Nodal analysis of density-wave oscillations in boiling water nuclear reactors

This work presents a model for density-wave oscillations in boiling water nuclear reactors (BWRs). A nodal approach is adopted to describe the power generation in the core. The nodal equations are derived within the framework of Avery's coupled-core kinetics theory. This method enables us to evaluate easily the internodal coupling coefficients using steady-state flux and importance distributions. Particular care is taken in treating the heater wall dynamics by using a distributed parameter model for the fuel rod. The flow is described by the homogeneous equilibrium conservation equations.A steady-state calculation is performed to determine the flow distribution and the neutronic coupling parameters in a core constituted by three hydrodynamic channel types and two neutronic radial nodes. The dynamics solution is based on linearization and use of the Nyquist stability criterion. The stability margin is found to be significantly affected by the coupling effect. The importance of a detailed representation of the neutronic behavior in BWR stability studies is demonstrated.     

Linear modal analysis of out-of-phase instability in boiling water reactor cores

An eigenvalue problem governing BWR core nuclear thermal-hydraulic modes which result in out-of-phase power oscillations is formulated. This formulation is based on the linearization approximation to nonlinear feedback terms and the very simple models for neutronics and thermal-hydraulics. The eigenvalue problem in 5 × 5 matrix formulation can be easily solved without using a computer. A series of the calculations are carried out, at a high-power and low-core-flow condition, to investigate the dependence of the eigenvalues and eigenfunctions on the void reactivity coefficient and the subcriticality of spatial neutronic modes, where the latter parameter is identical to the eigenvalue separation of the higher-harmonic neutronic mode. These results show that the threshold value of the void coefficient for initiating the unstable out-of-phase oscillation strongly depends on the subcriticality. The oscillation mode becomes more unstable with an increase in the absolute value of the negative void coefficient, whereas the mode becomes more stable, almost linearly, with increasing subcriticality. The resonant frequency of the oscillation and the phase shifts between the nuclear thermal-hydraulic variables are consistent with previous measured or calculated values.     

Low-quality and subcooled film boiling of water at elevated pressures

Stable film boiling heat transfer data have been obtained in an 8.9 mm ID tube at pressures from 2 to 9 MPa. These data were obtained at low-quality and subcooled conditions, over a mass flux range of 0.11 to 2.75 Mg m⁻² s⁻¹. Excessive film boiling surface temperatures were avoided by using the hot patch technique. Contrary to the high-quality data, the low-quality data showed a decrease in heat transfer coefficient with an increase in quality. The film boiling data were compared with existing film boiling correlations. None of these were found to be satisfactory.     

Influence of heat input waveform on transient critical heat flux of subcooled water flow boiling in a short vertical tube

The transient critical heat fluxes (CHFs) of the subcooled water flow boiling for ramp-wise heat input [Q = αt, α = 6.21 × 10⁸ to 1.63 × 10¹² W/m³ s, (q ≅ 1.08 × 10⁷ to 6.00 × 10⁷ W/m²)] and stepwise one [Q = Q_s, Q_s = 0 W/m³ at t = 0 s and Q_s = 2.95 × 10¹⁰ to 7.67 × 10¹⁰ W/m³ at t > 0 s, (q = 0 W/m² at t = 0 s and q ≅ 1.61 × 10⁷ to 3.87 × 10⁷ W/m² at t > 0 s)] with the flow velocities (u = 4.0-13.3 m/s), the inlet subcoolings (ΔT_sub,in = 86.8-153.3 K) and the inlet pressures (P_in = 742.2-1293.4 kPa) are systematically measured by an experimental water loop comprised of a pressurizer. The SUS304 tubes of inner diameters (d = 3, 6 and 9 mm), heated lengths (L = 33.15, 59.5 and 49.3 mm), L/d (=11.05, 9.92 and 5.48), and wall thickness (δ = 0.5, 0.5 and 0.3 mm) respectively with the rough finished inner surface (surface roughness, Ra = 3.18 μm) are used in this work. The experimental errors in the subcooling measure and the pressure one are ±1 K and ±1 kPa, while in the heat flux it is ±2%. The transient CHF data for the ramp-wise heat input and the stepwise one are compared with those for the exponentially increasing heat input (Q = Q₀ exp(t/τ), τ = 16.82 ms to 15.52 s) previously obtained and the dominant variables on transient CHF for heat input waveform difference are confirmed. The transient CHF data are compared with the values calculated by the steady state CHF correlations against inlet and outlet subcoolings, and the applicability of steady state CHF correlations is confirmed extending its possible validity for the reduced time, ω_p, down to 800 ms. The transient CHF data are compared with the values calculated by the transient CHF correlations against inlet and outlet subcoolings, and the influence of heat input waveform on transient CHF is clarified based on the experimental data for the ramp-wise heat input, the stepwise one and the exponentially increasing one. The dominant mechanisms of the subcooled flow boiling critical heat flux for the ramp-wise heat input, the stepwise one and the exponentially increasing one are discussed.     

Thermo-hydraulics during start-up in natural circulation boiling water reactors

The transient behavior of natural circulation for boiling two-phase flow was investigated by simulating normal and abnormal start-up conditions to research the feasibility of natural circulation BWRs such as the SBWR. It was found that the instabilities, which are out-of-phase geysering, in-phase natural circulation oscillation and out-of-phase density wave instability, may occur during the start-up when the vapor generation rate is insufficient. In this paper, the mechanism of in-phase natural circulation oscillation induced by hydrostatic head fluctuation in steam separators, which has never been understood well enough, is experimentally clarified. Next, the effect of system pressure on the occurrences of the geysering and the natural circulation oscillation are investigated. Finally, from the results, a recommendation is provided to establish the rational start-up procedure and reactor configuration for natural circulation BWRs.     

Interfacial area measurements in subcooled flow boiling

The purpose of the present study was to measure two-phase parameters in subcooled flow boiling. These parameters include void fraction distribution, interfacial area concentration distribution, Sauter mean diameter, and the interfacial velocity. A literature review was conducted and the results show that only three researchers have made local measurements in the subcooled boiling region. None of the previous have included results for interfacial area concentration distribution. To make these measurements an experimental facility was constructed that allows insertion of advanced local two-phase flow instrumentation. Experiments were performed for a number of conditions at atmospheric pressure.     

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.     

Vapour void fraction in subcooled flow boiling

Experimental data on steam void fraction and axial temperature distribution in an annular boiling channel for low mass-flux forced and natural circulation flow of water with inlet subcooling have been obtained. The ranges of variables covered are: mass flux = 1.4 × 10⁴−1.0 × 10⁵ kg/hr m²; heat flux = 4.5 × 10³−7.5 × 10⁴ kcal/hr m²; and inlet subcooling = 10–70°C. The present and literature data match well with the theoretical void predictions using a four-step method similar to that suggested by Zuber and co-workers.     

Dependence of bubble behavior in subcooled boiling on surface wettability

This paper presents the results of visualization experiments that were carried out to investigate the dynamics of vapor bubbles generated in water pool boiling. In the experiments, vapor bubbles were generated on a vertical circular surface of a copper block containing nine cartridge heaters, and the contact angle of the heated surface was used as a main experimental parameter. The experiments were performed under subcooled as well as nearly saturated conditions. To enable clear observation of individual bubbles with a high speed camera, the heat flux was kept low enough to eliminate significant overlapping of bubbles. When the contact angle was small, the bubbles were lifted-off the vertical heated surface within a short period of time after the nucleation. On the other hand, when the contact angle was large, they slid up the vertical surface for a long distance. When bubbles were lifted-off the heated surface in subcooled liquid, bubble life-time was significantly shortened since bubbles collapsed rapidly due to condensation. It was shown that this distinct difference in bubble dynamics could be attributed to the effects of surface tension force.     

Experimental and theoretical noise analysis investigations in boiling water reactors

The stochastic neutron flux fluctuations in boiling water reactors represent a significant source of informations about thermohydraulic parameters. In this paper, experimental results of a series of in core measurements are described together with the development of a theoretical model.     

Modeling of low-pressure subcooled boiling flow of water via the homogeneous MUSIG approach

Applying a three-dimensional two-fluid model coupled with homogeneous multiple size group (MUSIG) approach, numerical simulations of upward subcooled boiling flow of water at low pressure were performed on the computational fluid dynamics (CFD) code CFX-10 with user defined FORTRAN program. A modified bubble departure diameter correlation based on the Unal's semi-mechanistic model and the empirical correlation of Tolubinski and Kostanchuk was developed. The water boiling flow experiments at low pressure in a vertical concentric annulus from reference were used to validate the models. Moreover, the influences of the non-drag force on the radial void fraction distribution were investigated, including lift force, turbulent dispersion force and wall lubrication force. Good quantitative agreement with the experimental data is obtained, including the local distribution of bubble diameter, void fraction, and axial liquid velocity. The results indicate that the local bubble diameter first increases and then decreases due to the effect of bubble breakup and coalescence, and has the maximum bubble diameter along the radial direction. Especially, the peak void fraction phenomenon in the vicinity of the heated wall is predicted at low pressure, which is developed from the wall repulsive force between vapor bubbles and heated wall. Nevertheless, there is a high discrepancy for the prediction of the local axial vapor velocity.     

On the space cross-correlation of noise sources in boiling water reactors

In this paper, an attempt has been made to investigate the space cross-correlation of boiling noise in boiling water reactors by assuming the slip ratio to be unity and power profile to be constant along the core height of the reactor. Two cases have been considered: One, no flow fluctuations (ΔV = 0) caused by boiling noise and two, finite flow fluctuations (ΔV ≠ 0) caused by the boiling noise. It has been found that finite range space cross-correlation of noise sources exists only for ΔV ≠ 0 case and not for ΔV = 0 case. Auto power spectral density of steam content fluctuations, Δα have a break frequency which is directly related with the flow velocity and attenuation coefficient, μ of the cross-correlation of noise sources in BWRs. Normalized root mean square value of Δα is more sensitive to μ in the upper half of the core for μ ≤ 10. For $\text{μ ? 20}$, it is more sensitive to μ in the lower half of the core.     

An investigation of transition boiling mechanisms of subcooled water under forced convective conditions

A mechanistic model for forced convective transition boiling has been developed to investigate transition boiling mechanisms and to predict transition boiling heat flux realistically. This model is based on a postulated multi-stage boiling process occurring during the passage time of the elongated vapor blanket specified at a critical heat flux (CHF) condition. Between the departure from nucleate boiling (DNB) and the departure from film boiling (DFB) points, the boiling heat transfer is established through three boiling stages, namely, the macrolayer evaporation and dryout governed by nucleate boiling in a thin liquid film and the unstable film boiling characterized by the frequent touches of the interface and the heated wall. The total heat transfer rates after the DNB is weighted by the time fractions of each stage, which are defined as the ratio of each stage duration to the vapor blanket passage time. The model predictions are compared with some available experimental transition boiling data. The parametric effects of pressure, mass flux, inlet subcooling on the transition boiling heat transfer are also investigated. From these comparisons, it can be seen that this model can identify the crucial mechanisms of forced convective transition boiling, and that the transition boiling heat fluxes including the maximum heat flux and the minimum film boiling heat flux are well predicted at low qualities/high pressures near 10 bar. In future, this model will be improved in the unstable film boiling stage and generalized for high quality and low pressure situations.     

CFD analysis on subcooled boiling phenomena in PWR coolant channel

Eulerian two-fluid model coupled with wall boiling model was employed to calculate the three dimensional flow field and heat transfer characteristics in a hot channel with vaned spacer grid in PWR. The heat transfer from pellet-gap-cladding to coolant was also taken into account by a system coupled code MpCCI. The wall boiling model utilized in this study was validated by Bartolomei experiment data, and a good agreement can be observed. By solving the governing equation in a two-way coupled method, the distribution of temperature in the pellet-gap-cladding region and the distribution of temperature, void fraction and velocity of two-phase flow in coolant channel can be obtained. The influences of spacer grid and mixing vane on the thermal-hydraulic characteristics were analyzed. The heat transfer capacity was strongly improved by the spacer grid and mixing vane, while the flow resistance was also enlarged. Localized volume fraction of vapor phase decreased due to mixing vane, which will decrease the possibility of the departure from nucleate boiling (DNB) and increase the critical heat flux (CHF). By analyzing the temperature and void fraction at cladding outer surface, the critical regions where hot spot may occur were determined.     

Special features of external corrosion of fuel cladding in boiling water reactors

The cases of fuel failure caused by corrosion reactions have so far been a result of localized corrosion, and not of a uniform attack. This report describes special features of external corrosion which under certain circumstances can endanger the integrity of the cladding.     

Study of the radiotoxicity of actinides recycling in boiling water reactors fuel

In this paper the production and destruction, as well as the radiotoxicity of plutonium and minor actinides (MA) obtained from the multi-recycling of boiling water reactors (BWR) fuel are analyzed. A BWR MOX fuel assembly, with uranium (from enrichment tails), plutonium and minor actinides is designed and studied using the HELIOS code. The actinides mass and the radiotoxicity of the spent fuel are compared with those of the once-through or direct cycle. Other type of fuel assembly is also analyzed: an assembly with enriched uranium and minor actinides; without plutonium. For this study, the fuel remains in the reactor for four cycles, where each cycle is 18 months length, with a discharge burnup of 48 MWd/kg. After this time, the fuel is placed in the spent fuel pool to be cooled during 5 years. Afterwards, the fuel is recycled for the next fuel cycle; 2 years are considered for recycle and fuel fabrication. Two recycles are taken into account in this study. Regarding radiotoxicity, results show that in the period from the spent fuel discharge until 1000 years, the highest reduction in the radiotoxicity related to the direct cycle is obtained with a fuel composed of MA and enriched uranium. However, in the period after few thousands of years, the lowest radiotoxicity is obtained using the fuel with plutonium and MA. The reduction in the radiotoxicity of the spent fuel after one or two recycling in a BWR is however very small for the studied MOX assemblies, reaching a maximum reduction factor of 2.     

Contamination of the steam in boiling reactors from solution of water impurities

Raising the steam conditions in single loop atomic electric stations leads to more extensive transference of various water impurities to the water vapor as a result of solubility. In salt contamination of steam, the solubility must be taken into consideration at steam pressures above 140 absolute atmospheres, and in contamination by corrosion products of the structural materials (iron oxides, aluminum, cobalt, etc.), even as low as 40–60 absolute atmospheres.Experimental data are given on the distribution coefficients of the principal water contaminants. A calculation of steam contamination must include the distribution coefficients and the concen-trations of the water impurities actually dissolved.Translated from Atomnaya Énergiya, Vol. 15, No. 3, pp. 214–218, September, 1963