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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.

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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 estimateof the average characteristics of neutron population.

  • Numerical solutions.

  • Monte Carlo techniques.

MCNP

Nuclear Reactor Theory, JU, Second 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??

http://www.iop.org/EJ/article/0143-0807/26/5/023/ejp5_5_023.pdf

Nuclear Reactor Theory, JU, Second 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 Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh).


Fick’s Law

Nuclear Reactor Theory, JU, Second 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 Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh).


Fick’s Law

Number of neutrons scatteredper 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 Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh).


Fick’s Law

Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh).


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 Reactor Theory, JU, Second 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 Reactor Theory, JU, Second 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

Weekly absorbing t = s.

?

Nuclear Reactor Theory, JU, Second 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 Reactor Theory, JU, Second 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 10-3s(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 Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh).


Back to the Continuity Equation

Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh).


The 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 (and steady state)

Buckling equation.

Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh).


Steady State Diffusion Equation

Define L Diffusion Length

L2 Diffusion Area

Moderation Length

Boundary Conditions

  • Solve DE  get .

  • Solution must satisfy BC’s.

  • Solution should be real and non-negative.

Non-multiplying medium

Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh).


Steady State Diffusion Equation

One-speed neutron diffusion in infinite medium

Point source

HW 18

General solution

A, C determined from BC’s.

Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh).


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 Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh).


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 Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh).


Steady State Diffusion Equation

One-speed neutron diffusion in a finite medium

  • At the interface

  • What if A or B is a vacuum?

  • Linear extrapolation distance.

A B

x

Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh).


More realistic multiplying medium

One-speed neutron diffusion in a multiplying medium

  • The reactor core is a finite multiplying medium.

  • Neutron flux?

  • Reaction rates?

  • Power distribution in the reactor core?

  • Recall:

  • Critical (or steady-state):

  • Number of neutrons produced by fission = number of neutrons lost by:

  • absorption

  • and

  • leakage

Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh).


More realistic multiplying medium

Things to be used later…!

Recall:

For a critical reactor:

Keff = 1

K > 1

Steady state homogeneous reactor

multiplying medium

Material buckling

Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh).


More realistic multiplying medium

  • The buckling is a measure of extent to which the flux curves or “buckles.”

  • For a slab reactor, the buckling goes to zero as “a” goes to infinity. There would be no buckling or curvature in a reactor of infinite width.

  • Buckling can be used to infer leakage. The greater the curvature, the more leakage would be expected.

Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh).


More on One-Speed Diffusion

HW 20

Show that for a critical homogeneous reactor

Infinite Bare Slab Reactor (one-speed diffusion)

z

  • Vacuum beyond.

  • Return current = 0.

  • d = linear extrapolation distance

  • = 0.71 tr (for plane surfaces)

  • = 2.13 D.

Reactor

x

a/2

a

a0/2

d

d

Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh).


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