Diffusion Theory
J. Frybort, L. Heraltova
Department of Nuclear Reactors
November 28, 2014
Chapter 5
J. Frybort, L. Heraltova (CTU in Prague) Diffusion Theory November 28, 2014 1 / 58
Content
1 Fick’s Law
2 Continuity Equation
3 Diffusion EquationValidity and Boundary Conditions
4 Neutron Sources in Infinite Diffusive MediumPlanar Source in Infinite Diffusive MediumPoint Source in Infinite Diffusive MediumLine Source in Infinite Diffusive MediumDiffusion Length
5 Neutron Sources in Finite Diffusive Medium
6 Group and Thermal Neutron Diffusion
J. Frybort, L. Heraltova (CTU in Prague) Diffusion Theory November 28, 2014 2 / 58
Fick’s Law
Introduction
It is necessary to predict neutron distribution inside a nuclearreactorThe exact description of all the processes of neutrons (collisions,transport, nuclear reactions) is very difficultThe first approximation describes the movement of neutrons as akind of diffusionThis approximation is called diffusion approximation and was usedin development of the first types of nuclear reactorsMore advanced methods are developed now, but still, diffusiontheory is widely used for the analysis of large nuclear reactorsThe complete theory describing all neutron properties with littleapproximation is Transport theory solving Boltzmann transportequation
J. Frybort, L. Heraltova (CTU in Prague) Diffusion Theory November 28, 2014 3 / 58
Fick’s Law
Fick’s Law
The diffusion theory is based on Fick’s law originally used forchemical diffusionIt was observed in chemistry that if concentration of a solute inone region of solution is greater than in another, the solutediffuses from the region of higher concentration to the region oflower concentrationThe rate of solute flow is proportional to the negative of thegradient of the solute concentrationNeutrons behave to a good approximation in the same wayIf the density (neutron flux) of neutrons is higher in one part of areactor, there is a net flow of neutrons into a region with lowerneutron flux
J. Frybort, L. Heraltova (CTU in Prague) Diffusion Theory November 28, 2014 4 / 58
Fick’s Law
Neutron Current Density
If neutron density varies along x-direction, the net flow of neutronsthat pass per unit of time through a unit area perpendicular to thex-direction can be expressed as:
Jx “ ´DdΦ
dx(5-1)
Jx has the same unit as flux (neutrons/cm2-sec)Parameter D is called diffusion coefficient and has unit of cmThe flux is generally function of three spatial variables, therefore:
J “ ´D gradΦ “ ´D ∇Φ (5-2)
Here J is called neutron current density
J. Frybort, L. Heraltova (CTU in Prague) Diffusion Theory November 28, 2014 5 / 58
Fick’s Law
Diffusion Coefficient
We assume that D is not a function of spatial variablesThe diffusion coefficient can be approximately calculated as:
D “λtr
3, where λtr is transport mean free path (5-3)
λtr “1
Σtr“
1Σsp1´ µq
(5-4)
Transport mean free path (λtr) is an average distance a neutronwill move in its original direction after infinite number of collisionssµ “ Ěcosϑ is average value of the cosine of the angle at whichneutrons are scattered in the medium. It can be calculated formost of the neutron energies as:
µ “2
3A(5-5)
J. Frybort, L. Heraltova (CTU in Prague) Diffusion Theory November 28, 2014 6 / 58
Fick’s Law
Validity of Fick’s Law
Fick’s law is approximation which is not valid under the following condi-tions
1 In a medium that strongly absorbs neutrons2 Within about three mean free paths from either neutron source or
the outer surface of the diffusive medium3 When the scattering of neutrons is strongly anisotropic
These conditions are present in every practical reactor problemFick’s law and diffusion theory is therefore only the first estimateMore advanced methods must be used near sources, boundariesand in strongly absorbing media
J. Frybort, L. Heraltova (CTU in Prague) Diffusion Theory November 28, 2014 7 / 58
Continuity Equation
General Equation of Continuity
In an arbitrary volume V of a diffusive medium containing neutronsthe number of neutrons may changeThe change of the number of neutrons is a result of a net flow ofneutrons in or out of V, some neutrons are absorbed inside V andthere might be also neutron sources inside volume VThe equation of continuity is mathematical representation of thefact that neutrons cannot disappear unaccountably
«
rate of changein number ofneutrons in V
ff
“
»
—
–
rate ofproductionof neutronsin V
fi
ffi
fl
´
»
—
–
rate ofabsorptionof neutronsin V
fi
ffi
fl
´
»
—
–
rate ofleakageof neutronsfrom V
fi
ffi
fl
J. Frybort, L. Heraltova (CTU in Prague) Diffusion Theory November 28, 2014 8 / 58
Continuity Equation
Rate of Change of Neutrons in V
If n is density of neutrons at any point in time in V, the totalnumber of neutrons in V is then:
ż
Vn dV
The rate of change in number of neutrons is:
ddt
ż
Vn dV , which can be also written as:
ż
V
BnBt
dV
J. Frybort, L. Heraltova (CTU in Prague) Diffusion Theory November 28, 2014 9 / 58
Continuity Equation
Production and Absorption Rate in V
Let s be the rate at which neutrons are emitted from sources percm3/sec in VThe rate at which neutrons are produced through V is given as:
Production rate “
ż
Vs dV
The rate at which neutrons are lost by absorption per cm3/sec isequal to ΣaΦ
The total loss of neutrons through absorption in volume V is:
Absorption rate “
ż
VΣaΦ dV
J. Frybort, L. Heraltova (CTU in Prague) Diffusion Theory November 28, 2014 10 / 58
Continuity Equation
Leakage Rate out of V
J is current density vector on the surface of V and n is a unitnormal pointing outward from the surfaceThen Jn is the net number of neutrons passing outwards throughthe surface per cm2/secThe overall leakage through the surface A of the volume V is:
Leakage rate “
ż
AJn dA
This surface integral can be converted into a volume integral bythe divergence theorem:
ż
AJn dA “
ż
VdivJ dV
J. Frybort, L. Heraltova (CTU in Prague) Diffusion Theory November 28, 2014 11 / 58
Continuity Equation
Resulting Equation of Continuity
ż
V
BnBt
dV “
ż
Vs dV ´
ż
VΣaΦ dV ´
ż
VdivJ dV
Integral were carried out over the same volume, thus theirintegrands must also be equal:
BnBt“ s ´ ΣaΦ´ divJ (5-6)
The above equation is the equation of continuity, if the neutrondensity is not a function of time, this equation reduces to:
´divJ´ ΣaΦ` s “ 0
J. Frybort, L. Heraltova (CTU in Prague) Diffusion Theory November 28, 2014 12 / 58
Diffusion Equation
Diffusion Equation
The continuity equation has two unknowns – the neutron density(n) and neutron current density (J)There is a relation between neutron flux and neutron currentdensityOne of these unknowns can be eliminated by Fick’s lawSubstitution of (5-2) into (5-6) leads to:
´divp´D gradΦq ´ ΣaΦ` s “BnBt
Diffusion coefficient (D) is spatially independent
D divpgradΦq ´ ΣaΦ` s “BnBt
J. Frybort, L. Heraltova (CTU in Prague) Diffusion Theory November 28, 2014 13 / 58
Diffusion Equation
Diffusion Equation (cont’d)
The continuity equation can be further simplified by introducingsymbol ∇2 ” div grad called Laplacian operatorThe resulting equation is called diffusion equation
D∇2Φ´ ΣaΦ` s “1vBΦ
Bt(5-7)
If only time independent problems are considered, steady-statediffusion equation is formulated
D∇2Φ´ ΣaΦ` s “ 0 (5-8)
J. Frybort, L. Heraltova (CTU in Prague) Diffusion Theory November 28, 2014 14 / 58
Diffusion Equation
Laplacian Operator
Formula for Laplacian depends on used coordinate systemin rectangular coordinates:
∇2 “B2
Bx2 `B2
By2 `B2
Bz2
in cylindrical coordinates:
∇2 “1rB
Br
ˆ
rB
Br
˙
`1r2B2
Bϑ2 `B2
Bz2
in spherical coordinates:
∇2 “1r2B
Br
ˆ
r2 B
Br
˙
`1
r2 sinϑB
Bϑ
ˆ
sinϑB
Bϑ
˙
`1
r2 sin2 ϑ
B2
Bϕ2
J. Frybort, L. Heraltova (CTU in Prague) Diffusion Theory November 28, 2014 15 / 58
Diffusion Equation
1D Laplacian Operators
In the simplest examples in one-dimensional space, the Laplacianoperator reduces to the following formulas:
rectangular coordinates:
∇2 “B2
Bx2
cylindrical coordinates:
∇2 “1rB
Br
ˆ
rB
Br
˙
“B2
Br2 `1rB
Br
spherical coordinates:
∇2 “1r2B
Br
ˆ
r2 B
Br
˙
“B2
Br2 `2rB
Br
J. Frybort, L. Heraltova (CTU in Prague) Diffusion Theory November 28, 2014 16 / 58
Diffusion Equation
Diffusion Length
The equation (5-8) is often divided by D, resulting in:
∇2Φ´1L2 Φ`
sD“ 0 (5-9)
Parameter L2 is defined as:
L2 “DΣa
The quantity L is called diffusion length with unit cmQuantity L2 is diffusion area, unit cm2
Physical meaning of the diffusion length will be given later
J. Frybort, L. Heraltova (CTU in Prague) Diffusion Theory November 28, 2014 17 / 58
Diffusion Equation Validity and Boundary Conditions
Boundary Conditions
The diffusion equation was derived using Fick’s law, thereforeconditions for validity of Fick’s law are also valid for the diffusionequationSince the diffusion equation is a partial differential equation,boundary conditions are requiredThere are typical boundary conditions:
1 Neutron flux must be non-negative and finite:0 ď Φ ă 8
2 Both neutron flux and current must be continuous across boundaryof two diffusive media (A and B):
ΦA “ ΦB
pJAqn “ pJBqn
3 Boundary condition for an external boundary of a diffusive medium4 Source conditions
J. Frybort, L. Heraltova (CTU in Prague) Diffusion Theory November 28, 2014 18 / 58
Diffusion Equation Validity and Boundary Conditions
Extrapolation Distance
Fick’s law is not valid for area close to an external surfacebetween the diffusive medium and atmosphereIt was found that if the flux vanishes in a distance d from thesurface, then the flux calculated by diffusion theory is close to thereal fluxThe parameter d is known as extrapolation distance and in mostcases it is given by simple formula d “ 0.71 λtr, where λtr istransport mean free path of the mediumFrom relation for diffusion coefficient – D “ λtr{3 – results thatd “ 2.13 DThe extrapolation distance is usually in units of several cm andtherefore it can be in many cases neglected and assumed thatneutron flux diminishes at the physical boundary of the medium
J. Frybort, L. Heraltova (CTU in Prague) Diffusion Theory November 28, 2014 19 / 58
Diffusion Equation Validity and Boundary Conditions
Extrapolation Distance – Visualisation
If d is not negligible, physical dimensions of the reactor are increasedby d and extrapolated boundary is formulated with dimension a “ a` d
J. Frybort, L. Heraltova (CTU in Prague) Diffusion Theory November 28, 2014 20 / 58
Diffusion Equation Validity and Boundary Conditions
Source Condition
The diffusion equation is not valid for the neutron source location,but it is necessary to match the magnitude of the neutron flux tothe source intensityThe source is surrounded by area for which it holds that allneutrons flowing through this area must come from the neutronsource characterized by source emissivity SFormulation depends on the source geometry: planar (5-10a),point (5-10b), or line (5-10c)
limxÑ0
Jpxq “S2
(5-10a)
limrÑ0
4πr2Jprq “ S (5-10b)
limrÑ0
2πrJprq “ S (5-10c)
This condition will be illustrated by examples of neutron sources indiffusive media
J. Frybort, L. Heraltova (CTU in Prague) Diffusion Theory November 28, 2014 21 / 58
Neutron Sources in Infinite Diffusive Medium
Infinite Diffusive Medium
Spatial distribution of neutron flux in an infinite media will becalculated using diffusion equation and boundary conditionsBasic source geometries will be calculated – plane, point and lineOnly monoenergetic sources of neutrons are analysed
J. Frybort, L. Heraltova (CTU in Prague) Diffusion Theory November 28, 2014 22 / 58
Neutron Sources in Infinite Diffusive Medium Planar Source in Infinite Diffusive Medium
Planar Source in Infinite Diffusive Medium
Planar source emitting S neutrons per cm2/secThe flux is only function of distance from the plane, e.g. in xdirection. The source location is not part of the analysed areaThe plane is located in x “ 0 and there are two solution forpositive (x ą 0) and negative (x ă 0) directionsThe diffusion equation (5-9) has the following form:
d2Φ
dx2 ´1L2 Φ “ 0 , x ‰ 0
Solution for x ą 0 is expected in form:
Φpxq “ e´λx , second derivative:d2Φ
dx2 “ λ2e´λx
Substituted in the above equation leads to:
λ2e´λx “1L2 e´λx
J. Frybort, L. Heraltova (CTU in Prague) Diffusion Theory November 28, 2014 23 / 58
Neutron Sources in Infinite Diffusive Medium Planar Source in Infinite Diffusive Medium
Planar Source in Infinite Diffusive Medium (cont’d)
There are two possible solution for λ “ ˘1L
The diffusion equation for planar source has thus general solutionwith two constants to be determined by boundary conditions
Φpxq “ Ae´x{L ` Cex{L
The second term can be eliminated from the condition of finiteneutron flux, then the equation reduces to:
Φpxq “ Ae´x{L
The constant A is determined from the source conditionFrom Fick’s law:
J “ ´DdΦ
dx“
DAL
e´x{L
J. Frybort, L. Heraltova (CTU in Prague) Diffusion Theory November 28, 2014 24 / 58
Neutron Sources in Infinite Diffusive Medium Planar Source in Infinite Diffusive Medium
Planar Source in Infinite Diffusive Medium (cont’d2)
Source condition for the planar source:
limxÑ0
Jpxq “ limxÑ0
DAL
e´x{L “DAL“
S2
This gives constant A
A “SL2D
The final formula for spatial dependence of neutron flux from theplanar source in infinite medium is:
Φpxq “SL2D
e´x{L (5-11)
The solution is valid for x ą 0, but because of symmetry of theproblem similar formulation could be obtained for negativex-direction
J. Frybort, L. Heraltova (CTU in Prague) Diffusion Theory November 28, 2014 25 / 58
Neutron Sources in Infinite Diffusive Medium Point Source in Infinite Diffusive Medium
Point Source in Infinite Diffusive Medium
Point source emitting S neutrons/sec isotropically in an infinitemediumThe source is located in the centre of a spherical coordinatesystem and neutron flux depends only on distance r from thesourceThe diffusion equation (5-9) in spherical coordinate systembecomes for r ‰ 0:
d2Φ
dr2 `2r
dΦ
dr´
1L2 Φ “ 0
The equation is solved by introducing substitution uprq “ rΦprqSubstitution into the above equation results in:
d2udr2 ´
1L2 u “ 0
J. Frybort, L. Heraltova (CTU in Prague) Diffusion Theory November 28, 2014 26 / 58
Neutron Sources in Infinite Diffusive Medium Point Source in Infinite Diffusive Medium
Point Source in Infinite Diffusive Medium (cont’d)
The solution for function u is found in an identical way as in thecase of the planar source:
uprq “ Ae´r{L ` Cer{L
Transform to the original function gives:
Φprq “ Ae´r{L
r` C
er{L
rConstants A and C must be determined from boundary conditionsIt is clear that if the neutron density flux must remain finite, C mustequal zeroThe constant A is found from the source conditionFrom Fick’s law:
J “ ´DdΦ
dr“ DA
ˆ
1rL`
1r2
˙
e´r{L
J. Frybort, L. Heraltova (CTU in Prague) Diffusion Theory November 28, 2014 27 / 58
Neutron Sources in Infinite Diffusive Medium Point Source in Infinite Diffusive Medium
Point Source in Infinite Diffusive Medium (cont’d2)
The source condition for point source is:
limrÑ0
4πr2Jprq “ limrÑ0
4πDA´ r
L` 1
¯
e´r{L “ S
This gives constant A:
A “S
4πDResulting equation for neutron flux distribution is following:
Φprq “Se´r{L
4πDr(5-12)
J. Frybort, L. Heraltova (CTU in Prague) Diffusion Theory November 28, 2014 28 / 58
Neutron Sources in Infinite Diffusive Medium Line Source in Infinite Diffusive Medium
Line Source in Infinite Diffusive Medium
Line source emitting S neutrons/sec per unit length in an infinitemediumThe source is located in the centre of a cylindical coordinatesystem and neutron flux depends only on distance r from thesourceThe diffusion equation (5-9) in the cylindical coordinate systembecomes for r ‰ 0:
d2Φprqdr2 `
1r
dΦprqdr
´1L2 Φprq “ 0
The equation can be transformed by substitution u=r/L intomodified Bessel’s equation of the order zero:
u2 d2Φpuqdu2 ` u
dΦpuqdu
´ u2Φpuq “ 0
J. Frybort, L. Heraltova (CTU in Prague) Diffusion Theory November 28, 2014 29 / 58
Neutron Sources in Infinite Diffusive Medium Line Source in Infinite Diffusive Medium
Ordinary Bessel’s Functions
Bessel’s equation is:
x2 d2Φ
dx2 ` xdΦ
dx` pα2x2 ´ n2qΦ “ 0 , where α and n are constants
n is order of the equation, in practical problems it is usually zeroGeneral solution to Bessel’s equation is:
Φpxq “ AJnpαxq ` CYnpαxq
Functions Jn and Yn are called ordinary Bessel’s functions of thefirst and second kind, respectively
J. Frybort, L. Heraltova (CTU in Prague) Diffusion Theory November 28, 2014 30 / 58
Neutron Sources in Infinite Diffusive Medium Line Source in Infinite Diffusive Medium
Modified Bessel’s Functions
If α2 is negative, Bessel’s equation becomes:
x2 d2Φ
dx2 ` xdΦ
dx´ pα2x2 ` n2qΦ “ 0
General solution is in form of modified Bessel’s functions of thefirst and second kind, respectively – In and Kn
Φpxq “ AInpαxq ` CKnpαxq
The following figures will show Bessel’s functions used in reactorphysics
J. Frybort, L. Heraltova (CTU in Prague) Diffusion Theory November 28, 2014 31 / 58
Neutron Sources in Infinite Diffusive Medium Line Source in Infinite Diffusive Medium
Ordinary Bessel’s Functions Plot
J. Frybort, L. Heraltova (CTU in Prague) Diffusion Theory November 28, 2014 32 / 58
Neutron Sources in Infinite Diffusive Medium Line Source in Infinite Diffusive Medium
Modified Bessel’s Functions Plot
J. Frybort, L. Heraltova (CTU in Prague) Diffusion Theory November 28, 2014 33 / 58
Neutron Sources in Infinite Diffusive Medium Line Source in Infinite Diffusive Medium
Line Source in Infinite Diffusive Medium (cont’d)
General solution of this kind of Bessel’s equation is in form ofmodified Bessel’s function of the first (I) and second (K) kind of theorder zero
Φpuq “ AI0puq ` CK0puq
Φprq “ AI0pr{Lq ` CK0pr{Lq
Given the fact that function I0 Ñ8 for rÑ8, constant A mustequal 0 and above equation reduces to:
Φprq “ CK0pr{Lq
The constant C is found from the source conditionFrom Fick’s law, using dK0{dr “ ´K1:
J “ ´DdΦ
dr“ ´DC
dK0pr{Lqdr
“DCK1pr{Lq
L
J. Frybort, L. Heraltova (CTU in Prague) Diffusion Theory November 28, 2014 34 / 58
Neutron Sources in Infinite Diffusive Medium Line Source in Infinite Diffusive Medium
K1 Bessel’s Function Plot
J. Frybort, L. Heraltova (CTU in Prague) Diffusion Theory November 28, 2014 35 / 58
Neutron Sources in Infinite Diffusive Medium Line Source in Infinite Diffusive Medium
Line Source in Infinite Diffusive Medium (cont’d2)
K1(x) behaves similar to 1x function
It can be used in the source condition:
limrÑ0
2πpr{LqDCK1pr{Lq “ S
It can be written that limrÑ0rpr{LqK1pr{Lqs „ 1Resulting value for constant C is:
C “S
2πD
Final form of a neutron flux distribution from the line source is:
Φprq “S
2πDK0pr{Lq (5-13)
J. Frybort, L. Heraltova (CTU in Prague) Diffusion Theory November 28, 2014 36 / 58
Neutron Sources in Infinite Diffusive Medium Diffusion Length
The Diffusion Length
It is necessary to understand physical meaning of the diffusionlength L, which appears in solutions of neutron flux from neutronsourcesIn diffusive medium, neutrons are moving along complicatedpaths, however, every neutron is absorbed in the medium since itis infiniteNumber of neutrons absorbed from a point neutron source perunit path length can be expressed as:
dn “ ΣaΦprq dV “SΣa
Dre´r{L dr “
SL2 re´r{L dr
Probability that neutron is absorbed in dr is:
pprq dr “1L2 re´r{L dr
J. Frybort, L. Heraltova (CTU in Prague) Diffusion Theory November 28, 2014 37 / 58
Neutron Sources in Infinite Diffusive Medium Diffusion Length
The Diffusion Length (cont’d)
It is now possible to calculate a square of an average distancefrom the source at which a neutron is absorbed
sr2 “
ż 8
0r2pprq dr “
1L2
ż 8
0r3e´r{L dr “ 6L2
It means that L2 is one-sixth of the average of the square of thestraight distance a neutron travels from the point at which it isemitted to the point where it is finally absorbed
With higher diffusionlength, neutrons aremoving further. It meansthe diffusive medium isless absorbing
J. Frybort, L. Heraltova (CTU in Prague) Diffusion Theory November 28, 2014 38 / 58
Neutron Sources in Finite Diffusive Medium
Finite Diffusive Medium
Finite diffusive medium is a more realistic situationFor neutron flux spatial distribution computation it is necessary touse boundary conditionsFor this purpose, physical dimensions of the diffusive media willbe increased by extrapolation distance dNeutron flux will be calculated for monoenergetic neutron sourcesin basic geometries – slab, cylinder and sphereSince the method of solving these geometries is similar, it will beillustrated only for a bare slab with a plane source of neutrons
J. Frybort, L. Heraltova (CTU in Prague) Diffusion Theory November 28, 2014 39 / 58
Neutron Sources in Finite Diffusive Medium
Finite Bare Slab
The slab has thickness 2a, it is infinite in vertical direction and thesource of neutrons is located in its centreIt can be solved for positive and negative direction withsymmetrical resultsBoundary condition must be utilisedThe neutron flux is required to vanish at the extrapolated surfaceof the slab, at x “ a` d (or at x “ ´d ´ a)The boundary condition is:
Φpa` dq “ Φp´a´ dq “ 0
The general solution of the diffusion equation for the planar sourceof neutrons is the same as for infinite diffusive medium:
Φpxq “ Ae´x{L ` Cex{L
J. Frybort, L. Heraltova (CTU in Prague) Diffusion Theory November 28, 2014 40 / 58
Neutron Sources in Finite Diffusive Medium
Finite Bare Slab (cont’d)
Boundary condition at a` d is applied:
Φpa` dq “ Ae´pa`dq{L ` Cepa`dq{L “ 0
One constant can be expressed as the function of the other:
C “ ´Ae´2pa`dq{L
Substituting this into the general solution gives:
Φpxq “ A”
e´x{L ´ ex{L´2pa`dq{Lı
J. Frybort, L. Heraltova (CTU in Prague) Diffusion Theory November 28, 2014 41 / 58
Neutron Sources in Finite Diffusive Medium
Finite Bare Slab (cont’d2)
The constant A is found from the source condition in the usual wayand is:
A “SL2D
´
1` e´2pa`dq{L¯´1
For positive x-direction, function for Φ is given by:
Φpxq “SL2D
e´x{L ´ ex{L´2pa`dq{L
1` e´2pa`dq{L (5-14)
J. Frybort, L. Heraltova (CTU in Prague) Diffusion Theory November 28, 2014 42 / 58
Group and Thermal Neutron Diffusion
Group Neutron Diffusion
So far only monoenergetic neutron sources were considered andenergy dependence was neglectedCommonly used form of neutron energy spectrum description is agroup method
By this approach, neutrons aredivided into energy groupsdepending on their energyIf a neutron gains energy, itmoves to a higher energygroupBy loosing energy neutronmoves into a lower energygroup
J. Frybort, L. Heraltova (CTU in Prague) Diffusion Theory November 28, 2014 43 / 58
Group and Thermal Neutron Diffusion
Group Neutron Diffusion (cont’d)
Flux in a specific energy range g is calculated as:
Φg “
ż
gΦpEqdE
Neutrons can disappear from a group by absorption and byscattering from one group to another (g Ñ h):
absorption rate = ΣagΦg , group transfer rate = ΣgÑhΦg
Total transfer rate from one group to another is obtained bysumming over all groups that can scatter to the other groupThe group diffusion equation for the gth group can be written inform:
Dg∇2Φg ´ ΣagΦg ´
Nÿ
h“g`1
ΣgÑhΦg `
g´1ÿ
h“1
ΣhÑgΦh “ ´sg
J. Frybort, L. Heraltova (CTU in Prague) Diffusion Theory November 28, 2014 44 / 58
Group and Thermal Neutron Diffusion
Thermal Neutron Diffusion
Important aspect of neutron diffusion is diffusion of thermalneutronsEnergy distribution of thermal neutrons is given by Maxwelliandistribution:
npEq “2πn
pπkT q3{2E1{2e´E{kT (5-15)
Using the usual formula for speed vpEq “ p2E{mq1{2 anddefinition of neutron flux ΦpEq “ npEqvpEq an average thermalflux can be calculated as:
ΦT “
ż
TΦpEq dE
T represents all thermal energies up to 0.1 eV
J. Frybort, L. Heraltova (CTU in Prague) Diffusion Theory November 28, 2014 45 / 58
Group and Thermal Neutron Diffusion
Thermal Neutron Diffusion (cont’d)
Average thermal neutron flux is calculated as:
ΦT “2πn
pπkT q3{2
ˆ
2m
˙1{2 ż 8
0Ee´E{kT dE “
2n?π
ˆ
2kTm
˙1{2
(5-16)
Energy and velocity corresponding to energy kT is denoted as ETand vT
If temperature is given in Kelvins, it can be calculated as:
ET “ 8.617T ˆ 10´5 eV
vT “ 1.284T 1{2 ˆ 104 cm/sec
Equation (5-16) can be now written as:
ΦT “2?π
nvT (5-17)
J. Frybort, L. Heraltova (CTU in Prague) Diffusion Theory November 28, 2014 46 / 58
Group and Thermal Neutron Diffusion
Thermal Neutron Diffusion (cont’d2)
There is a concept of thermal energy of neutrons E0 = 0.0253 eV,which corresponds to energy of thermal motion for 20˝C(T0 = 293 K)Data are usually tabulated for this energy. Relation between ΦTand Φ0 can be written as:
ΦT
Φ0“
2?π
vT
v0
Given the mentioned relation for velocity, the above equationbecomes:
ΦT
Φ0“
2?π
ˆ
TT0
˙1{2
(5-18)
J. Frybort, L. Heraltova (CTU in Prague) Diffusion Theory November 28, 2014 47 / 58
Group and Thermal Neutron Diffusion
Thermal Weighted Cross-Section
Thermal group absorption cross-section is calculated as:
sΣa “1
ΦT
ż
TΣapEqΦpEq dE
This integral is total absorption rate equal to gapT qΣapE0qΦ0,where gapT q is non-1/v factor and ΣapE0q is absorptioncross-section for thermal energy 0.0253 eV and Φ0 thermal flux
sΣa becomes:
sΣa “ gapT qΣapE0qΦ0{ΦT “
?π
2gapT qΣapE0q
ˆ
T0
T
˙1{2
(5-19)
This is principle how to calculate thermal group cross-section fromdata tabulated at 0.0253 eV
J. Frybort, L. Heraltova (CTU in Prague) Diffusion Theory November 28, 2014 48 / 58
Group and Thermal Neutron Diffusion
Diffusion Equation for Thermal Energy Group
With thermal flux defined and possibility to calculate thermalgroup weighted cross-section, it is possible to formulate one-grouptime-independent diffusion equation for thermal neutrons:
sD∇2ΦT ´ sΣaΦT “ ´sT , where sT is thermal neutron source
Dividing by sD – thermal diffusion coefficient – gives:
∇2ΦT ´1
L2T
ΦT “ ´sTsD
(5-20)
LT is thermal diffusion length defined as L2T “
sDsΣa
J. Frybort, L. Heraltova (CTU in Prague) Diffusion Theory November 28, 2014 49 / 58
Group and Thermal Neutron Diffusion
Two-Group Diffusion Calculation
For most of the calculations, it is necessary to consider at leasttwo energetic groupsOne group describing thermal neutrons and the other groupdealing with neutron moderation (all neutrons above 5kT )The neutron source emits s neutrons of fast neutrons per secondThe source is placed inside infinite uniform diffusion medium
J. Frybort, L. Heraltova (CTU in Prague) Diffusion Theory November 28, 2014 50 / 58
Group and Thermal Neutron Diffusion
Diffusion Equations Formulation
The diffusion equation for the fast group has following shape (44):
Dg∇2Φg ´ ΣagΦg ´
Nÿ
h“g`1
ΣgÑhΦg `
g´1ÿ
h“1
ΣhÑgΦh “ ´sg
Absorption cross-section for fast neutrons is very small Ñ it ispossible to neglect the absorption altogetherWe are considering only 2 groups, therefore each neutronscattered in the fast group will reach thermal energy Ñ Σ1Ñ2 ” Σ1Possibility of scattering from the thermal group into the fast groupis neglectedThe equation is solved outside the point neutron source and thereare no other neutron sourcesThe diffusion equation is reduced to the following formulation:
D1∇2Φ1 ´ Σ1Φ1 “ 0 (5-21)
J. Frybort, L. Heraltova (CTU in Prague) Diffusion Theory November 28, 2014 51 / 58
Group and Thermal Neutron Diffusion
Diffusion Equations Formulation (cont’d)
Thermal neutron flux is described by equation (5-20)
∇2ΦT ´1
L2T
ΦT “ ´sT
D
Source of thermal neutrons are neutrons scattered from the fastgroup into the thermal group Ñ Σ1Φ1
Diffusion equation for thermal neutrons has the following form:
∇2ΦT ´1
L2T
ΦT “ ´Σ1Φ1
D(5-22)
For determination of the thermal flux, we must first determine thefast neutron flux
J. Frybort, L. Heraltova (CTU in Prague) Diffusion Theory November 28, 2014 52 / 58
Group and Thermal Neutron Diffusion
Neutron Age
We define parameter τT – neutron age:
τT “D1
Σ1(5-23)
Neutron age gives relation to the time required for slowing-downneutron to the thermal energyUnit of τT is cm2
By introducing neutron age into equation (5-21) we get:
∇2Φ1 ´1τT
Φ1 “ 0 (5-24)
J. Frybort, L. Heraltova (CTU in Prague) Diffusion Theory November 28, 2014 53 / 58
Group and Thermal Neutron Diffusion
Solution the Diffusion Equation for the Fast Group
After introducing the shape of Laplace operator for sphericalcoordinates, the diffusion equation for thermal neutrons has shapeidentical to equation (5-12)Neutron flux of fast neutrons is given by the following formula
Φ1 “Se´r{
?τT
4πD1r, r ‰ 0 (5-25)
The physical meaning of neutron age is similar to the case ofdiffusion lengthThe solution is identical except for the fact that neutrons do notdisappear by absorption, but only by scattering into the thermalgroup
J. Frybort, L. Heraltova (CTU in Prague) Diffusion Theory November 28, 2014 54 / 58
Group and Thermal Neutron Diffusion
Physical Meaning of Neutron Age
τT “16
r2 (5-26)
Neutron age gives 1/6 of square of the average distance neutron travels,from the place where it was released to the place where it was slowed-down to the thermal energy
J. Frybort, L. Heraltova (CTU in Prague) Diffusion Theory November 28, 2014 55 / 58
Group and Thermal Neutron Diffusion
Solution of Diffusion Equation for the Thermal Group
Introducing the solution for fast neutron flux (5-25) into thediffusion equation for thermal neutrons (5-22) a new equation withnonzero right side is obtainedThe solution is:
ΦT “S L2
T
4πDpL2T ´ τT q
´
e´r{LT ´ e´r{?τT
¯
(5-27)
This equation gives spatial distribution of thermal neutrons as aresult of presence of the source of fast neutrons
J. Frybort, L. Heraltova (CTU in Prague) Diffusion Theory November 28, 2014 56 / 58
Group and Thermal Neutron Diffusion
Important constants for moderators for the fast neutrongroup
Moderator D1 (cm) Σ1 (cm´1) τT (cm2)
H2O 0.92 0.0489 19D2O 1.26 0.0116 109Be 0.55 0.0090 61Graphite 1.04 0.0037 278
J. Frybort, L. Heraltova (CTU in Prague) Diffusion Theory November 28, 2014 57 / 58
Group and Thermal Neutron Diffusion
Diffusion parameters for moderators in the thermalneutrons group for temperature 20˝C
Moderator density (g/cm3) D (cm) LTM (cm)
H2O 1.00 0.13 2.62D2O 1.10 0.76 147Be 1.85 0.45 22Graphite 1.60 0.87 61
J. Frybort, L. Heraltova (CTU in Prague) Diffusion Theory November 28, 2014 58 / 58