What is the Geometric and Material Buckling in Diffusion Equation?

In summary: It also discusses the use of the diffusion equation for particular geometries (spheres, cylinders, slabs, etc.) and its application in finding the criticality of a reactor. As for the equation itself, the first term represents the time rate of change of the neutron population, the second term represents the rate of neutrons being absorbed by the material, and the third term represents the rate of neutrons being produced by fission. It is important to have a good understanding of partial differential equations and their solutions in order to use this equation effectively. The original source of the equation is usually attributed to E.E. Lewis in his book "Introduction to Nuclear Reactor Theory."
  • #1
ChangBroot
5
0
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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  • #2
ChangBroot said:
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.
 
  • #3
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.
 
  • #4
ChangBroot said:
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?
 
Last edited:

1. What is the Diffusion Equation?

The diffusion equation is a partial differential equation that describes the diffusion of a substance in a medium. It is a mathematical representation of Fick's first law of diffusion which states that the flux of a substance is proportional to its concentration gradient.

2. What is Flux in the context of the Diffusion Equation?

Flux, in the context of the diffusion equation, refers to the amount of a substance that moves through a unit area in a unit time. It is a measure of the rate at which a substance is diffusing in a given medium.

3. How is the Diffusion Equation used in scientific research?

The diffusion equation is used in scientific research to model and predict the movement and distribution of substances in various systems. It is commonly used in fields such as chemistry, physics, biology, and engineering to understand diffusion processes and make predictions about concentration changes over time.

4. What are the assumptions made in the Diffusion Equation?

The diffusion equation makes several assumptions, including that the substance being diffused is non-reactive, the medium is homogeneous and isotropic, and there are no external forces acting on the substance. These assumptions may not hold true in all situations, but the diffusion equation still provides a useful approximation for many systems.

5. How is the Diffusion Equation related to other mathematical equations?

The diffusion equation is a type of partial differential equation, which is a type of mathematical equation that describes the change of a function with respect to multiple variables. It is also closely related to the heat equation, as both describe the movement and distribution of substances through a medium. Additionally, the diffusion equation can be derived from the more general Navier-Stokes equation, which describes the movement of fluids.

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