Diffusion Equation and Flux


by ChangBroot
Tags: diffusion, equation, flux
ChangBroot
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#1
Jan23-13, 09:06 AM
P: 5
Hi,
I have read a lot about Diffusion Equation and solving neutron flux problems in different mediums, planes and groups, but I can't grasp this topic. In other words, I don't know why they mention:
1. Infinite/finite medium
2. Homogeneous/non-Homogenous medium
3. One/two or multi-group (what do they mean by group?)

I would appreciate it if someone could explain this topic with analogies. Thank you.
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Astronuc
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Jan23-13, 01:45 PM
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Quote Quote by ChangBroot View Post
Hi,
I have read a lot about Diffusion Equation and solving neutron flux problems in different mediums, planes and groups, but I can't grasp this topic. In other words, I don't know why they mention:
1. Infinite/finite medium
2. Homogeneous/non-Homogenous medium
3. One/two or multi-group (what do they mean by group?)

I would appreciate it if someone could explain this topic with analogies. Thank you.
Diffusion theory (diffusion equation) is an approximation of transport theory.

A finite medium has 'leakage' at the boundaries. An infinite medium has no boundaries at a free surface, although there could be boundaries between different volumes.

Homogenous means that the fuel (fissile)material is distributed through the moderator or coolant. An inhomogeneous system separates fuel from moderator, and the treatment is different.

Group refers to neutron energy group.

One group would assume a single energy group with macroscopic cross-sections determined as a weighted average over the energy range. Two group usually infers a fast energy group, up through the range of fission neutron energy (MeV) and a thermal neutron group, < 1 eV, and usually < 0.1 eV, for which most fissions occur.

More advanced methods use multigroup methods to account for fast neutron fissions in different transuranics, as well as resonance absorption and production of transuranics.
ChangBroot
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#3
Jan23-13, 02:36 PM
P: 5
Thanks a lot Astronuc. I really appreciate it. Also, if you could tell me about the application of the Diffusion equation (D∇2∅ + Ʃ∅ = 0) and some special cases (such as in vacuum, in a slab, infinite/finite plane etc). Basically, I don't know this diffusion equation, and therefore, don't know how to use it. For example, I know the formula for a circle, a cylinder, a cube etc, so if someone says find the volume of a cube, i know I have to take the product of the length, width and height (as opposed to using the formula of the cylinder or something else). Thanks.

I don't know when to use the following equation or which variable to set to zero. I know what each term is in the right, but I still don't know how to use it or what variable should be set to zero or should I take the partial derivative of the equation, or should I integrate it etc.

The time-dependent one-energy-group diffusion equation for a homogeneous reactor without delayed neutrons is written as:

∂n(r, t)/∂t = v∅(r, t) + D∇2∅(r, t) + Ʃ∅(r, t)


Where r is the position vector.

Thanks in advance.

Astronuc
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Jan23-13, 02:58 PM
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Diffusion Equation and Flux


Quote Quote by ChangBroot View Post
Thanks a lot Astronuc. I really appreciate it. Also, if you could tell me about the application of the Diffusion equation (D∇2∅ + Ʃ∅ = 0) and some special cases (such as in vacuum, in a slab, infinite/finite plane etc). Basically, I don't know this diffusion equation, and therefore, don't know how to use it. For example, I know the formula for a circle, a cylinder, a cube etc, so if someone says find the volume of a cube, i know I have to take the product of the length, width and height (as opposed to using the formula of the cylinder or something else). Thanks.

I don't know when to use the following equation or which variable to set to zero. I know what each term is in the right, but I still don't know how to use it or what variable should be set to zero or should I take the partial derivative of the equation, or should I integrate it etc.

The time-dependent one-energy-group diffusion equation for a homogeneous reactor without delayed neutrons is written as:

∂n(r, t)/∂t = v∅(r, t) + D∇2∅(r, t) + Ʃ∅(r, t)


Where r is the position vector.

Thanks in advance.
What is one's knowledge of partial differential equations and how to solve them? Can one describe the meaning/significance of each term in the differential equation?

What is the original source (reference) of the equation?
QuantumPion
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#5
Jan23-13, 04:33 PM
P: 733
I would recommend reading this article: http://en.wikipedia.org/wiki/Geometr...erial_Buckling

It describes how to solve the 1-group diffusion equation for a bare homogenous critical reactor.


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