1. 2010 January 1 R2: Neutron Current & Leakage B. Rouben
Course UN 0802
2. Review Material! All material in this presentation should really be review material, covered in the Physics Refresher course (and perhaps also in previous presentations) 2010 January 2
3. Neutron Flux 2010 January 3
4. Addition of Vectors Vectors can be added:
Or as arrows:
Vectors can also be integrated!
2010 January 4
5. Neutron Current Refer to the first slide, which defines the neutron flux.
If, instead of adding the arrow lengths, you add the arrows vectorially, you get a vector which is the net (or total) neutron current .
The neutron current tells you how the neutron population in the unit volume moves as an entity.
(Of course, within this entity, there are neutrons moving in various directions.) 2010 January 5
6. Number of Neutrons Crossing a Plane The magnitude J of the net current gives the number of neutrons crossing a unit area perpendicular to the direction ? of .
But if one wants to calculate the number of neutrons which cross a unit area on a plane “diagonal” to ? (? = angle between normal to plane and ?), then an additional factor cos ? needs to be applied. The number is J cos ? (see next 2 slides).
cont’d 2010 January 6
7. Number of Neutrons Crossing a Plane (Cont’d) This follows from the fact that the neutrons crossing an area 1 on the plane are those which cross a smaller area, 1*cos ?, perpendicular to their direction of motion (see illustration in next slide).
2010 January 7
8. Neutrons Crossing a “Diagonal” Plane 2010 January 8
9. 2009 September 9 Leakage in Neutron-Diffusion Equation
10. Physical Meaning of Divergence The physical meaning of the divergence is that it is the “leakage” of the vector function out of an infinitesimal volume around the point where the divergence is calculated, divided by the infinitesimal volume.
See proof of this “divergence theorem” in next slide.
Note: The proof is given in Cartesian co-ordinates, but holds for any shape of the infinitesimal volume.
2010 January 10
11. Divergence Theorem for Vector (Current) 2010 January 11
12. Leakage out of a Finite Volume Subdivide a finite volume into infinitesimal subvolumes; apply the divergence theorem in each subvolume, and “add” all (i.e., integrate).
The “internal” leakages (across internal surfaces out of one subvolume and into a neighbouring subvolume) obviously cancel out, leaving only the leakage out of the external surface.
Therefore the net leakage out of the finite volume = the volume integral of the divergence of the current. 2010 January 12
13. Leakage out of a Finite Volume We have just proved Gauss’s famous Theorem:
2010 January 13
14. 2009 September 14 Fick’s Law & Neutron Diffusion The neutron-diffusion equation is an approximation to the neutron-transport equation.
It is much simpler than the transport equation, because
it removes the neutron direction of motion from consideration, i.e., the dependent variable is the total flux at each energy regardless of neutron direction of motion, and
it is based on an approximate relationship between the neutron flux and the neutron current, Fick’s Law:
The proportionality constant D is called the diffusion coefficient.
15. 2009 September 15 Significance of Fick’s Law Fick’s Law expresses the expectation/fact that in regions of totally free neutron motion the overall net neutron current will tend to be from regions of high density to regions of low density.
Mathematically speaking, the net overall current should flow along the direction of greatest decrease in the neutron density (or, equivalently, of flux), i.e., it will be proportional to the negative of the gradient of the flux.
This is a consequence of the greater number of collisions in regions of greater density, with collisions allowing neutrons to go off equally in all directions.
16. 2009 September 16 Breakdown of Fick’s Law The approximation inherent in Fick’s Law breaks down near regions of strong sources or strong absorption, or near boundaries between regions with large differences in properties, or external boundaries, because the motion of neutrons is biased in or near such regions.
Here “near” a region or boundary means within, say, 2 or 3 neutron mean free paths of the region or boundary.
This is why diffusion theory cannot be used in lattice physics, as the fuel itself is a strong neutron absorber. Transport theory must be used to homogenize properties (and therefore weaken absorption, on the average) over (relatively large) lattice cells.
17. Leakage with Fick’s Law 2010 January 17
18. 2010 January 18