« can neglect the secondterm in this expression.

Next we consider the expression

To find an expression for c' we note that at the extrapolated radius H

x * •*«• y • •

Since the determinant of this set of equations does not, in general,vanish, it can have only the trivial solution X - Y - 0

X and Y in the outer zone are given by (11) and (12)

Writing out (26) in full- using (27) also

Comparing tbe two terns we see that as "K, »• o, all the terms in

the first one remain finite, whereas K.. M, -* m . and 9, •••.

..127.

. . 1 2 7 . .

Thus for sufficiently small H. we can gay that

— CXA1 <*6' L J

Similarly

All things put together, we can w i t e (25) as

Now «a substitue the expressions for all these quantities in terns of the' radial buckling of the unaubatitated reactor, and the perturbations ij%

and A/9 . The Bessel functions are expanded by Taylor series up to

first order terms. For « we gat from (27)

Since H is the extrapolated radius of the ^substituted reactor,

- 0 , and iB.R - J, the f irst zero of JQ.

• Th put

. . 12?

Remembering that the , e tc . refer to values at H_

J. C/»'"*O— - \ -A5 .0»"«

/ j , -j C/a

'"e oan write approxinately

From the recurrence relation

we have

L. J, 3. CO

Now (30) becomes

- 129 -

J. C^nO - •*.

c - f "*. a/S 3 .

- Q • -

Similarly

# • -

The second term here ia merely the Becond term in [3A\ multiplied by -—

a.. a/a 3".

y. J . -

Using the recurrence relation

a • * . ar. CA-"O

Substituting (2?), (31) and (J2) into (29)

. 3, Ga"»O

we have merely lumped a l l -the quantities into G.

. ( i . + ' . X

we have

• - T C f "

(8) and (10), we can write this as

B.-.ere ^ IB , given-by (9)» is the measured quantity, Ih is known from(7) . S eaa be calculated, because xe have merely substituted C for the followingexpression in (33)

3 too is unknown. Thus (34; contains two unknowns A 3J and 5, and 30 at

least two replace-ent experiments are needed. In actual practice a nuaberof replacements are done and the unknowns are evaluated by least square fitting.The value of G decreases fast as the radius of the test zone increases.

Thi3 French method has given good agreement for the experiments doneby the French. Iii the treatment given here, we have assumed that the reflectorsaving does not change on substitution. The French have tried to make a correctionfor this. The only difference observed in the final expression is. that there

is now a constant multiplying A 3 in (34) •

TTe had used a two group formalism because the one group method wasfelt to be rather uns at is factors'. Physically, one reason for the failureof the one group aodel can be viewed at as follows. One of the boundary conditionswe use is the equality of currents at the interface of the test and referenceregions. This might have been legitimate if the regions had been well-defined'with a sharp interface. But this is not so and there is a transition regionbetween the two Tith slowing down properties intermediate between those ofthe two regions. Thus in effect, the thermal current leaving one region willnot be the sane as the thenal current entering the other region becaus^so-ne neutrons may have been gained from or lost to the fast group. This factwas being overlooked in the one group tret.taent. A way of improving the

.. 152 ..

one group model would be to have a tid^tiin region which will help to settlethe currents properly by making it continuous from the test to the transitionand fron the transition to the reference sonea. Thia is the method adoptedby Per3 *on.

He defined the lattice cell in a way, by having rods at the corner3of the cell. There were thus three zone3, Ee used one group theory and thestatistical weight theorem and got fairly good agreement.

Lecture 13

The Swediah Method

According to the one -group perturbation theory, the average bucklingof a composite lattice is siven by

W 3kCO

where 7/? and ." are the s ta t i s t i ca l weights of the reference and teat regions

The deficiency' df the one group two region nodel has already been described.As an added exabiple we can consider a particular case. Let a composite corebe made of two lattices corresponding to the points 1 and 2. Here BI •= 3"

and the average buckling as calculated from (1), will also bs the same.We know that the buckling of the composite core should lie somewhere like

at 3» and so (1) has given an obviously wrong result. It turns, out that ifwe use a transition region the results are better.

"«r *

.. 134 ..

The subscript a ia uaed to denote the transition region. Since

we get

^ ™ * *

•Vs define the quantities

t c >M

How we can write(2) as

0If we assume that B ia independent of the number of rods replaced, we canm

** - «k W^ 2

see that - • is a linear punetion of — . Bl is obtained graphicallyw w i

by extrapolation of this curve, as the figure shows.The theory has also been extended to use two neutron groups,' but

the formalism is complicated and will not be given here. In effect, theidea is to use two group theory to improve the value, of EL obtained by

the one group model.

The objection to the use of the one group two region method hasalready been described in the last lecture. 3riefly the picture is this.In the two group model the thermal current may be written aa

H TL I Zl

2. i s a removal cross-section from fast to thermal. BothHT

- J * are continuous across tiie te3t -reference interface Tiut Zand

is not, and this makes the sum discontinuous. The transition zone net, helpsby serving as a source or sink of thermal neutrons.

The Canadians have analysed substitution erperinsnts by the heterogeneousmethod. If we turn back to lecture 4, we can conclude that 1 and L - - • •-•-

enter into the kernels j and f, and other lattice parameters likee » p and f should go into the calculation of 7*. a n* "*\SL •

given there any method for actually calculating y^ , *», e t c , nor dowe intend to give i t now. We merely assume that some method exists wiiohcan calculate the kernels from the lattice parameters.

Consider now the reference lat t ice , for thich complete flux mappinghas been done. por this lattice we now calculate a l l latt ice parametersexcept p using the code whose accuracy we want to tes t . I t need not bep of course, i t could be any other parameter t | , f, €. , but the panadians

had used p. The heterogeneous method is now used with trial values of puntil the measured value of B2 i s obtained.

m. .136.

. . 136 . .

In the actual experiment done by the Canadians, ttiers was a slightadded complication. I t is only a detail and of no significance in the generalanalysis of substitution ex-perinents using the heterogeneous method. I tca-ne in because of the following circumstance. In the heterogeneous theorythe reflector is taken into account in the derivation of the kernels j andf. So far this has been possible orly in the cases where the reflector andthe moderator are the sa^E, The actual substitution experiment however, wasdone Tith a 3J) reflector surrounded by a graphite reflector. The cri t icalheight vias measured in this assembly. How the heterogeneous model was appliedassuming that the reflector is fully 0-0 and the reflector thickness is

H. R and p are simultaneously adjusted to .jive the right c r i t i ca l height.This giW3 some elasticity in the values of R and p, but the permitted rangein p may not be large enough to make this a serious indeterminacy. For furthercalculations the value of R i s taken not to have changed.

Now m come to the actual substitution measurement, and repeat theheterogeneous calculation adjusting the value of p in the test zone to getthe correct cr i t ical height for the composite core. With this p value, the

B can be obtained strightaway from the or i t ioa l i ty equation

But what the Canadians-actually did was to repeat fee heterogeneous methodwith a full core of test rods using this p and adjust the height for

oc r i t i ca l i ty . The flux distribution is also calculated and B is obtained.

mOne point is worth noting. ~e have calculated p for a tes t rod

forming part of a small region in a replaced "ore, and assumed that thisvalue of p holds good for a full core of tes t rods. Now p is a functionof mutual shielding which depends upon whajt kini? of rods are surroundingthe rod under consideration. Evidently the error introduced in Bg by thisfact will be larger if the t e s t region is s aa l l .

">'ith this method, there is no need for progressive substitution.

The Infinite I.'ultiplioation Factor,

Another integral parameter *hich is eor:r.only measured is k w . 7/edescribe one of the methods commonly used in i t s measurement; the nullreactivity method.

. . 157 . .

r an infinite cr i t ica l reactor . Here k - 1. Now le t us intrrwlj.ee

a cavity in i t somewhere. This cavity will not cause an?.' ^ain or I033 inthe nunbar of neutrons and so the reactor will s t i l l remain c r i t i ca l . Towif the cavity i3 f i l led up by an unknown nediun whose Tc is greater than

unity, the systea will be supercri t ical . T7e can add a certain amount ofpoison to this unknown nediie and adjust the aaount such that the reactorstays c r i t i ca l ; we can then say that the k M of the pcisonsd test =ediu=

is unity. Knowing the aaount of poison and its thermal cross-section we cangat the k w of the test nediun. This is the principle of the null reactiviti-es thod.

In practice ho^revsr, an infinte c r i t i ca l reactor is not availableto as, and the experiment has to be done in a f inite reactor. In this casethe k ^ of the poisoned medium differs from unity by Z^k^ . A two group

perturbation theory can be used to evaluate & k a > The two group equationIn the finite med?ura ia given by

- Z, \

where Z., ia the removal cross-section from the fast group, and Z.fc

the thermal fission eross-section in the f ini te reactor, V ^ • 0.In the poisoned reactor this becomes

-z. "\«2*

2.\ is the absorption cross-section of the poison. The solution of (5)

. . 138 . .

The equation adjoint to (J) is

,*4 -i.-i*frors which

Since (3) hao non-trivial solutions, the determinant of the matrix shouldvanish

V f 1 ^ , - A C 2,we have

Prom (6) and (7)

(8) thus gives the amount of poison which when added to a material whoseinfinite multiplication factor is k,^, will give exact oriticality in an

infinite system.

let us imagine a finite reactor which has got a cavity inthe centre, and is critical with the cavity. Hext we fill this cavity withsome of the material described in the last paragraph, A suitable quantityof poison is adied to this material so that the reactor still remains critical.

The araount.of poison £j" in this case will in general be different from

-139 -

To this systaa *« now a-, ply perturbation theory. *e asaune thatthe reactor with cavity is the perturbei gystea, and the reactor with cavityf i l led with the poisoned material is the unperturbed system. We f i rs t writ-down the perturbed equation

L' and I' cay not be properly known for void, but aa we shall see, they

will not be needed. The unperturbed equation is

a O C "O

'.Ve confine our interest to the volume occupied by the cavity. The matrixoperators K of (10) and K*. of (9) act within this volume and are subjectto the boundary conditions of continuity of flux and current at the interfacebetween this volume and the remainder of the reactor. An operator K* adjointto M can be defined.by . , V

where the integration is performed over the given volume and <f and" -vyare any two functions satisfying the given boundary conditions. The equationadjoint to (10) is

a O C'O

Multiplying (9^ by m1 and (11) by «y* » integrating over the cavity volume,•and subtracting one from the other

.. 140 ..

The perturbed matrix X1 can be written as

where P is the perturbation. P oan be obtained by subtracting I', from II1.'e also note that V., P etc. are going to be used only in the form of integralsof the Icind appearing in (12). oo by taking the cavity volume as sufficientlysmall we can say that the spatial variations of ^ are small, therefore

V <j is snail and terns containing it nay be neglected.

-2, I*1*

Substituting (13) into (1.2)

J * J J

m' and y both satisfy the given boundary conditions. So we have

f 9* m"* W Av

This with (14) gives

"•«*' V * * 4.V("•

Considering the perturbation as small TO can replace «?* in (15) by theunperturbed flux <y* . If the cavity is very small, we can say that theintegrand of (15) itself i3 aero. Thus

- 141 -

C~'., -'O

k - T 2 .

-•pi.V, * c

Otir experiment essentially consists of determining Z t in

actual reactor and using thia value for Z ^ in (8) to give k — . ObvLously

the error in k ^ will be, from (a)

z!can evaluate ,. from (16)

- 142 -

t. i s obtained by substituting for k— from ( |) in (8)

— — v * - ' ~ — 'i T i •_

substituting (19) and (19) in (17)

Z. / J 1 Zi. - + 1 -r J i

Using (4)t

Using the value of p from (5)

If we want the error to be small, i.e., A * « -• o t *e shouldhavs either

*k ^a.

or J^L

. . 145 . .

In words, this means that the spectrum in t*ie poiaaned sanple in the cavityshould tie the sa:» as the apeetria in the "infinite crit ical poisoned mediumAlternatively, the adjoint spectrum should satisfy this condition, Usuallythis condition for the flux spectrum i s satisfied by the following experimentaltechnique. The reactor ia made ofifaur zones. In the centre there ia the testzone consisting of the poisoned tes t -aterial. Surrounding this there i s a bufferzone of unpoisoned test material. The-next outer zone is the same buffer zonebut so constituted this tine that i t s YjYf can be varied at will, ^he outermost

zone is a driver zone. It can be shown that the requisite condition is satisfiedi f the speetrun (or cadmium ratio) i s made oonstant over the test zone and into thebuffer zone.

Consider the fluxes in the test zone

Since the condition we are trying to attain is that of having k - 1 for

the poisoned test zone, we can write k' - 1 so that /** • 0. Then from(20) and (21) '

In the infinite medium

Our aim is to have — same as . Let us see what the error will/be at r - 0. <p[

. . 144 . .

- Vs.BA

a: i f I -A V

and 0 here.

aineeS.

oan write for using (22) and (23)

Ax. OV>

- 145 -

S, A A

A s. L

where C3 is the eadniua ratio at r - 0, Using (25) ind (26)

f o - i ) - -A s « c * • - 1 X.C.T»O' - f

Proa (24)

It follows that & ( « , / <»O - 0 if Ck (GR - 1) - 0. Thus i f we

could adjust the cadnium ratio to be spatially constant, the apeotrum atr - 0 will be the name as that in the infinite reactor. This is done bymaking adjustments in the buffer -zone. e.g. by changing the latt ioe.

The actual experiment oonaiata of three measurements. In the firstmeasurement, the reactor with cavity is aade exactly critical. In principlewe should now f i l l the cavity with the poisoned test medium and get criticality,However, changing the quantity of poison in the test zone is not an easytask and i t ia not practicable to repeat it until we get exact- oritioality.So a second measurement is done after filling the cavity with the unpois'onedtest mediu-n. Let us say the k -- in this case i9 1 + x... By a rough calculation

we oar. know what amount of poison will ba required, let it be M . Withthis poison the kg , f is measured again, let i t be 1 + x^. By linear extrapolationwe say that the amount of poisoning required is

B M• A.. —

MQ is taken to be the measured value of the amount of poisoning, k of

the *es.t medium can now be obtained from (8).

The Fast Fission Ratio

Another quantity which can be measured experimentally is the fastfission ratio. This is defined as the ratio between the total number of fissionsin U -238 and the total .number of thermal fissions. In a fresh uranium •rod we have

where F is the fission rate per atom.

The measurement is done in the following manner. Two foils have tobe irradiated, one of natural uranium and the other of depleted uranium. Afuel rod is cut and the foil is made of exactly the same dimensions as thetransverst section of the rod. Two cuts should be made in the rod, say about

• . 147 . .

1" apart and the two foils should be placed in- them. This is done in a mrr.berof rods, say in a 19 -rod cluster a natural choice will be the central rod,one from the outer six, and t-so fron the outermost 12. The cluster chosen i3normally the one at the centre of the reactor, since vre ant to simulate a cellin an infinite lattice. The foils are irradiated for some tine. They are thenreaoved and their activity is raeasured. This could be done by a f -counter.The count rate recorded by the natural uranium foil is

And for the depleted foil

no of atonsno. of f isai ias per atomis a correction to the flux arising from the fact 1iiat tiie flux in therod has been jHrturbed by the presence of the f o i l .fission product yield •. ;

counter efficiency

Since we are interested in finding for the natural.-. we have

• Let us substitute

Jipc.r-«n'- from becaiiae of two reasons. The first is that the perturbation

of flux due to the presence of ths foil will be different for the t«o fo i l s . .The oth^r is that the effective cro3s-aeetion3 are different since the neutronspectra are different. The former has been accounted for by the ,, and thelat te • we can asruse is orall enough to neglect.

. . - 5 - -5

(-•7) *nd (2>rv> can be rewritten as

• - /v «** F •* c « Ys a < °^t r« S c « Y« s s f s

From (29), (30) and (31)

4» •

3i;nilarly

H* •y ,

(32) »nd (33) can ba writtftn as

*t • r* *• V« c» - •» - N? P S *« Y,

* * Y C

or Croia (34)

. . 149 .

yf C *J * i c« - ^t

and ftfoa (35)

- 1(35) »tri (37)

r, * ; . . . v* • > • * • •

substitute

N - JJ

. . 150 . .

In order to get R(t) we have to know the Va and f ' s . Since theflux perturbation carunt be very great the 4's can be calculated roughly bysoT-.e approximate theoretical method. A.: very simple -nethod for instance, would

be to take the neutron density to dininah within the foil as I «

As for obtaining fgi — is known. -—— 0 : m b e approximately treated*•" a d Cfl "as equal to unity, since both CQ and Ca deal with U-238 fission and theo o

fission products will emit the same kind of *jr . To get f,., a subsidiary

experiment i3 performed. A couple of samples of the same kind as used earlierare irradiated together in a thermal column. Tjia will give

The G'a ax« salf-ahielding faotora la the thermal speotrum. These may be allghtlydifferent from the Q'a vhoioh are a«lf-shielding faotora in the fuel spectrum.Thus

It remains to know p ( t ) . For thia, <S i s first measured accuratelyin some reference position which oan physically accommodate a f ission chamber.Two fission chambers are placed back to back with a natural uranium foi l inone and a depleted foil In the other, so that the fo i l s are quite olose together.The counts R. and D, gin directly

(38) and (39) can be writt«B as

- 151 -

Continuing along preclaely the stun* lines aa before we end up with

-0, - -. ~.

"(f and «. being known, o . i te directly obtained. We call

It &*.

Hext we irradiate two foils in this reference position, measure their

activities, and calculate R(t) — let this be HB(t). Then P(t), which isIndependent of the foil and the spectrum is given be

The theoretical value for i is obtained during the €. calculations,where we had calculated the sources and currents for all groups and nil regions.

The total number of collisions in the n region In the j group is given

This is multiplied by — « ? i r " to **'* the total fl8sionB ** tfcat

group and region. When this quantity is sunmed over all J and n «• getthe total number of flsaiona per initial fast neutron. Multiplied by vthii gives &.

lecture 14

Init ial Conversion Ratio

A fourth quantity which is directly measurable is the conversionratio. Conventionally i t is measured at the start of l i fe of a reactor andis referred to as the initial conversion ratio. It is defined as the rationo. of Pu - 239 atoaa produced . .. . . . . - , *. , ., -no. of. tF - 235 atSms destroyed *» * • initial »**€* of the reactor. Inthe measurement of ICR what is actually measured is the value of the rationo. of P -258 atoms destroyed _ .. . , . , . .-,„„, „„.„.„no. of D - 235 atoms destroyed* I n t h e n o d e l m t i *» U T H E P o r D0MUC

we oan say that

1V C O

We can split this up into three parts corresponding to the thermal, resonance•ad fast energy- regions

*•«•.. 0 .

8 8C. and C. are captures in CT-2J8 and can be obtained during tne calculationof <5 . The subscripts refer to the groups 1 and 2 defined in the € £, J_^calculation. The expression (40), lecture 13, ha* to be multiplied by "*and summed over a l l n to give

The experiment consists of irradiating foils in the fuel rod bycutting the rod •to., exactly as it was done for the measurement of « .The foil should have the same composition as the fuel. In another experiment,exactly similar foils *-e irradiated in a thermal column. Both foils areremoved. The aotl-vit;- a thesee will now be due to the presence of twothings, vl»., Kp-259, and fission products. The ftp activity is determined*y oot#ting the coincidences between the 106 - ker J and fluorescenceX -rays, tdiioh are both emitted in Jfp deoay. The fission prodnact activity1* measured using a counter bias which screens out all the low energy f .ft)* experiment is then repeated in a reference spectrum, usually a thermaloolusm. Let

. . 1 5 4 . .

.. 154

Prom the definition of ICR,

Combining (2) and (3)

c • '"= c

OL4 3 S

The second term can be calculated in (g + rs) fashion on knowing the temperature

. . 155 . .

of the thermal column. «< can either be calculated from the code; or ifmeasurements have been done on the assembly for r and T , the (g + rs)expression can be used. n

There are two corrections to be applied. The first arises fromthe fact that IQR is defined for U-235 destruction alone, whereas toestimate i t we are measuring the total fissions which include the U-238fl89ions. If we assume that the activities of the 0-235 au*i U-258 fissionproducts are the same, we can make a very simple correction by multiplyingICH by

The second correction i s to account for accidental coiiicidencesdue to fission products. For this, a U, , - - Al Toil i s irradiated la athermal column and counted in the same way as the previous foi ls . A correctioncan then be applied assuming that the fission product coincidence ratesare proportional to the fission product f - ray rates.

Spectrum Parameters r and T -.n

These are the two parameters which decide the nature of the thermalspectrum• The method for measuring these will essentially consist of measuringthe activities of foils whose cross-sections show diffemt variations withenergy, r and T are measured simultaneously, and at least two foilshave to be irradiated. The reaction rate is given l(y «;- .

In principle, two measurements will be sufficient because r and TQ arethe only unknowns. In practice, n^Qt i s also unkowa and has ..to be eliminated

by using one more fo i l . We usually use Mn, a 1/v absorber. ^Anotheruncertainty is 1he efficiency of the counter used for measuring the activation.To eliminate this , ifae whole experiment iias to be'repeated in a known spectrum,usually a thermal column.

. . 1 5 6 . .

. . 156 . .

The normal procedure la to use a material for ^hich 3 is very. nail and g is a strong function of I • Thia i s used for the determinationof T . Some other material with g very nearly unity ani depending onlynireakly on T , but with large a, is used for finding r.

An approxioarfcion used in theae calculations ia to convert s toa simpler dependence on T . To a first approximation

*< •"• J L .

The noglection of the second term can 1» Justified on the grounds that when-s is large, the aecond term i3 small compared to tha first one; and when ala aoall, the whole thing ia small anyway. Thus we can writs

*h«r* I « <r- i 3 independ«nt of T .

<R . C TV

The natexial used for ciibiation shoud te a 1/v absorber, because tat thia

Thia oriterion is aatiafied by Mn *ioh haa

* 0 # 6 ' *100

-15 7 -

The subscript on in *C.

onThe second oaterial needed, with saall B* and g strongly dependent

is proTided by "

820 - M O -20 -

•lOO " * °'15

The third en«, with l u g e a, and g

^ - 1.02

«ioo *****

1 i s In, whioh has

s2Q - 19.8

'100 - 2 2 ' 4

These ths«e foils axe irzadiated both in the fuel spectrum and in therefexsnoe speotrua. le denote the fu»l position by the subscript X andthe xefexenoe position by d.

The aotiTitgr « i l l be given by an expression of the for*

•n. •». V

C and G are self-shielding factors for the thermal an4 the resonance

neutrons. They are obtained by calculation. One simple wa$ to do thi« wouldbe to take an exponential decrease in neutron density within the foil. Let

- 153 -

To obtain r at d, the Cd ratio of In is measured in the reference position.This is normally in the D_0 reflector so that i t van be asatasd that Ta i snearly the physical temperature;

h and 1', are transmission factors 'through the cadmium to the Uaxwellian

and epithermal components. Knowing the thickness of the cadmium coveringand the energy variation of i t s cross-section, these can be calculated nG is assumed fts 1 since r ia so small that a slight variation of

r • . i

G from unity will not oause muoh error. S. ia different from S_. Thisr o Qwe can see as follows. The form of the denominator of (7) should giva thereaction rate due to the neutrons which have passed through the cadmium filter.For the spectrum of these neutrons, eqn.(i), lecture 8 will take the fora

forking out in exaotly the same way an waa done there, we realise that thereaction rate does turn out to be of the form of the denominator of (7) withS£ given by

£k -j*r

All quantities in 47) are now known except for x. / — i . s and g are

obtained by assuming thaf T - phyaloal temperature. Thua C *. / "^

can be oaloulated. *

This done, we return to eqn. (<>). All quantities in the referenceposition are known. So are the G's and SQ, g(T )~ for In can be calculated

with Tft - moderator temperaure beoause g(T )' for In doeB not vary rapidly• " -'

with T . /* a / __2r j is the only unknown left, and oan be calculated.

•"76The next measurement is with Iw ' which has a g strongly depending

on Ta> We gt an expression aimilar to eqn. (6) for

-159 -

All quanities in this eqn. are known except for g(T Q) x i-i the numerator,

since this can no longer be obtained by an approximate value of T . Thus

we can obtain g(T n) x from (8). Using the tables of g for Lu as

a function of T , we can use our calculated g to obtain Tn<nThis gives both r and TQj but these are approximate because we

have made the approximation (4) and used (5) «hich follows from it. Writing

(5) *fr ,

s. . s —

aid considering S as belong defined by (9) we find from-. (4) that SQ i« t;, 00

constant whereas 1* reality 'it depends on T . We can now iterate as follow.Using tt. calculated * , we obtain a new So from (9) . All calculations arenow repeated with this S,. This" Tfl ca^ also be used to give better values •for g ( T j Y for Va'aA In . This will give us a second eatirate for the

values of r and TJJ.An alternative aethod of obtaining r and Tn tarn the measurements

is also often used. Guessed valuea of r and TQ are used in eo.n. (6) suchthat i t givss the correct A. A number of such pairs of r and Tn « e .btained

.. 160 ..

and plotted on a graph. The same thing is done with eon.(8) for Lu .The two graphs will intersect at a point. This is taken to,give r andT for our spectrum.

H P "T can also be measured by usingn

instead *f

This is because U-235 fission shares with Hn the quality that g «s 1and a is small, -and Pu-239 fission has a very string variation of iwith T .n

- 1.05

'- 1.115

- 2.33

- 2.47

The variation of g here is not as pronounced as it was for Lu.In CAJTDU, It was observed that with Lu the fuel T was 170°C, with

Fu it was 150°C. A difference of 20° is not a large error in the reactivitycalculations, but it serves to demonstrate -that the model is inaccurate.It naa already been mentioned that the use of the tfestcott model is justifiedonly in those systems where T does not change very much when different

materials are used for measuring it.

The Fine Structure RatioThe flux fine structure within the cluster can be measured with the

help of a 1/v absorber. Sinr»e the reaction rate is given by . n. .v - o-_ *TOT U ^ ,

its value at different spatial points will be simply proportional to the nneStron densities. However, since no absorber is strictly 1/v, we adoptthe following procedure. Ho measure the 3H at different positions x1t

. . 161 . .

r and T ** thaaa polata having already baan aeasured, «a know g, rCand a, and (10) atraJ^rt-amay givea

% first make the reactor critical and aesmtra ita critical heightand flux distribution. This gives ua fl . The k w of the lattice is assuaedto to known. In this situation

% now poiaon the lattice. Tha new Talma k'M can to calculated aince we

know the aaonnt of poiaon. The new critical height ia aeasured. Aaaomins

tha,t the eztrapolatad distance does not change, the new B? can to calculated.

that M «• M1. So w» eaa say tout

froa where IT eaa to calculate!.

* have briefly aumaarised the aeasuraaents of a, k — , «5, ICE,r, T , fins structure and fer. The quantitiea which eaa be calculated fxoa

2 2 _ 2our theeretie&l codes are C, p, f, L , L , k , ST. In order to eoapare,we eaa calculate the erers-seetloas using the aeasured r and T_. the finestruotnre can aow to used to give f. XC& and & can te theoreticallyebtaioed, as has alxoaiy been described. Coajariaons eaa kelp to decide•hsther the tfaoorotioal aodelc need iaprereoent sad i f so, to what extent.

In tfas nsxt leatuxo sa shall praaead to the calculation of burnup

Lecture 15

The Burn up Equations

% have seen bow to calculate the reactivity of a natural uranium -D-0 system, but so far we have confined ourselves to the clean Cord it ion of

the reactor. However, when a reactor has been in operation even for a fewdays, the composition of the fuel vill have become different due to the buildup of fission products aoi plutonium. This changes the reactivity of thesystem.

The U-238 present in the reactor undergoes a chain of reactions aashorn here:

C •<*«•<•.

U-235 also gives, a similar chaia. Ife can write a differential equationfor the number of atoms of U-235

i s Mm llestaott fltac and is th* Weotcott ereaa-aaellon. Since

.. 163 ..

<p i3 not spatially constant, I75C. too till VBJ - over the reactor.car. sr i te similar equations for a l l the i3cto;es. In cost cases, the r i r h t '-.ar.i side .vill contain a production tern as rel l

„*.„»„*„» -

A * COiv «- <? . v

AM * «

Zers i t should be reaenbered that the total absorption consists of threeparts; the fast , the resonance, and the thermal. In,case of al l isotopesothar than TJ-233 the fast and resonance components are included in the

J* but not so for U-238. la this ease -SB caa write for the total absorption,as we did in eqn.(1), lecture 14.

• z. , . , c c?i « ar

Comparing with (1) *e caa write

A _ a- Q -I- i - * "V*

• at flat r* <» «a s -w^

4Te assine that the deca^ of U-239 to 5p-239 i s isnediate. So we do notwrite an equation for i t .

• • 1 "3-T • •

^ have as3>jped that the decay of Kp-240 to Pu-240 i s instantaneous.

Thi3 set of sinultaneous equations, has to he solved to give the isotopieco--o3iti3n of t'-.e fuel. I-.it the flux varies with 'both space and tirse, namingthis diff ieal t . JO the procedure followed is to take a point aodel. 'Theequations can then hs solved with the fltcc at a particular point a t a particulartir.9.

In all the theory *hat has been covered over the las t 14 lectures,7T9 neeJe? anlr the relative f l i a . For thesa calculations however, us needthe absolute flux. Pi» 2uanit7 ^lich i s fixed in Tost reactors is the poorer,and this i3 relatad to the flux.

- . - eaer^.' produced/fission, in ?•

0 » eoovars'.oa factor ffsea fte» to ihr.Sec. •

- 1.6 x 10~13 Hw. seo/nsr

The total -our of the reactor *ill le

..165..

J T»C»O P d V s " P ^

' . is 5-iven. f is t!-.= ier.sitv of fie fael. (2} ard (33 could betot "

L.z Tlti V-.'J a.03ol:i-'? fT'ix if t'-.o r^i2.*.I'/e ri'ix '."er-3 t-nj.Tti, sz>.l the ;P-ux c;-r. "ce o'ut-.ir.al "sir- a f-ra--roj.r 2Lffusion th^or-/.

Tr.e ^ ' 3 here are the cell-averaged f lutes . Th "econi grcup flux ^

is t'ae "true flux" *itoh has 'been defined in lecta*. _• 8.

J- T. *

jbt=.ln»d slr.ce tha thermal u t i l i ac t l : ^ ficJ:;i

because

and w-.e 2 i are

.. 166 ..

.iestcott flux in the fuel is thus

The procedure to be foll^re! now has already 136311 explained, (o)is combined with (2) and (3) to give the absolute value of the fl'jx. Thisflux can now be used to salve the composition equations.

The correct ray to .50 about this will be to solve the diffusion equationsand th<? co-:?osit:or. equations simultaneously. This however is a fooidabletask? so numerical methods are adopted. One normally used is as follows.

1. In the initial sta~e all the compositions afe known. Ehus all thecoefficients in (4) and (5) are known at any point in the reactor. Ifotethat ?38 refer here only to the macroscopic variations. Fine structure isneglected.

2. The reactor equation" are solved to give the relative flux « C»O«

3. -Vith the help of (2) and (3), the absolute flux 9 C*O is calculated.

4. ^ C * O ig n<>w assunied to reaain constant for a snail space of

tLr.e and the burnup equations are solved at different values of r.

5. How w? know the new coinpositions at different values of r and soTO can go back to step 1. •

The fora of the burnup equations shows that there are two independant

variables y and t« It is possible to replace then by one independent

variable in ao3t of the equations if ve niake the substitution

•c '

X. i3 called the irrad!ati«n f and the composition of the fuel i s in mostcases a verr strong function of C and depends ora.7 ireakly on the historyof fluxes through ^hich this irradiation has been attained. The unit of-t is n/kb. ?uel *i«t i has been irradiated for 1 year = 3.15 x 10' sec

in a flux of <p » 3.2 x 10 n/c= /sec will have an irradiation

t - 1.01 i 1021 D,/ca2

- 1.01 n/kb

To replace ^ and t by « we divide all the b-^m, equationsby ^ an! obtain

d t ay c a s •-•"a*

"7e fird that $ has completely vanished except when there is decay, ie.,y- in1 (7) and (8) . . .Tq skrolifj further,.,we note, that . Itf-252 has a short decay

izi&t so that i-t saturates vei$- fast". (7) thus becones " :

. . 168 . .

Except ia th<* beginning when Ep-239 is building up, this holds. Ma canalso say tnat decay of Hp-239 is far aore probable than radiative capture,so that we can neglect the third term in (10)

=* "at y

Thus in the final farm of our set ef equations, (7) is absent, (8) i s replacedby

S» far there has baen ao referenoo to fisaian products. Va shall com* t*that later.

The initial rallies »f al l the H's being kno«n« «e ean now proceed%o solve tba bum-up equatio&s. The solutlona will give the H's as a function•f « \ . .

Obviously this proceauro i s Til id snly i f c»r th# eonoentratioa efNp-239 has attained saturation. During the period that!* **? buildingup, «s shall hava tf take the first set of e^i^tions cantatyiing ^ . Perthis purpose we substitute aa approximate value of ^ and assuoe i t tobe eonstani. This > i s obtained from the pover equation

This i s an avsrago flux •*•? the whol* reactor. So jastify uaing sish anapproximate Taluo on tin graunds that, the ffp-259 equation i s ono <fcioh may areabo neglected eoi»plotelyB and so very great aecuzasy. is not noeded fa? the Tainsof a sjoantity usod in i t .

He have mentioned earlier that ~$ can 'be obtained by solving (4)and (5) . The aethod given here is siopler. BUVI.AC and a»st ef the •thsrcodes folio* the siapler aethod.

After obtaining the ff's aa a function af t , we take a smalltime step Ac • Assuming <$> to remain constant ever this period wecalculated c at the end of i t . ffe can now calculate the R's correspondingto this t . These values of JT are substituted into (11) to give §which in turn is used to solve the turn-up equations. Thus process thusgoes on in discrete steps and we obtain the H values at irradiations of

2 DOMLAC calculates al l the lattice parameters like k — , p, L ,L , at discrete irradiation steps, p i s nearly constant since

2N2S fioea n o * o h a n S* such. L i s also constant because scattering cross-sections do not vary much frea nuclide to nuelide. Tcm and L are theparameters iftich change.

Products

The daughter nuclides produced in fission are numerous. Theyin mass numbers from 80 to 150, and have widely differing half live a andcapture cross-sections. If we were to try to selve the preblea exactly bywriting ene differential equation for each fission product, the task wouldbe f .-midable. So we take recourse to approximations.

Broadly the fission products fall into two categories. 1) The saturatingor high ar-033-section fission products. 2) The non-saturating or low cross-section fission products.

The saturating flas ion products are the following

Cd113 2 x to 4 stable

X*1?5 2.7 x 106 9.2 n«rs

S B * * 9 4 X 10*

Sn151 1.2 x 10*

£u155 1.4 x 10* 1.7 y»«»

G4155 5.6 x 10*

G4157 2.64 x 105

Hh 1 0 5 2.5 x 10* 36 lwur«

Tfa* halflife i s ispartant sine* i t detenaiaes the rat* «f decay, and i t is

. . 170 . .

d-cay rhich f inal ly leafs to the presence of $ In the equation. Forr-acii n-Jclide x::ere i s co=-.etition bet^e-:n the processes of decay and neutron

absorption. If a~ > > — , absorption wi l l predoninate, but i f •"

a -—• , the ra tes of decay and neutron absorption -sill be comparable.*¥ 135 105*

This s i tua t ion is found to ex is t in the case of Xe and- Rh . Ina l l other cases , absorption predominates and we can assune tha t the nuclideabsorbs a neutron the r.03«?nt that i t i s forced, so that

production = r e c v a l

0. This can also be seen as follows. Let us take the caseie.,

Solving- th i s vvith the i n i t i a l condition N = 0 when t = 0, we get

If 0- > > Tg , Tjhich i t i s for all saturating- fission products, thisreduces to

In other vrords, the t o t a l absorption is .e- .ual to the y ie ld .

In the case of Rh and Xe, (12) takes the farm

Solving- thia equation gives

-171 -

co «r y«- • t *

ie , ths total TH,imhPr of absorptions is exactly equal to the yieldfission jroduot.

?or Bh and Xe, (12) has to be written as

- N - v y - — - -r

Solving this equation gives

The total absorption due r.o a l l the saturating fission productsat the irradiation t ia given by the followingexpression. It gives the absorption for unit irradiation

Xe" ' as ws know, is produced in two 1373. About 10/J is produceddireetly, ard the rest aoaes from the decay of I135. The abovetreatoeat does not differsatiate between the

Lecture 16

The Saturating Fission Products

Last time we saw that for Hh and Xe, the absorption isof the fora

To calculate th i s , we have to know % . The flux variesspatially and the r i ^ i t thing to do will be to take the correctvalue at each space point. Tfe t ry to simplify natters by takingan effective flux irtiich is spatially f la t . In finding the valueof ?«tc » the quantity * i c h should be conserved is the reactivitychange aSie to absorption in the Xe. Sow

To obtain 4 for the reactor, wa will have to integrate(1) over the whole reactor using the appropriate s ta t i s t ica l weight.

The absorption due to Xe (or 3i) will show up as a change inso that

- 174 -

^ 5 can thua be calculated by using (3) in (2) with <9 asspatially variable or by substituting (3) in (1) with § replacedby <y 9 . 3oth should give the aame answer. Therefore

5TT31'* 4*.

KFrom this equality, T*U can be calculated assuming some <p(xThis has been done for CANXIEI, probably taking a flux distribution

.with some flattening and i t gives

Hon-3aturatini? fission products -. ' -"'~ ""•

These have cotaparatively low cross sections. Some of themdecay. Others are s table . Kany primary fission products give r iseto a chain, '.'•'e give below as an example, the chain started by

152 152Sm . Sm i3 produced both as a direct fission product and byradiative capture in Sm151

- 175 -

X 19 C

•fS"

Note that Eu , treated in the las t lecture, is produced herealso, but this time i t i s considered a non-saturating fissionproduct.

I'he procedure we follow is to find out the absorption at".the irradiation « due to the fissioning of one nucleus at n * o."Looking upon th is as a sort of kernel, we can find the absorptiondue to fission products at any irradiat ion.

There are about 90 fission product chains ^aid each chainwill have^abound six members - a set of nearly 500 equations.;equation for tiie f i r s t member of each chain is

-. 176 -

and for the subsequent meribers

•shere the subscript j refers to sone nuolide which producesthe i*11 nuolide by decay. The last two teraa of (4) are usallyneglected. This is achieved by making the assumption that a fiasionproduct is either stable or decays iEoediately. The half-lives ofmost of them are in the range 0-2JO days. If for a species, T^ ^150 days, TO asmine that it decays immediately. If Ti > 150 sdays,

we treat i t as stable.

A similar set of equations have to be written for Pu239 andPu-241, and the three sets Bhould be solved simultaneously.

'He aasiuse that at X. - 0r one U-235 fission has occurred.After that there are no further fissions. Let us study ho* the totalof the fission products varies with c . The equation far -the firstmember of the chain is

* « .

The initial condition is

«, C * • O - Yf

For the subsequent waters i . e . i

" 4 - 1 " * • -

Usually we do not take more than six cambers in one chain, this setof six equations can be solved and ire ean calculate £ , f u « \ f % 3 *The solution i s done naoarically. This is done fir a l l 'the 80 or 90chains and the *fct*>«i is eunned up far a l l the 300 or so nuelldes.It *as tBea fitted to an expression

- 177 --

This i3 tantamount to replacing a l l the fission products by orspseudo fi33ion product of yield Yp_ and crass section <r s .

The quantity of this fission product at any subsequent ti=e would _be y e~ ""f* * and 30 the absorption Till be y <r e

The number of absorptions a t t due- to fission producesfroni a l l fissions upto then can be obtained by integration.

- P , say

e ~\ a- Y *~ C O

The procedure then is th i s . T. . ."itting of Sqn. (5) to giveY._ ani <T is done once for a l l and the results preserved. For any

specific problem, S^n (6) can no* be solred numerically to give ?, theabsorption. I t is assumed that when a pseudo-fission product absorbs aneutron i t goes over into a nuclide with zero <r . ^hus ne have

eliminated 500 equations and replaced thea. by just one. However theyields and cross-sections of the real fis3ion products depend upon-thespectrum. So «"„ and Y p - should also be spectrua dependent. This

kind of f i t t ing was f i r s t done ty Walker. He gaire Y_ as depending

on the spectrua, but O~ as constant.

Walker found that f i t t ing the resu l t s to the fora

was d i f f icul t . But ha was able to f i t i t to the form

Z, • • . V •;,«i = i, a, 3

The three values of 0" were

ff" s So *•

. - 179.-

The y + * i are given by Walker as functions of *».The total absorption is now

talker has done this for other fissi le materials also,like U-235, ?u-239. Pu-241. Each tine he gets three pseudo-fissionproducts with r - 50, 300, 800, but the yield depends upon thef iss i le nueliae.

Thus the equation for non-saturating fission products is asex of three equations similar to (6), one for each pseudo-fissionproduct

If we want to account for fast fission, we can write eqn. (7) as

rr - y . . o- . o •*•s? - F i

Some fission products have large resonances which can. lead to tJBcomplication; because of self-shielding. Calculations showed thatin a typical D_0 reactor spectrua, the fast fission and eelf-shieldingeffects approximately cancel each other. So we neglect both and stickto eqn. (7) . - - \ '

In COKLAC, biker's results were used. An alternativeis to f i t the results to a form which used constant yields aiidspectrum-dependent V . The tqr was expressed in the form

- 180 -

"e have everrifhare used the nestcott fluxes <$ and orosa-

Reaction rate * r < ?

and r -a f, ( s + *--O

For the resonance nuclide IT-238, we do not use (8). We could have

+ t&e fast contribution

All nuclidse haviag re3onaace3 should get this treatment.Pu-240 has a reao^arice at 1.06 ev. ahere the cras3_3ection goesas high "as 10? b, aSt xhe infinite dilution 31 is 8J5O b.

The reason for preferring (9} to (3) is the following.The : * of ^3)t as Introduces in lecture 8, is giran by

I r defined by eqr. (5), lecture S, is fca infinite i lu t ion HI.Tfe$s is a i l i i ^ i t for raterUlJ? which have, low atadrption. îen aaaierial teai'.-.a •hiî-.%War?i£sai i t s t i l l ôea ndt natter if it?oorcentration i3 las, bat'••fees'the coajeiitratio'n id ilso hi@&, selffMeldisg'effects 3t*rt conlnt, ih.snd i t £3 no-longer sufficientto take the infinite dilatioa'resonance integral I . The actualresonance integral I ahs-44 be taken and this, and thereby A",trill depend upon the geometry. Tfe can approximately write

If 4 is calculated *ith the actual I for a cluster, the (g + ra) fcrnulacan be used without the fear that self-shielding is being ignored.

It aay be wandered *y this Dethod is not vised for U-2J2. ITnisis essentially because theaansity of U-258 atoas is ususaliy far greaterthan that of ?u-240 and so calls for nore accurate treatment.

In the SH approximation without Doppler broadening we have

* .. is. S*r *V

°"»« " potential scattering cross-section of the ?u-240

°"# " potential scattering cross-sect ion per pu-243 atoa.

Q~m - total cross-section of Pu-240 at 1.06 ev.

(11) can be written as

J • ' j ' - 1 '

. . 1 9 7 .

-- 162 -

The HI is normally expressed as a function of <T alone. 3ain (12), all quantities other than r can be luape-. into oneconstant.

The self shielded resonance integral fc clusters iscalculated using

Using (14) in (13)

4- < .V.

«t is a factor depending on the geometry. Substituting in (10}

Sfestcott

In a l l this, Ho i s thB number of Pu-2.40 atoms. This method is usedin LATRSP. Tney have "calculated the -value of «t as -

..198.

- 183 -

An alternative method is to use the expression

-*• 3>

nhere O"» ig given by (14). the second tern may be taken, to bethe sa=e as (13) • ^ f i r s t term may be considered the effect ofa l l other resonances. A'e can ay that when K -* 0, r -* a>%ao that

A, C and D can be obtained by calculating I for a hooogenepusfixture whose cocpoaition is varied to give different values of 9" •Thus we get the values of I for zany values of «* . "Se c«n f i t the.results to get k, C and 3, BUIGAC has the option to use either thismethod or the previous one.

Pu-240 self-shielding becomes- important when Pu is" recycled.For recycling, the burn up equations should be solved with differentin i t ia l conditions.

- 184 -

Lecture 17

Fnel Il ScheaesThe first o/aestion ifhich arises hers will be - •shat is the need

for fuel Dana.je-« t2 Te beg-in by tracing the effect of irradiation.

' Let us 1 'Ok at the way in vir.ich isototic composition changes sithirradiation

The saturation concentration of ?u-233 depends upon the IC3. The abo'W figurecorresponds to an ICH 0.8. Pu-240 concentration increases up to and beyond 2n/l<b. It aiT-ht be saturating at s:r:e later at age.

Let us see -a iat effect this change in composition will have on thereactivity of the reactor, -e know the two following equationg

1) is not s-fetctly true. T4 get the accurate value ws* should solve the•> . ' ' i .

diffusion equation. 3" gi^es a. aeasure o? the k ^ . So let us see how thisa . 2

varies with irradiation. The following cjrvs gives B aa a function of pointburn-up for different values of ICH.

- 135 -

These values are indicated on the curves.

-V fast capture

'.Then ICH increases, B decreases o*ing to increased resonancefli

absorption. The rise in the curve in the tejinnia? is due to theprod-action of Pu. L-vvn thouen ICR < 1, It w increases becauseof Pu i s larger.

3-, 233

a 235 650

These are not 2200 values. Tr.zj are the effective cro33-sectiona ina typieal reactor.

T» * "n T\+ *i

. . 201 ,

-186 -

where 2. "»«"•* is * e absorption due to a l l other nuclides. In thebeginning this is small, but as fiasien products b-iild up i t increases and

2"n cones down. The increase of 3_ in the beginning is due to the increase

in **) because It- increases and the effect of this is stronger in the9

n-iserator of (2) than in the denominator.

The thing of importance is always the discharge irradiation.ICE is larger, nore ?u is produced but the discharge irradiation might decrease.The folloirijig curve shows two cases. The decrease in ICH is usually achieved by

o- So?o-?6f

increasing the pitch, L.e. by increasing the aount of moderator. This willcause a decrease in r , since

flith decreasing r , theirradiation.

«t of Pu decreases so that we can get higher discharge

~*The actual conversion of TJ-23S to Pu-239 depends not only upon theirradiation but also on the effective flux under ahich this irradiation wasattained. This is because of the Np-259. Thus to find out the optimum ICS for

discharge irradiation, re have to aake exact calculations.

If we.plot.ik f - vs . irradiation we will get the saas curve shapes as wedid for - 3 . Point! 1 and 2 give the discharge irradiatio'h for the two cases.

Thug i t is- the 4c - . curve .which leads, directly, to the discharge irradiation. So

we come to the problem of how to calculate k -- . The simplest way ef doing this

is to use the point reactivi ty model. Any la t t i ce code, like DOILIAG will give th is .

- 187 -

k ,_ is here calculated froa Sqn. (1).

The discharge irradia-ion calculated with the point reactivity -a delis not of nnich si^nifisance, since this essentially- corresponds to a reactorin which the flux is spatial! - flat, let us try to see what hacrens in anactual reactor, Te begin by considering only the axial variation; radially *take the flux as f la t .

3e can change the variable from x to 9 wherefrom 0 to w .

• » « - *

varies

shown1?he viariation of point reactivi ty wi$h irradiation will be of the nature

%a = ft is-the irradiation in n/kb. The ordinate is a quantity

ex i - ke -(0). ,.?roa the share of the curve we can say that i t may be

possible to represent i t by an "expression of the fora ,.

This may not always be possible. For the GAI3U type fuel the jpe=k-is £.t about0.4 n/kb. Over the range O.o < «>» < 3, the curve is almost a straight l ine .So vte could f i t i t by two expressions. However, for the nousnt w. stick to Eqns(3).

ife have assured an axial sine variation of "fee flux. Tie furttierthat this regains constant with irradiation. This again, is not s t r i c t ly true.

'•••t :~ :". ~ o' sr-its the rector fir 5. certain length of ti-tfi t# ^inc?the C. :.- - i - :.:- KXi.-.Z variiiir.-., the irradiation ard the roint reactivity -sillW : vi :- ixL-1"*-. ' rh? reactivity cf the re.ic^r ha3 to b<? calculated Vy itsgratir^

-.';•'-r. i-. -he rea.^t-r isH

4 <*. * C O

Vh~ r e a c t i v i t y vt ary - - i r 1 : 9 i s , f ro- (3)

In zri* c--*-a r? •-; • 7? :<=zt a'll ths channels fiic?d f-r a perisd' ^^ ticst. ~- a r-^t ivi t ; ; o*" V-.e r'a^tcr at ths end cf this t i re c-.r. •» 3r-t.1ir.2f. by

- 149 -

n J V * - . . - -V

Norraall^, a. reactor continues to operate until the k .., ^ 1. If the reactor,

is such that kaf.(D) » 1, this will correspond to the case cf 7 « 0. In r.ost

reaator, k „ at start-of-life is always gr2?.tsr than 'inity, ba.t -ss vri.ll here

.take th* c iae of kef.f(q) • 1. 3o that Y = C TTLII jive iis the discharge irradiatior

la an actual case, ?re aeraly have to solve for Y » -C *here k __ (0) - 1 + C.

1)he results nay te different only to the extent of soae constants.

From (3), the discharge ixradia.tion for the point raactirlty ease is

Fox fixed channel, ve have from (7)

o - r *

«te see that the discharge irradiation has fallen to 72,5 cf ^*at re could have hadif the flux distribution had been f la t . Physically this can be explained toy?ayi:i£ that ths ^cre burnt-out fual is in places of higher statistical ifii^ii,so thai t;is r^ac',ivit:.' of *e reactor vfill te less than 'shat i i vould have, teenhad it rser. filled -rith fusl of uniforai composition whose ii-radiation is theover ihe reactor.

- 190 -

Let us see what we can do to improve the situation. He tora fuelling scheme i s ifaich an entire channel is refuelled at onetime, but different channels are refuelled at different tines in sucha way that at any instant, al l possible irradiations are present inthe reactor. The radial flux distribution is s t i l l assumed to bef lat . The irradiations of the different channels vary from zero to amaxiazua value of (ca ) , which is the discharge irradiation. If wo

H SAXassume that the number of channels containing fuel of any particularirradiation is the same for al l irradiations, we get the sberage irradiationof the reactor *9

kind of fuelling i s described as channel equilibrium.

To calculate the average Y of the reactor under conditions ofchannel equilibrium, we can take it that i f a channel has the averageirradiation^, the Y value in a reactor constituted of such channels

Since the average irradiations of different channels vary from zero toK ) max'

1.*-~1

"..207..

substituting from (7)i this becomes .

Thus the discharge irradiation in this case obtained by equatingS)to zero ia

I-OI —

- 192 -Lecture 18

Continuous Fuelling

The aiaplest kind of fuelling i s batch irradiation or fixedfuelling. In this wa operate the reactor until i t a It , - fall*-belowunity and then discharge a l l the fuel and f "»! up the whole reactorwith fre3h fuel.

As against this, let us consider continuous fuelling. Asaunethat ever7«40th day. for instance^ we refuel one channel. We could startdoin^ this frota the 3tarting of the reactor* that i s , the first channelire remove will contain nearly fresh fuel, nhen all the channels havebeen refuelled once, we reach a? equilibriua condition. If the reactorconsists of K channels and all of them get replaced after a tine T,

Sinca during any interval of duration. T all the channels getreplaced snce, at any instant of tine t , the only fuel present in thereactor is what was inserted after the tiae t - T. vfe now proceed tocalculate the y value for the reactor under equilibrium condition, 'fiecan find the irradiation v» of a particular channel. The y of thischannel is thsn known, being just the "Channel fixed" value. We canthen integrate over the reactor to find the y for the whole reactor

coler a

l e t us" asy that the average irradiation in the channel which ia justabout to be discharged is. ?3 .... "Foe;:±traoLiations of differentchannels *111 then vary^'unifoaaiy froa aero to u . For the wholereactor thsn

i6»re dn is the mciber of .shanriels ^ o s e lrxadialiolg^ie-ifeilwien«BdaB»,+-ito. tet ' t be:the?tiiae for which a"channei fids teen;^i ^reactor. The average ;irfatdiatiorr*»in i t ia given by sqn (5) ,lecture 17.

. . 210 . .

-195 -

Cabining this with »qn. (1)

the no. of enamels having Irradiations tetwaen o and

* < « .

substituting this (3)

1 " 5" J :*o *«

t"Chi* cwa* ia z*f«sx«d to as the channel eqo^libriidb ease

A3 in theixndia*£on ey

eases, we am get the 4isehargBto eese

••• C»^^>

. .211.

- 194 -

This is the saa» as •qa. (9), lecture 17f but the derivation hereis was more rigorous.

All these expressions for the discharge irradiation* withdifferent fuelling sehene3 in terms of the point value have beenobtained on tl* assumption that fc.«.(O) - 1. In general it is greater

than 1 o Then ire should use the condition

to get the discharge irradiation.

As the next step, we assuae that the channel is sot one singleentity but is aade of slugs. This ae call a point equilibrium aodel andin it ire discharge different slugs at different times, the slug Hhieh. isdischarged has an irradiation *».- and once equilibrium has -been

attained, the different slugs will have irradiations ranging front zeroto >o. with a uniform probability distribution

At point equilibriua y is kept constantly equal to zero.

€ «

..212.

- 195 -

In thia point equilibrium model, the procedure will be toidentify every slug which has reached *3^ and replace it at thatmoment. This oay be quite a complicated thing to do in practice.«e consider one aethoti "«hich has relatively simpler fuel handlingand lies sone where between the channel equilibrium and pointequilibrium cases. This is the bidirectional fuelling. It givesnearly a cosine distribution for the flux. Still assuaing radially flatflux, „

•,','e assume that the fuel is moving though the channel at a uniforc rateand it takes a tine T to move froa © » 0 to 8 • :.• So w 1 can say.that

Let us say that the slug is at S" for a tine d f and therefore duringthis period, i t s irradiation increases by <yC»~3 **"•Substituting for dt" from (5), this is

-=- «

Thus *hi irradiation in the channel at any, position 0' will be

..213.

-196 -

Using (4)

a. - —

The jr at tbl« point will t»

(•') a »•»' - V*i*

r .•( )

To gat 7 for th» ibole zeaetor, «a have to iattgrat* this

*T%£-

Cr -^OM**-^™

•."*"».

To diachargs Irradiation i s the yalvB of « at 9 «• *R. •fsoa (6) . ;

Alsff with (7) -

Slas* Awlliaff is esatimmu, th«r» is so reason for having k ^ > 'at aajr *!• • ; M that one* oquilitariua has b»en attaia»d «e bars

t y • 0

- 1 9 8 -

' • • c

So we see that by changing the fuelling scheme i t has been possible to increasethe discharge irradiation froo-722 of th-s point burn-up value for fixed fuelling160 a of the point Talue for bidirectional fuelling; — an increase by over afactor of two. It is interesting to study whether i t cm be iaproved further.Some theoretician has Calculated that the highs at value of the average dischargeirradiation is obtained if we use a scheme in ahich the fuel at every point i sdischarged at a different value of the irradiation, which is given by

3o auch for discharge irradiation. But another equally importantquantity in a reactor i s the power distribution. A fuelling scheme shouldnot be oblivious of this fact. Suppose we have a particular distributionto begin with. How does i t change with burn-up? ?or batch irradiation, theflux becomes flatter in the centre. If we subject a reactor of the CASD'Jtype to fixed fuelling, the flux distribution ia as shown here:

- - 1 9 9 -

: " • " :

', 1.57

', 1-52

; 1.28 :

To calculate the £ -* of the reactor with bidirectionalex i.

fuelling, the correct mthod will be to average the point valueby weighting i t with 1 . 3ut i f we consider two adjacentchannels tpgether, m can say that the average irradiation ofthe two taken together does not vary axial-ly. - So i t wa^'Joepermissible tc replace the whole reactor by a uniform isotopiccomposition ifaich is the average composition in the;reaeiior«.This aethpd for the reactivity calculation is followed quite ofen-In-giDJHBCT there i s an option for thiflv'aa .well as for thecalculation o£ X f f by flux w i

# • - . .

- 200 -Laeture 19

Let us say *« irradiate a oare with fixed fuelling untili t has beeoae subcritical. If we now break the fuel in the middle

and invert it axially, as shown ii* the figure, position 1 teposition 2, we get less depleted fuel in the centre, so that thereactor will be critical. However this will make the powerdistribution very peaked. At 180 days for CABOT, for factorincrease from 1.28 to 1.92 on inversion. At 270 days it is 1.98.Because of this disadvantage, axial inversion is rarely attempted.

Unidirectional Fuelling

This would give about the same discharge irradiation asbidirectional fuelling and has some practical advantage becauseof the simpler design of fuelling machines. We have ooa fuellingmachine at either end and in this case one can be made for pushingand the other for gripping; whereas for bidirectional fuelling,each machine should have both capabilities. The disadvantage isthat tha flux distribution becomes w r y distorted. The fora factorin a typical case is 2.26 as against 1.57 for bidirectional fuelling.

..218.

- 2 0 1 -

% aaw in the last lecture that with axial shuffling there Issignificant gain in discharge irradiation

I- ej *

•"- «£

For bidirectional fuelling we have to subdivide the fuel into slugs.If we use full length fuel we can only h=ive channel aquilibriua. Theratio of 3/2 in a CANDU type reactor seans a loaa.ef about 2000 i!*d/t.This could be made up by increasing the enrichment. But the over-allperformance of a reactcr using full length fuel or slug fuel will depend enaany things.

The advantage of the full length fuel is that it is easier to fabricate,there being fewer end fittings. Larger fission gas plena nay be provided sothat larger gas release can be aceoaodated. There is no flux peaking, and the axialneutron flux distribution beeore s progressively flatter as irradiation proceeds.

Among the advantages of ihe slugs nay be included the fact that a fabri-cated slug is significantly lifter than a full length fuel element, making shopshandling relatively easy. The reject rate is less, since if a finished productis seen to be substandard, we have to reject only one slug. This advantageextends to the case of fuel failure too, because if a fuel elenent fails we haveto discard only a slug. Fission product release following fuel failure is alsorestricted to the amount collected in one slug. The fuelling machine too, needhold only one slug.

The disadvantages are that every slug has to have its own end platesthus increasing the number of components, and its own fission gas plenua -nhichmeans a. lot of waste empty space inside the core. Also at every end plate thereis flux peaking.

Radial Fuelling

So far we have discussed only axial shuffling. Tfe have now cone to" radialshuffling. The core is divided into a nuaber of zones which are moved at regularintervals from out to in. The discharge irradiation can be expressed as:

where m is the minbsr of zones and C i s son® constant. Out-to-in shuffling i spreferred to in-»out because i t gives a flatter flux distribution. In-»out givesa higher discharge bumup since i t takes the burntout fuel from regions of higherto lower statistical weight.

- 202 -

An alternative scheme is to divide the cor* into a number ofsuper-cells each containing a definite number of channels. The figure

a supercell of four. Five supercells have been indicated in the figure. At3o-ie time all the channels marked 1 are replaced by fresh fuel. After anotherinterval of tin*, all the channels 2 are replaced. Thus at any tine eachsupercell contains fuel in four different stages of burn-up. This nay becompared to a kind of channel equilibrium, since there we assune ttiat the reactorcontains fuel in all stages of irradiation and here the supercell contains fuelin at least four stages of irradiation.

The Roundelay scheme

In this the core is divided into two radial zones, -the outer oneforming 1// ,x «f the core. The inner Tons is divided into supercells of n

channels each, numbered 1,2, n. At regular intervals of time, the outerzone is removed and inserted into the inner zone. At ihe end of the firstinterval they are inserted inte the channels 1. After the second interval theyare inserted inte the channels 2, and so on^

Problems of Initial Fuelling

In all that has gone before, one point is worth noticing; the fuellingscheme works fine in the equilibrium case, but if we decide to fallow the sameBcheae starting froa the first day that the reactor has started working withfresh fuel, the first few days we will be discarding fuel which has been veryinefficiently utilized. To prevent this waste, some scheme of initial fuellingwhich finally merges into the equilibrium scheme is a normally followed.

Let us consider the CANDU. % denote

-n- » discharge burn-up in Mud/teU

S - specific power in Mw/teU

F - total power Hw•p

Aneunt of fuel in the reactor - "• le TJS

- 203 -

In the equilibrium state, the rate at tfiich fuel is fed Into the reactor is~ Te U/day.

Total number ef days required to replve a full charge •

no. of days needed »

P - 700 w (th)

Hate of fuelling

Altogether -there jo».3G6O bundles in the core; o* i« have t« replace about~ 3 o 6 ° - 6 bundles/dayS*oo

Let us plot a curve of total uranium fuelled into CANDD vs. days. This

does not include the start-of-life inventory. Once equilibrium has beenattained, the curve has to be a straidjt lins rtiose slope is -2_ tons/day.

8cDifferent initial fuelling schemes can give curves 2, 3, etc. Obviously

3 is the best scheme here. 7e have to try different initial fuelling schemesif ire want to find the «ne * i c h gives maximum saving. Such schemes have t6 b*

-204 -

worked out subjeot to two constraints. The reactor should not becomesubcritical at any time, nor should the power distribution become very bad.

Lecture 20

Last tiae we discussed the ini t ia l fuelling of CABDDand considered the scheme of uai:og the equilibrium rate offuelling from the first day of operation.

In this case we can easily see that in. the first charge, theaverage discharge irradiation will he only half of the equilibriumvalue. Since the whole fir3t charge is removed in 500 days, secan say that during half this tiae we are discarding good fuel. Toprevent this waste then, we have to start equilibrium fuelling freethe 250th-day. This will give the curve labelled as "ideal" in thefigure. The question is can this be achieved?

Since i t is very difficult to decide analytically what theideal scheme will be, we will assume that we start the equilibriafuelling at 250 days and see what happens. We first describe theequilibrium fuelling/

In equilibrium fuelling we have two discharge irradiations.In CA1G3U. i t is around 10,000 £nd/t in the inner zone and 8000 in theouter zone. The rates of fuel movement in different channels areadjusted to satisfy this. The presence of two zones gives soniepower flattening and the total power i s , let us say, P.

So let us say we load the core thus, operate i t for 250 daysand then start the fuelling shown by the 'ideal' carve in the figure.To know whether the curve is acceptable, we have to know what are thecoiatraints on i t . If it fails to meet the constraints at any point,(and it nest probably will) we have to change the form of the curve.The actual curve ifoich gives the saxinuni saving while renainingwithin the frame work of the constraints can be found only by detailedcomputations using fuel management codes. Hers we will merely enlistthe constraints that the system has to satisfy.

. . 2 2 3 . .

— 206—

Now let us think of starting the reactor, ffe find one ofthe constraints violated the very first day. He f i l l i t up withfresh fuel. This will lead to a higher form factor and consequentlyloner power. .Ye can flatten the power by putting sorne absorber inthe centre. This could be fertile thorium or depleted uranium.

Different channels produce different power; so if we use thesane coolant flow rate for all channels, the outlet teaperature indifferent channels will be different. The maximum outlet temperatureis fixed by the system pressure since the coolant should not boil .

( ) ( )y y

Thus ( A T ) m a x i s -ft-16*1- ( A T ) a v

w i l 1 d-eci e "tbe average outletteaperature and thus the thermodynamic efficiency. 3ecuaae of th is ,the coolant flow in different channels is so throttled as to give thesame AT in all channels. The best situation will be

This may be too stringent a coalition. In CAKDU the designed value is

( * T > a v a 0.92

( & max a i l 0 u^ i i n o t "^c* 0" 1 oone v a l u e . The designed va lue f o r

CAHDU i s around 80°? .

The moderator tempera ture i s a l s o t o remain w i t h i n some l i m i t s ;otherwise the materials in contact with i t will have stress problems.

Then Keff £ 1.0

The power produced in any bundle should not exceed somedesigned value P

PBundle 4 PBC

..224..

- 207 -

where ff is the number of rods in the cluster and 1 is the length ofthe bundle.

Bven though the real l i a i t ia on the 'bundle power, i t is difficultto isonitor the bundle power. The reactor operator can only check th?channel power and so a mairinua channel power is calculated from the axialflux distribution and the known bundle power l i a i t . Nowhere shc-ald thisl in i t be exceeded.

Pchannel ^ PCC

where £„- is the wa-yiTmri) value as designed.

'xv is denoted by A , and the reactor

is designed for soae value A_f which is 0.92 for CAIUHJ.

7/hen A = A , the reactor runs at full power

These are the constraints. It should be remembered that thereis saving in cost if fuel is saved, but there will be loss if powerdecreases. If saving-in fuelling cost warrants it, slight falls inthe powsr level may be tolerated.

He have to fuel the reactor whenever one of the constraintsis violated. In CAH1. fuelling is done whenever.

Keff < U 0

or Ar < Ac

During the in i t i a l fuelling, sometimes ramping is also done, whichis an arrangement to sinulate the equilibrium condition. But usuallywhenever the constraints are violated, one of the channels is refuelled.The channel selected may be the one which contains the most burnt bundle,or the one in which the temperature rise is nriniiEwn. Usually these twoconditions point to the same channel.

Reactor operation over a number of years is simulated us in- fuelmanagenEnt codes Ii3os STOKB or TRTVSNI. Some studies were made forC4HIW for two years of simulated operation. The f i rs t time a channelwas fuelled, i t -was ramped. About 1510 bundles are discharged per yearover the two years of in i i t ia l fuelling, as against 2250 at equilibrium.There is some loss of powsr due to burdle limit, leading to a loss of$ 25,000.

..225..

- 208 -

A couple of other schemes were also studied. One of thesedid not use depleted fuel to flafte-. the power in the beginning.This gave a loss of $ 600,000.

In these two schemes, the coolant flow distribution hadbeen calculated such that we should get uniform tesperature r i s ewith a power form factor of 0.8. Another schene -sas studied forwhich this value was 0.773. The loss accrued to 3 100,000.

These figures serve to demonstrate that the saving obtainedby a proper in i t ia l fuelling scheme is not insignificant.

Appendix

This is pexlodio in x sad y with a period, s, so we oan expand

i t in a Fourier serlBB.

Since f (x,y) i s an eren function of x and 7, there v i l l be

no sine texns

- ( f 1 ft-t

- -v \ 1 • — - r - 1J

» 210 -

To evaluate the integral with respect to y, w put y - na • y

We know that

and toy summing over n, the l imits of the integral beconfi — *>

to + • .

<R* \ e.

. .230 .

- 211 -

> • " - i X .

<R«. c A *

The Integral w.r.t . x i s exactly similar

= -L. 4-n-e e

Substituting th is , we get

-- 212 -

Befegencea

1. Huelear Reactor Optlmisatioie, by P.M. Margea, Huolear bgiaeer iagMonograph (19^0).

2 . "Interpretation of Crit ical Experimemta with Heterogeneous ReactorTheory," by S.M. Peinberg. The seooed HPY iatexnatitm ldTueedSusmer School on Heactor Physios, Kjeller Report SB - 117. (1966).

3 . "A Computer Program for Heactor Studies," by M.P. Dorst and B.Marriott. ABCL - 911 (1959).

4 . "Kataral Uraaiua Heavy la ter Lattices, Ebcperiaest and Theory,?by D.W. none at a l . ABCL - 622 (1958).

5. "The Physics of LATBEP" by I.H. Gibsoa ABCL-2548 (1966).

6. "Methods and Cross-Sections for calculating Fast Effect" byM.R. Fleishman and H. Soodak. HSE, 7, 217 - 227 (1960)

7. "A Generalised Method for calculating Fast Fiss ion Effect incoaxial cylindrical l a t t i c e ce l l s" by H. Markl and A.G. Fowler.Hukleonik 6, 39-51 0 9 & 0 .

8. "Comparison of theory and experiment for (a) l a t t i c e properties ofD»0 - U reactor (b) central rod experiments (c) foreign rodexperiments" by E. Critoph, ABCL - 350 (1956).

9. "Resonance absorption in £90 l a t t i c e reactors11 by V.L. Brooks and H.Soodak, HQA - 2131 - 19 (1961).

10. "RESP - A program for the calculation of resonance escape probabilityfor D-0 moderated l a t t i c e s , " by A.N. Nakra, Internal reoort of ABCL.CHHL-T32 (1968).

11. "Neutron Capture by long cylindrical bodies surrounded byPredominantly scattering Media," by S.A. Koahneriu* AB3L-462(1957).

12. "The Calculation of Thermal Uti l isation by a Coll ision ProbabilityMethod <««o«<Hwg scattering and sources in the amnnli surreuadiagthe Fuel," br CJ). Mckay, ABCL - 1474 (1961).

13. "Effeoti-ra Cross-section and Cadmium Ratios for Beutroa spectra ofThermal Beactora," by C.E. Vesteott, W.H. Walker, and T.K. Alexander,A/Coaf. 15/P/202, second Geaeva Conference V o l . 16 U958).

. . 2 1 4 . .

- 214 -

14. "Effective Cross-seotion fc tfell-aoderated A t m L Reactor Speotra,"by C.H. Westoott, AECL-1101 (.. '.

15. "General Formulation oi' ths Diffusion Coefficient in a aediuawhinh may contain saTities," by P. Benoist AEHE Trans 842 (1959).Original French version Eapport CEA No. 1354 (1959).

16. nLecture on experimental methods", by B.E. Green Sdited byB.F., Rastogi and K.H. SriniTasao.

17. "Determination of Radial Buckling in Reflected Systen," byK.J. Serdula, NSE, 26, 1-12 ( ^ 6 )

18. "Meaure de Laplaoiena par la Itethode de Reaplaeeiaent Pragressif,"by P. Bacher and H. Naudet. Journal of Nuclear Energy (Fart AtReader Soienoe) 13, 112-12? (1961).

19. "One Group Perturbation theory applied to substitution Measurementswith Void," by R. Persson, AB-72 1962).

20. "MICHOTE: 1 G 20 Prograa far Calculation of f inite latt ice bythe Hiorosoopio Diaorate Theory," by J.D. Stewart ABCL-2547(1966).

21. "Evaluation of Buckling and other Parameters from two or mUti-regionexperiments," by E.. . Person, The Second SPY Intexmatioaal AdvancedSunmer School in Reactor Physics, Kjeller Report KR-117 (1966).

22. ^Experience in the use of the Physical Constants Testing Reactor,"by H.E. Heiaeaan, Second Genera Conference, Yol. 12, Pages, 650-661,

23. "Measurement of Fast Fission Ratios In Natural Uraniaa," byC.B. Bigbam, AH3L-2285 ( ^ )

24. "A Method for the Accurate Determination of Relative In i t ia lConversion Ratios," by P.R. Turniicliffe e t a l . USE, 15, 268-285 O963).,

25. "Experimental Effective Fission Cross-Seetions aad Heutroa Spaatrain a Uranium Fuel Rod," by C.B. Bigham e t a l Part I .AECL-1350 (1?61). Part III ABGl-14 19 (1951).

26. "Fuel Burnup and ReaotiTl^r Changea," by X.F. Duret rt a l ,Proo. of Third Geneva Coafexenee, v o l . 3 , 347-356 (1964).

27. "Calculated Crsss-Seetlon of Irrtdiated f iss ion Produrts", byD.G. Burst, ASCI.-346 ( 1 6 )

28. "Cross-seotions and Yields of Psendo-fissloa Products", byB.G. Burst, ABCL-715 (1958).

. .215.

- 2 1 5 -

"Yields and Effective Crosa«Se±tioa of ?isaiom Fredoats artFaeudo-fiaBioa Fxoduota," by W.H. Walker, A2CL-1054, (i960).

nTh'"' Effect of a«w data on Beactor Faiaoning by Hon-aatnratimgFieaion Producta," by W.H. Wai)c<sr. ABCL-2111. (1964).

"Theoretical Aapeeta of Fuel Cyoj..'ng in Indue tr ia l Calder Halltype Reactors," by S.E. Lewia and U.S. Levthiaa, Second G«aeraConfereace, Vol. 13, 229-236 (1958).

"Fuel Management in SGESR'S," by F.F.O1 Dell et a l . Faper BO. 7of Conference on Steam Generating and other D,0 Seaetors argasdaedby British Bualear Energy Sooiaty (I96S).

"The Effect of Bidirectional Fnellisg oa Seaotirity," by K.P.ABCL-1235 (1961).

"Initial Foelling of Fewer Beaotora" by W.B. Lewia ABCI-696 (1958).

"A Coapute? atudy of Beaetar Poalling," ty WJI. Baxaa, 1BCL-1528(1962).

"The STOKE Program for Beaotor Fuelling," by WJI. Barea andP.M. Attree, ABCL-2O58 (1964).

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