One-dimensional Two-fluid Model Research Articles

Effect of void fraction covariance on relative velocity in gas-dispersed two-phase flow

In the two-fluid model the dependence between the phases is given in the field equations by interaction terms which become a key focus for improving the overall model performance. Of the interfacial terms in the one-dimensional two-fluid model, the most important is the constitutive relation for the interfacial momentum transfer, specifically the steady-state drag force. The one-dimensional steady-state drag force is a function of area-averaged local relative velocity. This area-averaged local relative velocity can be derived from the drift-flux general expression and the relation between drift velocity and relative velocity. However, due to area averaging there is a void fraction covariance which current and past researchers have assumed to be one. Similarly, in the one-dimensional modified two-fluid model which divides the gas phase into two-groups (i.e. spherical/distorted bubbles as group-1, and cap/slug/churn-turbulent bubbles as group-2) the group-1 and group-2 area-averaged local relative velocity is required for the group-1 and group-2 steady-state drag force. These relative velocities introduce three covariance terms: group-1 void fraction, group-2 void fraction, and an inter-group covariance between group-1 and group-2 void fraction. The covariance terms have been analyzed with a substantial database from the literature including upward flow in pipe diameters of 1.27 cm–15.2 cm, downward flow in pipe diameters of 2.54 cm and 5.08 cm, and upward flow in a 1.90 cm hydraulic diameter annulus channel. Simple relations are proposed to specify the covariance in order to improve the prediction of area-averaged local relative velocity in the classical two-fluid model and the modified two-fluid model. These relations are shown to have good agreement with the experimental data in predicting the effect on the area-average relative velocity with an average relative error of 5% over the data range.

Modeling and simulation of circulating fluidized bed reactors applied to a carbonation/calcination loop

A fluid dynamic model for a gas-solid circulating fluidized bed (CFB) designed using two coupled riser reactors is developed and implemented numerically with code programmed in Matlab. The fluid dynamic model contains heat and species mass balances to calculate temperatures and compositions for a carbonation/calcination loop process.Because of the high computational costs required to resolve the three-dimensional phenomena, a model representing a trade-off between computational time requirements and accuracy is developed. For dynamic processes with a solid flux between the two reactor units that depends on the fluid dynamics of both risers, a dynamic one-dimensional two-fluid model is sufficient.A two-fluid model using the constant particle viscosity closure for the stress term is used for the solid phase, and an algebraic turbulence model is applied to the gas phase. The numerical model implementation is based on the finite volume method with a staggered grid scheme. The exchange of solids between the reactor units constituting the circulating fluidized bed (solid flux) is implemented through additional mass source/sink terms in the continuity equations of the two phases.For model validation, a relevant experimental analysis provided in the literature is reproduced by the numerical simulations. The numerical analysis indicates that sufficient heat integration between the two reactor units is important for the performance of the circulating fluidized bed system.The two-fluid model performs fairly well for this chemical process operated in a CFB designed as two coupled riser reactors. Further analysis and optimization of the solution algorithms and the reactor coupling strategy is warranted.

Interfacial-tension-force model for the wavy-stratified liquid–liquid flow pattern transition

The flow of two immiscible liquids is of common occurrence in a wide range of natural and industrial processes. The interest in liquid–liquid flow has recently increased mainly due to the petroleum industry, where oil and water are often transported together for long distances. In the current Brazilian offshore scenario, significant amount of water is being produced, and it tends to increase. The one-dimensional two-fluid model is used to model the wavy stratified liquid–liquid flow. A stability analysis is carried out, including the interfacial tension force and a single transition criterion is proposed. A new destabilizing term arises, which is a function of the cross-section curvature of the interface. The existence of short interfacial waves is considered and the effect of a concave or convex cross-section interface shape is included in the analysis. It is shown that the new interfacial tension term plays an important role in regions of extreme in situ volume fractions. The kinematic wave theory is used to model the observed interfacial wave. New geometrical and kinematic wave data are used to validate the proposed model. Transition boundaries are drawn on flow maps of the superficial velocities and the agreement with present data and data from literature is encouraging. The results help to elucidate the actual nature of the typical wavy structure observed in stratified flow and can be used for the proposition of more accurate flow-pattern transition models.

A one-dimensional multicomponent two-fluid model of a reacting packed bed including mass, momentum and energy interphase transfer

In this paper a multicomponent two-phase flow numerical model for reacting packed beds is presented, where one phase is compressible. The model is applied to a catalyst bed which decomposes highly concentrated hydrogen peroxide into oxygen and water vapour. Using dimensional analysis a number of simplifications to the governing equations are made. A pressure based finite volume approach is used to solve the resulting equations where the pressure–velocity–density coupling is established with a SIMPLE algorithm for two-fluid flows. The model is validated against experimental data and shows in general a reasonable agreement given the complexity of the physics. It is shown that with the two-fluid formulation characteristics are predicted that cannot be reproduced with mixture models. Simulations reveal that the largest part of the total mass transfer takes place in a very small section of the catalyst bed at the point where the liquid reaches the boiling point and that most of this transfer is due to evaporation. It is further shown that a thermal description of the packing of the catalyst bed would improve the prediction of temperature distribution in the two-phase regime.

Verification of a Higher-Order Finite Difference Scheme for the One-Dimensional Two-Fluid Model

The one-dimensional two-fluid model is widely acknowledged as the most detailed and accurate macroscopic formulation model of the thermo-fluid dynamics in nuclear reactor safety analysis. Currently the prevailing one-dimensional thermal hydraulics codes are only first-order accurate. The benefit of first-order schemes is numerical viscosity, which serves as a regularization mechanism for many otherwise ill-posed two-fluid models. However, excessive diffusion in regions of large gradients leads to poor resolution of phenomena related to void wave propagation. In this work, a higher-order shock capturing method is applied to the basic equations for incompressible and isothermal flow of the one-dimensional two-fluid model. The higher-order accuracy is gained by a strong stability preserving multi-step scheme for the time discretization and a minmod flux limiter scheme for the convection terms. Additionally the use of a staggered grid allows for several second-order centered terms, when available. The continuity equations are first tested by manipulating the two-fluid model into a pair of linear wave equations and tested for smooth and discontinuous initial data. The two-fluid model is benchmarked with the water faucet problem. With the higher-order method, the ill-posed nature of the governing equations presents severe challenges due to a growing void fraction jump in the solution. Therefore the initial and boundary conditions of the problem are modified in order to eliminate a large counter-current flow pattern that develops. With the modified water faucet problem the numerical models behave well and allow a convergence study. Using the L1 norm of the liquid fraction, it is verified that the first and higher-order numerical schemes converge to the quasi-analytical solution at a rate of O(1/2) and O(2/3), respectively. It is also shown that the growing void jump is a contact discontinuity, i.e. it is a linearly degenerate wave. The sub-linear convergence rates are in exact agreement with the theory for a contact discontinuity.

Simulations of Steam Methane Reforming/Sorption-Enhanced Steam Methane Reforming Bubbling Fluidized Bed Reactors by a Dynamic One-Dimensional Two-Fluid Model: Implementation Issues and Model Validation

Multiphase flows and in particular dispersed flows are frequently encountered in a variety of important process facilities in the chemical industry; herein, the fluidized bed reactors. Whereas the internal and external solid fluxes in the fluidized bed reactors are generally determined by empirical correlations or prescribed in the Kunii–Levenspiel type of models, the two-fluid models serve as highly relevant in the progress of commercial reactors for processes such as the novel sorption-enhanced steam methane reforming (SE-SMR) technology because the solid flux incorporated in the two-fluid model allows for dynamic modeling of interconnected fluidized bed reactors. Nevertheless, the two- and three-dimensional two-fluid models are yet too computationally demanding for chemical process evaluation studies due to the complexity of the gas–solid flow in the bed. Hence, these models are not efficient for studies of the processes such as the SE-SMR technology where the solid particles are transported between reactor units for utilization and recovery of the characteristic solid property, i.e. CO2-capture and CO2-release. Thus, the aim of the present study is to derive a model that allows for a more complex description of the fluidized bed reactors relative to the frequently used Kunii–Levenspiel type of models. On the other hand, the model should not predict details in the flow, as the two- and three-dimensional two-fluid models, in order to ensure reasonable simulation costs. Hence, in this study, the classical SIMPLE algorithm is extended to compressible two-phase reactive flows. The governing equations are cross-sectional averaged to smooth out the details in the flow. The suggested dynamic one-dimensional Eulerian–Eulerian two-fluid model is applied to investigate the reactive gas–solid flows of the SMR and SE-SMR processes; and moreover, the simulation results are compared with the results of the more complex two-dimensional model with a solid stress closure based on kinetic theory of granular flow. The one-dimensional model predictions of the chemical process performance are in good agreement with the corresponding profiles predicted with the two-dimensional model. The deviations are larger comparing the internal flow details but these do not owe significant impact on the chemical process which to a large extent is determined by the imposed temperature in the reactor.

Dual-Frequency Severe Slugging in Horizontal Pipeline-Riser Systems

A new type of severe slugging is found that can occur in two-phase flow of gas and liquid in pipeline-riser systems. This instability, which will be referred to as dual-frequency severe slugging (DFSS), generates a different class of flow oscillations compared to the classical severe slugging cycle, having a dominant single frequency that is commonly found in a pipe downward inclined by a few degrees from the horizontal connected to a vertical riser. The DFSS flow pattern was found in laboratory experiments carried out in a 100 m long, 50.8 mm diameter horizontal pipeline followed by a 15 m high, 50.8 mm diameter vertical riser operating at atmospheric end pressure. The experimental facility also included a 400 liter gas buffer vessel, placed upstream of the pipeline, to obtain extra pipeline compressibility. Air and water were used as the experimental fluids. At constant inflow conditions, we observed a type of severe slugging exhibiting a dual-frequency behavior. The relatively high-frequency fluctuations, which are in the order of 0.01 Hz, are related to the classical severe slugging cycle or to an unstable oscillatory process. The relatively low-frequency fluctuations, which are in the order of 0.001 Hz, are associated with the gradual cyclic transition of the system between two metastable states, i.e., severe slugging and unstable oscillations. Numerical simulations were performed using OLGA, a one-dimensional two-fluid flow model. The numerical model predicts the relatively low-frequency fluctuations associated with the DFSS flow regime. The laboratory experiments and the numerical simulations showed that the evolution of the DFSS is proportional to the length of the pipeline.

Severe slugging in a long pipeline–riser system: Experiments and predictions

At constant inflow conditions, large-amplitude pressure and flow rate fluctuations may occur in a pipeline–riser system operating at relatively low liquid and gas flow rates. This cyclic flow instability has been referred to as severe slugging. This study is an experimental, theoretical and numerical investigation of severe slugging in a relatively long pipeline–riser system. The experiments were carried out in a 65m long, 50.8mm diameter horizontal steel pipeline connected to a 35m long, 50.8mm diameter Perspex pipeline which is inclined to −2.54° from the horizontal, followed by a 15.5m high, 45mm vertical PVC riser operating at atmospheric end pressure. The experimental facility also included a 250l gas buffer vessel, placed upstream of the pipeline, to obtain extra pipeline compressibility. Air and water were used as the experimental fluids. Five types of flow regimes were found and characterized based on visual observation and on the measured pressure drop over the riser. It was found that transient slugs were generated in the pipeline upstream of the riser base and they effectively contributed to the initial blockage of the riser base. An existing model for the prediction of the flow behaviour in the pipeline–riser system was modified. The modified model, which was tested against new experimental results obtained in this study, showed a better performance than previously published models. Numerical simulations were also performed using a one-dimensional two-fluid model. A good agreement between the numerical simulations and the experimental data was found.

Two-group drift-flux model for closure of the modified two-fluid model

In an effort to improve the prediction of void fraction and heat transfer characteristics in two-phase systems, closure relations to the one-dimensional modified two-fluid model are addressed. The drift-flux general expression is extended to two bubble groups in order to describe the void weighted mean gas velocities of spherical/distorted (group-1) bubbles and cap/slug/churn-turbulent (group-2) bubbles. Therefore, correlations for group-1 and group-2 distribution parameters and drift velocities are proposed and evaluated with experimental data. Furthermore, the covariance in the convective flux of the one-dimensional two-fluid model is addressed and interpreted with the available database. The dataset chosen for evaluation of the two-group drift-flux general expression contains 126 total data points taken in an annulus geometry. The proposed distribution parameters show an agreement within ±4.9% and ±1.2% for group-1 and group-2 data, respectively. The overall estimation of group-1 and group-2 void weighted mean gas velocity calculated with the newly proposed two-group drift-flux general expression shows an agreement of ±11.8% and ±17.7%, respectively.

Slug modeling with 1D two-fluid model

Abstract Simulations of condensation-induced water hammer with one-dimensional two-fluid model requires explicit modeling of slug formation, slug propagation, and in some cases slug decay. Stratified flow correlations that are more or less well known in 1D two-fluid models, are crucial for accurate description of the initial phase of the slug formation and slug propagation. Slug formation means transition to other flow regime that requires different set of correlations. To use such two-fluid model for condensation induced water hammer simulations, a single slug must be explicitly recognized and captured. In the present work two cases of condensation-induced water hammer simulations performed with WAHA code, are described and discussed: injection of cold liquid into horizontal pipe filled with steam and injection of hot steam into horizontal pipe partially filled with cold liquid.

Modeling and experimental investigation on water-driven steam injector for waste heat recovery

Abstract This paper discusses a proposed model of the water-driven steam injector that can be used to recover waste heat in many industries. A mathematical model was developed for evaluating the injector recovery performance in this study; the results indicated that a higher inlet steam pressure, lower inlet-driven water temperature or lower inlet-driven water pressure improved the entrainment ratio, whereas the back pressure of the discharge water had little effect on the injector performance. Moreover, a one-dimensional two-fluid analytical model of two-phase flow was presented with a set of closure equations for estimating flow parameters of the whole injector. The one-dimensional model involved a single distinct flow regime, inverted annular flow in the mixing nozzle and throat tube of the injector. The condensation heat transfer and interfacial shear stress were found to be two main factors affecting the static pressure profiles along the injector. A series of experiments were conducted to investigate the performance of the injector and the pressure distribution along the injector under different operating conditions. The entrainment ratio and flow parameters along the condensing section of the models were reasonably consistent with the experimental data.

One Dimensional Two-Fluid Model Simulations of the SE-SMR Process Operated in a Circulating Fluidized Bed Reactor

Abstract A novel one-dimensional two-fluid model for Sorption-Enhanced Steam Methane Reforming operating in a Circulating Fluidized Bed Reactor is introduced. Constitutive relationships as well as reaction kinetics are taken from the literature. The model is compared with experimental data and correlated to a similar model from the literature. Finally, there is an assessment on heat integration and solid flux sensitivity.

Numerical Prediction of Flow Patterns in Bubble Pumps

In the present study, the ammonia-water mixing flow in a bubble pump is numerically simulated. The flow patterns of a two-phase flow in a bubble pump were studied under different conditions of heat flux and tube diameter. A one-dimensional two-fluid model was developed under constant heat flux. This model was used to predict the variations in void fraction and liquid and vapor velocities throughout the tube. Then, the void fraction profile and the curve of liquid velocity versus vapor velocity were used to predict the flow patterns along the tube length. It was found that at heat fluxes below 15 kW m−2, bubbly, slug, and churn flows are the dominating regimes, and the length of these flow regimes depends on the tube diameter. For heat fluxes higher than 15 kW m−2, the bubble pump operates under the churn and annular regimes, and the bubble pump performance is improved when the tube diameter increases.

Statistical characterization of two-phase slug flow in a horizontal pipe

The present paper reports the results of an ongoing project aimed at providing statistical information on slugs in two-phase flow in a horizontal pipe. To this end, the flow was examined experimentally and numerically. On the experimental side, three non-intrusive optical techniques were combined and employed to determine the velocity field and bubble shape: particle image velocimetry (PIV), Pulsed Shadow Technique (PST) and Laser-Induced Fluorescence technique (LIF). Statistical information was provided by photogate cells installed at two axial positions. The flow was numerically determined based on the one-dimensional Two-Fluid Model. The tests were conducted on a specially built transparent pipe test section, using air and water as the working fluids. The velocity fields were obtained for flow regimes where the slugs were slightly aerated to facilitate the utilization of the optical methods employed. The main parameters for characterizing the statistically steady flow regime such as slug length and velocity obtained numerically were compared with the experimental data and good agreement was obtained.

Effect of operating conditions on the performance of the bubble pump of absorption-diffusion refrigeration cycles

The mathematical model will be able to predict the operated condition (required tube diameters, heat input and submergence ratio?.). That will result in a successful bubble pump design and hence a refrigeration unit. In the present work a one-dimensional two-fluid model of boiling mixing ammonia-water under constant heat flux is developed. The present model is used to predict the outlet liquid and vapor velocities and pumping ratio for different heat flux input to pump. The influence of operated conditions such as: ammonia fraction in inlet solution and tube diameter on the functioning of the bubble pump is presented and discussed. It was found that, the liquid velocity and pumping ratio increase with increasing heat flux, and then it decreases. Optimal heat flux depends namely on tube diameter variations. Vapour velocity increases linearly with increasing heat flux under designed conditions.

Study of Interfacial Friction Force for Bubble Flows in a 2×1 Rod Channel Simplifying BWR

Abstract Most of the recent subchannel analysis codes are based on a multifluid model, and an accurate evaluation of the constitutive equations in the model is essential. In order to get an accurate interfacial friction force in two-phase bubble flows, experimental data on drag coefficient and interfacial area concentration have been obtained for air-water flows in a 2×1 rod channel simplifying a boiling water nuclear reactor fuel rod bundle. In order to know the effects of liquid properties on the data, the temperature of the test water was changed from 18°C to 50°C. The data are compared with the existing correlations reported in literatures. As a result, the semitheoretical correlation of Hibiki and Ishii (2001, “Interfacial Area Concentration in Steady Fully-Developed Bubbly Flow,” Int. J. Heat Mass Transfer, 44, pp. 3443–3461) was found to give the best prediction against the present interfacial area concentration data. The correlation of Delhaye and Bricard (1994, “Interfacial Area in Bubbly Flow: Experimental Data and Correlations,” Nucl. Eng. Des., 151, pp. 65–77) also gave a reasonably good prediction if their correlation was modified by incorporating liquid property effects. As for the drag coefficient, no correlation exists, which can predict the present data well. Therefore, we developed a new correlation, including three dimensionless numbers, i.e., bubble capillary number, Morton number, and Eötvös number. The correlation predicted the data of Liu et al. (2008, “Drag Coefficient in One-Dimensional Two-Group Two-Fluid Model,” Int. J. Heat Fluid Flow, 29, pp. 1402–1410) as well as the present data well.

Natural gas hydrate shell model in gas-slurry pipeline flow

A hydrate shell model coupled with one-dimensional two-fluid pipe flow model was established to study the flow characteristics of gas-hydrate slurry flow system. The hydrate shell model was developed with kinetic limitations and mass transfer limitations, and it was solved by Euler method. The analysis of influence factors was performed. It was found that the diffusion coefficient was a key parameter in hydrate forming process. Considering the hydrate kinetics model and the contacting area between gas and water, the hydrate shell model was more close to its practical situations.

Stability analysis of inclined stratified two-phase gas–liquid flow

The present investigation involves the modeling of gas–liquid interface in a two-phase stratified flow through a horizontal or nearly-horizontal circular duct. The most complete and fundamental model used for these calculations is known as the one-dimensional two-fluid model. It is the most accurate of the two-phase models since it considers each phase independently and links both phases with six conservation equations. The mass and momentum balance equations are written in dimensionless form. The dimensionless mass and momentum balance equations are combined with the method of characteristics and an explicit method to simulate the flow. At first, the linear stability of the flow is investigated by disturbing the liquid flow with a small perturbation. An improved version of the one-dimensional two-fluid model for horizontal flows is developed as a set of non-linear hyperbolic governing equations. The importance of this research lies in obtaining a model that accounts for the effects of flow and geometrical conditions (such as liquid viscosity, surface tension). It is shown that, for positive values of the slope angle (upward inclination), the slug flow becomes more probable, whereas negative values of the slope angle (downward inclination) induce a more stable stratified flow.

Coaxial discharge with axial magnetic field: Demonstration that the Boltzmann relation for electrons generally does not hold in magnetized plasmas

A one-dimensional two-fluid model is used to describe the quasineutral plasma of a discharge formed between coaxial cylinders under the influence of an axial magnetic field. The geometry treated in this paper is symmetric about the z-axis and is radially varying. The nested cylinders are necessarily different in size, leading to a potential difference between the sheath edges of the discharge plasma. This can be removed by applying a strong enough magnetic field, which also has the effect of flattening the potential profile, i.e., reducing the electric field in the plasma volume. In a previous publication [T. M. G. Zimmermann et al., Phys. Plasmas 16, 043501 (2009)], the authors examined the validity of the Boltzmann relation for electrons when applied to a similar geometry. When the magnetic field becomes strong enough to affect the electron flow in the radial direction, this expression breaks down. It was further discovered that certain situations require a self-consistent treatment of magnetic fields, since significant azimuthal currents can arise in such geometries. This work is applied and extended to offer a complete description of the electron density.

Numerical simulation of transient roll-waves in two-phase pipe flow

This paper presents a transient roll-wave simulator based on a one-dimensional incompressible two-fluid model. Using an efficient numerical method and a grid cell length of one pipe diameter, the simulator is able to accurately predict experimental data in the roll-wave regime. Essential to obtaining the stable roll-wave solutions is the use of a modified version of the Biberg friction model. Based on an algebraic eddy viscosity model, the Biberg model yields mechanistic expressions for the wall and interfacial shear stresses in stratified two-phase flow. Input to the model is closure relations for the turbulence levels at either side of the interface. In order to model the increased dissipation of energy in breaking wave fronts, we propose to modify the original closures by adding a term proportional to the negative gradient of the liquid height. As the waves grow and come close to breaking, this term redistributes the shear stresses such that the waves are stabilized. The numerical method used in the simulator is an extension of a previously presented pseudospectral Fourier method. In particular, spectral vanishing viscosity terms are introduced to stabilize short wavelengths inconsistent with the long wavelength approximation of the two-fluid model. The result is a simulator that provides reliable and convergent numerical solutions for all stratified flow conditions. In the last part of the paper, the simulator is tested against roll-wave experiments with horizontal and upward inclined flows of water and a dense gas in a 10 cm pipe. Wave heights and speeds as well as mean pressure drops and holdups are predicted within 20% accuracy when compared with 474 experiments. Furthermore, the transient simulations enable us to study the dynamics and interactions between waves. In agreement with experimental observations, two different wave growth mechanisms are identified, and their influence on the evolution of the roll-wave flow regime is analyzed.