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 nots (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 0.1x1x105 1x106
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
4D
e
HW 19 (continued)
r / L
r
r / L
S e
4D r
L2
D
a
4r 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
11