DOCSLIB.ORG

Chaotic Mixing Analyses by Distribution Matrices

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Chaotic Mixing Analyses by Distribution Matrices Bilder 14.02.2001 9:34 Uhr Seite 119 CHAOTIC MIXING ANALYSES BY DISTRIBUTION MATRICES Patrick D. Anderson and Han E. H. Meijer Materials Technology, Dutch Polymer Institute, Eindhoven University of Technology P. O. Box 513, 5600 MB Eindhoven, The Netherlands Fax: x31.40.2447355 E-mail: [email protected] and [email protected] Received: 31.5.2000, Final version: 7.6.2000 ABSTRACT Distributive fluid mixing in laminar flows is studied using the concept of concentration distribution mapping matri- ces, which is based on the original ideas of Spencer & Wiley [1], describing the evolution of the composition of two fluids of identical viscosity with no interfacial tension. The flow domain is divided into cells, and large-scale variations in composition are tracked by following the cell-average concentrations of one fluid using the mapping method of Kruijt et al. [2]. An overview of recent results is presented here where prototype two- and three-dimensional time- periodic mixing flows are considered. Efficiency of different mixing protocols are compared and for a particular ex- ample the (possible) influence of fluid rheology on mixing is studied. Moreover, an extension of the current method including the microstructure of the mixture is illustrated. Although here the method is illustrated making use of these simple flows, more practical, industrial mixers like twin screw extruders can be studied using the same approach. ZUSAMMENFASSUNG Die Vermischung von Flüssigkeiten in laminarer Strömung wird anhand der ”Concentration Distribution Mapping Matrice“-Methode, basierend auf den Arbeiten von Spencer & Wiley, untersucht [1]. Durch diese Methode wird die zeitliche Entwicklung der Zusammensetzung zweier Flüssigkeiten gleicher Viskosität ohne Grenzflächenspannung beschrieben. Hierfür wird der Strömungsbereich zunächst in Zellen unterteilt und anschließend wird die Veränderung der Zusammensetzung, durch das Verfolgen der zellengemittelten Konzentrationen eines der beiden Fluide mittels der ”Mapping Method“ nach Kruijt et al., beobachtet [2]. Dieser Beitrag gibt ein Überblick über neuere Resultate zu typischen zwei- und dreidimensionalen, zeitperiodischen Mischungströmungen. Die Effizienz verschiedener Misch- methoden wird verglichen und für ein ausgewähltes Beispiel wird der (mögliche) Einfluss der rheologischen Eigen- schaften der Flüssigkeiten auf den Mischungsvorgang untersucht. Ausserdem wird eine Erweiterung der Methode vorgestellt, die die Mikrostruktur der Mischung miteinbezieht. Obwohl diese Methode hier an einfachen Strö- mungssituationen illustriert wird, kann sie auch zur Untersuchung von mehr anwendungsorientierten, industriellen Mischern, wie dem Doppelwellenextruder verwendet werden. RÉSUMÉ Le mélange distributif de fluide dans les écoulements laminaires est étudié au moyen du concept de matrices de traçage de distributions de concentration, qui est basé sur les idées originales de Spencer & Wiley [1], décrivant l’évo- lution de la composition de deux fluides avec des viscosités identiques et sans tension interfaciale. Le domaine d’écoulement est divisé en cellules et les variations à grande échelle en composition sont suivies en estimant les con- centrations moyennes d’un des deux fluides à l’aide de la méthode de tracé de Kruijt et al. [2]. Une revue des résul- tats récents est ici présentée, où des écoulements de mélange prototypes à deux et trois dimensions et avec une dépendance temporelle périodique sont considérés. Les efficacités des différents protocoles de mélange sont com- parées et pour un exemple particulier, l’influence (éventuelle) de la rhéologie du fluide sur l’action de mélange est étudiée. De plus, une extension de la méthode, où la microstructure du mélange est incorporée, est présentée. Mal- gré le fait que nous donnions ici des exemples où la méthode est appliquée dans le cas d’écoulements simples, d’un point de vue plus pratique, des mélangeurs industriels, comme les extrudeurs à vis jumelles, peuvent être étudiés en utilisant une approche similaire. 1 INTRODUCTION study these flows are overviewed in this paper. It is well known that even for laminar flow at very Fluid mixing processes receive a considerable low Reynolds numbers mixing can lead to com- amount of recognition because of their import- plex flow patterns, and that the exposure of such ance in nature and industry. Although in many patterns may be understood by imploring the cases mixing is associated with turbulent fluid theory of dynamical systems [3]. The motion of motions, mixing of very viscous fluids constitutes passive particles in such a flow is described by the an important class of mixing occurrences. These set of ordinary differential equations, are typical for polymer blending, compounding, food processing etc, and some new models to xu˙ = (,)x t 1 © Appl. Rheol. 10, 3, 119-133 (2000) DOI: 10.1515/arh-2000-0008 Applied Rheology May/June 2000 119 Bilder 14.02.2001 9:34 Uhr Seite 120 Figure 1: Geometries of the 2D y and 3D cavity. T In the upper figure 1 during the first half of the period the top (back) wall is moved from left to right, -h 0 h x and in the second half of the period the opposite (front) wall moved from left to right. where x = (x, y, z) denotes the position vector and -1 This protocol is denoted as u the local Eulerian velocity. The amount of B the TB protocol. stretching is often used to characterise the qual- The scheme of the flow domain for the cubic cavity ity of mixing: and notation for possible motions of front and back Bu walls is denoted in the lower 2 figure; see text for Br explanation of the where I and L are the present and initial length Bl notation used. Fu of a material filament, respectively [4]. The effi- ciency of stretching in a two-dimensional flow Bd Fr ranges from 1/t where the flow is steady and the domain is closed, to exponential for a hyperbolic, Fl unbounded, chaotic, flow. In chaotic flows the Fd reorientation prevents the material lines from full alignment with the streamlines, which would lead to linear stretching, and this general phenomena is known as chaotic advection. It is relatively straightforward to generate a flow that original ideas of Spencer and Wiley [1]. Instead of can generate chaos; a necessary condition is in trying to understand all details of mixing, by general the crossing of streamlines. For three- using abovementioned tools from chaos theory, dimensional flows, the efficiency of stretching we use the concept of concentration distribution can even be exponential for steady flow. matrices to study the advection of fluid. Once Computational analysis is an important tool such a distribution matrix is constructed it is very in the quantitative estimation of stretching in simple (fast) to study different mixing protocols practical mixers. Most flows are fully three or the influence of initial conditions. dimensional and time-dependent and numerical Although the method will be exemplified methods are required to obtain the velocity field making use of relatively straightforward two- and the resulting stretching. An example is the and three-dimensional time-periodic prototype Kenics static mixer, a spatially-periodic duct flow, flows, see Figure 1, in our laboratory at present studied by Avalosse [5] using a finite element also more practical, industrial, efficient mixers method. Hobbs and Muzzio [6] showed that the like 3D space-periodic static mixing devices, as stretching distributions in the Kenics mixer are e.g. the multiflux mixer, and 3D time- or space self-similar. Another example is the partitioned- periodic dynamic mixing devices, as e.g. the co- pipe mixer [7] consisting of a rotating pipe parti- rotating, closely intermeshing, self-wiping twin tioned with a sequence of orthogonally places screw extruder, are being studied using the same rectangular plates. The system is characterised approach. Results, that allow for fast optimis- by a parameter B, the ratio of cross-sectional ation processes, e.g. concerning screw design, fall twist to axial stretching, and for b = 0 the flow is outside of the scope of this paper and will be pre- regular and becomes chaotic with increasing sented elsewhere. values of b. Meleshko et al. [8] demonstrated, The following Section discusses some using this system, the importance of an accurate results based on the application of ”classical“ velocity field in studying chaotic mixing flows. dynamical methods applied to a three-dimen- Modern tools of chaos theory – Poincaré sec- sional prototype flow. Results will show that it is tions [4], periodic point analysis [9, 10], and possible to analyse this type of flow, but that it stretching distributions [11, 25] – provide a deep is hard to make a quantitative comparison understanding of the mechanisms that make between mixing protocols. In Sections 3 and 4 the chaotic mixing so effective, even though they do same flow is studied using a technique based on not give a direct description of the final mixture. distribution matrices and now actual compari- The goal of this paper is to present an overview sons are more straightforward. Finally, in Section of some recent results where mixing is studied 5 it is shown that microstructure of the mixture by using a more explicit approach based on the can be added to the computations. Applied Rheology 120 May/June 2000 Bilder 14.02.2001 10:20 Uhr Seite 121 2 DISTRIBUTIVE MIXING IN TWO- AND The flow in cavity is described by the Stokes THREE-DIMENSIONAL FLOWS equations: Characteristics of chaotic mixing periodic flows are the determination
Load more

Recommended publications
  • Mixing, Chaotic Advection, and Turbulence
    Annual Reviews www.annualreviews.org/aronline Annu. Rev. Fluid Mech. 1990.22:207-53 Copyright © 1990 hV Annual Reviews Inc. All r~hts reserved MIXING, CHAOTIC ADVECTION, AND TURBULENCE J. M. Ottino Department of Chemical Engineering, University of Massachusetts, Amherst, Massachusetts 01003 1. INTRODUCTION 1.1 Setting The establishment of a paradigm for mixing of fluids can substantially affect the developmentof various branches of physical sciences and tech- nology. Mixing is intimately connected with turbulence, Earth and natural sciences, and various branches of engineering. However, in spite of its universality, there have been surprisingly few attempts to construct a general frameworkfor analysis. An examination of any library index reveals that there are almost no works textbooks or monographs focusing on the fluid mechanics of mixing from a global perspective. In fact, there have been few articles published in these pages devoted exclusively to mixing since the first issue in 1969 [one possible exception is Hill’s "Homogeneous Turbulent Mixing With Chemical Reaction" (1976), which is largely based on statistical theory]. However,mixing has been an important component in various other articles; a few of them are "Mixing-Controlled Supersonic Combustion" (Ferri 1973), "Turbulence and Mixing in Stably Stratified Waters" (Sherman et al. 1978), "Eddies, Waves, Circulation, and Mixing: Statistical Geofluid Mechanics" (Holloway 1986), and "Ocean Tur- bulence" (Gargett 1989). It is apparent that mixing appears in both indus- try and nature and the problems span an enormous range of time and length scales; the Reynolds numberin problems where mixing is important varies by 40 orders of magnitude(see Figure 1).
    [Show full text]
  • Chaotic Mixing in Microchannels Via Low Frequency Switching Transverse Electroosmotic flow Generated on Integrated Microelectrodes
    View Article Online / Journal Homepage / Table of Contents for this issue PAPER www.rsc.org/loc | Lab on a Chip Chaotic mixing in microchannels via low frequency switching transverse electroosmotic flow generated on integrated microelectrodes Hongjun Song,a Ziliang Cai,a Hongseok (Moses) Nohb and Dawn J. Bennett*a Received 3rd September 2009, Accepted 6th November 2009 First published as an Advance Article on the web 5th January 2010 DOI: 10.1039/b918213f In this paper we present a numerical and experimental investigation of a chaotic mixer in a microchannel via low frequency switching transverse electroosmotic flow. By applying a low frequency, square-wave electric field to a pair of parallel electrodes placed at the bottom of the channel, a complex 3D spatial and time-dependence flow was generated to stretch and fold the fluid. This significantly enhanced the mixing effect. The mixing mechanism was first investigated by numerical and experimental analysis. The effects of operational parameters such as flow rate, frequency, and amplitude of the applied voltage have also been investigated. It is found that the best mixing performance is achieved when the frequency is around 1 Hz, and the required mixing length is about 1.5 mm for the case of applied electric potential 5 V peak-to-peak and flow rate 75 mLhÀ1. The mixing performance was significantly enhanced when the applied electric potential increased or the flow rate of fluids decreased. Introduction flow on heterogeneous surface charges with non-uniform, time- independent zeta potentials.19,20 The heterogeneous surface can Micromixing is a critical process in miniaturized analysis systems be generated by coating the microchannel walls with different 1–4 such as lab-on-a-chip and microfluidic devices.
    [Show full text]
  • Chaos and Chaotic Fluid Mixing
    Snapshots of modern mathematics № 4/2014 from Oberwolfach Chaos and chaotic fluid mixing Tom Solomon Very simple mathematical equations can give rise to surprisingly complicated, chaotic dynamics, with be- havior that is sensitive to small deviations in the ini- tial conditions. We illustrate this with a single re- currence equation that can be easily simulated, and with mixing in simple fluid flows. 1 Introduction to chaos People used to believe that simple mathematical equations had simple solutions and complicated equations had complicated solutions. However, Henri Poincaré (1854–1912) and Edward Lorenz [4] showed that some remarkably simple math- ematical systems can display surprisingly complicated behavior. An example is the well-known logistic map equation: xn+1 = Axn(1 − xn), (1) where xn is a quantity (between 0 and 1) at some time n, xn+1 is the same quantity one time-step later and A is a given real number between 0 and 4. One of the most common uses of the logistic map is to model variations in the size of a population. For example, xn can denote the population of adult mayflies in a pond at some point in time, measured as a fraction of the maximum possible number the adult mayfly population can reach in this pond. Similarly, xn+1 denotes the population of the next generation (again, as a fraction of the maximum) and the constant A encodes the influence of various factors on the 1 size of the population (ability of pond to provide food, fraction of adults that typically reproduce, fraction of hatched mayflies not reaching adulthood, overall mortality rate, etc.) As seen in Figure 1, the behavior of the logistic system changes dramatically with changes in the parameter A.
    [Show full text]
  • Chaotic Mixing Induced Transitions in Reaction–Diffusion Systems Zolta´N Neufeld and Peter H
    CHAOS VOLUME 12, NUMBER 2 JUNE 2002 Chaotic mixing induced transitions in reaction–diffusion systems Zolta´n Neufeld and Peter H. Haynes Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, United Kingdom Tama´sTe´l Eo¨tvo¨s University, Department of Theoretical Physics, H-1518 Budapest, Hungary ͑Received 6 November 2001; accepted 7 March 2002; published 20 May 2002͒ We study the evolution of a localized perturbation in a chemical system with multiple homogeneous steady states, in the presence of stirring by a fluid flow. Two distinct regimes are found as the rate of stirring is varied relative to the rate of the chemical reaction. When the stirring is fast localized perturbations decay towards a spatially homogeneous state. When the stirring is slow ͑or fast reaction͒ localized perturbations propagate by advection in form of a filament with a roughly constant width and exponentially increasing length. The width of the filament depends on the stirring rate and reaction rate but is independent of the initial perturbation. We investigate this problem numerically in both closed and open flow systems and explain the results using a one-dimensional ‘‘mean-strain’’ model for the transverse profile of the filament that captures the interplay between the propagation of the reaction–diffusion front and the stretching due to chaotic advection. © 2002 American Institute of Physics. ͓DOI: 10.1063/1.1476949͔ The problem of chemical or biological activity in fluid Equation ͑1͒ can be nondimensionalized by the transfor- flows has recently become an area of active research.1–8 mations Apart from theoretical interest this problem has a num- x tU v k ber of industrial9 and environmental10,11 applications.
    [Show full text]
  • Chaotic Mixing of Viscous Fluids
    Why study fluid mixing - Context Mechanisms of mixing What is the speed of dye mixing in closed flows ? Mixing in open flows Chaotic mixing of viscous fluids Topological entanglement and transport barriers Emmanuelle Gouillart Joint Unit CNRS/Saint-Gobain, Aubervilliers Olivier Dauchot, CEA Saclay Jean-Luc Thiffeault, University of Wisconsin Why study fluid mixing - Context Mechanisms of mixing What is the speed of dye mixing in closed flows ? Mixing in open flows Outline 1 Why study fluid mixing - Context 2 Mechanisms of mixing 3 What is the speed of dye mixing in closed flows ? 4 Mixing in open flows Why study fluid mixing - Context Mechanisms of mixing What is the speed of dye mixing in closed flows ? Mixing in open flows Plan 1 Why study fluid mixing - Context 2 Mechanisms of mixing 3 What is the speed of dye mixing in closed flows ? 4 Mixing in open flows Why study fluid mixing - Context Mechanisms of mixing What is the speed of dye mixing in closed flows ? Mixing in open flows Fluid mixing is ubiquituous in the industry... Mixing step in many processes Low Reynolds number : no turbulence Closed (batch) or open flows Why study fluid mixing - Context Mechanisms of mixing What is the speed of dye mixing in closed flows ? Mixing in open flows ... or in the environment Why study fluid mixing - Context Mechanisms of mixing What is the speed of dye mixing in closed flows ? Mixing in open flows What we would like to know about fluid mixing Why study fluid mixing - Context Mechanisms of mixing What is the speed of dye mixing in closed flows ? Mixing in open flows What
    [Show full text]
  • Parametric Investigation on Mixing in a Micromixer with Two-Layer Crossing Channels
    Hossain and Kim SpringerPlus (2016) 5:794 DOI 10.1186/s40064-016-2477-x RESEARCH Open Access Parametric investigation on mixing in a micromixer with two‑layer crossing channels Shakhawat Hossain and Kwang‑Yong Kim* *Correspondence: [email protected] Abstract Department of Mechanical This work presents a parametric investigation on flow and mixing in a chaotic micro‑ Engineering, Inha University, Incheon 402‑751, Republic mixer consisting of two-layer crossing channels proposed by Xia et al. (Lab Chip 5: of Korea 748–755, 2005). The flow and mixing performance were numerically analyzed using commercially available software ANSYS CFX-15.0, which solves the Navier–Stokes and mass conservation equations with a diffusion–convection model in a Reynolds number range from 0.2 to 40. A mixing index based on the variance of the mass fraction of the mixture was employed to evaluate the mixing performance of the micromixer. The flow structure in the channel was also investigated to identify the relationship with mixing performance. The mixing performance and pressure-drop were evaluated with two dimensionless geometric parameters, i.e., ratios of the sub-channel width to the main channel width and the channels depth to the main channel width. The results revealed that the mixing index at the exit of the micromixer increases with increase in the channel depth-to-width ratio, but decreases with increase in the sub-channel width to main channel width ratio. And, it was found that the mixing index could be increased up to 0.90 with variations of the geometric parameters at Re 0.2, and the pressure drop was very sensitive to the geometric parameters.
    [Show full text]
  • Laminar Mixing of High-Viscous Fluids by a Novel Chaotic Mixer
    Laminar mixing of high-viscous fluids by a novel chaotic mixer S. M. Hosseinalipour*, P. R. Mashaei, M. E. J. Muslomani School of Mechanical Engineering, Iran University of Science and Technology, Tehran, Iran Article info: Abstract Received: 08/07/2016 In this study, an experimental examination of laminar mixing in a novel chaotic mixer was carried out by blob deformation method. The mixer was a circular Accepted: 17/11/2018 vessel with two rotational blades which move along two different circular paths Online: 17/11/2018 with a stepwise motion protocol. The flow visualization was performed by Keywords: coloration of the free surface of the flow with a material tracer. The effects of Chaotic mixer, significant parameters such as rotational speed of blades, blades length, and rotational speed amplitude on mixing efficiency and time were analyzed by Laminar flow, measuring of the area covered by the tracer. The results revealed that increasing Average rotational rotational speed intensifies stretching and folding phenomena; consequently, speed, better mixing is obtained. Also, a better condition in flow kinematic was Mixing index. provided to blend, as stepwise motion protocol with wider amplitude applied. A reduction in mixing time could be observed while the blades with longer length were used. In addition, it was also found that the promotion of mixing by rotational speed is more effective than that of two other parameters. The quantitative data and qualitative observations proved the potential of the proposed chaotic mixer in the wide range of industrial processes including chemical reaction and food processing in which laminar mixing is required.
    [Show full text]
  • Chaotic Dynamics of Particle Dispersion in Fluids L
    Chaotic dynamics of particle dispersion in fluids L. P. Wang and M. R. Maxey Center for Fluid Mechanics. Turbulence, and Computation, Brown University, Providence, Rhode Island 02912 T. D. Burton and D. E. Stock Department of Mechanical and Materials Engineering, Washington State University, Pullman, Washington 99164 (Received 7 January 1992; accepted 16 April 1992) An analysis of the Lagrangian motion for small particles denser than surrounding fluid in a two-dimensional steady cellular flow is presented. The Stokes drag, fluid acceleration, and added mass effect are included in the particle equation of motion. Although the fluid motion is regular, the particle motion can be either chaotic or regular depending on the Stokes number and density ratio. The implications of chaotic motion to particle mixing and dispersion are discussed. Chaotic orbits lead to the dispersion of particle clouds which has many of the features of turbulent dispersion. The mixing process of particles is greatly enhanced since the chaotic advection has the property of ergodicity. However, a high dispersion rate was found to be correlated with low fractal dimension and low mixing efficiency. A similar correlation between dispersion and mixing was found for particles convected by a plane shear mixing layer. 1. INTRODUCTION can be characterized by a positive largest Lyapunov expo- Chaotic dynamics is now recognized as a common nent. They also showed that a dispersion coefficient can be foundation of understanding nonlinear systems. In a spec- used to quantify the Lagrangian motion of particle clouds ified Eulerian flow field, the response of Lagrangian fluid for some parameter ranges. Using numerical simulations elements which simply move with the local fluid velocity is they found a similarity between the particle motion and the described by a set of nonlinear ordinary differential equa- motion of fluid elements in the presence of molecular dif- tions, i.e., a dynamical system.
    [Show full text]
  • Mirofluidic Mixing
    Microfluidic Mixing Synonyms Blending, homogenization. Definition Mixing is the process by which uniformity of concentration is achieved. Depending on the context, mixing may refer to the concentration of a particular component or set of components in the fluid. Overview Importance of mixing A mixer is one of the basic building blocks in microfluidics, along with components such as pumps and valves, and is a critical component in several microfluidic devices. For example, mixing of reaction components is essential for providing homogeneous reaction environments for chemical and biological reactions. The efficiency of many devices such as biosensors depends on mixing. In other applications, rapid and controlled mixing is essential for studying reaction kinetics with much better time resolution as compared to microscale techniques. Microfluidic mixers are thus integral components essential for proper functioning of microfluidic devices for a wide range of applications. Fundamentals of mixing In the context of microfluidics, mixing is the process through which uniformity of concentration is achieved. Depending on the application, the concentration may refer to that of solutes (ions, small molecules, biomolecules, etc.), solvents, or suspended particles such as colloids. Microfluidics typically involves incompressible aqueous or organic solutions, and we will consider only these systems here. Molecules in solution undergo random motions, giving rise to the process of diffusion. Under a concentration gradient, diffusion results in flux (J) of molecules that tends to homogenize the concentration (c) of that molecular species. J = −D∇c (1) - Here D is diffusivity of the species under consideration, and varies from approximately 10 9 2 -11 2 m /s for small molecules and ions to 10 m /s for large biomolecules.
    [Show full text]
  • Enhancement of Mixing Performance of Two-Layer Crossing Micromixer Through Surrogate-Based Optimization
    micromachines Article Enhancement of Mixing Performance of Two-Layer Crossing Micromixer through Surrogate-Based Optimization Shakhawat Hossain 1,*, Nass Toufiq Tayeb 2, Farzana Islam 3, Mosab Kaseem 3, P.D.H. Bui 4, M.M.K. Bhuiya 5, Muhammad Aslam 6 and Kwang-Yong Kim 7,* 1 Department of Industrial and Production Engineering, Jashore University of Science and Technology, Jashore 7408, Bangladesh 2 Gas Turbine Joint Research Team, University of Djelfa, Djelfa 17000, Algeria; toufi[email protected] 3 Department of Nanotechnology and Advanced Materials Engineering, Sejong University, Seoul 05006, Korea; [email protected] (F.I.); [email protected] (M.K.) 4 Department of Mechanical Engineering, University of Tulsa, Tulsa, OK 74104, USA; [email protected] 5 Department of Mechanical Engineering, Chittagong University of Engineering & Technology (CUET), Chittagong 4349, Bangladesh; [email protected] 6 Department of Chemical Engineering, Lahore Campus, COMSATS University Islamabad (CUI), Lahore 53720, Pakistan; [email protected] 7 Department of Mechanical Engineering, Inha University, 100 Inha-ro, Michuhol-gu, Incheon 22212, Korea * Correspondence: [email protected] (S.H.); [email protected] (K.-Y.K.); Tel.: +880-8810308-526191 (S.H.); +82-32-872-3096 (K.-Y.K.); Fax: +82-32-868-1716 (K.-Y.K.) Abstract: Optimum configuration of a micromixer with two-layer crossing microstructure was performed using mixing analysis, surrogate modeling, along with an optimization algorithm. Mixing performance was used to determine the optimum designs at Reynolds number 40. A surrogate modeling method based on a radial basis neural network (RBNN) was used to approximate the value Citation: Hossain, S.; Tayeb, N.T.; of the objective function.
    [Show full text]
  • Lecture 5 Mixing in Microfluidics 2016A
    ! E 155! Mixing in Microfluidics! Fawwaz Habbal! 1 Reference Book “Micromixers – Fundamentals, Design and Fabrications” Author: Nam-Trung Nguyen William-Andrew Inc., 2008 2 Mixing Need to mix multiple streams of fluids in micro channels or chambers with: – short mixing length (and time) – operational simplicity Methods for Mixing Methods for mixing fluids in macro scale channels have been studied over many years. Mixing comprises three mechanisms of mass transfer: – molecular diffusion, – eddy diffusion, – bulk diffusion. Mechanisms for Mixing Macroscale: turbulence can generate large eddy and bulk diffusions while the molecular diffusion is negligible. Microscale: the molecular diffusion more important while the eddy diffusion and the bulk diffusion components are limited under the low Reynolds number condition. § Mixing at the microscale is generally slow and time inef<icient. Types of Mixing Methods Passive mixing Accomplished by driving <luids through channels with three-dimensional channel geometries. Active mixing Based on chaotic advection Achieved by periodic perturbation of the low ields. Chaos can improve the mixing ef<iciency Mixing Mixing occurs due to: • Brownian motion - Diffusion • Taylor Dispersion • Chaotic Advection 7 1 - Diffusion 8 Diffusion • Particles start moving at random • Assumption: each step is equally probable • The direction is random and is not related to the previous step 9 Diffusion Stokes - Einstein Equation Diffusion of a particle (gas, fluid), of size (a) Translational Diffusivity K BT Dt = 6πηa Rotational Diffusivity K T D = B r 8πηa3 10 Diffusion in Fluids Very short diffusion time 1 x2 τ = 2 D Or: < x2 >= 2Dτ And approximately: D = diffusion constant x = 2Dτ X = diffusion length τ = diffusion rate! 11 Diffusion in Fluids BUT: Laminar flow limits fluid mixing.
    [Show full text]
  • A Mapping Method for Distributive Mixing with Diffusion: Interplay Between Chaos and Diffusion in Time-Periodic Sine flow Conor P
    PHYSICS OF FLUIDS 25, 052102 (2013) A mapping method for distributive mixing with diffusion: Interplay between chaos and diffusion in time-periodic sine flow Conor P. Schlick,1,a) Ivan C. Christov,1,2,b) Paul B. Umbanhowar,3 Julio M. Ottino,2,3,4 and Richard M. Lueptow3,4,c) 1Department of Engineering Sciences and Applied Mathematics, Northwestern University, Evanston, Illinois 60208, USA 2Department of Chemical and Biological Engineering, Northwestern University, Evanston, Illinois 60208, USA 3Department of Mechanical Engineering, Northwestern University, Evanston, Illinois 60208, USA 4The Northwestern Institute on Complex Systems (NICO), Northwestern University, Evanston, Illinois 60208, USA (Received 1 October 2012; accepted 13 February 2013; published online 14 May 2013) We present an accurate and efficient computational method for solving the advection- diffusion equation in time-periodic chaotic flows. The method uses operator splitting, which allows the advection and diffusion steps to be treated independently. Taking advantage of flow periodicity, the advection step is solved using a mapping method, and diffusion is “added” discretely after each iteration of the advection map. This approach results in the construction of a composite mapping matrix over an entire period of the chaotic advection-diffusion process and provides a natural framework for the analysis of mixing. To test the approach, we consider two-dimensional time- periodic sine flow. By comparing the numerical solutions obtained by our method to reference solutions, we find qualitative agreement for large time steps (structure of concentration profile) and quantitative agreement for small time steps (low error). Further, we study the interplay between mixing through chaotic advection and mixing through diffusion leading to an analytical model for the evolution of the intensity of segregation with time.
    [Show full text]