Experimental and Numerical Investigation of Porous

Diss. ETH No. 14836

Experimental and Numerical Investigation of Porous Media Flow with regard to the Emulsion Process A dissertation submitted to the ¨ SWISS FEDERAL INSTITUTE OF TECHNOLOGY ZURICH for the degree of Doctor of Technical Sciences

presented by Tobias Benedikt Hövekamp Dipl.-Ing. born June 18, 1970 citizen of Germany

accepted on the recommendation of Prof. Dr.-Ing. E. Windhab, examiner Prof. Dr. K. Feigl, co-examiner.

2002

c 2002 Tobias Hövekamp

Laboratory of Food Process Engineering (ETH Zu¨ rich) All rights reserved.

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

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

Published and distributed by: Laboratory of Food Process Engineering Swiss Federal Institute of Technology (ETH) Zu¨ rich ETH Zentrum, LFO CH-8092 Zurich Switzerland http://www.vt.ilw.agrl.ethz.ch

Printed in Switzerland by: bokos druck GmbH Badenerstrasse 123a CH-8004 Zürich

Rien n’est plus fort qu’une idée dont l’heure est venue Victor Hugo

To Barbara

Danksagung Die vorliegende Dissertation wurde erst durch die Mithilfe und Unterst u¨ tzung vieler Menschen möglich. Gerne möchte ich mich an dieser Stelle bei allen bedanken, die zum Gelingen dieser Arbeit beigetragen haben. Mein besonderer Dank gilt: Prof. Dr.-Ing. Erich Windhab, der mich in sein Team aufnahm und mir eine grosse akademische Freiheit bei meiner Promotion einräumte. Darüberhinaus bedanke ich mich herzlich fu¨ r die anregenden – teils weit u¨ ber das fachliche hinausgehenden – Gespräche. Prof. Dr. Kathleen Feigl, die die numerischen Aspekte meiner Arbeit souverän begleitete und das Koreferat u¨ bernahm. Gerne blicke ich auf die Zeit zuru¨ ck, in der sie noch im Büro nebenan sass. Dem gesamten Team des Laboratoriums fu¨ r Lebensmittelverfahrenstechnik fu¨ r die sehr angenehme Arbeitsatmosphäre, die vielen Anregungen, die ich erhalten habe, und die gemeinsamen 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 Arbeit wichtige Resultate und Einsichtigen lieferten: Paul Bannister, Daniela Brauss, Adrian D u¨ rig, Elia Herklotz, Fabien Rubli und Luzian Tobler. Der Informatik-Support-Group, insbesondere den Mitarbeitern der ‘ersten Stunde’ Peter Bircher und Roland Wernli. Gerne bedanke ich mich auch bei den Mitgliedern des Akademischen Chors Z u¨ rich und des Zürcher Studenten Skiklub, mit denen ich zusammen als Ausgleich musikalische und sportliche Gipfel erklimmen durfte. Adrian Whatley, für die gewissenhafte Durchsicht des Manuskripts und die geduldige Verbesserung 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 unterstützt hat. Meinen Eltern Thea und Theo Hövekamp, die meinen Lebensweg mit viel Liebe und Hingabe geebnet und begleitet haben. Ein ganz spezieller Dank geht an Barbara Meier fu¨ r den steten Ansporn zur Durchfu¨ hrung dieser Arbeit und die sehr schöne, gemeinsame Zeit. Zürich, 30. September 2002

v

Contents Notation

xi

Abstract

xix

Zusammenfassung

xxi

1 Introduction 1.1 Dispersing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Porous Media Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Aim of this Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Background 2.1 Flow through Porous Media and Nozzles . . . . . . . . . . . 2.1.1 Porous Media Flow . . . . . . . . . . . . . . . . . . 2.1.1.1 Introduction . . . . . . . . . . . . . . . . 2.1.1.2 Flow Behavior in Sphere Packings . . . . 2.1.1.3 Characteristics of Sphere Packing Flow . . 2.1.1.4 Regularly Arranged Porous Media . . . . 2.1.1.5 Representative Capillary Diameter . . . . 2.1.1.6 Compressible Porous Media . . . . . . . . 2.1.2 Model Geometries for Porous Media . . . . . . . . . 2.1.2.1 Orifice Geometry . . . . . . . . . . . . . 2.1.2.2 Nozzle with Constant Elongation Rate . . 2.1.3 Comparison of Geometries . . . . . . . . . . . . . . 2.1.4 Viscoelastic Flow in Porous Media and Nozzles . . . 2.1.5 Computational Fluid Dynamics . . . . . . . . . . . 2.1.6 Velocity Gradient . . . . . . . . . . . . . . . . . . . 2.1.6.1 Shear and Elongation Rates . . . . . . . . 2.1.6.2 Predefined Velocity Gradients . . . . . . . 2.1.6.3 Strain . . . . . . . . . . . . . . . . . . . . 2.2 Dispersing . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Single Droplet Break-up . . . . . . . . . . . . . . . 2.2.1.1 Steady Flow Conditions . . . . . . . . . . 2.2.1.2 Unsteady Flow Conditions . . . . . . . . 2.2.1.3 Numerical Simulation of Droplet Break-up 2.2.2 Emulsions . . . . . . . . . . . . . . . . . . . . . . . vii

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1 1 2 2 3 3 3 3 4 5 7 8 9 10 11 12 12 13 14 14 14 15 15 16 16 16 18 18 19

CONTENTS

viii 2.2.2.1 2.2.2.2 2.2.2.3

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . Emulsion Processes . . . . . . . . . . . . . . . . . . . . . Emulsion Rheology . . . . . . . . . . . . . . . . . . . . .

3 Material and Methods 3.1 Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Calculation of Macroscopic Flow Field . . . . . . . . . . . . 3.1.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . 3.1.1.2 Sepran . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Calculation of Drop Deformation . . . . . . . . . . . . . . . 3.1.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . 3.1.2.2 BIM program . . . . . . . . . . . . . . . . . . . . 3.1.3 Statistical Analysis . . . . . . . . . . . . . . . . . . . . . . . 3.1.3.1 Strategy for Establishing Models . . . . . . . . . . 3.1.3.2 Model Naming Conventions . . . . . . . . . . . . . 3.2 Analytical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Fluid Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Fluid Density . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Particle Size Distribution . . . . . . . . . . . . . . . . . . . . 3.3 Characterization of Fluids . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 PEG – SDS – H2 O Solutions . . . . . . . . . . . . . . . . . . 3.3.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . 3.3.1.2 Polyethylene Glycol (PEG) . . . . . . . . . . . . . 3.3.1.3 Viscosity Variation with Temperature . . . . . . . . 3.3.1.4 Density Variation with Temperature . . . . . . . . . 3.3.2 Xanthan Gum . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Silicone Oils . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.4 Rape Seed Oil . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.5 Emulsions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.5.1 Surfactant . . . . . . . . . . . . . . . . . . . . . . 3.3.5.2 Interfacial Tension . . . . . . . . . . . . . . . . . . 3.3.5.3 Preparation of Pre-emulsions . . . . . . . . . . . . 3.3.5.4 Stability of Emulsions . . . . . . . . . . . . . . . . 3.4 Experimental Setups and Procedures . . . . . . . . . . . . . . . . . . 3.4.1 Process Unit with Flow-Through Cell . . . . . . . . . . . . . 3.4.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . 3.4.1.2 Data Acquisition . . . . . . . . . . . . . . . . . . . 3.4.2 Sphere Packings . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2.1 Packing Structures . . . . . . . . . . . . . . . . . . 3.4.2.2 Types of Flow-Through Cells . . . . . . . . . . . . 3.4.2.3 Incompressible Spheres . . . . . . . . . . . . . . . 3.4.2.4 Incompressible Sphere Packing Flow Characteristics 3.4.2.5 Compressible Spheres . . . . . . . . . . . . . . . . 3.4.3 Orifices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3.1 Orifice Geometries . . . . . . . . . . . . . . . . . . 3.4.3.2 Droplet Break-up within Orifice Flows . . . . . . .

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19 19 20 23 23 23 23 23 24 24 25 26 26 26 27 27 27 27 27 28 28 28 28 29 30 30 31 31 31 31 32 32 33 33 33 33 34 34 35 36 36 37 38 38 38

CONTENTS 3.4.4

ix Experimental Procedures . . . . . . . . . . . . . . . . . . . . . . . .

38

4 Results and Discussion 4.1 Numerical Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Adjoint Converging Diverging Nozzles . . . . . . . . . . . . . . . . 4.1.1.1 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1.2 Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1.3 Annulus Probability . . . . . . . . . . . . . . . . . . . . . 4.1.2 Flow field within Converging-Diverging Nozzles . . . . . . . . . . . 4.1.2.1 Reynolds-number Re = 100 . . . . . . . . . . . . . . . . . 4.1.2.2 Reynolds-number Re = 1000 . . . . . . . . . . . . . . . . 4.1.3 Droplet Deformation and Break-up . . . . . . . . . . . . . . . . . . 4.1.3.1 Droplet Break-up . . . . . . . . . . . . . . . . . . . . . . 4.1.3.2 Shear and Elongation Rate . . . . . . . . . . . . . . . . . 4.1.3.3 Periodicity . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.3.4 Droplet Size . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.3.5 Particle Track . . . . . . . . . . . . . . . . . . . . . . . . 4.1.3.6 Entrance Flow . . . . . . . . . . . . . . . . . . . . . . . . 4.1.3.7 Cumulative Effects . . . . . . . . . . . . . . . . . . . . . 4.1.4 Orifice Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Dispersing Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Dispersing in Sphere Packing Flow . . . . . . . . . . . . . . . . . . 4.2.1.1 Energy and Power Input . . . . . . . . . . . . . . . . . . . 4.2.1.2 Packing Length and Viscosity Ratio . . . . . . . . . . . . . 4.2.1.3 Mean Diameter Model for Sphere Packing Flow (x50,3 – pack – IV) . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1.4 Influence of Dispersed Phase Volume Fraction . . . . . . . 4.2.1.5 Width of Particle Size Distribution (span –pack – IV) . . . 4.2.1.6 Comparison with Numerical Simulations . . . . . . . . . . 4.2.2 Dispersing in Orifice Flow . . . . . . . . . . . . . . . . . . . . . . . 4.2.2.1 Mean Diameter Model for Orifice Flows (x50,3 –orif) . . . . 4.2.2.2 Width of Particle Size Distributions (span –orif) . . . . . . 4.3 Compressible Porous Media Flows . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Packing Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1.1 Flow and Compressibility Characteristics . . . . . . . . . . 4.3.1.2 Influence of Packing Type . . . . . . . . . . . . . . . . . . 4.3.1.3 Influence of Material Strength . . . . . . . . . . . . . . . . 4.3.1.4 Influence of Packing Length . . . . . . . . . . . . . . . . . 4.3.1.5 Non-Newtonian fluid (watery Xanthan solution) . . . . . . 4.3.2 Emulsification in Compressible Porous Media . . . . . . . . . . . . . 4.3.2.1 Result of Emulsification Process . . . . . . . . . . . . . . 4.3.2.2 Comparison with Incompressible Porous Media . . . . . .

39 39 39 39 40 41 42 42 44 45 46 48 48 48 50 51 52 55 57 57 59 61 63 64 64 65 66 67 68 69 69 69 72 75 76 77 77 78 79

CONTENTS

x 5 Conclusions 5.1 Viscosity Ratio . . . . . . . . . . . 5.2 Physical Parameter Models . . . . . 5.3 Compressible Porous Media . . . . 5.4 Capabilities and Limitations of CFD

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81 81 81 82 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 A a b B c d E f G K K1 , K2 , K3 k k L p pL P P Q Q0 Q1 Q2 Q3 R Rc Rm Sv span r T

SI-Units [m2 ] [m] [–] [–] [–] [m] [J] [M L T−2 ] [s−1 ] [–] [–] [–] [m2 ] [m]

Meaning area undeformed droplet radius exponent in dispersing model Andrade Law coefficient concentration diameter (activation) energy external force field (body force) sum of shear and elongation rate Andrade Law coefficient porous media flow coefficients coordination number permeability packing length, length of deformed droplet [Pa] pressure [Pa] Laplace pressure [Pa] generalized Pressure (Sepran) [–] probability [–] cumulative distribution [–] cumulative number distribution [–] cumulative distribution based on particle length [–] cumulative surface distribution [–] cumulative volume distribution [J K−1 mol−1 ] universal gas constant [m−2 ] filter cake resistance [m−2 ] filter medium resistance −1 [m ] specific surface [–] width of particle size distribution [m] radius, radial position [o C] temperature ... xi

NOTATION

xii Symbol (cont’d) u v vi x x50,3 y y0

SI-Units (cont’d) NA [m s−1 ] [m s−1 ] [m] [m] [–] [–]

Meaning (cont’d) solution vector mean velocity velocity vector particle diameter mean diameter of volume distribution relative radial position wall region thickness

Greek letters Symbol α∗ β∗ Γ γ˙ ε δ ε˙ η ηr µ µ ξ ρ τ φ φm ω Ω

SI-Units [–] [–] [–] [s−1 ] [–] [–] [s−1 ] [–] [Pa s] [–] [Pa s] [–] [–] [kg m s−3 ] [Pa] [–] [–] [–] [–]

Meaning porous media flow coefficient porous media flow coefficient Strain shear rate porosity of compressible porous media compactability coefficient elongation rate strain of compressible porous media dynamic viscosity emulsion relative viscosity dynamic viscosity (Newtonian fluids) tortuosity factor loss factor Density deviatoric stress tensor volume fraction of dispersed phase maximum packing volume fraction relaxation factor (Sepran) angular velocity

Indices Symbol subscripts: 0 ac b c d hyd

Meaning initial condition above critical bulk region continuous phase dispersed phase hydraulic ...

xiii Symbol (cont’d) l min o p r s uc v, vol superscripts: *

Meaning (cont’d) loss minimum orifice packed bed, pipe radial sphere, spanwise unit cell volume specific normalized

Dimensionless numbers Symbol C Ca De Eu fk Λ Re We

Meaning C-value (centrifugal acceleration) capillary number Deborah number Euler number friction factor friction coefficient Reynolds number Weber number

Operator · × :

Meaning dot product cross product double dot products (of dyads)

Operators

Abbreviations Symbol Meaning AK product name of silicon oils made by Wacker BIM Boundary Integral Method, program name (droplet deformation) CCP Cubic-Close Packing CFD Computational Fluid Dynamics ...

NOTATION

xiv Symbol (cont’d) CMC DSR EFM FDM FEM FVM FTC GPL HCP NPT PDE PEG PSD RSO SDS

Meaning (cont’d) Critical Micelle Concentration Dynamical Stress Rheometer Extensional Flow Mixer Finite Difference Method Finite Element Method Finite Volume Method Flow-Through Cell GNU General Public License Hexagonal-Close Packing Numerical Particle Tracking Partial Differential Equation Poly-Ethylene Glycol Particle Size Distribution Rape Seed Oil Sodium Dodecyl Sulfate

List of Tables 2.1 2.2 2.3

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

7 8 18

3.1 3.2 3.3 3.4 3.5 3.6 3.7

Significance codes used within statistical analysis. . . . . . . . . . . . . . . . Viscosities and densities of silicon oils. . . . . . . . . . . . . . . . . . . . . Interfacial tension between AK silicon oils and PEG – 2% SDS – H2 O solutions. Incompressible sphere characteristics. . . . . . . . . . . . . . . . . . . . . . Sphere packing porosities for packing structures investigated. . . . . . . . . . Physical properties of elastic spheres materials. . . . . . . . . . . . . . . . . Loss factors, ξ, for orifice geometries. . . . . . . . . . . . . . . . . . . . . .

26 31 32 36 36 37 38

Strain above the critical capillary number Γac along track 3. . . . . . . . . . . Droplet deformation and break-up characteristics over a variety of initial droplet radii, a. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Units used for physical quantities within our statistical models. . . . . . . . . 4.4 Estimated coefficients of the statistical models for four dispersing experiments with varying viscosity ratio and packing length. . . . . . . . . . . . . . . . . 4.5 Data range for fit of dispersing model in sphere packing flows. . . . . . . . . 4.6 Model predictions for the dispersing process within incompressible porous media. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Data range for fit of dispersing model in orifice flows. . . . . . . . . . . . . . 4.8 Parameters of compressible sphere packings trials along with flow and compressibility characteristics. . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.9 Comparison of mean diameter, x50,3 , with model predictions for incompressible porous medium. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.10 Comparison of the width of particle size distribution, span, with model predictions for an incompressible porous medium. . . . . . . . . . . . . . . . .

4.1 4.2

47 49 57 62 63 66 67 73 80 80

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

92

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

94 95 96

xv

List of Figures 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.1 3.2 3.3 3.4 3.5

Porous medium from randomly arranged polydispersed spheres . . . . . . . . Microscopic, mesoscopic and macroscopic flow patterns within porous media. Porosity variation within columns of randomly packed monodispersed spheres. Flow characteristics for regularly and randomly arranged porous media. . . . Flow behavior in compressible porous media. . . . . . . . . . . . . . . . . . Loss factor versus area ratio for orifice flows. . . . . . . . . . . . . . . . . . Nozzle geometry with constant elongation rate along its centerline. . . . . . . Normalized area porosity in the spanwise direction versus normalized streamwise position for sphere packings and Drost’s nozzle geometry. . . . . . . . . Droplet break-up criteria in terms of critical capillary number for various flow types and viscosity ratios. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dynamic viscosity of PEG – SDS – water solutions in terms of PEG concentration and temperature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Density of PEG – SDS – water solutions in terms of PEG concentration and temperature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Process unit with perforated screen flow through cell. . . . . . . . . . . . . . Regularly arranged sphere packings. . . . . . . . . . . . . . . . . . . . . . . Friction coefficient versus Reynolds number for sphere packings. . . . . . . .

4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10

Geometry of eight adjoint converging-diverging nozzles studied numerically. Partial mesh of the converging-diverging nozzle geometry. . . . . . . . . . . Annulus probability in developed pipe flow. . . . . . . . . . . . . . . . . . . Shear and elongation rates within nozzles at Re = 100. . . . . . . . . . . . . Shear and elongation rates along 3 tracks at Re = 100. . . . . . . . . . . . . Shear and elongation rates within nozzles at Re = 1000. . . . . . . . . . . . Shear and elongation rates along 3 tracks at Re = 1000. . . . . . . . . . . . . Particle tracks within the converging-diverging nozzle geometry at Re = 1000. Droplet break-up along track 3 for λ = 1. . . . . . . . . . . . . . . . . . . . Influence of droplet size on its deformation and break-up within the adjoined nozzle geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.11 Maximum deformations of droplets along tracks 1, 2, and 3. . . . . . . . . . 4.12 Maximum droplet deformation within the throat of the first nozzle moving along track 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.13 Droplet deformation within eight adjoint converging-diverging nozzles along track 2 for λ = 1 at Rec = 1000. . . . . . . . . . . . . . . . . . . . . . . . . xvii

4 5 6 9 10 11 12 13 17 29 30 34 35 37 40 40 41 42 43 44 45 45 47 49 50 52 53

xviii

LIST OF FIGURES

4.14 Droplet deformation within eight adjoint converging-diverging nozzles along track 2 for λ = 3 at Rec = 1000. . . . . . . . . . . . . . . . . . . . . . . . . 4.15 Droplet deformation within eight adjoint converging-diverging nozzles along track 2 for λ = 5 at Rec = 1000. . . . . . . . . . . . . . . . . . . . . . . . . 4.16 Shear and elongation rates within adjoint die entries for Rec = 100. . . . . . 4.17 PSD for the dispersing process of 2% RSO in 10% PEG – SDS – H2 O solution in a packed bed of spheres. . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.18 Particle size characteristics vs. specific energy input. . . . . . . . . . . . . . 4.19 Comparison between experimental data and model predictions for dispersing in porous media. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.20 Compressed porous medium made of cubically arranged spheres. . . . . . . . 4.21 Flow and compression characteristics for a cubically arranged sphere packing. 4.22 Compression characteristics for three Newtonian fluids. . . . . . . . . . . . . 4.23 Flow and compression characteristics for rhombohedrally arranged sphere packing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.24 Dimensionless flow characteristics for both packing types. . . . . . . . . . . 4.25 Flow and compression characteristics for a long cubic sphere packing of hard material. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.26 Flow and compression characteristics for a short rhombohedral sphere packing from soft material. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.27 Flow and compression characteristics for a non-Newtonian fluid. . . . . . . . 4.28 Dispersing in a compressible porous medium. . . . . . . . . . . . . . . . . .

60 70 71 72

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

98 98

54 55 56 58 59

74 75 76 77 78 79

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

Abstract Dispersing an oily liquid in a non miscible aqueous liquid – with the resulting product being denoted as an emulsion – is the connecting theme of investigations performed within this work. Emulsion production is an important process, in the food industry, where milk, salad dressings, margarine, fat spreads and mayonnaise are prominent examples of emulsions, and elsewhere. Rotor-stator systems or high pressure homogenizers are typical types of processes which are used, with flow often being turbulent in such processes. Lately, the importance of elongational effects in dispersing processes has been recognized as witnessed by patents granted in the field of dispersing. With well-balanced contributions of shear and elongational flow to the overall flow field in porous media, partially periodically recurring, beneficial emulsion production was envisioned for porous media flow. Such flows were therefore chosen to form the basis of this work. Within this work, a process unit was designed and built, holding various flow-through cells. Within those flow through cells, porous media in the form of sphere packings of glass or steel spheres, ranging from 70 µm to 4 mm in diameter, were either randomly filled into the cell or in case of the largest spheres, regularly arranged. Three arranged sphere packing structures were studied, cubic, orthorhombic, and rhombohedral. The investigation of flow and compression behavior in compressible regularly arranged sphere packings was another part of this work, along with the evaluation of the suitability of compressible porous media as a dispersing device. Such compactible porous media were built up 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. PEG concentration ranged from 0% to 19% and the concentration of the emulsifying agent SDS was 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 orifice geometries – modeling porous media flow – was experimentally investigated. This was accompanied by applying computational fluid dynamics (CFD) to droplet deformation and break-up. Besides experimentally investigated orifice geometries, an adjoint convergingdiverging nozzles geometry, designed such that constant elongation rates are present along its center-line, was also numerically studied. In the numerical investigation, flow fields within nozzles and orifices were calculated by means of the finite element method (FEM), and particle tracks were calculated according to a numerical particle tracking algorithm. The boundary integral method (BIM) was applied to droplet deformation calculations along given particle tracks. Particle size distributions (PSD) of emulsions were determined by laser diffraction spectroscopy. 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 diameters and narrower particle size distributions. Experimental data were analyzed by means of the statistics 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 on the dispersing result were considered. The power input is given by the pressure drop across the flow through cell, whereas the energy input is related to the pressure drop times the number of passages through the flow through cell. It was found that regardless of sphere packing or orifice 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 the package. Given the task of producing an emulsion of a certain quality one has to take both aspects 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 phase and that of the continuous phase revealed that along with the expected influence of the viscosity ratio on the overall dispersing result as main effect, an interaction between the viscosity ratio and the packing length was found. Two levels of viscosity ratio were considered, 1.7 and 6.9. It was concluded from investigating the above interaction that the inflow becomes more important to the overall droplet break-up with higher viscosity ratio. Finally, dispersing models 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 rate to pressure drop relations reported in literature. Moreover, we were able to point out packing internal deformation characteristics. A comparison between dispersing within such compressible porous media and predictions made by our model for an incompressible porous medium using otherwise matching parameters gave finer emulsions in case of compressible porous media. An extension of the mean diameter model for incompressible porous media such that porous media compressibility could be accounted for, also desirable seemed to be too unrealistic, 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 with our experimental data. Nevertheless, one-to-one simulation of droplet break-up within complex 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 o¨ ligen in einer wässrigen Phase war der verbindende Prozessaspekt in dieser Arbeit. Das Herstellen solcher Dispersionen wird als Emulgieren bezeichnet und ist ein wichtiger Prozess-Schritt, nicht nur in der Lebensmittelindustrie. Milch, SalatSaucen, Mayonnaise und Margarine sowie Brotaufstriche sind typische Beispiele f u¨ r Emulsionen. Zur Herstellung werden oft Rotor-Stator-Systeme oder Hochdruckhomogenisatoren verwendet. Dabei ist die Strömung im Dispergierprozess oft turbulent. Die Bedeutung von Dehnströmungseffekten im Dipsergierprozess ist in den vergangenen Jahren zunehmend herausgestellt worden, wie dies in erteilten Patenten dokumentiert ist. Ausgewogene Scher- und Dehnstro¨ mungsanteile am Gesamtströmungsfeld bei der Durchströmung von porösen Haufwerken, die darüberhinaus noch teilweise periodisch wiederkehren, liessen solche Strömungen für das Emulgieren als günstig erscheinen. Deshalb wurden derartige Strömungen als Grundlage für diese Arbeit gewählt. Im Rahmen dieser Arbeit wurde eine Prozess-Einheit entwickelt und aufgebaut, die verschiedene Strömungszellen aufnehmen konnte. In diesen Stro¨ mungszellen wurden poröse Haufwerke in Form von Zufallsschu¨ ttungen aus Glas- und Stahlkugeln mit Durchmessern von 70 µm bis 4000 µm realisiert. Aus den 4 mm grossen Kugeln konnten dar u¨ berhinaus auch noch regelmässig angeordnete Kugelpackungen erstellt werden. Kubische, orthorhombische und rhombohedrale Packungsstrukturen wurden dabei ber u¨ cksichtigt. Ein weiterer Bestandteil dieser Arbeit war die Untersuchung des Kompressionsverhaltens von kompressiblen, regelmässig angeordneten Kugelpackungen sowie der Abschätzung zur Eignung solcher Kugelpackungen im Dispergierprozess. Die kompressiblen Kugelpackungen wurden aus Silikonkautschukkugeln hergestellt und in kubischer und rhombohedraler Anordnung untersucht. Für die Emulgierexperimente wurden Polyethylenglykol (PEG) in 2%iger Natriumlaurylsulfat – Wasser Lösungen als kontinuierliche Phase mit PEG Konzentrationen zwischen 0% und 19% verwendet. Die SDS Konzentration von 2% war so gewählt, dass sie weit oberhalb der kritischen Mizellenbildungskonzentration lag. Silikono¨ le mit unterschiedlichen Viskositäten und handelsübliches Rapsöl wurden als disperse Phase eingestezt. Neben dem Dispergieren in Kugelschu¨ ttungen wurde auch das Emulgierverhalten in mehreren hintereinandergereihten, Du¨ sengeometrien mit abrupten Querschnittsänderungen experimentell untersucht. Die Düsengeometrien wurden als Modell fu¨ r Kugelpackungen gewählt. Begleitet wurden diese Untersuchungen von numerischer Stro¨ mungssimulation mittels derer die Tropfendeformation und der allfällige Tropfenaufbruch simuliert werden konnte. Neben den experimentell untersuchten, abrupten” Düsengeometrien wurden auch aneinandergereih” te konvergierende-divergierende Du¨ sen betrachtet, deren Geometrie so gewählt war, dass entlang der Mittelline konstante Dehngeschwindigkeiten angetroffen wurden. In unseren numerischen Untersuchungen wurden die Stro¨ mungsfelder innerhalb der xxi

xxii

ZUSAMMENFASSUNG

Düsengeometrieen mittels Methode der Finiten Elemente (FEM) berechnet und Stromlinien u¨ ber einen Algorithmus zur Bestimmung dergleichen ermittelt. Die ’Boundary Integral Method’ (BIM) wurde angewandt, um Tropfendeformation und -aubruch entlang der Stromlinien zu bestimmen. Die Emulsionsqualität, die sich u¨ ber den mittleren Tropfendurchmesser und die Breite der Tropfendurchmesserverteilung bestimmt, wurde mittels Laserbeugungsspektroskopie durchgeführt, wobei eine höhere Qualität mit kleinerem mittleren Durchmesser und reduzierter Breite der Tropfengrössenverteilung einher geht. Experimentell gewonnene Daten wurden unter Verwendung des Statistikpakets ’R’ ausgewertet. Es wurden solche Modelle aufgestellt, die die Zielgrössen wie mittlerer Durchmesser und Verteilungsbreite, durch rein physikalische bzw. Prozess- Parameter erklären. Bei der Entwicklung unserer Modelle wurde zurerst der Einfluss von Energie- und Leistungseintrag auf das Emulgierergebnis betrachtet. Der Leistungseintrag ist dabei durch den Druckabfall u¨ ber der Durchströmungszelle gegeben und der Energieeintrag durch den Druckabfall mal der Anzahl an Durchläufen durch die Strömungszelle. Es konnte herausgestellt werden, dass der Leistungseintrag erwartungsgemäss einen stärkeren Einfluss auf die Feinheit der Emulsion hat als der Energieeintrag. Eine Erho¨ hung des Energieeintrags ist jedoch mit einer Verringerung der Verteilungsbreite verbunden. Die Aufgabe, eine Emulsion mit einer bestimmten Qualität zu erzeugen, wobei die Qulität durch den mittleren Tropfendurchmesser und die Verteilungsbreite bestimmt ist, muss unter Beru¨ cksichtigung sowhol des Energie- als auch des Leistungseintrags gelo¨ st werden. Mit der Erhöhung der Anzahl der untersuchten Parameter ging eine Erho¨ hung der Komplexität der gefundenen Modelle einher. Bei der Ausdehnung der Untersuchung auf den Einfluss der Packungslänge und des Viskositätsverhältnisses zwischen disperser und kontinuierlicher Phase wurde neben dem erwarteten Einfluss des Viskositätsverhältnisses als Haupteffekt auch noch eine Interaktion zwischen diesen beiden Parametern aufgezeigt. Dabei wurden Viskositätsverhältnisse von 1,7 und 6,9 betrachtet. Aus der Interaktion konnte abgeleitet werden, dass bei höherem Viskositätsverältnis die Einlaufeffekte stärker zum Gesamttropfenaufbruch beitragen als dies beim niedrigeren Viskositätsverhältniss der Fall war. Zum Abschluss der Modellentwicklung für das Dispergieren konnten erfolgreich Modelle aufgestellt werden, die alle untersuchten Parameter als erklärende Grössen enthielten. Diese Parameter bestanden aus Prozess- und Fluidparametern sowie Geometriegro¨ ssen. Die Durchströmung von kompressiblen poro¨ sen Medien konnte mittels aus der Literatur bekannten Zusammenhängen zwischen Durchfluss durch die Packung und Druckverlust u¨ ber derselben characterisiert werden. Daru¨ berhinaus wurden packungsinterne Deformationen aufzeiget und quantifizieret. Ein Vergleich zwischen dem Dispergierergebnis in einem kompressiblen porösen Kugelhaufwerk und dem aus unserem Modell erwarteten, mittleren Tropfendurchmesser bei sonst gleichen Bedingungen ergab feinere Emulsionen im Falle der kompressiblen Packung. Einer Erweiterung der fu¨ r die inkompressiblen Packungen gefundenen Modelle um den Einfluss der Kompressibilität erschien wegen des – immer noch – beschränkten Wissens um das Dispergieren in inkompressiblen Packung noch nicht angezeigt. Das Tropfenaufbruchverhalten, das mittels numerischer Stro¨ mungssimulation evaluiert ¨ wurde, war in guter Ubereinstimmung zu dem auf Experimenten basierten Modellen zum mitteleren Tropfendurchmesser. Trotzdem ist eine exakte Simulation von Tropfenaufbruch in komplexen Strömungen aufgrund von Einschränkungen, die in dieser Arbeit herausgestellt wurden, noch nicht umfassend mo¨ glich.

Chapter 1 Introduction 1.1

Dispersing

Dispersing is a process not only important to the food industry but also applied within the pharmaceutical, cosmetics and polymer industries among others. Typical foodstuff emulsions, where liquid droplets are dispersed in another immiscible liquid, are milk, margarine, mayonnaise and salad dressings. Those products are mainly processed within rotor-stator systems or high pressure homogenizers under the addition of emulsifying agents, used to facilitate the disruption of droplets and stabilize the drop interface. The quality of emulsions is often defined by two parameters, the mean diameter of the dispersed phase droplets and the width of their particle size distribution (PSD). The smallest possible droplets with the narrowest possible PSDs are desirable for emulsion quality in general and emulsion stability in particular. The mean size of the droplets can in general be adjusted by the volume specific energy applied with higher energy inputs resulting in smaller droplets. Correlations between PSD widths and process and fluid parameters are much more difficult to establish and PSDs ranging over one order of magnitude are already considered to be narrowly distributed. Within the dispersing process, droplet break-up results from shear and normal forces acting upon the droplet. These forces are typically of a transient nature as the droplet moves through the dispersing device. Flow fields within the droplet and the droplet shape itself also contribute to its break-up behavior. Furthermore, flow field fluctuations, along with turbulent flow behavior, as is often present in dispersing devices, increases the complexity of the breakup mechanisms. However, turbulent dispersing processes occur with higher specific energy input and broader particle size distributions. Break-up of single droplets under stationary flow conditions has been the subject of often cited investigations reported in literature. In these investigations, droplet break-up was studied for 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 the flow type, but also strongly on the ratio between the dispersed phase viscosity and the continuous phase viscosity. For viscosity ratios above about 4, droplets can not be broken up under simple shear flow. With increasing elongational flow contributions, droplets under higher viscosity ratios can be broken up if sufficient deformation is reached. Then, the required specific energy input is reduced. 1

CHAPTER 1. INTRODUCTION

2

1.2

Porous Media Flow

Laminar flow through porous media, with packed beds of spheres randomly distributed being one example, are characterized by a well-balanced composition of domains dominated by shear and elongational flow, partially even periodically recurring. Porous media made of spheres can be either built up randomly or in the case of monodispersed spheres, structured packings such as cubically arranged or hexagonal-close packings (HCP) can be realized. Flow through such packings can be characterized in terms of dimensionless numbers including the packing Reynolds-number, the friction coefficient and the packing porosity. Packing parameters comprise mean sphere diameter, packing porosity, and packing length. Fluid density and fluid dynamic viscosity function account for the fluid behavior. Pressure drop and volumetric flow rate account for the process parameters. Packing structures resulting from randomly packed or arranged spheres strongly influence the flow characteristics given by those dimensionless numbers stated above. Friction coefficients deviating by more than half an order of magnitude were reported for such packings in the literature. Moreover, large deviations are not only attributed to the packing structure, but also, 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, the porosity within such porous media depends on the pressure drop across the packing length and varies spatially over the flow in streamwise direction. This typically goes along with strong deviations 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 dispersing result within flows through porous media, striving for an optimization of the process given the optimization criteria of minimizing energy input and maximizing product, i. e. emulsion, quality. In order to achieve this goal, model geometries facilitating a comparison between numerical simulations of single droplet break-up and emperical results were considered. Secondly, we aimed at a characterization of regularly arranged compressible porous media. This comprised the flow rate dependence on the pressure drop across the porous medium and spatial porosity information for various flow-through conditions. Finally, the usability of such compressible porous media were assessed. In order to investigate the above problems, various experimental, analytical, and numerical tools were available at the Laboratory of Food Process Engineering at ETH Zurich but had to be adapted to the given task. Those tools included data acquisition systems used on pilot plant scale pressure filtration facilities, light spectroscopy for particle size measurements, rheometers, and computational fluid dynamics tools based on the finite element and boundary integral method.

Chapter 2 Background Various aspects centering around porous media flow applied to dispersing processes were studied as part of this work. First, flow characteristics through regularly and randomly arranged porous media and model systems thereof will be introduced in this chapter. Dispersing will then be considered for single and multiple droplets suspended in an immiscible fluid experiencing simple and mixed, steady and unsteady flow fields. The numerical methods applied to 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 nozzle flow, with the latter forming a model of the former, along with a description of their numerical treatments. Randomly arranged, regularly arranged and compressible porous media will be considered. 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 the direction of decreasing pressure will result. Parameters governing this flow comprise fluid and porous medium properties as depicted in Figure 2.1. Dating back to 1856, Darcy [Dar56, Appendix D] was the first to study flows through porous media, setting up the famous Darcy Law for laminar flow of Newtonian fluids through packed beds, A · ∆p · k V˙ ∝ L

(2.1)

with volumetric flow rate V˙ dependent on the packing cross-sectional area A, packing length L, pressure drop over the packing length ∆p and packing permeability k. The latter parameter correlates with primary packing characteristics such as the shape of the packed bed particles, 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 fluid viscosity η. In case of generalized non-Newtonian fluids the dynamic fluid viscosity is a 3

CHAPTER 2. BACKGROUND

4

. V

Fluid properties: ρ, η

L, ∆p

Packed bed properties: ε, d s (PSD), k

. V A

Figure 2.1: Porous medium of randomly arranged polydispersed spheres through which flows a fluid of density ρ and dynamic viscosity η over cross-section area A. Further parameters are explained in the text. function of shear rate γ, ˙ as given by η = η(γ). ˙ It has to be noted that viscosities of nonNewtonian fluids used within integral equations relating the volumetric flow rate to packing and 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 ratio of packing void volume to total packing volume. This parameter is widely used within porous media flow analysis. The volumetric flow rate V˙ is often expressed in terms of the superficial fluid velocity based on an empty column cross-section v¯. For a detailed introduction to porous media 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 resistance Rc and filter medium resistance Rm along with fluid viscosity η. This is indicated in the following equation. V˙ =

A∆p η(Rc L + 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 scale considered. As depicted in Figure 2.2 on the left (A), microscopic flow denotes the flow between 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 Reynoldsnumber, not only quantitatively but also qualitatively due to the formation of jet-like flow with

2.1. FLOW THROUGH POROUS MEDIA AND NOZZLES A) Mircoscopic

B) Mesoscopic

5 C) Macroscopic

Figure 2.2: Microscopic, mesoscopic and macroscopic flow patterns within porous media according 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 which partially rearrange under flow conditions forming areas with even lower porosity and likewise allowing 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 the right of Figure 2.2 (C) shows a somewhat unexpected flow profile compared to pipe flow in which highest flow rates occur along the centerline. In porous media flow, high flow rates close to the wall stem from a variation of porosity over the cross-section. In Figure 2.3, experimental data for such porosity variation of randomly arranged monodispersed spheres (◦) is given as radial porosity εr , over a normalized radial position, with 0 being at the column wall and increasing radial position pointing towards the column center. These data were reported by Benenati and Brosilow [BB62] obtained for a sphere diameter to column diameter ratio of ds /D = 0.07092. Figure 2.3 shows the radial porosity to be equal to 1 at the wall (εr = 1). With increasing distance from the wall, the radial porosity initially decreases reaching a minimum at about half the sphere diameter with radial porosity of about εr = 0.22. Further on, the radial porosity oscillates around the mean porosity of randomly packed monodispersed spheres ε with 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 radial porosity as follows: ( y y 1+ε : y ≤ y0 − (1 − ε) 1 − 2 2y0 y0 (2.3) εr = ε : y > y0 with relative radial position y = (R − r)/ds and a wall region thickness y0 . The model is included in Figure 2.3 as a dashed line with the wall region thickness chosen to be y 0 = 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


6

Radial porosity εr [-]

1.0 Model given by Liu and Masliyah Experimental data of Benenati and Brosilow 0.8

0.6

0.4

0.2

PSfrag replacements 0

1

2

3

4

5

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 (dashed line) according to Liu and Masliyah [LM96]. Figure 2.1. These are – given along with their dimensions in terms of mass M, time T, and length 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 over the 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 between those parameters can likewise be expressed in terms of four dimensionless parameters with their number (4) given by the number of initial parameters (7) minus the number of parameter dimensions (M, T and L, i. e. 3). The reader is referred to Stichlmair [Sti90] for an extensive introduction to dimensional analysis in the field of engineering. Two of the four resulting dimensionless numbers are trivial: the packing porosity ε, and the ratio between sphere diameter and packing length ds /L. The other two are most often expressed in terms of Reynolds-number Re and friction coefficient Λ: Re =

ρ¯ v ds η(1 − ε)

(2.4)

ε3 ∆p d2s · · (2.5) L η¯ v (1 − ε)2 It should be noted that the friction coefficient is sometimes expressed in terms of a friction factor fk times the Reynolds-number Re, and the friction factor itself can be expressed in terms of a Euler-number Eu2 as given in Eq. (2.6). Λ=

Λ = fk · Re = Eu2 ·

∆pds ε3 ε3 · Re = · Re · 1−ε Lρ¯ v2 1 − ε

(2.6)

2.1. FLOW THROUGH POROUS MEDIA AND NOZZLES

7

Table 2.1: Coefficients for porous media flow characteristics according to Eq. (2.7) relating friction coefficient Λ to Reynolds number Re. Reference Ergun, [Erg52] Haas and Durst, [HD82] Vorwerk and Brunn, [VB94]

K 1 K2 K3 150 1.75 1.0 185 1.75 1.0 181 2.01 0.96

Remark spheres, granular particles spheres spheres

For flow through sphere packings of mono-dispersed and poly-dispersed spheres and granular particles a relation between friction coefficient and Reynolds-number was found. This relation 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 at higher Reynolds-numbers, various definitions such as Forchheimer flow, Burke-Plumer flow and turbulent flow exist. Partially controversial discussions on the existence and cause for such flows are found in Nield [Nie01], Liu and Masliyah [LM96], as well as Masuoka and Takatsu [MT96]. Porous media flow coefficients as given in table 2.1 assume inflow and wall effects to be negligible. This assumption is valid in the case of inflow effects for packing length to sphere diameter ratios L/ds greater than 10, and in the case of wall effects for column diameter to sphere diameter ratios D/ds also greater than 10. Related investigations were reported by Pahl [Pah75]. 2.1.1.4 Regularly Arranged Porous Media Martin et al. [MMM51] and Franzen [Fra79b, Fra79a] investigated flow characteristics in regularly arranged porous media of monodispersed spheres. They studied cubic, orthorhombic, and rhombohedral packing structures, which would be denoted in terms of Bravais lattice types – as given in Appendix A – as cubic primitive (cP), hexagonal primitive (hP), and cubic 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 spheres for each sphere and is 6, 8 and 12 for cubic, orthorhombic, and rhombohedral packing structures respectively. Packing porosities ε for those packing structures are 0.476, 0,3954 and 0.2595. As an aside, Hales [Hal97b, Hal97a] recently suggested a proof for the Kepler Conjecture, that there are no packings of monodispersed spheres with porosities smaller than 0.2595. In the case of orthorhombic and rhombohedral packing structures, Franzen found the flow characteristics to depend on sphere packing orientation. Orientations were denoted by I and II. Franzen based his investigations on a modified Hagen-number Ha p and Reynolds-number Rep , as given in Eqs. (2.8, and 2.9) and found a relation between the two terms provided in Eq. (2.10).


8

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) as used within this work. Type cubic orthorhombic I orthorhombic II rhombohedral I rhombohedral II

K1 K2 K3 α∗ β∗ k 164.97 1.976 0.9 420 9.0 6 574.98 10.913 0.9 3400 101.5 8 101.47 2.925 0.9 600 27.2 8 168.90 2.412 0.9 5300 99.2 12 140.22 2.106 0.9 4400 86.6 12

Hap =

Used within this work yes yes no no yes

∆p d2s · L v·η

(2.8)

vρds η

(2.9)

Rep =

Hap = α∗ + β ∗ · Re0.9 p

(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 and Brunn [VB94]. Characteristics depend strongly on the packing structure and in the case of orthorhombic packings even more on their orientation with orthorhombic I denoting a packing with an unobstructed straight passage. No orientation has to be given for cubically arranged packings since flow always goes through unobstructed straight passages, given flow in the direction 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 provide a representative capillary radius for a given porous medium, the hydraulic radius concept of Debbas and Rumpf [DR66], with rhyd as the ratio of the porous medium void volume to the wetted surface area, is used. In terms of porous medium porosity ε and specific surface area Sv 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 cubically arranged sphere packings with a porosity of ε = 0.476, the hydraulic diameter becomes rhyd ≈ 0.151 · ds . Other capillary models state the minimum diameter of a stream-tube through a porous medium as the representative capillary radius rmin . For the cubically arranged sphere packing this minimum radius becomes rmin = 0.207 · ds . A capillary with the porous medium void


random packing cubic (k=6) orthorhombic I (k=8) orthorhombic II (k=8) rhombohedral I (k=12) rhombohedral II(k=12)

5000

Friction coefficient Λ [–]

9

2000 1000 500

200 100

PSfrag replacements

1e−01

1e+00

1e+01

1e+02

1e+03

Reynolds-number Re [–] Figure 2.4: Flow characteristics for regularly and randomly arranged porous media composed of monodispersed spheres. The characteristics are given in terms of friction coefficient Λ and Reynolds-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 r max 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 filtration processes. Recent advances were reported by Windhab et al. [WFM96] and Friedmann [Fri99], investigating hyperbaric filtration within centrifugal fields. Friedmann established a model for such flows fiven various process and material parameters. Other recent advances were those given by Tiller and co-workers in the field of highly compactible filter cakes with variable flow rates and filtration with sedimentation [LJKT00, TLKL99, THC95]. In an earlier work, Tiller and Hsyung [TH93] categorized the flow through compressible porous media according to the level of compactability into incompressible, low, moderate, high, and super-compactible media. Flow characteristics for such compressible porous media are shown in the left-hand graph of Figure 2.5 in terms of volumetric flow rate V˙ and the pressure drop over the porous medium ∆p. The compactability coefficient δ accounts for the ease of filter cake and porous media compressibility. Its value represents the negative slope of a logarithmic plot of permeability k, 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 between volumetric flow rate V˙ and pressure drop ∆p, according to their categorization as given in Eqs. (2.12, 2.13, and 2.14) for low, high, and super-compactible porous media, respectively.


10

incompressible δ = 0

low low δ = 0.5

Permeability k

⋅ Volumetric flow rate V

incompressible

high

moderate δ=1

super (super)

Pressure drop ∆p

high δ=2

Compressive pressure p

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

∆p1−δ V˙ ∝ L

(2.12)

ln(∆p) V˙ ∝ L

(2.13)

1 V˙ ∝ L

(2.14)

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

2.1.2 Model Geometries for Porous Media The porous media flow models based on representative capillaries previously introduced have the shortcoming that they disregard elongational flow behavior within porous media. Periodically expanding and contracting model pore geometries account for such elongational flow behavior. Aspects of shear and elongation within flows are treated in section 2.1.6 below.


11

0.5

0.3

A2

0.2 A1

Loss factor ξ [–]

0.4

0.1

0.0

PSfrag replacements 0.0

0.2

0.4

0.6

0.8

1.0

Area ratio A2 /A1 [–] Figure 2.6: Loss factor ξ versus area ratio A2 /A1 for orifice flows according to Beiz and Dubbel [BD90]. The terms orifice, die entry and sudden contraction will be used synonymously within this work. It has to be noted that contrary to real porous media flow, flow within such model pore geometries 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 pressure loss due to the contraction ∆pl , as given by ∆pl = ξρv 2 /2

(2.15)

with fluid density ρ and fluid velocity v. Expressing the fluid velocity by means of the Reynolds-number Reo = ρvdo /η with orifice diameter do and fluid viscosity η, the pressure loss can be written as ξ ∆pl = 2ρ

Reo · η do

2

(2.16)

The critical Reynolds-number, indicating the transition from laminar to turbulent flow within pipe and orifice flow is about 2300. This is about an order of magnitude above the critical Reynolds-number within porous media flow defined in Eq. (2.4). Therefore, Reynoldsnumbers 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 the cross-sectional area upstream of the sudden contraction A1 and the cross-sectional area downstream of the sudden contraction A2 .


x ro

r

ri

12

ln

Figure 2.7: Nozzle geometry with constant elongation rate along its centerline as given by Eq. (2.17) for inlet radius ri , outlet radius ro and nozzle length ln chosen to be 2, 0.2 and 4 respectively. The nozzle region is indicated by dark grey and the in and outflow regions by light grey. 2.1.2.2 Nozzle with Constant Elongation Rate Drost [Dro99] designed a nozzle with constant elongation rates along its center-line under the assumption of developed flow profiles within the nozzle. The nozzle geometry is described by

r =

ro−2 − ri−2 · x + ri−2 ln

−1/2

(2.17)

with inlet radius ri , outlet radius ro , and nozzle length ln . Such a nozzle is depicted in Figure 2.7. Drost [Dro99] reported on numerical investigations of the flow field within such a nozzle at a Reynolds-number of Re = 200, defined in terms of the outlet radius r o . Good agreement was found between the expected and calculated elongation rate along the center-line, with constant 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 geometries, modeling the former ones in terms of normalized area porosity in the spanwise direction ε∗s . It was normalized by the unit cell area in case of porous media and by the area in spanwise direction at the inflow in case of nozzle and orifice geometries. Normalized area porosity in the spanwise direction ε∗s is depicted versus the normalized streamwise position x∗ in Figure 2.8 for the geometries studied within this work. Large differences in this porosity can be attributed to the various geometries. Streamwise positions were normalized in the case of the porous media by the sphere diameter which is identical to the unit cell length of the cubically and orthorhombically (I) arranged sphere packings. The unit cell length luc of the rhombohedrally arranged sphere packing is shorter than the sphere diameter by a factor of 0.71. Therefore – disregarding the tortuosity factor µ – the normalized spanwise porosity (dotted line) is given along a shorter streamwise position as indicated by the vertical dotted lines. The minimum spanwise porosity was aligned with the minimum porosities of the other nozzles at x∗ = 0.5.


13

Normalized area porosity in the spanwise direction ε∗s [–]

1.0

PSfrag replacements

k=6

0.8

0.6 k=8 0.4

k = 12 nozzle

0.2 orifice

0.0 0.0

0.2

0.4

0.6

0.8

1.0 ∗

Normalized streamwise position x [–] Figure 2.8: Normalized area porosity in the spanwise direction ε∗s versus normalized streamwise position x∗ for regularly arranged porous media, an orifice with a 4:1 contraction, and a 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 rhombohedrally (II) arranged sphere packings, respectively. The spanwise porosity of the latter packing is shown as a dotted line, that of the nozzle as a dashed line, and that of the orifice as a 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 used to reveal and characterize the elastic properties of such fluids. This was described in Durst et al. [DHI87], Drost [Dro99], and Della Valle et al. [DVTC00]. Drost investigated the flow of beer through packings of spheres. He found an increase in the friction coefficient Λ of about an order of magnitude over a range of Reynolds-numbers of about two orders of magnitude compared with the respective characteristics for Newtonian fluids as given in Eq. (2.7). This behavior was attributed to elongational effects acting upon macromolecular proteins in beer. Introducing an experimental Deborah number, De0 = tε /(1/ε) ˙

(2.18)

with the time of strain tε , and characteristic elongation time 1/ε˙ in terms of elongation rate 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 porous media. They proposed that the larger part of the pressure loss of the flow of dilute polymer solutions through porous media is possibly caused by the elongational strains macromolecular fuid components experience.


14

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 governing equations are the conservation equations for mass and momentum, given in Eq. (2.19) and in Euler-Cauchy formulation in Eq. (2.20), respectively ∇ · ρv = 0

(2.19)

∂v ρ + (v · ∇) v + 2Ω × v + ∇P − ∇ · τ = ρf ∂t

(2.20)

τ = η ∇v + ∇vT

(2.21)

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

where η is usually specified to be a function of the second invariant of the rate of strain tensor. Further details on the underlying formulations can be found in Panton [Pan84]. Various methods exist for discretizing the (steady state) governing equations in order to solve 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 the case 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, numerical treatment of certain fluid flows is still limited. This is particularly obvious in simulations of turbulent flow or once visco-elastic fluid properties are considered. Limitations can also result from difficulties in mapping microscopic flow geometries, as present in randomly arranged porous media flow, due to limitations in computer memory capacities. Volume averaging techniques are applied to model randomly arranged porous media as described by Liu and 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. Shear and elongation rates γ˙ and ε˙1 respectively, are characteristics of flow fields derived from the velocity gradient tensor ∇v as follows according to VanderWal et al. [VGK + 96]: r 1 (∇v + ∇vT ) : (∇v + ∇vT ) (2.22) γ˙ = 2 ε˙1 =

v v 1 ∇v + ∇vT : 2 kvk kvk

(2.23)


15

2.1.6.2 Predefined Velocity Gradients In order to investigate particular flow fields with certain, predefined contributions of shear and elongation rates, velocity gradients can be ‘assembled’. Therefore, a parameter α is introduced with which the transformed velocity gradient tensor can be written as follows:



     ε˙1 γ˙ 0 0 1 0 1 0 0 0  ∇v =  0 ε˙2 0  = (1 − α)G  0 0 0  + αG  0 ε˙2 /ε˙1 0 0 ε˙3 0 0 0 0 0 ε˙3 /ε˙1

(2.24)

with shear rate γ, ˙ and elongation rate in the k-direction ε˙ k . G is the sum of shear rate and elongation rate in the 1-direction G = γ˙ + ε˙1 and the aforementioned parameter α is defined by α=

ε˙1 ε˙1 = γ˙ + ε˙1 G

(2.25)

Simple shear flow is given by α = 0, simple elongation flow by α = 1, and mixed flow for values in between, assuming γ˙ ≥ 0 and ˙ 1 ≥ 0. In planar flow, Eq. (2.24) reduces to

     1 0 0 0 1 0 ε˙ γ˙ 0 ∇v =  0 −ε˙ 0  = (1 − α)G  0 0 0  + αG  0 −1 0  0 0 0 0 0 0 0 0 0 

(2.26)

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



     ε˙ γ˙ 0 0 1 0 1 0 0 ˙ 0  = (1 − α)G  0 0 0  + αG  0 −0.5 0  (2.27) ∇v =  0 −ε/2 0 0 −ε/2 ˙ 0 0 0 0 0 −0.5 with ε˙ = ε˙1 . 2.1.6.3 Strain As an integral measure for shear and elongation rate variations over time, the strain Γ can be used: Γ=

Z

Gdt =

Z

(γ˙ + ε˙1 )dt

(2.28)

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


16

2.2

Dispersing

Within the following sections, the deformation and break-up of droplets suspended in another immiscible fluid will be treated. Underlying mechanisms of single droplet disrupture as well as multiple droplet disrupture (emulsification) will be given along with a numerical procedure to model such deformation and break-up. We will focus on emulsions, although the concepts introduced also apply to other types of dispersions. 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 investigations since the early work of Taylor [Tay34]. Publications of Grace [Gra82] and Bentley and 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 acting upon it. Different break-up behaviors can be attributed to different kinds of continuous phase flow, i.e. laminar or turbulent flow with the former being the focus of this work. Dispersing in cavitation has also to be mentioned. Binary break-up, in which a droplet splits into two equally sized droplets is desirable, but this is often superseded by break-up into more droplets often accompanied by large variations in droplet size. Forming of satellite drops and tip-dropping can be regarded as mechanisms resulting in rather small droplets with the latter being caused by a non-uniform surfactant distribution 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 no time to adapt its shape to the rapidly varying flow field. This results in a highly elongated shape 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 as mixed flows have been studied extensively experimentally and numerically in order to specify break-up criteria. Three dimensionless parameters found to primarily account for break-up are the capillary number Ca, the viscosity ratio between the dispersed and continuous phase viscosity 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 values in between resulting in mixed flow. The capillary number Ca relates the viscous forces acting on a droplet to the Laplace pressure acting against the droplet deformation. The Laplace pressure is due to the curvature of the droplet surface and its interfacial tension. Ca is given by ηc Ga (2.29) σ with the radius of the undeformed, spherical droplet a, the continuous phase viscosity η c , the sum of the shear- and elongation-rate G as given in section 2.1.6.2, and the interfacial tension σ. It should be noted that the capillary number is often given in terms of Webernumber W e, which is essentially the same except for a factor of 2. Ca =

Critical capillary number Cacrit [mPa s]

2.2. DISPERSING

17

20.0

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

10.0 5.0 simple shear flow

2.0 1.0 0.5 0.2 0.1

simple elongational flow

PSfrag replacements 1e−03

1e−02

1e−01

1e+00

1e+01

1e+02

Viscosity ratio λ [-] Figure 2.9: Droplet break-up criteria in terms of critical capillary number Ca crit for simple shear (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 the largest droplet not broken up given certain, steady flow condition. This implies that droplets deformed under capillary numbers greater than the critical capillary number (Ca > Ca crit ) break up. Droplets deformed under capillary numbers below the critical value (Ca < Ca crit ) will deform but no break-up occurs. Experimentally and numerically determined critical capillary numbers Ca crit are depicted in terms of flow-type parameter α and viscosity ratio λ in Figure 2.9. Experimental data on simple shear flow (∗, α = 0) was reported by Grace [Gra82] for Couette flow with data points being smoothly connected by a dashed line known also as Grace curve. The dotted vertical line was added to indicate the viscosity ratio of about 4. It was found that under simple shear flow, droplets with viscosity ratios greater than 4 can not be broken up, regardless of the shear rates imposed; such droplets will only rotate but not break-up. Experimental data on elongational and mixed flow (open symbols) was provided by Bentley and Leal [BL86] measured by means of a Four-Roller-Mill apparatus. The dotted line was added to guide the eye along the simple elongational flow data points. Numerical data provided 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 were calculated by Feigl et al. [FW01, FKFW02] and Kaufmann [Kau02] for Newtonian fluids under the assumption of negligible inertia effects and constant interfacial tension, indicated by filled triangles and circles. Critical capillary numbers for simple shear flow are also given in table 2.3. Care has to be taken when comparing numerically and experimentally gained data from mixed flow fields due to different definitions of the flow field parameter α as pointed out


18

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 0.42 2 0.61 3 1.27 5 (∞) in section 2.1.6.2. Two major conclusions can be drawn from Figure 2.9. Firstly, with an increase in elongational flow, smaller critical capillary numbers are present, thus droplet break-up occurs at lower deformation rates. Secondly, droplets with a viscosity ratio greater than 4 can be broken up, once elongational flow is present. With higher values of α and thus a larger elongational flow contribution, the critical viscosity ratio, λcrit , below which break-up is possible given a certain 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 found droplet break-up to occur at lower capillary numbers compared to single droplet break-up and suggested 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 depend on the level of deformation rates G applied, but also on the time that such deformation rates last. In the recent work of Ha and Leal [HL01], a high degree of sensitivity to the details of the deformation process and history were acknowledged. Lately, attempts have been made by Feigl et al. [FKFW02] to relate the droplet break-up to the Strain Γ(λ) above the critical capillary number Cacrit (λ) as given in Eq. (2.28). With all the implications that go along with unsteady flow conditions – even when disregarding turbulent 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 field are governed by Eqs. (2.19 - 2.21) introduced in section 2.1.5 along with boundary conditions that are enforced on the continuous phase fluid, the interface condition to describe the jump in velocity and stress along the droplet surface, and the kinematic condition to describe the evolution of the interface in the flow field. Under the assumptions of incompressible Newtonian fluids and small droplet Reynoldsnumbers Red = ρGa2 /ηc 1 the boundary integral method (BIM) as described by Ladyzhenskaya [Lad69] and Pozrikidis [Poz92] can be applied and the derived problem contains only surface integrals. Thus, the dimensionality of the problem is reduced by one. The interested 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 calculations performed within this work.

2.2.2 Emulsions Having considered the break-up of single droplets we will now look at multiple droplet disruptions. Within emulsion generation, a multitude of droplets are simultaneously broken up. 2.2.2.1 Introduction Emulsions are dispersed, multi-phase systems of at least two almost immiscible phases with the dispersed phase – also known as the inner phase – embedded as droplets within the continuous phase, also known as the matrix or outer phase. Emulsions are thermodynamically unstable resulting in the droplets tending to re-coalesce. This is due to a reduction of the interface 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 size distributions (Qi versus xi ) and characteristics thereof. The index i in the cumulative distribution Qi and in the diameter xi indicates the ’Mengenart’ upon which the distribution is based. 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 limiting diameters x90,i and x10,i . The latter diameters represent those diameters at which 10% of the 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 on emulsions. Emulsions with particle diameters ranging over an order of magnitude or less are considered to be narrowly distributed. A measure for the width of particle size distributions is given by 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 emulsions and stabilize them. The former is achieved by reducing the interfacial tension σ and the latter aims at the prevention of sedimentation, aggregation and recoalescence. In emulsion processes, recoalescence is difficult to quantify due to the short time scale on which the droplet disruption and possible recoalescence takes place. Adsorption kinetics of emulsifiers also 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 dispersing devices like rotor-stator systems and homogenizers as described by Schubert [SA89]. The latter are basically comprised of nozzle systems. Processes studied within this work can be


20

regarded as such. A patent, granted to Kahl et al. [KKS+ 98] describes such a homogenizer which 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 pressure 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 x 3,2 to the energy input, according to: x3,2 = C · Ev−b

(2.31)

with b being a parameter dependent on the flow type and C an empirically determined coefficient representing fluid and geometry characteristics. In the case of laminar flow, Karbstein and Stang found b = 1 and in turbulent flow 0.25 < b < 0.4. Lately, particularly elongational effects within emulsion processes have been subject to extensive 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 mixing process. Both inventions build upon the work by Suzaka [Suz82] on a nozzle type mixing device. Windhab [Win97] was granted a registered design for a rotor stator system with a k-shaped geometry. Emulsions are also the target of forced recoalescence in order to separate phases. Spielmann 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 that behavior, 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 φ m : 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 viscoelastic fluid properties. Single-parameter viscosity-concentration equations for emulsions have been formulated to describe the rheological behavior. They take into account the viscosity ratio λ between the dispersed phase viscosity and the continuous phase viscosity and have the general form

2.2. DISPERSING

21

ηr = f (λ, φ)

(2.35)

The Taylor equation [Tay32] is the best known example: ηr = f (φ, λ) 5λ + 2 = 1+ φ 2λ + 2

(2.36)

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

Chapter 3 Material and Methods After presenting the numerical methods utilized within this work, analytical methods, used to characterize fluids and emulsions, are described. Subsequently, the process unit and various flow-through cells containing rigid or compressible spheres and orifices modeling sphere packings are introduced. Finally, experimental procedures are explained.

3.1

Numerical Methods

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

3.1.1 Calculation of Macroscopic Flow Field 3.1.1.1 Introduction Stationary isothermal fluid flow is governed by the Navier-Stokes equations as given in Section 2.1.5, Eqs. (2.19 – 2.21). Various numerical discretization schemes exist for solving such partial differential equations (PDEs), such as the finite difference method (FDM), the finite volume method (FVM), or the finite element method (FEM). We use the FEM within this work. The main constituents of the finite element method are the variational or weak statement of the problem and the approximate solution of the variational equations through the use of so called ’finite element functions’. Sometimes however, the term ’finite elements’ is used to denote the elements forming the mesh. For an introduction to FEM, the interested reader is referred to Hughes [Hug87]. 3.1.1.2 Sepran Sepran is a proprietary computational analysis package based on the finite element method and was used for this work to calculate macroscopic flow fields. It was provided by Inge23

CHAPTER 3. MATERIAL AND METHODS

24

nieursbureau SEPRA, The Netherlands (See [Seg84]). Sepran has been developed for and applied to a variety of two and three-dimensional problems including second order elliptic and parabolic equations, mechanical problems, flow problems, solidification problems, lubrication and coupled problems. It comprises pre-processing, processing and post-processing tools. Within this work, Crouzeix-Raviart elements were chosen along with extended quadratic triangles for calculating flow fields. Choosing Crouzeix-Raviart elements implies that swirl is neglected within axisymmetric calculations and pressures being discontinuous over elements. With penalty function formulation applied, the calculation of pressure and velocity were decoupled. Thus, the system of equations to be solved was reduced as described in Girault and Raviart [GR79]. The penalty function parameter was chosen to be ε = 1 × 10 −7 . For details on three-dimensional elements suited for fluid flow calculations, the reader is referred to Tanguy et al. [BGT92]. The systems of equations were solved iteratively. In a first iteration, Stokes flow was assumed. Secondly, one Picard iteration was calculated, followed by Newton iterations until convergence was reached. Within the Picard iteration, the convective term is approximated by a successive substitution according to Eq. (3.1). The Newton approximation is given in Eq. (3.2). (v · ∇v)n+1 ≈ vn · ∇vn+1

(3.1)

(v · ∇v)n+1 ≈ vn · ∇vn+1 + vn+1 · ∇vn − vn · ∇vn

(3.2)

Iterative solver parameters were the accuracy and the relaxation factor, ω. The accuracy was set to 1×10−4 . The relaxation factor influences the rate at which the solution of a problem progresses by splitting the new solution u∗ into a contribution of the last solution un and the last 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 be reduced. In our calculations, the relaxation factor was set to ω = 0.2. Sepran calculations were run on a PC-workstation with two 1.7 GHz Pentium 4 Processors and 1024 MByte RAM. One of Sepran’s post-processing features is the calculation of particle tracks along with shear and elongation rates within given flow fields. Shear and elongation rates are calculated in terms of tangentially aligned coordinate systems moving along the particle tracks.

3.1.2 Calculation of Drop Deformation 3.1.2.1 Introduction The Boundary Integral Method (BIM) was applied for studying the deformation and breakup 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 transforming a three-dimensional problem into a two dimensional problem forms the cornerstone of this method.

3.1. NUMERICAL METHODS

25

The governing equations solved are, as described in section 2.2.1.3, the conservation equations of mass and momentum for both fluids at creeping flow, the constitutive equation for the stress, boundary conditions that are enforced on the continuous phase, interface conditions to describe the jump in velocity and stresses along the interface, and a kinematic condition to describe the evolution of the interface in the flow field. Further details on BIM are given by Feigl 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 also be called BIM, the same as the method it is based upon. It is obvious from the context whether the boundary integral method itself or the program is being referred to. Investigations by Loewenberg and co-workers based on BIM include investigations into concentrated emulsion flows and coalescence behavior [LH96, Loe98, LH97, CBL01, CBL98]. Feigl and Windhab [FW01] extended Loewenberg’s BIM such that droplet deformation calculations for transient droplet histories along particle tracks were possible. Within this work the extended BIM has been used. Calculations were performed under the assumptions that both fluids exhibit Newtonian flow behavior with equal density and constant interfacial tension. It was also assumed that shear and elongation rates are constant over the droplet, thus requiring the length scale of the macroscopic flow field to be much larger than that of the droplet. Moreover, elongation rates in the spanwise direction were considered to be zero. This was an acceptable approximation since elongation rates in the spanwise direction are, under the assumption of continuity, small compared to those along streamwise directions. Moreover, elongation rates in the streamwise direction were found to be small compared to the shear rates for most of the particle tracks investigated. The extended BIM program takes a list of particle track information in terms of dimensionless travel time, t∗ , track coordinates, dimensionless shear rates, γ˙ ∗ , and dimensionless elongation rates ε˙∗ , as input. The characteristic time, Tchar = aµc /σ, where the radius a of the initial spherical undeformed drop, the viscosity of the continuous phase µ c , and the interfacial tension σ, was used to non-dimensionalize the travel time and the shear and elongation rates provided by Sepran calculations. The maximum length of the particle track list was set to 1000. The influence of the suspended droplet on the macroscopic flow field is small and can therefore be disregarded. Calculations were run on a HP 9000 enterprise server (Superdome, Rechenzentrum ETH Zurich, 48 PA8600 550 MHz processors). Each droplet deformation calculation was performed on a single processor with calculations running up to several weeks. Calculation time strongly depended on the number of mesh points to be calculated and the viscosity ratio. The number of mesh points is initially set by the user and then adapted by the program itself. We restricted the maximum number to 25000 grid points. The BIM program is sensitive to high gradients of shear and elongation rates. This was even more strongly pronounced at increasing viscosity ratios. It has to be noted that the graphical output of the droplet shapes, as reproduced within this work, is produced at time intervals fixed prior to the calculations. The time intervals were set such 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.


26

Table 3.1: Significance codes used within statistical analysis. P-value Significance 0.05 < P not significant 0.01 < P ≤ 0.05 significant 0.001 < 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 taken at fixed intervals, the images of droplet break-up are unlikely to represent the last calculated droplet 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 statistical computation and graphics distributed under the GNU General Public License (GPL). R was initially written by Ross Ihaka and Robert Gentleman [IG96] at the Department of Statistics of the University of Auckland, New Zealand and is very similar in appearance to S, which is a high level language and an environment for data analysis and graphics. R is considered the free counterpart to the proprietary S-PLUS package which is based on S. Further details on 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 best strategy for establishing models depends on the knowledge about the physical correlation between dependent and explanatory variables. This is reflected in the models developed within this work. Some of our models were based on models given in literature, such as those that are presented in sections 3.3.1 and 4.2.1.1. Other models were built up from scratch following an iterative trial and error approach, in which linear models based on logarithmic expressions of all possible explanatory variables were used as a starting point such that all variables were collected together in a similar manner to the approach used in dimensional analysis. Non significant 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 named according to the following naming convention. The first part of the model name indicates the dependent variable (i.e. mean particle diameter x50,3 , width of particle size distributions span, fluid density ρ, or dynamic viscosity η). The second part gives either the geometry under 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 Roman numeral with the same numerals for the x50,3 and span models indicating the same underlying experiments. ’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 measurements. The particle size distributions of emulsions were investigated by laser diffraction spectroscopy and the interfacial tension between the two phases in emulsion by means of the drop detachment method for predefined drop formation time (DD-PDFT). The analytical methods 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 for rheological analysis of fluids used within this work. The DSR is a stress controlled rheometer with torque being applied by the measuring head. Applied stresses and resulting strains are analyzed by the software ‘Orchestrator’. Concentric cylinders with a rotating inner cylinder (Searle-type) were used, with the cup being 32.0 mm and the bob 29.5 mm in diameter. The bob with recessed ends had a length of 44.3 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 Density Meter DMA 38 (Anton Paar, Graz, Austria). The measuring temperature can be adjusted between 15 and 40 o C with an accuracy of ± 0.3 o C. 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 sample passes through the beam of a monochromatic laser (wavelength λ = 633 nm) and the light diffracted by the droplets is detected by a photo-diode array with 31 light sensors. Several thousand measurement sweeps are performed within a few seconds and the scattering patterns were 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 and characterized. Aqueous solutions including a shear-thinning fluid formed the basis for almost


28

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

3.3.1 PEG – SDS – H2 O Solutions 3.3.1.1 Introduction Solutions of polyethylene glycol (PEG), sodium dodecyl sulphate (SDS), and (demineralized) water formed the basis for most of our experiments. Solution viscosities were adjusted by PEG concentrations and SDS was added as an emulsifying agent. Following a description of PEG, variations in solution viscosity and density are given in terms of PEG concentration and solution 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(OCH2 CH2 )n OH and are synthesized by polycondensation from ethylene glycol. The PEG used within this work was PEG 35000 from Hoechst, Germany, supplied by Plu¨ ss-Staufer AG, Oftringen, Switzerland. It was supplied as powder and had a molecular weight of approximately 35000 g mol −1 . PEG – SDS – H2 O solutions investigated showed Newtonian behavior over a wide shear rate range (0.1–2000 s−1 ). No elastic properties were detected by means of oscillatory rheometric measurements. 3.3.1.3 Viscosity Variation with Temperature According to Partington [Par51], almost 50 models for the viscosity variation with temperature 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 in Kelvin. This model is also known by other names [Bla49], for instance the Arrhenius law where B is replaced by E/R with E being an activation energy for viscous flow and R the universal gas constant. The reader is referred to Barnes [Bar00] for a more detailed introduction to the topic of viscosity variation with temperature. For PEG – SDS – H2 O solutions, we established a model taking PEG concentrations into account. Model η – PEG – SDS, given in Eq. (3.5), is based on the Andrade law with coefficient K and B expressed as exponential functions of PEG concentrations. ln(η) = k1 + k2 · cP EG k3 + (b1 + b2 · cP EG b3 )T

(3.5)

With temperature T in ◦ C and PEG concentration, cP EG , in % (w/w) the coefficients were estimated 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 strongly significant expept b3 , with a p-value of p = 0.0531 being almost significant. Trials with 22 distinct pairs of temperature and PEG concentration were taken into account with a residual

3.3. CHARACTERIZATION OF FLUIDS

29

5000 40% PEG

Viscosity η [mPa s]

1000 500 20% PEG

100 50 10% PEG

10

5% PEG

5

2% PEG 0% PEG

1


25

30

35

◦

Temperature T [ C] Figure 3.1: Dynamic viscosity, η, of PEG – SDS – water solutions in terms of PEG concentration, cP EG , 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 difference between the number of trials and the degrees of freedom stems from viscosity data taken at about 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 our model for those six concentration levels (solid lines) are also given. A very good agreement between experimental data and model predictions can be seen. 3.3.1.4 Density Variation with Temperature The density of PEG – SDS – H2 O solutions exhibit a similar characteristic in terms of PEG concentration and temperature compared to the logarithmic values of the respective viscosities. The density increases super-proportionally with the concentration of PEG and decreases with 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 · cP EG m3 )T + (b1 + b2 · cP EG b3 )

(3.6)

The coefficients were estimated with a residual standard error of 0.00017 over 37 degrees of 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 3.2 shows the measured densities (◦) at the same levels of PEG concentration as given above and the model predictions (solid lines).


30 1.08

Density ρ [103 kg m−3 ]

40% PEG

1.06

1.04

20% PEG

10% PEG

1.02

5% PEG 2% PEG

1.00

0% PEG


20

25

30

35

40

◦

Temperature T [ C] Figure 3.2: Density, ρ, of PEG – SDS – water solutions in terms of PEG concentration, c P EG , and temperature, T . Measurements (◦) and model predictions (ρ – PEG – SDS, solid lines) are shown.

3.3.2 Xanthan Gum 0.2% (w/w) Xanthan gum in 0.1M NaCl solution was chosen as a shear-thinning fluid. Xanthan gum is used in the food industry as a stabilizer and thickening agent. It is a heteropolysaccharide consisting of D–glucose, D–mannose, and D–glucuronic acid residues with a molecR ular weight between five and ten million Dalton. In this study, Rhodigel Easy from Rhône Poulenc, France, supplied by Meyhall AG, Kreuzlingen, Switzerland was used. Friedmann [Fri99] measured the rheological properties of this solution. The zero shear viscosity 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 the transition 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 referred to Rochefort and Middleman [RM87].

3.3.3 Silicone Oils The silicon oils used within this study as the dispersed phase of emulsions were polydimethylsiloxanes. Their macromolecule backbones are built up of a chain of alternating silicon and oxygen atoms with each silicon atom being bound to two methyl groups. Viscosity ranges from 0.65 mPa s to 1000 Pa s and higher are achieved by mixing macromolecules of different chain lengths. Silicon oils with viscosities up to 1 Pa s exhibit Newtonian flow behavior under shear 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, Germany, at 25 ◦ C. Names given are Wacker’s product names. Silicone oil AK 10 AK 20 AK 50 AK 100 AK 200 AK 500 AK 1000

Dynamic viscosity η [Pa s] 0.0093 0.019 0.048 0.96 0.193 0.485 0.97

Density ρ [kg m−3 ] 930 945 960 963 966 969 970

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, was another oily phase used in our dispersing experiments. The dynamic viscosity of RSO at 25 ◦ C was 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 in this work. It should be noted, that PEG, Xanthan, and SDS concentrations are given in weight percent (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 processes. With fast interface adsorption kinetics, the likelihood of droplet recoalescence in the dispersing process is reduced. Texapon K 1296, produced by Henkel KGaA, D u¨ sseldorf, Germany, was the SDS used in our study. All continuous phases utilized in our dispersing experiments were based on 2% SDS demineralized water solutions. A concentration of 2% SDS is well above the critical micelle concentration (CMC) expected for emulsions with dispersed phase volume fractions of less than 10% and the mean particle diameters envisioned. Quantitative data on SDS micelle break-up kinetics can be found at Oh and Shah [OS94]. 3.3.5.2 Interfacial Tension Interfacial tension σ between PEG – 2% SDS – H2 O solutions and rape seed or silicon oils was measured by means of the drop detachment for predefined drop formation time method (DDPDFT). Interfacial tension were found to be independent of the drop formation time which was varied between 7 and 300 s. Table 3.3 lists measured interfacial tensions for various


32

Table 3.3: Interfacial tension, σ, between AK silicon oils (dispersed phase) and PEG – 2% SDS – H2 O solutions at three levels of PEG concentration, cP EG = 0, 5.5, and 10% (w/w). Values are given in 10−3 N m−1 . Silicone oil

AK 10 AK 100 AK 250 AK 1000

PEG concentration cP EG [%] 0 5.5 10 – 9.71 – 10.02 9.94 – – – 10.93 – 10.3 –

combinations of silicon oil drops in PEG – 2% SDS – H2 O solutions. Each interfacial tension given is an averaged value from 7–10 trials. The interfacial tension between AK 250 and 10% PEG – H2 O solution (without the surfactant SDS) was 25.4 × 10−3 N m−1 . The interfacial tension was reduced by a factor of about 2.5 with added SDS. Contrary to the silicone oil versus PEG – SDS –water solution, the surface tension of rape seed oil versus 10% PEG – 2% SDS – H2 O solution was dependent on the drop formation time with values between 3.1 and 4.7 × 10−3 N m−1 . The former value was measured at a drop formation time of 20 s and the latter at 120 s. This is due to surface active substances within rape 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 measured with drop formation times of 7–300 s are several orders of magnitude above a typical disruption time scale in dispersing flow. The same interfacial tensions for silicone oil versus PEG – SDS –water solution as those given within table 3.3 under typical dispersing flow conditions 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. They were 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 were used 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 – H2 O emulsion measured less than one hour after their production was compared to that measured after 5 days. No significant difference was found between the particle size distributions. Therefore, the stability of emulsions over the time range in which particle size distributions were measured with much smaller volume fractions of up to 10% can be affirmed.

3.4. EXPERIMENTAL SETUPS AND PROCEDURES

3.4

33

Experimental Setups and Procedures

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

3.4.1 Process Unit with Flow-Through Cell 3.4.1.1 Introduction Figure 3.3 shows the process unit with a flow-through cell holding four perforated screens as depicted in the inset image. Reservoir I was filled with pre-emulsions which were forced under pressure through pipe-work holding the flow-through cell into reservoir II against atmospheric pressure. 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 released and the emulsion was allowed to flow back into reservoir I through the bypass hose. With the emulsion 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 pressure of about 6 bar. The pressure up and down-stream of the flow-through cell was measured by two pressure sensors (Hänni Type ZED 501/873.111/075, range 0–2.5, and 0–6 bar, Hänni AG, Jegenstorf, Switzerland). Subtracting the downstream pressure and the pressure loss of the flow-through an empty flow through cell from the upstream pressure gives the effective pressure difference over the packing, ∆p. Magnetic inductive flow meters (60 and 1000 l h−1 , Type PICOMAG DMI 6530, Endress and Hauser Metso AG, Rheinach, Switzerland) were used. Fluid temperature was measured between 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 and four to ten consecutive data points were averaged in a post-processing step to give the data used in the analysis of this work. Fluid viscosities used within the analysis of this work were adjusted according to the measured temperatures. Note that the process unit was modified within the course of this study in order to simplify the experimental procedure. Earlier versions of the process unit include horizontally aligned flow-through cells with emulsions being pumped backwards and forwards through the flowthrough cell. With horizontally aligned flow through cells, emulsions were regularly mixed manually in order to reduce particle separations due to buoyancy. No significant influence of the flow-through cell alignment on the dispersing result was found for experiments analyzed within this work. Process units without a flow meter were also used; this will be indicated in the respective discussions.


34

reservoir II 246 mm

flow through cell

pressure controller 3−way valve pressure supply pressure sensor I bypass hose

∆p

main valve pressure sensor II temperature sensor flowmeter

reservoir I T . V inspection glass drainage

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

3.4.2 Sphere Packings Experiments with sphere packings were conducted within the process unit over a range of parameters including the packing structure, the material of the spheres (i. e. spheres made of elastic 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 packings and one being a random packing. The cubically arranged sphere packing with a coordination number k = 6 and the rhombohedral II packing with k = 12 is shown in Figure 3.4. The third arranged packing (not shown) was the orthorhombic I packing which is built up by stacking hexagonal 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

3.4. EXPERIMENTAL SETUPS AND PROCEDURES k=6

35 k = 12

Figure 3.4: Regularly arranged sphere packings made of incompressible spheres with d s = 4 mm. Left: cubically arranged sphere packing (coordination number, k = 6), right: rhombohedrally (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 commonly known as cubic-close packing (CCP). The latter packing is similar to the hexagonal-close packing (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 ‘results and 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 shown in Figure 3.4 and the other with a circular cross-section. The former consisted of transparent cylinders, 20 mm high, 50 mm in diameter with a quadratic clearance of 28 mm × 28 mm. Up to 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 the arranged sphere packing structures within the stacked cylinders. Each mount comprised 14 cylindrical pins, 1.2 mm in diameter, with seven of them being below the other seven arranged by right angles. Flow-through cells with circular cross-sections were used for random sphere packings constructed of 25 mm or 12.5 mm diameter pipes. Perforated screens were used to fix the spheres within the flow-through cell. In the case of small sphere diameters, the screens were covered by filters or stratified layers of spheres with decreasing sphere diameters toward the


36

Table 3.4: Incompressible sphere characteristics. Mean diameter ds [µm] 4000 2000 339 70

Width of PSD Material span [–] 0 steel 0 steel 0.369 glass 0.345 glass

Table 3.5: Sphere packing porosities ε for the packing structures investigated. The porosity of random packings is given for monodispersed spheres. Packing structure cubic orthorhombic rhombohedral random

Coordination Number Porosity k [–] ε [–] 6 0.4765 8 0.3954 12 0.2595 – 0.376

center of the sphere packing. In every case, screens and filters were chosen such that the clearance within the perforated screens and the filters were significantly larger than a representative capillary diameter within the sphere packing. 3.4.2.3 Incompressible Spheres Ball bearings made of stainless steel (provided by Springer, Zurich, Switzerland) and glass beads (supplied by Merck AG, Dietikon, Switzerland) were used within this work as incompressible spherical particles. Sphere characteristics are given in table 3.4, showing monodispersed ball bearings and glass beads with narrow particle size distribution. Arranged sphere packings were built up of 4000 µm spheres. The porosities of such arranged sphere packings are given in table 3.5 along with the porosity of random packings of monodispersed spheres as used in the analysis of this work. Note that porosities of random sphere 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 friction coefficient Λ and packing Reynolds number Re as given in Section 2.1.1.3 with Eqs. (2.5 and 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 to compile this figure. Each data point shown is an averaged value of four to ten single data points taken at intervals of one second as described in section 3.4.1.2. With dispersed phase volume fractions of less than 10%, elastic effects were disregarded. Inflow and wall effects were found to be

3.4. EXPERIMENTAL SETUPS AND PROCEDURES

37

Friction coefficient Lambda [−]

1000 orthorhombic I 500

200

random cubic rhombohedral II

100

random cubic (k=6) orthorhombic I (k=8)

orthorhombic II 5

10

20

50

100

200

500

1000

Reynolds number Re [−]

Figure 3.5: Friction coefficient Λ versus Reynolds number Re along with expected values according 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”were used within this study to denote the respective materials. Data as provided by Dow Corning. Physical property Durometer Tensile Strength, Die C Elongation at Break, Die C Tear Strength, Die B

R R Silastic S Silastic J “soft” “hard” Shore A 25 55 MPa 7.0 5.5 % 850 250 kN m−1 23 15

Unit

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 R R Silicon rubbers Silastic S and Silastic J (Dow Corning, Midland, Michigan, USA, supplied by Omya AG, Oftringen, Switzerland) were used as elastic sphere materials. The physical properties are given in table 3.6. Mats of 8 × 8 quadratically arranged 4 mm diameter spheres were moulded from these materials. A vacuum was applied to eliminate air mixed into the silicon rubber during mixing of the cross linking agent with the main component. Two mats at a time were solidified within the mold between two plungers at 130 ◦ C by applying a force of 120 kN for 3 minutes.


38

Table 3.7: Loss factors ξ for various orifice geometries. The loss factor marked by an asterisk (∗) indicates the flow from the mount into the first perforated cylinder. In that case, the inflow diameter d1 was replaced by the length l1 of a representative quadratic inflow area. Number of orifices 9 9 1 1

d2 [mm] 1.0 1.0 2.4 8.8

d1 , l1 [mm] 2.0 8.0 4.5 25

A2 /A1 ξ [–] [–] 0.25 0.40 0.0123 0.49∗ 0.284 0.39 0.124 0.46

Remark 1 mm and 2 mm holes alternating mount to 1 mm hole cylinder flow meter (60 l h−1 ) 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 high with a diameter of 50 mm. Cylinders with 3 × 3 holes of 1 mm or 2 mm were stacked together and fixed in the mount described in section 3.4.2.2 and depicted in figure 3.4. Three different arrangements were studied: (i) cylinders with 1 mm and 2 mm holes alternating; (ii) cylinders with 1mm holes alternating with transparent 20 mm high cylinders with a quadratic clearance of 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 connected to the measuring section of the flow meters form sudden contractions similar to those studied with the perforated cylinders. Table 3.7 list the geometry parameters of the orifices studied along with the loss factor ξ as given in section 2.1.2.1. Since loss factors are approximately the same for all the investigated orifice flows, pressure losses can also be assumed to be the same. 3.4.3.2 Droplet Break-up within Orifice Flows Within our dispersing experiments, only the flow meter with a maximum flow rate of 1000 l h−1 was used. Using the other flow meter with a maximum flow rate of 60 l h −1 significantly 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 pressure supplied, 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 a procedure consisting of three main steps: pre-emulsion preparation, processing in the process unit, and analyzing the generated emulsions using a Malvern Mastersizer X. Pre-emulsions were generated immediately prior to the processing and were manually stirred after filling the process unit. Processing in the process unit was described in section 3.4.1.1. Particle size measurements were performed either directly after all runs within the process unit were finished or on the next day. The stability of our emulsions was confirmed as described in chapter 3.3.5.4.

Chapter 4 Results and Discussion The goal of this work was subdivided into three parts with the dispersing process being the connecting theme. Numerically, drop deformation and break-up was studied for single drops moving along particle tracks within model geometries resembling representative capillaries of packed beds of spherical particles. In the second part, the process of dispersing emulsions in packed beds and through orifices was studied and compared to the numerical results. Finally, the compression behavior of deformable regularly arranged packed beds was investigated and we looked at the suitability of those packed beds for the dispersing process.

4.1

Numerical Simulation of Droplet Deformation and Break-Up

Investigating droplet deformation and break-up was carried out in two steps, firstly calculating the velocity field for laminar flows within model geometries and secondly calculating droplet deformations as they move within the velocity field along particle tracks. SEPRAN was used to calculate the velocity fields and BIM to investigate droplet deformation and break-up. Two axisymmetric geometries were considered, one modeled on regular packed beds of monodispersed spheres and the other being adjoint 4:1 die-entry flows. Reynolds-numbers Re were varied between 100 and 1000. BIM calculations were performed over a wide range of parameters including the initial radial position of the particle tracks, the undeformed droplet radius 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 spherical particles as shown in Figure 4.1. Each converging-diverging nozzle has a length of 8mm with diameters of 1mm and 4mm for the narrowest and widest cross-sections respectively. The length of the section between the inflow and the first nozzle was chosen to be long enough such that the prescribed zero velocity in the spanwise direction was valid given certain Reynolds numbers. However, the section between the last nozzle and the outflow could not be chosen 39

1 mm 15 mm

4 mm

CHAPTER 4. RESULTS AND DISCUSSION

40

8 mm

Figure 4.1: Geometry of eight adjoint converging-diverging nozzles with the prescribed inflow velocity profile given on the left as studied numerically. The length of the inflow region prior to 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 geometry.

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

4.1.1.2 Mesh For our calculations, we took advantage of the axisymmetric shape of the geometry thus greatly reducing the size of the numerical problem. The converging part of the first nozzle within our 2-dimensional axisymmetric mesh totaling 22560 elements on 46109 nodes is depicted in Figure 4.2. The triangulation with variations in element length and aspect ratio was done 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 velocity in the spanwise direction was prescribed at the inlet and outlet as well as along the centerline. With no-slip conditions fixed for nodes at the wall, a problem with 89253 degrees of freedom had to be solved. On our machines (specified in Section 3.1.1.2), calculations lasted about an hour regardless of Reynolds-number.

1

41

2

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Discrete annulus probability distribution P [−]

Normalized velocity v* [−] and annulus area dA* [−]

4.1. NUMERICAL SIMULATION

1.0

Normalized radial position r * [−]

Figure 4.3: Normalized radial velocity v ∗ for fully developed pipe flows (), normalized differential annuli area dA∗ (4) at normalized radial position r ∗ , and the annulus probability for infinitesimal volume elements to pass through certain annuli (solid line). The dotted lines are 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 differential annulus area dA∗ (4). The discrete probability P of a random infinitesimal volume element passing through an annulus at a certain radial position is shown in discrete form for 20 equally wide annuli as a solid line. This annulus probability is proportional to the velocity ∗ ∗3 times the differential p annulus area, P ∝ v · dA ∝ r − r . The maximum annulus probability ∗ is 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 inlet of the adjoint nozzle geometry for particle tracks investigated within this work. They were chosen to be at 0, 0.33, 0.67, 0.8, and 0.9. Within our droplet deformation and break-up investigations, our main focus was directed towards tracks 2 and 3 due to their high annulus probabilities. Although the annulus probability tends towards zero at the centerline, this track was also considered due to its absence of shear.


42 A: Elongation rates 1 2 3

4

5

B: Shear rates 1 2 3

4

5

Elongation rates -0.100 38.800 77.700 116.600 155.500

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

Elongation rates 194.400 233.300 272.200 311.100 350.000

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

Figure 4.4: Elongation and shear rates within the first one and a half converging-diverging nozzles 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 the converging-diverging nozzle geometry presented in Figure 4.1 for Re = 100. Tracks 1–5 with initial radial positions as given in the previous section are overlayed. In the area between track 5 and the meniscus between the diverging and converging part of the nozzles, recirculation occurs which is not shown in this graph. Due to inflow effects, the flow field within the converging part of the first nozzle differs from that within the following nozzles where, in the case of low Reynolds-numbers, the flow field is almost macroscopically periodical. The elongation rates indicate that particles are exposed to positive elongation rates for longer within the first nozzle than in the following nozzle. This is confirmed in Figure 4.5 which shows the shear and elongation rates for particles 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 macroscopically developed after the first nozzle as indicated by constant peak shear rates for track 2 and 3 as well as the similarity between the second and third elongation rate peaks. The maximum values of the shear rates increase approximately linearly with radial position, from zero along the center line as expected by Newton’s Law (τ = η · γ), ˙ though the maximum values for tracks close to the wall have to be treated with caution due to limitations in the fineness of the mesh. The shear rates exhibit a saw tooth like behavior with minimum values still exceeding those of the respective elongation rates. Contrary to shear rates, elongation rates maintain about the same maximum values over a wide range of initial radial positions with a slight decrease towards the pipe wall. As shown


43

Track 1

3000

2000

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4000


4000


Track 3

Track 2

3000

2000

3000

2000

1000

1000

1000

0

0

0

0.00

0.02 time [s]

0.04

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time [s]

Figure 4.5: Shear and elongation rates (dashed and solid lines, respectively) along particle tracks 1, 2, and 3 at Re = 100 for the first two and a half nozzles. Time was set to zero for each particle track 1mm prior to the beginning of the first nozzle. Dotted lines indicate the throats 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 compression is about half the amount of the corresponding elongation. Towards the wall, the ratio of compression 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 profiles at each cross-section within the nozzle, as described in Section 2.1.2.2. It was aimed at keeping the elongation rates along the centerline as constant as possible. It can be seen in the particle track calculations that the goal was achieved for at least the last half of the converging part of the nozzle. Moreover, it is interesting to note that the design goal is even better fulfilled on 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 based solely upon the elongation rate. This is in accordance with extra pressure losses at the inflow of porous media. In the annulus between track 2 and 3, which accounts for about 50% of the overall flow rate, the ratio between maximum shear and elongation rates is between 6 and 220. This ratio is smaller towards the centerline and greater towards the wall. The contribution of the elongation rate to the overall strain is almost constant between the radial position of 0 and 0.67 (track 1 and 3, respectively), whereas the contribution of the shear increases super-proportionally with the 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 maximum shear rates occur shortly after the maximum elongation rates is also advantageous for the


44 A: Elongation rates 1 2 3

4

5

B: Shear rates 1 2 3

4

5

Elongation rates

Shear rates

Elongation rates

Shear rates

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

-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

Figure 4.6: Elongation and shear rates within the first one and a half converging-diverging nozzles 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 main differences observed in this figure are threefold. Firstly, the elongation and shear rates are higher by an order of magnitude. Secondly, the particle tracks between two nozzles barely follow 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 between track 5 and the wall, with a maximum value of ε˙ = 8.48 × 103 . However, the likelihood for infinitesimal 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 passing through all eight nozzles. It reveals another major difference between the two different Reynolds-number flows. Contrary to the case with Re = 100, in which the flow is macroscopically developed after the first nozzle, it takes the flow 7 to 8 nozzles to develop at Re = 1000 as is best seen along track 2 for the shear rate. The shear rate of track 2 starts out close to zero in the throat of the first nozzle and reaches its final maximum value after about 7 nozzles due to 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 as pronounced as along track 2. Furthermore, the flow development is indicated by the elongation rates with a decrease in the maximum elongation rate with travel time, particularly for tracks 1 and 2. Although contributions of the elongation rates to the overall strain Γ is similarly small compared to the case with Re = 100, it is instructive to look at qualitative differences between

4.1. NUMERICAL SIMULATION 40000

Track 1

30000 shear and elongation rates [1/s]

shear and elongation rates [1/s]

30000

40000

Track 2

20000

10000

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30000 shear and elongation rates [1/s]

40000

45

20000

10000

0

0.000

0.004

0.008

time [s]

0.000

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0.008

time [s]

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

4 5

Figure 4.8: Particle tracks within the converging-diverging nozzle geometry at a Reynolds number 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 higher elongation rate strains within the first nozzle compared to the following nozzles. Thus, inflow effects become more pronounced with an increase in the Reynolds-number. Moreover, the peak elongation rates within the first nozzle is less dependent on the particle track and the ratio between positive and negative peak elongation rate values is much more favorable for the positive values in case of higher Reynolds-numbers. Figure 4.8 shows the particle tracks at Re = 1000 indicating the jet formed in this geometry 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 particle tracks described in the previous section. We focused on calculations along tracks 2 and 3 at

46


Re = 1000 because those tracks represent annuli with a large annulus probability and the high Reynolds-number was chosen since the maximum shear and elongation rates are almost directly proportional to the Reynolds-number and thus the break-up of smaller droplets is expected with greater Reynolds-numbers. Drop deformation and break-up within a mixed flow field gives rise to the definition of two Reynolds-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 second and third components ε˙2 and ε˙3 of the elongation rate vector within the velocity gradient in axisymmetric flow as given in Eq. (2.24) are small compared to its first component ε˙1 . We therefore neglected ε˙2 and ε˙3 . It should also be noted that the principal axes of the droplets are 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 prior to the first nozzle oscillates on its way through the adjoint nozzle geometry forming a typical dumbbell. Its deformation increases over the oscillations in a cumulative manner and it finally breaks-up within the fifth nozzle at a droplet deformation of L/a ≈ 4.4 as shown in Figure 4.9. The numerical calculation for this problem started out with 412 points, took a day, and the final mesh at break-up consisted of 8529 points. The viscosity ratio λ between the dispersed phase viscosity ηd and the viscosity of the continuous phase ηc is one. Shear and elongation rates as well as travel time are given in non-dimensional form made dimensionless with the characteristic time of T char = aηc /σ = 0.0125 mm · 0.01 Pa s /10.0 × 10−3 N m−1 = 12.5 × 10−6 s. The continuous and dispersed phase viscosities were chosen to match a typical combination of dispersed and continuous phases within our experimental studies. In simple shear flow with a viscosity ratio of λ = 1, droplets will not break-up at capillary numbers below 0.42. This critical capillary number is included as a dashed line at Ca crit = 0.42. It is obvious from Figure 4.9 that exceeding the critical capillary number alone is not a sufficient criterion for droplet break-up. Otherwise, the droplet would have broken up within the first nozzle. This implies that additional break-up criteria must be taken into account, one being the strain above the critical capillary number Γac and another one the droplet shape and its internal flow field. According to our observations, the latter factor is strongly influenced by the periodicity within the flow field. The strain above the critical capillary number is shown qualitatively as the shaded area within Figure 4.9 and given quantitatively in Table 4.1. It is interesting to note that the first three maximum droplet extensions are reached shortly after the dimensionless shear rate drops below 0.42. Likewise, minimal extensions are reached when the shear rates exceed the value of 0.42. Slight deviations from the exact positions given by the intersection of Cacrit = 0.42 with the shear rate curve are due to the graphical droplet shape output occurring at constant time intervals and inertia effects.

Dimensionless Shear and Elongation rate [−] 0.0 0.2 0.4


47

Cacrit = 0.42 Shear rate [−] Strain Elongation rate [−] 600

800

1000

Time [−]

1200

Figure 4.9: Break-up of a droplet along track 3 with an initial radius of a = 0.0125 mm, a viscosity ratio λ = 1, and Reynolds numbers for the continuous phase and dispersed phase of Rec = 1000 and Red = 0.625, respectively. All maximum and minimal droplet extensions are shown. The strains above the critical capillary number, Γac , are indicated as shaded areas below the shear rate curve. Table 4.1: Strain, Γac , above the critical capillary number (Cacrit = 0.42 for simple shear flow and λ = 1) along track 3 for each nozzle, along with the maximum critical capillary number Camax within each nozzle, and the time tac for which droplets are exposed to capillary numbers above the critical value. Nozzle 1 2 3 4 5 6 7 8

Strain (Γac ) [–] 0.485 1.162 1.525 1.770 1.958 2.089 2.183 2.151

Camax [–] 0.4524 0.4753 0.4825 0.4868 0.4895 0.4917 0.4934 0.4950

tac [–] 21.44 34.96 41.68 51.12 55.04 56.40 57.76 48.40

48


4.1.3.2 Shear and Elongation Rate The contributions of the shear and elongation rates to the droplet break-up in the example previously discussed were investigated. Two variations were studied. First, the elongation rate along track 3 was set to zero whereby the droplet break-up occurred 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 overall droplet deformation and breakup – at least at a viscosity ratio of λ = 1 and in flows with shear 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 inflow effects, this also lead to a break-up within the fifth nozzle at a position almost identical to the initially investigated case (t = 958 versus 955), again indicating the minor influence of elongation, 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 adjoint nozzle geometry was pointed out. In particular, the periodicity allowing for cumulative effects was mentioned. In order to illuminate the latter aspect, the converging-diverging nozzle geometry was scaled up in the streamwise direction by a factor of 2, thereby doubling the length 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 droplet oscillations being reduced by a factor of 2. Although the overall flow field was kept, certain aspects 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 Ca max = 0.50 compared to 0.452 with the unscaled geometry. Therefore, a comparison between the scaled and unscaled 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 of the first nozzle. The strain along this track within the first nozzle is about 2.0. A somewhat contrived approach to investigate the influence of periodicity might be to simply scale the travel 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 radii a, under the same conditions as discussed in Section 4.1.3.1. Droplets with an initial radius of a = 0.01 mm do not break-up. Such a drop is shown with its maximum extension of L/amax = 1.87 reached at the fourth nozzle. As indicated in Table 4.2, droplets up to an initial radius of a = 0.012 mm do not break-up although the maximum capillary number partly reaches 0.475. Along with an increase in the initial undeformed radius goes a strong increase of the maximum deformation L/amax at break-up. The maximum deformation amounts to 4.44 for a drop with a = 0.0125 mm and 15.3 for a = 0.02 mm. This also influences the break-up behavior insofar as binary break-up is observed for a = 0.0125 mm, ternary break-up for a = 0.015 mm, and a break-up into possibly even more droplets for a = 0.02 mm which cannot be predicted by the used BIM code. It can also be observed that the break-up occurs


49

a = 0.01 mm x = 31.7 mm a = 0.0125 mm x = 34.5 mm a = 0.015 mm x = 14.3 mm a = 0.02 mm x = 15.1 mm

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

Table 4.2: Characteristics for droplet deformation and break-up within the adjoint convergingdiverging nozzles over a variety of initial undeformed radii, a. Given are the maximum capillary number Camax along track 3, the maximum droplet deformation ratio L/amax , the number of initial points within the calculation npoints, 0 , and the maximum number of points npoints, max . a Camax [mm] [−] 0.01 0.396 0.012 0.475 0.0125 0.495 0.015 0.593 0.02 0.791

L/amax [−] 1.87 2.88 4.44 7.2 15.3

Break-up

npoints, 0

npoints, max

no no yes yes yes

252 412 412 642 252

607 1418 8529 13442 18590


50 track 1

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1

2 3 4

5

Figure 4.11: Maximum deformations of droplets along tracks 1, 2, and 3 within eight adjoint nozzles for droplets with initial undeformed radii of a = 0.0125 mm at λ = 1 and Re c = 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-up almost 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 rate G. We have also seen that the maximum values of shear rate within our adjoint nozzles are almost linearly dependent on the initial radial position of our particle tracks. Therefore, we do not expect droplet break-up for droplets moving along tracks closer to the centerline than track 3. Nevertheless, it is instructive to compare droplet deformation along different tracks, as this shows the strong dependence of the droplet behavior on the initial radial position. Figure 4.11 depicts the maximum droplet deformations over their positions within the geometry. Along tracks 1 and 2, maximum droplet extensions occur within the throat of the nozzles indicating that elongation within the flow field is the major cause for this behavior. A droplet moving along track 1 reaches maximum deformations of L/a max = 1.09 and always relaxes back to an undeformed shape between these maxima. A particle moving along track 2 reaches maximum deformations of L/amax = 1.35 and does not relax back to an undeformed shape between two nozzles. We have seen, that the elongation rates along tracks 1 and 2 can be considered to be almost identical. Therefore, the larger deformations along track 2 can be attributed solely to the shear rates. In addition, the shear rates cause the droplets to tilt towards the centerline. As noted before, the locations of the maximum deformations along track 3 are not directly coupled to the geometry but – at least for the first three extensions – rather to the strain above the critical


51

capillary number Cacrit . These findings give rise to two questions. What happens on tracks closer to the wall where higher shear rates are present? It is less likely that particles will move along tracks closer to the wall, but allowing droplets to shift between tracks – as can be expected in real porous media – will eventually lead them more frequently into annuli close to the wall, provided the real porous medium is long enough. The BIM program is very sensitive to the gradients along particle 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 shear and elongation rates are constant over the droplet. This assumption holds for infinitesimal volume elements but it would be interesting to know to what extent droplet break-up is influenced by 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 the first nozzle along track 2. This track does not, as we have seen before, significantly contribute to the overall dispersing process within the flow through our nozzle geometry at a viscosity ratio of λ = 1. Nevertheless, with an increase in viscosity ratio, elongation rates are expected to become more relevant for break-up mechanisms. We have seen that elongation rates are not as dependent on the particle track as is the case with shear rates. Since shear rates are almost zero within the first nozzle along track 2, and in view of the following Section on cumulative effects, track 2 was chosen for the study on entrance flows. Figure 4.12 shows the maximum droplet deformations within the first nozzle along track 2 for viscosity ratios λ = 1, 3, and 5 over undeformed droplet radius a. The respective shear and 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 1.09 and 1.21 respectively, these drops are far from being broken up. With larger drops we see a strong dependence of the maximum deformations on the viscosity ratio. For droplets of radius a = 0.1 mm, maximum deformations amount to 3.38, 2.29 and 1.87 for λ = 1, 3 and 5 respectively. The dotted line at a = 0.1 mm indicates the undeformed droplet radius a above which the assumptions for BIM calculations regarding the ratio of droplet size to narrowest geometry cross-section are no longer met. Nevertheless, calculations above drop radii of a = 0.1 mm allow for qualitative interpretations. Noteworthy is the exponential increase in the maximum droplet deformation between droplet radii of 0.05mm and 0.1mm for all viscosity ratios and in the case of the viscosity ratios of 1 and 3 even above 0.1mm. It should also be mentioned that BIM calculations for a droplet size of a = 0.25 mm in the case of a viscosity ratio of λ = 1 and a = 0.3 mm in the case of λ = 3 terminated during the calculations prior to the specified finish time due to re-meshing difficulties.


PSfrag replacements

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Figure 4.12: Maximum droplet deformation within the throat of the first nozzle for viscosity ratios λ = 1, 3, and 5 of droplets moving along particle track 2 at Re c = 1000. The dotted line is explained in the text. 4.1.3.7 Cumulative Effects After looking at deformation characteristics for droplets moving along track 2 within the first nozzle we now extend this investigation in the flow through all eight adjoint nozzles. The fact that the maximum shear rate increases over the nozzles due to the macroscopic flow development adds another aspect being considered in our investigations. The maximum capillary numbers, found within the last nozzle, range from Camax = 0.225 to 8.99 for drop radii of a = 0.0125 mm and 0.5 mm respectively. In the previous section we pointed out that the assumption for our BIM calculations in terms of the ratio between the droplet radii and the narrowest cross-section was not met for radii above a = 0.1. In the case of the flow through eight consecutive nozzles with large droplets, and particularly within the last nozzles where shear rates reach a plateau, we also stress the Stokes flow assumption in the BIM formulation since droplet Reynolds-numbers Red exceed 1. Nevertheless, qualitative conclusions appear to be valid also for large droplets within the last nozzles. Again, the dotted line at a droplet radius of a = 0.1 mm indicates the radius 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 extension and relaxation sequence of droplets injected on track 2 for a viscosity ratio of λ = 1. After the extension of the droplet within the throat of the first nozzle (◦), the drop relaxes almost completely back to its initial shape between the first and the second nozzles (•). Within the second 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 streamwise position until a plateau at about the seventh nozzle is reached, the droplet extension is also

4.1. NUMERICAL SIMULATION breakup

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0.10

0.20


Figure 4.13: Droplet deformation within eight adjoint converging-diverging nozzles along track 2 for λ = 1 at Rec = 1000. The arrows at droplet radius a = 0.05 mm are added to indicate the chronological order of the droplet extensions and relaxation along its path through the 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 occurs between the second, third, and fourth nozzles, though the rate at which the droplet extends slows down between adjoint nozzles. The BIM calculations finally terminate close to the throat 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 droplet as 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 the size 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 Ca max = 0.50 within the last nozzle. The minimum drop size that will be broken up along track 2 is therefore between a = 0.025 and 0.05 mm. The fact that in both cases the droplets relax back almost to their initial shape between the first two nozzles confirms our previous finding, that for λ = 1 the inflow effects do not influence the break-up. Nonetheless, it must be pointed out that droplet deformations within the second nozzle are always smaller than within the first nozzle, indicating the strong impact of the inflow. For any drop diameter, the deformation within the first nozzle is overcome for the first time within the third nozzle, with maximum shear rates exceeding those of the maximum elongation rates within the first nozzle by a factor of 3. For drop radii of a = 0.0125 mm and 0.025 mm the deformation maxima were observed at the throats of the nozzles or at positions slightly further downstream. BIM calculations for


54

Droplet deformation L/a [−]

7 6


5 4 3

2

1 0.01

0.02

0.05

0.10

0.20


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

drop radii larger than or equal to a = 0.05 mm terminated due to large drop deformations at positions further upstream with greater drop sizes. Figures 4.14 and 4.15 show results for the same parameters except for a variation in the viscosity ratio λ. None of the calculations performed with λ = 3 or 5 indicates droplet breakup. 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 the droplet deformation and break-up. In the case of λ = 3, a similar sequence can be observed as with λ = 1, with deformations within the second nozzle being smaller than those within the first nozzle. Furthermore, the deformation within the last nozzle always exceeds that within the first nozzle. The shoulder on the curve for the deformation within the eighth nozzle for droplet radii of a = 0.07 mm to 0.25 mm seems to be worth mentioning though it might be related to the non-compliance to the BIM assumptions. For a viscosity ratio of λ = 5 (Figure 4.15), maximum droplet deformations within the last nozzle can even be found to be smaller than deformations within the first nozzle. Droplets with a radius of a = 0.075 mm or above deform to a larger extent within the first nozzle than they do within the last nozzle. This holds true even though the shear rates within the last nozzle are much higher compared to the elongation rates within the first nozzle. This suggests, that inflow 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 the last nozzles, has to be taken into account.


Droplet deformation L/a [−]

7 6

55


5 4 3

2

1 0.01

0.02

0.05

0.10

0.20

0.50


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

4.1.4 Orifice Flow Besides converging-diverging nozzle flow calculations, the flow within adjoint die entries was calculated with SEPRAN at various Reynolds-numbers Rec . The axisymmetric geometry is depicted 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-diverging nozzles at a Reynolds-number of Rec = 1000, which is one order of magnitude above the Reynolds-number considered here. The formation of the jet is as pronounced as within the converging-diverging nozzle at Rec = 1000. Except that the maximum value of the shear rates are comparable to the corresponding ones in the nozzle flow and the fact that the flow is macroscopically developed after the first die, the shear and elongation rates differ from those observed in the nozzles at the same Reynolds-number (compare Figure 4.4). Firstly, the peak values of the elongation rates of track 2 are of the same magnitude as the respective shear rates although the exact value of the maximum elongation rate of track 3 has to be treated with caution due to limitations with regard to the discrete numerical calculations. Secondly, shear rates show a pronounced negative peak within the vicinity of the first die entry. Shear and elongation rates likewise exhibit very high gradients which are even more pronounced at Reynolds-numbers of Rec = 1000. Therefore, BIM calculations were at best difficult and at the worst impossible to perform. This gives rise to the assumption, that droplets within die entry flows are not broken up as smoothly as in the nozzle flows, resulting presumably in broader particle size distributions.


56

4000 2000 1

2

3

0 −2000

4 mm

4 mm 4 mm

3

−4000

1 mm


6000

2 1

−6000

0.04

0.06

0.08

0.10

time [s]

Figure 4.16: Shear (dashed lines) and elongation rates (solid lines) along particle tracks 1, 2 and 3 for adjoint die entries at Rec = 100. Time was set to zero for each particle track 4mm prior 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 rows show the dependent variables, the other ones give explanatory variables. Physical quantity Notation Mean droplet diameter x50,3 width of PSD span number of runs nruns pressure differences ∆p viscosity ratio λ length L continuous phase viscosity ηc diameters ds and do porosity ε volume fraction of dispersed phase Ψd Reynolds numbers Rep and Reo interfacial tension σ

4.2

Unit µm – – bar [–] m Pa s m [–] [–] [–] 10−3 N m−1

Dispersing Process

The dispersing process was studied within two sets of flow geometries related by the fact that a periodic strain is imposed upon droplets passing through them. One comprised packed beds of spherical particles and the other, a model geometry for the former, consisting of adjoint die entries. The goals of this section are twofold. One is the elucidation of the droplet breakup mechanisms within periodic flows. The other is to find statistical models to predict dispersing results over a wide range of fluid and geometry parameters with the models based solely upon physical parameters. The statistical models derived within this section are based upon the numerical 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 100 times at one of four distinct pressure differences across the bed. The packing Reynoldsnumber Rep increased, along with the pressure difference, from 19.8 to 77.9, with the friction coefficient Λ for this packed bed given in Figure 3.5. The viscosity ratio λ between dispersed and continuous phase viscosities was 2.26. The PSD of the pre-emulsion (◦, solid line) is depicted on the right hand side of Figure 4.17. The particle size characteristics x90,3 , x50,3 , x10,3 , and span for the pre-emulsion are 146.3 µm, 80.3 µm, 33.2 µm, and 1.41 µm, 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 curve towards 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 x 50,3 and also reduced the span of the emulsions as the volume of the larger particles is reduced more than the

CHAPTER 4. RESULTS AND DISCUSSION Cumulative volume distribution Q3 [–]

58 1.2

preemulsion 0.4 bar, 10

0.4 bar, 20x 0.4 bar, 50x

0.4 bar, 100x 0.8 bar, 10x

1.4 bar, 10x 2.2 bar, 10x

1.0 0.8 0.6 0.4 0.2 0.0


5

10

20

50

100

200

500

Particle diameter x [µm] Figure 4.17: Particle size distributions (PSD) for dispersing 2% (v/v) rape seed oil in 10% (w/w) – PEG in 2% (w/w) SDS – de-mineralized water solution within an orthorhombically (I) arranged 80 mm long sphere packing of ds = 4 mm spheres at four pressure differences and a variety of number of runs nruns .

volume of smaller particles being generated. A minimum particle diameter of x min = 6 µm regardless of the number of runs indicates a lower limit for particle sizes at this pressure difference. The emulsion generated with 100 runs at a pressure difference of ∆p = 0.41 bar was used as the pre-emulsion for the runs at the following pressure difference of ∆p = 0.79 bar which 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 was likewise 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, and they are distinguished solely by their position along the abscissa. The intervals between the four pressure differences were chosen to be high enough such that the same emulsion would have been generated regardless of whether the emulsions from the last run of the preceding pressure difference or the very first pre-emulsion was used. Note that the PSDs for the first 1-3 runs sometimes indicated that particularly large particles were not passed through the sphere packings, and therefore significantly influenced the particle size distribution. This must be attributed to the batch-wise processing of the emulsions.

PSfrag replacements

PSD parameters x90,3 , x50,3 , x10,3 , [µm]


59

35 30 25 20

10, 1.106 20, 0.943 50, 0.804

15

100, 0.733

10, 0.902

10

10, 1.034 5

10, 0.994

x90,3 x50,3 x10,3 5

10

15

20

25

30

40

Specific energy input Evn = ∆p × nruns [bar] Figure 4.18: Particle size characteristics vs. specific energy input for the results of the dispersing 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 number of runs nruns , and the energy input Ev = ∆p becomes obvious when plotting the particle size characteristics versus the specific energy input Evn = Ev · nruns . Figure 4.18 shows this for the previously presented results. The solid line connects the mean diameters over the energy input E vn . Instead of symbols indicating the size of the mean diameters, two numbers are given representing the number of runs nruns and the span. The latter number can also be deduced from the x 90,3 and x10,3 values which are indicated by symbols ( and 4) connected via dotted and dashed lines respectively. Pre-emulsion data is not shown. The curves are best read by moving along the z-shaped curves beginning on the left-hand side. This corresponds to the shift of the curves in Figure 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 x 50,3 decreases almost linearly with the number of runs at a given pressure difference. According to our interpretation of the previous figure, we expect the mean diameter to be bounded by a lower limit, though we have never been able to show this. Much higher numbers of passes are needed to explore this assumption. Secondly, an increase in pressure difference for a constant number of runs also results in a linear decrease in the mean particle diameter. The gradient for this 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 is almost 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. Logarithmic values of model variables were used resulting in better fits than with the untransformed values.


60

Mean diameter x50,3 [µm]

25 20

runs = 10

runs = 20

runs = 50

runs = 100

∆p = 0.41 bar

15 ∆p = 0.79 bar

10 ∆p = 1.40 bar

∆p = 2.23 bar

5


10

15

20

25 30

40

50

Specific energy input Ev = ∆p × nruns [bar] Figure 4.19: Mean diameters x50,3 from the dispersing experiment characterized within the previous two figures along with model predictions according to Eq. (4.3). For pressure differences ∆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). The values for the coefficients c1 , c2 , and c3 were estimated by the statistics package ‘R’ with a residual standard error of 0.06083 for 4 degrees of freedom, an adjusted R 2 of 0.9849 and an average deviation of 0.0387. They are given in Eq. (4.3). All explanatory variables within the model are significant and the former two are even very strongly significant. ln(x50,3 ) = ln(c1 ) + c2 · ln(∆p) + c3 · ln(nruns ) x50,3 = c1 · ∆pc2 · nruns c3 x50,3 = 13.54 · ∆p−0.789 · nruns −0.142

(4.1) (4.2) (4.3)

Figure 4.19 shows the predictions from our statistical model overlayed with experimental data. The accuracy of the model can be seen in the deviations of the experimental data from the intersections of the dashed and dotted lines, which represent constant pressure and constant numbers of runs respectively. The coefficients in the model Eq. 4.3 can be interpreted as the efficiency of the energy input versus the power input. An increase in energy input Evn as given by the abscissa of Figure 4.19 results in a decrease of the mean diameter by a power of -0.142, given constant pressure differences. Increasing the power input by increasing the pressure difference for a constant number of runs leads to a decrease of mean diameters by a power of -0.789. Therefore, an increase 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 emulsion of a certain quality, defined by the mean droplet diameter and the width of the particle size


61

distribution, at the lowest possible energy, the task is to find an appropriate balance between the pressure difference and the number of runs, since the width of the particle size distribution was found to be dependent on the number of runs. The width of particle size distributions will be 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 importance 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 levels each. One parameter was the viscosity ratio between the continuous phase, chosen to be a 19% PEG – 2% SDS – H2 O solution, and the dispersed phase. Using the silicone oil AK 250 as the dispersed phase provided a viscosity ratio of λ = 1.71, AK 1000 a viscosity ratio of 6.93. The length of the sphere packing L was the second variable parameter with values of L taken to be 20 mm and 100 mm. The packing structure of cubically arranged spheres with diameters of d s = 4 mm was kept for all experiments. Two of the experiments were performed twice and showed good repeatability. 32 data points were considered with Reynolds-numbers (Re p ) ranging between 3.96 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 experimental data 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 coefficient Λ being itself a function of the Reynolds-number Rep . In order to decouple the length L from the pressure difference ∆p, the Reynolds-number Rep was chosen in Eq. (4.4) instead of the pressure difference ∆p as used within the model of the previous section (Eq. 4.1). The Reynolds-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) x50,3 = c1 · Rep c2 · nruns c3 · Lc4

(4.4) (4.5)

For both viscosity ratios, all parameters were very strongly significant and the estimated coefficients are listed in the second and third columns of Table 4.4. The average deviation between 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, comparing single coefficients is difficult due to the dependence of the explanatory variable (i.e. the mean diameter 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 variable. 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 ratio λ into account. This model (x50,3 –pack – III) is given in Eq. (4.6) along with its estimated coefficients in the last column of Table 4.4. All parameters are again very strong significant and the adjusted R2 was 0.9817.


62

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

Coefficient c1 (Intercept) c2 (Re − number) c3 (nruns ) c4 (L) c5 (λ) c6 (L : λ) average deviation

Model x50,3 –pack – II x50,3 –pack – III λ = 1.7 λ = 6.9 λ = 1.7, 6.9 116.7 407.5 37.3 -1.07 -0.710 -0.925 -0.248 -0.186 -0.209 -0.483 -0.152 -0.589 n/a n/a 1.45 n/a n/a 0.227 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) x50,3 = c1 · Rep c2 · nruns c3 · Lc4 +c6 ·ln(λ) · λc5

(4.6) (4.7)

The coefficients of the latter model prove our assumption that the increase in the viscosity ratio from 1 to 5 goes along with the elongational flow in the inflow region becoming more important for the dispersing process. This can be seen by looking at the exponent of the packing 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 the number of runs c3 = −0.209, giving the relative importance of those two parameters on the overall dispersing result. In the λ = 1 case, the length of the packing has a greater influence on the size of the particles, as seen in the smaller coefficient of -0.589 compared with the coefficient of -0.224 for the number of runs. However, in the λ = 5 case, the two coefficients are -0.224 and -0.209, i. e. approximately the same, showing that the relative importance of the number of runs increases, thus showing the importance of inflow effects. Besides this conclusion, there is a significant overall influence of the viscosity ratio on the mean diameter x50,3 . The main effect of the viscosity ratio as given by coefficient c5 is somewhat disguised by the interaction coefficient c6 , though with c5 being much greater than c6 , we can concentrate on the main effect, which shows a strong increase in diameter with an increase in viscosity ratio. Note that the model could have been improved slightly in terms of average deviation by including an interaction between the Reynolds-number Rep and the packing length L. As this would not have affected the conclusions drawn from our experiments, we neglected this interaction for the sake of clarity.


63

Table 4.5: Range of the dependent variable x50,3 and the explanatory variables over the trials used 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 appended for completeness. Units used are those applicable within the model. Variable mean diameter x50,3 pressured difference ∆p number of runs nruns viscosity ratio λ packing length L continuous phase viscosity ηc sphere diameter ds packing porosity ε interfacial tension σ packing Reynolds-number Rep friction coefficient Λ

Range 5.49 – 166.1 µm 0.08 – 5.93 bar 1 – 105 1.71 – 10.43 0.01 – 0.40 m 0.0011 – 0.14 Pa s 70×10−6 – 0.004 m 0.2595 – 0.476 4.0 and 10.0 ×10−3 N m−1 3.96 – 1003 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 specific coefficients, predicting mean diameters of emulsions processed through porous media under various process and fluid conditions. Therefore, 26 trials were conducted over a wide range of parameters. The range of parameters is shown in Table 4.5 and a list of geometry and fluid parameters 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 analysis whereby all explanatory parameters are furnished with a power coefficient and are multiplied together. The explanatory variables were chosen to be primary and independent. Our resulting 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) (4.8) x50,3 = c1 · ∆pc2 · nruns c3 · λc4 · Lc5 · ηc c6 · ds c7 · εc8 · σ c9 The coefficients for this model were estimated over 125 degrees of freedom with an adjusted 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, and c9 = 0.530, with all parameters being very strongly significant. Comparing the coefficients for the pressure difference and the number of runs (c2 = −0.679 and c3 = −0.151 respectively) to those of the previous investigations confirms the same qualitative influence of the power input versus the energy input with the former being approximately 4.5 times more efficient. A positive viscosity ratio again indicates the need for higher energies to break up droplets with similarly higher viscosity ratios. The coefficient c6 = 0.234 for the packing length has to be judged in relation to the the pressure difference, since they are coupled via Darcy’s Law.

64


As pointed out with Eq. (2.31) in Section 2.2.2.2, the dispersing result within orifice flows was found to be a function of an empirical geometry coefficient and the pressure difference to the power of b, where b is a function of the Reynolds-number. Karbstein [Kar94] reported values of b = 1 and 0.25 ≤ b ≤ 0.4 for laminar and turbulent flows, respectively. The coefficient b can be qualitatively compared to c2 in model Eq. (4.8). Our experimental data mainly represented laminar flow conditions with a maximum packing Reynolds-number of Rep = 1003. Therefore, it is admissible to choose coefficient c2 to be constant and to ignore the 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 viscosity ratio. When this interaction is included in the current model, it also proves to be very strongly significant. However, the coefficient is very small compared to the corresponding value in the last section (0.04 versus 0.227) and the average deviation becomes worse (0.198 compared to 0.163). The fact that in some cases stratified packings with different layers of various monodispersed spheres were used might have blurred the significance of the inflow effects. The interfacial tension (σ) used within the model is based on two fluid type combinations, one being various silicone oils in PEG – SDS – H2 O solutions with varying PEG content and the other rape seed oil in 10% PEG – SDS – H2 O solution. While the former exhibits constant interfacial tension, that of the latter depends on the droplet break-up kinetics and time. Therefore, our model must be considered as a model for two levels of interfacial tension and is restricted to fluids with a constant interfacial tension of 10.0 × 10−3 N m−1 or the combination of rape seed oil in 10% PEG – SDS – H2 O solutions as we used them. Our model tries to describe break-up mechanisms that have been extensively studied in the literature for simple flow fields. Break-up investigations for such flow fields, e.g. as given in Figure 2.9, imply that the droplet break-up (in terms of a critical capillary number Ca crit ) can not be given as a simple exponential function in terms of the viscosity ratio (λ). Therefore, assuming such a simple exponential relationship as is implicit in our model is a good approximation, proved overall by the good predictions made by the model, although it does not 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 φ d , there is a change in the rheological behavior of the emulsion as well as an increase in the recoalescence probability during the dispersing process as described in sections 2.2.2.2 and 2.2.2.3. With dispersed phase volume fractions ranging from 2% to 10% in our experiments, we did not observe changes in flow characteristics and therefore we observed no influence of viscoelastic emulsion properties within porous media flow. Nor was the dispersed phase volume fraction significant 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 particularly sensitive to large droplets being present during the first two runs through the porous media. Therefore, the model developed in this section is based on the same data as used in section 4.2.1.3 for model x50,3 –pack – IV except, that data for the first two runs and two


65

outlying data points are omitted. Therefore, 100 data points are available with particle size distribution 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 model Eq. (4.8) and successively removing non-significant variables as well as adding the packing Reynolds-number as an explanatory variable resulted in model span –pack – IV best fitting our 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 = nruns c1 · λc2 · ηc c3 · εc4 · σ c5 · Rep c6

(4.9)

With an adjusted R2 value of 0.8579, and an average deviation of 0.22, all parameters were found to be very strongly significant, with the coefficients being c 1 = −0.0689, c2 = 0.169, c3 = 0.185, c4 = −0.354, c5 = 0.173 and c6 = 0.0598. Note that the packing length was not found to significantly contribute to the width of the particle size distribution, thus indicating that the width of the PSD is governed by inflow effects regardless of the viscosity ratio. This is surprising since the packing length was found to significantly influence the mean diameter for processes with viscosity ratios of λ = 5. As pointed out earlier, the width of the PSD decreases with the number of runs as indicated by a negative exponent (c1 = −0.0689). Moreover, higher viscosity ratios correspond with broader particle size distributions which is in accordance with the findings for the mean diameter. Another surprising finding is that the PSD width depends on the continuous phase viscosity, such that higher viscosities result in broader particle size distributions. Furthermore, the widths of the PSDs become narrower as packings become less porous and is broader with silicon oil as the dispersed phase compared with rape seed oil. With packing Reynoldsnumber between 3.95 and 545 for the data points considered within this investigation, higher Reynolds-number correspond to broader PSDs. In Section 4.1.3.4, we pointed out that the number of droplets an initially spherical droplet is broken-up into along certain particle tracks depends on the initial droplet diameter. Droplet sizes resulting in capillary numbers slightly above the critical capillary number were expected to undergo binary break-up. With an increase in droplet size, the number of resulting droplets increased. The generation of small satellite drops also influences the size distribution of the broken-up droplets. Moreover, the track along which a particle passes through the porous medium 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 radius a = 12 µm moving along track 3 was the largest drop not being broken up. In those calculations, the viscosity of the continuous phase was chosen to be η c = 10mP as, the viscosity ratio to be λ = 1, and the interfacial tension was set to σ = 10.0 × 10−3 N m−1 thus matching that between silicone oil and PEG – 2% SDS – H2 O solutions as used within our experiments. The flow through the 64 mm long converging-diverging nozzle at Reynolds-number 1000 resulted in a pressure loss of ∆p = 1.73 bar. The shape of the orthorhombic (I) sphere packing is closest to that of our convergingdiverging nozzle in terms of normalized void area over normalized streamwise position as


66

Table 4.6: Model predictions for the dispersing process within incompressible porous media according to model x50,3 –pack – IV (Eq. 4.8) for fluid and geometry parameters matching those in the numerical investigations. Sphere size ds [mm] 1 2 3 4

Predicted mean diameter x50,3 [µm] 5.32 4.48 4.05 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-diverging nozzle 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 a comparison for several sphere diameters. Table 4.6 gives predicted mean diameters in terms of 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 coupling of the sphere diameter to the fluid velocity within the porous medium and the pressure drop across it. With a decrease in sphere diameter, the volume specific surface area increases and thus more pressure is needed to overcome wall friction resulting in slower flow and thus in larger droplets. The numerical result matches the experimental data well if the following two two points are taken into account. Firstly, the numerically determined drop radius of a = 12 µm represents the largest drop which will not be broken up whereas the diameters given in Table 4.6 represent the mean diameter of an emulsion. Secondly, it is expected that numerical simulations of droplet break-up along tracks closer to the wall will predict that smaller droplets will be broken up, although no quantitative data is available. Overall, the comparison is very satisfactory.

4.2.2 Dispersing in Orifice Flow In this section, dispersing emulsions within single and adjoint orifice flows through one orifice and through nine parallel orifices was studied. Again, we focused on the mean diameter of the droplets in the emulsions generated and the width of their particle size distributions. Our analysis was based on the trials detailed in Appendix B, representing 56 data points, where data 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 considered for 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 well above the critical Reynolds-number for pipe flow of Recrit, pipe = 2300, only two out of 56 data points were above the critical Reynolds-number, indicating laminar flow conditions for almost all of the experimental data. In all trials, 5.5% – PEG – SDS – H2 O solutions were used. Therefore, continuous phase


67

Table 4.7: Range of the dependent variables x50,3 and span along with the explanatory variables used to estimate the coefficients in model Eqs. (4.10 and 4.11). The units used are those applicable to the model. Variable mean diameter x50,3 width of PSD span Reynolds-number Reo number of runs nruns viscosity ratio λ orifice diameter do number of dies ndies continuous phase viscosity ηc interfacial tension σ

Range 2.58 – 118.9 µm 0.970 – 2.65 µm 461 – 2935 3 – 40 1.05, 10.43 0.001, 0.0024, 0.0088 m 1,2,4 0.0092 Pa s (5.5% PEG – 2% SDS - H2 O) 10.0 ×10−3 N m−1

viscosity was not considered to be a main effect within our investigations. Choosing silicon oils AK 10 and AK 100 as the dispersed phase, viscosity ratios of λ = 1.05 and 10.43 were achieved 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 than pressure 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. On the other hand, the mean diameter model derived for the packing flow could also have been stated in terms of the Reynolds-number without a significant loss in accuracy. Assuming similar behavior in orifice flow seems valid, though we cannot prove it due to a partial lack of pressure loss data for the orifice flow. Including interaction terms in the models formulated below would have slightly improved the models at the cost of clarity. Interactions like that between Reynolds-number and viscosity ratio were found to be significant for the model for the width of the PSD but difficulties in interpretation 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 (x 50,3 –orif) is given in Eq. (4.10). All variables as given in Table 4.7 were included in the model and proved to be very 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 · Reo c2 · nruns c3 · λc4 · do c5 · ndies c6

(4.10)

The coefficients were estimated with an adjusted R 2 of 0.985 and an average deviation of 0.0844 to be c1 = 1.894 × 109 , c2 = −1.277, c3 = −0.2608, c4 = 0.1258, c5 = 1.454 and c6 = −0.2147. Similar to our observations for dispersing flows in porous media, where increasing the pressure drop strongly reduced the mean diameter, we found that the mean diameter in orifice

68


flow showed a similar dependence on the Reynolds-number with a coefficient of c 2 = −1.277. For laminar flow, the Reynolds-number is approximately a linear function of the pressure drop. Thus, a comparison with the coefficient of pressure drop in the mean diameter model for packing 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 coefficient c3 = −0.2608 which can be also be interpreted as an energy input coefficient. It is interesting to note that the ratio of 4.9 between the coefficients of the power input and the energy input is similar to the value of 4.5 found in the case of flow through a packed bed of spheres 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 droplets will be generated. It is also interesting to note, that the number of dies significantly influences the dispersing result, although we had pointed out in our numerical investigations that even for adjoint orifice flows at Reynolds numbers of 100, and thus far below the smallest Reynoldsnumber found within our experiments, jets form within adjoint nozzles. The number of dies might 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 model predicting 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 · Reo c2 · nruns c3 · λc4 · ndies c5 (4.11) All parameters are significant and the intercept, the Reynolds-number and the viscosity are 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 R 2 of 0.719 indicates a rather poor fit to the data. Therefore, care has to be taken in interpreting the results. To features can in any case be mentioned, one being the decrease in the PSD width with increasing numbers of runs and the second being the dependence of the span on the viscosity ratio, with the span increasing as the viscosity ratio increases. Both findings accord well with those for dispersing within porous media.

4.3. COMPRESSIBLE POROUS MEDIA FLOWS

4.3

69

Compressible Porous Media Flows

This section deals with flow investigations of flow through compressible periodically arranged porous media. The goal was twofold: firstly to characterize packing compressibility; secondly to estimate the usability of such compressible porous media for dispersing processes, hence closing the circle of our explorations. The former goal was accomplished by systematically investigating the flow through packings under a variation of five parameters. Those parameters were the packing length, the packing structure, the material of the spheres, the viscosity of Newtonian fluids, and the pressure drop across the porous medium. The first three of theses parameters were each assigned values which in the following discussion will be denoted as short and long, cubic and rhombohedral, and soft and hard respectively. The flow of a non-Newtonian fluid was also studied qualitatively. Fluid flow rate and packing deformation length data were acquired for all chosen pressure differences. For the second goal, a pre-emulsion was chosen as the fluid to be processed through the compressible porous medium. The resulting particle size distributions were compared to those expected 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 arranged elastic spheres depicted at rest on the left hand side and compressed by a flow through the medium on the right hand side. The length of the undeformed packing is L = 96 mm and the total deformation of the packing is ∆L as shown, resulting in a total strain over the whole packing of total = ∆L/L. Dividing the packing into four equally sized sections as indicated in the figure by the 1st , 2nd , 3rd and 4th quarter labels, provided strain information for each section, denoted 1st quarter , 2nd 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 of 0.5 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 negligible wall friction, although some sphere layers, particularly the ones furthest upstream, tended to loose contact with the bulk packing at low pressure differences. The type of fluid used influenced this behavior. As pointed out in chapter 3, we always forced a complete packing relaxation prior to each flow at distinct pressure differences. Above a certain pressure difference, cubically arranged packings made of soft material partially lost their structure resulting in different flow behavior. In theses cases, the data obtained were disregarded. 4.3.1.1 Flow and Compressibility Characteristics A typical experimental result for a 16% PEG solution flowing through a long, cubically arranged 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 by a dashed line. With Reynolds-numbers, Re∗ , in terms of undeformed packing parameters ranging from 2.6 to 15.1 the flow can be considered laminar. Therefore, a linear relationship


th

4 quarter 3rd quarter 2nd quarter 1st quarter

70

L

∆L

. V

Figure 4.20: Compressible porous medium made of cubically arranged spheres without fluid flow (left) and deformed by fluid flow from bottom to top (right). Explanations of the annotations are given in the text. between flow rate and pressure drop would be expected for an incompressible porous medium according to Darcy’s Law. Accordingly, the decrease of the flow curve gradient can be solely attributed to the compression of the porous medium. With flow rates above those presented here, the packing structure became disarranged. The packing compression is given on the right-hand ordinate with a total packing strain (total ) indicated by asterisks (∗) linked by a dotted line. Along with the total strain, strains are given for each quarter of the packing. These differ greatly from the total strain with respect to their magnitudes but otherwise have similar curve shapes. The strain of the 1 st quarter (4) exceeds that of the total strain by a factor of about 2 whereby those of the 3 rd and 4th quarter ( and ∇ respectively) fall short by a factor of about 2. The strains of the 2 nd quarter (◦) go well along with the total strain. The difference between the strain of the first and the fourth quarter amount to a factor of about 5. This pronounced dependence of the packing compression on the streamwise position within the packing gives rise to the assumption that local deformations, close to the mount of the packing will even be higher than the strains within the first section. The maximum possible strain is given by the porosity of a packing, being, in the case of the cubically arranged sphere packings, max = ε = 0.476, and resulting in a stall of the flow, which was not observed within our experiments. The data was tested against the three compressibility models (Eqs. 2.12, 2.13, 2.14) given in Section 2.1.1.6 and due to Tiller [TH93]. The best fit was obtained with the model for low compressibility (V˙ ∝ ∆p1−δ /L) with δ = 0.273 and a constant of proportionality of c = 51.8 with units for the pressure difference ∆p and the packing length L chosen to be bar and m

71 0.25

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Pressure drop ∆p [bar] Figure 4.21: Flow and compression characteristics for 16% PEG solution flowing through a long (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 viscosity. This is illustrated in Figure 4.22 where packing deformations are plotted versus pressure differences for three Newtonian fluids passing through the packing previously studied. The 16% 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 relationship between strain and pressure difference. Besides the total strain, data for the strain of the first quarter (dashed line) and the second half (dotted line) are given. Since third and fourth quarter strains were found to be almost identical, these strains were combined to form the second half strain for sake of clarity. It has to be pointed out that in case of water (, 4, ♦) strain values were below those for the PEG solutions. The difference is small, though significant, and is due to PEG acting as lubricant between the spheres and the wall. The fluid independence of the compressibility can also be seen in the similar compressibility coefficients δ estimated for each fluid. They were found to be 0.553, 0.523, and 0.273 for water, 5.5% and 16% PEG solutions respectively. Although the latter value deviates from the 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 obvious once different packing structures are considered. Trials performed for and discussed within this section are listed in Table 4.8. Trial parameters were the strength of the packing material (Mat.), the packing length (L), the packing structure (k, with values of 12 and 6 indicating rhombohedrally and cubically arranged sphere


72 0.25

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water 5.5% PEG 16% PEG

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Pressure drop ∆p [bar] Figure 4.22: Compression characteristics for a cubically arranged, 96 mm long packing made of 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 based on undeformed packing parameters. Compressibility characteristics according to Eq. (2.12) include a constant of proportionality (c), the compressibility coefficient (δ) and the adjusted R2 for each fit. 4.3.1.2 Influence of Packing Type Two packing types were investigated, cubically arranged spheres (k = 6) as discussed previously, and rhombohedrally arranged spheres (k = 12). The latter have a porosity in the undeformed state and a maximum strain of max = ε = 0.2595. Figure 4.23 depicts the flow and compression characteristics for such a rhombohedrally arranged packing with the maximum 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 cubically arranged sphere packings. Due to the dense packing of rhombohedrally arranged spheres, rearrangement is not possible. Therefore, much higher pressure differences could be applied, leading to a pronounced leveling off in strain starting at a pressure difference of about ∆p = 1.0 bar, along with an almost linear increase in flow rate, as expected from Darcy’s Law. Below 1.0 bar, almost linear correlations between the strains and the pressure drop can be observed, matching the result for cubically arranged sphere packings. Note that some of the furthest upstream sphere layers lost contact with the bulk packing as reflected in the lack of data for the total and the fourth quarter strain below 1.0 bar. Overall, the ratios between the strains of the various sections are


73

Table 4.8: Parameters of trials on compressible sphere packings along with flow characteristics in terms of Reynolds-number compressibility characteristics. The last three columns give the 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 bar and m respectively) and the R2 value for each fit. Further explanations are given in the text. Trial 1206B 1206C 1206D 1206E 1207A 1207B 1207C 1207D 1207E 1210A 1210B 1210C 1210D 1210E 1212A 1212B 1212C 1214A 1214B 1214C

Mat. [–] hard hard hard hard hard soft soft soft soft soft soft soft hard hard soft soft soft hard hard hard

L [mm] 95.4 95.4 48.3 48.3 48.3 95.4 95.4 48.3 48.3 48.3 48.3 95.4 95.4 48.3 96 96 96 96 96 96

k [–] 12 12 12 12 12 12 12 12 12 12 12 12 12 12 6 6 6 6 6 6

Fluid [–] H2O 5.5%-P 5.5%-P 16%-P H2O H2O 5.5%-P 5.5%-P 16%-P H2O 0.2%-X 0.2%-X 0.2%-X 0.2%-X H2O 5.5%-P 16%-P H2O 5.5%-P 16%-P

Visc. [Pa s] 0.00119 0.01098 0.01095 0.10010 0.00113 0.00114 0.01137 0.01131 0.10351 0.00113 NA NA NA NA 0.00112 0.01228 0.11148 0.00109 0.01259 0.10823

Re∗min [–] 1089 58.2 67.6 1.26 1373 901 25.5 29.1 1.10 1230 NA NA NA NA 1945 108 2.61 1468 114 2.29

Re∗max [–] 1695 96.3 163 3.94 2472 1170 48.1 84.4 1.76 1582 NA NA NA NA 2896 262 15.1 2881 255 29.5

c [–] 55.7 28.3 23.5 4.07 41.1 46.7 17.5 13.2 2.42 31.6 25.2 34.9 45.1 30.9 130 112 51.8 155 144 66.5

δ [–] 0.779 0.729 0.696 0.557 0.761 0.9 0.817 0.729 0.763 0.894 0.843 0.762 0.693 0.72 0.553 0.523 0.273 0.519 0.459 0.212

R2 [–] 1.000 1.000 0.998 0.998 1.000 1.000 0.993 0.995 1.000 1.000 0.998 0.857 0.979 0.911 1.000 1.000 0.993 1.000 1.000 0.995

closely comparable to those found with cubically arranged packings. The two distinct pressure regions found can be attributed to different deformation mechanisms. Firstly, at low pressure differences, small deformation theory applies with a linear relation between strain and force. Higher strains are no longer governed by linear elasticity and exponential relationships apply, thus increasingly large imposed forces lead to only small changes in strain. A total strain value of approximately 0.05 can be regarded as a threshold in the case of rhombohedral sphere packings. We noted before that the reading accuracy was 0.5 mm. This manifests itself in levels of the strain rates shown in Figure 4.23. A compressibility coefficient δ estimated to be 0.817, as given in Table 4.8 indicates a higher compactability compared to the cubically arranged packing (δ = 0.523) for otherwise identical parameters. Reynolds-numbers ranging from 25.5 to 48.1 accord well with the respective cubically arranged packing values of 108 to 262. Differences in the packing structure are 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 packing lengths and fluid viscosities. Comparing the compactability coefficient over the same range


0.12 220

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0.04 160

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Pressure drop ∆p [bar] Figure 4.23: Flow and compression characteristics for rhombohedrally arranged sphere packing. 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 packings and thus indicating still higher compressibility compared to the cubically arranged packing. The difference in the degree of compressibility becomes even more obvious by looking at Figure 4.24, which depicts the dimensionless flow characteristics for water flowing through each of the compressible porous media flows discussed so far, including the rhombohedral sphere arrangement. The friction coefficient (Λ∗ ) and the Reynolds-number (Re∗ ) are based on parameters of undeformed packings including the initial packing porosity. Data points for the cubically arranged packings, denoted by open symbols, are clustered according to fluid viscosity with decreasing viscosities resulting in a shift towards the right hand side. All of the points relate fairly well to the baseline indicating the flow characteristic through random sphere packings. The overall differences between the friction coefficients found and the baseline are most likely caused by production imperfections in the spheres used. Nevertheless, with increasing viscosity, friction coefficients tend to deviate more strongly from the baseline with increasing Reynolds-numbers. This can be attributed to the higher pressure drops imposed resulting in higher strains with higher viscosity ratios. Rhombohedrally arranged sphere packings (solid symbols) show a much more pronounced increase in friction coefficients with Reynolds-numbers. Looking at the data points for 5.5% PEG solution (•), one can see a slow increase in friction coefficients for the four leftmost data points which can be attributed to the linear deformation regime. The following data points are within the non-linear regime and have almost the same Reynolds-number, because this is based on the initial packing porosity.


Friction coefficient Λ∗ [–]

10000

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2000 1000 500

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Reynolds number Re [–] Figure 4.24: Dimensionless flow characteristics for long packings of cubically and rhombohedrally arranged spheres with varying fluids. The solid line indicates the flow characteristic for random sphere packings according to Λ = 181 + 2.01 · Re 0.96 . 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, cubic sphere packing from hard material are given in Figure 4.25. All data points indicate the same compression and flow behavior as found in all previously discussed trials with first quarter strains being much higher than the total strain, second quarter strains similar to the total ones and the strains of the last two quarters being about the same. Comparing these results to the corresponding soft material experiment (Figure 4.21) gives ratios of strain rates at the same pressure difference of about 2. For example, the first quarter strain at a pressure difference of ∆p = 1.25 bar was found to be 0.25 in the case of the soft material and 0.12 in the case of the hard material. This corresponds well with the compressibility 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 ∆p max = 1.25 bar in the soft material case, the spheres did not rearrange and deformations are still in the linear regime with the maximum total strain found to be 0.11 at a pressure difference of 2.0 bar. This agrees well with total strains up to a maximum value of 0.11 in the case of soft material being in the linear regime. For rhombohedral packings a threshold total strain for the transition from the linear to the non-linear regime was found to be at about 0.05. Extrapolating a corresponding threshold value for cubic packing according to the ratio between the porosities of the cubic and rhombohedral packings of 0.476 and 0.2595 respectively gives an approximate threshold value


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Pressure drop ∆p [bar] Figure 4.25: Flow and compression characteristics for 16% PEG solution flowing through a long cubic sphere packing of hard material. (Compare to results for soft material with otherwise identical parameters given in Figure 4.21.) of 0.10. This is about the maximum total strain observed in our investigation of cubic sphere packings. 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 in Figure 4.26. It covers flow and compression characteristics for 5.5% PEG solution flowing through a soft, rhombohedral sphere packing. Short packings were about half as long as long packings and were divided into two sections for strain analysis. Therefore, the sections called the first and second quarters in the previous discussions are now called the first and second halves respectively. Again, similar flow and almost identical compression behavior compared to the corresponding long packing data is found, with a total strain threshold of about 0.05 at a pressure difference 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 long packing. The transition from linear to non-linear compression behavior occurs coincident with the beginning of a Darcian flow regime found at pressure differences above 1.5 bar in the corresponding long packing experiment. The flow rates are about twice as high compared with the long packing, which is in good agreement with expectations. It should be noted that negative strain values are again present at low pressure differences due to contact being lost between the layers of spheres.

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Pressure drop ∆p [bar] 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 results for 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 initial packing heights were extended over their regular uncompressed height as indicated by the negative strains in Figure 4.27. Therefore, the results presented are of a more qualitative nature. In any case, the curves compare well with the previous findings, although the flow rates do not obviously reflect the shear-thinning fluid behavior. Tiller’s compressibility model (see Eq. 2.12) is given in terms of the volumetric flow rate (V˙ ) with the fluid viscosity accounted for by the coefficient of proportionality (c). Therefore, non-Newtonian flow behavior with flow-field dependent viscosity, is not covered by this model. Some poorly adjusted R 2 -values for fitting experimental data from trials with Xanthan to the model equation as given in Table 4.8 indicate the influence of the shear-thinning behavior, although the influence of packing extensions 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 within a compressible porous medium. The porous medium used was 95 mm long, built up from rhombohedrally arranged soft spheres. Compression characteristics for this packing were shown in Figure 4.23. 5% silicone oil AK 1000 was dispersed in 5.5% PEG – SDS solution with a viscosity ratio of λ = 10.4.


78

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Pressure drop ∆p [bar] 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 with the results for the same packing with a Newtonian fluid flowing through it as given in the previous 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 across the 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 one represents the non-linear regime with a total strain total = 0.06 and the strain of the first quarter being 1st quarter = 0.11. Results are shown in Figure 4.28 in terms of particle size characteristics versus the volume specific energy input times the number of runs Evn = ∆p · nruns = Ev · nruns . Symbols denoting the mean diameter of emulsions (x50,3 ) are again replaced by two numbers indicating the number of runs and the width of the particle size distribution. The same behavior compared to dispersing within incompressible sphere packings can be observed. Firstly, an increase in energy input going along with the increase in the number of runs results in an exponential decrease of emulsion mean diameters. Secondly, the width of the particle size distribution, particularly for the runs at the higher pressure drop, decreases with the number of runs. Finally, power input is again more efficient than energy input in terms 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 · nruns c3

(4.12)

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Specific energy input Ev = ∆p × nruns [bar] Figure 4.28: Dispersing 5% (v/v) silicone oil AK 1000 in 5.5% PEG – SDS solution in a long rhombohedral sphere packing made of soft material. Particle size characteristics x 90,3 , 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 an average 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 found within 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 compressible porous medium is superior in terms of efficiency. This becomes even more obvious when comparing the mean diameters measured to model predictions based on the parameters of the compressible porous medium experiment. Measured and predicted mean diameters are given in Table 4.9 two pressure differences and varying numbers of runs. Mean diameters measured are always smaller than those predicted by the model for the incompressible porous medium. This difference is greatest at high pressure difference with relative differences of almost 2. This could possibly be attributed to the reduction of capillaries due to the compression of the porous medium. A strong influence of the number of runs was also observed, emphasizing the importance of inflow effects as an explanation for dispersing mechanisms within compressible porous media. The width of the particle size distributions was found to be comparable to that expected by our model for incompressible porous media (span –pack – IV, Eq. 4.9) as given in Table 4.10. Reynolds-numbers used within the model are given in terms of incompressible porous medium parameters. Therefore, a direct comparison between span-values must be treated carefully.


80

Table 4.9: Comparison of mean diameter, x50,3 , with model predictions for incompressible porous medium. ∆p [bar] 0.84 0.84 4.76 4.76 4.76

nruns [–] 5 10 5 10 20

x50,3 measured [µm] 74.40 66.01 10.88 8.24 7.01

x50,3 predicted [µm] 82.23 74.05 25.20 22.70 20.44

Relative difference [–] -0.105 -0.122 -1.32 -1.75 -1.91

Table 4.10: Comparison of the width of particle size distribution, span, with model predictions for an incompressible porous medium. ∆p [bar] 0.84 0.84 4.76 4.76 4.76

nruns [–] 5 10 5 10 20

span measured [–] 1.34 1.35 2.43 2.05 1.82

span predicted [–] 1.63 1.56 1.67 1.59 1.51

Relative difference [–] -0.22 -0.15 0.31 0.23 0.16

It has to be noted that the investigations reported on in this section were based on a single experiment. Therefore, care has to be taken in interpreting these findings. Nevertheless, they are 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 in the previous chapter. We will focus on main findings and augment them with descriptions of possible future developments.

5.1

Viscosity Ratio

We were able to show that droplet break-up mechanisms in emulsions flowing through porous media depend on the interaction between viscosity ratio and packing length as reported in section 4.2.1.2. Droplet disruption at a dispersed to continuous phase viscosity ratio, λ, of 6.9 in flow through cubically arranged sphere packings can be attributed to stronger inflow effects and higher elongational flow field contributions, compared with the case when the viscosity ratio λ is 1.7. In this case the packing length becomes more important for the dispersing result than is the case for the higher viscosity ratio λ = 6.9. These findings accord well with our numerical investigations and expectations for single droplet break-up under steady flow conditions. In any case, an increase in packing length, under otherwise constant flow conditions (constant Reynolds-number), benefits the production of small mean emulsion drop diameters regardless of the viscosity ratios studied. This indicates the relevance of cumulative dispersing effects within porous media flow. Further investigations on dispersing multiple droplets in complex flow fields like those which occur in porous media flows must go hand-in-hand with studies on single droplet breakup in complex flow fields, performed experimentally as well as numerically. Perturbing ripples have not been addressed in this work but should also be considered and investigated as breakup 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 with parameters which must be empirically determined, in view of ease of process design. For the dispersing process in porous media and orifice flows, models with emulsion mean diameter and width of particle size distributions as dependent variables were established, based on 81

CHAPTER 5. CONCLUSIONS

82

a wide range of packing, process and fluid parameters. Almost all parameters within our models were very strongly significant. Droplet break-up within porous media was found to be of the same nature as that in orifice flow, as indicated by similar mean droplet diameters and PSD widths. Power input, which increases with an increase in the pressure drop across the packing was found to be much more efficient 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 decisive parameter for narrowing the width of the particle size distribution. Therefore, power input and 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 be needed and could possibly include further packing parameters such as the orientation of structured packings.

5.3

Compressible Porous Media

Non-linear deformation of compressible porous media when liquids were flowing through them 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 within such porous media produced finer emulsions compared with those processed through an incompressible porous medium, for otherwise identical process, fluid, and geometry parameters. A challenging, but promising task, would be the extension of mean diameter and width of PSD models established for incompressible porous media. Such an extension would have to take 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 arranged porous media, was determined by means of computational fluid dynamics (CFD). Therefore, flow field characteristics along particle tracks were used as the input for droplet deformation calculations based on the boundary integral method (BIM). The numerically determined critical diameter for break-up along a representative track was found to be in good agreement with the diameter predicted by our mean diameter model. Nevertheless, one-to-one modeling of break-up in such complex flow fields is still limited and it goes without saying that break-up within turbulent flow is beyond todays realm of possibility. One aspect which it would be desirable to investigate in the future is the break-up behavior on particle tracks closer to the wall, where droplet deformation calculations failed due to high shear and elongation rate gradients. Furthermore, periodicity and cumulative effects seem to be worthwhile targets for further investigations.

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Appendix A Crystal Families and Bravais Lattice Types Crystals consist of atoms, ions, and molecules arranged symmetrically and in a well ordered manner. Their structure can be divided into identical, repeating adjoint unit cells called crystal lattices. According to their geometry, such lattices can be divided into crystal families. Four types 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 the engineering literature. The orthorhombic and rhombohedral sphere packings which are used in this work are described in terms of Bravais lattice types as hexagonal primitive (hP), and cubic 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 Wolff et al. [DBB+ 85]. Their respective symbols are given in parentheses. Unit cell parameters are given in terms of unit cell edge lengths, a, b and c, and the angles between unit cell edges, α, β and γ. Crystal families Cubic (c) a=b=c α = β = γ = 90o Tetragonal (t) a = b 6= c α = β = γ = 90o Hexagonal (h) a = b 6= c α = β = 90o , γ = 120o Orthorhombic (o) a 6= b 6= c α = β = γ = 90o Monoclinic (m) a 6= b 6= c α = γ = 90o , β 6= 90o Triclinic (anorthic, a) a 6= b 6= c α 6= β 6= γ = 90o

Bravais lattice types Cubic primitive (cP) Cubic body-centered (cI) Cubic face-centered (cF) Tetragonal primitive (tP) Tetragonal body-centered (tI) Hexagonal primitive (hP) Rhombohedral (hR) Orthorhombic primitive (oP) Orthorhombic single-face centered (oS) Orthorhombic body-centered (oI) Orthorhombic all faces centered (oF) Monoclinic primitive (mP) Monoclinic centered (mS) Triclinic (aP)

Appendix B Parameters of Dispersing Experiments In table B.1, the trials employed for model estimations in the various sections of this work are listed. Note that the mean diameter model for the sphere packing flow was based on the same trials as the respective width of PSD model. Since data stemming from the first and second runs were not being considered in the model of the PSD width, some trials used for the mean diameter model were not used for the latter model. Process, fluid, and geometry parameters for all trials used in the analysis of this work are listed in table B.2 and continued in table B.3. ηd and ηc denote the dynamic viscosities of the dispersed and continuous phases, respectively, and their ratio is given by λ. Continuous phase density is given by ρc , sphere and orifice diameter by ds and do respectively, packing length by L, packing structure in terms of coordination number k (‘rand.’ indicates random packings), packing porosity by ε, dispersed phase volume fraction by φ d , the type of flow meter used by its maximum flow rate (with zero indicating measurements done without a flow meter), and interfacial tension by σ.

93

APPENDIX B. PARAMETERS OF DISPERSING EXPERIMENTS

94

Table B.1: Trials employed for model estimations in the various sections of this work. Trial parameters are listed in the following tables B.2 and B.3. Section 3.4.2.4

4.2.1.1 4.2.1.2 4.2.1.3

4.2.1.5

4.2.2

4.3.2

Trials Incompressible Sphere Packing Flow Characteristics 010120A, 010129A, 010130A, 010208A, 010208B, 010221A, 010222A 010227A, 010228A, 010308A, 010309A, 010411A, 010416A, 010418B 010423A, 010425A, 010520A Energy and Power Input 010120A Packaging Length 010208A, 010208B, 010221A, 010222A, 010227A, 010228A Mean Diameter Model for Sphere Packing Flow 001115A, 001128A, 001204A, 001207A, 001214A, 010120A, 001120A 001123A, 010129A, 010130A, 010208A, 010208B, 010221A, 010222A 010227A, 010228A, 010308A, 010309A, 010331A, 010411A, 010416A 010418B, 010423A, 010425A, 011119A Width of Particle Size Distribution 001115A, 001120A, 001123A, 001128A, 001204A, 001207A, 001214A 010120A, 010129A, 010130A, 010208A, 010208B, 010221A, 010222A 010227A, 010228A, 010308A, 010309A, 010331A, 010411A, 010416A 011119A Dispersing in Orifice Flow 011004A, 011011A, 011015B, 011016B, 011114B, 011116C, 011121C 011121D, 011126B, 011128B Emulsification in Compressible Porous Media 011205B

Trial

2%RSO 2%AK250 2%AK250 10%RSO 2%RSO 2%RSO 2%RSO 2%RSO 2%AK250 2%AK50 2%AK1000 2%AK1000 2%AK1000 2%AK1000 2%AK250 2%AK250 2%AK50 2%AK250 5%AK10 5%AK10 5%AK10 5%AK10 5%AK10 5%AK10

ηd [Pa s] 0.0600 0.2400 0.2400 0.0600 0.0600 0.0600 0.0600 0.0600 0.2400 0.0480 0.9700 0.9700 0.9700 0.9700 0.2400 0.2400 0.0480 0.2400 0.0093 0.0093 0.0093 0.0093 0.0093 0.0093

P hasecont 10%PEG 19%PEG 10%PEG 10%PEG 10%PEG 10%PEG 10%PEG 10%PEG 10%PEG 10%PEG 19%PEG 19%PEG 19%PEG 19%PEG 19%PEG 19%PEG 10%PEG 10%PEG 2%SDS-H2O 2%SDS-H2O 2%SDS-H2O 2%SDS-H2O 2%SDS-H2O 2%SDS-H2O

ηc [Pa s] 0.0265 0.1400 0.0265 0.0265 0.0265 0.0265 0.0265 0.0265 0.0265 0.0265 0.1400 0.1400 0.1400 0.1400 0.1400 0.1400 0.0265 0.0265 0.0011 0.0011 0.0011 0.0011 0.0011 0.0011

ρc kg [m 3]

1020 1035 1020 1020 1020 1020 1020 1020 1020 1020 1035 1035 1035 1035 1035 1035 1020 1020 1003 1003 1003 1003 1003 1003

ds , d o [mm] 4.000 4.000 4.000 4.000 4.000 4.000 4.000 4.000 4.000 4.000 4.000 4.000 4.000 4.000 4.000 4.000 2.000 2.000 0.339 0.070 0.339 2.000 0.339 0.339

L [mm] 100 20 20 100 80 80 100 80 80 80 100 20 20 100 100 20 40 40 300 10 160 400 110 110

k [–] 6 6 6 6 12 12 6 8 8 8 6 6 6 6 6 6 rand. rand. rand. rand. rand. rand. rand. rand.

ε φ [–] % 0.4760 2 0.4760 2 0.4760 2 0.4760 10 0.2595 2 0.2595 2 0.4760 2 0.3950 2 0.3950 2 0.3950 2 0.4760 2 0.4760 2 0.4760 2 0.4760 2 0.4760 2 0.4760 2 0.3760 2 0.3760 2 0.3760 5 0.3760 5 0.3760 5 0.3760 5 0.3760 5 0.3760 5

F low [ hl ] 0 0 0 0 0 0 0 1000 1000 1000 1000 1000 0 0 0 0 1000 1000 1000 1000 1000 1000 1000 1000

σ [ mN ] m 4 10 10 4 4 4 4 4 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10

λ [–] 2.26 1.71 9.05 2.26 2.26 2.26 2.26 2.26 9.05 1.81 6.92 6.92 6.92 6.92 1.71 1.71 1.81 9.05 8.45 8.45 8.45 8.45 8.45 8.45

Table B.2: Dispersing trial parameters. See text for explanations.

001115A 001120A 001123A 001128A 001204A 001207A 001214A 010120A 010129A 010130A 010208A 010208B 010221A 010222A 010227A 010228A 010308A 010309A 010331A 010411A 010416A 010418B 010423A 010425A

P hasedisp

95

5%AK10 5%AK100 5%AK100 5%AK100 5%AK100 5%AK100 5%AK100 5%AK100 5%AK10 5%AK100 5%AK100 5%AK100

P hasecont 5.5%PEG 5.5%PEG 5.5%PEG 5.5%PEG 5.5%PEG 5.5%PEG 5.5%PEG 5.5%PEG 5.5%PEG 5.5%PEG 5.5%PEG 5.5%PEG

ηc [Pa s] 0.0092 0.0092 0.0092 0.0092 0.0092 0.0092 0.0092 0.0092 0.0092 0.0092 0.0092 0.0092

ρc kg [m 3] 1012 1012 1012 1012 1012 1012 1012 1012 1012 1012 1012 1012

ds , d o [mm] 2.4 1.000 1.000 1.000 1.000 8.8 4.000 1.000 1.000 8.8 1.000 4.000

L [mm] 100 20 5 40 20 200 40 40 40 200 100 98

k ε [–] [–] NA NA NA NA NA NA NA NA NA NA NA 0.4760 6 0.4760 NA NA NA NA NA NA NA NA 12 0.2595

φ % 5 5 5 5 5 5 5 5 5 5 5 5

F low [ hl ] 60 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000

σ [ mN ] m 10 10 10 10 10 10 10 10 10 10 10 10

λ [–] 1.01 10.43 10.43 10.43 10.43 10.43 10.43 10.43 1.08 10.43 10.43 10.43

APPENDIX B. PARAMETERS OF DISPERSING EXPERIMENTS

011004A 011011A 011015B 011016B 011114B 011116C 011119A 011121C 011121D 011126B 011128B 011205B

ηd [Pa s] 0.0093 0.0960 0.0960 0.0960 0.0960 0.0960 0.0960 0.0960 0.0100 0.0960 0.0960 0.0960

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P hasedisp

Table B.3: Dispersing trial parameters (cont’d). Trial 011205B was conducted using a compressible porous medium.

Trial

Appendix C Adjoint Nozzle Flow Field Droplet deformation calculations presented in this work were accomplished with the adjoint nozzle geometry along particle tracks 1 to 3 as given in Figure 4.6. High shear rate gradients along 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.1 and elongation rates (without shear rates) in Figure C.2. Paths of tracks 4 and 5 were included in Figure 4.6.

97

APPENDIX C. ADJOINT NOZZLE FLOW FIELD

98

5e+04

0e+00


1e+05

Track 5

Track 4



Track 3

1e+05

5e+04

0e+00

0.000

0.010

0.020

0.030

1e+05

5e+04

0e+00

0.000

time [s]

0.010

0.020

0.030

0.000

time [s]

0.010

0.020

0.030

time [s]

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

3000

3000

3000 Track 4

Track 5

2000

2000

1000

1000

1000

0

−1000

−2000


2000



Track 3

0

−1000

−2000

0

−1000

−2000

−3000

−3000

−3000

−4000

−4000

−4000

0.000

0.010

0.020

time [s]

0.030

0.000

0.010

0.020

time [s]

0.030

0.000

0.010

0.020

0.030

time [s]

Figure C.2: Elongation rates along tracks 3, 4, and 5. Elongation rates shown are identical to those 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 the adjusted R2 -value and the average deviation. Model quality was also assessed based upon plots providing additional information on the fit. A typical set of plots, representing all model estimations performed in this work, is given in Figure D.1 for the mean diameter model of dispersing in sphere packing flow, as given in Eq. (4.8). Each data point is represented by an open circle (◦) or a vertical line. Essential conditions for obtaining good models are the presence of good characteristics in all four plots. Within the residuals versus fitted values plot (top left), data points should be likewise randomly distributed around the zero residual and along the abscissa. Within the normal Q-Q plot (top right) the points should be close to the diagonal. The data points being close to the diagonal indicates that the errors are randomly distributed. The scale – location plot (bottom left) is comparable to the residuals versus fitted values plot. The Cooks’s distance plot indicates the influence that each individual data point imposes on the estimated coefficients. Evenly distributed Cook’s distances are desirable.

99

APPENDIX D. STATISTICAL ANALYSIS – MODEL QUALITY

100

Residuals vs Fitted

Normal Q−Q plot 4

7 8

0.5 Residuals

7 8

3 Standardized residuals

4

0.0

−0.5

4

2 1 0 −1 −2 −3

2.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

−2

−1

0

1

Fitted values

Theoretical Quantiles

Scale−Location plot

Cook’s distance plot

2

8

7

0.10

4

1.5

8

Cook’s distance

(abs(Standardized residuals))0.5

7

1.0

0.08

75

0.06 0.04

0.5 0.02 0.0

0.00 2.0

2.5

3.0

3.5

4.0

Fitted values

4.5

5.0

5.5

0

20

40

60

80

100 120

Obs. number

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

Curriculum Vitae

Tobias Hövekamp born June 18, 1970 in Beckum, Germany

1/1996 – 10/2002

Ph. D. student and research assistant at the Swiss Federal Institute of Technology (ETH Zürich), Switzerland, Institute of Food Science and Nutrition, Laboratory of Food Process Engineering

10/1990 – 12/1995

RWTH Aachen, Germany, Dept. of Mechanical Engineering Dipl.-Ing.

9/1993 – 4/1995

Oregon State University, USA, Dept. of Mechanical Engineering Master of Science

6/1989 – 9/1990

Military service Grundwehrdienst

8/1980 – 5/1989

Thomas Morus Gymnasium, Oelde, Germany Abitur

8/1976 – 7/1980

St. Vitus Primary School, Oelde, Germany Grundschule

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