DIFFUSION OF NEUTRONS OVERVIEW • Basic Physical Assumptions • Generic Transport Equation • Diffusion Equation • Fermi’s Age Equation • Solutions to Reactor Equation
Dec 18, 2015
DIFFUSION OF NEUTRONS
OVERVIEW
• Basic Physical Assumptions• Generic Transport Equation• Diffusion Equation• Fermi’s Age Equation• Solutions to Reactor Equation
HT2005: Reactor Physics T10: Diffusion of Neutrons 2
Basic Physical Assumptions
• Neutrons are dimensionless points
• Neutron – neutron interactions are neglected
• Neutrons travel in straight lines
• Collisions are instantaneous
• Background material properties are isotropic
• Properties of background material are known and time-independent
HT2005: Reactor Physics T10: Diffusion of Neutrons 3
Ep E
E
10
9
-7
4.55 10cm; is in eV
0.01eV 4.55 10 cm
D(H) 10 cm
(a) (b)
Physical Model
HT2005: Reactor Physics T10: Diffusion of Neutrons 5
Initial Definitions
3 3
3 3 3 3
;
( , , ) Expected number of neutrons in within
x y zd dxdydz d dv dv dv
N t d d d d
r v
r v r v r v
2
1;
2( , )
vvmv
E
Ω v v Ω
Ω
r
ex
x
y
z
ey
v
ey
HT2005: Reactor Physics T10: Diffusion of Neutrons 6
3
22
0 0 0 0 4
( , ) ( , , ) ( , , )
( , , , ) sin ( , , , )
sin
x y zn t N t dv dv dv N t d
N v t v dvd d N E t d dE
d d d
r r v r v v
r Ω r Ω Ω
Ω
2
( , , , ) ( , , ) ( , , , ) ( , , )
( , , , ) ( , , )
1( , , , ) ( , , , )
vN E t N t N E t d dE N t d
mN v t v N t
N E t N v tmv
r Ω r v r Ω Ω r v v
r Ω r v
r Ω r Ω
Neutron Density
HT2005: Reactor Physics T10: Diffusion of Neutrons 7
Angular Flux and Current Density
( , , , )
( , , ) ( , , )
( , , , ) ( , , , ) ( , , , )
( , , , ) ( , , , )
( , , , ) ( , , , )
Et
t N t
E t N E t vN E t
E t vN E t
E t E t
r Ω
J r v v r v
J r Ω v r Ω Ω r Ω
r Ω r Ω
J r Ω Ω r ΩdS
J
number of neutrons
crossing per 1 second
d
d
J S
S
HT2005: Reactor Physics T10: Diffusion of Neutrons 8
Generic Transport Equation
.
changedue changedue changeduetimerate
of change to leakage to tomacro sources
collisions forcesof N throughS
We ignore macroscopic forces
Arbitrary volume V
3 3 3( , , ) ( , , ) ( , , )collV S V V
NN t d t d d Q t d
t t r v r J r v S r r v r
HT2005: Reactor Physics T10: Diffusion of Neutrons 9
3 3 3( , , ) ( , , ) ( , , ) ( , , )S V V V
t d t d N t d N t d J r v S J r v r v r v r v r v r
x y zx y z
e e er
Generic Transport Equation
0collV
N NN Q d
t t v r
( , , )( , , ) ( , , )
coll
N t NN t Q t
t t
r vv r v r v
Gauss Theorem:
HT2005: Reactor Physics T10: Diffusion of Neutrons 10
Substantial Derivative
r
xy
z
Leonhard Euler's (1707-1783) description:
We fix a small volumeNt
We let a small volume move
Joseph Lagrange's (1736-1813) description
dNdt
( , , )dN N N N
N N tdt t t t
N N Nt m
r vr v
r vF
vr v
( , , )coll
dN NQ t
dt t
r v
HT2005: Reactor Physics T10: Diffusion of Neutrons 11
( , , )coll
dN NQ t
dt t
r v
( , , )( , , ) ( , , )
coll
N t NN t Q t
t t
r vv r v r v
coll
N N N NQ
t m t
Fv
r v
Transport (Boltzmann) Equation
HT2005: Reactor Physics T10: Diffusion of Neutrons 12
Collision Term ,E Ω
r
x
y
z
,EΩ
2cm
, ,sterad eVs E E
Ω Ω
, , , ,s s BE E E E N Ω Ω Ω Ω
0 4
( )
( , , , ) ( , , , ) ( , ) ( , , , )tcoll Total absorption
Scatteringtothecurrent directionandenergy
NE E vN E t d dE E vN E t
t
r Ω Ω r Ω Ω r r Ω
HT2005: Reactor Physics T10: Diffusion of Neutrons 13
Neutron Transport Equation
( , , )( , , ) ( , , )
coll
N t NN t Q t
t t
r vv r v r v
0 4
, , ,1( , , , ) ( , , , )t
E tE E E t d dE Q
v t
r Ω
Ω r Ω Ω r Ω Ω
0( , , ,0) ( , , ) :
( , , , ) 0 0 : ( )s s
E E initial condition
E t boundary freesurface condition
r Ω r Ω
R Ω Ω n
( , , , ) ( , , , )E t vN E t r Ω r Ω
HT2005: Reactor Physics T10: Diffusion of Neutrons 14
Boundary Condition
( , , , ) 0 when 0sSE t
rr Ω n Ω
Ω
ns
xy
z
V
Volume V
Surface S
Outgoing direction
Outward normal
r
Ω
Incoming direction
Rs
HT2005: Reactor Physics T10: Diffusion of Neutrons 15
Difficulties• Mathematical structure is too involved• Mixed type equation (integro-differential), no way
to reduce it to a differential equation• Boundary conditions are given only for a halve of
the values• Too many variables (7 in general)• Angular variable
( , ) ( , , , ) ; ( , ) ( , , , )n t N E t d dE t E t d dE r r Ω Ω r r Ω Ω
HT2005: Reactor Physics T10: Diffusion of Neutrons 17
φ
RCR
Plane Anglesd dss n
nr
cos r ddsd
r r
e s
re
HT2005: Reactor Physics T10: Diffusion of Neutrons 18
2
AR
Solid Angles
d dAA n
nr e Ω
r
2 2
cosdA dd
r r
Ω A
Ω
HT2005: Reactor Physics T10: Diffusion of Neutrons 19
• Infinite homogeneous and isotropic medium• Neutron scattering is isotropic in Lab-system
• Weak absorption Σa << Σs
• All neutrons have the same velosity v. (One-Speed Approximation)
• The neutron flux is slowly varying function of position
One-Group Diffusion Model
HT2005: Reactor Physics T10: Diffusion of Neutrons 20
x
y
z ZJ J J
Derivation
J
Number of collisions s dV
2
cos
4 4Number of neutrons scattered within
Ω
Ω
s s
d dAdV dV
rd
dA
2
cos
4Number of neutrons reaching
srs
dAdV e
rdA
2 2
0 0 0
( ) cos sin4
srs
r
J r e d drd
Isotropic scattering
r = 0 is most important
HT2005: Reactor Physics T10: Diffusion of Neutrons 21
Taylor’s series at the origin: 00 00
...x y zx y z
sin cos ; sin sin ; cosx r y r z r
00 00
( ) sin cos sin sin cosr r rx y z
r
2 2
000 0 0
cos sin4
srs
r
J r e d drdz
Derivation II
HT2005: Reactor Physics T10: Diffusion of Neutrons 22
0
0
0
0
0
1
4 6
1
4 6
1
3
s
s
zs
Jz
Jz
Jz
Derivation III
0 0 0
1 1 1; ;
3 3 3z x ys s s
J J Jz x y
1
3x x y y z z x y zs
J J Jx y z
J e e e e e e
HT2005: Reactor Physics T10: Diffusion of Neutrons 23
Fick’s Law
1( ) ( ); ( )
3 x y zs x y z
J r r r e e e
1( ) ( );
3 3s
s
D D
J r r
CM-System → Lab-System:1
(1 );tr s trtr
1( ) ( );
3 3tr
tr
D D
J r r
HT2005: Reactor Physics T10: Diffusion of Neutrons 24
scos scos
tr s s s s s
n cos cos cos . . . . . cos
2 3
Transport Mean Free Path
s
tr
; 1 cos ; 1 cos1 cos
str tr s tr s
Transport correction =
A number of anisotropic collisions is replaced by one isotropic
Information about the original direction is lost
HT2005: Reactor Physics T10: Diffusion of Neutrons 25
Diffusion Equation
Change rate Production Leakage Absorption
of rate rate raten
3
Production( , ) ( , ) ( , )
rate f
nt t Q t
cms
r r r
3
Absorption( , ) ( , )
rate a
nt t
cms
r r
HT2005: Reactor Physics T10: Diffusion of Neutrons 26
Leakage Rate
(x,y,z)
x
y
z
dx
dy
dz
Jz2
2
( , , ) ( , , )z z z
z dz z
L J x y z dz dxdy J x y z dxdy
D dxdy D dxdydzz z z
2
2
2
2
2
2
x
y
z
L D dxdydzx
L D dxdydzy
L D dxdydzz
2 2 22
2 2 2Leakage from a unit volume D D
x y z
HT2005: Reactor Physics T10: Diffusion of Neutrons 27
Change rate Production Leakage Absorption
of rate rate raten
21;a ext fD Q Q Q
v t
Time-dependent:
Time-independent: 2 0aD Q
Time-independent from a steady source
2
2 22
0
1 10;
3 3
a
a s a tr
a
D Q
DQ L
L D
Diffusion Equation
HT2005: Reactor Physics T10: Diffusion of Neutrons 28
2 2 22
2 2 2
2 2
2 2 2
22
2 2 2 2 2
1 1
1 1 1sin
sin sin
x y z
rr r r r z
rr r r r r
Cartesian geometry
Cylindrical geometry
Spherical geometry
Laplace’s Operator
HT2005: Reactor Physics T10: Diffusion of Neutrons 29
Symmetries
2 1 nn
d dr
r dr dr
n = 0 for slab
n = 1 for cylindrical
n = 2 for spherical
x
y
z
Slab geometry
22
2x
r
Spherical geometry
2 22
1r
r r r
r
Cylindrical geometry
2 1r
r r r
z
HT2005: Reactor Physics T10: Diffusion of Neutrons 30
General Properties
• Flux is finite and non-negative
• Flux preserves the symmetry
• No return from a free surface
• Flux and current are continues
• Diffusion equation describes the balance of neutrons
HT2005: Reactor Physics T10: Diffusion of Neutrons 31
0 0
0 0
1 1,
4 6 4 6s s
J Jz z
for +z - direction: A trA A B trB B
z z4 6 4 6
A trA A B trB B
z z4 6 4 6
for -z - direction:
BA
Dz
DzA
AB
B
A B
z
AB
Interface Conditions
HT2005: Reactor Physics T10: Diffusion of Neutrons 32
00
0
00
0; 04 6
3
2
tr
tr
Jx
x
Straight line extrapolation from x = 0 towards vacuum: 0 0
3( )
2 tr
x x
2( ) 0 ( 0.71)
3 trx for x exact
extrapolation length = 0.71 tr
Free surface
Diffusion eq.
Transport equation
0.66 tr 0.71 tr
0 00 0
1 1
0.66 0.71tr trx x
Boundary Condition
HT2005: Reactor Physics T10: Diffusion of Neutrons 33
x = 0
( )x
2
2
20
2 2
( ) ( ) 0
( )1 ( )( )
a
dD x Q xdx
Q xd Q xx
dx L D D
2
2 2
0 0
0
1( ) 0 ( )
lim ( ) 0 0
lim ( )2 2
x L x L
x
x
dx x Ae Be
dx Lx B
Q Q LJ x A
D
Q0
0( )2
x LQ Lx e
D
Transport equation
3 s
Plane Infinite Source in Infinite Medium
HT2005: Reactor Physics T10: Diffusion of Neutrons 34
Point Source in Infinite Medium
r
22
22 2
1( ) ( ) 0
1 1( ) 0 0
a
d dD r r Q xr dr drd dr r r
r dr dr L
2 00
0
( )
lim ( ) 0
lim 4 ( )4
r L r L
r
r
e er A B
r rr B
Qr J r Q A
D
0( )4
r LQ er
D r
2
20 0
( )4n abs. ( , )( ) r La r r drr r dr rp r dr e dr
Q Q L
2 2 2
0
( ) 6r r p r dr L
HT2005: Reactor Physics T10: Diffusion of Neutrons 35
Plane Infinite Source in Slab Medium
( )x
Q0
0
2sinh
2( )2 cosh
2
a xQ L Lx
aDL
x = 0
2 22
1 1
0.71 a aa trx a
x = a/2x = -a/2
0( )2
x LQ Lx e
D
Slab:
Infinite:
HT2005: Reactor Physics T10: Diffusion of Neutrons 36
Plane Infinite Source with Reflector
Q0
a
21
12 21
1( ) 0
dx
dx L
12 21
22
22 22
1( ) 0
dx
dx L
Reflector Reflector
Bare slab
HT2005: Reactor Physics T10: Diffusion of Neutrons 37
• q(E) - number of neutrons, which per cubic-centimeter and second pass energy E.
• q(E) = [ncm-3 s-1]• X-sections depend on E: D(E),Σs(E),...
Energy
E q(E)
E0 Q0
2( )( )
( )
E
tE
D E dEE
Ecm
E
Slowing down medium: s a s t
log( )( )
( )
f
th
Ef th
th tht sE
E ED E dE DE
E E
1 ln1
21
6th s s mts sD n D L r
Mean Total Slowing down distance
Can be shown
Age of Neutrons
HT2005: Reactor Physics T10: Diffusion of Neutrons 38
2( ) ( , ) ( , ) ( , ) 0aD E E dE E dE Q E dE r r r
( , ) is the number of neutrons at with energies in ( , )E dE E E dE r r
( , )( , ) ( , ) ( , )
q EQ E dE q E dE q E dE
E
r
r r r
E+dE
E
q(E+dE)
q(E)
2 ( , )( ) ( , ) ( , ) 0a
q ED E E dE E dE dE
E
r
r r
( )Continuous slowing down: ( )
( )t
q E dEE dE
E E
2 ( , )( )( )( , ) ( , ) 0
( ) ( )a
t t
q EED Eq E q E
E E E E E
rr r
Fermi’s Age Equation
( ) ( ) ( )
( ) ( )
t
duu u du qu
u du E dE
HT2005: Reactor Physics T10: Diffusion of Neutrons 39
0 ( )ˆ ˆ( , ) ( , ) exp ; ( , ) ( , ) ( ) 0
( )
Ea
atE
E dEq E q E q E q E E
E E
r r r r
2 ( , )( )( )( , ) ( , ) 0
( ) ( )a
t t
q EED Eq E q E
E E E E E
rr r
2ˆ( , ) ( )ˆ( , )
( )t
q E D Eq E
E E E
r
r
0 ( )new variable: ( )
( )
E
tE
D E dEE
E E
2ˆ( , )
ˆ( , )q
q
r
r
Fermi’s Age Equation II
τ ~ time
HT2005: Reactor Physics T10: Diffusion of Neutrons 40
2
2
q qx
x = 0
No absorption
2
0 1 2
exp4
( , )4
pl
x
q x Q
r
22
1q qr
r r r
No absorption
2
0 3 2
exp4
( , )4
pt
r
q r Q
Solutions to the Age Equation
HT2005: Reactor Physics T10: Diffusion of Neutrons 41
-6 -4 -2 0 2 4 60.00
0.02
0.04
0.06
0.08
=0.5 =1.0 =1.5
q(r,
)
r
2
0 3 2
exp4
( , )4
pt
r
q r Q
Slowing Down Density for Different Fermi’s Ages
0 ( ) ( )Q r
HT2005: Reactor Physics T10: Diffusion of Neutrons 42
Migration Area (Length)Fast neutron borne
Thermal neutron absorbed
Fast neutron thermalized
r
rs rth
N N N
i s s i th th ii i i
r r r r r rN N N
2 2 2 2 2 2, ,
1 1 1
1 1 1; ;
2 21
6 thL r
M r2 216
s th
s th s s th th
s s th th s th
r r r
r r r r r
22 2 2
2 2 2 2 2
2
2
r r r
r r r r
r r
2 2
2 20
2
0
( , )4
6 6
( , )4
pt
s th
pt
r q r r dr
r r
q r r dr
th sM L L L 2 2 2 2
HT2005: Reactor Physics T10: Diffusion of Neutrons 43
Diffusion and Slowing Down Parameters for Various Moderators
Moderator g/cm3 tr
cmL
cmtth
ms
tss
0
cm2
H2O 1.0 0.43 2.7 0.21 0.92 1 27
D2O(pure)
1.1 2.5 165 130 0.51 8 131
D2O(normal)
1.1 2.5 100 50 0.51 8 115
Be 1.8 1.5 22 3.8 0.21 10 102
BeO 2.96 1.4 31 8.1 0.17 12 100
C (puregraphite)
1.6 2.6 59 17 0.158 24 368
C (normal.graphite)
1.6 2.6 50 12 0.158 24 368
HT2005: Reactor Physics T10: Diffusion of Neutrons 44
Neutrons in Multiplying Medium2
a
nD Q
t
2( , , )( ) ( , , ) ( ) ( , , ) ( , , )a
th th th th
n E tdE D E E t dE E E t dE Q E t dE
t
r
r r r
( , , ) ( ) ( ) ( )E t F G E T t r r
( , )( , , ) ( , ); ( , , ) ;
( ) ( , , ) ( , );
thth
avth th
a ac thth
tE t dE t n E t dE
v
E E t dE t
rr r r
r r
( ) ( , , )th
cth
D E E t dE
D
r
2( , )1( , ) ( , ) ( , )th
c th ac th thav
tD t t Q t
v t
rr r r
Assumption:
HT2005: Reactor Physics T10: Diffusion of Neutrons 45
Principles of a Nuclear Reactor
1
2
N
Nk
n/
fissi
onN1
N2Leakage
Fast fission
Resonance abs.
Non-fuel abs.
Leakage
Non-fissile abs.
Fission
Slo
win
g d
ow
n
Ene
rgy
E
2 MeV
1 eV
200 MeV/fission
ν ≈ 2.5
HT2005: Reactor Physics T10: Diffusion of Neutrons 46
Total number of fission neutronsFast fission factor 1.02
Number of fission neutrons from thermal neutrons
Ea
a sE
dEE
p E e
0
Resonance escape probability ( ) 0.87
F Fff
f F Fa a
P
Conditional probability
Ff
f Fa
P
Number of neutrons per absorption in fuel 1.65
Fa
a
f
Thermal utilization 0.71
FNL
TNL
NL FNL TNL
P
P
P P P
Fast non-leakage probability 0.97
Thermal non-leakage probability 0.99
Non-leakage probability
k fp
NL FNL TNLk k P fp P P
HT2005: Reactor Physics T10: Diffusion of Neutrons 47
f thCore
a thCore
p dV
kdV
Rate of neutron production in coreRate of neutron absorption in core
a f th f th a thk p Q p k
2( , )1( , ) ( , ) ( , )th
c th ac th thav
tD t t Q t
v t
rr r r
22
( , ) 11( , ) ( , )th
th thcav c
t kt t
t Lv D
rr r 2 c
cac
DL
2
2
1m
c
kB
L
22 ( )
In the stationary case: ( )th
mth
B
r
r
HT2005: Reactor Physics T10: Diffusion of Neutrons 48
Buckling as Curvature
B
2
aB 0
bBcB
a b cB B B
Large core
Small core
L SB B
HT2005: Reactor Physics T10: Diffusion of Neutrons 49
2 2( , )1( , ) ( , )th
th m thav c
tt B t
tv D
r
r r
( , ) ( ) ( )th t F T t r r
221 ( ) ( )
( )( )m
av c
dT t FB
dt Fv DT t
r
r
22 2 2( )
( ) ( )( ) g g
FB F B F
F
r
r rr
2 22
1In a critical reactor: g m
c
kB B
L
2 ( ) ( )F F r r
Criticality Condition
HT2005: Reactor Physics T10: Diffusion of Neutrons 50
2
2
1 2
1 2
2
2 2
12
matrix:
Differential operator: ( ) ( )
0 ( )
0 ( ) sin cos
BC1: (0) 0 0
BC2: ( ) 0 sin 0 ; 1,2,
; ( ) sin ; 1,2, ,
x x
n n
n n
dy x y x
dx
y x Ce C e
y x C x C x
y C
y a a n na
n ny x C x n
a a
Ax x
Eigenvalues
Transport operator
Differential operator
Matrix
HT2005: Reactor Physics T10: Diffusion of Neutrons 51
Eigenfunctions
0 a
2
1 1 12; ( ) siny x C x
a a
Only one is physically meaningful
HT2005: Reactor Physics T10: Diffusion of Neutrons 52
Solution of a Reactor Equation1
tr
tr
λ0.71RR
λ1.42LL
2 22
2 2
Φ 1 Φ ΦB Φ 0
r r r zΦ(r, z) F(r)G(z)
2( ) ( ) ( )2 2
2 2
2 22 2
2 2
1d F r 1 dF r 1 dG zB 0
F dr Fr dr G dz
1d F 1 dF 1 dGα β
F dr Fr dr G dz
2 2 2B α β
( ) sin cosG z A z C z
Symmetry: 0A
HT2005: Reactor Physics T10: Diffusion of Neutrons 53
( ) n
L π πzG(z) Ccos z z β n G(z) Ccos
2 L L
22 2
2
d F dFx αr x x x F 0
dx dx
0 0
0 0
( ) ( ) ( )
( ) ( ) ( )
F x DJ x EY x
or
F r DJ r EY r
0
00( )
Rr
F r DJR
1 2 3 4 5 6 7 8
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0 ( )J x
0 ( )Y x
0 2.405
HT2005: Reactor Physics T10: Diffusion of Neutrons 54
00
max
0max 0
222 0
rπzΦ(r, z) A cos J
L R
A Φ
rπzΦ(r, z) Φ cos J
L R
πB
L R
B C
0max 0
rπzΦ(r, z) Φ cos J
L R
222 0π
BL R
max
ππ πΦ(r, z) Φ cos cos cos
yx za b c
2 2 22 π π π
Ba b c
max
πsin
Φ( ) Φ
rRrr
22 π
BR
Rectangular
Cylinder
Sphere
HT2005: Reactor Physics T10: Diffusion of Neutrons 55
Critical Size of a ReactorWe assume bare homogenous reactorFor thermal neutrons we get:
2 ( ) ( ) ( , ) 0a thD r r qr
Slowing down neutrons:2 ( , )
( , )qr
qr
Assumption:Reactor is sufficiently big to treat neutron spectrum independently of space variables
2
2
2
22
0
2 2
0
( )( , ) ( ) ( ) ( ) ( ) ( )
( ) 1 ( )( )
( ) ( )
0
( , ) ( )
B
B
Tqr R r T T R r R r
R r dTB T T e
R r T d
R BR
qr R r T e
At the beginning slowing down density is =0 aR r T q r f p 0( ) ( ,0)
p1
HT2005: Reactor Physics T10: Diffusion of Neutrons 56
For > 0 one has to take into account resonance capture through p – resonance passage factor.
2 2 2
2 2
B τ B τ B τ0 a a
R(r) Φ(r) Φ B Φ 0
q(r,τ) R(r)T pe Σ Φ(r)f ηpe Σ Φ(r)k e
2aD Σ Φ q 0
2
2
2
2 B τa a
B τ2
2 2
2 2 B τ
DB Σ Φ Σ Φk e 0
1 k eB 0
L L
(B L 1) k e 0
or
2
2 2 11
Bk eB L
HT2005: Reactor Physics T10: Diffusion of Neutrons 57
Non-Leakage Probability
1NL FNL TNLk k P k P P 2
2 2 11
Bk eB L
2 2 22
1
1
a thV a
TNLaa th th
V V
dVA
PA L DB LBdV D dV
2
2 2
1
1
thBFNL
TNL
P e
PLB
HT2005: Reactor Physics T10: Diffusion of Neutrons 58
Volume of an cylindrical reactor with buckling derived from a critical equation – the smallest critical size:
2 22
22 2
2
min 3
22
2.405We assume that L L and R R
(2.405)
3 2.405 3 1480 gives ; gives
2
1(s )
Generally:big reactor small B-value
BL R
LV R L V R L
BL
dVL R V
dL B B B
Bidesize
HT2005: Reactor Physics T10: Diffusion of Neutrons 59
Minimum Volume
R
L
L = L(R)
V = V(R)
L
D = 1.08 L
BL R
RV
B R
222 0
2 2
220
HT2005: Reactor Physics T10: Diffusion of Neutrons 60
Optimum Core Dimensions
Core shape
Optimum dimensions
Minimal volume
Cube
Cylinder
Sphere
3a b c
B
3
161V
B
03 3;
2L R
B B
3
148V
B
RB
3
130V
B
HT2005: Reactor Physics T10: Diffusion of Neutrons 61
2
2 2
2 2 2 2 2 2 2 22
11
(1 )(1 ) 1 ( ) 11
6
Bk eB L
k k k kB L B B L BM r
B
Migration Area
11
1xe x
x
HT2005: Reactor Physics T10: Diffusion of Neutrons 62
Improved Diffusion
(1) Isotropic Scattering: 1
(2) Boundary Condition: 0.66 0.71
(3) Migration Length:
s s
tr tr
L M
HT2005: Reactor Physics T10: Diffusion of Neutrons 64
CRITICALITY EQUATION - physical interpretation
reactorinfiniteinrateproduction ka
aBk e
2 production rate in the FINITE reactor
2
2 2
2 2 2 2 2 2 2 22
11
(1 )(1 ) 1 ( ) 11
6
Bk eB L
k k k kB L B B L BM r
B
HT2005: Reactor Physics T10: Diffusion of Neutrons 65
e PBs
2 non leakage factor for all epithermal neutrons
Thermal leakage:
D
D a
Thermal non - leakage factor:
1
1
1
11
2
2 2
2 2
2
D
D DB
B LP
k e
B Lk P P
a
a
a
t
B
s t
for critical reactor
HT2005: Reactor Physics T10: Diffusion of Neutrons 66
Z Z ZJ J J
;n vn J v
Derivation
Number of collisions in dV s dV
Neutrons scattered towards dA2
cos
4 4
Ω
s s
d dAdV dV
r
Neutrons through dA per 1 second 2
cos
4
srs
dAdV e
r2 2
0 0 0
( ) cos sin4
srs
r
J r e d drd
HT2005: Reactor Physics T10: Diffusion of Neutrons 67
( ) n
L π πzG(z) Ccos z z β n G(z) Ccos
2 L L
22 2
2
d F dFx αr x x x F 0
dx dx
0 0
0 0
( ) ( ) ( )
( ) ( ) ( )
F x DJ x EY x
or
F r DJ r EY r
1
Rr
DJrF
R405.2
)(
405.2
0
HT2005: Reactor Physics T10: Diffusion of Neutrons 68
Delayed Neutrons( , , , ) ( , , , ) ( , , , )f scE E E E E E r Ω Ω r Ω Ω r Ω Ω
6
10 4 0 4
1(1 )t sc fi i
i
d dE d dE C Qv t
Ω Ω Ω
0 4
6
1
0.0065
ii i i f
ii
CC d dE
t
Ω
( , , , ) ( , ) ( ; , , )
1( ; , , ) ( ; )
4( ; ) ( ; ) ( ; )
( ; ) 1; ( ; ) ( , )
( , )( , , , ) ( ; ) ( , )
4
r Ω Ω r r Ω Ω
r Ω Ω r
r r r
r r r
rr Ω Ω r r
f f f
f
f f
E E E f E E
f E E E E
E E E E E
E E dE E E E
EE E E E