T.H.J.J. van der Hagen

Why Yule-Walker should not be used for autoregressive modelling

Autoregressive modelling of noise data is widely used for system identification, surveillance, malfunctioning detection and diagnosis. Several methods are available to estimate an autoregressive model. Usually, the so-called Yule-Walker method is employed. The various estimation methods generally yield comparable parameter estimates. In some special cases however, involving nearly periodic signals, the Yule-Walker approach may lead to incorrect parameter estimates. Burgs method offers the best alternative to Yule-Walker. In this paper a theoretical explanation of this phenomenon is given, while the 1994 IAEA Benchmark test is presented as a practical example of Yule-Walker yielding poor parameter estimates.

Methods for the determination of the in-phase and out-of-phase stability characteristics of a boiling water reactor

Measurements, taken in the Ringhals-1 boiling water reactor after revision in 1990, showed that instability occurred at high power and low core flow. Measurements in several points of the power-flow map showed that the decay ratio (DR), obtained by conventional methods, jumps from a moderate value directly to unity. This was valid for DR values calculated from both average power range monitor (APRM) and local power range monitor (LPRM) signals. Thus, the conventional DR cannot be used as a measure of the margin to instability. It was found that both global (in-phase) and regional (out-of-phase) oscillations occur, the global with low DR but large signal amplitude, and the regional with high DR but low signal amplitude. The former dominates the DR calculated from both APRMs and LPRMs, except when the instability is fully developed and impedes detection of the actual margin to instability. Methods for obtaining the stability characteristics of both modes separately from neutron noise signals were developed. The DR of the out-of-phase mode appears to be a good indicator of the margin to instability.

Analytical modeling of flashing-induced instabilities in a natural circulation cooled boiling water reactor

A dynamic model for natural circulation boiling water reactors (BWRs) under low-pressure conditions is developed. The motivation for this theoretical research is the concern about the stability of natural circulation BWRs during the low-pressure reactor start-up phase. There is experimental and theoretical evidence for the occurrence of void flashing in the unheated riser under these conditions. This flashing effect is included in the differential (homogeneous equilibrium) equations for two-phase flow. The differential equations were integrated over axial two-phase nodes, to derive a nodal time-domain model. The dynamic behavior of the interface between the one and two-phase regions is approximated with a linearized model. All model equations are presented in a dimensionless form. As an example the stability characteristics of the Dutch Dodewaard reactor at low pressure are determined.

Subchannel void-fraction measurements in a 6 × 6 rod bundle using a simple gamma-transmission method

A relatively simple subchannel void-fraction measurement system was developed and applied to a scaled model of a BWR fuel assembly. The measurement system is based upon the gamma-transmission technique and uses a tomographic reconstruction procedure to calculate the void-fractions in the subchannels. Data pre-processing is used to remove positioning errors of the gamma source and detector. Measurements are performed under different operating conditions and radial power profiles. The measurements are compared with the drift-flux two-phase flow model. The void-fraction in the wall and corner subchannels is lower than average. By applying the drift-flux model to each subchannel separately and comparing this calculation with measurements the presence of lateral void-drift should be detectable. However, the data do not show evidence for lateral void-drift. By comparing two chordal void-fractions, one through the fuel rods and one in between the fuel rods, a flow pattern transition could clearly be seen between a void-fraction of 30% and 40%.

Exploring the dodewaard type-I and type-II stability; from start-up to shut-down, from stable to unstable

Abstract Stability measurements on the Dodewaard natural circulation boiling water reactor have been performed for five years, covering the entire operational map. A data base of stability characteristics for testing of analytical models and for computer-code benchmarking has been set up. Type-I natural-circulation-flow stability and Type-II reactor stability have been investigated. Flashing turned out to be an important phenomenon. At a Type-II instability, the oscillating reactor power showed a peculiar modulation before the divergence to the scram level. The decay ratio was found to be an unpractical and unsafe stability indicator for reactors with a decay ratio larger than 0.6.

Nonlinear Analysis of a Natural Circulation Boiling Water Reactor

A dynamic model of natural circulation boiling water reactors (BWRs) is analyzed using a bifurcation code and numerical simulations. The two fundamental bifurcation types relevant to BWRs, the supercritical and the subcritical Hopf bifurcations, are first studied in natural circulation systems without nuclear feedback. The effect of nodalization approximation in the riser on stability and bifurcation characteristics of the system is determined. The strong effect of the nuclear-thermohydraulic interaction on the nonlinear characteristics of a natural circulation BWR is then explored in a parametric study. Supercritical bifurcations become dominant in the (high-power) Type-Il region for small values of the subcooling number and a strong nuclear-thermohydraulic coupling. A cascade of period-doubling pitchfork bifurcations (deep in the unstable region) is also predicted by the model under these conditions. Subcritical bifurcations in the Type-Il instability region were found for larger values of the subcooling number. Both Hopf-bifurcation modes were also encountered in the Type-1 instability region (low power or high power/ high subcooling ). Finally, the nonlinear reactor model was validated successfully compared with nonlinear power oscillations measured in a natural circulation BWR.

Reactor experiments on type-I and type-II BWR stability

Investigations on Type-I (hydrostatic head) and Type-II (density wave) BWR stability characteristics are reported. Experimental data are obtained from the Dodewaard natural-circulation BWR at conditions far away from nominal operating points. The Type-I stability boundary was reached; the Type-II stability boundary was crossed. At this latter unstable condition the reactor power showed peculiar oscillation modulations. Experimentally obtained decay ratios and resonance frequencies are put into context using a physical model. Flashing is shown to be an important phenomenon for driving the coolant flow at start-up conditions and for Type-I stability features. From low-pressure experiments, it appears that the decay ratio is an unsafe measure for Type-II stability close to the stability boundary.

Validation of the DALTON-THERMIX Code System with Transient Analyses of the HTR-10 and Application to the PBMR

Abstract The DALTON-THERMIX code system has been developed for safety analysis and core optimization of pebble-bed reactors. The code system consists of the new three-dimensional diffusion code DALTON, which is coupled to the existing thermal-hydraulic code THERMIX. These codes are linked to a database procedure for the generation of neutron cross sections using SCALE-5. The behavior of pebble-bed reactors during a loss of forced cooling (LOFC) transient is of particular interest since the absence of forced cooling could lead to a significant increase of the temperature of the coated particle fuel. Therefore, the reactor power may be constrained during normal operation to limit the temperature. For validation purposes, calculation results of normal operation, an LOFC transient, and a control rod withdrawal transient without SCRAM have been compared with experimental data obtained in the High Temperature Reactor–10 (HTR-10). The code system has been applied to the 400-MW(thermal) pebble bed modular reactor (PBMR) design, including the analysis of three different LOFC transients. Theses results are verified by a comparison with the results of the existing TINTE code system. It was found that the code system is capable of modeling both small (HTR-10) and large (PBMR) pebble-bed reactors and therefore provides a flexible tool for safety analysis and core optimization of future reactor designs. The analyses of the LOFC transients show that the peak fuel temperature is only slightly elevated (less than +100°C) as compared to its nominal value in the HTR-10 but reaches a maximum value of 1648°C during the depressurized LOFC case of the PBMR benchmark, which is significantly higher than the peak fuel temperature (976°C) during normal operation.

Feasibility study of a time-resolving x-ray tomographic system

This paper discusses the possibilities for developing a tomographic scanner for studying the phase distribution of fluidized beds. The system is based on a medical x-ray source equipped with 30 CdWO4 detectors. We mimic a five-source system via simulation and in experiments of voids in a 23 fluidized bed filled with polystyrene particles. Both static voids and moving ones are studied. The reconstruction uses the simultaneous algebraic reconstruction technique with regularization. We find that it is possible to reconstruct objects with a spatial resolution of about 5 mm at a frame rate of 200 Hz. It is concluded that noise levels should be kept below 2%.

Gamma radiation densitometry for studying the dynamics of fluidized beds

The purpose of this research project is to investigate the applicability of gamma radiation densitometry in fluidized bed behaviour research, and to obtain information about the turbulent fluidization of beds with coarse particles. Gamma radiation densitometry determines the chordally averaged void fraction in the bed by measuring the transmission of gamma radiation through the bed accurately. A dual beam setup was used with two 137^Cs gamma radiation sources (662 keV) and detectors, placed 0.12 m apart and vertically above each other. The technique has been applied to a fluidized bed with an inner diameter of 0.384 m filled with Geldart B or Geldart D particles. The Geldart D bed is operated at high gas velocities resulting in operation in the turbulent regime and in a high average void fraction. A linear least squares algorithm for the reconstruction of time averaged void fraction profiles has been developed. It approximates the profile with a linear combination of basis functions. The influence of asymmetry, drift and stochastic errors on the reconstruction accuracy has been investigated. Dominant error sources are gas flow fluctuations, and inaccuracies in the gamma beam-bed and gamma beam-bed wall intersection lengths. Profiles in the Geldart D bed can be measured with up to six reconstruction points, while those in the Geldart B bed have limited reliability. Structure velocity measurements in the turbulent bed have carried out using generalized crosscorrelation techniques on the void fraction signals from the dual beam setup. The techniques give estimates for both the upflow and downflow (Geldart D only) velocities. The dependence of these velocities on gas velocity and measurement height has been studied. Downflow detection is difficult in the freeboard zone, and at low gas velocities where the slugging regime is approached. Structures in the bed have been identified by analysis of time signals, spectra, correlation functions and visual observations.