Interfacial Area Transport Equation Research Articles

Three-dimensional simulations of air–water bubbly flows

Interfacial area transport equation (IATE) is considered promising to evaluate dynamic changes of the interfacial area concentration in gas–liquid two-phase flows, which is of significance in characterizing the interfacial structure of the flows. Efforts were made by the authors in the past on the implementation of the IATE into computational fluid dynamics codes, such as Fluent. However, it remained unclear whether the IATE model coefficients derived from one-dimensional IATE model calibrations can be applied to three-dimensional simulations. The current study aimed to examine, primarily by investigating the lateral profiles of phase distributions, the applicability of the coefficients obtained from the one-dimensional IATE model calibration to a three-dimensional simulation of bubbly flow in a pipe. In addition, effects of the lift force on the lateral phase distributions were studied. A new set of the IATE model coefficients was suggested for a three-dimensional bubbly flow simulation. Good agreement was obtained with the updated coefficients between the predicted and measured flow parameters.

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.

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.

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%.

Development of Research on Interfacial Area Transport

Interfacial area concentration and its transport model are quite important in two-fluid model analysis of gas-liquid two-phase flow, which is widely used for the thermal hydraulic analysis of nuclear reactors. Recent developments and the present status of research on interfacial area transport model were reviewed. The definition and rigorous formulation of interfacial area concentration were introduced. Based on this rigorous formulation, basic transport equations of interfacial area concentration were developed by several research groups. In the interfacial area transport equation, source and sink terms of interfacial area concentration are important, and detailed analyses of these terms were carried out, and constitutive equations based on bubble coalescence, bubble breakup, phase change, bubble expansion, and shrinkage due to pressure change were developed. These interfacial area transport equations and constitutive equations were installed in two-fluid model codes and the validities of these models were confirmed by experimental data. Experimental research studies on interfacial area concentration and its transport were also reviewed. Measurement methods of interfacial area concentration were reviewed and multiple-sensor probe methods were established and commonly used in experimental research. Considerable experimental works have been carried out for air–water two-phase flows, and detailed measurements of distributions on interfacial area concentration and their development in the flow direction were reported. For boiling twophase flow, experimental works on interfacial area concentration and its transport have been carried out and data are being accumulated. These data are used for the validation of interfacial area transport models.

A Characteristic Analysis of One‐Dimensional Two‐Fluid Model with Interfacial Area Transport Equation

An interfacial area transport equation (IATE), proposed to dynamically describe the interfacial structure evolution of two‐phase flows, could help improve the predictive capability of the two‐fluid model. The present study aims to investigate the well‐posedness issue of a one‐dimensional two‐fluid model with the IATE (named “two‐fluid‐IATE model” hereafter) using a characteristic analysis. The momentum flux parameters, which take into account the coupling of the volumetric fraction of phase and velocity distributions over the cross‐section of a flow passage, are employed. A necessary condition for the system to achieve hyperbolicity under an adiabatic flow condition is identified. A case study is performed for an adiabatic liquid‐liquid slug flow, which shows that the hyperbolicity of the two‐fluid‐IATE model is guaranteed if appropriate correlations of the momentum flux parameters are applied in the two‐fluid‐IATE model.

CFD Simulation of Phase Distribution in Adiabatic Upward Bubbly Flows Using Interfacial Area Transport Equation

In the study of gas-liquid two-phase flows, one challenge is to describe the dynamic changes in flow structure, which can be considerably affected by bubble coalescence and/or disintegration in addition to bubble nucleation and condensation processes. The interfacial structure, to a first-order approximation, may be characterized by the void fraction and a geometric parameter named “interfacial area concentration,” the evolution of which can be modeled by an interfacial area transport equation (IATE). A one-group IATE has been developed for bubbly flows in the literature, accounting for three dominant mechanisms: coalescence of bubbles due to random bubble collisions driven by turbulence, coalescence of bubbles due to wake entrainment, and disintegration of bubbles caused by turbulent-eddy impact. The current study is aimed at examining the capability of a computational fluid dynamics code, namely, FLUENT, with the one-group IATE implemented, in predicting two-phase-flow phase distributions. Simulations using the Eulerian multiphase model in FLUENT 6.2.16 have been performed for adiabatic upward bubbly flows in a pipe of 50.8-mm inner diameter with a range of void fractions from 4.9 to 23.1%. The predicted phase distributions yield satisfactory agreement with available experimental data, demonstrating that FLUENT with the IATE can provide a valuable simulation tool for two-phase bubbly flows.

Interfacial Area Transport in Horizontal Bubbly Flow with 90-deg Elbow

The present study develops an interfacial area transport equation applicable to an air-water horizontal bubbly flow with a flow restriction. The experiments are performed in a round glass pipe of 50.3-mm inner diameter, along which a 90-deg elbow is installed at L/D = 206.6 from the two-phase mixture inlet. In total, 15 different flow conditions in the bubbly flow regime are studied. The detailed local two-phase flow parameters are acquired by a double-sensor conductivity probe at four different axial locations. The effect of the elbow is evident in the distribution of local parameters as well as in the development of interfacial structures. The elbow clearly promotes bubble interactions resulting in significant changes in both the void fraction and interfacial area concentration. In the present study, the elbow is found to promote the coalescence mechanism while reducing the disintegration mechanism. These geometric effects are also reflected in the axial development of one-dimensional two-phase flow parameters. In the present analysis, the interfacial area transport equation is developed in one-dimensional form via area-averaging based on the existing model for vertical flow. In the averaging process, characteristic nonuniform distributions of the two-phase flow parameters in horizontal two-phase flow are treated mathematically by covariance calculations. Furthermore, the change in pressure due to the minor loss of the elbow is taken into consideration by using a newly developed correlation analogous to Lockhart and Martinelli’s. In total, 60 area-averaged data points are employed to benchmark the present model. The present model predicts the data well with an average percent difference of approximately ±10%.

Interfacial Area Transport Equation and Implementation Into Two-Fluid Model

A dynamic treatment of interfacial area concentration has been studied over the last decade by employing the interfacial area transport equation. When coupled with the two-fluid model, the interfacial area transport equation replaces the flow regime dependent correlations for interfacial area concentration and eliminates potential artificial bifurcation or numerical oscillations stemming from these static correlations. An extensive database has been established to evaluate the model under various two-phase flow conditions. These include adiabatic and heated conditions, vertical and horizontal flow orientations, round, rectangular, annulus, and 8×8 rod-bundle channel geometries, and normal-gravity and reduced-gravity conditions. Currently, a two-group interfacial area transport equation is available and applicable to comprehensive two-phase flow conditions spanning from bubbly to churn-turbulent flow regimes. A framework to couple the two-group interfacial area transport equation with the modified two-fluid model is established in view of multiphase computational fluid dynamics code applications as well as reactor system analysis code applications. The present study reviews the current state-of-the-art in the development of the interfacial area transport equation, available experimental databases, and the analytical methods to incorporate the interfacial area transport equation into the two-fluid model.

Benchmarking of the one-dimensional one-group interfacial area transport equation for reduced-gravity bubbly flows

In order to properly design and safely operate two-phase flow systems, especially those deployed on future space missions, it is necessary to have accurate predictive capabilities. The application of a novel predictive method, the interfacial area transport equation (IATE), to dynamically predict the change of interfacial area concentration for reduced-gravity two-phase flows is described in this paper. Fluid particle interaction mechanisms such as coalescence and breakup that are present in reduced-gravity two-phase flows have been studied experimentally as reported in a previous paper by the current authors [Vasavada et al., 2007]. These mechanisms represent the source and sink terms in the IATE and their mechanistic models are benchmarked using experimental data obtained in a 25 mm inner diameter ground-based test section wherein reduced-gravity conditions were simulated. The comparison of the predictions from the model against experimental data shows good agreement. It has been found that, in contrast to the hypothesis extended in the literature, the wake entrainment based coalescence mechanism is present in reduced-gravity two-phase flows and in some cases is more important than coalescence due to random collision. Physics based arguments are extended to support this conclusion.

Modeling of isothermal bubbly flow with ınterfacial area transport equation and bubble number density approach

In this study, isothermal turbulent bubbly flow is mechanistically modeled. For the modeling, Fluent version 6.3.26 is used as the computational fluid dynamics solver. First, the mechanistic models that simulate the interphase momentum transfer between the gas (bubbles) and liquid (continuous) phases are investigated, and proper models for the known flow conditions are selected. Second, an interfacial area transport equation (IATE) solution is added to Fluent’s solution scheme in order to model the interphase momentum transfer mechanisms. In addition to solving IATE, bubble number density (BND) approach is also added to Fluent and this approach is also used in the simulations. Different source/sink models derived for the IATE and BND models are also investigated. The simulations of experiments based on the available data in literature are performed by using IATE and BND models in two and three-dimensions. The results show that the simulations performed by using IATE and BND models agree with each other and with the experimental data. The simulations performed in three-dimensions give better agreement with the experimental data.

Interfacial Area Transport Equations for Gas-Liquid Flow

In relation to the modeling of the interfacial transfer terms in the two-fluid model, the concept of the interfacial area transport equation has been proposed to develop a constitutive relation for the interfacial area concentration. The interfacial area transport equation can replace the traditional flow regime maps and regime transition criteria. The changes in the two-phase flow structure can be predicted mechanistically and the effects of the boundary conditions and flow development are efficiently modeled by this transport equation. A successful development of the interfacial area transport equation can make a significant improvement in the two-fluid model formulation and the prediction accuracy of three-dimensional two-phase flow simulations. In this paper, the basic concept of the interfacial area transport equation and its formulation have been briefly described. Available models of the interfacial area sink and source terms and existing databases have been reviewed and the interfacial area transport equation has been benchmarked using adiabatic bubbly flow and condensation bubbly flow data. Future direction for this research has been suggested.

Drag coefficient in one-dimensional two-group two-fluid model

Recently, the concept of the interfacial area transport equation has been proposed to predict the dynamic change of interfacial structure in transient and developing flows. In this approach, bubbles are categorized into two groups (group 1: spherical/distorted bubbles; group 2: cap/slug/churn-turbulent bubbles) due to the considerable difference in their transport characteristics. In this paper, the equations for calculating drag coefficients of both groups under developing flow conditions are derived based on the momentum equations in the one-dimensional two-group two-fluid model. It is found that void fraction of both groups should be taken into account in determining drag coefficient of each group. The shape factor is important for group 2 bubbles even though it can be approximated to be unity for group 1 bubbles. Experimental data of air–water upward bubbly flows in various sizes of pipes are used to examine the existing drag coefficient model of group 1 bubbles. It is shown that the Ishii and Chawla’s models for spherical and distorted bubbles can predict the experimental data in the forced convective flow systems satisfactorily, which confirms their applicability to bubbly flow systems.

Computational Analysis of a Subcooled Boiling Flow with a One-group Interfacial Area Transport Equation

For a multidimensional analysis of a two-phase flow, a computational fluid dynamics (CFD) code was developed with the implementation of an interfacial area transport equation that is beneficial for dynamically estimating the interfacial area concentration (IAC). The code structure was based on the two-fluid model and the Simplified Marker and Cell (SMAC) algorithm. The SMAC algorithm was extended to a two-phase flow simulation with a phase change. Various well-known constitutive models regarding boiling, condensation, and nondrag forces have been implemented into the code. To verify the robustness of the code to predict wall boiling and void propagation phenomena, a subcooled boiling test in a vertical annulus channel was analyzed as a benchmark problem. As the analysis results, a model for bubble departure diameter on the heated wall was identified as the principal factor for subcooled boiling phenomena, and the limitation of the current departure diameter models under a low-pressure condition resulted in a deviation of the void fraction and IAC when compared with the results of the experiment. It is necessary that the research on the interfacial area transport equation focuses on modeling reliable source terms for the boiling mechanism as a future work.

Computational Analysis of a Subcooled Boiling Flow with a One-group Interfacial Area Transport Equation

Computational Analysis of a Subcooled Boiling Flow with a One-group Interfacial Area Transport Equation

Effect of Gravity on Flow Characteristics of Developing Vertical Upward Bubbly Flow (1st Report, Measurement of Local Flow Parameter)

In relation to the development of the interfacial area transport equation, axial developments of void fraction profile, interfacial area concentration and Sauter mean diameter of adiabatic nitrogen-water bubbly flows in a 9 mm-diameter pipe were measured by using a stereo image-processing method at normal-and micro-gravity conditions. The flow measurements were performed at four axial locations (axial distance from the inlet normalized by the pipe diameter, z/D=5, 20, 40 and 60) at various flow conditions of superficial gas velocity (0.008 40-0.029 8 m/s) and superficial liquid velocity (0.138-0.914 m/s). The effect of gravity on radial distribution of bubbles and the axial developments of two-phase flow parameter was discussed in detail based on the obtained data and the visual observation.

Modeling of the condensation sink term in an interfacial area transport equation

A model for the condensation sink term in an interfacial area transport equation (IATE) was developed. In the model, a bubble nucleation due to a wall surface boiling and a bubble collapse due to a condensation were assumed to be symmetric phenomena. Based on this consideration the condensing region for a subcooled condition can be divided into two regions: the heat transfer-controlled region and the inertia-controlled region. In the heat transfer-controlled region, the condensation Nusselt number approach is appropriate. On the other hand, in the inertia-controlled region, the resultant mechanical force may be balanced through an interface between a bubble and an ambient liquid. The modeled condensation sink term in an IATE in this study was evaluated against existing data which had been obtained from a bubble condensation in a subcooled water flow through a non-heated annulus. The evaluation result showed that the present model could predict the axial distribution of the interfacial area concentration accurately.

Study of two-phase flows in reduced gravity using ground based experiments

Experimental studies have been carried out to support the development of a framework of the two-fluid model along with an interfacial area transport equation applicable to reduced gravity two-phase flows. The experimental study simulates the reduced gravity condition in ground based facilities by using two immiscible liquids of similar density namely, water as the continuous phase and Therminol 59® as the dispersed phase. We have acquired a total of eleven data sets in the bubbly flow and bubbly to slug flow transition regimes. These flow conditions have area-averaged void (volume) fractions ranging from 3 to 30% and channel Reynolds number for the continuous phase between 2,900 and 8,800. Flow visualization has been performed and a flow regime map developed which is compared with relevant bubbly to slug flow regime transition criteria. The comparison shows that the transition boundary is well predicted by the criterion based on critical void fraction. The value of the critical void fraction at transition was experimentally determined to be approximately 25%. In addition, important two-phase flow local parameters, including the void fraction, interfacial area concentration, droplet number frequency and droplet velocity, have been acquired at two axial locations using state-of-the-art multi-sensor conductivity probe. The radial profiles and axial development of the two-phase flow parameters show that the coalescence mechanism is enhanced by either increasing the continuous or dispersed phase Reynolds number. Evidence of turbulence induced particle interaction mechanism is highlighted. The data presented in this paper clearly show the marked differences in terms of bubble (droplet) size, phase distribution and phase interaction in two-phase flow between normal and reduced gravity conditions.

On the numerical study of isothermal vertical bubbly flow using two population balance approaches

Two population balance approaches based on the MUltiple-SIze-Group (MUSIG) model and one-group average bubble number density (ABND) model for handling the bubble size distribution of gas–liquid bubbly flows under isothermal conditions are assessed. Three forms of coalescence and breakage mechanisms by Wu et al. [1998. One-group interfacial area transport in vertical bubbly flow. International Journal of Heat Mass Transfer 41, 1103–1112], Hibiki and Ishii [2002. Development of one-group interfacial area transport equation in bubbly flow systems. International Journal of Heat Mass Transfer 45, 2351–2372] and Yao and Morel [2004. Volumetric interfacial area prediction in upwards bubbly two-phase flow. International Journal of Heat Mass Transfer 47, 307–328] are incorporated in the ABND model. To examine the relative merits of both approaches, local radial distributions of five primitive variables in bubbly flows: void fraction, Sauter mean bubble diameter, interfacial area concentration, and gas and liquid velocities, are compared against the experimental data of Liu and Bankoff [1993a. Structure of air–water bubbly flow in a vertical pipe—I. Liquid mean velocity and turbulence measurements. International Journal of Heat Mass Transfer 36, 1049–1060; 1993b. Structure of air–water bubbly flow in a vertical pipe—II. Void fraction, bubble velocity and bubble size distribution. International Journal of Heat Mass Transfer 36, 1061–1072] and Hibiki et al. [2001. Axial interfacial area transport of vertical bubble flows. International Journal of Heat Mass Transfer 44, 1869–1888]. In general, both of the ABND model and MUSIG model predictions yield close agreement with experimental results. To account for the range of different bubble sizes in the gas–liquid bubbly flows, the resolution required is achieved through the application of the MUSIG model. Nevertheless, computational times increase by a factor of two when compared to applying the simpler ABND model. To further exploit the models’ capabilities, investigations are carried out by extending the two population approaches beyond the bubbly flow regime of higher void fraction, particularly in the transition regime. The numerical results are found to be grossly over-predicted, which expose the inherent limitations of the models. It is known that bubbles in this regime are generally highly distorted and closely packed instead of spherical shape and allowed to move freely in bubbly flow regime.

Effect of Gravity on Interfacial Area Transport of Vertical-upward Bubbly Flow

In relation to the development of the interfacial area transport equation, axial developments of void fraction profile, bubble number density, interfacial area concentration and Sauter mean diameter of adiabatic nitrogen-water bubbly flows in a 9 mm-diameter pipe were measured by using a stereo image-processing method under normal- and micro-gravity environment. The flow measurements were performed at four axial locations (axial distance from the inlet normalized by the pipe diameter, z/D = 5, 20, 40 and 60) at various flow conditions of superficial gas velocity (0.00823-0.0303 m/s) and superficial liquid velocity (0.147-0.907 m/s). The effect of gravity on radial distribution of bubbles and axial developments of two-phase flow parameter was discussed in detail based on the obtained data and the visual observation.