One-dimensional Two-fluid Model Research Articles

Numerical simulation of stratified-pattern two-phase flow in gas pipelines using a two-fluid model

Two-phase flows in pipelines occur in a variety of processes in the nuclear, petroleum and gas industries. Because of the practical importance of accurately predicting steady and unsteady flows along the line, two-fluid models have been extensively used in numerical simulations. These models are usually written as a system of non-linear partial-differential equations. A reliable prediction of such flows is a difficult task to accomplish due to the numerous sources of uncertainties, such as the basic two-phase flow model, the flow-pattern models, the initial and boundary conditions and the numerical method used to solve the initial-boundary-value problem. Several numerical methods, conservative or not, may be used to discretize the problem. In this paper we analyze the global performance of the numerical model, obtained when we couple a classical one-dimensional single-pressure four-equation two-fluid model to a numerical discretization based on the flux-corrected transport (FCT) method, to solve the stratified-pattern two-phase flow that occurs in typical gas-gathering offshore pipelines. The gas phase is considered to be compressible and the liquid phase is incompressible, whereas the flow is assumed to be isothermal. We show that the FCT method is second-order in space and that our numerical kernel produces accurate simulations for steady, as well as unsteady, simulations. We also analyze the hyperbolicity property of the two-fluid model and the adequate prescription of the boundary conditions. The results show that hyperbolicity is lost in part of the area of the stratified-pattern region of a classical flow-pattern diagram. Moreover, the eigenvalues alternate signs at the boundaries, which has implications on how to impose the boundary conditions. Numerical simulations of typical offshore gas pipeline flows show very good agreement when compared to a commercial software.

One-dimensional two-fluid model for wavy flow beyond the Kelvin–Helmholtz instability: Limit cycles and chaos

A 1D TFM numerical simulation of near horizontal stratified two-phase flow is performed where the TFM, including surface tension and viscous stresses, is simplified to a two-equation model using the fixed-flux approximation. As the angle of inclination of the channel increases so does the driving body force, so the flow becomes KH unstable, and waves grow and develop nonlinearities. It is shown that these waves grow until they reach a limit cycle due to viscous dissipation at wave fronts. Upon further inclination of the channel, chaos is observed. The appearance of chaos in a 1D TFM implies a nonlinear process that transfers energy intermittently from long wavelengths where energy is produced to short wavelengths where energy is dissipated by viscosity, so that an averaged energy equilibrium in frequency space is attained. This is comparable to the well-known turbulent stability mechanism of the multi-dimensional Navier–Stokes equations, i.e., chaos implies Lyapunov stability, but in this case it is strictly a two-phase phenomenon.

Ion temperature profiles in front of a negative planar electrode studied by a one-dimensional two-fluid model

Plasma-wall transition is studied by a one-dimensional steady state two-fluid model. Continuity and momentum exchange equations are used for the electrons, while the continuity, momentum exchange, and energy transport equation are used for the ions. Electrons are assumed to be isothermal. The closure of ion equations is made by the assumption that the heat flux is zero. The model equations are solved for potential, ion and electron density, and velocity and ion temperature as independent variables. The model includes coulomb collisions between ions and electrons and charge exchange collisions between ions and neutral atoms of the same species and same mass. The neutral atoms are assumed to be essentially at rest. The model is solved for finite ratio ε=λDL between the Debye length and λD and ionization length L in the pre-sheath and in the sheath at the same time. Charge exchange collisions heat the ions in the sheath and the pre-sheath. Even a small increase of the frequency of charge exchange collisions causes a substantial increase of ion temperature. Coulomb collisions have negligible effect on ion temperature in the pre-sheath, while in the sheath they cause a small cooling of ions. The increase of ε causes the increase of ion temperature. From the ion density and temperature profiles, the polytropic function κ is calculated according to its definition given by Kuhn et al. [Phys. Plasmas 13, 013503 (2006)]. The obtained profiles of κ indicate that the ion flow is isothermal only in a relatively narrow region in the pre-sheath, while close to the sheath edge and in the sheath it is closer to adiabatic. The ion sound velocity is space dependent and exhibits a maximum. This maximum indicates the location of the sheath edge only in the limit ε→0.

Revisiting low-fidelity two-fluid models for gas–solids transport

Two-phase gas–solids transport models are widely utilized for process design and automation in a broad range of industrial applications. Some of these applications include proppant transport in gaseous fracking fluids, air/gas drilling hydraulics, coal-gasification reactors and food processing units. Systems automation and real time process optimization stand to benefit a great deal from availability of efficient and accurate theoretical models for operations data processing. However, modeling two-phase pneumatic transport systems accurately requires a comprehensive understanding of gas–solids flow behavior. In this study we discuss the prevailing flow conditions and present a low-fidelity two-fluid model equation for particulate transport. The model equations are formulated in a manner that ensures the physical flux term remains conservative despite the inclusion of solids normal stress through the empirical formula for modulus of elasticity. A new set of Roe–Pike averages are presented for the resulting strictly hyperbolic flux term in the system of equations, which was used to develop a Roe-type approximate Riemann solver. The resulting scheme is stable regardless of the choice of flux-limiter. The model is evaluated by the prediction of experimental results from both pneumatic riser and air-drilling hydraulics systems. We demonstrate the effect and impact of numerical formulation and choice of numerical scheme on model predictions. We illustrate the capability of a low-fidelity one-dimensional two-fluid model in predicting relevant flow parameters in two-phase particulate systems accurately even under flow regimes involving counter-current flow.

Modeling of void fraction covariance in two-phase flows with phase change

The void fraction covariance, which accounts for non-uniformity in the void fraction distribution in the one-dimensional two-fluid model, has been shown to be critical for accurate prediction of the area-averaged interfacial drag force. Defined as the ratio of area-averaged square of void fraction to the square of the area-averaged void fraction, the void fraction covariance, which historically has been treated as unity, is analyzed for gas-dispersed flows undergoing phase change. The covariance is shown to be very large in subcooled boiling where the void fraction is highly non-uniform, which highlights the benefit of the bubble layer thickness averaged two-fluid model. In condensing and flashing flow the void fraction covariance is shown to be significant, having a very large impact on the interfacial drag force as the void fraction increases. The void fraction covariance is studied for its impact on the classical area-averaged two-fluid model, bubble-layer averaged two-fluid model, and area-averaged multiple bubble group two-fluid model. A simple set of correlations are proposed for easy implementation into the existing drag equations, and are shown to agree very well with experimental data.

A solver for the two-phase two-fluid model based on high-resolution total variation diminishing scheme

Finite volume techniques with staggered mesh are used to develop a new numerical solver for the one-dimensional two-phase two-fluid model using a high-resolution, Total Variation Diminishing (TVD) scheme. The solver is implemented to analyze numerical benchmark problems for verification and testing its abilities to handle discontinuities and fast transients with phase change. Convergence rates are investigated by comparing numerical results to analytical solutions available in literature for the case of the faucet flow problem. The solver based on a new TVD scheme is shown to exhibit higher-order of accuracy compared to other numerical schemes. Mass errors are also examined when phase change occurs for the shock tube problem, and compared to those of the 1st-order upwind scheme implemented in the nuclear thermal-hydraulics code TRACE. The solver is shown to exhibit numerical stability when applied to problems with discontinuous solutions and results of the new solver are free of spurious oscillations.

Numerical simulation of terrain-induced severe slugging coupled by hydrodynamic slugs in a pipeline–riser system

A numerical study based on a one-dimensional two-fluid model is carried out to describe the transient hydrodynamic slugging and terrain-induced severe slugging in a pipeline–riser system. The system of equations is rendered well-posed by interfacial pressure model for the riser. The selected flow conditions are restricted in the well-posed region for the horizontal and the downward inclined pipes to ensure the hydrodynamic slug characteristics are predicted correctly. The validity of the model is examined by water faucet problem and horizontal slug flow experiments. Simulations with and without slug capturing are conducted to address the effect of hydrodynamic slugs on severe slugging. It has been found that more accurate predictions are obtained by taking hydrodynamic slugs into account. At low superficial gas velocity, the simulation without slug capturing tends to overestimate the severe slugging period. When hydrodynamic slugs are captured, the upstream gas expansion is suppressed by the hydrodynamic slugs. At relatively high superficial gas velocity, the simulation without slug capturing tends to underestimate the severe slugging period. When hydrodynamic slugs are captured, the upstream compressible volume is greatly enlarged by the blowout of the hydrodynamic slugs. In both situations, the influences of the hydrodynamic slugs can reduce the errors of the predicted severe slugging characteristics.

NUMERICAL PREDICTION OF AIR-PARTICLES FLOW IN VERTICAL PIPES

This paper presents a steady state one-dimensional two-fluid model for gas-solid two-phase flow in a vertical raiser. The model is solved using conservative variable approach for the gas phase and fourth order Runge-Kutta method is used for the solid phase. The model predictions for pressure drop are compared with available experimental data and with Eulerian-Lagrangian predictions and a good agreement is obtained. The results indicate that the pressure drop increases as the solid mass flow rate, particle size and particles density increase. In addition, the model predictions for minimum pressure drop velocity are compared with experimental data from literature and the mean percentage error, MPE for minimum pressure drop velocity is -9.89%. It is found that the minimum pressure drop velocity increases as the solid mass flow, particle size and particle density increase while it decreases as the system total pressure increases.

A self-consistent two-fluid model of a magnetized plasma-wall transition

A self-consistent one-dimensional two-fluid model of the magnetized plasma-wall transition is presented. The model includes magnetic field, elastic collisions between ions and electrons, and creation/annihilation of charged particles. Two systems of differential equations are derived. The first system describes the whole magnetized plasma-wall transition region, which consists of the pre-sheath, the magnetized pre-sheath (Chodura layer), and the sheath, which is not neutral, but contains a positive space charge. The second system of equations describes only the neutral part of the plasma-wall transition region—this means only the pre-sheath and the Chodura layer, but not also the sheath. Both systems are solved numerically. The first system of equations has two singularities. The first occurs when ion velocity in the direction perpendicularly to the wall drops below the ion thermal velocity. The second occurs when the electron velocity in the direction perpendicularly to the wall exceeds the electron thermal velocity. The second system of differential equations only has one singularity, which has also been derived analytically. For finite electron to ion mass ratio, the integration of the second system always breaks down before the Bohm criterion is fulfilled. Some properties of the first system of equations are examined. It is shown that the increased collision frequency demagnetizes the plasma. On the other hand, if the magnetic field is so strong that the ion Larmor radius and the Debye length are comparable, the electron velocity in the direction perpendicularly to the wall reaches the electron thermal velocity before the ion velocity in the direction perpendicularly to the wall reaches the ion sound velocity. In this case, the integration of the model equations breaks down before the Bohm criterion is fulfilled and the sheath is formed.

An Assessment of the Virtual Mass Force in RELAP5/MOD3.3 for the Bubbly Flow Regime

One-dimensional two-fluid models used for the simulation of large, industrial-scale problems require many simplifying assumptions to make a closed model that is tractable, applicable to a wide variety of flow regimes, and computationally efficient. Of particular interest here is the virtual mass force and the simplified form used in the RELAP5/MOD3.3 model. Comparison of the characteristics of the simplified model with a more complete two-fluid model for bubbly two-phase flow shows a remarkable similarity. Comparison to experimental data is also surprisingly favorable—provided that the flow conditions are determined appropriately. Namely, the characteristic analysis determined that a drift velocity for distorted bubbly flow, rather than for churn-turbulent flow, matches the data more accurately. The study is concluded by implementing a distorted bubbly drift velocity correlation into the RELAP5/MOD3.3 code. A comparison of the void wave speeds with the data confirms the results of the characteristic analysis.

A Physically Based, One-Dimensional Two-Fluid Model for Direct Contact Condensation of Steam Jets Submerged in Subcooled Water

A simulation model for the direct contact condensation of steam in subcooled water is presented that allows determination of major parameters of the process, such as the jet penetration length. Entrainment of water by the steam jet is modeled based on the Kelvin–Helmholtz and Rayleigh–Taylor instability theories. Primary atomization due to acceleration of interfacial waves and secondary atomization due to aerodynamic forces account for the initial size of entrained droplets. The resulting steam-water two-phase flow is simulated based on a one-dimensional two-fluid model. An interfacial area transport equation is used to track changes of the interfacial area density due to droplet entrainment and steam condensation. Interfacial heat and mass transfer rates during condensation are calculated using the two-resistance model. The resulting two-phase flow equations constitute a system of ordinary differential equations, which is solved by means of the explicit Runge–Kutta–Fehlberg algorithm. The simulation results are in good qualitative agreement with published experimental data over a wide range of pool temperatures and mass flow rates.

Viscous Kelvin–Helmholtz instability analysis of liquid–vapor two-phase stratified flow for condensation in horizontal tubes

Based on the viscous Kelvin–Helmholtz theory, an instability analysis of stratified flow during condensation inside horizontal tubes is conducted to predict the transition from stable two-phase stratified flow to unstable flow. Taking into account the condensation heat transfer, a linear analysis of the one-dimensional two-fluid model is employed to calculate the amplification factor, which is used to quantify the instability of a liquid–vapor two-phase stratified flow. It emphasizes the response of the amplitude of the disturbance wave at the liquid–vapor interface to condensation heat transfer. The results show that mass transfer originating from condensation heat transfer is apt to increase this amplitude and ultimately accelerates the transition from stable stratified flow to an unstable transitional flow regime bounded between stratified and slug flow. In a liquid–vapor two-phase flow regime map the phenomenon is depicted as a much narrower stable stratified region than in the case of the adiabatic flow. Meanwhile, the stable flow region will shrink rapidly with increasing the saturation-to-wall temperature difference. All of results are found to be in agreement with the available experimental data very well.

Modeling and simulation of bubbling fluidized bed reactors using a dynamic one-dimensional two-fluid model: The sorption-enhanced steam–methane reforming process

A dynamic one-dimensional multicomponent model for two-phase flows which includes heat- and mass transfer processes are studied in the Euler framework. The model is intended for reactive gas–solid flows in bubbling fluidized bed reactors. A model is desired that allows for a more complex description of the fluidized bed reactors (e.g. prediction of the bed expansion) relative to the conventional fluidized bed reactor models such as, e.g., Kunii–Levenspiel type of models. The model should not predict details in the flow as the two- and three-dimensional Euler two-fluid models in order to ensure reasonable simulation costs. In particular, the two- and three-dimensional Euler two-fluid models challenges the current available computational capacity for studies of reactive flows.The novel sorption-enhanced steam methane reforming (SE-SMR) technology is simulated in the bubbling bed regime. Simulation results of the one-dimensional Euler two-fluid model is compared to both a two-dimensional Euler model and a conventional fluidized bed model consisting of mass and heat balances. Furthermore, a sensitivity study to operation conditions and transport coefficients is performed for the one-dimensional Euler two-fluid model.The present simulation results reveal that the chemical process performance of the reactor is to a large extent determined by the imposed temperature in the reactor. Further, the one-dimensional Euler model provides an improvement of the simpler conventional fluidized bed reactor models by prediction of the bed expansion. Compared with the two-dimensional Euler model, cross-sectional averaging results in a significant reduction in the computational time but on the cost of loss of flow details.

Analysis of stability, verification and chaos with the Kreiss–Yström equations

A system of two coupled PDEs originally proposed and studied by Kreiss and Yström (2002), which is dynamically similar to a one-dimensional two-fluid model of two-phase flow, is investigated here. It is demonstrated that in the limit of vanishing viscosity (i.e., neglecting second-order and higher derivatives), the system possesses complex eigenvalues and is therefore ill-posed. The regularized problem (i.e., including viscous second-order derivatives) retains the long-wavelength linear instability but with a cut-off wavelength, below which the system is linearly stable and dissipative. A second-order accurate numerical scheme, which is verified using the method of manufactured solutions, is used to simulate the system. For short to intermediate periods of time, numerical solutions compare favorably to those published previously by the original authors. However, the solutions at a later time are considerably different and have the properties of chaos. To quantify the chaos, the largest Lyapunov exponent is calculated and found to be approximately 0.38. Additionally, the correlation dimension of the attractor is assessed, resulting in a fractal dimension of 2.8 with an embedded dimension of approximately 6. Subsequently, the route to chaos is qualitatively explored with investigations of asymptotic stability, traveling-wave limit cycles and intermittency. Finally, the numerical solution, which is grid-dependent in space–time for long times, is demonstrated to be convergent using the time-averaged amplitude spectra.

Numerical investigation of steam methane reforming with CO2-capture in bubbling fluidized bed reactors

The aim of the present study is to investigate the performance of a dynamic one-dimensional multi-component model for two-phase flows which includes heat- and mass transfer processes. The model is intended for reactive gas–solid flows in bubbling fluidized bed reactors. In particular, the model is applied to the steam-methane reforming (SMR) process and the novel sorption-enhanced steam-methane reforming (SE-SMR) technology.The one-dimensional two-fluid model is subjected to a sensitivity study of important operation conditions. Moreover, the effect of model reduction is revealed by comparing the one-dimensional model predictions to the simulation results of a two-dimensional model. The effect of model reduction is investigated for more complex flows than previous studied for the present model. Even for more complex flows in bubbling bed reactors, the results reveal that the chemical process performance of the reactor to a large extent is determined by the imposed temperature in the reactor. As opposite to the Kunii–Levenspiel type of models, the one-dimensional model provides a prediction of the bed expansion, and thus provides an improvement compared to the fundamental models. Moreover, model reduction with respect to the number of independent space variables is favorable considering computational cost, but on the cost of loss in the flow details.

An Artificial Viscosity for the III-Posed One-Dimensional Incompressible Two-Fluid Model

Methods to remedy the ill-posedness of the basic one-dimensional two-fluid model, which is widely used in nuclear reactor safety codes, have been the subject of considerable study. Both of the two prevalent methods have drawbacks. Unconditional hyperbolization uses nonphysical constitutive relations to create a well-posed two-fluid model that is hyperbolic over all flow conditions. However, when the model is hyperbolized, it is also stabilized, which is not a universal property of two-phase flows. The second method, the preferred method of the U.S. Nuclear Regulatory Commission safety codes, is to simply use a first-order upwind numerical method that relies on numerical viscosity to regularize the ill-posedness of the model by damping the short-wavelength instabilities. Unfortunately, the scale of the “short wavelength” is related to a particular numerical grid or discretization. Because of the consistency of the numerical method, in the limit of an infinitely resolved grid, i.e., Δx, Δt→0, the numerical viscosity vanishes, as does its regularization effect. This results in a somewhat heuristic user guideline that suggests a lower limit on the grid size based on a cross-sectional dimension that is a combination of the long-wavelength assumption and experience. However, a cutoff wavelength achieved by numerical viscosity is not set by the grid size alone but also depends on the time step, the material, and the flow properties, as demonstrated with a von Neumann stability analysis. This can create poor resolution in areas where numerical stability may not be a substantial problem, unless the guideline is intentionally violated. Additionally, strict observance of this limit makes verification by convergence difficult or impossible. Therefore, it is proposed that an artificial viscosity be prescribed explicitly, i.e., independently of any particular numerical method or grid. An artificial viscosity model is derived that prescribes exactly a cutoff in the linear stability growth rate at a specified wavelength, e.g., λ0 = 2DH consistent with the aforementioned user guideline. It is shown, using the water faucet problem, that the proposed artificial viscosity model can be used to remove the high-frequency component of the solution without limiting the resolution of the grid. Furthermore, the solution also converges, which was not the case without the artificial viscosity.

Linear and nonlinear analysis of an unstable, but well-posed, one-dimensional two-fluid model for two-phase flow based on the inviscid Kelvin–Helmholtz instability

For nuclear reactor safety analysis, the one-dimensional two-fluid model equations are solved numerically with the first order upwind method because of its robust stability. In the present two-fluid model for horizontal stratified flow, surface tension is included because it makes the model well-posed. However, this is not done in industrial applications and numerical viscosity provides linear stabilization even when the model is ill-posed. It is now shown that numerical viscosity also provides nonlinear stabilization; meaning that the wave growth is bounded when the flow is unstable (e.g., in case of the Kelvin–Helmholtz instability). The formation of kinematic shocks in the presence of numerical viscosity provides the dissipation mechanism needed to stop the wave growth. However, numerical viscosity varies with the mesh size, which means that even though the unstable model is well-posed and meets von Neumann and nonlinear stabilization requirements, the solution does not converge for some short wavelengths greater than 2Δx. Furthermore, when the mesh size is large, significant artificial viscosity is added to the continuity and momentum equations and the model may become over-stabilized. A more scientific approach is proposed: to add a physical dissipation mechanism instead of numerical viscosity, i.e., Reynolds stresses. This approach has two advantages: first, the one-dimensional two-fluid model converges under severe dynamic conditions, such as the Kelvin–Helmholtz instability, and second, a higher order numerical method may be used instead of the first order upwind scheme without experiencing numerical excursions.

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.