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Population balance modelling of phase inversion in liquid–liquid pipeline flows

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Abstract

Population balance equations (PBEs) along with the equal surface energy criterion are used to predict phase inversion in liquid–liquid dispersed pipeline flows. Good agreement was found between theory and experiment. Our results suggest that an ambivalent range exists in terms of distance from the inlet (rather than volume fraction) which depends on system parameters.

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... In order to describe and predict the dynamics of the layers that form when dispersed liquid-liquid flows separate, in either batch separators or continuous flow pipes, modeling work can be carried out (e.g., Hartland and Jeelani, 1988;Jeelani and Hartland, 1998;Jeelani et al., 1999;Henschke et al., 2002;Pereyra et al., 2013;Weber, 2021), and population balance equations (PBE) can be used to incorporate the effects of droplet size distribution (Cunha et al., 2008;Grimes, 2012;Grimes et al., 2012). Hu et al. (2006) developed a method to predict phase inversion in liquid-liquid flows by combining an equal surface energy criterion with PBE modelling. Weber (2021) also proposed energy consumption as a criterion for modeling liquid-liquid separation or other gravity-driven multiphase flows. ...
... For the hyperbolic blending, all model parameters were set at a constant value of 0.5, while for linear blending, when one phase reached a volume fraction of 0.7, it was considered continuous. This shifting is consistent with that of the phase inversion reported in (Hu et al., 2006). Fig. 7a shows the concentration profiles along the pipe, where only marginal changes are observed for different cases between the two blending methods. ...
Article
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The separation of liquid-liquid dispersions in horizontal pipes is common in many industrial sectors. It remains challenging, however, to predict the separation characteristics of the flow evolution due to the complex flow mechanisms. In this work, Computational Fluid Dynamics (CFD) simulations of the silicone oil and water two-phase flow in a horizontal pipe are performed. Several cases are explored with different mixture velocities and oil fractions (15%-60%). OpenFOAM (version 8.0) is used to perform Eulerian-Eulerian simulations coupled with population balance models. The ‘blending factor’ in the multiphaseEulerFoam solver captures the retardation of the droplet rising and coalescing due to the complex flow behaviour in the dense packed layer (DPL). The blending treatment provides a feasible compensation mechanism for the mesoscale uncertainties of droplet flow and coalescence through the DPL and its adjacent layers. In addition, the influence of the turbulent dispersion force is also investigated, which can improve the prediction of the radial distribution of concentrations but worsen the separation characteristics along the flow direction. Although the simulated concentration distribution and layer heights agree with the experiments only qualitatively, this work demonstrates how improvements in drag and coalescence modelling can be made to enhance the prediction accuracy.
... Nevertheless, first we studied dispersed phase flows in a pipe with hypothetical immiscible liquids whose physical properties are identical to the water. The reason to study turbulent pipe flow is: The accurate steady state result of numerical simulation is available for certain Reynolds number flows [34,41]. This ensured us that the results involve the possible least error due to the computation of flow field variables; since there was no available experimental or numerical result to compare results of PBE computations for the pipe flow, we numerically and experimentally studied flow through a static mixer with the oil-in-water dispersion of which the density and the viscosity can be assumed to be the same with the water for low holdup values of the secondary phase. ...
... The flow is characterized by the Reynolds number, Re = dw ν = 114, 000 (w stands for the bulk velocity), that is influenced by the study of Hu et al. [41] who focused on one dimensional dispersed pipe flow modeling. All presented computational results in this section have been obtained by means of an extruded (2D to 3D) unstructured mesh employing 1344 hexahedral elements at each layer. ...
Article
It can be the motion of clouds, the movement of a smoke plume, or the dynamics of fluids in processes which are interesting to food, petroleum, chemical, pharmaceutical and many other industries; they are all governed by the same physical laws: fluid dynamics and population balances. Numerical solution of Population Balance Equations (PBE) coupled to Computational Fluid Dynamics (CFD) is a promising approach to simulate liquid/gas-liquid dispersed flows, for which the governing physical phenomena are breakup and coalescence of bubbles/droplets, additional to transport phenomena of fluids. In the literature, there are many breakup and coalescence models to close the PBE. Unfortunately, there is no unified framework for these closures; and, it is one of our objectives: to determine appropriate coalescence and breakage kernels for liquid/gas-liquid dispersions. Another objective is to investigate numerical techniques for one-way coupled CFD and PBE, and to develop a computational tool. The developed tool is based on the incompressible flow solver FeatFlow which is extended with Chien's Low-Reynolds number k-epsilon turbulence model and PBE. The presented implementation ensures strictly conservative treatment of sink and source terms which is enforced even for geometric discretization of the internal coordinate. The validation of our implementation which covers a wide range of computational and experimental problems enables us to proceed into three-dimensional applications as, turbulent flows in a pipe and through static mixers. Regarding the studies on static mixers, not only we have obtained numerical results; we have conducted comprehensive experimental studies in the Sulzer Chemtech Ltd. laboratories (Winterthur, Switzerland). The inclusive experimental results has offered a good ground for verifying the adopted mathematical models and numerical techniques. The obtained satisfactory results in the studies for one-way coupled CFD and PBE has motivated us to study two-way coupled CFD-PBE models. The so far developed numerical recipe of which main ingredients are the method of classes, positivity-preserving linearization and the high-order FEM-AFC with FeatFlow including the standard k-epsilon solver has been extended to cover bubble induced turbulence and mixture-model with algebraic slip relation. A smart algorithm is developed, offering a compromise between the computational cost and the accuracy. Numerical simulation of air-in-water dispersed phase systems in a flat bubble column which is, numerically, a very challenging case-study and is experimentally studied by Becker et al. has been performed with the developed computational tool. The dynamic movement of the bubble swarm which is observed in the experiments have been successfully simulated. Keywords: computational fluid dynamics (CFD), population balances, coalescence, breakage, numerical solution, method of classes, parallel parent daughter classes, simulation, static mixers, multiphase flows.
... The hampering issue in development of phase inversion theoretical modeling is the difficulty into identifying the leading mechanism responsible for the inversion. As pointed out by Hu et al.[169], one key-step could be identified in the evolution of drop size during the process. Population balance equations (PBE)[170]describe how populations of " entities " interact with their environment, generally assumed to be continuous. ...
... Usually, detailed description of the fluid dynamics and mixing is required to solve PBE within computational fluid dynamics code[172]. Moving from this considerations, Hu et al.[169]applied population balance equations to model phase inversion in tube thus showing a qualitative agreement of the simulated droplet size with the experimental one. The existence of the ambivalent region in terms of the distance from the tube inlet, rather than function of the dispersed volume fraction, has been also predicted by their simulations. ...
Article
This review is addressed to the phase inversion process, which is not only a common, low-energy route to make stable emulsions for a variety of industrial products spanning from food to pharmaceuticals, but can also be an undesired effect in some applications, such as crude oil transportation in pipelines. Two main ways to induce phase inversion are described in the literature, i. e., phase inversion composition (PIC or catastrophic) and phase inversion temperature (PIT or transitional). In the former, starting from one phase (oil or water) with surfactants, the other phase is more or less gradually added until it reverts to the continuous phase. In PIT, phase inversion is driven by a temperature change without varying system composition. Given its industrial relevance and scientific challenge, phase inversion has been the subject of a number of papers in the literature, including extensive reviews. Due to the variety of applications and the complexity of the problem, most of the publications have been focused either on the phase behavior or the interfacial properties or the mixing process of the two phases. Although all these aspects are quite important in studying phase inversion and much progress has been done on this topic, a comprehensive picture is still lacking. In particular, the general mechanisms governing the inversion phenomenon have not been completely elucidated and quantitative predictions of the phase inversion point are limited to specific systems and experimental conditions. Here, we review the different approaches on phase inversion and highlight some related applications, including future and emerging perspectives.
... Ioannou et al. studied phase inversion and its eff ect on pressure gradient during the dispersed fl ow of two immiscible liquids for two pipe materials (steel and acrylic) and two pipe sizes [26]. In another study, in order to predict the phase inversion, the population balance equations with the equal surface energy in liquid-liquid dispersed pipeline fl ows was utilized [27]. Piela et al. performed an experimental study about phase inversion in an oil-water fl ow through a horizontal pipe loop [28]. ...
Article
Phase inversion phenomenon is important in design of liquid-liquid extraction systems. In this study, the effects of the impeller speed, the hold up of dispersed phase, and surface active agent concentration on phase inversion phenomenon for toluene-water and butanol-water with sodium dodecyl sulfate (SDS) as surface active agent, were investigated. Based on the experiments in this work, some models are developed for the impeller speeds when the phase inversion phenomenon has been occurred (NPI). Comparison between models for toluene-water and butanol-water systems with paddle and propeller blades, with and without SDS has been done. The dispersed to continuous phase ratios (volumetric ratio of toluene to water) were divided into three regions VT/VW < 0.5, 0.5 < VT/VW < 1, and VT/VW > 1. It was found that by increasing the impeller speed and also the ratio of dispersed to continuous phase the separation time and the occurrence probability of the phase inversion phenomenon was increased. Moreover, by increasing the volumetric ratio of the dispersed to the continuous phase the separation time was raised and at a specific impeller speed the separation time did not depend on the impeller speed. By increasing the concentration of SDS up to 40 mg, the phase inversion phenomenon occurred at higher impeller speeds. In the same surface active agent concentration, the impeller speed for the phase inversion occurrence by paddle blade was higher than propeller blade.
... The literature involving two phases, that is, gas-liquid flows [53,58,83,[181][182][183][184][185], liquid-liquid flows [73,159,[186][187][188][189][190][191][192][193] and solid-liquid flows [194][195][196][197][198][199][200][201], is considerably well advanced. The studies nowadays focus on enhancing closure correlations for scenarios closer to oil production (e.g., larger pipelines, viscous or non-Newtonian fluids, higher pressures) and on enhancing mathematical solution procedures. ...
Thesis
A worldwide problem reported by oil companies is the plugging of flow lines because of gas hydrates, a crystal that forms and agglomerates causing partial of complete obstructions. The hydrate management strategy consists of letting hydrates to form, but assuring its stable flow. This strategy is not yet fully applicable due to a considerable lack of description in the non-equilibrium processes of hydrate formation under multiphase flow. This thesis quantitatively describes part of these processes considering several multiscale concepts overlooked in literature so far. Hydrates are porous, hydrophilic particles that act as sponges entrapping water, where crystallization occurs mainly in the capillary walls (1st new assumption). Permeation through the porous particles furnishes water to its outer surface, promoting liquid bridge formation after particles’ collision, which leads to agglomeration (2nd new assumption). Higher subcoolings are shown to promote faster sealing-up of the particles, decreasing permeation rates and causing the particles to be inert in the agglomeration-sense, called dry particles. Furthermore, additives with surfactant properties decrease the permeation rate, which explains their anti-agglomerant effects. Several mechanisms are discussed upon modeling growth kinetics and agglomeration using population balance and further coupling with a steady-state multiphase flow model. For engineering purposes, the model is simplified into a criterion that determines stable production in oil-dominant systems, giving rise to a new dimensionless number that relates the Damköhler and the Reynolds numbers.
... A mathematical tool potentially capable of reproducing such a behavior is the population balance modeling (PBM), which has already been employed in both traditional and polymeric systems (Alvarez et al., 1994;Oshinowo et al., 2016;Li et al., 2017;Aryafar et al., 2017;Xie et al., 2018) to study particle size distributions (PSDs), but not as a tool to assess a phase inversion process. On this matter, a two-region model by Hu et al. (Hu et al., 2006) predicts phase inversion curves in O/W systems assuming the coalescence vs breakage imbalance. Regarding the PI point in HIPS, some researchers have analyzed the evolution of particles before (Díaz de León et al., 2010) and after (Soriano-Corral et al., 2013;Leal and Asua, 2009) the inversion point, but from an experimental viewpoint. ...
Article
The mechanism of the phase inversion (PI) process that occurs during the bulk polymerization of High Impact Polystyrene (HIPS) is studied in this article. Transmission electron micrographs (TEM) were obtained for different operating conditions, varying initiator concentration, temperature, and stirring speed from a previous work. Particle size distributions were retrieved from such micrographs, and were compared to theoretical predictions. To this end, a population balance model, coupled with a heterogeneous polymerization module, was developed. The evolution of particle growth, break-up and coalescence is discussed to assess the breakage/coalescence imbalance that is thought to occur at the inversion point. Results indicate that a different criterion for PI seems to be needed in this system.
... Few mathematical models have been developed to predict the PI point, the most interesting being those by Hu et al. (2005Hu et al. ( , 2006 or Brauner and Ullmann (2002) for traditional O/W systems (the former for stirred tanks and the latter for pipe flow), and perhaps a simple rheological model for polymer blends (Mekhilef and Verhoogt 1996). ...
Article
In agitated systems, the phase inversion (PI) phenomenon-the mechanism by which a dispersed phase becomes the continuous one-has been studied extensively in an empirical manner, and few models have been put forward through the years. The underlying physics are still to be fully understood. In this work, the experimental evidence published in literature is used to train machine learning models that may infer the inherent rules that lead to a given dispersion type (O/W or W/O), as well as predict the value of the dispersed phase volume fraction at the edge of the inversion point. Decision trees, bagged decision trees, support-vector machines, and multiple perceptrons are implemented and compared. Results show that it is possible to infer an ensemble of physical rules that explain why a given dispersion is O/W or W/O, where a strong "turbulence constraint" is identified. The intuitive rule that PI occurs at 50% dispersed phase almost never holds. Moreover, neural networks have shown a better performance at predicting the PI point than the other algorithms tested. Finally, a theoretical study is performed in an effort to produce a phase inversion map with the relevant operating variables. This study showed a strong nonlinear effect of the impeller-to-vessel size ratio and an asymmetrical behavior of the interfacial tension on the phase inversion points.
... Research on this line has been conducted by Arashmid and Jeffreys, [21] Bouchama et al, [14] Groeneweg et al, [92] Hu et al, [4,119] and Liu et al, [37] among others, all focusing in traditional O/W dispersions. In polymeric systems, especially in the case of polymer compounding and blending, this is also the commonly suggested mechanisms as seen in the works by Shih, [120] Mekhilef and Verhoogt, [8] Sundararaj et al, [121] and Kitayama et al. [29] An example of this mechanism in polymer-aqueous emulsions is presented in Zerfa et al. [122] As explained previously, the PI point may strongly depend on the emulsification method (gradually adding dispersed phase to a mixture is essentially different from stirring a predefined volume of two separate phases). ...
Article
The phenomenon of phase inversion occurs in liquid‐liquid dispersions found in a variety of chemical engineering fields. From simple oil‐water mixtures to complex polymeric systems, the operating variables that affect this physical phenomenon are discussed in this work. The contribution on this matter by a large number of researchers is critically assessed, outlining both coherent and conflicting results. A detailed review of the mechanisms by which phase inversion takes place is also provided. While this subject has been studied for the past fifty years, this multivariate nonlinear process is not yet comprehensively understood, and this review article aims to describe the conclusions so far reached to provide insight for future research. This article is protected by copyright. All rights reserved.
... Regarding the theoretical prediction of the inversion point, some mathematical models have been developed [32][33][34][35] but are almost exclusively for traditional oil/water, non-reacting systems. ...
Article
The phase inversion (PI) during the bulk polymerization of the styrene–polybutadiene system (high impact polystyrene manufacturing process) is empirically and theoretically studied in this article. In the experimental work, a series of reactions were performed with benzoyl peroxide as initiator and at temperatures considered of industrial interest (80°C and 90°C), varying also the reactor stirring level. PI was determined by offline viscosity measurements and verified by scanning transmission electron microscopy. The rheological behavior of each reacting system was analyzed and an empirical correlation to predict its apparent viscosity from fundamental reaction parameters was derived. This was achieved successfully for both before and after the PI point. POLYM. ENG. SCI., 2019.
... Due to the multiscale nature of this system, researchers have previously studied emulsions at macroscopic, mesoscopic and molecular scales using both deterministic and stochastic models (Gillespie 1975;Janssen & Anderson 2011). At the macroscopic level, population balance models have been employed to study the stability of emulsions and drop size distributions using semi-empirical models for drop collision rates and coalescence probabilities (Bajpai et al. 1976;Tobin et al. 1990;Taylor & Tavlarides 1994;Zhang et al. 1995;Hu et al. 2006). However, the dynamics of two drops approaching one another and possibly coalescing are studied separately by usually assuming little or no influence from other drops and particles that are present in the emulsion. ...
Article
Full-text available
The fluid dynamics of the collision and coalescence of liquid drops has intrigued scientists and engineers for more than a century owing to its ubiquitousness in nature, e.g. raindrop coalescence, and industrial applications, e.g. breaking of emulsions in the oil and gas industry. The complexity of the underlying dynamics, which includes occurrence of hydrodynamic singularities, has required study of the problem at different scales – macroscopic, mesoscopic and molecular – using stochastic and deterministic methods. In this work, a multi-scale, deterministic method is adopted to simulate the approach, collision, and eventual coalescence of two drops where the drops as well as the ambient fluid are incompressible, Newtonian fluids. The free boundary problem governing the dynamics consists of the Navier–Stokes system and associated initial and boundary conditions that have been augmented to account for the effects of disjoining pressure as the separation between the drops becomes of the order of a few hundred nanometres. This free boundary problem is solved by a Galerkin finite element-based algorithm. The interplay of inertial, viscous, capillary and van der Waals forces on the coalescence dynamics is investigated. It is shown that, in certain situations, because of inertia two drops that are driven together can first bounce before ultimately coalescing. This bounce delays coalescence and can result in the computed value of the film drainage time departing significantly from that predicted from existing scaling theories.
... In fact, PBE has been used to predict phase inversion for non-reactive systems. [33,34] In agitated vessels, the criterion adopted to estimate the nascent stages of the phase inversion process is when the coalescence rate in the circulation region exceeds the breakup rate in the impeller region. [33] In this work, a new criterion needs to be formulated to describe the main phenomena involved during the phase inversion of a typical dispersion polymerization. ...
Article
A detailed model is presented to analyze the phase inversion in the dispersion polymerizations of methyl methacrylate (MMA) in non-polar solvents. The reaction conditions under which the polymer particles lose stability and the reaction system phase inverts are investigated. At high solvent/monomer ratios, well-defined micron-sized polymer particles are produced even in the absence of stabilizer. However, low solvent/monomer ratios or stabilizer concentrations yield porous structures after a massive agglomeration of the precipitating particles. To quantify the phase inversion, a population balance equation (PBE) is derived and combined with a kinetic/thermodynamic model to develop a procedure capable of predicting the system phase inversion curve. The model predictions agree well with the experimental results.
... To make the calculations possible it should be provided with accordingly defined boundary and initial conditions. As a main numerical module it was applied modified PDECOL system (van der LINDE et al. 1997; WANG et al. 2004WANG et al. , 2004aHU et al. 2006). Adapted numerical system solves nonlinear partial differential equations of the second order for multidimensional fields. ...
Article
The need to simulate the conditions of transport of the pollutants and the dissolved oxygen in the flux of sewage arises in the modelling of sewage biodegradation process in gravitational sewer system. In the properly kept sewers, sewage flow velocity is generally uniform, which allows the model describing the pollutant transport to be simplified. The application of further simplifying assumptions, justified by the conditions inside the modelled object, allows us to get the credible results with the acceptable efforts. These simplifications can be considered, depending on the costs of data for model calibration measurement and the numerical calculation costs. The present work is focused on finding the most convenient description of transport and transformation of dissolved and suspended substances by the advection–dispersion formulas, being one-dimensional description of sewage flow. The literature review concerning various treatment of longitudinal dispersion coefficient was presented. The influence of flow conditions in the modelled system on the dispersion coefficient was also shown. Finally, the matrix description of pollutant fraction transport and biodegradation in gravitational sanitation conduit based on advection–dispersion equation, along with the example of numerical procedure, was described.
... Hence the phase inversion was assumed to occur when the coalescence frequency exceeded that of breakup. Hu et al. [16] recommended population balance equations (PBEs) along with the equal surface energy criterion to predict the phase inversion in liquid-liquid dispersed pipeline flows. The mathematical model predicted the phase inversion at b ¼ 71%, and this value is in excellent agreement with that from the experimental investigations (i.e. ...
Article
The kerosene–water flow in vertical and inclined pipes of 77.8 mm inner diameter and 4500 mm length has been investigated numerically. Simulations were carried out at three inclination angles of 0°, 5° and 30° from vertical for superficial water velocities in the range of 0.29–1.6 m/s and volumetric qualities in the range of 9.2–65.5%. Results from the simulations have been compared with the corresponding experimental data from our previous study to check the suitability of meshing, physical models and boundary conditions used. The results from CFD predictions show a reasonable agreement with the experimental data for the vertical pipe. Some discrepancies have been observed near the walls for inclined pipes as the flow becomes more complex with the appearance of drops swarms. The same computational domain was then used to investigate the effect of a higher volumetric quality and superficial water velocity on the flow characteristics. The axisymmetrical distribution of the volume fraction, water velocity and drops velocity which were observed in the vertical pipe has changed into asymmetrical distribution in the inclined pipe. It has also been observed that increasing the superficial water velocity and volumetric quality modifies the distributions of flow parameters due to the movement of kerosene drops toward the lower part of the pipe. The results from CFD predictions on the volume fraction distribution at 30° inclination indicate the appearance of phase inversion phenomenon when the volumetric quality becomes greater than 65%.
... The bubble size distribution in bubble columns is largely due to breakage, which in turn determines the character-istics of the flow field in the column (Jakobsen, Lindborg, & Dorao, 2005 ). Furthermore, breakage is very important in emulsion technology determining not only the droplet size distribution but in some cases even the phase inversion point (Hu, Matar, Hewit, & Angeli, 2006). The dynamics of a particle population undergoing breakage is described by the breakage equation that belongs to the more general class of the population balance equations. ...
Article
Computers and Chemical Engineering j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / c o m p c h e m e n g a b s t r a c t The breakage equation is of great significance for modeling many physicochemical processes. The need for its extension to more than one internal coordinates in spatially distributed environment renders crucial the development of efficient numerical methods for its solution. In the present work two new sectional methods (Cell Average Technique and Extended Cell Average Technique) recently applied to the coagula-tion equation are implemented to breakage equation and tested extensively against the well-established Fixed Pivot Technique. The results of the analysis show that whereas the new methods cannot predict the complete particle size distribution better than the Fixed Pivot Technique (despite their superiority in the case of coagulation), they are very successful in predicting the moments of the distribution even for coarse grids. Thus, especially the Extended Cell Average Technique can be considered as a refinement of the moments method with increased number of degrees of freedom but also increased accuracy.
... For this reason, the open-source software package FeatFlow extended with Chien's Low-Reynolds number k − ε model was utilized to perform the flow simulations, which has already been successfully validated for channel flow problems (Re τ = 395) [20]. The flow considered here is characterized by the Reynolds number, Re = dw ν = 114, 000 (w stands for the bulk velocity), what was influenced by the study of Hu et al. [14] focused on one dimensional dispersed pipe flow modeling. All computational results presented in this section are obtained by means of an extruded (2D to 3D) unstructured mesh employing 1344 hexahedral elements in each of its layers. ...
Article
Full-text available
In this paper, we present numerical techniques for one-way coupling of CFD and Population Balance Equations (PBE) based on the incompressible flow solver FeatFlow which is extended with a) Chien's Low-Reynolds number k − ε turbulence model [4], b) breakage kernel model of Lehr et al. [34, 35], c) coalescence kernel model of Lehr et al. [34]. The presented implementation ensures a strictly conservative treatment of sink and source terms – arising due to breakage and/or coalescence – which is enforced even for geometric discretization of the internal coordinate. Direct enforcement of the aforementioned property is achieved by formulating the population balance equation with respect to class holdups. The validation of our implementation which covers a wide range of computational and experimental problems enables us to proceed into a three dimensional application for a turbulent pipe flow. The aim of this paper is to highlight the influence of different formulations of the novel theoretical breakage [29] and coalescence models on the equilibrium distribution of the population, to reveal some misinterpretations in the literature and, ultimately, to propose an implementation strategy for the three-dimensional one-way coupled CFD-PBE model.
... The flow considered here is characterized by the Reynolds number, Re = dw ν = 114, 000 (w stands for the bulk velocity), what was influenced by the study of [14] focused on one dimensional dispersed pipe flow modeling. All computational results presented in this section are obtained by means of an extruded (2D to 3D) unstructured mesh employing 1344 hexahedral elements in each of its layers. ...
Article
In this paper, we present numerical techniques for one-way coupling of CFD and Popu-lation Balance Equations (PBE) based on the incompressible flow solver FeatFlow which is extended with Chien's Low-Reynolds number k − ε turbulence model, and breakage and coalescence closures. The presented implementation ensures strictly conservative treatment of sink and source terms which is enforced even for geometric discretization of the internal coordinate. The validation of our implementation which covers wide range of computa-tional and experimental problems enables us to proceed into three-dimensional applications as, turbulent flows in a pipe and through a static mixer. The aim of this paper is to high-light the influence of different formulations of the novel theoretical breakage and coalescence models on the equilibrium distribution of population, and to propose an implementation strategy for three-dimensional one-way coupled CFD-PBE model.
Article
Horizontal and nearly horizontal liquid-liquid two-phase flows are widely encountered in petroleum, chemical, and other important industrial production fields. The three-dimensional (3D) flow visualization and the local parameter measurement are of great significance for understanding the flow dynamics and revealing the flow-pattern transition mechanism. In this paper, experiments of horizontal and nearly horizontal (+5°) liquid-liquid flow are carried out in an acrylic pipe with an inner diameter of 20 mm. Silicone oil (organic phase) and water/glycerol mixture (aqueous phase) are used as the test fluid. A novel planar laser-induced fluorescence (PLIF) system is designed to detect the flow structures. The incident direction of the laser sheet is along the radial section of the pipe, and the PLIF camera is positioned at 60° to the plane of the laser sheet, referred to as PLIF-60. In the experiment, the two-dimensional (2D) flow images at the pipe radial section are captured by the PLIF-60 system. Considering the effect of the angle between the laser sheet and the PLIF camera, the flow images are corrected to remove the horizontal and vertical distortions. The liquid-liquid stratified interface and the entrained droplets are identified in the corrected flow images. A droplet tracking method is proposed to derive the droplet diameter and its quantity distribution. Quasi-3D flow structures are reconstructed upon the accurate detection of the stratified interface and the entrained droplets. The evolution characteristics of the interface structures and the spatial distributions of droplets are studied. In addition, aqueous phase holdup and droplet entrainment ratios at pipe axial and radial sections are extracted based on the quasi-3D flow visualizations. The effects of the flow conditions and the pipe inclinations on the profiles of the aqueous phase holdup and the droplet entrainment ratio are uncovered.
Conference Paper
In this Thesis a novel approach is followed to facilitate the experimental investigations on the flow pattern transitions from separated to dispersed flows using a cylindrical bluff body in horizontal oil-water flows. A transverse cylindrical rod is used as a bluff body which is placed under the interface of the two immiscible liquids and near the test section inlet to passively generate interfacial perturbations and breaking waves. This approach was inspired from the use of hydrofoils in ships that reduce frictional drag via increased air entrainment. Studies are carried out using two flow facilities and high speed imaging combined with laser based measurements are performed at two axial locations along the test section, immediately after the cylinder and at large distance away from the cylinder. The effect of a confined geometry on the characteristics of the von Karman vortices and on the general flow behaviour immediately downstream of the cylinder are investigated in single phase water flows. It is found that the 3D pipe geometry does not affect significantly the vortex shedding behind the cylinder at least in the central plane of the pipe. The frequencies of the vortex shedding were comparable to those from a cylinder in an unconfined liquid. The results from two phase flows reveal that the cylinder reduces the mixture velocity for the transition separated to dispersed flows. It also actuates interfacial waves that are found to be non-linear and convective. In many cases the waves have the same frequencies as the von Karman vortices depending on the submergence depth of the cylinder underneath the oil-water interface and on the Froude number of the water layer. The observations suggest that strongly non-linear waves are responsible for forming thin ligaments that eventually break up into droplets.
Article
Full-text available
The effect of a cylindrical bluff body on the interface characteristics of stratified two-phase, oil-water, pipe flows is experimentally investigated with high speed Particle Image Velocimetry (PIV). The motivation was to study the feasibility of flow pattern map actuation by using a transverse cylinder immersed in water in the stratified pattern, and particularly the transition from separated to dispersed flows. The cylinder has a diameter of 5 mm and is located at 6.75 mm from the pipe top in a 37 mm ID acrylic test section. Velocity profiles were obtained in the middle plane of the pipe. For reference, single phase flows were also investigated for Reynolds numbers from 1550 to 3488. It was found that the flow behind the cylinder was similar to the two dimensional cases, while the presence of the lower pipe wall diverted the vorticity layers towards the top. In two-phase flows, the Froude number (from 1.4 to 1.8) and the depth of the cylinder submergence below the interface affected the generation of waves. For high Froude numbers and low depths of submergence the counter rotating von Karman vortices generated by the cylinder interacted with the interface. In this case, the vorticity clusters from the top of the cylinder were seen to attach at the wave crests. At high depths of submergence, a jet like flow appeared between the top of the cylinder and the interface. High speed imaging revealed that the presence of the cylinder reduced to lower mixture velocities the transition from separated to dual continuous flows where drops of one phase appear into the other.
Chapter
This chapter is concerned with the design of high-resolution finite element schemes satisfying the discrete maximum principle. The presented algebraic flux correction paradigm is a generalization of the flux-corrected transport (FCT) methodology. Given the standard Galerkin discretization of a scalar transport equation, we decompose the antidiffusive part of the discrete operator into numerical fluxes and limit these fluxes in a conservative way. The purpose of this manipulation is to make the antidiffusive term local extremum diminishing. The available limiting techniques include a family of implicit FCT schemes and a new linearity-preserving limiter which provides a unified treatment of stationary and time-dependent problems. The use of Anderson acceleration makes it possible to design a simple and efficient quasi-Newton solver for the constrained Galerkin scheme. We also present a linearized FCT method for computations with small time steps. The numerical behavior of the proposed algorithms is illustrated by a grid convergence study for convection-dominated transport problems and anisotropic diffusion equations.
Article
Phase inversion is the phenomenon where the continuous phase of a liquid-liquid dispersion changes to become dispersed and the dispersed becomes continuous. Phase inversion has important implications for a number of industrial applications where liquid-liquid dispersions are used, since the change in the mixture continuity affects drop size, settling characteristics, heat transfer and even the corrosion behaviour of the mixture. In pipeline flows, phase inversion is usually accompanied by a step change or a peak in pressure drop. The chapter reviews the work on phase inversion during the pipeline flow of liquid-liquid mixtures when no surfactants are present. Investigations have revealed that in pipes a transitional region occurs during inversion from one phase continuous to the other, characterized by complex flow morphologies (multiple drops, regions in the flow with different continuity) and even stratification of the two phases over a range of dispersed phase volume fractions. The observations on the phase inversion process in pipelines are discussed and the parameters which affect the phenomenon are summarized. In addition, the various models available for predicting phase inversion are analyzed, as well as the methodologies developed to account for the transitional region with the complex morphologies and the flow stratification and to predict pressure drop during inversion.
Chapter
The quality of emulsions and the efficiency of emulsification processes depends on the composition and physical/interphasial properties of both phases, in other words on formulation, and processing/hydrodynamic conditions. In surfactant free systems, physical properties are practically independent of hydrodynamic conditions and the coalescence/breakage models are sufficient for many applications. In the presence of surface active agents the interfacial properties become dynamic and they frequently vary with intensity of processing conditions and with the processing time. These variations are result of adsorption/desorption of a surfactant and they affect interfacial rheology and stability of interface. Current understanding of the interactions between hydrodynamic conditions in the bulk of both continuous and dispersed phase and conditions on the interface separating them is very limited and further research in this area is necessary. This chapter discusses the aspects of formulation engineering related to the manufacturing of emulsions and the equipment used in industrial emulsification processes.
Article
This work investigated the effects of breakage and coalescence on de-oiling hydrocyclone performance utilizing Computational Fluid Dynamics (CFD). It was found that an increase in the entrance flow rate with low entrance oil concentration not only did not increase the separation performance of the hydrocyclone but it also decreased the separation efficiency. On the contrary, the hydrocyclone performance was enhanced with increasing the inlet velocity. Furthermore, a comparison between the standard design and the conical inlet chamber design was drawn in terms of separation efficiency for low entrance oil concentration; the results depicted that the conical design had higher separation efficiency.
Article
Phase inversion refers to the phenomenon whereby a small change in operational flow conditions causes an oil-in-water dispersed flow pattern to suddenly switch to a water-in-oil flow pattern, and viceversa. This paper proposes an interpretation of phase inversion in terms of minimal dissipation rate. To this end, the dissipation rate is computed by a simple homogeneous model together with available correlations for effective viscosity in dispersed flows. It is shown that the data available in the literature can be reasonably interpreted as a manifestation of minimal dissipation rate. Furthermore, if the assumed effective viscosity correlations take into account pipe wettability, the minimal dissipation rate approach is capable to interpret also the so-called ambivalent range (hysteresis effect) and correlate the available data.
Article
In oil-water two-phase dispersed flow, phase inversion may occur when the continuous phase becomes dispersed. This phenomenon, which controls the nature of the phase in contact with the pipe, has a great importance on the corrosion and on the pressure drop, which dramatically affects the delivery ability and operational modality. It is therefore imperative for the phase inversion research to be taken into consideration. However, most of the knowledge on phase inversion is for light mineral oil with low viscosity, few research focuses on high viscosity oil-water phase inversion. Arirachakaran et al. (1989, "An Analysis of Oil/Water Flow Phenomena in Horizontal Pipes," SPE Professional Product Operating Symposium, Oklahoma, SPE Paper No. 18836) found that critical water fraction when inversion occurred was dramatically reduced with the increment of oil viscosity, and the existing phase inversion models are invalidated. In this paper, an experimental study has been made of high viscosity mineral oil-water flow through a horizontal pipe loop. Results indicate that phase inversion for oil phase with high viscosity occurs much earlier than low viscosity oil, and phase inversion tends to be delayed, with the increment in experimental temperature. The influence of mixture velocities on the inversion process could be neglected in the range of mixture velocities that we studied. As well, inversion point obtain by our experiment are best predicted by the correlation of Arirachakaran et al. (1989, "An Analysis of Oil/Water Flow Phenomena in Horizontal Pipes," SPE Professional Product Operating Symposium, Oklahoma, SPE Paper No. 18836). Models of Decarre and Fabre (1997, "Phase Inversion Prediction Study," Rev. Inst. Fr. Pet., 52, pp. 415-424) and Braunerand Ullmann (2002, "Modeling of Phase Inversion Phenomenon in Two-Phase Pipe Flows," Int. J. Multiph. Flow, 28, pp. 1177-1204), based on minimization of system total energy, seem to be invalidated for high viscosity oil.
Article
The breakage of droplets in turbulent pipe flow is a subject of great technological interest. The purpose of this work is to take advantage of state-of the-art theoretical developments and obtain quantitative predictions enhancing our physical understanding of this problem. From this perspective, the subject of the “exact” solutions to an appropriate mathematical model, being recently an issue in the literature, is clarified. Furthermore, the simplest possible mathematical model of the breakage process is derived that retains the complete process parameter dependencies. To implement the model, an appropriate algebraic approach is proposed for the prediction of the radial profiles of key turbulent flow field parameters. This mathematical model adequately relates the breakage pattern in the pipe with the operating process parameters. The results suggest that droplet breakage occurs almost exclusively in an annular region at the periphery of the pipe. Droplets larger than a certain critical size initially have a tendency to break quite rapidly in that region, whereas turbulent diffusion seems to be comparatively slow and cannot spread the fragments into the bulk. This interplay of breakage and diffusion mechanisms seems to lead to initially nonuniform lateral droplet number concentration profiles with rather strong axial (or equivalent time) dependence, over distances of practical significance for process plants. This work clearly suggests that the often-made assumption of uniform conditions in the pipe cross section, regarding size distribution of the droplets that are undergoing breakage, should be reconsidered.
Article
Liquid–liquid flow literature proposes models developed to predict quantities and phenomena of interest, once given fluid properties and the features of the flow domain. The validity of any model should be verified through experimental observations, being this practice an effective way to evaluate the model conditions of applicability and possible limitations. Despite the fact that several works have already been proposed on the validation of theoretical models, most of them concern liquids characterised by low viscosity ratio , while in industrial realities (such as petroleum or food ones) the liquids involved are often characterised by high viscosity ratios. The extension of low- results to high- flows is not straightforward, so that it is necessary to validate the models for the latter case specifically. This work presents experimental pressure drops and flow-pattern maps associated to the flow of oil and water in horizontal and slightly inclined pipe, where the chosen liquids are characterised by an oil-to-water viscosity ratio of about 800:1 at 20 °C. Various theoretical models have been considered, with particular attention to core-annular flow two-fluid model and oil-in-water dispersion homogeneous no-slip model for the prediction of associated pressure drops, and flow-pattern map transition criteria involving the regimes encountered in the experimental tests. The theoretical predictions have been then compared to the experimental results. A satisfactory agreement has been found especially as concerns pressure drop comparisons. As regards the predicted transition boundaries superimposed on the corresponding flow-pattern maps, the ‘free’ parameters have been fitted on the basis of experimental results and observations, and the final agreement is good in the prediction of both the core-annular flow region of existence and the transition to oil-in-water dispersion. No conclusion can be expressed on transition criteria involving stratified flow, which only seldom has been observed in the performed experiments.
Article
We report a simple evaporation process using liquid marbles as precursors to produce high sphericity, precisely diameter controlled polyelectrolyte microspheres. We use poly(diallyldimethylammonium chloride) (PDDA) as the test polyelectrolyte for this experimental study. We present measurements of the rate of mass loss during evaporation to demonstrate evidence of two limiting physical processes. At short times, the rate of mass loss is well described by the "D(2) law" regime, which is vapor diffusion limited. At long times, the rate of water diffusion inside the nearly solid polyelectrolyte microsphere becomes the rate-limiting step. The transition between these two limiting processes is accompanied by changes in the physical morphology inside the microsphere. We compare the estimated values of the water diffusion coefficients with the values reported in the literature to demonstrate good agreement.
Article
Sectional (zero order) methods constitute a very important class of methods for the solution of the population balance equation offering distinct advantages compared to their competitors, namely, higher order and moment methods. For the last ten years a particular sectional method, the so-called fixed pivot technique has been the most extensively used in the scientific community for the solution of the coagulation equation because it offers arbitrary grid choice and conservation of two moments of the particle size distribution. Very recently, a new method (called cell average technique) has been developed which gives more accurate results than the fixed point technique. In the present work, the extension of this new method in order to conserve three moments is attempted. A stable algorithm for the solution of the coagulation equation is developed. Although the new method allows improved computation of moments of practical interest, this is not always the case with respect to complete particle size distribution.
Article
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Phenomenological models are proposed to describe drop breakup and coalescence in a turbulently agitated liquid-liquid dispersion. Based on these models, breakage and coalescence rate functions are developed and used to solve the general population balance equation describing drop interactions in a continuous flow vessel. Parameters of the models are evaluated by comparison with experimental data on drop size distributions and mixing frequencies obtained in a continuous flow vessel over a range of operating conditions. The favorable agreement between experimental observation and the model are encouraging that the model is suitable for predicting dispersion properties such as drop size distributions, interfacial areas and mixing frequencies.
Article
A review is made on the significant developments over the past 40 years of phase inversion research. The main mechanisms of drop coalescence and break-up are discussed in detail. Further, studies investigating the effect of various physical, geometrical and flow parameters on phase inversion are covered.
Article
The population balance equation considers the change of the bubble size and the bubble number density due to coalescence and break-up in bubbly flows. Although a number of theoretical models exist for coalescence and break-up rates, nearly no experimental data for the validation of these models exist for turbulent flow with high void fraction. To overcome this lack, vertical air–water pipe flows were examined. A comprehensive validation of the population balance equation is based on the knowledge of the development of the number density distribution function and of the liquid flow turbulence along the pipe flow. The experimental and analytical procedure to obtain these parameters is presented. The void fraction, interfacial area density and mean bubble volume were calculated as statistical moments of the number density distribution function. For its measurement a combination of single and double fibre optical probes was used that allowed measurement in high void fraction flows. X-hotfilm probes were used to measure the liquid flow turbulence level of the two-phase flow. The flow inlet condition was influenced with the aid of turbulence grids. The results show that the number of coalescence and break-up events depend strongly on the number of bubbles per unit volume and the on bubble volume that differ even for the same gas and liquid superficial velocities. From the axial development of the number density distribution function coalescence and break-up events could be identified within certain bubble volume ranges and the source terms of the population balance equation could be quantified. In a first step the data were used to check two existing theoretical models with respect to coalescence and break-up rates.
Article
Phase inversion and its associated phenomena are experimentally investigated in co-current upward and downward oil-water flow in a vertical stainless steel test section (38 mm I.D.). Oil (ρo=828 kg/m³, µo=5.5 mPa s) and tap water are used as test fluids. Two inversion routes (w/o to o/w and o/w to w/o) are followed in experiments where either the mixture velocity is kept constant and the dispersed phase fraction is increased (type I experiments), or the continuous phase flow rate is kept constant and that of the dispersed phase is increased (type II experiments). By monitoring phase continuity at the pipe centre and at the wall it was found that phase inversion does not happen simultaneously at all locations in the pipe cross-section. In type I experiments, the velocity ratios (Uo/Uw) where complete inversion appeared acquired the same constant value in both flow directions, although the phase inversion points, based on input phase fractions, were different. In contrast to previous results in horizontal flows, frictional pressure gradient was found to be minimum at the phase inversion point. The interfacial energies of the two dispersions before and after phase inversion, calculated from the measured drop sizes, were found to be different in contrast to the previously suggested criterion of equal energies for the appearance of the phenomenon. In type II experiments the phase inversion point was found to depend on mixture velocity for low and medium velocities but not for high ones. In all cases studied an ambivalent region, commonly reported for inversion in stirred vessels, was not observed.
Article
Breakage processes are considered in the absence of agglomeration or coagulation. A new method is proposed, based on a population balance type of formulation, applicable to systems (such as dispersions) that may be characterized by a maximum stable particle size. In this method, considerable simplification is achieved by means of a transformation that effectively eliminates the breakage frequency, thus allowing the convenient computation of steady state through solution of an integral equation. To compute the steady state, apart from the maximum size and the breakage kernel, only an estimate of the initial distribution is required. Two functional forms of binary breakage kernels which can represent a large variety of possible breakage mechanisms are proposed (by an appropriate selection of parameter values). For the sake of completeness, analytical solutions are also presented for several, relatively simple kernels. Finally, a study is made to assess the influence of initial conditions on the steady-state size distribution, which is helpful in tackling the inverse problem of determining the breakage kernel using limited experimental data.
Article
A theoretical model for the prediction of drop and bubble (fluid-particle) breakup rates in turbulent dispersions was developed. The model is based on the theories of isotropic turbulence and probability and contains no unknown or adjustable parameters. Unlike previous work, this model predicts the breakage rate for original particles of a given size at a given combination of the daughter particle sizes and thus does not need a predefined daughter particle size distribution. The daughter particle size distribution is a result and can be calculated directly from the model. Predicted breakage fractions using the model for the air–water system in a high-intensity pipeline flow agree very well with the available 1991 experimental results of Hesketh et al. Comparisons of the developed model for specific particle breakage rate with earlier models show it to give breakage-rate values bracketed by other models. The spread in predictions is high, and improved experimental studies are recommended for verification.
Article
The evolution of droplet or bubble size distribution in turbulent flow is of great significance in a variety of technological fields. Modeling this evolution by employing a population balance approach requires knowledge of the so-called breakage functions (rate and kernel). Over the years, a large number of phenomenological breakage functions with various degrees of sophistication have been proposed in the literature. Among them, those based on the statistical theory of turbulence are of particular interest, in that they attempt to take into account the structure of the flow field responsible for breakage. The purpose of the present work is to present a unified framework for developing this type of breakage functions and to show how existing models can be derived in a systematic and consistent way. The key parameters in this modeling approach are identified, which have to be determined by comparison with experimental data. It is also shown that the breakage functions, obtained within the framework presented here, lead to predictions of a droplet size evolution whose main features are consistent with experimental observations. It is suggested that this framework is an important step toward the development of a standard approach for modeling droplet size evolution in turbulent flow.
Article
A non-intrusive dye tracing technique, laser-induced fluorescence (LIF), has been applied to investigate the co-current flow of two immiscible organic–aqueous liquid flows in a vertical pipe. This technique allowed detailed visualization of the dynamic evolution of the flows. Flow structures in liquid–liquid flows at high dispersed phase fraction were revealed which had not been seen before. In pipe flow, an unstable range was found in the flow pattern map in which oil-in-water (o/w) and water-in-oil (w/o) dispersions could co-exist. Secondary dispersions (o/w/o and w/o/w) were observed for most volume fractions and velocities, especially in the unstable range. It was observed that, when the flow is in the unstable region, both w/o/w and o/w/o secondary dispersions could appear in the same set of experiments. It was found that this unstable range in the pipe flow, in spite of the similar appearance, was different to the ambivalent range seen in agitated systems. The one-dimensional drift flux model of Wallis (1969) for dispersed flow and a laminar model for co-current downward annular flow, were also applied to predict the in situ oil holdup; good agreement was obtained.
Article
Phase inversion in oil–water flow systems corresponds to the transitional boundary between oil-in-water dispersion and water-in-oil dispersion. In this study, the criterion of minimum of the system free energy is combined with a model for drop size in dense dispersions to predict the critical conditions for phase inversion. The model has been favorably compared with available data on the critical holdup for phase inversion. It also provides explanations of features of phase inversion phenomena in liquid–liquid pipe flows and in static mixers.
Article
There is growing experimental and theoretical evidence that in common flow fields, such as stirred vessels and pipelines, the steady state of the dispersed phase size distribution (including the maximum stable size Xm) may be unattainable over a time period of practical interest. Therefore, computation of the temporal evolution of the dispersion (through the breakage population balance equation) is indispensable. The necessary breakage rate and breakage kernel can be determined from experimental data by solving the so-called inverse problem. To tackle the latter, an improvement of the method originally developed by Sathyagal et al. (Comput. Chem. Eng. 19 (1995) 437) is presented in this paper. The method is based on the concept of self-similarity of size distributions, which is shown here to hold even if the evolving maximum particle size is relatively close to an existing maximum stable size Xm. The proposed improved inversion procedure relies on the observation that the form of the breakage kernel can be inferred from the form of the self-similar distribution representing the experimental data. The new method is very stable with respect to noise in the experimental data.
Article
Phase inversion in agitated vessels was studied using a two-region model. In this model, breakup and coalescence were assumed to take place in the vicinity of the impeller and away from that region, respectively. The mechanism of phase inversion was regarded as the result of an imbalance between the breakup and coalescence processes. Hence phase inversion was assumed to occur when the coalescence frequency exceeded that of breakup. In addition, the concept of a radial distribution function was adopted in the model in order to account for droplet coalescence in concentrated dispersions. Using the two-region model, the effect of interfacial tension, viscosity, density and impeller size on the width of the ambivalent range was investigated. The predictions agree well with experimental data particularly for the upper curve of the ambivalent range; however, the organic phase fraction of the lower curve is in some cases underestimated by the model.
Article
A simulation model has been developed to model drop populations in a stirred tank. A multiblock stirred tank model has been used with the drop population balance equations developed in the literature. The stirred tank is modeled separately so that local turbulent energy dissipation values and fluid flows are used in the drop breakage and coalescence functions. This model has several attractive features, e.g. it can predict the inhomogenity of dispersions and some scale-up phenomena. Because local conditions can be used in the drop rate functions needed in the population balances, it is possible to take these fundamental processes into closer examination. It seems that the parameter values in the drop breakage and coalescence models depend on flow and turbulence averaging for the vessel. This proposes that for “intrinsic” drop breakage and coalescence rates, a multiblock model for the stirred tank is needed in parameter estimation as well. The stirred tank flow model may be obtained from measurements or from computational fluid dynamics simulations in a straightforward manner.
Article
Phase inversion and its effect on pressure gradient during the dispersed flow of two immiscible liquids have been studied for two pipe materials (steel and acrylic) and two pipe sizes (60 and 32 mm ID). Water and oil (796 kg m−3 density and 2.19 mm2 s−1 viscosity) were used as test fluids while the appearance of phase inversion in the acrylic pipes was confirmed with the use of impedance ring probes. In the large pipes (steel and acrylic with 60 mm ID) it was found that the phase inversion point (oil volume fraction where inversion appears) depended on whether the inversion was from oil to water continuous mixture or from water to oil. The difference in the phase inversion points between the two dispersion initialisation conditions increased with increasing mixture velocity. No effect of initial conditions on the inversion point was found for the small acrylic pipe.Phase inversion was in all cases preceded by a large increase in pressure gradient, which was sharply reduced immediately after the new continuous phase was established. The pressure gradient peak was sharper and larger at high mixture velocities than at low ones and in the acrylic pipe compared to the steel one. The change in phase continuity lasted a few minutes and was accompanied by large fluctuations in pressure gradient and mixture impedance.
Article
A speculative study on the conditions under which phase inversion occurs in agitated liquid-liquid dispersions is conducted using a Monte Carlo technique. The simulation is based on a stochastic model, which accounts for fundamental physical processes such as drop deformation, breakup, and coalescence, and utilizes the minimization of interfacial energy as a criterion for phase inversion. Profiles of the interfacial energy indicate that a steady-state equilibrium is reached after a sufficiently large number of random moves and that predictions are insensitive to initial drop conditions. The calculated phase inversion holdup is observed to increase with increasing density and viscosity ratio, and to decrease with increasing agitation speed for a fixed viscosity ratio. It is also observed that, for a fixed viscosity ratio, the phase inversion holdup remains constant for large enough agitation speeds. The proposed model is therefore capable of achieving reasonable qualitative agreement with general experimental trends and of reproducing key features observed experimentally. The results of this investigation indicate that this simple stochastic method could be the basis upon which more advanced models for predicting phase inversion behavior can be developed.
Theoretical prediction with PBE of phase inversion in dispersed two-phase liquid systems
  • B Hu
  • K Ioannou
  • O K Matar
  • G F Hewitt
  • P Angeli
Hu, B., Ioannou K., Matar, O.K., Hewitt, G.F., Angeli, P., 2004. Theoretical prediction with PBE of phase inversion in dispersed two-phase liquid systems. Third International Symposium on Two-Phase Flow Modelling and Experimentation, Pisa, 22–24 September 2004.
Intermittent three-phase flow of oil, water and gas in horizontal pipes
  • M Nädler
  • D Mewes
Nädler, M., Mewes, D., 1995. Intermittent three-phase flow of oil, water and gas in horizontal pipes. Proceedings of the Fifth International Offshore and Polar Engineering Conference, pp. 72–80.
Phase inversion in dispersed liquid-liquid pipe flows
  • K Ioannou
  • B Hu
  • O K Matar
  • G F Hewitt
  • P Angeli