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

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TIME-DEPENDENT  SHEAR FLOW OF  A R T I F I C I A L SLURRIES  '  by  MICHIHIKO|HORIE  ,  B . E n g . , Gumma U n i v e r s i t y , J a p a n , 1968 M . S c , N i i g a t a U n i v e r s i t y , J a p a n , 1970 M.Eng., M c M a s t e r U n i v e r s i t y , C a n a d a , 1972  A  THESIS  SUBMITTED IN.-PARTIAL F U L F I L L M E N T OF  . THE REQUIREMENTS DOCTOR OF  FOR THE DEGREE  OF  PHILOSOPHY in  THE FACULTY OF GRADUATE The  Department  We the  accept  of Chemical  this  required  thesis  STUDIES Engineering  as conforming t o  standard  THE U N I V E R S I T Y OF B R I T I S H March,  COLUMBIA  1978  ( c T ) M i c h i h i k o H o r i e , 1978  In p r e s e n t i n g t h i s  thesis  an advanced degree at  further  agree  fulfilment  of  the requirements  the U n i v e r s i t y of B r i t i s h Columbia,  the L i b r a r y s h a l l make i t I  in p a r t i a l  freely  available  for  I agree  r e f e r e n c e and  t h a t p e r m i s s i o n f o r e x t e n s i v e copying o f  this  of  i s understood that copying or  It  this thesis for financial  thesis  written permission.  Department of  The  Chemical  U n i v e r s i t y o f B r i t i s h Columbia  2075 Wesbrook P l a c e V a n c o u v e r , Canada V6T 1W5  Date  Engineering  March 23, 1978.  or  publication  gain s h a l l not be allowed without my  Michihiko  that  study.  f o r s c h o l a r l y purposes may be granted by the Head of my Department by h i s r e p r e s e n t a t i v e s .  for  Horie  ABSTRACT  An  e x p e r i m e n t a l m e t h o d was  dependent  slurries  of  elongated  v i s c o m e t e r w h i c h has and  the  stationary  consisted  of  dispersing  slurry shear  medium o f  the  well  found  layer  as  the  on  flows  increases  An  shear  the  time-variation  of  order-zero  experimental  the  equilibrium  two  rate  under  value of k,/B  of  be  standing. the  a part  of  shear,  d e c a y was  as  were c o r r e l a t e d  fraction  of  rotating  cylinder.  a  the  and  t i m e and  thickness  the  flowing  three  fitting  the  thickness  and  B/A.  of  layer  R  factors  of  flowing (R  a  the value.  (t)  as  slurries. for  (t).  the  A  the  the  model were  layer  R  , K../B, x°°  s l u r r y , and  R  constructed  with p a r t i c l e length-to-diameter  p a r t i c l e s i n the  of  equilibrium  fifteen  These f a c t o r s  at  cylinders  thickness  layer  of  curve.  model f i t t e d  the  The  rotating  decay  flowing  reversible reaction The  the  for  with  time-  between the  t y p e m o d e l was the  the  a  time-variation  approaches an  of  glycol  separately.  stress  that  i t  f i b e r s and  cylinder  gap  measured  of  nylon  The  inner  cylinder  studied;  polyethylene  examined  r B/A)  rotating  sized  time-  cylinder  whose e f f e c t s o n  recorded  thickness  order  parameters  of  of  reaction-rate  data w e l l .  the  only  the  on  wall  with  stress  empirical  each  viscometer  of  coaxial  inner  regularly  s l u r r y could  that  time-variation  second  phase of  a n g u l a r v e l o c i t y was was  between the  aqueous s o l u t i o n  the  exerted  viscometer  flowing The  of  characterizing  A m o d e l s l u r r y s y s t e m was  an  i n the  stress  It of  nature  for  particles in a  gap  sodium c h l o r i d e ,  gelled  constant  cup.  a dispersed  d e x t r o s e and dependent  a wide  developed  xoo  >  ar  >d  and  r ratio,  angular v e l o c i t y of  volume the  i i i  T A B L E OF CONTENTS  Chapter 1  INTRODUCTION  1  2  LITERATURE  4  3  2-1.  Experimental  2- 2.  Theoretical  THEORETICAL  Interpretations  4 11 16  3- 1.  Newton's P o s t u l a t e  16  3-2. 3-3. 3-4. 3-5. 3-6.  Bingham P l a s t i c s Power Law F l u i d s General F l u i d Simple Shear Flow Couette Flow ( C o a x i a l C y l i n d e r Viscometer) Complicating E f f e c t s i n Suspension Rheology 3-7-1. Continuum 3-7-2. Steady 3-7-3. Wall Effects 3-7-4. End E f f e c t s 3-7-5. Homogeneous 3-7-6. Incompressible 3-7-7. Isothermal 3-7-8. Laminar . . . . . 3-7-9. Taylor's Instability  18 18 19 20  D E F I N I T I O N OF THE PROBLEM AND METHOD OF ATTACK 4- 1. 4-2. 4-3. 4- 4.  5  Observations  BACKGROUND  3- 7.  4  SURVEY  Recommendations o f Other Workers D e f i n i t i o n o f the Problem System Chosen Method o f A t t a c k  EXPERIMENTAL 5- 1.  MATERIALS  AND TECHNIQUES  Materials 5-1-1. D i s p e r s e d Phase 5-1-2. D i s p e r s i n g Medium 5-1-3. Technique of Density Matching 5-1-4. Slurry Preparation o f Apparatus  22 28 29 30 30 32 33 33 33 33 34 . . . 35 35 36 37 37 39 39 39 43 . . . 43 44  5-2.  Description  5-3.  Torque C a l i b r a t i o n  55  5-4.  Experimental  57  Procedure  50  div  6  EXPERIMENTAL RESULTS AND D I S C U S S I O N  60  6-1. 6-2. 6-3. 6-4.  Reproducibility 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 . . . . 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 . 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 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 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 o n  60 62 62  Yield Stress 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 o n Y i e l d S t r e s s . . . . 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 o n E q u i l i b r i u m S t r e s s . E f f e c t of Angular V e l o c i t y of Inner C y l i n d e r  68  6-5. 6-6. 6-7. 6-8. 6- 9.  7  on Y i e l d S t r e s s E f f e c t s of P a r t i c l e Concentration, 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 Thickness of Flowing Layer . .  ANALYSES 7- 1. 7-2. 7-3. 7-4. 7-5.  •.  A Model P r e d i c t i n g Thickness of Flowing Layer D e t e r m i n a t i o n o f m and n i n t h e Model . . . . Two F i t t i n g P a r a m e t e r s ( k /B) a n d (B/A) o f the Model 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 Increase of Thickness of Flowing Layer . . 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 o n R a t e o f I n c r e a s e of Flowing Layer  7-6. 7-7.  Rheological Consideration of the Thickness of Flowing Layer P r a c t i c a l Use o f t h e Model  8  CONCLUSIONS  9  RECOMMENDATIONS  NOMENCLATURE  65  71 71 76  76 80  80 86 90 92  95 102 104 109 I l l 112  Roman L e t t e r s  112  Greek L e t t e r s M a t h e m a t i c a l Symbols and N o t a t i o n A b b r e v i a t i o n s and Other N o t a t i o n  114 115 116  BIBLIOGRAPHY  117  V  APPENDICES  122  A p p e n d i x A.  Flow Measurement w i t h D i f f e r e n t R a d i u s R of Inner C y l i n d e r at E q u i l i b r i u m  122  Appendix Appendix  B. C.  S u p p l e m e n t t o F i g . 6-6 i n S e c t i o n C o m p u t e r Programmes C-l. Programme " B E S T F I T " C-2. Programme " S U P L P L T " C-3. Programme "TAUDECAY"  126 131 131 139 148  Appendix  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 Results D-l. Shear S t r e s s a t t = 0 and t = t  C-4.  D-2.  Programme  R  x  6-5  . . .  "THICKNESS"  and S t r e s s  154  CO  Decay  Appendix E.  Solutions (7-5)  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 a n d Sum o f Squares  Appendix  G.  Decay G-l. G-2.  Appendix  H.  for Differential  . .  160 160 186  Equation 202  209  Models Evaluation of Expressions Predicting Shear S t r e s s Decay 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  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 and 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  . .  225 225 230 242  vi  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 Nylon F i b e r  41  5-2.  Length-to-Diameter Ratio  Fibers  41  5- 3.  Angular  Cylinder  56  6- 1. 7- 1.  V a r i a b l e s and Ranges I n v e s t i g a t e d V a r i o u s O r d e r s (m, n) and R e s u l t o f L e a s t Squares F i t 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 E x p e r i m e n t a l D a t a 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 E x p e r i m e n t a l D a t a and R e s u l t s o f C a l c u l a t i o n . . . . V a r i o u s O r d e r s (m, n) and R e s u l t s o f L e a s t  A-l. D-l. D-2. F. G-l. G—2. H.  V e l o c i t y of  Inner  of  Squares F i t Parameters of V a r i o u s Decay Models V a r i o u s O r d e r s o f G o v i e r ' s M o d e l and R e s u l t s Least Squares F i t V a r i o u s O r d e r s o f G o v i e r M o d e l and R e s u l t o f Least Squares F i t  61 89 123 161 187 210 227  of 233 243  vii  LIST  OF  FIGURES  No. 3-1. 3-2.  F o r c e Balance between P a r a l l e l P l a t e s D e f i n i n g Sketch f o r the Treatment of C o a x i a l  17  5-1.  C y l i n d e r Flow Data Photograph of 5 D i f f e r e n t l y  23 42  5-2. 5-3.  Sized  Fibers  5-4. 5-5.  D e n s i t y Match E f f e c t o f PEG C o n c e n t r a t i o n o n S u s p e n d i n g Medium V i s c o s i t y E f f e c t o f N a C l on S u s p e n d i n g Medium V i s c o s i t y E f f e c t o f D e x t r o s e on S u s p e n d i n g Medium  5-6.  Viscosity . . . . 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 Head  5-7. 5-8. 5- 9. 6- 1. 6-2. 6-3. 6-4. 6-5. 6-6.  C o n f i g u r a t i o n of C y l i n d e r s Grooves of t h e C y l i n d e r s C a l i b r a t i o n Curve of Torque versus V o l t a g e . . . . E f f e c t o f Age o f S l u r r y on Y i e l d S t r e s s T y p i c a l Decay Curve Thickness of Flowing Layer E f f e c t o f 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 . . . 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 . 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  (B)  45  . . .  . . . . .  Stress  46 47 48 51 53 54 58 63 64 66 67 69 70  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 on Y i e l d S t r e s s 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 on Y i e l d S t r e s s 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 l e on E q u i l i b r i u m S t r e s s 6-10. 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 on E q u i l i b r i u m S t r e s s 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 Yield Stress 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 Thickness of Flowing Layer 7- 1. Bundle Sizes i n Flowing S o l 7-2. E f f e c t o f P a r a m e t e r ( k / B ) on t h e M o d e l f  7-3. 7-4.  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 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 Flowing Layer •. . .  7-5.  E f f e c t of Angular V e l o c i t y (k /B) f  72 73 74 75 77  78 84 91 93 94  on t h e P a r a m e t e r 96  viii  7-6. 7-7. 7-8. 7-9. 7-10. 7-11. 7-12. 7-13. A-l. A-2. B-l. B-2. B-3. B-4. G-l. G-2. G-3. G-4.  Effect  of Angular  V e l o c i t y Q on t h e P a r a m e t e r  (B/A) 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 on Rate o f Increase of Thickness of Flowing Layer . . . . 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 of Thickness of Flowing Layer 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 o n t h e P a r a m e t e r ( k f / B ) 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 o n t h e P a r a m e t e r (B/A) Dependence o f A p p a r e n t S h e a r R a t e on Time . . . . Apparent V i s c o s i t y o f Time-Dependent A r t i f i c i a l Slurries Flow Curve of Time-Dependent Artificial Slurries E q u i l i b r i u m Flow Measurement w i t h Different Radius o f Inner C y l i n d e r E q u i l i b r i u m Flow Curve 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 o n Yield Stress 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 o n Yield Stress 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 o n Yield Stress 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 Yield Stress 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 D e c a y D a t a E f f e c t o f Parameter (k^/k^) on t h e Decay Model E f f e c t o f P a r a m e t e r (ki+/Po) o n t h e D e c a y Model 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 P a r a m e t e r (k /ki+) f  G-5.  Effect  of Angular  G-7.  98 99 100 101 103 105 106 124 125 127 128 129 130 228 235 236 238  V e l o c i t y on t h e P a r a m e t e r  (kVPo) G-6.  97  239  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 L/D R a t i o o n t h e P a r a m e t e r ( k ^ / k i | )  Particle  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 L/D R a t i o o n t h e P a r a m e t e r ( k ^ / P o )  Particle  240 241  ACKNOWLEDGEMENTS  The  author  Professor  K. L . P i n d e r ,  enthusiasm patience  throughout  N. E p s t e i n , their  committee  for their  willingness  the  graduate  comments a n d h e l p f u l  students  to discuss  author  discussions  Department  i n trouble-shooting for their  a l l aspects  as  of the study,  to the National  O i l Company  of B r i t i s h  of Chemical Engineering,  for financial  useful  t h e equipment,  suggestions  and  a n d t o Ms.  A.  the manuscript.  i s grateful  the Standard  Columbia,  Professors  A. P. W a t k i n s o n , a n d E . L . W a t s o n ,  assistance  f o r proof-reading  Canada,  study.  t h a n k s a r e d u e t o Mr. J . G. B a r a n o w s k i a n d h i s m a c h i n e -  fellow  The  for his  forhis infinite  a n d a p p r e c i a t i o n a r e owed  R. M. R. B r a n i o n ,  supervisor,  members.  shop s t a f f  Nokony  gratitude  guidance,  and e s p e c i a l l y  the course! of t h i s  many c o n s t r u c t i v e  Many  t o my  to the research  f o r h i s invaluable  and encouragement,  Special  for  i s profoundly' indebted  support.  Research Council of Columbia L i m i t e d ,  the University of  and  British  CHAPTER 1  INTRODUCTION  Scientific particles study  i n fluids  Stirring,  common p r o c e s s  materials.  trial  d i s p e r s i o n s began.  i s of engineering i n t e r e s t ,  t r a n s p o r t through  process  The  since  solid when  rheological  industrial  and f l o w o f  suspensions.  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 s t e p s w h i c h depend  a few  on t h e f l o w p r o p e r t i e s o f  P a p e r p r o d u c t i o n i s a well-known example o f an i n d u s -  involving  the flow of suspensions  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 understanding  of  the turn of the century,  frequently involves the deformation  t h e many  such  around  of c o l l o i d a l  of suspensions  processing  i n the flow p r o p e r t i e s of suspensions  had i t s s t a r t  of the v i s c o s i t y  behavior  of  interest  the behavior  of elongated  of suspensions  of b i o l o g i c a l  systems,  particles.  i s important f o r  f o r example,  blood,  (13) which  contains elongated  particles  known a s r o u l e a x  .  Furthermore, (22)  the  addition  and  polymer  study  of fibers  strengthening^ '  of rheology  of f i b e r  Time-dependent sions  of shear  liquid-like  thixotropy as  flow,  t o how  provides another  one o f t h e r h e o l o g i c a l  reason  behaviors  to the v a r i a t i o n  o f changes  Although  i s generally accepted, the rheological  reduction  ' f o rthe  suspensions.  as a r e s u l t  material.  f o r t u r b u l e n t drag '  known a s t h i x o t r o p y , r e f e r s  duration the  to f l u i d s  (53)  this  of v i s c o s i t y  i n the i n t e r n a l qualitative  there i s s t i l l  of suspenwith  structure of  explanation f o r  a general uncertainty  behavior o f time-dependent m a t e r i a l s can best 1  2 be  characterized  particles. defined  with  r e l a t i o n t o t h e shape and s i z e o f suspended  I t should  here.  be noted  Such a term u s u a l l y  flocculation,  or molecular  is  that  recognized  structure  a n d may  Thixotropic slurries system  consists  ing  structure  i n the l i q u i d - l i k e  i s probably  systems a r e n o r m a l l y  o f two p h a s e s ,  quite  clay  i . e . , dispersed  between t h e d i s p e r s e d  sensitive to trace  surfactants,  effect  i n some o f t h e n a t u r a l  p a r t i c l e s and  further  complicates  must  restrict  aqueous  glycol.  e.g.,  systems  i s correspondingly  present  could  one's a t t e n t i o n  i n natural  The r e s u l t The  phase i s  such as l u b r i c a n t s slurries.  This  dextrose,  information,  slurries  of this thesis i s :  and  to control  This  i n this  one  fibers i n  polyethylene t h e shape and s i z e  nature of the dispersing  and i o n i c f o r c e .  are called a r t i f i c i a l  such  from a study o f  t o a model system such as n y l o n  p a r t i c l e s and t h e p h y s i c a l  objective  material,  To o b t a i n  I n such a system i t i s p o s s i b l e  The  contains  complex.  be g a i n e d , however,  flow of s l u r r i e s .  v i s c o s i t y , density,  medium o f t e n  across  or p o l y e l e c t r o l y t e s .  s o l u t i o n o f sodium c h l o r i d e ,  dispersed  range  slurries  the problem.  information  time-dependent  dispersing  p a r t i c l e s i n those n a t u r a l  amounts o f a t h i r d  solid  ata l l .  p a r t i c l e s and t h e d i s p e r s i n g  which are u s u a l l y  Much u s e f u l the  system  It  i n w a t e r , e t c . , where t h e  The d i s p e r s i n g  of a thixotropic  material.  d i f f e r e n t from  as a s t r u c t u r e  encountered  components o f i o n i c s u b s t a n c e s  rheology  very  scale.  i s not p r e c i s e l y  to p a r t i c l e orientation,  i n f a c t not be r e c o g n i z e d  to millimeter  interface  of  entanglement  The shape o f t h e d i s p e r s e d  various  an  refers  "structure"  i r r e g u l a r and t h e 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  micron  or  liquid  the term  such as crude o i l , bentonite  medium. is  that  medium,  i s t h e r e a s o n why t h e  thesis.  to  examine the  particle  effects  of s u s p e n s i o n  (particle size,  concentration, electrolyte concentration,  medium v i s c o s i t y )  on t i m e - d e p e n d e n t s h e a r  to c o n s t r u c t a model e q u a t i o n dependent c h a r a c t e r i s t i c s of and  variables  to c o r r e l a t e  suspension  which best  suspending  flow, defines the  f l o w of the s l u r r i e s ,  the parameters of the model equation  variables.  time-  with  the  CHAPTER 2  LITERATURE SURVEY There i s much confusion i n the l i t e r a t u r e because of disagreements i n the terminology used to describe time-dependent rheology. thesis the following d e f i n i t i o n s are used.  In this  "Thixotropy" i s an i s o -  thermal and comparatively slow recovery, on standing of a material, of a consistency that was lost through shearing.  "Anti-thixotropy" i s 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" i s the  s o l i d i f i c a t i o n of a thixotropic system by gentle and regular movement. "Dilatancy" i s an increase i n volume caused by shear.  "Shear thinning"  i s a univalued reduction of the consistency with increasing rate of shear.  "Shear thickening" i s a univalued increase of the consistency  with increasing rate of shear.  "Shear softening" i s a decrease of  consistency, the r e s u l t of shearing deformation.  "Shear hardening" i s  an increase of consistency, the r e s u l t of shearing deformation. The l i t e r a t u r e review i s limited to a survey of pertinent l i t e r a t u r e concerning thixotropic behavior and a presentation of the rheological theories based mainly on chemical k i n e t i c s .  2-1.  Experimental Observations Green^"^ was one of the e a r l i e s t researchers to study thixotropic  behavior i n a systematic way.  His data were obtained by shearing the  f l u i d f o r a set time at a given shear rate, observing the shear stress, 4  5 and  then  shear  changing  rate,  repeated data  and  the d e s i r e d  shear  procedure  stress  was  bottom point  was  "down-curve". of  the degree l o o p s was into  rate  top  versus  c o n t i n u e d by  s t e p w i s e manner and  consisted  shear  shear  The  and  of t h i x o t r o p y . original;  shear plot  t h i c k n e s s of  stress.  r a t e was  r a t e was  the  to the next  shear  progressively  measuring  the up-curve  very  the  shear  obtained. The  physical  immediately  then re-measuring  until  as  the  called  of t h i s  d o w n - c u r v e , was  The  plot  "up-curve".  lowering the stress  shear  rate  in a  each  step u n t i l  part  of  t h e d a t a was  the called  loop, which  to the area of  i t is difficult  the  The  at  hysteresis  was  of  c o n s i d e r e d a measure  Green's a t t e n t i o n  of  This procedure  reached.  the r e s u l t i n g  however,  higher level  of  hysteresis  to t r a n s l a t e  the  area  meanings.  (44) Following index  Green's method,  S i m o n e t al.  f o r bentonite suspensions,  the h y s t e r e s i s meter.  The  thixotropic In  loop of the  d e f i n e d by  their  the area  a  thixotropic  c i r c u m s c r i b e d by  flow curves, using a cone-and-plate  type of p r e p a r a t i o n given the index, but  measured  the reason  cone-and-plate  suspensions  f o r t h i s was  not  affects  viscothe  clear.  v i s c o m e t e r measurements w i t h  polyvinyl  (23) chloride in  (PVC)  m e l t s , K h a n n a e t al.  observed  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  stress  breakdown".  They  tanglements could effect  not, of  suggested  of polymer  however, d i s t i n g u i s h the  termed  i t "rheological  entanglements  and  disen-  thixotropy clearly  from  the  They  elastic  melts.  Billington stress  molecular  reversible reduction  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.  (2)  shear  that  the  et  of an  al.  (3) '  aluminum  measured  i n a Couette-type viscometer  laurate-Newtonian  suspension of g r a p h i t e p a r t i c l e s  o i l system  and  a  d i s p e r s e d i n a Newtonian o i l .  the  colloidal It  was  6 found  that,  in addition  a Weisenberg To  effect,  test  al.  the  examined  oil.  The  to  the  thixotropic behavior,  while  the  latter  feasibility  of  experimentally  crude o i l exhibited  compressibility. influenced  by  They  found  does  the the  not  only  that  the  exhibits  p i p e l i n e , Perkins  strength  of  thixotropy yield  i t s temperature h i s t o r y ,  former  not.  Trans-Alaska yield  the  P r u d o e Bay  but  strength  et  crude  also of  the  s h e a r h i s t o r y , and  o i l is  composition.  (54)  Vocadlo stress  of  et  al.  reviewed  fluid-like  eliminate  the  g r o o v e s be  substances.  e f f e c t of  used  the  methods f o r m e a s u r i n g  They emphasized  anomalous  slip,  in a co-axial cylinder  that,  cylindrical  the  yield  i n order  surfaces  to  with  deep  viscometer.  (13) M a s o n et Canada a r e written  a  particle not,  al.  most large  number o f  i n shear  flow,  dispersions  observed  the  solution  any  cement sions  Newtonian  Research  d i l u t e to  of  the  that  cause  of  I n s t i t u t e of  dispersions.  basic  behavior  They of  dispersions.  a  I t seems  thixotropy.  But  they  relatively  high  of  have  single  They  thixotropy.  effect for dispersions  focussed He  c h a r g e on  on  the  found  e f f e c t of anionic  acid  condensates  cement  active particle  sites.  b e h a v e as  surfactant fluids.  surfactants  that  Portland  suspensions with  to  on  i n v e s t i g a t i o n s on  were too  cement.  onto  Paper  publications  commercial naphthalenesulfonic attracting  and  did that  instead.  P e t r i e ^ ^ Portland  Pulp  in addition  Weisenberg  concentration  the  concerned with microrheology  however, r e p o r t  their  of  of  surfactants appear  p a r t i c l e s by His  on  the v i s c o s i t y  based  sufficient  strength  i s that  behave  the  from normal  non-Newtonian t h i x o t r o p i c f l u i d s ;  s o l u t i o n of  certain  to n e u t r a l i z e  adsorption  conclusion  on  as  suspen-  Enoksson^^ of  solution  discovered with  of c e l l u l o s e that  Couette rate,  nitrate  o f temperature  on t h e t h i x o t r o p y  (Mn = 2 5 0 , 0 0 0 ) i n n i t r o b e n z e n e a n d properties  of the solution diminish  temperature.  (29)  Mercer  the effect  the thixotropic  increasing  viscosity  examined  of bentonite-water  viscometer.  They  t h e time-dependency  (30)  e t al.  and Mercer  investigated  suspensions  found  that,  of their  the  time-dependent  u s i n g a Haake R o t o v i s c o  after  a sudden change i n shear  apparent  viscosity  c a n be d e s c r i b e d  by n ( t ) = A exp [ - ( t / i ! ) ' ^ ] 0  where n ( t ) i s apparent istic  times,  viscosity,  a  increasing  and Ree  thixotropy  suspension,  on  ionic  (36)  , and P a r k ,  a  r  character-  e  2  The c h a r a c t e r i s t i c  rates  times  became s m a l l e r , d e c r e a s e d temperature  Ree, and E y r i n g  (37)  effect f o r  observed the Couette-type  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  i . e . , Newtonian, non-Newtonian n o n - t h i x o t r o p i c , and non-  thixotropic.  They a l s o  the apparent  viscosity  ( d i - and t r i - )  then decrease  examined  of the suspensions.  c o n c e n t r a t i o n was f u r t h e r  cations  i n shear  and x  change.  the flow properties  cations,  T\  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  viscometer.  Newtonian  t i s time,  c o n c e n t r a t i o n , a n d showed l i t t l e  20°C temperature Park  2  a n d A, B, a n d C a r e c o n s t a n t s .  became l o n g e r as t h e d i f f e r e n c e with  + B exp [ - t / r ] + C  caused  decreased, increased.  the v i s c o s i t y  the effect  of electrolytes  On a d d i t i o n  o f monovalent  f o l l o w e d by a r i s e Addition  as t h e  of multivalent  t o i n c r e a s e t o a maximum  to a constant value. (49)  Umeya et pensions  al.  and found  observed that  the flow behavior  the shear  stress  a t steady  of clay-water state  is a  sus-  function  8 of  pH.  The  expect  the  (which  is a  shear  s t r e s s was  internal structure loose  card  card  i n the  pack s t r u c t u r e Weyman^"^  (1)  Gel  of  In  a  like  house  a pH  the  other  structure  basic  long  rods  particles,  at  sponge-like  t o be  a  i n the  acidic  would  chain  structure  h a n d , i t seemed  previous  that  region  the  and  a  i n v e s t i g a t i o n s on  I t i s summarized  un-isodimensional,  thin discs.  which are  suspension or  or  w h e r e one  region.  review of  highly  =7.5,  suspension  t h i x o t r o p i c suspensions.  form of  (2)  the On  gave a good  p a r t i c l e s are  black  of  structure).  s u s p e n s i o n became a  viscosity  minimum a t  (An  as  follows:  i . e . , they  exception  the  have  i s the  the  carbon  spherical.)  r e s t , these  p a r t i c l e s form open, s c a f f o l d i n g -  structures  i n which the  liquid  phase  is  trapped. (3)  Thixotropic  suspensions  centrations. particle  This  density.  increasing  can  form gels  indicates that The  yield  over  there  value  a wide range of  i s no  increases  preferred strongly  con-  average  with  concentration.  (41) Pinder  investigated  tetrahydrofuran-hydrogen new  technique  to  give  interception  of  time,  almost  by  i t was  the  extrapolation.  viscosity  versus  first  to  ten  culation short  of  period  yield  gas  value.  hydrate Due  to  behavior  of  slurries,  employing  the  angle  small  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 axis  impossible  the  accurate  to  found  time y i e l d e d  yield  time-dependent  sulphide  Pinder  fifteen  of  a  the  readings.  a  determine that  a plot  reasonably This  value  and  of  rate  t i m e by  the  method  of of  of  the  straight line  technique allowed change of least  at  the the  f o r the  He  zero  value  of  over  fluidity  squares.  of  yield  inverse  a  after  cala  proposed  a  9 new m o d e l o f t h e s e c o n d o r d e r - z e r o f o r t h e time-dependent behavior viscosity (1)  data.  order  of the s l u r r y , which best  H i s experimental  The y i e l d v a l u e  r e v e r s i b l e r e a c t i o n mechanism f i t his  r e s u l t s a r e as f o l l o w s :  decreases with r o t a t i o n a l  speed f o r any g i v e n  spindle. (2)  The s m a l l e r t h e v i s c o m e t e r  spindle, the higher  i s the y i e l d  f o r t h e same d i a m e t e r c o n t a i n e r a n d t h e same r o t a t i o n a l (3)  The h i g h e r  (4)  T h e 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 , yield  (5)  the consistency, the higher  i s the y i e l d  value  speed.  value.  the lower  i s the  value.  The l a r g e r t h e l e n g t h - t o - w i d t h the y i e l d  ratio  of the p a r t i c l e s ,  the higher  value.  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 Newtonian medium.  The t i m e - d e p e n d e n c y o f h i s d i s p e r s i o n s was f o u n d t o i n c r e a s e  rapidly with increasing particle concentration. order  H i s decay curves  r e a c t i o n model.  size,  length-to-width  fitted  Pinder's  r a t i o , and  second  order-zero  He a l s o o b s e r v e d a r a g g e d d e c a y c u r v e a n d  suggested t h a t a breakdown o f an e l e c t r o s t a t i c  structure or particle  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  behavior. (35)  In order out  t o e l u c i d a t e Brown's s u g g e s t i o n ,  experiments using  s a l t s of d i f f e r e n t  w i t h t h e same p a r t i c l e s . the decay curve  Nicholson  cation valence  carried  f r o m Brown's  He f o u n d t h a t t h e d e g r e e o f r a g g e d n e s s o f  increased with  increased  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  c a t i o n s t r e n g t h and  concluded  s t r u c t u r e must be 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 raggedness of t h e decay  curve.  These two i n v e s t i g a t o r s u s e d 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  10 match the d e n s i t y of Therefore affected inate  i t was by  this  behavior  the  salt  rate.  the m o l a l i t y strength salt,  u n c l e a r as  i t would be  He  salt  found  of  the  that  The  CeCl3  viscosity  interesting  medium.  t o examine t h e  To  is  elim-  time-dependent  salt concentrations.  ability  suspensions,  the y i e l d stress  of the  i n this  order.  stress  to b u i l d  of  He  effects  at  zero  the  also  up  to  gel  cation  i n the  collected  data  of mixtures, which i s  i n terms o f the p r e s e n t  study.  or PEG-Dextrose/Fructose  artificial  the  i s proportional  salt  to the valence  of a wide range  system  behavior  particles  He  suggested  system  w o u l d be  a  to g i v e a wide range  of  characteristics. From t h i s  understood.  review,  The  the purpose  of  information  that  Furthermore,  early  i t i s apparent experimental  quantitatively  the  i s useful  types  of  g r e a t need  exhibit  received  liquids.  that  has  and  d e t r a c t e d from  real  value  Consequently, for research  provide  characterizing  S i n c e most p o l y m e r m e l t s  and  the  for  concepts.  rheological  behavior,  well-  were d e v e l o p e d  t h i x o t r o p y d i d not  f o r understanding  appreciable thixotropic  a r e o f any  t h i x o t r o p y i s not  techniques  measuring  much r e c e n t a t t e n t i o n .  liquids  that  i n developing theoretical  f l o w p r o p e r t i e s of polymers  few  necessary  proportional  MnCl2>  a CMC-Corn S y r u p  not  time-dependent  t h e maximum y i e l d  g o o d medium f o r s u s p e n d i n g  do  the  c o n c e n t r a t i o n s on  salt.  i s inversely  particularly  other  t o how  d i s p e r s e d phase.  c o n c e n t r a t i o n of the d i s p e r s i n g  and  i . e . , NaCl,  slurry  medium w i t h  investigated, with bentonite clay  t h e d e n s i t y and  that  dispersing  of d i s p e r s i o n s with d i f f e r e n t  cation valence  shear  on  left  problem  Hu^^ of  the  research  polymer  purposes.  on  solutions  s u b j e c t has  o n l y a few  the  not  s t u d i e s on  a  2-2.  Theoretical The  explained  r e v e r s i b l e n a t u r e o f time-dependency has been  qualitatively  b y a number o f a u t h o r s  down a n d  subsequent several of  rebuilding  quantitative  the chemical  models w i l l The tropic to  i n terms o f a b r e a k i n g  o f some f o r m o f s t r u c t u r e .  k i n e t i c models w i l l  be d i s c u s s e d  and shear  stress  softening  section  i n detail,  some  and o t h e r  researchers  f o r system e x h i b i t i n g  non-Newtonian b e h a v i o r  t h e time v a r i a t i o n of concentration  describe  In t h i s  hand  briefly.  time v a r i a t i o n of shear  several  On t h e o t h e r  models have been p u b l i s h e d .  be mentioned  behavior  Hence, to  Interpretations  thixo-  i s similar  i n r e v e r s i b l e chemical  reactions.  h a v e made u s e o f t h e same t y p e o f e q u a t i o n s  the thixotropic  systems. (1 fi^  The  model a t t r i b u t e d  application that  of the theory  are believed  of  the flow  of  two k i n d s  Thixotropic flow units Newtonian. rate  t o Hahn, Ree, and E y r i n g of rate  to play  an important  of materials.  part  They assumed  of flow u n i t s , behavior  processes  i s based  to the relaxation i n determining a general  processes  the nature  fluid  consists  i . e . , Newtonian and non-Newtonian  species.  i s supposed  that  on t h e  t o b e due t o t h e i n t e r n a l change o f  from non-Newtonian t o Newtonian, o r from Newtonian t o nonThey a r r i v e d  at the following  expression  fora  constant  of shear,  * - (TO " O e " where T  0  i s initial  equilibrium  stress  condition.,  of a gelled  F o r growth  at  +  (2" ) 1  structure  and x^ s t r e s s  from a completely  of the  deteriorated  state, x  =  (x, - x ) e "  \y where  represents  stress  co  a t  (2-2)  + x co  from a completely  deteriorated  structure.  12 It  Is  possible  by  using  to  three  parameter  decay of  point  shear  rate  tional  to  is  point  equation  the  of  (b  -  the  shear  T>  T  N  i n equations  , and  T,  y  .  to  They a l s o  amount  of  viscosity  representing  the  and  (2-1)  to  assumed  their  rate  and  obtain  curve).  of  that  (n  the  -  (2-2)  the  any  time  -  2  i s given 2cn (b  4ac)  and  n^)™  -  of  a  n  that  a  n ) , n  t  =  point  assumed the  some power propor-  where  shearing.  for m  +  t  -  2  They  ng  (tin  t, after start  -1 - tanh.  a  g r o w t h r a t e was  structural deterioration at  model on  gelled value  product  t h i x o t r o p i c decay  (b  4ac)  from a  2CTIQ + b  -1  to base  stress-shear  decayed  proportional  tank 2  given  have chosen  viscosity  r a t e was  the  Brodkey  (slope  the  model as  experimentally.  Denny and  that  the  measurable s t r e s s e s  "a"  viscosity  employ  n  t  Their  1 and  n  =  2  as  b  V  4ac)  (2-3)  where a = Ky  n  + "  ^0  b  =-(K  c  =  Y  nn - n u  these n  are  counting  )  oo  : growth r e a c t i o n  k^  : decay r e a c t i o n r a t e  K m  : equilibrium and  at  are  completely  rate  constant  ni, w h i c h a r e  i n v e s t i g a t o r s , there measured  +  k^  and Without  \  five  taken  constant constant (k^/k^). t o be  constants  g e l l e d and  1 and  to  be  completely  2 respectively evaluated.  deteriorated  no  by and  condi-  00 tions,  respectively.  Use  of  an  equilibrium  line  allows  the  estimation  13 of  K  and  P.  This  leaves  t o be  determined  from the  decay  data.  (41) Pinder state  of  up  a new  ing  the  assumed  randomness. position, two  change can  Shearing  which  extremes  equilibrium  order, be  that, i n i t i a l l y , slurry  i s i n a lesser  of  as  the  orientation,  n^  compared  and  ri i  s t a t e of  particles  the  are  in a  to  take  particles  randomness.  i . e . , complete  chemical  k  slurry  causes  terms of  n  2  to a  the  Designat-  randomness  apparent  and  viscosity,  the  reaction  f  \ with  forward  various  rate constant  reaction orders  equations  can  be  k^  obtained.  For  integration, (n  +  (n -  For  the  other  order  which have the  will n) (n s /  logioT  example,  e  also  been  second  various  order-zero  order  e  "  4)  o  =  cim  2.303  + n ) y  ^  (2-5)  e f  expressions  can  be  obtained  f o l l o w i n g form  order found  The For  i s the to  best  f o l l o w the  c t  (2-6)  versus  straight  Pinder's  best.  = f(n)  Plotting  curves.  mechanism.  order-zero  r e a c t i o n s , the  assuming  (2  reactions, similar  where c i s a c o n s t a n t .  reaction  By  2  r~  0  various  the  n )  ,  logi [f(Tl)]  yield  reverse  k^.  be -  y  n ) (n  of  and  rate constant  = V? - \  dt  .dm upon  reverse  f o r forward  reaction  which,  and  slurry  t on  line  i t was  order-zero  semi-log  represents found  Brown's a r t i f i c i a l second  a  the  that  paper best  the  s l u r r y ^ ' ^  order  will  second has  mechanism.  (43) Along  similar  lines  to Pinder's  work, R a t t e r  and  Govier  assumed  14 that  t h e measured  stress  stress  components.  reversible  reaction  i s t h e sum o f s t r u c t u r a l s t r e s s  They  also  assumed a f i r s t  f o r the rate  equation.  and Newtonian  order-second  They a r r i v e d  order  at the  equation  - T  SO  where  sO S T  i s a reduced  T  the final  T  s°°  . initial sO  similar  value  the rate  state  shear  stress  of T s o f network  concentration on c h e m i c a l  i n two r e s p e c t s .  some k i n d  constant  of increase  models based  occurrence  second  rate  steady  P Q the i n i t i a l The  s°°  structural•stress  s  k^ and  (2-7)  sO  k, A  x  the  M  .T .— .T S S°°  In  of a rate  process  o f network  reaction  The f i r s t  structure  kinetics  common e l e m e n t  consisting  i s the designation  discussed  and r e f o r m i n g o f  i n the l i q u i d .  of a relationship  of reaction  used  simple  actually  combinations  occurring of forward  Since the  was unknown, t h e v a r i o u s and r e v e r s e  The  between t h e  number o f e x i s t i n g b o n d s a n d t h e v i s c o s i t y o f t h e l i q u i d . type  here a r e  i s t h e assumption of  of breaking  o f bond between p a r t i c l e s o r m o l e c u l e s  common e l e m e n t  structure  reactions  theorists  i n their  rate  equations. The the  c h e m i c a l k i n e t i c s m o d e l s seem t o b e s u f f i c i e n t  time-dependent behavior  a p p r o a c h , however, w h i c h terms o f c o n s t i t u t i v e logical  approach.  of l i q u i d s quantitatively.  i s t o express  equations.  There  the characteristics  This  Some o f t h e i d e a s  to represent  method  i s called  are discussed  i s another  of liquids i n  t h e phenomeno-  below.  15 C h e n g and thixotropic tions  Evans  fluids  that  i s described  c o n s i s t i n g of  equation of  assumed  state  an  of  which  the  structural  viscosity is a  parameter  "X",  of  states  both  are  that  shear  supposed The  rate to  i s given,  from  equation  of  one  this  Along  the  this  y(t).  the  The  is  generally  (x,  The  sense,  y,  X)  equation  of  =  (x,  well  as  i s given  of  the  i n the  )  form  of  (2-9)  structure  i s explained  (2-9) If t  changes i s a  The  forms of  as  follows.  0  can  (x, y)  surface The t)  line  a  thixotropic fluid by  a  a  n  e  function  n and  g^  the  way  The  determined  X)  (x,  =  not  material  i n the  and  and  projection  i n the  f ( x , y,  i s therefore  surface  X(t)  from y ( t )  line  the  Supposing  for X =  -  a  i n the  space are by  p l  defines  g^  not  also  solve  i s eliminated  i n the  y,  but  one  =  and  as  surface. 0  only is  by  sheared  given  by  y,  space,  t)  a  line as  . (31),(52) thought  f o r flow the  idea  x(y) X)  i n the  surface  ultimate and  aim  of  "rheology"  deformation  i s to  thixotropy.  f a r from i t s  aim.  construct  i n terms of m a t e r i a l  p h e n o m e n o l o g i c a l a p p r o a c h may  phenomena o f stands  line  Y  rate  experimentally.  i s a parameter.  behavior  the  x =  The  of  shear  the  x(t).  f ( x , y,  e q u a t i o n dX/dt  in  tions  t  x =  equation.  (2-8)  rate  at which  their  line  the  a rate  s t r u c t u r a l parameters.  for  a  of  g.U,  from equation  (2-8)  on  the  =  determined  then  line  and  rate  and  be  obtains  line  surface  rate  and  n(A, Y ) Y  while  p e c u l i a r i t y of  Y'(t)  x(t),  the  of  s c a l a r c o n s t i t u t i v e equa-  form  function  f  which  state  i n the  x = in  rheological property  i n terms of  equation  i s expressed  the  Unfortunately,  be  a  the  c o n s t i t u t i v e equaparameters.  good method present  to  In  attack  s t a t e of  the  this the  art  CHAPTER 3  THEORETICAL BACKGROUND  The  literature  phenomena o f concepts  thixotropy.  concerning  which w i l l  reviewed  be  i n the  The  present  useful  in later  such  Complicating tions  will  tioned  chapters.  be  focussed  cover  fluids  rheology  the  some o f  the  and  These formulae  artificial  on  the  a r e based  particularly  slurries  are  the  on  a  when  dealt  associated with  methods t o c o r r e c t  formulae  with.  those  errors w i l l  be  assumpmen-  possible.  Newton's P o s t u l a t e When a  fluid  moving w i t h  motionless up  i n suspension  d e s c r i b e d and  of  subject to error,  time-dependent  effects  wherever  3-1.  one  as  chapter  chapter w i l l  the b a s i c c l a s s i f i c a t i o n  number o f a s s u m p t i o n s w h i c h a r e suspensions  previous  as  by  i s situated  a constant  a restraining  shown i n F i g . The  f o r c e per  b e t w e e n two  velocity  V c m / s e c , and  force  dyne-,  flat  the  plates,  lower  a velocity  one  the  top  held  gradient i s set  3-1. unit  area  r e q u i r e d to r e s t r a i n  (equal  to that r e q u i r e d to maintain  called  the  shear  parallel  stress  and  has  the  u p p e r one  been d e s i g n a t e d  by  the  lower  plate  i n m o t i o n ) has x  been  , i . e . , the  X-  yx directed ally  force acting  expressed  on  a plane  as  p e r p e n d i c u l a r t o Y.  I t i s mathematic-  . F T  yx  S  16  (3-1)  17  Fig.  3-1.  Force Balance between P a r a l l e l  Plates  18 The  r a t e of  change of v e l o c i t y  AV /AY,  in this  usually  expressed  x  V/H,  case  has  with  d i s t a n c e from  been d e s i g n a t e d  in differential  form  x  =  x  the  fluid  i s designated  Newtonian v i s c o s i t y .  3-2.  Bingham  most  shear  fluids  TQ,  are  the  Materials  which d i s p l a y  3-3.  are  Power Law  linear  the  constant  u  is called i t s  i s Newton's P o s t u l a t e .  overcome  represented  this  cases.  i n order  from  behavior  by  (3-3),  If, in a material,  residual  Newtonian b e h a v i o r are  equation  that flow occur,  p r o p o r t i o n a l to the  referred  to as  and  shear is  i f  stress  encountered.  Bingham  by - x ) 0  = ny  (3-4)  viscosity.  to c h a r a c t e r i z e r e a l logy,  from which  a wide range of y  empirical  relation  fluids  this  of  2  Fluids  versus  over  3  required i n other  described  i s the p l a s t i c  logt  <->  (3-3)  (3-3)  simplest departure  In order plot  TT  m  are adequately  (T where n  i  Newtonian and  rate i s effectively  and  is  that  = y y  yx  must b e  (T - TQ)>  Plastics  l  Equation  formulae  a minimum s t r e s s , the  r a t e and  Plastics  Although more complex  as  shear  plate,  viz.,  i s p r o p o r t i o n a l to y,  yx  lower  AV  A  T = T-  If  so  as  the  known a s  type.  fluids,  i t i s o f t e n found  f o r many f l u i d s .  t h e power l a w  This relation T  i t i s common p r a c t i c e  = ky  n  may  As  i s widely be w r i t t e n  that the  curve  a result, used  to  to is  an  describe  as (3-5)  19  where k the  and  n are  constants  c o n s i s t e n c y of  fluid; the  n  the  fluid,  i s a measure of  greater the  f o r the p a r t i c u l a r  the  departure  The the  shear  the  i s a measure  t h e more v i s c o u s  u n i t y t h e more p r o n o u n c e d  fluid.  ones w i t h  "apparent  viscosity"  stress  the  to  the k  k  shear  The n  fluids  > 1 are  n of a rate,  with n  called  fluid  are  and  the  < 1 are  dilatant  i s d e f i n e d as  of  definition  a Newtonian  i s consistent with fluid  [equation  fluids.  the  ratio  of a non-Newtonian f l u i d .  law  may  fluid  be  expressed  General  the  rheological  the  accuracy  i t must  These data  serves  The  as  apparent  the  viscosity  a definition  viscosity  of  for a  the  power  k  jn-l  ( 3  _  7 )  Y  i t i s convenient behavior  with which equation  of  to use  a  fluid,  of  state  i n a l l cases are u s u a l l y of  be  obtained  characterization  not  reasonable,  aid  o f more c o n v e n t i o n a l f l o w  the  actual  i s necessarily  r e p r e s e n t a t i o n of  of  i s limited  behavior. the  A  by  on  general  (3-8) e x p e r i m e n t a l l y measured  the use  large parallel  s i n c e t h e measurement  by  form  f(T)  based  The  a mathematical  i t s applicability  i t represents  Y = and  and  of  Fluid  Although  rheological  the u s u a l d e f i n i t i o n  = ^ = Y  3-4.  (3-6)  as  =X  n  of  or  (3-3)],  viscosity  non-  called  n = £ . Y This  of  the  degree of non-Newtonian b e h a v i o r  from  Newtonian p r o p e r t i e s of the pseudoplastic, while  the higher  fluid:  data.  of a v i s c o m e t r i c d e v i c e .  p l a t e s d e s c r i b e d i n F i g . 3-1 of  geometry.  shear  stress  can  occur  with  is the  20 3-5.  Simple A  Shear  fluid  process  Flow  responds  i n which  t o s t r e s s by f l o w i n g .  the material  deforms a t a f i n i t e  k i n e m a t i c measure o f t h e response o f a f l u i d tensor  A, w h o s e C a r t e s i a n 3v.  i s essentially a  rate.  i s the rate  The b a s i c of deformation  components a r e  .3v.  A. . = - r - ^ + T ij  Flow  3x.  1  3x.  = A. . j i  (3-9) (32)  The  components o f A i n t h e c y l i n d r i c a l  coordinates  (r,6,s)  are given  by A  2 r — 8  rr =  1  V  (3-10)  3r  3v  A B8  = 2 ~ 3 S  re  " V  A.  = A  A  (3-12)  = ^  <T>  3v„ a  «9  J  gr  3a  M o t i o n may element each  of f l u i d  "ra  3r  so  r  + —  3z  I  "  1  3  )  (3-15)  might  b e t r a n s l a t i n g a t t h e same l i n e a r  element might,  3  (3-14)  30  in a fluid  even i f A i s i d e n t i c a l l y  zero.  Each  v e l o c i t y , and  i n a d d i t i o n , h a v e t h e same a n g u l a r v e l o c i t y a b o u t rotation of the f l u i d .  a n d r o t a t i o n do n o t c o n t r i b u t e  a r e not associated  which  + I —^  exist  some a x i s , d u e t o a r i g i d tion  (  3v 8  = A  7 i f  , 3v  3v  A  +  with  i s of i n t e r e s t to us.  that  part  But u n i f o r m  transla-  to the deformation of the f l u i d , of the response of the m a t e r i a l  and  21 A particular devices zero  is  the  component  scripts  direction shear  of  flow  flow which w i l l  so-called of  "simple  velocity  2, a n d  1,  simple  in  3 denote,  shear  only  defined  x  "shear" with  shear The  same m a n n e r .  2  . i s  the  equations In  of  rate  neutral  a  the  non-  sub-  the  d i r e c t i o n , then  a  simple  A  x -coordinate.  the  shear  usually  Couette is  ( 1 , 2, 3) = ( 0 , r ,  «)  a n c i  2  components is  shearing  stress  *  A  of the  2 1  a  only  (-  A  1  2  flow  ) ,  a  s  shear  stress  whose n o t a t i o n  consistent  rate". to  the  geometry  components  required flow in  has  a  more  where  the  (3-18)  non-zero  simple  normal  "shear  is  flow notation  used,  the  and  is  j  2  called  example  definitions,  (3-17)  been  for  are  (3-16)  tensor,  notation,  Thus  flow,  If  flow direction,  deformation  simple  addition,  is  direction.  the  the  (3-1) a n d  Poiseuille  has  of  of  flow are  stress  x  the  (3-2), and  these  there  ( 0 1 0 ) 10 0 0 0 0  y( 2)  function  coordinates  to  viscometric  0, 0)  l 5  scalar of  coordinate  According  1  e  correspondence  cylindrical  x  m  equation  The  simple  o  component  the  common  s  various  i n which  the  the  in  by  A =  y( 2)  single  and  v = (v  where  flow",  respectively,  velocity variation,  is  a  appear  are  the  by  of  A in  equation  labeled  already  been  (3-17).  in  d i r e c t i o n of  the flow  used  in  found  in  (3-3). other  simple  parallel plate  shear  flow  torsion,  geometries  and  cone  and  are  plate  a  torsion.  and  22 3-6.  Couette A  Flow  familiar  (Coaxial  type  cylinders,  with  between.  Mechanical  cylinders. the  the  There  cylinders.  In  this  in  Couette  such  Viscometer)  rheological  fluid  t o be  instrument  studied  construction  held  allows  the  shows a s c h e m a t i c  formulae  consists  i n the  of  the  torque  drawing  for calculating  two  annular  angular motion  i s a means f o r m e a s u r i n g  F i g . 3-2  section,  One  of  Cylinder  space  of  one  of  e x e r t e d on  of  shear  coaxial  such  stress  an and  the or  by  instrument. shear  rate  flow are described. assumes  that  a simple  shear  flow exists  i n the  annular region,  that v  =  ( v , 0,  0)  Q  (3-19)  and v where w  i s the  The the  rate  Q  = v ( r ) = r ui(r)  (3-20)  Q  angular v e l o c i t y of deformation  (radian/sec).  tensor for this  simple  shear  f l o w has  only  components do)  And  one  can  define a  function  f ( T ) = -r~  where x  i s the  shear  Assume t h a t cylinder torque the  the  stress outer  i s driven with  T.  face of  an  and  component cylinder  the  inner cylinder,  R are  (3-21)  x  1 2  .  i s stationary  angular v e l o c i t y  T h i s t o r q u e must b a l a n c e  T where L  = y  so  = 2TTR L X  the  Q, by  torque  and  the  that  the  application  e x e r t e d by  the  of  a  fluid  that (3-22)  2  t h e h e i g h t and  inner  r  r a d i u s of  the  inner cylinder,  and  xR  on  Fig.  3-2.  Defining Coaxial  Sketch f o r the Treatment Cylinder  Flow  Data  of  24 is  the  one  shear  can  stress  exerted  on  inner  cylinder.  relates  measured dependent cylinder In  in  the  shear  at  R - 2^k  stress  at  the viscometer.  f l o w as any  i t  equations  are  principle  of  long  given  order  viscometer,  as  the  This  x  'is  inner  formula  constant  cylinder is  to  valid  over  the  the  even  torque  in  position  2 3 )  a  time-  on  the  inner  time.  to  derive  is  necessary  a  formula to  for  employ  shear  conservation  of  rate  momentum  mathematical formulations  cylindrical  "  ( 3  of  momentum.  in  a  coaxial  equations.  the  fundamental  Their  most  coordinates  cylinder  Momentum physical  general  form  p ~ ~ = - VP - V.x + Z p F In  (3-22)  From e q u a t i o n  find  T  which  the  is  (3-24)  (6,  r,  a),  (3-24)  equation  can  be  (4) transformed the  to  the  coordinate  following  system,  equations  assuming  in  gravity  to  terms  of  the  only  be  the  components  field  of  force  present, r-components: o M  /  9v  (  r  3t  3r  3v  ,  v  1-  r  ^r  r  3r  3r  ,  1  'v„ 6 r  3v 39  rr'  K  v„ 6 r  r  r  36  ,  1-  3v v  a  3a  r\ ) 1  r  3a  '  P  6  r  v  '  9-components:  0 M  ( v  3t  = _I r  — + V r  l £ 36  _  1-  1  3r  r  (1-1- ( 2  r  z  3r  r  36  T  r6  r  )+I r  r V a  ^68. + 30  3a  )  •fle« 3a  '  )+  p  g  M 6  e  (3-26)  25 a-components: ,  3v  a  D ( M  The (1)  1- V  3t  V  _ 9 P _  =  3v  ,  3a  a  v  .  l J _  r 3r  (  T  i s steady  36  r«  f o l l o w i n g assumptions  The f l o w  a, ) 3a  r V r  r  3v  a  1  r 3r  (  3v  8  a  ) + I  r  ;  + ^55.) + 9a '  30  are usually  ( a l lpartial  ( _ )  g y&  3  s  made f o r C o u e t t e  derivatives with  27  flow:  r e s p e c t t o t h e time  are zero). (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 e n d e f f e c t  (4)  The f l u i d  (5)  Isothermal  (6)  The f l o w  Then e q u a t i o n  of the cylinders.  i s homogeneous and conditions prevail  (3-26) l e a d s i m m e d i a t e l y  This  allows  one f i n d s  r dx 2  so e q u a t i o n  2rxdr  = 0  (3-29)  _  2 _d_ r dx  ( )^  = ^ ~ 2- T ~7~ dx  x  (  _  3  3  0  )  (3-21) becomes f  It  +  (3-28)  the transformation d dr  and  to  = constant  2  which  throughout.  i s laminar.  r x from  incompressible.  T  (3-31)  T  follows that = i  f  L  ' X: 0n  f(x) - dx S  1  (3-32)  J  TX  when t h e b o u n d a r y c o n d i t i o n to = 0 a t r = R is  used  and t h e s h e a r  Equation  stress  at the outer  (3-32) i s v a l i d ,  (3-33)  0  cylinder  i n particular  i s designated  a t r = R,  by  and becomes  XQ'  1 2  R f(T) ,  tt = ^ J  Now  r  — — T  Q  one d i f f e r e n t i a t e s  dtt dT  But,  T  J  from equation  equation f ( T  1  f ( T  Q  )  dT  finds  Q  (3-35)  dT.  R  (3-28)  0  =  2 0  T R  (3-36)  2  R  hence d  T  1T Thus  )  R  (3-34) and  R  R  T R  and  (3-34)  dx  equation  Vn  "Ro  r  (3-37)  S  >  (3-35) becomes (3-38)  2T  Since appear  i n equation  present the  form,  special  body  equation  (3-23) g i v e s  as a f u n c t i o n  (3-38) a r e d e t e r m i n e d  equation  case of s  of f l u i d ,  T_  = 0,  equation  t h e terms which  f r o m tt v s . T d a t a .  (3-38) i s a d i f f e r e n c e  2  o f T,  corresponding  equation  for f(x ). R  t o measurement  (3-38) c a n be r e d u c e d  But i n i t s  i n an  In  infinite  to (3-39)  since  f ( 0 ) = 0.  If T  i s introduced  f  Equation The Equation write  S T 2  (3-40) would difference  < R> -  m  be a good equation  (3-38) i s v a l i d f o r T „ and  T  into  result,  one  finds  d lntt d InT  (3-40)  approximation  c a n be  s o l v e d by  f o r a l l values  find  this  of T . r  for s <  0.1.  an i t e r a t i v e  procedure.  F o r example,  one  may  27  «.  and,  RJs x z  R  i n general  If tuting  S T  t h e complete  2  1 3  together,  , S^T-,  s e t o f equations  obtained  by s u c c e s s i v e l y  ... , s ^ ^ ^ T . f o r T „ i n e q u a t i o n  (3-38)  substi-  i s added  the r e s u l t i s  f(x ) R  Now  - f(s  2N  N-l T-n) =  dft  2P 2s^x  E P=0  R/  1  l e t N go t o i n f i n i t y ,  2P S  < " 3  T  and t a k e n o t e  4 3 )  R  of the fact  that  s  2(N-l)  (9) vanishes  i n this  f(x  w h e r e ft' = It  limit  since  ) = j = 2  T  s < 1.  E. s P=0  r  The r e s u l t  2P •[fi'l  i s  (3-44)  2  S  T  R  dft/dx,,. 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  glO  T R  >  f  r  o  m  which the  slope d  log  1 0  ft  m = —  (3-45)  d  can  b e more a c c u r a t e l y  constant  over  calculates  f(x ) R  = 2ft  rate  E s  R  t h e s l o p e ft', s i n c e m i s n e a r l y f o r many f l u i d s .  at the inner  m(s  P=0  R  8lOT  than  a wide range of x  t h e shear  Since  found  l o  L  cylinder  In that  a  c a s e one  using  T J  (3-46)  R  s i s l e s s than u n i t y ,  this  s e r i e s c o n v e r g e s , b u t does so (24)  slowly an  when s i s n e a r l y  approximation  unity.  to equation  In addition, Krieger  (3-38)  f o r -m  and E l r o d  found  l n s < 0 . 5 , and t h e r e s u l t i s  28  f ( 0 In  = -In  the  case  [1 - m  s  ln s +  ~r(m l n  of p u r e l y Newtonian  fluid  (3-47)  s) ] 2  the  shear  rate i s given  (3-48)  Y = For where n  a power l a w  i s given  fluid,  i n equation  m  of  equation  (3-5)],  and  fluid, late  the  the  3-7.  t h e momentum e q u a t i o n same r e s u l t  shear  Complicating As  to  s t a t e d i n the  derive equations  assumptions fluids  and an  the  average  stress  velocity  over  value  the  few  of  the  reasonable  of  the  the  fluid  (3-48),  the area  stress  i s constant  i s related  to  The  and  (3-49).  the  velocity  and  of  the  These  The  the  over  of  calculated  from  the  angular  a i d of  i s not  the  Consequently,  some m a t h e m a t i c a l  analysis  The  gradient  calculated  r a t e a r e unknown.  so  average.  be  cylinder.  to  a homo-  the w a l l ,  the  velocity  cylinder  i s related For  area  the  inner  torque  the  the  viscometer  g r a d i e n t cannot  shear  calcu-  homogeneous  The  of the w a l l .  rate with  gradient of  r a d i u s , so  and  shear  be  to  approximation.  the w a l l of  inner cylinder.  t h e w a l l may  used  law  many a s s u m p t i o n s h a v e b e e n made  on  shear  e x t r a measurements  be  f o r time-independent  over  velocity  f o r a power  ( 3 - 4 9 ) may i s a good  section,  stress  angular  angular  very  to  Rheology  shear  velocity.  constant  i n Suspension  e x e r t e d by  a t a p o i n t on  angular  the  the  t h e power l a w  flow,  i s , therefore, well-understood.  velocity  of the  Equation  previous  in fact,  torque  fluid  angular the  the  angular  geneous the  are,  Couette  (3-23), (3-46), (3-47),  whose v i s c o m e t r y  measures  of  Effects  (3-46) r e d u c e s  1/n  (3-49)  for this  i s found.  r a t e whenever  equation  [m r-  2m  1-s solving  (3-45) i s a c o n s t a n t  2mQ,  Y =  By  by  from the  of  always the gradient  Fortunately, a  a l l o w us  to  calculate  29 the  shear  rate  at  the w a l l ,  calculated.  Since  of  at  shear  rate  annular  not  so  former flow  simple.  Torque  yields  of  of  the  steady  the  the  ties  represented  pension  inherent  by  the  same  in  as  that  the  p r o f i l e may  stress the  and  be  wall  as  a  interior  can  be  function  of  calculated  This  relation  results  velocity  at  are  the w a l l  angular  homogeneity  velocity which  are  even  not  the  from  the  so  that  what  happens  to  t h e many  in  in  assumptions  The  time-dependent constant,  requires  the  particles near  in  more  properthe  the w a l l  fluid.  These  made  discussed  are  for  is  material  the  assump-  valid  particular,  happens  the bulk  measured.  kept  the  The  what  is  suspensions  generally  having  functions.  in  profile  inhomogeneity,  for  s t i l l  velocity  d i f f i c u l t y of  the w a l l ,  in  viscometer  the  discontinuous  ing  susis  not  complicatin  the  sub-sections.  3-7-1.  Continuum T h e momentum as  a  to  the  materials e.g.,  dextrose.  equations  continuum.  requires  relative  medium,  at  the v i s c o s i t y  angular  mathematical  t y p i c a l of  behaves  rate  and  necessarily  following  shear  shear  interact with  effects  and  suspensions.  than  use  the v i s c o s i t y  homogeneous,  Couette  Although  state  time-dependent  to  the  time  the v e l o c i t y  suspensions.  calculation  system  is  i n t e r p r e t a t i o n of  s t i l l  of  tions  Hence,  is  which  equations. The  is  fluid  the w a l l  space.  momentum  the  at  are  Treating  that  the molecular  size  of  with  suspended  relatively  polyethylene  based  on  the  suspensions  as  size  suspending  of  the  particles. low molecular  glycol,  assumptions  sodium  It  a  continuous  is  as  the  the  water,  fluid  two-phase  medium be  therefore  weight  chloride,  that  small  desirable suspending  and/or  30 3-7-2.  Steady It  carried  i s obvious  out  ibrium.  while  dependent the  Couette  is  be  of  sufficiently  this  large  case n e i t h e r  annular  nor  that  since  constant, the w a l l  3-7-3.  equation  shear  Wall Wall  these  center than  effects  space  space  that  the  the  of shear  variation with  time.  which  throughout  requires  refer  The  that  instruments  i s s m a l l compared w i t h  the  that  time.  of  the  shear  However,  suspended  rate,  velocity  time  a r e known.  of the  We  cylinder  rate  thus  viscomcylinders In  i n the  do  c a n be  know, kept  shear  employed  the  rate  particles.  profile  the presence  are reviewed  effective  by  is  rate  at  called  of  effect,  of the viscometer w a l l .  mechanical cannot  the p a r t i c l e .  to the w a l l w i t h a  of the bulk  fluid.  i n suspension Many  Mashmeyer  i n suspension  size  phenomena o b s e r v e d  lower  exclusion,  approach The  result  average  Maude and  Whitmore  occurs because  the w a l l is a  any thin  the  closer layer  c o n c e n t r a t i o n of ( 27 ^  than  time-  c l e a r a n c e between t h e  size  shear  i s not  to c a l c u l a t e  a c e r t a i n measure of the  to s e v e r a l  from  simplest wall  suspension next  equilibrium  equil-  Effects  of a p a r t i c l e  the  of  t h e y have assumed  (3-48) g i v e s us the  rate  at  rate."  effects The  shear  the angular v e l o c i t y  viscometry which a r i s e of  1 ) , and  compared w i t h  their  throughout  "apparent  annular  distribution  space,  however,  often  of  viscometry i s  f o r the part  impossible at present  the annular  suspensions  time-dependent  H e n c e , many w o r k e r s h a v e e m p l o y e d  (s = R / R Q =  constant over  etry  rate.  on  c o n d i t i o n s except  calculation  t y p e w i t h an  radii  experiment  state  i t i s almost  t r u e shear  cylinder  the  under unsteady  Therefore, the  difficult,  of  that  of  particles  (28^ '  calculated  the  31 magnitude spheres  of  in  measured not  creeping  enough to  exclusion  closeness sion.  of  No  made  wall  for  may  exclusion  f l o w and  viscosities.  large  ical  mechanical  be  compared  Their  explain  the  is  the  their  conclusion  expected  approach  on  viscosity  a  suspension  calculation with  was  that  experimental for  of  suspensions  experimentally  mechanical  results. of  A  exclusion  similar  fibers,  except  determined  by  o r i e n t a t i o n as  theoretical calculations  or  e x p e r i m e n t a l measurements  the magnitude  of  this  well  of  as  was  mechanthat  fiber  the  dimen-  have  been  effect.  (19) Karnis, capillary profiles the  flow are  dilute  are  become  flow,"  of  parabolic  particles  files  Goldsmith,  and Mason  suspensions  and  the  the  same.  blunted.  The  because  the  showed of  that  spheres  and  local velocities At  higher  blunted  p r o f i l e may  not  concentrations,  complete  plug  Blunted  velocity  profiles  flow  of  low Reynolds  fibers, the  is  the  designated  completely  the  fluid  concentrations  region be  in  velocity  and  those  velocity  "partial  flat.  At  number  even  of  pro-  plug higher  develops. have  been  observed  in  concentrated  (45) suspensions them  in  in  of  fibers  and  flow.  There  several  are  In  can be  First,  this  the v e l o c i t y  file  well.  capillary  suspensions.  stress.  as  case  caused  by  possible i t  the  profile  Stankoi  is  can  reasons  occur  term "plug  p a r t i c l e - d e p l e t e d suspension near  the  concentrated  presence  Experimental of  a wall  at  the  axis,  evidence^ ^  are  a  2  important  is  an  and  the  this  case.  a  yield  description, a blunted  away  the  a  velocity that  exhibits  profile  Second,  particles  the w a l l has  observing  velocity  accurate  flat.  indicates in  a blunted  f l u i d which  flow"  of  reported  al.  for  mathematically  The  this.  in  radial migration  region  et  both  from  lower  the  than  reflects  i n e r t i a and  Third,  wall.  viscosity  profile  pro-  the  suspension  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 . Another w a l l e f f e c t stems from p a r t i c l e i n t e r a c t i o n w i t h the  The  t h i c k n e s s of a l a y e r i n which  wall.  t h i s i n t e r a c t i o n o c c u r s i s supposed  to  (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 richer  .  The  s u s p e n d i n g medium c a n  i n t h i s l a y e r than i n the r e g i o n f a r from the w a l l .  that the f l u i d expected.  T h i s means  v e l o c i t y near the w a l l i n P o i s e u i l l e flow i s higher  V a n d ' (51)  p r e c  iicted  be  than  t h a t the p r e s s u r e drop v e r s u s f l o w r a t e  curves f o r d i f f e r e n t viscometer geometries would not W a l l e f f e c t s are of fundamental  superimpose.  significance for rheologists  b e c a u s e 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 d e p e n d o n v i s c o m e t e r geometry.  Theories  have been developed  a f f e c t e x p e r i m e n t a l d a t a and w i t h a decrease  they p r e d i c t that t h e i r  of the s i z e f o r f i b e r  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 the f i b e r  l e n g t h and  I t was be used  recommended  End The  t o l i e somewhere b e t w e e n  the  that c y l i n d r i c a l  by  orientation.  s u r f a c e s w i t h deep  end  be u n o b s e r v a b l e  practical  grooves  wall within  importance.  Effects  f o r m u l a e f o r s h e a r s t r e s s and  derived e a r l i e r , the f l a t  i t i s not  one  "somewhere" d e t e r m i n e d (54)  t h u s be o f l i t t l e  not  Intuitively,  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  e x p e r i m e n t a l e r r o r and  will  t a k e n as i t s c h a r a c t e r i s t i c  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 .  diameter;  Yet  i n a Couette type viscometer i n order to eliminate the  effects.  3-7-4.  effects  effects w i l l increase  suspensions.  s h o u l d be  might expect the c h a r a c t e r i s t i c dimension fiber  wall  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  predict the e f f e c t  size;  t o show how  do n o t a l l o w any  shear r a t e , which have been  s h e a r i n g a c t i o n between suspensions  surfaces of the c y l i n d e r s i n a Couette flow.  The  end  and  33 effects  can  cylinders can  be  of  checked  different  eliminate  3-7-5.  and  the  end  length. effects.  the  density  of  d i s p e r s i n g medium, t h e suspensions  lengths  are  of  the  well  This w i l l  be  of  the  explained  inner  inner  cylinder  later.  the  phase  i s matched w i t h  i s homogeneous on  same o r d e r the  the  o f m a g n i t u d e as  s u s p e n s i o n s may  not  the  that  be  the  s c a l e of  fibers, the  of  density.  some o f  length  of  whose  gap  h o m o g e n e o u s on  the  Incompressible  density be  as  dispersed  experiment.  Suspensions  should  design  work, however, c o n t a i n  c y l i n d e r s , so  3-7-6.  their  careful  suspension  in this  nearly  b e t w e e n two scale  A  making measurements w i t h  Homogeneous If  The  c o r r e c t e d by  does not  taken  be  treated  change w i t h  i n preparing  i n aging  3-7-7.  can  suspensions  as  incompressible  t i m e and  suspensions to prevent  to  fluids,  position. avoid  In  trapping  as  long  as  addition,  care  a i r bubbles  as  evaporation.  Isothermal (32) Middleman  presented  temperature  rise  heat  is negligible  effect  coaxial 3-7-8.  an  i n Couette flow. under  c y l i n d e r viscometer  with  approximate  estimation  It i s reasonable  the  usual  a device  operating  to  f o r the  assume t h a t  conditions  t o keep w a l l  maximum  of  the  a  temperature  constant,  Laminar (4) Bird  the  critical  e t al.  showed a n  R e y n o l d s number  KM  approximate  for a  Couette  expression  flow  for  estimating  as  ,41.3  trans.  (TUP/2  (  3  "  5  0  )  In  addition,  cometer  can  observation  confirm whether  Taylor's  3-7-9. The  a visual  Taylor  the  of  flow  can be  defined, R  The  on-set  rotating  of (the  =  T  (Rfl)(R  0  instability outer  - R>(^)( for  cylinder  <V All  the viscometric  laminar  region.  is  flow of laminar  suspensions or  in  a  vis-  not.  Instability  number  N  the  the at  0  with -  R  R—)  should  notation,  as  x (3-51) flow with  may  critical  present  2  Couette  rest)  measurements  the  = be  occur  4  at  the the  inner Taylor  1  cylinder number  (  carried  out  in  the  3  "  5  2  stable  )  CHAPTER 4  D E F I N I T I O N OF  Before specifically used  THE  discussing,  focussed  on  PROBLEM AND  in detail,  in this  for i t s solution w i l l  be  useful gives  f o r that the  slurries  definition chosen  are  stated  the  experiments  4-1.  and  of  ground  little  fiber  i s known a b o u t  suspensions seemingly his  slurries  that  the  approach  often valuable  systems  ranges  i n S e c t i o n 4-4  Section  of  t o be  and 4-2  artificial investigated  t h e methods used  in  mentioned.  Workers  the  literature  are h i g h l y the  r e s e a r c h on  endless  the  i n S e c t i o n 4-1.  survey  t h e phenomenon o f t h i x o t r o p y and  concentrated So  from  d e f i n e d and  problem  chapter.  Then the  Finally,  of Other  i s apparent  that  are outlined  variables with  are b r i e f l y  be  in this  the problem.  their  Recommendations  thesis w i l l  of other workers are  they  i n S e c t i o n 4-3.  It  And  purpose,  ATTACK  t h e e x p e r i m e n t a l work, t h e  outlined  S i n c e recommendations  METHOD OF  the  of  theoretical  the r h e o l o g i c a l  complex.  rheology  and  According  concentrated  s u b j e c t c o u l d be  back-  behavior  of"  to Mashmeyer^  fiber  continued  in  ways.  recommendation:  A s t u d y o f t h e e f f e c t o f f i b e r l e n g t h w o u l d be among t h e m o s t i n t e r e s t i n g and u s e f u l , b u t s u c h a s t u d y would be d i f f i c u l t both 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 b e e n d e v e l o p e d and b e c a u s e t h e 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 alumina  examined  the  i n propylene  thixotropic  glycol  and  behavior  recommended: 35  of a suspension  of  colloidal  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 s h a p e 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 a p p e a r 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 m i g h t 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 a n d t h e i r d i m e n s i o n s a r e more i m p o r t a n t f a c t o r s t h a n weight per cent i n determining r h e o l o g i c a l behavior.  (6) Brown  wrote:  Only  two  both  o f which were of t h e r i g i d  which can  particle  c a n be  be  prepared  made by  suggested  system  would  a  plate  type.  are rod-like  fibers  area into  that  be  were c o n s i d e r e d i n the p r e s e n t  easily  cutting  cross-sectional Hu^"^  shapes  Other  particles,  of c y l i n d r i c a l  the desired  a CMC-Corn S y r u p  and  study,  shapes which  elliptical  lengths.  system  good medium f o r s u s p e n d i n g  or  PEG-Dextrose/Fructose  artificial  particles.  (29) Mercer  used  a Haake R o t o v i s k o V i s c o m e t e r  dependent v i s c o s i t y  of t h i x o t r o p i c  forth  recommendations:  the f o l l o w i n g  For  any  future mechanical  should  be  designed  bentonite-water  experiments,  especially  f o r h i s study  the  on  the  suspensions  time-  and  set  viscometer  f o r time-dependent  measure-  ments. For  example,  purpose, 4-2.  Definition  workers  on  do  the  literature  (e.g.,  yield  variation What  t o be  particle  suspending  survey  treated  and  the recommendations  in this  of a r t i f i c i a l  s t u d y may  slurries  concentration, salt  medium, e t c . ) a f f e c t stress,  equilibrium  of t h i c k n e s s of  the  now  of  of  other  stated. ratio  of  concentration, viscosity time-dependent  stress,  stress  flowing layer with  flow of the  be  ( e . g . , L/D  decay  time,  i s the model e q u a t i o n which b e s t d e f i n e s the  characteristics  motions".  Problem  the v a r i a b l e s  particles,  (2)  of the  the problem  How  of  s h o u l d b e made f o r t h e  i n h i s o p i n i o n , of "minimization of o s c i l l a t o r y  Based  (1)  the m o d i f i c a t i o n s of the viscometer  slurries?  shear  flow  curve,  etc.)? time-dependent  (3)  What k i n d  of  have w i t h  the v a r i a b l e s  4-3.  Systems In  this  to  i s carried  out  dextrose,  and  polyethylene  The  L/D  ratio  from  from  10  to  It effect  of  single Also, on the  wt  %.  found  Dextrose  i n an  varied  from  ionic  slurries  details will  Method  be  of  There are  of  the  given  0.04 and  to a d j u s t  the d i s p e r s e d n y l o n  the  work of  slurries  sodium  L/D  series  of  t h a t of  of  chloride,  +  larger  ratios  i n the next  t o 156.  0.17.  i t s concentration was  than so  the  of  and  mole NaCl  the  runs  of the  the  dispersing  out  i n 10%  was  and  t h e make-up o f  with  a  PEG-H^O).  identical  analysis 37.4,  that  time-dependent  almost  22.9,  the  fibers.  experimental  69.8  was  varied  the d e n s i t y of  the v a r i a b l e s  1.0  The  Sodium  a n a l y s e s were c a r r i e d  DEX  selection  22.9 to  d i s p e r s e d p h a s e on  (i.e.,  ratios  particle  of  from  force,  used  time-dependency were o b s e r v e d , with  experimental  c o n c e n t r a t i o n of" E E C  experimental  d i s p e r s i n g medium L/D  varied  was  much g r e a t e r t h a n the  the  i.e., artifical  was  The  initial of  effectively  fibers  mole/1.  the v a r i a b l e s  results  equation  slurries?  i n aqueous s o l u t i o n s  f i b e r s was  2.0  Therefore  The  4-4.  the  to  for particle  the  the model  glycol.  medium t o t h a t o f  was  artificial  model systems,  to produce the  0.00  f l o w was  medium.  fibers  of n y l o n  of  used  50  dispersing  shear  nylon  fraction  varied  the  problem on  sized  c h l o r i d e was  the parameters of  of  solve the  regularly  volume  do  Chosen  order  thesis  correlation  effects  limited  to  69.8.  those  artificial  chapter.  Attack several kinds  of viscometers  available  on  the market  (52) they  are  described i n detail  i n the  literature  ;  f o r example,  the  and  capillary  type,  coaxial  measurement  of  viscometers  a r e most  viscometers  and  how  long  of  comparatively  cone-and-plate  The inner The to  gap  thickness  also of  of  can  be  plate.  take.  types  of  kept  in  be  Since a r t i f i c i a l  i n the  Consequently,  f o r the  the  the  present  slurries  study,  the very  a coaxial  the  small  space  cylinder  type  investigation. torque  at a constant the  two  For  c o n t i n u o u s l y measured  a p p l i c a b l e because of  exerted  angular  c y l i n d e r was  between the-two  the  of that  the  measured  tracer  artificial  the  f o r the  was  the  slurries of  t h i c k n e s s of  time-dependent as  cylinders  flowing layer  on  t h e . s u r f a c e of  velocity,  varied  in this  on  the  problem d e f i n e d i n S e c t i o n  t o be  kept  comparatively  t h e m a t e r i a l may  shear  flow.  time by  surface of  photographed.  flowing slurry  of  had  with  the  time.  work f r o m  0.116  The gap 4-2.  a very  Therefore  the  the m a t e r i a l under of  enough'  vary with  a movie camera.  results  will  wide  to  long f i b e r s .  f l o w i n g l a y e r would be  a function of  particles  viscometer width.of  nature  were used  measures the  rotating  velocity  accommodate  factor  the  last  type.  rad./sec.  The  felt  not  chosen  viscometer  cylinder,  6.283  was  was  was  the  s i n c e the m a t e r i a l can  long particles  cone and  angular  suitable,  cone-and-plate  flow  t h e m e a s u r e m e n t may  type  viscometer  and  shear  i t s time-dependent  matter  of  type,  a time-dependent  no  between the  cylinder  The  time.  important  thickness The  in  the  t h e measurement  of  the  role  was  movement  test  p l a y a key  It  i n solving  the  CHAPTER  5  EXPERIMENTAL MATERIALS AND  The  definition  chapter  and  the  present  chapter  of  the  p r o b l e m has  experimental describes  method has  the  TECHNIQUES  been given  i n the  been o u t l i n e d  experimental  materials  previous  there. and  The  techniques  in  detail. In  Section  mentioned be  and  5-1  then the  presented.  f o r the  Section  will  measure the meter by  5-1.  with  will  thickness  of  cine  the  the  i n the  describe be  experimental  flowing  artificial  the  given  layer  experiment w i l l  be  slurries  apparatus.  A  will  torque  i n Section  5-3.  procedure,  i n c l u d i n g how  i n the  annulus  Finally,  of  the  to  visco-  photographs.  Materials This  phase,  2)  section describes: the  liquids  employed  i n matching  slurries  were  5-1-1.  following  utilized  1)  the  for a  d e n s i t i e s of  particles  used  as  a  dispersed  d i s p e r s i n g medium, 3)  b o t h p h a s e s , and  4)  how  the the  technique artificial  prepared.  Dispersed  Selection  (1)  5-2  used  f o r m a k i n g up  apparatus w i l l  deal  means o f  materials  methods  Section  calibration 5-4  the  of  Phase materials  f o r the  particles  i s governed  by  the  properties:  Ease with  which  desired  s h a p e s and 39  s i z e s can  be  cut  or  punched.  40 (2)  I n e r t n e s s , w h i c h means t h a t no c h e m i c a l the  dispersing  (3)  The a b i l i t y  (4)  Density, which dispersing  (5)  takes  place  with  medium.  of the p a r t i c l e s  t o be wetted  by t h e d i s p e r s i n g  medium.  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  medium.  Flexibility,  i . e . , not too r i g i d  so t h a t  damage t h e w a l l o f t h e v i s c o m e t e r , they don't  reaction  lose  the property  the particles  do n o t  and n o t t o o f l e x i b l e  of their  original  so t h a t  shape under a  shear  flow. Because i t s a t i s f i e s for  the p a r t i c l e s .  The n y l o n  T y p e 120 b y E . I . du P o n t of  the reaction  diamine  acid  25,000 a n d a d e n s i t y ^  from  regular lated  4 7  shape-and-size  by a v e r a g i n g  ^ o f 1.14  the  the physical  length-to-diameter  different  sized  nylon  The  sized  fibers  i n which  they  plastic of  the nylon  fibers  hexamethylene (47) of about  five  The l e n g t h s o f t h e f i b e r s  through  from  a calibrated  were  graticule.  ratios  A photograph  c a n be seen  were i n e r t had been  Table of  Table  5-2 shows five  i n F i g . 5-1.  and were s t o r e d i n t h e bags o f v i n y l  shipped.  c a n be a v o i d e d  calcu-  magnified  fibers.  of the fibers.  lengths  kinds of  properties of the nylon  fibers  the threads  have been c u t i n t o  o f 50 f i b e r s  as  i s the product  weight  by F i b r e t e x L t d . , p r o d u c i n g  t h e measured v a l u e s  chosen  were manufactured  The d i a m e t e r o f  The t h r e a d s  flock.  f l o c k was  This nylon  molecular  g/cc.  photographs which had been t a k e n 5-1 g i v e s  here  p o l y m e r i z a t i o n between  t o an average  t o 6.72 mm  used  de Nemours & C o .  a t 43.1 m i c r o n s .  0.987 mm  threads  of condensation  and a d i p i c  was u n i f o r m  t h e c o n d i t i o n s above, n y l o n  I n h a l a t i o n and s k i n  by c a r e f u l  handling.  contact  TABLE  5-1  P H Y S I C A L PROPERTIES OF NYLON  FIBER  (47) 1.14  Density Molecular  Weight  g/  c c  25,000  (34) Shear Modulus Young's M o d u l u s Poisson's Glass  3.55  ( 3 4 )  Ratio^  Transition  Diameter  x 101°  1.22  3 4  dynes/cm  1010  x  dynes/cm  0.4  ^  Temperature^ ^ 3  50  °C  43.1  ( 4 8 )  TABLE  m  i  c  r  0  n s  5-2  LENGTH-TO-DIAMETER RATIO OF  FIBERS  Standard Length(mm)  L/D  Deviation  1.  0.987  22.9  0.168  2.  1.62  37.4  0.0828  3.  3.01  69.8  0.0877  4.  5.03  116.  0.108  5.  6.72  156.  0.106  2  2  42  Fig.  5-1.  Photograph of [scale: mm]  5  Differently  Sized  Fibers  43 5-1-2.  D i s p e r s i n g Medium  The by  the  selection  of m a t e r i a l s f o r the  requirements  be  a Newtonian  (2)  have v a r i a b l e v i s c o s i t y ,  (3)  h a v e same d e n s i t y a s  (4)  wet The  ethylene is  glycol,  very  dextrose,  (PEG)  manufactured  is  by  used  The  dependent  was  used  of nylon wet  on  To  The  of  Poly-  the water-soluble  9,500.  The  polymers  P o l y g l y c o l E9000  viscosity  o f PEG-H^O  with  solution  Sodium c h l o r i d e i s a  i s a l s o s o l u b l e i n PEG-H^O-NaCl  wellsolution  d e n s i t y o f d i s p e r s i n g medium t o has  and  been  that  found  to  well. nor  PEG  the  dextrose  requires special  i s a non-volatile, non-toxic,  cardboard  carton i n which solid,  but  i t had  care  in  storage  stable solid  been  and  shipped.  i t i s hygroscopic  and  so  The i t  was  bottles.  Technique  of Density  settling  a d e n s i t y m a t c h was  a p p r o p r i a t e amount  to  a 10  solution  or  carried  an  cc  sodium c h l o r i d e .  Company a s  i s also a non-toxic  avoid  work c o n s i s t s of  T h i s PEG-H^O-NaCl-Dextrose s o l u t i o n  stored i n the  5-1-3.  i s one  to adjust the  t h e PEG  i n glass  w a t e r , and  i t s concentration.  fibers  or handling.  stored  and  Chemical  Dextrose  fibers.  the nylon  dextrose  Dow  i n order  Neither  was  particles,  here  w e i g h t a b o u t  known e l e c t r o l y t e . and  nylon  particles.  glycol  molecular  liquid,  d i s p e r s i n g medium c h o s e n f o r t h e p r e s e n t  polyethylene  restricted  that i t :  (1)  nylon  d i s p e r s i n g medium i s  of  Matching  floating out  the  i n this  dextrose  was  of PEG-H 0-NaCl. 2  of  particles  i n the  investigation.  measured The  by  The  a balance  solution  of  suspensions, weight and  was  PEG-H 0-NaCl2  of added  44 D e x t r o s e was mixed w e l l u s i n g a magnetic  stirrer.  The d e n s i t y o f t h e  s o l u t i o n was measured by means o f 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 c o n s t a n t temperature b a t h a t 30.0°C.  T h i s procedure was r e p e a t e d by a d d i n g  d i f f e r e n t w e i g h t s of d e x t r o s e t o t h e same s o l u t i o n o f PEG-R^O-NaCl.  The  d e n s i t y o f t h e PEG-H^O-NaCl-Dextrose s o l u t i o n was p l o t t e d a g a i n s t t h e w e i g h t o f d e x t r o s e added as shown i n F i g . 5-2.  From t h e graph t h e  weight o f d e x t r o s e , which must be added t o make t h e d i s p e r s i n g medium d e n s i t y e q u a l t o t h a t of t h e n y l o n f i b e r s ( p = 1.14 mined.  g / c c ) , was d e t e r -  T h i s t e c h n i q u e was employed f o r a l l t h e s o l u t i o n s of PEG-R^O-  N a C l - D e x t r o s e w h i c h were p r e p a r e d f o r t h e p r e s e n t s t u d y . V i s c o s i t y o f t h e 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 a t 30.0°C by means of a Cannon-Fenske c a p i l l a r y v i s c o m e t e r Type 300, w h i c h had been c a l i b r a t e d by u s i n g v i s c o s i t y s t a n d a r d purchased  from Cannon Instrument Co.  F i g s . 5-3 and 5-4 show t h e  r e s u l t s of t h e v i s c o s i t y measurements. i s a l s o p l o t t e d i n these f i g u r e s .  oils  The weight of d e x t r o s e added  F i g . 5-5 shows e f f e c t o f d e x t r o s e  on t h e 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 t h a t t h e v i s c o s i t y o f t h e 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 n o r d e x t r o s e a f f e c t s t h e v i s c o s i t y much. 5-1-4.  Slurry Preparation  B e f o r e t h e f i b e r s can be d i s p e r s e d i n a d e n s i t y - m a t c h e d  dispersing  medium, they must be washed t o f r e e them from s a l t s and w e t t i n g a g e n t s . A p p r o x i m a t e l y 50 grams o f s i z e d f i b e r s were soaked 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 w a t e r .  The f i b e r s were p e r m i t t e d t o 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 t h e  1.15  1.10  0  I  2  3  4  Mass  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 0 2  Fig.  5-2.  Density Match  46  200 150 CL  o  >.I00 h o o  t/5  >  5 0  Y 0  10 2 0 3 0 4 0 5 0 Concentration o f P E G in H 0 ( % ) 2  Fig.  5-3.  E f f e c t of PEG C o n c e n t r a t i o n on Medium V i s c o s i t y  Suspending  47  Fig.  5-4.  Effect  of  NaCl  on  Suspending  Medium  Viscosity  48  W e i g h t of D e x t r o s e A d d e d to I 0 c c ( 3 0 % P E G - I - L O + N a C l ) (g) Fig.  5-5.  Effect  of Dextrose  on Suspending  Medium  Viscosity  49  f r e s h w a t e r was i n t r o d u c e d a  glass  tube connected  overflowed  continued and  measured  until  arm.  using  nylon  was t h e n fibers  deposit  filtered  were d r i e d  while  by i n s e r t i n g  the supernatent  The c o n d u c t i v i t y o f t h e l i q u i d  dipping  electrodes.  This  i n a Buchner Table-Type a t room t e m p e r a t u r e  through an impinger  due t o e v a p o r a t i o n  the impinger  fibers  weighed  was f o u n d  which had been  were s t o r e d As  i n another  The washing  was  weighed  i n a 500 c c v o l u m e t r i c  o f PEG-H^O.  t h e PEG-H 0 s o l u t i o n . 2  weighed  stirrer  i n a 1-liter  by a magnetic  solution  To fibers  nitrogen.  stirrer  flask.  tube  The d r i e d  o f PEG was water t o  amount o f N a C l  T h e f l a s k was t h e n  The a p p r o p r i a t e  i n t o which  filled  amount o f  dextrose  500 c c o f PEG-H^O-NaCl  f o r a day and a n i g h t ,  viscous,  the appropriate  i n a 100 c c b e a k e r ,  particles  until  c l e a r , and t r a n s p a r e n t  were c a r r i e d o u t a t room  The n y l o n  amount  The a p p r o p r i a t e  the f i n a l  liquid.  A l l  temperature. amount  of the dried  nylon  i n t o which t h e measured volume o f  P E G - H ^ O - N a C l - D e x t r o s e s o l u t i o n was s l o w l y  burette.  of the inner  2  flask,  produce the s l u r r y  was w e i g h e d  no  T h e s o l u t i o n o f P E G - H ^ O - N a C l - D e x t r o s e was a g i -  became a v e r y  these preparations  desiccator  T h e s o l u t i o n o f P E G - H 0 - N a C l was m i x e d b y  f o r 30 m i n u t e s .  s o l u t i o n was p o u r e d . tated  The f i l t e r 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  concentration  a magnetic  The n y l o n  t o a v a c u u m pump, u n t i l  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 and d i s s o l v e d  was  desiccator.  desired  with  i n a vacuum  i n liquid  liquid  p r o c e d u r e was  funnel.  on t h e s u r f a c e  immersed  the  the  line,  fibers  t h e c o n d u c t i v i t y showed n o f u r t h e r c h a n g e .  w h i c h was c o n n e c t e d  was  of the settled  c o n d u c t i v i t y measurement were done a t room t e m p e r a t u r e .  slurry  of  t o the water  through a side  periodically  to the area  were c o m p l e t e l y  added  through  wetted  a 50 c c  by t h e s o l u t i o n and  50 the r e s u l t i n g stainless the  slurry  steel  was  mixed  spatula.  The  aging  head,  t i m e was  which  kept  was  measured  In a d d i t i o n ,  s l o w l y by  s l u r r y was  removed  The  i n each  fibers  were,  The  (p  5-2.  = 1.14  diagram  of  of the  The  bob  slurries  jacket  of  diameter which  The  bob  bearings, portion. g e a r box  and  The  always  i n the  volume  fraction  and  mixed.  the d i s p e r s i n g  medium  volume o f d i s p e r s i n g of fibers  and  stress  of the  medium dispers-  time-  a Haake R o t o v i s k o V i s c o m e t e r cylinder  cylinder Inside  rotates  viscometer.  A  Model  schematic  5-6.  (cup)  i s surrounded  t h e cup  to shear  i s an  inner  the m a t e r i a l  by  a  temperature-  cylinder i n the  (bob)  gap  bob.  This  to a shaft shaft,  i s separated into The  was  of water.  i s fixed  the viscometer.  The  t h e known d e n s i t y  s e t - u p s i s shown i n F i g .  and  r u n was  b e f o r e t h e s l u r r y was  of f i b e r s  type coaxial  controlled  between cup  (30.0°C).  of the s l u r r y  t o measure the shear  outer stationary  smaller  into i t s  Apparatus  artificial  a rotating  f o r each  measured by w e i g h t ,  from  to  g/cc).  v i s c o m e t e r used  dependent RV1,  easily  clean  the beaker  t h e n mounted  cylinder.  determined  calculated  Description The  was  from  a  point.  set the l e v e l  proper weight  therefore,  i n g medium  this  edge o f t h e i n n e r  slurry  a r e most  volume.  from  was  hand u s i n g  at a constant temperature  cc which  to the upper  fibers  held  the volume of t h e s l u r r y  c o n s t a n t a t 37.0  annulus  by  and  sample-container of the viscometer which  measuring  of  gently  upper  portion  a transmission  extending from  which  an upper  i s mounted on d r y - r u n n i n g driving  i s d r i v e n by shaft.  the measuring  The  portion  and  a synchronous lower  shaft  head roller  a lower motor  of  driving  through  i s c o n n e c t e d by  a a  DRIVING UNIT  51  MEASURING HEAD  RECORDER  CO-AXIAL CYLINDERS  TEMP CONTROL BATH (A)  Thermostat  Rotary Test  bob  sample  Container  Measuring Head with Measuring Outfit M V attached Fig.  5-6.  Instrumentation  jacket  (outer  cylinder)  (B)  (A)  and Measuring  Head  (B)  52  torsion shaft the  spring  t o t h e upper  i s a wire-wound  lower  against  shaft.  shaft.  Fixed  potentiometer with the contact  A s t h e two s h a f t s  the torsion  spring,  circuit  which  on t h e c o n t r o l  into  recorder.  this viscometer.  because  i t could  the a r t i f i c i a l  cup  had an i n n e r  matching  bottom  slurries  diameter  wide  S V I I bob h a d a n o u t e r d i a m e t e r  The  resulting  longest  fibers  these cylinders  of  i n this  end e f f e c t s  two c y l i n d e r s was l e f t  o f 20.2 mm  artificial  under  t h e bob a n d o n l y  contact  with the material  is  had grooves  shown i n F i g . 5-8.  with the steel.  a n d a h e i g h t o f 19.6  slurries  mm.  I t was  because t h e  The c o n f i g u r a t i o n o f  between t h e bottoms reservoir  of material.  end so a n a i r b u b b l e  t h e s h a r p edge o f t h e bottom  was  was i n  tested.  a t t h e walls,,  i n the axial  The MVII  o f 43.0 mm  t o p r o v i d e an un-sheared  trapped  slip  chosen  shown i n F i g . 5-7.  t h e bob i s h o l l o w o n t h e l o w e r  bob  o f bobs and cups  was s h e a r e d was 10.9 mm.  this,  prevent  may b e f e d  f o r t h i s work, o f s t a i n l e s s  Besides  To  to a  may b e r e a d  fibers.  a c l e a r a n c e o f 8.4 mm  t o be  to the  gap b e t w e e n t h e b o b a n d t h e c u p  w o r k w e r e 6.72 mm.  i s schematically  To m i n i m i z e  i n sizes  and a d e p t h  the slurry  f o r the current  used  output which  of comparatively long  made, s p e c i f i c a l l y  wide  change p r o p o r t i o n a l  possible  o f 42.0 mm  gap i n w h i c h  end o f  t o one a n o t h e r  T h e s e t o f M V I I c u p a n d S V I I b o b was  The  sufficiently  t o t h e upper  box o f t h e v i s c o m e t e r o r which  provide a f a i r l y  for  end o f t h e upper  The p o t e n t i o m e t e r i s connected  T h e r e w e r e many c o m b i n a t i o n s for  relative  p r o v i d e s an e l e c t r i c a l  from a meter a chart  turn  a resistance  a n g u l a r d i s p l a c e m e n t i s measured. bridge  t o the! lower  the surfaces  direction.  of both  t h e cup and t h e  The d i m e n s i o n o f t h e s e  grooves  53  Fig.  5-7.  Configuration  of  Cylinders  54  0.19mm 0.38 mm  2TT/38 rad  2?r/90  (9.474°)  (4.00 ) e  Fig.  5-8.  Grooves  on t h e  Cylinders  In a d d i t i o n , surfaces  from  The for  t h e bob  corrosion  due  Haake R o t o v i s k o  transmitting  and  the  cup  to the  were g o l d - p l a t e d t o p r o t e c t  salt  Viscometer,  the d r i v i n g  used  as  i n the  purchased  power f r o m  the motor  artificial had  cable-drive,  operation,  and  however, caused  system  a flexible  '  was  cable head.  (29)  oscillations  t h e r e f o r e the d r i v i n g  slurries.  to the measuring  (17) This  the  during  modified  steady  to a  shaft-  drive. The apparent  viscometer shear  rate  with  the present  range  o f 0.101  combination  t o 16.3  sec  of cylinders  ^ and  a  shear  has  an  stress  2 range  of  45  t o 6Q,000 d y n e s / c m  cylinder  i s controlled  position  and  The  torque  by  the  To inside  Packard  prevent  Torque The  by a  head  experiment  was  The  t o r q u e w h i c h was  connected  the  Chart  of  Recorder  inner  5-3  shows t h e  the  inner  i n n e r c y l i n d e r was  tight  Model  gear  cylinder. recorded  7100B.  plexiglass  i n place during aging  l i d was and  was  aging  conremoved  started.  fixed  recorded on  the  to the  as  a voltage  s u r f a c e of the  surface of  the  t o a number o f known w e i g h t s .  electrical on  Table  of the  the m a t e r i a l during the period of  a w h e e l w h o s e b e a r i n g - f r i c t i o n was  were read  velocity  Calibration  s t r i n g w h i c h was  by  box.  30.0°C, a  always  g e n e r a t i n g a known f o r c e  was  Strip  at  angular  angular v e l o c i t y  surface of  Moseley  T h i s l i d was  when t h e d e c a y 5-3.  the  The  a gear  evaporation of  the measuring  structed.  through  corresponding  e x e r t e d on  a Hewlett  .  outputs  associated with  t h e r e c o r d e r and  plotted  (mV),  calibrated  inner cylinder  inner cylinder The  string  c o n s i d e r e d t o be torques  was  was  with  and  which  supported  negligible.  generated  in this  a g a i n s t t h e known t o r q u e s .  manner The  TABLE 5 - 3 ANGULAR V E L O C I T Y OF INNER  Gear Position  162  81  Angular  Velocity  CYLINDER  (rad/sec)  TT  0.0388  -|- TT  0.0775  ol  ol  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  54  plot  i s a straight  l i n e through  t h e f o l l o w i n g e q u a t i o n was  Experimental  (4.46  x IO )  (1)  x  4  Hence  (Voltage i n  mV).  Procedure  L i s t e d below i s the step-by-step mental  shown i n F i g . 5-9.  obtained:  ( T o r q u e i n dyne.cm) =  5-4.  t h e o r i g i n as  program f o l l o w e d i n a l l e x p e r i -  runs:  The  viscometer  r e c o r d e r was allowed  was  turned  turned  on  on.  ( w i t h gear box  then  turned  The  A f i f t e e n - m i n u t e warm-up p e r i o d  f o r the e l e c t r o n i c s .  30.0°C a n d  i n neutral).  on.  The The  w a t e r c i r c u l a t o r was  bob  was  m o u n t e d on  was  set  the  chart  to  measuring  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 g e a r was t o a l l o w t h e bob shifted  (3)  to r o t a t e f r e e l y i n a i r .  t o n e u t r a l and  The  cup  was  then  i n place to prevent desired period at (4)  The  l i d was  The  s l u r r y was  placed  in  t h e l i d was aged f o r  the put  the  30°C. and  the r e c o r d e r were a l l o w e d  t h e g e a r was  shifted  to the  to  be  desired  the decay experiment w i t h the r e c o r d e r  chart  inch/sec.  A f t e r 5 m i n u t e s t h e c h a r t s p e e d was  s l o w e d t o 0.1  to r e c o r d the e q u i l i b r i u m v a l u e of torque.  were l e f t  was  calibrated.  s l u r r y was  i n p o s i t i o n and  evaporation.  r e m o v e d and  s p e e d o f 0.1  order  Then, the gear  27  again.  p o s i t i o n to s t a r t  (6)  to the p o s i t i o n  z e r o p o i n t was  artificial  fastened  A f t e r aging, the viscometer warmed-up  (5)  the torque  A q u a n t i t y of newly prepared cup.  shifted  running u n t i l  t h e r e was  i n the s c a l e reading over  no  a 5-minute  inch/min. The  instruments  d e t e c t a b l e change of interval.  in  torque  58  50  0  2  4  6  8  Voltage (mV) Fig.  5-9.  C a l i b r a t i o n Curve  of  Torque  versus  Voltage  10  (7)  The  g e a r was  value  of  repeated The  (2)  position  the gear  flowing  particles  a  radially  1.5  record While  a mirror,  were t h e n and  the  of  by  close  stable  Taylor  procedure  f o r t h e measurement  was  of  the  slurry: a b o v e , a s m a l l amount  about  1 mm  l o n g ) was  of  placed i n  slurry.  a movie camera  taken at the time  the photographs  rotational  only  This  equilibrium  reached.  surface of the and  the  (Braun N i z o ,  Type  the approximate o f each  marker pen  on  rate  photograph  the  o f one  was  same c h a r t  picture  manually  b e i n g used  to  torque.  shifting  the procedures  To  o f 1 was  (red nylon f i b e r s  seconds,  started and  recorded.  removed, as mentioned  marked w i t h t h e event  (4)  of the  and  were s e t i n p l a c e .  Photographs per  layer  across the  illuminator,  S56)  was  p r o c e d u r e s were added  tracer  An  (3)  until  t h e l i d was  line  to a h i g h e r speed  speed  of the  After  switehed  torque at t h i s  following  thickness (1)  then  this  were b e i n g t a k e n ,  the gear outlined  the decay  from n e u t r a l  to a d e s i r e d  used  i n these studies,  laminar flow should occur  instabilities.  was  position  b e f o r e were f o l l o w e d .  c h a p t e r i t s h o u l d be m e n t i o n e d  speeds  measurement  that  over  calculations  i n t h e gap  and  that  full  showed  range  that  t h e r e were  no  CHAPTER  6  EXPERIMENTAL RESULTS AND  The  variables  summarized slurry of  i n Table  variables  flowing  noted of  and  layer  that  the  t h e i r ranges  6-1.  As  investigated  shown t h e r e ,  (independent v a r i a b l e s ) (dependent v a r i a b l e s )  latter  DISCUSSION  variables  are  the  on  i n t h i s study  influence  shear  of  stress  were measured.  b o t h dependent  on  are  several  and  thickness  It should arid a  be  function  time. The  6-1.  computer  output  i n Appendix  D.  Reproducibility Measurements of  value  of  the  check the Measured  yield  of  while  relatively  the  thickness  of  the  considered  inner  cylinder  of  the  In  such  gel.  error to  flow of  be a  l a y e r was  associated inherent.  with  i s suddenly  a phenomenon t h e  poor.  60  five  times  within  samples.  ±12%  obtained  to  of  torque  the  reproducibility is  ±7%.  of  i s the  What h a p p e n s destroying  the  within  the. measurement  Yield  gelled material.  viscometer  equilibrium  f o r newly-prepared  reproducible  large  the  l a y e r were r e p e a t e d  be  the  the  t o r q u e and  to  initiate  of  of  these values  found  to  ture  value  flowing  t o r q u e was  t o r q u e was  needed  yield  r e p r o d u c i b i l i t y of yield  The  the  thickness  average value,  the  f o r a l l r e s u l t s i s given  force is  that  strucusually  61 TABLE VARIABLES  INDEPENDENT I.  AND  RANGES  Dispersed  Phase  Length  L/D R a t i o Fiber  Concentration  Dispersing  I. II.  -  6.72  (- )  22.9  -  156  0.04  -  0.17  (CP.)  7.45  -  220  of NaCl  (mole/£)  0- •  -  2.0  (rad./sec.)  0.116  -  6.28  Parameter  Angular DEPENDENT  0.987  Medium  Concentration Flow  (mm)  (cc/cc)  Viscosity  III.  INVESTIGATED  VARIABLES  Fiber  II.  6-1  Velocity  of Cylinder  VARIABLES  Shear S t r e s s Thickness  of Flowing  (dyne/cm ) 2  Layer  (cm)  7.10X10 - 4.62><10' 1  0  +  -  0.83  62 6-2.  Effect The  of  Slurry  Age  rheological  characterized  i n two  parts.  s t r u c t u r e break-down.  on  standing,  stress aging All at  aging  for various time  the  of  the  samples  a particle  dextrose  +1.0  cylinder  was  plotted  on  6-1.  test  aged  be  concluded  time  and  has  an  Billington  6-3.  (39)  with  experimental  Decay A  Fig.  viscometer inner  C u r v e and  typical  6-2.  cylinder  0.12  PEG-H^O.  Thickness  the  temperature  w h o s e L/D  curve sample  other  yield  of  30°C. to  ratio  The  32.  hours.  was  37.4  velocity  graph that  time the  as  was  the  f o r 20  inner cylinder  actually  the  t i m i n g f o r the  d e c a y m e a s u r e m e n t was  inner  build-up  increasing function  t  .  This  conclusion  bentonite  W a t a n a b e and the  determined  clay  Takasaki  aging  time  t o be  in (55)  needed  20  hours.  Layer recorder hours  chart  i s shown i n  i n the  annulus  gear  to r o t a t e  30°C t h e  from n e u t r a l to the p o s i t i o n  The  the  shown i n F i g .  structural  with  graph,  Flowing  of  t =  7  thesis  on  at  Hu^ ^ and  of  of  the measurements were  aging  From the  of  the  the  v a r i e d f r o m 0.  of  versus  temperature  shifted  measuring  i s a monotonically  this  and  structure build-up  Angular  results  the  sol.  i s generally  ( c c / c c ) i n a d i s p e r s i n g medium  H u x l e y a n d  work of  aging  by  fibers  graphite i n water,  a constant was  out  asymptotical value  torque-decay  After at  examine t h e  s a m p l e s was  stress  and  a suspension  a constant  stress from  a vanadium pentoxide the  at  The  of y i e l d  of  for  to  c o n s i s t e d of d i s p e r s e d  mole N a C l i n 30%  of  i s structure build-up,  newly-prepared  a graph  water, Peter  nature  carried  samples  i n terms of y i e l d  with  was  0.233 r a d . / s e c .  with  Stress  In order  measured  agrees  One  concentration of  I t can  aging  Yield  time-dependent  is  an  on  b e g a n t o r o t a t e a t p o i n t B, started.  The  27  of  the  the  at point  A.  at which  point  value  the  of  Fig.  6-1.  Effect  of  Age  of  Slurry  on Y i e l d  Stress  (£=0.15,  (B)  0  10  20  L = 0.987(mm),  30  40  T i m e (sec.) Fig.  6-2.  Typical  Decay  Curve  L/D=22.9  50  60  65 sharp  peak at  point  B  (at t =  torque decay  recording  curve.  curve  The  0)  i s the  i s rougher  t j .  =  decreased monotonically  in  30  Practically  t o 40 The  the  tracer  minutes i n the  thickness  decay  viscometer.  the  roll  the  of  radius  of  The  value  x  the  approached values The  at  6-4.  inner  inner  8 mm  large  labelled  as  A  R  c y l i n d e r , i s the  of  x  Effect  and R^.  t = t to  the  are  equilibrium value  i n t h i s work t h a n  will  be  of  NaCl  that  further discussed  Concentration  on  of  at  the  i n the  Yield  the  - R),  the cine*  of  the  measured  from  where R  is  flowing  layer.  of  at  of  i s shown i n  value  (= R)  was  during  movement  thickness  result  (R  the  time  developed,  i s the  1.01  marked w i t h  obtained  annulus of  actual thickness from  (at  i n the  typical  (cm)  C  the  where the  Therefore,  increased  point  record  f i l m was  screen  cylinder.  monotonously  t = 0  R  material  m e a s u r e d by, a r u l e r .  the  measured  value.  a  the  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  approach  ries  on  of  approached  measured w i t h  a movie camera to  the  smoothed  e q u i l i b r i u m t o r q u e was  l a y e r was  After  ordinate  shown a t  flowing  of  actual  is a  t i m e and  experiments.  using  projected  with  torque  The  which  present  surface  The  center  the  the  l a y e r was  6-3.  R  the  of  on  p h o t o g r a p h was  Fig.  of  e x p e r i m e n t by  particles  flowing  speaking,  torque.  t h a n F i g . 6-2  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 value t  yield  t =  t =  t^.  symbol  (+)  0  and These  i n the  figure.  f a s t e r f o r a l l the  torque next  Stress  to  the  slur-  i t s equilibrium  chapter,  and  on  Equilibrium  Stress I n F i g . 6-4 is L/D  a p l o t of y i e l d  shown, w h e r e t h e ratio  i s L/D  =  particle 37.4,  and  stress versus  concentration  concentration  i s tj) =  0.14  d i s p e r s i n g medium i s DEX  of  (cc/cc), +  NaCl  in  NaCl particle 30%  9 L / D =  37.4  (£> = 0 . 1 2  (cc/cc)  D E X + I.Omole NaCl  inlO%PEG-H 0 2  X  X  xx  x  x* X X  Angular Velocity '.0.2 3 3 (rad / s e c ) o'i -3.28  1 2.343  1 7.966  Fig.  6-3.  Thickness  1 13.589 TJME of  1 19.21! (SEC)  Flowing  Layer  1 2A.63A  1 36  67  1  T  r  Mole-MaCl / Liter Fig.  6-4.  Effect  of  NaCl  Concentration  on  Yield  Stress  68  PEG-H^O.  The y i e l d  strongly  on t h e s a l t  higher  the salt  result  agrees with  s t r e s s o f t h e aged a r t i f i c i a l concentration  concentration, Hu^ ^  depends  o f t h e d i s p e r s i n g medium.  the higher  (on b e n t o n i t e  7  slurry  i s the y i e l d  The  stress.  c l a y suspension  with  This  N a C l ) and  (37) P a r k e t al. stress  that  the i o n i c  stress  of e q u i l i b r i u m stress versus  a l s o depends on t h e s a l t  concentration, 6-5.  Effect Fig.  particle  the higher  with  yield  respons-  concentration  i n F i g . 6-4.  concentration.  a third L/D r a t i o  be found It  o f 22.9 a n d N a C l particle  concentration  concentration,  role  w h i c h was u s e d organic  observed  L/D r a t i o s  was p r e s u m e d  important  way  Stress particle  concentra-  f o r the  1.0 m o l e .  I t can be  the y i e l d  stress  decreases  T h i s may  appear  unusual;  f o r a l l the other  artificial These  slurries graphs  B.  that  t h e d i s p e r s i n g medium v i s c o s i t y  i n the strength to give  the salt  o f 37.4, 6 9 . 8 , 1 1 6 . , a n d 1 5 6 .  of the g e l structure.  would p l a y  an  H o w e v e r , PEG,  a w i d e r a n g e o f d i s p e r s i n g medium v i s c o s i t y , i s  s u r f a c t a n t whose e f f e c t  opposite  on Y i e l d  p a r a m e t e r o f d i s p e r s i n g medium v i s c o s i t y  i n Appendix  i s shown  The e q u i l i b r i u m  The h i g h e r  stress versus  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 .  the p a r t i c l e  o f NaCl  i s the equilibrium s t r e s s .  h o w e v e r , t h e same t r e n d was with  used  shows a g r a p h o f y i e l d  seen t h a t , a t a f i x e d  the  The  i s one o f t h e f o r c e s  o f D i s p e r s i n g Medium V i s c o s i t y  6-6  tion with  an  KCl).  o f t h e g e l s t r u c t u r e , so i t can be  force of the s a l t  F i g . 6-5 f o r t h e same s l u r r i e s  can  with  f o r the g e l structure of the s l u r r i e s . A plot  in  c l a y suspension  i s a measure o f t h e s t r e n g t h  concluded ible  (on b e n t o n i t e  to NaCl.  on g e l s t r e n g t h was  Therefore  the higher  found  to act i n  concentration  o f PEG  69  CM  e  1  to  1  i  Particle: = 0.14 Dispersing Medium-.  >»  L / D =374 O  Dextrose+ 3 0 % P E G -  4 CO CO  0>  :  £ =3  H  2°  /  0.5  3  1.0  i  1.5  2.0  Mole-NaCl/Liter  cr Fig.  6-5.  Effect  of  NaCl  Equilibrium  Concentration  Stress  on  1  1  Particle  1  r  1  1  1  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 ) Fig.  6-6.  Effect  of  Dispersing  Medium V i s c o s i t y  on Y i e l d  Stress  r  cancels  o u t more s t r o n g l y t h e e f f e c t  detailed NaCl  i s known.  forces but  mechanism o f n e i t h e r  Whatever to  that  artificial  6-6.  Yield Fig.  than  with  This  the particle result  the yield  there  length,  Equilibrium  of y i e l d  concentration,  parameter  L/D R a t i o  on  L/D r a t i o .  value  (i.e.,  I t can be seen  the higher  the higher  i s the yield Ree a n d E y r i n g ^  stress  o f t h e L/D r a t i o  6-8  independent  greater  t h a n 37.4  L/D = 6 9 . 8 , 1 1 6 , a n d 1 5 6 ) .  Concentration  and P a r t i c l e  of p a r t i c l e  the higher  L/D r a t i o  particle  4 2  t h e L/D r a t i o , t h e  I t seems f r o m F i g .  o f t h e maximum y i e l d  since the points  particle  L/D R a t i o  on  Stress  the particle  well,  interaction.  Braune and R i c h t e r ^ \  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  As  interaction.  stress against  s t r e s s a t any c o n c e n t r a t i o n .  Effects of Particle  higher  of dispers-  i s more d o m i n a n t f o r  and P a r t i c l e  I t c a n a l s o be s e e n t h a t  on a s i n g l e curve  6-7.  interaction  of particle  agrees with  i s a limiting  particle  lie  Concentration  a parameter  Weymann^^.  that  the effect  on p a r t i c l e - l i q u i d  the p a r t i c l e - l i q u i d  6-7 a n d 6-8 a r e g r a p h s  stress.  of  slurries  the higher  higher  a c t merely  although  interaction  Stress  concentration  and  interaction,  the p a r t i c l e - p a r t i c l e  Effects of Particle  that  on p a r t i c l e - l i q u i d  ionic  t h e s t r u c t u r e o f t h e g e l l e d m a t e r i a l may b e , i t i s r e a s o n a b l e  conclude  the  could  however, t h a t ' t h e e f f e c t s o f b o t h  a c t not only  a l s o on p a r t i c l e - p a r t i c l e  i n g medium v i s c o s i t y  Unfortunately, the  r e a c t i o n n o r i n t e r a c t i o n b e t w e e n PEG a n d  It i s certain,  and s u r f a c t a n t s c o u l d  of NaCl.  concentration  a r e shown i n F i g . 6-9 a n d 6-10.  with  a  The  concentration,  the higher  i s the equilibrium stress.  t h e L/D r a t i o ,  the higher  i s the equilibrium stress.  D E X + 1.0 mole NaCl in 10% P E G - H 0  O L/D =22.9  40  CM  h  A  37.4  •  69.8  2  Angular Velocity.-. 0.233  (rad/sec)  e A  C >»  TO  O 20  35  >0.5  0.10  0.15  Volume Fraction of Fibers Fig.  6-7.  Effects Yield  of  Concentration  Stress  a n d L/D  Ratio  of  cf> ( c c / c c ) Particle  on  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 ) Fig.  6-8.  Effects  of  Concentration  and  L/D  Ratio  of  P a r t i c l e on Y i e l d  Stress  DEX+ 1.0 mole NaCl in Angular Velocity 6 000  O  O  10% P E G - H 0 2  0.233 (rac^ec)  L / D =22.7  A  374  •  69.8  3 000  • 0  _j  0.15  0.10  0.05  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 ) Fig.  6-9.  E f f e c t s of Concentration Equilibrium Stress  a n d L/D  Ratio  of  Partical  on  .  ,  ,  , — .  Dispersing Medium: Dextrose +1.0 mole NaCl in 3 0 % O  L/D=22.9  •  A  37.4  •  69.8  ®  116.  000  ^  0.05  0.10  0.15  Particle Concentration Fig.  6-10.  Effects  of  Equilibrium  Concentration Stress  a n d L/D  Ratio  <f> ( c c / c c ) of  Particle  on  PEG-H2O  The  former  result  Brown^^,  Brown  agreement  with  particle  L/W  Fig. the  inner  the  of  the  measurement  Fig.  of  the  (R  inner  of (2)  The of  the  and  Eyring^ ^, 4  latter  and  stress  result  Pinder^^  with  is  on  in  their  is  an  that  this  has  The  not  of  to  method  field  been  at  angular  compare  standard  the  of  dependence  order  a  in  velocity  increase  in  materials, required  Stress  angular  B r o w n .  suggests  stress  on Y i e l d  versus  and  thixotropic  yield  Cylinder  stress  4  yield  a  graph  -  R)/(R_  O  00  Thickness  of  -  thickness  (R  from  =  the  of  the  the  P a r t i c l e L/D of  equilibrium  on the  the  particle  flowing  flowing  R)  of  a l l  for  timeestablished  and  graph  Flowing  Ratio,  thickness  of  flowing  equilibrium  thickness  the  ordinate  the  flowing  R^ i t  of  layer  the  is can  of  concentration,  of  seen the  is  graph,  the  cylinder,  radius be  in  and  Angular  Layer  The  center  1.01 cm),  From  smaller the  Ree  The  Brown  decreases  Unfortunately,  measured  lower  Inner  yield  concentration.  x»  cylinder  The  and  Pinder^ "^,  Equilibrium  (RQ = 2 . 1 0 c m ) . (1)  of  stress  various  is  equilibrium  viscometer  of  with  the  on  6-12  of  Weymann^^.  Particle Concentration,  particle  terms  R i c h t e r ^ ,  time.  Velocity  in  of of  present  versus  a plot  on  rheology.  Effects  6-9.  and  Brown^^,  Velocity  agrees  stress  dependent  and  of  Yield  velocity  yield  the  Angular  which  the  at  findings  cylinder.  angular  Braune  Pinder ^ \  6-11 shows  velocity, of  and  with  ratio.  Effect  6-8.  agrees  expressed  where  annulus R  the  is  layer  of  radius  outer  R  Xoo  the of  the  cylinder  that: greater  is  the  thickness  layer. p a r t i c l e L/D layer.  ratio,  the  is  greater  is  the  thickness  CM  E O  L / D = 22.7,  O  12,000  0=0.14  D E X + I.O mole  CO  NqCI  in 1 0 %  PEG-H 0 2  m  O U>  <o  •O.  6,000  (/>  >-  o 0.2  0-4  Angular Velocity Fig.  6-11.  E f f e c t of  Angular  on Y i e l d  Stress  0.6  (rad/sec)  V e l o c i t y of  Inner  Cylinder  0.7  1  i O  Angular  Velocity  6.28  B  2.09  A  0.698  ©  Q233  78  (rad/sec)  (  0.6  n  o  ^ B  ^  0.5  ®^ A  A  V  o.  A  ^B.  \  A  A  A  A  \  V  0.4  L/D=69.8  cr cr i  8 XI  cr  L/D=374  v-  L/D = 22.9  I  DEX+ 1.0 mole NaCl  o rr  0.3  0.05  Volume Fig.  6-12.  0.10  in 1 0 % PEG-H^O  0.15  i b e r s <f> ( c c / c c )  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 o n E q u i l i b r i u m T h i c k n e s s o f Flowing Layer  79 (3)  The  higher  the  flowing  Results the  (1)  larger  slurry. inner the in  and  (3)  ratio  destroys  with  discussed  the  The the  in  that  the  that  gel  rate  greater  next  higher in  the  a  is  the  thickness  of  of  particle concentration  stronger  higher  structure  structural  the  the  result  indicates  cylinder.  connection  velocity,  indicate  (2)  p a r t i c l e L/D  cylinder  be  angular  layer.  Result  outer  w i l l  the  in of  decay  the  chapter.  of  angular  the  increase  gel  structure velocity  annulus with  further  time  is  time-dependent  and  of of  the the  toward  important slurry  and  CHAPTER  7  ANALYSES  The mainly (a  previous  the e f f e c t s  chapter  of various  dependent v a r i a b l e ) .  variable,  discussed  independent  This  i . e . , the thickness  the experimental  chapter  v a r i a b l e s on s h e a r  deals with  of flowing  layer  A new m o d e l t o p r e d i c t R^ i s c o n s t r u c t e d the  concepts of r a t e processes.  discussed analyzed  i n Section i n Section  a b l e s was n a r r o w e d 69.8, 10%  7-2. 7-3.  dependent  R^. i n Section  7-1  applying  of r e a c t i o n i n t h e model a r e  parameters o f t h e model a r e  The e x p e r i m e n t a l to the p a r t i c l e  range o f independent  vari-  a n d t h e s i n g l e d i s p e r s i n g medium o f d e x t r o s e  + 1.0 m o l e N a C l i n  A comparison of the experimental  i n Sections  R^ i n t e r m s o f s h e a r viscosity  and f l o w  7-7 d i s c u s s e s  7-4 a n d 7-5.  s t r e s s , apparent  curve  are briefly  the p r a c t i c a l  h a s b e e n shown i n F i g . 6-3  the time-dependent  cylinder  viscometer  with  Rheological  shear  considerations of  rate at the w a l l ,  mentioned  of Flowing that  a r t i f i c i a l slurry  i n Section  apparent  7-6. i n this  Section chapter.  Layer  the thickness  of flowing  layer  i n the annulus of the c o a x i a l  increases monotonically 80  t h e model and  parameters of the  use o f t h e model c o n s t r u c t e d  A Model P r e d i c t i n g Thickness It  data  o f t h e i n d e p e n d e n t v a r i a b l e s o n two f i t t i n g  model a r e g i v e n  of  fitting  stress x  o f 2 2 . 9 , 37.4, and  PEG-H2O.  down  The o r d e r s  another  regarding  L/D r a t i o s  effects  7-1.  Two  results  with  t i m e and a p p r o a c h e s an  equilibrium value in  the  fact  internal The  that  asymptotically. the material  structure  of  one  end If  the  the  chemical  of  phenomenon i s rates  associated  with  i t .  Processes,  Glasstone,  is  shear-  and  time-dependent;  the  duration  of  of  the  rate  processes,  between  the  gel-structure  the  a rate be  their  Laidler,  terms  with of  one  structure  at  might  In  in  the  sol-structure  reaction  of  itself  explained  transformation  and  origin  the material varies  phenomenon can be  reversible  The  phenomenon  lies the  shear. a at  other.  process, useful  famous and  this  the  in  classical  dealing  textbook  Eyring  with  The  (12)  theory the  Theory  of  problems  of  Rate  emphasized:  . . . the theory of absolute r e a c t i o n r a t e i s not merely a theory of the k i n e t i c s of chemical r e a c t i o n s ; i t i s one t h a t c a n , i n p r i n c i p l e , be a p p l i e d t o any 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 of m a t t e r , t h a t i s t o say, t o any " r a t e process". As  a matter  chemical stress  of  f a c t , many r e s e a r c h e r s  reaction  decay  described  or  in  their  constructing  layer,  i t  is  useful  the  theory  of  chemical  for  the  Suppose  the  concepts  predicting  materials  as  of shear  was  review  reaction  to  predict  some o f  rates.  the  the  thickness  fundamental  Consider  of  the  flow-  principles  a reversible  =  0;  B  forward the  rate  i n i t i a l  a  mole/£  b  mole/%  a  - x  b  +  constant  k  f  concentrations  and of  the  x  backward  the materials  rate  constant  A and  B  are  of  chemical  of  t with  of  models  time-dependent  A  t  of  use  2.  a model  to  made  development  change  Chapter  ing  reaction  in  viscosity  earlier  Before  rates  have  k^.  82 a mole/5- and of  b mole/£,  A reacts  into  and  the  the  reaction  B  respectively,  so  that  concentration is  concentration  the  of  of  (b  order-n  +  consider  -  f  a rate  x)  order,  the m a t e r i a l B w i l l  k (a  time  t  concentration  B becomes  m  j£ = Now  the  at  x )  process  of  0.  t  =  A becomes  mole/Z  the  At  rate  t,  (a  -  at  that  of  increase  x  mole/1  x)  mole/£  time.  If  of  the  be  k^Cb + x )  -  m  =  of  a  (7-1)  n  gel-sol  transformation  f —»-  such  that  k  gel  sol  K t  =  0;  TT(R  t  =  t;  M R  2  - R )  - i*(\  2  - R ) x  L  2  2  u  the  Suppose {TT(RQ is L  2  forward the  is  of  the  the  L}  2  radius  of  cm  the  length  gel  rate  i n i t i a l  - R )  volume  of  transforms  the  volume  of  - R  2  sol  the  the  gel  in  2 x  the  )  -  that  t i m e , where  a  volume is  sol  radius  so  {TT(R  of  2  At  the of  t  that  the  - R )  L}  2  =  constant  viscometer sol  the t,  2  rate  Couette  L  - R )  is  {^(R^  volume  is  zero,  inner 2  of  k^.  where  cylinder, - R ) 2  gel  R  n  and  L}  cm  3  becomes  L  becomes TT(R  at  R  backward  cylinder.  U  2  0  and  i n i t i a l  inner  - R )  2 0  L  kf  cylinder,  into  )>  r 2  2  of  the  the  = TT(R  and  and  3  -  2  TT(R x  constant  outer  {TT(R  0  2 0  = TT ( R n with  L  - R )  2 0  R  is  - R )  2  2  x  thickness  of  L  the  flowing  layer  measured  from  til the  center  of  order,  the  the  inner  rate  of  cylinder. increase  of  If  the  the  sol  transformation volume  can  be  is  m  order-  expressed  as  chr(R ^  -  2  Equation  (7-2)  reaction  A %  chemically in  the  and  uniform  has  takes  uniform  here  been  obtained  A^.  Any  Since  shear  7-1.  at  sol  t  larger  0,  i n i t i a l  layer  is  As  the t  or  time  average ),  both  a the  be  the  shear an  not  is  stress  in  unstable  of  though  the  average  of  =  the  t  (t  =  product  B  flowing  of  rate  of  the  the  of  of  the  the  a  term  there  of as  thickness bundles  a  which  is  shown  of  the  by the  an  bundle  under  the  is  At  the  such  structure.  occupied  of  visco-  possess  down  sizes  the  and  However,  undeformed  gap  is  chemical  possess  in  shear  smaller.  layer  the  sol.  breaks  thickness  gets  the  and  annulus  size  ) ,  2  flowing  bundle  t^),  (7-2)  n  to  introducing  across  the  bundles  thickness  2  necessarily  the  or  varies  space  (t  R )}  structures.  m o d i f i e d by  a d i s t r i b u t i o n of  on  the  -  2  similar  equilibrium size  flow  the  does  stress  the  size  and  change  of  goes  argument  the m a t e r i a l  be  this  whole  k^TTLCR^  completely deteriorated  must  than  -  m  homogeneous  sol  structure  shear  stage  small,  large.  and  o r i e n t a t i o n or  rate  =  as  given  ) }  from an  sol,  (7-2)  any  2 x  reaetarft A  gel  would  the  there w i l l  At  an  of  equation  size  material  the  - R  2 o  compositions  account  equilibrium  f  both  o r i e n t a t i o n as  into  =  k {7rL(R  represented  Assume- t h a t  (t  =  B where  Accordingly,  and  2  transformation  meter  At  R )L  shear.  flowing in  Fig.  the  gel.  flowing  comparatively  layer  increases,  equilibrium  average  size  of  the  oo  bundles  do  tion  bundle  of  Let the  not  a be  following  change sizes  t  is  the  time,  but  across  the  gap.  a mean v a l u e expression a  where  with  = A  duration  0 0  at +  of  of  the  any  bundle  is  an  equilibrium distribu-  size.  Assume  a  is  given  by  instant  'I  . A +  there  Bt  shear  _  (sec)  (cm)  and  A  is  the  (7-3)  equilibrium  Fig.  7-1.  Bundle  Sizes  in  Flowing  Sol  85 size is  (cm) o f f i b e r  the value  bundle at a given  shear  rate,  o f a a t t = 0, a n d B i s a c o n s t a n t  Assuming t h a t the f u r t h e r bundle s i z e v a l u e A^, rate  t h e more r a p i d  and  one c a n p r e d i c t  that the - A^),  2  (a - A )  (7-4)  A + Bt  (7-2)  i s modified d?r(R  account  the c o r r e c t i o n  =  dt  [k.{ir(R  I  A  A + B t  second  term i s the b u i l d - u p Equation  m and n .  )  L }  =  M  x L} ]  (7-5)  N  on t h e r i g h t - h a n d  of g e l s t r u c t u r e from  f o r second  K^.log|f  1 |Y - D l o g l ^ T o  order-zero  f o r various orders  t + 1|  order  F o r example, t h e  (2, 0)] i s  k  n  2  - 4c  of  (7-6)  [(m, n) =  b + D| f•. ,B • ^7^1 = DTTL — l o g |  = /b  equation  form  T  x  t + 1|  Y = 2X + b D  s i d e of  thes o l .  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 . x x  solution  equation  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  The s o l u t i o n s a r e i n t h e x  where  2  (7-5) c a n be s o l v e d a n a l y t i c a l l y  log|f(R )| where  2  t h e break-down  - R  0  - R )  2  term of the bracket  represents  2  f  - ^ { T T ^  (7-5)  [1/(A + B t ) ] ,  .  2  The f i r s t  term  to  - R )L  2  i.e.  - R )L  2  x  dt  Hence, t a k i n g i n t o  (cm)  the e q u i l i b r i u m  o f i n c r e a s e o f t h e f l o w i n g v o l u m e i s p r o p o r t i o n a l t o (2 div(R  = AQ  (cm "'".sec ^ ) .  i s from  i s t h e break-down,  + 1/A  r i  . (7-7) (7-8)  •  (7-9)  86 b = -2(R c = and  with  (R  X = R  the  2 Q  2 0  initial  2  the  experimental  7-2.  The  t =  (7-11)  2  derivation  g e n e r a l model  and  of m  other E.  for predicting and  the present  solutions  the  thickness  n were determined  artificial  of for  slurries  as  section.  and  and  from  the  i n Appendix  empirical constants m  obtained  constants m  0.  are presented  subsequent  Determination The  n)  is a  data  i n the  )  2 x o o  for this  (m,  (7-5)  flowing layer.  -R  c o n d i t i o n of  = R at  x  details  the  2 Q  (7-12)  various orders  described  (R  boundary  The. m a t h e m a t i c a l  Equation  -  2  2  R  with  (7-10)  2  -R )  - R  2  x  -R )  n  i n the  n are  the  Model  orders  of  the  gel-sol  transformation.  (14) Goodeve  considered  concentration rate  their  n)  decay Oil.  they  ( 1 , 2)  and  found  dependent  fits  the  a priori that  on  recombination  the  and  their  data  of  i s similar  frequency  analogous  equation  concepts  (m = 1 ) , w h i l e  to a  n)  =  ( 1 , 2)  the  in his  n)  =  (1,  Govier  assumed  for structural data  stress  of Pembina  Crude  of these workers i s t h a t  to  the  reaction  the mechanism of of p a i r s second  (m,  in  the n a t u r a l t h i x o t r o p i c (43)  and  experimental  among t h e  (m,  that the order  Moreover, R i t t e r  the breakdown  process  proposed  confirmed  in their-rate  collision  t r a n s f o r m a t i o n i n terms of  constructed a rate equation  experimental  i t fits  decomposition  and  Brodkey^^  examined.  common e l e m e n t  the mechanism of chemical  of p a r t i c l e s  point viscosity  =  One  general gel-sol  Denny and  equation  suspensions (m,  links  equations.  terms o f the in  of  a  of  order  of  a  simple  the b u i l d - u p i s  degraded chemical  units,  a  reaction  2)  87 (n =  2). (Al) On t h e o t h e r h a n d , P i n d e r time-dependency between the  the  particles,  i.e.,  constructed second data  c o m p l e t e l y randomness  his  model  order-zero  for  Brown's  the  in  order  terms [(m,  n)  =  (2,  order  mechanism  of  these  researchers  viscosity  or  shear  stress  took  consideration  m o d e l may  represent,  predicted  a priori  the  the  and  in  model  also  found  on a  constructing thickness of  of  found  his  hydrate  to  that  the  experimental slurries.  f o l l o w the  time v a r i a t i o n  t h e i r models.  of  the  the values  gas  He  second  model.  focussed  order  and  fitted  sulphide  Pinder's  of  viscosity  0)]  was  exhibits the slurry  equilibrium order.  apparent  s l u r r y ^ ^ ' ^ ^  All  into  of  and  tetrahydrofuran-hydrogen  a r t i f i c i a l  order-zero  considered that the s l u r r y extremes of o r i e n t a t i o n of  two  flowing  gel-sol m and  of  None  layer.  be  of  them  Whatever  transformation  n must  measurable  the  cannot  determined  be  experi-  mentally. All versus model rate an  time using  integer,  that  experimental data  t  obtained  a wide  processes  however,  the  the  one  the  examined As fitting  this of  orders  order  of  the  any  real  of  the  primary  should  number  this  shown  (m,  n).  In  transformation can  possibly  objectives  of  of  in  (m,  n)  from  flowing  fitted  view  of  (m, n)  be  by  the  the  is  simple 1)  as to  R  present  concepts  not  It  a model  was, here  possible. (2,  2)  in  equation  means  (k^/B) of  the  (7-6), and least  a l l  (B/A). square  of  the model  Hence,  have  These  two  equations parameters  method m i n i m i z i n g  the  of  necessarily  acceptable.  constructing  (0,  layer  been  work.  parameters,  e v a l u a t e d by  thickness  be m a t h e m a t i c a l l y as  different orders in  the  i n v e s t i g a t i o n were  range  but  the model eight  in  on  have  two  were sum  of  88 squares.  The  Table  c o m p u t e r programme i s g i v e n  7-1  concentration  shows t h e  $ = 0.09  length-to-diameter dextrose inner the to  1.0  of  the  ratio  L/D  the  =  37.4,  (rad./sec).  squares  of 0 - l s t  equation  order,  f o r these  The  the  -  R  slurry 1.62  angular  I t may  lst-0  C.  dispersing  be  order,  particle  medium  of  velocity  of  i n Table  lst-lst  the  that  came  the l e f t - h a n d same, t h a t i s  2  1 -  (7-13) R  X  the  7-1  order  x  remains  particle  (mm),  noted  and  of  f u n c t i o n f ( R ) of  three orders 2  R f (R ) =  the  PEG-H 0.  T h i s i s because  (7-6)  length L =  and  was  0.233  f i t f o r the  particle  i n 10%  same v a l u e .  of  (cc/cc),  of  mole NaCl  cylinder  sum  side  +  results  i n Appendix  -  R  Xco  Hence t h e s e  t h r e e models of  0-lst  predict  phenomenon w i t h  essentially  the  Although with  the  sum  of squares  the v a r i o u s orders  of  o r d e r model does have an results out  were found  that  i n terms of  [(m,  n)  =  thickness  note  of  t o be of  listed  of  a l l the listed  flowing  the  the  the  and  lst-lst  accuracy. 7-1  the  does not  second  experimental F.  much  Similar runs  I t can  order-zero  approximation  vary  order-zero  other models.  second  order  carried  be  concluded  order  for predicting  the  layer.  Pinderconsidered  a measure of fact  same  i n Appendix  squares  order,  i n Table  other  (2, 0)] model g i v e s t h e b e s t  Brown and model  sum  the  advantage over  They a r e the  lst-0  the model e q u a t i o n s ,  f o r almost  i n t h i s work.  order,  the  second  order  the d i s t a n c e between bonding  that the  electrostatic  f o r c e was  term  of  points.  their Taking  strongly related  to  (35) the  gel structure  proportional Law.  I t was  to  the  then  , they square  analyzed of  stated that  the the  that the bonding  d i s t a n c e , as  f o r c e was  suggested  r a t e of breakdown of  inversely  by  Coulomb's  the  electro-  89 TABLE VARIOUS ORDERS  Order (m,n)  (m, n ) AND  Resultant k./B  (cm)  U  m  ;  7-1  RESULT OF L E A S T  Parameters B/A  (sec)  SQUARES F I T  , Sum o f S q u a r e s  0 - 1st  3.40  11.0  6.20 X  0 - 2nd  1.86  25.9  8.98 X  1st  0.448  11.0  6.20  lst-lst  0.163  11.0  6.20 X  lst-2nd  0.106  20.2  8.07  2nd  0.0162  8.13  5.47 X  2nd-lst  0.00878  9.29  5.77 X  2nd-2nd  0.00634  rif  - 0  - 0  = 0.09  4)EX + 1 . 0  mole N a C l  X  7.04 X  14.8  ( c c / c c ) , L = 1.62  X  (mm),  io" io" io" io" io" io" io" io"  2  2  2  2  2  2 2 2  L/D = 37.4, ft = 0.233  i n 10% PEG-H 0 o  (rad/sec)  static rate the be  structure  of  was  breakdown,  distance. of  the  second  was  was  the  distance treated  7-3.  the  layer  present  The  Two  not  be  empirical  Fitting  I t was  f o r the  related  directly  constant  Parameters  shown i n t h e  i.e.,  i o .| ^ |  . ^  S  'Y + D 2X  D  =  /b  +  X = R  2  x  breakdown  " b -  as  current  the  to  to  2 o  2 Q  slurries,  thickness  of  the e l e c t r o s t a t i c  the  present  section  Brown  artificial  throughout  (B/A)  by  of  that  the the  (m,  n)  are  model.  Model second  model e q u a t i o n has  order-zero  already  been  k,  ao.ll t  D'  +  1|  (7-7) ( "8) 7  (7-9)  - R )  - R 2 )  - R  (7-10)  2  2  -  ( R  2 0  - R  2 X  M  )  (7-1D  2  (7-12)  2  _3 The B/A  model e q u a t i o n (sec  (7-7)  "*") w h i c h c a n  Fig.  7-2  of  present  implied  orders  The  I -= ™.4 TWT..  of  b  b = -2(R  and  square  - 4c  2  ( R  the  The  the e l e c t r o s t a t i c  the  previous  7-1,  i n Section  =  the  Therefore  f  presented  Y  force.  to  model p r e d i c t i n g  ( k / B ) and  f i t .  =  strength  the  best  c  gel  form of  o r d e r model gave the  where  for considering  electrostatic structure  present  could  the  bonding  r e s u l t shown i n F i g . 6-4  between bonding p o i n t s . as  the  proportional  i s a l s o very p l a u s i b l e f o r the  Unfortunately, flowing  to  order.  slurries.  Pinder^  directly  t h e i r reason  i s certainly responsible  artificial and  therefore,  This  In view of force  inversely proportional  be  shows t h e  has  two  p a r a m e t e r s , viz.,  regarded e f f e c t of  as  the  the  fitting  parameter  Kji/B  (  )  c m  parameters. ( k / B ) on f  the  model  *  k / B = 0.007(cm" )  B / A = 5.0(sec') R = 1.5 (cm)  3  f  0-005 0.003 0.001  X  +  4>  X o o  R = 2.1 (cm) 0  R = 1.01 (cm)  /t^y^  . . . . . A  A A A A A A A / W V X W  0  B+HIMIMIIMMMMMHMM+  + —  1  -10.0  7.143  1  24.286  :  1  41.429  1  58.571  TIME ISEC) Fig.  7-2.  Effect  of Parameter  ( k /B) o n t h e M o d e l  T  75.714  T  92.857  110  92  equation [B/A  =  The  ( 7 - 7 ) , where t h e  5.0].  The  other  parameter  various constants  parameter  ( k / B ) was  (B/A)  varied  f  was  kept  constant.  from  0.001  to  0.007.  i n t h e e q u a t i o n were s e t as f o l l o w s :  and  R  = 1.01  cm  Ro  =  cm  R  =1.50  2.10  cm.  x°° All  four curves  R  =  1.50  at  approach a s y m p t o t i c a l l y the  t =  t  Xco  .  I t can  CO  the parameter  (k /B),  the  f  be  faster  seen the  common e q u i l i b r i u m  from  R  I  the  figure  approaches  o  that  value  the  of  greater o  i t s equilibrium  X  value. A was  similar  kept  varied  plot  constant  from  0.05  i s shown i n F i g . 7-3,  (k^/B  = 0.005) and  t o 50.0.  The  the  values  where the parameter  other  o f R,  R  those  (B/A),  i n F i g . 7-2.  the By  faster  the  comparing  of  i n c r e a s e of R  than  to  the parameter  exponent  is  just  7-4.  of  Fig.  can  be  (B/A).  of  shows a  data with  cylinder. seen  with  that  the  This  i n equation  the  time  t.  Velocity  on  same  to  parameter  value.  be  concluded  that  the parameter  the  parameter  (7-7) , w h i l e  Rate  the  g r e a t e r the  i t can  i s because  are  was  v  i t s equilibrium  F i g . 7-3  Rn  3  (B/A)  (k^/B)  (k^/B) i s  the parameter  of Increase  the  of Thickness  (B/A)  of  Layer  7-4  experimental  seen  i s much more s e n s i t i v e  x  of Angular  Flowing  be  approaches  t-term  a multiplier  Effect  inner  the  R  F i g . 7-2  rate  an  I t can  , and Xco  3  as  parameter  (k^/B)  from  The the  comparison between the model e q u a t i o n s different  angular  velocities  model  fits  the  experimental  figure  that  the higher  the  of  the  and  rotating  data very w e l l .  angular  velocity,  It the  4*  B / A = 5 0 . 0 (sec )  •  5 0  -°  0  •f  0  3  3  f  R .... .(cm)  5  x  *  5  R = 2 . l (cm)  5  0  R = l.0l.(cm)  1  -10.0  k / B = 5.0 X I 0 ( c m " )  -1  1  7.143  Fig.  7-3.  24.286  Effect  — i  1  4J .429  58.571  TIME (SEC)  of Parameter  r  75.714  (B/A) on t h e M o d e l  1  92.857  110  0  20  40  60  80  100  120  T i m e (sec.) Fig.  7-4.  Effect of  of  Flowing  Angular Layer  V e l o c i t y on  Rate  of  Increase  of  Thickness  140  95 faster  i s the rate of increase Fig.  fitting the  7-5 a n d F i g . 7-6 show t h e e f f e c t  parameters  data  may  fact lar  (k^/B) and  be e x p l a i n e d  reproducibility expected  that  velocity A may  k^, A, a n d B s h o u l d  means  graphs that  be s l i g h t l y  Effects of P a r t i c l e  generally  of  i t i s expected  data.  layer.  F i g . 7-7  Fig.  A  experimental  i t would  angular  velocity. tends  of Flowing  velocity  t h e slower  than  B  velocity  to zero.  L/D R a t i o  the p a r t i c l e  the higher  than  that  concentration  the p a r t i c l e of flowing  on Rate  and t h e  and  (B/A) d e c r e a s e w i t h  be e x p e c t e d  that  the effect  in Fig.  the slower i s  By c o m p a r i n g F i g . of p a r t i c l e  L/D  concentration  and  k^/B a n d B/A, r e s p e c t i v e l y .  again,  be e x p l a i n e d  the r e p r o d u c i b i l i t y .  concentration  and L/D r a t i o .  k , A, a n d B s h o u l d f  of the thickness  i s changed  layer.  the  concentration.  parameters,  errors associated with particle  the higher  L/D r a t i o ,  of p a r t i c l e  s c a t t e r i n t h e g r a p h s may,  experimental  that  of p a r t i c l e  on t h e f i t t i n g  that  i s the rate of increase  of the thickness  the  i t would  angu-  Layer  i t c a n f u r t h e r be n o t e d  L/D r a t i o  be  velocity, the  As a n g u l a r  and P a r t i c l e  7-9 a n d 7-10 show t h e e f f e c t s  considerable  model  scatter i n  ( k ^ / B ) a n d (B/A) i n c r e a s e w i t h  Concentration  Although  i s more d o m i n a n t  particle  Since  I t c a n b e s e e n f r o m F i g . 7-7  concentration,  rate of increase  ratio  both  the  on t h e  T h e r e a r e a few s c a t t e r e d p o i n t s , b u t t h e f i t s a r e  7-8, i t c a n b e s e e n t h a t  7-8 w i t h  velocity  The l a r g e  6-1.  a l l increase with  t h a t k^ a l s o  Thickness  satisfactory.  flowing  the  that  i n Section  layer.  7-7 a n d 7-8 show c o m p a r i s o n s b e t w e e n t h e m o d e l e q u a t i o n s  experimental  particle  respectively.  a f f e c t e d by a n g u l a r  7-5.  Fig.  of angular  k^ i s m o r e s e n s i t i v e t o a n g u l a r  to zero,  Increase'of  of flowing  by t h e e r r o r s a s s o c i a t e d w i t h  tends  of  (B/A),  w h i c h was m e n t i o n e d  shown i n t h e s e  is.  of the thickness  i n terms o f Both  (k^/B)  From t h e  a l l decrease  i  vr * i  N i  °o  c  o o cn  ..... ,  ,  T  o  L  = 0.987 mm 10 _ L / D = 22.9  o  j&  9  GO  ^> q>  •«=-  o  i  'E  •  &  i  _ DEX+I.O mole NaCl in 10% PEG-r^O II 11 <f> = 0.14 (cc/cc)  ro g o o  i  -  O  y  8  —  °  Q>  E  o fees  "5  •  /  7  6  O  5  1  0.1  1  1  1  i  i  0.2  0.3  0.4  0.5  0.6  1  0.7  Angular Velocity XI (rod/sec) a-. Fig.  7-5.  Effect  of Angular  Velocity  Q on the Parameter  (k,/B)  DEX+ 1.0 mole NaCl in 10% <f> =0.14 (cc/cc) L  = 0 9 8 7 (mm)  L/D=22.9  O  I  I  I  01  02  I  I  I  I  0.3  0.4  0.5  0.6  A n g u l a r V e l o c i t y Si ( r a d / s e c ) -6.  Effect  of  Angular Velocity  0, o n t h e P a r a m e t e r  (B/A)  L_  Q7  ,  •  0  20  ,  1  1  1  40  60  80  T i m e (sec) Fig.  7-7.  Effect  of  Thickness  P a r t i c l e Concentration of  Flowing  Layer  on  Rate  $  jj-  00  of  Increase  of  I  1  i  i  r  1  1  T i m e (sec) Fig.  7-8.  E f f e c t of Thickness  P a r t i c l e L/D R a t i o of F l o w i n g Layer  on R a t e  of  Increase  of  vo  i  1  i  DEX + I.O mole NaCl in 1 0 % PEG - H 0 L= 0.987 (mm) £ 1 = 0.233 (rad/sec) A ^ A ^ \ _ 0 L/D = 22.9 2  A •  37.4 69.8  o  10 A  •  A  \  \  \  ° o  \  5  o  i 0.10  1  0.05  1  0.15  Volume Fro c t i o n of F i b eirs (cc/cc) Effects on  of  Particle  the Parameter  Concentration  (k./B)  and P a r t i c l e  L/D  Ratio  10 *-O  i  I—i  0  v-  0) W  8  -  0 I  1  DEX+1.0 mole NaCl in 10% L = 0.987 (mm) £1 =0233 ( r a d / s e c )  PEG-H 0 2  A A  i  i  i  0.05  0.10  0.15  Volume Fraction of Fibers (cc/cc) Fig.  7-10.  E f f e c t s of P a r t i c l e Concentration on t h e Parameter B/A  and  P a r t i c l e L/D  Ratio  102 with  these  tion  and  L/D  No flowing From  one, layer  the R  is  x  of  such a  R  in  where  s  =  Since  the  r  in  Now t h e  shear  with  shear  rate  r  =  equation RQ  would be more apparent  may 7-11  of  cannot  equation be  has  flow  chapter,  question  of  Flowing  studied in  a  considering  rate  as an  a  the  can be  as  to  seen  of  time-  encountered  time.  shear  rate  is  shear  not  true  (7-15)  is  the  shows  It  in  zero.  e x t r a p o l a t i o n of a plot  of  = R/R the  should  of  the  rate  at  the w a l l  as  a  system.  the whole  gap  has  In  the  from  as  y)  In  equation  (7-14)  f u n c t i o n of  time,  since  that  0 the  apparent  at  curve  apparent  t  since  A p r a c t i c a l method  time versus  determined.  i t  (7-16)  manner,  (t,  condition  flows,  x  be noted  the  defined  shearing shear  be  c o r r e c t , because  current  this  a  simple  cannot  The v a r i a b l e  t h e n becomes  determined is  to  shear not  flowing.  rate  time.  (7-14)  certainly characterizes  the  appropriate  be  that  how p r a c t i c a l  difficulties  f u n c t i o n of  of  viscometer.  rheology  raised  the  apparent  i t  Layer  thickness  Couette  however,  can be  the p a r t i c u l a r geometry  s  rate  concentra-  (7-15)  apparent  case,  function  Thickness  c h a r a c t e r i z i n g the  can d e f i n e  o f t e n used whenever  = R to  the  this in  a f f e c t e d by  R/RQ.  the wall  present  the  time-dependent  true  strongly  knowledge,  importance  one  more  A.  of  c h a r a c t e r i z a t i o n can be,  Fortunately,  been  the  flow.  c a l c u l a t i n g the  at  B and  author's  presented  great  shear  are  is  Consideration the  x  Hence  than  to  results  dependent  in  ratio  Rheological  7-6.  the  parameters.  to to  shear  = the  denominator  determine"y the rate  t  =  at  R  is  x  shear  of t  0 axis.  calculated  a  =  0  Fig. by  L/D=22.9 &=0,233(rad/sec) DEX + I.OmoleNaCI  Ar v  + +  inlO%PEG-H 0 2  = 0.16 ( c c / c c )  *  0.15  O  0.14  ^  0.13  x + X  +  <&  x  +  .+  «  +  4> : - 9 24  Fig.  1 6.6  7-11.  +  4> 1 22.44  Dependence  4> 1 38.28  of Apparent  1 54.12  TIME (SEC) Shear  Rate  on Time  1 69.96  1 85.8  104 equations  shear  ( 7 - 1 4 ) and  (7-16).  Now,  i t i s clear  that  rate  at the w a l l  are  Apparent  viscosity  apparent  shear  c a n be  rate.  of  apparent Shear  as  viscosity and  parameter  t from  Fig.  shows t h o s e  7-7.  the very  f a r as  suspensions 1.0 of  (7-18)  f  =  fco  =  =  the r a t i o  by  flowing layer  viscosity  stress  n is  ( 7  of  time.  equation  shear  over  rate  (7-18).  F i g . 7-12  I f one  flow curve  1 9 )  shows a  (7-19) v e r s u s  are both  "  plot  time.  functions of  time  e l i m i n a t e s the ( y , T ) may  be  obtained.  Model  special  PEG-^O R  shear  curves.  of the  i n 10%  of  h ( t )  a function  case of Couette  of elongated p a r t i c l e s  mole NaCl the  g(t)  d e f i n e d as  flow  As  apparent  j =  equations, the  Use  and  (7-17)  (7-17) and  Practical  the w a l l  f(t)  apparent  these  at  T =  calculated  given i n equations  7-13  stress  f u n c t i o n s of time, i . e . ,  n i s also  stress  shear  Hence t h e a p p a r e n t  11  w h i c h shows t h a t  both  and  dispersing  i s concerned,  w i t h time  flow of  t c a n be  time-dependent  medium o f D e x t r o s e  +  t h e change of the t h i c k n e s s predicted  by  the  present  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 thickness three 7-9,  the layer  factors and  i.e.,  of  can be  F i g . 7-10  particle  R  xoo  >  a  n  d  two  e s t i m a t e d by  i n terms of the  c o n c e n t r a t i o n <)j,  and  angular v e l o c i t y  the  independent  of  factors  parameters F i g . 6-12,  particle  investigated  0,.  slurries,  (k^/B) and F i g . 7-5,  independent  inner cylinder  variables  of the  i . e . , equilibrium  (B/A).  F i g . 7-6,  variables  examined  length-to-diameter r a t i o The  number and  were l i m i t e d ;  These Fig. here, L/D,  the range  t h e r e f o r e the  of  L/D=22.9 ft =0.233 (rad/sec) DEX + I.OmoleNaCI i n l 0 % P E G - H 0  03  2  + +  CM " O  + $  ID lO  s  X  0.15  O  0.14 0.13  LU tn-4  cn&i  »—^  X  +  **x ^in  I—  on"  3  > °cn  in. ru  +  *x  -  O CJ CO  0.16 ( c c / c c )  +  X  \  X X +  m oo. -9.24  6.6 Fig.  +  +  7-12.  <!>  t 22.44  Apparent  Viscosity  38.28  54.12  TIME (SEC)  o f Time-Dependent  69.96 Artificial  85.8  101.64  Slurries  o  L/D = 22.9 £2 = 0.233 (rad/sec) DEX+I.OmoleNaCI in 1 0 % P E G - H 0 + 3> = 0.16 ( c c / c c ) X 0.15 2  0.14 0.13  0  f  +  + +  X  X  1  0.727 Fig.  1  0.975  1.224  7-13.  Flow Curve  1  1.472 SHEAR RATE of Time-Dependent  1  1  1.72 1.968 (1.0/SEC) A r t i f i c i a l  Slurries  1 —  2.217  107 e s t i m a t i o n may the  not  be  conclusive.  c o r r e l a t i o n between  these  It  three  should  be  noted,  however,  characteristic factors  (R  that  ,  k,./B, r  x°o  and  B/A)  and  the  successfully  independent  used  for  variables  p r e d i c t i n g the  investigations  to  correlate  variables  as  temperature  are  such  recommended Since  along  this  is  the  the  the  and  ft)  can  time v a r i a t i o n  of  R  factors  and  lines  first  (<f),  w i t h many  various  of  L/D,  this  kinds  .  other  of  be Future  independent  dispersing  medium  work.  f l o w model  in  terms  of  R  , i t  cannot  be  x compared  with  other  shear  stress  at  decay  models  can be  used for  for  decay  the w a l l  apparent  shear  or  found  predicting  predicting the  models  the  apparent  in  shear  y(t)  at  G.  stress of  can  predict  viscosity.  Appendix  thickness  rate  which  It  decay  flowing  the w a l l  An  is  time v a r i a t i o n  evaluation  suggested  t ( t ) , and  layer  as  the  the  R (t)  and  x  discussed  in  of  that  hence  the  those  they  present  of  be  model  the  previous  section. Transport slurry  form)  present  of  through  work.  geometry,  pulp  In  where  (fiber  pipes  those  the  shear  may  suspension), be  systems stress  one  of  crude  the  o i l , or  possible  the material flows at  the wall  is  in  coal  sure  R  is  drop  radius in  Poiseuille  the  of  the  axial  geometry  is  pipe  the  expressed  defined  Q is  express  the  the volumetric true  shear  in  the  by  -  is  the  apparent  gradient shear  of  rate  in  20)  the  pres-  the  by • &  flow rate.  rate  The  of  Poiseuille  (7  (-dP/da)  direction.  t where  and  a  applications  v= -1 €> where  (as  this  (7-21) There geometry,  is  a  too,  correlation i.e.,  the  to Rabinowitsch  (32) equation  ;  materials.  however, If a flow  and  t r u e shear  can  be  such  flow  Section shear to  equation  i n other  equation  3-6.  On  specific  the  to the  annular  core which  the  other  central  in  a direct  shear  manner  value  of  thickness  with  one  that of as  the  a wide  industrial  gap  gap  of  geometry  geometry does (22) (25)  of  shear  the  can  since the part of  to another,  work l i e s  of  '  the  i n such  the  , the  central  core.  plug  could  However,  r a t e cannot because  geometry.  be  correspond  be  the  the  applied apparent  Therefore,  the  qualitative contribu-  time-dependent m a t e r i a l s i n a  this  however,  necessarily  as u s e d  present  R be  in this are not  '.is.  i n such  Couette  a very  more o f t e n i n a  I t i s hoped  measurements.  the  flow viscometer  work, b e c a u s e d i s p e r s e d p h a s e s a l w a y s s m a l l enough t o be  funda-  important  I t seems t h a t s l u r r y  undertaken  the u s u a l viscometer.  t h i s work a r e u s e f u l  work i s , i n d e e d ,  s t r e s s e d i n t h i s w o r k how  flowing layer  suspensions  gap  i n one  the annular  on  apparent  knowledge.  I t was,  of  size  obtain  described i n  conducted.  geometry  apparent  geometry  present  a wide  measurements w i l l  the  the  current  m e n t a l one.  of  from  t o t h e measurement  viscometer  with  same a s  i n f o r m a t i o n such  to the As  with  the  of the  to  was  example,  flowing i n a Couette  the v a r i a t i o n  difficult  i t i s pertinent only  experiment  For  r a t e i s , i n general,dependent  practical tion  geometries.  stress  information  i n terms of  section,  information obtained  i s not  of  in principle,  quantitative  the  shear  m a t e r i a l as  i f i t i s given  i n which  time-dependent  entire  I t i s not  plug part i n a Poiseuille  characteristics be,  geometries.  for  i n terms of  m a t e r i a l , the  i n the previous  H o w e v e r , much q u a l i t a t i v e  to  i s obtained  o t h e r hand  geometry  applied  to use  f o r a time-independent  r a t e as mentioned  the  difficult  rate for a particular  invariable  a  i t i s very  of  compatible  that the  results  CHAPTER 8  CONCLUSIONS  Time-dependent nylon and  fibers  artificial  From t h e r e s u l t s  obtained,  consisting of regularly-sized  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  sodium c h l o r i d e , have been  meter.  slurries,  based  studied  glycol with  i n a co-axial cylindrical  on t h e p r e s e n t  experimental  stress increases  strongly with  salt  (2)  Yield  stress increases  with  concentration  (4)  Yield  s t r e s s decreases with  inner  cylinder of the  (6)  an i n c r e a s e  and p a r t i c l e  Equilibrium thickness particle  velocity  of inner  Employing  behavior  particle  of flowing  slurries.  velocity of  concentration,  layer decreases with L/D  layer  particle  an  increase  ratio.  increases with  angular  viscometer.  of r a t e processes,  (k^/lTand  of the thickness  of angular  ratio.  and p a r t i c l e  an e m p i r i c a l model w i t h  two  B/A) t o c h a r a c t e r i z e t h e t i m e - d e p e n d e n t  of flowing  between c y l i n d e r s o f t h e v i s c o m e t e r , dependent  salt  c y l i n d e r of the  a concept  parameters  L/D  of flowing  concentration  Equilibrium thickness  fitting  and  viscometer.  Equilibrium stress increases with  of  concentration.  ratio.  concentration, (5)  particle  were  conditions.  Yield  (3)  visco-  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  (1)  L/D  dextrose  By a n a l y z i n g 109  layer i n the r e l a t i v e l y was c o n s t r u c t e d  t h e model  w i d e gap  f o r the time-  further conclusions  were  110 drawn. (7)  The s e c o n d o r d e r - z e r o best  approximation f o r p r e d i c t i n g time-dependent  the f l o w i n g (8)  order r e v e r s i b l e r e a c t i o n model g i v e s thickness  the of  layer.  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  in  the model. (9)  The p a r a m e t e r s inner  k^/B  a n d B/A  increase with angular  c y l i n d e r , and d e c r e a s e w i t h a n i n c r e a s e  t r a t i o n a n d p a r t i c l e L/D  ratio.  v e l o c i t y of  the  of p a r t i c l e concen-  CHAPTER 9  RECOMMENDATIONS  It  is  recommended  research  be v a r i e d ,  research  be  be  the  among  ing  the  is  on  the Even  parameters error,  should their  of  that  the  ranges  extended  or  filled  easiest  to  study:  the  forward  rate  though of  i t  parameters b e made  to  has  conducted should  in  that  be u n d e r t a k e n rising  pipe  of  flow  of  such  the  been  a  the  that study  present  suggested  show l e s s  separate  on  a Couette  currently  In  in  some r e f i n e m e n t s  experiments  of  held  ones  varied  regard, would  constant in  be  useful  this  the  temperature  correlation for  model,  in  would  for  test-  further  especially  k^/B  where  scatter  the  fitting  constant.  individual behavior All  in.  the model presented be  the variables  the Arrhenius-type  parameters  t h e r e may  fitting  any  a p p l i c a b i l i t y of  analyses k^  and  that  the may  the  multi-phase  in  of  7 was  due  in  to  experimental  the model which w i l l  the  In  that  fitting  s t r u c t u r e breakdown  a Poiseuille  interest  to  the  regard,  result  in  attempts  parameters  so  that  slurries  were  observed.  f l o w geometry  in  chapter  scatter.  parts be  that  problems  systems.  I l l  in  this  for  work.  f l o w geometry of  start-up  the  Further because and  experiments  of  the  control  for  the  NOMENCLATURE  Roman  a  i n i t i a l  a  constant  concentration of reactant  (mole.£  (2-1)  ( s e c """)  i n equation  (7-3)  ( c m ^")  constant  A  constant  Ag  i n i t i a l  A  equilibrium value  b  i n i t i a l  B  A  i n equation  A  B  Letters  value  of bundle  size  of bundle  "S  (cm) size  concentration of product  (cm) B  (mole.£  """)  constant constant  C  constant  D  diameter  F  force  i n equation  of  (cm "'".sec  (7-3)  fiber  "*")  (mm)  in x-direction  (dyne) -2  g  acceleration of gravity  (cm.sec  H  height  (cm)  k  constant  k^  rate  k^  growth  k^  decay  K  equilibrium rate  L  height  L  fiber  length  m  order  of decay  m  slope  d e f i n e d by  of  of  power  increase  law f l u i d  i n equation  of network  reaction rate reaction rate  structure  constant constant  constant  of rotating  (3-5)  (=k^/k^)  cylinder  (cm) (mm)  reaction rate equation  (3-45) 112  (-)  )  order of growth reaction rate  (-)  power law exponent i n equation (3-5)  (-)  exponent i n equation (3-42)  (-) -2  pressure i n equation (3-24)  (dyne.cm  )  parameter exponent of shear rate i n 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 i n pipe  (cm .sec  radius of inner cylinder  (cm)  radius of pipe i n 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)  r a t i o of inner cylinder radius over outer cylinder radius  (-)  area  (cm )  time  (sec)  time at equilibrium  (sec)  torque  (dyne,, cm)  velocity  (cm. sec "*")  velocity  (cm. sec "*")  v e l o c i t y i n x-direction  (cm. sec "*")  mean value of bundle size  (cm)  2  114 Greek  (-)  Y  strain  Y  shear  n  apparent  no  yield  ni  apparent v i s c o s i t y orientation  of  n2  apparent  of  (sec  rate  value  of  point  viscosity  yield  value  of  structural Newtonian of  of  in  completely  parameter  TO  T2  T  T  o R  yield  . s e c -1  viscosity  (g.cm  viscosity  equation  circumference to  its  _  1  .sec  t(.g.cm~  .sec  (g.cm  .sec  X  diameter  stress stress  1  (-) )  (dyne.cm  2  )  (dyne.cm  2  )  characteristic  time  (sec)  characteristic  time  (sec)  a  completely  shear stress structure  of  deteriorated  shear  stress  at  the  outer  cylinder  wall  (dyne.cm  2  )  shear  stress  at  the  inner  cylinder  wall  (dyne.cm  2  )  (dyne.cm  2  )  (dyne.cm  2  )  (dyne.cm  2  )  (dyne.cm  2  )  (dyne.cm  stress  )  S  T  shear  stress  at  i n i t i a l  shear  stress  at  steady  shear  stress  at  the  state state  S°° pipe  wall  -  (2-8)  (g.cm  structural  T  _ 1  .sec  t( g . c m  viscosity  shear  _ 1  (2-3)  viscosity  density shear  t( g . c m  . s e c "I  1  . s e c "I  point in  order  _  t( g . c m " I  apparent  equation  of  .sec  random  equilibrium  apparent  equilibrium value  (g.cm  viscosity  t (g.cm  viscosity  point  ^)  . -1 -1 (g.cm .sec  viscosity  equilibrium value  ratio  Letters  ;  115  x ^ ^ X  shear s t r e s s of x - d i r e c t i o n a c t i n g surface perpendicular to y-axis equilibrium  shear  on  the (dyne.cm  stress  (dyne.cm  3 <j)  volume  fraction  of  03  angular  velocity  tt  angular  velocity  particles  of  inner  cylinder Symbols  (x,y,a)  rectangular  coordinates  (r,0,a)  cylindrical  coordinates  ordinary  _9_ 3x  partial  D Dt  substantial  derivative  derivative  f  function  g  function  g. Y  function  h  function  e  exponential  exp  exponential  time  common  log  natural  logarithm  In  natural  logarithm  ^  with  with  respect  logarithm  arc-hyperbolic  F  force  v  velocity  A  rate  x  stress  tangent  vector  of  (cm  )  -3 .cm  vector deformation tensor  tensor  (rad.sec  and  respect  derivative  logj Q  tanh  suspension  -2  )  (rad.sec  Mathematical  _d_ dx  in  )  Notation  to  to  x  x  "'")  116  Vf  gradient  of  V.f  divergence  E  summation  °°  infinity  lim  f(x)  limit  of  f of  f  f(x)  as  x  approaches  a  x->-a Abbreviations CMC  carboxy  methyl  and  Other  Notation  cellulose  DEX  dextrose  JLjO  water  NaCl  sodium  PEG  polyethylene  pH  index  k^/B  fitting  parameter  of  the model  for  B/A  fitting  parameter  of  the model  for  k^/kit  fitting  parameter  of  Govier's  stress  decay  model  k^/Po  fitting  parameter  of  Govier's  stress  decay  model  chloride  of  glycol  concentration  of  hydrogen  ion,  i.e.,  login{l/[H  R  ]}  BIBLIOGRAPHY  1.  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"The V i s c o s i t y o f S u s p e n s i o n s  Science, 34(2),  on R h e o l o g y ,  Interscience  (1965).  s p h e r i c a l P a r t i c l e s i n Polymer Interface  Gel  Suspensions",  of the F o u r t h I n t e r n a t i o n a l Congress  (1963),  Publishers, 57.  "On  the Process of Thixotropic 4 5 ( 1 4 ) , 334-335 ( 1 9 5 8 ) .  of Large,  Fluids", Journal  185-196  (1970).  Non-  of C o l l o i d  and  APPENDICES  Appendix A  Flow Measurement w i t h D i f f e r e n t Cylinder  An-.equilibrium slurry  consisting  concentration mole NaCl and  1.20  plot  of the fibers  o f L = 3.01 mm,  and t h e d i s p e r s i n g  cm.  The r e s u l t s velocity  there that  were l i s t e d  there  i s a scatter  inner  cylinders  versus  the curves  were  apparent  here  shear  rate  so s m a l l t h a t state.  of l i t t l e  f o r each  = 69.8 a n d  earlier  importance  122  stress  x.  1.10,  In  shear  errors.  a  I t can  v a l u e o f R.  stress  although A l l the  surfaces to eliminate The w a l l  within experimental that  1.0  F i g . A - l shows  i n F i g . A-2,  i n F i g . 5-8.  I t c a n be concluded  particle  o f R = 1.01,  y and e q u i l i b r i u m  on t h e i r  they were u n o b s e r v a b l e  practical  shear  a s shown  artificial  of dextrose +  due t o e x p e r i m e n t a l  were grooved  a t t h e w a l l s a s shown  equilibrium fact,  L/D  medium  equilibrium  are distinct  i n t h e graph  used  Inner  f o r the  i n Table A - l .  x, h o w e v e r , t h e p o i n t s a r e o n o n e c u r v e ,  slip  undertaken  i n 1 0 % PEG-B^O b y u s i n g t h e i n n e r c y l i n d e r s  terms o f e q u i l i b r i u m  the  R of  Equilibrium  f l o w m e a s u r e m e n t was  <j> = 0.08,  of angular  be seen  at  Radius  the w a l l e f f e c t s  i n the present  system.  effects error at are, i n  123 TABLE A - l E Q U I L I B R I U M FLOW DATA WITH D I F F E R E N T R  R (cm)  (rad/sec) 0.233  Torque  (mV)  2 Shear  R  Stress  (dyne/cm  - R  RQ  "  R  R  (cm) X  oo  Apparent Rate  Shear  (1/sec)  )  0.698  2.09  6.28  1.01  0.84  0.86  0.88  0.89  1.10  1.01  1.02  1.04  1.06  1.20  1.20  1.22  1.25  1.26  1.01  2.99 10  s  1.10  3.01xl0  3  1.20  3.02x10  s  1.01  0.410  0.438  0.467  0.486  1.10  0.383  0.424  0.476  0.567  1.20  0.373  0.424  0.488  0.587  1.01  1.46  1.49  1.52  1.54  1.10  1.48  1.52  1.58  1.67  1.20  1.54  1.58  1.64  1.73  1.01  0.898  2.59  7.49  22.0  1.10  1.04  2.91  8.15  22.2  1.20  1.19  3.28  9.01  24.7  x  - 3.05xl0  3.13x103  3.15x103  3.06x103  3.13x10  s  3.17x103  3.07x10  3.14x10  s  3.18x10  3  s  s  UJ  Q2  0.3  0.4  0.5  1.0  2.0  A n g u l a r Velocity £1  3.0  4.0  5.0  (rad./sec) i-  Fig.  A-l.  Equilibrium Inner  Flow Measurement  Cylinder  with  Different  Radius  of  1  ^  "i—i  3.20  ro O CVJ  £  i— i—  ii i  L = 3.01 ( m m ) , L / D = 6 9 . 8 , < £ = 0 . 0 8 D E X + I . O mole NaCl in 10% P E G - H 0  i i  2  3.15 R = 1.20 (cm) 1.10'  (/)  1.01  C  >s  "O  — </>  3.10  O  3.05  CO  E .O 3  3.00 h J  111  LU  J  1.0  I  L  I  I  I  10.0  Equilibrium Apparent Shear Rate y (1/sec.) Fig.  A-2.  E q u i l i b r i u m Flow  Curve  Appendix B The on  yield  S u p p l e m e n t t o F i g . 6-6 i n S e c t i o n following  stress  figures  f o r various  6-5  show e f f e c t o f d i s p e r s i n g particle  126  lengths.  medium v i s c o s i t y  0.08  0.10  0.12  0.14  Particle C o n c e n t r a t i o n ^ Fig.  B-l.  Effect  of  Dispersing  Medium V i s c o s i t y  0.16  (cc/cc)  on Y i e l d  Stress  Yield Stress ( x l O  4  2 dynes/cm )  ro  o b  OQ  -4>  bd  I  ho  w l-h l-h CD O O f-h  01  (D H  D> •  cn w O O O  3 OQ  s C a  cn  o o  Q  CD  O O  N  CT)  Q O  o b oo  W rt i-i CD cn cn  \\  p o  O  8ZT  ^  w 8 p •  O  o o o o  Q  o o  to —  m  — rO \  X  3  ?  o  o o  O CD  cn  (D  "  o u  CO  ft) CO  (D H-  5  TJ  cn  H-  O  Ul  OJ  >1  &tn  Q  o  00  0.04  0.05  0.06  0.07  0.08  Particle Concentration  0.09  <& (cc/cc) r—  1  VO  Fig.  B-3.  Effect  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  Stress  >-  0  I  I  i  i  0.04  0.05  0.06  i  i  0.07  0.08  I  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 ) Fig.  B-4.  Effect  of  Dispersing  Medium V i s c o s i t y  on Y i e l d  Stress  o  Appendix C-l  C  Computer Programme  Programmes  "BESTFIT"  131  13 2 C c C C C C C C C CC C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C  . MAIN  .  PROGRAM  .  "BESTFIT"  T H I S PROGRAM E V A L U A T E S F I T T I N G P A R A M E T E R S OF V A R I O U S R E A C T I O N O R D E R S O F F I N D E R ' S M O D E L , E Y R I N G* S M O D E L , AND GOVIER'S O R I G I N A L F I R S T ORDER - SECOND ORDER MODEL I N T E R M S OF T I M E AND S H E A R S T R E S S AT T H E W A L L BY M E A N S O F L E A S T S Q U A R E S M E T H O D W I T H A S E T OF E X P E R I M E N T A L D A T A O F A S I N G L E R U N . HOW  TO C O M P I L E T H I S P R O G R A M ? I N ORDER TO C O M P I L E T H E PROGRAM " B E S T F I T " I N T O THE I B M V E R S I O N F O R T R A N G - L E V E L C O M P I L E R UNDER C O N T R O L MICHIGAN TERMINAL SYSTEM ( M T S ) FOR T H E I B M 3 7 0 M O D E L 1 6 8 C O M P U T E R AT T H E 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 , T f i E COMMAND $RUN SHOULD  HOB  *FTN BE  SCARDS=BESTFIT  SPUNCH=-OBJECT  USED.  TO E X E C U T E T H E T H E COMMAND $RDN  OF  OBJECT  -OBJECT+*PRPLOT  DECK?  4=BFORDER  5=1NPDTDATA  SHOULD B E U S E D , WHERE " B F O R D E R " I S T H E F I L E WHICH S H O O I D E E R E A D T H R O U G H L O G I C A L I / O U N I T 4 BY A - T Y P E F O R M A T , AND A L L T H E I N P U T D A T A MUST B E C O N S I S T E N T W I T H , AND C O R R E S P O N D T O T H E O R D E R AND F O R M A T I N D I C A T E D I N T H E M A I N PROGRAM. D E S C R I P T I O N OF  PARAMETERS.  IEND JEND KEND LEND MEND TORQT TORQZ  - NUMBER OF E X P E R I M E N T A L DATA. - NUMBER O F P I N D E R ' S M O D E L - N U M B E R O F L E T T E R S I N T H E A - F O R M AT. - MAXIMUM LENGTH / KEND. - N U M B E R O F T H E O R E T I C A L P O I N T S TO B E P L O T T E D . - E Q U I L I B R I U M V A L U E OF T O R Q U E ( D Y N E S C M . ) . - I N I T I A L TORQUE OF GEL STRUCTURE ( D Y N E S CM.) . TAI - E Q U I L I B R I U M V A L U E OF S H E A R S T R E S S { D Y N E S / S Q . C M . ) . TAZ - I N I T I A L SHEAR S T R E S S OF GEL STRUCTURE (DYNES / SQ.CM. ) . TORQ(I) - E X P E R I M E N T A L V A L U E S OF T O R Q U E ( D Y N E S CM.) . T(I) - TIME (SEC.), TAU(I) - E X P E R I M E N T A L V A L U E S OF S H E A R STRESS ( D Y N E S / SQ.CM.) . O R D E R ( J , L ) - R E A C T I O N O R D E R S OF P I N D E R » S MODEL. RCONS(J) - F O R W A R D R A T E C O N S T A N T S O F P I N D E R ' S MODEL,. TAUT(M) - P I N D E R ' S SHEAR S T R E S S (DYNES / SQ.CM.). TAWEYR(M) - EYRING'S SHEAR S T R E S S (DYNES / SQ.CM.). TAWGOV(M) - G O V I E R ' S SHEAR S T R E S S (DYNES / S Q . C M . ) . •.  ..  133 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 ) , E 2 ( 8 ) , NDQ <8) , P ( 8 ) , O R D E R ( 8 , 6 ) , 2RCONS{8), T T ( 5 1 ) , 0 ( 5 1 ) , TAUT{51), TAUTL{51), T A U L { 1 0 ) , TCBQ{10) 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 JEND = 8 KEND = 4 LEND = 6 MEND = 5 1 C C READ READ READ READ C C  I N P U T O F DATA . . . . . ( 5 , 1 0 0 ) T O R Q I , TORQZ (5,102) ( T ( I ) , 1=1,IEND) (5,103) ( T O R Q ( I ) , 1=1,TEND) ( 4 , 1 0 9 ) ( (ORDER ( J , L ) , L = 1 , L E N D ) ,  . . . . . 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 * 3.1416 * (R**2) TAZ = TORQZ / ( 2 . 0 * 3 . 1 4 1 6 * ( R * * 2 )  J= 1,JEND)  * RL) * RL)  C DO 1 1 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 - TAU ( I ) ) V ( 3 , I ) = (TAZ - T A I ) / (TAU (I) - T A I ) V ( 4 , I ) = (TAZ - TAX) / ( T A I - TAU ( I ) ) V ( 5 , I ) = (-TAU(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 ) / {(TAZ* 1*2) * ( T A I - T A U ( I ) ) ) V ( 8 , I ) = ( T A Z * T A I + ( T A Z - 2. 0 * T A I ) * T A U ( I ) ) / ( T A Z * ( T A I 1 TAU ( I ) ) ) 11 C O N T I N U E C  21 20  DO 2 0 J = 1 , J E N D DO 2 1 1 = 1 , I E N D H(J,I) = ALOG(ABS(V(J,I))) CONTINUE CONTINUE  C WZ = 0 . 0 EP = 0 . 0 0 0 5 WRITE (6,112) C DO 2 2 J = 1 , J E N D WRITE (6,123) WRITE (6,105) ( O R D E R { J , L ) , L = 1, L E N D )  134  23  DO 2 3 I=1,IEND WL ( I ) = W ( J , I ) CONTINUE WRITE  25  (6,106)  DO 2 5 I = 1 , I E N D WRITE (6,104) J , T ( I ) , CONTINUE  (I)  P L (1) = 0.0 NI = +20 ..... L E A S T S Q U A R E S 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 , 1X) E1 ( J ) = E 1 L (1) E2 ( J ) = E 2 L (1) N D Q ( J ) = ND P(J) =PL(1) 2 2 CONTINUE  1,N.I, N D , E P , A U  C A L C U L A T I O N O F R A T E C O N S T A N T S FOR P I N D E R ' S MODEL ECCNS(1) = - P ( 1 ) * (TAZ - T A I ) R C O N S ( 2 ) = P ( 2 ) * ( T A Z - TA I ) / (-2.0) RCGNS(3) = P ( 3 ) R C C N S ( 4 ) = P ( 4 ) * (TAZ ~ T A I ) / TAZ RCONS(5) = - P ( 5 ) * (TAZ - T A I ) / (TAZ + T A I ) RCONS(6) = P ( 6 ) / 2.0 / T A I RC0NS(7) = P ( 7 ) * (TAZ - T A I ) / ( T A I * ( 2 . 0 * TAZ - T A I ) ) RCCNS(8) = P ( 8 ) * (TAZ - T A I ) / ( 2 . 0 * T A I * TAZ) WRITE (6,101) DO 2 4 J = 1 , J E N D W R I T E ( 6 , 1 0 7 ) ( O R D E R ( J , L ) , L = 1, L E N D ) , J , E1 ( J ) , E 2 ( J ) , NDQ {J) , 1 ) , RCONS(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 ) / TAUT(1) = TAZ  26  (TAZ - T A I )  DO 2 6 M = 2 , M E N D T T (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 ) * T T (M) ) T A U T (M) = { (Q (M) + 1.0 ) / (Q (M) - 1 . 0 ) ) * T A I CONTINUE  C WRITE  (6,110)  C DO 2 7 M=1,MEND T A U T L ( H ) = A L O G 1 0 ( A B S ( T A U T (M) ) ) W R I T E ( 6 , 1 1 1 ) T T (M) , Q (M) , T A U T (M) , T A U T L (M)  P ( J  135 27 C C  CONTINUE M A X I M U M AND M I N I M U M T A Z L = ALOG10 (ABS (TAZ) ) T A I L = ALOG 1 0 { A B S ( T A I ) ) DAI = TAZL SflO = T A I L S A = A B S ( D A I - SHO) D D A I = D A I + 0 . 1 * SA S S H O = SHO - 0 . 1 * S A  C C  FOB  THE  PLOTS  OF  THE  RESULTS  P L O T 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 (» TIME ( S E C . ) ' , 16, • LOG10 1ES / S Q . C M . ) ' , 4 0 ) 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 A X , S S H O , D D A I ) CALL ALGRAF ( T , TAUL, - ( I E N D ) , -4)  C C C  CALCULATION CALL  c  c  EYRING  FOR  ( T A U , T,  CALCULATION  FOR  SHEAR  STRESS  (DYN  EYRING T A I , T A Z , P E YR , E 2 E Y R ) GOVIER  ,,  C CALL  GOVIER  (TAU, T,  T A I , TAZ, PGOVI,  E2GOVI,  AKKA)  C  30  DO 3 0 M= 1 ,59END T A W E Y R (M) = ( T A Z - T A I ) * E X P ( ( - P E Y R ( 1) ) * T T (M) ) + T A I P1 = -AKKA * (TAZ + T A I ) / {TAZ - T A I ) P2(M) = P G O V I ( 2 ) * T T (M) + 1.0 P3(M) = A L O G ( A B S ( P 2 (M) ) ) P4(M) = P1 * P 3 ( M ) P 5 (M) = E X P ( P 4 (M) ) T A W G O V ( M ) = ( T A I + T A Z * P 5 (M) ) / ( 1 . 0 + ( T A I / T A Z ) * P 5 (fl) ) T A W E L ( M ) = A L O G 1 0 ( A B S ( T A W E Y R (M) ) ) T A WGL (M) = A L O G 1 0 ( A B S ( T A W G O V (M) ) ) CONTINUE  C WRITE  (6,121)  C DO 3 1 M=1,MEND W R I T E ( 6 , 1 2 0 ) T T (M) , T A U T (M) , T A W E Y R ( M ) , 1 TAW E L (M) , T A W G L (M) 31 CONTINUE C C  TAWGOV(M),  P L O T OT 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 (• TIME ( S E C . ) ' , 16, ' 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 , S S H O , D D A I ) C A L L A L G R A F ( T T , T A W E L , - (MEND) , - 2 ) C A L L A L S C A L ( 0 . 0 0 , TM A X , S S H O , D D A I )  T A U T L (M) ,  ..... LOG 1 0 S H E A S  STRESS  (DYN  C C  136  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 FORMAT (/, 24X, «J», 25X, »T,' 19X 107 (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'TAWGOVV, 9.X ' TAWTL * , 10X, TAWE •L 1 2 3 F O R M A T ( 1 H 1 ) C TO TERMINATE PLOTTING C  16X, ' TAUTL •)  #  CALL PLOTND STOP END FUNCTION  AUX  ( P , D, X, L)  C C C C c c c  THIS FUNCTION DEFINES THE FORM OF PINDER'S EQUATION.  c c c c c c c c  THIS SUBROUTINE EVALUATES THE PARAMETER OF EYRING'S MODEL BY MEANS OF LEAST SQUARES FITTING.  DIMENSION P (1) , D(1) AUX = P (1) * X D(1) = X RETURN END SUBROUTINE EYRING (TAU, T, TAI, TAZ, P, E2)  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  (6,103)  C  10  DO 10 1 = 1 , T E N D R 1 ( I ) = (TAU ( I ) - T A I ) / (TAZ - T A I ) R2 ( I ) = ALOG (R1(I)) WRITE ( 6 , 1 0 4 ) T ( I ) , R2 ( I ) , TAU ( I ) CONTINUE  C P ( 1 ) = 0.0 WZ = 0 . 0 EPS = 0.0005 N I = +20 C A L L L Q F ( T , R 2 , Y F , WT, E 1 , E 2 , P , WZ, 1AUXEYR) WRITE (6,101) W R I T E ( 6 , 1 0 2 ) E 1 { 1 ) , E 2 ( 1 ) , ND, P { 1 ) C C 101 102 103 104  C C C C C C  c  1,  N I , ND,  EPS,  . . . . . F O R M A T S FOR I N P U T AND O U T P U T S T A T E M E N T S F O R M A T (//////, 2 8 X , V E T , 1 3 X , * E 2 ' , 1 3 X , ND' , 1 3 X , » P ' ) F O R M A T ( 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 ) FORMAT ( 1 H 1 , / / / , 3 0 X , »T», 1 9 X , » R 2 » , 1 8 X , ' T A U , //) FORMAT ( 2 0 X , 3 E 2 0 . 4 ) RETURN END FUNCTION AUXEYR ( P , D, X , L ) 1  1  .. THIS FUNCTION  D E F I N E S THE  FORM  OF  EYRING'S  EQUATION.  . 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,  C C C C C C C C  I END,  THIS SUBROUTINE EVALUATES BY M E A N S O F L E A S T S Q U A R E S  T A I , TAZ,  P,  E2,  THE PARAMETERS FITTING.  DIMENSION T A U { 1 0 ) , T ( 1 0 ) , Y 1 { 1 0 ) , 1 WT ( 1 0 ) , P ( 2 ) , E l ( 2 ) , E 2 ( 2 ) E X T E R N A L AUXGOV I E N D = 10 WRITE (6,101) WRITE (6,105) C DO 1 0 1 = 1 , I E N D Y1 ( I ) - T A U ( I ) - T A I Y 2 ( I ) = TAZ - TAU ( I ) * T A I /  TAZ  Y2(10),  OF  AKKA)  GOVIER'S  Y3(10),  MODEL  Y4(10),  YF(10),  138  10  Y3 ( I ) = Y1 ( I ) / Y2 ( I ) Y4 ( I ) = ALOG ( Y 3 {I) ) WRITE (6,102) T ( I ) , Y 1 ( I ) , CONTINUE  Y4 ( I ) , TAU (I)  C P ( 1 ) = 0.5 P { 2 ) = 0.5 WZ = 0 . 0 NI = +50 M = 2 EPS = 0 . 0 0 0 5 C A L L LQ.F ( T , Y 4 , Y F , WT, E 1 , E 2 , P , WZ, I E N D , fl, N I , ND, E P S , 1 AUXGOV) WRITE (6,103) AKKA = - P ( 1 ) * (TAZ - T A I ) / (TAZ + T A I ) W R I T E ( 6 , 1 0 4 ) E 1 { 1 ) , E 1 { 2 ) , E 2 ( 1 ) , E 2 (2) , ND, P ( 1 ) , P (2) , A K K A C C  F O R M A T S FOR I N P U T AND O U T P U T S T A T E M E N T S FORMAT ( 1 H 1 , / / / , 3 0 X , * T', 1 9 X , ' 1 1 ' , 1 8 X , ' Y 4 ' , 1 8 X , » T A 0 » ) FORMAT ( 2 0 X , 4 E 2 0 . 4 ) 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)«, 10X, «ND«, 13X, ' P ( 1 ) ' , 1 1 X , • P {2) », 1 1 X , ' A K K A ' ) 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 ) 105 FORMAT (1H0) RETURN END F U N C T I O N A U X G O V ( P , D, X , L )  101 102 103  C C C C C C C  THIS FUNCTION  D E F I N E S THE  DIMENSION P ( 2 ) , D(2) F X Y Z = A B S ( 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  FORM  OF  GOVIER'S  EQUATION.  139  C-2  Programme  "SUPLPLT"  c C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C  MAIN  PROGRAM  "SUPLPLT"  T H I S PROGRAM C A L C U L A T E S T H I C K N E S S O F F L O W I N G L A Y E R RX AND S H E A R S T R E S S AT W A L L F R O M A S E T O F E X P E R I M E N T A L D A T A O F A S I N G L E R U N , AND T H E N T R I E S TO F I T T H E D A T A B Y 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 OF F L O W I N G L A Y E R AND B Y 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 M O D E L , BY M E A N S OF L E A S T S Q U A R E S F I T E V A L U A T I N G T H E F I T T I N G PARAMETERS I N EACH OF THE MODELS. HOW  ,  TO C O M P I L E T H I S P R O G R A M ? I N ORDER TO C O M P I L E THE PROGRAM " S U P L P L T " I N T O T H E I B M V E R S I O N F O R T R A N G - L E V E L C O M P I L E R UNDER C O N T R O L M I C H I G A N T E R M I N A L S Y S T E M (MTS) FOR T H E I B M 3 7 0 M O D E L 1 6 8 C O M P U T E R AT T H E 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 , T H E COMMAND $RUN SHOULD  HOW  *FTR BE  SCARDS=SUPLPLT  SPUNCH=-OBJECT  USED.  TO E X E C U T E T H E T H E COMMAND $RUN  OF  OBJECT  DECK?  -OBJECT+*PRPLOT  5=INPUTDATA  S H O U L D B E U S E D , W H E R E " I N P U T DAT A" I S T H E F I L E W H I C H C O N T A I N S T H E I N P U T D A T A , W H C S E O R D E R AND F O R M A T MUST B E C O N S I S T E N T W I T H , A N D C O R R E S P O N D TO T H O S E O F 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. D E S C R I P T I O N OF IEND MEND NEND DATE(M) RATIOZ RATIOQ TIME (I) MV(I) RATIO(I) RO R RX(I) S (I) TAU ( I )  -  TORQ(I) GAMMA(I)  -  PARAMETERS NUMBER OF D E C A Y P O I N T S P I C K E D U P FROM T H E DECAY CURVE (-) . N U M B E R O F B L O C K S O F A- FORM AT FOR I N P U T (-) . NUMBER OF T H E O R E T I C A L P O I N T S ( - ) . D A T E O F E X P E R I M E N T {-) . ( F X - R) / ( R O - R) AT T I M E = 0 . 0 (-) . ( R X - R) / (RO - R) AT T I M E = I N F I N I T Y ( - ) . T I M E OF S H E A R (SEC). R E A D I N G S ON R E C O R D E R (Mill VOLT). ( R X ( I ) - R) / (RO - R) ( - ) . R A D I U S OF OUTER C Y L I N D E R ( C M ) . R A D I U S OF I N N E R C Y L I N D E R ( C M ) . R A D I U S OF M O V I N G L A Y E R ( C M ) . R / RX ( I ) {-) . S H E A R S T R E S S AT T H E S U R F A C E O F I N N E R C Y L I N D E R ( D Y N E S / S Q . CM) . TORQUE (DYNES C M ) . APPARENT SHEAR RATE AT T H E S U R F A C E OF I N N E R C Y L I N D E R (1/SEC).  141 H E A L M V Z , MVQ, MV(10) DIMENSION DATE ( 5 ) , TIME ( 1 0 ) , R X ( 1 0 ) , S ( 1 0 ) , RATIO (10) 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 ) , SGOV(51) 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 ) , Z 6 ( 5 1 ) 4 , GAMMA ( 1 0 ) , B U M B O ( 1 0 ) , 5PPGOV{2), E2PGOV(2), TAUGOV(51), PP2(51), PP3{51), PP4(51), 6 P P 5 (51) , V I S C O (10) C C  READ E X P E R I M E N T A L DATA ..... I E N D = 10 MEND = 5 NE N D = 5 1 R E A D ( 5 , 1 0 1 ) ( D A T E { M) , M=1,MEND) RATIOQ READ (5,102) R A T I O Z , MVQ READ ( 5 , 1 0 2 ) MVZ, R E A D ( 5 , 1 0 3 ) ( T I M E ( I ) , 1 = 1 , I END) R E A D ( 5 , 1 0 4 ) (MV ( I ) , 1 = 1 , I E N D ) READ ( 5 , 1 0 5 ) (RATIO (I) , 1=1,IEND) R E A D ( 5 , 1 0 6 ) R, O M E G A READ ( 5 , 1 0 6 ) PLENGT, P H I  C C  11  C A L C U L A T I O N FOR R X ( I ) RO = 2 . 1 E X Q = R A T I O Q * ( S O - R) + R SQ = R / R X Q DO 11 1 = 1 , I E N D R X ( I ) = R A T I O ( I ) * (RO - H) S ( I ) = R / RX ( I ) CONTINUE TIMEZ =0.0 SZ = 1.0 TIMESA = TIME(IEND) T I M A X = T I M E (IEND) + 0.1 TIMIN = -0.1 * TIMESA S S A = 1.0 - SQ S M A X = 1.0 + 0 . 1 * S S A S M I N = SQ - 0. 1 * S S A  C C  C C  *  AND  •  S (I)  R  TIMESA  C A L C U L A T E S H E A R S T R E S S TAD ( I ) RL = 1.96 T O R Q Z = ( 4 . 4 6 E + 0 4 ) * MVZ T O R Q Q = ( 4 . 4 6 E + 0 4 ) * MVQ T A U Z = TORQZ / ( 2 . 0 * 3 . 1 4 1 6 * ( R * * 2 ) * RL) T A U Q = TORQQ / ( 2 . 0 * 3 . 1 4 1 6 * ( R * * 2 ) * R L ) DO 1 2 1 = 1 , 1 END TORQ(I) = (4.46E + 04) * MV(I) TAU (I) = TORQ(I) / ( 2 . 0 * 3.1416 * (R**2) * 12 CONTINUE ..... PLOT T I M E VS TAU ..... T A U S A = T A U Z - TAUQ T A U M A X = T A U Z + 0.1 * T A U S A T A U M I N = T A U Q - 0.1 * T A U S A C A L L A L S I Z E ( 7 . 0 , 5.0)  RL)  142 C A L L A L A X I S (» T I M E ( S E C ) », 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) CALL ALGRAF (TIME, TAU, ( 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 ALGRAF (TIMEZ, TAUZ, - 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 ALGRAF ( 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 1 3 N = 2 , N E N D T T (N) = T I N C * ( F L O A T (N - 1 ) ) 13 C O N T I N U E  C R X Q = R A T I O Q * ( R O - R) + R X S A = RXQ - R R X M A X = RXQ + 0 . 1 * RXSA RXMIN = R - 0.1 * RXSA C C  C C C  R  L E A S T S Q U A R E S F I T TO M O D I F I E D CALL GOVIER (TAU, T I M E , TAUQ, TAUZ, DO 18 N= 1 , N EN D PP 1 = - 2 . 0 * TAUQ * AKP P P 6 = (TAUZ + TAUQ) / (TAUZ -TAUQ) P P 2 ( N ) = P P G O V ( 2 ) * T T ( N ) + 1.0 P P 3 ( N ) = ALOG ( A B S ( P P 2 (N) ) ) P P 4 ( N ) = PP 1 * P P 3 ( N ) P P 5 (N) = E X P ( P P 4 (N) ) T A U G O V ( N ) = T A U Q * ( P P 6 + P P 5 (N) ) 1/ (PP6 - P P 5 ( N ) ) 18 CONTINUE .....  GOVIER'S EQUATION ..... PPGOV, E2PGOV, AKP)  P L O T E X P E R I M E N T A L D A T A AND M O D I F I E D G O V I E R ' S T E R M S OF T I M E V S . T A U C A L L A L S I Z E ( 7 . 0 , 5.0) C A L L A L A X I S {' TIME (SEC)', 13, 1' T A U ( D Y N E S / S Q . CM) ', 2 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 ALGRAF ( 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 , TIMAX, TAUMIN, TAUMAX) C A L L ALGRAF (TIMEZ, TAUZ, - 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 ALGRAF ( 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 , T A U G O V , - ( N E N D ) , 0)  C C CALL  P L O T T I M E V S RX A L S I Z E ( 7 . 0 , 5.0)  LINE  IN  CALL CALL CALL CALL CALL CALL CALL CALL CALL  ALAXIS ALSCAL ALGRAF ALSIZE ALSCAL ALGRAF ALSIZE ALSCAL ALGRAF  (' TIME ( S E C ) ' , 13, ' RX (CM) (TIMIN, TI MA X, RXMIN, RXMAX) (TIME, RX, (IEND) , -4) ( 7 . 0 , 5.0) (TIMIN, TI M AX, RXMIN, RXMAX) (TIMEZ, R, - 1 , -3) (7.0, 5.0) (TIMIN, TI MAX, RXMIN, RXMAX) (TIMESA, RXQ, - 1 , -3)  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), Z 4 ( N ) , Z 5 ( N ) , Z6(N) 1, TAUGOV(N) CONTINUE CALL ALSIZE (7.0, 5.0) CALL ALAXIS (V (\ ( S E C )•' ,, 13, ' RX (CM)', 10) TIME (SEC) RX CALL ALSCAL (TIMIN, TIMAX, RXMIN, RXMAX) CALL ALGSAF (TIME, RX, (IEND) , -4) CALL ALSIZE (7.0, 5.0) CALL ALSCAL (TIMIN, TIMAX, RXMIN, RXMAX) CALL ALGRAF (TIMEZ, R, - 1 , -3) CALL ALSIZE (7.0, 5.0) CALL ALSCAL (TIMIN, TI MAX, RXMIN, RXMAX) CALL ALGRAF (TIMES A, RXQ, - 1 , -3) CALL ALSIZE (7.0, 5.0) CALL ALSCAL (TIMIN, TIMAX, RXMIN, RXMAX) CALL ALGRAF (TT, Z 6 , - (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  17 C C  OUTPUT OF THE RESULTS WRITE (6,133) DO 17 1 = 1 , I E N D WRITE (6,134) TIME ( I ) , T O R Q ( I ) , T A U ( I ) , CONTINUE WRITE ( 6 , 1 4 0 ) RXQ, T A U Z , TAUQ  . . . . . P L O T GAMMA V S T A U GAMQ = 2 . 0 * O M E G A / ( 1 . 0 - S Q * * 2 ) GA MSA = GAMMA ( 1 ) - GAMQ GAMAX = G A M M A ( 1 ) + 0 . 1 * GAMSA G A M I N = GAMQ - 0 . 1 * G A M S A 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 CALL ALSCAL ( G A M I N , G A M A X , T A U M I N , T A UMAX) C A L L ALGRAF (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) CALL ALGRAF (GAMQ, T A U Q , - 1 , - 3 )  C C CALL CALL CALL CALL CALL CALL CALL C C  GAM MA ( I ) , V I S C O ( I ) ,  P L O T O F T I M E V S GAMMA . . . . . A L S I Z E ( 7 . 0 , 5.0) A L A X I S (' TIME (SEC)', 13, » GAMMA ALSCAL ( T I M I N , T I M A X , G A M I N , GAMAX) ALGRAF ( T I M E , GAMMA, ( I E N D ) , - 4 ) A L S I Z E ( 7 . 0 , 5.0) A L S C A L ( T I M I N , T I M A X , G A M I N , GAMAX) ALGRAF ( T I M E S A , GAMQ, - 1 , - 3 )  RX ( I )  (D Y N E S / S Q .CM) » , 2 0 )  (1/SEC)*,  16)  F O R M A T S FOR I N P U T AND O U T P U T S T A T E M E N T S FORMAT ( 5 A 4 ) FORMAT (2F10.3) FORMAT (8F10.1) FORMAT (8F10.2) FORMAT (8F10.3) FORMAT (2F10.5) FORMAT ( 1 H 1 , 7 X , »TT», 1 3 X , » Z 1 » , 1 3 X , ' Z 2 « , 1 3 X , »Z3*, 1 3 X , *Z4* 1, 1 3 X , »Z5*, 1 3 X , * Z 6 ' , 1 3 X , 'TAUGOV*) 132 FORMAT ( 2 X , 8 E 1 5 . 4 ) 1 3 3 F O R M A T ( 1 H 1 , / / / , 1 0 X , ' T I M E ' , 1 6 X , * TORQ *, 1 6 X , » T A U » , 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 ) 1 4 0 F O R M A T (////, 1 0 X , ' R X Q ' , 1 7 X , * T A U Z ' , 1 6 X , * T A U Q * , / / , 3 E 2 0 . 4 )  101 102 103 104 105 106 131  C C CALL  TERMINATE PLGTND  PLOTTING  .....  C STOP END SUBROUTINE C C C C C  GOVIER  ( T A U , T, T A I , T A Z , P, E 2 , A K K A )  T H I S SUBROUTINE EVALOATES THE PARAMETERS M O D E L BY M E A N S O F L E A S T S Q U A R E S F I T T I N G .  OF  MODIFIED  GOVIER * S  145 C  c c  . DIMENSION T A U ( 1 0 ) , T ( 1 0 ) , Y 1 ( 1 0 ) , Y 2 ( 1 0 ) , 1WT ( 1 0 ) , P ( 2 ) , E1 ( 2 ) , E 2 ( 2 ) E X T E R N A L AUXGOV I E N D = 10 WRITE (6,101) WRITE (6,105) PP6 = (TAZ + T A I ) / (TAZ - T A I ) DO 1 0 1 = 1 , I E N D Y1 ( I ) = T A U ( I ) - T A I Y2(I) = TAU(I) + TAI Y3 ( I ) = P P 6 * Y1 ( I ) / Y 2 ( I ) Y4 ( I ) = ALOG ( A B S ( Y 3 ( I ) ) ) W R I T E ( 6 , 1 0 2 ) T ( I ) , 1 1 ( 1 ) , Y4 ( I ) , T A U ( 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 EPS = 0.0005 C A L L L Q F ( T , Y 4 , Y F , WT, 1 AUXGOV)  C C  C C.  E 1 , E 2 , P,  OUTPUT OF THE R E S U L T S WRITE (6,103) AKKA = - P ( 1 ) / ( 2 . 0 * T A I ) WRITE (6,104) E 1 ( 1 ) , E 1 ( 2 ) ,  E2(1),  WZ,  Y3{10),  IEND,  E2 (2) , ND,  M,  Y4(10),  N I , ND,  P ( 1 ) , P (2) ,  YF(10),  EPS,  AKKA  F O R M A T S F O R I N P U T AND O U T P U T S T A T E M E N T S FORMAT ( 1 H 1 , / / / , 3 0 X , «T», 1 9 X , * Y 1 » , 1 8 X , » Y 4 ' , 1 8 X , *TAU') FORMAT ( 2 0 X , 4 E 2 0 . 4 ) F O R M A T (//////, 1 7 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 , * N D « , 1 3 X , » P ( 1 ) » , 1 1 X , • P ( 2 ) • , 1 1 X , * A K K A *) 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 A U X G O V ( P , D, X , L ) 101 102 103  C C  ..  c C C C C C C C C C  THIS FUNCTION  D E F I N E S THE  D E S C R I P T I O N OF AUXGOV X P(I) D(I)  FORM  OF  MODIFIED GOVIER'S  EQUATION.  PARAMETERS  - THE F U N C T I O N . - INDEPENDENT VARIABLES. - P A R A M E T E R S OF T H E M O D E L E Q U A T I O N T O B E E V A I U A T E D . - P A R T I A L D E R I V A T I V E S OF T H E F U N C T I O N WITH R E S P E C T TO E A C H P A R A M E T E R D ( I ) = ( D ( A U X G O V ) ) / ( D ( P ( I ) ) .  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(FXYZ) 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,  THIS SUBROUTINE EVALUATES BY M E A N S O F L E A S T S Q U A R E S  R,  RO,  P,  THE PARAMETERS FITTING.  E2,  OF  AK)  MIT  MODEL  DIMENSION R X ( 1 0 ) , Y 1 ( 1 0 ) , Y 2 ( 1 0 ) , Y 3 ( 1 0 ) , F ( 1 0 ) , 1, E 1 ( 2 ) , E 2 ( 2 ) , Y F ( 1 0 ) , WT(10) EXTERNAL AUXMIT I E N D = 10 WRITE (6,101) RL = 1.96 Q = (3.1416 * RL * (RO**2 BXQ**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 = S Q R T ( B * * 2 - 4. 0 * C ) DQ 1 = A B S ( { B + D) / ( B - D) )  T I M E ( 1 0 ) , P (2)  DQ2 = A L O G (DQ1) DO 1 0 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) / ( Y 1 ( I ) + D) ) Y3 ( I ) = A L O G ( Y 2 ( I ) ) F ( I ) = Y 3 ( I ) + DQ2 WRITE ( 6 , 1 0 2 ) TIME ( I ) , Y 1 ( I ) , Y2 (I) , 1 3 ( 1 ) , F ( I ) 10 CONTINUE L E A S T SQUARES F I T ..... P ( 1 ) = 0.5 P ( 2 ) = 0.5 WZ = 0 . 0 NI = 50 M = 2 EPS = 0.0005 C A L L L Q F ( T I M E , F , Y F , WT, E 1 , E 2 , P , 1, AUXMIT) ..... OUTPUT OF THE R E S U L T S ..... WRITE (6,103) AK = P ( 1 ) / (D * 3 . 1 4 1 6 * R L ) WRITE (6,104) E 1 ( 1 ) , E 1 ( 2 ) , ,E2(1),  101  FORMAT  F O R M A T S FOR (1H1, ///,  WZ,  I END,  E2 (2) , ND,  I N P U T AND O U T P U T S T A T E M E N T S 2 0 X , * T I M E' , 1 6 X , » Y 1 » , 1 8 X ,  M,  N I , ND,  P ( 1 ) , P (2) ,  »Y2»,  18X,  EPS  AK  »Y3•  147 1 , 1 8 X , * F*) FORMAT ( 1 0 X , 5 E 2 0 . 4 ) FORMAT {//////, 1 4 X , » E 1 ( 1 ) « , 1 0 X , » E 1 ( 2 ) ' , 1 0 X , * E 2 { 1 ) ' , 1 ' E 2 ( 2 ) ' , 1 0 X , * ND* , 1 3 X , » P ( 1 ) * , 1 1 X , * P ( 2 ) ' , 1 1 X , ' A K ' ) 1 0 4 F O R M A T (( 11HH00, , 4 X , ** MMIITT'' , 4 E 1 5 . 4 , 1 1 5 , 3 E 1 5 . 4 ) RETURN END F O N O T I O N A U X M I T ( P , D, X , L ) 102 103  10X,  C C  c c c c c  c c c c c c c c c  THIS  FUNCTION  DESCRIPTION AUXMIT X P(I) D(I)  OF  DEFINES  THE  FORM  OF  MIT  EQUATION.  PARAMETERS THE FUNCTION. INDEPENDENT VARIABLES. P A R A M E T E R S O F T H E M O D E L E Q U A T I O N TO B E 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 T H E F U N C T I O N H I T H R E S P E C T TO E A C H P A R A M E T E R 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 ) AUXMIT = P (1) * D (1) D ( 2 ) = ( P ( 1 ) * X) / F X Y Z RETUEN END  •148  C-3  Programme  "TAUDECAY"  149 C  .  ..  c C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C  MAIN  PROGRAM  "TAUDECAY"  T H E P U R P O S E OF T H I S P R O G R A M I S T O F I N D OUT W H I C H O N E O F T H E V A R I O U S O R D E R S ( F R O M ( 0 , 1 ) TO ( 2 , 2 ) ) O F T H E M O D I F I E D G O V I E R ' S S T R E S S DECAY MODELS C A N B E S T - F I T THE E X P E R I M E N T A L DECAY C U R V E I N T E R M S OF S H E A R S T R E S S M E A S U R E D AT THE WALL G F T H E C O - A X I A L 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 O F T H E E X P E R I M E N T A L RUNS. T H E P R O G R A M C A L C U L A T E S T H E SUM O F S Q U A R E S A N D T R I E S TO M I N I M I Z E I T B Y A D J U S T I N G T H E F I T T I N G P A R A M E T E R S O F E A C H OF T H E M O D E L S , AFTER E V A L U A T I N G THE F I T T I N G PARAMETERS, THE COMPUTER WILL L I S T T H E F I N A L V A L U E S OF T H O S E F I T T I N G P A R A M E T E R S OF E A C H MODEL. HOW  TO C O M P I L E T H E P R O G R A M " T A U D E C A Y " ? I N ORDER TO C O M P I L E T H E PROGRAM " T A U D E C A Y " I N T O T H E I E H V E R S I O N F O R T R A N 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 T E R M I N A L S Y S T E M (MTS) FOR T H E I B M 3 7 0 MODEL 1 6 8 COMPUTER 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 $RUN * F T N SHOULD  HOW  SC A R D S = T A U D E C A Y S P U N C H = - O B J E C T  BE USED.  TO E X E C U T E T H E T H E COMMAND $RUN - O B J E C T  OBJECT  DECK  "-OBJECT"?  4-=BFOSDEF 5 = E X P D A T A  SHOULD B E U S E D , WHERE " B F O R D E R " I S T H E F I L E WHICH C O N T A I N S T H E N A M E S O F V A R I O U S O R D E R S OF M O D E L S A N D S H O U L D B E R E A D T H R O U G H L O G I C A L I / O U N I T 4 BY A - T Y P E F O R M A T . T H E " E X P D A T A " I S T H E F I L E C O N T A I N I N G T H E E X P E R I M E N T A L RAW D A T A , W H O S E O R D E R A N D F O R M A T M U S T C O R R E S P O N D T O , AND E E C O N S I S T E N T WITH T H O S E S P E C I F I E D I N T H E M A I N PROGRAM OF "TAUDECAY". D E S C R I P T I O N OF P A R A M E T E R S . IEND - NUMBER OF E X P E R I M E N T A L DATA. JEND - NUMBER OF MODEL E Q U A T I O N S , LEND - N U M B E R OF B L O C K S O F A - F O R M AT F O R O R D E R ( J , L ) . MEND - N U M B E R O F B L O C K S O F A - F O R M AT FOR D A T E {M) . 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. D A T E (M) - DATE OF E X P E R I M E N T . RATIOZ - RAW D A T A O F ( R X - R ) / ( R O - R ) AT T - 0 . 0 M E A S U R E D WITH C I N E PHOTOGRAPH { - ) . RATIOQ - RAW D A T A O F ( R X - R ) / ( R O - R ) A T E Q U I L I B R I U M MEASURED WITH C I N E PHOTOGRAPH ( - ) . DAMVZ - RAW D A T A O F T O R Q U E ON T H E R E C O R D E R C H A R T AT T = 0 . 0 ( M V ) . DAMVQ - RAW D A T A O F T O R Q U E ON T H E R E C O R D E R C H A R T AT E Q U I L I B R I U M ( M V ) . TIME (I) - T I M E (SEC) . DAMV(I) - RAW D A T A O F T O R Q U E O N T H E R E C O R D E R C H A R T  150 C C C C C C C C C C C C C C C C C C C C C C C C  RATIO(I) R OMEGA PLENGT PHI BL BC TOBQ(I) TAU(I) FF(I) AKGOV(J) PP(J,2)  (MV) . - RAW DATA OF {RX-R) / ( R O - R ) M E A S U R E D WITH C I N E P H O T O G R A P H (-) . - R A D I U S OF I N N E R C Y L I N D E R O F T H E VISCOMETER (CM) . - ANGULAR V E L O C I T Y OF THE INNER C Y L I N D E R (RAD/SEC). - L E N G T H O F F I B E R P A R T I C L E (MM) . - V O L U M E F R A C T I O N OF F I B E R S I N T H E ARTIFICIAL SLURRY (CC/CC) . - H I G H T OF T H E I N N E R C Y L I N D E R ( C M ) . - R A D I U S OF T H E OUTER C Y L I N D E R ( C M ) . - TORQUE (DYNES CM). - S H E A R 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 (DYNES/SQ.CM). - D E P E N D E N T V A R I A B L E O F T H E DATA TO B E F I T T E D BY SUBROUTINE 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 M O D I F I E D G O V I E R ' S MODEL. - T H E S E C O N D F I T T I N G P A R A M E T E R S OF M O D I F I E D G O V I E R ' S MODEL.  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 ) , E 1 ( 2 ) , 2E2(2), P ( 2 ) ,PP(8,2), AKGOV(8), YF(10) C EXTERNAL  AUX  C IEND JEND LEND MEND C C  = = = =  10 8 6 5  ..... READ VARIOUS R E A C T I O N ORDERS READ ( 4 , 1 0 9 ) ( (ORDER ( J , L ) , L= 1,LEND) ,  J=1,JEND)  C 1  CONTINUE  C C READ C C C  C  EXPERIMENT M= 1 , MEND)  ..... IF  C C  R E A D T H E DATE OF ( 5 , 1 0 1 ) ( D A T E (M) ,  I F NO M O R E E X P E R I M E N T A L D A T A THE CALCULATION ( D A T E ( 1 ) . E Q . OWAR) GO TO 2  ..... READ READ READ READ READ READ READ  R E A D E X P E R I M E N T A L RAW D A T A (5,102) R A T I O Z , RATIOQ ( 5 , 1 0 3 ) DAM V S , DA MVQ (5,104) (TIME ( I ) , 1=1, IEND) (5,105) (DAMV(I), 1=1,IEND) (5,106) (RATIO ( I ) , 1=1,IEND) ( 5 , 1 0 7 ) R, OMEGA (5,108) PLENGT, P H I  TO  BE  READ,  TERMINATE  151 C  C A L C U L A T E SHEAR STRESS RL = 1 . 96 RO = 2. 1 TORQZ = ( 4 . 4 6 E + 04) * DAMVZ T O B Q Q = ( 4 . 4 6 E + 0 4 ) * DAMVQ T A U Z = TORQZ / ( 2 . 0 * 3 . 1 4 1 6 T A U Q = TOBQQ / ( 2 . 0 * 3 . 1 4 1 6  C C  11  (R**2) (R**2)  * *  RL) RL)  C A L C U L A T E S H E A R S T R E S S AT ANY T I M E DO 11 1 = 1 , I E N D T O E Q ( I ) = ( 4 . 4 6 E + 04) * DAMV(I) TAU(I) = TORQ(I) / (2.0 * 3.1416 * (R**2) CONTINUE  C C BAKA6 B7 = D7 = BAKA7 Q8 = A8 = B8 = C8 = D8 = BAKA8 C C C  * *  *  RL)  C A L C U L A T E V A R I O U S C O N S T A N T S ..... = ( T A U Z • TAUQ) / (TAUZ - TAUQ) (TAUQ**2) / (TAUZ - TAUQ) TAUQ * ( 2 . 0 * TAUZ - TAUQ) / (TAUZ - TAUQ) = ( 2 . 0 * T A U Z + B 7 + D7) / ( 2 . 0 * T A U Z + B 7 - D 7 ) (TAUQ / ( T A U Z - TAUQ) ) **2 Q8 - 1.0 - 2 . 0 * Q8 * T A U Z Q8 * ( T A U Z * * 2 ) 2.0 * TAUQ * T A U Z / ( T A U Z - T A U Q ) = ( 2 . 0 * A 8 * T A U Z + B 8 • D8) / ( 2 . 0 * A 8 * T A U Z +  C A L C U L A T E V A R I O U S C O N S T A N T S FOR DEPENDENT V A R I A B L E S F (J , I ) ..... DO 1 2 1 = 1 , I E N D DEV (1,1) = ABS ( ( T A U ( I ) - TAUQ) / ( T A U Z - TAUQ) ) DEV (2,1) = A B S { ( T A U ( I ) • T A U Q - 2.0 * T A U Z ) / ( T A U ( I )  DEV ( 3 , 1 )  =  DEV(1,1)  D E V ( 4 , 1 ) = DEV ( 1 , 1 ) DEV ( 5 , 1 ) = ABS { (TAU (I) - TAUQ) / DEV ( 6 , 1 ) = ABS ( B A K A 6 * ( T A U ( I ) DEV (7,1) = ABS (BAKA7 * (2. 0 * TAU 1<2.0 * T A U ( I ) + B7 + D 7 ) ) DEV ( 8 , 1 ) = ABS ( B A K A 8 * ( 2 . 0 * A8 1 ( 2 . 0 * A 8 * T A U ( I ) + B 8 + D8) ) 12 C O N T I N U E C C  14 13  ..... C A L C U L A T E THE D E P E N D E N T DO 1 3 J = 1 , J E N D DO 1 4 1 = 1 , I E N D F ( J , I ) = ALOG (DEV ( J , I ) ) CONTINUE CONTINUE  C C  * TAU (I)  V A R I A B L E TO  P R I N T OUT T H E T I T L E F O R T H E R E S U L T S J=1,JEND (6,137) ( D A T E (M) , M = 1 , M E N D ) ( 6 , 1 3 1 ) (ORDER ( J , L ) , L= 1,LEND)  DO  CONVERT THE FORM 1=1,IEND  16  DEPENDENT  D8)  TAUQ) )  (TAUZ - TAU (I) * TAUQ / T A U Z ) ) TAUQ) / (TAD ( I ) + T A U Q ) ) ( I ) + B 7 - D7) /  DO 1 5 WRITE WRITE C C C  -  E8 -  VARIABLES INTO  +  B8  BE  -  D8)  /  FITTED LATER  .....  ONE  DIMENSIONAI  ON  ....  152  C  16  FF(I) = F(J,I) CONTINUE WRITE  C C C  C H E C K I N D E P E N D E N T AND D E P E N D E N T BY T H E S U E R O U T I N E L Q F DO 1 7 1 = 1 , I E N D WRITE (6,133) T I H E ( I ) , F F ( I ) 17 C G N T I N H E  C C  CONSTANTS P ( 1 ) = 0.5 P ( 2 ) = 0.5 WZ = 0 . 0 NI = 50 M = 2 EPS = 0.0005  C C  FOR  THE  LEAST  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, 1, AUX)  C C  15 C C C  C C  (6,132) V A R I A B L E S TO  SQUARES  FIT  E 1 , E2, P,  . . . . . O U T P U T OF T H E R E S U L T S OF W R I T E ( 6 , 1 3 4 ) ND, P ( 1 ) , P ( 2 ) PP(J,1) = P(1) PP(J,2) = P(2) CONTINUE  THE  WZ,  BE  .....  IEND,  M,  N I , ND,  FITTING  C A L C U L A T E THE F I R S T F I T T I N G P A R A M E T E R S I N TERMS M O D I F I E D G O V I E R ' S FORM . . . . . AKGOV(1) = - P P ( 1 , 1 ) * (TAUZ - TAUQ) A K G O V ( 2 ) = P P ( 2 , 1 ) * ( T A U Z - T A U Q ) / 2. 0 AKGOV(3) = -PP(3,1) A K G 0 V ( 4 ) = - P P ( 4 , 1 ) * (TAUZ - TAUQ) / T A U Z 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 , M E N D ) WRITE (6,135) PRINT THE RESULTS DO 18 J = 1 , J E N D WRITE (6,136) ( O R D E R ( J , L ) , L = 1, L END) , A K G O V ( J ) , 18. C O N T I N U E  C C C GO 2 C C 101 102  TO  REPEAT THE PROCEDURE EXPERIMENTAL RUN 1  DESCRIBED  ABOVE  FOR  CONTINUE . . . . . F O R M A T S FOR FORMAT (5A4) FORMAT (2F10.3)  INPUT  AND  OUTPUT  FITTED  STATEMENTS  OF  PP(J,2)  THE  NEXT  THE  EPS  153 103 104 105 106 107 108 109 131 132 133 134  FORMAT (2F10.3) FOBMAT (8F10.1) FORMAT (8F10.2) FORMAT (8F10.3) FORMAT (2F10.5) FORMAT (2F10.5) FORMAT ( 6 A 4 ) F O R M A T {//, 2 X , 6 A 4 ) F O R M A T {/, 1 5 X , ' T I M E ( I ) ' , 6 X , • F F ( I ) ' , / ) FORMAT ( 1 0 X , 2 E 1 3 . 4 ) F O R M A T (////, 6 X , ND' , 3 X , » P ( 1 ) » , 9 X , * P { 2 ) « , / / , 5 X , 1 2 , 2 E 1 3 . 4 1) 1 3 5 F O R M A T {//, 2 X , * OR DER * , 2 2 X , ' A K G O V ( J ) * , 5 X , ' P P ( J , 2 ) » , /} 1 3 6 FORMAT ( 2 X , 6 A 4 , 2 E 1 3 . 4 ) 137 FORMAT ( 1 H 1 , / / , 1 5 X , 5A4) f  STOP END FUNCTION C C C C C  c c c c c c c c c c  AUX  ( P , D,  X, I )  T H I S F U N C T I O N D E F I N E S T H E FORM THE SUBROUTINE LQF. DESCRIPTION AUX X P(I) D(I)  -  OF E Q U A T I O N  TO  BE F I T T E D  OF P A R A M E T E R S THE FUNCTION. INDEPENDENT VARIABLE. P A R A M E T E R S O F T H E M O D E L TO B E 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 T H E F U N C T I O N WITH 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  BY  RESPECT  154  C-4 Programme  "THICKNESS"  155 C  ,  c C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C^ C C C C C C C C C C C C C C C C C C C  MAIN  PRGGRAM  ,  "THICKNESS"  THE MODEL WHICH P R E D I C T S T H E T H I C K N E S S OF 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 B E E N  THE FLOWING LAYER CONSTRUCTED.  RX  T H E P U R P O S E O F T H I S P R O G R A M I S TO F I N D OUT W H I C H ONE OF T H E V A R I O U S O R D E R S ( F R O M ( 0 , 1 ) TO ( 2 , 2 ) ) OF T H E M O D E L C A N B E S T - F I T T H E E X P E R I M E N T A L D A T A W H I C H H A V E B E E N O B T A I N E D FROM T H E 8MM C I N E P H O T O G R A P H S FOR A L L S E T S O F T H E E X P E R I M E N T A L R U N S C A R R I E D OUT BEFORE. T H E P R O G R A M C A L C U L A T E S T H E SUM O F S Q U A R E S AND T R I E S TO M I N I M I Z E 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 E A C H ORDER OF T H E MODEL. A F T E R 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 , T H E C O M P U T E R W I L L L I S T T H E F I N A L V A L U E S O F T H O S E F I T T I N G P A R A M E T E R S OF E A C H ORDER OF T H E MODEL. HOW  TO C O M P I L E T H E P R O G R A M " T H I C K N E S S " ? I N O R D E R TO C O M P I L E T H E P R O G R A M " T H I C K N E S S " I N T O T H E I B M V E R S I O N F O R T R A N G - L E V E L C O M P I L E R UNDER C O N T R O L OF M I C H I G A N TERMINAL SYSTEM ( M T S ) F O R T H E I B M 3 7 0 M O D E L 1 6 8 C O M P U T E R AT T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A , T H E COMMAND $RUN SHOULD  HOW  *FTN  BE  SPUNCH=-OBJ  ECT  USED.  TO E X E C U T E T H E T H E COMMAND $RUN  SCARDS=THICKNESS  OBJECT  -OBJECT  DECK  4=BFORDE R  "-OBJECT"?  5=EXPDATA  SHOULD B E U S E D , WHERE " B F O R D E R " I S T H E F I L E T H E N A M E O F V A R I O U S O R D E R S O F T H E M O D E L AND T H R O U G H L O G I C A L I / O U N I T 4 EY A - T Y P E F O R M A T . I S T H E F I L E C O N T A I N I N G T H E E X P E R I M E N T A L RAW O R D E R AND F O R M A T MUST C O R R E S P O N D T O , AND B E THOSE S P E C I F I E D I N THE MAIN PROGRAM. D E S C R I P T I O N OF IEND JEND LEND MEND ORDER(J,L) B A T E (M) RATIOZ RATIOQ DAHVZ DAMVQ  WHICH C O N T A I N S S H O U L D BE R E A D THE " E X P L A T A " D A T A , WHOSE C O N S I S T E N T WITH  PARAMETERS. - NUMBER OF E X P E R I M E N T A L DATA. - NUMBER OF MODEL E Q U A T I O N S . - N U M B E R O F B L O C K S O F A - F O R M AT FOR O R D E R ( J , 1 ) . - N U M B E R O F B L O C K S O F A - F O R M AT FOR D A T E { M ) . - REACTION ORDERS. - D A T E OF E X P E R I M E N T . - RAW D A T A O F ( R X - R ) / ( R O - R ) AT T = 0 . 0 M E A S U R E D WITH C I N E PHOTOGRAPH ( - ) . - RAW D A T A O F ( R X - R ) / ( R O - R ) AT E Q U I L I B R I U M MEASURED WITH C I N E PHOTOGRAPH ( - ) . - RAW D A T A O F T O R Q U E ON T H E R E C O R D E R C H A R T AT T = 0 . 0 (MV) . - RAW D A T A O F T O R Q U E ON T H E R E C O R D E R C H A R T AT E Q U I L I B R I U M ( M V ) .  156 TIME(I) DAMV ( I ) RATIO ( I ) R OMEGA PLENGT PHI RL RO RXQ RX{I) FF ( I ) AK{J) PP(J,2)  TIME (SEC) . RAW D A T A O F T O R Q U E ON T H E R E C O R D E R C H A R T (MV) . RAW D A T A O F ( R X - R ) / ( R O - R ) M E A S U R E D W I T H C I N E P H O T O G R A P H {-) , R A D I U S OF I N N E R C Y L I N D E R O F T H E VISCOMETER (CM) . ANGULAR V E L O C I T Y OF THE INNER C Y L I N D E R (RAD/SEC). L E N G T H O F F I B E R P A R T I C L E (MM) . V O L U M E F R A C T I O N OF F I B E R S I N T H E ARTIFICIAL SLURRY (CC/CC) . H I G H T OF T H E I N N E R C Y L I N D E R ( C M ) . R A D I U S OF T H E O U T E R 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 F L O W I N G L A Y E R M E A S U R E D FROM T H E C E N T E R O F T H E C Y L I N D E R S OF T H E V I S C O M E T E R ( C M ) . D E P E N D E N T V A R I A B L E O F T H E D A T A TO B E F I T T E D B Y THE SUBROUTINE 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 T H E M O D E L .  DIMENSION 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 (10) 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 ) , YF(10) 2 , F F ( 1 0 ) , W T ( 1 0 ) , E 1 ( 2 ) , E 2 ( 2 ) , P ( 2 ) , P P { 8 , 2 ) , AK ( 8 ) , D A T E ( 8 ) 3, DAMV(10) C EXTERNAL  AUX  C IEND JEND LEND MEND  10 8 6 5  READ VARIOUS REACTION ORDERS R E A D { 4 , 1 0 9 ) ( (ORDER ( J , L) , L= 1 ,LEND) , 1 CONTINUE C C  C C  J=1,JEND)  . . . . . R E A D E X P E R I M E N T A L RAW DATA R E A D ( 5 , 1 0 1 ) ( D A T E (M) , M= 1 , MEND) READ ( 5 , 102) R A T I O S , RATIOQ R E A D ( 5 , 1 0 3 ) D A M V Z , DAMVQ READ ( 5 , 1 0 4 ) ( T I M E ( I ) , 1=1,IEND) READ ( 5 , 105) (DAMV(I) , 1=1,IEND) R E A D ( 5 , 1 0 6 ) ( R A T I O ( I ) , 1=1, I E N D ) R E A D ( 5 , 1 0 7 ) R, O M E G A READ ( 5 , 108) PLENGT, P H I C A L C U L A T E THE T H I C K N E S S OF RL = 1.96 RO = 2. 1 R X Q = R A T I O Q * (RO - R) + R DO 1 1 1 = 1 , I E N D RX(I) = RATIO (I) * ( R O - R ) + R  THE  FLOWING  LAYER  RX  11  X ( I ) = RX ( I ) **2 CONTINUE  -  R**2  C A L C U L A T E V A R I O U S C O N S T A N T S FOR D E P E N D E N T V A R I A B L E S F(J,I) RXQR RXQ**2 - R**2 P 0 1 = 1.0 / < 3 . 1 4 1 6 * R L * R X Q R ) P 0 2 = 1.0 / { ( 3 . 1 4 1 6 * R L * R X Q R ) * * 2 ) AO 2 = 1.0 / ( 3 . 1 4 1 6 * R L * S Q R T ( P 0 2 ) ) P10 = 3.1416 * RL * (RO**2 - RXQ**2) A10 = P10 / (3.1416 * RL) - (RO**2 - R**2) P 1 1 = ( R O * * 2 - R X Q * * 2 ) / RXQR a n = ( R O * * 2 - R**2) / ( P 1 1 + 1.0) P 1 2 -- ( R O * * 2 - R X Q * * 2 ) / ( 3 . 1 4 1 6 * R L * ( R X Q R * * 2 ) ) A 1 2 = P 1 2 * 3. 1 4 1 6 * R L B12 '- R 0 * * 2 - R * * 2 S Q R T ( 1 . 0 + 4.0 * A12 * B 1 2 ) D12 P20 = ( 3 . 1 4 1 6 * RL * ( R O * * 2 - R X Q * * 2 ) ) * * 2 B20 = - 2 . 0 * ( R O * * 2 - R * * 2 ) C 2 0 -- ( R O * * 2 - R * * 2 ) * * 2 - P 2 0 / ( { 3 . 1 4 1 6 * R L ) * * 2 ) D 2 0 = S Q R T ( B 2 0 * * 2 - 4.0 * C 2 0 ) P2 1 = 3 . 1 4 1 6 * R L * ( { R 0 * * 2 - R X Q * * 2 ) * * 2 ) / RXQR B 2 1 = - P 2 1 / ( 3 . 1 4 1 6 * RL) - 2.0 * ( R O * * 2 - R * * 2 ) C21 = (RG**2 - R * * 2 ) * * 2 D 2 1 = S Q R T ( B 2 1 * * 2 - 4.0 * C 2 1 ) P 2 2 = C (110**2 - R X Q * * 2 ) * * 2 ) / ( R X Q R * * 2 ) A 2 2 = P 2 2 - 1.0 B 2 2 - 2.0 * ( R O * * 2 - R * * 2 ) C 2 2 = - { (RO**2 - R**2) **2) D 2 2 = S Q 1 T ( B 2 2 * * 2 - 4.0 * A 2 2 * C 2 2 ) DO 12 1 = 1 , I E N D D E V (1,1) = A B S ( 1 . 0 - P 0 1 * 3 . 1 4 1 6 * R L * X ( I ) ) DEV ( 2 , 1 ) = A B S ( ( X ( I ) - A02) / ( X ( I ) + A 0 2 ) ) D E V (3,1) = A B S (X ( I ) / A 1 0 + 1.0) D E V (4,1) = A B S { 1 . 0 - X { I ) / A 1 1 ) Y 1 2 ( I ) = 2. 0 * A 1 2 * X ( I ) + 1.0 D E V (5,1) = A B S ( ( Y 1 2 ( I ) - D 1 2 ) * ( 1 . 0 + D 1 2 ) / ( ( Y 1 2 ( I ) + 1* ( 1 . 0 - D12)) ) Y 2 0 ( I ) = 2.0 * X ( I ) + B 2 0 D E V (6,1) = A B S ( ( Y 2 0 ( I ) D20) * ( B 2 0 + D20) / ( ( 7 2 0 ( 1 ) • 1*(B20 - D20))) Y21 (I) = 2 . 0 * X ( I ) + B21 D E V (7,1) = A B S { ( Y 2 1 ( I ) - D 2 1 ) * ( B 2 1 • D 2 1 ) / { ( Y 2 1 ( I ) + (B21 - D 2 1 ) ) ) 1* Y 2 2 ( I ) = 2.0 * A22 * X ( I ) + B 2 2 D E V (8,1) = ABS((Y22(I) - D22) * ( B 2 2 + D22) / ( ( Y 2 2 ( I ) + 1* ( B 2 2 - D 2 2 ) ) ) 12 CONTINUE DO 1 3 J = 1 , J E N D DO 14 1=1, I E N D F ( J , I ) = ALOG (DEV ( J , I ) ) 14 CONTINUE 13 CONTINUE  DO  15  P R I N T OUT J=1,JEND  THE  TITLE  FOR  THE  RESULTS  .....  D12)  D20)  D21)  D22)  158 WRITE{6,137) RRITE(6,131) C C C  ( D A T E ( M ) , M= 1, M E N D ) ( O R D E R ( J , L ) ,L= 1 , L E N D )  .....  16  CONVERT DEPENDENT FORM . . . . . DO 16 1 = 1 , 1 END. FP{I) = F(J,I) CONTINUE  VARIABLES  I N T O ONE  DIMENSIONAL  C C C  .....  17 C C  ..... CONSTANTS P ( 1 ) = 0.5 P { 2 ) = 0.5 WZ = 0 . 0 NI = 50 M = 2 EPS = 0.0005.  C C  15 C C  THE  DEPENDENT  LEAST  VARIABLES  SQUARES  ..... E 1 , E 2 , P,  . . . . . O U T P U T OF T H E R E S U L T S OF W R I T E { 6 , 1 3 4 ) ND, P ( 1 ) , P ( 2 ) PP(J,1) = P(1) PP ( J , 2 ) = P ( 2 ) CONTINUE  c  18  WZ,  IEND,  AK ( J ) ,  I F T H E R E A R E NO MORE E X P E R I M E N T A L RAW T E R M I N A T E THE C A L C U L A T I O N ..... I F ( D A T E ( 1 ) . E Q . OWAR) GO TO 2 ARE  STILL  DATA,  BE  M,  FITTED  N I , ND,  FITTING  ..... P R I N T THE R E S U L T S ..... W R I T E ( 6 , 1 3 7 ) ( D A T E ( M) , M= 1 , M E N D ) W R I T E ( 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) , CONTINUE  ..... I F THERE  TO  FIT  C A L C U L A T E THE F I R S T F I T T I N G P A R A M E T E R S AK ( 1 ) - P P ( 1 , 1) / P 0 1 AK ( 2 ) = - P P ( 2 , 1 ) / ( 2 . 0 * 3 . 1 4 1 6 * R L * P 0 2 * AK ( 3 ) - P P ( 3 , 1) AK ( 4 ) = - P P ( 4 , 1) / ( P 1 1 + 1 .0) AK{5) - P P ( 5 , 1) / D 1 2 AK ( 6 ) ( 3 . 1 4 1 6 * RL * D20) PP(6,1) / 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 C  FOR  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, 1, AUX)  C C  C C C  C H E C K I N D E P E N D E N T AND BY THE S U B R O U T I N E L Q F WRITE (6,132) DO 1 7 1 = 1 , I E N D WRITE (6,133) T I M E ( I ) , F F ( I ) CONTINUE  REPEAT THE  A02)  PP(J,2)  DATA,  CALCULATION  EPS  159  GO  101 102 103 104 105. 106 107 108 109 137 131 132 133 134  DESCRIBED TO 1  ABOVE  . . . . . F O R M A T S FOR I N P U T A N D O U T P U T S T A T E M E N T S FORMAT ( 5 A 4 ) FORMAT (2F10.3) FORMAT (2F10.3) FORMAT (8F10.1) FORMAT (8F10.2) FORMAT (8F10.3) FORMAT (2F10.5) FORMAT (2F10.5) FORMAT ( 6 A 4 ) FORMAT ( 1 H 1 , / / , 1 5 X , 5A4) FORMAT ( / / , 2 X , 6 A 4 ) F O R M A T (/, 1 5 X , ' T I M E ( I ) ' , 6 X , ' F F ( I ) ' , /) FORMAT(10X, 2E13.4) P(2) F O R M A T (////, 6 X , ' N D • , 3 X , • P ( 1 ) 9X, ,  #  D 135 136 2  FORMAT FORMAT  ( / / , 2 X , 'ORDER', (2X, 6A4, 2E13.4)  CONTINUE STOP END F U N C T I O N AUX  THIS  ( P , D,  FUNCTION  DESCRIPTION AUX X P{I) D(I)  OF  X,  22X,  »AK(J)», 8X,  //,  5X, 1 2 , 2E13.4  » P P < J , 2 ) « , /)  L)  D E F I N E S T H E FORM OF  MIT  EQUATION.  PARAMETERS.  - THE FUNCTION. - INDEPENDENT VARIABLES. - P A R A M E T E R S O F T H E M O D E L E Q U A T I O N T O B E E V A L DAT ED, - P A R T I A L D E R I V A T I V E S OF T H E F U N C T I O N W I T H R E S P E C T TO E A C H 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  Appendix D-l  D  Shear  Experimental Stress  Description  at  of  t  =  Raw D a t a 0 and  Variables  t  in  and  =  Calculated  t  m  Table  D-l  PHT(CC/CC)  -  Particle Concentration  L(MM)  -  P a r t i c l e Length Angular  OMEGA  (RAD/SEC)  -  YIELD  MV(MV)  -  Yield volt  E Q U I L I B R I U M MV(MV)  -  of  Recorder  volt  160  of  Inner  Torque  Cylinder  Measured  on  (rad./sec.) the  m i l l i  (mV).  Equilibrium Value m i l l i  (cc/cc).  (mm).  Velocity  Value  Results  of  Recorder  Torque (mV).  Measures  on  the  TABLE  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 T I M E = 0 A N D A T E Q U I L I B R I U M  DEX  +  1.0MOLE NACL  PHI(CC/CC) 0.04  L (MM) 5.030  OMEGA ( R A D / S E C ) 0.2327  Y I E L D MV (MV) 0.167E+01  TORQUE ( D Y N E S CM) 0.745E+05  SHEAR STRESS ( D Y N E S / S Q CM) 0.593E+04  TORQUE ( D Y N E S CM) 0.219E+05  SHEAR STRESS ( D Y N E S / S Q CM) 0.174E+04  EQUILIBRIUM (MV) 0.491E+00  DEX  +  MV  1.0MOLE NACL  I N 10.0%  PEG-H20  PHI(CC/CC) 0.05  L (MM) 5. 0 3 0  OMEGA ( R A D / S E C ) 0.2327  Y I E L D MV (MV) 0 . 2 9 8 E + 01  TORQUE ( D Y N E S CM) 0.133E +06  SHEAR S T R E S S ( D Y N E S / S Q CM) 0. 1 0 6 E + 0 5  TORQUE ( D Y N E S CM) 0.468E+05  SHEAR STRESS ( D Y N E S / S Q CM) 0.373E+04  EQUILIBRIUM (MV) 0.105E+01  DEX  I N 1 0 . 0 % PEG-H20  +  MV  1.0MOLE NACL  PHI (CC/CC) 0. 0 6 Y I E L D MV (MV) 0.735E+01 EQUILIBRIUM (MV) 0.190E+01  MV  I N 1 0 . 0 % PEG-H20 L (MM) 5.030  OMEGA ( R A D / S E C ) 0.2327  TORQUE ( D Y N E S CM) 0.328E+06  SHEAR STRESS ( D Y N E S / S Q CM) 0.261E+O5  TORQUE { D Y N E S CM) 0.847E+05  SHEAR STRESS ( D Y N E S / S Q CM) 0.674E+04  TABLE  DEX  + 1.0MOLE  NACL  1  IN  1 0 . 0 % PEG-H 20 L (MM) 5. 0 3 0  OMEGA ( B A D / S E C ) 0. 2 3 2 7  Y I E L D MV (MV) 0.127E+02  TORQUE ( D Y N E S CM) 0.565E+06  SHEAR STRESS ( D Y N E S / S Q CM) 0.450E+05  TORQUE ( D Y N E S CM) 0.259E+05  SHEAR STRESS ( D Y N E S / S Q CM) 0.2Q6E+04  +  MV  1.0MOLE NACL  IN  PHI(CC/CC) 0.04 YIELD (MV)  MV  EQUILIBRIUM (EV)  MV  0 . 3 2 1 E + 0 0  + 1.0MOLE  1 0 . 0 % PEG-H20 L (MM)  0.111E+01  DEX  162  (. . . C O N T . )  P H I (CC/CC) 0.07  EQUILIBRIUM (MV) 0.581E+00  DEX  D -  NACL  6.72 0  OMEGA ( R A D / S E C ) 0. 2 3 2 7  TORQUE ( D Y N E S CM) 0.495E+05  SHEAR STRESS ( D Y N E S / S Q CM) 0.3 9 4 E + 0 4  TORQUE ( D Y N E S CM) 0.143E+05  SHEAR STRESS ( D Y N E S / S Q CM) 0.114E+04  IN  1 0 . 0 % PEG-H20  PHI(CC/CC) 0.05  L (MM) 6. 7 2 0  OMEGA ( R A D / S E C ) 0.2327  Y I E L D MV (MV) 0.181E+01  TORQUE ( D Y N E S CM) 0.807E+05  SHEAR STRESS ( D Y N E S / S Q CM) 0.642E+04  TORQUE ( D Y N E S CM) 0.245E+05  SHEAR STRESS ( D Y N E S / S Q CM) 0.195E+04  EQUILIBRIUM (MV) 0.549E+00  MV  163 TABLE  DEX  D -  • 1.0HOLE NACL PHI(CC/CC)  L(MM)  OMEGA ( R A D / S E C )  6.720  Y I E L D MV (MV) 0.614E+01  TORQUE ( D Y N E S CM) 0.274E+06  SHEAR STRESS ( D Y N E S / S Q CM) 0.218E+05  MV T O R Q U E ( D Y N E S CM) 0.825E+05  SHEAR STRESS (DYNES/SQ CH) 0.657E+04  + 1.0HOLE  NACL  0.2327  I N 10.0% PEG-H20  PHI(CC/CC) 0.07  L (MM) 6. 7 2 0  OMEGA ( R A D / S E C ) 0. 2 3 2 7  Y I E L D MV (MV) 0.119E+02  TORQUE ( D Y N E S CM) 0.530E+06  SHEAR STRESS (DYNES/SQ CM) 0.422E+05  MV T O R Q U E ( D Y N E S CM) 0.201E+06  SHEAR STRESS (DYNES/SQ CM) 0.160E+05  EQUILIBRIUM (MV) 0.451E+01  DEX  I N 1 0 . 0 % PEG-H20  0.06  EQUILIBRIUM (MV) 0.185E+01  DEX  1 (. . . C O N T . )  • 1.0MOLE NACL PHI(CC/CC) 0. 14 Y I E L D MV (MV) 0.679E+00 EQUILIBRIUM (MV) 0.500E-01  I N 3 0 . 0 % PEG-H20 L (MM) 0.987  OMEGA ( R A D / S E C ) 0. 2 3 2 7  TORQUE ( D Y N E S CM) 0.303E+05  SHEAR STRESS ( D Y N E S / S Q CM) 0.241E+04  MV T O R Q U E ( D Y N E S CM) 0.223E+04  SHEAR STRESS ( D Y N E S / S Q CM) 0.178E+03  TABLE  DEX  +  D -  1.0MOLE NACL  PHI(CC/CC) 0 . 16 Y I E L D HV (MV) 0.388E+01 EQUILIBRIUM (MV) 0.220E+00  DEX  MV  + 1.0MOLE NACL PHI(CC/CC) 0 . 18 Y I E L D MV (MV) 0.130E+02 EQUILIBRIUM (MV) 0.601E+00  DEX  MV  + 1.0MOLE NACL PHI (CC/CC) 0. 10 Y I E L D MV (MV) 0.175E+00 EQUILIBRIUM (MV) 0.500E-01  MV  1  {. ...CONT.).  I N 30.0%  PEG-H20  L (MM) 0.987  OMEGA ( R A D / S E C ) 0. 2 3 2 7  TORQUE ( D Y N E S CM) 0.173E+06  SHEAR STRESS ( D Y N E S / S Q CM) 0.138E+05  TORQUE ( D Y N E S CM) 0.98 1E+04  SHEAR STRESS ( D Y N E S / S Q CM) 0.781E+03  I N 3 0 . 0 % PEG-H20 L (MM) 0. 9 8 7  OMEGA ( R A D / S E C ) 0.2327  TORQUE ( D Y N E S CM) 0.580E+06  SHEAR STRESS ( D Y N E S / S Q CM) 0.462E+05  TORQUE ( D Y N E S CM) 0.268E+05  SHEAR STRESS ( D Y N E S / S Q CM) 0.213E+04  I N 3 0 . 0 % PEG-H20 L(MM) 1. 6 2 0  OMEGA ( R A D / S E C ) 0.23 27  TORQUE ( D Y N E S CM) 0.78 1E+0U  SHEAR STRESS ( D Y N E S / S Q CM) 0.622E+03  TORQUE ( D Y N E S CM) 0.223E+04  SHEAR STRESS ( D Y N E S / S Q CM) 0. 1 7 8 E + 0 3  165 TABLE  DEX  +  1.0MOLE  NACL  IN  (. . . C O N T . )  30.0%  PEG-H20  L (MM) 1. 6 2 0  OMEGA ( R A D / S E C ) 0. 2 3 2 7  Y I E L D MV (MV) 0.197E+01  TORQUE ( D Y N E S CM) 0.878E+05  SHEAR STRESS ( D Y N E S / S Q CM) 0.699E+04  TORQUE ( D Y N E S CM) 0.401E+04  SHEAR STRESS ( D Y N E S / S Q CM) 0.319E+03  +  1.0MOLE  MV  NACL  IN  30.0%  PEG-H20  PHI(CC/CC) 0 . 13  L (MM) 1. 6 2 0  OMEGA ( R A D / S E C ) 0. 2 3 2 7  Y I E L D MV (MV) 0.283E+01  TORQUE ( D Y N E S CM) 0.126E+06  SHEAR S T R E S S ( D Y N E S / S Q CM) 0.100E+05  TORQUE ( D Y N E S CM) 0.401E+04  SHEAR STRESS ( D Y N E S / S Q CM) 0.319E+03  EQUILIBRIUM (MV) 0.899E-01  DEX  1  PHI{CC/CC) 0. 12  EQUILIBRIUM (MV) 0.899E-01  DEX  D -  +  1.0MOLE  MV  NACL  IN  30.0%  PEG-H20  PHI (CC/CC) 0,14  L (MM) 1. 6 2 0  OMEGA ( R A D / S E C ) 0.2327  Y I E L D MV (MV) 0.787E+01  TORQUE ( D Y N E S CM) 0.351E+06  SHEAR STRESS ( D Y N E S / S Q CM) 0.279E+05  TORQUE ( D Y N E S CM) 0.473E+05  SHEAR STRESS ( D Y N E S / S Q CM) 0.377E+04  EQUILIBRIUM (MV) 0.106E+01  MV  TABLE  DEX  *  1.0MOLE NACL  166  (. . . C O N T . )  I N 3 0 . 0 % PEG-H20 L (MM) 3. 0 1 0  OMEGA ( R A D / S E C ) 0. 2 3 2 7  Y I E L D MV (MV) 0.330E+00  TORQUE ( D Y N E S CM) 0.147E+05  SHEAR STRESS ( D Y N E S / S Q CM) 0.117E+0U  TORQUE ( D Y N E S CM) 0.357E+04  SHEAR STRESS ( D Y N E S / S Q CM) 0.284E+03  *  MV  1.0MOLE NACL  I N 3 0 . 0 % PEG-H20  PHI (CC/CC) 0.06  L (MM) 3. 0 1 0  OMEGA ( R A D / S E C ) 0. 2 3 2 7  Y I E L D MV (MV) 0.256E+01  TORQUE ( D Y N E S CM) 0.114E+06  SHEAR STRESS ( D Y N E S / S Q CM) 0. 9 0 7 E + 0 4  TORQUE ( D Y N E S CM) 0.357E+05  SHEAR STRESS ( D Y N E S / S Q CM) 0. 2 8 4 E + 0 U  EQUILIBRIUM (MV) 0.800E+00  DEX  1  P H I (CC/CC) 0.05  EQUILIBRIUM (UV) 0.800E-01  DEX  D -  +  MV  1.0MOLE NACL  I N 3 0 . 0 % PEG-H20  PHI(CC/CC) 0.07  L (MM) 3.010  OMEGA ( R A D / S E C ) 0.23 27  Y I E L D MV (MV) 0.601E+01  TORQUE ( D Y N E S CM) 0.268E+06  SHEAR STRESS ( D Y N E S / S Q CM) 0.213E+05  TORQUE ( D Y N E S CM) 0.535E+05  SHEAR STRESS ( D Y N E S / S Q CM) 0.426E+04  EQUILIBRIUM (MV) 0.120E+01  MV  TABLE  DEX  +  D -  1.0MOLE NACL  PHI(CC/CC) 0. 0 8 Y I E L D MV (MV) 0.113E+02 EQUILIBRIUM (MV) 0.251E+01  DEX  +  1.0MOLE  NACL  167  (. . . C O N T . )  I N 3 0 . 0 % PEG-H20 L(MM) 3.010  OMEGA ( R A D / S E C ) 0. 2 3 2 7  TORQUE ( D Y N E S CM) 0.502E+06  SHEAR STRESS ( D Y N E S / S Q CM) 0. 4 0 0 E + 0 5  TORQUE ( D Y N E S CM) 0.112E*06  SHEAR STRESS ( D Y N E S / S Q CM) 0.892E+04  I N 3 0 . 0 % PEG-H20  PHI (CC/CC) 0.04  L (MM) 5.030  OMEGA ( R A D / S E C ) 0.2327  Y I E L D MV (MV) 0.100E+01  TORQUE ( D Y N E S CM) 0.446E+G5  SHEAR STRESS ( D Y N E S / S Q CM) 0.355E+04  TORQUE ( D Y N E S CM) 0.669E+04  SHEAR STRESS ( D Y N E S / S Q CM) 0.533E+03  EQUILIBRIUM (MV) 0.150E+00  DEX  MV  1  +  1.0MOLE  MV  NACL  I N 3 0 . 0 % PEG-H20  PHI(CC/CC) 0.05  L (MM) 5. 0 3 0  OMEGA ( R A D / S E C ) 0. 2 3 2 7  Y I E L D MV (MV) 0.170E+01  TORQUE ( D Y N E S CM) 0.758E+05  SHEAR STRESS ( D Y N E S / S Q CM) 0.603E+04  EQUILIBRIUM (MV) 0.500E+00  TORQUE ( D Y N E S CM) 0.223E+05  SHEAR STRESS ( D Y N E S / S Q CM) 0.178E+04  TABLE  DEX  +  1.0MOLE NACL  {. . . C O N T . )  I N 30.  PEG-H20  L (MM) 5.030  OMEGA ( R A D / S E C ) 0.2327  Y I E L D MV <MV) 0.480E+01  TORQUE ( D Y N E S CM) 0.214E+06  SHEAR STRESS ( D Y N E S / S Q CM) 0.170E+05  TORQUE ( D Y N E S CM) 0.731E+05  SHEAR STRESS ( D Y N E S / S Q CM) 0. 5 8 2 E + 0 4  MV  1.0MOLE NACL  I N 3 0 . 0 % PEG-H20  PHI(CC/CC) 0.07  L (MM) 5. 0 3 0  OMEGA ( R A D / S E C ) 0. 2 3 2 7  Y I E L D MV (MV) 0.691E+01  TORQUE ( D Y N E S CM) 0.308E+06  SHEAR STRESS ( D Y N E S / S Q CM) 0.245E+05  TORQUE ( D Y N E S CM) 0.825E+05  SHEAR STRESS ( D Y N E S / S Q CM) 0.657E+04  EQUILIBRIUM (MV) 0.185E+01  DEX  1  PHI (CC/CC) 0.06  EQUILIBRIUM (MV) 0.164E+01  DEX  D -  MV  + 1.0MOLE NACL PHI(CC/CC) 0.04 Y I E L D MV (MV) 0. 951E«-00 EQUILIBRIUM (MV) 0.240E+00  MV  I N 3 0 . 0 % PEG-H20 L (MM) 6. 7 2 0  OMEGA ( R A D / S E C ) 0. 2 3 2 7  TORQUE ( D Y N E S CM) 0.424E+05  SHEAR S T R E S S ( D Y N E S / S Q CM) 0.338E+04  TORQUE ( D Y N E S CM) 0.107E+05  SHEAR STRESS ( D Y N E S / S Q CM) 0.852E+03  TABLE  DEX  +  1.0HOLE  D -  NACL  PHI (CC/CC) 0.05  DEX  +  1.0 M O L E  NA  I N 3 0 . 0 % PEG-H20  L  OMEGA ( R A D / S E C ) 0. 2 3 2 7  TORQUE ( D Y N E S CM) 0.812E+05  SHEAR STRESS ( D Y N E S / S Q CM) 0.646E+04  TORQUE ( D Y N E S CH) 0.312E+05  SHEAR STRESS ( D Y N E S / S Q CM) 0.248E+04  I N 3 0 . 0 % P E G H20  PHI(CC/CC) 0. 0 6  L (MM) 6. 7 2 0  OMEGA ( R A D / S E C ) 0. 2 3 2 7  Y I E L D MV (MV) 0.307E+01  TORQUE ( D Y N E S CM) 0. 137E+-06  SHEAR S T R E S S ( D Y N E S / S Q CM) 0.109E+05  TORQUE ( D Y N E S CM) 0.535E+05  SHEAR STRESS ( D Y N E S / S Q CM) 0. U 2 6 E + 0 4  EQUILIBRIUM (MV) 0.120E+01  DEX  MV  169  (. . . C O N T . )  L (MM) 6. 7 2 0  Y I E L D MV (MV) 0.182E+01 EQUILIBRIUM (HV) 0.700E+00  1  +  MV  1.0MOLE NACL  I N 3 0 . 0 % PEG-H20  PHI (CC/CC) 0.07  L (MM) 6.720  OMEGA ( R A D / S E C ) 0.23 27  Y I E L D MV (MV) 0.53UE+01  TORQUE ( D Y N E S CM) 0.238E+06  SHEAR STRESS ( D Y N E S / S Q CM) 0.189E+05  TORQUE ( D Y N E S CM) 0.892E+05  SHEAR STRESS ( D Y N E S / S Q CM) 0.710E+04  EQUILIBRIUM (MV) 0.200E+01  MV  TABLE  DEX  +  0.25MOLE  D -  NACL  P H I (CC/CC) 0. 14  DEX  +  0.50MOLE  MV  NACL  MV  0.525E+01 EQUILIBRIUM (MV) 0.850E+00  DEX  *•  MV  2.00MOLE NACL  PHI(CC/CC) 0 . 14 Y I E L D MV (MV) 0.106E+02 EQUILIBRIUM (MV) 0.144E+01  I N 3 0 . 0 % PEG-H20 OMEGA ( R A D / S E C )  1.620  PHI(CC/CC) 0 . 14 YIELD (MV)  170  (. . . C O N T . )  L (MM)  Y I E L D MV (MV) 0.401E+01 EQUILIBRIUM (MV) 0.751E+00  1  MV  0. 2327  TORQUE ( D Y N E S CM) 0.179E+06  SHEAR STRESS ( D Y N E S / S Q CM) 0.142E+05  TORQUE ( D Y N E S CM) 0.335E+05  SHEAR STRESS ( D Y N E S / S Q CM) 0.267E+04  I N 3 0 . 0 % PEG-H20 L (MM) 1. 6 2 0  OMEGA ( R A D / S E C ) 0. 2 3 2 7  TORQUE ( D Y N E S CM) 0.234E+06  SHEAR STRESS ( D Y N E S / S Q CM) 0. 1 8 6 E + 0 5  TORQUE ( D Y N E S CM) 0.379E+05  SHEAR STRESS ( D Y N E S / S Q CM) 0. 3 0 2 E + 0 4  I N 3 0 . 0 % PEG-H20 L (MM) 1. 6 2 0  OMEGA ( R A D / S E C ) 0.2327  TORQUE ( D Y N E S CM) 0.473E+06  SHEAR STRESS ( D Y N E S / S Q CM) 0.377E+05  TORQUE ( D Y N E S CM) 0.642E+05  SHEAR STRESS ( D Y N E S / S Q CM) 0.511E+04  TABLE  DEX  +  0.25MOLE  D -  1  HNCL2 I N 3 0 . 0 % PEG-H20  PHI(CC/CC) 0. 14 Y I E L D MV (MV) 0.283E+01 EQUILIBRIUM (MV) 0.751E+00  DEX  +  MV  L (MM) 1. 6 2 0  OMEGA ( R A D / S E C ) 0.2327  TORQUE ( D Y N E S CM) 0.126E+06  SHEAR S T R E S S ( D Y N E S / S Q CM) 0. 1 0 0 E + 0 5  TORQUE ( D Y N E S CM) 0.335E+05  SHEAR STRESS ( D Y N E S / S Q CH) 0.267E+04  0.25MOLE C E C L 3  IN 3 0 . 0 % PEG-H20  PHI (CC/CC) 0. 14 Y I E L D MV (MV) 0.294E+01 EQUILIBRIUM (MV) 0.751E+00  DEX  +  0.0MOLE  171  (...CONT.)  MV  NACL  PHI (CC/CC) 0. 1 4 Y I E L D MV (MV) 0.404E+01 EQUILIBRIUM (MV) 0.500E+00  MV  L (MM) 1.620  OMEGA ( R A D / S E C ) 0.23 27  TORQUE ( D Y N E S CM) 0.131E+06  SHEAR STRESS ( D Y N E S / S Q CM) 0.104E+05  TORQUE ( D Y N E S CM) 0.335E+05  SHEAR STRESS ( D Y N E S / S Q CM) 0.267E+04  IN  30.  PEG-H20  L (MM) 1. 6 2 0  OMEGA ( R A D / S E C ) 0. 2 3 2 7  TORQUE ( D Y N E S CM) 0.180E+06  SHEAR S T R E S S ( D Y N E S / S Q CM) 0.143E+05  TORQUE ( D Y N E S CM) 0.223E+05  SHEAR STRESS ( D Y N E S / S Q CM) 0. 1 7 8 E + 0 4  TABLE  DEX  +  D -  1.0MOLE NACL  PHI (CC/CC) 0. 16 Y I E L D HV (HV) 0.549E+00 EQUILIBRIUM (MV) 0.800E-01  DEX  +  MV  1.0MOLE NACL  P H I (CC/CC) 0. 1 7 Y I E L D MV (MV) 0.789E+00 EQUILIBRIUM (MV) 0.100E+00  DEX  *  MV  1.0MOLE NACL  PHI (CC/CC) 0. 18 Y I E L D MV (MV) 0.240E+01 EQUILIBRIUM (MV) 0.100E+00  1  172  {. . . C O N T . )  I N 4 0 . 0 % PEG-H20 L (MM) 0. 9 8 7  OMEGA ( R A D / S E C ) 0.2327  TORQUE ( D Y N E S CM) 0.245E+05  SHEAR STRESS ( D Y N E S / S Q CM) 0.195E+04  TORQUE ( D Y N E S CM) 0.357E+04  SHEAR STRESS ( D Y N E S / S Q CM) 0.2 8 4 E + 0 3  I N 4 0 . 0 % PEG-H20 L (MM) 0. 9 8 7  OMEGA(RAD/SEC) 0. 2 3 2 7  TORQUE ( D Y N E S CM) 0.352E+05  SHEAR STRESS ( D Y N E S / S Q CM) 0.280E+04  TORQUE ( D Y N E S CM) 0.446E+04  SHEAR STRESS ( D Y N E S / S Q CM) 0.355E+03  I N 40.0%  PEG-H20  L (MM) 0. 9 8 7  OMEGA ( R A D / S E C ) 0. 2 3 2 7  TORQU E ( D Y N E S CM) 0.107E+06  SHEAR STRESS ( D Y N E S / S Q CM) 0.852E+04  MV T O R Q U E ( D Y N E S CM) 0.446E+04  SHEAR STRESS ( D Y N E S / S Q CM) 0.355E+03  TABLE  DEX  +  D -  1.0MOLE NACL  PHI<CC/CC) 0. 1 4 Y I E L D MV (MV) 0.800E+00 EQUILIBRIUM (MV) 0.899E-01  DEX  +  MV  1.0MOLE NACL  PHI (CC/CC) 0. 15 Y I E L D MV (MV) 0.192E+01 EQUILIBRIUM (MV) 0.500E+00  DEX  +  MV  1.0MOLE NACL  PHI(CC/CC) 0. 16 Y I E L D MV (MV) 0.424E+01 EQUILIBRIUM (MV) 0.110E+01  MV  1  173  {. . . C O N T . )  I N 4 0 . 0 % PEG-H20 L (MM) 1. 620  OMEGA ( B A D / S E C )  TORQUE { D Y N E S CM) 0.357E+05  SHEAR STRESS ( D Y N E S / S Q CM) 0.284E+04  TORQUE ( D Y N E S CM) 0.401E+04  SHEAR STRESS ( D Y N E S / S Q CM) 0. 3 1 9 E + 0 3  0.23 27  I N 4 0 . 0 % PEG-H20 L (MM) 1. 6 2 0  OMEGA ( R A D / S E C ) 0. 2 3 2 7  TORQUE ( D Y N E S CM) 0.856E+05  SHEAR STRESS ( D Y N E S / S Q CM) 0.681E+04  TORQUE ( D Y N E S CM) 0.223E+05  SHEAR STRESS ( D Y N E S / S Q CM) 0.178E+04  IN 40.  PEG-H20  L (MM) 1.620  OMEGA ( R A D / S E C ) 0.2327  TORQUE ( D Y N E S CM) 0.189E+06  SHEAR STRESS ( D Y N E S / S Q CM) 0.150E+05  TORQUE ( D Y N E S CM) 0.491E+05  SHEAR STRESS ( D Y N E S / S Q CM) 0.391E+04  174 TABLE D - 1 {. . . CONT.)  DEX  * 1 . 0 H O L E NACL IN 40.0% PEG-H20 PHI(CC/CC) 0. 17  L (MM) 1.620  OMEGA (RAD/SEC) 0.2327  YIELD HV (MV) 0.890E+01  TORQUE (DYNES CM) 0.397E+06  SHEAR STRESS (DYNES/SQ CM) 0.316E+05  TORQUE (DYNES CM) 0.847E+05  SHEAR STRESS (DYNES/SQ CM) 0.674E+04  EQUILIBRIUM (MV) 0.190E+01  MV  DEX + 1.0HOLE NACL IN 40.0% PEG-H20 PHI(CC/CC) 0.08  L (MM) 3. 010  YIELD MV (MV) 0.193E+01 EQUILIBRIUM (MV) 0.300E+00  DEX  MV  OMEGA(RAD/SEC) 0.23 27  TORQUE (DYNES CM) 0.861E+05  SHEAR STRESS (DYNES/SQ CM) 0.685E+04  TORQUE (DYNES CM) 0.134E+05  SHEAR STRESS (DYNES/SQ CM) 0.107E+04  + 1.0MOLE NACL IN 40.0% PEG-H20 PHI (CC/CC) 0.09  L (MM) 3.010  OMEGA (RAD/SEC) 0. 2327  YIELD MV (HV) 0.359E+01  TORQUE (DYNES CM) 0. 160E+06  SHEAR STRESS (DYNES/SQ CM) 0.127E+05  TORQUE (DYNES CM) 0.285E+05  SHEAR STRESS (DYNES/SQ CM) 0.227E+04  EQUILIBRIUM (MV) 0.639E+00  MV  TABLE  DEX  +  1.0MOLE  D -  NACL  PHI (CC/CC) 0. 10 Y I E L D MV (MV) 0.637E+01 EQUILIBRIUM (MV) O.899E+00  DEX  +  1.0MOLE  HV  NACL  P H I (CC/CC) 0. 1 1 Y I E L D MV (MV) 0.702E+01 EQUILIBRIUM (MV) 0.120E+01  DEX  •  1.0MOLE  MV  NACL  1 {. . . C O N T . )  I N 4 0 . 0 % PEG-H20 L (MM) 3.010  OMEGA ( B A D / S E C ) 0. 2 3 2 7  TOBQUE ( D Y N E S CM) 0.284E+06  SHEAR STRESS ( D Y N E S / S Q CM) 0. 2 2 6 E + 0 5  TORQUE ( D Y N E S CM) 0.401E+05  SHEAR STRESS ( D Y N E S / S Q CM) 0.319E+04  I N 4 0 . 0 % PEG-H20 L (MM) 3.010  OMEGA ( B A D / S E C ) 0.2327  TORQUE ( D Y N E S CM) 0.313E+06  SHEAB STBESS ( D Y N E S / S Q CM) 0. 2 4 9 E + 0 5  TOBQUE ( D Y N E S CM) 0.535E+05  SHEAR STRESS ( D Y N E S / S Q CM) 0. 4 2 6 E + 0 4  I N 4 0 . 0 % PEG-H20  PHI(CC/CC) 0. 0 6  L (MM) 5. 0 3 0  OMEGA ( R A D / S E C ) 0. 2 3 2 7  Y I E L D MV (MV) 0.121E+01  TORQUE ( D Y N E S CM) 0.540E+05  SHEAR S T R E S S ( D Y N E S / S Q CM) 0.430E+04  TORQUE ( D Y N E S CM) 0.129E+05  SHEAR STRESS ( D Y N E S / S Q CM) 0.103E+04  EQUILIBRIUM (MV) 0.289E+00  MV  TABLE  DEX  •  1.0MOLE  NACL  IN  {.. . C O N T . )  40.0%  PEG-H20  L (MM) 5.03 0  OMEGA ( R A D / S E C ) 0.23 27  Y I E L D MV (MV) 0.224E+01  TORQOE ( D Y N E S CM) 0.100E+06  SHEAR STRESS ( D Y N E S / S Q CM) 0.796E+04  TORQUE ( D Y N E S CM) 0.303E+05  SHEAR STRESS ( D Y N E S / S Q CM) 0.241E+04  +  1.0MOLE  MV  NACL  IN  40.0%  PEG-H20  PHI (CC/CC) 0.08  L (MM) 5. 0 3 0  OMEGA ( R A D / S E C ) 0. 2 3 2 7  Y I E L D MV (MV) 0.379E+01  TORQUE ( D Y N E S CM) 0.169E+06  SHEAR STRESS ( D Y N E S / S Q CM) 0.135E+05  TORQUE ( D Y N E S CM) 0.290E+05  SHEAR STRESS ( D Y N E S / S Q CM) 0.231E*04  EQUILIBRIUM (MV) 0.650E+00  DEX  1  PHI (CC/CC) 0.07  EQUILIBRIUM (MV) 0.679E+00  DEX  D -  +  1.0MOLE  MV  NACL  IN  40.0%  PEG-H20  PHI (CC/CC) 0.09  L (MM) 5. 0 3 0  OMEGA ( R A D / S E C ) 0. 2 3 2 7  Y I E L D MV (MV) 0.935E+01  TORQUE ( D Y N E S CM) 0.417E+06  SHEAR STRESS ( D Y N E S / S Q CM) 0.332E+05  TORQUE ( D Y N E S CM) 0.607E+05  SHEAR STRESS ( D Y N E S / S Q CM) 0.483E+04  EQUILIBRIUM (MV) 0.136E+01  MV  TABLE  DEX  +  D -  1.0MOLE NACL  PHI (CC/CC)  +  I N 40.0%  PEG-H20  L (MM)  OMEGA ( R A D / S E C )  0.2327  6. 7 2 0  Y I E L D MV (MV) 0.980E*00  TORQUE ( D Y N E S CM) 0.437E*05  SHEAR STRESS ( D Y N E S / S Q CM) 0.348E+04  TORQUE ( D Y N E S CM) 0.357E+04  SHEAR STRESS ( D Y N E S / S Q CM) 0.284E+03  1.0MOLE  MV  NACL  I N 4 0 , 0 % PEG-H20  PHI (CC/CC) 0.06  L (MM) 6.720  . OMEGA ( R A D / S E C ) 0. 2 3 2 7  Y I E L D MV ,(MV) 0.260E+01  TORQUE ( D Y N E S CM) 0.116E+06  SHEAR STRESS ( D Y N E S / S Q CM) 0.923E+04  TORQUE ( D Y N E S CM) 0.312E+05  SHEAR STRESS ( D Y N E S / S Q CM) 0. 2 4 8 E + 0 4  EQUILIBRIUM (MV) 0.700E+00  DEX  177  {. . . C O N T . )  0.05  EQUILIBRIUM (MV) 0.800E-01  DEX  1  MV  + 1.0MOLE NACL PHI (CC/CC) 0.07  L (MM) 6. 7 2 0  Y I E L D MV (MV) 0.359E+01 EQUILIBRIUM (MV) 0.700E+00  I N 4 0 . 0 % PEG-H20  MV  OMEGA ( R A D / S E C ) 0.2327  TORQUE ( D Y N E S CM) 0.160E+06  SHEAR STRESS ( D Y N E S / S Q CM) 0.127E+05  TORQUE ( D Y N E S CM) 0.312E+05  SHEAR STRESS ( D Y N E S / S Q CM) 0.248E+04  TABLE  DEX  +  1.0MOLE  D -  NACL  1  IN  PHI(CC/CC) 0. 0 8 Y I E L D MV (MV) 0.437E+01 EQUILIBRIUM (MV) 0.110E+01  DEX  +  1.0MOLE  NACL  40.0%  PEG-H20  L (MM) 6. 7 2 0  OMEGA ( R A D / S E C ) 0.2327  TORQUE ( D Y N E S CM) 0.195E+06  SHEAR STRESS ( D Y N E S / S Q CM) 0. 1 5 5 E + 0 5  TORQUE ( D Y N E S CM) 0.491E+05  SHEAR STRESS ( D Y N E S / S Q CM) 0.391E+04  IN  50.0%  PEG-H20  PHI(CC/CC) 0.12  L (MM) 0.987  OMEGA ( R A D / S E C ) 0. 2 3 2 7  Y I E L D MV (MV) 0.330E+00  TORQUE ( D Y N E S CM) 0.147E+05  SHEAR STRESS ( D Y N E S / S Q CM) 0,117E+04  TORQUE ( D Y N E S CM) 0.669E+04  SHEAR STRESS ( D Y N E S / S Q CM) 0.533E+03  EQUILIBRIUM (MV) 0.150E+00  DEX  MV  178  (. . . C O N T . )  +  1.0MOLE  MV  NACL  IN  50.0%  PEG-H20  PHI(CC/CC) 0. 16  L (MM) 0.987  OMEGA ( R A D / S E C ) 0.2327  Y I E L D MV (MV) 0.630E+00  TORQUE ( D Y N E S CM) 0.281E+0 5  SHEAR STRESS ( D Y N E S / S Q CM) 0.224E+04  TORQUE ( D Y N E S CM) 0.40 1E+04  SHEAR STRESS ( D Y N E S / S Q CM) 0.319E+03  EQUILIBRIUM (MV) 0.899E-01  MV  179  TABLE  DEX  +  D -  1.0MOLE NACL  PHI(CC/CC) 0. 17 Y I E L D MV (MV) 0.731E+00 EQUILIBRIUM (MV) 0.700E-01  DEX  +  MV  1.0MOLE NACL  PHI (CC/CC) 0. 1 8  DEX  •  1.OMOLE  I N 5 0 . 0 % PEG-H20 L (MM) 0. 9 8 7  OMEGA ( R A D / S E C ) 0. 2 3 2 7  TORQUE ( D Y N E S CM) 0.326E+05  SHEAR STRESS ( D Y N E S / S Q CM) 0.260E+04  TORQUE ( D Y N E S CM) 0.312E+04  SHEAR STRESS ( D Y N E S / S Q CM) 0.2 48E+03  I N 50.0%  PEG-H20 OMEGA ( R A D / S E C ) 0.2327  TORQUE ( D Y N E S CM) 0.598E+05  SHEAR STRESS ( D Y N E S / S Q CM) 0. 4 7 6 E + 0 4  MV T O R Q U E ( D Y N E S CM) 0.446E+04  SHEAR STRESS ( D Y N E S / S Q CM) 0.355E+03  NACL  PHI (CC/CC) 0.19  I N 5 0 . 0 % PEG-H20 L (MM) 0.987  Y I E L D MV (MV) 0.379E+01 EQUILIBRIUM (MV) 0.951E+00  (. . . C O N T . )  L (MM) 0. 9 8 7  Y I E L D MV (MV) 0.134E+01 EQUILIBRIUM (MV) 0.100E+00  1  MV  OMEGA ( R A D / S E C ) 0. 2 3 2 7  TORQUE ( D Y N E S CM) 0. 1 6 9 E + 0 6  SHEAR STRESS ( D Y N E S / S Q CM) 0.135E+05  TORQUE ( D Y N E S CM) 0.424E+05  SHEAR STRESS ( D Y N E S / S Q CM) 0.338E+04  180 TABLE  DEX  D -  + 1.0MOLE NACL PHI (CC/CC) 0 . 10  DEX  MV  • 1.0MOLE NACL PHI(CC/CC) 0 . 14 Y I E L D MV (MV) 0.500E+00 EQUILIBRIUM (MV) 0.130E+00  DEX  +  (. . . C O N T . )  I N 5 0 . 0 % PEG-H20 L (MM) 1 .620  Y I E L D MV (MV) 0.300E+00 EQUILIBRIUM (MV) 0.899E-01  1  MV  1.0MOLE NACL  OMEGA ( R A D / S E C ) 0.23 27  TORQUE ( D Y N E S CM) 0.134E+05  SHEAR STRESS ( D Y N E S / S Q CM) 0.107E+04  TORQUE ( D Y N E S CM) 0. 4 0 1 E + 0 4  SHEAR STRESS ( D Y N E S / S Q CM) 0.319E+03  I N 5 0 . 0 % PEG-H20 L (MM) 1. 6 2 0  OMEGA ( R A D / S E C ) 0.2327  TORQUE ( D Y N E S CM) 0.223E+05  SHEAR STRESS ( D Y N E S / S Q CM) 0.178E+04  TORQUE ( D Y N E S CM) 0.580E+04  SHEAR STRESS ( D Y N E S / S Q CM) 0.462E+03  I N 5 0 . 0 % PEG-H20  PHI (CC/CC) 0 . 15  L (MM) 1.620  OMEGA ( R A D / S E C ) 0.23 27  Y I E L D MV (MV) 0.821E+00  TORQUE ( D Y N E S CM) 0.366E+05  SHEAR STRESS ( D Y N E S / S Q CM) 0.291E+04  EQUILIBRIUM (MV) 0.200E+00  TORQUE ( D Y N E S CM) 0.892E+04  SHEAR STRESS ( D Y N E S / S Q CM) 0.710E+03  TABLE  DEX  +  1.0MOLE NACL  (...COMT.)  I N 5 0 . 0 % PEG-H20 L(NM) 1. 6 2 0  OMEGA ( R A D / S E C ) 0.23 27  Y I E L D MV (HV) 0. 1 0 2 E + 0 1  TORQUE { D Y N E S CM) 0.455E+05  SHEAR STRESS ( D Y N E S / S Q CM) 0.362E+04  TORQUE ( D Y N E S CM) 0.178E+05  SHEAR STRESS ( D Y N E S / S Q CM) 0.142E+04  + 1.0MOLE  MV  NACL  PHI(CC/CC) 0. 17 Y I E L D MV (MV) 0.160E+01 EQUILIBRIUM (MV) 0.899E+00  DEX  1  PHI (CC/CC) 0 . 16  EQUILIBRIUM (MV) 0.399E+00  DEX  D -  +  1.0MOLE  MV  NACL  I N 5 0 . 0 % PEG-H20 L (MM) 1. 6 2 0  OMEGA ( R A D / S E C ) 0. 2 3 2 7  TORQUE ( D Y N E S CM) 0.71DE+05  SHEAR S T R E S S ( D Y N E S / S Q CM) 0.568E+04  TORQUE ( D Y N E S CM) 0.401E +05  SHEAR STRESS ( D Y N E S / S Q CM) 0. 3 1 9 E + 0 4  I N 5 0 . 0 % PEG-H20  PHI(CC/CC) 0. 0 6  L (MM) 3.010  OMEGA ( R A D / S E C ) 0.23 27  Y I E L D MV (MV) 0.300E+00  TORQUE ( D Y N E S CM) 0.134E+05  SHEAR STRESS ( D Y N E S / S Q CM) 0. 1 0 7 E + 0 4  EQUILIBRIUM (MV) 0.500E-01  TORQUE ( D Y N E S CM) 0.223E+04  SHEAR STRESS ( D Y N E S / S Q CM) 0.178E+03  TABLE  DEX  D -  + 1.0HOLE N A C L I N 5 0 . 0 % P E G - H 2 0 PHI(CC/CC) 0. 0 7 YIELD (MV)  HV  0.520E+00  EQUILIBRIUM (MV) 0.120E+00  DEX  * 1.0MOLE  MV  NACL  PHI (CC/CC) 0. 0 8 Y I E L D MV (MV) 0.165E+01 EQUILIBRIUM (MV) 0.280E+00  DEX  1 (. . . C O N T . )  MV  + 1.0MOLE NACL  L (MM) 3. 0 1 0  OMEGA ( H A D / S E C ) 0. 2 3 2 7  TOBQUE ( D Y N E S CM)  0. 23 2E+05  SHEAR STRESS ( D Y N E S / S Q CM) 0.185E+04  TORQUE ( D Y N E S CM) 0.535E+04  SHEAR STRESS ( D Y N E S / S Q CM) 0.426E+03  I N 5 0 . 0 % PEG-H20 L (MM) 3. 0 1 0  OMEGA ( R A D / S E C ) 0. 2 3 2 7  TORQUE ( D Y N E S CM) 0.736E+05  SHEAR STRESS ( D Y N E S / S Q CM) 0.586E+0U  TORQUE ( D Y N E S CM) 0.125E+05  SHEAR STRESS ( D Y N E S / S Q CM) 0. 9 9 5 E + 0 3  I N 5 0 . 0 % PEG-H20  P H I (CC/CC) 0.09  L (MM) 3. 0 1 0  OMEGA ( R A D / S E C ) 0. 2 3 2 7  Y I E L D MV (MV) 0.171E+01  TORQUE ( D Y N E S CM) 0.763E+05  SHEAR STRESS ( D Y N E S / S Q CM) 0 . 6 0 7 E + 04  TORQUE ( D Y N E S CM) 0.201E+05  SHEAR STRESS ( D Y N E S / S Q CM) 0.160E+04  EQUILIBRIUM (MV) 0 . 4 5 1 E + 00  MV  TABLE  DEX  +  D -  EQUILIBRIUM (MV) O.130E+01  +  1.0 M O L E  3.010  MV  NACL  P H I (CC/CC)  +  O M E G A (R A D / S E C )  0. 2327  TORQUE ( D Y N E S CM) 0.240E+06  SHEAR STRESS ( D Y N E S / S Q CM) 0.191E+05  TORQUE ( D Y N E S CM) 0.580E+05  SHEAR STRESS ( D Y N E S / S Q CM) 0.462E+04  I N 5 0 . 0 % PEG-H20  0. 06  L (MM) 5. 0 3 0  OMEGA ( R A D / S E C ) 0.2327  Y I E L D MV (MV) 0.841E+00  TORQUE ( D Y N E S CM) 0.375E+05  SHEAR STRESS ( D Y N E S / S Q CM) 0.299E+04  TORQUE ( D Y N E S CM) 0.446E+04  SHEAR STRESS ( D Y N E S / S Q CM) 0.355E+03  EQUILIBRIUM (MV) 0. 1 0 0 E + 0 0  DEX  PEG-H20  L (MM)  Y I E L D MV (MV) 0.538E+01  183  (. . . C O N T . )  1.0MOLE NACL I N 50.OX  PHI(CC/CC) 0 . 10  DEX  1  1.0MOLE  MV  NACL I N 5 0 . 0 % P E G - H 2 0  PHI(CC/CC) 0.07  L (MM) 5. 0 3 0  OMEGA ( R A D / S E C ) 0. 2 3 2 7  Y I E L D MV (MV) 0.149E+01  TORQUE ( D Y N E S CM) 0.665E+05  SHEAR STRESS ( D Y N E S / S Q CM) 0.529E+04  TORQUE ( D Y N E S CM) 0.156E+05  SHEAR STRESS ( D Y N E S / S Q CM) 0.124E+04  EQUILIBRIUM (MV) 0.350E+00  MV  184  TABLE  DEX  D -  + 1.0MOLE NACL PHI(CC/CC) 0.08  DEX  MV  + 1.0MOLE NACL PHI(CC/CC) 0.09  DEX  MV  + 1.0MOLE NACL PHI (CC/CC) 0. 0 6  OMEGA ( R A D / S E C ) 0. 2 3 2 7  TORQUE ( D Y N E S CM) 0.678E+05  SHEAR STRESS ( D Y N E S / S Q CM) 0.540E+04  TORQUE ( D Y N E S CM) 0.214E+05  SHEAR STRESS ( D Y N E S / S Q CM) 0. 1 7 0 E + 0 4  I N 5 0 . 0 % PEG-H20  MV  OMEGA ( R A D / S E C ) 0.2327  TORQUE ( D Y N E S CM) 0.156E+06  SHEAR STRESS ( D Y N E S / S Q CM) 0.124E+05  TORQUE ( D Y N E S CM) 0. 5 1 3 E + 0 5  SHEAR STRESS ( D Y N E S / S Q CM) 0.408E+04  I N 5 0 . 0 % PEG-H20 L (MM) 6. 7 2 0  Y I E L D MV (MV) 0.769E+00 EQUILIBRIUM (MV) 0.100E+00  5. 03 0  L (MM) 5. 0 3 0  Y I E L D MV (MV) 0.350E+01 EQUILIBRIUM (MV) 0.115E+01  I N 5 0 . 0 % PEG-H20 L (MM)  Y I E L D MV (MV) 0.152E+01 EQUILIBRIUM (MV) 0.480E+00  1 (. . . C O N T . )  OMEGA(RAD/SEC) 0.2327  TORQUE ( D Y N E S CM) 0.343E+05  SHEAR STRESS ( D Y N E S / S Q CM) 0 . 2 7 3 E + 04  TORQUE ( D Y N E S CM) 0.446E+04  SHEAR STRESS ( D Y N E S / S Q CM) 0.3 5 5 E + 0 3  185  TABLE D - 1 {...CONT.)  DEX + 1.0 HOLE NACL IN 50. 0%  PEG-H20  PHI(CC/CC) 0.07  L (MM) 6.720  YIELD MV (MV) 0. 144E+01  TORQUE {DYNES CM) 0.642E+05  SHEAR STRESS (DYNES/SQ CM) 0.511E+04  MV TORQUE (DYNES CM) 0.201E+05  SHEAR STRESS (DYNES/SQ CM) 0.160E+04  EQUILIBRIUM (MV) 0.451E+00  OMEGA (HAD/SEC) 0.2327  DEX + 1.0 HOLE NACL IN 50.0% PEG-H20 PHI (CC/CC) 0. 08  L (MM) 6.720  OMEGA (RAD/SEC) 0. 2327  YIELD MV (MV) 0.240E+01  TORQUE (DYNES CM) 0.107E+06  SHEAR STRESS (DYNES/SQ CM) 0.8 52E+04  MV TORQUE (DYNES CM) 0.312E+05  SHEAR STRESS (DYNES/SQ CM) 0. 248E+04  EQUILIBRIUM (MV) 0.700E+00  DEX + 1.0MOLE NACL IN 50.0% PEG-H20 P H I (CC/CC) 0.09  L (MM) 6.720  OMEGA (RAD/SEC) 0.2327  YIELD MV (MV) 0.291E+01  TORQUE (DYNES CM) 0.130E+06  SHEAR STRESS (DYNES/SQ CM) 0.103E+05  MV TORQUE (DYNES CM) 0.268E+05  SHEAR STRESS (DYNES/SQ CM) 0.213E+04  EQUILIBRIUM (MV) 0.601E+00  186  D-2  and  Stress  Description  of  Decay Variables  in  Table  PHI  -  Particle  Concentration  L  -  Particle  Length  OMEGA -  Angular  RX  The  -  Velocity  Thickness  of  D-2  (cc/cc).  (mm). of the  Inner  Cylinder  Flowing  Center  of  the  Cylinders  Layer  (cm).  R  -  Radius  of  the  Inner  Cylinder  (cm).  RO  -  Radius  of  the  Outer  Cylinder  (cm).  (rad./sec). R  Measured  from  the  TABLE EXPESIMENTAL PHI DECAY  =  DATA  0.16  D - 2  -  AND R E S U L T S  (CC/CC)  1 OF  L =  187  CALCULATION 0.987  (MM)  DATA OMEGA  = 0.2327  (RAD/SEC)  TIME  TORQUE  TORQUE  (SEC) 0.0 1.0 1.8 2.6 3.7 4.8 9.0 13.0 18.0 41.0 86. 7 EQUI  (MV) 13. 20 9.80 8.18 7.06 5. 9 9 5.24 3.73 3.09 2. 6 7 2.2 0 2. 1 0 2.00  ( D Y N E S CM) 0 . 5 8 9 E + 06 0.437E+06 0.365E+06 0.315E+06 0.267E+06 0.234E+06 0.166E+06 0. 1 3 8 E + 0 6 0.119E+06 0.981E+05 0.937E+05 0.892E+05  TIME (SEC) 0.0 1.0 1.8 2.6 3. 7 4.8 9.0 13.0 18.0 41.0 86. 7 EQUI  RX/R  RX  (") 1.000 1.119 1. 1 5 1 1 . 173 1.183 1. 1 9 4 1.216 1. 2 2 7 1.23 7 1.270 1. 2 9 1 1.453  (CM) 1.01 1. 13 1. 16 1. 18 1.20 1. 2 1 1.23 1.24 1.25 1.28 1.30 1.47  EQUILIBRIUM  TORQUE  (RAD/SEC) 0.2327 0.6981 2. 0 9 4 0 6. 2 8 3 0  ( D Y N E S CM) 0.892E+05 0. 9 8 1 E + 0 5 0. 1 0 7 E + 0 6 0.112E+06  (RAD/SEC) 0. 2 3 2 7 0.6981 2. 0 9 4 0 6. 2 8 3 0  APPARENT SHEAR R A T E (1/SEC) 0.232E+01 0.190E+01 0.171E+01 0.163E+01 0.156E+01 0.144E+01 0.139E+01 0.134E+01 0.123E+01 0.116E+01 0.884E+00  (RX-R) / (RO-R) (-) 0.0 0.11 0.14 0.16 0.17 0.18 0.20 0.21 0.22 0.25 0.27 0.42 APPARENT VISCOSITY (POISE)  •  0 . 1 5 0 E + 05 0 . 153E + 05 0 . 147E + 05 0 .131E+05 0.119E+05 0.921E+04 0.791E+04 0 .707E+04 0.637E+04 0 .641E+04 0 .803E+04  DATA  OMEGA  OMEGA  SHEAR STRESS ( D Y N E S / S Q CM) 0.469E+05 0.348E+05 0.290E+05 0.251E+05 0.213E+05 0. 1 8 6 E + 0 5 0.132E+05 0. 1 1 0 E + 0 5 0.948E+04 0.781E+04 0.746E+04 0. 7 1 0 E + 0 4  RX/R (") 1. 4 5 3 1. 5 2 9 1 .5 8 3 1. 6 5 3  RX (CM) 1.47 1. 5 4 1. 6 0 1. 6 7  SHEAR STRESS ( D Y N E S / S Q CM) 0.71OE+04 0.78 1E+04 0.852E+04 0.888E+04 APPARENT SHEAR RATE ( 1 / S EC) 0. 8 8 4 E + 0 0 0.244E+01 0.697E+01 0. 1 9 8 E + 0 2  (RX-R) /(RO-R) (-) 0.42 0.49 0.54 0.61 APPARENT VISCOSITY (POISE) 0 .803E+04 0 3 2 0 E + 04 0.122E+04 0.448E+03  TABLE EXPERIMENTAL PHI DECAY  =  DATA  2  D - 2 -  AND R E S U L T S  0.15 (CC/CC)  OF  L =  188  CALCULATION 0.987  (MM)  DATA OMEGA  = 0.2327  (RAD/SEC)  TIME  TORQUE  TORQUE  (SEC) 0.0 0.2 1.4 2. 3 3.2 4.5 5.5 7. 8 11.8 31.0 64.2 EQUI  (MV) 4.81 4. 4 7 3.86 3. 3 7 2.91 2.50 2.29 1 .98 1.67 1.09 0.68 0.30  ( D Y N E S CM) 0.215E+06 0.199E+06 0.172E+06 G.150E+G6 0. 1 3 0 E + 0 6 0.1 1 2 E + 0 6 0.102E+06 0.883E+05 0.745E+05 0.486E+05 0.303E+05 0.134E+05  TIME (SEC) 0.0 0.2 1. 4 2. 3 3.2 4.5 5.5 7.8 11.8 31.0 64.2 EQUI  RX/R  RX  (-) 1 .000 1. 1 4 7 1 . 196 1.216 1.235 1. 2 4 5 1 . 264 1.304 1 .333 1. 3 4 3 1 .363 1. 4 6 1  (CM) 1.01 1. 16 1.21 1.23 1.25 1.26 1.28 1.32 1,35 1. 3 6 1. 3 8 1.48  !QUI L I B R I U M  SHEAR STRESS ( D Y N E S / S Q CM) 0. 1 7 1 E + 0 5 0. 1 5 9 E + 0 5 0. 1 3 7 E + 0 5 0. 1 2 0 E + 0 5 0.103E+05 0.888E+04 0.813E+0 4 0.703E+04 0.593E+04 0.387E+04 0.241E+04 0.107E+04  (SX-R) /{RO-R)  (-)  0.0 0.14 0.18 0.20 0.22 0.23 0.25 0.28 0.31 0.32 0.34 0.43  APPARENT SHEAR RATE (1/SEC)  APPARENT VISCOSITY (POISE)  0.194E+01 0.154E+01 0.144E+01 0.135E+01 0.131E+01 0.124E+01 0.113E*01 0.106E+01 0.104E+01 0.101E+01 0.876E+00  0. 8 1 7 E * 0 4 0. 8 8 7 E + 0 4 0. 8 3 2 E + 0 4 0. 7 6 5 E + 0 4 0. 6 7 7 E + 0 4 0. 6 5 4 E + 0 4 0 .6 2 3 E + 0 4 0. 5 5 7 E + 0 4 0.3 7 1 E + 0 4 0.239E+04 0. 1 2 2 E + 04  DATA  OMEGA  TORQUE  (RAD/SEC) 0.2327 0. 6 9 8 1 2. 0 9 4 0 6. 2 8 3 0  ( D Y N E S CM) 0. 1 3 4 E + 0 5 0. 1 4 3 E + 0 5 0.152E+05 0. 1 5 6 E + 0 5  OMEGA  RX/R  RX  ( R A D / S EC) 0.2327 0. 6 9 8 1 2. 0 9 4 0 6. 2 8 3 0  ("} 1. 4 6 1 1. 5 4 0 1 .5 8 8 1. 7 8 5  (CM) 1.48 1.55 1. 6 0 1. 8 0  SHEAR STRESS ( D Y N E S / S Q CM) 0.107E+04 0.114E+04 0.121E+04 0. 1 2 4 E + 0 4 APPARENT SHEAR RATE (1/SEC) 0. 8 7 6 E + 0 0 0. 2 4 2 E + 01 0.694E+01 0.183E+02  (RX-R) /{RO-R)  M  0.43 0.50 0.55 0.73 APPARENT VISCOSITY (POISE) 0. 1 2 2 E + 0 4 0. 4 7 0 E + 0 3 0.174E+03 0. 6 7 8 E + 0 2  TABLE EXPERIMENTAL PHI = DECAY  DATA  0.14  AND  D  -  2 -  RESULTS  (CC/CC)  3 OP  L =  189 CALCULATION  0.987  (MM)  DATA OMEGA  = 0.2327  (RAD/SEC)  TIME  TORQUE  TORQUE  (SEC) 0.0 1.7 2.9 4.0 5.2 6.6 7. 5 8.7 17. 8 3 1 .3 58.2 EQUI  (MV) 2. 19 1.53 1.27 1 . 15 1.00 0.88 0.85 0.76 0.55 0.42 0. 3 2 0.07  ( D Y N E S CM) 0.977E+05 0.682E+05 0.566E+05 0.513E+05 0. 4 4 6 E + 0 5 0.392E+05 0.379E+05 0.339E+05 0.245E+05 0. 1 8 7 E + 0 5 0.143E+05 0. 3 1 2 E + 0 4  TIME  RX/R  RX  <") 1.000 1 . 248 1. 2 8 1 1. 2 9 1 1. 3 0 2 1.313 1.324 1.324 1. 3 5 6 1 .367 1.3 8 9 1.475  (CM) 1.01 1.26 1.29 1. 30 1. 3 2 1.33 1. 3 4 1.34 1. 3 7 1.38 1.40 1.49  (SEC) 0.0 1.7 2.9 4.0 5. 2 6.6 7. 5 8.7 17.8 31.3 58.2 EQUI  EQUILIBRIUM OMEGA  SH EAR STRESS ( D Y N E S / S Q CM) 0. 7 7 7 E + 0 4 0.543E+04 0.451E+04 0.408E+04 0.355E+04 0.312E+04 0.302E+04 0.270E+04 0.195E+04 0.149E+04 0.114E+04 0.249E+03  APPARENT VISCOSITY (POISE)  0. 1 3 0 E + 0 1 0. 1 1 9 E + 0 1 0.116E+01 0. 1 1 3 E + 0 1 0.111E+01 0. 1 0 8 E + 0 1 0. 1 0 8 E + 0 1 0. 1 0 2 E + 0 1 0. 1 0 0 E + 0 1 0. 9 6 7 E + 0 0 0. 8 6 1 E + 0 0  0. 4 1 8 E + 0 4 0.3 7 8 E + 0 4 0. 3 5 1 E + C 4 0.3 1 3 E + 04 0.282E+04 0. 2 7 8 E + 0 4 0. 2 4 9 E + 0 4 0. 1 9 1 E + 0 4 0. 1 4 9 E + 0 4 0. 1 1 7 E + 0 4 0. 2 8 8 E + 0 3  DATA SHEAR  TORQUE  (RAD/SEC) 0.2327 0.6981 2.0940 6.2 8 3 0  (RAD/SEC) 0. 2 3 2 7 0.6981 2. 0 9 4 0 6. 2 8 3 0  (-) 0.0 0.23 0.26 0.27 0.28 0.29 0.30 0.30 0.33 0.34 0.36 0.44  APPARENT SHEAR RATE ( 1/SEC)  STRESS  OMEGA  (RX-R) / (RO-R)  ( D Y N E S CM) 0.312E+04 0. 3 5 7 E + 0 4 0.401E+04 0.424E+04 RX/R (-) 1.475 1.550 1.604 1.669  RX (CM) 1.49 1. 5 7 1. 6 2 1. 6 9  ( D Y N E S / S Q CM) 0.249E+03 0. 2 8 4 E + 0 3 0. 3 2 0 E + 0 3 0. 3 3 7 E + 0 3 APPARENT SHEAR RATE ( 1 / S EC) 0. 8 6 1 E + 0 0 0. 2 3 9 E + 0 1 0. 6 8 5 E + 01 0. 1 9 6 E + 0 2  (RX-R) /(SO-R)  (-) 0.44 0.51 0.56 0.62 APPARENT VISCOSITY (POISE) 0.288E+03 0.119E+03 0.467E+02 0.172E+02  EXPERIMENTAL PHI DECAY  =  (SEC) 0.0 1.0 2.4 4.2 6.0 7.9 9.3 19.2 27.9 59.5 92.4 EQUI  D - 2  AND  RESULTS  DATA  0.13 (CC/CC)  -  4 OF  L =  190  CALCULATION 0.987  (MM)  DATA OMEGA  T I ME  TABLE  =  0.2327  TORQUE  (RAD/SEC)  TORQUE  (MV) 0.43 0.39 0.35 0.31 0.29 0.27 0.26 0,22 0.18 0. 16 0.14 0.04  ( D Y N E S CM) 0. 1 9 0 E + 0 5 0.174E+05 0. 1 5 6 E + 0 5 0.138E+05 0.129E+05 0.120E+05 0.116E+05 0.981E+04 0.803E+04 0.714E+04 0.624E+04 0.178E+04  TIME  RX/R  RX  (SEC) 0.0 1.0 2.4 4.2 6.0 7.9 9.3 19.2 27.9 59.5 92.4 EQUI  1 .000 1.302 1 . 356 1 .378 1 .399 1.410 1,421 1.432 1 ,442 1.453 1.464 1 .507  (")  (CM) 1.01 1.32 1. 3 7 1.39 1. 4 1 1.42 1.44 1. 4 5 1. 4 6 1.47 1.48 1. 5 2  EQUILIBRIUM  SHEAR STRESS ( D Y N E S / S Q CM) 0. 1 5 1 E + 0 4 0. 1 3 8 E + 0 4 0. 1 2 4 E + 0 4 0.110E+04 0.103E+04 0.959E+03 0.923E+03 0.781E+03 0.639E+03 0,568E+03 0.497E+03 0.142E+03 APPARENT SHEAR RATE (1/SEC) 0.113E+01 0.102E+01 0. 9 8 4 E + 0 0 Q.951E+00 0.936E+ 00 0.922E+00 0.909E+ 00 0.896E+00 0.884E+00 0.872E+00 0.831E+00  (RX-R) / (RO-R) (") 0.0 0.28 0.33 0.35 0.37 0.38 0.39 0.40 0.4 1 0.42 0.43 0.47 APPARENT VISCOSITY (POISE) 0. 1 2 2 E + 0 4 0, 122E+04 0. 1 1 2 E + 0 4 0. 1 0 8 E + 0 4 0. 1 0 2 E + 0 4 0. 1 0 0 E + 0 4 0. 8 5 9 E + 0 3 0.7 1 3 E + 0 3 0. 6 4 3 E + 0 3 0. 5 7 0 E + 0 3 0. 1 7 1 E + 0 3  DATA  OMEGA  TORQUE  ( R A D / S EC) 0.2327 0. 6 9 8 1 2. 0 940 6. 2 8 3 0  ( D Y N E S CM) 0.178E+04 0. 2 2 3 E + 0 4 0. 2 6 8 E + 0 4 0.290E+04  OMEGA  RX/R  RX  ( R A D / S EC) 0.2327 0.6 9 8 1 2. 0 9 4 0 6. 2 8 3 0  ("} 1. 5 0 7 1. 5 7 2 1. 6 4 8 1. 6 8 0  (CM) 1. 5 2 1. 5 9 1. 6 6 1.70  SHEAR STRESS ( D Y N E S / S Q CM) 0.142E+03 0. 1 7 8 E + 0 3 0.21 3E+0 3 0.23 1E+03 APPARENT SHEAR RATE (1/SEC)  0 . 8 3 IE+00 0. 2 3 5 E + 0 1 0.663E+01 0.195E+02  (RX-R) /(RO-R) (-) 0.47 0.53 C.60 0.63 APPARENT VISCOSITY (POISE) 0. 1 7 1 E + 0 3 0 ,7 5 7 E + 0 2 0. 3 2 1 E + 0 2 0. 1 1 9 E + 0 2  TABLE EX P E R I M E N T A L PHI DECAY  =  DATA  0.12  AND  D -  2 -  RESULTS  (CC/CC)  5 OF  L =  191 CALCULATION  1.520  (MM)  DATA OMEGA  TIME (SEC) 0.0 0.9 2.0 3.1 4.3 5.9 7.0 8. 1 9.6 20.9 32. 8 EQUI  =  0.2327  TORQUE  TORQ UE  (MV) 8.54 6.22 4.79 3.91 3.37 2.83 2.60 2.40 2. 24 1.56 1.33 0.95  TIME (SEC) 0.0 0.9 2.0 3.1 4.3 5.9 7.0 8.1 9. 6 20.9 32. 8 EQUI  (RAD/SEC)  ( D Y N E S CM) 0.381E+06 0.277E+06 0.214E+06 0. 1 7 4 E + 0 6 0. 1 5 0 E + 0 6 0.126E+06 0.116E+06 0. 1 0 7 E + 0 6 0.999E+05 0.696E+05 0.5 9 3 E + 0 5 0.424E+05  RX/R  RX  (-) 1 .000 1.151 1. 1 9 4 1.216 1 . 227 1 .248 1. 2 4 8 1 .259 1.270 1.291 1.313 1.432  (CM) 1.01 1. 16 1.21 1.23 1.24 1.26 1.26 1.27 1.28 1. 30 1.33 1. 4 5  EQUILIBRIUM  TORQUE  (RAD/SEC) 0.2 3 2 7 0.6981 2.0940 6. 2 8 3 0  ( D Y N E S CM) 0. 4 2 4 E + 0 5 0. 4 3 7 E + 0 5 0. 4 5 0 E + 0 5 0.455E+05  (RAD/SEC) 0. 2 3 2 7 0.6 981 2. 0 9 4 0 6.2830  APPARENT SHEAR RATE ( 1/SEC) 0. 1 9 0 E + 0 1 0. 1 5 6 E * 01 0. 1 4 4 E + 0 1 0.139E*01 0.130E+01 0.130E+01 0.126E+01 0.123E*01 0. 1 1 6 E + 0 1 0.111E+ 01 0.909E+00  (RX-R) / (RO-R) (") 0 .0 0.1 4 0.18 0.20 0.21 0.23 0.23 0.24 0.25 0.27 0.29 0.40  APPARENT VISCOSITY (POISE) 0.116E+05 0. 1 0 9 E + 0 5 0.965E+04 0. 8 6 2 E + 0 4 0.773E+04 0.710E+04 0.676E+04 0.649E+04 0.476E+04 0 . 4 2 6 E + 04 0.371E+04  DATA  OMEGA  OMEGA  SHEAR STRESS ( D Y N E S / S Q CM) 0. 3 0 3 E + O 5 0.221E+0 5 0.170E+05 0. 1 3 9 E + 0 5 0.120E+05 0.100E+05 0.923E+04 0.852E+04 0.795E+04 0.554E+04 0.472E+04 0.337E+04  RX/R  RX  {-) 1.432 1.529 1. 5 8 3 1.626  (CM) 1. 4 5 1. 5 4 1. 6 0 1. 6 4  SHEA R STRESS ( D Y N E S / S Q CM) 0.337E+04 0.34 8 E + 0 4 0 . 3 5 9 E + 04 0.362E+04 APPARENT SHEAR RATE (1/SEC) 0. 9 0 9 E + 0 0 0. 2 4 4 E + 0 1 0. 6 9 7 E + 01 0. 2 0 2 E + 0 2  (HX-E) / (RO-R) (-) 0.40 0.49 0.54 0.58 APPARENT VISCOSITY (POISE) 0.371E+04 0.143E+04 0.514E+03 0.179E+03  EXPERIMENTAL PHI DECAY  TABLE  D - 2  AND  RESULTS  DATA  =0.11  (CC/CC)  -  192  6 OF  L =  CALCULATION 1 . 6 2 0 (MM)  DATA OMEGA  =  0. 2 3 2 7  (RAD/SEC)  TIME  TORQUE  TORQUE  (SEC) 0.0 0.4 1.3 3.0 3.6 5.8 10.5 18.3 28.8 58.8 129.0 EQUI  (MV) 2.80 2.58 2.20 1.77 1.71 1.43 1.12 0.87 0.73 0. 5 6 0.45 0.26  ( D Y N E S CM) 0.125E+06 0.115E+06 0.981E+05 0.789E+05 0.763E+05 0.638E+05 0.500E+05 0.388E+05 0.326E+05 0.250E+05 0.201E+05 0.116E+05  TIME  RX/R  RX  (SEC) 0.0 0.4 1.3 3.0 3.6 5.8 10.5 18.3 28.8 58.8 129.0 EQUI  (") 1 .000 1.205 1 .227 1. 2 6 8 1 .288 1. 2 9 8 1 .309 1.328 1.339 1. 3 5 9 1.411 1. 4 6 3  (CM) 1.01 1.22 1. 2 4 1.28 1. 3 0 1.31 1. 3 2 1.34 1. 3 5 1. 3 7 1. 4 3 1.48  IQUILIBRIUM  SHEAR STRESS ( D Y N E S / S Q CM) 0.994E+04 0.916E+04 0.781E+04 0.628E+04 0.607E+04 0.508E+04 0,398E+04 0.309E+04 0.259E+04 0.199E+04' 0.160E+04 0.923E+03 APPARENT SHEAR RATE (1/SEC) 0.149E+01 0.139E+01 0.123E+O1 0.117E+01 0.115E+01 0. 1 1 2 E + 0 1 0.107E+01 0.105E+01 0.101E+01 0.935E+00 0.874E+00  (RX-R) / (RO-R) (-) 0.0 0.19 0.21 0.25 0.27 0.28 0.29 0.30 0.31 0.33 0.38 0.43  APPARENT VISCOSITY (POISE) 0.613E+04 0.563E+04 0.510E+04 0 .518E+04 0 .443E+04 0.355E+C4 0 .287E+04 0.246E+04 0. 196E+0 4 0. 171E + 04 0 . 1 0 6 E + 04  DATA  OMEGA  TORQUE  ( R A D / S EC) 0. 2 3 2 7 0. 6 9 8 1 2. 0 9 4 0 6. 2 8 3 0  ( D Y N E S CM) 0. 1 1 6 E + 0 5 0. 1 2 5 E + 0 5 0.134E+05 0.138E+05  OMEGA  RX/R  RX  { RAD/S EC) 0. 2 3 2 7 0. 6 9 8 1 2. 0 9 4 0 6. 2 8 3 0  (") 1. 4 6 3 1. 5 6 6 1.616 1. 6 6 8  (CM) 1. 4 8 1. 5 8 1.63 1. 6 8  SHEAR STSE SS ( D Y N E S / S Q CM) 0.92 3E+0 3 0.994E+03 0. 1 0 7 E + 0 4 0.11OE+04 APPARENT SHEAR RATE (1/SEC) 0. 8 7 4 E + 0 0 0 . 2 3 6 E + 01 0.679E+01 0 . 1 9 6 E + 02  (RX-E) / (RO-R) (") 0.43 0,52 0.57 0.62 APPARENT VISCOSITY (POISE) 0 . 1 0 6 E + 04 0.421E+03 0.157E+03 0.561E+02  TABLE EXPERIMENTAL PHI DECAY  =  D - 2  -  D ST A A N D R E S U L T S  0. 10  (CC/CC)  L  7 OF =  193 CALCULATION  1 . 6 2 0 (MM)  DATA OMEGA  TIME (SEC) 0.0 1.1 2.7 4.1 5.9 7.2 8.7 10.3 33.2 71.7 147.8 EQUI  =  0.2327  TORQUE  (RAD/SEC)  TORQUE  (HV)  ( D Y N E S CM) 0.500E+05 0.428E+05 0.366E+05 0.326E+05 0.299E+05 0 . 2 7 7 E + 05 0.259E+05 0.245E+05 0.178E+05 0.134E+05 0.112E+05 0.357E+04  1. 1 2 0.96 0.82 0.73 0.67 0.6 2 0.58 0.55 0.40 0.30 0.25 0.08  TIME  RX/R  RX  (SEC) 0.0 1. 1 2.7 4. 1 5.9 7.2 8.7 10.3 33.2 71.7 147.8 EQUI  (-) 1.000 1. 2 8 1 1.335 1 .345 1.367 1.378 1.378 1 .389 1.410 1.432 1.442 1 .464  (CM) 1.01 1.29 1. 3 5 1.36 1.38 1.39 1.39 1. 4 0 1.42 1. 4 5 1.46 1.48  EQUILIBRIUM  SHEAR STRESS ( D Y N E S / S Q CM) 0. 398E+-04 0.341E+04 0.291E+04 0.259E+04 0.238E+04 0.220E+04 0.206E+04 0.195E+04 0.142E+04 0.107E+04 0.888E+03 0.284E+03  (SX-R) / (RO-R) (-) 0.0 0.26 0.31 0.32 0.34 0.35 0.35 0.36 0.38 0.40 0.4 1 0.43  APPARENT SHEAR RATE ( 1/SEC)  APPARENT VISCOSITY (POISE)  0 .119E+01 0 .106E+01 0.104E+01 0.100E+01 0.984E+00 0. 984E+00 0.967E+00 0.936E+00 0.909E+00 0.896E+00 0.872E+00  0. 2 8 6 E + 04 0. 2 7 4 E + 0 4 0. 2 4 9 E + 0 4 0.2 3 8 E + 0 4 0. 2 2 4 E + 0 4 0 .2 0 9 E + 0 4 0. 2 0 2 E + 0 4 0 . 1 5 2 E + 04 0. 1 1 7 E + 0 4 0. 9 9 1 E + 0 3 0.3 2 6 E + 0 3  DATA  OMEGA  TORQUE  (RAD/SEC) 0.2327 0.6981 2.0940 6.2830  (DYNES" CM) 0.357E+04 0.401E+04 0. 4 4 6 E + 0 4 0. 4 6 8 E + 0 4  OMEGA  RX/R  RX  (RAD/SEC) 0.2327 0.6981 2.0940 6.2830  {-) 1. 4 6 4 1.550 1.615 1.648  (CM) 1.48 1.57 1. 6 3 1. 6 6  SHEAR STRESS ( D Y N E S / S Q CM) 0.284E+03 0.320E+03 0.355E+03 0. 3 7 3 E + 0 3 APPARENT SHEAR RATE (1/SEC) 0. 8 7 2 E + 0 0 0.239E+01 0 . 6 7 9 E + 01 0. 1 9 9 E + 0 2  (RX-R) / (RO-R) (") 0.43 0,51 0.57 0.60 APPARENT VISCOSITY (POISE) 0.326E+03 0.134E+03 0.523E+02 0.187E+02  TABLE EXPERIMENTAL PHI DECAY  =  DATA  0.09  D - 2 -  AND R E S U L T S  (CC/CC)  8 OF  L =  194 CALCULATION  1 . 6 2 0 (MM)  DATA OMEGA  = 0.2327  (RAD/SEC)  TIME  TORQUE  TORQUE  (SEC) 0.0 0.2 2.0 3. 9 5.8 7.7 9.3 11.1 13.4 48.8 93.7 EQUI  (MV) 0. 2 1 0. 2 1 0. 19 0. 1 8 0. 17 0 . 17 0. 16 0. 16 0. 15 0. 1 3 0. 1 2 0. 02  ( D Y N E S CM) 0.937E+04 0.937E+04 0.847E+04 0.803E+04 0 . 7 5 8 E + 04 0.758E+04 0.714E+04 0.714E+04 0.669E+04 0.580E+04 0.535E+04 0.892E+03  TIME  RX/R  RX  (SEC) 0.0 0.2 2.0 3.9 5.8 7. 7 9.3 11.1 13.4 48,8 93.7 EQUI  (-)  (CM) 1.01 1. 2 2 1.39 1.42 1.44 1.45 1. 4 5 1.46 1.46 1.48 1. 4 8 1.50  1 .000 1.205 1.378 1 . 410 1.421 1. 4 3 2 1.432 1. 4 4 2 1.442 1. 4 6 4 1 .464 1. 4 8 6  EQUILIBRIUM OMEGA (RAD/SEC) 0.2327 0.6981 2.0 940 6. 2 8 3 0  SHEAR STRESS ( D Y N E S / S Q CM) 0.746E+03 0.746E+03 0.675E+03 0.639E+03 0.604E+03 0.604E+03 0.568E+03 0.568E+03 0.533E+03 0.462E+03 0.426E+03 0.710E+02  (RX-R) /(RO-R) (-) , 0.0 0.19 0.35 0.38 0.39 0.40 0.40 0.41 0.41 0.43 0.43 0.45  APPARENT SHEAR RATE (1/SEC)  APPARENT VISCOSITY (POISE)  0. 1 4 9 E + 0 1 0.984E+00 0.936E+ 00 0.922E+00 0.909E+00 0.909E+00 0.8 9 6 E + 0 0 0.896E+00 0.872E*00 0.872E+00 0.851E+00  0.499E+03 0.686E+03 0.683E+03 0.654E+03 0.664E+03 0.625E+03 0.634 E+03 0.594E+03 0.529E*03 0.488E+03 0.834E+02  DATA TORQUE ( D Y N E S CM) 0.892E+03 0. 1 3 4 E + 0 4 0. 1 7 8 E + 0 4 0. 2 0 1 E + 0 4  OMEGA  RX/S  RX  (RAD/SEC) 0.2327 0. 6 9 8 1 2.0940 6.2830  (~) 1. 4 8 6 1. 5 8 3 1. 6 4 8 1. 6 6 9  (CM) 1. 50 1. 6 0 1. 6 6 1. 6 9  SHEAR STRE SS ( D Y N E S / S Q CM) 0.710E*02 0.107E+03 0. 1 4 2 E + 0 3 0.160E+03 APPARENT SHEAR RATE ' (1/SEC) 0. 8 5 1 E + 00 0.232E*01 0.663E+01 0. 1 9 6 E + 0 2  (RX-R) / (RO-R) (-) 0.45 0.54 0.60 0.62 APPARENT VISCOSITY (POISE) 0.834E+02 0 .458E+02 0.214E + 02 0.815E+01  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  (SEC) 0.0 0.9 2. 1 3.2 4.4 5.5 6.8 8.0 16. 3 36.5 63.0 EQUI  (MV) 11.26 10.34 9.56 8.94 8. 54 8. 14 7.84 7.60 6. 24 5.00 4.40 0.84  (DYNES CH) 0.502E+06 0.461E+06 0.426E+06 0.399E+06 0.381E+06 0.363E+06 0.350E+06 0.339E+06 0.278E+06 0.223E+06 0.196E+06 0.375E+05  TIME (SEC) 0.0 0.9 2. 1 3.2 4. 4 5.5 6.8 8.0 16.3 36.5 63.0 EQUI  RX/R  RX  {") 1.000 1 .045 1. 077 1.091 1.108 1.119 1. 130 1. 140 1. 162 1 . 205 1. 227 1 .442  (CM) 1.01 1.06 1. 09 1. 10 1. 12 1. 13 1. 14 1. 15 1.17 1.22 1.24 1, 46  EQUILIBRIUM OMEGA (RAD/SEC) 0. 2327 0. 6981 2. 0940 6. 2 830  SHEAR STRESS (DYNES/SQ CM) 0.400E+05 0.367E+05 0.339E+05 0,317E+05 0.303E+05 0.289E+05 0.278E+05 0.270E+05 0.222E*05 0. 178E+05 0. 156E+05 0.298E+04  (RX-R) / (RO-R) (") 0.0 0.04 0.07 0.08 0.10 0.11 0.12 0.1.3 0.15 0.19 0.21 0.41  APPARENT SHEAR RATE ( 1/SEC)  APPARENT VISCOSITY (POISE)  0.549E+01 0.339E+01 0.292E+01 0.251E+01 0.232E+01 0.2 15E+01 0.2 02E+01 0.180E+01 0. 149E+01 0.139E+01 0.896E+00  0 .669E+04 0 . 100E+05 0 . 109E+05 0 .121E+05 0 .125E+05 0 . 129E+05 0 .134E+05 0 .123 E+05 0 .119E+05 0 . 113E*05 0 .333E+04  DATA TORQUE (DYNES CM) 0.375E+05 0. 384E+05 0.392E+05 0.397E+05  OMEGA  RX/R  RX  (RAD/SEC) 0.2327 0.6 981 2. 0940 6.2 830  (-) 1. 442 1. 473 1. 504 1. 524  (CM) 1. 46 1. 49 1. 52 1. 54  SHEA R STRESS (DYNES/SQ C M ) 0.298E+04 0. 305E+04 0.312E*04 0.316E+04 APPARENT SHEAR RATE ( V S EC) 0. 896E+00 0.259E+01 0.751E + 01 0.221E+02  (RX-R) /(RO-R) (-) 0.41 0.44 0.47 0.49 APPARENT VISCOSITY (POISE) 0 .333E+04 0 .118E+04 0 .416E+03 0 . 143E + 03  TABLE D - 2 - 1 0 EXPERIMENTAL PHI DECAY  =  DATA  0,07  AND  RESULTS  L =  (CC/CC)  CALCULATION  3.010  (MM)  DATA OMEGA  =  0. 2 3 2 7  (RAD/SEC)  TIME  TORQUE  TORQUE  (SEC)  (MV) 7. 2 0 7. 0 9  (DYNES CM) 0.321E+06 0.316E+06  0.0 0. 1  OF  1.2 2.7 3.9 5.3 6.8 8. 2 13.8 28.7 88.8 EQUI TIME (SEC) 0.0  6. 13  0.273E+06  5. 4. 4. 4.  25 85 47 08 3. 84 3. 3 5  0.234E+06 0.216E+06 0.199E+06 0.182E+06  2. 6 5  0.171E+06 0. 1 4 9 E + 0 6 0. 1 1 8 E + 06  1. 9 5 0. 3 0  0.870E +05 0.134E+05 RX  RX/R ( 1.000  (CM) 1.01 1.02 1. 0 9  2. 7 3.9 5.3  1.011 1.076 1. 119 1 . 140 1. 151  1. 1 3 1. 1 5 1.16  6.8 8.2 13.8 28.7 88.8 EQUI  1 . 162 1. 173 1. 194 1. 227 1.281 1. 463  1. 1 7 1. 1 8 1.21 1.24 1.29 1,48  0.1 1.2  IQUI L I B R I U M  SHEAR STRESS (DYNES/SQ CM) 0.256E+05 0.252E+05 0. 2 1 8 E + 0 5 0.186E+05 0. 1 7 2 E + 0 5 0.159E+05 0.145E+05 0. 1 3 6 E + 0 5 0. 1 1 9 E + 0 5 0.941E+04 0.692E+04 0. 1 0 7 E + 0 4  0.21 0.26 0.43  VISCOSITY (POISE)  0. 2 1 9 E + 0 2 0.3 43E+ 01 0.232E+ 01 0.202E+01 0. 1 9 0 E + 0 1  0.115E+04 0.634E+04 0. 8 0 5 E + 0 4 0.854E+C4 0. 8 3 6 E + 0 4  0. 1 8 0 E + 0 1 0. 1 7 1 E + 0 1  0.807E+04 0.799E+04  0. 1 5 6 E + 0 1 0. 139E+ 01 0.119E+01 0. 8 7 4 E + 0 0  0.764E+04 0.678E+04 0.580E+04 0.122E+04  DATA  2.0940 6. 2 8 3 0  0. 1 5 2 E + 0 5 0.156E+05  0. 1 2 1 E + 0 4 0. 1 2 4 E + 0 4  0.6981 2.0940 6.2830  0.16 0.18  APPARENT  ( D Y N E S CM) 0.134E+05 0. 1 4 3 E + 0 5  (RAD/SEC) 0.2327  0.11 0.13 0.14 0.15  APPARENT  (RAD/SEC) 0.2327 0. 6 9 8 1  OMEGA  (") 0.0 0.01 0.07  SHEAR RATE { 1/SEC)  SHEAR STRESS { D Y N E S / S Q CM.) 0, 1 0 7 E + 0 4 0.114E+04  OMEGA  {RX-R) /(RO-R)  TORQUE  RX/R (") 1. 4 6 3 1. 4 8 3 1. 5 1 4 1. 5 3 4  RX (CM) 1. 4 8 1. 5 0 1. 5 3 1. 5 5  (RX-R) / (RO-R)  0.48 0.50  (-) 0.43 0.45  APPARENT  APPARENT  SHEAR RATE (1/SEC) 0.874E+00 0,256E+01 0 . 7 4 3 E +0 1 0.218E+02  VISCOSITY (POISE) 0.122E+04 0.444E+03 0. 1 6 2 E + 0 3 0.569E+02  TABLE EXPERIMENTAL PHI DECAY  =  D -  2 - 11  D A T A AND R E S U L T S  0.06  (CC/CC)  OF  L =  197  CALCULATION 3 . 0 1 0 (MM)  DATA OMEGA  = 0.2327  (RAD/SEC)  TIME  TORQUE  TORQUE  (SEC) 0.0 1.0 2.0 3.0 5.0 7.0 10.0 21.0 44.0 50.0 70.0 EQUI  (HV) 4. 8 7 3. 10 2. 35 1. 9 6 1. 5 9 1. 3 0 1. 1 1 0. 84 0. 6 4 0. 6 2 0. 5 6 0. 14  (DYNES CM) 0.217E+06 0.138E+06 0.105E+06 0.874E+05 0.709E+05 0.580E+05 0.495E+05 0.375E+05 0.285E+05 0.277E+05 0.250E+05 0.624E+04  TIME (SEC) 0.0 1.0 2.0 3.0 5.0 7.0 10.0 2 1 .0 44.0 50.0 70.0 EQUI  RX/R  RX  (-)  (CM) 1.01 1. 1 4 1. 17 1.20 1.23 1. 2 4 1.26 1. 2 8 1.32 1. 3 2 1. 3 3 1. 4 9  1.000 1 . 130 1. 162 1 .183 1.216 1. 227 1.248 1 .270 1 .302 1.302 1.313 1 .473  EQUILIBRIUM  SHEAR (RX-R) / (RO-R) STRESS ( D Y N E S / S Q CM) (") 0. 1 7 3 E + 0 5 0.0 0. 1 1 0 E + 0 5 0.12 0.834E+04 0. 1 5 0.17 0.696E+04 0.564E+04 0.20 0.462E+04 0.21 0.394E+04 0.23 0.298E+04 0.25 0.227E+04 0.28 0. 2 2 0 E + 0 4 0.28 0.199E+04 0.29 0.497E+03 0.44 APPARENT SHEAR RATE { 1/SEC) 0. 2 1 5 E + 0 1 0.180E+ 01 0.163E+0 1 0.144E+01 0.139E+01 0.130E+01 0. 1 2 3 E + 0 1 0. 1 1 3 E + 0 1 0. 1 1 3 E + 0 1 0. 1 1 1 E + 0 1 0.864E+00  APPARENT VISCOSITY (POISE) 0.511E+04 0.465E+04 0.428E+04 0.392E+04 0.333E+04 0.303E+04 0.243E+04 0.200E+04 0. 1 9 4 E + 0 4 0.179E+04 0.576E+03  DATA  OMEGA  TORQUE  (RAD/SEC) 0.2327 0.6981 2. 0 9 4 0 6. 2 8 3 0  ( D Y N E S CM) 0.624E+04 0.714E+04 0. 8 0 3 E + 0 4 0. 8 4 7 E + 0 4  OMEGA  RX/R  RX  (RAD/SEC) 0. 2 3 2 7 0.6981 2. 0 9 4 0 6. 2 8 3 0  (~)  (CM) 1. 4 9 1. 5 1 1. 5 4 1. 5 8  1. 4 7 3 1. 4 9 3 1. 5 2 4 1. 5 6 6  SHEA R STRESS ( D Y N E S / S Q CM) 0.497E+03 0.568E+03 0.639E+03 0.67 5E+03 APPARENT SHEAR RATE { 1 / S EC) 0. 8 6 4 E + O 0 0.253E+01 0. 7 3 5 E + 0 1 0.212E+02  (RX-R) / {RO—R)  (-)  0.44 0.46 0.49 0.52  APPARENT VISCOSITY (POISE) 0.576E+03 0.224E+03 0.869E+02 0.318E+02  TABLE EXPERIMENTAL PHI DECAY  =  DATA  0.05  AND  2 -  D -  RESULTS  (CC/CC)  12 OF  L =  198 CALCULATION  3 . 0 1 0 (MM)  DATA OMEGA  = 0. 2 3 2 7  (RAD/SEC)  TIME  TORQUE  TORQUE  (SEC) 0.0 1.0 2.0 3.0 4.2 5. 4 12.5 20.0 30.0 47.3 100.0 EQUI  (MV) 3.00 1.94 1.56 1. 2 3 1.10 1.03 0.7 1 0.60 0.52 0. 4 6 0.37 0. 0 7  ( D Y N E S CM) 0.134E+06 0.865E+05 0.696E+05 0.549E+05 0.491E+05 0.459E+05 0. 3 1 7 E + 0 5 0.268E+05 0.232E+05 0.205E+05 0.165E+05 0.312E+04  TIME  RX/R  RX  (SEC) 0.0 1.0 2.0 3.0 4.2 5. 4 12.5 20.0 30.0 47. 3 100.0 EQUI  1 .000 1.173 1.216 1. 2 3 7 1 .259 1. 2 7 0 1.313 1. 3 3 5 1 . 356 1.367 1 .389 1. 5 1 3  (-)  (CM) 1.01 1.18 1. 2 3 1.25 1.27 1.28 1.33 1. 3 5 1.37 1.38 1.40 1.53  !QUI L I B R I U M OMEGA (RAD/SEC) 0.2327 0. 6 9 8 1 2. 0 9 4 0 6. 2 8 3 0  SHEAR STRESS ( D Y N E S / S Q CM) 0. 1 0 7 E + 0 5 0.689E+04 0.554E+04 0.437E+04 0.391E+04 0.366E+04 0.252E+04 0.213E+04 0. 1 8 5 E + 0 4 0, 1 6 3 E + 0 4 0. 1 3 1 E + 0 4 0.249E*03  (RX-R) / (RO-R) (") 0.0 0.16 0.20 0.22 0.24 0.25 G.29 0.31 0.33 0.34 C.36 0.48  APPARENT SHEAR RATE (1/SEC)  APPARENT VISCOSITY (POISE)  0.171E+01 0.144E+01 0.134E+01 0.126E+01 0.123E+01 0.111E+01 0.106E+01 0.102E+01 0.100E+01 0.967E+00 0.827E+00  0. 4 0 4 E + 0 4 0. 3 8 5 E + 0 4 0.3 2 6 E + 0 4 0. 3 1 0 E + 0 4 0.2 9 8 E + 0 4 0. 2 2 7 E + 0 4 0.2 0 1 E + 0 4 0. 1 8 1 E + 0 4 0. 1 6 3 E + 0 4 0. 1 3 6 E + 0 4 0 .3 0 1 E + 0 3  DATA TORQUE ( D Y N E S CM) 0.312E*04 0.357E+04 0.401E+04 0.424E+04  OMEGA  RX/R  RX  (RAD/SEC) 0. 2 3 2 7 0.6981 2. 0 9 4 0 6. 2 8 3 0  (-) 1 .4 9 3 1. 5 1 4 1. 5 3 4 1. 6 1 6  (CM) 1. 5 1 1.53 1. 5 5 1.63  SHEAR STRE SS ( D Y N E S / S Q CM) 0.249E+03 0.28 4 E + 0 3 0.320E+03 0.337E+03 APPARENT SHEAR RATE (1/SEC) 0. 8 4 4 E + 0 0 0 . 2 4 8 E + 01 0. 7 2 8 E + 0 1 0.204E+02  (RX-R) /{RO-R) (") 0.46 0.48 0.50 0.57 APPARENT VISCOSITY (POISE) 0.2 9 4 E + 0 3 0. 1 1 5 E + 0 3 0. 4 3 9 E + 0 2 0. 1 6 6 E + 0 2  TABLE EXPERIMENTAL PHI = DECAY  DATA  0.14  D -  2 -  ANL RESULTS  (CC/CC)  13 OF  1 =  199 CALCULATION  0.987  (MM)  DATA OMEGA  TIME (SEC) 0.0 1.7 3. 2 5.0 6.3 8.2 10.0 11.4 13.0 27.2 66. 1 EQUI TIME  =  0.6981  (RAD/SEC)  TORQUE  TORQUE (M V ) 1. 1 7 0.69 0.53 0. 4 2 0. 3 8 0.33 0.30 0.28 0.27 0.21 0. 1 5 0.08  ( D Y N E S CM) . 0.522E+05 0.308E+05 0.236E+05 0.187E+05 0.169E+05 0. 1 4 7 E + 0 5 0. 1 3 4 E + G 5 0.125E+0 5 0. 1 2 0 E + 0 5 0.937E+04 0.669E+04 0.357E+04  RX/R  RX  (-) 1.000 1.410 1. 4 5 3 1 .486 1. 4 9 6 1 .507 1. 5 1 8 1 .529 1.540 . 1.561 1. 5 7 2 1.691  (CM) 1.01 1. 4 2 1.47 1. 5 0 1.51 1. 5 2 1.53 1.54 1.55 1. 5 8 1.59 1,71  (SEC) 0. 0 1.7 3. 2 5.0 6.3 8.2 10.0 11.4 13.0 27.2 66. 1 EQUI  EQUILIBRIUM  TORQUE  (RAD/SEC) 0. 6 9 8 1 1.0470 2.0940 6.2830  ( D Y N E S CM) 0.357E+04 0. 3 7 9 E + 0 4 0.401E+04 0.424E+04  ( R A D / S EC) 0.6981 1.0 4 7 0 2.0940 6.2830  APPARENT SHEAR RATE { 1/SEC) 0.281E+01 0.265E+01 0.255E+01 0 . 2 5 2 E + 01 0.249E+01 G.247E+G1 0.2 4 4 E + 0 1 0 . 2 4 2 E + 01 0.237E+01 0.235E+01 0.215E+01  (RX-R) / (RO-R) (") 0.0 0.38 0.42 0.45 0.46 0.47 0.48 0.49 0.50 0.52 0.53 0.64 APPARENT VISCOSITY (POISE) 0.872E+03 0.710E+03 0.584E+03 0.535E+03 0.470E+03 0.432E+03 0. 4 0 7 E + 0 3 0.397E+03 0.315E+03 0.227E+03 0.132E+03  DATA  OMEGA  OMEGA  SHEAR STRESS ( D Y N E S / S Q CM) 0.415E+04 0.245E+04 0.188E+G4 0. 1 4 9 E + 0 4 0. 1 3 5 E + 0 4 0. 1 1 7 E + 0 4 0.107E+04 0.994E+03 0.959E+03 0.746E+03 0.533E+03 0.284E+03  RX/R (") 1. 6 9 1 1. 7 2 3 1. 7 6 6 1. 8 2 0  RX (CM) 1.71 1. 7 4 1.78 1. 8 4  SHEA R STRESS ( D Y N E S / S Q CM) 0.284E+03 0.302E+03 0.32 0 E + 0 3 0. 3 3 7 E + 0 3 APPARENT SHEAR RATE ( 1 / S EC) 0.215E+01 0. 3 1 6 E + 0 1 0 . 6 1 6 E + 01 0. 1 8 0 E + 0 2  (RX-R) /(RO-R)  (") 0.64 0.67 0.71 0.76 APPARENT VISCOSITY (POISE) 0.132E+03 0.956E+G2 0.518E+02 0. 1 8 7 E + 0 2  TABLE EXPERIMENTAL PHI DECAY  =  DATA  0.14  D -  2 -  AND R E S U L T S  (CC/CC)  14 OF  L =  200 CALCULATION  0.987  (MM)  DATA OMEGA =  TIME (SEC) 0.0 1.2 2.3 3. 4 4.4 5. 5 6.7 7.8 8.7 25. 3 57.2 EQUI  0. 3 4 9 1  (RAD/SEC)  TORQUE  TORQUE  (MV) 1.19 0.90 0.74 0. 6 5 0.58 0.52 0.48 0. 4 5 0.43 0.26 0. 19 0.07  ( D Y N E S CM) 0.531E+05 0.401E+05 0 . 3 30 E + 0 5 0.290E+05 0.259E+05 0.232E+05 0.214E+05 0.201E+05 0.192E+05 0.116E+05 0.847E+04 0.334E+04  TIME  RX/R  RX  (SEC) 0.0 1.2 2.3 3. 4 4.4 5.5 6.7 7.8 8.7 25. 3 57.2 EQUI  (-) 1 .000 1.259 1.302 1.324 1.335 1. 3 4 5 1 .356 1.367 1 .378 1. 4 2 1 1 . 453 1.615  (CM) 1.01 1.27 1. 3 2 1. 3 4 1. 3 5 1. 3 6 1. 3 7 1.38 1.39 1.44 1. 4 7 1.63  EQUILIBRIUM  TORQUE  (RAD/S EC) 0. 3 4 9 0 0. 6 9 8 1 2. 0 9 4 0 6. 2 8 3 0  ( D Y N E S CM) 0.334E+04 0.379E+04 0.401E+04 0. 4 2 4 E + 0 4  { R A D / S EC) 0. 3 4 9 0 0. 6 9 8 1 2. 0 9 4 0 6. 2 8 3 0  APPARENT SHEAR RATE (1/SEC) 0. 1 8 9 E + 0 1 0. 1 7 0 E + 0 1 0.163E+01 0. 1 5 9 E + 0 1 0. 1 5 6 E + 0 1 0.153E+01 0 . 1 5 0 E + 01 0.148E+01 0. 1 3 8 E + 0 1 0. 1 3 3 E + 0 1 0. 1 1 3 E + 0 1  (RX-R) /{RO-R) (-) 0.0 0.24 0.28 0.30 0.31 0.32 0.33 0.34 0.35 0.39 0.42 0.57 APPARENT VISCOSITY (POISE) 0. 1 6 9 E + 0 4 0. 1 5 4 E + 0 4 0. 1 4 2 E + 0 4 0. 1 2 9 E + 0 4 0. 1 1 8 E + 0 4 0. 1 1 1 E + 0 4 0. 106 E+04 0. 1 0 3 E + 0 4 0. 6 6 7 E + 0 3 0. 5 0 9 E + 0 3 0. 2 3 5 E + 0 3  DATA  OMEGA  OMEGA  SHEAR STRESS ( D Y N E S / S Q CM) 0.422E+04 0.320E+04 0.263E+04 0.231E+04 0. 2 0 6 E + 0 4 0.185E+04 0.170E+04 0.160E+04 0.153E+04 0.923E+03 0.675E+03 0.266E+03  RX/R (-) 1. 6 1 5 1. 6 4 8 1. 7 0 1 1. 7 6 6  RX (CM) 1.63 1.66 1.72 1.78  SHEAR STRESS ( D Y N E S / S Q CM) 0. 2 6 6 E + 0 3 0. 3 0 2 E + 0 3 0.320E+03 0.337E+03 APPARENT SHEAR RATE (1/SEC) 0.113E+01 0.221E+01 0.640E+01 0.185E+02  (RX-R) /(RO-R) (-) 0.57 0.60 0.65 0.71 A P P A H ENT VISCOSITY (POISE) 0. 2 3 5 E + 0 3 0. 1 3 7 E + 0 3 0. 4 9 9 E + 0 2 0. 1 8 2 E + 0 2  TABLE EXPERIMENTAL PHI DECAY  =  DATA  0.14  AND  D - 2 RESULTS L  {CC/CC)  15 OF =  201 CALCULATION  0.987  {MM)  DATA OMEGA  TIME {SEC) 0.0 0.4 1.7 3.1 4. 5 5.8 7.2 8.7 10.0 36.0 138.6 EQUI  =  0.1164  TORQUE  {RAD/SEC)  TORQUE  (M V) 3. 4 6 3. 17 2.5 0 2.07 1.74 1.62 1.40 1.33 1. 1 8 0.59 0. 3 2 0,06  ( D Y N E S CM) 0. 1 5 4 E + 0 6 0. 1 4 1 E + 0 6 0.112E+06 0.923E+05 0.776E+05 0.723E+05 0.624E+05 Q.593E+05 0.526E+05 0.263E+05 0.143E+05 0.268E+04  TIME  RX/R  RX  (SEC) 0.0 0.4 1.7 3. 1 4. 5 5.8 7.2 8.7 10.0 36.0 138.6 EQUI  {") 1.000 1.097 1. 1 6 2 1 .183 1.194 1.205 1.216 1.227 1. 2 2 7 1.270 1. 3 0 2 1.421  (CM) 1.01 1. 11 1.17 1.20 1.21 1. 2 2 1.23 1. 2 4 1.24 1. 2 8 1. 3 2 1. 4 4  EQUILIBRIUM  TORQUE  (RAD/SEC) 0. 1 1 6 4 0.6 9 8 1 2. 0 9 4 0 6.2830  ( D Y N E S CM) 0.268E+04 0.357E+04 0.401E+04 0.424E+04  (RAD/SEC) 0. 1 1 6 4 0.6981 2. 0 9 4 0 6.2830  (RX-R) / (RO-R) {-) 0.0 0.09 0.15 0. 1 7 0.18 0.19 0.20 0.21 0.21 0.25 0,28 0.39  APPARENT SHEAR RATE { 1/SEC)  APPARENT VISCOSITY (POISE)  0 .138E+01 0 .8 9 8 E + 0 0 0.8 1 4 E + 0 0 0.779E+00 0.748E+00 0.7 2 0 E + 0 0 0.694E+00 0. 6 94E+00 0 .6 1 3 E + 0 0 0. 567E+00 0.461E+00  0 .818E+04 0 .988E+04 0 .903E+04 0 .793E+04 0.769E+04 0 .691E+04 0.680E+04 0. 6 0 4 E + 0 4 0.342E+04 0. 2 0 0 E + 0 4 0.462E+03  DATA  OMEGA  OMEGA  SHEAR STRESS ( D Y N E S / S Q CM) 0.123E+05 0. 1 1 3 E + 0 5 0.888E+04 0.735E+04 0.618E+04 0.575E+04 0.497E+04 0.472E+04 0.419E+04 0.209E+04 0.114E+04 0.213E+03  RX/R  {") 1. 4 2 1 1.442 1.464 1. 4 8 6  RX (CM) 1. 4 4 1. 4 6 1. 4 8 1. 5 0  SHEAR STRESS ( D Y N E S / S Q CM) 0.213E+03 0.284E+03 0.32 0Et-03 0.337E+03 APPARENT SHEAR RATE (1/SEC) 0.461E+00 0.269E+01 0.785E+01 0.230E+02  (RX-R) / (RO-R)  (-) 0.39 0.41 0.43 0.45 APPARENT VISCOSITY (POISE) 0.46 2E+03 0. 1 0 6 E + 0 3 0.407E+02 0. 1 4 7 E + 02  Appendix  E  Solutions  A model in  Section  equation  7-1  dir(R  f o r D i f f e r e n t i a l Equation f o r the thickness  and i t was shown -  2  At  i n equation  layer  was  constructed  (7-5), i . e . ,  2  * T ¥ IV*< » " > > K  2  R 2  l  m  x  -  dR  flowing  R )L  ^  from which  of  (7-5)  \ { * ( \  -  2  ,n 1} J  R ) 2  (7-5)  one can w r i t e k  2  equilibrium state  one knows  dR  2  x dt  =  0,  and R  = R  at t  Xoo  X  = t  (E-2)  Hence,  - V»" - r+V <* *» - »"  xinsr  < *  Solving  equation  l(E  L  (  E  O  2  CO  2 e2  (e  CO  (E-3) f o r  {  \  ~  f  k  ,L(R  {TTL(R  one -  2 0  2  -  R  "  3)  gets  R^ )} * 2  1  2 ) n  ^  }  Let {TTL(R  2 Q  -  R ^  2  ) }  1  = { L T  L(R  2  - R  2  ) }  N  P M  (> N  0)  (E-5)  "-  then k, = P k, D mn f Substituting  equation dR  (E-6) back  to equation  (E-6)  ( E - l ) one  obtains  k  2  -  P {^L(R mn x  2  -  202  R )} ] 2  n  (E-7)  203 Let  R  - R  2 x  then R  - R  2 0  dR x — dt  =  Therefore,  = R  2 x  - (R  2 0  - R ) - R  2  2  x  = - X +  2  (R  2 Q  - R ) 2  , dx — dt  2  . and  (E-8)  = X  2  equation  (E-7) becomes  - P {uLX} ] mn n  or  ^ d f  =  " AT^BT  Equation  mn  [ P  "  { l T L X } n  -  {  0  R  of X with  With  (m, n ) = ( 2 , 0 ) , e q u a t i o n  2  =  0  thebasic equation  L  - TT^BT  =  1  irLk  P  2  ^  L  2  rank f i r s t  order  (7-5).  differential  equa-  0).  -  R  2  {( O  2  2 0  J  >  2  0  (E-10)  (E-9) i s  [ ? 2  ° -^  L  2  R  - )- > i R 2  x  2  0  = " A T W  l  +  to equation  (E-5) i s  ( R  2  (E-9)  identical  r e s p e c t t o t f o r (m, n ) = ( 2 ,  P  * fr  R 2  equation  first  tion  " ) " X]}" ]  2  (E-9) i s t h e b a s i c model  Now o n e c a n s o l v e t h i s  and  [ (  L  (R  2 0  {  x  2  "  2  (  R  °  2  2  )  "  r  2  )  x  - R ) }] 2  2  from which ,  If  TTLk =  x r i t ] u  2  L  2  [  x  2  -  2  (  R  ° 2  r  x  +  (  r  °  2  "  r  2  )  2  (E-ll)  204  Let  b = -2(R  and  C =  then  equation  2  -  -  R )  0  (E-12)  R ) 2  2 0  (R  2 0  2  (E-ll)  becomes  TrLk,.  dX dt Separate  (E-13)  -  2  A +  (X  Bt  the variables  in  dX X  from which  2  b  f  c  (E-14)  A +  and  find  Bt  write  «* / f  2  (E-14)  C)  dt  TTLk  ^ X  here  +  bX +  equation  =  x  o n e may  / Note  +  +  2  + bX +  dt A +  (E-15)  Bt  C  that 4P o 2  b  -  2  4C =  4(R  2 0  -  R ) 2  4(R  2  2 0  -  R ) 2  +  2  ^ L 2  4P  2 0  2  Recall  in  >  0  a  2  shown i n  s  general  (E-16)  > 0  TT L since P 2 0  equation  (E-10)  that  dx ax  with  b  2  -  Therefore  log  + bx +  4ac >  c  /  4ac  +  b -  /b  2  2ax  + b + /b  2  -  4a c  -  4a c  + bX +  (E-17)  (E-18) side  of  1  dX 2  /, 2 vb -  2ax  0  the left-hand  X  2  c  equation  log 4c  (E-15)  2X + b -  is /b  2  2X + b + / b  2  -  4c  -  4c  (E-19)  205 4c  Let and  2X +  (E-20)  = D  (E-21)  b = Y  then dX X The  z  right-hand  + bX + side  of  ^ A+Bt  =  D  c  Y -  log  B  (E-22)  Y + D  equation  f t  D  (E-15)  is  "  ^ ~7~A B  =  k TTL— l o g | t  Therefore  log  D where The  CI  equation  Y - D Y + D is  =  (E-15)  has been  TTL-^ l o g |t +  boundary R  is  (E-23)  the integral  initial  which  A -|  +  2  equivalent  - R  Y = 2X + b ,  Substituting b  -  CI  from  as  at  given  t =  0  i n equation  (E-26)  into  (E-21).  equation  (E-24)  o n e may  find  k.  D  = TiL-j- l o g if 1 + C I  yields  = -  log  Substituting  D  is  to  equation  |b + D which  (E-24)  CI  = X = 0 a t t = 0  2  Y = b since  | +  as  constant.  condition  x  f  integrated  log  b  |b +  D D  l - T T L — il o gl |-|  equation  Y - D Y + D  which  -  b + D b - D  A  f  (E-25)  T  (E-25)  =  back  into  equation  k TrL-y- l o g |-| t +  1  (E-24)  one c a n  find  206  Y  log  where  -  D  b +  D  Y +  D  b -  D  Y  =  2X +  D  = /b  b  =  -2(R  c  =  (R  D T T L - J - l o g |f t  -  (E-21)  (E-20)  4c 2  -  -  R )  0  (E-12)  R ) 2  J 2 0  2  20  -  2  u L 2  X  and  (R  =  R  Equation (2, of  0)  and  this  has  2 0  - R  -  2  x  (E-26)  1  b  2  =  +  R  )  2  (V "  -  2  2  \J)  (E-13)  2  (E-8)  2  (E-26)  is  already  the  been  solution given  in  of  equation  equation  (7-5)  (7-7)  in  with  (m,  Section  n)  =  (7-1).  thesis.  The  solutions  summarized  below.  of  equation  It  should  (7-5)  be  for  noted  various  that  X =  values  R  -  2  of  (m,  n)  are  R . 2  x i.  (m,  n)  =  (0,  0) X  =  const  for  any  does  not  satisfy  This  conditions X  (m,  n)  log  =  1  (0,  X 2  Xoo  the  (E-27) boundary  =  0 at 2  t  =  -  R  0 2  H  at  t  =  t  1)  " R  t.  of  X = R i i .  time  = -  R  2  -  P  oiT  (E-28)  l o g  where l J  01  = TTL(R  2  -  R ) 2  207  i n .  (m,  n)  log  =  (0,  |X -  a  X +  a  2)  =  -  P TTL2a-—• lo£ T A 0 2  t  +  (E-29)  1  where 02  ^2 2(R L  X  and  a  (m,  n)  =  1  log  v.  (m,  n)  (1,  =  log  1  -  2)2  0)  R  2 -  (1,  1)  log R  2 _  R  t  +  (E-30)  1  2  X R  R  /F  X  -  2 _  1  = TTL  iv.  oo  = - (?io +  2  l o  t  §  +  1  (E-31)  Xoo  where R 2  -  0  R  2 Xoo  u  R  2  _  R  2  Xoo  VI.  (m,  n)  log  =  (1,  2)  |Y -  D  1 +  D  Y +  D  1 -  D  where  log  y  =  2aX +  D  =/l  a  =  b  = R  + P  1 2  Z 0  1  4ab frl -  R  z  t  +  1  (E-32)  208 v i i .  (m, n )  log  =  |Y -  (2,  1)  D  b +  D  Y + D  b-'-  D  where  a  Y =  D  2X +  t  log  TTLD—  +  (E-33)  1  b  = /b  -  4c  P21  b = -[—+ c  =  (Ro  2(R ^  -  0  -  2  R ) 2  R ).] z  2  2N2 P 1  =  2  R (m,  V l l l ,  log  n)  =  (2,  |Y - D Y + D  b + b. -  D  =  D  Y =  -  and  D  = /b  a  = P22  b  =  c  = -(R  P22  (0,  1)  log  This  has  interesting to  (2,  f (R  been  )  2)  =  2  2(R  4ac  "  1 -  2 0  -  2  R R  2  x°°  to note  -  2N2  )  x°°  ?)  2  (E-35)  R ) 2  that  are  the form  K—  loj T  mentioned  (E-34)  1  R )  -  2 0  2  +  b  -  0  t  =  (R  are  2  log  TTLD-^-  2aX +  (R  is  R  2)  where  It  -  2  2  a l l the solutions  for  which  of  (E-36)  A t + 1  in  (m, n )  equation  (7-6)  in  Section  7-1  of  this  thesis,  Appendix  All dextrose  F  Various  Orders  Flowing  Layer  the +1.0  Description  of  data  in  of and  these  the Model Sum  of  tables  mole NaCl  in  10%  Variables  in  Table  Predicting Thickness  Squares  are  for  the  dispersing  medium  PEG-H 0. 2  F  PHI(CC/CC)  -  Particle  Concentration  L(MM)  -  Particle  Length  L/D  -  Particle  Length-to-Diameter  OMEGA(RAD/SEC)- A n g u l a r  of  (mm)  V e l o c i t y of  209  (cc/cc)  Inner  Ratio  Cylinder  (-) (rad./sec.)  210  TABLE VARIOUS  ORDERS  (M,N)  ORDER (M,N)  ZERO-FIRST  1  AND R E S U L T  OF  RESULTANT KF/B  ORDER  ZERO-SECOND FIRST-ZERO  F -•  ORDER ORDER  FIRST-FIRST  ORDER  FIRST-SECOND SECOND-ZERO  ORDER ORDER  SECOND-FIRST  ORDER  SECOND-SECOND  ORDER  PHI (CC/CC) 0.160  L (MM) 0.987  LEAST  SQUARES F I T  PARAMETERS B/A  SUM O F SQUARES  0.104E+01  0.4 91 E + 0 1  0. 1 6 1 E - 0 2  0.715E+00  0. 9 8 8 E + 0 1  0. 2 8 8 E - 0 2  0. 14 9 E + 0 0  0.491E+01  0. 1 6 1 E - 0 2  0.500E-01  0. 4 91 E + 0 1  0.161E-02  0.382E-01  0. 8 0 4 E + 0 1  0.238E-02  0. 4 7 5 E - 0 2  0.410E+01  0. 1 4 0 E - 0 2  0.253E-02  0.445E+01  0. 1 5 0 E - 0 2  L/D 22.9  OM E G A (R A D / S EC) 0.2327  211  TABLE VARIOUS  ORDERS  (M,N)  ORDER (M,N)  ZERO-FIRST  ORDER  ZERO-SECOND FIRST-ZERO  ORDER ORDER  FIRST-FIRST  ORDER  FIRST-SECCND SECOND-ZERO  ORDER ORDER  SECOND-FIRST  ORDER  SECOND-SECOND  ORDER  PHI (CC/CC) 0. 1 5 0  L (MM) 0.987  F AND  2 RESULT  OF  LEAST  SQUARES F I T  RESULTANT KF/B  PARA METERS B/A  SUM OF SQUARES  0. 1 8 2 E + 0 1  0.420E+01  0.790E-01  0. 1 0 6 E + 0 1  0.118E+02  0.138E+00  0.255E+00  0. 420E+-C1  0.7 9 0 E - 0 1  0.870E-01  0.420E+01  0.790E-0 1  0.593E-01  0.874E+C1  0. 1 2 2 E + 0 0  0.861E-02  0.3 1 8 E + 0 1  0.618E-0 1  0.454E-02  0. 3 6 0 E + 0 1  0.695E-0 1  0.3U0E-O2  0.615E+01  0.102E+00  L/D 22. 9  OMEGA(RAD/S EC) 0.2327  212  TABLE VARIOUS  ORDERS  ( M , N ) AND  ORDER (M,N)  ZERO-FIRST  ORDER  ZERO-SECOND FIRST-ZERO  ORDER ORDER  FIRST-FIRST  ORDER  FIRST-SECOND SECOND-ZERO  ORDER ORDER  SECOND-FIRST  F -  ORDER  SECOND-SECOND  ORDER  PHI (CC/CC) 0.140  L(MM) 0.987  3 RESULT  OF  LEAST  SQUARES F I T  RESULTANT KF/B  PARAMETERS B/A  0.193E+01  0. 6 0 8 E + 0 1  0.382E- 02  0. 1 1 5 E + 0 1  0. 1 5 8 E + 0 2  0.4 7 4 E - 0 2  0.262E+O0  0. 6 0 8 E + 0 1  0.382E- 02  0.926E-01  0. 6 0 8 E + G 1  0.38 2E- 0 2  0.642E-01  0. 1 1 8 E + 0 2  0.4 4 5 E - 0 2  0.895E-02  0, 4 5 8 E + 0 1  0.353E- 02  0.485E-02  0. 5 1 8 E + 0 1  0.367E- 0 2  0. 36 9 E - 0 2  0. 8 4 5 E + 0 1  0.413E- 02  L/D 22.9  O M E G A (R A D / S EC) 0.2327  SUM O F SQUARES  TABLE VARIOOS  ORDERS  (M,N)  ORDER (H,N)  ZERO-FIRST  AND  4 RESULT  OF  RESULTANT KF/B  ORDER  ZERO-SECOND FIRST-ZERO  F -  ORDER ORDER  FIRST-FIRST  ORDER  FIRST-SECOND SECOND-ZERO  ORDER ORDER  SECOND-FIRST  ORDER  SECOND-SECOND  ORDER  PHI(CC/CC) 0. 1 3 0  L (MM) 0.987  L E A S T SQUARES F I T  PARAMETERS B/A  SUM O F SQUARES  0. 2 6 3 E + 0 1  0. 1 1 0 E + 0 2  0.159E-01  0.147E+01  0.313E+02  0.255E-01  0.329E+00  0. 1 1 0 E + 0 2  0.159E-01  0.126E+00  0.110E+02  0. 1 5 9 E - 0 1  0.848E-01  0.223E+02  0.224E-0 1  0. 1 2 0 E - 0 1  0.763E+01  0.130E-01  0.684E-02  0.884E+01  0.141E-01  0. 5 0 9 E - 0 2  0. 1 4 8 E + 0 2  0. 1 8 6 E - 0 1  L/D 22. 9  OMEGA(RAD/SEC) 0.2327  TABLE VARIOUS  ORDERS  (M,N) AND  ORDER (M,N)  ZERO-FIRST  ORDER  ZERO-SECOND FIRST-ZERO  ORDER ORDER  FIRST-FIRST  ORDER  FIRST-SECOND SECOND-ZERO  ORDER ORDER  SECOND-FIRST  F -  ORDER  SECOND-SECOND  ORDER  P H I (CC/CC) 0.120  L (MM) 1.620  5 RESULT  OF  LEAST  SQUARES F I T  RESULTANT KF/B  PARAMETERS B/A  SUM O F SQUARES  0.154E+01  0.397E+01  0.349E-02  0.981E+00  0.833E+01  0.451E-02  0.23 4E+00  0.397E+01  Q.349E-02  0.739E-01  0.397E+C1  0.349E-02  0.530E-01  0.682E+01  0.425E-02  0.748E-02  0.329E+01  0.316E-02  0.376E-02  0.361E+01  0.333E-02  0,292E-02  0.5 4 3 E + C 1  0.3 9 5 E - 0 2  L/D 37.4  O M E G A (R A D / S EC) 0.2327  215  TABLE VARIOUS  ORDERS  (M,N)  ORDER (M,N)  ZERO-FIRST  ORDER  ZERO-SECOND FIRST-ZEEO  ORDER ORDER  FIRST-FIRST  ORDER  FIRST-SECOND SECOND-ZERO  ORDER ORDER  SECOND-FIRST  ORDER  SECOND-SECOND  ORDER  PHI{CC/CC) 0.110  L (MM) 1.620  F AND  6 RESULT  OF  LEAST  SQUARES F I T  RESULTANT KF/B  PARAMETEBS B/A  0. 1 8 0 E + 0 1  0. 6 7 6 E + 0 1  0.205E + 0C  0.104E+01  0.218E+02  0. 2 4 0 E + O 0  0.252E+00  0 . 6 7 6 E+01  0.205E+00  0.86 4E-01  0. 6 76 E+ 0 1  0.205E+00  0.579E-01  0. 1 5 7 E + 0 2  0.233E+0G  0.870E-02  0.466E+01  0.187E+00  0.456E-02  0. 5 5 2 E + G 1  0.196E + 00  0.3 3 H E - 0 2  0.106E + 02  0.222E+00  L/D 37. U  OMEGA ( R A D / S E C ) 0.2327  SUM OF SQUARES  216  TABLE VARIOUS  ORDERS  (M,N)  ORDER  (H.N)  ZERO-FIRST  ORDER  ZERO-SECOND FIRST-ZERO  ORDER ORDER  FIRST-FIRST  ORDER  FIRST-SECOND SECOND-ZERO  ORDER ORDER  SECOND-FIRST  ORDER  F AND  7 RESULT  OF L E A S T  SQUARES F I T  RESULTANT KF/B  PARAMETERS B/A  SUM O F SQUARES  0.31 1E+01  0.438E+01  0.465E-01  0.167E+01  0.119E+02  0.445E-01  0.433E+00  0.438E+01  0.465E-01  0.149E+00  0.438E+01  0.465E-01  0.951E-01  0. 8 9 7 E + 0 1  0.450E-01  0.153E-01  0.320E+01  0.467E-01  0.796E-02  0.369E+01  0.467E-01  0. 5 6 2 E - 0 2  0.641E+G1  0. 4 5 8 E - 0 1  SECOND-SECOND  ORDER  P H I (CC/CC)  L (MM)  L/D  OMEGA (RAD/S EC)  1.620  37.4  0.2327  0. 1 0 0  217  TABLE VARIOUS  ORDERS  (M,N)  ORDER (M,N)  ZERO-FIRST  ORDER  ZERO-SECOND FIRST-ZERO  ORDER ORDER  FIRST-FIRST  ORDER  FIRST-SECOND SECOND-ZERO  ORDER ORDER  SECOND-FIRST  ORDER  SECOND-SECOND  ORDER  PHI (CC/CC) 0.090  L (MM) 1.620  F AND  8 RESULT  OF  LEAST  SQUARES F I T  RESULTANT ,KF/B  PARAMETERS B/A  0. 3 4 0 E + 0 1  0. 1 1 0 E + 0 2  0.620E- 01  0.186E+01  0.259E+02  0 . 8 9 8 E - •0 1  0. 4 4 8 E + 0 0  0. 1 1 0 E + 0 2  0.620E- 01  0. 1 6 3 E + 0 0  0, 1 1 0 E + 0 2  0 . 6 2 0 E - •0 1  0. 1 0 6 E + 0 0  0.202E+02  0.807E- 0 1  0. 1 6 2 E - 0 1  0.813E+01  0.547E- 0 1  0. 8 7 8 E - 0 2  0.929E+C1  0 . 5 7 7 E - 01  0. 6 3 4 E - 0 2  0. 1 4 8 E + 0 2  0 . 7 0 4 E - •0 1  L/D 37. 4  OMEGA ( R A D / S E C ) 0.2327  SUM OF SQUARES  TABLE VARIOUS  ORDERS  (M,N) AND  ORDER (M,N)  ZERO-FIRST  ORDER  ZERO-SECOND FIRST-ZERO  ORDER ORDER  FIRST-FIRST  ORDER  FIRST-SECOND SECOND-ZERO  ORDER ORDER  SECOND-FIRST  F -  ORDER  SECOND-SECOND  ORDER  PHI(CC/CC) 0.080  L(MM) 3.010  9 RESULT  OF  LEAST  SQUARES F I T  RESULTANT KF/B  PARAMETERS B/A  SUM O F SQUARES  0. 1 0 8 E + 0 1  0.774E+00  0.89 4E-0 3  0.789E+00  0.121E+01  0. 1 7 3 E - 0 2  0. 15 9 E + 0 0  0.774E+00  0.894E-03  0.518E-01  0. 7 7 4 E + G 0  0.894E-03  0.414E-01  0,106E+01  0.147E-0 2  0.493E-02  0. 6 9 4 E + 0 0  0.707E-03  0.259E-02  0.730E+00  0.794E-0 3  0.219E-02  0.920E+00  0. 1 2 0 E - 0 2  L/D 69. 8  O M E G A (R A D / S EC) 0.2327  219  TABLE VARIOUS  ORDERS  (M , N) / AND  ORDER («,N)  ZERO-FIRST  ORDER  ZERO-SECOND FIRST-ZERO  ORDER ORDER  FIRST-FIRST  ORDER  FIRST-SECOND SECOND-ZERO  ORDER ORDER  SECOND-FIRST  F -  ORDER  SECOND-SECOND  ORDER  PHI (CC/CC) 0.070  L (MM) 3.010  10 RESULT  OF  LEAST  SQUARES F I T  RESULTANT KF/B  PARAMETERS B/A  SUM OF SQUARES  0. 1 2 6 E + 0 1  0. 1 0 7 E + 0 1  0. 1 7 0 E - 0 2  0. 8 7 4 E + 0 0  0. 1 8 7 E + 0 1  0. 2 1 3 E - 0 2  0. 1 7 6 E + 0 0  0. 1 0 7 E + 0 1  0. 17 0 E - 0 2  0. 6 0 3 E - 0 1  0. 1 0 7 E + 0 1  0. 1 7 0 E - 0 2  0. 4 6 7 E - 0 1  0. 1 5 8 E + 0 1  0. 2 1 0 E - 0 2  0.562E-02  0. 9 2 2 E + 0 0  0. 1 4 9 E - •0 2  0. 3 0 5 E - 0 2  0. 9 8 6 E + 0 0  0.159E- 02  0. 2 5 3 E - 0 2  0. 1 3 1 E + 0 1  0. 1 8 9 E - 0 2  L/D 69.8  OMEGA ( R f l D / S E C ) 0.2327  220  TABLE VARIOUS  ORDERS  (M,N)  ORDER (M,N)  ZERO-FIRST ZERO-SECOND FIRST-ZERO FIRST-FIRST  ORDER ORDER ORDER ORDER  FIRST-SECOND SECOND-ZERO  ORDER ORDER  SECOND-FIRST  ORDER  SECOND-SECOND  ORDER  P H I (CC/CC) 0.060  L (MM) 3.010  F AND  11 RESULT  OF  LEAST  SQUARES F I T  RESULTANT KF/B  PARAMETERS B/A  0. 1 3 0 E + 0 1  0. 3 4 6 E + 0 1  0 . 1 3 4 E - •0 2  0.870E+00  0. 6 8 3 E + 0 1  0.335E- 02  0. 1 7 7 E + 0 0  0. 3 4 6 E + 0 1  0.134E- 0 2  0.621E-01  0. 3 4 6 E + C 1  0.131E- 02  0.U71E-01  0. 5 5 3 E + 0 1  0 . 2 5 6 E - •0 2  0. 5 7 5 E - 0 2  0. 2 8 7 E + C 1  0.101E- 02  0.317E-02  0. 3 1 1 E + 0 1  0 . 1 1 4 E - •0 2  L/D 69. 8  OMEGA(R AD/S EC) 0.2327  SUM O F SQUARES  221  TABLE VARIOUS  ORDERS  (M,N)  ORDER (M,N)  ZERO-FIRST ZERO-SECOND FIRST-ZERO FIRST-FIRST  ORDER ORDER ORDER ORDER  FIRST-SECOND SECOND-ZERO  ORDER ORDER  SECOND-FIRST  ORDER  SECOND-SECOND  ORDER  P H I (CC/CC) 0.050  L (MM) 3.010  F AND  12 RESULT  OF L E A S T  SQUARES F I T  RESULTANT KF/B  PARAMETERS B/A  SUM OF SQUARES  0. 1 7 6 E + 0 1  0. 3 6 2 E + 0 1  0 . 1 5 1 E - G2  0.112E+01  0. 7 8 6 E + 0 1  0.285E- 0 2  0.218E+00  0. 3 6 2 E + 0 1  0 . 1 5 1 E - 02  0,844E-01  0. 3 6 2 E + 0 1  0.624E-01  0.605E+01  0.2 2 5 E - 0 2  0.755E-02  0. 2 8 1 E + 0 1  0.129E- 0 2  0. 4 4 4 E - 0 2  0. 3 1 0 E + 0 1  0.137E- 02  0, 3 5 8 E - 0 2  0. 4 4 5 E + 0 1  0.175E- 02  L/D 69.8  r  OMEGA ( R A D / S E C ) 0.2327  0.151E- 02  222  TABLE VARIOUS  ORDERS  ( R , N ) AND  ORDER <M,N)  ZERO-FIRST ZERO-SECOND FIRST-ZERO FIRST-FIRST  ORDER  13 RESULT  OF  LEAST  SQUARES F I T  RESULTANT KF/B  PARAMETERS B/A  SUM O F SQUARES  0.275E+01  0. 1 6 7 E + 0 2  0. 16 8 E -•0 1  0.236E+00  0. 1 6 7 E + 0 2  0 . 1 6 8 E - •0 1  0.  0. 1 6 7 E + 0 2  0.168E- 0 1  0.101E+00  0. 2 8 8 E + 0 2  0.222E- 0 1  0.108E-01  0. 8 9 6 E + 0 1  0.105E- 01  0.808E-02  0. 1 0 3 E + 0 2  0 . 1 1 9 E - •0 1  0.688E-02  0. 1 4 0 E + C 2  0.151E- 01  ORDER ORDER ORDER  FIRST-SECOND SECOND-ZERO  F -  ORDER ORDER  SECOND-FIRST  ORDER  SECOND-SECOND  ORDER  P H I (CC/CC) 0. 1 4 0  L (MM) 0.987  132E+00  L/D 22. 9  OM E G A (R A D / S E C ) 0.6981  223  TABLE VARIOUS  ORDERS  (M,N)  ORDER (M,N)  ZERO-FIRST  ORDER  ZERO-SECOND FIRST-ZERO  ORDER ORDER  FIRST-FIRST  ORDER  FIRST-SECOND SECOND-ZERO  ORDER ORDER  SECOND-FIRST  ORDER  SECOND-SECOND  ORDER  P H I {CC/CC} 0.140  L (MM) 0.987  F -•  14  AND R E S U L T  OF L E A S T  SQUARES F I T  RESULTANT KF/B  PARAMETERS B/A  SDM OF SQUARES  0. 1 9 5 E + 0 1  0.732E+01  0.133E-02  0. 1 2 4 E + 01  0.177E+02  0.142E-02  0. 1 9 3 E + 0 0  0.732E+01  0. 1 3 3 E - 0 2  0.933E-01  0.732E+01  0.133E-0 2  0. 7 1 9 E - 0 1  0.121E+C2  0. 1 3 6 E - 0 2  0.749E-02  0.4 9 7 E + 0 1  0. 1 2 4 E - 0 2  0.518E-02  0.555E+01  0. 1 2 8 E - 0 2  L/D 22.9  OMEGA ( R A D / S E C ) 0.3491  TABLE VARIOUS  ORDERS  ( M , N)  ORDER <H,N)  ZERO-FIRST  AND  15 RESULT  OF  LEAST  SQUARES F I T  RESULTANT KF/B  PARAMETERS B/A  SUM O F SQUARES  0.171E+00  0.576E+01  0.171E-02  0.523E-01  0.576 E+C1  0.17IE-0 2  0.383E-01  0 .1 0 5 E+ 0 2  0.174E-02  0.536E-02  0. 4 7 2 E + 0 1  0.170E-02  0.26 4E-02  0.521E+01  0.171E-02  ORDER  ZERO-SECOND FIRST-ZERO  F -  ORDER ORDER  FIRST-FIRST  ORDER  FIRST-SECOND SECOND-ZERO  ORDER ORDER  SECOND-FIRST  ORDER  SECOND-SECOND  ORDER  PHI(CC/CC) 0.140  L(MM) 0.987  L/D 22. 9  OMEGA(RAD/SEC) 0.1 1 6 4  Appendix  G-l.  G  of  In  2-2  Section  Expressions various  m a t e r i a l have  terms  present has  Models  Evaluation  dependent in  Decay  of  shear  stress  experiment.  been  been  shown i n  for  given.  this  with  equation (x  of  viscosity  Denny  (slope  of  a  shear  viscosity,  so  that  the  their  section  curves  they  are  measured  Hahn,  Ree,  of  in  and  time-  compared the E y r i n g ^ ^  (2-1)  x oo  the  convert  +  t  co  B r o d k e y  to  Decay  as  and  impossible  Stress  s t r u c t u r a l decay  a t t r i b u t e d to  T ) e "  u  theory  In  torque-decay  (2-1) -  0  Shear  expressions  The model  T =  The  Predicting  w  a s  stress  data  of  theory  expressed  vs.  the  could  shear  present not  be  in  rate  terms  of  curve).  experiment evaluated  point It  into  here.  was  point The  (41) model  described  viscosity;  for (n  1  Apparent  0  by  8  1  (n  0  Pinder  example, +  e  -  n)(n  second e  e  is  his  e  +  n ) y  related  shear  the  equation  (2-5)  given  in  order-zero  terms order  of  apparent  equation  as  2  =  to  n = Hence,  been  ~ n )  n )(n  viscosity  has  2^03  "eV  shear  stress  (  by  equation  (3-6),  "  5  )  i.e., (3-6)  T/Y  can be  2  r e - w r i t t e n under  a  constant  rate  of  as (T log  1 0  7  +  T ) (TO - T  oo  <  T  - T  O O  r-j )(T  )  oo  u  O Q  +  r = k T )  N  t  G-l)  0  where  k  For  the  other  order  of  = 2303 T  reactions, 225  k  f  similar  ( G _ 2 )  expressions  can  be  226 obtained which  have  the  following  logi [f(T)]  =  0  form  c  t  (G-3)  (43) The model  of  structure  stress  current  Ritter  and  in  notation  equation  ln  T  o  the  and  -  and  TO  =  x  4.00 10 the of  tions  ten  l n ( — P  A  A  TQ  - T  expressions  and  one  by  (eight  T)  in  Table the  +  1.0  t +  of the  1)  (G-4)  data of  for  from the  Appendix  C.  to  the  decay  G-l.  As  far  as  second  order-zero  be  the  in  10%  L/D  curve  was  method  PEG-H^O)  Pinder's order  of  the  the best  values  of  f i t  concerned, i t  i t  of  obtained and  the  are is  the  equaof  programme  square  because  time  the value(s)  computer  least are  each  the  aging  equations  by  nylon  in  The  the  fitted  The  models  is  a l l  with  of  69.8) with  Hahn,  here  which were  2  evaluating  equations. results  in  by  slurry  ratio  dynes/cm  3  one  compared  a r t i f i c i a l  particle  constant  square  the The  were  2.98xl0  set  of  Govier)  for  Pinder,  v e l o c i t y 0.233 rad./sec. =  least  each  by  mole NaCl  and  2  run were  b y means  listed  terms  re-written with  equations  and  obtained  angular  dynes/cm  t f  decay  (t,  in  0  Ritter  data  (Dex  constant  parameter(s)  that  k  o  medium  hours  set  can be  ( p a r t i c l e concentration 0.08,  20  from  and  (2-7)  expressed  T  Eyring,  dispersing  in  =  experimental decay  fibers  been  - T- 0  Actually Ree,  has  as  T — T  T  Govier  is shown found  gives  the  least  (41) value  of  sum  B r o w n ^ , models also  of  and  (Harn,  shown i n Fig.  G-l  squares.  Brown Ree,  and and  Table shows  This  conclusion  P i n d e r ^ \ Eyring  The  [H-R-E],  agrees  results  and  with for  R i t t e r and  Pinder  the  ,  other  Govier  two  [R-G])  are  G-l. a comparison  of  Pinder's  second  order-zero  order  227 TABLE G - l PARAMETERS (j) = 0 . 0 8 ( c c / c c ) ,  L = 3.01 (mm), L/D = 6 9 . 8 , Q = 0.233 ( r a d / s e c )  DEX + 1 . 0 m o l e N a C l  Reaction Order  OF VARIOUS DECAY MODELS  i n 1 0 % PEG-H 0 o  Sum o f Square  Parameter  0-lst 0-2nd  k Y f  r  lst-0 k  lst-lst  f1  _2  (g/cm.sec)  7. 9 0 x l 0  2  4. 7 8 x l 0  (sec)-2(g/cm.sec)  5. 9 9 x l 0  2  1. 52  (sec)  2. 1 4 x l 0 ~  2  (sec)  (sec)  ^  1. 9 8 x l 0 ~  2  (sec)  ^  5. 7 l x l 0 ~  2  7. 4 7 x l 0 ~  7  "'"  7. 4 1 x l 0 ~  7  ^  7. 3 8 x l 0 ~  k  lst-2nd  fI  k  f  2nd-0  k /Y  (g/cm.sec)  2nd-lst  k /Y  (g/cm.sec)  2nd-2nd  k /Y r a  (g/cm.sec)  f  f  f  H-R-E*  *Model **Model  by Hahn, by R i t t e r  _ 1  4. 7 8 x l 0  - 1  1  1. 0 5 x l 0 "  2  1. 1 3 x l 0 ~  2  7  1. 2 2 x l 0 ~  2  2. 1 4 x l 0 "  2  4. 7 8 x l 0  (-)  3. 2 1 x l 0  -  1  (sec)  3. 1 9 x l 0  _ 1  (sec)  k^/Po  4. 7 8 x l 0  4 . 21X10  ^  9. 8 2 x l 0 ~  Ree, and E y r i n g and Govier  _ 1  "\  R-G** 1  _ 1  [equation  [equation  (2-1)]  (G-4)]  J  4  LOG10 SHEAR STRESS (DYNES / SQ.CM.)  o 3 M  3.362  3.632  3.903  4.173  4.444  4.  J  o  ^  Q  I  to M  X  V  CD i-i H-  3 CD 3 rt  — I  o  o  3  fu H HCO O  3  CD  1  °D C O Bi  VI  o  co  s;2  CD CD  3  cn cn  o CD o  Cu  • V!  I  CD o  TJ  m  <  Q. CD  —»  m  m  m  C Q  c  Q  o  O  o"  -—«  -—-  ,„  |  1  CD*  JQ  C J CO P rt r t i-i &) CD  a a —*-  -Ho  m  CD  ^ w. i-i  O  O  -Q  CD  x m X  T3 CD  3 CD  3  o Q  ro  m  i  ro •  CD  cn cn  OJ  O  —,  o  w 3  - H-  o 3 cn  s « o  cu p.  «  o  „ o CD O  _  ~ 5 TJ  m CD i X  cn  LO"  2ZZ  ro O  o P .. O CD CO CD co o o o o  I  229 m o d e l , Hahn, Ree, and E y r i n g ' s m o d e l , a n d R i t t e r with are  the experimental  One t h i n g common  assumption earlier  of stress  i n Chapter  viscometer employed  i s a very  that  instruments  the shear  time.  change w i t h definitely angular  throughout cylinder rate  task.  o f the Couette  because  velocity t h e decay  of this  difficulty constant decay  with  assumption  F i g . G - l which  experiment,  the other  correct  introducing  order  i s the  As mentioned  r a t e f o r Couette have  annular  i n their space  gap  experiment  throughout  the  i s doubtful f o r the present  Therefore,  stress  c a n be expanded and G o v i e r ' s  decay models.  cannot  The p r e d i c t i o n  exerted  constant on t h a t  o f constant  shear  to that of constant m o d e l seems t o h a v e a n In b r i e f ,  i s that they  easily  i s the best.  h o w e v e r , was k e p t  I f the assumption  decay models  r a t e which  i t i s not f a i r to  one o f t h e e x p r e s s i o n s  and t h e s h e a r  time.  stress  the stress  experiment.  assumed  of the inner cylinder,  therefore, Ritter  t r u e shear  model  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 was f o u n d t o  was m e a s u r e d w i t h  velocity,  rate.  a thin  the annular  i n the viscometer.  say from  advantage over  type with  over  i n t h e models examined here  angular  shear  H e n c e , many w o r k e r s  have simply  rate i s constant  time  t h r e e models, however,  decay under a constant  and t h e y  The v a l i d i t y  experiment,  among t h e s e  difficult  model  R-G m o d e l i s t h e t w o - p a r a m e t e r  3 the estimation of true shear  between c y l i n d e r s ,  the  P i n d e r ' s m o d e l a n d H-R-E  t h e one-parameter models, w h i l e  model.  The  decay data.  and G o v i e r ' s  be kept  t h e main  introduce the constant  i nthe  by a l l t h e models, however, a r e o f  o f m a g n i t u d e and c a n p o s s i b l y  an a p p r o p r i a t e measure o f shear  be f u r t h e r  rate.  improved by  230 G-2.  A Modification to As  a  first  Their  discussed  earlier  order-second  basic  rate  TQ is  backward network  the  rate  equation  structure. their  i n i t i a l  They  and  the  must  orders  equation  be  [k  -  f T  of  \ ( T  T , k^  Model  Ritter  k^  i n i t i a l  that  the  first  on Pembina orders  Q  and  the  determined is  Decay  reaction  respectively,  the  (G-5)  2-2  PQ i s  found  assume  Section  t  value  experimental data to  Govier's  reversible  H  +  0  and  and  for  Govier  the  (43)  decay  assumed  equation.  was  constants,  reasonable  fore  P  structure,  in  order  dt where  Ritter  T )  k^  are  is  (G-5)  ]  the  the rate  order-second o i l .  e m p i r i c a l l y by to  forward of  take  order  However,  reversible  now m o d i f i e d  2  concentration of  crude  of  "  and  increase network model i t  is  fits not  r e a c t i o n a priori  and  experimental data. into  account  of  There-  various  orders  as dr  _  =  dt where  m is  the  reaction.  (m,  order  (G-6) 0  of  Equation  analytically 1.  P  forward  (G-6)  with various  n)  =  (0,  (m,  n)  = T  (0, —  (m,  = T  log  the  a d i f f e r e n t i a l equation  orders  =  of  m and  n.  The  order and  of  backward  can be  solutions  solved  are  const  for  any  (G-7)  t  T  Tn  n)  is  1)  log  i i i .  n  0) T  i i .  is  r e a c t i o n and  (0, +  T  T  -  -  kit  T  t  log  +  (G-8)  1  2) . -  T  ,2tn T  0  " ™ T  K  log  t  +  1  (G-9)  231 iv.  (m, n ) =  (1,  T -  0)  T  log |  v.  0  T  T  t +  log  k4  o  (G-10)  1  (m, n ) = ( 1 , 1) T  log vx.  -  (m,  -  T "  Tn  -  Tg  T  t +  log  k  T  (G-ll)  1  L  n) = ( 1 , 2) T  -  T  log T  T  This  -  0  T  -  T  log  t +  (G-12)  1  TT  is  0  +  0  t h e o r i g i n a l model by R i t t e r  and G o v i e r a n d ' h a s been  .given i n equation (2-7). vii.  (m, n ) =  log  —  T  TO +  T  T  +  T  t(i  T  oo  (m, n ) =  log  0)  T  1  viii.  (2,  2 2  "  (2,  T  2  T  log  oo  t +  (G-13)  1  1) 2x  + b - D + b + D  T  = -  u  2  0  + b + D  T q  + b - D  log ~— t +  (G-14)  1  where  (G-15)  b =  c =  (G-16)  -  T  D = /b  and  ix.  (m, n ) =  log  (2,  - 4c =  (2x Tn  n  -  - x  (G-17)  T  2)  2a-_ + b - - D 2a + b + D T  2  2a  T n  2ax  n  + b + D + b - D  log  t +  1  (G-18)  232  where  2 a  -  = ( T  "  0  T . . )  .  b  T  T  c  =  T  )  "  0  T T  )  (G-21)  2 0  2  CO  U  2T  and  D = /b  -  2  4ac  of  the  solutions  above  T = T  and  T  =  are  (m, be  n)  view  is  of  not  the  (G-22) 0  at  T  t  =  different  orders  of  T  R  O  with  the  boundary  conditions  of  t  . CO  concepts  For  "  a t t = 0  O  necessarily  acceptable.  0  obtained  CO  In  T  = T  A l l  (G-20)  0  2  2  ( T  t  _ \2  "  n  (G-19)  2  = - 2 — ^ ( x 0  1  :  the  an  of  reaction  integer,  sake  of  rate  but  any  real  mathematical  (m,  n)  from  (0,  1)  interesting  to  note  that  to  the  (2,  order  of  the  reaction  number  can  possibly  simplicity,  the  eight  2)  have  been  considered  here. It  is  a l l  the  equations  are  of  the  form  of  f = K£k  log|f(x)|  log  k^ p  where  K  (k^/k^) the  is  a  and  least  programme  that  (k^/Pg).  square is  Table found  constant.  method,  given G-2  the  in  shows second  (G-23)  1  equation two  has  two  parameters  minimizing  Appendix an  +  0  Each These  t  fitting  were  the  sum  of  the  results  parameters  evaluated  squares.  by The  means  of  computer  C.  example  order-zero  of  order  model  is  of  the  f i t .  better  than  It any  was other  233  TABLE G-2 VARIOUS ORDERS OF GOVIER'S MODEL AND RESULTS OF LEAST SQUARES FIT  „ , r  (m, n)  R e s u l t a n t Parameters 2 1-m k /kit (dyne/cm ) kit/Po  -1 (sec)  3.03xl0  0-lst  1.34X10  4  0-2nd  8.43xl0  3  lst-0  3.62xl0  lst-lst  3 . 3 5 X 1 0 "  1  1st-2nd  3 . 2 1 X 1 0 "  2nd-0  1.83xl0"  2nd-lst  1 . 8 1 X 1 0 "  2nd-2nd  1.79xl0~  5 . 2 0 X 1 0  _ 1  _ 1  -  1  3.03X10"  1  Sum o f Squares  8.86xl0~  4  2.23xl0~  3  8.86X10"  4  8 . 8 6 X 1 0 "  3 . 0 3 X 1 0 "  1  1  3 . 1 9 X 1 0 "  1  9 . 8 2 X 1 0 "  5  1 . 2 5 X 1 0 "  1  2.98xl0~  5  5  1 . 2 6 X 1 0 "  1  1 . 2 7 X 1 0 "  1  4  4  5  3 . 1 8 X 1 0 "  5  3.41xl0~  5  <j) = 0.08 ( 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 0 9  orders  including  order  model  obtained  terms (2,  listed  the  sum  of  The (G-13),  gives for  has G-2  (G-13),  T  two  — +  the  =  in  G-l  as  Appendix  H.  It  approximation  best  second  time-dependent order  first  R-G.  carried  be  concluded  order-zero  decay  order  predicting  a r t i f i c i a l  results  runs  can  in  order-second  Similar  experimental  the  T T  Tn  CO  T[) "  no  out  were  in  this  that,  [(m,  n)  the  shear  in  =  slurries.  e q u a t i o n was  effect other  curves  asymptotically from  the  the  rate the  parameter  other  parameter are  T  shown  in  equation  0  of  =  that  parameter (k^/Pg)  in  l.OxlO  greater By  was  parameter  comparing  (k^/Pg),  Q  Fig.  G-2  kept  with  equation  constant  equation  were  set  follows:  dyne/cm  3  yield  similar  the  model  -1  Q  ~* t o  the  in  /P (sec)  as  2  greater  those  the  [ +  1.0x10  the  varied  kept  k  from  stress  was  on  and  dyne/cm -  common  A  2 -1 )  2  .  stress  Tg and  at  T  t  It  p a r a m e t e r (k^/kt^)-, t h e plot  constant  from  0.05  Fig.  G-2.  is  shown  (k^/B to  =  in  Fig.  3x10  "*)  50.0. It  The  can be  can  faster G-3,  and  values  seen  be  that  the of the  ,  faster  Fig.  approach  = t^.  _  the  (G-13)  1  varied  equilibrium  was  +  (k^/k^)  was  the  4  l.OxlO  the  (k^/k^)  t  log  (dyne/cm  (k^/k^  decay.  same as  1  k^/^.  the  =  T  f  T-  T  °° k^  oo  u  2  o  from  (k^/Pg)  the  -  constants  common  stress  where  and  decay  figure  of  T  parameter  various  the  =  parameter  and four  k  c  parameters,  The  The  T  u  the the  +  V  T  Tn  other  order-zero  5.0].  4.0x10  is  Table  Govier's  the  fitting  shows where  [k^/Pg  seen  in  and  i.e.,  iT  All  Ritter  squares,  the  second  log  Fig.  a l l  are  decay  which  earlier  almost  model  stress  original  They of  0)]  listed  for  work.  the  G-3  is  the  rate  i t  can be  of  stress  concluded  decay.  that  the  £  k /P =5.0 4  T T  ^  +  k /k =  ^  ^  ^  0  4  (dynes/cm )  co= I.Ox I O  3  (dynes/cm )  x  2.0  xlO"  0 *f  3.0 4.0  xld^ xlO  1 7.143  -10.0  Fig.  G-2.  Effect  2  2  ^  x IO (dyne,/ c m ) "  4  1  = 1.0 x I O  0  1.0  f  (sec )  5  2  1  5  5  x  1 24.286  of Parameter  1 41.429  1 58.571  TIME (SEC)  (k /k^) f  on t h e Decay  Model  1 ' 75.714  1 92.657  110  k /k = 3x I0" (dyne / cm )" 5  2  f  +  T = I.Ox I 0 0  "+++-  X  +  k /P = 4  0  X  3  (dynes/cm ) 2  (dynes/cm ) 2  0 . 0 5 (sec" ) 1  0.5  O  5.0  "r  50.0 ~1 24.286  -10.0  Tco=I.Oxl0  4  ~l •7.143  F i g . G-3.  X T  41.429 TIME  T  58.571 ISEC)  Effect of Parameter (k^/V ) on the Decay Model  "I 75.714  -T 92.857  110  (k^/ki+)  parameter than is  exponent  (k^/Pg)  is  Fig. fitting  just G-4  the  and  particle  L/D  respectively. errors  that  concluded and  G-7  ratio  on  The  L/D  ratio,  and  show t h e  to  because  equation the  the rate  e f f e c t of  stress  while  the  (k^/Pn), the  angular  It  velocity,  the  is  the  found greater  (ki+/Po). e f f e c t of  particle concentration  scatter  in  the  k^/k^,  d a t a may b e  and  that  the higher  are  the  and  k^/Po,  e x p l a i n e d by  experimental reproducibility.  smaller  parameter  v e l o c i t y on  respectively.  large  figures  decay  (k^/k^)  parameter  angular  parameters,  the  of  t.  fitting  the  the  (G-13),  the  with  from these  and  is  the higher  (k^/kit)  and  associated  show t h e  (k^/k^)  figures  G-6  This  t-term in  G-5  the parameters Fig.  sensitive  a m u l t i p l i e r of  parameters  from these  tion  of  much more  (kij/Po).  the parameter  an  are  is  It  can  the be  the p a r t i c l e concentra-  parameters  (kp/ki+ •  and  ki+/Po)  Second Order - Zero Order Parameter k / k f  > 3 C  5*  P  -?  ro  cf O  o  co CD  o  o  *""-** CD  2  4  -I  -4  ( d y n e / c m ) (xlO )  Second Order-Zero Order Parameter k / P 4  —r-  0  oo "T~  (sec)  (x  ro  , r f c , c  CD  ~T~  ° o p + " CD "j> ro oo  ^ O ro  o °  « 3 2.  a  O  O  O TJ  O  m o i X PO  o  o  O  Second Order - Zero Order Parameter k / k f  2 -I  4  (dyne/cm ) (xlO  -5  )  241  L= 0.987 (mm) & = 0.233 (rad./sec.) DEX + 1.0 mole NaCl in 1 0 % P E G - H 0  •  2  O I  o  v  T CO  or  \  I  w Q>  •  \ \  0)  XJ  £ o o a.  c  o o if)  37.4 69.8  \  o  N  = 22.9  •  X  •o O o  L/D  G  o  \  \  Q  0.05  0.10  0.15  Particle Concentration (cc/cc) Fig.  G-7.  Effects on  of  Particle  the Parameter  Concentration  (k./P»)  and  Particle  L/D  Ratio  Appendix  H  Various Model  All dextrose  the +1.0  Description  of  data  Orders  and in  of  Sum o f these  Modified  Ritter  and  Govier's  Decay  Squares  tables  mole NaCl  in  10%  Variables  in  Table  are  for  the  dispersing  medium  PEG-H 0. 2  H  PHI(CC/CC)  -  Particle  Concentration  L(MM)  -  Particle  Length  L/D  -  Particle  Length-to-Diameter  OMEGA(RAD/SEC)  -  Angular  Velocity  242  (cc/cc)  (mm)  of  Inner  Ratio  Cylinder  (-) (rad./sec.)  of  TABLE VARIOUS  ORDERS  OF  ORDER  ZERO-SECOND FIRST-ZERO  ORDER ORDER  FIRST-FIRST  ORDER  FIRST-SECOND SECOND-ZERO  ORDER ORDER  SECOND-FIRST  1  G O V I E R M O D E L AND R E S U L T LEAST SQUARES F I T  ORDER  ZERO-FIRST  H -  ORDER  SECOND-SECOND  ORDER  P H I (CC/CC) 0.160  L (MM) 0.987  OF  RESULTANT KF/K4  PARAMETERS K4/P0  0.598E+05  0.2 8 4 E + 0 0  0.872E-01  0.288E+05  0.527E+00  0.755E-01  0. 1 5 1 E + 01  0.2 8 4 E + 0 0  0.872E-01  0. 1 2 8 E + 0 1  0.2 8 4 E + 0 0  0.872E-01  0.110E+01  0.322E+00  0.844E-01  0, 1 1 6 E - 0 3  0. 8 8 4 E - 01  0.9 9 5 E - 0 1  0.106E-03  0.933E-01  0.1O0E+00  0.970E-04  0. 1 0 0 E + 0 0  0. 1 0 0 E + 0 0  L/D 22. 9  OMEGA ( R A D / S E C ) 0.2327  SUM O F SQUARES  TABLE H -  2  VARIOUS ORDERS OF GOVIER MODEL AND RESULT OF LEAST SQUARES FIT ORDER  RESULTANT PARAMETERS KF/K4 K4/P0  SUM OF SQUARES  ZERO-FIRST ORDER  0. 129E+05  0.2 95E+0 0  0.274E-0 1  ZERO-SECOND  0.669E+04  0.589E+00  0.398E-01  FIRST-ZERO ORDER  0.803E+00  0.295E+00  0.274E-01  F I R S T - F I R S T 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  0.338E-03  0.319E-01  0.497E-02  0.317E-03  0.341E-01  0.538E-02  ORDER  ORDER  SECOND-FIRST ORDER SECOND-SECOND  ORDER  PHI (CC/CC) 0.150  L (MM) 0.987  L/D 22.9  OMEGA (RAD/SEC) 0.2327  T A B L E fl VARIOUS  3  O R D E R S O F G O V I E R M O D E L AND R E S U L T LEAST SQUARES F I T  ORDER (M,N)  ZERO-FIRST  ORDER  ZERO-SECOND FIRST-ZERO  ORDER ORDER  FIRST-FIRST  ORDER  FIRST-SECOND SECOND-ZERO  ORDER ORDER  SECOND-FIRST  ORDER  SECOND-SECOND  ORDER  PHI (CC/CC) 0.140  L (MM) 0.987  OF  RESULTANT KF/K4  PARAMETERS K4/P0  0.471E+04  0.529E+00  0.593E-02  0.258E+04  0.103E+01  0. 1 3 1 E - 0 1  0.626E+00  0.529E+00  0.593E-02  0.606E+00  0.529E+00  0.593E-02  0.590E+00  0.545E+00  0.619E-0 2  0.440E-03  0. 8 C 3 E - 0 1  0.545E-04  0.437E-03  0. 8 0 8 E - 0 1  0.558E-04  0.435E-03  0. 8 1 2 E - 0 1  0. 5 7 3 E - 0 4  L/D 22. 9  OMEGA ( R A D / S E C ) 0.2327  SUM O F SQUARES  246  TABLE VARIOUS  ORDERS  OF  ORDER  ZERO-SECOND FIRST-ZERO  ORDER ORDER  FIRST-FIRST  ORDER  FIRST-SECOND SECOND-ZERO  ORDER ORDER  SECOND-FIRST  4  G O V I E R M O D E L AND R E S U L T LEAST SQUARES F I T  ORDER |M,N)  ZERO-FIRST  H --  ORDER  SECOND-SECOND  ORDER  PHI(CC/CC) 0.130  L (MM) 0.987  OF  RESULTANT KF/K4  PARAMETERS K4/P0  0.530E+03  0.349E+00  0.1 1 1 E - 0 1  0.32 5E+03  0.616E+00  0.221E-01  0.388E+00  0.3 4 9 E + 0 0  0.111E-0 1  0.351E+00  0.349E+00  0.111E-01  0.331E+00  0. 3 7 4 E + 0 0  0.122E-01  0.535E-03  0. 1 2 2 E + 0 0  0. 2 0 0 E - 0 2  0.526E-0 3  0. 1 3 5 E + CO  0.117E-02  0.519E-03  0.138E+00  0.126E-0 2  L/D 22.9  OMEGA(RAD/SEC) 0.2 3 2 7  SUM OF SQOARES  247  TABLE VARIOUS  ORDERS OF  ORDER  ZERO-SECOND FIRST-ZERO  ORDER ORDER  FIRST-FIRST  ORDER  FIRST-SECOND SECOND-ZERO  ORDER ORDER  SECOND-FIRST  5  G O V I E R M O D E L AND R E S U L T L E A S T SQUARES F I T  ORDER  ZERO-FIRST  H -  ORDER  SECOND-SECOND  ORDER  PHI(CC/CC) 0. 1 2 0  L (MM) 1.620  OF  RESULTANT KF/K4  PARAMETERS K4/P0  0.296E+05  0.429E+00  0.106E-02  0.146E+05  0.861E+00  0.160E-02  0.11OE+01  0.429E+00  0. 1 0 6 E - 0 2  0.977E+00  0.429E+00  0.106E-02  0.877E+00  0.477E+00  0. 1 1 3 E - 0 2  0.18 1E-03  0.798E-01  0.237E-03  0, 16 9 E - 0 3  0.841E-01  0.252E-03  0. 1 5 9 E - 0 3  0. 8 9 5 E - 01  0.271E-03  L/D 37. U  OMEGA ( R A D / S EC) 0.2327  SUM O F SQUARES  248  TABLE H -• VARIOUS OBDEBS  6  OF GOVIER MODEL AND RESULT OF LEAST SQUARES F I T  ORDER (H,N)  RESULTANT PARAMETERS KF/K4 K 4/P 0  SUM OF SQUARES  ZERO-FIRST ORDER  0.599E+04  0. 3 94E+00  0.337E-02  ZERO-SECOND  0.322E+04  0.778E+00  0.843E-02  FIRST-ZERO ORDER  0.664E+00  0.3 94E+00  0.337E-02  F I R S T - F I R S T 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  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  ORDER  ORDER  L/D 37. 4  OMEGA(RAD/SEC) 0.2 327  249  TABLE H -  7  VARIOUS ORDERS OF GOVIER MODEL AND RESULT OF LEAST SQUARES F I T ORDER {M, N)  RESULTANT KF/K4  PARAMETERS K4/P0  SUM OF 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) 0.100  L (MM)  L/D  1.620  37.4  OMEGA (R AD/SEC) 0.2327  250  TABLE H -  8  VARIOUS ORDERS OF GOVIER MODEL AND RESULT OF LEAST SQUARES FIT ORDER (M,N)  RESULTANT PARAMETERS KF/K4 K4/P0  SUM OF 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) 0.090  L (MM) 1.620  L/D 37.4  OMEGA (RAD/SEC) 0.2327  251  TABLE VARIOUS  O R D E R S OF  H -  9  G O V I E R MODEL LEAST SQUARES  ORDER  AND R E S U L T FIT  OF  RESULTANT KF/K4  PARAMETERS ,K4/P0  ZERO-FIRST O R D E R  0.134E+05  0.303E+Q0  0.886E-03  ZERO-SECOND  0. 8 4 3 E + 0 4  0.520E+00  0.223E-02  0.36 2E+00  0.3 0 3 E + 0 0  0.886E-03  0.335E+00  0.3 0 3 E + 0 0  0.886E-03  0.321E+00  0.319E+00  0.982E-0 3  0. 1 8 3 E - 0 4  0,125E+00  0.298E-04  0.181E-04  0. 1 2 6 E + 0 0  0.318E-0 4  0. 1 7 9 E - 0 4  0. 1 2 7 E + 0 0  0.34  (H.N)  FIRST-ZERO  ORDER ORDER  FIRST-FIRST  ORDER  FIRST-SECOND SECOND-ZERO  ORDER ORDER  SECOND-FIRST  ORDER  SECOND-SECOND  ORDER  PHI (CC/CC) 0. 0 8 0  L (MM) 3.010  L/D 69. 8  O M E G A ( R A D / S EC) 0.2327  SUM O F SQUARES  IE-0 4  252  TABLE VARIOUS  ORDERS  OF  G O V I E R M O D E L AND R E S U L T LEAST SQUARES F I T  ORDER  ZERO-FIRST  ORDER  ZERO-SECOND FIRST-ZERO  ORDER ORDER  FIRST-FIRST  ORDER  FIRST-SECOND SECOND-ZERO  ORDER ORDER  SECOND-FIRST  H -• 1 0  ORDER  SECOND-SECOND  ORDER  P H I (CC/CC) 0.070  L (MM) 3.010  OF  RESULTANT KF/K4  PARAMETERS K4/P0  SOM O F SQUARES  0.899E+04  0.581E+00  0.33 4E-0 2  0.551E + 04  0. 1 0 5 E + 0 1  0.118E-01  0.367E+00  0.581E+00  0.334E-02  0.352E+00  0 .5 81 E + 0 0  0.334E-02  0.342E+00  0.600E+00  0.363E-02  0.384E-04  0.168E+00  0.139E-04  0. 3 8 2 E - 0 4  0.169E+00  0.143E-04  0.381E-04  0. 1 7 0 E + 0 0  0.149E-0 4  L/D 69.8  OMEGA(RAD/SEC) 0.2327  253  TABLE VARIOUS  O R D E R S OF  ZERO-SECOND FIRST-ZERO FIRST-FIRST  ORDER ORDER ORDER ORDER  FIRST-SECOND SECOND-ZERO  ORDER ORDER  SECOND-FIRST  11  G O V I E R M O D E L AND R E S U L T LEAST SQUARES F I T  ORDER <M,N)  ZERO-FIRST  H -  ORDER  SECOND-SECOND  ORDER  PHI(CC/CC) 0.060  L(MM) 3.010  OF  RESULTANT KP/K4  PARAMETERS K4/P0  SUM O F SQUARES  0.848E+04  0 . 1 91 E + 0 1  0 . 1 9 1 E - •0 1  0.469E+04  0. 3 9 9 E + 0 1  0 . 4 0 5 E - 01  0.505E+00  0. 1 91 E + 0 1  0.191E- 0 1  0.490E+00  0. 1 9 1 E + 0 1  0 . 1 9 1 E - 01  0.479E+00  0. 1 97 E + 0 1  0.198E- 0 1  0.158E-03  0. 2 4 0 E + 0 0  0 . 7 4 4 E - 04  0. 1 5 8 E - 0 3  0. 2 4 1 E + 0 0  0 . 7 6 1 E - •0 4  0.157E-03  0. 2 4 2 E + 0 0  0.779E- 04  L/D 69.8  OMEGA(RAD/SEC) 0.2327  254  TABLE VARIOUS  ORDERS  OF  ZERO-SECOND FIRST-ZERO FIRST-FIRST  OF  RESULTANT PARAMETERS SUM OF KF/K4 K4/P0 SQUARES ORDER ORDER ORDER ORDER  FIRST-SECOND SECOND-ZERO  12  G O V I E R M O D E L AND R E S U L T LEAST SQUARES F I T  ORDER (8,N) ZERO-FIRST  3 -  ORDER ORDER  SECOND-FIRST  ORDER  SECOND-SECOND  ORDER  P H I (CC/CC) 0.050  L (MM) 3.010  0,450E+04  0.229E+01  0 . 2 2 9 E - 01  0.255E+04  0. 4 8 4 E + 01  0.4 9 0 E - 0 1  0.433E+00  0. 2 2 9 E + 0 1  0.229E- 0 1  0.42 3E+00  0. 2 2 9 E + 0 1  0.22 9 E - 01  0.415E+00  0. 2 3 4 E + 0 1  0.236E- 0 1  0.196E-03  0. 2 9 5 E + 0 0  0.75 4 E - 04  0. 1 9 6 E - 0 3  0. 2 9 6 E * 0 0  0.767E- 0 4  0.195E-03  0. 2 9 7 E + 0 0  0 . 7 7 9 E - 04  L/D 69. 8  O M E G A (R A D / S EC) 0.2327  TABLE VARIOUS  ORDERS  OF  13 OF  RESULTANT KF/K4  PARAMETERS K4/P0  S U M OF SQUARES  0.137E+04  0.193E+01  0.262E-0 1  0. 7 5 6 E - 0 3  0. 1 5 4 E + 0 0  0.242E-02  0. 7 3 7 E - 0 3  0, 1 5 8 E + 0 0  0.248E-02  0.720E-03  0. 1 6 2 E + 0 0  0. 2 5 5 E - 0 2  ORDER  ZERO-SECOND FIRST-ZERO  -  G O V I E R M O D E L AND R E S U L T LEAST SQUARES F I T  ORDER <H,N)  ZERO-FIRST  H  ORDER ORDER  FIRST-FIRST  ORDER  FIRST-SECOND SECOND-ZERO  ORDER ORDER  SECCND-FIRST  ORDER  SECOND-SECOND  ORDER  PHI(CC/CC) 0.140  L(MM) 0.987  L/D 22. 9  OMEGA(RAD/SEC) 0.6981  256  TABLE VARIOUS  O R D E R S OF  ORDER  ZERO-SECOND FIRST-ZERO  ORDER ORDER  FIRST-FIRST  ORDER  FIRST-SECOND SECOND-ZERO  ORDER ORDER  SECOND-FIRST  14  G O V I E R M O D E L AND R E S U L T LEAST SQUARES F I T  ORDER (M,N)  ZERO-FIRST  H -  OF  RESULTANT KF/K4  PARAMETERS K4/P0  SUM O F SQUARES  0. 2 6 5 E + 0 4  0. 5 1 8 E + 0 0  0. 2 2 8 E -•0 2  0.143E+04  0. 1 0 2 E + 0 1  0. 6 3 3 E - 0 2  0.670E+00  0. 5 1 8 E + 0 0  0. 2 2 8 E - •0 2  0.628E+00  0. 5 1 8 E + 0 0  0. 2 2 8 E - 0 2  0.595E+00  0, 5 4 9 E + 0 0  0. 2 5 2 E - 0 2  0.712E-03  0. 9 5 8 E - 0 1  0. 3 5 9 E - •04  0.686E-03  0. 9 9 3 E - 0 1  0. 3 9 5 E - 0 4  ORDER  SECOND-SECOND  ORDER  PHI(CC/CC) 0. 1 4 0  L(MM) 0.987  L/D 22. 9  OMEGA(RAD/SEC) 0.3491  TABLE VARIOUS  O R D E R S OF  ORDER  ZERO-SECOND FIRST-ZERO  ORDER ORDER  FIRST-FIRST  ORDER  FIRST-SECOND SECOND-ZERO  ORDER ORDER  SECOND-FIRST  15  G O V I E R M O D E L AND R E S U L T L E A S T SQUARES F I T  ORDER (M,N)  • ZERO-FIRST  H -  ORDER  SECOND-SECOND  ORDER  PHI(CC/CC) 0. 1 4 0  L (MM) 0.987  OF  RESULTANT KF/K4  PARAMETERS K4/P0  0.757E+04  0.455E+00  0.108E-01  0.411E+04  0. 8 9 9 E + 0 0  0.224E-01  0.628E+00  0.455E+00  0.108E-01  0.617E+00  0. 4 5 5 E + 0 0  0.108E-01  0.608E+00  0.463E+00  0. 1 1 0 E - 0 1  0. 4 0 2 E - 0 3  0. 4 6 7 E - 0 1  0.175E-C4  0.401E-03  0.469E-01  0.178E-04  0.400E-03  0. 4 7 0 E - 0 1  0. 1 8 1 E - 0 4  L/D 22.9  OMEGA ( R A D / S E C ) 0.1164  SUM O F SQUARES  258  ATOGAKI Chosha hayaimono  de  ga  1971-nen  7-nen amari  Syushikatei dewa,  Nogami  uke,  o  imadewa y a t t o  Shitencho  no  kawaigatte to,  sono  no  goon wa  Mata,  soshite  Kono  Daigaku deno  no  kanitori,  picnic,  golf,  wasureru  Maishu ni  made,  house,  oyobi  o nagashita ba  san,  jinsei  no  no  Suzuki  o k a r i t e o r e i moshi Suzie  Chung  san,  san,  Toshiaki  no Matsumoto  yoru, ni  tachi  Reiko  sensei  Sakura  golf  Daigaku  osewa no  ni  tehodoki  narimashita. Okura kazoku  Shoji no  soreni  Toshi  Vancouver  1-in  no  okunb  Tokyo san  Daigaku  tachito  Moritaka  tono  Lederer  yoni  party  san  s h i t a m a j a n wa  tennis  mo,  no  niwa,  tokubetsu  Fuke  tanoshinda  sensei  tenkeiteki shiroto  deno  McMaster  ya,  Shinya golf  Waseda  iwayuru  Vancouver  narimashita.  hassei  Orpheum T h e a t r e Vancouver  mo  naidesho.  A m e r i c a de  Toshihiro  sensei  teinei  ga  Tacoma no  nari,  Furumori UBC  dekiru yoni  koto  no, h i t o k o m a n i  kinyobi  shinsetsu  Association  ni  no Maeda  Hirata Akira  yugana  niwa  ni,  JAL no  Takeshi  hakase  gofusai  narimashita.  ya,  Toshiyuki  Yoshiko  Iku,  osewa  irai,  hitobito ni  toji  no  hakase,  oku no  u t s u t t e karawa,  ni  eiga,  kite  H a m i l t o n no  UBC  issho  Seishi  ni  o kiroku  gotoni  Jujo  o hajime,  America Washington-shu  ni  Canada  80-dai  l o n g weekend omoide  ni  Ontario-shu  Ishige  Takamatsu  itadaki,  20-ka  narimasu.  sensei  Tokuni,  Hakasekatei  ni  sugoshita  Yukihisa  narimashita. o  1-gatsu  no  chosha  o  oshiete  itadaita  ya,  Queen  Elizabeth  concert  Singers  de,  no member  issho no  ni  no  yona  Vancouver  Opera  Theatre/Playstage  minasama  de  hiyaase  nimo,  agemasu. Minami  Akiko  san,  Iida  Toshie  mono  san,  soshite  kono  259 Higashigawa ga, o  aisuru  oshiete  Michiko koto  aisareru koto  no  Canada  n i ,  kokoro  Ningen naru  to  ga  jinsei  ikigai  wa n a n i d e s u k a ? "  katsudo  no  katsudo  nimo  Kofukukan  uchino  nai  yodesu.  ga,  sore  Kono  kotonaku Sono  kofuku kono  Sumi,  kibishisa  Kunihiko,  Noriko  hagemashi tachi  OSK  no  to,  Lines  no  Horie  itashimasu.  surukoto  o okiku wakeru  ni  gasshodan dato  ga  shohi  (2)  sosaku  kikuto,  to,  tsugi  no  2-tsu  katsudo katsudo.  sono  naru  no  su-byo  to  koto  (1)  no  katsudo kan (2)  yodesu.  shohi  o kangaeru  mo,  sosaku  omoimasu.  no  yakimono  o utau koto  Soyu yona  omo mono  no  wa  wakarimasen  Yahari,  nandemo  kachi,  n a r a n a i monoda  kekka desu  tanka ya haiku mo,  munashii  wa j i n s e i  no  arimasu.  nanimono  desu.  no  to  tsukimashita.  katsudo  koto  concert de uta  ho  dochira  t o m o n a u kamo  minogashitewa  kiga  dewa n a k u t e , e o kaku  ga  sosaku katsudo  rombun o k a i t e i t e  hodo  no  2-tsu  j u j i t s u k a n ga  katsudo  niwa kaikan  toki,  kono  K o r e r a no  Shikashi,  tsuzukeba yoi no  aruiwa "Anata  k o t a e wa t a i t e i  y o r o k o b i ya manzokukan  owaru  shohi  bakari  koto  (1)  to  rombun wa sono  o kaku  o kaku  katsudo  mashita  soshite  kara himpan n i  mo y o i k a m o w a k a r i m a s e n .  tsukuridasu  n i ,  de  dochiraka n i  itte  wa h o n  mokuteki,  no  tsurasa,  " J i n s e i no m o k u t e k i wa n a n i k a ? "  sorezore  to  tsukuridasu  shi  yorokobi,  tate  omowaremasu;  hitobito ni  rombun  Horie  kara kansha  Oku no  yukoto  namikaze o  no 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  Norihiko  yoikara  no  t a i z a i e h u , Nippon  o morai mashita.  sekaiju  ni  t a c h i niwa, mawari n i  itadaki mashita.  Chosha tegami  to  san  mo,  ga,  o tsukuru  nanimo koto  o yaku  k o t o mo,  subete  rippana  sosaku katsudo  ga  okane  ni  mo, shiroto  sosaku naru  ka  260 d o k a wa b e t s u m o n d a i  ni  iru  sosaku  toki,  mata  soyu  jujitsukan  manzokukan  nodewa  de  nai  shoka.  shite,  ni  soyu  katsudo  koso,  sosaku  katsudochu  o nashitogeta  honto  no  i i r i i no  ni  toki ni  j i n s e i no  zammai n i nomi  natte  ajiwau  ikigai  ga  aru  

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