A Mathematical Model for Colloidal Aggregation. Colleen S. O Brien

Transcription

1 A Mathematcal Model for Collodal Aggregaton by Colleen S. O Bren A thess submtted n partal fulfllment of the reurements for the degree of Master of Scence n Chemcal Engneerng Department of Chemcal Engneerng College of Engneerng Unversty of South Florda Maor Professor: Lus H. GarcaRubo, Ph.D. John T. Wolan, Ph.D. Jule P. Harmon, Ph.D. Date of Approval: November, 003 Keywords: Brownan, model, agglomeraton, repulson, partcles, knetcs, populaton Copyrght 003, Colleen S. O Bren

2 DEDICATION I would lke to dedcate ths work to my fancé, Bran. Hs encouragement, patence, and love have enabled me to pursue my passon, and contnue to strve for excellence n all areas of my lfe.

3 ACKNOWLEDGEMENTS I would lke to use ths opportunty to thank everybody wthout whose help ths work would have not been possble. I would lke to thank Dr. GarcaRubo for hs support and gudance over the past several years. I would also lke to thank the Chemcal Engneerng faculty at USF for ther support n my endeavors. I apprecate the help, support, and advce of my workmates at Bausch & Lomb, Santos Vscasllas, Gabrel Salmon, and Don Herber, and also my old workmates at Custom Manufacturng & Engneerng Scott Eler, Tammy Huggns, and Chrs Plonsky.

4 TABLE OF CONTENTS LIST OF TABLES v LIST OF FIGURES LIST OF SYMBOLS v x CHAPTER : INTRODUCTION 3 CHAPTER : AGGREGATION THEORY 6. Introducton 6. Collodal Partcles 6.3 Surface Charge 7.4 GouyChapman Model 8.4. Dffuse Layer Model.4. Electrophoretc Moblty and Zeta Potental 3.5 Aggregaton 4.6 Models for Collson Rates 5.6. Perknetc Collson Mechansm 6.6. Orthoknetc Collson Mechansm Dfferental Settlng Mechansm.7 Comparson of Rates.8 Collson Effcences 4.8. Perknetc Collson Effcences (Stablty Rato) 4.8. Orthoknetc Collson Effcences Hydrodynamc Interacton 7.9 Interpartcle Forces 9

5 .9. Electrostatc Repulsve Forces SphereSphere Interactons 3.9. Van der Waals Hamaker Expressons for Interactng Spheres Other Interpartcle Forces Born Repulson Sterc Interacton 35.0 Populaton Balances 35. Aggregate Structure 39.. Fractal Geometry 40.. Fractal mass scalng 4..3 Packng Densty Coordnaton Numbers Fractals n Partcle Aggregaton 47. Summary 48 CHAPTER 3: MATHEMATICAL MODEL FOR AGGREGATION KINETICS Introducton Interpartcle Forces Van der Waals Attractve Interacton Energy Electrostatc Repulson Interacton Energy Lnearzed PBE Nonlnear PBE Comparson of Lnear and Nonlnear PBE Collson Mechansms Stablty Ratos Perknetc Stablty Ratos Orthoknetc Stablty Ratos Map of Aggregaton Coeffcents Populaton Balance Formulaton Dscretzaton Method Comparson wth Analytcal Solutons SzeIndependent Kernal SzeDependent Kernal Error Levels n Partcle Sze Dstrbuton Mathematcal Model Descrpton 9

6 3.8 Summary 93 CHAPTER 4: EXPERIMENTS CONDUCTED BY MATHEMATICAL MODEL Introducton The Effect of Dscretzng Parameters on the Model The Effect of Temperature and Vscosty on the Model The Effect of Dameter Sze on the Model The Effect of Salt Concentraton on the Model The Effect of Hamaker Constants on the Model The Effect of Fractal Dmenson and Prmary Partcle Szes on the Model The Effect of Surface Potental on the Model The Effect of Shear Rate on the Model The Effect of Smultaneous Perknetc and Orthoknetc Collson Mechansms 4. Summary CHAPTER 5: COMPARISION OF MATHEMATICAL MODEL WITH EXPERIMENTS 3 5. Introducton 3 5. Materals Partcle Sze Dstrbutons Measurement of Surface Potentals Measurement of Aggregaton Phenomenon Expermental Runs Comparson Experment A Experment B Experment C Experment D 43

7 5.7.5 Experment E Experment F Concluson 5 CHAPTER 6: CONCLUSIONS 5 6. Mathematcal Model 5 6. Experments Conducted wth Mathematcal Model Comparson of Mathematcal Model wth Experments Future Work 53 REFERENCES 54 APPENDICES 59 APPENDIX : CRITERION FOR MONOTONIC OR CONCAVE POTENTIAL 60 APPENDIX : BEHAVIOR OF CONCAVE ELECTROSTATIC POTENTIAL AS C VARIES. 6 APPENDIX 3: DERIVATION OF K TERMS FOR ADJUSTABLE DISCRETIZATION OF LITSTER ET AL. 66 APPENDIX 4: PROOF OF VOLUME CORRECTION FACTORS FOR DISCRETIZATION OF LITSTER ET AL. 68 APPENDIX 5: MATLAB CODE FOR MATHEMATICAL MODEL 7 v

8 LIST OF TABLES Table 3.. Table of Aggregaton Events for the Method of Ltster et al. 80 Table 3.. Table of Aggregaton Events for the Method of Ltster et al. (Updated) 85 Table 4. Dscretzng Parameter versus Total Number of Bns 94 Table 4.. Expermental Parameters for Perknetc and Orthoknetc Experments 3 Table 5.. Propertes of Partcle Standards 4 Table 5. Propertes of Partcle Stock Solutons 4 Table 5.3 Summary of the Intal Partcle Sze Dstrbuton for Standards 5 Table 5.4. Expermental Parameters for Experment A 8 Table 5.5. Degree of Aggregaton for Experment A 8 Table 5.6. Expermental Parameters for Experment B 33 Table 5.7. Degree of Aggregaton for Experment B 33 Table 5.8. Expermental Parameters for Experment C 38 Table 5.9. Degree of Aggregaton for Experment C 38 Table 5.0. Expermental Parameters for Experment D 43 Table 5.. Degree of Aggregaton for Experment D 43 Table 5.. Expermental Parameters for Experment E 48 Table 5.3. Degree of Aggregaton for Experment E 48 Table 5.4. Expermental Parameters for Experment F 50 Table 5.5. Degree of Aggregaton for Experment F 50 v

9 LIST OF FIGURES Fgure.. The GouyChapman Model 9 Fgure.. Double Layer Structure 0 Fgure.3. Potental Profle for GouyChapman Model 0 Fgure.4. Perknetc Aggregaton 6 Fgure.5. Boundary Condtons for Smoluchowsk s Fast Coagulaton Euaton 7 Fgure.6. Movement of Partcles n Shear Flow 9 Fgure.7. Movement of Partcles n Extensonal Flow 0 Fgure.8. Comparson of Collson Rate Constants for Dfferent Transport Mechansms.3 Fgure.9. Dfference Between Rectlnear and Curvlnear Traectores 5 Fgure.0. Collson Effcency as a Functon of Shear Rate and Partcle Sze 6 Fgure.. Lmtng Stablty Rato W lm, as a Functon of the Hamaker Constant (). 8 Fgure.. Potental Energy Curve for Collodal Interacton. 30 Fgure.3. Populaton Balance Modelng 39 Fgure.4. Arrangement for Dfferent Number of Prmary Partcles 40 Fgure.5. Constructon Process of the Prefractal Koch Curves. 4 Fgure.6. Schematc Illustraton of SelfSmlar Aggregate Structure 43 Fgure.7. 3D Representaton of Several Aggregates wth D f = Fgure.8. Comparson between the Coalescence and Fractal Aggregate Model 45 Fgure 3.. Contact of Two Dssmlar Spheres 50 Fgure 3.. Potental Energy of Interacton for Dfferent Partcle Szes 5 Fgure 3.3. Potental Energy of Attracton for Dfferent Partcle Szes 5 Fgure 3.4. Potental Energy of Attracton for Dfferent Partcle Szes 5 Fgure 3.5. Potental Energy of Repulson for Dfferent Partcle Szes 54 Fgure 3.6. Potental Energy of Repulson for Dfferent Partcle Szes 54 Fgure 3.7. Graphcal Depcton of the Method of Papadopoulos and Cheh 56 Fgure 3.8. Flowsheet for Numercal Integraton of NonLnear PBE 6 Fgure 3.9. PBE Interacton Energy Comparson 63 Fgure 3.0. PBE Interacton Energy Comparson 64 Fgure 3.. PBE Interacton Energy Comparson 65 Fgure 3.. PBE Interacton Energy Comparson 66 Fgure 3.3. PBE Interacton Energy Comparson 67 Fgure 3.4. Perknetc Rate Constants 68 Fgure 3.5. Orthoknetc Rate Constants 69 Fgure 3.6. Dfferental Settlng Rate Constants 70 Fgure 3.7. Stablty Ratos for HHF 7 Fgure 3.8. Stablty Ratos for RKF and Papadopoulos 7 v

10 Fgure 3.9. Stablty Ratos for Orthoknetc Collson Mechansm 73 Fgure 3.0. Stablty Ratos for Orthoknetc Collson Mechansm 74 Fgure 3.. Log of Aggregaton Coeffcents for Perknetc Collson Mechansm 75 Fgure 3.. Log of Aggregaton Coeffcents for Orthoknetc Collson Mechansm 76 Fgure 3.3. Log of Aggregaton Coeffcents for Dfferental Settlng Mechansm 76 Fgure 3.4. Dscrete Bns to Cover a Large Range of Partcle Szes 78 Fgure 3.5. Moments versus I agg 88 Fgure 3.6. Error n the th Moment for SzeDependent Case 90 Fgure 3.7. Error n the th Moment for SzeIndependent Case 9 Fgure 3.8. Flow Dagram of Mathematcal Model 9 Fgure 4.. Comparson of Dscretzng Parameters for Perknetc Collson Mechansm 95 Fgure 4.. Comparson of Dscretzng Parameters for Orthoknetc Mechansm 96 Fgure 4.3. Temperature Comparson 97 Fgure 4.4. Vscosty Comparson 98 Fgure 4.5. Dameter Sze Comparson for Perknetc Aggregaton 99 Fgure 4.6. Dameter Sze Comparson for Orthoknetc Aggregaton 99 Fgure 4.7. Effect of Ionc Strength on Interacton Energy for Clay Mneral Kaolnte 0 Fgure 4.8. Effect of Ionc Strength on Clay Mneral Kaolnte. 0 Fgure 4.9. Ionc Concentraton Comparson for the Model 03 Fgure 4.0. Potental Energy of Interacton for Varous Values of Hamaker Constant 04 Fgure 4.. Hamaker Constant Comparson 05 Fgure 4.. Comparson of Prmary Partcle Sze for Perknetc Collson Mechansm 06 Fgure 4.3. Comparson of Prmary Partcle Sze for Orthoknetc Collson 07 Fgure 4.4. Fractal Dmenson Comparson for Perknetc Collson 07 Fgure 4.5. Fractal Dmenson Comparson for Orthoknetc Collson Mechansm 08 Fgure 4.6. Constant Surface Potental Between Two Plates 09 Fgure 4.7. Scalng Surface Potental Between Two Plates 09 Fgure 4.8. Surface Potental Versus Dameter 0 Fgure 4.9. Comparson of Shear Rates Fgure 4.0. Peclet Number Versus Partcle Dameter Fgure 4.. Log of Aggregaton Coeffcent, G =, Peclet < 4 Fgure 4.. Log of Stablty Ratos, G =, Peclet < 4 Fgure 4.3. Log of Stablty Ratos, G = 70, Peclet < 5 Fgure 4.4. Log of Stablty Ratos, G = 70, Peclet < 5 Fgure 4.5. Perknetc and Orthoknetc Aggregaton 6 Fgure 4.6. Log of Aggregaton Coeffcent for G = 0 and the Peclet number <90 7 Fgure 4.7. Log of Stablty Ratos for G = 0 and the Peclet number <90 7 Fgure 4.8. Log of Aggregaton Coeffcent for G=0 8 Fgure 4.9. Log of Stablty Ratos for G=0 8 Fgure Perknetc and Orthoknetc Aggregaton 9 Fgure 4.3. Log of Perknetc and Orthoknetc Collson Mechansm where G = 0 0 Fgure 4.3. Log of Stablty Ratos for G = 0 0 Fgure Log of Aggregaton Coeffcents for G = 0 Fgure Addton of Perknetc and Orthoknetc Aggregaton Fgure 5.. Zeta Potental for Polystyrene 6 v

11 Fgure 5.. Log of Perknetc Collson Mechansm for Experment A 9 Fgure 5.3. Log of Stablty Ratos for Experment A 9 Fgure 5.4. Log of Aggregaton Coeffcent for Experment A 30 Fgure 5.5. Partcle Concentraton versus Tme for Experment A 30 Fgure 5.6. Partcle Concentraton versus Partcle Dameter for Experment A 3 Fgure 5.7. Actual versus Mathematcal Model Experments (Experment A) 3 Fgure 5.8. Log of Perknetc Collson Mechansm for Experment B 34 Fgure 5.9. Log of Stablty Ratos for Experment B 34 Fgure 5.0. Log of Aggregaton Coeffcents for Experment B 35 Fgure 5.. Partcle Concentraton versus Tme for Experment B 35 Fgure 5.. Partcle Concentraton versus Partcle Dameter for Experment B 36 Fgure 5.3. Actual versus Mathematcal Model Experments (Experment B) 37 Fgure 5.4. Log of Orthoknetc Collson Mechansm for Experment C 39 Fgure 5.5. Log of Aggregaton Coeffcent for Experment C 39 Fgure 5.6. Partcle Concentraton versus Tme for Experment C 40 Fgure 5.7. Partcle Concentraton versus Partcle Dameter for Experment C 40 Fgure 5.8. Log of Stablty Ratos for Experment C 4 Fgure 5.9. Actual versus Mathematcal Model Experments (Experment C) 4 Fgure 5.0. Stablty Ratos for Experment D 44 Fgure 5.. Log of Orthoknetc Collson Mechansm for Experment D 44 Fgure 5.. Log of Aggregaton Coeffcents for Experment D 45 Fgure 5.3. Partcle Concentraton versus Tme for Experment D 45 Fgure 5.4. Partcle Concentraton versus Partcle Dameter for Experment D 46 Fgure 5.5. Actual versus Mathematcal Model Experments (Experment D) 47 Fgure 5.6. Actual versus Mathematcal Model Experments (Experment E) 49 Fgure 5.7. Actual versus Mathematcal Model Experments (Experment F) 5 v

12 LIST OF SYMBOLS Alphanumerc Symbols A a a,b a a B C C C,C D D f D e F F g effectve Hamaker constant prmary partcle radus constants n Cauchy euaton actvty coeffcent of on radus of partcle wth sze brth functon concentraton of partcles Constant volume correcton factors n dscretzaton method death functon mass fractal dmenson dffuson coeffcent of partcle wth sze electron charge Faraday Constant growth rate of partcles along the sze axs acceleraton due to gravty x

13 G G Iagg I J k k (l) k f K k n l L l p M m m m ~ average shear rate shear rate degree of aggregaton modfed Bessel functon of the frst knd order one flux of partcle of sze Boltzmann constant absorpton coeffcent for partculate matter as a functon of wavelength fast aggregaton rate constant fracton of successful collsons for each type n dscretzaton procedure rate constant of aggregaton between partcle of sze and knematc vscosty center to center partcle separaton length sze coordnate path length mass of aggregate complex refractve ndex th moment of the partcle sze dstrbuton dmensonless th moment of the partcle sze dstrbuton n n(l,t) n(v,t) n + number concentraton of negatve ons populaton densty functon n length coordnates populaton densty functon n volume coordnates number concentraton of postve ons x

14 n o N 0 n 0 (l) n (l) N D (D,t) N n k N p N v (v,t) P R r number concentraton of ons total ntal number of partcles refractve ndex of medum as a functon of wavelength refractve ndex of the partculate matter as a functon of wavelength number densty functon on a dameter bass number concentraton of partcles n sze range kolmogoroff mcroscale total number of partcles n turbdty euaton number densty functon on a volume bass combned regonal and property space adustable dscretzaton parameter Ideal Gas Constant partcle radus R regon of property space for all x m r g R R S S(P) S A T geometrc rato radus of partcle wth sze collson radus between partcles of sze and arc length n Papadopolous ntegraton technue functon to relate mportant nterval lmts n dscretzaton procedure Surface Area Temperature V regon of dmensonal space x

15 V A ve V f v v o V part V R V T ve r v W x,y,z Xd X ref Z z van der Waals attractve nteracton energy propagaton through property coordnate nteracton energy per unt area of between two parallel plates volume of partcle n n sze range mean ntal partcle volume volume of partcles electrostatc repulsve nteracton energy total nteracton energy average propagaton velocty n all property space dmensonless volume stablty rato between partcles of sze and spatal coordnates separaton between plates dstance between plates correspondng to a partcular soluton total number of dscretzaton bns valence of on Greek Symbols a a m b o collson effcency Me sze parameter sze ndependent aggregaton constant x

16 b e e o e p k l F o aggregaton coeffcent between partcles of sze and alternate volume coordnate delectrc permttvty power nput per unt volume DebyeHuckel parameter alternate length coordnate or wavelength Electrostatc potental ntal aggregate densty, (porosty) y o y r r f r s t t r t(l o ) G,f m n x m aggregate densty of partcle wth radus R Surface potental reduced surface potental densty densty of flud densty of partcle dmensonless aggregaton tme resdence tme turbdty for wavelength l populaton densty functon angles used n Papadopolous ntegraton technue moment generatng functon vscosty property space x

17 A MATHEMATICAL MODEL OF COLLOIDAL AGGREGATION Colleen O Bren ABSTRACT The characterzaton of fne partcles s an area of mmense sgnfcance to many ndustral endeavors. It has been estmated that 70% of all ndustral processes deal wth fne partcles at some pont n the process. A natural phenomenon occurrng n these processes s collodal aggregaton. Ths study examnes aggregaton n collodal systems n order to characterze, examne, and control ths occurrence n ndustral processes. The study of partcle aggregaton has been broken nto many dfferent areas, such as collson mechansms, nteracton energy etc, but a complete model that ntegrates these dfferent aspects has never been fully realzed. A new model s reured to accurately predct the aggregaton behavor of collodal partcles. In ths work, a new model s developed that ntegrates Smoluchowsk knetcs, total nteracton energy between partcles, and stablty ratos for perknetc and orthoknetc collson mechansms. The total partcle nteracton energy necessary for the calculaton of stablty ratos s represented by the summaton of electrostatc and van der Waals

18 nteractons. The electrostatc nteractons are modeled usng DLVO theory, the lnear PossonBoltzmann euaton, and a numercal soluton for the nonlnear Posson Boltzmann Euaton, whle the van der Waals nteractons are represented by Hamaker theory. The mathematcal model s solved usng an adustable dscreton technue, whch s tested aganst a specfc analytc soluton, and yelds an assessment of the error ntrnsc n the dscretzaton method. The bass of the mathematcal model s a populaton balance framework. The model developed n ths study s general n many respects, but could be readly appled to many dfferent aggregaton systems wth mnor modfcaton. A comparson of the mathematcal model wth prevous experments conducted by Scott Fsher (998) s carred out for the perknetc and orthoknetc transportlmted aggregaton regmes. The fractal nature of soldsphere aggregates s consdered when comparng the mathematcal model predctons wth expermental measurements. The prevous experments that are used for comparson utlzed polystyrene partcles rangng from 00nm to 500nm n ntal dameter, several ntal partcle concentratons, and varous strrng rates. Zeta potental measurements are presented n order to set the range of transportlmted aggregaton. An assessment of the results of the mathematcal model wth the expermental results show good agreement for transportlmted aggregaton wthn the perknetc and orthoknetc transportlmted aggregaton, wth average partcle szes rangng from 00nm to well over mm.

19 CHAPTER : INTRODUCTION Aggregaton has become ncreasngly sgnfcant durng the last twentyfve years. Industry has become progressvely more nterested n controllng the mcroscopc propertes of partcles, such as composton, shape, surface roughness, surface characterstcs, and porosty. It has been estmated that 70% of all ndustral processes nvolve dealng wth fne partcles at some pont n the process (Bushell 998). Characterzaton of these partcles help us to understand and predct or control ther behavor n many processes. In ndustry, the propertes of partcles determne whether or not a dust s a respraton hazard, whether granular materals wll mx or segregate when agtated, and whether materal n a hopper wll flow n a controllable fashon, behave lke a lud, or not flow at all. Many of the tradtonal partcle characterzaton technues make assumptons about the shape or physcal structure of the partcles beng measured. The extracton of lnearsze parameters from laser scatterng measurements generally assume that the partcles are sphercal. Ths assumpton s normally made because t probably s not mportant to the technue beng used, but often t s because t s a dffcult problem. One class of partcles that has great mportance s aggregates. Almost every partculate system nvolves to a greater or lesser extent some partcles that are aggregates of smaller partcles n the system. Ths may be unmportant for systems such as the handlng of bulk ores, but n processes such as drnkng water fltraton, t s the domnant structure. These aggregates are often wspy, tenuous enttes that are absolutely unlke spheres, plates, or other famlar geometrc forms. The euatons used n ths model are for sphercal partcles, but the sphercal partcles are translated nto ther assocate fractal dmenson at the end of the model. 3

20 The begnnng of the current understandng of collodal aggregaton dates back to the work of Smoluchowsk (97). He dentfed aggregaton as a secondorder process dependent on the concentraton of aggregatng speces. The early work n ths area conssted of measurng and predctng rates of aggregaton based upon sngle euatons. Up untl the mdseventes, the study of aggregaton consdered only the formaton of dstnct aggregates from a monodspersed (unform n shape and sze) ntal partcle system, or the calculaton of rate constants of aggregaton from doublet formaton. The nature and magntude of varous forces actng on the partcles determne the stablty of the suspenson. Generally, n case of a sold dsperson n an aueous medum, the forces actng on a partcle nclude van der Waals forces, electrostatc forces and hydrodynamc forces. Of these, the van der Waals forces are attractve n nature and favor aggregaton. Electrostatc forces exst due to the presence of an electrc doublelayer around the partcles and cause repulson between the partcles. Ths has the effect of opposng aggregaton. Both of these forces are smlar n magntude, and act over comparable dstances from the partcle surface. If the electrc doublelayer repulson domnates, the suspenson remans stable, and does not agglomerate. However, suppressng the electrc doublelayer can destablze a suspenson makng the van der Waals attractve forces domnant, and creatng aggregaton. Ths knd of aggregaton s known as coagulaton. When partcles collde wth each other, not all collsons result n the formaton of agglomerates. Instead, only a fracton of the total collsons lead to the formaton of agglomerates. Ths fracton s known as the collson effcency, and t determnes the overall rate of aggregaton. When a suspenson s sheared, the partculate sze ncreases as aggregaton takes place. However, stresses develop at the same tme due to flud shearng and tend to break up the agglomerates nto a smaller sze. Hence, durng the flow of a suspenson, partcle enlargement and breakage take place smultaneously. Conseuently, agglomerates do not contnue to grow ndefntely n sze durng prolonged shearng; nstead they attan an eulbrum sze. 4

21 Ths paper s a summaton of the work of Lus H. GarcaRubo s collodal partcle students over the last decade. The maor contrbutons are by Esteban MaruezRuelme (994) n the area of the nonlnear PossonBoltzmann euaton to obtan the repulsve nteracton energes between sphercal partcles, and Scott Fsher (998) who researched and mplemented the populaton balance euatons wth the Ltster et al. (995) method of dscretzaton. The three maor sectons of ths paper are () the mathematcal model descrpton, () The nvestgaton of the mathematcal model, and (3) a valdaton of the model aganst a set of aggregaton experments that were performed by Scott Fsher (998). In accomplshng these goals, the work s outlned n the followng matter: Chapter deals wth the background nformaton that s necessary to understand the model. Chapter 3 outlnes the general mathematcal model for aggregaton phenomenon. Chapter 4 deals wth an nvestgaton of the model parameters. Chapter 5 provdes a comparson between the mathematcal model predctons and the measured aggregaton phenomena. Chapter 6 deals wth a general summary and conclusons, as well as future development of the model. 5

22 CHAPTER : AGGREGATION THEORY. Introducton Ths chapter descrbes the necessary background topcs reured to understand the mathematcal model for aggregaton. The frst secton descrbes the basc propertes of collodal partcles. The next few sectons descrbe aggregaton knetcs, collson mechansms, partcle stablty, and aggregate structure. The attractve and repulsve forces, such as electrostatc repulsve and van der Waals attractve forces, wll then be dscussed. Next, a mathematcal ntroducton to the populaton balance model s ncluded. The theory presented n ths chapter ntroduces the basc concepts necessary to understand the mathematcal model.. Collodal Partcles Collods form heterogeneous mxtures that are large enough to scatter lght. Collods usually consst of two phases, or one contnuous phase n whch the other phase s dspersed. These partcles are larger than the sze of molecules, but small enough for the dspersed phase to stay suspended for a long perod of tme. Collodal systems contan at least one or more substances that have at least one dmenson n the range between 0 9 m (0 A) and 0 6 m ( mm) n sze (Hemenz, 986). On the smaller end of ths scale, there are no dstnct boundares between the phases, and the system s consdered a soluton. On the larger end of ths scale, partcles wll begn to fall to the bottom due to gravtatonal force, and the phases are separated. Aggregaton nvolves the assocaton of partcles to form clusters, and depends on two dstnct nfluences: () partcles must move n a way that collsons occur, and () partcles that repel each other are sad to be stable, snce they do not form aggregates. Collods nteract wth each other at 6

23 an extremely short range, (usually much less than the partcle sze), so that partcles have to approach very close to each other before any sgnfcant nteracton s felt. The nteracton may be attractve (van der Waals) or repulsve (electrostatc repulson, sterc). There are many mportant propertes of collodal systems that are determned drectly or ndrectly by the nteracton forces between partcles. These collodal forces consst of the electrcal double layer, van der Waals, Born, hydraton, and sterc forces. Collodal partcles are domnated by surface propertes. If the surface area to volume, or surface area to mass of a sphercal partcle s looked at, the dependence on the partcle radus s S A /V µ /r. Ths relatonshp shows that as partcles decrease n sze, the surface propertes of the partcle become ncreasngly mportant (Fsher, 998). The measurement of partcle sze s also a defnng property. Optcal mcroscopy reles on vsble lght, whch renders collodal partcles largely nvsble to optcal technues. (Vsble lght lmts measurements to about 0.5 mm). Sedmentaton cannot be used to characterze partcles because the partcles need to be about.5 mm n sze (Fsher, 998)..3 Surface Charge Collodal suspensons usually consst of charged partcles dspersed throughout a contnuous solvent phase. When two phases are n contact, a separaton of charge wll occur whch causes a dfference n electrcal potental. If ths phase separaton s restrcted to a sold nterface wth an aueous electrolyte system, there are several possble mechansms for the separaton of charge: ) a dfference n affnty of ons for the two phases ) onzaton of surface groups 3) physcal restrcton of certan ons to one phase. The frst case of separaton of charge s usually found n metal haldes, calcum carbonate, and metal oxdes. The bestknown example s slver odde (Elmelech et al., 995). When slver odde s n contact wth pure water, slver ons have a tendency to escape from the 7

24 crystal lattce, leavng a crystal wth an excess negatve charge. If the concentraton of slver ons s ncreased, a pont s reached where the hgher escapng tendency of the slver ons s balanced by ther hgher concentraton n soluton, and the sold does not have a net charge (Elmelech et al., 995). For these cases, the surface potental, y 0, can be modeled by the Nerst euaton: Ê RT ˆ y 0 = constant + Á ln a z F Ë (.3.) where R s the gas constant, T s the absolute temperature, F s the Faraday constant, z s the valence, and a s the actvty of the escapng ons (Elmelech et al., 995). An example of phdependent surface charge s the case of metal oxdes. When metal oxdes are n contact wth water, the oxde surfaces become hydroxylated, gvng the possblty of surface onzaton to yeld ether postve or negatve groups. The onzaton of such groups can be wrtten as: S OH + S OH S O _ Where S denotes a sold surface (Elmelech et al., 995). Ths process nvolves the loss of two protons, whch are defned by approprate eulbrum constants. The degree of protonaton depends on the values of these eulbrum constants and the soluton propertes..4 GouyChapman Model If the collodal partcles n soluton are charged, and the soluton s electrcally neutral, the balancng charge s accounted for by an excess number of oppostely charged ons or counterons n soluton adacent to the charged surface and a defct of smlarly charged ons. In ths electrcal double layer, the counterons are dstrbuted accordng to a balance between ther thermal moton and the forces of electrcal attracton. The GouyChapman 8

25 model characterzes ths arrangement of charged speces around the collodal partcle. The electronc double layer s composed of two layers, the nner layer (Stern layer), and the outer layer (dffuse layer). The Stern layer conssts manly of oppostely charged ons adsorbed to the collod surface. The dffuse layer conssts of a mxture of ons extendng some dstance away from the collod. The Shear surface s the surface between the fxed and dffuse layer and defnng the moble porton of the collod. A dagram of the electrc double layer s shown n Fgures. (Anderson et al., 975) and. (Maruez, 994). A dagram of the potental profle for GouyChapman Model n shown n Fgure.3 (Anderson et al., 975). Fgure.. The GouyChapman Model 9

26 Fgure.. Double Layer Structure y 0 Potental Zeta Potental, z y d Potental at DffuseLayer Boundary d k Dstance Fgure.3. Potental Profle for GouyChapman Model 0

27 The Gouy and Chapman model s based on a number of smplfyng assumptons: ) an nfnte, flat mpenetrable nterface ) ons n soluton are pont charges, able to approach rght up to the nterface. 3) the surface charge and potental are unformly smeared out over the surface. 4) the solvent s a unform medum wth propertes (especally permttvty) that are ndependent of dstance from the surface. The relatonshp between charge densty, r (cm 3 ), and potental, y, at any pont s the Posson euaton: r y = (.4.) e where e s the permttvty of the medum. The Boltzmann Dstrbuton gves the dstrbuton of postve and negatve ons away from the partcle surface: n n + = n = n 0 0 Ê zef ˆ expá Ë kt Ê zef ˆ expá Ë kt (.4.) where n + and n are the number of catons and anons per unt volume wth charge +e and e respectvely. N 0 s the number of anons or catons far from the surface where the average electrostatc potental f s zero. T s the absolute temperature and k the Boltzmann constant, and z s the valence of electrons (Elmelech et al., 995).

28 Snce a flat nterface s beng consdered, the PossonBoltzmann expresson wll be used: d f = dx 0 zen e Ê snhá Ë zef ˆ kt (.4.3) The dmensonless parameters are: y = zef/kt and X = kx, where k s gven, for zz electrolytes, by: k e n0z = (.4.4) ekt The DebyeHuckel parameter, k, has the dmensons of recprocal length and plays a very mportant part n the electrcal nteracton between collodal partcles. Substtutng y and X: d y dx = snh y (.4.5) The GouyChapman theory has several shortcomngs. For example, measured capactances at certan nterfaces can be much lower than predcted by theory. Also, counterons concentratons close to charged nterfaces can become unreasonably hgh, even for moderate values of surface potental (Maruez, 994)..4. Dffuse Layer Model The surface charge densty s gven by: egtot s = (.4.6) S + a / K H where e s the protonc charge, G tot the total densty of chargeable stes, S a H demotes the surface actvty of the protons, and K s the dssocaton constant (Behrens et al., 999).

29 The charge of a latex surface can be related to the soluton propertes wth the socalled dffuse layer model (DLM). Ths model assumes that all the surface charge s located at the soluton nterface, whch s characterzed by the electrostatc surface potental y 0. The proton actvty at the surface s evaluated as: a = a exp( bey ) 0 (.4.7) S H H where ph=log a H, and b =k B T the thermal energy (Behrens et al., 999). These euatons defne a relaton between the surface charge and surface potental. Eulbrum reures the smultaneous fulfllment of a second chargepotental relaton that follows from the dstrbuton of moble ons n the dffuse layer. In a descrpton based upon the Posson Boltzmann euaton for electrolytes, the surface charge densty s of an solated partcle wth radus, R, can be expressed n terms of the surface potental y 0 as: ee0k È s = Ísnh( bey 0 / ) + tanh( bey 0 / 4) be Î kr (.4.8) where ee 0 s the total permttvty of the soluton and / k = ee0 /(N A be I) the Debye length, further nvolvng the onc strength I (N A s Avogadro s number). Wthout the second term on the rght sde, ths s ust the classcal GouyChapman result. The addtonal term was proposed by Loeb, Overbeek, and Wersema, and gves a frst order correcton for the surface curvature, accurate to wthn 5% of the true charge densty for any surface potental whenever the Debye length s smaller than the partcle dameter (Loeb, Overbeek, Wersema, 96). The last three euatons determne the eulbrum value of the surface charge s and potental y 0 at a gven ph (Behrens et al., 000)..4. Electrophoretc Moblty and Zeta Potental For a flat surface n a monovalent electrolyte, the electrostatc potental at a dstance x from the surface s related to the surface potental y 0 va: 3

30 4 y ( x) = arctan h[exp( kx) tanh( bey 0 / 4) (.4.9) be as follows from the ntegraton of the PossonBoltzmann euaton. The zeta potental was computed as the electrostatc potental z = y(x s ) at some dstance x s from the surface, correspondng to the thckness of an mmoble flud layer, adacent to the partcle surface (Behrens et al., 000). The outer end of ths mmoble layer, where the moton of flud relatve to the partcle sets n, s commonly referred to as the surface of shear..5 Aggregaton Three of the most basc propertes of a partculate system are the partcle composton, partcle sze dstrbuton, and partcle shape. As partcles undergo aggregaton, the partcle sze dstrbuton and shape of the partcle can change dramatcally dependng on the fractal nature of the aggregate structure. The foundatons of the rate of aggregaton start from the classc work of Smoluchowsk (97). It s convenent to thnk n terms of a dsperson of ntally dentcal partcles, whch, after a perod of aggregaton, contans aggregates of varous szes and dfferent concentratons. A fundamental assumpton s that aggregaton s a secondorder rate process, n whch the collson s proportonal to the product of concentratons of two colldng speces (Elmelech et al., 995). Threebody collsons are usually gnored n treatments of aggregaton they usually become mportant at very hgh partcle concentratons. The number of collsons occurrng between and partcles n unt tme and unt volume, J, s gven by: J = k n n (.5.) Where k s a second rate order constant, whch depends on a number of factors, such as partcle sze and transport mechansm. In consderng the rate of aggregaton, t must be recognzed that not all collsons may be successful n producng aggregates. The fracton of successful collsons s called the collson effcency and s gven the symbol a. If there s strong repulson between partcles, there wll not be any collsons that gve aggregates and a = 0. When there s no 4

31 sgnfcant net repulson or attracton between partcles, then the collson effcency can approach unty (Elmelech et al., 995). It s normal to assume n aggregaton modelng that the collson rate s ndependent of collod nteractons and depends only on partcle transport. Ths assumpton can be ustfed on the bass of the shortrange nature of nterpartcle forces, whch operate over a range whch s usually much less than the partcle sze, so that the partcles are nearly n contact before these forces come nto play. For the present, t shall be assumed that every collson s effectve n formng an aggregate (collson effcency, a=), so that the aggregaton rate constant s the same as the collson rate constant. The rate of change of concentraton of k fold aggregates, where k = + : dn dt k = = k  + > k = k n n n k  kk k = n (.5.) The frst term on the rghthand sde represents the rate of formaton of k aggregates by collson of any par of aggregates, and, such that + = k. Carryng out the summaton by ths method would mean countng each collson twce and hence the factor _ s ncluded. The second term accounts for the loss of k aggregates by collson, and aggregaton, wth any other aggregates. The terms k and k k are the approprate rate constants (Elmelech et al., 995). The above euaton s for rreversble aggregaton..6 Models for Collson Rates The determnaton of rate constants for aggregaton events s dependent on two factors: () the mechansm by whch partcle collsons occur, and () the presence of nterpartcle nteractons. In consderng the nature of partcle transport, and correspondngly partcle collson, there are three maor mechansms () Brownan moton (Perknetc aggregaton), () flud moton (Orthoknetc aggregaton), and (3) dfferental sedmentaton (Elmelech et al., 995). These mechansms wll be dscussed n the next few sectons. In all cases t s 5

32 assumed that partcles are sphercal and that the collson effcency s unty. Hydrodynamc nteracton wll be neglected n the next few sectons..6. Perknetc Collson Mechansm Small partcles n suspenson can be seen to undergo contnuous random movements called Brownan moton. Ths phenomenon occurs prmarly n partcles 00 nm to 000 nm (Peltomak, 00). Brownan moton s temperature dependent, and becomes ncreasngly mportant when partcles are one mcron or smaller. It s also mportant under condtons of hgh partcle concentraton (greater than 0 g/l), and low or no shear (Ernest, 995). The moton of perknetc aggregaton s shown n Fgure.4 (Ernest, 995). Fgure.4. Perknetc Aggregaton Smoluchowsk derved an expresson for collson freuency n ths case by consderng the dffusve flux of the partcles towards a statonary partcle (Agarwal, 00). Usng Fck s law for the number of partcles J gong through a unt area toward a reference partcle per unt tme: dn J ' = D (.6.) dr where D s the dffuson coeffcent of partcles, N s the number concentraton, and r s the radal dstance from the center of the reference partcle. Smoluchowsk defned a sphercal surface around a reference partcle (Fgure.5) so that any other partcle whose center 6

33 passes through that boundary wll be consdered to collde effectvely and produce coagulaton (Maruez, 994). R r R R Fgure.5. Boundary Condtons for Smoluchowsk s Fast Coagulaton Euaton The number of partcles gong through a sphere of radus r n unt tme s: dn J ' = (4pr ) D (.6.) dr 7

34 The dffuson coeffcent of a sphercal partcle s gven by the StokesEnsten euaton: D kt = (.6.3) 6 a m p where k s the Boltzmann s constant, T s the absolute temperature, a the partcle radus, and m the vscosty of the suspendng flud (Elmelech et al., 995). Smoluchowsk (97) calculated the rate of dffuson of sphercal partcles of type to a fxed sphere : J = 4pR D n (.6.4) Where D s the dffuson coeffcent of partcles of type and n s ther concentraton n the bulk expresson. Ths collson radus can be consdered the centertocenter dstance at whch a contact takes place. Ths s smply the sum of the partcle rad: R =a + a. (Elmelech et al., 995). In realty, the central sphere s not fxed, but s tself subect to Brownan dffuson. If the concentraton of partcles s n, then the number of I collsons occurrng n the unt volume per unt tme s: The rate constant for aggregaton s now: J = 4pR D n n (.6.5) k kt ( a + a ) = (.6.6) 3 m a a Ths euaton gves the rate constant for perknetc collsons. For partcles of approxmately eual sze, the collson rate constant becomes almost ndependent of partcle sze. 8

35 .6. Orthoknetc Collson Mechansm Collsons brought about by Brownan moton do not usually lead to the rapd formaton of large aggregates. Partcle transport brought about by flud moton can gve an enormous ncrease n the rate of nterpartcle collsons, and aggregaton brought about n ths way s known as orthoknetc collson. Ths type of collson becomes relevant between and 0 mcrons (Peltomak, 00). Smoluchowsk (97) also was the frst to study the rate of orthoknetc aggregaton due to unform lamnar shear forces. A dagram of the movement of partcles n shear flow s shown n Fgure.6 (Agarwal, 00). Z u dz b G=du/dz a Start Vew End Vew Fgure.6. Movement of Partcles n Shear Flow A unform lamnar shear feld s one n whch the flud velocty vares lnearly n only one drecton, perpendcular to the drecton of flow. Smoluchowsk assumed that partcles would flow n straght lnes and collde wth partcles movng on dfferent streamlnes accordng to ther relatve poston. The collson freuency depends on the szes of the partcles and on the velocty gradent or shear rate G (Elmelech et al., 995). By consderng a fxed central sphere of radus a and flowng partcles of radus a, t can be assumed that those movng partcles on streamlnes that brng ther centers wthn a dstance (a + a ). The collson freuency can then be calculated by consderng the flux of 9

36 partcles through a cylnder of radus R, the axs whch passes through the fxed sphere. The total flux towards the center partcle, J s ust twce that n one half of the cylnder and s gven by: J R 3 4Gn z ( R z ) dz = 4Gn R 0 = Ú (.6.7) The total number of collsons occurrng between and partcles n unt volume and unt tme s then smply: 4 J + 3 The rate constant for orthoknetc collsons between and partcles s: 3 = nn G( a a ) (.6.8) 4 k = G( a a ) (.6.9) Ths euaton shows that the rate s proportonal to the cube of the collson radus, whch has a maor effect on aggregate growth rate. As aggregaton proceeds and aggregate sze ncreases, the chance of capture becomes greater. The other maor flow feld of nterest s extensonal flow, whch s shown n Fgure.7 (Agarwal, 00). a Fgure.7. Movement of Partcles n Extensonal Flow 0

37 In ths case the collson freuency s gven by: J 6p 3 3 = g ext a N (.6.0) where g ext s the stran rate..6.3 Dfferental Settlng Mechansm Another mportant collson mechansm arses whenever partcles of dfferent szes and densty are settlng from a suspenson. Partcles of dfferent dameters settle at dfferent veloctes causng the faster movng partcles to collde wth slower movng partcles leadng to aggregaton. Ths type of collson mechansm usually becomes relevant at partcles of 000 mcrons n sze and larger. By balancng the forces of gravty, buoyancy and drag, the sedmentaton velocty of a partcle of radus a and densty r s n a medum of densty r s gven by Stokes euaton (Agarwal, 00): v g ( r s r) a = (.6.) 9 h The relatve velocty between two partcles of dameters a and a would be u = v v. The rate of N partcles through a cylndrcal cross secton of (a + a ) s gven by: dn dt = N p ( a + a ) ( v v ) (.6.) Usng the last two euatons, the resultng collson freuency, for partcles of eual densty s: Ê pg ˆ J ( ) ( ) 3 = Á r s r nn a + a ( a a Ë 9m ) (.6.3) where g s the acceleraton due to gravty, r s s the densty of the partcles and r s the densty of the flud.

38 .7 Comparson of Rates A summary of the collson mechansms descrbed n the prevous sectons are summarzed below: Perknetc: k = kt ( a + a ) 3 m a a Orthoknetc: 4 k = G( a + a 3 ) 3 Ê Dfferental Settlng: pg ˆ k ( )( ) 3 = Á r s r a + a ( a + a ) Ë 9m Usually t s assumed that the three mechansms of nterpartcle collsons are ndependent and when they operate smultaneously the aggregaton rates are addtve: J total =J Br +J Sedmentaton +J shear (.7.) The relatve magntudes of each contrbuton depend on the characterstcs of the suspenson and flow condtons. If the denstes of the partcles and the dspersng medum are nearly the same, contrbuton due to sedmentaton can be neglected. Other factors that lmt the effect of sedmentaton are hgh vscosty of the dspersng medum and the relatvely small sze of partcles (Bushell, 998). For a comparson of rates, t s convent to take one partcle of fxed sze and compute the varous rate constants as a functon of the sze of the second partcle. To compare the collson freuences due to shear flow that wth due to Brownan moton, ther rato s characterzed by the Peclet number (Agarwal, 00). If Pe >>, shear flow domnates, but f Pe<<, Brownan moton wll domnate. 3 4hga Pe = (.7.) k T B

39 As an example, one partcle s taken to have a dameter of mm and the other dameter vares between 0.0 and 0 mm. The shear rate s assumed to be 50 s and the densty of the partcles g/cm 3. All other values are approprate for aueous dspersons at 5 C. 0 Comparson of Collson Mechansm Rates] 0 m [ t n] a 3 t /s s n o C e t a R Perknetc Orthoknetc Sedmentaton Dameter of Partcle [mcron] Fgure.8. Comparson of Collson Rate Constants for Dfferent Transport Mechansms. As shown n Fgure.8 (Fsher, 998), t s clear that the perknetc mechansm gves the hghest collson rates for partcles less than 0.6 mm n dameter, but for larger partcles orthoknetc collsons and dfferental settlng become more mportant. As the sze of the second partcle becomes greater than a few mcrospheres, the collson rate due to sedmentaton ncreases sharply and becomes comparable to the shearnduced rate (Elmelech et al., 995). 3

40 .8 Collson Effcences In the collsons mechansms that were dscussed so far, t was assumed that all partcle collsons are successful n producng aggregates. In realty, ths s not the case, and a reduced collson effcency must be factored n. All that s needed to factor n ths rate s to nclude the collson effcency, a, nto the rate expressons. There remans the problem of assgnng a value to a, and ths wll be dscussed n the next few subsectons..8. Perknetc Collson Effcences (Stablty Rato) The effect of repulsve collodal nteractons on perknetc aggregaton s to gve a reducton n rate. In ths approach, a stablty rato, W, s used and s expressed as W = /a (the recprocal of the collson effcency). The stablty rato s smply the rato of the aggregaton rate n the absence of collodal nteractons. For cases where only van der Waals attracton and electrcal repulson need to be consdered, there s a energy barrer when the partcles approach the partcles. The stablty rato can be calculated by treatng the problem as one of dffuson n a force feld: W Ú 0 exp( ft / kt) = du ( u + ) (.8.) where f T s the total nteracton at a partcle separaton dstance d, and u s a functon of d and partcle sze. For eual partcles, u = d/a. It s also been cted as (Maruez, 63): ÊV ( l) ˆ dl W = + Ú Á ( R R ) exp (.8.) ( R + ) Ë kbt l R where V(l) s the total nteracton energy between partcles. To evaluate W, the ntegral n the last euaton has to be evaluated numercally, usng approprate expressons for the electrcal and van der Waals nteractons. Due to the exponental term, most of the contrbuton to the ntegral comes from a regon close to the maxmum. 4

41 .8. Orthoknetc Collson Effcences For collsons of nonbrownan partcles (> mcron), the Fuchs concept of dffuson n a force feld s not approprate and the relatve moton of partcles nduced by flud shear, or external gravty have to be consdered. It has been observed that aggregaton of otherwse stable collods can sometmes be acheved by the applcaton of suffcently hgh shear. For a gven suspenson, the collson effcency for Brownan aggregaton could be very dfferent from that of orthoknetc collsons. Smoluchowsk s theory makes the assumpton that partcles travel n straght traectores along streamlnes (streamlnes are not dsturbed by the presence of partcles. However, due to van der Waals, electrc doublelayer and hydrodynamc nteractons, partcle traectores devate from a straght lne as partcles approach each other. The dfference between rectlnear and curvlnear traectores s shown n Fgure.9 (Agarwal, 00). Fgure.9. Dfference Between Rectlnear and Curvlnear Traectores 5

42 The net velocty of a partcle s gven by the sum of velocty felds, whch are ndependent and superposable (Zechner and Schowalter, 977): U = u flow +u c (.8.3) The velocty feld u flow s due to the hydrodynamc flow whle u c s due to the presence of collodal forces. It should be noted that traectory analyss s vald for calculatng the collson effcency of doublet formaton resultng from two prmary partcles. No theory s avalable for the calculaton of the collson effcency of aggregated grow from doublets to trplets and larger aggregated because of the complexty of the hydrodynamcs nvolved. The collson effcency s shown as a functon of shear rate n Fgure.0 (Agarwal, 00). Partcle Dameter of Colldng Partcles, mcrons + + Collson Effcency Shear Rate (/s) Fgure.0. Collson Effcency as a Functon of Shear Rate and Partcle Sze 6

43 If the electrc double layer s completely suppressed by addng an electrolyte, numercal calculatons result n an easy expresson for collson effcency (Van de Ven and Mason, 977): 0.8 Ê A ˆ a = KÁ (.8.4) 3 Ë 36phga where K s a constant whose value s close to unty. Potann (99) suggested the followng expresson for aggregaton collson effcency: Ê ˆ ÁÊ d ˆ a ª ( ln( / )) Á (.8.5) d a ËË a 3 / where d s the dameter of the aggregate. Ths expresson shows that collson effcency decreases wth ncreasng d wth respect to prmary partcle sze a (Potann, 99). The collson effcency for orthoknetc collsons cannot be adeuately dscussed wthout reference to hydrodynamc nteracton..8.. Hydrodynamc Interacton The Smoluchowsk approach to aggregaton knetcs takes no account of the effect of the vscosty on the suspendng medum. The hydrodynamc or vscous effects can have a great effect on the aggregaton rates. As partcles approach very close t becomes ncreasngly dffcult for the lud between them to dran out of the gap, whch slows the aggregaton process. For orthoknetc and perknetc aggregaton, ths resstance wll prevent partcle contact completely unless a rapdly ncreasng attractve force such as van der Waals nteractons brngs the partcles together (Spelman, 978). The combned effect of van der Waals and hydrodynamc nteractons on the lmtng stablty rato of sphercal partcles n water s shown n Fgure. (Elmelech, 995). For comparson, the fnely dotted lne also shows the result n the absence of hydrodynamc nteracton. 7

44 3 Lmtng Stablty Rato Hamaker Constant/ 0 J Fgure.. Lmtng Stablty Rato W lm, as a Functon of the Hamaker Constant () It has been suggested that the collodal forces between agglomerates are determned by a couple of prmary partcles whereas hydrodynamc forces are euvalent to forces one would expect between two partcles of the sze of complete aggregates (Agarwal, 00). Thus, as the aggregate sze ncreases, hydrodynamc forces ncrease much more rapdly than collodal forces resultng n a much lower aggregate collson effcency than the prmary partcle collson effcency (Agarwal, 00). From the basc rate of aggregaton and the orthoknetc euaton, the orthoknetc rate of aggregaton becomes (Agarwal, 00): dn dt = an b (.8.6) 8

45 If the volume fracton of partcles f s assumed to reman constant then at any nstant, the number concentraton of partcles can be related to the partcle sze by 00). The last euaton can be ntegrated to obtan: 4 p b 3 N (Agarwal, 3 ln N N 0 4ayf = t (.8.7) p where N 0 s the number concentraton at tme t=0 and N s the number concentraton at any tme t. Ths euaton forms the bass for expermentally determnng the collson effcency. By followng the number concentraton wth respect to tme, the last euaton gves a straght lne durng the early stages of aggregaton. From the slope of ths lne, the expermental value of the collson effcency can be determned (Agarwal, 00)..9 Interpartcle Forces In collodal systems, there are three basc types of ntermolecular forces actng between molecules: () Van der Waals forces, () Electrostatc forces, and (3) sterc hndrance. The combnatons of these forces control the type and rate of coagulaton n partculate systems. Fgure. (Agarwal, 00) shows the potental energy curve for collodal nteracton. 9

46 Fgure.. Potental Energy Curve for Collodal Interacton.9. Electrostatc Repulsve Forces The GouyChapman model explans that collodal partcles are surrounded by an electrc double layer, where ons are dstrbuted such that the average electrostatc potental s represented by PossonBoltzmann euaton (PBE). When two partcles approach each 30

47 other, there s nterference between the electrostatc double layers whch wll result n an ncrease n energy for the two partcles..9.. SphereSphere Interactons A general formula for the force per unt area between two flat approachng double layers n a symmetrcal (zz) electrolyte was gven by Langmur (938) and Bell and Peterson (97): f R È Ê df ˆ = n kt Ícosh F Á ÍÎ Ëkdx (.9.) where n s the bulk number densty of ons, F = ze J / kt, and J denotes the potental at a dstance x from the plate. Integratng the force over dstance then gves the potental energy per unt area u R : u = f dx (.9.) R h Ú R where h s the separaton of the two surfaces. At eulbrum, f R should be eual everywhere. In the spheresphere double layer nteractons, the most common method for determnng nteracton energy, based upon the above euatons (Deragun, 934) s: V R Ú h paa = u Rdx a + a (.9.3) where h denotes the mnmum surfacetosurface separaton between the spheres. No analytcal expresson exsts for the electrostatc nteracton on the PossonBoltzman level. Whle computaton of the exact double layer energy stll reures a consderable numercal effort, the Deragun approxmaton s much more straghtforwardly obtaned as: 3

48 Ú Ú prn V ( h) = {cosh[ bey m( x)] } dxdy b h y (.9.4) where n s the number densty of partcles, and y m (h) s the electrostatc potental n the mdplane between flat surfaces wth the same densty of chargeable stes G tot. Ths mdplane potental can be calculated by solvng the set of euatons for s, y 0, and y m that s gven by : ee0k exp(bey m ) sn( v m) s = (.9.5) be exp( bey / ) cn( v m) dn( v m) m and y 0 = y m + ln cd( v m) (.9.6) be The functons sn(v m), cn(v m), dn(v m), and cd(v m) are Jacoban ellptc functons of argument v and parameter m n standard notaton, at v = kh /[ 4 exp( bey m / )] and m = exp( bey ). The above procedure yelds the electrostatc nteracton wth the surface m chemcal eulbrum mantaned at all partcle separatons,.e., full charge regulaton (Behrens et al., 000). When the surface potental s low (y 0 < 5 mv), the PossonBoltzmann euaton may be replaced by ts lnearzed verson, the DebyeHuckel euaton. The par nteracton energy then has the analytcal form: ( y 0 ) V ( h) = pree0 ln[ + D exp( kh)] D (.9.7) wth D = C C ) /( C + C ) takng values between and dependng on the ablty ( reg dl reg dl of the surfaces to adust ther charge densty upon approach (Behrens et al., 000) 3

49 .9. Van der Waals Van der Waals forces are present between all collodal partcles. Many physcal phenomena are conseuence of these forces. Examples nclude: the behavor of real gases, surface tenson of luds, adsorpton of gases on solds and aggregaton of collodal partcles. These forces arse from spontaneous electrcal and magnetc polarzatons, gvng a fluctuatng electromagnetc feld wthn the meda and n the gap between them. Van der Waals forces have several components that correspond to dfferent molecular nteractons, but the most relevant are (Maruez, 994): ) Debye or Inducton nteracton whch results from the attracton of permanent and nduced dpoles. ) Keesom or dpole orentaton that acts between permanent dpoles. 3) London or dsperson energy between nduced dpoles. There are two approaches to dervng van der Waals forces: () the classcal (mcroscopc) approach, and () the macroscopc approach. The classcal approach s due to Hamaker (937), the nteracton between macroscopc bodes s obtaned by the summaton of all the relevant ntermolecular nteractons (Elmelech et al., 995)..9.. Hamaker Expressons for Interactng Spheres The Hamaker expressons are based the assumpton of parwse addtvty of ntermolecular forces. The nteracton between two partcles s calculated by summng the nteractons of all molecules n one partcle wth all of the molecules n the other (Elmelech et al., 995). Each van der Waals expresson can be splt nto a geometrcal part and a constant A, the Hamaker constant, whch s related only to the propertes of the nteractng macroscopc bodes and the medum. A s usually n the range of 0 J to 0 9 J (Elmelech et al., 995). Assumng that two eual spheres of radus a are mmersed n a vacuum, the result s gven by: 33

50 V A = A a / h (.9.8) These expressons apply at close approach, and become ute naccurate at separatons greater than 0% of the partcle radus (Elmelech et al., 995). In many cases the nteracton energy s neglgble at larger dstance so that euatons lke the one above are acceptable for practcal purposes. For nteracton of meda through a lud the same expressons can be used, but wth a modfed Hamaker constant. A useful approxmaton for Hamaker constants of dfferent meda s the geometrcal mean assumpton: A = (.9.9) / ( AA ).9.3 Other Interpartcle Forces There are stuatons where two prncple forces alone (double layer and van der Waals nteractons) does not gve satsfactory agreement wth expermental results. For collodal partcles carryng adsorbed polymers, the forces are sterc or osmotc forces. The presence of an adsorbed layer can sometmes have a sgnfcant nfluence on the stablty of collodal dspersons through one or more of the followng mechansms: () by changng the electrcal double layer force ether drectly, n the case of polyelectrolytes or by causng a dsplacement of the Stern surface () by alterng the nterpartcle van der Waals attracton by modfyng the effectve Hamaker constant (3) by generatng addtonal nteractons ether due to desorpton of adsorbed molecules, or compresson and nterpenetraton of adsorbed layers Born Repulson Ths shortrange repulson (wthn nm) orgnates from the strong repulsve forces between atoms as ther electron shells nterpenetrate each other. A precse descrpton of the nteratomc structure must be based on uantum mechancal consderatons. However, a 34

51 number of smplfed approxmate euatons have been proposed, and of these, the most wdely used s the LennardJones mn potental: V LJ m n m n Ê n ˆ nm ÈÊ s Ê ˆ c ˆ s c = e E Á ÍÁ Á (.9.0) n m Ë m ÍÎ Ë r Ë r where r s the nteratomc separaton, s c the collson dameter, and e E the depth of the prmary energy well. The attractve part s due to van der Waals nteracton and the repulsve contrbuton s known as Born repulson Sterc Interacton Adsorbed layers can play a very mportant role n aggregaton phenomena. In the case of collodal partculate dspersons wth larger adsorbed amounts, polymers can gve great stablty, by an effect known as sterc stablzaton. The stablzng acton of such materals can be nterpreted n smple terms. As partcles approach suffcently close, the adsorbed layers come nto contact and any closer approach would nvolve some nterpretaton of the hydrophlc chans. Snce these chans are hydrated, overlap of these layers would cause some dehydraton and an ncrease n free energy and a repulson between partcles. For an ntal approxmaton, the repulson can be assumed to become nfnte as soon as the adsorbed layers begn to overlap, but zero at greater separatons..0 Populaton Balances Populaton balance models are used n stuatons where large numbers of enttes need to be tracked and modfed n the course of smulaton. For ths reason, populaton balance models have hstorcally been used n studes of crystallzaton, aerosol dynamcs, communton, heterogeneous phases, proten precptaton, latex reactors, and partculate systems. Even though the concept of contnuty can be extended to heterogeneous systems, many tmes the mathematcal descrpton cannot. Contnuous formulaton of a populaton balance for aggregaton produced n a suspenson can be formulated as a system of partal 35

52 ntegrodfferental euatons, but the soluton of such a soluton s dffcult numercally. Hounslow (988) proposed a general approach to crcumvent ths dffculty. He proposed a general populaton balance n dscrete form that can be expressed as a set of dfferental euatons as opposed to a system of PartalIntegroDfferental euatons (PIDs) for contnuous populaton balances (Maruez, 994). The dscrete populaton balance (DPB) ntroduces the dscretzaton at the formulaton stage rather than dscretze a contnuous formulaton (Maruez, 994). There are three sgnfcant drawbacks to the use of populaton balances. The frst drawback s the numercal dffculty nvolved wth usng the PIDs, whch result from usng populaton balances. The second drawback s that boundary condtons have to be known and set before a soluton can be attempted. The thrd drawback s that populaton balance models sometmes may agree wth certan expermental sets, but do not represent the true fundamentals of the system under study (Fsher, 998). Consder a dstrbuton of enttes, G, dstrbuted throughout an arbtrary regon of dmensonal space (xyz) known as V, and through some regon R of property space, x m. G = G x, y, z, x, x,..., x, ) (.0.) ( n t where x, y, z are spatal coordnates, and x,, x,..., x m represent property coordnates over whch the enttes are dstrbuted. (Fsher, 998). The property coordnates could represent sze, age, or composton of partcles (Maruez, 994). One way of lookng at the pont populaton densty functon, G, over V and R s that: GdR' V ' YdR' V ' Ú R' + V ' (.0.) s the fractonal porton of the total populaton dstrbuton whch fnd themselves n the regons x to x + dx, y to y + dy, z to z + dz, x to x + dx,, x n to x n + dx n. It s 36

53 assumed that the varous components of the spatal and property regons R and V are contnuous n nature and the propagaton of these spatal coordnates and propertes can be defned as ve x, ve y, ve z, ve, ve,, ve m, where: ve Ê dx = Á Ë dt ˆ (.0.3) Contnung, t s necessary to defne a manner n whch to modfy the number densty dstrbuton G. Ths manner of modfcaton s through the brth and death functons operatng through the space and property regon V and R. The brth functon s the number of enttes created at a pont V and R s: B = B x, y, z, x, x,..., x, ) (.0.4) ( n t The unts of the brth term are enttes per volume per second per unt property enttes. The death functon s the number of enttes destroyed at a pont n V and R s: D = D x, y, z, x, x,..., x, ) (.0.5) ( n t The brth and death functons are pont functons of the state of the system. The brth and death functons are governed by the physcal state of the system, and are used to characterze condtons wth the changng number dstrbuton G (Fsher, 998). Let us redefne the combned regonal and property space, P, as the partcle state vector, whch conssts of both R and V. The conservaton relatonshp s: Accumulaton=Input Output + Net Generaton (.0.6) In terms of the nomenclature defned above: d dt Ú GdP = Ú ( B D ) dp + Ú ( B D ) dp Ú ( Bn P P Dn ) dp (.0.7) P P 37

54 where dp = dxdydzdx dx... dx m The last euaton can be expanded to: d dt Ú P GdP = d dt Ú x Ú y Ú z Ú x... x Ú m Gd x m... dx dzdydx (.0.8) Ths euaton can be expanded further to a more useful term by repeatedly dfferentatng nsde the ntegrals usng the general rule of Lebntz. The euaton now becomes: m Ï T Ú Ì + ( vex, G) + ( ve y, G) + ( vez, G) + Ó t x y z = P  ( ve, G) + D B dp = 0 x (.0.9) Consderng that the regon P was arbtrarly defned, t s more compact to rewrte the euaton n vector notaton: G r + ( veg) + t m  = ( veg) + D B = 0 x (.0.0) where the vector ve s the average velocty vector of the partcles n the spatal regon V. The thrd term n the euaton does not show an absolute conservaton n accordance wth ther defntons as propertes. However, the combnaton of the thrd term wth the brth and death terms can be consdered much lke a source and snk for entty propertes. Fgure.3 (Bggs, 000) shows a pctoral representaton of populaton balance modelng. 38

55 Fgure.3. Populaton Balance Modelng. Aggregate Structure In the smplest case of eual spheres, a par of partcles would form a dumbbell. A thrd partcle can attach n several dfferent ways, and the greater number of aggregates, the number of possble structures rapdly ncreases. Fgure.4 (Elmelech, 995) shows dfferent partcle confguratons for an ncreasng number of prmary partcles. In most aggregatve processes, the number of partcles wthn a partcle cluster may be on the order of hundreds or thousands, therefore, an understandng of the nature of the cluster structure s mportant. 39

56 Doublet Trplets Quadruplets Fgure.4. Arrangement for Dfferent Number of Prmary Partcles Aggregates are recognzed as fractal obects. Fractal aggregates wll be dscussed n detal n the next few sectons... Fractal Geometry Eucldean geometry s prmarly the study of straght lnes and smooth curves. In nature, (the earth) straght lnes and smooth curves are the excepton rather than the rule. Rugged profles exst everywhere, from the clouds n the sky, to the outlnes of trees and mountan ranges, and the stars n the sky. Fractals are characterzed by a nonnteger power law dependence of a measurable uantty upon the length of the obect (or upon tme) (Mandelbrot, 983). The areas and permeters of fractals are nfnte. The farther one zooms n on the permeter of a fractal, the more bumps, curves, folds, spkes, and curls you see. Mandelbrot (983) argued that n cases such as ths, the exponent can and should be nterpreted as a dmensonalty. To smplfy ths concept, consder the Koch curve (Mandelbrot, 983). The steps shown n generatng a Koch curve are shown n Fgure.5 (Shabarshn, 00). One begns wth a straght lne shown at n = 0 n the fgure, known as the ntator. Each secton s then 40

57 replaced by decsve element defned on the fgure as n =. The straghtlne sectons of the frst generaton curve are then replaced by copes of the generator, and have been shrunk by a factor of three. Four copes are reured, leadng to the secondgeneraton prefractal curve shown at n =. The operaton s repeated ad nfntum, replacng straght sectons by copes of the generator, scaled down by a factor of three wth every succeedng generaton. Fgure.5. Constructon Process of the Prefractal Koch Curves The Koch curve s the lmtng curve generated as n. It s easy to see that the topologcal dmenson of all generatons of the prefractal Koch curves (and by extenson of the Koch curve also) s eual to one. Ths s done by settng <h> = w a where the exponent a s between 0 and (Russ, 994). The fractal dmenson can be calculated, and the rougher values correspondng to the smaller values of a have hgher fractal dmensons. The uanttatve relatonshp s D = a (Russ, 994)... Fractal mass scalng The frst scalng feature of aggregates that receved wde attenton n lterature, and the one that remans of great nterest to those attemptng to characterze aggregaton s the mass (or 4

58 number) scalng (Bushell, 998). For a large number of aggregates, the mass s plotted aganst aggregate sze, the plot may be lnear, but wth a nonnteger slope. For aggregates, the slope of the lne d F s called a mass fractal dmenson The lower the fractal dmenson, the more open the aggregate structure. The relatonshp between aggregate mass, M, and lnear measure of sze, R, s: D M µ R f (..) The mass fractal dmenson can range between and 3, 3 beng the deal stuaton of coalescng lud drops (Elmelech, 995). Fgure.7 shows a threedmensonal representaton of several aggregates wth a fractal dmenson rangng from.5 to.5 (Bushell, 998). Fgure.8 (Fsher, 998) shows the relatonshp between the coalescence model and the fractal aggregate model for varous values of D f wth a begnnng porosty of zero, whch would relate to the stuaton of aggregaton from a monodspersed orgnal populaton (Fsher, 998). As the fgure shows, the relatvely loosely packed fractal aggregates that the effectve aggregate radus s greatly ncreases. If the relatonshp n the above euaton apples over a wde range of aggregate szes, then ths mples that aggregates have a selfsmlar structure, whch s ndependent of the scale of observaton. The selfsmlar structure means that a small part of the fractal contans nformaton about the whole (Shabarshn, 00). Ths concept s llustrated n the Fgure.6 (Elmelech, 995) below, where the fundamental unt s assumed to be a trplet of eual spheres. 4

59 Fgure.6. Schematc Illustraton of SelfSmlar Aggregate Structure The essence of selfsmlar aggregates s that there s a contnuum of levels from largescale structures down to ndvdual prmary partcles. Early studes on aggregaton were based only on the random addton of sngle partcles to growng clusters. These gave farly dense structures wth d F of about.5 (Elmelech, 995). The early studes were not realstc because snglepartcle addton does not occur n nature. In realty, aggregaton occurs as a result of clustercluster encounters. Computer smulatons and expermental studes on a range of model collods show much more open structures wth a fractal dmenson of about.8 (Ln et al., 989). These smulatons assume that partcles attach permanently to other partcles at frst contact; the process s called dffusonlmted aggregaton (DLA). 43

60 Fgure.7. 3D Representaton of Several Aggregates wth D f =

61 Fgure.8. Comparson between the Coalescence and Fractal Aggregate Model Euaton.. can take on two subtly dfferent meanngs wthout losng ts valdty. The frst meanng s the scalng of the mass (number of partcles) contaned wthn aggregates of dfferent szes wthn a cluster polydsperse aggregatng system. As a conseuence of the (dentcal) fractal structure of the aggregates, the mass of each aggregate s related to ts lnear sze by the euaton. The second meanng s n terms of the structure wthn any ndvdual aggregate. If we pck an arbtrary partcle wthn an aggregate and center an magnary sphere upon t, the number of other partcles enclosed wthn the magnary sphere s related to the lnear sze of the sphere, R, by the last euaton. Ths relatonshp s only asymptotcally correct n the lmt of large aggregates and s ute naccurate for aggregates of only a few partcles. To be a bt more specfc about the frst meanng of euaton, t s often wrtten as: 45

62 N Ê Rg ˆ = kg Á r Ë 0 D f (..) where R g s the radus of gyraton of the aggregate, r 0 s the radus of the prmary partcles and k g s known as the power law prefactor (Bushell, 998). The subscrpt g s added to the power law prefactor to clearly assocate t wth lnear aggregate sze defned n terms of the radus of gyraton, whch s the rootmeansuare dstance of the mass elements from ther center of mass. Real processes that form natural fractals probably mpose even more strct lmtatons on the range of possble fractal dmensons. A smple computer smulaton of collodal aggregaton that has been extensvely studed s the clustercluster aggregaton model. Ths type of smulaton allows partcles and clusters to dffuse accordng to specfed traectory (usually Brownan or lnear) and stck rreversbly wth no restructurng at ther pont of contact. Ths type of smulaton mposes natural lmts on the resultng fractal dmenson such that ~.8. Df. ~.. The lower value comes about when a collson between clusters always results n the formaton of a bond. Ths s known as the dffusonlmted cluster aggregaton or DLCA lmt and produces ute tenuous, wspy structures. The hgher value s a result of collsons almost never formng bonds, so that all physcally possble conformatons between clusters have an eual chance of formng a bond and thus a new aggregate. Ths s known as the reactonlmted cluster aggregaton or RLCA lmt and produces structures that are stll ute tenuous and rugged but notceablely more compact and strongerlookng than DLCA aggregates (Bushell, 998). Despte the smplstc algorthm, there s good evdence that ths type of model descrbes ute accurately a range of collodal aggregates that can be observed n the laboratory. Thouy and Jullen (996) reported that ther clustercluster aggregaton model wth fractal dmenson as a tunable parameter could not produce structures wth a fractal dmenson hgher than about.55 n 3D space because the geometry of the clusters prevented them beng placed close enough together to produce hgher fractal dmensons. 46

63 ..3 Packng Densty An nterestng conseuence of fractal mass scalng s that the densty of the fractal obect s not constant as s usually consdered the case for everyday sold structures. The mass of the obect s gven by the last euaton. One way of defnng the volume of the obect s the volume of the smallest sphere (radus R e ) that encloses the structure. Ths bulk volume s smply 4/3ϖR 3 e. The packng densty s gven by the aggregate mass dvded by bulk volume (Bushell, 998): D f e 3 e R D f 3 µ = Re (..3) R..4 Coordnaton Numbers The mass fractal dmenson s far from suffcent for complete characterzaton of aggregates. Aggregates of fne partcles are not really fractal n the strct sense of the word, snce ther fractal scalng s only observed over a fnte range of length scales. They can be called natural fractals (Mandelbrot, 983). At small scales we observe the nonfractal subunts of whch the aggregate s composed. Here we encounter complextes assocated wth the shortrange nonfractal order of adacent partcles. Even when the prmary partcles are as smple as monodsperse spheres we stll have to specfy frst, and dependng on the purpose at hand possbly second and thrd coordnaton shell numbers to adeuately uantfy the aggregate structure. The mathematcal model does not nclude coordnaton numbers, but t s mentoned to provde more a more thorough explanaton of fractal nature...5 Fractals n Partcle Aggregaton The mass wthn a dstance l of the surface, or for a threedmensonal partcle, wthn a radus l of the center of gravty, ncreases as l E for a sold obect, where E s the Eucldean dmenson of the space ( for a surface, 3 for a sold). For a fractal structure, the exponent s less. For a classc dffusonlmted aggregate as defned above, the relatonshp s mass µ l.73 n D, mass µ l.5 n 3D (Russ, 994). 47

64 Changes n the rules for moton of the partcles or ther stckng probabltes alter the appearance and dmenson of the clusters. If the stckng probabltes are altered from 00% to 5%, 5%, and % respectvely, the cluster becomes progressvely more compact, and the mass dmenson ncreases (Russ, 994). Measurng the real fractal dmenson of the cluster, or surface of the cluster s a dffcult problem. It has been proposed that as the stckng probablty becomes low, the proecton of such a cluster wll have a boundary that s fractal. A modfcaton of the aggregaton modelng technue s to use weghtng functon. Each tme a random partcle reaches a ste adacent to the partcle that s already part of the cluster, t s counted, but not necessarly added to the cluster. The number of tmes a partcle must reach a ste before one s allowed to stck there s a weghtng value that can be adusted to alter the appearance and dmenson of the growng cluster. Larger weght values produce ncreasngly needlelke dendrtc shapes.. Summary Ths chapter has examned the basc theory of aggregaton processes n collodal systems. The nature and defnton of collodal systems, model structure for collodal partcles, the nature of surface chargng, and a dscusson of nterpartcle forces were covered n ths chapter. The theoretcal bass for aggregaton ncludes the topcs of Smoluchowsk aggregaton knetcs, descrpton of collson mechansms, and the defnton of the stablty ratos. Ths secton contaned the background nformaton that s necessary to understand the theory behnd the mathematcal model presented n Chapter 3. 48

65 CHAPTER 3: MATHEMATICAL MODEL FOR AGGREGATION KINETICS 3. Introducton Chapter presented an overvew of the basc topcs for modelng aggregaton phenomena. The model presented n ths chapter can be appled to a varety of aggregaton processes, and t focuses on the actual mplementaton of the deas presented n Chapter. A more detaled descrpton and comparson of the collson mechansms, nterpartcle forces, stablty ratos, and populatons balance model s ncluded n ths chapter. The dscreton method mplemented for the populaton balance model s also dscussed. 3. Interpartcle Forces The two maor contrbutors to the nterpartcle nteractons are van der Waals attractve forces and electrostatc repulson nteracton energy. The total nteracton energy between the two aggregatng partcles wll be represented by the sum of these two forces, whch can be expressed mathematcally as: V T (H) = V R (H) + V A (H) (3..) Where V T s the total nteracton energy for the calculaton of the stablty rato, V R s the electrostatc repulsve nteracton energy, and V A s the van der Waals attractve nteracton energy. The accuracy of the stablty ratos and the correspondng the aggregaton rate constants wll be determned by the nterpartcle forces used for ther calculaton. 3.. Van der Waals Attractve Interacton Energy Ths mathematcal model wll use the Hamaker theory to model the van der Waals forces. Hamaker obtaned the followng expressons for the potental energy of attracton resultng 49

66 50 from van der Waals forces for dfferent geometres. Interacton of plates of eual thckness (Maruez, 994): Í Î È = ) ( l l p l l H l V A (3..) where H s the Hamaker constant for the materal, l s the plate thckness, l s the plate separaton, and V A (l) s the attractve potental of dfferent rad. Hamaker derved the attractve potental energy for eual spheres of radus a and a and surfacetosurface dstance H s gven by: ˆ Á Á Ë Ê ˆ Á Á Ë Ê = 4 ln 4 6 ) ( a a h a h a h h a h a h a a h a h a h a a h a h a h a a A H V A (3..3) where s d d h + + = s the dstance between the partcle center, and s s the separaton dstance between surfaces as shown n Fgure 3. (Ahmad, 00). The assumpton of complete addtvty s a serous defcency and the resultng expressons always overestmate the nteracton. h d s d Fgure 3.. Contact of Two Dssmlar Spheres

67 Fgure 3. shows the potental energy of nteracton between partcles of dssmlar rad (Fsher, 998). The xaxs of the graph shows the partcle separaton, whle the yaxs shows the radus of one of the partcles whle the other s held constant at 00 nm. As the partcle msmatch becomes greater, the nteracton energy tends to saturate whle the nteracton s domnated by the larger partcle. Fgures 3.3 shows the nteracton energy as a functon of partcle surface separaton. Fgure 3.4 shows the nteracton energy as a functon of partcle dameter. Fgure 3.. Potental Energy of Interacton for Dfferent Partcle Szes 5

68 Fgure 3.3. Potental Energy of Attracton for Dfferent Partcle Szes Fgure 3.4. Potental Energy of Attracton for Dfferent Partcle Szes 5

69 3.. Electrostatc Repulson Interacton Energy In order to calculate the nteracton energy between two partcles due to the overlap of ther double layers, t s necessary to know the electrostatc potental profle that develops between the nteractng partcles. Ths s accomplshed wth the PossonBoltzmann euaton (PBE), whch was ntroduced n Chapter. Soluton of PBE wll yeld the electrostatc potental profle between two surfaces for the gven boundary condtons. Adeuate ntegraton of the electrostatc potental wll provde the repulsve potental energy between the nteractng surfaces. Unfortunately, the PBE does not have an analytcal exact soluton for even the smplest system of two plates wth constant surface potentals. Analytcal solutons to the PossonBoltzmann euaton are only avalable for certan values of the surface potental and doublelayer thckness (Fsher, 998). Fortunately, aggregaton processes are conducted wthn the range where these analytcal values are vald. The nteracton energy s calculated n two dfferent ways n the mathematcal model: lnearzed PossonBoltzmann euaton, and the nonlnearzed PossonBoltzmann euaton. Fgures 3.5 and 3.6 show the potental energy of repulson for dfferent partcle szes wth the lnearzed PossonBoltzmann euaton, whch s descrbed n Secton

70 Fgure 3.5. Potental Energy of Repulson for Dfferent Partcle Szes Fgure 3.6. Potental Energy of Repulson for Dfferent Partcle Szes 54

71 3... Lnearzed PBE The lnearzed PossonBoltzmann euaton s based on the soluton of Hogg, Healy, and Fuerstenau (Hogg et al., 966), and provdes a soluton for the electrostatc nteracton energy between two dssmlar sphere and uneual surface potentals (Fsher, 998): e a + È a( y y ) y y H ) = Í 4( a + a) Î( y + y Ê+ exp( kh ) ˆ ln Á + ln( exp( k ) Ë exp( kh ) ( 0 V R H )) (3..4) Ths expresson s lmted to ka 5, and values of the surface potentals up to 70 mv (Fsher, 988). Ths form of the electrostatc nteracton energy euaton wll be abbrevated HHF from now on Nonlnear PBE The nonlnear PBE form of electrostatc repulson wll be used when the soluton and system parameters are not wthn the range of the HHF soluton. The orgnal algorthm was based upon the work of Chan, Pashley, and Whte (980). Ths algorthm has several dsadvantages such as the fact that t was vald only for dentcally charged surfaces, where the electrostatc potental goes through a mdpont between plates (Fsher, 998). Ths method was mproved by Maruez (994), and s based on the numercal soluton usng the RuttaKutta ntegraton of the PossonBoltzmann euaton for the flat plate coupled wth the method of Papadopoulos and Cheh (984). A uck overvew of the method of Papadopoulos and Cheh (984) s presented below. Papadopoulos and Cheh (984) solved the lnearzed PossonBoltzmann euaton usng a modfed form of Deragun s approxmaton. A smlar system of rngs was used to represent the sphercal surfaces of partcles, but nstead of usng the shortest dstance between plates, they consdered them separated by a dstance eual to the arc of a crcle perpendcular to both rngs. For two spheres of rad a and a, they obtan the followng expresson: 55

72 V R max = ( H ) pa Ú V s d + pa ÚV fd f f ( )sn f sn (3..5) 0 where V f s the nteracton energy per unt area of two parallel plates separated by a dstance eual to the length of the arc of a crcle, s, perpendcular to both rngs. S, max, and f max are represented by: max + f max = p (3..6) + f Ê sn a ˆ a a s = Áa + + R (3..7) sn Ë cos R cos A graphcal representaton s shown n Fgure 3.7 (Maruez, 994). arclength (s) r f r f H R Fgure 3.7. Graphcal Depcton of the Method of Papadopoulos and Cheh Maruez (994) used exact numercal solutons based on a method of collocaton of fnte elements nstead of the lnearzed form of the PossonBoltzmann euaton for flat plates (Fsher, 998). A uck overvew of the dervaton of the formula s presented below. For a more detaled dscusson of the nonlnear PBE, refer to Appendx A. 56

73 The dervaton starts wth a dmensonless form of the PossonBoltzmann euaton n one dmenson: d Y dy = k snh( Y) (3..8) where Y s the dmensonless surface potental, and y s the dstance from the orgn. The boundary condtons for the case of a constant surface potental (Fsher, 998): Y(0) = Y Y(Xd) = Y (3..9) where Xd s the separaton between plates. Integratng euaton gves: Ê dy ˆ Á Ë dx = cosh( Y) + C (3..0) where C s an unknown ntegraton constant and x = y / k. If Y has a mnmum Y mn the value of the ntegraton constant s (Fsher, 998): mn C = cosh( Y ) (3..) The last euaton s solved wth the boundary condtons for both the monotonc and concave case. The lmtaton on the soluton s that Y > Y > 0. For every value of C there s a dstance between the plates gven by (Fsher, 998): Y Y cosh( Y) + C Y cosh( Y + Y dy dy Xd = Ú + Ú (3..) C ) mn mn f Y has a mnmum and by Ú Y dy Y cosh( Y + C ) f Y s monotonc. 57

74 Let x ref be the dstance between the plates resultant to a partcular case n whch (Fsher, 998): dy dx ( Y ) = 0 Wth Y mn =Y (3..3) Whch can be thought of as Y reachng a mnmum exactly at the boundary condton. In order to compute Xd for the monotonc case, the followng dfferental euaton needs to be ntegrated: dy dx = cosh( Y) + C Y(0) = Y (3..4) untl Y Y < TOL, whle n the convex case the next two dfferental euatons have to be ntegrated: dy dx = cosh( Y) + C Y(0) = Y (3..5) untl dy/dx < TOL and dy dx = cosh( Y) + C untl Y Y < TOL. TOL s a small value that s chosen as the lmt of accuracy to whch the computatons are conducted (Fsher, 998). In both cases C has to be determned usng a search algorthm untl the separaton between plated correspondng to the ntegraton chosen matches wthn a gven tolerance the separaton between the sets set by the problem (Fsher, 998). The followng lmts are used: Concave Case: cosh (Y ) C (3..6) Monotonc Case: cosh (Y ) C 0cosh (Y ) (3..7) 58

75 The selecton of these lmts s dscussed n Appendx (Fsher, 998). The lmt for the monotonc case s selected on the bass of the hghest potentals and the shortest dstances most lkely encountered (n actualty the upper lmt s nfnty) (Fsher, 998). The numercal ntegraton method that s used s RungeKuttaFehlberg wth error control (Burder and Fares, 985), whle the search algorthm for the ntegraton constant C s Brent s algorthm (Press et al., 984). A flowsheet of the entre method s presented n Fgure 3.8 (Maruez, 994). 59

76 Input Xd, ps, ps C = cosh(ps) Integrate euaton 3..4 representaton by RKF monotoncally to get x_ref Cmn=C cmax=0cosh(ps) xmax = 5 TOL =.0e6 er=0 yes Xmax < xref no c_r =.0 c_l = cmn c_r = cmax c_l = cmn Whle er > TOL Whle er > TOL Compute next C usng Bsecton Compute next C usng Bsecton Integrate euaton 3..4 by RKF for concave case to get xc Integrate euaton 3..4 by RKF for concave case to get xc yes Xc > xmax no yes Xc > xmax no C_r = C C_r = C C_r = C C_r = C er = xmax xc c_max = C er = xmax xc c_max = C er = 0.0 Contnue to next page 60

77 Contnue from prevous page er = 0.0 Xd < x_ref c_r = cmax x_r = xmax c_l = cmn C = 0.5 * (c_r + c_l) c_r = cmax c_l = cmn Whle er > TOL Whle er > TOL Integrate euaton 3..4 by RKF for concave case to get xc Compute next C usng bsecton er = Xd xc Integrate euaton 3..4 by RKF for concave case to get xc yes xc > xd no C_r = C C_r = C er = Xd xc Integrate euaton 3..5 by RKF for Concave case to get Vf Fgure 3.8. Flowsheet for Numercal Integraton of NonLnear PBE 6

78 3...3 Comparson of Lnear and Nonlnear PBE Ths secton wll nvestgate the range of applcablty for the lnear and nonlnear PBE. Three dfferent solutons wll be compared:. OHW: Deragun s ntegraton wth seres approxmaton presented by Ohma, Healy, and Whte (Ohsma et al., 994).. PHHF: Papadopoulos ntegraton wth Hogg, Healy, and Fuerstenau (966). 3. PRKF: Papadopoulos ntegraton wth numercal soluton. The results are plotted wth dmensonless dstance as the xaxs, and the dmensonless potental as the yaxs. The dmensonless dstance s the partcle surfacetosurface separaton, H, tmes the doublelayer thckness, K. The dmensonless potental s the potental dvded by kt, where k s the Boltzmann constant and T s the absolute temperature. These dmensonless groups are used so that comparsons can be made wthout takng nto account the effects of onc concentraton. HHF has lttle loss of accuracy f the values of both surface potentals are low, and the doublelayer thckness (Ka) s greater than or eual to 5. When surface potentals are large, or the partcle szes are small, the method of Papadopoulos makes a sgnfcant dfference. The Papadopoulos technue s needed for smaller partcles. Fgure 3.9 shows the varaton between HHF and Papadopoulos wth larger surface potentals. Fgures show examples under whch the HHF approxmaton can be used wth lttle loss of accuracy. Fgure 3.3 shows both eual and uneual small partcle szes as well as eual and uneual low surface potentals wth the lnear and nonlnear forms of the PossonBoltzmann euaton. 6

79 Fgure 3.9. PBE Interacton Energy Comparson 63

80 Fgure 3.0. PBE Interacton Energy Comparson 64

81 Fgure 3.. PBE Interacton Energy Comparson 65

82 Fgure 3.. PBE Interacton Energy Comparson 66

83 Fgure 3.3. PBE Interacton Energy Comparson 3.3 Collson Mechansms The three collson mechansms that were presented n Chapter are: perknetc, orthoknetc, and sedmentaton. They are summarzed below: Perknetc: k kt ( a + a ) = (3.3.) 3 m a a Orthoknetc: 4 k = G( a a ) (3.3.) Ê Dfferental Settlng: pg ˆ k ( )( ) 3 = Á r s r a + a ( a + a ) Ë 9m (3.3.3) 67

84 Fgures 3.3 through Fgures 3.5 shown the general form for the three collson mechansms. All of these fgures used the case of polystyrene n water at 5 C for demonstraton purposes. Fgure 3.3 shows the perknetc rate constant as a functon of partcle sze. From euaton 3.3., t s apparent that the aggregaton rate constant for the perknetc case vares proportonally wth temperature and nverse proportonally to solvent vscosty. Fgure 3.4. Perknetc Rate Constants Fgure 3.5 shows the orthoknetc case as a functon of partcle sze for a shear rate of 50 s. If euaton 3.3. s examned, t s obvous that the aggregaton rate constant vares proportonally wth average shear rate G. 68

85 Fgure 3.5. Orthoknetc Rate Constants Fgure 3.6 shows the dfferental settlng rate constants as a functon of partcle sze wth a partcle densty of.5 g/cm 3. Ths fgure demonstrated that the aggregaton rate constant approaches zero for partcles of the same sze. Euaton shows that the rate s nversely proportonal to the solvent vscosty, and vares proportonally to the dfference of partcle densty and soluton densty. 69

86 Fgure 3.6. Dfferental Settlng Rate Constants The dfferental settlng collson mechansm wll not be used n the experments conducted wth the mathematcal model because of the range of partcle szes s 0 9 m (0 A) to 9 * 0 6 m ( mm) n sze. The dfferental settlng collson mechansm s prmarly used for partcle szes of 00 mcrons and above. 3.4 Stablty Ratos The stablty ratos dscussed n Chapter are summarzed below: Perknetc: W Ú 0 exp( ft / kt) = du ( u + ) (3.3.4) Orthoknetc: 3 / Ê ˆ ÁÊ d ˆ a ª ( ln( / )) Á (3.3.5) d a ËË a 70

87 3.4. Perknetc Stablty Ratos The ntegraton of the stablty ratos for the perknetc collson mechansm s computatonally dffcult n two ways: () the lmts to whch the ntegraton must be carred out, and () the general form of the ntegrand. The lmts of ntegraton cover the range from partcle contact to nfnty. Tradtonal technues such as the trapezodal rule would reure many ntegraton steps to cover the entre range of ntegraton (Fsher, 998). To decrease the computatonal tme, and stll mantan a hgh degree of accuracy, Gaussan uadrature s used. Breakng up the range nto many subntervals usng a geometrc dstrbuton can also ncrease the computatonal effcency. The functon wll be ntegrated as follows: b Ú a a+ h Ú L Â Ú = L f ( x) dx = f ( x) dx + f ( x) dx where L = a + hâ a r g = 0 (3.3.6) r g s the geometrc rato used to ncrease the sze of the ntervals progressvely. Each subnterval uses Gaussan uadrature to ncrease computatonal accuracy. Fgures 3.7 and 3.8 show a map of the stablty ratos calculated usng the HHF approxmaton and the RKF numercal ntegraton soluton for the PossonBoltzmann euaton respectvely. For these examples, polystyrene n water s used agan wth a 5 mv Zeta potental. 7

88 Fgure 3.7. Stablty Ratos for HHF Fgure 3.8. Stablty Ratos for RKF and Papadopoulos 7

89 3.4. Orthoknetc Stablty Ratos The orthoknetc stablty ratos take hydrodynamc forces nto account n the calculaton of the aggregaton coeffcents based on the orthoknetc collson mechansm. Fgure 3.9 shows the map of stablty ratos wth a prmary partcle sze of 0.0 mcrons. Fgure 3.9. Stablty Ratos for Orthoknetc Collson Mechansm 73

90 Fgure 3.0 shows the map of stablty ratos wth a prmary partcle sze of 0.55 mcrons. Fgure 3.0. Stablty Ratos for Orthoknetc Collson Mechansm 3.5 Map of Aggregaton Coeffcents In Chapter, the aggregaton kernal s defned n terms of the contnuous populaton balance model for aggregaton. The aggregaton coeffcent, b, s the rate constant for two partcles. The aggregaton coeffcent s defned as (Fsher, 998): k f b, = (3.4.) W, where k f s the rate of aggregaton derved from the approprate collson mechansm, and W, s the stablty rato calculated from the models for nterpartcle forces. Fgures 3. through 3.4 show examples of the fnal map of aggregaton coeffcents used n the soluton of mathematcal model for the correspondng collson mechansm. The rate of aggregaton used for ths calculaton was calculated the Fgures 3.4 through 3.6, and the stablty ratos were calculated n Fgures 3.7 and 3.9. Notce that as the partcle sze 74

91 ncreases, the rate constant for aggregaton decreases, whch cause the selfsmlar sze dstrbuton often found n collodal systems. Fgure 3.. Log of Aggregaton Coeffcents for Perknetc Collson Mechansm 75

92 Fgure 3.. Log of Aggregaton Coeffcents for Orthoknetc Collson Mechansm Fgure 3.3. Log of Aggregaton Coeffcents for Dfferental Settlng Mechansm 76

93 3.6 Populaton Balance Formulaton The populaton balance euaton from Chapter s: dn( L) dt = B( L) D( L) (3.5.) where the brth and death functons are: B( L) = L L Ú 0 b ( L l ) /, l n ( L l ) 3 3 / 3 ( L l ) / 3, l n( l) dl (3.5.) Ú 0 D ( L) = n( L) b ( L, l) n( l) dl (3.5.3) The dscretzaton of Ltster et al. (995), and Wynn (996) s used to complete the mathematcal model Dscretzaton Method The accuracy of the populaton balance models are very mportance for the modelng of the aggregaton process because they are the bass of the mathematcal model. The methods of dscreton used for ths mathematcal model are more general n applcaton than other methods such as orthogonal collocaton, and rely less on the sgnfcant tunng of the algorthms for the dfferent ntal dstrbutons and aggregaton kernels (Fsher, 998). The dscretzaton method also s relatvely easy to use, therefore, t s an excellent choce for a general mathematcal model. Snce the dscretzaton method plays an mportant role n ths mathematcal model, a dervaton of the numercal ntegraton wth a thorough error analyss s necessary. The method of dscretzaton used n ths mathematcal model was orgnally proposed by Hounslow (988, 990). Ths method s based upon geometrc dscretzaton of the partcles n terms of volume or length (Fsher, 998): 77

94 v + = v or L L + = / 3 (3.5.4) where v s the volume, and L s the length of each dscretzaton bn. Each ndex represents a dscrete bn of partcle concentraton, N, whch s contaned wthn the range of partcle sze between v and v +. Fgure 3.4 shows the dscrete bns to cover an arbtrary partcle sze dstrbuton (Hounslow, 00). Ths type of geometrc dscretzaton allows a relatvely small number of dscrete bns to cover a large range of partcle szes (Fsher, 998). Interval n(v) n = N Dv v v + v Fgure 3.4. Dscrete Bns to Cover a Large Range of Partcle Szes Hounslow derved the followng dfferental euaton whch relates the change n partcle characterzaton n each bn, N, to the concentratons n surroundng bns and the aggregaton coeffcent, b (Fsher, 998): Ê dn Á Ë dt ˆ agg = Â b N N + b, N Â b, NN N N (3.5.5) Â b, = 78

95 One man dsadvantage of ths type of dscretzaton s that the partcle sze dvsons are fxed. Due to ths lmtaton, Ltster et al. (995) has mproved on the method by proposng an adustable dscretzaton method: v / + = or v L 3 / + = (3.5.6) L where s an nteger greater than or eual to one, known as the adustable dscretzaton parameter. The number of dscretzaton bns for a gven partcle sze doman can be ncreased by ncreasng the adustable parameter. Ths type of dscretzaton stll covers a wde range of partcle szes wth fewer dscretzng bns than lnear dscretzaton, and the fneness of dscretzaton can be customzed for comparson wth expermental results (Fsher, 998). A general development of the dervaton of Ltster et al. dscretzaton method s presented below. The development starts by defnng the basc aggregaton events between sets of partcles. The event fall nto fve categores (Fsher, 998): Partcles are created wthn the nterval wth the followng events: Type : Some of the nteractons gve partcles n the th nterval and gve some partcles smaller than the th nterval. Type : All nteractons gve partcles n the th nterval. Type 3: Some nteractons gve partcles n the th nterval and some gve nteractons larger than the th nterval. Partcles are removed from the nterval by the followng events: Type 4: Some nteractons remove partcles from the th nterval. Type 5: All nteractons remove partcles from the th nterval. 79

96 A summary of nteractons s provded n Table 3.. It s evdent whch of these nteractons fall nto the fve categores by examnng the partcle nteractons nvolvng partcles boundng each nterval (Fsher, 998). Sze Interval Table 3.. Table of Aggregaton Events for the Method of Ltster et al Sze Interval Type (Brth) Type (Brth) 80 Type 3 (Brth) Type 4 (Death) Type 5 (Death) S() S()+ S() S() S() S() S() S() S() 3 S() S() S()3 S(3) S()+ S()

97 8 To understand Table 3., the defnton of S() needs to be explaned. S() relates the mportant subntervals whch aggregate wth nterval to form new partcles wthn the th nterval (Fsher, 998): Â = = p p S ) ( (3.5.7) Inspecton of Table 3. gves the rate of change for N, the number concentraton n any dscretzaton bn, gven any value of (Fsher, 998):, ) ( ) ( ) (,, S k k k S k k S k k agg N N N K N N K dt dn = = = Â Â Â + + = ˆ Á Ë Ê b b b (3.5.8) Â Â Â Â = = + = = + + = ˆ Á Ë Ê k S S k k S k k S k k agg N N N N K N N K dt dn ) ( ) (,, 4 ) ( ) (, 3 b b b (3.5.9) The terms K, K, K 3, K 4, K 5 are the fractonal portons that are added or removed from the th nterval. In the above euatons, each term corresponds to a specfc aggregaton event: Type : Frst and second terms Type : Thrd term Type 3: Fourth term Type 4: Ffth term Type 5: Sxth term In contnung ths dervaton, Ltster et al. follows the same method as presented by Hounslow (988, 990) for the case of = for evaluatng the K terms (Fsher, 998). If v = /, the th nterval s defned as / < v < (I+)/.

98 The factors K and K for Type aggregatons events wll determned usng ths assumpton. The nteractons between the partcles n the th and th nterval where some events gve partcles n the th nterval and gve some partcles smaller than the th nterval. A partcle of sze between a and a + da must aggregate wth a partcle n the sze range / a < v < (Ip+)/ from the pth nterval to gve a partcle n the th nterval wth respect to a s (Fsher, 998): dr + a da / ( p+ ) / / [] = p, b p, N p p p N ( + ) / ( ) / ( + ) (3.5.0) ( p+ ) / ( p) / [] ( ) + a dr p, = b p, N p N da (3.5.) ( p+ ) / / ( ) The total rate of successful events can be found by ntegratng ths expresson between the Ipth and th ntervals (Fsher, 998): The values for K and K are: K / + ( + ) / / = (3.5.) ( ) K / k / / + ( + k ) / + / / = (3.5.3) ( ) For the case of Type and Type 5 nteractons, all events lead to changes n the th subnterval and are no fractonal events. For Type 3 and Type 4 events the values of K 3 and K 4 must be determned (Fsher, 998). Ths s done n a smlar manner as the case for Type events and can be found n Appendx 3. 8

99 83 Substtutng these K terms nto euaton yelds the followng expresson for the rate of change of partcle number wthn the th dscretzaton bn. Â Â Â Â Â Â Â + = = + = = + + = = ˆ Á Á Ë Ê + + ˆ + Á Á Ë Ê + + ˆ Á Á Ë Ê ˆ Á Á Ë Ê + = ˆ Á Ë Ê ) (, / ) ( / / ) (, / ) ( / / / / ) ( ) (,, / ) ( / / / / / ) ( ) (, / ) ( / / ) (, ) ( ) ( ) ( ) ( S S k k k k k S k k S k k k k k k k k S k k S k S agg N N C N N C N N N C N N C N N dt dn b b b b b b (3.5.4) C and C are the volume correcton factors. The euaton accurately represents the zero moment for any values of C and C. The thrd moment s not predcted as a drect result of the dscretzaton method. Consder a partcle n the th nterval that nteracts wth a partcle n the th nterval to produce a partcle that s stll n the th nterval. Events lke ths do not affect the number of partcles n the th nterval and are therefore not counted by euaton, but for however change the total volume of partculate materal present n ths nterval (Fsher, 998). In order to account for those nteractons whch do not affect the zeroth moment, but do not affect other moments n the th nterval, the ncluson of the correcton factors C and C s necessary. Appendx 4 shows that the values of C and C are: / ) ( C + = (3.5.5)

100 84 / / / / ) ( / ) ( / / / ) ( + + ˆ Á Á Ë Ê + = + k k k C (3.5.6) Combnng these volume correcton factors wth euaton yelds the followng: Â Â Â Â Â Â Â + = = = = + + = = ˆ Á Á Ë Ê ˆ Á Á Ë Ê ˆ Á Á Ë Ê + + ˆ Á Á Ë Ê = ˆ Á Ë Ê ) (, / / ) ( ) (, / / ) ( / / ) ( ) ( ) (,, / / ) ( / ) ( ) ( ) (, / / ) (, ) ( S S k k k k S k k S k k k k k k k S k k S k S agg N N N N N N N N N N N dt dn b b b b b b (3.5.7) Ths euaton reduces to the orgnal case of Hounslow shown n euaton for the case of =. If, only two addtonal summaton terms are necessary (Fsher, 998). Ths last euaton s only correct for cases up to =4, but for hgher values of Wynn (996) has provded a corrected verson of ths euaton. In the orgnal work of Ltster et al., the summaton lmts for the dfferental euaton for =,,3,4, and the general case were assumed by nspecton. Wynn provdes a more rgorous dervaton usng the propertes of sze dscretzaton. The corrected table s shown n Table 3..

101 Table 3.. Table of Aggregaton Events for the Method of Ltster et al (Updated) Sze Interval Sze Interval Type (Brth) Type (Brth) Type 3 (Brth) Type 4 (Death) Type 5 (Death) S()+ S()+ p;( p S(p) +S(p) ) S(p) +S(p+) S() S() The dervaton for the K terms and the volume correcton factors reman unchanged, and only the lmts of summaton for euaton need correcton. The functon S() changes, thus t wll not be referred to as S(p). The value off S(p) s not dependent on I, but does depend prmarly on. The corrected functon S(p) s gven by (Fsher, 998): p / È ln( ) S ( p) = IntÍ (3.5.8) Î ln 85

102 86 where Int(x) returns the fractonal part from x, makng t an nteger. The corrected number dfferental for any dscretzaton bn now becomes (Fsher, 998): Â Â Â Â Â Â Â + = + = = = + + = = + = ˆ Á Á Ë Ê ˆ Á Á Ë Ê ˆ Á Á Ë Ê + + ˆ Á Á Ë Ê = ˆ Á Ë Ê ) (, / / ) ( (), / / ) ( / / ) ( ) ( ) (,, / / ) ( / ) ( ) ( ) (, / / ) ( (), ) ( S S k k k S k S k k k k k k S k S k S agg N N N N N N N N N N N dt dn b b b b b b (3.5.9) In ths method of dscretzaton, no ntegraton wthn subntervals s necessary unlke the methods of Gelbard and Senfeld (980) and Landgrebe and Pratsns (990). The mplementaton of the dscretzaton method presented, results n a setoff ordnary dfferental euatons, one dfferental euaton per dscretzaton bn. These dfferental euatons can be evaluated wth relatve ease, and can be mplemented on any personal computer. The current algorthm s mplemented wth Matlab 5.0 usng a RungeKutta Fehlberg ntegraton routne wth error control Comparson wth Analytcal Solutons Analytcal solutons to the contnuous populaton balance for aggregaton are lmted to specalzed forms of the ntal partcle sze dstrbuton and aggregaton kernels. The comparson wll be carred out on the bass of actual number densty dstrbuton as a functon of tme as well as the error n the frst sx moments of the partcle sze dstrbuton (PSD). The th moment of the PSD at any tme s gven by:

103 Ú 0 m = L n( L) dl (3.5.0) or n the dmensonless form: Ú L n( L) dl ~ m 0 m = = m t = 0 Ú L n( L) dl (3.5.) 0 t = 0 where m s the th moment of the dstrbuton, m ~ s the dmensonless th moment of the dstrbuton, L s the partcle length, and n(l) s the partcle number densty functon (Fsher, 998). The two cases for comparson wll be those of the sze ndependent kernal and the szedependent sum kernal, both wth exponental ntal dstrbuton terms of volume. Another term that s useful n graphng and comparng aggregaton, s the degree of aggregaton, I agg whch s defned as: m 0 I agg = (3.5.) m0 t = 0 If m 0 s the total number of partcles, the value of the I agg ranges from zero to one. In ths study, hgh degrees of aggregaton are greater then 0.8, where the total number of partcles has been reduced by 80% (Fsher, 998). Fgure 3.5 (Fsher, 998) shows a comparson for the moments versus I agg. 87

104 Moments versus Iagg Percent Error =3 M0 M M M3 M4 M5 M Iagg Fgure 3.5. Moments versus I agg SzeIndependent Kernal Gelbard and Senfeld (978) provde analytcal solutons to the populaton balance model for aggregaton based on the ntal dstrbuton: N Ê ˆ = 0 v Á v ( v) exp (3.5.3) = 0 v0 Ë v0 n t where N 0 s the total ntal number of partcles, v 0 s the ntal mean volume. The partcle volumes must be converted to length because ths s the most common method of vewng partcle sze dstrbutons. To convert the number densty dstrbuton on a volume bass to that of length, or dameter, the followng converson s necessary (Fsher, 998): pd nd( D, t) = nv ( v, t) (3.5.4) 88

105 Gelbard and Senfeld (978) present solutons for the szendependent kernal defned as: b = v ( v, v) b0 (3.5.5) where b 0 s some constant. They present an analytcal soluton obtaned by the method of Laplace Transform (Fsher, 998): 4N 0 Ê v ˆ n ( v, t) = expá (3.5.6) v ( t + ) Ë t + 0 where t = N 0 b 0 t s the dmensonless tme value, and v s v/v 0. Ths euaton s referred to as the szendependent (SI) case SzeDependent Kernal Gelbard and Senfeld also present solutons for the szedependent sum kernal defned as: b v, v ) = b ( v ) (3.5.7) v ( 0 + v where b 0 s some constant value. An analytcal soluton s presented by the method of Laplace Transforms (Fsher, 998): N0( T ) / n ( v, t) = exp( ( + T ) v) I(vT ) (3.5.8) / v T 0 where T = exp(t), I s the modfed Bessel functon of the frst knd order one, and t = N 0 n 0 b 0 t, also known as the dmensonless tme value. Ths euaton wll now be referred to as the szedependent (SD) case (Fsher, 998) Error Levels n Partcle Sze Dstrbuton The levels of error ntrnsc n the soluton of the populaton balance for aggregaton n comparson wth the szendependent and szedependent case wll be nvestgated n ths secton. Fgures 3.6 and 3.7 show the relatve error n moments zero through sx for the 89

106 SD and SI case for a large value of I agg (Fsher, 998). From these two fgures, a reducton n error s shown wth ncreasng. For the SI case, the value of =, the error n the zero through sx moments s less than one percent. The error level for all moments n the SD case s below fve percent. Error n the th Moment SzeDependent Case Percent Error Iagg = 0.8 M0 M M M3 M4 M5 M Dscretzng Parameter Fgure 3.6. Error n the th Moment for SzeDependent Case 90

107 Relatve Error n the Moments for Szendependent Kernel 0 Iagg= M0 M M M3 M4 M5 M Values of Fgure 3.7. Error n the th Moment for SzeIndependent Case 3.7 Mathematcal Model Descrpton All of the deas and concepts presented n ths chapter are put together nto a sngle mathematcal model to predct the partcle sze dstrbuton over a perod of tme. A map of aggregaton coeffcents s created to match the dscrete bn szes used n the dscretzaton method. Fgure 3.6 shows a flow dagram of the complete mathematcal model. 9

108 Parameters: ) Partcle Sze ) Boltzmann Constant 3) Vscosty of Medum 4) Average Shear Rate 5) Temperature 6) Densty of Partcle 7) Densty of medum 8) Delectrc Permttvty 9) Ionc Concentraton 0) Valence of onc Speces ) Electronc Charge ) Surface Charge 3) Hamaker Constant Input Parameters:, 3 Hamaker Theory van der Waals Attractve Energy Total Interacton Energy Input Parameters:,, 5, 8, 9, 0,,, 3 Gouy Chapman Theory Electrostatc Repulsve Interacton Energy Input Parameters:,, 3, 5 Perknetc Collson Mechansm Stablty Rato Input Parameters:,, 5 Input Parameters:, 4 Orthoknetc Collson Mechansm Transport Lmted Rate Constant Map of Aggregaton Coeffcent Input Parameters:, 3, 6, 7 Dfferental Sedmentaton Collson Mechansm Populaton Balance Framework Intal Partcle Sze Dstrbuton Partcle Sze Dstrbuton Fgure 3.8. Flow Dagram of Mathematcal Model 9

109 3.8 Summary Ths chapter has presented a general mathematcal model for collodal aggregaton. A populaton balance framework s the bass of ths model. The Smoluchowsk knetcs are used wth the perknetc and orthoknetc collson mechansms. These rates are modfed to nclude collodal stablty usng stablty ratos. The sum of the van der Waals and electrostatc nteracton energy are used to evaluate the stablty ratos. Van der Waals nteracton energy s computed usng the Hamaker theory, and the electrostatc nteracton s computed usng the lnear and nonlnear PossonBoltzmann euaton for flat plates coupled wth the ntegraton method of Papadopoulos and Cheh (984). Dscretzaton and soluton of the populaton balance euatons s solved usng the Ltster et al (995), and the Wynn (996) methods. Models of the dscretzaton error were presented whch shows the dscretzaton method to provde adeuate levels of accuracy. The general nature of ths model makes t sutable for both ndustral and academc applcatons. The followng chapters present the mathematcal model experments, and a comparson of experments wth mathematcal model. 93

110 CHAPTER 4: EXPERIMENTS CONDUCTED BY MATHEMATICAL MODEL 4. Introducton Ths chapter wll show the expermental results from the mathematcal model. Dscretzng parameters, surface charge, fractal dmenson, temperature, vscosty, onc concentraton, shear rate, dameter szes and smultaneous perknetc and orthoknetc collson mechansms are studed to learn the capabltes of the model and to ensure agreement wth current lterature on collodal aggregaton. 4. The Effect of Dscretzng Parameters on the Model As the dscretzng parameter ncreases, there s also an ncrease n the total number of bns to cover a gven partcle sze range. For example, to cover a sze range between 0.0 to 0 mcrons, the followng total number of bns, Z, are needed, as shown n Table 4.. Table 4. Dscretzng Parameter versus Total Number of Bns Dscretzng Parameter, Q Total Number of Bns, Z

111 Increasng ncreases the accuracy of the model because there are more bns to represent the actual partcle concentraton. Refer to Secton to revew the reducton n error wth ncreasng. Fgure 4.. Comparson of Dscretzng Parameters for Perknetc Collson Mechansm 95

112 Fgure 4.. Comparson of Dscretzng Parameters for Orthoknetc Mechansm 4.3 The Effect of Temperature and Vscosty on the Model As the temperature of the suspenson ncreases, aggregaton becomes more rapd. Hgher temperatures accelerate partcle formaton and growth. J Br 4kbT = N 0 (4.) 3h The above euaton shows that ncreasng the temperature causes the perknetc collson freuency to ncrease whereas ncreasng the vscosty of the medum reduces the collson freuency. 96

113 Fgure 4.3. Temperature Comparson In the case of orthoknetc aggregaton, the temperature ncrease also ncreases natural partcle Brownan moton, and enhances the effcency of the mxng (especally n hgher vscostes). 97

114 Fgure 4.4. Vscosty Comparson The model shows that aggregaton slows rapdly wth ncreasng vscosty. The ncreasng vscosty of the medum prevents the partcles from undergong natural Brownan moton. In the case of the orthoknetc collson mechansm, when the vscosty ncreases, t becomes harder to mx the soluton, whch mpedes aggregaton. 4.4 The Effect of Dameter Sze on the Model In perknetc and orthoknetc aggregaton, aggregates rapdly form when larger and smaller sze partcles are n a suspenson together. As shown n Fgure 4.5 and 4.6, the dameter range of 0.0 * 0 6 to 5.0 * 0 6 and 0. * 0 6 to 0.0 * 0 6 had the most rapd aggregaton due to the large varaton of partcle szes n ths group. The smaller partcle sze group (0.0 * 0 6 to 0.0 * 0 6 ) dd not aggregate much under the perknetc or orthoknetc collson mechansm. 98

115 Fgure 4.5. Dameter Sze Comparson for Perknetc Aggregaton Fgure 4.6. Dameter Sze Comparson for Orthoknetc Aggregaton 99

116 4.5 The Effect of Salt Concentraton on the Model The addton of a salt to a suspenson produces charged ons that reduce the effectve dstance of the electrostatc nteractons. Ths occurs by ncreasng the soluton onc strength and suppressng the thckness of the layer of ons surroundng the partcles. Around an onc concentraton of 0. M, the thckness of the double layer s reduced to the extent that partcles can approach close enough for the attractve van der Waals forces to domnate (Spcer, 997). The suspenson s destablzed at ths pont, and the partcles wll aggregate f brought together by thermal moton or flud shear. Partcles can also be destablzed wth charged polymers that adsorb to the partcle surface and create a brdge between partcles to form flocs. In ether case, the result s a decreased electrostatc repulson between partcles. The destablzng agent allows the partcles to come close enough together to adhere. In most practcal cases, the reducton n collson effcency resultng from electrostatc effects s neglgble f suffcent flocculant has been added (Spcer, 997). As a functon of onc strength, the overall nteracton undergoes a sharp transton from repulsve to attractve as shown n Fgure 4.7 (McDuff and Heath, 00). Fgure 4.8 shows the behavor of the clay mneral kaolnte partcles that were montored at dfferent onc strengths (A = onc strength of 0.036; B = onc strength of 0.087, and C = onc strength of Clay Mneral Kaolnte) (McDuff and Heath, 00). 00

117 Fgure 4.7. Effect of Ionc Strength on Interacton Energy for Clay Mneral Kaolnte 0

118 Fgure 4.8. Effect of Ionc Strength on Clay Mneral Kaolnte As shown n Fgure 4.9, the onc concentraton was adusted from.0 * 0 to 000 mol/m 3, but only a small effect n aggregaton rate s seen. Ths may be due to an already hgh degree of aggregaton at the onc concentraton of.0 * 0 mol/m 3. If the aggregaton rate s reduced by adustng other parameters (such as temperature), the salt concentraton may have a larger nfluence on the partcle aggregaton accordng to ths model. 0

119 Fgure 4.9. Ionc Concentraton Comparson for the Model 4.6 The Effect of Hamaker Constants on the Model As Hamaker constants become larger, the nteracton energy becomes smaller as shown n Fgure 4.0 (Fsher, 998). Fgure 4. shows a partcle sze ncrease wth ncreasng Hamaker constant values. 03

120 Fgure 4.0. Potental Energy of Interacton for Varous Values of Hamaker Constant 04

121 Fgure 4.. Hamaker Constant Comparson 4.7 The Effect of Fractal Dmenson and Prmary Partcle Szes on the Model If one consders the prmary partcle as a pont obect, then the fractal dmenson of the growng aggregate should ncrease from a zerodmensonal pont to a onedmensonal lnear structure, to a branched chan of dmenson on the order of two as the growth proceeds. The ncrease n dmenson wth growth s a natural conseuence of the persstence of velocty for nanopartcle Brownan moton combned wth the random path of the collodng partcles. As growth proceeds, the presence of branches and the convoluted shape of the growng aggregate, remnscent of the Brownan path of the colldng partcles, shelds the nteror bondng stes from further growth. For ths reason, threedmensonal growth s not possble, except by nternal arrangement. The overall densty of the aggregate, N/R 3 ~ N (3/df) dmnshes wth tme n the asymptotc range snce N s a monotoncally ncreasng functon of tme (Bushell, 998). 05

122 Fgures 4. and 4.3 show the effect of prmary partcle sze for the aggregaton model. As the prmary partcle sze ncreases, the aggregate growth decreases. The smaller prmary partcles have a hgh rate of Brownan moton, and are more lkely to aggregate to larger collodal molecules. Fgures 4.4 and 4.5 show an ncreasng rate of aggregaton wth smaller fractal dmenson. The smaller fractal dmensons have more rregular structures wth more vod space, whch enable these partcles to attract oppostely charged molecules more readly. As the fractal dmenson approaches a fractal dmenson of 3.0, the partcle aggregate becomes more compact and sphercal n nature, whch s harder to attract other partcles. There are no dendrtc arms to attract and catch partcles. Fgure 4.. Comparson of Prmary Partcle Sze for Perknetc Collson Mechansm 06

123 Fgure 4.3. Comparson of Prmary Partcle Sze for Orthoknetc Collson Fgure 4.4. Fractal Dmenson Comparson for Perknetc Collson 07

124 Fgure 4.5. Fractal Dmenson Comparson for Orthoknetc Collson Mechansm 4.8 The Effect of Surface Potental on the Model The electrc potental of the plane s eual to the work aganst electrostatc forces reured to brng a unt electrcal charge from nfnty (from the bulk of soluton) to that plane. If the plane s the surface of the partcle, the potental s called surface potental, whch measures the total potental of the double layer. If the surface potental s kept constant, the surface charge densty must approach zero, as shown n Fgure 4.6 (Pelton, 003). Ths ndcates that the surface s loosng charged groups. Fgure 4.7 (Pelton, 003) shows that wth scalable potental, the ntal slopes reman constant, and the correspondng potental drops. Ths gves lower nteracton energy for the model, whch gves a lower osmotc pressure between partcle surfaces. 08

125 Fgure 4.6. Constant Surface Potental Between Two Plates Fgure 4.7. Scalng Surface Potental Between Two Plates 09

126 Fgure 4.8 shows the average scalng surface potental used n the model expermental runs. Fgure 4.8. Surface Potental Versus Dameter 4.9 The Effect of Shear Rate on the Model As the shear rate ncreases, partcle aggregaton also ncreases untl the shear rate gets hgh enough that there s smultaneous aggregaton and deaggregaton. Fgure 4.9 shows that the aggregaton rate ncreases wth ncreasng shear rate, but after the shear rate s hgher than forty, t s obvous that there s also deaggregaton occurrng because the partcle dameter s slowly startng to decrease agan sometme durng the 0 50 hour tme perod. Based upon the graph, t seems that the hgher the shear rate, the faster deaggregaton wll begn. 0

127 Fgure 4.9. Comparson of Shear Rates 4.0 The Effect of Smultaneous Perknetc and Orthoknetc Collson Mechansms Swft and Fredlander (964) analyzed the knetcs of smultaneous orthoknetc and perknetc coagulaton by assumng the two mechansms were addtve. They found good agreement of a monodsperse model wth expermental data for the aggregaton of polystyrene partcles. To compare the collson freuences due to shear flow that wth due to Brownan moton, ther rato s characterzed by the Peclet number whch s obtaned by dvdng euaton.3 by.7 (Agarwal, 00). 4hGa Pe = k T B 3 (4.)

128 where m s the flud vscosty, k s Boltzmann s constant and T s absolute temperature. Accordng to Agarwal (00), f Pe >>, shear flow domnates, but f Pe<<, Brownan moton wll domnate. Fgure 4.0. Peclet Number Versus Partcle Dameter Some experments were conducted wth the mathematcal model wth the perknetc and orthoknetc collson mechansm. The parameters for these experments are lsted Table 4..

129 Table 4.. Expermental Parameters for Perknetc and Orthoknetc Experments Expermental Parameter Value Smallest Partcle Dameter (m) 0.0E6 Largest Partcle Dameter (m) 5.0E6 Number of Euatons Coverng partcle range 8 Expermental Intal Condton (parts per ml) 8.399e+0 Mean of Intal Dstrbuton 0.0E06 Standard Devaton of Intal Dstrbuton E6 Vscosty.00 cp Temperature 93.5 K G vares Fractal Dmenson.7 The frst set of experments nvolve usng the perknetc collson mechansm and stablty rato for Peclet numbers less than, and usng the orthoknetc collson mechansm and ts stablty rato for Peclet numbers greater than. Fgures 4. through 4.5 show the results of these experments wth dfferent shear rates to change the Peclet numbers. 3

130 Fgure 4.. Log of Aggregaton Coeffcent, G =, Peclet < Fgure 4.. Log of Stablty Ratos, G =, Peclet < 4

131 Fgure 4.3. Log of Stablty Ratos, G = 70, Peclet < Fgure 4.4. Log of Stablty Ratos, G = 70, Peclet < 5

132 Fgure 4.5. Perknetc and Orthoknetc Aggregaton Zechner and Schowalter (979) found that for a rato of shear to Brownannduced collson freuences < 5 and G < 400 s, Brownan aggregaton affected (enhanced) shearnduced aggregaton. They dd, however, note that shear controlled the aggregaton rate for all shear rates (00800 s ) and concluded that Brownan collsons were mportant only for partcles brought close together by shearng. Feke and Schowalter (983) consdered sheardomnated aggregaton when small amounts of Brownan coagulaton are present and concluded that Euaton 4. could underpredct the shearnduced coagulaton rate for values of the Peclet number, Pe < 90 and overpredct t for Pe > 90. The next set of experments nvolve usng the perknetc collson mechansm and stablty rato for Peclet numbers less than 90, and usng the orthoknetc collson mechansm and ts stablty rato for Peclet numbers greater than 90. Fgures 4.6 through 4.30 show the results of these experments wth dfferent shear rates to change the Peclet numbers. 6

133 Fgure 4.6. Log of Aggregaton Coeffcent for G = 0 and the Peclet number <90 Fgure 4.7. Log of Stablty Ratos for G = 0 and the Peclet number <90 7

134 Fgure 4.8. Log of Aggregaton Coeffcent for G=0 Fgure 4.9. Log of Stablty Ratos for G=0 8

135 Fgure Perknetc and Orthoknetc Aggregaton Han and Lawler (99) assumed addtvty of dfferent aggregaton collson mechansms and calculated that Brownan moton was relevant durng aggregaton only when at least one of the colldng partcles s less than µm n dameter. Kusters et al. (996) found that polydsperse coagulaton models assumng addtvty most accurately matched expermental data for turbulent coagulaton. The fnal set of experments nvolve the addton of the perknetc aggregaton coeffcents wth the orthoknetc aggregaton coeffcents. Fgures 4.3 through 4.34 show the results of these experments wth dfferent shear rates. 9

136 Fgure 4.3. Log of Perknetc and Orthoknetc Collson Mechansm where G = 0 Fgure 4.3. Log of Stablty Ratos for G = 0 0

137 Fgure Log of Aggregaton Coeffcents for G = 0 Fgure Addton of Perknetc and Orthoknetc Aggregaton

138 4. Summary Ths chapter showed the expermental results from the mathematcal model. Dscretzng parameters, surface charge, fractal dmenson, temperature, vscosty, onc concentraton, shear rate, dameter szes and smultaneous perknetc and orthoknetc collson mechansms were vared to learn the capabltes of the model. There was good agreement wth current lterature on collodal aggregaton.

139 CHAPTER 5: COMPARISION OF MATHEMATICAL MODEL WITH EXPERIMENTS 5. Introducton Ths chapter dscusses the experments that were performed by Scott Fsher (998) to valdate the mathematcal model. A descrpton of the materals, expermental procedure, and the setup used n the aggregaton experments wll be descrbed. Experments were conducted for the perknetc and orthoknetc transport mechansms. Partcle szes were measured usng a partcle analyzer, electron mcroscopy and UV/Vsble spectroscopy to provde partcle sze analyss throughout the course of the aggregaton process. 5. Materals The general class of collods was represented by polystyrene latex because ts physcal propertes, () densty, (3) refractve ndex, and (4) surface charge are all well known. Polystyrene was also chosen because t has a densty close to that of salt solutons used for suspenson, whch lmts the effects of sedmentaton. Monodspersed polystyrene standards from Duke Scentfc Corporaton were used, and the partcle propertes are dsplayed n Table 5.. 3

140 Table 5.. Propertes of Partcle Standards Sample Nomnal Mean Standard Densty Index of Name Dameter Dameter Devaton (g/ml) Refracton at [nm] [nm] [nm] 589 nm PS PS PS These concentrated dspersons were dluted by three orders of magntude so that UV/Vs spectroscopy could be used for partcle sze analyss. The same stock soluton s used for all expermental runs to ad n reproducblty. Table 4. shows the volumes used to create the stock solutons, and relevant soluton propertes. Table 5. Propertes of Partcle Stock Solutons Partcle Volume of Volume of Mass Partcle ph Standard Standard Soluton Concentraton Concentraton (ml) (ml) (g/ml) (part/ml) PS x * PS x * PS x *

141 5.3 Partcle Sze Dstrbutons The partcle szes for the threepartcle standard used n experments were measured usng three partcle sze unts: electron mcroscopy, a Mcrotrac UPA 50 Ultrafne Partcle Analyzer, and a Hewlett Packard 8453 UV/Vsble Spectrophotometer (Fsher, 998). Table 5.3 shows a comparson of the mean and standard devaton for each of the three partcle standards. Table 5.3 Summary of the Intal Partcle Sze Dstrbuton for Standards Mcroscopy Mcrotrac UPA 50 UV/Vs Spectroscopy Partcle Mean Standard Mean Standard Mean Standard Standard (nm) Devaton (nm) Devaton (nm) Devaton (nm) (nm) (nm) PS PS PS Measurement of Surface Potentals Surface charge s mportant n modelng aggregaton phenomena because t domnates partcle dsperson and aggregaton. Surface potental measurements were taken usng a Zeta Plus Zeta Potental Analyzer manufactured by Brookhaven Instruments Lmted. Zeta potental measurements were conducted for three concentratons of potassum chlorde: 0. M, x 0 3 M, and x 0 4 M (Fsher, 998). The samples were prepared by mxng 00 ml of the approprate potassum chlorde soluton, addng 0.5 ml of concentrated partcle 5

142 standard, and then adustng the ph wth the addton of ether concentrated acd, HCl or base NaOH (Fsher, 998). 0 Zeta Potental for Polystyrene 0 Zeta Potental [mv] PS 0.00M PS 0. M PS M ph Fgure 5.. Zeta Potental for Polystyrene The salt concentraton presented n Fgure 5. agrees wth the GouyChapman theory n the fact when the salt concentraton s ncreased, the electrc double layer s compressed, yeldng lower effectve surface charge (Fsher, 998). The zeta potental also follows the theory because when the ph s decreased, the onzable groups on the surface of the polystyrene partcles become protonated, whch reduces the surface charge. 5.5 Measurement of Aggregaton Phenomenon The experments were conducted at varous ph and salt concentratons, and allowed to aggregate by perknetc or orthoknetc aggregaton. Partcle aggregaton was then characterzed by average partcle sze as a functon of expermental tme. 6

143 The expermental data was calculated from the turbdty measurements (Fsher, 998). Refer to Fsher (998) for the algorthms used to produce average partcle sze, and the two populaton average partcle sze calculaton. The mathematcal model wll use the same parameters and ntal condtons as were used and measured n the experments. The fractal dmenson used for ths comparson wll ether be.7 for dffusonlmted experments or.3 for reactonlmted or mxed systems. These values are the most commonly reported values from other expermental technues (Fsher, 998). 5.6 Expermental Runs The experments that were conducted were performed to test the valdty of the perknetc and orthoknetc collson mechansms. Experments A and B are used to determne the valdty of the perknetc collson mechansm for polystyrene partcles of varous szes. Experments C, D, E, and F wll be used to test the applcablty of the orthoknetc mechansm. To ensure that the experment conducted was n the transportlmted aggregaton regme, the soluton condtons, ph, and KCl concentraton were chosen to mnmze the surface potental. (Fsher, 998). Partcle szes between 00 nm and 503 nm were used as the partcle sze ntal condtons. For comparson, the followng notaton wll be used for the Fgures: EXP Davg s the expermental number average partcle dameter, EXP D s the expermental number average partcle dameter for the smaller of the two populaton analyss, and EXP D s the expermental number average partcle dameter for the larger of the two populaton analyss. MODEL Davg, MODEL D, and MODEL D are the results of the mathematcal model presented n ths paper. SF MODEL Davg, SF MODEL D, ans SF MODEL D are the results of Fsher s mathematcal model (998) whch excluded attracton and repulson forces and stablty ratos. 7

144 5.7 Comparson 5.7. Experment A Table 5.4. Expermental Parameters for Experment A Expermental Parameter Value Collson Mechansm Perknetc Smallest Partcle Dameter (m) 0.06E6 Largest Partcle Dameter (m) 4.0E6 Number of Euatons Coverng partcle range 9 Expermental Intal Condton (parts per ml) E+0 Mean of Intal Dstrbuton 0.0E06 Standard Devaton of Intal Dstrbuton E6 Vscosty (cp).00 cp Temperature (K) 93.5 K Fractal Dmenson.7 Table 5.5. Degree of Aggregaton for Experment A I agg Fsher Model Scott I agg Model

145 Fgure 5.. Log of Perknetc Collson Mechansm for Experment A Fgure 5.3. Log of Stablty Ratos for Experment A 9

146 Fgure 5.4. Log of Aggregaton Coeffcent for Experment A Fgure 5.5. Partcle Concentraton versus Tme for Experment A 30

147 Fgure 5.6. Partcle Concentraton versus Partcle Dameter for Experment A 3

148 Fgure 5.7. Actual versus Mathematcal Model Experments (Experment A) 3

149 5.7. Experment B Table 5.6. Expermental Parameters for Experment B Expermental Parameter Value Collson Mechansm Perknetc Smallest Partcle Dameter (m) 0.3E6 Largest Partcle Dameter (m).0e6 Number of Euatons Coverng partcle range 50 Expermental Intal Condton (parts per ml).475e+0 Mean of Intal Dstrbuton 0.503E06 Standard Devaton of Intal Dstrbuton E6 Vscosty (cp).00 cp Temperature (K) 93.5 K Fractal Dmenson.7 Table 5.7. Degree of Aggregaton for Experment B I agg Fsher Model Scott I agg Model

150 Fgure 5.8. Log of Perknetc Collson Mechansm for Experment B Fgure 5.9. Log of Stablty Ratos for Experment B 34

151 Fgure 5.0. Log of Aggregaton Coeffcents for Experment B Fgure 5.. Partcle Concentraton versus Tme for Experment B 35

152 Fgure 5.. Partcle Concentraton versus Partcle Dameter for Experment B 36

153 Fgure 5.3. Actual versus Mathematcal Model Experments (Experment B) 37

154 5.7.3 Experment C Table 5.8. Expermental Parameters for Experment C Expermental Parameter Value Collson Mechansm Orthoknetc Smallest Partcle Dameter (m) 0.06E6 Largest Partcle Dameter (m) 3.0E6 Number of Euatons Coverng partcle range 5 Expermental Intal Condton (parts per ml) 8.399E+0 Mean of Intal Dstrbuton 0.0E06 Standard Devaton of Intal Dstrbuton E6 Vscosty (cp).00 cp Temperature (K) 93.5 Fractal Dmenson.7 Table 5.9. Degree of Aggregaton for Experment C I agg Scott Fsher Model I agg Model

155 Fgure 5.4. Log of Orthoknetc Collson Mechansm for Experment C Fgure 5.5. Log of Aggregaton Coeffcent for Experment C 39

156 Fgure 5.6. Partcle Concentraton versus Tme for Experment C Fgure 5.7. Partcle Concentraton versus Partcle Dameter for Experment C 40

157 Fgure 5.8. Log of Stablty Ratos for Experment C 4

158 Fgure 5.9. Actual versus Mathematcal Model Experments (Experment C) 4

159 5.7.4 Experment D Table 5.0. Expermental Parameters for Experment D Expermental Parameter Value Collson Mechansm Orthoknetc Smallest Partcle Dameter (m) 0.06E6 Largest Partcle Dameter (m) 4.0E6 Number of Euatons Coverng partcle range 55 Expermental Intal Condton (parts per ml) 6.E+0 Mean of Intal Dstrbuton 0.0E06 Standard Devaton of Intal Dstrbuton E6; Vscosty (cp).00 Temperature (K) 93.5 Fractal Dmenson.5 Table 5.. Degree of Aggregaton for Experment D I agg Scott Fsher Model I agg Model

160 Fgure 5.0. Stablty Ratos for Experment D Fgure 5.. Log of Orthoknetc Collson Mechansm for Experment D 44

161 Fgure 5.. Log of Aggregaton Coeffcents for Experment D Fgure 5.3. Partcle Concentraton versus Tme for Experment D 45

162 Fgure 5.4. Partcle Concentraton versus Partcle Dameter for Experment D 46

163 Fgure 5.5. Actual versus Mathematcal Model Experments (Experment D) 47

164 5.7.5 Experment E Table 5.. Expermental Parameters for Experment E Expermental Parameter Value Collson Mechansm Orthoknetc Smallest Partcle Dameter (m) 0.08E6 Largest Partcle Dameter (m).0e6 Number of Euatons Coverng partcle range 4 Expermental Intal Condton (parts per ml) 3.546e+9 Mean of Intal Dstrbuton 0.55E06 Standard Devaton of Intal Dstrbuton 0.003E6 Vscosty (cp).00 cp Temperature (K) 93.5 K Fractal Dmenson.3 Table 5.3. Degree of Aggregaton for Experment E I agg Scott Fsher Model I agg Model

165 Fgure 5.6. Actual versus Mathematcal Model Experments (Experment E) 49

166 5.7.6 Experment F Table 5.4. Expermental Parameters for Experment F Expermental Parameter Value Smallest Partcle Dameter (m) 0.E6 Largest Partcle Dameter (m) 3.0E6 Number of Euatons Coverng partcle range Expermental Intal Condton (parts per ml).5e+8 Mean of Intal Dstrbuton 0.503E06 Standard Devaton of Intal Dstrbuton E6 Vscosty (cp).00 cp G 30 Temperature (K) 93.5 K Fractal Dmenson 3.0 Table 5.5. Degree of Aggregaton for Experment F I agg Scott Fsher Model I agg Model

167 Fgure 5.7. Actual versus Mathematcal Model Experments (Experment F) 5.8 Concluson From transportlmted experments presented n ths chapter, t s evdent that the mathematcal model does an excellent ob of predctng the average and two populaton average partcle szes. The perknetc experments show some devatons at long tmes, whle at shorter tmes, the model predctons are n agreement wth the expermental results. The orthoknetc collson mechansm shows agreement wth expermental results. The general trends n both partcle sze and strrng rate support the correctness of the orthoknetc mechansm. From the expermental range used, t can be stated that the mathematcal model hold true for strrng rates between 0 and 550 rpm, however, there mght be evdence of the onset of breakup n the hgher strrng range. Also ntal partcle szes and fnal szes stretch a range of 0. mcrons to.5 mcrons. Ths adeuately covers the range of applcablty for most collodal systems. 5

168 CHAPTER 6: CONCLUSIONS 6. Mathematcal Model The mathematcal model presented n ths work provdes a descrpton of aggregaton n a batch unt. The model ntegrates several dfferent mechansms and deas that are presented n lterature. Ths model s flexble because t can nclude the addton of many parameters that pertan to aggregaton of a specfc system. The populaton balance s necessary for a full descrpton of the aggregaton process. The ncluson of the van der Waals attractve and electrostatc repulsve forces and the stablty ratos provded a more accurate descrpton and model for collodal partcle aggregaton. The mathematcal model can easly be adusted and updated to nvestgate modfcatons to the collson mechansms, stablty ratos, nteracton energy forces etc. 6. Experments Conducted wth Mathematcal Model A number of experments were conducted wth the mathematcal model. Dscretzng parameters, surface charge, fractal dmenson, temperature, vscosty, onc concentraton, shear rate, dameter szes and smultaneous perknetc and orthoknetc collson mechansms were vared to learn the capabltes of the model. There was good agreement wth current lterature on collodal aggregaton. 6.3 Comparson of Mathematcal Model wth Experments The mathematcal model agrees well wth expermental results over a sze range of 0. to.5 mcrons for both the orthoknetc and perknetc models. The model provdes level of accuracy hgher than those of the partcle sze measurements when consderng the average and the two populaton average sze partcle szes. One hndrance n the comparson s the fractal nature of the fractal aggregates n that the true fractal dmenson s not known. 5

169 6.4 Future Work Ths mathematcal model sheds some basc nsght nto the nature of collodal aggregates, but there are many assumptons and technues that may mpose error nto the mathematcal model. Several factors should be addressed n the future: ) The ncluson of addtonal forces such as Born repulson and sterc nteracton, ) Surface propertes, such as surface roughness (especally for the smaller collodal partcles, where surface propertes begn to domnate the partcle s propertes), 3) Stablty ratos need to be determned for dfferental sedmentaton. The stablty ratos for dfferental sedmentaton are sometmes estmated n lterature, but there s currently no defnte mathematcal relaton to descrbe the stckness factor for dfferental settlng, 4) More accurate stablty ratos need to be determned for orthoknetc sedmentaton. The orthoknetc stablty rato used n ths mathematcal model has good agreement wth the actual experments conducted, but the stablty rato used s an estmate stablty rato from Potann (99), 5) Clustercluster aggregaton, or collsons between multple partcles, 6) Better measurement for the nput parameters to the model. An analyss of the propagaton of error n the model should be conducted n order to dentfy the most mportant process parameters. For example, the calculaton of average shear rate s a parameter wth sgnfcant uncertanty because t depends upon the vessel used. 53

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175 APPENDICES 59

176 Appendx : Crteron for Monotonc or Concave Potental An electrostatc potental profle f s defned by the followng dfferental euaton: df = ± cosh( f) + C dx A.. and the boundary condtons: f ( 0) = f f ( ) = f Xd A.. where X d s the dstance of separaton between plates. Notce that the suare root n the RHS of euaton A.. forces the neualty: cosh( f ) + C 0 A..3 If f s monotonc, the dstance of separaton between two plates can be expressed by ntegratng A.. as: f df Xd = Ú A..4 cosh( f + C f ) Let Xd a and Xd b be the dstances of seperaton correspondng to two monotonc electrostatc potental profles between plates wth boundary condtons A... Let Ca and Cb be the constants correspondng to Xd a and Xd b. If Ca > Cb, f Ú > Ú cosh( f) + Ca cosh( f + f df f ) f df Cb A..5 hence Xd a < Xd b and as a constant C decreases the dstance of seperaton between the plates Xd ncreases. A drect conseuence of A..3 s that: C cosh(f) A..6 60

177 Appendx : (Contnued) And therefore the mnmumvalue of C, whch corresponds to the maxmum value of Xd s cosh (f ). If x ref s the maxmum value of X d, f df x ref = Ú A..7 cosh( f) cosh( f f ) As a concluson, Xd > x ref the dstrbuton of the electrostatc potental s not monotonc and therefore must be concave. Moreover, some conclusons may be drawn about the range of values of C for the concave case. If f s concave, t reaches ts mnmum when: df = 0 = cosh( fmn ) + C dx A..8 or C = cosh(f mn ) A..9 In general, cosh(f), hence from A..9 t follows that C. It was seen above that f was monotonc f C > cosh(f ) and therefore t must be concave otherwse. Ths sets the range [cosh(f ), ] for C f f s concave. 6

178 Appendx : Behavor of Concave Electrostatc Potental as C Vares. The dstance of separaton between twp plates when the electrostatc potental between tam s concave can be expressed as (From Appendx ): f Ú df df Xd = + Ú cosh( f) cosh( f ) cosh( f) A.. cosh( f f ) mn mn fmn mn f where f mn s an ntegraton constant whch corresponds physcally to the mnmum of the electrostatc surface potentals on the plates. The obectve of ths appendx s to show that Xd ncreases as f mn decreases. Ths property of euaton A.. s mportant n order to mprove the effcency of the algorthm employed to solve the PBE and compute nteracton potental energes between plates. Establshng the sgn of the dervatve wll begn the proof n the nterval of nterest: dxd df mn A.. In order to accomplsh ths task, euaton A.. should be dfferentated by means of Lebntz rule for dfferentaton under the ntegral. If x( e ) Ú I ( e ) = f ( x, e ) dx A..3 x( e ) then di( e ) de = dx f ( x, e ) de dx f ( x, e ) + de x( e ) Ú x( e ) df ( x, e ) dx de A..4 Furthermore, t s only necessary to carry out the proof for one of the two ntegrals n A.. snce the result wll be dentcal for both. If A..4 s appled to one of the ntegrals n A.. 6

179 Appendx : (Contnued) the followng result s obtaned: di df mn = cosh( f) cosh( f mn ) df df mn mn + f f Ú mn [ cosh( f) cosh( f mn )] 3 / snh( f mn ) df A..5 The frst term of the RHS of A..5 s an ndetermnaton. Therefore the dervatve searched for cannot be determned analytcally by means of ths method. In order to crcumvent ths stuaton, onr of the ntervals n Euaton A.. can be rewrtten as: I = A Lm æææ f mn f Lm Ú + A æææf mn cosh( f) cosh( f ) Ú cosh( f) A df A mn df mn f cosh( fmn) A..6 Snce eaton A..6 converges as wll be shown below, n the lmt as A f the second ntegral n euaton A..6 wll approach zero. Applyng euaton A..4 to euaton A..6 result n the followng: di df mn = Lm A æææ fmn cosh( A) cosh( f f da Lm 3 / + A æææ fmn [cosh( f) cosh( fmn)] ) df Ú mn mn A snh( f ) df m A..7 In order to determne the sgn of A..7 the followng theorem from elementary calculus wll be used: 63

180 Appendx : (Contnued) If g(x) f(x) h(x), then b Ú b Ú b Ú M = g( x) dx f ( x) dx h( x) dx A..8 a a a Snce the followng relatonshp holds: f f f mn 0, also holds: snh( f) snh( f ) mn A..9 Therefore, di df mn < A Lm æææ fmn f Ú A [ snh( f) cosh( A) cosh( f mn )] 3 / df + z A..0 A Lm da z= æææ A.. cosh( f ) df A fmn Ú f cosh( A) mn mn mn The ntegral n A..0 can be ntegrated analytcally by means of transformaton u = cosh(f). Euaton A..0 becomes: di df mn Lm cosh( f ) < A æææ fmn cosh( A) + z [ u) cosh( f )] mn A.. Evaluaton of A..: di df mn Lm cosh( f ) < A æææ fmn cosh( A) + z z [ cosh( f ) cosh( f )] mn mn A..3 Takng the lmt of A..3 results n: 64

181 Appendx : (Contnued) di df mn < [ cosh( f mn < ) cosh( f mn )] 0 A..4 It s evdent that the sgn of I wll mantan for I and therefore t was establshed that: dxd df mn < 0 A..5 and therefore t was proved that as f mn decreases Xd ncreases n the range set n A..9. It stll must be proved that A..6 converges. TO accomplsh ths the same procedure s used n A..8 wll be appled. In fact, the followng neualty follows from euaton A..: f snh( f) I < Ú df A..6 snh( fmn) f cosh( f) cosh( f ) mn mn The same transformaton shown above [u=cosh(f)] s used to ntegrate A..6 and renders: I < cosh( f) cosh( f snh( f mn ) mn ) A..7 whch means that X d converges and sets an upper lmt for t that may be useful n numercal evaluatons of the ntegrals nvolved. 65

182 Appendx 3: Dervaton of K terms for Adustable Dscretzaton of Ltster et al. Type Interactons: Ths dervaton was performed n Chapter 3. Type Interactons: The only combnaton for all nteractons gve partcles n the th nterval s shown both partcles are from the Ith nterval. It then follows that: = b A.3. R,, N The factor _ ensures that the nteracton s not counted twce. Type 3 Interactons: Consder nteractons between partcles n the Ipth and the th nterval where some of the resultng partcles fall nto the th nterval and some are too large. A partcle of sze between a and a + da n the th nterval must aggregate wth a partcle n the sze range (Ip)/ < v < (I+)/ a from the Ipth nterval to gve a partclen the th nterval. The rate of formaton of partcles n the th nterval by such stuatons s: ( + ) / ( p) / [3] a dr p, = b p, N pn da A.3. ( + p) / / ( ) Integratng to get the total rate of successful nteractons between the Ipth and the th ntervals gves: R / ( p+ ) / ( + p) / ˆ ( ) Ê + = b Á p, N pn A.3.3 Ë [3] p, 66

183 Appendx 3: (Contnued) whch yelds the followng value for K 3 : K k / / + ( + k ) / 3 / / = A.3.4 ( ) Type 4 Interactons: Consder the nteractons between partcles n the th and th ntervals where some remove partcles from the th nterval. A partcle of sze between a and a + da n the th nterval must nteract wth a partcle n the sze range (I+)/ a < v < (I+)/ to remove partcles n the th. The total rate of successful collsons of ths type s: dr b N N [4] =,,, ( + ) / / ada ( ) A.3.5 Integraton over the th nterval gves: dr / [4] + ( ) /,, N, N / = b A.3.6 ( ) Type 5 Interactons: All combnatons are successful n removng partcles from the th nterval, and therefore: [5] dr, b, N, = N A

184 68 Appendx 4: Proof of Volume Correcton Factors for Dscretzaton of Ltster et al. Ths proof shows that the choce of correcton factors gven n euatons (3.3.4) meet the necessary reurements for conservaton of the zeroth and thrd moments of the dstrbuton m dt dm b = A.4. 3 = 0 dt dm A.4. Euaton from the text: Â ˆ Á Ë Ê = agg dt dn v dt dm 3 Â Â Â Â Â Â Â + = = + = = + + = = ˆ Á Á Ë Ê + + ˆ + Á Á Ë Ê + + ˆ Á Á Ë Ê ˆ Á Á Ë Ê + = ˆ Á Ë Ê ) (, / ) ( / / ) (, / ) ( / / / / ) ( ) (,, / ) ( / / / / / ) ( ) (, / ) ( / / ) (, ) ( ) ( ) ( ) ( S S k k k k k S k k S k k k k k k k k S k k S k S agg N N C N N C N N N C N N C N N dt dn b b b b b b (A 4.7)

185 69 Appendx 4: (Contnued) In the brackets, the summaton of terms and 5 eual 0: ˆ Á Á Ë Ê + = Â / / ) ( ) (, S N N b ˆ Á Á Ë Ê = Â / / ) ( ) (, S N N b =0 A.4.4 term + term 3 = Â Â = = k k S k S N N ) ( ) ( b, Â + = = ) (, S N N b A.4.5 Fnally, combnng all terms: N N N N N N dt dm,,, 0 b b b = ˆ Á Ë Ê = Â Â = A.4.6 Proof for the thrd moment usng euaton 3.3.4: Â ˆ Á Ë Ê = agg dt dn v dt dm 3

186 70 Appendx 4: (Contnued) Â Â Â Â Â Â Â + = = + = = + + = = ˆ Á Á Ë Ê + + ˆ + Á Á Ë Ê + + ˆ Á Á Ë Ê ˆ Á Á Ë Ê + = ˆ Á Ë Ê ) (, / ) ( / / ) (, / ) ( / / / / ) ( ) (,, / ) ( / / / / / ) ( ) (, / ) ( / / ) (, ) ( ) ( ) ( ) ( S S k k k k k S k k S k k k k k k k k S k k S k S agg N N C N N C N N N C N N C N N dt dn b b b b b b A 4.7 Addng terms and 4 wthn the brackets yelds: S N N dt dm / ) ( ) (, 3 = Â = b A.4.8 Summng terms and 3 yelds: ( ) / ) ( ) ( ) (, 3 + = Â Â = = k k S k S N N dt dm b (( ) / ) ( ) (, 3 + = + = Â S N N dt dm b A.4.9 Summng terms 4, A.4.8, and A.4.9 yelds: Â Â + = = =, / ) (, 3 N N N N dt dm b b A.4.0

187 Appendx 4: (Contnued) Fnally, Ê dm ˆ 3 ( ) / = Â v Á Â b, NN Â b, NN dt A.4. Ë = = + Rememberng that, =,I, and v + / v = dm 3 = 0 dt A.4. 7

188 Appendx 5: MATLAB Code for Mathematcal Model Populaton Balance Soluton Program (pbesol.m) %Orgnally wrtten by Scott Fsher (998), modfed by C. O Bren (003) %functon [Iagg,momentserror]=pbesoltstper(x) % fle used to set up and solve the Populaton Balance euatons % and solve them. Hamaker=0.79e0; %/* Joules *//* Eff. Hamaker Const. eps_=80; % /* Delectrc Const. medum eps_o=8.854e; %/* Delectrc Const. vac. boltzmann=.38044e3; %/* J/K */ Boltzmann Const. vscosty=.9e4; %/* Newton.sec/m */ e=.6097e9; %/* Coulombs */ /* electron charge [ N_Avo=6.069e3; %/* /mol *//* Avogadro's Num. [ Ic=.0e; %/* mole/m3 Impurtes Conc. */ (salt)[mole/m3] t=93.5; % Temperature %/* Calculate debye length */ debye_length = srt(.0*ic*e*e*n_avo/(eps_o*eps_*boltzmann*t)); 7

189 Appendx 5: (Contnued) %Input feld: % %Partcle Dameter Range: Ds=0.3E6; %Smallest Dameter n dscretzaton range Dl=.0E6; %Largest Dameter n dscretzaton range % % Enter dscretzng uantty global ; =6; % %Calculate upper and lower volume; Vs=4/3*p*(Ds/)^3; Vl=4/3*p*(Dl/)^3; %Calculate number of euatons needed to cover range global Z Z=cel((/log())*log(Vl/Vs)); %Create the lst of volumes; Vm=Vs; 73

190 Appendx 5: (Contnued) for I= :Z, V(I)=Vs*^(I/); end %calculate dameters; D=*(V*3/4/p).^(/3); %Convert ths to dmensonless length form; L=(V/Vm).^(/3); % Calculate Do for later Do=(6*Vm/p)^(/3); %make the ntal PSD; %Create ntal PSD % for test purposes ths wll be gven here but n realty % ths wll be read from a fle and the paramters n the % INPUT feld above wll be calculated from the read fle % No=.475E+0; %part per ml from experment ntal condton; mean=0.503e06 ; %mean of ntal dstr e05 74

191 Appendx 5: (Contnued) sgma=0.0063e6; %std dev of ntal dst; N=zeros(Z,); for I=:Z, %N(I)=gaussnt('ntaldst',D(I)*^(/(*3*)),D(I)*^(/(*3*)),500,mean,sgma,No); %N(I)=uad8('ntaldst',D(I)*^(/(*3*)),D(I)*^(/(*3*)),[],[],mean,sgma,No); N=No.*normpdf(D,mean,sgma)./max(normpdf(D,mean,sgma)); end %n part cm^3 %N=N.*000; %convert to part/cm^3 Nd=N; %save to look at % %Create kernel euaton; % global F F=zeros(Z); global W W=zeros(Z); 75

192 Appendx 5: (Contnued) global B pot_a=zeros(z,); pot_b=zeros(z,); tot_e=zeros(z,); h=zeros(z,); %h=zeros(z,); % constants for aggregaton kernel vs=.00; %vscosty n cp T=93.5; % Temperature %h=lnspace(5,500,z).*e9; %G=9; for I=:Z, for J=:Z, F(I,J)=cmper(D(I),D(J),vs,T); [W(I,J),pot_a(I),pot_b(I),h(I),tot_e(I)]=man_repulson(D(I),D(J)); Dz(I)=D(I); Dx(J)=D(J); 76

193 Appendx 5: (Contnued) %B(I,J)=cmper(D(I),D(J),vs,T)+cmdsed(D(I),D(J),vs,.05,); %B(I,J)=cmortho(D(I),D(J),G); end end F=F.*000000; %convert from m^3/s to cm^3/hr %W=W.*000000; %convert from m^3/s to cm^3/hr test % %Collson Mechansm fgure(); mesh(dz/e6,dx/e6,log(f)); axs eual axs normal ht=ttle('log of Perknetc Collson Mechansm for HHF'); hx=xlabel('partcle Dameter [mcron]'); hy=ylabel('partcle Dameter [mcron]'); hz=zlabel('stablty Rato'); set([ht,hx,hy,hz],'unts','normalzed'); 77

194 Appendx 5: (Contnued) set(gca,'xscale','log','yscale','log'); drawnow; %Stablty Ratos fgure(); mesh(dz/e6,dx/e6,log(w)); axs eual axs normal ht=ttle('log of Stablty Ratos ); hx=xlabel('partcle Dameter [mcron]'); hy=ylabel('partcle Dameter [mcron]'); hz=zlabel('stablty Rato'); set([ht,hx,hy,hz],'unts','normalzed'); set(gca,'xscale','log','yscale','log'); drawnow; % %Map of Aggregaton Coeffcent % 78

195 Appendx 5: (Contnued) %ww=log(w)./00; B=F./W; fgure(3); mesh(dz/e6,dx/e6,log(b)); axs eual axs normal ht=ttle('log of Aggregaton Coeffcent ); hx=xlabel('partcle Dameter [mcron]'); hy=ylabel('partcle Dameter [mcron]'); hz=zlabel('stablty Rato'); set([ht,hx,hy,hz],'unts','normalzed'); set(gca,'xscale','log','yscale','log'); drawnow; % %Integraton of the dfferental euaton; OPTIONS=odeset('RelTol',e4,'Stats','on'); TSPAN=[ ]; 79

196 Appendx 5: (Contnued) TSPAN=[ ]; [T,Y]=ode45('dffvec',TSPAN,N,OPTIONS); [lnenum,k]=sze(y); fgure(4); plot(t,y); %calculate Iagg; M0T0=moment(L,Y(,:),0); for I=:lnenum, Iagg(I)=moment(L,Y(I,:),0)/M0T0; end %calculate Moments from data for J=0:6, for I=:lnenum, resultmoments(j+,i)=moment(l,y(i,:),j)/moment(l,y(,:),j); end end Yn=Y; 80

197 Appendx 5: (Contnued) %fname=strcat('psdper',numstr()); fname=strcat('psd',numstr()); save(fname) fgure(5); semlogx(d,yn(,:),d,yn(,:),d,yn(3,:),d,yn(4,:),d,yn(5,:),d,yn(6,:),d,yn(7,:)); %axs([ ]); %hold on; Man repulson program (Orgnally created by Maruez (994), updated by C. O Bren (003) functon [w,pot_a,pot_b,h,tot_e]=man_repulson(d,d) Dam_o=0.0e6; %dameter of smallest partcle epslon=.0e5; hmn=.0e9; %/* Relatve Error %/* Mn nteg. length for forces n stab. p= ; % /* P Hamaker=0.79e0; %/* Joules *//* Eff. Hamaker Const. eps_=80; % /* Delectrc Const. medum eps_o=8.854e; %/* Delectrc Const. vac. 8

198 Appendx 5: (Contnued) boltzmann=.38044e3; %/* J/K */ Boltzmann Const. vscosty=.00; %/* Newton.sec/m */ e=.6097e9; %/* Coulombs */ /* electron charge [ N_Avo=6.069e3; %/* /mol *//* Avogadro's Num. [ Ic=.0e; %/* mole/m3 Impurtes Conc. */ (salt)[mole/m3] MAXIMO=90; t=93.5; cmap=zeros(,50); xmap=zeros(,50); s=zeros(,30); we=zeros(,30); %/* *************************************** */ for = :MAXIMO5; ndex() = ; end %/* *************************************** */ % /*Defne some relavent constants */ 8

199 Appendx 5: (Contnued) %t = 308.0; %/*Temperature */ % /*beleved to be converson constants.*/ a4 = 0.; c =.0; c = c/35.0; b = 6.0*c; c =.0; c = c/85.0; b = *c; c =.0; c = c/ ; b3 = 856.0*c; c =.0; c = c/50.0; b4 = 9.0*c; c =.0; c = c/55.0; b5 =.0*c; c_3 = 3.0*c; c =.0; c = c/3.0; c_3 = 3.0*c; c_33 = 9.0 *c; c =.0; c = c/3.0; 83

200 Appendx 5: (Contnued) c_4 =.0*c; c =.0; c = c/97.0; c_4 = 93.0*c; c_43 = 700.0*c; c_44 = 796.0*c; c =.0; c = c/6.0; a = 5.0*c; c_5 = 439.0*c; c =.0; c = c/53.0; c_5 = *c; c =.0; c = c/404.0; a3 = 97.0*c; c_53 = 845.0*c; c_63 = 859.0*c; c =.0; c = c/7.0; c_6 = 8.0*c; c =.0; c = c/565.0; a = 408.0*c; 84

201 Appendx 5: (Contnued) c_6 = *c; c =.0; c = c/40.0; c_64 =.0*c; %/* Roots for Gaussan Quadrature */ s() = ; s() = ; s(3) = ; s(4) = ; s(5) = ; s(6) = ; s(7) = ; s(8) = ; s(9) = ; s(0)= ; for = :0 s(0+) = s(0+); end %/*symetrc about zero*/ 85

202 Appendx 5: (Contnued) % /* Weghts for Gaussan Quadrature */ we() = ; we() = ; we(3) = ; we(4) = ; we(5) = ; we(6) = ; we(7) = ; we(8) = ; we(9) = ; we(0) = ; for = :0 we(0+) = we(0+); %/*symetrc about zero */ end %/* Surface potental scalng factor */ Cp = 0.6; 86

203 Appendx 5: (Contnued) Co = 6.5e8*exp( *Cp); %Cp=0.0; %/* Calculate debye length */ debye_length = srt(.0*ic*e*e*n_avo/(eps_o*eps_*boltzmann*t)); %fprntf(' Debye Length = %0e\n ',debye_length); %fprntf(' Cp = %4.f Temp = %5.føC \n\n',cp,(t 73)); %fprntf(' D D HHF SR OHW Rep Maruez SR\n\n'); _o = Co*exp(Cp*log(0.5*d)); %_o= Total Charge on partcle enter n Coulombs*/ ps_do = _o/4.0/p/eps_/eps_o/(0.5*d); %;/*Unts are Volt*/ %/* Actual surface scalng porton*/ = Co*exp(Cp*log(0.5*d)); ps = /4.0/p/eps_/eps_o/(0.5*d); f ps_do > ps psmax = ps_do; psmn = ps; 87

204 Appendx 5: (Contnued) do=d; do=d; else psmax = ps; psmn = ps_do; do=d; do=d; end psmax = psmax*e/boltzmann/t; psmn = psmn*e/boltzmann/t; cmn =.0*cosh(psmn);%mnmum case cmap()= cmn; cmax = 0.0*cosh(psmax); permso = 0; x_ref = sep_ref(psmax,psmn,cmn,permso); %ntegrate e. 6.4 by RKF monotoncally xmap() = x_ref; xmn = xmap(); 88

205 Appendx 5: (Contnued) xmap(4) = 5.0; xmap() = (xmap(4)xmap())*0.05; for =3:40 xmap() = ()*xmap() + x_ref; end xmap() =xmap()+ x_ref; cmax =.0; for =:4 xd = xmap(); c_der = cmax; c_z = cmn; permso = 0; er = 0.0; er = 0.0; f (er > epslon) & (er > 0.0*epslon) ps_o = 0; x0 = 0.0; 89

206 Appendx 5: (Contnued) xc = 0.0; c = 0.5*(c_der + c_z); [ps_o,x0,xc,x_cen]=sep_placas(psmax,psmn,c,permso,xd); f xc > xd c_der = c; else c_z = c; end er = abs(xdxc); er = abs(c_derc_z); end %/* f */ cmap() = c; cmn = c; end %/* next */ %/* Map formed */ cmax = 0.0*cosh(psmax); cmn = cmap(); 90

207 Appendx 5: (Contnued) xmn = xmap(); xmax = xmn; v = flat_ocfe(xmap(4),t,psmax,psmn,x_ref,cmap,cmax,xd,xmax,cmn,xmap); h = xmap(4); cmax = 0.0*cosh(psmax); cmn = cmap(); xmn = xmap(); v = flat_ocfe(xmap(35),t,psmax,psmn,x_ref,cmap,cmax,xd,xmax,cmn,xmap); h = xmap(35); f do > do [w,w3,pot_a,pot_b,h,tot_e]=w_per((0.5*do),(0.5*do),t,psmax,psmn,cmn,xmn,h,h,xd,x_ref); else [w,w3,pot_a,pot_b,h,tot_e]=w_per((0.5*do),(0.5*do),t,psmn,psmax,cmn,xmn,h,h,xd,x_ref); end 9

208 Appendx 5: (Contnued) Calculaton of Perknetc Stablty Ratos (w_per.m) (Orgnally created by Maruez (994), updated by C. O Bren (003)) functon [w,w3,pot_a,pot_b,h,tot_e]=w_per(r,r,t,ps,ps,cmn,xmn,h,h,xd,x_ref) %Calculaton of Stablty rato %defne constants epslon=.0e5; %/* Relatve Error hmn=.0e9; %/* Mn nteg. length for forces n stab. p= ; % /* P Hamaker=0.79e0; %/* Joules *//* Eff. Hamaker Const. eps_=80; % /* Delectrc Const. medum eps_o=8.854e; %/* Delectrc Const. vac. boltzmann=.38044e3; %/* J/K */ Boltzmann Const. vscosty=.00; %/* Newton.sec/m */ e=.6097e9; %/* Coulombs */ /* electron charge [ N_Avo=6.069e3; %/* /mol *//* Avogadro's Num. [ Ic=.0e; %/* mole/m3 Impurtes Conc. */ (salt)[mole/m3] %H=0:000; %H=H./e9; 9

209 Appendx 5: (Contnued) debye_length = srt(.0*ic*e*e*n_avo/(eps_o*eps_*boltzmann*t)) ; f=zeros(,0); f=zeros(,0); f3=zeros(,0); r=zeros(,0); s=zeros(,5); we=zeros(,5); s=zeros(,5); we=zeros(,5); % /* Integrate on the nonasymptotc regon of the functon */ s() =.0; s() = ; s(3) = ; s(4) = ; for = :4 s(4+) = s(4+); end 93

210 Appendx 5: (Contnued) we() = 0.0; we() = ; we(3) = ; we(4) = ; for = :4 we(4+) = we(4+); end factor = 4.0*p*r*r*Ic*N_Avo*boltzmann*t/(r+r)/debye_length/debye_length; ntegral = 0.0; %ntegral = 0.0; ntegral3 = 0.0; %fprntf(' Ps = %f Ps = %f \n',ps,ps); %fprntf(' R = %f R = %f \n',r*debye_length,r*debye_length); %fprntf(' DL h HHF Pot OCFE Pot Maruez\n'); %fprntf(' ========== ========== ========== =========\n'); nc = 6; ho =.0e9; %Mnmum length : 0 angstroms 94

211 Appendx 5: (Contnued) b = r + r + ho; for =:6 ho = ho * 3.0; a = b; b = b + ho; for =:(nc+) r() = 0.5*((ba)*s() + b + a); h = r() (r+r); pot_a = attracton(r,r,h); pot_b = hhf_rep(r,r,h,t,ps,ps)*factor; dl_r = debye_length*r; dl_r = debye_length*r; dl_h = debye_length*h; dl_ps = ps; dl_ps = ps; [p,p]=feld_potoc(dl_r,dl_r,dl_ps,dl_ps,dl_h,t,cmn,xmn,v,v,h,h,xd,x_ref); 95

212 Appendx 5: (Contnued) tot_e=(pot_a + pot_b); %pot_c = p; pot_d = p; pot_c =pot_c.* factor; pot_d = pot_d.* factor; ar = (pot_a + pot_b)./t./boltzmann; f() = exp(ar)/(r().*r()); %ar = (pot_a + pot_c)./t./boltzmann; %f() = exp(ar)./(r().*r()); ar = (pot_a + pot_d)./t./boltzmann; f3() = exp(ar)./(r().*r()); %fprntf(' %0e %0e %0e %0e \n',dl_h,pot_b,pot_c,pot_d); end % % Next par_nt = 0.0; for = :(nc+) par_nt = par_nt + we().*f(); ntegral = ntegral + 0.5*(ba).*par_nt; 96

213 Appendx 5: (Contnued) par_nt = 0.0; end %for = :(nc+) % par_nt = par_nt + we().*f(); %ntegral =ntegral + 0.5*(ba).*par_nt; % par_nt = 0.0; %end for = :(nc+) par_nt = par_nt + we().*f3(); ntegral3 = ntegral *(ba).*par_nt; end end %} %/* Next */ %/* Integrate on the asymptotc regon of the functon */ % /*Laguerre roots and weghts.*/ s() = ; s() = ; s(3) = ; 97

214 Appendx 5: (Contnued) s(4) = ; s(5) = ; s(6) = ; s(7) = ; s(8) = ; s(9) = ; s(0) = ; s() = ; s() = ; s(3) = ; s(4) = ; s(5) = ; we() = ; we() = ; we(3) = ; we(4) = ; we(5) = ; 98

215 Appendx 5: (Contnued) we(6) = ; we(7) = ; we(8) = ; we(9) = e5; we(0) = e7; we() = e9; we() = e; we(3) = e3; we(4) = e6; we(5) = e0; ntegral = (r+r)*ntegral; %ntegral = (r+r)*ntegral; ntegral3 = (r+r)*ntegral3; w = ntegral; %w = ntegral; 99

216 Appendx 5: (Contnued) w3 = ntegral3; aggdff.m (Orgnally wrtten by Scott Fsher (998), modfed by C. O Bren (003)) functon ret=aggdff(t,n,i) % dfferental for growth t s tme, N s the populaton % I s the ndex %defne globals whch should be n place global Z B ; %ntalze summaton terms; S=0; S=0; S3=0; S4=0; S5=0; S6=0; %buld aggregaton terms %S for J=:(IS()), 00

217 Appendx 5: (Contnued) f (((I)<) (J<)) S=S; else S=S+(B(I,J)*N(I)*N(J)*(^((JI+)/))/(^(/))); end end %S for P=:, for J=(IS(P)):(IS(P)), f (((IP)<) (J<)) S=S; else S=S+(B(IP,J)*N(IP)*N(J)*((^((JI+)/)+^((P)/))/(^(/)))); end end end %S3 0

218 Appendx 5: (Contnued) f (I)< S3=S3; else S3=((/)*B(I,I)*N(I)^); end %S4 for P=:(), for J=((I+)S(P)):((I+)S(P+)), f (((IP)<) (J<)) S4=S4; else S4=S4+(B(IP,J)*N(IP)*N(J)*((^((JI)/)+^(/)^((P)/))/(^(/)))); end end end %S5 for J=:(IS()+), 0

219 Appendx 5: (Contnued) f ((I<) (J<)) S5=S5; else S5=S5+(B(I,J)*N(I)*N(J)*((^((JI)/))/(^(/)))); end end %S6 for J=(IS()+):(Z), f ((I<) (J<)) S6=S6; else S6=S6+(B(I,J)*N(I)*N(J)); end end %fprntf('s,s,s3,s4,s5,s6: %.0e %.0e %.0e %.0e %.0e %.0e \n',s,s,s3,s4,s5,s6); ret=s+s+s3+s4s5s6; 03

220 Appendx 5: (Contnued) attracton.m (Hamaker Calculaton for van der Waals Forces) (Orgnally wrtten by Scott Fsher (998), modfed by C. O Bren (003)) functon ret=attracton(r,r,h) Hamaker=0.79e0; %/* Eff. Hamaker Const. %/* NOTICE : r,r,h may be expressed n any unts */ den = h.*(h + (.0*(r + r))); den = den + (4.0.*r.*r); ret= (Hamaker*(((.0.*r.*r)./den) + ((.0.*r.*r)./den)+ (log(den./den)))./6.0); avgd.m (Orgnally wrtten by Scott Fsher (998), modfed by C. O Bren (003)) lb=5; %set ths number on where to break the dstrbuton for I=:lnenum, %volume average %Vavg(I)=sum(4./3.*p.*((D/).^3).*Yn(I,:))./sum(Y(I,:)); %Vavg(I)=sum(4./3.*p*((D(:lb)/).^3).*Yn(I,:lb))./sum(Y(I,:lb)); %Vavg(I)=sum(4/3*p*((D(up:length(D))/).^3).*Yn(I,up:length(D)))./sum(Y(I,up:length (D))); %Vavg=.*((Vavg.*(3/4/p)).^(/3)); 04

221 Appendx 5: (Contnued) %Vavg=.*((Vavg.*(3/4/p)).^(/3)); %Vavg=.*((Vavg.*(3/4/p)).^(/3)); %Davg(I)=(Mavg*3/.05/4/p).^(/3)*; %length average Davg(I)=sum(Dn.*Yn(I,:))./sum(Y(I,:)); temp=fnd(dn<=davg(i)); %lb=cel((temp()+length(dn))/); lb=temp(length(temp)); up=lb; Davg(I)=sum(Dn(:lb).*Yn(I,:lb))./sum(Y(I,:lb)); Davg(I)=sum(Dn(up:length(D)).*Yn(I,up:length(D)))./sum(Y(I,up:length(D))); end Davg=polyval(polyft(TSPAN,Davg,3),TSPAN); Davg=polyval(polyft(TSPAN,Davg,3),TSPAN); Orthoknetc Stablty Rato (ceortho.m) Wrtten by C. O Bren (003) functon ret=ceortho(d) 05

222 Appendx 5: (Contnued) %Calculaton of Stablty rato %d=dameter of agglomerate %a=prmary partcle sze a=0.0e6; %calculate radus %a0=d0/; %calculate number of prmares at each dameter %a=((d./).^3)./(a0.^3); term=./((log(d./(*a)))^0.9); term=(((d./(*a))^0.075)0.)^(3/); ret=term*term; Dfferental Sedmentaton (cmdsed.m) Orgnally wrtten by Scott Fsher (998), modfed by C. O Bren (003) functon ret=cmdsed(d,d,vs,ps,p) %mfle to calculate dfferental settlng collson rate constant %unts needed %D,D meters 06

223 Appendx 5: (Contnued) %vs cp %ps,p Kg/m^3 Psdensty of sold, p densty of flud %gravtatonal constant g= ; %m/s^ %calculate t ret=(*p*g/(9*vs*0.00))*(psp)*((d/+d/).^3).*(dd); ret=abs(ret); Orthoknetc Collson Mechansm (cmortho.m) Orgnally wrtten by Scott Fsher (998), modfed by C. O Bren (003) functon ret=cmortho(d,d,g) %mfle to calculate orthoknetc collson rate constant %unts needed %D,D meters % G /s average shear rate %calculate t ret=(4/3)*g*((d/+d/).^3); 07

224 Appendx 5: (Contnued) Perknetc Collson Mechansm (cmper.m) Orgnally wrtten by Scott Fsher (998), modfed by C. O Bren (003) functon ret=cmper(d,d,vs,t) %mfle to calculate perknetc collson rate constant %unts needed %D,D meters %vs cp %T Kelvn %defne constants kb=.38066e3; %J/K %calculate t %ret=(/3)*(kb*t/(vs*0.00))*((d/+d/).^)./(d.*d/4); ret=(/3)*(kb*t/(vs*0.00))*((d/+d/).^)./(d.*d/4); control.m (Orgnally wrtten by E. Maruez (994), modfed by C. O Bren (003)) functon ret=control(x,c) f (.0*cosh(x) + c) >= 0.0 ret=; 08

225 Appendx 5: (Contnued) else ret=0; end dfracd.m (Orgnally wrtten by S. Fsher (998), modfed by C. O Bren (003)) functon ret=dfracd(d,df,ps,d0) %calculate radus a0=d0/; %calculate number of prmares at each dameter num=((d./).^3)./(a0.^3); %relaton from Kuster for I=:length(D), f (num(i)>. & num(i)<.8) Dn(I)=*.8*a0; else Dn(I)=*a0*(num(I)./ps).^(/Df); end end 09

226 Appendx 5: (Contnued) ret=dn; dffvec.m (Orgnally wrtten by E. Maruez (994), modfed by C. O Bren (003)) functon ret=dffvec(t,n) %functon to buld the vector for the dfferental euaton %solver to use; global Z; for I=:(Z), temp(i)=aggdff(t,n,i); end %fprntf('temp: %.0e \n',temp); ret=temp'; f.m (Orgnally wrtten by E. Maruez (994), modfed by C. O Bren (003)) functon ret=f(t,ps,c) arg =.0*cosh(ps) + c; f (arg < 0) fprntf (' message = %d arg f = %0f\n',message,arg); end 0

227 Appendx 5: (Contnued) ret= srt(arg); f.m (Orgnally wrtten by E. Maruez (994), modfed by C. O Bren (003)) functon ret=f(t,ps,c) arg =.0*cosh(ps) + c; f (arg < 0) fprntf (' message = %d arg f = %0f\n',message,arg); end ret = srt(arg); hhfrep.m (Orgnally wrtten by E. Maruez (994), modfed by C. O Bren (003)) %fle used to calculate hhf_repulson functon ret=hhf_rep(r,r,h,t,ps,ps) p= ; % /* P eps_=80; % /* Delectrc Const. medum eps_o=8.854e; %/* Delectrc Const. vac. boltzmann=.38044e3; %/* J/K */ Boltzmann Const. e=.6097e9; %/* Coulombs */ /* electron charge [ N_Avo=6.069e3; %/* /mol *//* Avogadro's Num. [

228 Appendx 5: (Contnued) Ic=.0e; %/* mole/m3 Impurtes Conc. */ (salt)[mole/m3] debye_length = srt(.0*ic*e*e*n_avo/(eps_o*eps_*boltzmann*t)); ys = ps; ys = ps; yp = 0.5*(ys + ys); y_ = 0.5*(ys ys); term = (yp*yp)*log(.0 + exp(debye_length*h))+ (y_*y_)*log(.0exp( debye_length*h)); ret = term*.0; modelcomp.m (Orgnally wrtten by S. Fsher (998), modfed by C. O Bren (003)) %model compare %plot dameters plotdam; load psd6; %load psd4co; %convert fractonal dmenson Dn=dfracd(D,.7,,0.503e6);

229 Appendx 5: (Contnued) %Dn=D; avgd; hold on; plot(tspan,davg./e6,'k'); plot(tspan,davg./e6,'k:'); plot(tspan,davg./e6,'k'); load psd6stco; Dn=dfracd(D,.7,,0.503e6); %Dn=D; avgd; hold on; plot(tspan,davg./e6,'r'); plot(tspan,davg./e6,'r:'); plot(tspan,davg./e6,'r'); %axs([ ]); legend('exp Davg','EXP D','EXP D','SF MODEL Davg','SF MODEL D','SF MODEL D','MODEL Davg','MODEL D','MODEL D',) 3

230 Appendx 5: (Contnued) moment.m (Orgnally wrtten by S. Fsher (998), modfed by C. O Bren (003)) functon ret=moment(l,y,mom) %for szendependent r=l()/l(); ret=sum(((((+r)/).*l).^mom).*y); %for sze dependent %ret=trapz(l,l.^mom.*y); plotdam.m (Orgnally wrtten by S. Fsher (998), modfed by C. O Bren (003)) % mfle to plot partcle szes for Exp C load sp.dat; X=sp; %plot(tspan,davg/e6,'b',tspan,davg/e6,'r',tspan,davg/e6,'g') %hold on plot(x(:,),x(:,)/e4,'ko') hold on; plot(x(:,),x(:,4)/e4,'kv') plot(x(:,),x(:,6)/e4,'kd') 4

231 Appendx 5: (Contnued) xlabel('tme [hr]') ylabel('partcle Dameter [mcron]') ttle('partcle Dameter versus Tme') S.m (Orgnally wrtten by S. Fsher (998), modfed by C. O Bren (003)) functon ret=s(x) global ; ret=floor((*log(^(x/))/log())); sep_mon.m (Orgnally wrtten by E. Maruez (994), modfed by C. O Bren (003)) functon ret=sep_mon(ps,ps,c,permso,xd) epslon=.0e5; %/* Relatve Error hmax = 5.0e3; h = hmax; u = ps; ps = ps; x = 0.0; valdo = ; a4 = 0.; 5

232 Appendx 5: (Contnued) g =.0; g = g/35.0; b = 6.0*g; g =.0; g = g/85.0; b = *g; g =.0; g = g/ ; b3 = 856.0*g; g =.0; g = g/50.0; b4 = 9.0*g; g =.0; g = g/55.0; b5 =.0*g; c_3 = 3.0*g; g =.0; g = g/3.0; c_3 = 3.0*g; c_33 = 9.0 *g; g =.0; g = g/3.0; c_4 =.0*g; g =.0; g = g/97.0; c_4 = 93.0*g; c_43 = 700.0*g; 6

233 Appendx 5: (Contnued) c_44 = 796.0*g; g =.0; g = g/6.0; a = 5.0*g; c_5 = 439.0*g; g =.0; g = g/53.0; c_5 = *g; g =.0; g = g/404.0; a3 = 97.0*g; c_53 = 845.0*g; c_63 = 859.0*g; g =.0; g = g/7.0; c_6 = 8.0*g; g =.0; g = g/565.0; a = 408.0*g; c_6 = *g; g =.0; g = g/40.0; c_64 =.0*g; 7

234 Appendx 5: (Contnued) f (permso == ) fprntf('error n sep_mon: %9.6f %0.8f %0.8f\n',x,ps,f(x,ps,c)); end f absval(psps) >= epslon & (valdo ==) f ps > ps %/* */ k = h*f(x,ps,c); xo = x + 0.5*h; u = ps + 0.5*k; f u > ps %/* */ k = h*f(xo,u,c); xo = x + c_3*h; u = ps + c_3*k + c_33*k; f u > ps %/* 3 */ k3 = h*f(xo,u,c); 8

235 Appendx 5: (Contnued) xo = x + c_4*h; u = ps + c_4*k c_43*k + c_44*k3; f u > ps %/* 4 */ k4 = h*f(xo,u,c); xo = x + h; u = ps + c_5*k 8.0*k + c_5*k3 c_53*k4; f u > ps % /* 5 */ k5 = h*f(xo,u,c); xo = x + 0.5*h; u = ps c_6*k +.0*k c_6*k3 + c_63*k4 c_64*k5; f u > ps %/* 6 */ k6 = h*f(xo,u,c); r = absval(k/ *k3/ *k4/ k5/ *k6/55.0); f r < epslon/54.9 9

236 Appendx 5: (Contnued) er_n = 4.5; else er_n = 0.84*srt(srt((epslon/r))); end f r < epslon u = ps + a*k + a*k3 + a3*k4 a4*k5; f control(u,c)== %/* 7 */ ps = u; x = x+h; u = f(x,ps,c); f (permso == ) fprntf('sep_mon error : %9.6f %0.8f %0.8f %0.8f %0.8f %0f\n ',x,ps,u,er_n,r,h); end f er_n < 0. h = h*0.; else f er_n > 4.0 0

237 Appendx 5: (Contnued) h = h*4.0; else h = h*er_n; end end %/* 7 */ else h = h*0.; end else h = h*0.; f (h < hmn) fprntf('sep_mon error 3: h < hmn.!!!\n'); valdo = 0; end end %/* 6 */ else h = h * 0.; end %/* 5 */ else h = h * 0.; end %/* 4 */ else h = h*0.;

238 Appendx 5: (Contnued) end %/* 3 */ else h = h*0.; end %/* */ else h = h*0.; end %/* */ else h = h*0.; end f h > hmax h = hmax; end end %/* End Whle */ ret=x; sep_placas.m (Orgnally wrtten by E. Maruez (994), modfed by C. O Bren (003)) functon [ps_o,x0,xc,x_cen]=sep_placas(ps,ps,c,permso,xd) a4 = 0.; g =.0; g = g/35.0;

239 Appendx 5: (Contnued) b = 6.0*g; g =.0; g = g/85.0; b = *g; g =.0; g = g/ ; b3 = 856.0*g; g =.0; g = g/50.0; b4 = 9.0*g; g =.0; g = g/55.0; b5 =.0*g; c_3 = 3.0*g; g =.0; g = g/3.0; c_3 = 3.0*g; c_33 = 9.0 *g; g =.0; g = g/3.0; c_4 =.0*g; g =.0; g = g/97.0; c_4 = 93.0*g; c_43 = 700.0*g; c_44 = 796.0*g; 3

240 Appendx 5: (Contnued) g =.0; g = g/6.0; a = 5.0*g; c_5 = 439.0*g; g =.0; g = g/53.0; c_5 = *g; g =.0; g = g/404.0; a3 = 97.0*g; c_53 = 845.0*g; c_63 = 859.0*g; g =.0; g = g/7.0; c_6 = 8.0*g; g =.0; g = g/565.0; a = 408.0*g; c_6 = *g; g =.0; g = g/40.0; c_64 =.0*g; hmax = 5.0e3; 4

241 Appendx 5: (Contnued) h = hmax; u = ps; ps = ps; x = 0.0; epslon=.0e5; %/* Relatve Error f permso == fprntf('error n sep_placas: %9.6f %0.8f %0.8f\n',x,ps,f(x,ps,c)); end valdo = ; %f ((absval(u) >= epslon) & (absval(psps) > epslon) & (valdo ==) ) f (abs(u) >= epslon) & (valdo ==) f control(ps,c) == %{ /* */ k = h*f(x,ps,c); xo = x + 0.5*h; u = ps + 0.5*k; 5

242 Appendx 5: (Contnued) f control(u,c) == % { /* */ k = h*f(xo,u,c); xo = x + c_3*h; u = ps + c_3*k + c_33*k; f control(u,c)== % { /* 3 */ k3 = h*f(xo,u,c); xo = x + c_4*h; u = ps + c_4*k c_43*k + c_44*k3; f control(u,c)== %{ /* 4 */ k4 = h*f(xo,u,c); xo = x + h; u = ps + c_5*k 8.0*k + c_5*k3 c_53*k4; f control(u,c)== %{ /* 5 */ 6

243 Appendx 5: (Contnued) k5 = h*f(xo,u,c); xo = x + 0.5*h; u = ps c_6*k +.0*k c_6*k3 + c_63*k4 c_64*k5; f ( absval(u) > 705.0) fprntf('error n sep_placas: u = %0f h = %0f xo = %0f x = %0f\n',u,h,xo,x); %message = 6; end f control(u,c)== % { /* 6 */ k6 = h*f(xo,u,c); r = absval(k/ *k3/ *k4/ k5/ *k6/55.0); f r < (epslon/54.9) er_n = 4.5; else er_n = 0.84*srt(srt((epslon/r))); end f r < epslon 7

244 Appendx 5: (Contnued) u = ps + a*k + a*k3 + a3*k4 a4*k5; f control(u,c)== % { /* 7 */ ps = u; x =x + h; u = f(x,ps,c); f (permso == ) fprntf('error n sep_placas: %9.6f %0.8f %0.8f %0.8f %0.8f %0f\n ',x,ps,u,er_n,r,h); end f er_n < 0. h = h * 0.; else f er_n > 4.0 h =h * 4.0; else h = h *er_n; end end %}/* 7 */ 8

245 Appendx 5: (Contnued) else h = h * 0.; end %} else h =h * 0.; f (h < hmn) fprntf('error n sep_placas: h < hmn.!!!\n'); valdo = 0; end end %} /* 6 */ else h =h * 0.; end %} /* 5 */ else h = h *0.; end %} /* 4 */ else h =h * 0.; end %} /* 3 */ else h =h * 0.; end %} /* */ 9

246 Appendx 5: (Contnued) else h =h * 0.; end %} /* */ else h =h * 0.; end f h > hmax h = hmax; end %} /* End Whle */ end ps_o = ps; f ps > ps x_sum = sep_mon(ps,ps,c,permso,xd); else x_sum = 0.0; end x0 = x + x_sum; xc =.0.*x + x_sum; x_cen=.0.*x + x_sum; %ps_o=dr; 30

247 Appendx 5: (Contnued) %x0=dr; %xc=dr3; %x_cen=dr3; %fprntf('xc n sep_placas: %f \n', xc); sep_ref.m (Orgnally wrtten by E. Maruez (994), modfed by C. O Bren (003)) functon ret=sep_ref(ps,ps,c,permso) a4 = 0.; g =.0; g = g/35.0; b = 6.0*g; g =.0; g = g/85.0; b = *g; g =.0; g = g/ ; b3 = 856.0*g; g =.0; g = g/50.0; b4 = 9.0*g; g =.0; g = g/55.0; b5 =.0*g; 3

248 Appendx 5: (Contnued) c_3 = 3.0*g; g =.0; g = g/3.0; c_3 = 3.0*g; c_33 = 9.0 *g; g =.0; g = g/3.0; c_4 =.0*g; g =.0; g = g/97.0; c_4 = 93.0*g; c_43 = 700.0*g; c_44 = 796.0*g; g =.0; g = g/6.0; a = 5.0*g; c_5 = 439.0*g; g =.0; g = g/53.0; c_5 = *g; g =.0; g = g/404.0; a3 = 97.0*g; c_53 = 845.0*g; c_63 = 859.0*g; 3

249 Appendx 5: (Contnued) g =.0; g = g/7.0; c_6 = 8.0*g; g =.0; g = g/565.0; a = 408.0*g; c_6 = *g; g =.0; g = g/40.0; c_64 =.0*g; hmax = 5.0e3; h = hmax; u = ps; ps = ps; x = 0.0; epslon=.0e5; %/* Relatve Error f permso == fprntf('sep_ref error : %9.6f %0.8f %0.8f\n',x,ps,f(x,ps,c)); end valdo = ; 33

250 Appendx 5: (Contnued) f ((abs(u) >= epslon) & (abs(psps) > epslon) & (valdo ==) ) %{ f control(ps,c) == %{ /* */ k = h*f(x,ps,c); xo = x + 0.5*h; u = ps + 0.5*k; f control(u,c) == % { /* */ k = h*f(xo,u,c); xo = x + c_3*h; u = ps + c_3*k + c_33*k; f control(u,c)== % { /* 3 */ k3 = h*f(xo,u,c); xo = x + c_4*h; u = ps + c_4*k c_43*k + c_44*k3; 34

251 Appendx 5: (Contnued) f control(u,c)== %{ /* 4 */ k4 = h*f(xo,u,c); xo = x + h; u = ps + c_5*k 8.0*k + c_5*k3 c_53*k4; f control(u,c)== %{ /* 5 */ k5 = h*f(xo,u,c); xo = x + 0.5*h; u = ps c_6*k +.0*k c_6*k3 + c_63*k4 c_64*k5; f control(u,c)== % { /* 6 */ k6 = h*f(xo,u,c); r = abs(k/ *k3/ *k4/ k5/ *k6/55.0); f r < (epslon/54.9) er_n = 4.5; else 35

252 Appendx 5: (Contnued) er_n = 0.84*srt(srt((epslon/r))); end f r < epslon % { u = ps + a*k + a*k3 + a3*k4 a4*k5; f control(u,c)== % { /* 7 */ ps = u; x =x + h; u = f(x,ps,c); f (permso == ) fprntf('sep_ref error : %9.6f %0.8f %0.8f %0.8f %0.8f %0f\n ',x,ps,u,er_n,r,h); end f er_n < 0. h = h * 0.; else f er_n >

253 Appendx 5: (Contnued) h =h * 4.0; else h = h *er_n; end end %}/* 7 */ else h = h * 0.; end %} else h =h * 0.; f (h < hmn) fprntf('sep_ref error 3: h < hmn.!!!\n'); valdo = 0; end end %} /* 6 */ else h =h * 0.; end %} /* 5 */ else h = h *0.; end %} /* 4 */ 37

254 Appendx 5: (Contnued) else h =h * 0.; end %} /* 3 */ else h =h * 0.; end %} /* */ else h =h * 0.; end %} /* */ else h =h * 0.; end f h > hmax h = hmax; end %} /* End Whle */ end ret=x; flat_ocfe.m (Orgnally wrtten by E. Maruez (994), modfed by C. O Bren (003)) functon ret=flat_ocfe(h,t,ps,ps,x_ref,cmap,cmax,xd,xmn,cmn,xmap) epslon=.0e5; %/* Relatve Error Dam_o=0.0e6; 38

255 Appendx 5: (Contnued) Q=; Z=0; e=.6097e9; boltzmann=.38044e3; %cmap=zeros(,50); %xmap=zeros(,50); xmax=xmn; %Begn flat_ocfe%%% f ps > ps ps_temp = ps; ps = ps; ps = ps_temp; end f h <= x_ref caso = 0; else caso = ; %caso almost always = end 39

256 Appendx 5: (Contnued) %fprntf('caso: %e \n',caso); er = 0.0; er = 0.0; permso = 0; f caso == xmax=xmap(); cmax = cmap(); = ; f (h>xmax)&(<=4) =+; xmax=xmap(); cmax=cmap(); end f xmax > xmap(4) fprntf('error n flat_ocfe: h > xmax!!!\n'); end c_der = cmax; 40

257 Appendx 5: (Contnued) x_der = xmax; c_z = cmn; x_z = xmn; c_cen = 0.5*(c_der+c_z); ps_o = 0; x0 = 0.0; xc = 0.0; %dr = ps_o; %dr = x0; %dr3 = x_cen; permso = 0; [ps_o,x0,xc,x_cen]=sep_placas(ps,ps,c_cen,permso,xd); %/*put xd n here because there was no argument*/ %fprntf('drr.mat: %f %f %f \n', ps_o,x0,xc); nt = 0; f (er > epslon) & (er > 0.0*epslon) nt=nt+; 4

258 Appendx 5: (Contnued) opcon = ; f ( absval(c_cenc_z) >.0e6 ) & ( absval(c_zc_der) >.0e6 ) & ( absval(c_derc_cen) >.0e6 ) al = ((x_derx_z)./(c_derc_z)(x_cenx_z)./(c_cenc_z))./(c_derc_cen); al=al+(al==0)*eps; be = (x_cenx_z)./(c_cenc_z) al*(c_cen+c_z); be=be+(be==0)*eps; ga = x_z al*c_z*c_z be*c_z; else opcon = 0; end f opcon == dsc = be*be 4.0*al*(gah); f dsc >= 0.0 c = 0.5*(be srt(dsc))./al; f ((c > c_der) (c < c_z)) c = 0.5*(be + srt(dsc))./al; f ((c > c_der) (c < c_z)) 4

259 Appendx 5: (Contnued) opcon = 0; end end end else opcon = 0; end f opcon == %dr = ps_o; %dr = x0; %dr3 = xc; permso = 0; [ps_o,x0,xc,x_cen]=sep_placas(ps,ps,c,permso,xd); %/*put n last argument because left out*/ %fprntf('drr.mat: %f %f %f \n', ps_o,x0,xc); f h > x_cen f xc > x_cen x_z = x_cen; 43

260 Appendx 5: (Contnued) c_z = c_cen; x_cen = xc; c_cen = c; else x_z = xc; c_z = c; end else f xc < x_cen x_der = x_cen; c_der = c_cen; x_cen = xc; c_cen = c; else x_der = xc; c_der = c; end end %/* Endf Opcon */ else 44

261 Appendx 5: (Contnued) c = c_cen; xc=x_cen; f xc > h c_der = c; x_der = xc; else c_z = c; x_z = xc; end c_cen = 0.5*(c_der+c_z); %dr = ps_o; %dr = x0; %dr3 = x_cen; permso = 0; [ps_o,x0,xc,x_cen]=sep_placas(ps,ps,c_cen,permso,xd); %/*put xd n last argument because left out*/ xc = x_cen; c = c_cen; 45

262 Appendx 5: (Contnued) end er = absval(hxc); er = absval(c_derc_z); cmn = c; xmn = xc; end %/* f */ %end %/* f */ else c_z = cmap(); c_der = cmax; f (er > epslon) & ( er > 0.0*epslon) c = 0.5*(c_der + c_z); xc = sep_mon(ps,ps,c,permso,xd); %/*put n last argument because left out*/ f (xc > h) c_z = c; else c_der = c; end 46

263 Appendx 5: (Contnued) er = absval(hxc); er = absval(c_derc_z); end %/* f */ cmax = c; end permso = 0; f caso == pot= RKF_Int(ps,ps,ps_o,h,c,permso); else pot = RKF_Intmon(ps,ps,ps_o,h,c,permso); end ret=pot; arc.m (Orgnally wrtten by E. Maruez (994), modfed by C. O Bren (003)) functon ret=arc(r,r,h,ph) %ac=zeros(4,4); %f=zeros(,4); %b=zeros(,4); eps =.0e6; 47

264 Appendx 5: (Contnued) d = h + r + r; x = r *h; y =.0; x = 0.0; y = r; x3 = r.*cos(ph); y3 = r.*sn(ph); const=0.5*p+ph; m = sn(const)./cos(const); error =.0; ac(,)= 0.0; ac(,3) = 0.0; ac(3,) = 0.0; ac(3,4) = 0.0; ac(3,) = m; ac(3,3) =.0; f absval(error) > eps 48

265 Appendx 5: (Contnued) f = (x3x).*(x3x) (xx).*(xx)+(y3y).*(y3y) (yy).*(yy); f = (dx).*(dx)+(y.*y)(r.*r); f3 = y3ym.*(x3x); f4 = y.*(yy)+(xx).*(xd); f() = f; f() = f; f(3) = f3; f(4) = f4; ac(,) =.0.*(xx3); ac(,) =.0.*(xx); ac(,3) =.0.*(yy3); ac(,4) =.0.*(yy); ac(,) =.0.*(xd); ac(,4) =.0.*y; ac(4,) = dx; ac(4,) =.0.*xxd; ac(4,3) = y; 49

266 Appendx 5: (Contnued) ac(4,4) =.0.*yy; [,r]=r_fact(4,ac); %=ones(4); for k = :4 %{ sum = 0.0; for = :4 sum = sum + (,k).*f(); end b(k) = sum; end % } /* next k */ b=back_subs(4,r,b,f); %f=f+(f==0)*eps; %fprntf('back_subs output: %e, %e \n',f,z); x = x + f(); x = x + f(); y = y + f(3); 50

267 Appendx 5: (Contnued) y = y + f(4); error = vecnorm(4,f); end %} /* End Whle */ f (y < 0.0) fprntf('!!error n arc!!: %f \n',y); end r3 = (x3 x).*(x3x)+(y3y).*(y3y); m = (x3 x).*(x3x)+ (y3y).*(y3y); l = (.0.*r3m)./(.0.*r3); %/* Cosne Theorem */ l=l+(l==0)*eps; f (.0 abs(l)) <.0e0 l = p; else f abs(l) <.0e0 l = 0.5*p; else l = srt(.0(l.*l))./l; 5

268 Appendx 5: (Contnued) l = atan(l); end f l < 0.0 l = p+l; end end xx=srt(r3).*l; ret=xx; back_subs.m (Orgnally wrtten by E. Maruez (994), modfed by C. O Bren (003)) functon ret=back_subs(n,u,y,f) z=zeros(,4); MAXIMO=90; for = :MAXIMO5; ndex() = ; end nan_locaton=fnd(snan(u)); 5

269 Appendx 5: (Contnued) u(nan_locaton)= ; z(ndex(n)) = y(ndex(n))./(u(ndex(n)).*(ndex(n))); %z(ndex(n))=z(ndex(n))+(z(ndex(n))==0)*eps; for k = (n):: %{ = ndex(k); sum = 0.0; for =(k+):n %{ = ndex(); sum = sum + u(,).*z(); z() = (y()sum)./u(,); %z()=z()+(z()==0)*eps; end %} /* next */ end %} /* next */ ret=z; 53

270 Appendx 5: (Contnued) cube.m (Orgnally wrtten by E. Maruez (994), modfed by C. O Bren (003)) functon ret=cube(z) ret = z*z*z; feld_potoc.m (Orgnally wrtten by E. Maruez (994), modfed by C. O Bren (003)) functon[p,p]=feld_potoc(r,r,ps,ps,h,t,cmn,xmn,v,v,h,h,xd,x_ref,cmap,xm ap) Dam_o=0.0e6; Q=; Z=0; e=.6097e9; boltzmann=.38044e3; epslon=.0e5; t = 308.0; %/*Temperature %s=zeros(,30); %we=zeros(,30); %cmap=zeros(,50); %xmap=zeros(,50); 54

271 Appendx 5: (Contnued) %fprntf('cmap feld_potoc: %e \n',cmap); %/* Roots for Gaussan Quadrature */ s() = ; s() = ; s(3) = ; s(4) = ; s(5) = ; s(6) = ; s(7) = ; s(8) = ; s(9) = ; s(0)= ; for g = :0 s(0+g) = s(0g+); end %/*symetrc about zero*/ % /* Weghts for Gaussan Quadrature */ we() = ; 55

272 Appendx 5: (Contnued) we() = ; we(3) = ; we(4) = ; we(5) = ; we(6) = ; we(7) = ; we(8) = ; we(9) = ; we(0) = ; for g = :0 we(0+g) = we(0g+); %/*symetrc about zero */ end %%begn feld potoc %%% %fprntf('h: %e \n',h); d = h + r + r; ngp = 0; gp = ; 56

273 Appendx 5: (Contnued) %/* calculate values of phmax and thetamax */ thetamax = maxangle(r,r,d); %swtches between.57(90 degrees) and.57 %thetamax=.0; %fprntf('thetamax: %e \n',thetamax); phmax = p thetamax; %/* Calculate frst ntegral usng Gaussan Quadrature */ %/* Integraton cycle */ theta = 0.5*thetamax*(s()+.0); l = arc(r,r,h,theta); % fprntf('l: %e \n',); sn_ = 0.0; snp = 0.0; sn_ = 0.0; snp = 0.0; %/* Integrate for dstances up to 4 wth flat_ocfe */ cmax = 0.0*cosh(ps); cmap()=cmn; 57

274 Appendx 5: (Contnued) xmap()=xmn; f (l <= 5.0) & (gp <= ngp) % {/* */ tterm = flat_hhf(l,t,ps,ps); term=tterm*sn(theta)*we(gp); tterm = flat_ocfe(l,t,ps,ps,x_ref,cmap,cmax,xd,xmn,cmn,xmap); %fprntf('tterm: %e \n',tterm); term=tterm*sn(theta)*we(gp); f term < 0.0 sn_ =sn_ + term ; else snp = snp + term; end f term < 0.0 sn_ = sn_ + term ; else snp = snp + term; 58

275 Appendx 5: (Contnued) end gp=gp+; f gp <= ngp %{ theta = 0.5.*thetamax.*(s(gp) +.0); l = arc(r,r,h,theta); f (lh) < 0.0 fprntf(' Error n feld_potoc: arclength < 0.0!!! %f %f %f %f\n',theta,l,h); end end %} end %} /* End Whle */ %/* Integrate for dstances > 5.0 wth flat_ocfe */ c = (log(v/v)./(hh)); c=c+(c==0)*eps; c = log(v)c*h; f gp <= ngp %{/* 3 */ 59

276 Appendx 5: (Contnued) tterm = flat_hhf(l,t,ps,ps); %where problem occurs term=tterm*sn(theta)*we(gp); term = exp(c*l + c)*sn(theta)*we(gp); f term < 0.0 sn_ = sn_ + term ; else snp = snp + term; end f term < 0.0 sn_ = sn_ + term ; else snp = snp + term; end gp=gp+; f gp <= ngp %{ theta = 0.5.*thetamax.*(s(gp) +.0); 60

277 Appendx 5: (Contnued) l = arc(r,r,h,theta); % f (lh) < 0.0 fprntf(' Error n feld_protoc: arclength < 0.0!!! %f %f %f %f\n',theta,l,h); end end %} end % } /* End Whle 3 */ nthhf = 0.5*thetamax*(sn_ + snp) ; ntoc = 0.5*thetamax*(sn_ + snp) ; %/* End Integraton */ %/* Calculate second ntegral usng Gauss' uadrature */ %/* Integraton Cycle */ f (r == r)&(ps == ps) %{ p = (r+r)*nthhf; p = (r+r)*ntoc; %fprntf('p,p feld_potoc: %e %e \n',p,p); %} 6

278 Appendx 5: (Contnued) else %{ gp = ; cmax = 0.0*cosh(ps); cmn = cmap(); xmn = xmap(); ph = 0.5*phmax.*(s()+.0); l = arc(r,r,h,ph); sn_ = 0.0; snp = 0.0; sn_ = 0.0; snp = 0.0; f (l <= 5.0) & (gp <= ngp) %{/* */ tterm = flat_hhf(l,t,ps,ps); term=tterm*sn(ph)*we(gp); tterm = flat_ocfe(l,t,ps,ps,x_ref,cmap,cmax,xd,xmn,cmn,xmap); 6

279 Appendx 5: (Contnued) term =tterm*sn(ph)*we(gp); f term < 0.0 sn_ = sn_ + term ; else snp = snp + term; end f term < 0.0 sn_ = sn_ + term ; else snp = snp + term; end gp=gp+; f gp <= ngp % { ph = 0.5*phmax*(s(gp) +.0); l = arc(r,r,h,ph); f (lh) <

280 Appendx 5: (Contnued) fprntf(' Error 3 n feld_protoc : arclength < 0.0!!! %f %f %f %f %f %f %f \n',theta,ph,l,h,thetamax,r,r); end end %} end %} /* End Whle */ %/* Integrate for dstances > 5.0 wth flat_ocfe */ c = log(v/v)./(hh); c = log(v)(c*h); f gp <= ngp %{/* 3 */ tterm = flat_hhf(l,t,ps,ps); term=tterm*sn(ph)*we(gp); term = exp(c.*l + c)*sn(ph)*we(gp); f term < 0.0 sn_ = sn_ + term ; else snp = snp + term; end 64

281 Appendx 5: (Contnued) f term < 0.0 sn_ = sn_ + term ; else snp =snp + term; end gp=gp+; f gp <= ngp %{ ph = 0.5*phmax*(s(gp) +.0); l = arc(r,r,h,ph); f (lh) < 0.0 fprntf(' Error 4 n feld_protoc: arclength < 0.0!!! %f %f %f %f %f %f %f \n',theta,ph,l,h,thetamax,r,r); end end % } end %} /* End Whle 3 */ %/* End Integraton */ nthhf = 0.5*phmax*(sn_ + snp) ; 65

282 Appendx 5: (Contnued) ntoc = 0.5*phmax*(sn_ + snp) ; p = (0.5*(r+r)*(r*nthhf/r + r*nthhf/r)); p = (0.5*(r+r)*(r*ntoc/r + r*ntoc/r)); %fprntf('p,p feld_potoc: %e %e \n',p,p); % no zero elements end flat_hhf.m (Orgnally wrtten by E. Maruez (994), modfed by C. O Bren (003)) %fle used to calculate flat_hhf functon ret=flat_hhf(z,t,ps,ps) ys = ps; ys = ps; %z = h; yp = 0.5.*(ys + ys); y_ = 0.5.*(ys ys); thz_ = ((.0 exp(z))./(.0 + exp(z))); cthz_ = ((.0 + exp(z))./(.0 exp(z))); ret = (yp.*yp).*(.0 thz_) (y_.*y_).*(cthz_.0); 66

283 Appendx 5: (Contnued) maxangle.m (Orgnally wrtten by E. Maruez (994), modfed by C. O Bren (003)) functon ret=maxangle(r,r,d) x = 0.0; error = 00.0; epslon=.0e5; %/* Relatve Error f error > epslon alpha = (r.*x.*x)./r + (d.*x) (r.*r); beta =.0((x.*x)./r./r); gamma = (r.*r) (x.*x); f = (alpha.*alpha) (r.*r.*beta.*gamma); df =.0.*alpha.*(((.0.*r.*x)./r) + d)+.0.*x.*r.*r.*(beta+gamma./r./r); x_new = xf./df; f abs(x_new) >.0e0 error = abs((x_new x)./x_new); else error = abs((x_new x)); end x = x_new; 67

284 Appendx 5: (Contnued) end %} /* end f */ alpha = x./r; f abs(alpha) <.0e4 value = 0.5*p; else sss=.0 (power(,alpha)); ttt=srt(sss)/alpha; value = atan(ttt); end ret=value; peclet.m (Orgnally wrtten by C. O Bren (003)) functon ret=pecet(d,t,vs,g) %calculate peclet number %G=average shear rate %vs=vscosty of medum %defne constants kb=.38066e3; %J/K 68

285 Appendx 5: (Contnued) ret=(3*p*g*vs*d^3)/(t*kb) power.m (Orgnally wrtten by E. Maruez (994), modfed by C. O Bren (003)) functon ret=power(n,z) f n == 0 value =.0; else value =.0; for = :n value = value * z; end end ret=value; r_fact.m functon [,r]=r_fact(n,ac) %t=zeros(4,4); %p=zeros(4,4); h=zeros(4,4); for = :n 69

286 Appendx 5: (Contnued) for = :n f == p(,) =.0; else p(,) = 0.0; end t(,) = ac(,); (,)= 0.0; r(,) = 0.0; end %} end = 0; for m = :(n) for =:n z() = t(,m); end s = 0.0; for = m:n 70

287 Appendx 5: (Contnued) s = s + z()*z(); end s = srt(s); f s > 0.0 %{Column Structure Check */ =+; den = s*(s+abs(z())); den=den+(den==0)*eps; for =:() z() = 0.0; end f z() > 0.0 z() = z() + s; else z()= z() s; end for = :n 7

288 Appendx 5: (Contnued) for k = :n h(,k) = (z()*z(k))./den; end for =:n h(,) =h(,)+.0; end end for = :n for k = m:n sum = 0.0; for l=:n sum = sum + h(,l)*t(l,k); end r(,k) = sum; r(,k)=r(,k)+(r(,k)==0)*eps; end %} /* next k */ end 7

289 Appendx 5: (Contnued) for =:n for k=:n %{ sum = 0.0; for l=:n sum = sum + p(,l)*h(l,k); end (,k) = sum; end %} /* next */ end for =:n for k=:n p(,k) = (,k); t(,k) = r(,k); end end end %} /* End column structure check */ 73

290 Appendx 5: (Contnued) %save rfact r; end %}/* next m */ RKF_Int.m (Orgnally wrtten by E. Maruez (994), modfed by C. O Bren (003)) functon ret=rkf_int(ps, ps, ps_o, xd, c, permso) hmn=.0e9; %* Mn nteg. length for forces n stab. epslon=.0e5; %/* Relatve Error a4 = 0.; g =.0; g = g/35.0; b = 6.0*g; g =.0; g = g/85.0; b = *g; g =.0; g = g/ ; b3 = 856.0*g; g =.0; g = g/50.0; b4 = 9.0*g; g =.0; g = g/55.0; b5 =.0*g; 74

291 Appendx 5: (Contnued) c_3 = 3.0*g; g =.0; g = g/3.0; c_3 = 3.0*g; c_33 = 9.0 *g; g =.0; g = g/3.0; c_4 =.0*g; g =.0; g = g/97.0; c_4 = 93.0*g; c_43 = 700.0*g; c_44 = 796.0*g; g =.0; g = g/6.0; a = 5.0*g; c_5 = 439.0*g; g =.0; g = g/53.0; c_5 = *g; g =.0; g = g/404.0; a3 = 97.0*g; c_53 = 845.0*g; c_63 = 859.0*g; 75

292 Appendx 5: (Contnued) g =.0; g = g/7.0; c_6 = 8.0*g; g =.0; g = g/565.0; a = 408.0*g; c_6 = *g; g =.0; g = g/40.0; c_64 =.0*g; hmax =.0e4; h = hmax; u = ps; ps = ps; x = 0.0; sum = 0.0; valdo = ; = ; term = 0.5*cosh(ps); termp = snh(ps)*f(x,ps,c)/0.0; 76

293 Appendx 5: (Contnued) termpp = (cosh(ps)*sr(f(x,ps,c))+sr(snh(ps)))/0.0; valdo = ; f (permso == ) prntf('error n RKF_Int: %9.6f %0.8f %0.8f\n',x,ps,f(x,ps,c)); end f (abs(psps) >= epslon) & (valdo ==) f ps > ps %/* */ k = h*f(x,ps,c); xo = x + 0.5*h; u = ps + 0.5*k; f u > ps %/* */ k = h*f(xo,u,c); xo = x + c_3*h; u = ps + c_3*k + c_33*k; f u>ps 77

294 Appendx 5: (Contnued) % { /* 3 */ k3 = h*f(xo,u,c); xo = x + c_4*h; u = ps + c_4*k c_43*k + c_44*k3; f u > ps %{ /* 4 */ k4 = h*f(xo,u,c); xo = x + h; u = ps + c_5*k 8.0*k + c_5*k3 c_53*k4; f u > ps %{ /* 5 */ k5 = h*f(xo,u,c); xo = x + 0.5*h; u = ps c_6*k +.0*k c_6*k3 + c_63*k4 c_64*k5; f u > ps % { /* 6 */ k6 = h*f(xo,u,c); 78

295 Appendx 5: (Contnued) r = abs(k/ *k3/ *k4/ k5/ *k6/55.0); f r < epslon/54.9 er_n = 4.5; else er_n = 0.84*srt(srt((epslon/r))); end f r < epslon u = ps + a*k + a*k3 + a3*k4 a4*k5; f control(u,c)== %{ /* 7 */ ps = u; x = x+h; u = f(x,ps,c); term = 0.5*cosh(ps); termp = snh(ps)*u/0.0; termpp = (cosh(ps)*u*u+sr(snh(ps)))/0.0; sum = sum + h*(term + term); sum = sum + h*h*(termp termp); 79

296 Appendx 5: (Contnued) sum = sum + h*h*h*(termpp + termpp); term = term; termp = termp; termpp = termpp; f (permso == ) fprntf('error n RKF_Int: %9.6f %0.8f %0.8f\n',x,ps,f(x,ps,c)); end end %}/* 7 */ else h = h*0.; end %} f er_n < 0. h = h*0.; else f er_n > 4.0 h = h*4.0; else h = h*er_n; end f h < hmn 80

297 Appendx 5: (Contnued) fprntf('error n RKF_Int: h < hmn.!!!\n'); valdo = 0; end end %} /* 6 */ else h = h*0.; end %} /* 5 */ else h = h*0.; end %} /* 4 */ else h = h*0.; end %} /* 3 */ else h = h*0.; end %} /* */ else h = h*0.; end %} /* */ else h = h*0.; end f h > hmax 8

298 Appendx 5: (Contnued) h = hmax; end end %} /* End Whle */ sum = sum; sum = 0.0; f (permso == ) fprntf('error n RKF_Int: %9.6f %0.8f %0.8f\n',x,ps,f(x,ps,c)); end f (absval(u) >= epslon) & (valdo ==) f (control(ps,c) ==) %{ /* */ k = h*f(x,ps,c); xo = x + 0.5*h; u = ps + 0.5*k; f control(u,c) == %{ /* */ k = h*f(xo,u,c); 8

299 Appendx 5: (Contnued) xo = x + c_3*h; u = ps + c_3*k + c_33*k; f control(u,c) == %{ /* 3 */ k3 = h*f(xo,u,c); xo = x + c_4*h; u = ps + c_4*k c_43*k + c_44*k3; f control(u,c)== %{ /* 4 */ k4 = h*f(xo,u,c); xo = x + h; u = ps + c_5*k 8.0*k + c_5*k3 c_53*k4; f control(u,c)== %{ /* 5 */ k5 = h*f(xo,u,c); xo = x + 0.5*h; u = ps c_6*k +.0*k c_6*k3 + c_63*k4 c_64*k5; 83

300 Appendx 5: (Contnued) f control(u,c)== %{ /* 6 */ k6 = h*f(xo,u,c); r = absval(k/ *k3/ *k4/ k5/ *k6/55.0); f r < (epslon/54.9) er_n = 4.5; else er_n = 0.84*srt(srt((epslon/r))); end f r < epslon %{ u = ps + a*k + a*k3 + a3*k4 a4*k5; f control(u,c)== %{ /* 7 */ ps = u; x = x+h; u = f(x,ps,c); term = 0.5*cosh(ps); 84

301 Appendx 5: (Contnued) termp = snh(ps)*u/0.0; termpp = (cosh(ps)*u*u+sr(snh(ps)))/0.0; sum = sum+h*(term + term); sum = sum+h*h*(termp termp); sum = sum+h*h*h*(termpp + termpp); term = term; termp = termp; termpp = termpp; f (permso == ) fprntf('error n RKF_Int: %9.6f %0.8f %0.8f %0.8f %0.8f %0f\n ',x,ps,u,er_n,r,h); end end %}/* 7 */ else h = h*0.; end %} f er_n < 0. h = h*0.; 85

302 Appendx 5: (Contnued) else f er_n > 4.0 h = h*4.0; else h = h*er_n; end f h < hmn fprntf('error n RKF_Int: h < hmn.!!!\n'); valdo = 0; end end %} /* 6 */ else h = h*0.; end %} /* 5 */ else h = h*0.; end %} /* 4 */ else h = h*0.; end %} /* 3 */ else h = h*0.; end %} /* */ 86

303 Appendx 5: (Contnued) else h = h*0.; end %} /* */ else h = h*0.; end f h > hmax h = hmax; end end %} %/* End Whle */ sum =.0*sum + sum; sum =.0*sum + xd*(cosh(ps_o)+.0)+ 4.0*(cosh(0.5*ps)+cosh(ps*0.5).0); ret=sum; RKF_Intmon.m (Orgnally wrtten by E. Maruez (994), modfed by C. O Bren (003)) functon ret=rkf_intmon(ps, ps, ps_o, xd, c, permso) hmn=.0e9; %* Mn nteg. length for forces n stab. hmax =.0e4; h = hmax; 87

304 Appendx 5: (Contnued) u = ps; ps = ps; x = 0.0; sum = 0.0; = ; epslon=.0e5; %/* Relatve Error a4 = 0.; g =.0; g = g/35.0; b = 6.0*g; g =.0; g = g/85.0; b = *g; g =.0; g = g/ ; b3 = 856.0*g; g =.0; g = g/50.0; b4 = 9.0*g; g =.0; g = g/55.0; b5 =.0*g; 88

305 Appendx 5: (Contnued) c_3 = 3.0*g; g =.0; g = g/3.0; c_3 = 3.0*g; c_33 = 9.0 *g; g =.0; g = g/3.0; c_4 =.0*g; g =.0; g = g/97.0; c_4 = 93.0*g; c_43 = 700.0*g; c_44 = 796.0*g; g =.0; g = g/6.0; a = 5.0*g; c_5 = 439.0*g; g =.0; g = g/53.0; c_5 = *g; g =.0; g = g/404.0; a3 = 97.0*g; c_53 = 845.0*g; c_63 = 859.0*g; 89

306 Appendx 5: (Contnued) g =.0; g = g/7.0; c_6 = 8.0*g; g =.0; g = g/565.0; a = 408.0*g; c_6 = *g; g =.0; g = g/40.0; c_64 =.0*g; term = 0.5*cosh(ps); termp = snh(ps)*f(x,ps,c)/.0; termpp = ( cosh(ps)*sr(f(x,ps,c))+sr(snh(ps)))/0.0; valdo = ; f permso == fprntf('error n RKF_Intmon: %9.6f %0.8f %0.8f\n',x,ps,f(x,ps,c)); end f absval(psps) >= epslon & (valdo ==) %{ f ps > ps 90

307 Appendx 5: (Contnued) %{ /* */ k = h*f(x,ps,c); xo = x + 0.5*h; u = ps + 0.5*k; f u > ps %{ /* */ k = h*f(xo,u,c); xo = x + c_3*h; u = ps + c_3*k + c_33*k; f u>ps %{ /* 3 */ k3 = h*f(xo,u,c); xo = x + c_4*h; u = ps + c_4*k c_43*k + c_44*k3; f u > ps %{ /* 4 */ k4 = h*f(xo,u,c); 9

308 Appendx 5: (Contnued) xo = x + h; u = ps + c_5*k 8.0*k + c_5*k3 c_53*k4; f u > ps %{ /* 5 */ k5 = h*f(xo,u,c); xo = x + 0.5*h; u = ps c_6*k +.0*k c_6*k3 + c_63*k4 c_64*k5; f u > ps %{ /* 6 */ k6 = h*f(xo,u,c); r = absval(k/ *k3/ *k4/ k5/ *k6/55.0); f r < epslon/54.9 er_n = 4.5; else er_n = 0.84*srt(srt((epslon/r))); end f r < epslon %{ 9

309 Appendx 5: (Contnued) u = ps + a*k + a*k3 + a3*k4 a4*k5; f control(u,c)== %{ /* 7 */ ps = u; x = x+h; u = f(x,ps,c); term = 0.5*cosh(ps); termp = snh(ps)*u/0.0; termpp = (cosh(ps)*u*u+sr(snh(ps)))/0.0; sum = sum + h*(term + term); sum = sum + h*h*(termp termp); sum = sum + h*h*h*(termpp + termpp); term = term; termp = termp; termpp = termpp; f (permso == ) 93

310 Appendx 5: (Contnued) fprntf('error n RKF_Intmon: %9.6f %0.8f %0.8f %0.8f %0.8f %0f\n ',x,ps,u,er_n,r,h); end end %}/* 7 */ else h = h * 0.; end %} f er_n < 0. h =h * 0.; else f er_n > 4.0 h =h * 4.0; else h = h * er_n; end f h < hmn fprntf('error n RKF_Intmon: h < hmn.!!!\n'); valdo = 0; end end %} /* 6 */ 94

311 Appendx 5: (Contnued) else h = h * 0.; end %} /* 5 */ else h = h * 0.; end %} /* 4 */ else h =h * 0.; end %} /* 3 */ else h = h * 0.; end %} /* */ else h = h * 0.; end %} /* */ else h =h * 0.; end f h > hmax h = hmax; end end %} /* End Whle */ 95

312 Appendx 5: (Contnued) an=0.5*ps; an=ps*0.5; sum =.0*sum + xd*(.00.5*c) + 4.0*(cosh(an)+cosh(an).0); ret=sum; vecnorm.m (Orgnally wrtten by E. Maruez (994), modfed by C. O Bren (003)) functon ret=vecnorm(n,z) norm = absval(z()); for =:n f absval(z()) > norm norm = absval(z*); end end ret=norm; eucnorm.m (Orgnally wrtten by E. Maruez (994), modfed by C. O Bren (003)) functon ret=eucnorm(n,z) norm = z()*z(); for =:n norm = norm + z()*z(); 96

313 Appendx 5: (Contnued) end ret=norm; 97