One-dimensional Interfacial Area Transport Equation Research Articles

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.

Improving the One-Dimensional Interfacial Area Transport Equation Using a Physics-Guided Machine Learning Method

Improving the One-Dimensional Interfacial Area Transport Equation Using a Physics-Guided Machine Learning Method

Evaluation of typical interfacial area transport models for bubbly flow

Interfacial area concentration is one of the most important parameters in two-phase flow. This parameter also reflects the interfacial area of mass, momentum and energy transfer in two-fluid model. The prediction of interfacial area concentration is essential to close the interfacial transfer terms in the two-fluid model. For the modeling of the one-dimensional interfacial area transport equation for adiabatic bubbly flow, existing typical sink and source terms have been reviewed and evaluated by the experimental data of vertical pipes and rod bundles. The interactive mechanisms of bubbles and turbulent eddies have also been analyzed, including bubbles breakup and bubbles coalescence. The evaluation results show the interfacial area transport model of Hibiki and Ishii (Liu and Hibiki) can provide the most accurate prediction of interfacial area concentration for pipes with the relative error of 8.22% for bubbly flow. Their model has also been evaluated by the experiment data of rod bundle with the relative error of 21.89% for bubbly flow.

Interfacial area transport due to shear collision of bubbly flow in small-diameter pipes

In view of the practical importance of the interfacial area transport equation in flow analysis based on the two-fluid model, especially for flow analysis in a reactor core which has a relatively small channel configuration, the dominant mechanism in the interfacial area transport in small diameter pipes was firstly identified by evaluating the experimental data obtained for pressurized bubbly flows under microgravity, where the source and sink terms of the interfacial area concentration due to bubble expansion and wake entrainment collision are negligible. The sink term of the interfacial area concentration of bubbly flow in small diameter pipe was modeled in terms of the shear collision mechanism. The one-dimensional interfacial area transport equation with the developed sink term was evaluated by using 74 datasets obtained for adiabatic gas-liquid bubbly flows, including upward flows and microgravity flows in pipes with inside diameters of 9.0, 5.0 and 3.0mm. A detailed discussion was provided on the interfacial area transport mechanism in small-diameter pipe. The overall standard deviation of the absolute error between predicted and measured interfacial area concentrations was 11.8%. The present model will reproduce the interfacial area transport of bubbly two-phase flows in small-diameter pipes with inside diameters of 3.0–9.0mm, at superficial liquid and gas velocities over the ranges 〈jf〉=0.073–0.949m/s, and 〈jg〉=0.00434–0.0500m/s, and also over a wide range of axial distances, z/D=5–188.

Experimental study of bubbly flow using image processing techniques

This paper presents an experimental study of bubbly flows at relatively high void fractions using an advanced image processing method. Bubble overlapping is a common problem in such flows and the past studies often treat the overlapping bubbles as a whole, which introduces considerable measurement uncertainties. In this study, a hybrid method combining intersection point detection and watershed segmentation is used to separate the overlapping bubbles. In order to reconstruct bubbles from separated segments, a systematic procedure is developed which can preserve more features captured in the raw image compared to the simple ellipse fitting method. The distributions of void fraction, interfacial area concentration, number density and velocity are obtained from the extracted bubble information. High-speed images of air-water bubbly flows are acquired and processed for eight test runs conducted in a 30mm×10mm rectangular channel. The developed image processing scheme can effectively separate overlapping bubbles and the results compare well with the measurements by the gas flow meter and double-sensor conductivity probe. The development of flows in transverse and mainstream directions are analyzed and compared with the prediction made by the one-dimensional interfacial area transport equation (IATE) and the bubble number density transport equation.

Interfacial area transport of subcooled water–steam condensing bubbly flow in vertical pipe

The interfacial area transport of a subcooled water–steam condensing bubbly flow in a vertical pipe with inner diameter of 29mm was investigated. Three fundamental parameters, the void fraction, interfacial area concentration (IAC) and bubble Sauter mean diameter, were experimentally obtained using a double-sensor conductivity probe technique. The radial and axial developments of local flow structure were interpreted based on the phase change and bubble interaction mechanisms, such as the bubble condensation, coalescence and break-up. Based on the experimental data sets, a critical Weber number range to identify the core peak was developed for one-component steam–water systems. Furthermore, the theoretical modeling of the one-dimensional interfacial area transport equation (IATE), including the condensation sink term, was discussed and compared with the experimental data. The axial distributions of area-averaged IAC profiles were best computed via the use of the IATE. The evaluation results showed that the interfacial area transport was dominated by the heat-mass transfer mechanism causing bubbles condensation in the flow.

Mechanistic modeling of interfacial area transport in large diameter pipes

Flow in large pipes is important in a wide variety of applications. In the nuclear industry in particular, understanding of flow in large diameter pipes is essential in predicting the behavior of reactor systems. This is especially true of natural circulation Boiling Water Reactor (BWR) designs, where a large-diameter chimney above the core provides the gravity head to drive circulation of the coolant through the reactor. The behavior of such reactors during transients and during normal operation will be predicted using advanced thermal-hydraulics analysis codes utilizing the two-fluid model. Essential to accurate two-fluid model calculations is reliable and accurate computation of the interfacial transfer terms. These interfacial transfer terms can be expressed as the product of one term describing the potential driving the transfer and a second term describing the available surface area for transfer, or interfacial area concentration. Currently, the interfacial area is predicted using flow regime-dependent semi-empirical correlations; however the interfacial area concentration is best computed through the use of the one-dimensional interfacial area transport equation (IATE). To be useful in practical applications the IATE requires mechanistic models for the change in interfacial area concentration due to interactions between bubbles. Therefore using previously collected data and proper scaling methods, current models for bubble interactions for the IATE are briefly discussed and new models applicable to large pipes are developed and compared with the existing data.

Two-phase flow structure in large diameter pipes

Flow in large pipes is important in a wide variety of applications. In the nuclear industry in particular, understanding of flow in large diameter pipes is essential in predicting the behavior of reactor systems. This is especially true of natural circulation Boiling Water Reactor (BWR) designs, where a large-diameter chimney above the core provides the gravity head to drive circulation of the coolant through the reactor. The behavior of such reactors during transients and during normal operation will be predicted using advanced thermal–hydraulics analysis codes utilizing the two-fluid model. Essential to accurate two-fluid model calculations is reliable and accurate computation of the interfacial transfer terms. These interfacial transfer terms can be expressed as the product of one term describing the potential driving the transfer and a second term describing the available surface area for transfer, or interfacial area concentration. Currently, the interfacial area is predicted using flow regime dependent empirical correlations; however the interfacial area concentration is best computed through the use of the one-dimensional interfacial area transport equation (IATE). To facilitate the development of IATE source and sink term models in large-diameter pipes a fundamental understanding of the structure of the two-phase flow is essential. This understanding is improved through measurement of the local void fraction, interfacial area concentration and gas velocity profiles in pipes with diameters of 0.102m and 0.152m under a wide variety of flow conditions. Additionally, flow regime identification has been performed to evaluate the existing flow regime transition criteria for large pipes. This has provided a more extensive database for the development and evaluation of IATE source and sink models. The data shows the expected trends with some distortion in the transition region between cap-bubbly and churn-turbulent flow. The flow regime map for the 0.102m and 0.152m diameter test sections agree with the existing flow regime transition criteria. It may be necessary to perform further experiments in larger pipes and at higher gas flow rates to expand the range of conditions for which models can be developed and tested.

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 (25.4mm and 48.3mm) 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.

Interfacial area transport of vertical upward bubbly two-phase flow in an annulus

In relation to the development of the interfacial area transport equation in a subcooled boiling flow, the one-dimensional interfacial area transport equation was evaluated by the data taken in the hydrodynamic separate effect tests without phase change, or an adiabatic air–water bubbly flow in a vertical annulus. The annulus channel consisted of an inner rod with a diameter of 19.1 mm and an outer round tube with an inner diameter of 38.1 mm, and the hydraulic equivalent diameter was 19.1 mm. Twenty data sets consisting of five void fractions, about 0.050, 0.10, 0.15, 0.20, and 0.25, and four superficial liquid velocities, 0.272, 0.516, 1.03, and 2.08 m/s were used for the evaluation of the one-dimensional interfacial area transport equation. The one-dimensional interfacial area transport equation agreed with the data with an average relative deviation of ±8.96%. Sensitivity analysis was also performed to investigate the effect of the initial bubble size on the interfacial area transport. It was shown that the dominant mechanism of the interfacial area transport was strongly dependent on the initial bubble size.

Modeling of bubble-layer thickness for formulation of one-dimensional interfacial area transport equation in subcooled boiling two-phase flow

In relation to the formulation of one-dimensional interfacial area transport equation in a subcooled boiling flow, the bubble-layer thickness model was introduced to avoid many covariances in cross-sectional averaged interfacial area transport equation in the subcooled boiling flow. The one-dimensional interfacial area transport equation in the subcooled boiling flow was formulated by partitioning a flow region into two regions; boiling two-phase (bubble layer) region and liquid single-phase region. The bubble-layer thickness model assuming the square void peak in the bubble-layer region was developed to predict the bubble-layer thickness of the subcooled boiling flow. The obtained model was evaluated by void fraction profile measured in an internally heated annulus. It was shown that the bubble-layer thickness model could be applied to predict the bubble-layer thickness as well as the void fraction profile. In addition, the constitutive equation for the distribution parameter of the boiling flow in the internally heated annulus, which was used for formulating the bubble-layer thickness model, was developed based on the measured data. The model developed in this study will eventually be used for the development of reliable constitutive relations, which reflect the true transfer mechanisms in subcooled boiling flows.

Interfacial Area Transport of Bubbly Flow in a Small Diameter Pipe

In relation to the development of the interfacial area transport equation, this study focused on modeling of the interfacial area transport mechanism of vertical adiabatic air-water bubbly flows in a relatively small diameter pipe where the bubble size-to-pipe diameter ratio was relatively high and the radial motion of bubbles was restricted by the presence of the pipe wall. The sink term of the interfacial area concentration was modeled by considering wake en-trainment as a possible bubble coalescence mechanism, whereas the source term was neglected by assuming negligibly small bubble breakup for low liquid velocity conditions based on visual observation. One-dimensional interfacial area transport equation with the derived sink term was evaluated by using five datasets of vertical adiabatic air-water bubbly flows measured in a 9.0mm-diameter pipe (superficial gas velocity: 0.013–0.052 m/s, superficial liquid velocity: 0.58–1.0m/s). The modeled interfacial area transport equation could reproduce the proper trend of the axial interfacial area transport and predict the measured interfacial area concentrations within an average relative deviation of 1±11.1%. It was recognized that the present model would be promising for predicting the interfacial area transport of the examined bubbly flows.

Interfacial Area Transport of Bubbly Flow in a Small Diameter Pipe.

In relation to the development of the interfacial area transport equation, this study focused on modeling of the interfacial area transport mechanism of vertical adiabatic air-water bubbly flows in a relatively small diameter pipe where the bubble size-to-pipe diameter ratio was relatively high and the radial motion of bubbles was restricted by the presence of the pipe wall. The sink term of the interfacial area concentration was modeled by considering wake en-trainment as a possible bubble coalescence mechanism, whereas the source term was neglected by assuming negligibly small bubble breakup for low liquid velocity conditions based on visual observation. One-dimensional interfacial area transport equation with the derived sink term was evaluated by using five datasets of vertical adiabatic air-water bubbly flows measured in a 9.0mm-diameter pipe (superficial gas velocity: 0.013–0.052 m/s, superficial liquid velocity: 0.58–1.0m/s). The modeled interfacial area transport equation could reproduce the proper trend of the axial interfacial area transport and predict the measured interfacial area concentrations within an average relative deviation of 1±11.1%. It was recognized that the present model would be promising for predicting the interfacial area transport of the examined bubbly flows.

One-group interfacial area transport of bubbly flows in vertical round tubes

In relation to the development of the interfacial area transport equation, the sink and source terms in an adiabatic bubbly flow system were modeled based on the mechanisms of bubble–bubble and bubble–turbulent eddy random collisions, respectively. The interfacial area transport mechanism was discussed based on the derived model. One-dimensional interfacial area transport equation with the derived sink and source terms was evaluated by using the area averaged flow parameters of adiabatic air–water bubbly flows measured in 25.4 mm and 50.8 mm diameter tubes. The flow conditions of the data set covered most of the bubbly flow regime, including finely dispersed bubbly flow (inlet superficial gas velocity: 0.0414–3.90 m/s, superficial liquid velocity: 0.262–5.00 m/s, void fraction: 1.27–46.8%). Excellent agreement was obtained between modeled and measured interfacial area concentrations within the average relative deviation of 11.6%. It was recognized that the present model would be promising for the interfacial area transport of the examined bubbly flows.

Foundation of the interfacial area transport equation and its closure relations

The interfacial area transport equation is derived from the statistical model of fluid particle number transport equation. The resulting equation includes the source and sink terms due to the particle interactions and interfacial phase change. The consistency of this new approach is demonstrated in terms of the macroscopic continuity equation of a bubbly flow field. The basic mechanisms affecting these source and sink terms are discussed. The general framework to develop the closure relations for the fluid particle interaction and phase change terms is presented. Finally, the one-dimensional interfacial area transport equation is used to identify possible methods to relate the source and sink terms to experimentally measurable interfacial parameters, such that experiments can be used to establish models for these terms.