Modeling Reactor Noise due to Rod and Thermal Vibrations with Thermal Feedback Using Stochastic Differential Equations

Abstract

Fluctuations associated with power and detector readings in a nuclear reactor, commonly known as reactor noise, are of great importance in nuclear science and engineering. Two different types of noise are described in the literature: internal noise, which is associated with the inherent stochasticity of fission chains, and external noise, which is governed by physical fluctuations of the macroscopic system. The latter may include temperature fluctuations, vibration of regulation rods, fluent turbulence, bubble formation, and more. It is generally true that in power reactors, where high temperatures and strong hydrodynamic flows are characteristic, the external noise is dominant. The goal of this paper is to propose a stochastic differential equation (SDE) that models the effect of two types of external noise terms: the inlet temperature variations, which affect the power through reactivity feedback, and rod vibrations, which affect the reactivity directly. Although these aspects were studied in the past, they were only treated via nonstochastic equations. It is argued that the SDE approach, previously used only for modeling the effect of internal noise on nuclear reactor dynamics, is also highly suitable for modeling external noise. The main advantage of our approach is the ability to arrive at analytic formulas. The contributions presented in this paper based on the SDE approach are as follows. Under a linear approximation of thermal feedback, the stabilizing effect of thermal feedback is explained and quantified, and a limiting distribution is analyzed in full. An analysis of the detector response on a finite time interval is carried out, leading to a version of the Feynman variance-to-mean-ratio formula in the presence of external noise. Finally, a calculation of the eigenvalues associated with the linearized system alluded to above is performed, showing that in practical cases the rod vibrations and inlet temperature fluctuations correspond to eigenvalues in distinct timescales. The significance of these finding is discussed.