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。     

Balance of Liquid-phase Turbulence Kinetic Energy Equation for Bubble-train Flow

In this paper the investigation of bubble-induced turbulence using direct numerical simulation (DNS) of bubbly two-phase flow is reported. DNS computations are performed for a bubble-driven liquid motion induced by a regular train of ellipsoidal bubbles rising through an initially stagnant liquid within a plane vertical channel. DNS data are used to evaluate balance terms in the balance equation for the liquid phase turbulence kinetic energy. The evaluation comprises single-phase-like terms (diffusion, dissipation and production) as well as the interfacial term. Special emphasis is placed on the procedure for evaluation of interfacial quantities. Quantitative analysis of the balance equation for the liquid phase turbulence kinetic energy shows the importance of the interfacial term which is the only source term. The DNS results are further used to validate closure assumptions employed in modelling of the liquid phase turbulence kinetic energy transport in gas-liquid bubbly flows. In this context, the performance of respective closure relations in the transport equation for liquid turbulence kinetic energy within the two-phase k—epsilon and the two-phase k—l model is evaluated.     

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.     

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.     

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.     

Two fluid model for two-phase turbulent jets

Eulerian two-fluid models are widely used in nuclear reactor safety and CFD. In these models turbulent diffusion of a dispersed phase must be formulated in terms of the fluctuating interfacial force and the Reynolds stresses. The interfacial force is obtained using the probability distribution function approach by Reeks (1992). This paper is the first application of this force to a case of engineering interest outside homogeneous turbulence. An Eulerian multidimensional two-fluid model for a cylindrical two-phase dispersed particle jet is proposed and compared with experimental data. The averaged conservation equations of mass and momentum are solved for each phase and the turbulent kinetic energy equation is solved for the continuous phase. The turbulent diffusion force and the Reynolds stresses are constituted within the context of the k- model of turbulence. A dissipation term has been added to the k- model for the turbulence modulation by the particles. Once the constitutive relations have been defined, the two-fluid model is implemented in a computational fluid dynamics code. It is shown that when the particles are very small the model is consistent with a convection-diffusion equation for particle transport where the diffusivity is defined according to Taylor's model (Taylor, G.I., 1921. Diffusion by continuous movements. Proc. London Math. Society, A20, pp. 196–211). The two-fluid model is also compared against two experimental data sets. Good agreement between the model and the data is obtained. The sensitivity of the results to various turbulent mechanisms is discussed.     

Model evaluation of two-group interfacial area transport equation for confined upward flow

The bubble interaction mechanisms have been analytically modeled in the first paper of this series to provide mechanistic constitutive relations for the two-group interfacial area transport equation (IATE), which was proposed to dynamically solve the interfacial area concentration in the two-fluid model. This paper presents the evaluation approach and results of the two-group IATE based on available experimental data obtained in confined upward flow, namely, 11 data sets in or near bubbly flow and 13 sets in cap-turbulent and churn-turbulent flows. The two-group IATE is evaluated in steady-state, one-dimensional (1D) form. To account for the inter-group bubble transport, the void fraction transport equation for Group-2 bubbles is also used to predict the void fraction for Group-2 bubbles. Agreement between the data and the model predictions is reasonably good and the average relative difference for the total interfacial area concentration between the 24 data sets and predictions is within 7%. The model evaluation demonstrates the capability of the two-group IATE focused on the current confined flow to predict the interfacial area concentration over a wide range of flow regimes.     

Modeling and validation of interfacial area transport equation in subcooled boiling flow

The first comprehensive validation of the interfacial area transport equation in subcooled boiling is presented and shown to perform exceptionally when compared with experimental data. The formulation and closure of the bubble layer averaged interfacial area transport equation is reviewed along with the treatment of the two-fluid model in subcooled boiling. Interfacial area concentration source and sink terms in subcooled boiling are presented including the bubble interaction mechanisms (random collision and turbulent impact), as well as phase change terms (wall nucleation and condensation). Additionally, the volume source terms from phase change are described and discussed in terms of their significance to the interfacial area transport equation. The validation of the interfacial area transport equation with a recently proposed wall nucleation source term is shown to have excellent prediction at low and elevated pressure, as well as a wide range of mass flux. With new confidence in the wall nucleation source term, the interfacial area concentration in subcooled boiling can be accurately predicted. Due to its strong dependence in the modeling of active nucleation site density, bubble departure frequency, and departure diameter, the calculation is shown to be very sensitive to wall temperature.     

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.     

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.     

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.     

Local two-phase flow measurements using sensor techniques

Radial profiles of various local parameters in bubbly two-phase flow were obtained. Measurements of the local void fraction, the local interfacial area concentration, the bubble interfacial velocity and Sauter mean diameter were made using the double sensor probe method. At the same locations, local liquid velocity and turbulence intensity measurements were made using a hotfilm anemometer. Data was taken at three different axial locations (L/D=2, L/D=32 and L/D=62) along a 3.24 m test section with an inner diameter of 0.0508 m. In comparison to previous data sets, the following data is more complete in the sense that both interfacial area measurements are combined with one of the local driving forces for interfacial transfer, namely the liquid turbulent diffusion. There have been few, if any, studies done combining local liquid turbulence and the local interfacial area concentration. The data taken will eventually be applied to the closure relations required by the one-dimensional, time-averaged interfacial area transport equation.     

Numerical evaluation of interfacial area concentration using the immiscible lattice gas

The phase separation in a Couette flow and the mixing of two phases in a cavity flow are simulated numerically using the immiscible lattice gas, which is one of the discrete methods of using particles to simulate two-phase flows. The interface is defined as the lattice sites between two phases, and the interfacial area concentration is evaluated in the steady state. In the Couette flow, the interfacial area concentration increases slightly with an increase in the wall speed. It is shown in the cavity flow that the interfacial area concentration increases largely with an increase in the wall speed. Macroscopic velocity fields in the two flows are in good agreement with analytical or numerical solutions of the Navier–Stokes equations. The interfacial area concentration is found to be correlated with the wall speed for the two flows, and the applicability of particle simulation methods to the numerical evaluation of the interfacial area concentration is indicated.     

Impedance imaging relative to gas-liquid systems

Determining interfacial area density in two-phase, gas-liquid flows is one of the major elements impeding significant development of predictive tools based on the two-fluid model. Currently, these models require coupling of liquid and vapor at interfaces using constitutive equations which do not exist in any but the most rudimentary form. This paper describes the theory of impedance imaging of two-phase mixtures and summarizes bench-type experiments utilized in its development, testing, and feasibility evaluation.     

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.     

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%.     

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.     

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.