Boundary Conditions – Diffusion Equation

To solve the diffusion equation, which is a second-order partial differential equation throughout the reactor volume, it is necessary to specify certain boundary conditions. It is very dependent on the complexity of certain problem. One-dimensional problems solutions of diffusion equation contain two arbitrary constants. Therefore, in order to solve one-dimensional one-group diffusion equation, we need two boundary conditions to determine these coefficients. The most convenient boundary conditions are summarized in following few points:
extrapolated length - boundary condition

Neutron flux as a function of position near a free surface according to diffusion theory and transport theory.

The diffusion equation is mostly solved in media with high densities such as neutron moderators (H2O, D2O or graphite). The problem is usually bounded by air. The mean free path of neutron in air is much larger than in the moderator, so that it is possible to treat it as a vacuum in neutron flux distribution calculations. The vacuum boundary condition supposes that no neutrons are entering a surface.

If we consider that no neutrons are reflected from the vacuum back to the volume, the following condition can be derived from the Fick’s law:

extrapolated length - equation

Where d ≈ ⅔ λtr is known as the extrapolated length. For homogeneous, weakly absorbing media, an exact solution of the mono-energetic transport equation in this case yields d ≈ 0.7104 λtr. The geometric interpretation of the previous equation is that the relative neutron flux near the boundary has a slope of -1/d, i.e., the flux would extrapolate linearly to 0 at a distance d beyond the boundary. This zero flux boundary condition is more straightforward and is can be written mathematically as:

extrapolated length - equation2

If d is not negligible, physical dimensions of the reactor are increased by d and extrapolated boundary is formulated with dimension Re = R + d and this condition can be written as Φ(R + d) =  Φ(Re) = 0.

It may seem the flux goes to 0 at an extrapolated length beyond the boundary. This interpretation is not correct. The flux cannot go to zero in a vacuum, because there are no absorbers to absorb the neutrons. The flux only appears to be heading to the zero value at the extrapolation point.

Note that, the equation d ≈ 0.7104 λtr is applicable to plane boundaries only. The formulas for curved boundaries can differ slightly, however, the difference is small unless the radius of curvature of the boundary is of the same order of magnitude as the extrapolated length.

Typical values of the extrapolated length:

The most common moderators have following diffusion coefficients (for thermal neutrons):

D(H2O) = 0.142 cm

D(D2O) = 0.84 cm

D(Be) = 0.416 cm

D(C) = 0.916 cm

The thermal neutron extrapolated lengths are given by:

d ≈ 0.7104 λtr = 0.7104 x 3 x D

therefore:

H2O: d ≈ 0.30 cm

D2O: d ≈ 1.79 cm

Be: d ≈ 0.88 cm

C: d ≈ 1.95 cm

As can be seen, this approximation is valid when the dimension L of the diffusing medium is much larger than the extrapolated length, L >> d.

This condition is determined from obvious requirement. The flux in the medium can reach only reasonable values i.e. must be real, non-negative, and single valued. Also 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 - equation

This conditions are often used to eliminate unnecessary functions from solutions.

It is also necessary to specify boundary conditions at an interface between two different media. 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.

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.

diffusion equation - two media

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

It was stated the diffusion equation is not valid near the neutron source. But 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 can be written mathematically as:

source condition - diffusion

It is well known that each reactor core is surrounded by a neutron reflector. The reflector reduces the non-uniformity of the power distribution in the peripheral fuel assemblies, reduces neutron leakage and reduces a coolant flow bypass of the core. The neutron reflector is a non-multiplying medium, whereas the reactor core is a multiplying medium.

On this special interface we shall apply an albedo boundary condition to represent the neutron reflector. Albedo, the latin word for “whiteness”, was defined by Lambert as the fraction of the incident light reflected diffusely by a surface.

In reactor engineering, albedo, or the reflection coefficient, is defined as the ratio of exiting to entering neutrons and we can express it in terms of neutron currents as:

albedo - reflection coefficient - equation

For sufficiently thick reflectors, it can be derived, that albedo becomes

albedo - reflection coefficient - equation2

where Drefl is the diffusion coefficient in the reflector and the Lrefl is the diffusion length in the reflector.

If we are not interested in the neutron flux distribution in the reflector (let say in the slab B) but only in the effect of the reflector on the neutron flux distribution in the medium (let say in the slab A), the albedo of the reflector can be used as a boundary condition for the diffusion equation solution. This boundary condition is similar to the vacuum boundary condition, i.e. Φ(Ralbedo) = 0, where Ralbedo = R + de and

albedo boundary condition - extrapolated

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: