Natural Modes of the Xenon Problem with Flow Feedback — An Example

The physical question of the spatial stability of a reactor with respect to xenon oscillations corresponds to a mathematical question regarding the location in the complex plane of the roots of a certain eigenvalue problem. The introduction of feedback controllers corresponds to the imposition of constraints on the eigenvalue problem. The effect of certain such constraints on the locations of the eigenvalues is examined in this paper for the idealized case of a one-group uniform-ring reactor. It is found that the eigenvalues obey a rule related to Rayleigh's separation theorem for vibrating mechanical systems. A numerical example is given in which the solutions of the constrained eigenproblem are displayed, interpreted physically, and compared with those of the unconstrained problem.