Lecture 12
Neutron Diffusion Equation
Objectives
In this lecture you will learn the following
We shall understand the essence of the movement of the slowed down neutrons.
We shall introduce the concept of neutron diffusion.
We shall then derive the neutron balance equation with diffusion.
Then we shall discuss the common boundary conditions that are used to solve problems.
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Lecture 12
Neutron Diffusion Equation
Definitions
We had seen previously in the course that the intensity of neutrons in a beam was defined as
the number of neutrons crossing a plane per unit area and unit time and this was shown to be
Since the beam travels in the same direction unless it is absorbed, the problem is a scalar as
there is no change of direction is involved.
However, when we deal with neutrons in a reactor, they are born at different places and
travel in different directions.
Further their directions are modified due to the process of scattering.
To predict the neutron population density variation in a reactor we need a more complex
treatment.
It will be practically impossible to track every neutron and study its motion as there are very
large number of neutrons present in the system.
To handle this problem in a simple manner, the concept of diffusion is introduced.
Before we formally look at the concept of diffusion, we shall define a few terms.
The term Neutron Flux, φ, is defined as the product of neutron density and the speed of the
neutron and it is a scalar quantity.
Another term Neutron Current, J, is defined as number of neutrons crossing a unit area per
unit time in a specific direction.
J is a vector quantity and shall have components such as Jx , Jy, etc.
Consider the case of several beams intersecting at a point as shown in the figure below.
At the point of intersection
When several beams cross, it is easy to visualise that, the former is a scalar sum, while the
latter is a vector addition.
If all the neutrons travel in the same direction as in a single beam, both φ and J are identical
in a non scattering system.
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Lecture 12
Neutron Diffusion Equation
Diffusion-I
Consider two nearby points A and B such that the neutron density is high at A and low at B.
Since the population density is more at A, the number of scattering reactions at A will be
larger than the number of scattering at B.
Assuming that all directions are equally likely as neutrons are travelling in random directions,
the number of neutrons that will migrate from A to B will be larger than number of neutrons
migrating from B to A.
Thus the population density at A will decrease and at B will increase.
This process by which the population migrates from higher density to lower density is called
diffusion.
This can be mathematically treated as follows:
The current
will be proportional to the gradient of n
The negative sign signifies that the direction of current is opposite to the direction of gradient.
Assuming that all neutrons are moving at one speed, we can write
In the last equation D is the proportionality constant and is called diffusion coefficient and the
equation is called Fick’s Law of Diffusion.
It should be clear that the process that influences diffusion is scattering and hence D will
depend on Σs .
In this introductory course we shall assume that the value of D is available from text
books/hand books.
D for some common materials are summarised in the table shown below.
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Material
H2O
D (cm)
0.16
D2O
0.87
Be
0.50
C
0.84
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Lecture 12
Neutron Diffusion Equation
Neutron Balance Equation
We shall now derive the neutron balance equation with diffusion also present.
We will first do this in one dimension Cartesian and then extend to three dimensions.
Subsequently we will generalise for other coordinates.
Consider a small strip of area, A, and thickness, Δx, sliced from a media that is infinitely
large.
The rate of increase of neutron population in the control volume (CV) has 4 components as
shown in the figure below
Now each of the componants can be quantified in the figure below
In the above figure S"' is the source strength of neutrons (number of neutrons emitted per
second) per unit volume.
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Lecture 12
Neutron Diffusion Equation
Thus the neutron population balance can be written as,
The above equation can be simplified as
This can be modified using Fick’s law as
By using the definition of flux we can rewrite the equation as
The only difference when we extend to three dimensions is the diffusion occurs in all three
directions and the equation can be modified as
In vectorial form we can write the same as
Where, the Laplacian term can be represented as
Cartesian
Cylindrical
Spherically Symmetric
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Lecture 12
Neutron Diffusion Equation
General Boundary Conditions
The following are the general boundary conditions. These are schematically shown in the
figure below.
Symmetric Boundary Condition
This implies that J=0 at the symmetric plane.
Vacuum Boundary Condition
When the boundary of a system ends in a non-interacting media, it is called
vacuum boundary condition.
Such boundaries are approximated by using φ=0 at a distance approximately 2.1
D away from the boundary.
Since the value of D is small in comparison with system dimensions, often flux is
assumed to vanish at the system boundary itself.
Interface Boundary Condition
Wherever the property of a media changes, the plane separating the two media is
termed as the interface boundary.
Every interface satisfies two continuity conditions viz, both flux and current are
continuous across the boundary.
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