19-09-2012, 04:51 PM
Neutron Diffusion Equation and Steady State Equation
[attachment=34103]
Let n be the density of neutrons at any point and time in V,
The total number of neutrons in V is then
where the subscript on the integral indicates that the integration is to be performed throughout V
The rate of change in the number of neutrons is
Neutrons do disappear, of course, when they undergo ,B-decay. However, this process has a comparatively long half-life and need not be taken into consideration.
Which also can be written as,
In moving the time derivative inside the integral, it is necessary to change to partial derivative notation because “n” may be a function of space variables as well as time.
Next, let S be the rate at which neutrons are emitted from sources per cm3 in V The rate at which neutrons are produced throughout V is given
The rate at which neutrons are lost by absorption per cm3/sec is equal
Where a is the macroscopic absorption cross-section and is the neutron flux.
Throughout the volume V, the total loss of neutrons per second due to absorption is then
THE DIFFUSION EQUATION
Unfortunately, the continuity equation has two unknowns-the neutron density n, and the neutron current density vector, J.
To eliminate one of these requires a relationship between them.
The relationship is based on the approximation that the current and flux are related by Fick's law.
On substitution of Fick's law into the equation of continuity , one obtains the neutron diffusion equation.
Assuming that D is not a function of position, this gives
where the symbol is called the Laplacian. Since = nv, where v is the neutron speed, Eq. (5 . 1 6) can also be written as
If it is a time dependant problem then the equation becomes
BOUNDARY CONDITIONS
The neutron flux can be found by solving the diffusion equation. Since the diffusion equation is a partial differential equation, it is necessary to specify certain boundary conditions that must be satisfied by the solution.
Some of these are determined from obvious requirements for a physically reasonable flux. For example, since a negative or imaginary flux has no meaning, it follows that must be a real, nonnegative function.
In many problems, neutrons diffuse in a medium that has an outer surface that is, a surface between the medium and the atmosphere.
It was pointed out that Fick's law is not valid in the immediate vicinity of such a surface, and it follows that the diffusion equation is not valid there either.
Higher order methods show, however, that if the flux calculated from the diffusion equation is assumed to vanish at a small distance d beyond the surface, then the flux determined from the diffusion equation is very nearly equal to the exact flux in the interior of the medium.
The assumption that the flux vanishes a small distance d beyond the surface is clearly nonphysical. Rather, it is a convenient mathematical approximation that provides a high degree of accuracy for estimates of the flux inside the medium.
[attachment=34103]
Let n be the density of neutrons at any point and time in V,
The total number of neutrons in V is then
where the subscript on the integral indicates that the integration is to be performed throughout V
The rate of change in the number of neutrons is
Neutrons do disappear, of course, when they undergo ,B-decay. However, this process has a comparatively long half-life and need not be taken into consideration.
Which also can be written as,
In moving the time derivative inside the integral, it is necessary to change to partial derivative notation because “n” may be a function of space variables as well as time.
Next, let S be the rate at which neutrons are emitted from sources per cm3 in V The rate at which neutrons are produced throughout V is given
The rate at which neutrons are lost by absorption per cm3/sec is equal
Where a is the macroscopic absorption cross-section and is the neutron flux.
Throughout the volume V, the total loss of neutrons per second due to absorption is then
THE DIFFUSION EQUATION
Unfortunately, the continuity equation has two unknowns-the neutron density n, and the neutron current density vector, J.
To eliminate one of these requires a relationship between them.
The relationship is based on the approximation that the current and flux are related by Fick's law.
On substitution of Fick's law into the equation of continuity , one obtains the neutron diffusion equation.
Assuming that D is not a function of position, this gives
where the symbol is called the Laplacian. Since = nv, where v is the neutron speed, Eq. (5 . 1 6) can also be written as
If it is a time dependant problem then the equation becomes
BOUNDARY CONDITIONS
The neutron flux can be found by solving the diffusion equation. Since the diffusion equation is a partial differential equation, it is necessary to specify certain boundary conditions that must be satisfied by the solution.
Some of these are determined from obvious requirements for a physically reasonable flux. For example, since a negative or imaginary flux has no meaning, it follows that must be a real, nonnegative function.
In many problems, neutrons diffuse in a medium that has an outer surface that is, a surface between the medium and the atmosphere.
It was pointed out that Fick's law is not valid in the immediate vicinity of such a surface, and it follows that the diffusion equation is not valid there either.
Higher order methods show, however, that if the flux calculated from the diffusion equation is assumed to vanish at a small distance d beyond the surface, then the flux determined from the diffusion equation is very nearly equal to the exact flux in the interior of the medium.
The assumption that the flux vanishes a small distance d beyond the surface is clearly nonphysical. Rather, it is a convenient mathematical approximation that provides a high degree of accuracy for estimates of the flux inside the medium.