Citation |

- Permanent Link:
- http://ufdc.ufl.edu/AA00029983/00001
## Material Information- Title:
- Dispersion of flocculated particles in simple shear and elongational flows
- Creator:
- Zhang, Xueliang, 1962-
- Publication Date:
- 1998
- Language:
- English
- Physical Description:
- vii, 140 leaves : ill. ; 29 cm.
## Subjects- Subjects / Keywords:
- Couette flows ( jstor )
Diameters ( jstor ) Flow velocity ( jstor ) Fluid shear ( jstor ) Orifice flow ( jstor ) Shear stress ( jstor ) Simple shear ( jstor ) Size distribution ( jstor ) Velocity ( jstor ) Viscosity ( jstor ) Aerospace Engineering, Mechanics and Engineering Science thesis, Ph. D ( lcsh ) Dissertations, Academic -- Aerospace Engineering, Mechanics and Engineering Science -- UF ( lcsh ) Fluid dynamics -- Mathematical models ( lcsh ) - Genre:
- bibliography ( marcgt )
non-fiction ( marcgt )
## Notes- Thesis:
- Thesis (Ph. D.)--University of Florida, 1998.
- Bibliography:
- Includes bibliographical references (leaves 136-139).
- General Note:
- Typescript.
- General Note:
- Vita.
- Statement of Responsibility:
- by Xueliang Zhang.
## Record Information- Source Institution:
- University of Florida
- Holding Location:
- University of Florida
- Rights Management:
- The University of Florida George A. Smathers Libraries respect the intellectual property rights of others and do not claim any copyright interest in this item. This item may be protected by copyright but is made available here under a claim of fair use (17 U.S.C. Â§107) for non-profit research and educational purposes. Users of this work have responsibility for determining copyright status prior to reusing, publishing or reproducing this item for purposes other than what is allowed by fair use or other copyright exemptions. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder. The Smathers Libraries would like to learn more about this item and invite individuals or organizations to contact the RDS coordinator (ufdissertations@uflib.ufl.edu) with any additional information they can provide.
- Resource Identifier:
- 029248741 ( ALEPH )
39705947 ( OCLC )
## UFDC Membership |

Downloads |

## This item has the following downloads: |

Full Text |

DISPERSION OF FLOCCULATED PARTICLES IN SIMPLE SHEAR AND ELONGATIONAL FLOWS By XUELIANG ZHANG A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1998 ACKNOWLEDGMENTS I would like to express my sincere appreciation to the following people for their help and support throughout my study at the University of Florida: Dr. Renwei Mei, my advisor, constantly gave me his trust, advice, guidance and patience. Without his commitment to the project, this dissertation couldn't have been completed. Dr. Roger Tran-Son-Tay, my co-advisor, offered me a great deal of helpful advice and expert guidance in my experimental work. Dr. James F. Klausner, as a member of my supervisory committee, gave me a lot of guidance and advice, especially in my early experimental work. Dr. Wei Shyy and Dr. Corin Segal served as members on my supervisory committee, reviewed my proposal and dissertation, and made valuable comments. Dr. Brij M. Moudgil and Dr. Hassan El-Shall gave me their guidances and helps. Ms. Emmanuelle Demay and Mr. Philippe Vigneron aided me in some of my experiments. Mr. Ron Brown helped me in the setup of experimental apparatuses. Dr. A. Zamam, Mr. J. Adler, Dr. J. S. Zhu, and Dr. S. Mathur assisted me in making flocs and use some instruments. The Engineering Research Center (ERC) for Particle Science & Technology at the University of Florida, the National Science Foundation (Grant number: EEC-9402989), and the industrial partners of the ERC provided the financial resources for the project. In addition, my colleagues and friends at the University of Florida, Dr. Jian Liu, Dr. Guobao Guo, Dr. Hong Oyang, Ms Hong Shang, and Mr. Cunko Hu, provided me with various helps in my study and life in Gainesville, Florida. I am also grateful to the staff at the AeMES departmental office for their helps. Special thanks are given to Mr. Darrell D. Williams at Bristol, England for his help and encouragement during the past ten years. Last but most important, I am deeply indebted to my wife, Jie, for her understanding, encouragement, patience, and love. ii TABLE OF CONTENTS Pages A CKN O W LED G M EN TS ................................................................................................ ii A B ST R A C T ..................................................................................................................... vi CHAPTERS 1. IN TR O D U C TIO N ..................................................................................................... 1 1.1 B ackground .................................................................................................. . 1 1.2 Literature R eview .. ........................................... ........................................ 2 1.3 O bjectives and Scope .................................................................................. 6 2. VISUALIZATION OF FINE FLOC BREAKUP PROCESS .................................. 9 2.1 Experim ental D evices .................................................................................. 9 2.1.1 Cone-plate Device and Flow Description ........................................... 9 2.1.2 Hyperbolic Flow Device and Flow Characteristics ........................... 13 2.1.3 Contractile Flow Chamber ................................................................ 19 2.2 Experimental Devices and Materials ......................................................... 20 2.2 .1 F locs ........................ .. ..................... ............................................ . 20 2.2.2 Suspending Fluids .............................................................................. 22 2.3 Results and D iscussions .............................................................................. 23 2.3.1 Floc Breakup in Cone-plate Shear Flow ........................................... 23 2.3.2 Floc Deformation and Breakup in Contractile Flow .......................... 26 2.3.3 Floc Breakup in Hyperbolic Flow .................................................... 33 2.4 Sum m ary ........... -.. ---------------. -----............. .............................................. 37 3. FLOC BREAKUP IN SIMPLE SHEAR FLOW AND FLOC STRENGTH ...... 38 3.1 Introduction .-----------------....................-- - - -..................................... ... 38 3.2 Experimental Procedure and Data Processing ........................................... 39 iii 3.3 Results and Discussions .............................................................................. 42 3.3.1 Variation of Floc Mass with Time under a Constant Shearing ......... 42 3.3.2 Variation of Floc Size with Time at Constant Shear Rates .............. 50 3.3.3 Variation of Floc Size and Size Distribution with Shear Stress ..... 54 3.3.4 Change of Floc Shape with Time and Stress .................................... 58 3.4 Sum m ary .................................................................................................... . 62 4. FLOC BREAKUP IN ORIFICE FLOW--PART 1 FLOW CHARACTERIZATION ......................................................................... 63 4.1 Introduction ................................................................................................ 63 4.2 Form ulation ................................................................................................ 65 4.2.1 Governing Equation and Boundary Conditions ................................. 65 4.2.2 Grid Arrangement and Numerical Schemes ...................................... 68 4.2.3 Validation of the Numerical Method ................................................ 72 4.3 Results and Discussions ........................................................................... 75 4.3.1 Basic Features of Orifice Flow Field ................................................ 75 4.3.2 Strain Rate Characteristics of Orifice Flow ...................................... 79 4.3.3 Maximum Centerline Velocity Gradient ........................................... 81 4.3.4 Comparison Between axisymmetric Flow and Two-dimensional F low ................................................................................................ . . 87 4.4 Sum m ary ..................................................................................................... 87 5. FLOC BREAKUP IN ORIRICE FLOW--PART 2 MEASUREMENTS ......................................................................................... 90 5.1 Introduction ................................................................................................ 91 5.2 Experimental Apparatus and Procedure .................................................... 92 5.2.1 Orifice Setup and Procedure .............................................................. 92 5.2.2 Couette Shear Device ......................................................................... 94 5.2.3 Particle Size Analyzer ....................................................................... 98 5.2.4 Estimate of Reflocculation in Couette Flow and Orifice Flow ........... 102 5.3 Results And Discussions .............................................................................. 104 5.3.1 Effect of Flow Condition on Floc Size Distribution ........................... 107 5.3.2 Dependence of Mean Floc Size and Maximum Floc Size on Flow Rate .......................................................................................111 5.3.3 Comparison with the Result from Uniform Cone-plate Sim ple Shear Flow . --....... ....... ............... ................................... 115 iv 5.3.4 Comparison of Floc Dispersion between Orifice Flow and Cylindrical Couette Flow ............................................................. 119 5.3.5 Re-Examination of Sonntag's Experimental Data .............................. 123 5.3.6 Floc Strength A ssessm ent ................................................................... 125 5.4 Sum m ary ...................................................................................................... 129 6. SUM M AR Y ............................................................................................................. 131 6.1 Sum m ary and Conclusions .......................................................................... 131 6.2 Suggestions for Future Studies ................................................................... 134 REFEREN CE ..............--.. ... ............................... .................................................. 136 BIO GRA PH ICA L SK ETCH ......................................................................................... 140 V Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy DISPERSION OF FLOCCULATED PARTICLES IN SIMPLE SHEAR AND ELONGATIONAL FLOWS By Xueliang Zhang May 1998 Chairman: Dr. Renwei Mei Co-Chairman: Dr. Roger Tran-Son-Tay Major Department: Aerospace Engineering, Mechanics and Engineering Science Experimental studies on the dispersion process of fine flocculated particles in different flows are carried out through visual image analyses and particle size measurements. The flows investigated include a cone-plate shear flow, a cylindrical Couette flow, an orifice contractile flow, and a hyperbolic flow. Visual studies on the mechanisms of floc breakup in different flows are first conducted through a video image acquisition and analysis system. A variety of dynamic processes of the deformation and breakup of fine flocs of size from 3mm to 30mm in the contractile flow, hyperbolic flow, and simple shear flow are visualized. The breakup and erosion process of flocs subjected to a constant shear stress in the cone-plate flow is analyzed based on the changes of floc mass, size, and shape with shear stress and shearing time through the image analysis. A significant portion of the breakup, or size reduction, of the vi fine flocs takes place upon the application of the shear stress. Floc size continues to decrease through erosion mechanism. The erosion rate depends on the applied shear stress, the floc size, and the floc shape. An orifice flow is applied to break flocs and determine floc strength. The flow field before an orifice of high area ratio is first numerically simulated and analyzed in order to characterize the flow and stress field. The dependence of the maximum centerline velocity gradient on orifice area ratio and Reynolds number is obtained and its asymptotic behavior in high Reynolds number regime is analyzed. The dispersion of flocs in the orifice flow is analyzed based on the floc size distribution measured using a particle size analyzer. Due to the rapid rise of the axial velocity gradient near the orifice entrance, the floc breakup in the orifice flow is instantaneous and the floc erosion mechanism can be excluded. The centerline maximum shear stress in the orifice flow thus gives the floc strength of the resulting flocs whose average size is subsequently measured. The floc strength determined from the short-time shearing in a cylindrical Couette flow at lower shear stresses follows essentially the same power law dependence on the floc size as determined in the orifice flow. Thus, floc strength measured in different flows can be unified using the maximum shear stress of the flow. vii CHAPTER 1 INTRODUCTION 1.1 Background Many modem advanced materials, such as electronic, magnetic, optic, and fine ceramic materials, are produced from suspensions of colloidal particles. Flocs or aggregates are loose, irregular, three-dimensional clusters of particles in such suspensions. The words, floc and aggregate usually are both used to refer to the wet powder structure in liquids. High performance of materials requires sufficient dispersion of the flocs in suspensions, that is, sufficient breakup of flocs into smaller flocs or constituent particles. Although this dispersion process is actually the result of a number of different steps including milling, mixing, stirring, and so on, hydrodynamic shearing plays an important role in controlling the stability and uniformity of the suspension since the dispersion process is usually carried out in a hydrodynamic environment with or without the aid of dispersants. The flocculation (particle size enlargement) of particles and redispersion (particle size reduction) of flocculated particles take place simultaneously and constantly in the flow environment of the solid-liquid suspension. An important characteristic of flocs is their binding force, that is, the ability of the aggregate structure to resist deaggregation. As a measure of this binding force, floc strength can be defined as resistance to breakup by shear forces induced by fluid velocity gradients. The quantitative evaluation of floc strength is important to both dispersion 1 2 process and flocculation process. However, it is understood that the strength of flocs in a suspension cannot be measured directly due to its spatially irregular structure and the random characteristic in its formation but must be deduced from the evaluation of other measurable parameters. Because the concept of "strength" for flocs is always associated with their breakup which involves different mechanisms, the study on the floc strength should include the mechanisms of floc breakup and the force which causes this breakup. 1.2 Literature Review Thomas (1964) gave the first analysis on the mechanisms of floc breakup and floc strength. He proposed that large flocs in a turbulent flow field break in the forms of bulgy deformation and rupture. He assumed that the pressure difference on the opposite sides of a floc causes its bulgy deformation and eventual rupture and that the pressure difference is due to the random velocity fluctuations of turbulent flow. His work formed the basis for a number of experimental investigations to determine floc strength since then. Based on Thomas' models for floc rapture mechanism and isotropic turbulence theory, several experimental studies of floc breakup in turbulent flows have been conducted to determine the floc strength by relating the floc size to the turbulent flow conditions. Tambo and Hozumi (1979) devised a special flocculator experiment to study floc strength by measuring the maximum floc diameter under a weak agitation. Matsuo and Unno (1981) used a turbulent pipe flow to evaluate floc strength. Bache and Al-Ani (1989) used a vertical pulsating water column driven by an oscillating plunger to relate the floc size to the turbulence energy dissipation. Moudgil, Springgate, and Vasudevan (1989) experimentally studied the strength of kaolinite, dolomite, and Al203 flocs in a stirred 3 tank. The results for floc strength obtained by the application of isotropic turbulence theory provide some qualitative understandings of floc characteristics. However, the shear field is spatially nonuniform in a stirring tank and only the overall mean energy dissipation rate can be estimated for flow description based on the power input. Floc breakup and reflocculation are usually present simultaneously. Therefore, the results obtained from such experiments do not suffice for the purposes of determining floc strength. Parker, Kauflian, and Jenkins (1972) derived a model for the breakup of complex activated sludge flocs and inorganic chemical flocs based on the breakup mode of surface erosion suggested by Argaman and Kaufman (1970). They proposed that the primary particles are stripped from the surface of a floc by fluid shear at a rate that is proportional to the floc surface area and the surface shearing stress. Kao and Mason (1975) and Powell and Mason (1982) used a four-roller device in their experiments of floc deformation and breakup in an elongational flow. This may be the first systematic visual work to study aggregate dispersion in fluid flows. Couette apparatus had also been used in their study for the case of simple shear. Quigley and Spielman (1977, see Lu and Spielman, 1985) conducted similar experiments for ferric hydroxide agglomerates in a four-roller device. It is important to note that in these experiments the size of primary particles from which the flocs or aggregates are generated ranges from 20im to 400ptm and the size of flocs or aggregates is about 3mm ~ 5mm. Sonntag and Russel (1986, 1987) investigated experimentally the structure and properties of flocculated suspensions in a simple shear flow of cylindrical Couette flow 4 device and in a contractile flow of syringe apparatus using small-angle light scattering to monitor the breakup. The average number of particles per floc and mean radius of gyration were related to the flow conditions. They proposed a mechanism of floc shape change during breakup in contractile flow. However, because of the incomplete flow characterization for the contractile flow, the experiment data in the orifice flow and the comparison with that in Couette shear flow were inconclusive. Their experimental data is re-examined based on the completed flow characterization in this work, and a more reasonable interpretation of their results is obtained. Lee and Brodkey (1987), and Wagle, Lee, and Brodkey (1988) studied various pulp floc dispersion mechanisms by performing a visual study in a turbulent shear flow between the two moving walls. Different pulp floc breakup modes were sketched according to their experimental observation and the effect of flow shear level on dispersion time and rate was proposed. The pulp floc size used in their experiment is about 5mm. Higashitani, Inada, and Ochi (1991) also employed a contractile flow ahead of an orifice in a pipe to directly observe the breakup process of flocs. No breakup images were reported and only the variations of floc size along the flow direction were reported. The analyses were mainly based on the variations in the average numbers of constituent particles in a floc going through the orifice. The primary particles in their early experiment are about 90 pm in diameter. Higashitani et al. (1992) completed experiments on flocs whose constituent particles were about 1p m in diameter using a Coulter counter to size flocs, and analyzed the variation of floc mean size with applied pressure across the orifice. No flow characterization was given. It is noted here that the overall pressure drop 5 across the orifice cannot be explicitly related to the stress which causes the breakup of flocs. Glasgow & Luecke (1980), Glasgow & Hsu (1982), and Glasgow & Liu (1991) have experimentally investigated the floc breakup mechanisms and floc strength in turbulent flow including impeller-stirred turbulence, turbulent jet, and turbulent channel flow. The methods for floc sizing were based on the manual measurement of the photographs of the flocs. The floc size in their work ranged from hundreds of micrometers to a few millimeters. The mechanical interaction between flocs and solid meshes (Glasgow and Liu 1991) was not considered. In their experiment with turbulent jet flow (Glasgow and Hsu 1982), the breakup process and trajectory of individual flocs were clearly filmed and measured to determine the strength of flocs. This work was very significant in the subject for flocs whose size is larger than 100 micron meters. Navavrrete, Scriven, and Macosko (1996) visually showed the effects of shear, extention, and vorticity on the microstructure of iron oxide suspensions and the effects on their deformation and breakup using cryogenic scanning electron microscopy and videoenhanced light microscopy. Jiang and Logan (1996) investigated the structure changes, breakup, and coagulation of aggregates both in a laminar shear flow in a concentric cylinder device and a turbulent shear in paddle mixer. Both particle counter and image analyzer were used to obtain the changes in aggregate structure. Jung, Amal, and Raper (1996) studied the effect of turbulent shearing induced by stirring on the restructure and breakup of hydroxide flocs using small angle light scattering technique. Spicer, Keller, and Pratsinis (1996) investigated the effects of hydrodynamic conditions on the evolution 6 of floc size and structure of polystyrene particles with aluminum sulfate in a stirred tank flow via the image analysis system. Yeung and Pelton (1996) used micromechanical technique to directly measure the extentional force by elongating flocs of 6 to 40 micron in size into breakup using micropipettes. They found that there was no correlation between rupture strength in terms of force and floc size. However, the large scattering in their data due to a small sample collection may prevent one from seeing weak correlation. 1.3 Objective and Scope Although the mechanisms of floc breakup, characterization of floc dispersion in various flows, and the strength of flocs have been the subjects of those numerous theoretical and experimental studies, many issues remain. Floc breakup observations were made only for large flocs (mostly, 3-5mm in diameter) and little is available for fine flocs. The methodology for determining fine floc strength is not well developed. Results obtained from different methods cannot be related to each other. Most of those studies were carried out in turbulent flows and focused on the effects of overall hydrodynamic conditions (such as nominal turbulent shear rate) on the structure of flocs and aggregates. Although relative effects resulted from the changes of the overall flow conditions can provide some insights into the dispersion process, the relationship between floc properties and the actual force acting on flocs remains uncertain due to the lack of knowledge on the flow and the stress field. As an effort to address these issues, the present study focuses on visualizing the dynamic breakup process of fine flocs whose size ranges from a few microns to tens of 7 microns, relating the floc dispersion results obtained in different shear flows, and developing a simple, reliable and practical technique for the determination of the strength of flocs. The basic approach is to apply a shear flow to fine flocs to deform and break them, and then measure the floc size distribution subjected to the flow. Two techniques are employed to size the flocs. One is based on the visual image analysis of the deformation and breakup of individual flocs via microscope-CCD camera image acquisition and analysis system. The other is to measure the floc size distributions using a Coulter particle size analyzer. To eliminate the uncertainty in the flow characterization, the following four different well-defined laminar shear flows are used to disperse flocs: (1) a simple shear flow generated with a counter rotating cone-plate device where the shear rate is constant in the entire flow field but there exists a rotation of fluid; (2) an elongational flow formed with two opposing jet flows where a hyperbolic flow forms around the stagnation point when the two opposing flows meet, the shear rate is constant near the stagnation point and the flow is irrotational; (3) a contractile flow near an orifice in a pipe in which the high orifice area ratio causes an extremely high velocity gradient before the orifice. The shear rate is non-uniform and the rotation is nonzero in the flow; (4) a simple shear flow in the annular gap between two concentric cylinders. The shear rate in this flow is uniform and the rotation is nonzero. The present dissertation work consists of two main parts. The first is the visual study on the mechanisms of floc breakup and erosion processes in different flows. In order to further understand the mechanisms of fine floc dispersion in different flows, the dynamic deformation and breakup processes of individual flocs of size ranging from a few microns to tens of microns in the cone-plate simple shear flow, hyperbolic flow, and contractile 8 flow are visualized. The breakup and erosion process of flocs in cone-plate shear flow is quantitatively analyzed to determine the effects of shearing time and applied shear stress on floc mass, size, and shape. The second part is the study on the floc dispersion and floc strength determination by an orifice flow. Based on the results from the visual observation and the cone-plate shear flow, an orifice flow device is designed and fabricated to break up flocs and to determine the strength of the flocs. The flow approaching the orifice of high orifice area contraction ratio is numerically simulated to characterize the flow. The size distribution of the flocs after breakup is measured using a Coulter particle size analyzer for floc size distribution measurement. The instantaneous breakup of flocs in the orifice flow is characterized based on the stress field near the orifice entrance. In order to compare the floc breakup in the orifice flow with that in uniform shear flow, floc breakup in a cylindrical Couette flow with a short shearing time is also conducted. A rational basis for the floc strength and floc dispersion process under a shearing is established based on dispersion results from various flows. CHAPTER 2 VISUALIZATION OF FINE FLOC BREAKUP PROCESS The visualization of floc deformation and breakup process was made previously only for flocs of size in the millimeter range. A visual study for the dynamic deformation and breakup process of fine flocs whose size is in the order of micron meters is first conducted to qualitatively investigate the mechanisms of fine floc dispersions in a simple shear flow, a pure shear flow, and an elongational flow. 2.1 Experimental Devices 2.1.1 Cone-plate Device and Flow Description The experimental facilities used in this work were originally set up by Tran-Son-Tay (1984) for the investigation of the motion of blood cells and recently modified for studying highly viscous cells and living cells (Henderson and Tran-Son-Tay, 1997). They include a flow chamber with flow tubing and a control system and a optical image acquisition system. Different flows can be formed with different flow chamber configurations. The image system consists of an inverted interference contrast microscope Olympus IMT-2, a CCD camera, a monitor, a video recorder, and a Macintosh computer. The magnification of lens used in the microscope is 40 and the resolution of the CCD camera is 720X360. A schematic of the system with a cone-plate flow chamber for generating simple shear flow is shown in Fig. 2.1. The shear flow is formed between a transparent glass plate and 9 10 a Plexiglas cone of small cone angle S= 1.26 * Both plate and cone can rotate and can be adjusted both in direction and in magnitude independently. The cone and the plate can be controlled through a differential drive mechanism to keep the differential rotation velocity between them constant. The cone-plate setting can be moved in a radial direction so that the microscope can focus on a small region at any radial location. In the experiment, we choose R=3mm so that the gap size is about 66pm. This is large enough to exclude possible mechanical effects of the surfaces of the cone and plate for flocs of maximum size of about 15pim in the present experiments. The gap between the cone and plate can be adjusted and measured by focusing on the cone surface and plate, respectively. The cone tip is truncated, so that there is a gap of 20pm in the center region. A more detailed description of the rheoscope system can be found in Tran-Son-Tay, 1984. In a simple shear flow, the velocity field is linear, hence the shear rate is constant in the field. In a simple shear flow, there exists a rigid body rotation of the fluid. In the absence of rotation, the flow is called a pure shear flow. In the low Reynolds viscous laminar flow regime, the flow field in the gap between the cone and the plate as shown in Fig. 2.1 is given as uo= 2Kz-z), Ur=O, u,=O (2.1) for small angle, 5. In the above equations, z is in the direction of rotating axis of the cone and the plate and zo is the location of velocity component uo=O. The shear rate, y, in the flow field can be obtained from the angular velocities of the cone and plate as below: I1 I ac P (2.2) 2 tan. where >, and w,, are the angular velocities of the cone and the plate, respectively, and the cone angle, is 1.26 'in the present setup. The strain rate tensor for the simple shear flow r=3mm Light Condenser 17.8 mm Plexiglas cone Floc suspension Object n tive Wate CCD Camera Glass plate r Video recorder NIH 1.6 Monitor Macintosh Fig. 2.1 A Schematic of Cone-Plate Flow Chamber and Image Acquisition System 12 can be expressed in terms of y as 0 Y Z-Z 0 r D= -y r 0 Y (2.3) 0 y 0 When the cone angle Sis very small, (z-z0)/r < tang<< 1.0 (2.4) so that the strain rate component DrO then is negligible compared to y. The strain rate tensor thus reduces to the constant tensor: 0 0 0~ D= 0 0 y (2.5) _0 Y 0The stress field accordingly is by Navier-Poison law for incompressible Newtonian fluids: T=2uD (2.6) where p is dynamic viscosity of the fluid. For this stress tensor T, we can calculate three principal stresses as o-=2py, U2=0, o-=-2uy (2.7) The maximum shear stress is: = 022 = 2p (2.8) Hence, the stress state in the flow field is determined by the shear rate y for a given fluid. It is noted that the rotation tensor for the flow d2-0. 13 2.1.2 Hyperbolic Flow Device and Flow Characteristics A hyperbolic flow (or alternatively, stagnation flow) is usually a local flow that exists in a small region around the stagnation point in a flow field. The devices mostly used to generate this flow are four roller device and two opposite jet impinging flow device. Four roller device has been widely used (Kao and Mason, 1975, Powell and Mason, 1982) in breaking droplets and flocs since G. I. Taylor's famous experiment (Taylor, 1934) on the deformation of a liquid drop in a viscous flow in 1934. Impinging flow was applied to the studies of drop deformation (Janssen et al., 1993) and blood cell deformation (Knoblock, 1996). It has not been used in the studies of breaking flocs or aggregates. The flow chamber for generating hyperbolic flow was initially built and used for the study of blood cell deformation by Knoblock (1996). It has been modified to improve the image quality and to readily produce either hyperbolic flow or contractile flow. The flow generator and the image system are shown in Fig. 2.2. The image acquisition system is identical to that described in the above section. The flow is self-driven by gravity via the height difference between the flow reservoir and the flow chamber. The flow chamber has two pairs of tubing ports. The flow channels are formed by sandwiching between two parallel plates a rubber with channels. When it is used to generate a hyperbolic flow, the rubber has two channels perpendicular to each other and crossed. Two opposite streams of flow coming from two entrance ports at the ends of a channel meet in the cross region of the two channels and flow out from the two exit ports at the both ends of the other channel. A stagnation point is developed in the central point of the chamber. The flow velocity is very low near the stagnation point while the shear rate is approximately 14 Reservoir Outflow for Orifice Flow Inflow for Hyperbolic Flow I n f l o w t u b in g L g Condenser Flow Chamber Syringe for floc injection Wate bjective Image Processin --+ro I f 3mm Hyperbolic Flow Chamber Fig. 2.2 A Schematic of Hyperbolic Flow Setup 15 (exactly in theory) constant everywhere in the central region. Flocs are introduced into the inflow channel through a syringe located in one of the two entry tubing ports. A hyperbolic flow is a linear two dimensional flow. In general, a linear 2-D flow can be described as (Janssen et al., 1993): u = Lox (2.9) in which u is the velocity vector and x the position vector and L the gradient tensor G ]+a 1-a 01 L= - -]+ a -1 -a 0], with -1 5 a _<1 (2.10) 2 0 0 0 where G, a are constants. In a Cartesian coordinate x(x, y, z) the components of u(u, v, w) are u = [( + a)x + (1- a)yl (2.11) v = G[(- I+ aix+ -1afy] (2.12) w=0 (2.13) The value of a indicates the flow properties. For the simple shear flow in cone-plate device, a = -1. For the hyperbolic flow, a = 1, and the flow field becomes u=Gx (2.14) v=-Gy (2.15) w=0. (2.16) The strain rate tensor thus is D =0 - G 0 (2.16) 0 0 0 16 and the rotation tensor is .2= 0 (2.17) Hence, the hyperbolic flow is a pure shear flow, that is, without rotation in flow field. Both the maximum extentional rate of strain and the maximum shear rate of strain are G everywhere in the flow field. That is, s =y=G (2.18) and -= r = 2uG (2.19) The flow in this experiment is in the creeping flow regime since the reduced Reynolds number based on the geometry is less than 10, that is, Re* = UL h2 <102 (2.20) v L where U. is the average velocity of inflow, h half the thickness of the channel, v the kinematics viscosity, and L denotes a characteristic length of interested region in the x,yplane of the flow field. Using the flowing typical values U"'=10 mm/s v=6x1a3 m2/s (p 1 1200 kg/n, u - 7.4 Ns/m2) h=0.6mm L=O.1mm it is estimated that Re* = 6x1 03. Two important characteristics of the creeping flow between parallel plates are that the streamlines for all parallel layers (z = constant) are 17 congruent and the flow has the same streamlines as a potential flow (Schlichting, 1979). Thus, the flow field can be expressed as follows: u= uo(x,y) 1 - ) (2.21) S Z2 v = vO(x, y)1 - h (2.22) where uO and v0 denote the velocity field of two-dimensional potential flow. For the stagnation flow generated by two opposing jets, the well known solution for u0 and vo is uo=Gx (2.23) vo=-Gy (2.24) which holds only in the small region near the stagnation point (xy)=(O, 0) in the flow. The only task in flow characterization is to determine G in this experimental study. In this study, the image of the trajectory of a very small constituent particle is used to determine the shear rate. G can be calculated as follows: G = 2 In = 1 In (2.25) Z2 X1 Z2 Y2 (t2 - t, ) )- (t2 - ti ) (Iby measuring the positions (x,, y,) and (x2, y.) of the particle at t=t, and t=t, at a given z. It completely relies on the particle trajectories. The trajectory of a moving particle and its location can be obtained by videotaping its motion and using an image analysis software. Several frames of pictures at a constant time interval can be digitized. The coordinates of the particle at each picture can be read. However, the coordinates are relative to the frame defined by the software. The origin of the hyperbola in the coordinates is unknown. To 18 obtain the origin for the hyperbolic trajectory of the particle, and hence the coordinates of these points relative to the origin, a best fitting of these points is done. Once the origin is found, Eq. 2.25 is used to calculate the shear rate, assuming a vertical position z=O, that is, particles move in the middle layer. The value of z at which a particle moves cannot be determined in the present experimental setup. This precludes any quantitative measurement of the flow and quantitative analysis of the interaction between the particles and the flow. However, the values of G calculated at two extreme positions, z=O and z=0.5h, differs only by a factor of 1.333. Since two points on a trajectory can give a value of G, several values of G based on different pairs of points on the trajectory are averaged to give the final value of shear rate G. Figure 2.3 shows six points on a particle trajectory at 0.5 second time interval and the hyperbola from the best fitting over these points. y 100 * Particle trajectory 80 - Best fitting G=0.00322 s' 60 40 20 00 050 100 150 200 Fig. 2.3 Best Fitting to a Particle Trajectory in Hyperbolic Flow 19 2.1.3 Contractile Flow Chamber When the system is used to generate a contractile flow, the sandwiched rubber between the two parallel plates has one channel connected to one pair of tubing ports in the flow chamber. The other pair of the ports are blocked by the rubber. A schematic for the flow channel is shown in Fig. 2.4. The flow channel has a blockage with a small slot to form the contractile flow. Flow comes from one end of a channel and goes out from the other end of the channel. The two-dimensional channel used in this experiment is 0.8 mm thick and 12 mm wide. The slot opening is 0.8 mm in width, so that the contraction ratio is 15. 0.8mm 12mm Fig. 2.4 2D Contractile Flow Channel 20 Although the flow channel section is rectangular, the flow is not two-dimensional because of the side wall effects. However, the shearing along flow direction is dominant since the shear rate due to the contraction of the flow is much higher than other components of the rate of strain tensor. In addition, the flow in the middle region is approximately two-dimensional since the flow at the middle layer is a theoretically perfect two-dimensional flow. The region of the two-dimensionality of the flow is very small compared with the channel thickness, but is very large compared with the size of the flocs. Therefore, the flocs moving in the middle region are subjected to a twodimensional contractile elongation. The characterization of the contractile flow is much more difficult than the cone-plate simple shear flow and the hyperbolic flow. A numerical simulation of the axisymmetric orifice flow and two-dimensional flow and the analyses of the flow characteristics will be presented in Chapter 4. In the preliminary experimental investigation in this chapter, the primary focus is to observe the dynamic process of deformation and breakup of individual flocs. The quantitative characterization of floc breakup in an axisymmetric contractile flow will be given in Chapter 5. 2.2 Flocs and Fluids 2.2.1 Flocs The flocs used in this study are flocculated silica particles. They are generated from monodisperse spherical silica particles. Different diameters of silica particles are used to obtain different types of flocs. The complete procedure for making flocs involves 21 polymer preparation and particle flocculation. Given below is the detailed procedure employed throughout all the experiments in the project. (1) Preparation of ethyleneoxide polymer: Step 1. In a flask, make a mother polymer solution with a concentration of 500ppm (1 ppm = one millionth). Step 2. Add 0.5g ethyleneoxide polymer of 5 million molecular weight. Step 3. Add 1000 ml deionized water. Step 4. Stir the solution for at least 10 hours with a stirring bar at a stirring machine. Cover the flask with a box to shield the light. (2) Flocculation of particles Step 1. Choose dry powder: monodisperse spherical silica particles. Three different diameters of particles are used: 0.5im, 1 .0pim, 1.5pim . Step 2. Select the solids loading: 0.5% (Ig powder + 200ml deionized water). Step 3. Sonicate the slurry of particle and water to break any possible agglomerates in the powder using a sonic dismemberator (Fisher Model 300). Step 4. Add 5ml of PEO into the slurry. Step 5. Agitate the slurry for a few minutes. The time period of agitation may affect the size of resulting flocs, but does not affect the chemical properties of flocs. Step 6. Drain the water to form a floc suspension of volume fraction at 2% after sedimentation of the flocs. The mechanical properties of flocs resulting from the flocculation depend on the properties of the materials, the dosages of polymer and solids loading, the instruments, 22 and the operations. To obtain flocs of constant mechanical properties for every flocculation in the study, special care in operation has to be exercised in the making of the flocs. The same dosages of materials such as polymer, water, and powders have been used. The instruments used such as flasks, beakers, stirring bars, and stirring settings are the same for every flocculation. It is crucial to have flocs made at different times possessing the same mechanical properties in order to isolate the hydrodynamic effects of fluid shear on the dispersion of flocs. The floc suspension is stored at normal laboratory temperature for future use. It is gently agitated before being added into suspending medium. Although there might be changes of the flocs in their size and properties due to their aging and the agitation (Hannah et al., 1967), the effects are insignificant because the tests for floc breakup are always conducted within a few hours on the same day on the same mixture of flocs and fluid for all different flow conditions in a flow to minimize all other unexpected factors. 2.2.2 Suspending Fluids Two million M.W. dextran solution is used in the experimental study. The main reason for the use of dextran is the requirement of an extremely high viscosity of the fluid to break flocs at low shear rates. Low shear rate is essential because the breakup process of fine flocs can only be visualized at very low flow velocity and a large velocity gradient at high shear rate often leads to difficulties in visualizing moving flocs. In fact, the limit on the shear rate is very strict, once the shear rate surpasses 4 s' in the cone-plate device, the floc image becomes very fuzzy. Solutions of highly viscous fluids have been widely used in previous experiments for the same purpose. Sonntag and Russel (1986) used a 55.2 % 23 (by volume) glycerol-water solution of 0.1 Poise viscosity. Higashitani et al (1991) also used a 24 % (by weight) glycerol solution with a viscosity of 0.017 Poise. Powell and Mason used silicone oil solution at 60% volume fraction. The mechanical properties of 2-million-molecular-weight dextran solution are investigated to determine its characteristics. The viscosities of the different concentration dextran solutions are measured using a Wells-Brookfield cone-plate digital viscometer. The relationships between the stress and strain rate at five concentrations are shown in Fig. 2.5a, from which we can see that the solutions are Newtonian for the ranges of concentration and shear rate considered. The viscosity of the dextran solution increases with increasing concentration almost quadratically as shown in Fig. 2.5b. 2.3 Results and Discussions 2.3.1 Floc Breakup in Cone-plate Shear Flow Figure 2.6a shows the detachment process of a primary silica particle of 1pm diameter from a triplet floc in the simple shear flow. This triplet is initially composed of three primary particles that form a straight line. As shearing continues, the floc rotates and one of the primary particles detaches from the floc gradually. This triplet breaks at a shear stress of r = 6.23 Pa. Fig. 2.6b shows the breakup process for a different triplet floc. The initial shape is a triangular type, so that each primary particle is in contact with two other particles. The floc ruptures, that is, the primary particles detach from each other simultaneously. The stress level in the shear flow field at which the triangular floc breaks up is 9.35 Pa. The results indicate that the floc strength depends strongly on the floc structure. The triangular triplet has more contacts and therefore more effective binding a(N/m2) concentr Ea 0.-. 0. -Y- 0. -.- - 0. 0. 0 10' (a) Stress - Rate of Strain Relationship [t (Ns/m2) Concenti 0.4 0.5 0.6 (b) Variation of Viscosity with Concentration 10c y(s-') ation (by weight) 0.8 Fig. 2.5 Mechanical Properties of Dextran Solution 24 nation 1 5 6 102 101 10 1 10 20 15 10 5 0. 0. 3 . . . . . . . . . . . . 0.7 25 (t=O1 (t= Is) (t=2s) (t=2.5) (t=3s) (t=3.5s) (a) Breakup Process of a Line-shape Fine Floc (y=0.842 s-1) (t= Is) (t=1.5) (t=2s) (t=3s) (b) Breakup Process of a Triangular-shape fine Floc (y=1.257 s-) (The edges of primary particles in this figure are manually traced for better visualization.) Fig.2.6 Two Breakup Processes of Flocs Consisting of Three Particles in Cone-plate Simple Shear Flow (p=7.4 Ns/m2, Primary particle diameter do=.Opm) (t=0) (t=0.5s) 26 among primary particles. Thus, it is not surprising that a higher stress is required for breaking a triangular floc than a linear floc. Therefore, the strength of flocs in a suspension can only be related to the statistic average of floc size since a suspension is composed of flocs with a large number of different structures. Figure 2.7 shows the breakup process of a larger and more complicated floc which is about 8ptm in equivalent diameter. The floc breaks up in the middle into two smaller flocs. One of them further breaks up in the middle to form two smaller flocs. The shear stress at which this process is observed is also 9.35 Pa and this process lasts for about 10 seconds. It seems that the fragmentation of a floc into smaller ones is a short time sequence process at the low shear rate for the flocs investigated. It is expected that when the shear rate is increased and the shear stress is kept the same, the time period will decrease. 2.3.2 Floc Deformation and Breakup in Contractile Flow The deformation and breakup process of flocs approaching to the entrance of an orifice have been investigated by other researchers. Sonntag and Russel's deformation model was based on the analysis of the variation of the average number of primary particles per floc and the variation of the average radii of gyration of flocs with shear rate. They found that the radii of gyration vary weakly with shear rate while the average number of primary particles per floc decreases exponentially. Hence, it was postulated that the flocs change shape in the orifice flow from spherical to elongated because of the continuous removal of fragments from the same part of the floc by the irrational flow near the orifice entrance, not because of the deformation. Higashitani, et al F- - I 27 (t=0) (t=5s) (t=8s) (t12) (t=15) (t= 16) Fig. 2.7 Deformation and Breakup Process of a Floc in a Simple Shear Flow (p=7.4 Ns/m2, y=1.257 s-1, Primary particle diameter dO=. Opm) 28 (1991) measured the variation of the length of flocs consisting of small number of particles, whose size is 90pm in diameter, in the flow direction as they approaches the entrance of an orifice using a stroboscope photograph technique. They proposed that the flocs were elongated and lined up towards the flowing direction, and then broken before entering into the orifice. The present experimental work is to examine those proposed scenarios by visualizing the dynamic deformation and breakup process of fine flocs subjected to a contractile flow approaching to a small opening in a two-dimensional channel. To obtain a clear images of flocs when they move into the focal visual region, the velocity of the flow near the slot entrance must be very low. This requires the use of a highly viscous suspending fluid in order to create a shear stress that is sufficiently high to deform and breakup the flocs. The dextran solution with viscosity of 7.4 sN/m2 is used. The flocs of 1.5 pm silica particles are used to obtain better visual images. The motion of flocs is clearly visualized in the present system. Unfortunately, the shear rate in the flow cannot be obtained in the present experimental conditions because the exact location of the flocs cannot be determined. For the two-dimensional slot with contraction area ratio of 15, the maximum dimensionless shear rate, normalized by the average channel velocity and the half channel width, can reach about 225 near the entrance of the slot, as will be quantitatively shown in Chapter 4. However, the absolute value of shear rate is very low, because the flow rate in the system is only 20 milliliter per hour. 29 Figure 2.8 shows the final moments of a dynamic elongation process of a floc. They are captured by manually moving the flow chamber along the centerline flow direction since the fixed region that the microscope can cover is very small (68pmX90pm). This floc whose image is captured is located at the centerline and near the entrance, as suggested by the orientation of the floc. Its two ends are elongated in axial direction prior to their detachments. The rest of the floc is also elongated and seems to be broken in the middle as shown in the last picture. The most frequently observed mode of floc breakup in the contractile flow is the one where flocs are elongated, then broken from the ends of the rod-like shape. Figure 2.9 shows a processes of disintegration of two flocs approaching to the slot. The two flocs are deformed and completely ruptured in less than one second. Elongation usually takes place before the disintegration. In Fig. 2.10, a cluster of a few flocs is followed. The cluster of flocs is captured somewhere upstream in the flow field, then is followed by manually adjusting the flow chamber along the centerline and lateral directions. It can be seen that the cluster of flocs is flowing toward the central region of the flow and its orientation is gradually shifting to align with the centerline direction as they approach to the orifice. The flocs undergo deformation and breakup in the process. On the last picture frame at t=17 second, the cluster of flocs becomes much thinner, consisting mainly of two or three particles in the width. During the initial phase, there are some small fragments removed from the main floc body by the flow. The major breakup takes place at the final moments around t=13 second. 30 0 t=O t=O. 125s 4 ~ t=0.250s t=0.375s t=0.5s t=0.625s Fig. 2.8 Deformation and Breakup Process of a Floc in a Contractile Flow (pi=7.4 Ns/m2, Primary particle diameter d0=1.5pm, area ratio P=/5) '1 31 * 4 7> :-Zj7j24; ,~ S -~ t2-tk$~ ~ t=o t=O. 125s t=0.125s t=0.875s t-O 25s t=1s Fig. 2.9 Rupture of a Floc in a Two-dimensional Contractile Flow (p=7.4 Ns/m2, Primary particle diameter do=1.Spm, area ratio 0=15) 32 68 pim WVV~r- t=Os t=3 s t=7s t=lls t=13s t=17s Fig.2.10 Motion and Breakup of Floes in a Two-dimensional Contractile Flow (p=7.4 Ns/m2, Primary particle diameter do=1.5pm, area ratio =15) Ae. 33 The visual monitoring of floc motion also reveals that the flocs tend to move toward to centerline in contractile flow. In particular, flocs detour the corner region of the flow when they move toward the orifice wall due to the large lateral velocity. This observation is significant for our study of the effective shear stress that flocs experience in orifice flow in Chapter 5. 2.3.3 Floc Breakup in Hyperbolic Flow As we have seen, in the simple shear flows of the cone-plate, flocs translate and rotate with the fluid. In the two-dimensional contractile flow approaching the slot, the flocs move through a high shear section for a short period, undergo deformation and then break up. In a hyperbolic flow, flocs move very slowly near the stagnation point and be exposed to the shear force for a much longer period. This is the case when the hyperbolic flow is used to break and deform bubbles, drops, and blood cells. However, flocs usually cannot remain near the stagnation point even for a short period because the irregular shape of flocs causes a non-zero resultant force on the flocs by the fluid. In fact, even for a spherical particle, the stagnation point is unstable point. Once it is perturbed, it will move away with the outgoing flow. The rotation speed of the four rollers has to be adjusted to bring the drops back to the stagnation point. For flocs, the incoming flow rate is controlled to an extremely low value, about 20ml/h, to obtain clear picture of moving flocs in the flow around the stagnation point. Such a low incoming flow velocity produces a very low shear rate, G=0.0032 s'. Although the dextran solution at concentration of 60% with the viscosity u=7.4 sN/m2 is used, it is still not sufficient to deform and break most of the flocs used in the study. 34 Hence, it is more difficult to obtain good pictures of floc deformation and breakup in the hyperbolic flow than in the contractile flow. Nevertheless, the breakup of some large flocs are observed and recorded. Two such processes are shown in Fig. 2.11 and Fig. 2.12. The same batch of flocs used in contractile flow tests are used in the present test. Figure 2.11 shows a typical process of deformation and breakup in the hyperbolic flow. When captured, the floc is already elongated to about 10 pm. The flocs in the hyperbolic flow experience a pure shear and an elongation similar to that in the contractile flow. The breakup process in the hyperbolic flow is very similar to the one observed in the contractile flow (Fig. 2.8), except that the floc in the hyperbolic flow experiment moves at a much smaller speed over a small distance since the velocity of flow in the stagnation region is near zero. This leads to a longer shearing time and the better image quality. In contractile flow, the flocs travel very fast, and experience a high shear due to the high flow velocity at the entrance region. Figure 2.12 shows another dynamic breakup process of a floc in the same flow conditions. A neck region forms in the elongated floc. Further necking leads to the breakup in the middle while no apparent deformation is observed in its shape. It is noticed that individual flocs have very different structures than the one captured in this case, even though they were generated from the same batch by using identical procedures and chemicals. However, Fig. 2.11 and 2.12 clearly show the simultaneous presence of deformation-breakup mode and breakup without deformation mode in the hyperbolic flow. v9 t=0.25s t=2.25s Fig. 2.11 Floc Deformation and Breakup in a Hyperbolic Flow ( G=0.00322 1/s, =7.4 Ns/m2,do=1.5pm ) 35 15[im U t U t = t=0.5s t-1.25s t=0.75s II~ . ' j t=1.Os t=2.Os t=1.5s 36 Ii . s s t=O.5s t=1.Os I I 444 S t=1.25s t-l.5s Fig. 2.12 Breakup Process of a Floc in a Hyperbolic Flow (G=0.00322s-, p=7.4Ns/m2, do=1.5 pm) 15pm t=o = 4 t-O .75s *5~ t 0.25 t=1.75s Zfl- 37 2.4 Summary This chapter presents the visual results for the dynamic deformation and breakup process of fine flocs of size ranging from a few microns to tens of microns in different flows. The main conclusions and significance of this work are summarized as follows. (1) Dynamic process of deformation and fragmentation of fine flocs in the size range of 3 - 30pm is clearly visualized for the first time in different flows including a simple shear flow, a pure shear (hyperbolic) flow, and an elongational (contractile) flow. (2) In hyperbolic flow, flocs can stay in the region of constant shear stress for a longer period of time compared to contractile flow where flocs experience a high shear stress only in a very short period of time. In both contractile flow and hyperbolic flow, the more frequently observed mode of breakup is that flocs are elongated, then broken into several smaller flocs simultaneously. (3) In simple shear flow, a floc is usually broken into two flocs at a time. Compared to breakup in contractile flow and hyperbolic flow, the breakup process in the simple shear flow is usually much longer than that in the elongational flows. (4) The visualizations of detailed dynamic floc deformation and breakup process reveal various modes of the dispersion of fine flocs. Such information is helpful to the understanding and the modeling of this complicated floc dispersion process. CHAPTER 3 ANALYSIS OF FINE FLOC BREAKUP IN CONE-PLATE SIMPLE SHEAR FLOW 3.1 Introduction Simple shear flow has been widely used to quantitatively study floc dispersions, using three devices: (1) Cylindrical Couette flow apparatus with two concentric cylinders. When the gap between the two cylinders is sufficiently small compared to the radius of inner cylinder, the velocity distribution in the viscous flow between them due to the rotation of either cylinder or both in opposite directions is linear. Hence, the shear field in the flow is constant when the Reynolds number is smaller than the critical Reynolds number. This apparatus has been widely used to generate either a laminar simple shear flow or turbulent flow for the experimental study of particle dispersions and aggregations in shear flow (Jiang and Logan, 1996; Kao and Mason, 1974; Patterson and Kamal, 1974; Powell, and Mason, 1982; Potanin et al., 1997; Serra et al, 1997; Sonntag and Russel, 1986). (2) Flow between two parallel walls moving in opposite directions. This apparatus was used in visualizing the pulp floc breakup process (Lee and Brodkey, 1987) and deforming liquid drop in another fluid (Taylor, 1934). (3) Flow between a counter-rotating plate and cone with a very small angle. This device has been used to deform blood cells (Tran-Son-Tay, 1984; Henderson and TranSon-Tay, 1997). This same device is used in the present study. 38 39 The simple shear flow device and image acquisition system used in the last chapter is employed to quantitatively study floc breakup process, and floc strength in this chapter. This is done by visualizing the deformation and breakup of individual flocs and analyzing the floc mass, size, and shape. Floc mass refers to the number of constituent particles of a floc. Since the constituent particles are spherical and the density of the particle is known, the mass of the flocs can be related to the number of constituent particles. The main objectives of this study are to further understand the breakup mechanisms of fine flocs of a few microns, analyze the effect of shearing time on floc dispersion process and evaluate the mechanical strength properties of flocs. 3.2 Experimental Procedure and Data Processing In order to analyze the change of floc mass, size, and shape with shearing time and shear stress, the flocs are sheared in the cone-plate shear flow. The experimental device was described in the last chapter. The flow chamber is centered and leveled before the fluid and flocs are put in. About 0.5 ml dextron is first dropped on the surface of the plate, then a small amount (no more than 105 ml) of wet flocs prepared previously is added to the drop of fluid using a needle. The flocs are initially concentrated at a small region in the fluid. The mixture of flocs and fluid is left undisturbed for several minutes before the cone is positioned to its preset position so that the flocs could spread with the very slow and free flowing of the drop of fluid. It can be seen that the volume fraction of the flocs in the fluid is about 10', when the solid volume concentration of the wet flocs added into the fluid is assumed 50%. 40 The shear rate is controlled by adjusting the differential rotation speed between the cone and the plate. To follow the motion of a floc in a limited range, the rotation speed of the cone or the plate can be adjusted while the differential speed is kept constant. The focus of the microscope is adjusted to change the focal plane during the experiment so that flocs at different heights can be observed. The focused region is 68pm x 90pm. All the flocs moving in the region on the focal plane are continuously filmed by a CCD camera after magnification through the microscope, and the images are monitored and videotaped using a monitor and a VCR connected to the microscope-CCD camera system. Then the movie is digitized with a given time interval on a Macintosh computer using the public domain NIH Image program. Figure 3.1 shows a frame of a typical floc picture. The averages of equivalent size of flocs, shape, and mass (number of constituent 68 pm K 90 urn Fig. 3.1 Typical Floc Sampling for Size and Shape Analysis 90 pM 41 particles of a floc) over all flocs in a picture represent an instantaneous values of size, shape, and mass of the flocs that move in the focal region at that particular instant. The flocs move in and out of the focused region from the two boundaries in circumferential direction following the fluid motion. It can be shown that the radial movements of flocs are random and negligibly small, and there does not exist an outward flow of flocs resulting from the centrifugal force since the fluid has an extremely high viscosity. Assume that a spherical floc of radius a has a radial velocity v, at a radial position r, the equation of motion for the floc may be approximated as, 4 4 3 2 3 "p'a dt 6xpatv,.+-ffppa rw (3. 1) dr where Vr =-, p, is the average density of the floc and a the angular velocity. Stokes drag is used in the equation since the Reynolds number for the flow is in the order of 10' based on the experimental conditions, floc size a=10-5 m or less, fluid viscosity p=1.0 9pu Ns/m2 or higher, fluid density p=I100 kg/i3, o < 0.Is-'. Let P= 22 , it is noted that /-10's', the Eq. (3.1) takes the following form: r+/3r-r)2 =0 (3.2) When w/<< 1.0, the solution given by r(0)=r,, and (O)= 0 is: 2 W2 W2 r=ro(.2 e- +ef )~roe P . (3.3) 42 For the conditions in the study, w/pl - o(Ift8), and /8 - o(10-9). The radial movement of flocs due to the centrifugal force in 3 hours (104 second), the an extreme case, is eventually zero in the highly viscous fluid. In the vertical direction, the possible sedimentation is examined experimentally. The size distribution of the flocs suspended in the dextran solution in a high tube is measured by a Coulter particle size analyzer at time interval of 1.5 hour. Samples are taken from the top of the tube container. Fig. 3.2 shows the two measured floc size distributions. It is found that when the concentration of dextran solution is 40%, the floc size distribution difference disappears. The concentration of dextran solution used in the study is higher than 50%. Thus there is no systematic depletion of the flocs in one specific region and the sample in the visual region is representative of the entire flow. 3.3 Results and Discussion 3.3.1 Variation of Floc Mass with Time Under a Constant Shearing In order to study the effect of shearing time on floc structure in a simple shear flow, an experiment on the change of floc mass under a constant shearing is conducted. Each floc consists of only small number (typically less than or equal to 6) of primary or constituent particles. Such flocs are highly irregular in shape. Such flocs are encountered in the last stage of dispersion process in which larger flocs (with diameters 10 or 100 times the primary particles) have been broken down and the remaining flocs of smaller size are difficult to break. An understanding of the floc breakup mechanism on this small scale is essential to the dispersion of colloidal particles. Flocs are sheared at a constant low shear rate y=1.91 1/s in the highly viscous dextran fluid of viscosity u=15.1 Nos/m2 for two 43 Sedimentation time 5- / t=0 - \-t=93 mimute 0) 0 7 0) I E 3 LA 2 1* 10' 10 Floc Size (pm) (a) Dextran Concentration 30% dz8 Sedimentation time 02 6L 4 0 10 0*11)0 0lcSz pn (b 4eta ocnrto 0 3 i.32Efc fSdmnaio nFo ieDsrbto 44 hours. The entire dispersion process is recorded by the video system as described above. After the experiment is done, the video is played back and digitized at twelve time segments apart approximately 10 minutes. At each time segment, about 5 frames of digitized pictures are taken every 5 or 10 seconds, depending on the image quality and the number of flocs in a picture, since the flocs do not distribute uniformly in the flow field. The pictures including too few flocs or those of poor quality are usually skipped. The number of visible primary particles in each of all the flocs in one frame of picture is counted. Some primary particles could be hidden in large flocs consisting of four or more particles. Hence, the floc structure observed is two-dimensional. The variation of floc mass, which is represented by the average number of primary particles of one floc over all flocs at a time point, with time is obtained. The changes of number fractions of flocs consisting of certain primary particles are also analyzed. To better visualize the floc structure, the flocs generated with silica particles of 1.5 pm in diameter are used. Experiments indicated that when this type of flocs is used, the individual particles in a floc are clearly visible. The flocs generated with this size of silica particles by the procedure described earlier are composed of small number of constituent particles. Figure 3.3 shows four frames of pictures illustrating typical floc structures under shear at four different moments. The decreasing tendency of large flocs consisting of more than 5 primary particles can be seen in the pictures. It is noted that the pictures taken at initial phase have more flocs and less flocs at final phase, in other words, the population of flocs t=40 minute t=80 minute t=120 minute Fig. 3.3 Floc Structures at Four Different Times Under Continuous Shearing (y=1.91 s-1, p=15J.Ns/m2-, Primary particle diameter do=1.5pm) t-o 46 is not balanced in these pictures. The reason is that the flocs are relatively concentrated spatially initially during the shearing process because the flocs are put in a small region in the fluid and the flocs are more and more distributed spatially as the continuous shearing proceeds because of the circumferential fluid flow. Therefore, the number fraction of flocs is used in the following analyses. Number fraction of some specified flocs is the ratio of the number of these flocs to the total number of flocs. Figure 3.4 shows the variation of the average number of primary particles in one floc with time. It can be seen that the average number decreases fast at the initial stage, indicating the breakup of larger flocs composed of large number of primary particles into flocs of smaller number of primary particles. At this initial phase, the breakup mechanism is dominant. This period is short, compared with the long shearing process. For the rest of the shearing, the average number of constituent particles decreases very slowly, which indicates that the erosion process of flocs dominates. This experiment clearly illustrates that the breakup of flocs in short time makes a major contribution of floc size reduction in floc dispersion by shear flows and the continuous shearing will result in the erosion of particles, but makes a small contribution to floc size reduction. Figure 3.5a shows the variations of number fractions of flocs consisting of one, two and three particles with time and Fig. 3.5b shows those of flocs consisting of four, five and six or more particles. For the sake of convenience, the individual primary particles in the pictures are considered as flocs consisting of one constituent particle. They are, of course, not flocs. It can be found that the number fraction of largest flocs ( composed of 6 or more visible particles) drops rapidly in the initial shearing phase ( from 12% to 3% in about 6 minutes). The number fractions for 4-particle flocs and 5-particle flocs also 47 o 3 0 0 C, S2.5 0 o .0 E 2 1.5 1 I . , , I . I . . I . I 0 20 40 60 80 100 120 Time (Minute) Fig. 3.4 Variation of Average Number of Constituent Particles in One Floc (y=1.91s', g=15.06 Ns/m2, do=J.5 m) 48 E) I-particle floc - - 2-particle floc 0.6 - - -A- 3-particle floc 0 o 0.5 .E 0.4 - _0 . 0.3 -0 0.2 0.1 A 0 20 40 60 80 100 120 Time (Minute) (a) Variation of Number Fractions of Small Flocs 0.12 0 0.1 -- 4-particlefloc --- - 5-particlefloc - - -- 6 and more-particle floc 0.08 E z 0.06 0.04 0.02 - 0 0 20 40 60 80 100 1 Time (Minute) (b) Variation of Number Fractions of Large Flocs Fig.3.5 Variation of Number Fractions of Large Flocs with Time 49 decrease, from 9% to 3.5% and 7% to 4% in 12minutes, respectively. It is noted that the bigger the flocs are, the larger the decreasing rates, for these three sizes of flocs. This quantitatively indicates that the large flocs are easier to break. On the other hand, the fractions of single particle flocs and doublet flocs have an increase of about 8% in the initial 6 minutes, and the number fraction for triplet flocs increases slightly, that is, the number of triplet flocs increases keeping pace with the increase of total floc numbers during this period due to the breakup of larger flocs. The strips of single particle or doublet flocs obviously take place in the initial breakup phase. Then what follows is the slow decrease in number fractions of large flocs with time during the shearing process and the slow increase in number fractions of single particle flocs. At the end of the shearing process, the largest flocs (composed of five and more primary particles) completely disappear. Although there are some deficiencies in these curves due to the nonuniform space distribution of flocs in the flow field and possible sampling bias, these curves clearly illustrate the breakup mechanisms from the point of view in floc mass changes. It is suggested that a continuous shearing does not result in a significant floc size reduction and a sufficiently high shear stress that overcomes the strength of flocs can break flocs almost instantaneously in the flow of shearing domination. The initial breakup phase in this experiment lasts about 6 minute because the extremely low shear rate is used. If the shear rate is increased by 500 times, up to about 1000 l/s, and correspondingly, a much lower viscosity is used to obtain the same shear stress, the initial breakup can be finished in a very short time, because the 50 nondimensional parameter yt usually is considered to the quantity that essentially governs the time effect (Powell and Mason, 1982). 3.3.2 Variation of Floc Size with Time at Constant Shear Rates Based on the conclusion that a continuous shearing makes little contribution to the floc size change, an experiment is performed to analyze the effect of shear stress on the floc size and shape, as well as the effect of shear stress on the erosion rate of flocs at a constant shearing. The experiment is done by the same procedure as above, but four shear rates are used for the shearing process. The flocs used in the experiment are those consisting of primary silica particles of 1.0ptrm in diameter. The digitization is made at a constant time interval for a given shear rate. The interval varies for different shear rate since the flocs move faster at a higher shear rate. The projected area of a floc in the focal plane is measured and an equivalent diameter is calculated. This diameter corresponds to the diameter of a circle that has the same floc projected area. The equivalent diameter of a floc provides an effective measure of the floc size since the flocs translate and rotate in the flow field. Each data sample of digitization is a frame of picture having a number of flocs. The average of equivalent diameters in a sample represents an instantaneous average size of the flocs that move in the focal region at that particular instant as shown in Fig. 3.1. The analyses of the floc size and shape are based on the measurements of the projected shape of flocs. It is done by tracing the profile of a floc image. The area of the outlined shape and the length of the profile edge are measured by the image software. Figure 3.6 shows the variation of the instantaneous average floc size with time for the entire shearing process of about 36 minutes. During this period, the shear stress is I I I I 2000 Time (Second) Fig. 3.6 Variation of Floc Size with Time and Applied Shear Stress 16 N C.) 0 2 0 0 T=6.2 N/M2 500 1000 x=20.6 N/M22 7.6 N/2 T=13.3 N/M2 1500 52 increased from 6.2Pa, to 13.3Pa, then to 20.6Pa, and finally to 27.6Pa. The data exhibit a large fluctuation, because the amount of flocs captured in the small focal region at a moment represents flocs which have a large random distribution in size. Since the flocs are randomly distributed in the flow field, the floc structures (size and shape) captured at a fixed small region in the flow appear to be random. A linear regression analysis is conducted to analyze the time effect at a given shear rate. That is, the averaged floc size d is expressed in terms of the linear function of dimensionless parameter yt as follows: d=do(J +kyft-t)) (3.4) Figure 3.7 shows the variation of the averaged instantaneous floc size change with nondimensional parameter, yt, and the linear regression results at four shear stress levels. The values of k and do for the four shear stresses are given in Table 1. The constant k is the linear rate of change in floc size. It is a measure for the erosion process during the constant shearing. It is seen that the k is very small at a constant shear rate, that is the floc size decreases with 't very slowly. The absolute value of k decreases as the shear rate increases for the first three shear levels. The positive k for the fourth shear level only indicates that the effect of slow erosion is very small so that k results from statistical fluctuations. The dimensionless values of kyT for the last two shear stress level is so small that the floc size can be considered not changing with time during the shearing. For the first two shear stress level, the erosion process contributes about 26% and 13% to the overall size reduction during the shearing. (a) T=6.2 N/M U .2 0.) 15 14 13 12 11 10 1000 (b) T=13.3 N/M2 2400 2600 2800 yt (c) T=20.6 N/M2 6000 yt 15 14 13 12 11 10 9 8 7 6 5 4 8Isu (d) T=27.6 N/n2 8100 8200 8300 Fig. 3.7 Effect of Time on Floc Size at Constant Shear Stresses 0) 15 14 13 12 10 9 8 7 6 5 4 3- 500 0.) 08 U 15 14 13 12 11 10 9 8 7 6 5 4 3000 3200 yt 5000 5500 8400 8500 yt 3 54 Table 3.1 Linear Regression Results For Floc Size Variation z(Pa) 2(1/s) to(S) Duration, T(s) k do( m) kyT 6.2 0.8899 0 1200 -0.000239 9.37 -0.255 13.3 1.8981 1200 500 -0.000139 5.95 -0.132 20.6 2.9407 1700 380 -0.0000156 4.90 -0.017 27.6 3.9346 2080 80 0.0000222 4.03 0.0070 Since the experiment is conducted on the same batch of flocs in a continuous operation for the four shear rates, there are actually two factors that affect the result. One is the shear rate, the other is the initial floc size and shape. As shear continues, the flocs become smaller, their shape becomes less irregular, and the erosion of flocs with time becomes more difficult. The characteristic of floc size change with time at different shear rates essentially gives the information for the strength of the flocs. For the flocs of the same chemical and physical properties, average floc size and size distribution vary when the applied shear stress changes. At a constant shear stress, the floc size does not change much with time since the strength of the flocs balances the external applied shear stress, unless a higher shear stress is applied to overcome their strength. The time effect exists in the size reduction and depends on floc initial size, shape, and applied shear stress. 3.3.3 Variation of Floc Size and Size Distribution with Applied Shear Stress If the time effect is neglected since it is small compared with that due to the increasing shear stress, the statistics of the floc size distribution can be obtained during continuous shearing under a given stress. Each individual floc that is sampled is treated as an independent element in the statistical analysis. The probability density function (PDF) of the floc size, the mean floc size, and its standard deviation are thus obtained for each 55 given shear stress. Figure 3.8a shows the mean equivalent diameter of flocs, d as a function of the applied shear stress. Although only four shear stresses are used, the data suggests a power law dependence of d on r, d C T-0.4 (3.5) -0.35 -3 -1.05 It should be noted that Sonntag and Russel (1987) obtained dC T or R OCT for flocs of 0.14 pm in diameter of polystyrene latex dispersed in glycerin-water mixture in a sheared Couette flow, in which R is the gyration radius of flocs. This similarity suggests that the different flocs share a similar dispersion behavior in the shear flow. Fig. 3.8b shows the variation of the standard deviation of the floc equivalent diameter, o-, with the applied shear stress. The standard deviation also decreases as the applied shear stress increases following the power law given below, U- C T-0.72 (3.6) This decrease is more rapid than that of the mean floc diameter, which indicates that the distribution of floc size is in a size band that becomes more narrow as the shear increases. The PDF's for the distributions of the logarithm of floc size at various shear stresses are shown in Fig. 3.9. It is seen that the PDF's are approximately logarithmic-normal. At a low shear stress, r = 6.23 Pa, floc sizes are widely distributed. As shear stress increases, the resulting flocs not only become smaller but also become more uniform in size which corresponds to a more narrow PDF distribution curve. 56 109 8 7 6 5 4 0 . 10 15 20 25 30 Shear Stress (N/m2) (a) Variation of Mean Floc Diameter with Applied Shear Stress 5 S4 3 4-2 UO 5 10 15 20 25 30 Shear Stress (N/M2) (b) Variation of Standard Deviation for Floc Size with Applied Shear Stress Fig. 3.8 Statistics of Floc Size at Various Applied Shear Stresses 57 0.5 -T=6.2 N/rM - =13.3 N/2 -=20.6 N/M2 - - =27.6 N/M2 0.4 0 C.3 0.2 0.1 0 10* 10, 102 Floc Size (pm) Fig. 3.9 PDF of Floc Size Distribution at Different Applied Shear Stresses 58 3.3.4 Change of Floc Shape with Time and Stress One of the advantages of the present study is the visualization and characterization of the floc shape. The spherical floc assumption has been widely used in theoretical models and measurement techniques in the particle sizing. The video images of the flocs during motion clearly show that the resulting flocs under shearing are rather irregular in the present study, however. As the floc size decreases with increasing shear stress, the flocs appear to have less irregularity on the surfaces, but are definitely non-circular for fine flocs composed of large number of particles. In order to quantify the floc shape, both the projected area S and perimeter P of a floc image on the focal plane are measured. A shape factor (sJ) is defined as follows to analyze the shape change, P sf = 2 (3.7) For a circular shape, sf = ;r. The averaged shape factor over all flocs in one sampled frame of picture is plotted with time and shear rate in Fig. 3.10. While the mean value of sf decreases slightly with increasing shear stress, the fluctuation of sf is noticeably smaller at higher shear stress. To see the variation of shape factor with time at a constant shear rate, the linear regression analysis is performed on the curve in terms of the following equation: sf=sfo(J +kyft-t)) (3.8) The coefficients in the regression equation are given in Table 2. Also given in the table are the statistical data for the shape factor obtained over all flocs at a given shear rate. It can been seen that the change rate of time, k, is small at all the four shear rates, while the 4.5 _s 3 =. N/m2 6.2 N/m =13.3 N/M2 0. 500 1000 1500 2000 Fig. 3. 10 Variation of Floc Shape Factor with Time and Shear StressTme(cnd 60 mean shape factor, sf, becomes closer and closer to r and its standard deviation decreases as the shear rate increases. The changes of mean shape factor and its standard deviation suggest that flocs become less irregular in their shapes. However, the deviation from ir indicates that the flocs are far from spherical. Table 3.2 Linear Regression Results and Statistics For Floc Shape Factor '(Pa) A(i/s) to(s) k sfO ,- C6.2 0.8899 0 -0.0000296 3.658 3.535 0.307 13.3 1.8981 1200 -0.0000229 3.526 3.452 0.241 20.6 2.9407 1700 0.0000061 3.395 3.390 0.197 27.6 3.9346 2080 -0.0000507 3.390 3.373 0.193 In order to illustrate the average shape of flocs in the shear flow, suppose an ellipse with a major axis 2a and minor axis 2b is used to fit the image of the projected floc crosssection, the area is given by S= ; a b (3.9) and the perimeter is approximately P ~ [ (1.5(a+b)- ] (3.10) The major-to-minor axis ratio of the ellipse can be expressed in terms of the shape factor as a ] 2 b 0.5(q+q2 -4) (3.11) with q= 2(+2 ). Using the measured values of sf and the above expression, we can a calculate bto estimate the degree of non-circularity of the flocs. To quantify the 61 a deviation of the floc across-section shape from a circle, the major-to-minor axis ratio, , is calculated using Eq. 3.11 and the measured sf. Figure 3.11 shows the mean values of a and sf under four different shear stresses. Even at the highest imposed shear stress of r a = 27.6 Pa, the ratio, b, is about 2. This clearly suggests that the flocs generated under simple shearing are not close to sphere. 3.6 sf=P/d alb- 2.4 - 2.3 3.5 - 2.2 3.4 - 2.1 2 3.3 1.9 3.2 I . . . . . . 1.8 5 10 15 20 25 30 T (N/M2 Fig. 3.11 Variation of Mean Shape Factor with Applied Shear Stress 62 3.4 Summary A visual study is carried out to help us understand the mechanism of dispersion of fine flocs in a simple shear flow and floc strength. The variations of floc size, shape, and mass with time and shear stress are analyzed. The main results and conclusions are summarized as follows: (1) In shear flow, rapid breakup of fine flocs occurs when the applied shear stress increases and it gives a significant contribution to the floc size reduction. In the absence of erosion, the shear stress required to break up flocs increases with decreasing floc size following a power law dependence. (2) The long-time constant-stress shearing results in slow erosions of flocs in a simple shear flow. Whether the erosion contributes a small part or a large part to the size reduction depends on the duration of shearing and the shear rate. The erosion rate mainly depends on the shear stress, floc size, and floc shape. The erosion is faster for a larger floc size with an irregular shape than that of a smaller floc size and a regular shape. The results in the part of the work are the base for other experimental setup and the determination of experimental conditions for the work in later chapters. CHAPTER 4 FLOC BREAKUP IN ORIFICE FLOW--PART I FLOW CHARACTERIZATION 4.1 Introduction Orifice flow device refers to the axisymmetric duct flow blocked using a plate with an orifice. The area ratio of an orifice flow device is defined as the ratio of the upstream duct section area to the orifice area. For more than 100 years, orifice device has been extensively used in hydrodynamic engineering as one of a family of differential pressure flowmeters due to its simplicity and reliability. The orifice area ratio is usually no more than 20 in flowmeters. Since the late 1970's, orifice with very high area ratio has been applied to study deflocculation and deaggregations (Kousaka et al, 1979, Yuu and Oda 1983, Sonntag and Russel 1987, Higashitani et al 1991,1992). Because the flow accelerates as it approaches the orifice, a very large velocity gradient is generated near the orifice entrance. When flocs and aggregates in the flow approach the orifice, they experience a high rate of strain or stress so that they may deform and break up. Clearly, a complete knowledge of the behavior of the stress field near the orifice entrance is essential for understanding floc dispersion process. Because of the complexity of the flow field near the orifice and the difficulty in the flow simulation and measurement for high area ratio (>100), very little attention has been paid to the flow characterization of high area orifice flow. Only limited information such as velocity along centerline was reported (Higashitani, 1991). Some overall flow parameters (e.g. pressure 63 64 loss estimate across the orifice or energy dissipation based on experience) in the flow are used to evaluate the hydrodynamic conditions. Unlike Couette flow and hyperbolic flow, in an orifice flow, the stress field is highly nonuniform in the region of interest and no simple methods are available to describe the flow field and stress field other than numerical simulation. Previous numerical simulations for orifice flow were for flowmeters with orifices of low area ratios ( < 16 ) and high Reynolds numbers (Re > J' ) (Giovannini & Gagnon, 1993; Morrison & DeOtte et al., 1990; Morrison & Panak et al., 1993; Patel & Sheikholeslami, 1986). The flow discharge coefficient was usually of interest. In the experimental investigations of particle dispersions, the orifices used have much higher area ratios (> 100 ) in order to obtain high strain rate. To observe and visualize the motion of particles approaching an orifice, the experiment is usually operated at a very low average velocity using highly viscous fluids to achieve sufficient stress to break the flocs. In the work of Higashitani et al., (1991,1992) the Reynolds numbers based on the diameter of flow pipe and the average velocity are less than 500. In the syringe experiment (Sonntag and Russel, 1987), Reynolds numbers ranged from 267 to 2667. To recapitulate, low to moderate Reynolds number and high area ratio are used for floc dispersion, in contrast to high Reynolds number and low area ratio of flow metering. In the work presented in this chapter, a numerical simulation for the flow inside a pipe with an orifice area ratio up to 567 is performed at finite Reynolds number ranging from 0.01 to 3200, depending on the orifice area ratio, using a stream function-vorticity 65 formulation for axisymmetric laminar flow. The velocity field, the rate of strain tensor field near orifice entrance are analyzed in detail. 4.2 Formulation 4.2.1 Governing Equations and Boundary Conditions The schematic for an orifice flow field and coordinates are shown in Fig. 4.1. The only important geometric parameter for the orifice is its area ratio: 1 2= (4.1) where r, is the ratio of orifice radius to upstream pipe radius R. r Wall Ti=d Inlet boundary Exit boundary = .. .-... . . . ................ .. .. . . . . . . . rfc ~ n e ln ' I Orifice Centerline I I x x Xo x=0 x=x, . 4=0 4=41 4= 2 =I Fig. 4.1 Orifice Schematic and Coordinates System 66 Stream function-vorticity equations for axisymmetric flow in cylindrical coordinates (x, r) are derived from the continuity equation and N-S equations by defining a vorticity 4 and a stream function y(x, r): 4 =V,-Ur (4.2) 1 1 U= rY , v=-r X (4.3) where u, v are the velocity components in x direction and r direction. The subscript represents partial derivative. The transport equation for vorticity in nondimensional form is 2 + + Re r r )44 in which all the parameters are normalized based on the radius of the pipe R and the average upstream velocity U, Reynolds number Re is based on the diameter of pipe D and U. The equation governing the stream function is 1 V. + y,, - y, = -r; (4.5) The region of interest is before the entrance of the orifice, since our visualization of floc breakup in contractile flow has indicated that the deformation and breakup of flocs occur when they approach the orifice. The inlet boundary are fixed at 1.OD from the orifice entrance face. The inlet condition is a fully developed pipe with the velocity profile given by u=2(1-r2), v=O (4.6) 67 Hence, at inlet, {=4r, y,=r2(1-0.5r2) 4.7) Along the centerline, v=0, Vi=0, (=0 (4.8) On the wall, v=O, u=O, y=0.5 (4.9) The exit boundary is located far downstream of the orifice and a zero derivative condition is assumed for the vorticity, with vanishing transversal velocity: Q=O, v=O (4.10) For comparison purpose, inviscid flow computations are also carried out on the same geometry. The governing equation is 1 Yxx + Yrr YIr 0 (4.11) and the inlet condition is '2 r . (4.12) On the wall, Y/=O. 5 (4.13) At the exit, Ox =~0 (4.14) 69X is enforced. 68 Meanwhile, computation for two dimensional channel with a small slot, as used in Chapter two for the test of the visualization of floc breakup in contractile flow is conducted to quantitatively compare the difference between an axisymmetric orifice flow and a two-dimensional channel flow through a small slot. 4.2.2 Grid Arrangement and Numerical Schemes Due to the large velocity gradient near the orifice, fine grids must be used and coordinate stretching is necessary. The stretching in x coordinate upstream of orifice corresponding to x, x 0 is x = x01 c atan (-tan I for 0 :s- , (4.15) as shown in Fig. 4.1, denotes the uniform coordinate of computational domain in axial direction. Downstream of orifice, x, x x , the following is used: x = x2+ A (e"'- 1) for 2 J5-1 (4.16) The grids within orifice, 0 J+r0 1-r0 (2q-1-q 1 r = + c atan tan for ,.51 (4.17) 69 where 7 denotes the uniform coordinate of computational domain in radial direction. The grids in the core region (0 The governing equations and boundary conditions in physical domain (r, x)then are transformed into computational domain (7 q). Equations (4.4) and (4.5) are transformed to the following equations: S+ h 7]=h +2(4.18) -[(L2 1 + 7rV,,)= -rq (4.19) where = (x +r) = (4.20) 2 (4.21) In the implementation of the finite difference method, the second order central difference scheme is used for interior points. The following difference scheme (Lugt and Haussling, 1974) for vorticity on the wall are employed: _ Aij, - i,j,-1 -4yi,j-2 + 4ij,.-3 I~ijw4A772 rj h -2 (4.22) jwrjW Orifice 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.5 Fig. 4.2 Coordinates and Mesh for an Orifice Flow (81X41) (P=400, rO=0.05) r -0.5 -1 C x Centerline I _- - - - - --- ----- L A 71 for horizontal wall wherej=j,, and 4=/j, - Vi" -,I -4Vi,-2,j + Yiw-_,, 2wj gAg2r h-2 (4.23) for the front wall of the orifice at x=O where i=i,. On the back side, x=x2, similar expression can be easily derived. These formulae are derived from the boundary conditions and making use of Taylor's expansion of the first order derivatives of stream function at (ijs) and (ij42) as follows: 2+ -Aii+j2+9 (4.24) + (-2A)+4A772+o(Aq3) (4.25) Since (yvq, ,=O and -r i4jhZ according to Eq. (3.19) and boundary conditions Eq. (4.9), taking 4xEq. (4.24) - Eq. (4.25), then the following equation for the wall vorticity, 46, is obtained: C I- 4 = -2r h1A + o(A (4.26) After the first order derivatives at interior point in the above equation are replaced with the central difference: + i"b,-1 2A,j,-3 + 2A ) 0V) 2A + (4.27) ,62- 2 72 (9V/ = - ijw-2 + o(A 2 2) (4.28) 1 -;)w-l q the Eq. (4.22) is obtained. Similarly, expanding the first derivative of stream function at = j, the expression for the vorticity on the vertical wall can be obtained. Time marching procedure is applied to solve the discretized equations for 4(ij) and y(ij). The velocity field u(ij) and v(ij) are then derived from Yi(ij) by Eq. (4.3) using central difference: 1 Vli,j+ - V/ij-1 ui' r 2A77 hrj (4.29) 1 Vl+.,j - Y(i-4 vi, = -rj 2A hxi (4.30) 4.2.3 Validation of the Numerical Method Grid independence is first examined for the case of 8=400, Re=200. Figure 4.3a shows the computed vorticity distributions along the vertical wall of the orifice at x=0, using six different grids. Figure 4.3b shows the computed velocity gradient along centerline using the same set of grids. It can be seen that when the grid is not less than 81X81, the results become independent of grid size. The discrepancy near the comer always exists because of the corner singularity. To further validate the numerical solution, the centerline velocity is compared to the experimental data of Higashitani (1991) and is shown in Fig. 4.4. The Reynolds number is 64, r0=0. 1 and a 81x81 grid is used in the computation. The good agreement between the data indicates the reliability of the code. 73 1 - r 0.9 0.8 Grid size 0.7 0.6 - 161X81 0.5 -- - - - 121X81 0.4 - 81X81 - - - 121X41 0.3 - 81X41 41X41 0.2 - 4 P=400, R =200 0.1 (a) Vorticity Distributions on Orifice Entry Wall 5000 du/dx Grid size 161X81 4000 - - --121X81 ------- 81X8I 3000 - 121X41 81X41 2000 41X41 1000 0 P3=400, R,=200 -1000 -2000 i | x -0.05 0 0.05 (b) Velocity Grendient Distributions along Orifice Centerline Fig. 4.3 Effect of Grid Size on Vorticity and Velocity Gradient 74 10 u (cm/s) 9 8 0 Experimental(Higashitani, 1990) Computational Re =64 7 6 5 4 3 2 0 1 -2 -1.5 -1 -0.5 0 x (cm) Fig.4.4 Comparison of Computational Centerline Velocity with Experimental Data (P=100, Re=64) 75 4.3 Results and Discussions Since the present numerical study is to aid the interpretation of the floc breakup experiment, the interest is focused only on the entrance region of the orifice ( x 4.3.1 Basic Features of Orifice Flow Field Figure 4.5a and 4.5b show the contours of stream function (i.e. streamlines) and the contours of vorticity at Re=] and ro=0.05 ( 8=400 ). When Re reaches some critical value, Rec for a given p, the flow after orifice becomes unsteady. Re, is very small for a high area ratio orifice. For example, flow behind the orifice becomes unsteady when Re >1 for p=400. Computational results indicate that the flow field before the orifice reaches a steady state while the flow field in the downstream region of the orifice is unsteady. Since we focused on the entrance region of the flow the unsteadiness in the downstream region is not of concern. The inclusion of a long downstream region in the computation is to guarantee that the exit boundary condition, Eq. (4.10) has no effect on the entrance region. Figure 4.6a and Fig. 4.6b show the distributions of axial velocity and its gradient in axial direction along centerline for the case of Re=1600 and different orifice area ratios. A 161xl51 grid is used for the computation of the case. Large variations in the velocity gradient are observed. It is also seen that the velocity gradients reach its the maximum values before the orifice. Figure 4.7a and 4.7b show the axial velocity and its velocity r C 3 enterline 2 3 1 (a) Streamline 2 1 (b) Contour of Vorticity Fig.4.5 Steady Solution of Vorticity for P=400 and R,=1 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0~. -N1\-- - -/1 ..-.Z --..... 4 OA ___ _____*" -- Centrlin N N' N. N,' -1 r 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 01 x / -4 0 x 77 500 400 300 200 -0.1 0 0 -U Orifice Entrance p=400 - - - -- p=256 - --- - 0=156 P=100 - =64 j3=25 -. . -V -/ (a) Centerline Velocity - du/dx I i Orifice E -( P=400 - - - - P=256 --- - - P=156 - P=100 -=64 P=25 - - ' -0.1 entrance 0 jX .1 0 (b) Centerline Velocity Gradient Fig. 4.6 Centerline Velocity and Velocity Gradient Distributions at Different Area Ratios ( Re=1600 ) x .1 100 0 6000 5000 4000 3000 2000 1000 0 - - . . 78 800 U 700 Re=10 Orifice Entrance - --- - Re=200 600 - - - - Re=400 Re=800 500 - - - Re=1600 Re=3200 400 300 200 100 0 I x -0.1 0 0.1 (a) Centerline Velocity du/dx 8000 - Orifice Entrance Re=10 7000 - - - - - - Re=200 Re=400 - - - Re=800 6000 - - - Re=1600 Re=3200 \ 5000 4000 - \ 3000 2000 1000 0 -0.1 0 0.1 X (b) Centerline Velocity Gradient Fig.4.7 Centerline Velocity and Velocity Gradient Distributions at Different Reynolds Numbers ( P=400 ) 79 gradient distributions at centerline for the orifice of area ratio p=400 at different Reynolds numbers. It can be seen that the location of the maximum velocity gradient is moving upstream as the Reynolds number increases. 4.3.2 The Strain Rate Characteristics Of Orifice Flow The rate of strain tensor D in cylindrical coordinates (x, r, 0) for axisymmetric flow can be determined using the gradients of two velocity components: U pu +V 0 U + V v 0 (4.31) If the eigenvalues of the symmetric rate of strain tensor at a point in the flow field are A,, A2, A, and: 2? A22 A3 (4.32) then the maximum extensional strain rate, S.,, maximum shear strain rate, y., can be expressed in terms of them as follows: S.= A, (4.33) 21-22 7n= 2 (4.44) They are useful to quantify the stress field exerted on the flocs. Figure 4.8a and 4.8b show the contours of maximum principal strain rate S. and the contours of maximum shear rate y. near orifice entrance region for /=156 at Re=100. The contours of S. and y. are circle-like around the orifice corner since they are dominated by the singularity at the sharp corner. The values of S. and y. increase near 80 1 72708 r (a) Maximum Principal Strain Rate ____________________1.7845 1-62152 1. x 900 7827 Centerline -1 -0.5 (b) Maximum Shear Strain Rate Fig.4.8 Contours of Maximum Strain Rate Before Orifiec (P3=400 and Re =100) 1.5~~2 31-. ~ 0.5 0 r 0.5 -1. 0 Orifice X Centerl5 ; I I rifice I 1 5 81 the corner, but change relatively slowly near centerline at the inlet face. They also change rapidly in the wall boundary layer. The resolution of the grids has some effect near the corner. The effect will always exists near the corner since the vorticity is mathematically infinite at the point. On the other hand, the changes in Sm and y in core region near the centerline are relatively small before orifice. Their isarithms normal to the centerline in the core region suggest that Sm and y,mt are radially uniform in the region and equal to their values at centerline. Similar behavior can be observed for the flow at other / and Re. It is also found that the size of the core region is approximately equal to the radius of orifice. 4.3.3 Maximum Centerline Velocity Gradient As we have seen from the experimental visualization of floc motion in contractile flow in Chapter 2, the flocs in the flow always tend to move toward the centerline in the contractile flow due to the radial velocity in the flow. Most of the flocs passing through an orifice will experience the stress in the core region. Therefore, the rate of strain at centerline is important for our study of floc breakup in orifice flow. At centerline all the off-diagonal elements of the strain rate tensor given by Eq. (4.31) are zero. The three eigenvalues of the strain rate tensor are a -a 1 1-Ia a,' 2=2&,A (4.45) which corresponds to a uniaxial extension. The maximum principal strain rate Sm, and maximum shear strain rate y. at any point along the centerline are determined by the axial velocity gradient at centerline as follows: 82 S.(x, r=0) = u (4.46) x, r=O max(x, r = 0) - u (4.47) x, r=O They vary along the centerline and reach their maximum values S,. and r,. somewhere before the orifice entrance face. Since y.ma is 3/4 of Sc,. and S,. is equal to the maximum centerline axial velocity gradient, the effects of Reynolds number and orifice area ratio on Sc,. are examined. Figure 4.9 shows the effect of Re on maximum centerline gradient velocity, S,. for five orifice area ratios. It can be seen that S,., which is already normalized by U/R, decreases as Re increases. As Reynolds number approaches to 3000, it tends to the value for inviscid flow of the corresponding orifice area ratio. The asymptotic behavior of S'. at large Re is obtained based on curve fit, S,.(Re, )= SO(f) + a(3) Re213 for Re > 200 (4.48) as shown in Fig. 4.10. The values of SO and a at several p's are given in Table 4-1. The centerline maximum velocity gradient Sc. ,, are also given in the table at corresponding /8. It can be see that SO is actually equal to Scm,, at moderate area ratios. The differences at two highest area ratio are less than 1%. Table 4.1 Coefficients in Eq. 4.48 and Sc. for inviscid flow P 64 100 156.25 256 400 SCO 243 500 1050 2280 4550 S cmi 243 500 1068 2303 4602 a 5198.3 7366.6 21067 28454 40439 83 10- S ------- P=567 - A- - P=400 - - - P=256 e P=156.25 _--- - =100 P 1=64 ------- -- - p =6 101 El6- 010, 10 10.2 10-1 100 10 102 103 104 Re Fig.4.9 Effect of Reynolds Number on the Centerline Maximum Velocity Gradient 84 S -S, 104 - - 0=400 -- A - 1=256 -- =- 156.25 --- - =100 -o--- 0=64 10, 102 Re 4 1 ~ 101 101 102 103 104 Re Fig.4. 10 Asymptotic Property of Maximum Centerline Velocity Gradient at Large Reynolds Number 85 In the range of 10 < Re < 200, a small flow separation bubble, whose length depends on the thickness of the orifice plate, occurs near the orifice front corner. This separation bubble affects S,,. so that the asymptotic value S,,() is established only at higher Reynolds number when this separation bubble merges with the main separation bubble after the orifice plate. The effects of the orifice area ratio on the maximum centerline shear rate are shown in Fig. 4.11. in log-log plot. Straight lines for creeping flow and for inviscid flow are obtained. The finite Reynolds number results are approximately linear between these two lines for the range of 8 investigated. For the Reynolds numbers investigated, the relation may be expressed as follows R du Scmax -: R A0Ck (4.49) Sc" U dx -S where k ~ 1.54 in the creeping flow regime and k ~ 1.61 based on an inviscid flow computation. Since the cross-sectional averaged velocity at the orifice increases linearly with 8 based on mass conservation, the length scale at the orifice scales with o C (4.50) a simple dimensional analysis indicates that S, /c 2 (4.51) This exponent 3/2 is in agreement with the numerical results. It is also seen that at Re=1600, the Navier-Stokes equation based result still differs than that of the inviscid flow solution by 5 - 16%, depending on the area ratio , indicating that the viscous flow computation is needed for determining S.... even at this Reynolds number. 86 105 S. Re=0.01 - ~- Re=100 -- -s- Re=1600 --_~ -Inviscid 10 10, 102 I I , , 200 400 600 800 1000 Fig.4.11 Variation of Maximum Centerline Strain Rate with Orifice Area Ratio 87 4.3.4 Comparisons Between Axisymmetric Flow and Two-dimensional Flow A numerical simulation for flow passing a two-dimensional slot is also performed to compare the rates of strain in 2-D flow with that in a 3-D orifice. Figure 4.12 compares among four configurations. First, for a 2-D slot and a 3-D orifice with the same area ratio ,8=64, 2-D slot generates much higher velocity gradient. Next, when the radius ratio of 3D orifice is the same as the height ratio of 2-D slot, maximum velocity gradient in the 3D orifice is 29 times that of the 2-D slot. For the two dimensional flow, a similar dimensional analysis shows that S.. c /8'. The numerical results agree with the analysis very well. 4.4 Summary (1) The flow through an orifice with high area ratio up to 400 is simulated numerically for finite Reynolds number. Orifice flow for /-3567 is also conducted at very small Reynolds numbers ( Re=O. 1) for our study of floc breakup in orifice flow. The rate of strain field of the flow is analyzed. The strain rate field in the core region is radially uniform and can be described by the value along the centerline. This core region before the orifice is of interest for the analysis of floc deformation and breakup. (2) The dependence of maximum centerline velocity gradient on Reynolds number and orifice area ratio is analyzed based on the numerical simulation and dimensional analysis. The variation of the maximum centerline velocity gradient scales with the orifice area ratio, 8, as S. ~ p''. The asymptotic behavior of the maximum centerline velocity gradient with Reynolds number is obtained. These relations can give a good estimation 88 S 1m0 Radius Ratio = 0.05 (3D) - Aera Ratio O = 64 (2D) 10 - Aera Ratio f = 64 (3D) H-A- - = - Height Ratio = 0. 05 (2D) A A A I I I I II 101 100 10' 102 103 Re Fig.4.12 Comparison of Maximum Centerline Velocity Gradient between a 2-D Slot and a 3-D Orifice 102 11 10 .2 104 89 for an orifice of given geometry configurations and flow conditions considered in this work. (3) The comparative numerical simulations for the flow of two-dimensional slot and for the inviscid orifice flow have also been conducted to compare their characteristics with axisymmetric flow and evaluate the validity of inviscid calculation of the flow characteristics. CHAPTER 5 FLOC BREAKUP IN ORIFICE FLOW--PART 2 MEASUREMENTS 5.1 Introduction Based on the flow simulation and characterization results for the flow of high area ratio orifice given in the last chapter, this chapter will present and discuss the experimental results on the floc breakup in an elongational flow generated by an orifice. These results include the measured variation of floc size distribution and mean size with flow conditions. In order to determine the effective stress that causes the breakup of flocs in the orifice flow, a comparative test of the floc breakup in a uniform simple shear flow is also conducted and presented in this chapter. The experimental devices and techniques used in this study are described in detail. There have been a few studies of the floc breakup in orifice flow. Early applications of orifices for deaggregation involve air stream. Kousaka et al., (1979) compared the deaggregations in flows generated by a variety of devices to investigate the possible dispersion mechanisms of CaCO3 and F.2O, aggregate in air stream. They used an orifice with diameter ratio r, = 0.15 as one of the dispersers in their experiments. The aggregate size distributions were obtained at different mean velocities near the orifice entrance. Effects of orifice flow on the dispersion of the aggregates were demonstrated. Yuu and Oda (1983) measured the changes of the fly ash particle size distributions due to the disruption in air stream through orifices of ro = 0.4 and 0.6 with various far upstream 90 91 velocities to verify their particle population balance equation. The results obtained in these studies were not well-established because of the lack of the precise control over experimental conditions. As mentioned in previous chapters, Sonntag and Russel (1986) used a common plastic syringe of r,= 0.083 to conduct their floc breakup experiments. A large acceleration of the flow was generated when a syringe was driven by a constant force, and the flocculated suspension of monodisperse polystyrene latex was expelled from the needle of the syringe. They obtained the variations of the floc mass and size with an overall nominal strain rate in the flow. A numerical analysis for an inviscid flow passing through an orifice of r,= 0.27 (8 = 13.7) was performed to characterize the flow in the syringe of ro = 0.083 (,8 = 144) used in their experiment. They assumed a floc deformation model based on the size variation with flow condition. Since the stress field was incorrectly characterized, large uncertainties exist in the interpretation of their results. Besides the breakup process of flocs along the centerline of contractile flow in an orifice reviewed in Chapter 2, Higashitani et al. (1992) investigated the breakup of flocs composed of a small number of polystyrene latex particles below 1 pm by measuring the floc size changes across the orifice using a Coulter counter. The diameter ratios of the three orifice used are r, = 0.002325, 0.0116 and 0.02325. The average size of broken flocs and the maximum number of constituent particles in a broken floc were expressed as function of the estimated energy dissipation at the orifice. Although the floc breakup in the orifice flow can be characterized by the power input to the system, the determination 92 of floc strength by the orifice flow needs the evaluation of the flow stress and the effective stress that causes the breakup of the flocs when they move in the flow. 5.2 Experimental Apparatus and Procedure 5.2.1 Orifice Setup and Procedure Figure 5.1 shows the schematic of the experimental apparatus. The flow is generated in a vertical syringe tube by a piston moving at constant velocity. The piston is driven by a syringe pump. The orifice plate is located at the bottom of the tube. The orifice diameter is d=J.1 mm and the internal diameter of the syringe tube is D=26.2 mm; hence the orifice area ratio is 8=567.3. The flow rate is controlled by the syringe pump, which can be adjusted continuously from 0 to 389 ml/h, corresponding to a maximum upstream average velocity of U=O. 02 cm/s in the tube. If water is used as the working fluid, the Reynolds number based on the D and U would be 5.24, since the dimensionless maximum velocity gradient of the centerline is SCmx dx -1500 (5.1) c max based on the numerical simulation results given in Chapter 4, the instantaneous centerline maximum shear stress for the setup is 3 ( du = du TcaX P du) x 1.72x102 (Pa) . (5.2) C max C max Obviously, this stress is too small to deform the flocs of interest. To achieve a sufficiently high shear stress to break flocs, a 2 million molecular weight at concentration 50% dextran solution with a viscosity of 2.9 Ns/m2 is used. The maximum shear stress can 4U d Sample Collection Orifice: D=26.2mm, d=1.1mm '.0 S Y R N G E P U M P I,. * Floc Size Distribution Fig. 5.1 A Schematic of Experimental Setup for the Floc Breakup in an Orifice Flow r Coulter LS-230 Particle Size Analyzer |

Full Text |

PAGE 1 DISPERSION OF FLOCCULATED PARTICLES IN SIMPLE SHEAR AND ELONGATIONAL FLOWS By XUELIANG ZHANG A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1998 PAGE 2 ACKNOWLEDGMENTS I would like to express my sincere appreciation to the following people for their help and support throughout my study at the University of Florida: Dr. Renwei Mei, my advisor, constantly gave me his trust, advice, guidance and patience. Without his commitment to the project, this dissertation couldn't have been completed. Dr. Roger Tran-Son-Tay, my co-advisor, offered me a great deal of helpful advice and expert guidance in my experimental work. Dr. James F. Klausner, as a member of my supervisory committee, gave me a lot of guidance and advice, especially in my early experimental work. Dr. Wei Shyy and Dr. Corin Segal served as members on my supervisory committee, reviewed my proposal and dissertation, and made valuable comments. Dr. Brij M. Moudgil and Dr. Hassan El-Shall gave me their guidances and helps. Ms. Emmanuelle Demay and Mr. Philippe Vigneron aided me in some of my experiments. Mr. Ron Brown helped me in the setup of experimental apparatuses. Dr. A. Zamam, Mr. J. Adler, Dr. J. S. Zhu, and Dr. S. Mathur assisted me in making floes and use some instruments. The Engineering Research Center (ERC) for Particle Science & Technology at the University of Florida, the National Science Foundation (Grant number: EEC-9402989), and the industrial partners of the ERC provided the financial resources for the project. In addition, my colleagues and friends at the University of Florida, Dr. Jian Liu, Dr. Guobao Guo, Dr. Hong Oyang, Ms Hong Shang, and Mr. Cunko Hu, provided me with various helps in my study and life in Gainesville, Florida. I am also grateful to the staff at the AeMES departmental office for their helps. Special thanks are given to Mr. Darrell D. Williams at Bristol, England for his help and encouragement during the past ten years. Last but most important, I am deeply indebted to my wife, Jie, for her understanding, encouragement, patience, and love. PAGE 3 TABLE OF CONTENTS Pages ACKNOWLEDGMENTS ii ABSTRACT vi CHAPTERS 1. INTRODUCTION 1 1 . 1 Background 1 1 .2 Literature Review 2 1.3 Objectives and Scope 6 2. VISUALIZATION OF FINE FLOC BREAKUP PROCESS 9 2.1 Experimental Devices 9 2.1.1 Cone-plate Device and Flow Description 9 2.1.2 Hyperbolic Flow Device and Flow Characteristics 13 2.1.3 Contractile Flow Chamber 19 2.2 Experimental Devices and Materials 20 2.2.1 Floes 20 2.2.2 Suspending Fluids 22 2.3 Results and Discussions 23 2.3.1 Floe Breakup in Cone-plate Shear Flow 23 2.3.2 Floe Deformation and Breakup in Contractile Flow 26 2.3.3 Floe Breakup in Hyperbolic Flow 33 2.4 Summary 37 3. FLOC BREAKUP IN SIMPLE SHEAR FLOW AND FLOC STRENGTH 38 3.1 Introduction 3g 3.2 Experimental Procedure and Data Processing 39 iii PAGE 4 3.3 Results and Discussions 42 3.3.1 Variation of Floe Mass with Time under a Constant Shearing 42 3.3.2 Variation of Floe Size with Time at Constant Shear Rates 50 3.3.3 Variation of Floe Size and Size Distribution with Shear Stress 54 3.3.4 Change of Floe Shape with Time and Stress 58 3.4 Summary 62 FLOC BREAKUP IN ORIFICE FLOW-PART 1 FLOW CHARACTERIZATION 63 4. 1 Introduction 63 4.2 Formulation 65 4.2.1 Governing Equation and Boundary Conditions 65 4.2.2 Grid Arrangement and Numerical Schemes 68 4.2.3 Validation of the Numerical Method 72 4.3 Results and Discussions 75 4.3.1 Basic Features of Orifice Flow Field 75 4.3.2 Strain Rate Characteristics of Orifice Flow 79 4.3.3 Maximum Centerline Velocity Gradient 81 4.3.4 Comparison Between axisymmetric Flow and Two-dimensional Flow g7 4.4 Summary gy FLOC BREAKUP IN ORIRICE FLOW-PART 2 MEASUREMENTS 90 5.1 Introduction 91 5.2 Experimental Apparatus and Procedure .92 5.2.1 Orifice Setup and Procedure 92 5.2.2 Couette Shear Device 94 5.2.3 Particle Size Analyzer 9g 5.2.4 Estimate of Reflocculation in Couette Flow and Orifice Flow 102 5.3 Results And Discussions 1 04 5.3.1 Effect of Flow Condition on Floe Size Distribution 107 5.3.2 Dependence of Mean Floe Size and Maximum Floe Size on Flow Rate 1 1 1 5.3.3 Comparison with the Result from Uniform Cone-plate Simple Shear Flow j j 5 iv PAGE 5 5.3.4 Comparison of Floe Dispersion between Orifice Flow and Cylindrical Couette Flow 119 5.3.5 Re-Examination of Sonntag's Experimental Data 123 5.3.6 Floe Strength Assessment 125 5.4 Summary 129 6. SUMMARY 131 6.1 Summary and Conclusions 131 6.2 Suggestions for Future Studies 134 REFERENCE 136 BIOGRAPHICAL SKETCH 140 V PAGE 6 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy DISPERSION OF FLOCCULATED PARTICLES IN SIMPLE SHEAR AND ELONGATIONAL FLOWS By Xueliang Zhang May 1998 Chairman: Dr. Renwei Mei Co-Chairman: Dr. Roger Tran-Son-Tay Major Department: Aerospace Engineering, Mechanics and Engineering Science Experimental studies on the dispersion process of fine flocculated particles in different flows are carried out through visual image analyses and particle size measurements. The flows investigated include a cone-plate shear flow, a cylindrical Couette flow, an orifice contractile flow, and a hyperbolic flow. Visual studies on the mechanisms of floe breakup in different flows are first conducted through a video image acquisition and analysis system. A variety of dynamic processes of the deformation and breakup of fine floes of size from 3mm to 30mm in the contractile flow, hyperbolic flow, and simple shear flow are visualized. The breakup and erosion process of floes subjected to a constant shear stress in the cone-plate flow is analyzed based on the changes of floe mass, size, and shape with shear stress and shearing time through the image analysis. A significant portion of the breakup, or size reduction, of the vi PAGE 7 fine floes takes place upon the application of the shear stress. Floe size continues to decrease through erosion mechanism. The erosion rate depends on the applied shear stress, the floe size, and the floe shape. An orifice flow is applied to break floes and determine floe strength. The flow field before an orifice of high area ratio is first numerically simulated and analyzed in order to characterize the flow and stress field. The dependence of the maximum centerline velocity gradient on orifice area ratio and Reynolds number is obtained and its asymptotic behavior in high Reynolds number regime is analyzed. The dispersion of floes in the orifice flow is analyzed based on the floe size distribution measured using a particle size analyzer. Due to the rapid rise of the axial velocity gradient near the orifice entrance, the floe breakup in the orifice flow is instantaneous and the floe erosion mechanism can be excluded. The centerline maximum shear stress in the orifice flow thus gives the floe strength of the resulting floes whose average size is subsequently measured. The floe strength determined from the short-time shearing in a cylindrical Couette flow at lower shear stresses follows essentially the same power law dependence on the floe size as determined in the orifice flow. Thus, floe strength measured in different flows can be unified using the maximum shear stress of the flow. vii PAGE 8 CHAPTER 1 INTRODUCTION 1.1 Background Many modern advanced materials, such as electronic, magnetic, optic, and fine ceramic materials, are produced from suspensions of colloidal particles. Floes or aggregates are loose, irregular, three-dimensional clusters of particles in such suspensions. The words, floe and aggregate usually are both used to refer to the wet powder structure in liquids. High performance of materials requires sufficient dispersion of the floes in suspensions, that is, sufficient breakup of floes into smaller floes or constituent particles. Although this dispersion process is actually the result of a number of different steps including milling, mixing, stirring, and so on, hydrodynamic shearing plays an important role in controlling the stability and uniformity of the suspension since the dispersion process is usually carried out in a hydrodynamic environment with or without the aid of dispersants. The flocculation (particle size enlargement) of particles and redispersion (particle size reduction) of flocculated particles take place simultaneously and constantly in the flow environment of the solid-liquid suspension. An important characteristic of floes is their binding force, that is, the ability of the aggregate structure to resist deaggregation. As a measure of this binding force, floe strength can be defined as resistance to breakup by shear forces induced by fluid velocity gradients. The quantitative evaluation of floe strength is important to both dispersion l PAGE 9 2 process and flocculation process. However, it is understood that the strength of floes in a suspension cannot be measured directly due to its spatially irregular structure and the random characteristic in its formation but must be deduced from the evaluation of other measurable parameters. Because the concept of "strength" for floes is always associated with their breakup which involves different mechanisms, the study on the floe strength should include the mechanisms of floe breakup and the force which causes this breakup. 1.2 Literature Review Thomas (1964) gave the first analysis on the mechanisms of floe breakup and floe strength. He proposed that large floes in a turbulent flow field break in the forms of bulgy deformation and rupture. He assumed that the pressure difference on the opposite sides of a floe causes its bulgy deformation and eventual rupture and that the pressure difference is due to the random velocity fluctuations of turbulent flow. His work formed the basis for a number of experimental investigations to determine floe strength since then. Based on Thomas' models for floe rapture mechanism and isotropic turbulence theory, several experimental studies of floe breakup in turbulent flows have been conducted to determine the floe strength by relating the floe size to the turbulent flow conditions. Tambo and Hozumi (1979) devised a special flocculator experiment to study floe strength by measuring the maximum floe diameter under a weak agitation. Matsuo and Unno (1981) used a turbulent pipe flow to evaluate floe strength. Bache and Al-Ani (1989) used a vertical pulsating water column driven by an oscillating plunger to relate the floe size to the turbulence energy dissipation. Moudgil, Springgate, and Vasudevan (1989) experimentally studied the strength of kaolinite, dolomite, and A1 2 0 3 floes in a stirred PAGE 10 tank. The results for floe strength obtained by the application of isotropic turbulence theory provide some qualitative understandings of floe characteristics. However, the shear field is spatially nonuniform in a stirring tank and only the overall mean energy dissipation rate can be estimated for flow description based on the power input. Floe breakup and reflocculation are usually present simultaneously. Therefore, the results obtained from such experiments do not suffice for the purposes of determining floe strength. Parker, Kaufman, and Jenkins (1972) derived a model for the breakup of complex activated sludge floes and inorganic chemical floes based on the breakup mode of surface erosion suggested by Argaman and Kaufman (1970). They proposed that the primary particles are stripped from the surface of a floe by fluid shear at a rate that is proportional to the floe surface area and the surface shearing stress. Kao and Mason (1975) and Powell and Mason (1982) used a four-roller device in their experiments of floe deformation and breakup in an elongational flow. This may be the first systematic visual work to study aggregate dispersion in fluid flows. Couette apparatus had also been used in their study for the case of simple shear. Quigley and Spielman (1977, see Lu and Spielman, 1985) conducted similar experiments for ferric hydroxide agglomerates in a four-roller device. It is important to note that in these experiments the size of primary particles from which the floes or aggregates are generated ranges from 20um to 400um and the size of floes or aggregates is about 3mm ~ 5mm. Sonntag and Russel (1986, 1987) investigated experimentally the structure and properties of flocculated suspensions in a simple shear flow of cylindrical Couette flow PAGE 11 device and in a contractile flow of syringe apparatus using small-angle light scattering to monitor the breakup. The average number of particles per floe and mean radius of gyration were related to the flow conditions. They proposed a mechanism of floe shape change during breakup in contractile flow. However, because of the incomplete flow characterization for the contractile flow, the experiment data in the orifice flow and the comparison with that in Couette shear flow were inconclusive. Their experimental data is re-examined based on the completed flow characterization in this work, and a more reasonable interpretation of their results is obtained. Lee and Brodkey (1987), and Wagle, Lee, and Brodkey (1988) studied various pulp floe dispersion mechanisms by performing a visual study in a turbulent shear flow between the two moving walls. Different pulp floe breakup modes were sketched according to their experimental observation and the effect of flow shear level on dispersion time and rate was proposed. The pulp floe size used in their experiment is about 5mm. Higashitani, Inada, and Ochi (1991) also employed a contractile flow ahead of an orifice in a pipe to directly observe the breakup process of floes. No breakup images were reported and only the variations of floe size along the flow direction were reported. The analyses were mainly based on the variations in the average numbers of constituent particles in a floe going through the orifice. The primary particles in their early experiment are about 90 urn in diameter. Higashitani et al. (1992) completed experiments on floes whose constituent particles were about lum in diameter using a Coulter counter to size floes, and analyzed the variation of floe mean size with applied pressure across the orifice. No flow characterization was given. It is noted here that the overall pressure drop PAGE 12 5 across the orifice cannot be explicitly related to the stress which causes the breakup of floes. Glasgow & Luecke (1980), Glasgow & Hsu (1982), and Glasgow & Liu (1991) have experimentally investigated the floe breakup mechanisms and floe strength in turbulent flow including impeller-stirred turbulence, turbulent jet, and turbulent channel flow. The methods for floe sizing were based on the manual measurement of the photographs of the floes. The floe size in their work ranged from hundreds of micrometers to a few millimeters. The mechanical interaction between floes and solid meshes (Glasgow and Liu 1991) was not considered. In their experiment with turbulent jet flow (Glasgow and Hsu 1982), the breakup process and trajectory of individual floes were clearly filmed and measured to determine the strength of floes. This work was very significant in the subject for floes whose size is larger than 100 micron meters. Navavrrete, Scriven, and Macosko (1996) visually showed the effects of shear, extention, and vorticity on the microstructure of iron oxide suspensions and the effects on their deformation and breakup using cryogenic scanning electron microscopy and videoenhanced light microscopy. Jiang and Logan (1996) investigated the structure changes, breakup, and coagulation of aggregates both in a laminar shear flow in a concentric cylinder device and a turbulent shear in paddle mixer. Both particle counter and image analyzer were used to obtain the changes in aggregate structure. Jung, Amal, and Raper (1996) studied the effect of turbulent shearing induced by stirring on the restructure and breakup of hydroxide floes using small angle light scattering technique. Spicer, Keller, and Pratsinis (1996) investigated the effects of hydrodynamic conditions on the evolution PAGE 13 6 of floe size and structure of polystyrene particles with aluminum sulfate in a stirred tank flow via the image analysis system. Yeung and Pelton (1996) used micromechanical technique to directly measure the extentional force by elongating floes of 6 to 40 micron in size into breakup using micropipettes. They found that there was no correlation between rupture strength in terms of force and floe size. However, the large scattering in their data due to a small sample collection may prevent one from seeing weak correlation. 1.3 Objectiv e and Scope Although the mechanisms of floe breakup, characterization of floe dispersion in various flows, and the strength of floes have been the subjects of those numerous theoretical and experimental studies, many issues remain. Floe breakup observations were made only for large floes (mostly, 3-5mm in diameter) and little is available for fine floes. The methodology for determining fine floe strength is not well developed. Results obtained from different methods cannot be related to each other. Most of those studies were carried out in turbulent flows and focused on the effects of overall hydrodynamic conditions (such as nominal turbulent shear rate) on the structure of floes and aggregates. Although relative effects resulted from the changes of the overall flow conditions can provide some insights into the dispersion process, the relationship between floe properties and the actual force acting on floes remains uncertain due to the lack of knowledge on the flow and the stress field. As an effort to address these issues, the present study focuses on visualizing the dynamic breakup process of fine floes whose size ranges from a few microns to tens of PAGE 14 microns, relating the floe dispersion results obtained in different shear flows, and developing a simple, reliable and practical technique for the determination of the strength of floes. The basic approach is to apply a shear flow to fine floes to deform and break them, and then measure the floe size distribution subjected to the flow. Two techniques are employed to size the floes. One is based on the visual image analysis of the deformation and breakup of individual floes via microscope-CCD camera image acquisition and analysis system. The other is to measure the floe size distributions using a Coulter particle size analyzer. To eliminate the uncertainty in the flow characterization, the following four different well-defined laminar shear flows are used to disperse floes: (1) a simple shear flow generated with a counter rotating cone-plate device where the shear rate is constant in the entire flow field but there exists a rotation of fluid; (2) an elongational flow formed with two opposing jet flows where a hyperbolic flow forms around the stagnation point when the two opposing flows meet, the shear rate is constant near the stagnation point and the flow is irrotational; (3) a contractile flow near an orifice in a pipe in which the high orifice area ratio causes an extremely high velocity gradient before the orifice. The shear rate is non-uniform and the rotation is nonzero in the flow; (4) a simple shear flow in the annular gap between two concentric cylinders. The shear rate in this flow is uniform and the rotation is nonzero. The present dissertation work consists of two main parts. The first is the visual study on the mechanisms of floe breakup and erosion processes in different flows. In order to further understand the mechanisms of fine floe dispersion in different flows, the dynamic deformation and breakup processes of individual floes of size ranging from a few microns to tens of microns in the cone-plate simple shear flow, hyperbolic flow, and contractile PAGE 15 8 flow are visualized. The breakup and erosion process of floes in cone-plate shear flow is quantitatively analyzed to determine the effects of shearing time and applied shear stress on floe mass, size, and shape. The second part is the study on the floe dispersion and floe strength determination by an orifice flow. Based on the results from the visual observation and the cone-plate shear flow, an orifice flow device is designed and fabricated to break up floes and to determine the strength of the floes. The flow approaching the orifice of high orifice area contraction ratio is numerically simulated to characterize the flow. The size distribution of the floes after breakup is measured using a Coulter particle size analyzer for floe size distribution measurement. The instantaneous breakup of floes in the orifice flow is characterized based on the stress field near the orifice entrance. In order to compare the floe breakup in the orifice flow with that in uniform shear flow, floe breakup in a cylindrical Couette flow with a short shearing time is also conducted. A rational basis for the floe strength and floe dispersion process under a shearing is established based on dispersion results from various flows. PAGE 16 CHAPTER 2 VISUALIZATION OF FINE FLOC BREAKUP PROCESS The visualization of floe deformation and breakup process was made previously only for floes of size in the millimeter range. A visual study for the dynamic deformation and breakup process of fine floes whose size is in the order of micron meters is first conducted to qualitatively investigate the mechanisms of fine floe dispersions in a simple shear flow, a pure shear flow, and an elongational flow. 2.1 Experimen tal Devices 2.1,1 Cone-plate Device and Flow Description The experimental facilities used in this work were originally set up by Tran-Son-Tay (1984) for the investigation of the motion of blood cells and recently modified for studying highly viscous cells and living cells (Henderson and Tran-Son-Tay, 1997). They include a flow chamber with flow tubing and a control system and a optical image acquisition system. Different flows can be formed with different flow chamber configurations. The image system consists of an inverted interference contrast microscope Olympus IMT-2, a CCD camera, a monitor, a video recorder, and a Macintosh computer. The magnification of lens used in the microscope is 40 and the resolution of the CCD camera is 720X360. A schematic of the system with a cone-plate flow chamber for generating simple shear flow is shown in Fig. 2.1. The shear flow is formed between a transparent glass plate and 9 PAGE 17 10 a Plexiglas cone of small cone angle 8= 1 .26 Â° Both plate and cone can rotate and can be adjusted both in direction and in magnitude independently. The cone and the plate can be controlled through a differential drive mechanism to keep the differential rotation velocity between them constant. The cone-plate setting can be moved in a radial direction so that the microscope can focus on a small region at any radial location. In the experiment, we choose R=3mm so that the gap size is about 66um. This is large enough to exclude possible mechanical effects of the surfaces of the cone and plate for floes of maximum size of about 15|a.m in the present experiments. The gap between the cone and plate can be adjusted and measured by focusing on the cone surface and plate, respectively. The cone tip is truncated, so that there is a gap of 20um in the center region. A more detailed description of the rheoscope system can be found in Tran-Son-Tay, 1984. In a simple shear flow, the velocity field is linear, hence the shear rate is constant in the field. In a simple shear flow, there exists a rigid body rotation of the fluid. In the absence of rotation, the flow is called a pure shear flow. In the low Reynolds viscous laminar flow regime, the flow field in the gap between the cone and the plate as shown in Fig. 2.1 is given as ue= 2y(z-Z(), u=0, u=0 (2.1) for small angle, & In the above equations, z is in the direction of rotating axis of the cone and the plate and z 0 is the location of velocity component ug=0. The shear rate, /, in the flow field can be obtained from the angular velocities of the cone and plate as below: PAGE 18 where co c and a> p are the angular velocities of the cone and the plate, respectively, and the cone angle,Â£, is 1.26 Â°in the present setup. The strain rate tensor for the simple shear flow r=3mm Light Tit Condenser 17.8 mm Plexiglas cone Floe suspension Objective Glass plate Water Video recorder 00 NIH 1.6 CCD Camera Monitor Macintosh Fig. 2.1 A Schematic of Cone-Plate Flow Chamber and Image Acquisition System PAGE 19 12 can be expressed in terms of /as D = 0 -y -y z-zÂ„ 0 r o y o (2.3) When the cone angle 5 is very small, (z-zj/r PAGE 20 13 2.1.2 Hyperbolic Flow Devic e and Flow Characteristics A hyperbolic flow (or alternatively, stagnation flow) is usually a local flow that exists in a small region around the stagnation point in a flow field. The devices mostly used to generate this flow are four roller device and two opposite jet impinging flow device. Four roller device has been widely used (Kao and Mason, 1975, Powell and Mason, 1982) in breaking droplets and floes since G. I. Taylor's famous experiment (Taylor, 1934) on the deformation of a liquid drop in a viscous flow in 1934. Impinging flow was applied to the studies of drop deformation (Janssen et al., 1993) and blood cell deformation (Knoblock, 1996). It has not been used in the studies of breaking floes or aggregates. The flow chamber for generating hyperbolic flow was initially built and used for the study of blood cell deformation by Knoblock (1996). It has been modified to improve the image quality and to readily produce either hyperbolic flow or contractile flow. The flow generator and the image system are shown in Fig. 2.2. The image acquisition system is identical to that described in the above section. The flow is self-driven by gravity via the height difference between the flow reservoir and the flow chamber. The flow chamber has two pairs of tubing ports. The flow channels are formed by sandwiching between two parallel plates a rubber with channels. When it is used to generate a hyperbolic flow, the rubber has two channels perpendicular to each other and crossed. Two opposite streams of flow coming from two entrance ports at the ends of a channel meet in the cross region of the two channels and flow out from the two exit ports at the both ends of the other channel. A stagnation point is developed in the central point of the chamber. The flow velocity is very low near the stagnation point while the shear rate is approximately PAGE 21 14 Fig. 2.2 A Schematic of Hyperbolic Flow Setup PAGE 22 (exactly in theory) constant everywhere in the central region. Floes are introduced into the inflow channel through a syringe located in one of the two entry tubing ports. A hyperbolic flow is a linear two dimensional flow. In general, a linear 2-D flow can be described as (Janssen et al., 1993): u = LÂ»x (2.9) in which u is the velocity vector and x the position vector and L the gradient tensor L = 1 + a 1 + a 0 l-a l-a 0 with-7 PAGE 23 16 and the rotation tensor is 0=0 (2.17) Hence, the hyperbolic flow is a pure shear flow, that is, without rotation in flow field. Both the maximum extentional rate of strain and the maximum shear rate of strain are G everywhere in the flow field. That is, s=y=G (2.18) and cy=T = 2fiG (2.19) The flow in this experiment is in the creeping flow regime since the reduced Reynolds number based on the geometry is less than 10" 2 , that is, Â„ . U n L h 2 , RS =^r-l? <10 (2-20) where UÂ„ is the average velocity of inflow, h half the thickness of the channel, v the kinematics viscosity, and L denotes a characteristic length of interested region in the x,yplane of the flow field. Using the flowing typical values U oo=10 mm/s v=6xia 3 m 2 /s (p ~ 1200 kg/m 3 , ju ~ 7.4 Ns/m 2 ) h=0.6mm L=0.1mm it is estimated that Re = 6x1 a 3 . Two important characteristics of the creeping flow between parallel plates are that the streamlines for all parallel layers (z = constant) are PAGE 24 17 congruent and the flow has the same streamlines as a potential flow (Schlichting, 1979). Thus, the flow field can be expressed as follows: u = u 0 (x,y) 1v = v 0 (x,y) h 2 (2.21) (2.22) where u 0 and v 0 denote the velocity field of two-dimensional potential flow. For the stagnation flow generated by two opposing jets, the well known solution for u 0 and v 0 is u 0 =Gx (2.23) v 0 =-Gy (2.24) which holds only in the small region near the stagnation point (x,y)=(0,0) in the flow. The only task in flow characterization is to determine G in this experimental study. In this study, the image of the trajectory of a very small constituent particle is used to determine the shear rate. G can be calculated as follows: G = v h , ln^ = ('2-'/) 2 In A yi (2.25) J by measuring the positions (x, , y,) and (xÂ„ y 2 ) of the particle at t=t, and t=t 2 at a given z. It completely relies on the particle trajectories. The trajectory of a moving particle and its location can be obtained by videotaping its motion and using an image analysis software. Several frames of pictures at a constant time interval can be digitized. The coordinates of the particle at each picture can be read. However, the coordinates are relative to the frame defined by the software. The origin of the hyperbola in the coordinates is unknown. To PAGE 25 18 obtain the origin for the hyperbolic trajectory of the particle, and hence the coordinates of these points relative to the origin, a best fitting of these points is done. Once the origin is found, Eq. 2.25 is used to calculate the shear rate, assuming a vertical position z=0, that is, particles move in the middle layer. The value of z at which a particle moves cannot be determined in the present experimental setup. This precludes any quantitative measurement of the flow and quantitative analysis of the interaction between the particles and the flow. However, the values of G calculated at two extreme positions, z=0 and z=0. 5h, differs only by a factor of 1.333. Since two points on a trajectory can give a value of G, several values of G based on different pairs of points on the trajectory are averaged to give the final value of shear rate G. Figure 2.3 shows six points on a particle trajectory at 0.5 second time interval and the hyperbola from the best fitting over these points. Fig. 2.3 Best Fitting to a Particle Trajectory in Hyperbolic Flow PAGE 26 19 2.1.3 Contractile Flow Chamber When the system is used to generate a contractile flow, the sandwiched rubber between the two parallel plates has one channel connected to one pair of tubing ports in the flow chamber. The other pair of the ports are blocked by the rubber. A schematic for the flow channel is shown in Fig. 2.4. The flow channel has a blockage with a small slot to form the contractile flow. Flow comes from one end of a channel and goes out from the other end of the channel. The two-dimensional channel used in this experiment is 0.8 mm thick and 12 mm wide. The slot opening is 0.8 mm in width, so that the contraction ratio is 15. 12mm Fig. 2.4 2D Contractile Flow Channel PAGE 27 20 Although the flow channel section is rectangular, the flow is not two-dimensional because of the side wall effects. However, the shearing along flow direction is dominant since the shear rate due to the contraction of the flow is much higher than other components of the rate of strain tensor. In addition, the flow in the middle region is approximately two-dimensional since the flow at the middle layer is a theoretically perfect two-dimensional flow. The region of the two-dimensionality of the flow is very small compared with the channel thickness, but is very large compared with the size of the floes. Therefore, the floes moving in the middle region are subjected to a twodimensional contractile elongation. The characterization of the contractile flow is much more difficult than the cone-plate simple shear flow and the hyperbolic flow. A numerical simulation of the axisymmetric orifice flow and two-dimensional flow and the analyses of the flow characteristics will be presented in Chapter 4. In the preliminary experimental investigation in this chapter, the primary focus is to observe the dynamic process of deformation and breakup of individual floes. The quantitative characterization of floe breakup in an axisymmetric contractile flow will be given in Chapter 5. 2.2 Floes and Fluids 2.2.1 Floes The floes used in this study are flocculated silica particles. They are generated from monodisperse spherical silica particles. Different diameters of silica particles are used to obtain different types of floes. The complete procedure for making floes involves PAGE 28 21 polymer preparation and particle flocculation. Given below is the detailed procedure employed throughout all the experiments in the project. (1) Preparation of ethyleneoxide polymer: Step 1 . In a flask, make a mother polymer solution with a concentration of 500ppm (1 ppm = one millionth). Step 2. Add 0.5g ethyleneoxide polymer of 5 million molecular weight. Step 3. Add 1000 ml deionized water. Step 4. Stir the solution for at least 10 hours with a stirring bar at a stirring machine. Cover the flask with a box to shield the light. (2) Flocculation of particles Step 1. Choose dry powder: monodisperse spherical silica particles. Three different diameters of particles are used: 0.5um, l.Oum, 1.5um . Step 2. Select the solids loading: 0.5% (lg powder + 200ml deionized water). Step 3. Sonicate the slurry of particle and water to break any possible agglomerates in the powder using a sonic dismemberator (Fisher Model 300). Step 4. Add 5ml of PEO into the slurry. Step 5. Agitate the slurry for a few minutes. The time period of agitation may affect the size of resulting floes, but does not affect the chemical properties of floes. Step 6. Drain the water to form a floe suspension of volume fraction at 2% after sedimentation of the floes. The mechanical properties of floes resulting from the flocculation depend on the properties of the materials, the dosages of polymer and solids loading, the instruments, PAGE 29 22 and the operations. To obtain floes of constant mechanical properties for every flocculation in the study, special care in operation has to be exercised in the making of the floes. The same dosages of materials such as polymer, water, and powders have been used. The instruments used such as flasks, beakers, stirring bars, and stirring settings are the same for every flocculation. It is crucial to have floes made at different times possessing the same mechanical properties in order to isolate the hydrodynamic effects of fluid shear on the dispersion of floes. The floe suspension is stored at normal laboratory temperature for future use. It is gently agitated before being added into suspending medium. Although there might be changes of the floes in their size and properties due to their aging and the agitation (Hannah et al., 1967), the effects are insignificant because the tests for floe breakup are always conducted within a few hours on the same day on the same mixture of floes and fluid for all different flow conditions in a flow to minimize all other unexpected factors. 2.2.2 Suspending Fluids Two million M.W. dextran solution is used in the experimental study. The main reason for the use of dextran is the requirement of an extremely high viscosity of the fluid to break floes at low shear rates. Low shear rate is essential because the breakup process of fine floes can only be visualized at very low flow velocity and a large velocity gradient at high shear rate often leads to difficulties in visualizing moving floes. In fact, the limit on the shear rate is very strict, once the shear rate surpasses 4 s 1 in the cone-plate device, the floe image becomes very fuzzy. Solutions of highly viscous fluids have been widely used in previous experiments for the same purpose. Sonntag and Russel (1986) used a 55.2 % PAGE 30 23 (by volume) glycerol-water solution of 0.1 Poise viscosity. Higashitani et al (1991) also used a 24 % (by weight) glycerol solution with a viscosity of 0.017 Poise. Powell and Mason used silicone oil solution at 60% volume fraction. The mechanical properties of 2-million-molecular-weight dextran solution are investigated to determine its characteristics. The viscosities of the different concentration dextran solutions are measured using a Wells-Brookfield cone-plate digital viscometer. The relationships between the stress and strain rate at five concentrations are shown in Fig. 2.5a, from which we can see that the solutions are Newtonian for the ranges of concentration and shear rate considered. The viscosity of the dextran solution increases with increasing concentration almost quadratically as shown in Fig. 2.5b. 2.3 Results and Discussions 2.3.1 Floe Breakup in Cone-plate Shear Flow Figure 2.6a shows the detachment process of a primary silica particle of lum diameter from a triplet floe in the simple shear flow. This triplet is initially composed of three primary particles that form a straight line. As shearing continues, the floe rotates and one of the primary particles detaches from the floe gradually. This triplet breaks at a shear stress of t = 6.23 Pa. Fig. 2.6b shows the breakup process for a different triplet floe. The initial shape is a triangular type, so that each primary particle is in contact with two other particles. The floe ruptures, that is, the primary particles detach from each other simultaneously. The stress level in the shear flow field at which the triangular floe breaks up is 9.35 Pa. The results indicate that the floe strength depends strongly on the floe structure. The triangular triplet has more contacts and therefore more effective binding PAGE 31 24 y(s') (a) Stress Rate of Strain Relationship r \i (Ns/m 2 ) (b) Variation of Viscosity with Concentration Fig. 2.5 Mechanical Properties of Dextran Solution PAGE 32 25 0=2.5) (t=3s) (t=3.5s) (a) Breakup Process of a Line-shape Fine Floe (7 =0.842 r 1 ) (t=0) (t=0.5s) (t=ls) (fl.5) (t=2s) (t=3s) (b) Breakup Process of a Triangular-shape fine Floe (y= 1.257 s-') (The edges of primary particles in this figure are manually traced for better visualization.) Fig.2.6 Two Breakup Processes of Floes Consisting of Three Particles in Cone-plate Simple Shear Flow (u=7.4 Ns/m 2 , Primary particle diameter d 0 =1.0\im) PAGE 33 26 among primary particles. Thus, it is not surprising that a higher stress is required for breaking a triangular floe than a linear floe. Therefore, the strength of floes in a suspension can only be related to the statistic average of floe size since a suspension is composed of floes with a large number of different structures. Figure 2.7 shows the breakup process of a larger and more complicated floe which is about 8um in equivalent diameter. The floe breaks up in the middle into two smaller floes. One of them further breaks up in the middle to form two smaller floes. The shear stress at which this process is observed is also 9.35 Pa and this process lasts for about 10 seconds. It seems that the fragmentation of a floe into smaller ones is a short time sequence process at the low shear rate for the floes investigated. It is expected that when the shear rate is increased and the shear stress is kept the same, the time period will decrease. 2.3.2 Floe Deformatio n and Breakup in Contractile Flow The deformation and breakup process of floes approaching to the entrance of an orifice have been investigated by other researchers. Sonntag and Russel's deformation model was based on the analysis of the variation of the average number of primary particles per floe and the variation of the average radii of gyration of floes with shear rate. They found that the radii of gyration vary weakly with shear rate while the average number of primary particles per floe decreases exponentially. Hence, it was postulated that the floes change shape in the orifice flow from spherical to elongated because of the continuous removal of fragments from the same part of the floe by the irrational flow near the orifice entrance, not because of the deformation. Higashitani, et al PAGE 34 27 (t=15) (t=16) Fig. 2.7 Deformation and Breakup Process of a Floe in a Simple Shear Flow (u=7.4 Ns/m 2 , y 1.257 r 1 , Primary particle diameter d 0 =1.0\wi) PAGE 35 28 (1991) measured the variation of the length of floes consisting of small number of particles, whose size is 90um in diameter, in the flow direction as they approaches the entrance of an orifice using a stroboscope photograph technique. They proposed that the floes were elongated and lined up towards the flowing direction, and then broken before entering into the orifice. The present experimental work is to examine those proposed scenarios by visualizing the dynamic deformation and breakup process of fine floes subjected to a contractile flow approaching to a small opening in a two-dimensional channel. To obtain a clear images of floes when they move into the focal visual region, the velocity of the flow near the slot entrance must be very low. This requires the use of a highly viscous suspending fluid in order to create a shear stress that is sufficiently high to deform and breakup the floes. The dextran solution with viscosity of 7.4 sN/m 2 is used. The floes of 1.5 urn silica particles are used to obtain better visual images. The motion of floes is clearly visualized in the present system. Unfortunately, the shear rate in the flow cannot be obtained in the present experimental conditions because the exact location of the floes cannot be determined. For the two-dimensional slot with contraction area ratio of 15, the maximum dimensionless shear rate, normalized by the average channel velocity and the half channel width, can reach about 225 near the entrance of the slot, as will be quantitatively shown in Chapter 4. However, the absolute value of shear rate is very low, because the flow rate in the system is only 20 milliliter per hour. PAGE 36 29 Figure 2.8 shows the final moments of a dynamic elongation process of a floe. They are captured by manually moving the flow chamber along the centerline flow direction since the fixed region that the microscope can cover is very small (68p.mX90p.rn). This floe whose image is captured is located at the centerline and near the entrance, as suggested by the orientation of the floe. Its two ends are elongated in axial direction prior to their detachments. The rest of the floe is also elongated and seems to be broken in the middle as shown in the last picture. The most frequently observed mode of floe breakup in the contractile flow is the one where floes are elongated, then broken from the ends of the rod-like shape. Figure 2.9 shows a processes of disintegration of two floes approaching to the slot. The two floes are deformed and completely ruptured in less than one second. Elongation usually takes place before the disintegration. In Fig. 2.10, a cluster of a few floes is followed. The cluster of floes is captured somewhere upstream in the flow field, then is followed by manually adjusting the flow chamber along the centerline and lateral directions. It can be seen that the cluster of floes is flowing toward the central region of the flow and its orientation is gradually shifting to align with the centerline direction as they approach to the orifice. The floes undergo deformation and breakup in the process. On the last picture frame at t=17 second, the cluster of floes becomes much thinner, consisting mainly of two or three particles in the width. During the initial phase, there are some small fragments removed from the main floe body by the flow. The major breakup takes place at the final moments around t=13 second. PAGE 37 30 Fig. 2.8 Deformation and Breakup Process of a Floe in a Contractile Flow {\i=7.4 Ns/m 2 , Primary particle diameter d 0 =1.5\im, area ratio P=/ 5) PAGE 38 31 t=0.875s t =ls Fig. 2.9 Rupture of a Floe in a Two-dimensional Contractile Flow (H=7.4 Ns/m 2 , Primary particle diameter d 0 =1.5\im, area ratio P=/5) PAGE 39 32 Fig.2.10 Motion and Breakup of Floes in a Two-dimensional Contractile Flow (\i=7.4 Ns/m 2 , Primary particle diameter d 0 =1.5\im, area ratio P= 15) PAGE 40 33 The visual monitoring of floe motion also reveals that the floes tend to move toward to centerline in contractile flow. In particular, floes detour the corner region of the flow when they move toward the orifice wall due to the large lateral velocity. This observation is significant for our study of the effective shear stress that floes experience in orifice flow in Chapter 5. 2.3.3 Floe Breakup in Hyperbolic Flow As we have seen, in the simple shear flows of the cone-plate, floes translate and rotate with the fluid. In the two-dimensional contractile flow approaching the slot, the floes move through a high shear section for a short period, undergo deformation and then break up. In a hyperbolic flow, floes move very slowly near the stagnation point and be exposed to the shear force for a much longer period. This is the case when the hyperbolic flow is used to break and deform bubbles, drops, and blood cells. However, floes usually cannot remain near the stagnation point even for a short period because the irregular shape of floes causes a non-zero resultant force on the floes by the fluid. In fact, even for a spherical particle, the stagnation point is unstable point. Once it is perturbed, it will move away with the outgoing flow. The rotation speed of the four rollers has to be adjusted to bring the drops back to the stagnation point. For floes, the incoming flow rate is controlled to an extremely low value, about 20ml/h, to obtain clear picture of moving floes in the flow around the stagnation point. Such a low incoming flow velocity produces a very low shear rate, G=0.0032 s" 1 . Although the dextran solution at concentration of 60% with the viscosity fi=7.4 sN/m 2 is used, it is still not sufficient to deform and break most of the floes used in the study. PAGE 41 34 Hence, it is more difficult to obtain good pictures of floe deformation and breakup in the hyperbolic flow than in the contractile flow. Nevertheless, the breakup of some large floes are observed and recorded. Two such processes are shown in Fig. 2.11 and Fig. 2.12. The same batch of floes used in contractile flow tests are used in the present test. Figure 2.11 shows a typical process of deformation and breakup in the hyperbolic flow. When captured, the floe is already elongated to about 10 um. The floes in the hyperbolic flow experience a pure shear and an elongation similar to that in the contractile flow. The breakup process in the hyperbolic flow is very similar to the one observed in the contractile flow (Fig. 2.8), except that the floe in the hyperbolic flow experiment moves at a much smaller speed over a small distance since the velocity of flow in the stagnation region is near zero. This leads to a longer shearing time and the better image quality. In contractile flow, the floes travel very fast, and experience a high shear due to the high flow velocity at the entrance region. Figure 2.12 shows another dynamic breakup process of a floe in the same flow conditions. A neck region forms in the elongated floe. Further necking leads to the breakup in the middle while no apparent deformation is observed in its shape. It is noticed that individual floes have very different structures than the one captured in this case, even though they were generated from the same batch by using identical procedures and chemicals. However, Fig. 2.11 and 2.12 clearly show the simultaneous presence of deformation-breakup mode and breakup without deformation mode in the hyperbolic flow. PAGE 42 Fig. 2.1 1 Floe Deformation and Breakup in a Hyperbolic Flow ( G=0.00322 1/s, n=7.4Ns/m 2 ,d 0 =1.5\m ) PAGE 43 36 Fig. 2.12 Breakup Process of a Floe in a Hyperbolic Flow (G=0.00322s>, \L=7.4Ns/m 2 , d 0 =1.5\im) PAGE 44 37 2.4 Summary This chapter presents the visual results for the dynamic deformation and breakup process of fine floes of size ranging from a few microns to tens of microns in different flows. The main conclusions and significance of this work are summarized as follows. (1) Dynamic process of deformation and fragmentation of fine floes in the size range of 3 30um is clearly visualized for the first time in different flows including a simple shear flow, a pure shear (hyperbolic) flow, and an elongational (contractile) flow. (2) In hyperbolic flow, floes can stay in the region of constant shear stress for a longer period of time compared to contractile flow where floes experience a high shear stress only in a very short period of time. In both contractile flow and hyperbolic flow, the more frequently observed mode of breakup is that floes are elongated, then broken into several smaller floes simultaneously. (3) In simple shear flow, a floe is usually broken into two floes at a time. Compared to breakup in contractile flow and hyperbolic flow, the breakup process in the simple shear flow is usually much longer than that in the elongational flows. (4) The visualizations of detailed dynamic floe deformation and breakup process reveal various modes of the dispersion of fine floes. Such information is helpful to the understanding and the modeling of this complicated floe dispersion process. PAGE 45 CHAPTER 3 ANALYSIS OF FINE FLOC BREAKUP IN CONE-PLATE SIMPLE SHEAR FLOW 3.1 Introduction Simple shear flow has been widely used to quantitatively study floe dispersions, using three devices: (1) Cylindrical Couette flow apparatus with two concentric cylinders. When the gap between the two cylinders is sufficiently small compared to the radius of inner cylinder, the velocity distribution in the viscous flow between them due to the rotation of either cylinder or both in opposite directions is linear. Hence, the shear field in the flow is constant when the Reynolds number is smaller than the critical Reynolds number. This apparatus has been widely used to generate either a laminar simple shear flow or turbulent flow for the experimental study of particle dispersions and aggregations in shear flow (Jiang and Logan, 1996; Kao and Mason, 1974; Patterson and Kamal, 1974; Powell, and Mason, 1982; Potanin et al, 1997; Serra et al, 1997; Sonntag and Russel, 1986). (2) Flow between two parallel walls moving in opposite directions. This apparatus was used in visualizing the pulp floe breakup process (Lee and Brodkey, 1987) and deforming liquid drop in another fluid (Taylor, 1934). (3) Flow between a counter-rotating plate and cone with a very small angle. This device has been used to deform blood cells (Tran-Son-Tay, 1984; Henderson and TranSon-Tay, 1997). This same device is used in the present study . 38 PAGE 46 39 The simple shear flow device and image acquisition system used in the last chapter is employed to quantitatively study floe breakup process, and floe strength in this chapter. This is done by visualizing the deformation and breakup of individual floes and analyzing the floe mass, size, and shape. Floe mass refers to the number of constituent particles of a floe. Since the constituent particles are spherical and the density of the particle is known, the mass of the floes can be related to the number of constituent particles. The main objectives of this study are to further understand the breakup mechanisms of fine floes of a few microns, analyze the effect of shearing time on floe dispersion process and evaluate the mechanical strength properties of floes. 3.2 Experim ental Procedure and Data Processing In order to analyze the change of floe mass, size, and shape with shearing time and shear stress, the floes are sheared in the cone-plate shear flow. The experimental device was described in the last chapter. The flow chamber is centered and leveled before the fluid and floes are put in. About 0.5 ml dextron is first dropped on the surface of the plate, then a small amount (no more than 10" 5 ml) of wet floes prepared previously is added to the drop of fluid using a needle. The floes are initially concentrated at a small region in the fluid. The mixture of floes and fluid is left undisturbed for several minutes before the cone is positioned to its preset position so that the floes could spread with the very slow and free flowing of the drop of fluid. It can be seen that the volume fraction of the floes in the fluid is about 10" 5 , when the solid volume concentration of the wet floes added into the fluid is assumed 50%. PAGE 47 40 The shear rate is controlled by adjusting the differential rotation speed between the cone and the plate. To follow the motion of a floe in a limited range, the rotation speed of the cone or the plate can be adjusted while the differential speed is kept constant. The focus of the microscope is adjusted to change the focal plane during the experiment so that floes at different heights can be observed. The focused region is 68p.m x 90^im. All the floes moving in the region on the focal plane are continuously filmed by a CCD camera after magnification through the microscope, and the images are monitored and videotaped using a monitor and a VCR connected to the microscope-CCD camera system. Then the movie is digitized with a given time interval on a Macintosh computer using the public domain NIH Image program. Figure 3.1 shows a frame of a typical floe picture. The averages of equivalent size of floes, shape, and mass (number of constituent 68 nm Fig. 3.1 Typical Floe Sampling for Size and Shape Analysis PAGE 48 41 particles of a floe) over all floes in a picture represent an instantaneous values of size, shape, and mass of the floes that move in the focal region at that particular instant. The floes move in and out of the focused region from the two boundaries in circumferential direction following the fluid motion. It can be shown that the radial movements of floes are random and negligibly small, and there does not exist an outward flow of floes resulting from the centrifugal force since the fluid has an extremely high viscosity. Assume that a spherical floe of radius a has a radial velocity v r at a radial position r, the equation of motion for the floe may be approximated as, drag is used in the equation since the Reynolds number for the flow is in the order of 10" 6 based on the experimental conditions, floe size a=10" 5 m or less, fluid viscosity ju=1.0 (3. 1) dr where v r ^ , p p is the average density of the floe and co the angular velocity. Stokes Ns/m 2 or higher, fluid density p=1100 kg/m\ a < 0.1s 1 . Let P = 9M 2p p a 2 , it is noted that P ~10 7 s~', the Eq. (3.1) takes the following form: r + fir-rto 2 = 0 (3.2) When co/p Â« 1.0, the solution given by r(0)=r 0 , and r(0) = 0 is: CO 2 P (3.3) PAGE 49 42 For the conditions in the study, (a/p~o(l(T s ), and (O 2 /p~o(10~ 9 ). The radial movement of floes due to the centrifugal force in 3 hours (10 4 second), the an extreme case, is eventually zero in the highly viscous fluid. In the vertical direction, the possible sedimentation is examined experimentally. The size distribution of the floes suspended in the dextran solution in a high tube is measured by a Coulter particle size analyzer at time interval of 1.5 hour. Samples are taken from the top of the tube container. Fig. 3.2 shows the two measured floe size distributions. It is found that when the concentration of dextran solution is 40%, the floe size distribution difference disappears. The concentration of dextran solution used in the study is higher than 50%. Thus there is no systematic depletion of the floes in one specific region and the sample in the visual region is representative of the entire flow. 3.3 Results and Discussion 3.3,1 Variation of Floe Mass with Ti m e Under a Constant Shearing In order to study the effect of shearing time on floe structure in a simple shear flow, an experiment on the change of floe mass under a constant shearing is conducted. Each floe consists of only small number (typically less than or equal to 6) of primary or constituent particles. Such floes are highly irregular in shape. Such floes are encountered in the last stage of dispersion process in which larger floes (with diameters 10 or 100 times the primary particles) have been broken down and the remaining floes of smaller size are difficult to break. An understanding of the floe breakup mechanism on this small scale is essential to the dispersion of colloidal particles. Floes are sheared at a constant low shear rate y=1.9J 1/s in the highly viscous dextran fluid of viscosity ^=15.1 NÂ«s/m 2 for two PAGE 50 43 Sedimentation time Â— t=93 mimute Floe Size (urn) (a) Dextran Concentration 30% Sedimentation time Â— Â— 1=79 minute Roc Size (urn) (b) Dextran Concentration 40% Fig. 3.2 Effect of Sedimentation on Floe Size Distribution PAGE 51 44 hours. The entire dispersion process is recorded by the video system as described above. After the experiment is done, the video is played back and digitized at twelve time segments apart approximately 10 minutes. At each time segment, about 5 frames of digitized pictures are taken every 5 or 10 seconds, depending on the image quality and the number of floes in a picture, since the floes do not distribute uniformly in the flow field. The pictures including too few floes or those of poor quality are usually skipped. The number of visible primary particles in each of all the floes in one frame of picture is counted. Some primary particles could be hidden in large floes consisting of four or more particles. Hence, the floe structure observed is two-dimensional. The variation of floe mass, which is represented by the average number of primary particles of one floe over all floes at a time point, with time is obtained. The changes of number fractions of floes consisting of certain primary particles are also analyzed. To better visualize the floe structure, the floes generated with silica particles of 1.5 urn in diameter are used. Experiments indicated that when this type of floes is used, the individual particles in a floe are clearly visible. The floes generated with this size of silica particles by the procedure described earlier are composed of small number of constituent particles. Figure 3.3 shows four frames of pictures illustrating typical floe structures under shear at four different moments. The decreasing tendency of large floes consisting of more than 5 primary particles can be seen in the pictures. It is noted that the pictures taken at initial phase have more floes and less floes at final phase, in other words, the population of floes PAGE 52 45 PAGE 53 46 is not balanced in these pictures. The reason is that the floes are relatively concentrated spatially initially during the shearing process because the floes are put in a small region in the fluid and the floes are more and more distributed spatially as the continuous shearing proceeds because of the circumferential fluid flow. Therefore, the number fraction of floes is used in the following analyses. Number fraction of some specified floes is the ratio of the number of these floes to the total number of floes. Figure 3.4 shows the variation of the average number of primary particles in one floe with time. It can be seen that the average number decreases fast at the initial stage, indicating the breakup of larger floes composed of large number of primary particles into floes of smaller number of primary particles. At this initial phase, the breakup mechanism is dominant. This period is short, compared with the long shearing process. For the rest of the shearing, the average number of constituent particles decreases very slowly, which indicates that the erosion process of floes dominates. This experiment clearly illustrates that the breakup of floes in short time makes a major contribution of floe size reduction in floe dispersion by shear flows and the continuous shearing will result in the erosion of particles, but makes a small contribution to floe size reduction. Figure 3.5a shows the variations of number fractions of floes consisting of one, two and three particles with time and Fig. 3.5b shows those of floes consisting of four, five and six or more particles. For the sake of convenience, the individual primary particles in the pictures are considered as floes consisting of one constituent particle. They are, of course, not floes. It can be found that the number fraction of largest floes ( composed of 6 or more visible particles) drops rapidly in the initial shearing phase ( from 12% to 3% in about 6 minutes). The number fractions for 4-particle floes and 5-particle floes also PAGE 54 47 1 I 1 1 1 I ' ' i i I i i ' ' I ' > Â• < I i | I , , . I 0 20 40 60 80 100 120 Time (Minute) Fig. 3.4 Variation of Average Number of Constituent Particles in One Floe (y=1.91s', \i=15.06Ns/m 2 , d=1.5\un) PAGE 55 48 o o c 5 I u E Z 0 1 -particle floe Â— -aÂ— 2-particle floe g g A 3-particle floe 0.5 0.4 0.3 0.2 0.1 A^ -A. A. 20 ^A Â— l Â— . Â— . , , i A -A^ y Â— A 40 60 80 100 120 Time (Minute) (a) Variation of Number Fractions of Small Floes 0.12 r Â— 4-particle floe ~0Â— 5 -particle floe -O 6 and more-particle floe 100 12 Time (Minute) (b) Variation of Number Fractions of Large Floes Fig. 3. 5 Variation of Number Fractions of Large Floes with Time PAGE 56 49 decrease, from 9% to 3.5% and 7% to 4% in 12minutes, respectively. It is noted that the bigger the floes are, the larger the decreasing rates, for these three sizes of floes. This quantitatively indicates that the large floes are easier to break. On the other hand, the fractions of single particle floes and doublet floes have an increase of about 8% in the initial 6 minutes, and the number fraction for triplet floes increases slightly, that is, the number of triplet floes increases keeping pace with the increase of total floe numbers during this period due to the breakup of larger floes. The strips of single particle or doublet floes obviously take place in the initial breakup phase. Then what follows is the slow decrease in number fractions of large floes with time during the shearing process and the slow increase in number fractions of single particle floes. At the end of the shearing process, the largest floes (composed of five and more primary particles) completely disappear. Although there are some deficiencies in these curves due to the nonuniform space distribution of floes in the flow field and possible sampling bias, these curves clearly illustrate the breakup mechanisms from the point of view in floe mass changes. It is suggested that a continuous shearing does not result in a significant floe size reduction and a sufficiently high shear stress that overcomes the strength of floes can break floes almost instantaneously in the flow of shearing domination. The initial breakup phase in this experiment lasts about 6 minute because the extremely low shear rate is used. If the shear rate is increased by 500 times, up to about 1000 1/s, and correspondingly, a much lower viscosity is used to obtain the same shear stress, the initial breakup can be finished in a very short time, because the PAGE 57 50 nondimensional parameter yt usually is considered to the quantity that essentially governs the time effect (Powell and Mason, 1982). 3.3,2 Variation of Floe Size with Tim e at Constant Shear Rates Based on the conclusion that a continuous shearing makes little contribution to the floe size change, an experiment is performed to analyze the effect of shear stress on the floe size and shape, as well as the effect of shear stress on the erosion rate of floes at a constant shearing. The experiment is done by the same procedure as above, but four shear rates are used for the shearing process. The floes used in the experiment are those consisting of primary silica particles of 1.0p.m in diameter. The digitization is made at a constant time interval for a given shear rate. The interval varies for different shear rate since the floes move faster at a higher shear rate. The projected area of a floe in the focal plane is measured and an equivalent diameter is calculated. This diameter corresponds to the diameter of a circle that has the same floe projected area. The equivalent diameter of a floe provides an effective measure of the floe size since the floes translate and rotate in the flow field. Each data sample of digitization is a frame of picture having a number of floes. The average of equivalent diameters in a sample represents an instantaneous average size of the floes that move in the focal region at that particular instant as shown in Fig. 3.1. The analyses of the floe size and shape are based on the measurements of the projected shape of floes. It is done by tracing the profile of a floe image. The area of the outlined shape and the length of the profile edge are measured by the image software. Figure 3.6 shows the variation of the instantaneous average floe size with time for the entire shearing process of about 36 minutes. During this period, the shear stress is PAGE 58 51 CD (wtI) azis oo| j PAGE 59 52 increased from 6.2Pa, to 13.3Pa, then to 20.6Pa, and finally to 27.6Pa. The data exhibit a large fluctuation, because the amount of floes captured in the small focal region at a moment represents floes which have a large random distribution in size. Since the floes are randomly distributed in the flow field, the floe structures (size and shape) captured at a fixed small region in the flow appear to be random. A linear regression analysis is conducted to analyze the time effect at a given shear rate. That is, the averaged floe size d is expressed in terms of the linear function of dimensionless parameter yt as follows: d=d 0 (l+ky(t-t 0 )) (3.4) Figure 3.7 shows the variation of the averaged instantaneous floe size change with nondimensional parameter, yt, and the linear regression results at four shear stress levels. The values ofk and d 0 for the four shear stresses are given in Table 1. The constant k is the linear rate of change in floe size. It is a measure for the erosion process during the constant shearing. It is seen that the k is very small at a constant shear rate, that is the floe size decreases with yt very slowly. The absolute value of k decreases as the shear rate increases for the first three shear levels. The positive k for the fourth shear level only indicates that the effect of slow erosion is very small so that k results from statistical fluctuations. The dimensionless values of kyT for the last two shear stress level is so small that the floe size can be considered not changing with time during the shearing. For the first two shear stress level, the erosion process contributes about 26% and 13% to the overall size reduction during the shearing. PAGE 61 54 Table 3.1 Linear Regression Results For Floe Size Variation PAGE 62 55 given shear stress. Figure 3.8a shows the mean equivalent diameter of floes, ^ as a function of the applied shear stress. Although only four shear stresses are used, the data suggests a power law dependence of " on t, dcCT (3.5) It should be noted that Sonntag and Russel (1987) obtained d qc t~ 035 or oc r~ 105 for floes of 0.14 um in diameter of polystyrene latex dispersed in glycerin-water mixture in a sheared Couette flow, in which R is the gyration radius of floes. This similarity suggests that the different floes share a similar dispersion behavior in the shear flow. Fig. 3.8b shows the variation of the standard deviation of the floe equivalent diameter, a d , with the applied shear stress. The standard deviation also decreases as the applied shear stress increases following the power law given below, _ Â„ --0.72 d (3.6) This decrease is more rapid than that of the mean floe diameter, which indicates that the distribution of floe size is in a size band that becomes more narrow as the shear increases. The PDF's for the distributions of the logarithm of floe size at various shear stresses are shown in Fig. 3.9. It is seen that the PDF's are approximately logarithmic-normal. At a low shear stress, r = 6.23 Pa, floe sizes are widely distributed. As shear stress increases, the resulting floes not only become smaller but also become more uniform in size which corresponds to a more narrow PDF distribution curve. PAGE 63 56 S Q s 10 9 8 7 10 15 _i_ 20 25 30 2 Â• Shear Stress (N/m 2 ) (a) Variation of Mean Floe Diameter with Applied Shear Stress c .2 1 Q I u PL, I 20 25 30 2 N 5 10 15 Shear Stress (N/m") (b) Variation of Standard Deviation for Floe Size with Applied Shear Stress Fig. 3.8 Statistics of Floe Size at Various Applied Shear Stresses PAGE 64 57 i=6.2N/m Â— Â— i=13.3 N/m 2 1=20.6 N/m 2 1=27.6 N/m 2 Floe Size (urn) Fig. 3.9 PDF of Floe Size Distribution at Different Applied Shear Stresses PAGE 65 58 3.3.4 Change of Floe Shape with Time and Stress One of the advantages of the present study is the visualization and characterization of the fioc shape. The spherical floe assumption has been widely used in theoretical models and measurement techniques in the particle sizing. The video images of the floes during motion clearly show that the resulting floes under shearing are rather irregular in the present study, however. As the floe size decreases with increasing shear stress, the floes appear to have less irregularity on the surfaces, but are definitely non-circular for fine floes composed of large number of particles. In order to quantify the floe shape, both the projected area S and perimeter P of a floe image on the focal plane are measured. A shape factor (sf) is defined as follows to analyze the shape change, For a circular shape, sf ' = n . The averaged shape factor over all floes in one sampled frame of picture is plotted with time and shear rate in Fig. 3.10. While the mean value of sf decreases slightly with increasing shear stress, the fluctuation of sf is noticeably smaller at higher shear stress. To see the variation of shape factor with time at a constant shear rate, the linear regression analysis is performed on the curve in terms of the following equation: The coefficients in the regression equation are given in Table 2. Also given in the table are the statistical data for the shape factor obtained over all floes at a given shear rate. It can been seen that the change rate of time, k, is small at all the four shear rates, while the P (3.7) sf=sf 0 (J+kr(t-to)) (3.8) PAGE 66 59 PAGE 67 60 mean shape factor, s f , becomes closer and closer to n and its standard deviation decreases as the shear rate increases. The changes of mean shape factor and its standard deviation suggest that floes become less irregular in their shapes. However, the deviation from vindicates that the floes are far from spherical. Table 3.2 Linear Regression Results and Statistics For Floe Shape Factor PAGE 68 61 a deviation of the floe across-section shape from a circle, the major-to-minor axis ratio, T , is calculated using Eq. 3.1 1 and the measured sf. Figure 3.11 shows the mean values of a 7 and sf under four different shear stresses. Even at the highest imposed shear stress of r a = 27.6 Pa, the ratio, 7, is about 2. This clearly suggests that the floes generated under simple shearing are not close to sphere. x (N/m 2 ) Fig. 3.11 Variation of Mean Shape Factor with Applied Shear Stress PAGE 69 62 3.4 Summary A visual study is carried out to help us understand the mechanism of dispersion of fine floes in a simple shear flow and floe strength. The variations of floe size, shape, and mass with time and shear stress are analyzed. The main results and conclusions are summarized as follows: (1) In shear flow, rapid breakup of fine floes occurs when the applied shear stress increases and it gives a significant contribution to the floe size reduction. In the absence of erosion, the shear stress required to break up floes increases with decreasing floe size following a power law dependence. (2) The long-time constant-stress shearing results in slow erosions of floes in a simple shear flow. Whether the erosion contributes a small part or a large part to the size reduction depends on the duration of shearing and the shear rate. The erosion rate mainly depends on the shear stress, floe size, and floe shape. The erosion is faster for a larger floe size with an irregular shape than that of a smaller floe size and a regular shape. The results in the part of the work are the base for other experimental setup and the determination of experimental conditions for the work in later chapters. PAGE 70 CHAPTER 4 FLOC BREAKUP IN ORIFICE FLOW-PART 1 FLOW CHARACTERIZATION 4.1 Introduction Orifice flow device refers to the axisymmetric duct flow blocked using a plate with an orifice. The area ratio of an orifice flow device is defined as the ratio of the upstream duct section area to the orifice area. For more than 100 years, orifice device has been extensively used in hydrodynamic engineering as one of a family of differential pressure flowmeters due to its simplicity and reliability. The orifice area ratio is usually no more than 20 in flowmeters. Since the late 1970's, orifice with very high area ratio has been applied to study deflocculation and deaggregations (Kousaka et al, 1979, Yuu and Oda 1983, Sonntag and Russel 1987, Higashitani et al 1991,1992). Because the flow accelerates as it approaches the orifice, a very large velocity gradient is generated near the orifice entrance. When floes and aggregates in the flow approach the orifice, they experience a high rate of strain or stress so that they may deform and break up. Clearly, a complete knowledge of the behavior of the stress field near the orifice entrance is essential for understanding floe dispersion process. Because of the complexity of the flow field near the orifice and the difficulty in the flow simulation and measurement for high area ratio (>100), very little attention has been paid to the flow characterization of high area orifice flow. Only limited information such as velocity along centerline was reported (Higashitani, 1991). Some overall flow parameters (e.g. pressure 63 PAGE 71 64 loss estimate across the orifice or energy dissipation based on experience) in the flow are used to evaluate the hydrodynamic conditions. Unlike Couette flow and hyperbolic flow, in an orifice flow, the stress field is highly nonuniform in the region of interest and no simple methods are available to describe the flow field and stress field other than numerical simulation. Previous numerical simulations for orifice flow were for flowmeters with orifices of low area ratios ( < 16 ) and high Reynolds numbers (Re > 10 4 ) (Giovannini & Gagnon, 1993; Morrison & DeOtte et al., 1990; Morrison & Panak et aL, 1993; Patel & Sheikholeslami, 1986). The flow discharge coefficient was usually of interest. In the experimental investigations of particle dispersions, the orifices used have much higher area ratios (> 100 ) in order to obtain high strain rate. To observe and visualize the motion of particles approaching an orifice, the experiment is usually operated at a very low average velocity using highly viscous fluids to achieve sufficient stress to break the floes. In the work of Higashitani et al., (1991,1992) the Reynolds numbers based on the diameter of flow pipe and the average velocity are less than 500. In the syringe experiment (Sonntag and Russel, 1987), Reynolds numbers ranged from 267 to 2667. To recapitulate, low to moderate Reynolds number and high area ratio are used for floe dispersion, in contrast to high Reynolds number and low area ratio of flow metering. In the work presented in this chapter, a numerical simulation for the flow inside a pipe with an orifice area ratio up to 567 is performed at finite Reynolds number ranging from 0.01 to 3200, depending on the orifice area ratio, using a stream function-vorticity PAGE 72 65 formulation for axisymmetric laminar flow. The velocity field, the rate of strain tensor field near orifice entrance are analyzed in detail. 4.2 Formulation 4.2.1 Governing Equati ons and Boundary Conditions The schematic for an orifice flow field and coordinates are shown in Fig. 4.1. The only important geometric parameter for the orifice is its area ratio: P=J (4.1) where r 0 is the ratio of orifice radius to upstream pipe radius R. Fig. 4.1 Orifice Schematic and Coordinates System PAGE 73 Stream function-vorticity equations for axisymmetric flow in cylindrical coordinates (x, r) are derived from the continuity equation and N-S equations by defining a vorticity ^ and a stream function yrfx, r)\ Â£= V U r (4.2) 1 1 u= ~Vr , v=--V, (4.3) where u, v are the velocity components in x direction and r direction. The subscript represents partial derivative. The transport equation for vorticity in nondimensional form is fi + (UQ X + = + Cr + ^ (4.4) in which all the parameters are normalized based on the radius of the pipe R and the average upstream velocity U, Reynolds number Re is based on the diameter of pipe D and U. The equation governing the stream function is V** + Yrr1 -V'r=-rZ (4.5) The region of interest is before the entrance of the orifice, since our visualization of floe breakup in contractile flow has indicated that the deformation and breakup of floes occur when they approach the orifice. The inlet boundary are fixed at 1.0D from the orifice entrance face. The inlet condition is a fully developed pipe with the velocity profile given by u=2(l-r>), v=0 (4.6) PAGE 74 67 Hence, at inlet, Â£=4r, yr=T*(l-0.5t>) 4.7) Along the centerline, v=0, y^O, Â£=0 (4.8) On the wall, v=0, u=0, y^0.5 (4.9) The exit boundary is located far downstream of the orifice and a zero derivative condition is assumed for the vorticity, with vanishing transversal velocity: Â£=0, v=0 (4.10) For comparison purpose, inviscid flow computations are also carried out on the same geometry. The governing equation is Vxx+Yrr--Yr=0 (4.11) and the inlet condition is 1 2 V = y . (4.12) On the wall, At the exit, y*=0.5 (4.13) <*L_n dx~ 0 (4-14) is enforced. PAGE 75 68 Meanwhile, computation for two dimensional channel with a small slot, as used in Chapter two for the test of the visualization of floe breakup in contractile flow is conducted to quantitatively compare the difference between an axisymmetric orifice flow and a two-dimensional channel flow through a small slot. 4.2.2 Grid Arrangement and Numerical Schemes Due to the large velocity gradient near the orifice, fine grids must be used and coordinate stretching is necessary. The stretching in x coordinate upstream of orifice corresponding to x g PAGE 76 69 where tj denotes the uniform coordinate of computational domain in radial direction. The grids in the core region (0 PAGE 78 71 for horizontal wall where j=j w , and (4.23) for the front wall of the orifice at x=0 where /=/ w . On the back side, x=x 2 , similar expression can be easily derived. These formulae are derived from the boundary conditions and making use of Taylor's expansion of the first order derivatives of stream function at (ij w .,) and (ij w . 2 ) as follows: + Kcftjj cry Atj 2 +o(Atj 3 ) (4 24) Jw Jw-2 -r+ \dri) . . (-2A7) + |f0J 4At 1 2 + o(at J 3 ) ( 4. 25) 'Jw ^ ij w Since (y/ n ) ijw =0 and ( *2 ^ <7 ^ ^2 J r j w Cj w h 2 according to Eq. (3.19) and boundary Jw conditions Eq. (4.9), taking 4xEq. (4.24) Eq. (4.25), then the following equation for the wall vorticity, is obtained: (*} -4 'Jw-2 I eft]) iw-\ (4.26) After the first order derivatives at interior point in the above equation are replaced with the central difference: V di]) . . ' >,JwJw-2 2Atj (4.27) PAGE 79 72 2Atj + o\ \dn) i (4.28) >Jw-\ the Eq. (4.22) is obtained. Similarly, expanding the first derivative of stream function at the expression for the vorticity on the vertical wall can be obtained. Time marching procedure is applied to solve the discretized equations for and yK}j). The velocity field u(i,j) and v(i,j) are then derived from yjjj) by Eq. (4.3) using central difference: 4.2.3 Validation of the Numerical Method Grid independence is first examined for the case of f3=400, Re=200. Figure 4.3a shows the computed vorticity distributions along the vertical wall of the orifice at x=0, using six different grids. Figure 4.3b shows the computed velocity gradient along centerline using the same set of grids. It can be seen that when the grid is not less than 81X81, the results become independent of grid size. The discrepancy near the corner always exists because of the corner singularity. To further validate the numerical solution, the centerline velocity is compared to the experimental data of Higashitani (1991) and is shown in Fig. 4.4. The Reynolds number is 64, r 0 =0.1 and a 81x81 grid is used in the computation. The good agreement between the data indicates the reliability of the code. r, 2Atj r J (4.29) rj 2A4 (4.30) PAGE 80 73 (a) Vorticity Distributions on Orifice Entry Wall (b) Velocity Grendient Distributions along Orifice Centerline Fig. 4.3 Effect of Grid Size on Vorticity and Velocity Gradient PAGE 81 74 Fig.4.4 Comparison of Computational Centerline Velocity with Experimental Data (p=100, R e =64) PAGE 82 75 4.3 Results and Discussions Since the present numerical study is to aid the interpretation of the floe breakup experiment, the interest is focused only on the entrance region of the orifice ( x<0 ). The flows in the downstream region are more complicated and more difficult to compute accurately at high Reynolds number. However, they have no impact on the computation of the upstream region. 4.3.1 Basic Features of Orifice Flow Field Figure 4.5a and 4.5b show the contours of stream function (i.e. streamlines) and the contours of vorticity at Re=/ and r 0 =0.05 ( p=400 ). When Re reaches some critical value, Re c for a given /?, the flow after orifice becomes unsteady. Re c is very small for a high area ratio orifice. For example, flow behind the orifice becomes unsteady when Re >1 for p=400. Computational results indicate that the flow field before the orifice reaches a steady state while the flow field in the downstream region of the orifice is unsteady. Since we focused on the entrance region of the flow the unsteadiness in the downstream region is not of concern. The inclusion of a long downstream region in the computation is to guarantee that the exit boundary condition, Eq. (4.10) has no effect on the entrance region. Figure 4.6a and Fig. 4.6b show the distributions of axial velocity and its gradient in axial direction along centerline for the case of Re =1600 and different orifice area ratios. A 161x151 grid is used for the computation of the case. Large variations in the velocity gradient are observed. It is also seen that the velocity gradients reach its the maximum values before the orifice. Figure 4.7a and 4.7b show the axial velocity and its velocity PAGE 83 76 PAGE 84 77 500 400 300 200 100 Orifice Entrance (a) Centerline Velocity 6000 r du/dx 5000 4000 3000 2000 1000 Orifice Entrance -0.1 o (b) Centerline Velocity Gradient j x 0.1 Fig. 4.6 Centerline Velocity and Velocity Gradient Distributions at Different Area Ratios ( Re=1600 ) PAGE 85 78 800 700 600 500 400 300 200 100 h 0 Re=10 Re=200 Re=400 Re=800 Re=1600 Re =3 200 Orifice Entrance (a) Centerline Velocity du/dx 8000 7000 6000 5000 4000 3000 2000 1000 H 0 Re=10 Re=200 Re=400 Re=800 Re=1600 Re=3200 . Orifice Entrance -0.1 o (b) Centerline Velocity Gradient Fig.4.7 Centerline Velocity and Velocity Gradient Distributions at Different Reynolds Numbers ( p=400 ) PAGE 86 79 gradient distributions at centerline for the orifice of area ratio P=400 at different Reynolds numbers. It can be seen that the location of the maximum velocity gradient is moving upstream as the Reynolds number increases. 4.3.2 The Strain Rate Characteristics Of Orifice Flow The rate of strain tensor D in cylindrical coordinates {x, r, 6) for axisymmetric flow can be determined using the gradients of two velocity components: D = u Uu +v ) 0 4:(u +v | v 2\ r x) r 0 o o yr (4.31) If the eigenvalues of the symmetric rate of strain tensor at a point in the flow field are XÂ„ X 2 , X 3 , and : X,>X 2 >X 3 (4.32) then the maximum extensional strain rate, S max , maximum shear strain rate, , can be expressed in terms of them as follows: Aj (4.33) /max = ~2 (4.44) They are useful to quantify the stress field exerted on the floes. Figure 4.8a and 4.8b show the contours of maximum principal strain rate S max and the contours of maximum shear rate near orifice entrance region for /3=156 at Re=100. The contours of 5^ and y max are circle-like around the orifice corner since they are dominated by the singularity at the sharp corner. The values of and y max increase near PAGE 87 80 Orifice (a) Maximum Principal Strain Rate Orifice (b) Maximum Shear Strain Rate Fig.4.8 Contours of Maximum Strain Rate Before Orifiec ($=400andRe=100) PAGE 88 81 the comer, but change relatively slowly near centerline at the inlet face. They also change rapidly in the wall boundary layer. The resolution of the grids has some effect near the comer. The effect will always exists near the comer since the vorticity is mathematically infinite at the point. On the other hand, the changes in S max and y max in core region near the centerline are relatively small before orifice. Their isarithms normal to the centerline in the core region suggest that S max and y max are radially uniform in the region and equal to their values at centerline. Similar behavior can be observed for the flow at other /? and Re. It is also found that the size of the core region is approximately equal to the radius of orifice. 4.3.3 Maximum Centerline Velocity Gradient As we have seen from the experimental visualization of floe motion in contractile flow in Chapter 2, the floes in the flow always tend to move toward the centerline in the contractile flow due to the radial velocity in the flow. Most of the floes passing through an orifice will experience the stress in the core region. Therefore, the rate of strain at centerline is important for our study of floe breakup in orifice flow. At centerline all the off-diagonal elements of the strain rate tensor given by Eq. (4.31) are zero. The three eigenvalues of the strain rate tensor are j _ di . 1 du I du which corresponds to a uniaxial extension. The maximum principal strain rate S max and maximum shear strain rate y max at any point along the centerline are determined by the axial velocity gradient at centerline as follows: PAGE 89 82 Smax( x > r=0 ) = ^ OX x, r=0 , * 3 du x, r=0 (4.46) (4.47) They vary along the centerline and reach their maximum values and y cmax somewhere before the orifice entrance face. Since y cmax is 3/4 of S cmax and S cnax is equal to the maximum centerline axial velocity gradient, the effects of Reynolds number and orifice area ratio on S cmax are examined. Figure 4.9 shows the effect of Re on maximum centerline gradient velocity, S cmax for five orifice area ratios. It can be seen that S cmax , which is already normalized by U/R, decreases as Re increases. As Reynolds number approaches to 3000, it tends to the value for inviscid flow of the corresponding orifice area ratio. The asymptotic behavior of S cmax at large Re is obtained based on curve fit, S^Re, 0)= SJJ3) + a(P) Re 2 ' 3 for Re > 200 (4.48) as shown in Fig. 4.10. The values of S c0 and a at several /?s are given in Table 4-1. The centerline maximum velocity gradient S cmax h are also given in the table at corresponding p. It can be see that S c0 is actually equal to S cmaxin at moderate area ratios. The differences at two highest area ratio are less than 1%. Table 4.1 Coefficients in Eq. 4.48 and S__ for inviscid flow p 64 100 156.25 256 400 243 500 1050 2280 4550 c Â° cmax in 243 500 1068 2303 4602 a 5198.3 7366.6 21067 28454 40439 PAGE 90 83 Â• $=567 Â— A$=400 Â— -B $=256 6 $=156.25 Â— v $=100 Â— --Â•e $=64 AÂ— _ ""A AAÂ— A-AAa --A B Ck^, -eABDÂ— . Q V Â— 0 ^ -oÂ— ^7 ^0. Â— V o -I -icr I I I H ill 1 1 I I I I III 1 I 10" 2 W 1 10 u 10 1 10 2 10 3 10 4 Re Fig.4.9 Effect of Reynolds Number on the Centerline Maximum Velocity Gradient PAGE 91 84 Fig.4.10 Asymptotic Property of Maximum Centerline Velocity Gradient at Large Reynolds Number PAGE 92 85 In the range of 10 < Re < 200, a small flow separation bubble, whose length depends on the thickness of the orifice plate, occurs near the orifice front corner. This separation bubble affects S cmax so that the asymptotic value SJP) is established only at higher Reynolds number when this separation bubble merges with the main separation bubble after the orifice plate. The effects of the orifice area ratio on the maximum centerline shear rate are shown in Fig. 4.11. in log-log plot. Straight lines for creeping flow and for inviscid flow are obtained. The finite Reynolds number results are approximately linear between these two lines for the range of f5 investigated. For the Reynolds numbers investigated, the relation may be expressed as follows R du k cmax ~Ufc CCP (4-49) where k ~ 1.54 in the creeping flow regime and k ~ 1.61 based on an inviscid flow computation. Since the cross-sectional averaged velocity at the orifice increases linearly with /? based on mass conservation, the length scale at the orifice scales with r Â«*P' Yl (4.50) a simple dimensional analysis indicates that 3/ 5 <:max 00 P 2 (4.51) This exponent 3/2 is in agreement with the numerical results. It is also seen that at Re=1600, the Navier-Stokes equation based result still differs than that of the inviscid flow solution by 5-16%, depending on the area ratio J3, indicating that the viscous flow computation is needed for determining S cmax even at this Reynolds number. PAGE 93 86 Fig.4. 1 1 Variation of Maximum Centerline Strain Rate with Orifice Area Ratio PAGE 94 87 4.3.4 Comparisons Between Axisymmetric Flow and Two-dimensional Flow A numerical simulation for flow passing a two-dimensional slot is also performed to compare the rates of strain in 2-D flow with that in a 3-D orifice. Figure 4.12 compares among four configurations. First, for a 2-D slot and a 3-D orifice with the same area ratio P=64, 2-D slot generates much higher velocity gradient. Next, when the radius ratio of 3D orifice is the same as the height ratio of 2-D slot, maximum velocity gradient in the 3D orifice is 29 times that of the 2-D slot. For the two dimensional flow, a similar dimensional analysis shows that S cmax PAGE 95 88 Height Ratio = 0.05 (2D) 10 2 10" 1 PAGE 96 89 for an orifice of given geometry configurations and flow conditions considered in this work. (3) The comparative numerical simulations for the flow of two-dimensional slot and for the inviscid orifice flow have also been conducted to compare their characteristics with axisymmetric flow and evaluate the validity of inviscid calculation of the flow characteristics. PAGE 97 CHAPTER 5 FLOC BREAKUP IN ORIFICE FLOW-PART 2 MEASUREMENTS 5.1 Introduction Based on the flow simulation and characterization results for the flow of high area ratio orifice given in the last chapter, this chapter will present and discuss the experimental results on the floe breakup in an elongational flow generated by an orifice. These results include the measured variation of floe size distribution and mean size with flow conditions. In order to determine the effective stress that causes the breakup of floes in the orifice flow, a comparative test of the floe breakup in a uniform simple shear flow is also conducted and presented in this chapter. The experimental devices and techniques used in this study are described in detail. There have been a few studies of the floe breakup in orifice flow. Early applications of orifices for deaggregation involve air stream. Kousaka et al., (1979) compared the deaggregations in flows generated by a variety of devices to investigate the possible dispersion mechanisms of C a C0 3 and F e2 0 3 aggregate in air stream. They used an orifice with diameter ratio r 0 = 0.15 as one of the dispersers in their experiments. The aggregate size distributions were obtained at different mean velocities near the orifice entrance. Effects of orifice flow on the dispersion of the aggregates were demonstrated. Yuu and Oda (1983) measured the changes of the fly ash particle size distributions due to the disruption in air stream through orifices of r 0 = 0.4 and 0.6 with various far upstream PAGE 98 91 velocities to verify their particle population balance equation. The results obtained in these studies were not well-established because of the lack of the precise control over experimental conditions. As mentioned in previous chapters, Sonntag and Russel (1986) used a common plastic syringe of r 0 = 0.083 to conduct their floe breakup experiments. A large acceleration of the flow was generated when a syringe was driven by a constant force, and the flocculated suspension of monodisperse polystyrene latex was expelled from the needle of the syringe. They obtained the variations of the floe mass and size with an overall nominal strain rate in the flow. A numerical analysis for an inviscid flow passing through an orifice of r 0 = 0.27 (J3= 13.7) was performed to characterize the flow in the syringe of r 0 = 0.083 {fi = 144) used in their experiment. They assumed a floe deformation model based on the size variation with flow condition. Since the stress field was incorrectly characterized, large uncertainties exist in the interpretation of their results. Besides the breakup process of floes along the centerline of contractile flow in an orifice reviewed in Chapter 2, Higashitani et al. (1992) investigated the breakup of floes composed of a small number of polystyrene latex particles below 1 um by measuring the floe size changes across the orifice using a Coulter counter. The diameter ratios of the three orifice used are r 0 = 0.002325, 0.0116 and 0.02325. The average size of broken floes and the maximum number of constituent particles in a broken floe were expressed as function of the estimated energy dissipation at the orifice. Although the floe breakup in the orifice flow can be characterized by the power input to the system, the determination PAGE 99 92 of floe strength by the orifice flow needs the evaluation of the flow stress and the effective stress that causes the breakup of the floes when they move in the flow. 5.2 Experimental Apparatus and Procedure 5.2.1 Orifice Setup and Procedure Figure 5.1 shows the schematic of the experimental apparatus. The flow is generated in a vertical syringe tube by a piston moving at constant velocity. The piston is driven by a syringe pump. The orifice plate is located at the bottom of the tube. The orifice diameter is d=l.l mm and the internal diameter of the syringe tube is D=26.2 mm; hence the orifice area ratio is fi=567.3. The flow rate is controlled by the syringe pump, which can be adjusted continuously from 0 to 389 ml/h, corresponding to a maximum upstream average velocity of U=0.02 cm/s in the tube. If water is used as the working fluid, the Reynolds number based on the D and U would be 5.24, since the dimensionless maximum velocity gradient of the centerline is c c mat c max based on the numerical simulation results given in Chapter 4, the instantaneous centerline maximum shear stress for the setup is 3 (du) U ( du =-R^T x (5.2) c max c max Obviously, this stress is too small to deform the floes of interest. To achieve a sufficiently high shear stress to break floes, a 2 million molecular weight at concentration 50% dextran solution with a viscosity of 2.9 Ns/m 2 is used. The maximum shear stress can PAGE 101 94 reach 50 Ns/m 2 . This is a reasonable stress level for breaking floes according to our results from the simple shear experiment. The high-stress, low-velocity and low-shear-rate setup has other advantages which are critical to the determination of floe strength. First, the use of low shear rate to a large extents eliminates the reflocculation of floes. Second, no jet after orifice is generated with such a low upstream velocity. Jet impinging on the sampling container wall could have an unpredictable effect on the floes since it may lead to additional breakup or aggregation of floes. Thirdly, in the Stokes flow condition, the floes completely follow the motion of the fluid. The additional shear stress on particle resulting from the relative velocity between the floe and the fluid is avoided. The floes are made of silica particles with diameters dÂ„=0.5p.m, 1.0p.m and 1.5|im in this study. The floe synthesis procedures and the mechanical properties of dextran solution are described in Chapter 2. In the experiment, the floes of volume fraction at about 5% are added to the viscous fluid and mixed gently to form a uniform suspension of floes at volume fraction PAGE 102 95 concentric cylinders when one of them rotating or both of them rotating in opposing directions. This flow device has been widely used in the investigations of flocculation and deflocculation. The apparatus used in this test is shown in Fig. 5.2. The diameter of the inner cylinder is 36.7mm, the inner diameter of the wall cylinder is 41.7mm, hence the width of the gap between the two cylinders is 2.5mm. The height of the cylindrical shear device is 61 mm, so the volume of fluid needed for one run of the test is about 19 ml. The inner cylinder is connected to a shaft and driven by a motor. The outer cylinder is fixed in this test. After shearing, the fluid is collected for floe size measurement using the same Coulter particle size analyzer used in orifice flow experiments. The rotation of inner cylinder is controlled by a control system which can preset the rotation velocity of the cylinder and the time duration of acceleration when starting and deceleration when stopping, as well as the duration of stable shearing. The flow between cylinders becomes unstable when the angular velocity of the inner cylinder, co, exceeds a critical angular value, co c . The following expression for the critical angular velocity is used (Oles, 1992, Serra, et al, 1997) for the cylindrical Couette flow generated with inner cylinder rotating. 0) C = V\ 339o[r\ (5.3) where R, is inner diameter, R 2 outer diameter. If water is used, then co = 0.6282 s', that is, when the rotation of inner cylinder exceeds n = 6 rpm, the flow will become unstable. To maintain a well defined laminar flow for floe disperion and floe strength evaluation as have been done throughout the study in this project, dextran solution with high viscosity PAGE 103 96 Control System Floe Sample Collection For Size Measurement 36.7mm < 41.7mm Simple Shear Flow Fig 5.2 Schematic of Couette Device PAGE 104 97 again is used for a sufficient critical rotation velocity and a stress level high enough to break floes when a low shear rate is applied. We will see in the next section that the low shear rate is necessary for the test to exclude floe reflocculation and the high viscosity is beneficial to avoiding the sedimentation of floes. The flow field in the gap between two cylinders as illustrated in Fig. 5.2 can be described as follows: which can be obtained by solving N-S equation for laminar flow in cylindrical coordinates. The constants C, and C 2 are functions of the angular velocities of the two cylinders co, and co 2 : Howevr, co 2 =0 for the device used in this test. The only nonzero component of the rate of strain tensor, E, is E lr It can be seen that the shear rate field is not constant. However, when the gap is small, hence the variation of EÂ„ is small, the shear rate can be considered constant and expressed in terms of its spatial average over the gap: u r =0 C, Â„ u 9 = Â— + C 2 r (5.4) (5.5) (5.6) PAGE 105 98 Then the average shear stress in the field is x=2nr (5.8) It is also the maximum shear stress and equals to the maximum principal stress in the flow. 5.2.3 Particle Size Analyzer Coulter particle size analyzer (LS230, Coulter Co.) is employed to measure the floe size distribution in this work. The instrument is a light scattering particle size analyzer, which uses the diffraction of laser light by particles as the main source of information on particle size. The LS230 Coulter uses a laser light with a wavelength of 750nm to size particles with diameters from 0.4-prn to 2000pm by light diffraction. The laser radiation passes through a spatial filter and projection lens to form a beam of light. The beam passes through the sample cell where particles suspended in liquid or air scatter the incident light in characteristic patterns which depend on their sizes. Fourier optics collect the diffracted light and focus it onto three sets of detectors, one for the low-angle scattering, the second for mid-angle scattering, and the third for high angle scattering (Fig. 5.3). The instrument measures particle size distributions by measuring the pattern of light scattered by the constituent particles in the sample. This pattern of scattered light is called a diffraction pattern. Specifically, a diffraction pattern is the scattered light intensity as a function of scattering angle. Each particle's diffraction pattern is characteristic of its size. PAGE 106 99 o -*-Â» u Q o O 1) u QfflQO 7 a 8 5 S S 1 3 3 * K od a O I 00 o o I I Qd Â§ o 5 CD CO 0 g Is ^ is a IX, o o 0 "el o 2 o 6 Q Â« D fc .2 -3 ^ J i c Â« .2, C/5 CU, O 9 Q g K J ft -h N (f| /HO I o U m / / o PAGE 107 100 The pattern measured by the LS is the sum of the patterns scattered by each constituent particle in the sample. The Fourier lens is an important component in making this measurement. It focuses any light striking on any part of the lens at a given angle onto a single annular area on its plane of focus, the Fourier plane. The Fourier lens is sensitive only to the angle of the light rays incident on it and not to the position or velocity of the source of light. The result is that the Fourier lens forms an image of the entire diffraction pattern of each particle, the image being centered at a fixed spot on the Fourier plane. This image is centered at the same fixed spot regardless of the position or velocity of the particle in the diffraction sample cell. The individual diffraction pattern from many moving particles in the sample cell are therefore superimposed, creating a single composite diffraction pattern that reflects the contributions from all the particles in the sample cell. This composite diffraction pattern can be accurately sensed by detectors judiciously placed on the Fourier plane. Over the course of a measurement, a running average is computed of the flux patterns at every instant. When the duration of the measurement is long enough that the flux pattern accurately represents the contributions from all sample particles, an analysis of the resulting pattern will yield the true particle size distribution of the sample. The composite diffraction pattern is measured by 126 detectors placed at angles to approximately 35 degree from the optical axis. The intensity in flux units (light intensity per unit area) is the diffraction pattern. When the sizes are computed, the composite diffraction pattern is decomposed into a number of diffraction patterns, one for each size PAGE 108 classification, and the relative amplitude of each pattern is used to measure the relative volume of spherical particles of that size. Data processing program that comes with the instrument calculates a number of statistical quantities to characterize the particle size distribution. The statistics are calculated as they would be for a frequency distribution, with the volume percent (or surface area percent or number percent) in a certain size channel being analogous to the frequency of occurrence of a certain value. Volume percent, q>, is defined as where V t is the total volume of all particles in ith size channel and V, is total volume of floes. The size channel are spaced logarithmically, and are therefore progressively wider in span toward larger sizes. Statistical calculations are made based on the logarithmic center of each channel, or diameter of a particle in that channel, d, c i , in um: The arithmetic mean of the volume over all particles is weighted based on spherical particles as below: The mean diameter of particles is then obtained based on the mean volume and spherical particle assumption: (5.9) (5.10) (5.11) PAGE 109 102 It can be seen that the sizing in Coulter is based on the frequency of particle occurrence detected during the run time and particles are assumed to be spherical. Although the floes in the present experiment is not spherical, the measurement can give an equivalent diameter since a floe can be detected for many times in different orientations. 5.2.4 Estimate of Reflo cculation in Couette Flow and Orifice Flow It is well known that reflocculation and breakup take place in shear flow simultaneously. The stable size distribution of floes in the flow is governed by a dynamic equilibrium between aggregation and breakup. For the purpose of assessing floe strength, the reflocculation of floes should be avoided. Reflocculation of floes is a result of collisions of individual floes as particles form aggregates by collisions. There are three different mechanisms that can account for the aggregation of particles (Serra et al., 1997): Brownian motion, shear rate, and sedimentation. Coagulation due to Brownian motion is important for particles with diameter smaller than 1p.m. Collision frequency is proportional to shear rate, it is significant for particle size in the range 1-40 urn. The difference in sedimentation velocity due to different size of floes increases the chance for particle collision. In the present study, floe size is larger than 1 um, thus reflocculation due to Brownian motion is negligible. Because of the wide size distribution of floes, the collision due to sedimentation may occur in general. It can be avoided, however, by the use of fluids with high viscosity as indicated in Chapter 3. At 40% concentration of dextran solution, little sedimentation is observed. Therefore, the only relevant mechanism for reflocculation of PAGE 110 103 floes is shear rate. The number of collisions experienced by a typical floe, N c , can be estimated by the formula below (Sonntag, 1987): * * f (5.13) where Â« 3x10^. The effective floe volume fraction in Eq. (5.13) is usually much higher than this nominal value because floes retain substantial water within their interstices. The following is an estimate given by Sonntag and Russel (1987) according to their light scattering measurement for their latex floes: PAGE 111 104 0.409 s" 1 to 16.353 s' 1 . The rotating cylinder is started and accelerating in 30 seconds to its preset rotation for 300 minutes, then decelerating and stopped with a 30 second transient process. The floes with fl? 0 =1.0pm is used. The size distribution of the floes subjected to such a shearing history is measured and given in Fig. 5.4b. It is found that the given shearing history resulted in a reflocculation of floes, that is, an increase in overall mean floe size as shown in Fig. 5.4c with increasing shear rate. It can be seen from the changes of size distribution in Fig. 5.4b that the small floes with size around 2pm, and mostly, the primary particles of size around lpm decrease, forming larger floes as shear rate increases. At first three shear rates, the tendency is small. When shear rate increases from ^=8.18 s" 1 to 12.17 s" 1 , the reflocculation of small floes increases rapidly, resulting in an increase of mean floe size from 2.47pm to 2.63p.m. When shear rate further increases to ^=16.35 s" 1 , the dispersion and reflocculation of different size of floes exhibit. The primary particles decrease, small floes of size around 3 pm increase while the large floes of size beyond 4pm decrease, compared to the size distribution at last shear rate level Y=\2.\l %\ This generates a small decrease in overall mean size from 2.63pm to 2.51pm because of the breakup of large floes. This result was discouraging, because we could not compare it to those from the orifice flow where no reflocculation can be found in measured size distributions. Table 4.1 shows the data for N c calculated with Eq. (5.13) for the reflocculation. Based on the experimental results and these data, reflocculation at the same test settings is expected to be avoided by shortening the shearing time. PAGE 112 105 (a) Shear History Time ( Second ) cylinder rotation(rpm) (b) Floe Reflocculation in Simple Shear Flow 4 5 6 7 8 910 Floe Size (urn) 0 10 20 30 40 50 Shear Stress (N/m 2 ) (c) Variation of Mean Floe Size Fig. 5.4 Reflocculation of Floes in Couette Flow (H=1.32Ns/m 2 , d=1.0\im) PAGE 113 106 Table 5.1 Estimate of Particle Collision Rate n, (rpm) Y (s1 ) x (N/m 2 ) T(s)* N c 1 0.4088 1.076 330 7.5x1 0" 3 2.58 10 4.0883 10.76 330 7.5xl0" 3 25.77 20 8.1766 21.51 330 7.5x1 0" 3 51.54 30 12.265 32.27 330 7.5x1 0" 3 77.31 40 16.353 43.03 330 7.5x1 0" 3 103.08 *The effective time for corresponding shear rate, considering the transient process. If the shearing time is decreased by 100 times, to about 3.3 second, which means a very rapid start and stop of the inner cylinder in Couette flow, the particle collision frequency could be low enough to completely eliminate floe reflocculation for the same test conditions as used above. However, because the time resolution of the control system for the transient process and shearing process is 0.1 minute, that is, 6 seconds, a longer effective is used. To alleviate the unsteady effect in the flow during start and stop, a 0.4 minute accelerating transient process and a 0.2 minute decelerating transient process are used. In total, the effective shearing time for a corresponding shear rate is 24 seconds. This generates a particle collision frequency of 7.3% of the value in Table 4-1 at corresponding conditions. It should be pointed out that the reflocculation is minimized in the Cone-plate experiments in Chapter 3, since the maximum collision number estimated with the conditions used in that experiment is N c =0.71, which is attributable to the lower shear rate, lower nominal floe volume fraction used, and reasonable shearing time, although much longer than the shearing time in Couette flow test (maximum y= 3.93 s" 1 for 80 second, $ = 10 s ). PAGE 114 107 The number of particles collisions along the centerline in the orifice flow is estimated using the following integration since the shear rate is nonuniform. where x, is the location of floe injection and x 2 is the location of the orifice exit. The value of the integral depends on the Reynolds number Re and the orifice area ratio /?. For the present experimental conditions, N =0.0924. This ensures that very little reflocculation takes place in the orifice experiments for floe breakup. 5.3.1 Effect of Flow C ondition on Floe Size Distribution The floe size distributions in terms of volume percentage at different flow conditions are given in Figs. 5.5-5.7, for the three types of monodisperse floes whose primary particle are d 0 = 0.5jjm, l.Oym and 1.5/jm in diameter, respectively. Each curve corresponds to a given flow rate. The fluid is 50% dextran-water solution and has a viscosity /j.=2.9 sN/m 2 . For floes with d 0 =1.5jum, large floes break at a low flow rate as shown in Fig. 5.5. The size distribution changes at flow rate Q=100 ml/h. The curves for flow rate Q=200, 300, 389ml '/h overlap and correspond to the uniform distribution, indicating that the floes can be completely dispersed at the flow condition of Q=200ml/h in the present settings. Figure 5.6 shows the size reduction at various flow rates for the floes with d 0 =1.0/jm. A very uniform floe size distribution is reached at the maximum flow rate Q= 389ml/h. It (5.15) 5.3 Experimental Result s and Discussions PAGE 115 108 Fig. 5.5 Floe Size Distributions at Various Flow Rates (Primary particle d 0 =1.5 fim) PAGE 116 109 Fig. 5.6 Size Distributions at Various Flow Rates (Primary Particle Size d 0 =1.0um) PAGE 117 110 Floe Diameter (yim) Fig. 5.7 Size Distributions at Various Flow Rates (Primary Particle Size d 0 =0.5 u.m) PAGE 118 Ill is noticed that there is a noticeable secondary peak at floe diameter d 0 =2.5fjm in the floe size distribution curve when flow rate Q=300 ml/h, while the maximum floe size survived in this flow condition is about 4.5pm in diameter. This suggests that the segmentation into smaller floes of big floes whose size is between 5-1 lpm in the flow. At the maximum flow rate, the floe size reaches a very uniform distribution. It indicates that the silica floes of 1 micron diameter constituent particles can be completely dispersed with the present orifice setting (area ratio /?= 567) and flow conditions. For floes with d 0 =0.5jum, the initial distributions do not tend to an uniform distribution with the increasing shear rate in the given flow stress region, which indicates that the given flow condition is not strong enough to break the floes consisting of 0.5pm particles generated at conditions described in Chapter 2. However, the larger floes is broken noticeably when the flow rate reaches to Q=200 ml/h and the large floes of size ranging between 10pm and 20pm reduce in the volume percentage as the flow rate increases. Figure 5.7 clearly illustrates the size reduction of large floes, although the size distribution at the highest flow rate is still nonuniform. The floe size will further reduce if a higher stress level is applied. 5.3.2 Dependence of Mean Floe Size and Maximum Floe Size on Flow Rate Figure 5.8 shows the variations of mean floe size with flow rate for the three types of floes. Dextran at different concentrations are used as working fluid, corresponding to different fluid viscosity and density. The effects of hydrodynamic shear stress on floe breakup can be demonstrated through the use of different values of viscosity. For the floes with d 0 =1.0 pm, the results for the two fluids with ju=2.90 Ns/m 2 and ju=1.32 Ns/m 2 PAGE 119 112 1 i i i i I i i i i I i i i i I i i i i I 0 100 200 300 400 Flow Rate (ml/h) Fig. 5.8 Variation of Mean Floe Size with Flow Rate and Effect of Viscosity PAGE 120 113 are compared. While the mean floe size changes with flow rate for Q < lOOml/h at the higher viscosity, it keeps unchanged until Q > 200ml for the lower viscosity. For the floes with d 0 =1.5fim, when the fluid of /j=0.57 Ns/m 2 is used, the mean floe size retains constant, which indicates the viscosity is too low to break this type of floes. But when the fluid of n=2.90 Ns/m 2 is applied to the test, the breakup of floes takes place at Q < 200ml fh, the mean size no longer changes when flow rate increase furthermore and the mean size is about 1 .7 urn, just a little larger than primary particles of which the floes are made. For floes with d 0 =0.5/jm, the viscosity is // = 2.90 Ns/m 2 , the mean floe size do not change when Q < 200 ml/h at this viscosity. When Q increases from 200 ml/h, the floe mean size decreases. To further break the floes of this type, higher viscosity is required at high flow rate. However, the syringe pump cannot be normally operated at a viscosity higher than this value at high flow rates because of the large wall friction drag. The maximum floes that can exist at a given flow rate are given in Fig. 5.9 for the case of /i=2.90 Ns/m 2 . The maximum size remains to be 2.92p.m for Q > 200ml/h for the floes with d 0 =1.5fjm. Compared with the size of initial unaffected floes, 6.76p.m at Q=0, the floes are almost completely broken at the given conditions except very few floes (about 0.0014% in number fraction), which possibly are doublets consisting of two primary particles. In contrary to the case of d 0 =1.5jum, the floes generated with particles of d 0 =0.5 jum are much stronger. With the same fluid of /x=2.90 Ns/m 2 , the breakup just begins in the same flow rate level. To disperse this type of floes, reducing the mean size and maximum floe size further, a much larger stress is required. For the floes with d 0 =1.0fjm, both the mean floe size and the maximum floe size have been significantly reduced. The PAGE 121 Fig. 5.9 Variation of Maximum Floe Size with Flow Rate (\i=2.9Ns/m 2 , $=567) PAGE 122 115 fact that the mean size has reached 1.2um and the maximum 2.66p.m indicate this type of floes has be sufficiently dispersed in the same flow conditions used in the experiment. 5.3.3 Comparison with the Result from Uniform Cone-Plate Simple Shear Flow If the floe mean size changes with flow conditions are plotted against the maximum principal stress at centerline in the orifice flow field: where S cmajl is the maximum dimensionless centerline velocity gradient, then above curves at different values of viscosity are converted to Fig. 5.10 for the three types of floes. It is found that the floe size is single-valued function of the stress. In the low stress region the floe mean size are constant for all three types of floes. The fitting straight lines in log-log coordinates have increasing absolute values of slopes as the primary particle diameter increases. The floe mean size does not reduce further for the floes of the largest primary particle d 0 =1.5jjm while the mean size is still decreasing for the floes of primary particle d 0 =0.5^m. It is unlikely that the mean size for the floes with d 0 =1.0/jm will continue decreasing when a higher stress level is applied since its mean size has reduced to a value near the primary diameter. We also can plot the curves against the maximum centerline shear stress: c max (5.16) t. cmax 3 du ~4~dx cmax (5.17) and the nominal average stress defined as below(Sonntag and Russel, 1987): PAGE 123 116 Fig. 5. 10 Variation of Mean Floe Size with Centerline Maximum Stress PAGE 124 117 where (5.19) Because the Reynolds number is very small in the experiment (Re Â« 1.0), dimensionless centerline velocity gradient S cmax is independent of Reynolds number as dimensionless E . Hence the three parameters, PAGE 125 118 Stress (N/m 2 ) Fig. 5.1 1 Stress-Floe Size Relationship in Orifice Flow and in Cone-plate Flow (Primary Particle Size d 0 =1.0(im) PAGE 126 119 the value of maximum shear stress in the core region of orifice flow that fiocs will experience instantaneously. 5.3.4 Comparison of Floe Dispersion between O rifice Flow and Cylindrical Couette Flow A comparative test on the breakup of floes in cylindrical Couette flow is conducted. The shearing history is determined based on the theoretical estimate of reflocculation and the test result in section 5.2.4. The shearing history used is shown in Fig. 5.12a. When this shearing history is applied at five different shear rates, 4.09s" 1 , 8.18s" 1 , 12.27s" 1 , 24.54s" 1 , the resulting floe size distributions are shown in Fig. 5.12b. It is found that the breakup dominates for the first three shear rates, which correspond to a maximum N=5.6. When the shear rate reaches 24.54s" 1 , corresponding to N=10.12, the reflocculation takes over. Due to the unexpected contamination of collected samples, no data is available at the two shear rates of 12.27 s" 1 and 24.54s" 1 . Nevertheless, it could be concluded from the test that when the shear rate y < 12.27 s' 1 , corresponding to an inner cylinder velocity n, < 30 rpm, the floe breakup is dominant. The comparison of the results of floe breakup in the orifice flow with that from the Couette constant shear flow can be made in the range of y< 12.27 s" 1 . Figure 5.13 gives the comparison of the result in the test with the result obtained from Cone-plate constant shear flow in Chapter 3. It is seen that the two results are parallel to each other in logarithmic coordinates with floe size being larger in Couette flow. There could be two major factors that are responsible for the minor disagreement. One is the effect of erosions of floes in cone-plate experiment owing to long shearing. The other is PAGE 127 0 10 20 30 40 50 Time ( Second) (a) Shearing History For Floe Dispersion Floe Diameter (nm) (b) Floe Size Distributions at Different Shear Stresses Fig. 5.12 Floe Breakup and Reflocculation in Cylindrical Couette Flow PAGE 128 121 ~ 10 r E 8 c 0! 8 10 u 10 1 Couette flow O Cone-plate flow 10 s " 10 3 Shear Stress (N/m 2 ) Fig. 5.13 Floe Breakup and Reflocculation in Simple Shear Flow that there could be some reflocculation of floes in the Couette flow experiment, although the breakup of floes dominated. To find the connection between the breakup of floes subjected to a spatially uniform shearing and the breakup in the spatially nonuniform orifice elongational flow, the variation of floe size with flow stress from all the three experiments are shown in Fig. 5.14. Once again, the result from the orifice experiment is expressed in terms of four different centerline maximum stresses: a cmax , r cnax , x Ax cmax , and r Â£ . It is clearly shown that the effective stress that results in the floe size reduction in orifice flow can be expressed in terms of the centerline maximum shear stress, r__ as follows: T eff aT cmax (5.20) PAGE 129 122 1 I i Â— i Â— i Â— i Â— i i i I 1 i i i i i i i I i i 10Â° 10' 10 2 10 s Stress (N/m 2 ) Fig. 5.14 Comparison of Floe Dispersion in Orifice Flow with Simple Shear Flows PAGE 130 123 To fit the long shearing results in cone-plate experiment, the maximum centerline shear stress in the orifice flow must be scaled down by a factor a Â« 0.5. To fit the short shearing time results from the cylindrical Couette flow, the factor must be a* 0.83. It will be see from the following comparison that when the uniform shearing is instantaneous, a Â« 1.0, since the shearing history that floes moving in the orifice flow experience resembles a instantaneous process. 5.3.5 Re-Examination of Sonntag's Experimental Data If the long shearing in the cone-plate test gives the lowest limit for the effective shear stress, the highest limit can be determine with a test using instantaneous shearing. However, the device available to this work cannot be controlled to achieve this shearing history due to its minimum resolution of 0.1 minute in shearing time and accelerating and decelerating process. Fortunately, there is an experimental data resource that can be used to serve this purpose. Sonntag and Russel (1987) conducted experimental studies using both an orifice flow and a Couette flow, as mentioned above. The uniform shear in their Couette flow actually is temporally a ramping history for the purpose to minimize reflocculation. With very high acceleration of shear rate 1200 1/s 2 , and very high deceleration of shear rate at 600 1/s 2 , the shearing history is very much like a spatial shear distribution in orifice flow shown in Fig. 5.15a. The results from the two flows were not compared intrinsically in their publications, perhaps because of the lack of the orifice flow characterization. They only plotted the data from orifice flow against r E and found the nominal stress has to be scaled up by 4.6 to fit the results from the Couette flow. PAGE 131 124 (b) Variation of Floe Size with Stress Fig.5.15 Comparison of Floe Dispersion in Couette Flow with Result in Orifice Flow, Experimental Data From Sonntag(1985) PAGE 132 125 In Sonntag's work (1985, 1986, 1987), the orifice area ratio is J3=144 and the Reynolds numbers range from 267 to 2667. According to our results in Chapter 4, the maximum centerline velocity gradient for the orifice is S cmax =915.13+16453R; 2/} (5.21) thus a cmax , r cmax , ViT cmax , can be obtained from S cmax . The data for the average number of particles per floe then are plotted against G cmax , r cmax , l /2T cmax , and t e , as shown in Fig 5.15b. The results from their Couette shear experiment are also plotted in the same figure. It is seen that the r cmax in orifice flow fits well to the uniform shear stress in Couette flow. That is, for their experimental data, a & 1.0 in Eq. (5.20). 5.3.6 Floe Strength As sessment It is assumed that floes can break instantaneously upon the exerting of a step external force as shown in Fig. 5.16a and 5.16b (Sonntag and Russel, 1987). According to our results of floe breakup obtained from the three devices, the cone-plate flow in Chapter 3, the cylindrical Couette flow, and orifice flow, the floe dispersion process under a step constant shearing can be considered to be composed of three part, the instantaneous breakup, the slow erosion, and the transition between the them, which features the more difficult breakup of the smaller floes resulting from the instantaneous breakup and the easier erosion due to the irregular floe shape. This transition process essentially is very short, depending on the shear rate and the shear stress. The instantaneous breakup makes major contribution to the total floe size reduction. The orifice flow gives this part. The transition part is captured in our cylindrical Couette shear experiment. It gives a significant part of total size reduction. The slow erosion is shown in the continuous PAGE 133 126 (a) Step External shear Force Time Instantaneous breakup Size reduction due to Erosion (b) Instantaneous Breakup Assumption Time Fig. 5.16 Response of Floe Size to a Step Shear (Sonntag, 1986) PAGE 134 127 shearing experiment in Chapter 3. Its contribution to the total floe size reduction mainly depends on the shear rate and shear stress. The response of floe breakup (size reduction) to a step shear stress and the contributions from the orifice flow, cylindrical Couette flow, and cone-plate shear flow are illustrated in Fig. 5.17, according to this analysis. Note that aÂ«0.5 for the constant long shearing time in the cone-plate experiment and a&0.6 for short shearing time in the Couette flow in our test, and a & 1.0 for impulsive shear. This suggests the effect of shear time on floe strength. In other word, if the value of shear stress for impulsively breaking floes of given chemical conditions is requested, then r cmax should be given when orifice flow is used to determine the value. On the other hand, if the value of shear stress for breaking floes by long shearing time is requested, then 0.5T cmax is the correct value. With the establishment of the effective shear stress for floe breakup in orifice flow, the average mechanical strength of floes can be assessed according to the relationship between the effective shear stress and the mean floe size, which is usually power function of the following form: d = d, or T ~ T i (5.22) \djj where d, and t, correspond to a reference status. Constant / is intrinsically the chemicalphysical properties of floes, which can be determined by the orifice flow methods. For the silica floes made in the present study, f=0.25 for floes with d 0 =0.5jum;f=0.490 for floes with d 0 =1.0^m and 0.55, for floes with d 0 =1.5fjm. The value off for floes with d 0 =1.5jum is not sufficiently determined because of the insufficient number of points in PAGE 135 128 PAGE 136 129 the narrow size reduction region. The floes for different primary particle size are generated in exactly the same conditions, so the variation of / with d 0 demonstrated the effect of primary size of silica floes on their strength. The smaller the primary particle is, the stronger the floes is, as illustrated in the change of floe size distribution with flow conditions. 5.4 Summary This chapter focuses on the dispersion of floes in orifice flow by measuring the floe size distribution and comparing the breakup of floes in nonuniform contractile orifice flow with the floe breakup in uniform Couette simple shear flow. The experimental work is based on the well controlled experimental conditions and the detailed flow characterization presented in the last chapter. The following conclusions are drawn from this part of work: (1) A simple and reliable orifice flow device is set up and applied to break floes. The dispersion of floes in the orifice flow is characterized based on the complete numerical flow characterization and floe size distributions measured with a Coulter particle size analyzer at various flow conditions. The change of the floe size distributions with changing flow conditions demonstrates the variation of floe strength with floe size. The relationship between average floe size and flow shear stress is established for the three types of floes generated respectively with primary particles of three different diameters at the same chemical conditions. (2) Due to the rapid rise of the axial velocity gradient near the orifice entrance, the floe breakup in the orifice flow is instantaneous and the floe erosion mechanism can be PAGE 137 excluded. The centerline maximum shear stress in the orifice flow thus gives the floe strength of the resulting floes whose average size is subsequently measured. The floe strength determined from the short-time shearing in a cylindrical Couette flow at lower shear stresses follows essentially the same power law dependence on the floe size as determined in the orifice flow. Thus, floe strength measured in different flows can be unified using the maximum shear stress of the flow. The floe dispersion results of similar experiments (Sonntag, 1987) in orifice flow and in a cylindrical Couette flow are then reinterpreted using the computational results for the flow near the orifice. The reinterpreted data give excellent agreement for the strength-size relationship between their dispersion results in two sets of experiment. (3) Because all floes must pass through a high shear section a very uniform particle size distribution can be obtained at a sufficiently high flow condition when orifice flow is used to disperse floes. Hence, it is an effective tool for particle dispersion and for floe strength determination. Its simplicity in construction and reliability in control and operation are also advantages over other devices. PAGE 138 CHAPTER 6 SUMMARY 6.1 Summary and Conclusions The dispersions of floes whose size ranges from a few microns to tens of microns in four different flows, cone-plate simple shear flow, cylindrical Couette flow, contractile orifice flow, and hyperbolic flow, are studied by visual image analyses and particle size measurements. The results from different flows are compared and related to each other. The visual observations and quantitative measurements are made to characterize and understand the dispersion process of fine floes in hydrodynamic environments and floe strength. All flows used in the present study are well-defined or computationally analyzed, so the uncertainties in the flow characterization and the interactions between the floes and the flows, which can be found in most previous work, are excluded. The main results are summarized as follows: 1 . In a uniform simple shear flow, the breakup of fine floes takes place upon the onset of the applied shear stress and the main contribution to the floe size reduction is from the increase in the shear stress. The average fine floe size under a constant applied shear stress decreases with the increasing shear stress following a power law. The continuous constant shearing results in slow erosions of floes in a simple shear flow. The erosion rate mainly depends on the shear stress, the floe size, and the floe shape. For larger size 131 PAGE 139 132 of floes with more irregular shapes, the erosion will generate a large fraction of floe size reduction. The floe strength is interpreted as the resistance to its breakup under an applied shear stress. Typically, the floes of smaller size have a higher strength. When the external shear stress is larger than the floe strength at a given size, the breakup will rapidly take place to reach a relatively stable smaller floe size. The subsequent change in the floe size under the applied external shear stress is due to slow erosion. 2. A simple orifice flow device is set up to study the breakup process of floes in the elongational flow near an orifice and assess floe strength. The flow through an orifice with high area ratio up to 567 is simulated numerically for finite Reynolds numbers and creeping flow regime. The rate of strain field near the orifice is analyzed. Although the rate of strain field near the sharp corner of the orifice is highly nonuniform, the maximum centerline velocity gradient can be used to represent the maximum strain rate in a radially uniform core region in the flow. The dependence of maximum centerline strain rate on Reynolds number and the orifice area ratio is analyzed based on the results of the numerical simulation. The variation of the maximum centerline strain rate with the orifice area ratio and its asymptotic behavior at finite Reynolds number are established. 3. The dispersion of floes in the orifice flow is characterized based on the numerical solution for the flow and the size distribution of floes just downstream of the orifice measured using a Coulter particle size analyzer. The floe size variation as a function of shear stress applied over a short period of time in a cylindrical Couette flow is also obtained. These results are compared with that obtained in the cone-plate Couette flow PAGE 140 133 sheared over a long period of time. Due to the rapid rise of the axial velocity gradient near the orifice entrance, the floe breakup in the orifice flow is instantaneous and the floe erosion mechanism can be excluded. The centerline maximum shear stress in the orifice flow thus gives the floe strength of the resulting floes whose average size is subsequently measured. The floe strength determined from the short-time shearing in a cylindrical Couette flow at lower shear stresses follows essentially the same power law dependence on the floe size as determined in the orifice flow. Thus, floe strength measured in different flows can be unified using the maximum shear stress of the flow. The floe dispersion results of similar experiments (Sonntag, 1987) in orifice flow and in a cylindrical Couette flow are then reinterpreted using the computational results for the flow near the orifice. The reinterpreted data give excellent agreement for the strengthsize relationship between their dispersion results in two sets of experiment. Because all floes must pass through a high shear section a very uniform particle size distribution can be obtained at a sufficiently high flow condition when orifice flow is used to disperse floes. Hence, it is an effective tool for particle dispersion and for floe strength determination. Its simplicity in construction and reliability in control and operation are also advantages over other devices. 4. Dynamic process of deformation and fragmentation of fine floes of about 3 30 um is visualized for the first time in different flows including a simple shear flow in cone-plate device, a pure shear flow generated with two impinging jets, and a contractile flow approaching a slot in a rectangular channel. In the hyperbolic pure shear flow, floes can stay in the region of constant stress for a longer period of time compared to the PAGE 141 134 contractile flow where floes experience a very high stress in a very short period of time. In both the contractile flow and the hyperbolic flow, the most frequently observed mode of breakup is the one in which a floe is elongated, then broken into several smaller floes simultaneously. In the simple shear flow, a floe is usually broken into two floes at a time. The dynamic visualizations elucidate the floe deformation mechanism and breakup process. The physical pictures can help advance the understanding and the modeling of this complicated process. 6.2 Suggestions for Future Studies Based on the conclusions of this study, the following suggestions are given for further studies. 1 . Study the dispersion process of a variety of other floes with different physiochemical properties using the orifice flow device and techniques used in this work. Simultaneously, visual observation can be also made to easily understand the dispersion mechanism for a type of specific floes in flow. 2. The results of visualization of the dynamic deformation and breakage process can be of help in the physical concept formation and the establishment of models. More visualization results can be obtained using the technique and the system used in the work for assistance and verification of computer simulation and modeling of the complex process. 3. Study the erosion rate of floes in a shear flow. As shown in this study, the erosion rate is dependent of the shear stress, floe size, and floe shape. Quantitative studies of erosion PAGE 142 135 process for floes with different structures will provide information for the effective control of the dispersion process. 4. More closely examine the floe disperion in cylindrical Couette shear flow and orifice flow. Analyze and compare quantitatively the efficiencies and other characteristics of the floe dispersions in the two flows. 5. Determine strength of a variety of floes using the orifice flow. The mechanical properties of floes can be obtained by the method developed in this study. Furthermore, the behavior of various floes subjected hydrodynamic force can be predicted. PAGE 143 REFERENCES Argaman, Y. and Kaufman, W. J., Turbulence and flocculation Journal of the Sanitary Engineering Division, ASCE, V.96 No.SA2, Apr., 1970, pp. 223-241 Bache, D. H. and Al-Ani, D. H., Development of a system for evaluating floe strength Wat. Sci. Tech., Vol. 21, 1989, pp. 529-537 Clark, M. M. and Flora, J. R. V., Floe restructuring in varied turbulent mixing, Journal of Colloid Interface Science, Vol. 147, No.2, December 1991, pp. 407-421 Giovannini, A. and Gagnon, Y. Vortex simulation of the flow inside a pipe containing an orifice plate with different blockage ratios. Experimental and numerical flow visualization, ASME, Vol. 172, 1993, pp. 101-109 Glasgow, L. A. and Hsu, P., An experimental study of floe strength. AIChE Journal, Vol. 28, 1982, pp. 779-785 Glasgow, L. A. & Liu, X., Response of aggregate structure to hydrodynamic stress. AIChE Journal, Vol. 37, 1991, pp. 1411-1414 Glasgow, L. A. and Luecke, R. H., Mechanisms of deaggregation for clay-polymer floes in turbulent systems. Industrial & Engineering Chemistry Fundamentals, Vol.19, 1980, pp. 148-156 Hannah, S. A., Cohen, J. M., and Robeck, G. G., Measurement of floe strength by particle counting. AWWA Journal, July 1967, pp. 843-858 Henderson, M. and Tran-Son-Tay, R., A differential drive rheoscope for studying living cells under shear flow. (1997) Higashitani, K. Inada, N. and Ochi, T., Floe breakup along centerline of contractile flow to orifice. Colloids and Surfaces, Vol.56, 1991, pp. 13-23 Higashitani, K., Tanise, N., and Murata H., Dispersion of coagulated particles by contractile flow to orifice. Journal of Chemical Engineering of Japan, Vol. 25, No.5, 1992, pp. 502-507 Hogg, R., Hydrodynamic effects in flocculation and dispersion of fine particle suspensions. Dispersion and aggregation, 1993, pp. 20-31 136 PAGE 144 137 Janssen, J. M. H., Peters, G. W. M., and Meijer, H. E. H., An opposed jets device for studying the breakup of dispersed liquid drops, Chemical Engineering Science, Vol.48, No. 2, 1993, pp. 255-256 Jiang, Q. and Logan, B. E., Fractal dimensions of aggregates from shear devices, Journal AWWA, February 1996, pp. 100-113 Jung, S. J. Amal, R. and Raper, J. A., Monitoring effects of shearing on floe structure using small-angle light scattering, Powder Technology, 88, pp. 51-54, 1996 Kamal, M. R. and Patterson, I., Transient deagglomeration of solid aggregates in sheared viscous liquids. The Canadian Journal of Chemical Engineering, Vol. 52, December, 1974, pp. 707-714 Kao, S. V., and Mason, S. G., Dispersion of particles by shear. Nature(London) 253, 1975, pp.619-621 Kerekes, R. J., Pulp floe behavior in entry flow to constrictions. Papermakers Conference, 1983, pp. 129-136 Knoblock, N. R, A hyperbolic flow channel design for studying micron size particles under elongational stresses. Master Thesis, University of Florida, 1996 Kousaka, Y., Okuyama, K., Shimizu, A., and Yoshida, T., Dispersion mechanism of aggregate particles in air. Journal of Chemical Engineering of Japan, 1979, Vol. 12, No. 12, pp. 152-159 Lee, C. W. and Brodkey, R. S., A visual study of pulp floe dispersion mechanisms. AIChE Journal, Vol. 33, No. 2, 1987, pp. 297-302 Lu, C. F. and Spielman, L. A., Kinetics of floe breakage and aggregation in agitated liquid suspensions. Journal of Colloid and Interface Science, Vol. 103, No. 1, January 1985, pp. 95-105 Lugt, H. J., and Haussling, H. J., Laminar flow past an abruptly accelerated elliptic cylinder at 45Â° incidence. Journal of Fluid Mechanics, Vol. 65, Part 4, 1974, pp. 711134. Matsuo,T. and Unno, H., Forces acting on floe and strength of floe. Journal of the Environmental Engineering Division, Vol.107, No. EE3, 1981, pp. 528-549 Morrison, G. L., DeOtte, R. E. Jr., Panak, D. L., and Nail, G. H., The flow field inside an orifice flow meter. Chemical Engineering progress, July 1990, pp. 75-80 PAGE 145 138 Morrison, G. L., Panak, D. L., and DeOtte, R. E. Jr., Numerical study of the effects of upstream flow condition upon orifice flow meter performance. Journal of Offshore Mechanics and Arctic Engineering, Vol. 115, November 1993, pp. 213-218 Moudgil, B. M., Springgate, M. E., and Vasudevan, T. V., Characterization of floes for solid/liquid separation processes. Solid/liquid Separation: Waste Management and Productivity Enhancement, International Symposium, 1989, pp. 245-253 Navarrete, R. C, Scriven, L. E. and Macosko, C. W., Rheology and structure of flocculated iron oxide suspensions. Journal of Colloid Interface Science, 180, 1996, pp. 200-211 Nelson, R. D., Jr. Dispersing Powders in Liquids, Elsevier, 1988 Oles, V., Shear-induced aggregation and breakup of polystyrene latex particles. Journal of Colloid Interface Science, 154, No.2, 1992, pp. 351-358 Parker, D. S. Kaufman, W. J. and Jenkins, D., Floe breakup in turbulent flocculation processes. Journal of the Sanitary Engineering Division, Proc. Of ASCE, 1972, pp. 79-99 Pandya, J. D. & Spielman, L. A. Floe breakage in agitated suspensions: theory and data processing strategy. Journal of Colloid Interface Science, Vol. 90, 1982, pp. 517-531. Patterson, I. and Kamal, M. R., Shear deagglomeration of solid aggregates suspended in viscous liquids. The Canadian Journal of Chemical Engineering, Vol. 52, June, 1974, pp. 306-315 Peng, S.J. and Williams, R.S., Direct measurement of floe breakage in flowing suspensions. Journal of Colloid and Interface Science, Vol. 166, 1994, pp. 321-332. Potanin, A. A., Verkhusha, V. V., and Muller, V. M., Disaggregation of particles with Biospecific interactions in shear flow. Journal of Colloid and Interface Science, Vol. 188, 1997, pp. 251-256. Powell, R. L., Mason, S. G. Dispersion by laminar flow. AIChE Journal, Vol. 28, 1982, pp. 286-293 Schlichting, H., Boundary-layer theory, McGraw-Hill Incorporated, New York, 1979 Serra, T., Colomer, J. and Casamitjana, X., Aggregation and breakup of particles in a shear flow, Journal of Colloid and Interface Science, 187, 1997, pp. 466-473 Sonntag, R. C. Structure and Breakup of Floes Subjected to Fluid Stresses. University of Princeton, Ph.D. Dissertation, 1985 PAGE 146 139 Sonntag, R.C. and Russel, W. B., Structure and breakup of floes subjected to fluid stress 1. shear experiments. Journal of Colloid Interface Science, Vol. 113, 1986, pp. 399-413 Sonntag, R.C. and Russel, W. B., Structure and breakup of floes subjected to fluid stress 3. converging flow. Journal of Colloid Interface Science, Vol. 115, 1987, pp390-395 Spicer, P. T., Keller, W. and Pratsinis, S. E., The effect of impeller type on floe size and structure during shear-induced flocculation, Journal of Colloid Interface Science, 184, 1996, pp. 112-122 Tambo, N. and Hozumi, H., Physical characteristics of flocs-2 strength of floe. Water Research Vol. 13, 1979, pp. 421-427 Taylor, G. I., The formation of emulsion in definable fields of flow, Proc. Roy. Soc, Vol. 146, 1934, pp. 501-523 Thomas, D. G., Turbulent disruption of floes in small particle size suspensions. A.I.Ch.E. journal, Vol.10, No.4, 1964, pp517-523 Tomi, D. T. and Bagster, D. F., The behavior of aggregates in stirred vessels, Part 1. Theoretical considerations on the effects of agitation, Trans IChemE, Vol.56, 1978, pp. 1-8 Tran-Son-Tay, A study of the tank-treading motion of red blood cells in shear flow. Ph.D dissertation, Washington University, May 1983. Tran-Son-Tay, R., Sutera, S. P. & Rao, P. R. Determination of red blood cell membrane viscosity from rheoscopic observations of tank-treading motion. Biophysical Journal, Vol. 46, July 1984, pp. 65-72. Wagle, D.G., Lee, C.W. & Brodkey, R. S., Further comments on a visual study of pulp floe dispersion mechanisms. Tappi Journal, Vol. 71, 1988, pp. 137-141 Williams, R. A., Peng, S. J. and Naylor, A., In situ measurement of particle aggregation and breakage kinetics in a concentrated suspension. Powder Technology, 73, 1992, pp. 75-83 Yeung, A. K. C. and Pelton, R., Micromechanics: a new approach to studying the strength and breakup of floes. Journal of Colloid and Interface Science, 184, pp. 579-585, 1996 Yuu, S., Oda, T. Disruption mechanism of aggregate aerosol particles through an orifice. AIChE Journal, Vol. 29, 1983, pp. 191-198. PAGE 147 BIOGRAPHICAL SKETCH Xueliang Zhang was born on November 2, 1962, in Chengdu City, Sichuan Province, China. He earned his bachelor's degree from the Department of Applied Mechanics and Vehicle Design at Beijing University of Aeronautics and Astronautics, Beijing, China, in July 1982 and master's degree from the Department of Jet Propulsion at the university in December 1985. From February 1986 to July 1994, he worked as a Senior Reserach Engineer in the Department of Research and Development of Chengdu Aircraft Industrial Corporation, Chengdu City, China. Since August 1994, he has been pursing the Ph.D. degree in the Department of Aerospace Engineering, Mechanics and Engineering Science at the University of Florida. 140 PAGE 148 I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. lenwei Mei, Chairmaf Associate Professor of Aerospace Engineering, Mechanics and Engineering Science I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Roger Tra#-Son-Tay, Co-Chair Associate Professor of Aerospace Engineering, Mechanics and Engineering Science I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. James F. Klausner Associate Professor of Mechanical Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Corin Segal Assistant Professor of Aerospace Engineering, Mechanics and Engineering Science PAGE 149 I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy ilosopny. Wei Shyy VV Professor of Aerospace Engineering, Mechanics and Engineering Science This dissertation was submitted to the Graduate Faculty of the College of Engineering and to the Graduate School and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. May, 1998 Winfred M. Phillips Dean, College of Engineering Karen A. Holbrook Dean, Graduate School |