Diffusion Theory

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Del curso dictado por National Research Nuclear University MEPhI

Nuclear Reactor Physics Basics

calificaciones

National Research Nuclear University MEPhI

Nuclear Reactor Physics Basics

calificaciones

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.

De la lección

Neutrons space behaviour by diffusion aproximation

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.

Diffusion Theory5:47

Fick’s Law4:43

Diffusion Equation3:18

Conoce a los instructores

Yury Volkov
Senior lecturer, PhD in Nuclear Engineering
Department of Theoretical and Experimental Physics of Nuclear Reactors

After studding this lecture, the students should be able to:

define main approximations of the diffusion theory —

Fick’s Law and the diffusion equation,

explain each term in the diffusion equation.

In this module we develop a one-speed diffusion theory.

Such a relatively simple description

has a great advantage

of illustrating many of the important features

of nuclear reactors without a complexity

introduced by the treatment of

important effects associated

with the neutron energy spectrum

and with highly directional neutron transport,

This topics are the subjects of subsequent chapters.

Moreover, diffusion theory is sufficiently accurate to provide

a quantitative understanding of many physical features

of nuclear reactors and is, in fact,

the workhorse computational method of nuclear reactor physics.

First, neutrons have the same energy,

that is the case when energy of neutrons does not change

in the interaction with nuclei.

It is possible only if there are processes of absorption

or second we considered potential scattering

on heavy nuclei (the atomic mass more than 10).

Third, there is a source of monoenergetic

and isotropic neutrons and the scattering on heavy nuclei

is isotropic in lab coordinates.

Now our functions vary depending on only space and time positions.

That means that we consider the global flux,

neutron density and others.

Let's consider volume unit at the point with radius vector r.

Calculate the balance in the volume

as a change of the neutron density per time unit.

Gain means the quantity of neutrons

to be gained in the volume by the process of generation

by fission and by outer sources.

Loss means the quantity of neutrons

to be lost in the volume by the process of leakage and absorption.

Thus, the balance equations looks like this.

Consider all members of the equation separately.

Take a small volume ΔV near the point r with

the total surface area S and calculate the number of neutrons

leaking from the volume

in a moment t.

The number of neutrons crossed through small area dS

in any directionn: it's a product of neutron current and dS.

It's by the definition of neutron current.

For the whole surface it is needed to integrate over the area.

By using the divergence theorem the leakage in unit volume

equals the divergence of the current vector.

How to find the reaction rate of x process

(x — the absorption, scattering, fission etc.)

Let imagine that v be the neutron velocity,

λx — mean free path for the process x.

Then the ratio λx/v is the mean time between two interactions;

ratio v/λx is the average number of process x

for one neutron in a volume unit.

Multiplying by neutron density

we obtain the number of process x in a volume unit.

And, finally: multiplying by the volume we

get the reaction rate in a volume ΔV.

So, in the end, we got that the reaction rate

of thr process x is thr product of macroscopic cross section

and the neutron flux.

Finally, the absorption rate is the product of the absorption cross section and the neutron flux.

The fission rate is the product of

the fission macroscopic cross section and the flux,

and multiplying by the average number

of neutrons per one fission (the nu f)

we can get the neutron generation by fission.

In a result, we can get the balance equation.

The first member is the neutron leakage from the unit volume.

The second is the loss by absorption,

the third is the generation by fission

and the last one is the outer source.

The equation is obtained within the most general assumptions:

all functions are statistical quantities,

the neutron is considered as a point particle,

neutron-neutron interactions aren't considered,

the neutron is a stable particle,

and the last — all neutrons have the same energy

that is the same as we integrate the flux by energy —

the global flux.

The equation that is written down

contains two unknown functions — the neutron flux and the current vector;

therefore it is necessary to derive one more equation

connecting these two functions.

This relation is called Fick’s Law.

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