Page 39 - Dynamics and Control of Nuclear Reactors

30 CHAPTER 3 The point reactor kinetics equations

logical to add the feedback to the input. The feedback may be positive or negative,
depending on the process responsible for the feedback. For example, in a reactor with
a positive temperature coefficient of reactivity, the feedback from the process that
determines that a temperature increase would be positive and destabilizing. How-
ever, feedbacks from other processes in the system would also contribute positive
or negative feedback. The total feedback would be the net feedback from all of
the system feedbacks. Therefore, a positive feedback from one of the processes is
acceptable for a system if other stabilizing negative feedbacks dominate. Later chap-
ters show that some reactors have a positive feedback component, but are stable
because of other, stronger negative feedbacks.
Negative feedbacks are usually stabilizing and, therefore, desirable. However,
negative feedbacks can also be destabilizing. This happens if the negative feedback
is delayed before adding to the input and if the feedback gain exceeds a certain limit.
Section 7.6 addresses the issue of destabilizing negative feedback and Section 13.6
addresses the issue of destabilizing negative feedback in BWRs.
Nonlinear systems are a different story. They can have multiple equilibrium
points. A transient can involve jumping from one equilibrium point to another
and each equilibrium point can be stable or unstable. Nonlinear stability can depend
on the magnitude of the input disturbance. Nonlinear systems can demonstrate limit
cycles, a continuous oscillatory response that may be non-symmetric in shape.
Efforts to develop useful and practical stability analysis methods for nonlinear
systems have occupied well-qualified mathematicians for many years, but the efforts
failed. Stability analysis, especially analysis of nonlinear systems, usually relies on
numerical solutions of the governing differential equations. Because the nature of the
response can depend on the magnitude and form of the input disturbance, it is nec-
essary to simulate a number of different scenarios. The characteristics of lineariza-
tion of a nonlinear system depend on the nominal state about which the model is
linearized. Thus, the linearized form of a nonlinear equation generally varies from
one equilibrium point to another equilibrium point.
In a reactor, it is possible for the flux shape to be unstable. That is, a disturbance in
one part of the reactor can stimulate cyclical flux increases in one part of the reactor
while other parts experience flux decreases. Generally, spatial stability is a concern
mainly in large, loosely-coupled reactors (reactors in which individual regions are
almost independently critical). Spatial stability is addressed in Chapter 9.
Flow instabilities are an issue in some reactors, especially boiling water reactors.
BWRs are operated so as to avoid conditions in which flow instability are likely
Chapter 13 addresses this issue.

3.9 Fluid-fuel reactors
In the 1960s, two types of fluid-fuel reactors (aqueous homogeneous and molten
salt) were considered as candidates for future power reactors, and small test reac-
tors were built and operated. Subsequently, these concepts were abandoned in favor