2. Nuclear Reactor Theory

Nuclear Reactor Theory

J. Frybort, L. Heraltova

Department of Nuclear Reactors

June 13, 2011

Chapter 5

J. Frybort, L. Heraltova (CTU in Prague) Nuclear Reactor Theory June 13, 2011 1 / 52

Content

1 One-Group Reactor Equation2 Basic Reactor Geometries

Slab ReactorSphere ReactorInfinite Cylinder ReactorFinite Cylinder Reactor

3 Maximum to Average Flux and Power4 Critical Equations

One-Group Critical Equation5 Thermal Reactors

Criticality Calculation6 Reflected Reactors

One-Group MethodTwo-Group Method


Introduction

In a critical nuclear reactors, there is balance between number ofproduced and lost neutrons

neutrons are produced by fissionneutrons are lost by either absorption or escape

Central problem in nuclear reactor designing is determination ofits dimensions and compositionTechniques for calculation of size and material composition of ahomogeneous nuclear reactor are introduced hereCriticality calculations are usually carried out by a group diffusionmethodThe first part will deal with one-group method suitable for fastreactors and to some extend even for thermal systems


One-Group Reactor Equation

Fast Homogeneous Reactor

Fast critical reactor with homogeneous mixture of fuel and coolantis consideredThe reactor has only one region, no reflector, it is so-called barereactorThe reactor is described by one-group diffusion equation:

D∇2Φ´ ΣaΦ` s “1vBΦ

Bt(5-1)

This equation is time-dependent and power of the reactor mightincrease or decrease



Fission Source of Neutrons

Fission neutrons are the source of neutrons (s) in a nuclearreactorIf Σf is fission cross-section of the fuel and ν number of neutronsemitted per one fission, source s can be expressed as:

s “ νΣfΦ

If fission source does not balance neutron absorption andleakage, then right-hand side of equation (5-1) is nonzeroParameter k can be used to adjust the source strength and toreach a steady state diffusion equation:

D∇2Φ´ ΣaΦ`1kνΣfΦ “ 0 (5-2)




Quantity geometric buckling (B2) is defined as:

B2 “1D

´ν

kΣf ´ Σa

¯

(5-3)

Then previous equation (5-2) can be rewritten in form ofone-group reactor equation:

∇2Φ` B2Φ “ 0 (5-4)

The formula for buckling (5-3) can be solved for the constant k:

k “νΣf

DB2 ` Σa(5-5)



Multiplication Factor

k “νΣf

DB2 ` Σa“

νΣfΦ

DB2Φ` ΣaΦ“

νΣfΦ

´D∇2Φ` ΣaΦ

Physical interpretation of the previous equation is following:numerator is the number of neutrons born in fission in the currentgenerationdenominator represents neutrons lost from the previous generationsince all neutrons must be absorbed or leak from the reactor, thedenominator must be also equal to the number of neutrons born inthe previous generation

This is definition of multiplication factor for a finite reactorIt can be also defined as a neutron birth rate divided by sum ofneutron absorption and leakage rate



Multiplication Factor for Infinite Reactor

The neutron source term can be rewritten with neutron absorption.Let ΣaF be cross-section for neutron absorption in fuel, then:

s “ ηΣaFΦ

Quantity η is called neutron reproduction factor and meansnumber of produced neutrons per a single neutron absorbed inthe fuel

It can be further adjusted to:

s “ ηΣaF

ΣaΣaΦ “ ηfΣaΦ , where f is

ΣaF

Σa

Quantity f is called fuel utilisation factor and means a fraction ofneutrons absorbed in the fuel from all neutrons absorbed in thereactor



Multiplication Factor for Infinite Reactor (cont’d)

There is no escape of neutrons in an infinite nuclear reactor, allneutrons are either absorbed in fuel or coolantIn one generation, a certain number of neutrons is born related toν, all these neutrons must be absorbed expressed as ΣaΦ

Of these neutrons fΣaΦ are absorbed in fuel and this leads toproduction of ηfΣaΦ neutronsAll these neutrons must be again absorbedIt means that absorption of ΣaΦ neutrons in one generation leadsto absorption of ηfΣaΦ neutrons in the following generation

Absorption of neutrons is directly related to production ofneutrons, therefore multiplication factor in an infinite reactor isdefined as:

k8 “ηfΣaΦ

ΣaΦ“ ηf (5-6)



Buckling for Critical Reactor

Value of η and f depend only on material composition and k8 istherefore identical for all infinite bare reactors with the samematerial compositionThe neutron source term can be written as:

s “ k8ΣaΦ

And one-group reactor equation (5-2) can be transformed to:

DB2Φ` ΣaΦ´k8k

ΣaΦ “ 0

If the reactor is just critical (k = 1) the right-hand side is zero and:

DB2Φ´ pk8 ´ 1qΣaΦ “ 0



Buckling for Critical Reactor (cont’d)

Dividing by D and introducing one-group diffusion area L2 “ D{Σaleads to:

B2Φ´k8 ´ 1

L2 Φ “ 0

The above equation can be solved for geometric buckling factor ofa critical reactor

B2 “k8 ´ 1

L2

It will be further shown that buckling factor determines shape ofneutron flux and sets a condition for a reactor to be critical


Basic Reactor Geometries Slab Reactor

Slab Reactor

First example is a critical infinite slab reactor with thickness a. Thereactor equation (5-4) is:

B2Φ

Bx2 ` B2Φ “ 0 (5-7)

The neutron flux within the reactor will be determined usingboundary conditionThe neutron flux vanishes on extrapolated surface ra “ a` 2dThe boundary condition becomes:

Φ

ˆ

ra2

˙

“ Φ

ˆ

´ra2

˙

“ 0

It is also obvious that because of the problem symmetry(Φpxq “ Φp´xq), there will be maximum neutron flux density andno net flow in the reactor centre:

dΦdx

“ 0 , for x “ 0J. Frybort, L. Heraltova (CTU in Prague) Nuclear Reactor Theory June 13, 2011 12 / 52


Slab Reactor (cont’d)

General solution to equation (5-7) is:

Φpxq “ A cospBxq ` C sinpBxq (5-8)

Constant A and C are to be determined by boundary conditionsPlacing the derivative of (5-8) equal zero at x = 0 givesimmediately C = 0The general solution reduces to:

Φpxq “ A cospBxq (5-9)

Introducing the boundary condition gives:

Φ

ˆ

ra2

˙

“ A cosˆ

Bra2

˙

“ 0



Slab Reactor (cont’d2)

Solution to the above equation is either trivial A = 0, or mustsatisfy equation:

cosˆ

Bra2

˙

“ 0

There are Bn solutions:

Bn “n πra

, n is odd integer

The various Bn constants are known as eigenvaluesIt can be shown that only the first eigenfunction is solution ofneutron flux in a steady state critical reactorFunction describing neutron flux in a bare critical slab reactor is:

Φpxq “ A cos B1x “ A cos´πxra

¯



Reactor Buckling

The square of the lowest eigenvalue B21 is called reactor buckling

It is found by solving the equation

d2Φ

dx2 ` B21Φ “ 0

B21 “ ´

1Φ

d2Φ

dx2

The right-hand side is proportional to the curvature of the neutronflux in the reactorSince in the slab reactor:

B21 “

´π

ra

¯2

Buckling decreases as ra is increasingIn the limit, B2

1 is approaching zero and flux is constant and doesnot buckle



Neutron Flux Magnitude

The constant A has not been determined yet. It is related to thereactor power and not material compositionReactor power is determined as P “ VEfΣfΦAV

With recoverable energy from fission 200 MeV (3.2ˆ10´11 J), thetotal power of the slab reactor can be calculated as:

P “ EfΣf

ż a{2

´a{2Φpxq dx

The integration is carried out for physical dimensions of the reactorInserting the previously calculated function for Φ (5-9) andperforming the integration gives:

P “2raEfΣf A sin

´

πa2ra

¯

π



Neutron Flux in a Bare Slab Reactor

The final formula for neutron flux in a bare slab reactor is:

Φpxq “πP

2raEfΣf sin´

πa2ra

¯ cos´πxra

¯

(5-10)

If d is small compared to physical dimensions of the reactor, theabove formula reduces to:

Φpxq “πP

2aEfΣfcos

´πxa

¯

(5-11)


Basic Reactor Geometries Sphere Reactor

Sphere Reactor

The spherical reactor has radius R and neutron flux inside thereactor depends only on distance rReactor equation (5-4) in spherical coordinates is:

d2Φ

dr2 `2Φ

dΦdr` B2Φ “ 0 (5-12)

The neutron flux must satisfy boundary condition ΦprRq “ 0By substituting Φ “ u{r into (5-12) and solving the resultingequation for u leads to a general solution for neutron flux:

Φprq “ Asin pBrq

r` C

cos pBrqr

, where A and C are constants

The second term becomes infinite when r goes to zero, thus Cmust be zero and resulting equation is:

Φprq “ Asin pBrq

rJ. Frybort, L. Heraltova (CTU in Prague) Nuclear Reactor Theory June 13, 2011 18 / 52


Sphere Reactor (cont’d2)

Boundary condition is satisfied if B is one of the eigenvalues

sinpBrRq “ 0 ùñ Bn “nπrR

, where n is any integer

Only the first eigenvalue is relevant for a critical reactor, thusbuckling is

B21 “

ˆ

π

rR

˙2

Flux becomes:

Φprq “ Asinpπr{rRq

r(5-13)



Sphere Reactor (cont’d3)

Constant A is determined by the reactor power

P “ EfΣf

ż

VΦprq dV

Volume element is calculated as dV “ 4πr2dr and previousequation changes to:

P “ 4πEfΣf

ż R

0r2Φprq dr

Introducing neutron flux (5-13) and integration leads to:

P “ 4πEfΣf ArRπ

«

rRπ

sinˆ

πRrR

˙

´ R cosˆ

πRrR

˙

ff

If d is small, neutron flux can be written in form:

Φprq “P

4EfΣf R2sinpπr{Rq

r(5-14)


Basic Reactor Geometries Infinite Cylinder Reactor

Infinite Cylinder Reactor

Consider a critical infinite bare cylinder with radius RIn such a system, neutron flux depends only on the distance rfrom the cylinder axisThe reactor equation (5-4) in cylindrical coordinates is:

d2Φ

dr2 `1r

dΦdr` B2Φ “ 0 (5-15)

Neutron flux must satisfy not only this equation, but also theboundary condition ΦprRq “ 0The equation (5-15) is an ordinary Bessel equation of the orderzero. Its general solution is in form of the zero order ordinaryBessel functions of the first (J0) and second (Y0) kind:

Φprq “ AJ0pBrq ` CY0pBrq

Function Y0 is infinite at r = 0, thus constant C must equal zero



Infinite Cylinder Reactor (cont’d)

Neutron flux reduces to:

Φprq “ AJ0pBrq

Based on function J0pxq shape, boundary conditionΦprRq “ AJ0pBrRq “ 0 is satisfied at several values of xn

It means that the boundary condition is satisfied if B is one thevalues:

Bn “xn

rRAgain, only the first eigenvalue is valid for a critical reactor andbuckling must equal:

B21 “

ˆ

x1rR

˙2

“

ˆ

2.405rR

˙2

(5-16)



Infinite Cylinder Reactor (cont’d2)

One-group flux in an infinite bare reactor is

Φprq “ AJ0

ˆ

2.405rrR

˙

(5-17)

The constant A is determined from the reactor powerVolume element in cylindrical reactor is dV “ 2πrdr , thus powerper unit length of the reactor is:

P “ 2πEfΣf

ż R

0rΦprq dr “ 2πEfΣf

ż R

0rJ0

ˆ

2.405rrR

˙

dr (5-18)



Infinite Cylinder Reactor (cont’d3)

The equation (5-18) can be evaluated using formulaş

J0px 1qx 1 dx 1 “ xJ1pxq, which gives for small d

P “ 2πEfΣf R2AJ1p2.405q

2.405“ 1.35EfΣf R2A

Final expression for the neutron flux is:

Φprq “0.738PEfΣf R2 J0

ˆ

2.405rR

˙

(5-19)


Basic Reactor Geometries Finite Cylinder Reactor

Finite Cylinder Reactor

Finite cylinder reactor with radius R and height H is close to a realreactor systemIn this reactor, the neutron flux depends on the distance r from theaxis and the distance z from the midpoint of the cylinderThe reactor equation (5-4) is following:

B2Φpr , zqBr2 `

1rBΦpr , zqBr

`B2Φpr , zqBz2 ` B2Φpr , zq “ 0

The solution must satisfy boundary conditions ΦprR, zq “ 0 andΦpr , rH{2q “ 0The solution is obtained by assuming separation of variablesΦpr , zq “ RprqZ pzq


Basic Reactor Geometries Finite Cylinder Reactor

Finite Cylinder Reactor(cont’d)

Separation of variables leads to two equations, which can besolved independently:

d2Rdr2 `

1r

dRdr` B2

r R “ 0

d2Zdz2 ` B2

z Z “ 0

The buckling B2 “ B2r ` B2

zThe final solution and application of boundary conditions is similarto the previously solved slab and infinite cylinder reactors:

Φpr , zq “ AJ0

ˆ

2.405rrR

˙

cosˆ

πzrH

˙

(5-20)

rH “ H ` 2d and rR “ R ` d are extrapolated boundariesConstant A can be determined from the reactor power


Maximum to Average Flux and Power

Maximum to Average Flux and Power

Maximum neutron flux (reactor power) is found in the centre of thereactorIt is useful to known the ratio of the average neutron power to themaximum valueFor example, in a spherical nuclear reactor the maximum flux is:

Φmax “P

4EfΣf R2 limrÑ0

sinpπr{Rqr

“πP

4EfΣf R3

Average flux is calculated as:

Φav “P

EfΣf VDividing the previous equations gives power peaking factor K for asphere reactor

K “Φmax

Φav“π2

3“ 3.29

Power distribution is kept uniform in the reactor coreJ. Frybort, L. Heraltova (CTU in Prague) Nuclear Reactor Theory June 13, 2011 27 / 52

Critical Equations

Critical Equations

It was shown previously that it is necessary for a reactor to becritical that B2 must be equal to the first eigenvalue B2

1

Critical equations will be determined for one-group and two-groupreactor equations


Critical Equations One-Group Critical Equation

One-Group Critical Equation

Critical reactor follows equation (5-5), which solved for B2 gives:

B2 “νΣf ´ Σa

D, or the critical buckling must be:

B2c “

νΣfΣa´ 1

DΣa

The right-hand side must equal to the first eigenvalue B21

depending only on dimensions and geometry of the reactorUsing definition of k8 and L2 it can be written:

k8 ´ 1L2 “ B2

1

It can be interpreted that in order to reach critical state, physicaldimensions and geometry must by appropriate to fuel composition



Neutron Leakage

The previous equation can be rearranged in the following form(from this moment B2 refers to buckling of a critical reactor):

k81` B2L2 “ 1 (5-21)

This form is usually known as one-group critical equation

Rate of neutron leakage from volume V through area A is:ż

AJn dA “

ż

VdivJ dV “ ´D

ż

V∇2Φ dV

From reactor equation (5-4), this can be rewritten as:

´Dż

V∇2Φ dV “ DB2

ż

VΦ dV



Neutron Leakage (cont’d)

Neutrons can either escape from the reactor or be absorbedinside with no other alternativeProbability of neutron non-leakage (PNL) can be calculated as:

PNL “Σa

ş

V Φ dVΣa

ş

V Φ dV ` DB2ş

V Φ dV“

Σa

Σa ` DB2 “1

1` B2L2

(5-22)From comparing previous equations (5-21) and (5-22) it can beconcluded that critical equation can be rewritten in form:

k8PNL “ 1 (5-23)

This result has following interpretation



Neutron Leakage (cont’d2)

Total Σaş

V Φ dV neutrons are absorbed in the reactor everysecond and this leads to release of fission neutrons:

ηfΣa

ż

VΦ dV “ k8Σa

ż

VΦ dV

Due to leakage, only PNLk8Σaş

V Φ dV initiate new generation ofneutronsFrom definition of multiplication factor follows that:

k “PNLk8

ş

V ΣaΦ dVΣa

ş

V Φ dV“ k8PNL “ ηfPNL (5-24)

Thus, the left-hand side of the critical equation (5-21) is actuallythe multiplication factor for the reactorThe critical equation follows by placing k = 1


Thermal Reactors

Thermal Reactors

Most of the currently operated nuclear reactors are thermalreactorsThermal reactors consist of fuel, constructional materials,moderator and coolantOnly fuel and moderator will be distinguished in the currentanalysisAll materials apart from fuel are considered as moderatorPurpose of moderator is to slow-down neutrons to reach thermalenergy range


Thermal Reactors Criticality Calculation

Thermal Reactors Difference

Previous criticality calculations were performed for one neutrongroup – suitable for fast reactorsOne-group method is not sufficient for thermal reactorcalculations, because neutrons can diffuse for a considerabledistance while slowing downAt least two neutron groups must be considered:

fast group for neutrons released during fissionthermal group for neutrons at energies with high probabilities toinitiate fission

It can be assumed that there is no absorption in the fast group andin this group neutrons are only lost as a result of slowing-downinto the thermal group



Two-Group Critical Calculation

Σ1Φ1 neutrons are scattered out of the fast group per cm3/sec,where Φ1 is fast neutron fluxFission is initiated mostly by thermal neutrons, the few fissions byfast neutrons are taken into account by fast fission factorIt follows that ηT fεsΣaΦT “ pk8{pqsΣaΦT fission neutrons areemitted per cm3/secThese neutrons appear in the fast group as a source of neutronsFormulation of reactor equations in two energy groups arefollowing:

D1∇2Φ1 ´ Σ1Φ1 `k8p

sΣaΦT “ 0, fast group (5-25)

sD∇2ΦT ´ sΣaΦT ` pΣ1Φ1 “ 0, thermal group (5-26)



Two-Group Critical Calculation (cont’d)

Both thermal and fast neutron flux follow the same spatialdistribution and can be written as:

Φ1 “ A1Φ

ΦT “ A2Φ

A1 and A2 are constants and Φ satisfies the equation:

∇2Φ` B2Φ “ 0

Substituting the last three equation into (5-25) and (5-26) yields:

´pD1B2 ` Σ1qA1 `k8p

sΣaA2 “ 0 (5-27)

pΣ1A1 ´ psDB2 ` sΣaqA2 “ 0 (5-28)



Two-Group Critical Calculation (cont’d2)

According to Cramer’s rule equations (5-27) and (5-28) havenon-trivial solution if system determinant equals zero:

ˇ

ˇ

ˇ

ˇ

ˇ

´pD1B2 ` Σ1qk8

psΣa

pΣ1 ´psDB2 ` sΣaq

ˇ

ˇ

ˇ

ˇ

ˇ

“ 0

Multiplying out the determinant gives:

k8sΣaΣ1 ´ psDB2 ` sΣaqpD1B2 ` Σ1q “ 0

Rearranging and dividing by Σ1sΣa finally yields:

Two-group critical equation for a bare homogeneous reactor

k8p1` B2L2

T qp1` B2τT q“ 0 (5-29)



Two-Group Critical Equation

In the previous equation (5-29) were used L2T – thermal diffusion

area and neutron age – τT defined as:

L2T “

sDsΣa

and τT “D1

Σ1

The two-group critical equation (5-29) contains probability thatthermal neutron will not leak from the reactor – PTNL andprobability of not escaping while slowing-down – PFNL

PTNL “1

1` B2L2T

and PFNL “1

1` B2τT

Multiplication factor of thermal reactor is k “ k8PTNLPFNLIf denominator in the two-group critical equation is multiplied out,term B4L2

T τT can often be ignored resulting in expression:

11` B2pL2

T ` τT q“ 1 (5-30)



Modified One-Group Critical Equation

It is possible to define thermal migration area – M2T “ L2

T ` τT ,then equation (5-30) can be written as:

11` B2M2

T“ 1 (5-31)

This form is known as modified one-group critical equation

It is clear that if τT is much smaller than L2T , then the reactor is

sufficiently described by the one-group critical equationThis approach is suitable for graphite and D2O moderated reactorTwo-group theory is necessary for reactors moderated by H2OThe resulting equations (5-29) and (5-31) are used to calculatecritical mass of bare thermal reactors


Reflected Reactors

Reactor Reflector

Neutron economy is improved if the reactor core is surrounded bya reflectorReflector is realized by a thick unfueled region of moderatorNeutrons escaping from the core must pass through thismoderator region and some of these diffuse backThe net result is that critical dimensions of the reactor are reducedReflector parameters will be derived using one-group method for asphere reactor of radius RThe following analysis deals only with an infinite reflector


Reflected Reactors One-Group Method

Reflected Sphere Reactor

In the following analysis, parameters referring to the reactor coreand reflector will have subscripts c and r, respectivelyNeutron flux in the reactor core must follow the equation:

∇2Φc ` B2Φc “ 0 , where for critical reactor (5-32)

B2 “k8 ´ 1

L2c

Since there are no neutron sources in the reflector, neutron flux inthis area must satisfy the following equation:

∇2Φr ´1L2

rΦr “ 0 (5-33)

The previous equations for Φc and Φr must be solved with respectto boundary conditions



Reflected Sphere Reactor (cont’d)

General solution for equation (5-32) was derived previously and itis in form:

Φc “ AcsinpBrq

r` Cc

cospBrqr

Because neutron flux must be finite in the whole reactor volume(including r = 0), constant Cc must equal 0 and Φc reduces to:

Φc “ AcsinpBrq

r(5-34)

General solution to equation (5-33) is:

Φr “ Are´r{Lr

r` Cr

er{Lr

rConstant Cr must equal zero to keep neutron flux finite as r goesto infinity and the equation reduces to:

Φr “ Are´r{Lr

r(5-35)



Reflected Sphere Reactor (cont’d2)

Boundary conditions for neutron flux and neutron current on thecore/reflector interface (r = R) are used:

ΦcpRq “ Φr pRq and (5-36)

´Dc

ˆ

dΦc

dr

˙

R“ ´Dr

ˆ

dΦr

dr

˙

R(5-37)

Introducing equations (5-34) and (5-35) into (5-36) gives:

AcsinpBRq

R“ Ar

e´R{Lr

R(5-38)

Next, differentiating equations (5-34) and (5-35) and insertingresults into (5-37) yields:

AcDc

ˆ

B cospBRqR

´sinpBRq

R2

˙

“ ´Ar Dr

ˆ

1RLr

`1R

˙

e´R{Lr

(5-39)



Critical Equation of a Reflected Sphere Reactor

System of equations (5-38) and (5-39) for unknowns Ac and Arhas nontrivial solution if determinant of the system equals zero.This is satisfied if

Dc

ˆ

B cotpBRq ´1R

˙

“ ´Dr

ˆ

1Lr`

1R

˙

(5-40)

The equation (5-40) is usually rearranged in the following form:

BR cotpBRq ´ 1 “ ´Dr

Dc

ˆ

RLr` 1

˙

(5-41)

It is critical equation for sphere reactor with infinite reflector

It must be satisfied for a reactor to be criticalFor given R, B2 must be calculated from equation (5-41) andcritical composition can be determined



Solution of Critical Equation of a Reflected SphereReactor

Critical equation (5-41) is transcendental and cannot be solvedanalytically, but graphical solution is possibleSuppose that the core composition and hence B is specifiedReactor core radius can then be found by plotting the left-handside (LHS) and the right-hand side (RHS) of the equationseparately as functions of BRAs shown later, LHS has infinite number of branches and RHSforms a straight line with slope ´Dr{pDcBLr q

Every value of BR corresponding to intersection of LHS and RHSforms an infinite number of solutionsOnly the first solution is relevant to a critical reactorIt is to be noted that the solution is smaller that π, which meansthat critical radius of reflected sphere is smaller than for a barespherical reactor



Graphical Solution of Critical Equation of a ReflectedSphere Reactor

Graphical solution to the critical equation of a reflected sphere reactor



Special Solution of Critical Equation of a ReflectedSphere Reactor

In a special case in which moderator in the core and reflector arethe same, Dc “ Dr and equation (5-41) reduces to:

B cotpBRq “ ´1Lr

(5-42)

This equation is not transcendental in R, thus if B is known, criticalradius R can be calculated directly



Determination of Power of a Critical Reflected SphereReactor

Having obtained the criticality conditions, it is necessary toevaluate constants Ac and Ar from the reactor powerConstants Ac and Ar are related by equation (5-38) and it ispossible to find for instance Ar in terms of Ac

Ar “ AceR{Lr sinpBRq

The reactor power is determined as P “ EfΣfş

V Φc dV withdV “ 4πr2dr

P “ 4πEfΣf Ac

ż R

0r sinpBRqdr “

4πEfΣf Ac

B2 psinpBRq´BR cospBRqq

Solution for Ac gives:

Ac “PB2

4πEfΣf psinpBRq ´ BR cospBRqq



Reflected Reactor Discussion

The presented procedure of critical equation determination isanalytically possible only for spherical and fully reflectedparallelepiped and cylindrical reactorIn practise, critical conditions of these reactors are evaluated bytransforming to a spherical reactor of the same composition andvolumeThe whole calculation was carried using one-group methodwithout regard for whether the reactor is fast or thermalIt is therefore valid for both energy groupsOne-group method is sufficient for critical parameterscomputation, but for neutron flux distribution it is necessary to usetwo-group calculation


Reflected Reactors Two-Group Method

Flux in a Reflected Thermal Reactor

In two-group approximation, it is necessary to solve a two-groupreactor equation for the reactor core and a two-group diffusionequation for the reflectorThe evaluation will not be presented here, only the most importantresultsThe most striking result is that thermal neutron flux rises near thecore-reflector interface and exhibits a peak in the reflectorIt is caused by fast neutrons thermalisation in the reflectorThermal neutrons are not much absorbed in the reflector,therefore they tend to accumulate there before they escapethrough the outer surface or diffuse back into the coreThis leads to flattening of the neutron flux distribution in the core



Plotted Neutron Flux a in Reflected Thermal Reactor



Reflector Savings

Reactor reflector reduces critical dimensions of the core andreduces maximum-to-average power ratioReflector savings is defined as:

δ “ rR0´R , where rR0 is critical diameter of a bare reactor (5-43)

Following formula can be used to roughly estimate reflectorsavings for D2O and graphite moderated/reflected reactor:

δ »sDcsDr

LTr (5-44)

The equation (5-44) is valid if reflector is several diffusion lengthsthick, then it can be considered effectively infinite and furtherthickness increase does not reduce critical size of the coreIn practise all reactors have sufficiently thick reflectors to beconsidered infinite


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2. Nuclear Reactor Theory