Diffusion Equation – Two Different Media

Solutions of the Diffusion Equation – Non-multiplying Systems

As was previously discussed the diffusion theory is widely used in core design of the current Pressurized Water Reactors (PWRs) or Boiling Water Reactors (BWRs). This section is not about such calculations, but provides an illustrative insights, how can be the neutron flux distributed in any diffusion medium. In this section we will solve diffusion equations:

solution of diffusion equation - equations

in various geometries that satisfy the boundary conditions discussed in the previous section.

We will start with simple systems and increase complexity gradually. The most important assumption is that all neutrons are lumped into a single energy group, they are emitted and diffuse at thermal energy (0.025 eV).

In the first section, we will deal with neutron diffusion in non-multiplying system, i.e., in system where fissile isotopes are missing and therefore the fission cross-section is zero. The neutrons are emitted by external neutron source. We will assume that the system is uniform outside the source, i.e. D and Σa are constants.

Solution of diffusion equation
Neutron flux distribution in various geometries of non-multiplying system.

Solution in two different diffusion media

A diffusion environment may consist of various zones of different composition. The consequence of this is that the diffusion coefficient, absorption macroscopic cross-section, and therefore, the neutron flux distribution, will vary per zone. For the determination of the flux distribution in various zones, the diffusion equations in zone 1 and zone 2 need to be solved:

diffusion equation - two different media

where a is the real width of zone 1 and b the outer dimension of the diffusion environment including the extrapolated distance. With problems involving two different diffusion media, the interface boundary conditions play crucial role and must be satisfied:

At interfaces between two different diffusion media (such as between the reactor core and the neutron reflector), on physical grounds the neutron flux and the normal component of the neutron current must be continuous. In other words, φ and J are not allowed to show a jump.

1., 2. Interface Conditions

interface condition - equations

It must be added, as J must be continuous, the flux gradient will show a jump if the diffusion coefficients in both media differ from each other. Since the solution of these two diffusion equations requires four boundary conditions, we have to use two boundary conditions more.

3. Finite Flux Condition

The solution must be finite in those regions where the equation is valid, except perhaps at artificial singular points of a source distribution.  This boundary condition can be written mathematically as:

finite flux condition - equation4. Source Condition

The presence of the neutron source can be used as a boundary condition, because it is necessary that all neutrons flowing through bounding area of the source must come from the neutron source. This boundary condition depends on the source geometry and for planar sourve can be written mathematically as:

source condition - planar source2

For x > 0, these diffusion equations have the following appropriate solutions:

Φ1(x) = A1exp(x/L1) + C1exp(-x/L1)

and

Φ2(x) = A2exp(x/L2) + C2exp(-x/L2)

where the four constants must be determined with use of the four boundary conditions. The typical neutron flux distribution in a simple two-region diffusion problem is shown at the picture below.

diffusion equation - two media
Solution of diffusion equation in two different non-multiplying diffusion media
 
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:

Diffusion Theory