Periodic and chaotic oscillations in the nuclear-coupled boiling channel system subject to strong void-reactivity feedbacks

The present study investigates the nonlinear dynamics and possible oscillation types of a single or multiple nuclear-coupled boiling channel system subject to strong void-reactivity feedbacks. The results demonstrate a route from periodic oscillations to chaotic oscillations through period-doubled bifurcation for a single nuclear-coupled boiling channel by strengthening the nuclear-coupled effects (3C_α). Moreover, complex interesting nonlinear phenomena, i.e. unbounded, limit cycle, period-doubled, period-N and complex Rossler type chaotic oscillations, may appear in the system of three boiling channels coupled with three subcores under the condition of strong void-reactivity feedbacks (4C_α) and weak subcore-to-subcore neutron interactions (ε_ij = 10).     

Dynamic analysis of multiple nuclear-coupled boiling channels based on a multi-point reactor model

This work investigates the non-linear dynamics and stabilities of a multiple nuclear-coupled boiling channel system based on a multi-point reactor model using the Galerkin nodal approximation method. The nodal approximation method for the multiple boiling channels developed by Lee and Pan [Lee, J.D., Pan, C., 1999. Dynamics of multiple parallel boiling channel systems with forced flows. Nucl. Eng. Des. 192, 31–44] is extended to address the two-phase flow dynamics in the present study. The multi-point reactor model, modified from Uehiro et al. [Uehiro, M., Rao, Y.F., Fukuda, K., 1996. Linear stability analysis on instabilities of in-phase and out-of-phase modes in boiling water reactors. J. Nucl. Sci. Technol. 33, 628–635], is employed to study a multiple-channel system with unequal steady-state neutron density distribution. Stability maps, non-linear dynamics and effects of major parameters on the multiple nuclear-coupled boiling channel system subject to a constant total flow rate are examined. This study finds that the void-reactivity feedback and neutron interactions among subcores are coupled and their competing effects may influence the system stability under different operating conditions. For those cases with strong neutron interaction conditions, by strengthening the void-reactivity feedback, the nuclear-coupled effect on the non-linear dynamics may induce two unstable oscillation modes, the supercritical Hopf bifurcation and the subcritical Hopf bifurcation. Moreover, for those cases with weak neutron interactions, by quadrupling the void-reactivity feedback coefficient, period-doubling and complex chaotic oscillations may appear in a three-channel system under some specific operating conditions. A unique type of complex chaotic attractor may evolve from the Rossler attractor because of the coupled channel-to-channel thermal-hydraulic and subcore-to-subcore neutron interactions. Such a complex chaotic attractor has the imbedding dimension of 5 and the fractal dimension ranging from 1.26 to 1.35.     

Analytical study of nuclear-coupled density-wave instability in a natural circulation pressure tube type boiling water reactor

An analytical model has been developed to study the nuclear-coupled density-wave instability in the Indian advanced heavy water reactor (AHWR) which is a natural circulation pressure tube type boiling water reactor. The model considers a point kinetics model for the neutron dynamics and a lumped parameter model for the fuel thermal dynamics along with the conservation equations of mass, momentum and energy and equation of state for the coolant. In addition, to study the effect of neutron interactions between different parts of the core, the model considers a coupled multipoint kinetics equation in place of simple point kinetics equation. Linear stability theory was applied to reveal the instability of in-phase and out-of-phase modes in the boiling channels of the AHWR. The results indicate that the stability behavior of the reactor is greatly influenced by the void reactivity coefficient, fuel time constant, radial power distribution and channel inlet orificing. The delayed neutrons were found to have a strong influence on the Type I and Type II instabilities observed at low and high channel powers, respectively. Also, it was found that the coupled multipoint kinetics model and the modal point kinetics model predict the same threshold power for out-of-phase instability if the coupling coefficient in the former model is half the eigen value separation between the fundamental and the first harmonic mode in the latter model. Decay ratio maps were predicted considering various operating parameters of the reactor, which are useful for its design.     

The development of a closed-form analytical model for the stability analysis of nuclear-coupled density-wave oscillations in boiling water nuclear reactors

A state-of-the-art one-dimensional thermal-hydraulic model has been developed to be used for the linear analysis of nuclear-coupled density-wave oscillations in a boiling water nuclear reactor (BWR). This model accounts for phasic slip, distributed spacers, subcooled boiling, space/time-dependent power distributions and distributed heated wall dynamics. In addition to a parallel channel stability analysis, a detailed model was derived for the BWR loop analysis of both the natural and forced circulation modes of operation.The model for coolant thermal-hydraulics has been coupled with the point kinetics model of reactor neutronics. Kinetics parameters for use in the neutronics model have been obtained by utilizing self-consistent nodal data and power distributions.The computer implementation of this model, NUFREQ-N, was used for the parametric study of a typical BWR/4, as well as for comparisons with existing in-core and out-of-core data. Also, NUFREQ-N was applied to analyze the expected stability characteristics of a typical BWR/4.     

“Delta-Wrinkly Zone” Observed Near Crack of Ductile Zircaloy-2 Cladding during Impact Loading Test

An equivalent linearization method has been introduced for density-wave oscillation analysis. Unlike the frequency domain approach with linearized equations assuming an infinitely small amplitude, the present method can deal with large amplitude oscillations with the nonlinearity effect on limit cycle amplitudes taken into account in the frequency domain.

Equivalent linear equations have been derived based on a one-dimensional separated flow model. A profile fit model was adopted for subcooled boiling. The characteristics of the equivalent linear equations and factors to limit the amplitudes were phenomenologically examined.

Analyses of the experimental data by Saha et al. using the present method revealed that, given an appropriate set of two-phase flow correlations, the equivalent linearization method can predict limit cycle amplitudes after instability initiation as well as the stability boundary.     

Analysis of chaotic instabilities in boiling systems

An analytic model for the investigation of non-linear dynamics in boiling systems has been developed. This model is comprised of a nodal formulation that uses one-dimensional homogeneous equilibrium assumptions for diabatic two-phase flow, a lumped parameter approach for heated wall dynamics, and point neutron kinetics for the consideration of nuclear feedback in a boiling water reactor (BWR) loop. This model indicates that a boiling channel coupled with a riser may experience chaotic oscillations. In contrast, a boiling channel without a riser that is subjected to a constant pressure drop (i.e. a parallel channel) may undergo a supercritical bifurcation (i.e. may experience a limit cycle), but chaos was not found. Flow instabilities in a two-phase natural circulation loop have been verified using the model presented in this paper. The predictions of the effects of the channel inlet resistance, outlet resistance and liquid level in the downcomer agree with the data of Kyung and Lee. Finally, an analysis of nuclear-coupled density-wave instabilities in a simplified BWR (SBWR) was performed. Significantly, even for low pressure conditions, a simplified SBWR appears to be stable during start-up and normal operations; however, a limit cycle may occur for abnormal operating conditions.     

Stability Analysis of BWRs Using Bifurcation Theory

Abstract

This paper presents a new approach using the bifurcation theory for the stability analysis of BWRs. In this approach, the dependencies of the equilibrium states on the parameters that have a large influence on the stability are investigated topological over a wide range of phase space. The stability information can be derived from the analysis of the bifurcation phenomena on the equilibrium states. This investigation enabled us to obtain qualitative and global information on the stability of a nonlinear system. The new approach was applied to the analysis of the stability associated with in-phase power oscillation (core reactivity stability). The loss of linear stability took place at a lower reactor power as the coolant flow rate decreased, and this instability occurs at the Hopf bifurcation point. The sensitivity analysis of the stability boundary for the various parameters revealed that the channel hydrodynamics heavily play a significant role in the stability. The Hopf bifurcation analysis proved that the periodic state bifurcating at the Hopf bifurcation point was orbitally unstable and a limit cycle attractor did not exist in the vicinity of the bifurcation point. This fact led to the conclusion that a limit cycle in-phase power oscillation observed in BWR instability was not excited directly by the bifurcation of the orbitally unstable periodic state.     

Dynamics of multiple parallel boiling channel systems with forced flows

This work investigates the nonlinear dynamics of multiple parallel boiling channels with forced flows by using the Galerkin nodal approximation method. The stability maps for multiple-channel systems subject to a constant total mass flow rate are constructed on the basis of the average heat flux and mass flow rate on the parameter plane of the subcooling and phase change numbers. The limit cycle oscillations for 3-, 4- and 5-channel systems indicate that the most heated channel exhibits the largest magnitude of oscillation, and is out-of-phase with the other channels. Owing to that the boiling assembly contains only a small number of channels, the above system becomes unstable due to channel-to-channel interaction with an increase of the channel number. This work also elucidates the dynamics of a 3-channel system with a periodic total mass flow rate. In addition, complicated nonlinear oscillations are predicted for different operating points with high inlet subcooling numbers in the stable region. The system dynamics are highly sensitive to the frequency of the forcing function. Periodic, quasi-periodic and chaotic oscillations may appear in the forcing function system. Unstable oscillations appear when either the frequency of the forcing function is equal or extremely close to the system’s natural frequency or the two frequencies are frequency-locking.     

Stability analysis and nonlinear dynamics of natural circulation boiling water reactors

Numerous experimental and numerical investigations have been reported in literature on the effect of various parameters on the dynamics of natural circulation boiling water reactors (NCBWRs). Different findings were reported in literature regarding the effect of void reactivity coefficient and fuel time constant on Type-I and Type-II instabilities. The present work contains stability analysis and nonlinear dynamics of NCBWR using a lumped parameter model. The purpose of the stability analysis is to resolve different, seemingly contradictory parametric trends reported by previous authors. Stability analysis for three different system configurations has been done using the same mathematical model and compared with the findings reported in literature. It has been observed that the effects of void reactivity coefficient and fuel time constant are influenced by geometrical parameters and are different for different systems. Nonlinear dynamics and bifurcations have been studied numerically, for one of the system configurations, for boiling channels with and without riser. The possibility of existence of stable and unstable limit cycles, period doublings, and chaotic oscillations has been investigated. The effect of void reactivity coefficient on the nonlinear dynamics has been studied in both Type-I and Type-II regions. In the Type-I region, subcritical Hopf bifurcation leading to unstable limit cycle, as well as supercritical Hopf bifurcation leading to stable limit cycle have been found. The boiling channels with and without riser have been observed to undergo chaotic oscillations under strong reactivity feedback in the Type-II region. It has been found that the presence of riser aggravates chaotic oscillations.     

Instabilities in Parallel Channel of Forced-Convection Boiling Upflow System, (V)

The density wave instability in a parallel boiling channel system heated electrically has been studied experimentally and analytically by the authors. In our country, the steam generator for LMFBR has been investigated with Power Reactor and Nuclear Fuel Development Corp. as the central figure for its development, and many results of this instability were reported. Their results were different from our ones as regard to the governing factor of the period of flow oscillation in the unstable region and to the effect of the slip ratio on the stability in analysis. A new linear analytical model is proposed in this paper and the analytical results are compared with ones of two-phase analyses based on the same linear method as this model. Subsequently, the effect of the slip ratio on the stability is studied analytically by this model. The parallel boiling channel system is studied experimentally and analytically, using Freon-113 as test fluid heated by hot water as simulation of the SG for LMFBR. The governing factor of the period of flow oscillation is made clear.     

Study on the stability behaviour of a natural circulation pressure tube type boiling water reactor

The stability behaviour of a natural circulation pressure tube type boiling water reactor (BWR) has been investigated analytically. The analytical model considers homogeneous two-phase flow, a point kinetics model for the neutron dynamics and a lumped heat transfer model for the fuel dynamics. The results indicate that both Type I and Type II density-wave instabilities can occur in the reactor in both in-phase and out-of-phase mode of oscillations in the boiling channels of the reactor. The delayed neutrons were found to have strong influence on the stability of Type I and Type II density-wave instabilities. Also, the stability of the reactor is found to increase with increase in negative void reactivity coefficient unlike that observed previously in vessel type BWRs. Decay ratio map was predicted considering the effects of channel power, channel inlet subcooling, feed water temperature and channel exit quality, which are useful for the design of the reactor.     

气空间对自然循环两相流动系统稳定性影响实验研究

微沸腾工况运行是核供热堆实现热电联供的关键性问题之一，微沸腾运行工况下，两相流系统稳定性更加不利和复杂。通过实验研究，揭示了气空间对两相流系统稳定性的影响，研究提出通过气空间改性来抑制系统不稳定。实验结果表明，在气空间加装隔离孔板，对两相流系统不稳定振幅有明显的抑制作用，对两相流系统不稳定边界也有改善。      

An analytical study on excitation of nuclear-coupled thermal hydraulic instability due to seismically induced resonance in BWR

A core-wide in-phase neutron flux oscillation, which took place, for example, at LaSalle-2 in the USA in 1988, is one of the nuclear-coupled thermal hydraulic instabilities in boiling water reactors (BWRs). In this study, an analysis has been performed focusing on the excitation of this type of instability in BWRs due to seismically induced resonance, within the scope of a point kinetics model. For this purpose, the trac-bf1 code has been modified to take into account the external acceleration in addition to gravity. As a result of this analysis, it is shown that reactivity insertion can occur accompanied by in-surge of the coolant into the core resulting from excitation. It is also shown that the amount of reactivity inserted largely depends on the degree of stability of the initial state and the amplitude of the seismic wave, whose frequency is the same as the characteristic frequency of the instability.     

Stability analysis of BWR nuclear-coupled thermal-hydraulics using a simple model

A simple mathematical model is developed to describe the dynamics of the nuclear-coupled thermal-hydraulics in a boiling water reactor (BWR) core. The model, which incorporates the essential features of neutron kinetics and single-phase and two-phase thermal-hydraulics, leads to a simple dynamical system comprised of a set of nonlinear ordinary differential equations (ODEs). The stability boundary is determined and plotted in the inlet-subcooling-number (enthalpy)/external-reactivity operating parameter plane. The eigenvalues of the Jacobian matrix of the dynamical system also are calculated at various steady-states (fixed points); the results are consistent with those of the direct stability analysis and indicate that a Hopf bifurcation occurs as the stability boundary in the operating parameter plane is crossed. Numerical simulations of the time-dependent, nonlinear ODEs are carried out for selected points in the operating parameter plane to obtain the actual damped and growing oscillations in the neutron number density, the channel inlet flow velocity, and the other phase variables. These indicate that the Hopf bifurcation is subcritical, hence, density wave oscillations with growing amplitude could result from a finite perturbation of the system even when it is being operated in the parameter region thought to be safe, i.e. where the steady-state is stable. Finally, the power-flow map, frequently used by reactor operators during start-up and shut-down operation of a BWR, is mapped to the inlet-subcooling-number/neutron-density (operating-parameter/phase-variable) plane, and then related to the stability boundaries for different fixed inlet velocities corresponding to selected points on the flow-control line. Also, the stability boundaries for different fixed inlet subcooling numbers corresponding to those selected points, are plotted in the neutron-density/inlet-velocity phase variable plane and then the points on the flow-control line are related to their respective stability boundaries in this plane. The relationship of the operating points on the flow-control line to their respective stability boundaries in these two planes provides insight into the instability observed in BWRs during low-flow/high-power operating conditions. It also shows that the normal operating point of a BWR is very stable in comparison with other possible operating points on the power-flow map.     

Study of flow instabilities in double-channel natural circulation boiling systems

Natural circulation boiling systems consisting of parallel channels can undergo different types of oscillations (in-phase or out-of-phase) depending on the geometric parameters and operating conditions. Disturbances in one channel affect the flow in other channels, which triggers thermal–hydraulic oscillations. In the present work, the modes of oscillation under different operating conditions and channel-to-channel interaction during power fluctuations and on-power refueling in a double-channel natural circulation boiling system are investigated. The system is modeled using a lumped parameter mathematical model and RELAP5/MOD3.4. Parametric studies are carried out for an equal-power double-channel system, at different operating conditions, with both the models, and the results are compared. Instabilities, non-linear oscillations, and effects of recirculation loop dynamics and geometric parameters on the mode of oscillations, are studied using the lumped model. The two channels oscillate out-of-phase in Type-I region, but in Type-II region, both the modes of oscillation are observed under different conditions. Channel-to-channel interaction and on-power refueling studies are carried out using the RELAP model. At high powers, disturbances in one channel significantly affect the stability of the other channel. During on-power refueling, a near-stagnation condition or low-velocity reverse flow can occur, the possibility of reverse flow being higher at lower pressures.     

Thermo-hydraulic instability of boiling natural circulation loop induced by flashing (analytical consideration)

An analytical study is presented on the thermo-hydraulic stability of a boiling natural circulation loop with a chimney at low pressure start-up. The effect of flashing induced by the pressure drop in the channel and the chimney due to gravity head on the instability is considered. A method to analyze linear stability is developed, in which a drift-flux model is used. The analytical result of a stability map agrees very well with the experimental one obtained in a previous report. Instability does not occur when the heater power is too low to generate voids in the chimney and only natural circulation of single phase can be induced. Instability tends to occur when boiling occurs only near the chimney exit due to flashing. This instability phenomenon has some similarities with density wave oscillation, such as the phase difference of temperature between the boiling region and non-boiling region, and the oscillation period which is near to the time required for fluid to pass through the chimney. However, there are also some differences from density wave oscillation, such as the boiling region is very short, and pressure fluctuation can affect void fraction fluctuation.     

Nonlinear effects in two-phase flow dynamics

Thermally induced two-phase flow oscillations in uniformly heated boiling channels have been analyzed numerically using a one-dimensional model of two-phase flow. Two different approaches to modeling of subcooled boiling have been considered: a mechanistic model and a profile-fit model. The overall model has been numerically implemented as a computer code, DYNOBOSS, which has been validated against a linear stability analysis code and experimental data.The effects of both modeling assumptions and numerical methods of solution have been studied. It has been shown that the calculated transient response of the boiling channel may be very sensitive to the numerical scheme and spatial discretization, especially for operating conditions in the linearly unstable region. For the range of operating parameters studied, phasic slip has shown a significant stabilizing effect on the system, whereas subcooled boiling has indicated smaller influence. Furthermore, it has been shown that the rate of increase of limit cycle amplitude with channel exit quality is higher for low than high inlet subcoolings.     

Modeling and numerical simulation of oscillatory two-phase flows, with application to boiling water nuclear reactors

This paper is concerned with the analysis of dynamics and stability of boiling channels and systems. The specific objectives are 2-fold. One of them is to present the results of a study aimed at analyzing the effects of various modeling concepts and numerical approaches on the transient response and stability of parallel boiling channels. The other objective is to investigate the effect of closed-loop feedback on stability of a boiling water reactor (BWR). Various modeling and computational issues for parallel boiling channels are discussed, such as: the impact of the numerical discretization scheme for the node containing the moving boiling boundary on the convergence and accuracy of computations, and the effects of subcooled boiling and other two-phase flow phenomena on the predictions of marginal stability conditions. Furthermore, the effects are analyzed of local loss coefficients around the recirculation loop of a boiling water reactor on stability of the reactor system. An apparent paradox is explained concerning the impact of changing single-phase losses on loop stability. The calculations have been performed using the DYNOBOSS computer code. The results of DYNOBOSS validation against other computer codes and experimental data are shown.     

低压自然循环复合流量脉动实验研究

在低压流动沸腾不稳定性实验中,研究了自然循环流动在不同入口过冷度下的演化过程。对实验中的流动沸腾不稳定性入口流量信号进行快速傅里叶变换,基于振幅和频率特性区分了3种流动脉动模式：小幅流量脉动、复合流量脉动和逆流。分析了加热功率和入口过冷度对自然循环不稳定性的影响。根据加热段出口水温变化得到了出口的流型变化,当流量波动振幅较小时加热段出口流体始终是饱和状态,而当流量波动振幅较大时,加热段出口为单相液体和两相混合物交替通过。给出了这3种流量脉动的边界图,分析了热流密度和入口过冷度对流量脉动模式的影响。结果表明：出口含气率大于0时发生流动不稳定性,热流密度达到间歇干涸型临界热流密度时发生逆流。     

An equivalent linearization technique for two-phase flow instability analysis

An equivalent linearization technique has been developed to analyze two-phase flow instability oscillation with finite amplitude in the frequency-domain. A one dimensional slip flow model has been used to formulate equivalent linear equations, and implemented into a computer program.A simple boiling channel has been analyzed by the program. Results for very small amplitude oscillations agreed well with results by a detailed linear frequency-domain program.Analyses for larger amplitude oscillations gave more stable results. Limit cycle amplitudes have been calculated from the amplitudes at which channel characteristics change from unstable to stable as the amplitude is increased.