1. Equation of Continuity
Net flow of neutrons per second per unit area normal to the x direction:
In general:
Equation of Continuity
Rate of change in number of neutrons
Production rate
Absorption rate
“Leakage in/out” rate
Normal to A (outwards)
Source distribution function
Surface area bounding 
Volume
Nuclear Reactors, BAU, 1st Semester, (Saed Dababneh).

2. Fundamental equation in Reactor Theory
Equation of Continuity
Using Gauss’ Divergence Theorem
Recall:
Both flux and current!!
Convert current to flux? 
Fundamental equation in Reactor Theory
Equation of Continuity
Nuclear Reactors, BAU, 1st Semester, (Saed Dababneh).

3. Equation of Continuity
Steady state
Non-spacial dependence
Delayed sources?
Nuclear Reactors, BAU, 1st Semester, (Saed Dababneh).

4. Nuclear Reactors, BAU, 1st Semester, 2008-2009 (Saed Dababneh).
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.
MCNP
Nuclear Reactors, BAU, 1st Semester, (Saed Dababneh).

5. Nuclear Reactors, BAU, 1st Semester, 2008-2009 (Saed Dababneh).
Fick’s Law
Assumptions:
The medium is infinite.
The medium is uniform
There are no neutron sources in the medium.
Scattering is isotropic in the lab coordinate system.
The neutron flux is a slowly varying function of position.
The neutron flux is not a function of time.
Restrictive! Applicability??
Nuclear Reactors, BAU, 1st Semester, (Saed Dababneh).

6. Nuclear Reactors, BAU, 1st Semester, 2008-2009 (Saed Dababneh).
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 Reactors, BAU, 1st Semester, (Saed Dababneh).

7. Nuclear Reactors, BAU, 1st Semester, 2008-2009 (Saed Dababneh).
Fick’s Law
Nuclear Reactors, BAU, 1st Semester, (Saed Dababneh).

8. Nuclear Reactors, BAU, 1st Semester, 2008-2009 (Saed Dababneh).
Fick’s Law
Diffusion: random walk of an ensemble of particles from region of high “concentration” to region of small “concentration”.
Flow is proportional to the negative gradient of the “concentration”.
Recall:
From larger flux to smaller flux!
Neutrons are not pushed!
More scattering in one direction than in the other.
Nuclear Reactors, BAU, 1st Semester, (Saed Dababneh).

9. Nuclear Reactors, BAU, 1st Semester, 2008-2009 (Saed Dababneh).
Fick’s Law
Number of neutrons scattered per second from d at r and going through dAz z d  r
dAz
Removed en route
(assuming no buildup) y 
Slowly varying x
Isotropic
Nuclear Reactors, BAU, 1st Semester, (Saed Dababneh).

10. Nuclear Reactors, BAU, 1st Semester, 2008-2009 (Saed Dababneh).
Fick’s Law
Nuclear Reactors, BAU, 1st Semester, (Saed Dababneh).

11. Diffusion coefficient
Fick’s Law
HW 14
and show that
and generalize ?
Fick’s law
Diffusion coefficient
Total removal
The current density is proportional to the negative of the gradient of the neutron flux.
Nuclear Reactors, BAU, 1st Semester, (Saed Dababneh).

12. Nuclear Reactors, BAU, 1st Semester, 2008-2009 (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 15
Nuclear Reactors, BAU, 1st Semester, (Saed Dababneh).

13. Nuclear Reactors, BAU, 1st Semester, 2008-2009 (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 16
Isotropic tr = t.
Weekly absorbing t = s.
Recall: ?
Nuclear Reactors, BAU, 1st Semester, (Saed Dababneh).

14. Nuclear Reactors, BAU, 1st Semester, 2008-2009 (Saed Dababneh).
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 Reactors, BAU, 1st Semester, (Saed Dababneh).

15. Nuclear Reactors, BAU, 1st Semester, 2008-2009 (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).

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

17. The Diffusion Equation The Steady State Diffusion Equation
If D is independent of r (uniform medium)
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).

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

19. The Diffusion Equation
Repeated!!
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 Reactors, BAU, 1st Semester, (Saed Dababneh).

20. 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).

21. Steady State Diffusion Equation
One-speed neutron diffusion in infinite medium
Point source 
HW 18
General solution
A, C determined from BC’s.
Nuclear Reactors, BAU, 1st Semester, (Saed Dababneh).

22. Steady State Diffusion Equation
HW 18 (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).

23. Scalar flux, vector current.
Steady State Diffusion Equation
Scalar flux, vector current.
HW 19
Study example 5.3 and solve problem 5.8 in Lamarsh.
Multiple Point Sources?
Nuclear Reactors, BAU, 1st Semester, (Saed Dababneh).