DIFFUSION OF NEUTRONS
OVERVIEW
•
•
•
•
•
Basic Physical Assumptions
Generic Transport Equation
Diffusion Equation
Fermi’s Age Equation
Solutions to Reactor Equation
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
2
Physical Model
4.55  10 10
 
cm;
p
E
E is in eV
E  0.01eV    4.55  10 9 cm
D(H) 10 -7 cm
(a)
HT2005: Reactor Physics
(b)
T10: Diffusion of Neutrons
3
Collision Model
v
χm
θ
v´
rm
b
rc
rc
HT2005: Reactor Physics
T10: Diffusion of Neutrons
4
Initial Definitions
ey
z
v
 W
ey

ex
r
1
v; v  vΩ
v
mv 2
E
2
Ω  ( ,  )
Ω
y
x
d3r  dxdydz;
d3 v  dvx dvy dvz
N (r, v, t )d3rd3 v  Expected number of neutrons in d3r within d3 v
HT2005: Reactor Physics
T10: Diffusion of Neutrons
5
Neutron Density
  
n(r, t ) 

  
N (r, v, t )dvx dvy dvz   N (r, v, t )d3 v 
2  



2
N
(
r
,
Ω
,
v
,
t
)
v
sin  dvd d  

0 0 0
 N (r, Ω, E, t)dΩdE
04
dΩ  sin  d d
v
N (r, v, t )  N (r, Ω, E, t )dΩdE  N (r, v, t )d v
m
N (r, Ω, v, t )  v 2 N (r, v, t )
N (r, Ω, E, t ) 
N (r, Ω, E, t ) 
1
N (r, Ω, v, t )
mv
HT2005: Reactor Physics
T10: Diffusion of Neutrons
6
Angular Flux and Current Density
J (r , v , t )  v N (r , v , t )
J (r, Ω, E, t )  vN (r, Ω, E, t )  Ω vN(r, Ω, E, t )
 ( r ,Ω ,E,t )
 (r, Ω, E, t )  vN(r, Ω, E, t )
J
J (r, Ω, E, t )  Ω (r, Ω, E, t )
dS
J  dS  number of neutrons
crossing dS per 1 second
HT2005: Reactor Physics
T10: Diffusion of Neutrons
7
Generic Transport Equation
 time rate   change due   change due   change due  

 
 
 
 
of
change

to
leakage

to

to
macro
.

 
 
 
   sources
 
 of N
  through S   collisions   forces
 
 

 
 





We ignore
macroscopic forces
Arbitrary volume V

 N 
3
3
3
N
(
r
,
v
,
t
)
d
r


J
(
r
,
v
,
t
)

d
S

d
r

Q
(
r
,
v
,
t
)
d
r






t V
t coll
S
V
V
HT2005: Reactor Physics
T10: Diffusion of Neutrons
8
Generic Transport Equation
  ex




 ey
 ez

x
y
z r
Gauss Theorem:
3
3
3
J
(
r
,
v
,
t
)

d
S



J
(
r
,
v
,
t
)
d
r



v
N
(
r
,
v
,
t
)
d
r

v

N
(
r
,
v
,
t
)
d
r




S
V
V
V
 N

 N 

v


N


Q
 dr  0


V  t
 t coll

N(r, v, t)
 N 
 v  N(r, v, t)  
  Q(r, v, t)
t
 t coll
HT2005: Reactor Physics
T10: Diffusion of Neutrons
9
Substantial Derivative
Leonhard Euler's (1707-1783) description:
N
We fix a small volume 
t
dN  N 

  Q(r, v, t )
dt  t coll
z
r
dN
We let a small volume move 
dt
Joseph Lagrange's (1736-1813) description
y
x
N  N (r , v, t ) 
HT2005: Reactor Physics
dN N r N v N

 
 

dt
t t r t v
N
N
F N

 v
 
t
r
m v
T10: Diffusion of Neutrons
10
Transport (Boltzmann) Equation
dN  N 

  Q(r, v, t )
dt  t coll
N
N
F N  N 
 v
 

 Q
t
r
m v  t coll
N(r, v, t)
 N 
 v  N(r, v, t)  
  Q(r, v, t)
t
 t coll
HT2005: Reactor Physics
T10: Diffusion of Neutrons
11
Collision Term
 cm2 
 s  Ω, E  Ω, E   

sterad

eV


z  Ω, E
 Ω, E
s  Ω, E  Ω, E   s  Ω, E  Ω, E NB
r
y
x

 N 

    (r, Ω, E  Ω, E)vN (r, Ω, E, t)dΩdE  t (r, E)vN (r, Ω, E, t)
 t coll 0 4
Total absorption
(Scattering to the current direction and energy )
HT2005: Reactor Physics
T10: Diffusion of Neutrons
12
Neutron Transport Equation
N(r, v, t)
 N 
 v  N(r, v, t)  
  Q(r, v, t)
t
 t coll
(r, Ω, E, t)  vN(r, Ω, E, t)
1   r, Ω, E, t 
 Ω   t     (r, Ω, E  Ω, E)(r, Ω, E, t)dΩdE  Q
v
t
0 4

: initial condition
(r, Ω, E,0)   0 (r, Ω, E)

(R s , Ω, E, t)  0 Ω  n s  0 : boundary ( free surface) condition
HT2005: Reactor Physics
T10: Diffusion of Neutrons
13
Boundary Condition
Outgoing direction
Ω
Outward normal
ns
Incoming direction
r
Volume V
Ω
z
Rs
V
Surface S
y
x
(r, Ω, E, t) rS  0 when ns  Ω  0
HT2005: Reactor Physics
T10: Diffusion of Neutrons
14
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(r, t )   N (r, Ω, E, t )dΩdE;
HT2005: Reactor Physics
 (r, t )   (r, Ω, E, t )dΩdE
T10: Diffusion of Neutrons
15
Angular Measures
180 Solar disks
HT2005: Reactor Physics
T10: Diffusion of Neutrons
16
Plane Angles
ds  nds
er

n
r

C
R
HT2005: Reactor Physics
R
φ
T10: Diffusion of Neutrons
d 
ds cos  er  ds

r
r
17
Solid Angles
dA  ndA
er  Ω
 n
r
dΩ 
A
 2
R
HT2005: Reactor Physics
T10: Diffusion of Neutrons
dA cos  Ω  dA

2
r
r2
18
One-Group Diffusion Model
•
•
•
•
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
HT2005: Reactor Physics
T10: Diffusion of Neutrons
19
Derivation
Isotropic scattering
JZ  J   J 
s dV  Number of collisions
z

J
dA
y

x
2
dΩ
dA cos 
  s dV
4
4 r 2
Number of neutrons scattered within dΩ
s dV

dA cos  s r
 s dV
e
2
4 r
Number of neutrons reaching dA

2
s
 s r
J 

(
r
)
cos

e
sin  d dr d



4  0  0 r 0
HT2005: Reactor Physics
T10: Diffusion of Neutrons
r = 0 is most important
20
Derivation II
  
  
  
  0  x    y    z    ...
  x 0   y 0   z 0
Taylor’s series at the origin:
x  r sin  cos ; y  r sin  sin ; z  r cos 
  
  
  
 (r )  0  r sin  cos 

r
sin

sin


r
cos







x

y

z

0

0

0

J  s
4
HT2005: Reactor Physics
2



    sr


r
cos


  e sin  d dr d
 0



  z 0 
 0  0 r 0 
2
T10: Diffusion of Neutrons
21
Derivation III
J 
0
4

1   


6 s   z  0
0
1   
J  


4 6 s   z  0
1   
Jz  


3 s   z 0
HT2005: Reactor Physics
Jz  
1   
1   
1   
;
J


;
J




x
y




3 s   z 0
3 s   x 0
3 s   y 0
1  

 
J  ex Jx  e y J y  ez Jz  
 ey
 ez
 ex

3s   x
y
z
T10: Diffusion of Neutrons
22
Fick’s Law
J(r)  
1



 (r);  (r)  e x
 ey
 ez
3s
x
y
z
J (r )   D (r );
CM-System → Lab-System:

1
 s
3s 3
tr   s (1   );
J (r)   D (r);
HT2005: Reactor Physics
D
D
tr 
1
tr

1
 tr
3tr
3
T10: Diffusion of Neutrons
23
Transport Mean Free Path
Information about the
original direction is lost

Transport correction =
A number of anisotropic
collisions is replaced by
one isotropic


s
scos
scos2
t
 tr   s   s cos    s cos    s cos  . . .. .  s cos 
2
r
tr 
s
1  cos 
HT2005: Reactor Physics
;

3


n
 tr   s 1  cos  ; tr  s 1  cos 
T10: Diffusion of Neutrons

24
Diffusion Equation
 Change rate   Production   Leakage   Absorption 


 
 

of
n
rate
rate
rate

 
 
 

 Production 
 n 



(
r
,
t
)

(
r
,
t
)

Q
(
r
,
t
)
f


 cm3s 
rate


 Absorption 
 n 


(
r
,
t
)

(
r
,
t
)
a


 cm3s 
rate


HT2005: Reactor Physics
T10: Diffusion of Neutrons
25
Leakage Rate
Lz  J z ( x, y, z  dz)dxdy  J z ( x, y, z)dxdy 
z
  
 2
   
  D  
    dxdy   D 2 dxdydz
z
 z  z  dz  z  z 
Jz
dz
dx
(x,y,z)
dy
y
x
 2
Lx   D 2 dxdydz
x
 2
Ly   D 2 dxdydz
y
 2
Lz   D 2 dxdydz
z
  2  2  2 
Leakage from a unit volume   D  2  2  2    D 2
 x y z 
HT2005: Reactor Physics
T10: Diffusion of Neutrons
26
Diffusion Equation
 Change rate   Production   Leakage   Absorption 


 
 

of
n
rate
rate
rate

 
 
 

Time-dependent:
Time-independent:
Time-independent
from a steady source
HT2005: Reactor Physics
1 
 D 2   a  Q;
v t
Q  Qext   f 
D2  a  Q  0
D 2   a  Q  0
 2 
1
1
D a s  a tr 
2


Q

0;
L



2

L
D
a
3
 3 
T10: Diffusion of Neutrons
27
Laplace’s Operator
2
2
2



  2 

 2 
2
2
x y z
Cartesian geometry
1  
1 2
2

r
 2
 2 
2
r  r  r r 
z
1  2 
1


1
2
 2
r
 2
sin 
 2 2
r  r  r r sin  
 r sin   2
HT2005: Reactor Physics
T10: Diffusion of Neutrons
Cylindrical geometry
Spherical geometry
28
Symmetries
Slab geometry
Spherical geometry
Cylindrical geometry

z
r
y
r
x
z

2
 
 x2
2 
2
1  2 
r
2
r r r

2 
1  
r
r r r
n = 0 for slab
1 d n d
  n
r
r dr dr
2
HT2005: Reactor Physics
n = 1 for cylindrical
n = 2 for spherical
T10: Diffusion of Neutrons
29
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
30
Interface Conditions

A
B
B
A
z
0
0
1  
1  
J 



 , J 


4 6s   z 0
4 6s   z 0
for +z - direction:
for -z - direction:
A
4

 trA   A  B  trB   B


6 z
4
6 z

 trA   A  B  trB   B


6 z
4
6 z
A
4
HT2005: Reactor Physics
T10: Diffusion of Neutrons
 A  B
DA
 A

 DB B
z
z
31
Boundary Condition
Transport equation

Free surface
Diffusion eq.
J 
0
4

 tr    

  0; 0  0
6   x 0
3
  

0
 
2 tr
  x 0
0.66tr 0.71tr
Straight line extrapolation from x = 0 towards vacuum:
 ( x)  0
for
2
x  tr (exact 0.71)
3
extrapolation length = 0.71  tr
HT2005: Reactor Physics
 ( x)  0 
3
2tr
0  x
1
1
  
  






0
0





x
0.66


x
0.71


0

0
tr
tr
T10: Diffusion of Neutrons
32
Plane Infinite Source in Infinite Medium
Transport equation
Q0
d 2
D 2   a ( x)  Q( x)  0
dx
Q0 ( x)
d 2 1
Q( x)
 2  ( x)  

2
dx L
D
D
 ( x)
3s
x=0
d 2 1
x L
x L


(
x
)

0


(
x
)

Ae

Be
dx 2 L2
lim  ( x)  0  B  0
x 
 ( x) 
Q0
Q0 L
lim J ( x) 
A
x 0
2
2D
HT2005: Reactor Physics
T10: Diffusion of Neutrons
Q0 L  x
e
2D
L
33
Point Source in Infinite Medium
1 d 2 d
r
  a (r )  Q( x)  0
2
r dr dr
1 d 2 d 1
r
 2  (r )  0 r  0
2
r dr dr L
D
r
e r L
e r L
 (r )  A
B
r
r
lim  (r )    B  0
r 
Q0
lim 4 r J (r )  Q0  A 
r 0
4 D
Q0 e r L
 (r ) 
4 D r
2

n abs. (r, r  dr) a (r)4 r 2dr r r L
2
p(r)dr 

 2 e dr  r   r 2 p(r )dr  6L2
Q0
Q0
L
0
HT2005: Reactor Physics
T10: Diffusion of Neutrons
34
Plane Infinite Source in Slab Medium
1
1
  





a 2
a2



x
0.71

a

a 2
tr
Q0
 ( x) 
Slab:
a2 x
sinh
QL
2L
 ( x)  0
2D cosh a
2L
 ( x)
x = -a/2
x=0
HT2005: Reactor Physics
Q0 L  x
e
2D
Infinite:
x = a/2
T10: Diffusion of Neutrons
L
35
Plane Infinite Source with Reflector
d21 1
 2 1 (x)  0
2
dx
L1
2
1
Q0
1
Reflector
2
Reflector
a
d22 1
 2 2 (x)  0
2
dx
L2
HT2005: Reactor Physics
T10: Diffusion of Neutrons
Bare slab
36
Age of Neutrons
• q(E) - number of neutrons, which per cubiccentimeter and second pass energy E.
• q(E) = [ncm-3 s-1]
• X-sections depend on E: D(E),Σs(E),...
Energy
Q
E0
D(E) dE
E t (E) E   (E)
E0
 cm2 
E
Slowing down medium: s
a  s  t

Ef
D( E) dE D log E f Eth
 th   ( Eth )  

t ( E) E s

Eth
  1

 th  D  s  ns  D  Lmts
Mean Total Slowing
down distance
HT2005: Reactor Physics
T10: Diffusion of Neutrons

1
q(E)
ln 
1 2
 rs
6
Can be shown
37
Fermi’s Age Equation
 (r, E)dE is the number of neutrons at r with energies in (E, E  dE)
D(E)2 (r, E)dE  a (r, E)dE  Q(r, E)dE  0
q(E+dE)
q(r, E)
Q(r, E)dE  q(r, E  dE)  q(r, E) 
dE
E
D( E) 2 (r, E)dE   a (r, E)dE 
q(E)
E+dE
E
q(r, E)
dE  0
E
du 
q(E) dE

Continuous
slowing
down:

(
E
)
dE

 
t (E) E

 (u)du   (E)dE

t (u) (u)du  q(u)
q(r, E)
a ( E)
D(E)
2
 q(r, E) 
q(r, E) 
0
t (E)E
t (E)E
E
HT2005: Reactor Physics
T10: Diffusion of Neutrons
38
Fermi’s Age Equation II
q(r, E)
a ( E)
D(E)
2
 q(r, E) 
q(r, E) 
0
t (E)E
t (E)E
E
a (E) dE
qˆ (r, E)  q(r, E)  exp 
;
t (E) E
E
E0
qˆ(r, E)  q(r, E)  a ( E)  0
qˆ (r, E)
D(E)

2qˆ (r, E)
E
t (E)E
D(E) dE
new variable:  (E)  
t (E) E
E
E0
qˆ (r, )
  2 qˆ (r, )

τ ~ time
HT2005: Reactor Physics
T10: Diffusion of Neutrons
39
Solutions to the Age Equation
q  2 q
 2
 x
No absorption
 x2 
exp   
4 

qpl ( x, )  Q0
12
 4 
x=0
r
q 1  2 q
 2 r
 r r r
HT2005: Reactor Physics
No absorption
T10: Diffusion of Neutrons
 r2 
exp   
4 

q pt (r , )  Q0
32
 4 
40
Slowing Down Density for
Different Fermi’s Ages
Q (r) ( )
q(r,)
 r2 
exp   
4 

q pt (r , )  Q0
32
 4 
 0
0.08
 =0.5
 =1.0
 =1.5
0.06
0.04
0.02
0.00
-6
HT2005: Reactor Physics
-4
-2
T10: Diffusion of Neutrons
r
0
2
6
4
41
Migration Area (Length)
Fast neutron
borne
r
1
L2  rth2
6
M2 
rs
rth
1 2
r
6
1 N 2
r   ri ;
N i 1
2
1 N 2
r   rs ,i ;
N i 1
2
s
1 N 2
r   rth ,i
N i 1
2
th
r  rs  rth
r   rs  rth   r  2rs  rth  r
2
s
2
th
r  r  2rs  rth  r  r  r
2
2
s
Fast neutron
thermalized

2
2
Thermal neutron
absorbed
2
th
2
s
2
th
r2 
2
2
r
q
(
r
,

)4

r
dr
 pt
0

q
pt
(r, )4 r 2dr
 6  rs2  6 th
0
M2   th  L2  L2s  L2
HT2005: Reactor Physics
T10: Diffusion of Neutrons
42
Diffusion and Slowing Down
Parameters for Various Moderators
Moderator
g/cm3
1.0
tr
cm
0.43
L
cm
2.7
tth
ms
0.21
H2O
D 2O
(pure)
D 2O
(normal)
Be
1.1
2.5
165
1.1
2.5
1.8
BeO
C (pure
graphite)
C (normal.

0.92
ts
s
1
0
cm2
27
130
0.51
8
131
100
50
0.51
8
115
1.5
22
3.8
0.21
10
102
2.96
1.4
31
8.1
0.17
12
100
1.6
2.6
59
17
0.158
24
368
1.6
2.6
50
12
0.158
24
368
graphite)
HT2005: Reactor Physics
T10: Diffusion of Neutrons
43
Neutrons in Multiplying Medium
n
 D 2    a   Q
t
n(r, E, t)
2
dE

D
(
E
)

(r, E, t )dE   a (E)(r, E, t )dE   Q(r, E, t )dE
th t
th
th
th
Assumption:
 (r, E, t)dE  th (r, t);
th
  (E)(r, E, t)dE  
a
ac
(r, E, t)  F(r)  G( E)  T (t)
 n(r, E, t)dE 
th
 th (r, t )
;
vav
 th (r, t );
Dc 
 D(E)(r, E, t )dE
th
 th
th
1 th (r, t )
 Dc2th (r, t )  ac th (r, t )  Qth (r, t )
vav
t
HT2005: Reactor Physics
T10: Diffusion of Neutrons
44
Principles of a Nuclear Reactor
E
Leakage
N2
2 MeV
N2
k
N1
Fast fission
Resonance abs.
ν ≈ 2.5
Non-fissile abs.
1 eV
Slowing down
 n/fission
Energy
N1
Non-fuel abs.
Fission
200 MeV/fission
HT2005: Reactor Physics
Leakage
T10: Diffusion of Neutrons
45
Total number of fission neutrons
Fast fission factor  
Number of fission neutrons from thermal neutrons
Resonance escape probability p( E)  e

E0

1.02
dE
a
 a 
s E
E
0.87
 Ff
Conditional probability Pf  F  F
a  a
Ff
 Ff
Number of neutrons per absorption in fuel    Pf   F
a
aF
Thermal utilization f 
a
0.71
Fast non-leakage probability
PFNL
0.97
Thermal non-leakage probability PTNL
0.99
Non-leakage probability PNL  PFNL  PTNL
HT2005: Reactor Physics
1.65
T10: Diffusion of Neutrons
k   fp
k  k  PNL   fp PFNL PTNL
46

p  f  th dV
Rate of neutron production in core Core
k 

Rate of neutron absorption in core
 ath dV
Core
k a  p  f  Qth  p  f th  k ath
1 th (r, t )
 Dc2th (r, t )  ac th (r, t )  Qth (r, t )
vav
t
k 1
1 th (r, t)
 2th (r, t)   2 th (r, t)
t
Lc
vav Dc
L2c 
Dc
 ac
k  1
B  2
Lc
2
m
2

th (r)
In the stationary case: Bm2  
th (r)
HT2005: Reactor Physics
T10: Diffusion of Neutrons
47
Buckling as Curvature
Large core
 
B 

2
Bc
Bb
BL
Ba  0
BS
Small core
Ba  Bb  Bc
HT2005: Reactor Physics
T10: Diffusion of Neutrons
48
Criticality Condition
1 th (r, t)
 2th (r, t)  Bm2 th (r, t)
t
vav Dc
th (r, t)  F(r)  T (t)
1
dT (t) 2 F(r)

 Bm2
F (r)
vav DcT (t) dt
2 F (r)
 Bg2  2 F(r)  Bg2 F(r)
F (r)
2 F(r)   F(r)
k  1
In a critical reactor: B  B  2
Lc
2
g
HT2005: Reactor Physics
2
m
T10: Diffusion of Neutrons
49
Eigenvalues
Matrix
n  n matrix: Ax   x
d2
Differential operator:
y ( x)   y ( x)
2
dx
  0  y ( x)  C1e
x
  0  y ( x)  C1 sin
 C2e 

Differential operator
x

 x  C 2 cos

 x

Transport operator
BC1: y (0)  0  C 2  0
BC2: y (a)  0  sin


 a  0    n
n2 2
 n 
n   2 ; yn ( x)  C1 sin 
x ;
a
 a 
HT2005: Reactor Physics

a
n  1, 2,
; n  1, 2,
,
T10: Diffusion of Neutrons
50
Eigenfunctions
Only one is
physically
meaningful
0
a
1 
HT2005: Reactor Physics
2

;
y
(
x
)

C
sin
1
1

a2
a
T10: Diffusion of Neutrons

x

51
Solution of a Reactor Equation
1
L  L 1.42λtr
R  R 0.71λtr
2Φ 1 Φ 2Φ
2



B
Φ0
2
2
r
r r z
Φ(r, z)  F(r) G(z)
1 d 2 F (r ) 1 dF (r ) 1 d 2G( z)
2



B
 0 B2  α2  β2
2
2
F dr
Fr dr
G dz
G(z)  A sin  z  C cos  z
1 d 2 F 1 dF
2



α
F dr 2 Fr dr
HT2005: Reactor Physics
1 d 2G
2


β
G dz 2
Symmetry: A  0
T10: Diffusion of Neutrons
52
L
G(z)  Ccos(  z)  z  

2
βn  n
F ( x)  DJ0 ( x)  EY0 ( x)
2
d
F
dF
2
2
x  αr  x

x

x
F 0
2
dx
dx
1
π
 πz 
 G(z)  Ccos  
L
 L 
or
F (r )  DJ0 ( r )  EY0 ( r )
J 0 ( x)
0.8
0.6

0.4
0.2
0
R
 0r 
F (r )  DJ 0 

 R 
0  2.405
-0.2
0
-0.4
-0.6
-0.8
Y0 (x)
-1
1
2
3
HT2005: Reactor Physics
4
5
6
7
8
T10: Diffusion of Neutrons
53
 πz    0 r 
Φ(r, z)  Acos 
J0 


 L   R  
A  B  C  Φ max
 πz    0 r 
Φ(r, z)  Φ max cos 
 J0   

L   R 
 
 π 
B2      0 
 L 
 R 
2
Rectangular
Cylinder
Sphere
2
 πy
 πx
 πz
Φ(r, z)  Φ max cos 
cos
cos 




 a 
 c 
 b 
 πz    0 r 
Φ(r, z)  Φ max cos 
 J0 

 L   R  
Φ(r )  Φ max
HT2005: Reactor Physics
 πr 
sin 

 R 
r
T10: Diffusion of Neutrons
2
2
π π π
B       
 a   b   c 
2
 π   
B     0 
 L   R 
2
2
2
 π 
B  
 R 
2
2
54
2
Critical Size of a Reactor
We assume bare homogenous reactor
For thermal neutrons we get:
Slowing down neutrons:
D2(r)  a(r)  q(r , th )  0
 q(r , )
 q(r , ) 

2
Assumption:
Reactor is sufficiently big to treat neutron spectrum independently of space variables
 T ( )
q(r , )  R(r )T ( )  T ( ) R(r )  R(r )

 2 R(r )
1 dT ( )

  B2  T ( )  T0 e  B 
R(r )
T ( ) d
2
2
 2 R  B2 R  0
p1
q(r , )  R(r )T0 e  B 
2
At the beginning slowing down density is
=0
HT2005: Reactor Physics
R(r)T0  q(r ,0)  af p
T10: Diffusion of Neutrons
55
For > 0 one has to take into account resonance
capture through p – resonance passage factor.
Φ(r)   2 Φ  B2Φ  0
R(r)
q(r, τ)  R(r) T0 pe
 ΣaΦ(r) f  ηpe
 B2 τ
 B2τ
 ΣaΦ(r) k e
 B2τ
D 2  ΣaΦ q  0
 DB   Σa Φ  Σa Φk  e
or
2
 B2 
 B2 τ
1 k e

2
L
L2
 (B L  1)  k  e
2
 B2 τ
2
HT2005: Reactor Physics
 B2 τ
0
0
0
 B2
k e
1
2 2
1 B L
T10: Diffusion of Neutrons
56
Non-Leakage Probability
k e  B 
1
2 2
1 B L
2
k  k  PNL  k  PFNL  PTNL  1
PTNL 
A

A L
 
a
th
dV
V
2


dV

D

 th dV
a
th


V
a
1

 a  DB2 1  L2 B2
V
PFNL  e
PTNL 
HT2005: Reactor Physics

 B2 th
1
1  L2 B2
T10: Diffusion of Neutrons
57
Volume of an cylindrical reactor with
buckling derived from a critical
equation – the smallest critical size:

2.405 
B2     

 L  R 
2
2
We assume that L  L and R   R
 L(2.405)2
V  R L 
2
 
V R L
2
2
B  
 L
dV
0
dL
L
gives
B2 
 3
B
; R
2.405 3
B
2
gives
Vmin 
148
B3
1
(side size)2
Generally : big reactor  small B-value
HT2005: Reactor Physics
T10: Diffusion of Neutrons
58
Minimum Volume
   0 

2
B     
2
2
L
 L
V
R
 2 R2
B   0 R 
2
V = V(R)
2
L = L(R)
L
D = 1.08 L
HT2005: Reactor Physics
T10: Diffusion of Neutrons
R
59
Optimum Core Dimensions
Core
shape
Cube
Cylinder
Sphere
HT2005: Reactor Physics
Optimum
dimensions
Minimal
volume
3
B
161
V  3
B
abc
L
3
; R
B
R
3 0
2 B

B
T10: Diffusion of Neutrons
V 
148
B3
V 
130
B3
60
Migration Area
 B2
k e
1
2 2
1 B L

k
k
k



2 2
2
2
2
2
2
(1  B L )(1  B  ) 1  B ( L   ) 1  B M
ex  1  x 
k
r2
1 B
6
2
1
1 x
HT2005: Reactor Physics
T10: Diffusion of Neutrons
61
Improved Diffusion
 s   s 1   
(1)
Isotropic Scattering:
(2)
Boundary Condition: 0.66  tr  0.71 tr
(3)
Migration Length:
HT2005: Reactor Physics
LM
T10: Diffusion of Neutrons
62
The END
HT2005: Reactor Physics
T10: Diffusion of Neutrons
63
CRITICALITY EQUATION - physical interpretation
a k productionrateininfinite
reactor
 a ke
 B 2
 production rate in the FINITE reactor
 B2
k e
1
2 2
1 B L

k
k
k



2 2
2
2
2
2
2
(1  B L )(1  B  ) 1  B ( L   ) 1  B M
HT2005: Reactor Physics
T10: Diffusion of Neutrons
k
r2
1 B
6
2
64
e
 B 2
 Ps  non leakage factor for all epithermal neutrons
Thermal leakage:
 D
 D   a
Thermal non - leakage factor:
 D
 a
1

 D   a DB2   a
1
 Pt
2 2
B L 1
 B 2
k e

 k Ps Pt  1 for critical reactor
2 2
1 B L
HT2005: Reactor Physics
T10: Diffusion of Neutrons
65
Derivation
JZ  JZ  JZ
J  vn;
 s dV
Number of collisions in dV
Neutrons scattered towards dA
Neutrons through dA per 1 second
s
J 
4
HT2005: Reactor Physics
2
  vn

2
 
dΩ
dA cos 
  s dV
4
4 r 2
dA cos   s r
 s dV
e
2
4 r
 s dV

 r

(
r
)
cos

e
sin  d dr d

s
0 0 r 0
T10: Diffusion of Neutrons
66
L
G(z)  Ccos(  z)  z  

2
2
d
F
dF
2
2
x  αr  x

x

x
F 0
2
dx
dx
1
βn  n
π
 πz 
 G(z)  Ccos  
L
 L 
F ( x)  DJ0 ( x)  EY0 ( x)
or
F (r )  DJ0 ( r )  EY0 ( r )

2.405
R
 2.405r 
F (r )  DJ0 


 R 
HT2005: Reactor Physics
T10: Diffusion of Neutrons
67
Delayed Neutrons
(r, Ω, E  Ω, E)   f (r, Ω, E  Ω, E)  sc (r, Ω, E  Ω, E)


6
1 
 Ω   t     sc dΩdE  (1   )   f dΩdE   iCi  Q
v t
i 1
0 4
0 4

Ci
 iCi   i    f dΩdE
t
0 4
6
    i  0.0065
i 1
 f (r, Ω, E   Ω, E )   f (r, E ) f f (r; Ω, E   Ω, E )
1
(r; E   E )
4
(r; E   E )  (r; E   E )(r; E )
f f (r; Ω, E   Ω, E ) 
 (r; E   E )dE  1;
 f (r, Ω, E   Ω, E ) 
HT2005: Reactor Physics
T10: Diffusion of Neutrons
(r; E   E )  (r, E )
(r, E )
(r; E ) f (r, E )
4
68
1
Optimum dimensions and critical mass of a
cylindrical core
HT2005: Reactor Physics
T10: Diffusion of Neutrons
69