Transcript SLOWING DOWN - Royal Institute of Technology

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
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2
4.55  1010
 
cm;
p
E
E is in eV
E  0.01eV    4.55  10 9 cm
(a)
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(b)
Diffusion of Neutrons
3
v
χm
θ
v´
rm
b
rc
rc
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Diffusion of Neutrons
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Initial Definitions
ey
z
v
 W
ey

ex
r
1
v; v  vΩ
v
mv 2
E
2
Ω  ( ,  )
Ω
y
x
dr  dxdydz; dv  dvxdvydvz
N(r, v, t)drdv  Expected number of neutrons in dr within dv
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  
n(r, t ) 
   N(r, v, t)dv dv dv   N(r, v, t)dv 
x
y
z
  
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 )dv
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
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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
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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 
N
(
r
,
v
,
t
)
d
r


J
(
r
,
v
,
t
)

d
S

S
V  t coll dr  V Q(r, v, t)dr
t V
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Generic Transport Equation
  ex




 ey
 ez

x
y
z r
Gauss Theorem:
 J(r, v, t)  dS     J(r, v, t)dr     vN(r, v, t)dr   v  N(r, v, t)dr
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
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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 ) 
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dN N r N v N

 
 

dt
t t r t v
N
N
F N

 v
 
t
r
m v
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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
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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 )
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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 
 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
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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
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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 ) 
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Diffusion of Neutrons
(r; E   E )  (r, E )
(r, E )
(r; E ) f (r, E )
4
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(r, t )   N (r, Ω, E, t )dΩdE;
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 (r, t )   (r, Ω, E, t )dΩdE
Diffusion of Neutrons
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Angular Measures
180 Solar disks
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17
Plane Angles
ds  nds
er

n
r

C
R
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R
d 
φ
Diffusion of Neutrons
ds cos  er  ds

r
r
18
Solid Angles
dA  ndA
er  Ω
 n
r
dΩ 
A
 2
R
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Diffusion of Neutrons
dA cos  Ω  dA

2
r
r2
19
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
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Derivation
JZ  JZ  JZ
J  vn;
 s dV
Number of collisions in dV
Neutrons through dA per 1 second
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2

2

  
dΩ
dA cos 
  s dV
4
4 r 2
dA cos   s r
 s dV
e
2
4 r
 s dV
Neutrons scattered towards dA
s
J 
4
  vn
r

s
 (r ) cos  e
sin  d dr d

 0   0 r  0
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21
  
  
  
  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
sin

sin


  r cos  



y

z
  x 0

0

0
 (r )  0  r sin  cos 
J


2
s
4
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
2
 
 0
0


      s r


r
cos

sin  d  d r d
 0

 e

  z 0 
 0 
Diffusion of Neutrons
22
Jz 
Jz 
0
4
0
4


1
  


  z 0
1
  


  z 0
6 s
6 s
Jz  Jz  Jz  
Jx  
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1
3 s
1
3 s
  


  z 0
  

 och
x
Jy  
Diffusion of Neutrons
1
3 s
  



y


23
Fick’s Law
J (r )  
1



 (r );  (r )  i
j z
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 );
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D
D
tr 
1
tr

1
 tr
3tr
3
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24
Transport Mean Free Path



s
scos
scos2
t
 tr   s   s cos    s cos    s cos  . . . . .  s cos 
r
tr 
s
1  cos 
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2
3

 tr   s 1  cos 
Diffusion of Neutrons
n

25
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

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Diffusion of Neutrons
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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
 2
 2
Lx   D 2 dxdydz; Ly  D 2 dxdydz
x
y
x
  2  2  2 
Leakage from a unit volume  D  2  2  2    D2
 x y z 
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Diffusion Equation
 Change rate   Production   Leakage   Absorption 





of
n
rate
rate
rate

 
 
 

Time-dependent:
Time-independent:
Time-independent
from a steady source
HT2004: 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 
Diffusion of Neutrons
28
Laplace’s Operator
2
2
2
 
 2 2 
2
x y z
2
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
2 

1  
1  2 
1 d n d

r

r

r
2
2
n
 x r  r  r r  r  r r dr dr
Cylindrical geometry
Spherical geometry
n = 0 for slab
2
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Diffusion of Neutrons
n = 1 for cylindrical
n = 2 for spherical
29
1.
2.
3.
4.
CONDITIONS:
The neutron flux finite and non-negative.
 - symmetric if there is any symmetry in the system
Boundary conditions for interface between two different media:
neutron flux and neutron current density are continuous
No return from a free surface - the flux becoming zero at extrapolated length.
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Diffusion of Neutrons
30
Interface between 2 different
media:
1
Jz  
0
4

1 
0
1  
,
J






z
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
HT2004: Reactor Physics
Diffusion of Neutrons
 A  B
DA
 A

 DB B
z
z
31
Boundary Condition
Transport eq.

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
HT2004: Reactor Physics
 ( x)  0 
3
2tr
0  x
1
1
  
  






0
0





x
0.66


x
0.71


0

0
tr
tr
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
HT2004: Reactor Physics
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  6 L2
Q0
Q0
L
0
HT2004: Reactor Physics
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
QL
2L
 ( x)  0
2D cosh a
2L
 ( x)
x = -a/2
Q0 L  x
e
2D
Infinite:
L
sinh
x=0
HT2004: Reactor Physics
x = a/2
Diffusion of Neutrons
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
HT2004: Reactor Physics
Diffusion of Neutrons
Bare slab
36
Age of Neutrons
Energy
• 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),...
E0
E0
D( E) dE
E t (E) E   (E)
E
Slowing down medium: s
Ef
 a   s  t

D( E) dE D log E f Eth
 th   ( Eth )  

t ( E) E  s

Eth
  1

 th  D  s  ns  D  Lmts
HT2004: Reactor Physics
Diffusion of Neutrons
Q
 cm2 
q(E)

ln 
1
1 2
 rs
6
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
q( E) dE
Continuous slowing down:  (E)dE 
t (E) E
q(r, E)
a (E)
D( E)
2
 q(r, E) 
q(r, E) 
0
t (E)E
t (E)E
E
HT2004: Reactor Physics
Diffusion of Neutrons
38
Fermi’s Age Equation (cont)
q(r, E)
 (E)
D( E)
2q(r, E)  a
q(r, E) 
0
t (E)E
t (E)E
E
 a ( E) dE
;
t ( E) E
E
E0
qˆ (r, E)  q(r, E)  exp 
qˆ (r, E)  q(r, E)   a ( E)  0
qˆ (r, E)
D(E)

2qˆ (r, E)
E
t (E)E
new variable:  ( E) 
HT2004: Reactor Physics
E0
D( E) dE
E t (E) E
qˆ (r, )
 2qˆ (r, )

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
HT2004: Reactor Physics
No absorption
Diffusion of Neutrons
 r2 
exp   
4 

qpt (r , )  Q0
32
 4 
40
Slowing Down Density for
Different Fermi’s Ages
q(r,)
 r2 
exp   
4 

qpt (r , )  Q0
32
 4 
0.08
 =0.5
 =1.0
 =1.5
0.06
0.04
0.02
0.00
-6
HT2004: Reactor Physics
-4
-2
Diffusion of Neutrons
r
0
2
4
6
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
Fast neutron
thermalized

r   rs  rth   r  2rs  rth  r
2
Thermal neutron
absorbed
2
2
s
2
th
r 2  rs2  2rs  rth  rth2  rs2  rth2
r2 
2
2
r
q
(
r
,

)4

r
dr
 pt
0

q
pt
(r, )4 r 2dr
 6  rs2  6 th
0
M 2   th  L2  L2s  L2
HT2004: Reactor Physics
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
D2O
(pure)
D2O
(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)
HT2004: Reactor Physics
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
(r, E, t )  F (r )  G( E)  T (t )
 n(r, E, t)dE 
th
 th (r, t )
;
vav
 a (E)(r, E, t)dE  acth (r, t);
 D(E)(r, E, t)dE
Dc  th
th
th
1  th (r, t )
 Dc  2  th (r, t )   ac  th (r, t )  Qth (r, t )
vav
t
HT2004: Reactor Physics
Diffusion of Neutrons
44
Principles of a Nuclear Reactor
E
Leakage
N2
2 MeV
N2
k
N1
Resonance abs.
ν ≈ 2.5
Non-fissile abs.
1 eV
Fast fission
Slowing down
 n/fission
Energy
N1
Non-fuel abs.
Fission
200 MeV/fission
HT2004: Reactor Physics
Leakage
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

dE
 a a s  E
E
1.02
0.87
 Ff
Conditional probability Pf  F  F
a  a
 Ff
 Ff
Number of neutrons per absorption in fuel    Pf   F 1.65
a
aF
Thermal utilization f 
0.71
a
k   fp
Fast non-leakage probability
PFNL 0.97
Thermal non-leakage probability PTNL 0.99
k  k  PNL   fp PFNL PTNL
Non-leakage probability PNL  PFNL  PTNL
HT2004: Reactor Physics
Diffusion of Neutrons
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 )
HT2004: Reactor Physics
Diffusion of Neutrons
47
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)  0
F (r)
In a critical reactor: Bg2  Bm2
HT2004: Reactor Physics
Diffusion of Neutrons
48
Solution of a Reactor Equation
1
L  L  1.42 λtr
R  R  0.71 λtr
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 1 dF 1 d 2G
2



B
0
2
2
F dr
Fr dr G dz
B2  α 2  β 2
1 d 2 F 1 dF
2


α
F dr 2 Fr dr
Symmetry: A  0
HT2004: Reactor Physics
1 d 2G
2


β
G dz 2
G( z)  A sin  z  C cos  z
Diffusion of Neutrons
49
L
G(z)  Ccos(  z)  z  

2
2
d
F
dF
x  αr  x 2 2  x
 x2F  0
dx
dx
1
β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 )  DJ 0 

 R 
HT2004: Reactor Physics
Diffusion of Neutrons
50
 πz   2.405 r 
Φ(r, z)  Acos 
J0 


 L   R  
A  B  C  Φ max
 πz   2.405 r 
Φ(r, z)  Φ max cos 
J0 


 L   R  
2
 π 
 2.405 
B2     

 L 
 R 
2
Rectangular
πy
 πx
 πz
Φ(r, z)  Φ max cos 
cos
cos 




 a 
 c 
 b 
π
π
π
B       
 a 
 b 
 c 
Cylinder
 πz   2.405 r 
Φ(r, z)  Φ max cos 
 J0 

 L   R  
 π 
 2.405 
B    

 L 
 R 
Sphere
Φ(r )  Φ max
HT2004: Reactor Physics
 πr 
sin 

 R 
r
Diffusion of Neutrons
2
2
2
2
2
2
 π 
B 

 R 
2
2
51
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
q(r , )  R(r )T0 e
 B2
At the beginning slowing down density is
=0
HT2004: Reactor Physics
R(r)T0  q(r ,0)  af
Diffusion of Neutrons
52
For > 0 one has to take into account resonance
capture through p – resonance passage factor.
R(r)
Φ(r)   2 Φ  B2Φ  0
q(r, τ)  R(r) T0 pe
 Σa Φ(r) f  ηpe
 B2 τ
 B2 τ
 Σ aΦ(r) k e
 B2 τ
D  2  Σa Φ  q  0
 DB2   Σa Φ  Σa Φk  e  B τ  0
2
or
 B2 
 B2 τ
1 k e

2
L
L2
 (B L  1)  k  e
2
2
HT2004: Reactor Physics
 B2 τ
0
0
 B2
k e
1
2 2
1 B L
Diffusion of Neutrons
53
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
HT2004: Reactor Physics
Diffusion of Neutrons
54
1
Optimum dimensions and critical mass of a
cylindrical core
HT2004: Reactor Physics
Diffusion of Neutrons
55
CRITICALITY EQUATION - physical interpretation
 a k production rate in infinite 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
HT2004: Reactor Physics
Diffusion of Neutrons
k
r2
1 B
6
2
56
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
HT2004: Reactor Physics
Diffusion of Neutrons
57