CHE 3804 NUCLEAR ENGINEERING SECTION 4 REACTORTHEORY ... Library/20041706.pdf · CHE 3804 NUCLEAR...

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CHE 3804 NUCLEAR ENGINEERING

SECTION 4

REACTOR THEORY

ONE GROUP REACTOR EQUATION

Previously the equation for the diffusion of monoenergetic neutrons in a medium wasderived. This one group steady state diffUsion equation is given by:

The first term is related to the diffusion of neutrons in the medium while the second term onthe left hand side is related to the absorption ofneutrons in the medium. The term on theright hand side is t.'le source term related to the production ofneutrons.

In a critical reactor, where the number ofneutrons in each succeeding generation is constant,that is k is equal to unity, the number ofneutrons absorbed is equal to the nU1T,ber produced.

Here 11 is the number ofneutrons emitted per neutron absorbed in the fuel. A further factorf may be defined as the ratio of neutrons absorbed in the fuel to those absorbed in wholereactor.

This factor f is known as the fuel utilization factor. Substituting this into the above equationgives

For an infinite reactor with no surface from which neutrons can leak k.. is used to define thenumber ofneutrons in the current generation over the number ofneutrons in the previousgeneration. Thus k.. may be defined as follows:

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k. = 1] f (Eareactor ) / (E"eactor <1»

k.= 1]f

The source term may thus be redefmed in terms of k.

s = k. Ea reactor 

Substituting this into the one group diffusion equation gives:

D va<l> - Ea=ctor = - l~ Eareactor 

This may then be rearranged to give the following equation:

The diffusion length L may bc incorporated since:

Ea/D=U

Furthermore a new term B2 known a~ t.~c Buckling may be incorporated. Buckling is definedas follows:

Theone group steady state diffusion equation then reduces to the one group reactor equation

The first term is related to the diffusion ofneutrons while the second term is related to thcproduction and absorption ofneutrons.

REACTOR EQUATION APPLICATIONS

The one group reactor equation may be applied to a variety of different geometries.Applications requiring solutions in three dimensions are quite complex so a one dimensionalapplication only will be considered. Similar principles apply to all geometries.

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The Infinite Slab Reactor

For an infinite slab ofthickness a the one group reactor equation simplifies as follows:

V2</> + B2 </> = 0

d2</>/dx + B2 </> =0

Ifx is measured from the centre ofthe slab there is symmetry in the solution and boundaryconditions may be fixed as follows:

</> is zero at the surface</> is maximum at the centre

The solution to the above differential equation is:

</>x =A cos Bx

To satisfy the boundary conditions:

B= rt/a

</>x = A cos (rtx / a)

The buckling B2 is given by

The constant A may be solved by considering the total power produced in the reactor. Thisis obtained by multiplying the macroscopic fission cross section by the neutron flux,integrating over the whole reactor and applying a conversion factor for the amount ofenergy

produced in fission.

jP = Econversionfoetor ~I x dx

-.12

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The complete solution for the infinite slab reaction is t.1}us

<l>x = (n: P 12 a ~R ~r) cos (n:x / a)

<l>x = (1.57 P / a ~R ~r) ccs (1tX / a)

REACTOR EQUATION SOLUTIONS

The practical reactor shapes are the rcctangular parallelepiped, the finite cylinder and thesphere. By following mathematical procedures similar to that above, equations for theneutron flux at any point in the reactor may be obtained. The results are as follows:

Rectangular Reactor

 = (3.87 P / ER ~rV) cos (n:x / a) cos (1ty / b) cos (1tZ / c)

Cylindrical Reactor

 = (3.63 P / ER ~f V) Jo (2.405 r / R) cos (1tZ / H)

Spherical Reactor

 = (0.25 P / ER ~fR2) (l / r) sin (1tr / R)

In the above Jo is a Bessel Function which may be obtained from tables.

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NEUTRONFLUXANDPO~R

Using the above solutions of the reactor equation the relationship between maximum fluxand average flux for different reactor shapes may be obtained. This is important in thedesign of a reactor since the power produced is proportional to the neutron flux. Themaximum power and maximum flux at any point is limited by the fuel design while theaverage power and average flux of the reactor determines the overall power output.

Spherical Reactor

The maximum flux occurs in the centre oft.'le reactor that is when the radius is equal to zero.

<l>max = (P / 4 ER ~fR2) Limit ,-0 {sin (nr / R) / r}

The average flux is given by the flux in any elemental volume integrated over the wholevolume and divided by the total volume

<l>average = f dV / V

The total reactor power is obtained by multiplying the macroscopic fission cross section bythe neutron flux, integrating over the whole reactor and applying a conversion factor for theamount ofenergy produced in fission

If this is substituted into the above equation an expression for the average flux is obtained

<l>average = P / ER ~f V

The ratio ofmaximum flux to average flux may then be obtained

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=3.29

MAXIMUM TO AVERAGE FLUX

The ratios ofmaximum to average flux may be determi..,ed for different reactor shapes andare given below

Rectangular Reactor

<Pmax/ <Pave = 3.88

Spherical Reactor

<P01ax / <Pave = 3.29

Cylindrical ReactOl'

<P""", / <Pave = 3.64

ONE GROUP CRITICAL EQUATION

From the one group reactor equation already derived the buckling may be expressed in termsof the neutron multiplication factor Ie.. for an infinite reactor:

This is the condition for a reactor in the steady state or critical condition. The buckling B2depends upon the dimensions and geometry ofthe reactor while Ie.. and L depend upon thephysical properties ofthe reactor. Thus to create a critical condition in a reactor one or otherside of the equation must be adjusted to become equal to the other side. If the physicalproperties of the reactor are given then, to obtain a critical condition the dimensions of itmust be selected appropriately. Alternatively if the reactor geometry is known then thephysical properties need to be adjusted to obtain criticality.

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When considering a reactor ofgiven geometry and hence finite dimensions there is inevitablya certain leakage of neutrons from its surface. The degree of leakage or absorption may bedetermined by comparing the number of neutrons leaking from the system with the numberof neutrons absorbed in the system. The number leaking from the system is the flow ofneutrons across an elemental surface integrated over the whole surface.

Number leaking = fA J n d A

= fvdiv J dV

This may be transformed using the one group reactor equation.

Number leaking = DJv B2 dV

Thc number absorbed in the system is given by the macroscopic absorption cross-section andthe tlux integrated over the whole volume.

Number absorbed = ~a Jv dV

The probability ofabsorption or non-leakage PL is the ratio ofthe number absorbed over thetotal number absorbed and leaking.

The one group reactor equation may be rewritten to isolate the neutron multiplication factork..:

If this is substituted into the equation for the probability ofnon-leakage then:

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In a critical system the number ofneutrons absorbed is:

Number absorbed = :E. Jv 4> dV

This leads to the number ofneutrons released:

Number released = T\ f:E. Jv <!> dV

Of this number a certain proportion are lost by leakage. The number remaining on notleaking from the system is given by:

Number not leaking: PL k., :E Jv dV

This is the number in the next generation of neutrons. From tht: definition of the neutronmultiplication factor k the following is obtained

k = PL k., :E. Jv 4> dV / :E. Jv 4> dV

k=k.,PL

Here k is the neutron multiplication factor for a finite reactor and k., the neutronmultiplication factor for an infinite reactor. PL is the probability on non-leakage through thesurface of the reactor.

THE FOUR FACTOR FORMULA

So far consideration has been given only to mono-energetic neutrons presumed to havethermalised immediately after having been produced. On this assumption the neutronmultiplication factor k is given by two factors namely the number of neutrons emitted perneutron absorbed T\ and the thermal utilization factor f

T\ = v :Er fuel / :E. wei

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f = ~. fuel / ~. «.cw

In a reactor the neutrons produced by fission at high energy must be moderated to reducetheir energies. If the moderation occurs entirely within the moderator the above factors givea satisfactory result. In reality however some high energy neutrons cause fission in U-238before entering the moderator. This creates a few additional neutrons not previouslyanticipated. Furthem,ore some neutrons may interact with the fuel while still in theintermediate or epithermal energy range and be captured by the U-238 resonances. Twoadditional factors are therefore required. The fast fission factor E takes account of theadditional neutrons produced by the fissioning ofU-238 by fast neutrons. This factor isslightly greater than unity. The resonance t'scape probability p is the probability ofnot beingabsorbed in the resonances ofU-238 while slowing down to thermal conditions. This factoris less than unity since some are always absorbed. These four factors may be summarisedas follows:

1<.. =pET)f

l' = Thermal utilization factorT) = Neutron~ emitted per neutron absorbedE = Fast fission factorp = Resonance escape probability

THE SIX FACTOR FORMULA

The four factor formula applies to an infinite reactor with no free surface from whichneutrons may be lost. In a real reactor leakage takes place such that k is less than 1<.. by thenon-leakage probability PL

It is convenient to divide the non-leakage factor into two separate factors to take accountof fast neutron leakage and thermal neutron leakage. Thus two new factors namely thefast neutron non-leakage probability I\r and the thermal neutron non-leakage probability1\, are obtained

k = 1<.. /\r /\,

This is the six factor formula for neutron multiplication.

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NEUTRON CYCLE

The four or six factor fonnula may be applied to the neutron cycle. In each phase theappropriate factor may be applied and the total numbe! ofneutrons tracked through the cycle.In the algebraic example given the cycle starts with N thennal neutrons absorbed and endswith N e p T] f /\f 1\ neutrons available to be absorbed in the next generation giving amultiplication of e p T] f /\f /\t. From the defInition of the neutron multiplication factor thisis equal to k. Ir. the numerical example given the cycle starts with 1000 fast neutrons beingproduced from fIssion and ends with 1000 being produced in the next generation. Note thatthe two examples start at different points in the cycle but this is of little consequence.

REACTOR SHAPES

The non-leakage probability PL varies with shape and size of the reactor. This is due to h'lediffering surface to volume ratio in different shapes of the same volume and a changingsurface to volume ratio in different sizes of the same shape. For the most common shapesa rectangular block has a greater surface to volume ratio than a cylinder and a cylinder agreater surface to volume ratio than a sphere. Any particular shape will have a decreasingsurface to volume ratio as the size or volume increascs since the surface is proportional tothe square of the linear dimension and the volume proportional to the cube of the samedimension. The smaller the surface to volume ratio the smaller the value ofthe non-leakageprobability PL'

NEUTRON LEAKAGE

At the boundary of a reactor neutrons leak from the surface such that the neutronmultiplication factor k for an actual reactor is less than the neutron multiplication factor k..for an infInite reactor. This loss ofneutron'i is detrimental to the overall perfonnance ofthereactor since they have come from the fIssioning of fuel. Since unnecessary consumptionof fuel is undesirable it is advantageous to try to reduce this leakage.

One way of reducing the leakage is ofcourse to minimise the surface area of the reactor byselecting an appropriate geometry for the reactor. Further savings can be achieved byinstalling a reflector around the core. This does not actually reflect the neutrons as a mirrorwould reflect light Rather it allows neutrons to diffuse far enough into the reflectingmedium so that there is a signifIcant probability that some will diffuse back into the core.

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REACTOR REFLECTORS

The properties of a good reflector are the same as those of a good moderator. A reflectormust have a high neutron scattering cross-section and a low neutron absorption cross-section.Obviously neutrons should not be absorbed in the reflector while a high probability ofscattering ensures that the neutrons are soon deflected from their original path. After anumber of random scattering collisions the neutrons could be travelling in any directionincluding back towards the reactor core. In a flat reflector of adequate thickness and withperfect neutron properties mounted on the side ofa reactor having a plane surface halfoftheneutrons would leave the reflector on the inside to return to the reactor and half would leaveon the outside to be lost forever. More loss would occur at a comer or edge so that areflector can never be as good as 50% efficient. Nevertheless this represents a substantialsaving in neutrons and nearly all reactors make use ofa reflector. The reflector is invariablyan extension ofthe moderator a short distance beyond the outermost fuel elements.

NEUTRON FLUX

The neutron flux at the boundary of a bare reactor (no reflector) falls to a low value at thesurface. It is assumed to fall to zero a short distance beyond the surface at the extrapolationdistance to permit the application of diffusion theory. This compensates for the surfaceeffects and allows the neutron flux to be predicted using the diffusion equation.

With a reflected reactor neutrons diffuse through the reflector and some are returned to thereactor core. This has the effect ofenhancing the neutron flux near the surface ofthe reactorcore. Neutrons that would otherwise be lost are returned and added to the existing neutronfluxjust inside the boundary. Furthermore the neutron flux in the reflector may actually begreater than that in the core near the boundary since there is no fuel to absorb neutrons andthe neutrons tend to accumulate while diffusing in the reflector. The neutron flux thusdecreases with distance from the centre of the reactor but this decrease levels off and mayreverse near the boundary. In the reflector the neutron flux actually rises with increasingdistance from the centre but then falls to a very low value near the outer surface of thereflector.

FLUX FLATTENING

A reflector on a reactor has a further advantage besides improving the neutron utilization inthe reactor. It increases the <Paverage / <Pmaximum ratio in the reactor. Since power is proportionalto neutron flux this means that more power is obtained from the reactor without increasing

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the maximum flux. Thus the highest rated fuel elements can still operate within the designpower, and hence temperature, limitations. Flux flattening is highly desirable in all powerreactors and other methods are also employed to obtain the most uniform neutron flux acrossthe reactor as is possible.

FLUX FLATTENING IN CANDU REACTORS

CANDU reactors employ four methods of flux flattening to increase the ratio of maximl;m'

Reflector

A reflector returns some neutrons to the reactor core so enhancing the neutron flux near thereactor surface. In a CANDU reactor this is done around the cylindrical core and thus hasan effect in a radial direction only. Refuelling is done from the ends ofthe reactor so thereis no space for a reflector ofadequate thickness.

Adjuster Rods

Adjuster rods which absorb neutrons weakly are inserted into the centre of the reactor toreduce the neutron flu.x in that region. Although inserted from the top they can be designedto be most effective in the very centre ofthe reactor core. The effect is in all directions bothaxial and radial. Once the neutron flux in the centre has been suppressed the overall reactorpower can be increased to bring the maximum neutron flux back up to its maximunlallowable value. The average neutron flux is then higher than prior to the insertion of theadjuster rods since the flux profile has been changed into a flatter shape.

Bi-directional FueUing

In a CANDU reactor the horizontal fuel channels each contain twelve fuel bundles. If onlyeight are removed by inserting eight new ones from the other end the effect is to skew theneutron flux towards the end having the fresh fuel. If the adjacent fuel channel is refuelledin the same way but from the opposite side the neutron flu.x will be skewed the other way.The net effect is a sUlllIDation of the two flux profiles giving a flux profile which issomewhat flatter than the cosine shape that would occur ifthe whole ofeach fuel channel hadbeen fuelled with new fuel. This effect naturally only occurs in the axial direction.

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Differential Fuel Burnup

By dividing the reactor core fuel channels into an inner zone and an outer zone and allowingthe fuel in the inner zone to bum up to a greater degree than that in the outer zone the neutronflux near the centre of the reactor can be suppressed. This allows the fuel chaImels in theouter zone to operate at higher neutron fluxes than would otherwise be possible. The netresult is a flatter neutron flux profile across the reactor in a radial direction.

The overali effect in CANDU reactors is that the average neutron flux is about 60% of themaximum neutron flux. The ratio of<!>.vaage! «I>maximum is thus 0.60. Note that for a cylindricalreactor with no flux flattening the value of «I>average i 4l..ximum would be 0.27. This is theinverse of «I>maximum / «I>.verage which is 3.64. This means that a CANDU reactor can producejust over twice the amount of power as a simple cylindrical reactor of the same size anddesign but without flux flattening devices. There is thus a great economic benefit derivedfrom neutron flux flattenL'lg.

EFFECT OF FUEL RODS

The effect of fuel rods on the neutroll flux is to suppress the thermal flux since thermalneutrons are absorbed 1be thermal neutron flux profile thus varies between the fuel and themoderator with a series ofdips wherever there is a fuel rod and a series of peaks whereverthere is moderator. The inverse occurs with the fast neutron flux. Fast neutrons are createdin the fuel so that fast neutron flux peaks in the fuel. In the moderator the fast neutrons arethermalised and the fast neutron flux dips while thermal neutron flux builds up.

EFFECT OF CONTROL RODS

The effect ofcontrol rods is to suppress the neutron flux since neutrons are strongly absorbedby them. This can have the effect ofenhancing the surrounding neutron flux if adjustmentsare made to the overall reactor power. For this reason it is undesirable to have only a fewcontrol rods. Most reactors have a large number ofcontrol rods so that the overall effect onthe neutron flux is more uniform across the reactor. The PWR in fact has a cluster ofcontrolrods which are inserted into each fuel assembly to distribute the effect as smoothly aspossible across the reactor.

CHEMICAL SHIM

In water reactors it is convenient to add a neutron absorber to the moderator or coolant to

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absorb neutrons unifonnly. The reactor may then operate generally with the control rods intheir optimum position so as to minimise neutron flux distortions. Soluble neutron absorbersor reactor poisons, such as boric aid, lose their effectiveness as they absorb neutrons. Theburnup of the reactor poison is compensated by the burnup of reactor fuel. As the fuel isdepleted less soluble poison is required but it itself is becoming,depleted so the two effectscounterbalance one another. Slight changes in poison concentration are made continuouslyby the reactor chemical treatment facility hence the term chemical shim when reference ismade to reactivity control using soluble neutron absorbers. This method is widely used onPWR's which are refuelled once a year. CANDU reactors which utilise on-line refuellingdo not need such treatment except after their first commissioning with a complete charge ofnew fuel.

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- "

c

°c;OQ....._0:I'"eu.l:lz<illS

... .c01:.c.-'lijCl:lc.el

r.!I ..~

6

'"1..<9.. ....c Q-." - 1;V I; '" /' ~

j~ <II =S Cl:: ...... .."

, a:• ~ •

;I, z: <II

T---- 1 v !l<II.§ li

.,J; ~e

j Cl::<3

0 S8 .5 l! • ~I E

'.,g z: c i' . § • Qc_ ;:

to'-~ l! .5'e =i .. Ee is II JQ

~~ ~ ..:~ ~ '" - ,

'".. '" .. is ",s = i• Ii "'," ...

~§• ~,

~ = , 'i."ll ii: a:.. ; z: .• ltng uonn;ilN

~ Ii::J r..-!Z .q 0;~

":i'...... 10:<9

Effect of Reflectcr on Shapeof Radial Flux

Flux Flattening Producedby Adjuster Rods

Rux without-witll.........

wn

100 200 300 _ 500 600 700 600

Distance along chord AB (em)

Fluxwitll

I "'"\-; II ;;-:djustoIsor \;.. inse_t

/ /I I I "- \ Flux'

" "' -.) I ,........: 1\\-II I I \

( )"mB

Tt End \\o

'.0

0.2

1.2

~ o.e

1 0-6

J 0.'

..30() -200 ·100 0 100 200 300

RetIector Reflector

llistanca along radialdIracIian (an)

FIux_reftOdOr

(ot same power)

.... u F1g.U

Effect of Bi-Directional Refuelling inFlattening Axial Flux Shape

2 3 .. 5 , 7 8 8 10 11 12

Theoretical flux/ shape with u."iform fuelling

Bundle posltlons slonll channels....

..

.

>-~.Q .--c=II II~.E:= II

ogClll:0.. IIIc.. c~.2-tiII IIE.!::~!!<II c<;;;,

.... ..,

Flux Flattening Produced byDifferential Fuelling

flux Flattening

III

Radial only

Rscllelend wei

.-------- Wlti10utftattening

----- WIIh flaItenIngWith nattening and power increaSe

.\xle' only

........I••

/', ••.,l ~~ ••,. . . •.. . . •• •

"\ •I • • •l • • I• •: . I ••

Radlel

Bt-directionaJIuolling

Inner zone

Aodlal pos!lJon

_lace

....u OMU

_...

.."E"><

~.,E0":l

5"~!!.,...

II, ,

Flux Flattening in CANDU Reactors

RofllCl Bl-dl....uonal Ad) "'''''' 01110..,.1101 .""9or fuelling U bumup +....

NPD ""Iol & rldlal x 42%

x 50%

Pk:ko~"l!'A radlol •• 57%

l&.f.'!!!iI.~8!1.'ft~.::,~&"ll • - -60%

Br,JC8 • A radlol. • -59%

~. , .:f .....'·~..~" .. "·, a~

•.".'fc',' -A""'lI'¥"'" -UU~'- ..... -

Do~I"ljton <adlol • • • -60%

ltr1l.....i..·"'~':,;;:I~·"·'~"• .' -60%WW'IlU'..Ii::D":J!I~",;;;~·~~.J·:.•..' ....

lSl

ZIIIC.,o:l.."C><5"/"')o.,III

~ ~!l

(-+ (:3I

0,/

i',tUi

. ! III

~ LI _

.., .. till"

Depression of the Thermal Neutron Flux in theInterior of Fuel Bundle

1.9

I.eRelativeRadiel 1.3Aux

1.0

0.7

CHE 3804 NUCLEAR ENGINEERING SECTION 4 REACTORTHEORY ... Library/20041706.pdf · CHE 3804 NUCLEAR...

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