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.     

Nonlinear stability analysis of a reduced order model of nuclear reactors: A parametric study relevant to the advanced heavy water reactor

In this paper, we perform a parametric study of the nonlinear dynamics of a reduced order model for boiling water reactors (BWR) near the Hopf bifurcation point using the method of multiple scales (MMS). Analysis has been performed for general values of the parameters, but the results are demonstrated for parameter values of the model corresponding to the advanced heavy water reactor (AHWR). The neutronics of the AHWR is modeled using point reactor kinetic equations while a one-node lumped parameter model is assumed both for the fuel and the coolant for modeling the thermal-hydraulics. Nonlinearities in the heat transfer process are ignored and attention is focused on the nonlinearity introduced by the reactivity feedback. It is found that the steady-state operation of the AHWR mathematical model looses stability via. a Hopf bifurcation resulting in power oscillations as some typical bifurcation parameter like the void coefficient of reactivity is varied. The bifurcation is found to be subcritical for the parameter values corresponding to the AHWR. However, with a decrease in the fuel temperature coefficient of reactivity the bifurcation turns to supercritical implying global stability of the steady state operation in the linear stability regime. Moreover slight intrusion into the instability regime results in small-amplitude limit cycles leaving the possibility of retracting back to stable operation.     

Nonlinear analysis of nuclear coupled density wave instability in time domain for a boiling water reactor core undergoing core-wide and regional modes of oscillations

The objective of the paper is to develop a nuclear coupled thermal-hydraulic model in order to simulate core-wide (in-phase) and regional (out-of-phase) stability analysis in time domain within the limitation of desktop research facility for a boiling water reactor subjected to operational transients. The integrated numerical tool, which is a combination of thermal-hydraulic, neutronic and fuel heat conduction models, is used to analyze a complete boiling water reactor core taking into account the strong nonlinear coupling between the core neutron dynamics and primary circuit thermal-hydraulics via the void-temperature reactivity feedback effects. The integrated model is validated against standard benchmark and published results. Finally, the model is used for various parametric studies and a number of numerical simulations are carried out to investigate core-wide and regional instabilities of the boiling water reactor core with and without the neutronic feedback effects. Results show that the inclusion of neutronic feedback effects has an adverse effect on boiling water reactor core by augmenting the instability at lower power for same inlet subcooling during core-wide mode of oscillations, whereas the instability is being suppressed during regional mode of oscillations in presence of the neutronic feedback. Dominance of core-wide instability over regional mode of oscillations is established for the present case of simulations which indicates that the preclusion of the former will automatically prevent the latter at the existing working condition.     

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.     

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.     

Investigation of chaotic oscillations in boiling-water reactors

The chaotic dynamics of boiling-water reactors is investigated on the basis of a one-dimensional integral model of momentum for the boiling-water channel and point equation of kinetics. It is shown that chaotic oscillations during which the sign of the coolant velocity in the boiling channel changes occur in the case of strong feedback on steam content with the parameters of the boiling channel deep in the region of instability occur in boiling-water reactors with natural and forced circulation of the coolant. It is determined that such oscillations can occur with the standard reactor arrangement when the core entrance is open for water to enter the core and for back circulation of the coolant as well as with an arrangement where the entrance is half open – closed for back circulation of the coolant. A numerical calculation of the chaotic oscillations is performed. The mechanism of pulsed chaos is described. Regions of stability and stochasticity are separated in the plane of the parameters characterizing the underheating of water to the saturation temperature at the entrance to the reactor and stationary average steam content in the core. One-dimensional point mappings determining the chaotic dynamics of the boiling water reactor are constructed. The properties of the mappings and the bifurcation of their stationary points are investigated.     

Influence of subcooled boiling on out-of-phase oscillations in boiling water reactors

In this paper, we develop a reduced order model with modal kinetics for the study of the dynamic behavior of boiling water reactors. This model includes the subcooled boiling in the lower part of the reactor channels. New additional equations have been obtained for the following dynamics magnitudes: the effective inception length for subcooled boiling, the average void fraction in the subcooled boiling region, the average void fraction in the bulk-boiling region, the mass fluxes at the boiling boundary and the channel exit, respectively, and so on. Each channel has three nodes, one of liquid, one with subcooled boiling, and one with bulk boiling. The reduced order model includes also a modal kinetics with the fundamental mode and the first subcritical one, and two channels representing both halves of the reactor core. Also, in this paper, we perform a detailed study of the way to calculate the feedback reactivity parameters. The model displays out-of-phase oscillations when enough feedback gain is provided. The feedback gain that is necessary to self-sustain these oscillations is approximately one-half the gain that is needed when the subcooled boiling node is not included.     

Simulation of an hypothetical out-of-phase instability case in boiling water reactor by RELAP5/PARCS coupled codes

A very complex type of power instability occurring in boiling water reactor (BWR) consists of out-of-phase regional oscillations, in which normally subcritical neutronic modes are excited by thermal-hydraulic feedback mechanisms. The out-of-phase mode of oscillation is a very challenging type of instability and its study is relevant because of the safety implications related to the capability to promptly detect any such inadvertent occurrence by in-core neutron detectors, thus triggering the necessary countermeasures in terms of selected rod insertion or even reactor shutdown. In this work, simulations of out-of-phase instabilities in a BWR obtained by assuming an hypothetical continuous control rod bank withdrawal are being presented. The RELAP5/Mod3.3 thermal-hydraulic system code coupled with the PARCS/2.4 3D neutron kinetic code has been used to simulate the instability phenomenon. Data from a real BWR nuclear power plant (NPP) have been used as reference conditions and reactor parameters. Simulated neutronic power signals from local power range monitors (LPRM) have been used to detect and study the local power oscillations. The decay ratio (DR) and the natural frequency (NF) of the power oscillations (typical parameters used to evaluate the instabilities) have been used in the analysis. The results are discussed also making use of two-dimensional plots depicting relative core power distribution during the transient, in order to clearly illustrate the out-of-phase behavior.     

Linear Analysis of Thermo-hydraulic Instabilities of the Advanced Heavy Water Reactor (AHWR)

Analysis was carried out to predict the threshold of instability for Ledinegg type and density wave oscillations for the Indian Advanced Heavy Water Reactor (AHWR) which is a Natural Circulation Pressure Tube Type Boiling Water Reactor. The mathematical model considers homogeneous two-phase flow and the conservation equations are solved analytically to obtain the steady state thermo-hydraulic characteristics and flow stability map. The model was applied for the AHWR concept after it had been validated with the test data obtained from a simple forced circulation loop with small parallel boiling channels and from the High Temperature Loop (PNC). The results indicate that the proposed design configuration of the AHWR may experience both Ledinegg type (static instability) and Type-I and Type-II density wave oscillations depending on the operating condition. The effects of various geometric and operating parameters on these types of instabilities were studied. It can be seen from the results that the Ledinegg type instability is suppressed with an increase in pressure and disappears when the operating pressure is higher than 0.7MPa. But the density wave instability may occur even at 7 MPa. In addition, it is found that for parallel multiple channels operating under natural circulation condition, the stability of density wave instability is not always enhanced by increasing the throttling coefficients at the inlet of channels.     

基于IQS/MC方法的ADS次临界反应堆中子时空动力学模拟分析

采用改进准静态近似与蒙特卡罗中子输运程序相结合(IQS/MC)的方法实现了加速器驱动的次临界系统(ADS)中子时空动力学模拟计算。以加速器驱动嬗变研究装置的靶堆耦合参考方案物理模型为例,通过对束流瞬变引入和燃料组件提升两种工况进行动态模拟,计算得到了堆芯总的相对功率、分能群相对中子注量率及相对功率三维网格分布随时间的变化。将IQS/MC方法计算结果与点堆计算结果进行了对比分析,模拟结果符合物理规律,两种方法对比结果与国外相关文献一致,表明IQS/MC方法适用于ADS次临界反应堆中子时空动力学过程的瞬态安全分析。     

Mass flow rate sensitivity and uncertainty analysis in natural circulation boiling water reactor core from Monte Carlo simulations

Our aim was to evaluate the sensitivity and uncertainty of mass flow rate in the core on the performance of natural circulation boiling water reactor (NCBWR). This analysis was carried out through Monte Carlo simulations of sizes up to 40,000, and the size, i.e., repetition of 25,000 was considered as valid for routine applications. A simplified boiling water reactor (SBWR) was used as an application example of Monte Carlo method. The numerical code to simulate the SBWR performance considers a one-dimensional thermo-hydraulics model along with non-equilibrium thermodynamics and non-homogeneous flow approximation, one-dimensional fuel rod heat transfer. The neutron processes were simulated with a point reactor kinetics model with six groups of delayed neutrons. The sensitivity was evaluated in terms of 99% confidence intervals of the mean to understand the range of mean values that may represent the entire statistical population of performance variables. The regression analysis with mass flow rate as the predictor variable showed statistically valid linear correlations for both neutron flux and fuel temperature and quadratic relationship for the void fraction. No statistically valid correlation was observed for the total heat flux as a function of the mass flow rate although heat flux at individual nodes was positively correlated with this variable. These correlations are useful for the study, analysis and design of any NCBWR. The uncertainties were propagated as follows: for 10% change in the mass flow rate in the core, the responses for neutron power, total heat flux, average fuel temperature and average void fraction changed by 8.74%, 7.77%, 2.74% and 0.58%, respectively.     

Numerical study of nuclear coupled two-phase flow instability in natural circulation system under low pressure and low quality

Based on the one-dimension two-phase drift flow model, the numerical simulation of two-phase flow stability characteristic on the test loop (HRTL-5) for 5 MW heating reactor (developed by the Institute of Nuclear and New Energy Technology of Tsinghua University, Beijing) is performed with and without coupled point neutron kinetics. The density wave oscillation instability is analyzed in the system under low pressure at 1.5 MPa and low steam quality less than 10%. The effect of inlet subcooling and heating flux on the system instability is simulated under the system pressure P_sys = 1.5 MPa. The numerical results show that there exist two instability inlet subcooling boundaries at different heat flux. The numerical results show good agreement with the experimental results on HRTL-5 without consideration of point neutron kinetics. If coupled with point neutron kinetics, the system will exhibit little difference on instability boundaries from that without considering the nuclear characteristics. But the amplitude and the phase of the oscillation of the thermal hydraulic parameters of the system will be somehow affected in unstable zone if the system is coupled with point neutron kinetics.     

Fractional neutron point kinetics equations for nuclear reactor dynamics

The fractional point-neutron kinetics model for the dynamic behavior in a nuclear reactor is derived and analyzed in this paper. The fractional model retains the main dynamic characteristics of the neutron motion in which the relaxation time associated with a rapid variation in the neutron flux contains a fractional order, acting as exponent of the relaxation time, to obtain the best representation of a nuclear reactor dynamics. The physical interpretation of the fractional order is related with non-Fickian effects from the neutron diffusion equation point of view. The numerical approximation to the solution of the fractional neutron point kinetics model, which can be represented as a multi-term high-order linear fractional differential equation, is calculated by reducing the problem to a system of ordinary and fractional differential equations. The numerical stability of the fractional scheme is investigated in this work. Results for neutron dynamic behavior for both positive and negative reactivity and for different values of fractional order are shown and compared with the classic neutron point kinetic equations. Additionally, a related review with the neutron point kinetics equations is presented, which encompasses papers written in English about this research topic (as well as some books and technical reports) published since 1940 up to 2010.     

Parametric effects on reactivity instabilities and nonlinear oscillations in a nuclear-coupled double channel natural circulation boiling system

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. The coupling between the neutronics and thermal-hydraulics has a strong influence on the modes of oscillations in a multi-channel system. In the present study a natural circulation double channel system is modeled. The reactor kinetics is represented by multi-point neutron kinetics model which includes the spatial variation of neutrons. Parametric effects on stability of the system, frequency, and the oscillation modes (reactivity instabilities) are investigated. It is found that at high powers compact cores will be more stable compared to larger cores, while the opposite will be the case at low powers. Further, nonlinear analysis is carried out to investigate the parametric effects on the bifurcation characteristics, transition from one mode to the other mode and chaotic oscillations. The delay in heat transfer and strong neutron interactions between the subcores delays the occurrence of chaotic oscillations.     

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.     

液体燃料熔盐堆三维稳态分析程序开发

液体燃料熔盐堆的物理热工特性与固体燃料反应堆有很大的不同,在分析计算中必须考虑燃料流动特性的影响,一般分析固体反应堆的程序均不能直接用于分析液体燃料熔盐堆。根据熔盐堆的流动特性,建立了液体燃料熔盐堆的三维中子动力学模型和流动传热模型,开发了针对液体燃料熔盐堆的三维稳态核热耦合程序,并以此分析了稳态情况下MOSART堆的物理热工特性。结果表明,堆芯流速对快中子和热中子影响较小,对堆芯温度和缓发中子分布影响较大。     

Nonlinear dynamics of a nuclear-coupled boiling channel with forced flows

The nonlinear dynamics of a nuclear-coupled boiling channel with forced flows was explored on the basis of the Galerkin nodal approximation method for the channel fluid flow and point kinetics for the neutron field dynamics. The marginal stability boundary (MSB) for a Freon boiling channel predicted by the model agreed well with experimental data from previous available literature, thereby verifying the model accuracy. A strip of limit cycle oscillation on the unstable side of the boundary was found to be system dependent. The marginal stability boundaries for boiling water channels with/without nuclear-coupled effects were also obtained. Moreover, the characteristics of limit cycle oscillations on the boundary were investigated. Routes from limit cycle oscillation to chaotic oscillation were identified under high inlet subcooling conditions. The strange attractor found in this study was characterized by a correlation dimension of 1.69±0.01.     

基于波形逼近的快中子脉冲堆耦合计算

快中子脉冲堆在爆发脉冲过程中的中子输运与热弹性力学相互耦合,该耦合作用过程决定了脉冲特性。基于绝热近似下燃料元件温升始终正比于系统总裂变数的事实,提出了通过调整参数使温升随时间变化的曲线逼近裂变率曲线的耦合计算方法。在迭代逼近过程中,采用了有限元商业软件ANSYS处理力学建模和热弹性力学求解,利用点堆方程描述中子学行为,两者利用基于微扰理论的反应性反馈方程进行耦合。通过调整参数使力学模型的温升加载函数波形逼近通过输运计算得到的裂变率波形,直至两者一致。以Lady Godiva脉冲堆为例的裂变产额计算结果与实验结果一致,该计算方法有望用于快中子脉冲堆的研究和设计。     

Spectrum behavior for the nonlinear fractional point reactor kinetics model

The nonlinear fractional point reactor kinetics equation in the presence of Newtonian temperature reactivity feedback with a multi-group of delayed neutrons,which describes the spectrum behavior of neutron density into the homogenous nuclear reactors, is developed. This system is one of the most important stiff coupled nonlinear fractional differentials for nuclear reactor dynamics. The generalization of Taylor's formula that involves Caputo fractional derivatives is developed in an attempt to overcome the difficulty of the stiffness of the nonlinear fractional differential model. Moreover, the general fractional derivatives are calculated analytically throughout this work. Furthermore, the local and global estimated errors were analyzed, which suggest that the error quantification should take into account the possible grow in time of the error. This observation provides a motivation for going beyond more classical local-in-time concepts of error(local truncation error). The neutron density response with time is analyzed for the anomalous diffusion, sub-diffusion, and super-diffusion processes.     

Modeling and simulation of coupled nuclear heat energy deposition and transfer in the fuel assembly of the Ghana Research Reactor-1 (GHARR-1)

Monte Carlo N-Particle (MCNP) code coupled with PLTEMP/ANL code were used to model and simulate the heat transfer problems in the fuel elements assembly of the Ghana Research Reactor-1 (GHARR-1) by solving Boltzmann transport approximation to the heat conduction equation. Coupled neutron radiation-thermal codes were used to determine the spatial variations of thermal energy in the fuel channels, the heat energy distribution in the radial and axial segments of the fuel assembly and the convective heat transfer processes in the entire core of the reactor. The thermal energy at maximum reactivity load of 4 mk, reactor power of 30 kW and inlet system pressure of 101.3 kPa were found to be 8.896 × 10⁻¹⁶ J for a single fuel pin, and 1.104 × 10⁻¹⁵ J and 7.376 × 10⁻¹⁶ J, for the radial and axial sectioning of the core respectively. Using the PLTEMP/ANL V4.0 code and given that the inlet coolant temperature was 30 °C, the maximum outlet coolant temperature was 51 °C. The energy values were obtained using the following thermodynamic parameters as maximum pressure drop of 0.7 MPa and mass flow rate of 0.4 kg/s. Neutronics point kinetics model and Safety Analysis Report used to validate the results confirmed that the heat distribution in the core did not exceed 100 °C. The heat energy profiles based on the data suggested no nucleate boiling at the simulated energies, and since the melting point of U–Al alloy fuel material is 640 °C, the reactor was considered to be inherently safe during normal or steady state operations.