1. Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
Fick’s Law
Validity:
1. The medium is infinite. Integration over all space.
 after few mean free paths  0  corrections at the surface are still required.
2. The medium is uniform.
  and  are functions of space  re-derivation of Fick’s law?  locally larger s  extra J cancelled by iff ???
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.
HW 16
Nuclear Reactors, BAU, 1st Semester, (Saed Dababneh).

2. Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
Fick’s Law
4. Scattering is isotropic in the lab. coordinate system.
If  reevaluate D.
For “practical” moderators:
5. The flux is a slowly varying function of position.
a   variation in  .
HW 17
Isotropic tr = t.
Weekly absorbing tr = s. ?
Nuclear Reactors, BAU, 1st Semester, (Saed Dababneh).

3. Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
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, (Saed Dababneh).

4. Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
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!
Very small fractional change during the time needed for the neutron to travel this “significant” distance.
Nuclear Reactors, BAU, 1st Semester, (Saed Dababneh).

5. Back to the Continuity Equation
Nuclear Reactors, BAU, 1st Semester, (Saed Dababneh).

6. The Diffusion Equation The Steady State Diffusion Equation
If D is independent on r
Laplacian
The Diffusion Equation
The Steady State Diffusion Equation
or scalar Helmholtz equation.
Non-multiplying medium
Buckling equation.
Nuclear Reactors, BAU, 1st Semester, (Saed Dababneh).

7. Non-multiplying medium
Steady State Diffusion Equation
Define L  Diffusion Length
L2  Diffusion Area 
Non-multiplying medium
Nuclear Reactors, BAU, 1st Semester, (Saed Dababneh).

8. 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, (Saed Dababneh).

9. Steady State Diffusion Equation
Boundary Conditions
Solve DE  get .
Solution must satisfy BC’s.
Solution should be real and non-negative.
Nuclear Reactors, BAU, 1st Semester, (Saed Dababneh).

10. Steady State Diffusion Equation
One-speed neutron diffusion in infinite medium
Point source 
HW 19
General solution
A, C determined from BC’s.
Nuclear Reactors, BAU, 1st Semester, (Saed Dababneh).

11. Steady State Diffusion Equation
HW 19 (continued)
BC r      0  C = 0.
Show that 
neutrons per second absorbed in the ring.
Show that dr r
Nuclear Reactors, BAU, 1st Semester, (Saed Dababneh).