Two-group interfacial area transport in vertical air-water flow -II. Model evaluation

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 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.     

Modeling of bubble coalescence and disintegration in confined upward two-phase flow

This paper presents the modeling of bubble interaction mechanisms in the two-group interfacial area transport equation (IATE) for confined gas–liquid two-phase flow. The transport equation is applicable to bubbly, cap-turbulent, and churn-turbulent flow regimes. In the two-group IATE, bubbles are categorized into two groups: spherical/distorted bubbles as Group 1 and cap/slug/churn-turbulent bubbles as Group 2. Thus, two sets of equations are used to describe the generation and destruction rates of bubble number density, void fraction, and interfacial area concentration for the two groups of bubbles due to bubble expansion and compression, coalescence and disintegration, and phase change. Five major bubble interaction mechanisms are identified for the gas–liquid two-phase flow of interest, and are analytically modeled as the source/sink terms for the transport equation in the confined flow. These models include both intra-group and inter-group bubble interactions.     

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.     

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.     

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.     

Code performance with improved two-group interfacial area concentration correlation for one-dimensional forced convective two-phase flow simulation

In gas–liquid two-phase flow simulation for reactor safety analysis, interfacial momentum transfer in two-fluid model plays an important role in predicting void fraction. Depending on flow conditions, a shape of the two-phase interface complicatedly evolves. One of the proposed approaches is to quantify the gas–liquid interface information using interfacial area transport equation. On the other hand, a more simplified and robust approach is to classify bubbles into two-groups based on their transport characteristics and utilize constitutive equations for interfacial area concentration for each group. In this paper, interfacial drag model based on the two-group interfacial area concentration correlations is implemented into system analysis code, and void fractions were calculated for the evaluation of numerical behaviors. The present analysis includes (1) comparison of one-group and two-group relative velocity models, (2) comparison with separate effect test database, (3) uncertainty evaluation of drag coefficient, (4) numerical stability assessment in flow regime transition, and (5) transient analysis for simulating the prototypic condition. Results showed that utilization of interfacial drag force term using constitutive equations of two-group interfacial area concentration yields satisfactory void fraction calculation results. The proposed solution technique is practical and advantageous in view of reducing the computational cost and simplifying the solution scheme.     

The use of MRI to quantify multi-phase flow patterns and transitions: an application to horizontal slug flow

In this paper we discuss the current benefits and limitations of using magnetic resonance imaging (MRI) to examine multi-phase fluid flow patterns and transitions. The advantages and disadvantages are highlighted in the context of an ongoing collaborative research effort between the University of California, San Francisco (UCSF) School of Medicine MRI Center and the Department of Nuclear Engineering at Oregon State University (OSU). Of particular interest, are the MRI measurements of the liquid volumetric flux distribution, the void fraction and the interfacial area concentration for slug flow in a horizontal air–water system. The data presented herein is unique relative to the slug flow conditions examined and the method implemented for data acquisition. The special scanning sequences designed by UCSF were capable of imaging at repetition intervals as fast as 7 ms.     

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.     

Study on interfacial structures in slug flows using a miniaturized four-sensor conductivity probe

A miniaturized four-sensor conductivity probe is designed to effectively minimize the reported limitations of the previous designs. The new probe is capable of measuring both large and small bubbles. The signal processing scheme is constructed for the probe in such a way that the two-phase parameters of different types of bubbles can be identified and categorized. Image analysis is employed to benchmark the new probe. A good agreement between the experimental data and the theoretical calculation is obtained, which assesses both the measurement principle and the capability of the signal processing scheme. The experimental data are obtained in a 5.08-cm ID vertical co-current air/water loop at two different axial locations of L/D=32 and 64 in slug flow conditions. The local time-averaged two-phase parameters obtained by the probe include the interfacial area concentration, void fraction, interface velocity, chord length, and Sauter mean diameter for various types of bubbles. The measured parameters are categorized in two groups in view of the development of a two-group interfacial transport equation. The development of geometric two-phase flow parameters of each group along the axial direction of the flow duct is well demonstrated.     

Geometric effects of 90-degree Elbow in the development of interfacial structures in horizontal bubbly flow

Present study investigates the geometric effects of flow obstruction on the distribution of local two-phase flow parameters and their transport characteristics in horizontal bubbly flow. The round glass tubes of 50.3 mm in inner diameter are employed as test sections, along which a 90-degree Elbow is located at L/D = 206.6 from the two-phase mixture inlet. In total, 15 different flow conditions are examined within the air–water bubbly flow regime. The detailed local two-phase flow parameters are acquired by the double-sensor conductivity probe at four different axial locations. The effect of elbow is found to be evident in both the distribution of local parameters and their development. The elbow clearly promotes bubble interactions resulting in significant changes in interfacial area concentration. It is also found that the elbow-effect propagates to be more significant further downstream (L/D = 250) than immediate downstream (L/D = 225) of the elbow. Furthermore, it is shown that the elbow induces significant oscillations in the flow in both vertical and horizontal directions of the tube cross-section. Characteristic geometric effects due to the existence of elbow are also shown clearly in the transport of one-dimensional interfacial area concentration and void fraction along the flow.     

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.     

基于竖直圆管空气-水两相流实验的相间曳力模型研究

研究两相流相间阻力特性对系统程序关键本构模型封闭具有重要意义。本文基于竖直圆管开展了空气-水两相流实验,采用四探头电导探针对空泡份额、气泡弦长和界面面积浓度等气泡参数的径向分布进行了测量。结果表明空泡份额和气泡弦长呈现“核峰型”分布,而界面面积浓度并没有表现出随流速的单调关系。进一步开发了泡状流和弹状流的相间曳力模型,考虑了液相表观流速与管径对气泡尺寸分布的影响,建立了临界韦伯数与不同液相流速的关系。计算得到的空泡份额和界面面积浓度与实验数据整体符合较好,验证了模型的可靠性,为两相流相间阻力特性研究提供参考意义。     

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.     

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.     

Interfacial area density, mean radius and number density measurements in bubbly two-phase flow

This paper presents a new experimental method for measuring the local interfacial area density, average bubble radius and number density for bubbly two-phase flow. Experiments were performed using a KfK dual-sensor resistivity probe. The results obtained for interfacial area density using this method agreed with other methods previously proposed, except for high void fractions, close to the transition to slug flow, where prior methods are known to no longer be valid.     

Horizontal bubbly flow with elbow restrictions: Interfacial area transport modeling

The present study develops an interfacial area transport equation applicable to an air-water horizontal bubbly flow, along which two types of horizontal elbows are installed as flow restrictions. Two sets of experiments are performed in a round glass tube of 50.3 mm inner diameter. Along the test section, a 90-degree elbow is installed at L/D = 206.6 from the two-phase mixture inlet and then a 45-degree elbow is installed at L/D = 353.5. In total, 15 different flow conditions in the bubbly flow regime for each of the two flow restriction experiments are studied. Detailed local two-phase flow parameters are acquired by a double-sensor conductivity probe at four different axial locations in the 90-degree experiment and three different axial locations in the 45-degree experiment. The effect of the elbows is found to be evident in the distribution of local parameters as well as in the development of interfacial structures. It is clear that the elbows make an effect on the bubble interactions resulting in significant changes to both the void fraction and interfacial area concentration. In the present analysis, the interfacial area transport equation is developed in one-dimensional form via area averaging. In the averaging process, characteristic non-uniform distributions of the flow parameters in horizontal two-phase flow are treated mathematically through a distribution parameter. The mechanistic models for the major bubble interaction phenomena developed in vertical flow analysis are employed in the present study. Furthermore, the change in pressure due to the minor loss of an elbow is taken into consideration by using a newly developed correlation analogous to Lockhart and Martinelli's. In total, 105 area-averaged data points are employed to benchmark the present model. The present model predicts the data relatively well with an average percent difference of approximately ±20%.