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Effect of bubble size and sparging frequency on the power transferred onto membranes for fouling control Jankhah, Sepideh 2013

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  EFFECT OF BUBBLE SIZE AND SPARGING FREQUENCY ON THE POWER TRANSFERRED ONTO MEMBRANES FOR FOULING CONTROL  by  Sepideh Jankhah  B.Sc. Shiraz University, 2003 M.Sc. Universit? de Sherbrooke, 2007  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF  DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES (Civil Engineering)  THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver)  December, 2013  ? Sepideh Jankhah, 2013  Abstract Fouling control through air sparging in membrane systems is governed by the hydrodynamic conditions in the system and the resulting shear stress induced onto membranes. However, the relationship between hydrodynamic conditions and the extent of fouling control is not well understood. As a result, sparging approaches are designed using a capital and time intensive empirical trial-and-error approach that does not guarantee that optimal conditions are identified. To address this knowledge gap, the present research focused on characterizing the hydrodynamic conditions in a membrane system under different sparging conditions (bubble size and frequency) and on finding a correlation between the induced hydrodynamic conditions and fouling control efficiency. New concepts of zone of influence of bubbles and power transferred were defined to characterise the hydrodynamic conditions in the system. A non-homogenous fouling distribution was observed in the zone of influence of bubbles due to a non-homogenous distribution of velocity and shear stress in this zone.  Fouling rates generally decreased with an increase in the area of the zone of influence, the root mean square of shear stress induced onto membranes and the rise velocity of bubbles. However, none of these parameters on their own could accurately describe the effect of the hydrodynamic conditions on fouling rate. On the other hand, power transferred onto fibers, which incorporates the effect of all the three parameters, could more effectively describe the effect of the hydrodynamic conditions on the rate of fouling.  Power transfer efficiency into the system, defined as the ratio of power transferred onto membranes to the power input in the system, was used to identify optimal sparging approaches. For all cases investigated, the power transfer efficiency to the system was consistently much higher for pulse bubble than for coarse bubble sparging. The results also indicated that as sparging frequency and the size of the bubbles increased, the width of zone of influence increased, suggesting that the spacing between the spargers could be increased when sparging with larger bubbles or at higher frequencies. Increasing the spacing would not only decrease the number of spargers, but also the volume of the gas required for sparging.  ii  Preface I am the principal investigator of this research project, in charge of developing the research proposal, i.e. identifying the research questions, planning the research program, performing research experimental work, and analysing the research data. The following manuscripts were submitted to peer reviewed journals summarizing some of the outcomes of this research. I am the principal author of the first three manuscripts under supervision of Professor Pierre B?rub?. The fourth manuscript was written in collaboration with Mr. Lutz Boehm, PhD student at Technical University of Berlin, Germany for which I was responsible for 50% of the work. In addition, I am the principal author of 10 conference publications and a supporting author on one. Journal papers: ? Sepideh Jankhah, Pierre R. B?rub? (2013) ?Power induced by bubbles of different sizes and frequencies onto hollow fibers in submerged membrane system?, Water Research, DOI: 10.1016/j.watres.2013.08.020. ? Sepideh Jankhah, Pierre R. B?rub? (2013) ?Fouling control in submerged hollow fiber membrane systems with pulse and coarse bubble sparging?, Submitted. ? Sepideh Jankhah, Pierre R. B?rub? (2013) ?Efficiency of coarse and pulse bubble sparging in terms of fouling control in submerged hollow fiber membrane systems?, Submitted. ? Lutz B?hm, Sepideh Jankhah, Jaroslav Tihon, Pierre R. B?rub?, Matthias Kraume (2013) ?Application of the electrodiffusion method to measure wall shear stress: integrating theory and practice?, Submitted. Podium presentations / Conference proceedings: ? Jankhah S, B?rub? P.R. (August, 2013) Efficiency of coarse and pulse bubble sparging in terms of fouling control in submerged hollow fiber membrane systems, podium presentation at the American Water Works Association (AWWA) Water Quality Technology Conference, Toronto, Canada. ? Jankhah S, B?rub? P.R. (Nov., 2012) Designing High Performance Air Spargers For Minimizing The Fouling Rate In Submerged HF Membranes, podium presentation at the American Water Works Association (AWWA) Water Quality Technology iii  Conference, Toronto, Canada. ? Jankhah S, B?rub? P.R. (April, 2011) Investigation of the relationship between variable hydrodynamic conditions at the surface of air sparged submerged membrane systems and membrane fouling rate. podium presentation at the BCWWA Annual Conference & Trade Show/Canadian Association on Water Quality (CAWQ), Kelowna, Canada ? Jankhah S, B?rub? P.R. (March, 2011) Behaviour of Foulants under Different Hydrodynamic Conditions at the surface of submerged Membranes, podium presentation at the AWWA Membrane Technology Conference, Long Beach, California ? Jankhah S, B?rub? P.R. (Oct. 2010) Characterizing Hydrodynamics at Membrane Surfaces in Air Sparged Submerged Systems through Direct Observation and Particle Image Velocimetry, podium presentation at the IWA World Water Conference, Montreal, Canada. ? Jankhah S, B?rub? P.R. (May 2010) Characterizing Hydrodynamic Conditions at Membrane Surface in Air Sparged Membrane Systems through Direct Observation ? Development of the Technique, podium presentation at the BCWWA Annual Conference & Trade Show, Whistler, Canada.  ? Jankhah S., B?rub? P.R., Y.Ye, P. Le-Clech, V. Chen (Sept. 2009) Investigation of Fouling Mechanisms in Submerge Membrane Systems, Proceedings of the 5th International Water Association Membrane Technology Conference and Exhibition, Beijing, China. ? N. Ratkovich, C.C.V. Chan, S. Jankhah, P.R. B?rub? and I. Nopens (Sept. 2009) Analysis of shear stress and energy consumption in a tubular airlift membrane system, podium presentation at 5th International Water Association Membrane Technology Conference and Exhibition, Beijing, China.  ? Jankhah S., and B?rub? P.R. (April 2009) Investigation of Fouling Mechanism in Submerged Membrane Systems, podium presentation at the BCWWA Annual Conference & Trade Show, Penticton, Canada  ? Jankhah S., B?rub? P.R. and Chan C.C.V. (July 2008) Shear Forces and Fouling Control in Membrane Systems, 2nd Workshop on CFD Modeling for MBR iv  applications, European MBR-Network, Gent, Belgium.  ? Jankhah, S., B?rub?, P., Mavinic. D. S., Andrews, S. A., Gagnon, G. A.,Walsh, M.  (April 2008)  Addressing Water Quality Concerns Associated with Disinfection By-Products in Drinking Water Systems for Small and Rural Communities, podium presentation at the BCWWA Annual Conference & Trade Show, Whistler, Canada   v  Table of Contents Abstract ...................................................................................................................................... i Preface ...................................................................................................................................... ii Table of Contents ...................................................................................................................... v List of Tables ........................................................................................................................... ix List of Figures ........................................................................................................................... x Nomenclature .......................................................................................................................... xii Acknowledgements................................................................................................................ xvi Dedication ............................................................................................................................. xvii 1 Introduction....................................................................................................................... 1 1.1 Relationship between sparging approaches and hydrodynamic conditions induced on submerged hollow fiber membrane systems .............................................................. 6 1.2 Effect of hydrodynamic conditions on fouling rate in submerged hollow fiber membranes ................................................................................................................................................... 10 1.3 Approaches to investigate the effect of sparging scenarios on the hydrodynamic conditions, induced shear stress, and fouling control .................................................................. 12 1.4 Research tasks ............................................................................................................................... 14 2 Experimental setup and measurement approaches ......................................................... 15 2.1 Experimental setup and experimental conditions investigated ................................ 15 2.2 Filtration setup .............................................................................................................................. 20 2.3 Measurement approaches ......................................................................................................... 20  Imaging of sparged bubbles ................................................................................................ 20 2.3.1 Measurement of shear stress induced onto membranes ......................................... 21 2.3.2 Particle Image Velocimetry (PIV) ...................................................................................... 24 2.3.33 Bubble characteristics obtained using imaging .............................................................. 26 vi  3.1 General physical characteristics of sparged bubbles investigated ........................... 26 3.2 General behavior of sparged bubbles investigated ......................................................... 32  Coarse bubble sparging ......................................................................................................... 32 3.2.1 Small pulse bubble sparging ............................................................................................... 33 3.2.2 Medium pulse bubble sparging .......................................................................................... 35 3.2.3 Large pulse bubble sparging ............................................................................................... 36 3.2.43.3 Conclusion ....................................................................................................................................... 38 4 Characterisation of the hydrodynamic conditions induced by sparged bubbles............. 39 4.1 Distribution of vorticity and velocity for discrete sparging ........................................ 39  Vertical distribution of velocity for discrete bubble sparging ............................... 41 4.1.1 Horizontal distribution of velocity for discrete bubble sparging ......................... 42 4.1.2 Zone of influence ...................................................................................................................... 44 4.1.3 Vertical distribution of shear stress for discrete bubble sparging ....................... 53 4.1.4 Horizontal distribution of shear stress for discrete bubble sparging ................. 55 4.1.54.2 Effect of sparging frequency on the distribution of velocity, vorticity and shear stress 57  Effect of sparging frequency on the vertical distribution of vorticity, velocity 4.2.1and shear stress .......................................................................................................................................... 57  Effect of sparging frequency on the horizontal distribution of velocity and the 4.2.2shear stress ................................................................................................................................................... 74 4.3 Summary of the hydrodynamic conditions induced by bubbles of different sizes and sparging frequencies ........................................................................................................................ 86 4.4 Conclusion ....................................................................................................................................... 93 5 Relationship between the induced hydrodynamic conditions and power transfer efficiency in the system .......................................................................................................... 95 5.1 Power transfer and power transfer efficiency per bubble for discrete bubble sparging .......................................................................................................................................................... 96 5.2 Power transfer and power transfer efficiency per bubble for sparging at higher frequencies .................................................................................................................................................. 101 5.3 System-wide power transfer and power transfer efficiency at different sparging flow rates ..................................................................................................................................................... 102 5.4 Conclusion ..................................................................................................................................... 106 6 Effect of induced hydrodynamic conditions on the fouling rate .................................. 107 vii  6.1 Effect of bubble size and sparging frequency on fouling rate ................................... 107 6.2 Effect of bubble size and sparging frequency on the spatial distribution of fouling rate in the system ..................................................................................................................................... 120 6.3 Conclusion ..................................................................................................................................... 123 7 Conclusions and recommendation ................................................................................ 125 7.1 Overall conclusions .................................................................................................................... 125 7.2 Engineering significance.......................................................................................................... 127 7.3 Recommendations for future work ..................................................................................... 128 References............................................................................................................................. 130 APPENDIX A Calibration of the electrochemical shear probes .......................................... 142 Appendix B Application of the electrodiffusion method (EDM) to measure wall shear stress: integrating theory and practice ............................................................................................. 145 B-1 Introduction........................................................................................................................................ 145 B.2 Electrodiffusion Method (EDM): theory .................................................................................. 147 B.2.1 The basic electrical circuit ................................................................................................. 147 B.2.2 The electrodes ........................................................................................................................ 147 B.2.3 The electrolytic solution ..................................................................................................... 148 B.2.4 Limiting diffusion current .................................................................................................. 148 B.2.5 Steady state flow conditions ............................................................................................. 150 B.2.6 Dynamic flow conditions .................................................................................................... 153 B.3 Electrodiffusion Method (EDM): application ......................................................................... 156 B.3.1 Experimental setup ............................................................................................................... 156 B.3.2 Practical aspects influencing the measurement ........................................................ 156 B.3.3 Data conditioning .................................................................................................................. 158 B.3.4 Wall shear rate calculation and correction ................................................................. 162 B.4 Conclusions ......................................................................................................................................... 165 Appendix C : Matlab codes developed to process voltage signals ....................................... 166 V-step in-situ calibration of the shear probes ............................................................................... 166 Correction of data under transient flow condtions ..................................................................... 171 viii  Appendix D Matlab codes developed to process images and the data obtained from PIV .. 190 Appendix E Filtration data.................................................................................................... 206 Appendix F Horizontal distribution of the shear stress for medium and large pulse bubble sparging................................................................................................................................. 211 Appendix G: Correlation between cut off velocity and rate of fouling ................................ 218    ix   List of Tables Table 2-1 Sparging conditions investigated ........................................................................... 19 Table 3-1 General characteristics of studied bubbles ............................................................. 30 Table 4-1 General characteristics of studied bubbles and the induced zone of influence ...... 50    x  List of Figures Figure 1-1 Typical shear stress profile in a confined (tubular) membrane system .................. 8 Figure 1-2 Typical shear stress profile in unconfined (hollow fiber) membrane systems ....... 8 Figure 2-1 Picture of the system tank with membrane module .............................................. 16 Figure 2-2 Experimental system ............................................................................................. 17 Figure 2-3 Electrical circuit used for measurement of shear stress with EDM ...................... 22 Figure 2-4 A shear probe fixed on a test fiber shown on top of a ZW-500 hollow fiber membrane ............................................................................................................................... 23 Figure 2-5 Typical shear stress profile for coarse bubble sparging ........................................ 24 Figure 2-6 Typical 2 dimensional velocity map generated from the PIV data ....................... 25 Figure 3-1 Typical images of bubbles generated by coarse and pulse sparging .................... 27 Figure 3-2 Bubble rise velocity for bubble size and frequencies investigated ....................... 31 Figure 3-3 Typical images of bubbles generated by coarse bubble sparging ......................... 33 Figure 3-4 Typical images of bubbles generated by small (150 mL) pulse sparging ............. 34 Figure 3-5 Typical images of bubbles generated by medium (300 mL) pulse ....................... 36 Figure 3-6 Typical images of bubbles generated by large (500 mL) pulse sparging ............. 37 Figure 4-1 Typical vorticity and velocity distributions induced by discrete rising bubbles .. 40 Figure 4-2 Typical horizontal distributions of velocity across the width of system tank....... 43 Figure 4-3 Typical distribution of a) velocity and b) vorticity for discrete bubble sparging with the coarse bubble sparger ............................................................................................... 45 Figure 4-4 Typical distribution of a) velocity and b) vorticity for discrete bubble sparging with the small pulse bubble sparger........................................................................................ 46 Figure 4-5 Typical distribution of a) velocity and b) vorticity for discrete bubble sparging with the medium pulse bubble sparger ................................................................................... 47 Figure 4-6 Typical distribution of velocity a), and vorticity d) for discrete bubble sparging with the large pulse bubble sparger ........................................................................................ 48 Figure 4-7 Dimensionless area of zone of influence for discrete bubbles .............................. 52 Figure 4-8 Typical shear stress distribution induced by discrete rising bubbles at vertical centreline of the tank .............................................................................................................. 54 Figure 4-9 Horizontal distributions of shear stress across the width of system tank for discrete sparging frequency .................................................................................................... 56 Figure 4-10 Typical distribution of hydrodynamic conditions induced by coarse bubble sparging at discrete sparging frequency ................................................................................. 58 Figure 4-11 Typical distribution of hydrodynamic conditions induced by coarse bubble sparging at 0.25 Hz sparging frequency ................................................................................. 59 Figure 4-12 Typical distribution of hydrodynamic conditions induced by coarse bubble sparging at 0.5 Hz sparging frequency ................................................................................... 60 Figure 4-13 Typical distribution of hydrodynamic conditions induced by small (150 mL) pulse bubble sparging at discrete sparging frequency ............................................................ 63 Figure 4-14 Typical distribution of hydrodynamic conditions induced by small (150 mL) pulse bubble sparging at 0.25 Hz sparging frequency ............................................................ 64 Figure 4-15 Typical distribution of hydrodynamic conditions induced by small (150 mL) pulse bubble sparging at 0.5 Hz sparging frequency .............................................................. 65 Figure 4-16 Typical distribution of hydrodynamic conditions induced by medium (300 mL) pulse bubble sparging at discrete sparging frequency ............................................................ 68 xi  Figure 4-17 Typical distribution of hydrodynamic conditions induced by medium (300 mL) pulse bubble sparging at 0.25 Hz sparging frequency ............................................................ 69 Figure 4-18 Typical distribution of hydrodynamic conditions induced by medium (300 mL) pulse bubble sparging at 0. 5 Hz sparging frequency ............................................................. 70 Figure 4-19 Typical distribution of hydrodynamic conditions induced by large (500 mL) pulse bubble sparging at discrete sparging frequency ............................................................ 71 Figure 4-20Typical distribution of hydrodynamic conditions induced by large (500 mL) pulse bubble sparging at 0.25 Hz sparging frequency ............................................................ 72 Figure 4-21 Typical distribution of hydrodynamic conditions induced by large (500 mL) pulse bubble sparging at 0.5 Hz sparging frequency .............................................................. 73 Figure 4-22 Typical horizontal distributions of velocity across the width of system tank..... 75 Figure 4-23 Typical vertical distribution of shear stress induced by coarse bubble sparging at different locations for discrete sparging frequency ................................................................ 77 Figure 4-24 Typical vertical distribution of shear stress induced by coarse bubble sparging at different locations for 0.25Hz sparging frequency ................................................................. 78 Figure 4-25 Typical vertical distribution of shear stress induced by coarse bubble sparging at different locations for 0. 5Hz sparging frequency .................................................................. 79 Figure 4-26 Typical vertical distribution of shear stress induced by small (150 mL) at different locations for pulse bubble sparging at discrete sparging frequency ........................ 81 Figure 4-27 Typical vertical distribution of shear stress induced by small (150 mL) at different locations for pulse bubble sparging at 0.25Hz sparging frequency ......................... 82 Figure 4-28 Typical vertical distribution of shear stress induced by small (150 ml) at different locations for pulse bubble sparging at 0. 5 Hz sparging frequency ......................... 83 Figure 4-29 Horizontal distributions of shear stress across the width of system tank ........... 85 Figure 4-30 Area of zone of influence .................................................................................... 87 Figure 4-31 Average width of zone of influence .................................................................... 89 Figure 4-32 System wide RMS of bulk velocity .................................................................... 90 Figure 4-33 RMS of shear stress ............................................................................................ 92 Figure 5-1 Power transferred onto membranes per bubble .................................................... 97 Figure 5-2 Force per bubble ................................................................................................... 98 Figure 5-3 Power transfer efficiency per bubble .................................................................. 100 Figure 5-4 Relationship between power transfer and air sparging conditions ..................... 103 Figure 5-5 Power transfer efficiency for the sparging conditions investigated .................... 105 Figure 6-1 Typical results from filtration experiments......................................................... 108 Figure 6-2 System average fouling rate constant for different sparging conditions ............. 109 Figure 6-3 Relationship between fouling rate and power transferred onto membranes ....... 111 Figure 6-4 Relationship between fouling rate and root mean square of shear stress in the system ................................................................................................................................... 114 Figure 6-5 Relationship between fouling rate and root mean square of shear stress for individual fibers .................................................................................................................... 116 Figure 6-6 Power transfer efficiency with respect to power transferred onto membrane surface ................................................................................................................................... 117 Figure 6-7 Power cost distribution for MBR systems .......................................................... 119 Figure 6-8 Distribution of fouling rate in the system ........................................................... 121  xii  Nomenclature A  Electrode area (m2) Ab   The area of the bubbles (m2) Az  Area of zone of influence of a rising bubble (m2)  Az,bubble Area of zone of influence per bubble (m2) ap  Particle diameter (m) C0  Concentration of the oxidizing ion at the wall (mol/ m3) Cb  Concentration of the oxidizing ion in the bulk solution (mole/ m3) ic?   Concentration gradient of the ferricyanide (mol/m4) ci  Concentration of the ferricyanide (mol/m3) D   Diffusion coefficient (m2/s) d  Diameter of the probe (m) ds   Diameter of the spherical cap (m) de   Bubble equivalent diameter; de = (6V/?)1/3 E0  E?tv?s number; E0= g??de2/? [unitless] f  Frequency (1/s) f*  Dimensionless frequency of the flow change ???Wff =* [unitless] F   Faraday constant (A s/V) Fb  Buoyancy force Fdrag, CF  Drag force induced by the liquid cross flow (N) Fdrag, permeations Drag force due to the permeation (N) Flift,shear  Lift force induced by the shear stress (N) Fg  Gravity force (N) G  Gravitational acceleration (9.82 m/s2) h  Height of the spherical cap (m) H  modified Peclet number [unitless] I  Current (A) I0  Current correction for edge effects (A) j  Specific current (A/m2) xiii  J   permeate flux (L/m2/h) kCot   Cottrell coefficient ( A 21s ) kLev  Leveque coefficient (A 31s ) km  Mass transfer coefficient (m/s) Lchar  Characteristic length of the cathode (m) Lz  Length of zone of influence (m) n  Amount of exchanged electrons during the reaction [unitless] n? i,conv  Molar flux due to convection (mol/(s m2)) n? i,migr  Molar flux due to migration (mol/(s m2)) n? i,diff  Molar flux due to diffusion ( mol/(s m2)) in?   Molar flux (mol/(s m2)) N  Number of exchanged electrons during the reaction [unitless] iN?   Molar flow rate (mol/s) P  Perimeter of the circular electrode (m) Pe  Peclet number Pe=? d2/D [unitless] Ptrans   Power transferred onto membranes (watts) P0  Initial pressure (Pa) Pn  Normalized pressure (Pa) Q  Electrical charge (A s) r?   Homogeneous reaction term (mol/(m3 s)) r   Radius of curvature (m) Reb   Bubble Reynolds number [unitless] Rec   Bubble corrected Reynolds number [unitless] Refibers Reynolds for smooth cylinders, i.e. fibers, immersed in-line with the flow[unitless] Sc  Schmidt number [unitless] Sh  Sherwood number [unitless] t  time (s) t0  characteristic time of the probe (s) ?RMS  Root mean square of shear stress (Pa) xiv  ui  Electrical mobility (m2/(V s)) v?  Velocity vector (m/s) vx  Velocity vector in x direction (m/s) vy  Velocity vector in y direction (m/s) Vm  Voltage (V) V   Volume of the bubble (m3) Vb   Rise velocities of the bubbles (m/s) Vbc   Velocity predicted based on literature (m/s) Wz   Width of zone of influence  x  x-coordinate (m) y  y-coordinate (m) *y   Dimensionless distance from the wall [unitless] z  z-coordinate, m ?   Relative shear rate fluctuation amplitude [unitless] ?   Velocity gradient or shear rate respectively (1/s) c?   Transient corrected velocity gradient or shear rate respectively (1/s) s?   Velocity gradient or shear rate respectively at steady state (1/s) ?~   The amplitude of wall shear rate fluctuations (1/s)  ?    The mean value of wall shear rate (1/s) ?  channel half width in a rectangular channel (m) ?c  concentration boundary layer thickness (m) ?  variable for the edge effect correction function [unitless] ?   Specific conductivity (1/( ? m)) ?   Dynamic viscosity (Pa s) ?   Kinematic viscosity (m2/s) ?   Density (kg/m3) ?p   Particle density (kg/m3) ??  Difference between water and air density at 17?C (kg/m3) ?  Water-air surface tension (N/m)  ?    Shear stress (Pa) xv  ??   Electrical potential gradient (V/m) ?  correction function for edge effects [unitless]xvi  Acknowledgements  I would like to acknowledge the people without whom this achievement would not be possible. Firstly, I would like to thank my supervisor, Dr. Pierre B?rub?, for his guidance and support during this project, and especially for being such a great mentor throughout my PhD program. I would also like to thank my supervisory committee, Dr. Eric Hall, Dr. Greg Lawrence, and Dr. Pierre Le-Clech, for their feedback and guidance throughout this project. I have also been very lucky to have the support of great colleagues and friends through these years, including Colleen Chan, Kelley Hishon, Lisa Walls, Isabelle Londonio, Mona, Negar and Arezoo. I also want to thank Kaveh, for patiently encouraging me when I was writing my thesis and for his endless support.  Last but not least, I would like to express my gratitude to my parents, Firouzeh and Hesam, for supporting me and my decisions throughout these years, and my sisters, Sanaz and Golnaz, for their love and moral support. And a final note to my dad who passed away last week and was looking forward to seeing my graduation: you will always be in my mind and heart. This project was partially supported by the Canadian Natural Sciences and Engineering Research Council (NSERC).   xvii  Dedication       To my parents, Firouzeh and Hesam 1  1 Introduction Submerged membrane systems provide effective treatment meeting stringent guidelines for the quality of drinking water or/and the discharge to surface waters. Compared to the conventional treatment systems, membrane systems offer advantages such as higher treated water quality, smaller footprint, higher volumetric loadings and lower sludge production rates. However their relatively higher operating costs compared to the conventional treatment systems can limit their use. Currently, the cost associated with membrane fouling control accounts for more than 30% of the operation costs in membrane bioreactor systems [1] .  Fouling of membranes, defined as accumulation of particles and organic matter at the surface or in the pores of membranes, significantly affects membrane performance by increasing the resistance to the permeate flow. Membrane fouling can be divided into three types: hydraulically reversible, chemically reversible and irreversible fouling. Hydraulically reversible fouling can be removed physically (e.g. by introducing turbulence at the proximity of a membrane surface) or through backwashing of membranes. On the other hand, chemically reversible fouling can only be removed by chemical cleaning methods, while irreversible fouling is permanent.  Unless otherwise indicated, in the present dissertation, fouling refers to hydraulically reversible fouling.   Four mechanisms are typically used to describe the progression of membrane fouling: complete blocking, standard fouling, intermediate blocking, and cake filtration. Depending on the membrane pore size distribution and morphology, and foulant size distribution, one or more of the above mechanisms may be dominating the fouling [2]. The hydrodynamic conditions at the proximity of membrane also affect the rate of fouling of membranes.  However, the hydrodynamic conditions do not affect all of the components of a fouling matrix similarly [3]. The transport of different foulants at a membrane surface is the result of the balance of forces exerted on foulants such as buoyancy forces (Fb = 1.33? ?gap3) , gravifty forces (Fg = 1.33??pgap3), drag forces incurred by the cross flow velocity Fdrag,CF = (6.325?????? ), drag forces exerted due to the permeation flow through the membrane 2  (Fdrag,permeation = 3?????) and the lift forces exerted by the gradient of the velocity at the membrane surface, i.e. shear stress (Flift, shear =  0.761 ?1.5ap3??0.5? ), where ap is the particle diameter, and ? is shear stress at the wall, i.e. membrane surface, ?p is particle density , g is the acceleration due to gravity, ? is the dynamic viscosity, vy is the cross flow velocity, and J is the permeate flux [4].  If the membrane surface is installed vertically in a module and a cross flow is applied along the membrane surface, the buoyancy force, the gravity force, and the drag force are exerted on the foulants parallel to the membrane surface. The balance of these forces results in transport of foulants at the membrane surface. If the difference between the density of the foulants and the density of the solution is very small (such as in MBR systems), the force resulting from the balance between the buoyancy force and the gravity force is negligible. The drag force exerted by the liquid cross flow results in the transport of foulants parallel to the cross flow along the membrane surface. Perpendicular to the membrane surface, the drag forces exerted by the permeation flow through the membrane result in accumulation of the foulants at the membrane surface, and the lift force exerted by the velocity gradient, i.e. shear stress, at the membrane surface results in the back transport of the foulants into the bulk solution. The resulting balance between the permeation drag and the back transport of the foulants into the bulk solution perpendicular to the membrane surface is presented in Equation 1.1, when the first term corresponds to the permeation drag and the second term corresponds to the lift forces [4]. F = 3????? - 0.761 ?1.5ap3??0.5?                                                                             (1.1) Based on the equation 1.1, a ?critical permeation flux? in membrane filtration systems is defined as the permeation flux for which the rate of accumulation of foulants at the membrane surface is equal to the rate of back transport of the foulants into the bulk solution [4]. If the operating permeation flux is higher than the critical flux, the rate of accumulation of foulants at the membrane surface due to the permeation flow is larger than the rate of back transport of the particles due to the shear lift forces and therefore membrane fouling occurs. 3  Equation 1.1 suggests that depending on the hydrodynamic conditions in the system, e.g. permeate flux and the shear stress at the membrane surface, particles of different sizes will tend to accumulate at membrane surfaces [5, 6]. For instance, smaller foulants may preferentially accumulate at the membrane surfaces at low permeate flux and high cross-flow velocities, i.e. high surface shear stress. Although the critical flux concept suggested by Equation 1.1 has been observed experimentally [7, 8], a simple force balance approach cannot be used to comprehensively describe the rate and direction of transport of particulate foulants. Knutsen and Davis [8] observed that particles at a membrane surface do not travel easily along the membrane surface at a higher permeate flux and therefore are not readily removed as suggested by Equation 1.1. This discrepancy was attributed to the interactions that can exist between particles and rough membrane surfaces at a higher flux. Knutsen and Davis [8] also reported that although the permeate drag remains constant, particles decelerate as they travel through the shear layer at a membrane surface. The back transport of small particles has also been reported to be enhanced by the presence of larger particles [7]. This was attributed to additional surface shear and boundary layer disturbances caused by larger particles. It should be noted that the hydrodynamic conditions also affect the structure of the cake layer formed by the accumulation of particulate material. Tarabara et al. [3] reported that cakes formed at lower Peclet numbers (Pe) or at higher collision efficiency are expected to have lower hydrodynamic resistance and a more open morphology. Also, over time, restructuring of the fouling layer can modify both its resistance and morphology. Restructuring can occur when a porous cake, formed at the beginning of a filtration cycle, reaches a critical thickness, at which time the cake may collapse. If this process repeats several times the result will be a more compact cake at the base and more porous structure at the surface of the foulant layer [3]. Unlike colloidal fouling, biofouling is typically not homogeneously distributed on a membrane surface, but tends to occur at discrete locations which can change over time [9]. A number of studies suggest that the solution chemistry and surface interactions, along with hydrodynamic conditions at the membrane surface, also control the rate of biofouling, i.e. accumulation of biological foulants [10-13]. These surface interactions can promote biofouling even in the absence of permeation flux through the membrane [13]. Soluble 4  organic foulants such as biopolymers can also accumulate at a membrane surface, increasing the hydraulic resistance to permeate flow and the membrane fouling rate. Le-Clech et al. [9] observed a concentration polarization layer of soluble alginate at the surface of the bentonite cake layer formed on a membrane surface when filtering a solution of bentonite and alginate; alginate was used as a model organic foulant to investigate the effect of biopolymers on fouling rate. The layer could be observed as bentonite traveled through the accumulated alginate towards the membrane surface. The bentonite velocity in this layer was inversely related to the concentration of the accumulated alginate. Le-Clech et al. [9] also observed that the alginate increased the specific resistance and decreased the compressibility of the bentonite cake layer that formed on the membrane surface The rate of fouling can be minimized by promoting the back transport of foulants away from a membrane surface. A number of mechanisms have been suggested as contributing to the back transport of foulants. Belfort et al. [6] suggested that, depending on the hydrodynamic conditions and the size of the material in the solution being filtered, different back transport mechanisms were likely to dominate fouling control. Molecular diffusion dominates at low shear rates and when filtering molecular size material. Inertial lift (due to the velocity gradient imposed on a foulant) dominates at high shear rates when filtering large particles and shear-induced diffusion dominates at intermediate shear rates and when filtering intermediate size material. The surface transport of particles, i.e. rolling or sliding of the foulants along the membrane surface due to bulk tangential flow, can also contribute to the transport of particles away from a membrane surface, affecting the rate of fouling. It should be noted that for smaller particles other factors such as membrane surface charge, Van der Waals forces, and physical-chemical properties of the membrane can also affect the back transport of particles [13]. Although the lateral flow, i.e. permeation flow perpendicular to the membrane surface, is generally small compared to the tangential flow parallel to the membrane surface in membrane systems operated with cross flow, permeation through a membrane does affect the near-surface mean velocities and the instantaneous velocity and shear force profiles at the proximity of the membrane surface. The effect of surface suction, i.e. permeate flux, on the hydrodynamics of the flow was studied numerically by Sofialidis and Prinos [14]. They 5  observed that the turbulence at the wall decreased with increasing permeate flux. Beavers and Joseph [15] theoretically demonstrated the existence of a non-zero tangential velocity on the surface of a permeable boundary. Therefore, although the permeation may not affect the mean flow velocities and bulk Reynolds number, it has been suggested by Gaucher et al. [16] that it may affect the velocity profile at the membrane surface, changing the surface tangential velocity and consequently the shear force profiles at the membrane surface. Gaucher et al. also suggested that high permeate flux increases the rate of fouling by decreasing the turbulence and the variability of the shear forces at a membrane surface [16, 17]. However, in these studies, electrochemical shear probes were installed on the surface of ceramic membranes to measure the shear stress [16, 17]. As a consequence of the geometry and the installation technique of these probes (as described in Section 2.3.2 and Appendix B), no permeation actually occurred at the surface of the electrochemical shear probes. Therefore, the effect of different permeation fluxes on the shear stress induced on to the shear probes cannot be properly investigated using this technique. Different methods have been developed to minimize fouling in submerged membrane systems. Gas sparging is one of the most common methods applied to control fouling in submerged hollow fiber membrane systems [18, 19] and can reduce the rate of fouling by 30 to 100%, depending on the applications, operating conditions, membrane configuration and characteristics of the foulants [20-26]. Sparging induces hydrodynamic conditions near the membrane surface which promote the back transport of foulants. Gas sparging may also physically remove the fouling layer if the bubbles contact and scour the fouling layer [27, 28]. However, the relationship between sparging conditions, bubble size and frequency, and efficiency of fouling control is not well understood [27, 28]. As a result, sparging approaches are designed using a capital and time intensive trial-and-error approach that does not guarantee that optimal conditions are identified. To address this knowledge gap, a comprehensive understanding of the relationship between bubble size and frequency and induced hydrodynamic conditions at a membrane surface and the effect of these conditions on the efficiency of fouling control is essential. The following sections reviews the work that has been done prior to this research to investigate the relationship between bubble size and 6  frequency and the hydrodynamic conditions induced in the system, and their effects on the efficiency of fouling control.   1.1 Relationship between sparging approaches and hydrodynamic conditions induced on submerged hollow fiber membrane systems Although the mechanisms of fouling control through air sparging are not fully understood, a number of models have been suggested to describe the effect of air sparging on fouling control. In general, the models are based on the force balance between the back transport of foulants away from the membranes (e.g. through shear-induced diffusion, inertial lift, scouring and etc.) and the transport of foulants towards the membrane by the drag introduced by permeate flux [5], similar to the models described in Equation 1.1.  The bulk liquid velocity induced by air sparged bubbles has been reported to contribute to fouling control in air sparged submerged membrane systems [19]. However, B?rub? and Lei [23] observed that the fouling rate for a given bulk cross flow velocity in a submerged hollow fiber module was substantially lower for two-phase flow, i.e. with air sparging, than for single-phase flow, i.e. without air sparging, for conditions where the bulk liquid velocities were similar. In addition, for a given bulk cross flow velocity, the magnitudes of both the average and maximum shear stress induced onto membranes were observed to be substantially greater for two-phase flow than for single-phase flow [29]. Similar observations were made by others when using flat sheet [30] and tubular membranes [31]. These observations suggest that bulk liquid movement on its own does not significantly contribute to fouling control. Pressure instabilities caused by sparged bubbles at the proximity of membranes have also been suggested as another mechanism of fouling control through air sparging [24, 32]. However, the potential beneficial impact of pressure instabilities in either confined or unconfined membrane systems has not yet been experimentally quantified.    Secondary oscillating flows induced in the wake of sparged bubbles have been suggested as contributing to fouling control. These secondary flows result in a highly variable shear stress of relatively high magnitude at the membrane surface, which prevents 7  the accumulation of retained material on membrane surfaces [6, 20, 21, 33, 34], and/or reduces the thickness of the mass transfer-limiting layer [30, 35].  In submerged membrane systems, a lower fouling rate was observed at a higher variable liquid velocity at the proximity of a membrane surface compared to a constant liquid velocity [36]. A lower fouling rate was also observed at higher variable shear stress than in constant shear stress [37, 38]. These observations suggest that oscillating flows induced by rising air sparged bubbles significantly contribute to fouling control in membrane systems [39]. However, to date, the secondary oscillating flows induced in the wake of sparged bubbles in hollow fiber membrane systems have not been fully characterized. In addition, no information is available regarding the relationship between the sparging conditions, i.e. bubble size and frequency, and the characteristics of these secondary oscillating flows. Furthermore, the relationship between the characteristics of these oscillations and fouling control efficacy is not known.  The shear stress induced at a membrane surface by gas sparging and the resulting secondary flows has been recognized as one of the most significant parameters governing fouling control [6, 19, 39-43]. Historically, it was assumed that because the packing density of fibers in a submerged hollow fiber membrane system is relatively high, the magnitude, variability and distribution of shear stress induced onto membrane surfaces in these types of systems were similar to those induced by an air slug in a confined system [19, 44, 45]. However, recent studies demonstrated that shear stress induced by sparged bubbles in unconfined systems, such as submerged hollow fiber membranes, is different from that observed in confined systems [29, 46, 47]. In confined systems most of the shear stress induced by slug flow is due to the flow reversal within the falling film between the slug and the membrane surface [48]. Although shear stress induced by flow reversal has been observed in unconfined systems, it is not common [38, 49]. Typical shear stress profiles in confined and unconfined systems are presented in Figure 1-1 and Figure 1-2, respectively. In unconfined systems, oscillatory flows in the wake of rising bubbles generated by gas sparging are largely responsible for highly variable shear stress induced onto the membranes (Figure 1-2) and therefore shear stress events (peaks) occur more frequently compared to the shear events occurring in the confined systems (Figure 1-1)  [50]. Also, in submerged hollow fiber systems, the sway of fibers can also contribute to the variable shear stress [19, 51]. In 8  addition to the lateral movement of the fibers, physical contact of loosely held fibers could also potentially scour the membrane surface and remove accumulated foulant [52]. Higher frequencies of shear events [29], and higher magnitudes of shear stress induced on to a membrane surface due to fiber contact were reported for loosely held fibers in comparison to those of tightly held fibers [53]. As such, fouling control is likely to be greatly enhanced in loosely-held systems where physical contact between fibers is promoted.   Figure 1-1 Typical shear stress profile in a confined (tubular) membrane system (Adopted from [54])   Figure 1-2 Typical shear stress profile in unconfined (hollow fiber) membrane systems  (Coarse bubble sparging in full scale membrane systems, adopted from [46]) 9    The magnitude, variability and distribution of shear stress induced onto membranes in submerged hollow fiber membrane systems is affected by the sparging conditions and the membrane module configuration [46]. Increasing sparging flow rates generally increases the bubble frequency and as a result, affects the number of shear events, i.e. peaks in time variable shear stress, and the variability of shear stress induced on to membrane surface [20, 33, 46, 55]. The size and geometry of sparged bubbles also affect the magnitude, variability and distribution of the shear stress induced onto membrane surface [35, 38].The magnitude of shear stress induced by larger bubbles tends to be greater than those induced by smaller bubbles. However, at a given sparging flow rate, sparging with larger bubbles decreases the number of shear events compared to sparging with smaller bubbles [30, 35]. The optimal conditions are likely a balance of the shear stress events of greater magnitude, achievable using large bubbles, and more frequent shear stress of lower magnitude, achieved using small bubbles. The magnitude, variability and distribution of shear stress induced onto membrane surfaces have also been suggested to be affected by the configuration of the membrane module. The shear stress experienced by membrane fibers is dependent on the location of the fibers in relation to the location of the sparged bubbles, and the fiber packing density in the module.  Fibers that are located closer to the sparged bubbles, such as those in the outer sections of a module are exposed to higher bulk velocities [56] and highly variable shear stress  magnitude [46, 49]. Chan [49] reported that the amplitude of the shear stress was not homogeneously distributed around a hollow fiber, where the amplitude of shear stress on the fibers facing the bubbles was approximately three times greater than that on the other sides of the fibers. The fiber packing density also affects the size and the rise velocity of bubbles in a module and the resulting shear stress induced onto membranes. Chang and Fane [57] observed smaller bubbles and lower bubble velocities in a high packing density module compared to those of a low packing density module.  Yeo et al. [36] reported that the axial velocities inside a hollow fiber module were up to ten times lower than those outside of the module where sparged air bubbles were introduced.    10  Although the characteristics of the secondary flows trailing sparged bubbles, and the resulting shear stresses induced onto membrane surfaces have been recognized as two of the most, if not the most, significant parameters governing fouling control in air sparged membrane systems, the effect of the sparging conditions, i.e. bubble size and frequency, on the characteristics of the secondary flows, as well as on the magnitude, variation and distribution of shear stress induced onto membranes, have not yet been comprehensively investigated.  The first research question (presented below) considered as part of the present dissertation was selected to address this knowledge gap.  Question 1: How do the sparging approaches affect the hydrodynamic conditions and the resulting shear stress in a membrane system? To answer to this research question, the secondary flows induced under different sparging conditions, i.e. bubble size and frequency, were characterized based on the distribution of liquid velocity and the vorticity in the system  The shear stresses induced onto the membrane surface by the sparging were also characterized for different sparging conditions. 1.2 Effect of hydrodynamic conditions on fouling rate in submerged hollow fiber membranes  The rate of fouling control in air-sparged submerged membrane systems is dependent on the hydrodynamic conditions and the resulting shear stress induced onto membranes under different sparging conditions, i.e. bubble size and frequency, and the module configurations (as discussed in Section 1.1). However, the effects of hydrodynamic conditions and resulting shear stress on fouling rate remain poorly understood.  A number of studies have suggested that the variations in the shear stress over time have a significant effect on fouling rate [11, 18, 36, 49]. In general, a lower fouling rate can be achieved by inducing highly variable shear stress on membranes rather than inducing constant shear conditions [39, 52, 57-61]. However, Chan [49] reported that above a given frequency of shear events, fouling control could be inhibited.   11  A number of summative parameters have been considered to relate time-variable shear stresses to fouling control, such as average shear stress, root mean square (RMS) of shear stress, and the standard deviation of shear stress [36, 49, 58]. Of these, the RMS of the shear stress has been reported to be most strongly correlated to the rate of fouling in submerged hollow fiber membrane systems [49, 62].  However none of the summative parameters considered to date can, on their own, be used to consistently relate the effect of time-variable shear stresses to fouling control [49, 62].  The rate of fouling also generally decreases with increasing the air sparging flow rate in unconfined systems [20, 33, 55]. However, a critical air sparging flow rate is observed above which a further increased in the flow does not further decrease the rate of fouling [23, 26, 55, 63]. At this critical condition, the shear stress controlling particle back-transport is likely high enough to prevent any particle deposition on membrane surfaces [23, 27, 28]. Therefore, further increases in the sparging flow rate (and therefore increase in shear stress induced onto membranes) have no additional effect on the prevention of particle deposition and hydraulically reversible fouling [30].  Although the hydrodynamic conditions generated by air sparging and the resulting shear stresses induced onto membrane surfaces have been recognized as one of the most significant parameters governing fouling control, the relationship between the hydrodynamic conditions and fouling rate is poorly understood. In addition, the optimum sparging conditions, in terms of power requirements, to reach a certain level of fouling control has not been identified. The second research question (presented below) considered as part of the present dissertation was selected to address this knowledge gap. Question 2: How do the induced hydrodynamic conditions affect the rate of fouling? To answer this question, the relationship between both the characteristics of the secondary flows and the shear stress induced onto membranes and the rate of fouling was studied.    12   1.3 Approaches to investigate the effect of sparging scenarios on the hydrodynamic conditions, induced shear stress, and fouling control  A number of different approaches have been applied by others to study the effect of sparging on the hydrodynamic conditions, induced shear stress, and fouling control in membrane systems. A summary of these approaches, along with their limitations, are discussed below.  a. Direct observation methods Different optical tools such as Direct Observation Through Membrane (DOTM) and Direct Visual Observation (DVO) have been used for investigating the effect of hydrodynamic conditions in the system on the structure of the fouling layer [9]. Direct Observation through Membrane (DOTM) was recently developed as a non-destructive online method providing information about the behaviour of foulants at the membrane surface [9]. However, it is only possible to observe the first fouling layer formed on a membrane surface with DOTM. The subsequent fouling layers, and consequently the cake structure, cannot be observed. On the other hand, the Direct Observation Technique (DOT) enables online observation of the behaviour of foulants at the proximity of membrane surface [45, 64]. However, due to the limitations of the setup used by others to date [45, 64], such as line of sight limitations, as well as limitations on scale, fouling could not be investigated for the hydrodynamic conditions that are relevant to full scale/commercial membrane applications.  b. Electrodiffusion method (EDM) Electrochemical probes, hereafter referred to as shear probes, can be used to measure the shear stress on a submerged hollow fiber membrane [23]. It is also possible to detect flow reversal using a double shear probe [38]. Shear probes have been widely used to measure surface shear forces in steady state and transient flows. However, the response time of the shear probe must be known in a transient flow before being used for shear measurements. The response time of the probe is important because the limiting current of the shear probe is calculated assuming a quasi-steadystate condition in the mass transfer boundary layer where Pe is high enough to neglect the longitudinal and transverse diffusion. Therefore, the shear probe could be used for shear force measurements in transient flow if the assumption of the 13  electrochemical reaction being faster than the fluctuation in shear force is valid. The response time of shear probes can be determined by different models [65-68]. Consequently, it is essential to develop an approach for correction and interpretation of data collected under transient flow conditions. This limitation is addressed in the present research (Section 2.3.2 and Appendix B). c. Image Velocimetry Tools Particle Image Velocimetry (PIV) is a non-intrusive tool for quantifying shear stress.  In addition, PIV can be applied to quantify the distribution of velocity and vorticity induced in the system by sparged bubbles. PIV could also be used to investigate particle trajectories close to a membrane surface. A number of studies have investigated the mechanism of fouling using PIV. The effects of bubble frequency and size on the fouling rate of hollow fiber membrane systems under different hydrodynamic conditions were recently studied using PIV for a biological model solution with two-phase flow [69]. Yang et al. [70] used PIV to study fouling of hollow fiber membrane systems using Rhodamine B (fluorescent) particle tracers. In their study, transparent fibers were used since real hollow fiber membranes would have blocked the light sheet used for PIV. Gimmelshtein et al. [71] studied the flow in membranes using PIV when filtering a solution containing fresh seeding of yeast particles of approximately 5 ?m diameter. However, the study did not consider the effect of permeate flux on shear forces and flow distribution at the proximity of the membrane. Yeo et al. [58] used PIV to investigate the effect of different air sparging conditions, i.e. different bubble sizes, between 5 mm and 20 mm, and frequencies) on the hydrodynamics of flow, i.e. bulk turbulence, in proximity to a hollow fiber. They reported that PIV can overestimate the velocity at the surface of hollow fiber membranes by as much as 30%. In addition, air sparging causes the fibers to sway. Swaying fibers change the geometry of the system, making PIV analyses difficult [58]. However, one of the limitations of this method is that it can only be applied where a sheet of light can be created at the membrane surface and no obstacles are in the line of sight of the high speed camera. Therefore, this method cannot be used to quantify the velocity and the shear stress at the surface of hollow fiber membranes installed inside a packed hollow fiber module.    14  1.4  Research tasks  As discussed in the previous sections, the hydrodynamic conditions and the resulting surface shear stress induced onto submerged hollow fiber membranes are significantly affected by the sparging approach. However, the link between these conditions and fouling rate remain unclear. To address this knowledge gap, the proposed research focused on addressing two major questions outlined in sections 1.1 and 1.2 by performing the following overall tasks. Task 1. Characterize the hydrodynamic conditions in a membrane system under different sparging conditions, i.e. bubble size and frequency. The following intermediate tasks were performed to enable research question 1 to be addressed:  Task 1.1. Design a system that mimics the secondary flows that are representative of the conditions in real size submerged membrane systems (Chapter 2). Task 1.2. Develop a method to characterize the secondary flows, i.e. velocity and vorticity, and the induced shear stress onto membranes in the system (Chapter 2 and Chapter 3). Task 2. Find the correlation between induced hydrodynamic conditions and fouling control efficiency in the system. The following intermediate tasks were performed to enable research question 2 to be addressed: Task 2.1. Investigate the relationship between hydrodynamic conditions, i.e. velocity and vorticity, and shear stress induced onto membranes in the system and the rate of fouling (Chapter 4). Task 2.2. Develop parameters that can accurately correlate the effects of bubble size and frequency on the induced secondary flows in the system to the rate of fouling (Chapters 4 and 5). Task 2.3. Develop an approach to identify optimal sparging conditions, i.e. bubble size and frequency, that induce optimal fouling control conditions (Chapter 6).  15  2 Experimental setup and measurement approaches Chapter 2 presents the design of the experimental setup and the measurement approaches developed for investigating the hydrodynamic conditions induced under different sparging conditions and their effect on fouling rate. 2.1 Experimental setup and experimental conditions investigated  The system tank and the spargers designed in this research enabled the hydrodynamic conditions that are representative of full size submerged membrane systems to be mimicked (See section 2.3.2). All experiments were performed in a 2 m high, 1 m wide and 15 cm thick rectangular Plexiglas tank (Figure 2-1 and Figure 2-2). A module containing seven fibers, each 174 cm long, was placed vertically at the center of the tank. The fibers used when filtering and when measuring shear stress are described in section 2.2 and 2.3.2, respectively. The space between the fibers in the module was 7 cm. The spacing between the top and the bottom module bulkheads was 172 cm, allowing the fibers to sway in the system. The dimensions of the system tank and the module allowed commercially available, i.e. ZW500, GE Water and Process Technologies) full length hollow fiber membranes to be used.     16   a Figure 2-1 Picture of the system tank with membrane module  17     a b Figure 2-2 Experimental system  (a: schematic of front view of system, probe locations identified with numbers [1 to 4]; b:  schematic of side view of system)   Air spargers were fixed at the bottom of the center of the tank (Figure 2-1 and Figure 2-2).  Two types of spargers were used: a coarse bubble sparger that generated small bubbles of 0.73 mL to 2.5 mL in volume and a pulse bubble sparger that generated 150, 300 and 500 mL bubbles. The coarse sparger was a perforated pipe with three holes of 0.5 cm diameter, one at the centerline of the tank, and the other two on either side, each spaced 5 cm apart. The coarse bubble sparger generated small bubbles characteristic of those generally used in MBR systems [1]. The pulse bubble sparger is a proprietary design provided by GE Water and Process Technologies and therefore the details about the pulse bubble sparger design cannot be disclosed in the present thesis. The pulse bubble sparger was selected because sparging with relatively large pulse bubbles, i.e. 150 mL, has been reported to result in less fouling compared to sparging with coarse bubbles [62]. Pulse bubble spargers are commercially used by some membrane manufacturers (e.g. Siemens, Samsung, and GE Water and Process Technologies) with claims of better performance, in terms of fouling control, than coarse sparging. However, no 18  investigations have been done to characterize the hydrodynamic conditions induced by pulse bubble spargers. Since both bubble size and frequency have been reported to affect fouling, three sparging flow rates were selected to generate three different sparging frequencies: 1) discrete pulse bubbles, 2) pulse bubbles at a frequency of 0.25 Hz, and 3) pulse bubbles at a frequency of 0.5 Hz. The discrete sparging frequency was selected to generate single pulse bubbles. The sparging frequency of 0.25 Hz was selected to generate a series of pulse bubbles, with a distance between successive bubbles that was sufficiently large so that two successive bubbles did not interact (See section 4.2 for more details). The sparging frequency of 0.5 Hz was selected to generate a series of pulse bubbles where the distance between two successive bubbles was short enough so that the two successive bubbles could interact, i.e. the distance between the bubbles was shorter than the length of the wake of descrete bubbles. For the coarse bubble sparger a sparging frequency could not be defined. Therefore, three sparging flow rates of low, medium, and high were selected. The low sparging flow rate for coarse bubble sparging was selected to be equivalent to that for discrete sparging of pulse bubbles (996 mL/min). The medium flow rate for coarse sparging was selected to be equivalent to that for the small pulse bubbles, i.e. 150 mL, at a 0.25Hz sparging frequency. The high sparging flow rate for coarse bubble sparging was selected to be equivalent to the sparging flow rate for the medium pulse bubble, i.e. 300 mL, at a 0.5Hz sparging frequency. The sparging conditions investigated in the present research are summarized in Table 2.1. 19     Table 2-1 Sparging conditions investigated  Sparger type Coarse Small pulse Medium pulse Large pulse Nominal Sparging frequency Discrete 0.25 Hz 0.5 Hz Discrete 0.25 Hz 0.5 Hz Discrete 0.25 Hz 0.5 Hz Discrete 0.25 Hz 0.5 Hz Bubble volume [mL] 0.73 0.75 2.5 150 150 150 300 300 300 500 500 500 Sparging flow rate [mL/min] 996 2600 9200 996 2600 4300 996 4700 9200 996 8100 13500  The sparging flow rate for discrete pulse bubble sparging resulted in bubble frequency of less than 0.06 Hz such that successive bubbles do not interact; for course bubble sparging, nominal frequency of Discrete, 0.25 Hz and 0.5 Hz corresponded to 996, 2600, and 9200 mL/min flow rates respectively. 20   2.2 Filtration setup  When filtering, the module contained hollow fiber membranes (ZW500, GE Water and Process Technologies). The fibers had a 1.8 mm outer diameter with a normal pore size of 0.04 mm. Each hollow fiber in the module was connected to a separate permeate line, each with an individual peristaltic pump, and a pressure transducer to measure the trans-membrane pressure. The permeate flux collected from each fiber was monitored over time enabling the fouling rate in each hollow fiber to be assessed independently. The fouling rate was quantified based on the rate of change of normalized trans-membrane pressure, Pn, (defined as the ratio of the trans-membrane pressure at a given time to the initial trans-membrane pressure) with respect to the volume filtered. The total fouling rate in the system was estimated as an average of the fouling rate of four fibers combined. Filtration was performed at a constant permeate flux of 100 L/m2.h when filtering a solution containing 750 mg/L of bentonite with the average particle size of 3 ?m in water (size distributions of the particles in the bentonite solution were analyzed using a laser particle size analyzer, Mastersizer Hydro 2000S, Malvern, with an average of 3 ?m and the smallest particle of 0.3 ?m) which corresponds to an overall solid mass flux of 75 g/m2.hr, which is typical for MBR systems. All filtration experiments were performed in duplicate. Each filtration experiment was terminated when the trans-membrane pressure (TMP) reached 60 KPa. Data obtained from filtration experiments are presented in Appendix E.  2.3 Measurement approaches  Imaging of sparged bubbles 2.3.1 Imaging of bubbles was performed using a high-speed high-resolution camera (Phantom Miro 4, with 800 x 640 pixel resolution) and VidPIV software (Oxford Lasers). Two high intensity light sources were used to create a vertical thin (approximately 1 cm) sheet of light (Figure 2-2a). The light sheet was created orthogonal to the focus axis of the high-speed camera (Figure 2-2b). A video of 2 minutes duration was captured at each set of sparging conditions. Recordings were repeated in duplicate for each sparging condition.   21   Measurement of shear stress induced onto membranes 2.3.2 When measuring shear stress, the module contained test fibers made of flexible Teflon tubes with a diameter similar to that of hollow fiber membranes used in the present study [29].   The shear probes were fixed half way along the length of the test fibers in the module. The shear stresses induced by sparged bubbles at the surface of the test fibers were measured using an electrodiffusion method (EDM) [29, 72]. The reagent used for the electrochemical measurements contained 0.003 M ferricyanide, 0.006 M ferrocyanide, and 0.3 M potassium chloride in deoxygenated, de-chlorinated tap water [46, 48]. A limiting diffusion current of 500 mV was selected as described by [72]. Measurements were collected at a frequency of 200 Hz and a water temperature of 170C. Experimental temperaute affects the physical characteristics of the water, e.g. diffusion coefficient, see Appendix B for correction of data for different temperatures. A stainless steel anode was used in all experiments. The magnitude of the shear stress was obtained from the current measured at the probes using the quasi-steady state Leveque relationship presented in Equation 2.1 [73]:  313132862.0 ??= dDcFAnI b                                                                                 (2.1) where D = diffusion coefficient (m2/s), d = diameter of the probe (m), ? = shear stress (Pa), F = Faraday constant (A s/V), A = electrode area (m2), n = number of exchanged electrons during the reaction [-], Cb = concentration of the oxidizing ion in the bulk (mole/ m3), and I = current (A).  Figure 2-3 illustrates the electrical circuit used for shear measurements. Figure 2-4 illustrates the shear probes used in this study [29]. 22   Figure 2-3 Electrical circuit used for measurement of shear stress with EDM  (Adapted from [46])   23   Figure 2-4 A shear probe fixed on a test fiber shown on top of a ZW-500 hollow fiber membrane  (Adopted from [46])  Calibration of the probes was done exsitu prior to all experiments as presented in [45] (See Appendix A for detailed calculations). Due to the turbulent nature of the hydrodynamic conditions in air-sparged membrane systems, the flow conditions at the proximity of the probes are highly transient. Measurement of shear stress using EDM under highly transient conditions, such as the hydrodynamic conditions induced in the present system, requires V-step insitu calibration and correction of the signal [72, 73]. Therefore, the magnitude of the shear stress calculated by the steady state solution (Equation 2.1) was corrected to account for the non-steady state, i.e. transient, conditions [72, 73].  Extensive literature exists on the EDM technique. Some reviews focusing on different aspects of the technique have been published by [78-81], but none of these present the derivations of the underlying theory and the correction necessary for transient flows. To address this gap, a reference document was developed. The theoretical assumptions and hypotheses used in developing the equations that are used in the post-processing to calculate the shear stress under transient conditions were reviewed in detail. The calibration and correction methods for the data collected under transient conditions were optimized, and challenges regarding the calibration of this technique and the care that must be taken before 24  using the technique were also investigated as presented in Appendix B. Matlab codes developed for correction of the data using the V-Step insitu calibration method are presented in Appendix C. Figure 2-5 presents a typical shear profile obtained in the present study for coarse bubble sparging. The order of magnitude and the variation of the shear stress measured for coarse bubbles observed in the present study (Figure 2-5) were similar to those measured in full scale membrane modules (as presented in Figure 1-2), confirming that the designed experimental system generated shear stress conditions similar to those in a full-scale system (e.g. ZW500 systems). Measurements of shear stress were made in triplicate, with measurements recorded for 2 minutes.   Figure 2-5 Typical shear stress profile for coarse bubble sparging (Coarse bubble sparging at 9200 mL/min [results from the present research)   Particle Image Velocimetry (PIV)  2.3.3 Particle Image Velocimetry (PIV) was performed to track seeding particles in the flow and develop maps of the distribution of velocity and the distribution of vorticity. Seeding particles with mean size of 0.490-0.690 mm, and relatively neutral density, i.e. 1.02 mg/L, stained with Rodamine B were used. A cross-correlation algorithm with 50% overlap for a 32 x 32 pixel interrogation area, followed by a second cross-correlation with an interrogation area of 16 x 16 pixels was used for PIV analyses. Local filters were applied to detect and 25  eliminate invalid velocity vectors using a local median filter. The filtered velocity vectors were replaced using a median interpolation algorithm in a 3 x 3 matrix. The images were captured at a frequency of 200 frames per seconds (fps), consistent with the rate at which shear stress measurements were collected. At this frequency, each particle traveled less than 25% of the interrogation area during the time between subsequent images. The concentration of particles was chosen to ensure that sufficient number of particles were present in each interrogation zone, i.e. minimum of 4 per interrogation area [74]. A trigger was used to synchronize the signals obtained by the electrochemical shear probes and the videos captured by the high-speed camera. Matlab codes were developed to process the data generated by the PIV software to enable further statistical analyses and to produce time resolved velocity and vorticity maps (Appendix D). Figure 2-6 shows a typical image analysed by the PIV, the vectors show the velocity in the system. The resolution of the captured images was not high enough to calculate shear stress using the PIV.    Figure 2-6 Typical 2 dimensional velocity map generated from the PIV data    26   3 Bubble characteristics obtained using imaging 3.1 General physical characteristics of sparged bubbles investigated  Typical images of the bubbles generated for the sparging conditions investigated are presented in Figure 3-1. The characteristics of bubbles, i.e. geometric shape and behavior, were determined using the images captured by the high speed camera. Based on the 2-D images obtained, the coarse bubbles were observed to be predominantly spherical although some ellipsoidal bubbles were also observed (Figure 3-1) while the pulse bubbles were observed to be spherical cap (Figure 3-1b).   27    a  b Figure 3-1 Typical images of bubbles generated by coarse and pulse sparging (a: coarse sparging [insert is the magnification of coarse bubbles], b: 500 mL pulse sparging; each square in images is 2cmx2cm)  28  The radius of curvature of pulse bubbles was r = (h2 + (ds/2)2)/2h, where ds and h are the diameter and height of the spherical cap. The rise velocities (Vb) of the bubbles were estimated based on the vertical distance traveled over a given time.  The bubble Reynolds number (Reb) was calculated as Reb = deVb/? where (Vb) is the bubble rise velocity, de is bubble equivalent diameter (de = (6V/?)1/3), V is the volume of the bubble, and ? is water dynamic viscosity at 17?C. The bubble corrected Reynolds number (Rec= (deVbc/?)) was calculated using the bubble equivalent diameter and the velocity of discrete bubbles predicted based on literature (Vbc) [75] for a bubble with the corresponding equivalent diameter. The projected area of the bubbles (Ab) was calculated with respect to the plane orthogonal to the camera. The E?tv?s number (E0) was calculated according to (E0= g??de2/?), where g is the gravitational acceleration (9.82 m/s2), ?? is the difference between water density at 17 ?C and air density at the same temperature, and ? is the water-air surface tension. The Froude number was calculated as ??????2 . E?tv?s number and Reynolds number can be used to predict the bubble characteristics, i.e. geometric shape, and behavior, under different experimental conditions, such as for different solutions, bubble sizes and different temperatures [75]. Non dimensional numbers such as E?tv?s number and Reynolds number can also be used to compare the data obtained under conditions investigated in the present study to the data in the literature or with the future research.  The general physical characteristics of the sparged bubbles considered are summarized in Table 3.1. The rise velocities of the bubbles were generally higher than expected [75-77]. This was likely due to the upward liquid flow induced by the sparging at the center of the system tank, and the corresponding downward liquid flow at the sides of the system tank (See Figure 4-2 for details).  As a result, the Reynolds numbers of the entrained bubbles were also higher than expected for their size. As previously indicated, the bubbles generated by coarse sparging were observed to generally behave as wobbling spherical bubbles although ellipsoidal bubbles were also observed, while those generated by pulse sparging behaved as spherical cap bubbles. These observations were expected based on the Re and Eo numbers of the bubbles (Table 3.1) [75]. For the large pulse bubbles, i.e. 300 and 500 mL, small satellite bubbles were generally observed at the edges of the rising pulse bubbles, which is consistent with observations by others [76]. The Froude number for single pulse bubbles, i.e. at discrete 29  sparging frequency, was between 0.9 and 1.1, which agrees well with data reported by others for single bubbles rising in stationary liquids [77, 78].  30  Table 3-1 General characteristics of studied bubbles   Coarse Small pulse Medium pulse Large pulse Nominal sparging frequency Discrete 0.25 Hz 0.5 Hz Discrete 0.25 Hz 0.5 Hz Discrete 0.25 Hz 0.5 Hz Discrete 0.25 Hz 0.5 Hz Bubble volume 0.73ml 0.75 mL 2.5 mL 150 mL 150 mL 150 mL 300 mL 300 mL 300 mL 500 mL 500 mL 500 mL Sparging flow rate [mL/min] 996 2600 9200 996 2600 4300 996 4660 9200 996 8100 13500 Bubble per minute 1330 3600 10830 5.48 15.38 27.15 2.94 14.08 27.33 2.03 15.83 26.9 Interaction No No yes No No yes No No yes No No yes ra[m] 0.008 ?0.001 0.008 ?0.003 0.0125 ?0.001 0.050 ?0.004 0.041 ?0.006 0.042 ?0.004 0.058 ?0.003 0.051 ?0.006 0.053 ?0.006 0.092 ?0.008 0.074 ?0.001 0.071 ?0.057 de [m] 0.011 0.015 0.023 0.066 0.066 0.066 0.083 0.083 0.083 0.098 0.098 0.098 Vb [m/s] 0.52 ?0.02 0.61 ?0.02 0.77 ?0.05 0.61 ?0.01 0.68 ?0.01 0.78 ?0.02 0.59 ?0.01 0.77 ?0.01 0.87 ?0.02 0.69 ?0.01 0.82 ?0.02 1.02 ?0.03 Fr [-] 1.8 2.6 2.6 1.07 1.19 1.37 0.92 1.20 1.36 0.99 1.18 1.47 Reb [-] 7700 9090 12100 37200 41700 47700 45300 59500 67200 62900 74800 93000 Rec [-] 3200 N/A N/A 33583 N/A N/A 47447 N/A N/A 60764 N/A N/A Eo [-] 28 34 38 583 583 583 926 926 926 1300 1300 1300 Notes: ? corresponds to the standard error of repeated measurements; ra: radius of curvature; de: equivalent diameter; Vb: bubble rise velocity; Reb: bubble Reynolds number; Rec: corrected Reynolds number; E0: E?tv?s number; for course bubble sparging, nominal frequency of Discrete, 0.25 Hz and 0.5 Hz correspond to 996, 2600, and 9200 mL/min sparging flow rates, respectively31   The bubble rise velocities for the bubble sizes and frequencies investigated in the present study are summarized in Figure 3-2. As expected, the bubble rise velocity increased with size and frequency of sparged bubbles [75]. In a bubble swarm, i.e. at gas sparging frequencies of 0.25 and 0.5 Hz, the trailing bubbles may accelerate due to the interaction with the wake of the preceding bubble [78]. This interaction may also result in the rupture of the successive bubbles [78].    Figure 3-2 Bubble rise velocity for bubble size and frequencies investigated  (Error bars correspond to the standard error of repeated measurements, for course bubble sparging, discrete, 0.25 Hz and 0.5 Hz frequencies correspond to 996, 2600, and 9200 mL/min sparging flow rates, respectively)32   3.2 General behavior of sparged bubbles investigated   Coarse bubble sparging 3.2.1 Bubbles generated with the coarse sparger at the lowest sparging frequency (corresponding to discrete) ascended as individual bubbles on a vertical path in the center of the system tank where the spargers were installed (Figure 3-3a). Bubbles sparged with the coarse sparger at an intermediate flow (corresponding to a frequency of 0.25 Hz) ascended on a vertical path in the center of the system tank (Figure 3-3b); however, the successive bubbles interacted with each other. The same trend was observed for bubbles sparged with the coarse sparger at high flow (corresponding to a frequency of 0.5 Hz) (Figure 3-3c). At the highest sparging flow, the number of sparged bubbles was high, and as a result, successive bubbles were generally observed to coalesce and form larger bubbles (Table 3.1).    33    a   b  c Figure 3-3 Typical images of bubbles generated by coarse bubble sparging (a: Discrete; b: 0.25 Hz; and c: 0.5 Hz , For course bubble sparging, discrete, 0.25 Hz and 0.5 Hz frequencies correspond to 996, 2600, and 9200 mL/min sparging flow rates, respectively)    Small pulse bubble sparging 3.2.2 Small pulse bubbles, i.e. 150 mL, generated with the pulse bubble sparger at the lowest sparging frequency, i.e. discrete, were wobbling spherical cap bubbles which ascended on a vertical path with slight wobbling in the center of the system tank (Figure 3-4a).  Satellite bubbles were generally observed at the edges of the rising pulse bubbles, which was consistent with observations by others [76]. Bubbles sparged at the higher frequency of 0.25 Hz were also wobbling spherical cap bubbles, but unlike those observed at the lower frequencies, they ascended following a zigzag path in the system tank (Figure 34  3-4b). The same trend was observed for bubbles sparged at the highest sparging frequency, i.e. 0.5 Hz, however, breakage of large bubbles into small bubbles was periodically observed at the sparging frequency of 0.5 Hz. In a bubble swarm, i.e. at a gas sparging frequency of 0.5 Hz, the trailing bubble may accelerate due to the interaction with the wake of the preceding bubble [78]. This interaction may also result in the rupture of the successive bubbles [78].    a b  c Figure 3-4 Typical images of bubbles generated by small (150 mL) pulse sparging  (a: Discrete; b: 0.25 Hz; and c: 0.5 Hz )       35    Medium pulse bubble sparging 3.2.3 Medium pulse bubbles, i.e. 300 mL, generated with the pulse bubble sparger at the lowest sparging frequency, i.e. discrete, ascended following a zigzag path in the system tank (Figure 3-5a). A zigzag path for the bubbles with this size and Reynolds number was expected due to the vortex shedding of their wakes [75]. Similar to the small pulse bubbles, medium pulse bubbles were wobbling spherical cap bubbles and satellite bubbles were generally observed at the edges of the rising pulse bubbles. However, a larger number of satellite bubbles followed the medium pulse bubbles, i.e. 300 mL, in comparison to the number of satellite bubbles observed following small pulse bubbles (150 mL). Similar behaviours were observed for medium pulse bubbles sparged at higher sparging frequencies of 0.25 Hz (Figure 3-5b) and 0.5 Hz (Figure 3-5c). However, the breakage of large bubbles into small bubbles was observed more frequently at the sparging frequency of 0.5 Hz. In a bubble swarm, i.e. at gas sparging frequency of 0.5 Hz, the trailing bubble may accelerate due to the interaction with the wake of the preceding bubble [78]. This interaction may also result in the rupture of the successive bubbles [78].      36    a b  c Figure 3-5 Typical images of bubbles generated by medium (300 mL) pulse (a: Discrete; b: 0.25 Hz; and c: 0.5 Hz)    Large pulse bubble sparging 3.2.4 Similar trends to those observed for medium pulse bubbles were observed for large pulse bubbles, i.e. 500 mL. Bubbles generated with the pulse bubble sparger at the lowest sparging frequency, i.e. discrete, were wobbling spherical cap bubbles which ascended on a zigzag path in the system tank (Figure 3-6a). Satellite bubbles were generally observed at the edges of the rising pulse bubbles. As the sparging frequency was increased, the width of the zigzag path of the bubbles increased and a larger number of satellite bubbles followed the pulse bubbles (Figure 3-6b and Figure 3-6c). A zigzag path for the bubbles with this size and Reynolds number was also expected due to the vortex shedding of their wakes [75]. A larger number of satellite bubbles followed the large pulse bubbles, i.e 500 mL, in comparison to the number of satellite bubbles following the small and medium pulse bubbles. The breakage 37  of large bubbles into small bubbles was observed more frequently than for sparging at 150 and 300 mL. This could be explained because the largest stable bubble diameter in water is predicted to be about 0.049 cm. At larger diameter bubbles breakage will be observed [75].  In addition, in a bubble swarm, i.e. at gas sparging frequency of 0.5 Hz, the trailing bubble may accelerate due to the interaction with the wake of the preceding bubble [78]. This interaction may also result in the rupture of the successive bubbles [78].    a b  c Figure 3-6 Typical images of bubbles generated by large (500 mL) pulse sparging  (a: Discrete; b: 0.25 Hz; and c: 0.5 Hz)        38  3.3 Conclusion  Characteristics of the sparged bubbles, geometrics shapes and behaviour, were identified both qualitatively and quantitatively. The small bubbles generated by coarse bubble sparging were observed to behave predominantly as wobbling spherical bubbles. Their geometric shapes and behaviour were consistent with the work done in the literature. Large bubbles generated by pulse sparging behaved as spherical cap bubbles. Limited data exist about the geometric shapes and behavior of large bubbels with the volumes studied in the present study. The results indicated that bubbles generated at the discrete sparging frequency ascended on a vertical path in the center of the system tank. However, as the sparging frequency was increased, the interactions between successive bubbles caused them to wobble and move on a zigzag path. A zigzag path for the bubbles in this range of size and Reynolds number was expected due to the vortex shedding of their wakes. The width of the zigzag path increased with sparging frequency. This could be explained by the effect of interaction of the succeeding bubbles with the wake of the preceding bubbles at higher sparging frequencies. Bubble break up was observed when sparging with the pulse sparger at the sparging frequencies of 0.25 Hz and 0.5 Hz. The breakage of large bubbles into small bubbles was observed more frequently at the sparging frequency of 0.5 Hz. In a bubble swarm, i.e. at gas sparging frequency of 0.5 Hz, the successive bubble may accelerate due to the interaction with the wake of the preceding bubble. This interaction may also result in the rupture of successive bubbles. The breakage of large bubbles into small bubbles was also observed more frequently as the size of bubbles increased from 150 mL to 500 mL.  Breakage of the bubbles could also be the result of the interaction of bubbles with the fibers installed in the system. The effect of these characterisitcs is investigated as discussed in Chapters 4 to 7.   39  4 Characterisation of the hydrodynamic conditions induced by sparged bubbles As discussed in Chapter 1, secondary oscillating flows induced in the wake of the sparged bubbles have been suggested to contribute to the fouling control [6, 20, 21, 33, 34]. These secondary flows result in high velocities and vorticities as well as highly variable shear stress of high magnitude, which prevents the accumulation of retained material on membrane surfaces. The rate of fouling in air-sparged submerged membranes, i.e. unconfined systems, has been reported to be related to the local liquid velocity and vorticity induced by the sparged bubbles near the membrane surface [16, 25, 58, 62, 79, 80]. In addition, the RMS of the shear stress induced onto membranes by the secondary flows has been reported to be the parameter that is most correlated to the rate of fouling in submerged hollow fiber membrane systems [49, 62].  However, the effects of bubble size and sparging frequency on the characteristics of the secondary flows and the resulting shear stress induced at the membrane surface have not been yet been comprehensively investigated. The effects of bubble size and frequency on the distribution of liquid velocity and vorticity as well as on the distribution of the shear stress induced onto membranes for the sparged bubbles conditions described in Chapter 3 are presented in the sections which follow.  4.1 Distribution of vorticity and velocity for discrete sparging  The secondary flows induced by sparged bubbles in the system can be characterized based on the distribution of the local liquid velocity and vorticity in the system. PIV was used to quantify the liquid velocity and the vorticity over time in the system tank. Figures Figure 4-1a and b present typical 2-dimensional vorticity distributions in the system for the discrete sparging for coarse and pulse sparging, respectively.  Qualitatively, it can be observed that the fraction of the system tank with high vorticity, for a pulse bubble was much larger than that for the coarse bubbles even though the volume of gas delivered to the system for both sparging conditions was similar.  40     a  b  c  d Figure 4-1 Typical vorticity and velocity distributions induced by discrete rising bubbles (a: distribution of vorticity for multiple coarse bubble (in 1/s); b:  distribution of vorticity for a single pulse bubble (150 mL bubble) (in 1/s); distributions based on vorticity measured at a fixed horizontal axis at probe location over time; c: distribution of velocity at vertical centerline of tank for multiple coarse bubble [insert illustrates vertical distribution for a single coarse bubble]; d: distribution of velocity at vertical centerline for single pulse bubble)  41    Vertical distribution of velocity for discrete bubble sparging 4.1.1 The vertical distributions of the velocity induced by coarse bubbles were characterized by a rapid rise in the velocity followed by a rapid decrease in the wake trailing the bubbles (Figure 4-1c). The vertical distributions of the velocity induced by pulse bubbles were also characterized by a rapid rise in the velocities; however, this was followed by a gradual decrease in the wake trailing the bubble (Figure 4-1d). A similar trend was observed by Bhaga and Weber [81] when investigating the velocity distribution in wakes of small bubbles. When comparing the velocity (and vorticity) measurements collected, for different conditions investigated, to the images collected for the same conditions (Chapter 3), the following observations could be made.   The magnitude of the liquid velocity (and vorticity) increased rapidly to a maximum velocity (and vorticity) observed at the tail end of the bubble (between 3 and 4 seconds on Figure 4-1 d). The magnitude of the liquid velocity (and vorticity) gradually decreases from the tail end of the bubble to the bottom edge of the wake, i.e secondary flows, behind the bubble where it reached the magnitude of the velocity (and vorticity) of the background bulk liquid flow. These observations can be explained by the fact that the wake behind a rising bubble is known to be in the form of power functions, with the maximum magnitude of the velocity at the tail end of the bubble and a gradual decrease to the bottom edge of the wake where the magnitude of the velocity in the wake is equal to that of the bulk liquid fluid [82, 83].   As illustrated in Figure 4-1d, the magnitude and the duration of the peaks observed in the area of the zone of influence were a function of the pulse bubble size; the larger the pulse bubble, the higher the magnitude of the velocity and the longer the duration of the peak.  This was expected because the magnitude of the maximum velocity, i.e peaks, and the dimension of the wake generally increase with the bubbles size [82]. As the pulse bubble size increased, the bubble rise velocity increased and as a result, the maximum velocity in the wake and the dimension of the wake increased [82].    42   Horizontal distribution of velocity for discrete bubble sparging 4.1.2 The horizontal distributions of the velocity induced by sparging within the system were bell shaped, with a maximum at the vertical centerline along the bubble rise path and a rapid decrease to the side edges of the zone of influence (Figure 4-2). These results are similar to those reported by others when investigating velocity distributions in wakes of small bubbles [85] and are consistent with empirical models developed to describe the  velocity profile in the wake behind a rising bubble [82, 84]. For the horizontal distribution of velocity, the magnitude of maximum velocity increased as the size of the bubbles increased (Figure 4-2b).  These results suggest that within the zone of influence, fouling control is likely to be heterogeneous, with the lowest fouling occurring at the centerline of the system along the bubble rise path, and the extent of fouling increasing towards the edges of the system. The relationship between the hydrodynamic conditions, induced by sparging and fouling control is discussed in Chapter 6.  Negative (downward) velocities were observed at the edges of the system tank. This likely resulted from the upward liquid flow entrained by the bubbles rising at the centre of the system and the resulting downward liquid flow at the edges of the system (the video captures images in the centre and for only 60% of the width of the tank. It is likely that  negative velocities close to the walls of the tank occurred and were not captured by the images presented for the bubbles of 150mL and 300 mL).  43    a b Figure 4-2 Typical horizontal distributions of velocity across the width of system tank (a: coarse bubbles and b: pulse bubbles). 44    Zone of influence 4.1.3 Figure 4-3 to Figure 4-6 illustrate typical changes in liquid velocity and the vorticity over time at the vertical center line of the system tank at the height of the probes (see Figure 2-2a) for different sparging conditions. Figure 4-3 illustrates the distributions of the liquid velocity and vorticity for coarse bubble sparging for the discrete sparging frequency. As presented, the trends in the liquid velocity and the vorticity over time are similar to each other. A similar trend between vorticity and the velocity profile in the wake behind a single rising bubble is expected because vorticity is defined as  ??????????? where vy and vx correspond to the velocity in the y and x directions, respectively.  The peaks observed in both liquid velocity and vorticity profiles correspond to the passage of bubbles at the centerline of the system tank at the height of the probes.  The distributions of the liquid velocity and vorticity over time for small pulse bubbles (150 mL) sparged at the discrete sparging frequency is presented in Figure 4-4 . Again, similar trends are observed for the liquid velocity and vorticity over time. The duration of the periods over which elevated liquid velocities and the vorticities, i.e. velocities and vorticities higher than the those for the bulk, were observed for small pulse bubbles (Figure 4-4) were by an order of magnitude longer than those observed for coarse bubbles (Figure 4-3) and as a result, the fraction of the system with a higher magnitude of velocity and vorticity induced by pulse bubble was much larger than that induced by the coarse bubbles.  Similar trends to those observed for the small pulse bubbles were observed for the larger pulse bubbles with respect to the liquid velocity and vorticity over time (Figure 4-5 and Figure 4-6). The magnitude and the duration of the peaks observed in the liquid velocity and the vorticity profiles increased with pulse bubble size, as expected. Since the liquid velocity and the vorticity profiles exhibit the same trend in terms of magnitude and variation, the liquid velocity was selected in the present study to furher investigate the effect of bubble characteristics, i.e. bubble size and frequency, on the induced hydrodynamic conditions.   45          a  b Figure 4-3 Typical distribution of a) velocity and b) vorticity for discrete bubble sparging with the coarse bubble sparger (Measurements were made at the vertical centerline of the system at the height of the probes)    46   a   b Figure 4-4 Typical distribution of a) velocity and b) vorticity for discrete bubble sparging with the small pulse bubble sparger (Measurements were made at the vertical centerline of the system at the height of the probes)    47    a   b Figure 4-5 Typical distribution of a) velocity and b) vorticity for discrete bubble sparging with the medium pulse bubble sparger (Measurements were made at the vertical centerline of the system at the height of the probes)   48    a   b Figure 4-6 Typical distribution of velocity a), and vorticity d) for discrete bubble sparging with the large pulse bubble sparger (Measurements were made at the vertical centerline of the system at the height of the probes) 49   The distributions of velocity and vorticity in the system delineate a zone of secondary flows over which liquid velocities and vorticities are high. The size of the zone of influence for different sparging conditions was defined as the area orthogonal to the imaging plane (Az) where sparged bubbles induce secondary flows with a velocity greater than 0.2 m/s. The cut-off velocity of 0.2 m/s was selected because it corresponded to that of the background liquid movement induced by the rising bubble and did not consistently generate vorticities that were greater than those associated with background bulk liquid movement. Selection of 0.2 m/s as the cut off velocity was also confirmed by statistical analyses of the effect of cut off velocity on the area of zone of influence and its correlation to rate of fouling (See Appendix G). The width, length and area of the zone of influence for different sparging conditions investigated are summarized in Table 4.1. When bubbles interacted, the length of the zone of influence, (Lz), was defined as the distance between two successive bubbles (Table 4.1). 50  Table 4-1 General characteristics of studied bubbles and the induced zone of influence Sparger type Coarse Small pulse Medium pulse Large pulse Nominal sparging frequency Discrete 0.25 Hz 0.5 Hz Discrete 0.25 Hz 0.5 Hz Discrete 0.25 Hz 0.5 Hz Discrete 0.25 Hz 0.5 Hz Bubble volume 0.75ml 0.73 mL 2.5 mL 150 mL 150 mL 150 mL 300 mL 300 mL 300 mL 500 mL 500 mL 500 mL Sparging flow rate [mL/min] 996 2600 9200 996 2600 4300 996 4700 9200 996 8100 13515 Interaction No Yes Yes No No/minimal Minimal No No/minimal Yes No No/minimal Yes Az [m2] 0.0024 ?0.003 5.23 ?0.077 17.3 ?0.105 0.178 ?0.004 12.42 ?0.066 15.89 ?0.083 0.463 ?0.104 14.47 ?0.0799 24.94 ?0.169 0.825 ?0.003 18.95 ?0.081 30.61 ?0.097 Az,bubble/Ab [-] 22.3 14 8 71 338 179 123 165 246 131 72 145 Az/Sparging volume[1/m] 2.41 2011 1880 178 4776 3695 464 3078 2710 828 2339 2264 Wz [m] 0.01 0.14 0.37 0.17 0.303 0.34 0.24 0.311 0.47 0.3 0.385 0.5 Lz [m] 0.24 0.015 0.012 1.04 2.66 1.72 1.95 3.3 1.92 2.77 3.1 2.27 Notes: ? corresponds to the standard error of repeated measurements; Az: system average area of zone of influence; Az,bubble: area of zone of influence per bubble; Ab: bubble area; Wz: width of zone of influence, Lz:  length of zone f influence; for course bubble sparging, nominal frequency of Discrete, 0.25 Hz and 0.5 Hz correspond to 996, 2600, and 9200 mL/min sparging flow rates respectively. 51  A non-dimensional scaling was used to compare the results from the present study to those reported by the others. The ratio of Az,bubble/Ab was defined as the ?dimensionless area? of the zone of influence, where Az,bubble is the area of the zone of influence per bubble, and Ab is the area of the bubble itself, orthogonal to the imaging plane. The dimensionless area of the zone of influence was substantially affected by the size and frequency of bubbles (Figure 4-7).  For coarse bubbles, the dimensionless area of the zone of influence was approximately 20 (Figure 4-7 and Table 4.1), which is consistent with results reported by Komasawa et al. [78]. However, Komasawa et al. suggested that beyond a given Re, the dimensionless area of the zone of influence remains constant, which is not consistent with the observations from the present study (Figure 4-7). For pulse bubbles the dimensionless area of the zone of influence ranged from 71 to 131 for bubble sizes investigated (Table 4.1). Unfortunately, limited quantitative published data exist on the dimensionless area of the zone of influence for large bubbles at high Reynolds numbers. Dimensions of the zone of influence for small single rising bubbles can be estimated using empirical models that describe the velocity profile in the wake of the bubbles and which can be evaluated computationally using CFD [75]. However, the spherical cap bubbles investigated in the present study had large diameters, high Reynolds numbers, and a wobbling behavior. Extensive CFD analysis would be required to estimate the dimension of their zone of influence [84, 86, 87], which is beyond the scope of the present research.  The results from the present study indicate that the size of zone of influence of secondary flows induced by a pulse bubble can be as much as an order of magnitude larger than that induced by a coarse bubble, suggesting that for a given volume of sparged gas added to a system, pulse bubbles could be more effective in control of fouling rate in submerged membrane systems. The effect of the induced hydrodynamic conditions on the fouling rate is presented in Chapter 6.     52   Figure 4-7 Dimensionless area of zone of influence for discrete bubbles  (Open shapes: experimental results from present study, lines and solid squares adapted from Komasawa et al. [78] )   53   Vertical distribution of shear stress for discrete bubble sparging 4.1.4 Figure 4-8 illustrates the vertical distribution of the shear stress measured at probe 1, located in the middle of the system tank (See Figure 2-2). The vertical distribution of the shear stress was characterized by a rapid rise from the nose of the bubble to the wake area immediately downstream of the tail of the bubble, followed by a gradual decrease to the bottom edge of the zone of influence. These observations were expected because the liquid velocity and kinetic energy are at their highest magnitudes in the wake immediately behind the bubble and decrease gradually to the bottom edge of the zone of influence [82, 84].   The magnitude and duration of the peaks increased with the size of pulse bubbles (Figure 4-8b).  This was expected because as described in Section 4.1.1, the rise velocity of the bubbles, and as a result, the liquid velocity (See figures 4.5 to 4.6) and the kinetic energy in the wake immediately behind the bubble, increase with the size of sparged bubbles [83]. The higher kinetic energy in the wake of the larger pulse bubbles dissipates over a longer time (or distance), resulting in longer duration of the peaks observed in the shear stress profiles measured for larger pulse bubbles. 54         a b Figure 4-8 Typical shear stress distribution induced by discrete rising bubbles at vertical centreline of the tank  (a: for multiple coarse bubbles [ insert illustrates distribution for a single bubble ]; b: for a single pulse bubble) 55   Horizontal distribution of shear stress for discrete bubble sparging 4.1.5 In order to characterize the horizontal distribution of shear stress in the system tank, shear stresses induced onto membranes by sparged bubbles were measured at 4 locations as illustrated in Figure 2-2.  Probe 1 was installed on the fiber positioned in the center of the system tank, and probes 2, 3 and 4 were installed 7, 14 and 21cm from the centerline, respectively (Figure 2-2). As described in Section 2, four shear probes were installed on one side of the system tank (Figure 2-2). It was assumed that the shear stress in the tank is symmetrical, and therefore, the magnitudes of the shear stress presented for the other side of the system tank are a mirrored image from the magnitude of shear stress measured. Of the different summative parameters that have been used to express time variable shear stress induced by gas sparging, the root mean square (RMS) of shear stress has been reported to be most correlated to the rate of fouling in submerged hollow fiber membrane systems [49, 62]. For this reason, RMS of the shear stress measured at each probe was used in the analysis which follows.  The horizontal distribution of RMS of the shear stress is compared in Figure 4-9 for discrete sparging and for coarse and pulse bubble spargers. The horizontal distributions of shear stress in the system were bell shaped, with a maximum at the vertical centerline of the system along the bubble rise path and a rapid decrease to the side edges of the system (Figure 4-9). This was expected because, as described in Section 4.1.2, the velocity profile in the zone of influence of a rising bubble exhibited a bell shape with the highest magnitude of velocity (and kinetic energy) in the center of the zone of influence. The higher liquid velocities (and kinetic energy) induced in the zone of influence of larger bubbles resulted in an increase in the magnitude of the shear stress with the size of pulse bubbles. The above results suggests that fouling control over the width of the tank is likely to be non-homogeneous, with the lowest fouling occurring at the centerline of the zone of influence, the extent of fouling increasing towards the edges of the zone of influence, and fouling being highest outside the zone of influence. This is investigated further in Chapter 6.  56      a b  Figure 4-9 Horizontal distributions of shear stress across the width of system tank for discrete sparging frequency (a: coarse bubbles and b: pulse bubbles; For course bubble sparging, discrete corresponds to 996 mL/min sparging flow rate)57   4.2 Effect of sparging frequency on the distribution of velocity, vorticity and shear stress   The information presented in Section 4.2 is qualitative.  The figures presented in this section are important because they provide two dimensional maps for vorticity, and illustrate how the magnitude of velocity and shear stress changes with time when a bubble 1) approaches the probe, 2) is on contact with the probe or 3) passes by the probes. Using these figures, qualitative comparison of the maps of vorticity with velocity and shear stress profiles and for different sparging conditions is also possible.  A quantitative analysis of the data extracted from section 4.2 is presented in Section 4.3.  Effect of sparging frequency on the vertical distribution of vorticity, velocity and 4.2.1shear stress   Figure 4-10 to Figure 4-12 illustrate the vertical distribution of maximum velocity and maximum shear stress in the system for coarse bubble sparging at discrete, 0.25 Hz and 0.5 Hz sparging frequencies. For comparison purposes, the 2-D distribution of vorticity is also presented. Qualitatively, it can be observed that the area of zone of influence, i.e. the area over which high velocities and high vorticies were induced by sparged bubbles, increased with the sparging frequency (Figure 4-10a, Figure 4-11a, and Figure 4-12a). The length of the zone of influence for pulse bubbles at the discrete sparging frequency was delineated based on the threshold of 0.2 m/s for the local velocity, as defined in Section 4.1.3. A peak in the velocity and shear stress could be observed every time a bubble rises through the system. As a result, the frequency of the peaks increased with the sparging frequency. The magnitude of the peaks increased with the increase in the sparing frequency (Figure 4-10 to Figure 4-12). Also, the magnitude of the baseline in the profiles increased with an increase in the sparging frequency. This was likely due to the higher bulk liquid velocity in the system at higher sparging frequencies. The quantitative analysis of the data is presented in Section 4.3.   58     A   B   c Figure 4-10 Typical distribution of hydrodynamic conditions induced by coarse bubble sparging at discrete sparging frequency  (a: distribution of vorticity for multiple coarse bubbles (in 1/s); distributions based on vorticities measured at a fixed horizontal axis at probe location over time; b: velocity distribution at vertical centerline of tank; c: shear stress distribution at vertical centerline of tank; for course bubble sparging, discrete corresponds to 996 mL/min sparging flow rate)59     a   b   c Figure 4-11 Typical distribution of hydrodynamic conditions induced by coarse bubble sparging at 0.25 Hz sparging frequency  (a: distribution of vorticity fro multiple coarse bubbles (in 1/s); distributions based on vorticities measured at a fixed horizontal axis at probe location over time; b: velocity distribution at vertical centerline of tank; c: shear stress distribution at vertical centerline of tank; for course bubble sparging, nominal frequency of 0.25 Hz corresponds to 2600 mL/min sparging flow rate)60     a   b   c Figure 4-12 Typical distribution of hydrodynamic conditions induced by coarse bubble sparging at 0.5 Hz sparging frequency  (a: distribution of vorticity for multiple coarse bubbles (in 1/s); distributions based on vorticities measured at a fixed horizontal axis at probe location over time; b: velocity distribution at vertical centerline of tank; c: shear stress distribution at vertical centerline of tank; for course bubble sparging, nominal frequency of 0.5 Hz corresponds 9200 mL/min sparging flow rate)61   Figure 4-13 to Figure 4-15 illustrate the distribution of vorticity, as well as the vertical distribution of maximum velocity and maximum shear stress in the system for small pulse bubble sparging at discrete, 0.25 Hz and 0.5Hz sparging frequencies. Qualitatively, it can be observed that the area of the zone of influence, i.e. where high vorticies were induced by sparged bubbles, increased with the sparging frequency (Figure 4-13a, Figure 4-14a, and Figure 4-15a). The distance between successive bubbles at discrete sparging was much larger than their length of zone of influence. Therefore, it was confirmed that the zones of influence of successive pulse bubbles at discrete sparging did not interact.  The vertical distribution of maximum velocity at the sparging frequency of 0.25 Hz is illustrated in Figure 4-14b.  The same threshold of 0.2 m/s as for the pulse bubbles at discrete sparging (defined in Section 4.1.3) was applied to identify the zone of influence. At this sparging frequency, the distance between successive bubbles was smaller than the distance between the successive bubbles at discrete sparging. However, the distance between successive bubbles was larger than their length of zone of influence (Table 4.1). Therefore, it was confirmed that no/minimal interaction occured between successive pulse bubbles at sparging frequency of 0.25 Hz.   The vertical distribution of velocity at the sparging frequency of 0.5 Hz is illustrated in Figure 4-15b. Again, the threshold of 0.2 m/s was applied to obtain the length of the zone of influence. At the highest sparging frequency of 0.5Hz, the distance between successive pulse bubbles was larger than the length of their zones of influence, but the difference was very small (Table 4.1). Therefore, it was confirmed that the zone of influence of successive bubbles could interact at the sparging frequency of 0.5Hz (Figure 4-15b). At the discrete sparging, bubbles ascended on a relatively vertical path in the center of the system tank, i.e. where the sparger was installed (Figure 4-13). As a result, maximum velocity and shear stress was measured at the centreline of the system tank. However, at higher sparging frequencies (Figure 4-14 and Figure 4-15) bubbles moved on a zigzag path and therefore, the maximum velocity and shear stress were measured on the path of the bubbles. This was consistent with observations of bubble behaviour under these conditions as discussed in section 3.2.2. 62  A peak in the velocity and shear stress could be observed every time a bubble rrose through the system. As a result, the frequency of the peaks increased with the sparging frequency. The magnitude of the peaks increased with the increase in the sparing frequency (Figure 4-13 to Figure 4-15). Also, the magnitude of the baseline in the profiles increased with an increase in the sparging frequency. This was likely due to the higher bulk liquid velocity in the system at higher sparging frequencies (See Section 4.3).  As discussed in Sections 4.1 and 4.2, the vertical distributions of the velocity and shear stress within the zone of influence were characterized by a rapid rise from the nose of the bubble to the wake area immediately downstream of the tail of the bubble, followed by a gradual decrease to the bottom edge of the zone of influence (Figure 4-13b and c). At 0.25 Hz, the vertical velocity and the shear stress also were characterized by a rapid rise from the nose of the bubble to the wake area immediately downstream of the tail of the bubbles, the velocity and shear stress gradually decreased from the tail end of the bubble to the bottom edge of the zone of influence (Figure 4-14b and c). However, at 0.5 Hz, no gradual decrease was observed in the distribution of vertical velocity and shear stress over time. Because bubbles are rising closely at 0.5 Hz, the gradual decrease is interrupted by the velocity and shear stress induced by the trailing bubble.  At higher sparging frequencies of 0.25 Hz and 0.5 Hz (Figure 4-14 and Figure 4-15), the trend observed in the vertical distribution of the shear stress periodically differed from the trend observed in the vertical distribution of velocity due to the fiber sway. If the fibers sway, this may induce additional shear stress onto the fibers, which will result in the measurements of higher magnitudes of shear stress than expected. It may also cause the fibers to move in the tank in three dimensions and this could move them out of the zone of influence trailing the bubbles. This would result in the measurements of lower magnitudes of shear stress than expected.     63    a   b   c Figure 4-13 Typical distribution of hydrodynamic conditions induced by small (150 mL) pulse bubble sparging at discrete sparging frequency  (a: distribution of vorticity for pulse bubbles (in 1/s); distributions based on vorticities measured at a fixed horizontal axis at probe location over time; b: velocity distribution at vertical centerline of tank; c: shear stress distribution at vertical centerline of tank)   64    a    b   c Figure 4-14 Typical distribution of hydrodynamic conditions induced by small (150 mL) pulse bubble sparging at 0.25 Hz sparging frequency  (a: distribution of vorticity for pulse bubbles (in 1/s); distributions based on vorticities measured at centerline of tank; b: velocity distribution at vertical centerline of tank; c: shear stress distribution at vertical centerline of tank)   65    a   b   c Figure 4-15 Typical distribution of hydrodynamic conditions induced by small (150 mL) pulse bubble sparging at 0.5 Hz sparging frequency  (a: distribution of vorticity for pulse bubbles (in 1/s); distributions based on vorticites measured at a fixed horizontal axis at probe location over time; b: velocity distribution at vertical centerline of tank; c: shear stress distribution at vertical centerline of tank)   66  The same trend was observed for medium and large pulse bubbles as for small pulse bubbles; qualitatively, it can be observed that the area of the zone of influence, i.e. where high vorticies were induced by sparged bubbles, increased with the sparging frequency (Figure 4-16a to Figure 4-21a).  The distance between successive bubbles at discrete sparging was much larger than the length of their zone of influence. Therefore, it was confirmed that the zone of influence of successive pulse bubbles at discrete sparging did not interact.   At a sparging frequency of 0.25 Hz, the distance between successive bubbles was smaller than the distance between the successive bubbles with discrete sparging. However, the distance between successive bubbles was larger than the length of their zone of influence (Table 4.1). Therefore, it was confirmed that no/minimal interaction existed between successive pulse bubbles at sparging frequency of 0.25 Hz.    At the highest sparging frequency of 0.5 Hz (Figure 4-18b and Figure 4-21b), the distance between successive pulse bubbles was smaller than the length of the zone of influence but the difference was very small (Table 4.1). Therefore, it was confirmed that the zone of influence of successive bubbles could interact at the sparging frequency of 0.5Hz .At the discrete sparging, bubbles ascended on a vertical path in the center of the system tank where the sparger was installed (Figure 4-16 and Figure 4-19). However, at higher sparging frequencies (Figure 4-17, Figure 4-18, Figure 4-20 and Figure 4-21) bubbles moved on a zigzag path. This was consistent with observations of bubble behaviour under these conditions as discussed in section 3.2.3 and 3.2.4.  A peak in the velocity and shear stress could be observed every time a bubble rose through the system. As a result, the frequency of the peaks increased with the sparging frequency. The magnitude of the peaks increased with the increase in the sparing frequency (Figure 4-16 to Figure 4-21). Also, the magnitude of the baseline in the profiles increased with an increase in the sparging frequency. This was likely due to the higher bulk liquid velocity in the system at higher sparging frequencies.  As discussed in Section 4.1, the vertical distributions of the velocity and shear stress within the zone of influence were characterized by a rapid rise from the nose of the bubble to the wake area immediately downstream of the tail of the bubble, followed by a gradual decrease to the bottom edge of the zone of influence (Figure 4-16 and Figure 4-19). At 0.25 67  Hz, the vertical velocity and the shear stress also gradually decreased from the tail end of the bubble to the bottom edge of the zone of influence (Figure 4-17 and Figure 4-20). However, at 0.5 Hz, no gradual decrease was observed in the distribution of vertical velocity and shear stress over time. Because bubbles were rising in close proximity at 0.5 Hz, the gradual decrease is interrupted by the velocity and shear stress induced by the trailing bubble. At the higher sparging frequencies of 0.25 Hz and 0.5 Hz (Figure 4-17 and Figure 4-18), the trend observed in the vertical distribution of the shear stress periodically differed from the trend observed in the vertical distribution of velocity due to the fiber sway. If the fibers sway, this may induce additional shear stress onto the fibers, which will result in the measurements of higher magnitudes of shear stress than expected. It may also cause the fibers to move in the tank in three dimensions such that they could move out of the zone of influence trailing the bubbles. This would result in the measurements of lower magnitudes of shear stress than expected.     68    a   b   c Figure 4-16 Typical distribution of hydrodynamic conditions induced by medium (300 mL) pulse bubble sparging at discrete sparging frequency  (a: distribution of vorticity for pulse bubbles (in 1/s); distributions based on vorticities measured at a fixed horizontal axis at probe location over time; b: velocity distribution at vertical centerline of tank; c: shear stress distribution at vertical centerline of tank)   69    a   b   c Figure 4-17 Typical distribution of hydrodynamic conditions induced by medium (300 mL) pulse bubble sparging at 0.25 Hz sparging frequency  (a: distribution of vorticity for pulse bubbles (in 1/s); distributions based on vorticities measured at a fixed horizontal axis at probe location over time; b: velocity distribution at vertical centerline of tank; c: shear stress distribution at vertical centerline of tank)   70    a   b   c Figure 4-18 Typical distribution of hydrodynamic conditions induced by medium (300 mL) pulse bubble sparging at 0. 5 Hz sparging frequency  (a: distribution of vorticity for pulse bubbles (in 1/s);  distributions based on vorticities measured at a fixed horizontal axis at probe location over time; b: velocity distribution at vertical centerline of tank; c: shear stress distribution at vertical centerline of tank)   71    a   b   c Figure 4-19 Typical distribution of hydrodynamic conditions induced by large (500 mL) pulse bubble sparging at discrete sparging frequency  (a: distribution of vorticity for pulse bubbles (in 1/s); distributions based on vorticities measured at a fixed horizontal axis at probe location over time; b: velocity distribution at vertical centerline of tank; c: shear stress distribution at vertical centerline of tank)   72    a   b   c Figure 4-20Typical distribution of hydrodynamic conditions induced by large (500 mL) pulse bubble sparging at 0.25 Hz sparging frequency  (a: distribution of vorticity for pulse bubbles (in 1/s); distributions based on vorticities measured at a fixed horizontal axis at probe location over time; b: velocity distribution at vertical centerline of tank; c: shear stress distribution at vertical centerline of tank)    73    a   b  c Figure 4-21 Typical distribution of hydrodynamic conditions induced by large (500 mL) pulse bubble sparging at 0.5 Hz sparging frequency  (a: distribution of vorticity for pulse bubbles (in 1/s); distributions based on vorticities measured at a fixed horizontal axis at probe location over time; b: velocity distribution at vertical centerline of tank; c: shear stress distribution at vertical centerline of tank)        74   Effect of sparging frequency on the horizontal distribution of velocity and the 4.2.2shear stress  Horizontal distributions of the velocity induced by sparging within the system at higher sparging frequencies are illustrated in Figure 4-22. The magnitudes of velocity were generally higher within the zone of influence induced by sparged bubbles and gradually decreased to the sides. The magnitude of velocity increased with an increase in the sparging frequency. In addition, as discussed in Section 4.1, the area of zone of influence increased with the sparging frequencies. With an increase in the area of the zone of influence, the fraction of the system covered with secondary flows increased and therefore, higher velocities were measured over a wider width within the system tank at higher sparging frequencies. These results suggest that within the system, fouling control is likely to be heterogeneous, with the lowest fouling occurring within the zone of influence, the extent of fouling increasing towards the edges of the zone of influence and the fouling being highest outside the zone of influence. The relationship between the hydrodynamic conditions, induced by sparging and fouling control is discussed in Chapter 6. Negative (downward) velocities were observed at the edges of the system. This likely resulted from the upward liquid flow entrained by the bubbles rising at the center of the system and the resulting downward liquid flow at the edges of the system.  The velocity distribution presented in Figure 4-22 is measured at the tail end of the bubbles. At higher sparging frequencies of 0.25 Hz and 0.5 Hz, pulse bubbles ascended on a zigzag path, as described in Chapter 3. As a result, the maximum velocity (peaks) is not observed at the centerline of the system consistently (Figure 4-22). Rather, the maximum velocity was observed at the centerline of the zone of influence trailing the sparged bubbles. 75   a  b Figure 4-22 Typical horizontal distributions of velocity across the width of system tank (a: 0.25 Hz and b: 0.5 Hz frequencies)    76    In order to investigate the effect of sparging frequency on the horizontal distribution of the shear stress within the system, shear stress was measured at 4 probe locations, as described in Section 4.1.5, at discrete, 0.25 Hz and 0.5 Hz sparging frequencies. Shear stress induced onto the fibers for sparging with the coarse sparger at discrete, 0.25 and 0.5 Hz frequencies is illustrated in Figure 4.23 to Figure 4.25.  A peak in the shear stress was observed every time a bubble rose through the system. As a result, the frequency of the peaks increases with the sparging frequencies. The magnitude of shear stress measured at probe location 1 (located in the centerline of the system tank), was consistently higher in comparison to the magnitude of shear stress measured at probe locations 2, 3 and 4 (located further to the side of the system tank). As a result, the horizontal distribution of the shear stress was not homogenous. This was expected considering that bubbles sparged under these conditions were rising along the centerline of the system tank, as previously discussed in section 3.2.1. A bubble rising along the centerline of the system tank induced higher velocity (and kinetic energy) at the centre line of the system tank in comparison to the sides of the system tank.  The magnitude of the shear stress generally increased with the increase in the sparging frequency (Figure 4-24 and Figure 4-25). This was expected because, as discussed above, at higher sparging frequencies bubbles induced higher velocities (Figure 4-22) and higher kinetic energy and therefore, they were generally expected to induce higher shear stress on to the membranes.   77    a   b    c   d Figure 4-23 Typical vertical distribution of shear stress induced by coarse bubble sparging at different locations for discrete sparging frequency (a: shear stress at position 1; b: shear stress at position 2; c:shear stress at position 3; d: shear stress at position 4; for course bubble sparging, discrete corresponds to 996 mL/min sparging flow rate)78     a   b   c   d  Figure 4-24 Typical vertical distribution of shear stress induced by coarse bubble sparging at different locations for 0.25Hz sparging frequency (a: shear stress at position 1; b: shear stress at position 2; c:shear stress at position 3;d: shear stress at position 4; for course bubble sparging, nominal frequency of 0.25Hz corresponds to 2600 mL/min sparging flow rate)79     a  b   c   d Figure 4-25 Typical vertical distribution of shear stress induced by coarse bubble sparging at different locations for 0. 5Hz sparging frequency  (a: shear stress at position 1; b: shear stress at position 2; c:shear stress at position 3;d: shear stress at position 4; for course bubble sparging, nominal frequency of 0.5 Hz corresponds to 9200 mL/min sparging flow rate)80 The shear stress measured for sparging with small pulse bubble (150 mL) at discrete sparging frequency is presented in Figure 4-26. A peak in the shear stress was observed every time a bubble rose through the system. As a result, the frequency of the peaks increases with the sparging frequencies. Shear stress measured at the probe location 1 (located in the center of the system tank) had the highest magnitude in comparison to the shear stress measured at probe locations 2, 3 and 4 (located further to the side of the system tank). As a result, the horizontal distribution of the shear stress was not homogenous. This was expected considering that bubbles sparged under these conditions were rising along the centerline of the system tank, as previously discussed in section 3.2.2. A bubble rising along the centerline of the system tank induced higher velocity (and kinetic energy) at the centre of the system tank in comparison to the sides of the tank (Figure 4-22).  However, at higher sparging frequencies, the shear stress measured at probe location 1 (located in the center of the system tank) was not consistently higher than the magnitude of shear stress measured at probe locations 2, 3 and 4 (located further to the side of the system tank) as illustrated in Figure 4-27 and Figure 4-28. This was expected because at higher sparging frequencies, sparged bubbles rose along zigzag paths (Section 3.2.2) and therefore, higher magnitudes of the shear stress, i.e. the peaks, were measured at the probes located on the path of the bubbles. For example, as presented in Figure 4-28, the shear stress was greater at probe locations 2 and 3 compared to probe location 1 for the period examined.  The magnitude of the shear stress generally increased with the increase in the sparging frequency (Figure 4-27 and Figure 4-28). This was expected because, as discussed above, at higher sparging frequencies, bubbles induced higher velocities and higher kinetic energy and therefore, they were generally expected to induce higher shear stress on to the membranes (Figure 4-22).  Similar trends to those observed for small pulse bubbles were observed for medium and large pulse bubbles. The typical vertical distributions of shear stress induced by medium and large pulse bubbles at different locations are presented in Appendix F.   81  a   b   c   d Figure 4-26 Typical vertical distribution of shear stress induced by small (150 mL) at different locations for pulse bubble sparging at discrete sparging frequency (a: shear stress at position 1; b: shear stress at position 2; c:shear stress at position 3;d: shear stress at position 4)   82   a   b   c   d Figure 4-27 Typical vertical distribution of shear stress induced by small (150 mL) at different locations for pulse bubble sparging at 0.25Hz sparging frequency (a: shear stress at position 1; b: shear stress at position 2; c:shear stress at position 3;d: shear stress at position 4)   83    a   b   c   d Figure 4-28 Typical vertical distribution of shear stress induced by small (150 ml) at different locations for pulse bubble sparging at 0. 5 Hz sparging frequency (a: shear stress at position 1; b: shear stress at position 2; c:shear stress at position 3;d: shear stress at position 4)   84 To obtain the horizontal distribution of the RMS of the shear stress in the system, the RMS of the shear stress measured at each probe location was plotted versus the width of the system tank for the sparging frequencies of 0.25 Hz and 0.5 Hz (Figure 4-29). The results were mirrored for the left side of the system tank (See Figure 2-2).  The magnitude of maximum RMS of the shear stress was generally higher in the centerline of the system tank and gradually decreased to the edges of the system tank (Figure 4-9). The magnitude of maximum RMS of the shear stress induced onto the fibers increased with an increase in the sparging frequency. This behavior is explained by higher liquid velocities induced at higher sparging frequencies as described above (Figure 4-22).  Higher velocities result in higher kinetic energy in the zone of influence following the bubbles and therefore, induce higher shear stress on to the fibers immersed in their zone of influence [84].   In addition, with an increase in the sparging frequency, the horizontal distribution of shear stress was flattened (Figure 4-29) in comparison to the profile observed at the discrete sparging frequency (Figure 4-9). This observation was expected because, as discussed in Section 4.1 and also illustrated in Figure 4-22, the area of zone of influence increased at higher sparging frequencies. With an increase in the area of the zone of influence, the fraction of the system covered with secondary flows, i.e. higher local velocity and therefore higher kinetic energy, increased. As a result, it was expected that higher shear stress would be observed over a wider width within the system tank. The effect of the horizontal distribution of the shear stress on the fouling control is discussed in Chapter 6. 85   a  b Figure 4-29 Horizontal distributions of shear stress across the width of system tank (a: 0.25 Hz and b: 0.5 Hz frequencies; fibers inside zone of influence are open symbols, while those outside the zone of influence are solid symbols; Error bars corresponds to minimum and maximum measurements as measurements were done in triplicates)  86  4.3 Summary of the hydrodynamic conditions induced by bubbles of different sizes and sparging frequencies  Previous sections presented a semi-qualitative comparison of the hydrodynamic conditions induced by bubbles for the different sizes and frequencies investigated. In the discussions which follows, the hydrodynamic conditions induced by bubbles of different sizes and frequencies are quantitatively compared. The total system area of zone of influence, i.e. summation of the zones of influence for all bubbles in the system at a given time, increased with bubble size and frequency (Figure 4-30a). However, the total system area of zone of influence induced at 0.5 Hz was not two times larger than the total system area of zone of influence at 0.25 Hz, even though the number of bubbles in the system at any given time was greater. At a bubble frequency of 0.5 Hz, the distance between successive bubbles was less than the length of the zones of influence of the bubbles rising discretely (Table 4.1), indicating that the zone of influence of successive bubbles overlapped. As a result, the area of the zone of influence per bubble was smaller (Figure 4-30b). The zone of influence per bubble was calculated based on the total system zone of influence devided by the number of bubbles in the system.   87     a   b Figure 4-30 Area of zone of influence (a: total system area of zone of influence; b: area of zone of influence per bubble; area of zone of influence of coarse bubbles was very small in comparison to the area of zone of influence of pulse bubbles and therefore is not visible on the graph; for course bubble sparging, discrete, 0.25 Hz and 0.5 Hz frequencies correspond to 996, 2600, and 9200 mL/min sparging flow rates respectively; Error bars correspond to the standard errors in the measurements)  88  The increase in the dimensionless area of zone of influence per bubble with bubble size resulted in an increase in the average width of the zone of influence (Wz) (Table 4.1 and Figure 4-31). The average width of the zone of influence for pulse bubbles was much larger than that of coarse bubbles. The average width of the zone of influence also increased with the increase in the sparging frequency. These observations were expected because of the zigzag movement of the pulse bubbles at the higher sparging frequency which resulted in a wider area of zone of influence, as well as the higher rise velocities of the bubbles (Table 4.1).   These results are of significant importance because they suggest that the spacing between spargers can be greater when sparging with larger pulse bubbles. For instance increasing the sparging frequency from 0.25 Hz to 0.5 Hz increases the width of influence by a factor of approximately 1.3 for large pulse bubbles (500 mL), and therefore, the spacing between the spargers could be increased by the same factor, reducing the number of air spargers. Reducing the overall number of air spargers could result in reducing the overall power requirement for air sparging in the system by 30%.   89     Figure 4-31 Average width of zone of influence   (For course bubble sparging, discrete, 0.25 Hz and 0.5 Hz frequencies correspond to 996, 2600, and 9200 mL/min sparging flow rates respectively)90  System wide RMS of the bulk liquid velocity (Figure 4.32) was calculated as the average of RMS of the liquid velocity measured at the probe locations 1 to 4 (See probe locations on Figure 2-2). The RMS of the velocity generally increased with the size of sparged bubbles, although the increase was not statistically significant. The RMS of the velocity also generally increased with sparging frequency for a given bubble size (again, not consistently statistically significant). This increase was likely due to the larger number of bubbles presented in the system at a higher frequency (and therefore more secondary flows in the system).    Figure 4-32 System wide RMS of bulk velocity  (Error bars correspond to the standard errors in the measurements)       91     System wide RMS of the shear stress (Figure 4.33) was calculated as the average of the RMS of the shear stress for all probes. The RMS of the shear stress increased with sparging frequency for a given bubble size (Figure 4-33a). This increase was likely due to the greater number of bubbles in the system at a higher frequency and therefore, more secondary flows in the system, as well as the higher RMS of bulk velocity. The increase in the system wide RMS of shear stress is consistent with the increase in the RMS of bulk velocity (Figure 4-33). However, as presented in Figure 4-33a, the RMS of the system wide shear stress at 0.5 Hz was not twice the RMS of the system wide shear stress at 0.25 Hz. As a result of the bubble interactions, the area of the zone of influence per bubble decreased (Figure 4-30b). As a result, the RMS of the shear stress per bubble (Figure 4-33b), which was calculated by dividing the RMS of the system wide shear stress by the number of bubbles in the system, decreased at higher frequencies.    92      a     b Figure 4-33 RMS of shear stress (a: system wide average RMS of shear stress;  b: system wide average RMS of the shear stress per bubble; RMS of the shear stress per bubble of coarse bubbles was very small in comparison to the RMS of the shear stress per bubble of pulse bubble sparging and therefore is not visible on the graph; for course bubble sparging, discrete, 0.25 Hz and 0.5 Hz frequencies correspond to 996, 2600, and 9200 mL/min sparging flow rates respectively; Error bars correspond to the standard errors in the measurements)93   4.4 Conclusion The rate of fouling in air-sparged submerged membranes has been reported to be related to the liquid velocity and the vorticities as well as to the shear stress induced by the secondary flows induced by the sparged bubbles. Therefore, the fraction of the system where high vorticities and velocities are induced by the secondary flows trailing a sparged bubble was defined as the area of ?zone of influence? of a bubble. The two dimensional maps for vorticity compared the size, the shape, and the location of the zone of influence of a bubble in the system for different sparged bubble sizes and frequencies. The information provided by the graphs of velocity and shear stress illustrated how the magnitude of velocity and shear stress changed with the time when a bubble 1) approached the probe, 2) was in contact with the probe or 3) passed by the probes. They also qualitatively illustrated how the change of the magnitude of velocity and shear stress with the time was affected by the sparged bubble size and frequency. Using the figures presented in Section 4.2, qualitative comparison of the maps of vorticity with velocity and shear stress profiles and for different sparging conditions was also possible.  The results indicated that the system-wide area of the zone of influence was substantially affected by the size and frequency of bubbles induced. The system-wide area of the zone of influence increased with bubble size and frequency. The results also indicate that the zone of influence induced by pulse bubbles is an order of magnitude larger than that induced by coarse bubbles.  The average width of the zone of influence became larger as the bubble size and sparging frequency increased. These results are of significant importance because they suggest that the spacing between spargers can be greater when sparging with larger bubbles. This can reduce the overall number of air spargers and therefore, reducing the overall power requirement for the system. The velocities and the shear stresses within the zones of influence of bubbles were not homogenously distributed. The vertical distributions of the velocity and shear stress within the zone of influence for discrete bubbles were characterized by a rapid rise from the nose of the bubble to the wake area immediately downstream of the tail of the bubble, followed by a gradual decrease to the bottom edge of the zone of influence.   94  Velocity and system-wide RMS of the shear stress increased with bubble size and frequency. For interacting bubbles, the velocity and shear stress were characterized by a rapid rise from the nose of the bubble to the zone of influence immediately downstream of the tail end of the bubble, followed by a gradual decrease. However, at higher sparging frequencies, the decrease in the velocity and shear stress was interrupted by the trailing bubbles. The horizontal distributions of the velocity and the shear stress within the zone of influence were bell shaped, with a maximum at the vertical centerline of the zone of influence and a rapid decrease to the side edges of the zone of influence. The magnitude of maximum velocity and shear stress increased with the size of the pulse bubbles.  The horizontal distribution of the shear stress in the system was flattened at higher sparging frequencies. The magnitude of the shear stress also increased with sparging frequency. Because the horizontal distribution of the velocity and shear stress in the system is non-homogenous, fouling control over the width of the flow cell is likely not to be even, as discussed in Chapter 6. These results also indicated that the system-wide area of zone of influence did not only increase with the size of pulse bubbles but also the maximum velocity and the shear stress in the zone of influence increased with the size of pulse bubbles. The larger area of system-wide zone of influence and greater magnitude of velocity observed for larger pulse bubbles are expected to result in better fouling control in the system. The reported improvement in fouling control that has been achieved using pulse bubble sparging compared to that which can be achieved with coarse bubble sparging, as claimed by commercial membrane manufacturers such as GE Water and Process Technologies, is likely due to the difference in the characteristics of the induced zones of influence (i.e. their size), induced RMS of shear stress, and rise velocity, between pulse bubble and coarse bubble sparging. This hypothesis is considered in Chapter 6.    95  5 Relationship between the induced hydrodynamic conditions and power transfer efficiency in the system   As discussed in Chapter 1, depending on the characteristics of the solution being filtered and the permeation flux, a certain amount of force is required for the transport of the foulants away from the membrane surface to minimize the rate of fouling. The transport of the foulants away from the membrane surface is induced by the forces generated by secondary flows (characterized by liquid velocities and vorticities) and the shear stress induced at the surface of the membranes (as described in Equation 1.1). However, as discussed in Chapter 1, none of these parameters, i.e. liquid velocity and shear stress, on their own can fully characterize the hydrodynamic conditions induced by sparged bubbles in membrane systems, and the resulting effect of fouling control. To assess the efficiency of different sparging scenarios in terms of fouling control, for the first time a new parameter was in the present thesis to quantify the power transferred onto the fibers by sparged bubbles.  Power transfer was defined as the product of the force induced onto the fibers, estimated as the RMS of the shear stress induced on the fibers multiplied by the area over which the shear stress is applied, i.e. zone of influence, and the rise velocity of the bubbles, assuming that the zone of influence rises at the same velocity as the bubbles, as presented in Equation 5.1. The root mean square (RMS) was used to quantify the time variable shear force, as this parameter has been demonstrated to be correlated to fouling control [49, 62].  Ptrans = (?RMS Az)Vb                                                                                                                                                      (5.1)  where ? RMS is the root mean square shear stress for all fibers in the system [Pa], Az is the system wide area of zone of influence induced by the bubbles [m2], and Vb is the average rise velocity of the bubbles in the system [m/s]. Values of the system wide area of the zone of influence (Az) and the RMS of the shear stress (?RMS) for the different sparging conditions investigated are summarized in Figure 4-30 and Figure 4-33, respectively.   96    5.1 Power transfer and power transfer efficiency per bubble for discrete bubble sparging   As illustrated in Figure 5-1, the power transferred onto the membranes per sparged bubble, when sparging with discrete pulse bubbles, was significantly higher than when sparging with coarse bubbles. In addition, the power transferred onto the membranes per sparged bubble generally increased with the size of sparged bubbles. This was expected because the area of zone of influence and the magnitude of shear stress induced onto the membranes increased with the size of pulse bubbles as discussed in Chapter 4. As a result, the force induced per bubble, estimated as the RMS of the shear stress induced on the fibers multiplied by the area over which the shear stress is applied, increased with the size of pulse bubbles (Figure 5-2). In addition, the rise velocity of the bubbles increased with the size of pulse bubbles as discussed in Chapter 4.   97     Figure 5-1 Power transferred onto membranes per bubble  (Power transfer per bubble of coarse bubbles was very small in comparison to the power transfer per bubble of pulse bubble sparging and therefore is not visible on the graph; for course bubble sparging, discrete, 0.25 Hz and 0.5 Hz frequencies correspond to 996, 2600, and 9200 mL/min sparging flow rates, respectively; Error bars correspond to the standard errors in the measurements)  98   Figure 5-2 Force per bubble  (Force per bubble of coarse bubbles was very small in comparison to the force per bubble of pulse bubble sparging and therefore is not visible on the graph; for course bubble sparging, discrete, 0.25 Hz and 0.5 Hz frequencies correspond to 996, 2600, and 9200 mL/min sparging flow rates respectively; Error bars correspond to the standard errors in the measurements)99   To compare the different sparging conditions, a power transfer efficiency term for fouling control was defined as the ratio of the power transferred to the fibers to the actual total power added to the system. The total power added to the system was directly proportional to the sparging flow rate [88] and therefore, was similar for all the sparged bubble sizes at a given sparging flow rate.  The power transfer efficiency per bubble for pulse bubble sparging was significantly higher than the power transfer efficiency per bubble of coarse bubble sparging when sparging at discrete sparging frequency (Figure 5-3). The power transfer efficiency per bubble also increased with the size of pulse bubbles at discrete sparging frequency. These results indicated that when sparging at a very low sparging frequency (discrete), large pulse bubbles transfer a larger portion of the total power input to the system onto the fibers in comparison to sparging with coarse bubble sparging or smaller pulse bubbles. These results suggest that large pulse bubbles are more efficient in terms of delivering power for fouling control at discrete sparging. However, as discussed in the next section, interaction between bubbles can significantly affect the power transfer efficiency.     100     Figure 5-3 Power transfer efficiency per bubble   (Power transfer efficiency per bubble of coarse bubbles was very small in comparison to the power transfer efficiency per bubble of pulse bubble sparging and therefore is not visible on the graph; for course bubble sparging, discrete, 0.25 Hz and 0.5 Hz frequencies correspond to 996, 2600, and 9200 mL/min sparging flow rates, respectively; Error bars correspond to the standard errors in the measurements)101  5.2 Power transfer and power transfer efficiency per bubble for sparging at higher frequencies  When the sparging flow rate increases, the frequency at which the bubbles are released into the system also increases. At a given frequency, when bubbles interact (Table 4.1) the bubbles can no longer be considered to be rising discreetly.  The higher gas sparging flow rates required to achieve the higher frequencies increased the upward liquid flow at the center of the system tank, and as a result, the rise velocity of the bubbles (Figure 3-2). However, due to the decrease in the RMS of shear stress per bubble at higher sparging frequencies compared to the discrete sparging (Figure 4.31), the force induced onto the fibers per bubble (defined as ? RMS*Az per bubble) also decreased at higher sparging frequencies (Figure 5-2). The combined effect of the decrease in the force induced by the pulse bubbles and the increase in the rise velocity of bubbles on the power transferred to the fibers resulted in a decrease in the power transfer per bubble at higher sparging frequencies compared to discrete bubble sparging (Figure 5-1).  As was observed for discrete bubbles, the power transferred per bubble increased with bubble size when bubbles interacted at 0.5 Hz sparging frequency, i.e. when trailing bubbles rise within the wake of leading bubbles (see Table 4.1). However, the difference between small pulse bubbles and large pulse bubbles was not as pronounced for interacting bubbles as for discrete bubbles. At the sparging frequency of 0.25 Hz, when no, or minimal interaction of bubbles was observed, no consistent trend existed between bubble size and the power transferred. The higher gas sparging flow rate required to achieve the higher frequencies combined with a decrease in the power transferred onto the fibers per bubble at higher sparging frequencies compared to those for discrete sparging resulted in an overall decrease in the power transfer efficiency per bubble at higher sparging frequencies (Figure 5-3).      102  5.3 System-wide power transfer and power transfer efficiency at different sparging flow rates  The power transfer and the power transfer efficiency per bubble only provide insights into the ability of individual bubbles to contribute to fouling control for each sparging frequency. In terms of practical applications, the total power and power transfer efficiency to the system when sparging with bubbles of different sizes and sparging frequencies at given sparging flows are of interest.  As presented in Figure 5-4, for the conditions studied, the total power transferred to the system by coarse bubbles increased linearly with the sparging flow rate while that for pulse bubble sparging increased exponentially. This indicates that for the same incremental increase in the sparging flow rates, the power transfer increases more rapidly for pulse bubble sparging than for coarse bubble sparging. For small and medium pulse bubbles, the power was generally greater than for coarse bubble sparging. At low sparging flows, larger pulse bubbles transferred less power compared to the small and medium pulse bubbles or coarse bubbles at a given sparging flow rate. This was because of the low bubble release frequency for larger pulse bubbles which resulted in longer periods when no air was added to the system. However, at higher sparging flow rates, small and medium pulse bubbles may not be able to induce large magnitudes of power required for fouling control. This is discussed in Chapter 6.     103      Figure 5-4 Relationship between power transfer and air sparging conditions  (Dashed lines correspond to the linear and exponential relationships fitted to the data; for course bubble sparging, discrete, 0.25 Hz and 0.5 Hz frequencies correspond to 996, 2600, and 9200 mL/min sparging flow rates respectively)104   As observed for power transfer, for consistency, trend lines for coarse and pulse bubbles were also assumed to be linear and exponential respectively, for the relationship between the system wide power transfer efficiency and the sparging flow rate (Figure 5-5). These trend lines imply that the power transfer efficney for pulse bubbles increases exponentially with an increase in the sparging flow rate (over the range investigated), however, the power transfer efficiney for coarse bubbels increased linearly with the increase in the sparging flow rate. System-wide power transfer efficiency (Figure 5-5) is different from the power transfer efficiency per bubble because of the time gaps between the bubbles (See Figure 5-3 for power transfer efficieny per bubble).  In terms of power transfer efficiency into the system, over the range of conditions investigated, pulse bubble sparging with small bubbles, i.e. 150 mL, was consistently the most efficient condition at inducing power onto the membranes for fouling control at low and intermediate sparging flow rates (Figure 5-5). Medium pulse bubbles, i.e. 300 mL, were most efficient in terms of power transfer efficiency onto the system over the intermediate sparging flow rates. If a large amount of power is required for fouling control, it may not be possible to generate the required amount of power for fouling control with small and medium pulse bubbles (e.g. if require greater than 15 watts). In such a case, pulse bubble sparging with larger bubbles may be required. Pulse bubble sparging with large pulse bubbles, i.e. 500 mL, was most efficient at inducing a given amount of power to the system for fouling control. This will be discussed in more details in the next Chapter.   105      Figure 5-5 Power transfer efficiency for the sparging conditions investigated   (Solid lines represents trends; dashed lines represents linear and exponential relationships fitted to the data; for course bubble sparging, discrete, 0.25 Hz and 0.5 Hz frequencies correspond to 996, 2600, and 9200 mL/min sparging flow rates respectively) 106   5.4 Conclusion  To assess the efficiency of different sparging scenarios in terms of fouling control, a new parameter was defined in the present study to quantify the power transferred onto the fibers by bubbles.  The hydrodynamic conditions and the resulting power induced on to the membranes for fouling control were substantially affected by the sparged bubble size and frequency, i.e. sparging flow rate. The power transfer per bubble increased with bubble size when sparging with discrete bubbles. However, the extent of the increase in power with bubble size was not as pronounced when sparging at higher frequencies.  For small and medium pulse bubbles, the power transfer was generally greater than for coarse bubble sparging. At low sparging flows, larger pulse bubbles transferred less power than the small pulse bubbles or coarse bubbles at a given sparging flow rate. Power transfer and power transfer efficiency were defined to quantify the hydrodynamic conditions induced by sparged bubbles of different sizes and frequencies by incorporating the area of the zone of influence, the liquid velocity, and the RMS of the shear stress induced in the system. However, of interest is the relationship between the fouling rate and the power transferred onto the membranes. This is discussed in Chapter 6.  107  6  Effect of induced hydrodynamic conditions on the fouling rate As discussed in Chapter 5, power transfer was defined to quantify the hydrodynamic conditions induced by sparged bubbles of different size and frequency by incorporating the area of zone of influence, the liquid velocity, and the RMS of the shear stress induced in the system (Equation 5.1). For the sparging conditions investigated, at high sparging flows, the power transfer efficiency to the system was higher for pulse bubble sparging than for coarse bubble sparging. The present chapter summarizes the results of the filtration experiments conducted with different sparging approaches. The different sparging conditions were compared in terms of 1) power transfer and 2) power transfer efficiency and 3) rate of fouling control.   6.1 Effect of bubble size and sparging frequency on fouling rate  The filtration experiments were conducted for the sparging conditions, i.e. bubble size and frequency, presented in Table 4.1. Typical trans-membrane pressure measurements collected during filtration are presented in Figure 6-1 (filtration data for all investigated sparging conditions are presented in Appendix E). For all cases, the normalized trans-membrane pressure could be modeled with the exponential relationship presented in equation 6.1.   P=PoeKV                                                                                                                 (6.1) where P is the trans-membrane pressure (kPa), K is the fouling rate constant (1/mL) , V is the volume filtered (mL), and the subscript ?o? corresponds to initial conditions.  The exponential increase in TMP observed in the present study was consistent with the exponential increases in TMP observed by others for filtration of solutions containing bentonite through commercially available hollow fiber membranes [52, 62]. For this reason, in the discussion which follows, the rate of fouling is expressed in terms of the exponential fouling rate constant.     108         a b Figure 6-1 Typical results from filtration experiments (a: coarse bubble sparging (results presented for 0.75 mL bubbles and discrete sparging); b: pulse bubble sparging ( results presented for 500 mL pulse bubbles and discrete sparging); HF1: Hollow Fiber at location 1; HF2: Hollow Fiber at location 2; HF3: Hollow Fiber at location 3; HF4: Hollow Fiber at location 4) 109   The fouling rate constants calculated for all experimental conditions investigated are summarized in Figure 6-2. The system average fouling rate constant was calculated by averaging the fouling rate constants measured for each fiber. The average fouling rate is representative of the overall fouling rate in the system because all fibers were individually connected to an individual pump which enabled the flux in all fibers to be similar.    Figure 6-2 System average fouling rate constant for different sparging conditions  (Error bars correspond to minimum and maximum measurements as filtration tests were done in triplicates; for course bubble sparging, discrete, 0.25 Hz and 0.5 Hz frequencies correspond to 996, 2600, and 9200 mL/min sparging flow rates respectively)110   When sparging with discrete bubbles, the rate of fouling remained relatively constant as the size of the sparged bubbles increased. This was somewhat expected because the volume of sparged gas per unit time added to the system was similar for all sparged bubble sizes. When sparging at an intermediate frequency of 0.25 Hz, the rate of fouling was lower than that observed when sparged with discrete bubbles. Additionally, the rate of fouling generally decreased as the size of the sparged bubbles increased. Again, this was expected because the volume of sparged gas per unit time added to the system at a given frequency increases with bubble size. When sparging at the highest frequency of 0.5 Hz, the rate of fouling was again lower than those observed at the lower frequencies. However, as observed for discrete bubbles, no clear trend was observed between bubble size and fouling rate. As previously discussed in Chapter 5, the sparging conditions significantly affected the shear stress induced onto the hollow fibers, the zone of influence of secondary flows and the bubble rise velocities and, as the result, the power transferred onto the membrane surface. Power transferred onto the membrane surface was defined as presented in Equation 5.1 to characterise the hydrodynamic conditions induced by sparged bubbles of different sizes and frequencies.  Power transfer efficiency was defined to quantify the percentage of the total power input the system through air sparging that was transferred onto the membrane surface for fouling control.  The relationship between the fouling rate and the power transferred is presented in Figure 6-3. The fouling rate was observed to be significantly affected by the power transferred. 1) At low power transfer values (<2 w), the fouling rates were high and decreased linearly with an increase in the power transferred. 2) At intermediate power transfer values (2-35 w), the incremental change in the fouling rate decreased as the power increased. 3) At high power transfer values (>35 w), the fouling rate was essentially zero.   No other studies have investigated the relationship between the fouling rate and the power transferred onto the membranes prior to the present study. However, when investigating the relationship between the fouling rate in hollow fiber membranes and coarse 111  bubble sparging, similar trends, i.e. an initial fast decrease in the rate of fouling with sparging flow, a transition range at intermediate sparging flows, and a range over which fouling is essentially zero, have been reported by others  [52, 90].    Figure 6-3 Relationship between fouling rate and power transferred onto membranes  (Error bars correspond to minimum and maximum measurements as filtration tests were done in triplicates; for course bubble sparging, discrete, 0.25 Hz and 0.5 Hz frequencies correspond to 996, 2600, and 9200 mL/min sparging flow rates respectively)112   As previously discussed in Chapter 1, the accumulation of foulants on a membrane surface is the result of the force balance between the drag forces towards the membrane surface due to the permeation flow, and shear stress-induced back transport of foulants away from the membrane surface. As presented in Equation 1.1, this force balance is a function of both the size of the foulants and the applied shear stress. Therefore, for a given shear stress, there is a critical particle size for which all the particles with a larger size than the critical size will be transported back into the solution through the lift forces exerted by the shear stress [4].  Therefore, depending on the power transferred, three fouling behaviors are possible. 1) When the power applied is high, the shear stress induced onto the membrane is high enough for the back transport of foulants of all sizes. Under these conditions Fdrag,permeation < Flift, shear for foulants of all sizes and therefore, minimal, or no fouling is observed. This likely corresponds to the fouling rates observed when the power transfer was larger than 35 w. 2) When the power applied is lower, the back transport forces exerted by the shear stress may not be high enough to overcome the permeation drag forces for small foulants. In this transition range a foulant layer forms that is likely to be populated mainly by smaller foulants, for which the back transport forces are lower [4]. When the power applied further decreased, the accumulation of foulants increased. Also, when the power applied is decreased, the fouling layer is expected to be composed of increasingly larger particle sizes [5, 52]. This likely corresponds to the fouling rates observed when the power transfer was in the range of 2-35 w. 3) When the power applied is low, the back transport forces are not large enough to overcome the permeation drag forces for most of the foulants with different sizes. Under these conditions, the foulant layer is likely to be populated by foulants of all sizes. Under these conditions Fdrag,permeation > Flift, shear for foulants of all sizes and therefore, high fouling rates are observed. This likely corresponds to the fouling rates observed when the power transfer was less than 2w.  113  When considering power transfer, above a given value, the rate of fouling was negligible (Figure 6-3). These results indicate that for a given water matrix, negligible fouling occurs when sufficient power is transferred to the membranes to prevent the accumulation of hydraulically reversible fouling. For the solution filtered, fouling could be effectively controlled by transferring approximately 5 watts of power to the system by sparging (Figure 6-3). This threshold of power, i.e. 5 watts, is the point of diminishing returns for the data presented in Figure 6-3. Other studies have also reported limited benefits of increased sparging above a given sparging intensity (the threshold sparging intensity differed in different experimental setups) [23, 39, 55].   A threshold, was also observed for the RMS of shear stress, below which fouling rate generally decreased with an increase in RMS of the shear stress. Fouling was essentially zero above this critical RMS of the shear stress (Figure 6-4). However, in contrast to the relationship observed between the fouling rate and the power transfer, substantial discontinuities were observed when considering the relationship between fouling rate and the RMS of shear stress, such that substantially different fouling rates were observed at a given RMS of shear stress (e.g. at a RMS of 0.9 Pa, the fouling rate ranged from essentially 0 to over 0.0002/mL). These results are consistent with those from previous studies that have concluded that single summative parameters, such as RMS of the shear stress, cannot consistently describe the effect of hydrodynamic conditions on the rate of fouling [49]. 114     Figure 6-4 Relationship between fouling rate and root mean square of shear stress in the system (Error bars correspond to minimum and maximum measurements as filtration tests were done in triplicates; for course bubble sparging, discrete, 0.25 Hz and 0.5 Hz frequencies correspond to 996, 2600, and 9200 mL/min sparging flow rates respectively115   The minimum average constant shear stress required for all particles in a minimal to no fouling condition for the solution of bentonite particles with an average diameter of 2 um and a constant permeation flux of 100 L/m2.hr was calculated using Equations 1.1 to be 4 Pa. However, as presented in Figure 6-4 (Relationship between fouling rate and root mean square of shear stress for individual fibers), the threshold of shear stress for which no or minimum fouling was observed in the present research was 1 Pa. This discrepancy is likely due to the fact that Equation 1.1 assumes constant shear stress conditions, while those in the present study were variable. As previously discussed, variable shear stress is more efficient in terms of fouling control than constant shear stress. Also, the values presented in Figure 6-5, are RMS values of time-variable shear stress, while the peak shear stress values were much higher.    116      Figure 6-5 Relationship between fouling rate and root mean square of shear stress for individual fibers (For course bubble sparging, discrete, 0.25 Hz and 0.5 Hz frequencies correspond to 996, 2600, and 9200 mL/min sparging flow rates respectively117   The power transfer efficiency, defined as the ratio of power transferred onto membranes to the actual power added to the system, was used to identify the optimal sparging approach to effectively control fouling. The relationship between power transfer efficiency and power transfer is presented in Figure 6-6. Also presented in Figure 6-6 is the amount of power selected in the present study for effective fouling control (i.e. 5 watts).   For the present study, sparging with small pulse bubbles was the most efficient approach to transfer the required amount of power, i.e. 5 watts, for fouling control (Figure 6-6).  For the sparging conditions investigated, it was not possible to generate more than about 15 watts with small pulse bubble sparging. Therefore, if more than 15 watts of power is needed for fouling control, pulse bubble sparging with small pulse bubbles at a frequency greater than 0.5 Hz or sparging with larger bubbles may be required.    Figure 6-6 Power transfer efficiency with respect to power transferred onto membrane surface (Dashed lines correspond to power required in the present study for effective fouling control and corresponding power transfer efficiency for coarse and small pulse bubble sparging; solid lines present overall trends,for course bubble sparging, discrete, 0.25 Hz and 0.5 Hz frequencies correspond to 996, 2600, and 9200 mL/min sparging flow rates respectively118   These results indicate that pulse bubble sparging is substantially more efficient at transferring power to a membrane than coarse bubble sparging. For the solution filtered, pulse sparging with small bubbles is expected to be approximately two times more efficient than coarse bubble sparging for fouling control at low-to-intermediate sparging flow rates.  Considering that coarse bubble sparging in MBR systems accounts for over % 30 of the total operating cost of these systems (Figure 6-7a), the use of pulse sparging instead of coarse sparging can therefore reduce the overall power requirements from approximately 0.62 kWh/m3 of permeate to less than 0.53 kWh/m3 of permeate (Figure 6-7b).     119      a  b  Figure 6-7 Power cost distribution for MBR systems (a: for continuous coarse bubble sparging; b: with small pulse bubble sparging. Assuming a 50% reduction in power cost for small pulse bubble sparging compared to coarse bubble sparging; Figure 6-7a was adapted from [1]) 120   6.2 Effect of bubble size and sparging frequency on the spatial distribution of fouling rate in the system  The spatial distribution of the fouling rate in the system was not homogeneous, where the fouling rate generally was lower at the centerline of the system tank directly above the spargers (Figure 6-8).  Fibers located in the zone of influence are marked with clear symbols and fibers out of zone of influence are marked with solid symbols. It was observed that fouling rate was generally lower in the zone of influence induced by sparged bubbles.  This observation could be explained by the non-homogeneous distribution of velocity and shear stress in the system, as described in Chapter 4. Higher velocity and higher kinetic energy in the zone of influence of sparged bubbles result in higher shear stress induced onto the membranes located in the zone of influence of sparged bubbles compared to the membranes located at the edges (as discussed in Chapter 4). Higher shear stress induced onto the membranes could result in a higher rate of back transport of foulants from the membrane surface and therefore, a lower fouling rate (Equation 1.1). As a result the lower fouling rates were observed in the zone of influence of sparged bubbles than at the edges of the zone of influence of sparged bubbles.         121   a  b  c Figure 6-8 Distribution of fouling rate in the system (a: discrete bubble sparging; b: sparging at 0.25 Hz; c: sparging at 0.5 Hz; fibers inside zone of influence marked clear, fibers outside zone of influence marked solid; error bars corresponds to minimum and maximum measurements as filtration tests were done in triplicates; for course bubble sparging, discrete, 0.25 Hz and 0.5 Hz frequencies correspond to 996, 2600, and 9200 mL/min sparging flow rates respectively 122    As the size and frequency of bubbles increased, the zone of influence became wider, flattening the distribution of the fouling rate (as discussed in Chapter 4). These results suggest that as the size and frequency of the bubbles increase, the spacing between the spargers could be increased reducing the overall number of spargers and therefore reducing the volume of gas required for sparging the system. For instance increasing the sparging frequency from 0.25 Hz to 0.5 Hz increases the average width of influence by approximately a factor of 1.3, and therefore the spacing between the spargers could be increased by the same factor, reducing the number of air spargers. Reducing the overall number of air spargers results in reducing the overall power requirement for air sparging in the system by % 30. When considering individual fibers, it is not possible to estimate the power transferred because the membrane area over which the shear stress is induced by gas sparging cannot be accurately estimated. For this reason, the RMS of the shear stress was used as a surrogate to characterize the spatial distribution of the power transferred. The spatial distribution of shear stress in the system tank is illustrated in Figure 4-9 and Figure 4-29 for the different sparging conditions investigated. As observed, the distribution of shear stress in the system tank was characterized with the highest magnitude of shear stress generally in the middle of the system tank and gradual decrease to the sides of the system tank. As previously discussed, athough the correlation between the RMS of the shear stress and fouling rate for individual fibers is scattered, a lower fouling rate was consistently observed for fibers which were exposed to higher RMS of the shear stress (Figure 6-4 and Figure 6-5). Also, as observed for the system-wide RMS of shear stress, when considering individual fibers, for the solution filtered, when applying more than approximately more than 1Pa, the rate of fouling was negligible. 123  6.3 Conclusion  Power transferred, which considers the combined effects of the bubble rise velocity, the area of the zone of influence and the RMS of the shear stress could be used to accurately characterise the relationship between the hydrodynamic conditions induced by gas sparging and the rate of fouling observed. When sparging with discrete bubbles, the rate of fouling remained relatively constant as the size of the sparged bubbles increased. When sparging at an intermediate frequency of 0.25 Hz, the rate of fouling was lower than that observed when sparging with discrete bubbles. Additionally, the rate of fouling generally decreased as the size of the sparged bubbles increased. When sparging at the highest frequency of 0.5 Hz, the rate of fouling was again lower than those observed at the lower frequencies. However, as was observed for discrete bubbles, no clear trend was observed between bubble size and fouling rate. The higher power transfer onto the membranes using pulse bubble spargers in comparison to that using coarse bubble spargers resulted in better fouling control. This is consistent with the claims made by commercial membrane manufacturers such as Siemens, Samsung, and GE Water and Process Technologies and two recent studiedies [62,147] The fouling rate was observed to be significantly affected by the power transferred. 1) At a low power transfer (< 2 watts), the fouling rates were high and decreased linearly with an increase in the power transferred. 2) At an intermediate power transfer (2-35 watts), the incremental change in the fouling rate decreased as the power increased. 3) At a high power transfer (> 35 watts), the fouling rate was essentially zero. For the solution filtered in the present research, pulse sparging with small bubbles is expected to be approximately two times more efficient than coarse bubble sparging for fouling control.  Considering that coarse bubble sparging in MBR systems accounts for over 30% of the total operating cost of these systems, the use of pulse sparging instead of coarse sparging can reduce the overall power requirements from approximately 0.62 kWh/m3 of permeate to less than 0.53 kWh/m3 of permeate .  For the first time it is demonstrated that the spatial distribution of the fouling rate in the system was not homogeneous. The fouling rate generally was lower in the zone of influence of bubbles.  This observation could be explained by the non-homogeneous distribution of 124  velocity, vorticity, and shear stress in the zone of influence. Higher velocity and higher kinetic energy in the zone of influence of sparged bubbles resulted in higher shear stress induced onto the membranes located in the zone of influence of sparged bubbles compared to the membranes located at the edges. Higher shear stress induced onto the membranes resulted in a lower fouling rate (due to the increase in the rate of backtransport of the foulants). As the size of the bubbles was increased, the horizontal distribution of fouling was flattened and the fouling rate decreased. The width of the relatively flat portion of the distribution corresponded to the width of zone of influence of the sparged bubbles. The width of the zone of influence increased with bubble size and frequency (Chapter 4), suggesting that as the size and frequency of the bubbles increase, the spacing between the spargers could be increased, reducing the overall number of spargers, and therefore reducing the volume of gas required for sparging the system.           125   7 Conclusions and recommendation 7.1 Overall conclusions  The present research identified the optimum sparging conditions in terms of power requirement for fouling control in submerged membrane systems. The investigations focused on addressing two main questions: 1) how do the sparging approaches affect the hydrodynamic conditions and the resulting shear stress in a membrane system, and 2) how do the induced hydrodynamic conditions affect the rate of fouling?    The designed experimental system mimicked the hydrodynamic conditions that are representative of full size submerged membrane systems. Direct measurement of the shear stress induced onto membranes was made using an Electrodiffusion Method (EDM).  A procedure was developed for correction and interpretation of the data collected under transient flow conditions (which occur in submerged membrane systems). The results of this investigation were compiled comprehensively in a form that can be used as a reference for future work that applies EDM in practical applications under steady state or transient conditions. An approach was developed to measure velocity and vorticity of the liquid in the system as well as bubble characteristics using a high speed camera, high intensity light sources, and particle image velocimetry.  For the first time, the zone of influence and the power transferred by bubbles onto fibers were defined as parameters that could be used to characterize the complex hydrodynamic conditions induced under different sparging conditions. The zone of influence was defined as the fraction of the system in which high velocities and high vorticities are induced by the bubbles.   The velocity and the shear stress within the zones of influence of bubbles were not homogeneously distributed. The results also indicated that the area of the zone of influence was not only larger for larger pulse bubbles (dimonsionless zone of influence was 10 times larger for the pulse bubbles), but also the maximum velocity and the shear stress in the zone 126  of influence were greater for larger pulse bubbles. The larger area of the zone of influence and the larger magnitude of velocity observed for larger pulse bubbles are expected to result in better fouling control in the system.  For the first time, the horizontal distribution of the velocity and the shear stress induced within the zone of influence of sparged bubbles in a submerged membrane system was characterized. The horizontal distributions of the velocity and the shear stress within the zone of influence indicated that the maximum velocity and shear stress occurred at the vertical centerline of the zone of influence with a rapid decrease to the side edges of the zone of influence. The knowledge regarding the horizontal distribution of the velocity and shear stress in the system is of importance because as it was demonstrated in the present research that non-homogeneous distribution of the velocity and shear stress in the system induced non-homogeneous fouling control over the width of the system tank.  Results from this investigation indicated that as the size and frequency of the bubbles increased, the average width of the zone of influence increased. These results suggest that as the size and frequency of the bubbles increase, the spacing between the spargers could be increased. The latter could reduce the overall number of spargers and therefore, reducing the volume of gas required for sparging the system.   For all cases investigated, a clear relationship was observed between the fouling rate and the power transferred onto membranes. Fouling rate decreased consistently with an increase in the magnitude of the power transferred onto membranes, until the fouling rate reached a minimum above which no further improvement in fouling control was achieved.   For the solution filtered in the present research, pulse sparging with small bubbles is expected to be approximately two times more efficient than coarse bubble sparging for fouling control.  Considering that coarse bubble sparging in MBR systems accounts for over 30% of the total operating cost of these systems, the use of pulse sparging instead of coarse sparging can reduce the overall power requirements from approximately 0.62 kWh/m3 of permeate to less than 0.53 kWh/m3 of permeate .  127   With the insight into the hydrodynamic conditions induced under different sparging conditions, i.e. bubble size and frequency, (which can be obtained by using the concepts of the zone of influence, power transfer onto membranes, and power transfer efficiency developed in the present research) the optimal sparging conditions in submerged membrane systems can be identified without solely relying on an empirical pilot-testing approach which will result in significant savings in time and cost.  7.2 Engineering significance  This research addressed the knowledge gap that existed in determining the optimum sparging conditions, i.e. bubble size and frequency, in air sparged submerged membrane systems. For the first time, the methods developed in this research enabled the multiple effects of sparged bubbles on the hydrodynamic conditions in submerged membrane systems to be quantitatively characterized. The zone of influence and power transfer concepts developed made it possible to optimize the sparging approach for fouling. Optimizing sparging conditions can reduce power requirements for fouling control by more than 50%, significantly reducing the operation costs of membrane systems, and making their widespread adoption more likely.  Moreover, the zone of influence could be used to design the spacing between the spargers. Optimal spacing of spargers could reduce the volumetric flowrate of gas required for sparging the system.   With the insight into the hydrodynamic conditions induced under different sparging conditions (which can be obtained by using the concepts of the zone of influence, power transfer onto membranes, and power transfer efficiency developed in the present research), the optimal sparging conditions in submerged membrane systems can be identified without solely relying on an empirical pilot-testing approach which will result in significant savings in time and cost. The knowledge gained from the present research is being used by industrial collaborators (GE Water and Process Technologies) to design their next generation of sparging systems  128  The EDM reference manual is the first document to cover the theoretical aspects of EDM, the practical aspects of EDM, and the limitations of EDM for practical applications. This document can be used as a reference for application of EDM in practice under steady state and transient conditions. 7.3 Recommendations for future work  The result from this research opens the opportunity for further investigation on the items listed below.  ? The effect of packing density on the behavior of sparged bubbles and the induced hydrodynamic conditions in a full module. Physical characteristics of bubble (size, path, rise velocity, and etc.) may change when sparged in a packed module. This may affect the hydrodynamic conditions induced in the system and as a result the efficiency of sparging conditions. This is currently being investigated as part of an independent study. ? Characterizing the hydrodynamic conditions induced in the system and the power transfer efficiency at sparging frequencies higher than those investigated in the present research for the small and medium pulse bubbles. ? The contribution of fiber contact or fiber sway to the magnitude and time variation of shear stress induced on the fibers in comparison to the contribution of the turbulence induced by sparged bubbles. ? The effect of multiple spargers or the spacing of spargers on the induced hydrodynamic conditions. This can lead to an optimum design with minimum power requirements. ? The relationship between solution physical and chemical characteristics and the behavior of bubbles and the induced hydrodynamic conditions in the system. This is of significance considering that solutions filtered in real systems generally contain organic materials in addition to particulate matter. ? Different combinations of sparging flow rates and bubble sizes that haven?t been studied in the present study. One of the main challenges in this research was time constrains. PIV technique is a time consuming process that generates a great amount 129  of data to analyze. 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Lighthill, "Contributions to the theory of heat transfer through a laminar boundary layer,"  Proc.R.Soc.London, Sera, vol. 202, pp. 359-377, 1950.  [132] D. J. Pickett, Electrochemical Reactor Design. Elsevier Publishing Company, 1979. [133] A. Acrivos, "Solution of the laminar boundary layer energy equation at high prandtl numbers,"  Phys.  Fluids, vol. 3, pp. 657-658, 1960.  [134] H. G. Dimopoulos and T. J. Hanratty, "Velocity gradients at the wall for flow around a cylinder for Reynolds numbers between 60 and 360,"  J.  Fluid Mech., vol. 33, pp. 303-319, 1968.  [135] W. K. Lewis and W. G. Whitman, "Principles of Gas Absorption."  Ind.Eng.Chem., vol. 16, pp. 1215-1220, 1924.  [136] M. Kraume, Transportvorg\ange in Der Verfahrenstechnik (in German). Springer, 2012. [137] L. B?hm, A. Drews and M. Kraume, "Bubble induced shear stress in flat sheet membrane systems - Serial examination of single bubble experiments with the electrodiffusion method,"  J.  Membr.  Sci., vol. 437, pp. 131-140, 2013.  [138] R. Higbie, The Rate of Absorption of a Pure Gas into Still Liquid during Short Periods of Exposure. The University of Michigan, 1935. [139] J. Crank, The Mathematics of Diffusion. Clarendon Press, 1975. [140] O. Wein, "Convective diffusion from convex microprobes into colloidal suspensions: The edge effects,"  Int.  J.  Heat Mass Transfer, vol. 53, pp. 1868, 2010.  141  [141] O. Wein, V. V. Tovcigrecko and V. Sobolik, "Transient convective diffusion to a circular sink at finite Peclet number,"  Int.  J.  Heat Mass Transfer, vol. 49, pp. 4596, 2006.  [142] P. Kaiping, "Unsteady forced convective heat transfer from a hot film in non-reversing and reversing shear flow,"  Int.  J.  Heat Mass Transfer, vol. 26, pp. 545, 1983.  [143] V. Sobolik, O. Wein and J. Cermak, "Simultaneous measurement of film thickness and wall shear-stress in wavy flow of non-newtonian liquids,"  Collect.  Czech.  Chem.  Commun., vol. 52, pp. 913-928, 1987.  [144] D. M. Wang and J. M. Tarbell, "An approximate solution for the dynamic response of wall transfer probes,"  Int.  J.  Heat Mass Transfer, vol. 36, pp. 4341, 1993.  [145] F. Rehimi, F. Aloui, S. B. Nasrallah, L. Doubliez and J. Legrand, "Inverse method for electrodiffusional diagnostics of flows,"  Int.  J.  Heat Mass Transfer, vol. 49, pp. 1242, 2006.  [146] R. J. Marks II, Introduction to Shannon Sampling and Interpolation Theory. New York, NY, USA: Springer-Verlag New York, Inc, 1991. [147] H. Wray, A. Robert, and P.R. B?rub?, ?Surface shear stress and membrane fouling when considering natural water matrices?, Desalination, vol 330, pp 22-27, 2013142  APPENDIX A Calibration of the electrochemical shear probes  Electrochemical shear probes were fixed vertically in the middle of a cylinder with internal radius of r0=0.025m as illustrated in Figure A.1. The magnitude of shear stress induced onto the test fiber under a laminar flow condition was calculated analytically using Equations A.1 to A.5.    Figure A.1 Experimental set up used for calibration of the probes    143    ?(?) =  14???????2 ? ?02 +??02??12?ln????0?ln ???0??                                                                            (A.1) ? = ? ?(?)2?????2?1                                                                                                             (A.2) ? =  ?  ?8??????+ ??? ??04 ? ??4 ???02???2?2ln??0?1??                                                                         (A.3) ? =?????                                                                                                                                 (A.4) ????? =  ?  2?? ?2?+??2??02ln??0???????04???4???02???2?2ln??0????                                                                                             (A.5) where r corresponds to radius; ri= 0.0018m, radius of fibers; ro= 0.025m, radius of the cylinder; u(r) :velocity at any given r [m/s]; p: pressure; Q: liquid flow rate [m3/s]; and ? : shear rate [1/s]. Using the electrochemical shear probes installed onto the fibers and the EDM method, the current induced in the system was measured using Equation A.6. ???? =  ???????                                                                                                                            (A.6) where Iexp corresponds to current measured in the system[A] ; Vexp, to voltage [volts] ; R, to resistance of the resister in the circuit, i.e100 ohms, and A,to amplification (1000). Using the theoretical Leveque Equation for steady state conditions (Equation E.7), Klev_theo is calculated as illustrated in Equation A.8.  ? = 0.862 ??????23? ??13? ?13?                                                                                          (A.7) ????_???? = 0.862 ??????23? ??13?                                                                                    (A.8)  Using Equation (A.6) and (A.8) Klev_exp can be calculated as in Equation A.9. ????_??? =  ?????????13?                                                                                                                  (A.9) Using Equation (A.9) and (A.8) correction factor for each probe is calculated as follows:  144  ????????? ?????? =????_???????_????                                                                                            (A.10) Correction factors for the four probes used in this study are summarized in Table A.1.   Table A. 0-1 Calibration parameters  Probe 1 Probe 2 Probe 2 Probe 3 I analytical [A] 2.39E-06 2.39E-06 2.39E-06 2.39E-06 V analytical [V] 0.23904198 0.23904198 0.23904198 0.23904198 V measured [V] 0.303 0.347 0.314 0.345 alpha 0.79 0.69 0.76 0.69 C correction factor 4.44 2.94 4 3.11 t0 [s] 1.2 4.06E-01 5.52E-01 1.67E-01    145  Appendix B Application of the electrodiffusion method (EDM) to measure wall shear stress: integrating theory and practice B-1 Introduction   This chapter describes the research done on integrating the theory and application of the electrodiffusion method for measurements of shear stress at the membranes under transient flow conditions.  Various techniques have been developed for the measurement of wall shear stress by mechanical, thermal, optical or chemical methods (see Table 3.1). In general, these techniques are non-intrusive at a macro scale and relatively complex data processing is required to obtain shear stress values from the measured parameter (for more information about the different techniques see also [91-93].  Of the measurement techniques listed in Table B.1, the electrodiffusion method (EDM) is of particular interest due to its high sensitivity to near-wall flow fluctuations and its ability to detect local flow phenomena that appear only on small areas. Extensive literature exists on the EDM technique. Major Reviews of the techniques focussing on different aspects of the technique have been published by [94-97] but none of these present the derivations of the underlying theory and the correction necessary for transient flows. To address this gap, the present reference document is developed. The theoretical assumptions and hypotheses used in developing the equations that are used in the post-processing to calculate the shear stress under transient conditions are reviewed in detail. The calibration and correction methods for the data collected under transient conditions are optimized. Challenges regarding the calibration of this technique and the care that must be taken before using the technique are also discussed.        146  Table B-1 Shear stress measurement techniques sorted by their measurement principle Type Principle Measured Parameter Exemplary Publication Micro Electromechanical Sensor mechanical / thermal / optical shear stress / temperature / particle movement [98] Piezo foils mechanical pressure [99] Pressure sensitive copolymers mechanical pressure [100]  Pressure transducer mechanical pressure [101]  Preston pipe mechanical pressure [102]  Shear force scale mechanical pressure [37]  Surface fence cathode mechanical pressure [103] Hot wire/film anemometry thermal temperature [104] Infrared thermography thermal temperature [105] Laser-2-focus anemometry optical particle movement [106]  Laser-oil-film interferometry optical oil film slope [107] Laser Doppler Anemometry optical particle movement [108] Laser Speckle Anemometry optical particle movement [109] Particle Image Velocimetry optical particle movement [110]] Liquid crystal chemical molecular changes [111]  Electrodiffusion Method chemical mass transfer [112]  147  B.2 Electrodiffusion Method (EDM): theory  For measurement of shear stress with the EDM two electrodes and an electrolytic solution are necessary when a voltage is applied between the cathode and the anode, a heterogeneous reaction takes place at the electrodes. The transfer of the reducing ions to the cathode and the electron exchange leads to charge equalization between anode and cathode which induces a measurable current. The stronger the mass transfers of the ions, the higher the measured value of the current. Because the rate of mass transfer of ions at the cathode is directly related to the hydrodynamic conditions at the proximity of the cathode in the system which are governed by local shear stress, the magnitude of current induced at the cathode can be used to measure the magnitude of shear stress. B.2.1 The basic electrical circuit Reiss and Hanratty [112] show the basic electrical circuit of an EDM system. The signal is amplified and an ohmic resistance is used in the circuit (about 100 ?). This should be two orders of magnitude higher than the ohmic resistance of the electrochemical system so that the latter one does not affect the signal.  B.2.2 The electrodes Both circular [73] and rectangular strip [96] cathodes have been used. Platinum or Nickel is often used as anode and cathode material (see e.g. [112] and [29]). Application of stainless steel as an anode has also been reported [49]. The cathode has to be very small in contrast to the anode which needs to have a much larger surface area so that the oxidizing reaction is not limiting the process. By installing two or more cathodes each with its own circuit with a very small distance from each other in the flow it is possible to determine the direction of the flow [72, 92, 113, 114]. Having multiple cathodes very close to each other in the flow, the concentration boundary layer of the upstream cathode influences the concentration boundary layer formed on the downstream flow and therefore the signal measured by the downstream cathode is smaller than the signal measured by the upstream cathode. Using a two segmented cathodes, which is most common in literature, the direction can only be obtained in a range of 0 to 180? relative to the alignment axis of the cathode. Using a multi-segmented cathode with proper calibration (the 148  measurement of the all signals in a defined flow while turning the combination of cathodes to defined angles) both the direction and the shear rate can be determined quantitatively [114].  B.2.3 The electrolytic solution There are various combinations of electrolytes mentioned in literature (Table B.2). For most of these solutions, ferri- and ferrocyanide are used as oxidizing and reducing ions. The reaction that takes place at the cathode is: ( ) ( ) 4636 ??? ?+ CNFeeCNFe   Oxygen existing in the system can cause side reactions and influence the induced current. Therefore, oxygen should be purged from the used deionized water. An inert electrolyte needs to be added to the solution to avoid electrical migration. Potassium compounds are the most commonly used electrolyte. As one example, potassium sulphate has the additional beneficial effect, that it can also suppress the solubility of oxygen in the solution [115] and therefore it is commonly used in cases of long usage of the solution to ensure, no side reactions with oxygen occurs. As the EDM is based on mass transfer, the diffusion coefficient which is a function of the temperature is of interest. The Stokes-Einstein relationship, i.e. .constTD =? , is valid for solutions with a viscosity similar to that of water [95, 116]. Values of the diffusion coefficient and viscosity of the electrolyte solution at 30?C have been reported to be 8.36*10-10 m2/s and 8.33*10-4 Pa s, respectively [16, 117]. The viscosity of the electrolyte solution at 20?C is comparable to that of pure water, i.e sPa10 3?=? . According to the Stokes-Einstein Equation mentioned above, for a temperature of 25?C, the diffusion coefficient then has a value of 7.4*10-10m?/s. Other Equations have to be used for high viscosity or non-Newtonian liquids [118].  B.2.4 Limiting diffusion current For measurement of wall shear stress using the EDM, a threshold for the applied voltage exists at which the rate of reaction of oxidizing ions at the cathode is a function of the mass transfer of the ions and is not limited by the number of electrons available at the cathode. This well-known effect is often described by polarization curves in electrochemistry [92, 94, 119, 120]. The plateau in these curves indicates the limiting current conditions and the current 149  measure is called the limiting diffusion current [73, 112].  This range of limiting current conditions should be obtained for each investigated electrochemical system.   Table B-2 Materials used in literature for the electrolytic solution  Publication Electrolyte [M] Comment [113, 116]  Potassium ferricyanide 0,0028-0,01 equimolar Potassium ferrocyanide 0,0028-0,01 Potassium chloride 1  [41, 121, 122] Potassium ferricyanide 0,003  Potassium ferrocyanide 0,006 Potassium chloride 0,3-0,33 [73, 114, 118, 123, 124]  Potassium ferricyanide 0,003-0,025 equimolar Potassium ferrocyanide 0,003-0,025 Potassium sulfate 0,057-0,25  [79, 80, 125] Potassium ferricyanide 0,002-0,004  Potassium ferrocyanide 0,05 Potassium sulfate 0,1 [72, 126]  Potassium ferricyanide 0,01 equimolar Potassium ferrocyanide 0,01 Sodium hydroxide 2  [127]  Potassium ferricyanide 0,02  Potassium ferrocyanide 0,05 Sodium hydroxide 0,5 [128, 129]  Iode 0,0038  Potassium iodide 0,1 [130]  Oxygen 9.5-97*10-4  Potassium sulfate 0,01   150  B.2.5 Steady state flow conditions A theoretical relationship can be derived to calculate the magnitude of wall shear stress from the magnitude of current measured by EDM when the following conditions apply [92]: - The concentration boundary layer at the cathode is within the region where the velocity gradient is linear. - The flow is homogeneous over the cathode. - The concentration boundary layer thickness at the cathode is thin compared to the width of the electrode. - Diffusion in the direction of the bulk convective flow in negligible at the cathode. - Flow normal to the surface of the cathode is negligible. - The reacting ion is completely consumed at the cathode. - No reaction happens in the bulk of the electrolyte solution. - Steady state conditions are apparent. The Planck-Nernst-Equation can be used to describe the specific molar flux of ions at the cathode based on diffusion, migration and convection (Equation (B.1) and (B.2)).  conv,imigr,idiff,ii nnnn ???? ++=  (B.1) vccucDn iiiiii?? +?????=    . (B.2) If the conditions listed above are negligible and the specific conductivity of the solution is high, the electrical potential gradient is negligible. Assuming no-slip conditions at the surface of the cathode, the Planck-Nernst-relationship can be simplified as presented in Equation (B.3): 0yiii dydcDn=?????????=?  (B.3) Because the mass flux of ions at the cathode results from reduction, Equation (B.3) can be equated to Faraday?s law, yielding Equation (B.4). Assuming limiting current conditions are applied, this is equal to the product of the mass transfer and the concentration in the bulk solution. bm0yii ckdydcDFAnI=?????????==   . (B.4) 151  In Equation (B.4), the concentration gradient as well as the mass transfer coefficient is unknown. To find a describing Equation for the concentration gradient, the general mass balance with the conditions described above can be used which yields in Equation (B.5) [72, 73, 131-134]. 22yx ycDycvxcv??=????????+?? (B.5) Keeping in mind that the actual goal is to determine the wall shear stress, the velocity terms vx and vy in Equation (B.5) are the relation to Newton?s law for viscosity. For the relatively high Schmidt numbers typically found in aqueous solutions (1500-2000), as mentioned above the velocity profile can be assumed to have a linear slope in the concentration boundary layer, and therefore can be defined using the relationship presented in Equation (B.6). Note that the assumption of linearity is valid regardless of the flow regime for relatively high Schmidt numbers. y)x(yyvv xx ?=??=  (B.6) Assuming no dependency in the z-direction, i.e. no flow normal to the cathode, Equation (B.6) can be substituted into the continuity Equation yielding the relationship for vy presented in Equation (B.7) 2y yx)x(21v????=  (B.7) Substituting Equation (B.6) and (B.7) in Equation (B.5) and applying the boundary conditions c=0 at all x and y=0, c=cb at all x and y?? and c=cb at x=0 and all y based on the boundary layer and film theory [135, 136] yields in Equation (B.8) [72, 131, 133, 134] describing the concentration gradient at the surface of the cathode which still depends on x. ( )( )31x02121b310ydx)x()x(893.0cD91yc?????? ????????=?????????= (B.8) For a very small cathode, a very thin concentration boundary layer is established on the cathode, making the solution presented in Equation (B.8) independent of the geometry of the system being investigated. The mean mass transfer coefficient over the entire cathode can be described using Equation (B.9). 152  ?=????????=charL0 0ycharbm dxycL1Dck  (B.9) Introducing Equation (9) into the relationship for the Sherwood number yields Equation (B.10)  dxdx893.0cD91c1DLkShcharL0 31x02121b31bcharm ?? ????????????????==  (B.10) which when solved yields in Equation (B.11). 312charcharmDL807.0DLkSh ???????? ?==  (B.11) Lchar for a rectangular cathode is the length of the cathode in the main flow direction. For circular electrodes the characteristic length Lchar is equal to the diameter multiplied by a factor of 0.82 [92, 126] as presented in Equation (B.12): 3122mDd82.0807.0Dd82.0kSh ???????? ?==  (B.12) Therefore, for a circular cathode, the mass transfer coefficient can be estimated using Equation (B.13). 313132m dD862.0k ?=? . (B.13) Combining Equations (B.4) and (B.13) and rearranging yields Equation (B.14) that describes the relationship between the current and the shear rate for a circular cathode which can be simplified to Equation (B.15) where the Leveque coefficient describes the relationship between the shear rate at the surface of the cathode and current measured through the electrical system. 313132b dDcFAn862.0I ?=? (B.14) 31LevkI ?=    . (B.15) Because many of the parameters in Equation (B.14) are not accurately known for a given system (e.g. exact cathode surface and diameter) the Leveque coefficient can be estimated by 153  applying a known shear rate at the cathode and measuring the current through the electrochemical system [e.g. [137]]. B.2.6 Dynamic flow conditions - Voltage step response The relationship developed for steady state conditions are valid if the concentration boundary layer at the cathode establishes itself more rapidly than changes in the velocity boundary layer [94]. If not, the mass transfer relationship presented in Equation (B.4) must be modified to take into account that the mass transfer is time dependent as presented in Equation (B.16): ( ) ( ) bmt,0yii ctkdydcDFAntI=?????????== (B.16) The development of the concentration profile as a function of time and distance from the cathode can be described by using Fick?s second law of diffusion [120] presented in Equation (B.17) with c=cb at t=0 and all y, c=0 at y=0 and t>0 and c=cb at t>0 and y??:  22ycDt)t,y(c??=??????????    . (B.17) Solving Equation (B.17) based on the penetration theory which is only valid for a short time after the change [138, 139], for a circular cathode, the mass transfer coefficient can be estimated using Equation (18) tDk m ?=    . (B.18) Combining Equations (B.16) and (B.18) and rearranging yields Equation (B.19) that describes the relationship between the current and the time for a circular cathode which can be simplified to Equation (B.20) where the Cottrell coefficient describes the relationship between the time and current measured through the electrical system 212121b tDcFAnI???=  (B.19) 21Cot tkI?=    . (B.20) 154  The Cottrell coefficient can be estimated by applying a voltage step and measuring the current through the electrical system over time [73, 115]. The characteristic time of the EDM system, which is the time it takes for the current from an applied voltage step (Equation (B.20)) to reach steady state conditions (Equation (B.15)), can be estimated by equating Equations (B.15) and (B.20) yielding Equation (B.21) which describes the characteristic response time of the cathode 322Lev2Cot0 kkt??=    . (B.21) The edge effect, i.e. augmenting the diffusion with additional mass transport from the sides, can also change the behaviour of the cathode under the transient condition [133, 134]. One approach to include the edge effects is to describe this effect with the help of an additional term in Equation (B.20).  021Cot21Cot Itk2FnPDtkI +=+=??   , (B.22) where the intercept I0 stands for the correction for edge effects. As another approach to consider the spatial diffusion related to the edge effect, a numerical model for the solution to a 3 dimensional mass transfer model over the surface of the cathode was developed [140]. Based on this model, a correction of Sh/ShDLA =1+? is suggested, where Sh is the actual Sherwood number, ShDLA is the Sherwood number for diffusion layer approximation where the effect of streamwise and lateral diffusion is neglected (1 dimensional model). For a circular cathode, ? was estimated for a range of modified Pe numbers between 1 and 100.  The edge effect can be neglected at high Pe numbers where 1) the area of the spatial diffusion is very small compared to the total area of the cathode or 2)  the velocity and therefore the convection is high so that the spatial diffusion is negligible [140, 141] (see also section B.2.6). - Approximate model of the cathode dynamic response Knowing the characteristic time of the system, it is possible to correct the wall shear rate measured at conditions when the concentration boundary layer is not able to follow rapid changes in the velocity boundary layer. To take into account that the mass transfer is time dependent, Equation (B.6) is modified as presented in (B.23).  155  y)t,x(vx ?=    . (B.23) As described above, the mass transfer relationship presented in Equation (B.4) must be modified to take into account that the mass transfer is time dependent as presented in Equation (B.16). In Equation (B.16), the concentration gradient as well as the mass transfer coefficient is unknown. To find a describing Equation for the concentration gradient, the general mass balance with the conditions described above can be used in combination with Equation (B.23) which yields in Equation (B.24). ??????????+??=???+??2222ycxcDxcy)t,x(tc   . (B.24) The concentration profile at the surface of the cathode can be approximated using Equation (B.25)[142-145].  ??????????????????????????=31b xl)t(yG1c)t,y,x(c  (B.25) where G=f(?) is a decreasing function assuring G(0)=1, G(? )=0, G?(0)=-1. By substituting Equation (B.25) into Equation (B.24), assuming axial diffusion is negligible and integrating near the cathode surface in the viscous boundary layer yields Equation (B.26) that can be used to calculate a transient, i.e corrected, shear rate from the shear rate obtained assuming steady state conditions (Equation(B.15)). ???????????+?=?tt32)t()t( s0sc    . (B.26) Substituting Equation (B.15) and (B.21) into Equation (26) yields a relationship similar to that presented in Equation (15) but that can be used for steady and unsteady flow conditions as presented in Equation (B.27). ??????????+=? ?tIk2Ik)t( 2Cot33Levc  (B.27) Note that Equation (B.27) is not valid at conditions with very large or rapid flow fluctuations. Under these conditions, Equation (B.27) provides only rough estimates of extreme values, with maxima determined more precisely (as probe response is better at high wall shear rates). Similarly, when large wall shear rate fluctuations with dimensionless amplitudes 156  1~ ???=?  lead to the near-wall flow reversal, negative values obtained using Equation (B.27) can be used only as qualitative indicators of the flow reversal phenomenon [123]. B.3 Electrodiffusion Method (EDM): application The theory presented in section B.2 was applied to measure the shear rate induced by a gas bubble rising in a vertical flow cell. This application is motivated by the measurement of wall shear stress in membrane systems where aeration is used to detach fouling layers from the membranes. Several groups in this field worked with EDM [137]. The experimental setup, the procedure of data processing and the resulting wall shear rate obtained in this study are discussed in the following sections. B.3.1 Experimental setup The experimental apparatus used consisted of a vertical flow cell, a liquid recirculation system and a gas bubble release mechanism. Details of the system are presented in [137] and summarize as follows. The flow cell consisted of a thin vertical acrylic glass tank (height: 1200mm; width: 160mm; thickness: 7mm). An electrochemical solution [38] was circulated through the flow cell at an average upward velocity of 0.2m/s. Single 3 mm diameter bubbles were introduced into the upflowing liquid at the base of the flow cell. An array of 8 x 0.5 mm diameter platinum EDM cathodes were installed along the width of the flow cell on the inside wall, perpendicular to the liquid and bubble rise path 800 mm from the base of the flow cell. Data from the EDM cathodes was collected and conditioned using a custom electrical circuit and data collection system. Nitrogen gas was used to generate the bubble and purge oxygen from the electrochemical solution used. Other than the cathode and anode, all system components in contact with the electrochemical solution were non-conductive. B.3.2 Practical aspects influencing the measurement Knowing the theoretical background of the EDM, it is an obvious fact that it is a very delicate measurement technique to work with. This section summarises the main factors influencing the results of measurements. There are several factors influencing the calibration: ? Surface area of the cathode o Not perfectly circular 157  o Scratches o Breakages or air bubbles in the epoxy resin near the surface o Oxidized layer on the cathode / poisoned cathode ? Concentrations of the electrolytes o degeneration due to photo catalytic reaction ? Concentration of oxygen in the solution as it can cause side reactions ? Material properties o Mainly influenced by the temperature of the solution During the measurement the following hardware related factors can affect the measured signal: ? Current o Magnetic field / Electrostatic field ? from equipment ? e.g. frequency converter of the pump ? mobile phone ? electrolytic solution flowing in (long) tubing o Galvanic cell o resistance in the system ? Conductivity of the electrolyte ? Distance between cathode and anode o Size of the anode o Anything that might influence the circuit (power supply) of the equipment o Amplification (linear/non-linear) o Ohmic resistance in the system o Grounding Chan [49] gives a sensitivity study of the signal to selected parameters from the list above. This list doesn?t claim completeness as there are always factors specific to the used setup and its surroundings. 158  B.3.3 Data conditioning Data conditioning consisted of first establishing a relationship between the measured current and the imposed wall shear rate, i.e. calibration, and then correcting the acquired wall shear rate signal in respect to the unsteady near-wall flow conditions observed during the bubble rise. - Calibration Both direct and indirect calibration is possible. For direct calibration, a known shear rate is applied at the surface of the cathode and a resulting current through the EDM system measured. Equation (B.15) is then applied to determine the relationship between the measured current and the imposed wall shear rate. Ideally, direct calibration is done in-situ. However, for some more complex flow systems where it is not possible to achieve well defined flow conditions in-situ, the calibration should be performed ex-situ. In this case, a versatile removable probe is temporarily moved from the system of interest into a separate ex-situ calibration system where the Leveque coefficient is estimated. Care must be taken to ensure that the temperature and composition of the electrolyte solution in the ex-situ calibration system are the same as those in the system of interest. For indirect calibration, a voltage step approach described in section 2.6.1 is used not only to determine the Cottrell coefficient but also to estimate the Leveque coefficient (see [73]). A typical result obtained from such voltage step experiments for our cathodes is presented in Figure B.1. The Cottrell coefficient is estimated from the slope of the log-log plot of the current measured over time during the transient period of the voltage step. As the current at the beginning and ending of the transient process is influenced by additional effects (such as cathode surface roughness or gradual approaching the magnitude of steady current), only a middle linear part of the transient response is considered for data regression. Our experience indicates that such a relevant time interval is ranging from 0.01 s to 0.5 t0. As the characteristic response time t0 is not known beforehand, it has to be estimated by an iterative procedure schematically illustrated in Figure B.2. Here t0 value is determined by the interception between the transient and steady state asymptotes. For the data presented in Figure B.1, this procedure provides for the Cottrell coefficient a value of kCot=1.13*10-6 As1/2. The theoretical relationship for the Cottrell coefficient (compare Equations (B.19) and (B.20)) is presented in Equation (B.28). 159  2121bCot DcFAnk??=  (B.28) Equation B.28section can be used to obtain an estimate of the diffusion coefficient D. For the known parameters of applied electrochemical system (n=1, F=96485 C/mol, cb=3 mol/m3, d=0.0005 m, and A=?d2/4= 0.196 mm2), it provides a value of 1.24*10-9 m2/s. This value is in good agreement with those measured in previous experiments (see chapter B.2.3) and therefore it can be also applied to determine indirectly the Leveque coefficient. Combining the Equations (B.14) and (B.15) klev can be calculated using Equation B.29. 3132bLev dDcFAn862.0k?=  (B.29) The theoretical value calculated was kLev=6.9*10-7 As1/3. For comparison, the Leveque coefficient obtained by direct calibration done in our experimental set-up under conditions of the laminar single-phase channel flow has a similar value of 6.3*10-7 As1/3.   Figure B-1  Voltage step data and the different regression lines    10-410-310-210-110010110-610-510-410-3t [s]I [A]measured currentsteady statetransient (iterations)t0loglog160   Figure B-2  Structured flow chart for the iteration to determine kCot   Considering the influence of edge effects as discussed in section B.2.6, the Peclet number calculated for typical wall shear rate in our experimental set/up (?=100 s-1) has a value of  Pe=? d2/D = 31000. The modified Peclet number H= 52 then provides a small edge-effect correction factor of ? = 0.02 (see [141] for details), which means that a correction factor of 2% would be needed to find the actual Sherwood number considering the edge effects. - Signal acquisition and pre-processing The accuracy of the correction for dynamic flow conditions can be affected by the signal sampling rate. If the sampling frequency is too small, peaks, i.e. maximums and minimums, in the signal can be damped, whereas background noise in the measured signal can be amplified if the sampling rate is too large. In general, the sampling frequency should be at least twice that of t0,counter=10-1si,0i,01i,00 tttt?=? +steady2i,Cot1i,0 Ikt =+?t0<0.1%i,CotCot kk =yesno1ii +=kCot,i from linear regression of I=kCot,ix+n with x=t-? for interval t=[10-2s 0.5t0,i]161  the frequency of expected flow fluctuations in the system [146]. For the investigated flow system, this frequency can be estimated from the time necessary for a bubble to pass the sensor (bubble size/bubble velocity). Considering 3 mm bubbles rising in co-current liquid flow of 0.2 m/s, the minimal sampling frequency is estimated to be 200 Hz. To obtain wall shear rate response to a rising bubble in more detail, the sampling frequency ranging from 500 to 750 Hz have been used [137]. The advantage of using higher frequencies is that a moving average can be used to remove background noise without substantially dampening the peaks in the signal. Figure B.3a presents typical results for a signal sampling frequency of 500 Hz, with and without averaging. As presented, averaging over up to 8 sampling events significantly reduces the background noise of the signal (Figure B.3b) without significantly affecting the overall profile of the signal (Figure B.3a). Taking into account the characteristic response time of the sensor (estimated to be in the order of 0.1 s), the signal pre-processing is necessary also for the correct calculation of time derivatives needed for the signal correction. 162   Figure B-3 Current generated by a bubble with moving averages of different averaging ranges  (a) and standard  deviations of the signal for different averaging ranges (b)   B.3.4 Wall shear rate calculation and correction With the Leveque and Cottrell coefficient determined from direct or indirect calibration, the properly measured current signal through the electrochemical system can be converted to wanted wall shear rate course. When time variations of the current are slow, i.e. dI/dt values become negligibly small, the quasi-steady Equation (B.15) can be applied to calculate the actual wall shear rates. As a general rule of thumb based on experience, a criterion for such quasi-steady measurement conditions can be expressed by an equality tIk2 2Cot ??< 0.05 I?, holding for the whole time of measurement. Under unsteady flow conditions, as in the case of near-wall flow induced by rising bubbles, this condition is not fulfill, the frequency response of electrodiffusion 400 420 440 460 480 5004567x 10-6current [A]0 10 20 30 40 5000.511.52standard deviation [%]a)b)20 100sampling eventsaveraging range0 60 803           original signal           averaging range              2  4  8  16   163  sensors should be taken into account, and Equation (B.27) has to be used for wall shear rate calculations. Typical results from wall shear rate measurements are presented in Figure B.4. The shear rate calculated with Equation (B.15) is illustrated in Figure B.4a is demonstrated for two conditions, first condition where the steady state flow conditions prevail and second where the dynamic flow conditions prevail. Before and after the data shown here a steady state flow was apparent as well. Figure B.4b shows the same data treated with a moving average with an averaging range of 10 data points. Here the noise is reduced and the peak value is marginally lowered as well. Figure B.4c and Figure B.4.d show the data from Figure B.4b treated with Equation (B.27) for two different linearization ranges. In Figure B.4c a rather large linearization range of 400 data points, i.e. ?t=?t=0.8 s, was chosen. As expected with such a large linearization range, only minor changes to the data appear. The peak value is slightly increased approximately back to the value before the noise reduction step without increasing the noise as well. In Figure B.4.d on the other hand, a smaller linearization range of 30 data points, i.e. ?t=?t=0.06 s, was chosen. Here, a strong enhancement of the fluctuations is visible. The peak value is increased by more than 100%, a negative peak directly following the positive peak value is apparent and the fluctuations after the peak are enhanced as well. As mentioned earlier in section 2.6.2, the actual value of the negative peak should not be used in the analysis but it should rather be seen qualitatively as an indicator for flow reversals. This is reasonable here as, when the bubble passes by the cathodes, only a thin liquid film between bubble and cathode exists in which a flow reversal due to displacement of the liquid around the bubble can happen. 164   Figure B-4 Shear rate from the raw data (Equation (B.15)) (a), treated with a moving average with an averaging range of 10 (b), then corrected with Equation (B.27) with a linearization range of 0.8 s (=400 sampling events) (c) and 0.06 s (=30 sampling events) (d)     0 200 400-1000010002000sampling eventsshear rate [1/s]0 200 400-1000010002000sampling eventsshear rate [1/s]0 200 400-1000010002000sampling eventsshear rate [1/s]0 200 400-1000010002000sampling eventsshear rate [1/s]a) b)c) d)165   B.4 Conclusions The purpose of this chapter was to study the theoretical background necessary to interpret the data provided by the EDM in practical applications. Complete derivations of the steady flow equations that govern the mass transfer induced by the electrochemical reaction in the system are gathered from the available EDM literature. Methods used for transient flow conditions are introduced.  A procedure for correction of the signal under transient conditions is developed.  A new algorithm is suggested to evaluate transient calibration data where the Cottrell calibration factor is determined iteratively. Practical aspects and limitations of the in-situ calibrations are discussed.  This optimized method is then applied to a system where a single bubble rises in a vertical, narrow rectangular channel with a co-current flow of the electrolyte solution. These unsteady wall shear stress data are used here as an example to demonstrate the practical aspects of the data interpretation. As illustrated in the case of resulting wall shear rate response to a rising bubble, the optimized EDM method and the correction procedure enables measuring shear stress under transient flow conditions and revealing a short-time near-wall flow reversal even if the measurement is carried out with a single-segment probe.   166   Appendix C : Matlab codes developed to process voltage signals   V-step in-situ calibration of the shear probes  clear all; close all; % find the directory where the files are in % there should be no text header for the files, i.e. only containe numbers direc='D:\jan-30-2012-probe calibration\'; %Load bias for each channel fileIn=strcat('BiasAverage.txt'); O(:,:)=load(fileIn); %Load files for probe calibration Format "calibration_.txt" %_ is the name of the txt file=probe identification\   for a =1:4; number=num2str(a);     file=[direc '150-50_trigger_1_',number,'.txt']; T(:,:)=load(file); [m,n]=size(T); S(m,2)=0; %correct data for channel bias     %put corrected data in matrix S     % c is the number of columns (n-1, channel, or probe)here is 2 because     % only one channel (zero on the box) was used for calibration     %     %0.00025 because data was collected at 4000Hz         for c=2;             S(:,c)=T(:,c)-O(1,c-1);         end                      %plot((1:m),squeeze(S(:,2))); %%         %%IMPORTANT   %Stop here plot S and find the time where the graph starts to rise: 1700 is %the difference between the max and min where the graph is linear              %create a matrix (S) where the calibration resutls will be stored     %Add time to the first column of matrix      167      %%%%%%%duration of the rise of V signal from infinity, it can be     %%%%%%%less than nop!!!            %0.01 because data was collected at 1000Hz      % number of points=nop      nop=6000;     for d=1:nop         M(d,1)=0.001*d;     end              %add t^(-0.5) to column 2     %data is fitted to this value V=(AR)at^(-05)     %V from the probes, AR is amplification x resistance (1000x100)     M(1:nop,2) = (M(1:nop,1)).^(-0.5);                        %Cycle through different columns (column 1 is time so start at 2)     %c is the column number    % for         c=2;           %set threshold above which data of interest starts         %this corresponds to point when current is applied         %t is the threshold value         %sign needs to be reversed since data in file is negative         t=-min(S(:,c));           %cycle through each row         %%%% important : r is the row number        %        for r=1:m;                          %v is the value in row r and column c             %sign needs to be reversed since data in file is negative             v=-S(r,c);                         %is current value greater than the threshhold             %if yes, collect data for 0.75 seconds             if v==t              %if abs(v-t)<=abs(t)*0.001                 %collect all data for  rows below v                 168                  %save to column C+1 since column 2 in M is for t^-.5                 %sign needs to be reversed since data in file is negative                 M(1:nop,c+1)=-S(r:r+nop-1,c);                                  %exit from the current for loop that cycles through rows                 break                end                    end         %end         %fit line to data V=a(AR)t^(-0.5)         %V is y value and t^(-0.5) is x value         %two first parameters in polytif function are x and y         %ignore the first data points i.e start at 10.         %consider points until the 200                 % IMPORTANT**********************                  %% for some reason the graph is inveresly ploted so the data should         %% be taken from the end                  st=145;         fn=1030;         %% same comment we measure only one channel         c=2;                  P=polyfit(M(st:fn,2),M(st:fn,c+1),1);                  %save resutls in a matrix R         %firt row = fitted a[AR], slope         %second row=residuals         R(a,c-1)=P(1,1);         R(a,c)=P(1,2); %% Signal correction         %f is the slope of quasi-steady shear signal, taken from the last         %points of the graph         f=mean (M(2000:6000,c+1));         %find the Equation for cottrelle part          PP=polyfit(M(st:fn,1),M(st:fn,c+1),1);                            %save resutls in a matrix R         %firt row = fitted a[AR], slope         %second row=residuals 169          RR(a,c-1)=PP(1,1);         RR(a,c)=PP(1,2);         %calculate the intersection of Cottrelle and Leveque asymptots         t0(a,1)= (f-RR(a,c))/RR(a,c-1);         % M is plotted versus (t^-0.5)        % t0(a,2)= (t0(a,1))^(-2);         Ave_t0=mean(t0(a,1)); %%         %%plot fitted results         %plot of  data V vs. t^(-0.5)         subplot(2,3,a)         plot(M(st:fn,2),M(st:fn,c+1))         hold on;         %Plot of fitted data V vs. t^(-0.5)         X = linspace(min(M(st:fn,2)),max(M(st:fn,2)),500);          Y = polyval(P,X);   % values for fit curve          plot(X,Y,'--k');    % draw fit curve          xlabel('t^ (-0.5) (1/s ^ (-0.5))');         ylabel('V(volts)');         hold off;                % a;              %Generate correction factors  ????? U(a,:)=R(a,:)/(1000*100);  % 100 is the resistance of the first channel % IMPORTANT***************** resistancew should be checked for diff % channels end     %% you can change the parameters for T and viscosity correction %  b=0.67521; % is the constant 1/(Pi*k^-2) d=0.000000000676; %Diffusion coefficent m2/s ra=0.0005; %diameter or radius , m u=0.00097;  % viscosity %eq 5-Sobolik: K/a=((b^(-1/2))*(d^(1/6))*(ra^(-1/3))); %Klev: K(:,:)=U(:,:).*((b^(-1/2))*(d^(1/6))*(ra^(-1/3))); clear C1;      Klev=mean(K(:,1));   170    C1(:,1)=u*((K(:,1).*100000).^(-3));     %Calculate mean of all replicate calibration tests %only aerage between first and last file (in case first file in not 1) Ca=mean(C1(:,1))     %% signal correction % Cottrell Coeff. = slope of V versus t^(-0.5)/RA kc= mean(R(:,1))./(100*1000);   % Leveque Coeff.= (Ca*(RA)^3)/viscosity   k1(1,1)= ((100000^3)*Ca(1,1))/u;   fileOut = strcat('Leveque_Coeff_probe0.txt'); save( fileOut ,'Klev','-ascii');   fileOut = strcat('(Leveque_Coeff)^-3_probe0.txt'); save( fileOut ,'k1','-ascii');   fileOut = strcat('t0_probe0.txt'); save( fileOut ,'Ave_t0','-ascii');   fileOut = strcat('cottrell coeff_probe0.txt'); save( fileOut ,'kc','-ascii'); %%  fileOut = strcat('CalibrationFactorAverage_probe0.txt'); save( fileOut ,'Ca','-ascii');     %% figure (2) for d=1:10000 S(d,1)=0.001*d; end plot(S(:,1),-S(:,c),'. b'); %hold on %plot(M(:,1),M(:,c+1),'. k'); %hold on %plot(M(st:fn,1),M(st:fn,c+1),'. r'); %plot(M(:,2),M(:,c+1),'. k');   figure (3) plot(M(:,1),M(:,c+1),'. k'); 171  hold on plot(M(st:fn,1),M(st:fn,c+1),'. r'); figure(4); plot(M(:,2),M(:,c+1),'. k'); hold on plot(M(st:fn,2),M(st:fn,c+1),'. r'); %Save matrix Ca to File %% change the name of the file % fileOut = strcat('CalibrationFactorAverageDouble.txt'); % save( fileOut ,'Ca','-ascii');     Correction of data under transient flow condtions clear all; close all; % find the directory where the files are in % there should be no text header for the files, i.e. only containe numbers direc='D:\January-29-2012-shear data\'; %Load bias for each channel fileIn=strcat('BiasAverage.txt'); Q(:,:)=load(fileIn); %% %viscosity u=0.00097; %% there are four Leveque Coefficients, one for each channel fileIn = strcat('(Leveque_Coeff)^-3_probe0.txt'); k1(1,1)=load(fileIn);   fileIn = strcat('(Leveque_Coeff)^-3_probe1.txt'); k1(1,2)=load(fileIn);   fileIn = strcat('(Leveque_Coeff)^-3_probe2.txt'); k1(1,3)=load(fileIn); fileIn = strcat('(Leveque_Coeff)^-3_probe3.txt'); k1(1,4)=load(fileIn); %theretical 1/Klev ^3 %k1(1,1)=1.05e19; %% there are four t0, one for each channel fileIn = strcat('t0_probe0.txt'); Ave_t0(1,1)=load(fileIn); fileIn = strcat('t0_probe1.txt'); Ave_t0(1,2)=load(fileIn); fileIn = strcat('t0_probe2.txt'); Ave_t0(1,3)=load(fileIn); fileIn = strcat('t0_probe3.txt'); 172  Ave_t0(1,4)=load(fileIn);   %% %% there are four Cottrell Coefficients , one for each channel fileIn = strcat('cottrell coeff_probe0.txt'); kc(1,1)=load(fileIn); fileIn = strcat('cottrell coeff_probe1.txt'); kc(1,2)=load(fileIn); fileIn = strcat('cottrell coeff_probe2.txt'); kc(1,3)=load(fileIn); fileIn = strcat('cottrell coeff_probe3.txt'); kc(1,4)=load(fileIn);   k2=2*kc.^2; %Cottrelle theory %k2(1,1)=1.2e-12; j=0; for f=1:3;  number=num2str(f);   %O is the correction factor for  channels O(:,:)=[0.49 0.33  0.44  0.33];   % P2-test-F=1CC=8.txt is the name of the shear measurements fileIn=strcat('150-10_',number,'.txt'); V(:,:)=load(fileIn); [m,n]=size(V); for c=2:5;             G(:,c)=V(:,c)-Q(1,c-1); end   %remove the first part of the graph before trigger %t= mean(V(:,6)); % for c=2:5;  %    for r=1:m;   %          if V(r,6)< (t-0.5)                 %if abs(v-t)<=abs(t)*0.001                 %collect all data for 0.75 seconds in rows below v                 %data acquired at 4000 hz so 3000 rows)                 %save to column C+1 since column 2 in M is for t^-.5                 %sign needs to be reversed since data in file is negative    %             T(1:(m-r+1),c)=G(r:m,c);                                 %exit from the current for loop that cycles through rows     %   break                   %       end 173                %end  %end    %move time scale   %nop=m-r+1;    % for d=1:nop     %    T(d,1)=0.01*d;     %end            T(:,:)=G(:,:);  %change m and n to new values        [m,n]=size(T);      U(:,2)=-(T(:,2).^3); U(:,3)=-(T(:,3).^3); U(:,4)=-(T(:,4).^3); U(:,5)=-(T(:,5).^3); %for s=1:m; S(:,2)=U(:,2).*O(1,1); S(:,3)=U(:,3).*O(1,2); S(:,4)=U(:,4).*O(1,3); S(:,5)=U(:,5).*O(1,4); %%converting voltage signal to current I=-V/RA T(:,:)=-T(:,:)./100000;   for c=2:5;      a=mean(S(:,c));    s=std(S(:,c));   % use z factor for 90 % will be 1.64    e=1.64*s;  %Save resutls in a matrix format with first row having averages         %and second row having errors (rows= txt files= replicates,         %columns= probes=channels)         %means stored in matrix R    R(f,c-1)=a;       %errors stored in matrix P,(rows= txt files= replicates,         %columns= probes=channels)    P(f,c-1)=e;    %% 0.00025 because data was collected at 4000Hz %% removing outliers, larger than 90% in a normal distribution        % check for the number of changed values due to a too high change of the    % value from one time step to the other      for dm=4 174              for i=1:m       if i>0 andand i<dm+1         if ((abs(S(i,c)-mean(S((1:dm),c))))> 2.*std(S((1:dm),c)))             S_clean(i,c)= mean (S(1:dm,c));         else             S_clean (i,c)= S(i,c);         end         elseif i>dm andand i<m-dm           dev=std(S((i-dm):(i+dm),c));          if abs(S(i,c)- (mean(S((i-dm):(i+dm),c))))> 2*dev             S_clean(i,c)= mean(S((i-dm):i+dm,c));             else             S_clean (i,c)= S(i,c);         end                    elseif i>m-(dm+1) andand i<m+1           dev=std(S((m-dm):m,c));        if abs(S(i,c)-mean(S((m-dm):m,c)))> 2*dev           S_clean(i,c)=mean(S((m-dm):m,c));             else             S_clean (i,c)= S(i,c);       end        end                 end       a_clean=mean(S_clean(:,c));    s_clean=std(S_clean(:,c));   % use z factor for 90 % will be 1.64    e_clean=1.64*s_clean;  %Save resutls in a matrix format with first row having averages         %and second row having errors (rows= txt files= replicates,         %columns= probes=channels)         %means stored in matrix R    R_clean(f,c-1)=a_clean;       %errors stored in matrix P,(rows= txt files= replicates,         %columns= probes=channels)    P_clean(f,c-1)=e_clean;        figure (6);   subplot(2,3,f)   plot( dm, s_clean, '*r');   xlabel('number of points');         ylabel('STD for cleaned shear stress');    %hold on     175  %% Exponentially weighted moving average constant Q=0.3 QQ=0.3        i=1;             for i=1:m       if i>0 andand i<dm+1          S_moving_ave(i,c)=mean(S_clean(1:dm,c)) ;           S_ave(i,c)=mean(S_clean(1:dm,c));       elseif i>dm andand i<m-dm          S_ave(i,c)=mean(S_clean((i-dm):(i+dm),c));          S_moving_ave(i,c)= (1-QQ).*S_clean(i,c)+ (1-QQ)*(QQ)*S_clean (i-1,c)+(1-QQ)*(QQ^2)*S_clean (i-2,c)+(1-QQ)*(QQ^3)*S_clean (i-3,c)+(1-QQ)*(QQ^4)*S_clean (i-4,c);       elseif i>m-(dm+1) andand i<m+1          S_ave(i,c)=mean(S_clean((m-dm):m,c));            S_moving_ave(i,c)= ((1-QQ).*S_clean(i,c))+ ((1-QQ)*(QQ)*S_clean (i-1,c))+((1-QQ)*(QQ^2)*S_clean (i-2,c))+((1-QQ)*(QQ^3)*S_clean (i-3,c)+(1-QQ))*((QQ^4)*S_clean (i-4,c));       end          end a_ave=mean(S_ave(:,c));    s_ave=std(S_ave(:,c));   % use z factor for 90 % will be 1.64    e_ave=1.64*s_ave;  %Save resutls in a matrix format with first row having averages         %and second row having errors (rows= txt files= replicates,         %columns= probes=channels)         %means stored in matrix R    R_ave(f,c-1)=a_ave;       %errors stored in matrix P,(rows= txt files= replicates,         %columns= probes=channels)    P_ave(f,c-1)=e_ave;      a_moving_ave=mean(S_moving_ave(:,c));    s_moving_ave=std(S_moving_ave(:,c));   % use z factor for 90 % will be 1.64    e_moving_ave=1.64*s_moving_ave;  %Save resutls in a matrix format with first row having averages         %and second row having errors (rows= txt files= replicates,         %columns= probes=channels)         %means stored in matrix R    R_moving_ave(f,c-1)=a_moving_ave;       %errors stored in matrix P,(rows= txt files= replicates,         %columns= probes=channels)    P_moving_ave(f,c-1)=e_moving_ave; figure (8);   subplot(2,3,f)    plot( dm, s_ave, '*b'); 176     %hold on        xlabel('number of points averaged');         ylabel(' STD for averaged shear stress');    figure (18);   subplot(2,3,f)    plot( dm, s_moving_ave, '*b');    %hold on        xlabel('number of points averaged');         ylabel(' STD for averaged shear stress');         figure (17);    plot((1:m).*0.005,squeeze(S_clean(:,c)-S_ave(:,c)),' r');  xlabel('time');         ylabel(' S-S_ave');    end %% put the first column as time for r=1:m S(r,1)=0.005*r;   end % S_clean:  shear stress without outiers   for r=1:m S_clean(r,1)=0.005*r;   end % S_correct: corrected shear stress   for r=1:m S_correct(r,1)=0.005*r;   end  % S_ave is the  shear stress smooth  for r=1:m S_ave(r,1)=0.005*r;   end  %T_clean: smooth curve for voltage signal    for r=1:m T_clean(r,1)=0.005*r;   end % T_correct: corrected voltage signal   for r=1:m T_correct(r,1)=0.005*r;   end %current   for r=1:m T(r,1)=0.005*r;   end % smooth current   for r=1:m 177  T_ave(r,1)=0.005*r;   end   %% % Correcting the signal based on personal communication with Dr. Tihon  changes=1;  i=1;      for dt=10     % dt=50;     for i=1:m       if i>0 andand i<dt+1          S_correct(i,c)=S_ave(i,c);       elseif i>dt andand i<m-dt           slope(i,c)=regress(S_ave((i-dt):(i+dt),c),S_ave(((i-dt):(i+dt)),1));         % S_correct(i,c)= S_ave(i,c)+((2/3).*Ave_t0(1,c-1).*(S_ave(i-dt,c)-S_ave(i+dt,c))/(S_ave(i-dt,1)-S_ave(i+dt,1)));        % S_correct(i,c)= S_ave(i,c)+((2/3).*Ave_t0(1,c-1).*slope(i,c));                % dI/dt from polyfit.m with y=mx+n             param3=polyfit(S_ave(((i-dt):(i+dt)),1),S_ave((i-dt):(i+dt),c),1);             slope3(i,c)=param3(1);                          S_correct(i,c)= S_ave(i,c)+((2/3).*Ave_t0(1,c-1).*slope3(i,c));          elseif i>m-dt-1 andand i<m+1          S_correct(i,c)=S_ave(i,c);         end      end      a_correct=mean(S_correct(:,c));    s_correct=std(S_correct(:,c));   % use z factor for 90 % will be 1.64    e_correct=1.64*s_correct;  %Save resutls in a matrix format with first row having averages         %and second row having errors (rows= txt files= replicates,         %columns= probes=channels)         %means stored in matrix R    R_correct(f,c-1)=a_correct;       %errors stored in matrix P,(rows= txt files= replicates,         %columns= probes=channels)    P_correct(f,c-1)=e_correct;    figure (9);   subplot(2,3,f)    plot( dt, s_correct, '*b');    hold on          xlabel('number of points'); 178          ylabel('STD for corrected shear stress');  %figure (f) %subplot(8,8,changes) %plot((1:m).*0.005,squeeze(S(:,c)),'- b'); %hold on %plot((1:m).*0.005,squeeze(S_clean(:,c)),' y');  %hold on; %plot((1:m).*0.005,squeeze(S_ave(:,c)),'k');  %hold on %plot((1:m).*0.005,squeeze(S_correct(:,c)),'- r'); %hold on %plot((1:m).*0.005,squeeze(S_moving_ave(:,c)),'- g'); %hold on %changes = changes+1;      end     %% i=1; changes =1;    dm=4             for i=1:m       if i>0 andand i<dm+1         if ((abs(T(i,c)-mean(T((1:dm),c))))> 2.*std(T((1:dm),c)))             T_clean(i,c)= mean (T(1:dm,c));         else             T_clean (i,c)= T(i,c);         end         elseif i>dm andand i<m-dm           dev=std(T((i-dm):(i+dm),c));          if abs(T(i,c)- (mean(T((i-dm):(i+dm),c))))> 2*dev             T_clean(i,c)= mean(T((i-dm):i+dm,c));             else             T_clean (i,c)= T(i,c);         end                    elseif i>m-(dm+1) andand i<m+1           dev=std(T((m-dm):m,c));        if abs(T(i,c)-mean(T((m-dm):m,c)))> 2*dev           T_clean(i,c)=mean(T((m-dm):m,c));             else             T_clean (i,c)= T(i,c);       end        end         179          end       a_clean=mean(T_clean(:,c));    s_clean=std(T_clean(:,c));   % use z factor for 90 % will be 1.64    e_clean=1.64*s_clean;  %Save resutls in a matrix format with first row having averages         %and second row having errors (rows= txt files= replicates,         %columns= probes=channels)         %means stored in matrix R    RT_clean(f,c-1)=a_clean;       %errors stored in matrix P,(rows= txt files= replicates,         %columns= probes=channels)    PT_clean(f,c-1)=e_clean;        figure (7);   subplot(2,3,f)   plot( dm, s_clean, '*r');   xlabel('number of points');         ylabel('STD for cleaned current');    hold on     %%  i=1;         for i=1:m       if i>0 andand i<dm+1          T_ave(i,c)=mean(T_clean(1:dm,c));       elseif i>dm andand i<m-dm          T_ave(i,c)=mean(T_clean((i-dm):(i+dm),c));       elseif i>m-(dm+1) andand i<m+1          T_ave(i,c)=mean(T_clean((m-dm):m,c));         end          end a_ave=mean(T_ave(:,c));    s_ave=std(T_ave(:,c));   % use z factor for 90 % will be 1.64    e_ave=2*s_ave;  %Save resutls in a matrix format with first row having averages         %and second row having errors (rows= txt files= replicates,         %columns= probes=channels)         %means stored in matrix R    RT_ave(f,c-1)=a_ave;       %errors stored in matrix P,(rows= txt files= replicates,         %columns= probes=channels)    PT_ave(f,c-1)=e_ave;   figure (10); 180    subplot(2,3,f)    plot( dm, s_ave, '*b');    xlabel('number of points');         ylabel('STD for averaged current');    hold on    %changes=changes+1;            % Correcting the signal  changes=1; i=1;      for dt=10;     % dt=50     for i=1:m       if i>0 andand i<dt+1          T_correct(i,c)=u*((k1(1,c-1)).^(1)).*((T_ave(i,c)).^3);       elseif i>dt andand i<m-dt         %T_correct(i,c)= u*((k1(1,c-1)).^(-3)).* (((T_ave(i,c)).^3) + (k2(1,c-1)).*((T_ave(i-dt,c)-T_ave(i+dt,c))/(T_ave(i-dt,1)-T_ave(i+dt,1))));         %slope_T(i,c)=regress(T_ave((i-dt):(i+dt),c),T_ave(((i-dt):(i+dt)),1));                 %T_correct(i,c)= u*((k1(1,c-1)).^(-3)).* (((T_ave(i,c)).^3) + (k2(1,c-1)).*(slope_T(i,c)));                            param4=polyfit(T_ave(((i-dt):(i+dt)),1),T_ave((i-dt):(i+dt),c),1);             slope4(i,c)=param4(1);         T_correct(i,c)= u*((k1(1,c-1)).^(1)).* (((T_ave(i,c)).^3) + (k2(1,c-1)).*(slope4(i,c)));                elseif i>m-dt-1 andand i<m+1          T_correct(i,c)=u*(k1(1,c-1).^(1)).*((T_ave(i,c)).^3);         end      end     a_correct=mean(T_correct(:,c));    s_correct=std(T_correct(:,c));   % use z factor for 90 % will be 1.64    e_correct=1.64*s_correct;  %Save resutls in a matrix format with first row having averages         %and second row having errors (rows= txt files= replicates,         %columns= probes=channels)         %means stored in matrix R    RT_correct(f,c-1)=a_correct;       %errors stored in matrix P,(rows= txt files= replicates,         %columns= probes=channels)    PT_correct(f,c-1)=e_correct;    figure (16);   subplot(2,3,f) 181     plot( dt, s_correct, '*b');    hold on         xlabel('number of points');         ylabel('STD for corrected current');        % figure (f) %subplot(8,8,changes)   %plot((1:m).*0.005,squeeze((T_correct(:,c))),'b'); %hold on %xlabel('time(s)');   %      ylabel('current');    %      changes = changes+1;     % end             %% put a line where the video is recorded at      %plot ([7752.*0.005  7752.*0.000667], [0 0.4]);   %hold on  %plot ([7752.*0.005  7752.*0.000667], [0 0.4]);                 end              %% put all S corrected data in one matrix % j number of columns j=j+1; TotalS_correct(:,j)= S_correct(:,c); TotalT_correct(:,j)=T_correct(:,c);   end % as f changes j needs to change %j=j+1;      figure(1); subplot(2,3,f) %plot((1:m).*0.005,squeeze(S(:,2)),'. y'); %hold on %plot((1:m).*0.005,squeeze(S_clean(:,c)),' y');  %hold on; 182  %plot((1:m).*0.005,squeeze(S_ave(:,c)),'r');  %hold on %plot((1:m).*0.005,squeeze(S_moving_ave(:,c)),'- g'); %hold on   plot((1:m).*0.005,squeeze(T_correct(:,2)),'. k');   hold on  plot((1:m).*0.005,squeeze(T_correct(:,3)),'. r');   hold on   plot((1:m).*0.005,squeeze(T_correct(:,4)),'. b');   hold on   plot((1:m).*0.005,squeeze(T_correct(:,5)),'. g');   hold on    xlabel('Time(s)');         ylabel('T correct-shear stress (Pa)');                  figure(45); subplot(2,3,f) plot((1:m).*0.005,squeeze(S(:,2)),'. y'); hold on %plot((1:m).*0.005,squeeze(S_clean(:,c)),' y');  %hold on; %plot((1:m).*0.005,squeeze(S_ave(:,c)),'r');  hold on %plot((1:m).*0.005,squeeze(S_moving_ave(:,c)),'- g'); %hold on   plot((1:m).*0.005,squeeze(S_correct(:,2)),'. k');   hold on  plot((1:m).*0.005,squeeze(S_correct(:,3)),'. r');   hold on   plot((1:m).*0.005,squeeze(S_correct(:,4)),'. b');   hold on   plot((1:m).*0.005,squeeze(S_correct(:,5)),'. g');   hold on    xlabel('Time(s)');         ylabel('S correct-shear stress (Pa)'); %% end   fileOut = strcat(' shear profiles.txt'); save( fileOut ,'T_correct','-ascii'); % plot the trigger %figure(1) %hold on  %plot((1:m).*0.005,squeeze(V(:,6).*(10)),'-r'); %% % area under shear profiles 183    area_S_correct (:,:)= (sum(TotalS_correct(:,j),1)).*0.005;  area_T_correct (:,1:12)= (sum(TotalT_correct(:,1:12))).*0.005;  %std_area_T_correct(:,1:12)=std((TotalS_correct(:,1:12))); fileOut = strcat('area under shear profiles.txt'); save( fileOut ,'area_T_correct','-ascii'); %fileOut = strcat('std_area under shear profiles.txt'); %save( fileOut ,'std_area_T_correct','-ascii');  %%  %total data for each probe  probe0(1:m,1)= TotalT_correct(:,1);  probe0(m+1:2*m,1)= TotalT_correct(:,5);  probe0(2*m+1:3*m,1)= TotalT_correct(:,9);    probe1(1:m,1)= TotalT_correct(:,2);  probe1(m+1:2*m,1)= TotalT_correct(:,6);  probe1(2*m+1:3*m,1)= TotalT_correct(:,10);    probe2(1:m,1)= TotalT_correct(:,3);  probe2(m+1:2*m,1)= TotalT_correct(:,7);  probe2(2*m+1:3*m,1)= TotalT_correct(:,11);    probe3(1:m,1)= TotalT_correct(:,4);  probe3(m+1:2*m,1)= TotalT_correct(:,8);  probe3(2*m+1:3*m,1)= TotalT_correct(:,12);    %%  %RMS    n=length(probe0(:,:));  rms= norm(probe0)./sqrt(n);  fileOut = strcat('RMS of shear-150_10_replicates-probe0.txt'); save( fileOut ,'rms','-ascii');   n=length(probe0(:,:));  rms= norm(probe1)./sqrt(n);  fileOut = strcat('RMS of shear-150_10_replicates-probe1.txt'); save( fileOut ,'rms','-ascii');   n=length(probe0(:,:));  rms= norm(probe2)./sqrt(n);  fileOut = strcat('RMS of shear-150_10_replicates-probe2.txt'); save( fileOut ,'rms','-ascii');   n=length(probe0(:,:));  rms= norm(probe3)./sqrt(n); 184   fileOut = strcat('RMS of shear-150_10_replicates-probe3.txt'); save( fileOut ,'rms','-ascii');   %% % threshold valuse for shear clear Thresh clear rms  n=length(probe0(:,:)); T=1; t=1; r=1; for Th=0.3:0.1:1.5 for T=1:n     if probe0(T,:)>=Th         Thresh(t,1)=probe0(T,:);         t=t+1;     end end for T=1:n     if probe1(T,:)>=Th         Thresh(t,1)=probe1(T,:);         t=t+1;     end end for T=1:n     if  probe2(T,:)>=Th         Thresh(t,1)=probe2(T,:);         t=t+1;     end end  for T=1:n           if   probe3(T,:)>=Th         Thresh(t,1)=probe3(T,:);         t=t+1;     end end nn=length(Thresh(:,:));  rms(r,:)= norm(Thresh(:,:))./sqrt(nn);  r=r+1; end fileOut = strcat('RMS of shear-150-10_trigger_1_between_0.3_to_1.5_Pa.txt'); save( fileOut ,'rms','-ascii');   %% % STD and Avergae 185       Probe0total_average=mean(probe0(:,:));    Probe0total_std=std(probe0(:,:));    fileOut = strcat('Total Average Shear-Probe0-150_10_replicates_trigger_1.txt'); save( fileOut ,'Probe0total_average','-ascii'); fileOut = strcat('Total STD of Shear-Probe0-150_10_replicates_trigger_1.txt'); save( fileOut ,'Probe0total_std','-ascii');      Probe1total_average=mean(probe1(:,:));    Probe1total_std=std(probe1(:,:));    fileOut = strcat('Total Average Shear-Probe1-150_10_replicates_trigger_1.txt'); save( fileOut ,'Probe1total_average','-ascii'); fileOut = strcat('Total STD of Shear-Probe1-150_10_replicates_trigger_1.txt'); save( fileOut ,'Probe1total_std','-ascii');      Probe2total_average=mean(probe2(:,:));    Probe2total_std=std(probe2(:,:));    fileOut = strcat('Total Average Shear-Probe2-150_10_replicates_trigger_1.txt'); save( fileOut ,'Probe2total_average','-ascii'); fileOut = strcat('Total STD of Shear-Probe2-150_10_replicates_trigger_1.txt'); save( fileOut ,'Probe2total_std','-ascii');      Probe3total_average=mean(probe3(:,:));    Probe3total_std=std(probe3(:,:));    fileOut = strcat('Total Average Shear-Probe3-150_10_replicates_trigger_1.txt'); save( fileOut ,'Probe3total_average','-ascii'); fileOut = strcat('Total STD of Shear-Probe3-150_10_replicates_trigger_1.txt'); save( fileOut ,'Probe3total_std','-ascii');   %%  figure; hist(probe0(:,:),100); title('Total shear-T-correct- probe0'); kk=0:0.001:10; n_elements=histc(probe0(:,:),kk); c_elements= cumsum(n_elements); figure plot(kk,c_elements./(max(c_elements))); title('Total shear-T-correct- probe0'); xlabel('Shear stress (Pa)');  ylabel('cumulative frequency');   186    figure; hist(probe1(:,:),100); title('Total shear-T-correct- probe1'); kk=0:0.001:10; n_elements=histc(probe1(:,:),kk); c_elements= cumsum(n_elements); figure plot(kk,c_elements./(max(c_elements))); title('Total shear-T-correct- probe1'); xlabel('Shear stress (Pa)');  ylabel('cumulative frequency');     figure; hist(probe2(:,:),100); title('Total shear-T-correct- probe2'); kk=0:0.001:10; n_elements=histc(probe2(:,:),kk); c_elements= cumsum(n_elements); figure plot(kk,c_elements./(max(c_elements))); title('Total shear-T-correct- probe2'); xlabel('Shear stress (Pa)');  ylabel('cumulative frequency');     figure; hist(probe3(:,:),100); title('Total shear-T-correct- probe3'); kk=0:0.001:10; n_elements=histc(probe3(:,:),kk); c_elements= cumsum(n_elements); figure plot(kk,c_elements./(max(c_elements))); title('Total shear-T-correct- probe3'); xlabel('Shear stress (Pa)');  ylabel('cumulative frequency');           %% cumulative histograms (if change 2 to c, then it will be for each %% channel) for c=2:5;   figure; hist(S_ave(:,c),100); title('shear-S-Ave-probe'); 187  xlabel('shear stress (Pa)');         ylabel('frequency'); kk=0:0.001:50; n_elements=histc(S_ave(:,c),kk); c_elements= cumsum(n_elements); figure plot(kk,c_elements./(max(c_elements))); title('shear-S-Ave-probe'); xlabel('shear stress (Pa)');         ylabel('frequency'); figure; hist(S_correct(:,c),100); title('shear-S-correct');xlabel('shear stress (Pa)');         ylabel('frequency'); kk=0:0.001:50; n_elements=histc(S_correct(:,c),kk); c_elements= cumsum(n_elements); figure plot(kk,c_elements./(max(c_elements))); title('shear-S-correct-probe'); xlabel('shear stress (Pa)');         ylabel('frequency');   figure; hist(T_correct(:,c),100); title('shear-T-correct-probe');xlabel('shear stress (Pa)');         ylabel('frequency'); kk=0:0.001:50; n_elements=histc(T_correct(:,c),kk); c_elements= cumsum(n_elements); figure plot(kk,c_elements./(max(c_elements))); title('shear-T-correct');xlabel('shear stress (Pa)');         ylabel('frequency'); end         %% %Average and STD of corrected shear values with Klev fileOut = strcat('T_correct_flowcell-150-15psi_50_trigger.txt'); save( fileOut ,'RT_correct','-ascii'); fileOut = strcat('T_std_probe_flowcell-150-15psi_50_trigger.txt'); save( fileOut ,'PT_correct','-ascii');   %Average and STD of shear values with Calibration factor   fileOut = strcat('S_ave_probe_flowcell-150-15psi_50_trigger.txt'); save( fileOut ,'R_ave','-ascii'); 188  fileOut = strcat('S_std_probe_flowcell-150-15psi_50_trigger.txt'); save( fileOut ,'P_ave','-ascii'); %Average and STD of shear values with Calibration factor   fileOut = strcat('S_correct_probe_flowcell-150-15psi_50_trigger.txt'); save( fileOut ,'R_correct','-ascii'); fileOut = strcat('S_correct_STD_probe_flowcell-150-15psi_50_trigger.txt'); save( fileOut ,'P_correct','-ascii');   %Average and STD of ave current signal values with Klev   fileOut = strcat('T_ave-current signal_probe_flowcell-150-15psi_50_trigger.txt'); save( fileOut ,'RT_ave','-ascii'); fileOut = strcat('T_std-current signal_probe_flowcell-150-15psi_50_trigger.txt'); save( fileOut ,'P_ave','-ascii'); %hold on %plot((1:m).*.00025,squeeze(S(:,3)));   %fileOut = strcat('Average-Shear_0.081m_s_F1C5.txt'); %save( fileOut ,'average','-ascii'); %fileOut = strcat('STD-Shear_0.081m_s_F1C5.txt'); %save( fileOut ,'Deviation','-ascii');       %% Histogram of corrected S for each probe but total of f files instead of %% c 2:5, choose c =2 ; total=f*m; Total=reshape(TotalS_correct,total,1); figure; hist(Total(:,:),100); title('shear-S-correct-'); kk=0:0.02:1; n_elements=histc(Total(:,:),kk); c_elements= cumsum(n_elements); figure plot(kk,c_elements./(max(c_elements))); title('shear-S-correct'); xlabel('Shear stress (Pa)');  ylabel('cumulative frequency');   %%       %% 189  % total histogram total=f*m; Total=reshape(TotalS_ave,total,1); figure; hist(Total(:,:),100); title('shear-S-ave-'); kk=0:0.02:1; n_elements=histc(Total(:,:),kk); c_elements= cumsum(n_elements); figure plot(kk,c_elements./(max(c_elements))); title('shear-S-ave'); xlabel('Shear stress (Pa)');  ylabel('cumulative frequency');       190  Appendix D Matlab codes developed to process images and the data obtained from PIV   clear all file='D:\Jan-29-2012\300-10-trigger2-full\300-10-trigger2avi'; nframes=aviinfo(file);   nframes=nframes.NumFrames; nx=103;%number of int zones in x ny=78;%number of int zones in y n1=142252;%number of first file nend=144050;%number of last file n=nend-n1+1;% the total number of files bigim2(1:ny,1:nx,1:n)=0; bigim4(1:ny,1:nx,1:n)=0; for i=1:n; im=aviread(file,i); im=double(im.cdata); im2=imresize(im,[ny nx],'bilinear'); im3=colfilt(im2,[5 5],'sliding',@min); im4=colfilt(im3,[5 5],'sliding',@max); bigim2(:,:,i)=im2; bigim4(:,:,i)=im4; i; end save video_Jan_29_2012_300-10-trigger2.mat clear all direc='D:\Jan-29-2012\300-10-trigger2-8888\velocity2\'; nx=103;%number of int zones in x ny=78;%number of int zones in y n1=165639;%number of first file nend=167325;%number of last file n=nend-n1+1;% the total number of files bigdata(1:(nx*ny),1:4,1:n)=0; for i=1:n; file=[direc 'A' num2str(n1+i-1) '.DAT']; bigdata(:,:,i)=load(file); i; end save velocity_Jan_29_2012_300-10-trigger2-full.mat     %%  %vorticity clear all 191  direc='D:\Jan-29-2012\300-10-trigger2-8888\vorticity3\'; nx=103;%number of int zones in x ny=78;%number of int zones in y n1=165639;%number of first file nend=167325;%number of last file n=nend-n1+1;% the total number of files vorticity(1:(nx*ny),1:3,1:n)=0; for i=1:n; file=[direc 'A' num2str(n1+i-1) '.DAT']; bigdata(:,:,i)=load(file); i; end save vorticity_Jan_29_2012_300-10-trigger-8800_996ml_min.mat %% %vorticiy analysis load vorticity_Jan_29_2012_300-10-trigger-8800_996ml_min.mat name='300-10-trigger-8800-996ml/min-trigger3'; uu=bigdata(:,3,:);   x=squeeze(bigdata(:,1,1)); y=squeeze(bigdata(:,2,1)); X=reshape(x,nx,ny); Y=reshape(y,nx,ny); x1=X(:,1); y1=Y(1,:); clear bigdata UU=reshape(uu,nx,ny,n);   clear uu  UU2(1:size(UU,2),1:size(UU,1),1:size(UU,3))=0;   for i=1:size(UU,3); UU2(:,:,i)=squeeze(UU(:,:,i))';   end UU=UU2; %% plot vorticity magnitude at image #i figure; i=1140; Vel=((((UU(:,:,:))).^2)).^0.5; imagesc(-x1,-y1,squeeze(Vel(:,:,i))); xlabel('width(mm) ');title(name);  ylabel('height (mm)');     figure imagesc((1:n)./200,-x1,squeeze(nanmean(Vel(41:43,:,:),1))) 192  ylabel('channel width(Vy) (mm)');title(name);  xlabel('Time(s) ');   % contours  figure  contour(UU(:,:,1140));    %% plot vorticity magnitude versus width of the channel   Vel=(((nanmean(UU(41:43,:,:),1)).^2)).^0.5;    z=squeeze(nanmean(Vel(:,10:12,:),2)); zfilt1=colfilt(z,[10 1],'sliding',@mean); figure plot((1:n)./200, zfilt1,'g')% hold on    z=squeeze(nanmean(Vel(:,20:22,:),2)); zfilt1=colfilt(z,[10 1],'sliding',@mean); plot((1:n)./200, zfilt1,'b')% hold on    z=squeeze(nanmean(Vel(:,30:32,:),2)); zfilt1=colfilt(z,[10 1],'sliding',@mean);   plot((1:n)./200, zfilt1,'r')% hold on     z=squeeze(nanmean(Vel(:,40:42,:),2)); zfilt1=colfilt(z,[10 1],'sliding',@mean);   plot((1:n)./200, zfilt1,'k')% hold on     z=squeeze(nanmean(Vel(:,50:52,:),2)); zfilt1=colfilt(z,[10 1],'sliding',@mean);   plot((1:n)./200, zfilt1,'.k')% hold on     z=squeeze(nanmean(Vel(:,60:62,:),2)); zfilt1=colfilt(z,[10 1],'sliding',@mean);   plot((1:n)./200, zfilt1,'.r')% hold on     z=squeeze(nanmean(Vel(:,70:72,:),2)); zfilt1=colfilt(z,[10 1],'sliding',@mean); 193    plot((1:n)./200, zfilt1,'b')% hold on      z=squeeze(nanmean(Vel(:,80:82,:),2)); zfilt1=colfilt(z,[10 1],'sliding',@mean);   plot((1:n)./200, zfilt1,'.g')% hold on     z=squeeze(nanmean(Vel(:,90:92,:),2)); zfilt1=colfilt(z,[10 1],'sliding',@mean);   plot((1:n)./200, zfilt1,'-g')% hold on        z=squeeze(nanmean(Vel(:,100:102,:),2)); zfilt1=colfilt(z,[10 1],'sliding',@mean);   plot((1:n)./200, zfilt1,'*g')% hold on  xlabel('Time(s) ');title(name);  ylabel('averaged vorticity profile'); %% %velocity analysis % changing the two long columns of U and V  %into 3-d matrices that match the images loaded %from the video above load velocity_Jan_29_2012_300-10-trigger-8800_996ml_min.mat name='300-10-trigger2-full'; u=bigdata(:,3,:); v=bigdata(:,4,:); x=squeeze(bigdata(:,1,1)); y=squeeze(bigdata(:,2,1)); X=reshape(x,nx,ny); Y=reshape(y,nx,ny); x1=X(:,1); y1=Y(1,:); clear bigdata U=reshape(u,nx,ny,n); V=reshape(v,nx,ny,n); clear u v U2(1:size(U,2),1:size(U,1),1:size(U,3))=0; V2(1:size(V,2),1:size(U,1),1:size(V,3))=0; 194  for i=1:size(U,3); U2(:,:,i)=squeeze(U(:,:,i))'; V2(:,:,i)=squeeze(V(:,:,i))'; end U=U2;V=V2;     %%   %% %check for exact place of the probes on the image z=(nanmean(V(41,:,:),3)); % 41 is the Y level where the probes are fixed at figure (1); plot(z); figure; %imagesc(squeeze(bigim2(:,:,i))); %f is the  X position of the probe 0 (middle probe) f=41; l=42; % V profile (average over time at the Y level of the probe ) z=(nanmean(V(41,:,:),3)); figure (1) plot(-x1, z); title(name); xlabel('Width of Channel (mm)');  ylabel('average velocity profile in the middle - Vy (m/s)'); % U profile at one point  %z=(squeeze(U(41,50,:))); %zfilt1=colfilt(z,[10 1],'sliding',@mean); %figure(2) %plot((1:n)./200,zfilt1); %title(name); %ylabel('Average velocity profile at the probe- Ux(m/s)'); % xlabel('Time(s) ');   % U profile versus x z=(nanmean(U(41,:,:),3)); figure plot(-x1,z); xlabel('Width of Channel (mm)');  ylabel('avergae velocity profile in the middle - Ux (m/s)'); title(name);         195    %% % plot velocity at one point (probe) versus time   %figure %plot((1:n)./200,squeeze(V(41,50,:))); % xlabel('Time(s) ');title(name); % ylabel('Velocity at the probe 0 in Y direction'); %velocity  profile averaged  z=(nanmean(V(:,f:l,:),2)); z=squeeze(nanmean(z(41:43,1,:),1)); zfilt1=colfilt(z,[10 1],'sliding',@mean); figure plot((1:n)./200, zfilt1,'b')% hold on  xlabel('Time(s) ');title(name);  ylabel('averaged velocity profile at the probe 0');      % Plot Vy versus time versus X  figure imagesc((1:n)./200,-x1,squeeze(nanmean(V(41:43,:,:),1))) ylabel('channel width(Vy) (mm)');title(name);  xlabel('Time(s) ');   % plot on log scale (absolute value of Vy)  logVy(:,:)=nanmean(V(41:43,:,:),1);  figure; imagesc((1:n)./200,-x1,squeeze(log(abs(logVy(:,:))))); ylabel('channel width(Vy) (mm)');title(name);  xlabel('Time(s) ');       %Plot Ux versus time versus X   figure imagesc((1:n)./200,-x1,squeeze(nanmean(U(41:43,:,:),1))) ylabel('channel width(Ux) (mm)');title(name);  xlabel('Time(s) ');         logUx(:,:)=nanmean(U(41:43,:,:),1);  figure; imagesc((1:n)./200,-x1,squeeze(log(abs(logUx(:,:))))); ylabel('channel width(Ux) (mm)');title(name);  xlabel('Time(s) ');     %%  %find interval between bubbles   196   %bulk velocity profile     kkk=(10.8*200):(34*200); z=squeeze(V(41,:,kkk)); %z=squeeze(V(41,:,:));   [mmm,nnn]=size(z); Vtotal=reshape(z,1,(mmm*nnn));     z=squeeze(U(41,:,kkk)); %z=squeeze(U(41,:,:)); Utotal=reshape(z,1,(mmm*nnn));   %figure %z=(nanmean(V(:,f,:),2)); %z=squeeze(nanmean(z(41,1,:),1)); %zfilt1=colfilt(z,[10 1],'sliding',@mean);   %hist(zfilt1, 100);  %xlabel(' velocity at the probe 0  (m/s)');title(name); % ylabel('frequency');  figure;  hist(Vtotal, 50);  xlabel('velocity in the middle over time and width(Vy,m/s)');title(name);  ylabel('frequency');   figure;  hist(Utotal, 50);  xlabel('velocity in the middle over time and width(Ux,m/s)');title(name);  ylabel('frequency'); % cumulative histograms max_c_elements =n; %h=abs((1-0)./((max_c_elements)^0.5)); %kk=-2:0.001:2; %z=(nanmean(V(:,f,:),2)); %z=squeeze(nanmean(z(41,1,:),1)); %zfilt1=colfilt(z,[10 1],'sliding',@mean);   %n_elements=histc(zfilt1(:,:),kk); %c_elements= cumsum(n_elements); %figure  %plot(kk,c_elements./(max(c_elements))); %xlabel(' velocity at probe(m/s)');title(name); % ylabel('cumulative frequency');  %hh=abs((1-0)./((max(c_elements))^0.5)); max_Vtotal=max(Vtotal); min_Vtotal=min(Vtotal); %kk= min_Vtotal:((max_Vtotal-min_Vtotal)/50):max_Vtotal; 197  kk= min_Vtotal:0.005:max_Vtotal;   %kk=-1.1:0.001:1.1;   n_elements=histc(Vtotal(:,:),kk); c_elements= cumsum(n_elements); figure  plot(kk,c_elements./(max(c_elements))); xlabel('vertical velocity along the middle line (Vy,m/s)');title(name);  ylabel('cumulative frequency');   max_Utotal=max(Utotal); min_Utotal=min(Utotal); %kk= min_Utotal:((max_Utotal-min_Utotal)/50):max_Utotal; kk= min_Utotal:0.005:max_Utotal; n_elements=histc(Utotal(:,:),kk); c_elements= cumsum(n_elements); figure  plot(kk,c_elements./(max(c_elements))); xlabel('horizontal velocity along the middle line (Ux, m/s)');title(name);  ylabel('cumulative frequency');    %%  % plot magnitude of velocity= (Vy^2+Ux^2)^0.5      % plot magnitude of velocity= (Vy^2+Ux^2)^0.5       Vel(:,:,:)=(((nanmean(U(41:43,:,:),1)).^2)+((nanmean(V(41:43,:,:),1)).^2)).^0.5;  % Vel(:,:)=(((nanmean(U(41:43,:,:),1)).^2)+((nanmean(V(41:43,:,:),1)).^2)).^0.5;           z=squeeze(nanmean(Vel(:,10:12,:),2)); zfilt1=colfilt(z,[10 1],'sliding',@mean); figure plot((1:n)./200, zfilt1,'g')% hold on    z=squeeze(nanmean(Vel(:,20:22,:),2)); zfilt1=colfilt(z,[10 1],'sliding',@mean); plot((1:n)./200, zfilt1,'b')% hold on    z=squeeze(nanmean(Vel(:,30:32,:),2)); zfilt1=colfilt(z,[10 1],'sliding',@mean); 198    plot((1:n)./200, zfilt1,'r')% hold on     z=squeeze(nanmean(Vel(:,40:42,:),2)); zfilt1=colfilt(z,[10 1],'sliding',@mean);   plot((1:n)./200, zfilt1,'k')% hold on     z=squeeze(nanmean(Vel(:,50:52,:),2)); zfilt1=colfilt(z,[10 1],'sliding',@mean);   plot((1:n)./200, zfilt1,'.k')% hold on     z=squeeze(nanmean(Vel(:,60:62,:),2)); zfilt1=colfilt(z,[10 1],'sliding',@mean);   plot((1:n)./200, zfilt1,'.r')% hold on     z=squeeze(nanmean(Vel(:,70:72,:),2)); zfilt1=colfilt(z,[10 1],'sliding',@mean);   plot((1:n)./200, zfilt1,'b')% hold on      z=squeeze(nanmean(Vel(:,80:82,:),2)); zfilt1=colfilt(z,[10 1],'sliding',@mean);   plot((1:n)./200, zfilt1,'.g')% hold on     z=squeeze(nanmean(Vel(:,90:92,:),2)); zfilt1=colfilt(z,[10 1],'sliding',@mean);   plot((1:n)./200, zfilt1,'-g')% hold on        z=squeeze(nanmean(Vel(:,100:102,:),2)); zfilt1=colfilt(z,[10 1],'sliding',@mean);   plot((1:n)./200, zfilt1,'*g')% 199  hold on  xlabel('Time(s) ');title(name);  ylabel('averaged velocity profile');    %%  %plot vertical velocity   z=squeeze(nanmean(V(41,40:42,:),2)); figure plot((1:n)./200, z,'k')% hold on  xlabel('Time(s) ');title(name);  ylabel('averaged velocity profile at the probe 0');    z=squeeze(nanmean(V(41,50:52,:),2)); plot((1:n)./200, z,'r')% hold on  xlabel('Time(s) ');title(name);  ylabel('averaged velocity profile at the probe 1');    z=squeeze(nanmean(V(41,60:62,:),2)); plot((1:n)./200, z,'b')% hold on  xlabel('Time(s) ');title(name);  ylabel('averaged velocity profile at the probe 2');    z=squeeze(nanmean(V(41,70:72,:),2)); plot((1:n)./200, z,'b')% hold on  xlabel('Time(s) ');title(name);  ylabel('averaged velocity profile at the probe 3');       z=squeeze(nanmean(V(41,80:82,:),2)); plot((1:n)./200, z,'.g')% hold on  xlabel('Time(s) ');title(name);  ylabel('averaged velocity profile at the probe 3');     z=squeeze(nanmean(V(41,90:92,:),2)); plot((1:n)./200, z,'*g')% hold on  xlabel('Time(s) ');title(name);  ylabel('averaged velocity profile at the probe 3');  %% %plot velocity map  figure; 200   imagesc((1:n)./200,-x1,squeeze(Vel(:,:))); ylabel('channel width(Vel) (mm)');title(name);  xlabel('Time(s) ');    % plot log scale of Velocity magnitude  figure;  imagesc((1:n)./200,-x1,squeeze(log((Vel(:,:))))); ylabel('channel width(Vel) (mm)');title(name);  xlabel('Time(s) ');    %%  % Magnitude of Velocity in the  middle between two bubbles z=squeeze(Vel(:,2208:6833)); [mmm,nnn]=size(z) Veltotal=reshape(z,1,(mmm*nnn));    figure;  hist(Veltotal, 50);  xlabel('velocity in the middle over time and width(Vel,m/s)');title(name);  ylabel('frequency');  max_Veltotal=max(Veltotal); min_Veltotal=min(Veltotal); %kk= (0.00001+min_Veltotal):((max_Veltotal-min_Veltotal)/500):max_Veltotal; kk= (0.00001+min_Veltotal):0.005:max_Veltotal;      n_elements=histc(Veltotal(:,:),kk); c_elements= cumsum(n_elements); figure  plot(kk,c_elements./(max(c_elements))); xlabel('velocity along the middle line (Vel,m/s)');title(name);  ylabel('cumulative frequency');    % Magnitude of Velocity in the  middle between two bubbles z=squeeze(V(41,:,2208:6833));   [mmm,nnn]=size(z) Vtotal=reshape(z,1,(mmm*nnn));    figure;  hist(Vtotal, 50);  xlabel('velocity in the middle over time and width(V,m/s)');title(name);  ylabel('frequency');  max_Vtotal=max(Vtotal); min_Vtotal=min(Vtotal); %kk= (0.00001+min_Veltotal):((max_Veltotal-min_Veltotal)/500):max_Veltotal; kk= (0.00001+min_Vtotal):0.005:max_Vtotal;   201     n_elements=histc(Vtotal(:,:),kk); c_elements= cumsum(n_elements); figure  plot(kk,c_elements./(max(c_elements))); xlabel('velocity along the middle line (V,m/s)');title(name);  ylabel('cumulative frequency');  %%    %% % velocity for one image includes bubble       Vel2=((((U(:,:,6874))).^2)+(((V(:,:,6874))).^2)).^0.5; kk=0; for ii=10:10:60     kk=kk+1; z=squeeze(nanmean(Vel2(ii:ii+2,:),1)); zfilt1=colfilt(z,[1 10],'sliding',@mean); figure (100); %subplot(2,3,kk) plot(-x1, zfilt1,'b')%   xlabel('velocity versus x at 6874 (Vel,m/s)');title(name);  ylabel('velocity');  hold on z=squeeze(nanmean(V(ii:ii+2,:,6874),1)); zfilt1=colfilt(z,[1 10],'sliding',@mean); figure (101); %subplot(2,3,kk) plot(-x1, zfilt1,'b')% xlabel('vertical velocity versus x at 6874 (Vel,m/s)');title(name);  ylabel('velocity'); hold on   z=squeeze(nanmean(U(ii:ii+2,:,6874),1)); zfilt1=colfilt(z,[1 10],'sliding',@mean); figure (102); %subplot(2,3,kk) plot(-x1, zfilt1,'b')% xlabel('horizontal velocity versus x at 6874 (Vel,m/s)');title(name);  ylabel('velocity'); hold on   end 202      %% %matrix with probe velocities [m,n]=size(Vel); i=1; for i=1:n new(i,1)=i./200; end   new(:,2)=(nanmean(Vel(50:52,:),1));   new(:,3)=(nanmean(Vel(60:62,:),1));   new(:,4)=(nanmean(Vel(70:72,:),1));   new(:,5)=(nanmean(Vel(80:82,:),1));   new(:,6)=(nanmean(Vel(90:92,:),1)); fileOut = strcat('velocity at 5 position-500-10.txt'); save( fileOut ,'new','-ascii');     %%ArcTan(:,:)= atand(((nanmean(U(41:43,:,:),1))./ (nanmean(V(41:43,:,:),1))));     figure;  hist(new(:,2), 10);  xlabel('50 velocity(m/s)');title(name);  ylabel('frequency');     figure;  hist(new(:,3), 10);  xlabel('60 velocity(m/s)');title(name);  ylabel('frequency');   figure;  hist(new(:,4), 10);  xlabel('70 velocity(m/s)');title(name);  ylabel('frequency');   figure;  hist(new(:,5), 10);  xlabel('80 velocity(m/s)');title(name);  ylabel('frequency');   figure;  hist(new(:,6), 10);  xlabel('90 velocity(m/s)');title(name);  ylabel('frequency'); 203     figure  kk=0:0.001:1;  n_elements=histc(new(:,2),kk); c_elements= cumsum(n_elements);    plot(kk,c_elements./(max(c_elements)),'g');  hold on   kk=0:00.001:1;  n_elements=histc(new(:,3),kk); c_elements= cumsum(n_elements);    plot(kk,c_elements./(max(c_elements)),'b');  hold on   kk=0:0.001:1;  n_elements=histc(new(:,4),kk); c_elements= cumsum(n_elements);    plot(kk,c_elements./(max(c_elements)),'b');  hold on  kk=0:0.001:1;  n_elements=histc(new(:,5),kk); c_elements= cumsum(n_elements);    plot(kk,c_elements./(max(c_elements)),'p');  hold on  kk=0:0.001:1;  n_elements=histc(new(:,6),kk); c_elements= cumsum(n_elements);    plot(kk,c_elements./(max(c_elements)),'r');        title(name); xlabel(' (Vel,m/s)');title(name);  ylabel('cumulative frequency');  %%  %Avergae and STD   Veltotal_average=nanmean(Veltotal(:,kkk));    Veltotal_std=nanstd(Veltotal(:,kkk));    fileOut = strcat('Average V in the middle-300-10-trigger2-full.txt'); save( fileOut ,'Veltotal_average','-ascii'); fileOut = strcat('STD of V in the middle_300-10-trigger2-full.txt'); save( fileOut ,'Veltotal_std','-ascii');   204     %RMS    n=length(Veltotal(:,kkk));  rms= norm(Veltotal)./sqrt(n);  fileOut = strcat('RMS of V in the middle_300-10-trigger2-full.txt'); save( fileOut ,'rms','-ascii');     %% % Arctan angle Utry(:,kkk)=nanmean(U(41,:,kkk),1); Vtry(:,kkk)=nanmean(V(41,:,kkk),1); i=0; j=0; [mm,nn]=size(Vtry); for i=1:mm     for j=1:nn  if Utry(i,j)>0 andand Vtry(i,j)>0 andand Utry(i,j)~=0         ArcTan(i,j)= atand(Vtry(i,j)./Utry(i,j));   elseif  Utry(i,j)<0 andand Vtry(i,j)>0 andand Utry(i,j)~=0     ArcTan(i,j)= (atand(Vtry(i,j)./Utry(i,j)))+180;        elseif  Utry(i,j)<0 andand Vtry(i,j)<0 andand Utry(i,j)~=0     ArcTan(i,j)= (atand(Vtry(i,j)./Utry(i,j)))+180;         elseif Utry(i,j)>0 andand Vtry(i,j)<0 andand Utry(i,j)~=0     ArcTan(i,j)= (atand(Vtry(i,j)./Utry(i,j)))+360;   elseif Utry(i,j)==0 andand Vtry(i,j)>0      ArcTan(i,j)=90; elseif Utry(i,j)==0 andand Vtry(i,j)<0      ArcTan(i,j)=270;     elseif Vtry(i,j)==0 andand Utry(i,j)>0 andand Utry(i,j)~=0      ArcTan(i,j)=0; elseif Vtry(i,j)==0 andand Utry(i,j)<0 andand Utry(i,j)~=0      ArcTan(i,j)=180;  end       end       end                                 %%ArcTan(:,:)= atand(((nanmean(U(41:43,:,:),1))./ (nanmean(V(41:43,:,:),1)))); z=squeeze(ArcTan(:,:)); 205  [mmm,nnn]=size(ArcTan(:,:)); AngelTotal(:,:)=reshape(z,1,(mmm*nnn));    figure;  hist(AngelTotal(:,:), 50);  xlabel('Angle velocity in the middle over time and width(Vel,m/s)');title(name);  ylabel('frequency');  kk=0.1:0.1:360;        n_elements=histc(AngelTotal(:,:),kk); c_elements= cumsum(n_elements); figure  plot(kk,c_elements./(max(c_elements))); xlabel('Angle velocity along the middle line (Vel,m/s)');title(name);  ylabel('cumulative frequency');       %Avergae and STD   AngelTotal_average=nanmean(AngelTotal(:,:));    AngelTotal_std=nanstd(AngelTotal(:,:));    fileOut = strcat('Average V Angle in the middle_300-10-trigger2-full.txt'); save( fileOut ,'AngelTotal_average','-ascii'); fileOut = strcat('STD of V Angle in the middle_300-10-trigger2-full.txt'); save( fileOut ,'AngelTotal_std','-ascii');      %RMS    n=length(AngelTotal(:,:));  rms= norm(AngelTotal(:,:))./sqrt(n);  fileOut = strcat('RMS of V Angle in the middle_300-10-trigger2-full.txt'); save( fileOut ,'rms','-ascii');          206  Appendix E Filtration data In this chapter typical results from filtration experiments are presented for the sparging conditions studied.   207    a   b   c Figure D.1 Typical results from filtration experiments for coarse bubble sparging (a: discrete; b: 0.25 Hz;c:0.5Hz)    208   a   b    c Figure D.2 Typical results from filtration experiments for pulse bubble sparging at 150 mL (a: discrete; b: 0.25 Hz;c:0.5Hz)   209   a     b   c Figure D.3 Typical results from filtration experiments for pulse bubble sparging at 300 mL (a: discrete; b: 0.25 Hz;c:0.5Hz)  210   a   b   c Figure D.4 Typical results from filtration experiments for pulse bubble sparging at 500 mL (a: discrete; b: 0.25 Hz;c:0.5Hz)    211  Appendix F Horizontal distribution of the shear stress for medium and large pulse bubble sparging The same trend was observed for medium and large pulse bubbles; with an increase in bubble size and sparging frequency the magnitude of shear stress increased (Figures F.1 to F.6). At the discrete sparging frequency, bubbles ascended on a vertical path in the center of the system tank where the sparger was installed and therefore the highest magnitude of shear stress was measured in the center of the system tank (Figure F.1 and Figure F.4). However, at higher sparging frequencies (Figures F.2, F.3, F.5 and F.6) bubbles moved on a zigzag path and therefore the highest magnitude of shear stress was measured on the probes that were on the moving path of the ascending bubbles. The magnitude of velocity and shear stress was observed to increase with the increase in the sparing frequency and bubble size (Figures F.2, F.3, F.5 and F.6).   212     a   b   c   d Figure F-1 Typical distribution of shear stress induced by medium (300 mL) pulse bubble sparging at discrete sparging frequency (a: shear stress at position 1; b: shear stress at position 2; c:shear stress at position 3;d: shear stress at position 4)   213    a   b   c   d Figure F-2 Typical distribution of shear stress induced by medium (300 mL) pulse bubble sparging at 0.25Hz sparging frequency (a: shear stress at position 1; b: shear stress at position 2; c:shear stress at position 3;d: shear stress at position 4)    214   a   b   c   d Figure F-3  Typical distribution of shear stress induced by medium (300 mL) pulse bubble sparging at 0. 5Hz sparging frequency (a: shear stress at position 1; b: shear stress at position 2; c:shear stress at position 3;d: shear stress at position 4)    215   a   b   c   d Figure F-4 Typical distribution of shear stress induced by medium (500 mL) pulse bubble sparging at discrete sparging frequency (a: shear stress at position 1; b: shear stress at position 2; c:shear stress at position 3;d: shear stress at position 4)   216    a   b   c   d Figure F-5  Typical distribution of shear stress induced by medium (500 mL) pulse bubble sparging at 0.25 Hz sparging frequency (a: shear stress at position 1; b: shear stress at position 2; c:shear stress at position 3;d: shear stress at position 4)   217    a   b   c   d Figure F-6  Typical distribution of shear stress induced by medium (500 mL) pulse bubble sparging at 0. 5 Hz sparging frequency (a: shear stress at position 1; b: shear stress at position 2; c:shear stress at position 3;d: shear stress at position 4)    218  Appendix G: Correlation between cut off velocity and rate of fouling As described in Section 4.1.3, a cut off velocity of 0.2 m/s was selected for the zone of influence induced by bubbles, based on the velocity and vorticity measurements. This was confirmed by comparing the effect of cut of velocity on the area of zone of influence and its correlation with the fouling rate. The area of zone of influence was calculated for different cut off velcoities of 0.1, 0.15, 0.2, 0.25, 0.3, and 0.4 m/s. The correlation between the area of zone of influence at each cut off velocity and rate of fouling was found using curve fitting. The coefficient of determination (R2) of each fitted curve was compared, as illustrated in Figure G-1. Figure G-1 illustrates that cut off velocities of 0.2 m/s and lower resulted in the same correlation between fouling rate and area of zone of influence (using a cut off velocity of 0.2 m/s and lower will result in a zone of influence that covers all the system).    Figure G ? 1 Statitical analyses of the effect of cut off velocity on the area of zone of influence and the induced fouling rate  

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