Reflected Reactor – One-group Diffusion Method

Flux in a Reflected Thermal Reactor – One-group Method

The mathematical treatment of a reflected thermal reactor can be illustrated most simply by considering a slab reactor of thickness a extending from x= -a/2 to x= +a/2 reflected on both sides by a non-multiplying slab (neutron reflector) of thickness b. Neutrons will be considered as monoenergetic with thermal energy. Although the one-group method may provide reasonable results for the reactor, this method does not accurately predict the flux.

For the determination of the flux distribution in the reactor and in the reflector, the following diffusion equations in these zones need to be solved:

reflected-reactor-equations

where a is the real width of the reactor and b the outer dimension of the reflector (bex is the outer dimension including the extrapolation distance). With problems involving two different diffusion media, the following boundary conditions must be satisfied:

1,2. Interface Conditions

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. 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 - equation

4. Vacuum Boundary Condition

The vacuum boundary condition requires the relative neutron flux near the boundary to have a slope of -1/d, i.e., the flux would extrapolate linearly to 0 at a distance d beyond the boundary (d is the extrapolated length). This zero flux boundary condition is more straightforward and is can be written mathematically as:

reflected-reactor-boundary

If d is not negligible, physical dimensions of the reactor are increased by d and this condition can be written as Φ(a/2 + bex) =  0.

The solution in the core satisfying these boundary conditions is:

reflected-reactor-solution

where

reflected-reactor-geometric-factor

The solution in the reflector satisfying these boundary conditions is:

reflected-reactor-solution-reflector

Neutron flux in a finite slab reactor with and without reflector.
Neutron flux in a finite slab reactor according to the one-group method (with and without reflector).
 
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. Kenneth S. Krane. Introductory Nuclear Physics, 3rd Edition, Wiley, 1987, ISBN: 978-0471805533
  7. G.R.Keepin. Physics of Nuclear Kinetics. Addison-Wesley Pub. Co; 1st edition, 1965
  8. Robert Reed Burn, Introduction to Nuclear Reactor Operation, 1988.
  9. 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:

Reflected Reactor