Two-group interfacial area transport in vertical air–water flow: I. Mechanistic model

In a companion paper, mechanistic models of major fluid particle interaction phenomena involving two bubble groups have been proposed. The prediction of interfacial area concentration evolution using the one-dimensional two-group transport equation and evaluation with experimental results are performed in the paper. These evaluations are based on solid databases for a 2-inch air–water loop with sufficient information on the axial development and the radial distribution of the local parameters. Model evaluation strategies are systematically analyzed. The predictions for the interfacial area concentration evolution demonstrate satisfactory accuracy. The proposed model predicts a smooth transition across the bubbly-to-slug flow regime boundary and demonstrates mechanisms for the generation and development of the cap/slug bubble group. The two-group interfacial area transport equation covers a wide range from bubbly, slug, to churn turbulent flow regimes for adiabatic air–water upward flow in moderate diameter pipes. The generality of the interfacial transport model is also discussed.     

Two-group interfacial area transport equations at bubbly-to-slug flow transition

In relation to nuclear reactor accident and safety studies, experiments on hot-leg U-bend two-phase natural circulation in a loop with a relatively large diameter pipe (10.2 cm inner diameter) were performed for understanding the two-phase natural circulation and flow termination during a small break loss of coolant accident in LWRs. The loop design was based on the scaling criteria developed under this program and the loop was operated either in a natural circulation mode or in a forced circulation mode using nitrogen gas and water. Various tests were carried out to establish the basic mechanism of the flow termination as well as to obtain essential information on scale effects of various parameters such as the loop frictional resistance, thermal center and pipe diameter. The void distribution in a hot-leg, flow regime and natural circulation rate were measured in detail for various conditions. The termination of the natural circulation occurred when there was insufficient hydrostatic head in the downcomer side. The superficial gas velocity at the flow termination could be predicted well by the simple model derived from a force balance between the frictional pressure drop along the loop and the hydrostatic head difference. The bubbly-to-slug flow transition was found to be dependent on axial locations. It turned out that the formation of cap bubbles in the large diameter pipe caused the increased drift velocity, which would affect the prediction of the void fraction in the hot leg.     

Two-group interfacial area transport in vertical air-water flow: I. Mechanistic model

The prediction of the dynamical evolution of interfacial area concentration is one of the most challenging tasks in two-fluid model application. This paper is focused on developing theoretical models for interfacial area source and sink terms for a two-group interfacial area transport equation. Mechanistic models of major fluid particle interaction phenomena involving two bubble groups are proposed, including the shearing-off of small bubbles from slug/cap bubbles, the wake entrainment of spherical/distorted bubble group into slug/cap bubble group, the wake acceleration and coalescence between slug/cap bubbles, and the breakup of slug/cap bubbles due to turbulent eddy impacts. The existing one-group interaction terms are extended in considering the generation of cap bubbles, as well as different parametric dependences when these terms are applied to the slug flow regime. The complete set of modeling equations is closed and continuously covers the bubbly flow, slug flow, and churn-turbulent flow regimes. Prediction of the interfacial area concentration evolution using a one-dimensional two-group transport equation and evaluation with experimental results are described in a companion paper.     

Model evaluation of two-group interfacial area transport equation for confined upward flow

The bubble interaction mechanisms have been analytically modeled in the first paper of this series to provide mechanistic constitutive relations for the two-group interfacial area transport equation (IATE), which was proposed to dynamically solve the interfacial area concentration in the two-fluid model. This paper presents the evaluation approach and results of the two-group IATE based on available experimental data obtained in confined upward flow, namely, 11 data sets in or near bubbly flow and 13 sets in cap-turbulent and churn-turbulent flows. The two-group IATE is evaluated in steady-state, one-dimensional (1D) form. To account for the inter-group bubble transport, the void fraction transport equation for Group-2 bubbles is also used to predict the void fraction for Group-2 bubbles. Agreement between the data and the model predictions is reasonably good and the average relative difference for the total interfacial area concentration between the 24 data sets and predictions is within 7%. The model evaluation demonstrates the capability of the two-group IATE focused on the current confined flow to predict the interfacial area concentration over a wide range of flow regimes.     

Measurement of three-dimensional wave form and interfacial area in an air-water stratified flow

A method of measuring three-dimensional wave forms in a thin liquid film is presented, with experimental data. Water film thicknesses were measured instantaneously across the width direction by a newly developed multi-conductance probe, along nearly horizontal (4.1°) and near vertical (87.0°) flat plates with air - water concurrent stratified flow. The multi-conductance probe was composed of a large flush electrode and eleven wire electrodes, made of fine platinum wire of 0.05 mm diameter to minimize the flow disturbance. The measured data were processed to show the three-dimensional interface shapes, using wave celerity and computer graphics, and the reconstructed wave shape is compared with the real photographic image. The measuring technique is applicable for the shapes of both two and three-dimensional waves. The interfacial areas were also calculated from the measured three-dimensional wave shape. The increase in interfacial area is less than a few percent in a moderate range of the water film flow (500 < Re₁ < 1250).     

Implementation and evaluation of one-group interfacial area transport equation in TRACE

The present study implements the one-dimensional interfacial area transport equation into the TRACE code, being developed by the U.S. Nuclear Regulatory Commission. The interfacial area transport equation replaces the conventional flow regime dependent correlations and the regime transition criteria for furnishing the interfacial area concentration in the two-fluid model. This approach allows dynamic tracking of the interfacial area concentration by mechanistically modeling bubble coalescence and disintegration mechanisms. Thus, it eliminates potential artificial bifurcations or numerical oscillations stemming from the use of conventional static correlations. To implement the interfacial area transport equation, a three-field version of TRACE is utilized, which is capable of tracking both the continuous liquid and gas fields as well as a dispersed gas field. To demonstrate the feasibility of the present approach, the steady-state one-group interfacial area transport equation applicable to adiabatic air-water bubbly two-phase flow is first tested in the present study. Data obtained in 18 different flow conditions from two vertical co-current upward air-water bubbly two-phase flow experiments performed in round pipes (25.4 mm and 48.3 mm) are used to help evaluate the implementation. Results obtained from TRACE with the interfacial area transport equation (TRACE-T) and those from TRACE without the transport equation (TRACE-NT) are compared to demonstrate the enhancement in prediction accuracy. The predictions made by TRACE-T agree well with the data with an average percent difference of approximately ±8%. It is also evident from the results that while TRACE-T accounts for dynamic interaction of bubbles along the flow field, the predictions made by TRACE-NT are attributed primarily to the pressure change.     

Modeling and validation of interfacial area transport equation in subcooled boiling flow

The first comprehensive validation of the interfacial area transport equation in subcooled boiling is presented and shown to perform exceptionally when compared with experimental data. The formulation and closure of the bubble layer averaged interfacial area transport equation is reviewed along with the treatment of the two-fluid model in subcooled boiling. Interfacial area concentration source and sink terms in subcooled boiling are presented including the bubble interaction mechanisms (random collision and turbulent impact), as well as phase change terms (wall nucleation and condensation). Additionally, the volume source terms from phase change are described and discussed in terms of their significance to the interfacial area transport equation. The validation of the interfacial area transport equation with a recently proposed wall nucleation source term is shown to have excellent prediction at low and elevated pressure, as well as a wide range of mass flux. With new confidence in the wall nucleation source term, the interfacial area concentration in subcooled boiling can be accurately predicted. Due to its strong dependence in the modeling of active nucleation site density, bubble departure frequency, and departure diameter, the calculation is shown to be very sensitive to wall temperature.     

Wall pressure measurements of flooding in vertical countercurrent annular air-water flow

An experimental study of flooding in countercurrent air-water annular flow in a large diameter vertical tube using wall pressure measurements is described in this paper. Axial pressure profiles along the length of the test section were measured up to and after flooding using fast response pressure transducers for three representative liquid flow rates representing a wide range of liquid Reynolds numbers (Re_L = 4Γ/μ; Γ is the liquid mass flow rate per unit perimeter; μ is the dynamic viscosity) from 3341 to 19,048. The results show that flooding in large diameter tubes cannot be initiated near the air outlet and is only initiated near the air inlet. Fourier analysis of the wall pressure measurements shows that up to the point of flooding, there is no dominant wave frequency but rather a band of frequencies encompassing both the low frequency and the broad band that are responsible for flooding. The data indicates that flooding in large diameter vertical tubes may be caused by the constructive superposition of a plurality of waves rather than the action of a single large-amplitude wave.     

Analysis of subcooled boiling flow with one-group interfacial area transport equation and bubble lift-off model

To enhance the multi-dimensional analysis capability for a subcooled boiling two-phase flow, the one-group interfacial area transport equation was improved with a source term for the bubble lift-off. It included the bubble lift-off diameter model and the lift-off frequency reduction factor model. The bubble lift-off diameter model took into account the bubble's sliding on a heated wall after its departure from a nucleate site, and the lift-off frequency reduction factor was derived by considering the coalescences of the sliding bubbles. To implement the model, EAGLE (elaborated analysis of gas-liquid evolution) code was developed for a multi-dimensional analysis of two-phase flow. The developed model and EAGLE code were validated with the experimental data of SUBO (subcooled boiling) and SNU (Seoul National University) test, where the subcooled boiling phenomena in a vertical annulus channel were observed. Locally measured two-phase flow parameters included a void fraction, interfacial area concentration, and bubble velocity. The results of the computational analysis revealed that the interfacial area transport equation with the bubble lift-off model showed a good agreement with the experimental results of SUBO and SNU. It demonstrates that the source term for the wall nucleation by considering a bubble sliding and lift-off mechanism enhanced the prediction capability for the multi-dimensional behavior of void fraction or interfacial area concentration in the subcooled boiling flow. From the point of view of the bubble velocity, the modeling of an increased turbulence induced by boiling bubbles at the heated wall enhanced the prediction capability of the code.     

Measurement of interfacial area concentration in subcooled liquid-vapour flow

In two-fluid modelling, accurate prediction of the interfacial transport of mass, momentum and energy is required. Experiments were carried out to obtain a database for the development of interfacial transport models, or correlations, for subcooled water-steam flow in vertical conduits. The experimental data of interest included the interfacial area concentration, interfacial condensation heat transfer and bubble relative velocity. This paper focuses on the interfacial area concentration. The interfacial area concentration was obtained by measuring the distributions of bubble volume and surface area as well as the area-averaged void fraction at various axial locations in subcooled water-steam condensing vertical upward flow under low flow rate and low pressure conditions. The bubble size and surface area were determined using high-speed photography and digital image processing techniques. The area-averaged void fraction was measured by a single-beam gamma densitometer. The results were compared with existing correlations, which were developed on the basis of data obtained for air-water adiabatic flows. Poor agreement between the present data and the existing correlations was obtained. Accordingly, new correlations suitable for subcooled liquid-vapour bubbly flow are proposed.     

Local measurement of interfacial area, interfacial velocity and liquid turbulence in two-phase flow

Double sensor probe and hotfilm anemometry methods were developed for measuring local flow characteristics in bubbly flow. The formulation for the interfacial area concentration measurement was obtained by improving the formulation derived by Kataoka and Ishii. The assumptions used in the derivation of the equation were verified experimentally. The interfacial area concentration measured by the double sensor probe agreed well with one by the photographic method. The filter to validate the hotfilm anemometry for measuring the liquid velocity and turbulent intensity in bubbly flow was developed based on removing the signal due to the passing bubbles. The local void fraction, interfacial area concentration, interfacial velocity, Sauter mean diameter, liquid velocity, and turbulent intensity of vertical upward air–water flow in a round tube with an inner diameter of 50.8 mm were measured by using these methods. A total of 54 data sets were acquired consisting of three superficial gas flow rates, 0.015–0.076 m s⁻¹, and three superficial liquid flow rates, 0.600, 1.00, and 1.30 m s⁻¹. The measurements were performed at the three locations: L/D=2, 32, and 62. This data is expected to be used for the development of reliable constitutive relations which reflect the true transfer mechanisms in two-phase flow.     

Modeling and measurement of interfacial area concentration in two-phase flow

This paper presents experimental and modeling approaches in characterizing interfacial structures in gas-liquid two-phase flow. For the modeling of the interfacial structure characterization, the interfacial area transport equation proposed earlier has been studied to provide a dynamic and mechanistic prediction tool for two-phase flow analysis. A state-of-the-art four-sensor conductivity probe technique has been developed to obtain detailed local interfacial structure information in a wide range of flow regimes spanning from bubbly to churn-turbulent flows. Newly obtained interfacial area data in 8 × 8 rod-bundle test section are also presented. This paper also reviews available models of the interfacial area sink and source terms and existing databases. The interfacial area transport equation has been benchmarked using condensation bubbly flow data.     

Evolution of interfacial area concentration in a vertical air–water flow measured by wire–mesh sensors

An application of wire–mesh sensors to obtain the interfacial area concentration in vertical pipes is presented as an alternative to the widely used multiple-tip electrical or optical fibre probes. The measuring data of a mesh sensor consists of a three-dimensional matrix of local instantaneous gas fractions measured at each crossing point of the wires and recorded as a time sequence. Bubbles are clearly distinguishable in this data matrix, since they represent regions of interconnected elements containing the gaseous phase. The method to deduce the interfacial area concentration from this data is based on a full reconstruction of the gas–liquid interface, where the interfacial area of each bubble is recovered as the sum of the surface area of all surface elements belonging to the given bubble. The new method can be applied to large bubbles with an arbitrary shape. To study the change of the interfacial area concentration along the pipe the distance between sensor and gas injection was varied. The axial development of the interfacial area density measured in the test pipe of 195.3 mm inner diameter was compared to the measurements carried out by Sun et al. [Sun, X., Smith, T., Kim, S., Ishii, M., Uhle, J., 2002. Interfacial area of bubbly flow in a relatively large diameter pipe. Exp. Thermal Fluid Sci. 27, 97–109] in a pipe of 101.6 mm diameter, which is the largest pipe for which interfacial area densities are presented in literature. An acceptable agreement was found, whereas deviations are consistent with the differences in the boundary conditions of both experiments.     

Prediction of interfacial area transport in a coupled two-fluid model computation

A computer code has been written to predict interfacial area transport within the framework of the two-fluid model. The suitability of various constitutive models was evaluated from a scientific and numerical standpoint, and selected models were used to close the two-fluid model. The resulting system was then used to optimize the empirical constants in the interfacial area transport equation for large diameter pipes. The optimized model was evaluated based on comparison with the data of Shen et al. and Schlegel et al. The optimization shows agreement with previous research conducted by Dave et al. and Talley et al. using TRACE-T, and reduced the RMS error in the interfacial area concentration prediction for the large diameter pipe data from 52.3% to 34.9%. The results also highlight a need for additional high-resolution data at multiple axial locations to provide a more detailed picture of the axial development of the flow. The results also indicate a need for improved modeling of the interfacial drag, especially for Taylor cap bubbles under relatively low void fraction conditions.     

Modified two-fluid model for the two-group interfacial area transport equation

This paper presents a modified two-fluid model that is ready to be applied in the approach of the two-group interfacial area transport equation. The two-group interfacial area transport equation was developed to provide a mechanistic constitutive relation for the interfacial area concentration in the two-fluid model. In the two-group transport equation, bubbles are categorized into two groups: spherical/distorted bubbles as Group 1 while cap/slug/churn-turbulent bubbles as Group 2. Therefore, this transport equation can be employed in the flow regimes spanning from bubbly, cap bubbly, slug to churn-turbulent flows. However, the introduction of the two groups of bubbles requires two gas velocity fields. Yet it is not practical to solve two momentum equations for the gas phase alone. In the current modified two-fluid model, a simplified approach is proposed. The momentum equation for the averaged velocity of both Group-1 and Group-2 bubbles is retained. By doing so, the velocity difference between Group-1 and Group-2 bubbles needs to be determined. This may be made either based on simplified momentum equations for both Group-1 and Group-2 bubbles or by a modified drift-flux model.     

Characteristics and behavior of the interfacial wave on the liquid film in a vertically upward air-water two-phase annular flow

The behavior of individual interfacial waves on liquid film in vertically upward air-water annular flows has been visualized, observed and analyzed by a pigment luminance method(PLM) which was calibrated with a fiber-optic liquid film sensor. By means of this technique, we distinguished three different types of interfacial waves, i.e. the ripple wave, the ring wave and the disturbance wave. Furthermore we measured the characteristics of these three different kinds of waves, and in particular those of the disturbance wave: i.e. its propagation velocity, its frequency in passing and the distance between two adjacent waves, and then obtained the dependency of these characteristics on the air and water volumetric fluxes j_g and j_l. These results agreed well with the results obtained by other investigators, using an electric needle contact method. A probable mechanism of the occurrence of the ring and the disturbance waves was posited.     

Numerical evaluation of interfacial area concentration using the immiscible lattice gas

The phase separation in a Couette flow and the mixing of two phases in a cavity flow are simulated numerically using the immiscible lattice gas, which is one of the discrete methods of using particles to simulate two-phase flows. The interface is defined as the lattice sites between two phases, and the interfacial area concentration is evaluated in the steady state. In the Couette flow, the interfacial area concentration increases slightly with an increase in the wall speed. It is shown in the cavity flow that the interfacial area concentration increases largely with an increase in the wall speed. Macroscopic velocity fields in the two flows are in good agreement with analytical or numerical solutions of the Navier–Stokes equations. The interfacial area concentration is found to be correlated with the wall speed for the two flows, and the applicability of particle simulation methods to the numerical evaluation of the interfacial area concentration is indicated.     

Pressure wave propagation in air-water bubbly and slug flow

Related to nuclear reactor safety problems, such as the loss of coolant accident caused by some small crevasses in nuclear reactor, choked flows after postulated breaks of hot and cold legs of pressurized water reactors and the boiling flow instability in parallel channels, the characteristics of pressure wave propagation were investigated experimentally for the air-water bubbly and slug two-phase flow in a vertical pipe. Pressure wave was generated from the small pressure disturbance by the up-and-down movement of piston in the test section. Air void fraction was up to 0.7 and superficial liquid velocity was up to 1.5 m/s as experimental conditions. The experimental results show that the pressure wave propagation velocity in bubbly flow decreases acutely with the increase of air void fraction from 0 to 0.05. In slug flow, it is constant when the air void fraction is less than 0.5 but increases gradually when the void fraction increases beyond 0.5. The attenuation coefficient of pressure wave increases with the increase of air void fraction in bubbly flow. The dependency of pressure wave propagation velocity on angle frequency ω in air-water flow shows the dispersion characteristic. The propagation velocity and attenuation coefficient increases gradually with the increase of angle frequency. However, the increase vanishes slowly as the angle frequency reaches 250 Hz in bubbly flow. The propagation of pressure wave in bubbly flow is independent of the superficial velocity of fluids in the range of experiment.     

Flow patterns and their transition characteristicsof the air-water two-phase flow in a horizontal pipe with asudden-changed cross-section area

Flow patterns in upstream and downstream straight tubes of suddenchanged areas in a horizontal straight pipe were experimentally examined.Both sudden-expansion cross-section(SECS)and sudden-contraction cross-section(SCCS) were inverstigated.The flow pattern maps upstream and downstream were delineated and compared with those in straight tubes with uniform cross-sections.The effects of the SECS and SCCS on flow patterns were discussed and analyzed.Furthermore.flow pattern transition mechanisms resulting in occurrences of different flow patterns were simply discussed and some transition criteria for the flow pattern transitions were deduced by using the non-dimensionlized analysis method.     

Interfacial area transport and evaluation of source and sink terms for confined air–water bubbly flow

The interfacial area transport equation applicable to the bubbly flow is presented. The model is evaluated against data acquired by a state-of-the-art miniaturized double-sensor conductivity probe in an adiabatic air–water co-current vertical test loop under atmospheric pressure condition. In general, a good agreement, within the measurement error of ±10%, is observed for a wide range in the bubbly flow regime. The evaluation of the individual particle interaction mechanisms demonstrates the active interactions between the bubbles and highlights the mechanisms playing the dominant role in interfacial area transport. The analysis employing the drift flux model is also performed for the data acquired. Under the given flow conditions, the distribution parameter of 1.076 yields the best fit to the data.