Page 115 - Dynamics and Control of Nuclear Reactors

References 109

9.6.1 Finite difference methods
Finite difference methods use discrete approximations to the space derivatives. This
results in a set of ordinary differential equations that can be solved numerically. See
Appendix F for a description of the finite difference method.

9.6.2 Finite element method (FEM)
See Appendix F for a brief discussion of the finite element method. The finite ele-
ment method was developed initially for structural analysis of objects with complex
geometry. Its use later expanded to include other disciplines, including heat transfer
and fluid mechanics analysis. FEM has also been used for reactor analysis.

9.6.3 Modal methods
The neutron population is represented with specified shape functions multiplied by
time-dependent amplitudes. The algorithm solves for the amplitude functions.

9.6.4 Quasi-static methods
The flux shape is assumed to be slowly varying. Point kinetics solutions provide the
response between re-evaluations of the flux shape using steady-state diffusion theory.

9.6.5 Nodal methods
The reactor is divided into sub-regions, each of which is assumed to have uniform
nuclear properties. Node-to-node coefficients define the neutron flow [2].
There have been many reported developments of methods for space-time kinet-
ics. The literature contains many reports of efforts to reduce computation times,
increase accuracy, and provide accuracy assessments.

Exercises
9.1 Explain why there is no such thing as an exact space-time simulation.
9.2 How does the flux shape affect economics?
9.3 How would a space-time model be modified to deal with Xe-135?
9.4 What would happen to the local Xe-135 concentration if an off-center control rod were moved
out by a small distance? How would it affect the local power? How would it affect total power? If
the reactor returned to a constant power without any control action, explain how this happened.

References
[1] J.H. Ferziger, P.F. Zweifel, The Theory of Neutron Slowing Down in Nuclear Reactors,
The MIT Press, Cambridge, MA, 1966.
[2] J.J. Duderstadt, L.J. Hamilton, Nuclear Reactor Analysis, John Wiley & Sons, New York,
1976.