Page 45 - Dynamics and Control of Nuclear Reactors

36 CHAPTER 4 Solutions of the point reactor kinetics equations

For a positive reactivity step, s is positive, and the neutron density increases indef-
initely. The reciprocal of s is the time required for the response to increase by a factor
of e. This quantity is called the reactor period, T, where T ¼ 1/s.
For a negative reactivity step, all of the exponentials are negative and the neutron
density decreases indefinitely. In this case the terms with large negative exponentials
decrease faster with time than the slower exponentials, leaving a decrease deter-
mined largely by the slowest exponential term. The smallest value for s (in a light
1
water reactor) is around 1/80 s . So, the response to a negative reactivity step even-
tually settles out to an exponential with a negative period of around 80 s. But this
result is not quite exact because delayed neutron precursors continue to be produced
as the neutron density decreases. But the resulting delayed neutron production is
small compared to production from precursors that existed before the initiation of
the reactivity decrease.
Available software packages usually solve state variable formulations of a
dynamic model. These include MATLAB/Simulink, MAPLE, MATHEMATICA,
Modelica, and others. Students would benefit by familiarity with one of these or a
similar software system. Appendix G provides a brief description of MATLAB/
Simulink.

4.3 Maneuvers in a zero power reactor
Simulations presented here are for a U-235 fueled reactor with a generation time of
5
10 s. This is called the reference model in subsequent sections of this chapter.
Numerical solutions for the reference model and various perturbations are obtained
and are shown in subsequent sections. A table of parameters related to delayed neu-
tron precursors is given in Chapter 3.
The most common perturbation used to illustrate the behavior of a zero-power
reactor is the step change in reactivity. Fig. 4.1 shows the responses of the reference
model to a positive step change of ρ ¼ 0.00067 (10¢), as the solution of Eqs. (3.22)
and (3.23). This simulation illustrates the initial prompt jump, the sudden increase in
reactor power that follows a step increase. This is the initial response to an immediate
and constant change in reactivity.
In order to demonstrate the effect of the magnitude of reactivity on response,
responses at a selected time into the transient following a reactivity step are calcu-
lated. Fig. 4.2 shows the response at one second into the transient following various
reactivity steps. The dramatic increase in the response as the reactivity approaches a
value of ρ ¼ β, the delayed neutron fraction, is apparent.
The rapid rise when reactivity exceeds the delayed neutron fraction occurs
because the reactor no longer depends on delayed neutron contributions to grow.
The prompt neutrons alone are sufficient to cause a positive rate of change. This con-
dition is called prompt critical. However, it should be noted that delayed neutrons
still influence the response. They are no longer essential, but they reduce the rate
of change relative to the response if there were no delayed neutrons.