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.     

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.     

Effects of non-uniform inlet boundary conditions and lift force on prediction of phase distribution in upward bubbly flows with Fluent-IATE

This study investigates the profile effects of the boundary conditions in two-phase flows, such as the inlet void fraction, interfacial area concentration, and phase velocity, on the predictions of flow behaviors downstream. Simulations are performed for upward air-water bubbly flows in a 48.3-mm inner diameter pipe by employing Fluent's two-fluid model together with an interfacial area transport equation (IATE) model. The IATE was developed in the literature to model the interfacial area concentration by taking into account the bubble coalescence and disintegration, and phase change effects.In this study, two types of inlet boundary conditions are considered, one being a uniform-profile boundary condition in the radial direction with area-averaged experimentally measured values while the other being a non-uniform profile condition based on the actual measured profiles at the inlet. The numerical predictions of downstream profiles of the phase distributions indicate that the two types of boundary conditions yield similar results for the downstream flow behaviors for the bubbly flow conditions investigated. In addition, the results with and without the lift force demonstrated that the lift force is essential to obtain accurate lateral phase distribution.     

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.     

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.     

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.     

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

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.     

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.     

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。     

相界面浓度输运方程在一维两流体模型中的应用研究

为解决一维两流体模型核电厂系统分析程序中使用流型图所带来的缺陷,提高系统分析程序计算的准确性,探索在一维两流体模型中应用相界面浓度输运方程（IATE）对两相流动进行预测。采用FORTRAN语言开发耦合了IATE的一维两流体模型求解器（Solver-IATE）,并对其进行验证。基于SolverIATE对小直径绝热圆管内向上泡状流进行了数值模拟,并与采用流型图的计算结果进行了对比。研究结果表明：采用IATE计算的相界面浓度结果比采用流型图的计算结果更接近实验值。因此,在一维两流体模型中使用IATE可以提高其计算相界面浓度的准确性,进而提高一维两流体模型核电厂系统分析程序计算两相间相互作用项的准确性,能更准确预测反应堆的瞬态响应特性。     

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.     

Characteristics of two-phase condensing flow by visualization using computed image processing

The mechanics of the condensing behavior of vapor bubbles in a subcooled bulk flow is complicated and influenced by both heat and mass transfer. To examine the characteristics of such thermal-nonequilibrium two-phase flow, experimental and analytical researches have been made. In the experiment, the movement of each vapor bubble in a flowing channel was recorded on video tapes and analyzed by an image processing system. As result, the distributions of void fraction along the test section were obtained. In the analysis, a simple analytical model was introduced to predict the distributions of void fraction and liquid subcooling temperature. By considering the rate of vapor condensation along the flow direction, the differential equation of energy balance between two phases was obtained. Integration of this equation yielded the void fraction and bulk liquid subcooling at any position. The condensation rate was estimated as a function of the local liquid subcooling, interfacial area and mass velocity. Finally, a close fit between calculated results and experimental data was obtained.     

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.     

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.     

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.     

Local measurement of bubbly flow in helically coiled tubes using double-sensor conductivity probe

ABSTRACT

The two-phase flow in helically coiled tubes (HCTs) is rather important in many industries, such as the heat exchange facility in nuclear power plant. In this work, a double-sensor conductivity probe was used to study the air/water bubbly flow in HCTs. The cross-sectional distribution profile of the interfacial parameters (void fraction, interfacial area concentration, bubble size, etc.) of air–water bubbly flow were systematically studied. Through carefully processing the raw data collected by the double-sensor conductivity probe, the distribution of the void fraction, interfacial area concentration, the bubbles number frequency over the cross-section are demonstrated, as well as the bubble velocities and sizes vertically in the dense region. Some statistical parameters of cross-sectional-averaged quantities, coefficients of variation, and bubble aggregation core coordinates are defined to quantitatively describe the distribution characteristics of interfacial parameters. The measured data are helpful for improving the understanding of two-phase flow characteristics in HCTs.     

竖直下降两相流相界面结构输运特性

竖直下降两相流具有与竖直上升两相流不同的相界面结构特征及输运特性。本文对竖直下降管内的气水两相流进行了实验研究,运用微型四头电导探针对7.5、31.5及55.5倍管径横截面处的空泡份额、相界面浓度、气泡直径、气泡频率及气泡速度等相界面结构参数的局部分布进行了测量。分析获得了相界面结构参数的沿程变化规律,并研究了气相表观流速对相界面结构发展的影响及一维相界面结构输运特性。发现竖直下降泡状流的升力指向管中心,导致相界面结构参数基本呈中心峰值分布;气相表观流速的增大会提高空泡份额和相界面浓度分布的峰度;竖直下降两相流在距入口31.5倍管径处基本达到充分发展。     

An application of the method of moments to the modeling of bubbly flow

Bubbly flows are relevant in nuclear reactors thermalhydraulics and safety analysis. Regularly, empirical constitutive laws are required to close the two-fluid equations, particularly in relating the interfacial area and the bubble number densities to the local void fraction. In this article, starting from a generalized Boltzmann transport equation for the bubble size spectrum, a convection equation for the bubble number density is derived using the method of moments. The equation is analyzed for a vertical bubbly flow in stagnated liquid, showing excellent agreement with experimental data. The model is useful as a mean to provide conservation-based correlations to complement the existing two-fluid models.     

Measurement and modeling of average void fraction, bubble size and interfacial area

The local void fraction, bubble size and interfacial area concentration for co-current air-water bubbly flow through a horizontal pipe of 50.3 mm internal diameter were investigated experimentally using the double-sensor resistivity probe method. The local and area-averaged void fractions and interfacial area concentrations were analyzed as a function of liquid and gas flow rates. These parameters were found to increase systematically with decreasing liquid flow and increasing gas flow. However, variations with the liquid flow were not as significant as with the gas flow. A consistent variation of the gas phase drift velocity and distribution parameter with the liquid flow rate was observed. It was demonstrated that presentation of the average void fraction in terms of flowing volumetric concentration was more appropriate for horizontal bubbly flow. Several bubble break-up mechanisms were discussed. It was concluded that average pressure fluctuations generated by the turbulent liquid fluctuations acting across a bubble diameter are the only mechanism which causes distortion of a bubble. Based on this force and the competing surface tension force, a theoretical model was developed for mean bubble size and interfacial area concentration. The theoretically predicted mean bubble size and interfacial area concentration were found to agree reasonably well with those measured by the double-sensor resistivity method.