Transcript Document

Fick’s Law
• The exact interpretation of neutron transport in
heterogeneous domains is so complex.
• Assumptions and approximations.
• Simplified approaches.
• Simplified but accurate enough to give an estimate of
the average characteristics of neutron population.
• Numerical solutions.
• Monte Carlo techniques.
Nuclear Reactor Theory, JU, Second Semester, 2008-2009
(Saed Dababneh).
1
Fick’s Law
Assumptions:
1. The medium is infinite.

2. The medium is uniform  not (r )
3. There are no neutron sources in the medium.
4. Scattering is isotropic in the lab coordinate system.
5. The neutron flux is a slowly varying function of
position.
6. The neutron flux is not a function of time.
http://www.iop.org/EJ/article/0143-0807/26/5/023/ejp5_5_023.pdf
Nuclear Reactor Theory, JU, Second Semester, 2008-2009
(Saed Dababneh).
2
Fick’s Law
Lamarsh puts it more bluntly:
“Fick’s Law is invalid:
a) in a medium that strongly absorbs neutrons;
b) within three mean free paths of either a neutron
source or the surface of a material; and
c) when neutron scattering is strongly anisotropic.”
Nuclear Reactor Theory, JU, Second Semester, 2008-2009
(Saed Dababneh).
3
Fick’s Law
Nuclear Reactor Theory, JU, Second Semester, 2008-2009
(Saed Dababneh).
4
Fick’s Law
• Diffusion: random walk of(x)
Negative Flux Gradient
Current J x
Current Jx
an ensemble of particles
from region of high
“concentration” to region
High flux
of small “concentration”.
dC/dx
• Flow is proportional to
More collisions
the negative gradient of
Low flux
the “concentration”.
Recall:
• From larger flux to smaller
x flux!
• Neutrons are not pushed!
• More scattering in one direction
than in the other.
Nuclear Reactor Theory, JU, Second Semester, 2008-2009
(Saed Dababneh).
Less collisions
x

J x  D
x
5
Fick’s Law
z
d

r
dAz
Number of neutrons scattered per
second from d at r and going
through dAz
 cos dAz t r
 s  (r )
e d
2
4r
y


s not s (r )
x
Removed
en route
(assuming no
buildup)
Slowly varying
Isotropic
Nuclear Reactor Theory, JU, Second Semester, 2008-2009
(Saed Dababneh).
6
Fick’s Law
Nuclear Reactor Theory, JU, Second Semester, 2008-2009
(Saed Dababneh).
7
Fick’s Law
HW 14
 s dAz
J dAz 
4

z
2  / 2 
 t r
  (r )e cos  sin drdd 
 
0 0 r 0

z
J dAz  ?
  s   
and show that J z  J  J   2 

 3t  z 0

s
Diffusion
and generalize J   D
coefficient D 
2
3t

z

z
D
1
3 s
Total removal
The current density is proportional to the negative of the gradient
of the neutron flux.
Nuclear Reactor Theory, JU, Second Semester, 2008-2009
(Saed Dababneh).
8
Fick’s Law
Validity:
1. The medium is infinite. Integration over all space.
e  t r  after few mean free paths  0 
corrections at the surface are still required.

2. The medium is uniform. s nots (r )
s (r )   and  are functions of space  rederivation of Fick’s law?  locally larger s  extra
J cancelled by e t r  e( a   s ) r iff ???
HW 15
Note: assumption 5 is also violated!
3. There are no neutron sources in the medium.
Again, sources are few mean free paths away and
corrections otherwise.
Nuclear Reactor Theory, JU, Second Semester, 2008-2009
(Saed Dababneh).
9
Fick’s Law
4. Scattering is isotropic in the lab. coordinate system.
2


cos
(

)

 0  reevaluate D.
If
HW 16
3A
tr
1
1
D


3(t   s  ) 3 tr
3
Weekly absorbing t = s.
s
For “practical” moderators: tr 
1 
5. The flux is a slowly varying function of position.
a   variation in  .
 2 
r
2
(r )
Nuclear Reactor Theory, JU, Second Semester, 2008-2009
(Saed Dababneh).
?
10
Fick’s Law
HW 17
Estimate the diffusion coefficient of graphite at 1 eV.
The scattering cross section of carbon at 1 eV is 4.8 b.
Scattering
Other
materials?
Absorption
Nuclear Reactor Theory, JU, Second Semester, 2008-2009
(Saed Dababneh).
11
Fick’s Law
6. The neutron flux is not a function of time.
Time needed for a thermal neutron to traverse 3
mean free paths  1 x 10-3 s (How?).
If flux changes by 10% per second!!!!!!


1ms
 / 

1ms  0.1x103  1x104
1s
Very small fractional change during the time
needed for the neutron to travel this “significant”
distance.

J  D
Nuclear Reactor Theory, JU, Second Semester, 2008-2009
(Saed Dababneh).
12
Back to the Continuity Equation
  

 
1  
 (r , t )  S (r , t )   a (r ) (r , t )    J (r , t )
v t


 

 
1  
 (r , t )  S (r , t )   a (r ) (r , t )    D (r , t )
v t
Nuclear Reactor Theory, JU, Second Semester, 2008-2009
(Saed Dababneh).
13
The Diffusion Equation

 

 
1  
 (r , t )  S (r , t )   a (r ) (r , t )    D (r , t )
v t
If D is independent of r (uniform medium)
Laplacian

 

1  
2
 (r , t )  S (r , t )   a (r ) (r , t )  D  (r , t )
v t
or scalar Helmholtz equation.

 

2
0  S (r )  a (r ) (r )  D  (r )
 

2
0   a (r ) (r )  D  (r )
Nuclear Reactor Theory, JU, Second Semester, 2008-2009
(Saed Dababneh).
Buckling equation.
14
Steady State Diffusion Equation

 

2
0  S (r )  a (r ) (r )  D  (r )
D
2
Define
L  Diffusion Length
L 
L2  Diffusion Area
a

1
S
  2 
L
D
1
2
  2 0
L
Moderation Length
2
Nuclear Reactor Theory, JU, Second Semester, 2008-2009
(Saed Dababneh).
Boundary Conditions
• Solve DE  get .
• Solution must satisfy BC’s.
• Solution should be real and
non-negative.
15
Steady State Diffusion Equation
One-speed neutron diffusion in infinite medium
Point source

1 
2
  (r )  2  (r )  0
L

HW 18
2
d
2 d
1
 (r ) 
 (r )  2  (r )  0
2
dr
r dr
L
General solution
A
e
r / L
r
C
A, C determined from BC’s.
Nuclear Reactor Theory, JU, Second Semester, 2008-2009
(Saed Dababneh).
e
r/L
r
16
Steady State Diffusion Equation
r      0  C = 0.
BC
A
S
Show that A 
4D
e
HW 18 (continued)
r / L
r

r / L
S e

4D r
L2 
D
a
4r 2 dr a  neutrons per second absorbed in the ring.
dr
r
Show that
r
Nuclear Reactor Theory, JU, Second Semester, 2008-2009
(Saed Dababneh).
r  6L
2
2
17
Steady State Diffusion Equation
HW 19
Study example 5.3 and solve problem 5.8 in Lamarsh.
Multiple Point Sources?
Nuclear Reactor Theory, JU, Second Semester, 2008-2009
(Saed Dababneh).
18
Steady State Diffusion Equation
One-speed neutron diffusion in a finite medium
• At the interface
A  B
A
B
d A
dB
J A  J B   DA
  DB
dx
dx
x
• What if A or B is a vacuum?
• Linear extrapolation distance.
Nuclear Reactor Theory, JU, Second Semester, 2008-2009
(Saed Dababneh).
19
More realistic multiplying medium
One-speed neutron diffusion in a multiplying medium
The reactor core is a finite multiplying medium.
• Neutron flux?
• Reaction rates?
• Power distribution in the reactor core?
Recall:
• Critical (or steady-state):
Number of neutrons produced by fission = number
of neutrons lost by:
neutronproductionrate(S)
k 
absorption
neutronabsorptionrate( A)
and
neutronproductionrate( S )
k

leakage
eff
neutronabsorptionrate( A)  neutronleakage rate( LE )
Nuclear Reactor Theory, JU, Second Semester, 2008-2009
(Saed Dababneh).
20
More realistic multiplying medium
Things to be used later…!
keff
A

 Pnon leak
k A  LE
LE  SA
surface area
S V
Volum e
non- leakageprobability
Recall:
For a critical reactor:
Keff = 1
K > 1
LE SA a 2 1

 3 
S
V
a
a
Steady state homogeneous reactor



2
0  a k (r )  a  (r )  D  (r )


k  1
2
2
2
  (r )  B  (r )  0
B  2
L
Nuclear Reactor Theory, JU, Second Semester, 2008-2009
(Saed Dababneh).
Material buckling
21
More realistic multiplying medium


2
  (r )  B  (r )  0

2
  (r )
2
B 

 (r )
2
• The buckling is a measure of extent to which the flux
curves or “buckles.”
• For a slab reactor, the buckling goes to zero as “a”
goes to infinity. There would be no buckling or curvature
in a reactor of infinite width.
• Buckling can be used to infer leakage. The greater the
curvature, the more leakage would be expected.
Nuclear Reactor Theory, JU, Second Semester, 2008-2009
(Saed Dababneh).
22
More on One-Speed Diffusion
HW 20
Show that for a critical homogeneous reactor
Pnon leak
a 
a 
1
 2 2


2
2
B L  1  a   D   a   B D
Infinite Bare Slab Reactor (one-speed diffusion) z

• Vacuum beyond.
• Return current = 0.
Reactor
x
d = linear extrapolation distance
a/2
= 0.71 tr (for plane surfaces)
= 2.13 D.
a0/2 a
d
d
Nuclear Reactor Theory, JU, Second Semester, 2008-2009
(Saed Dababneh).
23