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The formation and properties of coherent flocs in fibre suspensions Soszyński, Robert Marian 1987

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THE FORMATION AND PROPERTIES OF COHERENT FLOCS IN FIBRE SUSPENSIONS by ROBERT MARIAN SOSZYNSKI B.D. Tech. School Energ., Warsaw, 1966 Mgr. Inz., Warsaw Polytechnic, 1973 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES Department of Chemical Engineering We accept t h i s thesis as conforming to the required standard UNIVERSITY OF BRITISH COLUMBIA June 1987 Robert Marian Soszynski, 1987 In presenting t h i s thesis in p a r t i a l f u l f i l l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the Head of my Department or by his or her representatives. It i s understood that copying or publication of the thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of Chemical Engineering University of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date: June, 1987 i i ABSTRACT F i b r e s i n c o n c e n t r a t e d s u s p e n s i o n s a r e i n c o n t i n u o u s c o n t a c t w i t h o t h e r f i b r e s a n d may i n t e r l o c k t h r o u g h e l a s t i c b e n d i n g t o f o r m c o h e r e n t n e t w o r k s . S u c h i n t e r l o c k i n g i s t e r m e d T y p e - C c o h e s i o n . T h e p r o c e s s by w h i c h T y p e - C c o h e s i o n f o r m s among f i b r e s a n d t h e r e s u l t i n g s t r u c t u r e a n d t e n s i l e s t r e n g t h o f i n d i v i d u a l f l o e s o f f i b r e s h a v e b e e n e x a m i n e d i n e x p e r i m e n t a l s t u d y i n w h i c h r e l a t i v e l y s t r a i g h t , s m o o t h n y l o n ( 6 - 6 ) f i b r e s o f a s p e c t r a t i o s f r o m 65 t o 189 were s u s p e n d e d i n a q u e o u s - s u g a r s o l u t i o n s . T h e f i b r e s were i n most c a s e s n e u t r a l l y b u o y a n t . T h e s u s p e n s i o n s were c a u s e d t o f l o w i n a p a r t i a l l y f i l l e d , i n c l i n e d - t o - t h e - h o r i z o n t a l o r h o r i z o n t a l l y - o r i e n t e d c y l i n d e r r o t a t e d a b o u t i t s p r i n c i p a l a x i s t o p r o d u c e a r e c i r c u l a t i n g a n d m o d e r a t e l y u n s t e a d y f l o w . A t a w e l l - d e f i n e d a n d r e p r o d u c i b l e " t h r e s h o l d c o n c e n t r a t i o n " T y p e - C c o h e r e n t f l o e s f o r m e d . T h e f l o e s were v e r i f i e d t o be o f T y p e - C by h e a t t r e a t m e n t . The h e a t t r e a t m e n t c a u s e d s t r e s s r e l a x a t i o n i n e l a s t i c a l l y b e n t f i b r e s r e s u l t i n g i n r e d u c e d f l o e s t r e n g t h . V i s u a l o b s e r v a t i o n s o f f l o e f o r m a t i o n a n d v e l o c i t y m e a s u r e m e n t s w i t h L a s e r D o p p l e r Anemometer i n d i c a t e d t h a t t h e f l o e s o r i g i n a t e d i n t h e z o n e i n w h i c h f l o w d e c e l e r a t e d . I n t h i s z o n e f l o e s f o r m e d by c o m p a c t i o n o f c r o w d e d f i b r e s . T h e t h r e s h o l d c o n c e n t r a t i o n d e p e n d e d on f i b r e g e o m e t r y a n d v i s c o s i t y o f t h e s u s p e n d i n g l i q u i d . B e l o w an a s p e c t r a t i o o f a p p r o x i m a t e l y 50 a n d a b o v e a s u s p e n d i n g l i q u i d v i s c o s i t y o f a p p r o x i m a t e l y 0 . 0 1 3 P a « s , T y p e - C c o h e r e n t f l o e s d i d n o t f o r m a t a n y c o n c e n t r a t i o n o f f i b r e s . U n d e r t h e t e s t c o n d i t i o n s o f t h i s w o r k , t h e t h r e s h o l d c o n c e n t r a t i o n was u n a f f e c t e d by t h e c y l i n d e r r o t a t i o n a l speed, cylinder diameter, and angle of i n c l i n e to the horizontal, provided that s u f f i c i e n t shear was induced in the cylinder to create r e c i r c u l a t i n g flow. The structure and strength of Type-C coherent floes were examined. The number of contact points per f i b r e was less than values estimated from t h e o r e t i c a l , s t a t i s t i c a l models in the l i t e r a t u r e . The t e n s i l e strength of individual Type-C floes measured in a tester with a unique comb support showed values larger than strengths reported in the l i t e r a t u r e for either man-made or wood-pulp f i b r e networks. A mathematical model developed to describe t e n s i l e strength based on f r i c t i o n a l f i b r e - t o - f i b r e interaction accounted for only a part of the t o t a l floe strength. iv Table of Contents 1 INTRODUCTION 1 2 LITERATURE REVIEW 7 2.1 Flocculation in Dilute Fibre Suspensions 7 2.2 Flocculation in Semiconcentrated Fibre Suspensions 11 2.2.1 Fibre interactions and c o l l i s i o n s 11 2.2.2 Fibre cohesion 18 2.3 Flocculation in Concentrated Fibre Suspensions 22 2.3.1 Process of network formation 23 2.3.2 Structure of coherent networks 33 2.3.3 Network strength 38 2.3.4 Cohesion forces 54 2.4 C l a s s i f i c a t i o n of Findings from the Literature Review ...59 2.5 Summary of Literature Review 63 2.6 Thesis Objectives 68 3 EXPERIMENTAL PROGRAM 70 3.1 Preparation of f i b r e s 70 3.1.1 Choice of fibrous material 70 3.1.2 Determination of fibr e geometry 72 3.1.3 Moisture absorption by nylon f i b r e s 73 3.1.4 Determination of fibr e e l a s t i c properties 74 3.1.5 Determination of wet-friction c o e f f i c i e n t 75 3.2 Creation of Coherent Floes 75 3.2.1 Observation of the phenomenon of f l o c c u l a t i o n 78 3.2.2 Sediment concentration 78 3.3 Experimental Variables 79 3.3.1 Fibre concentration 80 V 3.3.2 Density of the suspending l i q u i d 81 3.3.3 Cylinder diameter, angle of i n c l i n e , and rotational speed 81 3.3.4 Fibre dimensions - length and diameter 82 3.3.5 V i s c o s i t y of the suspending l i q u i d 82 3.4 Flow Patterns in Horizontal Rotating Cylinder 83 3.5 Studies of Floe Structure 85 3.6 Floe Strength Measurement 86 4 RESULTS AND DISCUSSION 88 4.1 Ef f e c t of Fibre Concentration on Floe Formation 88 4.2 Observations of Origin and Growth of Type-C Floes 94 4.3 Ef f e c t of Key Variables on Threshold Concentration 98 4.3.1 Cylinder (flow) variables 100 4.3.2 Suspending l i q u i d v i s c o s i t y 104 4.3.3 Fibre geometry 109 4.4 Evaluation of C r i t e r i a for Threshold Concentration 113 4.5 Flow Conditions under which Coherent Floes Form 124 4.5.1 D i s s i m i l a r i t i e s between flows in horizontal cylinder 126 4.5.2 Other types of flow 135 4.6 Observations of Floe Structure 136 4.7 Tensile Strength of Type-C Floes 139 4.5 Mathematical Model of Tensile Strength of Type-C Floes .150 5 SUMMARY 156 6 RECOMMENDATIONS FOR FURTHER WORK 159 NOMENCLATURE 160 LIST OF REFERENCES 163 GLOSSARY 173 v i APPENDIX I. Water Retention Ratio of Wood Pulp Fibres 176 APPENDIX I I . Apparent Density of Wood Pulp Fibres in Aqueous Suspensions 179 APPENDIX I I I . Estimates of Magnitudes of Various Cohesion Forces 185 APPENDIX IV. Filament Cutting Procedure 187 APPENDIX V. Measurement of Fibre Length and Curvature 189 APPENDIX VI. Water Absorption by Nylon Fibres 197 APPENDIX VII. Relationships Between Mass and Volume Concentration of Hydrophilic P a r t i c l e s Suspended in Aqueous-Solute Solutions 201 APPENDIX VIII. Measurement of the E l a s t i c Moduli of Fibres ..205 APPENDIX IX. F r i c t i o n Between Wet Fibre Surfaces 215 APPENDIX X. Rotating Cylinder Apparatus 219 APPENDIX XI. Suspension Preparation and Type-C Floe Formation 221 APPENDIX XII. Sediment Concentration Measurements 223 APPENDIX XIII. Flow Velocity Measurement in Fibre Suspensions in Horizontal Rotating Cylinder 224 APPENDIX XIV. Floe Preparation and Contact Counting Procedures 236 APPENDIX XV. Tensile Strength of Wet Nylon Floes 239 APPENDIX XVI. Computer Model of Fibre Deposition into 3-D networks 245 APPENDIX XVII. Mathematical Analysis of the Tensile Strength of Type-C Floes 248 APPENDIX XVIII. Conductivity and pH Measurements 254 v i i APPENDIX XIX. Data Col l e c t i o n Program For Fibre Length and Fibre Curvature Measurements 258 APPENDIX XX. Data C o l l e c t i o n Program for Water Absorption by Nylon Fibres 261 APPENDIX XXI. Fibre Bending. Data Processing Program, Raw Data F i l e , Averages and Standard Deviations of Fibre Deflections, Bulk Reynolds Numbers. Eye-Piece-Micrometer and GILMONT Flowmeter Calib r a t i o n Data 263 APPENDIX XXII. Data Acquisition Program for Flow Velocity Measurements 270 APPENDIX XXIII. Data Acquisition Program for Tensile Tests of Wet, Type-C Nylon Floes 276 APPENDIX XXIV. Data C o l l e c t i o n Program for Determination of Break Area 280 APPENDIX XXV. FORTRAN Program Processing Conductivity Data. Data L i s t i n g 283 APPENDIX XXVI. Data from Threshold Concentration Measurements 286 APPENDIX XXVII. Data from Velocity Measurements in Horizontal Rotating Cylinder 295 APPENDIX XXVIII. Data from Tensile Strength Measurements ....303 APPENDIX XXIX. Data from Sedimentation Experiments 307 APPENDIX XXX. Data from Co e f f i c i e n t of F r i c t i o n Measurements 315 APPENDIX XXXI. Average V e l o c i t i e s and Root-Mean-Squares of Velocity Fluctuation 317 v i i i L i s t of Tables Table Page I. Factors A f f e c t i n g Flocculation of Wood-Pulp Fibres 5 I I . Number of Fibre Contacts per Fibre in Compacted Mats [E4] and Calculated from Equation (2) and (3) 39 I I I . Y i e l d Stress of Nylon Fibre S l u r r i e s 42 IV. Shear Strength of Fibre Suspensions. Direct Methods. Quasi-Static Tests 46 V. Shear Strength of Fibre Suspensions. Indirect Methods. Quasi-Static Tests 47 VI. Tensile Strength of Fibre Networks. Quasi-Static Tests ..49 VII. Tensile Strength of Dry Wood-Pulp Floes 53 VIII. Threshold Concentration for 15 Denier Nylon Fibres in Water and in Aqueous-Sugar Solution 99 IX. System Cha r a c t e r i s t i c s for Suspensions of Nylon Fibres (L=4.97 mm, d=0.0442 mm) in the Suspending Media of Various V i s c o s i t i e s 106 X. Nylon Fibre Dimensions in Wet State 110 XI. Threshold Concentrations and Sediment Concentrations ....114 XII. Crowding Factors Calculated from Equation (6) and (7) at the Onset of Type-C Floe Formation 121 XIII. Limits of Equations (14) to (17) 123 XIV. F i t C o e f f i c i e n t s and Limits for Number of Fibres in a Cubical Volume 124 XV. Power F i t Parameters of Tensile Strength Data 148 XVI. Evaluation of the Mathematical Model of Tensile Strength 153 I- XVII. Average Dimensions of Wood Fibres and Pulp Fibres ...178 I I - XVIII. WRR at the "Knee" for Various Never-dried Pulps ...181 I I - XIX. Apparent Density of Wood Pulp Fibres at Selected WRRk Values 1 83 I I I - XX. Forces Deflecting Simply Supported Fibres 186 V-XXI. Nylon Fibres in Wet State. Fibre Length S t a t i s t i c s ..194 V- XXII. Nylon Fibres in Wet State. Curvature S t a t i s t i c s ....195 VI- XXIII. Water Absorption by Nylon Fibres in 98% Relative Humidity Environment 1.200 VII- XXIV. Relationships Between Mass and Volume Concentration of Hydrophilic P a r t i c l e s in Suspensions 204 VIII- XXV. E l a s t i c Moduli of 3 and 6 Denier Nylon Filaments ..209 VIII- XXVI. E l a s t i c Moduli and S t i f f n e s s of Wet Nylon Fibres .213 IX- XXVII. S t a t i c and Dynamic C o e f f i c i e n t of F r i c t i o n for Wet Nylon Fibres 218 XIV-XXVIII. Contact Points in Type-C Floes 238 XVIII-XXIX. F i t Parameters for Two Straight Lines Shown in Figure XVIII-65 256 X L i s t of F i g u r e s Figure Page 1. The P r i n c i p l e of Type-C (a) and Type-B (b) Cohesion 3 2. Shear Modulus versus Suspending Liquid V i s c o s i t y 25 3. Concepts of Fibre Crowding in Suspensions and Experimental Investigations with Man-made Fibres 60 4. Concepts of Fibre Crowding in Suspensions and Experimental Investigations with Wood-pulp Fibres 62 5. Surface Texture and Shape of Nylon (dark) and Wood-pulp Fibres. Magnification 120x 71 6. Rotating Cylinder Apparatus 77 7. Experimental Setup for Velocity Measurements in a Horizontal Rotating Cylinder 84 8. Schematic of I n i t i a l Comb Positioning in Tensile Tests ....86 9. Increasing Cloudiness with an Increase in Fibre Concentration 89 10. a. Cloudiness; b. Granularity; c. Granules in rediluted Suspension ....91 11. Floe Concentration versus Suspension Concentration 96 12. Floe Weighing 0.12 g Suspended in the A i r by Few Fibres ..97 13. E f f e c t of Cylinder Diameter on Threshold Concentration ..101 14. E f f e c t of Cylinder Incline on Threshold Concentration ...102 15. E f f e c t of Angular Speed on Threshold Concentration 103 16. Threshold Concentration versus V i s c o s i t y of the Suspending Liquid 105 17. E f f e c t of Fibre Length and Diameter on Threshold Concentration 111 x i 18. Threshold Concentration versus d" 112 19. Threshold Concentrations and Sediment Concentrations versus Fibre Aspect Ratio 115 20. Sediment Concentrations and Mathematical Models 116 21. Mathematical Models and Concentrations of Fibres in Suspensions at the Onset of Type-C Floe Formation 118 22. Flow Pattern in a Horizontal Rotating Cylinder 125 23. Velocity Vectors for the Case 1 Flow 127 24. Velocity Vectors for the Case 2 Flow 128 25. Velocity Vectors for the Case 3 Flow 130 26. Flow V e l o c i t i e s at r/R=0.85 and 0 from 0 to 100 degrees. Cylinder peripheral v e l o c i t y i s 0.295 m/s 131 27. Flow V e l o c i t i e s at r/R=0.96 and 0 from 0 to 100 degrees. Cylinder peripheral v e l o c i t y i s 0.295 m/s 132 28. Number of Contacts per Fibre versus Fibre Concentration in Floes (L=6.26 mm, d=44.2 /xm) 137 29. Number of Contacts per Fibre versus Fibre Concentration in Floes (L=45.9 mm, d=559 Mm) 138 30. Typical Shape of Load-separation Curve 140 31. Tensile Stress versus Floe Apparent Volume Concentration (L=6.2 mm, d=44.2 um) 141 32. Tensile Stress versus Floe Apparent Volume Concentration (L=4.9 mm, d=44.2 am) 142 33. Tensile Stress versus Floe Apparent Volume Concentration (L=2.9 mm, d= 44.2 Mm) 142 34. Tensile Stress versus Floe Apparent Volume Concentration (L=4.6 mm, d=27.9 Mm) 144 35. Tensile Stress versus Floe Apparent Volume Concentration (L=3.7 mm, d=27.9 Mm) 145 36. Tensile Stress versus Floe Apparent Volume Concentration (L=2.7 mm, d=27.9 Mm) 146 37. Tensile Stress versus Floe Apparent Volume Concentration (L=3.7 mm, d=l9.7 Mm) 146 38. Breaking Stress versus Suspension Apparent Volume Concentration 149 39. Concepts of Fibre Crowding in Suspensions and F i t t e d Lines to the Threshold Concentration Data 156 1-40. Apparent Volume Concentration versus Mass Concentration for Wood-pulp Fibres 176 IV- 41. Multifilament Preparation for Cutting 187 V- 42. Sample of Fibres deposited on the Slide Mount and Closed in It 189 V-43. P r i n c i p l e of Fibre Length and Radius of Curvature Measurements 190 V- 44. Data Acquisition System for Fibre Length and Fibre Curvature Measurements 192 VI- 45. Experimental Setup for Water Absorption by Nylon Fibres 197 VI-46. Water Absorption by Nylon Fibres in 98% RH Environment 198 VIII-47. Typical Load-elongation Curves for 3 Denier Nylon Filaments in Wet and Dry State 207 VIII-48. Simply supported Fibre def l e c t s in a Cross-flow ....210 VIII-49. Fibre Deflection versus Bulk Reynolds Number 212 xi i i IX-50. Method of F r i c t i o n Measurement 215 IX-51. Wet-friction C o e f f i c i e n t s versus Sled Loading for 3 Denier Nylon Filaments 216 IX- 52. Wet-friction C o e f f i c i e n t Versus Sled Loading for 6 Denier Nylon Filaments 217 X- 53. Rotating Cylinder Apparatus 219 XIII-54. Experimental Setup for Ve l o c i t y Measurements in a Horizontal Rotating Cylinder 224 XIII-55, Beam Crossing Geometry at an Instant in Time 226 XIII-56. Locations of Measuring Points within the Central Plane of the Cylinder 228 XIII- 57. Change in Focal Length caused by Various Indices of Refraction 230 XIV- 58. Progressive Dismemberment of Type-C Floe 237 XV- 59. Coherent Floe removed from the Suspension by the Fork 239 XV-60. Tensile Test Equipment 240 XV-61. Device for Tensile Testing of Nylon Floes 241 XV- 62. Plane View of the Break Zone 243 XVI- 63. Parallelepiped in which Fibre deposition was Modeled 246 XVII- 64. Bending Forces acting on Fibre 251 XVIII- 65. Standard and Seibold Conductivities versus KC1 Concentration in Solutions 255 ACKNOWLEDGEMENTS xiv This d i s s e r t a t i o n i s , apart from e x p l i c i t reference, the o r i g i n a l work of the author. However, I am deeply indebted to my supervisor, Dr. Richard J. Kerekes, Honorary Professor of Chemical Engineering, and to my colleague Dr. Peter A. Tarn Doo for many hours of valuable discussion. I thank the People of Canada who contributed to the existence and operation of the University of B r i t i s h Columbia, the Pulp and Paper Research Institute of Canada, the Natural Sciences and Engineering Research Council of Canada, and MacMillan Bloedel Limited. By doing t h i s , they provided me with a place of study and f i n a n c i a l assistance. DEDICATION I dedicate t h i s work to my wife, Irena, and my son, Antoine. 1 INTRODUCTION Flocculation of wood-pulp f i b r e s in suspension i s an important factor in pulp rheology, cleaning, screening, washing, beating, and mixing as well as in r e l a t i o n to the design of the wet end of paper machines. In consequence, f l o c c u l a t i o n a f f e c t s the properties of the end product-paper. The competitive environment demands improved paper qu a l i t y and production e f f i c i e n c y . The improvement of both cannot happen without understanding the factors a f f e c t i n g them. This i s the rationale for doing research in the area of f l o c c u l a t i o n of wood-pulp f i b r e s . From past work, fi b r e f l o c c u l a t i o n has emerged as a complex phenomenon that has to date defied precise d e f i n i t i o n . Nevertheless, three general aspects of f i b r e f l o c c u l a t i o n have become distinguishable: state, process and nature. The state of f l o c c u l a t i o n relates to the degree of suspension nonuniformity at a given instant of time. Hence, the state of f l o c c u l a t i o n can be observed in decaying turbulence, fully-developed flow, or in an immobile suspension. The process of f l o c c u l a t i o n relates to the dynamics of the suspension between two f l o c c u l a t i o n states separated by an in t e r v a l of time. It deals with r e l a t i v e f i b r e motion and interaction between suspending medium and f i b r e s . The nature of f l o c c u l a t i o n i s uniquely related to forces of interaction at f i b r e contact points in a given state of f l o c c u l a t i o n . 2 Broadly speaking, the f l o c c u l a t i o n of f i b r e s in suspension consists of r e l a t i v e f i b r e motion, f i b r e c o l l i s i o n s , and cohesion. The forces of cohesion can be of a c o l l o i d a l or mechanical nature. C o l l o i d a l forces are the result of molecular and surface charge interactions whereas mechanical forces the result of physical entanglement/interlocking of f i b r e s . In a wood-pulp fi b r e suspension of papermaking consistency, the forces of mechanical cohesion dwarf the e f f e c t s of c o l l o i d a l a t t r a c t i o n . H i s t o r i c a l l y , an electro-chemical cohesion was recognized f i r s t [B5,B6,B12,E6,R9,W8]; then mechanical cohesion was shown to be more s i g n i f i c a n t [A5,A8,J3,J4,J5,J6,M1,P3,S12]. While mechanical cohesion i s now accepted as being dominant, i t s precise nature i s not known. A recent review of current knowledge of f i b r e f l o c c u l a t i o n [K7] pointed out that two d i s t i n c t l y d i f f e r e n t forms of mechanical cohesion may exist - f i b r e l i n k i n g and f i b r e i n t e r l o c k i n g . For sake of convenience these have been c a l l e d Type-B and Type-C cohesion: Type-B cohesion occurs as a result of l i n k i n g or hooking between f i b r e s ; Type-C cohesion i s the result of constraint imposed on e l a s t i c a l l y bent f i b r e s in a f i b r e network. Figure 1 shows the p r i n c i p l e s of these two cohesions. In Type-C cohesion, the normal forces at f i b r e contact points develop along with the network and give r i s e to the surface f r i c t i o n forces which prevent r e l a t i v e f i b r e movement. In Type-B cohesion, the normal forces at contact points are unnecessary for the network to exist but they would develop along with the f r i c t i o n a l forces i f the network was strained. The subject of 3 Figure 1. P r i n c i p l e of Type-C (a) and Type-B (b) Cohesion. In (a), normal forces are represented by l e t t e r N and f r i c t i o n a l forces by F. Figure taken from [K7] t h i s d i s s e r t a t i o n i s Type-C cohesion, i t s nature and formation. The two pa r t i c u l a r approaches to studies of f i b r e f l o c c u l a t i o n are s t a t i s t i c a l and phenomenological. The s t a t i s t i c a l approach inquires exclusively about variations in the mass d i s t r i b u t i o n of f i b r e s and/or variations in the number of floes [A3,A4,A6,A8,A10,E7,H5,H8,I2,J1,J2,L1,M17,N3,N4,R4,R5,T3, W2]. The phenomenological approach, on the other hand, which sees f l o c c u l a t i o n as a process having various f i b r e - t o - f i b r e or 4 f i b r e - t o - l i q u i d interactions [A5,H8,J3,J4,J5,J6,S12] i s much more demanding than the s t a t i s t i c a l approach because i t requires understanding and control of a greater number of factors. Table I l i s t s the majority of known factors that a f f e c t f l o c c u l a t i o n . The phenomenological approach was chosen for this study because improved understanding of the process and the nature of f l o c c u l a t i o n could r e s u l t . Numerous factors were eliminated or controlled through use of a model f i b r e system -nylon f i b r e s suspended in an aqueous-sugar solution. Type-C cohesion was isolated experimentally to determine the conditions under which i t occurs. The Type-C coherent floes were dismantled to explore th e i r structure and were strained to measure their strength. This d i s s e r t a t i o n begins with a systematic review of the l i t e r a t u r e (Section 2) on the f l o c c u l a t i o n of wood-pulp f i b r e s 1 and man-made fib r e s in suspension. The process and the nature of f l o c c u l a t i o n for man-made fib r e s and wood-pulp fi b r e s are discussed separately in three sub-chapters corresponding to the three sub-ranges of concentration: d i l u t e , semiconcentrated and concentrated. The boundaries of these sub-ranges which depend on fi b r e aspect r a t i o and f i b r e concentration are well defined only for the fib r e s of uniform length and diameter. This review shows that, despite considerable research, mechanical cohesion has not been investigated and remains unexplored. The l i t e r a t u r e i s replete with suppositions and guesses on f i b r e 1 This term refers exclusively to pulps obtained through the pulping processes which produce undamaged, elongated p a r t i c l e s from wood fi b r e s or tracheids. T a b l e I. F a c t o r s A f f e c t i n g F l o c c u l a t i o n o f W o o d - p u l p F i b r e s . S U S P E N S I O N C O N C E N T R A T I O N P H Y S I C A L AND C H E M I C A L F A C T O R S HYDRODYNAMIC F A C T O R S F I B R E C H A R A C T E R I S T I C S WATER C H A R A C T E R I S T I C S C O N C E N T R A T I O N F I B R E L E N G T H p H OF WATER C H A N N E L GEOMETRY [ C 1 , F 1 . G 5 . H 8 . W 8 ] [ F 1 , H 8 , M 6 , T 8 ] [ A 5 . C 1 ] [ A 9 . K 5 . K 6 . K 7 ] C R I T I C A L C O N C E N T R A T I O N F I B R E D I A M E T E R ION T Y P E S AND FLOW V E L O C I T Y [ B 1 1 . M 5 , M G ] [ T 8 ] C O N C E N T R A T I O N [ F 1 . H 8 . R 5 ] [C1 , R 2 ] S E D I M E N T C O N C E N T R A T I O N WALL T H I C K N E S S , T U R B U L E N C E ( S C A L E , [ A 5 . T 8 . T 9 ] S W E L L I N G A D D I T I V E S I N T E N S I T Y ) [ 0 3 , J 4 ] [ B 6 . C 1 , E 6 , H 8 , R 2 , W 8 ] [ A 9 , G 5 , K 7 , R 5 ] L I M I T I N G C O N C E N T R A T I O N [ M 1 0 . T 9 ] F I B R E S H A P E (HOOKS , P R E S E N C E OF A I R SHEAR R A T E K I N K S , B E N D S ) [ B 6 , G 4 , H 9 , K 2 . M 4 , T 1 2 ] SHEAR D U R A T I O N [M5 ] [ H 8 . K 1 , K 7 , M 4 , W 8 ] T E M P E R A T U R E F I B R I L L A T I O N , S U R F A C E [ E 6 . U 1 , W 8 ] WATER V I S C O S I T Y ROUGHNESS [ 0 1 ] [ E 6 . F 1 , M 5 , J 5 . J 6 . T 8 . W 8 ] F I B R E - W A T E R R E L A T I V E F I B R E F L E X I B I L I T Y V E L O C I T Y [ E 6 , F 1 , M 6 , J 3 . J 4 . T 8 ] S U R F A C E C H E M I S T R Y [ C 1 , J 1 , J 2 , J 5 , 0 6 , M 6 ] 6 cohesion in general and Type-C cohesion in p a r t i c u l a r . The Summary of the Literature Review (Section 2.5) discusses experimental findings and focusses on the subject of the d i s s e r t a t i o n . The Literature Review concludes with the statement of objectives for t h i s research work. Section 3 presents an experimental program for study of a f i b r e system whose main c h a r a c t e r i s t i c s are: 1. The nylon f i b r e s suspended in an aqueous-sugar solution are neutrally buoyant. 2. The v i s c o s i t y of the suspending medium i s 3.7 times that of water. 3. The f i b r e length d i s t r i b u t i o n s are narrow. The range of lengths was selected to correspond to the naturally occurring lengths of wood-pulp f i b r e s . 4. The f i b r e diameters were selected to match the f l e x i b i l i t y of wood-pulp f i b r e s . The experimental program i s divided into three topics: Type-C floe formation, floe structure, and floe strength. Section 4 reports and discusses the experimental resu l t s ; Section 5 summarizes the new findings; Section 6 recommends further work. The numerous appendices which are referred to throughout the text were introduced to record information which would obscure the c l a r i t y of the main text. This information i s , nevertheless, of primary importance, and the subjects discussed in them are i n t e r r e l a t e d . 2 LITERATURE REVIEW Many concepts concerning f i b r e f l o c c u l a t i o n were published in the 1950's and 1960's. Some of these studies have been referred to extensively in the l i t e r a t u r e in the years since, but misinterpretations which subsequently have been repeated have often resulted. In some cases, concepts originating from one work have been extrapolated beyond their a p p l i c a b i l i t y so that an erroneous picture of f l o c c u l a t i o n emerged. In other cases, concepts not c l e a r l y stated at f i r s t were c l a r i f i e d l a t e r or not c l a r i f i e d at a l l . Frequently, reported observations have been wrongly interpreted or have remained unexplained. The absence of a global view on the process and nature of f i b r e f l o c c u l a t i o n in part accounts for t h i s . For minimal misinterpretation, of the kind that has occurred, the l i t e r a t u r e review that follows d i r e c t l y quotes o r i g i n a l publications and provides a un i f i e d picture of f i b r e behaviour in suspension. 2.1 F l o c c u l a t i o n i n D i l u t e F i b r e Suspensions A suspension is defined here as d i l u t e when fibres are free to move without interaction with other f i b r e s . This does not mean that interaction cannot happen. It means only that the distances between fibres are several times larger than the fi b r e length and, i f the interaction occurs, i t i s a chance inte r a c t i o n . This regime of f i b r e concentration i s reviewed to i l l u s t r a t e the complexity of fi b r e motion under the simplest flow conditions, i . e . , sedimentation or simple shear. The studies of suspensions of fi b r e s which are of well-defined regular shape 8 (man-made fibres) are addressed separately from the studies of suspensions of wood-pulp fibres which are of irregular shape. Man-made f i b r e s . Mason and his colleagues studied motion of individual c y l i n d r i c a l p a r t i c l e s in the corn syrup subjected to uniform shear [M5,M7,T11]. Observations were made in a Couette system consisting of two concentric glass cylinders rotated in the opposite directions by the independent mechanical drives having continuously variable speeds. The p a r t i c l e s were made of glass filament 9.5 um in diameter and were cut to lengths which gave a range of f i b r e aspect r a t i o s , L/d's, from 17.8 to 132 [T11]. The applied rates of shear were less than 2 s~ 1. The periodic rocking motion of individual p a r t i c l e s was observed. The p a r t i c l e s rotated i r r e g u l a r l y around the axis perpendicular to the plane of shear with the periods of rotation varying inversely with the rate of shear. The ends of a single p a r t i c l e described t r a j e c t o r i e s the projections of which on the plane of shear were e l l i p s e s . At the same time, the p a r t i c l e was spinning about i t s p r i n c i p a l axis. In another study [N2], the motion of longer p a r t i c l e s was observed. Rayon fi b r e s of 9.5 um diameter and aspect ratios of 43.2, 113, 173, 240 and 356 were suspended in castor o i l of 2.5 Pa«s v i s c o s i t y . Uniform, 10 s 1 rate of shear was applied to the dispersion medium. Short p a r t i c l e s (L/d=43.2 & 113) rotated in spherical e l l i p t i c a l o r b i t s described in the preceding paragraph. Longer f i b r e s were seen to undergo increasing amounts 9 of bending during rotations. Fibres of L/d=1?6 showed s l i g h t springiness; f i b r e s of L/d=240 showed increased springiness; f i b r e s of L/d=356 showed springiness and f l e x i b i l i t y during rotations. Thus, f i b r e length adds new features to the already complex motion of short and s t i f f f i b r e s . The interaction between c y l i n d r i c a l p a r t i c l e s was observed in another similar study [M8]. Dacron Polyester cylinders were suspended in a corn syrup ( v i s c o s i t y about 5 Pa«s and density about 1300 kg/m ) which was uniformly sheared with rates lower than 2.5 s 1. The Dacron cylinders had 12.4 um mean diameter. Very d i l u t e suspensions were prepared having volume concentrations, C v's, of 0.0000026, 0.0000062, 0.0000031 for fi b r e aspect r a t i o of 20, 68 and 115 respe c t i v e l y . 1 As a result of the velocity gradient in the liquid, two rotating cylinders can be carried by translation into proximity and thus interact. On approaching one another, the two participating cylinders become associated for a time, separate and rotate in orbits which differ from those which prevailed before i nt er act i on. Such sudden changes in orbi t s were always associated with the close approach of another p a r t i c l e . The interaction did not involve apparent collision of the two particles but merely a close approach and possibly an i nt er penet r at i on of their orbits. [M8] Andersson [A2] studied interactions of pairs of sedimenting c y l i n d r i c a l p a r t i c l e s . He observed in a t r a v e l l i n g microscope the pairs of glass f i b r e s of equal diameter but of di f f e r e n t 1 In th i s d i s s e r t a t i o n a l l concentrations are reported as fr a c t i o n s . Nomenclature gives d e f i n i t i o n s of symbols. 10 length sedimenting in a pure glycerol (p=1260 kg/m , M=1.76 Pa«s). The motion of glass f i b r e s during interaction was complicated. No c o l l i s i o n between interacting p a r t i c l e s was ever observed although e f f o r t s to attain a physical contact were made. Andersson t r i e d to simulate c o l l i s i o n s in other sedimentation systems. Such model systems were nylon fibres suspended in organic liquids swelling nylon, rayon fibres suspended in CMC solution or viscose solution, but in no case was any flocculating tendency observed. [A8] Wood-pulp f i b r e s . Motion of wood-pulp fibres in a uniform shear flow was also studied [A11,F1,F2,M7]. In [A11] the suspending medium was a corn syrup of 5 Pa«s v i s c o s i t y . Douglas f i r pulps of various y i e l d s were fractionated (20/28-mesh f r a c t i o n ) . F i f t y fibres of a mean length close to 2 mm were selected from each y i e l d f r a c t i o n . The fibres were v i s u a l l y observed during rotations induced by 3.5 s 1 rate of shear. Visual observations showed that in each pulp there existed a spectrum of flexibility of individual fibres; it was therefore not surprising that in any sample a multitude of rotation patterns was observed [A11]. A simple c l a s s i f i c a t i o n of f i b r e orbits was made [A11]: 1. Rigid Fibres. Rigid fibres executed the same type of rotations as r i g i d cylinders. 2. F l e x i b l e Fibres. a. Group 1. The f i b r e undergoes a x i a l spin in the course of which i t p e r i o d i c a l l y straightens and bends into an arc. 11 b. Group 2. The f l e x i b l e spin described i s superimposed on a spherical e l l i p t i c a l o r b i t . c. Group 3. This group i s preferred by f l e x i b l e f i b r e s . The modes of rotation in t h i s group may be further subdivided in order of increasing f l e x i b i l i t y : Springy Rotation, Snake Turn and S-Turn. It i s noteworthy that, because the corn syrup used was about 5000 times more viscous than water i s , the motions of f i b r e s observed by Mason et a l . [A11,F1,F2,M7] in the corn syrup may not be encountered in water. From the nature of these rotations, i t i s apparent that the volume swept out by one rotating f i b r e i s considerably greater than f i b r e ' s actual volume. When two swept volumes intercept, f i b r e c o l l i s i o n can occur. 2.2 F l o c c u l a t i o n i n S e m i c o n c e n t r a t e d F i b r e Suspensions A suspension i s defined here as semiconcentrated when the distances between f i b r e centres are smaller than the average f i b r e length. In such suspensions, the fibres move r e l a t i v e to one another with interactions and/or c o l l i s i o n s . The review of the l i t e r a t u r e pertinent to t h i s concentration regime w i l l be made to i l l u s t r a t e how experimental findings from t h i s regime influenced understanding of f i b r e f l o c c u l a t i o n in general. 2 . 2 . 1 F i b r e i n t e r a c t i o n s and c o l l i s i o n s Man-made f i b r e s . Blakeney [B11] studied the e f f e c t of f i b r e concentration on the interaction of straight, r i g i d f i b r e s . He used nylon f i b r e s 12 16.9 and 43.1 Atm in diameter with aspect ra t i o s of 19.2 and 20.3 respectively. Measurements were conducted in a concentric cylinder viscometer at shear rates between 0.1 and 0.5 s 1. The fibr e s were suspended in tetrachloroethane-paraffin o i l solution adjusted to the density of f i b r e s . Analysis of the orientation fact or1 revealed that i t remained constant up to the concentration Cv=0.0042. Above t h i s concentration, the formation of groups of two or three f i b r e s increased rapidly. Between concentrations of Cv=0.0042 and 0.0050, the interacting fibres changed th e i r o r b i t s to more horizontal, i . e . , closer to the plane of shear. It was concluded that below Cv=0.0042 the interaction between fi b r e s was n e g l i g i b l e . This concentration agreed c l o s e l y with Mason's estimate of the critical concent rat i on which was defined as: The concentration above which unrestrained motion of all the fibres is no longer possible is defined as the critical concent ral i on [M6]. Mason la t e r estimated the c r i t i c a l concentration quanti t a t i v e l y as: An estimate of the order of magnitude of critical concentration can be made by assuming that the statistically averaged shape of the space enclosed by a rotating fibre is a sphere whose diameter equals the particle length. [Ml] This d e f i n i t i o n leads to an expression for the c r i t i c a l volumetric concentration of c y l i n d r i c a l p a r t i c l e s C v c r : C v , c r = i<E>2 "» 1 See Glossary. 13 Attanasio et a l . [A18] studied v i s c o s i t y of suspensions of rayon fi b r e s in the c a p i l l a r y (3 and 120 s 1 rate of shear) and Couette (100 to 500 s 1 rate of shear) viscometers. The aqueous-sucrose solution with 67% sucrose content by weight was used as the suspending medium (p=1300 kg/m , M=0.18 Pa«s). The volumetric concentration of f i b r e s ranged from CV=0.0015 to 0.01. The f i b r e aspect ra t i o s were 11.3, 16.2, 17.7, 18.2 and 69.5. The analysis of experimental results revealed an almost linear r e l a t i o n s h i p between the limiting viscosity number1 and the aspect r a t i o for a l l aspect r a t i o s but for 69.5. This i r r e g u l a r i t y indicates the presence of intense f i b r e interaction between the longest f i b r e s for which the c r i t i c a l concentration C v c r=0.0003 i s well below the range of experimental concentrations. Since the c r i t i c a l concentration for the second longest fibres is C y c r=0.0045 and i s only s l i g h t l y less than the upper l i m i t of experimental concentrations, the l i n e a r r e l a t i o n s h i p between the limiting viscosity number and the aspect r a t i o appears to occur below the c r i t i c a l concentration defined by equation (1) . Numerous publications relate to the rheology of f i b r e suspensions to which Attanasio et a l . [A18] work belongs. The l a t e s t review of the l i t e r a t u r e in t h i s area was published by Ganani and Powell [GI]. Most rheological works, however, say nothing about the process and the nature of f i b r e interactions. 1 See Glossary. 1 4 Wood-pulp f i b r e s . In a shear flow and at very low concentrations, wood-pulp fi b r e s can rotate independently of one another and can undergo chance c o l l i s i o n s , i . e . , c o l l i s i o n s of low frequency [M6]. As the concentration i s increased, the frequency of two-body encounters increases as the fib r e orbits become more and more crowded u n t i l a point i s reached at which a rotating f i b r e i s under more or less continuous interaction of forced c o l l i s i o n s . These were described in [M7]: At very low concentrations of wood-pulp fibres, e . g . , less than 0.0005 by weight, which is the upper limit of concentration at which single fibres can be followed in the microscope field, the encounters are very frequent and cause the particles to move in erratic orbits. For example, an average f i b r e length for never-dried softwoods i s 3.37 mm and an average width i s 30.3 M m giving Lw/w=111.1 The c o l l e c t i o n of such fibres would have the c r i t i c a l volumetric concentration of C „=0.000154. This would v,cr correspond to the consistency of C m cr=0.000054 i f the water retained by fi b r e s was 2 g per 1 g of dry fi b r e mass [E5]. An average, never-dried, hardwood f i b r e i s 1.23 mm long and 17.6 M m wide giving Lw/d=70. The suspension of such fi b r e s would have C =0.000391. This would correspond to the consistency of Cm c r = u « 0 0 0 1 3 6 i f the water retained by fibres was 1.7 g per 1 g of dry f i b r e mass [E5]. These simple calculations reveal how small the c r i t i c a l concentrations for wood-pulp fibres are. Thus, in any suspension of p r a c t i c a l consistency, the 1 See Table I-XVII in Appendix I. 15 interactions and/or c o l l i s i o n s between fibres are forced. Arlov et a l . [A11] measured the steady-state d i s t r i b u t i o n of o r b i t s in a suspension of 28/48-mesh fra c t i o n (L=1.6 mm [T7]) of an unbeaten, unbleached softwood sulphite pulp at consistency less than C =0.0005 after shearing for 15 hours at 2 s 1 rate of m shear. The or b i t s of wood-pulp f i b r e s , as those of smooth r i g i d cylinders [M8], had preferred instantaneous orientations in the plane of shear under forced f i b r e interactions. The or b i t s changed frequently as a result of " c o l l i s i o n s . " 1 The d i s t r i b u t i o n of o r b i t s was said to have reached a dynamic equilibrium. Andersson [A8] conducted sedimentation experiments of wood-pulp f i b r e s in aqueous-glycerol mixtures having v i s c o s i t i e s from 0.02 to 0.2 Pa«s (region of "creeping motion"). I n i t i a l f i b r e consistencies ranged from Cm=0.000l to 0.0006. Fibre approaches within p a r t i c l e pairs occurred due to d i f f e r e n t i a l speed of sedimentation. Hydrodynamic repulsion counteracted such approaches; however, due to the f i b r e s ' irregular shape, they approached each other close enough to establish physical contact. In an experiment where the approaching fibre was shaped as a bow, concave downwards, the hydrodynamic forces acted in vain, and the approaching fibre landed right on the other fibre, which was slightly curved in the opposite direction [A2]. In d i l u t e f i b r e suspensions, the man-made fib r e s which were r i g i d and straight did not c o l l i d e . On the other hand, wood-pulp f i b r e s , which were neither s t i f f nor straight, were found to c o l l i d e . 1 Quotations marks around the word c o l l i s i o n s were used in [A11] without explanation. 16 In 1938, Wollwage, in a thesis t i t l e d "The Flocculation of Papermaking Fibres" [W7] described experiments [W7,W8] with wood-pulp fi b r e suspensions having consistencies, C m, from 0.00005 to 0.0002, and flowing with bulk speeds from 0.005 to 0. 0274 mm/s ' (Re from 3.8 to 20.8) 1 in a v e r t i c a l glass pipe. Estimated shear rates based on the maximum speeds were from 0.26 to 1.4 s 1 . The pipe was 8-foot long and 3-inch in diameter. A controlled laminar flow of a we 11 - di s pe r s e d suspension, free from eddy currents, was v i s u a l l y observed and photographed. The distance from the top of the pipe to the point at which f l o c c u l a t i o n occurred was a dependent variable. Evaluations of the flocculation end-points were made where definite fibre bunches could be seen, as well as small portions of the dispersing medium which are devoid of fibres. Above this point there existed a relatively complete degree of dispersion of the fibres, but below it, the condition was one of definite flocculation. [W8] This equipment and technique permitted a dynamic study of the f l o c c u l a t i o n of wood-pulp f i b r e s . Erspamer [E6] and Beasley [B6] used the same experimental setup though Beasley ran i t under s l i g h t l y d i f f e r e n t conditions. Beasley's experimental conditions were: C m from 0.00002 to 0.0005, 0.0168 m/s bulk speed (Re=12.8) and 0.88 s 1 estimated shear rate. A summary of common, non-contradictory results from these three investigations shows: 1. Factors Enhancing F l o c c u l a t i o n : Alum flock (in small amount), hydrophilic betonite clay, starch, a i r , increasing consistency, beating action and increasing f i b r e length. 2. Factors Suppressing F l o c c u l a t i o n : Alum flock (in large 1 Reynolds numbers are based on the bulk speed, pipe inner diameter and v i s c o s i t y of water at 25°C. 17 amount), deacetylated Karaya gum, Locust bean gum, methylcellulose, mannogalactan, decreasing consistency, low water temperature. 3. Factors Having No E f f e c t : calcium carbonate, surface-active agents (surface tension), sodium hexametaphosphate, water pH. Wollwage concluded that non-mechanical factors play important roles in the phenomenon of fibre flocculation. Wollwage's and Erspamer's publications (1939,1940) [W8,E6] popularized the subject of c o l l o i d a l cohesion. In 1953, Beasely received the Shibley Award from TAPPI for his paper. This event indicates the popularity of c o l l o i d a l cohesion among papermakers in the 1940's and 1950's. Though cohesion between fibres was considered to be dominantly c o l l o i d a l in nature, i t was recognized that the mechanism by which the flocculating fibres are brought together differs fundamentally from that of colloid sol in that an entirely different type of particle motion is involv ed. [M4] Wollwage was f u l l y aware of th i s fact in 1939. As the dispersion moves down the column, fibres are acted upon by viscous forces due to relative motions in the fluid, individual fibres being rotated, moved about, and brought into contact, and groups of two or more fibres which may be adhering together tending to be sheared apart. At some point down the flow tube, the number of fibres col I i di ng with each other and tending to adhere will exceed the number being sheared apart and separated by the fluid motion and, at this point, fibre bunching or flocculation should be observed. Since the shearing forces will vary with changes in fluid velocity in the tube, the flocculation point will depend on the flow rate. [W8] 18 -c Wood-pulp f i b r e s which have dimensions between 10 to -3 10 m are thus outside the actual c o l l o i d a l region of p a r t i c l e sizes which range from 10 to 10 m [Nl], They, contrary to c o l l o i d p a r t i c l e s , cannot be brought into contact by Brownian motion because, in the case of suspension of large p a r t i c l e s such as c e l l u l o s e f i b r e s , t h i s motion i s non-detectable and therefore i n s i g n i f i c a n t . The distances between fi b r e s are also large, - 3 about one fi b r e length, 10 m, while the interaction range of electro-chemical forces i s about 10 m. The importance of fib r e motion in suspension as an element of fl o c c u l a t i o n was recognized by Mason in 1948 [M4]. Under Mason's leadership, a series of rigorous studies of fi b r e motion was i n i t i a t e d . Some of those studies have already been reviewed in thi s chapter. 2 .2 .2 Fibre cohesion When c o l l i s i o n has taken place, fibres may cohere to form floes which may subsequently disperse in a shear flow [K1,W8]. Mason et a l . [M4] described t h i s process: Shear motion provides a mechanism whereby fibres can be br ought i nt o cont act and, if an at t r act i v e force exists, can adhere to form the nucleus of a floe which can grow by further collisions. The floe itself will also tend to rotate and in the process will be subjected to shear and tensile stresses which tend to stretch it and disrupt it, the disruptive stresses increasing with increased rate of shear. Thus the motion which allows floes to form will also tend to destroy them. If s u f f i c i e n t time i s allowed, a dynamic equilibrium i s established between floe formation and floe destruction. At low concent r at i ons where individual floes could be observed they were continuously formed and broken up and . . . the size of the floes at equilibrium appeared to increase with decreasing shear rates. [M4] The 19 equilibrium moreover could be reproduced repeatedly by restoring a given rate of shear even though the equilibrium was disturbed in the interval. [H8] The existence of dynamic equilibrium was shown experimentally [H8] when pulp suspension (100/200-mesh, bleached sulphite, L=0.45 mm [T7]) was subjected to varying rates of shear, from 19 to 70 s 1 in a Couette type apparatus. Fibre mass concentration was Cm=0.0004. During dynamic equilibrium two counteracting elements c l e a r l y are in balance: the rate of floe dispersion and the rate of floe formation. The cohesion forces developed at fi b r e contact points found at the floe peripheries equal the dispersion forces r e s u l t i n g from the drag imposed on fi b r e s . Andersson [A5] observed that, under d i f f e r e n t i a l sedimentation conditions, the fi b r e s have frequently remained in contact af t e r c o l l i s i o n . He speculated on the possible nature of cohesion: This suggests the existence of adhesion between fibre surfaces. Such adhesion may be due to protruding fibre elements being like barbs, or it may also be due to adhesive forces of an electrical or chemical nature. [A5] Mason [M6] envisaged f i b r e cohesion: When the fibres come into contact with one another t her e ar e t wo possible ways by whi c h t he y can stick together. Per haps t he mos t obvious is by i nt e r-par t i cl e attraction in the same way, for example, as the particles of coagulated sol are held together. For convenience we shall call this chemical flocculation. The other is by i nt erme s hi ng or mechanical entanglement, and t o distinguish it from the first, will be called mechani cal  fl occul at i on. From the nature of the processes involved one would anticipate chemical flocculation to be determined in part by the surface condition and mechani cal flocculation by the geometric shape of fibre. 20 Here, Mason should have used the word "cohesion" instead of " f l o c c u l a t i o n " since he had already distinguished " c o l l i s i o n " from "cohesion" as two steps of wood-pulp f i b r e f l o c c u l a t i o n [M4], This d i s t i n c t i o n could have eliminated much of the ambiguity from his discussions of f l o c c u l a t i o n . If f i bres have strong e l e c t r o s t a t i c surface charge, there i s normally a repulsion force between the f i b r e s . However, i f they are f o r c i b l y brought into intimate contact, e.g., through c o l l i s i o n in the shear flow, the repulsion barrier can be overcome and a force of a t t r a c t i o n i s generated between them. The f i b r e s are held together by van der Waals forces. Erspamer's and Wollwage's [E6,W8] experiments l e f t no doubt that electro-chemical cohesion occurred when fibres had been brought into contact by hydrodynamic forces and varied, within l i m i t s , with addition of certain materials. Their experiments, however, were conducted at shear rates not larger than 1.4 s 1 . Hubley et a l . [H8] studied the e f f e c t s of the addition of varying amounts of thorium chloride, aluminum sulphate, deacetylated Karaya gum and methyl c e l l u l o s e to the suspensions of pulp-fines (L<0.2 mm) of consistency Cm=0.0007. The applied rate of shear was 16 s 1. The observed changes in the flocculation index^ were small, from 0.55 to 0.76. An exception was the excessive addition of deacetylated Karaya gum. The f l o c c u l a t i o n index in t h i s case changed from 0.411 to 2.06. The e f f e c t of shear rate on the f l o c c u l a t i o n of 100/200-mesh (L=0.45 mm [T7]) bleached sulphite pulp at Cm=0.0004 was also 1 See Glossary. 21 studied [H8]. The change of shear rate from 70 to 1 9 s 1 increased the fl o c c u l a t i o n index from 0.470 to 2.66. The increase in the fl o c c u l a t i o n index was also produced by the increased presence of longer f i b r e s in the suspension. Suspension concentration was maintained constant, at Cm=0.0008, but the proportion of 48/100-mesh fibres (L=0.7 mm) and pass-200-mesh fines (L<0.2 mm) was varied. At 100% fines, the flo c c u l a t i o n index was 0.540 and, at 20% fines, i t rose to 2.54. The effects of shear rate and f i b r e length may be regarded as purely mechanical in nature [M4]. Hence, the eff e c t s of electro-chemical interaction between fibres of finer pulp fr a c t i o n were, in most cases, small compared with the variations res u l t i n g from varied shear and fi b r e length [H8], The whole-pulp f i b r e s were observed to form large stable f l o e s , presumably because of increased mechanical cohesion [H8]. Shear rates greater than 50 s 1 were necessary to break up these floes appreciably. This strong cohesion rendered the whole-pulp fi b r e s unsuitable for studying the electro-chemical e f f e c t s . How thi s strong cohesion developed was not described. The picture presented by Mason [M5,M6] i s in part speculative. In pulp suspensions we deal with particles which possess a measure of flexibility, and are bent and twisted. Fibrillation provides surfaces with innumerable hooks which can join fibres together. All of these factors will tend to favor interlocking of the particles. The tendency to clot mechanically will increase markedly as the aspect ratio is increased. It will be aided further by fibrillation, which will facilitate interlocking, and by increasing flexibility, which will allow the fibres to bend and interweave. [M5] 22 No d i r e c t investigation has ever been conducted to evaluate the importance of f i b r e f i b r i l l a t i o n , f l e x i b i l i t y or aspect r a t i o on f l o c c u l a t i o n . 2 . 3 F l o c c u l a t i o n i n C o n c e n t r a t e d F i b r e S u s p e n s i o n s In semiconcentrated suspension as the concentration i s increased, the f i b r e s form larger floes and the space occupied by the r e l a t i v e l y free fibres diminishes. At some point, large floes merge to form a continuous network [M5] or connect ed structure [H8], A suspension i s defined here as concentrated when a l l f i b r e s are in continuous contact with other f i b r e s and form one network between confining boundaries. This network i s sometimes c a l l e d a "plug." The lowest concentrations at which plug flows of wood-pulp fibres occur are s l i g h t l y below sediment concentrations [A9]. The connected networks of measurable strength were produced at concentrations as low as Cm=0.002 [G5,V1]. Andersson [A5] measured consistencies of sedimented f i b r e and defibrated Swedish softwood pulp laps. The reported sediment consistencies were: mats formed from C =0.0005 consistency suspensions of rewetted m C m 1. Spruce, unbleached sulphite 0.0032 2. Spruce, bleached sulphite 0.0030 3. Pine, unbleached kraft 0.0032 4. Pine, bleached kraft 0.0028 23 Thalen and Wahren [T8] reported sediment consistencies for much wider spectrum of pulp f i b r e s . They used the same i n i t i a l consistency as in [A5]. The sediment consistencies for hardwood fib r e s are only c i t e d here to demonstrate that shorter f i b r e s sediment into denser mats. Cm 1. Birch, NSSC, 80% y i e l d 0.0046 2. Aspen, sulphite, beaten 0.0058 3 . Birch, bleached kraft, beaten 0 . 0 0 3 3 2.3.1 Process of network formation Man-made f i b r e s . Meyer and Wahren [M10] noted that elongated p a r t i c l e s such as glass, Teflon and Perlon f i b r e s may form a unique type of mechanically entangled, 3-dimensional (3-D), coherent network. The coherence of such a network was not ascribed to chemical bonding but was primarily due to normal forces associated with the bending stresses in fib r e s and to f r i c t i o n a l forces produced by these normal forces acting at contact points between fi b r e s [S12]. Meyer and Wahren [M10] c a r e f u l l y worded the concept of such network formation and cohesion: When a fibre suspension is agitated, the fibres are exposed to viscous and dynamic forces, which bend and twist fibres. When agitation ceases, the fibres tend to regain their original unstrained shape. However, if there are many fibres per unit volume, the fibres cannot straighten out freely but will come to rest in contact with other fibres. A fraction of the fibres will come in contact with so many other fibres that they will come to rest in strained positions, and forces will be transmitted from fibre to fibre. These fibres become interlocked by normal and frictional forces and 24 constitute a fibre network, where forces can be transmitted through the fibres and from fibre to fibre. This concept remained unaltered in la t e r publications by Wahren et a l . [S12,W1,W3,W4] and has been assumed to apply to wood-pulp fi b r e s [C1,E2,P3,W1,W3,W4,W10]. The existence of such a network has not been v e r i f i e d experimentally. Steenberg et a l . [S12] have presented some evidence of i t by describing the circumstances of suspension preparation: // bundles of straight Perlon fibres or any other man-made fibres of a suitable I engt h-1 o~r adi us ratio are put into water, they settle to the bottom of the container and do not form a network, even after gentle stirring with a spoon. If, however, the suspension is vigorously agitated - for instance, with a propeller -the fibres are dispersed randomly and form a network that fills the whole available volume and possesses elastic properl i es . [S12] Steenberg et a l . [S12] witnessed the change in suspension appearance and described conditions under which this change occurred, but they never d i r e c t l y v e r i f i e d the process of network format ion. Steenberg et a l . [S12] showed that, using d i f f e r e n t v i s c o s i t i e s of the suspending medium, networks varying in r i g i d i t y were formed. Equal amounts of fib r e s (Perlon) were dispersed in aqueous-sugar solutions of d i f f e r e n t v i s c o s i t i e s and the shear modulus was measured. The volumetric concentration of fib r e s was CV=0.015. It was observed that the shear modulus decreased as the v i s c o s i t y of the suspending medium increased. This i s i l l u s t r a t e d in Figure 2. The authors concluded that no fi b r e networks formed at very high v i s c o s i t i e s of the suspending medium. They s a t i s f i e d themselves with the following 25 Steenberg et alii (1966) .001 .01 .1 1. Viscosity of sugar solutions, Pa-s Figure 2. Shear Modulus versus Suspending Liquid V i s c o s i t y . Figure redrawn from [S12]. explanation: It takes longer for fibres to come to rest after agitation in a more viscous medium; hence, the elastic energy of many of them may be dissipated and unstrained configurations will result. Once the fibres have lost their elastic energy, they cannot become actively engaged in the network however much time elapses. [S12] This explanation i s incomplete because f i b r e - t o - f i b r e interactions were not mentioned. At the concentration used, CV=0.015, there i s ubiquitous contact between fi b r e s during and after a g i t a t i o n . L o g i c a l l y , less time should be required for fi b r e s and the entire suspension to come to rest after a g i t a t i o n in a more viscous medium since the only d i s s i p a t i n g forces are the viscous forces. Further comments on thi s work cannot be made because no information on fi b r e geometry was given. Attanasio 26 e t a l . [A17] p r o v i d e d a l t e r n a t e e x p l a n a t i o n t o the e f f e c t of v i s c o s i t y on network r i g i d i t y : This effect, observed with suspensions of Perlon fibres in glucose solutions (the drop is found at a values between 0.01 and 0.05 Pa«s^ shows that the medium has a lubricating effect on the contact points. [A17] T h a l e n and Wahren [T9] p r e s e n t e d some t e s t i m o n y t o the e v e r - p r e s e n t n o n u n i f o r m i t y of f i b r e networks and t o the f r i c t i o n a l c h a r a c t e r of f i b r e - t o - f i b r e c o n t a c t : Considerable difficulties were encountered when trying to get uniform networks of the Perlon and glass fibres, especially at high concentrations and with long (6 mm) fibres. The networks formed at high concentrations were uneven also after intense stirring. Although the Teflon fibres were of the order of 6 mm long, they were much easier to disperse and the network that formed had an even appearance. The networks of fibres having low coefficient of friction (Teflon) are easier to disperse evenly, thus giving higher values for the shear modulus. [T9] Not o n l y does f i b r e m a t e r i a l a f f e c t the f i b r e - t o - f i b r e i n t e r a c t i o n s but a l s o the type of suspending l i q u i d . E l i a s [E4] has found t h a t c o m p r e s s i o n b e h a v i o u r of the g l a s s f i b r e mats was s i g n i f i c a n t l y a f f e c t e d by the immersion l i q u i d . I t i s p o s s i b l e t h a t the immersion l i q u i d s c o u l d a l t e r f i b r e s u r f a c e p r o p e r t i e s and t h e r e b y a f f e c t the f r i c t i o n a l i n t e r a c t i o n a t f i b r e c o n t a c t p o i n t s . Wood-pulp f i b r e s . As the c o n c e n t r a t i o n i s i n c r e a s e d w e l l above the c r i t i c a l c o n c e n t r a t i o n , the interweaving of f i b r e s becomes more pronounced and, e v e n t u a l l y , more o r l e s s continuous network of entangled f i b r e s w i l l be formed, p o s s e s s i n g a s t r u c t u r a l r i g i d i t y [M5]. 27 Forgacs, Robertson and Mason [F1] were f i r s t to demonstrate that f i b r e s in pulp suspensions form coherent fibre networks which exhibit t e n s i l e strength at concentrations as low as Cm=0.003. They coined the term "coherent f i b r e networks" and suggested that networks are held together by inter-fibre friction. A term "interlocked networks" was introduced by Chang & Robertson [C1] in t h e i r studies of t e n s i l e strength of pendent plugs. The network preparation was done by fast r e c i r c u l a t i o n of pulp suspension around a closed flow loop and sudden stoppage of flow. The formation of plugs was promoted by increased f i b r e length, f i b r e f l e x i b i l i t y , number of hooks and bends, f i b r i l l a t i o n , surface roughness and concentration of fib r e s [F1]. When concentrated suspension i s pumped through the tubes or pipes, i t exhibits three flow regimes: plug, mixed, and distributed [R4] or turbulent [C4,F1], There are two t r a n s i t i o n v e l o c i t i e s between these regions. Robertson et a l . [R4] described the regimes as follows: Below the first transition velocity, the stock could be observed to flow as a solid plug with the entire velocity gradient concentrated in a narrow boundary at the wall. The plug may be regarded as a continuous network of interlocked fibres whose independent movement (rotational or translational) is inhibited by the close packing; this network is rigid in the sense that it resists disruption by the shear stress at the wall. Above the second transition velocity, the network was disrupted into a series of fibres and fibre floes with velocity gradients extending across the tube. [R4] In studies in which plugs were formed for t e n s i l e strength tests [A9,C1,F1,G5,R2], the objective was to disperse fibres as uniformly as possible in the turbulent type of flow and freeze t h i s uniform dispersion by suddenly stopping the flow. 28 In the case of abruptly stopping the circulation process, the turbulent state of flow becomes, to a certain extent, "frozen" into the just formed network of the plug but only when the transition from turbulent flow to the state of rest or, at least, to laminar flow takes place suddenly. [G5] Giese and Giese [G5] experimented with two pumps of di f f e r e n t dispersing potentials and showed that the degree of homogenization i s c r i t i c a l to the plug t e n s i l e strength. The p e r i s t a l t i c pump, working on the displacement p r i n c i p l e , had no mixing and no deflocculating e f f e c t s . The agi t a t i o n of the suspension in the r e c i r c u l a t i n g tube was gentle because of low flow rates. The cen t r i f u g a l pump gave good mixing and faster flow which produced more agitation in the r e c i r c u l a t i n g tube. The plugs produced with the ce n t r i f u g a l pump exhibited much higher t e n s i l e strength than those produced with the p e r i s t a l t i c pump. Thus, the appearance and strength of plugs depends on the conditions of plug formation, more uniform plug giving higher strength [G5] or exhibiting better e l a s t i c properties [T8]. It was not clear how cohesion occurred but there was unanimous consensus that f i b r e - t o - f i b r e cohesion i s predominantly of a mechanical nature [C1,F1,G5,R2]. Two important properties of a coherent plug were i d e n t i f i e d : f i b r e - t o - f i b r e cohesion and the plug structure which includes plug nonuniformity. A series of experiments of great importance in the study of wood-pulp f i b r e f l o c c u l a t i o n was ca r r i e d out by Jacquelin [J3,J4,J5,J6]. He exposed pulp suspensions to unique flow conditions in partly f i l l e d , rotating c y l i n d r i c a l vessels. The vessels were rotated about their longitudinal axis while 29 remaining horizontal or inc l i n e d at 45 degree to the horizontal. Jacquelin agitated about 4 l i t e r s of suspension in a 20 cm diameter cylinder at the cylinder l i n e a r speed of about 40 m/min (about 42 rad/s) for a few hours and obtained very dense, almost spherical floes which he named "coherent f l o e s " [J3]. He used the r a t i o between the weight of dry fib r e s constituting the floes and the t o t a l weight of dry f i b r e s from a given batch as a dependent variable. He c a l l e d t h i s r a t i o "taux des floes coherent" 1 and denoted i t for convenience as T.F.C. The maximum reported T.F.C. was 94%. Observing the evolution of the number of coherent floes Jacquelin saw the number s t a b i l i z e d more quickly than T.F.C, in one case less than 5 hours instead of more than 15 hours for T.F.C. This, he concluded, shows that, after an i n i t i a l period of aggregation of coherent floes, the growth of floes happens mainly by capture of the suspended fibres having physical c h a r a c t e r i s t i c s suitable to enter into these f l o e s . The remaining f i b r e s in the suspension were progressively reduced in number and the process of capture slowed [J3]. Within the consistency range from Cm=0.0075 to 0.049, Jacquelin found the concentration sub-ranges related to f i b r e nature and conditions of agitation which favoured the formation of coherent f l o e s . The maxima in T.F.C. were attained at consistencies about three times larger than the sediment consistencies [ J 3 ] . 2 They were: 1 Mass content of coherent f l o e s . 2 Sediment consistencies were c i t e d at the beginning of Section 2.3. 30 from 0.01 to 0.012 for unbleached softwood monosulphite, from 0.013 to 0.015 for unbleached softwood kraft, from 0.015 to 0.017 for unbleached softwood b i s u l p h i t e , and from 0.026 to 0.028 for bleached birch kraft. The observed difference between maxima for softwoods and hardwood can be attributed to the difference in f i b r e lengths. This important point has not been explored by Jacquelin. His empirical investigations i n c l i n e d toward studying f i b r e f l e x i b i l i t y as a variable. He was influenced by the contemporary concept of f i b r e network formation which was postulated in 1964 by Meyer and Wahren [M10] and described e a r l i e r in t h i s section as one in which f i b r e e l a s t i c i t y played a key role in the formation process as well as in the mode of cohesion. Jacquelin found predominant quantities of thick-wall f i b r e s in coherent floes and the thin-wall fibres in the suspension. S i m i l a r l y , high-yield softwood fibres were more apt to form coherent floes than did low y i e l d f i b r e s . He concluded that the f l e x i b l e f i b r e s are unable to accumulate enough energy in their e l a s t i c deformation and do not aggregate. He also noted that very s t i f f f i b r e s , l i k e certain f i b r e s of semichemical pulps or mechanical pulps, are unable to form coherent networks. He concluded that these fi b r e s are unable to deform s u f f i c i e n t l y under e x i s t i n g conditions of agitation so that the optimum f i b r e f l e x i b i l i t y should exist at a given condition of agitation and that outside t h i s optimum the cohesion of networks cannot aris e from the e l a s t i c deformation of fi b r e s [J3,J4], 31 Modification of f i b r e f l e x i b i l i t y through r e f i n i n g brought unexpected r e s u l t s . It was observed that s l i g h t r e f i n i n g greatly reduced the a b i l i t y of fi b r e s to form coherent f l o e s . Determination of the f l e x i b i l i t y index by Mason's method [R7] revealed that, in the case of kraft pulp, the changes in f i b r e f l e x i b i l i t y were small between 13° and 18°SR [J6]. In addition, i t seemed that the surfaces of fi b r e s which were d r a s t i c a l l y modified showed f i b r i l s and m i c r o f i b r i l s . These gently refined f i b r e s formed r e l a t i v e l y weak aggregates and thei r existence impaired the development of coherent floes . Jacquelin r e a l i z e d that the nature of cohesion had changed and c l a s s i f i e d floes into two groups: (1) the fragile floes which impart to the suspension a cloudy appearance; (2) the coherent floes which form slowly under rather peculiar conditions and have a somewhat button-like appearance. He described them [J5]: Whereas the first and more common type of floe forms rapidly and has neither a defined structure nor cohesion, the second, of slow and progressive formation has good structure and cohesion. The important point is that all factors liable to favor the formation of the first type of flock impair the development of the second type, and vice versa. Particularly in the field which is of special interest here the "cloudy" non-cohesive flocculation, will interfere with the formation of cohesive floes and consequently reduce the T.F.C. In fact, when the attraction between fibres is very strong, a rapid formation of structureless units will ensue to the detriment of the slow bonding of cohesive networks. These l a t t e r require a certain liberty of individual fibre movement . Lee [L2] produced floes using Jacquelin's technique for his studies of indi v i d u a l floe dispersion. He described the mechanism of floe formation by the r o l l i n g method as a two-step 32 process: network truncation into floes and floe compaction [L2]: The secondary flow caused by the rolling motion separates the mat of fibres into large patches or lumps on the order of several centimeters. As time progresses, further fragmentation and flocculation take place. The patches larger than the final equilibrium size will undergo further breaking or shedding, while the ones smaller than the equilibrium size will undergo flocculation. At this stage, the aggregates are not highly entangled and the boundaries of the aggregates are not sharply defined. The shape of the aggregates is also irregular. These loose aggregates at their equilibrium size then undergo compaction as a result of their interactions. The compaction is a process in which the aggregates grind against one another. The surface fibres bend and adhere to the surface or detach from the aggregate. Thus the surface of the aggregate is smoothed and loose surface fibres are removed. At the same time, the grinding action compresses the aggregate causing rearrangement of the internal fibres and a spheroidal floe with a very strong network takes form. By increasing the rolling time, floes of more uniform shape and higher compaction can be obtained. Surface smoothness also increases as time progresses. Lee did not notice any fundamental difference in the form of cohesion or the structure of floes he studied. The author of thi s d i s s e r t a t i o n attempted to observe the formation of wood-pulp fi b r e networks in decaying turbulence at sediment concentration. Turbulence was created in a suspension by a gr i d being moved through 13 mm deep plexiglas channel [K7]. The g r i d speed of 0.28 m/s produced a jet speed of 1.3 m/s from the g r i d openings. With high-speed cinematography, the motion of dyed fi b r e s in decaying turbulent shear was recorded. An area of 15 mm by 10 mm was filmed to observe individual f i b r e motion. Suspensions of bleached softwood kraft f i b r e s having a consistency of Cm=0.003 and containing about 10% of dyed f i b r e s were used. Individual fibres experienced substantial bending which stopped in 0.13 s after passage of the g r i d . If fi b r e s 33 interlocked by regaining their unbent configurations, the process of network formation must have been fast. Complete cessation of motion in the suspension occurred after 0.84 s. Interactions between colored and non-colored f i b r e s could not be observed because of large amount of l i g h t scatter through the suspension. 2.3.2 Structure of coherent networks In 1962, Corte and Kallmes [C5] reported on a th e o r e t i c a l analysis of random 3-D structures of f i b r e s . They c i t e d a private communication with R.E. Miles on his Ph.D. thesis in which he showed that the volumetric concentration is related to the number of contacts per f i b r e , f i b r e length and fi b r e diameter: This model was developed for fi b r e s having uniform lengths and diameters. The second published analysis of 3-D fi b r e networks was that of Meyer and Wahren [M10]. They refined the s t a t i s t i c a l model of Onogi and Sasaguri [02] to account for f i b r e length. Their model assumed that the fibre length and the fibre radius have variable d i s t r i b u t i o n s expected to cover the normally occurring radius and fibre length distributions [M10]. The concepts of the number fi b r e f r a c t i o n at a given length with a given number of contact points and the p a r t i a l segment length were applied. The segment i s a part of an active f i b r e between two consecutive contact points. The segment length i s the distance along the fi b r e 34 c e n t e r l i n e between the segment ends. The p a r t i a l segment l e n g t h d i s t r i b u t i o n (or c o n t a c t p o i n t d i s t r i b u t i o n ) was assumed t o be a P o i s s o n d i s t r i b u t i o n s t a r t i n g a t n =3 and n o r m a l i z e d from n =3 t o i n f i n i t y . The p a r t i a l segment l e n g t h d i s t r i b u t i o n has been chosen t o f i t the assumption t h a t e v e r y u n i t l e n g t h of f i b r e has the same p r o b a b i l i t y of r e c e i v i n g a c o n t a c t p o i n t . W i t h the f i n a l a ssumption t h a t e v e r y f i b r e i n the p o p u l a t i o n has the i d e n t i c a l l e n g t h and d i a m e t e r and i s i n c o n t a c t w i t h a t l e a s t n c o t h e r f i b r e s , the e x p r e s s i o n f o r the v o l u m e t r i c c o n c e n t r a t i o n r eaches a minimum. From t h e i r m a t h e m a t i c a l approach the r e l a t i o n s h i p has emerged between the f i b r e volume c o n c e n t r a t i o n , the number of c o n t a c t p o i n t s per f i b r e , and f i b r e l e n g t h - t o - d i a m e t e r r a t i o . T h i s r e l a t i o n s h i p i s : 1 6 • IT-L/d C — —^———^—————— v,min 0 r n (3) 2-L n 3 n • a n -1 c c c E q u a t i o n (3) i n d i c a t e s the e x i s t e n c e of a limiting concent r at i on [M10] f o r f i b r e s of u n i f o r m l e n g t h and d i a m e t e r a l l a c t i v e l y engaged i n t h e network. The f i b r e l e n g t h - t o - d i a m e t e r r a t i o e x e r t s a s t r o n g i n f l u e n c e on the l i m i t i n g c o n c e n t r a t i o n . The models of M i l e s and Meyer & Wahren a r e s u i t a b l e f o r e x p e r i m e n t a l e v a l u a t i o n of u n i f o r m l y l o n g and t h i c k f i b r e s . One d i f f i c u l t y , however, remains. An i m p l i c i t assumption was made i n the development of t h e s e models t h a t the network i s i s o t r o p i c and unbound. In r e a l i t y , l a r g e networks a r e always nonuniform [T9]. I t i s p o s s i b l e t h a t t h i s f a c t o r i n h i b i t e d d i r e c t e x p e r i m e n t a l 35 v e r i f i c a t i o n of equations (2) and (3). Thalen and Wahren [T9] detected the lowest concentrations at which a shear modulus of suspensions could be measured. These minimum concentrations were only s l i g h t l y higher than the corresponding sediment concentrations. The postulation that coherent networks are not formed at concentrations lower than the sediment concentrations seemed, therefore, natural. E l i a s [E4] elaborated on the number of contact points between f i b r e s in the sedimented networks:. During the formation of a sediment, as fibre is deposited upon a partly formed bed it will be supported at two points, as it would be a rare coincidence for three or more of points of suspension to occur in the same straight line. As each point of support on the bottom of a fibre is paired with one on the top of another fibre, each fibre will also, as a rule, contact two fibres along its top surface. Each fibre will then touch an average of four other fibres in a sediment. Since a l l fibres in sedimented networks have four contact points, the sediment concentration should exceed C at n =3. c v,min c This i s , in fact, what Thalen and Wahren observed by comparing sediment concentrations with the l i m i t i n g concentration calculated at nc=3 [T9]. In a more recent publication, Wahren [W4] stated that attempts at an experimental v e r i f i c a t i o n of the formula for the li m i t i n g concentration have indicated that nc=4 is closer to r e a l i t y . Indeed, better agreement i s observed between the sediment concentrations and the l i m i t i n g concentrations calculated at n c = 4 [T9]. This agreement i s surprising because the sedimented network i s not an i s o t r o p i c , 3-D network [E4]. 36 Whether suspended f i b r e s form a c o h e r e n t network or a r e i n some o t h e r s t a t e ( s e d i m e n t ) , t he number of f i b r e s i n the sus p e n s i o n i s f i n i t e . I f a l l f i b r e s a r e u n i f o r m l y d i s t r i b u t e d , the number of f i b r e s i n any volume s m a l l e r than t he t o t a l s u s p e n s i o n volume i s c o n s t a n t so t h a t the number of f i b r e s i n the volume V can be r e l a t e d t o the v o l u m e t r i c c o n c e n t r a t i o n C v when the average f i b r e l e n g t h L and d i a m e t e r d a r e known: 2 N F«7T«d «L C» - 4 - V ( 4 ) or N f - — f - ( 5> 7T«d *L The number of f i b r e s , N^, was used as an i n d i c a t o r of fibre crowding [K7] i n one i n s t a n c e and as a number density [B9] i n a n o t h e r . K e r e k e s e t a l . [K7] chose a sphere h a v i n g d i a m e t e r L f o r volume V: N f S • rV<a>2 ( 6 ) Bibbo e t a l . [B9], on the o t h e r hand, s e l e c t e d a cube w i t h s i d e L f o r V: " f c • rV<3>2 <7> In b o t h c a s e s was a u s e f u l parameter i n the d e s c r i p t i o n of t he s t a t e of f i b r e c r o w d i n g . For f i b r e s of i d e n t i c a l p r o p o r t i o n s , f o r example, t h e softwood f i b r e s from S e c t i o n 2.2 a t C =0.01 and L /w=111, t h e number of f i b r e s i n the sphere i s v w 37 N f s ~ 8 2 , and in the cube N£c==157. The potential for interaction between fi b r e s in such suspension i s great. Numbers of f i b r e contacts per f i b r e calculated from equation (2) and (3) are nc=*4.4 and nc=*4 respectively. Bibbo et a l . [B9], studying the shear properties of semiconcentrated suspensions, defined t h i s regime using N ^ as a parameter. For semiconcentrated suspensions, 1 /7r<N^c<L/(ir-d) was suggested; for d i l u t e suspension, N^C<1/7T. Doi and Kuzuu [D3] stated that a f i b r e suspension behaves as a v i s c o e l a s t i c l i q u i d i f N^ c<L/(7r»d), and as an e l a s t i c material i f N^ c>L/(7r»d). These concepts, though unrefined, are worthy of mention since they constitute the beginning of the systematic c l a s s i f i c a t i o n of f i b r e suspensions with respect to f i b r e behaviour. Man-made f i b r e s . No experimental work has been published on the structure of t r u l y random, three-dimensional (3-D) fibr e networks. E l i a s [E4] investigated the behaviour of individual glass f i b r e s during the mechanical compression of thick, filtration-formed mats. Though the mats were undeniably 3-D, the arrangement of f i b r e s in them was not random. More than 90% of fibres were in c l i n e d to the horizontal at angles of less than 10 degrees whereas in an isotropic network an average angle between fibres and horizontal i s 32.7 degree [W3]. From photographs showing the shapes of the f i b r e s in the loaded beds, E l i a s [E4] determined the minimum number of f i b r e s touched by a given s i l v e r e d f i b r e . The term "minimum" was used 38 because one f i b r e could touch another without producing a detectable bending of either f i b r e . The experimental data are given in the f i r s t three columns of Table I I . Volume concentrations shown in second column were calculated from E l i a s ' s solids fraction assuming a glass density of 2600 kg/m . The number of f i b r e contacts per f i b r e , n c, was calculated from equations (2) and (3) using E l i a s ' s data; the results are given in columns 4 and 5 of Table II. Cl e a r l y , the calculated numbers of contacts are greater than observed. Wahren considered [W3] the deflections of f i b r e s in the coherent network from their o r i g i n a l nearly straight shape as small. E l i a s ' s observations of f i b r e deflections in the compressed mats of glass fibres support Wahren's opinion. In mats, the deflections were about one f i b r e diameter [E4]. 2.3.3 Network strength Devising methods of e f f i c i e n t floe dispersion requires a knowledge of network strength. For a uniform suspension, floes must be dispersed into individual f i b r e s , i . e . , a suspension must be homogenized. Homogenized suspension i s required for f i b r e s to be evenly accessible to chemicals or to produce a uniform sheet of paper. Man-made f i b r e s . In 1958, Forgacs et a l . [F1] reported that, at Cm=0.008 concentration, the chopped rayon f i b r e s (d=l0 Aim, L=1.5 mm) had no t e n s i l e strength. The l i m i t i n g concentration calculated at T a b l e I I . Number of F i b r e C o n t a c t s per F i b r e 1n Compacted Mats [E4] and C a l c u l a t e d from E q u a t i o n (2) and ( 3 ) . ASPECT RATIO L/d VOLUME CONCENTRATION OF FIBRES C v OBSERVED NUMBER OF CONTACTS [E4] CALCULATED NUMBER OF CONTACTS eq.(2) CALCULATED NUMBER OF CONTACTS eq.(3) 313 0.0119 7.3 14.9 13.6 313 0.0187 11.4 23.5 17.3 313 0.0209 12.6 26.2 18.4 151 0.0231 6.5 13.9 9 . 1 151 0.0284 6.5 17.2 10.2 630 0.0221 21.6 55.7 38.6 179 0.0254 7.2 18.2 11.5 40 n =3 f o r thes e rayon f i b r e s i s C =0.003. F o r g a c s e t a l . c J v,min r were u n s u c c e s s f u l i n f o r m i n g a c o h e r e n t network a t a c o n c e n t r a t i o n of almost t h r e e t i m e s the l i m i t i n g c o n c e n t r a t i o n i n a d e v i c e which produced enough a g i t a t i o n t o e n t a n g l e wood-pulp f i b r e s of s i m i l a r d i m e n s i o n s and f l e x i b i l i t y . H o r i e and P i n d e r [H6,H7], e x p e r i m e n t i n g w i t h a r t i f i c i a l s l u r r i e s of r e g u l a r l y s i z e d n y l o n f i b r e s i n aqueous s o l u t i o n s of p o l y e t h y l e n e g l y c o l , sodium c h l o r i d e , and d e x t r o s e , employed a c o a x i a l c y l i n d e r v i s c o m e t e r w i t h a gap of 10.9 mm. The s u r f a c e s of b oth the cup and the bob w i t h groves i n the a x i a l d i r e c t i o n p r e v e n t e d s l i p of f i b r e s a t t h e w a l l s . The e f f e c t s of f i b r e a s p e c t r a t i o , f i b r e and s a l t c o n c e n t r a t i o n on t h e y i e l d s t r e s s , the e q u i l i b r i u m s t r e s s and v a r i a t i o n of t h i c k n e s s of f l o w i n g l a y e r w i t h time were examined. The f i b r e s of u n i f o r m d i a m e t e r of 43.1 nm and v a r i o u s l e n g t h s (0.987, 1.62, 3.01, 5.03 and 6.72 mm), gave a range of a s p e c t r a t i o s from 22.9 t o 156. The v o l u m e t r i c f i b r e c o n c e n t r a t i o n s i n s l u r r i e s ranged from C v=0.04 t o 0.17. Three of H o r i e and P i n d e r ' s [H7] numerous o b s e r v a t i o n s a r e of i n t e r e s t t o t h i s s t u d y : 1. The h i g h e r the f i b r e c o n c e n t r a t i o n , the h i g h e r the y i e l d s t r e s s . 2. The h i g h e r the a s p e c t r a t i o , t he h i g h e r the y i e l d s t r e s s a t any c o n c e n t r a t i o n . I t would appear, however, t h a t t h e r e i s a l i m i t i n g v a l u e of maximum y i e l d s t r e s s independent of f i b r e l e n g t h s i n c e the p o i n t s f o r L/d r a t i o s g r e a t e r than the 37.4 41 l i e on a single curve, i . e . , the points for L/d=68.8, 116, and 156. 3. At a fixed f i b r e concentration, the y i e l d stress decreased with increasing dispersing medium v i s c o s i t y which varied from 0.00745 to 0.22 Pa-s. These observations applied to the equilibrium stress as well. The second observation indicates that s l u r r i e s made of long fibres behave a l i k e within a range of C v from 0.04 to 0.11. The t h i r d observation i s in agreement with Steenberg et a l . [S12]. Relationships between the y i e l d stress and the volumetric concentration were analyzed by the o r i g i n a l data being f i t t e d to a power relationship of the form: ry = a-C v b (8) Since such relationships have been widely used in the comparison of shear strengths of wood-pulp f i b r e suspensions, t h i s relationship i s used here for the same reason. Values for a and b along with the o r i g i n a l data for L/d=69.8, 116, and 156 are shown in Table I I I . These values are used l a t e r in t h i s d i s s e r t a t i o n in the analysis and discussion of experimental r e s u l t s . Wood-pulp f i b r e s . Numerous researchers used two main approaches in studying network strength: shear strength and t e n s i l e strength measurements. These studies can be c l a s s i f i e d as direct and i n d i r e c t . The dir e c t methods comprise experimental techniques in CN I D * - CO « 1 Z O in CO CO CM U J I _ l O 1 r-in CN m TT E O in O U J ra ra \ O o 1-C L Z + O O o U J O CN CN io in o d in a c II Q m CN i - — 1 _ J L U E t- in io r- O O I O I O O CO CN O X U J CC \ O co co O * - CO CN T f t~ »- in co u> 1—1 h- Z in I D CD CN in in CN CM in co O Q > - l/> " CO CN co ( J CM CO (tj 13 l z o TT T t CO CM U J X _ i O 1 s 10 CN o o E in U J (0 <0 \ o <D C L C L Z + CO d d CN U J O o m tr o 6 t/> a: C II Q ( / ) CN t- — % _ J U J E m O O O O l t O O co co o O X L U CC \ CO I D 0) CO CD in C N T t CN r~ in U J t-H 1- Z I P CN C N T T t t- CO CN co co C N in a >- l/l C N CN CO CN in T t o 10 O (0 l z o CO CO C N U J I _ J O 1 in £ CN cs ra E o I D r-— U J C L 10 \ o O co Q . CO Z + in o CN T Ul O in O in co a 6 in cc c II O C O CN H a. _ l L U E r- r- O O in co o O CO ID O O X U J o: \ O co O in o o in CO T t O 0) Ul I - l Z T— 0) o CO I D CO (0 O CO Q >- </) CN T t - - CN CN CO O 1— T f (0 n l z o CO T t CN U J X _ i O 1 10 £ CN t -CO E T- T-T- U J C L CD \ T- i~ C L Z + co o U J O O (/) * -O d in cc c n Q m CN i - — 4 _ l U J E co O O O i f M O O X U J CC \ CD CD O CO CO CM U J t-H H- Z in o io in CO ID * - CM Q > - c/> T- CN TJ- CN T t ME in ID t- co cn O • • - •* in I D t- co CO •q- in io r- co co 3 C J 1 O O O O O -r- O O O O O O O O O O O O _1 Z > l O O C J o o o o o o o O O O O O O O O O O O O > O H C J O CO U J —1 T J a. H \ 1 0) I D I D (/> < _ J 1 10 in < cc 43 which the force i s applied to the body of the network so that the break area i s r e s t r i c t e d ; the force and the area are d i r e c t l y measured. The indi r e c t method has, in a l l cases, some uncertainty a r i s i n g from the estimates of either the breaking force or the load-bearing area or both. Studies of shear and t e n s i l e strength of f i b r e networks by the di r e c t methods are: 1. Shear strength studies of suspended f i b r e networks in shear testers under quasi-static test conditions (low shear rates) in which only network-network interface transmits the load. The applied torque i s measured d i r e c t l y and the break area i s predetermined [D4,D5,K10,T12]. 2. Tensile strength of dry wood-pulp f i b r e s . The load and break area of the network i s d i r e c t l y measured so that the breaking stress i s e a s i l y calculated [G2]. Strength studies by the i n d i r e c t methods are: 1. Tensile strength of f i b r e plugs. The plugs which are formed in tubes have a predefined cross-sectional area. However, because the load i s not applied at a well-defined cross-section, the plug breaks always in the weakest spot, i . e . , at the floe boundaries at which the network discontinuity may e x i s t . The length of the pendent plug after rupture, i s used in the c a l c u l a t i o n of the breaking stress [C1,E3,F1,G5,V1]. 2. Shear strength studies of f i b r e suspensions in the shear testers under dynamic flow conditions (large rates of shear) and/or where the network-solid boundary interface transmits the load under either q u a s i - s t a t i c or dynamic conditions. Under dynamic flow conditions when pulp suspension moves r e l a t i v e to the smooth walls in a shear tester, fibres migrating away from the walls form a clear water annulus between the walls and the pulp network surface [M16], S i m i l a r l y , when the pulp suspension flows in a pipe, fibres migrate away from the pipe wall [C4]. a. The shear stress i s evaluated at the inner cylinder surface and at a given cylinder speed [B8,M13,R1,T8]. b. The shear stress i s evaluated at the network (plug) dispersive surface at a given cylinder speed [G6,H3], c. The shear stress i s evaluated at the outer cylinder surface at the onset of turbulent a g i t a t i o n of the entire suspension [G8], 3. Shear strength studies of flowing f i b r e networks in pipes, i . e . , plug surface disruption in pipe flow. a. The shear strength was calculated at the pipe radius which was evaluated from the estimate of the water annulus thickness at the onset of turbulence in i t [D1,D2,G7,K4,M14,M15,R5,S9]. b. The shear strength was calculated at the v i s u a l l y assessed radius of the plug disruption surface [D4,D5]. c. The shear strength was calculated at the radius taken from the v e l o c i t y p r o f i l e studies [D4,D5,M12], d. The shear strength was assumed to be the same as the wall shear stress at the onset of drug reduction in pipe flow [B1,P2]. 45 Publications include enough data for v a l i d comparison of quasi-static test methods. The comparison can be made employing the mathematical relationship most commonly c i t e d in the l i t e r a t u r e . r = a'.C„,b' (9) Since t h i s r e l a t i o n s h i p has not been derived from basic p r i n c i p l e s i t does not have t h e o r e t i c a l j u s t i f i c a t i o n for i t s c o e f f i c i e n t s . The f i t c o e f f i c i e n t s a' and b' for the d i r e c t shear measurement studies (Table IV) are consistently greater than those for the indir e c t shear measurement studies (Table V). This implies that the ultimate shear strength of networks was probably measured by the application of the dir e c t methods. This finding should guide those who undertake any experimental investigations of network strength. The power c o e f f i c i e n t s obtained from the f i t s of o = a " -C m b' ' (10) to the data of t e n s i l e strength studies, shown in Table VI, are generally lower than those from the indir e c t methods of shear measurement shown in Table V. The reason could l i e in the d i f f e r e n t ranges of consistencies in which the experiments were performed. Since the f i t c o e f f i c i e n t s do not have clear physical meaning, t h i s observation cannot be interpreted. The t e n s i l e strength studies produced more experimental evidence on the complexity of f i b r e - t o - f i b r e interaction than did Table IV. Shear Strength of Fibre Suspensions. Direct Methods. Quasi-Static Tests. r(N/m 2) = a' C ( % ) b ' FIBRE TYPE AUTHORS PULP TYPE CONSISTENCY a' b' a' b' RANGE % N/m2 -- N/m2 --Long Fibre DUFFY [D5] Unbleached kraft from pine 0.9-3.1 24.6 2 21 Chemical Pulps Bleached kraft from pine 0.95-4.G 16.3 2 12 (Softwoods) P1ne kraft 1.5-4. 1 6.62 2 73 DUFFY, Unbleached, unrefined kraft TITCHENER from Plnus Radlata 2.4-3.9 22.4 2 27 [D4] Bleached kraft from Plnus 13. 1 2.46 Radlata 2.6-5.0 6.64 2 73 TURNER et a l . Unbleached kraft from pine 1.0-5.0 9 .80 2 83 [T12] Bleached kraft.from pine 1.0-5.0 7.87 2 69 KRYSKI [K10] Softwood kraft 0.94-12.3 10.8 2 14 Short Fibre DUFFY [05] Hardwood sulphite 1.7-5.7 3.37 3 03 Chemical Pulps Birch kraft 1.6-4.0 2.51 3 21 (Hardwoods) DUFFY, Semi-bleached aspen sulphite 2.2-3.5 3.50 3 01 TITCHENER [D4] 3.90 2.96 TURNER [T12] Bleached kraft from birch 1.0-5.0 5.42 2 60 Semi-bleached aspen sulphite 1.0-5.0 4.72 2 85 Groundwoods DUFFY [D5] Refiner groundwood 1.6-5.3 2.57 2 95 2.36 3.03 Stone groundwood 1.8-5.3 2. 15 3 1 1 Table V. Shear Strength of Fibre Suspensions. Indirect Methods. Quasi-Static Tests. 2 b' r(N/m ) = a' C (%) m FIBRE TYPE AUTHORS PULP TYPE CONSISTENCY a' b' a' b' RANGE 2 2 % N/m -- ,N/m --Long Fibre RAIJ, WAHREN High y i e l d spruce sulphite 2 .1-3. 2 0. 21 4 . 16 Chemical Pulps [R1] Unbleached sulphite from spruce 1 .3-1 . 9 2. 51 2 .42 (Softwoods) Bleached sulphite from spruce 0. 72-1 . 7 1 . 91 1 .77 Unbleached kraft from spruce C .6-1 . 9 2. 98 2 .63 THALEN, Spruce sulphite 15.9'SR 0. 55-4. 5 1 84 1 .62 WAHREN [T8] 68.0'SR 0. 64-5. 6 2 95 1 .81 Bleached sulphite from spruce 2.87 2.24 18' SR 0. 41-5. 5 2 18 1 .58 46' SR 0. 40-4 8 2 77 1 .38 84' SR 0. 45-5 7 4 22 1 .66 BERGMAN Spruce sulphite, 60.5% y i e l d 0. 81-2 21 5 56 2 .99 TAKAMURA [B8] 47.4% y i e l d 0. 63-1 78 4 50 2 .64 Short Fibre THALEN, Aspen sulphite, 27.5'SR 0. 82-3 9 2 13 1 .68 Chemical Pulps WAHREN [T8] Bleached sulphite from birch, 1 .96 1 .69 (Hardwoods) 17.0' SR 0. 78-4 0 1 80 1 .70 Groundwoods RAIU & WAHREN Spruce groundwood 102mL CSF 0. 98-5 3 1 01 3 . 10 [RI] 1 .78 2.25 THALEN. Birch groundwood 100mL CSF 1 . 10-7 2 1 62 2 .03 WAHREN [T8] Spruce groundwood 191mL CSF 1 . 10-5 0 1 62 1 .99 Spruce groundwood 91mL CSF 0. 68-5 1 2 88 1 .88 •f 48 the shear strength studies. For t h i s reason, the t e n s i l e strength studies are discussed here in more d e t a i l . Each t e n s i l e test consisted of two steps: the f i r s t step was plug preparation; the second step was plug extrusion. Section 2.3.1 describes the preparation of plugs. The majority of investigators allowed the downward extruded plug to break under i t s own buoyed weight. Andersson [A9], on the other hand, pushed the plug upward to a predetermined length and disrupted i t in an annular flow of water whose flow-rate increased l i n e a r l y with time. Only a few investigators used neutrally buoyant rings [C1,R2] to lower the f r i c t i o n between the plug column and the sides of the glass tube. In general, a f i b r e plug is pushed downward to protrude from the glass tube into another tube of larger diameter and con c e n t r i c a l l y arranged. U n t i l the plug breaks, i t emerges from the tube into water flowing at the same v e l o c i t y . Experiments show that the plug-breaking length, i s independent of the tube diameter when other variables are kept constant provided that the diameter is large enough to permit uniform f i b r e dispersion [F1], However, the plug-breaking length i s dependent on the speed of extrusion, water temperature, and presence of a i r [G5,R3]. A l l a i r bubbles must be evacuated. Speed of extrusion and water temperature should be kept constant. The extrusion speeds are l i s t e d in Table VI. If any changes in plug dimensions due to elongation of the network are neglected, the stress at the break i s readily shown Table VI. Tensile Strength of Fibre Networks. Quasi-Static Tests. <r(N/m2) = a " C ( % ) b " AUTHORS EXTRUSION SPEED mm/s PULP TYPE CONSISTENCY RANGE % a' ' N/m2 b' ' a' ' N/m2 b' ' FORGACS, ROBERTSON,. MASON [F1] 1. 13 68% y i e l d spruce sulphite, 740mL CSF 662mL CSF 539mL CSF 440mL CSF 318mL CSF 0.30-0.96 0.68-1.22 0.68-1.08 0.64-1.00 0.56-1.03 1 . 19 1 .84 1 .87 2.38 3.01 1 .25 1 . 25 1 . 55 1 . 47 1 . 74 1 .86 1 .59 CHANG, ROBERTSON [C1] 1 .0 Standard kraft pulp, 400mL CSF 0.30-1.30 2.49 1 .85 GIESE, GIESE [G5] 0.8 F1-S1-Cellulose, 47'SR F1-S1-Cellulose, 42'SR Mechanical pulp, 52'SR Mechanical pulp, 52'SR 0.40-1.60 0.18-0.99 0.56-1.33 0.32-1.06 1 .86 1 .36 1 . 75 1 . 20 1 .48 1 . 37 2.02 1 . 74 VEINOV et al . [VI] 1 . 1 Unknown pulp, 19'SR Unknown pulp, 42'SR 0.20-1.06 0.38-1.32 1.31 2 .08 1 . 36 2.05 50 to be: P„ a = g-Cm.PL.(1 - -2— ) (11) m A linear relationship was found between the measured plug length, PL, and the mass concentration of f i b r e s C m within a range of C m from 0.004 to 0.01 [C1]. At higher concentrations, r e l i a b l e values of PL were d i f f i c u l t to to obtain because increasingly poor f i b r e dispersion created weak spots in the plug structure [F1]. Furthermore, the fa u l t y spots existed at plug edges so that premature breaks resulted. Giese and Giese [G5] experimented with two pumps of d i f f e r e n t dispersing potentials and demonstrated that the degree of homogenization was c r i t i c a l to PL. The same pulp dispersed by ce n t r i f u g a l and p e r i s t a l t i c pumps exhibited di f f e r e n t strength relationships with consistency. PL did not increase monotonously with increased consistency of the long f i b r e pulps which were prepared with the p e r i s t a l t i c pump; rather, i t l e v e l l e d off and started to drop beyond 0.006 consistency [G5], Contrary to Forgacs et a l . [F1], Giese and Giese [G5] demonstrated that groundwood pulps develop networks whose strength can be measured. However, chemical pulps formed stronger plugs than groundwood pulps at the same consistency. A similar trend emanates from shear strength experiments. Softwood pulps (long fibres) formed stronger networks at the same consistency than hardwood pulps (short fibres) which, in turn, formed stronger networks than groundwood pulps (Table IV and V). 51 In the pendant plug t e s t s , d i f f e r e n t i a t i o n was made as far as the character of a break zone was concerned between smooth and fibrous breaks. A pulp with short and r i g i d f i b r e s , such as mechanical pulp, gave smooth break surfaces. On the other hand, pulps having long and f l e x i b l e f i b r e s which gave brushy break surfaces indicated that long f i b r e s were pulled from the plug over distances comparable to the fi b r e length. Andersson [A9] studied the disruptive v e l o c i t i e s of 20 mm long plugs of d i f f e r e n t mass concentration, from Cm=0.0046 to 0. 015. The median v e l o c i t y of water was plotted versus plug concentration and nearly l i n e a r relationship between the breaking v e l o c i t y of the l i q u i d and the plug concentration was observed. At t h i s plug length, the second order term of v e l o c i t y dominated the stress equation, and Andersson concluded that this result agrees well with the observation that the network strength varies c l o s e l y with the second power of concentration as indicated by Chang and Robertson [C1], Table VI. The dependence of t e n s i l e strength on various parameters considered in pendent plug tests i s summarized below: 1. The t e n s i l e strength increased as the beating l e v e l increased for black-spruce sulphite [F1], 2. Increase in strength was observed as y i e l d of Douglas-fir and black spruce sulphites decreased u n t i l an apparent maximum was reached below 60% y i e l d [FI], 3. Increase in strength was observed with the increased degree of beating applied to F i - S i - c e l l u l o s e u n t i l a maximum at 45°SR was reached [G5], 52 4. Moderate addition of el e c t r o l y t e s or polymers produced sizable changes in the te n s i l e strength [C1,R2], 5. Mixtures of softwood chemical and mechanical pulps produced stronger networks as the content of chemical fibres increased [G5]. 6. Strength of the plugs increased with the increased degree of plug homogenization [G5]. Some questions remain unanswered, e.g., 1, what are the mechanical cohesion forces that prevent or hamper the r e l a t i v e displacement of individual f i b r e s , and 2, were plugs of coherent or fragile nature in Jacquelin's sense [J5,J6]. Recently, a unique study of the t e n s i l e strength of dry wood-pulp floes has been published by Garner [G2]. This study i s unique for two reasons: i t i s the only study done on dry floes and one of only two devoted to the strength of single floes [G2,L2], For a l l pulps investigated, the floe strength exhibited a power-law relat i o n s h i p with the bulk density. The f i t s were made to the d i g i t i z e d data from Figure 3 of [G2], The exponents ranged from 2.29 to 3.60 as shown in Table VII. The exponents for two Refiner Mechanical Pulps were greater than for bleached hardwood and softwood kr a f t s . These res u l t s , similar to those found from the shear strength studies of pulp suspensions (Table IV, Direct Methods), suggest that similar mechanisms of fi b r e interaction during network disruption may be operating in "wet" and "dry" f i b r e networks. The most probable f i b r e - t o - f i b r e interaction i s of a f r i c t i o n a l nature. 53 Table VII. Tensile Strength of Dry Wood-pulp Floes. a(N/m2) = a"'-C b(kg/m3) b"' TYPE OF WOOD-PULP BULK DENSITY RANGE a ' " b ' " kg/m^ N/m2 Bleached softwood kraft 6 . 2 - 2 9 . 7 2 . 9 4 X 1 0 " 1 2 .29 Bleached hardwood kraft 1 4 . 4 - 3 9 . 1 2. 1 4 X 1 0 ~ 2 2.57 Refiner mechanical pulp 1 2 3 . 4 - 6 7 . 1 1 . 3 0 x 1 0 " 4 3.60 Refiner mechanical pulp 2 2 5 . 9 - 7 2 . 9 1 . 8 5 X 1 0 ~ 3 3.17 The e f f e c t of rate of straini n g on the t e n s i l e strength was undetected because the measurements were within the scatter of data. This wide scatter of data was attributed to the experimental d i f f i c u l t i e s , nonuniformity within f l o e s , and errors of measurement in the floe dimensions [G2], On the other hand, the effect of straini n g would be small i f the f i b r e - t o - f i b r e interaction was predominantly of a f r i c t i o n a l nature because the dynamic c o e f f i c i e n t of f r i c t i o n i s constant. Thus, the supposition of f r i c t i o n a l interactions between fibres during floe straining i s supported. 54 2.3.4 Cohesion forces The e l a s t i c interlocking of whole fi b r e s i s expected to occur in concentrated suspensions. Man-made f i b r e s . In 1964, Meyer and Wahren [M10] indicated that a fibre network is a system of fibres in contact where every fibre is locked in position in the network by contact with at least three other fibres in the network and in such a way as to be able to transmit forces. It was unclear how one fib r e can be locked in position and have at least three contacts with other f i b r e s . In 1972, Parker [P3] understood t h i s as interlocking by three alternate contact points and i l l u s t r a t e d i t using fingers for fibres (Figure 1.18 in [P3]). In 1979, Wahren [W3] agreed. The pr i n c i p l e of th i s interlocking i s shown in Figure 1a. Thalen and Wahren [T9] wrote: the results of the experiments indicate that specific attraction forces between fibres are not needed for the formation of coherent networks, but that the coherence of networks is due to physical bonding by entanglement of the fibres. It i s not clear on which experimental evidence t h i s statement was b u i l t since the magnitudes of the s p e c i f i c a t t r a c t i o n forces are not reported. Wood-pulp f i b r e s . Wollwage [W7] attempted to measure f i b r e - t o - f i b r e cohesion forces with a torsion balance. He used unbleached spruce sulphite f i b r e s . The results were not entirely satisfactory, 55 principally because of lack of sensitivity. The exploratory experiments made to date indicated that f i be r-1 o-f i be r adhesion forces do exist and are of the order of 10 4 dynes (10 ^ N). This was a direc t approach in the determination of a force normal to the plane of contact between the two f i b r e s . This method focussed solely on the a t t r a c t i v e electro-chemical interaction between two crossed f i b r e s . Without doubt the cohesion forces between wood-pulp fibres result from electro-chemical and/or mechanical interactions. The question of their r e l a t i v e importance was raised by Chang & Robertson [C1] and Reeves & Gerischer [R2]. Chang and Robertson published results of the dependence of the pendent plug strength and zeta potential of kraft-pulp f i b r e s on the concentration of additives: e l e c t r o l y t e s KC1, CaCl 2 , FeClg, and T h C l 4 and poly-(1,2 dimethyl-5-vinyl-piridinium methyl sulphate) or shortly poly-(DMVPMS). Changes in plug breaking length, PL, of the kraft pulp beaten to 400 mL CSF [C9,T5] and at Cm=0.007 were: about 2.5 cm due to charge neu t r a l i z a t i o n , about 1.5 cm due to ThCl 4 , and about 3 cm due to poly-(DMVPMS). Change in PL due to change in suspension concentration from Cm=0.003 to 0.011 was about 6 cm. Clear l y , electro-chemical interactions affected strength half as much as the change in suspension concentration. Reeves and Gerischer [R2], who used Alum (Al 2(S0 4) 3•18H 20) and modified polyethyleneimine (PEI) as additives to the bleached, long f i b r e , kraft pulp of Cm=0.006, noticed similar e f f e c t s . Addition of Alum produced about 1 cm change in PL, and addition of PEI only 0.3 cm. These changes constituted less than 30% of the i n i t i a l 56 breaking length of plugs. Moderate additions of e l e c t r o l y t e s and polymers producing noticeable changes in the strengths showed that the surface charge which i s so s i g n i f i c a n t in the s t a b i l i t y of c o l l o i d systems also has a measurable e f f e c t on f i b r e - t o - f i b r e interactions in f i b r e plugs. However, i t can also be concluded from these experiments that the chemical bonding between fibres i s not the primary cause of coherence in f i b r e plugs but that they are coherent mainly through entanglement of f i b r e s . It i s noteworthy that the strength of pendent plugs was the lowest ever recorded for f i b r e networks. Hence, the electro-chemical e f f e c t s were noticeable only in the weakest of produced networks. Studies of electro-chemical e f f e c t s on shear strength of f i b r e networks have never been reported, the possible reason being that these effects were unnoticeable. Jacquelin [J5,J6J observed the formation of two types of floes in wood-pulp f i b r e suspensions: fragile and coherent. Gentle mechanical treatment applied to f i b r e s which normally formed hard (coherent) floes changed them into f i b r e s having a soft (fragile), floc-forming tendency. Experiments with chemical additives shed more l i g h t on t h i s transformation and the importance of f i b r e surface interactions, e.g., addition of polyethyleneimine (PEI) in the presence of Alum to the gently-refined pulp produced large changes in T.F.C.1 whereas the same addition did not do so for unrefined pulp. This indicates 1 Mass content of coherent f l o e s . 57 that the extent of the electro-chemical effects can be altered by mechanical modification of fi b r e surfaces [J5], i . e . , i t increases with the creation of f i b r i l s and m i c r o f i b r i l s . Jacquelin also experimented with Alum in combination with sodium hydroxide s i z i n g of unrefined and gently refined pulps with and without PEI, galactomannan, starch and diastase (cellulose enzyme). These experiments confirmed the importance of the i n t e r f a c i a l physico-chemical actions on the behaviour of fi b r e in suspension. It appears that when wood-pulp fibres have developed surface features, the size of which f a l l s into the c o l l o i d region, e.g., f i b r i l s , the interaction of these added up to a si g n i f i c a n t electro-chemical force between f i b r e s . Surface tension forces deserve mention because numerous publications refer to small a i r bubbles as a cause of fi b r e f l o c c u l a t i o n [F3,G4,H9,K2]. The commonly accepted description of the phenomenon i s given in [G4]: Water repellent parts of fibres are easily contacted by gas bubbles and a bubble contacting two or several fibres at thi same time will hold them together with a force that is actually the surface tension of water. In dilute fibre suspension it is sometimes possible to see fibre floes built up around gas bubbles which seem to act as a kind of cement. Turner, Titchener and Duffy [T12] measured disruptive shear stress q u a s i - s t a t i c a l l y in the Fisher & Porter Tester to evaluate the contribution of a i r bubbles to the network strength. They used the technique described by Duffy and Titchener [D4] to test o r i g i n a l and deaerated pulps. The disruptive shear of the deaerated pulps did not appear to d i f f e r within experimental error from that for the o r i g i n a l pulp at the same consistency. 58 Undeniably, surface tension forces must be present since the bubbles attached to f i b r e surfaces e x i s t . However, how they could contribute to the network strength i s unknown. General c l a s s i f i c a t i o n of cohesion forces. The mechanical strength of f i b r e networks results from cohesion forces developed at the f i b r e - t o - f i b r e contacts. In a recent l i t e r a t u r e review, Kerekes et a l . [K7] c l a s s i f i e d these forces into four types and for convenience l a b e l l e d them Type-A, B, C and D: Type-A Colloidal: These are the well-known electrostatic and electrokinetic forces which exist between small particles. Their strength depends on intimacy of contact, which is difficult to achieve between the rough surfaces of fibres, and on chemical additives that modify the negative charge found on fibre surfaces. Type-B Mechanical Surface Linkage: This is a hooking force caused by mechanical entanglement of fibres having kinked or curled configurations, or fibrillated surfaces. When a force is applied to separate fibres, the reaction force caused by hooking at contact points opposes relative movement between them. This type of cohesion depends on surface fibrillation, the degree to which fibres are contorted, and fibre stiffness. Type-C Elastic Fibre Bending: Here the cohesive force between fibres is caused by frictional resistance induced by normal forces at fibre contact points. The normal forces arise when elastically bent fibres are restrained from straightening as a result of contact with other fibres. The ultimate source of t hi s cohesive force is t her e f or e an i nt ernal stress within individual fibres rather than an externally imposed stress on the network. Type-D Surface Tension: Bubbles of undissolved gas (usually air) at fibre interstices produce cohesive forces as a result of surface t ensi on. 59 This c l a s s i f i c a t i o n combined the experimental evidence and the postulates on cohesion of man-made and wood-pulp f i b r e s . Type-C cohesion was singled out from other mechanical cohesion because Kerekes et a l . [K7] anticipated the experimental findings of t h i s research. No dire c t study of Type-B cohesion has been published. The c l a s s i f i c a t i o n could be expanded by the inclusion of another type of mechanical cohesion: fibr e stringing [S5,V2,W9] which i s d i s t i n c t from either Type-B or Type-C. For the reminder of t h i s d i s s e r t a t i o n the term "Type-C cohesion" instead of somewhat lengthy wording "interlocking by the e l a s t i c f i b r e bending" i s used. 2 .4 C l a s s i f i c a t i o n of Findings from the Literature Review Fibre geometry and suspension concentration have the most profound effect on the process of f l o c c u l a t i o n . These two factors are used to c l a s s i f y f i b r e suspensions in terms of f i b r e behaviour into: d i l u t e , semiconcentrated and concentrated. Other parameters such as the physico-chemical properties of fibres or suspending medium are also important, but their significance i s predetermined primarily by the suspension concentration and f i b r e geometry. Thus, f i b r e geometry and concentration are the most suitable in the construction of a space for graphic representation of the regimes of fibr e f l o c c u l a t i o n . This i s shown in Figure 3 in which the aspect r a t i o (L/d) i s plotted on the v e r t i c a l axis and the volumetric concentration of fi b r e s on the horizontal axis. Both axes are logarithmic. The graph contains fiv e l i n e s which represent the geometric concepts on f i b r e crowding. From l e f t to right they are: 60 10"5 10"4 10"3 10"2 10"1 10° c v gure 3. Concepts of Fibre Crowding in Suspensions and Experimental Investigations with Man-made Fibres. The concentration below which suspensions are d i l u t e as defined by Bibbo et a l . [B9], The concentration at which only one fibre i s in a spherical volume having a diameter equal to the f i b r e length, as defined by Mason [M7] (equation (1)). The concentration which constitutes the upper boundary of semiconcentrated suspensions, as defined by Bibbo et a l . [B9]. Below t h i s concentration, the suspension behaved as v i s c o e l a s t i c l i q u i d but, above i t , as an e l a s t i c material, according to Doi and Kuzuu [D3]. The concentration below which continuous networks having three contact points per f i b r e should not ex i s t , as defined 61 by Meyer and Wahren [M10] (equation (3) at n c=3). 5. The concentration defined by equation (2) [C5] and calculated for nc=3 since nc=3 was suggested by others [M10]. Only analysis of experimental observations can show which one of these concepts i s v a l i d . The experimental investigations with man-made fib r e s are shown in Figure 3 to id e n t i f y the regimes of f l o c c u l a t i o n to which they belong. The experimental evidence, though scarce, i s s u f f i c i e n t to support some cautious conclusions. It could be suggested that the boundary between d i l u t e and semiconcentrated suspensions be delineated by Mason's c r i t i c a l concentration on the basis of Blakeney's [B11] and Attanasio et a l . [A18] re s u l t s , and the boundary between semiconcentrated and concentrated suspensions be delineated by Meyer-Wahren's l i m i t i n g concentration at nc=3 [M10]. In thi s case, the boundaries proposed by Bibbo et a l . [B9] seem to be out of place. V e r i f i c a t i o n of these conditions requires further research. C l a s s i f i c a t i o n of fi b r e suspensions into three types, d i l u t e , semiconcentrated, and concentrated, provides systematic organization of the scattered experimental data reported in the l i t e r a t u r e so that a useful basis for the understanding of fi b r e interactions in suspensions is provided. The experimental studies with wood-pulp fi b r e suspensions are presented in Figure 4 to ide n t i f y the regimes of flo c c u l a t i o n to which they belong. Calculation of volume concentration from the mass concentration was made under 62 10"4 10"3 10'2 10"1 10' Figure 4. Concepts of Fibre Crowding in Suspensions and Experimental Investigations with Wood-pulp Fibres. assumption that: the water retained by fi b r e s i s 2 g/g [E5]; the water density i s 998 kg/m [C10]; and the density of dry f i b r e matter i s 1500 kg/m [S4]. The ranges of fi b r e aspect r a t i o were not reported in the l i t e r a t u r e but are presented in Figure 4 with some degree of approximation. The data are i n s u f f i c i e n t to draw any general conclusions regarding the location of the boundaries separating the three regions of f i b r e behaviour. Location of sediment data indicates that Meyer-Wahren l i m i t i n g concentration may be accepted as a boundary between 63 semiconcentrated and concentrated suspensions. 2 . 5 Summary o f L i t e r a t u r e R e v i e w D i l u t e s u s p e n s i o n s . Short, r i g i d , c y l i n d r i c a l p a r t i c l e s spin, rotate and o s c i l l a t e in a simple shear flow. However, long, c y l i n d r i c a l p a r t i c l e s spin, rotate, o s c i l l a t e and flex in a simple shear flow. The motion of both types of p a r t i c l e s i s complex. In any case, the space swept by the single p a r t i c l e i s much larger than the space defined by the p a r t i c l e volume. Wood-pulp fib r e s exhibit more complex motion in a simple shear than man-made fi b r e s because of their irregular shape and higher f l e x i b i l i t y . Wood-pulp fib r e s deform in various ways. Their modes of deformation depend on the degree of f i b r e f l e x i b i l i t y . In simple shear flow, deformation of a f l e x i b l e f i b r e is superimposed on the basic rotational and t r a n s l a t i o n a l motion of a r i g i d f i b r e . The volume swept by the f l e x i b l e f i b r e may be smaller than the volume swept by the r i g i d f i b r e of i d e n t i c a l dimensions. Rigid, c y l i n d r i c a l p a r t i c l e s do not c o l l i d e in simple shear flow or under d i f f e r e n t i a l sedimentation conditions; they only interact on approach. These are chance interactions which manifest themselves by a drastic change in p a r t i c l e o r b i t s . Wood-pulp f i b r e s , on the other hand, c o l l i d e in a simple shear and under d i f f e r e n t i a l sedimentation conditions. These are chance c o l l i s i o n s which occur below the c r i t i c a l concentration. 64 Semiconcentrated suspensions. The interaction of straight, r i g i d f i bres increases with concentration in shear flows. It was found that the c r i t i c a l concentration delineates the regime of chance interactions and forced interactions. For the straight nylon f i b r e s of L/d^20 thi s concentration was Cv=0.0042 (Blakeney, 1966). S l i g h t l y above c r i t i c a l concentration, the formation of groups of two and three f i b r e s increased rapidly. The interaction of straight, r i g i d f i b r e s increases with increase in f i b r e length as well. Forced c o l l i s i o n s between wood-pulp fi b r e s occur at and above the c r i t i c a l concentration. When c o l l i s i o n occurs in a Couette type apparatus, the fib r e s may cohere and form a f l o e . Cohesion may be of a c o l l o i d a l or mechanical nature. For the short f i b r e f r a c t i o n of softwood-pulp, the forces of c o l l o i d a l nature are, in most cases, small in comparison with mechanical e f f e c t s , i . e . , f i b r e length v a r i a t i o n and shear rate changes. A dynamic equilibrium in the f l o c c u l a t i o n process was observed in a Couette-type apparatus with f i n e - f r a c t i o n of f i b r e s . The existing floes shed peripheral f i b r e s and acquired new f i b r e s at the same rate. The cohesion forces developed at the contact points found at floe peripheries equalled the dispersion forces which resulted from the viscous drag imposed on fi b r e s . In the semiconcentrated suspensions, only f l o c c u l a t i o n of wood-pulp f i b r e s was documented. The f l o c c u l a t i o n mechanism consisted of two steps; f i r s t , forced c o l l i s i o n , and second, 65 cohesion. C o l l i s i o n s occurred as the result of or b i t overlap in the shear a r i s i n g from suspension c i r c u l a t i o n . Cohesion resulted from chemical or mechanical interactions. The cohesion developed between the whole-pulp fibres was much stronger than the cohesion between the f i n e - f r a c t i o n of f i b r e s . This increase in strength was attributed to the mechanical entanglement. Whether the c o l l o i d a l or mechanical cohesion i s predominant depends not only on f i b r e geometry or suspension concentration but also on fibre surface properties ( f i b r i l s ) and flow conditions. The c o l l o i d a l cohesion i s predominant only in the d i l u t e suspensions and under low rates of shear. Mason termed mechanical cohesion of wood-pulp f i b r e s in many ways - c l o t t i n g , hooking, intermeshing, interlocking, interweaving and entanglement - and used t h i s terminology l i b e r a l l y to distinguish wood-pulp fi b r e f l o c c u l a t i o n from c o l l o i d a l aggregation [M4,M5,M6,M7]. None of these terms was precisely defined or documented in any investigation. Concentrated suspensions. It has been observed that coherent networks form when man-made fi b r e s of suitable length s e t t l e and are subsequently vigorously agitated. Such networks are always nonuniform. Better uniformity i s achieved i f the v i s c o s i t y of the suspending l i q u i d i s increased. However, at very high v i s c o s i t i e s , the network may not form at a l l . How these networks formed has never been documented. Inference not experiment indicated that f i b r e s interlocked e l a s t i c a l l y . 66 It has been demonstrated that wood-pulp f i b r e s form form continuous networks, e.g., a sediment mat or a plug in plug flow. The term "continuous" implies that the network e n t i r e l y f i l l s the volume of a container or a conduit. It has been also demonstrated that coherent networks form under favorable flow conditions, i . e . , through a s p e c i f i c process of plug formation. The structure of continuous coherent network depended on flow conditions, i . e . , well dispersed (homogenized) or poorly dispersed (floccy) networks were formed. Neither the process nor the structure of those networks has been studied. Jacquelin produced i n d i v i d u a l , well-defined networks which he c l a s s i f i e d as "coherent f l o e s . " These floes possessing organized structure and sizable strength, he distinguished from " f r a g i l e " floes with no v i s i b l e structure or s i g n i f i c a n t strength. The coherent floes could only be formed from the long f i b r e chemical-pulps never exposed to even gentle mechanical treatment (beating). Neither strength nor the structure of those floes was investigated by Jacquelin. Two models of 3-D, i s o t r o p i c f i b r e networks pertain to fi b r e s of uniform length and diameter. The Meyer-Wahren model developed for networks having three and more contact points per f i b r e has never been v e r i f i e d empirically. It has only been conjectured from the sediment and shear modulus experimental data. Neither Miles's model nor Meyer-Wahren model has been v e r i f i e d to apply to 3-D, i s o t r o p i c f i b r e networks. Thus far, the estimated numbers of contact points per f i b r e calculated from both models do not agree with those counted in the compressed 67 f i b r e mats. It was also postulated that the man-made fi b r e s interlock by the three alternate contact point arrangement. This d e t a i l of the form of interlocking has never been checked experimentally. The strength of f i b r e networks varies widely over several orders of magnitude. The lowest strength was recorded for wood-pulp f i b r e networks under tension. The highest strength was obtained from shearing the nylon f i b r e s l u r r i e s . The t e n s i l e o stress ranged from 0.2 to 4 N/m whereas the shear y i e l d stress ranged from 400 to 4500 N/m in suspending l i q u i d s of comparable v i s c o s i t y . The exponents from a l l power f i t s ranged from 1.59 to 4.43. A trend was observed for the exponents from the direct test methods in the case of wood-pulp f i b r e s (Table IV and VII). The exponents were the largest for mechanical pulps (3.03 & 3.6) followed by those for hardwood-pulps (2.96 & 2.57) and those for softwood-pulps (2.46 & 2.29). This trend indicates that the mechanism of f i b r e separation in tension and in shear may be si m i l a r . Nylon f i b r e s did not form coherent networks in a flow loop under conditions that produced coherent networks from wood-pulp fi b r e s [F1] although the volumetric concentration was almost three times larger than C v m ^ n « This behaviour seems contrary to the observations made in another case [M10] in which the continuous coherent networks were produced by vigorous s t i r r i n g of f i b r e s with an impeller and sudden decay of agitation at 68 concentrations corresponding to C . . . 3 v,mm The only estimate of the order of magnitude of cohesion force due to c o l l o i d a l a t t r a c t i o n between two crossed wood-pulp -9 f i b r e s i s that of Wollwage, 10 N. The electro-chemical forces, though smaller than mechanical entanglement forces, are nevertheless s i g n i f i c a n t in the cohesion of floes or continuous networks when myriads of f i b r i l s i n t e r a c t . The presence of small a i r bubbles enhances f l o c c u l a t i o n but shows no e f f e c t on the disruptive shear stress. 2.6 T h e s i s O b j e c t i v e s The p r i n c i p a l goal of t h i s thesis, that of answering major questions concerning the existence and properties of Type-C cohesion, i s pursued by meeting the following objectives: 1. To v e r i f y the existence of Type-C cohesion experimentally by i s o l a t i n g i t from other types of cohesion. 2. To i d e n t i f y the process of Type-C network formation by exploring the conditions under which Type-C networks exist and by paying p a r t i c u l a r attention to the l i m i t i n g concentration below which such networks do not form as postulated by Wahren et a l i i . a. To investigate f i b r e properties and flow conditions under which Type-C cohesion can or cannot e x i s t . b. To explain the ef f e c t of suspending l i q u i d v i s c o s i t y on the formation process of Type-C cohesion. c. To v e r i f y the mechanism of e l a s t i c f i b r e interlocking proposed by Wahren et a l i i . 69 d. To v e r i f y whether the sediment concentration approximates the l i m i t i n g concentration. 3. To i d e n t i f y the structure of 3-D, Type-C coherent networks. a. To v e r i f y experimentally the predictive power of two e x i s t i n g theories r e l a t i n g C v to L/d and n c . b. To gather d i r e c t evidence of alternate contact point i n t e r l o c k i n g . 4. To measure strength of Type-C networks. a. To reveal whether the r e l a t i o n s h i p between stress and concentration y i e l d s the same exponent as indicated by the shear and/or t e n s i l e strength studies. b. To v e r i f y whether f r i c t i o n forces developed at contact points are the source of network strength. 3 EXPERIMENTAL PROGRAM 3.1 Preparation of Fibres A study of fl o c c u l a t i o n of wood-pulp fi b r e s i s hampered by numerous independent variables created by tremendous heterogeneity of any c o l l e c t i o n of natural f i b r e s . These variables cannot be controlled, but can be eliminated by using substitute f i b r e s such as man-made f i b r e s . 3.1.1 Choice of fibrous material Wood tracheids and fibres which are elongated tubular objects tapered at both ends have irregular geometry and surface features due partly to morphology and partly to the pulping process by which they were separated from wood. Morphological c h a r a c t e r i s t i c s of deciduous tree fibres or coniferous tree tracheids vary between species as well as within one species. In addition, each tree has fibres or tracheids of various lengths, cross-sections and wall thicknesses [13,P1,R10,S18]. Some pulping processes can change a straight tubular shape into a kinked, contorted and collapsed shape. Other processes may roughen the tube wall or even tear the tube apart, in both cases exposing the f i b r i l l a t e d wall structure [C2]. This broad nonuniformity of shape and surface roughness creates various mechanical entanglements [M5] and enhances electro-chemical interaction [C1,J5]. Accordingly, no exclusive type of cohesion can be produced while wood-pulp fibres are used. For Type-C cohesion to be isolated , the features that produce other types of cohesion had to be eliminated. For thi s reason 7 1 the experimental work of t h i s thesis used model f i b r e s which had smooth surfaces and straight configuration. Apart from the need to produce exclusively Type-C cohesion, i t was important that the model fi b r e s required similar f l e x i b i l i t y and density as those of wood-pulp f i b r e s . Nylon 6-6 •3 was selected because i t has an apparent density of 1130 kg/m in water which i s close to that of wood pulp f i b r e s . 1 Nylon f i b r e s and wood-pulp f i b r e s are compared in Figure 5. Since f i b r e s t i f f n e s s was shown to be c r u c i a l to Type-C cohesion [J3,T9], nylon 6-6 filaments were obtained in various 1 See Appendix II for further d e t a i l s . Figure 5. Surface Texture and Shape of Nylon (dark) and Wood-pulp Fibres. Magnification 120x. 72 diameters to cover the range of wood-pulp f i b r e s t i f f n e s s . The s t i f f n e s s range for the wet nylon filaments was expected to be from 8•10~ 1 2 to 200*10~ 1 2 N-m2, based on e l a s t i c modulus 9 2 E=1.17-10 N/m [M9], whereas wood pulp f i b r e s t i f f n e s s was reported to range from 0.21 -10~ 1 2 to 157-10~ 1 2 N«m 2 [S1,S6,S11,T1,T2]. The nylon filaments were cut to the desired lengths with a technique that produced a narrow length d i s t r i b u t i o n , a c r u c i a l requirement to meet to test equations (2) and ( 3 ) . After nylon 6-6 filaments were cut, the fi b r e lengths spanned the f u l l spectrum of naturally-occurring lengths of softwood and hardwood fi b r e s [13,P1,R10,W6]. The choice of nylon 6-6 material made electro-chemical a t t r a c t i v e forces between fi b r e s in aqueous-sugar solution i n s i g n i f i c a n t compared to mechanical bending forces which are postulated to be the source of Type-C cohesion. The magnitude of the electro-chemical a t t r a c t i o n between two fib r e s was estimated to be 1000 times smaller than the interaction due to e l a s t i c f ibre bending. 1 3.1.2 Determination of f i b r e geometry Thalen and Wahren [T9], and Jacquelin [ J 3 , J 4 ] have observed that f i b r e geometry has strong influence on formation and properties of coherent networks. It was anticipated that f i b r e length and diameter would be primary independent variables. Fibres were cut from multifilaments with a technique described in Appendix IV. After being cut, the fibres were 1 Appendix III contains d e t a i l s of t h i s estimate. 73 washed with mild detergent in hot water for removal of impurities acquired during manufacturing process. Subsequently, they were thoroughly rinsed in deionized water, dewatered in Buchner funnel, and dried at 105°C for four hours. This time was s u f f i c i e n t for stable dry weight to be ascertained. At t h i s point, oven-dry samples were weighed for floe formation t e s t s . Much smaller samples were taken for length and curvature measurements. Length and curvature measurements followed drying because the l a t t e r shortens f i b r e s permanently by about 7 to 8% [D7,K9]. It i s understood that relaxation of latent stresses incurred during manufacture of filaments causes th i s shortening by rearrangement of molecules. Fibre samples were rewetted for 48 hours in preparation for length and curvature measurements. The measurement method r e l i e d on projecting a magnified f i b r e image on a d i g i t i z e r and processing the d i g i t a l information on the computer. The d e t a i l s of t h i s method are given in Appendix V. 3.1.3 Moisture absorption by nylon f i b r e s The t h e o r e t i c a l models of three-dimensional (3-D) f i b r e networks governed by equations (2) and (3) [C5,M10] can be v e r i f i e d only when accurate and precise data on their parameters are a v a i l a b l e . Two parameters, f i b r e length and diameter, have already been discussed. The t h i r d geometrical parameter, the apparent volumetric concentration, i s d i f f i c u l t to measure d i r e c t l y ; i t can however, be calculated from the mass concentration. Because th i s c a l c u l a t i o n requires knowledge of 74 the amount of water retained by saturated nylon 6-6, the water retention of nylon 6-6 was investigated by exposing nylon to a 98% r e l a t i v e humidity (RH) environment. Moisture uptake was continuously monitored with a computer-balance system u n t i l i t l e v e l l e d o f f . 1 The apparent volumetric concentration was determined from equations derived in Appendix VII by the substitution of the le v e l l e d off moisture uptake for WRRk.2 3.1.4 D e t e r m i n a t i o n of f i b r e e l a s t i c p r o p e r t i e s Jacquelin [J3] found optimum f i b r e f l e x i b i l i t y at a given intensity of agitation above and below which coherent floes formed with more d i f f i c u l t y . Since f i b r e f l e x i b i l i t y was shown to determine the network e l a s t i c properties [M10], a close estimate of the f l e x i b i l i t y of nylon fi b r e s was necessary. Fibre f l e x i b i l i t y i s defined as a reciprocal of fi b r e s t i f f n e s s in bending, S, which i s the product of e l a s t i c modulus, E, and the moment of i n e r t i a , I (S = E«I). The moment of i n e r t i a for a c i r c u l a r cross-section i s defined as I = 7rd /64, in which d denotes f i b r e diameter. Fibre diameters were calculated from the weight per standard length and the density data provided by DuPont Canada [D7]. The diameters were corrected for water swelling. The moment of i n e r t i a could then be calculated. However, since no r e l i a b l e data on the e l a s t i c modulus of water saturated nylon 6-6 were available, the modulus of e l a s t i c i t y was determined empirically. Two methods were used: a t e n s i l e method [A12,A14] and a bending method [T1,T2], Both tests are described 1 Details of th i s experimental approach are in Appendix VI. 2 Discussion of WRRk i s in Appendix I and I I . 75 in Appendix VIII. 3 . 1 . 5 Determination of wet-friction c o e f f i c i e n t As described e a r l i e r , Meyer and Wahren [M10] postulated that f i b r e s interlock through e l a s t i c bending and f r i c t i o n a l forces so that we t - f r i c t i o n c o e f f i c i e n t should be important to Type-C cohesion. Sta t i c and dynamic c o e f f i c i e n t s of f r i c t i o n were measured employing an i n c l i n e d plane method [A15,T4]. The experimental setup and procedures for t h i s test are described in Appendix IX. 3.2 Creation of Coherent Floes Experimental v e r i f i c a t i o n of the relationships (2) and (3) requires construction of large, uniform fibr e networks of a given f i b r e concentration. In practice, such networks cannot be formed because f i b r e s in suspension tend to flo c c u l a t e into mass concentrations (floes) [T9,G5]. The experimental e f f o r t , for th i s reason, was directed toward production of r e l a t i v e l y uniform f l o e s . The method of producing such floes from wood-pulp fi b r e s was described by Jacquelin [J3,J4,J7]. He applied continuous, moderate agitat i o n to the concentrated suspensions and produced unique, mechanically entangled, aggregates. They were formed in a p a r t i a l l y f i l l e d , rotating cylinder either h o r i z o n t a l l y oriented or i n c l i n e d at 45 degrees to the horiz o n t a l . Jacquelin c a l l e d them "coherent f l o e s . " Preliminary experiments of t h i s study showed that the agitat i o n of a suspension of nylon f i b r e s under the conditions 76 s p e c i f i e d by Jacquelin produced regularly-shaped floes suitable for studies of network properties. The f l o c c u l a t i o n phenomenon observed by Jacquelin seemed to be preserved despite the r a d i c a l change in the type of f i b r e s used, i . e . , nylon f i b r e s in place of wood-pulp f i b r e s . The nature of cohesion, though expected to be Type-C, was enigmatic. Qualitative experiments determined whether the nylon floes derive t h e i r strength from the energy stored in the e l a s t i c a l l y bent f i b r e s . A few floes were heated in water at a slow rate to 90°C temperature at which they remained for 5 minutes. The nylon temperature, raised above i t s glass t r a n s i t i o n point (65°C [H4]), permitted the bending stresses to relax. After subsequent cooling, the heat-treated floes were placed in a large volume of water with never-heated floes and t h e i r r e l a t i v e strength was evaluated by slow increase of the intensity of suspension a g i t a t i o n . The heat-treated floes dispersed e a s i l y under gentle agitation whereas the never-heated floes required intense agitation to be dispersed. The outcome of t h i s experiment j u s t i f i e d further systematic study. The nature of cohesion was of Type-C. The design of a v e r s a t i l e rotating cylinder apparatus was based on Jacquelin's device. In t h i s apparatus, shown in Figure 6, various cylinder diameters could be accommodated by the relocation of one r o l l e r (A). A l l cylinders were 164 mm long. The frame (B) that houses r o l l e r s (A,C) can be in c l i n e d to the horizontal at 15 degree increments from 0 to 60 degrees. The cylinder (D) which rests on r o l l e r s (A,C) and i s prevented from 77 Figure 6. Rotating Cylinder Apparatus. s l i d i n g off the frame by the t h i r d r o l l e r (E). Rotation of the cylinder (D) i s induced by the r o l l e r (C) which i s driven by the speed-controlled, e l e c t r i c motor (F). The suspension in the cylinder i s concentrated by gradual removal of the suspending l i q u i d with the pipet (G) and the pipet f i l l e r (H). This feature was e s p e c i a l l y useful in the experimental evaluation of the l i m i t i n g concentration, i . e . , the concentration below which coherent networks cannot e x i s t . The suspension volume was chosen so that the f l a t base was always covered with the suspension. This was necessary to assure only one type of r e c i r c u l a t i n g flow. D e t a i l s of the apparatus are given in Appendix X. 78 3.2.1 O b s e r v a t i o n of the phenomenon of f l o c c u l a t i o n The f i r s t steps in the exploration of the unknown process of floe formation were v i s u a l observations. The appearance of the suspension was assessed and photographed. The dynamic behaviour of floes was observed and in some cases videotaped. Changes in the suspension appearance with the increase in concentration were studied. These observations were f a c i l i t a t e d by the use of multicolored f i b r e s . Most tests were ca r r i e d out in an 82 mm internal diameter (ID), plexiglas cylinder i n c l i n e d at 45-degrees to the horizontal and rotated at the constant speed of 8.9 rad/s. The cylinder length was 164 mm. Details of f i b r e suspension preparation, test conditions, and experimental procedure are in Appendix XI. In addition to the production of floes in r e c i r c u l a t i n g flow in a rotating cylinder, the formation of Type-C floes in other flows was attempted. The other flows occurred: in the zone between rotating concentric cylinders under laminar and turbulent conditions; in a tank wich the suspension agitated by a s t i r r e r ; in a stationary channel through which a g r i d was passed c y c l i c a l l y . 3.2.2 Sediment c o n c e n t r a t i o n Thalen and Wahren [T9] observed that the lowest concentrations of Perlon f i b r e s in suspension at which a shear modulus could be measured were only s l i g h t l y higher than the corresponding sediment concentrations. It seemed natural for them to postulate that coherent networks are not formed at 79 concentrations lower than the sediment concentration. For t h i s study to test t h i s postulate, the sediment concentration for each type of f i b r e was determined and compared with the concentration at which nylon floes formed. The sediment pads were formed by the usual method of f i b r e s being allowed to s e t t l e gently from a d i l u t e suspension. The sedimentation took place in a one- l i t e r graduated glass cylinder. The detailed procedure i s described in Appendix XII. 3.3 Experimental V a r i a b l e s Since l i t t l e was known about the process of Type-C floe formation, each independent variable was investigated. The examined variables and th e i r ranges are: Fibre volume concentration (.0001-0.1) Cylinder rotational speed (1.5-22 rad/s) Cylinder i n c l i n e (0,15,30,45,60,90 deg.) Cylinder internal diameter (32-146 mm) Fibre length (0.91-6.26 mm) Fibre diameter (19.8, 27.9, 44.2 nm) V i s c o s i t y of the suspending l i q u i d (0.001-0.14 Pa«s) density (998-1315 kg/m3) The e f f e c t of these variables on the appearance and behaviour of the suspension, s p e c i f i c a l l y the formation of coherent f l o e s , was determined by v i s u a l observation and photographic techniques to be described l a t e r . 80 3.3.1 F i b r e c o n c e n t r a t i o n As described in Section 2.4, fibr e concentration i s a variable of the utmost importance in f l o c c u l a t i o n . The effect of t h i s variable was studied by a d i l u t e suspension being placed in the rotating cylinder and the suspension being thickened. In the thickening procedure, a sample of rewetted f i b r e s was f i r s t placed in a o n e - l i t e r graduated cylinder f u l l y f i l l e d with the suspending l i q u i d . The concentration was low, about one tenth of the sediment concentration. At t h i s concentration any f i b r e aggregates were e a s i l y dispersed with several strokes of a plunger having at one end a perforated rubber disk which f i t t e d loosely inside the graduated c y l i n d e r . Next, the suspension was thickened by slow removal of l i q u i d by suction into a 100 mL pipet. The pipet's t i p covered with a 200-mesh screen prevented f i b r e removal. When the suspension volume in the graduated cylinder reached 500 mL, the suspension was transferred to the plexiglas c y l i n d e r . The plexiglas cylinder was mounted on the r o l l i n g mechanism and rotated at constant speed. Further l i q u i d removal was achieved by suction into a pipet which was inserted through the open top-end of the cylinder. The suspending l i q u i d was removed in 5 mL increments u n t i l f i r s t occurrence of Type-C f l o e s , then the suspension volume was measured. The precision of volume measurement in a 500 mL graduated cylinder was ±5 mL. For recognition of Type-C floes from other floes to be enhanced, the removal of the l i q u i d was done in steps, i . e . , each time 25 mL 81 was removed, 20 mL was reintroduced. 3.3.2 D e n s i t y o f t h e s u s p e n d i n g l i q u i d The density difference between water and nylon which produced f i b r e sedimentation created v i s i b l e f i b r e concentration near the bottom of the i n c l i n e d cylinder so that the average suspension concentration did not represent the concentration at every zone in the suspension. This effect was eliminated by an aqueous-sugar solution that matched the apparent density of nylon fibres being used as the suspending l i q u i d . Hence, most of the experimental work used the neutrally buoyant fi b r e s in the aqueous-sugar solution. One exception was the experiment involving varying v i s c o s i t i e s of the suspending l i q u i d described in Section 3.3.5. 3.3.3 C y l i n d e r d i a m e t e r , a n g l e o f i n c l i n e , a n d r o t a t i o n a l s p e e d . The e f f e c t s of physical dimensions of the rotating cylinder, cylinder angle of i n c l i n e , and rotational speed on floe formation were examined. Flocculation behaviour was observed in eight cylinders of internal diameters (ID): 32, 45, 56, 70, 82, 96, 121, and 146 mm. The 82 mm ID cylinder was rotated under various angles of i n c l i n e to the horizontal: 0, 15, 30, 45, and 60 degrees. At a 45 degree i n c l i n e , the 82 mm ID cylinder was rotated at various speeds. The rotational speed, controlled by a variable speed-drive, was changed from 5.2 to 78 rad/s on the driving r o l l e r . This range 82 of r o t a t i o n a l speeds created v e l o c i t i e s ranging from 0.061 to 0.90 m/s at the inner cylinder surface. 3.3.4 F i b r e dimensions - l e n g t h and diameter Thalen and Wahren [T9] found that the f i b r e aspect r a t i o profoundly affected the sediment concentration which, in turn, corresponded to the concentration at which the shear modulus became a measurable quantity. A n t i c i p a t i n g a major ef f e c t of f i b r e geometry on the l i m i t i n g concentration below which Type-C floes do not form [J3,J4,M10,S12,T9], thirteen f i b r e samples of various lengths and diameters were prepared. In t h i s study, the aspect r a t i o s for the thirteen f i b r e samples ranged from 32.7 to 189.1. Detailed information on f i b r e geometry i s in Appendix V. 3.3.5 V i s c o s i t y of the suspending l i q u i d Steenberg et a l . [S12] observed that the v i s c o s i t y of the suspending l i q u i d greatly influenced the e l a s t i c properties of the Perlon f i b r e networks in suspension. When Perlon suspensions having 0.015 volume concentration were tested in a double cylinder viscometer, the f i b r e network suspended in a viscous l i q u i d showed much smaller shear modulus than the network suspended in a low v i s c o s i t y l i q u i d . Gradual change in the v i s c o s i t y of the suspending l i q u i d produced a step change in the shear modulus. This interesting behaviour required further investigation, s p e c i f i c a l l y , v e r i f i c a t i o n of Type-C floe existence in high v i s c o s i t y suspending media. This investigation varied the v i s c o s i t y of the suspending l i q u i d from 0.001 Pa«s to 0.14 Pa«s, covering the range in which a sudden drop in shear 83 modulus was reported by Steenberg et a l . [S12], Addition of sugar to water changes the v i s c o s i t y and density of the suspending l i q u i d , e.g., aqueous-sugar solution containing 64% sugar by weight has a density of 1310 kg/m and v i s c o s i t y 0.12 Pa-s [C10], It should be noted that Perlon [T9] and nylon 6-6 have the same density. 3.4 Flow P a t t e r n s i n H o r i z o n t a l R o t a t i n g C y l i n d e r The flow f i e l d in a partly f i l l e d , i n c l i n e d , rotating cylinder i s unique and d i f f i c u l t to be characterized and measured. No description of i t even for a single phase could be found in the f l u i d mechanics l i t e r a t u r e . A simpler flow which also produced coherent floes of wood-pulp fi b r e s occurs in a p a r t i a l l y f i l l e d , h o r i z o n t a l l y oriented, long rotating cylinder [J3]. A description of flow of l i q u i d s (not suspensions) in such a cylinder has been given by Haji-Sheiks et a l . [H1], The flow f i e l d in a horizontal cylinder was more eas i l y characterized but not measured. The length of the cylinder (172 mm) was too large to be penetrated by two beams of the Laser Doppler Anemometer. A shorter cylinder of length 36 mm was therefore used. Experiments showed that the flow in t h i s short cylinder produced Type-C nylon floes at the conditions found in the longer i n c l i n e d c y l i n d e r s . The v e l o c i t y p r o f i l e s in the central plane of the 36 mm deep, 94 mm ID cylinder were determined with the TSI Laser Doppler Anemometer (LDA). Figure 7 shows the experimental setup. Details are given in Appendix XIII. Three flow cases were studied: one produced by a high v i s c o s i t y suspending l i q u i d 84 Figure 7. Experimental Setup for Velocity Measurements in a Horizontal Rotating Cylinder. (0.14 Pa«s) where floes did not form, and two others involving low v i s c o s i t y suspending l i q u i d (0.00375 Pa»s) before and a f t e r the Type-C floes formed. These three flow cases were expected to reveal the mechanism by which the Type-C coherent floes form. The experimental approach included an analysis of suspension flow patterns recorded by video-taping and c i n e - f i l m i n g and was complemented by the measurements of the v e l o c i t y p r o f i l e s with the LDA. In a l l cases the cylinder was h a l f - f u l l of a suspension of 4.97 mm long, 44.2 nm diameter nylon f i b r e s . The cylinder was rotated at the constant angular speed of 2n rad/s. The low and high v i s c o s i t y suspending l i q u i d s had densities 1130 and 1315 kg/m respectively. 3.5 S t u d i e s of F l o e S t r u c t u r e 85 The physical properties of any 3-D f i b r e network are determined by the properties of i n d i v i d u a l f i b r e s and t h e i r arrangement in space. While f i b r e properties are r e l a t i v e l y e a s i l y measured, the s p a t i a l arrangement of fibres in the network i s not. This explains why the t h e o r e t i c a l models of f i b r e network proposed more than twenty years ago [M10,C5] have not been v e r i f i e d experimentally (equations (2) and (3)). This study examined r e l a t i o n s h i p between f i b r e concentration and the number of contact points per f i b r e in Type-C nylon floes which were formed in the rotating cylinder at increasing l e v e l s of f i b r e concentration. The selected floes removed from suspension were dried. During drying, sugar from the suspending l i q u i d deposited on f i b r e surfaces and at f i b r e - t o - f i b r e contact points. This deposition led to bonding between f i b r e s . Those v i s i b l e bonds which were close to the floe extremity were broken f i r s t . This freed some fib r e s which were subsequently removed from the f l o e . The newly exposed bonds were ruptured and next fib r e s were removed. The sequence of bond breaking and f i b r e removal continued u n t i l the floe was dismembered. The broken bonds and the removed f i b r e s were counted. This tedious, time-consuming procedure was c a r r i e d out on several floes as described in Appendix XIV. 86 fa. 3.6 Floe Strength Measurement The strength of ind i v i d u a l floes was measured. While floes can be ruptured in certain flows, e.g., in a shear flow [L2] or in an extensional flow [K1], the disruptive force and the break zone are never c e r t a i n . ,0n the other hand, applying rupture force d i r e c t l y by ph y s i c a l l y restraining the floe with the load-sensing elements [G2] res u l t s in easy control of the mode of f a i l u r e (shear, t e n s i l e or compression), the break zone, and the rate of floe deformation. This approach was used here. The t e n s i l e mode was . selected because i t seemed suitable for the v e r i f i c a t i o n of the concept of f r i c t i o n a l f i b r e interlocking due to e l a s t i c bending. beading needles rubber Figure 8. Schematic of I n i t i a l Comb Positioning in Tensile Tests. 87 The t e n s i l e strength of floes formed at various concentrations in the rotating cylinder was measured. The fl o e s , removed from the suspension and placed in a t e n s i l e tester, were pierced by two intermeshing combs of beading needles 1 as shown in Figure 8. The needles secured to two rubber supports formed a retractable comb-like structure. The needles of one comb evenly interspaced those of the other comb. The floe was pulled apart as the combs separated in the d i r e c t i o n indicated by the arrows. Details of t h i s test method are in Appendix XV. 1 Needles of small diameter used to s t r i n g small pieces of glass, wood, stone etc. for ornamentation. 4 RESULTS AND DISCUSSION The experimental investigation accentuates three aspects of Type-C cohesion: formation, structure, and strength of Type-C fl o e s . 4.1 Ef f e c t of Fibre Concentration on Floe Formation Preliminary experiments had shown that floes of nylon f i b r e s , s i m i l a r to wood-pulp fi b r e "coherent floes" formed by Jacquelin [J3,J4], could be produced in an incl i n e d rotating cylinder. Fibre concentration was increased by removal of the suspending l i q u i d with a pipet while the suspension was in motion. The suspension was observed v i s u a l l y for formation of coherent f l o e s . Although, v i s u a l observations are subjective and therefore biased, in t h i s case the observations were made simple through the car e f u l selection of the experimental system as described in Section 3.1.1. The subjective evaluation was limited to simple a "yes" or "no" decision. A strong spotlight illuminating the entire suspension enabled assessment of the suspension state at any depth. C y c l i c , unsteady flow produced good mixing which forced f i b r e s and floes to the surface or to approach transparent cylinder walls so that the detection of Type-C floes was f a c i l i t a t e d . The in c l i n e d cylinder arrangement made the suspension handling and the removal of the suspending l i q u i d easy. With increasing concentration, the suspension appearance changed from uniformly dispersed, (Figure 9a) to cloudy (Figure 9b). The cloudiness i n t e n s i f i e d with increasing Figure 9. Increasing Cloudiness with Increase in Fibre Concentration. 90 concentration, (Figure 9b,c,d), but the individual clouds (floes) formed and then disappeared. While floes were always present in the flow, they did not remain for long in the flow as e n t i t i e s . Rather, floes appeared to be in a state of continual dispersion and formation. This process was similar to the "dynamic equilibrium" described by Mason [M4], and experimentally demonstrated by Hubley [H8]. It should be noted that the dynamic state observed in thi s study occurred at concentrations of one order of magnitude greater than those of Hubley et a l . [H8], i. e . , C„ =0.02 versus C =0.001 for fib r e s of similar aspect va va r r a t i o . 1 The cloudy nature of the suspension could be diminished at any time by reintroduction of the suspending l i q u i d into the suspension. The cloudy appearance could be eliminated altogether i f the suspension was rediluted to the i n i t i a l concentration. This indicates a reversible process of floe (cloud) formation. Further increases of suspension concentration by removal of l i q u i d resulted in a condition in which the suspension appearance changed from cloudy to grainy, as shown in Figure 10a,b. This change in appearance was accompanied by the formation of nylon floes that persisted through the r e c i r c u l a t i n g flow as  i d e n t i f i a b l e e n t i t i e s , i . e . , coherent f l o e s . Moreover, these floes did not disperse even when the suspension was rediluted, as shown in Figure 10c, provided the rotational speed was not increased. In Figure 10c, the average f i b r e concentration, i s 1 Assuming WRRk=2 g/g for wood-pulp fi b r e s which had L=0.45 mm, d=30 nm giving an average L/d=52.5. WRRk i s discussed in Appendix I and I I . Figure 10. a. Cloudiness; b. Granularity; c. Granules in red i l u t e d Suspension. 92 the same as the i n i t i a l f i b r e concentration in Figure 9a. This i l l u s t r a t e s the permanent change in suspension appearance. Suspension r e d i l u t i o n helped v i s u a l i d e n t i f i c a t i o n of floes and demonstrated the i r r e v e r s i b l e process of floe formation. It had been shown e a r l i e r by selection of f i b r e s and the experimental conditions that only Type-C cohesion existed in the suspension of f i b r e s . Thus, these floes may be l a b e l l e d Type-C  coherent f l o e s . For convenience, they w i l l be c a l l e d Type-C  floes through the remainder of t h i s study. Experiments with dyed fibres confirmed the existence of the i r r e v e r s i b l e process of floe formation. F i r s t , two separate batches of f i b r e s were prepared. One batch contained translucent f i b r e s and the other red-dyed f i b r e s . Second, red Type-C floes were formed and i n d i v i d u a l l y transferred to the batch of translucent f i b r e s . The suspension rotation started and i t s concentration was gradually increased to produce translucent Type-C f l o e s . After f i v e P i i n u t e s , rotation was halted and a l l floes were examined by disse c t i o n . Not one translucent f i b r e was found inside the,red floe centres. This shows that the i n i t i a l l y formed floes always remained coherent. In another experiment, red f i b r e s were added to the suspension of translucent fibres at the cloudy state of suspension appearance, i . e . , at f i b r e concentration below the threshold concentration. Subsequently, concentration was increased to form Type-C floes. Examined floes were found to be a mixture of colored f i b r e s . 93 These observations indicate that floes of s i g n i f i c a n t strength can form and r e s i s t rupture by the hydrodynamic forces exerted upon them in their c y c l i c flow within the cylinder. As stated e a r l i e r , the floe strength must have come from e l a s t i c bending because the p o s s i b i l i t y of other forms of cohesion had been eliminated. With the procedure described in Section 3.2, the existence of e l a s t i c interlocking was v e r i f i e d by comparison of the r e l a t i v e strength of heat-treated (above the glass t r a n s i t i o n temperature) floes and never-heated f l o e s . Gentle s t i r r i n g e a s i l y dispersed the former floes but the l a t t e r floes did not disperse at a l l . In f a c t , vigorous a g i t a t i o n was required to disperse the never-heated f l o e s . It therefore appears that the source of floe strength i s t r u l y one from f i b r e interlocking by e l a s t i c bending. The tests involving increasing suspension concentration in the rotating cylinder also showed that, for a given f i b r e type and at given flow conditions, there exists a reproducible concentration at which Type-C floes form and do not disperse on continuous passage through the flow. This leads to the conclusion that Type-C floes have acquired s u f f i c i e n t strength to withstand hydrodynamic forces acting upon them at a reproducible l e v e l of f i b r e crowding. The concentration at which Type-C floes * form i s termed the threshold concentration, and marked C . 1 When va th i s concentration had been exceeded, the Type-C floes formed instantaneously, i . e . , within a period of time which was 1 Threshold concentration i s expressed as an apparent volumetric concentration throughout th i s d i s s e r t a t i o n . Its numerical values are reported as fract i o n s . 94 n e c e s s a r y t o c o n c e n t r a t e s u s p e n s i o n , between 10 t o 20 seconds. The p r o c e s s of Type-C f l o e f o r m a t i o n was f a s t , u n l i k e t h a t d e s c r i b e d by J a c q u e l i n [ J 3 ] f o r c o h e r e n t f l o e s of wood-pulp f i b r e s . H i s c o h e r e n t f l o e s formed over hours of s t i r r i n g . I t must be emphasized t h a t the t h r e s h o l d c o n c e n t r a t i o n i s not an a b s o l u t e c o n c e n t r a t i o n a t which Type-C c o h e s i o n i n i t i a t e s , but i t i s an e x p e r i m e n t a l c o n d i t i o n r e q u i r e d f o r Type-C f l o e s f o r m a t i o n i n a unique type of f l o w . Subsequent s e c t i o n s show t h a t the t h r e s h o l d c o n c e n t r a t i o n i s independent of some e x p e r i m e n t a l v a r i a b l e s and s t r o n g l y dependent on o t h e r s . 4.2 O b s e r v a t i o n s o f O r i g i n and Growth o f Type-C F l o e s E x p e r i m e n t s were c a r r i e d out t o d e t e r m i n e t h e mechanism by which Type-C f l o e s formed. The dyed f i b r e s h e l p e d v i s u a l o b s e r v a t i o n s . At low c o n c e n t r a t i o n s below th e t h r e s h o l d c o n c e n t r a t i o n , i t was observed t h a t , d u r i n g c i r c u l a t i o n , the s u s p e n s i o n t r u n c a t e d i n t o groups of f i b r e s , ( e a r l i e r termed ''clouds"). As the c o n c e n t r a t i o n was i n c r e a s e d , some of t h e s e c l o u d s become denser and t r a n s f o r m e d i n t o Type-C f l o e s . These f l o e s a c t e d as n u c l e i ; they grew by a c q u i s i t i o n of new f i b r e s and became denser w i t h i n c r e a s e d s u s p e n s i o n c o n c e n t r a t i o n . Thus, the f o r m i n g p r o c e s s of Type-C f l o e s appears t o c o n s i s t of two s t a g e s - n u c l e u s f o r m a t i o n and f i b r e a c c u m u l a t i o n . A s i m i l a r mechanism of c o h e r e n t f l o e f o r m a t i o n was d e s c r i b e d by J a c q u e l i n [ J 3 ] and Lee [L2] f o r wood-pulp f i b r e s . However, they formed c o h e r e n t f l o e s s t a r t i n g w i t h " c o n c e n t r a t e d " s u s p e n s i o n s , whereas i n t h i s s t u d y the i n i t i a l s u s p e n s i o n s were " s e m i c o n c e n t r a t e d " as d e f i n e d 95 i n the L i t e r a t u r e Review and i l l u s t r a t e d i n F i g u r e 4. The shape of Type-C f l o e s appeared t o be s p h e r o i d a l . A t a g i v e n c o n c e n t r a t i o n , a f t e r Type-C f l o e s formed and a f t e r p r o l o n g e d r o l l i n g over many m i n u t e s , a l l f l o e s adopted a smoother " s u r f a c e " and became more s p h e r i c a l . The number of f l o e s became f i x e d . That an i n c r e a s e i n the f l o e d e n s i t y was not d e t e c t a b l e s u g g e s t s t h a t the p r o l o n g e d r o l l i n g produced o n l y changes i n t h e f l o e ' s appearance. In c o m p a r i s o n , J a c q u e l i n found t h a t , f o r wood-pulp f i b r e s , s t a b i l i z a t i o n i n f l o e number o c c u r r e d a f t e r about f i v e hours and t h a t the maximum i n T.F.C. was reached a f t e r about 12 hours of r o l l i n g . These s e e m i n g l y d i v e r s e o b s e r v a t i o n s a r e the r e s u l t of d i f f e r e n t e x p e r i m e n t a l approaches. In t h i s s t u d y , s u s p e n s i o n c o n c e n t r a t i o n was g r a d u a l l y i n c r e a s e d from s e m i c o n c e n t r a t e d t o a l e v e l above the t h r e s h o l d c o n c e n t r a t i o n . In J a c q u e l i n ' s e x p e r i m e n t s , the i n i t i a l s u s p e n s i o n c o n c e n t r a t i o n was a l r e a d y w e l l above the t h r e s h o l d c o n c e n t r a t i o n . In t h i s s t u d y , the development of Type-C f l o e s was p a r a l l e l e d by the d i l u t i o n of the r e m a i n i n g s u s p e n s i o n so t h a t the f o r m a t i o n of a d d i t i o n a l Type-C f l o e s was h i n d e r e d . These e v e n t s o c c u r r e d w i t h i n the s h o r t time r e q u i r e d (about o n e - h a l f minute) f o r the s u s p e n s i o n t o be c o n c e n t r a t e d . In J a c q u e l i n ' s e x p e r i m e n t s , on the o t h e r hand, the f o r m a t i o n and d e n s i f i c a t i o n of c o h e r e n t f l o e s o c c u r r e d s i m u l t a n e o u s l y . As t h e s u s p e n s i o n t r u n c a t e d i n t o c o h e r e n t f l o e s which were g r a d u a l l y compacted and reshaped, the c o n c e n t r a t i o n of the s u s p e n s i o n e x t e r n a l t o t h e s e f l o e s remained h i g h so t h a t new f l o e s formed, changed shape and became den s e r . T h i s sequence of e v e n t s may 96 .07 o o .06 w CD Z .05 o c o «5 C CD O c o o CD E .04 .03 nylon fibres L= 6.2 mm d=44.2jim • - t • • • * • • • • • t _ • ± • t t t • • 1 1 .02 .0045 .005 .0055 Volumetric concentration of fibres in suspension Figure 1 1 . Floe Concentration versus Suspension Concentration. explain the long times (hours) needed for a stable number of coherent floes of wood-pulp f i b r e s to be reached. The maximum in T.F.C. was reached even l a t e r , probably due to the processes associated with the growth and compaction of floes that seem to be slower in the case of wood-pulp f i b r e s . The increase in suspension concentration when Type-C floes had formed led to the d e n s i f i c a t i o n of a l l Type-C floes. Some floes appearing more dense than the others indicated that floe 97 F i g u r e 12 F l o e Weighing 0.12 g Suspended i n t h e A i r by Few F i b r e s . f o r m a t i o n was not u n i f o r m . The d e n s i f i c a t i o n p r o c e s s o c c u r r e d i n s t a n t a n e o u s l y w i t h the 10 t o 30 second removal of the s u s pending l i q u i d . The d e n s i f i c a t i o n seems t o r e s u l t from g r a d u a l r e d u c t i o n of the space w i t h i n which a c e r t a i n number of f l o e s e x i s t s . S i n c e the s u s p e n s i o n volume i s r e d u c e d , the f l o e s g r i n d a g a i n s t s u r r o u n d i n g f i b r e s and a g a i n s t c y l i n d e r w a l l s , or p r o t r u d e t h r o u g h the f r e e s u r f a c e . Which of t h e s e a c t i o n s or which c o m b i n a t i o n of them c o n t r i b u t e most t o f l o e d e n s i f i c a t i o n i s not known. I t i s known t h a t f l o e d e n s i t y i n c r e a s e s w i t h i n c r e a s e d s u s p e n s i o n c o n c e n t r a t i o n , as shown i n F i g u r e 11. Three samples of t e n f l o e s each were drawn from the same sus p e n s i o n but a t d i f f e r e n t c o n c e n t r a t i o n s . A l t h o u g h v a r i a t i o n s 98 i n f l o e d e n s i t y a r e l a r g e w i t h i n each sample, the i n c r e a s e i n f l o e c o n c e n t r a t i o n w i t h an i n c r e a s e i n s u s p e n s i o n c o n c e n t r a t i o n i s c l e a r l y v i s i b l e . When the p r o l o n g e d r o l l i n g was combined w i t h the g r a d u a l i n c r e a s e i n s u s p e n s i o n c o n c e n t r a t i o n , v e r y s t r o n g c o h e s i o n d e v e l o p e d , e.g., by p u l l i n g of a few f i b r e s p r o t r u d i n g from th e s u r f a c e , Type-C f l o e c o u l d be l i f t e d from the s u s p e n s i o n , as shown i n F i g u r e 12. 4.3 E f f e c t of Key Variables on Threshold Concentration The next s t e p i n t h i s e x p e r i m e n t a l program was t o e s t a b l i s h the dependence of the t h r e s h o l d c o n c e n t r a t i o n on f i b r e and f l o w p r o p e r t i e s . As s t a t e d i n S e c t i o n 3.3.2, the d e n s i t y d i f f e r e n c e between n y l o n f i b r e s and water caused f i b r e s e d i m e n t a t i o n which c r e a t e d a zone of h i g h f i b r e c o n c e n t r a t i o n a t the l o w e s t p a r t of the c y l i n d e r . A c c o r d i n g l y , aqueous-sugar s o l u t i o n s of the same d e n s i t y as t h a t of n y l o n f i b r e s were used as the s u s p e n d i n g medium. The e f f e c t of the d e n s i t y d i f f e r e n c e on the t h r e s h o l d c o n c e n t r a t i o n was examined w i t h a l l o t h e r v a r i a b l e s c o n s t a n t . T a b l e V I I I shows the d i f f e r e n c e between the t h r e s h o l d c o n c e n t r a t i o n s f o r the n y l o n f i b r e s i n pure water and i n aqueous-sugar s o l u t i o n s . 1 N u l l h ypotheses f o r the averages were t e s t e d u s i n g S t u d e n t - t d i s t r i b u t i o n . C a l c u l a t e d ' t ' v a l u e s exceeded the t a b u l a t e d v a l u e s [B7] even at the 0.001 l e v e l of s i g n i f i c a n c e (see T a b l e V I I I ) . The n u l l h ypotheses were r e j e c t e d , i . e . , the e f f e c t of the suspending l i q u i d d e n s i t y on 1 T a b l e s of e x p e r i m e n t a l d a t a a r e i n Appendix XXVI. Table VIII. Threshold Concentrations for 15 Denier Nylon Fibres In Water and 1n Aqueous-sugar Solution. FIBRE LENGTH THRESHOLD CONCENTRATION STATISTICS FIBRES IN WATER FIBRES IN AQUEOUS-SUGAR SOLUTION t-TEST OF NULL HYPOTHESIS (MEAN 1-MEAN2) mm No. OF TESTS MEAN1 STANDARD DEVIATION No. OF TESTS MEAN2 STANDARD DEVIATION CALCULATED t DEGREES OF FREEDOM 2.947 6 .01892 .0005224 8 .02221 .0006022 -10.68 12 4 .973 8 .007004 .0001925 9 .007915 .0002300 - 8.780 15 G.2S1 8 .003620 .0001577 7 .004469 .0002267 - 8.518 13 100 the threshold concentration was s i g n i f i c a n t in a l l cases. In fact, f i b r e sedimentation resulted in l o c a l i z e d crowding of fibres in the lowest part of the cylinder. The average suspension concentration in such a case was lower than that recorded in a neutrally buoyant f i b r e suspension. The sedimentation e f f e c t was eliminated when the nylon f i b r e s were suspended in the aqueous-sugar solutions to render them neutrally buoyant in a l l experiments but one - that determining the effect of suspending l i q u i d v i s c o s i t y on the threshold concentration. 4.3.1 Cylinder (flow) variables The e f f e c t of flow variables on the threshold concentration was f i r s t examined with various cylinder diameters, angles of i n c l i n e and rotational speeds. Every possible variable was investigated since Type-C nylon floes have been formed for the f i r s t time in the p a r t i a l l y f i l l e d rotating cylinder. Fibres 4.97 mm long and 44.2 um diameter were used in these investigat ions. In c ylinders having internal diameters from 56 to 146 mm Type-C floes formed and the cylinder diameter had no ef f e c t on threshold concentration. This i s i l l u s t r a t e d in Figure 13 where 95% confidence l i m i t s are also shown.1 In these cylinders partly f i l l e d with suspensions, a complex r e c i r c u l a t i n g flow formed. In cylinders of internal diameter 32 mm and 45 mm, such flow did not e x i s t . Rather, fibres aggregated into large bodies which r o l l e d 1 Threshold concentration i s expressed as an apparent volumetric concentration throughout t h i s d i s s e r t a t i o n . Its numerical values are reported as f r a c t i o n s . 101 L = 4.97mm, d=44 .2Lim, Cylinder incline = 45 degree Rotational speed = 9.6 rad /s i n 11 * * I • o o c c •a T3 o o O O LL LL I ' I I I I 1 20 40 60 80 100 120 140 160 Cylinder I.D., mm Figure 13. Effe c t of Cylinder Diameter on Threshold Concentration. e r r a t i c a l l y within cylinders and Type-C floes did not form. It can be postulated that an internal diameter of a cylinder should be more than (2«R/L=9), i . e . , nine times larger than fibr e length for a proper r e c i r c u l a t i n g flow for Type-C floe formation to be produced. Angles of i n c l i n e from 0 to 45 degree had no effect on the threshold concentration, as shown in Figure 14. However, no floes formed at angles of 60 and 90 degrees at which the suspension was observed to adopt s o l i d body rotation, whereas at lower angles i t had a complex r e c i r c u l a t i n g flow pattern. Thus, the supposition based on the results of t h i s and the previous experiment can be made that a r e c i r c u l a t i n g flow is needed to produce Type-C nylon f l o e s . This flow i s analyzed in further c o (0 >_ •*-> C <D O C o o o CO 0) .c r-.01 102 c o ra 1_ c <D O c o o o SZ V) <D x: .01 .008 .006 .004 .002 L=4.97mm, d=44.2,ym, Rotational speed = 6.8 rad/s Cylinder I.D. :82 mm I 1 E o o c *6 w o o Li. 1 30 60 Angle of incline, degrees E o o c •a T3 <o u o u. 90 Figure 14. Ef f e c t of Cylinder Incline on Threshold Concentration. d e t a i l in Section 4.5. It was found that a minimum angular speed was necessary for r e l a t i v e motion in the suspension to be created. Below th i s minimum the suspension was r e l a t i v e l y stationary as the cylinder rotated. The v e l o c i t y gradient was confined to a thin annulus between the cylinder wall and the f i b r e suspension; there was no r e l a t i v e motion within the suspension. This flow condition i s simi l a r to the "plug flow" regime of pulp suspensions in pipes [C4,R6] but with one difference that the wall i s moving and the suspension i s stationary. At higher angular speeds above 4 103 4 8 12 16 20 Cylinder angular speed, rad/s 24 Figure 15. E f f e c t of Angular Speed on Threshold Concentration. rad/s, the r e c i r c u l a t i n g motion was produced within the f i b r e suspension in the 82 mm ID cylinder. Under such a condition, Type-C f l o e s formed as soon as the threshold concentration was reached. When the angular speed had exceeded 4 rad/s, the threshold concentration was unaffected by i t . This i s i l l u s t r a t e d i n Figure 15 where 95% confidence l i m i t s are also shown. As the r o l l i n g speed increased, the t o t a l number of Type-C flo e s v i s i b l y decreased, but the threshold concentration was unaffected. The shear layer at the cylinder walls at which 104 f i b r e s were well dispersed grew in thickness with increasing rot a t i o n a l speed. It seems that the floc-forming phenomenon was countered by the dispersive action of t h i s shear layer. Jacquelin [J3] also noted the reduction in the mass content of coherent floes (T.F.C.) as the r o l l i n g speed increased. The absence of coherent networks in Forgacs et a l . [F1] apparatus could have been caused by high shear rates present there. The data and s t a t i s t i c s for experiments described in t h i s chapter are in Appendix XXVI. 4.3.2 Suspending l i q u i d v i s c o s i t y The e f f e c t of suspending l i q u i d v i s c o s i t y on the threshold concentration was examined with aqueous-sugar solutions of varying sugar content and one type of fibres (length=4.97 mm and diameter=44.2 um). The results are shown in Figure 16. The data and s t a t i s t i c s are in Appendix XXVI. I n i t i a l l y , the threshold concentration increased with v i s c o s i t y . In conjunction with t h i s , smaller numbers of weaker floes formed. However, above 0.013 Pa«s, Type-C floes did not form at a l l . At v i s c o s i t y 0.033 Pa«s, the suspension adopted a cloudy appearance, but at 0.14 Pa«s the suspension appeared uniform. At v i s c o s i t i e s higher than 0.013 Pa«s, no Type-C floes formed. Indeed, the concentration was increased to the point at which a l l f i b r e s formed a " s o l i d , " i . e . , there was i n s u f f i c i e n t suspending l i q u i d to have a suspension. The magnitude of viscous interaction between f l u i d and fi b r e s can be i l l u s t r a t e d by the c a l c u l a t i o n of the terminal 105 L=4.97mm, d = 44.2jjm, Cylinder I.D. = 82mm, Incline = 45 degree .016 c o CO k. +•> c o o c o o o sz (Si 2> sz H .012 .008 I .004 J i i i i i i 11 £ o o c •g •a at o o u. £ o o c •o T3 (A O O £ o o c •g TJ cn o o I - l I I I 111 .001 .01 .1 Suspending liquid viscosity, Pa»s Figure 16. Threshold Concentration versus V i s c o s i t y of the Suspending Liquid. v e l o c i t y and the terminal Reynolds number for a single f i b r e [C6,C7] in the examined suspending l i q u i d s . The results of such ca l c u l a t i o n s , shown in Table IX, indicate that a single p a r t i c l e or a c o l l e c t i o n of p a r t i c l e s in a d i l u t e , stationary suspension would undergo creeping r i s e or sedimentation. Thus, the motion of f i b r e s under net buoyancy forces i s very slow. It i s a reasonable conclusion that f i b r e s follow the steady flow very c l o s e l y , i . e . , the r e l a t i v e v e l o c i t y between ind i v i d u a l f i b r e s and f l u i d i s very small. Table IX. System C h a r a c t e r i s t i c s for Suspensions of Nylon Fibres (L=4.97 mm, d=0.0442 mm) 1n the Suspending Media of Various V i s c o s i t i e s . SUSPENDING MEDIUM DENSITY RATIO P p / p ' TERMINAL VELOCITY TERMINAL REYNOLDS NUMBERS * DENSITY P kg/m3 VISCOSITY c Pa. s AXIAL MOTION UT1 mm/s CROSS-AXIAL MOTION UT2 mm/s AXIAL MOTION Re T 1 CROSS-AXIAL MOTION R e T 2 998 0.00100 1.13 1 .4565 0.8862 6.42x10 - 2 3.91x10~2 1 130 0.00336 1 .00 0.0 0.0 0.0 0.0 1 175 0.00602 0.96 0.0825 0.0502 7.11x10~4 4.33x10 - 4 1220 0.0129 0.926 0.0770 0.0468 3.22X10" 4 1.96x10"4 1265 0.0327 0.893 0.0455 0.0277 7.79x10~5 4.74x10~5 1280 0.0498 0.882 0.0332 0.0202 3.77X10 - 5 2.30x10"5 1315 0. 144 0.859 0.0142 0.0086 5.72x10~6 3.48X10"6 * Based on the terminal v e l o c i t y , f i b r e diameter, and kinematic v i s c o s i t y of suspending l i q u i d . 107 The flow in a partly f i l l e d , i n c l i n e d , rotating cylinder is unsteady; i t decelerates and accelerates while changing d i r e c t i o n s . The f i b r e s do not tend to follow such a flow c l o s e l y , they crowd where the flow decelerates and accelerates. The degree of crowding depends on the v i s c o s i t y of the suspending l i q u i d and the concentration of f i b r e s , both counteracting each other. As the v i s c o s i t y increased and the crowding tendency of f i b r e s decreased, t h i s had to be compensated for by increased f i b r e concentration for Type-C coherence to be produced, as is c l e a r l y v i s i b l e in Figure 16. There existed, however, a v i s c o s i t y above which the crowding did not result in Type-C floe formation. This v i s c o s i t y must be s l i g h t l y higher than 0.013 Pa*s under these experimental conditions. Comparison of these findings with those reported by Steenberg et a l . [S12] i s i n t e r e s t i n g . They examined the ef f e c t of suspending l i q u i d v i s c o s i t y on the shear modulus of suspensions of Perlon f i b r e s at the volumetric concentration of 0.015 and reported that the shear modulus decreased as the v i s c o s i t y of the suspending medium increased. Their data show a o dramatic drop in the shear modulus from 100 N/m in water (0.001 Pa«s), to less than 3 N/m in a very viscous medium (1.0 Pa«s). 1 The steep drop occurred within a v i s c o s i t y change from 0.01 to 0.05 Pa«s. Steenberg et a l . attributed the drop to the change in the r e l a t i o n s h i p between viscous forces from the suspending l i q u i d and the e l a s t i c forces of f i b r e s . Referring to intense a g i t a t i o n as the process of coherent network formation, 1 These data have been reproduced in Figure 3 of t h i s d i s s e r t a t i o n . 1 0 8 they reasoned that it takes longer for the f i b r e s to come to rest after a g i t a t i o n in more viscous medium; hence, the e l a s t i c energy of many of them may be dissipated and unstrained configurations w i l l r e s u l t . Once f i b r e s l o s t t h e i r e l a s t i c energy, they cannot become a c t i v e l y engaged in the network however much time elapses. Because t h i s explanation does not consider the restraining action of surrounding f i b r e s at the concentration in question, their explanation is incomplete. It is not clear how the f l u i d v i s c o s i t y can play the predominant role in d i s s i p a t i n g the e l a s t i c energy of fibres when the l a t t e r are in a continuous, multiple contact with one another. In summary, the gradual increase in the suspending l i q u i d v i s c o s i t y produced flow conditions that progressively eliminated the f i b r e interlocking into f l o e s . Clearly the v i s c o s i t y plays a strong role in Type-C floe formation in the r e c i r c u l a t i n g flow which is unsteady (this study) as well as in the highly agitated flow which decayed (other studies). In the r e c i r c u l a t i n g flow, the f i b r e s may align themselves to avoid formation of the isotropic 3-D networks or can follow the flow so cl o s e l y that the f i b r e crowding i s nonexistent. It i s also conceivable that both effects occur at the same time. In the highly agitated suspension the picture of f i b r e motion i s more complex, but i t may include these e f f e c t s . Although the density of the suspending l i q u i d changed with v i s c o s i t y , i t only changed by the factor of 1.32 whereas the v i s c o s i t y changed more than one hundred times. 109 4 . 3 . 3 Fibre geometry The threshold concentration was found to be profoundly affected by the dimensions of f i b r e s . The dimensions of wet f i b r e s used in these experiments are shown in Table X. As f i b r e length, L, increased, the threshold concentration decreased (Figure 17). Experimental data concerning the e f f e c t of f i b r e geometry on the threshold concentration are in Appendix XXVI. Moreover, for a given diameter, d, there appears to be a lower l i m i t of f i b r e length, L, below which no Type-C floes form in the c i r c u l a t i n g suspension regardless of how high f i b r e concentration i s . This suggests a lower l i m i t of f i b r e length or possibly lower l i m i t of f i b r e aspect r a t i o at which Type-C floes cannot form. This subject i s expanded in Section 4.4. Another rarely discussed variable related to the three alternate contact interlocking i s f i b r e d e f l e c t i o n . Intuition suggests that fibres deflect more as they are brought more cl o s e l y together in the Type-C network. S i m i l a r l y , short f i b r e s must bend more to interlock since the distances between their centers are smaller as indicated by the increased f i b r e concentration in Figure 17, i . e . , cohesion cannot develop i f the fibre-bending forces (compaction forces) are too low. This seems to be the case for floe formation in rotating cylinders because Type-C floes can be formed from the shortest f i b r e s of t h i s study by f i b r e s being compacted between fingers. Thus, i t may be concluded that the l i m i t i n g condition for floe formation in rotating cylinders was not fib r e length, but the intensity of f i b r e crowding. Nevertheless, i t i s reasonable to expect that T a b l e X. Nylon F i b r e D i m e n s i o n s In Wet S t a t e . TYPE OF FIBRE DIAMETER d SAMPLE SIZE FIBRE LENGTH ASPECT RATIO L /d CURVATURE AVERAGE LENGTH L STANDARD DEVIATION AVERAGE 1/R STANDARO DEVI AT I ON d e n i e r * lyll) mm mm mm* 1 mm" 1 3 19.76 1062 1054 123 1 1117 0 .9158 1 .875 2.815 3.737 0 .08484 0 .09244 0 .1019 0 .1122 4 6 . 3 5 94 . 88 142.5 189. 1 0 .3094 0 .2553 0. 28 1 1 0 . 2 8 5 5 0 .2132 0 . 1 7 19 0. 1933 0.1881 6 27.95 1237 1 138 1042 1049 1024 0 .9139 1 .832 2.757 3.718 4 .666 0 .08322 0 .09898 0 .1267 0 .1576 0 .1338 3 2 . 7 0 6 5 . 5 5 9 8 . 6 5 133 .0 166.9 0 .1696 0 . 1 5 2 2 0 . 1618 0 .1297 0 . 1 3 7 0 0 .1453 0 .1113 0 . 1 195 0 .1003 0 .1034 15 44. 19 1038 1 162 1001 1006 1 .560 2 .947 4 .973 6.261 0 .1009 0.1081 0 .1762 0 .1769 3 5 . 3 0 6 6 . 6 9 112.5 141 .7 0 .08366 0 . 0 7 2 4 0 0 . 0 7 107 0 . 0 6 0 6 0 0 .08033 0 .06239 0.05731 0 .05105 * D e n i e r i s a we ight i n grams of 9000 meter s l o n g m o n o f i l a m e n t . * * R deno t s a r a d i u s of c u r v a t u r e . 111 c o CO o o c o o 2 o CO Si .05 .04 .03 2 .02 .01 F i b r e d i a m e t e r , J i m • 1 9 . 8 • • • 2 7 . 9 4 4 . 2 •~ • form form • o c z • o c • •o '•5 '•5 tn — o o LU <fl / \ o iZ : i • I • • I • • • I 3 4 Fibre length, mm Figure 17. E f f e c t of Fibre Length and Diameter on Threshold Concentration. the lower l i m i t of f i b r e length or f i b r e aspect r a t i o should e x i s t below which fi b r e s cannot interlock e l a s t i c a l l y in a 3-D network. This interlocking cannot develop regardless of the magnitude of the f i b r e bending forces. A strong e f f e c t of f i b r e diameter, d, on the threshold concentration i s also v i s i b l e . As f i b r e diameter increases for a given f i b r e length, the threshold concentration increases. It should be noted, however, that doubling of fib r e diameter in a random 3-D network should increase the volumetric concentration about three times. The estimated increase, based on Figure 17, 112 i s s l i g h t l y less than three when the fibre diameter more than doubles: 19.8 to 44.2 /um. It must also be noted that the t o t a l number of f i b r e s in a unit volume should double i f the diameter is halved. Values close to that have been observed, and are discussed in the next section. Thus, the effect of f i b r e diameter on threshold concentration i s accounted for in large part on the basis of network geometry. Fibre diameter i s a key factor affecting f i b r e s t i f f n e s s , S, 4 where S=7r«E«d /64. Figure 18 shows the threshold concentration 4 plotted versus d for a constant f i b r e length, L, as well as for Figure 18. Threshold Concentration versus d". 1 13 a constant f i b r e aspect r a t i o , L/d. Such a plot i s j u s t i f i e d because the differences between the e l a s t i c moduli of the wet nylon f i b r e s are small. The e l a s t i c moduli were determined experimentally and are reported in Appendix VIII. At a constant L, the increasingly s t i f f e r f i b r e s produce cohesion at higher concentrations whereas at a constant L/d, the trend i s reversed. The f i b r e length and diameter c l e a r l y a f f e c t the threshold concentration in a complex way. 4 . 4 E v a l u a t i o n o f C r i t e r i a f o r T h r e s h o l d C o n c e n t r a t i o n The c r i t e r i a for threshold concentration are evaluated with the experimental observations of t h i s investigation. T h r e s h o l d a n d S e d i m e n t C o n c e n t r a t i o n s . It was proposed [T9] that the sediment concentration i s an estimate of the lowest concentration at which coherent network strength may be detected. This postulate can be tested by the sediment concentrations being compared to the threshold concentrations measured in t h i s study. For fibres having diameters 19.8 and 27.9 Mm, the sediment concentrations are systematically lower then the threshold concentrations, as shown in Figure 19 and in Table XI. 1 On the other hand, for the 44.2 Mm diameter f i b r e s , the s i t u a t i o n i s reversed. There i s no systematic agreement between the sediment and the threshold concentrations, though both have similar trends with respect to L/d. This finding indicates that the sedimented fi b r e s form 1 The data and s t a t i s t i c s from sedimentation experiments are in Appendix XXIX. Table XI. Threshold Concentrations and Sediment Concentrations FIBRE DIAMETER (WET) d AVERAGE FIBRE LENGTH (WET) L mm FIBRE ASPECT RATIO L/d THRESHOLD CONCENTRATION SEDIMENT CONCENTRATION APPARENT VOLUMETRIC * C v a STANDARD DEVIATION a APPARENT VOLUMETRIC <va STANDARD DEVIATION a 19.76 0.9158 1 .875 2.815 3.737 46.35 94.88 142.5 189 . 1 NO FLOCS .01972 .009334 .005791 .0005615 .0002908 .0004115 .03517 .01154 .005373 .003132 .0005677 .001637 .0006473 .0003314 27 .95 0.9139 1 .832 2.757 3.718 4.666 32.70 65 .55 98 .65 133.0 166 .9 NO FLOCS .03314 .01641 .008549 .005099 .001350 .001124 .0007753 .0001874 .04786 .02182 .008173 .006043 .003812 .003088 .001091 .001220 .0008337 .0003147 44 . 19 1 .560 2.947 4.973 6.261 35.30 66.69 112.5 141.7 NO FLOCS .02221 .007915 .004469 .0006021 .0002300 .0002266 .05572 .02427 .00976 .005416 .004084 .001249 .0007970 .0005382 115 Figure 19. Threshold and Sediment Concentrations versus Fibre Aspect Ratio. 1 1 6 d i s t i n c t l y d i f f e r e n t networks from the networks c i r c u l a t i n g i n an i n c l i n e d r o t a t i n g c y l i n d e r . Sediment C o n c e n t r a t i o n and Mathematical Models. In the l i t e r a t u r e [ T 9 ] , the l i m i t i n g c o n c e n t r a t i o n s c a l c u l a t e d from equation ( 3 ) were compared with the sediment c o n c e n t r a t i o n s of v a r i o u s man-made f i b r e s . The agreement of the sediment data with the equation (3) a t n c from 3 to 5 gave reason to advance the p o s t u l a t e t h a t , above the l i m i t i n g c o n c e n t r a t i o n , the f i b r e s i n t e r l o c k by t h r e e a l t e r n a t e c o n t a c t p o i n t s [P3,W3], F i g u r e 20. Sediment C o n c e n t r a t i o n s and Mathematical Models. 1 17 The experimental data for sediment concentrations from t h i s investigation (Appendix XXIX) indicate that, for large f i b r e aspect r a t i o s (L/d>l00), a l l sediment concentrations f a l l between the t h e o r e t i c a l l i n e s calculated from equation (3), with nc=3 and nc=4, as shown in Figure 20. Below the aspect r a t i o of 100, the theo r e t i c a l l i n e s s t a r t to overestimate the sediment concentrations. Similar results were obtained by Thalen and Wahren [T9], with the one difference that most of their sediment concentrations were located between n =4 and n =5 l i n e s . Miles's c c model, represented by dashed l i n e s (Figure 20), agrees with the experimental data to a lesser extent than does the Meyer-Wahren model, es p e c i a l l y at f i b r e aspect r a t i o s lower than 100. The sediment concentrations and the predictions from s t a t i s t i c a l models appear to agree for the aspect r a t i o s larger than 100. Below the aspect r a t i o of 100, only the Meyer-Wahren model at nc=3 agrees with the sediment data. This result i s surprising because the sedimented network is not an isotropic 3-D network; the fi b r e s are p r e f e r e n t i a l l y oriented at low angles to the horizontal in sedimented networks, esp e c i a l l y long f i b r e s (L/d>l00). In addition, in a l l sedimented networks, each f i b r e should have four ordinary contact points with other f i b r e s [E4]. Thus, a sedimented network has a larger number of contact points per f i b r e than an is o t r o p i c , 3-D network at the same concentration. This explains why both concentrations coincide. In general, the comparison of sedimented f i b r e networks with the is o t r o p i c , 3-D networks should not be made because they d i f f e r s i g n i f i c a n t l y in s p a t i a l f i b r e o r i e n t a t i o n . Threshold Concentration and Mathematical Models. 118 The threshold concentrations of this study were compared to the l i m i t i n g concentrations calculated from equation (2) and (3) at nc=4 since i t was said to be closer to r e a l i t y [W4]. The comparison i s shown in Figure 21. The s o l i d l i n e i s a plot of Meyer-Wahren t h e o r e t i c a l relationship for a continuous, i s o t r o p i c , 3-D network of uniformly long and thick f i b r e s . 40 80 120 160 200 L/d Figure 21. Mathematical Models and Concentrations of Fibres in Suspensions at the Onset of Type-C Floe Formation. 119 1 6 * 7T • L C (12) va The dotted l i n e i s a plot of Miles's model at n =4. C va d L (13) At high L/d, both l i n e s follow the trend of the experimental data points, but not completely. Overall, the Meyer-Wahren model f i t s the experimental data well. Miles's model agrees poorly with the data at L/d's less than 120 at which the most s i g n i f i c a n t changes in threshold concentration take place. It i s noteworthy that neither of these models accounts for the e f f e c t of f i b r e diameter evident from the experimental observations shown in Figure 21. This raises doubt about the a p p l i c a b i l i t y of both s t a t i s t i c a l models. Crowding Factor and Number Density. "Crowding factor" and "number density" are the alternate c r i t e r i a for the characterization of the threshold concentration. Kerekes et a l . [K7] proposed the "crowding factor" extrapolating from Mason's concept of " c r i t i c a l concentration" [M5,M6,M7], The "crowding factor," N^ g, represents the number of f i b r e s in a spherical volume having diameter equal to one f i b r e length as expressed by equation (6). Bibbo et a l . [B9] used a cube of side L instead and coined the term "number density," N f as 120 expressed by equation (7). Both concepts carry an assumption that the fib r e length i s f i n i t e , i . e . , i t i s not zero or * i n f i n i t y . At the threshold concentration, C , these terms are denoted and and are large as shown in Table XII. This unequivocally points to the presence of strong f i b r e i n t e r a c t i o n . * Examination of Table XII reveals that for a given diameter, * and N^ c are r e l a t i v e l y constant over a range of f i b r e lengths. This suggests that, for a given diameter, the threshold concentration occurs at a constant number of f i b r e s in a given volume. Moreover, the number of f i b r e s diminishes with increases f i b r e diameter. This result warrants further consideration of * the character of the r e l a t i o n s h i p between N^'s and f i b r e * diameter. For brevity, further discussion i s l i m i t e d to as def ined: With N^ c assumed to depend so l e l y on f i b r e diameter, the following hyperbolic, logarithmic and exponential relationships were selected to relate these two variables: N f C = a n q ( 1 5 ) * fc = K 3-ln(1+d«K 4) (16) Table XII. Crowding Factors'Calculated from Equations (6) and (7) at the Onset of Type-C Floe Formation. FIBRE DIAMETER (WET) d j/m AVERAGE FIBRE LENGTH (WET) L mm FIBRE ASPECT RATIO L/d THRESHOLD CONCENTRATION < a AVERAGE NUMBER OF FIBRES IN A SPHERE WHICH DIAMETER IS EQUAL TO THE AVERAGE FIBRE LENGTH * N f s AVERAGE NUMBER OF FIBRES IN A CUBE WHICH SIDE IS EQUAL TO THE AVERAGE FIBRE LENGTH N f c 19.76 0.9158 1 .875 2.815 3.737 46.35 94.88 142.5 189. 1 NO FLOCS .01972 .009334 .005791 1 18.4 126.3 141.0 226. 1 241 .2 269.3 27.95 0.9139 1 .832 2.757 3.718 4.666 32.70 65.55 98.65 133.0 166.9 NO FLOCS .03314 .01641 .008549 .005099 94.92 106.4 100.8 94.74 181 .3 203.3 192.6 180.9 44. 19 1 .560 2.947 4.973 6.261 35.30 66.69 112.5 141.7 NO FLOCS .02221 .007915 .004469 65.85 66.83 62.79 125.8 127.6 119.9 122 N ; c = 4i-d-K6) (17) Selection of these simple relationships respected geometrical l i m i t s shown in Table XIII because the domain of f i b r e diameter in these relationships i s (0,L] at any f i n i t e L. L i m i t (1). * As d approaches zero, N^ c approaches CONSTANT^ CONSTANT .j represents a number of straight l i n e segments * of length L in a cube of side L. C then, of course, V3 approaches zero. L i m i t ( 2 ) . As d approaches L, N^c approaches CONSTANT2 making * C approach (rr/i) • CONSTANT- . va z The geometric interpretation of the second l i m i t i s d i f f i c u l t . The l i m i t dependence on f i b r e length is cumbersome. Certainly the relationships (14) to (17) do not apply at small aspect r a t i o s , i . e . , where Type-C networks do not form. In t h i s case the upper l i m i t of f i b r e diameter must be °-=(^iimit* Regardless of whether <d/L> l i m i t or L l i m i t e x i s t , N * c > l i m i t represents the minimum number of f i b r e s required to form Type-C floe for given L and d. These experiments indicate that the l i m i t i n g L/d i s large, about 50, and in practice d never equals L, contrary to what mathematical l i m i t (2) s t i p u l a t e s . 123 T a b l e X I I I . L i m i t s o f E q u a t i o n s (14) t o ( 1 7 ) . LIMIT (1) LIMIT (2) d 0 d —> L c* va -» 0 -» |-CONSTANT2 N* f c —> CONSTANT1 -> CONSTANT2 Nonlinear f i t s made to a l l data from Table XII determine K-coefficients with the NL2S0L Fortran subprogram available from the UBC Computing Centre. The calculated parameters are shown in Table XIV. The second column containing the sum of least squares indicates the goodness of f i t . The f i t c o e f f i c i e n t s and l i m i t s are shown in the next two columns. To supplement these observations, the computer program described in Appendix XVI was written to model f i b r e sedimentation. In this program, a depositing f i b r e contacts the topmost f i b r e of previously deposited mat lying on i t s path of descent, then rotates and/or slides to rest on the top of the mat or penetrates the mat. The program was run with f i b r e diameter set to zero after an i n i t i a l sedimentation with f i b r e s of f i n i t e diameter. The number of sedimented l i n e segments of L=140 "distance units" in a cube of side L was 253. This number of l i n e segments i s comparable to the LIMIT (1) obtained from three f i t s , c f . Table XIV. This i s only an i l l u s t r a t i o n of 124 Table XIV. F i t Co e f f i c i e n t s and Limits for Number of Fibres in a Cubical Volume. * f c L squares COEFFICIENTS LIMIT ( 1 ) K1 d+K2 4.32-10~6 K!=6.8221mm K2=0.00933mm 731 K 3-ln(1+d-K 4) 1 . 17- 10~ 4 K3=170.71 K4=5000.1mm~1 171 (1-d-K 6) *5 4.02-10"6 K5=370.06 K6=4.1519mm"1 370 finite n e s s of'"CON ST ANT 1. 4.5 Flow Conditions under which Coherent Floes Form Observations and measurements of the flows were made to ide n t i f y the hydrodynamic conditions that lead to formation of Type-C fl o e s . The differences between flows that caused and did not cause Type-C floe formation were sought. Visual observations and LDA ve l o c i t y measurements were made under selected conditions in the 36 mm deep, 94 mm ID r o l l i n g cylinder described in Section 3.4. One type of nylon f i b r e s was used: 44.2 ixm in diameter and 4.97 mm long, (L/d=112.5). The r a t i o of cylinder depth to f i b r e length was 7.2. 125 Figure 22. Flow Pattern in a Horizontal Rotating Cylinder. A t y p i c a l flow pattern observed v i s u a l l y with the cylinder half f u l l and rotating at a constant angular speed of 2<T rad/s i s shown in Figure 22. The cylinder rotation i s counter-clockwise, as marked by the v e l o c i t y vector at the lowest point of the c y l i n d e r . The cylinder walls imparting motion to the suspension exploit the adhesive and viscous forces. The viscous forces combine with the g r a v i t a t i o n a l forces to accelerate suspension in the lower-left quadrant (-7r/2</3<0) of the cylinder cross-section. In the lower-right quadrant (O<0<VT/2) , the i n e r t i a and g r a v i t a t i o n a l forces oppose the viscous forces. In the zone near the free surface and close to the c y l i n d r i c a l wall (7T/4<0<7T/2 and 126 0.65R<r<R), the suspension decelerates in the tangential d i r e c t i o n , turns sharply towards the cylinder axis, and accelerates in the r a d i a l d i r e c t i o n across the cylinder. On reaching the opposite c y l i n d r i c a l wall, the suspension turns again to follow the downward d i r e c t i o n of t h i s wall movement. A small portion of the upwardly flowing suspension adheres to the cylinder walls and moves as a f i l m . This f i l m c a r r i e s some fib r e s and produces a d i l u t i o n zone at 0=-7r/2. In the lower half of the cylinder (-7r/2</3<7r/2) , a layer of zero v e l o c i t y (evaluated with respect to a stationary observer outside the cylinder) exists as i f a midfeather were placed there. Other features of t h i s general flow pattern are discussed in the following section. 4.5.1 D i s s i m i l a r i t i e s between flows in horizontal cylinder In a suspending l i q u i d of low v i s c o s i t y , Type-C floes formed when the threshold concentration was exceeded. In the suspending l i q u i d of high v i s c o s i t y , floes never formed. It was expected that studying the d i s s i m i l a r i t i e s between these flows could lead to understanding of the mechanism of coherent floe formation. Three flow cases were compared: Case 1. Low v i s c o s i t y suspending l i q u i d (M=0.00375 Pa«s) with f i b r e concentration (C v a=0.0076) s l i g h t l y below the threshold concentration. Type-C floes did not form. Case 2. Low v i s c o s i t y suspending l i q u i d , as in Case 1, with f i b r e concentration (C. =0.0114) above the threshold va concentration. Type-C floes formed. 127 1 CO-R Figure 2 3 . V e l o c i t y Vectors for the Case 1 Flow. Case 3 . High v i s c o s i t y suspending l i q u i d (M=0.14 Pa«s). The apparent volumetric concentration of fi b r e s was the same as in Case 2. Type-C floes did not form. The v e l o c i t y vectors for a l l three cases are shown in Figures 2 3 , 2 4 , and 2 5 . (The data obtained from v e l o c i t y measurements are in Appendix XXVII.) The velocity vectors 128 represent the average v e l o c i t i e s of the suspending l i q u i d and f i b r e s . The main differences between flows, based on the vi s u a l observations and v e l o c i t y measurements, are l i s t e d : 0)-R Figure 24. Ve l o c i t y Vectors for the Case 2 Flow. 129 V i s u a l observations Direct v i s u a l observations and v i s u a l analysis of films and video tapes aided assessment of the motion of f i b r e s and f l o e s . The observations are: 1. In Case 2, fibr e crowding in the zone between angles 0=50° and /3=90° at r/R>0.6 was observed. In t h i s zone, groups of f i b r e s moved unsteadily in a tangential d i r e c t i o n . At the end of t h i s zone, these groups turned and accelerated across the cyl i n d e r . Such f i b r e crowding had not been observed in the other two flow cases. This important feature of Case 2 flow i s discussed in the next subsection. 2. The suspensions of low v i s c o s i t y (Case 1 and Case 2) produced an almost horizontal free surface, as shown in Figure 23 and 24. In contrast, the suspension of high v i s c o s i t y (Case 3) produced a free surface i n c l i n e d to the horizontal as shown in Figure 25. 3. The l i q u i d f i l m thickness on the upper walls of the cylinder (7r/2<j3<3 • VT/2 ) increased with increased v i s c o s i t y of the suspending l i q u i d . 4. The d i l u t e zone marked in Figure 22 did not exist in the Case 3 flow. Ve l o c i t y measurements LDA measurements gave average v e l o c i t i e s of the suspending l i q u i d and f i b r e s at selected points in the central plane of the cyli n d e r . The findings are summarized: 1. In Case 2, the flow v e l o c i t y decreased sharply between 0=40° 130 (0-R g u r e 2 5 . V e l o c i t y V e c t o r s f o r t h e C a s e 3 F l o w . a n d 0 = 7 0 ° a t r / R = 0 . 8 5 , a n d b e t w e e n 0 = 5 0 ° a n d 0 = 8 0 ° a t r / R = 0 . 9 6 . I n C a s e 1, t h e f l o w v e l o c i t y d e c r e a s e d g r a d u a l l y b e t w e e n 0 = 4 0 ° a n d 0 = 8 0 ° a t r / R = 0 . 8 5 , a n d be tween 0 = 5 0 ° a n d 0 = 8 0 ° a t r / R = 0 . 9 6 ( F i g u r e 26 a n d F i g u r e 2 7 ) . H e n c e , t h e r a t e s o f d e c e l e r a t i o n d i f f e r g r e a t l y b e t w e e n C a s e 2 , i n w h i c h T y p e - C n e t w o r k s f o r m e d , a n d C a s e 1, i n w h i c h t h e y d i d n o t . 131 r/R = .85 24 20 XT 16 12 8 Case 1 -• Case 3 4 -Case 2 20 40 60 p, degrees 80 100 gure 26. Flow V e l o c i t i e s at r/R=0.85 and 0 from 0 to 100 degrees. Cylinder peripheral v e l o c i t y i s 0.295 m/s. In Case 1, the reduction in the flow velocity and the change in flow d i r e c t i o n were gradual as indicated by the v e l o c i t y vectors in Figure 23. The quantitative observations agree with the v i s u a l assessment of a smooth flow of f i b r e suspension. In Case 2, the reduction in the flow v e l o c i t y and the change in the flow di r e c t i o n were abrupt (Figure 24). This picture corresponds well with the v i s u a l observation of the unsteady flow of fibres in which 132 groups of f i b r e s moved e r r a t i c a l l y before being accelerated across the cy l i n d e r . In Case 3, the flow v e l o c i t y decreased and the flow d i r e c t i o n changed smoothly between 0=60° and 0=80°, as in the Case 1 flow. The fib r e s followed the flow clo s e l y , i . e . , they did not crowd. 2. In Case 1 and 2, the zero-velocity layer remained at the same ra d i a l l o c a t i o n , r/R=0.64 (Figure 23 and 24). In Case 3, Figure 27. Flow V e l o c i t i e s at r/R=0.96 and 0 from 0 to 100 degrees. Cylinder peripheral velocity i s 0.295 m/s. 133 the zero-velocity layer was further away from the axis of rotation, r/R=0.73 (Figure 25). The shear zone was approximately twice as long as the corresponding zones in two other flow cases. Calculations of the shear rates between the zero v e l o c i t y layer and the cylinder wall yielded: AV/Ar = 26 s 1 for high v i s c o s i t y l i q u i d , and AV/Ar = 7.8 s 1 for low v i s c o s i t y l i q u i d . Hence, in the Case 3 flow, the suspension passing between the wall and the zero v e l o c i t y layer i s subjected to a higher shear rate for longer periods of time. It . should be noted that the shear rates of similar magnitude dispersed wood-pulp f i b r e floes in water [H8]. 3. The viscous interaction between the l i q u i d and the fi b r e s was greater in Case 3 than in the other two cases. This subject has already been discussed in Section 4.3.2. Summary From the vi s u a l observations and the v e l o c i t y measurements, a unifying picture of Type-C f l o c c u l a t i o n in the p a r t i a l l y f i l l e d , rotating cylinder emerges. A c y c l i c flow having acceleration, deceleration, and low shear zones seems required. In addition, the suspension concentration, the properties of f i b r e s , and the properties of the suspending l i q u i d must f u l f i l l conditions which were discussed in Sections 4.3.2 and 4.3.3. Two consecutive events appear to take place in the r e c i r c u l a t i n g flow: f i b r e crowding in the deceleration zone, and floe dispersion in the shear zone. Above the threshold 134 concentration f i b r e crowding i s s u f f i c i e n t to create cohesion in low v i s c o s i t y f l u i d and the shear forces are not large enough to disperse such f l o e s . The high v i s c o s i t y f l u i d , on the other hand, appears to prevent network crowding and enhances floe dispersion in the shear zone. A higher average suspension concentration would be expected to be needed to reach the threshold concentration as the suspending l i q u i d v i s c o s i t y increases. This trend is shown in Figure 16. At low suspending l i q u i d v i s c o s i t i e s , the floes survive through the shear zone and return to the crowding zone where they undergo further compaction. C y c l i c exposure to compaction densities Type-C f l o e s . Floe d e n s i f i c a t i o n coexists with the d i l u t i o n of the inter-floc-space. In d i l u t e d inter-floc-space below the threshold concentration f i b r e s cannot crowd. Floe d e n s i f i c a t i o n i s stopped and formation of new floes is prevented. Only a portion of suspended fi b r e s ends up in floes, i . e . , the number of Type-C floes i s fixed. Wahren et a l . [M10,S12,T9] postulated that vigorous agita t i o n of a suspension and subsequent decay of agitation are needed to form Type-C fl o e s . The present work shows that Type-C floes form in a continuously agitated suspension. Work of others [A9,C1,F1,G5,R2,V1] points to intense a g i t a t i o n as a means of coherent network formation. It i s possible that neither high nor low intensity of agitation i s exclusively responsible for Type-C network formation, but that a spectrum of agitation i n t e n s i t i e s exist within which such floes form. If t h i s i s the case, the present work has been conducted at the lower end of t h i s 135 spectrum. For completeness of flow characterization the average v e l o c i t i e s and Root-Mean-Squares (RMS) of v e l o c i t y fluctuation are reported in Appendix XXXI for low v i s c o s i t y suspending l i q u i d only. The calculations of the RMS of v e l o c i t y fluctuation of f i b r e suspensions were unreliable because the LDA signal was validated only for a small fracti o n of time, e.g., less than 10% of the test time, in the f i b r e crowding zone. A suspending l i q u i d of high v i s c o s i t y causes fi b r e s to follow the flow more cl o s e l y , and thereby prevents l o c a l i z e d crowding of f i b r e s which i s a f i r s t requirement in the formation of Type-C f l o e s . It follows that the explanation of Attanasio et a l . [A17] ascribing a l u b r i c a t i n g property to the suspending l i q u i d may be incorrect. Neither the explanation of Steenberg et a l . [S12], based on longer decay times for more viscous l i q u i d s , seem to be a correct explanation for Type-C floe formation. 4.5.2 Other types of flow As described in Section 3.2.1, other flows were examined for production of Type-C f l o e s . Shearing a suspension in a gap between two concentric cylinders did not produce Type-C floes under either laminar, t r a n s i t i o n , or turbulent flow conditions with gradually increased f i b r e concentration from below to above threshold concentration. Type-C floes did not form in a channel through which a gri d was passed up and down in a c y c l i c manner. On the other hand, Type-C floes did form in a suspension agitated 136 by a s t i r r e r in a tank. These floes existed outside the zone of high shear which surrounds the impellers. High shear dispersed f l o e s . This explains why Forgacs et a l . [F1] did not produce coherent networks from rayon f i b r e s (L/d=150) although the l i m i t i n g concentration was greatly exceeded. These supplementary experiments confirm the observation that Type-C floes form in c y c l i c flows having acceleration and deceleration zone where crowding occurs. Another flow requirement appears to be that floes not be forced to pass through a zone of very high shear. Although high shear may exist l o c a l l y in the flow, the floes should be free to pass outside of t h i s zone. Such i s the case of c y c l i c flow in the rotating cylinder and mixing tank, but not in the case of repeated passage of suspension through the g r i d . In summary, the flow conditions should be suitable to impart strength to a floe during the crowding process and produce cohesive forces larger than the subsequent forces exerted on the floe to break i t apart during passage through the shear zone. 4.6 O b s e r v a t i o n s of F l o e S t r u c t u r e The mathematical models of past t h e o r e t i c a l studies describe randomly d i s t r i b u t e d straight f i b r e s segments in 3-D [C5,M10,01]. Two of these models [C5,M10] are of interest because they relate f i b r e geometry and the number of contacts per f i b r e to the volumetric concentration of f i b r e network. These models have been discussed e a r l i e r , in Section 2.3.2 as equations (2) and (3). 137 No experimental studies of t r u l y 3-D f i b r e networks can be found in the l i t e r a t u r e . E l i a s [E4], however, came the closest by studying compacted mats of glass fibres and counting f i b r e contacts. The mats merely had a 2-D structure and therefore did not provide good experimental results to test these models. Consequently, an experiment was designed to v i s u a l l y count number of contacts between f i b r e s in Type-C floes. The method r e l i e s on sugar deposition at the contact points as water evaporates from the aqueous-sugar solut i o n . The deposited sugar bonded the 8 Meyer & Wahren L/d=141.7 • • • 0 .1 0 Figure 28. Number of Contacts per Fibre versus Fibre Concentration in Floes (L=6.26 mm, d=44.2 nm). 138 f i b r e s . The f i b r e contact points were counted by progressive bond breaking and f i b r e removal. The numbers of contact points per f i b r e are shown in Figure 28 for fib r e s of L=6.26 mm and d=44.2 Mm and in Figure 29 for f i b r e s of L=45.9 mm and d=559 Mm. The threshold concentration i s marked in Figure 28. The predicted number of contacts from equations (2) and (3) for these conditions are also shown. Evidently both models greatly overestimate the number of Figure 29. Number of Contacts per Fibre versus Fibre Concentration in Floes (L=45.9 mm, d=559 Mm). 139 contacts per f i b r e in Type-C f l o e s . The discrepancy between the s t a t i s t i c a l models and the experiment i s large. Models' agreement with the threshold concentration data has already been questioned in Section 4.4. Here, the discrepancy between the s t a t i s t i c a l theories and experimental values i s larger. Observations of contact point location on f i b r e surfaces revealed that contacts rarely occur on alternate sides of f i b r e in a given plane, as shown in the s i m p l i s t i c i l l u s t r a t i o n s of Parker [P3] and Kerekes et a l . [K7], In r e a l i t y , most of the contacts are di s t r i b u t e d around a f i b r e perimeter. Moreover, i t was observed that, on average, less than three contact points per f i b r e exist in Type-C fl o e , c f . Figure 29. These observations indicate that fibres interlock in a more complex way than was previously understood [P3,K7]. Many fibres (a few hundred) constitute one coherent floe so that interlocking i s a three-dimensional event requiring s p a t i a l interaction of f i b r e s . 4.7 Tensile Strength of Type-C Floes After examination of the structure of coherent f l o e s , their t e n s i l e strength was measured. Measurements using the procedure described in Section 3.6 were c a r r i e d out in a THWING/ALBERT te n s i l e t e s t e r . The data from these measurements i s in Appendix XXVIII. A t y p i c a l load-separation curve i s shown in Figure 30. The load increased steadily from zero (A) to a maximum (B). The ordinate corresponding to the maximum on the curve represents a measure of floe strength. Beyond t h i s point (B), the load decreased jaggedly while the floe separated into 140 A B o A separation Figure 30. Typical Shape of Load-separation Curve. two segments. The t e n s i l e stress was calculated by the floe strength being divided by the cross-section area of a break zone. The t e n s i l e stress varied with floe concentration, as shown in Figures 31 to 37 and i t also varied with the length of fibres having the same diameter. The floes formed from long fibres are stronger than floes from short f i b r e s at the same concentration, c f . Figures 31 to 33 for d=44.2 Mm, and Figures 34 to 36 for d=27.9 ium. This e f f e c t was anticipated because longer fibres at a given volumetric concentration y i e l d a larger number of contacts with other f i b r e s . F i t s made by a power re l a t i o n s h i p to the data points allowed comparison of the t e n s i l e strength of nylon floes with the other strengths reported i n the l i t e r a t u r e . 141 10 CM / E / top y • 7. nylon fibres L= 6 . 2 m m d = 4 4 . 2 j j m -jl ' - i i i i i i i i I I i i i i i i '.01 .1 1 Figure 31. Tensile Stress versus Floe Apparent Volume Concentration. aT = a " " * C v a b " " ( 1 8 ) The f i t t e d l i n e s are shown in Figures 31 to 37 and the f i t c o e f f i c i e n t s are l i s t e d in Table XV along with the ranges of floe concentration. Whether f i b r e networks rupture in a t e n s i l e or shear mode, the microscopic mechanism of network disruption i s the same -f i b r e s are pulled apart. This j u s t i f i e s comparison of results 142 nylon fibres L= 4 . 9 m m d = 4 4 . 2 ( u m -| I I I I I I I I I 1 i i i i i i i i I '.01 .1 1 Figure 33. Tensile stress versus floe apparent volume concentration. from t e n s i l e strength experiments and shear strength measurements reported in the l i t e r a t u r e [H6]. Indeed, the strengths are of the same magnitude as i l l u s t r a t e d in Figure 38 by the group of li n e s marked (1) and (2). The average value of exponent of f i t t e d l i n e s to the nylon floe strength data is 2.65. It i s below exponent values from the y i e l d stress measurements of nylon f i b r e s l u r r i e s , c f . Tables XV and I I I . The ranges of concentrations investigated are also comparable. The t e n s i l e strength of nylon f l o e s i s somewhat larger than the y i e l d stress 143 -jl i I I I I I i I I I I I j I I I I I '.01 .1 1 ^va Figure 33. Tensile Stress versus Floe Apparent Volume Concentration. in shear. As already pointed out in Section 2.3.3, the experimental conditions may a l t e r the way in which forces are applied to the f i b r e network. The application of force through b r i s t l e s and vanes resulted in higher shear strengths at a given concentration than application of force through s o l i d , smooth surfaces, c f . l i n e (3) and l i n e (4) in Figure 38. However, the mode of application of the t e n s i l e force in t h i s study resulted in even higher strengths 10 1 44 nylon fibres L= 4.6 mm d= 27.9 jjm CM £ £ 1 / A. J r --J I I I I I I I I I I I I I I I I I 11 '.01 .1 1 ^va Figure 34. Tensile Stress versus Floe Apparent Volume Concentration. (group of l i n e s marked (1)). It i s possible that a l l shear methods, regardless of th e i r accuracy, recorded an average strength of the weak and strong zones in the f i b r e network [M16]. Such an average would be less than the strength of a r e l a t i v e l y uniform i n d i v i d u a l f l o e . Comparison of the t e n s i l e strength for nylon floes with the shear strength of wood-pulp f i b r e suspensions by the d i r e c t methods (line (3)) reveals a s i g n i f i c a n t difference in the magnitude of recorded stresses but nearly s i m i l a r slopes of stress dependence on concentration: 2.65 145 10 nylon fibres L=3.7mm d=27.9>jm E E 1 — o> r--| I I I I I I I I I I I I I I I I I I '.01 .1 1 Figure 35. Tensile Stress versus Floe Apparent Volume Concentration. versus 2.46. Two published works deal with the strength of individual floes of wood-pulp f i b r e s . Lee [L2,L3] studied floe dispersion in the turbulent shear, and Garner [G2] measured the t e n s i l e strength of air-formed floes of various moisture content. Lee's work which aimed so l e l y at the phenomena of floe dispersion did not lead to any r e l a t i o n s h i p between the floe strength and consistency. Garner's work, which was reviewed in Section 2.3.3, 146 10 CM £ £ 1 nylon fibres L= 2.7 mm d=,27.9;im -jl i i i i t i t i i | I I I I I I I I ".01 .1 1 Figure 37. Tensile Stress versus Floe Apparent Volume Concentration. produced a c o r r e l a t i o n between the bulk density of floes and their t e n s i l e strength. The data for softwood-pulp fi b r e s are included in Figure 38 as l i n e (5). The magnitude of the t e n s i l e strength of dry pulp floes i s below the t e n s i l e strength of Type-C floes ( l i n e s (1)). Garner's floes were formed from highly curled and contorted wood-pulp f i b r e s , whereas the nylon f i b r e s of t h i s study were r e l a t i v e l y s t r a i g h t . The contorted nature of Garner's f i b r e s probably introduced s i g n i f i c a n t Type-B cohesion. This i s suggested by the jagged shape of the f a l l i n g slope of 147 nylon fibres L=3.7 mm d= 19.7 pm • • m • /•. I I I I I I I I I I I I I I I '.01 .1 1 ^va Figure 37. Tensile Stress versus Floe Apparent Volume Concentration. Garner's t e n s i l e load-elongation curves as postulated by Kerekes et a l . [K7], The range of concentrations at which Garner's floes formed i s s u b s t a n t i a l l y lower than that for Type-C nylon f l o e s . These ranges r e f l e c t the basic difference in the process of floe formation. Type-B and Type-C cohesions are mutually exclusive; f i b r e s suitable for Type-B cohesion (short, kinked, contorted) are not l i k e l y to produce Type-C cohesion. Similar observations were made by Jacquelin [J5,J6] concerning fragile and coherent f l o e s . It i s possible that wood-pulp plugs were of CN £ DJ r-to 148 Table XV. Power F i t Parameters of Tensile Strength Data. aT(N/m2) = a"".C v a(%)b"" FIBRE DIAMETER d Mm AVERAGE FIBRE LENGTH L mm POWER FIT PARAMETERS APPARENT VOLUME CONCENTRATION C Va % a"" N/m2 b"" 19.76 3.737 67.35 2.68 4.06-6.86 27.95 2.757 3.718 4.666 14.49 100.1 12.71 2.86 2.23 3.31 6.08-11 .8 5.22-9.85 4.79-8.30 44.19 2.947 4.973 6.261 83.53 23.73 62.21 2.22 2.02 3.23 4.91-12.3 4.96-8.33 3.03-6.62 f r a g i l e nature ( l i n e ( 6 ) ) . Comparison of t e n s i l e strengths between softwood-pulp networks ( l i n e (6)) and Type-C floes ( l i n e (1)) reveals the largest difference in the magnitude of recorded strengths and the slopes of f i t t e d l i n e s : 1.59 (Table VI) versus 2.65 an average of b"" from Table XV. Although the plugs (line(6)) were coherent fibre networks [F1] or inrerlocked networks [C1], the nature of cohesion seems to be fragile [J5]. As described e a r l i e r , the strength of fi b r e networks in suspension has been experimentally determined by various methods and the results d i f f e r widely, c f . Figure 38, over the ranges of concentration and strength measured. The direc t test methods 149 10; CN 10' 10* £ z £ 10' CD c o C/) 101 10l IO'1 Softwood-pulp fibres @©(D© Nylon fibres ©@ © / WRRk = 2g/g p,=1.5g/cm3 p w=0.998g/cm 3 10" 10" 10 -1 10° Figure 38. ^ v a Breaking Stresses versus Suspension Concentration, Tensile of nylon floes (1). Shear of nylon fibr e s l u r r i e s (2); shear (3,4) and t e n s i l e (5,6) of softwood f i b r e networks from Tables III to VII. 150 (l i n e (3)) gave higher strengths than the i n d i r e c t test methods (li n e (4)) over the same range of f i b r e concentration and for the same type of f i b r e s . The spread between l i n e group (1) and l i n e (5) can only be attributed to the type of f i b r e s that form networks because the test methods were d i r e c t . The s i m i l a r i t y of the slopes of these l i n e s (2.65 versus 2.29) i s a sign of a common mechanism of strength buildup in the stressed nylon and wood-pulp f i b r e networks. If t h i s mechanism was purely f r i c t i o n a l , the magnitudes of those strengths would depend on the number of contacts between fi b r e s per unit area of the break zone, the normal forces at the contact points, and the c o e f f i c i e n t of f r i c t i o n . A mathematical model developed for the t e n s i l e strength of Type-C floes i s tested against experimental results in the next section. 4.8 Mathematical Model o f Tensile Strength o f Type-C Floes The mathematical model i s based on the assumption that the strength of Type-C network comes from f r i c t i o n forces developed at the contact points when e l a s t i c a l l y bent f i b r e s press against each other. Development of t h i s model i s given in Appendix XVII. This model predicts that floe strength, o^, w i l l behave according to: °? = I = T f - s i n ^ k c - k d - k f * d - E - f l * 6 m a x - C v a - n c { 1 9 ) The variables in t h i s equation f a l l into three groups. One 2 encompassing factors describing f i b r e properties contains (d/L) 151 which i s proportional to C„_ as indicated by equation (7). " a P F = ^ -E4 (20) L Another group contains network parameters, and i s related to f i b r e and network geometry, e.g., as L/d increases, the number of contact per f i b r e increases at a constant C v a » c f . Figure 27 and 28. This group i s : 5max' Cva' nc ( 2 1> The t h i r d group containing parameters constant for a given type of f i b r e network i s symbolized by P . P = -j-l-sinT-k -k.-k, (22) c 16 ' c d f Now, equation (19) may be concisely written: °T = P c - p f S m a x ' C v a ' n c ( 2 3 ) It appears that the r e l a t i o n s h i p between a T and C v a i s l i n e a r . However, since the parameters, P F, 8__ . and n_ are T. Tiia. X C related to C„_ in some unknown way, the following r e l a t i o n s h i p V a should be expected to hold: a T = D-Cj a (24) where D i s a constant. 152 The results of t e n s i l e tests of th i s investigation (Section 4.7) indicate that x i s 3.23 for fibres of L/d=141.7. The proposed mathematical model was tested against experiment while the exponent of C v a equal 3.4 was kept constant. This is explained further in assumption 5. Constants D and a"" were compared for a l l types of fi b r e tested. The results of th i s test are presented in Table XVI where the constants D and a"" are shown in the l a s t two rows. In this test the following assumptions have been made: 1. The we t - f r i c t i o n c o e f f i c i e n t for 15 denier nylon was assumed to be the same as that for 6 denier nylon. 2. The c o e f f i c i e n t s k Q and k^ were assumed to be at their maxima. 3. Fibres crossing the rupture plane did not change their orientation during floe separation into two parts, i . e . , they were randomly oriented. 4. The maximum f i b r e d e f l e c t i o n was assumed to be no more than one f i b r e diameter. 5. The re l a t i o n s h i p between the number of contact points per fi b r e and the apparent volumetric concentration was assumed to be the same for a l l f i b r e geometries. It was determined on the basis of the experimental data for 15 denier fibres having L/d=141.7. The f i t n c • X * C v a to the data shown in Figure 28 gave: (25) T a b l e XVI. E v a l u a t i o n of the Mathematical Model of T e n s i l e S t r e n g t h . VARIABLE OR CONSTANT 3 DENIER FIBRES 6 DENIER FIBRES 15 DENIER FIBRES L/d 189. 1 98 .65 133 .0 166.9 66.69 112.5 141.7 E, N/m2 *d d, m 1.69x10 9 0.273 19.76x10~ S 1.78x10 9 0. 196 2 7 . 9 5 x 1 0 _ G 1.76X10 9 0.196 44. 19x10" 6 L, m 3.737x10~ 3 2.757x10" 3 3.718x10~ 3 4 .666x10 - 3 2.947x10" 3 4.973x10~ 3 6.261X10" 3 P f, N/m2 3.452X10 6 13.01x10 6 5.303X10 6 2.583x10 S 26.32x10 S 5.477x10 S 2.744x10 S 3 s1n(0.57)/16 k f k c k .=s1n(0.57) d 0.1 0 3 0 7 0.5 0.25 0.540 0.1 0 3 0 7 0.5 0.25 0.540 0.10307 0.5 0. 25 0.540 6.957x10~ 3 6.957X10" 3 6 . 9 5 7 X 1 0 " 3 6 i m max n c Cva(W 19.76x10" 6 0.205 C 2 a 4 2 7 . 9 5 x 1 0 _ G 0.205 C 2 a 4 4 4 . 1 9 x 1 0 _ S 0.205 C 2 a 4 D, N/m2 0.0967 0.520 0.211 0. 103 1 .662 0.346 0. 173 a"", N/m2 27 . 29 8 .06 9.78 27.25 7.69 13 . 57 82 .06 1 54 n c = 2 . 1 2 9 - C v a ( % ) 0 ' 3 5 2 (26) The assumption that equation (26) holds for a l l f i b r e geometries i s the best in the present circumstances. The exponent x in equation (24) i s now 4-0.352+2=3.4 and i s close to the experimental value of 3.23. The predictions of the model turned out to be poor espe c i a l l y for fibres of large aspect r a t i o in which the calculated D's are much lower than the constants a"" obtained from experiments. This was expected because only part of the floe break phenomenon was modelled and many add i t i o n a l assumptions were necessary. Forces r e s i s t i n g f i b r e pullout from the floe can be of f r i c t i o n a l and interweaving nature. Only forces of f r i c t i o n a l nature and only those which result from f i b r e - t o - f i b r e interaction from e l a s t i c f i b r e bending were accounted for in thi s model. The maximum def l e c t i o n (5 ) of max each f i b r e was assumed to be only one f i b r e diameter (assumption 4). In r e a l i t y , f i b r e deflection can be many times larger than f i b r e diameter and extend beyond e l a s t i c deformation of f i b r e . Other e f f e c t s , such as fi b r e reorientation or f i b r e wedging at the crossings with other f i b r e s , were not accounted for ei t h e r . A closer study of the process of floe breakup can only determine the presence and importance of these unaccounted-for e f f e c t s . The last assumption i s the most risky and has probably caused the constant D to decrease with increasing L/d, whereas the trend of experimental parameter a"" is the reverse. More experimental 1 55 work i s needed to determine the relationships between n c and C v a for a wide range of L/d's. The dependence of $ m a x on floe concentration, also unknown, was not included in the model. This analysis, which has taken the knowledge of floe strength forward, shows experimental problems and indicates areas in which further research should be done. The most important finding i s that the f r i c t i o n a l mechanism of load buildup during floe s t r a i n i n g can be tackled from fundamental p r i n c i p l e s . 5 SUMMARY The findings of t h i s study are: 1 . The existence of Type-C cohesion between fibres has been experimentally v e r i f i e d for the f i r s t time. The cohesion between f i b r e s in a 3-dimensional fibr e structures has been shown to a r i s e from e l a s t i c a l l y bent f i b r e s that are prevented from straightening by the presence of other f i b r e s . 2. Type-C coherent fl o e s have been found to form at a well defined concentration for fi b r e s of r e l a t i v e l y uniform length and diameter in a unique r e c i r c u l a t i n g flow in a p a r t i a l l y f i l l e d r otating c y l i n d e r . This concentration was termed the 103 L d 10° 10 5 10 ,-4 10'3 10 -2 10 ,-1 10° Figure 39. Concepts of Fibre Crowding in Suspensions and F i t t e d Lines to the Threshold Concentration Data. 157 "threshold concentration." For the s p e c i f i c conditions of these te s t s , the threshold concentration appeared to be uniquely related to f i b r e geometry. The li n e s shown in Figure 39 correspond to the threshold concentrations for the three diameters of f i b r e s , d 1, d 2, d^. There i s a lower l i m i t of either f i b r e length or fi b r e aspect r a t i o (about L/d=50) below which Type-C cohesion does not occur at any concentration. The shaded area in Figure 39 marks t h i s l i m i t . It was v e r i f i e d that the sediment concentration i s an approximate measure of the threshold concentration. There i s an upper l i m i t of v i s c o s i t y of the suspending l i q u i d above which Type-C cohesion w i l l not form. This l i m i t was found to be close to 0.013 Pa«s for the experimental conditions of this study. The Meyer-Wahren model for predicting the l i m i t i n g concentration from f i b r e length, diameter, and number of contacts per f i b r e predicts the threshold concentration reasonably well when the number of contacts i s assumed to be 4. The model of Miles does not predict the threshold concentration well because i t depends on d/L to the f i r s t power (equation (2)). Both s t a t i s t i c a l models do not account for the ef f e c t of fi b r e diameter. Calculations of the number of fib r e s within a unit volume having the dimension of one f i b r e length indicate that a large number of fibres exist in the v i c i n i t y of one fi b r e at the threshold concentration. Under these conditions, fibres do not rotate freely when the suspension i s sheared. 158 8. A c y c l i c flow that imposes deceleration, acceleration and rotation without subjecting floes to disruptive shear appears to be required for Type-C floe formation. 9. Type-C coherent networks can be produced in flows other than ones having a cessation of vigorous agitation (Wahren et alii, quote on p.23). These networks can form in a moderately agitated suspension with decelerating/accelerating flow. The f i b r e s crowd l o c a l l y in the deceleration zone and interlock e l a s t i c a l l y as a result of compaction of crowded f i b r e s . An i r r e v e r s i b l e f r i c t i o n a l interlocking occurs and makes f i b r e networks strong enough to successfully r e s i s t the dispersing forces existing in the flow. 10. The existing mathematical models do not describe the i s o t r o p i c , 3-D networks well; they overestimate the number of contact points per f i b r e at a given f i b r e concentration. 11. The t e n s i l e strength of individual Type-C floes was greater than any other strength reported in the l i t e r a t u r e for either man-made or wood-pulp fibre networks. The power re l a t i o n s h i p between stress and concentration yielded similar exponents as indicated by the published shear and t e n s i l e studies. A mathematical model of t e n s i l e strength buildup based on f r i c t i o n a l f i b r e - t o - f i b r e interaction indicates that the phenomenon of strength development can be in part a t t r i b u t e d to f r i c t i o n a l forces that develop at f i b r e contact points during floe s t r a i n i n g . 6 RECOMMENDATIONS FOR FURTHER WORK This work has opened a new f i e l d of study for experimental s c i e n t i s t s and theoreticians. There i s room for considerable additional work on fi b r e f l o c c u l a t i o n and Type-C coherence in p a r t i c u l a r . S p e c i f i c a l l y , i t would be useful to es t a b l i s h whether wood-pulp f i b r e s interlock e l a s t i c a l l y and under what conditions. Investigation should start with f i b r e s as close as possible in geometry to nylon f i b r e s , i . e . , sodium c h l o r i t e (NaC102)-cooked fib r e s from softwood species. This pulping process produces stra i g h t , mechanically undamaged f i b r e s . When the f i r s t step has been accomplished, i . e . , floes are formed, the ef f e c t of other variables (Table I) may be studied. Extension of the experimental data on the structure of coherent networks to include a broad range of aspect ra t i o s would be useful. The derivation of the expression for the l i m i t i n g concentration from purely geometrical considerations should be attempted. No less important than the structure of floes i s their strength. A worthwhile e f f o r t would be to study the mechanism of floe breakup to determine the interplay of various factors. An attempt should be made to perfect the te n s i l e and shear tests for the evaluation of the magnitudes of network strength from various types of cohesion, e.g., Type-B and Type-C. NOMENCLATURE Greek l e t t e r s a - dimensionless factor determined by the shape, dimensions, and orientation of suspended p a r t i c l e s , a 1 - c o e f f i c i e n t , |3 - angular coordinate, 7 - average angle between f i b r e axis and the plane disecting a random 3-D f i b r e network, 5 m a x - maximum fi b r e d e f l e c t i o n in bending, 8 - spherical polar coordinate, angle, M - v i s c o s i t y , M Q - v i s c o s i t y of a suspending l i q u i d , ur=u/u0 ~ r e l a t i v e v i s c o s i t y of suspensions of c y l i n d r i c a l rods, rr - r a t i o between circumference and diameter of a c i r c l e , p - density, a - t e n s i l e stress, T - shear stress, ry ~ y i e l d stress in shear, <t>£ ~ dynamic c o e f f i c i e n t of f r i c t i o n . * - spherical polar coordinate, angle, Roman l e t t e r s a - power f i t c o e f f i c i e n t , b - power f i t exponent, d - f i b r e diameter, g - gr a v i t a t i o n a l acceleration=9.80665 m/s , k c - half of the fraction of s l i d i n g contacts, k^ - d i r e c t i o n c o e f f i c i e n t , kj - half of the fraction of fi b r e s crossing the break plane, n c - average number of contact points per f i b r e , r - r a d i a l coordinate, w - fi b r e width, x - exponent in equation (24), C - concentration, - bulk concentration, C m - mass concentration, m ' C v - volumetric concentration, C y m ^ n - l i m i t i n g volumetric concentration, C v a - apparent volumetric concentration, D - constant in equation (24), E - e l a s t i c modulus, G - shear rate, I - bending moment of i n e r t i a , K1 to Kg - parameters in equations (15), (16), and (17), L - f i b r e length, L w - average weighed f i b r e length, - number of f i b r e s , N^ c - number of fib r e s in a cube of side=L, N^ s - number of fibres in a sphere of diameter=L, P c - parameter in equation (22), Pf - parameter in equation (20), PL - plug length, R - internal radius of a cylinder, radius of curvature of f ibres Re - Reynolds number, S - f i b r e s t i f f n e s s in bending, V - volume, Subscripts a - apparent, b - bulk, c - cube, contact point, d - dynamic, cr - c r i t i c a l , fm - oven-dry f i b r e mass, m - mass, min - minimum, s - sphere, v - volumetric, w - water, weighed, y - y i e l d , T - t e n s i l e , Superscripts * - threshold, - average, ' - associated with shear strength of wood-pulp networks, '' - associated with t e n s i l e strength of wood-pulp networks, ''' - associated with t e n s i l e strength of floes made of "dry" wood-pulp f i b r e s , - associated with t e n s i l e strength of Type-C fl o e s . 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C , Ph.D. Thesis, "The Flocculation of Papermaking Fibres", Lawrence College, Appleton, Wisconsin (1938). WOLLWAGE, J . C , Paper Trade J., 108 (12):41 (1939), 108 (13):25 (1939), also, Tappi, 22:578 (1939). WRIST, P.E., "Fundamentals of Papermaking Fibres", Trans. Symp. Cambridge 1957, F. Bolam ed., T.S.B.P. & B.M.A., London, p.485 (1961). WRIST, P.E., in "Surfaces and Coatings Related to Paper and Wood", ed. R.H. Marchessault and C. Skaar, Syracuse University Press (1967). GLOSSARY APPARENT DENSITY i s the r a t i o of apparent mass to apparent volume of a hydrophilic p a r t i c l e . APPARENT MASS i s a sum of masses of a l l materials co n s t i t u t i n g a p a r t i c l e . APPARENT VOLUME is a space occupied by a l l materials constituting a p a r t i c l e . APPARENT VOLUMETRIC CONCENTRATION i s a r a t i o of the apparent volume of p a r t i c l e s to the suspension volume. It i s also a r a t i o of the apparent volume of fib r e s to the Type-C floe volume. COHERENT FLOC i s a denser part of fi b r e network having strength. COHERENT FIBRE NETWORK i s a physically interlocked system of fibr e s that exhibits strength. A coherent floe i s a coherent network, but not every part of a coherent network i s a f l o e , only the denser parts are. CONSISTENCY i s a term commonly used by paper makers with reference to PULP CONSISTENCY or PULP MASS CONCENTRATION. See term PULP CONSISTENCY. FIBRE NETWORK i s a phys i c a l l y interconnected system of fibr e s where forces can be transferred at contact points. FLOC i s a part of suspension having larger mass concentration than the surroundings. 174 FLOCCULATION INDEX i s a r a t i o a/NQ. a i s the spread of the experimental curve r e l a t i v e to the normalized d i s t r i b u t i o n function N=N Q«p(s)•[1-p(s)] in which p(s) i s the p r o b a b i l i t y of a deviation exceeding s when the standard deviation i s unity. N i s the number of pulses per centimeter at a given l e v e l of l i g h t transmission and N Q i s a constant. LIMITING CONCENTRATION i s defined by equation (3). LIMITING VISCOSITY NUMBER i s a term in the equation for reduced v i s c o s i t y of suspensions. Where, M S p / c v -*s fc^e reduced v i s c o s i t y , [u] i s the l i m i t i n g v i s c o s i t y number, a 1 i s a c o e f f i c i e n t , and C y represents volumetric concentration. NETWORK - see FIBRE NETWORK. ORIENTATION FACTOR i s sin 6• sin (24>). It i s a component of dimensionless factor a which i s shown below. a = [(L/d) 2/6(ln(2L/d)-1.80)]•sin 40•sin 2(2L*) The factor a i s in turn a term in the expression for the r e l a t i v e v i s c o s i t y of suspensions of c y l i n d r i c a l p a r t i c l e s , M r « M r = 1+a-Cv Where, L, d, C y are cylinder length, cylinder diameter, volumetric concentration, and 6, # are spherical polar 175 coordinates (angles). NYLON 6-6 - condensation polymer of hexamethylene diamine with adipic acid. (-NH-(CHoOc-NHC-(CH2)c-C-) \\ \\ n 0 0 PULP CONSISTENCY or, more properly "pulp mass concentration" i s defined as the weight in grams of oven dry f i b r e in 100 grams of pulp-water mixture [C8,T6], and reported in %. In t h i s d i s s e r t a t i o n , pulp consistencies are reported as fr a c t i o n s . SUGAR C12 H22°11' m o l e c u l a r weight=342.3, density p=1580.5 kg/m3. THRESHOLD CONCENTRATION i s an average suspension concentration at the onset of coherent floe formation under given flow conditions. In t h i s d i s s e r t a t i o n i t i s always expressed in terms of apparent volumetric concentration. TYPE-C COHESION - f i b r e interlocking by e l a s t i c bending. WRR - Water Retention Ratio i s the r a t i o of mass of the imbibed water to the dry mass of a p a r t i c l e . WRRk - Water Retention Ratio at the "knee" (see Appendix I and I I ) . For nylon i t i s simply a WRR at saturation. For wood-pulp f i b r e s , i t i s a WRR when lumen i s f u l l of water and fi b r e walls are at the saturation point. 176 A p p e n d i x I . W a t e r R e t e n t i o n R a t i o o f Wood P u l p F i b r e s . In the l i t e r a t u r e , the concentration of fi b r e s in water i s frequently given as the r a t i o of the dry fibre mass to the mass or volume of the whole suspension. Though such representation i s precise, i t gives only a fr a c t i o n of the f u l l picture. Knowledge of the volumetric concentration i s also needed to assess the degree of f i b r e crowding or to rel a t e f i b r e dimensions to the number of f i b r e - t o - f i b r e contacts. The results of studies on man-made f i b r e s should only be compared with those for wood-pulp fi b r e s on the basis of volumetric concentration. Determination Figure 1-40. Apparent Volume Concentration versus Mass Concentration for Wood-pulp Fibres. 1 77 of the volumetric concentration i s complicated by the fact that wood pulp f i b r e s are very hydrophilic. Often unknown are the volume of water held in the fi b r e lumen or by the f i b r e walls and the extent to which the fi b r e walls swell. The importance of the amount of water held by fib r e s cannot be overemphasized. This i s i l l u s t r a t e d in Figure 1-40 in which the relationships between mass concentration, C , and apparent volumetric concentration, C v a, are compared for wood-pulp f i b r e s . In the cal c u l a t i o n of the apparent volumetric concentration t y p i c a l Water Retention Ratios (WRRk) were used (see Appendix II for discussion of WRRk and Appendix VII for derivation of the rel a t i o n s h i p ) . It i s apparent that C„_ may vary considerably for any given C m. The most common value of WRRk i s 2 g/g (grams of water per one gram of dry fibre) for softwood-pulps [E5], At thi s WRRk, the value of C y a i s about 2.7 times larger than C^. For hardwood-pulps, WRRk=1.7 g/g and C„_ i s about 2.4 times larger than C m. WRRk can be calculated from the known dimensions of fi b r e s , the density of fi b r e walls and the amount of water absorbed by fib r e walls. The average dimensions of wood fib r e s are reported in several publications [I 3,P1,R10,S3]. The average dimensions for North American species are reproduced in Table I-XVII for softwoods tracheids and hardwood f i b r e s . The reduced dimensions after soda pulping to 60% y i e l d , a t y p i c a l y i e l d , are also given. The c a l c u l a t i o n of WRRk values assumed that a l l fi b r e s are regular tubes of rectangular cross-section having walls of 1500 kg/m density [S4] and that their walls absorb 0.40 g of water 178 Table I-XVII. Average Dimensions of Wood Fibres and Pulp Fibres. SOFTWOOD TRACHEIDS HARDWOOD FIBRES SOURCE L w t L w t mm <um Lim mm Mm Mm [P1] 3.53 36.2 1 .61 [R10] 3.89 1 .20 [13] 3.28 34. 1 3.75 1 .05 19.3 3.62 [13] 2.95 36.6 1 .27 26.6 [S4] 29.2 2.75 13.9 2.23 average 3.41 34.0 3.25 1 .28 19.9 2.92 [S4] 3.37 30.3 2.24 1 .23 17.7 2.00 pulped per 1g of dry wall material [S.I 7]. WRRk values were: 2.0 g/g for softwood-pulp and 1.3 g/g for hardwood-pulp. Although these two values were calculated from average f i b r e dimensions, they are in surp r i s i n g l y good agreement with the values reported in the E l l i s et a l . study [E5]. 179 Appendix I I . Apparent D e n s i t y of Wood Pulp F i b r e s i n Aqueous Suspensions. The selection of a substitute for wood-pulp fi b r e s requires knowledge of a l l f i b r e properties. One of these properties i s the apparent density of wood-pulp fibres in water. This topic has not been adequately addressed in the l i t e r a t u r e . Wood pulp fibres are hydrophilic and must be dealt with as e n t i t i e s having an apparent mass and volume. Their mass i s a sum of the mass of s o l i d material constituting f i b r e walls and the mass of imbibed water which i s defined as the water that can be quantita t i v e l y removed without changing the composition of f i b r e material [S13]. The quantity of imbibed water i s a function of fi b r e morphology and chemical or physical treatment to which fi b r e was exposed. The apparent density of wood pulp f i b r e i s defined as: °water" Vwater + ^wall* Vwall , T T p = == (11 -1 ) a Vwater wall where pwater " d e n s i t y o f water, PvaH ~ density of f i b r e wall s o l i d s , Vwater " v o l u m e o f water, V w a l l ~ v o-'- u m e °f fi b r e wall s o l i d s . Wood fib r e s are tube-like with closed and pointed ends. The space enclosed by the tube walls i s c a l l e d a "lumen." The tube walls are composed of c e l l u l o s e , hemicellulose and l i g n i n . Cellulose forms a skeleton of the f i b r e wall while hemicellulose 180 and l i g n i n f i l l up a space within the c e l l u l o s e f i b r i l matrix. Chemical pulping a l t e r s the chemical composition and physical structure of the f i b r e walls. Lignin and hemicellulose which are gradually removed leave open spaces between f i b r i l l a e [S14,S16]. The spaces can be enlarged or new spaces created through mechanical or chemical treatment. In the wet state, water f i l l s up these spaces and the lumen. From a hydrodynamic point of view, wood pulp f i b r e s are elongated, s o l i d p a r t i c l e s . Imbibed water that constitutes part of the p a r t i c l e is retained within the s o l i d in three d i s t i n c t ways: a) Fibre walls are swollen with water. The mass of water held by the walls, Mw, corresponds to the Fibre Saturation Point (FSP) [S17]. The degree of cooking a f f e c t s the amount of water imbibed by f i b r e walls. The FSP of kraft cooked black spruce was reported to be 0.67 g/g at 92.4% y i e l d and 1.14 g/g at 48.7% y i e l d as determined by porous plate method at the r e l a t i v e vapour pressure of water p/po=0.9975 [S17]. The larger water content of a c e l l wall was attributed to the increase in the wall thickness. Any changes to f i b r e length or f i b r e diameter were undetectable [S16,S15]. As the fibr e wall swelled inward, the space occupied by the lumen diminished [S17]. b) The lumen i s f i l l e d with water. A certain mass of water M^ , depending upon the morphology of the f i b r e s and the state of f i b r e collapse, is held in i t . c) Water mass Mf i s trapped by m i c r o f i b r i l s which develop on the 181 Table II-XVIII. WRR at the "knee" for Various Never-dried Pulps. PULP WRR at the "knee" g/g Unbeaten unbleached softwood kraft pulp 2.1 Unbeaten unbleached hardwood kraft pulp 1.8 Bleached softwood kraft pulp 670 mL CSF 2.0 605 mL CSF 2.2 302 mL CSF 2.4 200 mL CSF 2.4 100 mL CSF 2.5 Bleached hardwood kraft pulp 600 mL CSF 1 .7 300 mL CSF 2.1 external surfaces of f i b r e s p a r t i c u l a r l y after mechanical treatment. Determining the amount of imbibed water, i . e . , the sum of Mw' M l ' a n d M f * s e x t r e m e l y d i f f i c u l t . One r e l a t i v e l y fast method i s removal of water from the pulp pad by centrifugation. However, the i n t e r s t i c e s between the pad fibres form a c a p i l l a r y system that holds an additional mass of water . This mass augments the d i f f i c u l t i e s in interpretation of method's r e s u l t s . The amount of water retained by a pulp pad decreases with increasing applied c e n t r i f u g a l force. Water would f i r s t be forced out of i n t e r - f i b r e spaces and laminae at lower centrifugal force because of their large c a p i l l a r y sizes in comparison with 182 the intra-wall pores. The t r a n s i t i o n from one mechanism of water removal to another i s gradual within a range of ce n t r i f u g a l 2 2 acceleration from 3923 m/s to 29420 m/s [S2], and not precisely defined. The water i s l i k e l y p a rtly confined within p a r t i a l l y dried f i b r e walls and in i n t r a - f i b r e spaces or lumina. There was no well-defined c e n t r i f u g a l force where a point of i n f l e c t i o n might be i d e n t i f i e d . The point of i n f l e c t i o n was c l e a r l y noticeable when a 2 greater range of ce n t r i f u g a l acceleration, from 490 m/s to 2 490000 m/s [E5] was used. This point was defined as an intercept between two straight l i n e s drawn through the data points, and was c a l l e d "a knee in water retention curve" [E5]. "The knee" i s the best possible estimate of the r a t i o of the mass of imbibed water to the mass of dry, s o l i d , f i b r e material. Table II-XVIII shows Water Retention Ratios (WRR) at the "knee" for various never-dried pulps [E5]. WRR's were determined with the following mathematical expression: moist pad weight after centrifuging , / T T o X = oven-dry pad weight " 1 ( I I - 2 ) Knowing that WRRk = Pwater^water,k ( I I _ 3 ) ^wall* wall equation (11 — 1) can be transformed to: 183 Table II-XIX. Apparent Density of Wood Pulp Fibres at Selected WRRk Values. WRRk Pa g/g g/cm3 1.7 1 .152 1.8 1 .145 1.9 1 .140 2.0 1 .134 2.1 1 . 129 2.2 1 . 125 2.3 1.121 2.4 1.116 2.5 1.113 "water' ( 1 + W R R k )  p a (II-4) + WRRk p w a l l With c e l l u l o s e density = 1500 kg/m [S4,S15] and water density = 998 kg/m3 (at 20°C) [C10], values of the apparent density for di f f e r e n t WRRk's were calculated and are shown in Table II-XIX. These values change s l i g h t l y over a range of ty p i c a l WRRk. It should be noted that nylon 6-6 has 1140 kg/m density when dry. Its density at saturation i s s l i g h t l y lower, i . e . , about 1130 kg/m as i t absorbs about 7.4% of water. 1 Thus, nylon 1 Experimental determination of moisture absorption by nylon 6-6 is described in Appendix VI. 1 8 4 fibres and wood pulp fibres of similar shapes and dimensions should, from the hydrodynamic point of view, behave a l ike . 185 Appendix I I I . Estimates of Magnitudes of Various Cohesion Forces. The magnitude of electro-chemical a t t r a c t i v e forces between nylon 6-6 f i b r e s in the concentrated aqueous-sugar solutions i s unknown. It i s not even known whether the mechanism of f i b r e i nteraction involves repulsive double-layer forces, a t t r a c t i v e van der Waals forces, short distance repulsive forces [14] and s t e r i c forces [N1] at the same time. Only a d i r e c t measurement could c l a r i f y t h i s ; however, experimental work hitherto not done and using equipment capable of measuring very small forces [14] would be required. Because such equipment i s unavailable, t h i s study used a simpler approach. The net buoyant force on an individual f i b r e was used as the d r i v i n g force to separate the f o r c i b l y brought-to-contact f i b r e s with the permanently mounted monofilament. This force gave the estimate of the electro-chemical a t t r a c t i o n between f i b r e s and was compared to the forces needed to d e f l e c t (bend) simply supported f i b r e s . The d e f l e c t i o n forces applied at the centre of simply supported f i b r e s , as shown in Figure XVII-64, were calculated and are presented in Table III-XX. The shortest and the longest f i b r e s of each diameter were considered to be supported at their ends. 1 The maximum f i b r e d e f l e c t i o n was assumed to be equal to f i b r e diameter which in a l l cases did not exceed the e l a s t i c deformation regime [M2]. This approach provides a range of d e f l e c t i o n forces for a l l types and lengths of f i b r e s . The values of f i b r e s t i f f n e s s in a wet state were used. 2 1 Fibre dimensions are discussed in Appendix V. 2 See Appendix VIII. 186 Table III-XX. Forces Deflecting Simply Supported Fibres. NYLON FIBRE DIAMETER um FORCE, nN LONG SPAN SHORT SPAN 3 19.76 229 1813 6 27.95 704 1 1630 15 44.19 2847 27299 Nylon f i b r e s , 4.666 mm long and 27.95 um in diameter, were suspended in aqueous-sugar solutions of increasing sugar content. The solutions prepared had 1100, 1110, and 1120 kg/m density. 1 The apparent density of nylon fibres in pure water at 23°C was 3 3 1130 kg/m . 2 The difference of 10 kg/m between f i b r e and l i q u i d densities produced the net buoyancy force of 0.28 nN. Fibres brought into physical contact with the monofilament displayed no permanent stickiness in a l l three solutions. The conclusion i s that the forces of a t t r a c t i o n must be smaller than 0.28 nN. Clearly the forces presented in Table III-XX are at least three orders of magnitude larger than the upper l i m i t placed on the electro-chemical a t t r a c t i o n . If e l a s t i c f i b r e deflections in Type-C networks are about one f i b r e diameter, the normal forces at contact points are thousand times larger than the a t t r a c t i v e c o l l o i d a l forces. 1 Solution preparation i s described in Appendix XI. 2 See Appendix I I . 187 A p p e n d i x I V . F i l a m e n t C u t t i n g P r o c e d u r e . The fifteen-denier nylon 6-6 was obtained as f i b r e s . The s i x - and three-denier nylon 6-6 was provided by DuPont Canada in the form of multifilaments. These multifilaments had to be cut for f i b r e s to be produced. A cutting procedure which produced f i b r e s of r e l a t i v e l y uniform length was invented. A sheet of A7 size millimeter graph-paper was folded in half with the l i n e pattern on the outside. Two adhesive tabs were attached to the bottom part of the folded sheet at i t s opposite free edges so that half of the adhesive surfaces were exposed. The exposed surfaces coincided with the blank side of the folded Figure IV-41. Multifilament Preparation for Cutting. 188 sheet. A nylon multifilament l a i d across the piece of paper was secured to the adhesive tabs, as shown in Figure IV-41. The top side of the folded sheet covered the filaments and was pressed down by a f l a t piece of metal. The paper and sandwiched nylon multifilaments were cut on a hand-operated g u i l l o t i n e . The printed millimeter gri d helped alig n the sandwich and f a c i l i t a t e d even spacing between cuts. The shearing action of the g u i l l o t i n e and the top pressure on the sandwich kept the multifilaments stretched. 189 A p p e n d i x V . M e a s u r e m e n t o f F i b r e L e n g t h a n d C u r v a t u r e . Fibre samples were enclosed in slide-mounts for f i b r e length and curvature determination. F i r s t , f i b r e s were deposited on the wetted surface of the bottom glass window. For assurance that fibr e deposited f l a t on the surface and could be e a s i l y recognized, the number of f i b r e s per slide-mount was l i m i t e d . Second, the top window was placed over the bottom window. The glass-windows spaced about 0.4 mm apart enclosed a volume t o t a l l y f i l l e d with water. Two p a r a l l e l black l i n e s on the bottom glass window served as c a l i b r a t i o n marks. The distance between these marks had been determined by a traversing microscope and was Figure V-42. Sample of Fibres deposited on the Slide Mount and Closed in I t . 190 employed as a c a l i b r a t i o n constant. A photograph of an open'and closed slide-mount i s shown in Figure V-42. Each slide-mount was i n s t a l l e d in-the slide-projector, and the images of fi b r e s themselves were projected onto the d i g i t i z i n g pad. Magnification of about twelve times was achieved by the 24x36 mm frame being projected on the 300x450 mm active area of the pad. The shortest f i b r e s had about 10 mm of projected length and the longest about 75 mm. The resolution of the d i g i t i z e r which was set to 0.1 mm led to a maximum ±2% error for straight f i b r e s . Other sources of error arose from the fuzzy l i n e s of f i b r e ends and fi b r e curvature. The maximum ov e r a l l 1 9 1 error was estimated to be less than ±3%. Fibre length was approximated by the sum of s t r a i g h t - l i n e segments. The number of such segments per f i b r e , a r b i t r a r i l y chosen, depended on f i b r e length and f i b r e curvature. More curved f i b r e s needed more segments. Figure V-43 i l l u s t r a t e s the measuring p r i n c i p l e . The projected f i b r e image was delimited by two p a r a l l e l l i n e s on a millimeter graph paper. These l i n e s touched the f i b r e ends and established the f i r s t and the l a s t node. The millimeter paper f a c i l i t a t e d d i v i s i o n of the f i b r e image into sections of almost equal length. Thicker l i n e s on the millimeter paper, such as the one-centimeter or half-centimeter l i n e , crossed the f i b r e contour establishing additional nodes. The sum of straight l i n e distances between the nodes constituted an approximation of a projected f i b r e length. Each pair of straight l i n e segments was also employed in calculations of a l o c a l radius of curvature. These segments were a n a l y t i c a l l y halved. Straight l i n e s normal to these segments were a n a l y t i c a l l y constructed at the mid-points of each dis s e c t i o n . These li n e s intercepted i f the f i b r e was curved, as shown in Figure V-43. The distance between the intercept point and the common point of neighboring segments was taken as the l o c a l radius of curvature. Each f i b r e had several l o c a l r a d i i from which an average radius of curvature was calculated. The length and curvature data were stored on a floppy-disk in a d i g i t a l form. This d i g i t a l information was transferred to the main-frame computer for a n a l y s i s . The photograph of data 192 Figure V-44. Data Acquisition System for Fibre Length and Fibre Curvature Measurements. acq u i s i t i o n hardware i s shown in Figure V-44. The major components were: the IBM Personal Computer (8088 microprocessor) (A), the d i g i t i z e r (Summagraphics Microgrid, model MG1218) (B), and the slide-projector (C). 1 93 The data a c q u i s i t i o n program written in BASIC i s l i s t e d in Appendix XIX. The additional software packages that enabled data processing and transfer were: PC-DOS version 2 . 1 , Basic Interpreter version 1 . 0 , Michigan Communication Protocol (to transfer f i l e s between the UBC-MTS and IBM-PC), and *FREQ ( s t a t i s t i c a l program available from UBC-MTS l i b r a r y ) . The results of s t a t i s t i c a l analysis in terms of f i r s t four moments are presented in Table V-XXI and Table V-XXII. The c o e f f i c i e n t of skewness i s a measure of asymmetry of d i s t r i b u t i o n . If the c o e f f i c i e n t of skewness i s p o s i t i v e , an excess of po s i t i v e deviations from the mean i s indicated. In such a case, the d i s t r i b u t i o n i s said to be " p o s i t i v e l y skewed." If the c o e f f i c i e n t of skewness i s negative, the d i s t r i b u t i o n i s said to be "negatively skewed." The c o e f f i c i e n t of skewness i s zero when the d i s t r i b u t i o n i s symmetrical. The c o e f f i c i e n t s of skewness of fi b r e length d i s t r i b u t i o n s which are p o s i t i v e and negative indicate the excess of either longer or shorter than average f i b r e s in each sample. The c o e f f i c i e n t s of skewness of f i b r e curvature d i s t r i b u t i o n s , a l l p o s i t i v e , indicate excess of straight f i b r e s . Kurtosis refers to peakedness of a d i s t r i b u t i o n . If the d i s t r i b u t i o n i s very peaked and has r e l a t i v e l y wide t a i l s , i t i s referred to as "lep t o k u r t i c . " The c o e f f i c i e n t of kurtosis i s po s i t i v e . If the d i s t r i b u t i o n i s rather f l a t in the middle and has r e l a t i v e l y thin t a i l s , i t i s c a l l e d " p l a t y k u r t i c . " The c o e f f i c i e n t of kurtosis i s negative. For a Gaussian Table V-XXI. Nylon Fibres 1n Wet State. Fibre Length S t a t i s t i c s . TYPE OF FIBRE DIAMETER d SAMPLE SIZE FIBRE LENGTH ASPECT RATIO L/d SKEWNESS COEFFICIENT KURTOSIS COEFFICIENT AVERAGE LENGTH L STANDARD DEVIATION denier* mm mm 3 19.76 1062 1054 1231 1117 0.9158 1 .875 2.815 3.737 0.08484 0.09244 0.1019 0.1122 46 . 35 94.88 142 . 5 189 . 1 -0.06787 -1 .564 5.268 5.572 1 . 298 67 . 15 -297 . 2 -552 .9 6 27.95 1237 1 138 1042 1049 1024 0.9139 1 .832 2.757 3.718 4.666 0.08322 0.09898 0.1267 0.1576 0.1338 32.70 65.55 98.65 133.0 166.9 0.02546 -1.279 1 .642 2.354 -4.929 1 .926 49.87 -66.22 -138.9 403. 1 15 44 . 19 1038 1162 1001 1006 1 .560 2.947 4 .973 6.261 0.1009 0.1081 0. 1762 0. 1769 35.30 66.69 112.5 141 .7 -0.3115 4.566 -3.211 -13.38 1 .920 -249.1 235.7 729 .9 * Denier 1s a weight In grams of 9000 meters long monofilament. Table V-XXII. Nylon Fibres 1n Wet State. Curvature S t a t i s t i c s . TYPE OF FIBRE DIAMETER d SAMPLE SIZE ' FIBRE CURVATURE SKEWNESS COEFFICIENT KURTOSIS COEFFICIENT AVERAGE 1/R ** STANDARD DEVIATION denier* i/fn mm" 1 mm - 1 3 19.76 1062 1054 1231 1117 0.3094 0.25533 0.281 14 0.28547 0.21323 0. 17189 0. 19328 0.18807 0.7796 0.7765 0.8155 0.7543 0.3561 0.4457 0.4324 0.3127 6 27.95 1237 1 138 1042 1049 1024 0.16965 0.15223 0.16181 0.12968 0.13701 0.14534 0. 11 126 0.11952 0.10034 0.10336 1 .850 1 . 124 1 .250 1 .476 1 .259 7 . 250 1 .742 2.397 3 .660 2 .088 15 44 . 19 1038 1 162 1001 1006 0.08366 0.07240 0.07107 0.06060 0.08033 0.06239 0.05731 0.05105 3 . 384 3.289 1 .730 1 .983 25.30 27.45 4.297 7 .636 * Denier 1s a weight 1n grams of 9000 meters long monofilament. ** R denots a radius of curvature. 196 d i s t r i b u t i o n , the c o e f f i c i e n t of kurtosis i s zero. The d i s t r i b u t i o n s of f i b r e length are both lepto- and p l a t y k u r t i c whereas the d i s t r i b u t i o n s of f i b r e curvature are always l e p t o k u r t i c . The c o e f f i c i e n t s of skewness and kurtosis indicate that length and curvature d i s t r i b u t i o n s cannot be considered as Gaussian [B7], The standard deviations of the means of f i b r e length and curvature indicate very narrow d i s t r i b u t i o n s of both. Low f i b r e curvatures and narrow length d i s t r i b u t i o n s j u s t i f y the use of these f i b r e s in testing the s t a t i s t i c a l theories developed by Miles [C5] and Meyer and Wahren [M10]. Fibre diameters were calculated from the information on weight per unit length of multifilament, the number of monofilaments, and the material density. The diameters were corrected for water swelling e f f e c t under the assumption that fi b r e s swell uniformly in every d i r e c t i o n . Some evidence suggests t h i s i s not so. The nylon f i b r e s swelled more l a t e r a l l y (4.8%) than l o n g i t u d i n a l l y (1.2%), increasing f i b r e volume by 11.1% [S10]. If uniform swelling upon absorption of 7.4%1 of water i s assumed, the diameter increase of 2.4% i s calculated. Such a small discrepancy in the diameter estimate (2.4% vs. 3.2%) did not j u s t i f y further extensive experimental investigation. Hence, the 2.4% increase was used in calculations of wet f i b r e diameters. 1 Water absorption by nylon 6-6 i s described in Appendix VI. 197 Appendix VI. Water A b s o r p t i o n by Nylon F i b r e s . A l l three types of nylon were tested for water absorption. Figure VI-45 shows a photograph of the experimental setup. The oven-dry sample, weighing few grams, was placed in 98% r e l a t i v e humidity (RH) environment and i t s weight increase recorded. A r e l a t i v e humidity of 98% was created in the desiccator (A) containing a large amount of saturated aqueous solution of cupric sulphate (CuSC>4«5H20) (B) [C10]. The nylon sample (C) was placed in the glass dish (D) located immediately above the cupric sulphate. The dish was suspended in the wire-basket (E) hooked on to the AE183 Mettler balance (F) from underneath. The Figure VI-45. Experimental Setup for Water Absorption Fibres. by Nylon 198 suspending wire through a small hole in the desiccator 1 l i d reached the balance's hook. The balance was e l e c t r i c a l l y connected to the IBM-PC (G) through an Option-012 data interface. Control over data a c q u i s i t i o n was made possible through a BASIC program which i s l i s t e d in Appendix XX. Every increase in weight equal to or s l i g h t l y larger than 0.001 g was stored in a data f i l e along with the TIMER readings in seconds. The data were transferred to the UBC-MTS system through a modem for analysis and p l o t t i n g . The water absorption data for 15 denier 10 8 V-g m £ 6 §5 I-Z LU A O o v • t=23°C, 98%RH, 15 denier nylon fibres • • • • • • • • 8 _L X 16 24 32 . TIME, h J L 40 48 Figure VI-46. Water Absorption by Nylon Fibres in 98% RH Environment. 199 nylon are shown in Figure VI-46 in which only every tenth experimental point is plotted. Similar curves were obtained for 6 and 3 denier nylon f i b r e s . It can be seen that, after 48 hours of exposure, the water content l e v e l l e d o f f . Longer periods of exposure were also checked. Table VI-XXIII shows the weights of samples after 48, 72, and 96 hours of exposure. Variations in ambient temperature were small. These data indicate that no further water absorption took place beyond a 48 hour period. The average water content for 15 samples and three time periods was accepted as the representative water to nylon mass r a t i o at saturation. This r a t i o can be substituted as WRRk in the calculations of mass and volume concentrations. The water content of nylon material at 100% RH i s s l i g h t l y greater [S10], but the water content at 98% RH s a t i s f a c t o r i l y approximates i t . An advantage of experiments conducted at 98% RH i s that water condensation due to ambient temperature fluctuations i s avoided. Table VI-XXIII. Water Absorbtlon by Nylon Fibres 1n 98% Relative Humidity Environment. TYPE OF NYLON SAMPLE NUMBER SAMPLE WEIGHT MOISTURE RETENTION IN NYLON FILAMENTS OVEN -DRY EXPOSED TO 98% R.H. ENVIRONMENT AFTER 48 HOURS T=22.7°C AFTER' 72 HOURS T=21.7°C AFTER 96 HOURS T=21.8°C AVERAGE STANDARD DEVIATION denier* H g g g g g/g g/g 3 1 2 3 4 5 1.1189 1.5896 1.6870 1.7736 1.9166 1 .3172 1.7052 1.8098 1.9027 2.0546 1.3187 1.7073 1.8103 1.9042 2.0572 1.3158 1.7046 1.8077 1.9002 2.0548 0.07250 0.002423 6 6 7 8 9 10 2.9235 3.0768 3.3618 3.4039 3.5055 3.1415 3.3015 3.6120 3.6526 3.7649 3. 1352 3.2956 3.6086 3.6515 3.7649 3.1393 3.3007 3.6076 3.6540 3.7630 0.07329 0.002315 15 11 12 13 14 15 1.0274 1.7767 1.7897 1.9118 1.9887 1.1053 1.9138 1.9320 2.0621 2.1470 1.1053 1.9197 1.9251 2.0535 2.1423 1.1047 1.9122 1.9256 2.0564 2.1403 0.07689 0.004941 FOR ALL OBSERVATIONS 0.07423** 0.003478 * Denier 1s a weight In grams of 9000 meters long monofilament. ** This average value was accepted as WRRk in calc u l a t i o n s of mass or volume concentrations. 2 0 1 Appendix VII. Relationship Between Mass and Volume Concentration of Hydrophilic P a r t i c l e s Suspended in Aqueous-Solute Solutions. The majority of the experimental works in the l i t e r a t u r e contains findings reported in terms of suspension mass concentration or as a r a t i o of oven-dry f i b r e mass to the suspension volume. This i s so because the oven-dry f i b r e mass i s ea s i l y determined with commonly available equipment - an oven and a balance. Using the available information from the l i t e r a t u r e requires recalculation of mass concentration or mass content into volume concentration. The coherence and structure of f i b r e networks are c l o s e l y related to the s p a t i a l arrangement of fib r e s and the volume they occupy. Thus, comparison of experimental results or interpretation of th e o r e t i c a l calculations requires recalculation of mass concentrations into volume concentrations or vice versa. Such a task i s frequently repeated and, i f the necessary relationships are not readily available, becomes an unnecessary burden. If a l l possible relationships for hydrophilic p a r t i c l e s are derived from the following basic parameters compiled in th i s Appendix, the task i s avoided: Vp - moisture-free p a r t i c l e volume ( s o l i d substance), V w - volume of water absorbed by p a r t i c l e s , V^ - volume of suspending l i q u i d (aqueous-solvent solution), Vp+Vw - apparent p a r t i c l e volume, Pp - moisture-free p a r t i c l e density ( s o l i d substance), P w - water density, p^ - suspending l i q u i d density, C - mass concentration, m 202 C m a - apparent mass concentration, C v - volume concentration, C v a - apparent volume concentration, Wood-pulp fibres and nylon f i b r e s are hydrophilic. A wood-pulp f i b r e absorbs water into i t s c e l l u l o s i c wall and imbibes water into i t s lumen. A nylon f i b r e retains water solely by absorption. Regardless of the mechanism of absorption, both f i b r e s have a certain mass of water in them at saturation. A concept of Water Retention Ratio at saturation (WRRk) introduced in Appendix II can be used for any hydrophilic material. WRRk = P v y W k (VII - 1 ) P* P V «p r = E 2 (VII-2) Si, V -p +V . «p +V, -p, l v u ^  p 'p wk *w 1 K l V •p +V ,•p P P W k W (VII -3 ) 'ma V • p +V . • p +V. • p. p *p wk *w 1 ^1 c v = v + v P , + V l ( V I I " 4 ) p wk 1 V Vwk r = E (VII-5) u v a V +V , +V, v v i i o; p wk 1 The expression (VII - 1) introduced to equations (VII - 2 , V I I - 3 , V I I - 4 , VII - 5 ) yields the following relationships: 203 1 C = - — - — (V I I - 6 ) m l * p l 1 + WRRk + V «p P P 1 Cma = ( V ^ P ^ / t V p ' P ^ T ( V I I - 7 ) 1 + 1 + WRRk C v " p V — ( V I I " 8 ) 1 + _2.WRRk + ^f-»w V p C v a - V77V (VII"9) i + 1 P  1 •+ (P p/P w)'WRRk T a b l e V I I - X X I V p r o v i d e s r e l a t i o n s h i p s between mass and volume c o n c e n t r a t i o n s d e r i v e d from e q u a t i o n s ( V I I - 6 ) , ( V I I - 7 ) , ( V I I - 8 ) and ( V I I - 9 ) . In the case of n y l o n f i b r e s , the WRRk s h o u l d be r e p l a c e d by water t o n y l o n mass r a t i o a t the s a t u r a t i o n p o i n t . Appendix VI c o n t a i n s a d e s c r i p t i o n of the e m p i r i c a l e v a l u a t i o n of water a b s o r p t i o n by n y l o n f i b r e s . Table VII-XXIV. Relationships Between Mass and Volume Concentration of Hygrophlllc P a r t i c l e s in Suspensions. C = m C = ma V C v a = Cm ( 1+WRRk)C m 1 PP 1+—WRRk pp pp pp 1 1 + ( )WRRk+—( 1 ) pp "p pp 1 1+WRRM )+—( 1 ) pu p \ p \ Cm pw "1 p\ Cm C ma C ma 1 PP 1+—WRRk 'w 1+WRRk 1 p \ pp ( 1 )( 1+WRRk)+—( 1+—WRRk) Cma pp pu pp pp 1 pp pp 1+WRRk( )+ ( 1+WRRk) -°w p1 Cma "} p : 1 1 p\ 1 p\ 1+WRRk( 1 ) + ( 1 ) — pv C v pp 1 p] pp (1+—WRRk) C v pp p\l 1+ 1+WRRk C v ( 1+—WRRk)C pw 1 1 C v a 1 p\ p\ 1+WRRk+( 1)(—+—WRRk) p\ 1 pp — ( 1 )( 1+—WRRk) fiP ( 1+—WRRk) C va C v a pp pw pp C v a w^ 1+ 1+WRRk 'w 205 Appendix VIII. Measurement of the E l a s t i c Moduli of Fibres. Two methods of measurement were considered: f i b r e bending in a cross flow of water and t e n s i l e testing of wet filaments. The bending method has been used in studies of wood-pulp fi b r e f l e x i b i l i t y in the PAPRICAN laboratory at the UBC. The apparatus was readily a v a i l a b l e . The experimental technique and th e o r e t i c a l background are described elsewhere [T1,T2], This test method and computational method were applied without change. It was found that the calculated e l a s t i c modulus of a given nylon f i b r e varied nonlinearly with the degree of f i b r e d e f l e c t i o n . Accordingly, the computational part of this method was judged to be unreliable and was discarded. The experimentally determined deflections were kept and used to evaluate the r a t i o between e l a s t i c moduli. The t e n s i l e method, probably the most popular, r e l i e s on the load-elongation character of a material in i t s e l a s t i c regime. This method has been standardized in material testing [A12,A14]. Nylon 6-6 filaments were heat treated at 105°C for four hours and soaked in water for 48 hours before being tested. Groups of filaments or a single filament were elongated at a constant rate of 10.5 mm/min in the THWING/ALBERT t e n s i l e t e s t e r . However, the tester clamps were too coarse for adequate clamping of filaments, and therefore the tests were conducted with two d i s t i n c t l y d i f f e r e n t span lengths, one twice as long as the other. The rationale for such change which s t a r t s with the equation defining the e l a s t i c modulus i s : 206 p = 1 = F/A = F»L e AL7L A-AL where E - e l a s t i c modulus, a - stress corresponding to s t r a i n e, e - s t r a i n within e l a s t i c regime, A - cross-section area of a test sample, F = (T'A - force, L - test span, distance between clamps, AL - elongation under applied force F, Equation (VIII-l) can be transformed to: (VIII-1) AL = F-L A • E (VIII-2) Under imperfect clamping conditions, the observed elongation i s d i f f e r e n t from that expressed by equation (VIII-2) by the amount of s l i p , 8. AL' = AL + 6 (VIII-3) If experimentally obtained AL' was used in the cal c u l a t i o n of the e l a s t i c modulus, i t would result in an underestimate. Such an error can e a s i l y be eliminated by the testing of two sets of samples, one having substantially d i f f e r e n t test span from the other. In practice i t i s convenient for one test span to be twice the length of the other. The apparent elongations would then be: AL = AL + 8 = S A«E (VIII-4) for the short-span samples, and AL X = 2-AL + 6 = Jf|^ (VIII-5) for the long-span samples. The amount of s l i p should be i d e n t i c a l in both experiments, and the difference between apparent elongations y i e l d s the true elongation for short-span samples. elongation elongation Figure VIII-47. Typical Load-elongation Curves for 3 Denier Nylon Filaments in Wet and Dry State. 208 AL = AL X - AL g (VI11-6) The apparent elongations were recorded on charts along with the applied loads. Typical load-elongation curves for 3 denier nylon filaments are shown in Figure VIII-47. The d i f f e r e n t slopes of the i n i t i a l parts of load-elongation curves for wet and dry filaments are c l e a r l y distinguishable. The e l a s t i c modulus of 15 denier nylon was not measured because i t was pre-cut into f i b r e s too short to be clamped. The 3 and 6 denier filaments were tested. Table VIII-XXV shows the test conditions and the e l a s t i c moduli. Cle a r l y , air-dry filaments exhibit about three times higher e l a s t i c moduli than heat-treated, saturated-with-water filaments. These empirical findings agree with those c i t e d in the l i t e r a t u r e [S10,M9]. For example, at 50%RH the e l a s t i c modulus was reported to be 3.12-109 N/m2, and at 100%RH i t was only 1.1V•109 N/m2 [M9], Since 15 denier nylon could not be t e n s i l e tested, the results of f i b r e flexing were used to evaluate i t s e l a s t i c modulus. The de f l e c t i o n , y„,„„, of the fi b r e beam and the flow •'max rate of water through the glass c a p i l l a r y tube, Q, had been measured. P l o t t i n g the defle c t i o n and the bulk Reynolds number established a deflection r a t i o between 6 and 15 denier f i b r e s from which the ra t i o of e l a s t i c moduli could be obtained. When this r a t i o was known, the e l a s t i c modulus for 15 denier, heat-treated, wet nylon could be calculated from the e l a s t i c 209 T a b l e VIII-XXV. E l a s t i c M o d u l i of 3 and 6 D e n i e r N y l o n F i l a m e n t s . ELASTIC MODULUS N/m2 TEST CONDITIONS 3 d e n i e r 6 d e n i e r 4.95-10 9 5.45.10 9 23°C 22% RH 1.69.10 9 1.78-10 9 4 hours a t 105°C r e w e t t e d f o r 48h modulus of 6 d e n i e r n y l o n . The geometry of a bent f i b r e i s shown i n F i g u r e V I I I - 4 8 w i t h the shape of the v e l o c i t y p r o f i l e of the f l o w c a u s i n g the d e f l e c t i o n [ T 1 ] . The water f l o w r a t e i s r e l a t e d t o the v e l o c i t y d i s t r i b u t i o n as f o l l o w s : dQ = 2.7r«r.V(r) «dr ( V I I I - 7 ) I n t e g r a t i o n and d i v i s i o n of ( V I I I - 7 ) by the c r o s s - s e c t i o n a l a r e a of the g l a s s tube y i e l d s t h e b u l k v e l o c i t y , V^. 7 c/2 V, = — - — W V ( r ) . r - d r ( V I I I - 8 ) b ( c / 2 r 0 The b u l k R e y n o l d s number based on f i b r e d i a m e t e r i s d e f i n e d as p«V «d c/2 Re, = - — = — - — R e ( r ) « r . d r ( V I I I - 9 ) b M ( c / 2 ) 2 0 The maximum f i b r e d e f l e c t i o n i s governed by the e q u a t i o n 2 1 0 Figure V I I I - 4 8 . Simply supported Fibre deflects in a Cross-flow. Figure taken from [ T 1 ] . W = THTT ( V I I I - I O of the small-deflection beam theory. Where, c - beam span, i . e . , c a p i l l a r y tube diameter, k - constant depending on the type of beam support and the shape 21 1 of the v e l o c i t y d i s t r i b u t i o n across the c a p i l l a r y tube, F - the resultant force from the hydrodynamic drag which depends on the v e l o c i t y d i s t r i b u t i o n along f i b r e beam: c/2 F = C_.-p.d-/ V^(r)-dr (VIII-11) u 0 It can also be expressed in terms of Reynolds number: 2 c/2 F = r—K'f Re^(r)-dr (VIII-12) D p-d JQ Since the v e l o c i t y d i s t r i b u t i o n p r o f i l e s behind the f i b r e beam do not change over the range of Reynolds numbers used [T1], drag forces expressed in terms of Reynolds number can be compared. Substitution of (VIII-12) into (VIII-10) y i e l d s a relationship between the f i b r e d e f l e c t i o n and the Reynolds number: 6 4 - C n . M 2 - l 3 c/2 9 y-E = - ^- -J Re^(r)-dr (VIII-13) 7r- k- p-d 0 When Re(r) d i s t r i b u t i o n s for two d i f f e r e n t f i b r e s are i d e n t i c a l , the drag c o e f f i c i e n t s are equal and the r a t i o of y-E between 6 and 15 denier fibres depends solely on f i b r e geometry: (VII1-14) The same holds i f the bulk Reynolds numbers are i d e n t i c a l , c f . equation (VIII-9). The plot of f i b r e d e f l e c t i o n versus bulk Reynolds number i s shown in Figure VIII-49. S o l i d l i n e s 212 Figure VIII-49. Fibre Deflection versus Bulk Reynolds Number. represent the power curve f i t s of the form, y = a « ( R e b ) b (VIII-15) Sa t i s f a c t o r y agreement between the theory (equation (VIII-14)) and experiment exists (Appendix XXI, point f ) . The desired r a t i o of f i b r e deflections can be calculated from 213 U - ' "« f b 2 R e ' b 6 - b , 5 ) d R e K - ( V I I I - 1 6 ) 1 5 R eb2 + R eb1 a l 5 R eb1 for a chosen range of the bulk Reynolds numbers. With Rel:>1=20. and Rej 3 2 = 50. for the integral l i m i t s , the defl e c t i o n r a t i o i s 9.78. The e l a s t i c modulus for 15 denier, wet nylon f i b r e can be now calculated from the following expression. 5 E 1 C = E c • — ^ — • ( V I I I - 1 7 ) v6 r de '15 = *6' v15 V i s -i t s value i s 1.76-109 N/m2. A l l e l a s t i c moduli and s t i f f n e s s for wet nylon f i b r e s are shown in Table VIII-XXVI. Table VIII-XXVI. E l a s t i c Moduli and St i f f n e s s of Wet Nylon Fibres. NYLON denier DIAMETER Mm ELASTIC MODULUS N/m2 STIFFNESS N«m 2 3 19.76 1.69-109 12.6•10~ 1 2 6 27.95 1.78-109 53 . 3•10 _ 1 2 15 44.19 1.76-109 329.4-10 - 1 2 214 The FORTRAN program that processes raw data, c a l l s NL2SOL curve f i t t i n g subroutine, and plots the r e s u l t s , i s l i s t e d in Appendix XXI. The l i s t i n g of de f l e c t i o n data, and c a l i b r a t i o n f i t s for GILMONT flowmeter and micrometer eye-piece are also given there. 215 Appendix IX. F r i c t i o n Between Wet Fibre Surfaces. An i n c l i n e d plane method [A15,T4] was used to determine the f r i c t i o n c o e f f i c i e n t s between wet nylon f i b r e surfaces. Both c o e f f i c i e n t s , s t a t i c and dynamic, have been measured. Figure IX-50 i l l u s t r a t e s the experimental p r i n c i p l e . The a c r y l i c tank (A) f i l l e d with an aqueous sugar solution housed the i n c l i n e d plane (B) and the sled (C). The sled and the plane were covered with .tightly wound multifilaments of nylon (D). One end of the plane was pulled up by the st r i n g (E) at the constant speed of 10 mm/min. At the moment when the sled started to s l i d e down the slope, the p u l l was halted and the angle a Figure IX-50. Method of F r i c t i o n Measurement. 216 measured. The tangent of the s l i p angle, a, which i s the s t a t i c c o e f f i c i e n t of f r i c t i o n , <p , was calculated. <j> = T/N = tan(a) (XXI-1) This procedure has been repeated with various weights (F) placed on the sl e d . Figures IX-51 and IX-52 show plots of 3 denier filament O g A. A A E " J 1 ± A A -334 A _ £ u_ A A A S-3 : „ O J i E O ^ . 273 Z o 0.2 CC u. LU 0 J I L 0 50 100 WEIGHT, g Figure IX-51. Wet-friction C o e f f i c i e n t versus Sled Loading for 3 Denier Nylon Filaments. 217 f r i c t i o n c o e f f i c i e n t s versus sled loading. In both figures the lower groups of points represent the data for the dynamic c o e f f i c i e n t of f r i c t i o n <p^. The experimental setup for i t s measurement was s l i g h t l y d i f f e r e n t from that for the s t a t i c c o e f f i c i e n t of f r i c t i o n . A s t r i n g (H) attached to the front of the sled went around wire (W) to be connected to the constant speed drawing mechanism. During an experiment, both strings (E) and (H), were pulled simultaneously with a speed of 10 mm/min. UJ O LL LL O CC LLI £ . 1 .196 50 100 WEIGHT, g Figure IX-52. Wet-friction C o e f f i c i e n t s versus Sled Loading for 6 Denier Nylon Filaments. 218 Table IX-XXVII. S t a t i c and Dynamic C o e f f i c i e n t s of F r i c t i o n for Wet Nylon Fibres. NYLON FRICTION COEFFICIENT FILAMENT STATIC DYNAMIC AVERAGE S.D. AVERAGE S.D. 3 denier 0.334 0.0318 0.273 0.0267 6 denier 0.308 0.0256 0. 196 0.0090 Thus, the sled was pulled down the slope with a speed of about 10 mm/min. At the moment when the sled started to s l i d e down the plane by i t s e l f , the draw mechanism was stopped and the tangent of a calculated. The average values and the standard deviations of f r i c t i o n c o e f f i c i e n t s are shown in Table IX-XXVII. These values are comparable with the dry c o e f f i c i e n t s of f r i c t i o n reported elsewhere [H2,K3], In p r i n c i p l e the i n c l i n e plane method i s independent of the magnitude of the load placed on the sled, but in practice the compressibility of wound filaments affected the f r i c t i o n angle e s p e c i a l l y at the sled loadings of less than ten grams. These low loadings were excluded from data analysis. In a l l measurements, the ef f e c t of viscous drag on the sled can be neglected because i t was calculated to be six orders of magnitude smaller than the smallest sled loading, i . e . , 10 g. 219 Appendix X. Rotating Cylinder Apparatus. A r e l a t i v e l y simple mechanism, shown in Figure X-53, was devised to r o l l cylinders f i l l e d with suspensions. It consisted of an i n c l i n i n g table (A) with r o l l e r s (B,C) on stand (S). The table could be in c l i n e d at 0°, 15°, 30°, 45°, and 60° to the horizontal, and secured in position by the threaded pin (N). The r o l l e r s were grooved to accommodate "0"-rings (H) through which propelling torque was transmitted to the cylinder (D). The torque was applied from an e l e c t r i c motor (F) connected to one r o l l e r through the f l e x i b l e shaft ( J ) . The e l e c t r i c motor was driven by the so l i d - s t a t e power co n t r o l l e r (K) (COLE-PARMER INST. Co.) giving a 6 to 600 rpm speed range. The ro t a t i o n a l speed was Figure X-53. Rotating Cylinder Apparatus. 220 monitored by the d i g i t a l frequency meter (DARCY/TSI, model 460) (M) and a photo-reflective sensing pickup (model 863) consisting of a l i g h t source and a photodetector probe (G). The l i g h t pulses were picked up d i r e c t l y from the r o l l e r , transformed into e l e c t r i c pulses which were counted and their frequency displayed on the d i g i t a l frequency meter. The ro t a t i o n a l speed of the cylinder was calculated from: C J = 2-TT- (f/8) --5-!- (X-1) a2 where f - measured frequency, s 1, d 1 - "0"-ring outside diameter, mm, &2 _ cylinder outside diameter, mm. In equation (X-1), measured frequency i s divided by 8 because of 8 l i g h t pulses detected per one revolution of the r o l l e r . 221 Appendix XI. Suspension Preparation and Type-C Floe Formation. Fibres were f i r s t washed in hot water with a detergent to remove impurities acquired during the manufacturing process. Subsequently they were thoroughly rinsed in deionized water, dewatered in a Buchner funnel, and oven-dried at 105°C for four hours. Oven-dry nylon was weighed on a Mettler 163F balance with 0.0001 g precision to prepare f i b r e samples. The samples were then soaked in deionized and u l t r a f i l t e r e d water (conductivity ^0.2 M S ) . 1 After 24 hours soaking, f u l l saturation of nylon with water had occurred. 2 The suspending water was removed in the Buchner funnel. The f i b r e mat c o l l e c t e d on the f i l t e r paper was kept wet and t h i s mat was slowly immersed in the suspending l i q u i d (deionized, u l t r a f i l t e r e d water or aqueous-sugar sol u t i o n ) . The immersion technique of wetted f i b r e s insured complete exclusion of a i r bubbles from the suspension. The presence of a i r bubbles could introduce cohesion due to surface tension forces [H9,K7,M6]. A sample of rewetted fibres was f i r s t placed in a o n e - l i t e r graduated cylinder f u l l y f i l l e d with the suspending l i q u i d . The concentration was very low, about one tenth of the sediment concentration. At t h i s concentration any f i b r e aggregates were ea s i l y dispersed by several strokes of a plunger having a perforated rubber disk at one end. The plunger f i t t e d loosely inside the graduated cyl i n d e r . 1 Measurements of water conductivity are described in Appendix XVIII. 2 Appendix VI contains d e t a i l s of water absorption by nylon. 222 Next, the suspension was thickened by slow removal of the l i q u i d by suction into a 100 mL pipet. The pipet's t i p covered with a 200-mesh screen prevented f i b r e removal. When the suspension volume in the cylinder reached 500 mL, the suspension was transferred to the plexiglas c y l i n d e r . The cylinder was mounted on the r o l l i n g mechanism and rotated at constant speed. Further l i q u i d removal was achieved in 5 mL increments, i . e . , each time 25 mL was removed, 20 mL was reintroduced. When i t was observed that floes did not disperse upon r e d i l u t i o n , the procedure was halted and the suspension volume was measured. The precision of volume measurement in a 500 mL graduated cylinder was ±5 mL. The aqueous-sugar solution was prepared from deionized, u l t r a f i l t e r e d water and a grocery store sucrose. The e l e c t r i c conductivity for a l l sucrose solutions was less than 15 uS. In most instances the aqueous-sugar solutions matched the apparent nylon density of 1130 kg/m . The solution density was measured r 3 with the Universal Beaume Hydrometer graduated every 10 kg/m . 223 Appendix XII. Sediment Concentration Measurements. The sediment concentrations were measured by a simple procedure. The samples of oven-dried fi b r e s having weights from 0.3 g (long fibres) to 6 g (short fibres) were f i r s t soaked for 48 hours in water, dewatered in a Buchner funnel, and dispersed well in one l i t e r of aqueous-sugar solution. Dewatering in the Buchner funnel removed a l l a i r bubbles that attach to f i b r e surfaces and produce d i f f e r e n t i a l sedimentation. Subsequent immersion in the suspending l i q u i d prevented small bubbles from attaching themselves to f i b r e s . Uniform f i b r e dispersion was achieved in turbulent shear produced behind a plunger consisting of a perforated rubber disk fastened to the end of a glass rod. The disk f i t t e d loosely inside the graduated glass cylinder of 60.3 mm ID. Aqueous-sugar solutions of 1100 kg/m density used as the suspending medium allowed slow sedimentation with l i t t l e s e t t l i n g of the formed pad. Two hours after the fi b r e s began s e t t l i n g , the pad volume was read off with ±5 mL p r e c i s i o n . The known volume of f i b r e sample divided by the pad volume gave the sediment concentration. The pad volume determination was done with an error which was related to the wall e f f e c t . At the cylinder v e r t i c a l wall, in the annulus of about one half f i b r e length, the fibres form a less dense pad. The maximum estimated error i s 8.3% for the longest f i b r e s (6.26 mm) and 1.3% for the shortest f i b r e s (0.913 mm). 224 Appendix XIII. Flow Veloci t y Measurement in Fibre Suspensions in Horizontal Rotating Cylinder. A non-intrusive technique of f l u i d v e l o c i t y measurement, Laser Doppler Anemometry, was used in the study of flow patterns in a horizontal rotating cylinder. The experimental setup i s shown in Figure XIII-54. The plexiglas cylinder (D) was rotated by the mechanism described in Appendix X. The mechanism consisted of: a r o l l i n g table (A), a f l e x i b l e shaft (J), and a v e l o c i t y - c o n t r o l l e d motor (F). The f l u i d v e l o c i t y measurement system was the TSI dual beam laser Doppler system working in the transmission mode. The experimental setup consisted of: an Argon-ion laser (L) (LEXEL, model 85) a p o l a r i z e r , a beam Figure XIII-54. Experimental Setup for Velocity Measurements in a Horizontal Rotating Cylinder. 225 s p l i t t e r , a Bragg c e l l , and focusing optics (G); a photo-detector and receiving optics (H); a Bragg c e l l driver with the frequency downmix c i r c u i t (K); a signal tracker and the signal processor (M); a two-channel oscilloscope (N). The laser was mounted on the "UniSlide" assembly (T) which provided XYZ coordinate motion. The data a c q u i s i t i o n equipment consisted of the analog-to-digital conversion board ( S c i e n t i f i c Solutions Inc.) (Y), the IBM-PC (X), and the Epson-80 printer (Z). The p r i n c i p l e s of Laser Doppler Anemometry are described in numerous publications [D8]. Some p r i n c i p l e s b r i e f l y discussed here emphasize the important aspects of the exist i n g experimental setup. A basic p r i n c i p l e of dual beam anemometry i s the crossing of two laser beams of equal intensity at a measuring volume in the f l u i d . Where these beams cross, they interfere with each other to form "fringes." The fringes caused by the coherent l i g h t in the two beams cancel each other in some regions and complement each other in others. The beam-crossing geometry i s shown in Figure XIII-55. The distance between the brightest l i n e s of the two neighboring fringes, d^, measured within a cross-plane of the beams depends on the l i g h t wavelength, X, and the beam crossing angle, d f = 2.sin(»/2) = 2.sin(H J?3V2) = 2 ' 6 7 6 7 " m (XIII-1) A p a r t i c l e in the f l u i d moving through the measuring volume and in the cross-plane of two beams generates -light pulsing of 226 F i g u r e X I I I - 5 5 . Beam C r o s s i n g Geometry a t an I n s t a n t i n Time. c e r t a i n f r e q u e n c y , f , w h i l e i t c r o s s e s the " f r i n g e s . " A phot-ode t e c t o r senses the l i g h t p u l s e s and c o n v e r t s them i n t o an e l e c t r i c s i g n a l . The f r e q u e n c y of the s i g n a l i s measured and the p a r t i c l e v e l o c i t y c a l c u l a t e d . From the i n t e r s e c t i o n a n g l e , 4», the l a s e r w a v e l e n g t h , X, and the f r e q u e n c y , f , the p a r t i c l e v e l o c i t y can be c a l c u l a t e d from: U = f * d f '- 2 - s i n ( » / 2 ) ( X I I I " 2 ) The measurement d i r e c t i o n i s a v e c t o r a t r i g h t a n g l e t o the a x i s of t h e o p t i c a l u n i t and i n the c r o s s - p l a n e of two beams. The t r a n s m i s s i o n o p t i c s can be r o t a t e d and the measurement d i r e c t i o n t h e r e b y changed. The a c t u a l l o c a t i o n of the measuring 227 volume does not change d u r i n g o p t i c s r o t a t i o n . The LDA employed measured two o r t h o g o n a l v e l o c i t y components, one h o r i z o n t a l and the o t h e r v e r t i c a l . V e l o c i t y d i r e c t i o n a l a m b i g u i t y d i d not e x i s t i n the system. The system c o u l d d i s t i n g u i s h the " f o r w a r d " f l o w from the " r e v e r s e " because of a Bragg c e l l which i n t r o d u c e d a t r a v e l l i n g wave a c r o s s the beam. A 40 MHz fr e q u e n c y g e n e r a t e d by the Bragg c e l l i n the " r e f e r e n c e " beam produced "moving" f r i n g e s w i t h i n measuring volume. Hence, when a p a r t i c l e i s s t a t i o n a r y i n the measuring volume, i t g e n e r a t e s 40 MHz l i g h t p u l s e s . P a r t i c l e movement i n the same d i r e c t i o n as the "moving" f r i n g e s l o w e r s the f r e q u e n c y of the l i g h t p u l s e s , w h i l e movement i n the o p p o s i t e d i r e c t i o n r a i s e s i t . The d i r e c t i o n s of "moving" f r i n g e s were s e t t o be o p p o s i t e t o the p o s i t i v e d i r e c t i o n of x and y a x i s shown i n F i g u r e X I I I - 5 6 . The shape of the measuring volume and i t s main d i m e n s i o n s a r e shown i n F i g u r e X I I I - 5 5 . I f the l a s e r beam d i a m e t e r , D g. 2, i n c l u d e s a l l of the beam i n t e n s i t y g r e a t e r than e" 2 of the c e n t e r l i n e , t h e c o r r e s p o n d i n g d i a m e t e r a t the f o c a l p o i n t i s : d = ±.X.=^_ = i . 5 1 4 . 5 n m . 2 5 0 ; 4 m i n - 0.1491mm ( X I I I - 3 ) e it D . 2 it i • i mm e where F D i s the f o c a l l e n g t h of the l e n s . Two o t h e r d i m e n s i o n s of the measuring volume, d m and l m a r e r e l a t e d t o d„ as f o l l o w s : e 228 d = d e = °_j149lmm = 0.1498mm (XIII - 4 ) m c o s T * / 2 T cosTTl.03°/2) J3 , e 0.1491mm , C C 1 / V T T T C \ Xm = sin(»/2) = sin(H.03V2) = 1 ' 5 5 l m m (XIII-5) In the forward scatter mode, the dimensions lm and d m determine ' m m the size of the measuring volume and the s p a t i a l resolution of 229 the system. The number of fringes at the central section of the measuring volume i s : * T m .1498mm cc /VTTT r\ NFR = d7 = 2.676722Mm = 5 5 ' 9 6 (XIII-6) In other sections of the measuring volume, the number of fringes decreases as they approach each end of the length l m . If a single p a r t i c l e having a size smaller than d^ crossed the center of the measuring volume, i t would generate N p R of Doppler cycles. In r e a l i t y , the number of Doppler cycles seen by the photodetector i s smaller due to the small size of the aperture. This aperture blocks l i g h t a r r i v i n g from locations other than the beam crossing volume and thereby improves the signal to noise r a t i o . In the Thermo System Inc. photodetector, the e f f e c t i v e beam diameter i s : F d_ = 0.7-d = 0.7-0. 1873-IfS-^r = 0. 1 309mm (XIII-7) a ?'D _ b u. 4 Where, F A i s a focal length of the scattered l i g h t focusing lens and F D is a focal length of the c o l l e c t i n g lens, and 0.7 is a design constant. The number of fringes from the central section which i s seen by the photodetector i s : NEFR ' af - 2 ! n w « i " 4 8 - 9 ( X I I I " 8 ) The LDA system measures the ve l o c i t y of the p a r t i c l e s entrained by the f l u i d . For f l u i d v e l o c i t y to be measured with a 230 laser anemometer, the p a r t i c l e s must follow the f l u i d flow." An additional requirement i s that there be a s u f f i c i e n t number of p a r t i c l e s to produce an uninterrupted stream of Doppler cycles. Since the suspending l i q u i d was prepared from u l t r a f i l t e r e d water and pure sugar, the seed p a r t i c l e s had to be introduced. White polystyrene p a r t i c l e s having the average diameter of 1.091 um (S.D.=0.0082 um) were used [S8]. They had 1050 kg/m3 density, and were shown to follow c l o s e l y the flow of water [D8], Addition of one drop of suspended p a r t i c l e s (10% solids) to 125 mL of aqueous sugar solution was sati s f a c t o r y as indicated by the number of samples per second and the time fraction of the validated signal displayed by the signal tracker. lens TR t T position 1 position 2 Figure XII1-57. Change in Focal Length caused by Various Indices of Refraction. 231 The suspension in which v e l o c i t i e s were to be measured was confined between the cylinder walls. Two laser beams must pass through one plexiglas plate and the suspension before a r r i v i n g at the measuring locat i o n . The plexiglas and the aqueous-sugar solution have d i f f e r e n t indices of r e f r a c t i o n than has the a i r . This displaces the measuring volume location and changes the intersection angle of two beams, as i l l u s t r a t e d in Figure XIII-57 (position 2). This change in r e f r a c t i o n indices does not a f f e c t the c a l i b r a t i o n factor, i . e . , the spacing between fringes. The equation (XIII-1) holds because the wavelength and the sine of the angle depend equally on the r e f r a c t i o n indices. Since the measuring volume had to be located in the central plane of the cylinder, a r e l i a b l e method of finding t h i s plane was required. The laser and the optics were located on a three-directional traversing r i g which could be moved with respect to the stationary axis of the cylinder. The location of the measuring volume within the central plane of the cylinder was obtained by the coordinate setting which followed the predetermined pattern of measurement points shown in Figure XIII-56. For the measuring volume to be placed in the central plane of the cylinder, both beams had to go through 3 mm thick plexiglas and penetrate 18 mm into the suspension. The traversing r i g , however, had to be displaced by the distance TR from position 1 to position 2, as marked in Figure XIII-57. F_, i s an e f f e c t i v e focal length [D8]. 232 tanOy tant9 1 F E " ( T R " t " i i n ^ ) " i a n ^ + fc + FD " T R (XIII-9) but F E=F D-TR+t+T and TR obtained from equation (XIII-9) i s , tant?-, tant?~ T R " ^tlHef + t-tlHe; (xni-10) where 6^=9/2 - h a l f of the beam c r o s s i n g angle, B - sin-HSiS*/l) * M2 e = S i n - i 3 M 3 M1 _ index of r e f r a c t i o n of a i r , =1, 1*2 ~ index of r e f r a c t i o n of p l e x i g l a s , M 3 _ index of r e f r a c t i o n of 1.13 g/cm3 aqueous-sugar s o l u t i o n , F D - l e n s f o c a l d i s t a n c e i n the a i r , F E - l e n s e f f e c t i v e f o c a l d i s t a n c e , t - p l e x i g l a s t h i c k n e s s , T - p e n e t r a t i o n d i s t a n c e i n t o suspension. TR can be c a l c u l a t e d from (XIII-10) f o r F Q= 250.4 mm, M2=1.65, M3=1.382 [C10], ¥=11.03°, t=3 mm, T=18 mm. Thus, TR=14.8 mm. L i g h t p u l s e s having a frequency c l o s e to 40 MHz were transformed i n t o e l e c t r i c s i g n a l by the p h o t o d e t e c t o r . T h i s s i g n a l entered the frequency s h i f t c i r c u i t . The 40 MHz frequency s h i f t was u n d e s i r a b l y l a r g e and needed to be decreased through downmixing. The purpose of the downmix i s to b r i n g the frequency 233 into the range of the signal processor or, even better, to the optimum part of t h i s range. The horizontal cylinder was rotated with constant angular speed _)=27r rad/s. Its peripheral speed was R«_>=0.295 m/s where R denotes internal cylinder radius. The measuring volume could not be located closer to the cylinder wall than 3 mm due to the geometry of the crossing beams and the beam diameter. In consequence, the expected maximum speed to be measured was estimated to be about 0.240 m/s. The corresponding Doppler frequency from a stationary fringe pattern would be 90 kHz. The downmix frequencies were 100 kHz or 200 kHz so that only the lowest detection range (2-500 kHz) of the signal tracker was used. From the frequency downmix c i r c u i t , the LDA signal went to the signal tracker. Here, the signal was amplified to the optimum amplitude l e v e l and passed to the signal v a l i d a t i o n processor which eliminates spurious signals. The validation processor works on the p r i n c i p l e of a phase lock loop which locks onto the incoming frequency and can "track" the frequency variations within ±15% frequency window. The lock detector starts one of two counters of Doppler cycles. At a count of 8, the hold mode i s i n i t i a t e d and the second counter starts sampling. Though sampling was stopped by the f i r s t counter after 8 Doppler cycles, the signal was not validated and released to the output u n t i l 10 cycles. This important feature of the signal tracker means that only frequencies one tenth or less than the Doppler frequency can be measured as the validated signal, in 234 t h i s case 20 kHz or l e s s . The count of samples per second i s derived from the switching off of the sample-and-hold c i r c u i t s . The percent signal i s a measure of the f r a c t i o n of time the phase lock loop i s " i n lock." These two readouts give s u f f i c i e n t information to assess whether the v a l i d a t i o n c i r c u i t i s working properly. When this has been established, the readouts can be used to determine the optimum dose of the seed p a r t i c l e s . The phase lock c i r c u i t gives an output voltage which i s l i n e a r l y proportional to the frequency deviation from the center frequency of the detection range. The pr o p o r t i o n a l i t y constant i s 0.01 V/kHz for the detection range 2-500 kHz. If no new signal is validated, the l a s t voltage i s kept by the c i r c u i t . The validated signal i s passed on to the signal processor in which i t can be o f f s e t and/or f i l t e r e d . For convenience, the voltage was offset to read 0.0 v o l t s while the v e l o c i t y was 0.0 m/s. The signal was c o l l e c t e d by the IBM-PC through the analog to d i g i t a l converter (TM-40, S c i e n t i f i c Solutions Inc.). The program that acquired the data i s l i s t e d in Appendix XXII. At each measurement point, 25000 voltage readings were taken at the rate of 5000 readings per second. The average voltage, Volt, was used for the c a l c u l a t i o n of each orthogonal v e l o c i t y component. u = Volt-d Um)/0.01(V/kHz) = 0.26767•Volt(m/s) (XIII-11) The data were subsequently transferred to the UBC main-frame computer system with the help of MCP-WINDOW software. The experimental data are in Appendix XXVII. 236 Appendix XIV. Floe Preparation and Contact Counting Procedures. For the study of the rel a t i o n s h i p between floe concentration and the number of contact points per f i b r e , strong nylon floes were formed at increasing levels of fibre concentration in the rotating c y l i n d e r . Fibres of 44.19 jum diameter and 4.9 mm long were used (L/d=141.7). F i r s t , nuclei were formed of red f i b r e s . Next, the nuclei were introduced to suspension of translucent f i b r e s . A layer of translucent f i b r e s interlocked with and around n u c l e i . The suspension c i r c u l a t e d for two minutes at a given concentration and the floe was removed and dried in the ambient a i r . Slow motion movies of the drying process were made for observation of any changes in floe structure occurring during drying. No changes in floe shape or structure were observed. Since each floe had a spheroidal shape, the three major, orthogonal dimensions were measured. During drying, the sugar from the solution deposited on the fi b r e surfaces and p r e f e r e n t i a l l y at the fibre contact points. This deposition led to bonding between f i b r e s . Each contact-counting procedure consisted of the floe being dried and then in d i v i d u a l f i b r e s being freed from i t under microscope. The fi b r e s were freed by the indi v i d u a l bonds being progressively broken. The contacts made by the red fibres with the red and translucent f i b r e s were only counted. After the floe was completely dismantled, a l l red fi b r e s were counted and an average number of contact points was calculated. 237 S t u d i e s u n d e r m i c r o s c o p e p r o v e d v e r y d e m a n d i n g on t h e e y e s . C o n s e q u e n t l y , a s i m i l a r p r o c e d u r e was a d o p t e d f o r s c a l e d - u p f l o e s made o f a b o u t t e n t i m e s l a r g e r n y l o n f i b r e s , i . e . , 4 5 . 9 mm l o n g a n d 0 . 5 5 9 mm i n d i a m e t e r ( L / d = 8 2 . 3 ) . T h e f l o e s w e r e f o r m e d by h a n d i n a n a q u e o u s - s u g a r s o l u t i o n . I n c r e a s i n g f l o e d e n s i t y was a c h i e v e d by f l o e s b e i n g s q u e e z e d s i m i l a r l y t o a s n o w - b a l l b e i n g r o l l e d . T h r e e h u n d r e d d y e d f i b r e s , e a c h h u n d r e d w i t h a d i f f e r e n t c o l o u r , f o r m e d f l o e n u c l e u s . T h e n u c l e u s was s u r r o u n d e d by t h e t r a n s l u c e n t f i b r e s . P l o c d i s m e m b e r m e n t p r o c e e d e d t h e same as b e f o r e . The c o l o u r e d f i b r e s f a c i l i t a t e d c o n t a c t p o i n t r e c o g n i t i o n . T a b l e X I V - X X V I I I c o n t a i n s d a t a on f l o e F i g u r e X I V - 5 8 . P r o g r e s s i v e Di smemberment o f T y p e - C F l o e . Table X I V - X X V I I I . Contact Points m Type-C Floes. L/d FLOC NUMBER TOTAL NUMBER OF FIBRES NUMBER OF COLORED FIBRES VOLUMETRIC CONCENTRATION OF FIBRES NUMBER OF CONTACTS PER FIBRE 1 863 300 0.0445 2.26 82.3 2 1 154 300 0.0731 2.81 3 1205 300 0.1005 3.97 4 1370 300 0.1027 3 .68 1 — 332 0.0512 3.68 141.7 2 -- 623 0.0687 4 .40 3 -- 491 0.1083 4.84 3 uO c CD in OJ in tn CD 3 cr x <  < i oi > CO tn n> » Q c CD 3 O CD c Ui cr i-i 0) n-3 UQ O O G i H' Ui 3 CD 3 or CD •-j 3 CD 3 r r (A ut 3*' O 3 CO CO 239 Appendix XV. Tensile Strength of Wet Nylon Floes. Coherent floes of various fib r e concentration were produced in the i n c l i n e d rotating cylinder for t e n s i l e t e s t i n g . For increased floe density, the suspension concentration in the cylinder was gradually increased s t a r t i n g from the threshold concentration. At each concentration, r o l l i n g continued for two minutes. Only floes having a shape close to spherical were selected for t e s t i n g . Before t e n s i l e testing, the floe volume was determined. It was assumed that the floe boundaries were defined by the surface of the l i q u i d associated with the floe when the floe had been Figure XV-59. Coherent Floe removed from the Suspension by the Fork. 240 removed from the suspension. Each floe was removed by a fork made of two beading needles. Figure XV-59 depicts the fork and a floe transfixed on i t . This step did not damage the f l o e . The floe was weighed and i t s volume calculated by the floe's weight being divided by i t s density. The density of floe matter was uniform throughout since the fibres were rendered neutrally buoyant during floe formation. The floe was then ready to be mounted in the t e n s i l e tester. Figure XV-60 shows the assembly of the t e n s i l e test equipment. The equipment consisted of: a Thwing-Albert t e n s i l e tester (A), a load c e l l (B), a signal amplifier (C), an analog-to-digital converter (D), an IBM-PC (E), a plexiglas tank Figure XV-60. Tensile Test Equipment. 241 (F) and two combs with a floe (G). The plexiglas tank was f i l l e d with an aqueous sugar solution. The tank and the lower comb were attached to the cross-head (H) of the Thwing-Albert tester. The upper comb was suspended from the load c e l l on a f l e x i b l e s t r i n g ( J ) . Each comb was comprised of No.10 beading needles (D=0.4 mm) inserted h o r i z o n t a l l y through the holes d r i l l e d in the "L" shaped plexiglas f i x t u r e s . The needles were spaced evenly in a 16 mm long row. A closer photograph of the tank i s shown in Figure XV-61. One comb (K) was firmly secured to the bottom of the tank, while the other comb (L) was secured to the l o a d - c e l l by a long, f l e x i b l e s t r i n g ( J ) . The floe (M) and the combs were submerged in an aqueous-sugar solution during the t e s t . Figure XV-61. Device for Tensile Testing of Nylon Floes. 242 Floe mounting on the needles was straight-forward: (1) the floe was pierced in the middle by the bottom set of needles; (2) the top comb was put into place; (3) the top set of needles was inserted into the f l o e . The floe was ready to be strained. A constant speed of 10 mm/min was applied to the bottom comb by the driv i n g mechanism of the Thwing-Albert tester. The tank and the bottom comb moved downward. The upper comb remained immobile. The floe started to deform and i t s resistance to deformation grew ste a d i l y . The force opposing the p u l l was sensed by the load c e l l which converted the force to an e l e c t r i c s i g n a l . The signal from the load c e l l was amplified and sampled by the d i g i t a l to analog converter (TM-40, S c i e n t i f i c Solutions Inc.) at the rate of 30 readings per second. The force, in i t s d i g i t a l form, was stored in data f i l e s by the IBM-PC. Data a c q u i s i t i o n program i s l i s t e d in Appendix XXIII. The area of the break zone was photographed while the separated piece was s t i l l attached to the upper comb but the comb was removed from the suspending l i q u i d as shown in Figure XV-62. This was necessary for the size of the cross-sectional area to be determined so that the number of f i b r e s crossing the break plane could be calculated. Slides of each area were projected on the d i g i t i z e r and the area was measured by polygonal approximation. A data acqui s i t i o n setup similar to that for f i b r e length measurements was used. The BASIC program which accomplished t h i s i s l i s t e d in Appendix XXIV. Both separated pieces of floe were removed from combs, washed, dried at 105°C for four hours, and weighed. Washing was 243 F i g u r e X V - 6 2 . P l a n e V i e w o f t h e B r e a k Z o n e . c r u c i a l b e c a u s e t h e r e s i d u a l s u g a r c o u l d c o n t r i b u t e t o t h e d r y f i b r e w e i g h t . C o n s e q u e n t l y , t h e f i b r e s were w a s h e d f o u r t i m e s w i t h t o t a l d i l u t i o n r a t i o b e i n g 1:256 t o r e d u c e t h e amount o f r e s i d u a l s u g a r t o l e s s t h a n 0 .001 g . The w e i g h t o f a i r i n t h e w e i g h i n g b o t t l e s c o u l d c o n t r i b u t e t o i n a c c u r a t e r e s u l t s i f w e i g h i n g was d o n e b e f o r e b o t t l e s were c o o l e d . T h e d i f f e r e n c e i n w e i g h t o f 20 mL o f a i r when c o o l e d f r o m 1 0 0 ° C t o 2 3 ° C i s 0 . 0 0 5 g . T h i s amount i s n o t l a r g e w i t h r e s p e c t t o t h e d r y f l o e w e i g h t s r a n g i n g f r o m 0.1 t o 0 . 8 g , b u t c a n c a u s e an e r r o r o f 5% i n m e a s u r e m e n t s o f t h e w e i g h t o f l i g h t f l o e s . S i n c e t h e b o t t l e s were a l w a y s c o o l e d i n t h e d e s i c c a t o r , t h e t o t a l w e i g h i n g e r r o r was l e s s t h a n 2% e v e n i n t h e most u n f a v o r a b l e c a s e . T h e 244 concentration of fibres in the floe was calculated from i t s i n i t i a l volume and the dry-weight of f i b r e s . 245 Appendix XVI. Computer Model of Fibre Deposition into 3-D Networks. An understanding of 3-D f i b r e networks can be achieved through experimental studies of laboratory constructed networks. A faster procedure models such networks on a computer where the parameters (f i b r e length and diameter) can be e a s i l y changed. The creation of a computer program requires the choice of suitable programming language which uses an inexpensive compiler. The FORTRAN language was chosen. The program should also accommodate the size l i m i t a t i o n of the computer memory. Such a program has been written for deposition of f i b r e s because the process of sediment formation was known. The space in which f i b r e deposition was modelled was discrete and shaped as a square-base parallelepiped. The base side was three f i b r e lengths long as shown in Figure XVI-63. A two-dimensional array containing INTEGER*2 elements constituted the base of the p a r a l l e l e p i p e d . Each array element stored the height of the deposited mat of f i b r e s . This choice of space accommodating f i b r e s of a large aspect r a t i o (up to 1000) did not exceed the memory capacity of the UBC main-frame computer. Fibres were deposited at random within the parallelepiped, i . e . , t heir i n i t i a l o rientation and the location of their centres were determined by pseudo-random process. The fibres were oriented h o r i z o n t a l l y during approach to the mat; however, upon deposition on the mat, they could rotate and/or s l i d e to rest on the top of the mat or could penetrate the mat. The d i r e c t i o n of rotation was determined by the r e l a t i v e p o sition of the f i r s t 246 F i g u r e XVI-63. P a r a l l e l e p i p e d i n which F i b r e d e p o s i t i o n was Modeled. co n t a c t p o i n t with r e s p e c t t o the f i b r e c e n t r e . The s l i d e angle was set equal t o the arcus tangens of the s t a t i c c o e f f i c i e n t of f r i c t i o n determined e x p e r i m e n t a l l y (see Appendix I X ). A d e s i r e d number of f i b r e s i s d e p o s i t e d before the program c a l c u l a t e s the c o n c e n t r a t i o n of d e p o s i t e d f i b r e s , number of c o n t a c t s between f i b r e s , i n t e r - c o n t a c t d i s t a n c e s , angles of d e p o s i t i o n and f r a c t i o n of s u r f a c e coverage. These c a l c u l a t i o n s are l i m i t e d t o a space d e f i n e d by a cube shown i n Figure XVI-63. T h i s cube has the s i d e s of one f i b r e l e n g t h .and i s r a i s e d one-half of a f i b r e l e n g t h above the base of the l a r g e p a r a l l e l e p i p e d . 247 Before the s t r a i g h t l i n e segments were d e p o s i t e d , the bottom of the l a r g e p a r a l l e l e p i p e d was covered with f i b r e s forming a mat having 3-D s t r u c t u r e . On top of t h i s mat the s t r a i g h t l i n e segments were d e p o s i t e d and t h e i r number was c a l c u l a t e d . 248 Appendix XVII. Mathematical Analysis o f the Tensile Strength o f Type-C Floes. If i t i s assumed that the b u i l d up of load i s a process of gradual engagement of f i b r e s in f r i c t i o n a l i nteraction. The f i b r e s embedded in a floe above the rupture plane s l i d e along f i b r e s embedded in the part of the floe below the rupture plane in monotonously increasing numbers. If i t i s further assumed that only the dynamic f r i c t i o n a l interaction exists at the contact points, the t o t a l load developing in a strained floe can be expressed mathematically by: N f n c i F = k f.1 (k .-I T..) (XVII-1) f i=1 C 1 j=1 ^ where F - t o t a l load carried by the strained floe, - number of fibres crossing the "rupture plane", k^ - f r a c t i o n of fibres that carry the load, n . - number of contacts of j-th f i b r e with other f i b r e s , c 1 J k . - f r a c t i o n of n ., i . e . , f r a c t i o n of s l i d i n g contacts, c 1 c 1 T^j - f r i c t i o n a l force component p a r a l l e l to the d i r e c t i o n of floe separation produced at j-th contact on i - t h f i b r e , The f r i c t i o n a l forces opposing floe separation develop within the plane of each f i b r e - t o - f i b r e contact. Such planes are randomly oriented at the beginning of the deformation process. Not a l l contact points along one f i b r e are s l i d i n g . Most are firmly associated with either the top or the bottom part of 249 the f l o e . The c o e f f i c i e n t k c^ represents the fraction of contacts which s l i d e . The sum of "embedding" forces i s greater than the sum of " s l i d i n g " forces on each f i b r e . It i s possible for one half of the contact points to s l i d e and the other half to be embedded since the dynamic c o e f f i c i e n t of f r i c t i o n i s smaller than the s t a t i c . The sum of " s l i d i n g " forces i s summed up in equation (XVII-1). The assumption that each s l i d i n g f i b r e from the top part of the floe has i t s counterpart embedded in the bottom part of the floe i s reasonable. In other words, the fi b r e separation i s 50/50. This assumption would c e r t a i n l y hold for large and uniformly dense f l o e s . Moreover, not a l l the fibres that cross the break plane contribute to the load-bearing capacity of the fl o e . Those f i b r e s which l i e near the peripheries of the floe do not. Some fi b r e s in the core of the floe may not be active either, e s p e c i a l l y in less dense f l o e s . The c o e f f i c i e n t k^ i s introduced to account for t h i s . The f r i c t i o n a l component T^j can be expressed in terms of normal force, R-•, and the dynamic c o e f f i c i e n t of f r i c t i o n , 0 , . where k^j i s a d i r e c t i o n c o e f f i c i e n t , or sine of the angle between the f r i c t i o n a l force and the dir e c t i o n of floe separation. = k dj* R i j**d (XVII-2) If average quantities for ^ c ^ i n introduced, the following equality holds: c i ' and R— are 250 c c d d (XVII-3) where n c - average number of contact points per f i b r e , k c - half of the fraction of s l i d i n g contacts, R - average normal force at a contact point, k^ - average d i r e c t i o n c o e f f i c i e n t ; average sine of an angle between the f r i c t i o n force acting along fibr e p r i n c i p a l axis and the break plane. The average number of contact points per f i b r e can be estimated from the experimental findings presented in Section 4.6. Introducing (XVII-3) into (XVII-1) yie l d s The average normal force, R, can be expressed in terms of an average f i b r e d e f l e c t i o n , 5, the e l a s t i c modulus of wet f i b r e , E, the bending moment of i n e r t i a , I, and the average distance between the adjacent, alternate contact points, 1 [M2]. Here, only the e l a s t i c deformation of f i b r e s i s accounted for because such deformation was postulated do develop in Type-C networks. It i s possible that the f i b r e s , while the floe i s separated, bend more than i s necessary to interlock; t h i s i s not included in the present mathematical model. F i r s t , the case of nc=3 which i s depicted in Figure XVII-64 is considered. The expression for f f c c d v d (XVII-4) 2 5 1 the maximum f i b r e d e f l e c t i o n i s : R - l 3 R - l 3  5max = 3TETT = 67171 (XVII-5) and for force R1 i s : 3 - E - I . 6 . = max (XVII-6) 1 1 J If t h i s f i b r e had been pulled out as in a t e n s i l e test, most l i k e l y only one contact point would s l i d e . The other two contacts would be associated with one of two separating parts of the f l o e . Therefore, only R1 would contribute to the t o t a l force, F. The s i t u a t i o n becomes more complex when n c i s greater 252 than three. In th i s case, more contact points would s l i d e , each contributing to the t o t a l force. If i t i s assumed that every additional s l i d i n g contact develops the average force R^^'k^. The inter-contact distance, measured along f i b r e axis i s approximated by 1 = ^p- (XVII-7) The bending moment of i n e r t i a for a c i r c u l a r p r o f i l e i s ,4 I = ^ f - (XVII-8) In non-oriented f i b r e networks, the average angle between f i b r e longitudinal axis and the dissecting plane i s [W3] TT/2 7 = J 7 - c o S 7'd7 = 0.57 (XVII -9) 0 7 = 32.7° (XVII -10) An average f i b r e cross-section laying within the break plane i s ,2 A f = T'^TTT- (XVI 1-1 1 ) f 4 s i n 7 This equation i s v a l i d for 7 from arcus tangens (d/L) to TT/2 , i . e . , t h i s analysis excludes a l l fibres i n c l i n e d at angles smaller than arcus tangens(d/L) to the plane of break. The number of fi b r e s crossing the break cross-section can be estimated from: 253 N f = ^ ™ = i^lSJm.r (XVII-12) t "r" ^ c. va 7T'd Where, B i s a cross-section area of the break zone. Equation (XVII-4) can now be rewritten in terms of breaking stress. °T " I " l i - s i n ^ k c * k d * k f ^ d - E - f l - 5 m a x - C v a - n c UVII-13) The components of t h i s equation are further discussed in Section 4.8. 254 Appendix XVIII. Conductivity and pH Measurements. Conductivity Measurements. The water used in a l l experiments was deionized in the deionizer i n s t a l l e d in the Chemical Engineering building, and was further u l t r a f i l t e r e d by being passed through the Ultrapore Cartridge (Syborn/Barnstead). The conductivity of the deionized and u l t r a f i l t e r e d water was frequently measured with the Seibold (type LTB) conductivity meter. Before the conductivity measurements proceeded, the functioning of the instrument and the measuring c e l l constant was examined. This was necessary because the instrument was in considerable d i s r e p a i r . Determination of the c e l l constant required potassium chloride solutions of various concentrations [A13,S7]. For the measuring c e l l constant to be determined, the standard and the empirical c o n d u c t i v i t i e s were compared. Measuring conductivities of KC1 solutions followed the standard procedure [A13]; the r e s u l t s are shown in Figure XVIII-65. Since the average temperature of KC1 solutions was 22.98°C (S.D.=0.798°C), the standard c o n d u c t i v i t i e s at 23°C were taken from the tables in references [A13,S7], The s t r a i g h t - l i n e f i t s obtained by a least-squares method were made to the base-ten logarithms of concentrations and c o n d u c t i v i t i e s . LOG, n ( c o n d u c t i v i t y ) = a + b'LOG -.(concentration) (XVIII-1) 255 100 E E o 3 c o O 10 • 111 standard i m e t e r readings • i i i i i i i ' i i i i i i .01 Figure XVIII-65. KCI concentration, N Standard and Seibold Conductivities versus KCI Concentration in Solutions. where a and b are the f i t parameters. L i s t i n g of the FORTRAN program used for the lin e a r regression and p l o t t i n g i s given in Appendix XXV along with the l i s t i n g s of the data f i l e s . The f i t parameters are shown in Table XVI11-XXIX. The c e l l constant, K c, i s the r a t i o of standard to empirical c o n d u c t i v i t i e s over the range of concentrations measured. 256 Table XVIII-XXIX. F i t Parameters for Two Straight Lines Shown in Figure XVIII-65. a b SUM OF SQUARES SEIBOLD @ 22.98°C 1.919863 0.9473736 1.415678'10"2 STANDARD @ 23.0°C 2.052766 0.9592081 4.132784-10"6 CONDUCTIVITY STANDARD = R (XVII1-2) c CONDUCTIVITY S E T B O L D Two concentrations, 0.01N and 0.1N, were taken as the lower and the upper l i m i t of integration. . 1 a 1 b. J (10 + CONCENTRATION > dCONCENTRATI ON K = •— c (XVI 11-3) c 1 a- b ? J. (10 + CONCENTRATION ) dCONCENTRATI ON .01 Taking a4=2.052766, b1=0.9592081, a2=1.919863, and b2=0.9473736 and integrating equation (XVIII-3) y i e l d s Kc=1.3577. pH measurements Water and solutions pH were monitored with the Beckman combination electrode (531013) connected to the Beckman (Model 4500) d i g i t a l pH meter according to the standard procedure [A16]. 257 Prior to each new batch of experiments, the meter was checked and the slope was adjusted with standard buffer solutions of 4 and 7pH. Appendix XIX. Data C o l l e c t i o n Program for Fibre Length and Fibre Curvature Measurements. 10 REM THIS PROGRAM COLLECTS FIBRE LENGTH AND CURVATURE DATA USING 20 REM SUMMAGRAPHICS MICROGRID DIGITIZER. 30 REM FIBRE LENGTH IN mm, NUMBER OF POINTS ALONG THE FIBRE, AND 40 REM LOCAL RADII OF CURVATURE IN mm ARE STORED SEQUENTIALLY IN 50 REM A DISKETTE FILE. 60 CLS:KEY OFF:CLOSE 'Clear screen, soft keys off , close a l l f i l e s or devices 70 DEFINT I-K 'Integer I-K 80 OPTION BASE 1 'The lowest value an array subscript has is one 90 DIM XX(50),YY(50),FF(50),RC(50) 'Coordinates of points along f i b r e contour, cursor status, local radius of curvature 100 LOCATE 5,1,1,0,7 'Screen location, block cursor on 110 INPUT "INPUT THE NAME OF A NEW DATA FILE :" , F$ 120 LOCATE 25,1 'Screen location 130 INPUT "DO YOU WANT TO CHANGE THIS NAME, [Y] OR [N] ";A$ 140 A=ASC(A$):IF A=89 OR A=121 GOTO 100:A$="" 150 OPEN "C0M1:9600" AS #1 'Open communication port with SUMMAGRAPHICS MICROGRID tablet, OPEN statement is short because of the intentional DIP switches setting on the tablet to f i t BASICA's defaults 160 OPEN F$ FOR APPEND AS #2 'Open data f i l e named F$ for sequential output 170 PRINT #1,CHR$(27)"Z" 'Tablet reset 180 PRINT #1,CHR$(27) "M1",CHR$(27)"R3" 'Point mode, 10 coordinates/second 190 PRINT #1,CHR$(27)"C1" 'Resolution - metric-low; 10 points/mm 200 LOCATE 6,1 210 INPUT "INPUT THE REAL SIZE OF THE CALIBRATION SCALE IN MILIMETERS :",C 220 LOCATE 25, 1 230 NUM%=0 'Consecutive f i b r e number 240 INPUT "DO YOU WANT TO CHANGE THIS VALUE, [Y] OR [N] ";A$ 250 A=ASC(A$):IF A=89 OR A=121 GOTO 200:A$="" 260 CLS :LOCATE 7,1 270 PRINT "INPUT END COORDINATES OF THE CALIBRATION SCALE." 280 PRINT "PRESS ANY BUTTON ON THE CURSOR FOR EACH COORDINATE." 290 INPUT #1,XX(1),YY(1),FF(1) ,T 300 PRINT XX( 1 ) , YY( 1 ) , FF( 1 ) 310 PRINT #1,CHR$(27)"L11" 'Yellow LED on the cursor is on 320 INPUT #1,XX(2),YY(2),FF(2),T 330 PRINT XX(2),YY(2),FF(2) 340 PRINT #1,CHR$(27)"A",CHR$(27)"L10" 'Beeper, yellow LED off 350 LOCATE 25,1,1,4,7 360 INPUT "DO YOU WANT TO REPEAT THE CALIBRATION PROCEDURE, [Y] OR [N] ";A$ 370 A=ASC(A$):IF A=89 OR A=121 GOTO 260:A$="" 380 LOCATE 25,1 :PRINT " "; 390 CS=SOR((XX(1)-XX(2))"2+(YY(1)-YY(2))"2) 'Calibration distance, tablet units 400 CS=CS/C 'Calibration distance in units/mm 410 LOCATE 11,1 420 WRITE "CALIBRATION DISTANCES: REAL AND PROJECTED ON THE TABLET." 430 PRINT "C=";C;CHR$(109)CHR$(109),"CS=";CS;"units/mm" 440 LOCATE 13,1 450 PRINT "INPUT COORDINATES OF UP TO 50 POINTS ALONG THE FIBRE CONTOUR." 460 PRINT "PRESS BUTTON #5 FOR THE FINAL COORDINATES." [)j CO 'Reset cumulative f i b r e length to zero 'Yellow LED on the cursor is on 'Was Button #5 pressed? '50 points is maximum 'Yellow LED of, 'Line feed beeper 470 FL=0 480 INPUT tt\,XX( 1 ) , YY( 1 ) , FF( 1 ) ,T 490 PRINT XX(1 ) ,YY( 1 ) , FF( 1 ) 500 PRINT #1,CHR$(27) "L11 " 510 FOR 1=2 TO 50 520 INPUT #1,XX(I ) , YY(I ) ,FF(I ) ,T 530 PRINT XX(I),YY(I),FF(I) 540 IF FF(I)=6 GOTO 570 550 IF 1=50 GOTO 920 560 NEXT 570 PRINT #1,CHR$(27)"L10",CHR$(27)"A' 580 PRINT CHR$(10) "DO YOU ACCEPT THIS "IF NOT, PRESS "IF YES, PRESS ANY OTHER BUTTON." 620 INPUT #1,X1,Y1,F1,T 630 CLS :LOCATE 1,1,0 'Clear screen, cursor off 640 IF F1=16 GOTO 440 'Repeat last series of readings 650 NUM%=NUM%+1:PRINT "FIBRE NUMBER ";NUM% 660 FL = SQR((XX(1)-XX(2) ) ~2+(YY(1)-YY(2))"2) 670 FOR K=2 TO 1-1 680 KM=K-1:KP=K+1 690 X4=(XX(K)+XX(KM))/2:X5=(XX(KP)+XX(K))/2 'Middle point 700 Y4=(YY(K)+YY(KM))/2:Y5=(YY(KP)+YY(K))/2 'coordinates 710 A1=-(XX(KM)-X4)/(YY(KM)-Y4):A2=-(XX(K)-X5)/(YY(K)-Y5) 'Slopes 720 B1=Y4-A1*X4:B2=Y5-A2*X5 'Intercepts with y-axis 590 PRINT 600 PRINT 610 PRINT INPUT?" F ON THE CURSOR. 'Normal lines intercept coordinates 'Radius of curvature, tablet units 'Radius of curvature in millimeters Sum of straight line segments 'Total, approximated f i b r e length :KP 730 X6=(B2-B1)/(A1-A2):Y6=A1*X6+B1 740 RC(KM)=SQR((X6-XX(K))~2+(Y6-YY(K))"2) 750 RC(KM)=RC(KM)/CS 760 FL = SQR((XX(KP)-XX(K) )~2+(YY(KP)-YY(K))~2) + FL 770 NEXT 780 FL=FL/CS 790 PRINT CHR$(10) 800 PRINT "FIBRE LENGTH AND NUMBER OF DATA POINTS ALONG IT 810 PRINT "FL=";FL;CHR$(109)CHR$(109),"KP 820 PRINT "LOCAL RADII OF CURVATURE:" 830 FOR J=1 TO KM 840 PRINT "RC(";J; " ) = " ;RC(J) ; 850 NEXT 860 PRINT #2,FL;KM; 870 FOR J=1 TO KM 880 PRINT #2,RC(d); 890 NEXT 900 PRINT #2,CHR$(13) 910 GOTO 930 920 PRINT "THIS WAS THE 50-TH POINT ON THE CONTOUR.":GOTO 570 930 PRINT CHR$(10) 'Line feed 940 PRINT "DO YOU WANT TO CONTINUE WITH THE NEXT FIBRE?" 950 PRINT "IF NOT, PRESS F ON THE CURSOR." 960 PRINT "IF YES, PRESS ANY OTHER BUTTON." 'Carriage return 970 INPUT #1,X1,Y1,F1,T 980 IF F1=16 THEN CLOSE #1,#2 ELSE GOTO 440 990 LOCATE 24,1,0 :KEY ON 'Cursor off, soft keys on 1000 END O Appendix XX. Data Col l e c t i o n Program for Water Absorption by Nylon Fibres 10 REM THIS PROGRAM CONTROLS DATA ACQUISITION FROM 20 REM METTLER BALANCE AE-163 THROUGH DATA INTERFACE 30 REM CARD OPTION 012. 40 REM WEIGHT READING IN GRAMS AND TIMER READING IN SECONDS 50 REM ARE DISPLAYED ON THE SCREEN AND STORED IN A DATA FILE. 60 CLS :KEY OFF 'Clear screen, soft keys off 70 DEL=.001 'Weight increment is O.OOIg 80 LOCATE 4, 1 90 INPUT "INPUT THE NAME OF A NEW DATA FILE :",F$ 100 LOCATE 25,1 110 INPUT "DO YOU WANT TO CHANGE THE DATA FILE NAME";YY$ 120 Y=ASC(YY$):IF Y=89 OR Y=121 GOTO 80 130 OPEN F$ FOR APPEND AS #2 'Open communication with F$ data f i l e 140 OPEN "C0M1:2400,e,7,1,LF,PE" AS #1 'Open communication with the balance 150 PRINT #1,"R1" 'Disable the balance single control bar 160 CLS :LOCATE 8,15 170 PRINT "PREPARE BALANCE FOR TARE INPUT." 180 LOCATE 10,15 190 PRINT "PRESS ANY KEY TO INPUT TARE." 200 CLOSE #1:0PEN "COM 1:2400,e,7, 1,LF,PE" AS #1 210 A$=INKEY$: IF A$="" THEN 210 220 T=TIMER:PRINT #1,CHR$(84) 'Trigger tare procedure 230 INPUT #1,X$:WRITE #2,X$,T 'Store tare and time in f i l e 240 LOCATE 11,15 :PRINT X$,,T 250 CLOSE #1:0PEN "C0M1:2400,e,7,1,LF,PE" AS #1 260 PRINT #1,"D do i t ! " ; " " 'Display text on the balance display 270 LOCATE 13,15 280 PRINT B$ "PREPARE BALANCE FOR MEASUREMENTS." 290 LOCATE 15, 15 300 PRINT B$ "PRESS ANY KEY TO START DATA ACQUISITION." 310 A$=INKEY$: IF A$="" THEN 310 320 PRINT #1,"D" 'Clear the balance display 330 CLOSE #1:0PEN "C0M1:2400,e,7,1,LF,PE" AS #1 340 T=TIMER:PRINT #1,"SI" 'Send immediate value - dynamic measurement 350 INPUT #1,X$:L0CATE 16,15:PRINT X$,T ' F i r s t recorded load and time 360 Y$ = MID$(X$,4,9) :Z=VAL(Y$):WRITE <?2,Z,T 'Convert st r i n g to variable 370 CNT%=0 'Reading count 380 T=TIMER:PRINT #1,"SI" 'Send immediate value 390 INPUT #1,U$:V$=MID$(U$,4,9):W=VAL(V$) 400 CNT%=CNT%+1 'Increment the reading count 410 IF CNT%>900 GOTO 480 'Cycle of no change in weight 420 IF ABS(W-Z)<DEL GOTO 380 'Is weight change more than O.OOIg? 430 Z=W:WRITE #2,Z,T 'Store weight and time in data f i l e 440 LOCATE 17,15 :PRINT:LOCATE 17,15:PRINT U$,T 450 LOCATE 22,1 460 PRINT "TO STOP PROGRAM PRESS 'Ctrl-Break' THEN TYPE 'Close' AND PRESS Enter" 470 GOTO 370 480 LOCATE 18 ,15 :PRINT "WEIGHT HAS LEVELED O F F . " 490 WRITE # 2 , W , T : L 0 C A T E 19, 15:PRINT:LOCATE 19 ,15:PRINT U $ , T 500 GOTO 370 510 STOP Appendix XXI. Fibre Bending. Data Processing Program, Raw Data, Averages and Standard Deviations of Fibre Deflection. Bulk  Reynolds Numbers. Eye-Piece-Micrometer and GILMONT Flowmeter C a l i b r a t i o n Data. (a) FORTRAN program. N) V(200), IV(100) C This program calculates average Reynolds numbers and average C f i b r e d e f l e c t i o n s , uses NL2S0L to f i t def = A*Reynol ds**B, C makes a pl o t . C Input i s taken from (4). C Output is directed to (7). IMPLICIT REAL*8(A - H,0 - Z), INTEGER( I EXTERNAL CALCR, CALCJ INTEGER FLAG, LIST(1) COMMON XX(50), YY(50) DIMENSION A(50), B(50), P(2), REAL*4 C(100), D(100), X1, Y1 DATA LIST /'*'/ PI = 4. * ATAN(1.) M = 2 P(1) = 1.DO P(2) = 1.DO CALL AXCTRL(' CALL AXCTRL(' CALL AXCTRL( CALL AXCTRL( 'SYMS' 'LABE' 'DIGI' 'XORI' .2) 1) -1) 2.0) 2.0) REYNOLDS NO.; DEFLECTION UM; 0. 3) 0, 2) 0, 2) 10 20 30 CALL AXCTRL('YORI CALL AXPLOT('BULK CALL AXPLOT('MAX. CALL PL0T(2.O, 12 CALL PL0T(12.0, 12 CALL PLOT(12.0, FIND (4'7000) READ (4,LIST) X, Y, Z, W YOLD = Y J = O I = 0 READ (4,LIST,END=150,ERR= IF (X .EO. 0.0) GO TO 40 1 = 1 + 1 A(I) = .55068D-1*X - .360563DO A(I) = A(I) * 4 / PI / .152DO / A(I) = A(I) * RHO(W) / VIS(W) 0.0, 10., 0.0, 10.) , 90.0, 10., 0.0, 20.) 170) X, Y, Z, W 152D0 / 100 40 B(I) = GO TO IF (Y (Z 30 .GT. - Y) / .5932D0 YOLD) GO TO 100 Averages, bulk Reynolds number, maximum f i b r e deflection SUM1 = O.DO SUM2 = O.DO to DO 50 K = 1, I SUM1 = SUM1 + A(K) * Y / 100 SUM2 = SUM2 + B(K) 50 CONTINUE AVE 1 = SUM 1 / I AVE2 = SUM2 / I d = J + 1 C C Standard deviations XX(J) = AVE 1 YY(J) = AVE2 551 = O.DO 552 = O.DO DO GO K = 1 , I 551 = SS1 + (A(K) * Y / 100 - AVE 1) ** 2 552 = SS2 + (B(K) - AVE2) ** 2 60 CONTINUE SIGMA1 = DS0RT(SS1/(I - 1)) SIGMA2 = DSQRT(SS2/(I - 1)) WRITE (7,70) Y 70 FORMAT (1X, 'FIBRE DIAMETER = ', F8.6, 'CM'/) WRITE (7,80) AVE 1, SIGMA 1 80 FORMAT (1X, 'AVERAGE RE. H = ', E13.6, ' STD. DEV. =', E13.6/) WRITE (7,90) AVE2, SIGMA2 90 FORMAT (1X, 'AVERAGE DEFLEC. = ', E13.6, 'MICROMETERS 1 STD. DEV. =', E13.G/) GO TO 20 C C Set default values in IV() & V() 100 YOLD = Y CALL DFALT(IV, V) IV(1) =0 CALL NL2S0L(d, M, P, CALCR, CALCJ, IV, V, IPARM, RPARM, FPARM) C C Write the return code and solution WRITE (7,110) IV(1) 110 FORMAT (1X, ' RETURN CODE =', 12) WRITE (7,120) (P(K),K=1 ,M), V(10) 120 FORMAT (1X, ' SOLUTION:', 1P2G16.8/' SUM OF SQUARES/2 =', 1PG16.8) L = I FIX(SNGL(XX(J))) + 3 C(1) = 2 .0 D(1) = 2.0 DO 130 K = 1, L C(K + 1) = FLOAT(K) / 10 + 2. D(K + 1) = (P(1)*FL0AT(K)**P(2)) / 20 + 2. 130 CONTINUE CALL PL0T(2., 2., 3) CALL LINE(C, D, L, 1) DO 140 K = 1, J X1 = SNGL(XX(K)/10 +2.) ' C7\ Y1 = S N G L ( Y Y ( K ) / 2 0 + 2 . ) C A L L S Y M B 0 L ( X 1 , Y 1 , . 2 , 1, 0 . 0 , - 1 ) 140 C O N T I N U E GO TO 10 150 WRITE ( S , 1 6 0 ) 160 FORMAT ( 1 X , ' E N D OF DATA 4 E N C O U N T E R E D . ' ) C A L L PLOTND STOP 170 M = I + 6 WRITE ( 6 , 1 8 0 ) M 180 FORMAT ( I X , ' R E C O R D ' , 1 5 , ' IN F I L E 4 S K I P P E D . ' ) GO TO 2 0 END C F U N C T I O N RHO(W) RHO = 2 8 0 . 5 4 2 5 3 D - 1 2 * W RHO = ( 1 0 5 . 5 6 3 0 2 D - 0 9 - RHO) * W RHO = ( 4 6 . 1 7 0 4 6 9 D - 0 6 + RHO) * W RHO = ( 7 . 9 8 7 0 4 1 D - 0 3 - RHO) * W RHO = ( 1 6 . 9 4 5 1 7 6 D 0 - R H 0 ) * W RHO = ( 9 9 9 . 8 3 9 5 2 D 0 + R H 0 ) / ( 1 . + 1 6 . 8 7 9 8 5 D - 0 3 * W ) R E T U R N END C F U N C T I O N V I S ( W ) V I S = 1 . 3 2 7 2 D 0 * ( 2 0 . D 0 - W ) - . 0 0 1 0 5 3 D 0 * (W - 2 0 . D O ) * * 2 V I S = V I S / ( 1 0 5 . D 0 + W ) V I S = 1 . 0 0 2 D 0 * 1 0 . D O * * V I S / 1 0 0 0 R E T U R N END C C F U N C T I O N g ( t ) = f ( t ) - Y 1 c a l c u l a t i o n s S U B R O U T I N E C A L C R ( N , M, P, N F , R, IPARM, RPARM, FPARM) I M P L I C I T R E A L * 8 ( A - H , 0 - Z ) D I M E N S I O N P ( M ) , R ( N ) COMMON X X ( 5 0 ) , Y Y ( 5 0 ) C C T e s t f o r e x p o n e n t i a l o v e r f l o w I F ( P ( 2 ) . L T . 1 . D - 1 ) GO TO 2 0 C C P l a c e r e s i d u a l s i n R DO 10 I = 1 , N R ( I ) = P ( 1 ) * X X ( I ) * * P ( 2 ) - Y Y ( I ) 10 C O N T I N U E R E T U R N 2 0 NF = 0 R E T U R N END C C D e r i v a t i v e s d f / d A & d f / d T c a l c u l a t i o n s to Ln SUBROUTINE CALCJ(N, M, P, NF. D, IPARM, RPARM, FPARM) IMPLICIT REAL*8(A - H,0 - 2) DIMENSION P(M), D(N,M) COMMON XX(50), YY(50) C C Test for exponential overflow IF (P(2) .LT. 1.D-1) GO TO 20 C C Put derivatives into D DO 10 I = 1, N D(I, 1) = XX(I) ** P(2) D(I,2) = P(1) * D(I,1) * DLOG(XX(I)) 10 CONTINUE RETURN 20 NF = 0 RETURN END (b) Data on f i b r e flexing. Data for two types of nylon: 6 and 15 denier. Data i s l i s t e d in packets. Each l i n e 1n a packed contains Guilmont flowmeter reading (mL), i n i t i a l eye-piece reading, f i n a l eye-piece reading, and water temperature (degree Celsius). 6 DENIER FIBRES (d1ameter=0.0279mm) 0. .3000E+02 0. 9570E+02 0. 1292E+03 0. .2205E+02 0. .3000E+02 0, 9510E+02 0. 1254E+03 0. .2260E+02 0. .3000E+02 0. 9540E+02 0. 1248E+03 0. .2280E+02 0. .3000E+02 0. 9480E+02 0. 1154E+03 0. ,2310E+02 0. ,3000E+02 0. 9470E+02 0. 1208E+03 0. .2320E+02 0. .3000E+02 O. 9540E+02 0. 1179E+03 0. ,2330E+02 0. .3000E+02 0, 9460E+02 0. 1248E+03 0. .2330E+02 0. .3000E+02 0. 9500E+02 0. 1177E+03 0. . 2350E+02 o. .3500E+02 0. .9570E+02 0. 1391E+03 0. .2205E+02 0. 3500E+02 0. 9510E+02 0. 1404E+03 0. 2260E+02 0. ,3500E+02 0. 9540E+02 0. 1348E+03 0. 2280E+02 0. ,3500E+02 0. 9480E+02 0. 1245E+03 0. 2310E+02 0. .3500E+02 0. 9470E+02 0. 1296E+03 0. 2320E+02 0. .3500E+02 0. 9450E+02 0. 1223E+03 0. 2330E+02 0. .3500E+02 0. 9460E+02 0. 1337E+03 0. 2330E+02 0. 3500E+02 O. 9500E+02 0. 1262E+03 0. 2350E+02 0. 4000E+02 0. 9570E+02 0. 1514E+03 0. 2205E+02 0. ,4000E+02 0. 9510E+02 0. 1530E+03 0. 8000E+01 0. .4000E+02 0. 9540E+02 0. 1458E+03 0. 2280E+02 0. .4000E+02 0. 9480E+02 0. 1317E+03 0. 2310E+02 0. 4000E+02 0. 9470E+02 0. 1386E+03 0. 2320E+02 0. 4000E+02 0. 9540E+02 0. 1329E+03 0. 2330E+02 0. 4000E+02 0. 9460E+02 0. 1469E+03 0. 2330E+02 o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o ui U l U l J i & J i J> *. 4> J> J> CJ CJ CO CO CO CO CO CO CO O O O O O O O O O O o O O O O O O O O O O O O O O o O O o O O O o o O O O O O o O O O O O O O o O O o O o O o o O O O O o o O O O O m r n m m m m m m m m m m m m m m m m m m m m + + + + + + + + + + + + + + + + + + + + + + O O O O O O O O O O O O O O O O O O O O O O to IO M IO r o IO IO IO ro IO IO r o r o IO IO r o ro ro IO r o r o r o Ul a m o O O O O O O O O O O O O O O O O O O O O O z CO CO CD CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO m J i UI Ul Ul Ul Ul J i J > Ul Ul Ul Ul Ul J i J > J i J > Ul Ul Ul 72 CO r o CO CO O 00 Ul CO ro CO CO o CO Ul - J CO r o CO CO o O O o o O O O o O O O O o o O o o o O O O . - n m m m m m m rn m m m rn r n m m m m m m m m m m i—< + + + + + + + + + + + + + + + + + + + + + + 03 O O O O O O O O O O O O O O O O O O O O O O TO ro r o IO r o r o ro ro IO ro r o ro r o r o ro IO r o ro ro ro r o r o r o m U l U l U l U l oi U l U l U l U l U l U l U l U l U l U l U l J> -fc. .fc. -fc. J > J> J > J> - C U l U l U l U l u i U l U l U l O O O O O O O O U l U l U l U l U l U l U l u i O o O O O O O o O O O O O O O O O o O O O O O O O O O O O O O O o O O O O O O O O O o O o O O O O O O m m m m m m m m m m m m m m m m m m m m m m m m m + + + + + + +. + + + + + + + + + + + + + + + + + + O O O O O O O O O O O O O O O O O O O O O O O O O r o ro IO r o r o IO r o r o IO ro r o IO r o r o IO fO ro r o IO r o r o IO r o ro i o O O O O O O O O O O O O O O O O O O O O O O o O O CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO U l J > U l J > U l U l U l U l J > U l J i J > 01 U l U l U l -fc. U l U l U l U l U l O CD J > oo J i ~4 o cn j i - J oo - J o cn J i - j oo J > . - J O O o o O O o O O o O o o O O o o O O o O o O o O O m m m m m m m m m m m m m m r n m r n m m m m m m m m + + + + + + + + + + + + + + + + + + + + + + + + + O O O O O O O O O O O O O O O O O O O O O O O O O r o ro r o r o ro t o ro IO ro ro r o ro r o IO r o r o IO IO ro r o ro r o IO ro r o 0) O O O O O O O O O O O O O O O O O O O O O O 3 o o o o o o o o o o o o o o o o o o o o o o o o o CD O -i O _•. O O . A O O _k O O _k O CO O O O O O  00 -~1 CD ~ i oo j i _k cn oo j i •~1 U l cn CO CO J i cn J i U l J i U l ~4 j ) CO 03 03 oo oo 01 t o j i - j CO -k r o o O CO j i ro - J CO s CD CO CO r o - j j i U l ^1 oo j i -k -~1 j i cn O CD r o J i CO CO r o - J -~i J i CO J i O j i -k ^1 - j IO CO CO CO —k -~l O o CO -i O U l U l II -A CO CO U l O CO CO U l CO 00 U l -k 03 CO O ~ J CD r o U l o oo 03 IO CO m m m m m m m m m m m m m m m m m m m m m m O m m m m m r n m m m m r n m m m m m m r n rn m m rn m m + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + o O O O O O O O O O O O o O O O O O O O O O b O O O O O O O O O O O O O O O O O O O o O O O O CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO ro CO CO CO CO CO j i CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO O O O O O O O O O O O O O O O O O O O O O O ro ro i o ro ro co cn cn o O O O m m m + + + o o o ro ro r o r o r o ro ro r o ro ro ro co cn cn cn O O O O r n r n n m + + + -t-o o o o r o r o ro r o r o ro r o ro cn cn O O m m + + O O ro i o ro ro r o r o r o ro co r o cn cn o co O O O O m m m m + + + + o o o o i o i o r o r o r o ro r o ro cn cn O O m m + + O O ro i o i o i o i o i o ro i o cn cn cn O O O m m m + + + O O O r o ro r o ro r o ro r o i o r o co r o cn c n o co O O O O m m m m + + + + O O O O ro r o r o r o O O O O O O O O O O O O O O O O O O O O O O O O O r o r o r o r o r o r o i o r o r o N 3 i o i o r o i o i o r o i o r o r o r o i o r o r o i o r o c o c o c o c o c o r o r o r o c o c o c o c o c o i o i o r o c o c o c o c o c o r o r o r o c o u i c o c o r o - k O o c n o u i c o c o r o - k O o c n o u i c o c o r o - k O O c n o u i O O O O O O O U I O O O O O O O U I O O O O O O O U I O r T i r n r n r n r n r n r n r n r n r n r n r n r n r n i ^ r T i r n r n r n r n r n r h r n r n r n + + + + + + + + + + + + + + + + + + + + + + + + + O O O O O O O O O O O O O O O O O O O O O O O O O r o r o M r o r o i o r o r o r o r o r o r o r o r o r o r o i o i o r o r o r o r o r o i o r o L9Z 0.5000E+02 0.5000E+02 0.5000E+02 0.5000E+02 0.5000E+02 0.6000E+02 0.6000E+02 0.6000E+02 0.6000E+02 0.6000E+02 0.6000E+02 0.6000E+02 0.6000E+02 0.6000E+02 0.7000E+02 0.7000E+02 0.7000E+02 0.7000E+02 0.7000E+02 0.7000E+02 0.7000E+02 0.7000E+02 0.7000E+02 0.9470E+02 0.9450E+02 0.9480E+02 0.9500E+02 0.9510E+02 0.9530E+02 0.9530E+02 0.9520E+02 0.9430E+02 0.9470E+02 0.9450E+02 0.9480E+02 0.9500E+02 0.9510E+02 0.9530E+02 0.9530E+02 0.9520E+02 0.9430E+02 0.9470E+02 0.9450E+02 0.9480E+02 O.950OE+02 0.9510E+02 0.1197E+03 0.1093E+03 0.1067E+03 0.1222E+03 0.1158E+03 0.1299E+03 0.1258E+03 0.1164E+03 0.1288E+03 0.1345E+03 0.119GE+03 0.1146E+03 0.1361E+03 0.1299E+03 0.1490E+03 0.1432E+03 0.1248E+03 0.1414E+03 0.1522E+03 0.1532E+03 0.1294E+03 0.1488E+03 0.1433E+03 0.2260E+02 0.2260E+02 0.2260E+02 0.2260E+02 0.2260E+02 0.2290E+02 0.2300E+02 0.22GOE+02 0.2260E+02 0.2260E+02 0.2260E+02 0.2260E+02 0.2260E+02 0.2260E+02 0.2290E+02 0.2300E+02 0.2260E+02 0.22G0E+02 0.2260E+02 0.2260E+02 O.2260E+02 0.22G0E+02 0.2260E+02 (d) Micrometer eye-piece c a l i b r a t i o n . F i r s t l i n e - Real distance in mm on c a l i b r a t i o n scale. Second l i n e - Eyepiee readings in micrometer units. 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 59.4 120.0 178.9 237.6 296.1 356.2 413.4 474.4 533.9 594.0 The straight l i n e f i t by least squares regresion gave: R=0.999; intercept a=0.0pm; slope b=1.eSSS^m/unit. (e) GILMONT flowmeter c a l i b r a t i o n . FLOWMETER TIME INTERVAL FLOW RATE READING MIN:SEC mL/s 66 2:32 3.29 60 2:48 2 .97 55 3:07 2 .67 50 3:29 2.39 45 3:58 2. 10 40 4:34 1 .82 35 5:26 1 .53 30 6:33 1 .27 25 8: 13 1 .01 20 11:01 0.762 15 17: 16 0.482 10 40:57 0.203 The straight l i n e f i t by least squares regresion produced: R=0.999; intercept a=-.03606mL/s; slope b=0.05507mL/s/f1owmeter (f) Average deflections and bulk Reynolds numbers at each flow rate. Results of power curve f i t s . FIBRE DIAMETER fjtn AVERAGE FIBRE DEFLECTION y m^ AVERAGE BULK REYNOLDS NUMBER POWER CURVE FIT b y=a Re b a b sum of squares 27.95 45 . 37 61 . 28 78.53 103.4 129 . 3 177 . 3 21 .28 25.82 29. 17 34 .90 39.44 43.98 .10314 1 .9580 185 .5 44. 19 14.40 22.42 33.38 52.71 80.75 33.42 47.66 61.91 76. 16 90.41 .00745 2.0568 38 . 23 to 0-1 Appendix XXII. Data Acquisition Program for Flow Velocity Measurements a) Main program written in BASIC. 10 REM $LINESIZE: 132 $PAGESIZE: 55 20 WIDTH 80:CLS 30 PRINT "DATA COLLECTION PROGRAM FOR LDA ANALYSIS OF FLOW" 40 PRINT " IN THE HORIZONTAL ROTATING CYLINDER." 50 PRINT 60 PRINT "HARDWARE: Dual-beam LDA, Brag c e l l , frequency s h i f t , " 70 PRINT " IBM personal computer," 80 PRINT " Tecmar Lab Master A/D & D/A converter," 90 PRINT " (0 channel single-ended, gain=10 -I.V to +1.V)" 100 PRINT 110 DEFINT I,J,K 120 DIM A%(25000),L$(3) 130 INPUT "Enter your name. ",NAM$ 140 PRINT "Enter: suspending l i q u i d density, suspension concentration, cylinder" 150 PRINT "rotational speed, LDA orientation, tracker range, frequency s h i f t . " 160 INPUT L$(1) 170 INPUT "Enter the number of data points to c o l l e c t . (<=25000) ",N% 180 INPUT "Enter the number of data points/second desired.(<=30000) ",S 190 INPUT "Enter the name of the f i l e in which data w i l l be stored.",FILN$ 200 D$=DATE$: T$=TIME$ 210 OPEN FILN$ FOR OUTPUT AS #2 220 PRINT #2,"NAME:";NAM$,"DATE:";D$,"TIME:";T$ 230 PRINT #2,L$( 1 ) 240 PRINT #2,"NUMBER OF DATA POINTS:";N%,"SAMPLING RATE:";S;"PTS/SEC" 250 PRINT #2,"Data stored 1n lines containes: radial coordinate-mm," 260 PRINT #2,"angular coordinate-deg., average voltage, RMS of voltage" 270 PRINT #2,"fluctuations." 280 C%=0 'A/D channel number 290 S%=S: IF S<1 THEN S%=-1!/S 'Convert to proper format for timer routine 300 P%=1:IF S%>2000 THEN P%=0 'Plot i f < 2000 pts/sec 310 REM ***** CALIBRATION PROCEDURE 320 CLS 330 PRINT "Prepare tracker for c a l i b r a t i o n . " 340 PRINT "Set zero voltage on the tracker and press any key." 350 I$=INKEY$:IF 1$="" THEN 350 360 MUL%=INT(N%/15): NN%=MUL%*15 370 F%=0 ' I n i t i a l i z e overrun f l a g 380 CALL TIMER(A%( 1 ) ,F%,P%,NN%,C%,S%) 390 IF F%<>0 THEN PRINT "Warning--data taken too fast!":NN%=NN%-F% 400 SUMO=0 410 FOR 1=1 TO NN% STEP 15 420 SUM1=0 430 K=I: KK=I+14 440 FOR d=K TO KK ^ O 450 IF A%(J)>32767 THEN A%(d)=A%(d)-65535! 460 SUM1=SUM1+A%(d) 470 NEXT d 480 SUM1=SUM1/15 490 SUM0=SUM0+SUM1 500 NEXT I 510 SUM0=SUM0/MUL% 520 PRINT "SUM0=",SUM0 'di g i t s 530 PRINT "MUL%=",MUL% 'H of sets of 15 readings 540 INPUT "Do you wish to repeat calibration? ",Y$ 550 IF Y$="y" OR Y$="Y" GOTO 320 560 PRINT "Set -traversing r i g to new coordinates." 570 INPUT "Radius in milimeters. ",RAD% 580 INPUT "Angle FI in degrees. ",ANG% 590 PRINT "Press any key to start data c o l l e c t i o n . " 600 I$=INKEY$: IF 1$="" THEN 600 610 F%=0 ' I n i t i a l i z e overrun f l a g 620 CALL TIMER(A%(1),F%,P%,NN%,C%,S%) 630 IF F%<>0 THEN PRINT "Warning—data taken too fast!":NN% 640 SUM2=0 650 FOR 1=1 TO NN% STEP 15 660 SUM1=0 670 K=I: KK=I+14 680 FOR d=K TO KK 690 IF A%(d) >32767 THEN A%(d)=A%(d)-65535! 700 SUM1=SUM1+A%(d) 710 NEXT d 720 SUM1=SUM1/15 730 SUM2=SUM2+SUM1 740 NEXT I 750 SUM2=SUM2/MUL% 760 PRINT "SUM2=",SUM2 'di g i t s 770 PRINT "MUL%=",MUL% '# of sets of 15 readings 780 DIG=SUM2-SUM0 790 SUM3=0 800 FOR 1=1 TO NN% STEP 15 810 SUM1=0 820 K=I: KK=I+14 830 FOR d=K TO KK 840 SUM1=SUM1+(SUM2-A%(d))a2 850 NEXT d 860 SUM1=SUM1/15 870 SUM3=SUM3+SUM1 880 NEXT I 890 SUM3=SUM3/MUL% 900 SUM3=SQR(SUM3)/2048 910 DIG=DIG/2048 920 PRINT "DIGVOL=",DIG 930 PRINT "SORT=";SUM3;"VOLTS" 940 INPUT "Do you wish to store results in the f i l e ? ",Y$ N%-F% N> H 950 IF Y$="y" OR Y$="Y" GOTO 1000 960 Y$="" 970 INPUT "Next coordinates? ",Y$ 980 IF Y$="y" OR Y$="Y" GOTO 560 990 GOTO 1020 1000 PRINT #2,RAD%,ANG%,DIG,SUM3 1010 GOTO 960 1020 END b) Assembler subroutine for INTEL 8088 microprocessor. PAGE ,132 TITLE TIMER SUBROUTINE TO DO TIMED DATA COLLECTION FROM TECMAR A/D/A BOAR CALL FROM BASIC WITH CALL OF FORM: CALL TIMER (A%(1),F%,P%,N%,C%,S%) WHERE A% IS ARRAY WHERE DATA ARE TO BE STORED IS OVERRUN FLAG--SET TO ZERO UPON NORMAL EXIT OTHERWISE SET TO VALUE OF CX REGISTER TO GIVE NUMBER OF POINTS NOT COLLECTED IS 0 TO OMIT REAL-TIME PLOT, OTHER TO PLOT IS NUMBER OF POINTS TO BE COLLECTED IS CHANNEL NUMBER OF A/D IS NUMBER OF DATA POINTS PER SECOND S% MUST BE <= SPEED OF A/D IF S% <0 THAT MEANS WE WANT THAT MANY SEC/POINT F% P% N% C% S% CSEG HEADER: SEGMENT ASSUME CS:CSEG, DS:NOTHING DB OFDH CODE FOR BLOAD FILE DW 0 DW 0 DW RTNLEN TEMP DW ? TEMP. STORAGE PLOT DW ? PLOT FLAG TEMPSI DW ? TEMP. STORAGE FOR SI REGISTER OVRUN DW ? OVERRUN OF A/D FLAG ;DEFINITIONS: ADDO = 1808 I/O ADDRESS OF TECMAR BOARD ADD4 =ADD0+4 A/D CONTROL BYTE ADD5 =ADD0+5 A/D CHANNEL NUMBER ADD6 =ADD0+6 SOFTWARE START CONVERSION ADD8 =ADD0+8 TIMER 9513 DATA PORT ADD9 =ADD0+9 PAGE TIMER 9513 CONTROL PORT TIMER PROC FAR PUSH BP ;SAVE BP to MOV B P . S P ;SET BASE PARAMETER LIST MOV D I , [ B P ] + 6 ;GET DATA P O I N T S / S E C . MOV A X , [ D I ] ; INTO BX REGISTER MOV B X . A X MOV D I , [ B P ] + 8 ;GET CHANNEL NUMBER MOV A X , [ D I ] ; AND STORE AS AX MOV DX.ADD5 ; AND OUTPUT TO A / D OUT D X , A L ; (USE ONLY LOWER BYTE) MOV D I , [ B P ] + 1 0 ;GET NUMBER OF DATA POINTS MOV C X , [ D I ] ; STORE IN CX REGISTER MOV DI , [BP]+12 ;GET PLOT FLAG MOV A X , [ D I ] ; STORE IN MEMORY MOV P L O T , A X MOV A L . 1 2 8 ; S E L E C T A/D MODE (DISABLE AUTOINCREMENT, MOV DX.ADD4 ; EXTERN. START CONVERSION, ALL INTERRUPTS OUT D X , A L ; GAIN=1) MOV A X , 0 ;SI IS X-VALUE OF POINT TO BE MOV TEMPSI .AX ; P L O T T E D - - S A V E FOR LATER MOV A X , 6 ;SET UP HIGH RESOLUTION GRAPHICS MODE INT 10H MOV DX,ADDS ;RESET DONE F L I P - F L O P OF A / D IN A L , D X MOV DX,ADD9 ;SET DATA POINTER TO MASTER MODE REGISTER MOV A L , 2 3 OUT D X , A L MOV DX, ADD8 ;SET MASTER MODE REGISTER FOR SCALER CONTROL = MOV A L . O ; BCD DIVISION, ENABLE INCREMENT, 8 - B I T BUS, OUT D X , A L ; FOUT ON, DIVIDE BY 18, SOURCE=F1, MOV A L . 1 2 8 ; COMPARATORS DISABLED, TOD DISABLED OUT DX, AL MOV DX.ADD9 ;SET DATA POINTER TO COUNTER MODE OF MOV A L , 5 ; REGISTER 5 OUT D X , A L MOV DX.ADD8 ;SET COUNTER 5 FOR COUNT R E P E T I T I V E L Y , MOV A L , 3 3 ;BINARY COUNT, COUNT DOWN, ACTIVE HIGH OUT D X , A L ; T C , DISABLE SPECIAL GATE, RELOAD FROM LOAD, CMP BX.31 ;CHECK IF >=31 POINTS/SEC JGE FAST ; I F SO, JUMP TO FAST CMP B X , 0 ;CHECK IF >0 POINTS/SEC UG MED ; I F SO JUMP PAGE ; BRANCH TO HERE IF POINTS/SEC < 0 , IT MEANS THAT WE WANT ; LESS THAN ONE P O I N T / S E C . SLOW: MOV A L . 1 5 ;SET TO 100 Hz (NO G A T E , RISING EDGE OUT D X , A L ; OF F5) NEG BX ;GET ABSOLUTE VALUE OF BX MOV A X . B X ;AND MULTIPLY BY 100 TO GET COUNT MOV D I . 1 0 0 MUL D l JMP GO ;BRANCH TO HERE FOR 31 TO 20000 P O I N T S / S E C - - U S E 1 MHz CLOCK FAST : MOV A L , 1 1 COUNT AT 1 MHz (NO G A T E , RISING OUT DX, AL EDGE OF F1) MOV AX,10000 DIVIDE 1000000 BY P T S / S E C BY MOV D l , 1 0 0 GETTING 10F6 INTO DX+AX MUL D l DIV BX BX=PTS/SEC; RESULT IN DX+AX. BUT IGNORE DX, SINCE DX=0 CMP A X , 2 0 0 DISABLE INTERRUPTS IF >=5000 JG FAST2 POINTS/SEC CLI F A S T 2 : JMP GO ; BRANCH TO HERE FOR 1 TO 30 P O I N T S / S E C - - U S E 10kHz CLOCK MED: MOV A L , 13 COUNT AT 10kHz (NO G A T E , RISING OUT DX, AL EDGE OF F3) MOV AX,10000 CALCULATE NUMBER OF TICKS OF 10000 Hz CLOCK CWD PER DATA POINT BY DIVIDING DIV BX 10000 BY P T S / S E C ;START CLOCK TICKING AT DESIRED RATE GO: MOV DX,ADD8 AND LOAD COUNTER 5 WITH TICKS DEC AX (COUNT TO ZERO, DECREMENT AX OUT DX, AL FOR CORRECT COUNT) MOV A L , AH OUT DX, AL 8 BITS AT A TIME MOV D l , [ B P ] + 1 4 ;GET OVERRUN FLAG ADDRESS MOV WORD PTR [ D l ] , 0 ;ZERO THE FLAG MOV OVRUN.DI AND STORE THE FLAG ADDRESS MOV D l , [ B P ] + 1 6 GET ADDRESS OF DATA ARRAY MOV DX,ADD9 LOAD COUNTER 5 FROM LOAD REGISTER MOV A L , 1 12 AND ARM (START COUNTING) OUT DX, AL MOV DX.ADD4 ENABLE EXTERNAL START (PINES 3 ,4 OF MOV A L , 1 3 2 J2 CONNECTOR MUST BE CONNECTED) OUT DX , AL PAGE ;BEGIN DATA COLLECTION; COLLECT UPON EXTERNAL START TRIGGER DONE : MOV DX,ADD4 ;CHECK IF DATA READY IN A L . D X CMP A L , 1 2 8 BY CHECKING READY BIT (BIT 7) JB DONE LOOP UNTIL READY TEST A L , 6 4 SEE IF DATA OVERRUN FLAG SET JNE ERRMESS IF SO, NOTIrY BASIC PROGRAM AND EXIT MOV DX,ADD5 Y E S , DONE, SO GET LOW BYTE OF DATUM IN A L , DX MOV [ D l ] , A L AND STORE IT INC D l GO TO NEXT LOCATION IN ARRAY (1 BYTE LATER) MOV DX,ADDS GET HIGH BYTE AND STORE IT IN A L , DX MOV [ D l ] , A L INC D l to CMP PLOT.O ;DON'T PLOT IF PLOT FLAG=0 JZ NOPLOT ;PLOT ROUTINE STARTS HERE MOV TEMP.CX ;SAVE CX FIRST MOV AH,AL ;GET HIGH BYTE JUST TAKEN MOV AL,[DI-2] ;AND LOW BYTE FROM STORAGE SO AX=DATUM ADD AX,2047 ;CALCULATE Y-VALUE TO PLOT = CWD ; 199-((DATUM+2047)/21) MOV BX.21 ;DIVIDE BY 21--QU0TIENT IN AX DIV BX MOV DX.AX ;RESULT INTO DX NEG DX ;NEGATE AND ADD TO 199 ADD DX.199 MOV SI.TEMPSI ;GET X-VALUE OF LAST POINT ON SCREEN INC SI ;G0 TO NEXT LOCATION ON SCREEN CMP SI,640 ;TEST IF AT RIGHT EDGE OF 640x200 JL M1 ; SCREEN MOV SI.O ;IF SO, GO TO LEFT EDGE TO PLOT M1: MOV CX,SI ;GET X-VALUE INTO CX MOV TEMPSI,SI ;SAVE X-VALUE MOV AX,3073 ;AH=12,AL=1 TO WRITE DOT TO SCREEN INT 10H ;PLOT POINT MOV CX.TEMP ;RESTORE CX NOPLOT: LOOP DONE ;DECREMENT CX AND LOOP IF >0 ;BRANCH TO HERE UPON FINISH OR OVERRUN NOGO: MOV DX,ADD4 ;TURN OFF A/D MOV AL,0 OUT DX,AL STI ;RESTORE INTERRUPT SERVICE POP BP ;RESTORE BP RET 12 ;6 ARGUMENTS IN CALL X 2=12 ERRMESS: MOV DI.OVRUN ;SET OVERRUN FLAG SINCE A/D GOING MOV WORD PTR [DI],CX ;TOO FAST JMP NOGO TIMER ENDP RTNLEN EOU $-TEMP ;LENGTH OF "OUTINE FOR HEADER CSEG ENDS END HEADER ;NEEDED FOR A .BIN FILE CONVERSION This subroutine was taken from [G3] Appendix XXI11. Data Acquisition Program for Tensile Tests of Wet, Type-C Nylon Floes. a) Main program written in BASIC. 10 20 30 40 50 60 70 80 90 100 1 10 120 130 140 REM $LINESIZE: 132 $PAGESIZE: 55 WIDTH 80:CLS "DATA COLLECTION PROGRAM FOR WET TENSILE STRENGTH" OF NYLON FIBRE AGGREGATES." "HARDWARE: PRINT PRINT PRINT PRINT PRINT " PRINT " PRINT " PRINT " PRINT " PRINT " PRINT DEFINT I,J,K 150 DIM A%(4000) 160 INPUT "Enter INPUT INPUT INPUT INPUT Thwing-Albert t e n s i l e tester," IBM personal computer," 20 Newtons load c e l l , " B-2-F Bofors Electronic amplifier," (10V excitation, 0-15mV Tecmar Lab Master A/D & (O channel single-ended s e n s i t i v i t y range)" D/A converter," , gain=10 -1.V to +1.V)" 170 180 190 200 210 C%=0 220 S%=S: 230 240 250 260 270 280 310 320 330 340 350 360 370 380 390 400 B(4000),SG%(9),L$(3) your name. ",NAM$ the sample i d e n t i f i c a t i o n (type of nylon). ",S$ the number of data points to c o l l e c t . (<=4000) ",N% the number of data points/second desired.(<=30000) ",S cross-head speed in centimeters/minute. ",CHS channel number 'Convert to proper format for timer routine 'Plot i f < 2000 pts/sec PROCEDURE "Enter "Enter "Enter "Enter 'A/D IF S<1 THEN S%=-1!/S P%=1:IF S%>2000 THEN P%=0 REM ***** LOAD CALIBRATION CLS PRINT "Prepare tester for load c a l i b r a t i o n . " PRINT "Set zero load on the load c e l l and press I$=INKEY$:IF 1$="" THEN 280 290 NUM=N%/10: MUL%=INT(NUM/15+1): NN%=MUL%*15 300 F%=0 ' I n i t i a l i z e overrun f l a g CALL TIMER(A°/,( 1 ) , F%, P%, NN%, C%, S%) IF F%<>0 THEN PRINT "Warning--data taken too fast! SUMO=0 FOR 1=1 TO NN% STEP 15 SUM1=0 K=I: KK=I+14 FOR J = K TO KK IF A%(J)>32767 THEN A%(J)=A%(0)-65535 ! SUM1=SUM1+A%(J) NEXT J 410 SUM1=SUM1/15 420 SUM0=SUM0+SUM1 430 NEXT I 440 SUM0=SUM0/MUL% any key. :NN%=NN%-F% to 450 PRINT "SUM0=",SUM0 460 PRINT "MUL%=",MUL% 470 PRINT "Hang c a l i b r a t i o n load on the load c e l l and" 480 INPUT "enter this load in grams. ",FSL% 490 F%=0 ' I n i t i a l i z e overrun f l a g 500 CALL TIMER(A%(1),F%,P%,NN%,C%,S%) 510 IF F%<>0 THEN PRINT "Warning--data taken too fast!":NN%=NN%-F% 520 SUM2=0 530 FOR 1=1 TO NN% STEP 15 540 SUM1=0 550 K=I: KK=I+14 560 FOR d=K TO KK 570 IF A%(J)>32767 THEN A%(J)=A%(J)-65535! 580 SUM1=SUM1+A%(J) 590 NEXT J 600 SUM1=SUM1/15 610 SUM2=SUM2+SUM1 620 NEXT I 630 SUM2=SUM2/MUL% 640 PRINT "SUM2=",SUM2 650 PRINT "MUL%=",MUL% 660 CAL=(FSL%/(SUM2-SUM0))*100 'centigrams/one d i g i t 670 PRINT "CAL=",CAL 680 INPUT "Do you wish to repeat calibration? ",Y$ 690 IF Y$="y" OR Y$="Y" GOTO 240 700 PRINT "Run the cross-head and start data ac q u i s i t i o n as the upc 710 PRINT "Press any key to start the dead run data c o l l e c t i o n . " 720 I$=INKEY$:IF 1$="" THEN 720 730 F%=0 ' I n i t i a l i z e overrun f l a g 740 CALL TIMER(A%(1),F%,P%,NN%,C%,S%) 750 IF F%<>0 THEN PRINT "Warning--data taken too fast! ":NN%=NN%-F% 760 SUM3=0 770 FOR 1=1 TO NN% STEP 15 780 SUM 1=0 790 K=I: KK=I+14 800 FOR J=K TO KK 810 IF A%(d)>32767 THEN A%(d)=A%(d)-65535! 820 SUM1=SUM1+A%(d) 830 NEXT d 840 SUM1=SUM1/15 850 SUM3=SUM3+SUM1 860 NEXT I 870 SUM3=SUM3/MUL% 880 SU%=INT((SUM3+2048)*CAL) 890 INPUT "Do you wish to repeat the last run? ",Y$ 900 IF Y$="y" OR Y$="Y" GOTO 700 910 REM ***** DATA COLLECTION 920 CLS 930 PRINT "Prepare sample for testing." 940 INPUT "Enter aggregate description. ",L$(1) comb clears the lower comb." to 950 PRINT "Press any key to start data c o l l e c t i o n . " 960 I$=INKEY$:IF 1$="" THEN 960 970 F%=0 ' I n i t i a l i z e overrun f l a g 980 CALL TIMER(A%(1),F%,P%,N%,C%,S%) 990 IF F%<>0 THEN PRINT "Warning--data taken too fast!":N%=N%-F% 1000 FOR 1=1 TO N% 1010 IF A%(I)>32767 THEN A%(I)=A%(I)-65535! 1020 A%(I)=INT((A%(I)+2048)*CAL) 'optimum resolution data storage, centigrams at A/D s e n s i t i v i t y range -1.0V to +1.0V 1030 NEXT I 1040 INPUT "Enter comments about finished test. ",L$(2) 1050 CLS: PRINT "Enter a 1 i f you wish to recalibrate the load." 1060 PRINT "Enter a 2 to plot data on the screen." 1070 PRINT "Enter a 3 to store data in a f i l e . " 1080 PRINT "Enter a 4 to smooth the data." 1090 PRINT "Enter a 5 to do next test." 1100 PRINT "Enter a 6 to e x i t . " 1110 INPUT Y 1120 ON Y GOSUB 240,1160,1330,1500,910,1580 1130 GOTO 1050 1140 ***********SUBROUTINES********** 1150 REM Screen ploting routine. 1160 SCREEN 2: KEY OFF 1170 DEF FNSCALE(Z%)=180!-180!*(Z%-YMIN%)/(YMAX%-YMIN%) 1180 CLS:YMAX%=A%(1): YMIN%=A%(1) 1190 FOR 1=1 TO N% 1200 IF A%( I ) <YMIN% THEN YMIN%=A%(I) ELSE IF A%( I )>YMAX°/= THEN YMAX%=A%(I) 1210 NEXT I 1220 PRINT "YMIN=";YMIN°/o, "YMAX=";YMAX%, "ZER0=";SU% 1230 YPLOT=FNSCALE(A%(1))+10 1240 PSET (60,YPL0T),0 1250 FOR 1=2 TO N%: XPL0T=60!+579!*(I- 1)/(N%-1): YPLOT = FNSCALE(A%(I))+10: LINE -(XPLOT,YPLOT) : NEXT I 1260 LOCATE 25,20:PRINT "Samp1e-";L$(1),S%;"p/s",CHS;"cm/min";: PSET (60,0): LINE (60,10)-(639,190),,B 'Label and box plot 1270 YPL0T=FNSCALE(SU%)+10 1280 PSET(60,YPLOT),0: LINE -(639,YPLOT) 1290 LOCATE 25,8: PRINT "1";: LOCATE 25,75: PRINT N%;: LOCATE 2,1: PRINT YMAX%;:LOCATE 12,1: PRINT "cg";:LOCATE 24,1: PRINT Y 1300 Y$=INKEY$: IF Y$="" THEN 1300 1310 RETURN 1320 REM Subroutine to store data in f i l e 1330 INPUT "Enter the name of the f i l e in which data w i l l be stored. ",FILN$ 1340 D$=DATE$: T$=TIME$ 1350 OPEN FILN$ FOR OUTPUT AS #2 1360 PRINT #2,"NAME:";NAM$,"DATE:";D$,"TIME : " ; T$ 1370 PRINT #2, S$ 'Save sample description 1380 PRINT #2,"NUMBER OF DATA POINTS:";N%,"SAMPLING RATE:";S%;"PTS/SEC" 1390 PRINT #2,"CROSS-HEAD SPEED=";CHS;"cm/min","ZERO LOAD=";SU%,"CALIBRATION LOAD=";FSL%;"g" 1400 FOR 1=1 TO 2: PRINT #2,L$(I): NEXT I 1410 FOR 1=1 TO N%: WRITE #2,A%(I): NEXT I 1420 CLOSE HI: RETURN 1430 REM Subroutine to compute second-order 9-point Savitzk1-Golay smooth 1440 REM including smoths at both beginning and and of data ~ j oo 1450 REM It computes a "smoothed" value for each point by adding together 1460 REM the 4 points on either side of i t , plus i t s e l f , each multiplied 1470 REM by the corresponding c o e f f i c i e n t . 1480 REM It then computes the "smoothed value for each successive point 1490 REM using the or i g i n a l data array. 1500 DATA -21,14,39,54,59,54,39,14,-21: ' Savitzki-Golay c o e f f i c i e n t s 1510 RESTORE 1520 FOR 1=1 TO 9: READ SG%(I): NEXT I 1530 FOR 1=1 TO N%: B(I)=0: DF%=0: FOR d=-4 TO +4: IF I+d<1 OR I+d>N% THEN 1550 1540 ZZ=A%(I+d): B(I ) =B(I) + ZZ*SG%(d+5): DF%=DF%+SG%(d+5) 1550 NEXT J: B(I)=B(I)/DF%: NEXT I 'Divide by sum of the c o e f f i c i e n t s used 1560 FOR 1=1 TO N%: A%(I)=B(I): NEXT I 'Store back in the original array 1570 RETURN 1580 END b) The assembler subroutine TIMER c a l l e d from main program has already been l i s t e d in Appendix XXII. Appendix XXIV. Data C o l l e c t i o n Program for Determination of Break Area. 10 REM BASICA/D, LOAD AREA2, RUN or BASICA AREA2/D - Run commands 20 REM THIS PROGRAM CALCULATES AN AREA OF A REGULAR CLOSED SHAPE 30 REM BY POLYGONAL APPROXIMATION. 40 CLS:KEY OFF:CLOSE 'Clear screen, soft keys off, close a l l f i l e s or devices 50 DEFDBL 0-Z 'Double-precision variables 60 DEFINT I-N 'Integer variables 70 OPTION BASE 1 'The lowest value any array subscript has is one 80 DIM XX(100),YY(100) 'Coordinates of points along contour 90 LOCATE 4,1,1,0,7 'Screen location, block coursor on 100 INPUT "INPUT EXPERIMENT DESCRIPTION. ",F$ 110 OPEN "C0M1:9600" AS tt\ 'Open communication port with SUMMAGRAPHICS MICROGRID tablet, OPEN statement is short because of the intentional DIP switches setting on the tablet to f i t BASICA's defaults 120 PRINT #1,CHR$(27)"Z" 'Tablet reset 130 PRINT #1,CHR$(27)"M1",CHR$(27)"R3" 'Point mode, 10 coordinates/second 140 PRINT #1,CHR$(27)"FO" 'Origin in the lower l e f t corner 150 PRINT #1,CHR$(27)"C3" 'Resolution - metric-high, 40 points/m 160 LOCATE 6, 1 170 INPUT "INPUT THE SIZE OF THE CALIBRATION SCALE IN MILIMETERS : ",C 180 LOCATE 25,1 190 INPUT "DO YOU WANT TO CHANGE THIS VALUE, [Y] OR [N] ";A$ 200 A=ASC(A$):IF A=89 OR A=121 GOTO 160:A$="" 210 CLS :LOCATE 8,1 220 PRINT "INPUT END COORDINATES OF THE CALIBRATION SCALE." 230 PRINT "PRESS ANY BUTTON ON THE CURSOR FOR EACH COORDINATE." 240 INPUT #1,XX( 1 ) ,YY(1 ) ,FF,N 250 PRINT XX(1),YY(1),FF,N 260 PRINT #1,CHR$(27) "L11 " 'Yellow LED on the coursor is on 270 INPUT #1,XX(2) ,YY(2),FF,N 280 PRINT XX(2),YY(2),FF,N 290 PRINT #1,CHR$(27) "L10" 'Yellow LED is off 300 LOCATE 25, 1 310 INPUT "DO YOU WANT TO REPEAT THE CALIBRATION PROCEDURE, [Y] OR [N] ";A$ 320 A=ASC(A$):IF A=89 OR A=121 GOTO 210:A$="" 330 LOCATE 25.1 340 PRINT " 350 SS = SQR((XX(1 )-XX(2))**2+(YY(1)-YY(2))**2) 'Calibration distance, tablet units 360 SS=SS/C 'Calibration constant, units/mm 370 LPRINT 380 LPRINT F$ 390 LPRINT "SCALE LENGTH=";C;"mm","CALIBRATION CONSTANT=";SS;"units/mm" 400 LOCATE 11,1 410 PRINT "SCALE LENGTH=";C;"mm","CALIBRATION CONSTANT=";SS;"uni ts/mm" 420 LPRINT 430 INPUT "FLOC NUMBER: " ,FLOC$ 440 LPRINT "FLOC NUMBER:";FLOCS 450 CLS :LOCATE 14,1 460 PRINT "INPUT COORDINATES OF A POINT APPROXIMATELY AT THE CENTRE OF THE AREA" O 470 480 490 500 510 520 530 540 550 560 570 580 590 600 610 620 630 640 650 660 670 680 690 700 710 720 730 740 750 760 770 780 790 800 810 820 830 840 850 860 870 880 890 900 910 920 930 940 950 960 'Set the o r i g i n with the coursor 'Yellow LED on PRINT #1,CHR$(27)"F1" INPUT #1,XX(3) ,YY(3),FF,N PRINT #1,CHR$(27)"L11" PRINT XX(3),YY(3),FF,N PRINT CHR$(10) "INPUT COORDINATES OF POINTS ALONG.THE CONTOUR." "PLACE POINTS AT THE IMAGINARY CORNERS OF A POLYGON" "WHICH CLOSELY APPROXIMATES THE CONTOUR." PI=3.141592654**2: IFLAG=0 SAREA=0#: PRINT PRINT PRINT SANGLE=0# SBETA=0# 1=4 INPUT #1,XX(I ) , YY(I),FF,N PRINT #1,CHR$(27)"L10" 1 = 1 + 1 INPUT #1,XX(I ) , YY(I),FF,N J = I-1 SA = SQR((XX(J) )**2+(YY(J))**2) SB = SQR((XX(I ) )**2+(YY(I))**2) IF SA>SB GOTO 820 S = YY(I)/XX(I) : S=SGN(S)*S X1=SQR(SA**2/(1+S**2)) Y1=X1*S IF SGN(XX(I )) = -1 THEN X1 = -X1 IF SGN(YY(I))=-1 THEN Y1=-Y1 SD=SQR((XX(J)-X1)**2+(YY(d)-Y1)**2)/2 X2 = (X1+XX(d) )/2 Y2=(Y1+YY(J))/2 SE=S0R(X2**2+Y2**2) SBETA=ATN(SD/SE)*2 SH=SIN(SBETA)*SA SANGLE=SANGLE+SBETA IF IFLAG=1 GOTO 980 IF SANGLE>PI THEN XX(I)=XX(4): SAREA=SAREA+SB*SH/2 GOTO 600 S = YY(J)/XX(J) : S = SGN(S)*S X1=S0R(SB**2/( 1+S**2)) Y1=X1*S IF SGN(XX(J) ) = -1 THEN X1=-X1 IF SGN(YY(J ) ) = - 1 THEN Y1 = -Y1 SD = SQR((XX(I )-X1 )**2+(YY(I)-Y1) X2=(X1+XX(I))/2 Y2=(Y1+YY(I ) )/2 SE=S0R(X2**2+Y2**2) SBETA=ATN(SD/SE)*2 SH=SIN(SBETA)*SB SANGLE=SANGLE+SBETA IF IFLAG= 1 GOTO 1000 IF SANGLE>PI THEN XX(I)=XX(4): SAREA=SAREA+SA*SH/2 'F i r s t point 'Yellow LED off 'Distance from the o r i g i n to point A 'Distance from the o r i g i n to point B 'Slope of OB, always pos i t i v e 'Point on OB and OA distant from 'the o r i g i n 0 'Distance from A to C 'Distance from o r i g i n 0 to C 'Angle between OA and OB 'Distance between A and D 'Sum of angles YY(I)=YY(4): IFLAG=1 : GOTO 620 Slope of OB, always positive 'Point on OA and OB distant from the o r i g i n 2)/2 'Distance between B and C 'Distance from the o r i g i n 0 to C 'Angle between OA and OB 'Sum of angles YY(I)=YY(4): IFLAG=1: GOTO 620 CO 970 GOTO 600 980 K=I-3: SAREA=SAREA+SB*SH/2 990 GOTO 1010 1000 K=I-3: SAREA=SAREA+SA*SH/2 1010 PRINT #1,CHR$(27)"A" 'Beeper 1020 LPRINT "THERE WAS ";K;" POINTS ON THE CONTOUR." 1030 PRINT CHR$(10) 1040 SAREA=SAREA/SS/SS 'Area 1n mm**2 1050 AREA=CSNG(SAREA) 'Convert to single-precision 1060 PRINT "AREA IS " ; AREA ; " mm**2" 1070 LPRINT "AREA IS ";AREA; " mm**2" 1080 LPRINT 1090 PRINT "DO YOU WANT TO CONTINUE WITH THE NEXT AREA?" 1100 PRINT "IF NOT, PRESS F ON THE CURSOR." 1110 PRINT "IF YES, PRESS ANY OTHER BUTTON." 1120 INPUT #1,X,Y,F,N 1130 IF F=16 THEN CLOSE #1 ELSE GOTO 430 1140 LOCATE 24,1,0:KEY ON 'Coursor o f f , soft keys on 1150 END M oo to Appendi x XXV. FORTRAN Program Processing Conductivity Data. Data L i s t i n g . C This program f i t s straight line to conductivity data C and makes a plot of conductivity versus KCI solution C concentration. C Input is taken from (4). C Output is directed to (7). IMPLICIT REAL*8(A - H,0 - Z), INTEGER(I - N) INTEGER FLAG, LIST(1) DIMENSION A(100). B(100) DIMENSION S(3), SIGMA(3) REAL*4 X1, Y1 LOGICAL LK DATA LIST /'*'/ LK = .TRUE. CALL AXCTRL('SYMS', .2) CALL AXCTRL('LABE', 1) CALL AXCTRL('LOGS', 1) CALL AXCTRL( 'XORI', 2.0) CALL AXCTRL( ' YORI' , 2.0) CALL AXPLOT('CONCENTRATION; CALL AXPLOT('CONDUCTIVITY; CALL PL0T(2.0, 12.0, 3) CALL PLOT(12.0,12.0, 2) CALL PL0T(12.0, 2.0, 2) FIND (4'7000) I = 0 10 READ (4,LIST,END=80,ERR=100) X, Y, Z 1 = 1 + 1 A(I) = DL0G1O(1.DO/X) B(I) = DL0G10(Y) C(I) = Z GO TO 10 80 WRITE (6,90) 90 FORMAT (1X, ' C(100), Y F(100), YD(100), WT(100) XX(2), YY(2), P(2) , 0.0, 90.0, 10. 10. -2., 0.0) 0.0, 2.0) END OF DATA 4 ENCOUNTERED.') C Averages SUM 1 = O.O DO 20 K = 1, I SUM1 = SUM 1 + C(K) 20 CONTINUE AVE 1 = SUM1 / I C Standard deviations SS1 = 0.0 DO 30 K = 1, I SS1 = SS1 + (C(K) - AVE 1) ** 2 M 00 CO 30 CONTINUE SIGMA 1 = DS0RT(SS1/(I - 1 )) WRITE (7,40) AVE 1, SIGMA 1 40 FORMAT (1X, 'AVERAGE TEMP. = ', E13.G, ' STD. DEV. =', E13.6/) CALL DOLSF(1,I,A,B,YF,YD,WT,0,S,SIGMA,XX,YY,SS,LK,P) WRITE(7,50) P 50 FORMAT(1X,'PARAMETER ESTIMATES:',1P2E13.6) WRITE(7,55) SS 55 FORMAT(1X,'SUM OF SQUARES IS:',1PE13.6) DO 60 K = 1, I A(K) = (A(K)+2.D0)*5 + 2.DO B(K) = B(K) * 5 + 2.DO 60 CONTINUE DO 70 K = 1 , I X1 = SNGL(A(K)) Y1 = SNGL(B(K)) CALL SYMB0L(X1, Y1, .15, 2, 0.0, -1) 70 CONTINUE CALL PLOTND STOP 100 M = I + 6 WRITE (6,110) M 110 FORMAT (1X, 'RECORD', 15, ' IN FILE 4 SKIPPED.') GO TO 10 1000 WRITE(6,1001) 1001 FORMAT(1X,'ERROR: SINGULAR MATRIX. SOLUTION FAILED.') STOP END b) Data L i s t i n g . Readings of conductivity for c a l i b r a t i o n of Seibold conductivity c e l l . Each l i n e contains: denominator of Normal concentration, conductivity (mMhos), temperature of KCI solution (deg. Celsius). 0. 2000E+01 0. 4300E+02 0. .2130E+02 0. .5000E+01 0. .1800E+02 0. 2190E+02 0. .7000E+01 0. .1340E+02 0. 2210E+02 0. .1000E+02 0, 9600E+01 0. 2240E+02 0. 3000E+02 0. ,3400E+01 0. 2260E+02 0. 4000E+02 0. .2600E+01 0. .2270E+02 0. .5000E+02 0. .2030E+01 0. .2290E+02 0. 7000E+02 0. 1500E+01 0. 2300E+02 0. 1000E+03 0. .1080E+01 0. 2300E+02 0. 5000E+01 0. .1820E+02 0. 2280E+02 0. 1000E+02 0. 9580E+01 0. 2300E+02 0. 2500E+02 0. 4040E+01 0. 2340E+02 0. 5000E+02 0. .2040E+01 0. 2370E+02 o o o o o o o o o o o o j . U l - » _ * U l _ ^ _ k ^ l _ i . - L _ i . ~ J O O O O O O O O O I O O U 1 O O O O O O O O O 0 1 O O o o o o o o o o o o o o (n rn rnmrnrnr r i rn rnrT i rn rT i + + + + + + + + + + + + O O O O O O O O O O O O coiotocoioiocotorococoro O O O O O O O O O O O O -^ro-^oo-^oo-^-^CDOo-i.-' cocnrooa(oro-^cr)cnuiOJ> U O I U O M O C J I O O ^ B O (010C0OOOOOOOOO rnrr irr imrrirnrnrTirnrnrTirTi + + + + + + + + + + + + O O O O O O O O O O O O O O O O O O O O O O O O M u u u i o r o i o i o u i o i o u C0C0C0J>C0UIOK)IOJ>J>CJ O O O U l ~ 4 0 1 N ) f O ~ J O O l D O O O O O O O O O O O O r n m r n r n r n r n r r i r T i r n r T i r n m + + + + + + + + + + + + O O O O O O O O O O O O lOUMMfOMFOUIOUlOU S8Z Appendix XXVI. Data from Threshold Concentration Measurements. Data on threshold concentration from the experiments conducted in aqueous sugar solutions. Each l i n e contains: oven-dry nylon f i b r e weight (grams), suspension temperature (degrees C e l s i u s ) , and suspension volume ( m i l l i l i t e r s ) . 3 DENIER FIBRES (diameter=0.0197mm) F i bre 1ength=1.875mm 0.3770E+01 0.40O0E+01 0.4000E+01 0.4O0OE+01 0.4000E+01 0.2510E+02 0.2340E+02 0.2290E+02 0.2260E+02 O.2330E+02 0.1750E+03 0.1950E+03 0.1950E+03 0.2000E+03 0.1900E+03 Mean threshold concentrations.01979 Standard deviation=0.0005615 95% confidence 1imits=0.01979 ± 0.001787 Fibre 1ength=2.815mm O.1629E+01 0.2000E+01 0.2000E+01 0.2000E+01 0.2000E+01 0.2000E+01 0.200OE+01 0.2000E+01 0.2400E+02 O.2400E+02 O.229OE+02 0.2350E+02 0.2320E+02 0.2310E+02 0.2280E+02 0.2280E+02 O.1650E+03 0.2100E+03 0.2100E+03 0.2100E+03 O.2O5OE+03 0.2050E+03 0.195OE+03 0.1950E+03 Mean threshold concentration=0.009333 Standard deviation=0.0002908 95% confidence 1imits=0.009333 ± 0.0006878 Fibre 1ength=3.737mm 0.1OOOE+01 0.2000E+01 O.20O0E+01 0.1000E+01 0.1O0OE+01 0.2000E+01 0.2000E+01 0.2000E+01 O.2OOOE+01 0.2000E+01 0.2590E+02 0.2210E+02 O.2240E+02 0.2370E+02 0.2480E+02 0.2200E+02 0.2230E+02 0.2210E+02 0.2330E+02 0.2220E+02 0.1400E+03 0.3300E+03 0.3650E+03 0.1650E+03 0.1650E+03 0.3200E+03 0.3350E+03 0.3250E+03 0.3350E+03 0.3500E+03 M 00 Mean threshold concentration^.005791 Standard deviation=0.0004116 95% confidence 1imits=0.005791 ± 0.0009310 6 DENIER FIBRES (diameter=0.0279mm) Fibre 1ength=1.832mm O.80O0E+O1 0.8000E+01 0.8000E+01 0.8000E+01 0.8000E+01 0.8000E+01 0.2570E+02 0.2470E+02 0.2420E+02 0.2350E+02 0.2450E+02 0.2380E+02 0.2300E+03 O.2300E+03 0.2450E+O3 0.2200E+03 0.2350E+03 0.2200E+03 Mean threshold concentration=0.03314 Standard deviatlon=0.001350 95% confidence 1imits=0.03314 ± 0.003471 Fibre 1ength=2.757mm 0. .4000E+01 0. .2230E+02 0. 2250E+03 0. •4000E+01 0. .2230E+02 0. 2100E+03 0. 4000E+01 0. .2210E+02 0. .2500E+03 0. 4000E+01 0. 2200E+02 O. 2450E+03 0. 4000E+01 0. 2190E+02 0. 2200E+03 0. 4000E+01 0. 2150E+02 0. 2500E+03 0. .4000E+01 0. 2120E+02 0. 2300E+03 Mean threshold concentration=0.01641 Standard deviation=0.001124 95% confidence 1imits=0.01641 + 0.002751 Fibre 1ength=3.718mm 0.1000E+01 0.2000E+01 0.2000E+01 0.2000E+01 0.2000E+01 0.2000E+01 0.2000E+01 0.2000E+01 0.2000E+01 0.2000E+01 0.2000E+01 0.2300E+02 0.2000E+02 0.2070E+02 0.2100E+02 0.2100E+02 0.2070E+02 0.2070E+02 0.2060E+02 0.2060E+02 0.2060E+02 0.2070E+02 0.9000E+02 0.2350E+03 0.2200E+03 0.2250E+03 0.2200E+03 0.2100E+03 0.2250E+03 0.2300E+03 0.2300E+03 0.2500E+03 0.2400E+03 Mean threshold concentration=0.008549 Standard deviation=0.0007754 95% confidence 1imits=0.008549 ± 0.001728 CO —1 Fibre 1ength=4.666mm 0.1000E+01 0.1000E+01 0.1000E+01 O.1000E+01 0.1000E+01 0.1000E+01 O.1000E+01 0.1000E+01 0.1000E+01 0.1000E+01 O.1000E+01 0.2370E+02 0.2170E+02 0.2250E+02 0.2180E+02 0.2180E+02 0.2200E+02 0.2210E+02 0.2170E+02 0.2170E+02 0.2180E+02 0.2180E+02 0.1750E+03 O.1950E+03 0.1800E+03 O.1850E+03 0.1900E+03 O.1850E+03 0.1950E+03 0.1950E+03 0.1850E+03 0.1800E+03 0.1900E+03 Mean threshold concentration=0.005099 Standard deviation=0.0001875 95% confidence Hmits=0.005099 ± 0.0004176 15 DENIER FIBRES (diameter=0.0442mm) Fibre 1ength=2.947mm 0.3200E+01 0.6400E+01 0.6400E+01 0.6400E+01 0.5000E+01 0.5000E+01 0.5000E+01 0.5000E+01 0.2350E+02 0.2400E+02 0.2390E+02 0.2360E+02 0.2360E+02 0.2340E+02 0.2320E+02 0.2530E+02 0.1400E+03 0.2800E+03 0.2750E+03 0.2850E+03 0.2050E+03 0.2150E+03 0.2100E+03 0.2100E+03 Mean threshold concentration=0.02221 Standard deviation=0.0006022 95% confidence 1imits=0.02221 + 0.004176 Fibre 1ength=4.976mm 0.1600E+01 0.1600E+01 0.1600E+01 0.3200E+01 0.1600E+01 O.1600E+01 0.1600E+01 0.1600E+01 0.1600E+01 0.2280E+02 0.2250E+02 0.2300E+02 0.2320E+02 0.2300E+02 0.2310E+02 0.2330E+02 0.2340E+02 0.2360E+02 0.1850E+03 0.1900E+03 0.1950E+03 0.3850E+03 0.1850E+03 0.1900E+03 0.1950E+00 0.2000E+03 0.2000E+03 Mean threshold concentration=0.007915 Standard deviation=0.0002300 95% confidence 1imits=0.007915 ± 0.005304 oo 00 Fibre 1ength=6.261mm 0.8000E+00 0.8000E+00 0.8000E+00 0.8000E+00 0.8000E+00 0.8000E+00 0.8000E+00 0.2440E+02 0.2390E+02 0.2300E+02 0.2320E+02 0.234OE+02 0.2470E+02 0.2310E+02 O.1600E+03 O.1800E+03 O.1700E+03 0.1650E+03 0.1650E+03 O.1850E+03 0.1700E+03 Mean threshold concentration=0.004469 Standard deviat1on=0.0002267 95% confidence 1imits=0.004469 ± 0.0005547 Data on threshold concentration from experiments conducted in water as a suspending medium. Each l i n e contains: oven-dry nylon f i b r e weight (grams), suspension temperature (degrees Celsius), and suspension volume ( m i l l i l i t e r s ) . 15 DENIER FIBRES (diameter=0.0442mm) Fibre 1ength=2.947mm 0.5000E+01 0.5000E+01 0.5000E+01 0.5000E+01 0.5000E+01 0.5000E+01 0.2350E+02 0.2340E+02 0.2320E+02 0.2310E+02 0.2340E+02 0.2340E+02 0.2400E+03 0.2500E+03 0.2550E+03 0.25OOE+03 0.2600E+03 0.2550E+03 Mean threshold concentration=0.01892 Standard deviation=0.0005244 95% confidence 1imits=0.01892 + 0.0001348 Fibre 1ength=4.973mm 0.1600E+01 0.1600E+01 0.1600E+01 0.1600E+01 0.1600E+01 0.1600E+01 0.1600E+01 0.1600E+01 0.2350E+02 0.2290E+02 0.2330E+02 0.2310E+02 0.2190E+02 0.2270E+02 0.2260E+02 0.2170E+02 0.2200E+03 0.2200E+03 0.2100E+03 0.2150E+03 0.2100E+03 0.2150E+03 0.2250E+03 0.2250E+03 Mean threshold concentration=0.007004 Standard deviation=0.0001925 95% confidence 1imits=0.007004 + 0.0004553 to oo Fibre 1ength=6.261mm 0.8000E+00 0.8000E+00 0.8000E+00 0.8000E+00 0.8000E+00 0.8000E+00 0.8000E+00 0.8000E+00 0.2110E+02 0.2200E+02 0.2240E+02 0.2280E+02 0.2350E+02 0.2320E+02 0.2470E+02 0.2220E+02 0.2200E+03 0.2200E+03 0.2000E+O3 0.2000E+03 0.2150E+03 0.2000E+03 0.2150E+03 0.2150E+03 Mean threshold concentration=0.003620 Standard dev1 ation=0.0001577 95% confidence 1imits=0.003620 ± 0.0003727 Data on the ef f e c t of cylinder diameter on threshold concentration. Each l i n e contains: oven-dry sample weight (grams), suspension volume at the onset of Type-C fl o e formation (mL), suspension temperature (degree Celsius), 8 times the rotational speed of driving r o l l e r (Rev/s), and cylinder i n c l i n e to the horizontal (degrees). 15 DENIER FIBRES (d1ameter=0.0442mm), FIBRE LENGTH=4.973mm Cylinder ID=56mm. 0. , 7000E+00 0. ,2420E+02 0. 7600E+02 0. ,2860E+02 0. .4500E+02 0. ,7000E+00 0. ,2440E+02 0. 7200E+02 0. .2860E+02 0. .4500E+02 0. .7OO0E+OO 0. .2420E+02 0. 7000E+02 0. .2860E+02 0. 4500E+02 0. .7000E+00 0. .2400E+02 0. 7500E+02 0. 2860E+02 0. 4500E+02 0. 7000E+00 0. .2420E+02 0. 7500E+02 0. 2860E+02 0. 4500E+02 0. .7000E+00 0. .2420E+02 0. 8000E+02 0. 28G0E+02 0. 4500E+02 Mean threshold concentration=0.00894 Standard deviation=0.0004105 95% confidence 1imits=0.00894 ± 0.001056 Cylinder ID=69mm. 0. 1400E+01 0. . 2340E+02 0. 1400E+03 0. .3400E+02 0. 4500E+02 0. 1400E+01 0. 2350E+02 0. 1440E+03 0. .3400E+02 0. 4500E+02 0. 1400E+01 0. 2400E+02 0. 1420E+03 0. .3400E+02 0. 4500E+02 0. 1400E+01 0. 2320E+02 0. 1400E+03 0. ,3400E+02 0. 4500E+02 0. 1400E+01 0. 2340E+02 0. 1470E+03 0. .3400E+02 0. 4500E+02 0. 1400E+01 0. 2360E+02 0. 1470E+03 0. ,3400E+02 0. 4500E+02 Mean threshold concentration=0.00930 Standard deviat1on=0.0002073 95% confidence 1imits=0.00930 ± 0.000533 n o O O < < < —• —• —• — J c/> 3 o O O 3 CD CO 3 o O O O 3 CO c/) 3 o o o o o o 3 CO CO 3 o o o o o o o o o o 3 rt CD a U l r+ CD a U l r+ CD a U l r+ CD a ft) 0) —t. -fc —». ID ft) 0) cn - J - j - j ID s« ft) ft) CO CO CO CO CO CO ID s« ft) 0) _fc_fc-j.-j._fc_fcCO-fc-*-fc (D -4. —J. s 3 3 j i O O O ~y 3 3 O O O O O O S 3 3 c n c n c n c n c n c n r o c n c n c n s a o O O o a. o O o O 0 a O O O O O O 0 a o o o o o o o o o o 0) «+ o o O 1—1 0 0) i"+ o o o o 0 0) r f O O O O O O i—i 0 ft) r+ o o o o o o o o o o 1—1 ^ 3" m m rn o 3 "3 3* m m m rn a 3 S 3 1 m m m m m m o 3 S 3" m m m m m m m m m m a a. *3 + + + II -h a T + + + + II -h a "3 + + + + + + II -h a. - J + + + + + + + + + + II ID o O O —4. ID o O O O _t -». CD o o o o o o CO ->. CD o o o o o o o o o o 00 a ui r o ro I O J i a a cn _k. —1. to a. a co _k . 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II n O O O O O - O II 0 O O O O O O O O O O O CD m m m r l - O CD m m m m H - O CD m m m m m m r+ O CD m m m m m m m m m m • 3 + • l - + CO • 3 + + + + 10 • 3 + + + + + + CO • 3 + + + + + + + + + + O H - O O O o O O O O H O r+ O O O O O o II O rt- O O O O O O O O O O O "3 I O ro ro O O s M ro ro ro O O s to ro ro ro ro ro O O - ! r o r o r o r o r o r o r o r o t o r o o oi • o oi o oi O fl) CO r* O - H- O CO r+ O ro r* 0 0 O u i o -fc -1. O CO — O O O O oo o O o O O o 03 J i 0 o o o o o o -4 - J O O O O O O O O O O O 00 3 0 0 00 3 - J U l 3 03 O 3 II k —* L - J II - J ~ i - j ~ l CO II CO CO CO CO CO CO 00 II r o t o r o - f c - f c - f c C O - f c - f c - f c o —k _k I O O o CO U l U l o ro co to -fc j i to O O O O C O C 0 0 3 0 3 C O C 0 0 3 I O j i I O 1+ o U l U l O 1+ O O u i O u i O 1+ o o o u i o u i u i u t o u i b CJI o O O o O O O b O O O O O O b o o o o o o o o o o o m m m o o m m m m o o m m m m m m o o m m m m m m m m m m CD + + + 00 + + + + 00 + + + + + + - J + + + + + + + + + + o O O O O 00 O O O O b ~1 o o o o o o b 00 o o o o o o o o o o I O J i J i J i O CO CO CO CO o CD CO CO CO CO CO CO o 00 c o c o c o c o c o c o c o c o c o c o o o o J i oo U l 00 o CO o o o o O O O o CD o o o o O O cn O O O O O O O O O O CT) cn cn CO U l U l U l j i j i j i j i J> j i C O J i J i J i J i J i J i J i J i J i CD CO CO CO - J ^ 1 - ~ l U l U l U l U l U l U l C O O O O O O O O O O O O O - 4 - 1 ^ 1 ~1 U l U l O I U l OI 0 1 O O O O O O O O O O o O O o O O o O O O O O O o o o o o o o o o o m m m rn m m m m m m m m m m m m m m m m m m m + + + + + + + + + + + + + + + + + + + + + + O O O o O O O O O O O O o O O O O O O O O O O I O I O ro I O ro ro ro ro ro ro ro ro ro r o r o r o r o r o r o r o r o r o r o O O O O O O O O O O O O O O O O O O O O O O O J > J i J i J i J i J i J i j> j i j i j i j i j i j i j i j i j i j i j i j i j i j i j i 0 1 0 1 cn U l U l U l U l U l U l U l U l U l U l U I U I U I U I U I U I U I U I U I U I o o O O O O O O O O O O O O O O O O O O O O O o o O O O O O O O O O O O O O O O O O O O O O r n m m m m m m m m m m m m m m m m m m m m m m + + + + + + 4- + + + + + + + + + + + + + + + + O O O O O O o O O O O o o O O O O O O O O O O I O ro ro ro ro ro ro ro to to ro to to t o r o r o r o t o t o t o r o r o t o T6Z 95% confidence 1imits=0.00902 ± 0.001669 Data on the e f f e c t of cylinder rotational speed on threshold concentration. Each line contains: oven-dry sample weight (grams), suspension temperature (degree Celsius), and suspension volume (mL). Cylinder ID=82mm. 15 DENIER FIBRES (diameter=0.0442mm) FIBRE LENGTH=4.973mm Cylinder rotational speed=4.2 rad/s 0.1600E+01 0.1600E+01 0.1600E+01 0.1600E+01 0.1600E+01 0.1600E+01 0.2140E+02 0.2140E+02 O.2150E+02 0.2150E+02 0.2170E+02 O.2180E+02 0.2150E+03 0.2250E+03 0.2300E+03 0.2150E+03 0.1950E+03 0.2150E+03 Mean threshold concentrat 1on=0.00707 Standard deviation=0.0004097 95% confidence 1imits=0.00707 ± 0.001053 Cylinder rotational speed=6.8 rad/s 0.1600E+01 0.1600E+01 0.1600E+01 0.1600E+01 0.1600E+01 0.1600E+01 0.2160E+02 0.2160E+02 0.2170E+02 0.2180E+02 0.2170E+02 0.2170E+02 0.2000E+03 O.1900E+03 0.1900E+03 0.2050E+03 0.2150E+03 0.2050E+03 Mean threshold concentrat1on=0.00759 Standard deviation=0.0003661 95% confidence 1imits=0.00759 ± 0.000941 Cylinder rotational speed=11.3 rad/s 0.1600E+01 0.1600E+01 0.1600E+01 0.1600E+01 0.1600E+01 0.1600E+01 0.2150E+02 0.2150E+02 0.2150E+02 0.2160E+02 0.2170E+02 0.2170E+02 0.1950E+03 0.2050E+03 0.2050E+03 0.1850E+03 0.1800E+03 0.2000E+03 Mean threshold concentrat1on=0.00783 Standard deviation=0.0004307 95% confidence 11mits=0.00783 + 0.001107 Cylinder rotational speed=16.8 rad/s 0.1600E+01 0.1600E+01 0.1600E+01 0.1600E+01 0.1G00E+01 O.1600E+01 0.2130E+02 0.2130E+02 0.2150E+02 0.2160E+02 0.2170E+02 0.2170E+02 0.1850E+03 0.2050E+03 0.2100E+03 0.1850E+03 0.2050E+03 0.1850E+03 Mean threshold concentration=0.00780 Standard deviation=0.0004764 95% confidence 1imits=0.00780 ± 0.001225 Cylinder rotational speed=22.1 rad/s 0.1600E+01 0.1600E+01 0.1600E+01 0.1600E+01 0.1600E+01 0.1600E+01 0.2140E+02 0.2150E+02 0.2150E+02 0.2140E+02 0.2160E+02 0.2170E+02 0.1800E+03 0.2050E+03 0.1800E+03 0.2000E+03 0.1950E+03 0.1800E+03 Mean threshold concentration=0.00804 Standard deviat 1on=0.0004770 95% confidence 1imits=0.00804 ± 0.001226 Data on the e f f e c t of cylinder i n c l i n e on threshold concentration. Each l i n e contains: oven-dry sample weight (grams), suspension temperature (degree Celsius), suspension volume (mL), 8 times the rotational speed of the driving r o l l e r (Rev/s), and cylinder i n c l i n e (degree). Cylinder: ID=82mm; 0D=88mm. 15 DENIER FIBRES (diameter=0.0442mm), FIBRE LENGTH=4.973mm 0.16E+01 0.211E+02 0.190E+03 0.292E+02 0.0 Threshold concentration=0.00801 0.16E+01 0.214E+02 0.205E+03 0.304E+02 0.15E+02 Threshold concentrat1on=0.00743 0.16E+01 0.216E+02 0.200E+03 0.304E+02 0.30E+02 threshold concentrat1on=0.00761 Data on the e f f e c t of v i s c o s i t y of suspending l i q u i d on threshold concentration. Each line containes: oven-dry sample weight (grams), suspension temperature (degree Celsius), suspension volume (mL), and 8 times the rotational speed of the driving r o l l e r (Rev/s). 15 DENIER FIBRES (diameter=0.0442mm) FIBRE LENGTH=4.973mm Density of the suspending 1iquid=1.175g/cm**3 0.2000E+01 0.2000E+01 0.2000E+01 0.2000E+01 0.2000E+01 0.2000E+01 0.2300E+02 0.2300E+02 0.2310E+02 0.2350E+02 0.2370E+02 0.2380E+02 0.1800E+03 0.1750E+03 0.1900E+03 0.1750E+03 0.1850E+03 0.1700E+03 Mean threshold concentrations.010 c Standard deviation=0.0004323 95% confidence 1imits=0.0106 + 0.001111 Density of the suspending 1iquid=1.22g/cm**3 0.2000E+01 0.2000E+01 0.2000E+01 0.2000E+01 0.2000E+01 0.2000E+01 0.2500E+02 0.2510E+02 0.2540E+02 0.2580E+02 0.2570E+02 0.2570E+02 0.1450E+03 O.1550E+03 0.1400E+03 0.1350E+03 0.1500E+03 0.1400E+03 Mean threshold concentrations.0132 Standard deviation=0.0006662 95% confidence 1imits=0.0132 ± 0.001713 0.4000E+02 0.4000E+02 0.4000E+02 0.4000E+02 0.4000E+02 0.4000E+02 0.4010E+02 O.4010E+02 0.4010E+02 0.4010E+02 0.4010E+02 0.4010E+02 Appendix XXVII. Data from Velocity Measurements in Horizontal Rotating Cylinder a) Case 1 . NAME:Robert M. Soszynski, DATE:03-01-198o, TIME:23:06:18 1.13;1.0g in 125mL;1/s;norizontal;2-500kHz or 0.01V/kHz;200kHz NUMBER OF DATA POINTS: 25000, SAMPLING RATE: 2500 PTS/SEC Data stored in line s containes: radial coordinate - mm, angular coordinate - deg., average voltage for horizontal and v e r t i c a l v e l o c i t y componenets - vol t s . 10 0 -0 .470413E+00 0 .109549E+00 15 0 -0 .337353E+00 0 .122745E+00 20 0 -0 .210124E+00 0 .777233E-01 25 O -0 .778463E-01 0 .671929E-01 30 0 0. .724024E-01 0 .388355E-01 35 0 0, ,155700E+00 0 .162050E-01 40 0 0, .319181E+00 0. .390779E-02 45 0 0. .481168E+00 0. .951210E-02 25 10 O. .159210E-01 0. .329458E-01 30 10 0. 607521E-01 0 .613596E-02 35 10 0. .203023E+00 -0, .153780E-01 40 10 0. .330611E+00 -0. .636548E-01 45 10 0. .504228E+00 -0, .797557E-01 10 20 -0. ,483226E+00 0. .750384E-01 15 20 -0. . 347085E+00 0, .723528E-01 20 20 -0. .211551E+00 0. .602350E-01 25 20 -o. ,719233E-01 0. 519083E-01 30 20 0. .102826E+00 -0. .382016E-01 35 20 0. .200579E+00 -0. .804626E-01 40 20 0. .341528E+00 -0, ,106151E+00 45 20 0. 487241E+00 -0. .143197E+00 25 30 -0. 923697E-01 -0. 133527E-01 30 30 0. 122205E+00 -0. 767988E-01 35 30 0. 203152E+00 -0. 130044E+00 40 30 0. 304335E+00 -0. 193567E+00 45 30 0. 448712E+00 -0. 277052E+00 10 40 -0. 509067E+00 0. 788339E-01 15 40 -0. 434588E+00 0. 770602E-01 20 40 -0. 253258E+00 0. 477927E-01 25 40 -0. 958958E-01 0. 165657E-01 30 40 0. 675963E-01 -0. 984510E-01 35 40 0. 111578E+00 -0. 160057E+00 40 40 o. 263140E+00 -0. 234349E+00 45 40 0. 381765E+00 -0. 310228E+00 25 50 -0. 189692E+00 -0. 373345E-01 30 50 -o. 645996E-01 -0. 826580E-01 35 50 0. 859486E-01 -0. 107020E+00 40 50 0. 178760E+00 -0. 279601E+00 45 50 0, .307620E+00 -0. . 385633E+00 10 60 -0. 558540E+O0 0, .590906E-01 15 60 -0. .521508E+00 0. 679862E-01 20 60 -0. .405841E+00 0. .545692E-01 25 60 -0. ,278040E+00 0. .183709E-01 30 60 -0. ,176243E+00 -o. 896399E-01 35 60 -0. 447635E-01 -0, .182338E+00 40 60 0. .124532E+00 -o. .202449E+00 45 60 0. ,199724E+00 -0. 381822E+00 25 70 -0. .377341E+00 0. .456746E-02 30 70 -0. ,250210E+00 -0. .391865E-01 35 70 -0. .125346E+00 -o. ,102065E+00 40 70 0. .125453E-01 -0, . 234137E+00 45 70 0. .132134E+00 -0. . 347365E+00 15 80 -0. ,519270E+00 0, .416339E-01 20 80 -0. .599522E+00 0. 568729E-01 25 80 -0. ,530196E+00 0. .561977E-01 30 80 -0. 452165E+00 0. 319531E-01 35 80 -0. .193317E+00 -0. 462316E-01 40 80 -0. ,138606E+00 -o. , 1 15882E+00 45 80 -0. 312772E-02 -0. .289138E+00 15 280 -0. 754685E-01 0. 518916E-02 20 280 -0. . 178243E+00 0. 838600E-02 25 280 -o. 214551E+00 0. 909403E-02 30 280 -0. .261155E+00 0. 854780E-02 35 280 -0. 265733E+00 0. 481027E-02 40 280 -0. 233377E+00 -0. .180281E-01 45 280 -0. 908498E-01 0. .184510E-02 25 290 -0. 365059E+00 0. 218813E-01 30 290 -0. 292931E+00 0. .587402E-01 35 290 -0. 209492E+00 0. 341299E-01 40 290 -0. 128477E+00 0. 540412E-01 45 290 -0. 159699E-01 0. 974093E-01 10 300 -0. 483005E+00 0. 558972E-01 15 300 -0. 465948E+00 0. 551656E-01 20 300 -0. 419729E+00 0. 631033E-01 25 300 -0. 350098E+00 0. 649176E-01 30 300 -0. 240058E+00 0. 727243E-01 35 300 -0. 146114E+00 0. 741315E-01 40 300 -0. 843289E-01 0. 818717E-01 45 300 0. 719724E-01 0. 173261E+00 25 310 -0. 264593E+00 0. 101724E+00 30 310 -0. 162274E+00 0. 105546E+00 35 310 -0. 875978E-01 0. 112817E+00 40 310 -0. 277126E-01 0. 112306E+00 45 310 0. 162287E+00 0. 205543E+00 10 320 -0. 499105E+00 0. 763421E-01 15 320 -0. 413193E+00 0. 958417E-01 20 320 -0. 341397E+00 0. 102700E+00 25 320 -0. 211437E+00 0. 109140E+00 to VO 30 320 -0. .134655E+00 0. 920352E-01 35 320 -0. 957425E-02 0. ,115924E+00 40 320 0. 367763E-01 0. ,125297E+00 45 320 0. 264417E+00 0. ,204840E+00 25 330 -0. 142671E+00 0. 132521E+00 30 330 -0. 353827E-01 0. 137587E+00 35 330 0. ,204828E-01 0. 138561E+00 40 330 0. 136476E+00 0. 145843E+00 45 330 0. 307774E+00 0. 180249E+00 10 340 -0. 478917E+00 0. 783091E-01 15 340 -0. 350344E+00 0. 969390E-01 20 340 -0. 247264E+00 0. 964268E-01 25 340 -0. .150155E+00 0. 852638E-01 30 340 O. 149045E-01 0. 987043E-01 35 340 0. 936582E-01 0. 104633E+00 40 340 0. 191197E+00 0. 10G151E+00 45 340 0. 395405E+00 0. 141370E+00 25 350 -0. 119893E-01 0. 665227E-01 30 350 0. 505018E-01 0. 688755E-01 35 350 0. 105004E+00 0. 427272E-01 40 350 0. 258189E+00 0. 646885E-01 45 350 0. 436576E+00 0. 805734E-01 b) Case 2. NAME:Robert M. Soszynskl, DATE:02-22-1986, TIME:23:56:09 1.13;1.5g in 125mL;1/s;horizontal;2-500kHz so 0.01V/kHz;100kHz NUMBER OF DATA POINTS: 25000, SAMPLING RATE: 5000 PTS/SEC Data stored in lines containes: radial coordinate - mm, angular coordinate - deg., average voltage for horizontal and v e r t i c a l v e l o c i t y components - vol t s . 10 0 -0. 442774E+00 -0. 644879E -01 15 0 -o. ,301163E+00 0. 855581E -01 20 0 -0. .170778E+00 0. ,71 1411E -01 25 0 -0. 385428E-01 0. 478650E -01 30 0 0. 745004E-01 0. 466815E -01 35 0 0. 205431E+00 -0. 978420E -02 40 0 0. 340536E+00 0. 979175E -02 45 0 0. 494546E+00 0. 145979E -01 25 10 -0. 399374E-01 -0. 272714E -01 30 10 0. 123813E+00 -0. 289418E -01 35 10 0. 190380E+00 -0. 460816E -01 40 10 0. 358226E+00 -0. 694743E -01 45 10 0. 512884E+00 -0. 595649E -02 10 20 -0. 471556E+00 0. 590236E -01 15 20 -0. 353666E+00 0. 515908E -01 20 20 -0. 201871E+00 0. 184117E -01 25 20 -0. 538190E-01 0. 403428E -02 30 20 0. 963724E-01 -0. 421060E -01 35 20 0, .121188E+00 -0 .567951E-01 40 20 0 . 319095E+00 -0 .112149E+00 45 20 0 .506686E+00 -0 .178558E+00 25 30 -o .489481E-01 -0 .728648E-02 30 30 O .426044E-01 -0 .127048E+00 35 30 0 .169772E+00 -0 .123821E+00 40 30 0 .305983E+00 -0 .186748E+00 45 30 0 .447693E+00 -0 .239914E+00 10 40 -0 . 549264E+00 -0 .188259E-01 15 40 -0. . 338904E+00 -0 .150258E-01 20 40 -0. .281949E+00 0, .783234E-02 25 40 -0, .112632E+00 -0. .115184E-01 30 40 0. .157285E-01 -0. .359651E-01 35 40 0. .134934E+00 -0, .161129E+00 40 40 0. .243256E+00 -0. .177001E+00 45 40 0. .356741E+00 -0, .283984E+00 25 50 -0. .176273E+00 -0. .297930E-02 30 50 -0. .310040E-01 -o. .99G057E-02 35 50 0. .505428E-01 -0. 679876E-02 40 50 0. .168058E+00 -0. ,132484E-02 45 50 0. 325002E+00 -0. 334200E+00 10 60 -0. .545801E+00 0. ,130413E-01 15 60 -0. 529872E+00 -0. 965350E-02 20 60 -0. 407095E+00 -0. . 151016E-01 25 60 -0. . 257788E+00 -0. .142141E-01 30 60 -0. 120387E+00 -o. 200474E-01 35 60 -0. 406489E-01 -0. 175827E-01 40 60 0. 674721E-01 -0. 314138E-01 45 60 0. 166266E+00 -0. 2906G6E+00 25 70 -0. 348824E+00 -0. 184389E-01 30 70 -o. 229980E+00 -0. 186676E-01 35 70 -0. 132801E+00 -0. 182364E-01 40 70 0. 174301E-01 -o. 190713E-01 45 70 0. 133407E+00 -0. 161213E+00 20 80 -0. 514141E+00 0. 247231E-01 25 80 -0. 491947E+00 0. 320376E-01 30 80 -0. 369574E+00 -0. 168680E-01 35 80 -0. 246478E+00 -o. 337394E-01 40 80 -0. 126153E+00 -0. 192849E-01 45 80 -0. 339320E-01 0. 556065E-01 25 280 -0. 225822E+00 0. 773563E-02 30 280 -0. 221313E+00 0. 280551E-01 35 280 -0. 212055E+OO 0. 439743E-01 40 280 -o. 173442E+00 0. 333430E-01 45 280 -0. 650751E-01 -0. 678460E-01 25 290 -0. 334859E+00 -0. 180164E-01 30 290 -0. 228486E+00 -0. 156173E-01 35 290 -0. 209697E+00 -0. 155452E-01 40 290 -0. 100017E+00 0. 332284E-01 45 290 -0. 304766E-01 0. 414700E-01 ' 10 300 -0, .380916E+00 0. .352298E-02 15 300 -0. , 427326E+00 -0. .149615E-01 20 300 -0. .337120E+00 -0. .159127E-01 25 300 -0. .271237E+00 -0. .129686E-01 30 300 -0. .168917E+00 -0. .149692E-01 35 300 -0. .115265E+00 -0, .157204E-01 40 300 -0, .387061E-01 -0. .131860E-01 45 300 0. ,318077E-01 0, .183620E+00 25 310 • -0. ,201334E+00 0, .246496E-02 30 310 -0. ,126501E+00 0. .868433E-02 35 310 -0. 660401E-01 0. .100183E-01 40 310 -0. ,740724E-02 0. .764792E-02 45 310 0. ,107230E+00 0. ,224222E+00 10 320 -0. , 434412E+00 -0. ,169510E-01 15 320 -0. . 367634E+00 -0. ,225577E-01 20 320 -0. . 291579E+00 -0. 218048E-01 25 320 -0. .160952E+00 0. 453296E-01 30 320 -0. 893036E-01 0. 553187E-01 35 320 0. ,388267E-02 0. . 181333E-01 40 320 O. 107316E+00 O. .255200E-01 45 320 0. ,238240E+00 0. .231569E+00 25 330 -0. 166888E+00 0. 994360E-01 30 330 -0. 492859E-01 0. ,106515E+00 35 330 0. 673500E-01 0. 597215E-01 40 330 0. .142566E+00 0. 826791E-01 45 330 0. 300848E+00 0. 173862E+00 10 340 -0. 420787E+00 0. 642369E-01 15 340 -0. 347557E+00 0. 568977E-01 20 340 -0. 228392E+00 0. 658665E-01 25 340 -0. 104167E+00 0. 816137E-01 30 340 -0. 733647E-02 0. 704390E-01 35 340 0. 870856E-01 0. 879960E-01 40 340 0. 276608E+00 0. 843433E-01 45 340 0. 389298E+00 0. 132052E+00 25 350 -0. 761341E-01 0. 776763E-01 30 350 0. 324011E-01 0. 272753E-01 35 350 0. 145408E+00 0. 315864E-01 40 350 0. 285605E+00 0. 458640E-01 45 350 0. 431695E+00 o. 206854E-01 c) Case 3. NAME:Robert M. Soszynski, DATE:02-23-1986, TIME:23:53:58 1.456;1.5g 1n 125mL;1/s;norizontal;2-500kHz or O.01V/kHz;200kHz NUMBER OF DATA POINTS: 25000, SAMPLING RATE: 5000 PTS/SEC Data stored 1n line s containes: radial coordinate - mm, angular coordinate - deg., average voltage for horizontal and v e r t i c a l v e l o c i t y components - volts. 10 0 -0.451620E+00 0.340755E-01 15 0 -0. .427205E+00 0 .949689E-01 20 O -O. .231965E+00 0 .103178E+00 25 0 -0. . 228998E+00 0 .390461E-01 30 0 -0. .134877E+00 0 .326238E-01 35 0 -0. .829622E-01 0 .266744E-01 40 0 0. .375195E+00 0 .679225E-02 45 0 0, .757407E+00 0 .775583E-02 25 10 -0, .308750E+00 0 .101089E+00 30 10 -0, .963623E-01 0 .518279E-01 35 10 -o. 252236E-01 0 .110394E-O1 40 10 0, .359502E+00 -0 .391579E-01 45 10 0. ,747069E+00 -0, .103160E-01 10 20 -0. .470029E+00 0. .313528E-01 15 20 -0. 463933E+00 0, .120045E+00 20 20 -o. .423305E+00 0. .122707E+00 25 20 -o. , 239854E+00 0, .141191E+00 30 20 -0. 674376E-01 0. .111084E+00 35 20 0. 321433E-01 -0. .655071E-01 40 20 0. .381206E+00 -0. .127390E+00 45 20 0. 686768E+00 -0. ,257072E+00 25 30 -o. .281035E+00 0. .945972E-01 30 30 -0. .729544E-01 0. .370044E-02 35 30 0. 857815E-01 -0. ,118028E+00 40 30 0. 338983E+00 -0. ,220844E+00 45 30 0. 628721E+00 -0. .341985E+00 10 40 -0. 480043E+00 0. 743423E-01 15 40 -o. 541361E+00 o. 899879E-01 20 40 -o. 466018E+00 0. 672025E-01 25 40 -0. 266710E+00 0. 594738E-01 30 40 -0. 166596E+00 -o. 777107E-02 35 40 0. 286565E-01 -0. .170713E+00 40 40 0. 253929E+00 -0. ,251828E+00 45 40 0. 468575E+00 -0. 471401E+00 25 50 -0. 385922E+00 0. 145777E+00 30 50 -o. 227183E+00 0. 365051E-01 35 50 -0. 518643E-01 -o. 209074E+00 40 50 0. 195145E+00 -0. 318236E+00 45 50 0. 367002E+00 -0. 545870E+00 10 60 -0. 527398E+00 0. 100375E+00 15 60 -0. 532380E+00 0. 987549E-01 20 60 -0. 518619E+00 0. 167420E+00 25 60 -o. 424276E+00 0. 141389E+00 30 60 -0. 295802E+00 0. 692990E-01 35 60 -0. 488953E-01 -0. 111826E+00 40 60 0. 145312E+00 -0. 348889E+00 45 60 0. 248862E+00 -0. 565223E+00 25 70 -o. 497891E+00 0. 146312E+00 30 70 -0. 467352E+00 0. 921491E-01 35 70 -0. 274856E+00 -o. 541076E-01 40 70 -0. 114857E+00 -0. 962816E-01 to o 45 70 0. . 153395E+00 -0 . 522856E+00 15 80 -0 . 371542E+00 0 .566213E-01 20 80 -0. .495122E+00 0 .161321E+00 25 80 -0 .488336E+00 0 .132021E+00 30 80 -0 .441202E+00 0 .146161E+00 35 80 -0. .358351E+00 0 .574343E-02 40 80 -0, .190333E+00 -0 .233883E-01 45 80 -0, .259287E-01 -0 .467071E+00 25 90 -0, .155485E+00 0 .120882E-02 30 90 -o, .155598E+00 -0 .223294E-01 35 90 -o. ,144176E+00 0 .198558E+00 40 90 -0. .143074E+00 -0 .108312E+00 45 90 -o. ,572086E-01 -0 .136273E+00 35 100 -0. ,283926E-01 0 .535953E-01 40 100 0. .196207E-01 0 .593532E-01 45 100 -o. 454408E-01 -0 .434826E-01 30 290 -0. .303205E+00 0 .236720E-O1 35 290 -0. .325674E+00 0, .315893E-01 40 290 -0. .218251E+00 0 .368653E-01 45 290 -0. .346835E-01 0, .176446E+00 15 300 -0. 392113E+00 0. .562289E-01 20 300 -0. ,369683E+00 0. .397054E-01 25 300 -0. .362807E+00 0. .384319E-01 30 300 -0. 328171E+00 0, .379053E-01 35 300 -o. .230924E+00 0. 41OO70E-01 40 300 -0. 139634E+00 0. 654199E-01 45 ' 300 0. .146173E+00 0. . 356612E+00 25 310 -0. 347860E+00 0. 953447E-02 30 310 -0. 304884E+00 0. 963233E-02 35 310 -0. 130561E+00 0. 272321E-01 40 310 -0. 325129E-01 0. .153647E+00 45 310 0. 311833E+00 0. 373477E+00 10 320 -0. 499907E+00 0. 321594E-02 15 320 -0. 477663E+00 0. 597591E-02 20 320 -0. 356092E+00 -0. 426134E-03 25 320 -0. 336270E+00 0. 455189E-02 30 320 -0. 240083E+00 0. 753978E-02 35 320 -o. 144991E+00 0. 204481E-01 40 320 0. 226530E-01 0. 207729E-01 45 320 o. 351344E+00 0. 173123E+00 25 330 -0. 279009E+00 0. 486331E-01 30 330 -0. 203130E+00 0. 768448E-01 35 330 -o. 142827E+00 0. 105955E+00 40 330 0. 164156E+00 0. 141337E+00 45 330 o. 531916E+00 0. 326905E+00 10 340 -o. 454856E+00 0. 257170E-01 15 340 -0. 387750E+00 0. 459753E-01 20 340 -0. 365481E+00 0. 426211E-01 25 340 -0. 278020E+00 0. 659819E-01 30 340 -0. 195137E+00 0. 589397E-01 o o o o o o o o I I + I I I + I L U L U L U U J U J U J U J L 1 J i t i n o i c n o n K i r - ^ - C M C M c o c o r - m c M f - c N o o o c o c o - ' -a j i o m r o c n ' - O i n l D 0 0 C M O ) 0 0 C r ) ' - t ~ O O O O O O O O o o O H-CM O O O o O O o o O O 1 UJ + LU + LU + Ul 1 UJ 1 Ul + Ul + UJ to t CM t~ in 0) O LO in 10 CO to 0) CM CO *r CO -— in to O to <J- CO cn CO O) CM oo CO O in CN CD CN •<r co CO >-O O O O O O O O I I I I O O O O O O O O • q - i ^ i n i n i n i n i n cococoeococococo i n O i n i n o i n O i n Appendix XXVIII. Raw Data from Tensile Strength Measurements, This appendix contains the wet-tensile-test results. The data are grouped in packets corresponding to one type of nylon f i b r e s . Each l i n e of data contains from l e f t to right: breaking load (centigrams), "zero"-load (centigrams), oven-dry f l o e weight (grams), wet f l o e weight (grams), and break area (millimeters squared). e length=3.737mm, f i b r e • diameter= =0.0197mm. 1306 674 0. 510000E -02 0. 129700E+00 0. 198000E+02 1289 674 0. 560000E -02 0. 127800E+00 0. 226000E+02 1 194 674 0. 780000E -02 0. 168600E+00 0. 294000E+02 910 674 0. 300000E -02 0. 634000E-01 0. 133000E+02 1206 674 0. 510000E -02 0. 106200E+00 0. 165000E+02 1602 674 O. 610000E -02 0. 117300E+00 0. 183000E+02 1273 674 0. 500000E -02 0. 102000E+00 0. 126000E+02 1775 674 0. 600000E -02 0. 111200E+00 0. 150000E+02 1451 674 0. 440000E -02 0. 783000E-01 0. 130000E+02 1440 674 0. 520000E -02 0. 899000E-01 0. 124000E+02 1295 674 0. 390000E -02 0. 819000E-01 0. 150000E+02 1384 674 0. 430000E -02 0. 113600E+00 0. 148000E+02 3601 674 0. 750000E -02 0. 126500E+00 0. 198000E+02 1468 674 0. 490000E -02 0. 104200E+00 0. 170000E+02 1870 674 0. 550000E -02 0. 123600E+00 0. 179000E+02 1507 674 0. 900000E -02 0. 140800E+00 0. 1 13000E+02 1485 674 0. 560000E -02 0. 108200E+00 O. 154000E+02 Fibre 1ength=2.757mm, f i b r e diameter=0.0279mm. 1236 882 0. .600000E -02 0. ,753000E-01 0. 146000E+02 1319 779 0. 124000E -01 0. ,173600E+00 0. 265000E+02 1308 744 0. 940000E -02 0. .166000E+00 0. 176000E+02 1 135 740 0. .980000E -02 0. .132200E+00 0. 176000E+02 2355 890 0. ,236000E -01 0. .233900E+00 0. 322000E+02 1687 841 0. 123000E -01 0. ,17810CF+00 0. 281000E+02 3446 773 0. 156000E -01 0. 201100E+00 0. 286000E+02 1364 761 0. . 780000E -02 0. 113800E+00 v 0. 221000E+02 4153 723 0. 224000E -01 0. ,210600E+00 0. 342000E+02 3407 766 0. .209000E -01 0. , 260600E+00 0. 392000E+02 2321 741 0. 196000E -01 0. .221100E+00 0. 277000E+02 3986 712 0. .282000E -01 0. .301100E+00 0. 420000E+02 4883 719 0. 225000E -01 0. .211200E+00 0. 331000E+02 5267 722 0. 221000E -01 0. 207100E+00 0. 310000E+02 5312 703 O. .221000E -01 o. .200800E+00 0. 310000E+02 4098 703 0. .139000E -01 0. .140100E+00 0. 253000E+02 U) o to 304 CM CM CM O O O + + + LU LU LU o o O o o o o o o 01 LO CO LO T 0) CO CO CM 6 6 6 O O O O O O + + + E LU LU LU E O O o 01 O O O f-CM CO r- CM co O CM o o - LO CM CM T- o II 6 6 6 L. a -*-» CD E CO o o o 1 1 i V LU LU LU o o Q a O Q O t-O O O £> LO CM 0) •I— O 0) CM 4-CM *- •*-6 6 6 E" £ co ^~ r> 0) »- CM O 01 T~ CO r- 10 CO n £ •F 0) c 05 CO CD t- L0 CO i — r~ r- 0) LO CO CO CD c LL CM CM o o + + LU LU O O O O o o r- CM 1- 01 CM CM CM CM CM o o o + + + LU LU LU o o o o o o o o o CM TT LO CO t- CM CM CO CM CM o o + + LU LU O O O O 0 O 01 01 co 0) CM T-CM CM o o + + LU LU o o O O 0 O 01 0) co to CO CM CM CM CM CM o o o o + + + + LU LU LU LU o o o o O O O O O O O O CO CO LO L0 m *- r-co co co CM O O O O O O O O O O O O O O O O O + + LU LU o o 0 O CM CO 01 LO L0 r» CM CM o o o o o o + + + LU LU LU o o o o o o CO ID L0 co co to LO r- CM CM CM o o o o + + LU LU o o 0 o CM 10 01 CM CM o o o o + + LU LU o o o o CM 0) to L0 0) CM •»-o o o o o o o o + + + + LU LU LU LU o o o o o o o o •<* O LO tO cn t- •<-O CM CM f» C0 CM CM -1-o o o o o o o o o o o o o O O O O O O O O O O O O O i i i i i i i i i i i i i m H I i i i i n H I i n i i i n i n i n O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O l O t < q-0 ) l B ' q - - ' - C n c O C M O T l ' ' -C M C O ' a - T - ' ^ C O L O t B L O ^ ^ L O ' * 000606666660b 01OC0C0L0C0CMCDOt-~C00)O • s rOr - r -^ - ' -CMOCMO- ' -OCM O ! ' - L O ' - c o i 0 t " ~ r > i o c M c O T r O 0 1 l P C 0 C M r - C 0 t ^ C 0 l - - ' - C 0 L O T -t - c o c o c o O r f f - t O t T - r - t o r -' -CM-^'-COCMCMCMCMCOLOtCO 01 r~ CM O CD [_ JQ E E to to III 0) C CD L . CM CM o o + + LU LU o o O O O O •q- CO in T-•r- CM CM CM O O + + LU LU O O O O O O LO CM - O CO CO CM CM O O + + LU LU O O o o o o to *-LO co CM CM CM CM CM CM O O O O + + + + LU LU LU UJ O O O O O O O O O O O O r~ 01 CM CO f 0) IP CM co co co O O O O O O O O O O O O O O + + LU LU O O O O cn co O to CM tr O - --O O O O + + LU LU O O O O co r-CD co "» 01 *r CM O O O O + + LU LU O O O O ^ CO ~- O CO CM O O O O O O O O + + + + LU LU LU LU O O O O O O O C •^ r ••- to r CO -r- 1-•* CM CM to CM 1 CM O O O O O O O O O O CM CM ••- *-O O O O 1 1 1 1 LU LU LU LJ O O O O O O O O O O O O O O co O 0) *- 01 in to CM 1-0 O 1 1 LU LU O O O O O o CO t-O co CM --CM »- " O O O O I I I I LU LU LU LU O O O O O O O O O O O O 0 tO CD LO CO CM LO 01 *- co O O O O O O O O O O t o c o c n i n c o c o L O O t > c o t o r ^ t r - r ^ r - r - O l c - L O c o r - r - r - r - - r - r - - r - r - - r -Ot001t>C0C0C0r~Ot~ l O c o c c i i n c v o O ' - ' - P i •"-••-r-LOOCOCOCDCOCN •^i-i-CMCMCM'-C0<*C0 E E CM O CD L -E E cn 0) c CD CM CM O O + + LU LU o o o o o o to t" co in CM <tf CM CM o o + + LU LU O O O O O O cn cn r— co CO CM CM CM CM O O O + + + LU LU LU O O O O O O O O O CO CM to co to CM •tf CO CM CM O O + + LU LU o o O O O o •*r co to r-•5f CM CM CM CM O O O + + + LU LU LU O O O O O O O O O CM CM CO ••- •q- CM CO CM CM O O O O O O O O O O O O O O O O + + LU LU o o o o CM CM LO CO 01 CM CO o o o o + + LU LU o o o o t o *r ••-cn r-CM -r-O O O O + + LU LU O o O O CO CM r~ CM CO CO CO O O O O O O + + + LU LU LU O O O O O O r~ in 01 to 01 CO tO CM CM CO CM O O O O O O + + + LU LU LU O O O O O O t- ^ O CM to CM CM CM CM CM O O O O O O O O O O O O O O O O O O O O O O O O 1 1 1 1 1 1 1 1 1 1 1 1 i n i n i n i n i n i n 111 i n i n 111 i n 111 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 01COLOCMCM--0)COC»-' - - ' - lO C M ^ j - f - C M ' - t D r - c n c o c O ' ^ t -T - C N * - ' * - ' - C , > * - C M * - * - C N ' » -606666666666 COOOCOCOCOCOCOCOOOCOCOCO C S C O C O C O C 0 C O C O C O C O C O C 0 C 0 c o c p t o c o t o t o t o c o c o c p c o c o cncocncno-^coin i i-^coLO t---'-t0OC0CMLOL0-'-C001C0 • ^ t~CM - ^ L O t O C n C J)r~CN^CO --•'-CMC0CM1-CM1-CMC01-CM rocnco-»-.i>rocococororo->.roroco-»-ro-'. - ^ I O r O ~ 4 0 1 4 > C J J > O ( D M U l * . O 0 1 ( D U l ~ J u i - j . u i u i u i - » c n c J i J>~4 - » . o o c o r o ^ i o o " > o 0101OlO>Cncn01Cncn01cnO)01010)0 }Cncn OIUIUIUIUIUIUIOI^ICOCOCOCOCOCOCDCDCO o o o o o o o o o o o o o o o o o o ^ A - ' ( D M - ' J ^ ^ - » - i - - i M I O - » - i - i .i>cooico-'.coJ>cnoococoocn.b.~« ,ocDO cncooiO-i>^iro-"..i>ocDOOoioai~icoco O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O rnfnrnmmmiTimrnrnrnrnrnrnrnrnrnrn I I I I i I i i i i I I i i i i I i O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O CJ co co U U (D ~ i cn co •t* CO 01 O O O ooo rn rn rn + + + O O O ooo r o 4> -j -i. -»• J> ^ ca O O O O m m + + O O O O CO CO -- CD o o 00 CO O O O O m m + + O O o o co J> c a CD CO CD cn - j O O O O m m + + O O o o ui co -U CO cn o O O O O m m + + O O o o co J> c n 01 -•• cn J> ro J> u i u i co - j co u i o O O O O O O O O r n r n m m + + + + O O o o o o o o co J> co ro co CD O cn cn 4> o O O O O O O rn m m + + + O O O O O O CD 3 IO ro cn -t ID O J> •fc. ro 3 3 c n - j ~ j ~ j i i ~ i - j c o u i J > . o u i c o r o c o c o 4 > c o c o O c n o c o o - ' - J ~ 4 4 > - ^ i o-i . o o o r o c o c j i - j - ' j>_i.coJ>co-> .cno- ' - ' - - JOui - ioocDO- ' - ' co j^ooorooDJ>4>iDO- '~ i rococncoOoico cn-j~i~icn~io>oioicncncncn^jcn-j- i^ioo c o o r o o c o o c o t o o o o ^ c o c o o c o o r o ^ i o J>C0J>C0C0-^J>C0C0C0C0CO^lCDC0C0OO<0 O O O O O O O O O O O O O O O O O O O r o ^ c o r o - t foco - t f o r o c o c o r o r o r o c o r o i o r o co-4coco^iro--.uicocncocororocn.t*CDUi-i. O r o J > u i 4 i O o c o c o ~ J - ' C n o o u i r o u i o c o c D O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O rnmmmmmmrnrnrnrnrnnirnrnrnrnrnrn i I i I i i i i I i I I I I I i i i i O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O co cn o i -o. O O co oo o i co co -c ooo ooo rn m m + + + ooo ooo CO CO Ul •b CO O O O O m m + + O O O O CO J> O CO 00 CD Ul CO o o o o rn m + + o o o o r o co o i o i co o CD J> 00 03 CO O O O O O O m m m + + + O O O O O O J> O l J> U l co CD CD O l oo cn co O O O O O O m m m + + + O O O O O O co oo J> O O J> cn o o o o m m + + o o o o -j J> J> co J> ui cn oi cn <n oo oi co cn O O O O O O O O rn m m m + + + + O O O O O O O O ID 3 CQ r+ 3" II J> CD -4 CO 3 3 U -I O J> •o ro 3 3 O o - J cn co ro co - t co U l ^ . CO J> O cn ui 4> cn cn cn cn cn oo oo oo oo oo 00 00 00 00 00 O o o o o J> ro co co J> O O O o 0 O m m 1 i J> -•. CD CO ui CD cn O O O O O O O O O m m m i i O O o o o o o o o o co r o 4> co cn -J CD -•• -J J> r o co O O O O O O m m m + + + O O O O O O J> r o ro co J> J> j> co O O O O m m + + 88 O O O O O O O O O O O O O O O O O O CJ -J CO J> r o oo 00 ~ i ~ i O O O ooo ooo m m m + + + O O O i o r o r o ro co CD -j cn u i O O O O O O m m + + O O ro ro co co ro oo co ro O O O O O O m m + + O O ro ro ro ->• oi O O O O O O m m + + O O io ro j*. ro J> 0 ) j> co O O O O o o m m + + o o ro r o co co o i cn r o oo - J u i ro oo oo O O O O O O O O O O O O m m m m + + + + O O O O r o ro r o ro ro o i co co cn co co oo O O O O O O O O O m m rn + + + O O O r o r o ro O O O O O O O O O O O O O O O O O O O u c n o i £ * c o 4 > o i - b J > £ > o i c n c o c o o i £ > u i c o c o ~ i ~ j r o c o o i o c n - t - o c o r o c o c D C D O C D J > c o r o r o J > o o c n c n c o r o u i r o ^ i j > - ' C o o c o c D r o c D O i O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O m m m m m m m m m r i i r T i n i r n n n i n i n i m i T i + + + + + + + + + + + + + + + + + + + O O O O O O O O O O O O O O O O O O O r o r o r o r o r o r o r o r o r o i o r o r o r o r o r o r o r o r o r o O O O O O u i co o i u i O o i oo - g -». O O O O O O O O O m m m + + + O O O i o ro r o o i co O co J> J> O O O O O O m m + + O O ro r o CO (0 O CD Ol 01 CO CD CO J> CO J> Ol -J 01 (Jl 4> J> (0 Ol & J> 00 CD 01 A Ol CD S3 J> (0 -J J> 01 -J Ol CD O IO CD .!> -J ^1 -J 01 O J> 01 01 01 01 01 01 ~ J ~ J ~ J ~ J ~1 -4 CD CO (0 CO CD CD O) 01 01 01 01 01 ~I ~J ~1 O l 01 to CD CO CO IO IO O O O O O O O O O O O O roioro-t-t-tio-'.-tio-i.ro ACOOCDCD^llotOOl-^OllO - > t n > i O O c o t ( j ( J o i t D O ^ O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O r n m r n r n r n r n r n i n r n r n r n r n i i i i i i i i i i t i O O O O O O O O O O O O O O O O O O O O O O O O J> CO CO IO 0) o o o o o m m + + o o o o co Co co co .u O -•• - J - j oo ro 01 O -» oo o i O O O O O O O O m m m m + + + + O O O O O O O o -J. o O 01 CD CD O O O O m m + + o o o o J> J> Ol — O O M — 00 O 00 O "^ 1 -*• CO o o o o o o o o rn rn rn rn + + + + o o o o o o o o o o o o o o o o o o o o J > J ^ 4 ^ 4 ^ N > 4 ^ C J I J ^ J ^ J > C 0 0 1 fllfOC0OCD00C0(JlflllOOl~4 0 1 I O r O C O C O - J - . . t s O O O C O C O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O r n m r n r n r n m r n r n r n r n r n m + + + + + + + + + + + + O O O O O O O O O O O O rororororororofoMforofo Appendix XXIX . Data from Sedimentation Experiments This f i l e contains sediment volumes (mL) and oven dried weights (grams) of 3, 6, and 15 denier f i b r e s . Suspension temperature was close to 22°C and conductivity was less than 5.0*10' Mho. 3 DENIER FIBRES (diameter=0.0197mm) Fibre length=0.915mm 0.167000E+03 0.1S8000E+03 0.128000E+03 0.129000E+03 0.142000E+03 0.142000E+03 O.63O90OE+01 0.630900E+01 0.473760E+01 0.473760E+01 0.516430E+01 0.516430E+01 Mean sediment concentration=0.0352 Standard deviation=0.000568 95% confidence 1imits=0.0352 ± 0.00146 Fibre 1ength=1.875mm 0.188000E+03 0.189000E+03 0.151000E+03 0.159000E+03 0.140000E+03 0.141000E+03 0.115000E+03 0.110000E+03 0.110000E+03 0.112000E+03 0.115000E+03 0.115000E+03 0.130000E+03 0.129000E+03 0.940000E+02 0.920000E+02 0.135000E+03 0.143000E+03 0.130000E+03 0.130000E+03 0.102000E+03 0.100000E+03 O.105000E+03 0.980000E+02 0.250170E+01 0.250170E+01 O.231060E+01 0.231060E+01 0.190480E+01 0.190480E+01 0.117670E+01 0.117670E+01 0.102970E+01 0.102970E+01 0.112720E+01 0.112720E+01 0.162720E+01 0.162720E+01 0.116580E+01 0.116580E+01 0.190420E+01 0.190420E+01 0.169380E+01 0.169380E+01 0.120130E+01 0.120130E+01 0.107380E+01 0.107380E+01 Mean sediment concentration=0.0115 Standard deviation=0.00164 95% confidence 1imits=0.0115 ± 0.00339 Fibre 1ength=2.815mm 0.140000E+03 0.135000E+03 0.150000E+03 0.14700OE+O3 0.105000E+03 O.110000E+03 0.140000E+03 O.133000E+03 0.103000E+03 0.100000E+03 0.850000E+02 0.820000E+02 0.100000E+03 0.100000E+03 0.112000E+03 O.112000E+03 O.105000E+03 O.104000E+03 0.108000E+03 0.112000E+03 0.740000E+02 0.740000E+02 0.116000E+03 0.120000E+03 0.928900E+00 0.928900E+00 0.932000E+00 0.932000E+00 0.683900E+00 0.683900E+00 0.765700E+00 0.765700E+00 0.523400E+00 0.513400E+00 0.394200E+00 0.394200E+00 0.522200E+00 0.522200E+00 0.574000E+00 0.574000E+00 0.480100E+00 0.480100E+00 0.648300E+00 0.648300E+00 0.445100E+00 0.445100E+00 0.716000E+00 .0.716000E+00 Mean sediment concentration=0.00537 Standard deviat 1on=0.000647 95% confidence 1imits=0.00537 ± 0.00134 Fibre 1ength=3.737mm 0.135000E+03 0.138000E+03 0.950000E+02 0.950000E+02 0.980000E+02 0.100000E+03 0.140000E+03 0.145000E+03 0.130000E+03 O.135000E+03 O.115000E+03 0.110000E+03 O.125000E+03 0.523700E+00 0.523700E+00 0.355200E+00 0.355200E+00 0.336300E+00 0.336300E+00 O.515300E+00 0.515300E+00 0.448000E+00 0.448000E+00 0.363400E+00 0.363400E+00 0.434300E+00 O co O.135000E+03 0.185000E+03 0.195000E+03 0.165000E+03 0.170000E+03 0.145000E+03 0.145000E+03 0.11OOO0E+03 0.108000E+03 0.1G5000E+03 0.165000E+03 0.434300E+00 0.591200E+00 0.591200E+00 0.504600E+00 0.504600E+00 0.416900E+00 0.416900E+00 0.362400E+00 0.362400E+00 0.433700E+00 0.433700E+00 Mean sediment concentration=0.00313 Standard deviations.000332 95% confidence 1imits=0.00313 ± O.000686 6 DENIER FIBRES (diameter=0.0279mm) bre 1 engthS. 914mm 0.810000E+02 0.810000E+02 0.770000E+02 0.770000E+02 0.113000E+03 0.117000E+03 0.118000E+03 0.900000E+02 0.900000E+02 0.102000E+03 0.104000E+03 0.384340E+01 0.384340E+01 0.349680E+01 0.349680E+01 0.616910E+01 0.616910E+01 0.616910E+01 0.463690E+01 0.463690E+01 0.540560E+01 0.540560E+01 Mean sediment concentration=0.0479 Standard deviation=0.00309 95% confidence 1imits=0.0479 ± 0.00688 bre 1ength=1.832mm 0.136000E+03 0.136000E+03 0.120000E+03 0.121000E+03 0.110000E+03 0.110000E+03 0.150000E+03 0.154000E+03 0.122OOOE+03 0.121000E+03 0.138000E+03 0.138000E+03 0.330900E+01 0.330900E+01 0.279150E+01 0.279150E+01 0.255370E+01 0.255370E+01 0.364140E+01 0.364140E+01 0.271700E+01 0.271700E+01 0.291850E+01 0.291850E+01 o o o o o o o o o o _k CD O CD - o o o o o o o m m + + o o cd ro co co ro ro O O O O O O o o m m + + O O ro ro CO CO -k O CO J> O O cn O O O O O O O O O m m m + + + O O O ro 10 co J i O O U l CO 00 O O O O O O ooo m m m + + + O O O CO CO CO O O O O O O O O O O cn 01 A J> ro ro 01 01 O O O O m m + + O O O O J> J> U l co co ro co co ~ J ui ui 01 O O O O O O m rn rn + -I- + O O O O O O U l 03 ro oi -J -k cn ui o o O O m m + + O O o o 00 oo 03 01 u i u i -k CO CO u i ro ro O O O O O O m m m + + + O O O O O O CD 3 CQ 03 3 3 co in 2 U l H- CD s? D) 0) 3 3 o a O 0) 01 3 S CD -h a a a a 3 CD CD CD 3 < 3 O — r+CD 0) r+ 0 —• —j- 0 0 3 3 3 0 ll CD r+ Q 3 CO • r+ II O ~> ooo) r+ O ro O ro 0 03 3 . II o l + b o o 03 b o ro U l U l o o o o o o o o o o o o o o o o o o o o CDC0^.->.-fc-k-k->.-fc-k-fc->.-k-k->.-k-k-k-k-4. ^ icnoOt>J>4>(>>j> | i> i iS ) i i iACj roMcoro O O O - ' C O O ' J ^ i O - ' C D O O c n r o o o c o o i o c D O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O m m r i m n i r n i i i i n m r n r n m i T i i T i r n n i r n r i n i m + + + + + + + + + + + + + + + + + + + + O O O O O O O O O O O O O O O O O O O O rofococococococococococococdcocococococo O O O O O O O O O O O O O O O O O O O O oicn~i~icoco-k-'-k-*-k-k-k->.-k-k-k-k-k-k u i c n c » c»cno3-k - k 4 i * v c o c o J i J > u i u i-k-k-k-k ro ro ro rococou iu iooosoooo^ i - JOscncocD OOuiuio icnoocncr ) - j - - icocooooouiu ic i icn O O O O O O c o c o M r o O O ~ J ~ J J > J > c o c o r o r o O O O O O O O O O O O O O O O O O O O O m m m m m m m r r i m i T i m r n r n r n r n r T i m m m m + + + + + + + + + + + + + + + + + + + + O O O O O O O O O O O O O O O O O O O O CD 3 CQ e* 3-II ro ui 3 3 co in 2 O O O O O O 01 r+ CD SS 0) 0) ~ l -J 0 ) 0 ) 3 3 03 CO U l 01 o O o a O O O O ui ui O 0) U ) O O O O O O • S CD O O O O O O -i> a a O O O O O O m m m m m m a a 3 + + + + + + CD ID ID O O O O O O 3 < 3 ro ro to ro co co a -•• r+ CD 0) r+ CI - . ... 0 - 0 3 O O O O O O 3 3 0 - II It -k PO IO H- O 3 - J - J . J i J i U l U l CD • rt O O O O co co H O T U l U l J i J> CD 01 OOO) CO CO J i J i CO CO • -»• r+ O O O O O O O O - rn rn rn m m m ro co o + + + + + + -k 3 O O O O O O 03 IIo 1+ o O io O 03 o ro CO ro O.126000E+03 0.128000E+03 0.900000E+02 0.890000E+02 0.970000E+02 0.900000E+02 0.161000E+03 0.160000E+03 0.150000E+03 0.158000E+03 0.220000E+03 0.225000E+03 0.170000E+03 0.175000E+03 0.706000E+00 0.706000E+00 0.628400E+00 0.628400E+00 0.643200E+00 0.643200E+00 O.112690E+01 0.112G90E+01 0.110310E+01 0.110310E+01 0.138810E+01 0.138810E+01 0.945500E+00 0.945500E+00 Mean sediment concentrat1on=0.00604 Standard deviation=0.000834 95% confidence 1imits=0.00604 ± 0.00172 Fibre 1ength=4.666mm 0.128000E+03 O.130000E+03 0.900000E+02 0.850000E+02 0.170000E+03 0.165000E+03 0.175000E+03 0.180000E+03 O.115000E+03 0.115000E+03 0.110000E+03 0.110000E+03 0.165000E+03 0.160000E+03 0.980000E+02 0.900000E+02 0.110000E+03 0.105000E+03 0.140000E+03 0.150000E+03 0.105000E+03 0.110000E+03 0.900000E+02 0.950000E+02 0.588000E+00 0.588000E+00 0.374800E+00 0.374800E+00 0.729200E+00 0.729200E+00 0.736000E+00 0.736000E+00 0.501400E+00 0.501400E+00 0.428500E+00 0.428500E+00 0.648900E+00 0.648900E+00 O.356600E+00 O.356600E+00 0.396800E+00 O.396800E+00 0.538600E+00 0.538600E+00 0.387400E+00 0.387400E+00 0.338600E+00 0.338600E+00 Mean sediment concentration=0.00381 Standard deviation=0.000315 95% confidence 1imits=0.00381 + 0.000651 CT "3 ID OOOOOOOO CD CD CO 3 3 U l nr CD (0—«•.—•»_>>—•.—i—i.—*. (Q S« 0) 0) IDO-'-'OO-'-' r+ 3 O-^-'-'cnoioo 3 0 0 . OOOOOOOO II O 01 CO OOOOOOOO I O 3 T CD OOOOOOOO -* a a m m m r n m r n m r n CO —1. + + + + + + + + a a OOOOOOOO CD CD CD u u c o u u c o u u 3 3 < 3 0 i+ CD DJ O o OOOOOOOO -• o 3 3 3 0 r o r o r o r o r o r o r o r o n CD r+ O 3 - ' - ' 0 7 ) C D i > & C O C O to • |H> c o c o t n u i o a c o o o c o HOT OOOOOOOO OOO) OOOOOOOO • J> rf m m m r n m r n m r n O O - J -+ + + + + + + + U l 00 0 OOOOOOOO U l 3 - J II o 1+ b o U l U l b ^1 o 03 co t o oooooooooooo CO CO O U l O O o o o o o o m m + + o o M i o 00 oo 00 00 O O O O O O O O m m + + O O rO ro CO CO IO 01 o o O O O O O O m m + + O O r o r o O) -J oi o O o O O O O o o m m + + O O r o ro to to -•• -•• O) -0. - - O O o o o o o o o o o o o o m r n m r n + + +• + o o o o t o co ro r o oooooooooooo u i t n -~i - j O O O O O O m m + + O O t n t n u i CO CO CO OOO OOO OOO OOO m m m + + + OOO o i co CO u i O oo O O O O O O m m + + O O CO O) U l 00 00 ->. o o o o o o m m + + o o O l 01 O l co ro r o -•• u i u i OOO OOO OOO m m m + + + OOO CT s CD CD 3 ID r+ 3 " II U l O) o 3 3 CO 1/1 3 Ol r* CO 0) 0) 3 3 O a 0 0) Ul "3 ID -i> a a a a 3 CD CD ID 3 < 3 O -•• r+ CD fi) H - n —* — o -i. O 3 3 3 n -•- II CD r* O 3 01 • rf II O T OOO) •O rt-_h •ft. — ro to 0 CO 3 Ol II O i+ O ro b o CO oo CO OOOOOOOOOOOO o o - J 01 o o o o o o m m + + o o t o co CD A O O o o o o m m + + O O co t o - i 00 ooo) OOO OOO OOO OOO m m m + + + OOO co t o r o 03 - * 01 o -•• -ooo ooo ooo m m m + + + OOO ro co co ro r o O O O O O O m m + + O O co t o OOOOOOOOOOOO co t o Ol u i O O O O m m + + O O r o r o - o - J - 4 - i O O O O m m + + O O r o ro r o r o r o r o o i O O u i OOO OOO m m m + + + OOO r o r o r o 01 J > o i t o O O O O m m + + O O ro r o J > J > A to CO co co co OOO OOO m m m + + + OOO ro r o ro •n CT "3 CD U l _ o CD m 3 2 CQ r+ m 3 73 II O •n t o DO CO 73 —k m 3 t o 3 .—* a 0) 3 CD r+ CD "1 II o b J > I O 3 3 313 CT) ip in — CN o „ g co o "* • 01 O CN O o O O +1 +1 t o ^ o o i n ^ 2 2 — - cO c^ncncocn -H ° In r ^ScNCNCNCNCNCNOo22SLoSS i ^ l l i C N C N ^ - T r ~ r - r ~ r - t ~ t - r ~ r - r r ' J - r - - r - - C O +• O O ' - ' - O O O O O O i n i n t f - ' - n n ' - ' - c O •H C N C N C N C N C M C N C N C N C N C N C N C N C N C N C M C M CI) II — — — — — — — — — — — C T ) C T > C O C O C 0 C O O ) O > C 0 C 0 CD II — O O O O O O O O O O O O O O O O C O - O O O O O O O O O O O O O O O O O O O O C O — O-H (J V "~ %$mmiwm\ ill | ill I ELOcn S o o o o o d d d d d o . d d o o d d d d o E to c3 CU 0) i. i. U XI , — — 0) 6606660006006666 CO CO 3 U l <+ fD 0) 0) 3 3 0 a 0 0) 0) 3 T CD -l! a a a a 3 ID CD CD 3 < 3 0 r* CD fi) r+ O — i — 0 -j. 0 3 3 3 0 -J. II CD r* O 3 01 • r+II O ^ o o a i O H-o U l o CO 0 U l 00 3 J i II t o o 1+ b o o U l J i b ro o r o o o o o o o o o CO r o O u i O O O O O O rn m + + O O CO CO CO - J U l U l o o o o O O m m + + O O CO CO U l CO J i J i O U l U l O O O O O O O O O O O O O m rn r n n + + + + O O O O CO CO CO CO O O O O O O O O cocococooocococo O O O O O O O O O O O O O O O O O O O O O O O O o o o o o o o o o o o o o o o o r n m r n m m r n m m + + + + + + + + o o o o o o o o o o o o o o o o Appendix XXX. Data from Coeffi c i e n t of F r i c t i o n Measurements. Data from s t a t i c and dynamic f r i c t i o n measurements between two wet nylon surfaces. Each l i n e containes: load placed on a s l i d e (grams), and tangents of an in c l i n e angle for s t a t i c and dynamic cases. 3 DENIER FILAMENTS (diameter=0.0197mm) 0.100000E+02 0.100000E+02 0.100000E+02 0.200000E+02 0.200000E+02 0.200000E+02 0.500000E+02 0.500000E+02 0.500000E+02 0.100000E+03 0.100000E+03 0.100000E+03 0.100000E+02 0.100000E+02 0.100000E+02 0.200000E+02 0.200000E+02 0.200000E+02 0.300000E+02 0.300000E+02 0.300000E+02 0.4OO0OOE+O2 0.400000E+02 0.400000E+02 0.740000E+02 0.740000E+02 0.740000E+02 0.810000E+02 0.810000E+02 0.810000E+02 0.500000E+02 0.500000E+02 0.500000E+02 0.100000E+03 0.100000E+03 0.100000E+03 0.367400E+00 0.357500E+00 0.357500E+00 0.358400E+00 0.351200E+00 0.343200E+00 0.336900E+00 0.324400E+00 0.302000E+00 0.338700E+00 0.328800E+00 0.318100E+00 0.360000E+00 0.348000E+00 0.342000E+00 0.362000E+00 0.348000E+00 0.334000E+00 0.365000E+00 0.365000E+00 0.347000E+00 0.360000E+00 0.358000E+00 0.343000E+00 0.340000E+00 0.321000E+00 0.303000E+00 0.340000E+00 0.337000E+00 0.315000E+00 0.358000E+00 0.337000E+00 0.322600E+00 0.257OOOE+00 0.241000E+00 0.241000E+00 0.302900E+00 0.296600E+00 0.293000E+00 0.293000E+00 0.285800E+00 0.285800E+00 0.267900E+00 0.264300E+00 0.261600E+00 0.261600E+00 0.246400E+00 0.239200E+00 0.305000E+00 0.289600E+00 0.284000E+00 0.284000E+00 0.274000E+00 0.2S9000E+00 0.278000E+00 0.264000E+00 0.249000E+00 0.266000E+00 0.266000E+00 0.255000E+00 0.256000E+00 0.252000E+00 0.244000E+00 0.244000E+00 0.242000E+00 0.234000E+00 0.261000E+00 0.261000E+00 0.261000E+00 0.343200E+00 0.333000E+00 0.333000E+00 Mean s t a t i c c o e f f i c i e n t of friction=0.334 Standard deviat1on=0.0318 95% confidence 1imits=0.0334 ± 0.0647 CO H Ln Mean dynamic c o e f f i c i e n t of friction=0.273 Standard deviations.0267 95% confidence 1imits=0.273 + 0.0542 6 DENIER FILAMENTS (diameter=0.0279mm) 0.100000E+02 0.100000E+02 0.200000E+02 0.200000E+02 0.500000E+02 0.500000E+02 0.100000E+03 0.100000E+03 0.100000E+02 0.100000E+02 0.200000E+02 0.200000E+02 0.500000E+02 0.500000E+02 0.100000E+03 0.100000E+03 0.100000E+02 0.100000E+02 0.200000E+02 0.200000E+02 0.500000E+02 0.500000E+02 0.100000E+03 0.100000E+03 0.302700E+00 0.342800E+00 0.296400E+00 0.315200E+00 0.261600E+00 0.291100E+00 0.259800E+00 O.279500E+00 O.343700E+0O 0.325000E+00 0.335000E+00 0.324100E+00 O. 334000E+00 0.327000E+00 0.334800E+00 0.325000E+00 0.296100E+00 0.336900E+00 0.2885O0E+OO 0.288500E+00 0.267900E+00 0. 315400E+00 0.302000E+00 0.303700E+00 0.205300E+00 0.210700E+00 0.207100E+00 0.212500E+00 0.199100E+00 0.214200E+00 O.199100E+00 0.204800E+00 0.197300E+00 0.191100E+00 O.197300E+00 0.190200E+00 0.190200E+00 0.180400E+00 0.198600E+00 0.191100E+00 0.194400E+00 0.194400E+00 0.191700E+00 0.191700E+00 0.183700E+00 0.187500E+00 0.189100E+00 0.189100E+00 Mean s t a t i c c o e f f i c i e n t of f r i c t i o n s . 308 Standard deviations.0256 95% confidence limitsS.308 ± 0.0529 Mean dynamic c o e f f i c i e n t of f r i c t i o n s . 196 Standard dev iat i o n S . 00898 95% confidence limitsS.196 + 0.0186 c^ i Appendix XXXI. Average V e l o c i t i e s and Root-Mean-Squares of Velocity Fluctuation Data presented in this appendix are for the low v i s c o s i t y suspending l i q u i d discussed in Section 4.5. COORDINATES AVERAGE VELOCITIES RMS OF VELOCITIES RMS/AVERAGE VELOCITY r mm e deg. RADIAL TANGENTIAL m/s m/s RADIAL TANGENTIAL RADIAL TANGENTIAL m/s m/s 10 0 0 . 20E -02 -0 .61E-02 0 . 26E -01 0 . 20E -01 0 .13E+02 -0 .32E+01 15 0 -0 . 15E -01 0. .83E-02 0 . 19E -01 0. . 14E -01 -0 .13E+01 0 .17E+01 20 0 -0. . 13E -01 0. .18E-01 0, . 17E -01 0. . 13E -01 -0. .13E+01 0 .70E+00 25 0 -0. . 19E -01 0, .29E-01 0. . 15E -01 0 . 13E -01 -0 .76E+00 0 .46E+00 30 0 -0. . 15E -01 0. .19E-01 0, . 16E -01 0 . 15E -01 -0. .11E+01 0 .79E+00 35 0 -o. . 19E -01 0, .88E-02 0. . 18E -01 0. . 17E -01 -0. .97E+00 0 .19E+01 40 0 -0. . 14E -01 0. .18E-01 0, . 13E -01 0. .23E -01 -0 .93E+00 0 .13E+01 45 0 -0. . 29E -02 0. .11E+00 0. . 1 1E -01 0. . 17E -01 -0 .38E+01 0 .16E+00 25 10 -0. . 20E -01 0, .10E-01 0. . 16E -01 0. . 12E -01 -0. .81E+00 0 .12E+01 30 10 -0. ,44E -02 0. .15E-01 0. ,20E -01 0, . 13E -01 -0. .44E+01 0. .87E+00 35 10 -0. , 18E -01 0. 28E-01 0. . 17E -01 0. . 12E -01 -0. .94E+00 0. .44E+00 40 10 -0. . 15E -01 0. ,28E-01 0. . 15E -01 0. . 15E -01 -O. .11E+01 0, .54E+00 45 10 -0. . 73E -03 0. ,11E+00 0. . 16E -01 0. . 17E -01 -0. .22E+02 0, .16E+00 10 20 -o. 87E -03 -0. ,14E-01 0. . 20E -01 0. . 10E -01 -0. .23E+02 -o, .73E+00 15 20 -0. . 14E -01 0. .11E-01 0. 21E -01 0. 87E -02 -0. .15E+01 0. .76E+00 20 20 -0. 22E -01 0. .19E-01 0. 27E -01 0. 48E -02 -0. .12E+01 0. .25E+00 25 20 -0. . 18E -01 0. .22E-01 0. , 17E -01 0. . 58E -02 -0. ,96E+00 0. .26E+00 30 20 -0. 25E -01 0. 27E-01 0. 17E -01 0. 1 1E -01 -0. ,67E+00 0. .39E+00 35 20 -0. 26E -01 0. 27E-01 0. 19E -01 0. 1 1E -01 -0. 74E+00 0. .41E+00 40 20 -0. 16E -01 0. 30E-01 0. 18E -01 0. 1 1E -01 -0. 11E+01 0. 3GE+00 45 20 0. 22E -02 0. 12E+00 0. 18E -01 0. 84E -02 0. 81E+01 0. 69E-01 25 30 -0. 21E -01 0. 25E-01 0. 21E -01 0. 42E -02 -0. 98E+00 0. .17E+00 30 30 -0. 20E -01 0. 27E-01 0. 26E -01 0. 17E -03 -0. 13E+01 0. 62E-02 35 30 -0. 19E -01 0. 22E-01 0. 26E -01 0. 58E -02 -0. 14E+01 o. .26E+00 40 30 -0. 59E -02 0. 42E-01 0. 23E -01 0. 42E -02 -0. 40E+01 0. 10E+00 45 30 -0. 60E -02 0. 12E+00 0. 17E -01 0. 58E -02 -0. 29E+01 0. 49E-01 10 40 -0. 83E -02 -0. 98E-02 0. 18E -01 0. 66E -03 -0. 22E+01 -0. 68E-01 15 40 -0. 99E -02 0. 38E-02 0. 19E -01 -o. 16E -02 -0. 19E+01 -0. 42E+00 20 40 -0. 1 1E -01 0. 25E-01 0. 19E -01 0. 87E -03 -0. 17E+01 0. 35E-01 25 40 -0. 13E -01 0. 24E-01 0. 19E -01 0. 31E -02 -0. 15E+01 0. 13E+00 30 40 -0. 15E -01 0. 30E-01 0. 20E -01 -o. 31E -03 -0. 14E+01 -0. 10E-01 35 40 -0. 26E -01 0. 30E-01 0. 20E -01 0. 19E -02 -0. 78E+00 0. 62E-01 40 40 -0. 15E -01 0. 51E-01 0. 20E -01 0. 40E -02 -0. 13E+01 0. 79E-01 45 40 0. 17E -02 0. 14E+00 0. 18E -01 -o. 66E -03 0. 10E+02 -0. 48E-02 25 50 -0. 19E -01 0. 93E-02 0. 24E -01 0. 49E -03 -0. 12E+01 0. 53E-01 30 50 -0. 18E -01 0. 35E-01 0. 19E -01 -o. 15E -02 -0. 10E+01 -0. 44E-01 35 50 0. 29E -02 0. 24E-01 0. 26E -01 -o. 1 1E -01 0. 91E+01 -0. 48E+00 40 50 0. 39E -02 0. 21E-01 ' 0. 23E -01 -o. 47E -02 0. 60E+01 -0. 22E+00 45 50 0. 27E -02 0. 12E+00 0. 30E -01 -o. 93E -02 0. 11E+02 -0. 80E-01 10 60 15 60 20 60 25 60 30 60 35 60 40 60 45 60 25 70 30 70 35 70 40 70 45 70 20 80 25 80 30 80 35 80 40 80 45 80 25 280 30 280 35 280 40 280 45 280 25 290 30 290 35 290 40 290 45 290 10 300 15 300 20 300 25 300 30 300 35 300 40 300 45 300 25 310 30 310 35 310 40 310 45 310 10 320 15 320 20 320 25 320 30 320 35 320 40 320 45 320 -0 . 76E -01 -0, .73E -03 0 . 24E -01 0 . 19E -01 -0 . 26E -02 -0 . 35E -02 -o .38E -02 -o, .22E -02 -0. .41E -02 -0. .20E -02 -o. . 34E -02 -0. .65E -02 0. .21E -02 -o. .15E+00 -0. .94E -01 -0 .66E -01 -0. . 48E -01 -0. 20E -01 -0. 64E -02 0. . 20E -01 0. , 12E -01 0. . 12E -01 0. 41E -02 -0. 22E -01 0. 30E -01 0. 18E -01 0. 29E -01 0. 10E -01 0. 92E -02 0. 52E -01 0. 24E -01 0. 16E -01 0. 16E -01 0. 68E -02 0. 12E -01 0. 92E -02 0. 17E -01 0. 25E -01 0. 78E -02 0. 13E -01 0. 1 1E -01 0. 24E -03 0. 35E -01 0. 23E -01 0. 25E -01 0. 15E -01 o. 22E -01 0. 61E -02 o. 14E -02 -0. 42E -02 -0 .43E -01 -0 . 28E -01 -0 .32E -01 -0 . 15E -01 0 . 14E -01 0 . 19E -01 0 .53E -01 0 .14E+00 0 .67E -02 0 . 13E -01 0 . 20E -01 0 .55E -01 O .15E+00 -0 .35E -01 -0 . 20E -01 -0 .81E -02 0. .86E -02 0, .41E -01 0. . 14E+00 -0 .91E -01 -0. . 52E -01 -0. .43E -01 -0, . 50E -01 -0, .22E -01 -0. 47E -01 -0. . 20E -01 -0. . 39E -01 -0. . 16E -01 0. 26E -02 -0. 91E -01 -0. ,40E -01 -0. 28E -01 -0. 19E -01 -0. 72E -02 -0. 36E -02 0. 13E -01 0. 29E -01 -0. 45E -01 -0. 21E -01 -o. 24E -01 -o. 18E -01 0. 15E -02 -0. 35E -01 -0. 32E -01 -0. 29E -01 -0. 26E -01 -0. 28E -01 -o. 30E -01 -0. 20E -01 0. 19E -01 0 .41E -01 0 .27E -01 0 . 25E -01 0 .24E -01 0 . 18E -01 0 . 22E -01 0 . 21E -01 0 .24E -01 0 . 13E -01 0 . 14E -01 0 . 17E -01 0 . 19E -01 0 . 17E -01 0 . 23E -01 0 . 30E -01 0 . 20E -01 0. . 20E -01 0. . 15E -01 0, , 15E -01 o. .91E -02 0, 91E -02 0. . 13E -01 0. , 13E -01 0. . 19E -01 0. . 10E -01 0. 97E -02 0. 14E -01 0. 77E -02 0. 13E -01 -0. 60E -02 -0. 62E -02 0. 54E -02 0. 60E -02 0. 49E -02 0. 36E -02 0. 5«: -02 0. 68E -02 -0. 74E -02 -0. 18E -02 -0. 22E -02 -o. 41E -03 0. 65E -03 -o. 18E -01 -0. 16E -01 -0. 14E--01 -0. 84E -02 -0. 80E -02 -0. 42E--02 -0. 10E--02 -0. 35E--02 0 . 74E -02 -0 . 14E -01 -0 .22E -01 -0 . 25E -01 -0 .89E -02 -0 . 16E -01 -0 . 1 1E -01 -0 .94E -02 -0 .68E -02 -o .86E -02 -0 . 12E -01 -0 . 16E -01 -0 .99E -02 -0 . 12E -01 -0 . 10E -01 -0 . 13E -01 -0. . 13E -01 -0, . 20E -01 -0, . 16E -01 0, . 36E -01 0. .48E -01 0. . 34E -01 0, . 39E -01 0. . 22E -01 0. . 36E -01 0. 24E -01 0. 32E -01 0. 25E -01 0. 19E -01 0. 36E -01 0. 43E -01 0. 33E -01 0. 26E -01 0. 21E -01 0. 25E -01 0. 18E -01 0. 22E -01 0. 35E -01 0. 21E -01 0. 23E -01 0. 19E -01 0. 21E -01 0. 31E -01 0. 31E -01 0. 32E -01 0. 27E -01 0. 24E -01 0. 24E--01 0. 19E -01 0. 22E--01 -0.54E+00 -0.37E+02 0.10E+01 0.13E+01 -0.70E+01 -0.65E+01 -0.56E+01 -0.11E+02 -0.32E+01 -0.68E+01 -0.49E+01 -0.30E+01 0.79E+01 -0.16E+00 -0.32E+00 -0.30E+00 -0.42E+00 -0.75E+00 -0.23E+01 0.46E+00 0.77E+00 0.11E+01 0.31E+01 -0.83E+00 0.34E+00 0.55E+00 0.49E+00 0.74E+00 0.14E+01 -0.12E+00 -0.26E+00 0.33E+00 0.38E+00 0.73E+00 0.29E+00 0.58E+00 0.39E+00 -0.29E+00 -0.23E+00 -0.17E+00 -0.37E-01 0.27E+01 -0.51E+00 -0.71E+00 -0.56E+00 -0.56E+00 -0.36E+00 -0.69E+00 -0.75E+00 0.82E+00 -0.17E+00 0.50E+00 0.70E+00 0.16E+01 -0.66E+00 -0.85E+00 -0.21E+00 -0.70E-01 -0.10E+01 -0.66E+00 -0.57E+00 -0.29E+00 -0.64E-01 0.34E+00 0.51E+00 0.16E+01 -0.15E+01 -0.48E+00 -0.12E+00 -0.39E+00 -0.93E+00 -0.81E+00 -0.78E+00 -0.98E+00 -0.77E+00 -0.12E+01 -0.82E+00 -0.16E+01 0.74E+01 -0.39E+00 -0.11E+01 -0.12E+01 -0.14E+01 -0.29E+01 -O.70E+01 0.14E+01 0.74E+00 -0.79E+00 -0.97E+00 -0.98E+00 -0.11E+01 0.14E+02 -0.88E+00 -0.98E+00 -0.11E+01 -0.10E+01 -O.86E+OO -0.81E+00 -0.93E+00 0.12E+01 •^•^->-r-OCM'-'-'--^->--^O'-C0'-CN-^ o o o o o o o o o o o o o o o o o o + + + + + + + + + + + + + + + + + + III III I I I III I I I III III III ||l 1)1 |ll jl1 I I I |ll I I I III H I |1l c o i ^ c N C M L O - ' - L O C M U > c M - ' - » - L O t ~ c M c M T - ^ O O O O O O O O O O O O O O O O O O I i i i i I I o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o + + + + + + + + + + + + + + + + + + L U L U Q J L U L U L U L U U J L U I 1 J I 1 J 1 J I I J L L J L U L I J L I J L U O ^ O T i o c n c n m ^ ^ ^ ^ ^ m o a i h - L o c N i t O C M - ' - C M - ' - C D C D C O C O C M C O C M ' - r - T ' ' 3 - C O C M 600066666666060060 O O O O O O O O O O O O O O O O O O 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 111 in H I 111 in in 111 in in in in in 111 111 in ul 111 in c 0 L O < j 3 ^ ^ ^ t ~ c o c c c N c o c 0 ' " - ' * r c o L 0 t 0 L O • ^ • r - T - - » - C M C O C M * - - ^ - C N - ' - ' ' - C M C M C M - > - ' - ' i -O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 U J L U L U L U L U L U L U L J L U U J L U U J U J U J U J l l J L U L U c o c o c o O c o L O - ' ^ * j - - ' - r ~ c O L O ' < f r C O * - - ' - * - r ~ r - t r-- co *- LOI^ COCD-'-CO-'--'--'--'-*--'-O O O O O O O O O O O O O O O O O O 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 C M C M C M C M ' ' - C M C M C M C M - ' - - ' - C M - ' - C M t C M C O C M O O O O O O O O O O O O O O O O O O 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 in in in in in in in 1 ^ tii 1 A 1 H I in in in in in in in O O i n O ^ ^ ^ c o t ^ o o c n i ^ c M i o i p r o ' T c o c M t O L O r r c M L O t ^ c N ^ ^ c n c o c o c n i o c o c n o o o o o o o o o o o o o o o o o o 1 1 1 1 1 I I o o o o o o o o o o o o o o o o o o I 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 I 1 Ul Ul LU LU LU LU LU LU LU in in in in |i 1 in in in |i 1 CM CO LO CO CO •»- 1 - C O - ' - O t O C O C O C O C M I D * " * - CM CM «T CM 00 CM CM C M C M < 3 - r - * - C M C M C 0 r » 0 0 1 1 0 1 0 1 O O O 1 1 0 O 1 1 000000000 1 1 1 1 1 1 1 1 1 0 0 0 0 O O O O O 000000000 CO CO CO CO CO 'V • ^ • • " T ^ - T L O L O L O L O L O CO CO CO co CO CO CO CO CO C O C 0 C O C O C O C O O 3 C 0 C O LO O LO 0 LO O LO O LO O L O O L O L O O l O O L O CM CO CO Tr *- -r- CM CM C O C ' W ' t f C M C O C O ' s r ' * 

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