Investigation of rotating mode behavior in BWR out-of-phase limit cycle oscillations – Part 1: Reduced order model

Highlights

• Out-of-phase oscillations were simulated in a 4-channel reduced-order model.

• A novel approach was used to model thermal hydraulic channel coupling in the plena.

• With little or no neutronic feedback, a rotating mode limit cycle was observed.

• With strong neutronic feedback, a side-to-side limit cycle behavior was observed.

Abstract

Previous neutronic/thermal-hydraulic (TH) coupled numerical simulations using full-core TRACE/PARCS and SIMULATE-3K BWR models have shown evidence of a specific “rotating mode” behavior (steady rotation of the symmetry line, i.e. constant phase shift of approximately 90° between the first two azimuthal modes) in out-of-phase limit cycle oscillations, regardless of initial conditions and even if the first two azimuthal modes have different natural frequencies. This suggests a nonlinear coupling between these modes; otherwise, the phase shift between these modes would change at a constant rate during the limit cycle. The goal of the present work is to gain further insights on the rotating mode behavior using a simplified mathematical model which contains all of the important physics for this application while providing sufficient flexibility and simplicity to allow for in-depth understanding of the underlying phenomena. This was accomplished using a multi-channel, multi-modal reduced-order model, using a modification of the fixed pressure drop boundary condition to simulate channel coupling via the inlet and outlet plena, in order to destabilize the out-of-phase mode over the in-phase mode. Examination of the time-dependent solution of the nonlinear system showed a clear preference for rotating mode behavior in the four-channel model under stand-alone TH conditions and for conditions with weak neutronic feedback. When neutronic feedback was strengthened (i.e., larger reactivity feedback coefficients), the side-to-side mode (stationary symmetry line) was favored instead. Additional analyses using higher-fidelity numerical modeling, as well as a physical explanation for the rotating behavior seen in both sets of analyses, will be provided in a companion paper (“Part 2”).

Introduction

Instability events in operating boiling water reactors (BWRs) have most commonly been associated with the coupled neutronic-thermal-hydraulic (TH) instability phenomenon, involving density-wave oscillations coupled with neutron kinetic feedback (March-Leuba and Rey, 1993). Typically, the core will oscillate under one of two primary classifications: in-phase (core-wide) oscillations, in which the oscillations in all fuel bundles occur with the same phase, and out-of-phase (regional) oscillations, in which one-half of the core oscillates 180° out-of-phase relative to the opposite half at any given point in time. If unmitigated, the oscillations will typically grow until reaching an asymptotic limit cycle behavior¹, with a fixed amplitude and periodic behavior determined by the nonlinear dynamics of the underlying neutronic and TH equations².

The line of symmetry dividing the two halves during out-of-phase oscillations need not be stationary, as has been observed, e.g., in the Leibstadt (KKL) September 1990 stability test (Blomstrand, 1990) and detected in a natural circulation test performed in the Oskarshamn-3 BWR (Andersson and Stepniewski, 1990). In the latter study, according to the authors, a clear “rotating mode” behavior was observed, at least for a time, in which the symmetry line rotated with more or less constant angular velocity, until the behavior suddenly switched into a “side-to-side” behavior characterized by a stationary symmetry line. As has been elaborated by previous authors (Miro et al., 2000, Zinzani et al., 2008), the rotating case can be interpreted in terms of a superposition of the first two azimuthal modes of the static neutron flux distribution, where a 90° phase shift between the time-dependent amplitudes of these two modes is associated with a continuous rotation of the symmetry line over time (one complete rotation occurs over one oscillation period); this is what is meant by the “rotating mode” discussed above. Conversely, a 0° phase shift between the first and second azimuthal modes is associated with a stationary line of symmetry, herein referred to as the “side-to-side” oscillation mode. For intermediate phase shifts between these values, the orientation of the line of symmetry changes continuously with a non-constant angular velocity, with the superimposed behavior appearing more “side-to-side”-like or “rotating”-like depending on the proximity of the phase shift to $0°$ or $90°$, respectively.

In the literature, it has typically been assumed that the first two azimuthal modes oscillate independently of each other during out-of-phase oscillations, with neither mode influencing the time-dependent phase of the other. In this case, the time-dependent amplitude of each azimuthal mode would oscillate at its own natural frequency (which may be different for each mode due to asymmetric fuel loading or burnup conditions), with an arbitrary and continuously changing phase shift relative to the other mode. For example, consider a BWR core in which the first azimuthal mode has a 10% shorter natural oscillation frequency than the second azimuthal mode; if at a given point in time the two modes were to oscillate in-phase, then after the first azimuthal mode completes one full oscillation period, the second mode would have a 36° phase lag relative to the first mode. Every subsequent oscillation period would add an additional 36° phase lag between the modes, until after the 10th period the two modes would once again oscillate in-phase. In this hypothetical case, the core would oscillate in a predominantly side-to-side fashion at certain times and a predominantly rotating fashion at other times, with gradual transitions between the two.

However, two previous numerical studies—one by Wysocki et al. (2014) using the TRACE/PARCS coupled code system (U.S. NRC, 2012) and another by Dokhane et al. (2013) using the SIMULATE-3K code—indicated a specific tendency towards the rotating pattern under asymptotic limit cycle conditions, for at least some out-of-phase full-core conditions. In these studies, the phase shift between azimuthal modes remained at a constant value (approximately 90°) during the limit cycle, rather than gradually drifting over time as described in the previous paragraph. This was true even for cases where the first two azimuthal modes had different natural frequencies based on linear stability analysis. This suggested that during large-amplitude limit cycle oscillations, the first two azimuthal modes of a BWR do not oscillate independently of each other, as a linear analysis of the neutron flux modes would suggest. Rather, these studies suggest that a nonlinear coupling mechanism between the modes exists which promotes the specific 90° (rotating) behavior.

The primary focus of the current study is to provide further insights into the rotating mode limit cycle behavior from a phenomenological standpoint. To that end, the scope has been narrowed to smaller “N-channel” models including either $N=2$ or $N=4$ channels, rather than a full-core model. Primarily, the four-channel model will be used, as this gives the fewest number of channels while still allowing for two separate azimuthal modes (in the coupled case). In addition, a reduced-order model (ROM) is used, which provides a variety of advantages in simplifying the analysis and allowing for ease of physical insights. From this simplified model, insights have been gained into the physical causes for the oscillatory behavior, e.g., rotating versus side-to-side, and these insights may be easily extended for an understanding of the full-core results shown in the previous works.

Numerous previous studies have utilized ROMs to investigate unstable behavior in thermal hydraulic and neutronic systems representative of BWRs. One of the first of these was developed by March-Leuba et al. (1986), relying on point kinetics and simplified fuel heat transfer and void reactivity models. However, for the present study, a direct representation of the dynamic fluid mass, momentum, and energy transport behavior was required; thus, the authors selected the ROM by Karve et al. (1997) as a starting point; this model includes one of the simplest treatments of the dynamic fluid transport behavior, calculated in a single thermal-hydraulic channel (which was extended to multiple channels in the present work as described below).

Later, Lange (2009) and Dokhane (2004) were among the first to develop two-channel ROMs which allow for the studying of regional (out of phase) type oscillations in heated and neutronic-coupled channels corresponding to the first azimuthal mode, as well as core-wide oscillations corresponding to the fundamental mode. More recently, Dykin et al. (2013) further extended the application of ROMs to four parallel channels, which permit examination of the fundamental mode and the first two azimuthal modes. The work done by Dykin et al. includes investigation of a four-channel case in which all three of these modes are unstable. In the case analyzed, the first two azimuthal models appeared to oscillate independently, with a constantly changing phase shift between these modes. In other words, a specific rotating mode limit cycle behavior was not observed. This is contrary to the findings of the ROM presented in the current work; however, the current work investigates the case of identical TH channels and symmetric neutronic conditions (identical azimuthal modes) while significant asymmetry was present in the model of Dykin et al., which points to the possibility that the azimuthal modes must be sufficiently similar (by some as-yet undetermined criterion) to exhibit the rotating behavior with a fixed phase shift between modes. However, it is also possible that differences in, for example, boundary condition and inlet plenum treatment between the two models could attribute to the difference in behavior as well. However, the authors again point out that the rotating mode behavior was first observed in full-core BWR simulations (Dykin et al., 2013, Wysocki et al., 2014), which indicates that the rotating mode behavior presented in the current work is not purely an artifact of the particular assumptions used for the ROM.

A secondary aim of the current study is to present a new approach for boundary condition treatment in multi-channel systems which, to the authors’ knowledge, has not been done previously in quite the same way. This approach freely allows the user to control the preferred oscillation type (in-phase versus out-of-phase) by simple adjustment of inlet and outlet plena loss factors, while maintaining consistency across cases for clear comparison if done properly.

Previous authors have examined, both experimentally and theoretically, the behavior of systems of $N$ identical TH channels connected in parallel via common inlet and outlet plena, in particular when the total flow rate among channels is held constant. For $N=2$, only the $(0°,180°$) pattern is possible (i.e., phase shift of $180°$ between the individual channel oscillations). For $N=4$, Nakanishi et al. (1983) reported experimental results in which two pairs of channels form, with the two channels of each pair oscillating counter-phase to each other but with an arbitrary phase shift for one pair relative to the other. This can be expressed as $\left(0°,\varphi ,180°,180°+\varphi \right)$, where each element within the parentheses denotes the phase of oscillations in each of the four channels (with the first channel being given a phase of $0°$ by convention) and $\varphi$ is the arbitrary phase shift between the two pairs.

This is sufficient to ensure that the total flow rate remains constant in the linear case (i.e., perfectly sinusoidal oscillations, which occur for small-amplitude oscillations); indeed, as will be shown, examination of the eigenvalues and eigenvectors of the linearized form of the ROM in this paper indicates no preference in the phase shift between counter-oscillating channel pairs. However, for nonlinear oscillations (i.e., oscillations that deviate from simple sinusoidal behavior, as occurs particularly for large-amplitude limit cycle oscillations in BWRs), the total flow rate cannot remain strictly constant and this gives rise to a preference for rotating-mode behavior (at least in terms of thermal hydraulics), as will be demonstrated in this paper.