Experimental and Numerical Investigation of Porous Media Flow

Research Collection

Doctoral Thesis

Experimental and numerical investigation of porous media flowwith regard to the emulsion process

Author(s): Benedikt Hövekamp, Tobias

Publication Date: 2002

Permanent Link: https://doi.org/10.3929/ethz-a-004511272

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Diss. ETH No. 14836

Experimental and NumericalInvestigation of Porous Media Flow with

regard to the Emulsion Process

A dissertation submitted to theSWISS FEDERAL INSTITUTE OF TECHNOLOGY ZURICH

for the degree ofDoctor of Technical Sciences

presented byTobias Benedikt Hovekamp

Dipl.-Ing.born June 18, 1970citizen of Germany

accepted on the recommendation ofProf. Dr.-Ing. E. Windhab, examiner

Prof. Dr. K. Feigl, co-examiner.

2002

c© 2002 Tobias HovekampLaboratory of Food Process Engineering (ETH Zurich)All rights reserved.

Experimental and Numerical Investigation ofPorous Media Flow with regard to the Emulsion Process

ISBN: 3-905609-17-7LMVT Volume: 16

Published and distributed by:

Laboratory of Food Process EngineeringSwiss Federal Institute of Technology (ETH) ZurichETH Zentrum, LFOCH-8092 ZurichSwitzerland

http://www.vt.ilw.agrl.ethz.ch

Printed in Switzerland by:

bokos druck GmbHBadenerstrasse 123aCH-8004 Zurich

Rien n’est plus fort qu’une ideedont l’heure est venue

Victor Hugo

To Barbara

Danksagung

Die vorliegende Dissertation wurde erst durch die Mithilfe und Unterstutzung vieler Men-schen moglich. Gerne mochte ich mich an dieser Stelle bei allen bedanken, die zum Gelingendieser Arbeit beigetragen haben. Mein besonderer Dank gilt:

Prof. Dr.-Ing. Erich Windhab, der mich in sein Team aufnahm und mir eine grosse akademi-sche Freiheit bei meiner Promotion einraumte. Daruberhinaus bedanke ich mich herzlich furdie anregenden – teils weit uber das fachliche hinausgehenden – Gesprache.

Prof. Dr. Kathleen Feigl, die die numerischen Aspekte meiner Arbeit souveran begleitete unddas Koreferat ubernahm. Gerne blicke ich auf die Zeit zuruck, in der sie noch im Buro neben-an sass.

Dem gesamten Team des Laboratoriums fur Lebensmittelverfahrenstechnik fur die sehr ange-nehme Arbeitsatmosphare, die vielen Anregungen, die ich erhalten habe, und die gemeinsa-men Erlebnisse. Insbesondere gilt mein Dank den Mitarbeitern der Werkstatt: Ulrich Glunk,Dani Kiechl, Jan Corsano und Peter Bigler. Sie standen stets mit Rat und Tat zur Seite.

Den Semester- und Diplomarbeitern sowie Hilfsassistenten, die durch ihre wertvolle Arbeitwichtige Resultate und Einsichtigen lieferten: Paul Bannister, Daniela Brauss, Adrian Durig,Elia Herklotz, Fabien Rubli und Luzian Tobler.

Der Informatik-Support-Group, insbesondere den Mitarbeitern der ‘ersten Stunde’ Peter Bir-cher und Roland Wernli.

Gerne bedanke ich mich auch bei den Mitgliedern des Akademischen Chors Zurich und desZurcher Studenten Skiklub, mit denen ich zusammen als Ausgleich musikalische und sportli-che Gipfel erklimmen durfte.

Adrian Whatley, fur die gewissenhafte Durchsicht des Manuskripts und die geduldige Verbes-serung meiner sprachlichen Unreinheiten.

Dem Schweizerischen Nationalfonds, der im Rahmen des Projekts”Investigation of flow

through compressible porous media” (21-50622.97) die vorliegende Arbeit finanziell un-terstutzt hat.

Meinen Eltern Thea und Theo Hovekamp, die meinen Lebensweg mit viel Liebe und Hingabegeebnet und begleitet haben.

Ein ganz spezieller Dank geht an Barbara Meier fur den steten Ansporn zur Durchfuhrungdieser Arbeit und die sehr schone, gemeinsame Zeit.

Zurich, 30. September 2002

v

Contents

Notation xi

Abstract xix

Zusammenfassung xxi

1 Introduction 11.1 Dispersing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Porous Media Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Aim of this Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Background 32.1 Flow through Porous Media and Nozzles . . . . . . . . . . . . . . . . . . . . 3

2.1.1 Porous Media Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.1.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 32.1.1.2 Flow Behavior in Sphere Packings . . . . . . . . . . . . . 42.1.1.3 Characteristics of Sphere Packing Flow . . . . . . . . . . . 52.1.1.4 Regularly Arranged Porous Media . . . . . . . . . . . . . 72.1.1.5 Representative Capillary Diameter . . . . . . . . . . . . . 82.1.1.6 Compressible Porous Media . . . . . . . . . . . . . . . . . 9

2.1.2 Model Geometries for Porous Media . . . . . . . . . . . . . . . . . . 102.1.2.1 Orifice Geometry . . . . . . . . . . . . . . . . . . . . . . 112.1.2.2 Nozzle with Constant Elongation Rate . . . . . . . . . . . 12

2.1.3 Comparison of Geometries . . . . . . . . . . . . . . . . . . . . . . . 122.1.4 Viscoelastic Flow in Porous Media and Nozzles . . . . . . . . . . . . 132.1.5 Computational Fluid Dynamics . . . . . . . . . . . . . . . . . . . . 142.1.6 Velocity Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.1.6.1 Shear and Elongation Rates . . . . . . . . . . . . . . . . . 142.1.6.2 Predefined Velocity Gradients . . . . . . . . . . . . . . . . 152.1.6.3 Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2 Dispersing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.2.1 Single Droplet Break-up . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2.1.1 Steady Flow Conditions . . . . . . . . . . . . . . . . . . . 162.2.1.2 Unsteady Flow Conditions . . . . . . . . . . . . . . . . . 182.2.1.3 Numerical Simulation of Droplet Break-up . . . . . . . . . 18

2.2.2 Emulsions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

vii

viii CONTENTS

2.2.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 192.2.2.2 Emulsion Processes . . . . . . . . . . . . . . . . . . . . . 192.2.2.3 Emulsion Rheology . . . . . . . . . . . . . . . . . . . . . 20

3 Material and Methods 233.1 Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.1.1 Calculation of Macroscopic Flow Field . . . . . . . . . . . . . . . . 233.1.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 233.1.1.2 Sepran . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.1.2 Calculation of Drop Deformation . . . . . . . . . . . . . . . . . . . 243.1.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 243.1.2.2 BIM program . . . . . . . . . . . . . . . . . . . . . . . . 25

3.1.3 Statistical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.1.3.1 Strategy for Establishing Models . . . . . . . . . . . . . . 263.1.3.2 Model Naming Conventions . . . . . . . . . . . . . . . . . 26

3.2 Analytical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.2.1 Fluid Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.2.2 Fluid Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.2.3 Particle Size Distribution . . . . . . . . . . . . . . . . . . . . . . . . 27

3.3 Characterization of Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.3.1 PEG – SDS – H2O Solutions . . . . . . . . . . . . . . . . . . . . . . 28

3.3.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 283.3.1.2 Polyethylene Glycol (PEG) . . . . . . . . . . . . . . . . . 283.3.1.3 Viscosity Variation with Temperature . . . . . . . . . . . . 283.3.1.4 Density Variation with Temperature . . . . . . . . . . . . . 29

3.3.2 Xanthan Gum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.3.3 Silicone Oils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.3.4 Rape Seed Oil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.3.5 Emulsions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.3.5.1 Surfactant . . . . . . . . . . . . . . . . . . . . . . . . . . 313.3.5.2 Interfacial Tension . . . . . . . . . . . . . . . . . . . . . . 313.3.5.3 Preparation of Pre-emulsions . . . . . . . . . . . . . . . . 323.3.5.4 Stability of Emulsions . . . . . . . . . . . . . . . . . . . . 32

3.4 Experimental Setups and Procedures . . . . . . . . . . . . . . . . . . . . . . 333.4.1 Process Unit with Flow-Through Cell . . . . . . . . . . . . . . . . . 33

3.4.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 333.4.1.2 Data Acquisition . . . . . . . . . . . . . . . . . . . . . . . 33

3.4.2 Sphere Packings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.4.2.1 Packing Structures . . . . . . . . . . . . . . . . . . . . . . 343.4.2.2 Types of Flow-Through Cells . . . . . . . . . . . . . . . . 353.4.2.3 Incompressible Spheres . . . . . . . . . . . . . . . . . . . 363.4.2.4 Incompressible Sphere Packing Flow Characteristics . . . . 363.4.2.5 Compressible Spheres . . . . . . . . . . . . . . . . . . . . 37

3.4.3 Orifices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.4.3.1 Orifice Geometries . . . . . . . . . . . . . . . . . . . . . . 383.4.3.2 Droplet Break-up within Orifice Flows . . . . . . . . . . . 38

CONTENTS ix

3.4.4 Experimental Procedures . . . . . . . . . . . . . . . . . . . . . . . . 38

4 Results and Discussion 394.1 Numerical Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.1.1 Adjoint Converging Diverging Nozzles . . . . . . . . . . . . . . . . 394.1.1.1 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . 394.1.1.2 Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.1.1.3 Annulus Probability . . . . . . . . . . . . . . . . . . . . . 41

4.1.2 Flow field within Converging-Diverging Nozzles . . . . . . . . . . . 424.1.2.1 Reynolds-number Re = 100 . . . . . . . . . . . . . . . . . 424.1.2.2 Reynolds-number Re = 1000 . . . . . . . . . . . . . . . . 44

4.1.3 Droplet Deformation and Break-up . . . . . . . . . . . . . . . . . . 454.1.3.1 Droplet Break-up . . . . . . . . . . . . . . . . . . . . . . 464.1.3.2 Shear and Elongation Rate . . . . . . . . . . . . . . . . . 484.1.3.3 Periodicity . . . . . . . . . . . . . . . . . . . . . . . . . . 484.1.3.4 Droplet Size . . . . . . . . . . . . . . . . . . . . . . . . . 484.1.3.5 Particle Track . . . . . . . . . . . . . . . . . . . . . . . . 504.1.3.6 Entrance Flow . . . . . . . . . . . . . . . . . . . . . . . . 514.1.3.7 Cumulative Effects . . . . . . . . . . . . . . . . . . . . . 52

4.1.4 Orifice Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.2 Dispersing Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.2.1 Dispersing in Sphere Packing Flow . . . . . . . . . . . . . . . . . . 574.2.1.1 Energy and Power Input . . . . . . . . . . . . . . . . . . . 594.2.1.2 Packing Length and Viscosity Ratio . . . . . . . . . . . . . 614.2.1.3 Mean Diameter Model for Sphere Packing Flow (x50,3 –

pack – IV) . . . . . . . . . . . . . . . . . . . . . . . . . . 634.2.1.4 Influence of Dispersed Phase Volume Fraction . . . . . . . 644.2.1.5 Width of Particle Size Distribution (span –pack – IV) . . . 644.2.1.6 Comparison with Numerical Simulations . . . . . . . . . . 65

4.2.2 Dispersing in Orifice Flow . . . . . . . . . . . . . . . . . . . . . . . 664.2.2.1 Mean Diameter Model for Orifice Flows (x50,3 –orif) . . . . 674.2.2.2 Width of Particle Size Distributions (span –orif) . . . . . . 68

4.3 Compressible Porous Media Flows . . . . . . . . . . . . . . . . . . . . . . . 694.3.1 Packing Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.3.1.1 Flow and Compressibility Characteristics . . . . . . . . . . 694.3.1.2 Influence of Packing Type . . . . . . . . . . . . . . . . . . 724.3.1.3 Influence of Material Strength . . . . . . . . . . . . . . . . 754.3.1.4 Influence of Packing Length . . . . . . . . . . . . . . . . . 764.3.1.5 Non-Newtonian fluid (watery Xanthan solution) . . . . . . 77

4.3.2 Emulsification in Compressible Porous Media . . . . . . . . . . . . . 774.3.2.1 Result of Emulsification Process . . . . . . . . . . . . . . 784.3.2.2 Comparison with Incompressible Porous Media . . . . . . 79

x CONTENTS

5 Conclusions 815.1 Viscosity Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 815.2 Physical Parameter Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 815.3 Compressible Porous Media . . . . . . . . . . . . . . . . . . . . . . . . . . 825.4 Capabilities and Limitations of CFD . . . . . . . . . . . . . . . . . . . . . . 82

6 Bibliography 83

Appendices 90

A Crystal Families and Bravais Lattice Types 91

B Parameters of Dispersing Experiments 93

C Adjoint Nozzle Flow Field 97

D Statistical Analysis – Model Quality 99

Notation

Latin letters

Symbol SI-Units MeaningA [m2] areaa [m] undeformed droplet radiusb [–] exponent in dispersing modelB [–] Andrade Law coefficientc [–] concentrationd [m] diameterE [J] (activation) energyf [M L T−2] external force field (body force)G [s−1] sum of shear and elongation rateK [–] Andrade Law coefficientK1, K2, K3 [–] porous media flow coefficientsk [–] coordination numberk [m2] permeabilityL [m] packing length,

length of deformed dropletp [Pa] pressurepL [Pa] Laplace pressureP [Pa] generalized Pressure (Sepran)P [–] probabilityQ [–] cumulative distributionQ0 [–] cumulative number distributionQ1 [–] cumulative distribution based on particle lengthQ2 [–] cumulative surface distributionQ3 [–] cumulative volume distributionR [J K−1 mol−1] universal gas constantRc [m−2] filter cake resistanceRm [m−2] filter medium resistanceSv [m−1] specific surfacespan [–] width of particle size distributionr [m] radius,

radial positionT [oC] temperature

. . .

xi

xii NOTATION

Symbol SI-Units Meaning(cont’d) (cont’d) (cont’d)u NA solution vectorv [m s−1] mean velocityvi [m s−1] velocity vectorx [m] particle diameterx50,3 [m] mean diameter of volume distributiony [–] relative radial positiony0 [–] wall region thickness

Greek letters

Symbol SI-Units Meaningα∗ [–] porous media flow coefficientβ∗ [–] porous media flow coefficientΓ [–] Strainγ [s−1] shear rateε [–] porosity of compressible porous mediaδ [–] compactability coefficientε [s−1] elongation rateε [–] strain of compressible porous mediaη [Pa s] dynamic viscosityηr [–] emulsion relative viscosityµ [Pa s] dynamic viscosity (Newtonian fluids)µ [–] tortuosity factorξ [–] loss factorρ [kg m s−3] Densityτ [Pa] deviatoric stress tensorφ [–] volume fraction of dispersed phaseφm [–] maximum packing volume fractionω [–] relaxation factor (Sepran)Ω [–] angular velocity

Indices

Symbol Meaningsubscripts:0 initial conditionac above criticalb bulk regionc continuous phased dispersed phasehyd hydraulic

. . .

xiii

Symbol Meaning(cont’d) (cont’d)l lossmin minimumo orificep packed bed,

piper radials sphere,

spanwiseuc unit cellv, vol volume specificsuperscripts:* normalized

Dimensionless numbers

Symbol MeaningC C-value (centrifugal acceleration)Ca capillary numberDe Deborah numberEu Euler numberfk friction factorΛ friction coefficientRe Reynolds numberWe Weber number

Operators

Operator Meaning· dot product× cross product: double dot products (of dyads)

Abbreviations

Symbol MeaningAK product name of silicon oils made by WackerBIM Boundary Integral Method,

program name (droplet deformation)CCP Cubic-Close PackingCFD Computational Fluid Dynamics

. . .

xiv NOTATION

Symbol Meaning(cont’d) (cont’d)CMC Critical Micelle ConcentrationDSR Dynamical Stress RheometerEFM Extensional Flow MixerFDM Finite Difference MethodFEM Finite Element MethodFVM Finite Volume MethodFTC Flow-Through CellGPL GNU General Public LicenseHCP Hexagonal-Close PackingNPT Numerical Particle TrackingPDE Partial Differential EquationPEG Poly-Ethylene GlycolPSD Particle Size DistributionRSO Rape Seed OilSDS Sodium Dodecyl Sulfate

List of Tables

2.1 Coefficients for porous media flow characteristics. . . . . . . . . . . . . . . . 72.2 Coefficients for flow through regularly arranged sphere packings. . . . . . . . 82.3 Critical capillary numbers for droplet break-up in simple shear flow. . . . . . 18

3.1 Significance codes used within statistical analysis. . . . . . . . . . . . . . . . 263.2 Viscosities and densities of silicon oils. . . . . . . . . . . . . . . . . . . . . 313.3 Interfacial tension between AK silicon oils and PEG – 2% SDS – H2O solutions. 323.4 Incompressible sphere characteristics. . . . . . . . . . . . . . . . . . . . . . 363.5 Sphere packing porosities for packing structures investigated. . . . . . . . . . 363.6 Physical properties of elastic spheres materials. . . . . . . . . . . . . . . . . 373.7 Loss factors, ξ, for orifice geometries. . . . . . . . . . . . . . . . . . . . . . 38

4.1 Strain above the critical capillary number Γac along track 3. . . . . . . . . . . 474.2 Droplet deformation and break-up characteristics over a variety of initial

droplet radii, a. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.3 Units used for physical quantities within our statistical models. . . . . . . . . 574.4 Estimated coefficients of the statistical models for four dispersing experiments

with varying viscosity ratio and packing length. . . . . . . . . . . . . . . . . 624.5 Data range for fit of dispersing model in sphere packing flows. . . . . . . . . 634.6 Model predictions for the dispersing process within incompressible porous

media. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664.7 Data range for fit of dispersing model in orifice flows. . . . . . . . . . . . . . 674.8 Parameters of compressible sphere packings trials along with flow and com-

pressibility characteristics. . . . . . . . . . . . . . . . . . . . . . . . . . . . 734.9 Comparison of mean diameter, x50,3, with model predictions for incompress-

ible porous medium. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 804.10 Comparison of the width of particle size distribution, span, with model pre-

dictions for an incompressible porous medium. . . . . . . . . . . . . . . . . 80

A.1 Three dimensional crystal families and Bravais lattice types. . . . . . . . . . 92

B.1 Trials employed for model estimations within respective sections. . . . . . . 94B.2 Dispersing trial parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . 95B.3 Dispersing trial parameters (cont’d). . . . . . . . . . . . . . . . . . . . . . . 96

xv

List of Figures

2.1 Porous medium from randomly arranged polydispersed spheres . . . . . . . . 42.2 Microscopic, mesoscopic and macroscopic flow patterns within porous media. 52.3 Porosity variation within columns of randomly packed monodispersed spheres. 62.4 Flow characteristics for regularly and randomly arranged porous media. . . . 92.5 Flow behavior in compressible porous media. . . . . . . . . . . . . . . . . . 102.6 Loss factor versus area ratio for orifice flows. . . . . . . . . . . . . . . . . . 112.7 Nozzle geometry with constant elongation rate along its centerline. . . . . . . 122.8 Normalized area porosity in the spanwise direction versus normalized stream-

wise position for sphere packings and Drost’s nozzle geometry. . . . . . . . . 132.9 Droplet break-up criteria in terms of critical capillary number for various flow

types and viscosity ratios. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.1 Dynamic viscosity of PEG – SDS – water solutions in terms of PEG concen-tration and temperature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.2 Density of PEG – SDS – water solutions in terms of PEG concentration andtemperature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.3 Process unit with perforated screen flow through cell. . . . . . . . . . . . . . 343.4 Regularly arranged sphere packings. . . . . . . . . . . . . . . . . . . . . . . 353.5 Friction coefficient versus Reynolds number for sphere packings. . . . . . . . 37

4.1 Geometry of eight adjoint converging-diverging nozzles studied numerically. 404.2 Partial mesh of the converging-diverging nozzle geometry. . . . . . . . . . . 404.3 Annulus probability in developed pipe flow. . . . . . . . . . . . . . . . . . . 414.4 Shear and elongation rates within nozzles at Re = 100. . . . . . . . . . . . . 424.5 Shear and elongation rates along 3 tracks at Re = 100. . . . . . . . . . . . . 434.6 Shear and elongation rates within nozzles at Re = 1000. . . . . . . . . . . . 444.7 Shear and elongation rates along 3 tracks at Re = 1000. . . . . . . . . . . . . 454.8 Particle tracks within the converging-diverging nozzle geometry at Re = 1000. 454.9 Droplet break-up along track 3 for λ = 1. . . . . . . . . . . . . . . . . . . . 474.10 Influence of droplet size on its deformation and break-up within the adjoined

nozzle geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.11 Maximum deformations of droplets along tracks 1, 2, and 3. . . . . . . . . . 504.12 Maximum droplet deformation within the throat of the first nozzle moving

along track 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524.13 Droplet deformation within eight adjoint converging-diverging nozzles along

track 2 for λ = 1 at Rec = 1000. . . . . . . . . . . . . . . . . . . . . . . . . 53

xvii

xviii LIST OF FIGURES

4.14 Droplet deformation within eight adjoint converging-diverging nozzles alongtrack 2 for λ = 3 at Rec = 1000. . . . . . . . . . . . . . . . . . . . . . . . . 54

4.15 Droplet deformation within eight adjoint converging-diverging nozzles alongtrack 2 for λ = 5 at Rec = 1000. . . . . . . . . . . . . . . . . . . . . . . . . 55

4.16 Shear and elongation rates within adjoint die entries for Rec = 100. . . . . . 564.17 PSD for the dispersing process of 2% RSO in 10% PEG – SDS – H2O solution

in a packed bed of spheres. . . . . . . . . . . . . . . . . . . . . . . . . . . . 584.18 Particle size characteristics vs. specific energy input. . . . . . . . . . . . . . 594.19 Comparison between experimental data and model predictions for dispersing

in porous media. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604.20 Compressed porous medium made of cubically arranged spheres. . . . . . . . 704.21 Flow and compression characteristics for a cubically arranged sphere packing. 714.22 Compression characteristics for three Newtonian fluids. . . . . . . . . . . . . 724.23 Flow and compression characteristics for rhombohedrally arranged sphere

packing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744.24 Dimensionless flow characteristics for both packing types. . . . . . . . . . . 754.25 Flow and compression characteristics for a long cubic sphere packing of hard

material. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 764.26 Flow and compression characteristics for a short rhombohedral sphere pack-

ing from soft material. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 774.27 Flow and compression characteristics for a non-Newtonian fluid. . . . . . . . 784.28 Dispersing in a compressible porous medium. . . . . . . . . . . . . . . . . . 79

C.1 Additional shear and elongation rate information. . . . . . . . . . . . . . . . 98C.2 Elongation rates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

D.1 Indications for statistical model quality. . . . . . . . . . . . . . . . . . . . . 100

Abstract

Dispersing an oily liquid in a non miscible aqueous liquid – with the resulting product beingdenoted as an emulsion – is the connecting theme of investigations performed within thiswork. Emulsion production is an important process, in the food industry, where milk, saladdressings, margarine, fat spreads and mayonnaise are prominent examples of emulsions, andelsewhere. Rotor-stator systems or high pressure homogenizers are typical types of processeswhich are used, with flow often being turbulent in such processes.

Lately, the importance of elongational effects in dispersing processes has been recognizedas witnessed by patents granted in the field of dispersing. With well-balanced contributionsof shear and elongational flow to the overall flow field in porous media, partially periodicallyrecurring, beneficial emulsion production was envisioned for porous media flow. Such flowswere therefore chosen to form the basis of this work.

Within this work, a process unit was designed and built, holding various flow-throughcells. Within those flow through cells, porous media in the form of sphere packings of glassor steel spheres, ranging from 70 µm to 4 mm in diameter, were either randomly filled intothe cell or in case of the largest spheres, regularly arranged. Three arranged sphere packingstructures were studied, cubic, orthorhombic, and rhombohedral.

The investigation of flow and compression behavior in compressible regularly arrangedsphere packings was another part of this work, along with the evaluation of the suitability ofcompressible porous media as a dispersing device. Such compactible porous media were builtup from silicone rubber spheres, cubically and rhombohedrally arranged.

Within our experiments on dispersing of model o/w emulsions, polyethylene glycol (PEG)in 2% sodium dodecyl sulfate (SDS) water solutions were used as the continuous phase. PEGconcentration ranged from 0% to 19% and the concentration of the emulsifying agent SDSwas chosen to be well above the critical micelle concentration (CMC). As dispersed phases,silicone oils of various viscosities and common rape seed oil were employed.

Along with dispersing in sphere packings, dispersing behavior in multiple adjoint ori-fice geometries – modeling porous media flow – was experimentally investigated. This wasaccompanied by applying computational fluid dynamics (CFD) to droplet deformation andbreak-up. Besides experimentally investigated orifice geometries, an adjoint converging-diverging nozzles geometry, designed such that constant elongation rates are present alongits center-line, was also numerically studied.

In the numerical investigation, flow fields within nozzles and orifices were calculated bymeans of the finite element method (FEM), and particle tracks were calculated according toa numerical particle tracking algorithm. The boundary integral method (BIM) was applied todroplet deformation calculations along given particle tracks.

Particle size distributions (PSD) of emulsions were determined by laser diffraction spec-troscopy. Emulsion quality was characterized by the mean droplet diameter and the width of

xix

xx ABSTRACT

the particle size distribution, with higher quality being associated with smaller mean diame-ters and narrower particle size distributions. Experimental data were analyzed by means of thestatistics package ’R’ whereby models were established, solely based on physical parameters,with emulsion quality variables as dependent variables.

When developing our dispersing models, first, the influence of power and energy input onthe dispersing result were considered. The power input is given by the pressure drop across theflow through cell, whereas the energy input is related to the pressure drop times the numberof passages through the flow through cell. It was found that regardless of sphere packing ororifice geometry, higher power input is better suited to gaining finer emulsions. Nevertheless,the width of the emulsion PSD can be reduced by increasing the number of runs through thepackage. Given the task of producing an emulsion of a certain quality one has to take bothaspects into account.

Extending the number of parameters investigated led to an increase of model complexity.Including packing length and the viscosity ratio between the viscosities of the dispersed phaseand that of the continuous phase revealed that along with the expected influence of the viscos-ity ratio on the overall dispersing result as main effect, an interaction between the viscosityratio and the packing length was found. Two levels of viscosity ratio were considered, 1.7 and6.9. It was concluded from investigating the above interaction that the inflow becomes moreimportant to the overall droplet break-up with higher viscosity ratio. Finally, dispersing mod-els were successfully established, accounting solely for the physical parameters investigated,comprising process, fluid and geometry parameters.

Flow through our compressible porous media were characterized according to flow rateto pressure drop relations reported in literature. Moreover, we were able to point out packinginternal deformation characteristics. A comparison between dispersing within such compress-ible porous media and predictions made by our model for an incompressible porous mediumusing otherwise matching parameters gave finer emulsions in case of compressible porousmedia. An extension of the mean diameter model for incompressible porous media such thatporous media compressibility could be accounted for, also desirable seemed to be too unreal-istic, given the – still limited – knowledge of dispersing in incompressible porous media.

Break-up behavior modelled by means of CFD was found to be in good agreement withour experimental data. Nevertheless, one-to-one simulation of droplet break-up within com-plex flow fields as present in dispersing devices – even when disregarding turbulent flow –was found to be not yet fully feasible due to limitations pointed out in this work.

Zusammenfassung

Das Dispergieren von einer oligen in einer wassrigen Phase war der verbindende Prozessa-spekt in dieser Arbeit. Das Herstellen solcher Dispersionen wird als Emulgieren bezeichnetund ist ein wichtiger Prozess-Schritt, nicht nur in der Lebensmittelindustrie. Milch, Salat-Saucen, Mayonnaise und Margarine sowie Brotaufstriche sind typische Beispiele fur Emul-sionen. Zur Herstellung werden oft Rotor-Stator-Systeme oder Hochdruckhomogenisatorenverwendet. Dabei ist die Stromung im Dispergierprozess oft turbulent.

Die Bedeutung von Dehnstromungseffekten im Dipsergierprozess ist in den vergange-nen Jahren zunehmend herausgestellt worden, wie dies in erteilten Patenten dokumentiertist. Ausgewogene Scher- und Dehnstromungsanteile am Gesamtstromungsfeld bei der Durch-stromung von porosen Haufwerken, die daruberhinaus noch teilweise periodisch wiederkeh-ren, liessen solche Stromungen fur das Emulgieren als gunstig erscheinen. Deshalb wurdenderartige Stromungen als Grundlage fur diese Arbeit gewahlt.

Im Rahmen dieser Arbeit wurde eine Prozess-Einheit entwickelt und aufgebaut, die ver-schiedene Stromungszellen aufnehmen konnte. In diesen Stromungszellen wurden poroseHaufwerke in Form von Zufallsschuttungen aus Glas- und Stahlkugeln mit Durchmessernvon 70 µm bis 4000 µm realisiert. Aus den 4 mm grossen Kugeln konnten daruberhinaus auchnoch regelmassig angeordnete Kugelpackungen erstellt werden. Kubische, orthorhombischeund rhombohedrale Packungsstrukturen wurden dabei berucksichtigt.

Ein weiterer Bestandteil dieser Arbeit war die Untersuchung des Kompressionsverhaltensvon kompressiblen, regelmassig angeordneten Kugelpackungen sowie der Abschatzung zurEignung solcher Kugelpackungen im Dispergierprozess. Die kompressiblen Kugelpackungenwurden aus Silikonkautschukkugeln hergestellt und in kubischer und rhombohedraler Anord-nung untersucht.

Fur die Emulgierexperimente wurden Polyethylenglykol (PEG) in 2%iger Natriumlauryl-sulfat – Wasser Losungen als kontinuierliche Phase mit PEG Konzentrationen zwischen 0%und 19% verwendet. Die SDS Konzentration von 2% war so gewahlt, dass sie weit ober-halb der kritischen Mizellenbildungskonzentration lag. Silikonole mit unterschiedlichen Vis-kositaten und handelsubliches Rapsol wurden als disperse Phase eingestezt.

Neben dem Dispergieren in Kugelschuttungen wurde auch das Emulgierverhalten in meh-reren hintereinandergereihten, Dusengeometrien mit abrupten Querschnittsanderungen expe-rimentell untersucht. Die Dusengeometrien wurden als Modell fur Kugelpackungen gewahlt.Begleitet wurden diese Untersuchungen von numerischer Stromungssimulation mittels dererdie Tropfendeformation und der allfallige Tropfenaufbruch simuliert werden konnte. Nebenden experimentell untersuchten,

”abrupten” Dusengeometrien wurden auch aneinandergereih-

te konvergierende-divergierende Dusen betrachtet, deren Geometrie so gewahlt war, dass ent-lang der Mittelline konstante Dehngeschwindigkeiten angetroffen wurden.

In unseren numerischen Untersuchungen wurden die Stromungsfelder innerhalb der

xxi

xxii ZUSAMMENFASSUNG

Dusengeometrieen mittels Methode der Finiten Elemente (FEM) berechnet und Stromlinienuber einen Algorithmus zur Bestimmung dergleichen ermittelt. Die ’Boundary Integral Me-thod’ (BIM) wurde angewandt, um Tropfendeformation und -aubruch entlang der Stromlinienzu bestimmen.

Die Emulsionsqualitat, die sich uber den mittleren Tropfendurchmesser und die Breite derTropfendurchmesserverteilung bestimmt, wurde mittels Laserbeugungsspektroskopie durch-gefuhrt, wobei eine hohere Qualitat mit kleinerem mittleren Durchmesser und reduzierterBreite der Tropfengrossenverteilung einher geht. Experimentell gewonnene Daten wurdenunter Verwendung des Statistikpakets ’R’ ausgewertet. Es wurden solche Modelle aufgestellt,die die Zielgrossen wie mittlerer Durchmesser und Verteilungsbreite, durch rein physikalischebzw. Prozess- Parameter erklaren.

Bei der Entwicklung unserer Modelle wurde zurerst der Einfluss von Energie- und Lei-stungseintrag auf das Emulgierergebnis betrachtet. Der Leistungseintrag ist dabei durch denDruckabfall uber der Durchstromungszelle gegeben und der Energieeintrag durch den Druck-abfall mal der Anzahl an Durchlaufen durch die Stromungszelle. Es konnte herausgestelltwerden, dass der Leistungseintrag erwartungsgemass einen starkeren Einfluss auf die Feinheitder Emulsion hat als der Energieeintrag. Eine Erhohung des Energieeintrags ist jedoch miteiner Verringerung der Verteilungsbreite verbunden. Die Aufgabe, eine Emulsion mit einerbestimmten Qualitat zu erzeugen, wobei die Qulitat durch den mittleren Tropfendurchmesserund die Verteilungsbreite bestimmt ist, muss unter Berucksichtigung sowhol des Energie- alsauch des Leistungseintrags gelost werden.

Mit der Erhohung der Anzahl der untersuchten Parameter ging eine Erhohung der Komple-xitat der gefundenen Modelle einher. Bei der Ausdehnung der Untersuchung auf den Einflussder Packungslange und des Viskositatsverhaltnisses zwischen disperser und kontinuierlicherPhase wurde neben dem erwarteten Einfluss des Viskositatsverhaltnisses als Haupteffekt auchnoch eine Interaktion zwischen diesen beiden Parametern aufgezeigt. Dabei wurden Visko-sitatsverhaltnisse von 1,7 und 6,9 betrachtet. Aus der Interaktion konnte abgeleitet werden,dass bei hoherem Viskositatsveraltnis die Einlaufeffekte starker zum Gesamttropfenaufbruchbeitragen als dies beim niedrigeren Viskositatsverhaltniss der Fall war. Zum Abschluss derModellentwicklung fur das Dispergieren konnten erfolgreich Modelle aufgestellt werden, diealle untersuchten Parameter als erklarende Grossen enthielten. Diese Parameter bestanden ausProzess- und Fluidparametern sowie Geometriegrossen.

Die Durchstromung von kompressiblen porosen Medien konnte mittels aus der Litera-tur bekannten Zusammenhangen zwischen Durchfluss durch die Packung und Druckverlustuber derselben characterisiert werden. Daruberhinaus wurden packungsinterne Deformatio-nen aufzeiget und quantifizieret. Ein Vergleich zwischen dem Dispergierergebnis in einemkompressiblen porosen Kugelhaufwerk und dem aus unserem Modell erwarteten, mittlerenTropfendurchmesser bei sonst gleichen Bedingungen ergab feinere Emulsionen im Falle derkompressiblen Packung. Einer Erweiterung der fur die inkompressiblen Packungen gefunde-nen Modelle um den Einfluss der Kompressibilitat erschien wegen des – immer noch – be-schrankten Wissens um das Dispergieren in inkompressiblen Packung noch nicht angezeigt.

Das Tropfenaufbruchverhalten, das mittels numerischer Stromungssimulation evaluiertwurde, war in guter Ubereinstimmung zu dem auf Experimenten basierten Modellen zummitteleren Tropfendurchmesser. Trotzdem ist eine exakte Simulation von Tropfenaufbruch inkomplexen Stromungen aufgrund von Einschrankungen, die in dieser Arbeit herausgestelltwurden, noch nicht umfassend moglich.

Chapter 1

Introduction

1.1 Dispersing

Dispersing is a process not only important to the food industry but also applied within thepharmaceutical, cosmetics and polymer industries among others. Typical foodstuff emulsions,where liquid droplets are dispersed in another immiscible liquid, are milk, margarine, may-onnaise and salad dressings. Those products are mainly processed within rotor-stator systemsor high pressure homogenizers under the addition of emulsifying agents, used to facilitate thedisruption of droplets and stabilize the drop interface.

The quality of emulsions is often defined by two parameters, the mean diameter of thedispersed phase droplets and the width of their particle size distribution (PSD). The small-est possible droplets with the narrowest possible PSDs are desirable for emulsion quality ingeneral and emulsion stability in particular. The mean size of the droplets can in general beadjusted by the volume specific energy applied with higher energy inputs resulting in smallerdroplets. Correlations between PSD widths and process and fluid parameters are much moredifficult to establish and PSDs ranging over one order of magnitude are already considered tobe narrowly distributed.

Within the dispersing process, droplet break-up results from shear and normal forces act-ing upon the droplet. These forces are typically of a transient nature as the droplet movesthrough the dispersing device. Flow fields within the droplet and the droplet shape itself alsocontribute to its break-up behavior. Furthermore, flow field fluctuations, along with turbulentflow behavior, as is often present in dispersing devices, increases the complexity of the break-up mechanisms. However, turbulent dispersing processes occur with higher specific energyinput and broader particle size distributions.

Break-up of single droplets under stationary flow conditions has been the subject of oftencited investigations reported in literature. In these investigations, droplet break-up was studiedfor simple shear, simple elongational, and mixed flow fields with the latter being ’assembled’from the former two flow types. It was found that droplet break-up depends not only on theflow type, but also strongly on the ratio between the dispersed phase viscosity and the contin-uous phase viscosity. For viscosity ratios above about 4, droplets can not be broken up undersimple shear flow. With increasing elongational flow contributions, droplets under higher vis-cosity ratios can be broken up if sufficient deformation is reached. Then, the required specificenergy input is reduced.

1

2 CHAPTER 1. INTRODUCTION

1.2 Porous Media Flow

Laminar flow through porous media, with packed beds of spheres randomly distributed beingone example, are characterized by a well-balanced composition of domains dominated byshear and elongational flow, partially even periodically recurring. Porous media made ofspheres can be either built up randomly or in the case of monodispersed spheres, structuredpackings such as cubically arranged or hexagonal-close packings (HCP) can be realized.

Flow through such packings can be characterized in terms of dimensionless numbers in-cluding the packing Reynolds-number, the friction coefficient and the packing porosity. Pack-ing parameters comprise mean sphere diameter, packing porosity, and packing length. Fluiddensity and fluid dynamic viscosity function account for the fluid behavior. Pressure drop andvolumetric flow rate account for the process parameters.

Packing structures resulting from randomly packed or arranged spheres strongly influencethe flow characteristics given by those dimensionless numbers stated above. Friction coeffi-cients deviating by more than half an order of magnitude were reported for such packings inthe literature. Moreover, large deviations are not only attributed to the packing structure, butalso, in case of arranged packings, to their orientation.

Porous media – particularly in the food industry / biotechnology where they appear e. g.in filtration processes and packed bed reactors – are often compressible. Consequently, theporosity within such porous media depends on the pressure drop across the packing length andvaries spatially over the flow in streamwise direction. This typically goes along with strongdeviations from the flow characteristics for incompressible porous media.

1.3 Aim of this Work

One aim of this work was the investigation of packing and fluid parameters on the dispersingresult within flows through porous media, striving for an optimization of the process giventhe optimization criteria of minimizing energy input and maximizing product, i. e. emulsion,quality. In order to achieve this goal, model geometries facilitating a comparison betweennumerical simulations of single droplet break-up and emperical results were considered.

Secondly, we aimed at a characterization of regularly arranged compressible porous me-dia. This comprised the flow rate dependence on the pressure drop across the porous mediumand spatial porosity information for various flow-through conditions. Finally, the usability ofsuch compressible porous media were assessed.

In order to investigate the above problems, various experimental, analytical, and numericaltools were available at the Laboratory of Food Process Engineering at ETH Zurich but hadto be adapted to the given task. Those tools included data acquisition systems used on pilotplant scale pressure filtration facilities, light spectroscopy for particle size measurements,rheometers, and computational fluid dynamics tools based on the finite element and boundaryintegral method.

Chapter 2

Background

Various aspects centering around porous media flow applied to dispersing processes werestudied as part of this work. First, flow characteristics through regularly and randomly ar-ranged porous media and model systems thereof will be introduced in this chapter. Dispersingwill then be considered for single and multiple droplets suspended in an immiscible fluid ex-periencing simple and mixed, steady and unsteady flow fields. The numerical methods appliedto the various aspects are also reviewed.

2.1 Flow through Porous Media and Nozzles

In the following, a basis will be provided for understanding porous media flow and nozzleflow, with the latter forming a model of the former, along with a description of their numericaltreatments. Randomly arranged, regularly arranged and compressible porous media will beconsidered. Finally, the behavior of viscoelastic fluids in such flows is examined.

2.1.1 Porous Media Flow

2.1.1.1 Introduction

Given a certain pressure difference over a porous medium filled with a fluid, fluid flow in thedirection of decreasing pressure will result. Parameters governing this flow comprise fluidand porous medium properties as depicted in Figure 2.1.

Dating back to 1856, Darcy [Dar56, Appendix D] was the first to study flows throughporous media, setting up the famous Darcy Law for laminar flow of Newtonian fluids throughpacked beds,

V ∝A · ∆p · k

L(2.1)

with volumetric flow rate V dependent on the packing cross-sectional area A, packinglength L, pressure drop over the packing length ∆p and packing permeability k. The latterparameter correlates with primary packing characteristics such as the shape of the packed bedparticles, the particle size distribution of these particles (PSD) and the packed bed porosity ε.

Fluid properties affecting porous media flow are the fluid density ρ and the dynamic fluidviscosity η. In case of generalized non-Newtonian fluids the dynamic fluid viscosity is a

3

4 CHAPTER 2. BACKGROUND

V.

V.

L, p∆

A

ρ, ηFluid properties:

ε,Packed bed properties: d (PSD), ks

Figure 2.1: Porous medium of randomly arranged polydispersed spheres through which flowsa fluid of density ρ and dynamic viscosity η over cross-section area A. Further parameters areexplained in the text.

function of shear rate γ, as given by η = η(γ). It has to be noted that viscosities of non-Newtonian fluids used within integral equations relating the volumetric flow rate to packingand fluid parameters, such as Darcy’s Law, are based upon averaged shear rates.

Packing parameters, as introduced above, include the packing porosity ε, denoting the ratioof packing void volume to total packing volume. This parameter is widely used within porousmedia flow analysis. The volumetric flow rate V is often expressed in terms of the superficialfluid velocity based on an empty column cross-section v. For a detailed introduction to porousmedia flow, the interested reader is referred to Bear [Bea72].

In filtration technology, Darcy’s Law (Eq. 2.1) is usually given in terms of filter cake re-sistance Rc and filter medium resistance Rm along with fluid viscosity η. This is indicated inthe following equation.

V =A∆p

η(RcL + Rm)(2.2)

2.1.1.2 Flow Behavior in Sphere Packings

According to Tsotsas [Tso92], flow patterns through porous media can be categorized into(i) microscopic, (ii) mesoscopic, and (iii) macroscopic, depending on the the length scaleconsidered. As depicted in Figure 2.2 on the left (A), microscopic flow denotes the flowbetween individual spheres, which do not necessarily have to be regularly arranged as shown.Even in laminar flow, individual profiles at any cross-section depend strongly on the Reynolds-number, not only quantitatively but also qualitatively due to the formation of jet-like flow with

2.1. FLOW THROUGH POROUS MEDIA AND NOZZLES 5

A) Mircoscopic B) Mesoscopic C) Macroscopic

Figure 2.2: Microscopic, mesoscopic and macroscopic flow patterns within porous mediaaccording to Tsotsas [Tso92]. Flow profiles are given qualitatively.

higher – still laminar – Reynolds-numbers as opposed to the flow pattern shown.Mesoscopic flow is due to bypass channels possibly forming within the porous media.

This can be observed in improperly fixed packings or in the case of dense packings whichpartially rearrange under flow conditions forming areas with even lower porosity and likewiseallowing for bypass channels to form. With less resistance to the flow in such bypass areas,flow rates within them are higher.

The macroscopic flow profile over the whole porous medium cross-section depicted on theright of Figure 2.2 (C) shows a somewhat unexpected flow profile compared to pipe flow inwhich highest flow rates occur along the centerline. In porous media flow, high flow rates closeto the wall stem from a variation of porosity over the cross-section. In Figure 2.3, experimentaldata for such porosity variation of randomly arranged monodispersed spheres () is given asradial porosity εr, over a normalized radial position, with 0 being at the column wall andincreasing radial position pointing towards the column center. These data were reported byBenenati and Brosilow [BB62] obtained for a sphere diameter to column diameter ratio ofds/D = 0.07092.

Figure 2.3 shows the radial porosity to be equal to 1 at the wall (εr = 1). With increasingdistance from the wall, the radial porosity initially decreases reaching a minimum at abouthalf the sphere diameter with radial porosity of about εr = 0.22. Further on, the radialporosity oscillates around the mean porosity of randomly packed monodispersed spheres εwith amplitudes tapering off. From about 5 sphere diameters away from the column wall,porosity variations become negligible. Liu and Masliyah [LM96] stated a model for the radialporosity as follows:

εr =

1+ε2

− (1 − ε)(

1 − y2y0

)

yy0

: y ≤ y0

ε : y > y0

(2.3)

with relative radial position y = (R − r)/ds and a wall region thickness y0. The model isincluded in Figure 2.3 as a dashed line with the wall region thickness chosen to be y0 = 0.75.

2.1.1.3 Characteristics of Sphere Packing Flow

The flow of a Newtonian fluid through porous media is mainly governed by seven parameters,two fluid parameters, three packing parameters, and two process parameters as denoted in


0 1 2 3 4 5

0.2

0.4

0.6

0.8

1.0Model given by Liu and MasliyahExperimental data of Benenati and Brosilow

PSfrag replacements

Rad

ialp

oros

ityε r

[-]

Relative radial position y [-]

Figure 2.3: Porosity variation within columns of randomly packed monodispersed spheres.Experimental data () from Benenati and Brosilow [BB62] along with model data (dashedline) according to Liu and Masliyah [LM96].

Figure 2.1. These are – given along with their dimensions in terms of mass M, time T, andlength L – the fluid density ρ [M L−3], the fluid viscosity η [M L−1 T−1], the packing length L[L], the mean sphere diameter ds [L], the packing porosity ε [–], the pressure difference overthe packing ∆p [M L−1 T−2], and the velocity in a void column v [M T−1].

According to the Buckingham – π – theorem [Buc14], the functional relationship betweenthose parameters can likewise be expressed in terms of four dimensionless parameters withtheir number (4) given by the number of initial parameters (7) minus the number of param-eter dimensions (M, T and L, i. e. 3). The reader is referred to Stichlmair [Sti90] for anextensive introduction to dimensional analysis in the field of engineering. Two of the fourresulting dimensionless numbers are trivial: the packing porosity ε, and the ratio betweensphere diameter and packing length ds/L. The other two are most often expressed in terms ofReynolds-number Re and friction coefficient Λ:

Re =ρvds

η(1 − ε)(2.4)

Λ =∆p

L·d2

s

ηv·

ε3

(1 − ε)2 (2.5)

It should be noted that the friction coefficient is sometimes expressed in terms of a frictionfactor fk times the Reynolds-number Re, and the friction factor itself can be expressed interms of a Euler-number Eu2 as given in Eq. (2.6).

Λ = fk · Re = Eu2 ·ε3

1 − ε· Re =

∆pds

Lρv2·

ε3

1 − ε· Re (2.6)


Table 2.1: Coefficients for porous media flow characteristics according to Eq. (2.7) relatingfriction coefficient Λ to Reynolds number Re.

Reference K1 K2 K3 Remark

Ergun, [Erg52] 150 1.75 1.0 spheres, granular particlesHaas and Durst, [HD82] 185 1.75 1.0 spheres

Vorwerk and Brunn, [VB94] 181 2.01 0.96 spheres

For flow through sphere packings of mono-dispersed and poly-dispersed spheres and gran-ular particles a relation between friction coefficient and Reynolds-number was found. Thisrelation is given in Eq. (2.7). A selection of the coefficients K1, K2, and K3 in this equation,as reported in literature, is provided in table 2.1.

Λ = K1 + K2 · ReK3 (2.7)

At low Reynolds-numbers (Re < 1), flow is said to be in the Darcian regime. For flow athigher Reynolds-numbers, various definitions such as Forchheimer flow, Burke-Plumer flowand turbulent flow exist. Partially controversial discussions on the existence and cause forsuch flows are found in Nield [Nie01], Liu and Masliyah [LM96], as well as Masuoka andTakatsu [MT96].

Porous media flow coefficients as given in table 2.1 assume inflow and wall effects to benegligible. This assumption is valid in the case of inflow effects for packing length to spherediameter ratios L/ds greater than 10, and in the case of wall effects for column diameter tosphere diameter ratios D/ds also greater than 10. Related investigations were reported byPahl [Pah75].

2.1.1.4 Regularly Arranged Porous Media

Martin et al. [MMM51] and Franzen [Fra79b, Fra79a] investigated flow characteristics inregularly arranged porous media of monodispersed spheres. They studied cubic, orthorhom-bic, and rhombohedral packing structures, which would be denoted in terms of Bravais latticetypes – as given in Appendix A – as cubic primitive (cP), hexagonal primitive (hP), and cu-bic face-centered (cF), respectively. The latter structure is also know as cubic-close packing(CCP). Images of cubic and rhombohedral packings are presented in the following chapter(see Figure 3.4).

The coordination number of packing structures k gives the number of neighboring spheresfor each sphere and is 6, 8 and 12 for cubic, orthorhombic, and rhombohedral packing struc-tures respectively. Packing porosities ε for those packing structures are 0.476, 0,3954 and0.2595. As an aside, Hales [Hal97b, Hal97a] recently suggested a proof for the Kepler Con-jecture, that there are no packings of monodispersed spheres with porosities smaller than0.2595.

In the case of orthorhombic and rhombohedral packing structures, Franzen found the flowcharacteristics to depend on sphere packing orientation. Orientations were denoted by I andII. Franzen based his investigations on a modified Hagen-number Hap and Reynolds-numberRep, as given in Eqs. (2.8, and 2.9) and found a relation between the two terms provided inEq. (2.10).


Table 2.2: Coefficients for flow through regularly arranged sphere packings in terms of α∗

and β∗ as given by Franzen [Fra79b] (Eq. 2.10) and K1, K2, and K3 according to Eq. (2.7) asused within this work.

Type K1 K2 K3 α∗ β∗ k Used within this work

cubic 164.97 1.976 0.9 420 9.0 6 yesorthorhombic I 574.98 10.913 0.9 3400 101.5 8 yesorthorhombic II 101.47 2.925 0.9 600 27.2 8 norhombohedral I 168.90 2.412 0.9 5300 99.2 12 norhombohedral II 140.22 2.106 0.9 4400 86.6 12 yes

Hap =∆p

L·

d2s

v · η(2.8)

Rep =vρds

η(2.9)

Hap = α∗ + β∗ · Re0.9p (2.10)

For the arranged sphere packings studied by Franzen, table 2.2 lists coefficients α∗ and β∗

along with coefficients in terms of K1, K2, and K3 as given in Eq. (2.7).Flow characteristics for such regularly arranged sphere packings are shown in Figure 2.4

along with the characteristic for randomly arranged sphere packings according to Vorwerk andBrunn [VB94]. Characteristics depend strongly on the packing structure and in the case oforthorhombic packings even more on their orientation with orthorhombic I denoting a packingwith an unobstructed straight passage. No orientation has to be given for cubically arrangedpackings since flow always goes through unobstructed straight passages, given flow in thedirection of the packing’s principal axes.

2.1.1.5 Representative Capillary Diameter

Attempts have been made to model porous media flow using capillaries. In order to providea representative capillary radius for a given porous medium, the hydraulic radius concept ofDebbas and Rumpf [DR66], with rhyd as the ratio of the porous medium void volume to thewetted surface area, is used. In terms of porous medium porosity ε and specific surface areaSv the hydraulic radius for a representative capillary is:

rhyd =ε

(1 − ε) · Sv(2.11)

with Sv = 6/ds for sphere packings of monodispersed spheres. For such cubicallyarranged sphere packings with a porosity of ε = 0.476, the hydraulic diameter becomesrhyd ≈ 0.151 · ds.

Other capillary models state the minimum diameter of a stream-tube through a porousmedium as the representative capillary radius rmin. For the cubically arranged sphere packingthis minimum radius becomes rmin = 0.207 · ds. A capillary with the porous medium void


1e−01 1e+00 1e+01 1e+02 1e+03

100

200

500

1000

2000

5000 random packingcubic (k=6)orthorhombic I (k=8)orthorhombic II (k=8)rhombohedral I (k=12)rhombohedral II(k=12)

PSfrag replacements

Fric

tion

coef

ficie

ntΛ

[–]

Reynolds-number Re [–]

Figure 2.4: Flow characteristics for regularly and randomly arranged porous media composedof monodispersed spheres. The characteristics are given in terms of friction coefficient Λ andReynolds-number Re. Packing orientations are denoted by ‘I’, and ‘II’.

volume of a representative pore is also used as a model capillary with a radius denoted rmax

taking the value rmax = 0.389 · ds in the case of cubically arranged sphere packings.

2.1.1.6 Compressible Porous Media

Compressible porous media flows have been studied in the past chiefly with regard to filtra-tion processes. Recent advances were reported by Windhab et al. [WFM96] and Friedmann[Fri99], investigating hyperbaric filtration within centrifugal fields. Friedmann established amodel for such flows fiven various process and material parameters.

Other recent advances were those given by Tiller and co-workers in the field of highlycompactible filter cakes with variable flow rates and filtration with sedimentation [LJKT00,TLKL99, THC95]. In an earlier work, Tiller and Hsyung [TH93] categorized the flow throughcompressible porous media according to the level of compactability into incompressible, low,moderate, high, and super-compactible media. Flow characteristics for such compressibleporous media are shown in the left-hand graph of Figure 2.5 in terms of volumetric flow rateV and the pressure drop over the porous medium ∆p.

The compactability coefficient δ accounts for the ease of filter cake and porous mediacompressibility. Its value represents the negative slope of a logarithmic plot of permeabilityk, versus compressive pressure p, as shown in the right-hand graph of Figure 2.5.

Following Darcy’s Law (Eq. 2.1), Tiller and Hsyung proposed functional relations be-tween volumetric flow rate V and pressure drop ∆p, according to their categorization as givenin Eqs. (2.12, 2.13, and 2.14) for low, high, and super-compactible porous media, respectively.


(super)super

⋅

Pressure drop ∆p

Per

mea

bilit

y k

Compressive pressure p

incompressible δ = 0incompressible

low δ = 0.5

low

highδ = 2

high moderateδ = 1

Vol

umet

ric fl

ow r

ate

V

Figure 2.5: Flow behavior in compressible porous media with various levels of compactabilityaccording to Tiller and Hsyung [TH93]. Compactability is given in terms of compactabilitycoefficient δ.

V ∝∆p1−δ

L(2.12)

V ∝ln(∆p)

L(2.13)

V ∝1

L(2.14)

Up to this point, only reversible porous media deformations have been considered. Takingirreversible deformations into account, caused either by rearrangement or disrupture of theporous media matrices, flow characteristics can deviate significantly from those observed forreversible porous media. Friedmann [Fri99] presented data on porous media flow exhibiting adecrease of volumetric flow-rate V with an increase in pressure drop. This was related to thebreak-down of pore structure.

2.1.2 Model Geometries for Porous Media

The porous media flow models based on representative capillaries previously introduced havethe shortcoming that they disregard elongational flow behavior within porous media. Period-ically expanding and contracting model pore geometries account for such elongational flowbehavior. Aspects of shear and elongation within flows are treated in section 2.1.6 below.


0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.1

0.2

0.3

0.4

0.5

A2

A1

PSfrag replacements

Loss

fact

orξ

[–]

Area ratio A2/A1 [–]

Figure 2.6: Loss factor ξ versus area ratio A2/A1 for orifice flows according to Beiz andDubbel [BD90].

The terms orifice, die entry and sudden contraction will be used synonymously within thiswork. It has to be noted that contrary to real porous media flow, flow within such model poregeometries is unaffected by flow through neighboring pores.

2.1.2.1 Orifice Geometry

Flow through orifices can be characterized in terms of a loss factor ξ, defined via the pressureloss due to the contraction ∆pl, as given by

∆pl = ξρv2/2 (2.15)

with fluid density ρ and fluid velocity v. Expressing the fluid velocity by means of theReynolds-number Reo = ρvdo/η with orifice diameter do and fluid viscosity η, the pressureloss can be written as

∆pl =ξ

2ρ

(

Reo · η

do

)2

(2.16)

The critical Reynolds-number, indicating the transition from laminar to turbulent flowwithin pipe and orifice flow is about 2300. This is about an order of magnitude above thecritical Reynolds-number within porous media flow defined in Eq. (2.4). Therefore, Reynolds-numbers for porous media and orifice flows showed not be compared directly.

Figure 2.6 shows the loss factor ξ as a function of the contraction ratio in terms of thecross-sectional area upstream of the sudden contraction A1 and the cross-sectional area down-stream of the sudden contraction A2.


xr

ln

r i

r o

Figure 2.7: Nozzle geometry with constant elongation rate along its centerline as given byEq. (2.17) for inlet radius ri, outlet radius ro and nozzle length ln chosen to be 2, 0.2 and 4respectively. The nozzle region is indicated by dark grey and the in and outflow regions bylight grey.

2.1.2.2 Nozzle with Constant Elongation Rate

Drost [Dro99] designed a nozzle with constant elongation rates along its center-line under theassumption of developed flow profiles within the nozzle. The nozzle geometry is describedby

r =

(

r−2o − r−2

i

ln· x + r−2

i

)−1/2

(2.17)

with inlet radius ri, outlet radius ro, and nozzle length ln. Such a nozzle is depicted inFigure 2.7.

Drost [Dro99] reported on numerical investigations of the flow field within such a nozzleat a Reynolds-number of Re = 200, defined in terms of the outlet radius ro. Good agreementwas found between the expected and calculated elongation rate along the center-line, withconstant elongation rates along two-thirds of the center-line.

2.1.3 Comparison of Geometries

In this section, regularly arranged porous media will be compared to orifice and nozzle geome-tries, modeling the former ones in terms of normalized area porosity in the spanwise directionε∗s. It was normalized by the unit cell area in case of porous media and by the area in spanwisedirection at the inflow in case of nozzle and orifice geometries.

Normalized area porosity in the spanwise direction ε∗s is depicted versus the normalizedstreamwise position x∗ in Figure 2.8 for the geometries studied within this work. Large dif-ferences in this porosity can be attributed to the various geometries.

Streamwise positions were normalized in the case of the porous media by the sphere diam-eter which is identical to the unit cell length of the cubically and orthorhombically (I) arrangedsphere packings. The unit cell length luc of the rhombohedrally arranged sphere packing isshorter than the sphere diameter by a factor of 0.71. Therefore – disregarding the tortuosityfactor µ – the normalized spanwise porosity (dotted line) is given along a shorter streamwiseposition as indicated by the vertical dotted lines. The minimum spanwise porosity was alignedwith the minimum porosities of the other nozzles at x∗ = 0.5.


0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

k = 6

k = 8

k = 12

nozzle

orificePSfrag replacements

Nor

mal

ized

area

poro

sity

inth

esp

anw

ise

dire

ctio

nε∗ s

[–]

Normalized streamwise position x∗ [–]

Figure 2.8: Normalized area porosity in the spanwise direction ε∗s versus normalized stream-wise position x∗ for regularly arranged porous media, an orifice with a 4:1 contraction, anda nozzle with constant elongation rates along its centerline for the same contraction ratio.Coordination number k = 6, 8, and 12, denote cubically, orthorhombically (I), and rhom-bohedrally (II) arranged sphere packings, respectively. The spanwise porosity of the latterpacking is shown as a dotted line, that of the nozzle as a dashed line, and that of the orifice asa dashed and dotted line.

2.1.4 Viscoelastic Flow in Porous Media and Nozzles

The flow of viscoelastic fluids through porous media and nozzle geometries have been usedto reveal and characterize the elastic properties of such fluids. This was described in Durst etal. [DHI87], Drost [Dro99], and Della Valle et al. [DVTC00].

Drost investigated the flow of beer through packings of spheres. He found an increase inthe friction coefficient Λ of about an order of magnitude over a range of Reynolds-numbersof about two orders of magnitude compared with the respective characteristics for Newtonianfluids as given in Eq. (2.7). This behavior was attributed to elongational effects acting uponmacromolecular proteins in beer. Introducing an experimental Deborah number,

De′ = tε/(1/ε) (2.18)

with the time of strain tε, and characteristic elongation time 1/ε in terms of elongationrate ε as defined below, gave rise to qualitative conclusions for the effects seen.

Durst et al. [DHI87] investigated the flow of dilute polymer solutions through porousmedia. They proposed that the larger part of the pressure loss of the flow of dilute polymersolutions through porous media is possibly caused by the elongational strains macromolecularfuid components experience.


2.1.5 Computational Fluid Dynamics

With the emergence of modern computers, the field of computational fluid dynamics arose,solving the equations governing fluid flow. Considering a fluid as a continuum, the governingequations are the conservation equations for mass and momentum, given in Eq. (2.19) and inEuler-Cauchy formulation in Eq. (2.20), respectively

∇ · ρv = 0 (2.19)

ρ

[

∂v

∂t+ (v · ∇)v + 2Ω× v

]

+ ∇P −∇ · τ = ρf (2.20)

with fluid density ρ, velocity vector v, angular velocity Ω, deviatoric stress tensor τ andbody force vector f . Constitutive equations relate the deviatoric stress tensor τ , to the flowfield and the fluid viscosity, with the latter possibly being a function of the flow field itself.This is given by

τ = η(

∇v + ∇vT)

(2.21)

where η is usually specified to be a function of the second invariant of the rate of straintensor. Further details on the underlying formulations can be found in Panton [Pan84].

Various methods exist for discretizing the (steady state) governing equations in order tosolve fluid flow problems numerically. Among others are the finite difference method (FDM)and the finite element method (FEM), with the latter being applied within this work. In thecase of unsteady flow problems, a discretization scheme for time dependence has to be added.A brief overview of discretization schemes for fluid flow problems is given by Dervieux[Der96].

Although computer capacities increased exponentially over the past few decades, numeri-cal treatment of certain fluid flows is still limited. This is particularly obvious in simulationsof turbulent flow or once visco-elastic fluid properties are considered. Limitations can alsoresult from difficulties in mapping microscopic flow geometries, as present in randomly ar-ranged porous media flow, due to limitations in computer memory capacities. Volume aver-aging techniques are applied to model randomly arranged porous media as described by Liuand Masliyah [LM96].

2.1.6 Velocity Gradient

2.1.6.1 Shear and Elongation Rates

In case of non-Newtonian fluid behavior, fluid viscosity depends on the velocity field. Shearand elongation rates γ and ε1 respectively, are characteristics of flow fields derived from thevelocity gradient tensor ∇v as follows according to VanderWal et al. [VGK+96]:

γ =

√

1

2(∇v + ∇vT ) : (∇v + ∇vT ) (2.22)

ε1 =1

2

(

∇v + ∇vT)

:v

‖v‖

v

‖v‖(2.23)


2.1.6.2 Predefined Velocity Gradients

In order to investigate particular flow fields with certain, predefined contributions of shearand elongation rates, velocity gradients can be ‘assembled’. Therefore, a parameter α isintroduced with which the transformed velocity gradient tensor can be written as follows:

∇v =

ε1 γ 00 ε2 00 0 ε3

= (1 − α)G

0 1 00 0 00 0 0

+ αG

1 0 00 ε2/ε1 00 0 ε3/ε1

(2.24)

with shear rate γ, and elongation rate in the k-direction εk. G is the sum of shear rate andelongation rate in the 1-direction G = γ + ε1 and the aforementioned parameter α is definedby

α =ε1

γ + ε1

=ε1

G(2.25)

Simple shear flow is given by α = 0, simple elongation flow by α = 1, and mixed flowfor values in between, assuming γ ≥ 0 and ε1 ≥ 0.

In planar flow, Eq. (2.24) reduces to

∇v =

ε γ 00 −ε 00 0 0

= (1 − α)G

0 1 00 0 00 0 0

+ αG

1 0 00 −1 00 0 0

(2.26)

and in flow with elongational flow being uniaxial, Eq. (2.24) becomes

∇v =

ε γ 00 −ε/2 00 0 −ε/2

= (1 − α)G

0 1 00 0 00 0 0

+ αG

1 0 00 −0.5 00 0 −0.5

(2.27)

with ε = ε1.

2.1.6.3 Strain

As an integral measure for shear and elongation rate variations over time, the strain Γ can beused:

Γ =

∫

Gdt =

∫

(γ + ε1)dt (2.28)

We will come back to this definition when droplet deformations along particle tracks areto be considered.


2.2 Dispersing

Within the following sections, the deformation and break-up of droplets suspended in anotherimmiscible fluid will be treated. Underlying mechanisms of single droplet disrupture as wellas multiple droplet disrupture (emulsification) will be given along with a numerical procedureto model such deformation and break-up.

We will focus on emulsions, although the concepts introduced also apply to other types ofdispersions. Finally, the rheological properties of emulsions are described.

2.2.1 Single Droplet Break-up

Break-up of single droplets suspended in another liquid has been the subject of many inves-tigations since the early work of Taylor [Tay34]. Publications of Grace [Gra82] and Bentleyand Leal [BL86] are often regarded as further milestones.

In order to break up a droplet, its Laplace pressure pL has to be overcome by stresses actingupon it. Different break-up behaviors can be attributed to different kinds of continuous phaseflow, i.e. laminar or turbulent flow with the former being the focus of this work. Dispersing incavitation has also to be mentioned.

Binary break-up, in which a droplet splits into two equally sized droplets is desirable, butthis is often superseded by break-up into more droplets often accompanied by large variationsin droplet size. Forming of satellite drops and tip-dropping can be regarded as mechanismsresulting in rather small droplets with the latter being caused by a non-uniform surfactantdistribution as stated by Jansen et al. [JAM01] and de Bruijn [dB99].

Capillary droplet break-up resulting in a multitude of droplets occurs if the drop has notime to adapt its shape to the rapidly varying flow field. This results in a highly elongatedshape on which perturbing ripples develop.

2.2.1.1 Steady Flow Conditions

Under steady flow conditions, droplet break-up in simple shear and elongation flows as well asmixed flows have been studied extensively experimentally and numerically in order to specifybreak-up criteria. Three dimensionless parameters found to primarily account for break-upare the capillary number Ca, the viscosity ratio between the dispersed and continuous phaseviscosity λ = ηd/ηc, and a parameter α indicating the flow type as defined in section 2.1.6.2,with α = 0 indicating simple shear, α = 1 standing for simple elongational flow and valuesin between resulting in mixed flow.

The capillary number Ca relates the viscous forces acting on a droplet to the Laplacepressure acting against the droplet deformation. The Laplace pressure is due to the curvatureof the droplet surface and its interfacial tension. Ca is given by

Ca =ηcGa

σ(2.29)

with the radius of the undeformed, spherical droplet a, the continuous phase viscosity ηc,the sum of the shear- and elongation-rate G as given in section 2.1.6.2, and the interfacialtension σ. It should be noted that the capillary number is often given in terms of Weber-number We, which is essentially the same except for a factor of 2.

2.2. DISPERSING 17

1e−03 1e−02 1e−01 1e+00 1e+01 1e+02

0.1

0.2

0.5

1.0

2.0

5.0

10.0

20.0Exp: α’ = 0.0Exp: α’ = 0.2Exp: α’ = 0.4Exp: α’ = 0.6Exp: α’ = 0.8Exp: α’ = 1.0Num: RallisonNum: α = 0.0Num: α = 1.0

simple elongationalflow

simple shearflow

PSfrag replacements

Crit

ical

capi

llary

num

berC

acr

it[m

Pa

s]

Viscosity ratio λ [-]

Figure 2.9: Droplet break-up criteria in terms of critical capillary number Cacrit for simpleshear (α = 0), simple elongational (α = 1), and mixed flow (0 < α < 1) at viscosity ratios λ.Further explanations are given in the text.

Looking at droplet break-up, the critical capillary number Cacrit is used to indicate thelargest droplet not broken up given certain, steady flow condition. This implies that dropletsdeformed under capillary numbers greater than the critical capillary number (Ca > Cacrit)break up. Droplets deformed under capillary numbers below the critical value (Ca < Cacrit)will deform but no break-up occurs.

Experimentally and numerically determined critical capillary numbers Cacrit are depictedin terms of flow-type parameter α and viscosity ratio λ in Figure 2.9. Experimental data onsimple shear flow (∗, α = 0) was reported by Grace [Gra82] for Couette flow with data pointsbeing smoothly connected by a dashed line known also as Grace curve. The dotted verticalline was added to indicate the viscosity ratio of about 4. It was found that under simple shearflow, droplets with viscosity ratios greater than 4 can not be broken up, regardless of the shearrates imposed; such droplets will only rotate but not break-up.

Experimental data on elongational and mixed flow (open symbols) was provided by Bent-ley and Leal [BL86] measured by means of a Four-Roller-Mill apparatus. The dotted linewas added to guide the eye along the simple elongational flow data points. Numerical dataprovided for the same flow types as given above by Bentley and Leal was reported by Rallison[Ral81] (filled box) for a viscosity ratio of λ = 1.

Numerical data for simple shear and simple elongation flow at various viscosity ratios werecalculated by Feigl et al. [FW01, FKFW02] and Kaufmann [Kau02] for Newtonian fluidsunder the assumption of negligible inertia effects and constant interfacial tension, indicatedby filled triangles and circles. Critical capillary numbers for simple shear flow are also givenin table 2.3. Care has to be taken when comparing numerically and experimentally gained datafrom mixed flow fields due to different definitions of the flow field parameter α as pointed out


Table 2.3: Critical capillary numbers Cacrit for droplet break-up in simple shear flow (α = 0)as calculated by Feigl et al. [FW01, FKFW02].

Viscosity ratio λ Cacrit

1 0.422 0.613 1.275 (∞)

in section 2.1.6.2.Two major conclusions can be drawn from Figure 2.9. Firstly, with an increase in elon-

gational flow, smaller critical capillary numbers are present, thus droplet break-up occurs atlower deformation rates. Secondly, droplets with a viscosity ratio greater than 4 can be brokenup, once elongational flow is present. With higher values of α and thus a larger elongationalflow contribution, the critical viscosity ratio, λcrit, below which break-up is possible given acertain value of α, will also increase.

So far, we have only examined single droplet deformations and break-up. Jansen et al.[JAM01] investigated droplet break-up in concentrated emulsions and shear flow. They founddroplet break-up to occur at lower capillary numbers compared to single droplet break-up andsuggested a shift of the Grace curve toward lower critical capillary numbers.

2.2.1.2 Unsteady Flow Conditions

As early as 1972, Torza et al. [TMC72] proposed, that droplet break-up does not only dependon the level of deformation rates G applied, but also on the time that such deformation rateslast. In the recent work of Ha and Leal [HL01], a high degree of sensitivity to the details ofthe deformation process and history were acknowledged.

Lately, attempts have been made by Feigl et al. [FKFW02] to relate the droplet break-upto the Strain Γ(λ) above the critical capillary number Cacrit(λ) as given in Eq. (2.28). Withall the implications that go along with unsteady flow conditions – even when disregardingturbulent flow – break-up mechanisms of droplets are still far from being understood.

2.2.1.3 Numerical Simulation of Droplet Break-up

The deformation of droplets suspended in another immiscible fluid experiencing a flow fieldare governed by Eqs. (2.19 - 2.21) introduced in section 2.1.5 along with boundary conditionsthat are enforced on the continuous phase fluid, the interface condition to describe the jumpin velocity and stress along the droplet surface, and the kinematic condition to describe theevolution of the interface in the flow field.

Under the assumptions of incompressible Newtonian fluids and small droplet Reynolds-numbers Red = ρGa2/ηc 1 the boundary integral method (BIM) as described by La-dyzhenskaya [Lad69] and Pozrikidis [Poz92] can be applied and the derived problem containsonly surface integrals. Thus, the dimensionality of the problem is reduced by one. The in-terested reader is referred to Loewenberg and Hinch [LH96] and Feigl et al. [FKFW02] for

2.2. DISPERSING 19

further details on the numerical method applied to droplet deformation and break-up calcula-tions performed within this work.

2.2.2 Emulsions

Having considered the break-up of single droplets we will now look at multiple droplet dis-ruptions. Within emulsion generation, a multitude of droplets are simultaneously broken up.


Emulsions are dispersed, multi-phase systems of at least two almost immiscible phases withthe dispersed phase – also known as the inner phase – embedded as droplets within the con-tinuous phase, also known as the matrix or outer phase. Emulsions are thermodynamicallyunstable resulting in the droplets tending to re-coalesce. This is due to a reduction of theinterface area between inner and outer phases leading to a reduction of droplet surface energy.

Droplet diameters in emulsions are characterized in terms of their droplet or particle sizedistributions (Qi versus xi) and characteristics thereof. The index i in the cumulative dis-tribution Qi and in the diameter xi indicates the ’Mengenart’ upon which the distribution isbased. i = 0, 1, 2, 3 denote distributions based on number, length, surface area and volume,respectively. The latter two are the most commonly used in emulsion technology.

Characteristics of particle size distributions (PSD) are the mean diameter x50,i and limitingdiameters x90,i and x10,i. The latter diameters represent those diameters at which 10% ofthe emulsion in terms of the ’Mengenart’ are still larger or smaller than the given diameter.The Sauter diameter x3,2 is a surface weighted mean diameter, often found in literature onemulsions.

Emulsions with particle diameters ranging over an order of magnitude or less are consid-ered to be narrowly distributed. A measure for the width of particle size distributions is givenby

span =x90,3 − x10,3

x50,3

(2.30)

in terms of volume based distribution characteristics as used within this work.Emulsifying agents are added to emulsions in order to facilitate the generation of emul-

sions and stabilize them. The former is achieved by reducing the interfacial tension σ andthe latter aims at the prevention of sedimentation, aggregation and recoalescence. In emul-sion processes, recoalescence is difficult to quantify due to the short time scale on which thedroplet disruption and possible recoalescence takes place. Adsorption kinetics of emulsifiersalso influence the emulsifying result.

For a detailed discussion on emulsions, the interested reader is referred to Becher [Bec83],[Bec85], [Bec88], Dickinson and Stainsby [DS88] and Walstra [Wal93].

2.2.2.2 Emulsion Processes

Several emulsion processes exist and can be categorized according to the used dispersingdevices like rotor-stator systems and homogenizers as described by Schubert [SA89]. Thelatter are basically comprised of nozzle systems. Processes studied within this work can be


regarded as such. A patent, granted to Kahl et al. [KKS+98] describes such a homogenizerwhich was developed for the preparation of an aqueous two-component poly-urethane coating.

The volume specific energy input Ev within homogenizers can be expressed as the pres-sure drop across the nozzle system ∆p, with Ev = ∆p. Karbstein [Kar94] and Stang [Sta98]were able to relate the dispersing result, given in terms of Sauter diameter x3,2 to the energyinput, according to:

x3,2 = C · E−bv (2.31)

with b being a parameter dependent on the flow type and C an empirically determined co-efficient representing fluid and geometry characteristics. In the case of laminar flow, Karbsteinand Stang found b = 1 and in turbulent flow 0.25 < b < 0.4.

Lately, particularly elongational effects within emulsion processes have been subject to ex-tensive investigations. This is documented in patents granted to Nguyen and Utracki [NU95]for their extensional flow mixer (EFM) and Kurtz [Kur99] on a continuous squeeze flow mix-ing process. Both inventions build upon the work by Suzaka [Suz82] on a nozzle type mixingdevice. Windhab [Win97] was granted a registered design for a rotor stator system with ak-shaped geometry.

Emulsions are also the target of forced recoalescence in order to separate phases. Spiel-mann and Su [SS77] studied the breaking of emulsions within porous media.

2.2.2.3 Emulsion Rheology

Rheological properties of emulsions deviate significantly from that of their continuous phases(ηc) once dispersed phase volume fractions φ exceed about 10%. In order to describe thatbehavior, the emulsion relative viscosity ηr is introduced:

ηr =η

ηc

(2.32)

with η being the dispersion viscosity.For dilute and moderately concentrated suspensions of rigid, non-colloidal particles, the

relative viscosity ηr can be written as a zero-parameter model:

ηr = f (φ) (2.33)

One example for such a model is the famous Einstein equation [Ein06, Ein11].For concentrated dispersions, the relative viscosity is not only a function of the volume

fraction φ but also depends on the maximum packing volume fraction φm:

ηr = f(φ, φm) (2.34)

with the Krieger - Dougherty equation [KD59] being one of the best known examples.Looking at emulsions instead of suspensions complicates the description of the rheological

behavior as droplets can deform and therefore give rise to shear-thinning and viscoelasticfluid properties. Single-parameter viscosity-concentration equations for emulsions have beenformulated to describe the rheological behavior. They take into account the viscosity ratio λbetween the dispersed phase viscosity and the continuous phase viscosity and have the generalform

2.2. DISPERSING 21

ηr = f(λ, φ) (2.35)

The Taylor equation [Tay32] is the best known example:

ηr = f (φ, λ)

= 1 +

[

5λ + 2

2λ + 2φ

]

(2.36)

The Taylor equation has been the basis for further development of single-parameter equa-tions by Schowalter et al. [SCB68], Frankel and Acrivos [FA70], Oldroyd [Old53], Phan-Thien and Pham [PP97] and Pal [Pal01]. A theory of the linear viscoelastic behavior of con-centrated emulsions has been developed recently by Palierne [Pal90]. A general introductionto emulsion rheology can be found in Barnes [Bar94].

Chapter 3

Material and Methods

After presenting the numerical methods utilized within this work, analytical methods, usedto characterize fluids and emulsions, are described. Subsequently, the process unit and vari-ous flow-through cells containing rigid or compressible spheres and orifices modeling spherepackings are introduced. Finally, experimental procedures are explained.

3.1 Numerical Methods

Numerical methods introduced in this section comprise computational fluid dynamics andstatistical analysis methods. The former are treated by means of finite element (FEM) andboundary integral methods (BIM) which are applied to macroscopic flow field and dropletdeformation calculations respectively. Fluid flow and droplet deformation calculations arecoupled via numerical particle tracking. At the end of this section, details on the statisticalpackage used for data analysis is given.

3.1.1 Calculation of Macroscopic Flow Field


Stationary isothermal fluid flow is governed by the Navier-Stokes equations as given in Sec-tion 2.1.5, Eqs. (2.19 – 2.21). Various numerical discretization schemes exist for solving suchpartial differential equations (PDEs), such as the finite difference method (FDM), the finitevolume method (FVM), or the finite element method (FEM). We use the FEM within thiswork.

The main constituents of the finite element method are the variational or weak statementof the problem and the approximate solution of the variational equations through the use ofso called ’finite element functions’. Sometimes however, the term ’finite elements’ is used todenote the elements forming the mesh. For an introduction to FEM, the interested reader isreferred to Hughes [Hug87].

3.1.1.2 Sepran

Sepran is a proprietary computational analysis package based on the finite element methodand was used for this work to calculate macroscopic flow fields. It was provided by Inge-

23

24 CHAPTER 3. MATERIAL AND METHODS

nieursbureau SEPRA, The Netherlands (See [Seg84]). Sepran has been developed for andapplied to a variety of two and three-dimensional problems including second order ellipticand parabolic equations, mechanical problems, flow problems, solidification problems, lubri-cation and coupled problems. It comprises pre-processing, processing and post-processingtools.

Within this work, Crouzeix-Raviart elements were chosen along with extended quadratictriangles for calculating flow fields. Choosing Crouzeix-Raviart elements implies that swirl isneglected within axisymmetric calculations and pressures being discontinuous over elements.With penalty function formulation applied, the calculation of pressure and velocity were de-coupled. Thus, the system of equations to be solved was reduced as described in Girault andRaviart [GR79]. The penalty function parameter was chosen to be ε = 1 × 10−7. For de-tails on three-dimensional elements suited for fluid flow calculations, the reader is referred toTanguy et al. [BGT92].

The systems of equations were solved iteratively. In a first iteration, Stokes flow wasassumed. Secondly, one Picard iteration was calculated, followed by Newton iterations untilconvergence was reached. Within the Picard iteration, the convective term is approximatedby a successive substitution according to Eq. (3.1). The Newton approximation is given inEq. (3.2).

(v · ∇v)n+1 ≈ vn · ∇v

n+1 (3.1)

(v · ∇v)n+1 ≈ vn · ∇v

n+1 + vn+1 · ∇v

n − vn · ∇v

n (3.2)

Iterative solver parameters were the accuracy and the relaxation factor, ω. The accuracywas set to 1×10−4. The relaxation factor influences the rate at which the solution of a problemprogresses by splitting the new solution u

∗ into a contribution of the last solution un and thelast but one un−1 according to Eq. (3.3).

u∗ = ω · un + (1 − ω) · un−1 (3.3)

With a careful choice of relaxation factors, the likelihood that solutions diverge can bereduced. In our calculations, the relaxation factor was set to ω = 0.2. Sepran calculationswere run on a PC-workstation with two 1.7 GHz Pentium 4 Processors and 1024 MByteRAM.

One of Sepran’s post-processing features is the calculation of particle tracks along withshear and elongation rates within given flow fields. Shear and elongation rates are calculatedin terms of tangentially aligned coordinate systems moving along the particle tracks.

3.1.2 Calculation of Drop Deformation


The Boundary Integral Method (BIM) was applied for studying the deformation and break-up of a liquid drop suspended in another fluid with the latter fluid experiencing a flow field.Mapping the velocity field onto the boundary between the two phases and thus transforminga three-dimensional problem into a two dimensional problem forms the cornerstone of thismethod.

3.1. NUMERICAL METHODS 25

The governing equations solved are, as described in section 2.2.1.3, the conservation equa-tions of mass and momentum for both fluids at creeping flow, the constitutive equation for thestress, boundary conditions that are enforced on the continuous phase, interface conditions todescribe the jump in velocity and stresses along the interface, and a kinematic condition todescribe the evolution of the interface in the flow field. Further details on BIM are given byFeigl et al. [FKFW02], Cristini et al. [CBL98, CBL01], and Stone and Leal [SL89].

3.1.2.2 BIM program

The program used within this work was developed by Loewenberg and co-workers. It is alsobe called BIM, the same as the method it is based upon. It is obvious from the context whetherthe boundary integral method itself or the program is being referred to.

Investigations by Loewenberg and co-workers based on BIM include investigationsinto concentrated emulsion flows and coalescence behavior [LH96, Loe98, LH97, CBL01,CBL98]. Feigl and Windhab [FW01] extended Loewenberg’s BIM such that droplet defor-mation calculations for transient droplet histories along particle tracks were possible. Withinthis work the extended BIM has been used.

Calculations were performed under the assumptions that both fluids exhibit Newtonianflow behavior with equal density and constant interfacial tension. It was also assumed thatshear and elongation rates are constant over the droplet, thus requiring the length scale of themacroscopic flow field to be much larger than that of the droplet. Moreover, elongation ratesin the spanwise direction were considered to be zero. This was an acceptable approximationsince elongation rates in the spanwise direction are, under the assumption of continuity, smallcompared to those along streamwise directions. Moreover, elongation rates in the streamwisedirection were found to be small compared to the shear rates for most of the particle tracksinvestigated.

The extended BIM program takes a list of particle track information in terms of dimen-sionless travel time, t∗, track coordinates, dimensionless shear rates, γ∗, and dimensionlesselongation rates ε∗, as input. The characteristic time, Tchar = aµc/σ, where the radius a ofthe initial spherical undeformed drop, the viscosity of the continuous phase µc, and the inter-facial tension σ, was used to non-dimensionalize the travel time and the shear and elongationrates provided by Sepran calculations. The maximum length of the particle track list was setto 1000. The influence of the suspended droplet on the macroscopic flow field is small andcan therefore be disregarded.

Calculations were run on a HP 9000 enterprise server (Superdome, Rechenzentrum ETHZurich, 48 PA8600 550 MHz processors). Each droplet deformation calculation was per-formed on a single processor with calculations running up to several weeks. Calculation timestrongly depended on the number of mesh points to be calculated and the viscosity ratio. Thenumber of mesh points is initially set by the user and then adapted by the program itself. Werestricted the maximum number to 25000 grid points. The BIM program is sensitive to highgradients of shear and elongation rates. This was even more strongly pronounced at increasingviscosity ratios.

It has to be noted that the graphical output of the droplet shapes, as reproduced within thiswork, is produced at time intervals fixed prior to the calculations. The time intervals were setsuch that the number of droplet images generated along one particle track was about 100.

The times at which BIM calculations terminated are approximate droplet break-up times.


Table 3.1: Significance codes used within statistical analysis.

P-value Significance

0.05 < P not significant0.01 < P ≤ 0.05 significant0.001 < P ≤ 0.01 strong significant

P ≤ 0.001 very strong significant

Those times are simply called break-up times within this work. Due to droplet images takenat fixed intervals, the images of droplet break-up are unlikely to represent the last calculateddroplet shape but rather the droplet shape slightly earlier.

3.1.3 Statistical Analysis

Statistical analysis within this work was done using R which is a free system for statisticalcomputation and graphics distributed under the GNU General Public License (GPL). R wasinitially written by Ross Ihaka and Robert Gentleman [IG96] at the Department of Statisticsof the University of Auckland, New Zealand and is very similar in appearance to S, whichis a high level language and an environment for data analysis and graphics. R is consideredthe free counterpart to the proprietary S-PLUS package which is based on S. Further detailson R can be found at the R homepage (http://www.r-project.org/) or in the R-FAQ by Hornik[Hor02].

3.1.3.1 Strategy for Establishing Models

R is capable of estimating linear and non-linear model parameters for given data. The beststrategy for establishing models depends on the knowledge about the physical correlationbetween dependent and explanatory variables. This is reflected in the models developed withinthis work.

Some of our models were based on models given in literature, such as those that arepresented in sections 3.3.1 and 4.2.1.1. Other models were built up from scratch followingan iterative trial and error approach, in which linear models based on logarithmic expressionsof all possible explanatory variables were used as a starting point such that all variables werecollected together in a similar manner to the approach used in dimensional analysis. Nonsignificant variables were excluded step by step. As soon as main effects were recognized,interactions between variables were investigated.

Standard significance codes as used within this work are listed in table 3.1.

3.1.3.2 Model Naming Conventions

In order to allow for easier comparison, models established within this study were namedaccording to the following naming convention. The first part of the model name indicatesthe dependent variable (i.e. mean particle diameter x50,3, width of particle size distributionsspan, fluid density ρ, or dynamic viscosity η). The second part gives either the geometryunder which the emulsions were dispersed (i.e. ’orif’ for orifice flow or ’pack’ for packing

3.2. ANALYTICAL METHODS 27

flow) or the fluid upon which density and viscosity data was based. The third part is a Romannumeral with the same numerals for the x50,3 and span models indicating the same underlyingexperiments. ’x50,3 – pack – IV’ is an example of the naming convention.

3.2 Analytical Methods

Fluids used as part of this work were analyzed by means of viscosimetry and density mea-surements. The particle size distributions of emulsions were investigated by laser diffractionspectroscopy and the interfacial tension between the two phases in emulsion by means ofthe drop detachment method for predefined drop formation time (DD-PDFT). The analyticalmethods used are described below except for DD-PDFT which is described in Gunde et al.[GDHK92, GKLb+01].

3.2.1 Fluid Viscosity

The dynamic stress rheometer DSR (Rheometric Scientific, Piscataway, USA) was utilized forrheological analysis of fluids used within this work. The DSR is a stress controlled rheometerwith torque being applied by the measuring head. Applied stresses and resulting strains areanalyzed by the software ‘Orchestrator’.

Concentric cylinders with a rotating inner cylinder (Searle-type) were used, with the cupbeing 32.0 mm and the bob 29.5 mm in diameter. The bob with recessed ends had a length of44.3 mm. Sample temperatures were controlled using an attached water bath.

3.2.2 Fluid Density

Fluid densities were measured by means of the oscillating U-tube method in the DensityMeter DMA 38 (Anton Paar, Graz, Austria). The measuring temperature can be adjustedbetween 15 and 40 oC with an accuracy of ± 0.3 oC. Densities measured have an accuracy of± 0.001 g cm−3 with a repeatability of ± 0.0002 g cm−3.

3.2.3 Particle Size Distribution

The method of laser diffraction spectroscopy was applied using a Mastersizer X instrument(Malvern Instruments, Malvern, UK) to determine the particle size distributions of emulsions.A diluted representative emulsion sample was circulated within the sample unit. The samplepasses through the beam of a monochromatic laser (wavelength λ = 633 nm) and the lightdiffracted by the droplets is detected by a photo-diode array with 31 light sensors. Severalthousand measurement sweeps are performed within a few seconds and the scattering patternswere analyzed according to the Mie theory with refractive indices provided by the user.

3.3 Characterization of Fluids

In this section, fluids employed within the experimental part of this work are introduced andcharacterized. Aqueous solutions including a shear-thinning fluid formed the basis for almost


all experiments. In the dispersing experiments, aqueous solutions served as the continuousphase and oils as the dispersed phase, thus forming oil in water emulsions (o/w).

3.3.1 PEG – SDS – H2O Solutions


Solutions of polyethylene glycol (PEG), sodium dodecyl sulphate (SDS), and (demineralized)water formed the basis for most of our experiments. Solution viscosities were adjusted byPEG concentrations and SDS was added as an emulsifying agent. Following a description ofPEG, variations in solution viscosity and density are given in terms of PEG concentration andsolution temperature. SDS specifications are presented later on in section 3.3.5 on emulsions.All concentrations are given in weight percent (w/w).

3.3.1.2 Polyethylene Glycol (PEG)

Polyethylene glycols have a general formula of H(OCH2CH2)nOH and are synthesized bypolycondensation from ethylene glycol. The PEG used within this work was PEG 35000 fromHoechst, Germany, supplied by Pluss-Staufer AG, Oftringen, Switzerland. It was supplied aspowder and had a molecular weight of approximately 35000 g mol−1.

PEG – SDS – H2O solutions investigated showed Newtonian behavior over a wide shearrate range (0.1–2000 s−1). No elastic properties were detected by means of oscillatory rheo-metric measurements.

3.3.1.3 Viscosity Variation with Temperature

According to Partington [Par51], almost 50 models for the viscosity variation with tempera-ture exist. The moost widely utilized is a model due to Andrade [dCA34]:

log(η) = K + B/T (3.4)

where K and B are fluid dependent coefficients and T is the absolute temperature inKelvin. This model is also known by other names [Bla49], for instance the Arrhenius lawwhere B is replaced by E/R with E being an activation energy for viscous flow and R theuniversal gas constant. The reader is referred to Barnes [Bar00] for a more detailed introduc-tion to the topic of viscosity variation with temperature.

For PEG – SDS – H2O solutions, we established a model taking PEG concentrations intoaccount. Model η – PEG – SDS, given in Eq. (3.5), is based on the Andrade law with coeffi-cient K and B expressed as exponential functions of PEG concentrations.

ln(η) = k1 + k2 · cPEGk3 + (b1 + b2 · cPEG

b3)T (3.5)

With temperature T in C and PEG concentration, cPEG, in % (w/w) the coefficients wereestimated with the statistics package R to be k1 = −6.248, k2 = 0.8505, k3 = 0.6125,b1 = −0.02174, b2 = −0.01155, and b3 = 0.05034. All coefficients were very stronglysignificant expept b3, with a p-value of p = 0.0531 being almost significant. Trials with 22distinct pairs of temperature and PEG concentration were taken into account with a residual

3.3. CHARACTERIZATION OF FLUIDS 29

20 25 30 35

1

5

10

50

100

500

1000

5000

0% PEG

2% PEG

5% PEG

10% PEG

20% PEG

40% PEG

PSfrag replacements

Vis

cosi

tyη

[mP

as]

Temperature T [C]

Figure 3.1: Dynamic viscosity, η, of PEG – SDS – water solutions in terms of PEG concen-tration, cPEG, and temperature, T . Measurements () and model predictions (η – PEG – SDS,solid lines) are shown.

standard error of 0.04456 over 291 degrees of freedom. An order of magnitude differencebetween the number of trials and the degrees of freedom stems from viscosity data taken atabout 10 different shear rates for each trial.

Figure 3.1 indicates the measured viscosities () at six PEG concentration-levels (c = 0,2.06, 5.03, 9.88, 19.85, and 40.04) and various temperatures. The viscosities predicted by ourmodel for those six concentration levels (solid lines) are also given. A very good agreementbetween experimental data and model predictions can be seen.

3.3.1.4 Density Variation with Temperature

The density of PEG – SDS – H2O solutions exhibit a similar characteristic in terms of PEGconcentration and temperature compared to the logarithmic values of the respective viscosi-ties. The density increases super-proportionally with the concentration of PEG and decreaseswith increasing temperature. Therefore, a similar, but non exponential, model (ρ – PEG –SDS) gave the best fit to the density data. The model is given by Eq. (3.6):

ρ = (m1 + m2 · cPEGm3)T + (b1 + b2 · cPEG

b3) (3.6)

The coefficients were estimated with a residual standard error of 0.00017 over 37 degreesof freedom to be m1 = −2.791 × 10−4, m2 = −5.888 × 10−6, m3 = 1.110, b1 = 1.007,b2 = 1.568× 10−3, and b3 = 1.064. All parameters were very strongly significant. Figure 3.2shows the measured densities () at the same levels of PEG concentration as given above andthe model predictions (solid lines).


15 20 25 30 35 40

1.00

1.02

1.04

1.06

1.08

0% PEG

2% PEG

5% PEG

10% PEG

20% PEG

40% PEG

PSfrag replacements

Den

sity

ρ[1

03kg

m−

3]

Temperature T [C]

Figure 3.2: Density, ρ, of PEG – SDS – water solutions in terms of PEG concentration, cPEG,and temperature, T . Measurements () and model predictions (ρ – PEG – SDS, solid lines) areshown.

3.3.2 Xanthan Gum

0.2% (w/w) Xanthan gum in 0.1M NaCl solution was chosen as a shear-thinning fluid. Xan-than gum is used in the food industry as a stabilizer and thickening agent. It is a heteropolysac-charide consisting of D–glucose, D–mannose, and D–glucuronic acid residues with a molec-ular weight between five and ten million Dalton. In this study, Rhodigel R© Easy from RhonePoulenc, France, supplied by Meyhall AG, Kreuzlingen, Switzerland was used.

Friedmann [Fri99] measured the rheological properties of this solution. The zero shearviscosity at 25 C and a density of ρ = 1.002 kg dm−3 was found to be η0 = 1.437 Pa s.Viscoelastic effects were also shown and attributed to the formation of a double helix or thetransition from helix to coiled polymeric structures as described in Kulicke et al. [KA97].For further information on the rheological properties of Xanthan gum, the reader is referredto Rochefort and Middleman [RM87].

3.3.3 Silicone Oils

The silicon oils used within this study as the dispersed phase of emulsions were polydimethyl-siloxanes. Their macromolecule backbones are built up of a chain of alternating silicon andoxygen atoms with each silicon atom being bound to two methyl groups. Viscosity rangesfrom 0.65 mPa s to 1000 Pa s and higher are achieved by mixing macromolecules of differentchain lengths. Silicon oils with viscosities up to 1 Pa s exhibit Newtonian flow behavior undershear rates below 1000 s−1.

In this work, silicon oils from Wacker Chemie, Germany, were used. Table 3.2 lists the

3.3. CHARACTERIZATION OF FLUIDS 31

Table 3.2: Dynamic viscosities, η, and densities, ρ, of silicon oils supplied by Wacker, Ger-many, at 25 C. Names given are Wacker’s product names.

Silicone oil Dynamic viscosity Densityη [Pa s] ρ [kg m−3]

AK 10 0.0093 930AK 20 0.019 945AK 50 0.048 960

AK 100 0.96 963AK 200 0.193 966AK 500 0.485 969

AK 1000 0.97 970

dynamic viscosities and densities of the silicone oils at 25 C used within this study.

3.3.4 Rape Seed Oil

Normal rape seed oil (RSO) produced by Lipton-Sais Food Service, Zug, Switzerland, wasanother oily phase used in our dispersing experiments. The dynamic viscosity of RSO at 25 Cwas η = 0.06 Pa s and the density ρ = 915 kg m−3.

3.3.5 Emulsions

Oil in water emulsions with dispersed phase volume fractions φd up to 10% were studied inthis work. It should be noted, that PEG, Xanthan, and SDS concentrations are given in weightpercent (w/w), whereas dispersed phase volume fractions are given in volume percent (v/v).

3.3.5.1 Surfactant

Sodium dodecyl sulphate was chosen as the emulsifying agent within our dispersing pro-cesses. With fast interface adsorption kinetics, the likelihood of droplet recoalescence in thedispersing process is reduced. Texapon K 1296, produced by Henkel KGaA, Dusseldorf,Germany, was the SDS used in our study.

All continuous phases utilized in our dispersing experiments were based on 2% SDS dem-ineralized water solutions. A concentration of 2% SDS is well above the critical micelle con-centration (CMC) expected for emulsions with dispersed phase volume fractions of less than10% and the mean particle diameters envisioned. Quantitative data on SDS micelle break-upkinetics can be found at Oh and Shah [OS94].

3.3.5.2 Interfacial Tension

Interfacial tension σ between PEG – 2% SDS – H2O solutions and rape seed or silicon oils wasmeasured by means of the drop detachment for predefined drop formation time method (DD-PDFT). Interfacial tension were found to be independent of the drop formation time whichwas varied between 7 and 300 s. Table 3.3 lists measured interfacial tensions for various


Table 3.3: Interfacial tension, σ, between AK silicon oils (dispersed phase) and PEG –2% SDS – H2O solutions at three levels of PEG concentration, cPEG = 0, 5.5, and 10% (w/w).Values are given in 10−3 N m−1.

Silicone oil PEG concentrationcPEG [%]

0 5.5 10

AK 10 – 9.71 –AK 100 10.02 9.94 –AK 250 – – 10.93AK 1000 – 10.3 –

combinations of silicon oil drops in PEG – 2% SDS – H2O solutions. Each interfacial tensiongiven is an averaged value from 7–10 trials.

The interfacial tension between AK 250 and 10% PEG – H2O solution (without the surfac-tant SDS) was 25.4 × 10−3 N m−1. The interfacial tension was reduced by a factor of about2.5 with added SDS.

Contrary to the silicone oil versus PEG – SDS –water solution, the surface tension of rapeseed oil versus 10% PEG – 2% SDS – H2O solution was dependent on the drop formation timewith values between 3.1 and 4.7 × 10−3 N m−1. The former value was measured at a dropformation time of 20 s and the latter at 120 s. This is due to surface active substances withinrape seed oil slowly adsorbing at the interface which are not present in silicon oils.

It must be noted, that the time scale under which the interfacial tensions were measuredwith drop formation times of 7–300 s are several orders of magnitude above a typical dis-ruption time scale in dispersing flow. The same interfacial tensions for silicone oil versusPEG – SDS –water solution as those given within table 3.3 under typical dispersing flow con-ditions can therefore only be assumed.

3.3.5.3 Preparation of Pre-emulsions

Pre-emulsions were generated with a perforated blade stirrer in a 5 l beaker with baffles. Theywere stirred for 5–6 min at 150–200 rpm. By doing so, 4–5 l of pre-emulsion were produced.Note that within our dispersing experiments, emulsions from the preceding experiments wereused as pre-emulsions. This is indicated in the respective discussions.

3.3.5.4 Stability of Emulsions

The particle size distributions of a 30% AK 10 in 2% SDS – H2O emulsion measured less thanone hour after their production was compared to that measured after 5 days. No significantdifference was found between the particle size distributions. Therefore, the stability of emul-sions over the time range in which particle size distributions were measured with much smallervolume fractions of up to 10% can be affirmed.

3.4. EXPERIMENTAL SETUPS AND PROCEDURES 33

3.4 Experimental Setups and Procedures

A process unit was designed and built on a pilot plant scale in order to investigate the differentaspects of this work. The unit was capable of holding various flow-through cells. Within thosecells, adjustable length elastic and inelastic sphere packings as well as single and multipleparallel orifices were mounted. The process unit was used within our dispersing experimentsand to study flow and deformation characteristics of deformable sphere packings.

3.4.1 Process Unit with Flow-Through Cell


Figure 3.3 shows the process unit with a flow-through cell holding four perforated screens asdepicted in the inset image. Reservoir I was filled with pre-emulsions which were forced underpressure through pipe-work holding the flow-through cell into reservoir II against atmosphericpressure. The pipe-work consisted of pipes with diameters ranging from 25 mm to 50 mm.Once reservoir I was empty, the flow was stopped, the pressure within reservoir I releasedand the emulsion was allowed to flow back into reservoir I through the bypass hose. With theemulsion back in reservoir I, the cycle is complete. Further cycles were processed similarly.Emulsion samples were drawn from reservoir I.

The flow rate was controlled by the pressure controller reducing the supplied pressureof about 6 bar. The pressure up and down-stream of the flow-through cell was measured bytwo pressure sensors (Hanni Type ZED 501/873.111/075, range 0–2.5, and 0–6 bar, HanniAG, Jegenstorf, Switzerland). Subtracting the downstream pressure and the pressure loss ofthe flow-through an empty flow through cell from the upstream pressure gives the effectivepressure difference over the packing, ∆p.

Magnetic inductive flow meters (60 and 1000 l h−1, Type PICOMAG DMI 6530, Endressand Hauser Metso AG, Rheinach, Switzerland) were used. Fluid temperature was measuredbetween reservoir I and the flow through cell by means of a thermocouple (Type K: NiCr-Ni,Thermocontrol GmbH, Dietikon, Switzerland).

3.4.1.2 Data Acquisition

The data were acquired by a computer equipped with a data acquisition board (PCI–20428W,Intelligent- Instrumentation Inc., Tucson, Arizona, USA). Data were read every second andfour to ten consecutive data points were averaged in a post-processing step to give the dataused in the analysis of this work. Fluid viscosities used within the analysis of this work wereadjusted according to the measured temperatures.

Note that the process unit was modified within the course of this study in order to simplifythe experimental procedure. Earlier versions of the process unit include horizontally alignedflow-through cells with emulsions being pumped backwards and forwards through the flow-through cell. With horizontally aligned flow through cells, emulsions were regularly mixedmanually in order to reduce particle separations due to buoyancy. No significant influence ofthe flow-through cell alignment on the dispersing result was found for experiments analyzedwithin this work. Process units without a flow meter were also used; this will be indicated inthe respective discussions.


reservoir I

flowmeter

pressure sensor I

bypass hose

flow through cell

pressure sensor II

temperature sensor

reservoir II

pressure controller

3−way valve

pressure supply

main valve

inspection glass

drainage

∆p

T

.V

246

mm

Figure 3.3: Process unit used for dispersing experiments in sphere packing and orifice flowsas well as trials with compressible porous media. The small image on the left shows a flow-through cell with perforated screens.

3.4.2 Sphere Packings

Experiments with sphere packings were conducted within the process unit over a range ofparameters including the packing structure, the material of the spheres (i. e. spheres made ofelastic and inelastic materials), the size of the spheres, and the length of the packings.

3.4.2.1 Packing Structures

Four different packing structures were studied, three of them being arranged sphere packingsand one being a random packing. The cubically arranged sphere packing with a coordinationnumber k = 6 and the rhombohedral II packing with k = 12 is shown in Figure 3.4. The thirdarranged packing (not shown) was the orthorhombic I packing which is built up by stackinghexagonal layers of spheres in the streamwise direction.

Sphere packing names used within this work were chosen in accordance to Franzen[Fra79b] and Martin et al. [MMM51] rather than Bravais lattice types as given in table A.1.The notation of Franzen is advantageous since packing names also reflect the orientation of the


k = 6 k = 12

Figure 3.4: Regularly arranged sphere packings made of incompressible spheres with ds =4 mm. Left: cubically arranged sphere packing (coordination number, k = 6), right: rhombo-hedrally (II) arranged spheres (k = 12). The transparent cylinders are 50 mm in diameter.

packings. The orthorhombic I packing represents a hexagonal primitive Bravais lattice type(hP) and the rhombohedral II packing a cubic face-centered type (cF), which is commonlyknown as cubic-close packing (CCP). The latter packing is similar to the hexagonal-closepacking (HCP) differing only in ordering of hexagonal layers.

Only one packing orientation was studied for each packing type investigated. Therefore,packing name appendices (I, and II) will be omitted for the sake of lucidity within the ‘resultsand discussion’ chapter.

3.4.2.2 Types of Flow-Through Cells

Two different flow-through cell types were used, one with a quadratic cross-section as shownin Figure 3.4 and the other with a circular cross-section. The former consisted of transparentcylinders, 20 mm high, 50 mm in diameter with a quadratic clearance of 28 mm × 28 mm. Upto 6 such cylinders could be stacked together.

Two mounts, one on top of the first and one below the last cylinder were attached to fix thearranged sphere packing structures within the stacked cylinders. Each mount comprised 14cylindrical pins, 1.2 mm in diameter, with seven of them being below the other seven arrangedby right angles.

Flow-through cells with circular cross-sections were used for random sphere packingsconstructed of 25 mm or 12.5 mm diameter pipes. Perforated screens were used to fix thespheres within the flow-through cell. In the case of small sphere diameters, the screens werecovered by filters or stratified layers of spheres with decreasing sphere diameters toward the


Table 3.4: Incompressible sphere characteristics.

Mean diameter Width of PSD Materialds [µm] span [–]

4000 0 steel2000 0 steel339 0.369 glass70 0.345 glass

Table 3.5: Sphere packing porosities ε for the packing structures investigated. The porosityof random packings is given for monodispersed spheres.

Packing structure Coordination Number Porosityk [–] ε [–]

cubic 6 0.4765orthorhombic 8 0.3954rhombohedral 12 0.2595

random – 0.376

center of the sphere packing. In every case, screens and filters were chosen such that the clear-ance within the perforated screens and the filters were significantly larger than a representativecapillary diameter within the sphere packing.

3.4.2.3 Incompressible Spheres

Ball bearings made of stainless steel (provided by Springer, Zurich, Switzerland) and glassbeads (supplied by Merck AG, Dietikon, Switzerland) were used within this work as incom-pressible spherical particles. Sphere characteristics are given in table 3.4, showing monodis-persed ball bearings and glass beads with narrow particle size distribution.

Arranged sphere packings were built up of 4000 µm spheres. The porosities of such ar-ranged sphere packings are given in table 3.5 along with the porosity of random packings ofmonodispersed spheres as used in the analysis of this work. Note that porosities of randomsphere packings of polydispersed spheres are different from that of monodispersed spheres.However, the range is small compared to the porosity variation of arranged sphere packings.

3.4.2.4 Incompressible Sphere Packing Flow Characteristics

Figure 3.5 shows the flow characteristics for sphere packings used in this study in terms of fric-tion coefficient Λ and packing Reynolds number Re as given in Section 2.1.1.3 with Eqs. (2.5and 2.4). Our characteristics are in good agreement with the findings of Franzen [Fra79b]which are included in the graph as curves. See Appendix B for a list of experiments used tocompile this figure.

Each data point shown is an averaged value of four to ten single data points taken atintervals of one second as described in section 3.4.1.2. With dispersed phase volume fractionsof less than 10%, elastic effects were disregarded. Inflow and wall effects were found to be


5 10 20 50 100 200 500 1000

100

200

500

1000

Reynolds number Re [−]

Fric

tion

coef

ficie

nt L

ambd

a [−

]

random

cubic

orthorhombic I

rhombohedral II

orthorhombic II

randomcubic (k=6)orthorhombic I (k=8)

Figure 3.5: Friction coefficient Λ versus Reynolds number Re along with expected valuesaccording to Franzen [Fra79b] for various sphere packings. (See section 2.1.1.4.)

Table 3.6: Physical properties of elastic spheres materials. The terms “soft”and “hard”wereused within this study to denote the respective materials. Data as provided by Dow Corning.

Physical property Unit Silastic R© S Silastic R© J“soft” “hard”

Durometer Shore A 25 55Tensile Strength, Die C MPa 7.0 5.5

Elongation at Break, Die C % 850 250Tear Strength, Die B kN m−1 23 15

negligible. Experiments with rhombohedral sphere packings were done without a flow meter,therefore, no data are shown for this packing structure.

3.4.2.5 Compressible Spheres

Silicon rubbers Silastic R© S and Silastic R© J (Dow Corning, Midland, Michigan, USA, suppliedby Omya AG, Oftringen, Switzerland) were used as elastic sphere materials. The physicalproperties are given in table 3.6.

Mats of 8 × 8 quadratically arranged 4 mm diameter spheres were moulded from thesematerials. A vacuum was applied to eliminate air mixed into the silicon rubber during mixingof the cross linking agent with the main component. Two mats at a time were solidified withinthe mold between two plungers at 130 C by applying a force of 120 kN for 3 minutes.


Table 3.7: Loss factors ξ for various orifice geometries. The loss factor marked by an aster-isk (∗) indicates the flow from the mount into the first perforated cylinder. In that case, theinflow diameter d1 was replaced by the length l1 of a representative quadratic inflow area.

Number of d2 d1, l1 A2/A1 ξ Remarkorifices [mm] [mm] [–] [–]

9 1.0 2.0 0.25 0.40 1 mm and 2 mm holes alternating9 1.0 8.0 0.0123 0.49∗ mount to 1 mm hole cylinder1 2.4 4.5 0.284 0.39 flow meter (60 l h−1)1 8.8 25 0.124 0.46 flow meter (1000 l h−1)

3.4.3 Orifices

3.4.3.1 Orifice Geometries

The orifice flow experiments were mainly based on perforated aluminum cylinders 5 mm highwith a diameter of 50 mm. Cylinders with 3× 3 holes of 1 mm or 2 mm were stacked togetherand fixed in the mount described in section 3.4.2.2 and depicted in figure 3.4. Three differentarrangements were studied: (i) cylinders with 1 mm and 2 mm holes alternating; (ii) cylinderswith 1mm holes alternating with transparent 20 mm high cylinders with a quadratic clearanceof 28 mm × 28 mm; and (iii) stacks of up to 4 cylinders with 1 mm holes.

Orifice flows through a single orifice were realized using flow meters. Pipes connectedto the measuring section of the flow meters form sudden contractions similar to those studiedwith the perforated cylinders. Table 3.7 list the geometry parameters of the orifices studiedalong with the loss factor ξ as given in section 2.1.2.1. Since loss factors are approximatelythe same for all the investigated orifice flows, pressure losses can also be assumed to be thesame.

3.4.3.2 Droplet Break-up within Orifice Flows

Within our dispersing experiments, only the flow meter with a maximum flow rate of1000 l h−1 was used. Using the other flow meter with a maximum flow rate of 60 l h−1 signif-icantly influenced the dispersing processes due to the small diameter of its measuring section(d2 = 2.4 mm). Given the flow cells described in section 3.4.2.2 and the maximum pressuresupplied, the packing length was restricted to 400 mm.

3.4.4 Experimental Procedures

Dispersing in the process unit described in section 3.4.1 was always performed according to aprocedure consisting of three main steps: pre-emulsion preparation, processing in the processunit, and analyzing the generated emulsions using a Malvern Mastersizer X. Pre-emulsionswere generated immediately prior to the processing and were manually stirred after fillingthe process unit. Processing in the process unit was described in section 3.4.1.1. Particlesize measurements were performed either directly after all runs within the process unit werefinished or on the next day. The stability of our emulsions was confirmed as described inchapter 3.3.5.4.

Chapter 4

Results and Discussion

The goal of this work was subdivided into three parts with the dispersing process being theconnecting theme. Numerically, drop deformation and break-up was studied for single dropsmoving along particle tracks within model geometries resembling representative capillaries ofpacked beds of spherical particles. In the second part, the process of dispersing emulsions inpacked beds and through orifices was studied and compared to the numerical results. Finally,the compression behavior of deformable regularly arranged packed beds was investigated andwe looked at the suitability of those packed beds for the dispersing process.

4.1 Numerical Simulation of Droplet Deformation andBreak-Up

Investigating droplet deformation and break-up was carried out in two steps, firstly calculatingthe velocity field for laminar flows within model geometries and secondly calculating dropletdeformations as they move within the velocity field along particle tracks. SEPRAN was usedto calculate the velocity fields and BIM to investigate droplet deformation and break-up.

Two axisymmetric geometries were considered, one modeled on regular packed beds ofmonodispersed spheres and the other being adjoint 4:1 die-entry flows. Reynolds-numbers Rewere varied between 100 and 1000. BIM calculations were performed over a wide range ofparameters including the initial radial position of the particle tracks, the undeformed dropletradius a and the viscosity ratio λ.

4.1.1 Adjoint Converging Diverging Nozzles

4.1.1.1 Geometry

Stacking eight nozzles with a geometry determined according to Eq. (2.17) derived by Drost[Dro99] and given in section 2.1.2.2 resulted in our model for a regular packed bed of sphericalparticles as shown in Figure 4.1. Each converging-diverging nozzle has a length of 8mm withdiameters of 1mm and 4mm for the narrowest and widest cross-sections respectively. Thelength of the section between the inflow and the first nozzle was chosen to be long enough suchthat the prescribed zero velocity in the spanwise direction was valid given certain Reynoldsnumbers. However, the section between the last nozzle and the outflow could not be chosen

39

40 CHAPTER 4. RESULTS AND DISCUSSION

8 mm15 mm

1 m

m

4 m

m

Figure 4.1: Geometry of eight adjoint converging-diverging nozzles with the prescribed inflowvelocity profile given on the left as studied numerically. The length of the inflow region priorto the first nozzle is 15 mm, the length of the outflow region is 10 mm.

Figure 4.2: Partial 2-dimensional axisymmetric mesh of the converging-diverging nozzle ge-ometry.

to be long enough to meet the criterion of zero velocity in the spanwise direction at the outletas we will see later on.

4.1.1.2 Mesh

For our calculations, we took advantage of the axisymmetric shape of the geometry thusgreatly reducing the size of the numerical problem. The converging part of the first nozzlewithin our 2-dimensional axisymmetric mesh totaling 22560 elements on 46109 nodes is de-picted in Figure 4.2. The triangulation with variations in element length and aspect ratio wasdone accounting for the overall velocity field as well as for the gradients in the field.

The elements used were triangles formed by dividing quadrilaterals along their diagonals.At the inlet a parabolic velocity profile was specified in the streamwise direction. Zero ve-locity in the spanwise direction was prescribed at the inlet and outlet as well as along thecenterline. With no-slip conditions fixed for nodes at the wall, a problem with 89253 degreesof freedom had to be solved. On our machines (specified in Section 3.1.1.2), calculationslasted about an hour regardless of Reynolds-number.

4.1. NUMERICAL SIMULATION 41

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0N

orm

aliz

ed v

eloc

ity v

[−

] and

ann

ulus

are

a dA

[−

]

Normalized radial position r [−]

0.005

0.010

0.015

Dis

cret

e an

nulu

s pr

obab

ility

dis

trib

utio

n P

[−]

2 31 54

**

*

Figure 4.3: Normalized radial velocity v∗ for fully developed pipe flows (), normalizeddifferential annuli area dA∗ (4) at normalized radial position r∗, and the annulus probabilityfor infinitesimal volume elements to pass through certain annuli (solid line). The dotted linesare explained in the text.

4.1.1.3 Annulus Probability

In developed pipe flow a parabolic velocity profile is present. The normalized velocity v∗ ()is depicted over the normalized radial position r∗ in Figure 4.3 along with a normalized dif-ferential annulus area dA∗ (4). The discrete probability P of a random infinitesimal volumeelement passing through an annulus at a certain radial position is shown in discrete form for20 equally wide annuli as a solid line. This annulus probability is proportional to the velocitytimes the differential annulus area, P ∝ v ·dA ∝ r∗− r∗3. The maximum annulus probabilityis found at r∗ =

√

(1/3) ≈ 0.577 with the annulus probability being – generally speaking –higher towards the pipe wall.

The dotted vertical lines in Figure 4.3 indicate those initial radial positions along the inletof the adjoint nozzle geometry for particle tracks investigated within this work. They werechosen to be at 0, 0.33, 0.67, 0.8, and 0.9. Within our droplet deformation and break-upinvestigations, our main focus was directed towards tracks 2 and 3 due to their high annulusprobabilities. Although the annulus probability tends towards zero at the centerline, this trackwas also considered due to its absence of shear.


Elongation rates Shear ratesElongation rates Shear rates

A: Elongation rates

B: Shear rates

4 5

1

2

3

4 5

1

2

3

3.889E+03 4.667E+03 5.444E+03 6.222E+03 7.000E+03

-5.000E-03 7.778E+02 1.556E+03 2.333E+03 3.111E+03

-0.100 38.800 77.700 116.600 155.500

194.400 233.300 272.200 311.100 350.000

Figure 4.4: Elongation and shear rates within the first one and a half converging-divergingnozzles at Re = 100. Particle tracks 1–5 are overlayed.

4.1.2 Flow field within Converging-Diverging Nozzles

4.1.2.1 Reynolds-number Re = 100

Figure 4.4 gives the elongation and shear rates within the first one and a half nozzles of theconverging-diverging nozzle geometry presented in Figure 4.1 for Re = 100. Tracks 1–5 withinitial radial positions as given in the previous section are overlayed. In the area between track5 and the meniscus between the diverging and converging part of the nozzles, recirculationoccurs which is not shown in this graph.

Due to inflow effects, the flow field within the converging part of the first nozzle differsfrom that within the following nozzles where, in the case of low Reynolds-numbers, the flowfield is almost macroscopically periodical. The elongation rates indicate that particles areexposed to positive elongation rates for longer within the first nozzle than in the followingnozzle. This is confirmed in Figure 4.5 which shows the shear and elongation rates for parti-cles moving along tracks 1, 2, and 3 over their travel time for the first two and a half nozzles.

It can be seen in the latter figure that the flow at Re = 100 can be considered macroscopi-cally developed after the first nozzle as indicated by constant peak shear rates for track 2 and3 as well as the similarity between the second and third elongation rate peaks. The maximumvalues of the shear rates increase approximately linearly with radial position, from zero alongthe center line as expected by Newton’s Law (τ = η · γ), though the maximum values fortracks close to the wall have to be treated with caution due to limitations in the fineness of themesh. The shear rates exhibit a saw tooth like behavior with minimum values still exceedingthose of the respective elongation rates.

Contrary to shear rates, elongation rates maintain about the same maximum values over awide range of initial radial positions with a slight decrease towards the pipe wall. As shown


0.00 0.02 0.04

0

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r an

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rate

[1/s

]

time [s]

Track 3

Figure 4.5: Shear and elongation rates (dashed and solid lines, respectively) along particletracks 1, 2, and 3 at Re = 100 for the first two and a half nozzles. Time was set to zero foreach particle track 1mm prior to the beginning of the first nozzle. Dotted lines indicate thethroats of the first and second nozzles.

in Figure 4.4 and expected from theory, elongation rates are zero at the wall.Particles moving along tracks 1–3 experience an elongation within the converging part and

will be compressed within the diverging part. Along the centerline, the maximum compressionis about half the amount of the corresponding elongation. Towards the wall, the ratio ofcompression to elongation becomes greater and becomes almost 1 for track 3.

The design of the nozzle geometry was based on the assumption of developed flow pro-files at each cross-section within the nozzle, as described in Section 2.1.2.2. It was aimed atkeeping the elongation rates along the centerline as constant as possible. It can be seen in theparticle track calculations that the goal was achieved for at least the last half of the convergingpart of the nozzle. Moreover, it is interesting to note that the design goal is even better fulfilledon tracks off the centerline. It should also be noted, that the strain Γ as defined in Eq. (2.28)is much greater within the first nozzle than the strain within the consecutive nozzles if basedsolely upon the elongation rate. This is in accordance with extra pressure losses at the inflowof porous media.

In the annulus between track 2 and 3, which accounts for about 50% of the overall flowrate, the ratio between maximum shear and elongation rates is between 6 and 220. This ratio issmaller towards the centerline and greater towards the wall. The contribution of the elongationrate to the overall strain is almost constant between the radial position of 0 and 0.67 (track 1and 3, respectively), whereas the contribution of the shear increases super-proportionally withthe initial radial position of the track due to higher shear rate values and longer travel times.The fact that the peaks of shear and elongation rates are phase shifted such that maximumshear rates occur shortly after the maximum elongation rates is also advantageous for the


Elongation rates Shear ratesElongation rates

-1.000E-02 3.889E+02 7.778E+02 1.167E+03 1.556E+03

Shear rates

-5.000E-04 7.778E+03 1.556E+04 2.333E+04 3.111E+04

1.944E+03 2.333E+03 2.722E+03 3.111E+03 3.500E+03

3.889E+04 4.667E+04 5.444E+04 6.222E+04 7.000E+04

A: Elongation rates

B: Shear rates

4 5

1

2

3

4 5

1

2

3

Figure 4.6: Elongation and shear rates within the first one and a half converging-divergingnozzles at Re = 1000. Particle tracks 1-5 are overlayed. (Compare Figure 4.4.)

strain experienced by particles.

4.1.2.2 Reynolds-number Re = 1000

We will now look at Re = 1000 pointing out the differences to the former findings for Re =100. The elongation and shear rate plots for Re = 1000 are given in Figure 4.6. The maindifferences observed in this figure are threefold. Firstly, the elongation and shear rates arehigher by an order of magnitude. Secondly, the particle tracks between two nozzles barelyfollow the geometry of the nozzles but rather pass through the nozzles like a jet. Finally,the highest elongation rates are not along the centerline but in the converging part betweentrack 5 and the wall, with a maximum value of ε = 8.48 × 103. However, the likelihood forinfinitesimal volumes passing through this area is very small.

Figure 4.7 shows the shear and elongation rates for tracks 1 to 3 at Re = 1000 pass-ing through all eight nozzles. It reveals another major difference between the two differentReynolds-number flows. Contrary to the case with Re = 100, in which the flow is macroscop-ically developed after the first nozzle, it takes the flow 7 to 8 nozzles to develop at Re = 1000as is best seen along track 2 for the shear rate. The shear rate of track 2 starts out close to zeroin the throat of the first nozzle and reaches its final maximum value after about 7 nozzles dueto the development of a quasi-parabolic velocity profile over the jet.

The macroscopic flow development can also be seen along track 3 – though it is not aspronounced as along track 2. Furthermore, the flow development is indicated by the elonga-tion rates with a decrease in the maximum elongation rate with travel time, particularly fortracks 1 and 2.

Although contributions of the elongation rates to the overall strain Γ is similarly smallcompared to the case with Re = 100, it is instructive to look at qualitative differences between


0.000 0.004 0.008

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/s]

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0

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r an

d el

onga

tion

rate

s [1

/s]

time [s]

Track 3

Figure 4.7: Shear and elongation rates along particle tracks 1, 2 and 3 at Re = 1000. Timewas set to zero for each particle track 1mm prior to the beginning of the first nozzle. Dottedlines indicate the throat of the first and second nozzles.

543

21

Figure 4.8: Particle tracks within the converging-diverging nozzle geometry at a Reynoldsnumber of Re = 1000. The streamwise direction is scaled down by a factor of 6.

the elongation rates at Re = 100 and Re = 1000. In the latter case, we see much higherelongation rate strains within the first nozzle compared to the following nozzles. Thus, infloweffects become more pronounced with an increase in the Reynolds-number. Moreover, thepeak elongation rates within the first nozzle is less dependent on the particle track and theratio between positive and negative peak elongation rate values is much more favorable forthe positive values in case of higher Reynolds-numbers.

Figure 4.8 shows the particle tracks at Re = 1000 indicating the jet formed in this geom-etry at high Reynolds-numbers. The streamwise direction was scaled down by a factor of 6.Elongation and shear rates for tracks 4 and 5 are given in Figures C.1 and C.2 of Appendix C.

4.1.3 Droplet Deformation and Break-up

The basis for our numerical drop deformation and break-up simulations (BIM) was the particletracks described in the previous section. We focused on calculations along tracks 2 and 3 at


Re = 1000 because those tracks represent annuli with a large annulus probability and thehigh Reynolds-number was chosen since the maximum shear and elongation rates are almostdirectly proportional to the Reynolds-number and thus the break-up of smaller droplets isexpected with greater Reynolds-numbers.

Drop deformation and break-up within a mixed flow field gives rise to the definition of twoReynolds-numbers, one for the flow of the continuous phase as used in the previous sections,subsequently called Rec, and one for the drop itself. The latter is called Red.

Our droplet deformation calculations were performed under the assumption that the sec-ond and third components ε2 and ε3 of the elongation rate vector within the velocity gradientin axisymmetric flow as given in Eq. (2.24) are small compared to its first component ε1. Wetherefore neglected ε2 and ε3. It should also be noted that the principal axes of the dropletsare not necessarily aligned with the axes of the plotted droplets as will be shown later on.

4.1.3.1 Droplet Break-up

An initially undeformed droplet with radius of a = 0.0125 mm injected along track 3 priorto the first nozzle oscillates on its way through the adjoint nozzle geometry forming a typicaldumbbell. Its deformation increases over the oscillations in a cumulative manner and it finallybreaks-up within the fifth nozzle at a droplet deformation of L/a ≈ 4.4 as shown in Figure4.9. The numerical calculation for this problem started out with 412 points, took a day, andthe final mesh at break-up consisted of 8529 points.

The viscosity ratio λ between the dispersed phase viscosity ηd and the viscosity of thecontinuous phase ηc is one. Shear and elongation rates as well as travel time are given innon-dimensional form made dimensionless with the characteristic time of Tchar = aηc/σ =0.0125 mm · 0.01 Pa s /10.0 × 10−3 N m−1 = 12.5 × 10−6 s. The continuous and dispersedphase viscosities were chosen to match a typical combination of dispersed and continuousphases within our experimental studies.

In simple shear flow with a viscosity ratio of λ = 1, droplets will not break-up at capillarynumbers below 0.42. This critical capillary number is included as a dashed line at Cacrit =0.42. It is obvious from Figure 4.9 that exceeding the critical capillary number alone is not asufficient criterion for droplet break-up. Otherwise, the droplet would have broken up withinthe first nozzle.

This implies that additional break-up criteria must be taken into account, one being thestrain above the critical capillary number Γac and another one the droplet shape and its inter-nal flow field. According to our observations, the latter factor is strongly influenced by theperiodicity within the flow field.

The strain above the critical capillary number is shown qualitatively as the shaded areawithin Figure 4.9 and given quantitatively in Table 4.1.

It is interesting to note that the first three maximum droplet extensions are reached shortlyafter the dimensionless shear rate drops below 0.42. Likewise, minimal extensions are reachedwhen the shear rates exceed the value of 0.42. Slight deviations from the exact positions givenby the intersection of Cacrit = 0.42 with the shear rate curve are due to the graphical dropletshape output occurring at constant time intervals and inertia effects.


Time [−]

Shear rate [−]Strain

600 800 1000 1200

Elongation rate [−]

0.0

0.2

0.4

Cacrit= 0.42an

d E

long

atio

n ra

te [−

]D

imen

sion

less

She

ar

Figure 4.9: Break-up of a droplet along track 3 with an initial radius of a = 0.0125 mm, aviscosity ratio λ = 1, and Reynolds numbers for the continuous phase and dispersed phase ofRec = 1000 and Red = 0.625, respectively. All maximum and minimal droplet extensionsare shown. The strains above the critical capillary number, Γac, are indicated as shaded areasbelow the shear rate curve.

Table 4.1: Strain, Γac, above the critical capillary number (Cacrit = 0.42 for simple shearflow and λ = 1) along track 3 for each nozzle, along with the maximum critical capillarynumber Camax within each nozzle, and the time tac for which droplets are exposed to capillarynumbers above the critical value.

Nozzle Strain (Γac) [–] Camax [–] tac [–]

1 0.485 0.4524 21.442 1.162 0.4753 34.963 1.525 0.4825 41.684 1.770 0.4868 51.125 1.958 0.4895 55.046 2.089 0.4917 56.407 2.183 0.4934 57.768 2.151 0.4950 48.40


4.1.3.2 Shear and Elongation Rate

The contributions of the shear and elongation rates to the droplet break-up in the examplepreviously discussed were investigated. Two variations were studied.

First, the elongation rate along track 3 was set to zero whereby the droplet break-up oc-curred within the same nozzle, though slightly earlier than with elongation rates included(t = 923 versus 955). This indicates an inferior influence of the elongation rates on the over-all droplet deformation and breakup – at least at a viscosity ratio of λ = 1 and in flows withshear rates being about an order of magnitude greater than their elongational counterparts.

If the droplet was injected between the first and the second nozzles in order to omit infloweffects, this also lead to a break-up within the fifth nozzle at a position almost identical tothe initially investigated case (t = 958 versus 955), again indicating the minor influence ofelongation, at least for a viscosity ratio of λ = 1.

4.1.3.3 Periodicity

Previously, the importance of the transient history of a droplet passing through the adjointnozzle geometry was pointed out. In particular, the periodicity allowing for cumulative ef-fects was mentioned. In order to illuminate the latter aspect, the converging-diverging nozzlegeometry was scaled up in the streamwise direction by a factor of 2, thereby doubling thelength of each of the eight nozzles from 8 mm to 16 mm.

In doing so, the overall flow field was mainly preserved, with the frequency of dropletoscillations being reduced by a factor of 2. Although the overall flow field was kept, certainaspects changed, e.g. the maximum capillary number along certain tracks within each nozzle.The maximum capillary number along track 3 within the first nozzle was Camax = 0.50 com-pared to 0.452 with the unscaled geometry. Therefore, a comparison between the scaled andunscaled geometries in terms of cumulative effects within the dispersing process is difficult.

A droplet suspended on track 3 of the extended geometry already broke up at the end ofthe first nozzle. The strain along this track within the first nozzle is about 2.0. A somewhatcontrived approach to investigate the influence of periodicity might be to simply scale thetravel times within particle track data, without changing other data.

4.1.3.4 Droplet Size

Figure 4.10 shows the deformation and break-up of droplets with varying initial droplet radiia, under the same conditions as discussed in Section 4.1.3.1. Droplets with an initial radiusof a = 0.01 mm do not break-up. Such a drop is shown with its maximum extension ofL/amax = 1.87 reached at the fourth nozzle. As indicated in Table 4.2, droplets up to aninitial radius of a = 0.012 mm do not break-up although the maximum capillary numberpartly reaches 0.475.

Along with an increase in the initial undeformed radius goes a strong increase of themaximum deformation L/amax at break-up. The maximum deformation amounts to 4.44 fora drop with a = 0.0125 mm and 15.3 for a = 0.02 mm. This also influences the break-upbehavior insofar as binary break-up is observed for a = 0.0125 mm, ternary break-up fora = 0.015 mm, and a break-up into possibly even more droplets for a = 0.02 mm whichcannot be predicted by the used BIM code. It can also be observed that the break-up occurs


a = 0.01 mmx = 31.7 mm

x = 34.5 mma = 0.0125 mm

x = 14.3 mma = 0.015 mm

x = 15.1 mma = 0.02 mm

Figure 4.10: Influence of droplet size on its deformation and break-up within the adjoinednozzle geometry along track 3 with λ = 1 and Rec = 1000. x indicates the streamwiseposition beginning at the first nozzle.

Table 4.2: Characteristics for droplet deformation and break-up within the adjoint converging-diverging nozzles over a variety of initial undeformed radii, a. Given are the maximumcapillary number Camax along track 3, the maximum droplet deformation ratio L/amax, thenumber of initial points within the calculation npoints, 0, and the maximum number of pointsnpoints, max.

a Camax L/amax Break-up npoints, 0 npoints, max

[mm] [−] [−]

0.01 0.396 1.87 no 252 6070.012 0.475 2.88 no 412 1418

0.0125 0.495 4.44 yes 412 85290.015 0.593 7.2 yes 642 134420.02 0.791 15.3 yes 252 18590


track 1 track 2 track 3

5

43

2

1

Figure 4.11: Maximum deformations of droplets along tracks 1, 2, and 3 within eight adjointnozzles for droplets with initial undeformed radii of a = 0.0125 mm at λ = 1 and Rec = 1000.Dashed lines indicate the position within the geometry where maximum deformations occur.For the third deformation along track 3 only the contour is given.

further upstream with an increase in the droplet size, though the largest droplets break-upalmost at the same position close to the end of the second nozzle.

4.1.3.5 Particle Track

The capillary number Ca is directly proportional to the sum of the elongation and shear rateG. We have also seen that the maximum values of shear rate within our adjoint nozzles arealmost linearly dependent on the initial radial position of our particle tracks. Therefore, we donot expect droplet break-up for droplets moving along tracks closer to the centerline than track3. Nevertheless, it is instructive to compare droplet deformation along different tracks, as thisshows the strong dependence of the droplet behavior on the initial radial position. Figure 4.11depicts the maximum droplet deformations over their positions within the geometry.

Along tracks 1 and 2, maximum droplet extensions occur within the throat of the nozzlesindicating that elongation within the flow field is the major cause for this behavior. A dropletmoving along track 1 reaches maximum deformations of L/amax = 1.09 and always relaxesback to an undeformed shape between these maxima. A particle moving along track 2 reachesmaximum deformations of L/amax = 1.35 and does not relax back to an undeformed shapebetween two nozzles.

We have seen, that the elongation rates along tracks 1 and 2 can be considered to be almostidentical. Therefore, the larger deformations along track 2 can be attributed solely to the shearrates. In addition, the shear rates cause the droplets to tilt towards the centerline. As notedbefore, the locations of the maximum deformations along track 3 are not directly coupled tothe geometry but – at least for the first three extensions – rather to the strain above the critical


capillary number Cacrit.

These findings give rise to two questions. What happens on tracks closer to the wall wherehigher shear rates are present? It is less likely that particles will move along tracks closer tothe wall, but allowing droplets to shift between tracks – as can be expected in real porousmedia – will eventually lead them more frequently into annuli close to the wall, provided thereal porous medium is long enough. The BIM program is very sensitive to the gradients alongparticle tracks. Therefore, we were limited to tracks distant from the wall.

The second question is the validity of the assumption in the BIM formulation, that shearand elongation rates are constant over the droplet. This assumption holds for infinitesimal vol-ume elements but it would be interesting to know to what extent droplet break-up is influencedby changes of the shear and elongation rate over the droplet surface itself.

4.1.3.6 Entrance Flow

Within this section we will discuss the droplet deformation and break-up behavior within thefirst nozzle along track 2. This track does not, as we have seen before, significantly contributeto the overall dispersing process within the flow through our nozzle geometry at a viscosityratio of λ = 1. Nevertheless, with an increase in viscosity ratio, elongation rates are expectedto become more relevant for break-up mechanisms.

We have seen that elongation rates are not as dependent on the particle track as is the casewith shear rates. Since shear rates are almost zero within the first nozzle along track 2, andin view of the following Section on cumulative effects, track 2 was chosen for the study onentrance flows.

Figure 4.12 shows the maximum droplet deformations within the first nozzle along track2 for viscosity ratios λ = 1, 3, and 5 over undeformed droplet radius a. The respective shearand elongation rates for track 2 were presented in Figure 4.7.

Deformations in the case of undeformed droplets of radii a = 0.0125 mm and a =0.025 mm do not depend on the viscosity ratio. But with maximum deformations of 1.09and 1.21 respectively, these drops are far from being broken up. With larger drops we seea strong dependence of the maximum deformations on the viscosity ratio. For droplets ofradius a = 0.1 mm, maximum deformations amount to 3.38, 2.29 and 1.87 for λ = 1, 3 and 5respectively.

The dotted line at a = 0.1 mm indicates the undeformed droplet radius a above which theassumptions for BIM calculations regarding the ratio of droplet size to narrowest geometrycross-section are no longer met. Nevertheless, calculations above drop radii of a = 0.1 mmallow for qualitative interpretations.

Noteworthy is the exponential increase in the maximum droplet deformation betweendroplet radii of 0.05mm and 0.1mm for all viscosity ratios and in the case of the viscosityratios of 1 and 3 even above 0.1mm. It should also be mentioned that BIM calculations fora droplet size of a = 0.25 mm in the case of a viscosity ratio of λ = 1 and a = 0.3 mm inthe case of λ = 3 terminated during the calculations prior to the specified finish time due tore-meshing difficulties.


0.01 0.02 0.05 0.10 0.20 0.50

1

2

3

4

5

6

7M

axim

um d

ropl

et d

efor

mat

ion

L/a_

max

[−]

Undeformed droplet radius a [mm]

PSfrag replacements

λ = 1λ = 3λ = 5

Figure 4.12: Maximum droplet deformation within the throat of the first nozzle for viscosityratios λ = 1, 3, and 5 of droplets moving along particle track 2 at Rec = 1000. The dottedline is explained in the text.

4.1.3.7 Cumulative Effects

After looking at deformation characteristics for droplets moving along track 2 within the firstnozzle we now extend this investigation in the flow through all eight adjoint nozzles. The factthat the maximum shear rate increases over the nozzles due to the macroscopic flow devel-opment adds another aspect being considered in our investigations. The maximum capillarynumbers, found within the last nozzle, range from Camax = 0.225 to 8.99 for drop radii ofa = 0.0125 mm and 0.5 mm respectively.

In the previous section we pointed out that the assumption for our BIM calculations interms of the ratio between the droplet radii and the narrowest cross-section was not met forradii above a = 0.1. In the case of the flow through eight consecutive nozzles with largedroplets, and particularly within the last nozzles where shear rates reach a plateau, we alsostress the Stokes flow assumption in the BIM formulation since droplet Reynolds-numbersRed exceed 1. Nevertheless, qualitative conclusions appear to be valid also for large dropletswithin the last nozzles. Again, the dotted line at a droplet radius of a = 0.1 mm indicates theradius above which the BIM assumptions are no longer met.

Looking at a drop with radius a = 0.05 mm in Figure 4.13 shows a typical extensionand relaxation sequence of droplets injected on track 2 for a viscosity ratio of λ = 1. Afterthe extension of the droplet within the throat of the first nozzle (), the drop relaxes almostcompletely back to its initial shape between the first and the second nozzles (•). Within thesecond nozzle (4), the drop is elongated again, though not as much as within the first nozzle.With an increase in the shear rate, which for track 2 corresponds with increasing streamwiseposition until a plateau at about the seventh nozzle is reached, the droplet extension is also


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Figure 4.13: Droplet deformation within eight adjoint converging-diverging nozzles alongtrack 2 for λ = 1 at Rec = 1000. The arrows at droplet radius a = 0.05 mm are added toindicate the chronological order of the droplet extensions and relaxation along its path throughthe nozzles. The droplet deformations for the first nozzle (), connected by the dashed line,are identical with the previous figure.

increased as seen at the throat of the third (+) and fourth (×) nozzles. No relaxation occursbetween the second, third, and fourth nozzles, though the rate at which the droplet extendsslows down between adjoint nozzles. The BIM calculations finally terminate close to thethroat of the seventh nozzle with a droplet deformation of L/a = 14.8 indicating break-up.

None of the calculations presented in this Section resulted in a dumbbell shaped dropletas seen in Figure 4.10 for droplets moving along track 3 with a viscosity ratio of λ = 1.However, some of the droplets were extended such that break-up could be predicted from thesize of their deformations.

With a drop diameter of a = 0.025 mm, the same sequence develops as with 0.05 mm,except that no break-up occurs, although the maximum capillary number amounts to Camax =0.50 within the last nozzle. The minimum drop size that will be broken up along track 2 istherefore between a = 0.025 and 0.05 mm. The fact that in both cases the droplets relax backalmost to their initial shape between the first two nozzles confirms our previous finding, thatfor λ = 1 the inflow effects do not influence the break-up.

Nonetheless, it must be pointed out that droplet deformations within the second nozzle arealways smaller than within the first nozzle, indicating the strong impact of the inflow. For anydrop diameter, the deformation within the first nozzle is overcome for the first time withinthe third nozzle, with maximum shear rates exceeding those of the maximum elongation rateswithin the first nozzle by a factor of 3.

For drop radii of a = 0.0125 mm and 0.025 mm the deformation maxima were observedat the throats of the nozzles or at positions slightly further downstream. BIM calculations for


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Figure 4.14: Droplet deformation within eight adjoint converging-diverging nozzles alongtrack 2 for λ = 3 at Rec = 1000.

drop radii larger than or equal to a = 0.05 mm terminated due to large drop deformations atpositions further upstream with greater drop sizes.

Figures 4.14 and 4.15 show results for the same parameters except for a variation in theviscosity ratio λ. None of the calculations performed with λ = 3 or 5 indicates droplet break-up. Maximum deformations found are L/amax = 5.69 and 2.87 for λ = 3 and 5 respectively,located within the last nozzle and showing the strong influence of the viscosity ratio on thedroplet deformation and break-up.

In the case of λ = 3, a similar sequence can be observed as with λ = 1, with deformationswithin the second nozzle being smaller than those within the first nozzle. Furthermore, thedeformation within the last nozzle always exceeds that within the first nozzle. The shoulderon the curve for the deformation within the eighth nozzle for droplet radii of a = 0.07 mm to0.25 mm seems to be worth mentioning though it might be related to the non-compliance tothe BIM assumptions.

For a viscosity ratio of λ = 5 (Figure 4.15), maximum droplet deformations within the lastnozzle can even be found to be smaller than deformations within the first nozzle. Droplets witha radius of a = 0.075 mm or above deform to a larger extent within the first nozzle than theydo within the last nozzle. This holds true even though the shear rates within the last nozzleare much higher compared to the elongation rates within the first nozzle. This suggests, thatinflow effects become increasingly important for higher viscosity ratios. It is also noteworthy,that deformation within the last nozzle approach an upper limit with increasing droplet sizes.Again, the non-compliance with BIM assumptions for large droplets, particularly within thelast nozzles, has to be taken into account.


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1st nozzle1st relaxation2nd nozzle3rd nozzle4th nozzle8th nozzle

Figure 4.15: Droplet deformation within eight adjoint converging-diverging nozzles alongtrack 2 for λ = 5 at Rec = 1000.

4.1.4 Orifice Flow

Besides converging-diverging nozzle flow calculations, the flow within adjoint die entries wascalculated with SEPRAN at various Reynolds-numbers Rec. The axisymmetric geometry isdepicted along with shear and elongation rates for Rec = 100 in Figure 4.16.

The flow pattern within the nozzles resembles that within the adjoint converging-divergingnozzles at a Reynolds-number of Rec = 1000, which is one order of magnitude above theReynolds-number considered here. The formation of the jet is as pronounced as within theconverging-diverging nozzle at Rec = 1000.

Except that the maximum value of the shear rates are comparable to the correspondingones in the nozzle flow and the fact that the flow is macroscopically developed after the firstdie, the shear and elongation rates differ from those observed in the nozzles at the sameReynolds-number (compare Figure 4.4). Firstly, the peak values of the elongation rates oftrack 2 are of the same magnitude as the respective shear rates although the exact value ofthe maximum elongation rate of track 3 has to be treated with caution due to limitationswith regard to the discrete numerical calculations. Secondly, shear rates show a pronouncednegative peak within the vicinity of the first die entry.

Shear and elongation rates likewise exhibit very high gradients which are even more pro-nounced at Reynolds-numbers of Rec = 1000. Therefore, BIM calculations were at best dif-ficult and at the worst impossible to perform. This gives rise to the assumption, that dropletswithin die entry flows are not broken up as smoothly as in the nozzle flows, resulting presum-ably in broader particle size distributions.


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Figure 4.16: Shear (dashed lines) and elongation rates (solid lines) along particle tracks 1, 2and 3 for adjoint die entries at Rec = 100. Time was set to zero for each particle track 4mmprior to the beginning of the first nozzle.

4.2. DISPERSING PROCESS 57

Table 4.3: Units used for physical quantities within our statistical models. The first two rowsshow the dependent variables, the other ones give explanatory variables.

Physical quantity Notation Unit

Mean droplet diameter x50,3 µmwidth of PSD span –number of runs nruns –pressure differences ∆p barviscosity ratio λ [–]length L mcontinuous phase viscosity ηc Pa sdiameters ds and do mporosity ε [–]volume fraction of dispersed phase Ψd [–]Reynolds numbers Rep and Reo [–]interfacial tension σ 10−3 N m−1

4.2 Dispersing Process

The dispersing process was studied within two sets of flow geometries related by the fact thata periodic strain is imposed upon droplets passing through them. One comprised packed bedsof spherical particles and the other, a model geometry for the former, consisting of adjoint dieentries.

The goals of this section are twofold. One is the elucidation of the droplet breakup mech-anisms within periodic flows. The other is to find statistical models to predict dispersingresults over a wide range of fluid and geometry parameters with the models based solely uponphysical parameters. The statistical models derived within this section are based upon thenumerical values of physical quantities given in terms of the units indicated in Table 4.3.

4.2.1 Dispersing in Sphere Packing Flow

Figure 4.17 shows a typical dispersing result in terms of the cumulative volume distribution,Q3, for emulsions that were produced by being passed through the packed bed up to 100times at one of four distinct pressure differences across the bed. The packing Reynolds-number Rep increased, along with the pressure difference, from 19.8 to 77.9, with the frictioncoefficient Λ for this packed bed given in Figure 3.5. The viscosity ratio λ between dispersedand continuous phase viscosities was 2.26.

The PSD of the pre-emulsion (, solid line) is depicted on the right hand side of Figure4.17. The particle size characteristics x90,3, x50,3, x10,3, and span for the pre-emulsion are146.3µm, 80.3µm, 33.2µm, and 1.41µm, respectively.

Passing this pre-emulsion ten times through the packed bed at a pressure difference of∆p = 0.41 bar (4) strongly reduced the size of the droplets as seen in a shift in the curvetowards the left. Increasing the number of passages nruns from 10 to 20, 50 and 100 (+, ×and ♦ respectively) further reduces the mean diameter of the emulsion droplets x50,3 and alsoreduced the span of the emulsions as the volume of the larger particles is reduced more than the


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Figure 4.17: Particle size distributions (PSD) for dispersing 2% (v/v) rape seed oil in10% (w/w) – PEG in 2% (w/w) SDS – de-mineralized water solution within an orthorhombi-cally (I) arranged 80 mm long sphere packing of ds = 4 mm spheres at four pressure differ-ences and a variety of number of runs nruns.

volume of smaller particles being generated. A minimum particle diameter of xmin = 6 µmregardless of the number of runs indicates a lower limit for particle sizes at this pressuredifference.

The emulsion generated with 100 runs at a pressure difference of ∆p = 0.41 bar wasused as the pre-emulsion for the runs at the following pressure difference of ∆p = 0.79 barwhich were repeated 10 times (∇). The product after ten runs at a pressure difference of∆p = 0.79 bar was itself used as the pre-emulsion for the runs at a pressure difference of∆p = 1.40 bar () and the emulsion from the tenth run at this pressured difference waslikewise used as the pre-emulsion for the runs at a pressure difference of ∆p = 2.23 bar (∗).It can be seen, that the shape of the PSD curves after 10 runs (dotted lines) are identical, andthey are distinguished solely by their position along the abscissa. The intervals between thefour pressure differences were chosen to be high enough such that the same emulsion wouldhave been generated regardless of whether the emulsions from the last run of the precedingpressure difference or the very first pre-emulsion was used.

Note that the PSDs for the first 1-3 runs sometimes indicated that particularly large parti-cles were not passed through the sphere packings, and therefore significantly influenced theparticle size distribution. This must be attributed to the batch-wise processing of the emul-sions.


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Specific energy input Evn = ∆p × nruns [bar]

x90,3x50,3x10,3

Figure 4.18: Particle size characteristics vs. specific energy input for the results of the dis-persing process within a packed bed of spheres shown in Figure 4.17.

4.2.1.1 Energy and Power Input

A correlation between the particle size characteristics (x90,3, x90,3, x90,3, and span), the num-ber of runs nruns, and the energy input Ev = ∆p becomes obvious when plotting the particlesize characteristics versus the specific energy input Evn = Ev · nruns. Figure 4.18 shows thisfor the previously presented results.

The solid line connects the mean diameters over the energy input Evn. Instead of symbolsindicating the size of the mean diameters, two numbers are given representing the numberof runs nruns and the span. The latter number can also be deduced from the x90,3 and x10,3

values which are indicated by symbols ( and 4) connected via dotted and dashed linesrespectively. Pre-emulsion data is not shown. The curves are best read by moving along thez-shaped curves beginning on the left-hand side. This corresponds to the shift of the curves inFigure 4.17 from the right-hand side towards smaller particles on the left-hand side.

Three notable findings emerge from studying Figure 4.18. Firstly, the mean diameter x50,3

decreases almost linearly with the number of runs at a given pressure difference. Accordingto our interpretation of the previous figure, we expect the mean diameter to be bounded by alower limit, though we have never been able to show this. Much higher numbers of passes areneeded to explore this assumption. Secondly, an increase in pressure difference for a constantnumber of runs also results in a linear decrease in the mean particle diameter. The gradient forthis decrease is steeper than for the decrease with the increase in the number of runs. Finally,the width of the particle size distribution (span) decreases with the number of runs and isalmost constant with the increase in the pressure difference for a constant number of runs.

Eq. (4.1) gives the model x50,3 –pack – I accounting for the first two findings. Logarithmicvalues of model variables were used resulting in better fits than with the untransformed values.


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Figure 4.19: Mean diameters x50,3 from the dispersing experiment characterized within theprevious two figures along with model predictions according to Eq. (4.3). For pressure dif-ferences ∆p = 0.41, 0.79, 1.40 and 2.23 bar (dashed lines) and for 10, 20, 50 and 100 runs(dotted lines).

For the sake of readability, this model is given with untransformed variables in Eq. (4.2). Thevalues for the coefficients c1, c2, and c3 were estimated by the statistics package ‘R’ with aresidual standard error of 0.06083 for 4 degrees of freedom, an adjusted R2 of 0.9849 and anaverage deviation of 0.0387. They are given in Eq. (4.3). All explanatory variables within themodel are significant and the former two are even very strongly significant.

ln(x50,3) = ln(c1) + c2 · ln(∆p) + c3 · ln(nruns) (4.1)

x50,3 = c1 · ∆pc2 · nrunsc3 (4.2)

x50,3 = 13.54 · ∆p−0.789 · nruns−0.142 (4.3)

Figure 4.19 shows the predictions from our statistical model overlayed with experimentaldata. The accuracy of the model can be seen in the deviations of the experimental data from theintersections of the dashed and dotted lines, which represent constant pressure and constantnumbers of runs respectively.

The coefficients in the model Eq. 4.3 can be interpreted as the efficiency of the energy inputversus the power input. An increase in energy input Evn as given by the abscissa of Figure4.19 results in a decrease of the mean diameter by a power of -0.142, given constant pressuredifferences. Increasing the power input by increasing the pressure difference for a constantnumber of runs leads to a decrease of mean diameters by a power of -0.789. Therefore, anincrease in power input is much more efficient for reducing the size of the particles generated.

Given our combination of fluids, the packed bed and the task of producing an emulsionof a certain quality, defined by the mean droplet diameter and the width of the particle size


distribution, at the lowest possible energy, the task is to find an appropriate balance betweenthe pressure difference and the number of runs, since the width of the particle size distributionwas found to be dependent on the number of runs. The width of particle size distributions willbe discussed in more detail below (in section 4.2.1.5).

4.2.1.2 Packing Length and Viscosity Ratio

In sections 4.1.3.2 and 4.1.3.7 we found indications that inflow effects are not of high impor-tance for a viscosity ratio of λ = 1 but have to be taken into account for viscosity ratios of 5.Within this section we compare this with our experimental findings.

Therefore, four experiments were considered with varying parameters on two levelseach. One parameter was the viscosity ratio between the continuous phase, chosen to be a19% PEG – 2% SDS – H2O solution, and the dispersed phase. Using the silicone oil AK 250as the dispersed phase provided a viscosity ratio of λ = 1.71, AK 1000 a viscosity ratio of6.93. The length of the sphere packing L was the second variable parameter with values of Ltaken to be 20 mm and 100 mm.

The packing structure of cubically arranged spheres with diameters of ds = 4 mm waskept for all experiments. Two of the experiments were performed twice and showed goodrepeatability. 32 data points were considered with Reynolds-numbers (Rep) ranging between3.96 and 17.7, the number of runs (nruns) ranging between 5 and 105, and pressure differences(∆p) between 0.08 bar and 1.55 bar.

As a first step, for each viscosity ratio, the packing length was varied and the experimentaldata was tested against the model x50,3 –pack – II given in Eq. (4.4). For any packed bed,the packing length L is coupled to the pressure difference ∆p via the friction coefficientΛ being itself a function of the Reynolds-number Rep. In order to decouple the length Lfrom the pressure difference ∆p, the Reynolds-number Rep was chosen in Eq. (4.4) insteadof the pressure difference ∆p as used within the model of the previous section (Eq. 4.1). TheReynolds-number Rep is independent of the viscosity ratio λ and of the packing length L,thus it is independent of the variable parameters in the experiments.

ln(x50,3) = ln(c1) + c2 · ln(Re) + c3 · ln(nruns) + c4 · ln(L) (4.4)

x50,3 = c1 · Repc2 · nruns

c3 · Lc4 (4.5)

For both viscosity ratios, all parameters were very strongly significant and the estimatedcoefficients are listed in the second and third columns of Table 4.4. The average deviationbetween experimental data and model predictions is given in the last row of the table.

The coefficients for the two viscosity ratios show similar values. Nevertheless, comparingsingle coefficients is difficult due to the dependence of the explanatory variable (i.e. the meandiameter x50,3) on the product of all the parameters and therefore on all of the coefficients.

Therefore, different models applicable to the data from all four experiments were tested(with two experiments performed twice), including the viscosity ratio as an explanatory vari-able. The best fit was achieved by considering all variables as main effects – as done before– and additionally taking an interaction between the packing length and the viscosity ratioλ into account. This model (x50,3 –pack – III) is given in Eq. (4.6) along with its estimatedcoefficients in the last column of Table 4.4. All parameters are again very strong significantand the adjusted R2 was 0.9817.


Table 4.4: Estimated coefficients of the statistical models x50,3 –pack – II and x50,3 –pack – IIIgiven by Eqs. (4.5 and 4.7) for four dispersing experiments with varying viscosity ratios andpacking lengths. The average deviations between experiment and model predictions are alsogiven.

Modelx50,3 –pack – II x50,3 –pack – III

Coefficient λ = 1.7 λ = 6.9 λ = 1.7, 6.9

c1 (Intercept) 116.7 407.5 37.3c2 (Re − number) -1.07 -0.710 -0.925c3 (nruns) -0.248 -0.186 -0.209c4 (L) -0.483 -0.152 -0.589c5 (λ) n/a n/a 1.45c6 (L : λ) n/a n/a 0.227average deviation 0.075 0.054 0.068

ln(x50,3) = ln(c1) + c2 · ln(Re) + c3 · ln(nruns) + c4 · ln(L) +

+c5 · ln(λ) + c6 · ln(L) · ln(λ) (4.6)

x50,3 = c1 · Repc2 · nruns

c3 · Lc4+c6·ln(λ) · λc5 (4.7)

The coefficients of the latter model prove our assumption that the increase in the viscosityratio from 1 to 5 goes along with the elongational flow in the inflow region becoming moreimportant for the dispersing process. This can be seen by looking at the exponent of thepacking length L in Eq. (4.7). It reads c4 + c6 · ln(λ). In the case of a viscosity ratio of λ = 1,this exponent becomes -0.589 whereas in the case of λ = 5 it becomes -0.224.

Our assumption is validated by comparing the latter two values to the coefficient of thenumber of runs c3 = −0.209, giving the relative importance of those two parameters on theoverall dispersing result. In the λ = 1 case, the length of the packing has a greater influenceon the size of the particles, as seen in the smaller coefficient of -0.589 compared with thecoefficient of -0.224 for the number of runs. However, in the λ = 5 case, the two coefficientsare -0.224 and -0.209, i. e. approximately the same, showing that the relative importance ofthe number of runs increases, thus showing the importance of inflow effects.

Besides this conclusion, there is a significant overall influence of the viscosity ratio onthe mean diameter x50,3. The main effect of the viscosity ratio as given by coefficient c5 issomewhat disguised by the interaction coefficient c6, though with c5 being much greater thanc6, we can concentrate on the main effect, which shows a strong increase in diameter with anincrease in viscosity ratio.

Note that the model could have been improved slightly in terms of average deviation byincluding an interaction between the Reynolds-number Rep and the packing length L. Asthis would not have affected the conclusions drawn from our experiments, we neglected thisinteraction for the sake of clarity.


Table 4.5: Range of the dependent variable x50,3 and the explanatory variables over the trialsused for our estimation of coefficients within the model x50,3 –pack – IV given by Eq. (4.8).The range of packing Reynolds-number (Rep) and friction coefficient (Λ) have been appendedfor completeness. Units used are those applicable within the model.

Variable Range

mean diameter x50,3 5.49 – 166.1 µmpressured difference ∆p 0.08 – 5.93 barnumber of runs nruns 1 – 105viscosity ratio λ 1.71 – 10.43packing length L 0.01 – 0.40 mcontinuous phase viscosity ηc 0.0011 – 0.14 Pa ssphere diameter ds 70×10−6 – 0.004 mpacking porosity ε 0.2595 – 0.476interfacial tension σ 4.0 and 10.0 ×10−3 N m−1

packing Reynolds-number Rep 3.96 – 1003friction coefficient Λ 118 – 1134

4.2.1.3 Mean Diameter Model for Sphere Packing Flow (x50,3 –pack – IV)

One of our mail goals was the establishment of a statistical model, free of geometry specificcoefficients, predicting mean diameters of emulsions processed through porous media undervarious process and fluid conditions. Therefore, 26 trials were conducted over a wide rangeof parameters. The range of parameters is shown in Table 4.5 and a list of geometry and fluidparameters for all trials conducted for this work is given in appendix B.

Our approach in setting up a statistical model followed the procedure in dimensional anal-ysis whereby all explanatory parameters are furnished with a power coefficient and are mul-tiplied together. The explanatory variables were chosen to be primary and independent. Ourresulting model x50,3 –pack – IV is given in Eq. (4.8).

ln(x50,3) = ln(c1) + c2 · ln(∆p) + c3 · ln(nruns) + c4 · ln(λ) + c5 · ln(L) +

c6 · ln(ηc) + c7 · ln(ds) + c8 · ln(ε) + c9 · ln(σ)

x50,3 = c1 · ∆pc2 · nrunsc3 · λc4 · Lc5 · ηc

c6 · dsc7 · εc8 · σc9 (4.8)

The coefficients for this model were estimated over 125 degrees of freedom with anadjusted R2 of 0.9419 and an average deviation of 0.163 to c1 = 2.94, c2 = −0.679,c3 = −0.151, c4 = 0.770, c5 = 0.371, c6 = 0.234, c7 = −0.248, c8 = −0.747, andc9 = 0.530, with all parameters being very strongly significant. Comparing the coefficientsfor the pressure difference and the number of runs (c2 = −0.679 and c3 = −0.151 respec-tively) to those of the previous investigations confirms the same qualitative influence of thepower input versus the energy input with the former being approximately 4.5 times more ef-ficient. A positive viscosity ratio again indicates the need for higher energies to break updroplets with similarly higher viscosity ratios. The coefficient c6 = 0.234 for the packinglength has to be judged in relation to the the pressure difference, since they are coupled viaDarcy’s Law.


As pointed out with Eq. (2.31) in Section 2.2.2.2, the dispersing result within orifice flowswas found to be a function of an empirical geometry coefficient and the pressure differenceto the power of b, where b is a function of the Reynolds-number. Karbstein [Kar94] reportedvalues of b = 1 and 0.25 ≤ b ≤ 0.4 for laminar and turbulent flows, respectively. Thecoefficient b can be qualitatively compared to c2 in model Eq. (4.8). Our experimental datamainly represented laminar flow conditions with a maximum packing Reynolds-number ofRep = 1003. Therefore, it is admissible to choose coefficient c2 to be constant and to ignorethe fact that this coefficient decreases for Reynolds-numbers in the turbulent regime.

In the previous section, we found an interaction between the packing length and the vis-cosity ratio. When this interaction is included in the current model, it also proves to be verystrongly significant. However, the coefficient is very small compared to the correspondingvalue in the last section (0.04 versus 0.227) and the average deviation becomes worse (0.198compared to 0.163). The fact that in some cases stratified packings with different layers ofvarious monodispersed spheres were used might have blurred the significance of the infloweffects.

The interfacial tension (σ) used within the model is based on two fluid type combinations,one being various silicone oils in PEG – SDS – H2O solutions with varying PEG content andthe other rape seed oil in 10% PEG – SDS – H2O solution. While the former exhibits constantinterfacial tension, that of the latter depends on the droplet break-up kinetics and time. There-fore, our model must be considered as a model for two levels of interfacial tension and isrestricted to fluids with a constant interfacial tension of 10.0×10−3 N m−1 or the combinationof rape seed oil in 10% PEG – SDS – H2O solutions as we used them.

Our model tries to describe break-up mechanisms that have been extensively studied inthe literature for simple flow fields. Break-up investigations for such flow fields, e.g. as givenin Figure 2.9, imply that the droplet break-up (in terms of a critical capillary number Cacrit)can not be given as a simple exponential function in terms of the viscosity ratio (λ). There-fore, assuming such a simple exponential relationship as is implicit in our model is a goodapproximation, proved overall by the good predictions made by the model, although it doesnot model the microscopic flow behavior within the porous media.

4.2.1.4 Influence of Dispersed Phase Volume Fraction

Accompanying an increase in the dispersed phase volume fraction φd, there is a change inthe rheological behavior of the emulsion as well as an increase in the recoalescence proba-bility during the dispersing process as described in sections 2.2.2.2 and 2.2.2.3. With dis-persed phase volume fractions ranging from 2% to 10% in our experiments, we did not ob-serve changes in flow characteristics and therefore we observed no influence of viscoelasticemulsion properties within porous media flow. Nor was the dispersed phase volume fractionsignificant when included in the statistical model of the previous section (p-value of 0.35).

4.2.1.5 Width of Particle Size Distribution (span –pack – IV)

The width of the particle size distributions obtained in the dispersing experiments is partic-ularly sensitive to large droplets being present during the first two runs through the porousmedia. Therefore, the model developed in this section is based on the same data as usedin section 4.2.1.3 for model x50,3 –pack – IV except, that data for the first two runs and two


outlying data points are omitted. Therefore, 100 data points are available with particle sizedistribution widths ranging from 0.705 to 2.13 and a mean value of 1.35.

Starting out with model equations containing all explanatory variables as used in modelEq. (4.8) and successively removing non-significant variables as well as adding the packingReynolds-number as an explanatory variable resulted in model span –pack – IV best fittingour data as given in Eq. (4.9).

ln(span) = c1 · ln(nruns) + c2 · ln(λ) + c3 · ln(ηc) +

c4 · ln(ε) + c5 · ln(σ) + c6 · ln(Rep)

span = nrunsc1 · λc2 · ηc

c3 · εc4 · σc5 · Repc6 (4.9)

With an adjusted R2 value of 0.8579, and an average deviation of 0.22, all parameters werefound to be very strongly significant, with the coefficients being c1 = −0.0689, c2 = 0.169,c3 = 0.185, c4 = −0.354, c5 = 0.173 and c6 = 0.0598. Note that the packing length was notfound to significantly contribute to the width of the particle size distribution, thus indicatingthat the width of the PSD is governed by inflow effects regardless of the viscosity ratio. Thisis surprising since the packing length was found to significantly influence the mean diameterfor processes with viscosity ratios of λ = 5.

As pointed out earlier, the width of the PSD decreases with the number of runs as indi-cated by a negative exponent (c1 = −0.0689). Moreover, higher viscosity ratios correspondwith broader particle size distributions which is in accordance with the findings for the meandiameter. Another surprising finding is that the PSD width depends on the continuous phaseviscosity, such that higher viscosities result in broader particle size distributions. Further-more, the widths of the PSDs become narrower as packings become less porous and is broaderwith silicon oil as the dispersed phase compared with rape seed oil. With packing Reynolds-number between 3.95 and 545 for the data points considered within this investigation, higherReynolds-number correspond to broader PSDs.

In Section 4.1.3.4, we pointed out that the number of droplets an initially spherical dropletis broken-up into along certain particle tracks depends on the initial droplet diameter. Dropletsizes resulting in capillary numbers slightly above the critical capillary number were expectedto undergo binary break-up. With an increase in droplet size, the number of resulting dropletsincreased. The generation of small satellite drops also influences the size distribution of thebroken-up droplets. Moreover, the track along which a particle passes through the porousmedium strongly influences the dispersing result.

4.2.1.6 Comparison with Numerical Simulations

In numerical simulations (Section 4.1.3.4), we found that a droplet of initial undeformed ra-dius a = 12µm moving along track 3 was the largest drop not being broken up. In those cal-culations, the viscosity of the continuous phase was chosen to be ηc = 10mPas, the viscosityratio to be λ = 1, and the interfacial tension was set to σ = 10.0× 10−3 N m−1 thus matchingthat between silicone oil and PEG – 2% SDS – H2O solutions as used within our experiments.The flow through the 64 mm long converging-diverging nozzle at Reynolds-number 1000 re-sulted in a pressure loss of ∆p = 1.73 bar.

The shape of the orthorhombic (I) sphere packing is closest to that of our converging-diverging nozzle in terms of normalized void area over normalized streamwise position as


Table 4.6: Model predictions for the dispersing process within incompressible porous mediaaccording to model x50,3 –pack – IV (Eq. 4.8) for fluid and geometry parameters matchingthose in the numerical investigations.

Sphere size Predicted mean diameterds [mm] x50,3 [µm]

1 5.322 4.483 4.054 3.77

shown in Figure 2.8. It was therefore chosen for our comparison and has a porosity of ε =0.395. The number of runs (nruns) was taken to be one.

Our model is based on sphere diameters, whereas the width of the converging-divergingnozzle is given in terms of pipe diameters. Several models relating these two diameters exist,but they vary considerably. We did not chose one of these models but rather conducted acomparison for several sphere diameters. Table 4.6 gives predicted mean diameters in termsof sphere diameters according to model Eq. (4.8) with parameters given above.

The fact that smaller sphere diameters result in larger mean diameters is due to the cou-pling of the sphere diameter to the fluid velocity within the porous medium and the pressuredrop across it. With a decrease in sphere diameter, the volume specific surface area increasesand thus more pressure is needed to overcome wall friction resulting in slower flow and thusin larger droplets.

The numerical result matches the experimental data well if the following two two pointsare taken into account. Firstly, the numerically determined drop radius of a = 12 µm repre-sents the largest drop which will not be broken up whereas the diameters given in Table 4.6represent the mean diameter of an emulsion. Secondly, it is expected that numerical simu-lations of droplet break-up along tracks closer to the wall will predict that smaller dropletswill be broken up, although no quantitative data is available. Overall, the comparison is verysatisfactory.

4.2.2 Dispersing in Orifice Flow

In this section, dispersing emulsions within single and adjoint orifice flows through one orificeand through nine parallel orifices was studied. Again, we focused on the mean diameter ofthe droplets in the emulsions generated and the width of their particle size distributions. Ouranalysis was based on the trials detailed in Appendix B, representing 56 data points, wheredata for the first two runs were omitted for reasons given in previous sections.

Table 4.7 lists the range of parameters varied over the trials. Three values were consideredfor the orifice diameter (do = 1, 2.4 and 8.8 mm) and for the number of adjoint orifices(ndies = 1, 2, 4). Although at Reo = 2935, the Reynolds-number reaches with a value wellabove the critical Reynolds-number for pipe flow of Recrit, pipe = 2300, only two out of 56data points were above the critical Reynolds-number, indicating laminar flow conditions foralmost all of the experimental data.

In all trials, 5.5% – PEG – SDS – H2O solutions were used. Therefore, continuous phase


Table 4.7: Range of the dependent variables x50,3 and span along with the explanatory vari-ables used to estimate the coefficients in model Eqs. (4.10 and 4.11). The units used are thoseapplicable to the model.

Variable Range

mean diameter x50,3 2.58 – 118.9 µmwidth of PSD span 0.970 – 2.65 µmReynolds-number Reo 461 – 2935number of runs nruns 3 – 40viscosity ratio λ 1.05, 10.43orifice diameter do 0.001, 0.0024, 0.0088 mnumber of dies ndies 1,2,4continuous phase viscosity ηc 0.0092 Pa s (5.5% PEG – 2% SDS - H2O)interfacial tension σ 10.0 ×10−3 N m−1

viscosity was not considered to be a main effect within our investigations. Choosing siliconoils AK 10 and AK 100 as the dispersed phase, viscosity ratios of λ = 1.05 and 10.43 wereachieved with a constant interfacial tension of σ = 10.0 × 10−3 N m−1.

In this section, the derived models are given in terms of Reynolds-number rather thanpressure difference ∆p. On one hand, we had found in the investigations of the packing flow,that the width of the PSD was predicted more accurately using the Reynolds-number. Onthe other hand, the mean diameter model derived for the packing flow could also have beenstated in terms of the Reynolds-number without a significant loss in accuracy. Assumingsimilar behavior in orifice flow seems valid, though we cannot prove it due to a partial lack ofpressure loss data for the orifice flow.

Including interaction terms in the models formulated below would have slightly improvedthe models at the cost of clarity. Interactions like that between Reynolds-number and viscosityratio were found to be significant for the model for the width of the PSD but difficulties ininterpretation made us disregard this interaction.

4.2.2.1 Mean Diameter Model for Orifice Flows (x50,3 –orif)

The mean diameter model for emulsions generated in orifice flows (x50,3 –orif) is given inEq. (4.10). All variables as given in Table 4.7 were included in the model and proved to bevery strongly significant.

ln(x50,3) = ln(c1) + c2 · ln(Reo) + c3 · ln(nruns) +

c4 · ln(λ) + c5 · ln(do) + c6 · ln(ndies)

x50,3 = c1 · Reoc2 · nruns

c3 · λc4 · doc5 · ndies

c6 (4.10)

The coefficients were estimated with an adjusted R2 of 0.985 and an average deviation of0.0844 to be c1 = 1.894 × 109, c2 = −1.277, c3 = −0.2608, c4 = 0.1258, c5 = 1.454 andc6 = −0.2147.

Similar to our observations for dispersing flows in porous media, where increasing thepressure drop strongly reduced the mean diameter, we found that the mean diameter in orifice


flow showed a similar dependence on the Reynolds-number with a coefficient of c2 = −1.277.For laminar flow, the Reynolds-number is approximately a linear function of the pressuredrop. Thus, a comparison with the coefficient of pressure drop in the mean diameter model forpacking flow seems appropriate and coefficient c2 can be regarded as a power input coefficient.

The mean diameter reduces with an increase in the number of runs as indicated by coef-ficient c3 = −0.2608 which can be also be interpreted as an energy input coefficient. It isinteresting to note that the ratio of 4.9 between the coefficients of the power input and theenergy input is similar to the value of 4.5 found in the case of flow through a packed bed ofspheres in Section 4.2.1.3.

Again, the viscosity ratio strongly influences the mean diameter of emulsions generated,with higher viscosity ratios resulting in larger particles. With larger orifices, larger dropletswill be generated. It is also interesting to note, that the number of dies significantly influencesthe dispersing result, although we had pointed out in our numerical investigations that even foradjoint orifice flows at Reynolds numbers of 100, and thus far below the smallest Reynolds-number found within our experiments, jets form within adjoint nozzles. The number of diesmight therefore rather be a variable representing the length of an orifice.

4.2.2.2 Width of Particle Size Distributions (span –orif)

The diameter of the orifices did not significantly influence the width of the PSDs. Our modelpredicting the width of PSDs for the emulsions processed (span –orif) is given in Eq. (4.11).

ln(span) = ln(c1) + c2 · ln(Reo) + c3 · ln(nruns) + c4 · ln(λ) + c5 · ln(ndies)

span = c1 · Reoc2 · nruns

c3 · λc4 · ndiesc5 (4.11)

All parameters are significant and the intercept, the Reynolds-number and the viscosityare very strongly significant with coefficients estimated to be c1 = 3.528, c2 = −0.120,c3 = −0.0532, c4 = 0.1382 and c5 = −0.0629.

Although the average deviation was found to be 0.079, an adjusted R2 of 0.719 indicatesa rather poor fit to the data. Therefore, care has to be taken in interpreting the results. To fea-tures can in any case be mentioned, one being the decrease in the PSD width with increasingnumbers of runs and the second being the dependence of the span on the viscosity ratio, withthe span increasing as the viscosity ratio increases. Both findings accord well with those fordispersing within porous media.

4.3. COMPRESSIBLE POROUS MEDIA FLOWS 69

4.3 Compressible Porous Media Flows

This section deals with flow investigations of flow through compressible periodically arrangedporous media. The goal was twofold: firstly to characterize packing compressibility; secondlyto estimate the usability of such compressible porous media for dispersing processes, henceclosing the circle of our explorations.

The former goal was accomplished by systematically investigating the flow through pack-ings under a variation of five parameters. Those parameters were the packing length, thepacking structure, the material of the spheres, the viscosity of Newtonian fluids, and the pres-sure drop across the porous medium. The first three of theses parameters were each assignedvalues which in the following discussion will be denoted as short and long, cubic and rhom-bohedral, and soft and hard respectively. The flow of a non-Newtonian fluid was also studiedqualitatively. Fluid flow rate and packing deformation length data were acquired for all chosenpressure differences.

For the second goal, a pre-emulsion was chosen as the fluid to be processed through thecompressible porous medium. The resulting particle size distributions were compared to thoseexpected for an equivalent porous medium made of incompressible spheres.

4.3.1 Packing Characteristics

Figure 4.20 shows a typical experimental set-up for porous media made of cubically arrangedelastic spheres depicted at rest on the left hand side and compressed by a flow through themedium on the right hand side. The length of the undeformed packing is L = 96 mm andthe total deformation of the packing is ∆L as shown, resulting in a total strain over the wholepacking of εtotal = ∆L/L.

Dividing the packing into four equally sized sections as indicated in the figure by the1st, 2nd, 3rd and 4th quarter labels, provided strain information for each section, denotedε1st quarter, ε2nd quarter, etc. Strain differences over the packing length, as seen in the figure,can than be evaluated. The length of each section was read off the scale in an accuracy of0.5 mm. Spheres were made of silicone rubber with material properties given in table 3.6.

Packing deformations being nearly constant over the channel cross-section imply negligi-ble wall friction, although some sphere layers, particularly the ones furthest upstream, tendedto loose contact with the bulk packing at low pressure differences. The type of fluid usedinfluenced this behavior. As pointed out in chapter 3, we always forced a complete packingrelaxation prior to each flow at distinct pressure differences.

Above a certain pressure difference, cubically arranged packings made of soft materialpartially lost their structure resulting in different flow behavior. In theses cases, the dataobtained were disregarded.

4.3.1.1 Flow and Compressibility Characteristics

A typical experimental result for a 16% PEG solution flowing through a long, cubically ar-ranged packing is given in Figure 4.21. On the left-hand ordinate, the volumetric flow rate,V , (•) is given in terms of the pressure drop, ∆p, across the porous medium, connected bya dashed line. With Reynolds-numbers, Re∗, in terms of undeformed packing parametersranging from 2.6 to 15.1 the flow can be considered laminar. Therefore, a linear relationship


V.

nd 2

quar

ter

1 q

uart

erst

rd 3 q

uart

erth

4 q

uart

er

L

∆ L

Figure 4.20: Compressible porous medium made of cubically arranged spheres without fluidflow (left) and deformed by fluid flow from bottom to top (right). Explanations of the annota-tions are given in the text.

between flow rate and pressure drop would be expected for an incompressible porous mediumaccording to Darcy’s Law. Accordingly, the decrease of the flow curve gradient can be solelyattributed to the compression of the porous medium. With flow rates above those presentedhere, the packing structure became disarranged.

The packing compression is given on the right-hand ordinate with a total packing strain(εtotal) indicated by asterisks (∗) linked by a dotted line. Along with the total strain, strains aregiven for each quarter of the packing. These differ greatly from the total strain with respectto their magnitudes but otherwise have similar curve shapes. The strain of the 1st quarter (4)exceeds that of the total strain by a factor of about 2 whereby those of the 3rd and 4th quarter( and ∇ respectively) fall short by a factor of about 2. The strains of the 2nd quarter () gowell along with the total strain.

The difference between the strain of the first and the fourth quarter amount to a factor ofabout 5. This pronounced dependence of the packing compression on the streamwise positionwithin the packing gives rise to the assumption that local deformations, close to the mount ofthe packing will even be higher than the strains within the first section. The maximum possiblestrain is given by the porosity of a packing, being, in the case of the cubically arranged spherepackings, εmax = ε = 0.476, and resulting in a stall of the flow, which was not observedwithin our experiments.

The data was tested against the three compressibility models (Eqs. 2.12, 2.13, 2.14) givenin Section 2.1.1.6 and due to Tiller [TH93]. The best fit was obtained with the model for lowcompressibility (V ∝ ∆p1−δ/L) with δ = 0.273 and a constant of proportionality of c = 51.8with units for the pressure difference ∆p and the packing length L chosen to be bar and m


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ε3rd−quarter

ε4th−quarter

Figure 4.21: Flow and compression characteristics for 16% PEG solution flowing through along (L = 96 mm) cubically arranged sphere packing made of soft material.

respectively.As expected from theory and implied by the compressibility models given by Tiller, the

compressibility coefficient δ is, given Newtonian fluids, not dependent on the fluid viscos-ity. This is illustrated in Figure 4.22 where packing deformations are plotted versus pressuredifferences for three Newtonian fluids passing through the packing previously studied. The16% PEG solution data is identical to that in the previous figure.

The solid line indicates a fit for the total packing strain data for all three fluids, water (),5.5% PEG solution (), and 16% PEG solution (∇), showing an almost linear relationshipbetween strain and pressure difference. Besides the total strain, data for the strain of the firstquarter (dashed line) and the second half (dotted line) are given. Since third and fourth quarterstrains were found to be almost identical, these strains were combined to form the second halfstrain for sake of clarity.

It has to be pointed out that in case of water (, 4, ♦) strain values were below those forthe PEG solutions. The difference is small, though significant, and is due to PEG acting aslubricant between the spheres and the wall.

The fluid independence of the compressibility can also be seen in the similar compress-ibility coefficients δ estimated for each fluid. They were found to be 0.553, 0.523, and 0.273for water, 5.5% and 16% PEG solutions respectively. Although the latter value deviates fromthe former ones, and Reynolds-numbers in terms of incompressible packing parameters, Re∗,reach 2900 in case of water, a strong similarity can be attested to which becomes more obviousonce different packing structures are considered.

Trials performed for and discussed within this section are listed in Table 4.8. Trial pa-rameters were the strength of the packing material (Mat.), the packing length (L), the packingstructure (k, with values of 12 and 6 indicating rhombohedrally and cubically arranged sphere


0.2 0.4 0.6 0.8 1.0 1.2

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water 5.5% PEG 16% PEG



PSfrag replacements


Str

ain

ε[–

]εtotal ε2nd−half

ε1st−quarter

Figure 4.22: Compression characteristics for a cubically arranged, 96 mm long packing madeof soft material for three different fluids (water, 5.5% PEG solution, and 16% PEG solution).

packings respectively) and the choice of fluid, each with its characteristic dynamic viscosity.Xanthan and PEG are abbreviated to X and P respectively.

Flow characteristics are given for each trial in terms of Reynolds-number ranges basedon undeformed packing parameters. Compressibility characteristics according to Eq. (2.12)include a constant of proportionality (c), the compressibility coefficient (δ) and the adjustedR2 for each fit.

4.3.1.2 Influence of Packing Type

Two packing types were investigated, cubically arranged spheres (k = 6) as discussed pre-viously, and rhombohedrally arranged spheres (k = 12). The latter have a porosity in theundeformed state and a maximum strain of εmax = ε = 0.2595. Figure 4.23 depicts theflow and compression characteristics for such a rhombohedrally arranged packing with themaximum strains of the first quarter being approximately half the maximum possible strain.A similar ratio of maxima was also observed in the case of the previously studied cubicallyarranged sphere packings.

Due to the dense packing of rhombohedrally arranged spheres, rearrangement is not pos-sible. Therefore, much higher pressure differences could be applied, leading to a pronouncedleveling off in strain starting at a pressure difference of about ∆p = 1.0 bar, along with an al-most linear increase in flow rate, as expected from Darcy’s Law. Below 1.0 bar, almost linearcorrelations between the strains and the pressure drop can be observed, matching the resultfor cubically arranged sphere packings. Note that some of the furthest upstream sphere layerslost contact with the bulk packing as reflected in the lack of data for the total and the fourthquarter strain below 1.0 bar. Overall, the ratios between the strains of the various sections are


Table 4.8: Parameters of trials on compressible sphere packings along with flow characteris-tics in terms of Reynolds-number compressibility characteristics. The last three columns givethe constant of proportionality c, the compactability coefficient δ according to Tiller’s [TH93]compressibility model (with units for pressure difference ∆p, and packing length L being barand m respectively) and the R2 value for each fit. Further explanations are given in the text.

Trial Mat. L k Fluid Visc. Re∗min Re∗max c δ R2

[–] [mm] [–] [–] [Pa s] [–] [–] [–] [–] [–]

1206B hard 95.4 12 H2O 0.00119 1089 1695 55.7 0.779 1.0001206C hard 95.4 12 5.5%-P 0.01098 58.2 96.3 28.3 0.729 1.0001206D hard 48.3 12 5.5%-P 0.01095 67.6 163 23.5 0.696 0.9981206E hard 48.3 12 16%-P 0.10010 1.26 3.94 4.07 0.557 0.9981207A hard 48.3 12 H2O 0.00113 1373 2472 41.1 0.761 1.0001207B soft 95.4 12 H2O 0.00114 901 1170 46.7 0.9 1.0001207C soft 95.4 12 5.5%-P 0.01137 25.5 48.1 17.5 0.817 0.9931207D soft 48.3 12 5.5%-P 0.01131 29.1 84.4 13.2 0.729 0.9951207E soft 48.3 12 16%-P 0.10351 1.10 1.76 2.42 0.763 1.0001210A soft 48.3 12 H2O 0.00113 1230 1582 31.6 0.894 1.0001210B soft 48.3 12 0.2%-X NA NA NA 25.2 0.843 0.9981210C soft 95.4 12 0.2%-X NA NA NA 34.9 0.762 0.8571210D hard 95.4 12 0.2%-X NA NA NA 45.1 0.693 0.9791210E hard 48.3 12 0.2%-X NA NA NA 30.9 0.72 0.9111212A soft 96 6 H2O 0.00112 1945 2896 130 0.553 1.0001212B soft 96 6 5.5%-P 0.01228 108 262 112 0.523 1.0001212C soft 96 6 16%-P 0.11148 2.61 15.1 51.8 0.273 0.9931214A hard 96 6 H2O 0.00109 1468 2881 155 0.519 1.0001214B hard 96 6 5.5%-P 0.01259 114 255 144 0.459 1.0001214C hard 96 6 16%-P 0.10823 2.29 29.5 66.5 0.212 0.995

closely comparable to those found with cubically arranged packings.The two distinct pressure regions found can be attributed to different deformation mech-

anisms. Firstly, at low pressure differences, small deformation theory applies with a linearrelation between strain and force. Higher strains are no longer governed by linear elasticityand exponential relationships apply, thus increasingly large imposed forces lead to only smallchanges in strain. A total strain value of approximately 0.05 can be regarded as a threshold inthe case of rhombohedral sphere packings.

We noted before that the reading accuracy was 0.5 mm. This manifests itself in levels ofthe strain rates shown in Figure 4.23.

A compressibility coefficient δ estimated to be 0.817, as given in Table 4.8 indicates ahigher compactability compared to the cubically arranged packing (δ = 0.523) for otherwiseidentical parameters. Reynolds-numbers ranging from 25.5 to 48.1 accord well with the re-spective cubically arranged packing values of 108 to 262. Differences in the packing structureare represented within Tiller’s compressibility model by the coefficient of proportionality (c)being 112 in the cubic arrangement and 17.5 in the rhombohedral case with the same packinglengths and fluid viscosities. Comparing the compactability coefficient over the same range


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1]


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ain

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]

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ε3rd−quarter

ε4th−quarter

Figure 4.23: Flow and compression characteristics for rhombohedrally arranged sphere pack-ing. Packing length L = 95 mm, soft sphere material, and 5.5% PEG solution.

of pressure differences results in δ = 0.661 in the case of rhombohedrally arranged packingsand thus indicating still higher compressibility compared to the cubically arranged packing.

The difference in the degree of compressibility becomes even more obvious by looking atFigure 4.24, which depicts the dimensionless flow characteristics for water flowing througheach of the compressible porous media flows discussed so far, including the rhombohedralsphere arrangement. The friction coefficient (Λ∗) and the Reynolds-number (Re∗) are basedon parameters of undeformed packings including the initial packing porosity.

Data points for the cubically arranged packings, denoted by open symbols, are clusteredaccording to fluid viscosity with decreasing viscosities resulting in a shift towards the righthand side. All of the points relate fairly well to the baseline indicating the flow characteristicthrough random sphere packings. The overall differences between the friction coefficientsfound and the baseline are most likely caused by production imperfections in the spheres used.Nevertheless, with increasing viscosity, friction coefficients tend to deviate more stronglyfrom the baseline with increasing Reynolds-numbers. This can be attributed to the higherpressure drops imposed resulting in higher strains with higher viscosity ratios.

Rhombohedrally arranged sphere packings (solid symbols) show a much more pro-nounced increase in friction coefficients with Reynolds-numbers. Looking at the data pointsfor 5.5% PEG solution (•), one can see a slow increase in friction coefficients for the four left-most data points which can be attributed to the linear deformation regime. The following datapoints are within the non-linear regime and have almost the same Reynolds-number, becausethis is based on the initial packing porosity.


2 5 10 20 50 100 200 500 1000

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water5.5% PEG

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tion

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ntΛ

∗[–

]cubic

rhombohedr.

Figure 4.24: Dimensionless flow characteristics for long packings of cubically and rhombo-hedrally arranged spheres with varying fluids. The solid line indicates the flow characteristicfor random sphere packings according to Λ = 181 + 2.01 · Re0.96.

4.3.1.3 Influence of Material Strength

So far, we considered only packings made of soft material. Now we consider hard material.Flow and compressibility characteristics for 16% PEG solution flowing through a long, cubicsphere packing from hard material are given in Figure 4.25. All data points indicate the samecompression and flow behavior as found in all previously discussed trials with first quarterstrains being much higher than the total strain, second quarter strains similar to the total onesand the strains of the last two quarters being about the same.

Comparing these results to the corresponding soft material experiment (Figure 4.21) givesratios of strain rates at the same pressure difference of about 2. For example, the first quarterstrain at a pressure difference of ∆p = 1.25 bar was found to be 0.25 in the case of the softmaterial and 0.12 in the case of the hard material. This corresponds well with the compress-ibility coefficient δ being found to be 0.273 and 0.212 for soft and hard materials, respectively,confirming that the hard material is indeed less compressible.

Although higher pressure drops could be imposed compared to ∆pmax = 1.25 bar in thesoft material case, the spheres did not rearrange and deformations are still in the linear regimewith the maximum total strain found to be 0.11 at a pressure difference of 2.0 bar. This agreeswell with total strains up to a maximum value of 0.11 in the case of soft material being in thelinear regime.

For rhombohedral packings a threshold total strain for the transition from the linear to thenon-linear regime was found to be at about 0.05. Extrapolating a corresponding thresholdvalue for cubic packing according to the ratio between the porosities of the cubic and rhom-bohedral packings of 0.476 and 0.2595 respectively gives an approximate threshold value


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ε3rd−quarter

ε4th−quarter

Figure 4.25: Flow and compression characteristics for 16% PEG solution flowing througha long cubic sphere packing of hard material. (Compare to results for soft material withotherwise identical parameters given in Figure 4.21.)

of 0.10. This is about the maximum total strain observed in our investigation of cubic spherepackings. To verify this threshold value, higher pressure differences would have to be applied,which would lead to rearrangements of cubic packings rather than ordered compression.

4.3.1.4 Influence of Packing Length

The packing length was the last packing parameter investigated. A typical result is given inFigure 4.26. It covers flow and compression characteristics for 5.5% PEG solution flowingthrough a soft, rhombohedral sphere packing. Short packings were about half as long as longpackings and were divided into two sections for strain analysis. Therefore, the sections calledthe first and second quarters in the previous discussions are now called the first and secondhalves respectively.

Again, similar flow and almost identical compression behavior compared to the corre-sponding long packing data is found, with a total strain threshold of about 0.05 at a pressuredifference of 1.5 bar. This is backed up by the values of the compression coefficients, δ,with 0.729 for the short packing being close to the corresponding value of 0.817 for the longpacking.

The transition from linear to non-linear compression behavior occurs coincident with thebeginning of a Darcian flow regime found at pressure differences above 1.5 bar in the corre-sponding long packing experiment. The flow rates are about twice as high compared with thelong packing, which is in good agreement with expectations. It should be noted that negativestrain values are again present at low pressure differences due to contact being lost betweenthe layers of spheres.


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Figure 4.26: Flow and compression characteristics for a short rhombohedral sphere packing(L = 48.3 mm) from soft material flown through by 5.5% PEG solution. (Compare to resultsfor the corresponding long packing given in Figure 4.23.)

4.3.1.5 Non-Newtonian fluid (watery Xanthan solution)

Investigations on flow and compression characteristics of packings through which a 0.2%-Xanthan solution was flowing were affected by fluid covering the spheres such that initialpacking heights were extended over their regular uncompressed height as indicated by thenegative strains in Figure 4.27. Therefore, the results presented are of a more qualitativenature.

In any case, the curves compare well with the previous findings, although the flow ratesdo not obviously reflect the shear-thinning fluid behavior. Tiller’s compressibility model (seeEq. 2.12) is given in terms of the volumetric flow rate (V ) with the fluid viscosity accountedfor by the coefficient of proportionality (c). Therefore, non-Newtonian flow behavior withflow-field dependent viscosity, is not covered by this model. Some poorly adjusted R2-valuesfor fitting experimental data from trials with Xanthan to the model equation as given in Table4.8 indicate the influence of the shear-thinning behavior, although the influence of packingextensions on the adjusted R2-value cannot be excluded.

4.3.2 Emulsification in Compressible Porous Media

Finally, various aspects of this thesis can be brought together by dispersing an emulsion withina compressible porous medium. The porous medium used was 95 mm long, built up fromrhombohedrally arranged soft spheres. Compression characteristics for this packing wereshown in Figure 4.23. 5% silicone oil AK 1000 was dispersed in 5.5% PEG – SDS solutionwith a viscosity ratio of λ = 10.4.


1 2 3 4 5

400

450

500

550

600

−0.05

0.00

0.05

0.10

0.15

−0.05

0.00

0.05

0.10

0.15

PSfrag replacements

Vol

umet

ricflo

wra

teV

[lh−

1]


Str

ain

ε[–

]

εtotal

V ε1st−halfε2nd−half

Figure 4.27: Flow and compression characteristics for a shear-thinning fluid (0.2% Xanthan –water solution) flowing through a short, soft, rhombohedral sphere packing. (Compare withthe results for the same packing with a Newtonian fluid flowing through it as given in theprevious figure.)

4.3.2.1 Result of Emulsification Process

Emulsions were generated at two distinct pressure drops of ∆p = 0.84 bar and 4.76 bar acrossthe packing, and for various numbers of runs through the packing. At the lower pressure drop,the packing deformation was still in the linear deformation regime, whereas the higher onerepresents the non-linear regime with a total strain εtotal = 0.06 and the strain of the firstquarter being ε1st quarter = 0.11.

Results are shown in Figure 4.28 in terms of particle size characteristics versus the volumespecific energy input times the number of runs Evn = ∆p · nruns = Ev · nruns. Symbolsdenoting the mean diameter of emulsions (x50,3) are again replaced by two numbers indicatingthe number of runs and the width of the particle size distribution.

The same behavior compared to dispersing within incompressible sphere packings can beobserved. Firstly, an increase in energy input going along with the increase in the number ofruns results in an exponential decrease of emulsion mean diameters. Secondly, the width ofthe particle size distribution, particularly for the runs at the higher pressure drop, decreaseswith the number of runs. Finally, power input is again more efficient than energy input interms of particle size reduction.

Fitting the data to the energy versus power input model (x50,3 –pack – I) given in Eq. (4.1)results in the following correlation:

x50,3 = c1 · ∆pc2 · nrunsc3 (4.12)


5 10 20 50 100

2

5

10

20

50

1005, 1.34 10, 1.35

5, 2.4310, 2.05

20, 1.82

PSfrag replacements

PS

Dpa

ram

eter

sx

90,3,

x50,3,

x10,3

[µm

]

Specific energy input Ev = ∆p × nruns [bar]

x90,3x50,3x10,3

Figure 4.28: Dispersing 5% (v/v) silicone oil AK 1000 in 5.5% PEG – SDS solution in a longrhombohedral sphere packing made of soft material. Particle size characteristics x90,3, x50,3,and x10,3 versus volume specific energy input Evn.

with c1 = 99.99, c2 = -1.15 and c3 = -0.288, and an adjusted R2 value of 0.998 and anaverage deviation of 0.032.

4.3.2.2 Comparison with Incompressible Porous Media

Comparing the power input coefficient c2 = −1.15 from the previous section to that foundwithin our mean diameter model for sphere packings (x50,3 –pack – IV) as given in Eq. (4.8)where c2 = −0.679 reveals that, at least in case of this experiment, dispersing in a compress-ible porous medium is superior in terms of efficiency. This becomes even more obvious whencomparing the mean diameters measured to model predictions based on the parameters of thecompressible porous medium experiment. Measured and predicted mean diameters are givenin Table 4.9 two pressure differences and varying numbers of runs.

Mean diameters measured are always smaller than those predicted by the model for theincompressible porous medium. This difference is greatest at high pressure difference withrelative differences of almost 2. This could possibly be attributed to the reduction of capillariesdue to the compression of the porous medium. A strong influence of the number of runs wasalso observed, emphasizing the importance of inflow effects as an explanation for dispersingmechanisms within compressible porous media.

The width of the particle size distributions was found to be comparable to that expectedby our model for incompressible porous media (span –pack – IV, Eq. 4.9) as given in Table4.10. Reynolds-numbers used within the model are given in terms of incompressible porousmedium parameters. Therefore, a direct comparison between span-values must be treatedcarefully.


Table 4.9: Comparison of mean diameter, x50,3, with model predictions for incompressibleporous medium.

∆p nruns x50,3 measured x50,3 predicted Relative difference[bar] [–] [µm] [µm] [–]0.84 5 74.40 82.23 -0.1050.84 10 66.01 74.05 -0.1224.76 5 10.88 25.20 -1.324.76 10 8.24 22.70 -1.754.76 20 7.01 20.44 -1.91

Table 4.10: Comparison of the width of particle size distribution, span, with model predic-tions for an incompressible porous medium.

∆p nruns span measured span predicted Relative difference[bar] [–] [–] [–] [–]

0.84 5 1.34 1.63 -0.220.84 10 1.35 1.56 -0.154.76 5 2.43 1.67 0.314.76 10 2.05 1.59 0.234.76 20 1.82 1.51 0.16

It has to be noted that the investigations reported on in this section were based on a singleexperiment. Therefore, care has to be taken in interpreting these findings. Nevertheless, theyare promising, particularly as a foundation for further investigations.

Chapter 5

Conclusions

In this chapter, we will present conclusions drawn from the results given and discussed inthe previous chapter. We will focus on main findings and augment them with descriptions ofpossible future developments.

5.1 Viscosity Ratio

We were able to show that droplet break-up mechanisms in emulsions flowing through porousmedia depend on the interaction between viscosity ratio and packing length as reported insection 4.2.1.2.

Droplet disruption at a dispersed to continuous phase viscosity ratio, λ, of 6.9 in flowthrough cubically arranged sphere packings can be attributed to stronger inflow effects andhigher elongational flow field contributions, compared with the case when the viscosity ratio λis 1.7. In this case the packing length becomes more important for the dispersing result than isthe case for the higher viscosity ratio λ = 6.9. These findings accord well with our numericalinvestigations and expectations for single droplet break-up under steady flow conditions.

In any case, an increase in packing length, under otherwise constant flow conditions (con-stant Reynolds-number), benefits the production of small mean emulsion drop diameters re-gardless of the viscosity ratios studied. This indicates the relevance of cumulative dispersingeffects within porous media flow.

Further investigations on dispersing multiple droplets in complex flow fields like thosewhich occur in porous media flows must go hand-in-hand with studies on single droplet break-up in complex flow fields, performed experimentally as well as numerically. Perturbing rippleshave not been addressed in this work but should also be considered and investigated as break-up mechanisms in porous media flow in following studies.

5.2 Physical Parameter Models

Models which are based solely on physical parameters are generally preferred over those withparameters which must be empirically determined, in view of ease of process design. For thedispersing process in porous media and orifice flows, models with emulsion mean diameterand width of particle size distributions as dependent variables were established, based on

81

82 CHAPTER 5. CONCLUSIONS

a wide range of packing, process and fluid parameters. Almost all parameters within ourmodels were very strongly significant.

Droplet break-up within porous media was found to be of the same nature as that in orificeflow, as indicated by similar mean droplet diameters and PSD widths. Power input, whichincreases with an increase in the pressure drop across the packing was found to be much moreefficient in generating fine emulsions compared with increases in energy input. Energy input,being directly related to the number of runs, was on the other hand found to be the decisiveparameter for narrowing the width of the particle size distribution. Therefore, power inputand energy input have to be well balanced in order to generate an emulsion of a given quality,defined by its mean droplet diameter and the width of its particle size distribution.

A refinement of the models seems to be desirable. More experimental data would beneeded and could possibly include further packing parameters such as the orientation of struc-tured packings.

5.3 Compressible Porous Media

Non-linear deformation of compressible porous media when liquids were flowing throughthem were observed and described within this work. Due to continuously narrowing pores,such packed beds look appealing for dispersing processes. It was shown that dispersing withinsuch porous media produced finer emulsions compared with those processed through an in-compressible porous medium, for otherwise identical process, fluid, and geometry parameters.

A challenging, but promising task, would be the extension of mean diameter and width ofPSD models established for incompressible porous media. Such an extension would have totake the non-linear deformation characteristics of compressible porous media into account.

5.4 Capabilities and Limitations of CFD

The critical droplet break-up diameter for the flow through a nozzle, modeling regularly ar-ranged porous media, was determined by means of computational fluid dynamics (CFD).Therefore, flow field characteristics along particle tracks were used as the input for dropletdeformation calculations based on the boundary integral method (BIM). The numerically de-termined critical diameter for break-up along a representative track was found to be in goodagreement with the diameter predicted by our mean diameter model.

Nevertheless, one-to-one modeling of break-up in such complex flow fields is still limitedand it goes without saying that break-up within turbulent flow is beyond todays realm ofpossibility. One aspect which it would be desirable to investigate in the future is the break-upbehavior on particle tracks closer to the wall, where droplet deformation calculations faileddue to high shear and elongation rate gradients. Furthermore, periodicity and cumulativeeffects seem to be worthwhile targets for further investigations.

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Appendix A

Crystal Families and Bravais LatticeTypes

Crystals consist of atoms, ions, and molecules arranged symmetrically and in a well orderedmanner. Their structure can be divided into identical, repeating adjoint unit cells called crystallattices. According to their geometry, such lattices can be divided into crystal families. Fourtypes of unit cells are primitive (P), body-centered (I), face-centered (F), and side-centered(S). 14 different Bravais lattice types can be distinguished as given in Table A.1.

In this work, sphere packing structures are described according to common practice in theengineering literature. The orthorhombic and rhombohedral sphere packings which are usedin this work are described in terms of Bravais lattice types as hexagonal primitive (hP), andcubic face-centered (cF) respectively. The latter type is also known as cubic-close packing(CCP).

91

92 APPENDIX A. CRYSTAL FAMILIES AND BRAVAIS LATTICE TYPES

Table A.1: Three dimensional crystal families and Bravais lattice types according to De Wolffet al. [DBB+85]. Their respective symbols are given in parentheses. Unit cell parameters aregiven in terms of unit cell edge lengths, a, b and c, and the angles between unit cell edges, α,β and γ.

Crystal families Bravais lattice types

Cubic (c) Cubic primitive (cP)a = b = c Cubic body-centered (cI)α = β = γ = 90o Cubic face-centered (cF)Tetragonal (t) Tetragonal primitive (tP)a = b 6= c Tetragonal body-centered (tI)α = β = γ = 90o

Hexagonal (h) Hexagonal primitive (hP)a = b 6= c Rhombohedral (hR)α = β = 90o, γ = 120o

Orthorhombic (o) Orthorhombic primitive (oP)a 6= b 6= c Orthorhombic single-face centered (oS)α = β = γ = 90o Orthorhombic body-centered (oI)

Orthorhombic all faces centered (oF)Monoclinic (m) Monoclinic primitive (mP)a 6= b 6= c Monoclinic centered (mS)α = γ = 90o, β 6= 90o

Triclinic (anorthic, a) Triclinic (aP)a 6= b 6= cα 6= β 6= γ = 90o

Appendix B

Parameters of Dispersing Experiments

In table B.1, the trials employed for model estimations in the various sections of this work arelisted. Note that the mean diameter model for the sphere packing flow was based on the sametrials as the respective width of PSD model. Since data stemming from the first and secondruns were not being considered in the model of the PSD width, some trials used for the meandiameter model were not used for the latter model.

Process, fluid, and geometry parameters for all trials used in the analysis of this work arelisted in table B.2 and continued in table B.3. ηd and ηc denote the dynamic viscosities of thedispersed and continuous phases, respectively, and their ratio is given by λ. Continuous phasedensity is given by ρc, sphere and orifice diameter by ds and do respectively, packing length byL, packing structure in terms of coordination number k (‘rand.’ indicates random packings),packing porosity by ε, dispersed phase volume fraction by φd, the type of flow meter used byits maximum flow rate (with zero indicating measurements done without a flow meter), andinterfacial tension by σ.

93

94 APPENDIX B. PARAMETERS OF DISPERSING EXPERIMENTS

Table B.1: Trials employed for model estimations in the various sections of this work. Trialparameters are listed in the following tables B.2 and B.3.

Section Trials

3.4.2.4 Incompressible Sphere Packing Flow Characteristics010120A, 010129A, 010130A, 010208A, 010208B, 010221A, 010222A010227A, 010228A, 010308A, 010309A, 010411A, 010416A, 010418B010423A, 010425A, 010520A

4.2.1.1 Energy and Power Input010120A

4.2.1.2 Packaging Length010208A, 010208B, 010221A, 010222A, 010227A, 010228A

4.2.1.3 Mean Diameter Model for Sphere Packing Flow001115A, 001128A, 001204A, 001207A, 001214A, 010120A, 001120A001123A, 010129A, 010130A, 010208A, 010208B, 010221A, 010222A010227A, 010228A, 010308A, 010309A, 010331A, 010411A, 010416A010418B, 010423A, 010425A, 011119A

4.2.1.5 Width of Particle Size Distribution001115A, 001120A, 001123A, 001128A, 001204A, 001207A, 001214A010120A, 010129A, 010130A, 010208A, 010208B, 010221A, 010222A010227A, 010228A, 010308A, 010309A, 010331A, 010411A, 010416A011119A

4.2.2 Dispersing in Orifice Flow011004A, 011011A, 011015B, 011016B, 011114B, 011116C, 011121C011121D, 011126B, 011128B

4.3.2 Emulsification in Compressible Porous Media011205B

95

TableB

.2:D

ispersingtrialparam

eters.See

textforexplanations.

Trial Phasedisp ηd Phasecont ηc ρc ds, do L k ε φ F low σ λ

[Pa s] [Pa s] [ kgm3 ] [mm] [mm] [–] [–] % [ l

h] [mN

m] [–]

001115A 2%RSO 0.0600 10%PEG 0.0265 1020 4.000 100 6 0.4760 2 0 4 2.26001120A 2%AK250 0.2400 19%PEG 0.1400 1035 4.000 20 6 0.4760 2 0 10 1.71001123A 2%AK250 0.2400 10%PEG 0.0265 1020 4.000 20 6 0.4760 2 0 10 9.05001128A 10%RSO 0.0600 10%PEG 0.0265 1020 4.000 100 6 0.4760 10 0 4 2.26001204A 2%RSO 0.0600 10%PEG 0.0265 1020 4.000 80 12 0.2595 2 0 4 2.26001207A 2%RSO 0.0600 10%PEG 0.0265 1020 4.000 80 12 0.2595 2 0 4 2.26001214A 2%RSO 0.0600 10%PEG 0.0265 1020 4.000 100 6 0.4760 2 0 4 2.26010120A 2%RSO 0.0600 10%PEG 0.0265 1020 4.000 80 8 0.3950 2 1000 4 2.26010129A 2%AK250 0.2400 10%PEG 0.0265 1020 4.000 80 8 0.3950 2 1000 10 9.05010130A 2%AK50 0.0480 10%PEG 0.0265 1020 4.000 80 8 0.3950 2 1000 10 1.81010208A 2%AK1000 0.9700 19%PEG 0.1400 1035 4.000 100 6 0.4760 2 1000 10 6.92010208B 2%AK1000 0.9700 19%PEG 0.1400 1035 4.000 20 6 0.4760 2 1000 10 6.92010221A 2%AK1000 0.9700 19%PEG 0.1400 1035 4.000 20 6 0.4760 2 0 10 6.92010222A 2%AK1000 0.9700 19%PEG 0.1400 1035 4.000 100 6 0.4760 2 0 10 6.92010227A 2%AK250 0.2400 19%PEG 0.1400 1035 4.000 100 6 0.4760 2 0 10 1.71010228A 2%AK250 0.2400 19%PEG 0.1400 1035 4.000 20 6 0.4760 2 0 10 1.71010308A 2%AK50 0.0480 10%PEG 0.0265 1020 2.000 40 rand. 0.3760 2 1000 10 1.81010309A 2%AK250 0.2400 10%PEG 0.0265 1020 2.000 40 rand. 0.3760 2 1000 10 9.05010331A 5%AK10 0.0093 2%SDS-H2O 0.0011 1003 0.339 300 rand. 0.3760 5 1000 10 8.45010411A 5%AK10 0.0093 2%SDS-H2O 0.0011 1003 0.070 10 rand. 0.3760 5 1000 10 8.45010416A 5%AK10 0.0093 2%SDS-H2O 0.0011 1003 0.339 160 rand. 0.3760 5 1000 10 8.45010418B 5%AK10 0.0093 2%SDS-H2O 0.0011 1003 2.000 400 rand. 0.3760 5 1000 10 8.45010423A 5%AK10 0.0093 2%SDS-H2O 0.0011 1003 0.339 110 rand. 0.3760 5 1000 10 8.45010425A 5%AK10 0.0093 2%SDS-H2O 0.0011 1003 0.339 110 rand. 0.3760 5 1000 10 8.45

96A

PPEN

DIX

B.

PAR

AM

ET

ER

SO

FD

ISPER

SING

EX

PER

IME

NT

S

TableB

.3:D

ispersingtrial

parameters

(cont’d).T

rial011205B

was

conductedusing

acom

-pressible

porousm

edium.

Trial Phasedisp ηd Phasecont ηc ρc ds, do L k ε φ F low σ λ

[Pa s] [Pa s] [ kgm3 ] [mm] [mm] [–] [–] % [ l

h] [mN

m] [–]

011004A 5%AK10 0.0093 5.5%PEG 0.0092 1012 2.4 100 NA NA 5 60 10 1.01011011A 5%AK100 0.0960 5.5%PEG 0.0092 1012 1.000 20 NA NA 5 1000 10 10.43011015B 5%AK100 0.0960 5.5%PEG 0.0092 1012 1.000 5 NA NA 5 1000 10 10.43011016B 5%AK100 0.0960 5.5%PEG 0.0092 1012 1.000 40 NA NA 5 1000 10 10.43011114B 5%AK100 0.0960 5.5%PEG 0.0092 1012 1.000 20 NA NA 5 1000 10 10.43011116C 5%AK100 0.0960 5.5%PEG 0.0092 1012 8.8 200 NA 0.4760 5 1000 10 10.43011119A 5%AK100 0.0960 5.5%PEG 0.0092 1012 4.000 40 6 0.4760 5 1000 10 10.43011121C 5%AK100 0.0960 5.5%PEG 0.0092 1012 1.000 40 NA NA 5 1000 10 10.43011121D 5%AK10 0.0100 5.5%PEG 0.0092 1012 1.000 40 NA NA 5 1000 10 1.08011126B 5%AK100 0.0960 5.5%PEG 0.0092 1012 8.8 200 NA NA 5 1000 10 10.43011128B 5%AK100 0.0960 5.5%PEG 0.0092 1012 1.000 100 NA NA 5 1000 10 10.43011205B 5%AK100 0.0960 5.5%PEG 0.0092 1012 4.000 98 12 0.2595 5 1000 10 10.43

Appendix C

Adjoint Nozzle Flow Field

Droplet deformation calculations presented in this work were accomplished with the adjointnozzle geometry along particle tracks 1 to 3 as given in Figure 4.6. High shear rate gradientsalong particle tracks close to the wall impeded deformation calculations along these tracks.Nevertheless, shear and elongation rates are given for particle tracks 4 and 5 in Figures C.1and elongation rates (without shear rates) in Figure C.2. Paths of tracks 4 and 5 were includedin Figure 4.6.

97

98 APPENDIX C. ADJOINT NOZZLE FLOW FIELD

0.000 0.010 0.020 0.030

0e+00

5e+04

1e+05

shea

r an

d el

onga

tion

rate

s [1

/s]

time [s]

Track 3

0.000 0.010 0.020 0.030

0e+00

5e+04

1e+05

shea

r an

d el

onga

tion

rate

s [1

/s]

time [s]

Track 4

0.000 0.010 0.020 0.030

0e+00

5e+04

1e+05

shea

r an

d el

onga

tion

rate

s [1

/s]

time [s]

Track 5

Figure C.1: Additional shear and elongation rate information for the flow through the adjointnozzle geometry at Re = 1000. The plot on the left-hand side (track 3) is identical to that onthe right-hand side of Figure 4.7. Particle track paths were given in Figure 4.6.

0.000 0.010 0.020 0.030

−4000

−3000

−2000

−1000

0

1000

2000

3000

shea

r an

d el

onga

tion

rate

s [1

/s]

time [s]

Track 3 Track 4 Track 5

0.000 0.010 0.020 0.030

−4000

−3000

−2000

−1000

0

1000

2000

3000

shea

r an

d el

onga

tion

rate

s [1

/s]

time [s]

0.000 0.010 0.020 0.030

−4000

−3000

−2000

−1000

0

1000

2000

3000

shea

r an

d el

onga

tion

rate

s [1

/s]

time [s]

Figure C.2: Elongation rates along tracks 3, 4, and 5. Elongation rates shown are identical tothose given in the previous Figure (C.1).

Appendix D

Statistical Analysis – Model Quality

The quality of the statistical models derived within this work were given in terms of theadjusted R2-value and the average deviation. Model quality was also assessed based uponplots providing additional information on the fit. A typical set of plots, representing all modelestimations performed in this work, is given in Figure D.1 for the mean diameter model ofdispersing in sphere packing flow, as given in Eq. (4.8). Each data point is represented by anopen circle () or a vertical line.

Essential conditions for obtaining good models are the presence of good characteristicsin all four plots. Within the residuals versus fitted values plot (top left), data points shouldbe likewise randomly distributed around the zero residual and along the abscissa. Within thenormal Q-Q plot (top right) the points should be close to the diagonal. The data points beingclose to the diagonal indicates that the errors are randomly distributed.

The scale – location plot (bottom left) is comparable to the residuals versus fitted valuesplot. The Cooks’s distance plot indicates the influence that each individual data point imposeson the estimated coefficients. Evenly distributed Cook’s distances are desirable.

99

100 APPENDIX D. STATISTICAL ANALYSIS – MODEL QUALITY

2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5

0.0

0.5

1.0

1.5

Fitted values

Scale−Location plot7

84

0 20 40 60 80 100 120

0.00

0.02

0.04

0.06

0.08

0.10

Obs. number

Coo

k’s

dist

ance

Cook’s distance plot

7

8

75

2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5

−0.5

0.0

0.5

Fitted values

Res

idua

ls

Residuals vs Fitted

7

84

−2 −1 0 1 2

−3

−2

−1

0

1

2

3

4

Theoretical Quantiles

Sta

ndar

dize

d re

sidu

als

Normal Q−Q plot

7

84

abs(

Sta

nd(

ardi

zed

resi

dua

0.5

l)

s)

Figure D.1: Plots indicating the quality of the data fitted to the mean diameter model fordispersing in sphere packing flow, given by Eq. (4.8).

Curriculum Vitae

Tobias Hovekamp

born June 18, 1970 in Beckum, Germany

1/1996 – 10/2002 Ph. D. student and research assistant at the Swiss Federal Instituteof Technology (ETH Zurich), Switzerland, Institute of Food Scienceand Nutrition, Laboratory of Food Process Engineering

10/1990 – 12/1995 RWTH Aachen, Germany, Dept. of Mechanical EngineeringDipl.-Ing.

9/1993 – 4/1995 Oregon State University, USA, Dept. of Mechanical EngineeringMaster of Science

6/1989 – 9/1990 Military serviceGrundwehrdienst

8/1980 – 5/1989 Thomas Morus Gymnasium, Oelde, GermanyAbitur

8/1976 – 7/1980 St. Vitus Primary School, Oelde, GermanyGrundschule

101

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Experimental and Numerical Investigation of Porous Media Flow