Gas-liquid Model Research Articles

Global solutions to a one-dimensional non-conservative two-phase model

In this paper we investigate a basic one-dimensional viscous gas-liquid model based on the two-fluid model formulation.The gas is modeled as a polytropic gas whereas liquid is assumed to be incompressible. A main challenge with this model is the appearance of a non-conservative pressure term which possibly also blows up at transition to single-phase liquid flow (due to incompressible liquid).We investigate the model both in a finite domain (initial-boundary value problem) and in the whole space (Cauchy problem).We demonstrate that under appropriate smallness conditions on initial data we canobtain time-independent estimates which allow us to show existence and uniqueness of regular solutions as well as to gain insight into the long-time behavior of the model. These results rely strongly on the fact that we can derive appropriate upper and lower uniform bounds on the gas and liquid mass. In particular, the estimates guarantee that gas does not vanish at any point for any time when initial gas phase has a positive lower limit. The discussion of the Cauchy problem is general enough to take into account the possibility that the liquid phase may vanish at some points at initial time.

Simulation of Water Level Fluctuations in a Hydraulic System Using a Coupled Liquid-Gas Model

A model for simulating vertical water level fluctuations with coupled liquid and gas phases is presented. The Preissmann implicit scheme is used to linearize the governing equations for one-dimensional transient flow for both liquid and gas phases, and the linear system is solved using the chasing method. Some classical cases for single liquid and gas phase transients in pipelines and networks are studied to verify that the proposed methods are accurate and reliable. The implicit scheme is extended using a dynamic mesh to simulate the water level fluctuations in a U-tube and an open surge tank without consideration of the gas phase. Methods of coupling liquid and gas phases are presented and used for studying the transient process and interaction between the phases, for gas phase limited in a chamber and gas phase transported in a pipeline. In particular, two other simplified models, one neglecting the effect of the gas phase on the liquid phase and the other one coupling the liquid and gas phases asynchronously, are proposed. The numerical results indicate that the asynchronous model performs better, and are finally applied to a hydropower station with surge tanks and air shafts to simulate the water level fluctuations and air speed.

On the large time behavior of the compressible gas–liquid drift-flux model with slip

The main purpose of this work is to study the long-time behavior of a compressible gas–liquid model based on the drift-flux formulation. The model is composed of two mass conservation equations and one mixture momentum equation. The flow domain is closed at one end and involves a free gas–liquid interface at the other where both phases vanish. The model includes a slip between the gas and liquid phases, i.e. they move with different velocities. This is a main reason why the model is useful for many industrial applications. We introduce a reformulation of the model based on the gas velocity. This gives rise to new nonlinear terms in the mixture momentum equation which account for the difference in the gas and liquid velocity. New challenges in the mathematical analysis will then appear. In particular, under appropriate smallness conditions on initial data (initial energy) various time-independent estimates of gas and liquid masses, as well as fluid velocities, are obtained. Novel upper and lower bounds on masses are provided that contain precise information about the time-dependent decay rate. Hence, the long-time behavior can be directly extracted from these estimates.

Numerical simulation of optimal submergence depth of impellers in an oxidation ditch

Submergence depth of impellers for an oxidation ditch (OD) has significant impact on energy consumption and effluent quality of wastewater treatment plants. The effect of the submergence depth of impellers on the structure of flow fields in an OD was studied using an experimentally validated numerical tool, based on computational fluid dynamics (CFD) model. The two-phase gas–liquid model and the 3D RNG k–ε turbulence model were used to describe flow in ODs. The pressure implicit with splitting of operators algorithm was used to solve velocity and pressure. The volume of fluid (VOF) method was used to simulate free-water surface. The concept of submergence depth ratio representing the impeller’s submergence depth h to the total water depth H, h/H, was introduced to analyze the simulation results. Under different submergence depth ratios of impellers in an OD, the velocity fields were computed and analyzed, which shows that with the submergence depth ratio 0.45, the percentage of VOF particles with velocity greater than 0.3 m/s to the volume of entire fluid particles is the greatest, and that the velocity distribution is more uniform along water depth, being much useful for preventing sludge from settling in ODs. Therefore, the submergence depth ratio 0.45 is called the optimal submergence ratio, and its corresponding operating condition will better improve the efficiency of an OD wastewater treatment system.

Global Solutions of a Viscous Gas-Liquid Model with Unequal Fluid Velocities in a Closed Conduit

We consider a compressible gas-liquid drift-flux model with a general slip law commonly used to describe realistic two-phase flow scenarios. The slip law will introduce a difference in the magnitude of the two fluid velocities, and they possibly will also have different sign. This allows the model to describe the effect of buoyant forces, for example, in a vertical conduit, where heavy liquid will move downward due to gravity whereas light gas will be displaced upwardly. Combining the two mass equations and the mixture momentum equation with the slip law, the model can be expressed in terms of the gas-fluid velocity and a generalized pressure function that depends on the common pressure and three new terms that depend on the liquid mass and the gas velocity. This generalized pressure function introduces new nonlinear effects and coupling mechanisms between the two mass equations and the mixture momentum equation which require careful refinements of techniques previously used for the analysis of the classical Navier--Stokes model. In this work we discuss global existence and uniqueness of strong solutions considered in a closed one-dimensional conduit subject to appropriate smallness assumptions and regularity assumptions on the initial data. The analysis provides insight into the special role played by the slip parameters. This work presents a first global existence result for the drift-flux model with a general slip law.

Review and extension of pressure drop models applied to Taylor flow regimes

This paper investigates the pressure drop induced by both liquid–liquid and liquid–gas segmented Taylor flow regimes. A comprehensive experimental programme was completed using four different liquid–liquid and three different liquid–gas combinations over dimensionless slug length, Reynolds and Capillary numbers that spanned several orders of magnitude. Comparisons between the liquid–liquid pressure drop data and the most referenced expressions in the literature highlighted their lack of robustness and demonstrated their inapplicability for use with most practical systems that incorporate liquid–liquid Taylor flow regimes. The experimental pressure drop values obtained for the liquid–gas flows agreed well with existing pressure drop correlations. Interpretation of the liquid–liquid data using liquid–gas models unearthed the existence of a threshold viscosity ratio. Above this threshold, experimental liquid–liquid data was found to agree well with existing liquid–gas models. Below this threshold, results showed that the dispersed slug velocity needed to be considered as the flows were subject to higher interfacial contributions and inertial effects. A modification is proposed to an existing liquid–gas pressure drop correlation. This proposed modification extends the applicability of the correlation to liquid–liquid flows, and furthermore extends the non-dimensional limits of the correlation.

Ozonation Kinetics of Acid Red 27 Azo Dye: A Novel Methodology Based on Artificial Neural Networks for the Determination of Dynamic Kinetic Constants in Bubble Column Reactors

A procedure for the determination of initial parameter values for quadratically convergent optimization methods is proposed using artificial neural networks coupled with a non-stationary gas-liquid reaction model. The evaluation of the regression and the mean squared error coefficients of the neural network during its training process allow the parameter sensitivity analysis of the gas-liquid model. This analysis examines how many and which parameters of the model will be available depending on the observable information of the mathematical model. Numerical simulations show the relevance of the initial values and the non-linearity of the objective function. The methodology has been applied to the study of the reaction of the azo-dye Acid Red 27 with ozone in acid media. The rate constant is in the order of (1.6 ± 0.1) 103 M−1 s−1 under the experimental conditions.

Asymptotic stability of the compressible gas–liquid model with well-formation interaction and gravity

In this paper we are concerned with the initial–boundary value problem of the compressible gas–liquid model with well-formation interaction and gravity. The asymptotic behavior of solutions to steady states is established. Also the time-decay rates of perturbed solutions in the sense of L∞ norm are obtained under some suitable assumptions on the initial date, if γ>1 (associated with pressure law of gas) and β∈(0,γ2]∩(0,γ−αγ)∩(0,γ+αγ3] where β characterizes the viscosity coefficient and α describes the mass decay rate at the boundary. A main purpose of this work is to clarify the role played by the well-reservoir interaction term. The analysis demonstrates that it is essential to take into account information about sign as well as size of the interaction term in order to obtain time-independent estimates when it operates in combination with gravity.

Nonlinear generalized functions and jump conditions for a standard one pressure liquid–gas model

The mixture of a liquid and a gas is classically represented by one pressure models. These models are a system of PDEs in nonconservative form and shock wave solutions do not make sense within the theory of distributions: they give rise to products of distributions that are not defined within distribution theory. But they make sense by applying a theory of nonlinear generalized functions to these equations. In contrast to the familiar case of conservative systems the jump conditions cannot be calculated a priori. Jump conditions for these nonconservative systems can be obtained using the theory of nonlinear generalized functions by inserting some adequate physical information into the equations. The physical information that we propose to insert for the one pressure models of a mixture of a liquid and a gas is a natural mathematical expression in the theory of nonlinear generalized functions of the fact that liquids are practically incompressible while gases are very compressible, and so they do not satisfy equally well their respective state laws on the shock waves. This modelization gives well defined explicit jump conditions. The great numerical difficulty for solving numerically nonconservative systems is due to the fact that slightly different numerical schemes can give significantly different results. The jump conditions obtained above permit to select the numerical schemes and validate those that give numerical solutions that satisfy these jump conditions, which can be an important piece of information in the absence of other explicit discontinuous solutions and of precise observational results. We expose with care the mathematical originality of the theory of nonlinear generalized functions (an original abstract analysis issued by the Leopoldo Nachbin team on infinite dimensional holomorphy) that permits to state mathematically physical facts that cannot be formulated within distribution theory, and are the key for the removal of “ambiguities” that classically appear when one tries to calculate on “multiplications of distributions” that occur in the differential equations of physics.

Smoothing effect on the damping mechanism for an inviscid two-phase gas—liquid model

In this paper, we consider the Cauchy problem for an inviscid two-phase gas—liquid model with external force, in order to demonstrate the smoothing effect on the damping mechanism. It is shown that the Cauchy problem admits a unique global smooth solution provided that the initial data are smooth and the C0-norm of the derivative of the initial data are small.

Research on optimal radius ratio of impellers in an oxidation ditch by using numerical simulation

Impellers are the main power source for oxidation ditches. Impeller radius has an important influence on the velocity distribution and the flow field structure in oxidation ditch channels. In this paper, the relation between the size of impeller radius and the structure of flow field in an oxidation ditch was studied by using the two-phase gas–liquid model and the 3D Realizable k–ε turbulence model. The pressure implicit with splitting of operators algorithm was used for the solution of velocity and pressure. The volume of fluid method was used to simulate the free surface. The research results show that when the ratio of the impeller radius to the diameter of the oxidation ditch channel bend, r/d, is 0.218, the percentage of the fluid with velocity greater than 0.3 m/s to the entire fluid is the greatest, and the length of the backflow region in straight channels is relatively shorter. The ratio of impeller radius to the diameter of an oxidation ditch channel bend, r/d, with a value of 0.218, is called the optimal impeller radius ratio.

Well-Posedness of A Compressible Gas-Liquid Model for Deepwater Oil Well Operations

The main purpose of this paper is two-fold: (i) to generalize an existence result for a compressible gas-liquid model with a friction term recently published by Friis and Evje [SIAM J. Appl. Math., 71 (2011), pp. 2014–2047]; (ii) to derive a uniqueness result for the same model. A main ingredient in the existence part is the observation that we can consider weaker assumptions on the initial liquid and gas mass, and still obtain an existence result. Compared to the above mentioned work, we rely on a more refined application of the estimates provided by the basic energy estimate. Concerning the uniqueness result, we borrow ideas from Fang and Zhang [Nonlinear Anal. TMA, 58 (2004), pp. 719–731] and derive a stability result under appropriate constraints on parameters that determine rate of decay toward zero at the boundary for gas and liquid masses, and growth rate of masses associated with the friction term and viscous coefficient.

Two-Phase Flow Model Applied in the Oxidation Ditch

The unique hydraulic characteristics in oxidation ditch have a close relation with the quality of treated water, the design and optimization of the oxidation ditch. The experimentally validated numerical tools, based on computational fluid dynamics(CFD), were proposed. Sewage- sludge two-phase model and liquid-gas two-phase model of an oxidation ditch were built through CFD numerical method. Meanwhile, a velocity field measurement was enforced on the ditch by Acoustic Doppler Velocimeter(ADV). The simulated results and experimental data were in good correspondence, which verify the simulation methods are reasonable and the simulation results are acceptable. The combination of simulation and experimrntal measurement has profound influence on the hydraulic optimization of oxidation ditch. 3D simulation could be a good supplement for improving the hydrodynamic performance in oxidation ditch designs.

Simulation Research of Small Holes by Combined Ultrasonic and Electrochemical Machining Based on CFX

A device was designed to study the small holes by the rotary combined ultrasonic and electrochemical machining, and the gap between cathode and anode in the processing was also analyzed. A three-dimensional model of flow field was developed in ANSYS CFX software based on FEM by the gas-liquid two-phase fluid cavitations model as well as the effect of rotary cathode and the vibrated cathode to the flow field was analyzed. The simulation showed that the pressure and the velocity of the electrolyte in the gap were oscillated by additional motion of cathode, which is helpful to the electrochemical machining. The comparison of rotary electrochemical machining and the rotary combined ultrasonic and electrochemical machining showed that the rotary combined ultrasonic and electrochemical machining has better ability of making small holes than that of rotary electrochemical machining

Modeling and Simulation of Non-Newtonian Fluid MoldFilling Process with Phase Change

A gas-liquid two-phase model for the simulation of a power-law fluid mold filling process with the consideration of phase change is proposed, in which the governing equations for the melt and air in the cavity, including the mass conservation, momentum conservation and energy conservation equations, are unified into one system of equation. A revised Enthalpy method, which can be used for both the melt and air in the mold cavity, is proposed to describe the phase change during the mold filling. Finite volume method on non-staggered grid is used to solve the system. The level set method is used to capture the interface evolution during the mold filling process. The interface evolution and the distributions of physical quantities such as velocity, pressure and temperature and so on are given. The “frozen skin” layers under different temperature and velocities conditions are discussed in detail. Numerical results show that increasing the temperatures of the melt and cavity is a better way to get rid of the “frozen skin” layer than increasing the injection velocity.

Analysis on Tow-Phase Flow of WJ-4FB Explosion-Proof Scraper Waste Gas Treatment Tank Based on CFD

This paper establishes internal thermodynamic model, taking the WJ-4FB explosion-proof scraper waste gas treatment tank as the research object. And establish the gas-liquid two-phase flow simplified model based on CFD. Through the numerical simulation of the gas-liquid two-phase flow transient flow pattern, draw two-phase flow figures and gas content distribution curve .Further analysis of the gas-liquid two-phase flow, this paper obtains initial gas velocity, static level, internal mounting member and other influencing factors. It provides a theoretical basis for the design of the waste gas treatment tank.

Asymptotic behavior of a compressible two-phase model with well–formation interaction

In this work we consider a compressible gas–liquid model with a well–reservoir interaction term that is relevant for coupled wellbore-reservoir flow systems involved in e.g. drilling operations. Main focus is on deriving estimates that are independent of time. Under suitable conditions on the well–reservoir interaction term we obtain such estimates which allow prediction of the long-time behavior of the gas and liquid masses. Moreover, we also obtain a quantification of the convergence rates as a function of time and gain some insight into the role played by the rate characterizing how fast the well–reservoir interaction must die out. The model is investigated in a free boundary setting where the initial mass is a mixture of both phases, i.e., no single-phase zone exists.

Calculation Analysis of Pressure Wave Velocity in Gas and Drilling Mud Two-Phase Fluid in Annulus during Drilling Operations

Investigation of propagation characteristics of a pressure wave is of great significance to the solution of the transient pressure problem caused by unsteady operations during management pressure drilling operations. With consideration of the important factors such as virtual mass force, drag force, angular frequency, gas influx rate, pressure, temperature, and well depth, a united wave velocity model has been proposed based on pressure gradient equations in drilling operations, gas-liquid two-fluid model, the gas-drilling mud equations of state, and small perturbation theory. Solved by adopting the Runge-Kutta method, calculation results indicate that the wave velocity and void fraction have different values with respect to well depth. In the annulus, the drop of pressure causes an increase in void fraction along the flow direction. The void fraction increases first slightly and then sharply; correspondingly the wave velocity first gradually decreases and then slightly increases. In general, the wave velocity tends to increase with the increase in back pressure and the decrease of gas influx rate and angular frequency, significantly in low range. Taking the virtual mass force into account, the dispersion characteristic of the pressure wave weakens obviously, especially at the position close to the wellhead.

Weak solutions of a gas-liquid drift-flux model with general slip law for wellbore operations

In this work we study a compressible gas-liquid models highly relevant for wellbore operations like drilling. The model is a drift-flux model and is composed of two continuity equations together with a mixture momentum equation. The model allows unequal gas and liquid velocities, dictated by a so-called slip law, which is important for modeling of flow scenarios involving for example counter-current flow. The model is considered in Lagrangian coordinates. The difference in fluid velocities gives rise to new terms in the mixture momentum equation that are challenging to deal with. First, a local (in time) existence result is obtained under suitable assumptions on initial data for a general slip relation. Second, a global in time existence result is proved for small initial data subject to a more specialized slip relation.

A Compressible Two-Phase Model with Pressure-Dependent Well-Reservoir Interaction

This paper deals with a two-phase compressible gas-liquid model relevant for modeling of gas-kick flow scenarios in oil wells. To make the model more realistic we include a natural pressure-dependent well-formation interaction term allowing for modeling of dynamic gas influx/efflux. More precisely, the interaction between well and surrounding formation is controlled by a term of the form $A=q_w(P_w-P)$ which appears in the gas continuity equation where $q_w$ is a rate constant, and $P_w$ is a critical pressure, whereas $P$ is pressure in the well. Consequently, an additional coupling mechanism is added to the mass and momentum equations. We obtain a global existence result for the new model. One consequence of the existence result is that as long as the well initially is filled with a mixture of gas and liquid, the system will regulate itself (in finite time) in such a way that there does not exist any point along the well where all the gas vanishes, e.g., by escaping into the formation. Similarly, the result guarantees that neither will any pure gas region appear in finite time, despite that gas is free to enter the well from the formation as long as the well pressure $P$ is lower than the critical pressure $P_w$.