Transcript Document

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 16
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 Reactors, BAU, 1st Semester, 2007-2008 (Saed
Dababneh).
1
Fick’s Law
4. Scattering is isotropic in the lab. coordinate system.
2


cos
(

)

 0  reevaluate D.
If
HW 17
3A
tr
1
1
D


3(t   s  ) 3 tr
3
• Isotropic tr = t.
• Weekly absorbing tr = s.
s
For “practical” moderators: tr 
1 
5. The flux is a slowly varying function of position.
2

a   variation in  .  (r )
r 2
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed
Dababneh).
?
2
Fick’s Law
HW 18
Estimate the diffusion coefficient of graphite at 1 eV.
The scattering cross section of carbon at 1 eV is 4.8 b.
Scattering
Absorption
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed
Dababneh).
3
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 10-5 s (How?).
If flux changes by 10% per second!
  / 

t  0.1x1x105  1x106

t
Very small fractional change during the time
needed for the neutron to travel this “significant”
distance.

J  D
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed
Dababneh).
4
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 Reactors, BAU, 1st Semester, 2007-2008 (Saed
Dababneh).
5
The Diffusion Equation

 

 
1  
 (r , t )  S (r , t )   a (r ) (r , t )    D (r , t )
v t
If D is independent on r
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 Reactors, BAU, 1st Semester, 2007-2008 (Saed
Dababneh).
Buckling equation.
6
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
2
1
  2 0
L
2
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed
Dababneh).
7
The Diffusion Equation
• The exact interpretation of neutron transport in
heterogeneous domains is so complex.
• Simplified approaches.
• Simplified but accurate enough to give an estimate of
the average characteristics of neutron population.
• Numerical solutions.
• Monte Carlo techniques.
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed
Dababneh).
8
Steady State Diffusion Equation

 

2
0  S (r )  a (r ) (r )  D  (r )
1
S
  2 
L
D
2
Boundary Conditions
• Solve DE  get .
• Solution must satisfy BC’s.
• Solution should be real and non-negative.
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed
Dababneh).
9
Steady State Diffusion Equation
One-speed neutron diffusion in infinite medium
Point source

1 
2
  (r )  2  (r )  0
L

HW 19
2
d
2 d
1
 (r ) 
 (r )  2  (r )  0
2
dr
r dr
L
General solution
A
e
r / L
r
A, C determined from BC’s.
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed
Dababneh).
C
e
r/L
r
10
Steady State Diffusion Equation
r      0  C = 0.
BC
A
S
Show that A 
4D
e
HW 19 (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 Reactors, BAU, 1st Semester, 2007-2008 (Saed
Dababneh).
r  6L
2
2
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