Derivation of One-group Diffusion Equation

Derivation of One-group Diffusion Equation

The derivation of the diffusion equation depends on Fick’s law, which states that solute diffuses from high concentration to low. But first, we have to define a neutron flux and neutron current density. The neutron flux is used to characterize the neutron distribution in the reactor and it is the main output of solutions of diffusion equations.  The neutron flux, φ, does not characterize the flow of neutrons. There may be no flow of neutrons, yet many interactions may occur (I = Σ.φ). The neutrons move in a random directions and hence may not flow. Therefore the neutron flux φ is more closely related to densities. Neutrons will exhibit a net flow when there are spatial differences in their density. Hence we can have a flux of neutron flux! This flux of neutron flux is called the neutron current density.

Fick’s Law

The derivation of the diffusion equation depends on Fick’s law, which states that solute diffuses (neutron current) from high concentration (high flux) to low concentration. As can be seen we have to investigate the relationship between the flux (φ) and the current (J). This relationship between the flux (φ) and the current (J) is identical in form to a law (the Fick’s law) used in the study of physical diffusion in liquids and gases.

In chemistry, Fick’s law states that:

If the concentration of a solute in one region is greater than in another of a solution, the solute diffuses from the region of higher concentration to the region of lower concentration, with a magnitude that is proportional to the concentration gradient.

In one (spatial) dimension, the law is:

Ficks Law - equation

where:

  • J is the diffusion flux,
  • D is the diffusion coefficient,
  • φ (for ideal mixtures) is the concentration.

The use of this law in nuclear reactor theory leads to the diffusion approximation.

The Fick’s law in reactor theory stated that:

The current density vector J is proportional to the negative of the gradient of the neutron flux. The proportionality constant is called the diffusion coefficient and is denoted by the symbol D.

In one (spatial) dimension, the law is:

Ficks Law - equation

where:

  • J is the neutron current density (neutrons.cm-2.s-1) along x-direction, the net flow of neutrons that pass per unit of time through a unit area perpendicular to the x-direction.
  • D is the diffusion coefficient,  it has unit of cm and it is given by:diffusion coefficient - equation
  • φ is the neutron flux, which is the number of neutrons crossing through some arbitrary cross-sectional unit area in all directions per unit time.

The generalized Fick’s law (in three dimension) is:
Ficks Law - 3D

where J denotes the diffusion flux vector. Note that the gradient operator turns the neutron flux, which is a scalar quantity into the neutron current, which is a vector quantity.

See also: Validity of Fick’s Law

Physical Interpretation

Ficks Law - physical interpretationThe physical interpretation is similar to fluxes of gases. The neutrons exhibit a net flow in the direction of least density. This is a natural consequence of greater collision densities at positions of greater neutron densities.

Consider neutrons passing through the plane at x=0 from left to right as the result of collisions to the left of the plane. Since the concentration of neutrons and the flux is larger for negative values of x, there are more collisions per cubic centimeter on the left. Therefore more neutrons are scattered from left to right, then the other way around. Thus the neutrons naturally diffuse toward the right.

Neutron Balance – Continuity Equation

The mathematical formulation of neutron diffusion theory is based on the balance of neutrons in a differential volume element. Since neutrons do not disappear (β decay is neglected) the following neutron balance must be valid in an arbitrary volume V.

rate of change of neutron density = production rate – absorption rate – leakage rate

where

neutron balance - components

Substituting for the different terms in the balance equation and by dropping the integral over (because the volume V is arbitrary) we obtain:

neutron balance - continuity equation

where

  • n is the density of neutrons,
  • s is the rate at which neutrons are emitted from sources per cm3 (either from external sources (S) or from fission (ν.Σf)),
  • J is the neutron current density vector
  • Ф is the scalar neutron flux
  • Σa is the macroscopic absorption cross-section

In steady state, when n is not a function of time:

neutron balance - general equation

The Diffusion Equation

In previous chapters we introduced two bases for the derivation of the diffusion equation:

Fick’s law:

Ficks Law - 3D

which states that neutrons diffuses from high concentration (high flux) to low concentration.

Continuity equation:

neutron balance - continuity equation

which states, that rate of change of neutron density = production rate – absorption rate – leakage rate.

We return now to the neutron balance equation and substitute the neutron current density vector by J = -D∇Ф. Assuming that ∇.∇ = ∇2 = Δ  (therefore div J = -D div (∇Ф) = -DΔФ) we obtain the diffusion equation.

diffusion equation - general

 
References:
Nuclear and Reactor Physics:
  1. J. R. Lamarsh, Introduction to Nuclear Reactor Theory, 2nd ed., Addison-Wesley, Reading, MA (1983).
  2. J. R. Lamarsh, A. J. Baratta, Introduction to Nuclear Engineering, 3d ed., Prentice-Hall, 2001, ISBN: 0-201-82498-1.
  3. W. M. Stacey, Nuclear Reactor Physics, John Wiley & Sons, 2001, ISBN: 0- 471-39127-1.
  4. Glasstone, Sesonske. Nuclear Reactor Engineering: Reactor Systems Engineering, Springer; 4th edition, 1994, ISBN: 978-0412985317
  5. W.S.C. Williams. Nuclear and Particle Physics. Clarendon Press; 1 edition, 1991, ISBN: 978-0198520467
  6. G.R.Keepin. Physics of Nuclear Kinetics. Addison-Wesley Pub. Co; 1st edition, 1965
  7. Robert Reed Burn, Introduction to Nuclear Reactor Operation, 1988.
  8. U.S. Department of Energy, Nuclear Physics and Reactor Theory. DOE Fundamentals Handbook, Volume 1 and 2. January 1993.

Advanced Reactor Physics:

  1. K. O. Ott, W. A. Bezella, Introductory Nuclear Reactor Statics, American Nuclear Society, Revised edition (1989), 1989, ISBN: 0-894-48033-2.
  2. K. O. Ott, R. J. Neuhold, Introductory Nuclear Reactor Dynamics, American Nuclear Society, 1985, ISBN: 0-894-48029-4.
  3. D. L. Hetrick, Dynamics of Nuclear Reactors, American Nuclear Society, 1993, ISBN: 0-894-48453-2. 
  4. E. E. Lewis, W. F. Miller, Computational Methods of Neutron Transport, American Nuclear Society, 1993, ISBN: 0-894-48452-4.

See above:

Neutron Diffusion