Page 42 - Dynamics and Control of Nuclear Reactors

CHAPTER

Solutions of the point reactor

kinetics equations and 4

interpretation

4.1 Evolution of simulation methods
In earlier days, analysts had to perform modeling, computer programming, and
numerical analysis to accomplish a reactor simulation. Often, one person had to han-
dle all three tasks. Now software packages are available that eliminate the need for
expertise in computer programming and numerical analysis. Even software that pro-
vides simulations for specific cases is available. These packages expand the ability to
perform reliable simulations, but the analyst still needs basic understanding in order
to interpret the results correctly.

4.2 Numerical analysis
Mathematically, the point reactor kinetics equations are a “stiff system”. Solutions
of the reactor kinetics equations have a very fast component (due to the small value of
generation time or lifetime) and a slow component (due to the much larger values
of the precursor half-lives). Consequently, any numerical method used to solve
the equations must have the ability to handle stiff systems. A number of differential
equation solvers are available in easy-to-use software packages, and most have good
methods for handling stiff systems. Generally, stiff systems require small time steps
in the solution. The more sophisticated solution methods can achieve suitable accu-
racy with varying time steps, small values at the beginning of a transient when the
response is changing rapidly, and larger time steps later when the speed of response
slows down. However, it should be noted that the speed of modern computers some-
times makes it feasible to use small computation increments needed in simpler
methods. The MATLAB/Simulink software platform is often used for numerical
solutions.
The simplest simulation involves a model consisting of linear, constant-
coefficient differential equations. The point reactor kinetics equations with constant
reactivity are a linear constant-coefficient set of equations. For simulations of the
response to time-varying reactivity perturbations, the point kinetics equations are
a set of variable-coefficient equations. As we shall see in later sections, reactivity
is a function of other dependent variables (such as component temperatures and sys-
tem pressure) in a power reactor. In this case, the model is nonlinear.
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