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Diffusion Equation

Course video 13 of 30

In this module students will get knowledge concerning neutrons space behavior by diffusion approximation. Students will learn about diffusion theory, diffusion equation and Fick’s Law.

Nuclear Reactor Physics Basics

Instituto de Ingeniería Física de Moscú de la Universidad Nuclear de Investigación Nacional

This engineering course is designed to introduce students to a range of concepts, ideas and models used in nuclear reactor physics. This course will focus on the physical theory of reactors and methods of experimental studies of the neutron field. This course is based on the course “Neutron transport theory” which has been taught at the National Research Nuclear University “MEPhI” for the past 20 years. What you'll learn: • Define basic processes that may occur in the reactor core, laws, equations, and the limits of applicability of models describing the neutron field in the reactor; • Demonstrate practical experience of calculating the distribution of neutrons in media; • Demonstrate the ability to analyze the process of slowing down neutrons in various media (typical for nuclear fission reactors) from the standpoint of understanding the physics of the process; • Evaluate important reactor parameters including performance and safety.

Impartido por:

Yury Volkov
Senior lecturer, PhD in Nuclear Engineering

Transcripción

The relation between the current vector and

the neutron flux is the Fick Law.

Fick's law is true for big,

weak absorbing medium consisting of heavy nuclei,

it is far from local heterogeneity

of medium and if the macroscopic

cross section of scattering

weakly depends on the space coordinate variables.

Diffusion does not work with the hydrogen medium.

The constant of proportionality

is a new constant — the diffusion coefficient.

One of the basic assumptions

was that the potential scattering is

isotropic in the laboratory frame.

In fact for a nucleus, except the heaviest ones, it is not true.

We will discuss this in the future.

That means we need the transport correction.

We introduce in neutron physics

the new macroscopic cross section —

the transport cross section

equals to sigma total minus average cosine

of the scattering angle multiplied by sigma S.

Say, roughly, one collision of the neutron

with a nucleus, which is anisotropic

changed by a combined collision,

which is isotropic.

That is why the mean free path has increased.

Fick's law means that the primary movement of the neutrons

is directed from the higher neutron density place

to the lower neutron place.

Fick's law works under the conditions of the complete

chaotic movement of neutrons in the medium

(similarity to the Brownian motion of molecules).

Where the chaos is broken

(local heterogeneities, strong absorption, etc.)

Fick's law does not work.

By using the Fick’s Law we can get

the one speed diffusion equation.

There is a number of simplifications of the equation.

We will consider stationary case

and the medium is homogeneous

which means that the diffusion coefficient

and cross sections don’t depend on the space coordinates.

And for a better understanding of how the neutron

is distributed in space we often use

a source of neutrons the only outer source

without neutron multiplication by the fission.

The diffusion coefficient can be pulled out of

the divergence symbol, and the divergence of a gradient

of function is a Laplacian applied to this function.

Divide the last equation by the coefficient of diffusion

and then, we will come to a new notion.

The diffusion length is the ratio of

the diffusion coefficient and the macroscopic

cross section of absorption.

The last equation is a differential equation

of the second order

regarding to the neutron flux function.

Therefore, in its solution there are two constants

which need to be defined

from the physical problem description.

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