Modelling Extraction in Microchannels with Stratified Flow: Channel Geometry, Flow Configuration and Marangoni Stresses
TL;DR: In this article, a hierarchy of mathematical models for extraction in microchannels with two-phase stratified flow is presented, and conditions under which a simpler 1D model can replace the complex 2D model are identified.
Abstract: In this work, we present a hierarchy of mathematical models for extraction in microchannels with two-phase stratified flow. A flat stable inter-fluid interface is considered. We present three models for rectangular channels, of varying degrees of complexity: 2D, 1D Laminar and 1D Plug models. The predictions of these models are compared with each other and with experiments reported in the literature. Conditions under which a simpler 1D model can replace the complex 2D model are identified. Next, a detailed model for channels with a circular cross-section is developed using bipolar cylindrical coordinates. We also show how a much simpler description can be obtained by applying the 1D rectangular channel models to a circular channel. The relative performance of co-current and counter-current flow configurations is analysed next, using the 1D Plug flow model. Finally, the model is extended to account for Marangoni stresses that result from the variation of interfacial tension with solute concentratio...
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TL;DR: In this article, the instability of layered two-phase flows caused by the presence of a soluble surfactant (or a surface-active solute) is studied, and a linear stability analysis is performed, using a combination of asymptotic and numerical methods.
Abstract: In this paper, the instability of layered two-phase flows caused by the presence of a soluble surfactant (or a surface-active solute) is studied. The fluids have different viscosities, but are density matched to focus on Marangoni effects. The fluids flow between two flat plates, which are maintained at different solute concentrations. This establishes a constant flux of solute from one fluid to the other in the base state. A linear stability analysis is performed, using a combination of asymptotic and numerical methods. In the creeping flow regime, Marangoni stresses destabilize the flow, provided that a concentration gradient is maintained across the fluids. One long-wave and two short-wave Marangoni instability modes arise, in different regions of parameter space. A well-defined condition for the long-wave instability is determined in terms of the viscosity and thickness ratios of the fluids, and the direction of mass transfer. Energy budget calculations show that the Marangoni stresses that drive long- and short-wave instabilities have distinct origins. The former is caused by interface deformation while the latter is associated with convection by the disturbance flow. Consequently, even when the interface is non-deforming (in the large-interfacial-tension limit), the flow can become unstable to short-wave disturbances. On increasing the Reynolds number, the viscosity-induced interfacial instability comes into play. This mode is shown to either suppress or enhance the Marangoni instability, depending on the viscosity and thickness ratios. This analysis is relevant to applications such as solvent extraction in microchannels, in which a surface-active solute is transferred between fluids in parallel stratified flow. It is also applicable to the thermocapillary problem of layered flow between heated plates.
18 citations
TL;DR: In this article, the instability of layered two-phase flows caused by the presence of a soluble surfactant (or a surface active solute) is studied, and a linear stability analysis is performed, using a combination of asymptotic and numerical methods.
Abstract: In this paper, the instability of layered two-phase flows caused by the presence of a soluble surfactant (or a surface active solute) is studied. The fluids have different viscosities, but are density matched to focus on Marangoni effects. The fluids flow between two flat plates, which are maintained at different solute concentrations. This establishes a constant flux of solute from one fluid to the other in the base state. A linear stability analysis is performed, using a combination of asymptotic and numerical methods. In the creeping flow regime, Marangoni stresses destabilize the flow, provided a concentration gradient is maintained across the fluids. One long wave and two short wave Marangoni instability modes arise, in different regions of parameter space. A well-defined condition for the long wave instability is determined in terms of the viscosity and thickness ratios of the fluids, and the direction of mass transfer. Energy budget calculations show that the Marangoni stresses that drive long and short wave instabilities have distinct origins. The former is caused by interface deformation while the latter is associated with convection by the disturbance flow. Consequently, even when the interface is non-deforming (in the large interfacial tension limit), the flow can become unstable to short wave disturbances. On increasing $Re$, the viscosity-induced interfacial instability comes into play. This mode is shown to either suppress or enhance the Marangoni instability, depending on the viscosity and thickness ratios. This analysis is relevant to applications such as solvent extraction in microchannels, in which a surface-active solute is transferred between fluids in parallel stratified flow. It is also applicable to the thermocapillary problem of layered flow between heated plates.
14 citations
28 Apr 2008
TL;DR: It is revealed that Marangoni’s daughter-in-law is expecting their first child.
Abstract: 传质在在二不溶混的液体之间的接口导致的 Marangoni 效果在实验室和溶剂萃取的工业操作上显示重要影响。在二个液层系统的二维的 Marangoni 效果的系统的数字研究被进行。界面张力对溶质集中的线性关系在两个阶段为液体流动和传质被合并到数学模型财务。典型盒子由 Sternling 与 Scriven 分析了(AIChE J. ， 1959 ) 使用线性不稳定性理论被在理论之间的有限差别方法和好同意模仿，数字模拟被观察。模拟建议 Marangoni 传送对流需要某些时间在力量和规模足够地发展提高分裂期间传质， Marangoni 效果动态、短暂，并且只要传质推动力被使经常，在某稳定的水平留下。什么时候某些水平砍作为在实际意义的大多数情况中在接口被强加，当 shear 逐渐地被增加， Marangoni 效果稍微，但是日益增多地被压制。Marangoni 效果的现在的二维的模拟提供某卓见进内在的机制并且也在真实世界上并且在化学工程应用程序的三维的 Marangoni 效果的进一步理论的学习的基础。
6 citations
TL;DR: In this article, a semi-analytical approach was adopted to model liquid-liquid extraction in the circular channel with stratified flow, where a bipolar cylindrical coordinate system was employed for an elegant description of the boundaries in the form of iso-coordinate surfaces.
5 citations
Journal Article•
TL;DR: In this article, the instability of layered two-phase flows caused by the presence of a soluble surfactant (or a surface-active solute) is studied, and a linear stability analysis is performed, using a combination of asymptotic and numerical methods.
Abstract: In this paper, the instability of layered two-phase flows caused by the presence of a soluble surfactant (or a surface-active solute) is studied. The fluids have different viscosities, but are density matched to focus on Marangoni effects. The fluids flow between two flat plates, which are maintained at different solute concentrations. This establishes a constant flux of solute from one fluid to the other in the base state. A linear stability analysis is performed, using a combination of asymptotic and numerical methods. In the creeping flow regime, Marangoni stresses destabilize the flow, provided that a concentration gradient is maintained across the fluids. One long-wave and two short-wave Marangoni instability modes arise, in different regions of parameter space. A well-defined condition for the long-wave instability is determined in terms of the viscosity and thickness ratios of the fluids, and the direction of mass transfer. Energy budget calculations show that the Marangoni stresses that drive long- and short-wave instabilities have distinct origins. The former is caused by interface deformation while the latter is associated with convection by the disturbance flow. Consequently, even when the interface is non-deforming (in the large-interfacial-tension limit), the flow can become unstable to short-wave disturbances. On increasing the Reynolds number, the viscosity-induced interfacial instability comes into play. This mode is shown to either suppress or enhance the Marangoni instability, depending on the viscosity and thickness ratios. This analysis is relevant to applications such as solvent extraction in microchannels, in which a surface-active solute is transferred between fluids in parallel stratified flow. It is also applicable to the thermocapillary problem of layered flow between heated plates.
3 citations
Cites background from "Modelling Extraction in Microchanne..."
...However, in a recent study, we have shown that this effect is not strong enough to impact the primary pressure driven flow for practical fluid–solute systems (Picardo et al. 2015)....
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...For dilute solutions these boundary conditions read as (Picardo et al. 2015) −η′xc′1,x + c′1,y =Dr(−η′xc′2,x + c′2,y), (2.4) c′1 =Kc′2....
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...However, in a recent study, we have shown that this effect is not strong enough to impact the primary pressure driven flow, for practical fluid solute systems (Picardo et al. 2015)....
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...For dilute solutions these boundary conditions read as (Picardo et al. 2015):...
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...It consists of a solute transport equation in each fluid and two interface boundary conditions that enforce equality of flux and local equilibrium at the interface (Picardo et al. 2015)....
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References
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01 Jun 2007
TL;DR: Advanced Transport Phenomena as mentioned in this paper provides a detailed discussion of modern analytic methods for the solution of fluid mechanics and heat and mass transfer problems focusing on approximations based on scaling and asymptotic methods, beginning with the derivation of basic equations and boundary conditions and concluding with linear stability theory.
Abstract: Advanced Transport Phenomena is ideal as a graduate textbook. It contains a detailed discussion of modern analytic methods for the solution of fluid mechanics and heat and mass transfer problems, focusing on approximations based on scaling and asymptotic methods, beginning with the derivation of basic equations and boundary conditions and concluding with linear stability theory. Also covered are unidirectional flows, lubrication and thin-film theory, creeping flows, boundary layer theory, and convective heat and mass transport at high and low Reynolds numbers. The emphasis is on basic physics, scaling and nondimensionalization, and approximations that can be used to obtain solutions that are due either to geometric simplifications, or large or small values of dimensionless parameters. The author emphasizes setting up problems and extracting as much information as possible short of obtaining detailed solutions of differential equations. The book also focuses on the solutions of representative problems. This reflects the book's goal of teaching readers to think about the solution of transport problems.
1,082 citations
"Modelling Extraction in Microchanne..." refers background in this paper
...This will result in stresses that act along the interface, directed towards regions of high interfacial tension, called Marangoni stresses [24]....
[...]
...The changes in interfacial concentration are governed by the interface species transport equation, which accounts for the adsorption–desorption processes, as well as transport along the interface [24], [25]....
[...]
...This leads to a lubrication approximation of the flow field, which is often used to describe flow in narrow gaps of varying width and flow of thin films [24]....
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TL;DR: In this article, a simplified mathematical model has been analyzed in order to detail the mechanism of the "interfacial engine" which supplies the mechanical energy of interfacial turbulence, which is a manifestation of hydrodynamic instability, touched off by ever present, small, random fluctuations about the interface.
Abstract: The origin of interfacial turbulence, spontaneous agitation of the interface between two unequilibrated liquids, has been explained in terms of classical flow, diffusion, and surface processes. The essence of the explanation is the long-known though much neglected Marangoni effect, wherein movement in an interface is caused by longitudinal variations of interfacial tension. It is proposed that interfacial turbulence is a manifestation of hydrodynamic instability, which is touched off by ever present, small, random fluctuations about the interface.
A simplified mathematical model has been analyzed in order to detail the mechanism of the “interfacial engine” which supplies the mechanical energy of interfacial turbulence. In its present form the analysis incorporates several drastic simplifications, though ways of removing some of these have been suggested. The groundwork has been laid for the more elaborate analyses that are needed for a decisive test of the theory.
The analysis shows how some systems may be stable with solute transfer in one direction yet unstable with transfer in the opposite direction, a striking result. It also suggests that interfacial turbulence is usually promoted by (1) solute transfer out of the phase of higher viscosity, (2) solute transfer out of the phase in which its diffusivity is lower, (3) large differences in kinematic viscosity and solute diffusivity between the two phases, (4) steep concentration gradients near the interface, (5) interfacial tension highly sensitive to solute concentration, (6) low viscosities and diffusivities in both phases, (7) absence of surface-active agents, and (8) interfaces of large extent.
That some of these effects have been observed in the laboratory lends credence to the theory.
818 citations
"Modelling Extraction in Microchanne..." refers background in this paper
...Examples include mass transfer between a slowly moving drop and a stationary fluid [28–30] and mass transfer between stationary fluid layers [26, 31, 32]....
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...The simplified model can be written in terms of bulk phase concentrations alone, if the interfacial tension is known empirically in terms of the bulk phase concentrations of the solute [26]....
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TL;DR: In this article, the shape of the interface between the immiscible liquids was controlled by a competition between the viscous forces and the local interfacial tension, and the mass transfer coefficients for parallel and slug flow were determined using instantaneous eutralisation (acidbase) reaction.
376 citations
"Modelling Extraction in Microchanne..." refers background in this paper
...In fact, there has been speculation in the literature about the possible influence of Marangoni stresses and how they might affect the system’s output [8, 27]....
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Book•
31 May 2001TL;DR: In this article, the authors provide an accessible introduction to the basic theory of fluid mechanics and computational fluid dynamics (CFD) from a modern perspective that unifies theory and numerical computation.
Abstract: This book provides an accessible introduction to the basic theory of fluid mechanics and computational fluid dynamics (CFD) from a modern perspective that unifies theory and numerical computation. Methods of scientific computing are introduced alongside with theoretical analysis and MATLAB codes are presented and discussed for a broad range of topics: from interfacial shapes in hydrostatics, to vortex dynamics, to viscous flow, to turbulent flow, to panel methods for flow past airfoils. The third edition includes new topics, additional examples, solved and unsolved problems, and revised images. It adds more computational algorithms and MATLAB programs. It also incorporates discussion of the latest version of the fluid dynamics software library FDLIB, which is freely available online. FDLIB offers an extensive range of computer codes that demonstrate the implementation of elementary and advanced algorithms and provide an invaluable resource for research, teaching, classroom instruction, and self-study. This book is a must for students in all fields of engineering, computational physics, scientific computing, and applied mathematics. It can be used in both undergraduate and graduate courses in fluid mechanics, aerodynamics, and computational fluid dynamics. The audience includes not only advanced undergraduate and entry-level graduate students, but also a broad class of scientists and engineers with a general interest in scientific computing.
226 citations
TL;DR: In this article, the mass transfer characteristics of immiscible fluids in the two kinds of stainless steel T-junction microchannels, the opposing-flow and the cross-flow Tjunction, are investigated experimentally.
Abstract: In this work, the mass transfer characteristics of immiscible fluids in the two kinds of stainless steel T-junction microchannels, the opposing-flow and the cross-flow T-junction, are investigated experimentally. Water-succinic acid-n-butanol is chosen as a typical example of liquid-liquid two-phase mass transfer process. In our experiments, the mixture velocities of the immiscible liquid-liquid two phases are varied in the range from 0.01 to 2.5 m/s for the 0.4 mm microchannel and from 0.005 to 2.0 m/s for the 0.6 mm microchannel, respectively. The Reynolds numbers of the two-phase mixture vary between 19 and 650. The overall volumetric mass transfer coefficients are determined quantitatively in a single microchannel, and their values are in the ranges of 0.067-17.35 s(-1), which are two or three orders of magnitude higher than those of conventional liquid-liquid contactors. In addition, the effects of the inlet configurations, the fluids inlet locations, the height and the length of the mixing channel, the volumetric flux ratio have been investigated. Empirical correlations to predict the volumetric mass transfer coefficients based on the experimental data are developed. (c) 2007 American Institute of Chemical Engineers AIChE J, 53:3042-3053, 2007.
207 citations