Interfacial Area Transport Equation Research Articles

Modeling and simulation of bubble condensation using polydisperse approach with bubble collapse model

An accurate modeling of the bubble condensation phenomena in sub-cooled water is developed in this work, using a two-fluid model with one disperse phase and one Interfacial Area Transport Equation (IATE). For this, the aspects of some models are investigated, such as the choice of population balance models, Nusselt closure model and bubble collapse model in IATE. The standard method of moments is formulated using two different bubble size distribution functions among all, namely Dirac and quadratic laws. Therefore, different models for mass and energy interfacial transfers, interfacial forces and source terms of the IATE are developed using both functions. While limited differences are noticeable for the mass and energy transfers obtained by both functions, the results are largely improved using quadratic function for the drag force term and for the IATE source terms. Moreover, the simulations results show that the widely used Ranz-Marshall correlation clearly underestimates the condensation rate. However, this work shows that Chen–Mayinger correlation is relevant to simulate this type of flow, regardless the distribution function. Afterwards, a model is introduced to take into account the effect of bubble collapse by condensation in the IATE. The numerical results obtained using the quadratic function, the collapse model in the IATE and the Chen–Mayinger correlation are comparable to those obtained by the inhomogeneous MUltiple Size Group (iMUSIG) approach, while being less time-consuming.

Validation of two-group interfacial area transport equation in boiling flow

The Two-Fluid Model is the backbone of thermal-hydraulics and system-analysis codes for nuclear design. The Two-Fluid Model tracks the transfer of conserved quantities—mass, momentum, and energy—without the need for bubble interface tracking. However, two-phase flows are characterized by their different flow regimes which change as more vapor is present in the flow, from bubbly flow to cap/slug flow to annular flow. The two-group Two-Fluid Model can track the progression of boiling flows beyond the bubbly region without the need for flow-regime maps. A two-group wall-boiling model is implemented and coupled with the two-group Interfacial Area Transport Equation. The resulting model is compared against experimental data and predicts the growth in cap/slug bubbles through interaction models rather than through flow-regime maps. The coupled model can predict the growth in void fraction and interfacial area concentration beyond the capability of the one-group model, demonstrating the applicability of a coupled two-group approach to modeling boiling flow.

Two-group drift-flux model for dispersed gas-liquid flows in annuli

Interfacial area concentration (IAC) plays a vital role in mass, momentum, and heat transfers between gas and liquid phases in two-phase flows. The IAC of group-one bubbles, including spherical and distorted bubbles, differs from the IAC of group-two bubbles, including cap, slug and churn-turbulent bubbles. The two-group interfacial area transport equation (IATE) can be used as a mechanistic model to obtain the IAC of each group. The equation demands the two-group gas velocity fields, leading to the need to solve an extra momentum equation, which is complicated. The two-group drift-flux model can prevent this complexity by providing two-group gas velocities without introducing the extra momentum equation. Despite the importance of gas-liquid flows in annuli, the drift-flux model for gas-liquid flows is not well-developed like it is for pipe flows. The main objective of this study is to develop a two-group drift-flux model for dispersed vertical flows in annuli. The two-group drift-flux model provides the flow characteristics of group-one and group-two bubbles, such as void fractions and velocities, which strongly affect the prediction of the available interfacial area concentration of the group-one and group-two bubbles for heat and mass transfer. Hence, after comprehensively investigating the available experimental data, this study first improves a one-group drift-flux model for dispersed gas and low-viscosity liquid upward flows in annuli and proposes a new two-group drift-flux model. Then, the models are evaluated using the respective group experimental data. The consistency between one-group and two-group drift-flux models is also investigated. This consistency helps to reduce the two gas momentum equations to one gas mixture momentum equation. The results of the comparison between model predictions and experimental data are satisfactory.

Two-group drift-flux model for dispersed gas-liquid flows in rod bundles

Interfacial transfer terms in gas-liquid two-phase flows are formulated as the product of interfacial area concentration (IAC) and flux. The interfacial area transport equation (IATE) is essential for obtaining IAC in transient and developing two-phase flows. Two-group IATE was developed to account for the difference in the interfacial drag force between two groups of bubbles, where spherical and distorted bubbles are categorized as group one, while cap, slug, and churn turbulent bubbles are categorized as group two. The rigorous two-group approach requires two-group gas momentum equations, resulting in one additional momentum equation in the two-fluid model. The mixture gas momentum equation is considered to avoid the additional momentum equation. The two-group drift-flux model is necessary to calculate the two-group gas velocity from the mixture velocity. The present study develops an approximation methodology to acquire subchannel average and rod bundle average two-group two-phase flow parameters based on the validated power law assumption and a limited number of local data. The two-group model is developed for the distribution parameter and drift velocity for dispersed two-phase flows in rod bundles. The total performance of the newly developed two-group drift-flux model is evaluated by two-phase flow data in rod bundles. The results show that the developed two-group drift-flux model could well predict two-group gas velocities in rod bundles. This model is useful for subchannel thermal hydraulic analysis codes as a complement to the conventional one-group drift-flux model.

Improved gas influx distribution estimation using interfacial area transport equation (IATE) enabled two-fluid model: An advanced modeling and full-scale experimental study

Gas influx management is a critical aspect of petroleum drilling operations, which aims to prevent the uncontrolled outflow of formation gas and the potential consequences caused by blowouts. This study investigates the distribution of gas influx and the interfacial area concentration (IAC) during gas influx circulation operations through a combination of full-scale experiments and advanced numerical simulations.Experiments were conducted to simulate downhole gas influxes by injecting Nitrogen gas into the base of a 5,160-foot-deep experimental wellbore, which was filled with water. A Distributed Fiber-Optic Sensing (DFOS) system was installed in the wellbore to obtain high-resolution acoustic and temperature data and monitor the gas influx behaviors when circulated out of the annulus. Various types of measurements were obtained to analyze the gas influx length, slip velocities, void fraction distribution, and dispersion behaviors. In addition, an advanced numerical simulator was developed based on a modified Two-Fluid Model and the Interfacial Area Transport Equation (IATE) to simulate the gas influx behaviors. The model estimations were compared to the experimental measurements for validation of accuracy and further insights.The numerical modeling framework, based on the interfacial area transport equation, effectively captured bubble interactions, including mechanisms including bubble breakage and coalescence, by estimating the interfacial area distribution. The gas influx dispersion during circulation was analyzed using both model estimations and experimental data. Good agreement was observed between the model predictions and experimental data, including multi-depths gauge measurements and the DFOS data.This is a novel application of the IATE in well-scale multiphase flow simulations, which has proved to be an effective tool for predicting the dynamics of gas influx distributions. The outcomes of this study provide critical insights into the design and optimization of gas influx management during Managed Pressure Drilling (MPD).

Two-group drift-flux model for dispersed gas-liquid flow in large-diameter pipes

Interfacial heat and mass transfer are prevalent in industrial processes. The interfacial transfer rate can be obtained by the product of their fluxes and interfacial area concentration (IAC) calculated by the interfacial area transport equation (IATE). Bubbles show different behavior according to their sizes. Hence, bubbles are classified into two groups. Consequently, two-group IATE is required causing to use of two gas momentum equations leading to more complexity. The present study suggests a new reliable two-group drift-flux modeling to reduce the two gas momentum equations to one gas mixture momentum equation for gas-liquid flow in large-diameter pipes. The model is developed based on the drift-flux model concept and experimental data. Group-one and group-two distribution parameters and drift velocities are validated through experimental data. The results show that the proposed two-group drift-flux model can support the concept of drift velocity from the bubbly to beyond the bubbly flow and consistency between the one-group and two-group drift-flux models. Moreover, steam-water data are used to validate the applicability of the model in steam-water flows condition. The developed two-group drift-flux model is indispensable for reducing the two gas momentum equations to one gas mixture momentum equation when two-group IATE is implemented into thermal-hydraulic codes to improve the prediction accuracy of IAC.

Numerical Simulation of Flashing Flows in a Converging–Diverging Nozzle with Interfacial Area Transport Equation

Flashing flows of initially sub-cooled water in a converging–diverging nozzle is investigated numerically in the framework of the two-fluid model (TFM). The thermal non-equilibrium effect of phase change is considered by an interfacial heat transfer model, while the pressure jump across the interface is ignored. The bubble size distribution induced by nucleation, bubble growth/shrinkage, coalescence, and breakup is described based on the interfacial area transport equation (IATE) and constant bubble number density model (CBND), respectively. The results are compared with the experimental data. Satisfactory prediction of the axial pressure distribution along the nozzle as well as the flashing inception, is achieved by the TFM-IATE coupling method. It was also found that the vapor production in the diverging section was overpredicted, and the radial gas volume fraction distribution deviated from the experiment. The radial diameter profiles exhibit opposite patterns at the nozzle throat and near the outlet, and similar trends can be observed for the superheated degree. A poly-disperse method is suggested to be introduced to describe the evolution of interfacial area concentration.

CFD Simulation of isothermal upward two-phase flow in a vertical annulus using interfacial area transport equation

This work presents a numerical simulation of a vertical, upward, isothermal two-phase flow of air bubbles and water in an annular channel applying a Computational Fluid Dynamics (CFD) code. For this, the Two-Fluid model is applied considering interfacial force correlations, namely: drag, lift, wall lubrication, turbulent dispersion, and virtual mass. The turbulence k-ε model effects and the influence of One-group Interfacial Area Transport Equation (IATE) are taken into account, in this case, the influence of two source term correlations for the bubble breakup and coalescence IATE is analysed. The work assesses whether the code properly represents the physical phenomenon by comparing the simulation results with experimental data obtained from the literature. Six flow conditions are evaluated based on two superficial liquid velocities and three void fractions in the bubbly flow regimen. The annular channel adopted has an outer pipe with an internal diameter of 38.1 mm and an inner cylinder of 19.1 mm. To represent this geometry, a three-dimensional mesh was generated with 160,000 elements, after a mesh sensitivity study. The void fraction distribution, taken radially to the flow section, is the main parameter analysed as well as interfacial area concentration, interfacial gas velocity, and bubble sizes distribution. The CFD model implemented in this work demonstrates satisfactory agreement with the reference experimental data but indicates the need for further improvement in the phase interaction models.

Effect of inlet gas and liquid velocity profiles on one-group interfacial area transport equation in a vertical large rectangular channel

Bubbly flows appear in many heat and mass transfer equipment. The interfacial area concentration is a critical parameter in determining the performance of the heat and mass transfer systems. The one-dimensional one-group interfacial area transport equation (IATE) was developed based on area-averaged bubbly flow data collected under uniform inlet flow boundary conditions. The applicability of the IATE to the flow systems with non-uniform inlet flow boundary conditions was not comprehensively examined. This study investigated the effect of inlet liquid and gas velocity profiles on measured interfacial area concentration changes along the flow direction. The data used in this study were collected for vertical upward air-water two-phase flow in a rectangular channel with a gap length of 10 mm and width of 200 mm. The test conditions were the superficial gas velocity ranging from 0.0870 to 0.288 m/s and the superficial liquid velocity ranging from 0.515 to 2.53 m/s. The interfacial area concentrations with uniform and non-uniform inlet flow boundary conditions measured in a large vertical rectangular channel were compared under the same area-averaged flow rate conditions. Then, the predictive capability of the IATE to the bubbly flow test data with uniform and non-uniform inlet flow boundary conditions was assessed. Some non-uniform inlet flow boundary conditions, including channel-center peaking of liquid flow (CPF), channel-center peaking of gas flow (CPG), single-side-wall peaking of liquid flow (SPF), and single-side-wall peaking of gas flow (SPG), significantly affected the interfacial area concentration changes along the axial location at relatively high superficial gas or liquid velocity conditions. The IATE's predictive capability deteriorated for several non-uniform inlet flow boundary conditions, including double-side-wall peaking of gas flow (DPG) and single-side-wall peaking of liquid flow (SPF), specifically at relatively high superficial liquid velocity conditions. The maximum percentage error of the interfacial area concentration prediction by the IATE for these non-uniform inlet flow boundary conditions reached 75%.

Comparison of bubbles interaction mechanisms of two-group Interfacial Area Transport Equation model

This work deals with 3D simulations of complex bubbly, cap-bubbly and churn regimes exhibiting bubbles of different shapes and with broad bubble size distribution. The first contribution of this work is to investigate and compare several bubble interaction mechanisms of coalescence and fragmentation for the 2-Group Interfacial Area Transport Equation (IATE) model. For two of these models, this is the first time their performances are assessed within a CFD code. The second contribution is to propose and assess a novel model of fragmentation and coalescence. Finally, a validation versus experimental data on three different configurations and three different regimes is performed. In particular, the interaction mechanisms are analyzed for one specific regime. The implementation of the two-group IATE model has been systematically performed in the 3D NEPTUNE CFD code.

The research on interfacial area transport equation in rectangular channels: Experimental research and models development of sink and source terms

Interfacial area concentration is a crucial constitutive parameter to close the two-fluid model, the accuracy of whose prediction is vital, so the interfacial area transport equation (IATE) has been proposed. According to the analysis of experimental results involving bubble interaction mechanisms in the rectangular channel from the bubbly-to-slug flow, new sink and source terms in a one-group IATE are put forward in this study. The new source term aroused by random collision is considered with relative velocity inducing collision and the effect of gap size on coalescence efficiency. Besides, the number density of small and large bubbles is considered respectively to model the source term of wake entrainment. Combined with the source terms of break up brought by turbulent impact and expansion caused by the pressure gradient, a new one-group IATE applicable to rectangular channels are completed. The coefficients in the newly-derived model are determined by experimental results of 4 × 66 mm and 8 × 66 mm vertical rectangular channels under co-current upward air-water two-phase flow, and the average relative error is 11.9%. Furthermore, the new model is also testified by the experimental result in the 6 × 66 mm rectangular channel independently, and the average relative error is 14.4%. Generally, the new model is feasible to predict the change of the interfacial area concentration in different rectangular channels.

The research on the interfacial area transport equation in rectangular channels: Interfacial characteristics distribution and bubbles interaction mechanism

The interfacial area transport equation (IATE) is employed to describe the evolution of interfacial structure and evaluate the interfacial area concentration of two-phase flow. However, the interaction mechanisms among bubbles change in different rectangular channels, significantly impacting interfacial characteristics and source terms in IATE. In order to study the interface characteristics and mechanism of bubble interaction in rectangular channels with different gap sizes, an experimental study was conducted on the upward mixture flow of air and water in an adiabatic vertical channel under atmospheric pressure. Two channels with cross-sections of 4 × 66 mm and 8 × 66 mm were selected. The superficial velocities of water and air were 0.5–3 and 0–1.12 m/s, respectively, almost at the transitional region of bubbly to slug flow. Flat electrodes and the four-sensor probes measured the interfacial characteristics. This study analyzes the development of interface and mechanism of interaction among bubbles. The result shows that the evolution in the distribution of void fraction and interfacial area concentration significantly depends on the flow regime, flow rate, and cross-sectional geometric size of channels. Under the condition of a lower liquid flow rate, the gap size of the channel plays a vital role in the efficiency of bubble coalescence, influencing the mechanisms of the random collision and wake entrainment among bubbles. However, the interfacial parameter distribution under higher liquid rate conditions is dominated by turbulence intensity and flow regime, and the mechanisms of bubble interaction become identical. Finally, the mechanisms of interaction among bubbles (random collision and wake entrainment) are analyzed based on the development of interfacial characteristics. In a word, based on the conclusions specified in this research, the new models of source terms of ITAE in rectangular channels will be predictably established in the future.

An experimental study on interfacial parameters in a narrow rectangular channel under different flow regimes

This study experimented with air-water up-flow in a 65 mm × 2 mm narrow rectangular duct. Forty-two flow conditions were acquired with the ⟨jg⟩: 0.47–8.58 m/s, the ⟨jf⟩: 0.1–1.5 m/s, involving the void fraction: 2.75%–85.7%. The high-speed camera was located at z/DH = 154.6, 309.2, and 463.8. For calculating the void fraction, interfacial area concentration (IAC), and bubble number density (BND) under various cases, an image processing scheme was proposed combining the non-overlapping characteristic of bubbles in the narrow gap. There are good consistencies between the void fraction of all conditions, the IAC of bubbly flow and the models, respectively. Group-1 and Group-2 parameters' evolution laws were discussed under the mechanisms of breakup, coalescence and pressure drop. Gas phase expansion was constantly influencing the parameters. The Group-1 bubbles’ substantial coalescence will decrease the total void fraction under the experimental cases located at the transition line of bubbly to slug or churn flow. Furthermore, the change of IAC influenced by the bubble breakup and coalescence was more remarkable than that of the void fraction, which the BND showed. Moreover, the change of IAC with the gas or liquid rate was not invariable under different superficial gas or liquid velocity, which was concerned with the transition mechanism of flow patterns. The database obtained will be helpful for the verification of the current interfacial area transport equation (IATE), especially for a higher void fraction.

Advanced Instrumentation and Modeling Framework for Two-Phase Flow

To fully realize the advantages of the two-fluid model, accurate prediction of the interfacial area concentration (IAC) is indispensable. Since conventional flow regime–based IAC correlations are not capable of dynamically describing the evolution of interfacial structure, the interfacial area transport equation (IATE) was developed to close the two-fluid model. In the past 30 years, intensive efforts have been made to improve the prediction performance of IATE and extend the experimental database for the IATE benchmark. Recent efforts of the IATE development and benchmark conducted by the Thermal-hydraulics and Reactor Safety Laboratory at Purdue University are reviewed in this paper. This review covers (1) the development of IATE; (2) the experimental database for IATE modeling, including instrumentation development, local measurement data of adiabatic/diabatic two-phase flow, and annular flow characterization; and (3) implementation and evaluation of IATE in one-dimensional/three-dimensional scenarios. Significant progress has been achieved since 2009, and future works required to advance the modeling of IATE are also suggested.

Effect of interfacial area modeling method on interfacial drag predictions for two-fluid model calculations in large diameter pipes

The nuclear industry requires accurate modeling efforts to ensure safety standards are consistently achieved. To that end, the sensitivity of void fraction and interfacial area concentration predictions to changes in the interfacial area concentration modeling methods was evaluated. Several interfacial area concentration correlations and interfacial area transport equation coalescence and breakup models have been compared to the data of Schlegel et al. and Shen et al. using a computational tool developed in MATLAB. Bubble coalescence and breakup empirical constants were optimized for the group void fractions and interfacial area concentrations using the computational tool and the surface response method. The results indicate that the transport equation approach has a higher accuracy for both the void fraction and interfacial area concentration predictions than the correlation approach. The Smith et al. model is shown to have the highest void fraction predictive accuracy. The optimized model is shown to successfully increase accuracy of the interfacial area concentration prediction. This is also the highest accuracy observed for interfacial area concentration predictions.

Modeling of turbulent diffusion terms for one-dimensional interfacial area transport equation in vertical round channels

The two-fluid model is comprised of the mass, momentum, and energy equations for liquid and gas phases separately. The terms of interfacial transfer govern mass, momentum, and energy transfer across the gas-liquid interface. The interfacial area concentration is critical to formulating the interfacial transfer terms. The one-dimensional Interfacial Area Transport Equation (IATE) has excellent potential in dynamically predicting the interfacial area concentration. The sink and source terms modeling for the interfacial area concentration is vital in the successful IATE development. The current IATE in bubbly flows considers the sink terms caused by bubble coalescence owing to random bubble collision and wake entrainment. The source term is caused by bubble breakup owing to turbulent impact. The turbulent diffusion term has not been considered in the current practice of one-dimensional IATE. This study developed the equation to predict the turbulent diffusion terms in the one-dimensional IATE for bubbly flow in vertical round channels. The predictive capability of the developed equation was validated with the collected database. The non-dimensionalized turbulent diffusion terms predicted by the developed equation agreed with those directly calculated by the experimental data. The developed turbulent diffusion term was also implemented in the one-dimensional IATE. The interfacial area concentrations predicted by the one-dimensional IATE with the turbulent diffusion term had a satisfactory agreement with the experimental results by the statistical parameters of the mean relative deviation of 3.76% and the mean absolute relative deviation of 10.1%.

Improving the one-dimensional interfacial area transport equation using a physics-informed machine learning method

The interfacial area concentration modeling has been greatly improved with the development of the interfacial area transport equation (IATE). The one-dimensional IATE, which is obtained through area-averaging the three-dimensional form of the model, has been adopted into many thermal hydraulic analysis codes. Towards improving the performance of one dimensional interfacial area transport equation, along with many of the existing studies on developing sophisticated source and sink constitutive models describing the liquid-bubble and bubble-bubble interactions, additional efforts should be made on improving the accuracies of the closure models supporting the calculation of these constitutive models. In this study, a physics-informed machine learning method was proposed that couples the machine learning and the IATE model for improving the performance of the existing IATE models. The machine learning model serves to predict the errors of the closure models that calculate the key parameters related to the interfacial area transports. The machine learning model is guided by the physical models, and its training is through an IATE model-based loss optimization. To effectively train the machine learning model, a data augmentation method was developed based on the existing IAC database. The newly proposed method that couples the trained machine learning model and the IATE model gives a 12% of improvement compared with the state-of-art IATE model. This new method and the approach on training the model provide good reference on how to effectively couple the machine learning and physical model for an improved modeling of the indirectly observable physical phenomena.

Characteristics of one-dimensional bubbly flow interfacial area transport in horizontal narrow rectangular channel

The experiments of horizontal adiabatic air-water flow in narrow rectangular channel with a cross-sectional area of 68 mm × 1.9 mm were conducted under an atmosphere pressure condition in order to study bubbly flow interfacial area transport phenomenon. The liquid film thickness was obtained by PCB liquid film sensor. The results show that the liquid film thickness can be correlated with the product of Capillary number Ca and jg/jl, so a correlation for liquid film thickness was proposed. The effect of liquid film thickness on IAC and void fraction was also evaluated. The results show that the average error caused by ignoring liquid film thickness is about 6.1% and 10.1%, respectively. The lateral profiles and axial direction development of the void fraction, interfacial area concentration, bubble Sauter mean diameter were obtained by high-speed camera at three axial locations of Z/D = 63, 241 and 510. The simplified one-dimensional, steady-state, one-group interfacial area transport equation for an adiabatic bubbly flow with six sets of IAC source and sink models was evaluated by the current experimental data. The evaluation results showed that the models of Yao and Morel (2004) and Shen and Hibiki (2013) performed better with mean prediction errors of 12.13% and 18.92%, respectively.

The Modeling of Bubble Lift-Off Diameter in Vertical Subcooled Boiling Flow

Bubble lift-off diameter is characterized as the size of a bubble rising from a wall, which is vital in the boundary condition of heat transfer model and interfacial area transport equation. In this paper, mechanistic force analysis was conducted to explore a predictive model for bubble lift-off diameter in a vertical channel of subcooled boiling flow. Specifically, the component of growth force normal to the wall and the shear lift force lead to the lift-off of a bubble on the vertical surface. Through force analysis, we found that bubble lift-off diameter is arranged to be related to wall superheat, latent heat, liquid velocity, fluid properties, bulk liquid subcooling, etc. To account for the contribution of the influencing factors, the dimensionless bubble lift-off diameter was correlated with dimensionless parameters, including the Prandtl number, the Reynolds number, the Jacob number, and dimensionless subcooling. The proposed correlation was assessed according to experimental data and the predictions showed good agreement with the data.

The study on the partition method of two-phase wall drag in the one-dimensional two-fluid model

The interfacial area transport equation(IATE) is being investigated to replace the flow regime map in the one-dimensional two-fluid model (1-D TFM). The partition of two-phase wall drag is a part of flow regime-dependent constitutive relations in the code using the 1-D TFM such as RELAP5. Hence, to develop the partition method for two-phase wall drag that is consistent with 1-D TFM using the IATE, the mechanism for the partition was investigated and clarified in this work. And the partition method for two-phase wall drag that can capture the dynamic change of two-phase flow structure was developed with the clarified mechanism. Furthermore, the method was preliminarily validated with the experimental data, which shows that the two-phase wall drag computed by the method can accurately capture the variation of flow structure and relative velocity in transient two-phase flow. Accordingly, the developed method is able to take advantage of the IATE.