Reactor Stability

Abstract

Understanding time-dependent behaviors of nuclear reactors and the methods of their control is essential to the operation and safety of nuclear power plants. This chapter provides researchers and engineers in nuclear engineering very general yet comprehensive information on the fundamental theory of nuclear reactor kinetics and control and the state-of-the-art practice in actual plants, as well as the idea of how to bridge the two. The dynamics and stability of engineering equipment affect their economical and operation from safety and reliable operation point of view. In this chapter, we will talk about the existing knowledge that is today practiced for the design of reactor power plants and their stabilities as well as available techniques to designers. Although, stable power processes are never guaranteed. An assortment of unstable behaviors wrecks power apparatus, including mechanical vibration, malfunctioning control apparatus, unstable fluid flow, unstable boiling of liquids, or combinations thereof. Failures and weaknesses of safety management and safety management systems are the underlying causes of most accidents.

Problems

Problem 13.1

Perform the Nyquist and Bode plots for first-order system using Scilab functions.

Problem 13.2

The Stability of Dynamic System

The equations which describe the dynamic behavior of a reactor are complicated and, in particular, are essentially nonlinear for the diffusion equation:

$$ l\frac{\partial P\left(\overrightarrow{\boldsymbol{r}},t\right)}{\partial t}={M}^2{\bigtriangledown}^2P\left(\overrightarrow{\boldsymbol{r}},t\right)+\left({k}_{\infty }-1\right)P\left(\overrightarrow{\boldsymbol{r}},t\right) $$

(1)

where

• l is prompt neutron life time (s)

• P is reactor power density (W/cm²)

• M is migration length (cm)

• k is multiplication factor

Equation (1) necessarily involves the product ${k}_{\infty }P\left(\overrightarrow{\boldsymbol{r}},t\right)$. Since k_∞ is itself a function of the reactor temperature, Xenon concentration, and so on, this term is essentially nonlinear as well. Consequently, it is not in general possible to solve the dynamic equation (1) in analytic terms, and extensive numerical integrations are normally necessary. Such calculations are carried out, for example, in the investigation of fault conditions. In stability studies, however, a number of simplifications are possible.¹ Assume that Eq. (1) reduces to form of Eq. (2) below, by setting up a mesh of points in three dimensions covering the reactor and defining each variable –power density, fuel temperature, xenon concentration, and so on – at each point of mesh. Terms in the equations like ∂²p(x)/∂x² can now be represented approximately as:

$$ \frac{\partial^2p(x)}{\partial {x}^2}=\frac{1}{h^2}\left\{p\left(x-h,y,z\right)-2p\Big(x,y,z\left)+p\right(x+h,y,z\Big)\right\} $$

(2)

and so on. This can be made accurate as one wishes by taking a fine enough mesh, provided one is not concerned with quantities that vary arbitrarily steeply in space. From physical considerations, however, such variations are of no interest in a stability study, for the stabilizing effect of neutron leakage must dominate all other effects in a highly localized disturbance. Thus, such a representation is legitimate. Then, Eq. (2) now takes the following form:

$$ \frac{dx_i}{dt}=\sum \limits_{j=1}^n{a}_{ij}{x}_j\kern1em i=1,2,\cdots, n $$

(3)

where x_i is the various representations of reactor temperature, power densities, and so on, at the different mesh points. The a_ij is independent of t. Equation (3) is a very simple set of differential equations, and they can, in principle, be solved easily. Find the general solution of this equation.