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UBC Theses and Dissertations

Time-dependent shear flow of artificial slurries Horie, Michihiko 1978

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TIME-DEPENDENT SHEAR FLOW OF ARTIFICIAL SLURRIES ' by MICHIHIKO|HORIE , B.Eng., Gumma U n i v e r s i t y , Japan, 1968 M . S c , N i i g a t a U n i v e r s i t y , Japan, 1970 M.Eng., McMaster U n i v e r s i t y , Canada, 1972 A THESIS SUBMITTED IN.-PARTIAL FULFILLMENT OF . THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES The Department of Chemical E n g i n e e r i n g We a c c e p t t h i s t h e s i s as conforming t o the r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA March, 1978 ( c T ) M i c h i h i k o H o r i e , 1978 In present ing th is thes is in p a r t i a l fu l f i lment of the requirements for an advanced degree at the Un ivers i ty of B r i t i s h Columbia, I agree that the L ibrary sha l l make i t f ree ly ava i lab le for reference and study. I fur ther agree that permission for extensive copying of th is thesis for scho la r ly purposes may be granted by the Head of my Department or by h is representat ives . It is understood that copying or pub l i cat ion of th is thes is f o r f i nanc ia l gain sha l l not be allowed without my writ ten permission. M i c h i h i k o H o r i e Department of C h e m i c a l E n g i n e e r i n g The Univers i ty of B r i t i s h Columbia 2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5 Date M a r c h 2 3 , 1 9 7 8 . ABSTRACT An e x p e r i m e n t a l method was d e v e l o p e d f o r c h a r a c t e r i z i n g t i m e -dependent s l u r r i e s o f e l o n g a t e d p a r t i c l e s i n 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 which has a wide gap between the i n n e r r o t a t i n g c y l i n d e r and the s t a t i o n a r y cup. A model s l u r r y system was s t u d i e d ; i t c o n s i s t e d of a d i s p e r s e d phase of r e g u l a r l y s i z e d n y l o n f i b e r s and a d i s p e r s i n g medium of an aqueous s o l u t i o n of p o l y e t h y l e n e g l y c o l w i t h d e x t r o s e and sodium c h l o r i d e , each of whose e f f e c t s on the t i m e -dependent n a t u r e o f the s l u r r y c o u l d be examined s e p a r a t e l y . The s l u r r y g e l l e d i n t h e v i s c o m e t e r on s t a n d i n g . The t i m e - v a r i a t i o n of shear s t r e s s e x e r t e d on the w a l l of the i n n e r c y l i n d e r r o t a t i n g a t a c o n s t a n t a n g u l a r v e l o c i t y was r e c o r d e d as a s t r e s s decay c u r v e . I t was found t h a t o n l y a p a r t of the gap between the c y l i n d e r s of t h e v i s c o m e t e r f l o w s under s h e a r , and t h a t the t h i c k n e s s of t h e f l o w i n g l a y e r i n c r e a s e s w i t h time and approaches an e q u i l i b r i u m v a l u e . The t i m e - v a r i a t i o n of the t h i c k n e s s of the f l o w i n g l a y e r R ( t ) as w e l l as the shear s t r e s s decay was measured f o r f i f t e e n s l u r r i e s . An e m p i r i c a l r e a c t i o n - r a t e t y p e model was c o n s t r u c t e d f o r t h e t i m e - v a r i a t i o n o f t h e t h i c k n e s s o f t h e f l o w i n g l a y e r R ( t ) . A second o r d e r - z e r o o r d e r r e v e r s i b l e r e a c t i o n model f i t t e d t h e e x p e r i m e n t a l d a t a w e l l . The t h r e e f i t t i n g f a c t o r s o f t h e model were th e e q u i l i b r i u m v a l u e o f the t h i c k n e s s o f t h e f l o w i n g l a y e r R x o o> ar>d two r a t e parameters k,/B and B/A. These f a c t o r s (R , K../B, and r x°° r B/A) were c o r r e l a t e d w i t h p a r t i c l e l e n g t h - t o - d i a m e t e r r a t i o , volume f r a c t i o n o f p a r t i c l e s i n t h e s l u r r y , and a n g u l a r v e l o c i t y o f t h e r o t a t i n g c y l i n d e r . i i i TABLE OF CONTENTS Chapter 1 INTRODUCTION 1 2 LITERATURE SURVEY 4 2-1. E x p e r i m e n t a l O b s e r v a t i o n s 4 2- 2. T h e o r e t i c a l I n t e r p r e t a t i o n s 11 3 THEORETICAL BACKGROUND 16 3- 1. Newton's P o s t u l a t e 16 3-2. Bingham P l a s t i c s 18 3-3. Power Law F l u i d s 18 3-4. G e n e r a l F l u i d 19 3-5. Simple Shear Flow 20 3-6. Co u e t t e Flow ( 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 ) 22 3- 7. C o m p l i c a t i n g E f f e c t s i n Suspension Rheology 28 3-7-1. Continuum 29 3-7-2. Steady 30 3-7-3. W a l l E f f e c t s 30 3-7-4. End E f f e c t s 32 3-7-5. Homogeneous 33 3-7-6. I n c o m p r e s s i b l e 33 3-7-7. I s o t h e r m a l 33 3-7-8. Laminar . . . . . 33 3-7-9. T a y l o r ' s I n s t a b i l i t y 34 4 DEFINITION OF THE PROBLEM AND METHOD OF ATTACK . . . 35 4- 1. Recommendations of Other Workers 35 4-2. D e f i n i t i o n o f the Problem 36 4-3. System Chosen 37 4- 4. Method o f A t t a c k 37 5 EXPERIMENTAL MATERIALS AND TECHNIQUES 39 5- 1. M a t e r i a l s 39 5-1-1. D i s p e r s e d Phase 39 5-1-2. D i s p e r s i n g Medium 43 5-1-3. Technique of D e n s i t y M a t c h i n g . . . 43 5-1-4. S l u r r y P r e p a r a t i o n 44 5-2. D e s c r i p t i o n o f Apparatus 50 5-3. Torque C a l i b r a t i o n 55 5-4. E x p e r i m e n t a l P r o c e d u r e 57 div 6 EXPERIMENTAL RESULTS AND DISCUSSION 60 6-1. R e p r o d u c i b i l i t y 60 6-2. E f f e c t of S l u r r y Age on Y i e l d S t r e s s . . . . 62 6-3. Decay Curve and T h i c k n e s s o f F l o w i n g L a y e r . 62 6-4. E f f e c t of NaCl C o n c e n t r a t i o n on Y i e l d S t r e s s and on E q u i l i b r i u m S t r e s s 65 6-5. E f f e c t o f D i s p e r s i n g Medium V i s c o s i t y on Y i e l d S t r e s s 68 6-6. E f f e c t s o f P a r t i c l e C o n c e n t r a t i o n and P a r t i c l e L/D R a t i o on Y i e l d S t r e s s . . . . 71 6-7. E f f e c t s o f P a r t i c l e C o n c e n t r a t i o n and P a r t i c l e L/D R a t i o on E q u i l i b r i u m S t r e s s . 71 6-8. E f f e c t of A n g u l a r V e l o c i t y of Inner C y l i n d e r on Y i e l d S t r e s s 76 6- 9. E f f e c t s of P a r t i c l e C o n c e n t r a t i o n , P a r t i c l e L/D R a t i o , and A n g u l a r V e l o c i t y on E q u i l i b r i u m T h i c k n e s s of F l o w i n g L a y e r . . 76 7 ANALYSES • . 80 7- 1. A Model P r e d i c t i n g T h i c k n e s s of F l o w i n g L a y e r 80 7-2. D e t e r m i n a t i o n of m and n i n t h e Model . . . . 86 7-3. Two F i t t i n g Parameters (k /B) and (B/A) of the Model 90 7-4. E f f e c t o f A n g u l a r V e l o c i t y on Rate o f I n c r e a s e of T h i c k n e s s of F l o w i n g L a y e r . . 92 7-5. E f f e c t s o f P a r t i c l e C o n c e n t r a t i o n and P a r t i c l e L/D R a t i o on Rate of I n c r e a s e of F l o w i n g L a y e r 95 7-6. R h e o l o g i c a l C o n s i d e r a t i o n o f t h e T h i c k n e s s of F l o w i n g L a y e r 102 7-7. P r a c t i c a l Use o f t h e Model 104 8 CONCLUSIONS 109 9 RECOMMENDATIONS I l l NOMENCLATURE 112 Roman L e t t e r s 112 Greek L e t t e r s 114 M a t h e m a t i c a l Symbols and N o t a t i o n 115 A b b r e v i a t i o n s and Other N o t a t i o n 116 BIBLIOGRAPHY 117 V APPENDICES 122 Appendix A. Flow Measurement w i t h D i f f e r e n t Radius R of Inner C y l i n d e r a t E q u i l i b r i u m 122 Appendix B. Supplement t o F i g . 6-6 i n S e c t i o n 6-5 . . . 126 Appendix C. Computer Programmes 131 C - l . Programme "BESTFIT" 131 C-2. Programme "SUPLPLT" 139 C-3. Programme "TAUDECAY" 148 C-4. Programme "THICKNESS" 154 Appendix D. E x p e r i m e n t a l Raw Data and C a l c u l a t e d R e s u l t s 160 D - l . Shear S t r e s s a t t = 0 and t = t . . 160 CO D-2. R x and S t r e s s Decay 186 Appendix E. S o l u t i o n s f o r D i f f e r e n t i a l E q u a t i o n (7-5) 202 Appendix F. V a r i o u s Orders o f t h e Model P r e d i c t i n g T h i c k n e s s o f F l o w i n g L a y e r and Sum of Squares 209 Appendix G. Decay Models 225 G - l . E v a l u a t i o n o f E x p r e s s i o n s P r e d i c t i n g Shear S t r e s s Decay 225 G-2. A M o d i f i c a t i o n t o R i t t e r and G o v i e r ' s Decay Model 230 Appendix H. V a r i o u s Orders of M o d i f i e d R i t t e r and G o v i e r ' s Decay Model and Sum of Squares . . 242 v i LIST OF TABLES No. 5-1. P h y s i c a l P r o p e r t i e s of N y l o n F i b e r 41 5-2. L e n g t h - t o - D i a m e t e r R a t i o of F i b e r s 41 5- 3. A n g u l a r V e l o c i t y of Inner C y l i n d e r 56 6- 1. V a r i a b l e s and Ranges I n v e s t i g a t e d 61 7- 1. V a r i o u s Orders (m, n) and R e s u l t o f L e a s t Squares F i t 89 A - l . E q u i l i b r i u m Flow Data w i t h D i f f e r e n t R 123 D - l . E x p e r i m e n t a l Data o f Shear S t r e s s e s a t Time = 0 and a t E q u i l i b r i u m 161 D-2. E x p e r i m e n t a l Data and R e s u l t s of C a l c u l a t i o n . . . . 187 F. V a r i o u s Orders (m, n) and R e s u l t s of L e a s t Squares F i t 210 G - l . Parameters of V a r i o u s Decay Models 227 G—2. V a r i o u s Orders of G o v i e r ' s Model and R e s u l t s of L e a s t Squares F i t 233 H. V a r i o u s Orders of G o v i e r Model and R e s u l t of L e a s t Squares F i t 243 v i i LIST OF FIGURES No. 3-1. F o r c e B a l a n c e between P a r a l l e l P l a t e s 17 3-2. D e f i n i n g Sketch f o r the Treatment of C o a x i a l C y l i n d e r Flow Data 23 5-1. Photograph o f 5 D i f f e r e n t l y S i z e d F i b e r s 42 5-2. D e n s i t y Match 45 5-3. E f f e c t of PEG C o n c e n t r a t i o n on Suspending Medium V i s c o s i t y 46 5-4. E f f e c t o f NaCl on Suspending Medium V i s c o s i t y . . . 47 5-5. E f f e c t of D e x t r o s e on Suspending Medium V i s c o s i t y . . . . - . 48 5-6. I n s t r u m e n t a t i o n (A) and M e a s u r i n g Head (B) . . . . 51 5-7. C o n f i g u r a t i o n of C y l i n d e r s 53 5-8. Grooves of t h e C y l i n d e r s 54 5- 9. C a l i b r a t i o n Curve of Torque v e r s u s V o l t a g e . . . . 58 6- 1. E f f e c t of Age of S l u r r y on Y i e l d S t r e s s 63 6-2. T y p i c a l Decay Curve 64 6-3. T h i c k n e s s o f F l o w i n g L a y e r 66 6-4. E f f e c t of NaCl C o n c e n t r a t i o n on Y i e l d S t r e s s . . . 67 6-5. E f f e c t o f NaCl C o n c e n t r a t i o n on E q u i l i b r i u m S t r e s s . 69 6-6. E f f e c t of D i s p e r s i n g Medium V i s c o s i t y on Y i e l d S t r e s s 70 6-7. E f f e c t s of C o n c e n t r a t i o n and L/D R a t i o o f P a r t i c l e on Y i e l d S t r e s s 72 6-8. E f f e c t s o f C o n c e n t r a t i o n and L/D R a t i o o f P a r t i c l e on Y i e l d S t r e s s 73 6-9. E f f e c t s of C o n c e n t r a t i o n and L/D R a t i o o f P a r t i c l e on E q u i l i b r i u m S t r e s s 74 6-10. E f f e c t s of C o n c e n t r a t i o n and L/D R a t i o o f P a r t i c l e on E q u i l i b r i u m S t r e s s 75 6-11. E f f e c t of A n g u l a r V e l o c i t y o f Inner C y l i n d e r on Y i e l d S t r e s s 77 6- 12. E f f e c t s o f P a r t i c l e C o n c e n t r a t i o n , P a r t i c l e L/D R a t i o , and A n g u l a r V e l o c i t y on E q u i l i b r i u m T h i c k n e s s o f F l o w i n g L a y e r 78 7- 1. Bundle S i z e s i n F l o w i n g S o l 84 7-2. E f f e c t of Parameter ( k f / B ) on the Model 91 7-3. E f f e c t of Parameter (B/A) on t h e Model 93 7-4. E f f e c t o f A n g u l a r V e l o c i t y on Rate o f I n c r e a s e of T h i c k n e s s of F l o w i n g Layer •. . . 94 7-5. E f f e c t of A n g u l a r V e l o c i t y on the Parameter ( k f / B ) 96 v i i i 7-6. E f f e c t of A n g u l a r V e l o c i t y Q on the Parameter (B/A) 97 7-7. E f f e c t of P a r t i c l e C o n c e n t r a t i o n on Rate of I n c r e a s e of T h i c k n e s s o f F l o w i n g L a y e r . . . . 98 7-8. E f f e c t of P a r t i c l e L/D R a t i o on Rate of I n c r e a s e o f T h i c k n e s s o f F l o w i n g L a y e r 99 7-9. E f f e c t s of P a r t i c l e C o n c e n t r a t i o n and P a r t i c l e L/D R a t i o on t h e Parameter ( k f / B ) 100 7-10. E f f e c t s o f P a r t i c l e C o n c e n t r a t i o n and P a r t i c l e L/D R a t i o on t h e Parameter (B/A) 101 7-11. Dependence of Apparent Shear Rate on Time . . . . 103 7-12. Apparent V i s c o s i t y o f Time-Dependent A r t i f i c i a l S l u r r i e s 105 7-13. Flow Curve of Time-Dependent A r t i f i c i a l S l u r r i e s 106 A - l . E q u i l i b r i u m Flow Measurement w i t h D i f f e r e n t R a d i u s o f Inner C y l i n d e r 124 A-2. E q u i l i b r i u m Flow Curve 125 B - l . E f f e c t of D i s p e r s i n g Medium V i s c o s i t y on Y i e l d S t r e s s 127 B-2. E f f e c t o f D i s p e r s i n g Medium V i s c o s i t y on Y i e l d S t r e s s 128 B-3. E f f e c t of D i s p e r s i n g Medium V i s c o s i t y on Y i e l d S t r e s s 129 B-4. E f f e c t o f D i s p e r s i n g Medium V i s c o s i t y on Y i e l d S t r e s s 130 G - l . Comparison between S t r e s s Decay E x p r e s s i o n s and E x p e r i m e n t a l Decay Data 228 G-2. E f f e c t o f Parameter (k^/k^) on the Decay Model 235 G-3. E f f e c t of Parameter (ki+/Po) on t h e Decay Model 236 G-4. E f f e c t of A n g u l a r V e l o c i t y on the Parameter (k f/ki+) 238 G-5. E f f e c t o f A n g u l a r V e l o c i t y on t h e Parameter ( k V P o ) 239 G-6. E f f e c t s of P a r t i c l e C o n c e n t r a t i o n and P a r t i c l e L/D R a t i o on t h e Parameter (k^/ki|) 240 G-7. E f f e c t s o f P a r t i c l e C o n c e n t r a t i o n and P a r t i c l e L/D R a t i o on t h e Parameter (k^/Po) 241 ACKNOWLEDGEMENTS The author i s p r o f o u n d l y ' i n d e b t e d t o t h e r e s e a r c h s u p e r v i s o r , P r o f e s s o r K. L. P i n d e r , f o r h i s i n v a l u a b l e g uidance, f o r h i s enthusiasm and encouragement, and e s p e c i a l l y f o r h i s i n f i n i t e p a t i e n c e throughout the course! o f t h i s s t u d y . S p e c i a l g r a t i t u d e and a p p r e c i a t i o n a r e owed P r o f e s s o r s N. E p s t e i n , R. M. R. B r a n i o n , A. P. Watkinson, and E. L. Watson, f o r t h e i r many c o n s t r u c t i v e comments and h e l p f u l d i s c u s s i o n s as committee members. Many thanks a r e due to Mr. J . G. Baranowski and h i s machine-shop s t a f f f o r t h e i r a s s i s t a n c e i n t r o u b l e - s h o o t i n g t h e equipment, to my f e l l o w g r aduate s t u d e n t s f o r t h e i r u s e f u l s u g g e s t i o n s and w i l l i n g n e s s t o d i s c u s s a l l a s p e c t s o f t h e study , and to Ms. A. Nokony f o r p r o o f - r e a d i n g t h e m a n u s c r i p t . The a u t h o r i s g r a t e f u l t o t h e N a t i o n a l R e search C o u n c i l o f Canada, t h e Standard O i l Company of B r i t i s h Columbia L i m i t e d , and the Department o f Chemical E n g i n e e r i n g , t h e U n i v e r s i t y o f B r i t i s h Columbia, f o r f i n a n c i a l s u p p o r t . CHAPTER 1 INTRODUCTION S c i e n t i f i c i n t e r e s t i n the f l o w p r o p e r t i e s o f s u s p e n s i o n s of s o l i d p a r t i c l e s i n f l u i d s had i t s s t a r t around t h e t u r n of the c e n t u r y , when stud y o f the v i s c o s i t y of c o l l o i d a l d i s p e r s i o n s began. The r h e o l o g i c a l b e h a v i o r o f s u s p e n s i o n s i s of e n g i n e e r i n g i n t e r e s t , s i n c e i n d u s t r i a l p r o c e s s i n g f r e q u e n t l y i n v o l v e s t h e d e f o r m a t i o n and f l o w o f s u s p e n s i o n s . S t i r r i n g , t r a n s p o r t through p i p e s , e x t r u s i o n , and s p i n n i n g a r e j u s t a few of t h e many common p r o c e s s s t e p s which depend on the f l o w p r o p e r t i e s of such m a t e r i a l s . Paper p r o d u c t i o n i s a well-known example of an i n d u s -t r i a l p r o c e s s i n v o l v i n g t h e f l o w of s u s p e n s i o n s of e l o n g a t e d p a r t i c l e s . Knowledge o f t h e f l o w c h a r a c t e r i s t i c s of s u s p e n s i o n s i s important f o r u n d e r s t a n d i n g the b e h a v i o r of b i o l o g i c a l systems, f o r example, b l o o d , (13) which c o n t a i n s e l o n g a t e d p a r t i c l e s known as r o u l e a x . Fur t h e r m o r e , (22) (53) the a d d i t i o n o f f i b e r s t o f l u i d s f o r t u r b u l e n t drag r e d u c t i o n ' and polymer s t r e n g t h e n i n g ^ ' ' p r o v i d e s anot he r r e a s o n f o r the stud y of r h e o l o g y of f i b e r s u s p e n s i o n s . Time-dependent f l o w , one o f t h e r h e o l o g i c a l b e h a v i o r s of suspen-s i o n s known as t h i x o t r o p y , r e f e r s t o the v a r i a t i o n o f v i s c o s i t y w i t h d u r a t i o n of shear as a r e s u l t o f changes i n the i n t e r n a l s t r u c t u r e o f t h e l i q u i d - l i k e m a t e r i a l . A l t h o u g h t h i s q u a l i t a t i v e e x p l a n a t i o n f o r t h i x o t r o p y i s g e n e r a l l y a c c e p t e d , t h e r e i s s t i l l a g e n e r a l u n c e r t a i n t y as t o how the r h e o l o g i c a l b e h a v i o r o f time-dependent m a t e r i a l s can b e s t 1 2 be c h a r a c t e r i z e d w i t h r e l a t i o n t o the shape and s i z e o f suspended p a r t i c l e s . I t s h o u l d be noted t h a t t h e term " s t r u c t u r e " i s not p r e c i s e l y d e f i n e d h e r e . Such a term u s u a l l y r e f e r s t o p a r t i c l e o r i e n t a t i o n , f l o c c u l a t i o n , o r m o l e c u l a r entanglement i n the l i q u i d - l i k e m a t e r i a l . I t i s r e c o g n i z e d t h a t l i q u i d s t r u c t u r e i s p r o b a b l y q u i t e d i f f e r e n t from s o l i d s t r u c t u r e and may i n f a c t not be r e c o g n i z e d as a s t r u c t u r e a t a l l . T h i x o t r o p i c systems a r e n o r m a l l y encountered i n some of t h e n a t u r a l s l u r r i e s s u ch as crud e o i l , b e n t o n i t e c l a y i n water, e t c . , where t h e system c o n s i s t s o f two phases, i . e . , d i s p e r s e d p a r t i c l e s and d i s p e r s i n g medium. The shape o f t h e d i s p e r s e d p a r t i c l e s i n those n a t u r a l s l u r r i e s i s i r r e g u l a r and the s i z e s o f t h e p a r t i c l e s f r e q u e n t l y range a c r o s s m i c r o n t o m i l l i m e t e r s c a l e . The d i s p e r s i n g medium o f t e n c o n t a i n s v a r i o u s components o f i o n i c s u b s t a n c e s or p o l y e l e c t r o l y t e s . The r e s u l t -i n g r h e o l o g y o f a t h i x o t r o p i c system i s c o r r e s p o n d i n g l y complex. The i n t e r f a c e between the d i s p e r s e d p a r t i c l e s and the d i s p e r s i n g phase i s v e r y s e n s i t i v e t o t r a c e amounts o f a t h i r d m a t e r i a l , such as l u b r i c a n t s o r s u r f a c t a n t s , which a r e u s u a l l y p r e s e n t i n n a t u r a l s l u r r i e s . T h i s e f f e c t f u r t h e r c o m p l i c a t e s t h e problem. Much u s e f u l i n f o r m a t i o n c o u l d be g a i n e d , however, from a study o f the time-dependent f l o w o f s l u r r i e s . To o b t a i n such i n f o r m a t i o n , one must r e s t r i c t one's a t t e n t i o n t o a model system such as n y l o n f i b e r s i n an aqueous s o l u t i o n o f sodium c h l o r i d e , d e x t r o s e , and p o l y e t h y l e n e g l y c o l . I n such a system i t i s p o s s i b l e t o c o n t r o l t h e shape and s i z e of d i s p e r s e d p a r t i c l e s and t h e p h y s i c a l n a t u r e o f the d i s p e r s i n g medium, e.g., v i s c o s i t y , d e n s i t y , and i o n i c f o r c e . T h i s i s the r e a s o n why t h e systems a r e c a l l e d a r t i f i c i a l s l u r r i e s i n t h i s t h e s i s . The o b j e c t i v e o f t h i s t h e s i s i s : t o examine t h e e f f e c t s of s u s p e n s i o n v a r i a b l e s ( p a r t i c l e s i z e , p a r t i c l e c o n c e n t r a t i o n , e l e c t r o l y t e c o n c e n t r a t i o n , s u s p e n d i n g medium v i s c o s i t y ) on time-dependent s h e a r f l o w , t o c o n s t r u c t a model e q u a t i o n w h i c h b e s t d e f i n e s t h e t i m e -dependent c h a r a c t e r i s t i c s of f l o w of t h e s l u r r i e s , and t o c o r r e l a t e t h e p a r a m e t e r s o f t h e model e q u a t i o n w i t h t h e s u s p e n s i o n v a r i a b l e s . CHAPTER 2 LITERATURE SURVEY There is much confusion in the literature because of disagreements in the terminology used to describe time-dependent rheology. In this thesis the following definitions are used. "Thixotropy" is an iso-thermal and comparatively slow recovery, on standing of a material, of a consistency that was lost through shearing. "Anti-thixotropy" is an isothermal and comparatively slow f a l l , on standing of the sample, of a consistency that was gained as a result of shearing. "Rheopexy" is the soli d i f i c a t i o n of a thixotropic system by gentle and regular movement. "Dilatancy" is an increase in volume caused by shear. "Shear thinning" is a univalued reduction of the consistency with increasing rate of shear. "Shear thickening" is a univalued increase of the consistency with increasing rate of shear. "Shear softening" is a decrease of consistency, the result of shearing deformation. "Shear hardening" is an increase of consistency, the result of shearing deformation. The literature review is limited to a survey of pertinent literature concerning thixotropic behavior and a presentation of the rheological theories based mainly on chemical kinetics. 2-1. Experimental Observations Green^"^ was one of the earliest researchers to study thixotropic behavior in a systematic way. His data were obtained by shearing the flu i d for a set time at a given shear rate, observing the shear stress, 4 5 and t h e n changing the shear r a t e immediately t o the next h i g h e r l e v e l of shear r a t e , and t h e n r e - m e a s u r i n g t h e shear s t r e s s . T h i s p r o c e d u r e was r e p e a t e d u n t i l t h e d e s i r e d top shear r a t e was r e a c h e d . The p l o t o f t h e d a t a as shear s t r e s s v e r s u s shear r a t e was c a l l e d "up-curve". The p r o c e d u r e was c o n t i n u e d by p r o g r e s s i v e l y l o w e r i n g t h e shear r a t e i n a s t e p w i s e manner and measuring t h e shear s t r e s s a t each s t e p u n t i l t h e bottom p o i n t was o b t a i n e d . The p l o t o f t h i s p a r t of the d a t a was c a l l e d "down-curve". The t h i c k n e s s of the r e s u l t i n g h y s t e r e s i s l o o p , which c o n s i s t e d of the up-curve and down-curve, was c o n s i d e r e d a measure of the degree of t h i x o t r o p y . Green's a t t e n t i o n t o t h e a r e a of h y s t e r e s i s l o o p s was v e r y o r i g i n a l ; however, i t i s d i f f i c u l t t o t r a n s l a t e t h e a r e a i n t o p h y s i c a l meanings. (44) F o l l o w i n g Green's method, Simon e t al. measured a t h i x o t r o p i c i n d e x f o r b e n t o n i t e s u s p e n s i o n s , d e f i n e d by t h e a r e a c i r c u m s c r i b e d by t h e h y s t e r e s i s l o o p of t h e f l o w c u r v e s , u s i n g a c o n e - a n d - p l a t e v i s c o -meter. The t y p e of p r e p a r a t i o n g i v e n t h e s u s p e n s i o n s a f f e c t s t h e t h i x o t r o p i c i n d e x , but t h e r e a s o n f o r t h i s was not c l e a r . I n t h e i r c o n e - a n d - p l a t e v i s c o m e t e r measurements w i t h p o l y v i n y l (23) c h l o r i d e (PVC) m e l t s , Khanna e t al. observed the r e v e r s i b l e r e d u c t i o n i n s t r e s s and r e c o v e r a b l e s t r a i n by s h e a r i n g and termed i t " r h e o l o g i c a l breakdown". They suggested t h a t m o l e c u l a r entanglements and d i s e n -tanglements of polymer c h a i n s a r e r e s p o n s i b l e f o r t h e s e phenomena. They c o u l d n o t , however, d i s t i n g u i s h t h i x o t r o p y c l e a r l y from the e l a s t i c e f f e c t o f the m e l t s . (2) (3) B i l l i n g t o n et al. ' measured i n a C o u e t t e - t y p e v i s c o m e t e r the shear s t r e s s of an aluminum l a u r a t e - N e w t o n i a n o i l system and a c o l l o i d a l s u s p e n s i o n of g r a p h i t e p a r t i c l e s d i s p e r s e d i n a Newtonian o i l . I t was 6 found t h a t , i n a d d i t i o n to the t h i x o t r o p i c b e h a v i o r , the former e x h i b i t s a Weisenberg e f f e c t , w h i l e the l a t t e r does n o t . To t e s t the f e a s i b i l i t y of t h e T r a n s - A l a s k a p i p e l i n e , P e r k i n s et al. examined e x p e r i m e n t a l l y the y i e l d s t r e n g t h of Prudoe Bay crude o i l . The c r u d e o i l e x h i b i t e d not o n l y t h i x o t r o p y but a l s o c o m p r e s s i b i l i t y . They found t h a t the y i e l d s t r e n g t h of the o i l i s i n f l u e n c e d by i t s temperature h i s t o r y , shear h i s t o r y , and c o m p o s i t i o n . ( 5 4 ) V o c a d l o et al. reviewed the methods f o r measuring t h e y i e l d s t r e s s of f l u i d - l i k e s u b s t a n c e s . They emphasized t h a t , i n o r d e r to e l i m i n a t e the e f f e c t of anomalous s l i p , c y l i n d r i c a l s u r f a c e s w i t h deep grooves be used i n 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 . (13) Mason et al. of the P u l p and Paper Research I n s t i t u t e o f Canada a r e most concerned w i t h m i c r o r h e o l o g y of d i s p e r s i o n s . They have w r i t t e n a l a r g e number of p u b l i c a t i o n s on the b a s i c b e h a v i o r of a s i n g l e p a r t i c l e i n shear f l o w , i n a d d i t i o n to t h a t of d i s p e r s i o n s . They d i d n o t , however, r e p o r t any i n v e s t i g a t i o n s on t h i x o t r o p y . I t seems t h a t t h e i r d i s p e r s i o n s were too d i l u t e t o cause t h i x o t r o p y . But they o b s e r v e d the Weisenberg e f f e c t f o r d i s p e r s i o n s o f r e l a t i v e l y h i g h c o n c e n t r a t i o n i n s t e a d . P e t r i e ^ ^ f o c u s s e d on the e f f e c t of s u r f a c t a n t s on t h e v i s c o s i t y of P o r t l a n d cement. He found t h a t a n i o n i c s u r f a c t a n t s based on c e r t a i n commercial n a p h t h a l e n e s u l f o n i c a c i d condensates appear to n e u t r a l i z e the a t t r a c t i n g charge on P o r t l a n d cement p a r t i c l e s by a d s o r p t i o n from s o l u t i o n onto a c t i v e p a r t i c l e s i t e s . H i s c o n c l u s i o n i s t h a t normal cement s u s p e n s i o n s behave as non-Newtonian t h i x o t r o p i c f l u i d s ; suspen-s i o n s w i t h s u r f a c t a n t s o l u t i o n of s u f f i c i e n t s t r e n g t h behave as Newtonian f l u i d s . E n o k s s o n ^ ^ examined t h e e f f e c t o f temperature on the t h i x o t r o p y of s o l u t i o n of c e l l u l o s e n i t r a t e (Mn = 250,000) i n n i t r o b e n z e n e and d i s c o v e r e d t h a t t h e t h i x o t r o p i c p r o p e r t i e s o f t h e s o l u t i o n d i m i n i s h w i t h i n c r e a s i n g temperature. (29) (30) Mercer and Mercer e t al. i n v e s t i g a t e d the time-dependent v i s c o s i t y o f b e n t o n i t e - w a t e r s u s p e n s i o n s u s i n g a Haake R o t o v i s c o C o u e t t e v i s c o m e t e r . They found t h a t , a f t e r a sudden change i n shear r a t e , t h e time-dependency o f t h e i r apparent v i s c o s i t y can be d e s c r i b e d by n ( t ) = A exp [ - ( t / i ! ) 0 ' ^ ] + B exp [ - t / r 2 ] + C where n ( t ) i s apparent v i s c o s i t y , t i s time, T\ and x 2 a r e c h a r a c t e r -i s t i c t i m e s , and A, B, and C a r e c o n s t a n t s . The c h a r a c t e r i s t i c times became l o n g e r as t h e d i f f e r e n c e i n shear r a t e s became s m a l l e r , d e c r e a s e d w i t h i n c r e a s i n g c o n c e n t r a t i o n , and showed l i t t l e t emperature e f f e c t f o r a 20°C temperature change. (36) (37) Park and Ree , and Park, Ree, and E y r i n g o b served t h e t h i x o t r o p y o f an aqueous b e n t o n i t e s u s p e n s i o n u s i n g a C o u e t t e - t y p e v i s c o m e t e r . They p o s t u l a t e d t h r e e k i n d s o f f l o w b e h a v i o r s i n the s u s p e n s i o n , i . e . , Newtonian, non-Newtonian n o n - t h i x o t r o p i c , and non-Newtonian t h i x o t r o p i c . They a l s o examined the e f f e c t o f e l e c t r o l y t e s on t h e f l o w p r o p e r t i e s of the s u s p e n s i o n s . On a d d i t i o n o f monovalent c a t i o n s , the apparent v i s c o s i t y d e c r e a s e d , f o l l o w e d by a r i s e as the i o n i c c o n c e n t r a t i o n was f u r t h e r i n c r e a s e d . A d d i t i o n o f m u l t i v a l e n t c a t i o n s ( d i - and t r i - ) caused t h e v i s c o s i t y t o i n c r e a s e t o a maximum th e n d e c r e a s e t o a c o n s t a n t v a l u e . (49) Umeya et al. observed the f l o w b e h a v i o r o f c l a y - w a t e r s u s -p e n s i o n s and found t h a t the shear s t r e s s a t ste a d y s t a t e i s a f u n c t i o n 8 of pH. The shear s t r e s s was minimum at a pH =7.5, where one would expect the i n t e r n a l s t r u c t u r e of the s u s p e n s i o n to be a c h a i n s t r u c t u r e (which i s a l o o s e s t r u c t u r e ) . On the o t h e r hand, i t seemed t h a t th e s u s p e n s i o n became a c a r d house s t r u c t u r e i n t h e a c i d i c r e g i o n and a c a r d pack s t r u c t u r e i n t h e b a s i c r e g i o n . Weyman^"^ gave a good r e v i e w of p r e v i o u s i n v e s t i g a t i o n s on t h e v i s c o s i t y o f t h i x o t r o p i c s u s p e n s i o n s . I t i s summarized as f o l l o w s : (1) G e l p a r t i c l e s a r e h i g h l y u n - i s o d i m e n s i o n a l , i . e . , they have the form of l o n g r o d s or t h i n d i s c s . (An e x c e p t i o n i s t h e c a r b o n b l a c k p a r t i c l e s , which a r e s p h e r i c a l . ) (2) In a s u s p e n s i o n a t r e s t , t h e s e p a r t i c l e s form open, s c a f f o l d i n g -l i k e o r s p o n g e - l i k e s t r u c t u r e s i n which t h e l i q u i d phase i s t r a p p e d . (3) T h i x o t r o p i c s u s p e n s i o n s can form g e l s over a wide range of con-c e n t r a t i o n s . T h i s i n d i c a t e s t h a t t h e r e i s no p r e f e r r e d average p a r t i c l e d e n s i t y . The y i e l d v a l u e i n c r e a s e s s t r o n g l y w i t h i n c r e a s i n g c o n c e n t r a t i o n . (41) P i n d e r i n v e s t i g a t e d the time-dependent b e h a v i o r of t e t r a h y d r o f u r a n - h y d r o g e n s u l p h i d e gas h y d r a t e s l u r r i e s , employing a new t e c h n i q u e t o g i v e a y i e l d v a l u e . Due t o t h e s m a l l a n g l e o f i n t e r c e p t i o n of the v i s c o s i t y - t i m e c u r v e on the v i s c o s i t y a x i s a t z e r o time, i t was almost i m p o s s i b l e t o d e t e r m i n e the a c c u r a t e y i e l d v a l u e by e x t r a p o l a t i o n . P i n d e r found t h a t a p l o t of t h e i n v e r s e of t h e v i s c o s i t y v e r s u s time y i e l d e d a r e a s o n a b l y s t r a i g h t l i n e o v e r the f i r s t t e n t o f i f t e e n r e a d i n g s . T h i s t e c h n i q u e a l l o w e d f o r the c a l -c u l a t i o n o f y i e l d v a l u e and of r a t e of change of f l u i d i t y a f t e r a s h o r t p e r i o d of time by the method of l e a s t s q u a r e s . He proposed a 9 new model of t h e second o r d e r - z e r o o r d e r r e v e r s i b l e r e a c t i o n mechanism f o r t h e time-dependent b e h a v i o r o f t h e s l u r r y , w h i c h b e s t f i t h i s v i s c o s i t y d a t a . H i s e x p e r i m e n t a l r e s u l t s a r e as f o l l o w s : (1) The y i e l d v a l u e d e c r e a s e s w i t h r o t a t i o n a l speed f o r any g i v e n s p i n d l e . ( 2 ) The s m a l l e r t h e v i s c o m e t e r s p i n d l e , t h e h i g h e r i s t h e y i e l d v a l u e f o r t h e same d i a m e t e r c o n t a i n e r and t h e same r o t a t i o n a l speed. (3) The h i g h e r t h e c o n s i s t e n c y , t h e h i g h e r i s t h e y i e l d v a l u e . (4) The l a r g e r t h e mean d i a m e t e r o f t h e p a r t i c l e s , t h e l o w e r i s t h e y i e l d v a l u e . ( 5 ) The l a r g e r t h e l e n g t h - t o - w i d t h r a t i o o f t h e p a r t i c l e s , t h e h i g h e r t h e y i e l d v a l u e . F o l l o w i n g P i n d e r ' s work, B r o w n ^ ^ ' ^ ^ used g e l s formed by suspend-i n g c e l l u l o s e a c e t a t e p a r t i c l e s o f r e g u l a r shape and s i z e i n a New t o n i a n medium. The time-dependency o f h i s d i s p e r s i o n s was found t o i n c r e a s e r a p i d l y w i t h i n c r e a s i n g p a r t i c l e s i z e , l e n g t h - t o - w i d t h r a t i o , and c o n c e n t r a t i o n . H i s decay c u r v e s f i t t e d P i n d e r ' s second o r d e r - z e r o o r d e r r e a c t i o n model. He a l s o o b s e r v e d a ragged decay c u r v e and s u g g e s t e d t h a t a breakdown o f an e l e c t r o s t a t i c s t r u c t u r e o r p a r t i c l e i n t e r f e r e n c e might be r e s p o n s i b l e f o r time-dependent b e h a v i o r . ( 3 5 ) I n o r d e r t o e l u c i d a t e Brown's s u g g e s t i o n , N i c h o l s o n c a r r i e d o u t e x p e r i m e n t s u s i n g s a l t s o f d i f f e r e n t c a t i o n v a l e n c e f r o m Brown's w i t h t h e same p a r t i c l e s . He found t h a t t h e degr e e o f raggedness o f t h e decay c u r v e i n c r e a s e d w i t h i n c r e a s e d c a t i o n s t r e n g t h and c o n c l u d e d t h a t t h e breakdown o f t h e e l e c t r o s t a t i c s t r u c t u r e must be one of t h e f o r c e s r e s p o n s i b l e f o r raggedness o f t h e decay c u r v e . These two i n v e s t i g a t o r s used a l a r g e c o n c e n t r a t i o n o f s a l t t o 1 0 match the d e n s i t y of the d i s p e r s i n g medium w i t h d i s p e r s e d phase. T h e r e f o r e i t was l e f t u n c l e a r as to how the time-dependent b e h a v i o r i s a f f e c t e d by t h e s a l t c o n c e n t r a t i o n of t h e d i s p e r s i n g medium. To e l i m -i n a t e t h i s problem i t would be n e c e s s a r y t o examine t h e time-dependent b e h a v i o r of d i s p e r s i o n s w i t h d i f f e r e n t s a l t c o n c e n t r a t i o n s . H u ^ ^ i n v e s t i g a t e d , w i t h b e n t o n i t e c l a y s u s p e n s i o n s , the e f f e c t s of c a t i o n v a l e n c e and s a l t c o n c e n t r a t i o n s on the y i e l d s t r e s s a t z e r o shear r a t e . He found t h a t the maximum y i e l d s t r e s s i s p r o p o r t i o n a l t o t h e m o l a l i t y o f t h e s a l t . The a b i l i t y of t h e s a l t t o b u i l d up g e l s t r e n g t h i s i n v e r s e l y p r o p o r t i o n a l t o the v a l e n c e of t h e c a t i o n i n t h e s a l t , i . e . , N a C l , MnCl2> CeCl3 i n t h i s o r d e r . He a l s o c o l l e c t e d d a t a on the d e n s i t y and v i s c o s i t y of a wide range o f m i x t u r e s , which i s p a r t i c u l a r l y i n t e r e s t i n g i n terms o f th e p r e s e n t s t u d y . He s u g g e s t e d t h a t a CMC-Corn Syrup system o r P E G - D e x t r o s e / F r u c t o s e system would be a good medium f o r s u s p e n d i n g a r t i f i c i a l p a r t i c l e s t o g i v e a wide range of s l u r r y c h a r a c t e r i s t i c s . From t h i s r e v i e w , i t i s apparent t h a t t h i x o t r o p y i s not w e l l -u n d e r s t o o d . The e a r l y e x p e r i m e n t a l t e c h n i q u e s t h a t were d e v e l o p e d f o r t h e purpose of q u a n t i t a t i v e l y measuring t h i x o t r o p y d i d not p r o v i d e i n f o r m a t i o n t h a t i s u s e f u l i n d e v e l o p i n g t h e o r e t i c a l c o n c e p t s . Furthermore, t h e g r e a t need f o r u n d e r s t a n d i n g and c h a r a c t e r i z i n g t h e f l o w p r o p e r t i e s o f polymers has d e t r a c t e d from r h e o l o g i c a l r e s e a r c h on o t h e r types of l i q u i d s . S i n c e most polymer m e l t s and polymer s o l u t i o n s do not e x h i b i t a p p r e c i a b l e t h i x o t r o p i c b e h a v i o r , the s u b j e c t has not r e c e i v e d much r e c e n t a t t e n t i o n . C o n s e q u e n t l y , o n l y a few s t u d i e s on a few l i q u i d s a r e o f any r e a l v a l u e f o r r e s e a r c h purposes. 2-2. T h e o r e t i c a l I n t e r p r e t a t i o n s The r e v e r s i b l e n a t u r e o f time-dependency has been q u a l i t a t i v e l y e x p l a i n e d by a number of a u t h o r s i n terms o f a b r e a k i n g down and subsequent r e b u i l d i n g of some form o f s t r u c t u r e . On t h e o t h e r hand s e v e r a l q u a n t i t a t i v e models have been p u b l i s h e d . In t h i s s e c t i o n some of t h e c h e m i c a l k i n e t i c models w i l l be d i s c u s s e d i n d e t a i l , and o t h e r models w i l l be mentioned b r i e f l y . The time v a r i a t i o n of shear s t r e s s f o r system e x h i b i t i n g t h i x o -t r o p i c b e h a v i o r and shear s o f t e n i n g non-Newtonian b e h a v i o r i s s i m i l a r to t h e time v a r i a t i o n o f c o n c e n t r a t i o n i n r e v e r s i b l e c h e m i c a l r e a c t i o n s . Hence, s e v e r a l r e s e a r c h e r s have made use o f t h e same ty p e o f e q u a t i o n s to d e s c r i b e t h e t h i x o t r o p i c systems. (1 fi^ The model a t t r i b u t e d t o Hahn, Ree, and E y r i n g i s based on the a p p l i c a t i o n o f t h e t h e o r y o f r a t e p r o c e s s e s t o the r e l a x a t i o n p r o c e s s e s t h a t a r e b e l i e v e d t o p l a y an im p o r t a n t p a r t i n d e t e r m i n i n g t h e n a t u r e of t h e f l o w of m a t e r i a l s . They assumed t h a t a g e n e r a l f l u i d c o n s i s t s o f two k i n d s o f f l o w u n i t s , i . e . , Newtonian and non-Newtonian s p e c i e s . T h i x o t r o p i c b e h a v i o r i s supposed t o be due to the i n t e r n a l change o f f l o w u n i t s from non-Newtonian t o Newtonian, o r from Newtonian t o non-Newtonian. They a r r i v e d a t t h e f o l l o w i n g e x p r e s s i o n f o r a c o n s t a n t r a t e of s h e a r , * - (TO " Oe" a t + ( 2 " 1 ) where T 0 i s i n i t i a l s t r e s s o f a g e l l e d s t r u c t u r e and x^ s t r e s s o f t h e e q u i l i b r i u m c o n d i t i o n . , For growth from a c o m p l e t e l y d e t e r i o r a t e d s t a t e , x = (x, - x ) e " a t + x (2-2) \y co co where r e p r e s e n t s s t r e s s from a c o m p l e t e l y d e t e r i o r a t e d s t r u c t u r e . 12 I t Is p o s s i b l e t o employ the model as g i v e n i n e q u a t i o n s (2-1) and (2-2) by u s i n g t h r e e measurable s t r e s s e s T N> T , and T, and to o b t a i n the parameter " a " e x p e r i m e n t a l l y . Denny and Brodkey have chosen to base t h e i r model on a p o i n t v i s c o s i t y ( s l o p e of t h e shear s t r e s s - s h e a r r a t e c u r v e ) . They assumed t h a t the p o i n t v i s c o s i t y decayed from a g e l l e d v a l u e ng and t h a t the decay r a t e was p r o p o r t i o n a l to the p r o d u c t of (n - n^)™ a n a some power of shear r a t e y . They a l s o assumed t h a t the growth r a t e was p r o p o r -t i o n a l t o the amount of s t r u c t u r a l d e t e r i o r a t i o n (tin - n t ) n , where n t i s t h e p o i n t v i s c o s i t y a t any time t , a f t e r s t a r t of s h e a r i n g . T h e i r e q u a t i o n r e p r e s e n t i n g t h i x o t r o p i c decay i s g i v e n f o r m = 1 and n = 2 as 2CTIQ + b ( b 2 - 4ac) where tank -1 ( b 2 - 4ac) - tanh. -1 2 c n t + b ( b 2 - 4ac) V (2-3) a = Ky n + ^ 0 " \ b =-(K Y + ) c = nn - n u oo k^ : growth r e a c t i o n r a t e c o n s t a n t k^ : decay r e a c t i o n r a t e c o n s t a n t and K : e q u i l i b r i u m c o n s t a n t ( k ^ / k ^ ) . Without c o u n t i n g m and ni, which a r e t aken t o be 1 and 2 r e s p e c t i v e l y by t h e s e i n v e s t i g a t o r s , t h e r e a r e f i v e c o n s t a n t s to be e v a l u a t e d . no and n a r e measured at c o m p l e t e l y g e l l e d and c o m p l e t e l y d e t e r i o r a t e d c o n d i -00 t i o n s , r e s p e c t i v e l y . Use of an e q u i l i b r i u m l i n e a l l o w s the e s t i m a t i o n 13 of K and P. T h i s l e a v e s t o be determined from the decay d a t a . (41) P i n d e r assumed t h a t , i n i t i a l l y , the s l u r r y p a r t i c l e s a r e i n a s t a t e of randomness. S h e a r i n g t h e s l u r r y causes the p a r t i c l e s t o take up a new p o s i t i o n , which i s i n a l e s s e r s t a t e o f randomness. D e s i g n a t -i n g t h e two extremes of o r i e n t a t i o n , i . e . , complete randomness and e q u i l i b r i u m o r d e r , as n^ and r i 2 i n terms of apparent v i s c o s i t y , t h e change can be compared to a c h e m i c a l r e a c t i o n k f \ w i t h f o r w a r d r a t e c o n s t a n t k^ and r e v e r s e r a t e c o n s t a n t k^. By assuming v a r i o u s r e a c t i o n o r d e r s f o r f o r w a r d and r e v e r s e r e a c t i o n s , t h e v a r i o u s e q u a t i o n s can be o b t a i n e d . F o r example, the second o r d e r - z e r o o r d e r r e a c t i o n .dm dt = V? - \ (2"4) which, upon i n t e g r a t i o n , w i l l be (n + n) (n - n ) 2 l o g i o T s / y , r~ = o c i m ^ ( 2 - 5 ) (n - n ) (n + n ) 2 . 3 0 3 e f e e y For the o t h e r o r d e r of r e a c t i o n s , s i m i l a r e x p r e s s i o n s can be o b t a i n e d which have t h e f o l l o w i n g form l o g i 0 [ f ( T l ) ] = c t (2-6) where c i s a c o n s t a n t . P l o t t i n g f ( n ) v e r s u s t on a s e m i - l o g paper w i l l y i e l d v a r i o u s c u r v e s . The b e s t s t r a i g h t l i n e r e p r e s e n t s t h e b e s t r e a c t i o n mechanism. F o r P i n d e r ' s s l u r r y i t was found t h a t the second o r d e r - z e r o o r d e r i s t h e b e s t . Brown's a r t i f i c i a l s l u r r y ^ ' ^ has a l s o been found to f o l l o w t h e second o r d e r - z e r o o r d e r mechanism. (43) A l o n g s i m i l a r l i n e s t o P i n d e r ' s work, R a t t e r and G o v i e r assumed 14 t h a t the measured s t r e s s i s the sum of s t r u c t u r a l s t r e s s and Newtonian s t r e s s components. They a l s o assumed a f i r s t o r d e r - s e c o n d o r d e r r e v e r s i b l e r e a c t i o n f o r the r a t e e q u a t i o n . They a r r i v e d at the e q u a t i o n M I n . T . — . T S S°° - T SO S T ( 2 - 7 ) sO s°° sO where k, i s a reduced r a t e c o n s t a n t A T t h e f i n a l s t e a d y s t a t e shear s t r e s s s°° x s t r u c t u r a l • s t r e s s s T . i n i t i a l v a l u e o f T sO s k^ t h e r a t e o f i n c r e a s e o f network s t r u c t u r e and P Q the i n i t i a l c o n c e n t r a t i o n o f network s t r u c t u r e The models based on c h e m i c a l r e a c t i o n k i n e t i c s d i s c u s s e d here a r e s i m i l a r i n two r e s p e c t s . The f i r s t common element i s t h e assumption of the o c c u r r e n c e o f a r a t e p r o c e s s c o n s i s t i n g o f b r e a k i n g and r e f o r m i n g of some k i n d of bond between p a r t i c l e s o r m o l e c u l e s i n the l i q u i d . The second common element i s the d e s i g n a t i o n o f a r e l a t i o n s h i p between the number of e x i s t i n g bonds and t h e v i s c o s i t y of the l i q u i d . S i n c e t h e type o f r e a c t i o n a c t u a l l y o c c u r r i n g was unknown, t h e v a r i o u s t h e o r i s t s used s i m p l e c o m b i n a t i o n s o f forw a r d and r e v e r s e r e a c t i o n s i n t h e i r r a t e e q u a t i o n s . The c h e m i c a l k i n e t i c s models seem t o be s u f f i c i e n t t o r e p r e s e n t the time-dependent b e h a v i o r of l i q u i d s q u a n t i t a t i v e l y . There i s another approach, however, which i s t o ex p r e s s the c h a r a c t e r i s t i c s o f l i q u i d s i n terms o f c o n s t i t u t i v e e q u a t i o n s . T h i s method i s c a l l e d the phenomeno-l o g i c a l approach. Some of t h e i d e a s a r e d i s c u s s e d below. 15 Cheng and Evans assumed t h a t th e r h e o l o g i c a l p r o p e r t y of t h i x o t r o p i c f l u i d s i s d e s c r i b e d i n terms of s c a l a r c o n s t i t u t i v e equa-t i o n s c o n s i s t i n g of an e q u a t i o n of s t a t e and a r a t e e q u a t i o n . The e q u a t i o n of s t a t e i s e x p r e s s e d i n th e form of x = n(A, Y)Y (2-8) i n which the v i s c o s i t y i s a f u n c t i o n of shear r a t e as w e l l as of t h e s t r u c t u r a l parameter "X", w h i l e t h e r a t e e q u a t i o n i s g i v e n i n t h e form o f f = g . U , Y ) (2-9) which s t a t e s t h a t t h e r a t e a t which the s t r u c t u r e changes i s a f u n c t i o n o f b o t h s h e a r r a t e and s t r u c t u r a l parameters. The forms o f n and g^ a r e supposed to be determined e x p e r i m e n t a l l y . The p e c u l i a r i t y of t h e i r i d e a i s e x p l a i n e d as f o l l o w s . Supposing Y'(t) i s g i v e n , t h e n from e q u a t i o n (2-9) one can s o l v e f o r X = X ( t ) and from e q u a t i o n (2-8) f o r x = x ( t ) . I f t i s e l i m i n a t e d from y ( t ) and x ( t ) , one o b t a i n s a l i n e x = x(y) i n t h e (x, y) p l a n e - The p r o j e c t i o n o f t h i s l i n e on the f ( x , y, X) = 0 s u r f a c e d e f i n e s a l i n e i n the s u r f a c e . A l o n g t h i s l i n e t i s a parameter. The l i n e i n the f ( x , y, X) = 0 s u r f a c e and the l i n e i n t h e (x, y, t ) space a r e determined not o n l y by t h e r a t e e q u a t i o n dX/dt = g^ but a l s o by the way the m a t e r i a l i s sheared y ( t ) . The b e h a v i o r of a t h i x o t r o p i c f l u i d i s t h e r e f o r e g i v e n by a l i n e i n t h e (x, y, X) s u r f a c e and not by a s u r f a c e i n the (x, y, t ) space, as . (31),(52) i s g e n e r a l l y thought The u l t i m a t e aim of " r h e o l o g y " i s to c o n s t r u c t c o n s t i t u t i v e equa-t i o n s f o r f l o w and d e f o r m a t i o n i n terms of m a t e r i a l parameters. In t h i s sense, the p h e n o m e n o l o g i c a l approach may be a good method to a t t a c k t h e phenomena of t h i x o t r o p y . U n f o r t u n a t e l y , t h e p r e s e n t s t a t e of t h e a r t s t a n d s f a r from i t s aim. CHAPTER 3 THEORETICAL BACKGROUND The l i t e r a t u r e reviewed i n t h e p r e v i o u s c h a p t e r f o c u s s e d on the phenomena o f t h i x o t r o p y . The p r e s e n t c h a p t e r w i l l c o v e r some o f the c o n c e p t s c o n c e r n i n g t h e b a s i c c l a s s i f i c a t i o n o f f l u i d s and the f o rmulae w hich w i l l be u s e f u l i n l a t e r c h a p t e r s . These formulae a r e based on a number o f assumptions which a r e s u b j e c t t o e r r o r , p a r t i c u l a r l y when su s p e n s i o n s such as time-dependent a r t i f i c i a l s l u r r i e s a r e d e a l t w i t h . C o m p l i c a t i n g e f f e c t s i n s u s p e n s i o n r h e o l o g y a s s o c i a t e d w i t h t h o s e assump-t i o n s w i l l be d e s c r i b e d and methods to c o r r e c t t h e e r r o r s w i l l be men-t i o n e d wherever p o s s i b l e . 3-1. Newton's P o s t u l a t e When a f l u i d i s s i t u a t e d between two p a r a l l e l f l a t p l a t e s , t h e top one moving w i t h a c o n s t a n t v e l o c i t y Vcm/sec, and the lower one h e l d m o t i o n l e s s by a r e s t r a i n i n g f o r c e dyne-, a v e l o c i t y g r a d i e n t i s s e t up as shown i n F i g . 3-1. The f o r c e per u n i t a r e a r e q u i r e d to r e s t r a i n the lower p l a t e ( e q u a l to t h a t r e q u i r e d to m a i n t a i n t h e upper one i n motion) has been c a l l e d t h e shear s t r e s s and has been d e s i g n a t e d by x , i . e . , t h e X-yx d i r e c t e d f o r c e a c t i n g on a p l a n e p e r p e n d i c u l a r to Y. I t i s mathematic-a l l y e x p r e s s e d as . F T yx S 16 (3-1) 17 F i g . 3-1. F o r c e B a l a n c e between P a r a l l e l P l a t e s 18 The r a t e of change of v e l o c i t y w i t h d i s t a n c e from the lower p l a t e , AVx/AY, i n t h i s case V/H, has been d e s i g n a t e d as shear r a t e and i s u s u a l l y e x p r e s s e d i n d i f f e r e n t i a l form so t h a t A AV T = T- = l i m TT <3-2> I f x i s p r o p o r t i o n a l to y, viz., yx x = y y (3-3) yx t h e f l u i d i s d e s i g n a t e d as Newtonian and the c o n s t a n t u i s c a l l e d i t s Newtonian v i s c o s i t y . E q u a t i o n (3-3) i s Newton's P o s t u l a t e . 3-2. Bingham P l a s t i c s A l t h o u g h most f l u i d s a r e a d e q u a t e l y r e p r e s e n t e d by e q u a t i o n ( 3 - 3 ) , more complex formulae a r e r e q u i r e d i n o t h e r c a s e s . I f , i n a m a t e r i a l , a minimum s t r e s s , TQ, must be overcome i n o r d e r t h a t f l o w o c c u r , and i f t h e s h e a r r a t e i s e f f e c t i v e l y p r o p o r t i o n a l t o t h e r e s i d u a l shear s t r e s s (T - TQ)> t h e s i m p l e s t d e p a r t u r e from Newtonian b e h a v i o r i s encountered. M a t e r i a l s which d i s p l a y t h i s b e h a v i o r a r e r e f e r r e d t o as Bingham P l a s t i c s and a r e d e s c r i b e d by (T - x 0 ) = ny (3-4) where n i s t h e p l a s t i c v i s c o s i t y . 3-3. Power Law F l u i d s In o r d e r to c h a r a c t e r i z e r e a l f l u i d s , i t i s common p r a c t i c e t o p l o t l o g t v e r s u s l o g y , from which i t i s o f t e n found t h a t the c u r v e i s l i n e a r o v e r a wide range o f y f o r many f l u i d s . As a r e s u l t , an e m p i r i c a l r e l a t i o n known as the power law i s w i d e l y used to d e s c r i b e f l u i d s of t h i s t y p e . T h i s r e l a t i o n may be w r i t t e n as T = k y n (3-5) 19 where k and n a r e c o n s t a n t s f o r t h e p a r t i c u l a r f l u i d : k i s a measure of t h e c o n s i s t e n c y of t h e f l u i d , t h e h i g h e r t h e k t h e more v i s c o u s t h e f l u i d ; n i s a measure of the degree of non-Newtonian b e h a v i o r and t h e g r e a t e r t h e d e p a r t u r e from u n i t y t h e more pronounced a r e the non-Newtonian p r o p e r t i e s of t h e f l u i d . The f l u i d s w i t h n < 1 a r e c a l l e d p s e u d o p l a s t i c , w h i l e the ones w i t h n > 1 a r e c a l l e d d i l a t a n t f l u i d s . The "apparent v i s c o s i t y " n of a f l u i d i s d e f i n e d as the r a t i o o f t h e shear s t r e s s to the shear r a t e , or n = £ (3-6) . Y T h i s d e f i n i t i o n i s c o n s i s t e n t w i t h the u s u a l d e f i n i t i o n o f the v i s c o s i t y o f a Newtonian f l u i d [ e q u a t i o n ( 3 - 3 ) ] , and s e r v e s as a d e f i n i t i o n of the v i s c o s i t y o f a non-Newtonian f l u i d . The apparent v i s c o s i t y f o r a power law f l u i d may be e x p r e s s e d as n = X = ^ = k j n - l ( 3 _ 7 ) Y Y 3-4. G e n e r a l F l u i d A l t h o u g h i t i s c o n v e n i e n t to use a m a t h e m a t i c a l r e p r e s e n t a t i o n of t h e r h e o l o g i c a l b e h a v i o r of a f l u i d , i t s a p p l i c a b i l i t y i s l i m i t e d by t h e a c c u r a c y w i t h which i t r e p r e s e n t s the a c t u a l b e h a v i o r . A g e n e r a l r h e o l o g i c a l e q u a t i o n of s t a t e i s n e c e s s a r i l y o f the form Y = f ( T ) (3-8) and i t must i n a l l c a s e s be based on e x p e r i m e n t a l l y measured d a t a . These d a t a a r e u s u a l l y o b t a i n e d by the use of a v i s c o m e t r i c d e v i c e . The c h a r a c t e r i z a t i o n o f l a r g e p a r a l l e l p l a t e s d e s c r i b e d i n F i g . 3-1 i s not r e a s o n a b l e , s i n c e the measurement of shear s t r e s s can o c c u r w i t h t h e a i d of more c o n v e n t i o n a l f l o w geometry. 20 3-5. Simple Shear Flow A f l u i d responds t o s t r e s s by f l o w i n g . Flow i s e s s e n t i a l l y a p r o c e s s i n which the m a t e r i a l deforms a t a f i n i t e r a t e . The b a s i c k i n e m a t i c measure o f t h e response o f a f l u i d i s the r a t e o f d e f o r m a t i o n t e n s o r A, whose C a r t e s i a n components a r e 3v. .3v. A. . = - r - ^ + T - 1 = A. . (3-9) i j 3x. 3x. j i (32) The components o f A i n the c y l i n d r i c a l c o o r d i n a t e s ( r , 6 , s ) a r e g i v e n by A 2 8 V r r r = 1 — (3-10) 3r 3v A = 2 ~ - (3-12) B 8 3 S A r e " V = ^ <T> + 7 i f ( 3 " 1 3 ) 3v„ , 3v A. = A a «9 3a r 30 + I — ^ (3-14) A = A 3v 3v 8 + — I (3-15) J g r " r a 3r 3z M o t i o n may e x i s t i n a f l u i d even i f A i s i d e n t i c a l l y z e r o . Each element o f f l u i d might be t r a n s l a t i n g a t t h e same l i n e a r v e l o c i t y , and each element might, i n a d d i t i o n , have t h e same a n g u l a r v e l o c i t y about some a x i s , due to a r i g i d r o t a t i o n o f t h e f l u i d . But u n i f o r m t r a n s l a -t i o n and r o t a t i o n do not c o n t r i b u t e t o the d e f o r m a t i o n o f the f l u i d , and so a r e not a s s o c i a t e d w i t h t h a t p a r t o f t h e response of the m a t e r i a l w hich i s of i n t e r e s t t o us. 21 A p a r t i c u l a r s i m p l e f l o w w h i c h w i l l a p p e a r i n v a r i o u s v i s c o m e t r i c d e v i c e s i s t h e s o - c a l l e d " s i m p l e s h e a r f l o w " , i n w h i c h t h e r e i s a n o n -z e r o c o m p o n e n t o f v e l o c i t y i n o n l y a s i n g l e d i r e c t i o n . I f t h e s u b -s c r i p t s 1, 2, a n d 3 d e n o t e , r e s p e c t i v e l y , t h e f l o w d i r e c t i o n , t h e d i r e c t i o n o f v e l o c i t y v a r i a t i o n , a n d t h e n e u t r a l d i r e c t i o n , t h e n a s i m p l e s h e a r f l o w i s d e f i n e d b y v = ( v l 5 0, 0) (3-16) A = y ( x 2 ) ( 0 1 0 ) 1 0 0 0 0 0 (3-17) w h e r e y ( x 2 ) s o m e s c a l a r f u n c t i o n o f t h e x 2 - c o o r d i n a t e . j i s a " s h e a r " c o m p o n e n t o f t h e r a t e o f d e f o r m a t i o n t e n s o r , a n d i s c o n s i s t e n t w i t h t h e e q u a t i o n ( 3 - 2 ) , a n d h a s b e e n c a l l e d t h e " s h e a r r a t e " . T h e c o r r e s p o n d e n c e o f t h e s i m p l e s h e a r f l o w n o t a t i o n t o t h e m o r e common c o o r d i n a t e n o t a t i o n , f o r e x a m p l e C o u e t t e f l o w g e o m e t r y w h e r e t h e c y l i n d r i c a l c o o r d i n a t e s a r e u s u a l l y u s e d , i s ( 1 , 2, 3) = ( 0 , r , «) (3-18) A c c o r d i n g t o t h e s e d e f i n i t i o n s , t h e o n l y n o n - z e r o c o m p o n e n t s o f A i n a s i m p l e s h e a r f l o w a r e A i 2 a n c * A 2 1 ( - A 1 2 ) , a s r e q u i r e d b y e q u a t i o n ( 3 - 1 7 ) . T h e s t r e s s c o m p o n e n t s o f a s i m p l e s h e a r f l o w a r e l a b e l e d i n t h e same m a n n e r . T h u s i s t h e n o r m a l s t r e s s i n t h e d i r e c t i o n o f f l o w a n d x 1 2 . i s t h e s h e a r i n g s t r e s s w h o s e n o t a t i o n h a s a l r e a d y b e e n u s e d i n e q u a t i o n s (3-1) a n d ( 3 - 3 ) . I n a d d i t i o n , o t h e r s i m p l e s h e a r f l o w g e o m e t r i e s a r e f o u n d i n P o i s e u i l l e f l o w , p a r a l l e l p l a t e t o r s i o n , a n d c o n e a n d p l a t e t o r s i o n . 22 3-6. C o u e t t e Flow ( 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 ) A f a m i l i a r t y p e of r h e o l o g i c a l i n s t r u m e n t c o n s i s t s of two c o a x i a l c y l i n d e r s , w i t h the f l u i d t o be s t u d i e d h e l d i n the a n n u l a r space between. M e c h a n i c a l c o n s t r u c t i o n a l l o w s a n g u l a r motion of one of t h e c y l i n d e r s . There i s a means f o r measuring the t o r q u e e x e r t e d on o r by t h e c y l i n d e r s . F i g . 3-2 shows a s c h e m a t i c drawing of such an i n s t r u m e n t . In t h i s s e c t i o n , t h e formulae f o r c a l c u l a t i n g shear s t r e s s and shear r a t e i n C o u e t t e f l o w a r e d e s c r i b e d . One assumes t h a t a s i m p l e shear f l o w e x i s t s i n the a n n u l a r r e g i o n , such t h a t v = ( v Q , 0, 0) (3-19) and v Q = v Q ( r ) = r ui(r) (3-20) where w i s t h e a n g u l a r v e l o c i t y ( r a d i a n / s e c ) . The r a t e o f d e f o r m a t i o n t e n s o r f o r t h i s s i m p l e shear f l o w has o n l y t h e components do) And one can d e f i n e a f u n c t i o n f (T) = -r~ = y (3-21) where x i s t h e shear s t r e s s component x 1 2 . Assume t h a t the o u t e r c y l i n d e r i s s t a t i o n a r y and t h a t the i n n e r c y l i n d e r i s d r i v e n w i t h an a n g u l a r v e l o c i t y Q, by t h e a p p l i c a t i o n o f a t o r q u e T. T h i s t o r q u e must b a l a n c e the t o r q u e e x e r t e d by the f l u i d on t h e f a c e of the i n n e r c y l i n d e r , so t h a t T = 2TTR 2L X r (3-22) where L and R a r e the h e i g h t and r a d i u s of the i n n e r c y l i n d e r , and x R F i g . 3-2. D e f i n i n g S ketch f o r the Treatment of C o a x i a l C y l i n d e r Flow Data 24 i s t h e s h e a r s t r e s s e x e r t e d o n t h e i n n e r c y l i n d e r . F r o m e q u a t i o n (3-22) o n e c a n f i n d TR - 2 ^ k ( 3 " 2 3 ) w h i c h r e l a t e s t h e s h e a r s t r e s s a t t h e i n n e r c y l i n d e r t o t h e t o r q u e m e a s u r e d i n t h e v i s c o m e t e r . T h i s f o r m u l a i s v a l i d e v e n i n a t i m e -d e p e n d e n t f l o w a s l o n g a s x ' i s c o n s t a n t o v e r t h e p o s i t i o n o n t h e i n n e r c y l i n d e r a t a n y g i v e n t i m e . I n o r d e r t o d e r i v e a f o r m u l a f o r s h e a r r a t e i n 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 , i t i s n e c e s s a r y t o e m p l o y momentum e q u a t i o n s . Momentum e q u a t i o n s a r e m a t h e m a t i c a l f o r m u l a t i o n s o f t h e f u n d a m e n t a l p h y s i c a l p r i n c i p l e o f c o n s e r v a t i o n o f momen tum. T h e i r m o s t g e n e r a l f o r m i s p ~~ = - VP - V.x + ZpF (3-24) I n c y l i n d r i c a l c o o r d i n a t e s ( 6 , r , a ) , e q u a t i o n (3-24) c a n b e (4) t r a n s f o r m e d t o t h e f o l l o w i n g e q u a t i o n s i n t e r m s o f t h e c o m p o n e n t s o f t h e c o o r d i n a t e s y s t e m , a s s u m i n g g r a v i t y t o b e t h e o n l y f i e l d f o r c e p r e s e n t , r - c o m p o n e n t s : 9v 3v ' v „ 3v v „ 3v / r , r , 6 r 6 , r \ o ( 1- v 1 1- v ) M 3 t r 3 r r 39 r a 3a 1 3 r ^ r 3 r K r r ' r 36 r 3a ' P 6 r v ' 9 - c o m p o n e n t s : 0 ( — + V 1 1- r V ) M v 3 t r 3 r r 36 r a 3a = _ I l £ _ ( 1 - 1 - ( r 2 T ) + I ^ 6 8 . + •fle«)+ p g ( 3 - 2 6 ) r 36 rz 3 r r 6 r 30 3a ' M 6 e 25 a-components: 3v 3v v 3v 3v , a , a . 8 a a, D ( 1- V 1 r V ) M V 3t r 3 r r 36 a 3a = _ 9 P _ ( l J _ ( r T ) + I + ^ 55.) + g ( 3_ 2 7) 3a r 3r r « ; r 30 9a ' y&s The f o l l o w i n g assumptions a r e u s u a l l y made f o r Couette f l o w : (1) The f l o w i s st e a d y ( a l l p a r t i a l d e r i v a t i v e s w i t h r e s p e c t t o t h e time a r e z e r o ) . (2) There i s no s l i p p a g e a t t h e w a l l . (3) There i s no end e f f e c t o f t h e c y l i n d e r s . (4) The f l u i d i s homogeneous and i n c o m p r e s s i b l e . (5) I s o t h e r m a l c o n d i t i o n s p r e v a i l t h r o u g h o u t . (6) The f l o w i s l a m i n a r . Then e q u a t i o n (3-26) l e a d s immediately t o r 2 x = c o n s t a n t (3-28) from which one f i n d s r 2 d x + 2rxdr = 0 (3-29) T h i s a l l o w s the t r a n s f o r m a t i o n d _ 2 x_d_ ( 3 _ 3 0 ) dr r dx and so e q u a t i o n (3-21) becomes ^ ~ -T ~7~ dx I t f o l l o w s t h a t f ( x ) f ( T ) = 2 T ^ (3-31) = i f L S 1 J - dx (3-32) ' X n X : 0 T when the boundary c o n d i t i o n to = 0 a t r = R 0 (3-33) i s used and the shear s t r e s s a t t h e o u t e r c y l i n d e r i s d e s i g n a t e d by XQ' E q u a t i o n (3-32) i s v a l i d , i n p a r t i c u l a r a t r = R, and becomes 1 r R f ( T ) , tt = ^ J — — dx 2 J T Q T (3-34) Now one d i f f e r e n t i a t e s e q u a t i o n (3-34) and f i n d s dtt 1 d T R f ( T R ) f ( T Q ) d T Q R dT. R (3-35) But, from e q u a t i o n (3-28) T 0 R 0 2 = T R R 2 (3-36) and hence d T Vn > S 1T r "Ro Thus e q u a t i o n (3-35) becomes 2T (3-37) (3-38) S i n c e e q u a t i o n (3-23) g i v e s T_ as a f u n c t i o n o f T, t h e terms which appear i n e q u a t i o n (3-38) a r e determined from tt v s . T d a t a . But i n i t s p r e s e n t form, e q u a t i o n (3-38) i s a d i f f e r e n c e e q u a t i o n f o r f ( x R ) . I n the s p e c i a l c ase o f s 2 = 0, c o r r e s p o n d i n g t o measurement i n an i n f i n i t e body o f f l u i d , e q u a t i o n (3-38) can be reduced t o (3-39) s i n c e f ( 0 ) = 0. I f T i s i n t r o d u c e d i n t o t h i s r e s u l t , one f i n d s d lntt f< TR> - m d InT (3-40) E q u a t i o n (3-40) would be a good a p p r o x i m a t i o n f o r s < 0.1. The d i f f e r e n c e e q u a t i o n can be s o l v e d by an i t e r a t i v e p r ocedure. E q u a t i o n (3-38) i s v a l i d f o r a l l v a l u e s of T r . F o r example, one may w r i t e S 2 T f o r T„ and f i n d 27 «. R J s z x R and, i n g e n e r a l I f t h e complete s e t o f e q u a t i o n s o b t a i n e d by s u c c e s s i v e l y s u b s t i -t u t i n g S 2 T 1 3 , S ^ T - , ... , s ^ ^ ^ T . f o r T „ i n e q u a t i o n ( 3 - 3 8 ) i s added t o g e t h e r , the r e s u l t i s dft N - l 2N 2P f ( x R ) - f ( s T-n) = E 2 s ^ x P=0 R/1 S T 2P <3 " 4 3 ) R 2 ( N - l ) Now l e t N go to i n f i n i t y , and t a k e note o f t h e f a c t t h a t s (9) v a n i s h e s i n t h i s l i m i t s i n c e s < 1. The r e s u l t i s 2P f ( x ) = j = 2 T r E. s • [ f i ' l 2 ( 3 - 4 4 ) P=0 S T R where ft' = dft/dx,,. I t i s common p r a c t i c e t o p l o t l o g . i o ^ v s - l o g l O T R > f r o m which the s l o p e d l o g 1 0 f t m = — ( 3 - 4 5 ) d l o 8 l O T R can be more a c c u r a t e l y found than the s l o p e ft', s i n c e m i s n e a r l y a c o n s t a n t over a wide range of x f o r many f l u i d s . I n t h a t case one c a l c u l a t e s t h e shear r a t e a t t h e i n n e r c y l i n d e r u s i n g f ( x R ) = 2ft E s m(s L T J ( 3 - 4 6 ) R P=0 R S i n c e s i s l e s s t h a n u n i t y , t h i s s e r i e s c o n v e r g e s , but does so (24) s l o w l y when s i s n e a r l y u n i t y . I n a d d i t i o n , K r i e g e r and E l r o d found an a p p r o x i m a t i o n t o e q u a t i o n ( 3 - 3 8 ) f o r -m l n s < 0.5, and the r e s u l t i s 28 f ( 0 = - I n s [1 - m l n s + ~r(m l n s )2 ] (3-47) In t h e case of p u r e l y Newtonian f l u i d t h e shear r a t e i s g i v e n by Y = (3-48) For a power law f l u i d , m o f e q u a t i o n (3-45) i s a c o n s t a n t [m r- 1/n where n i s g i v e n i n e q u a t i o n ( 3 - 5 ) ] , and e q u a t i o n (3-46) reduces to 2mQ, (3-49) Y = 2m 1-s By s o l v i n g t h e momentum e q u a t i o n f o r t h i s C o u e t t e f l o w , f o r a power law f l u i d , t h e same r e s u l t i s found. E q u a t i o n (3-49) may be used to c a l c u -l a t e the shear r a t e whenever the power law i s a good a p p r o x i m a t i o n . 3-7. C o m p l i c a t i n g E f f e c t s i n S u s p e n s i o n Rheology As s t a t e d i n t h e p r e v i o u s s e c t i o n , many assumptions have been made to d e r i v e e q u a t i o n s (3-23), (3-46), (3-47), (3-48), and (3-49). These assumptions a r e , i n f a c t , v e r y r e a s o n a b l e f o r time-independent homogeneous f l u i d s whose v i s c o m e t r y i s , t h e r e f o r e , w e l l - u n d e r s t o o d . The v i s c o m e t e r measures the t o r q u e e x e r t e d by the f l u i d on t h e w a l l of t h e i n n e r c y l i n d e r and the a n g u l a r v e l o c i t y o f t h e i n n e r c y l i n d e r . The t o r q u e i s r e l a t e d t o an average o f t h e s h e a r s t r e s s over t h e a r e a o f the w a l l . F o r a homo-geneous f l u i d the shear s t r e s s i s c o n s t a n t over the a r e a of the w a l l , so t h e s t r e s s a t a p o i n t on the w a l l may be c a l c u l a t e d from the average. The a n g u l a r v e l o c i t y i s r e l a t e d t o shear r a t e w i t h the a i d of the g r a d i e n t o f t h e a n g u l a r v e l o c i t y . The g r a d i e n t o f t h e a n g u l a r v e l o c i t y i s not always c o n s t a n t over t h e r a d i u s , so t h e g r a d i e n t cannot be c a l c u l a t e d from t h e v a l u e of the a n g u l a r v e l o c i t y o f the c y l i n d e r . C o n s e q u e n t l y , t h e g r a d i e n t of t h e a n g u l a r v e l o c i t y and the shear r a t e a r e unknown. F o r t u n a t e l y , a few e x t r a measurements and some ma t h e m a t i c a l a n a l y s i s a l l o w us to c a l c u l a t e 29 t h e s h e a r r a t e a t t h e w a l l , a t w h i c h t i m e t h e v i s c o s i t y a t t h e w a l l c a n b e c a l c u l a t e d . S i n c e t h e f l u i d i s h o m o g e n e o u s , t h e v i s c o s i t y a s a f u n c t i o n o f s h e a r r a t e a t t h e w a l l i s t h e same a s t h a t i n t h e i n t e r i o r o f t h e a n n u l a r s p a c e . H e n c e , t h e v e l o c i t y p r o f i l e may b e c a l c u l a t e d f r o m t h e momentum e q u a t i o n s . T h e i n t e r p r e t a t i o n o f C o u e t t e v i s c o m e t e r r e s u l t s f o r s u s p e n s i o n s i s n o t s o s i m p l e . T o r q u e a n d a n g u l a r v e l o c i t y a r e s t i l l m e a s u r e d . T h e f o r m e r s t i l l y i e l d s t h e s h e a r s t r e s s a t t h e w a l l e v e n i n t i m e - d e p e n d e n t f l o w o f s u s p e n s i o n s . A l t h o u g h t h e a n g u l a r v e l o c i t y i s k e p t c o n s t a n t , t h e c a l c u l a t i o n o f t h e s h e a r r a t e a n d t h e v e l o c i t y p r o f i l e r e q u i r e s t h e a s s u m p -t i o n s o f s t e a d y s t a t e a n d h o m o g e n e i t y w h i c h a r e n o t g e n e r a l l y v a l i d f o r t i m e - d e p e n d e n t s u s p e n s i o n s . T h i s i n h o m o g e n e i t y , i n p a r t i c u l a r , i s m o r e t h a n t h e i n h e r e n t m a t h e m a t i c a l d i f f i c u l t y o f h a v i n g t h e m a t e r i a l p r o p e r -t i e s r e p r e s e n t e d b y d i s c o n t i n u o u s f u n c t i o n s . T h e p a r t i c l e s i n t h e s u s -p e n s i o n i n t e r a c t w i t h t h e w a l l , s o t h a t w h a t h a p p e n s n e a r t h e w a l l i s n o t n e c e s s a r i l y t y p i c a l o f w h a t h a p p e n s i n t h e b u l k f l u i d . T h e s e c o m p l i c a t -i n g e f f e c t s i n r e l a t i o n t o t h e many a s s u m p t i o n s made a r e d i s c u s s e d i n t h e f o l l o w i n g s u b - s e c t i o n s . 3 - 7 - 1 . C o n t i n u u m T h e momentum e q u a t i o n s a r e b a s e d o n t h e a s s u m p t i o n s t h a t t h e f l u i d b e h a v e s a s a c o n t i n u u m . T r e a t i n g s u s p e n s i o n s a s a c o n t i n u o u s t w o - p h a s e s y s t e m r e q u i r e s t h a t t h e m o l e c u l a r s i z e o f t h e s u s p e n d i n g m e d i u m b e s m a l l r e l a t i v e t o t h e s i z e o f s u s p e n d e d p a r t i c l e s . I t i s t h e r e f o r e d e s i r a b l e t o u s e m a t e r i a l s w i t h r e l a t i v e l y l o w m o l e c u l a r w e i g h t a s t h e s u s p e n d i n g m e d i u m , e . g . , p o l y e t h y l e n e g l y c o l , s o d i u m c h l o r i d e , w a t e r , a n d / o r d e x t r o s e . 30 3-7-2. Steady I t i s o bvious t h a t t h e experiment on time-dependent v i s c o m e t r y i s c a r r i e d out under unsteady s t a t e c o n d i t i o n s except f o r the p a r t a t e q u i l -i b r i u m . T h e r e f o r e , t h e c a l c u l a t i o n o f shear r a t e o f e q u i l i b r i u m i s not d i f f i c u l t , w h i l e i t i s almost i m p o s s i b l e a t p r e s e n t t o c a l c u l a t e t i m e -dependent t r u e shear r a t e . Hence, many workers have employed i n s t r u m e n t s of t h e C o u e t t e t y p e w i t h an a n n u l a r space which i s s m a l l compared w i t h t h e c y l i n d e r r a d i i (s = R/RQ = 1 ) , and they have assumed t h a t t h e shear r a t e i s c o n s t a n t over the a n n u l a r space throughout t h e time. However, v i s c o m -e t r y o f s u s p e n s i o n s o f t e n r e q u i r e s t h a t the c l e a r a n c e between t h e c y l i n d e r s be s u f f i c i e n t l y l a r g e compared w i t h the s i z e of suspended p a r t i c l e s . In t h i s c ase n e i t h e r d i s t r i b u t i o n o f shear r a t e , v e l o c i t y p r o f i l e i n t h e a n n u l a r space, nor t h e i r v a r i a t i o n w i t h time a r e known. We do know, however, t h a t s i n c e t h e a n g u l a r v e l o c i t y o f t h e c y l i n d e r can be kept c o n s t a n t , e q u a t i o n (3-48) g i v e s us a c e r t a i n measure of the shear r a t e a t t h e w a l l throughout t h e time. The shear r a t e thus employed i s c a l l e d "apparent shear r a t e . " 3-7-3. W a l l E f f e c t s W a l l e f f e c t s r e f e r t o s e v e r a l phenomena observed i n s u s p e n s i o n v i s c o m e t r y which a r i s e from t h e p r e s e n c e of t h e v i s c o m e t e r w a l l . Many o f t h e s e e f f e c t s a r e reviewed by Mashmeyer The s i m p l e s t w a l l e f f e c t , m e c h a n i c a l e x c l u s i o n , o c c u r s because t h e c e n t e r o f a p a r t i c l e i n s u s p e n s i o n cannot approach the w a l l any c l o s e r t h a n t h e e f f e c t i v e s i z e of the p a r t i c l e . The r e s u l t i s a t h i n l a y e r of s u s p e n s i o n next to the w a l l w i t h a lower average c o n c e n t r a t i o n of p a r t i c l e s ( 27 ^  (28^ t h a n t h a t o f t h e b u l k f l u i d . Maude and Whitmore ' c a l c u l a t e d t h e 31 m a g n i t u d e o f m e c h a n i c a l w a l l e x c l u s i o n o n t h e v i s c o s i t y o f a s u s p e n s i o n o f s p h e r e s i n c r e e p i n g f l o w a n d c o m p a r e d t h e i r c a l c u l a t i o n w i t h e x p e r i m e n t a l l y m e a s u r e d v i s c o s i t i e s . T h e i r c o n c l u s i o n w a s t h a t m e c h a n i c a l e x c l u s i o n w a s n o t l a r g e e n o u g h t o e x p l a i n t h e e x p e r i m e n t a l r e s u l t s . A s i m i l a r m e c h a n -i c a l e x c l u s i o n may b e e x p e c t e d f o r s u s p e n s i o n s o f f i b e r s , e x c e p t t h a t t h e c l o s e n e s s o f a p p r o a c h i s d e t e r m i n e d b y o r i e n t a t i o n a s w e l l a s f i b e r d i m e n -s i o n . No t h e o r e t i c a l c a l c u l a t i o n s o r e x p e r i m e n t a l m e a s u r e m e n t s h a v e b e e n made f o r t h e m a g n i t u d e o f t h i s e f f e c t . (19) K a r n i s , G o l d s m i t h , a n d M a s o n s h o w e d t h a t i n l o w R e y n o l d s n u m b e r c a p i l l a r y f l o w o f d i l u t e s u s p e n s i o n s o f s p h e r e s a n d f i b e r s , t h e v e l o c i t y p r o f i l e s a r e p a r a b o l i c a n d t h e l o c a l v e l o c i t i e s o f t h e f l u i d a n d t h o s e o f t h e p a r t i c l e s a r e t h e s a m e . A t h i g h e r c o n c e n t r a t i o n s t h e v e l o c i t y p r o -f i l e s b e c o m e b l u n t e d . T h e b l u n t e d r e g i o n i s d e s i g n a t e d " p a r t i a l p l u g f l o w , " b e c a u s e t h e p r o f i l e may n o t b e c o m p l e t e l y f l a t . A t e v e n h i g h e r c o n c e n t r a t i o n s , c o m p l e t e p l u g f l o w d e v e l o p s . B l u n t e d v e l o c i t y p r o f i l e s h a v e b e e n o b s e r v e d i n c o n c e n t r a t e d (45) s u s p e n s i o n s o f f i b e r s a s w e l l . S t a n k o i e t al. r e p o r t e d o b s e r v i n g t h e m i n c a p i l l a r y f l o w . T h e r e a r e s e v e r a l p o s s i b l e r e a s o n s f o r a b l u n t e d v e l o c i t y p r o f i l e i n s u s p e n s i o n s . F i r s t , i t c a n o c c u r i n a f l u i d w h i c h e x h i b i t s a y i e l d s t r e s s . I n t h i s c a s e t h e t e r m " p l u g f l o w " i s a n a c c u r a t e d e s c r i p t i o n , a n d t h e v e l o c i t y p r o f i l e i s m a t h e m a t i c a l l y f l a t . S e c o n d , a b l u n t e d p r o -f i l e c a n b e c a u s e d b y r a d i a l m i g r a t i o n o f p a r t i c l e s away f r o m t h e w a l l . T h e p a r t i c l e - d e p l e t e d s u s p e n s i o n n e a r t h e w a l l h a s a l o w e r v i s c o s i t y t h a n t h e c o n c e n t r a t e d r e g i o n a t t h e a x i s , a n d t h e v e l o c i t y p r o f i l e r e f l e c t s t h i s . E x p e r i m e n t a l e v i d e n c e ^ 2 ^ i n d i c a t e s t h a t b o t h i n e r t i a a n d t h e p r e s e n c e o f a w a l l a r e i m p o r t a n t i n t h i s c a s e . T h i r d , t h e s u s p e n s i o n 32 may s i m p l y show s h e a r t h i n n i n g b e h a v i o r . A n o t h e r w a l l e f f e c t stems f r o m p a r t i c l e i n t e r a c t i o n w i t h t h e w a l l . The t h i c k n e s s o f a l a y e r i n w h i c h t h i s i n t e r a c t i o n o c c u r s i s supposed t o (26) be p r o p o r t i o n a l t o t h e p a r t i c l e s i z e . The s u s p e n d i n g medium can be r i c h e r i n t h i s l a y e r t h a n i n t h e r e g i o n f a r f r o m t h e w a l l . T h i s means t h a t t h e f l u i d v e l o c i t y n e a r t h e w a l l i n P o i s e u i l l e f l o w i s h i g h e r t h a n e x p e c t e d . V a n d ' (51) p r e c i i c t e d t h a t t h e p r e s s u r e drop v e r s u s f l o w r a t e c u r v e s f o r d i f f e r e n t v i s c o m e t e r g e o m e t r i e s would n o t s u p e r i m p o s e . W a l l e f f e c t s a r e o f f u n d a m e n t a l s i g n i f i c a n c e f o r r h e o l o g i s t s b e cause t h e y make t h e v i s c o s i t y o f s u s p e n s i o n s depend on v i s c o m e t e r geometry. T h e o r i e s have been d e v e l o p e d t o show how w a l l e f f e c t s w i l l a f f e c t e x p e r i m e n t a l d a t a and t h e y p r e d i c t t h a t t h e i r e f f e c t s w i l l i n c r e a s e w i t h a d e c r e a s e i n t h e r a t i o o f i n s t r u m e n t / p a r t i c l e s i z e , b u t do n o t p r e d i c t t h e e f f e c t o f t h e s i z e f o r f i b e r s u s p e n s i o n s . Y e t i t i s n o t c l e a r what d i m e n s i o n o f t h e f i b e r s h o u l d be t a k e n as i t s c h a r a c t e r i s t i c s i z e ; t h e f i b e r l e n g t h need n o t be c h a r a c t e r i s t i c . I n t u i t i v e l y , one m i ght e x p e c t t h e c h a r a c t e r i s t i c d i m e n s i o n t o l i e somewhere between t h e f i b e r l e n g t h and d i a m e t e r ; "somewhere" d e t e r m i n e d by o r i e n t a t i o n . (54) I t was recommended t h a t c y l i n d r i c a l s u r f a c e s w i t h deep grooves be used i n a C o u e t t e t y p e v i s c o m e t e r i n o r d e r t o e l i m i n a t e t h e w a l l e f f e c t s . I f w a l l e f f e c t s a r e s m a l l , t h e y may be u n o b s e r v a b l e w i t h i n e x p e r i m e n t a l e r r o r and t h u s be o f l i t t l e p r a c t i c a l i m p o r t a n c e . 3-7-4. End E f f e c t s The f o r m u l a e f o r s h e a r s t r e s s and s h e a r r a t e , w h i c h have been d e r i v e d e a r l i e r , do n o t a l l o w any s h e a r i n g a c t i o n between s u s p e n s i o n s and t h e f l a t end s u r f a c e s o f t h e c y l i n d e r s i n a C o u e t t e f l o w . The end 33 e f f e c t s can be checked and c o r r e c t e d by making measurements w i t h i n n e r c y l i n d e r s of d i f f e r e n t l e n g t h . A c a r e f u l d e s i g n of t h e i n n e r c y l i n d e r can e l i m i n a t e t h e end e f f e c t s . T h i s w i l l be e x p l a i n e d l a t e r . 3-7-5. Homogeneous I f t h e d e n s i t y o f d i s p e r s e d phase i s matched w i t h t h a t o f t h e d i s p e r s i n g medium, the s u s p e n s i o n i s homogeneous on the s c a l e of d e n s i t y . The s u s p e n s i o n s i n t h i s work, however, c o n t a i n the f i b e r s , some of whose l e n g t h s a r e n e a r l y the same o r d e r o f magnitude as the l e n g t h of gap between two c y l i n d e r s , so t h e s u s p e n s i o n s may not be homogeneous on the s c a l e o f the experiment. 3-7-6. I n c o m p r e s s i b l e Suspensions can be t r e a t e d as i n c o m p r e s s i b l e f l u i d s , as l o n g as t h e i r d e n s i t y does not change w i t h time and p o s i t i o n . I n a d d i t i o n , c a r e s h o u l d be t a k e n i n p r e p a r i n g s u s p e n s i o n s to a v o i d t r a p p i n g a i r b u b b l e s as w e l l as i n a g i n g s u s p e n s i o n s t o p r e v e n t e v a p o r a t i o n . 3-7-7. I s o t h e r m a l (32) Middleman p r e s e n t e d an approximate e s t i m a t i o n f o r the maximum temperature r i s e i n C o u e t t e f l o w . I t i s r e a s o n a b l e to assume t h a t the heat e f f e c t i s n e g l i g i b l e under the u s u a l o p e r a t i n g c o n d i t i o n s of 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 d e v i c e to keep w a l l temperature c o n s t a n t , 3-7-8. Laminar (4) B i r d e t al. showed an approximate e x p r e s s i o n f o r e s t i m a t i n g t h e c r i t i c a l Reynolds number f o r a C o u e t t e f l o w as K M ,41.3 t r a n s . (TUP/2 ( 3 " 5 0 ) I n a d d i t i o n , a v i s u a l o b s e r v a t i o n o f t h e f l o w o f s u s p e n s i o n s i n a v i s -c o m e t e r c a n c o n f i r m w h e t h e r t h e f l o w i s l a m i n a r o r n o t . 3 - 7 - 9 . T a y l o r ' s I n s t a b i l i t y T h e T a y l o r n u m b e r c a n b e d e f i n e d , w i t h t h e p r e s e n t n o t a t i o n , a s R 0 - R x N T = ( R f l ) ( R 0 - R>(^)( R — ) 2 ( 3 - 5 1 ) T h e o n - s e t o f i n s t a b i l i t y f o r t h e C o u e t t e f l o w w i t h t h e i n n e r c y l i n d e r r o t a t i n g ( t h e o u t e r c y l i n d e r a t r e s t ) may o c c u r a t t h e T a y l o r n u m b e r <V c r i t i c a l = 4 1 ( 3 " 5 2 ) A l l t h e v i s c o m e t r i c m e a s u r e m e n t s s h o u l d b e c a r r i e d o u t i n t h e s t a b l e l a m i n a r r e g i o n . CHAPTER 4 DEFINITION OF THE PROBLEM AND METHOD OF ATTACK B e f o r e d i s c u s s i n g , i n d e t a i l , t h e e x p e r i m e n t a l work, the problem s p e c i f i c a l l y f o c u s s e d on i n t h i s t h e s i s w i l l be d e f i n e d and the approach used f o r i t s s o l u t i o n w i l l be o u t l i n e d i n t h i s c h a p t e r . S i n c e recommendations of o t h e r workers a r e o f t e n v a l u a b l e and u s e f u l f o r t h a t purpose, they a r e o u t l i n e d i n S e c t i o n 4-1. S e c t i o n 4-2 g i v e s the d e f i n i t i o n o f t h e problem. Then t h e systems of a r t i f i c i a l s l u r r i e s chosen and t h e i r v a r i a b l e s w i t h the ranges to be i n v e s t i g a t e d a r e s t a t e d i n S e c t i o n 4-3. F i n a l l y , i n S e c t i o n 4-4 t h e methods used i n the experiments a r e b r i e f l y mentioned. 4-1. Recommendations of Other Workers I t i s apparent from the l i t e r a t u r e s u r v e y and t h e o r e t i c a l back-ground t h a t t h e phenomenon of t h i x o t r o p y and the r h e o l o g i c a l b e h a v i o r of" c o n c e n t r a t e d f i b e r s l u r r i e s a r e h i g h l y complex. A c c o r d i n g t o Mashmeyer^ So l i t t l e i s known about the r h e o l o g y of c o n c e n t r a t e d f i b e r s u s p e n s i o n s t h a t r e s e a r c h on t h e s u b j e c t c o u l d be c o n t i n u e d i n s e e m i n g l y e n d l e s s ways. And h i s recommendation: A study of t h e e f f e c t of f i b e r l e n g t h would be among t h e most i n t e r e s t i n g and u s e f u l , but such a s t u d y would be d i f f i c u l t b o t h because the n e c e s s a r y e x p e r i m e n t a l t e c h n i q u e s have not been develo p e d and because the b e h a v i o r i s i n h e r e n t l y c o m p l i c a t e d . Jones examined the t h i x o t r o p i c b e h a v i o r of a s u s p e n s i o n of c o l l o i d a l a l umina i n p r o p y l e n e g l y c o l and recommended: 35 36 S t u d i e s o f t h e e f f e c t s o f d i f f e r e n t p a r t i c l e s i z e and shape d i s t r i b u t i o n s on v i s c o m e t r i c b e h a v i o r appear t o be a p a r t i c -u l a r l y p r o m i s i n g a r e a of r e s e a r c h f o r the immediate f u t u r e . I t might w e l l be t h a t t h e number o f p a r t i c l e s p r e s e n t i n t h e l i q u i d and t h e i r dimensions a r e more im p o r t a n t f a c t o r s than weight p e r c e n t i n d e t e r m i n i n g r h e o l o g i c a l b e h a v i o r . (6) Brown wrote: Only two p a r t i c l e shapes were c o n s i d e r e d i n the p r e s e n t s t u d y , b o t h of which were of the r i g i d p l a t e t y p e . Other shapes which can be p r e p a r e d e a s i l y a r e r o d - l i k e p a r t i c l e s , w hich can be made by c u t t i n g f i b e r s o f c y l i n d r i c a l and e l l i p t i c a l c r o s s - s e c t i o n a l a r e a i n t o t h e d e s i r e d l e n g t h s . H u ^ " ^ s u g g e s t e d t h a t a CMC-Corn Syrup system or P E G - D e x t r o s e / F r u c t o s e system would be a good medium f o r suspending a r t i f i c i a l p a r t i c l e s . (29) Mercer used a Haake R o t o v i s k o V i s c o m e t e r f o r h i s study on t h e t i m e -dependent v i s c o s i t y o f t h i x o t r o p i c b e n t o n i t e - w a t e r s u s p e n s i o n s and s e t f o r t h t h e f o l l o w i n g recommendations: For any f u t u r e m e c h a n i c a l e x p e r i m e n t s , t h e v i s c o m e t e r s h o u l d be d e s i g n e d e s p e c i a l l y f o r time-dependent measure-ments. For example, the m o d i f i c a t i o n s of t h e v i s c o m e t e r s h o u l d be made f o r the purpose, i n h i s o p i n i o n , of " m i n i m i z a t i o n of o s c i l l a t o r y motions". 4-2. D e f i n i t i o n o f t h e Problem Based on the l i t e r a t u r e s u r v e y and t h e recommendations of o t h e r workers the problem to be t r e a t e d i n t h i s study may now be s t a t e d . (1) How do t h e v a r i a b l e s o f a r t i f i c i a l s l u r r i e s ( e . g . , L/D r a t i o o f p a r t i c l e s , p a r t i c l e c o n c e n t r a t i o n , s a l t c o n c e n t r a t i o n , v i s c o s i t y of s u s p e n d i n g medium, e t c . ) a f f e c t the time-dependent shear f l o w (e.g., y i e l d s t r e s s , e q u i l i b r i u m s t r e s s , s t r e s s decay c u r v e , v a r i a t i o n o f 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, e t c . ) ? (2) What i s the model e q u a t i o n which b e s t d e f i n e s t h e time-dependent c h a r a c t e r i s t i c s of f l o w of t h e s l u r r i e s ? (3) What k i n d o f c o r r e l a t i o n do the parameters of t h e model e q u a t i o n have w i t h the v a r i a b l e s of the a r t i f i c i a l s l u r r i e s ? 4-3. Systems Chosen In o r d e r to s o l v e t h e problem e f f e c t i v e l y t h e e x p e r i m e n t a l work of t h i s t h e s i s i s c a r r i e d out on model systems, i . e . , a r t i f i c a l s l u r r i e s o f r e g u l a r l y s i z e d n y l o n f i b e r s i n aqueous s o l u t i o n s o f sodium c h l o r i d e , d e x t r o s e , and p o l y e t h y l e n e g l y c o l . The L/D r a t i o of n y l o n f i b e r s was v a r i e d from 22.9 to 156. The volume f r a c t i o n o f t h e f i b e r s was v a r i e d from 0.04 to 0.17. Sodium c h l o r i d e was used to produce t h e i o n i c f o r c e , and i t s c o n c e n t r a t i o n was v a r i e d from 0.00 to 2.0 mole/1 . The c o n c e n t r a t i o n of" EEC was v a r i e d from 10 to 50 wt %. D e x t r o s e was used to a d j u s t t h e d e n s i t y o f t h e d i s p e r s i n g medium to t h a t o f the d i s p e r s e d n y l o n f i b e r s . I t was found i n an i n i t i a l s e r i e s o f e x p e r i m e n t a l runs t h a t the e f f e c t of t h e v a r i a b l e s o f t h e d i s p e r s e d phase on the time-dependent she a r f l o w was much g r e a t e r than t h a t of t h e v a r i a b l e s o f t h e d i s p e r s i n g medium. T h e r e f o r e t h e e x p e r i m e n t a l a n a l y s e s were c a r r i e d out w i t h a s i n g l e d i s p e r s i n g medium ( i . e . , DEX + 1.0 mole NaCl i n 10% PEG-H^O). A l s o , f o r p a r t i c l e L/D r a t i o s l a r g e r than 69.8 almost i d e n t i c a l e f f e c t s on t h e time-dependency were o b s e r v e d , so t h e a n a l y s i s was l i m i t e d to t h e r e s u l t s w i t h p a r t i c l e L/D r a t i o s of 22.9, 37.4, and 69.8. The d e t a i l s of the s e l e c t i o n and the make-up of t h o s e a r t i f i c i a l s l u r r i e s w i l l be g i v e n i n the next c h a p t e r . 4-4. Method of A t t a c k There a r e s e v e r a l k i n d s of v i s c o m e t e r s a v a i l a b l e on the market and (52) t h e y a r e d e s c r i b e d i n d e t a i l i n t h e l i t e r a t u r e ; f o r example, the c a p i l l a r y t y p e , c o a x i a l c y l i n d e r t y p e , and c o n e - a n d - p l a t e t y p e . F o r t h e measurement of a time-dependent shear f l o w the l a s t two types o f v i s c o m e t e r s a r e most s u i t a b l e , s i n c e the m a t e r i a l can be kept i n t h e v i s c o m e t e r s and i t s time-dependent n a t u r e can be c o n t i n u o u s l y measured no matter how l o n g t h e measurement may t a k e . S i n c e a r t i f i c i a l s l u r r i e s o f c o m p a r a t i v e l y l o n g p a r t i c l e s were used i n t h e p r e s e n t s t u d y , t h e c o n e - a n d - p l a t e t y p e was not a p p l i c a b l e because of the v e r y s m a l l space between the cone and the p l a t e . C o n s e q u e n t l y , a c o a x i a l c y l i n d e r t y p e of v i s c o m e t e r was chosen f o r t h e i n v e s t i g a t i o n . The v i s c o m e t e r measures t h e t o r q u e e x e r t e d on t h e . s u r f a c e of t h e i n n e r c y l i n d e r , r o t a t i n g a t a c o n s t a n t a n g u l a r v e l o c i t y , w i t h time. The a n g u l a r v e l o c i t y o f t h e c y l i n d e r was v a r i e d i n t h i s work from 0.116 t o 6.283 r a d . / s e c . The gap between the-two c y l i n d e r s had to be kept wide enough' t o accommodate the a r t i f i c i a l s l u r r i e s of c o m p a r a t i v e l y l o n g f i b e r s . The t h i c k n e s s of t h e f l o w i n g l a y e r o f the m a t e r i a l may v a r y w i t h time. I t was f e l t t h a t the t h i c k n e s s of f l o w i n g l a y e r would be a v e r y important f a c t o r f o r t h e time-dependent shear f l o w . T h e r e f o r e t h e t h i c k n e s s was a l s o measured as a f u n c t i o n o f time by a movie camera. The movement of t r a c e r p a r t i c l e s on t h e s u r f a c e of t h e m a t e r i a l under t e s t i n t h e v i s c o m e t e r was photographed. The r e s u l t s of t h e measurement of t h e w i d t h . o f t h e f l o w i n g s l u r r y gap w i l l p l a y a key r o l e i n s o l v i n g t h e problem d e f i n e d i n S e c t i o n 4-2. CHAPTER 5 EXPERIMENTAL MATERIALS AND TECHNIQUES The d e f i n i t i o n of t h e problem has been g i v e n i n the p r e v i o u s c h a p t e r and the e x p e r i m e n t a l method has been o u t l i n e d t h e r e . The p r e s e n t c h a p t e r d e s c r i b e s the e x p e r i m e n t a l m a t e r i a l s and t e c h n i q u e s i n d e t a i l . I n S e c t i o n 5-1 t h e m a t e r i a l s used i n the experiment w i l l be mentioned and t h e n the methods f o r making up a r t i f i c i a l s l u r r i e s w i l l be p r e s e n t e d . S e c t i o n 5-2 w i l l d e s c r i b e t h e a p p a r a t u s . A t o r q u e c a l i b r a t i o n f o r t h e a p p a r a t u s w i l l be g i v e n i n S e c t i o n 5-3. F i n a l l y , S e c t i o n 5-4 w i l l d e a l w i t h t h e e x p e r i m e n t a l p r o c e d u r e , i n c l u d i n g how t o measure the t h i c k n e s s of t h e f l o w i n g l a y e r i n the annulus of the v i s c o -meter by means of c i n e photographs. 5-1. M a t e r i a l s T h i s s e c t i o n d e s c r i b e s : 1) t h e p a r t i c l e s used as a d i s p e r s e d phase, 2) t h e l i q u i d s u t i l i z e d f o r a d i s p e r s i n g medium, 3) the t e c h n i q u e employed i n matching d e n s i t i e s of b o t h phases, and 4) how the a r t i f i c i a l s l u r r i e s were p r e p a r e d . 5-1-1. D i s p e r s e d Phase S e l e c t i o n of m a t e r i a l s f o r t h e p a r t i c l e s i s governed by t h e f o l l o w i n g p r o p e r t i e s : (1) Ease w i t h which d e s i r e d shapes and s i z e s can be c u t or punched. 39 40 (2) I n e r t n e s s , which means t h a t no c h e m i c a l r e a c t i o n t a k e s p l a c e w i t h t h e d i s p e r s i n g medium. (3) The a b i l i t y of t h e p a r t i c l e s t o be wetted by the d i s p e r s i n g medium. (4) D e n s i t y , which i s d e s i r e d t o be as c l o s e as p o s s i b l e t o t h a t o f t h e d i s p e r s i n g medium. (5) F l e x i b i l i t y , i . e . , not too r i g i d so t h a t t h e p a r t i c l e s do not damage t h e w a l l of t h e v i s c o m e t e r , and not too f l e x i b l e so t h a t they don't l o s e t h e p r o p e r t y o f t h e i r o r i g i n a l shape under a shear f l o w . Because i t s a t i s f i e s t h e c o n d i t i o n s above, n y l o n f l o c k was chosen f o r the p a r t i c l e s . The n y l o n t h r e a d s used h e r e were manufactured as Type 120 by E. I . du Pont de Nemours & Co. T h i s n y l o n i s the p r o d u c t of t h e r e a c t i o n o f c o n d e n s a t i o n p o l y m e r i z a t i o n between hexamethylene (47) diamine and a d i p i c a c i d t o an average m o l e c u l a r weight of about 25,000 and a d e n s i t y ^ 4 7 ^ o f 1.14 g/cc. The d i a m e t e r o f the t h r e a d s was u n i f o r m a t 43.1 m i c r o n s . The t h r e a d s have been c u t i n t o l e n g t h s from 0.987 mm t o 6.72 mm by F i b r e t e x L t d . , p r o d u c i n g f i v e k i n d s o f r e g u l a r s h a p e - a n d - s i z e f l o c k . The l e n g t h s o f t h e f i b e r s were c a l c u -l a t e d by a v e r a g i n g t h e measured v a l u e s o f 50 f i b e r s from m a g n i f i e d photographs which had been t a k e n t h r o u g h a c a l i b r a t e d g r a t i c u l e . T a b l e 5-1 g i v e s t h e p h y s i c a l p r o p e r t i e s o f t h e n y l o n f i b e r s . T a b l e 5-2 shows t h e l e n g t h - t o - d i a m e t e r r a t i o s o f t h e f i b e r s . A photograph o f f i v e d i f f e r e n t s i z e d n y l o n f i b e r s can be seen i n F i g . 5-1. The s i z e d f i b e r s were i n e r t and were s t o r e d i n the bags o f v i n y l p l a s t i c i n which they had been s h i p p e d . I n h a l a t i o n and s k i n c o n t a c t of t h e n y l o n f i b e r s can be a v o i d e d by c a r e f u l h a n d l i n g . TABLE 5-1 PHYSICAL PROPERTIES OF NYLON FIBER (47) D e n s i t y 1.14 g / c c M o l e c u l a r Weight 25,000 (34) Shear Modulus 1.22 x 101 ° d y n e s / c m 2 Young's M o d u l u s ( 3 4 ) 3.55 x 1010 d y n e s / c m 2 P o i s s o n ' s R a t i o ^ 3 4 ^ 0 .4 G l a s s T r a n s i t i o n T e m p e r a t u r e ^ 3 ^ 50 °C D i a m e t e r ( 4 8 ) 43.1 m i c r 0 n s TABLE 5-2 LENGTH-TO-DIAMETER RATIO OF FIBERS Sta n d a r d Length(mm) L/D D e v i a t i o n 1. 0.987 22.9 2. 1.62 37.4 3. 3.01 69.8 4. 5.03 116. 5. 6.72 156. 0.168 0.0828 0.0877 0.108 0.106 42 F i g . 5-1. Photograph of 5 D i f f e r e n t l y S i z e d F i b e r s [ s c a l e : mm] 43 5-1-2. D i s p e r s i n g Medium The s e l e c t i o n o f m a t e r i a l s f o r t h e d i s p e r s i n g medium i s r e s t r i c t e d by the r e q u i r e m e n t s t h a t i t : (1) be a Newtonian l i q u i d , (2) have v a r i a b l e v i s c o s i t y , (3) have same d e n s i t y as n y l o n p a r t i c l e s , and (4) wet n y l o n p a r t i c l e s . The d i s p e r s i n g medium chosen f o r t h e p r e s e n t work c o n s i s t s of p o l y e t h y l e n e g l y c o l , d e x t r o s e , water, and sodium c h l o r i d e . P o l y -e t h y l e n e g l y c o l (PEG) used h e r e i s one of t h e w a t e r - s o l u b l e polymers and i s manufactured by The Dow Chemical Company as P o l y g l y c o l E9000 w i t h m o l e c u l a r w e i g h t a b o u t 9,500. The v i s c o s i t y o f PEG-H^O s o l u t i o n i s v e r y dependent on i t s c o n c e n t r a t i o n . Sodium c h l o r i d e i s a w e l l -known e l e c t r o l y t e . D e x t r o s e i s a l s o s o l u b l e i n PEG-H^O-NaCl s o l u t i o n and was used i n o r d e r to a d j u s t t h e d e n s i t y of d i s p e r s i n g medium to t h a t o f n y l o n f i b e r s . T h i s PEG-H^O-NaCl-Dextrose s o l u t i o n has been found to wet t h e n y l o n f i b e r s w e l l . N e i t h e r t h e PEG nor t h e d e x t r o s e r e q u i r e s s p e c i a l c a r e i n s t o r a g e o r h a n d l i n g . The PEG i s a n o n - v o l a t i l e , n o n - t o x i c , s t a b l e s o l i d and was s t o r e d i n t h e c a r d b o a r d c a r t o n i n which i t had been s h i p p e d . The d e x t r o s e i s a l s o a n o n - t o x i c s o l i d , but i t i s h y g r o s c o p i c and so i t was s t o r e d i n g l a s s b o t t l e s . 5-1-3. Technique of D e n s i t y M a t c h i n g To a v o i d s e t t l i n g o r f l o a t i n g o f the p a r t i c l e s i n the s u s p e n s i o n s , a d e n s i t y match was c a r r i e d out i n t h i s i n v e s t i g a t i o n . The weight of an a p p r o p r i a t e amount of d e x t r o s e was measured by a b a l a n c e and was added to a 10 cc s o l u t i o n o f PEG-H 20-NaCl. The s o l u t i o n o f PEG-H 20-NaCl-44 Dextrose was mixed w e l l using a magnetic s t i r r e r . The d e n s i t y of the s o l u t i o n was measured by means of a s p e c i f i c g r a v i t y b o t t l e i n a constant temperature bath at 30.0°C. This procedure was repeated by adding d i f f e r e n t weights of dextrose to the same s o l u t i o n of PEG-R^O-NaCl. The de n s i t y of the PEG-H^O-NaCl-Dextrose s o l u t i o n was p l o t t e d against the weight of dextrose added as shown i n F i g . 5-2. From the graph the weight of dextrose, which must be added to make the d i s p e r s i n g medium de n s i t y equal to that of the nylon f i b e r s ( p = 1.14 g/cc), was d e t e r -mined. This technique was employed f o r a l l the s o l u t i o n s of PEG-R^O-NaCl-Dextrose which were prepared f o r the present study. V i s c o s i t y of the d i s p e r s i n g medium a f t e r d e n s i t y match was measured at 30.0°C by means of a Cannon-Fenske c a p i l l a r y viscometer Type 300, which had been c a l i b r a t e d by using v i s c o s i t y standard o i l s purchased from Cannon Instrument Co. F i g s . 5-3 and 5-4 show the r e s u l t s of the v i s c o s i t y measurements. The weight of dextrose added i s a l s o p l o t t e d i n these f i g u r e s . F i g . 5-5 shows e f f e c t of dextrose on the suspending medium v i s c o s i t y . I t can be concluded from these f i g u r e s that the v i s c o s i t y of the d i s p e r s i n g medium depends s t r o n g l y on the c o n c e n t r a t i o n of PEG, w h i l e n e i t h e r NaCl nor dextrose a f f e c t s the v i s c o s i t y much. 5-1-4. S l u r r y P r e p a r a t i o n Before the f i b e r s can be dispersed i n a density-matched d i s p e r s i n g medium, they must be washed to f r e e them from s a l t s and w e t t i n g agents. Approximately 50 grams of s i z e d f i b e r s were soaked i n 2 l i t e r s of d i s t i l l e d and d e i o n i z e d water. The f i b e r s were permitted to s e t t l e out i n a g e n t l y s t i r r e d , l a r g e d i s t i l l i n g f l a s k . A s m a l l stream of the 1.15 1.10 0 M a s s 4 I 2 3 D e x t r o s e ( g ) 10 c c o f 0.25 m o l e N a C l in 10%PEG H 2 0 Fig. 5-2. Density Match 46 2 0 0 1 5 0 C L o > . I 0 0 h o o t/5 > 5 0 Y 0 1 0 2 0 3 0 4 0 5 0 C o n c e n t r a t i o n o f P E G i n H 2 0 (%) F i g . 5 -3 . E f f e c t of PEG Concentration on Suspending Medium V i s c o s i t y 47 F i g . 5-4. E f f e c t o f N a C l o n S u s p e n d i n g M e d i u m V i s c o s i t y 48 W e i g h t o f D e x t r o s e A d d e d t o I 0 c c ( 3 0 % P E G - I - L O +NaCl ) (g) F i g . 5 - 5 . E f f e c t o f D e x t r o s e o n S u s p e n d i n g M e d i u m V i s c o s i t y 49 f r e s h water was i n t r o d u c e d t o t h e a r e a o f the s e t t l e d f i b e r s by i n s e r t i n g a g l a s s tube connected t o the water l i n e , w h i l e the s u p e r n a t e n t l i q u i d o v e r f l o w e d t h r o u g h a s i d e arm. The c o n d u c t i v i t y o f t h e l i q u i d was p e r i o d i c a l l y measured u s i n g d i p p i n g e l e c t r o d e s . T h i s p r o c e d u r e was c o n t i n u e d u n t i l t h e c o n d u c t i v i t y showed no f u r t h e r change. The washing and c o n d u c t i v i t y measurement were done a t room temperature. The n y l o n s l u r r y was the n f i l t e r e d i n a Buchner Ta b l e - T y p e f u n n e l . The f i l t e r e d n y l o n f i b e r s were d r i e d a t room temperature i n a vacuum d e s i c c a t o r which was connected t h r o u g h an impinger t o a vacuum pump, u n t i l no d e p o s i t due t o e v a p o r a t i o n was found on t h e s u r f a c e o f t h e i n n e r tube of t h e impinger which had been immersed i n l i q u i d n i t r o g e n . The d r i e d f i b e r s were s t o r e d i n a n o t h e r d e s i c c a t o r . As f o r t h e d i s p e r s i n g medium, t h e a p p r o p r i a t e amount o f PEG was weighed and d i s s o l v e d i n 2 l i t e r s o f d i s t i l l e d and d e i o n i z e d water t o the d e s i r e d c o n c e n t r a t i o n o f PEG-H^O. The a p p r o p r i a t e amount o f NaCl was weighed i n a 500 cc v o l u m e t r i c f l a s k . The f l a s k was the n f i l l e d w i t h t h e PEG-H 20 s o l u t i o n . The s o l u t i o n o f PEG-H 20-NaCl was mixed by a magnetic s t i r r e r f o r 30 minutes. The a p p r o p r i a t e amount o f d e x t r o s e was weighed i n a 1 - l i t e r f l a s k , i n t o which 500 cc of PEG-H^O-NaCl s o l u t i o n was poured. The s o l u t i o n o f PEG-H^O-NaCl-Dextrose was a g i -t a t e d by a magnetic s t i r r e r f o r a day and a n i g h t , u n t i l t h e f i n a l s o l u t i o n became a v e r y v i s c o u s , c l e a r , and t r a n s p a r e n t l i q u i d . A l l t h e s e p r e p a r a t i o n s were c a r r i e d out a t room temperature. To produce t h e s l u r r y t h e a p p r o p r i a t e amount of the d r i e d n y l o n f i b e r s was weighed i n a 100 cc beaker, i n t o which t h e measured volume o f th e PEG-H^O-NaCl-Dextrose s o l u t i o n was s l o w l y added through a 50 cc b u r e t t e . The n y l o n p a r t i c l e s were c o m p l e t e l y wetted by t h e s o l u t i o n and 50 the r e s u l t i n g s l u r r y was mixed g e n t l y and s l o w l y by hand u s i n g a c l e a n s t a i n l e s s s t e e l s p a t u l a . The s l u r r y was removed from t h e beaker to the s a m p l e - c o n t a i n e r of the v i s c o m e t e r which was t h e n mounted i n t o i t s measuring head, which was h e l d a t a c o n s t a n t temperature (30.0°C). The a g i n g time was measured from t h i s p o i n t . In a d d i t i o n , t h e volume of t h e s l u r r y f o r each r u n was always kept c o n s t a n t a t 37.0 cc which s e t t h e l e v e l of the s l u r r y i n t h e annulus t o the upper edge of th e i n n e r c y l i n d e r . The volume f r a c t i o n o f f i b e r s i n each s l u r r y was d e t e r m i n e d b e f o r e t h e s l u r r y was mixed. The f i b e r s a r e most e a s i l y measured by weight, the d i s p e r s i n g medium by volume. The p r o p e r weight o f f i b e r s and volume o f d i s p e r s i n g medium were, t h e r e f o r e , c a l c u l a t e d from t h e known d e n s i t y o f f i b e r s and d i s p e r s -i n g medium (p = 1.14 g / c c ) . 5-2. D e s c r i p t i o n o f Apparatus The v i s c o m e t e r used t o measure the shear s t r e s s o f t h e t i m e -dependent a r t i f i c i a l s l u r r i e s was a Haake R o t o v i s k o V i s c o m e t e r Model RV1, a r o t a t i n g bob t y p e 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 . A s c h e m a t i c diagram of t h e s e t - u p s i s shown i n F i g . 5-6. The o u t e r s t a t i o n a r y c y l i n d e r (cup) i s surrounded by a t e m p e r a t u r e -c o n t r o l l e d j a c k e t o f water. I n s i d e t h e cup i s an i n n e r c y l i n d e r (bob) o f s m a l l e r d i a m e t e r which r o t a t e s t o shear the m a t e r i a l i n the gap between cup and bob. The bob i s f i x e d t o a s h a f t e x t e n d i n g from t h e measuring head of t h e v i s c o m e t e r . T h i s s h a f t , which i s mounted on d r y - r u n n i n g r o l l e r b e a r i n g s , i s s e p a r a t e d i n t o an upper d r i v i n g p o r t i o n and a lower d r i v i n g p o r t i o n . The upper p o r t i o n i s d r i v e n by a synchronous motor through a gear box and a t r a n s m i s s i o n s h a f t . The lower s h a f t i s connected by a DRIVING UNIT MEASURING HEAD 51 RECORDER CO-AXIAL CYLINDERS TEMP CONTROL BATH (A) Thermostat jacket Rotary bob Test sample Container (outer cy l inder) Measuring Head with Measuring Outfit MV ( B ) attached F i g . 5 - 6 . I n s t r u m e n t a t i o n ( A ) a n d M e a s u r i n g H e a d ( B ) 52 t o r s i o n s p r i n g t o the upper s h a f t . F i x e d t o the! lower end of t h e upper s h a f t i s a wire-wound p o t e n t i o m e t e r w i t h t h e c o n t a c t t o t h e upper end of t h e lower s h a f t . As the two s h a f t s t u r n r e l a t i v e t o one another a g a i n s t t h e t o r s i o n s p r i n g , a r e s i s t a n c e change p r o p o r t i o n a l t o t h e a n g u l a r d i s p l a c e m e n t i s measured. The p o t e n t i o m e t e r i s connected t o a b r i d g e c i r c u i t which p r o v i d e s an e l e c t r i c a l o utput which may be r e a d from a meter on t h e c o n t r o l box of t h e v i s c o m e t e r or which may be f e d i n t o a c h a r t r e c o r d e r . There were many com b i n a t i o n s p o s s i b l e i n s i z e s o f bobs and cups f o r t h i s v i s c o m e t e r . The s e t of MVII cup and SVII bob was chosen because i t c o u l d p r o v i d e a f a i r l y wide gap between t h e bob and t h e cup f o r t h e a r t i f i c i a l s l u r r i e s o f c o m p a r a t i v e l y l o n g f i b e r s . The MVII cup had an i n n e r d i a m e t e r o f 42.0 mm and a depth of 43.0 mm w i t h t h e matching bottom made, s p e c i f i c a l l y f o r t h i s work, o f s t a i n l e s s s t e e l . The SVII bob had an o u t e r diameter of 20.2 mm and a h e i g h t o f 19.6 mm. The r e s u l t i n g gap i n which the s l u r r y was sheared was 10.9 mm. I t was s u f f i c i e n t l y wide f o r t h e c u r r e n t a r t i f i c i a l s l u r r i e s because t h e l o n g e s t f i b e r s used i n t h i s work were 6.72 mm. The c o n f i g u r a t i o n o f t h e s e c y l i n d e r s i s s c h e m a t i c a l l y shown i n F i g . 5-7. To m i n i m i z e end e f f e c t s a c l e a r a n c e o f 8.4 mm between t h e bottoms of two c y l i n d e r s was l e f t to p r o v i d e an un-sheared r e s e r v o i r of m a t e r i a l . B e s i d e s t h i s , the bob i s h o l l o w on t h e lower end so an a i r b u b b l e was t r a p p e d under the bob and o n l y t h e sharp edge o f the bottom was i n c o n t a c t w i t h t h e m a t e r i a l t o be t e s t e d . To p r e v e n t s l i p a t the walls,, t h e s u r f a c e s o f b o t h t h e cup and t h e bob had grooves i n t h e a x i a l d i r e c t i o n . The d i m e n s i o n of t h e s e grooves i s shown i n F i g . 5-8. 53 F i g . 5 -7 . C o n f i g u r a t i o n o f C y l i n d e r s 54 0.19mm 2TT/38 rad (9 .474°) 0.38 mm 2?r /90 ( 4 . 0 0 e ) F i g . 5-8. G r o o v e s o n t h e C y l i n d e r s In a d d i t i o n , t h e bob and the cup were g o l d - p l a t e d t o p r o t e c t the s u r f a c e s from c o r r o s i o n due to the s a l t used i n the a r t i f i c i a l s l u r r i e s . The Haake R o t o v i s k o V i s c o m e t e r , as purchased had a f l e x i b l e c a b l e f o r t r a n s m i t t i n g t h e d r i v i n g power from t h e motor t o t h e measuring head. (17) (29) T h i s c a b l e - d r i v e , however, caused o s c i l l a t i o n s ' d u r i n g s t e a d y o p e r a t i o n , and t h e r e f o r e t h e d r i v i n g system was m o d i f i e d t o a s h a f t -d r i v e . The v i s c o m e t e r w i t h t h e p r e s e n t c o m b i n a t i o n o f c y l i n d e r s has an apparent shear r a t e range o f 0.101 to 16.3 sec ^ and a s h e a r s t r e s s 2 range of 45 to 6Q,000 dynes/cm . The a n g u l a r v e l o c i t y o f t h e i n n e r c y l i n d e r i s c o n t r o l l e d through a gear box. T a b l e 5-3 shows the gear p o s i t i o n and t h e c o r r e s p o n d i n g a n g u l a r v e l o c i t y o f t h e i n n e r c y l i n d e r . The t o r q u e e x e r t e d on the s u r f a c e of t h e i n n e r c y l i n d e r was r e c o r d e d by a Hewlett P a c k a r d Moseley S t r i p C hart Recorder Model 7100B. To p r e v e n t e v a p o r a t i o n of t h e m a t e r i a l d u r i n g t h e p e r i o d of a g i n g i n s i d e the measuring head a t 30.0°C, a t i g h t p l e x i g l a s s l i d was con-s t r u c t e d . T h i s l i d was always i n p l a c e d u r i n g a g i n g and was removed when the decay experiment was s t a r t e d . 5-3. Torque C a l i b r a t i o n The t o r q u e which was r e c o r d e d as a v o l t a g e (mV), was c a l i b r a t e d by g e n e r a t i n g a known f o r c e on t h e s u r f a c e o f the i n n e r c y l i n d e r w i t h a s t r i n g w hich was f i x e d t o t h e s u r f a c e of the i n n e r c y l i n d e r and which was connected to a number of known w e i g h t s . The s t r i n g was s u p p o r t e d by a wheel whose b e a r i n g - f r i c t i o n was c o n s i d e r e d t o be n e g l i g i b l e . The e l e c t r i c a l o u t p u t s a s s o c i a t e d w i t h t o r q u e s g e n e r a t e d i n t h i s manner were r e a d on t h e r e c o r d e r and p l o t t e d a g a i n s t t h e known t o r q u e s . The TABLE 5 - 3 ANGULAR VELOCITY OF INNER CYLINDER Gear P o s i t i o n A n g u l a r V e l o c i t y ( r a d / s e c ) 162 TT 0.0388 o l 81 -|- TT 0.0775 o l 54 YJ TT 0.116 27 TT 0.233 18 "g~ TT 0.349 9 | TT 0.698 6 j TT 1.05 3 -| TT 2.09 2 1 TT 3.14 1 2 TT 6.28 p l o t i s a s t r a i g h t l i n e t h r o u g h t h e o r i g i n as shown i n F i g . 5-9. Hence t h e f o l l o w i n g e q u a t i o n was o b t a i n e d : (Torque i n dyne.cm) = (4.46 x I O 4 ) x ( V o l t a g e i n mV). 5-4. E x p e r i m e n t a l P r o c e d u r e L i s t e d below i s t h e s t e p - b y - s t e p program f o l l o w e d i n a l l e x p e r i -m e n t a l r u n s : (1) The v i s c o m e t e r was t u r n e d on ( w i t h gear box i n n e u t r a l ) . The c h a r t r e c o r d e r was t u r n e d on. A f i f t e e n - m i n u t e warm-up p e r i o d was a l l o w e d f o r t h e e l e c t r o n i c s . The w a t e r c i r c u l a t o r was s e t t o 30.0°C and t h e n t u r n e d on. The bob was mounted on t h e m e a s u r i n g head s h a f t . (2) A f t e r e l e c t r i c a l warm-up, t h e gear was s h i f t e d t o t h e p o s i t i o n 27 t o a l l o w t h e bob t o r o t a t e f r e e l y i n a i r . Then, t h e gear was s h i f t e d t o n e u t r a l and t h e t o r q u e z e r o p o i n t was c a l i b r a t e d . (3) A q u a n t i t y of newly p r e p a r e d a r t i f i c i a l s l u r r y was p l a c e d i n t h e cup. The cup was t h e n f a s t e n e d i n p o s i t i o n and t h e l i d was put i n p l a c e t o p r e v e n t e v a p o r a t i o n . The s l u r r y was aged f o r t h e d e s i r e d p e r i o d a t 30°C. (4) A f t e r a g i n g , t h e v i s c o m e t e r and t h e r e c o r d e r were a l l o w e d t o be warmed-up a g a i n . (5) The l i d was removed and t h e gear was s h i f t e d t o t h e d e s i r e d p o s i t i o n t o s t a r t t h e decay e x p e r i m e n t w i t h t h e r e c o r d e r c h a r t speed o f 0.1 i n c h / s e c . (6) A f t e r 5 m i n u t e s t h e c h a r t speed was slowed t o 0.1 i n c h / m i n . i n o r d e r t o r e c o r d t h e e q u i l i b r i u m v a l u e of t o r q u e . The i n s t r u m e n t s were l e f t r u n n i n g u n t i l t h e r e was no d e t e c t a b l e change o f t o r q u e i n t h e s c a l e r e a d i n g o v e r a 5-minute i n t e r v a l . 58 5 0 0 2 4 6 8 10 Voltage (mV) F i g . 5 - 9 . C a l i b r a t i o n C u r v e o f T o r q u e v e r s u s V o l t a g e (7) The gear was then switehed to a h i g h e r speed and t h e e q u i l i b r i u m v a l u e o f t o r q u e a t t h i s speed was r e c o r d e d . T h i s p r o c e d u r e was r e p e a t e d u n t i l the gear p o s i t i o n o f 1 was r e a c h e d . The f o l l o w i n g p r o c e d u r e s were added f o r the measurement of the t h i c k n e s s of t h e f l o w i n g l a y e r o f t h e s l u r r y : (1) A f t e r t h e l i d was removed, as mentioned above, a s m a l l amount of t r a c e r p a r t i c l e s ( r e d n y l o n f i b e r s about 1 mm l o n g ) was p l a c e d i n a l i n e r a d i a l l y a c r o s s the s u r f a c e o f the s l u r r y . (2) An i l l u m i n a t o r , a m i r r o r , and a movie camera (Braun N i z o , Type S56) were s e t i n p l a c e . (3) Photographs were th e n t a k e n a t t h e approximate r a t e o f one p i c t u r e per 1.5 seconds, and the time o f each photograph was manually marked w i t h t h e event marker pen on t h e same c h a r t b e i n g used to r e c o r d the t o r q u e . (4) While t h e photographs were b e i n g t a k e n , t h e decay measurement was s t a r t e d by s h i f t i n g the gear from n e u t r a l to a d e s i r e d p o s i t i o n and the p r o c e d u r e s o u t l i n e d b e f o r e were f o l l o w e d . To c l o s e t h i s c h a p t e r i t s h o u l d be mentioned t h a t over f u l l range o f r o t a t i o n a l speeds used i n t h e s e s t u d i e s , c a l c u l a t i o n s showed t h a t o n l y s t a b l e l a m i n a r f l o w s h o u l d o c c u r i n t h e gap and t h a t t h e r e were no T a y l o r i n s t a b i l i t i e s . CHAPTER 6 EXPERIMENTAL RESULTS AND DISCUSSION The v a r i a b l e s and t h e i r ranges i n v e s t i g a t e d i n t h i s s t u d y a r e summarized i n T a b l e 6-1. As shown t h e r e , t h e i n f l u e n c e of s e v e r a l s l u r r y v a r i a b l e s (independent v a r i a b l e s ) on shear s t r e s s and t h i c k n e s s o f f l o w i n g l a y e r (dependent v a r i a b l e s ) were measured. I t s h o u l d be n oted t h a t t h e l a t t e r v a r i a b l e s a r e b o t h dependent on arid a f u n c t i o n of time. The computer output f o r a l l r e s u l t s i s g i v e n i n Appendix D. 6-1. R e p r o d u c i b i l i t y Measurements o f t h e y i e l d v a l u e o f t o r q u e and th e e q u i l i b r i u m v a l u e of t h e t h i c k n e s s o f f l o w i n g l a y e r were r e p e a t e d f i v e t imes to check t h e r e p r o d u c i b i l i t y of t h e s e v a l u e s f o r n ewly-prepared samples. Measured y i e l d t o r q u e was found to be r e p r o d u c i b l e w i t h i n ±12% o f t h e average v a l u e , w h i l e the t h i c k n e s s of the l a y e r was o b t a i n e d w i t h i n ±7%. The r e l a t i v e l y l a r g e e r r o r a s s o c i a t e d w i t h the. measurement of y i e l d t o r q u e was c o n s i d e r e d to be i n h e r e n t . Y i e l d t o r q u e i s the f o r c e needed to i n i t i a t e the f l o w o f a g e l l e d m a t e r i a l . What happens i s t h a t the i n n e r c y l i n d e r of t h e v i s c o m e t e r i s suddenly d e s t r o y i n g the s t r u c -t u r e of t h e g e l . In such a phenomenon the r e p r o d u c i b i l i t y i s u s u a l l y poor. 60 61 TABLE 6-1 VARIABLES AND RANGES INVESTIGATED INDEPENDENT VARIABLES I. D i s p e r s e d Phase F i b e r L e n g t h (mm) 0.987 - 6.72 L/D R a t i o ( - ) 22.9 - 156 F i b e r C o n c e n t r a t i o n ( c c / c c ) 0.04 - 0.17 I I . D i s p e r s i n g Medium V i s c o s i t y ( C P . ) 7.45 - 220 C o n c e n t r a t i o n o f NaCl (mole/£) 0- • - 2.0 I I I . Flow Parameter A n g u l a r V e l o c i t y o f C y l i n d e r ( r a d . / s e c . ) 0.116 - 6.28 DEPENDENT VARIABLES I. Shear S t r e s s (dyne/cm 2) 7.10X101- 4.62><10'+ I I . T h i c k n e s s of F l o w i n g L a y e r (cm) 0 - 0.83 62 6-2. E f f e c t o f S l u r r y Age on Y i e l d S t r e s s The r h e o l o g i c a l time-dependent n a t u r e of a s u s p e n s i o n i s g e n e r a l l y c h a r a c t e r i z e d i n two p a r t s . One i s s t r u c t u r e b u i l d - u p , and the o t h e r i s s t r u c t u r e break-down. In o r d e r to examine the s t r u c t u r e b u i l d - u p on s t a n d i n g , an a g i n g t e s t was c a r r i e d out by measuring the y i e l d s t r e s s f o r v a r i o u s aged samples at a c o n s t a n t temperature of 30°C. The a g i n g time of the newly-prepared samples was v a r i e d from 0. to 32. h o u r s . A l l t h e samples c o n s i s t e d of d i s p e r s e d f i b e r s whose L/D r a t i o was 37.4 a t a p a r t i c l e c o n c e n t r a t i o n o f 0.12 ( c c / c c ) i n a d i s p e r s i n g medium o f d e x t r o s e + 1 . 0 mole NaCl i n 30% PEG-H^O. A n g u l a r v e l o c i t y o f t h e i n n e r c y l i n d e r was 0.233 r a d . / s e c . The r e s u l t s o f t h e measurements were p l o t t e d on a graph of y i e l d s t r e s s v e r s u s a g i n g time as shown i n F i g . 6-1. I t can be c o n c l u d e d from the graph t h a t the s t r u c t u r a l b u i l d - u p measured i n terms of y i e l d s t r e s s i s a m o n o t o n i c a l l y i n c r e a s i n g f u n c t i o n of a g i n g time and has an a s y m p t o t i c a l v a l u e a t t = t . T h i s c o n c l u s i o n agrees w i t h B i l l i n g t o n and H u x l e y a n d H u ^ 7 ^ w i t h b e n t o n i t e c l a y i n (39) (55) water, P e t e r w i t h g r a p h i t e i n water, and Watanabe and T a k a s a k i w i t h a vanadium p e n t o x i d e s o l . From the graph, the a g i n g time needed f o r t h e e x p e r i m e n t a l work of t h i s t h e s i s was determined to be 20 hours. 6-3. Decay Curve and T h i c k n e s s of F l o w i n g L a y e r A t y p i c a l t orque-decay c u r v e on t h e r e c o r d e r c h a r t i s shown i n F i g . 6-2. A f t e r a g i n g t h e sample f o r 20 hours i n the annulus of th e v i s c o m e t e r a t a c o n s t a n t temperature of 30°C the gear to r o t a t e t h e i n n e r c y l i n d e r was s h i f t e d from n e u t r a l to t h e p o s i t i o n 27 a t p o i n t A. The i n n e r c y l i n d e r a c t u a l l y began t o r o t a t e a t p o i n t B, a t which p o i n t t h e t i m i n g f o r the decay measurement was s t a r t e d . The v a l u e o f t h e F i g . 6 - 1 . E f f e c t o f A g e o f S l u r r y o n Y i e l d S t r e s s (B) ( £ = 0 . 1 5 , L = 0.987(mm), L /D=22.9 0 10 20 30 40 50 60 T i m e ( s e c . ) F i g . 6-2. T y p i c a l D e c a y C u r v e 65 sharp peak at p o i n t B (at t = 0) i s the y i e l d t o r q u e . The a c t u a l t o r q u e decay r e c o r d i n g i s rougher than F i g . 6-2 which i s a smoothed c u r v e . The c u r v e d e c r e a s e d m o n o t o n i c a l l y w i t h time and approached a s y m p t o t i c a l l y the e q u i l i b r i u m v a l u e of t o r q u e shown at p o i n t C (at t = t j . P r a c t i c a l l y s p e a k i n g , the e q u i l i b r i u m t o r q u e was o b t a i n e d i n 30 to 40 minutes i n the p r e s e n t e x p e r i m e n t s . The t h i c k n e s s of t h e f l o w i n g l a y e r was measured w i t h time d u r i n g the decay experiment by u s i n g a movie camera to r e c o r d the movement of t r a c e r p a r t i c l e s on t h e s u r f a c e of t h e m a t e r i a l i n t h e annulus of the v i s c o m e t e r . A f t e r the r o l l o f the 8 mm f i l m was d e v e l o p e d , the cine* photograph was p r o j e c t e d on a l a r g e s c r e e n where the t h i c k n e s s of t h e f l o w i n g l a y e r was measured by, a r u l e r . A t y p i c a l r e s u l t i s shown i n F i g . 6-3. The o r d i n a t e l a b e l l e d as R (cm) i s t h e v a l u e measured from the c e n t e r of t h e i n n e r c y l i n d e r . T h e r e f o r e , (R - R), where R i s t h e r a d i u s of the i n n e r c y l i n d e r , i s the a c t u a l t h i c k n e s s of f l o w i n g l a y e r . The R x v a l u e monotonously i n c r e a s e d from 1.01 (= R) a t t = 0 and approached a s y m p t o t i c a l l y the e q u i l i b r i u m v a l u e a t t = t ^ . These v a l u e s a t t = 0 and t = t a r e marked w i t h the symbol (+) i n t h e f i g u r e . The approach of R^ . t o t h e e q u i l i b r i u m v a l u e was f a s t e r f o r a l l t h e s l u r -r i e s measured i n t h i s work than t h a t of t h e t o r q u e to i t s e q u i l i b r i u m v a l u e . R w i l l be f u r t h e r d i s c u s s e d i n the next c h a p t e r , x 6-4. E f f e c t of NaCl C o n c e n t r a t i o n on Y i e l d S t r e s s and on E q u i l i b r i u m S t r e s s In F i g . 6-4 a p l o t o f y i e l d s t r e s s v e r s u s c o n c e n t r a t i o n of NaCl i s shown, where the p a r t i c l e c o n c e n t r a t i o n i s tj) = 0.14 ( c c / c c ) , p a r t i c l e L/D r a t i o i s L/D = 37.4, and d i s p e r s i n g medium i s DEX + N a Cl i n 30% 9 L / D = 3 7 . 4 (£> = 0.12 ( c c / c c ) D E X + I .Omole NaCl i n l O % P E G - H 2 0 X x x x x * X X Angular Velocity '.0.2 3 3 (rad / sec) X o ' i 1 1 1 1 1 1 -3 .28 2.343 7.966 13.589 19.21! 2A.63A 36 T J M E ( S E C ) F i g . 6 - 3 . T h i c k n e s s o f F l o w i n g L a y e r 67 T 1 r Mole-MaCl / Liter F i g . 6-4. E f f e c t o f N a C l C o n c e n t r a t i o n o n Y i e l d S t r e s s 68 PEG-H^O. The y i e l d s t r e s s o f t h e aged a r t i f i c i a l s l u r r y depends s t r o n g l y on t h e s a l t c o n c e n t r a t i o n o f t h e d i s p e r s i n g medium. The h i g h e r t h e s a l t c o n c e n t r a t i o n , the h i g h e r i s t h e y i e l d s t r e s s . T h i s r e s u l t a g r e e s w i t h H u ^ 7 ^ (on b e n t o n i t e c l a y s u s p e n s i o n w i t h NaCl) and (37) P a r k e t al. (on b e n t o n i t e c l a y s u s p e n s i o n w i t h K C l ) . The y i e l d s t r e s s i s a measure o f t h e s t r e n g t h o f the g e l s t r u c t u r e , so i t can be co n c l u d e d t h a t the i o n i c f o r c e o f the s a l t i s one o f t h e f o r c e s r e s p o n s -i b l e f o r t h e g e l s t r u c t u r e o f the s l u r r i e s . A p l o t o f e q u i l i b r i u m s t r e s s v e r s u s c o n c e n t r a t i o n o f NaCl i s shown i n F i g . 6-5 f o r the same s l u r r i e s used i n F i g . 6-4. The e q u i l i b r i u m s t r e s s a l s o depends on t h e s a l t c o n c e n t r a t i o n . The h i g h e r t h e s a l t c o n c e n t r a t i o n , t h e h i g h e r i s the e q u i l i b r i u m s t r e s s . 6-5. E f f e c t o f D i s p e r s i n g Medium V i s c o s i t y on Y i e l d S t r e s s F i g . 6-6 shows a graph o f y i e l d s t r e s s v e r s u s p a r t i c l e c o n c e n t r a -t i o n w i t h a t h i r d parameter o f d i s p e r s i n g medium v i s c o s i t y f o r t h e p a r t i c l e L/D r a t i o o f 22.9 and NaCl c o n c e n t r a t i o n 1.0 mole. I t can be seen t h a t , a t a f i x e d p a r t i c l e c o n c e n t r a t i o n , t h e y i e l d s t r e s s d e c r e a s e s w i t h i n c r e a s i n g d i s p e r s i n g medium v i s c o s i t y . T h i s may appear u n u s u a l ; however, t h e same t r e n d was observed f o r a l l the o t h e r a r t i f i c i a l s l u r r i e s w i t h t h e p a r t i c l e L/D r a t i o s o f 37.4, 69.8, 116., and 156. These graphs can be found i n Appendix B. I t was presumed t h a t the d i s p e r s i n g medium v i s c o s i t y would p l a y an imp o r t a n t r o l e i n t h e s t r e n g t h o f t h e g e l s t r u c t u r e . However, PEG, which was used t o g i v e a wide range o f d i s p e r s i n g medium v i s c o s i t y , i s an o r g a n i c s u r f a c t a n t whose e f f e c t on g e l s t r e n g t h was found t o a c t i n t h e o p p o s i t e way to N a C l . T h e r e f o r e the h i g h e r c o n c e n t r a t i o n o f PEG 6 9 CM e to >» CO CO 0> £ =3 3 c r 4 1 1 i i Particle: = 0.14 L/D =374 Dispersing Medium-. O Dextrose+ 30% PEG- H 2 ° :/ 0.5 1.0 1.5 2.0 M o l e - N a C l / L i t e r F i g . 6-5. E f f e c t o f N a C l C o n c e n t r a t i o n on E q u i l i b r i u m S t r e s s 1 1 1 r 1 1 1 r Particle L / D = 22.9 0.12 0.14 0.16 0.18 P a r t i c l e C o n c e n t r a t i o n cf) ( c c / c c ) F i g . 6-6. E f f e c t o f D i s p e r s i n g M e d i u m V i s c o s i t y o n Y i e l d S t r e s s c a n c e l s out more s t r o n g l y t h e e f f e c t of NaCl. U n f o r t u n a t e l y , t h e d e t a i l e d mechanism of n e i t h e r r e a c t i o n nor i n t e r a c t i o n between PEG and NaCl i s known. I t i s c e r t a i n , however, t h a t ' t h e e f f e c t s o f b o t h i o n i c f o r c e s and s u r f a c t a n t s c o u l d a c t not o n l y on p a r t i c l e - l i q u i d i n t e r a c t i o n but a l s o on p a r t i c l e - p a r t i c l e i n t e r a c t i o n , a l t h o u g h t h e e f f e c t o f d i s p e r s -i n g medium v i s c o s i t y c o u l d a c t mer e l y on p a r t i c l e - l i q u i d i n t e r a c t i o n . Whatever t h e s t r u c t u r e of t h e g e l l e d m a t e r i a l may be, i t i s r e a s o n a b l e t o c o n c l u d e t h a t t h e p a r t i c l e - p a r t i c l e i n t e r a c t i o n i s more dominant f o r the a r t i f i c i a l s l u r r i e s t h a n the p a r t i c l e - l i q u i d i n t e r a c t i o n . 6-6. E f f e c t s o f P a r t i c l e C o n c e n t r a t i o n and P a r t i c l e L/D R a t i o on Y i e l d S t r e s s F i g . 6-7 and 6-8 a r e graphs of y i e l d s t r e s s a g a i n s t p a r t i c l e c o n c e n t r a t i o n w i t h a parameter o f p a r t i c l e L/D r a t i o . I t can be seen t h a t the h i g h e r t h e p a r t i c l e c o n c e n t r a t i o n , the h i g h e r i s t h e y i e l d s t r e s s . T h i s r e s u l t agrees w i t h Braune and R i c h t e r ^ \ Ree and E y r i n g ^ 4 2 and Weymann^^. I t can a l s o be seen t h a t t h e h i g h e r t h e L/D r a t i o , t h e h i g h e r t h e 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 seems from F i g . 6-8 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 o f t h e maximum y i e l d s t r e s s independent of p a r t i c l e l e n g t h , s i n c e t h e p o i n t s of the L/D r a t i o g r e a t e r t h a n 37.4 l i e on a s i n g l e c u r v e ( i . e . , L/D = 69.8, 116, and 156). 6-7. E f f e c t s o f P a r t i c l e C o n c e n t r a t i o n and P a r t i c l e L/D R a t i o on E q u i l i b r i u m S t r e s s Graphs o f e q u i l i b r i u m s t r e s s a g a i n s t p a r t i c l e c o n c e n t r a t i o n w i t h a parameter of p a r t i c l e L/D r a t i o a r e shown i n F i g . 6-9 and 6-10. The h i g h e r t h e p a r t i c l e c o n c e n t r a t i o n , t h e h i g h e r i s the e q u i l i b r i u m s t r e s s . As w e l l , the h i g h e r t h e L/D r a t i o , the h i g h e r i s t h e e q u i l i b r i u m s t r e s s . 4 0 O L/D =22.9 A 37.4 h • 69.8 CM e C >» TO O 35 >-20 DEX+ 1.0 mole NaCl in 10% P E G - H 2 0 Angular Velocity.-. 0.233 (rad/sec) A 0.5 0.10 0.15 V o l u m e F r a c t i o n o f F i b e r s cf> ( c c / c c ) F i g . 6 - 7 . E f f e c t s o f C o n c e n t r a t i o n a n d L/D R a t i o o f P a r t i c l e o n Y i e l d S t r e s s tS3 0 0.05 0.10 0.15 P a r t i c l e C o n c e n t r a t i o n <f> ( c c / c c ) F i g . 6-8. E f f e c t s o f C o n c e n t r a t i o n a n d L/D R a t i o o f P a r t i c l e o n Y i e l d S t r e s s DEX+ 1.0 mole NaCl in 10% P E G - H 2 0 Angular Velocity 0.233 (rac^ec) O 6 000 3 0 0 0 0 O L /D =22.7 A 374 • 69.8 _ j • 0.05 0.10 0.15 V o l u m e F r a c t i o n o f F i b e r s <jf> ( c c / c c ) F i g . 6-9. E f f e c t s o f C o n c e n t r a t i o n a n d L/D R a t i o o f P a r t i c a l o n E q u i l i b r i u m S t r e s s . , , , — . Dispersing Medium: Dextrose +1.0 mole NaCl in 30% PEG-H2O O L /D=22 .9 • A 37.4 • 69.8 ® 116. ^ 000 0.05 0.10 0.15 P a r t i c l e C o n c e n t r a t i o n <f> ( c c / c c ) F i g . 6 - 1 0 . E f f e c t s o f C o n c e n t r a t i o n a n d L/D R a t i o o f P a r t i c l e o n E q u i l i b r i u m S t r e s s T h e f o r m e r r e s u l t a g r e e s w i t h B r a u n e a n d R i c h t e r ^ , R e e a n d E y r i n g ^ 4 ^ , B r o w n ^ ^ , B r o w n a n d P i n d e r ^ \ a n d W e y m a n n ^ ^ . T h e l a t t e r r e s u l t i s i n a g r e e m e n t w i t h t h e f i n d i n g s o f B r o w n ^ ^ , a n d B r o w n a n d P i n d e r ^ ^ o n t h e i r p a r t i c l e L/W r a t i o . 6-8. E f f e c t o f A n g u l a r V e l o c i t y o f I n n e r C y l i n d e r o n Y i e l d S t r e s s F i g . 6-11 s h o w s a p l o t o f y i e l d s t r e s s v e r s u s a n g u l a r v e l o c i t y o f t h e i n n e r c y l i n d e r . Y i e l d s t r e s s d e c r e a s e s w i t h a n i n c r e a s e o f a n g u l a r v e l o c i t y , w h i c h a g r e e s w i t h P i n d e r ^ 4 " ^ , a n d B r o w n . T h e d e p e n d e n c e o f a n g u l a r v e l o c i t y o n y i e l d s t r e s s s u g g e s t s t h a t i n o r d e r t o c o m p a r e t h e y i e l d s t r e s s o f v a r i o u s t h i x o t r o p i c m a t e r i a l s , a s t a n d a r d m e t h o d f o r t h e m e a s u r e m e n t o f t h e y i e l d s t r e s s i s r e q u i r e d i n t h e f i e l d o f t i m e -d e p e n d e n t r h e o l o g y . U n f o r t u n a t e l y , t h i s h a s n o t b e e n a t a l l e s t a b l i s h e d a t t h e p r e s e n t t i m e . 6-9. E f f e c t s o f P a r t i c l e C o n c e n t r a t i o n , P a r t i c l e L/D R a t i o , a n d A n g u l a r V e l o c i t y o n E q u i l i b r i u m T h i c k n e s s o f F l o w i n g L a y e r F i g . 6-12 i s a g r a p h o f e q u i l i b r i u m t h i c k n e s s o f f l o w i n g l a y e r v e r s u s p a r t i c l e c o n c e n t r a t i o n . T h e e q u i l i b r i u m t h i c k n e s s i s e x p r e s s e d i n t e r m s o f (R - R ) / ( R _ - R ) o n t h e o r d i n a t e o f t h e g r a p h , w h e r e R i s x » O0 0 Xoo t h e e q u i l i b r i u m t h i c k n e s s o f t h e f l o w i n g l a y e r i n t h e a n n u l u s o f t h e v i s c o m e t e r m e a s u r e d f r o m t h e c e n t e r o f t h e c y l i n d e r , R i s r a d i u s o f t h e i n n e r c y l i n d e r (R = 1.01 c m ) , a n d R^ i s r a d i u s o f t h e o u t e r c y l i n d e r (RQ = 2 . 1 0 c m ) . F r o m t h e g r a p h i t c a n b e s e e n t h a t : ( 1 ) T h e l o w e r t h e p a r t i c l e c o n c e n t r a t i o n , t h e g r e a t e r i s t h e t h i c k n e s s o f t h e f l o w i n g l a y e r . ( 2 ) T h e s m a l l e r t h e p a r t i c l e L/D r a t i o , t h e g r e a t e r i s t h e t h i c k n e s s o f t h e f l o w i n g l a y e r . CM E O 12,000 CO m U> 6 , 0 0 0 <o (/> o >-O L / D = 22.7 , 0=0.14 DEX + I.O mole NqCI in 1 0 % P E G - H 2 0 O • O . 0.2 0-4 0.6 A n g u l a r V e l o c i t y ( r a d / s e c ) F i g . 6-11. E f f e c t o f A n g u l a r V e l o c i t y o f I n n e r C y l i n d e r o n Y i e l d S t r e s s cr i 8 X I cr cr I o r r 0.7 0.6 i 1 O Angular Velocity 6.28 ( r a d / s e c ) B 2 .09 A 0 .698 © Q 2 3 3 ( 78 o 0.5 ®^ o. n ^ ^ B V A ^ B . \ A A V 0.4 L/D=69.8 L/D=374 A A A A \ v-L/D = 22.9 DEX+ 1.0 mole NaCl in 10% PEG-H^O 0.3 0.05 V o l u m e 0.10 0.15 i b e r s <f> ( c c / c c ) F i g . 6-12. E f f e c t s o f P a r t i c l e C o n c e n t r a t i o n , P a r t i c l e L/D R a t i o , and A n g u l a r V e l o c i t y on E q u i l i b r i u m T h i c k n e s s o f F l o w i n g L a y e r 79 (3) T h e h i g h e r t h e a n g u l a r v e l o c i t y , t h e g r e a t e r i s t h e t h i c k n e s s o f t h e f l o w i n g l a y e r . R e s u l t s (1) a n d (2) i n d i c a t e t h a t t h e h i g h e r p a r t i c l e c o n c e n t r a t i o n a n d t h e l a r g e r p a r t i c l e L/D r a t i o r e s u l t i n a s t r o n g e r g e l s t r u c t u r e o f t h e s l u r r y . R e s u l t (3) i n d i c a t e s t h a t t h e h i g h e r a n g u l a r v e l o c i t y o f t h e i n n e r c y l i n d e r d e s t r o y s t h e g e l s t r u c t u r e i n t h e a n n u l u s f u r t h e r t o w a r d t h e o u t e r c y l i n d e r . T h e r a t e o f i n c r e a s e o f w i t h t i m e i s i m p o r t a n t i n c o n n e c t i o n w i t h t h e s t r u c t u r a l d e c a y o f t h e t i m e - d e p e n d e n t s l u r r y a n d w i l l b e d i s c u s s e d i n t h e n e x t c h a p t e r . CHAPTER 7 ANALYSES The p r e v i o u s c h a p t e r d i s c u s s e d the e x p e r i m e n t a l r e s u l t s r e g a r d i n g m a i n l y the e f f e c t s o f v a r i o u s independent v a r i a b l e s on shear s t r e s s x (a dependent v a r i a b l e ) . T h i s c h a p t e r d e a l s w i t h a n o t h e r dependent v a r i a b l e , i . e . , t h e t h i c k n e s s o f f l o w i n g l a y e r R^. A new model to p r e d i c t R^ i s c o n s t r u c t e d i n S e c t i o n 7-1 a p p l y i n g the c o n c e p t s o f r a t e p r o c e s s e s . The o r d e r s of r e a c t i o n i n the model a r e d i s c u s s e d i n S e c t i o n 7-2. Two f i t t i n g parameters o f the model a r e a n a l y z e d i n S e c t i o n 7-3. The e x p e r i m e n t a l range o f independent v a r i -a b l e s was narrowed down to the p a r t i c l e L/D r a t i o s of 22.9, 37.4, and 69.8, and the s i n g l e d i s p e r s i n g medium of d e x t r o s e + 1.0 mole NaCl i n 10% PEG-H2O. A comparison o f t h e e x p e r i m e n t a l d a t a w i t h the model and e f f e c t s of t h e independent v a r i a b l e s on two f i t t i n g parameters of the model a r e g i v e n i n S e c t i o n s 7-4 and 7-5. R h e o l o g i c a l c o n s i d e r a t i o n s o f R^ i n terms o f shear s t r e s s , apparent shear r a t e a t the w a l l , apparent v i s c o s i t y and f l o w c u r v e a r e b r i e f l y mentioned i n S e c t i o n 7-6. S e c t i o n 7-7 d i s c u s s e s the p r a c t i c a l use of the model c o n s t r u c t e d i n t h i s c h a p t e r . 7-1. A Model P r e d i c t i n g T h i c k n e s s of F l o w i n g L a y e r I t has been shown i n F i g . 6-3 t h a t the t h i c k n e s s of f l o w i n g l a y e r o f t h e time-dependent a r t i f i c i a l s l u r r y i n t h e annulus o f t h e 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 i n c r e a s e s m o n o t o n i c a l l y w i t h time and approaches an 80 e q u i l i b r i u m v a l u e a s y m p t o t i c a l l y . T h e o r i g i n o f t h i s p h e n o m e n o n l i e s i n t h e f a c t t h a t t h e m a t e r i a l i t s e l f i s s h e a r - a n d t i m e - d e p e n d e n t ; t h e i n t e r n a l s t r u c t u r e o f t h e m a t e r i a l v a r i e s w i t h t h e d u r a t i o n o f s h e a r . T h e p h e n o m e n o n c a n b e e x p l a i n e d i n t e r m s o f o n e o f t h e r a t e p r o c e s s e s , a r e v e r s i b l e t r a n s f o r m a t i o n o f t h e s t r u c t u r e b e t w e e n t h e g e l - s t r u c t u r e a t o n e e n d a n d t h e s o l - s t r u c t u r e a t t h e o t h e r . I f t h e p h e n o m e n o n i s a r a t e p r o c e s s , t h e c l a s s i c a l t h e o r y o f c h e m i c a l r e a c t i o n r a t e s m i g h t b e u s e f u l i n d e a l i n g w i t h t h e p r o b l e m s a s s o c i a t e d w i t h i t . I n t h e i r f a m o u s t e x t b o o k T h e T h e o r y o f R a t e ( 1 2 ) P r o c e s s e s , G l a s s t o n e , L a i d l e r , a n d E y r i n g e m p h a s i z e d : . . . t h e t h e o r y o f a b s o l u t e r e a c t i o n r a t e i s n o t m e r e l y a t h e o r y o f t h e k i n e t i c s o f c h e m i c a l r e a c t i o n s ; i t i s o n e t h a t c a n , i n p r i n c i p l e , b e a p p l i e d t o a n y p r o c e s s i n v o l v i n g a r e a r r a n g e m e n t o f m a t t e r , t h a t i s t o s a y , t o a n y " r a t e p r o c e s s " . A s a m a t t e r o f f a c t , many r e s e a r c h e r s h a v e made u s e o f t h e c o n c e p t s o f c h e m i c a l r e a c t i o n r a t e s i n t h e i r d e v e l o p m e n t o f m o d e l s p r e d i c t i n g s h e a r s t r e s s d e c a y o r v i s c o s i t y c h a n g e f o r t i m e - d e p e n d e n t m a t e r i a l s a s w a s d e s c r i b e d e a r l i e r i n C h a p t e r 2 . B e f o r e c o n s t r u c t i n g a m o d e l t o p r e d i c t t h e t h i c k n e s s o f t h e f l o w -i n g l a y e r , i t i s u s e f u l t o r e v i e w some o f t h e f u n d a m e n t a l p r i n c i p l e s o f t h e t h e o r y o f c h e m i c a l r e a c t i o n r a t e s . C o n s i d e r a r e v e r s i b l e c h e m i c a l r e a c t i o n o f A B t = 0; a m o l e / £ b m o l e / % t a - x b + x w i t h t h e f o r w a r d r a t e c o n s t a n t k f a n d t h e b a c k w a r d r a t e c o n s t a n t k^. S u p p o s e t h e i n i t i a l c o n c e n t r a t i o n s o f t h e m a t e r i a l s A a n d B a r e 82 a m o l e / 5 - a n d b m o l e / £ , r e s p e c t i v e l y , a t t i m e t = 0 . A t t = t , x m o l e / 1 o f A r e a c t s i n t o B s o t h a t t h e c o n c e n t r a t i o n o f A b e c o m e s ( a - x ) m o l e / £ a n d t h e c o n c e n t r a t i o n o f B b e c o m e s (b + x ) m o l e /Z a t t h a t t i m e . I f t h e r e a c t i o n i s t h e m o r d e r - n o r d e r , t h e r a t e o f i n c r e a s e o f t h e c o n c e n t r a t i o n o f t h e m a t e r i a l B w i l l b e j£ = k f ( a - x ) m - k^Cb + x ) n ( 7 - 1 ) Now c o n s i d e r a r a t e p r o c e s s o f a g e l - s o l t r a n s f o r m a t i o n s u c h t h a t k f g e l —»- s o l K t = 0 ; T T ( R 0 2 - R 2) L 0 t = t ; M R 0 2 - R 2) - i*(\2 - r 2 ) > L = TT (Rn 2 - R 2 ) L TT(R 2 - R 2) L u x x w i t h t h e f o r w a r d r a t e c o n s t a n t k f a n d t h e b a c k w a r d r a t e c o n s t a n t k^ . S u p p o s e t h e i n i t i a l v o l u m e o f t h e g e l i n a C o u e t t e v i s c o m e t e r i s {TT(RQ 2 - R 2) L } c m 3 a n d t h e i n i t i a l v o l u m e o f t h e s o l i s z e r o , w h e r e R n i s r a d i u s o f t h e o u t e r c y l i n d e r , R i s r a d i u s o f t h e i n n e r c y l i n d e r , a n d L i s t h e l e n g t h o f t h e i n n e r c y l i n d e r . A t t = t , {^(R^ 2 - R 2) L } c m 3 o f t h e g e l t r a n s f o r m s i n t o t h e s o l s o t h a t t h e v o l u m e o f g e l b e c o m e s { T T (R 0 2 - R 2) U - {TT(R 2 - R 2) L } = T T ( R 0 2 - R x 2 ) L a n d t h e v o l u m e o f s o l b e c o m e s TT(R 2 - R 2) L x a t t h a t t i m e , w h e r e R i s t h i c k n e s s o f t h e f l o w i n g l a y e r m e a s u r e d f r o m t i l t h e c e n t e r o f t h e i n n e r c y l i n d e r . I f t h e t r a n s f o r m a t i o n i s m o r d e r -o r d e r , t h e r a t e o f i n c r e a s e o f t h e s o l v o l u m e c a n b e e x p r e s s e d a s chr(R 2 - R 2 ) L - ^ = k f { 7 r L ( R o 2 - R x 2 ) } m - k ^ T T L C R ^ 2 - R 2 ) } n ( 7 - 2 ) E q u a t i o n ( 7 - 2 ) h a s b e e n o b t a i n e d f r o m a n a r g u m e n t s i m i l a r t o t h e c h e m i c a l r e a c t i o n A % B w h e r e b o t h t h e r e a e t a r f t A a n d t h e p r o d u c t B p o s s e s s c h e m i c a l l y u n i f o r m c o m p o s i t i o n s a n d h o m o g e n e o u s s t r u c t u r e s . H o w e v e r , i n t h e t r a n s f o r m a t i o n o f g e l s o l , t h e m a t e r i a l f l o w i n g i n t h e v i s c o -m e t e r a n d h e r e r e p r e s e n t e d a s a s o l d o e s n o t n e c e s s a r i l y p o s s e s s s u c h a u n i f o r m o r i e n t a t i o n a s w o u l d t h e c o m p l e t e l y d e t e r i o r a t e d s t r u c t u r e . A c c o r d i n g l y , e q u a t i o n ( 7 - 2 ) m u s t b e m o d i f i e d b y i n t r o d u c i n g a t e r m w h i c h t a k e s i n t o a c c o u n t t h e s t r u c t u r e c h a n g e i n t h e f l o w i n g s o l . A s s u m e - t h a t a t a n y g i v e n s h e a r s t r e s s o r r a t e o f s h e a r t h e r e i s a n e q u i l i b r i u m s o l o r i e n t a t i o n o r a n e q u i l i b r i u m s i z e o f u n d e f o r m e d b u n d l e A ^ . A n y s i z e l a r g e r t h a n t h i s i s u n s t a b l e a n d b r e a k s down u n d e r s h e a r . S i n c e s h e a r r a t e o r s h e a r s t r e s s v a r i e s a c r o s s t h e g ap o f t h e f l o w i n g m a t e r i a l t h e r e w i l l b e a d i s t r i b u t i o n o f b u n d l e s i z e s a s s h o w n i n F i g . 7 - 1 . A t t = 0 , t h e w h o l e s p a c e o f t h e a n n u l u s i s o c c u p i e d b y t h e g e l . A t a n i n i t i a l s t a g e o f t h e f l o w ( t = t ^ ) , t h e t h i c k n e s s o f t h e f l o w i n g l a y e r i s s m a l l , t h o u g h t h e a v e r a g e s i z e o f t h e b u n d l e s i s c o m p a r a t i v e l y l a r g e . A s t i m e g o e s o n ( t = t 2 ) , t h e t h i c k n e s s o f t h e l a y e r i n c r e a s e s , a n d t h e a v e r a g e s i z e o f t h e b u n d l e s g e t s s m a l l e r . A t e q u i l i b r i u m ( t = t ) , b o t h t h e t h i c k n e s s o f t h e l a y e r a n d t h e a v e r a g e s i z e o f t h e o o b u n d l e s do n o t c h a n g e w i t h t i m e , b u t t h e r e i s a n e q u i l i b r i u m d i s t r i b u -t i o n o f b u n d l e s i z e s a c r o s s t h e g a p . L e t a b e a m e a n v a l u e o f t h e b u n d l e s i z e . A s s u m e a i s g i v e n b y t h e f o l l o w i n g e x p r e s s i o n a t a n y i n s t a n t a = A + . 'I _ ( cm) ( 7 - 3 ) 0 0 A + B t w h e r e t i s t h e d u r a t i o n o f s h e a r ( s e c ) a n d A i s t h e e q u i l i b r i u m F i g . 7 - 1 . B u n d l e S i z e s i n F l o w i n g S o l 85 s i z e (cm) of f i b e r b undle a t a g i v e n shear r a t e , and + 1/A = A Q (cm) i s t h e v a l u e of a a t t = 0, and B i s a c o n s t a n t (cm "'".sec ^ ) . Assuming t h a t t h e f u r t h e r bundle s i z e i s from t h e e q u i l i b r i u m v a l u e A^, t h e more r a p i d i s the break-down, one can p r e d i c t t h a t t h e r a t e of i n c r e a s e of the f l o w i n g volume i s p r o p o r t i o n a l t o (2 - A ^ ) , i . e . div(R 2 - R 2 ) L x dt (a - A ) A + Bt (7-4) Hence, t a k i n g i n t o account the c o r r e c t i o n term [1/(A + B t ) ] , e q u a t i o n (7-2) i s m o d i f i e d t o d?r(R 2 - R 2 ) L . = A I [ k . { i r ( R 2 - R 2 ) L } M dt A + B t f 0 x - ^ { T T ^ 2 - R 2) L } N ] (7-5) The f i r s t term o f t h e b r a c k e t on the r i g h t - h a n d s i d e of e q u a t i o n (7-5) r e p r e s e n t s t h e break-down o f g e l s t r u c t u r e t o f l o w i n g s o l and t h e second term i s t h e b u i l d - u p of g e l s t r u c t u r e from the s o l . E q u a t i o n (7-5) can be s o l v e d a n a l y t i c a l l y f o r v a r i o u s o r d e r s o f m and n. The s o l u t i o n s a r e i n the form l o g | f ( R x ) | = K ^ . l o g | f t + 1| (7-6) where f ( R ) i s a f u n c t i o n of R and K i s a c o n s t a n t . For example, t h e x x s o l u t i o n f o r second o r d e r - z e r o o r d e r [(m, n) = (2, 0 ) ] i s 1 |Y - D b + D| n T k f • . ,B r i . l o g l ^ T o • ^ 7 ^ 1 = DTTL — l o g | x t + 1| (7-7) where Y = 2X + b (7-8) D = / b 2 - 4c • (7-9) 86 b = - 2 ( R 0 2 - R 2 ) (7-10) c = ( R Q 2 - R 2 ) 2 - ( R Q 2 - R x o o 2 ) 2 (7-11) and X = R 2 - R 2 (7-12) x w i t h t h e i n i t i a l boundary c o n d i t i o n o f R = R a t t = 0. x The. m a t h e m a t i c a l d e t a i l s f o r t h i s d e r i v a t i o n and the o t h e r s o l u t i o n s w i t h v a r i o u s o r d e r s (m, n) a r e p r e s e n t e d i n Appendix E. E q u a t i o n (7-5) i s a g e n e r a l model f o r p r e d i c t i n g the t h i c k n e s s of t h e f l o w i n g l a y e r . The e m p i r i c a l c o n s t a n t s m and n were determined f o r t h e e x p e r i m e n t a l d a t a o b t a i n e d from the p r e s e n t a r t i f i c i a l s l u r r i e s as d e s c r i b e d i n the subsequent s e c t i o n . 7-2. D e t e r m i n a t i o n o f m and n i n t h e Model The c o n s t a n t s m and n a r e t h e o r d e r s of the g e l - s o l t r a n s f o r m a t i o n . (14) Goodeve c o n s i d e r e d a g e n e r a l g e l - s o l t r a n s f o r m a t i o n i n terms of the c o n c e n t r a t i o n o f l i n k s o f p a r t i c l e s and proposed (m, n) = (1, 2) i n h i s r a t e e q u a t i o n s . Denny and B r o d k e y ^ ^ c o n s t r u c t e d a r a t e e q u a t i o n i n terms o f t h e p o i n t v i s c o s i t y and c o n f i r m e d t h a t the o r d e r (m, n) = (1, 2) i n t h e i r e q u a t i o n f i t s the e x p e r i m e n t a l d a t a of the n a t u r a l t h i x o t r o p i c (43) s u s p e n s i o n s they examined. Moreover, R i t t e r and G o v i e r assumed (m, n) = (1, 2) a priori i n t h e i r - r a t e e q u a t i o n f o r s t r u c t u r a l s t r e s s decay and found t h a t i t f i t s t h e i r e x p e r i m e n t a l d a t a of Pembina Crude O i l . One common element among the c o n c e p t s of t h e s e workers i s t h a t t h e mechanism of the breakdown i s s i m i l a r to the r e a c t i o n o f a s i m p l e c h e m i c a l d e c o m p o s i t i o n (m = 1 ) , w h i l e the mechanism of the b u i l d - u p i s dependent on the c o l l i s i o n f r e q u e n c y of p a i r s of degraded u n i t s , a r e c o m b i n a t i o n p r o c e s s analogous to a second o r d e r c h e m i c a l r e a c t i o n 87 ( n = 2 ) . ( A l ) On t h e o t h e r h a n d , P i n d e r c o n s i d e r e d t h a t t h e s l u r r y e x h i b i t s t h e t i m e - d e p e n d e n c y b e t w e e n t h e t w o e x t r e m e s o f o r i e n t a t i o n o f t h e s l u r r y p a r t i c l e s , i . e . , c o m p l e t e l y r a n d o m n e s s a n d e q u i l i b r i u m o r d e r . He c o n s t r u c t e d h i s m o d e l i n t e r m s o f a p p a r e n t v i s c o s i t y a n d f o u n d t h a t t h e s e c o n d o r d e r - z e r o o r d e r [ ( m , n ) = ( 2 , 0 ) ] m o d e l f i t t e d h i s e x p e r i m e n t a l d a t a f o r t h e t e t r a h y d r o f u r a n - h y d r o g e n s u l p h i d e g a s h y d r a t e s l u r r i e s . B r o w n ' s a r t i f i c i a l s l u r r y ^ ^ ' ^ ^ w a s a l s o f o u n d t o f o l l o w t h e s e c o n d o r d e r - z e r o o r d e r m e c h a n i s m o f P i n d e r ' s m o d e l . A l l o f t h e s e r e s e a r c h e r s f o c u s s e d o n a t i m e v a r i a t i o n o f m e a s u r a b l e v i s c o s i t y o r s h e a r s t r e s s i n c o n s t r u c t i n g t h e i r m o d e l s . N o n e o f t h e m t o o k i n t o c o n s i d e r a t i o n t h e t h i c k n e s s o f f l o w i n g l a y e r . W h a t e v e r t h e m o d e l may r e p r e s e n t , t h e o r d e r o f t h e g e l - s o l t r a n s f o r m a t i o n c a n n o t b e p r e d i c t e d a priori a n d t h e v a l u e s o f m a n d n m u s t b e d e t e r m i n e d e x p e r i -m e n t a l l y . A l l t h e e x p e r i m e n t a l d a t a o n t h e t h i c k n e s s o f f l o w i n g l a y e r R v e r s u s t i m e t o b t a i n e d i n t h i s i n v e s t i g a t i o n w e r e f i t t e d b y t h e p r e s e n t m o d e l u s i n g a w i d e r a n g e o f o r d e r s (m, n ) . I n v i e w o f t h e c o n c e p t s o f r a t e p r o c e s s e s t h e o r d e r o f t h e t r a n s f o r m a t i o n (m, n ) i s n o t n e c e s s a r i l y a n i n t e g e r , b u t a n y r e a l n u m b e r c a n p o s s i b l y b e a c c e p t a b l e . I t w a s , h o w e v e r , o n e o f t h e p r i m a r y o b j e c t i v e s i n c o n s t r u c t i n g a m o d e l h e r e t h a t t h e m o d e l s h o u l d b e m a t h e m a t i c a l l y a s s i m p l e a s p o s s i b l e . H e n c e , t h e e i g h t d i f f e r e n t o r d e r s o f (m, n ) f r o m ( 0 , 1 ) t o ( 2 , 2 ) h a v e b e e n e x a m i n e d i n t h i s w o r k . A s s h o w n i n e q u a t i o n ( 7 - 6 ) , a l l o f t h e m o d e l e q u a t i o n s h a v e t w o f i t t i n g p a r a m e t e r s , ( k ^ / B ) a n d ( B / A ) . T h e s e t w o p a r a m e t e r s w e r e e v a l u a t e d b y m e a n s o f t h e l e a s t s q u a r e m e t h o d m i n i m i z i n g t h e sum o f 88 s q u a r e s . The computer programme i s g i v e n i n Appendix C. T a b l e 7-1 shows the r e s u l t s o f t h e f i t f o r the s l u r r y of p a r t i c l e c o n c e n t r a t i o n $ = 0.09 ( c c / c c ) , p a r t i c l e l e n g t h L = 1.62 (mm), p a r t i c l e l e n g t h - t o - d i a m e t e r r a t i o L/D = 37.4, and the d i s p e r s i n g medium of d e x t r o s e + 1.0 mole NaCl i n 10% PEG-H 0. The a n g u l a r v e l o c i t y o f t h e i n n e r c y l i n d e r was 0.233 ( r a d . / s e c ) . I t may be noted i n T a b l e 7-1 t h a t t h e sum of squares of 0 - l s t o r d e r , l s t - 0 o r d e r , and l s t - l s t o r d e r came to t h e same v a l u e . T h i s i s because the f u n c t i o n f ( R ) of the l e f t - h a n d x s i d e o f e q u a t i o n (7-6) f o r t h e s e t h r e e o r d e r s remains the same, t h a t i s 2 2 R - R f (R ) = 1 - (7-13) X R - R Xco Hence t h e s e t h r e e models of 0 - l s t o r d e r , l s t - 0 o r d e r , and l s t - l s t o r d e r p r e d i c t t h e phenomenon w i t h e s s e n t i a l l y the same a c c u r a c y . A l t h o u g h the sum o f squares l i s t e d i n T a b l e 7-1 does not v a r y much w i t h the v a r i o u s o r d e r s of the model e q u a t i o n s , t h e second o r d e r - z e r o o r d e r model does have an advantage over the o t h e r models. S i m i l a r r e s u l t s were found f o r almost a l l the o t h e r e x p e r i m e n t a l runs c a r r i e d out i n t h i s work. They a r e l i s t e d i n Appendix F. I t can be c o n c l u d e d t h a t i n terms of th e sum of squares t h e second o r d e r - z e r o o r d e r [(m, n) = (2, 0)] model g i v e s the b e s t a p p r o x i m a t i o n f o r p r e d i c t i n g t h e t h i c k n e s s o f f l o w i n g l a y e r . Brown and P i n d e r c o n s i d e r e d t h e second o r d e r term of t h e i r model to be a measure of the d i s t a n c e between bonding p o i n t s . T a k i n g n o t e of the f a c t t h a t t h e e l e c t r o s t a t i c f o r c e was s t r o n g l y r e l a t e d t o (35) t h e g e l s t r u c t u r e , they a n a l y z e d t h a t t h e bonding f o r c e was i n v e r s e l y p r o p o r t i o n a l to the square of t h e d i s t a n c e , as suggested by Coulomb's Law. I t was t h e n s t a t e d t h a t the r a t e of breakdown of the e l e c t r o -89 TABLE 7-1 VARIOUS ORDERS (m, n) AND RESULT OF LEAST SQUARES FIT R e s u l t a n t Parameters Order , (m,n) k./B (cm) U m ; B/A (sec) Sum of Squares 0 - 1 s t 3.40 11.0 6.20 X i o " 2 0 - 2nd 1.86 25.9 8.98 X i o " 2 1 s t - 0 0.448 11.0 6.20 X i o " 2 l s t - l s t 0.163 11.0 6.20 X i o " 2 l s t - 2 n d 0.106 20.2 8.07 X i o " 2 2nd - 0 0.0162 8.13 5.47 X i o " 2 2 n d - l s t 0.00878 9.29 5.77 X i o " 2 2nd-2nd 0.00634 14.8 7.04 X i o " 2 rif = 0.09 ( c c / c c ) , L = 1.62 (mm), L/D = 37.4, ft = 0.233 ( r a d / s e c ) 4)EX + 1 . 0 mole NaCl i n 10% PEG-H o0 s t a t i c s t r u c t u r e was i n v e r s e l y p r o p o r t i o n a l t o t h e b onding f o r c e . The r a t e of breakdown, t h e r e f o r e , was d i r e c t l y p r o p o r t i o n a l to the square of the d i s t a n c e . T h i s was t h e i r r e a s o n f o r c o n s i d e r i n g the breakdown to be of the second o r d e r . In view of the p r e s e n t r e s u l t shown i n F i g . 6-4 the e l e c t r o s t a t i c f o r c e i s c e r t a i n l y r e s p o n s i b l e f o r t h e g e l s t r e n g t h of t h e p r e s e n t a r t i f i c i a l s l u r r i e s . The e l e c t r o s t a t i c s t r u c t u r e as i m p l i e d by Brown and P i n d e r ^ i s a l s o v e r y p l a u s i b l e f o r t h e c u r r e n t a r t i f i c i a l s l u r r i e s , U n f o r t u n a t e l y , t h e p r e s e n t form of the model p r e d i c t i n g t h i c k n e s s of f l o w i n g l a y e r c o u l d not be r e l a t e d d i r e c t l y t o the e l e c t r o s t a t i c d i s t a n c e between bonding p o i n t s . T h e r e f o r e the o r d e r s (m, n) a r e t r e a t e d as t h e e m p i r i c a l c o n s t a n t throughout the p r e s e n t model. 7-3. Two F i t t i n g Parameters (k f/B) and (B/A) of t h e Model I t was shown i n the p r e v i o u s s e c t i o n t h a t the second o r d e r - z e r o o r d e r model gave the b e s t f i t . The model e q u a t i o n has a l r e a d y been p r e s e n t e d i n S e c t i o n 7-1, i . e . , k, . I = T W T . . ' Y + D " b - D' where Y = 2X + b ( 7"8) i o S | ^ | . ^ - ™.4 a o . l l t + 1| (7-7) D = / b 2 - 4c (7-9) b = - 2 ( R Q 2 - R 2) (7-10) c = ( R o 2 - R 2 ) 2 - ( R 0 2 - R X M 2 ) 2 ( 7 - 1 D and X = R 2 - R 2 (7-12) x _3 The model e q u a t i o n (7-7) has two p a r ameters, viz., Kji/B ( c m ) B/A (sec "*") which can be r e g a r d e d as the f i t t i n g p arameters. F i g . 7-2 shows the e f f e c t of t h e parameter ( k f / B ) on the model X + * k f / B = 0.007(cm" 3 ) 0 - 0 0 5 0 . 0 0 3 0 . 0 0 1 4> 0 B / A = 5.0(sec' ) R X o o = 1.5 (cm) R 0= 2.1 (cm) R = 1.01 (cm) / t ^ y ^ . . . . . A A A A A A A A / W V X W + B+HIMIMIIMMMMMHMM+ — 1 1 : 1 1 -10 .0 7.143 24.286 41.429 58.571 TIME ISEC) T T 75.714 92.857 110 F i g . 7 -2 . E f f e c t o f P a r a m e t e r ( k /B) o n t h e M o d e l 92 e q u a t i o n ( 7 - 7 ) , where the o t h e r parameter (B/A) was kept c o n s t a n t . [B/A = 5.0]. The parameter (k f/B) was v a r i e d from 0.001 to 0.007. The v a r i o u s c o n s t a n t s i n the e q u a t i o n were s e t as f o l l o w s : R = 1.01 cm Ro = 2.10 cm and R = 1 . 5 0 cm. x°° A l l f o u r c u r v e s approach a s y m p t o t i c a l l y t h e common e q u i l i b r i u m v a l u e of R = 1.50 a t t = t . I t can be seen from the f i g u r e t h a t the g r e a t e r X c o CO o o the parameter ( k f / B ) , the f a s t e r t h e R approaches i t s e q u i l i b r i u m I X v a l u e . A s i m i l a r p l o t i s shown i n F i g . 7-3, where the parameter (k^/B) was kept c o n s t a n t (k^/B = 0.005) and the o t h e r parameter (B/A) was v a r i e d from 0.05 t o 50.0. The v a l u e s o f R, R , and Rn a r e the same 3 X c o 3 v as t h o s e i n F i g . 7-2. I t can be seen t h a t the g r e a t e r the parameter (B/A), the f a s t e r t h e R approaches i t s e q u i l i b r i u m v a l u e . By comparing F i g . 7-2 w i t h F i g . 7-3 i t can be c o n c l u d e d t h a t the r a t e of i n c r e a s e of R x i s much more s e n s i t i v e t o the parameter (k^/B) t h a n t o t h e parameter (B/A). T h i s i s because the parameter (k^/B) i s an exponent o f th e t - t e r m i n e q u a t i o n (7-7) , w h i l e the parameter (B/A) i s j u s t a m u l t i p l i e r of the time t . 7-4. E f f e c t of A n g u l a r V e l o c i t y on Rate of I n c r e a s e o f T h i c k n e s s o f F l o w i n g L a y e r F i g . 7-4 shows a comparison between the model e q u a t i o n s and e x p e r i m e n t a l d a t a w i t h d i f f e r e n t a n g u l a r v e l o c i t i e s of the r o t a t i n g i n n e r c y l i n d e r . The model f i t s the e x p e r i m e n t a l d a t a v e r y w e l l . I t can be seen from t h e f i g u r e t h a t the h i g h e r the a n g u l a r v e l o c i t y , the 4* B /A=50 .0 (sec - 1) k f / B = 5.0 X I0 3(cm" 3) • 0 5 - ° 5 R x . . . . 5 . ( c m ) * 0 0 5 R 0 =2. l (cm) R = l.0l.(cm) •f 1 1 — i 1 r 1 -10 .0 7.143 24.286 4J .429 58.571 75.714 92.857 110 TIME (SEC) F i g . 7 - 3 . E f f e c t o f P a r a m e t e r ( B/A ) o n t h e M o d e l 0 20 40 60 80 100 120 140 T i m e ( s e c . ) F i g . 7-4. E f f e c t o f A n g u l a r V e l o c i t y o n R a t e o f I n c r e a s e o f T h i c k n e s s o f F l o w i n g L a y e r 95 f a s t e r i s the r a t e o f i n c r e a s e o f the t h i c k n e s s of f l o w i n g l a y e r . F i g . 7-5 and F i g . 7-6 show t h e e f f e c t of a n g u l a r v e l o c i t y on t h e f i t t i n g parameters (k^/B) and (B/A), r e s p e c t i v e l y . The l a r g e s c a t t e r i n the d a t a may be e x p l a i n e d by t h e e r r o r s a s s o c i a t e d w i t h the e x p e r i m e n t a l r e p r o d u c i b i l i t y which was mentioned i n S e c t i o n 6-1. S i n c e i t would be expec t e d t h a t k^, A, and B s h o u l d a l l i n c r e a s e w i t h a n g u l a r v e l o c i t y , t h e f a c t shown i n t h e s e graphs t h a t b o t h (k^/B) and (B/A) i n c r e a s e w i t h angu-l a r v e l o c i t y means t h a t k^ i s more s e n s i t i v e t o a n g u l a r v e l o c i t y than B i s . A may be s l i g h t l y a f f e c t e d by a n g u l a r v e l o c i t y . As a n g u l a r v e l o c i t y tends t o z e r o , i t i s expe c t e d t h a t k^ a l s o tends t o z e r o . 7-5. E f f e c t s o f P a r t i c l e C o n c e n t r a t i o n and P a r t i c l e L/D R a t i o on Rate of I n c r e a s e ' o f T h i c k n e s s of F l o w i n g L a y e r F i g . 7-7 and 7-8 show comparisons between the model e q u a t i o n s and the e x p e r i m e n t a l d a t a . There a r e a few s c a t t e r e d p o i n t s , but the f i t s a r e g e n e r a l l y s a t i s f a c t o r y . I t can be seen from F i g . 7-7 t h a t t h e h i g h e r t h e p a r t i c l e c o n c e n t r a t i o n , t h e slower i s t h e r a t e o f i n c r e a s e o f t h e t h i c k n e s s of f l o w i n g l a y e r . A l t h o u g h the p a r t i c l e c o n c e n t r a t i o n i s changed i n F i g . 7-8, i t can be seen t h a t the h i g h e r the p a r t i c l e L/D r a t i o , the slower i s the r a t e o f i n c r e a s e of the t h i c k n e s s o f f l o w i n g l a y e r . By comparing F i g . 7-8 w i t h F i g . 7-7 i t can f u r t h e r be noted t h a t the e f f e c t o f p a r t i c l e L/D r a t i o i s more dominant than t h a t o f p a r t i c l e c o n c e n t r a t i o n . F i g . 7-9 and 7-10 show the e f f e c t s o f p a r t i c l e c o n c e n t r a t i o n and p a r t i c l e L/D r a t i o on the f i t t i n g p arameters, k^/B and B/A, r e s p e c t i v e l y . A c o n s i d e r a b l e s c a t t e r i n the graphs may, a g a i n , be e x p l a i n e d i n terms o f the e x p e r i m e n t a l e r r o r s a s s o c i a t e d w i t h the r e p r o d u c i b i l i t y . Both (k^/B) and (B/A) d e c r e a s e w i t h p a r t i c l e c o n c e n t r a t i o n and L/D r a t i o . From the model i t would be expected t h a t k f , A, and B s h o u l d a l l d e c r e a s e i i i i ..... , , T II _ DEX+I.O mole NaCl in 10% PEG-r^O ro g 11 <f> = 0.14 (cc/cc) o L = 0.987 mm 10 _ L/D = 22.9 o o v ' E o r * i • & N i j& GO 9 - O y • o °o >^ q> •«=-Q> E o fe-es 8 7 / ° — c o o cn "5 6 5 O 1 1 1 1 i i 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Angular Velocity XI (rod/sec) a-. F i g . 7 - 5 . E f f e c t o f A n g u l a r V e l o c i t y Q o n t h e P a r a m e t e r ( k , / B ) DEX+ 1.0 mole NaCl in 10% <f> =0.14 (cc/cc) L =0 987 (mm) L/D=22.9 O I I I I I I I L _ 01 02 0.3 0.4 0.5 0.6 Q7 A n g u l a r V e l o c i t y Si ( r a d / s e c ) - 6 . E f f e c t o f A n g u l a r V e l o c i t y 0, o n t h e P a r a m e t e r ( B / A ) , , $ • 1 1 1 jj-0 20 4 0 60 80 T i m e ( s e c ) 0 0 F i g . 7-7. E f f e c t o f P a r t i c l e C o n c e n t r a t i o n o n R a t e o f I n c r e a s e o f T h i c k n e s s o f F l o w i n g L a y e r 1 I i i 1 1 r T i m e ( s e c ) F i g . 7 -8 . E f f e c t o f P a r t i c l e L/D R a t i o o n R a t e o f I n c r e a s e o f T h i c k n e s s o f F l o w i n g L a y e r vo 10 5 1 DEX + I.O mole NaCl L= 0.987 (mm) £ 1 = 0.233 (rad/sec) i in 10% PEG - H 2 0 A i _ 0 L/D = 22.9 A 37.4 • 69.8 ^ A ^ \ o \ • \ A A \ ° o \ 1 i o 1 0.05 0.10 0.15 Volume Fro c t i o n of F i b e irs (cc/cc) E f f e c t s o f P a r t i c l e C o n c e n t r a t i o n a n d P a r t i c l e L/D R a t i o o n t h e P a r a m e t e r ( k . / B ) 10 I—i 1  DEX+1.0 mole NaCl in 10% P E G - H 2 0 *- i 0 L = 0.987 (mm) -O 0 ) £1 =0233 (rad/sec) A v- W 8 - A 0 I i i i 0.05 0.10 0.15 V o l u m e F r a c t i o n o f F i b e r s ( c c / c c ) F i g . 7 - 1 0 . E f f e c t s o f P a r t i c l e C o n c e n t r a t i o n a n d P a r t i c l e L/D R a t i o o n t h e P a r a m e t e r B/A 1 0 2 w i t h t h e s e p a r a m e t e r s . H e n c e i s m o r e s t r o n g l y a f f e c t e d b y c o n c e n t r a -t i o n a n d L/D r a t i o t h a n a r e B a n d A . 7-6. R h e o l o g i c a l C o n s i d e r a t i o n o f t h e T h i c k n e s s o f F l o w i n g L a y e r No o n e , t o t h e a u t h o r ' s k n o w l e d g e , h a s s t u d i e d t h e t h i c k n e s s o f f l o w i n g l a y e r R x i n t h e t i m e - d e p e n d e n t f l o w i n a C o u e t t e v i s c o m e t e r . F r o m t h e r e s u l t s p r e s e n t e d i n t h i s c h a p t e r , h o w e v e r , i t c a n b e s e e n t h a t t h e R x i s o f g r e a t i m p o r t a n c e i n c h a r a c t e r i z i n g t h e r h e o l o g y o f t i m e -d e p e n d e n t s h e a r f l o w . Now t h e q u e s t i o n c a n b e r a i s e d a s t o how p r a c t i c a l s u c h a c h a r a c t e r i z a t i o n c a n b e , c o n s i d e r i n g t h e d i f f i c u l t i e s e n c o u n t e r e d i n c a l c u l a t i n g t h e t r u e s h e a r r a t e a s a f u n c t i o n o f t i m e . F o r t u n a t e l y , o n e c a n d e f i n e a n a p p a r e n t s h e a r r a t e a t t h e w a l l a s w h e r e s = R/RQ. ( 7 - 1 5 ) S i n c e t h e a p p a r e n t s h e a r r a t e c e r t a i n l y c h a r a c t e r i z e s a s h e a r i n g c o n d i t i o n a t t h e w a l l w i t h t h e p a r t i c u l a r g e o m e t r y o f t h e s i m p l e s h e a r f l o w s , i t h a s b e e n o f t e n u s e d w h e n e v e r t h e t r u e s h e a r r a t e c a n n o t b e d e t e r m i n e d . I n t h e p r e s e n t c a s e , e q u a t i o n ( 7 - 1 5 ) i s n o t c o r r e c t , b e c a u s e t h e w h o l e g a p f r o m r = R t o r = RQ i s n o t f l o w i n g . T h e v a r i a b l e d e f i n e d a s s = R / R x ( 7 - 1 6 ) w o u l d b e m o r e a p p r o p r i a t e t o t h e c u r r e n t s y s t e m . I n e q u a t i o n ( 7 - 1 4 ) t h e a p p a r e n t s h e a r r a t e t h e n b e c o m e s a f u n c t i o n o f t i m e , s i n c e R x i s a f u n c t i o n o f t i m e . I t s h o u l d b e n o t e d t h a t a t t = 0 t h e a p p a r e n t s h e a r r a t e c a n n o t b e d e t e r m i n e d i n t h i s m a n n e r , s i n c e t h e d e n o m i n a t o r o f e q u a t i o n ( 7 - 1 4 ) i s z e r o . A p r a c t i c a l m e t h o d t o d e t e r m i n e " y a t t = 0 may b e t h e e x t r a p o l a t i o n o f t h e ( t , y ) c u r v e t o t h e t = 0 a x i s . F i g . 7 - 1 1 s h o w s a p l o t o f t i m e v e r s u s a p p a r e n t s h e a r r a t e c a l c u l a t e d b y L /D=22 .9 &=0,233(rad/sec) DEX + I.OmoleNaCI i n l O % P E G - H 2 0 Ar = 0.16 ( c c / c c ) v * 0.15 O 0.14 ^ 0.13 + + x + X + <& x « + + + . + 4> 4> 4> : 1 1 1 1 1 1 - 9 24 6.6 22.44 38.28 54.12 69.96 85.8 TIME (SEC) F i g . 7 - 1 1 . D e p e n d e n c e o f A p p a r e n t S h e a r R a t e o n T i m e 104 e q u a t i o n s (7-14) and (7-16). Now, i t i s c l e a r t h a t b o t h shear s t r e s s at t h e w a l l and apparent s h e a r r a t e a t t h e w a l l a r e f u n c t i o n s of time, i . e . , T = f ( t ) (7-17) j = g ( t ) (7-18) Apparent v i s c o s i t y can be d e f i n e d as the r a t i o of shear s t r e s s over apparent s h e a r r a t e . Hence the apparent v i s c o s i t y n i s 1 1 = f= f c o = h ( t ) ( 7 " 1 9 ) which shows t h a t n i s a l s o a f u n c t i o n o f time. F i g . 7-12 shows a p l o t o f apparent v i s c o s i t y c a l c u l a t e d by e q u a t i o n (7-19) v e r s u s time. Shear s t r e s s and a p p a r e n t s h e a r r a t e a r e b o t h f u n c t i o n s o f time as g i v e n i n e q u a t i o n s (7-17) and (7-18). I f one e l i m i n a t e s the parameter t from t h e s e e q u a t i o n s , t h e f l o w c u r v e (y, T ) may be o b t a i n e d . F i g . 7-13 shows t h o s e f l o w c u r v e s . 7-7. P r a c t i c a l Use o f t h e Model As f a r as t h e v e r y s p e c i a l c a s e o f C o u e t t e f l o w o f time-dependent s u s p e n s i o n s of e l o n g a t e d p a r t i c l e s and d i s p e r s i n g medium of D e x t r o s e + 1.0 mole NaCl i n 10% PEG-^O i s concerned, t h e change of the t h i c k n e s s of t h e f l o w i n g l a y e r R w i t h time t can be p r e d i c t e d by the p r e s e n t X model w i t h t h e t h r e e c h a r a c t e r i s t i c f a c t o r s o f the s l u r r i e s , i . e . , e q u i l i b r i u m t h i c k n e s s o f t h e l a y e r R x o o> a n d two parameters (k^/B) and (B/A). These t h r e e f a c t o r s can be e s t i m a t e d by F i g . 6-12, F i g . 7-5, F i g . 7-6, F i g . 7-9, and F i g . 7-10 i n terms of th e independent v a r i a b l e s examined h e r e , i . e . , p a r t i c l e c o n c e n t r a t i o n <j), p a r t i c l e l e n g t h - t o - d i a m e t e r r a t i o L/D, and a n g u l a r v e l o c i t y of i n n e r c y l i n d e r 0,. The number and the range of t h e independent v a r i a b l e s i n v e s t i g a t e d were l i m i t e d ; t h e r e f o r e t h e 03 CM " O ID lO LU tn-4 cn&i »—^  ^ i n I— -on"3 O C J CO > ° cn in. ru m oo. L / D = 2 2 . 9 ft =0.233 (rad/sec) DEX + I.OmoleNaCI i n l 0 % P E G - H 2 0 + + $ s 0.16 (cc /cc ) + X O 0.15 0.14 0.13 X + **x + + * x + + X \ X X + t <!> - 9 . 2 4 6.6 22.44 38.28 54.12 69.96 TIME (SEC) F i g . 7 - 1 2 . A p p a r e n t V i s c o s i t y o f T i m e - D e p e n d e n t A r t i f i c i a l S l u r r i e s 85.8 101.64 o L/D = 22.9 £2 = 0.233 (rad/sec) DEX+I.OmoleNaCI in 1 0 % PEG-H 2 0 + 3> = 0.16 ( c c/cc ) X 0.15 0 0.14 f 0.13 + + X + X 1 1 1 1 1 1 — 0.727 0.975 1.224 1.472 1.72 1.968 2.217 SHEAR RATE ( 1 .0/SEC ) F i g . 7 -13. F l o w C u r v e o f T i m e - D e p e n d e n t A r t i f i c i a l S l u r r i e s 107 e s t i m a t i o n may n o t b e c o n c l u s i v e . I t s h o u l d b e n o t e d , h o w e v e r , t h a t t h e c o r r e l a t i o n b e t w e e n t h e s e t h r e e c h a r a c t e r i s t i c f a c t o r s (R , k , ./B, x ° o r a n d B/A) a n d t h e i n d e p e n d e n t v a r i a b l e s (<f), L / D , a n d ft) c a n b e s u c c e s s f u l l y u s e d f o r p r e d i c t i n g t h e t i m e v a r i a t i o n o f R . F u t u r e i n v e s t i g a t i o n s t o c o r r e l a t e t h e f a c t o r s w i t h many o t h e r i n d e p e n d e n t v a r i a b l e s s u c h a s t e m p e r a t u r e a n d v a r i o u s k i n d s o f d i s p e r s i n g m e d i u m a r e r e c o m m e n d e d a l o n g t h e l i n e s o f t h i s w o r k . S i n c e t h i s i s t h e f i r s t f l o w m o d e l i n t e r m s o f R , i t c a n n o t b e x c o m p a r e d w i t h o t h e r d e c a y m o d e l s w h i c h c a n p r e d i c t t h e t i m e v a r i a t i o n o f s h e a r s t r e s s a t t h e w a l l o r a p p a r e n t v i s c o s i t y . A n e v a l u a t i o n o f t h o s e d e c a y m o d e l s c a n b e f o u n d i n A p p e n d i x G . I t i s s u g g e s t e d t h a t t h e y b e u s e d f o r p r e d i c t i n g t h e s h e a r s t r e s s d e c a y t ( t ) , a n d t h e p r e s e n t m o d e l f o r p r e d i c t i n g t h e t h i c k n e s s o f f l o w i n g l a y e r R x ( t ) a n d h e n c e t h e a p p a r e n t s h e a r r a t e y ( t ) a t t h e w a l l a s d i s c u s s e d i n t h e p r e v i o u s s e c t i o n . T r a n s p o r t o f p u l p ( f i b e r s u s p e n s i o n ) , c r u d e o i l , o r c o a l ( a s a s l u r r y f o r m ) t h r o u g h p i p e s may b e o n e o f t h e p o s s i b l e a p p l i c a t i o n s o f t h e p r e s e n t w o r k . I n t h o s e s y s t e m s t h e m a t e r i a l f l o w s i n t h e P o i s e u i l l e g e o m e t r y , w h e r e t h e s h e a r s t r e s s a t t h e w a l l i s e x p r e s s e d b y v= -1 €> (7-20) w h e r e R i s r a d i u s o f t h e p i p e a n d ( - d P / d a ) i s t h e g r a d i e n t o f t h e p r e s -s u r e d r o p i n t h e a x i a l d i r e c t i o n . T h e a p p a r e n t s h e a r r a t e i n t h e P o i s e u i l l e g e o m e t r y i s d e f i n e d b y t • & (7-21) w h e r e Q i s t h e v o l u m e t r i c f l o w r a t e . T h e r e i s a c o r r e l a t i o n t o e x p r e s s t h e t r u e s h e a r r a t e i n t h i s g e o m e t r y , t o o , i . e . , t h e R a b i n o w i t s c h (32) e q u a t i o n ; however, i t i s v e r y d i f f i c u l t t o use f o r time-dependent m a t e r i a l s . I f a f l o w e q u a t i o n i s o b t a i n e d i n terms of shear s t r e s s and t r u e shear r a t e f o r a p a r t i c u l a r m a t e r i a l , t h e e n t i r e i n f o r m a t i o n can be i n v a r i a b l e i n o t h e r g e o m e t r i e s . I t i s not d i f f i c u l t t o o b t a i n such a f l o w e q u a t i o n f o r a time-independent m a t e r i a l as d e s c r i b e d i n S e c t i o n 3-6. On the o t h e r hand i f i t i s g i v e n i n terms of apparent s h e a r r a t e as mentioned i n th e p r e v i o u s s e c t i o n , i t i s p e r t i n e n t o n l y t o t h e s p e c i f i c geometry i n which the experiment was conducted. However, much q u a l i t a t i v e i n f o r m a t i o n o b t a i n e d i n one geometry can be a p p l i e d t o t h e o t h e r g e o m e t r i e s . F o r example, s i n c e t h e p a r t o f t h e a n n u l a r c o r e which i s n o t f l o w i n g i n a C o u e t t e geometry does c o r r e s p o n d (22) (25) to t h e c e n t r a l p l u g p a r t i n a P o i s e u i l l e geometry ' , th e c h a r a c t e r i s t i c s o f t h e v a r i a t i o n o f t h e s i z e o f the c e n t r a l p l u g c o u l d be, i n p r i n c i p l e , t h e same as t h a t of t h e a n n u l a r c o r e . However, the q u a n t i t a t i v e i n f o r m a t i o n such as apparent shear r a t e cannot be a p p l i e d i n a d i r e c t manner from one geometry to a n o t h e r , because the apparent shear r a t e i s , i n general,dependent on t h e geometry. T h e r e f o r e , t h e p r a c t i c a l v a l u e of t h e p r e s e n t work l i e s i n such q u a l i t a t i v e c o n t r i b u -t i o n t o t h e c u r r e n t knowledge. As to t h e measurement of time-dependent m a t e r i a l s i n a C o u e t t e v i s c o m e t e r w i t h a wide gap t h i s p r e s e n t work i s , i n d e e d , a v e r y f u n d a -m e n t a l one. I t was, however, s t r e s s e d i n t h i s work how important t h e t h i c k n e s s o f t h e f l o w i n g l a y e r R '.is. I t seems t h a t s l u r r y f l o w measurements w i l l n e c e s s a r i l y be u n d e r t a k e n more o f t e n i n a v i s c o m e t e r w i t h a wide gap as used i n t h i s work, because d i s p e r s e d phases of i n d u s t r i a l s u s p e n s i o n s a r e not always s m a l l enough t o be c o m p a t i b l e w i t h t h e gap of the u s u a l v i s c o m e t e r . I t i s hoped t h a t t h e r e s u l t s of t h i s work a r e u s e f u l i n such measurements. CHAPTER 8 CONCLUSIONS Time-dependent a r t i f i c i a l s l u r r i e s , c o n s i s t i n g o f r e g u l a r l y - s i z e d n y l o n f i b e r s and aqueous s o l u t i o n s o f p o l y e t h y l e n e g l y c o l w i t h d e x t r o s e and sodium c h l o r i d e , have been s t u d i e d i n a c o - a x i a l c y l i n d r i c a l v i s c o -meter. From t h e r e s u l t s and d i s c u s s i o n t h e f o l l o w i n g c o n c l u s i o n s were o b t a i n e d , based on the p r e s e n t e x p e r i m e n t a l c o n d i t i o n s . (1) Y i e l d s t r e s s i n c r e a s e s s t r o n g l y w i t h s a l t c o n c e n t r a t i o n . (2) Y i e l d s t r e s s i n c r e a s e s w i t h p a r t i c l e c o n c e n t r a t i o n and p a r t i c l e L/D r a t i o . (3) Y i e l d s t r e s s d e c r e a s e s w i t h an i n c r e a s e o f a n g u l a r v e l o c i t y o f i n n e r c y l i n d e r o f t h e v i s c o m e t e r . (4) E q u i l i b r i u m s t r e s s i n c r e a s e s w i t h s a l t c o n c e n t r a t i o n , p a r t i c l e c o n c e n t r a t i o n , and p a r t i c l e L/D r a t i o . (5) E q u i l i b r i u m t h i c k n e s s o f f l o w i n g l a y e r d e c r e a s e s w i t h an i n c r e a s e of p a r t i c l e c o n c e n t r a t i o n and p a r t i c l e L/D r a t i o . (6) E q u i l i b r i u m t h i c k n e s s o f f l o w i n g l a y e r i n c r e a s e s w i t h a n g u l a r v e l o c i t y o f i n n e r c y l i n d e r of t h e v i s c o m e t e r . Employing a concept of r a t e p r o c e s s e s , an e m p i r i c a l model w i t h two f i t t i n g parameters ( k ^ / l T a n d B/A) to c h a r a c t e r i z e the time-dependent b e h a v i o r o f the t h i c k n e s s o f f l o w i n g l a y e r i n the r e l a t i v e l y wide gap between c y l i n d e r s of t h e v i s c o m e t e r , was c o n s t r u c t e d f o r t h e tim e -dependent s l u r r i e s . By a n a l y z i n g t h e model f u r t h e r c o n c l u s i o n s were 109 110 drawn. (7) The s e cond o r d e r - z e r o o r d e r r e v e r s i b l e r e a c t i o n mode l g i v e s t h e b e s t a p p r o x i m a t i o n f o r p r e d i c t i n g t i m e - d e p e n d e n t t h i c k n e s s o f t h e f l o w i n g l a y e r . (8) The p a r a m e t e r k^/B i s more s e n s i t i v e t h a n t h e p a r a m e t e r B/A i n t h e m o d e l . (9) The p a r a m e t e r s k^/B and B/A i n c r e a s e w i t h a n g u l a r v e l o c i t y o f t h e i n n e r c y l i n d e r , and d e c r e a s e w i t h an i n c r e a s e o f p a r t i c l e c o n c e n -t r a t i o n and p a r t i c l e L/D r a t i o . CHAPTER 9 RECOMMENDATIONS I t i s r e c o m m e n d e d t h a t a n y o f t h e v a r i a b l e s h e l d c o n s t a n t i n t h i s r e s e a r c h b e v a r i e d , a n d t h a t t h e r a n g e s o f t h e o n e s v a r i e d i n t h e r e s e a r c h b e e x t e n d e d o r f i l l e d i n . I n t h a t r e g a r d , t e m p e r a t u r e w o u l d b e among t h e e a s i e s t t o s t u d y : s u c h a s t u d y w o u l d b e u s e f u l f o r t e s t -i n g t h e a p p l i c a b i l i t y o f t h e A r r h e n i u s - t y p e c o r r e l a t i o n f o r f u r t h e r a n a l y s e s o n t h e p a r a m e t e r s o f t h e p r e s e n t m o d e l , e s p e c i a l l y k^/B w h e r e k^ i s t h e f o r w a r d r a t e c o n s t a n t . E v e n t h o u g h i t h a s b e e n s u g g e s t e d t h a t t h e s c a t t e r i n t h e f i t t i n g p a r a m e t e r s o f t h e m o d e l p r e s e n t e d i n c h a p t e r 7 w a s d u e t o e x p e r i m e n t a l e r r o r , t h e r e may b e some r e f i n e m e n t s t o t h e m o d e l w h i c h w i l l r e s u l t i n f i t t i n g p a r a m e t e r s t h a t s h o w l e s s s c a t t e r . I n t h a t r e g a r d , a t t e m p t s s h o u l d b e made t o s e p a r a t e t h e p a r t s o f t h e f i t t i n g p a r a m e t e r s s o t h a t t h e i r i n d i v i d u a l b e h a v i o r may b e o b s e r v e d . A l l e x p e r i m e n t s o n t h e s t r u c t u r e b r e a k d o w n f o r t h e s l u r r i e s w e r e c o n d u c t e d i n a C o u e t t e f l o w g e o m e t r y i n t h i s w o r k . F u r t h e r e x p e r i m e n t s s h o u l d b e u n d e r t a k e n i n a P o i s e u i l l e f l o w g e o m e t r y b e c a u s e o f t h e c u r r e n t l y r i s i n g i n t e r e s t i n p r o b l e m s o f s t a r t - u p a n d c o n t r o l f o r t h e p i p e f l o w o f m u l t i - p h a s e s y s t e m s . I l l NOMENCLATURE Roman L e t t e r s a i n i t i a l c o n c e n t r a t i o n o f r e a c t a n t A ( m o l e . £ "S a c o n s t a n t i n e q u a t i o n ( 2 - 1 ) ( s e c """) A c o n s t a n t A c o n s t a n t i n e q u a t i o n (7-3) ( cm ^") A g i n i t i a l v a l u e o f b u n d l e s i z e (cm) A e q u i l i b r i u m v a l u e o f b u n d l e s i z e (cm) b i n i t i a l c o n c e n t r a t i o n o f p r o d u c t B ( m o l e . £ """) B c o n s t a n t B c o n s t a n t i n e q u a t i o n (7-3) ( cm " ' " . s e c "*") C c o n s t a n t D d i a m e t e r o f f i b e r (mm) F f o r c e i n x - d i r e c t i o n ( d y n e ) -2 g a c c e l e r a t i o n o f g r a v i t y ( c m . s e c ) H h e i g h t ( cm) k c o n s t a n t o f p o w e r l a w f l u i d i n e q u a t i o n (3-5) k^ r a t e o f i n c r e a s e o f n e t w o r k s t r u c t u r e k^ g r o w t h r e a c t i o n r a t e c o n s t a n t k^ d e c a y r e a c t i o n r a t e c o n s t a n t K e q u i l i b r i u m r a t e c o n s t a n t ( = k^/k^ ) L h e i g h t o f r o t a t i n g c y l i n d e r (cm) L f i b e r l e n g t h (mm) m o r d e r o f d e c a y r e a c t i o n r a t e ( - ) m s l o p e d e f i n e d b y e q u a t i o n (3-45) 112 order of growth reaction rate (-) power law exponent in equation (3-5) (-) exponent in equation (3-42) (-) -2 pressure in equation (3-24) (dyne.cm ) parameter exponent of shear rate in equation (2-3) ' (-) i n i t i a l concentration of network structure constant defined by equation (E-5) 3 -1 volumetric flow rate in pipe (cm .sec ) radius of inner cylinder (cm) radius of pipe in equation (7-20) (cm) radius of outer cylinder (cm) thickness of flowing layer measured from the center of the cylinders (cm) equilibrium value of (cm) ratio of inner cylinder radius over outer cylinder radius (-) area (cm2) time (sec) time at equilibrium (sec) torque (dyne,, cm) velocity (cm. sec "*") velocity (cm. sec "*") velocity in x-direction (cm. sec "*") mean value of bundle size (cm) 114 Y Y n no ni n2 TO T 2 T o T R T S T S°° G r e e k L e t t e r s s t r a i n s h e a r r a t e a p p a r e n t v i s c o s i t y y i e l d v a l u e o f p o i n t v i s c o s i t y a p p a r e n t v i s c o s i t y o f c o m p l e t e l y r a n d o m o r i e n t a t i o n a p p a r e n t v i s c o s i t y o f e q u i l i b r i u m o r d e r e q u i l i b r i u m v a l u e o f a p p a r e n t v i s c o s i t y p o i n t v i s c o s i t y i n e q u a t i o n ( 2 - 3 ) y i e l d v a l u e o f a p p a r e n t v i s c o s i t y e q u i l i b r i u m v a l u e o f p o i n t v i s c o s i t y s t r u c t u r a l p a r a m e t e r i n e q u a t i o n ( 2 - 8 ) N e w t o n i a n v i s c o s i t y r a t i o o f c i r c u m f e r e n c e t o i t s d i a m e t e r d e n s i t y s h e a r s t r e s s y i e l d s h e a r s t r e s s c h a r a c t e r i s t i c t i m e c h a r a c t e r i s t i c t i m e s h e a r s t r e s s o f a c o m p l e t e l y d e t e r i o r a t e d s t r u c t u r e s h e a r s t r e s s a t t h e o u t e r c y l i n d e r w a l l s h e a r s t r e s s a t t h e i n n e r c y l i n d e r w a l l s t r u c t u r a l s t r e s s s h e a r s t r e s s a t i n i t i a l s t a t e s h e a r s t r e s s a t s t e a d y s t a t e s h e a r s t r e s s a t t h e p i p e w a l l (-) ( s e c ^ ) . - 1 - 1 ( g . c m . s e c ( g . c m . s e c t _ 1 " I ( g . c m . s e c t _ 1 _ 1 ( g . c m . s e c t " I " I ( g . c m . s e c t _ 1 -1 ( g . c m . s e c ( g . c m . s e c t ~X -1 (.g.cm . s e c ( g . c m . s e c ; ( - ) ( g . c m ) ( d y n e . c m 2 ) ( d y n e . c m 2 ) ( s e c ) ( s e c ) ( d y n e . c m ) ( d y n e . c m 2 ) ( d y n e . c m 2 ) ( d y n e . c m 2 ) ( d y n e . c m 2 ) ( d y n e . c m 2 ) ( d y n e . c m 2 ) 1 1 5 x ^ s h e a r s t r e s s o f x - d i r e c t i o n a c t i n g o n t h e ^ X s u r f a c e p e r p e n d i c u l a r t o y - a x i s ( d y n e . c m ) -2 e q u i l i b r i u m s h e a r s t r e s s ( d y n e . c m ) 3 -3 <j) v o l u m e f r a c t i o n o f p a r t i c l e s i n s u s p e n s i o n (cm . cm ) 03 a n g u l a r v e l o c i t y ( r a d . s e c tt a n g u l a r v e l o c i t y o f i n n e r c y l i n d e r ( r a d . s e c "'") M a t h e m a t i c a l S y m b o l s a n d N o t a t i o n ( x , y , a ) r e c t a n g u l a r c o o r d i n a t e s ( r , 0 , a ) c y l i n d r i c a l c o o r d i n a t e s _d_ d x _9_ 3x D D t o r d i n a r y d e r i v a t i v e w i t h r e s p e c t t o x p a r t i a l d e r i v a t i v e w i t h r e s p e c t t o x s u b s t a n t i a l t i m e d e r i v a t i v e f f u n c t i o n g f u n c t i o n g . f u n c t i o n Y h f u n c t i o n e e x p o n e n t i a l e x p e x p o n e n t i a l l o g j Q common l o g a r i t h m l o g n a t u r a l l o g a r i t h m I n n a t u r a l l o g a r i t h m t a n h ^ a r c - h y p e r b o l i c t a n g e n t F f o r c e v e c t o r v v e l o c i t y v e c t o r A r a t e o f d e f o r m a t i o n t e n s o r x s t r e s s t e n s o r 116 V f g r a d i e n t o f f V . f d i v e r g e n c e o f f E s u m m a t i o n °° i n f i n i t y l i m f ( x ) l i m i t o f f ( x ) a s x a p p r o a c h e s a x->-a A b b r e v i a t i o n s a n d O t h e r N o t a t i o n CMC c a r b o x y m e t h y l c e l l u l o s e DEX d e x t r o s e JLjO w a t e r N a C l s o d i u m c h l o r i d e P E G p o l y e t h y l e n e g l y c o l p H i n d e x o f c o n c e n t r a t i o n o f h y d r o g e n i o n , i . e . , l o g i n { l / [ H ] } k ^/B f i t t i n g p a r a m e t e r o f t h e m o d e l f o r B/A f i t t i n g p a r a m e t e r o f t h e m o d e l f o r R k ^ / k i t f i t t i n g p a r a m e t e r o f G o v i e r ' s s t r e s s d e c a y m o d e l k^/Po f i t t i n g p a r a m e t e r o f G o v i e r ' s s t r e s s d e c a y m o d e l BIBLIOGRAPHY 1. B e l l , J . P., "Flow O r i e n t a t i o n of Short F i b e r Composites", J o u r n a l o f Composite M a t e r i a l s , _3 ( A p r i l ) , 244-253 (1969). 2. B i l l i n g t o n , E. W. , and A. S. Huxley, "Measurements of the Flow P r o p e r t i e s o f Two Types o f T h i x o t r o p i c F l u i d " , T r a n s a c t i o n s o f Faraday S o c i e t y , 61(12), 2784-2793 (1965). 3. B i l l i n g t o n , E. W., and A. S. Huxley, "Some Measurements of t h e Un-Steady M o t i o n of a T h i x o t r o p i c F l u i d C o n f i n e d between R o t a t i n g C o a x i a l C y l i n d e r s " , J o u r n a l o f C o l l o i d and I n t e r f a c e S c i e n c e , 22, 257-268 (1966). 4. B i r d , R. B., W. E. Stewart, and E. N. L i g h t f o o t , T r a n s p o r t Phenomena, John W i l e y & Sons, I n c . , New York, N.Y. (1960). 5. Braune,H. , and I. R i c h t e r , "Zur T h i x o t r o p i e von B e n t o n i t -s u s p e n s i o n e n " , K o l l o i d Z e i t s c h r i f t , 113(1), 20-28 (1949). 6. Brown, J . P., Time Dependent Rheology of A r t i f i c i a l S l u r r i e s , M.A.Sc. T h e s i s , The U n i v e r s i t y of B r i t i s h Columbia, Vancouver, B.C. (1965). 7. Brown, J . P., and K. L. P i n d e r , "Time Dependent Rheology of A r t i f i c i a l S l u r r i e s " , The Canadian J o u r n a l of Chemical E n g i n e e r i n g , 49, 38-43.(1971). 8. Cheng, D. C.-H., and F. Evans, "Phenomenological C h a r a c t e r i z a t i o n of t h e R h e o l o g i c a l B e h a v i o u r of I n e l a s t i c R e v e r s i b l e T h i x o -t r o p i c and A n t i t h i x o t r o p i c F l u i d s " , B r i t i s h J o u r n a l o f A p p l i e d P h y s i c s , 16, 1599-1617 (1965). 9. Coleman, B. D., and W. N o l l , "On C e r t a i n Steady Flows o f G e n e r a l F l u i d s " , A r c h i v e s of R a t i o n a l M e c h a n i c a l A n a l y s i s , _3, 289-303 (1959). 10. Denny, D. A., and R. S. Brodkey, " K i n e t i c I n t e r p r e t a t i o n of Non-Newtonian Flow", J o u r n a l o f A p p l i e d P h y s i c s , _33(7), 2269-2274 ' (1962). 11. Enoksson, B., "A R o t a t i o n a l Double 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 " , R h e o l o g i c a A c t a , 11, 275-285.(1972). 12. G l a s s t o n e , S., K. J . L a i d l e r , and H. E y r i n g , The Theory of Rate P r o c e s s e s , M c G r a w - H i l l Book Company, I n c . , New York, N.Y. (1941). 13. G o l d s m i t h , H. L., and S. G. Mason, "The M i c r o r h e o l o g y of D i s p e r s i o n s " , Rheology v o l . 4, E d i t e d by F. R. E i r i c h , Academic P r e s s I n c . , New York, N.Y. (1967). 117 118 14. Goodeve, C. F. , "A G e n e r a l Theory o f T h i x o t r o p y and V i s c o s i t y " , T r a n s a c t i o n s of Faraday S o c i e t y , 35., 342-358 (1939). 15. Green, H., I n d u s t r i a l Rheology and R h e o l o g i c a l S t r u c t u r e s , John W i l e y & Sons, I n c . , New York, N.Y. (1949). 16. Hahn, S. J . , T. Ree, and H. E y r i n g , "A Theory o f T h i x o t r o p y and i t s A p p l i c a t i o n t o Grease", N a t i o n a l L u b r i c a t i n g Grease I n s t i -t u t e Spokesman, 21(3)., 12-20 (1957). 17. Hu, C.-S., E f f e c t o f S a l t C o n c e n t r a t i o n and C a t i o n V a l a n c e on Maximum Y i e l d S t r e s s o f a B e n t o n i t e C l a y P a s t e , M.A.Sc. T h e s i s , The U n i v e r s i t y of B r i t i s h Columbia, Vancouver, B.C. (1972). 18. Jones, L. G., T h i x o t r o p i c B e h a v i o r o f a C o l l o i d a l S u s p e nsion, Ph.D. D i s s e r t a t i o n , The Ohio S t a t e U n i v e r s i t y , Columbus, Ohio (1968). 19. K a r n i s , A., H. L. G o l d s m i t h , and S. G. Mason, "The K i n e t i c s o f F l o w i n g D i s p e r s i o n s , I : C o n c e n t r a t e d Suspensions o f R i g i d P a r t i c l e s " , J o u r n a l o f C o l l o i d and I n t e r f a c e S c i e n c e , 22, 531-553 (1966). 20. K a r n i s , A., H. L. G o l d s m i t h , and S. G. Mason, "The Flow o f Sus-p e n s i o n s Through Tubes, V. I n e r t i a l E f f e c t s " , The Canadian J o u r n a l o f Chemical E n g i n e e r i n g , 44, 181-193 (1966). 21. K e n c h i n g t o n , J . M., V. P. Sin g h , and M. E. C h a r l e s , "Flow o f B l e a c h e d Hardwood P u l p Suspensions w i t h A d d i t i v e s P r e s e n t " , To be p u b l i s h e d . 22. K e r e k e s , R.J..E.., arid W.iJ.M. Douglas, " V i s c o s i t y P r o p e r t i e s of Suspensions a t t h e L i m i t i n g C o n d i t i o n s f o r T u r b u l e n t Drag R e d u c t i o n " , The Canadian J o u r n a l o f Chemical E n g i n e e r i n g , 50, 228-231 (1972). 23. Khanna, S. K., and W. F. 0. P o l l e t t , " R h e o l o g i c a l Breakdown i n the Continuous S h e a r i n g o f P l a s t i c i z e d P o l y v i n y l C h l o r i d e " , J o u r n a l o f A p p l i e d Polymer S c i e n c e , 9_, 1767-1785 (1965). 24. K r i e g e r , I . M., and H. E l r o d , " D i r e c t D e t e r m i n a t i o n o f t h e Flow Curves of Non-Newtonian F l u i d s , I I . S h e a r i n g Rate i n the C o n c e n t r i c C y l i n d e r V i s c o m e t e r " , J o u r n a l o f A p p l i e d P h y s i c s , 24^ ( 2 ) , 134-136 (1953). 25. Lee, P. F. W., and G. G. D u f f y , "An A n a l y s i s o f the T r a n s i t i o n Regime of F i b r e S u s p e n s i o n Flow i n P i p e s " , The Canadian J o u r n a l of Chemical E n g i n e e r i n g , 55, 361-362 (1977). 26. Maschmeyer, R. 0., Rheology o f C o n c e n t r a t e d S u s p e n s i o n o f F i b e r s , D.Sc. D i s s e r t a t i o n , Washington U n i v e r s i t y , S a i n t L o u i s , M i s s o u r i (1974). 119 27. Maude, A. D., and R. L. Whitmore, "The W a l l E f f e c t and t h e V i s c o -metry of S u s p e n s i o n s " , B r i t i s h J o u r n a l of A p p l i e d P h y s i c s , ]_, 98-102 (1956). 28. Maude, A. D., " T h e o r e t i c a l E v a l u a t i o n o f C a p i l l a r y V i s c o m e t e r s f o r the Measurement of the V i s c o s i t y o f Suspensions of Spheres", B r i t i s h J o u r n a l of A p p l i e d P h y s i c s , 10, 371-376 (1959). 29. Mercer, H.A, Time Dependent V i s c o s i t y o f T h i x o t r o p i c B e n t o n i t e -Water Su s p e n s i o n s , Ph.D. D i s s e r t a t i o n , The U n i v e r s i t y of R o c h e s t e r , R o c h e s t e r , N.Y. (1973). 30. Mercer, H.A.,and H. D. Weymann, " S t r u c t u r e o f T h i x o t r o p i c Suspen-s i o n s i n Shear Flow, I I I . Time-Dependent B e h a v i o r " , T r a n s -a c t i o n s o f t h e S o c i e t y o f Rheology, 1 8 ( 2 ) , 199-218 (1974). 31. Metzner, A. B., "Flow of Non-Newtonian F l u i d s " , Handbook of F l u i d Dynamics, E d i t e d by V. L. S t r e e t e r , M c G r a w - H i l l Book Company, I n c . , New York, N.Y..(1961). 32. Middleman, S., The Flow o f H i g h Polymers, Continuum and M o l e c u l a r Rheology, I n t e r s c i e n c e P u b l i s h e r s , New York, N.Y. (1968). 33. M i l l s , N. J . , "The Rheology of F i l l e d Polymers", J o u r n a l of A p p l i e d Polymer S c i e n c e , 15, 2791-2805 (1971). 34. Nakagawa, T., Rheology, Iwanami Shoten, Tokyo, Japan (1960). 35. N i c h o l s o n , R. D., C a t i o n E f f e c t i n A r t i f i c i a l T h i x o t r o p i c S l u r r i e s , B.A.Sc. T h e s i s , The U n i v e r s i t y o f B r i t i s h Columbia, Vancouver, B.C. (1966). 36. P a r k , K., and T. Ree, " K i n e t i c s of T h i x o t r o p y of Aqueous B e n t o n i t e S u s p e n s i o n " , J o u r n a l of the Korean Chemical S o c i e t y , 1 5 ( 6 ) , 293-303 (1971). 37. Park, K., T. Ree, and H. E y r i n g , " E f f e c t of E l e c t r o l y t e s on Flow P r o p e r t i e s o f Aqueous B e n t o n i t e S u s p e n s i o n " , J o u r n a l of t h e Korean Chemical S o c i e t y , 1 5 ( 6 ) , 303-312 (1971). 38. P e r k i n s , T. K., and J . B. T u r n e r , " S t a r t i n g B e h a v i o r of G a t h e r i n g L i n e s and P i p e l i n e s F i l l e d w i t h G e l l e d Prudhoe Bay O i l " , J o u r n a l o f P e t r o l e u m Technology, March, 301-308 (1971). 39. P e t e r , S., "Uber d i e T h i x o t r o p i e von G r a p h i t s u s p e n s i o n e n i n A b h a n g i g k e i t von der KorngroBe und dem S u s p e n s i o n s m i t t e l " , K o l l o i d Z e i t s c h r i f t , 113(1), 29-37 (1949). 40. P e t r i e , E. M., " E f f e c t o f S u r f a c t a n t on the V i s c o s i t y of P o r t l a n d Cement-Water D i s p e r s i o n s " , I n d u s t r i a l and E n g i n e e r i n g C h e m i s t r y , P r o d u c t Research and Development, L 5 ( 4 ) , 242-249 (1976). 120 41. P i n d e r , K. L., "Time Dependent Rheology of t h e T e t r a h y d r o f u r a n -Hydrogen S u l p h i d e Gas H y d r a t e S l u r r y " , The Canadian J o u r n a l of Chemical E n g i n e e r i n g , 42, 132-138 (1964). 42. Ree, T., and H. E y r i n g , "Theory of Non-Newt o n i a n Flow, I I . S o l u -t i o n System of H i g h Polymers", J o u r n a l of A p p l i e d P h y s i c s , 2 6 ( 7 ) , 800-809 (1955). 43. R i t t e r , R. A., and G. W. G o v i e r , "The Development and E v a l u a t i o n o f a Theory of T h i x o t r o p i c B e h a v i o r " , The Canadian J o u r n a l of Chemical E n g i n e e r i n g , 48, 505-513 (1970). 44. Simon, T. H., H. G. DeKay, and G. S. Banker, " E f f e c t s of P r o c e s s i n g on t h e Rheology of T h i x o t r o p i c S u s p e n s i o n s " J o u r n a l of P h a r m a c e u t i c a l S c i e n c e s , 50(10), 880-885 (1961). 45. S t a n k o i , G. G., E. B. T r o s t y a n s k a y a , Y. N. K a z a n s k i i , V. V. Okorokov, and 0. Y. Mikhasenok, "Flow of G l a s s F i b r e M o l d i n g M a t e r i a l s " , S o v i e t P l a s t i c s , September, 47-50 (1968). 46. The Dow Chemical Company, " I n f o r m a t i o n B o o k l e t on P o l y e t h y l e n e G l y c o l s " , M i d l a n d , M i c h i g a n (1962). 47. Thurlow, J . L. M., Du Pont of Canada L t d . , P r i v a t e communication to the a u t h o r . 48. T k a d l e t z , F. J . , F i b r e t e x L t d . , P r i v a t e communication to t h e a u t h o r . 49. Umeya, K., T. I s o d a , G. K o i z u m i , and K. Kato, "Some O b s e r v a t i o n s on the Flow B e h a v i o r of Clay-Water S u s p e n s i o n s " , Z a i r y o , 21(224), 469-475 (1972). 50. Vand, V., " V i s c o s i t y o f S o l u t i o n s and S u s p e n s i o n s , I . Theory", J o u r n a l o f P h y s i c a l C h e m i s t r y , _52, 277-299 (1948). 51. Vand, V., " V i s c o s i t y o f S o l u t i o n s and S u s p e n s i o n s , I I . E x p e r i -m ental D e t e r m i n a t i o n of t h e V i s c o s i t y - C o n c e n t r a t i o n F u n c t i o n of S p h e r i c a l S u s p e n s i o n s " , J o u r n a l of P h y s i c a l C h e m i s t r y , 52, 300-321 (1948). 52. Van Wazer, J . R., J . W. Lyons, K. Y. Kim, and R. E. C o l w e l l , V i s c o s i t y and Flow Measurement, A L a b o r a t o r y Handbook of  Rheology, I n t e r s c i e n c e P u b l i s h e r s , New York, N.Y. (1963). 53. V a s e l e s k i , R. C., and A. B. Metzner, "Drag R e d u c t i o n i n t h e T u r b u l e n t Flow of F i b e r S u s p e n s i o n s " , American I n s t i t u t e of Chemical E n g i n e e r s J o u r n a l , 2 0 ( 2 ) , 301-306 (1974). 54. V o c a d l o , J . J . , and M. E. C h a r l e s , "Measurement of Y i e l d S t r e s s of F l u i d - L i k e V i s c o p l a s t i c S u b stances", The Canadian J o u r n a l of Chemical E n g i n e e r i n g , 49, 576-582 (1971). 121 55. Watanabe, T., and Y. T a k a s a k i , "On t h e P r o c e s s o f T h i x o t r o p i c G e l F o r m a t i o n " , N a t u r w i s s e n s c h a f t e n , 45(14), 334-335 (1958). 56. Weymann, H. D., "On the V i s c o s i t y of T h i x o t r o p i c S u s p e n s i o n s " , P r o c e e d i n g s o f the F o u r t h I n t e r n a t i o n a l Congress on Rheology, P a r t 3 (1963), E d i t e d by E. H. Lee, 573-591, I n t e r s c i e n c e P u b l i s h e r s , New York, N.Y. (1965). 57. Z i e g e l , K. D., "The V i s c o s i t y o f Suspensions o f L a r g e , Non-s p h e r i c a l P a r t i c l e s i n Polymer F l u i d s " , J o u r n a l o f C o l l o i d and I n t e r f a c e S c i e n c e , 3 4 ( 2 ) , 185-196 (1970). APPENDICES Appendix A Flow Measurement w i t h D i f f e r e n t Radius R o f Inner C y l i n d e r a t E q u i l i b r i u m A n - . e q u i l i b r i u m f l o w measurement was u n d e r t a k e n f o r the a r t i f i c i a l s l u r r y c o n s i s t i n g o f t h e f i b e r s o f L = 3.01 mm, L/D = 69.8 and p a r t i c l e c o n c e n t r a t i o n <j> = 0.08, and the d i s p e r s i n g medium o f d e x t r o s e + 1.0 mole NaCl i n 10% PEG-B^O by u s i n g the i n n e r c y l i n d e r s o f R = 1.01, 1.10, and 1.20 cm. The r e s u l t s were l i s t e d i n T a b l e A - l . F i g . A - l shows a p l o t of a n g u l a r v e l o c i t y v e r s u s e q u i l i b r i u m shear s t r e s s x. I t can be seen t h e r e t h a t t h e c u r v e s a r e d i s t i n c t f o r each v a l u e o f R. In terms o f e q u i l i b r i u m apparent shear r a t e y and e q u i l i b r i u m shear s t r e s s x, however, t h e p o i n t s a r e on one c u r v e , as shown i n F i g . A-2, a l t h o u g h t h e r e i s a s c a t t e r i n t h e graph due to e x p e r i m e n t a l e r r o r s . A l l t h e i n n e r c y l i n d e r s used h e r e were grooved on t h e i r s u r f a c e s t o e l i m i n a t e the s l i p at the w a l l s as shown e a r l i e r i n F i g . 5-8. The w a l l e f f e c t s were so s m a l l t h a t they were u n o b s e r v a b l e w i t h i n e x p e r i m e n t a l e r r o r a t e q u i l i b r i u m s t a t e . I t can be c o n c l u d e d t h a t the w a l l e f f e c t s a r e , i n f a c t , o f l i t t l e p r a c t i c a l importance i n t h e p r e s e n t system. 122 123 TABLE A - l EQUILIBRIUM FLOW DATA WITH DIFFERENT R R (cm) ( r a d / s e c ) 0.233 0.698 2.09 6.28 Torque (mV) 1.01 1.10 1.20 0.84 0.86 0.88 0.89 1.01 1.02 1.04 1.06 1.20 1.22 1.25 1.26 2 Shear S t r e s s (dyne/cm ) 1.01 1.10 1.20 2.99 x10 s - 3 . 0 5 x l 0 3 3.13x103 3.15x103 3 . 0 1 x l 0 3 3.06x103 3.13x10 s 3.17x103 3.02x10 s 3.07x10 s 3.14x10 s 3.18x10 s R - R RQ " R 1.01 1.10 1.20 0.410 0.438 0.467 0.486 0.383 0.424 0.476 0.567 0.373 0.424 0.488 0.587 R (cm) X oo 1.01 1.10 1.20 1.46 1.49 1.52 1.54 1.48 1.52 1.58 1.67 1.54 1.58 1.64 1.73 Apparent Shear Rate (1/sec) 1.01 1.10 1.20 0.898 2.59 7.49 22.0 1.04 2.91 8.15 22.2 1.19 3.28 9.01 24.7 UJ Q2 0.3 0.4 0.5 1.0 2.0 3.0 4.0 5.0 A n g u l a r V e l o c i t y £ 1 ( r a d . / s e c ) i - 1 F i g . A - l . E q u i l i b r i u m F l o w M e a s u r e m e n t w i t h D i f f e r e n t R a d i u s o f ^ I n n e r C y l i n d e r 3.20 ro O CVJ £ 3.15 (/) C >s "O — 3.10 </> O 3.05 CO E .O 3.00 h 3 LU " i — i i i i i—i—i—i i i L= 3.01 (mm), L / D = 69 .8 , <£=0 .08 DEX+I .O mole NaCl in 10% P E G - H 2 0 R = 1.20 (cm) 1.10' 1.01 J 1 1 1 J I L I I I 1.0 10.0 Equilibrium Apparent Shear Rate y (1/sec.) F i g . A - 2 . E q u i l i b r i u m F l o w C u r v e A p p e n d i x B Supplement t o F i g . 6-6 i n S e c t i o n 6-5 The f o l l o w i n g f i g u r e s show e f f e c t o f d i s p e r s i n g medium v i s c o s i t y on y i e l d s t r e s s f o r v a r i o u s p a r t i c l e l e n g t h s . 126 0.08 0.10 0.12 0.14 0.16 P a r t i c l e C o n c e n t r a t i o n ^ ( c c / c c ) F i g . B - l . E f f e c t o f D i s p e r s i n g M e d i u m V i s c o s i t y o n Y i e l d S t r e s s 4 2 Y i e l d S t r e s s ( x l O d y n e s / c m ) ro OQ bd I ho w l-h l-h CD O O f-h 01 (D H cn H-3 OQ s (D H-C a cn o o cn O (D W rt i-i CD cn cn T J Q CD O O O CD Q O o o o o o b -4> o o CT) o b oo p o D> • O o u CO Q cn w 5 O O O " m X o N ^ U l >1 w 8 & p • OJ tn o o to — ft) CO — r-O \ 3 ? Q o 00 \\ O 8ZT 0.04 0.05 0.06 0.07 0.08 0.09 Particle Concentration <& (cc/cc) r—1 VO F i g . B - 3 . E f f e c t o f D i s p e r s i n g M e d i u m V i s c o s i t y o n Y i e l d S t r e s s >- 0 I I i i i i I 0.04 0.05 0.06 0.07 0.08 0.09 P a r t i c l e C o n c e n t r a t i o n <f> ( c c / c c ) F i g . B - 4 . E f f e c t o f D i s p e r s i n g M e d i u m V i s c o s i t y o n Y i e l d S t r e s s o A p p e n d i x C C o m p u t e r P r o g r a m m e s C - l P r o g r a m m e " B E S T F I T " 1 3 1 13 2 C . . . c C MAIN PROGRAM " B E S T F I T " C C T H I S PROGRAM E V A L U A T E S F I T T I N G P ARAMETERS OF V A R I O U S R E A C T I O N C ORDERS OF F I N D E R ' S MODEL, E Y R I N G* S MODEL, AND G O V I E R ' S C O R I G I N A L F I R S T ORDER - SECOND ORDER MODEL I N TERMS OF T I M E C AND SHEAR S T R E S S AT THE WALL BY MEANS OF L E A S T SQUARES METHOD C WITH A SET OF E X P E R I M E N T A L DATA OF A S I N G L E RUN. C-C HOW TO C O M P I L E T H I S PROGRAM? C I N ORDER TO C O M P I L E THE PROGRAM " B E S T F I T " INTO THE C I B M V E R S I O N FORTRAN G - L E V E L C O M P I L E R UNDER CONTROL OF C M I C H I G A N T E R M I N A L SYSTEM (MTS) FOR THE I B M 3 7 0 C MODEL 168 COMPUTER AT THE U N I V E R S I T Y OF B R I T I S H C C O L U M B I A , T f i E COMMAND C C $RUN * F T N S C A R D S = B E S T F I T SPUNCH=-OBJECT C C SHOULD BE USED. C C HOB TO E X E C U T E THE O B J E C T DECK? C THE COMMAND C C $RDN - O B J E C T + * P R P L O T 4=BFORDER 5=1NPDTDATA C C SHOULD B E USED, WHERE "BFORDER" I S THE F I L E WHICH SHOOID C EE READ THROUGH L O G I C A L I/O U N I T 4 BY A - T Y P E FORMAT, C AND A L L THE I N P U T DATA MUST BE C O N S I S T E N T WITH, AND C CORRESPOND TO THE ORDER AND FORMAT I N D I C A T E D I N THE MAIN C PROGRAM. C C D E S C R I P T I O N OF P A R A M E T E R S . C C I E N D - NUMBER OF E X P E R I M E N T A L DATA. C J E N D - NUMBER OF P I N D E R ' S MODEL C KEND - NUMBER OF L E T T E R S I N THE A-FORM AT. C LEND - MAXIMUM LENGTH / KEND. C MEND - NUMBER OF T H E O R E T I C A L POINTS TO BE P L O T T E D . C TORQT - E Q U I L I B R I U M V A L U E OF TORQUE (DYNES C M . ) . C TORQZ - I N I T I A L TORQUE OF GEL STRUCTURE C (DYNES CM.) . C T A I - E Q U I L I B R I U M V A L U E OF SHEAR S T R E S S {DYNES / S Q . C M . ) . C TAZ - I N I T I A L SHEAR S T R E S S OF GEL STRUCTURE C (DYNES / SQ.CM. ) . C T O R Q ( I ) - E X P E R I M E N T A L V A L U E S OF TORQUE (DYNES CM.) . C T ( I ) - T I M E ( S E C . ) , C T A U ( I ) - E X P E R I M E N T A L V A L U E S OF SHEAR S T R E S S C (DYNES / SQ.CM.) . C O R D E R ( J , L ) - R E A C T I O N ORDERS OF P I N D E R » S MODEL. C R C O N S ( J ) - FORWARD RA T E CONSTANTS OF P I N D E R ' S MODEL,. C TAUT(M) - P I N D E R ' S SHEAR S T R E S S (DYNES / S Q . C M . ) . C TAWEYR(M) - E Y R I N G ' S SHEAR S T R E S S (DYNES / S Q . C M . ) . C TAWGOV(M) - G O V I E R ' S SHEAR S T R E S S (DYNES / S Q . C M . ) . C C •. . . 1 3 3 C D I M E N S I O N T A U { 1 0 ) , T ( 1 0 ) , V { 8 , 1 0 ) , W { 8 , 1 0 ) , W L ( 1 0 ) , Y F L ( 1 0 ) , WWL{1 1 0 ) , E 1 I ( 1 ) , E 2 L ( 1 ) , P L ( 1 ) , E 1 { 8 ) , E2 (8) , NDQ <8) , P ( 8 ) , ORDER ( 8 , 6 ) , 2 R C O N S { 8 ) , T T ( 5 1 ) , 0 ( 5 1 ) , T A U T { 5 1 ) , T A U T L { 5 1 ) , T A U L { 1 0 ) , T C B Q { 1 0 ) 3 , P E Y R ( 1 ) , E 2 E Y R {1) , P G O V I ( 2 ) , E 2 G O V I (2) 4, TAWEYR ( 5 1 ) , P 2 { 5 1 ) , P 3 { 5 1 ) , P 4 { 5 1 ) , P 5 ( 5 1 ) , T A W G O V ( 5 1 ) , 5 T A W E L { 5 1 ) , TAWGL ( 5 1 ) , T I N F ( 1 ) , T A I L L (1) C E X T E R N A L AUX I E N D = 10 J E N D = 8 KEND = 4 LEND = 6 MEND = 51 C C I N P U T OF DATA ..... READ ( 5 , 1 0 0 ) T O R Q I , TORQZ READ ( 5 , 1 0 2 ) ( T ( I ) , 1 = 1,IEND) READ ( 5 , 1 0 3 ) ( T O R Q ( I ) , 1=1,TEND) READ ( 4 , 1 0 9 ) ( (ORDER ( J , L ) , L = 1 , L E N D ) , J = 1,JEND) C C ..... C A L C U L A T I O N FOR P I N D E R * S MODEL R = 1.010 RL = 1 . 9 6 T A I = TORQI / ( 2 . 0 * 3 . 1 4 1 6 * ( R * * 2 ) * RL) TAZ = TORQZ / ( 2 . 0 * 3 . 1 4 1 6 * ( R * * 2 ) * RL) C DO 11 1 = 1 , I E N D TAD ( I ) = T O R Q ( I ) / ( 2 . 0 * 3 . 1 4 1 6 * ( R * * 2 ) * RL) T A U L ( I ) = ALOG10 (ABS (TAU ( I ) ) ) V ( 1 , I ) = (TAU ( I ) - T A I ) / ( T A Z - T A I ) V ( 2 , I ) = ( T A I - TAU ( I ) ) / ( 2 . 0 * TAZ - T A I - T A U ( I ) ) V ( 3 , I ) = (TAZ - T A I ) / (TAU ( I ) - T A I ) V ( 4 , I ) = ( T A Z - TAX) / ( T A I - TAU ( I ) ) V ( 5 , I ) = ( - T A U ( I ) * T A I + T A I * * 2 ) / (TAZ * (TAU ( I ) - T A I ) ) V ( 6 , I ) = ( T A U ( I ) + T A I ) * ( T A Z - T A I ) / ( T A U ( I ) - T A I ) / (TAZ + TA 11) V ( 7 , I ) = (TAZ - T A I ) * ( ( T A Z - T A I ) * TAU ( I ) + TAZ * T A I ) / { ( T A Z * 1*2) * ( T A I - TAU ( I ) ) ) V ( 8 , I ) = (TAZ * T A I + (TAZ - 2. 0 * T A I ) * TAU ( I ) ) / ( T A Z * ( T A I -1 TAU ( I ) ) ) 11 C O N T I N U E C DO 20 J = 1 , J E N D DO 21 1 = 1 , I E N D H ( J , I ) = A L O G ( A B S ( V ( J , I ) ) ) 21 C O N T I N U E 2 0 C O N T I N U E C WZ = 0.0 EP = 0 . 0 0 0 5 WRITE ( 6 , 1 1 2 ) C DO 22 J = 1 , J E N D WRITE ( 6 , 1 2 3 ) WRITE ( 6 , 1 0 5 ) (ORDER { J , L) , L= 1, LEND) 134 C C DO 23 I=1,IEND WL ( I ) = W ( J , I ) 2 3 C O N T I N U E WRITE ( 6 , 1 0 6 ) DO 25 I = 1 , I E N D WRITE ( 6 , 1 0 4 ) J , T ( I ) , ( I ) 25 C O N T I N U E PL (1) = 0.0 NI = +20 ..... L E A S T SQUARES F I T FOR P I N D E R ..... C A L L L Q F ( T , WL, Y F L , W W L , E 1 L , E 2 L , P L , WZ, T E N D , 1,N.I, ND, E P , AU 1X) E1 ( J ) = E 1 L (1) E2 ( J ) = E 2 L (1) NDQ(J) = ND P ( J ) = P L ( 1 ) 2 2 C O N T I N U E C A L C U L A T I O N OF R A T E CONSTANTS FOR P I N D E R ' S MODEL E C C N S ( 1 ) = - P ( 1 ) * (TAZ - T A I ) R C O N S ( 2 ) = P ( 2 ) * ( T A Z - TA I ) / (-2.0) R C G N S ( 3 ) = P ( 3 ) R C C N S ( 4 ) = P ( 4 ) * (TAZ ~ T A I ) / TAZ R C O N S ( 5 ) = - P ( 5 ) * (TAZ - T A I ) / (TAZ + T A I ) R C O N S ( 6 ) = P ( 6 ) / 2.0 / T A I R C 0 N S ( 7 ) = P ( 7 ) * ( T A Z - T A I ) / ( T A I * ( 2 . 0 * TAZ - T A I ) ) R C C N S ( 8 ) = P ( 8 ) * (TAZ - T A I ) / ( 2 . 0 * T A I * TAZ) WRITE ( 6 , 1 0 1 ) DO 24 J = 1 , J E N D WRITE ( 6 , 1 0 7 ) (ORDER ( J , L) , L= 1, LEND) , J , E1 ( J ) , E2 ( J ) , NDQ {J) , P ( J 1 ) , R C O N S ( J ) 24 C O N T I N U E TM AX = T ( 1 0 ) T I N C = THAX / 5 0 . 0 T T ( 1 ) = 0.0 Q ( 1 ) = (TAZ + T A I ) / (TAZ - T A I ) T A U T ( 1 ) = TAZ DO 26 M=2,MEND TT (M) = T I N C * ( F L O AT {M - 1 ) ) Q(M) = ( { T A Z + T A I ) / ( T A Z - T A I ) ) * E X P ( P ( 6 ) * TT (M) ) TAUT (M) = { (Q (M) + 1.0 ) / (Q (M) - 1 . 0 ) ) * T A I 26 C O N T I N U E WRITE ( 6 , 1 1 0 ) DO 27 M=1,MEND T A U T L ( H ) = ALOG10 (ABS (TAUT (M) ) ) WRITE ( 6 , 1 1 1 ) TT (M) , Q (M) , TAUT (M) , T A U T L (M) 135 2 7 C O N T I N U E C C MAXIMUM AND MINIMUM FOB THE PLOTS OF THE R E S U L T S T A Z L = ALOG10 (ABS (TAZ) ) T A I L = ALOG 1 0 { A B S ( T A I ) ) D A I = T A Z L SflO = T A I L SA = ABS ( D A I - SHO) DDAI = DAI + 0.1 * SA SSHO = SHO - 0.1 * SA C C PLOT OF T I M E V S . L O G 1 0 ( S H E A R S T R E S S ) C A L L A L A X I S (» T I M E ( S E C . ) ' , 1 6 , • L O G 1 0 SHEAR S T R E S S (DYN 1ES / S Q . C M . ) ' , 40) C A L L A L S C A L ( 0 . 0 0 , TMAX, SSHO, DDAI) C A L L A L G R A F ( T T , T A U T L , MEND, 0) C A L L A L S C A L { 0 . 0 0 , TM AX, SSHO, DDAI) C A L L A L G R A F ( T , T A U L , - ( I E N D ) , - 4 ) C C C A L C U L A T I O N FOR E Y R I N G C c C A L L E Y R I N G ( T A U , T, T A I , T A Z , PE YR , E2EYR) c C A L C U L A T I O N FOR G O V I E R ,, C C C A L L G O V I E R ( T A U , T, T A I , T A Z , P G O V I , E 2 G O V I , AKKA) DO 30 M= 1 ,59END TAWEYR (M) = (TAZ - T A I ) * EXP ( (-PEYR ( 1) ) * TT (M) ) + T A I P1 = -AKKA * ( T A Z + T A I ) / {TAZ - T A I ) P 2 ( M ) = P G O V I (2) * TT (M) + 1.0 P3 ( M ) = ALOG ( A B S ( P 2 (M) ) ) P 4 ( M ) = P1 * P 3 ( M ) P5 (M) = EXP (P4 (M) ) TAWGOV(M) = ( T A I + TAZ * P5 (M) ) / ( 1 . 0 + ( T A I / TAZ) * P5 (fl) ) TAWEL(M) = ALOG10 (ABS (TAWEYR (M) ) ) TA WGL (M) = A L O G 1 0 ( A B S ( T A W G O V (M) ) ) 30 C O N T I N U E C WRITE ( 6 , 1 2 1 ) C DO 31 M=1,MEND WRITE ( 6 , 1 2 0 ) TT (M) , TAUT (M) , TAWEYR ( M ) , TAWGOV(M), T A U T L (M) , 1 TAW EL (M) , TAWGL (M) 31 C O N T I N U E C C PLOT OT T I M E VS. LOG 1 0 ( S H E A R S T R E S S ) ..... C A L L A L A X I S (• T I M E ( S E C . ) ' , 1 6 , ' LOG 10 SHEAS S T R E S S (DYN 1ES / S Q . C M . ) ' , 40) C A L L A L S C A L ( 0 . 0 0 , TMAX, SSHO, B E A I ) C A L L A L G R A F ( T T , T A U T L , MEND, 0) C A L L A L S C A L ( 0 . 0 0 , TMAX, SSHO, DDAI) C A L L ALGRAF" ( T , T A U L , - ( I E N D ) , - 4 ) C A L L A L S C A L ( 0 . 0 0 , TM A X , SSHO, DDAI) C A L L A L G R A F ( T T , TAWEL, - (MEND) , - 2 ) C A L L A L S C A L ( 0 . 0 0 , TM AX, SSHO, DDAI) 136 C C C C CALL ALGRAF (TT, TAWGL, -(MEND), -5) TINF(1) = TM AX TAIIL(1) = TAIL CALL ALSCAL (0.00, TM AX, SSHO, DDAI) CALL ALGRAF (TINF, TAILL, -1, -12) FORMATS FOR INPUT AND OUTPUT STATEMENTS 100 FORMAT (2E10.3) 101 FORMAT (1H1, //, 32X, »3', 8X , ' E1 1P', 14X , 'RCONS') 102 FORMAT (8F10. 1) 103 FORMAT (8E10.2) 104 FORMAT (20X, 15, 15X, 2E20.4) 105 FORMAT (//////////, 10X, 6A4) 106 FORMAT (/, 24X, «J», 25X, »T' , 19X 107 FORMAT (1H0, 5X, 6A4, 13, 2E15.4, 109 FORMAT (6A4) 110 FORMAT (1H1, ////, 33X, * TT* , 18X, 111 FORMAT (1H0, 2OX, 4E20.3) 112 FORMAT (1H1) 120 FORMAT (1H0, 20X, 7E15.4) 121 FORMAT (1H1, ///, 25X, 'TT* , 13X, 1'TAWGOV V, 9.X# ' TAWTL * , 10X, • TAWEL 123 FORMAT (1H1) 16X, ' TAUTL •) TO TERMINATE PLOTTING CALL PLOTND C C C C c c c c c c c c c c c STOP END FUNCTION AUX (P, D, X, L) THIS FUNCTION DEFINES THE FORM OF PINDER'S EQUATION. DIMENSION P (1) , D(1) AUX = P (1) * X D(1) = X RETURN END SUBROUTINE EYRING (TAU, T, TAI, TAZ, P, E2) THIS SUBROUTINE EVALUATES THE PARAMETER OF EYRING'S MODEL BY MEANS OF LEAST SQUARES FITTING. DIMENSION TAU{10), T{10), R1(10), R2(10), YF(10), WT(10), E1(1), 1E2(1), P(1) EXTERNAL AUXEYR TEND = 10 137 WRITE ( 6 , 1 0 3 ) C DO 10 1=1,TEND R 1 ( I ) = (TAU ( I ) - T A I ) / ( T A Z - T A I ) R2 ( I ) = ALOG ( R 1 ( I ) ) WRITE ( 6 , 1 0 4 ) T ( I ) , R2 ( I ) , TAU ( I ) 10 C ONTINUE C P ( 1 ) = 0.0 WZ = 0.0 E P S = 0 . 0 0 0 5 N I = +20 C A L L L Q F ( T , R 2 , Y F , WT, E 1 , E 2 , P, WZ, I END, 1, N I , ND, E P S , 1AUXEYR) WRITE ( 6 , 1 0 1 ) WRITE ( 6 , 1 0 2 ) E 1 { 1 ) , E 2 ( 1 ) , ND, P { 1 ) C C ..... FORMATS FOR I N P U T AND OUTPUT STATEMENTS 1 0 1 FORMAT (//////, 2 8 X , V E T , 1 3 X , * E 2 ' , 1 3 X , 1 ND' , 1 3 X , »P') 1 0 2 FORMAT ( 1 H 0 , 1 0 X , * E Y R I N G *, 4 X , 2 E 1 5 . 4 , 1 1 5 , E 1 5 . 4 ) 1 0 3 FORMAT ( 1 H 1 , / / / , 3 0 X , »T», 1 9 X , » R 2 » , 1 8 X , ' T A U 1 , //) 1 0 4 FORMAT ( 2 0 X , 3 E 2 0 . 4 ) RETURN END F U N C T I O N AUXEYR ( P , D, X, L ) C C . . C C T H I S F U N C T I O N D E F I N E S THE FORM OF E Y R I N G ' S EQUATION. C C . c D I M E N S I O N P (1) , D{1) AUXEYR = - P ( 1 ) * X D ( 1 ) = -X RETURN END S U B R O U T I N E G O V I E R ( T A U , T, T A I , T A Z , P, E 2 , AKKA) C C C C T H I S S U B R O U T I N E E V A L U A T E S THE PARAMETERS OF G O V I E R ' S MODEL C BY MEANS OF L E A S T SQUARES F I T T I N G . C C C D I M E N S I O N T A U { 1 0 ) , T ( 1 0 ) , Y 1 { 1 0 ) , Y 2 ( 1 0 ) , Y 3 ( 1 0 ) , Y 4 ( 1 0 ) , Y F ( 1 0 ) , 1 WT (10) , P ( 2 ) , E l (2) , E 2 ( 2 ) E X T E R N A L AUXGOV I E N D = 10 WRITE ( 6 , 1 0 1 ) WRITE ( 6 , 1 0 5 ) C DO 10 1 = 1 , I E N D Y1 ( I ) - TAU ( I ) - T A I Y 2 ( I ) = TAZ - TAU ( I ) * T A I / TAZ 138 Y3 ( I ) = Y1 ( I ) / Y2 ( I ) Y4 ( I ) = ALOG ( Y 3 {I) ) WRITE ( 6 , 1 0 2 ) T ( I ) , Y 1 ( I ) , Y4 ( I ) , TAU ( I ) 10 C O N T I N U E C P ( 1 ) = 0.5 P { 2 ) = 0.5 WZ = 0.0 NI = +50 M = 2 EP S = 0 . 0 0 0 5 C A L L LQ.F ( T , Y4 , Y F , WT, E1 , E 2 , P, WZ, I E N D , fl, N I , ND, E P S , 1 AUXGOV) WRITE ( 6 , 1 0 3 ) AKKA = - P ( 1 ) * (TAZ - T A I ) / (TAZ + T A I ) WRITE ( 6 , 1 0 4 ) E 1 { 1 ) , E 1 { 2 ) , E 2 ( 1 ) , E2 (2) , ND, P ( 1 ) , P (2) , AKKA C C FORMATS FOR I N P U T AND OUTPUT ST A T E M E N T S 101 FORMAT ( 1 H 1 , / / / , 3 0 X , * T', 1 9 X , ' 1 1 ' , 1 8 X , ' Y 4 ' , 1 8 X , »TA0») 10 2 FORMAT ( 2 0 X , 4 E 2 0 . 4 ) 1 0 3 FORMAT {//////, 1 7 X , « E 1 { 1 ) ' , 1 0 X , » E 1 { 2 ) ' , 1 0 X , V E 2 ( 1 ) » , 1 0 X , 1'E2(2)«, 1 0 X , «ND«, 1 3 X , ' P ( 1 ) ' , 11X, • P {2) », 1 1 X , 'AKKA') 104 FORMAT ( 1 H 0 , 4 X , ' G O V I E R ' , 4 E 1 5 . 4 , 1 1 5 , 3 E 1 5 . 4 ) 1 0 5 FORMAT (1H0) RETURN END F U N C T I O N AUXGOV ( P , D, X, L) C C C C T H I S F U N C T I O N D E F I N E S THE FORM OF G O V I E R ' S EQUATION. C C C D I M E N S I O N P ( 2 ) , D ( 2 ) F X Y Z = ABS (P (2) * X + 1.0) D ( 1 ) = ALOG ( F X Y Z ) AUXGOV = P { 1 ) * D ( 1 ) D ( 2 ) = <P{1) * X) / F X Y Z RETURN END 139 C - 2 P r o g r a m m e " S U P L P L T " c C MAIN PROGRAM " S U P L P L T " C C T H I S PROGRAM C A L C U L A T E S T H I C K N E S S OF FLOWING L A Y E R RX C AND SHEAR S T R E S S AT WALL FROM A SET OF E X P E R I M E N T A L DATA C OF A S I N G L E RUN, AND THEN T R I E S TO F I T THE DATA BY THE C MODEL P R E D I C T I N G T H I C K N E S S OF FLOWING L A Y E R AND BY C M O D I F I E D G O V I E R ' S (SECOND ORDER -ZERO ORDER) S T R E S S DECAY C MODEL, BY MEANS OF L E A S T SQUARES F I T E V A L U A T I N G THE F I T T I N G C PARAMETERS I N EACH OF THE MODELS. C C HOW TO C O M P I L E T H I S PROGRAM? C I N ORDER TO C O M P I L E THE PROGRAM " S U P L P L T " INTO THE C I B M V E R S I O N FORTRAN G - L E V E L C O M P I L E R UNDER CONTROL OF C M I C H I G A N T E R M I N A L SYSTEM (MTS) FOR THE I B M 3 7 0 C MODEL 168 COMPUTER AT THE U N I V E R S I T Y OF B R I T I S H C , C O L U M B I A , THE COMMAND C C $RUN * F T R S C A R D S = S U P L P L T SPUNCH=-OBJECT C C SHOULD BE USED. C C HOW TO E X E C U T E THE O B J E C T D E C K ? C THE COMMAND C C $RUN - O B J E C T + * P R P L O T 5 = I N P U T D A T A C C SHOULD BE USED, WHERE " I N PUT DAT A" I S THE F I L E WHICH C C O N T A I N S THE I N P U T DATA, WHCSE ORDER AND FORMAT MUST C BE C O N S I S T E N T WITH, AND CORRESPOND TO THOSE OF C THE D E S C R I P T I O N S T A T E D I N T H E MAIN PROGRAM. C C D E S C R I P T I O N OF P A R A M E T E R S C I E N D - NUMBER OF DECAY P O I N T S P I C K E D UP FROM THE C DECAY CURVE (-) . C MEND - NUMBER OF BLOCKS OF A- FORM AT FOR I N P U T (-) . C NEND - NUMBER OF T H E O R E T I C A L POINTS ( - ) . C DATE(M) - DATE OF E X P E R I M E N T {-) . C R A T I O Z - (FX - R) / (RO - R) AT T I M E = 0 . 0 (-) . C RATIOQ - (RX - R) / (RO - R) AT TIME= I N F I N I T Y ( - ) . C T I M E ( I ) - T I M E OF SHEAR ( S E C ) . C M V ( I ) - R E A D I N G S ON RECORDER ( M i l l V O L T ) . C R A T I O ( I ) - ( R X ( I ) - R) / (RO - R) ( - ) . C RO - R A D I U S OF OUTER C Y L I N D E R ( C M ) . C R - R A D I U S OF I N N E R C Y L I N D E R ( C M ) . C R X ( I ) - R A D I U S OF MOVING L A Y E R ( C M ) . C S ( I ) - R / RX ( I ) {-) . C TAU ( I ) - SHEAR S T R E S S AT THE S U R F A C E OF INNER C Y L I N D E R C (DYN ES/SQ. CM) . C T O R Q ( I ) - TORQUE (DYNES C M ) . C GAMMA(I) - APPARENT SHEAR R A T E AT THE S U R F A C E C OF I N N E R C Y L I N D E R ( 1 / S E C ) . C 141 HEAL MVZ, MVQ, M V ( 1 0 ) D I M E N S I O N DATE ( 5 ) , T I M E ( 1 0 ) , R X ( 1 0 ) , S ( 1 0 ) , R A T I O ( 1 0 ) 1, T O B Q ( 1 0 ) , T A U { 1 0 ) , P G O V I { 2 ) , E 2 G O V I ( 2 ) 2, T T ( 5 1 ) , P 2 ( 5 1 ) , P 3 ( 5 1 ) , P 4 { 5 1 ) , P 5 < 5 1 ) , S G O V ( 5 1 ) 3 , P M I T ( 2 ) , E 2 M I T ( 2 ) , Z 1 ( 5 1 ) , £ 2 ( 5 1 ) , Z 3 ( 5 1 ) , Z4 ( 5 1 ) , Z 5 ( 5 1 ) ,Z6 (51) 4, GAMMA ( 1 0 ) , BUMBO ( 1 0 ) , 5 P P G O V { 2 ) , E 2 P G O V ( 2 ) , T A U G O V ( 5 1 ) , P P 2 ( 5 1 ) , P P 3 { 5 1 ) , P P 4 ( 5 1 ) , 6 P P 5 (51) , V I S C O ( 1 0 ) C C READ E X P E R I M E N T A L DATA ..... ( D A T E { M) , M=1,MEND) R A T I O Z , RATIOQ MVZ, MVQ ( T I M E ( I ) , 1 = 1 , I END) (MV ( I ) , 1 = 1 , I E N D ) ( R A T I O ( I ) , 1=1,IEND) R, OMEGA P L E N G T , P H I C C C A L C U L A T I O N FOR RX ( I ) AND S ( I ) RO = 2.1 EXQ = R A T I O Q * (SO - R) + R SQ = R / RXQ DO 11 1 = 1 , I E N D R X ( I ) = R A T I O ( I ) * (RO - H) • R S ( I ) = R / RX ( I ) 11 C O N T I N U E I E N D = 10 MEND = 5 NE N D = 51 READ ( 5 , 1 0 1 ) READ ( 5 , 1 0 2 ) READ ( 5 , 1 0 2 ) READ ( 5 , 1 0 3 ) READ ( 5 , 1 0 4 ) READ ( 5 , 1 0 5 ) READ ( 5 , 1 0 6 ) READ ( 5 , 1 0 6 ) T I M E Z = 0 . 0 SZ = 1.0 T I M E S A = T I M E ( I E N D ) T I M A X = T I M E ( I E N D ) + 0.1 * T I M E S A T I M I N = - 0 . 1 * T I M E S A S S A = 1.0 - SQ SMAX = 1.0 + 0.1 * SSA S M I N = SQ - 0. 1 * S S A C C C A L C U L A T E SHEAR S T R E S S TAD ( I ) RL = 1.96 TORQZ = ( 4 . 4 6 E + 0 4 ) * MVZ TORQQ = ( 4 . 4 6 E + 0 4 ) * MVQ TAUZ = TORQZ / ( 2 . 0 * 3 . 1 4 1 6 * ( R * * 2 ) * RL) TAUQ = TORQQ / ( 2 . 0 * 3 . 1 4 1 6 * ( R * * 2 ) * RL) DO 12 1=1,1 END T O R Q ( I ) = ( 4 . 4 6 E + 0 4 ) * M V ( I ) TAU ( I ) = T O R Q ( I ) / ( 2 . 0 * 3 . 1 4 1 6 * ( R * * 2 ) * RL) 12 C O N T I N U E C C ..... PLOT T I M E VS TAU ..... TAUSA = TAUZ - TAUQ TAUMAX = TAUZ + 0.1 * TAUSA T A U M I N = TAUQ - 0.1 * TAUSA C A L L A L S I Z E ( 7 . 0 , 5.0) 142 C A L L A L A X I S (» T I M E (SEC) », 1 3 , 1« TAU ( D Y N E S / S Q . C M ) ' , 20) C A L L A L S C A L ( T I M I N , T I M AX, T A U M I N , TAUMAX) C A L L A L G R A F ( T I M E , T A U , ( I E N D ) , - 4 ) C A L L A L S I Z E ( 7 . 0 , 5.0) C A L L A L S C A 1 ( T I M I N , T I M A X , T A U M I N , TAUMAX) C A L L A L G R A F ( T I M E Z , T A U Z , - 1 , -3) C A L L A L S I Z E ( 7 . 0 , 5.0) C A L L A L S C A L ( T I M I N , T I M A X , T A U M I N , TAUMAX) C A L L A L G R A F ( T I M E S A , TAUQ, - 1 , -3) C C ..... C A L C U L A T E T H E O R E T I C A L T I M E TM AX = T I M E ( 1 0 ) T I N C = TMAX / 5 0 . 0 T T { 1 ) = 0.0 DO 13 N=2,NEND TT (N) = T I N C * ( F L O A T (N - 1 ) ) 13 C O N T I N U E C RXQ = RATIOQ * (RO - R) + R RXSA = RXQ - R RXMAX = RXQ + 0 . 1 * RXSA R X M I N = R - 0.1 * RXSA C C L E A S T SQUARES F I T TO M O D I F I E D G O V I E R ' S E Q U A T I O N ..... C A L L G O V I E R ( T A U , T I M E , TAUQ, T A U Z , PPGOV, E 2 P G O V , AKP) DO 18 N= 1 , N EN D PP 1 = - 2 . 0 * TAUQ * AKP P P 6 = (TAUZ + TAUQ) / (TAUZ -TAUQ) P P 2 ( N ) = PPGOV (2) * TT (N) + 1.0 P P 3 ( N ) = ALOG (ABS ( P P 2 (N) ) ) P P 4 ( N ) = PP 1 * P P 3 ( N ) P P 5 (N) = EXP ( P P 4 (N) ) TAUGOV(N) = TAUQ * ( P P 6 + P P 5 (N) ) 1/ ( P P 6 - P P 5 ( N ) ) 18 C O N T I N U E C C ..... PLOT E X P E R I M E N T A L DATA AND M O D I F I E D G O V I E R ' S L I N E I N C TERMS OF T I M E VS. TAU C A L L A L S I Z E ( 7 . 0 , 5.0) C A L L A L A X I S {' T I M E ( S E C ) ' , 1 3 , 1' TAU (DYNES/SQ. CM) ', 20) C A L L A L S C A L ( T I M I N , T I M A X , T A U M I N , TAUMAX) C A L L A L G R A F ( T I M E , T A U , ( I E N D ) , -4) C A L L A L S I Z E ( 7 . 0 , 5.0) C A L L A L S C A L ( T I M I N , T I M A X , T A U M I N , TAUMAX) C A L L A L G R A F ( T I M E Z , T A U Z , - 1 , - 3 ) C A L L A L S I Z E ( 7 . 0 , 5.0) C A L L A L S C A L ( T I M I N , T I M A X , T A U M I N , TAUMAX) C A L L A L G R A F ( T I M E S A , TAUQ, - 1 , - 3 ) C A L L A L S I Z E ( 7 . 0 , 5.0) C A L L A L S C A L ( T I M I N , T I M A X , T A U M I N , TAUMAX) C A L L A L G R A F ( T T , TAUGOV, - ( N E N D ) , 0) C C PLOT T I M E VS RX C A L L A L S I Z E ( 7 . 0 , 5.0) CALL ALAXIS (' TIME (SEC)' , 13, ' RX CALL ALSCAL (TIMIN, TI MA X, RXMIN, RXMAX) CALL ALGRAF (TIME, RX, (IEND) , -4) CALL ALSIZE (7.0, 5.0) CALL ALSCAL (TIMIN, TI M AX, RXMIN, RXMAX) CALL ALGRAF (TIMEZ, R, - 1 , -3) CALL ALSIZE (7.0, 5.0) CALL ALSCAL (TIMIN, TI MAX, RXMIN, RXMAX) CALL ALGRAF (TIMESA, RXQ, - 1 , -3) (CM) 10) . . . . . L E A S T SQUARES FIT OF THE EXPERIMENTAL VALUES OF RX TO THE MODEL EQUATION CALL MIT (RX, TIME, RXQ, R, RO, PMIT, E2MIT, AK) CALCULATE THEORETICAL VALUES OF RX PPL = (RO**2 - RXQ**2) **2 ROR = RO**2 - R**2 B = -2.0 * ROR C = ROR**2 - PPL D = SQRT (B**2 - 4.0 * C) DO 15 N= 1,NEND Z1(N) =" 1.0 + PMIT(2) * TT (N) Z2(N) = Z1(N) ** PMIT(1) Z3 (N) = Z2(N) * ABS((B - D) / (B + D) ) Z4<N) = D * (1.0 + Z3(N)) / (1.0 - Z3 (N) ) Z5 (N) = (Z4 (N) - B) / 2.0 Z6<N) = SQRT(Z5(N) + R**2) CONTINUE OUTPUT OF THE RESULTS AND PLOT OF EXPERIMENTAL VALUES AND THE MODEL LINE IN TERMS OF TIME VS RX . . . . . WRITE (6,131) DO 16 N=1,NEND WRITE (6,132) TT (N) , Z1(N), Z2(N), Z3(N), Z4(N), Z5(N), Z6(N) 1, TAUGOV(N) CONTINUE CALL ALSIZE CALL ALAXIS (\ TIME (SEC)' , 13, ' RX (CM)', 10) CALL ALSCAL CALL ALGSAF CALL ALSIZE CALL ALSCAL CALL ALGRAF CALL ALSIZE CALL ALSCAL CALL ALGRAF CALL ALSIZE CALL ALSCAL CALL ALGRAF (7.0, 5.0) V EC) •  RX (TIMIN, TIMAX, RXMIN, RXMAX) (TIME, RX, (IEND) , -4) (7.0, 5.0) (TIMIN, TIMAX, RXMIN, RXMAX) (TIMEZ, R, - 1 , -3) (7.0, 5.0) (TIMIN, TI MAX, RXMIN, RXMAX) (TIMES A, RXQ, - 1 , -3) (7.0, 5.0) (TIMIN, TIMAX, RXMIN, RXMAX) (TT, Z6, - (NEND) , 0) CALCULATE APPARENT SHEAR RATE GAMMA(I) DO 19 1=1,I END BUMBO (I) = 1.0 - (S(I)**2) GAMMA (I) = 2.0 * OMEGA / BUMBO (I) VISCO(I) = TAU (I) / GAMMA (I) CONTINUE 144 C OUTPUT OF THE R E S U L T S WRITE ( 6 , 1 3 3 ) DO 17 1 = 1 , I E N D WRITE ( 6 , 1 3 4 ) T I M E ( I ) , T O R Q ( I ) , T A U ( I ) , GAM MA ( I ) , V I S C O ( I ) , RX ( I ) 17 C O N T I N U E WRITE ( 6 , 1 4 0 ) RXQ, T A U Z , TAUQ C C ..... PLOT GAMMA VS TAU GAMQ = 2.0 * OMEGA / ( 1 . 0 - SQ**2) GA MSA = GAMMA (1) - GAMQ GAMAX = GAMMA(1) + 0 . 1 * GAMSA GAMIN = GAMQ - 0.1 * GAMSA C A L L A L S I Z E ( 7 . 0 , 5.0) C A L L A L A X I S (* GAMMA ( 1 / S E C ) * , 1 6 , ' TAU (D YNE S/SQ .CM) » , 20) C A L L A L S C A L ( G A M I N , GAMAX, T A U M I N , T A UMAX) C A L L A L G R A F (GAMMA, T A U , ( I E N D ) , - 4 ) C A L L A L S I Z E ( 7 . 0 , 5.0) C A L L A L S C A L ( G A M I N , GAMAX, T A U M I N , TAUMAX) C A L L A L G R A F (GAMQ, TAUQ, - 1 , -3) C C PLOT OF T I M E VS GAMMA ..... C A L L A L S I Z E ( 7 . 0 , 5.0) C A L L A L A X I S (' T I M E ( S E C ) ' , 1 3 , » GAMMA ( 1 / S E C ) * , 16) C A L L A L S C A L ( T I M I N , T I M A X , G A M I N , GAMAX) C A L L A L G R A F ( T I M E , GAMMA, ( I E N D ) , - 4 ) C A L L A L S I Z E ( 7 . 0 , 5.0) C A L L A L S C A L ( T I M I N , T I M A X , G A M I N , GAMAX) C A L L A L G R A F ( T I M E S A , GAMQ, - 1 , -3) C C FORMATS FOR I N P U T AND OUTPUT STATEMENTS 101 FORMAT ( 5 A 4 ) 1 0 2 FORMAT ( 2 F 1 0 . 3 ) 1 0 3 FORMAT ( 8 F 1 0 . 1 ) 1 0 4 FORMAT ( 8 F 1 0 . 2 ) 1 0 5 FORMAT ( 8 F 1 0 . 3 ) 1 0 6 FORMAT ( 2 F 1 0 . 5 ) 131 FORMAT ( 1 H 1 , 7 X , » T T » , 1 3 X , » Z 1 » , 1 3 X , ' Z 2 « , 1 3 X , » Z 3 * , 1 3 X , *Z4* 1, 1 3 X , » Z 5 * , 1 3 X , * Z 6 ' , 1 3 X , 'TAUGOV*) 132 FORMAT ( 2 X , 8 E 1 5 . 4 ) 1 3 3 FORMAT ( 1 H 1 , / / / , 1 0 X , ' T I M E ' , 1 6 X , * TORQ *, 1 6 X , »TAU», 1 7 X , 1*GAMMA*, 1 5 X , ' V I S C O * , 1 5 X , »RX») 134 FORMAT ( 2 X , 6 E 2 0 . 4 ) 140 FORMAT (////, 1 0 X , 'RXQ', 1 7 X , * T A U Z ' , 1 6 X , * TAUQ *, / / , 3 E 2 0 . 4 ) C C T E R M I N A T E P L O T T I N G ..... C A L L PLGTND C STOP END S U B R O U T I N E G O V I E R ( T A U , T, T A I , T A Z , P, E 2 , AKKA) C C C C T H I S S U B R O U T I N E E V A L O A T E S THE PARAMETERS OF M O D I F I E D G O V I E R * S C MODEL BY MEANS OF L E A S T SQUARES F I T T I N G . 1 4 5 C c . c D I M E N S I O N T A U ( 1 0 ) , T ( 1 0 ) , Y 1 ( 1 0 ) , Y 2 ( 1 0 ) , Y 3 { 1 0 ) , Y 4 ( 1 0 ) , Y F ( 1 0 ) , 1WT (10) , P ( 2 ) , E1 (2) , E 2 (2) E X T E R N A L AUXGOV I E N D = 10 WRITE ( 6 , 1 0 1 ) WRITE ( 6 , 1 0 5 ) P P 6 = (TAZ + T A I ) / (TAZ - T A I ) DO 10 1 = 1 , I E N D Y1 ( I ) = T A U ( I ) - T A I Y 2 ( I ) = T A U ( I ) + T A I Y3 ( I ) = P P 6 * Y1 ( I ) / Y 2 ( I ) Y4 ( I ) = ALOG (ABS (Y3 ( I ) ) ) WRITE ( 6 , 1 0 2 ) T ( I ) , 1 1 ( 1 ) , Y4 ( I ) , TAU ( I ) 10 C O N T I N U E C C ..... L E A S T SQUARES F I T P ( 1 ) = 0.5 P ( 2 ) = 0.5 WZ = 0.0 NI = +50 M = 2 E P S = 0 . 0 0 0 5 C A L L L Q F ( T , Y4 , Y F , WT, E 1 , E 2 , P, WZ, I E N D , M, N I , ND, E P S , 1 AUXGOV) C C OUTPUT OF THE R E S U L T S WRITE ( 6 , 1 0 3 ) AKKA = - P ( 1 ) / ( 2 . 0 * T A I ) WRITE ( 6 , 1 0 4 ) E 1 ( 1 ) , E 1 ( 2 ) , E 2 ( 1 ) , E2 (2) , ND, P ( 1 ) , P (2) , AKKA C C. FORMATS FOR I N P U T AND OUTPUT STATEMENTS 1 0 1 FORMAT ( 1 H 1 , / / / , 3 0 X , «T», 1 9 X , * Y 1 » , 1 8 X , » Y 4 ' , 1 8 X , *TAU') 102 FORMAT ( 2 0 X , 4 E 2 0 . 4 ) 1 0 3 FORMAT (//////, 17X , ' E 1 ( 1 ) « , 1 0 X , •E1(2)», 1 0 X , * E 2 { 1 ) » , 1 0 X , 1 » E 2 ( 2 ) « , 1 0 X , *ND«, 1 3 X , » P ( 1 ) » , 11X, • P (2) • , 1 1 X , * AKKA *) 104 FORMAT ( 1 H 0 , 4 X , » G O V I E R ' , 4 E 1 5 . 4 , 1 1 5 , 3 E 1 5 . 4 ) 1 0 5 FORMAT (1H0) RETURN END F U N C T I O N AUXGOV ( P , D, X, L ) C C . . c C T H I S F U N C T I O N D E F I N E S THE FORM OF M O D I F I E D G O V I E R ' S EQUATION. C C D E S C R I P T I O N OF P A R A M E T E R S C C AUXGOV - THE F U N C T I O N . C X - I N D E P E N D E N T V A R I A B L E S . C P ( I ) - PARAMETERS OF THE MODEL EQUATION TO BE E V A I U A T E D . C D ( I ) - P A R T I A L D E R I V A T I V E S OF THE F U N C T I O N WITH R E S P E C T C TO EACH PARAMETER D ( I ) = (D(AUXGOV) ) / ( D (P ( I ) ) . D I M E N S I O N P (2) , D<2) F X Y Z = ABS (P (2) * X «- 1.0) D ( 1 ) = A L O G ( F X Y Z ) AUXGOV = P (1) * D ( 1 ) D ( 2 ) = ( P ( 1 ) * X) / F X Y Z RETURN END S U B R O U T I N E MIT ( R X , T I M E , RXQ, R, RO, P , E 2 , AK) T H I S S U B R O U T I N E E V A L U A T E S THE PARAMETERS OF MIT MODEL BY MEANS OF L E A S T SQUARES F I T T I N G . D I M E N S I O N R X ( 1 0 ) , Y 1 ( 1 0 ) , Y 2 ( 1 0 ) , Y 3 ( 1 0 ) , F ( 1 0 ) , T I M E ( 1 0 ) , P (2) 1, E 1 ( 2 ) , E 2 ( 2 ) , Y F ( 1 0 ) , WT(10) E X T E R N A L AUXMIT I E N D = 10 WRITE ( 6 , 1 0 1 ) RL = 1.96 Q = ( 3 . 1 4 1 6 * RL * (RO**2 - B X Q * * 2 ) ) * * 2 B = - 2 . 0 * (RO**2 - R**2) C = (RO**2 - R * * 2 ) * * 2 - Q / ( ( 3 . 1 4 1 6 * R L ) * * 2 ) D = SQRT ( B * * 2 - 4. 0 * C) DQ 1 = A B S ( { B + D) / (B - D) ) DQ2 = ALOG (DQ1) DO 10 1 = 1 , I E N D Y 1 ( I ) = 2.0 * ( R X ( I ) * * 2 - R**2) + B Y 2 ( I ) = A B S ( ( Y 1 ( I ) - D) / (Y1 ( I ) + D) ) Y3 ( I ) = ALOG ( Y 2 ( I ) ) F ( I ) = Y 3 ( I ) + DQ2 WRITE ( 6 , 1 0 2 ) T I M E ( I ) , Y 1 ( I ) , Y2 ( I ) , 1 3 ( 1 ) , F ( I ) 10 C O N T I N U E L E A S T SQUARES F I T ..... P ( 1 ) = 0.5 P ( 2 ) = 0.5 WZ = 0.0 NI = 50 M = 2 E P S = 0 . 0 0 0 5 C A L L L Q F ( T I M E , F , Y F , WT, E 1 , E 2 , P, WZ, I END, M, N I , ND, EPS 1 , AUXMIT) ..... OUTPUT OF THE R E S U L T S ..... WRITE ( 6 , 1 0 3 ) AK = P ( 1 ) / (D * 3 . 1 4 1 6 * RL) WRITE ( 6 , 1 0 4 ) E 1 ( 1 ) , E 1 ( 2 ) , , E 2 ( 1 ) , E2 (2) , ND, P ( 1 ) , P (2) , AK FORMATS FOR I N P U T AND OUTPUT STATEMENTS 101 FORMAT ( 1 H 1 , / / / , 2 0 X , * T I M E' , 1 6 X , »Y1», 1 8 X , »Y2», 1 8 X , »Y3• 147 1 0 4 FORMAT ( 1 H 0 , 4 X , * M I T ' , 4 E 1 5 . 4 , 1 1 5 , 3 E 1 5 . 4 ) C C c c c c c c c c c c c c c c 1, 1 8 X , * F*) 102 FORMAT ( 1 0 X , 5 E 2 0 . 4 ) 1 0 3 FORMAT {//////, 1 4 X , » E 1 ( 1 ) « , 1 0 X , » E 1 ( 2 ) ' , 1 0 X , * E 2 { 1 ) ' , 1 0 X , 1 ' E 2 ( 2 ) ' , 1 0 X , * ND* , 1 3 X , » P ( 1 ) * , 1 1 X , * P ( 2 ) ' , 1 1 X , 'AK') H 0 , M I T ' RETURN END FO NOTION AUXMIT ( P , D, X, L ) T H I S F U N C T I O N D E F I N E S THE FORM OF MIT E Q U A T I O N . D E S C R I P T I O N OF P A R A M E T E R S AUXMIT X P ( I ) D ( I ) THE F U N C T I O N . I N D E P E N D E N T V A R I A B L E S . P A R A M E T E R S OF T H E MODEL EQUATION TO BE E V A L U A T E D . P A R T I A L D E R I V A T I V E S OF THE F U N C T I O N H I T H R E S P E C T TO EACH PARAMETER D ( I ) = ( D ( A U X M I T ) ) / < D { P ( I ) ) . D I M E N S I O N P (2) , D{2) F X Y Z = ABS ( P ( 2 ) * X + 1.0) D ( 1 ) = ALOG ( F X Y Z ) A U X M I T = P (1) * D (1) D ( 2 ) = ( P ( 1 ) * X) / F X Y Z RETUEN END •148 C-3 P r o g r a m m e " T A U D E C A Y " 1 4 9 C . .. c C MAIN PROGRAM " T A U D E C A Y " C C THE PURPOSE OF T H I S PROGRAM I S TO F I N D OUT WHICH ONE OF THE C V A R I O U S ORDERS (FROM ( 0 , 1 ) TO ( 2 , 2 ) ) OF THE M O D I F I E D G O V I E R ' S C S T R E S S DECAY MODELS CAN B E S T - F I T THE E X P E R I M E N T A L DECAY CURVE C I N TERMS OF SHEAR S T R E S S MEASURED AT THE WALL GF THE C O - A X I A L C C Y L I N D R I C A L V I S C O M E T E R AND T I M E FOR A L L S E T S OF THE C E X P E R I M E N T A L RUNS. C C THE PROGRAM C A L C U L A T E S THE SUM OF SQUARES AND T R I E S TO M I N I M I Z E C I T BY A D J U S T I N G THE F I T T I N G P A R A M E T E R S OF EACH OF THE MODELS, C A F T E R E V A L U A T I N G T H E F I T T I N G P A R A M E T E R S , THE COMPUTER W I L L L I S T C THE F I N A L V A L U E S OF THOSE F I T T I N G PARAMETERS OF EACH MODEL. C C HOW TO C O M P I L E THE PROGRAM "TAU D E C A Y " ? C I N ORDER TO C O M P I L E THE PROGRAM "TAUDECAY" INTO THE I E H C V E R S I O N FORTRAN G - L E V E L C O M P I L E R UNDER CONTROL OF M I C H I G A N C T E R M I N A L S Y S T E M (MTS) FOR THE I B M 3 7 0 MODEL 168 COMPUTER C AT THE U N I V E R S I T Y OF B R I T I S H C O L U M B I A , THE COMMAND C C $RUN * F T N SC ARDS=TAUDECAY SPUNCH=-OBJECT C C SHOULD BE USED. C C HOW TO EXECUTE THE O B J E C T DECK " - O B J E C T " ? C THE COMMAND C C $RUN - O B J E C T 4-=BFOSDEF 5 = EXP DATA C C SHOULD BE USED, WHERE "BFORDER" I S THE F I L E WHICH C O N T A I N S C THE NAMES OF V A R I O U S ORDERS OF MODELS AND SHOULD BE READ C THROUGH L O G I C A L I/O U N I T 4 BY A - T Y P E FORMAT. C THE " E X P D A T A " I S THE F I L E C O N T A I N I N G THE E X P E R I M E N T A L RAW C DATA, WHOSE ORDER AND FORMAT MUST CORRESPOND TO, AND EE C C O N S I S T E N T WITH THOSE S P E C I F I E D I N THE MAIN PROGRAM OF C "TAUDECAY". C C D E S C R I P T I O N OF P A R A M E T E R S . C I E N D - NUMBER OF E X P E R I M E N T A L DATA. C J E N D - NUMBER OF MODEL EQUATIONS, C LEND - NUMBER OF B L O C K S OF A-FORM AT FOR O R D E R ( J , L ) . C MEND - NUMBER OF BLOCKS OF A-FORM AT FOR DATE {M) . C ORDER ( J , L) - R E A C T I O N ORDERS OF M O D I F I E D G O V I E R ' S MODEL. C DATE (M) - DATE OF E X P E R I M E N T . C RATIOZ - RAW DATA OF ( R X - R ) / ( R O - R ) AT T - 0 . 0 MEASURED C WITH C I N E PHOTOGRAPH { - ) . C RATIOQ - RAW DATA OF ( R X - R ) / ( R O - R ) AT E Q U I L I B R I U M C MEASURED WITH C I N E PHOTOGRAPH ( - ) . C DAMVZ - RAW DATA OF TORQUE ON THE RECORDER CHART C AT T=0.0 (MV) . C DAMVQ - RAW DATA OF TORQUE ON THE RECORDER CHART C AT E Q U I L I B R I U M ( M V ) . C T I M E ( I ) - T I M E (S E C ) . C D A M V ( I ) - RAW DATA OF TORQUE ON T H E RECORDER CHART 150 C (MV) . C R A T I O ( I ) - RAW DATA OF {RX-R) / ( R O - R ) MEASURED WITH C C I N E PHOTOGRAPH (-) . C R - R A D I U S OF I N N E R C Y L I N D E R OF THE V I S C O M E T E R C (CM) . C OMEGA - ANGULAR V E L O C I T Y OF THE INNER C Y L I N D E R C ( R A D / S E C ) . C PLENGT - L E N G T H OF F I B E R P A R T I C L E (MM) . C P H I - VOLUME F R A C T I O N OF F I B E R S I N THE A R T I F I C I A L C S L U R R Y (CC/CC) . C BL - HIGHT OF THE I N N E R C Y L I N D E R ( C M ) . C BC - R A D I U S OF THE OUTER C Y L I N D E R ( C M ) . C T O B Q ( I ) - TORQUE (DYNES C M ) . C T A U ( I ) - SHEAR S T R E S S AT THE WALL OF T H E I N N E R C Y L I N D E R C ( D Y N E S / S Q . C M ) . C F F ( I ) - DEPENDENT V A R I A B L E OF THE DATA TO BE F I T T E D BY C S U B R O U T I N E L Q F . C A K G O V ( J ) - THE F I R S T F I T T I N G PARAMETERS OF M O D I F I E D C G O V I E R ' S MODEL. C P P ( J , 2 ) - T HE SECOND F I T T I N G PARAMETERS OF M O D I F I E D C G O V I E R ' S MODEL. C C C D I M E N S I O N ORDER ( 8 , 6 ) , DATE (8) , T I M E ( I O ) , D A M V ( 1 0 ) , R A T I O ( 1 0 ) , 1 T O R Q ( 1 0 ) , T A U { 1 0 ) , D E V ( 8 , 1 0 ) , F { 8 , 1 0 ) , F F ( 1 0 ) , WT ( 1 0 ) , E1 (2) , 2 E 2 ( 2 ) , P ( 2 ) , P P ( 8 , 2 ) , A K G O V ( 8 ) , Y F ( 1 0 ) C E X T E R N A L AUX C I E N D = 10 J E N D = 8 LEND = 6 MEND = 5 C C ..... READ V A R I O U S R E A C T I O N ORDERS READ ( 4 , 1 0 9 ) ( (ORDER ( J , L ) , L= 1,LEND) , J = 1 , J E N D ) C 1 C O N T I N U E C C READ THE DATE OF E X P E R I M E N T READ ( 5 , 1 0 1 ) ( D A TE (M) , M= 1 , MEND) C C ..... I F NO MORE E X P E R I M E N T A L DATA TO BE READ, T E R M I N A T E C THE C A L C U L A T I O N I F ( D A T E ( 1 ) .EQ. OWAR) GO TO 2 C C ..... READ E X P E R I M E N T A L RAW DATA READ ( 5 , 1 0 2 ) R A T I O Z , RATIOQ READ ( 5 , 1 0 3 ) DAM V S , DA MVQ READ ( 5 , 1 0 4 ) ( T I M E ( I ) , 1 =1, I E N D ) READ ( 5 , 1 0 5 ) ( D A M V ( I ) , 1 = 1 , I E N D ) READ ( 5 , 1 0 6 ) ( R A T I O ( I ) , 1 = 1 , I E N D ) READ ( 5 , 1 0 7 ) R, OMEGA READ ( 5 , 1 0 8 ) P L E N G T , P H I C 151 C C A L C U L A T E SHEAR S T R E S S RL = 1 . 96 RO = 2. 1 TORQZ = ( 4 . 4 6 E + 04) * DAMVZ TOBQQ = ( 4 . 4 6 E + 0 4 ) * DAMVQ TAUZ = TORQZ / ( 2 . 0 * 3 . 1 4 1 6 * ( R * * 2 ) * RL) TAUQ = TOBQQ / ( 2 . 0 * 3 . 1 4 1 6 * ( R * * 2 ) * RL) C C C A L C U L A T E SHEAR S T R E S S AT ANY T I M E DO 11 1=1,IEND T O E Q ( I ) = ( 4 . 4 6 E + 04) * D A M V ( I ) T A U ( I ) = T O R Q ( I ) / ( 2 . 0 * 3 . 1 4 1 6 * ( R * * 2 ) * RL) 11 C O N T I N U E C C C A L C U L A T E V A R I O U S CONSTANTS ..... B A K A 6 = (TAUZ • TAUQ) / (TAUZ - TAUQ) B7 = (TAUQ**2) / (TAUZ - TAUQ) D7 = TAUQ * ( 2 . 0 * TAUZ - TAUQ) / (TAUZ - TAUQ) BAKA7 = ( 2 . 0 * TAUZ + B7 + D7) / ( 2 . 0 * TAUZ + B7 - D7) Q8 = (TAUQ / (TAUZ - TAUQ) ) **2 A8 = Q8 - 1.0 B8 = - 2 . 0 * Q8 * TAUZ C8 = Q8 * ( T A U Z * * 2 ) D8 = 2.0 * TAUQ * TAUZ / (TAUZ - TAUQ) B A K A 8 = ( 2 . 0 * A8 * TAUZ + B 8 • D8) / ( 2 . 0 * A8 * TAUZ + E8 - D8) C C C A L C U L A T E V A R I O U S CONSTANTS FOR DEPENDENT C V A R I A B L E S F (J , I ) ..... DO 12 1=1,IEND DEV (1,1) = ABS ( ( T A U ( I ) - TAUQ) / (TAUZ - TAUQ) ) DEV (2 , 1 ) = ABS { (TAU ( I ) • TAUQ - 2.0 * TAUZ) / (TAU ( I ) - TAUQ) ) DEV (3 , 1 ) = D E V ( 1 , 1 ) DEV (4 , 1 ) = DEV (1 , 1 ) DEV (5 , 1 ) = ABS { (TAU ( I ) - TAUQ) / (TAUZ - TAU ( I ) * TAUQ / T A U Z ) ) DEV (6 , 1 ) = ABS ( B A K A 6 * ( T A U ( I ) - TAUQ) / (TAD ( I ) + T A U Q ) ) DEV (7 , 1 ) = ABS ( B A K A 7 * ( 2 . 0 * T A U ( I ) + B 7 - D7) / 1<2.0 * T A U ( I ) + B7 + D 7 ) ) DEV ( 8 , 1 ) = ABS ( B A K A 8 * ( 2 . 0 * A8 * TAU ( I ) + B8 - D8) / 1 ( 2 . 0 * A8 * TAU ( I ) + B8 + D8) ) 12 C O N T I N U E C C ..... C A L C U L A T E THE DEPENDENT V A R I A B L E TO B E F I T T E D L A T E R ON .... DO 13 J = 1 , J E N D DO 14 1=1,IEND F ( J , I ) = ALOG (DEV ( J , I ) ) 14 C O N T I N U E 13 C O N T I N U E C C P R I N T OUT THE T I T L E FOR THE R E S U L T S ..... DO 15 J = 1 , J E N D WRITE ( 6 , 1 3 7 ) (DATE (M) , M=1,MEND) WRITE ( 6 , 1 3 1 ) (ORDER ( J , L ) , L= 1,LEND) C C CONVERT T H E DEPENDENT V A R I A B L E S I N T O ONE D I M E N S I O N A I C FORM DO 16 1=1,IEND 1 5 2 F F ( I ) = F ( J , I ) 16 C O N T I N U E C WRITE ( 6 , 1 3 2 ) C C CHECK I N D E P E N D E N T AND DEPENDENT V A R I A B L E S TO BE F I T T E D C BY THE S U E R O U T I N E L Q F DO 17 1 = 1 , I E N D WRITE ( 6 , 1 3 3 ) T I H E ( I ) , F F ( I ) 17 C G N T I N H E C C CONSTANTS FOR THE L E A S T SQUARES F I T . . . . . P ( 1 ) = 0.5 P ( 2 ) = 0.5 WZ = 0.0 N I = 50 M = 2 E P S = 0 . 0 0 0 5 C C L E A S T SQUARES F I T T I N G C A L L L Q F ( T I M E , F F , Y F , WT, E 1 , E 2 , P , WZ, I E N D , M, N I , ND, E P S 1, AUX) C C ..... OUTPUT OF THE R E S U L T S OF THE F I T T I N G WRITE ( 6 , 1 3 4 ) ND, P ( 1 ) , P ( 2 ) P P ( J , 1 ) = P ( 1 ) P P ( J , 2 ) = P ( 2 ) 15 C O N T I N U E C C C A L C U L A T E THE F I R S T F I T T I N G P ARAMETERS I N TERMS OF THE C M O D I F I E D G O V I E R ' S FORM ..... A K G O V ( 1 ) = - P P ( 1 , 1 ) * (TAUZ - TAUQ) A K G O V ( 2 ) = P P ( 2 , 1 ) * (TAUZ - TAUQ) / 2. 0 A K G O V ( 3 ) = - P P ( 3 , 1 ) A K G 0 V ( 4 ) = - P P ( 4 , 1 ) * (TAUZ - TAUQ) / TAUZ A K G O V ( 5 ) = - P P ( 5 , 1 ) * (TAUZ - TAUQ) / (TAUZ + TAUQ) A K G O V ( 6 ) = - P P ( 6 , 1 ) / ( 2 . 0 * TAUQ) A K G O V ( 7 ) = - P P { 7 , 1 ) / D7 A K G O V ( 8 ) = P P ( 8 , 1 ) / D8 W R I T E ( 6 , 1 3 7 ) ( D A T E ( M) , M= 1 , MEND) WRITE ( 6 , 1 3 5 ) C C P R I N T T H E R E S U L T S DO 18 J = 1 , J E N D WRITE ( 6 , 1 3 6 ) (ORDER ( J , L ) , L= 1, L END) , AKGOV ( J ) , P P ( J , 2 ) 18. C O N T I N U E C C R E P E A T THE PROCEDURE D E S C R I B E D ABOVE FOR THE NEXT C E X P E R I M E N T A L RUN GO TO 1 2 C O N T I N U E C C ..... FORMATS FOR I N P U T AND OUTPUT STATEMENTS 101 FORMAT (5A4) 1 0 2 FORMAT ( 2 F 1 0 . 3 ) 153 1 0 3 FORMAT ( 2 F 1 0 . 3 ) 1 0 4 FOBMAT ( 8 F 1 0 . 1 ) 1 0 5 FORMAT ( 8 F 1 0 . 2 ) 1 0 6 FORMAT ( 8 F 1 0 . 3 ) 1 0 7 FORMAT ( 2 F 1 0 . 5 ) 1 0 8 FORMAT ( 2 F 1 0 . 5 ) 109 FORMAT (6A4) 131 FORMAT {//, 2 X , 6 A 4 ) 13 2 FORMAT {/, 1 5 X , ' T I M E ( I ) ' , 6X , • F F ( I ) ' , /) 1 3 3 FORMAT ( 1 0 X , 2 E 1 3 . 4 ) 134 FORMAT (////, 6 X , f ND' , 3 X , » P ( 1 ) » , 9 X , * P { 2 ) « , / / , 5 X , 1 2 , 2 E 1 3 . 4 1) 1 3 5 FORMAT {//, 2 X , * OR DER * , 2 2 X , ' A K G O V ( J ) * , 5 X , ' P P ( J , 2 ) » , /} 13 6 FORMAT ( 2 X , 6 A 4 , 2 E 1 3 . 4 ) 1 3 7 FORMAT ( 1 H 1 , / / , 1 5 X , 5A4) STOP END F U N C T I O N AUX ( P , D, X, I ) C C C C C c c c c c c c c c c T H I S F U N C T I O N D E F I N E S THE FORM OF E Q U A T I O N TO BE F I T T E D BY THE S U B R O U T I N E L Q F . D E S C R I P T I O N OF P A R A M E T E R S AUX - THE F U N C T I O N . X - I N D E P E N D E N T V A R I A B L E . P ( I ) - P A R A M E T E R S OF THE MODEL TO BE E V A L U A T E D . D ( I ) - P A R T I A L D E R I V A T I V E S OF THE FU N C T I O N WITH R E S P E C T TO EACH PARAMETER. D I M E N S I O N P (2) , D (2) F X Y Z = A B S ( P ( 2 ) * X + 1.0) D { 1 ) = ALOG ( F X Y Z ) AUX = P { 1 ) * D (1) D ( 2 ) = ( P { 1 ) * X) / F X Y Z RETURN END 154 C-4 Programme "THICKNESS" 155 C , , c C MAIN PRGGRAM " T H I C K N E S S " C C THE MODEL WHICH P R E D I C T S THE T H I C K N E S S OF THE FLOWING L A Y E R RX C I N fl 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 HAS BEEN CONSTRUCTED. C C THE PURPOSE OF T H I S PROGRAM I S TO F I N D OUT WHICH ONE OF THE C V A R I O U S ORDERS (FROM ( 0 , 1 ) TO ( 2 , 2 ) ) OF THE MODEL CAN C B E S T - F I T THE E X P E R I M E N T A L DATA WHICH HAVE BEEN O B T A I N E D FROM C THE 8MM C I N E PHOTOGRAPHS FOR A L L SETS OF THE E X P E R I M E N T A L RUNS C C A R R I E D OUT B E F O R E . C C THE PROGRAM C A L C U L A T E S THE SUM OF SQUARES AND T R I E S TO M I N I M I Z E C I T BY A D J U S T I N G THE F I T T I N G P A R A M E T E R S OF EACH ORDER OF T H E C MODEL. AFTER E V A L U A T I N G THE F I T T I N G P A R A M E T E R S , THE COMPUTER C W I L L L I S T THE F I N A L V A L U E S OF THOSE F I T T I N G P ARAMETERS OF EACH C ORDER OF THE MODEL. C C HOW TO C O M P I L E THE PROGRAM " T H I C K N E S S " ? C I N ORDER TO C O M P I L E THE PROGRAM " T H I C K N E S S " I N T O THE I B M C V E R S I O N FORTRAN G - L E V E L C O M P I L E R UNDER CONTROL OF M I C H I G A N C T E R M I N A L S Y S T E M (MTS) FOR THE I B M 3 7 0 MODEL 168 COMPUTER AT C THE U N I V E R S I T Y OF B R I T I S H C O L U M B I A , THE COMMAND C C $RUN * F T N S C A R D S = T H I C K N E S S SPUNCH=-OBJ ECT C C SHOULD BE USED. C C HOW TO E X E C U T E THE O B J E C T DECK " - O B J E C T " ? C THE COMMAND C C $RUN - O B J E C T 4=BFORDE R 5=EXPDATA C C SHOULD B E USED, WHERE "BFORDER" I S THE F I L E WHICH C O N T A I N S C THE NAME OF V A R I O U S ORDERS OF THE MODEL AND SHOULD BE READ C^ THROUGH L O G I C A L I/O U N I T 4 EY A - T Y P E FORMAT. THE " E X P L A T A " C I S THE F I L E C O N T A I N I N G THE E X P E R I M E N T A L RAW DATA, WHOSE C ORDER AND FORMAT MUST CORRESPOND TO, AND BE C O N S I S T E N T WITH C THOSE S P E C I F I E D I N THE MAIN PROGRAM. C C D E S C R I P T I O N OF PARAMETERS. C I E N D - NUMBER OF E X P E R I M E N T A L DATA. C J E N D - NUMBER OF MODEL EQUATIONS. C LEND - NUMBER OF B L O C K S OF A-FORM AT FOR ORDER (J,1) . C MEND - NUMBER OF BLOCKS OF A-FORM AT FOR DATE{M) . C O R D E R ( J , L ) - R E A C T I O N ORDERS. C BATE (M) - D A T E OF E X P E R I M E N T . C RATIOZ - RAW DATA OF (RX-R) / (RO-R) AT T=0.0 MEASURED C WITH C I N E PHOTOGRAPH ( - ) . C RATIOQ - RAW DATA OF ( R X - R ) / ( R O - R ) AT E Q U I L I B R I U M C MEASURED WITH C I N E PHOTOGRAPH ( - ) . C DAHVZ - RAW DATA OF TORQUE ON THE RECORDER CHART C AT T=0.0 (MV) . C DAMVQ - RAW DATA OF TORQUE ON THE RECORDER CHART C AT E Q U I L I B R I U M ( M V ) . 156 T I M E ( I ) DAMV ( I ) R A T I O ( I ) R OMEGA PLENGT P H I RL RO RXQ R X { I ) F F ( I ) A K { J ) P P ( J , 2 ) T I M E ( S E C ) . RAW DATA OF TORQUE ON THE RECORDER CHART (MV) . RAW DATA OF ( R X - R ) / ( R O - R ) MEASURED WITH C I N E PHOTOGRAPH {-) , R A D I U S OF I N N E R C Y L I N D E R OF THE V I S C O M E T E R (CM) . ANGULAR V E L O C I T Y OF THE INNER C Y L I N D E R ( R A D / S E C ) . L E N G T H OF F I B E R P A R T I C L E (MM) . VOLUME F R A C T I O N OF F I B E R S I N THE A R T I F I C I A L S L U R R Y (CC/CC) . HIGHT OF THE I N N E R C Y L I N D E R ( C M ) . R A D I U S OF THE OUTER C Y L I N D E R ( C M ) . E Q U I L I B R I U M V A L U E OF RX ( C M ) . T H I C K N E S S OF FLOWING L A Y E R MEASURED FROM THE CENTER OF THE C Y L I N D E R S OF THE V I S C O M E T E R ( C M ) . DEPENDENT V A R I A B L E OF THE DATA TO BE F I T T E D BY THE S U B R O U T I N E LQF. T H E F I R S T F I T T I N G P A R A M E T E R S OF THE MODEL, THE SECOND F I T T I N G PARAMETERS OF THE MODEL. C C D I M E N S I O N T I M E { 1 0 ) , R A T I O ( 1 0 ) , R X { 1 0 ) , O R D E R ( 8 , 6 ) , X ( 1 0 ) 1, D E V ( 8 , 1 0 ) , Y 1 2 ( 1 0 ) , Y 2 0 { 1 0 ) , Y 2 1 { 1 0 ) , Y 2 2 { 1 0 ) , F ( 8 , 1 0 ) , Y F ( 1 0 ) 2, F F ( 1 0 ) , W T ( 1 0 ) , E 1 ( 2 ) , E 2 ( 2 ) , P ( 2 ) , P P { 8 , 2 ) , AK (8) , DATE (8) 3, DAMV(10) E X T E R N A L AUX I E N D J E N D LEND MEND 10 8 6 5 C C C C READ VARIOUS R E A C T I O N ORDERS R E A D { 4 , 1 0 9 ) ( (ORDER ( J , L) , L= 1 ,LEND) , J = 1 , J E N D ) 1 C O N T I N U E ..... READ R E A D ( 5 , 1 0 1 ) READ ( 5 , 102) READ ( 5 , 103) READ ( 5 , 1 0 4 ) READ ( 5 , 105) R E A D ( 5 , 1 0 6 ) R E A D ( 5 , 107) READ ( 5 , 108) E X P E R I M E N T A L RAW DATA (DATE (M) , M= 1 , MEND) R A T I O S , RATIOQ DAMVZ, DAMVQ ( T I M E ( I ) , 1=1,IEND) ( D A M V ( I ) , 1=1,IEND) ( R A T I O ( I ) , 1=1, IEND) R, OMEGA P L E N G T , P H I C A L C U L A T E THE T H I C K N E S S OF THE FLOWING L A Y E R RX RL = 1.96 RO = 2. 1 RXQ = RATIOQ * (RO - R) + R DO 11 1 = 1 , I E N D R X ( I ) = R A T I O ( I ) * ( R O - R ) + R X ( I ) = RX ( I ) **2 - R**2 11 C O N T I N U E RXQR P 0 1 = P 0 2 = AO 2 = P 1 0 = A 1 0 = P 1 1 = a n = P 1 2 --A 1 2 = B12 '-D12 P2 0 = B20 = C 2 0 --D20 = P2 1 = B21 = C 2 1 = D21 = P2 2 = A 2 2 = B 2 2 -C 2 2 = C A L C U L A T E V A R I O U S CONSTANTS FOR DEPENDENT V A R I A B L E S F ( J , I ) RXQ**2 - R**2 1.0 / < 3 .1416 * RL * RXQR) 1.0 / { ( 3.1416 * RL * R X Q R ) * * 2 ) 1.0 / ( 3.1416 * RL * S Q R T ( P 0 2 ) ) 3 . 1 4 1 6 * RL * (RO**2 - RXQ**2) P 1 0 / ( 3.1416 * RL) - (RO**2 - R**2) (RO**2 - RXQ**2) / RXQR (RO**2 - R**2) / ( P 1 1 + 1.0) (RO**2 - RXQ**2) / ( 3.1416 * RL * (RXQR**2) ) P 1 2 * 3. 1 4 1 6 * RL R 0 * * 2 - R**2 S Q R T ( 1 . 0 + 4.0 * A12 * B 1 2 ) ( 3.1416 * RL * (RO**2 - R X Q * * 2 ) ) * * 2 - 2 . 0 * (RO**2 - R**2) (RO**2 - R * * 2 ) * * 2 - P 2 0 / ( { 3.1416 * R L ) * * 2 ) S Q R T ( B 2 0 * * 2 - 4.0 * C 2 0 ) 3.1416 * RL * ( { R 0 * * 2 - R X Q * * 2 ) * * 2 ) / RXQR - P 2 1 / ( 3.1416 * RL) - 2.0 * (RO**2 - R**2) (RG** 2 - R * * 2 ) * * 2 S Q R T ( B 2 1 * * 2 - 4.0 * C 2 1 ) C (110**2 - R X Q * * 2 ) * * 2 ) / (RXQR**2) P 2 2 - 1.0 2.0 * (RO**2 - R**2) - { (RO**2 - R**2) **2) S Q 1 T ( B 2 2 * * 2 - 4.0 * A22 * C 2 2 ) 1 = 1 , I E N D = A B S ( 1 . 0 - P 0 1 * 3 . 1 4 1 6 * R L * X ( I ) ) = A B S ( ( X ( I ) - A02) / ( X ( I ) + A 0 2 ) ) = ABS (X ( I ) / A10 + 1.0) = A B S { 1 . 0 - X { I ) / A 1 1 ) 2. 0 * A12 * X ( I ) + 1.0 D22 = DO 12 DEV (1,1) DEV ( 2 , 1 ) DEV (3,1) DEV (4,1) Y 1 2 ( I ) = DEV (5,1) 1* ( 1 . 0 -Y 2 0 ( I ) = DEV (6,1) 1 * ( B 2 0 - D 2 0 ) ) ) Y 2 1 ( I ) = 2 . 0 * DEV (7,1) 1* = ABS ( ( Y 1 2 ( I ) - D12) D 1 2 ) ) ) 2.0 * X ( I ) + B20 = ABS ( ( Y 2 0 ( I ) - D20) * ( 1 . 0 + D12) / ( ( Y 1 2 ( I ) + D12) * ( B 2 0 + D20) / ( ( 7 2 0 ( 1 ) • D20) X ( I ) + B21 = ABS { ( Y 2 1 ( I ) - D21) (B21 - D 2 1 ) ) ) Y 2 2 ( I ) = 2.0 * A22 * X ( I ) + B 2 2 DEV (8,1) = A B S ( ( Y 2 2 ( I ) - D22) * 1* ( B 2 2 - D 2 2 ) ) ) 12 C O N T I N U E DO 13 J = 1 , J E N D DO 14 1=1, I E N D F ( J , I ) = ALOG (DEV ( J , I ) ) 14 C O N T I N U E 13 C O N T I N U E * ( B 2 1 • D21) / { ( Y 2 1 ( I ) + D21) ( B 2 2 + D22) / ( ( Y 2 2 ( I ) + D22) P R I N T OUT THE T I T L E FOR THE R E S U L T S ..... DO 15 J = 1 , J E N D 1 5 8 W R I T E { 6 , 1 3 7 ) ( D A T E ( M ) , M= 1, MEND) R R I T E ( 6 , 1 3 1 ) (ORDER ( J , L ) ,L= 1,LEND) C C ..... CONVERT DEPENDENT V A R I A B L E S I N T O ONE D I M E N S I O N A L C FORM ..... DO 16 1 = 1 , 1 END. F P { I ) = F ( J , I ) 16 C O N T I N U E C C ..... CHECK I N D E P E N D E N T AND DEPENDENT V A R I A B L E S TO BE F I T T E D C BY THE S U B R O U T I N E LQF WRITE ( 6 , 1 3 2 ) DO 17 1 = 1 , I E N D WRITE ( 6 , 1 3 3 ) T I M E ( I ) , F F ( I ) 17 C O N T I N U E C C ..... CONSTANTS FOR THE L E A S T SQUARES F I T P ( 1 ) = 0.5 P { 2 ) = 0.5 WZ = 0.0 NI = 5 0 M = 2 E P S = 0 . 0 0 0 5 . C C L E A S T SQUARES F I T T I N G ..... C A L L L Q F ( T I M E , F F , Y F , WT, E 1 , E 2 , P, WZ, I E N D , M, N I , ND, EPS 1, AUX) C C ..... OUTPUT OF THE R E S U L T S OF F I T T I N G W R I T E { 6 , 1 3 4 ) ND, P ( 1 ) , P ( 2 ) P P ( J , 1 ) = P ( 1 ) PP ( J , 2 ) = P ( 2 ) 15 C O N T I N U E C C C A L C U L A T E THE F I R S T F I T T I N G P ARAMETERS AK (1) - P P ( 1 , 1) / P 0 1 AK (2) = - P P ( 2 , 1 ) / ( 2 . 0 * 3 . 1 4 1 6 * RL * P02 * A02) AK (3) - P P ( 3 , 1) AK (4) = - P P ( 4 , 1) / ( P 1 1 + 1 .0) A K { 5 ) - P P ( 5 , 1) / D12 AK (6) P P ( 6 , 1 ) / ( 3 . 1 4 1 6 * RL * D20) A K ( 7 ) — P P ( 7 , 1 ) / ( 3 . 1 4 1 6 * RL * D21) AK (8) — - P P ( 8 , 1 ) / ( 3 . 1 4 1 6 * RL * D22) c C ..... P R I N T T H E R E S U L T S ..... W R I T E ( 6 , 1 3 7 ) (DATE ( M) , M= 1 , MEND) WRITE (6 ,13.5) DO 18 J = 1 , J E N D W B I T E ( 6 , 1 3 6 ) (ORDER ( J , L) , L= 1 ,LEND) , AK ( J ) , P P ( J , 2 ) 18 C O N T I N U E C C I F THERE ARE NO MORE E X P E R I M E N T A L RAW DATA, C T E R M I N A T E THE C A L C U L A T I O N ..... I F (DATE (1) .EQ. OWAR) GO TO 2 C C ..... I F THERE ARE S T I L L DATA, REPEAT THE C A L C U L A T I O N 159 D E S C R I B E D ABOVE GO TO 1 ..... FORMATS FOR I N P U T AND OUTPUT STATEMENTS 101 FORMAT (5A4) 102 FORMAT ( 2 F 1 0 . 3 ) 1 0 3 FORMAT ( 2 F 1 0 . 3 ) 104 FORMAT ( 8 F 1 0 . 1 ) 1 0 5 . FORMAT ( 8 F 1 0 . 2 ) 1 0 6 FORMAT ( 8 F 1 0 . 3 ) 1 0 7 FORMAT ( 2 F 1 0 . 5 ) 1 0 8 FORMAT ( 2 F 1 0 . 5 ) 1 0 9 FORMAT (6A4) 13 7 FORMAT ( 1 H 1 , / / , 1 5 X , 5A4) 131 FORMAT ( / / , 2 X , 6 A 4 ) 132 FORMAT (/, 1 5 X , ' T I M E ( I ) ' , 6 X , ' F F ( I ) ' , /) 13 3 F O R M A T ( 1 0 X , 2 E 1 3 . 4 ) 134 FORMAT (////, 6 X , 'ND•, 3X, • P ( 1 ) , # 9 X , D 1 3 5 FORMAT ( / / , 2 X , 'ORDER', 2 2 X , »AK(J)», 1 3 6 FORMAT ( 2 X , 6 A 4 , 2 E 1 3 . 4 ) P ( 2 ) / /, 5 X , 1 2 , 2 E 1 3 . 4 8X, »PP <J,2) « , /) 2 C O N T I N U E STOP END F U N C T I O N AUX ( P , D, X, L) T H I S F U N C T I O N D E F I N E S T H E FORM OF MIT E Q U A T I O N . D E S C R I P T I O N OF P A R A M E T E R S . AUX - THE F U N C T I O N . X - I N D E P E N D E N T V A R I A B L E S . P { I ) - PARAMETERS OF THE MODEL EQUATION TO BE E V A L DAT ED, D ( I ) - P A R T I A L D E R I V A T I V E S OF THE F U N C T I O N WITH R E S P E C T TO EACH PARAMETER. D I M E N S I O N P (2) , D ( 2 ) F X Y Z = A B S ( P ( 2 ) * X + 1.0) D ( 1 ) = ALOG ( F X Y Z ) AUX = P (1) * D (1) D ( 2 ) = ( P { 1 ) * X) / F X Y Z RETURN END A p p e n d i x D E x p e r i m e n t a l Raw D a t a a n d C a l c u l a t e d R e s u l t s D - l S h e a r S t r e s s a t t = 0 a n d t = tm D e s c r i p t i o n o f V a r i a b l e s i n T a b l e D - l P H T ( C C / C C ) L (MM) OMEGA (RAD/SEC ) Y I E L D MV(MV) - P a r t i c l e C o n c e n t r a t i o n ( c c / c c ) . - P a r t i c l e L e n g t h (mm). - A n g u l a r V e l o c i t y o f I n n e r C y l i n d e r ( r a d . / s e c . ) - Y i e l d V a l u e o f T o r q u e M e a s u r e d o n t h e m i l l i v o l t R e c o r d e r ( m V ) . E Q U I L I B R I U M MV(MV) - E q u i l i b r i u m V a l u e o f T o r q u e M e a s u r e s o n t h e m i l l i v o l t R e c o r d e r ( m V ) . 1 6 0 T A B L E D - 1 E X P E R I M E N T A L DATA OF SHEAR S T R E S S E S AT TIME=0 AND AT E Q U I L I B R I U M DEX + 1.0MOLE NACL I N 1 0 . 0 % P E G - H 2 0 P H I ( C C / C C ) L (MM) OMEGA ( R A D / S E C ) 0.04 5 . 0 3 0 0 . 2 3 2 7 Y I E L D MV TORQUE SHEAR S T R E S S (MV) (DYNES CM) (DYNES/SQ CM) 0 . 1 6 7 E + 0 1 0 . 7 4 5 E + 0 5 0 . 5 9 3 E + 0 4 E Q U I L I B R I U M MV TORQUE SHEAR S T R E S S (MV) (DYNES CM) (DYNES/SQ CM) 0 . 4 9 1 E + 0 0 0 . 2 1 9 E + 0 5 0 . 1 7 4 E + 0 4 DEX + 1.0MOLE NACL I N 10 . 0 % PEG-H20 P H I ( C C / C C ) L (MM) OMEGA (RAD/SEC) 0.05 5. 0 3 0 0 . 2 3 2 7 Y I E L D MV TORQUE SHEAR S T R E S S (MV) (DYNES CM) (DYNES/SQ CM) 0 . 2 9 8 E + 01 0 . 1 3 3 E + 0 6 0. 1 0 6 E + 0 5 E Q U I L I B R I U M MV TORQUE SHEAR S T R E S S (MV) (DYNES CM) (D Y N E S / S Q CM) 0 . 1 0 5 E + 0 1 0 . 4 6 8 E + 0 5 0 . 3 7 3 E + 0 4 DEX + 1.0MOLE NACL I N 1 0 . 0 % P E G - H 2 0 P H I (CC/CC) L (MM) OMEGA ( R A D / S E C ) 0. 06 5 . 0 3 0 0 . 2 3 2 7 Y I E L D MV TORQUE SHEAR S T R E S S (MV) (DYNES CM) (DYNES/SQ CM) 0 . 7 3 5 E + 0 1 0 . 3 2 8 E + 0 6 0 . 2 6 1 E + O 5 E Q U I L I B R I U M MV TORQUE SHEAR S T R E S S (MV) {DYNES CM) (DYNES/SQ CM) 0 . 1 9 0 E + 0 1 0 . 8 4 7 E + 0 5 0 . 6 7 4 E + 0 4 T A B L E D - 1 (. . . CONT.) 162 DEX + 1.0MOLE NACL I N 1 0 . 0 % PEG-H 20 P H I (CC/CC) 0.07 L (MM) OMEGA ( B A D / S E C ) 5. 03 0 0. 23 27 Y I E L D MV (MV) 0 . 1 2 7 E + 0 2 TORQUE (DYNES CM) 0 . 5 6 5 E + 0 6 SHEAR S T R E S S (DYNES/SQ CM) 0 . 4 5 0 E + 0 5 E Q U I L I B R I U M MV TORQUE (MV) (DYNES CM) 0 . 5 8 1 E + 0 0 0 . 2 5 9 E + 0 5 SHEAR S T R E S S (DYNES/SQ CM) 0.2Q6E+04 DEX + 1.0MOLE NACL I N 1 0 . 0 % P E G - H 2 0 P H I ( C C / C C ) 0 . 0 4 Y I E L D MV (MV) 0 . 1 1 1 E + 0 1 E Q U I L I B R I U M (EV) 0 . 3 2 1 E + 0 0 L (MM) 6.72 0 TORQUE (DYNES CM) 0 . 4 9 5 E + 0 5 MV TORQUE (DYNES CM) 0 . 1 4 3 E + 0 5 OMEGA (RAD/SEC) 0. 2 3 2 7 SHEAR S T R E S S (DYNES/SQ CM) 0.3 94E+04 SHEAR S T R E S S (DYNES/SQ CM) 0 . 1 1 4 E + 0 4 DEX + 1.0MOLE NACL I N 1 0 . 0 % P E G - H 2 0 P H I ( C C / C C ) 0 . 05 L (MM) OMEGA (R A D / S E C ) 6. 7 2 0 0 . 2 3 2 7 Y I E L D MV (MV) 0 . 1 8 1 E + 0 1 TORQUE (DYNES CM) 0 . 8 0 7 E + 0 5 SHEAR S T R E S S (DYNES/SQ CM) 0 . 6 4 2 E + 0 4 E Q U I L I B R I U M MV TORQUE (MV) (DYNES CM) 0 . 5 4 9 E + 0 0 0 . 2 4 5 E + 0 5 SHEAR S T R E S S (DYNES/SQ CM) 0 . 1 9 5 E + 0 4 T A B L E D - 1 (. . .CONT.) 163 DEX • 1.0HOLE NACL I N 1 0 . 0 % P E G - H 2 0 P H I ( C C / C C ) L(MM) OMEGA (RAD/SEC) 0 . 0 6 6 . 7 2 0 0 . 2327 Y I E L D MV TORQUE SHEAR STRESS (MV) (DYNES CM) (DYNES/SQ CM) 0. 6 1 4 E + 0 1 0 . 2 7 4 E + 0 6 0 . 2 1 8 E + 0 5 E Q U I L I B R I U M MV TORQUE SHEAR S T R E S S (MV) (DYNES CM) (DYNES/SQ CH) 0. 1 8 5 E + 0 1 0 . 8 2 5 E + 0 5 0 . 6 5 7 E + 0 4 DEX + 1 . 0 H O L E NACL I N 10 . 0 % P E G - H 2 0 P H I ( C C / C C ) L (MM) OMEGA (RAD/SEC) 0.07 6. 7 2 0 0. 2 3 2 7 Y I E L D MV TORQUE SHEAR S T R E S S (MV) (DYNES CM) (DYNES/SQ CM) 0 . 1 1 9 E + 0 2 0 . 5 3 0 E + 0 6 0 . 4 2 2 E + 0 5 E Q U I L I B R I U M MV TORQUE SHEAR S T R E S S (MV) (DYNES CM) (DYNES/SQ CM) 0 . 4 5 1 E + 0 1 0 . 2 0 1 E + 0 6 0 . 1 6 0 E + 0 5 DEX • 1.0MOLE NACL I N 3 0 . 0 % PEG-H20 P H I ( C C / C C ) L (MM) OMEGA (RAD/SEC) 0. 14 0 . 9 8 7 0. 2 3 2 7 Y I E L D MV TORQUE SHEAR S T R E S S (MV) (DYNES CM) (DYNES/SQ CM) 0 . 6 7 9 E + 0 0 0 . 3 0 3 E + 0 5 0 . 2 4 1 E + 0 4 E Q U I L I B R I U M MV TORQUE SHEAR S T R E S S (MV) (DYNES CM) (DYNES/SQ CM) 0 . 5 0 0 E - 0 1 0 . 2 2 3 E + 0 4 0 . 1 7 8 E + 0 3 T A B L E D - 1 {. ...CONT.). DEX + 1.0MOLE NACL I N 30.0% P E G - H 2 0 P H I ( C C / C C ) 0. 16 Y I E L D HV (MV) 0 . 3 8 8 E + 0 1 E Q U I L I B R I U M (MV) 0 . 2 2 0 E + 0 0 L (MM) 0 . 9 8 7 TORQUE (DYNES CM) 0 . 1 7 3 E + 0 6 MV TORQUE (DY N E S CM) 0.98 1E+04 OMEGA (RAD/SEC) 0. 2 3 2 7 SHEAR S T R E S S (DYNES/SQ CM) 0 . 1 3 8 E + 0 5 SHEAR STRESS (DYNES/SQ CM) 0 . 7 8 1 E + 0 3 DEX + 1.0MOLE NAC L I N 3 0 . 0 % P E G - H 2 0 P H I ( C C / C C ) 0. 18 L (MM) OMEGA (RAD/SEC) 0. 9 8 7 0 . 2 3 2 7 Y I E L D MV (MV) 0 . 1 3 0 E + 0 2 TORQUE (DYNES CM) 0 . 5 8 0 E + 0 6 SHEAR S T R E S S (DYNES/SQ CM) 0 . 4 6 2 E + 0 5 E Q U I L I B R I U M MV TORQUE (MV) (DYNES CM) 0 . 6 0 1 E + 0 0 0 . 2 6 8 E + 0 5 SHEAR S T R E S S (DYNES/SQ CM) 0 . 2 1 3 E + 0 4 DEX + 1.0MOLE NACL I N 3 0 . 0 % P E G - H 2 0 P H I ( C C / C C ) 0. 10 Y I E L D MV (MV) 0 . 1 7 5 E + 0 0 L(MM) 1. 6 2 0 TORQUE (DYNES CM) 0.78 1E+0U OMEGA (RAD/SEC) 0.23 27 SHEAR S T R E S S (DYNES/SQ CM) 0 . 6 2 2 E + 0 3 E Q U I L I B R I U M MV TORQUE (MV) (DYNES CM) 0 . 5 0 0 E - 0 1 0 . 2 2 3 E + 0 4 SHEAR S T R E S S (DYNES/SQ CM) 0. 1 7 8 E + 0 3 T A B L E D - 1 (. . .CONT.) 165 DEX + 1.0MOLE NACL I N 3 0 . 0 % P E G - H 2 0 P H I { C C / C C ) L (MM) OMEGA ( R A D / S E C ) 0. 12 1. 6 2 0 0. 2 3 2 7 Y I E L D MV TORQUE SHEAR S T R E S S (MV) (DYNES CM) (DYNES/SQ CM) 0 . 1 9 7 E + 0 1 0 . 8 7 8 E + 0 5 0 . 6 9 9 E + 0 4 E Q U I L I B R I U M MV TORQUE SHEAR STRESS (MV) (DYNES CM) (DYNES/SQ CM) 0 . 8 9 9 E - 0 1 0 . 4 0 1 E + 0 4 0 . 3 1 9 E + 0 3 DEX + 1.0MOLE NACL I N 3 0 . 0 % PEG-H20 P H I ( C C / C C ) L (MM) OMEGA (RAD/SEC) 0. 13 1. 6 2 0 0. 2 3 2 7 Y I E L D MV TORQUE SHEAR S T R E S S (MV) (DYNES CM) (DYNES/SQ CM) 0 . 2 8 3 E + 0 1 0 . 1 2 6 E + 0 6 0 . 1 0 0 E + 0 5 E Q U I L I B R I U M MV TORQUE SHEAR S T R E S S (MV) (DYNES CM) (DYNES/SQ CM) 0 . 8 9 9 E - 0 1 0 . 4 0 1 E + 0 4 0 . 3 1 9 E + 0 3 DEX + 1.0MOLE NACL I N 3 0 . 0 % P E G - H 2 0 P H I (CC/CC) L (MM) OMEGA (RAD/SEC) 0,14 1. 6 2 0 0 . 2 3 2 7 Y I E L D MV TORQUE SHEAR S T R E S S (MV) (DYNES CM) (DYNES/SQ CM) 0 . 7 8 7 E + 0 1 0 . 3 5 1 E + 0 6 0 . 2 7 9 E + 0 5 E Q U I L I B R I U M MV TORQUE SHEAR S T R E S S (MV) (DYNES CM) (DYNES/SQ CM) 0 . 1 0 6 E + 0 1 0 . 4 7 3 E + 0 5 0 . 3 7 7 E + 0 4 T A B L E D - 1 (. . .CONT.) 166 DEX * 1.0MOLE NACL I N 3 0 . 0 % PEG-H20 P H I (CC/CC) 0.05 L (MM) OMEGA (RAD/SEC) 3. 0 1 0 0. 2 3 2 7 Y I E L D MV (MV) 0 . 3 3 0 E + 0 0 TORQUE (DYNES CM) 0 . 1 4 7 E + 0 5 SHEAR S T R E S S (DYNES/SQ CM) 0. 1 1 7 E + 0 U E Q U I L I B R I U M MV TORQUE (UV) (DYNES CM) 0 . 8 0 0 E - 0 1 0 . 3 5 7 E + 0 4 SHEAR S T R E S S (DYNES/SQ CM) 0 . 2 8 4 E + 0 3 DEX * 1.0MOLE NAC L I N 3 0 . 0 % PEG-H20 P H I (CC/CC) L (MM) OMEGA (RAD/SEC) 0.06 3. 0 1 0 0. 23 27 Y I E L D MV (MV) 0 . 2 5 6 E + 0 1 E Q U I L I B R I U M (MV) 0 . 8 0 0 E + 0 0 TORQUE (DYNES CM) 0 . 1 1 4 E + 0 6 MV TORQUE (DYNES CM) 0 . 3 5 7 E + 0 5 SHEAR S T R E S S (DYNES/SQ CM) 0. 9 0 7 E + 0 4 SHEAR S T R E S S (DYNES/SQ CM) 0. 2 8 4 E + 0 U DEX + 1.0MOLE NACL I N 3 0 . 0 % P E G - H 2 0 P H I ( C C / C C ) 0.07 L (MM) OMEGA (RAD/SEC) 3 . 0 1 0 0.23 27 Y I E L D MV (MV) 0 . 6 0 1 E + 0 1 TORQUE (DYNES CM) 0 . 2 6 8 E + 0 6 SHEAR S T R E S S (DYNES/SQ CM) 0 . 2 1 3 E + 0 5 E Q U I L I B R I U M MV TORQUE (MV) (DYNES CM) 0 . 1 2 0 E + 0 1 0 . 5 3 5 E + 0 5 SHEAR S T R E S S ( D Y N E S / S Q CM) 0 . 4 2 6 E + 0 4 T A B L E D - 1 (. . .CONT.) 167 DEX + 1.0MOLE NACL I N 3 0 . 0 % P E G - H 2 0 P H I ( C C / C C ) 0. 08 L(MM) OMEGA (RAD/SEC) 3 . 0 1 0 0. 2 3 2 7 Y I E L D MV (MV) 0 . 1 1 3 E + 0 2 TORQUE (DYNES CM) 0 . 5 0 2 E + 0 6 SHEAR S T R E S S (DYNES/SQ CM) 0. 4 0 0 E + 0 5 E Q U I L I B R I U M MV TORQUE (MV) (DYNES CM) 0 . 2 5 1 E + 0 1 0 . 1 1 2 E * 0 6 SHEAR S T R E S S (DYNES/SQ CM) 0 . 8 9 2 E + 0 4 DEX + 1.0MOLE NACL I N 3 0 . 0 % P E G - H 2 0 P H I ( C C / C C ) L (MM) OMEGA (RAD/SEC) 0.04 5 . 0 3 0 0 . 2 3 2 7 Y I E L D MV (MV) 0 . 1 0 0 E + 0 1 E Q U I L I B R I U M (MV) 0 . 1 5 0 E + 0 0 TORQUE (DYNES CM) 0.446E+G5 MV TORQUE (DYNES CM) 0 . 6 6 9 E + 0 4 SHEAR S T R E S S (DYNES/SQ CM) 0 . 3 5 5 E + 0 4 SHEAR S T R E S S (DYNES/SQ CM) 0 . 5 3 3 E + 0 3 DEX + 1.0MOLE NACL I N 3 0 . 0 % PE G - H 2 0 P H I ( C C / C C ) 0 . 05 Y I E L D MV (MV) 0 . 1 7 0 E + 0 1 E Q U I L I B R I U M (MV) 0 . 5 0 0 E + 0 0 L (MM) 5. 0 3 0 TORQUE (DYNES CM) 0 . 7 5 8 E + 0 5 TORQUE (DYNES CM) 0 . 2 2 3 E + 0 5 OMEGA (RAD/SEC) 0. 2 3 2 7 SHEAR S T R E S S (DYNES/SQ CM) 0 . 6 0 3 E + 0 4 SHEAR S T R E S S ( D Y N E S / S Q CM) 0 . 1 7 8 E + 0 4 T A B L E D - 1 {. . .CONT.) DEX + 1.0MOLE NACL I N 30. P E G - H 2 0 P H I (CC/CC) 0.06 L (MM) OMEGA (RAD/SEC) 5 . 0 3 0 0 . 2 3 2 7 Y I E L D MV <MV) 0 . 4 8 0 E + 0 1 TORQUE (DYNES CM) 0 . 2 1 4 E + 0 6 SHEAR S T R E S S (DYNES/SQ CM) 0 . 1 7 0 E + 0 5 E Q U I L I B R I U M MV TORQUE (MV) (DYNES CM) 0 . 1 6 4 E + 0 1 0 . 7 3 1 E + 0 5 SHEAR S T R E S S (DYNES/SQ CM) 0. 5 8 2 E + 0 4 DEX 1.0MOLE NACL I N 3 0 . 0 % P E G - H 2 0 P H I ( C C / C C ) 0.07 L (MM) OMEGA (RAD/SEC) 5. 0 3 0 0. 2 3 2 7 Y I E L D MV (MV) 0 . 6 9 1 E + 0 1 TORQUE (DYNES CM) 0 . 3 0 8 E + 0 6 SHEAR S T R E S S (DYNES/SQ CM) 0 . 2 4 5 E + 0 5 E Q U I L I B R I U M MV TORQUE (MV) (DYNES CM) 0 . 1 8 5 E + 0 1 0 . 8 2 5 E + 0 5 SHEAR S T R E S S (DYNES/SQ CM) 0 . 6 5 7 E + 0 4 DEX + 1.0MOLE NACL I N 3 0 . 0 % PEG-H20 P H I ( C C / C C ) 0.04 L (MM) OMEGA (RAD/SEC) 6. 7 2 0 0. 2 3 2 7 Y I E L D MV (MV) 0. 951E«-00 TORQUE (DYNES CM) 0 . 4 2 4 E + 0 5 SHEAR S T R E S S (DYNES/SQ CM) 0 . 3 3 8 E + 0 4 E Q U I L I B R I U M MV TORQUE (MV) (DYNES CM) 0 . 2 4 0 E + 0 0 0 . 1 0 7 E + 0 5 SHEAR S T R E S S (DYNES/SQ CM) 0 . 8 5 2 E + 0 3 T A B L E D - 1 (. . .CONT.) 169 DEX + 1.0HOLE NACL I N 3 0 . 0 % P E G - H 2 0 P H I ( C C / C C ) L (MM) OMEGA ( R A D / S E C ) 0.05 6. 7 2 0 0. 2 3 2 7 Y I E L D MV TORQUE SHEAR S T R E S S (MV) (DYNES CM) (DYNES/SQ CM) 0 . 1 8 2 E + 0 1 0 . 8 1 2 E + 0 5 0 . 6 4 6 E + 0 4 E Q U I L I B R I U M MV TORQUE SHEAR S T R E S S (HV) (DYNES CH) (DYNES/SQ CM) 0 . 7 0 0 E + 0 0 0 . 3 1 2 E + 0 5 0 . 2 4 8 E + 0 4 DEX + 1.0 MOLE NA P H I ( C C / C C ) 0. 06 Y I E L D MV (MV) 0 . 3 0 7 E + 0 1 E Q U I L I B R I U M (MV) 0 . 1 2 0 E + 0 1 L I N 3 0 . 0 % P E G L (MM) 6. 7 2 0 TORQUE (DYNES CM) 0. 137E+-06 MV TORQUE (DYNES CM) 0 . 5 3 5 E + 0 5 H20 OMEGA (RAD/SEC) 0. 23 27 SHEAR S T R E S S (DYNES/SQ CM) 0 . 1 0 9 E + 0 5 SHEAR S T R E S S (DYNES/SQ CM) 0. U26E+04 DEX + 1.0MOLE NACL I N 3 0 . 0 % P E G - H 2 0 P H I (CC/CC) L (MM) OMEGA ( R A D / S E C ) 0 . 0 7 6 . 7 2 0 0.23 27 Y I E L D MV TORQUE SHEAR S T R E S S (MV) (DYNES CM) ( D Y N E S / S Q CM) 0.53UE+01 0 . 2 3 8 E + 0 6 0 . 1 8 9 E + 0 5 E Q U I L I B R I U M MV TORQUE SHEAR S T R E S S (MV) (DYNES CM) (DYNES/SQ CM) 0 . 2 0 0 E + 0 1 0 . 8 9 2 E + 0 5 0 . 7 1 0 E + 0 4 T A B L E D - 1 (. . .CONT.) 170 DEX + 0.25MOLE NACL I N 3 0 . 0 % P E G - H 2 0 P H I (CC/CC) 0. 14 L (MM) OMEGA ( R A D / S E C ) 1.620 0. 2327 Y I E L D MV (MV) 0 . 4 0 1 E + 0 1 TORQUE (DYNES CM) 0 . 1 7 9 E + 0 6 SHEAR S T R E S S (DYNES/SQ CM) 0 . 1 4 2 E + 0 5 E Q U I L I B R I U M MV TORQUE (MV) (DYNES CM) 0 . 7 5 1 E + 0 0 0 . 3 3 5 E + 0 5 SHEAR S T R E S S (DYNES/SQ CM) 0 . 2 6 7 E + 0 4 DEX + 0.50MOLE NACL I N 3 0 . 0 % P E G - H 2 0 P H I ( C C / C C ) 0. 14 L (MM) OMEGA ( R A D / S E C ) 1. 6 2 0 0. 23 27 Y I E L D MV (MV) 0.525E+01 TORQUE (DYNES CM) 0 . 2 3 4 E + 0 6 SHEAR S T R E S S (DYNES/SQ CM) 0. 1 8 6 E + 0 5 E Q U I L I B R I U M MV TORQUE (MV) (DYNES CM) 0 . 8 5 0 E + 0 0 0 . 3 7 9 E + 0 5 SHEAR S T R E S S (DYNES/SQ CM) 0. 3 0 2 E + 0 4 DEX *• 2.00MOLE NA C L I N 3 0 . 0 % P E G - H 2 0 P H I ( C C / C C ) 0. 14 L (MM) OMEGA (R A D / S E C ) 1. 6 2 0 0 . 2 3 2 7 Y I E L D MV (MV) 0 . 1 0 6 E + 0 2 TORQUE (DYNES CM) 0 . 4 7 3 E + 0 6 SHEAR S T R E S S (DYNES/SQ CM) 0 . 3 7 7 E + 0 5 E Q U I L I B R I U M MV TORQUE (MV) (DYNES CM) 0 . 1 4 4 E + 0 1 0 . 6 4 2 E + 0 5 SHEAR S T R E S S (DYNES/SQ CM) 0 . 5 1 1 E + 0 4 T A B L E D - 1 (...CONT.) 171 DEX + 0.25MOLE H N C L 2 I N 3 0 . 0 % P E G - H 2 0 P H I ( C C / C C ) 0. 14 L (MM) OMEGA (RAD/SEC) 1. 6 2 0 0 . 2 3 2 7 Y I E L D MV (MV) 0 . 2 8 3 E + 0 1 TORQUE (DYNES CM) 0 . 1 2 6 E + 0 6 SHEAR S T R E S S (DYNES/SQ CM) 0. 1 0 0 E + 0 5 E Q U I L I B R I U M MV TORQUE (MV) (DYNES CM) 0 . 7 5 1 E + 0 0 0 . 3 3 5 E + 0 5 SHEAR STRESS (DYNES/SQ CH) 0 . 2 6 7 E + 0 4 DEX + 0.25MOLE C E C L 3 I N 3 0 . 0 % P E G - H 2 0 P H I (CC/CC) 0. 14 L (MM) OMEGA ( R A D / S E C ) 1.620 0.23 27 Y I E L D MV (MV) 0 . 2 9 4 E + 0 1 TORQUE (DYNES CM) 0 . 1 3 1 E + 0 6 SHEAR S T R E S S (DYNES/SQ CM) 0 . 1 0 4 E + 0 5 E Q U I L I B R I U M MV TORQUE (MV) (DYNES CM) 0 . 7 5 1 E + 0 0 0 . 3 3 5 E + 0 5 SHEAR S T R E S S (DYNES/SQ CM) 0 . 2 6 7 E + 0 4 DEX + 0.0MOLE NACL I N 3 0 . PEG - H 2 0 P H I (CC/CC) 0. 14 L (MM) OMEGA (RAD/SEC) 1. 6 2 0 0. 2 3 2 7 Y I E L D MV (MV) 0 . 4 0 4 E + 0 1 TORQUE (DYNES CM) 0 . 1 8 0 E + 0 6 SHEAR S T R E S S (DYNES/SQ CM) 0 . 1 4 3 E + 0 5 E Q U I L I B R I U M MV TORQUE (MV) (DYNES CM) 0 . 5 0 0 E + 0 0 0 . 2 2 3 E + 0 5 SHEAR STRESS (DYNES/SQ CM) 0. 178E+04 T A B L E D - 1 {. . . CONT. ) 172 DEX + 1.0MOLE N A C L I N 4 0 . 0 % P E G - H 2 0 P H I (CC/CC) 0. 16 Y I E L D HV (HV) 0 . 5 4 9 E + 0 0 E Q U I L I B R I U M (MV) 0 . 8 0 0 E - 0 1 L (MM) 0. 9 8 7 TORQUE (DYNES CM) 0 . 2 4 5 E + 0 5 MV TORQUE (DYNES CM) 0 . 3 5 7 E + 0 4 OMEGA (R A D / S E C ) 0 . 2 3 2 7 SHEAR S T R E S S (DYNES/SQ CM) 0 . 1 9 5 E + 0 4 SHEAR S T R E S S (DYNES/SQ CM) 0.2 8 4 E + 0 3 DEX + 1.0MOLE NACL I N 4 0 . 0 % P E G - H 2 0 P H I ( C C / C C ) 0. 17 Y I E L D MV (MV) 0 . 7 8 9 E + 0 0 E Q U I L I B R I U M (MV) 0 . 1 0 0 E + 0 0 L (MM) 0. 9 8 7 TORQUE (DYNES CM) 0 . 3 5 2 E + 0 5 MV TORQUE (DYNES CM) 0 . 4 4 6 E + 0 4 OMEGA(RAD/SEC) 0. 2 3 2 7 SHEAR S T R E S S (DYNES/SQ CM) 0 . 2 8 0 E + 0 4 SHEAR S T R E S S (DYNES/SQ CM) 0 . 3 5 5 E + 0 3 DEX * 1.0MOLE NACL I N 40 . 0 % P E G - H 2 0 P H I ( C C / C C ) 0. 18 Y I E L D MV (MV) 0 . 2 4 0 E + 0 1 L (MM) 0. 9 8 7 TORQU E (DYNES CM) 0 . 1 0 7 E + 0 6 OMEGA (R A D / S E C ) 0. 2 3 2 7 SHEAR S T R E S S (DYNES/SQ CM) 0 . 8 5 2 E + 0 4 E Q U I L I B R I U M MV (MV) 0 . 1 0 0 E + 0 0 TORQU E (DYNES CM) 0 . 4 4 6 E + 0 4 SHEAR S T R E S S (DYNES/SQ CM) 0 . 3 5 5 E + 0 3 T A B L E D - 1 {. . .CONT.) 173 DEX + 1.0MOLE NACL I N 4 0 . 0 % P E G - H 2 0 P H I < C C / C C ) 0. 14 L (MM) OMEGA ( B A D / S E C ) 1. 620 0.23 27 Y I E L D MV (MV) 0 . 8 0 0 E + 0 0 TORQUE { DYNES CM) 0 . 3 5 7 E + 0 5 SHEAR S T R E S S (DYNES/SQ CM) 0 . 2 8 4 E + 0 4 E Q U I L I B R I U M MV TORQUE (MV) (DYNES CM) 0 . 8 9 9 E - 0 1 0 . 4 0 1 E + 0 4 SHEAR S T R E S S (DYNES/SQ CM) 0. 3 1 9 E + 0 3 DEX + 1.0MOLE NACL I N 4 0 . 0 % PEG-H20 P H I ( C C / C C ) 0. 15 Y I E L D MV (MV) 0 . 1 9 2 E + 0 1 E Q U I L I B R I U M (MV) 0 . 5 0 0 E + 0 0 L (MM) 1. 6 2 0 TORQUE (DYNES CM) 0 . 8 5 6 E + 0 5 MV TORQUE (DYNES CM) 0 . 2 2 3 E + 0 5 OMEGA (RAD/SEC) 0. 2 3 2 7 SHEAR S T R E S S (DYNES/SQ CM) 0 . 6 8 1 E + 0 4 SHEAR S T R E S S (DYNES/SQ CM) 0 . 1 7 8 E + 0 4 DEX + 1.0MOLE NACL I N 4 0 . P E G - H 2 0 P H I ( C C / C C ) 0. 16 L (MM) OMEGA ( R A D / S E C ) 1 . 6 2 0 0 . 2 3 2 7 Y I E L D MV (MV) 0 . 4 2 4 E + 0 1 TORQUE (DYNES CM) 0 . 1 8 9 E + 0 6 SHEAR S T R E S S (DYNES/SQ CM) 0 . 1 5 0 E + 0 5 E Q U I L I B R I U M MV TORQUE (MV) (DYNES CM) 0.110E+01 0 . 4 9 1 E + 0 5 SHEAR S T R E S S (DYNES/SQ CM) 0 . 3 9 1 E + 0 4 TABLE D - 1 {. . . CONT.) 174 DEX * 1 .0HOLE NACL IN 40.0% PEG-H20 PHI(CC/CC) 0. 17 L (MM) OMEGA (RAD/SEC) 1.620 0.2327 YIELD HV (MV) 0.890E+01 TORQUE (DYNES CM) 0.397E+06 SHEAR STRESS (DYNES/SQ CM) 0.316E+05 EQUILIBRIUM MV TORQUE (MV) (DYNES CM) 0.190E+01 0.847E+05 SHEAR STRESS (DYNES/SQ CM) 0.674E+04 DEX + 1.0HOLE NACL IN 40.0% PEG-H20 PHI(CC/CC) 0.08 YIELD MV (MV) 0.193E+01 EQUILIBRIUM (MV) 0.300E+00 L (MM) 3. 010 TORQUE (DYNES CM) 0.861E+05 MV TORQUE (DYNES CM) 0.134E+05 OMEGA(RAD/SEC) 0.23 27 SHEAR STRESS (DYNES/SQ CM) 0.685E+04 SHEAR STRESS (DYNES/SQ CM) 0.107E+04 DEX + 1.0MOLE NACL IN 40.0% PEG-H20 PHI (CC/CC) 0.09 YIELD MV (HV) 0.359E+01 EQUILIBRIUM (MV) 0.639E+00 L (MM) 3.010 TORQUE (DYNES CM) 0. 160E+06 MV TORQUE (DYNES CM) 0.285E+05 OMEGA (RAD/SEC) 0. 2327 SHEAR STRESS (DYNES/SQ CM) 0.127E+05 SHEAR STRESS (DYNES/SQ CM) 0.227E+04 T A B L E D - 1 {. . .CONT.) DEX + 1.0MOLE NACL I N 4 0 . 0 % P E G - H 2 0 P H I (CC/CC) 0. 10 L (MM) OMEGA (BAD/SEC) 3 . 0 1 0 0. 2 3 2 7 Y I E L D MV (MV) 0 . 6 3 7 E + 0 1 TOBQUE (DYNES CM) 0 . 2 8 4 E + 0 6 SHEAR STRESS (DYNES/SQ CM) 0. 226E+ 0 5 E Q U I L I B R I U M HV TORQUE (MV) (DYNES CM) O . 8 9 9 E + 0 0 0 . 4 0 1 E + 0 5 SHEAR S T R E S S (DYNES/SQ CM) 0 . 3 1 9 E + 0 4 DEX + 1.0MOLE NACL I N 4 0 . 0 % P E G - H 2 0 P H I (CC/CC) 0. 1 1 Y I E L D MV (MV) 0 . 7 0 2 E + 0 1 E Q U I L I B R I U M (MV) 0 . 1 2 0 E + 0 1 L (MM) 3 . 0 1 0 TORQUE (DYNES CM) 0 . 3 1 3 E + 0 6 MV TOBQUE (DYNES CM) 0 . 5 3 5 E + 0 5 OMEGA ( B A D / S E C ) 0 . 2 3 2 7 SHEAB S T B E S S (DYNES/SQ CM) 0. 2 4 9 E + 0 5 SHEAR S T R E S S (DYNES/SQ CM) 0. 4 2 6 E + 0 4 DEX • 1.0MOLE NACL I N 4 0 . 0 % P E G - H 2 0 P H I ( C C / C C ) 0. 06 Y I E L D MV (MV) 0 . 1 2 1 E + 0 1 E Q U I L I B R I U M (MV) 0 . 2 8 9 E + 0 0 L (MM) 5. 0 3 0 TORQUE (DYNES CM) 0 . 5 4 0 E + 0 5 MV TORQUE (DYNES CM) 0 . 1 2 9 E + 0 5 OMEGA (RAD/SEC) 0. 2 3 2 7 SHEAR S T R E S S (DYNES/SQ CM) 0 . 4 3 0 E + 0 4 SHEAR S T R E S S (DYNES/SQ CM) 0 . 1 0 3 E + 0 4 T A B L E D - 1 {.. .CONT.) DEX • 1.0MOLE NACL I N 4 0 . 0 % P E G - H 2 0 P H I (CC/CC) L (MM) OMEGA (RAD/SEC) 0.07 5.03 0 0.23 27 Y I E L D MV TORQOE SHEAR S T R E S S (MV) (DYNES CM) (DYNES/SQ CM) 0 . 2 2 4 E + 0 1 0 . 1 0 0 E + 0 6 0 . 7 9 6 E + 0 4 E Q U I L I B R I U M MV TORQUE SHEAR S T R E S S (MV) (DYNES CM) (DYNES/SQ CM) 0 . 6 7 9 E + 0 0 0 . 3 0 3 E + 0 5 0 . 2 4 1 E + 0 4 DEX + 1.0MOLE NACL I N 4 0 . 0 % P E G - H 2 0 P H I (CC/CC) L (MM) OMEGA (RAD/SEC) 0.08 5. 0 3 0 0. 2 3 2 7 Y I E L D MV TORQUE SHEAR S T R E S S (MV) (DYNES CM) (DYNES/SQ CM) 0 . 3 7 9 E + 0 1 0 . 1 6 9 E + 0 6 0 . 1 3 5 E + 0 5 E Q U I L I B R I U M MV TORQUE SHEAR S T R E S S (MV) (DYNES CM) (DYNES/SQ CM) 0 . 6 5 0 E + 0 0 0 . 2 9 0 E + 0 5 0 . 2 3 1 E * 0 4 DEX + 1.0MOLE NACL I N 4 0 . 0 % PEG-H20 P H I ( C C / C C ) L (MM) OMEGA ( R A D / S E C ) 0.09 5. 0 3 0 0. 2 3 2 7 Y I E L D MV TORQUE SHEAR S T R E S S (MV) (DYNES CM) (DYNES/SQ CM) 0 . 9 3 5 E + 0 1 0 . 4 1 7 E + 0 6 0 . 3 3 2 E + 0 5 E Q U I L I B R I U M MV TORQUE (MV) (DYNES CM) 0 . 1 3 6 E + 0 1 0 . 6 0 7 E + 0 5 SHEAR S T R E S S (DYNES/SQ CM) 0 . 4 8 3 E + 0 4 T A B L E D - 1 {. . .CONT.) 177 DEX + 1.0MOLE NACL I N 40 . 0 % P E G - H 2 0 P H I ( C C / C C ) L (MM) OMEGA ( R A D / S E C ) 0.05 6. 7 2 0 0.2327 Y I E L D MV TORQUE SHEAR S T R E S S (MV) (DYNES CM) (DYNES/SQ CM) 0 . 9 8 0 E * 0 0 0 . 4 3 7 E * 0 5 0 . 3 4 8 E + 0 4 E Q U I L I B R I U M MV TORQUE SHEAR S T R E S S (MV) (DYNES CM) (DYNES/SQ CM) 0 . 8 0 0 E - 0 1 0 . 3 5 7 E + 0 4 0 . 2 8 4 E + 0 3 DEX + 1.0MOLE NACL I N 4 0 , 0 % PE G - H 2 0 P H I ( C C / C C ) L (MM) . OMEGA (R A D / S E C ) 0.06 6 . 7 2 0 0. 2 3 2 7 Y I E L D MV TORQUE SHEAR S T R E S S ,(MV) (DYNES CM) ( D Y N E S / S Q CM) 0 . 2 6 0 E + 0 1 0 . 1 1 6 E + 0 6 0 . 9 2 3 E + 0 4 E Q U I L I B R I U M MV TORQUE SHEAR S T R E S S (MV) (DYNES CM) ( D Y N E S / S Q CM) 0 . 7 0 0 E + 0 0 0 . 3 1 2 E + 0 5 0. 2 4 8 E + 0 4 DEX + 1.0MOLE NACL I N 4 0 . 0 % P E G - H 2 0 P H I ( C C / C C ) L (MM) OMEGA ( R A D / S E C ) 0.07 6. 7 2 0 0 . 2 3 2 7 Y I E L D MV TORQUE SHEAR S T R E S S (MV) (DYNES CM) (DYNES/SQ CM) 0 . 3 5 9 E + 0 1 0 . 1 6 0 E + 0 6 0 . 1 2 7 E + 0 5 E Q U I L I B R I U M MV TORQUE SHEAR S T R E S S (MV) (DYNES CM) (DYNES/SQ CM) 0 . 7 0 0 E + 0 0 0 . 3 1 2 E + 0 5 0 . 2 4 8 E + 0 4 T A B L E D - 1 (. . .CONT.) 178 DEX + 1.0MOLE NACL I N 4 0 . 0 % PEG-H20 P H I ( C C / C C ) 0. 08 L (MM) OMEGA (RAD/SEC) 6. 7 2 0 0 . 2 3 2 7 Y I E L D MV (MV) 0 . 4 3 7 E + 0 1 TORQUE (DYNES CM) 0 . 1 9 5 E + 0 6 SHEAR S T R E S S (DYNES/SQ CM) 0. 1 5 5 E + 05 E Q U I L I B R I U M MV TORQUE (MV) (DYNES CM) 0 . 1 1 0 E + 0 1 0 . 4 9 1 E + 0 5 SHEAR S T R E S S (DYNES/SQ CM) 0 . 3 9 1 E + 0 4 DEX + 1.0MOLE NACL I N 5 0 . 0 % P E G - H 2 0 P H I ( C C / C C ) 0 . 1 2 L (MM) OMEGA (RAD/SEC) 0 . 9 8 7 0. 2 3 2 7 Y I E L D MV (MV) 0 . 3 3 0 E + 0 0 TORQUE (DYNES CM) 0 . 1 4 7 E + 0 5 SHEAR S T R E S S (DYNES/SQ CM) 0 , 1 1 7 E + 0 4 E Q U I L I B R I U M MV TORQUE (MV) (DYNES CM) 0 . 1 5 0 E + 0 0 0 . 6 6 9 E + 0 4 SHEAR S T R E S S (DYNES/SQ CM) 0 . 5 3 3 E + 0 3 DEX + 1.0MOLE NACL I N 5 0 . 0 % P E G - H 2 0 P H I ( C C / C C ) 0. 16 L (MM) OMEGA (RAD/SEC) 0 . 9 8 7 0 . 2 3 2 7 Y I E L D MV (MV) 0 . 6 3 0 E + 0 0 TORQUE (DYNES CM) 0 . 2 8 1 E + 0 5 SHEAR S T R E S S (DYNES/SQ CM) 0 . 2 2 4 E + 0 4 E Q U I L I B R I U M MV TORQUE (MV) (DYNES CM) 0 . 8 9 9 E - 0 1 0.40 1E+04 SHEAR S T R E S S (DYNES/SQ CM) 0 . 3 1 9 E + 0 3 T A B L E D - 1 (. . .CONT.) 179 DEX + 1.0MOLE NACL I N 5 0 . 0 % P E G - H 2 0 P H I ( C C / C C ) 0. 17 Y I E L D MV (MV) 0 . 7 3 1 E + 0 0 E Q U I L I B R I U M (MV) 0 . 7 0 0 E - 0 1 L (MM) 0. 9 8 7 TORQUE (DYNES CM) 0 . 3 2 6 E + 0 5 MV TORQUE (DYNES CM) 0 . 3 1 2 E + 0 4 OMEGA (RAD/SEC) 0. 2 3 2 7 SHEAR S T R E S S (DYNES/SQ CM) 0 . 2 6 0 E + 0 4 SHEAR S T R E S S (DYNES/SQ CM) 0.2 4 8 E + 0 3 DEX + 1.0MOLE NACL I N 50 . 0 % P E G - H 2 0 P H I (CC/CC) 0. 18 Y I E L D MV (MV) 0 . 1 3 4 E + 0 1 E Q U I L I B R I U M MV (MV) 0 . 1 0 0 E + 0 0 L (MM) 0. 9 8 7 TORQUE (DYNES CM) 0 . 5 9 8 E + 0 5 TORQUE (DYNES CM) 0 . 4 4 6 E + 0 4 OMEGA (RAD/SEC) 0 . 2 3 2 7 SHEAR S T R E S S (DYNES/SQ CM) 0. 4 7 6 E + 0 4 SHEAR S T R E S S (DYNES/SQ CM) 0 . 3 5 5 E + 0 3 DEX • 1.OMOLE NACL I N 5 0 . 0 % P E G - H 2 0 P H I (CC/CC) 0.19 L (MM) OMEGA (RAD/SEC) 0 . 9 8 7 0. 2 3 2 7 Y I E L D MV (MV) 0 . 3 7 9 E + 0 1 TORQUE (DYNES CM) 0. 1 6 9 E + 06 SHEAR STRESS (DYNES/SQ CM) 0 . 1 3 5 E + 0 5 E Q U I L I B R I U M MV TORQUE (MV) (DYNES CM) 0 . 9 5 1 E + 0 0 0 . 4 2 4 E + 0 5 SHEAR S T R E S S (DYNES/SQ CM) 0 . 3 3 8 E + 0 4 T A B L E D - 1 (. . .CONT.) 180 DEX + 1.0MOLE NACL I N 5 0 . 0 % P E G - H 2 0 P H I ( C C / C C ) 0. 10 Y I E L D MV (MV) 0 . 3 0 0 E + 0 0 E Q U I L I B R I U M (MV) 0 . 8 9 9 E - 0 1 L (MM) 1 . 6 2 0 TORQUE (DYNES CM) 0 . 1 3 4 E + 0 5 MV TORQUE (DY N E S CM) 0. 4 0 1 E + 0 4 OMEGA (RAD/SEC) 0.23 27 SHEAR S T R E S S (DYNES/SQ CM) 0 . 1 0 7 E + 0 4 SHEAR STRESS (DYNES/SQ CM) 0 . 3 1 9 E + 0 3 DEX • 1.0MOLE NACL I N 5 0 . 0 % P E G - H 2 0 P H I ( C C / C C ) 0. 14 L (MM) OMEGA (RAD/SEC) 1. 6 2 0 0 . 2 3 2 7 Y I E L D MV (MV) 0 . 5 0 0 E + 0 0 TORQUE (DYNES CM) 0 . 2 2 3 E + 0 5 SHEAR S T R E S S (DYNES/SQ CM) 0 . 1 7 8 E + 0 4 E Q U I L I B R I U M MV TORQUE (MV) (DYNES CM) 0 . 1 3 0 E + 0 0 0 . 5 8 0 E + 0 4 SHEAR S T R E S S (DYNES/SQ CM) 0 . 4 6 2 E + 0 3 DEX + 1.0MOLE NACL I N 5 0 . 0 % PE G - H 2 0 P H I (CC/CC) 0. 15 Y I E L D MV (MV) 0 . 8 2 1 E + 0 0 E Q U I L I B R I U M (MV) 0 . 2 0 0 E + 0 0 L (MM) 1.620 TORQUE (DYNES CM) 0 . 3 6 6 E + 0 5 TORQUE (DYNES CM) 0 . 8 9 2 E + 0 4 OMEGA (R A D / S E C ) 0.23 27 SHEAR S T R E S S (DYNES/SQ CM) 0 . 2 9 1 E + 0 4 SHEAR STRESS (DYNES/SQ CM) 0 . 7 1 0 E + 0 3 T A B L E D - 1 (...COMT.) DEX + 1.0MOLE NACL I N 5 0 . 0 % P E G - H 2 0 P H I ( C C / C C ) 0. 16 L(NM) OMEGA (R A D / S E C ) 1. 6 2 0 0.23 27 Y I E L D MV (HV) 0. 102E + 01 TORQUE {DYNES CM) 0 . 4 5 5 E + 0 5 SHEAR S T R E S S (DYNES/SQ CM) 0 . 3 6 2 E + 0 4 E Q U I L I B R I U M MV TORQUE (MV) (DYNES CM) 0 . 3 9 9 E + 0 0 0 . 1 7 8 E + 0 5 SHEAR S T R E S S (DYNES/SQ CM) 0 . 1 4 2 E + 0 4 DEX + 1.0MOLE NACL I N 5 0 . 0 % PE G - H 2 0 P H I ( C C / C C ) 0. 17 L (MM) OMEGA (RAD/SEC) 1. 6 2 0 0. 2 3 2 7 Y I E L D MV (MV) 0 . 1 6 0 E + 0 1 TORQUE (DYNES CM) 0 . 7 1 D E + 0 5 SHEAR S T R E S S ( D Y N E S / S Q CM) 0 . 5 6 8 E + 0 4 E Q U I L I B R I U M MV TORQUE (MV) (DYNES CM) 0 . 8 9 9 E + 0 0 0 . 4 0 1 E + 0 5 SHEAR S T R E S S (DYNES/SQ CM) 0. 3 1 9 E + 0 4 DEX + 1.0MOLE NACL I N 5 0 . 0 % P E G - H 2 0 P H I ( C C / C C ) 0. 06 Y I E L D MV (MV) 0 . 3 0 0 E + 0 0 E Q U I L I B R I U M (MV) 0 . 5 0 0 E - 0 1 L (MM) 3 . 0 1 0 TORQUE (DYNES CM) 0 . 1 3 4 E + 0 5 TORQUE (DYNES CM) 0 . 2 2 3 E + 0 4 OMEGA ( R A D / S E C ) 0.23 27 SHEAR S T R E S S (DYNES/SQ CM) 0. 1 0 7 E + 0 4 SHEAR S T R E S S (DYNES/SQ CM) 0 . 1 7 8 E + 0 3 T A B L E D - 1 (. . .CONT.) DEX + 1.0HOLE NACL I N 5 0 . 0 % P E G - H 2 0 P H I ( C C / C C ) 0. 07 L (MM) OMEGA (HAD/SEC) 3. 0 1 0 0. 2 3 2 7 Y I E L D HV (MV) 0.520E+00 TOBQUE (DYNES CM) 0. 23 2E+05 SHEAR S T R E S S (DYNES/SQ CM) 0 . 1 8 5 E + 0 4 E Q U I L I B R I U M MV TORQUE (MV) (DYNES CM) 0. 1 2 0 E + 0 0 0 . 5 3 5 E + 0 4 SHEAR S T R E S S (DYNES/SQ CM) 0 . 4 2 6 E + 0 3 DEX * 1.0MOLE NACL I N 5 0 . 0 % PEG-H20 P H I ( C C / C C ) 0. 08 L (MM) OMEGA (R A D / S E C ) 3. 0 1 0 0. 2 3 2 7 Y I E L D MV (MV) 0 . 1 6 5 E + 0 1 TORQUE (DYNES CM) 0 . 7 3 6 E + 0 5 SHEAR S T R E S S (DYNES/SQ CM) 0. 5 8 6 E + 0 U E Q U I L I B R I U M MV TORQUE (MV) (DYNES CM) 0 . 2 8 0 E + 0 0 0 . 1 2 5 E + 0 5 SHEAR S T R E S S (DYNES/SQ CM) 0. 9 9 5 E + 0 3 DEX + 1.0MOLE NACL I N 5 0 . 0 % P E G - H 2 0 P H I (CC/CC) 0.09 L (MM) OMEGA (RAD/SEC) 3. 0 1 0 0. 2 3 2 7 Y I E L D MV (MV) 0 . 1 7 1 E + 0 1 TORQUE (DYNES CM) 0 . 7 6 3 E + 0 5 SHEAR S T R E S S (DYNES/SQ CM) 0 . 6 0 7 E + 04 E Q U I L I B R I U M MV TORQUE (MV) (DYNES CM) 0 . 4 5 1 E + 00 0 . 2 0 1 E + 0 5 SHEAR S T R E S S (DYNES/SQ CM) 0. 1 6 0 E + 0 4 T A B L E D - 1 (. . .CONT.) 183 DEX + 1.0MOLE NACL I N 50.O X P E G - H 2 0 P H I ( C C / C C ) 0. 10 L (MM) OMEGA (R AD/SEC) 3.010 0. 2327 Y I E L D MV (MV) 0 . 5 3 8 E + 0 1 TORQUE (DYNES CM) 0 . 2 4 0 E + 0 6 SHEAR STRESS (DYNES/SQ CM) 0 . 1 9 1 E + 0 5 E Q U I L I B R I U M MV TORQUE (MV) (DYNES CM) O.130E+01 0 . 5 8 0 E + 0 5 SHEAR S T R E S S (DYNES/SQ CM) 0 . 4 6 2 E + 0 4 DEX + 1.0 MOLE NACL I N 5 0 . 0 % P E G - H 2 0 P H I (CC/CC) 0. 06 L (MM) OMEGA ( R A D / S E C ) 5. 0 3 0 0 . 2 3 2 7 Y I E L D MV (MV) 0 . 8 4 1 E + 0 0 TORQUE (DYNES CM) 0 . 3 7 5 E + 0 5 SHEAR S T R E S S (DYNES/SQ CM) 0 . 2 9 9 E + 0 4 E Q U I L I B R I U M MV TORQUE (MV) (DYNES CM) 0. 1 0 0 E + 00 0 . 4 4 6 E + 0 4 SHEAR S T R E S S (DYNES/SQ CM) 0 . 3 5 5 E + 0 3 DEX + 1.0MOLE NACL I N 5 0 . 0 % P E G - H 2 0 P H I ( C C / C C ) 0.07 L (MM) OMEGA (R A D / S E C ) 5. 0 3 0 0. 2 3 2 7 Y I E L D MV (MV) 0 . 1 4 9 E + 0 1 TORQUE (DYNES CM) 0 . 6 6 5 E + 0 5 SHEAR S T R E S S (DYNES/SQ CM) 0 . 5 2 9 E + 0 4 E Q U I L I B R I U M MV TORQUE (MV) (DYNES CM) 0 . 3 5 0 E + 0 0 0 . 1 5 6 E + 0 5 SHEAR S T R E S S (DYNES/SQ CM) 0 . 1 2 4 E + 0 4 T A B L E D - 1 (. . .CONT.) 184 DEX + 1.0MOLE NACL I N 5 0 . 0 % PE G - H 2 0 P H I ( C C / C C ) 0 . 0 8 Y I E L D MV (MV) 0 . 1 5 2 E + 0 1 E Q U I L I B R I U M (MV) 0 . 4 8 0 E + 0 0 L (MM) 5. 03 0 TORQUE (DYNES CM) 0.678E+05 MV TORQUE (DYNES CM) 0.214E+05 OMEGA (RAD/SEC) 0. 2 3 2 7 SHEAR S T R E S S (DYNES/SQ CM) 0 . 5 4 0 E + 0 4 SHEAR S T R E S S (DYNES/SQ CM) 0. 1 7 0 E + 0 4 DEX + 1.0MOLE NACL I N 5 0 . 0 % P E G - H 2 0 P H I ( C C / C C ) 0.09 Y I E L D MV (MV) 0 . 3 5 0 E + 0 1 E Q U I L I B R I U M (MV) 0 . 1 1 5 E + 0 1 L (MM) 5. 0 3 0 TORQUE (DYNES CM) 0 . 1 5 6 E + 0 6 MV TORQUE (DYNES CM) 0. 51 3E+0 5 OMEGA (R A D / S E C ) 0 . 2 3 2 7 SHEAR S T R E S S (DYNES/SQ CM) 0 . 1 2 4 E + 0 5 SHEAR S T R E S S (DYNES/SQ CM) 0 . 4 0 8 E + 0 4 DEX + 1.0MOLE NACL I N 5 0 . 0 % P E G - H 2 0 P H I ( C C / C C ) 0. 06 Y I E L D MV (MV) 0 . 7 6 9 E + 0 0 E Q U I L I B R I U M (MV) 0 . 1 0 0 E + 0 0 L (MM) 6. 7 2 0 TORQUE (DYNES CM) 0 . 3 4 3 E + 0 5 MV TORQUE (DYNES CM) 0 . 4 4 6 E + 0 4 OMEGA(RAD/SEC) 0 . 2 3 2 7 SHEAR S T R E S S (DYNES/SQ CM) 0 . 2 7 3 E + 04 SHEAR S T R E S S (DYNES/SQ CM) 0.3 5 5 E + 0 3 TABLE D - 1 {...CONT.) 1 8 5 DEX + 1.0 HOLE NACL IN 50. 0% PEG-H20 PHI(CC/CC) 0.07 L (MM) OMEGA (HAD/SEC) 6.720 0.2327 YIELD MV (MV) 0. 144E+01 T O R Q U E { D Y N E S CM) 0 . 6 4 2 E + 0 5 S H E A R S T R E S S ( D Y N E S / S Q C M ) 0 . 5 1 1 E + 0 4 EQUILIBRIUM MV TORQUE (MV) (DYNES CM) 0.451E+00 0.201E+05 S H E A R S T R E S S ( D Y N E S / S Q CM) 0 . 1 6 0 E + 0 4 DEX + 1.0 HOLE NACL IN 50.0% PEG-H20 PHI (CC/CC) 0. 08 YIELD MV (MV) 0.240E+01 EQUILIBRIUM (MV) 0.700E+00 L (MM) 6.720 TORQUE (DYNES CM) 0.107E+06 MV TORQUE (DYNES CM) 0.312E+05 OMEGA (RAD/SEC) 0. 2327 SHEAR STRESS (DYNES/SQ CM) 0.8 52E+04 SHEAR STRESS (DYNES/SQ CM) 0. 248E+04 DEX + 1.0MOLE NACL IN 50.0% PEG-H20 P H I (CC/CC) 0.09 L (MM) OMEGA (RAD/SEC) 6.720 0.2327 YIELD MV (MV) 0.291E+01 TORQUE (DYNES CM) 0.130E+06 SHEAR STRESS (DYNES/SQ CM) 0.103E+05 EQUILIBRIUM MV TORQUE (MV) (DYNES CM) 0.601E+00 0.268E+05 SHEAR STRESS (DYNES/SQ CM) 0.213E+04 186 D-2 a n d S t r e s s D e c a y D e s c r i p t i o n o f V a r i a b l e s i n T a b l e D -2 P H I - P a r t i c l e C o n c e n t r a t i o n ( c c / c c ) . L - P a r t i c l e L e n g t h (mm). OMEGA - A n g u l a r V e l o c i t y o f I n n e r C y l i n d e r ( r a d . / s e c ) . RX - T h e T h i c k n e s s o f t h e F l o w i n g L a y e r R M e a s u r e d f r o m t h e C e n t e r o f t h e C y l i n d e r s ( c m ) . R - R a d i u s o f t h e I n n e r C y l i n d e r ( c m ) . RO - R a d i u s o f t h e O u t e r C y l i n d e r ( c m ) . T A B L E D - 2 - 1 187 E X P E S I M E N T A L DATA AND R E S U L T S OF C A L C U L A T I O N P H I = 0.16 (CC/CC) L = 0 . 9 8 7 (MM) DECAY DATA OMEGA = 0 . 2 3 2 7 (RAD/SEC) T I M E TORQUE TORQUE SHEAR (RX-R) S T R E S S / (RO-R) ( S E C ) (MV) (DYNES CM) (DYNES/SQ CM) (-) 0.0 1 3 . 2 0 0 . 5 8 9 E + 06 0 . 4 6 9 E + 0 5 0.0 1.0 9.80 0 . 4 3 7 E + 0 6 0 . 3 4 8 E + 0 5 0.11 1.8 8.18 0 . 3 6 5 E + 0 6 0 . 2 9 0 E + 0 5 0.14 2.6 7.06 0 . 3 1 5 E + 0 6 0 . 2 5 1 E + 0 5 0.16 3.7 5. 99 0 . 2 6 7 E + 0 6 0 . 2 1 3 E + 0 5 0.17 4.8 5.24 0 . 2 3 4 E + 0 6 0. 1 8 6 E + 0 5 0.18 9.0 3 . 7 3 0 . 1 6 6 E + 0 6 0 . 1 3 2 E + 0 5 0.20 1 3 . 0 3 . 0 9 0. 1 3 8 E + 0 6 0. 110E + 0 5 0.21 18. 0 2. 67 0 . 1 1 9 E + 0 6 0 . 9 4 8 E + 0 4 0.22 4 1 . 0 2.2 0 0 . 9 8 1 E + 0 5 0 . 7 8 1 E + 0 4 0.25 8 6 . 7 2. 10 0 . 9 3 7 E + 0 5 0 . 7 4 6 E + 0 4 0.27 EQUI 2 . 0 0 0 . 8 9 2 E + 0 5 0. 7 1 0 E + 0 4 0.42 T I M E RX/R RX APPARENT A P P A R E N T SHEAR R A T E V I S C O S I T Y (SEC) (") (CM) ( 1 / S E C ) ( P O I S E ) 0.0 1.000 1.01 1.0 1.119 1. 13 0 . 2 3 2 E + 0 1 • 0 . 1 5 0 E + 05 1.8 1. 151 1. 16 0 . 1 9 0 E + 0 1 0 . 1 5 3 E + 05 2.6 1 . 1 7 3 1. 18 0 . 1 7 1 E + 0 1 0 . 1 4 7 E + 05 3. 7 1 . 1 8 3 1.20 0 . 1 6 3 E + 0 1 0 . 1 3 1 E + 0 5 4.8 1. 194 1. 21 0 . 1 5 6 E + 0 1 0 . 1 1 9 E + 0 5 9.0 1 . 2 1 6 1.23 0 . 1 4 4 E + 0 1 0 . 9 2 1 E + 0 4 1 3 . 0 1. 2 27 1.24 0 . 1 3 9 E + 0 1 0 . 7 9 1 E + 0 4 1 8 . 0 1.23 7 1.25 0 . 1 3 4 E + 0 1 0 .707E+04 4 1 . 0 1 . 2 7 0 1.28 0 . 1 2 3 E + 0 1 0 . 6 3 7 E + 0 4 8 6 . 7 1. 291 1.30 0 . 1 1 6 E + 0 1 0 .6 4 1 E + 0 4 E Q U I 1 . 4 5 3 1.47 0 . 8 8 4 E + 0 0 0 . 8 0 3 E + 0 4 E Q U I L I B R I U M DATA OMEGA TORQUE SHEAR (RX-R) S T R E S S / ( R O - R ) ( R A D / S E C ) (DYNES CM) (DYNES/SQ CM) (-) 0 . 2 3 2 7 0 . 8 9 2 E + 0 5 0.71OE+04 0.42 0 . 6 9 8 1 0. 9 8 1 E + 0 5 0.78 1E+04 0.49 2. 0 940 0. 107E+06 0 . 852E+04 0.54 6. 2 830 0 . 1 1 2 E + 0 6 0 . 8 8 8 E + 0 4 0.61 OMEGA RX/R RX A P P A R E N T A P P A R E N T SHEAR R A T E V I S C O S I T Y ( R AD/SEC) (") (CM) ( 1 / S EC) ( P O I S E ) 0. 2 3 2 7 1. 4 5 3 1.47 0. 8 8 4 E + 0 0 0 .803E+04 0 . 6 9 8 1 1. 5 2 9 1. 54 0 . 2 4 4 E + 0 1 0 3 2 0 E + 04 2. 0 9 4 0 1 . 5 8 3 1. 60 0. 6 9 7 E + 0 1 0 . 1 2 2 E + 0 4 6. 2 830 1. 6 5 3 1. 67 0. 198E+02 0 . 4 4 8 E + 0 3 T A B L E D - 2 - 2 188 E X P E R I M E N T A L DATA AND R E S U L T S OF C A L C U L A T I O N P H I = 0.15 (CC/CC) L = 0.987 (MM) DECAY DATA OMEGA = 0 . 2 3 2 7 (RAD/SEC) T I M E TORQUE TORQUE SHEAR (SX-R) S T R E S S /{RO-R) (SEC) (MV) (DYNES CM) (DYNES/SQ CM) ( - ) 0.0 4.81 0 . 2 1 5 E + 0 6 0. 1 7 1 E + 0 5 0.0 0.2 4. 4 7 0 . 1 9 9 E + 0 6 0. 1 5 9 E + 0 5 0.14 1.4 3.86 0 . 1 7 2 E + 0 6 0. 1 3 7 E + 0 5 0.18 2. 3 3. 3 7 G.150E+G6 0. 120E + 05 0.20 3.2 2 . 9 1 0. 1 3 0 E + 0 6 0 . 1 0 3 E + 0 5 0.22 4.5 2 . 5 0 0.1 12E+06 0 . 8 8 8 E + 0 4 0.23 5.5 2 . 2 9 0 . 1 0 2 E + 0 6 0 . 8 1 3 E + 0 4 0.25 7. 8 1 .98 0 . 8 8 3 E + 0 5 0 . 7 0 3 E + 0 4 0.28 1 1 . 8 1.67 0 . 7 4 5 E + 0 5 0 . 5 9 3 E + 0 4 0.31 3 1 . 0 1.09 0 . 4 8 6 E + 0 5 0 . 3 8 7 E + 0 4 0.32 6 4 . 2 0.68 0 . 3 0 3 E + 0 5 0 . 2 4 1 E + 0 4 0.34 E Q U I 0 . 3 0 0 . 1 3 4 E + 0 5 0 . 1 0 7 E + 0 4 0.43 T I M E RX/R RX APPARENT A P P A R E N T SHEAR R A T E V I S C O S I T Y (SEC) (-) (CM) ( 1 / S E C ) ( P O I S E ) 0.0 1 . 0 0 0 1.01 0.2 1. 147 1. 16 0 . 1 9 4 E + 0 1 0. 8 1 7 E * 0 4 1 . 4 1 . 196 1.21 0 . 1 5 4 E + 0 1 0. 8 8 7 E + 0 4 2. 3 1 . 2 1 6 1.23 0 . 1 4 4 E + 0 1 0. 83 2 E + 0 4 3.2 1 . 2 3 5 1.25 0 . 1 3 5 E + 0 1 0. 7 6 5 E + 0 4 4.5 1. 2 4 5 1.26 0 . 1 3 1 E + 0 1 0. 6 7 7 E + 0 4 5.5 1 . 2 6 4 1.28 0 . 1 2 4 E + 0 1 0. 6 5 4 E + 0 4 7.8 1.304 1.32 0 . 1 1 3 E * 0 1 0. 6 2 3 E + 04 11 . 8 1 . 3 3 3 1,35 0 . 1 0 6 E + 0 1 0. 5 5 7 E + 0 4 3 1 . 0 1. 3 4 3 1. 36 0 . 1 0 4 E + 0 1 0. 3 7 1 E + 0 4 6 4 . 2 1 . 3 6 3 1. 38 0 . 1 0 1 E + 0 1 0 . 2 3 9 E + 0 4 E Q U I 1. 461 1.48 0 . 8 7 6 E + 0 0 0. 1 2 2 E + 04 !QUI L I B R I U M DATA OMEGA TORQUE SHEAR (RX-R) S T R E S S /{RO-R) (RAD/SEC) (DYNES CM) (DYNES/SQ CM) M 0 . 2 3 2 7 0. 1 3 4 E + 0 5 0 . 1 0 7 E + 0 4 0.43 0. 6 9 8 1 0. 1 4 3 E + 0 5 0 . 1 1 4 E + 0 4 0.50 2. 0 9 4 0 0 . 1 5 2 E + 0 5 0 . 1 2 1 E + 0 4 0.55 6. 2 830 0. 1 5 6 E + 0 5 0. 1 2 4 E + 0 4 0.73 OMEGA RX/R RX A P P A R E N T APPARENT SHEAR R A T E V I S C O S I T Y ( RAD/S EC) ("} (CM) ( 1 / S E C ) ( P O I S E ) 0 . 2 3 2 7 1. 461 1.48 0. 8 7 6 E + 0 0 0. 12 2 E + 0 4 0. 6 981 1. 540 1.55 0. 242E + 01 0. 4 7 0 E + 0 3 2. 0 9 4 0 1 . 5 8 8 1. 60 0 . 6 9 4 E + 0 1 0 . 1 7 4 E + 0 3 6. 2 830 1. 7 8 5 1. 80 0 . 1 8 3 E + 0 2 0. 6 7 8 E + 0 2 T A B L E D - 2 - 3 189 E X P E R I M E N T A L DATA AND R E S U L T S OP C A L C U L A T I O N PHI = 0.14 (CC/CC) L = 0 . 9 8 7 (MM) DECAY DATA OMEGA = 0 . 2 3 2 7 ( R A D / S E C ) T I M E TORQUE TORQUE SH EAR (RX-R) S T R E S S / (RO-R) ( S E C ) (MV) (DYNES CM) (DYNES/SQ CM) (-) 0.0 2. 19 0 . 9 7 7 E + 0 5 0. 7 7 7 E + 0 4 0.0 1.7 1.53 0 . 6 8 2 E + 0 5 0 . 5 4 3 E + 0 4 0.23 2.9 1.27 0 . 5 6 6 E + 0 5 0 . 4 5 1 E + 0 4 0.26 4.0 1 . 15 0 . 5 1 3 E + 0 5 0 . 4 0 8 E + 0 4 0.27 5.2 1.00 0. 4 4 6 E + 0 5 0 . 3 5 5 E + 0 4 0.28 6.6 0.88 0 . 3 9 2 E + 0 5 0 . 3 1 2 E + 0 4 0.29 7. 5 0.85 0 . 3 7 9 E + 0 5 0 . 3 0 2 E + 0 4 0.30 8.7 0.76 0 . 3 3 9 E + 0 5 0 . 2 7 0 E + 0 4 0.30 17 . 8 0.55 0 . 2 4 5 E + 0 5 0 . 1 9 5 E + 0 4 0.33 31 .3 0.42 0. 1 8 7 E + 0 5 0 . 1 4 9 E + 0 4 0.34 5 8 . 2 0. 32 0 . 1 4 3 E + 0 5 0 . 1 1 4 E + 0 4 0.36 E Q U I 0.07 0. 3 1 2 E + 0 4 0 . 2 4 9 E + 0 3 0.44 T I M E RX/R RX APPARENT A P P A R E N T SHEAR R A T E V I S C O S I T Y ( S E C ) <") (CM) ( 1/SEC) ( P O I S E ) 0.0 1 . 0 0 0 1.01 1.7 1 . 248 1.26 0. 1 3 0 E + 0 1 0. 4 1 8 E + 0 4 2.9 1. 281 1.29 0. 119E+01 0. 3 7 8 E + 0 4 4.0 1. 291 1. 30 0 . 1 1 6 E + 0 1 0. 3 5 1 E + C 4 5. 2 1. 3 0 2 1. 32 0. 1 1 3 E + 0 1 0. 3 1 3 E + 04 6.6 1 . 3 1 3 1.33 0 . 1 1 1 E + 0 1 0 . 2 8 2 E + 0 4 7. 5 1.324 1. 34 0. 1 0 8 E + 0 1 0. 2 7 8 E + 0 4 8.7 1.324 1.34 0. 1 0 8 E + 0 1 0. 2 4 9 E + 0 4 1 7 . 8 1. 3 5 6 1. 37 0. 102E+01 0. 1 9 1 E + 0 4 3 1 . 3 1 .367 1.38 0. 100E+01 0. 1 4 9 E + 0 4 5 8 . 2 1.3 89 1.40 0. 9 6 7 E + 0 0 0. 1 1 7 E + 0 4 EQUI 1 . 4 7 5 1.49 0. 8 6 1 E + 0 0 0. 2 8 8 E + 0 3 E Q U I L I B R I U M DATA OMEGA ( R A D / S E C ) 0 . 2 3 2 7 0 . 6 9 8 1 2 . 0 9 4 0 6.2 830 TORQUE (DYNES CM) 0 . 3 1 2 E + 0 4 0. 3 5 7 E + 0 4 0 . 4 0 1 E + 0 4 0 . 4 2 4 E + 0 4 SHEAR S T R E S S (DYNES/SQ CM) 0 . 2 4 9 E + 0 3 0. 28 4E+03 0. 3 2 0 E + 0 3 0. 3 3 7 E + 0 3 ( R X - R ) / ( S O - R ) (-) 0.44 0.51 0.56 0.62 OMEGA RX/R RX A P P A R E N T A P P A R E N T SHEAR R A T E V I S C O S I T Y ( R A D / S E C ) (-) (CM) (1/S EC) ( P O I S E ) 0. 2 3 2 7 1.475 1.49 0. 861E + 00 0 . 2 8 8 E + 0 3 0 . 6 9 8 1 1.550 1. 57 0. 2 3 9 E + 0 1 0 . 1 1 9 E + 0 3 2. 0 9 4 0 1.604 1. 62 0. 685E + 01 0 . 4 6 7 E + 0 2 6. 2 8 3 0 1.669 1. 69 0. 196E+02 0 . 1 7 2 E + 0 2 T A B L E D - 2 - 4 1 9 0 E X P E R I M E N T A L DATA AND R E S U L T S OF C A L C U L A T I O N P H I = 0.13 (C C / C C ) L = 0.987 (MM) DECAY DATA OMEGA = 0 . 2 3 2 7 ( R A D / S E C ) T I ME TORQUE TORQUE SHEAR (RX-R) S T R E S S / (RO-R) (SEC) (MV) (DYNES CM) (DYNES/SQ CM) (") 0.0 0.43 0. 1 9 0 E + 0 5 0. 1 5 1 E + 0 4 0.0 1.0 0.39 0 . 1 7 4 E + 0 5 0. 138E+04 0.28 2.4 0.35 0. 1 5 6 E + 0 5 0. 124E+04 0.33 4.2 0 . 3 1 0 . 1 3 8 E + 0 5 0 . 1 1 0 E + 0 4 0.35 6.0 0.29 0 . 1 2 9 E + 0 5 0 . 1 0 3 E + 0 4 0.37 7.9 0.27 0 . 1 2 0 E + 0 5 0 . 9 5 9 E + 0 3 0.38 9.3 0.26 0 . 1 1 6 E + 0 5 0 . 9 2 3 E + 0 3 0.39 1 9 . 2 0,22 0 . 9 8 1 E + 0 4 0 . 7 8 1 E + 0 3 0.40 2 7 . 9 0.18 0 . 8 0 3 E + 0 4 0 . 6 3 9 E + 0 3 0.4 1 5 9 . 5 0. 16 0 . 7 1 4 E + 0 4 0 , 5 6 8 E + 0 3 0.42 9 2 . 4 0.14 0 . 6 2 4 E + 0 4 0 . 4 9 7 E + 0 3 0.43 E Q U I 0.04 0 . 1 7 8 E + 0 4 0 . 1 4 2 E + 0 3 0.47 T I M E RX/R RX APPARENT A P P A R E N T SHEAR R A T E V I S C O S I T Y (SEC) (") (CM) ( 1 / S E C ) ( P O I S E ) 0.0 1 .000 1.01 1.0 1.302 1.32 0 . 1 1 3 E + 0 1 0. 1 2 2 E + 0 4 2.4 1 . 3 5 6 1. 37 0 . 1 0 2 E + 0 1 0, 12 2 E + 0 4 4.2 1 . 3 7 8 1.39 0. 9 8 4 E + 0 0 0. 11 2 E + 0 4 6.0 1 . 3 9 9 1. 41 Q. 9 5 1 E + 0 0 0. 108E+04 7.9 1.410 1.42 0.936E+ 00 0. 10 2 E + 0 4 9.3 1,421 1.44 0 . 9 2 2 E + 0 0 0. 100E+04 1 9 . 2 1.432 1. 45 0.909E+ 00 0. 8 5 9 E + 0 3 2 7 . 9 1 , 4 4 2 1. 46 0 . 8 9 6 E + 0 0 0. 7 1 3 E + 0 3 5 9 . 5 1.453 1.47 0 . 8 8 4 E + 0 0 0. 6 4 3 E + 0 3 9 2 . 4 1.464 1.48 0 . 8 7 2 E + 0 0 0. 5 7 0 E + 0 3 E Q U I 1 . 5 0 7 1. 52 0 . 8 3 1 E + 0 0 0. 1 7 1 E + 0 3 E Q U I L I B R I U M DATA OMEGA TORQUE SHEAR (RX-R) S T R E S S /(RO-R) (RAD/S EC) (DYNES CM) (DYNES/SQ CM) (-) 0 . 2 3 2 7 0 . 1 7 8 E + 0 4 0 . 1 4 2 E + 0 3 0.47 0. 6 9 8 1 0. 2 2 3 E + 0 4 0. 1 7 8 E + 0 3 0.53 2. 0 9 40 0. 2 6 8 E + 0 4 0.21 3E+0 3 C.60 6. 2 830 0 . 2 9 0 E + 0 4 0.23 1E+03 0.63 OMEGA RX/R RX A P P A R E N T APPARENT SHEAR R A T E V I S C O S I T Y (RAD/S EC) ("} (CM) ( 1 / S E C ) ( P O I S E ) 0 . 2 3 2 7 1. 507 1. 52 0 . 8 3 IE+00 0. 1 7 1 E + 0 3 0.6 981 1. 572 1. 59 0. 235E+01 0, 7 5 7 E + 0 2 2. 0 940 1. 6 4 8 1. 66 0 . 6 6 3 E + 0 1 0. 3 2 1 E + 0 2 6. 2 8 3 0 1. 6 8 0 1.70 0 . 1 9 5 E + 0 2 0. 1 1 9 E + 0 2 T A B L E D - 2 - 5 191 EX P E R I M E N T A L DATA AND R E S U L T S OF C A L C U L A T I O N P H I = 0.12 (CC/CC) L = 1.520 (MM) DECAY DATA OMEGA = 0 . 2 3 2 7 (RAD/SEC) T I M E TORQUE TORQ UE SHEAR (RX-R) S T R E S S / (RO-R) ( S E C ) (MV) (DYNES CM) (DYNES/SQ CM) (") 0.0 8.54 0 . 3 8 1 E + 0 6 0. 303E+O5 0 .0 0.9 6.22 0 . 2 7 7 E + 0 6 0 . 2 2 1 E + 0 5 0.1 4 2.0 4 . 7 9 0 . 2 1 4 E + 0 6 0 . 1 7 0 E + 0 5 0.18 3.1 3 . 9 1 0. 1 7 4 E + 0 6 0. 139E + 0 5 0.20 4.3 3.37 0. 150E+06 0 . 1 2 0 E + 0 5 0.21 5.9 2 . 8 3 0 . 1 2 6 E + 0 6 0 . 1 0 0 E + 0 5 0.23 7.0 2 . 6 0 0 . 1 1 6 E + 0 6 0 . 9 2 3 E + 0 4 0.23 8. 1 2 . 4 0 0. 107E+06 0 . 8 5 2 E + 0 4 0.24 9.6 2. 24 0 . 9 9 9 E + 0 5 0 . 7 9 5 E + 0 4 0.25 2 0 . 9 1.56 0 . 6 9 6 E + 0 5 0 . 5 5 4 E + 0 4 0.27 3 2 . 8 1.33 0.5 9 3 E + 0 5 0 . 4 7 2 E + 0 4 0.29 EQUI 0.95 0 . 4 2 4 E + 0 5 0 . 3 3 7 E + 0 4 0.40 T I M E RX/R RX APPARENT A P P A R E N T SHEAR RATE V I S C O S I T Y (SEC) (-) (CM) ( 1/SEC) ( P O I S E ) 0.0 1 . 0 0 0 1.01 0.9 1.151 1. 16 0. 1 9 0 E + 0 1 0 . 1 1 6 E + 0 5 2.0 1. 194 1.21 0. 1 5 6 E * 01 0. 1 0 9 E + 0 5 3.1 1 . 2 1 6 1.23 0. 1 4 4 E + 0 1 0 . 9 6 5 E + 0 4 4.3 1 . 2 2 7 1.24 0 . 1 3 9 E * 0 1 0. 8 6 2 E + 0 4 5.9 1 .248 1.26 0 . 1 3 0 E + 0 1 0 . 7 7 3 E + 0 4 7.0 1. 2 4 8 1.26 0 . 1 3 0 E + 0 1 0 . 7 1 0 E + 0 4 8.1 1 . 2 5 9 1.27 0 . 1 2 6 E + 0 1 0 . 6 7 6 E + 0 4 9. 6 1 . 2 7 0 1.28 0 . 1 2 3 E * 0 1 0 . 6 4 9 E + 0 4 2 0 . 9 1.291 1. 30 0. 1 16E+01 0 . 4 7 6 E + 0 4 3 2 . 8 1 . 3 1 3 1.33 0.111E+ 01 0 . 4 2 6 E + 04 EQUI 1.432 1. 45 0 . 9 0 9 E + 0 0 0 . 3 7 1 E + 0 4 E Q U I L I B R I U M DATA OMEGA ( R A D / S E C ) 0.2 3 2 7 0 . 6 9 8 1 2 . 0 9 4 0 6. 2 830 TORQUE (DYNES CM) 0. 4 2 4 E + 0 5 0. 4 3 7 E + 0 5 0. 4 5 0 E + 0 5 0 . 4 5 5 E + 0 5 SHEA R STRESS (DYNES/SQ CM) 0 . 3 3 7 E + 0 4 0.34 8E+04 0.359E+ 04 0 . 3 6 2 E + 0 4 (HX-E) / (RO-R) (-) 0.40 0.49 0.54 0.58 OMEGA RX/R RX A P P A R E N T SHEAR RATE (RAD/SEC) {-) (CM) ( 1 / S E C ) 0. 2 3 2 7 1.432 1. 45 0. 909E + 00 0.6 981 1.529 1. 54 0. 2 4 4 E + 0 1 2. 0 9 4 0 1. 5 8 3 1. 60 0. 697E + 01 6 . 2 8 3 0 1.626 1. 64 0. 2 0 2 E + 0 2 A P P A R E N T V I S C O S I T Y ( P O I S E ) 0 . 3 7 1 E + 0 4 0 . 1 4 3 E + 0 4 0 . 5 1 4 E + 0 3 0 . 1 7 9 E + 0 3 T A B L E D - 2 - 6 192 E X P E R I M E N T A L DATA AND R E S U L T S OF C A L C U L A T I O N P H I = 0 . 1 1 (CC/CC) L = 1.620 (MM) DECAY DATA OMEGA = 0. 2 3 2 7 (RAD/SEC) T I M E TORQUE TORQUE SHEAR (RX-R) S T R E S S / (RO-R) (SEC) (MV) (DYNES CM) (DYNES/SQ CM) (-) 0.0 2 . 8 0 0 . 1 2 5 E + 0 6 0 . 9 9 4 E + 0 4 0.0 0.4 2.5 8 0 . 1 1 5 E + 0 6 0 . 9 1 6 E + 0 4 0.19 1.3 2 . 2 0 0 . 9 8 1 E + 0 5 0 . 7 8 1 E + 0 4 0.21 3.0 1.77 0 . 7 8 9 E + 0 5 0 . 6 2 8 E + 0 4 0.25 3.6 1.71 0 . 7 6 3 E + 0 5 0 . 6 0 7 E + 0 4 0.27 5.8 1.43 0 . 6 3 8 E + 0 5 0 . 5 0 8 E + 0 4 0.28 1 0 . 5 1.12 0 . 5 0 0 E + 0 5 0 , 3 9 8 E + 0 4 0.29 1 8 . 3 0.87 0 . 3 8 8 E + 0 5 0 . 3 0 9 E + 0 4 0.30 2 8 . 8 0 . 7 3 0 . 3 2 6 E + 0 5 0 . 2 5 9 E + 0 4 0.31 5 8 . 8 0. 56 0 . 2 5 0 E + 0 5 0 . 1 9 9 E + 0 4 ' 0.33 1 2 9 . 0 0.45 0 . 2 0 1 E + 0 5 0 . 1 6 0 E + 0 4 0.38 E Q U I 0.26 0 . 1 1 6 E + 0 5 0 . 9 2 3 E + 0 3 0.43 T I M E RX/R RX APPARENT A P P A R E N T SHEAR RATE V I S C O S I T Y ( S E C ) (") (CM) ( 1 / S E C ) ( P O I S E ) 0.0 1 .000 1.01 0.4 1.20 5 1.22 0 . 1 4 9 E + 0 1 0 .613 E + 0 4 1.3 1 . 2 2 7 1. 24 0 . 1 3 9 E + 0 1 0 . 5 6 3 E + 0 4 3.0 1. 2 6 8 1.28 0.123E+O1 0 . 5 1 0 E + 0 4 3.6 1 . 2 8 8 1. 30 0 . 1 1 7 E + 0 1 0 . 5 1 8 E + 0 4 5.8 1. 2 9 8 1.31 0 . 1 1 5 E + 0 1 0 .443 E + 0 4 1 0 . 5 1 . 3 0 9 1. 32 0. 1 1 2 E + 0 1 0 .355 E + C 4 1 8 . 3 1 . 3 2 8 1.34 0 . 1 0 7 E + 0 1 0 .2 8 7 E + 0 4 2 8 . 8 1.339 1. 35 0 . 1 0 5 E + 0 1 0 . 2 4 6 E + 0 4 5 8 . 8 1. 3 5 9 1. 37 0 . 1 0 1 E + 0 1 0 . 196E+0 4 1 2 9 . 0 1.411 1. 43 0 . 9 3 5 E + 0 0 0 . 1 7 1 E + 04 E Q U I 1. 4 6 3 1.48 0 . 8 7 4 E + 0 0 0 . 1 0 6 E + 04 I Q U I L I B R I U M DATA OMEGA TORQUE SHEAR (RX-E) STSE SS / (RO-R) (RAD/S EC) (DYNES CM) (DYNES/SQ CM) (") 0. 2 3 2 7 0. 1 16E+05 0.92 3E+0 3 0.43 0. 6 9 8 1 0. 1 2 5 E + 0 5 0 . 9 9 4 E + 0 3 0,52 2. 0 9 4 0 0 . 1 3 4 E + 0 5 0. 1 0 7 E + 0 4 0.57 6. 2 8 3 0 0 . 1 3 8 E + 0 5 0.11OE+04 0.62 OMEGA RX/R RX A P P A R E N T A P P A R E N T SHEAR R A T E V I S C O S I T Y { RAD/S EC) (") (CM) ( 1 / S E C ) ( P O I S E ) 0. 2 3 2 7 1. 4 6 3 1. 48 0. 8 7 4 E + 0 0 0 . 1 0 6 E + 04 0. 6 9 8 1 1. 566 1. 58 0 . 2 3 6 E + 01 0 . 4 2 1 E + 0 3 2. 0 9 4 0 1.616 1.63 0 . 6 7 9 E + 0 1 0 . 1 5 7 E + 0 3 6. 2 8 3 0 1. 6 6 8 1. 68 0 . 1 9 6 E + 02 0 . 5 6 1 E + 0 2 T A B L E D - 2 - 7 193 E X P E R I M E N T A L D ST A AND R E S U L T S OF C A L C U L A T I O N P H I = 0. 10 (CC/CC) DECAY DATA OMEGA = 0 . 2 3 2 7 ( R A D / S E C ) L = 1.620 (MM) T I M E TORQUE TORQUE SHEAR (SX-R) S T R E S S / (RO-R) ( S E C ) (HV) (DYNES CM) (DYNES/SQ CM) (-) 0.0 1. 12 0 . 5 0 0 E + 0 5 0. 398E+-04 0.0 1.1 0.96 0 . 4 2 8 E + 0 5 0 . 3 4 1 E + 0 4 0.26 2.7 0.82 0 . 3 6 6 E + 0 5 0 . 2 9 1 E + 0 4 0.31 4.1 0.73 0 . 3 2 6 E + 0 5 0 . 2 5 9 E + 0 4 0.32 5.9 0.67 0 . 2 9 9 E + 0 5 0 . 2 3 8 E + 0 4 0.34 7.2 0.6 2 0 . 2 7 7 E + 05 0 . 2 2 0 E + 0 4 0.35 8.7 0.58 0 . 2 5 9 E + 0 5 0 . 2 0 6 E + 0 4 0.35 1 0 . 3 0.55 0 . 2 4 5 E + 0 5 0 . 1 9 5 E + 0 4 0.36 3 3 . 2 0.40 0 . 1 7 8 E + 0 5 0 . 1 4 2 E + 0 4 0.38 7 1 . 7 0.30 0 . 1 3 4 E + 0 5 0 . 1 0 7 E + 0 4 0.40 1 4 7 . 8 0 . 25 0 . 1 1 2 E + 0 5 0 . 8 8 8 E + 0 3 0.4 1 E Q U I 0.08 0 . 3 5 7 E + 0 4 0 . 2 8 4 E + 0 3 0.43 T I M E RX/R RX APPARENT A P P A R E N T SHEAR R A T E V I S C O S I T Y (SEC) (-) (CM) ( 1/SEC) ( P O I S E ) 0.0 1 . 0 0 0 1.01 1 . 1 1. 2 8 1 1.29 0 . 1 1 9 E + 0 1 0. 2 8 6 E + 04 2.7 1 . 3 3 5 1. 35 0 . 1 0 6 E + 0 1 0. 2 7 4 E + 0 4 4. 1 1 . 3 4 5 1.36 0 . 1 0 4 E + 0 1 0. 2 4 9 E + 0 4 5.9 1 . 3 6 7 1.38 0 . 1 0 0 E + 0 1 0. 2 3 8 E + 0 4 7.2 1 . 3 7 8 1.39 0 . 9 8 4 E + 0 0 0. 2 2 4 E + 0 4 8.7 1.378 1.39 0 . 9 8 4 E + 0 0 0. 2 0 9 E + 0 4 1 0 . 3 1 . 3 8 9 1. 40 0 . 9 6 7 E + 0 0 0. 2 0 2 E + 0 4 3 3 . 2 1 . 4 1 0 1.42 0 . 9 3 6 E + 0 0 0. 1 5 2 E + 04 7 1 . 7 1 . 4 3 2 1. 45 0 . 9 0 9 E + 0 0 0. 11 7 E + 0 4 1 4 7 . 8 1.442 1.46 0 . 8 9 6 E + 0 0 0. 9 9 1 E + 0 3 EQUI 1 . 4 6 4 1.48 0 . 8 7 2 E + 0 0 0. 3 2 6 E + 0 3 E Q U I L I B R I U M DATA OMEGA TORQUE ( R A D / S E C ) 0 . 2 3 2 7 0 . 6 9 8 1 2 . 0 9 4 0 6 . 2 8 3 0 (DYNES" CM) 0 . 3 5 7 E + 0 4 0 . 4 0 1 E + 0 4 0. 4 4 6 E + 0 4 0. 4 6 8 E + 0 4 SHEAR S T R E S S (DYNES/SQ CM) 0. 2 8 4 E + 0 3 0 . 3 2 0 E + 0 3 0 . 3 5 5 E + 0 3 0. 37 3E+0 3 (RX-R) / (RO-R) (") 0.43 0,51 0.57 0.60 OMEGA RX/R RX A P P A R E N T SHEAR R A T E ( R A D / S E C ) {-) (CM) ( 1 / S E C ) 0 . 2 3 2 7 1. 4 6 4 1.48 0. 872E + 00 0 . 6 9 8 1 1.550 1.57 0 . 2 3 9 E + 0 1 2 . 0 9 4 0 1.615 1. 63 0. 6 7 9 E + 01 6 . 2 8 3 0 1.648 1. 66 0. 199E+02 A P P A R E N T V I S C O S I T Y ( P O I S E ) 0 . 3 2 6 E + 0 3 0 . 1 3 4 E + 0 3 0 . 5 2 3 E + 0 2 0 . 1 8 7 E + 0 2 T A B L E D - 2 - 8 194 E X P E R I M E N T A L DATA AND R E S U L T S OF C A L C U L A T I O N P H I = 0.09 (CC/CC) L = 1.620 (MM) DECAY DATA OMEGA = 0 . 2 3 2 7 ( R A D / S E C ) T I M E TORQUE TORQUE (SEC) (MV) (DYNES CM) 0.0 0. 2 1 0 . 9 3 7 E + 0 4 0.2 0. 2 1 0 . 9 3 7 E + 0 4 2.0 0. 19 0 . 8 4 7 E + 0 4 3. 9 0. 18 0 . 8 0 3 E + 0 4 5.8 0. 17 0 . 7 5 8 E + 04 7.7 0. 17 0 . 758E+04 9.3 0. 16 0 . 7 1 4 E + 0 4 11.1 0. 16 0 . 7 1 4 E + 0 4 13.4 0. 15 0 . 6 6 9 E + 0 4 4 8 . 8 0. 13 0 . 5 8 0 E + 0 4 9 3 . 7 0. 12 0 . 5 3 5 E + 0 4 EQUI 0. 02 0 . 8 9 2 E + 0 3 SHEAR (RX-R) S T R E S S / ( R O - R ) (DYNES/SQ CM) (-) 0 . 7 4 6 E + 0 3 , 0.0 0 . 7 4 6 E + 0 3 0.19 0 . 6 7 5 E + 0 3 0.35 0 . 6 3 9 E + 0 3 0.38 0 . 6 0 4 E + 0 3 0.39 0 . 6 0 4 E + 0 3 0.40 0 . 5 6 8 E + 0 3 0.40 0 . 5 6 8 E + 0 3 0.41 0 . 5 3 3 E + 0 3 0.41 0 . 4 6 2 E + 0 3 0.43 0 . 4 2 6 E + 0 3 0.43 0 . 7 1 0 E + 0 2 0.45 T I M E RX/R RX APPARENT A P P A R E N T SHEAR R A T E V I S C O S I T Y (SEC) (-) (CM) ( 1 / S E C ) ( P O I S E ) 0.0 1 .000 1.01 0.2 1 . 2 0 5 1. 22 0. 149E+ 01 0 . 4 9 9 E + 0 3 2.0 1 . 3 7 8 1.39 0 . 9 8 4 E + 0 0 0 . 6 8 6 E + 0 3 3.9 1 . 4 1 0 1.42 0.936E+ 00 0 . 6 8 3 E + 0 3 5.8 1.421 1.44 0 . 9 2 2 E + 0 0 0 . 6 5 4 E + 0 3 7. 7 1. 4 3 2 1.45 0 . 9 0 9 E + 0 0 0 . 6 6 4 E + 0 3 9.3 1.432 1. 45 0 . 9 0 9 E + 0 0 0 . 6 2 5 E + 0 3 1 1 . 1 1. 4 42 1.46 0.8 96E+00 0.634 E+03 13.4 1 . 4 4 2 1.46 0 . 8 9 6 E + 0 0 0 . 5 9 4 E + 0 3 4 8 , 8 1. 4 6 4 1.48 0 . 8 7 2 E * 0 0 0 . 5 2 9 E * 0 3 9 3 . 7 1 . 4 6 4 1. 48 0 . 8 7 2 E + 0 0 0 . 4 8 8 E + 0 3 E Q U I 1. 4 8 6 1.50 0 . 8 5 1 E + 0 0 0 . 8 3 4 E + 0 2 E Q U I L I B R I U M DATA OMEGA TORQUE SHEAR (RX-R) STRE SS / (RO-R) (RAD/SEC) (DYNES CM) (DYNES/SQ CM) (-) 0 . 2 3 2 7 0 . 8 9 2 E + 0 3 0 . 7 1 0 E * 0 2 0.45 0 . 6 9 8 1 0. 134E+04 0 . 1 0 7 E + 0 3 0.54 2.0 940 0. 1 7 8 E + 0 4 0. 1 4 2 E + 0 3 0.60 6. 2 8 3 0 0. 2 0 1 E + 0 4 0 . 1 6 0 E + 0 3 0.62 OMEGA RX/S RX A P P A R E N T A P P A R E N T SHEAR R A T E ' V I S C O S I T Y (RAD/SEC) (~) (CM) ( 1 / S E C ) ( P O I S E ) 0 . 2 3 2 7 1. 4 8 6 1. 50 0. 8 5 1 E + 00 0 . 8 3 4 E + 0 2 0. 6 9 8 1 1. 5 8 3 1. 60 0 . 2 3 2 E * 0 1 0 . 4 5 8 E + 0 2 2 . 0 9 4 0 1. 648 1. 66 0 . 6 6 3 E + 0 1 0 . 2 1 4 E + 02 6 . 2 8 3 0 1. 6 6 9 1. 6 9 0. 196E + 02 0 . 8 1 5 E + 0 1 TABLE D - 2 - 9 195 EXPERIMENTAL DATA AND RESULT S OF CALCULATION PHI = 0.08 (CC/CC) L = 3.0 10 (MM) DECAY DATA OMEGA = 0.2327 (RAD/SEC) TIME TORQUE TORQUE SHEAR (RX-R) STRESS / (RO-R) (SEC) (MV) (DYNES CH) (DYNES/SQ CM) (") 0.0 11.26 0.502E+06 0.400E+05 0.0 0.9 10.34 0.461E+06 0.367E+05 0.04 2. 1 9.56 0.426E+06 0.339E+05 0.07 3.2 8.94 0.399E+06 0,317E+05 0.08 4.4 8. 54 0.381E+06 0.303E+05 0.10 5.5 8. 14 0.363E+06 0.289E+05 0.11 6.8 7.84 0.350E+06 0.278E+05 0.12 8.0 7.60 0.339E+06 0.270E+05 0.1.3 16. 3 6. 24 0.278E+06 0.222E*05 0.15 36.5 5.00 0.223E+06 0. 178E+05 0.19 63.0 4.40 0.196E+06 0. 156E+05 0.21 EQUI 0.84 0.375E+05 0.298E+04 0.41 TIME RX/R RX APPARENT APPARENT SHEAR RATE VISCOSITY (SEC) {") (CM) ( 1/SEC) (POISE) 0.0 1.000 1.01 0.9 1 .045 1.06 0.549E+01 0 .669E+04 2. 1 1. 077 1. 09 0.339E+01 0 . 100E+05 3.2 1.091 1. 10 0.292E+01 0 . 109E+05 4. 4 1.108 1. 12 0.251E+01 0 .121E+05 5.5 1.119 1. 13 0.232E+01 0 .125E+05 6.8 1. 130 1. 14 0.2 15E+01 0 . 129E+05 8.0 1. 140 1. 15 0.2 02E+01 0 .134E+05 16.3 1. 162 1.17 0.180E+01 0 .123 E+05 36.5 1 . 205 1.22 0. 149E+01 0 .119E+05 63.0 1. 227 1.24 0.139E+01 0 . 113E*05 EQUI 1 .442 1, 46 0.896E+00 0 .333E+04 EQUILIBRIUM DATA OMEGA TORQUE SHEA R (RX-R) STRESS /(RO-R) (RAD/SEC) (DYNES CM) (DYNES/SQ CM) (-) 0. 2327 0.375E+05 0.298E+04 0.41 0. 6981 0. 384E+05 0. 305E+04 0.44 2. 0940 0.392E+05 0.312E*04 0.47 6. 2 830 0.397E+05 0.316E+04 0.49 OMEGA RX/R RX APPARENT APPARENT SHEAR RATE VISCOSITY (RAD/SEC) (-) (CM) ( V S EC) (POISE) 0.2327 1. 442 1. 46 0. 896E+00 0 .333E+04 0.6 981 1. 473 1. 49 0.259E+01 0 .118E+04 2. 0940 1. 504 1. 52 0.751E + 01 0 .416E+03 6. 2 830 1. 524 1. 54 0.221E+02 0 . 143E + 03 TABLE D - 2 - 1 0 E X P E R I M E N T A L D A T A A N D R E S U L T S OF C A L C U L A T I O N P H I = 0 , 0 7 ( C C / C C ) D E C A Y D A T A O M E G A = 0 . 2 3 2 7 ( R A D / S E C ) L = 3.010 (MM) T I M E T O R Q U E T O R Q U E S H E A R { R X - R ) S T R E S S / ( R O - R ) ( S E C ) ( M V) ( D Y N E S CM) ( D Y N E S / S Q C M ) (") 0 . 0 7 . 2 0 0 . 3 2 1 E + 0 6 0 . 2 5 6 E + 0 5 0.0 0. 1 7 . 0 9 0 . 3 1 6 E + 0 6 0 . 2 5 2 E + 0 5 0 . 0 1 1.2 6. 1 3 0 . 2 7 3 E + 0 6 0. 2 1 8 E + 0 5 0 . 0 7 2 . 7 5 . 2 5 0 . 2 3 4 E + 0 6 0 . 1 8 6 E + 0 5 0 . 1 1 3 . 9 4. 8 5 0 . 2 1 6 E + 0 6 0. 1 7 2 E + 0 5 0 . 1 3 5 . 3 4. 4 7 0 . 1 9 9 E + 0 6 0 . 1 5 9 E + 0 5 0 . 1 4 6 . 8 4 . 0 8 0 . 1 8 2 E + 0 6 0 . 1 4 5 E + 0 5 0 . 1 5 8. 2 3 . 8 4 0 . 1 7 1 E + 0 6 0. 1 3 6 E + 0 5 0 . 1 6 1 3 . 8 3 . 3 5 0. 1 4 9 E + 0 6 0. 1 1 9 E + 0 5 0 . 1 8 2 8 . 7 2 . 6 5 0 . 1 1 8 E + 0 6 0 . 9 4 1 E + 0 4 0 . 2 1 8 8 . 8 1. 9 5 0 . 8 7 0 E + 0 5 0 . 6 9 2 E + 0 4 0 . 2 6 E Q U I 0. 3 0 0 . 1 3 4 E + 0 5 0. 1 0 7 E + 0 4 0 . 4 3 T I M E R X / R RX A P P A R E N T A P P A R E N T S H E A R R A T E V I S C O S I T Y ( S E C ) ( (CM) { 1 / S E C ) ( P O I S E ) 0 . 0 1 . 0 0 0 1 . 0 1 0 . 1 1 . 0 1 1 1 . 0 2 0. 2 1 9 E + 0 2 0 . 1 1 5 E + 0 4 1.2 1 . 0 7 6 1. 0 9 0 . 3 4 3 E + 0 1 0 . 6 3 4 E + 0 4 2. 7 1 . 1 1 9 1. 1 3 0 . 2 3 2 E + 0 1 0. 8 0 5 E + 0 4 3 . 9 1 . 1 4 0 1. 1 5 0 . 2 0 2 E + 0 1 0 . 8 5 4 E + C 4 5 . 3 1 . 1 5 1 1 . 1 6 0. 1 9 0 E + 0 1 0. 8 3 6 E + 0 4 6 . 8 1 . 1 6 2 1. 17 0. 1 8 0 E + 0 1 0 . 8 0 7 E + 0 4 8 . 2 1 . 1 7 3 1. 1 8 0. 1 7 1 E + 0 1 0 . 7 9 9 E + 0 4 1 3 . 8 1 . 1 9 4 1 . 2 1 0. 1 5 6 E + 0 1 0 . 7 6 4 E + 0 4 2 8 . 7 1 . 2 2 7 1 . 2 4 0 . 1 3 9 E + 0 1 0 . 6 7 8 E + 0 4 8 8 . 8 1 . 2 8 1 1 . 2 9 0 . 1 1 9 E + 0 1 0 . 5 8 0 E + 0 4 E Q U I 1 . 4 6 3 1 , 4 8 0. 8 7 4 E + 0 0 0 . 1 2 2 E + 0 4 I Q U I L I B R I U M D A T A O M E G A T O R Q U E S H E A R ( R X - R ) S T R E S S / ( R O - R ) ( R A D / S E C ) ( D Y N E S CM) { D Y N E S / S Q CM.) (-) 0 . 2 3 2 7 0 . 1 3 4 E + 0 5 0, 1 0 7 E + 0 4 0 . 4 3 0. 6 9 8 1 0. 1 4 3 E + 0 5 0 . 1 1 4 E + 0 4 0 . 4 5 2 . 0 9 4 0 0. 1 5 2 E + 0 5 0. 1 2 1 E + 0 4 0 . 4 8 6. 2 8 3 0 0 . 1 5 6 E + 0 5 0. 1 2 4 E + 0 4 0 . 5 0 O M E G A R X / R RX A P P A R E N T A P P A R E N T S H E A R R A T E V I S C O S I T Y ( R A D / S E C ) (") ( C M ) ( 1 / S E C ) ( P O I S E ) 0 . 2 3 2 7 1. 4 6 3 1. 4 8 0 . 8 7 4 E + 0 0 0 . 1 2 2 E + 0 4 0 . 6 9 8 1 1. 4 8 3 1. 5 0 0 , 2 5 6 E + 0 1 0 . 4 4 4 E + 0 3 2 . 0 9 4 0 1. 5 1 4 1. 5 3 0 . 7 4 3 E + 0 1 0 . 1 6 2 E + 0 3 6 . 2 8 3 0 1. 5 3 4 1. 5 5 0 . 2 1 8 E + 0 2 0 . 5 6 9 E + 0 2 T A B L E D - 2 - 11 E X P E R I M E N T A L DAT A AND R E S U L T S OF C A L C U L A T I O N PHI = 0.06 (CC/CC) L = 3.010 (MM) DECAY DATA OMEGA = 0 . 2 3 2 7 ( R A D / S E C ) 197 T I M E TORQUE TORQUE SHEAR (RX-R) S T R E S S / (RO-R) ( S E C ) (HV) (DYNES CM) (DYNES/SQ CM) (") 0.0 4. 87 0 . 2 1 7 E + 0 6 0. 173E + 05 0.0 1.0 3. 1 0 0 . 1 3 8 E + 0 6 0. 110E + 0 5 0.12 2.0 2. 35 0 . 1 0 5 E + 0 6 0 . 8 3 4 E + 0 4 0. 15 3.0 1. 96 0 . 8 7 4 E + 0 5 0 . 6 9 6 E + 0 4 0.17 5.0 1. 59 0 . 7 0 9 E + 0 5 0 . 5 6 4 E + 0 4 0.20 7.0 1. 3 0 0 . 5 8 0 E + 0 5 0 . 4 6 2 E + 0 4 0.21 10.0 1. 11 0 . 4 9 5 E + 0 5 0 . 3 9 4 E + 0 4 0.23 2 1 . 0 0. 84 0 . 3 7 5 E + 0 5 0 . 2 9 8 E + 0 4 0.25 4 4 . 0 0. 64 0 . 2 8 5 E + 0 5 0 . 2 2 7 E + 0 4 - 0.28 5 0 . 0 0. 6 2 0 . 2 7 7 E + 0 5 0. 2 2 0 E + 0 4 0.28 7 0 . 0 0. 56 0 . 2 5 0 E + 0 5 0 . 1 9 9 E + 0 4 0.29 E Q U I 0. 14 0 . 6 2 4 E + 0 4 0 . 4 9 7 E + 0 3 0.44 T I M E RX/R RX APPARENT APPARENT SHEAR RATE V I S C O S I T Y (SEC) ( -) (CM) { 1/SEC) ( P O I S E ) 0.0 1 . 0 0 0 1.01 1.0 1 . 130 1. 14 0. 2 15E+01 0 . 5 1 1 E + 0 4 2.0 1 . 162 1. 17 0.180E+ 01 0 . 4 6 5 E + 0 4 3.0 1 . 1 8 3 1.20 0 . 1 6 3 E + 0 1 0 . 4 2 8 E + 0 4 5.0 1 . 2 1 6 1.23 0 . 1 4 4 E + 0 1 0 . 3 9 2 E + 0 4 7.0 1 . 2 2 7 1. 24 0 . 1 3 9 E + 0 1 0 . 3 3 3 E + 0 4 1 0 . 0 1 . 2 4 8 1.26 0 . 1 3 0 E + 0 1 0 . 3 0 3 E + 0 4 21 .0 1 . 2 7 0 1. 28 0. 1 2 3 E + 0 1 0 . 2 4 3 E + 0 4 4 4 . 0 1 .302 1.32 0. 1 13E+01 0 . 2 0 0 E + 0 4 5 0 . 0 1 . 3 0 2 1. 32 0. 1 13E+01 0. 1 9 4 E + 04 7 0 . 0 1 . 3 1 3 1. 33 0. 1 11E+01 0 . 1 7 9 E + 0 4 E Q U I 1 . 4 7 3 1. 49 0 . 8 6 4 E + 0 0 0 . 5 7 6 E + 0 3 E Q U I L I B R I U M DATA OMEGA TORQUE SHEA R (RX-R) S T R E S S / {RO—R) ( R A D / S E C ) (DYNES CM) (DYNES/SQ CM) (-) 0 . 2 3 2 7 0 . 6 2 4 E + 0 4 0 . 4 9 7 E + 0 3 0.44 0 . 6 9 8 1 0 . 7 1 4 E + 0 4 0 . 5 6 8 E + 0 3 0.46 2. 0 9 4 0 0. 8 0 3 E + 0 4 0 . 6 3 9 E + 0 3 0.49 6. 2 8 3 0 0. 8 47E+04 0.67 5E+03 0.52 OMEGA RX/R RX A P P A R E N T APPARENT SHEAR RATE V I S C O S I T Y ( R A D / S E C ) (~) (CM) {1/S EC) ( P O I S E ) 0. 2 3 2 7 1. 4 7 3 1. 4 9 0. 864E+O0 0 . 5 7 6 E + 0 3 0 . 6 9 8 1 1. 4 9 3 1. 51 0 . 2 5 3 E + 0 1 0 . 2 2 4 E + 0 3 2. 0 9 4 0 1. 5 2 4 1. 54 0. 735E + 01 0 . 8 6 9 E + 0 2 6. 2 8 3 0 1. 5 6 6 1. 58 0 . 2 1 2 E + 0 2 0 . 3 1 8 E + 0 2 T A B L E D - 2 - 12 198 E X P E R I M E N T A L DATA AND R E S U L T S OF C A L C U L A T I O N P H I = 0.05 (CC/CC) L = 3.010 (MM) DECAY DATA OMEGA = 0. 2 3 2 7 (RAD/SEC) T I M E TORQUE TORQUE SHEAR (RX-R) S T R E S S / (RO-R) (SEC) (MV) (DYNES CM) (DYNES/SQ CM) (") 0.0 3 . 0 0 0 . 1 3 4 E + 0 6 0. 107E + 0 5 0.0 1.0 1.94 0 . 8 6 5 E + 0 5 0 . 6 8 9 E + 0 4 0.16 2.0 1.56 0 . 6 9 6 E + 0 5 0 . 5 5 4 E + 0 4 0.20 3.0 1. 23 0 . 5 4 9 E + 0 5 0 . 4 3 7 E + 0 4 0.22 4.2 1.10 0 . 4 9 1 E + 0 5 0 . 3 9 1 E + 0 4 0.24 5. 4 1.03 0 . 4 5 9 E + 0 5 0 . 3 6 6 E + 0 4 0.25 1 2 . 5 0.7 1 0. 3 1 7 E + 0 5 0 . 2 5 2 E + 0 4 G.29 2 0 . 0 0 . 6 0 0 . 2 6 8 E + 0 5 0 . 2 1 3 E + 0 4 0.31 3 0 . 0 0.52 0 . 2 3 2 E + 0 5 0. 185E+0 4 0.33 4 7 . 3 0. 46 0 . 2 0 5 E + 0 5 0, 163E+04 0.34 1 0 0 . 0 0.37 0 . 1 6 5 E + 0 5 0. 131E+04 C.36 E Q U I 0. 07 0 . 3 1 2 E + 0 4 0 . 2 4 9 E * 0 3 0.48 T I M E RX/R RX APPARENT A P P A R E N T SHEAR RATE V I S C O S I T Y (SEC) (-) (CM) ( 1 / S E C ) ( P O I S E ) 0.0 1 . 0 0 0 1.01 1.0 1 . 1 7 3 1.18 0 . 1 7 1 E + 0 1 0. 4 0 4 E + 0 4 2.0 1 . 2 1 6 1. 23 0 . 1 4 4 E + 0 1 0. 3 8 5 E + 0 4 3.0 1. 2 3 7 1.25 0 . 1 3 4 E + 0 1 0. 3 2 6 E + 04 4.2 1 . 2 5 9 1.27 0 . 1 2 6 E + 0 1 0. 3 1 0 E + 0 4 5. 4 1. 2 7 0 1.28 0 . 1 2 3 E + 0 1 0. 2 9 8 E + 0 4 1 2 . 5 1 . 3 1 3 1.33 0 . 1 1 1 E + 0 1 0. 2 2 7 E + 0 4 2 0 . 0 1. 3 3 5 1. 35 0 . 1 0 6 E + 0 1 0. 2 0 1 E + 0 4 3 0 . 0 1 . 3 56 1.37 0 . 1 0 2 E + 0 1 0. 181E+04 4 7 . 3 1. 3 6 7 1.38 0 . 1 0 0 E + 0 1 0. 16 3 E + 0 4 1 0 0 . 0 1 . 3 8 9 1.40 0 . 9 6 7 E + 0 0 0. 136E+04 E Q U I 1. 5 1 3 1.53 0 . 8 2 7 E + 0 0 0. 3 0 1 E + 0 3 !QUI L I B R I U M DATA OMEGA TORQUE SHEAR (RX-R) STRE SS /{RO-R) (RAD/SEC) (DYNES CM) (DYNES/SQ CM) (") 0 . 2 3 2 7 0 . 3 1 2 E * 0 4 0 . 2 4 9 E + 0 3 0.46 0. 6 9 8 1 0 . 3 5 7 E + 0 4 0.28 4E+03 0.48 2. 0 9 4 0 0 . 4 0 1 E + 0 4 0 . 3 2 0 E + 0 3 0.50 6. 2 8 3 0 0 . 4 2 4 E + 0 4 0 . 3 3 7 E + 0 3 0.57 OMEGA RX/R RX A P P A R E N T APPARENT SHEAR R A T E V I S C O S I T Y ( R AD/SEC) (-) (CM) ( 1 / S E C ) ( P O I S E ) 0. 2 327 1. 4 9 3 1. 51 0. 8 4 4 E + 0 0 0. 2 9 4 E + 0 3 0 . 6 9 8 1 1. 5 1 4 1.53 0 . 2 4 8 E + 01 0. 1 1 5 E + 0 3 2. 0 9 4 0 1. 534 1. 55 0. 7 2 8 E + 0 1 0. 4 3 9 E + 0 2 6. 2 8 3 0 1. 6 1 6 1.63 0 . 2 0 4 E + 0 2 0. 16 6 E + 0 2 T A B L E D - 2 - 13 199 E X P E R I M E N T A L DATA A N L R E S U L T S OF C A L C U L A T I O N P H I = 0.14 (CC/CC) 1 = 0.987 (MM) DECAY DATA OMEGA = 0 . 6 9 8 1 (RAD/SEC) T I M E TORQUE TORQUE SHEAR (RX-R) S T R E S S / (RO-R) ( S E C ) (M V) (DYNES CM) (DYNES/SQ CM) (") 0.0 1. 17 . 0 . 5 2 2 E + 0 5 0 . 4 1 5 E + 0 4 0.0 1.7 0.69 0 . 3 0 8 E + 0 5 0 . 2 4 5 E + 0 4 0.38 3. 2 0.5 3 0 . 2 3 6 E + 0 5 0.188E+G4 0.42 5.0 0. 42 0 . 1 8 7 E + 0 5 0. 149E+04 0.45 6.3 0. 38 0 . 1 6 9 E + 0 5 0. 135E+04 0.46 8.2 0.33 0. 1 4 7 E + 0 5 0. 1 1 7 E + 0 4 0.47 10.0 0.30 0. 134E+G5 0 . 1 0 7 E + 0 4 0.48 1 1 . 4 0.28 0 . 1 2 5 E + 0 5 0 . 9 9 4 E + 0 3 0.49 1 3 . 0 0 . 2 7 0. 1 2 0 E + 0 5 0 . 9 5 9 E + 0 3 0.50 2 7 . 2 0.21 0 . 9 3 7 E + 0 4 0 . 7 4 6 E + 0 3 0.52 6 6 . 1 0. 15 0 . 6 6 9 E + 0 4 0 . 5 3 3 E + 0 3 0.53 EQUI 0.08 0 . 3 5 7 E + 0 4 0 . 2 8 4 E + 0 3 0.64 T I M E RX/R RX A P P A R E N T A P P A R E N T SHEAR R A T E V I S C O S I T Y (SEC) (-) (CM) { 1/SEC) ( P O I S E ) 0. 0 1 . 0 0 0 1.01 1.7 1.410 1. 42 0 . 2 8 1 E + 0 1 0 . 8 7 2 E + 0 3 3. 2 1. 4 5 3 1.47 0 . 2 6 5 E + 0 1 0 . 7 1 0 E + 0 3 5.0 1 . 4 8 6 1. 50 0 . 2 5 5 E + 0 1 0 . 5 8 4 E + 0 3 6.3 1. 4 9 6 1.51 0.252E+ 01 0 . 5 3 5 E + 0 3 8.2 1 .507 1. 52 0 . 2 4 9 E + 0 1 0 . 4 7 0 E + 0 3 1 0 . 0 1. 5 1 8 1.53 G.247E+G1 0 . 4 3 2 E + 0 3 11.4 1 . 5 2 9 1.54 0.2 44 E + 0 1 0. 4 0 7 E + 0 3 1 3 . 0 1.540 . 1.55 0.242E+ 01 0 . 3 9 7 E + 0 3 2 7 . 2 1.561 1. 58 0 . 2 3 7 E + 0 1 0 . 3 1 5 E + 0 3 6 6 . 1 1. 572 1.59 0 . 2 3 5 E + 0 1 0 . 2 2 7 E + 0 3 E Q U I 1.691 1,71 0 . 2 1 5 E + 0 1 0 . 1 3 2 E + 0 3 E Q U I L I B R I U M DATA OMEGA ( R A D / S E C ) 0. 6 9 8 1 1 . 0 4 7 0 2 . 0 9 4 0 6 . 2 8 3 0 TORQUE (DYNES CM) 0 . 3 5 7 E + 0 4 0. 3 7 9 E + 0 4 0 . 4 0 1 E + 0 4 0 . 4 2 4 E + 0 4 SHEA R S T R E S S (DYNES/SQ CM) 0 . 2 8 4 E + 0 3 0 . 3 0 2 E + 0 3 0.32 0E+03 0. 3 3 7 E + 0 3 (RX-R) /(RO-R) (") 0.64 0.67 0.71 0.76 OMEGA RX/R RX A P P A R E N T APPARENT SHEAR RATE V I S C O S I T Y (RAD/S EC) (") (CM) (1/S EC) ( P O I S E ) 0 . 6 9 8 1 1. 691 1.71 0 . 2 1 5 E + 0 1 0 . 1 3 2 E + 0 3 1.0 47 0 1. 7 23 1. 7 4 0. 3 1 6 E + 0 1 0.956E+G2 2 . 0 9 4 0 1. 7 6 6 1.78 0 . 6 1 6 E + 01 0 . 5 1 8 E + 0 2 6 . 2 8 3 0 1. 820 1. 84 0. 180E+02 0. 1 8 7 E + 0 2 T A B L E D - 2 - 14 200 E X P E R I M E N T A L DATA AND R E S U L T S OF C A L C U L A T I O N P H I = 0.14 (CC/CC) L = 0.987 (MM) DECAY DATA OMEGA = 0. 3 4 9 1 (RAD/SEC) T I M E TORQUE TORQUE SHEAR (RX-R) S T R E S S /{RO-R) (SEC) (MV) (DYNES CM) (DYNES/SQ CM) (-) 0.0 1.19 0 . 5 3 1 E + 0 5 0 . 4 2 2 E + 0 4 0.0 1.2 0.90 0 . 4 0 1 E + 0 5 0 . 3 2 0 E + 0 4 0.24 2.3 0.74 0.3 30 E+05 0 . 2 6 3 E + 0 4 0.28 3. 4 0. 65 0 . 2 9 0 E + 0 5 0 . 2 3 1 E + 0 4 0.30 4.4 0.58 0 . 2 5 9 E + 0 5 0. 2 0 6 E + 0 4 0.31 5. 5 0.52 0 . 2 3 2 E + 0 5 0 . 1 8 5 E + 0 4 0.32 6.7 0.48 0 . 2 1 4 E + 0 5 0 . 1 7 0 E + 0 4 0.33 7.8 0. 45 0 . 2 0 1 E + 0 5 0 . 1 6 0 E + 0 4 0.34 8.7 0.43 0 . 1 9 2 E + 0 5 0 . 1 5 3 E + 0 4 0.35 2 5 . 3 0.26 0 . 1 1 6 E + 0 5 0 . 9 2 3 E + 0 3 0.39 5 7 . 2 0. 19 0 . 8 4 7 E + 0 4 0 . 6 7 5 E + 0 3 0.42 E Q U I 0.07 0 . 3 3 4 E + 0 4 0 . 2 6 6 E + 0 3 0.57 T I M E RX/R RX APPARENT APPARENT SHEAR R A T E V I S C O S I T Y (SEC) (-) (CM) ( 1 / S E C ) ( P O I S E ) 0.0 1 . 0 0 0 1.01 1.2 1.259 1.27 0. 189E+01 0. 1 6 9 E + 0 4 2.3 1.302 1. 32 0. 1 7 0 E + 0 1 0. 1 5 4 E + 0 4 3. 4 1.324 1. 34 0 . 1 6 3 E + 0 1 0. 1 4 2 E + 0 4 4.4 1.335 1. 35 0. 1 5 9 E + 0 1 0. 129E+04 5.5 1. 3 4 5 1. 36 0. 1 5 6 E + 0 1 0. 1 1 8 E + 0 4 6.7 1 . 3 5 6 1. 37 0 . 1 5 3 E + 0 1 0. 1 1 1 E + 0 4 7.8 1.367 1.38 0.150E+ 01 0. 106 E+04 8.7 1 . 3 7 8 1.39 0 . 1 4 8 E + 0 1 0. 103E+04 2 5 . 3 1. 4 2 1 1.44 0. 138E+01 0. 6 6 7 E + 0 3 5 7 . 2 1 . 4 5 3 1. 47 0. 1 3 3 E + 01 0. 5 0 9 E + 0 3 E Q U I 1 . 6 1 5 1.63 0. 1 13E+01 0. 2 3 5 E + 0 3 E Q U I L I B R I U M DATA OMEGA TORQUE SHEAR (RX-R) S T R E S S / ( R O - R ) (RAD/S EC) (DYNES CM) (DYNES/SQ CM) (-) 0. 3 4 9 0 0 . 3 3 4 E + 0 4 0. 26 6E+03 0.57 0. 6 9 8 1 0 . 3 7 9 E + 0 4 0. 3 0 2 E + 0 3 0.60 2. 0 9 4 0 0 . 4 0 1 E + 0 4 0 . 3 2 0 E + 0 3 0.65 6. 2 8 3 0 0. 4 2 4 E + 0 4 0 . 3 3 7 E + 0 3 0.71 OMEGA RX/R RX A P P A R E N T APPAH ENT SHEAR R A T E V I S C O S I T Y { RAD/S EC) (-) (CM) ( 1 / S E C ) ( P O I S E ) 0. 3 4 9 0 1. 6 1 5 1.63 0 . 1 1 3 E + 0 1 0. 2 3 5 E + 0 3 0. 6 9 8 1 1. 6 4 8 1.66 0. 2 2 1 E + 0 1 0. 1 3 7 E + 0 3 2. 0 9 4 0 1. 7 0 1 1.72 0 . 6 4 0 E + 0 1 0. 4 9 9 E + 0 2 6. 2 8 3 0 1. 766 1.78 0 . 1 8 5 E + 0 2 0. 1 8 2 E + 0 2 T A B L E D - 2 - 15 201 E X P E R I M E N T A L DATA AND R E S U L T S OF C A L C U L A T I O N P H I = 0.14 {CC/CC) DECAY DATA OMEGA = 0 . 1 1 6 4 {RAD/SEC) L = 0.987 {MM) T I M E TORQUE TORQUE SHEAR (RX-R) S T R E S S / (RO-R) {SEC) (M V) (DYNES CM) (DYNES/SQ CM) {-) 0.0 3. 46 0. 1 5 4 E + 0 6 0 . 1 2 3 E + 0 5 0.0 0.4 3. 17 0. 1 4 1 E + 06 0. 1 1 3 E + 0 5 0.09 1.7 2.5 0 0 . 1 1 2 E + 0 6 0 . 8 8 8 E + 0 4 0.15 3.1 2 . 0 7 0 . 9 2 3 E + 0 5 0 . 7 3 5 E + 0 4 0. 17 4. 5 1.74 0 . 7 7 6 E + 0 5 0 . 6 1 8 E + 0 4 0.18 5.8 1.62 0 . 7 2 3 E + 0 5 0 . 5 7 5 E + 0 4 0.19 7.2 1.40 0 . 6 2 4 E + 0 5 0 . 4 9 7 E + 0 4 0.20 8.7 1.33 Q . 5 9 3 E + 0 5 0 . 4 7 2 E + 0 4 0.21 1 0 . 0 1. 18 0 . 5 2 6 E + 0 5 0 . 4 1 9 E + 0 4 0.21 3 6 . 0 0.59 0 . 2 6 3 E + 0 5 0 . 2 0 9 E + 0 4 0.25 1 3 8 . 6 0. 32 0 . 1 4 3 E + 0 5 0 . 1 1 4 E + 0 4 0,28 EQUI 0,06 0 . 2 6 8 E + 0 4 0 . 2 1 3 E + 0 3 0.39 T I M E RX/R RX APPARENT A P P A R E N T SHEAR R A T E V I S C O S I T Y (SEC) {") (CM) { 1/SEC) ( P O I S E ) 0.0 1.000 1.01 0.4 1.097 1. 11 0 . 1 3 8 E + 0 1 0 . 8 1 8 E + 0 4 1.7 1. 162 1.17 0 .8 9 8 E + 0 0 0 .988E+04 3. 1 1 . 1 8 3 1.20 0.8 14E+00 0 .9 0 3 E + 0 4 4. 5 1.194 1.21 0 . 7 7 9 E + 0 0 0 .793 E + 0 4 5.8 1 . 2 0 5 1. 22 0 . 7 4 8 E + 0 0 0 . 7 6 9 E + 0 4 7.2 1 . 2 1 6 1.23 0.7 2 0 E + 0 0 0 .6 9 1 E + 0 4 8.7 1.227 1. 24 0 . 6 9 4 E + 0 0 0 . 6 8 0 E + 0 4 1 0 . 0 1. 2 2 7 1.24 0 . 6 94 E + 0 0 0 .604 E + 0 4 3 6 . 0 1 . 2 7 0 1. 28 0 .6 13E+00 0 . 3 4 2 E + 0 4 1 3 8 . 6 1. 3 0 2 1. 32 0 . 5 6 7 E + 0 0 0 .200E+04 E Q U I 1.421 1. 44 0 . 4 6 1 E + 0 0 0 . 4 6 2 E + 0 3 E Q U I L I B R I U M DATA OMEGA TORQUE ( R A D / S E C ) 0. 1 1 6 4 0.6 981 2. 0 9 4 0 6 . 2 8 3 0 OMEGA (RAD/SEC) 0. 1 164 0 . 6 9 8 1 2. 0 9 4 0 6 . 2 8 3 0 (DYNES CM) 0. 2 6 8 E + 0 4 0 . 3 5 7 E + 0 4 0 . 4 0 1 E + 0 4 0 . 4 2 4 E + 0 4 R X / R {") 1. 421 1.442 1.464 1. 4 8 6 RX (CM) 1. 44 1. 46 1. 48 1. 50 SHEAR S T R E S S (DYNES/SQ CM) 0 . 2 1 3 E + 0 3 0 . 2 8 4 E + 0 3 0.32 0Et-03 0 . 3 3 7 E + 0 3 A P P A R E N T SHEAR R A T E ( 1 / S E C ) 0 . 4 6 1 E + 0 0 0 . 2 6 9 E + 0 1 0 . 7 8 5 E + 0 1 0 . 2 3 0 E + 0 2 ( R X - R ) / ( R O - R ) (-) 0.39 0.41 0.43 0.45 APPARENT V I S C O S I T Y ( P O I S E ) 0.46 2E+03 0. 1 0 6 E + 0 3 0 . 4 0 7 E + 0 2 0. 1 4 7 E + 02 A p p e n d i x E S o l u t i o n s f o r D i f f e r e n t i a l E q u a t i o n ( 7 - 5 ) A m o d e l e q u a t i o n f o r t h e t h i c k n e s s o f f l o w i n g l a y e r w a s c o n s t r u c t e d i n S e c t i o n 7 - 1 a n d i t w a s s h o w n i n e q u a t i o n ( 7 - 5 ) , i . e . , d i r ( R 2 - R 2 ) L ^ * T ¥ IV*<K»2 " Rx2>l>m , n - \ { * ( \ 2 - R 2 ) 1 } J ( 7 - 5 ) f r o m w h i c h o n e c a n w r i t e dR 2 k A t e q u i l i b r i u m s t a t e o n e k n o w s dR 2 x d t X Xoo H e n c e , = 0 , a n d R = R a t t = t ( E - 2 ) xinsr < * L ( E O 2 - V»" - r+V <*l(E*»2- e 2»" ( e"3 ) CO CO S o l v i n g e q u a t i o n ( E - 3 ) f o r o n e g e t s { , L ( R 0 2 - R ^ 2 ) } 1 * \ ~ k f {TTL(R 2 - R 2 ) } n ^ L e t { T T L ( R Q 2 - R ^ 2 ) } 1  { L T L ( R 2 - R 2 ) } N M N "-= P (> 0 ) ( E - 5 ) t h e n k, = P k , ( E - 6 ) D mn f S u b s t i t u t i n g e q u a t i o n ( E - 6 ) b a c k t o e q u a t i o n ( E - l ) o n e o b t a i n s dR 2 k - P { ^ L ( R 2 - R 2 ) } n ] ( E - 7 ) mn x 202 203 L e t R x 2 - R 2 = X (E - 8 ) t h e n R 0 2 - R x 2 = R 0 2 - ( R x 2 - R 2) - R 2 = - X + ( R Q 2 - R 2) dR 2 , . x dx and — = — dt d t T h e r e f o r e , e q u a t i o n (E-7) becomes - P {uLX} n] mn or ^ d f = " AT^BT [ P m n { l T L X } n " { - L [ ( R 0 2 " R 2) " X]}"1] (E-9) E q u a t i o n (E-9) i s t h e b a s i c model e q u a t i o n i d e n t i c a l t o e q u a t i o n ( 7 - 5 ) . Now one can s o l v e t h i s f i r s t r a nk f i r s t o r d e r d i f f e r e n t i a l equa-t i o n o f X w i t h r e s p e c t t o t f o r (m, n) = (2, 0 ) . With (m, n) = (2, 0 ) , e q u a t i o n (E-5) i s P 2 0 = ^ 2 L 2 ( R 0 2 - R 2 J 2 > 0 (E-10) and the b a s i c e q u a t i o n (E-9) i s * L f r = - TT^BT [ ? 2° - ^ L 2 { ( R O 2 - R 2) - x> 2i i r L k P 2 0 = " A T W l { x 2 " 2 ( R ° 2 " r 2 ) x + ( R 0 2 - R 2 ) 2 } ] from which , TTLk If = x r i t - [ x 2 - 2 ( R ° 2 - r 2 ) x + ( r ° 2 " r 2 ) 2 ] ( E - l l ) u 2 L 2 204 L e t b = - 2 ( R 0 2 - R 2 ) 2 0 a n d C = ( R 0 2 - R 2 ) 2 -t h e n e q u a t i o n ( E - l l ) b e c o m e s TrLk, . d X ( X 2 + b X + C) d t A + B t S e p a r a t e t h e v a r i a b l e s i n e q u a t i o n ( E - 1 4 ) a n d f i n d d X d t = TTLk d t A + B t X 2 + b x + c f A + B t f r o m w h i c h o n e may w r i t e / ^ «* f / X 2 + b X + C N o t e h e r e t h a t b 2 - 4C = 4 ( R 0 2 - R 2 ) 2 - 4 ( R 0 2 - R 2 ) 2 + 4 P 2 0 > 0 4 P 2 o ^ 2 L 2 TT 2L 2 s i n c e P 2 0 > 0 a s s h o w n i n e q u a t i o n ( E - 1 0 ) R e c a l l i n g e n e r a l t h a t d x l o g a x + b x + c /, 2 / v b - 4 a c w i t h b 2 - 4 a c > 0 2 a x + b - / b 2 - 4 a c 2 a x + b + / b 2 - 4 a c T h e r e f o r e t h e l e f t - h a n d s i d e o f e q u a t i o n ( E - 1 5 ) i s 1 d X X 2 + b X + c l o g 4 c 2X + b - / b 2 - 4c 2X + b + / b 2 - 4c ( E - 1 2 ) ( E - 1 3 ) ( E - 1 4 ) ( E - 1 5 ) ( E - 1 6 ) ( E - 1 7 ) ( E - 1 8 ) ( E - 1 9 ) 205 L e t 4 c = D a n d 2X + b = Y t h e n d X X z + b X + c D l o g Y - D Y + D T h e r i g h t - h a n d s i d e o f e q u a t i o n ( E - 1 5 ) i s ^ A+Bt = f tB ^ ~7~A " B k A = T T L — l o g | t + - | T h e r e f o r e e q u a t i o n ( E - 1 5 ) h a s b e e n i n t e g r a t e d a s D l o g Y - D Y + D = TTL-^ l o g |t + f | + C I ( E - 2 0 ) ( E - 2 1 ) ( E - 2 2 ) ( E - 2 3 ) (E-24) w h e r e C I i s t h e i n t e g r a l c o n s t a n t . T h e i n i t i a l b o u n d a r y c o n d i t i o n i s R 2 - R 2 = X = 0 a t t = 0 x w h i c h i s e q u i v a l e n t t o Y = b a t t = 0 s i n c e Y = 2X + b , a s g i v e n i n e q u a t i o n ( E - 2 1 ) . S u b s t i t u t i n g e q u a t i o n ( E - 2 6 ) i n t o e q u a t i o n ( E - 2 4 ) o n e may f i n d k . b - D |b + D w h i c h y i e l d s = TiL-j- l o g if 1 + C I C I = - l o g b - D T f i l A l - T T L — l o g |-| ( E - 2 5 ) |b + D S u b s t i t u t i n g e q u a t i o n ( E - 2 5 ) b a c k i n t o e q u a t i o n ( E - 2 4 ) o n e c a n f i n d k D l o g Y - D b + D Y + D b - D = TrL-y- l o g |-| t + 1 f r o m w h i c h 2 0 6 l o g Y - D b + D w h e r e Y + D b - D Y = 2X + b D T T L -J - l o g |f t + 1 D = / b 2 - 4 c b = - 2 ( R 0 2 - R 2 ) c = ( R 0 2 - R 2 ) 2 -( E - 2 6 ) ( E - 2 1 ) ( E - 2 0 ) ( E - 1 2 ) J 2 0 u 2 L 2 a n d = ( R 0 2 - R 2 ) 2 - (V " \J)2 X = R 2 - R 2 x ( E - 1 3 ) ( E - 8 ) E q u a t i o n ( E - 2 6 ) i s t h e s o l u t i o n o f e q u a t i o n ( 7 - 5 ) w i t h (m, n ) = ( 2 , 0 ) a n d h a s a l r e a d y b e e n g i v e n i n e q u a t i o n ( 7 - 7 ) i n S e c t i o n ( 7 - 1 ) . o f t h i s t h e s i s . T h e s o l u t i o n s o f e q u a t i o n ( 7 - 5 ) f o r v a r i o u s v a l u e s o f (m, n ) a r e s u m m a r i z e d b e l o w . I t s h o u l d b e n o t e d t h a t X = R 2 - R 2 . x i . (m, n ) = ( 0 , 0 ) X = c o n s t f o r a n y t i m e t . ( E - 2 7 ) T h i s d o e s n o t s a t i s f y t h e b o u n d a r y c o n d i t i o n s o f X = 0 a t t = 0 X = R 2 - R 2 H a t t = t i i . (m, n ) = ( 0 , 1 ) X l o g 1 " R 2 - R 2 Xoo = - P w h e r e J 01 = o i T l o g l (E-28) TTL(R 2 - R 2 ) 207 i n . i v . v. V I . (m, n ) = ( 0 , 2 ) |X - a l o g X + a w h e r e = - P 0 2 T T L 2 a - — • l o £ T t + 1 A 02 a n d (m, n ) = ( 1 , 0 ) X l o g 1 -R 2 - R 2 (m, n ) = ( 1 , 1 ) X l o g 1 -R 2 _ R 2 Xoo w h e r e (m, n ) = ( 1 , 2 ) |Y - D 1 + D l o g Y + D 1 - D ^ 2 L 2 ( R 2 _ R 2 ) 2 X oo a = 1 TTL /F l o g t + 1 = - (?io + l o§ t + 1 R 0 2 - R 2 u Xoo R 2 _ R 2 Xoo l o g t + 1 w h e r e y = 2 a X + 1 D =/ l + 4 a b a = P 1 2 f r l ( E - 2 9 ) ( E - 3 0 ) ( E - 3 1 ) (E-32) b = R 0 Z - R z 208 v i i . (m, n ) = ( 2 , 1 ) |Y - D b + D V l l l , l o g Y + D b - ' - D w h e r e Y = 2 X + b T T L D — l o g a t + 1 D = / b - 4c P 2 1 b = -[—+ 2 ( R 0 ^ - R z ) . ] c = ( R o 2 - R 2 ) 2 2 N 2 P 2 1 = (m, n ) = ( 2 , 2 ) |Y - D b + D R 2 - R 2 l o g = - T T L D - ^ - l o g Y + D b. - D w h e r e Y = 2 a X + b t + 1 D = / b 2 - 4 a c a = P 2 2 " 1 b = 2 ( R 0 2 - R 2 ) 2 N 2 c = - ( R 0 2 - R ) a n d P 2 2 = ( R 0 2 - R ? ) 2  x°° (R 2 - R 2 ) 2 x°° ( E - 3 3 ) ( E - 3 4 ) ( E - 3 5 ) I t i s i n t e r e s t i n g t o n o t e t h a t a l l t h e s o l u t i o n s f o r (m, n ) w h i c h a r e ( 0 , 1 ) t o ( 2 , 2 ) a r e t h e f o r m o f l o g f (R ) = K — l o j T t + 1 A ( E - 3 6 ) T h i s h a s b e e n m e n t i o n e d i n e q u a t i o n ( 7 - 6 ) i n S e c t i o n 7 - 1 o f t h i s t h e s i s , A p p e n d i x F V a r i o u s O r d e r s o f t h e M o d e l P r e d i c t i n g T h i c k n e s s o f F l o w i n g L a y e r a n d Sum o f S q u a r e s A l l t h e d a t a i n t h e s e t a b l e s a r e f o r t h e d i s p e r s i n g m e d i u m d e x t r o s e + 1 . 0 m o l e N a C l i n 1 0 % P E G - H 2 0 . D e s c r i p t i o n o f V a r i a b l e s i n T a b l e F P H I ( C C / C C ) - P a r t i c l e C o n c e n t r a t i o n ( c c / c c ) L (MM) - P a r t i c l e L e n g t h (mm) L/D - P a r t i c l e L e n g t h - t o - D i a m e t e r R a t i o ( - ) O M E G A ( R A D / S E C ) - A n g u l a r V e l o c i t y o f I n n e r C y l i n d e r ( r a d . / s e c . ) 209 210 T A B L E F -• 1 VA R I O U S ORDERS (M,N) AND R E S U L T OF L E A S T SQUARES F I T ORDER RES U L T A N T PARAMETERS SUM OF (M,N) K F / B B/A SQUARES Z E R O - F I R S T ORDER 0 . 1 0 4 E + 0 1 0.4 91 E+01 0. 1 6 1 E - 0 2 ZERO-SECOND ORDER 0 . 7 1 5 E + 0 0 0. 9 8 8 E + 0 1 0. 28 8 E - 0 2 F I R S T - Z E R O ORDER 0. 14 9E+00 0 . 4 9 1 E + 0 1 0. 1 6 1 E - 0 2 F I R S T - F I R S T ORDER 0 . 5 0 0 E - 0 1 0. 4 91 E+01 0 . 1 6 1 E - 0 2 F I R S T - S E C O N D ORDER 0 . 3 8 2 E - 0 1 0. 8 04E+01 0 . 2 3 8 E - 0 2 SECOND-ZERO ORDER 0. 4 7 5 E - 0 2 0 . 4 1 0 E + 0 1 0. 1 4 0 E - 0 2 S E C O N D - F I R S T ORDER 0 . 2 5 3 E - 0 2 0 . 4 4 5 E + 0 1 0. 1 5 0 E - 0 2 SECOND-SECOND ORDER P H I (CC/CC) L (MM) L/D OM EGA (R AD/S EC) 0 . 1 6 0 0.987 2 2 . 9 0 . 2 3 2 7 211 T A B L E F - 2 V A R I O U S ORDERS (M,N) AND R E S U L T OF L E A S T SQUARES F I T ORDER R E S U L T A N T PARA METERS SUM OF (M,N) K F / B B/A SQUARES Z E R O - F I R S T ORDER 0. 1 8 2 E + 0 1 0 . 4 2 0 E + 0 1 0 . 7 9 0 E - 0 1 ZERO-SECOND ORDER 0. 1 0 6 E + 01 0 . 1 1 8 E + 0 2 0 . 1 3 8 E + 0 0 F I R S T - Z E R O ORDER 0 . 2 5 5 E + 0 0 0. 420E+-C1 0.7 9 0 E - 0 1 F I R S T - F I R S T ORDER 0 . 8 7 0 E - 0 1 0 . 4 2 0 E + 0 1 0 . 7 9 0 E - 0 1 F I R S T - S E C C N D ORDER 0 . 5 9 3 E - 0 1 0 . 8 7 4 E + C 1 0. 12 2E+0 0 SECOND-ZERO ORDER 0 . 8 6 1 E - 0 2 0.3 18E+01 0 . 6 1 8 E - 0 1 S E C O N D - F I R S T ORDER 0 . 4 5 4 E - 0 2 0. 3 6 0 E + 0 1 0 . 6 9 5 E - 0 1 SECOND-SECOND ORDER 0.3U0E-O2 0 . 6 1 5 E + 0 1 0 . 1 0 2 E + 0 0 P H I ( C C / C C ) 0. 1 5 0 L (MM) 0 . 9 8 7 L/D 2 2 . 9 OMEGA(RAD/S EC) 0 . 2 3 2 7 212 T A B L E F - 3 V A R I O U S ORDERS (M,N) AND R E S U L T OF L E A S T SQUARES F I T ORDER R E S U L T A N T P ARAMETERS SUM OF (M,N) K F / B B/A SQUARES Z E R O - F I R S T ORDER 0 . 1 9 3 E + 0 1 0. 6 08E+01 0 . 3 8 2 E - 0 2 ZERO-SECOND ORDER 0. 115E+01 0. 1 5 8 E + 0 2 0.4 7 4 E - 0 2 F I R S T - Z E R O ORDER 0.262E+O0 0. 6 08E+01 0 . 3 8 2 E - 0 2 F I R S T - F I R S T ORDER 0 . 9 2 6 E - 0 1 0. 608E+G1 0.38 2E- 0 2 F I R S T - S E C O N D ORDER 0 . 6 4 2 E - 0 1 0. 118E+02 0.4 4 5 E -0 2 SECOND-ZERO ORDER 0 . 8 9 5 E - 0 2 0, 4 58E+01 0 . 3 5 3 E - 0 2 S E C O N D - F I R S T ORDER 0 . 4 8 5 E - 0 2 0. 5 1 8 E + 0 1 0 . 3 6 7 E - 0 2 SECOND-SECOND ORDER 0. 36 9 E - 0 2 0. 8 4 5 E + 0 1 0 . 4 1 3 E - 02 P H I (CC/CC) L(MM) L/D OMEGA (R AD/S EC) 0 . 1 4 0 0.987 2 2 . 9 0 . 2 3 2 7 T A B L E F - 4 VARIOOS ORDERS (M,N) AND R E S U L T OF L E A S T SQUARES F I T ORDER R E S U L T A N T PARAMETERS SUM OF (H,N) K F / B B/A SQUARES Z E R O - F I R S T ORDER 0. 2 6 3 E + 0 1 0. 110E+02 0 . 1 5 9 E - 0 1 ZERO-SECOND ORDER 0 . 1 4 7 E + 0 1 0 . 3 1 3 E + 0 2 0 . 2 5 5 E - 0 1 F I R S T - Z E R O ORDER 0 . 3 2 9 E + 0 0 0. 1 10E+02 0 . 1 5 9 E - 0 1 F I R S T - F I R S T ORDER 0 . 1 2 6 E + 0 0 0 . 1 1 0 E + 0 2 0. 1 5 9 E - 0 1 F I R S T - S E C O N D ORDER 0 . 8 4 8 E - 0 1 0 . 2 2 3 E + 0 2 0 . 2 2 4 E - 0 1 SECOND-ZERO ORDER 0. 1 2 0 E - 0 1 0 . 7 6 3 E + 0 1 0 . 1 3 0 E - 0 1 S E C O N D - F I R S T ORDER 0 . 6 8 4 E - 0 2 0 . 8 8 4 E + 0 1 0 . 1 4 1 E - 0 1 SECOND-SECOND ORDER 0. 50 9 E - 0 2 0. 1 4 8 E + 0 2 0. 1 8 6 E - 0 1 P H I ( C C / C C ) L (MM) L/D OMEGA(RAD/SEC) 0. 130 0 . 9 8 7 2 2 . 9 0 . 2 3 2 7 T A B L E F - 5 V A R I O U S ORDERS (M,N) AND R E S U L T OF L E A S T SQUARES F I T ORDER RESULTANT PARAMETERS SUM OF (M,N) K F / B B/A SQUARES Z E R O - F I R S T ORDER 0 . 1 5 4 E + 0 1 0 . 3 9 7 E + 0 1 0 . 3 4 9 E - 0 2 ZERO-SECOND ORDER 0 . 9 8 1 E + 0 0 0 . 8 3 3 E + 0 1 0 . 4 5 1 E - 0 2 F I R S T - Z E R O ORDER 0.23 4E+00 0 . 3 9 7 E + 0 1 Q . 3 4 9 E - 0 2 F I R S T - F I R S T ORDER 0 . 7 3 9 E - 0 1 0 . 3 9 7 E + C 1 0 . 3 4 9 E - 0 2 F I R S T - S E C O N D ORDER 0 . 5 3 0 E - 0 1 0 . 6 8 2 E + 0 1 0 . 4 2 5 E - 0 2 SECOND-ZERO ORDER 0 . 7 4 8 E - 0 2 0 . 3 2 9 E + 0 1 0 . 3 1 6 E - 0 2 S E C O N D - F I R S T ORDER 0 . 3 7 6 E - 0 2 0 . 3 6 1 E + 0 1 0 . 3 3 3 E - 0 2 SECOND-SECOND ORDER 0 , 2 9 2 E - 0 2 0.5 43E+C1 0.3 9 5 E - 0 2 P H I (CC/CC) L (MM) L/D OMEGA (R AD/S EC) 0 . 1 2 0 1.620 3 7 . 4 0 . 2 3 2 7 215 T A B L E F - 6 V A R I O U S ORDERS (M,N) AND R E S U L T OF L E A S T SQUARES F I T ORDER R E S U L T A N T P A RAMETEBS SUM OF (M,N) K F / B B/A SQUARES Z E R O - F I R S T ORDER 0. 1 8 0 E + 0 1 0. 6 76E+01 0 . 2 0 5 E + 0C ZERO-SECOND ORDER 0. 1 0 4 E + 0 1 0 . 2 1 8 E + 0 2 0. 2 4 0 E + O 0 F I R S T - Z E E O ORDER 0 . 2 5 2 E + 0 0 0 . 6 7 6 E+01 0 . 2 0 5 E + 0 0 F I R S T - F I R S T ORDER 0.86 4 E - 0 1 0. 6 76 E+ 01 0 . 2 0 5 E + 0 0 F I R S T - S E C O N D ORDER 0 . 5 7 9 E - 0 1 0. 1 5 7 E + 0 2 0 . 2 3 3 E + 0 G SECOND-ZERO ORDER 0 . 8 7 0 E - 0 2 0 . 4 6 6 E + 0 1 0 . 1 8 7 E + 0 0 S E C O N D - F I R S T ORDER 0 . 4 5 6 E - 0 2 0. 552E+G1 0 . 1 9 6 E + 00 SECOND-SECOND ORDER 0.3 3H E - 0 2 0 . 1 0 6 E + 02 0 . 2 2 2 E + 0 0 P H I { C C / C C ) L (MM) 0 . 1 1 0 1.620 L/D 3 7 . U OMEGA (RAD/SEC) 0 . 2 3 2 7 216 T A B L E F - 7 V A R I O U S ORDERS (M,N) AND RE S U L T OF L E A S T SQUARES F I T ORDER R E S U L T A N T PARAMETERS SUM OF (H.N) K F / B B/A SQUARES Z E R O - F I R S T ORDER 0.31 1E+01 0 . 4 3 8 E + 0 1 0 . 4 6 5 E - 0 1 ZERO-SECOND ORDER 0 . 1 6 7 E + 0 1 0 . 1 1 9 E + 0 2 0 . 4 4 5 E - 0 1 F I R S T - Z E R O ORDER 0 . 4 3 3 E + 0 0 0 . 4 3 8 E + 0 1 0 . 4 6 5 E - 0 1 F I R S T - F I R S T ORDER 0 . 1 4 9 E + 0 0 0 . 4 3 8 E + 0 1 0 . 4 6 5 E - 0 1 F I R S T - S E C O N D ORDER 0 . 9 5 1 E - 0 1 0. 8 9 7 E + 0 1 0 . 4 5 0 E - 0 1 SECOND-ZERO ORDER 0 . 1 5 3 E - 0 1 0 . 3 2 0 E + 0 1 0 . 4 6 7 E - 0 1 S E C O N D - F I R S T ORDER 0 . 7 9 6 E - 0 2 0 . 3 6 9 E + 0 1 0 . 4 6 7 E - 0 1 SECOND-SECOND ORDER 0. 56 2 E - 0 2 0.641E+G1 0. 45 8 E - 0 1 P H I (CC/CC) 0. 1 0 0 L (MM) 1.620 L/D OMEGA (RAD/S EC) 3 7 . 4 0 . 2 3 2 7 217 T A B L E F - 8 VARIOUS ORDERS (M,N) AND R E S U L T OF L E A S T SQUARES F I T ORDER R E S U L T A N T PARAMETERS SUM OF (M,N) ,KF/B B/A SQUARES Z E R O - F I R S T ORDER 0. 3 4 0 E + 0 1 0. 1 10E+02 0 . 6 2 0 E - 0 1 ZERO-SECOND ORDER 0 . 1 8 6 E + 0 1 0 . 2 5 9 E + 0 2 0 . 8 9 8 E -•0 1 F I R S T - Z E R O ORDER 0. 4 4 8 E + 0 0 0. 110E+02 0 . 6 2 0 E - 01 F I R S T - F I R S T ORDER 0. 1 6 3 E + 0 0 0, 1 10E+02 0 . 6 2 0 E -•0 1 F I R S T - S E C O N D ORDER 0. 106E+00 0 . 2 0 2 E + 0 2 0 . 8 0 7 E - 0 1 SECOND-ZERO ORDER 0. 1 6 2 E - 0 1 0 . 8 1 3 E + 0 1 0 . 5 4 7 E - 0 1 S E C O N D - F I R S T ORDER 0. 8 7 8 E - 0 2 0 . 9 2 9 E + C 1 0 . 5 7 7 E - 01 SECOND-SECOND ORDER 0. 6 3 4 E - 0 2 0. 1 4 8 E + 0 2 0 . 7 0 4 E -•0 1 P H I ( C C / C C ) L (MM) 0 . 0 9 0 1 . 6 2 0 L/D 3 7 . 4 OMEGA (R A D / S E C ) 0 . 2 3 2 7 T A B L E F - 9 V A R I O U S ORDERS (M,N) AND R E S U L T OF L E A S T SQUARES F I T ORDER R E S U L T A N T PARAMETERS SUM OF (M,N) K F / B B/A SQUARES Z E R O - F I R S T ORDER 0. 108E+01 0 . 7 7 4 E + 0 0 0.89 4 E - 0 3 ZERO-SECOND ORDER 0 . 7 8 9 E + 0 0 0 . 1 2 1 E + 0 1 0. 1 7 3 E - 0 2 F I R S T - Z E R O ORDER 0. 15 9E+00 0 . 7 7 4 E + 0 0 0 . 8 9 4 E - 0 3 F I R S T - F I R S T ORDER 0 . 5 1 8 E - 0 1 0. 7 7 4 E + G 0 0 . 8 9 4 E - 0 3 F I R S T - S E C O N D ORDER 0 . 4 1 4 E - 0 1 0 , 1 0 6 E + 0 1 0 . 1 4 7 E - 0 2 SECOND-ZERO ORDER 0 . 4 9 3 E - 0 2 0. 6 94E + 00 0 . 7 0 7 E - 0 3 S E C O N D - F I R S T ORDER 0 . 2 5 9 E - 0 2 0 . 7 3 0 E + 0 0 0 . 7 9 4 E - 0 3 SECOND-SECOND ORDER 0 . 2 1 9 E - 0 2 0 . 9 2 0 E + 0 0 0. 1 2 0 E - 0 2 P H I ( C C / C C ) L(MM) 0 . 0 8 0 3 . 0 1 0 L/D OMEGA (R AD/S EC) 6 9 . 8 0 . 2 3 2 7 219 T A B L E F - 10 V A R I O U S ORDERS (M , N) / AND R E S U L T OF L E A S T SQUARES F I T ORDER R E S U L T A N T PARAMETERS SUM OF («,N) K F / B B/A SQUARES Z E R O - F I R S T ORDER 0. 126E+01 0. 1 0 7 E + 0 1 0. 17 0E- 02 ZERO-SECOND ORDER 0. 8 7 4 E + 0 0 0. 1 8 7 E + 0 1 0. 2 1 3 E - 0 2 F I R S T - Z E R O ORDER 0. 176E+00 0. 1 07E+01 0. 17 0E- 0 2 F I R S T - F I R S T ORDER 0. 6 0 3 E - 0 1 0. 1 07E+01 0. 1 7 0 E - 0 2 F I R S T - S E C O N D ORDER 0. 4 6 7 E - 0 1 0. 158E+01 0. 2 1 0 E - 02 SECOND-ZERO ORDER 0 . 5 6 2 E - 0 2 0. 9 2 2 E + 0 0 0. 1 4 9 E - •0 2 S E C O N D - F I R S T ORDER 0. 3 0 5 E - 0 2 0. 9 8 6 E + 0 0 0 . 1 5 9 E - 02 SECOND-SECOND ORDER 0. 2 5 3 E - 0 2 0. 1 3 1 E + 0 1 0. 1 8 9 E - 0 2 P H I ( C C / C C ) L (MM) L/D OMEGA ( R f l D / S E C ) 0 . 0 7 0 3 . 0 1 0 6 9 . 8 0 . 2 3 2 7 220 T A B L E F - 11 V A R I O U S ORDERS (M,N) AND R E S U L T OF L E A S T SQUARES F I T ORDER R E S U L T A N T P A R A M E T E R S SUM OF (M,N) K F / B B/A SQUARES Z E R O - F I R S T ORDER 0. 130E+01 0. 3 4 6 E + 0 1 0 . 1 3 4 E - •0 2 ZERO-SECOND ORDER 0 . 8 7 0 E + 0 0 0. 6 8 3 E + 0 1 0 . 3 3 5 E - 02 F I R S T - Z E R O ORDER 0. 1 7 7 E + 0 0 0. 3 4 6 E + 0 1 0 . 1 3 4 E - 0 2 F I R S T - F I R S T ORDER 0 . 6 2 1 E - 0 1 0. 3 4 6 E + C 1 0.131E- 02 F I R S T - S E C O N D ORDER 0 . U 7 1 E - 0 1 0. 5 5 3 E + 0 1 0 . 2 5 6 E -•0 2 SECOND-ZERO ORDER 0. 5 7 5 E - 0 2 0. 2 87E+C1 0 . 1 0 1 E - 0 2 S E C O N D - F I R S T ORDER 0.317E - 0 2 0. 3 1 1 E + 0 1 0 . 1 1 4 E -•0 2 SECOND-SECOND ORDER P H I (CC/CC) L (MM) L/D OMEGA(R AD/S EC) 0 . 0 6 0 3 . 0 1 0 6 9 . 8 0 . 2 3 2 7 221 T A B L E F - 12 VARIOUS ORDERS (M,N) AND R E S U L T OF L E A S T SQUARES F I T ORDER R E S U L T A N T PARAMETERS SUM OF (M,N) K F / B B/A SQUARES Z E R O - F I R S T ORDER 0. 176E+01 0. 3 6 2 E + 0 1 0 . 1 5 1 E - G2 ZERO-SECOND ORDER 0 . 1 1 2 E + 0 1 0. 7 8 6 E + 0 1 0 . 2 8 5 E - 0 2 F I R S T - Z E R O ORDER 0 . 2 1 8 E + 0 0 0. 3 6 2 E + 0 1 0 . 1 5 1 E - 02 F I R S T - F I R S T ORDER 0 , 8 4 4 E - 0 1 0. 3 6 2 E + 0 1 r 0 . 1 5 1 E - 0 2 F I R S T - S E C O N D ORDER 0 . 6 2 4 E - 0 1 0 . 6 0 5 E + 0 1 0.2 2 5 E - 0 2 SECOND-ZERO ORDER 0 . 7 5 5 E - 0 2 0. 2 8 1 E + 0 1 0 . 1 2 9 E - 0 2 S E C O N D - F I R S T ORDER 0. 4 4 4 E - 0 2 0. 3 1 0 E + 0 1 0 . 1 3 7 E - 02 SECOND-SECOND ORDER 0, 35 8 E - 0 2 0. 4 4 5 E + 0 1 0 . 1 7 5 E - 0 2 P H I (CC/CC) L (MM) L/D OMEGA (RA D / S E C ) 0 . 0 5 0 3 . 0 1 0 6 9 . 8 0 . 2 3 2 7 222 T A B L E F - 13 V A R I O U S ORDERS (R,N) AND R E S U L T OF L E A S T SQUARES F I T ORDER R E S U L T A N T PARAMETERS SUM OF <M,N) K F / B B/A SQUARES Z E R O - F I R S T ORDER 0 . 2 7 5 E + 0 1 0. 1 6 7 E + 0 2 0. 16 8 E -•0 1 ZERO-SECOND ORDER F I R S T - Z E R O ORDER 0 . 2 3 6 E + 0 0 0. 1 6 7 E + 0 2 0 . 1 6 8 E - •0 1 F I R S T - F I R S T ORDER 0. 132E+00 0. 1 6 7 E + 0 2 0 . 1 6 8 E - 0 1 F I R S T - S E C O N D ORDER 0 . 1 0 1 E + 0 0 0. 2 88E+02 0 . 2 2 2 E - 0 1 SECOND-ZERO ORDER 0 . 1 0 8 E - 0 1 0. 8 9 6 E + 0 1 0 . 1 0 5 E - 01 S E C O N D - F I R S T ORDER 0 . 8 0 8 E - 0 2 0. 1 03E+02 0 . 1 1 9 E - •0 1 SECOND-SECOND ORDER 0 . 6 8 8 E - 0 2 0. 140E+C2 0 . 1 5 1 E - 01 P H I (CC/CC) L (MM) L/D OM EG A (R AD/S EC) 0. 140 0.987 2 2 . 9 0 . 6 9 8 1 223 T A B L E F -• 14 VA R I O U S ORDERS (M,N) AND R E S U L T OF L E A S T SQUARES F I T ORDER R E S U L T A N T PARAMETERS SDM OF (M,N) K F / B B/A SQUARES Z E R O - F I R S T ORDER 0. 195E+01 0 . 7 3 2 E + 0 1 0 . 1 3 3 E - 0 2 ZERO-SECOND ORDER 0. 1 2 4 E + 01 0 . 1 7 7 E + 0 2 0 . 1 4 2 E - 0 2 F I R S T - Z E R O ORDER 0. 1 9 3 E + 0 0 0 . 7 3 2 E + 0 1 0. 1 3 3 E - 0 2 F I R S T - F I R S T ORDER 0 . 9 3 3 E - 0 1 0 . 7 3 2 E + 0 1 0 . 1 3 3 E - 0 2 F I R S T - S E C O N D ORDER 0. 7 1 9 E - 0 1 0 . 1 2 1 E + C 2 0. 1 3 6 E - 0 2 SECOND-ZERO ORDER 0 . 7 4 9 E - 0 2 0.4 97 E + 0 1 0. 1 2 4 E - 0 2 S E C O N D - F I R S T ORDER 0 . 5 1 8 E - 0 2 0 . 5 5 5 E + 0 1 0. 1 2 8 E - 0 2 SECOND-SECOND ORDER P H I {CC/CC} L (MM) L/D OMEGA (R A D / S E C ) 0 . 1 4 0 0 . 9 8 7 2 2 . 9 0 . 3 4 9 1 T A B L E F - 15 VARIOUS ORDERS (M, N) AND R E S U L T OF L E A S T SQUARES F I T ORDER R E S U L T A N T PARAMETERS SUM OF <H ,N ) K F / B B/A SQUARES Z E R O - F I R S T ORDER ZERO-SECOND ORDER F I R S T - Z E R O ORDER 0 . 1 7 1 E + 0 0 0 . 5 7 6 E + 0 1 0 . 1 7 1 E - 0 2 F I R S T - F I R S T ORDER 0 . 5 2 3 E - 0 1 0.576 E+C1 0 . 1 7 I E - 0 2 F I R S T - S E C O N D ORDER 0 . 3 8 3 E - 0 1 0 .1 05 E+ 02 0 . 1 7 4 E - 0 2 SECOND-ZERO ORDER 0 . 5 3 6 E - 0 2 0. 4 7 2 E + 0 1 0 . 1 7 0 E - 0 2 S E C O N D - F I R S T ORDER 0.26 4 E - 0 2 0 . 5 2 1 E + 0 1 0 . 1 7 1 E - 0 2 SECOND-SECOND ORDER P H I ( C C / C C ) L(MM) L/D OMEGA(RAD/SEC) 0 . 1 4 0 0.987 2 2 . 9 0.1 164 A p p e n d i x G D e c a y M o d e l s G - l . E v a l u a t i o n o f E x p r e s s i o n s P r e d i c t i n g S h e a r S t r e s s D e c a y I n S e c t i o n 2-2 v a r i o u s e x p r e s s i o n s f o r s t r u c t u r a l d e c a y o f t i m e -d e p e n d e n t m a t e r i a l h a v e b e e n g i v e n . I n t h i s s e c t i o n t h e y a r e c o m p a r e d i n t e r m s o f s h e a r s t r e s s w i t h t o r q u e - d e c a y c u r v e s m e a s u r e d i n t h e p r e s e n t e x p e r i m e n t . T h e m o d e l a t t r i b u t e d t o H a h n , R e e , a n d E y r i n g ^ ^ h a s b e e n s h o w n i n e q u a t i o n (2-1) a s T = ( x 0 - T ) e " a t + x (2-1) u co oo T h e t h e o r y o f D e n n y a n d B r o d k e y w a s e x p r e s s e d i n t e r m s o f p o i n t v i s c o s i t y ( s l o p e o f t h e s h e a r s t r e s s v s . s h e a r r a t e c u r v e ) . I t w a s i m p o s s i b l e t o c o n v e r t t h e d a t a o f t h e p r e s e n t e x p e r i m e n t i n t o p o i n t v i s c o s i t y , s o t h a t t h e i r t h e o r y c o u l d n o t b e e v a l u a t e d h e r e . T h e (41) m o d e l d e s c r i b e d b y P i n d e r h a s b e e n g i v e n i n t e r m s o f a p p a r e n t v i s c o s i t y ; f o r e x a m p l e , h i s s e c o n d o r d e r - z e r o o r d e r e q u a t i o n a s ( n e + n ) ( n ~ n e ) 2  1 0 8 1 0 ( n - n e ) ( n e + n y ) = 2 ^ 0 3 "eV ( 2 " 5 ) A p p a r e n t v i s c o s i t y i s r e l a t e d t o s h e a r s t r e s s b y e q u a t i o n ( 3 - 6 ) , i . e . , n = T/Y ( 3 - 6 ) H e n c e , t h e e q u a t i o n ( 2 - 5 ) c a n b e r e - w r i t t e n u n d e r a c o n s t a n t r a t e o f s h e a r a s (T + T) (TO - T ) oo u oo N l o g 1 0 7 r-j r = k t G - l ) < T - T O O ) ( T O Q + T 0 ) w h e r e k = 2 3 0 3 T kf ( G _ 2 ) F o r t h e o t h e r o r d e r o f r e a c t i o n s , s i m i l a r e x p r e s s i o n s c a n b e 2 2 5 226 o b t a i n e d w h i c h h a v e t h e f o l l o w i n g f o r m l o g i 0 [ f ( T ) ] = c t ( G - 3 ) ( 4 3 ) T h e m o d e l o f R i t t e r a n d G o v i e r h a s b e e n e x p r e s s e d i n t e r m s o f s t r u c t u r e s t r e s s i n e q u a t i o n ( 2 - 7 ) a n d c a n b e r e - w r i t t e n w i t h t h e c u r r e n t n o t a t i o n a s l n T — T T o T o - T-= - k A l n ( — t + 1 ) ( G - 4 ) A TQ - T P 0 T 0 A c t u a l l y t e n e x p r e s s i o n s ( e i g h t e q u a t i o n s b y P i n d e r , o n e b y H a h n , R e e , a n d E y r i n g , a n d o n e b y R i t t e r a n d G o v i e r ) w e r e c o m p a r e d h e r e w i t h t h e e x p e r i m e n t a l d e c a y d a t a o b t a i n e d f o r t h e a r t i f i c i a l s l u r r y o f n y l o n f i b e r s ( p a r t i c l e c o n c e n t r a t i o n 0 . 0 8 , p a r t i c l e L/D r a t i o 6 9 . 8 ) i n t h e d i s p e r s i n g m e d i u m ( D e x + 1 . 0 m o l e N a C l i n 1 0% P E G - H ^ O ) w i t h a g i n g t i m e 2 0 h o u r s a n d c o n s t a n t a n g u l a r v e l o c i t y 0 . 2 3 3 r a d . / s e c . T h e v a l u e s o f TO = 4 . 0 0 x 1 0 t f d y n e s / c m 2 a n d = 2 . 9 8 x l 0 3 d y n e s / c m 2 w h i c h w e r e o b t a i n e d f r o m t h e d e c a y r u n w e r e s e t t o b e c o n s t a n t i n a l l t h e e q u a t i o n s a n d t h e s e t o f ( t , T ) d a t a f r o m t h e d e c a y c u r v e w a s f i t t e d b y e a c h o f t h e e q u a -t i o n s b y m e a n s o f t h e l e a s t s q u a r e m e t h o d e v a l u a t i n g t h e v a l u e ( s ) o f p a r a m e t e r ( s ) f o r e a c h o f t h e e q u a t i o n s . T h e c o m p u t e r p r o g r a m m e i s l i s t e d i n A p p e n d i x C . T h e r e s u l t s o f t h e l e a s t s q u a r e f i t a r e s h o w n i n T a b l e G - l . A s f a r a s P i n d e r ' s m o d e l s a r e c o n c e r n e d , i t i s f o u n d t h a t t h e s e c o n d o r d e r - z e r o o r d e r i s t h e b e s t b e c a u s e i t g i v e s t h e l e a s t ( 4 1 ) v a l u e o f sum o f s q u a r e s . T h i s c o n c l u s i o n a g r e e s w i t h P i n d e r , B r o w n ^ , a n d B r o w n a n d P i n d e r ^ \ T h e r e s u l t s f o r t h e o t h e r t w o m o d e l s ( H a r n , R e e , a n d E y r i n g [ H - R - E ] , a n d R i t t e r a n d G o v i e r [ R - G ] ) a r e a l s o s h o w n i n T a b l e G - l . F i g . G - l s h o w s a c o m p a r i s o n o f P i n d e r ' s s e c o n d o r d e r - z e r o o r d e r 227 TABLE G - l PARAMETERS OF VARIOUS DECAY MODELS (j) = 0 . 0 8 ( c c / c c ) , L = 3.01 (mm), L/D = 69.8, Q = 0.233 ( r a d / s e c ) DEX + 1 . 0 mole NaCl i n 10% PEG-H o0 R e a c t i o n Sum of Order Parameter Square 0 - l s t _2 ( s e c ) ( g / c m . s e c ) 7 . 9 0 x l 02 4 . 7 8 x l 0 _ 1 0 - 2 n d k f Y ( s e c ) - 2 ( g / c m . s e c ) 5 . 9 9 x l 02 1 . 52 l s t - 0 r k f ( s e c ) 2 . 1 4 x l 0 ~ 2 4 . 7 8 x l 0 _ 1 l s t - l s t 1 k f ( s e c ) ^ 1 . 9 8 x l 0 ~ 2 4 . 7 8 x l 0 - 1 l s t - 2 n d I k f ( s e c ) ^ 5 . 7 l x l 0 ~2 4 . 21X101 2 n d - 0 k f / Y ( g / c m . s e c ) 7 . 4 7 x l 0 ~7 1 . 0 5 x l 0 " 2 2 n d - l s t k f / Y ( g / c m . s e c ) "'" 7 . 4 1 x l 0 ~7 1 . 1 3 x l 0 ~ 2 2 n d - 2 n d k f / Y ( g / c m . s e c ) ^ 7 . 3 8 x l 0 ~7 1 . 2 2 x l 0 ~ 2 H - R - E * r a ( s e c ) ^ 2 . 1 4 x l 0 " 2 4 . 7 8 x l 0 _ 1 R - G * * 1 k^/Po ( - ) ( s e c ) 3 . 3 . 2 1 x l 0 - 1 "\ 1 9 x l 0 _ 1 J 9 . 8 2 x l 0 ~ 4 * M o d e l b y H a h n , R e e , a n d E y r i n g [ e q u a t i o n ( 2 - 1 ) ] * * M o d e l b y R i t t e r a n d G o v i e r [ e q u a t i o n (G-4)] Q I M o X o V 3 CD i-i fu H- H 3 H-CD CO 3 O r t 3 I—1 co ° 2 CD s; O CD Bi CD VI 3 CJ CO P r t r t i-i &) CD cn cn o CD o Cu • V! ^ w. i-i CD cn cn - H-o 3 cn cu p . o o 3 M LOG10 SHEAR STRESS (DYNES / SQ.CM.) 3.362 3.632 3.903 4.173 4.444 4. J ^ I to CD o -Ho m « o c n LO" O I O x CD o < CD* TJ Q. CD m —» m m m J Q -Q a C c a Q Q —*-o O o" -—« -—- , „ CD | CD ro 1 i m X T3 CD 3 CD 3 o Q m o P .. O CD CO CD ro • OJ O w 3 — , o s « o „ o CD _ O ~ 5 TJ m CD i X ro O co o o o o 2ZZ 229 model, Hahn, Ree, and E y r i n g ' s model, and R i t t e r and G o v i e r ' s model w i t h t h e e x p e r i m e n t a l decay d a t a . P i n d e r ' s model and H-R-E model a r e t h e one-parameter models, w h i l e R-G model i s the two-parameter model. One t h i n g common among t h e s e t h r e e models, however, i s the assumption of s t r e s s decay under a c o n s t a n t shear r a t e . As mentioned e a r l i e r i n Chapter 3 the e s t i m a t i o n o f t r u e s h e a r r a t e f o r Cou e t t e v i s c o m e t e r i s a v e r y d i f f i c u l t t a s k . Hence, many workers have employed i n s t r u m e n t s o f t h e C o u e t t e type w i t h a t h i n a n n u l a r gap between c y l i n d e r s , and they have s i m p l y assumed i n t h e i r experiment t h a t the shear r a t e i s c o n s t a n t over the a n n u l a r space throughout the time. The v a l i d i t y o f t h i s assumption i s d o u b t f u l f o r the p r e s e n t experiment, because t h e t h i c k n e s s of the f l o w i n g l a y e r was found t o change w i t h time i n the v i s c o m e t e r . T h e r e f o r e , i t i s not f a i r t o d e f i n i t e l y say from F i g . G - l which one of t h e e x p r e s s i o n s i s the b e s t . The a n g u l a r v e l o c i t y o f the i n n e r c y l i n d e r , however, was kept c o n s t a n t throughout t h e decay experiment, and the shear s t r e s s e x e r t e d on t h a t c y l i n d e r was measured w i t h time. I f the assumption o f c o n s t a n t s h e a r r a t e i n the models examined here can be expanded to t h a t of c o n s t a n t a n g u l a r v e l o c i t y , t h e r e f o r e , R i t t e r and G o v i e r ' s model seems t o have an advantage over t h e o t h e r s t r e s s decay models. In b r i e f , t h e main d i f f i c u l t y w i t h the s t r e s s decay models i s t h a t they i n t r o d u c e t h e c o n s t a n t t r u e shear r a t e which cannot e a s i l y be kept c o n s t a n t i n the decay experiment. The p r e d i c t i o n by a l l the models, however, a r e o f the c o r r e c t o r d e r of magnitude and can p o s s i b l y be f u r t h e r improved by i n t r o d u c i n g an a p p r o p r i a t e measure of shear r a t e . 2 3 0 G - 2 . A M o d i f i c a t i o n t o R i t t e r a n d G o v i e r ' s D e c a y M o d e l ( 4 3 ) A s d i s c u s s e d e a r l i e r i n S e c t i o n 2 - 2 R i t t e r a n d G o v i e r a s s u m e d a f i r s t o r d e r - s e c o n d o r d e r r e v e r s i b l e r e a c t i o n f o r t h e d e c a y e q u a t i o n . T h e i r b a s i c r a t e e q u a t i o n w a s d t P 0 + H t [ k f T - \ ( T Q " T )2 ] ( G - 5 ) w h e r e T Q i s t h e i n i t i a l v a l u e o f T , k^ a n d k^ a r e t h e f o r w a r d a n d b a c k w a r d r a t e c o n s t a n t s , r e s p e c t i v e l y , k^ i s t h e r a t e o f i n c r e a s e o f n e t w o r k s t r u c t u r e , a n d PQ i s t h e i n i t i a l c o n c e n t r a t i o n o f n e t w o r k s t r u c t u r e . T h e y f o u n d t h a t t h e f i r s t o r d e r - s e c o n d o r d e r m o d e l f i t s t h e i r e x p e r i m e n t a l d a t a o n P e m b i n a c r u d e o i l . H o w e v e r , i t i s n o t r e a s o n a b l e t o a s s u m e t h e o r d e r s o f r e v e r s i b l e r e a c t i o n a priori a n d t h e o r d e r s m u s t b e d e t e r m i n e d e m p i r i c a l l y b y e x p e r i m e n t a l d a t a . T h e r e -f o r e e q u a t i o n ( G - 5 ) i s now m o d i f i e d t o t a k e i n t o a c c o u n t v a r i o u s o r d e r s a s d r = _ d t P 0 ( G - 6 ) w h e r e m i s t h e o r d e r o f f o r w a r d r e a c t i o n a n d n i s t h e o r d e r o f b a c k w a r d r e a c t i o n . E q u a t i o n ( G - 6 ) i s a d i f f e r e n t i a l e q u a t i o n a n d c a n b e s o l v e d a n a l y t i c a l l y w i t h v a r i o u s o r d e r s o f m a n d n . T h e s o l u t i o n s a r e 1 . (m, n ) = ( 0 , 0 ) T = c o n s t f o r a n y t (m, n ) = ( 0 , 1 ) ( G - 7 ) i i . l o g T — T i i i . (m, n ) = ( 0 , 2 ) T + T . - , 2 t n Tn - T kit l o g t + 1 ( G - 8 ) l o g T - T T 0 " T™ K l o g t + 1 ( G - 9 ) 231 i v . (m, n) = ( 1 , 0) v . v x . l o g T - T | T 0 - T o (m, n) = ( 1 , 1) k4 l o g t + 1 (G-10) l o g T - T Tn " T Tg - T kL l o g t + 1 ( G - l l ) (m, n) = ( 1 , 2) l o g T - T T 0 - T-+ T T 0 - T l o g t + 1 (G-12) T 0 T h i s i s t h e o r i g i n a l mode l by R i t t e r and G o v i e r a n d ' h a s been . g i v e n i n e q u a t i o n ( 2 - 7 ) . v i i . (m, n) = ( 2 , 0) l o g T — T TO + T T + T t ( i " T 1 oo u v i i i . (m, n) = ( 2 , 1) 2 x 0 + b + D = - 2 T l o g oo t + 1 l o g where 2 T + b - D 2 T + b + D 2 T q + b - D l o g ~ — t + 1 b = (G-13) (G-14) (G-15) and c = -T ( 2 x n - x D = / b 2 - 4c = Tn - T i x . (m, n) = ( 2 , 2) l o g 2a-_ + b - - D 2 a T n + b + D 2 a T + b + D 2 a x n + b - D l o g t + 1 (G-16) (G-17) (G-18) 2 3 2 w h e r e 2 a = ( T 0 " T . . ) : - 1 ( G - 1 9 ) . T 2 b = - 2 — ^ - t 0 ( G - 2 0 ) _ \ 2 0 ( x n " T ) 2 T 2 c = - T 0 2 ( G - 2 1 ) ( T 0 " T ) 2 U CO 2 T T 0 a n d D = / b 2 - 4 a c = ( G - 2 2 ) T 0 " T R O A l l o f t h e s o l u t i o n s a b o v e a r e o b t a i n e d w i t h t h e b o u n d a r y c o n d i t i o n s o f T = T O a t t = 0 a n d T = T a t t = t . CO CO I n v i e w o f t h e c o n c e p t s o f r e a c t i o n r a t e t h e o r d e r o f t h e r e a c t i o n (m, n ) i s n o t n e c e s s a r i l y a n i n t e g e r , b u t a n y r e a l n u m b e r c a n p o s s i b l y b e a c c e p t a b l e . F o r t h e s a k e o f m a t h e m a t i c a l s i m p l i c i t y , t h e e i g h t d i f f e r e n t o r d e r s o f (m, n ) f r o m ( 0 , 1 ) t o ( 2 , 2 ) h a v e b e e n c o n s i d e r e d h e r e . I t i s i n t e r e s t i n g t o n o t e t h a t a l l t h e e q u a t i o n s a r e o f t h e f o r m o f k f l o g | f ( x ) | = K £ - l o g k ^ p 0 t + 1 ( G - 2 3 ) w h e r e K i s a c o n s t a n t . E a c h e q u a t i o n h a s t w o f i t t i n g p a r a m e t e r s ( k ^ / k ^ ) a n d ( k^/Pg) . T h e s e t w o p a r a m e t e r s w e r e e v a l u a t e d b y m e a n s o f t h e l e a s t s q u a r e m e t h o d , m i n i m i z i n g t h e sum o f s q u a r e s . T h e c o m p u t e r p r o g r a m m e i s g i v e n i n A p p e n d i x C . T a b l e G - 2 s h o w s a n e x a m p l e o f t h e r e s u l t s o f t h e f i t . I t w a s f o u n d t h a t t h e s e c o n d o r d e r - z e r o o r d e r m o d e l i s b e t t e r t h a n a n y o t h e r 233 TABLE G-2 VARIOUS ORDERS OF GOVIER'S MODEL AND RESULTS OF LEAST SQUARES FIT „ , Resultant Parameters r 2 1-m -1 (m, n) k /kit (dyne/cm ) kit/Po (sec) Sum of Squares 0 - l s t 1.34X10 4 3 . 0 3 x l 0 _ 1 8.86xl0~ 4 0-2nd 8 . 4 3 x l 0 3 5 . 2 0 X 1 0 - 1 2 . 2 3 x l 0 ~ 3 l s t - 0 3 . 6 2 x l 0 _ 1 3 . 0 3 X 1 0 " 1 8 . 8 6 X 1 0 " 4 l s t - l s t 3 . 3 5 X 1 0 " 1 3 . 0 3 X 1 0 " 1 8 . 8 6 X 1 0 " 4 1st-2nd 3 . 2 1 X 1 0 " 1 3 . 1 9 X 1 0 " 1 9 . 8 2 X 1 0 " 4 2nd-0 1.83xl0" 5 1 . 2 5 X 1 0 " 1 2.98xl0~ 5 2nd-lst 1 . 8 1 X 1 0 " 5 1 . 2 6 X 1 0 " 1 3 . 1 8 X 1 0 " 5 2nd-2nd 1 . 7 9 x l 0 ~ 5 1 . 2 7 X 1 0 " 1 3 . 4 1 x l 0 ~ 5 <j) = 0.08 ( c c / c c ) , L = 3.01 (mm), L/D = 69.8, Q = 0.233 (rad./sec.) DEX +1.0 mole NaCl i n 10% PEG-H90 o r d e r s i n c l u d i n g t h e o r i g i n a l R i t t e r a n d G o v i e r ' s f i r s t o r d e r - s e c o n d o r d e r m o d e l l i s t e d e a r l i e r i n T a b l e G - l a s R - G . S i m i l a r r e s u l t s w e r e o b t a i n e d f o r a l m o s t a l l t h e o t h e r e x p e r i m e n t a l r u n s c a r r i e d o u t i n t h i s w o r k . T h e y a r e l i s t e d i n A p p e n d i x H. I t c a n b e c o n c l u d e d t h a t , i n t e r m s o f t h e sum o f s q u a r e s , t h e s e c o n d o r d e r - z e r o o r d e r [ ( m , n ) = ( 2 , 0 ) ] m o d e l g i v e s t h e b e s t a p p r o x i m a t i o n i n p r e d i c t i n g t h e s h e a r s t r e s s d e c a y f o r t h e t i m e - d e p e n d e n t a r t i f i c i a l s l u r r i e s . T h e s e c o n d o r d e r - z e r o o r d e r d e c a y e q u a t i o n w a s s h o w n i n e q u a t i o n ( G - 1 3 ) , i . e . , i T — T Tn + T CO V c l o g T + T T[) " T no u o k f = - 2 T T1- l o g °° k^ t + 1 ( G - 1 3 ) 2 - 1 - 1 w h i c h h a s t w o f i t t i n g p a r a m e t e r s , k^/^ . ( d y n e / c m ) a n d k [ + / P Q ( s e c ) F i g . G - 2 s h o w s t h e e f f e c t o f t h e p a r a m e t e r ( k ^ / k ^ ) o n t h e m o d e l e q u a t i o n ( G - 1 3 ) , w h e r e t h e o t h e r p a r a m e t e r ( k ^ / P g ) w a s k e p t c o n s t a n t [ k ^ / P g = 5 . 0 ] . T h e p a r a m e t e r ( k ^ / k ^ w a s v a r i e d f r o m 1 . 0 x 1 0 ~* t o 4 . 0 x 1 0 T h e v a r i o u s c o n s t a n t s i n t h e e q u a t i o n w e r e s e t a s f o l l o w s : T 0 = l . O x l O 4 d y n e / c m 2 -Q 2 a n d T = l . O x l O 3 d y n e / c m . A l l f o u r c u r v e s d e c a y f r o m t h e common y i e l d s t r e s s Tg a n d a p p r o a c h a s y m p t o t i c a l l y t h e common e q u i l i b r i u m s t r e s s T a t t = t ^ . I t c a n b e s e e n f r o m t h e f i g u r e t h a t t h e g r e a t e r t h e p a r a m e t e r ( k ^ / k t ^ ) - , t h e f a s t e r i s t h e r a t e o f s t r e s s d e c a y . A s i m i l a r p l o t i s s h o w n i n F i g . G - 3 , w h e r e t h e p a r a m e t e r ( k ^ / k ^ ) w a s k e p t c o n s t a n t ( k ^ / B = 3 x 1 0 "*) a n d t h e o t h e r p a r a m e t e r ( k ^ / P g ) w a s v a r i e d f r o m 0 . 0 5 t o 5 0 . 0 . T h e v a l u e s o f Tn a n d T a r e t h e s ame a s t h o s e i n F i g . G - 2 . I t c a n b e s e e n t h a t t h e u oo _ , g r e a t e r t h e p a r a m e t e r ( k ^ / P g ) , t h e f a s t e r i s t h e r a t e o f s t r e s s d e c a y . By c o m p a r i n g F i g . G - 2 w i t h F i g . G - 3 i t c a n b e c o n c l u d e d t h a t t h e £ k 4 / P 0 = 5 . 0 ( s e c 1 ) T 0 = 1.0 x I O 4 ( d y n e s / c m 2 ) Tco= I.Ox IO 3 ( d y n e s / c m 2 ) ^ ^ ^ ^ ^ + k f / k 4 = 1.0 x IO 5 (dyne,/ c m 2 ) " 1 x 2 . 0 xlO"5 0 3.0 x l d ^ *f 4 . 0 x l O 5 x 1 1 1 1 1 ' 1 -10 .0 7.143 24.286 41.429 58.571 75.714 92.657 110 TIME (SEC) F i g . G-2. E f f e c t o f P a r a m e t e r ( k f / k ^ ) o n t h e D e c a y M o d e l + X "+++-k f /k = 3x I0" 5 (dyne / cm 2)" T 0 = I.Ox I 0 4 ( dynes / cm 2 ) Tco= I .Ox l0 3 (dynes/cm 2 ) + k 4 / P 0 = 0 . 0 5 (sec" 1 ) X 0 . 5 O 5 . 0 "r 5 0 . 0 X ~l •7.143 ~1 24.286 T T "I 75.714 -T 92.857 -10.0 41.429 58.571 TIME ISEC) 110 Fig. G-3. Effect of Parameter (k^/V ) on the Decay Model p a r a m e t e r (k^ /ki+) i s m u c h m o r e s e n s i t i v e t o t h e r a t e o f s t r e s s d e c a y t h a n t h e p a r a m e t e r (kij/Po). T h i s i s b e c a u s e t h e p a r a m e t e r (k^/k^) i s a n e x p o n e n t o f t h e t - t e r m i n e q u a t i o n ( G - 1 3 ) , w h i l e t h e p a r a m e t e r (k^/Pg) i s j u s t a m u l t i p l i e r o f t h e t . F i g . G - 4 a n d G - 5 s how t h e e f f e c t o f a n g u l a r v e l o c i t y o n t h e f i t t i n g p a r a m e t e r s (k^/k^) a n d (k ^ / P n ) , r e s p e c t i v e l y . I t i s f o u n d f r o m t h e s e f i g u r e s t h a t t h e h i g h e r t h e a n g u l a r v e l o c i t y , t h e g r e a t e r a r e t h e p a r a m e t e r s (k^/kit) a n d (ki+/Po). F i g . G - 6 a n d G - 7 s how t h e e f f e c t o f p a r t i c l e c o n c e n t r a t i o n a n d p a r t i c l e L/D r a t i o o n t h e f i t t i n g p a r a m e t e r s , k^/k^, a n d k^/Po, r e s p e c t i v e l y . T h e l a r g e s c a t t e r i n t h e d a t a may b e e x p l a i n e d b y t h e e r r o r s a s s o c i a t e d w i t h t h e e x p e r i m e n t a l r e p r o d u c i b i l i t y . I t c a n b e c o n c l u d e d f r o m t h e s e f i g u r e s t h a t t h e h i g h e r t h e p a r t i c l e c o n c e n t r a -t i o n a n d L/D r a t i o , t h e s m a l l e r a r e t h e p a r a m e t e r s (kp/ki+ • a n d ki+/Po) S e c o n d O r d e r - Z e r o O r d e r 2 -I -4 P a r a m e t e r k f / k 4 ( d y n e / c m ) ( x l O ) > 3 C 5* P -? ro cf O o co CD o o *""-** CD Second Order-Zero Order Parameter k 4 / P 0 (sec) ( x , r f c , c —r- oo " T ~ ro ~T~ CD O ro O O ° o p + " CD "j> o ro oo ° ^ « 3 2. a O O TJ m o i X PO o o O S e c o n d O r d e r - Z e r o O r d e r 2 -I -5 P a r a m e t e r k f / k 4 ( d y n e / c m ) ( x l O ) 2 4 1 I X J c o o if) o X v T •o O o N I w Q> o CO o r 0 ) £ o o a. • L= 0.987 (mm) & = 0.233 (rad./sec.) DEX + 1.0 mole NaCl in 10% P E G - H 2 0 O L/D = 22.9 • 37.4 69.8 \ \ • \ \ o G \ \ Q 0.05 0.10 0.15 P a r t i c l e C o n c e n t r a t i o n ( c c / c c ) F i g . G-7. E f f e c t s o f P a r t i c l e C o n c e n t r a t i o n a n d P a r t i c l e L/D R a t i o o n t h e P a r a m e t e r ( k . / P » ) A p p e n d i x H V a r i o u s O r d e r s o f M o d i f i e d R i t t e r a n d G o v i e r ' s D e c a y M o d e l a n d Sum o f S q u a r e s A l l t h e d a t a i n t h e s e t a b l e s a r e f o r t h e d i s p e r s i n g m e d i u m o f d e x t r o s e +1.0 m o l e N a C l i n 10% P E G - H 20. D e s c r i p t i o n o f V a r i a b l e s i n T a b l e H P H I ( C C / C C ) - P a r t i c l e C o n c e n t r a t i o n ( c c / c c ) L (MM) - P a r t i c l e L e n g t h (mm) L/D - P a r t i c l e L e n g t h - t o - D i a m e t e r R a t i o ( - ) OMEGA(RAD/SEC) - A n g u l a r V e l o c i t y o f I n n e r C y l i n d e r ( r a d . / s e c . ) 242 T A B L E H - 1 VARIOUS ORDERS OF G O V I E R MODEL AND RESULT O F L E A S T SQUARES F I T ORDER RES U L T A N T P A R A M E T E R S SUM OF K F / K 4 K 4 / P 0 SQUARES Z E R O - F I R S T ORDER 0 . 5 9 8 E + 0 5 0.2 84E+00 0 . 8 7 2 E - 0 1 ZERO-SECOND ORDER 0 . 2 8 8 E + 0 5 0 . 5 2 7 E + 0 0 0 . 7 5 5 E - 0 1 F I R S T - Z E R O ORDER 0. 1 5 1 E + 01 0.2 84E+00 0 . 8 7 2 E - 0 1 F I R S T - F I R S T ORDER 0. 128E+01 0.2 84E+00 0 . 8 7 2 E - 0 1 F I R S T - S E C O N D ORDER 0 . 1 1 0 E + 0 1 0 . 3 2 2 E + 0 0 0 . 8 4 4 E - 0 1 SECOND-ZERO ORDER 0, 1 1 6 E - 0 3 0. 8 8 4 E - 01 0.9 9 5 E - 0 1 S E C O N D - F I R S T ORDER 0 . 1 0 6 E - 0 3 0 . 9 3 3 E - 0 1 0.1O0E+00 SECOND-SECOND ORDER 0 . 9 7 0 E - 0 4 0. 1 0 0 E + 0 0 0. 1 0 0 E + 0 0 P H I (CC/CC) L (MM) 0 . 1 6 0 0.987 L/D OMEGA ( R A D / S E C ) 2 2 . 9 0 . 2 3 2 7 TABLE H - 2 VARIOUS ORDERS OF GOVIER MODEL AND RESULT OF LEAST SQUARES FIT ORDER RESULTANT PARAMETERS SUM OF KF/K4 K4/P0 SQUARES ZERO-FIRST ORDER 0. 129E+05 0.2 95E+0 0 0.274E-0 1 ZERO-SECOND ORDER 0.669E+04 0.589E+00 0.398E-01 FIRST-ZERO ORDER 0.803E+00 0.295E+00 0.274E-01 FIRST-FIRST ORDER 0.753E+00 0.295E+00 0.274E-01 FIRST-SECOND ORDER 0. 71 1E+00 0.314E+00 0.283E-01 SECOND-ZERO ORDER 0.338E-03 0.319E-01 0.497E-02 SECOND-FIRST ORDER SECOND-SECOND ORDER 0.317E-03 0.341E-01 0.538E-02 PHI (CC/CC) L (MM) L/D OMEGA (RAD/SEC) 0.150 0.987 22.9 0.2327 T A B L E fl - 3 VA R I O U S ORDERS OF G O V I E R MODEL AND R E S U L T OF L E A S T SQUARES F I T ORDER R E S U L T A N T P A R A M E T E R S SUM OF (M,N) K F / K 4 K 4 / P 0 SQUARES Z E R O - F I R S T ORDER 0 . 4 7 1 E + 0 4 0 . 5 2 9 E + 0 0 0 . 5 9 3 E - 0 2 ZERO-SECOND ORDER 0 . 2 5 8 E + 0 4 0 . 1 0 3 E + 0 1 0. 1 3 1 E - 0 1 F I R S T - Z E R O ORDER 0 . 6 2 6 E + 0 0 0 . 5 2 9 E + 0 0 0 . 5 9 3 E - 0 2 F I R S T - F I R S T ORDER 0 . 6 0 6 E + 0 0 0 . 5 2 9 E + 0 0 0 . 5 9 3 E - 0 2 F I R S T - S E C O N D ORDER 0 . 5 9 0 E + 0 0 0 . 5 4 5 E + 0 0 0 . 6 1 9 E - 0 2 SECOND-ZERO ORDER 0 . 4 4 0 E - 0 3 0. 8 C 3 E - 0 1 0 . 5 4 5 E - 0 4 S E C O N D - F I R S T ORDER 0 . 4 3 7 E - 0 3 0. 8 0 8 E - 0 1 0 . 5 5 8 E - 0 4 SECOND-SECOND ORDER 0 . 4 3 5 E - 0 3 0. 8 1 2 E - 0 1 0. 5 7 3 E - 0 4 P H I (CC/CC) L (MM) 0 . 1 4 0 0.987 L/D OMEGA (RAD/SEC) 2 2 . 9 0 . 2 3 2 7 246 T A B L E H -- 4 VARIOUS ORDERS OF GOV I E R MODEL AND RESULT OF L E A S T SQUARES F I T ORDER R E S U L T A N T PARAMETERS SUM OF |M,N) K F / K 4 K 4 / P 0 SQOARES Z E R O - F I R S T ORDER 0 . 5 3 0 E + 0 3 0 . 3 4 9 E + 0 0 0.1 1 1 E - 0 1 ZERO-SECOND ORDER 0.32 5E+03 0 . 6 1 6 E + 0 0 0 . 2 2 1 E - 0 1 F I R S T - Z E R O ORDER 0 . 3 8 8 E + 0 0 0.3 4 9 E + 0 0 0 . 1 1 1 E - 0 1 F I R S T - F I R S T ORDER 0 . 3 5 1 E + 0 0 0 . 3 4 9 E + 0 0 0 . 1 1 1 E - 0 1 F I R S T - S E C O N D ORDER 0 . 3 3 1 E + 0 0 0. 3 7 4 E + 0 0 0 . 1 2 2 E - 0 1 SECOND-ZERO ORDER 0 . 5 3 5 E - 0 3 0. 1 2 2 E + 0 0 0. 2 0 0 E - 0 2 S E C O N D - F I R S T ORDER 0 . 5 2 6 E - 0 3 0. 135E+ CO 0 . 1 1 7 E - 0 2 SECOND-SECOND ORDER 0 . 5 1 9 E - 0 3 0 . 1 3 8 E + 0 0 0 . 1 2 6 E - 0 2 P H I ( C C / C C ) L (MM) 0 . 1 3 0 0 . 9 8 7 L/D OMEGA(RAD/SEC) 2 2 . 9 0.2 327 2 4 7 T A B L E H - 5 V A R I O U S ORDERS OF GOV I E R MODEL AND R E S U L T OF L E A S T SQUARES F I T ORDER R E S U L T A N T PARAMETERS SUM OF K F / K 4 K 4 / P 0 SQUARES Z E R O - F I R S T ORDER 0 . 2 9 6 E + 0 5 0 . 4 2 9 E + 0 0 0 . 1 0 6 E - 0 2 ZERO-SECOND ORDER 0 . 1 4 6 E + 0 5 0 . 8 6 1 E + 0 0 0 . 1 6 0 E - 0 2 F I R S T - Z E R O ORDER 0.11OE+01 0 . 4 2 9 E + 0 0 0. 1 0 6 E - 0 2 F I R S T - F I R S T ORDER 0 . 9 7 7 E + 0 0 0 . 4 2 9 E + 0 0 0 . 1 0 6 E - 0 2 F I R S T - S E C O N D ORDER 0 . 8 7 7 E + 0 0 0 . 4 7 7 E + 0 0 0. 1 1 3 E - 0 2 SECOND-ZERO ORDER 0.18 1 E - 0 3 0 . 7 9 8 E - 0 1 0 . 2 3 7 E - 0 3 S E C O N D - F I R S T ORDER 0, 16 9 E - 0 3 0 . 8 4 1 E - 0 1 0 . 2 5 2 E - 0 3 SECOND-SECOND ORDER 0. 1 5 9 E - 0 3 0. 8 9 5 E - 01 0 . 2 7 1 E - 0 3 P H I ( C C / C C ) 0. 120 L (MM) 1.620 L/D 3 7 . U OMEGA (RAD/S EC) 0 . 2 3 2 7 248 TABLE H -• 6 VARIOUS OBDEBS OF GOVIER MODEL AND RESULT OF LEAST SQUARES FIT ORDER RESULTANT PARAMETERS SUM OF (H,N) KF/K4 K 4/P 0 SQUARES ZERO-FIRST ORDER 0.599E+04 0. 3 94E+00 0.337E-02 ZERO-SECOND ORDER 0.322E+04 0.778E+00 0.843E-02 FIRST-ZERO ORDER 0.664E+00 0.3 94E+00 0.337E-02 FIRST-FIRST ORDER 0.603E+00 0.3 94E+00 0.337E-02 FIRST-SECOND ORDER 0.558E+00 0, 429E+00 0.386E-02 SECOND-ZERO ORDER 0.248E-03 0.843E-01 0. 1 12E-03 SECOND-FIRST ORDER 0.239E-03 0. 869E-01 0. 118E-03 SECOND-SECOND ORDER 0. 231E-03 0.9 01 E-01 0. 125E-03 PHI (CC/CC) 0. 110 L (MM) 1. 620 L/D 37. 4 OMEGA(RAD/SEC) 0.2 327 249 TABLE H - 7 VARIOUS ORDERS OF GOVIER MODEL AND RESULT OF LEAST SQUARES FIT ORDER RESULTANT PARAMETERS SUM OF {M, N) KF/K4 K4/P0 SQUARES ZERO-FIRST ORDER 0. 156E+04 0. 501 E+00 0.444E- 0 2 ZERO-SECOND ORDER 0.911E+03 0. 972E+C0 0.118E- 02 FIRST-ZERO ORDER 0.423E+00 0. 5 01E+00 0.4 44E-•0 2 FIRST-FIRST ORDER 0.392E+00 0. 501E+00 0.444E- 02 FIRST-SECOND ORDER 0.373E+00 0. 533E+00 0.485E- 0 2 SECOND-ZERO ORDER 0.304E-0 3 0. 127E+00 0.673E- 03 SECOND-FIRST ORDER 0.299E-03 0. 129E+00 0.700E-•0 3 SECOND-SECOND ORDER 0. 295E-03 0. 131E+00 0.731E- 03 PHI (CC/CC) L (MM) L/D OMEGA (R AD/SEC) 0.100 1.620 37.4 0.2327 2 5 0 TABLE H - 8 VARIOUS ORDERS OF GOVIER MODEL AND RESULT OF LEAST SQUARES FIT ORDER RESULTANT PARAMETERS SUM OF (M,N) KF/K4 K4/P0 SQUARES ZERO-FIRST ORDER 0.114E+03 0. 496E+00 0.3 00E- 02 ZERO-SECOND ORDER 0.831E+02 0. 751E+00 0.907E-•0 2 FIRST-ZERO ORDER 0.169E+00 0. 496E+ GO 0.300E- 02 FIRST-FIRST ORDER 0.153E+00 0. 4 96E+00 0.300E- 0 2 FIRST-SECOND ORDER 0. 147E+00 0. 520E+00 0.349E- 02 SECOND-ZERO ORDER 0.306E-03 0. 297E+00 0.125E-•0 3 SECOND-FIRST ORDER 0.302E-03 0. 2 99E+00 0.135E- 03 SECOND-SECOND ORDER 0.300E-03 0. 3 01 E+00 0.148E- 0 3 PHI(CC/CC) L (MM) L/D OMEGA (RAD/SEC) 0.090 1.620 37.4 0.2327 2 5 1 T A B L E H - 9 VARIOUS ORDERS OF G O V I E R MODEL AND RESULT OF L E A S T SQUARES F I T ORDER R E S U L T A N T P A R A M E T E R S SUM OF (H.N) K F / K 4 ,K4/P0 SQUARES ZERO-FIRST ORDER 0 . 1 3 4 E + 0 5 0.303E+Q0 0 . 8 8 6 E - 0 3 ZERO-SECOND ORDER 0. 84 3E+04 0 . 5 2 0 E + 0 0 0 . 2 2 3 E - 0 2 F I R S T - Z E R O ORDER 0.36 2E+00 0.3 03E+00 0 . 8 8 6 E - 0 3 F I R S T - F I R S T ORDER 0 . 3 3 5 E + 0 0 0.3 03E+00 0 . 8 8 6 E - 0 3 F I R S T - S E C O N D ORDER 0 . 3 2 1 E + 0 0 0 . 3 1 9 E + 0 0 0 . 9 8 2 E - 0 3 SECOND-ZERO ORDER 0. 1 8 3 E - 0 4 0 , 1 2 5 E + 0 0 0 . 2 9 8 E - 0 4 S E C O N D - F I R S T ORDER 0 . 1 8 1 E - 0 4 0. 1 2 6 E + 0 0 0 . 3 1 8 E - 0 4 SECOND-SECOND ORDER 0. 1 7 9 E - 0 4 0. 1 2 7 E + 0 0 0.34 IE-0 4 P H I (CC/CC) 0. 0 8 0 L (MM) 3 . 0 1 0 L/D 6 9 . 8 OMEGA (RAD/S EC) 0 . 2 3 2 7 2 5 2 T A B L E H -• 10 V A R I O U S ORDERS OF G O V I E R MODEL AND R E S U L T OF L E A S T SQUARES F I T ORDER R E S U L T A N T PARAMETERS SOM OF K F / K 4 K 4 / P 0 SQUARES Z E R O - F I R S T ORDER 0 . 8 9 9 E + 0 4 0 . 5 8 1 E + 0 0 0.33 4 E - 0 2 ZERO-SECOND ORDER 0 . 5 5 1 E + 04 0. 1 0 5 E + 0 1 0 . 1 1 8 E - 0 1 F I R S T - Z E R O ORDER 0 . 3 6 7 E + 0 0 0 . 5 8 1 E + 0 0 0 . 3 3 4 E - 0 2 F I R S T - F I R S T ORDER 0 . 3 5 2 E + 0 0 0 .5 81 E+00 0 . 3 3 4 E - 0 2 F I R S T - S E C O N D ORDER 0 . 3 4 2 E + 0 0 0 . 6 0 0 E + 0 0 0 . 3 6 3 E - 0 2 SECOND-ZERO ORDER 0 . 3 8 4 E - 0 4 0 . 1 6 8 E + 0 0 0 . 1 3 9 E - 0 4 S E C O N D - F I R S T ORDER 0. 3 8 2 E - 0 4 0 . 1 6 9 E + 0 0 0 . 1 4 3 E - 0 4 SECOND-SECOND ORDER 0 . 3 8 1 E - 0 4 0. 1 7 0 E + 0 0 0 . 1 4 9 E - 0 4 P H I (CC/CC) 0 . 0 7 0 L (MM) 3.01 0 L/D OMEGA(RAD/SEC) 6 9 . 8 0 . 2 3 2 7 253 T A B L E H - 11 V A R I O U S ORDERS OF GO V I E R MODEL AND R E S U L T OF L E A S T SQUARES F I T ORDER R E S U L T A N T P A R A M E T E R S SUM OF <M,N) KP/K4 K 4 / P 0 SQUARES Z E R O - F I R S T ORDER 0 . 8 4 8 E + 0 4 0. 1 91 E+01 0 . 1 9 1 E - •0 1 ZERO-SECOND ORDER 0 . 4 6 9 E + 0 4 0. 3 99 E+01 0 . 4 0 5 E - 01 F I R S T - Z E R O ORDER 0 . 5 0 5 E + 0 0 0. 1 91 E+01 0 . 1 9 1 E - 0 1 F I R S T - F I R S T ORDER 0 . 4 9 0 E + 0 0 0. 191 E+01 0 . 1 9 1 E - 01 F I R S T - S E C O N D ORDER 0 . 4 7 9 E + 0 0 0. 1 97 E+01 0 . 1 9 8 E - 0 1 SECOND-ZERO ORDER 0 . 1 5 8 E - 0 3 0. 2 4 0 E + 0 0 0 . 7 4 4 E - 04 S E C O N D - F I R S T ORDER 0. 1 5 8 E - 0 3 0. 2 4 1 E + 0 0 0 . 7 6 1 E - •0 4 SECOND-SECOND ORDER 0 . 1 5 7 E - 0 3 0. 2 4 2 E + 0 0 0 . 7 7 9 E - 04 P H I ( C C / C C ) L(MM) L/D OMEGA(RAD/SEC) 0 . 0 6 0 3 . 0 1 0 6 9 . 8 0 . 2 3 2 7 254 T A B L E 3 - 12 VARIOUS ORDERS OF G O V I E R MODEL AND RESULT OF L E A S T SQUARES F I T ORDER RESULTANT PARAMETERS SUM OF (8,N) KF/K4 K4/P0 SQUARES Z E R O - F I R S T ORDER 0 , 4 5 0 E + 0 4 0 . 2 2 9 E + 0 1 0 . 2 2 9 E - 01 ZERO-SECOND ORDER 0 . 2 5 5 E + 0 4 0. 484E+ 01 0.4 9 0 E - 01 F I R S T - Z E R O ORDER 0 . 4 3 3 E + 0 0 0. 2 2 9 E + 0 1 0 . 2 2 9 E - 0 1 F I R S T - F I R S T ORDER 0.42 3E+00 0. 2 2 9 E + 0 1 0.22 9 E - 01 F I R S T - S E C O N D ORDER 0 . 4 1 5 E + 0 0 0. 2 3 4 E + 0 1 0 . 2 3 6 E - 0 1 SECOND-ZERO ORDER 0 . 1 9 6 E - 0 3 0. 2 9 5 E + 0 0 0.75 4 E - 04 S E C O N D - F I R S T ORDER 0. 1 9 6 E - 0 3 0. 2 9 6 E * 0 0 0 . 7 6 7 E - 0 4 SECOND-SECOND ORDER 0 . 1 9 5 E - 0 3 0. 2 9 7 E + 0 0 0 . 7 7 9 E - 04 P H I (CC/CC) L (MM) L/D OMEGA (R AD/S EC) 0 . 0 5 0 3 . 0 1 0 6 9 . 8 0 . 2 3 2 7 T A B L E H - 13 VARIOUS ORDERS OF GOVIER MODEL AND RE S U L T O F L E A S T SQUARES F I T ORDER R E S U L T A N T PARAMETERS SUM OF <H,N) K F / K 4 K 4 / P 0 SQUARES Z E R O - F I R S T ORDER ZERO-SECOND ORDER 0 . 1 3 7 E + 0 4 0 . 1 9 3 E + 0 1 0 . 2 6 2 E - 0 1 F I R S T - Z E R O ORDER F I R S T - F I R S T ORDER F I R S T - S E C O N D ORDER SECOND-ZERO ORDER 0. 7 5 6 E - 0 3 0. 1 5 4 E + 0 0 0 . 2 4 2 E - 0 2 S E C C N D - F I R S T ORDER 0. 7 3 7 E - 0 3 0, 1 5 8 E + 0 0 0 . 2 4 8 E - 0 2 SECOND-SECOND ORDER 0 . 7 2 0 E - 0 3 0. 1 6 2 E + 0 0 0. 2 5 5 E - 0 2 P H I ( C C / C C ) L(MM) L/D OMEGA(RAD/SEC) 0 . 1 4 0 0.987 2 2 . 9 0 . 6 9 8 1 2 5 6 T A B L E H - 14 VA R I O U S ORDERS OF GOV I E R MODEL AND R E S U L T OF L E A S T SQUARES F I T ORDER R E S U L T A N T P ARAMETERS SUM OF (M,N) K F / K 4 K 4 / P 0 SQUARES Z E R O - F I R S T ORDER 0. 2 6 5 E + 04 0. 5 1 8 E + 0 0 0. 2 2 8 E -•0 2 ZERO-SECOND ORDER 0 . 1 4 3 E + 0 4 0. 1 0 2 E + 0 1 0. 6 3 3 E -02 F I R S T - Z E R O ORDER 0 . 6 7 0 E + 0 0 0. 5 1 8 E + 0 0 0. 2 2 8 E -•0 2 F I R S T - F I R S T ORDER 0 . 6 2 8 E + 0 0 0. 5 1 8 E + 0 0 0. 2 2 8 E - 0 2 F I R S T - S E C O N D ORDER 0 . 5 9 5 E + 0 0 0, 5 4 9 E + 0 0 0. 2 5 2 E - 02 SECOND-ZERO ORDER 0 . 7 1 2 E - 0 3 0. 9 5 8 E - 0 1 0. 3 5 9 E -•04 S E C O N D - F I R S T ORDER SECOND-SECOND ORDER 0 . 6 8 6 E - 0 3 0. 9 9 3 E - 0 1 0. 3 9 5 E - 0 4 P H I ( C C / C C ) L(MM) L/D OMEGA(RAD/SEC) 0. 140 0.987 2 2 . 9 0 . 3 4 9 1 T A B L E H - 15 VA R I O U S ORDERS OF GOV I E R MODEL AND R E S U L T OF L E A S T SQUARES F I T ORDER R E S U L T A N T P A R A M E T E R S SUM OF (M,N) K F / K 4 K 4 / P 0 SQUARES • Z E R O - F I R S T ORDER 0 . 7 5 7 E + 0 4 0 . 4 5 5 E + 0 0 0 . 1 0 8 E - 0 1 ZERO-SECOND ORDER 0 . 4 1 1 E + 0 4 0. 8 9 9 E + 0 0 0 . 2 2 4 E - 0 1 F I R S T - Z E R O ORDER 0 . 6 2 8 E + 0 0 0 . 4 5 5 E + 0 0 0 . 1 0 8 E - 0 1 F I R S T - F I R S T ORDER 0 . 6 1 7 E + 0 0 0. 455E+ 00 0 . 1 0 8 E - 0 1 F I R S T - S E C O N D ORDER 0 . 6 0 8 E + 0 0 0 . 4 6 3 E + 0 0 0. 1 1 0 E - 0 1 SECOND-ZERO ORDER 0. 4 0 2 E - 0 3 0. 4 6 7 E - 0 1 0 . 1 7 5 E - C 4 S E C O N D - F I R S T ORDER 0 . 4 0 1 E - 0 3 0 . 4 6 9 E - 0 1 0 . 1 7 8 E - 0 4 SECOND-SECOND ORDER 0 . 4 0 0 E - 0 3 0. 4 7 0 E - 0 1 0. 1 8 1 E - 0 4 P H I ( C C / C C ) 0. 140 L (MM) 0.987 L/D OMEGA ( R A D / S E C ) 2 2 . 9 0 . 1164 258 ATOGAK I C h o s h a g a 1 9 7 1 - n e n 1 - g a t s u 2 0 - k a n i C a n a d a n i k i t e i r a i , mo h a y a i m o n o d e 7 - n e n a m a r i n i n a r i m a s u . S y u s h i k a t e i o s u g o s h i t a O n t a r i o - s h u H a m i l t o n no M c M a s t e r D a i g a k u d e w a , N o g a m i Y u k i h i s a s e n s e i o h a j i m e , o k u n o h i t o b i t o n i o s e w a n i n a r i m a s h i t a . T o k u n i , I s h i g e T o s h i y u k i h a k a s e n i w a g o l f n o t e h o d o k i o u k e , i m a d e w a y a t t o 8 0 - d a i o k i r o k u d e k i r u y o n i n a r i m a s h i t a . H a k a s e k a t e i n o UBC n i u t s u t t e k a r a w a , t o j i O k u r a S h o j i V a n c o u v e r S h i t e n c h o no T a k a m a t s u I k u , Y o s h i k o g o f u s a i n i , k a z o k u n o 1 - i n n o y o n i k a w a i g a t t e i t a d a k i , e i g a , k a n i t o r i , p i c n i c , g o l f , s o r e n i o k u n b p a r t y t o , s o n o g o o n w a i s s h o w a s u r e r u k o t o g a n a i d e s h o . M a t a , A m e r i c a W a s h i n g t o n - s h u T a c o m a n o T o s h i L e d e r e r s a n n i w a , l o n g w e e k e n d g o t o n i o s e w a n i n a r i , A m e r i c a d e s h i t a m a j a n wa t o k u b e t s u no o m o i d e n i n a r i m a s h i t a . J u j o S e i s h i n o M a e d a T o s h i h i r o s a n , T o k y o D a i g a k u n o F u k e S h i n y a h a k a s e , s o s h i t e JAL n o F u r u m o r i T o s h i a k i s a n t a c h i t o t a n o s h i n d a g o l f y a , K o n o T a k e s h i s a n , UBC n o M a t s u m o t o M o r i t a k a s e n s e i y a , W a s e d a D a i g a k u no H i r a t a A k i r a s e n s e i t a c h i t o n o t e n n i s m o , i w a y u r u V a n c o u v e r d e n o y u g a n a j i n s e i no , h i t o k o m a n i n a r i m a s h i t a . M a i s h u k i n y o b i n o y o r u , t e n k e i t e k i s h i r o t o n o c h o s h a n o y o n a mono n i m a d e , s h i n s e t s u t e i n e i n i h a s s e i o o s h i e t e i t a d a i t a V a n c o u v e r O p e r a A s s o c i a t i o n no S u z u k i R e i k o s e n s e i y a , Q u e e n E l i z a b e t h T h e a t r e / P l a y -h o u s e , o y o b i O r p h e u m T h e a t r e d e n o c o n c e r t d e , i s s h o n i s t a g e d e h i y a a s e o n a g a s h i t a V a n c o u v e r S a k u r a S i n g e r s n o member no m i n a s a m a n i m o , k o n o b a o k a r i t e o r e i m o s h i a g e m a s u . S u z i e C h u n g s a n , M i n a m i A k i k o s a n , I i d a T o s h i e s a n , s o s h i t e 2 5 9 H i g a s h i g a w a M i c h i k o s a n t a c h i n i w a , m a w a r i n i n a m i k a z e o t a t e m a s h i t a g a , a i s u r u k o t o t o a i s a r e r u k o t o n o y o r o k o b i , t s u r a s a , s o s h i t e k i b i s h i s a o o s h i e t e i t a d a k i m a s h i t a . C h o s h a no C a n a d a t a i z a i e h u , N i p p o n k a r a h i m p a n n i h a g e m a s h i n o t e g a m i o m o r a i m a s h i t a . H o r i e S u m i , K u n i h i k o , N o r i k o t a c h i t o , s e k a i j u n o m i n a t o . k a r a t e g a m i o k a i t e k u r e t a M i t s u i OSK L i n e s n o H o r i e N o r i h i k o n i , k o k o r o k a r a k a n s h a i t a s h i m a s u . N i n g e n g a j i n s e i d e s u r u k o t o o o k i k u w a k e r u t o , t s u g i n o 2 - t s u n i n a r u t o o m o w a r e m a s u ; ( 1 ) s h o h i k a t s u d o ( 2 ) s o s a k u k a t s u d o . O k u no h i t o b i t o n i " J i n s e i n o m o k u t e k i w a n a n i k a ? " a r u i w a " A n a t a no i k i g a i w a n a n i d e s u k a ? " t o k i k u t o , s o n o k o t a e w a t a i t e i k o n o 2 - t s u no k a t s u d o n o u c h i n o d o c h i r a k a n i n a r u y o d e s u . K o r e r a no d o c h i r a n o k a t s u d o n i m o s o r e z o r e n i y o r o k o b i y a m a n z o k u k a n j u j i t s u k a n g a a r i m a s u . K o f u k u k a n t o i t t e mo y o i k a m o w a k a r i m a s e n . S h i k a s h i , n a n i m o n o omo t s u k u r i d a s u k o t o n a k u o w a r u ( 1 ) n o s h o h i k a t s u d o h o d o m u n a s h i i mono w a n a i y o d e s u . S o n o s h o h i k a t s u d o n i w a k a i k a n g a t o m o n a u kamo w a k a r i m a s e n g a , s o r e w a h o n no s u - b y o k a n t s u z u k e b a y o i h o d e s u . Y a h a r i , n a n d e m o y o i k a r a t s u k u r i d a s u k o t o ( 2 ) n o s o s a k u k a t s u d o wa j i n s e i n o k a c h i , m o k u t e k i , k o f u k u t o o k a n g a e r u t o k i , m i n o g a s h i t e w a n a r a n a i m o n o d a t o y u k o t o n i , k o n o r o m b u n o k a i t e i t e k i g a t s u k i m a s h i t a . K o n o r o m b u n w a s o n o s o s a k u k a t s u d o n o k e k k a d e s u g a , n a n i m o r o m b u n o k a k u b a k a r i d e w a n a k u t e , t a n k a y a h a i k u o t s u k u r u k o t o m o , s h i o k a k u k o t o mo , e o k a k u k o t o mo , y a k i m o n o o y a k u k o t o m o , s h i r o t o no g a s s h o d a n g a c o n c e r t d e u t a o u t a u k o t o mo , s u b e t e r i p p a n a s o s a k u k a t s u d o d a t o o m o i m a s u . S o y u y o n a s o s a k u k a t s u d o g a o k a n e n i n a r u k a 260 d o k a wa b e t s u m o n d a i n i s h i t e , s o y u s o s a k u k a t s u d o c h u n i z a m m a i n i n a t t e i r u t o k i , m a t a s o y u s o s a k u k a t s u d o o n a s h i t o g e t a t o k i n i n o m i a j i w a u j u j i t s u k a n m a n z o k u k a n n i k o s o , h o n t o no i i r i i no j i n s e i n o i k i g a i g a a r u n o d e w a n a i d e s h o k a . 

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