University of Southern California Dissertations and Theses

Investigations On The Flow Behavior Of Disperse Systems

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Investigations On The Flow Behavior Of Disperse Systems

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Content 72-11,934 KASEM, Ahmed, 1933- INVESTIGATIONS ON THE FLOW BEHAVIOR OF DISPERSE SYSTEMS. University of Southern California, Ph.D., 1971 Engineering, chemical University Microfilms, A X E R Q K Company, Ann Arbor. Michigan THIS DISSERTATION HAS BEEN MICROFILMED EXACTLY AS RECEIVED INVESTIGATIONS ON THE FLOW BEHAVIOR O F DISPERSE SYSTEMS by A hm ed K asem A D isse rta tio n P re s e n te d to the FACULTY O F THE GRADUATE SCHOOL UNIVERSITY O F SOUTHERN CALIFORNIA In P a rtia l F ulfillm ent of the R eq u irem en ts for the D egree DOCTOR O F PHILOSOPHY (C hem ical E ngineering) August 1971 UNIVERSITY O F SOUTHERN CALIFORNIA THE GRADUATE SCHOOL UNIVERSITY RARK LOS ANGELES. CALIFORNIA SOOC7 This dissertation, written by ^ m e d .K a se ,m under the direction of Dissertation Com mittee, and approved by all its members, has been presented to and accepted by The Gradu ate School, in partial fulfillment of require ments of the degree of D O C T O R OF P H IL O S O P H Y Dtmn Dnt* S ep tem b er 4. 1971 DISSERTATION COMMITTEE j C j . . c&a. . . .x ~ PHASE NOTE: So m p it* * hav* I n d ia tin e t p r i n t . F il« * d r*c*lv*d, UNIVERSITY MICROFILMS, TO MY W IFE ii A CKNOW LEDGM ENTS The author acknow ledges with thanks the a s s is ta n c e given by the following individuals and o rg an izatio n s: Dr, C. J. R e b e rt — for his in te re s t, advice, and guidance during the c o u rse of the p re s e n t investigation. The M e m b ers of the D isse rta tio n C om m ittee: Dr. W. R. W ilcox and Dr, M. P. E sc u d ie r — for th e ir c o n stru ctiv e c ritic is m and contributions, M r. H. M a sri — for help w ith the co m p u ter p ro g ram . M rs. A. K asen (the a u th o r's wife) — for p re p a ra tio n of the rough d raft of the m a n u sc rip t and h e r s a c rific e s and enco u rag em en t during the p erio d of the re s e a rc h . The C hevron Oil F ield R e se a rc h Com pany, La H abra, C alifornia — for furnishing data on in te rfa c ia l tension. The D ep artm en t of C hem ical E ngin eering, U n iv e rsity of Southern C alifornia — for the financial support of this w ork. Hi T A B L E O F C O N TE N T S P a g e DEDICATION............................................................................................................ ii A CKNOW LEDGM ENTS....................................................................................... iii LIST O F F IG U R E S ................................................................................................ vii LIST O F T A B L E S ................................................................................................ xii N O M E N C L A T U R E .....................................................................................................xvi A B STR A C T ................................................................................................................ xx C H A PTER I - IN T R O D U C T IO N ................................................................... 1 C H A PT E R II - T H E O R E T IC A L ................................................................... 6 A. P re v io u s W o r k ............................................................................ 6 B. P r e s e n t W o r k ............................................................................... 14 1. F u n d a m en tal C o n c e p t s .................................................... 16 Z. D iffere n tia l E quations of M otion w ith In te rp h a se D ra g ................................................................... 18 3. U n stead y R o ta to ry M o t i o n ............................................ Zl 3, 1 Solution for S m a ll V a lu es of the T i m e ........................................................................30 3 ,2 Solution fo r L a rg e V a lu es of the T i m e ....................................................................... 33 4. D e te rm in a tio n of the T an g e n tia l S tr e s s e s . . . 36 4. 1 S m all V alues of the T i m e .................................. 38 4. Z L a rg e V alu es of the T i m e ................................. 40 5. N o n -D im en sio n al F o rm of the D eveloped E q u a t io n s ............................................................................... 42 iv P ag e 5. 1 V elo city D istrib u tio n ........................................ 43 5. 2 W all S h ear S t r e s s .............................................. 44 6. D e riv a tio n of the C oefficient of V isc o sity of the M ix tu re ...................................................................... 45 6. 1 A G e n era l E x p re s s io n fo r M ix tu re V i s c o s i t y .............................................. 47 6. 2 An A lte rn a te E quation fo r M ix tu re V i s c o s i t y .............................................. 51 6. 3 M ethods for D e term in in g the P h y sic o M e c h a n ic a l C o n s t a n ts ..................... 53 C H A PTER III - E X P E R IM E N T A L ............................................................... 55 A. P r o c e d u r e ......................................................................................... 55 1. S am ple P re p a ra tio n ............................................................ 56 2. R h eo lo g ical M e a s u r e m e n ts ........................................... 59 3. P a r tic le S ize M e a s u r e m e n t s ........................................ 60 B. E q u ip m en t......................................................................................... 61 1. T he H aake R o to v is c o ........................................................ 61 2. M ic ro s c o p e ............................................................................. 63 3. P a r tic le Size A n a l y z e r .................................................... 67 4. H om ogenize r ......................................................................... 68 5. W estphal B a l a n c e ............................................................ 70 6. A u to m atic R e c o r d e r ........................................................ 71 1, C on stant T e m p e ra tu re C i r c u l a t o r ........................... 72 8, M e ttle r B a l a n c e ................................................................ 73 C H A PTER IV - E X P E R IM E N T A L RESU LTS AND A N A L Y S IS ........................................................................... 74 Pag© A. D ata R ed u ctio n ............................................................................ 74 1. C alcu latio n of V i s c o s i t y ........................................... 74 2. C alcu latio n of In te ra c tio n F re q u e n c ie s . . . . 78 3. The T w o -P h a se R ate of S h e a r ............................... 79 B. M ean P a r tic le Size and Shape D e te rm in a tio n . . , 81 C. C o r r e l a t i o n ................................................................................. 86 1. F u n ctio n al D ependence of and .................. 88 2. Influence of P a r tic le S i z e ....................................... 90 3. E ffects Due to C o n c e n t r a t i o n ............................... 92 4. A S e m i- E m p iric a l C o r r e l a t i o n .......................... 99 5. D e te rm in a tio n of the In te ra c tio n C o e f f i c i e n t s ................................................................... 101 D. Influence of S tab ilizin g A gent.......................................... 104 E. D evelopm ent of V elo city and S tr e s s P ro f ile s . . 108 1. V elo city D istrib u tio n ................................................... 108 2. W all S h e a r S t r e s s ....................................................... 109 C H A PTER V - S U M M A R Y ....................................................................... 119 R E F E R E N C E S ..................................................................................................... 128 A P PE N D IX A - T A B L ES O F DATA AND SHEAR DIAGRAM S....................................................................... 129 A P PE N D IX B - FO RTRAN PROGRAM FOR DETERM INING IN TER A CTIO N , V E L O C ITY D ISTRIBU TION, AND STRESS P R O F IL E S ......................... 199 vi LIST O F FIGURES F ig u re Page 1 C irc u la r Flow and C o o rd in ates.................................................. 22 2 D istilled W ater C alibration, T - 20° C .................................. 64 3 TGZ P a rtic le Size A nalyzer C a lib ra tio n .............................. 69 4 Shear D ia g ra m for 30 P e rc e n t B enzene /W a te r, T = 20° C .............................................................................................. 76 5 Shear D iag ram for 40 P e rc e n t B en ze n e/W a ter, T = 20f® C ............................................................................................. 77 6 P h o to m icro g rap h s of D isp e rse P h ase, 30 P e rc e n t (M ineral O il/W a te r and n -H ex ad ecan e/ W ater).................................................................................................... 82 7 Size F req u en cy Sum m ation Curve for 20 P e rc e n t c is -D e c a lin /W a te r (Sam ple 1 ) ....................... 84 8 Size F req u en cy Sum m ation C urve for 40 P e rc e n t o -X y le n e /W a te r (Sam ple 2 ) ............................. 85 9 Functional D ependence of and for 30 P e rc e n t o -X y len e /W a te r, T = 20° C................................. 89 10 F unctional D ependence of k_, and u )q for 30 P e rc e n t M in eral O il/W a te r, T = 20° C ........................ 89 11 Influence of D^ and u > q on In tera ctio n F requency, 30 P e rc e n t o - X y l e n e / W a t e r ..................................................... 91 12 Influence of D^ and on In teractio n F requency, 30 P e rc e n t M in eral O i l / W a t e r ................................................. 92 13 E ffect of P a rtic le Size and C oncentration on 0 ( B e n z e n e /W a te r ) ................................................................ 93 vii F ig u re P ag e 14 E ffect of P a r tic le Size and C o n c en tratio n on Cl- (o -X y le n e /W a te r)............................................................................... 94 15 E ffect of P a r tic le S ize and C o n c en tratio n on Q, (M in e ra l O i l / W a t e r ) ....................................................................... 95 16 E ffect of P a r tic le Size and C o n c en tratio n on f)2 (L in see d O i l / W a t e r ) ...................................................................... 96 17 E ffect of P a r tic le Size and C o n c en tratio n on Q2 ( c i s - D e c a l i n / W a t e r ) ...................................................................... 97 18 E ffect of P a rtic le Size and C o n c en tratio n on ft. ( n - H e x a d e c a n e /W a te r ) ................................................................... 98 19 A C o rre la tio n P lo t fo r the S y stem s M in e ra l O il/ W ater and L in see d O i l / W a t e r .................................................. 100 20 A D etailed C o rre la tio n P lo t fo r the S y stem s M in e ra l O il/W a te r and L in se e d O i l / W a t e r ..................... 102 21 Influence of S ta b iliz e r C o n cen tratio n on V isc o sity (M in e ra l O i l / W a t e r ) ....................................................................... 105 22 Influence of S a tb iliz e r C o n cen tratio n on V isc o sity (L in see d O i l / W a t e r ) ....................................................................... 105 23a Single C om ponent D ata, T = 20° C (n -H ex ad ecan e, c i s - D e c a l i n ) .................................................. 107 23b S h ear D ia g ra m w ithout S tab ilizin g A gent (n -H e x a d e c a n e /W a te r, c i s - D e c a l i n / W a t e r ) ..................... 107 23c S h ear D ia g ra m w ith 0, 001 M S tab ilizin g A gent......................................................................................................... 107 24 W eighted A v e rag e V e lo city D istrib u tio n , 30 P e r c e n t n -H e x a d e c a n e /W a te r (Sam ple 1)........................... 110 2 5 Individual P h a se V elo city D istrib u tio n , 30 P e r c e n t n -H e x a d e c a n e /W a te r (Sam ple 1 ) ......................... I l l 26 T ra n s ie n t S h earin g S tr e s s a t S u rface of C ylinder (30 P e r c e n t B e n z e n e /W a te r, S am ple 1)............................. 114 v iii F ig u re P age 27 T ra n sie n t Shearing S tre s s a t Surface of C ylinder (30 P e rc e n t n -H e x ad e ca n e/W ater, Sam ple 1)..........................115 28 T ra n sie n t S hearing S tre s s at Surface of C ylinder for a ll S y stem s, f., = 0. 3.............................................................. 116 29a Single Com ponents D ata, T = 20° C (50 gm cm M easu rin g H e a d ) ..................................................... 130 29b Single C om ponents D ata, T = 20P C ....................................... 130 30 S hear D iag ram for 10 P e rc e n t B e n ze n e/W a ter, T = 20° C .............................................................................................. 135 31 Shear D iagram for 20 P e rc e n t B e n ze n e/W a ter, T = 20° C ............................................................................................. 135 32 S hear D iag ram for 50 P e rc e n t B e n ze n e/W a ter, T = 20? C ............................................................................................. 143 33 E ffect of C oncentration on M ixture V iscosity, T = 20° C ( B e n z e n e /W a te r ......................................................... 143 34 Shear D iag ram for 10 P e rc e n t o -X y len e /W a te r, T = 2(f C ............................................................................................. 146 3 5 S hear D iagram for 20 P e rc e n t o -X y len e /W a te r, T = 2fl° C ............................................................................................. 146 36 S hear D iag ram for 30 P e rc e n t o -X y len e /W a te r, T = 20° C ............................................................................................. 151 37 S hear D iag ram for 40 P e rc e n t o -X y le n e /W a te r, T = 2<f C ............................................................................................. 151 38 Shear D iag ram for 50 P e rc e n t o-X ylene /W a te r, T = 2(f C ............................................................................................. 155 39 E ffect of C oncen tration on M ixture V iscosity, T = 20° C ( o - X y le n e /W a te r ) ..................................................... 155 40 S hear D iag ram for 10 P e rc e n t M in eral O il/W a te r, T = 2<f C ............................................................................................. 158 41 S hear D iagram for 20 P e rc e n t M in eral O il/W a te r, T = 20° C ............................................................................................. 158 ix F ig u re P age 42 S hear D ia g ra m for 30 P e rc e n t M in eral O il/W a te r, T = 20° C .....................................................................................................163 43 S hear D iag ram for 40 P e rc e n t M in eral O il/W a te r, T - 2 0 1 ° C .................... ............................................................................163 44 E ffect of C oncentration on M ixture V isco sity , T = 20° C (M in eral O i l / W a t e r ) ........................................................ 165 45 E ffect of C oncentration on M ixture V isco sity , T = 20° C (L inseed O i l / W a t e r ) .........................................................165 46 S hear D iag ram for 10 P e rc e n t L inseed O il/W a te r, T = 2(f C .....................................................................................................168 47 S h ear D iag ram for 20 P e rc e n t L in seed O il/W a te r, T = 2<f C .....................................................................................................168 48 S hear D iagram for 30 P e rc e n t L in seed O il/W a te r, T s 2 0 ° C .....................................................................................................173 49 S hear D iag ram for 40 P e rc e n t L in seed O il/W a te r, T = 20P C .....................................................................................................173 50 S h ear D iag ram for 10 P e rc e n t c is -D e c a lin /W a te r, T = 20° C .....................................................................................................177 51 S hear D iag ram for 20 P e rc e n t c is -D e c a lin /W a te r, T = 20° C .....................................................................................................177 52 S hear D iag ram for 30 P e rc e n t c is -D e c a lin /W a te r, T = 2tf* C .....................................................................................................182 53 S hear D iag ram for 40 P e rc e n t c is -D e c a lin /W a te r, T = Z(f C .....................................................................................................182 54 S hear D iag ram for 50 P e rc e n t c is -D e c a lin /W a te r, T = 2 ( f C .....................................................................................................186 55 E ffect of C oncentration on M ixture V isco sity , T = 20° C (c is -D e c a lin /W a te r).................................................... 1 86 56 S hear D iag ram for 10 P e rc e n t n -H e x a d e c a n e /W a te r, T = 20° C .....................................................................................................189 57 S hear D iag ram for 20 P e rc e n t n -H e x ad e ca n e/W ater, T » 2<f C .................................................................................................... 189 x F ig u re P ag e 56 S hear D iag ram for 30 P e rc e n t n -H e x a d e c a n e /W a te r, T = 20P C ............................................................................................. 194 59 S hear D iag ram for 40 P e rc e n t n -H e x ad e ca n e/W ater, T = 20P C ............................................................................................. 194 60 S hear D iagram for 50 P e rc e n t n -H e x a d e c a n e /W a te r, T = 20° C ............................................................................................. 198 61 E ffect of C oncentratio n on M ixture V isco sity , T = 2QP C (n -H e x ad e ca n e/W ater)............................................. 198 62 D iagram of C om puter P ro g ra m for C alculating In tera ctio n and T ra n sie n t R e s p o n s e ....................................... 200 xi LIST O F TABLES Table Page 1 Some H iy sic a l P ro p e rtie s of the D isp e rse P hase, T e m p e ra tu re = 20° C......................................................................... 57 2 Sam ple C oncentrations and Aqueous P hase P ro p e rtie s , T e m p e ra tu re = 20° C ............................................. 58 3 R otovisco C onstants w ith NV M easu rin g S ystem . . . . 65 4 V isc o m e te r Scale D eflection: Single Com ponent Data, T = 20° C ...................................................................................... 131 5 TGZ P a rtic le Size A nalyzer C ounter N um ber v e rs u s Size R a n g e ..................................................................................132 6 V isc o m e te r Scale D eflection and S hearing Data, T = 20P C {10 P e rc e n t B e n z e n e / W a t e r ) ......................................133 7 M easu red Size D istrib u tio n for 10 P e rc e n t B e n z e n e / W a t e r ...................................................................................... 134 8 V isc o m e te r Scale D eflection and S hearing Data, T = 20° C (20 P e rc e n t B e n z e n e /W a te r ) ......................................136 9 M e asu red Size D istrib u tio n for 20 P e rc e n t B e n z e n e / W a t e r ...................................................................................... 137 10 V isc o m e te r Scale D eflection and S hearing Data, T = 20° C (30 P e rc e n t B e n z e n e / W a t e r ) ......................................138 11 M easu red Size D istrib u tio n for 30 P e rc e n t B e n z e n e / W a t e r ...................................................................................... 139 12 V isc o m e te r Scale D eflection and S hearing Data, T = 20P C {40 P e rc e n t B e n z e n e / W a t e r ) ......................................140 13 M easu red Size D istrib u tio n for 40 P e rc e n t B e n z e n e / W a t e r ...................................................................................... 141 14 V isc o m eter Scale D eflection and S hearing D ata, T = 20° C (50 P e rc e n t B e n z e n e /W a te r ) ......................................142 x ii 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 144 145 147 148 149 1 50 152 153 154 156 157 159 160 161 162 164 V isc o m e te r Scale D eflection and S hearing Data, T = 20° C ( 10 P e rc e n t o -X y len e /W a te r) . . . . M e a su re d Size D istrib u tio n for 10 P e rc e n t o-X ylene / W a t e r ............................................................... V isc o m e te r Scale D eflection and Shearing Data, T = 20° C ( 20 P e rc e n t o -X y len e /W a te r) . . . . M easu red Size D istrib u tio n for 20 P e rc e n t o - X y l e n e / W a te r ............................................................... V isc o m e te r Scale D eflection and Shearing Data, T = 2 0 ^ 0 (3 0 P e rc e n t o -X y len e /W a te r) . . . . M e asu red Size D istrib u tio n for 30 P e rc e n t o - X y l e n e / W a t e r ................................................................. V isc o m e te r Scale D eflection and S hearing Data, T = 20? C (40 P e rc e n t o - X y le n e /W a te r ) ................ M e asu red Size D istrib u tio n for 40 P e rc e n t o - X y l e n e / W a t e r ................................................................. V isc o m e te r Scale D eflection and S hearing Data, T = 2(f C (50 P e rc e n t o - X y le n e /W a te r ) ................ V isc o m e te r Scale D eflection and S hearing D ata, T = 20° C (10 P e rc e n t M in eral O il/W a te r). . , . M e asu red Size D istrib u tio n for 10 P e rc e n t M in e ra l O i l / W a t e r ............................................................. V isc o m e te r Scale D eflection and Shearing Data, T = 20° C (20 P e rc e n t M in eral O il/W a te r ) . . . . M e asu red Size D istrib u tio n for 20 P e rc e n t M in eral O i l / W a t e r ............................................................. V isc o m e te r Scale D eflection and S hearing Data, T = 2 (f C (30 P e rc e n t M in eral O il/W a te r ) , . . . M e asu red Size D istrib u tio n for 30 P e rc e n t M in eral O i l / W a t e r ............................................................. V isc o m e te r Scale D eflection and S hearing Data, T = 20? C (40 P e rc e n t M in eral O il/W a te r ), . . . x iii 31 32 33 34 35 36 37 36 39 40 41 42 43 44 45 166 167 169 170 171 172 174 175 176 178 179 180 181 183 184 V is c o m e te r Scale D eflectio n and S h e a rin g D ata, T = Z(f C (10 P e r c e n t L in se e d O il/W a te r). . , . M e a s u re d Size D istrib u tio n for 10 P e rc e n t L in se e d O i l / W a t e r .............................................................. V is c o m e te r Scale D eflection and S h e a rin g D ata, T = 20^ C (20 P e r c e n t L in see d O il/W a te r ), , . . M e a s u re d Size D istrib u tio n fo r 20 P e r c e n t L in se e d O i l / W a t e r .............................................................. V is c o m e te r S cale D eflection and S h earin g D ata, T = 20° C (30 P e r c e n t L in se e d O il/W a te r). . , . M e a s u re d Size D istrib u tio n for 30 P e rc e n t L in see d O i l / W a t e r .............................................................. V is c o m e te r Scale D eflectio n and S h e a rin g D ata, T = 20° C (40 P e r c e n t L in see d O il/W a te r ) . . . . V is c o m e te r Scale D eflection and S h earin g D ata, T = 20° C (10 P e r c e n t c is - D e c a lin /W a te r ) , . . . M e a s u re d Size D istrib u tio n fo r 10 P e rc e n t c i s - D e c a l i n / W a t e r ............................................................... V is c o m e te r S cale D eflectio n and S h earin g D ata, T = 2 O P C (20 P e r c e n t c is - D e c a lin /W a te r ) , , , , M e a s u re d Size D istrib u tio n fo r 20 P e rc e n t c i s - D e c a l i n / W a t e r .............................................................. V is c o m e te r Scale D eflectio n and S h earin g D ata, T = 2 (f C (30 P e r c e n t c is - D e c a lin /W a te r ) . . . . M e a s u re d Size D istrib u tio n fo r 3 0 P e r c e n t c i s - D e c a l i n / W a t e r .............................................................. V is c o m e te r Scale D eflectio n and S h e a rin g D ata, T = 2 (f C (40 P e rc e n t c is - D e c a lin /W a te r ) . . . , M e a s u re d Size D istrib u tio n for 40 P e r c e n t c i s - D e c a l i n / W a t e r ............................................................... x iv T ab le P a g e 46 V is c o m e te r Scale D eflection and S h e a rin g D ata, T = 20° C ( 50 P e r c e n t c is - D e c a lin /W a te r ) ......................... 185 47 V is c o m e te r Scale D eflection and S h earin g D ata, T = Z(f C (10 P e rc e n t n - H e x a d e c a n e /W a te r ) ..................... 187 48 M e a su re d Size D istrib u tio n for 10 P e rc e n t n - H e x a d e c a n e / W a t e r .................................................................. 188 49 V is c o m e te r Scale D eflection and S h earin g D ata, T = 20° C (20 P e rc e n t n - H e x a d e c a n e /W a te r ) ..................... 190 50 M e a su re d Size D istrib u tio n fo r 20 P e rc e n t n - H e x a d e c a n e /W a te r ........................................................................ 191 51 V is c o m e te r Scale D eflectio n and S h earin g D ata, T - 20® C (30 P e rc e n t n -H e x a d e c a n e /W a te r )..................... 192 52 M e a su re d Size D istrib u tio n fo r 30 p e rc e n t n -H e x a d e c a n e /W a te r....................................................................... 193 53 V is c o m e te r Scale D eflection and S h earin g D ata, T = 20° C (40 P e r c e n t n -H e x a d e c a n e /W a te r..................... 195 54 M e a s u re d Size D istrib u tio n for 40 P e rc e n t n - H e x a d e c a n e /W a te r ...................................................................... 196 55 V is c o m e te r Scale D eflection and S h earin g D ata, T = Z (f C (50 P e r c e n t n - H e x a d e c a n e / W a t e r ) ................. 197 xv NOM ENCLATURE A s h e a r s tr e s s facto r B ra te of sh e a r (geom etry) factor C constant defined by Log C = 0, 5772, , , A, B, C co n stan ts in E in s te in 's hydro dynam ical th eo ry D p a rtic le d ia m e te r e rfc c o m p lem e n ta ry e r r o r function hydrodynam ic in te ra c tio n fo rce (in terp h ase friction) f frac tio n a l volum e (without su b sc rip t re f e r s to the d isp e rse d phase) 1^ m odified B e sse l function of f ir s t o rd e r and f ir s t kind K. m odified B e sse l (M acdonald) function of f ir s t o rd e r and second kind K v isc o sity c a lib ra tio n constant k in te ra c tio n frequency, re c ip ro c a l seconds K j 2 coefficient of inter action Log n a tu ra l lo g arith m , base e = 2. 7 1 8 2 8 .,, N constant defined by LogN = £ N q ro tatio n a l speed, rp m n in te g er; frequ ency of o b se rv atio n s p p re s s u re ; tra n s fo rm a tio n p a ra m e te r q y p /v R d im e n sio n less rad iu s = r / r Q; v a ria b le = q r; rad iu s of unit cell around a suspended sp h ere xvi r ra d iu s, m e a su re d fro m axis of ro tatio n Tq rad iu s of ro tatin g cylinder S sc a le read ing , d im e n sio n less; su rfa ce s ra te of sh e a r, re c ip ro c a l seconds T d im e n sio n less tim e v a ria b le s t tim e, seconds U speed fac to r, sec. V d im e n sio n less velocity; volum e of unit cell around a suspended sp h ere v^ in stan tan eo u s, local tangential velocity Vq c irc u m fe re n tia l velocity of ro tatin g cylin der = W^r^ W w ork, en erg y d issip ated u0 , v 0,w 0 velocity com ponents of u n p e rtu rb e d flow u /, v /, w / velocity com ponents of additional flows u, v, w velocity com ponents of p e rtu rb e d flow X c o rre la tio n p a ra m e te r x length; value of o b serv atio n X, Y, Z com ponents of s tr e s s te n s o r x, y, z C a rte sia n co ordinates 0 p a rtic le ellip ticity m u ltip lie r (J e ffe ry 's equation) r den o m in ato r of the d eriv ed v isc o sity equation; gam m a or g e n eralize d fa c to ria l function Y E u le r 's constant = 0. 5772. .. e ellipticity; in fin ite sim a l rad iu s ri (r - rg)A /4vt, d im e n sio n less x v ii A p a rtic le shape fa c to r and B ro w n ian m o v e m e n t m ultiply ing co efficien t ( B u rg e r's equation) X dum m y v a ria b le U v isc o sity (without s u b s c rip t r e f e r s to the e x te rn a l (suspending) fluid) v 't r u e 1 k in e m a tic v isc o sity rr c o n stan t = 3. 141 5 9 .. . Pq ra d iu s of su sp en d sp h e re 2 2 2 0 ^ 5 + rt + C > re d u c e d d en sity T' su m m atio n , sum of ^ lo g -s ta n d a rd g e o m e tric deviation S r s h e a r s t r e s s $ volum e of su sp en d ed sp h e re # sp h e ric ity , shape fac to r Q d im e n sio n le s s in te ra c tio n freq u e n cy = k/ut>Q (U q a n g u la r v elo city a t r= r^ 5, n, C C a rte s ia n c o o rd in a te s in E in s te in 's equations 2 7 L ap lace O p e ra to r SU BSCRIPTS 1 ,2 f i r s t p h a se, seco nd p h ase d d ra g g g e o m e tric m m ix tu re th n n o rm a l; n com ponent of a m u ltico m p o n en t s y s te m n n a s s o c ia te d w ith d e n sity to in d icate 't r u e ' value for the u x v iil pm a b b rev iatio n for p a rtic le m ean (used w ith d iam ete r) as steady state (m easu red ) value r - < p u sed to denote the d ire c tio n of sh earin g s t r e s s (p erp en d ic u la r and p a ra lle l axes resp ectiv ely ) a in te rfa c ia l ten sio n 5, n, C u sed with s t r e s s te n s o r to indicate com ponent in d ire c tio n of the co ordinates SU PERSCRIPT * indicates n o n -d im en sio n al (s tre s s ) ratio ABSTRA CT The p re s e n t w ork a r i s e s fro m an a tte m p t to dev elop new m eth o d s fo r analyzin g fluid m otion in m u ltico m p o n en t s y s te m s . A study of the phenom ena o c c u rrin g w hen two o r m o re m u tu a lly i m m is c ib le p h a se s flow conjointly is e s s e n tia l for the solutio n of a g re a t n u m b e r of in d u s tria l and e n g in e e rin g p ro b le m s. In the fol lowing the p rin c ip le s of c la s s ic a l h y d ro d y n a m ic s a r e applied to the m otion of a tw o -p h a se m ix tu re in the p re s e n c e of m u tu al in te ra c tiv e fo rc e s (hydrody nam ic d rag ) am ong the com ponents of the h e te r o geneous m edium . The m a th e m a tic a l m odel thus y ield s a g e n e ra l e x p re s s io n for the co efficien t of m ix tu re v isc o sity in te r m s of each co m p o n e n t's p h y sical p r o p e r tie s , fra c tio n a l volum e and in te rp h a s e d rag . The ensuing fo rm u la e and a p p ro a c h tak en w e re applied d ire c tly in an e x p e rim e n ta l r e s e a r c h p ro g ra m a im e d a studying v a rio u s flow a s p e c ts of finely d is p e rs e d e m u lsio n s. C o m p a riso n of the rh e o lo g ic a l (sh e arin g ) m e a s u re m e n ts and the e x p re s s io n d e veloped for v is c o s ity enabled d e te rm in a tio n of the h y p o th esized in te rp h a s e in te ra c tio n and su b seq u e n t c o rre la tio n s th e re o f w ith im p o rta n t s y s te m v a ria b le s . The p h y sic a l p ro b le m c o n sid e re d is the analogue for c ir c u la r flow of R a y le ig h 's p ro b le m , c h o sen for its sim p le g e o m e try and re la tiv e a m e n a b ility to m a th e m a tic a l a n a ly s is . S p ecifically , the flow g e n e ra te d by a suddenly a c c e le r a te d c y lin d e r im m e rs e d in an infinite m ed iu m of in te ra c tin g continua is an aly zed u n d e r the co n xx dition of constant fluid density. The basic po stulate introd uced and v e rifie d by e x p e rim e n ta l o b se rv atio n s is that the in te ra c tiv e f r i c tional effects a re e x p re ss ib le in te rm s of known sy ste m p a ra m e te r s , including size of the hom ogeneously d is p e rs e d p a rtic le s. An ap p ro x im ate solution of the posed bou n d ary -v alu e pro b lem is ob tained in the fo rm of a s e r ie s of c o m p lem e n ta ry e r r o r functions by the use of the asy m p to tic expansion of the B e s s e l re la tio n s. F ro m this solution the tw o-phase v isc o sity is deduced by analyzing the skin fric tio n developed at the su rfa c e of the ro tatin g cylinder. In C hapter I the b a sic flow p ro b lem is introduced and p r e vious p ertin en t r e s e a r c h e ffo rts a re b riefly d isc u sse d . E m p h asis h e re is placed on the n u m ero u s fac to rs influencing the flow and past m ethods of tre a tm e n t. The firs t p a rt of C hapter II is devoted m ain ly to the c la s sical hyd rodynam ical tre a tm e n t due to E in ste in and subsequent m odifications th e re to n e c e ssita te d by the id ealizatio n s and lim ita tions in h ere n t in the th e o re tic a l approach. In the second p art the m a th e m a tic a l pro b lem is fo rm u lated , som e postulates introduced, and the solution given for the tangential velocity of each phase. On the b a sis of this a n a ly sis, the com bined sh e a rin g s tr e s s caused by both fluids' m otion is found and an e x p re s s io n for the coefficient of m ix tu re v isc o sity d eriv ed . The e x p e rim e n ta l p ro c e d u re and a b rie f d e sc rip tio n of the equipm ent u sed a r e given in C hapter III, Sam ple p re p a ra tio n , te st conditions and rh eo lo g ical and p a rtic le size m e a su re m e n ts xxi p e rfo rm e d a re d isc u sse d in detail. F ro m th ese data b asic sh e ar d iag ra m s depicting s t r e s s , r a te - o f - s h e a r rela tio n sh ip a re ob tained. The ch ap ter on equipm ent is included for fu rth e r p ro c e d u ra l c la rific a tio n s as m ay re la te d ire c tly to the p re s e n t w ork. C hapter IV contains the a n aly sis and c o rre la tio n of the e x p erim e n tal re s u lts . F i r s t it is shown that m ix tu re s of fre sh ly p re p a re d sam p les of im m isc ib le liquids m ay be tre a te d as Newton ian up to 40 p e rc en t frac tio n a l volum e of the org an ic (d isp e rse) phase. In addition, p a rtic le size effects on v isc o sity re m a in consequential even at the low est c o n cen tratio n s. F u rth e r, it is noted that th ese effects a re in v e rse ly p ro p o rtio n al to m ean p a rtic le size. Finally, the in te ra c tio n te r m s a re c o rre la te d w ith physical p a ra m e te r s and the tra n s ie n t sh e a rin g s tr e s s and velocity d is tr i bution in the h etero g en eo u s m edium a re studied. The final ch ap ter of this d is s e rta tio n provides a su m m a ry of e a r lie r c h a p te rs and includes som e o b se rv atio n al re m a rk s and suggestions for continued future w ork. The m a te ria l given in the A ppendices is a com pilation of e x p e rim e n ta l and calculated data and contains also the com p u ter p ro g ra m and a sam ple output. x x ii CHAPTER I INTRODUCTION In the six and one h alf d ecad es since A lb e rt E in ste in firs t published his th e o re tic a l w o rk [1] "E in neue B eatim m ung d er M olektildim ensionen" ("A New D e term in a tio n of M o lecu lar D im en sions") and the extension th e re o f to the p red ic tio n of the rheolog- ica l b ehavior of a su sp en sio n of u n ifo rm rig id sp h e re s in a Newtonian d is p e rs io n m edium , a g re a t n u m b er of tech n ical p a p ers has a p p eared on the subject. In 1922, for exam ple, Je ffe ry [2] g e n eralize d E in s te in 's w ork to include the influence of ellip so id al p a rtic le s and th e ir o rien tatio n on the m otion of the suspending fluid. Ten y e a rs la te r , T ay lo r [3] extended the c la s s ic a l h y d ro - dynam ical th eo ry of E in stein to the case of one fluid containing sm a ll d rops of an o th e r, and a rr iv e d at a g e n eral e x p re ss io n for the ap p aren t in c re a s e of the v isc o sity of the liquid. In 1938, B u rg e rs [4] in v estig ated the com bined influence of the la m in a r m otion of the liquid and the d istu rb a n c e due to the m o le c u la r a g i tations (B row nian m ovem ent) of sm a ll Hookean p a rtic le s of e lo n gated fo rm s at v ario u s o rien tatio n s of th e ir axes of revo lutio n in the flow field. In m o re re c e n t y e a rs , in v e stig a to rs in this d iscip lin e have focused th e ir r e s e a r c h e ffo rts on stu d ies of m o re co n cen trated and p o ly d isp e rse sy ste m s a s opposed to the id ea liz atio n s in h ere n tly 1 2 c h a r a c te ris tic of a p u rely th e o re tic a l approach. Consequently, n u m ero u s th e o rie s w e re advanced and te ste d w ith varying d e g re es of su c c e ss . F o r exam ple, Sim ha [5] and H appel [6] m ade d e te r m inatio ns of p a rtic le -p a r tic le in te ra c tio n effects in c o n cen trated su sp en sio n s of u n ifo rm rigid sp h e re s , and M ooney [7] included a hydordynam ic in te ra c tio n coefficient which v a rie d w ith the concen tra tio n of the suspended p a rtic le s . F o r dilute sy s te m s w h ere the suspended p a rtic le s a re liquid, T a y lo r's equation is unique in that it c o n sid e rs the influence due to the v isc o sity of the d is p e rs e phase. The w ork of R ajagopal [8], R ich a rd so n [9] and S herm an [10] c o n cern ed itself, in p a rt, with the study of the effects of p a r ticle size and d istrib u tio n upon the flow p ro p e rtie s of the sy s te m . T hus, by introducing m odifications to E in s te in 's equation it was p o ssib le to account for som e of the in te ra c tin g v a ria b le s and d e te rm in e th e ir effects, though not e n tire ly in com plete isolation. U nfortunately, m uch of the e x p e rim e n ta l findings, e sp e c ia lly those co ncerned with p a rtic le size e ffe cts, have been co n trad icto ry . In the p re s e n t w ork, the m otion of a tw o-ph ase m edium is exam ined in the p re s e n c e of in te rp h a se d rag (interaction). The in te rn a l phase is c o n sid ere d in e rt, in that p a rtic le s e x e rt no a t tra c tiv e fo rc e s upon each o th e r, and th e ir ap p aren t w eight is su f ficiently sm a ll as to re n d e r sed im en tatio n effects negligible. When the d is p e rs e d p a rtic le s a re solid in stea d of liquid, the solid phase is tre a te d p ro v isio n ally as a Newtonian fluid by introducing a "p ro v isio n al coefficient of v isc o sity " [ll] that is only valid if the 3 solid p a rtic le s r e m a in in su sp en sio n . W hen such a tw o -p h ase m ix tu re is contain ed in the a n n u la r spacing b etw een v e rtic a l co a x ial c y lin e rs in re la tiv e m otion and su b je cted to a u n ifo rm im p u l siv e s h e a r , both the h y d ro d y n am ic d ra g and the p ro v isio n a l v i s c o sity m a y be d e te rm in e d e m p iric a lly fro m to rq u e (sh e a rin g s t r e s s ) m e a s u r e m e n ts ta k e n at s e v e r a l tim e in te rv a ls . F o r the c a s e of two im m is c ib le liquids w hose individual v is c o s itie s a re known, the tim e dependency m ay be e lim in a te d and the fric tio n a l d ra g e s t i m a te d fro m m e a s u re m e n ts taken a t ste ad y s ta te . Due to e x tre m e d iffic u lties e n co u n tere d in obtaining re lia b le m e a s u re m e n ts in the tra n s ie n t dom ain, it w as n e c e s s a r y to r e s t r i c t the e x p e rim e n tal p a rt of this w o rk to im m is c ib le liquid s y s te m s only. In o r d e r to effect a c lo s e d -fo rm so lu tio n of the N a v ie r- Stokes e q u atio n s, it w as n e c e s s a r y to fu rth e r lim it th is in v e s tig a tion to c a s e s w h e re the flow is la m in a r and the fluid s y s te m und er study is N ew tonian. A c co rd in g ly , e x c e s s iv e ly high s h e a rin g r a te s and high c o n c e n tra tio n s of the d is p e r s e p h ase w e re excluded fro m the e x p e rim e n ta l plan. Indeed, none of the s y s te m s studied (o il/w a te r type e m u lsio n s) have exh ibited non-N ew tonian c h a r a c t e r i s t i c s below 40 p e rc e n t volum e c o n c e n tra tio n of the in te rn a l ph ase. The s o - c a lle d phenom enon of phase in v e rs io n w hich m a n i fe sts its e lf w hen the p a r tic le s b ecom e too c lo se ly packed h as thus b e e n avoided. E m p iric a l d e te rm in a tio n of the p h y sic o m e ch a n ica l c o n stan t of each s y s te m re v e a le d that the h y d ro d y n am ic in te r a c tion is stro n g ly in fluenced by c o n ce n tra tio n and m e a n p a rtic le sia e 4 of the sh e a re d sam ple. The p re se n c e of a sm a ll film of s ta b ilis e r, in itially d isso lv ed in the aqueous ph ase, and its h ith e rto unknown influence on such su rfa c e phenom ena as in te r facial tension, e lec- tro v isc o u s effects, double la y e r, etc. undoubtedly h in d ere d effo rts to a rr iv e at a g e n eralize d e m p iric a l c o rre la tio n for the p ropo sed in te rac tio n . When c o n sid erin g sy ste m s w hose com ponents a re not too d is s im ila r, how ever, a g e n eralize d e m p iric a l c o rre la tio n h a s been possible. A solution of the N av ier-S to k es equations in cy lin d ric a l p o lar co o rd in ates w ith m utual hydrodynam ic d rag am ong the phases and which sa tisfie d the boundary and initial conditions is obtained in the form of infinite in te g ra ls in B e sse l functions. Due to con vergence c h a r a c te r is tic s of such s e r ie s , how ever, an a lte rn a te solution is given in the fo rm of a s e r ie s of c o m p lem e n ta ry e r r o r functions and th e ir in te g ra ls by p e rfo rm in g asy m ptotic expansion of the B e sse l functions fo r sm all and la rg e values of the tim e. Thus fro m the developm ent of the velocity p ro file s, the sh e a r s tr e s s is evaluated and an e x p re ssio n c h a ra c te riz in g the v isc o sity of the tw o-phase m ix tu re is deduced. So fa r, no m ention has been m ade of the effect of p a rtic le th e rm a l m o le c u la r m ovem ent on the hydrod ynam ics of the pro b lem co n sid ered . It h as long been reco gnized that p a rtic le s of ten m ic ro n s o r s m a lle r, e x p erien c e tra n s la to ry and ro ta to ry fo rm s of B row nian m ovem ent caused by p r e s s u r e fluctuations in the solvent. In his c la s s ic a l tre a tm e n t on the subject, E in stein ignored the sup p le m e n ta ry diffu sio n al fluxes p re s u m a b ly of the g re a t d ilution of the su sp e n sio n s c o n sid e re d . If the p a rtic le s a r e not p e rfe c t s p h e re s such a s p ro la te and o b late sp h e ro id s o r o th e r elong ated fo rm s , the flow w ill induce a n is o tro p y in the d is trib u tio n of the o rie n ta tio n of th e ir a x e s of rev o lu tio n . In the c a se of s p h e r ic a l g e o m e try , how e v e r, a ll d ire c tio n s of tr a n s la tio n and all a x e s of ro ta tio n th ro u g h the c e n te r of the p a rtic le a r e id en tica l; the o rie n ta tio n thus re m a in s c o n sta n t and the m e a n a n g u la r v e lo c ity of the flow is u n a lte re d . In the e x p e rim e n ts p e rfo rm e d , sp e c im e n s of the s h e a re d sa m p le w e re view ed u n d e r a L e itz O rth o lu x m ic ro s c o p e and p hoto g rap h ed fo r la te r a n a ly s is . T he globules show ed no d is to rtio n o r d e p a rtu re fro m s p h e ric a l g e o m e try th at could have fav o re d sp e cific o r ie n ta tion in the solvent. A cco rd in g ly , B ro w n ian m o v em en t, though f o r tuitous to a c e r ta in ex ten t as in m in im iz in g se d im e n ta tio n e ffe c ts, have b e en excluded fro m fu rth e r a n a ly sis. In the follow ing c h a p te rs , a b r ie f rev ie w of p a s t th e o re tic a l w o rk is f i r s t p r e s e n te d , follow ed by the d e v elo p m en t of the p r e s e n t th e o ry and so lu tio n of the m a th e m a tic a l m o d el. Included a lso in th is d is s e r ta tio n is a d e s c r ip tio n of the e x p e rim e n ta l p ro c e d u re , a n a ly s is and g ra p h ic a l r e p r e s e n ta tio n of the te s t r e s u lts and a s e m i- e m p ir ic a l c o r r e la tio n for in te rp h a s e d ra g . The IBM c o m p u te r p ro g r a m u s e d for d a ta red u c tio n , d e te rm in a tio n of the i n t e r - p h ase d ra g and c a lc u la tio n of the tr a n s ie n t v elo city p ro file s and s h e a r s t r e s s h i s t o r y to g e th e r w ith a sa m p le p rin to u t is given in A ppendix B. R e c o m m e n d a tio n s and su g g e stio n s fo r continuing e ffo rts in this field a r e m ade. CHA PTER II TH EO RETICA L A. P re v io u s W ork In his q u e st for a new d e te rm in a tio n of the actu al d im e n sions of m o lecu les fro m p h y sical phenom ena o b se rv ab le in liq uid s, E in ste in set fo rth the b a sic foundation for m any subsequent h y d ro - d y n am ical in v estig atio n s. N e a rly all of the ex istin g fo rm u lae and re la te d e x p re s s io n s depicting the a p p aren t in c r e a s e in v isc o sity of p ure solvent, due to the p re s e n c e of dilute am ounts of u n d isso - ciated solute m o le cu le s, a re but a m e re ex ten sio n or m odification of h is m o st c e le b ra te d h y d ro d y n am ical tre a tm e n t. B riefly, E in ste in c o n sid e re d the la m in a r m otion of a hom ogeneous New tonian liquid and the flow m o difications th e re to c re a te d by d istu rb a n c e s in the im m e d ia te neighborhood of a single, rig id p a rtic le suspended in that liquid. C onsidering only d ilatatio n al m o v em en t in th re e m u tu ally p e rp e n d ic u la r p rin c ip a l axes 5, r\, C. he thus w rite s : u 0 = A ? v0 = Bq (U w0 = C C for the u n p e rtu rb e d flow, and 6 7 u = + u ' v = Br| + v ' w = CC + w ' <Z) for the p e rtu rb e d flow. H e re u ', v ', w / a re the additional flows and A, B, C a re con stan ts w hich, for an in c o m p re s sib le liquid, m u st sa tisfy the condi tion F o r n ran d o m ly suspended n o n -in te ra c tin g sp h e re s , the p e rtu rb e d flows becom e W here the su m m atio n is extended o v e r a unit cell of fluid, s u rro u n d ing each sp h e re , w hose rad iu s R is la rg e co m p ared to th at of the sp h e re and outside w hich the flow is c o n sid e re d fric tio n le ss. If the s p h e re s a r e evenly d istrib u te d , the su m m atio n sign m ay be re p la c e d by an in te g ra l, and the functions u, v, and w m u st s a tis fy the h y d ro d y n am ical equations of c reep in g m otion: A + B + C = 0 (3) u = A ?+ E u * v - Bri + F v ' w = cr + £ w ' (4) 8 | | . U7*u | E = U7ZV (5) & p _2 SC = “ 7 w and continuity: I f + I ^ + ¥ T = ° (6) E qu ations (2), (3) and (4) a r e su b je c t to the bou n d ary conditions 1. p = Pq : u = v = w = 0 2. p » p q : u #= v ' = w ' = 0 = V?2+nZ +c: W here pg is the ra d iu s of the su sp en d e d s p h e re , and p Having obtained a solutio n fo r the m o tio n d e s c r ib e d by the p re v io u s se t of eq u atio n s, E in s te in p ro c e e d e d to c a lc u la te the r a te a t w hich e n e rg y is d is s ip a te d in th e m o v e m e n t of the fluid in th e unit cell. T his e n e rg y is equal to the m e c h a n ic a l w o rk done on the liquid by the s t r e s s com ponents a ctin g on the s u rfa c e of the sp h e re of ra d iu s R. Denoting th e s e co m p o n en ts by X^, Yn and Z ^, the w o rk W is then given by w = S I ( x „ “ + Ynv + Z nw ) dS (7) 9 w h e re the in te g ra tio n is ex ten d ed o v e r the su rfa c e of the sp h e re of ra d iu s R, and X = - ( x _ Z + n \ ? p X -3 + r) P C P / Y - ~ ( Y--£ + « ' ? P Y -2 + r) P Yr -C) C P / (8) Z = - ( z _ 4 + n \ ? P z -2 + T 1 P z i ) c p z X -, Y , Z a re the n o rm a l e le m e n ts of the s t r e s s te n s o r and 5 T [ t Y . = Z , Z _ - X ,, X = Y_ a r e the tan g e n tia l com p onents defined t t i 5 t t i S in the u su a l w ay, n a m e ly * s . p - 2 « H . Yc = - u ( l ? + 1 ? ) ■ 7 - « 9 W V . ( 3 U , C ' p ■ u ac ‘ r T _ u i a-n a ? ) In the a b se n c e of s p h e re s , E q u atio n (7) y ie ld s for the ra te of e n e rg y d is s ip a tio n p e r unit volum e V of fluid W = Z\jLtZ (10) and the ra te w hen only a sin g le s p h e re is p r e s e n t in the liquid W = 2U62 ( l + (11) 10 w h e re * 4 3 * * T ” < > o F o r n su sp en d ed s p h e r e s , the e n e rg y d is s ip a tio n r a te in a u nit of volum e is lik ew ise by f> the volum e fra c tio n occupied by the g ro u p of s p h e re s . If b y analogy w ith E q u atio n (10) we a re p e rm itte d to e x p re s s the m ix turefa r a te of e n e rg y d issip a tio n in the fo rm then E q u atio n s (12) and (13) y ield for the c o efficien t of v is c o s ity of the " m ix tu re " the follow ing r e s u lt ( 12 ) The s u b s c r ip t m r e f e r s to the " m ix tu re " and m a y be re p la c e d W m = u ( l + °* 5 f ) r z (14) 11 2 2 2 2 w h e re = A _ + B + C , and A . B , C _ a r e the v alu es of m m m m m m m the p rin c ip a l d ire c tio n s of d ila ta tio n of the m ix tu re p a ra lle l to A, B, C and a re re la te d in the m a n n e r A = A ( 1 - f ) m B = B ( 1 - f) (15) m Cm = C (1 - f ) N eglecting second and h ig h e r o r d e r te r m s in f, w e get fro m the p re c e d in g re la tio n sh ip s = ®2( 1 ‘ 2f) (16) and fro m E q u atio n s (14) and (16) m ( 1+0. 5f \ = u V i - i t ) (17) the fra c tio n a l p a rt in E qu atio n (17) m ay be expanded in the s e r i e s 1 + 0. 5f ' 1 - 2 / = 1 + 2. 5f + T hus, E quation (12) b e c o m e s finally Um = u ( 1 + 2. 5 f ) (18) 12 the w ell known equation of E instein, N otw ithstanding the id e a listic n atu re of the "m ix tu re " d e s c rib ed by this equation, th at of uniform sp h e ric a l and rig id p a rtic le s at infinite dilution, it n e v e rth e le s s has found extensive a p p lica b il ity to tw o-p hase s y s te m s ' rheology. Subsequent m odifications th e re to have included effects caused by n o n -sp h e ric a l p a rtic le g eo m etry , p a rtic le -p a r tic le in te ra c tio n a ris in g fro m reduced d ilu tion, and in te rn a l phase v isc o sity w hen the suspended p a rtic le s a re liquid. Thus J e ffe ry 's equation [2] for ellip so id al p a rtic le s is of the form M m = H<1 + Pf) <19) w h ere 0 is a m u ltip lie r w hich depends on ellip ticity e, and the o rien tatio n a ssu m e d in the flow as re q u ire d by his m inim um d is s i pation of en erg y hypothesis. Its m ax im u m value o c c u rs w hen e = 0, for sp h e re s , and is n u m e ric a lly equal to a s dem anded by E in s te in 's form ula. S im ilarly , sta rtin g from O s e e n 's fo rm u lae for the m otion of a liquid under the influence of e x te rn a l fo rc e s, B u rg e rs [4] deriv ed a re la tio n sh ip for the in c re a s e in v isco sity w hich takes into co n sid era tio n the p h y sical shape of the p a rtic le as w ell as the in ten sity of m o le c u la r ag itatio n s. The resu ltin g equa tion m ay be e x p re ss e d thusly: u m * u ( i + a h (20 ) 13 The co efficien t A h a s the in trin s ic p ro p e rty th at fo r d ilu te s u s p e n sio n s of s p h e r ic a l p a r tic le s w ith m in im a l m o le c u la r a g ita tio n s (low in te n sity B ro w n ian m o v em en t) its n u m e ric a l value a p p ro a c h e s th at given by E in s te in 's equation. The e x te n sio n by T a y lo r of the c l a s s ic a l h y d ro d y n a m ic a l tr e a tm e n t to the case w hen the su sp en d ed p a r tic le s a re s m a ll d ro p le ts of a n o th e r liquid haB y ield ed analogous r e s u lts . A ssu m in g for low sh e a rin g r a te s th at the d ro p le ts a re u n d e fo rm a b le and re m a in so by the a ctio n of in te rf a c ia l ten sio n , T a y lo r 's a n a ly s is [3] h a s given r i s e to th e equation: 2. 5 — + 1 lJl = M, 1 + --------- f m M 1 u- — + 1 (2 1 ) H e re the s u b s c rip ts 1 and 2 r e f e r to the e x te rn a l and in te rn a l (d is p e rs e ) p h a se s re s p e c tiv e ly . In the m o re re c e n t lite r a tu r e the e f fect of p a rtic le siz e and siz e d is trib u tio n and a h o st of su rfa c e p h enom ena e. g. e le c tro v is c o u s effectB, e le c tr o k in e tic (zeta) po te n tia l of the c h a rg e d p a r tic le s , d ie le c tr ic c o n sta n t e tc . have re c e iv e d c o n sid e ra b le a tten tio n . H ow ev er, m uch m o re r e s e a r c h e ffo rt m u s t be expended in th e se nebulous a r e a s b e fo re th e ir tru e in flu e n c e s, actin g individually o r sim u lta n e o u sly , a r e fully a s c e r tained. In the follow ing p ag es the p rin c ip le s of c la s s ic a l h y d ro d y - 14 n a m ics a r e applied to the m otion of a tw o -p h ase m ix tu re w ith the in tro d u ctio n of the concept of m utual h y d ro d y n am ical in te ra c tio n am ong the p h a se s. The solution to the flow equations thus leads to a g e n e ra l e x p re ss io n for the v isc o sity of a tw o -phase m ix tu re with the in te ra c tio n to be ev alu ated ex p erim en tally , B. The P re s e n t W ork The concepts, fo rm u lae and h y p o th eses elu cid ated in this sectio n re fle c t the risin g c u rre n t tre n d s tow ard the s e a rc h for new techniques for analyzing flow sy ste m s with two o r m o re im m isc ib le p h ases. The application of the e stab lish ed p rin c ip le s of continuum m ech an ics to each com ponent in the sy ste m h as given r is e to the developm ent of n u m ero u s a n aly tical m odels w ith varyin g d e g re e s of com plexity and so phistication. The sim p listic ap p ro ach tr e a ts the m ix tu re as a pseudofluid w ith a v erag e flow p a r a m e te r s (hom o geneous e q u ilib riu m flow). Subsequent d e p a rtu re s th e re fro m , a re focused on the m otion of the individual com ponents (se p a ra te d flow) o r rela tiv e m otion (drift-flux) with m utual h y d ro d y n am ical in te r action betw een the com ponents of the sy ste m . The contention that se p a ra te equations of m otion m ay be w ritte n for each phase in the ta m e sen se that continuity of m a tte r holds for e v ery com ponent of the sy ste m lends its o rig in to the m o re g e n eral m ethods of i r re v e rs ib le th erm o d y n am ics. F o r ex am p le, W allis [12] notes that entropy production in tw o-phase flow m u st include that g e n erate d 15 by the re la tiv e m otion and in te ra c tio n betw een the co n stitu en ts of the sy stem , B e a rm a n and Kirkw ood [13] have d eriv ed the equations of m otion for each com ponent of a m ulticom ponent sy ste m using the m ethods of s ta tis tic a l m ech an ics. Subsequently, they have shown that the fa m ilia r phenom enological rela tio n sh ip s of t r a n s p o rt p r o c e s s e s a re d e riv a b le fro m the m ic ro sc o p ic equations of h y d ro d y n am ics w hich include the fo rce s a ris in g fro m c o n sid era tio n of m utual in te rac tio n , F aizu llaev [ 11 ] applying the th e o ry of in te rp e n e tra tin g m otions of m u ltip h ase m edia and the concept of red u ced d en sity advanced by R akhm atulin [14] solved nu m ero u s p ro b lem s taking into account frictio n a l fo rc e s caused by m ed ia in te ra c tio n . In the m a th e m a tic a l developm ents to follow, the tw o-phase m ix tu re is co n sid ere d iso tro p ic and hom og eneously d istrib u te d w ith each com ponent re g a rd e d as a continuum . The la tte r p ro v iso is a d ire c t consequence of the afo rem en tio n ed red u c ed density concept w h e re in m otion in a hom ogeneously porous (m ultiphase) m edia with velocity v and tru e d en sity pn Q is id en tical to m otion in a fre e m edium with v elocity v and reduced d ensity pn> H ere the th su b s c rip t n r e f e r s to the n com ponent of the m u ltip h ase sy stem . A ccordingly, the ra tio of red u ced to tru e d en sity is the so -c a lle d volum e p o ro sity f o r m o re com m only, void (volum e) fraction. T hroughout the c o u rse of the p re s e n t m a th e m a tic a l d evelopm ents, no d istin ctio n is m ade in the sta te of m a tte r of the individual ph ases. Thus in the case of a su sp en sio n of solid p a rtic le s , the solid phase is tre a te d as a Newtonian fluid p o sse ssin g a p ro v isio n al v isc o sity coefficient which m u st be d e te rm in e d e x p erim e n tally and co n sid ere d only o p e ra tiv e w hile the p a rtic le s re m a in suspended. 1. F undam ental Concepts At the o u tset we do not know a p rio ri how the m u ltip lic ity of fa c to rs m ay com bine o r becom e in te rre la te d to produce the hypoth e sise d in te ra c tiv e fo rce s e x e rte d by one constituent of the sy ste m upon another. N e v e rth e le ss, it is re a so n a lb e to c o n sid er the dif fere n ce s in physical p ro p e rtie s of each com ponent as cau sativ e fa c to rs contributing to the e m e rg en c e of these additional fo rc e s. P a ram o u n t am ong these a r e the individual phase v isc o sitie s, size and shape of the hom ogeneously d is p e rs e d p a rtic le s o r d ro p le ts. Some influence of the configuration of the flow ap p aratu s m ay a lso be anticipated. A ccordingly, fo r m a th e m a tic a l expediency we se le c t an infinite g e o m e try to in v estig ate the ro ta to ry m otion of a cylinder im m e rs e d in a viscous tw o-p hase m edium w ith m utual hydro d y n am ical in te ra c tio n included in the m otion of the individual ph ases. In this re s p e c t, it is logical to introduce am ong the v a r ious v a rib a le s the angular v elocity U )q of rotation. Thus if we denote, for a tw o-com ponent sy ste m such in te ra c tio n fo rce by F j2 then fro m re c ip ro c ity of actio n and in te ra c tio n we m ay w rite 17 and inview of the p reced ing (23) H e re Uj and a re the v iaco aities of the individual p h a ses, Dpm ia a m ean p a rtic le d ia m e te r defined in any a p p ro p ria te m a n n e r, and t|r is a shape facto r (aphericity) being unity for p erfect sp h e re s. Im plicit in the above e x p re ss io n ia the dependence of the in te r a c tion upon the lo cal velocity of the com ponents, m an ifeste d by in clusion of the in v e rs e of the ra d ia l d istan c e, r, fro m the axia of rotation. This re c ip ro c a l re la tio n sh ip ia suggested by the fact that the in te ra c tio n fo rce s m u st n e c e s s a rily vanish as the m otion g radu ally decays w ith in c re a sin g ra d ia l d ista n c e s r e g a rd le s s of d ifferen ces in the physical p ro p e rtie s of the m ix tu re com ponents. If the p ro p o rtio n a lity sign in E quation (2) is re p la c e d by a constant of p ro p o rtio n ality , K ^ , then the in te ra c tio n force F ^ b eco m es for the f ir s t m edium , and analogously (24) for the second m edium 18 w h e re ~ ^21 an<* a re r e ^e r r e ^ to aB in te ra c tio n coefficients. It m u st be noted, h o w ev er, that w henever a fluid sy ste m c o n sists of two im m isc ib le p h a ses, we find in v ariab ly that one phase is d is p e rs e d , hom ogeneously or o th erw ise, in the oth er. Although p h y sical d istin ctio n betw een the individual p h ases is not a m a th e m a tic a l n e c e ssity , it n on etheless is im p e ra tiv e to do so for p ra c tic a l c o n sid era tio n s. A ccordingly, throughout the p re s e n t w ork the e x te rn a l fluid is r e f e r r e d to a s "phase 1" w h e re a s "phase 2" alw ays denotes the in te rn a l o r d is p e rs e phase. 2, D ifferential E quations of M otion w ith In terp h ase Drag We sh all now w rite the N av ier-S to k es equations for each com ponent se p a ra te ly applying the concepts and p o stu lates in tr o duced e a r lie r . Following R akhm atulin [14] we define the "p o ro sity " o r volum e fractio n , f , for the n ^ phase of the m ix tu re by the re la tio n sh ip H e re pn &nd a re the red u ced and tru e d e n sities re sp e c tiv e ly , and fi + f2 + * * * + fN = 1 ’ ^ being the n u m b er of com ponents in the m ulticom ponent m edium . L et F jn be the m utual in te ra c tio n fo rce e x e rte d by com ponent j upon com ponent n in the N -co m p o nent sy ste m . If r , tp and z denote the ra d ia l, azim u th al and 19 ax ial co o rd in ates re sp e c tiv e ly , of a th re e -d im e n s io n a l s y s te m of c o o rd in ates, and v , v , v denote the velocity com ponents in the r q p z resp e c tiv e d ire c tio n s, then for an in c o m p re ssib le , Newtonian fluid in la m in a r m otion in the constant "p o ro sity " c a se , we have for the th , n phase: f ^ vn r , 3vn r „ ^vn r vnqp , Svn r l r-M o m en tu m : Pn ^— + v „r — + — ----------^ — J _ 5 &V = + t~ U_l----- T- + r - g - [ ^2 5 v , « + 1 ar2 r v | av _ av a2v "I ^ r r r ^ <jz J j = 1 ■ jn r J 4 n [ Sv 3v v dv v v + v ntP + — 2® . — IW B + . n r - ot n r or r oqp t dv i p . p r ^ v + „ >!£ = p F - _ ” I | E + _ » U ------«s> nz an J n n® p ^ r a® pn 0 nL Sr2 ♦ - > ♦ 4 % * + ^ + t H r r 3cp r ^ dc J N + s F. j 4 n (26) j = l J 1 1 ^ 20 r dv <Jv v z-M om entum : p | — + v —5“ + -7 ® n L ot n r or r 2 P- P r & v- - 1 = p F - I E + _ S _ U r + I I ^ S E n* P„o 8* cn0 n L 1 7 ^ r 8r I ~T r 8Zv nz 82 v 3© 3z nz " 1 7~ J N + . £ . F. j = 1 jnz J * and the com ponents of the p a rtia l s t r e s s te n so r a ssu m e the form N o rm al C om ponents T angential Com ponents ° n r r = (n [ - e +2un - 5 T 1] Tn r ,T fn«n [ r ^ ( - p2 ) r 3© J ntpr dv_ < ’™ W = fn [ - p +2un ( r 9 ncp £ ) ] r 1 av„ . 1 T = f u ! —2S6 + — — — n©z n n L dz r 5© J (27) = T nz© a nzz “ *n T"P+ ^u n &z av nz ] T - f U ' + n rz nM n &z ” ] = T. n z r 3. U n stead y R o ta to ry M otion L e t a v isc o u s in c o m p re s s ib le and h o m o g en e o u sly d istrib u te d tw o -p h a se m e d iu m occupying the annulus fo rm e d by two c o n c e n tric c y lin d e r s , in fin itely s e p a ra te d (F ig u re 1), be given an im p u lsiv e m otion. A t tim e t= 0, the in n e r c y lin d er is suddenly a c c e le ra te d fro m r e s t and m o v es w ith a c o n stan t c irc u m fe r e n tia l v elo city v0 = ; (a lte rn a tiv e ly , a p r e d e te rm in e d a c c e le r a tio n ra te m ay be im p a rte d in w hich c a se e x p e rim e n ta l m e a s u r e m e n ts can be tak e n a t w id e ly sp a ce d tim e in te rv a ls ). L et the v elo city com po n e n ts, and h e n ce the in te ra c tio n , of the m e d ia along the rad iu s and along the ax is be z e ro . T hus, c o n sid e rin g only the tp-com po nent of v elocity, we obtain fro m the s y s te m of E quations (26) fo r the c a se n = 1,2: 2 v, r - M om entum : (28) C p- M om entum : pj — ^ cW 1 2 < p (29) z-Momentum: z X Figure 1. Circular Flow and Coordinates 23 E quations (28) and (30) d e te rm in e the ra d ia l and ax ial p r e s s u r e d istrib u tio n s resu ltin g fro m the m otion and g rav itatio n al fo rce s resp e c tiv e ly . In the p re s e n t w ork, h ow ever, we shall co n cern o u rse lv e s only w ith obtaining a solution for the developing tangential velocity p ro files and the co rre sp o n d in g sh e arin g s tr e s s e s . A c cordingly, E quations (29), a fte r substituting for the in te ra c tio n te rm s th e ir p ro p o sed equivalen ts, becom e (31) w ith K j2 = ^ 2 1 ’ an<* an<* boundary conditions: Initial Condition: F o r all values of r B oundary C onditions: t > 0 : 1. vl<p= % B v0 For r = r 0 For r •* ® 24 The com ponents of the p a rtia l s t r e s s te n s o r of consequence a re : Tj-rqp = f,Ul [ r ^ ( ^ 2 ) ] = - ^ * 1 (32) We now apply the L aplace T ra n sfo rm a tio n in t to Equation (31) using the form ulae v i» , e ) ' £ e P \ , , (t> dt vz » ,p) = I . e PS » (t)dt w h ere v (p) is the L aplace tra n s fo rm of v ft) and p is the n c p * r n q j tra n s fo rm a tio n p a ra m e te r. C onsequently, the su b sid ia ry equations c o rresp o n d in g to E quations (31) a re . , r d v i© . i d v i© vi © l . Ku r ~ . 1 1 pl p v Kp' flu l + 7 “ d r - ~ ^ 1 + “ F t U1 ^ 2 ' Dpm* *-u,o J 7 (33) 2— — — r d v_ dv* v_ K_. r P2Pv2 flp = f2u2 [ “^ + 7 “K® - ’ ~ [ > ‘l"u2 ' Dpm '*',lo ]7 the boundary conditions being 25 i - - v 0 *• r = r 0 : v l * = % ' - 5- 2- r : vl „ = v2<p= 0 E quations (3 3) can be w ritte n as d^ v ,„ , d v , „ / , x kj dr » * « : ) • lcp “ pr , d v , „ , , ^ x d r ( - w ) * 2(p ~ pr w here 2 p 2 p 0 . • ^ : o* ■ £ (34) * , , v . - k 2i r s ^ B « ± a ] 1 12- fjUj J 2 21. ^ 2^ 2 ^1 u 2 H e re v , = - — , V-, = - — a r e the tru e coefficients of v isc o sity of 1 OlO 2 p20 the two p h ases. F ro m the p rec ee d in g rela tio n sh ip s for we thus deduce the re c ip ro c ity condition: 26 V l * l * -k2f2“2 < 35> To solve the sy ste m of E quations (34), we p e rfo rm the substitution R = q ^ r into the f ir s t which then becom es , d v . t . v k. + 5 - a s 2 - ( ^ r + 1) vi«p = - < 36> The hom ogeneous d iffe re n tia l equation, the B e s s e l's equation in this case , AZ ~ d v “ j 2 + r “ a K 2 ■ + 0 % ' 0 <37) h a s the solution Vlqj = c i rj (R ) + C ^K ^R ) (38) w h e re Ij(R) and Kj(R) a re the m odified B e s s e l functions of the f ir s t o rd e r and of the f ir s t and second kinds resp e c tiv e ly . F o r a com plete so lu tio n o f E quation (36) we m u st have, acco rd in g to the m ethod of L agrange, the functions Cj and C^ sa tisfy the form u lae: 27 KjfRHtR) c i = J iJdOfejdO - - Ij(R)f(R) c z = J i ^ r w c ^ w - K i m ^ H e re f(R) ------= -& , the rig h t hand side of E qu atio n (36). Introdu - P9 i K cing the W ro n sk ian re la tio n Ij(R)Kj(R) - Kj(R)Ij(R) = into the d e n o m in a to r of the in te g ra n d s of E q u atio n (39) and p e r f o r m ing the in d efin ite in te g ra tio n , we find k k C, ------- f K. (R)dR = — — Kn(R) + C , p q ! J i pqj ° 3 (40) c z = ^ 7 X V R)dR ■ 1^7 V R>+ c4 C om bining E quation s (38) and (40), and u sin g the re la tio n I0(R)Kj(R) + Ij(R)K0(R) = ^ The com plete solution of E quation (36) becom es 28 T V 1 ^ + C3Il<R>+ C4 K 1<R > < 41> A solution w hich re m a in s finite as r * ♦ °»(C^ - 0) is ''lip = + C4 K 1(R) R eplacing R by q ^ r and fro m the boundary condition r = r^: V0 v i* = T ’ we ® et ki , r vo ki i Ki<V> V l g = ~ * ~ + L ~p TT K,|q,r.| (4Z> p q t r p p ^ ! r0 1 1 0 Using the In v e rsio n T h eo rem : i ra+ltu i t - V l< p (t> = 2 S Iff-to* V l< p U,dX T h ere fo re ... kl V . v0 r + -K i(pyx/M,).U dx vlip * ~ r ~ + 2^T . ! — T ( 4 3) C1V1 1 f +" K1<'A /V | ) .W V q " Z ttI J - » K ^ r ^ X / V j ) d\ 7 29 By m eans of the C alculus of R esid u es an infinite s e r ie s solution is possible. Thus, by choosing an a p p ro p ria te contour in te g ra tion path the contribution from all the sin g u la ritie s in the denom in ato r of the in teg ran d s can be found. Often, how ever, s e rie s solutions thus obtained a re slow ly convergent, inconvenient to use or seldom suitable for p ra c tic a l applications. Consequently, an a lte rn a te solution m u st be sought. This is d isc u sse d in the following sections. * By substitution of the identity K. (iz) = -£ tt[ J . (z) - iY. (z) ] and 2 ±iir letting \ = VjU e on e ith e r side of the cut along the negative re a l sem i ax is, the in te g ra ls m ay be tra n s fo rm e d into of Equation (43) in te rm s of B e sse l functions (in this case J^ , 2ni J -» X 2, and J ^ u r J Y ^ u r ^ - Yj(urJJ^urg) 0' du — X u 3. I Solution for S m all V alu es of the T im e 30 R easo n ab ly a c c u ra te so lu tions to m any p ro b le m s of h e a t conduction o r v isc o u s fluid m otion in c y lin d ric a l c o o rd in a te s a r e o btainable by m e a n s of the a sy m p to tic ex p an sio n s of the B e s s e l functions. F o r sm a ll v alu es of the tim e , we m ake use of the a sy m p to tic expansio n + • • n = 0, 1,2 • s « in the in te g ra n d s of the equation F o r the first w e have: -«*i<r - r 0> {**) 31 and for the second l Ki <qi r) _ i n /r r"q i (r‘ r °Vi , 3 3 1 ^ 1 7 ‘W 7 ' 7 L v ° / r e l1 + ^ + • • •) J The sy ste m of E quations (44) m ay now be in v erted , each te rm se p a ra te ly , using the In v ersio n T h eo re m o r any convenient ta b u la tion of the L aplace tra n s fo rm s . F ro m the la tte r, we find that the in v e rs e of the quantity p"*” T e ”* ^ is 4 t^ n ine rfc -£r- . Thus v p erfo rm in g the re q u ire d in v e rsio n and re a rra n g in g som e of the te rm s , we obtain a solution of Equation (36) su b ject to the initial and boundary conditions of E quation (31), suitable for sm a ll values of the tim e: k iV|‘ Vl(p V - + vo ( -f) [erfc,l i + V 4V (llr - TB^)Z ie r fc T li ] ( ^ ) V — , +V- M h ' , ] and com pletely analogously we w rite (45) 2< p * + v0 ( - r ) i [ erfcT> 2 +V4V ( t I f - )2i.rfcr,j ] — 7 ~ ( " r ) [ 4l2®rfcT' 2 +V 4v2t ( ? 7 - ? F ^ ) 63,*rfc,’2 ] 32 for the second (internal) phase, w h ere r “ ro r - r 0 ^2 = v/4v_t and erfcri is the c o m p lem e n ta ry Gauss-error function defined by 2 r " - e rf c T ' = V n J n e d? v / tt J ti the in te g ra l of the c o m p lem e n ta ry erro r function. inerfc<p, is given by i* ® i inerfcri = [ in” erfc^d? , n s 1,2,3,,, with terfcr) = erfcrj. and the g e n eral r e c u r re n c e fo rm u la 2ninerfcri = in ^erfcri - 2nin *erfcn, n= 0, 1,2 Applying the boundary conditions to E quations (45) and taking note of the following erfc(O) = 1 erfc(») = 0 33 we see at r = r^: v^ = = v^, and as r tends to infinity: = 0 as stated in the p re m ise . Also from the in itia l condition t = 0, we find: = 0. 3. 2 Solution for L arg e V alues of the T im e A solution to the co n sid ere d bou n d ary -v alu e p ro b lem s u it able for la rg e values of the tim e involves the asym ptotic expansion fottie B e s s e l functions in the ascending s e r ie s : 3 kjU) = ^ + r Log £ z + y] I j j — (l + l + i ) + ***] (46) ^ 2 * 4 w h e re y is E u le r 's constant = 0.5772... . F u rth e r sim p lificatio n 2 n _ is p o ssib le if it is reco gnized that a term q^ in w h e re n is in te g e r, w ill not con tribute to the re su lt. Hence Ki (qi r) ro T 1 + r2jlj0e r 1 Kl (ql ° * r L 1 + ^ q jr ^ L o g iq jr Q J = ^ + i q j r 2 L o g i q i f - frq2 34 By B rom w hich 'a In teg ral fcqJ^Logfcqjr = jki I X" ^ql f2 Lo* ^ i rdX w h ere C is a curve fro m < 3 - i" to cj+ i» and C D EF is the Integra* tion path, not containing the o rig in , form ed by the n a rro w slit on each side of the negative re a l sem i axis from - < ® to -€ and -e to - * ; e being the in fin itesim al radius of the c irc le draw n around the origin. A ccordingly, the final re s u lt given by B rom w hich [15] is | q j r 2 L o g i q i r = - ^ o r m o re g e n erally 2n (*q,r) Logiqjr = i(-?i-F )r(n) T hus, the first in te g ran d in E quation {43) can be e x p re s s e d as w hich fro m p re c e d in g a n a ly s e s , is e q u iv alen t to 35 2 2 S im ila rly , the seco nd in te g ra n d b eco m es I P - ( 1 + ^ q i r Z L o g * q i r “ * q i r o Lo« * q i r o ) ] and the in v e rs e th e re o f is -T rt - 4v,t ° Log a + ^ - ( r 2 L ° g ^ ~ - r0 Lo* ^ ' ) " ' w h e re log C = y = 0, 5772. . . . The c o m p lete solution for v elocity, a fte r sim plifying, is th e re fo re 2 2 r0 • - r ' r0 1 klT 2 4v.tC 2 4V C-. Vl<p = v 0 T L1 ‘ 4vjt J + 7 ? [ r Log — 1.~ ’ r 0 L ° b T ] and quite an alo g o u sly fo r the second p h a se, we w rite (47) 2 2 r o r . ' ’ r0 ] *2 r z . v2 2 . . . V2 1 '2 (P o T L - 4v,t J + — r ~ • r0 Log “"2 J 4 0 36 E q u atio n s (47) d e s c r ib e the d ev elo p m en t of the velocity p ro file s u n d e r the r e s tr ic tio n s dem an d ed by the a sy m p to tic ex p an sio n of the B e s s e l functions, n a m e ly w hen the d im e n sio n le s s tim e v a ria b le 1® la rg e . C onsequently, the s y s te m of E q u atio n s (47) tend to be r0 m o s t a c c u ra te in the im m e d ia te neighborhood of the in n e r ro ta tin g cyl in d e r i. e. w hen r is sm a ll. H o w ev er, as r in c r e a s e s o u tw ard , s u b s ta n tia l d ev iatio n s m a n ife s te d by d isc o n tin u itie s in the velo city p ro file s m ay o c c a sio n a lly develop depending upon the m agnitud e of r. T his is not a d is tr e s s in g r e s u lt as one m ig h t f i r s t tend to co n clude. Indeed, it is fo rtu ito u s th a t the expansion s e r i e s e x p re s s e d by E quation (46) le a d s to a c c u ra te re s u lts w h e re a c c u ra c y is d e m anded m o st, n a m e ly in the p ro x im ity of the in n e r c y lin d e r w a ll w h e re e x p e rim e n ta l s h e a rin g m e a s u re m e n ts a r e tak en a t p re c is e ly r = r^ . O ur next ta s k , th e re f o re , w ill be to c a lc u la te fro m the p rec ed in g r e s u lts the s h e a rin g s t r e s s e s e x e rte d conjointly by the two p h a se s on the in sid e c y lin d e r. 4. D e te rm in a tio n of the T an g e n tia l S tr e s s e s Having d e riv e d e x p re s s io n s for v e lo c ity d is trib u tio n along the ra d ia l c o o rd in a te , we sh a ll now ex am in e the d ev elo p m en t of the tan g e n tia l s t r e s s e s o rig in a tin g b e c a u se of the f i r s t and seco n d phase m o tio n s. The com ponents of the p a r tia l s t r e s s te n s o r of co n seq u en ce to the p re s e n t in v e stig a tio n have been p re v io u s ly defined by 37 Tl r -cp = for the fir at p h a se , and T2 r -cp = - f2u2r ^ ( _ | £ ) for the seco n d ph ase. The in tro d u c tio n of the negative sig n s in the above e x p re s s io n s is s tr ic tly a m a tte r of convention. F u r t h e r m o re , it should be noted th at w h en ev er the second p h ase is a s u s p en sio n of so lid p a rtic le s , the p h y sic o -m e c h a n ic a l c o n sta n ts i. e. , the co efficien t of v is c o s ity (i^ and the in te ra c tio n coefficient m u s t be d e te rm in e d fro m e m p iric a l o b se rv a tio n s. The com bined p h ase o r " m ix tu re " s h e a rin g s t r e s s r „ i s , th e re fo re , the r mr-cp a lg e b ra ic sum of the individual c o n trib u tio n s. H ence T = -f u r ±(Zl &)_ t u r “ f — \ m r-cp 1U1 3r \ r J W a r \ r J (48) The to rq u e tr a n s m itte d by the fluid to the ro ta tin g c y lin d er o r, m o re p e rtin e n tly in th is c a s e , the sh e a rin g s t r e s s e x e rte d on the c y lin d e r su rfa c e m a y be e v alu ated fro m E qu ations (48) fo r the condition r= r ^ . A gain we sh a ll c o n sid e r the two d o m ain s of the tim e fo r m e r ly in v e stig a te d . 4. 1 Sm all V alues of the T im e The tangential sh e arin g s tr e s s e x e rte d on the c y lin d er w all is thus by E quation (48) l \ , r^i® »i»i . rav2 < p 'Wi l T mr-<pJ = T m = _flul L“ a T ' r J ' f2u2 . “ a T ' r J r = r 0 r = r Q r (49) F o r sm a ll values of the tim e, we use Equations (45) and p e rfo rm the re q u ire d m a th e m a tic a l o p e ra tio n s in E quation (49) taking note of the following r_ro 1 1 = 7 ? v t ^ ( 2ie rf c ri) = - 2erfcri ^ ^4i2e rfc r) )a -4 ierfcr| ^ 6i ^ e r f c r | 6i2erfcr| a - -2. d r " 5ti 8r w hence 39 r r 3 . 2 i 3 1 * / 4V i* i = fiu i vo L ^ + v ^ r v ^ t " + k , v, t + l V ( 1 4 1 J 4 v l t ^ 1 ' 0 r 0 ^ " V ^ ^ V " 4 ? * n and s im ila r ly (50) p 3 2 3 1 /4 v2* t 2 = f2W 2v 0 ZTT +C74rTv7t + S T f T ” 0 2 r 0 , k 2 V2 fc / 1 4 1 1 v rtr ~ \ 2 r _ - v ' 4 T T \ u t ” 7" I f tt / J 0*0 ' “ *'0 4 r - o The s y s te m of E quations (50) d e s c rib e s the tim e -d e p e n d e n t s tr e s s e i e x e rte d by both p h a se s w hen t is sm a ll. T hus the com bined " m ix tu re " s h e a rin g s t r e s s t . a m e a s u ra b le q u an tity , a s s u m e s the fo rm £ , r 3 . 2 . 3 1 . / ‘ V Tm = n t : i fnu nvoL + ~ 4 ttvT + 8 , kn V / 1 4 i J I V ) - ] V o 4 ‘z \ n J J (31) w h e re vQ = (U qTq = 2TTNQrp/60, and N q is the c y lin d e r ro ta tio n a l sp eed in rev o lu tio n s p e r m inu te. It is in te re s tin g to note th at w hen t=0, the q u a n titie s T. and T_, and h en ce T , s t a r t fro m in- finity and then d e c r e a s e to th e ir final stead y sta te v a lu e s, a s itu a tion re m in is c e n t of skin fric tio n in the bou n d ary la y e r on the leading edge of a flat p late w h e re x »0. 4, 2 L a rg e V alu es of the T im e We have shown that for la rg e values of the tim e , the v elo city p ro file s a r e e x p re s s e d by for the f ir s t m ed iu m , and fo r the second m ed iu m . By E quation (49) the tan g e n tia l sh e a rin g s t r e s s at the w all is 4 V C 1 L o g j — J 0 P e rfo rm in g the d iffe re n tia tio n and lettin g r -* r^ , we obtain 41 w h e re Log N = - f c S um m ing up the in d iv id u al effects and applying the re c ip ro c ity condition fl u I k l = - f2uZk E we o b tain an e x p re s s io n d e s c rib in g the s h e a r s t r e s s h is to r y at the in n e r cy lin d er su rfa c e due to the co m b in ed m otion. Thus 2v0!\ A „ r0\. , . ft . r0 ^ f2U2k2 T V2 1 Tm " r fl L 1W 1 \ + W ^ t ) 2W 2 \ l + 4 ( |> 0 8 Vj J ( 5 3 ) 2T TN Or O w h e re , as b e fo re , = u ^ r^ = — — and is the ro to r speed in rev o lu tio n s p e r m inute. Thus fa r, we have not sta te d e x p lic itly and q u a n tita tiv e ly w hat c r i t e r i a one w ould u s e in d e te rm in in g th e o p e ra b le ran g e of the p re v io u s ly developed eq u atio n s and the point a t w hich the two tim e d o m ain s b eco m e coin cid ent. O bviously, it is in a d m is s ib le to u se the d e riv e d re la tio n s h ip s o u tsid e th e ir in ten d ed ran g e of a p p lica b ility - th a t w hich is co m p atib le w ith the c o n s tra in ts 42 im p o sed in the ex p an sio n of the B e s s e l functions. If, for ex am p le , a plot of w all s h e a r s t r e s s h is to r y is c o n stru c te d fro m E quations (51) and (53), it would then be p o ssib le to e s ta b lis h the co n v erg en c e point in tim e , and h e n ce the c o rre sp o n d in g ran g e of each , sin ce each equation in its re s p e c tiv e tim e d om ain d e s c r ib e s a w ell b e haved function and as such m u st n e c e s s a r ily yield a continuous p ro file . A convenient and freq u e n tly u sed c r ite r io n in such plots is the d im e n sio n le s s tim e v a ria b le T = — , the analogue of the F o u r ie r n u m b e r in h e a t conduction. T hus, fo r op tim um u se of the r e s u lts in su b seq u en t a n a ly se s, we m u st f ir s t tr a n s fo r m the equ ations into n o n -d im e n sio n a l fo rm . 5. N o n -D im en sio n al F o rm of the D eveloped E quations We sh a ll now tr a n s fo r m the v elo city d is trib u tio n and s h e a r s t r e s s eq uation s to n o n -d im e n sio n a l fo rm su itab le for fu rth e r a n a ly tic a l and e x p e rim e n ta l d e te rm in a tio n s. T h u s, v e lo c itie s and d ista n c e s sh a ll be r e f e r r e d to th e ir re s p e c tiv e v alu es at die s u r face of the c y lin d e r, and the d im e n sio n le ss tim e v a ria b le T sh a ll be as defined p rev io u sly . A cco rd in g ly , we in tro d u c e the follow ing notation D im e n sio n le ss D is ta n :e H = r 0 v D im e n sio n le ss V elo city V = ™ n vQ D im ension less T im e T = — r 0 Since the product f |i h a s die dim ensio ns fo rce p e r unit a re a , n n u a n o n -d im en sio n al sh e a rin g s tr e s s m ay be defined by * Tn T" = W o Note that the stead y state value of the skin fric tio n at the surface of the cylinder in single phase flow is equivalent to a p rim e facto r in the above definition. Finally, we define a n o n-dim en sional (interaction) frequen cy by w h ere n, in the above equations, denotes the n* * 1 phase of the two phase m edia. 5. 1 V elocity D istrib u tio n U sing the above notations for the sy ste m v a ria b le s , the v elocity p ro file s, w hen T is sm a ll, becom e 44 0 T V n = +v 75' [erfc + V4Tn(T5R- TC)ZlerfcJ n ^ l ■ 0 n T n r 4 , 2 e r f c > ^ ; r ’ + V 4 T n ( ? T r ‘ ? K , r £ c ^ T ] n = * > Z and w hen T is la rg e (54) 1 T, R 2 - 1 ( 1 n 4 T n C ) 1 . , Vn=RL1 ? \ tT ‘ °nLog ' J . n=I’2 w h e re 2 n ' - 2R R = ~ r R - 1 5. 2 W all S h e a r S tr e s s F o r s m a ll v alu es of T, the tim e -d e p e n d e n t (d im e n sio n less) skin fric tio n on the s u rfa c e of the c y lin d er m a y be w ritte n as Tn “ [ I V i T ^ + + ftn T n ( l ' C ^ ; ‘ 2' V ^ L ) ] (5 5 ) n = 1, 2 The tw o -p h a se sh e a rin g s t r e s s is c a lc u la te d fro m 45 (56) and £ o f u = o . n n n n= 1 F o r la rg e T, we have * _ 4T +1 0 m2 , = ! r T T - * T L» ' I R n n n - 1, 2 (57) w h e re , as b e fo re , T = _ £ , U)«T*f U m n = I O n n n (58) and Log N = £ Log C = v = °* 5772 2 , n nfn“ n = 0 n = i 6 . D e riv a tio n of the C o efficient of V is c o s ity of the M ix tu re We now p ro c e e d to develop a g e n e ra l e x p re s s io n fo r the c o efficien t of m ix tu re v is c o s ity fro m the r e s u lts obtained thus fa r. 46 We shall define a tw o-p hase or w eighted m ix tu re velocity by the form ula rrup " f 1 v lqp + *2 v 2tp (59) o r equivalently in n o n -d im en sio n al fo rm w h ere each te r m in E quation (59) has been divided by v^, the c i r c u m fe ren tia l velocity at the su rfa ce of the cylinder. Applying N ew ton's hypothesis to the com bined tw o-ph ase m ix tu re , we can w rite the sh e a rin g s t r e s s in cy lin d ric a l p olar co o rd in ates as which upon substituting E quation (60) and rep la cin g r by r^R b eco m es w here um is the coefficient of v isc o sity of the m ix tu re. In the i m m ed iate section to follow we sh all d e riv e a g e n e ra l e x p re ss io n for the coefficient of v isco sity , u m > r e g a rd le s s of the sta te of m a tte r of the d isp e rso id . In subsequen t c h a p te rs e m p iric a l m ethods to m r - c p (62) 47 d e te rm in e the p h y sic o -m e c h a n ic a l c o n sta n ts of the m ix tu re w ill be su g g e ste d . 6 . 1 A G e n e ra l E x p r e s s io n for M ix tu re V isc o sity E q u atio n (62) can be r e a r r a n g e d as dV, V, x / dV , V, T m r -<p = - “ m - r {R [fi (-a ir - if )+ f2 (nr - if ]} E quating the value of th is a t R = 1 to th at given by E quation (58) v0 and re p la c in g — by u)n , we get r o u -“mU lo{R[fl ( ' W ' l f ) + f2 ( n f - l f ) ] } R=1' ,J ,0ITlflUl+T2f2U 2 ] o r in s h o r t fo rm r 2 *V V 2 ’um“0 [ nPlRfn VHT' Tt )] = nPl“ V n fnun ♦ w h e re t , for re a s o n s w hich w ill b eco m e a p p a re n t, a r e those e x p re s s e d by E quations (54) and (57) re s p e c tiv e ly for the c a se of la rg e T. Solving for lim » the co efficien t of m ix tu re v isc o sity , and c an celin g like te r m s we obtain P e rfo rm in g the d ifferen tiatio n in the denom inator of Equation (63), letting R= 1 and taking note of the re c ip ro c ity condition in E quation (35), Equation (63) b eco m es 4 T .+ 1 4T ,+ 1 f7| T ? flu l T T - + f2u2 T T " - H -1 t ; --------- ! ----------- 1 r k « 64» fl f2 f2U2n 2 , (4 T2 C / n 2 ) ‘2 1 + 4T j + 4T2 3 L° g 1/u, (4TjC/N 2 ) ^2 w h ere ia the d im e n sio n less in te ra c tio n frequ ency, viz, 0 ~ = — l £ . U )Q and N and C a r e constants defined in E quation (58). A ccording to N ew ton's h y p o th esis, the coefficient of m ix tu re v isc o sity is constant for a given frac tio n a l volum e f below n a c e rta in c ritic a l value, w hereupon the m ix tu re begins to exhibit non-N ew tonian tr a its . T h e re fo re , the m e re a p p earan ce of the tim e v a ria b le , T in E quation (64) does not n e c e s s a rily re fle c t tim e dependency as such since as T is v a rie d both n u m e ra to r and denom inator w ill change p ro p o rtio n a te ly such that a constant o v e ra ll value is m aintained. F u rth e r sim p lificatio n of the re s u lt, h ow ever, is p o ssib le since under the conditions being in v estig ated 49 (T is larg e) rea so n a b le app ro x im atio n s can be m ade w ithout c a u s ing significant e r r o r . H ence, we let 4 T . + 1 — « 1 i 4 T , + 1 T T — ~ 1 F u rth e rm o re , f l <<: 4 T 1 ; f2 << 4T2 w hence W ~ v 0 2 Introducing these ap p roxim ation s into the e x p re ssio n for v isc o sity ^2 ^2 and noting that = — , we get L l V1 f2u2n 2 . v2 flu l + h H 1 ^ v f — ,6 5 ) W i . (^Tj C /N 2 ) 1 ------^----- Log (♦ T ^ C /N 2 ) UMi 50 A t this ju n c tu re , a fu rth e r s im p lific a tio n b e c o m e s a p p a re n t if we ex am in e the lo g a rith m ic t e r m in the d e n o m in a to r of E quation (65). R e a rra n g in g Log ( * t 2 c / n 2 ) * ^ T j C / N 2) 1 = Log 1 1 As t ten d s to infinity, le t us a p p ro x im a te the fra c tio n a l p a rt 1 1 / 4 C t W r0 NY ‘ H by a n u m e ric a l co n stan t, say Cq, T h is, h o w e v er, can be e ith e r a b s o rb e d in the fre e c o n stan t o r m &de equal to 1 w ithout lo ss of g e n e ra lity . A cco rd in g ly , the final r e s u lt w ill be given as f2u 2 ° 2 . v2 U m flUl + f2 ^ 2 ------- 1 — L°8 ^ 1 - f2U2 ° 2 Log o r eq u ally valid (66 ) 51 . , f2U2n 2 T v2 £1U1 + f2w2 ■ Lo« ^ ; Um = ----------------------------------------------------- < 6 6 > f,0 *2 2 r u 2 1 1 - - 5- [L ogV 2- — L o g v J 6 . 2 An A lte rn a te E q u atio n fo r M ix tu re V isc o sity It can be show n — in single phase flow s - that the w all sh e a rin g s t r e s s e x e rte d by the m o tio n of a fluid o u tsid e a ro ta tin g cy lin d er is e x p re s s e d by T = 2 |J tu 0 By analogy, we m ay choose to define a tw o*phase o r m ix tu re sh e a rin g s t r e s s as t = 2U_U)n (67) m m u T hus, equating th is to the e x p re s s io n given in E q u atio n (58) we obtain 2 2u tl)A = ^iU )rtT f U m 0 n - 1 0 n n^n w hence C hoosing T la rg e , and su b stitu tin g the e x p re s s io n given by E q uation (57), E quation (68 ) b e c o m e s 4 T 1+1 4 T 2+1 n Z n 2 u m = flUl 4 f ” + f2u2 4 T 2 + Q1 flu l L o g 4 T 1 £ + 0 2f2u2 L,Og? T ^ C (69) By analogous re a so n in g em p lo y ed p re v io u sly , w e m a y se t 4 T . + 1 T T T " 1 l 4T-+1 c » 1 ^2 V2 and since " ^ 2*2^2 ' 1 f r~ ~ “ *^e e x p re s s io n a s s u m e s the fo rm _ r ,, X f ,, ° 2f2u2 T __ V2 wm ' 11 2 2 3 g v7 (70) In c o n tra s t to E quation (66 ) the r e s u lt e x p re s s e d by E q u a tion (70) does not r e f le c t the e ffe cts of the h y p o th e s is e d h y d ro d y - n a m ic a l in te ra c tio n on the sh e a rin g ra te sin c e by v irtu e of the 53 definition in E quation (67) it h as been p u rp o se ly excluded, although h ow ever, it is evident in the e x p re ssio n s for sh e arin g s tr e s s and m ix tu re v isco sity . T hroughout the rem a in in g ch ap ters of this d is s e rta tio n e sp ec ially in the c h ap ter on the IBM co m p u ter p ro g ra m given in Appendix B, Equation (66) sh all have the desig nation of E quation 1 w hile E quation (70) shall be r e f e r r e d to as E quation 2. 6 . 3 M ethods for D eterm ining the P hysicoM ech anical C onstants E quation (66) contains, in the case of a solid d isp e rso id , two unknowns ft 2 ( ° r ^2 ^ an<* ^ 2 ' Provi fli ° n&l coefficient of v isc o sity of the solid phase. Consequently, it cannot be used d i re c tly to solve for both unknowns since, at fixed frac tio n a l volum e, is constant. H ow ever, if physical m e a su re m e n ts of the t r a n s ien t skin frictio n at the su rface of the c y lin d er a r e m ade at t = tg and t = tg + At, then by sim ultaneous solution of the ensuing s tr e s s equatio ns, n u m e ric a l values for the coefficients can thus be ob tained. M athem atically 2 T m ( t » * n ? l W n < 7 1 > w h e re t* is given by e ith e r E quation (55) o r E quation (57) d ep en ding on the m agnitude of the tim e. If, h ow ev er, the m ix tu re is fo rm ed fro m two im m isc ib le liquids as done in the c u rre n t e x p e ri m en ts, the coefficient of v isc o sity u-, i* known and hence only one m e a s u re d value of Tm w ill be re q u ire d a ssu m in g that 0 ^ 1* tim e-in d ep en d en t. F u rth e rm o re , if the in te ra c tio n freq u en cy is not c o n sid ere d a function of the physical dim ensio ns of the sy s te m - a not w holly u n reaso n ab le p re m is e - then E quation (66 ) becom es m o st advantageous to use in its e m p iric a l evaluation, for v isc o sity m e a s u re m e n ts m ay be m ade in any conveniently a rra n g e d flow a p p aratu s. Indeed, we m ay use E quation (66) for e x p erim e n tal d e te rm in a tio n of the p h y sic o -m e ch a n ic a l coefficients of a liquid- solid sy ste m if it is reco g n ized that by changing the fractio n al volum e fn by an in fin itesim al am ount the coefficient of m ix tu re v isc o sity changes p ro p o rtio n ately , w h e re a s the in te ra c tio n freq uency being le s s se n sitiv e , re m a in s n e a rly constant o v er this n a rro w range. C H A PTER III E X P E R IM E N T A L A. P ro c e d u re T he e x p e rim e n ta l p ro c e d u re c o n siste d of s e v e r a l ta s k s in w hich the sa m p le w as f i r s t p re p a re d , te s te d in a v is c o m e tric flow a p p a ra tu s , view ed u n d e r a m ic ro s c o p e and photo g rap h ed for su b seq u en t d e te rm in a tio n of p a rtic le siz e and siz e d istrib u tio n . The sa m p le s in v e stig a te d w e r e m ix tu re s of two im m is c ib le liquids o f the o il/w a te r v a rie ty s ta b iliz e d by a u n iv a le n t (sodium ) s a lt of o leic acid. By m ak in g c a re fu l a d ju s tm e n ts of the s m a ll o rific e of a h a n d -p o w e re d h o m o g e n iz e r, it w as p o ssib le to c o n tro l the d e g re e of d is p e rs io n so a s to fo rm th re e c a te g o rie s of e m u lsio n s a t any one given c o n c e n tra tio n . In re la tiv e te r m s th e s e w e re c la s s ifie d as c o a r s e , m ed iu m , and fine; although, in a b so lu te te r m s the c o a r s e s t sa m p le m a y qualify as "fin e". Upon changing sa m p le c o n c e n tra tio n , the fra c tio n a l volum e of the o rg a n ic (d is p e rs e ) phase w as g ra d u a lly in c re m e n te d u n til the s h e a re d sa m p le began to ex h ib it d e fin ite tre n d s c h a r a c te r is tic of n on-N ew ton ian fluids. In no c irc u m s ta n c e w a s th is n e a r the volum e c o n c e n tra tion c o rre sp o n d in g to c lo s e packing of u n ifo rm s p h e re s (approx. 74 p e rc e n t) and r e q u ir e d for the o n se t of p h ase in v e rsio n . 55 56 In T able 1, a re the im p o rtan t p h y sical p ro p e rtie s of the o rg anic com ponents se le c te d in the p re s e n t study. The v ario u s c o n cen tratio n s em ployed, the density, and the v isc o sity of the aqueous phase a re given in Table 2, The p e rc e n t co ncentrations by w eight of the s ta b iliz e r shown a re equivalent to 0. 001 m ole sodium o leate p e r 100 cc em u lsio n , w hich, in all c a s e s , w as suf ficient for com plete d isp e rsio n . 1, Sam ple P re p a ra tio n T h ree sa m p le s c o a rs e , m edium , and fine, each consisting of 2 5 cc of em ulsio n, w e re p re p a re d fo r each of the volum e f r a c tions: 0. 1, 0.2, 0. 3, 0,4 and 0. 5 of the organic phase. The aqueous phase w as fo rm ed a t ro o m te m p e ra tu re by dissolution of 76 m g of " p ra c tic a l" sodium o leate in an a c c u ra te ly m e a su re d volum e of d istilled w a te r that c o rre sp o n d s w ith each of the above c o n cen tratio n s. The org an ic phase w as then added slow ly but continuously to the aqueous phase, in itially contained in a 50 cc bottle, w ith a slight shake at the end of the addition, followed by the p a ssa g e of the m ix tu re once in a h a n d -o p e ra te d h o m ogenizer of the la b o ra to ry type. The issu in g m ix tu re w as co llected in a 150 cc b e a k e r, and then tr a n s f e r r e d to a n o th er 50 cc bottle fro m w hich a p p ro x im ate ly 8, 0 cc sam ple w as w ithdraw n for flow m e a su re m e n ts . T his p ro c e d u re w as re p e a te d th re e tim e s for each concen tration varying only the a d ju stm en t of the sp rin g loaded valve, acting as the h o m o g en izer o rific e , to obtain a c e rta in d e s ire d ran g e of TABLE 1 SOM E PHYSICAL PROPERTIES OF THE DISPERSE PHASE Temperature * 20°C Organic Component Dens Ity gm/cc Viscos i ty Cent!pot se Kinematic V iscosity Centistoke In te rfa c ia l Tens I on/Water dyne/cm SolubiIi ty In Water gm/100 gm Benzene 0.876 0.710 0.808 35-00 0.07 o-Xylene 0.878 0.837 0.950 36.06 Insoluble H1ne r a 1 Oil 0.834 21.700 25.740 52.50* insoluble Linseed Oil** 0.923 33.700 36.510 21.60* 1nsoluble cls-D ecalin 0.883 2.360 2.670 46.50* Insoluble n-Hexadecane 0.765 2.780 3-590 50.60* Insoluble **Purlfled ( a r t i s t ' s grade) * These data were furnished by Chevron Oil Field Research Company, La Habra C alifo rn ia TABLE 2 SAMPLE CONCENTRATIONS AND AQUEOUS PHASE PROPERTIES Temperature * 20 °C Fractional Volume Volume Volume S ta b iliz e r* V iscosity Density Disperse Phase Disperse Phase Aqueous Phase Wt./Vt. Aqueous Phase Aqueous Phase Aqueous Phase -cc- -c c - -C entlpoise- -gm/cm- 0 0.0 25.0 0.00 1.005 0.998 0 0.0 25.0 0.30 1.040 0.998 10 2.5 22.5 0.34 1.070 0.998 20 5.0 20.0 0.38 1.080 0.998 30 7-5 17.5 0.44 1.080 0.998 4o 10.0 15.0 0.51 1.090 0.998 50 12.5 12.5 0.61 1.100 0.998 * Each concentration is equ iv alen t to 0.001 M Sodium Oleate per 100 cc emulsion ut 00 59 p a rtic le s iz e s . T hus, Sam ple 1 (c o a rse ) w as p r e p a r e d w ith the sm a ll o rific e w ide open, S am ple 2 (m edium ) w ith an in te rm e d ia te p o sitio n , and S am ple 3 (fine) at the tig h te s t p o sitio n of the valve. 2, R h e o lo g ica l M e a s u re m e n ts F r o m th e fre s h ly p r e p a r e d e m u lsio n , a sa m p le m e a su rin g a p p ro x im a te ly 8. 0 cc w as w ith d raw n in a 10 m l p ip ette, placed in the m e a s u rin g s y s te m and te s te d at s e v e r a l sp e e d s in a Haake ro ta tio n a l v is c o m e te r (R otovisco) p ro v id e d w ith a te m p e rin g v e sse l. T he m e a s u rin g unit c o n siste d of a fixed o u te r cup and an in n er ro to r w hich w hen a s s e m b le d fo rm two c o n c e n tric annuli w hich a re fille d w ith the sa m p le . The te m p e r a tu re of the sa m p le throughout la m in a r s h e a r w a s m a in ta in e d c o n sta n t a t 20° C by m e a n s of a LAUDA C irc u la to r equipped w ith a th e r m o r e g u la to r and a so lid - sta te th y r is to r c o n tro l s y s te m , The sp e e d s s e le c te d ran g ed fro m 64. 8 rp m to 292 rp m c o rre sp o n d in g to s h e a rin g r a te s of 41, 8 and 1884. 0 in v e r s e seco n d s re s p e c tiv e ly . W hen the flow a p p a ra tu s w as tu rn e d on and the sam p le se t in m o tio n , the s c a le d e fle c tio n on the v is c o m e te r in d ic a to r m e te r w as re c o rd e d fo r each sp e ed and la te r tr a n s fo r m e d into a sh e a rin g d ia g ra m . At the end of each ru n , a p r e lim in a r y plot of sc a le d e fle c tio n a g a in s t sp e e d w a s c o n stru c te d to a s c e r ta in p o ssib le non lin e a r tre n d s in the s t r e s s - s t r a i n re la tio n sh ip . T im e dependency w a s in v e s tig a te d w ith the aid of an a u to m a tic r e c o r d e r , connected to the e le c tr ic a l output le a d s of the v is c o m e te r fro m w hich a 60 continuous plot of scale reading v e rs u s elap sed tim e w as obtained. In this m a n n e r, it w as p o ssib le to exclude sa m p le s which exhibited non-N ew tonian p ro p e rtie s fro m fu rth e r a n a ly sis. T hese sam p les contained, in v ariab ly , the h ig h est c o n ce n tra tio n s of the d is p e rs e d phase. At the conclusion of the rh eo lo g ical m e a s u re m e n ts , sp e cim ens w e re taken fro m the sh e a re d sam p le for m ic ro sc o p ic ob se rv a tio n s and photographing; the re m a in d e r d isc a rd e d and a new sam p le containing the next h ig h er c a ic e n tra tio n p re p a re d . 3. P a rtic le Size M e a su re m e n ts Im m ed iately following the v isc o m e tric flow m e a s u re m e n ts , the em ulsion w as exam ined for d ro p let siz e , shape, and size d is tribution. This w as acco m p lish ed by placing the sp e cim en on a c lean g lass slide under a L eitz O rtholux m ic ro sc o p e fitted w ith a L eica c a m e ra . A thin c o v e r-g la s s w as m ounted over the sp e cim en form ing an a ir - tig h t gap which p rev e n ted lea k ag e s. High in ten sity illu m in atio n w as provided fro m a xenon lam p so that v e ry fine p a rtic le s in the neighborhood of 1 m ic ro n in d ia m e te r re q u irin g e x tre m e ly fast e x p o su re tim e could be photographed d esp ite th e ir B row nian m otion. A wide v a rie ty of p a n ch ro m atic filte rs and an a s s o rtm e n t of eye p ie c e s, including an oil im m e rs io n lens with to tal m ag n ificatio n of 1050 tim e s, w e re availab le. The oil i m m e r sion lens w as u se d m ainly to photograph sp e cim en s fro m those sa m p le s p re p a re d w ith the h o m o g en izer valve a t its fin est setting. The im m e rs io n m edium w as an oil w hose index of re fra c tio n is 61 1.515, n e a rly the sa m e as th at of the co ver g lass. At the end of the visual o b se rv a tio n , p h o to m ic ro g ra p h s w e re taken using a plus X film , and all m agnifications and p e rtin e n t data re c o rd e d . The o rig in a l 3 5 m m n egatives w e re e n la rg e d onto 4 x 5 in. tra n s p a re n c ie s and re a d ie d for p a rtic le sizing. A ctual counting and classify in g of the p a rtic le s w e re sim u lta n eo u sly a cc o m p lish e d by using a se m i-a u to m a tic p a rtic le size a n a ly z e r d e sc rib e d by E n d te r and G ebauer (16] and m an u fa ctu red in G erm an y by Z e is s . In som e c a s e s , up to 400 p a rtic le s w e re counted and sized; w hile in o th e rs a few h u ndred w e re on the p h o to m icro g rap h . F ro m the in fo rm atio n thus d eriv e d , it w as p o ssib le to e s tim a te m ean p a rtic le d ia m e te r and sta n d a rd deviations. B. E quipm ent The b a sic in s tru m e n ts em ployed in the e x p e rim e n ta l phase of this r e s e a r c h c o n siste d of a ro ta tio n a l v is c o m e te r, a m ic r o scope, and a s e m i-a u to m a tic p a rtic le size a n a ly z e r. A u x iliary equipm ent included a h a n d -p o w ere d h o m o g en ize r, au to m atic r e c o rd e r, W estphal b alan ce, c a m e ra , and a th e rm a l co n tro l a p p a r atu s. In the following se c tio n s, we sh all give a b rie f account of the use and o p e ra tio n of th e se com ponents to the extent that m ay be rele v a n t to the p re s e n t w ork. 1. The Haake R otovisco The R otovisco is a ro tatin g v isc o m e te r m ade in W est 62 G erm an y by G e b ru d e r-H a a k e . The su b stan ce w hose rh eo lo g ical p ro p e rtie s a re to be m e a s u re d is intro d u ced into a gap betw een a ro tatin g (inner) and a fixed (outer) su rfa c e . The d riv e m ec h an ism of the R otovisco is a sy n ch ro n m o to r with ten b a sic speeds fro m 3. 6 rp m to 583, 2 rp m a t 60 c y cles A. C, . The v isco u s d rag on the ro to r caused by the sam p le m a te r ia l is m e a s u re d by the d e flection o r tw isting of the to rsio n spring. A high p re c isio n poten tio m e te r c o n v erts this action w hich is a m e a s u re of the tr a n s m it ted to rq u e into an e le c tr ic a l sig n al p ro p o rtio n a l to the angle of d isp lac em e n t, A m i r r o r d ial m a rk e d fro m 0 to 100 units and located on the c o n tro l panel r e g is te r s this value, o r a lte rn a tiv e ly re la y s it to a sta n d a rd 10 m v la b o ra to ry r e c o r d e r connected to the e le c tr ic a l output leads of the v is c o m e te r. The R otovisco is equipped w ith a te m p e rin g v e sse l, a dual m e a su rin g head and a v a rie ty of m e a su rin g s y s te m s which p e rm it in v estig atio n s in a wide range of sh e a r. In the e x p e rim e n ts con ducted, we u se d the 50 gm cm m e a su rin g head and the NV Couette m e a su rin g s y s te m d esig n ed fo r low v isc o sity m e a s u re m e n ts and c o n sistin g of a double gap ro to r and a double gap b r e a k e r w ith p r o v ision for a coolant. The 500 gm cm to rq u e m e a su rin g head w as u se d once in conjunction w ith the NV unit for m e a su re m e n t of the m o d e ra te v is c o s itie s of the two oils em ployed in th is study. C a lib ratio n of the r o to r sp eed s w as acc o m p lish e d using a stop w atch for sp eeds below 32 rp m and a G e n e ra l R adio Co, S trobotac type 1538-A for the rem a in in g h ig h e r sp eed s. The re s u lts 63 w e re found to be in e x cellen t a g re e m e n t with the m a n u fa c tu re r's re p o rte d values. The v isc o sity constant K of the R otovisco w as d e te rm in e d fro m flow m e a s u re m e n ts with a known stan d ard , in this c a se , d istille d w a te r at 20° C . A c a lib ra tio n plot is provided in F ig u re 2 fro m w hich the co n stan t K is obtained. T hus, to calcu late the v isc o sity of a given fluid, we m ake use of the fo rm u la U = Ux S x K (72) w h e re u is the v isc o sity , U is the speed facto r i. e. the g e ar position n u m b er in sc rib e d on the control unit and S is the scale deflection. The constant K re fle c ts the com bined e le c tric a l, m ech an ical and configuratio nal c h a ra c te ris tic s of the o v e ra ll m achine. C on v ersio n of the d im e n sio n less scale read ing s and ro to r speed s to the conventional s t r e s s - s t r a i n units is done through the s t r e s s facto r A and the ra te of s h e a r (geom etry) facto r B given in T able 3. 2. M icro sco p e P a rtic le size m e a s u re m e n ts w e re acco m p lish ed by using a L eitz O rtholux m ic ro sc o p e equipped w ith an M r L eica c a m e ra . The in s tru m e n t's b a sic fe a tu re s a re a b in o cu lar photo tube FS, for v isu al o b se rv a tio n as w ell as photographing, and a quintuple Scale Deflection, S, Dimenslonless 0 0 .2 0 .4 0 .6 0 .8 1.0 Turbulent Laminar Laminar 0.1 0 0.2 0 .3 0 . 5 R eciprocal o f Speed F a c to r, jj, Sec Figure 2. P i s t i l led Water C a li b r a t i o n , T * 20*C TABLE 3 ROTOVISCO CONSTANTS WITH N V MEASURING SYSTEM Constant Measuring Head 50 gm an 500 gw i cm K - dyne/cm2 0.000190 0.00368 8 - dfmensionless 3768 3768 A - dyne/cm2 0.7)6 13-87 2 V iscosity : w * U x S x K dyne-sec/cm (poise) Shear Rate s * B/U 1/sec • ji 2 2 Shear S tress : t * A * S dyne/cm S tress Factor: A * B * K dyne/cm O ' U l 66 revolving nosepiece objective which acco m m o d ates a filte r slide and capable of m agnification s up to 105x. Thus using a 10x eye piece, m agnificatio ns fro m 100x to 1050X a re attainable. O ptim um field illu m in atio n w as provided by a xenon lam p, a sw ingout con d en ser and a v a rie ty of n e u tra l density and p an ch ro m atic y e llo w -g re e n filte rs . F ocusing of the sp ecim en w as m ade f i r s t by a c o a rs e ad ju stm en t of the object stage followed by the final focusing of the m ic ro m e te r fine a d ju stm en t knob. With a 10x eye piece, g rey field m agnification s on a 35 m m still negative w e re calcu lated fro m the re la tio n sh ip T hus, for the ob jectives u sed in the c u rr e n t m ic ro sc o p ic studies the o v e ra ll m ag nifications w e re M agnification = 4, 24 X Pow er of O bjective P ow er of O bjective M agnification (35 m m still negative) 1 Ox eye piece 10x 20x 50x 105X 42. 4 84. 8 212. 2 445. 0 A ccordingly, we m ay calcu late the tru e size ^ tru e of a single d ro p le t w hose d ia m e te r on the 35 m m negative is dfllm fro m the fo rm u la 67 d = dfilm tru e - P ow er of O bjective w h ere dt and d,,, have the sam e units of m e a s u re m e n t. F u r- tru e film th e r e n la rg e m e n t of the 35 m m p h o to m ic ro g ra p h s w as n e c e ssita te d , how ever, by the m e a su rin g ran ge of the p a rtic le size a n a ly z e r d e s c rib e d in the following p a ra g ra p h s , 3. P a rtic le Size A nalyzer A se m i-a u to m a tic p a rtic le size a n a ly z e r d e sc rib e d by E n d te r and G ebauer [16] and developed by Z e is s w as used fo r sizing and counting of the d ro p le ts. E n la rg e m e n ts of the o rig in a l photom i c ro g ra p h s w e re m ade c o n siste n t w ith the m e a su rin g ran ge of the in stru m e n t. The a n a ly z e r, d esig n ated TGZ 3, provided a rapid and highly a c c u ra te technique for d e te rm in in g size and d istrib u tio n of the photographed p a rtic le s . B rie fly , the counting p ro c e ss con siste d of placing the e n la rg e d photographs in a p lex ig lass plate and m aking a spot of light coincide w ith the d ro p le t on the p rin t via an i r is d iap h rag m , illu m in a ted fro m one side, and im ag ed by a lens onto the plane of the p late. The d iap h ra g m , in tu rn , w as connected to a counting c irc u it and w hen the light w as equalized w ith the p a rtic le on the m ic ro g ra p h , a fo o t-sw itch w as d e p re s s e d and a count r e g is te re d . The p ro c e s s w as then re p e a te d until a ll p a rtic le s w e re counted. To avoid count duplication, a "p u n ch er" n eedle m a rk e d the counted p a rtic le , a s the foot sw itch w as d e p re s s e d by p ierc in g a tiny hole in its c e n te r and p e rm ittin g the light to shine through. P a rtic le siaea a re divided into 48 continuous c a te g o rie s which upon the sim p le o p e ra tio n of a sw itch, the d ia m e te r of the m e a su rin g m a rk can be m ade to in c re a s e lin e a rly (F ig u re 3) o r exponentially with co u n ter n u m b er. The device a lso r e g is te r s freq uency o r accu m u lativ e counts in two m e a su rin g ran g e s fro m 1 , 2 - 27. 7 m m sta n d a rd , and 0. 4 - 9. 2 m m red u c ed ran g e depend ing on the size of the p a rtic le s on the p rin t. T hus, the v a rio u s p o ssib le com binations of sw itching p o sitio n s provided a wide range of o p e ra tin g conditions w hich w e re ad ju sted acc o rd in g to the specific re q u ire m e n ts of each p h o to m icro g rap h . 4. H om ogenizer D isp e rsio n of the o rg an ic phase w as a c c o m p lish e d by the use of a p o rtab le, h a n d -p o w ered h o m o g en ize r of the la b o ra to ry type. The device, m ade by C. W, L ogem an Co, , B rooklyn, New Y ork, is a b ench-top unit em ploying a sp rin g loaded valve fed by a piston pum p connected to a hand o p e ra te d level. The su b sta n c es to be em u lsified w e re m ixed in the p ro p e r ra tio and placed In the 16-ounce sta in le s s ste e l bowl und er the p lunger. W hen the le v e r w as p r e s s e d dow nw ard, the liq uid s w e re fo rce d through the tiny o rific e under high p r e s s u r e . The a c c e le ra tio n and the tu rb u le n ce Particle Diameter 30 69 o 10 20 30 to 50 Counter Number Figure 3. TGZ P a r t i c l e S ize Analyzer C a lib r a tio n 70 that re s u lte d fro m the s h e a r of passin g through a n a rro w p a ssa g e and the sudden red u ctio n in p r e s s u r e p roduced p a rtic le s of n e a rly u n ifo rm siz e. B y the sim p le ad ju stm en t (tightening of th read ) of the p r e s s u r e at the o rific e , it w as p o ssib le to produce d isp e rsio n s w hich ran g ed in a v e ra g e sam p le d ia m e te r fro m 0. 5 to 18 m ic ro n s at the e x tre m e s of c o n ce n tra tio n em ployed. 5. W estphal B alance D ensity m e a s u re m e n ts of the p u re org an ic com ponents w e re taken a t 2 0° C by m ean s of a W estphal b alance d istrib u te d by W. M. W elch Scientific Com pany, Chicago, Illinois. The in stru m e n t depends on A rc h im e d e s ' P rin c ip le in d e te rm in in g the v a ria tio n in w eight of a sin k e r w hen im m e rs e d in liquids of d ifferen t density. The m e a s u re m e n ts involved w e re m ade in the following m a n n e r. The sin k e r, w hich en clo se s a th e rm o m e te r, w as suspended in 20° C d istille d w a te r by m ean s of a fine platinum w ire w hile a unit (stan d ard ) h o rs e s h o e w eight was attach ed to the sin k e r hook. A slight a d ju stm e n t of the th rea d ed c o u n terp o ise p roduced eq u ilib riu m . The sin k e r w as then suspended in the te s t liquid of the sa m e te m p e r a tu r e as the w a te r and th re e sm a ll r id e r s 0. 1, 0. 01, 0. 001 g ra m w e re p laced at the a p p ro p ria te p la c e s on the divided sc ale so as to s e c u re equipoise. T hus, the d en sity w as obtained in this m a n n e r w ith a c c u ra c y to the fourth d ecim al place. In each m e a s u r e m en t, the platinum w ire w as im m e rs e d to the sa m e liquid depth 71 and effects due to su rfa c e ten sio n of the w ire , g e n e ra lly s m a lle r than the in stru m e n ts e x p e rim e n ta l e r r o r , w e re neglected, 6. A utom atic R e c o rd e r In the developm ent of the p re s e n t th eo ry we have co n sid ere d only fluids w hose coefficient of v isc o sity is independent of tim e and ra te of sh e a r. It is known, h o w ev er, as the co n cen tratio n of the o rg an ic phase is in c re a s e d , m ix tu re s of two o r m o re im m isc ib le fluids show c o n sid e ra b le deviation fro m N ew ton's hy p o th esis of constant v isc o sity o r lin e a r s t r e s s - s t r a i n rela tio n . In o rd e r to p reclu d e fro m fu rth e r a n a ly sis d ata re p re s e n ta tiv e of these s a m p les, it w as n e c e s s a r y to e s ta b lis h the lim iting co n cen tratio n beyond w hich the sam p le is c o n sid e re d non-N ew tonian. To e x a m ine the tim e dependency, we have em ployed a S a rg en t M odel SR R e c o rd e r S- 72180 for continuous g ra p h ic a l re c o rd in g of angular d eflectio n as a v a ria b le po ten tial w ith tim e. The in s tru m e n t is a se lf-b a lan c in g p o te n tio m e te r w ith a range of 1-125 m v and a re p o rte d a c c u ra c y of ± 4%. The output signal fro m the R otovisco w as fed into the r e c o r d e r via the r e c o r d e r output te rm in a ls of the R otovisco. M atching of sig n als betw een the r e c o r d e r and R otovisco w a s provided by a ran g e a tten u a to r. C hart d riv e sp eeds of w hich 0. 2, 1, and 5 inches p er m inute w e re c a lib ra te d using a sto p w a tc h and a g re e m e n t with the m a n u fa c tu re r's re p o rte d sp eed s w as g e n e ra lly excellent. 7. C onstant T e m p e r a tu r e C irc u la to r 72 T e m p e r a tu re of the s h e a re d sa m p le w as m a in ta in e d co n sta n t by m ea n s of a c irc u la tio n a p p a ra tu s w hich d e liv e re d 20° C w a te r to the in le t n o z zle of the NV m e a s u rin g s y s te m via the te m p e r a t u r e c o n tro l a s s e m b ly (te m p e rin g v e s s e l) of the R o tov isco. The c ir c u la to r, a LAUDA M odel k -2 d is trib u te d by B rin k m a n n In s tru m e n ts Inc. , W estb u ry , New Y ork, em p lo y s a so lid sta te th y r is to r c o n tro l s y s te m w hich d e liv e rs ra p id ly p u lsa tin g e n e rg y to a 500 w att h e a te r . The bath ho u sin g co n tain s th e m o to r, cooling coil, and the c irc u la tio n pum p and n o z z le s fo r supply and r e tu r n of the c o n sta n t te m p e r a tu r e fluid. A su p p le m e n ta ry (se p a ra te ) cooling s y s te m supplied ch illed w a te r to the c ir c u la t o r 's cooling coil and re tu rn e d it to the m ain coolant r e s e r v o i r m ain ta in e d a p p ro x im a te ly a t 15° C by o c c a s io n a l ad d ition of s m a ll am o u n ts of ice . A th e rm o re g u la to r w hich p ro tru d e s into the l a te r a l ex te n sio n of the bath housing and equipped w ith a ro ta tin g m a g n e t fo r te m p e r a tu r e a d ju s tm e n t m a in ta in e d th e c irc u la tio n fluid te m p e ra tu re a t 20° C. S ta b iliza tio n of the te m p e r a tu re w a s in d ic ate d by a p e rio d ic a lly flic k e rin g h e a te r c o n tro l light. O v e ra ll te m p e r a tu re g ra d ie n t b etw een the c irc u la tio n fluid and the te s te d sa m p le w as n e g lig ib ly s m a ll (le s s than 0. 5° C) to w a r r a n t fu rth e r c o r r e c tio n of the re c o r d e d v a lu e s. 73 8. M e ttle r B alan ce A M e ttle r P -120 p re c is io n balance of the top loading v a rie ty w as u sed fo r w eighing ex act am ounts {76 mg) of p ra c tic a l g rad e sodium o leate for d isso lu tio n in the w a te r phase. The b alan ce, m an u factu red in Sweden and d istrib u te d in the U nited S tates by Van W aters and R o g e rs, Inc. , h a s a capacity of 120 g ra m s and depends on the p rin cip le of weighing by su bstitution. C onsequently the in stru m e n t p ro v id es a c c u ra c y independent of b eam d isto rtio n . An optical sc a le having a range of -0. 05 to 10. 5 g ra m s is brought to e q u ilib riu m rap id ly by m ea n s of a m ag n etic dam ping device w ith a re p o rte d a c c u ra c y of ±0. 003 g. CHA PTER IV EX PER IM EN TA L RESULTS AND ANALYSIS A. Data R eduction Data reduction and a n a ly sis of the ex p erim e n tal re s u lts w e re facilitated through the use of a co m p u ter p ro g ra m w ritte n for the IBM 360 and d e s c rib e d in A ppendix B. S h ear and p a rtic le size m e a s u re m e n ts along w ith the p h y sical p ro p e rtie s of the p u re com ponents constitute the p rin c ip a l input data. The p ro g ra m e m ploys an ite ra tiv e com putational p ro c e d u re b ased on the Newton- R aphson m ethod to solve for the m utual in te ra c tio n using the p r e viously d e riv e d fo rm u la e, E quations (66) and (70), for the coef ficient of m ix tu re v isc o sity . The re s u lts a re then u tilized to c a l culate the d evelopm en t of velocity p ro file s in the tw o -p h ase m e dium and to obtain the sh e a rin g s tr e s s h is to ry at the su rfa c e of the ro tating cylinder. The b a sic sy ste m p a ra m e te r s a re c a lc u lated fro m m e a su re d q u a n titie s a s d isc u sse d in the following s e c tions. 1. C alculation of V isco sity The coefficient of v isc o sity um of the sh e a re d sam p le is calcu lated fro m the re la tio n sh ip given in E quation (72), n am ely 74 75 U = U x S x K m w h e re , a s befo re, U is the speed fac to r, S is the sc ale deflection on the m i r r o r dial, and K is the w a te r c a lib ra tio n constant of the flow ap p aratu s given in T able 5. A coefficient of m ix tu re v isc o sity applicable throughout the e n tire ran g e of te s te d speeds is obtained fro m the lin e a r s t r e s s - s t r a i n re la tio n sh ip w hich is calcu lated a c co rdin g to the ra te of sh e a r form ula 4m * § (75) and the c o rre sp o n d in g sh e a rin g s tr e s s at the su rfa c e of the cylinder Tm = A x S (74) H e re B is the sh e a r ra te (geom etry) fa c to r and A is the s tr e s s facto r w hose values w e re given e a r lie r in Table 3. S hear d ia g ra m s calcu lated fro m the p rec ed in g equations have been c o n stru c te d for each sam ple co v erin g the e n tire range of co n ce n tra tio n s teste d . T ypical plots a r e shown in F ig u re s 4 and 5, the rem a in in g graphs a re in Appendix A, C learly , they d e m o n s tra te the influence of d is p e r s e phase fra c tio n a l volum e and p a rtic le size on the v isc o sity of the tw o -p h ase m ix tu re . It is thus evident fro m th ese d ia g ra m s that the coefficient of v is co sity of the m ix tu re in c re a s e s w ith d e c re a sin g p a rtic le size Shear Stress t , dyne/cm' 76 microns (sample 1) microns (sample 2) microns (sample 3) 20 2000 1500 1000 500 Rate o f Shear, s sec Figure k. Shear Diagram fo r 30 Percent Benzene/Water T - 20*C 77 « c > » T > V > L . m v JC < /» □ 0 - 6 .0 microns (sample I) 9 A O >2. 4 microns (sample 2) 9 OD ■ 2 .0 microns (sample 3L 60 40 20 0 1500 2000 1000 500 0 Rate o f Shear, s . sec -I m F igure 5. Shear 0 fagram f o r 40 Percent Benzene/Water T - 20*C 78 until a c e rta in c o n ce n tra tio n is re a c h e d beyond which the m ix tu re is c o n sid e re d non-N ew tonian and p a rtic le siz e effects no longer follow a fixed p a tte rn . F u rth e rm o re , the dependence of v isc o sity on o rg an ic phase c o n ce n tra tio n is shown to follow som e exponen tia l ru le as evidenced by the v isc o sity -c o n c e n tra tio n plot d e s c rib in g the b eh av io r of each sy ste m (Appendix A). 2. C alculation of In te ra c tio n F re q u e n c ie s Having obtained n u m e ric a l values of v isc o sity , we now p ro ceed to calcu late the in te ra c tio n fro m the v isc o sity e x p re s sion given in E quation (66) w hich is r e f e r r e d to as E quation 1 in the co m p u ter p ro g ra m , nam ely c e ss iv e ap p licatio n of the N ew ton-R aphson m ethod, we obtain le s s of the final a ssu m e d o r tru e values. C o rrespo nding values of k j a re d e te rm in e d a cc o rd in g to the re c ip ro c ity condition a d vanced e a r lie r : 2 w h e re — h as been su b stitu ted for the d im e n sio n less f B y sue w 0 2 - 5 values of k^ w hich a re rap id ly co n v erg en t and w ithin 10 o r k l f l u l + k 2f2u 2 = 0 79 w hence ^2^2 k . - - k . (75) 1 2 S im ila r calcu latio n s a re p e rfo rm e d using the a lte rn a te v is c o sity e x p re ss io n given by E quation (71), d esig n ated as E quation 2 in the IBM p ro g ra m . H ow ever, in o rd e r to m ain tain co n sisten cy w ith the p re s e n t concept of m utual in te ra c tio n (re fle c te d both in the s t r e s s and s tr a in te rm s ) and to avoid confusion, we have r e s tr ic te d the c o rre la tio n e ffo rt to those re s u lts obtained fro m E q u a tion (66), Suffice to say that both equations show s im ila r c o r r e lativ e tre n d s , differing m ain ly in se n sitiv ity to the v a rie d p a r a m e te r e. g, p a rtic le size o r co n cen tratio n . Since k j and k^ p o ss e s s d im en sio n s of re c ip ro c a l tim e a p p earin g c o n sisten tly with the te r m u>_, we have given th em the desig n atio n of in te ra c tio n fre - k k 1 2 q u e n cies, and the ra tio s — and — a re thus called d im e n sio n less *0 “ 0 in te ra c tio n fre q u e n c ie s. P h y sically , h o w ev er, k j and k j m ay be in te rp re te d as the " e x c e s s " sh e a r due to in te rp h a se friction. 3, The T w o -P h a se R ate of Shear R e fe rrin g to E quation (57) and (58), the skin fric tio n e x e rte d by the tw o-p hase m ed ia on the su rfa c e of the ro tatin g cy linder w hen T is c o n sid ere d la rg e m ay be given by eo T = 2(1) m T, . , f2U2k2 T v2 1 0 _ 1U1 2U2 - “ * ^ r L o g ^ r J (76) and the co rresp o n d in g ra te of s h e a r from N ew ton's hypo thesis is m m u (77) m com bining E quations (76) and (77) with the v isc o sity e x p re ssio n s fro m E quations (66) and (71), it is then p ossible to w rite the s h e a r ing ra te in the fo rm *m * 2»0r (78) w h ere r is defined by f2u 2k 2 r = i - -j 4(s Log o r equally c o rr e c t (79) r= i - f2k2 [Logv2 - ^ Logv,] and that based on E quation (71) is 81 r = 1.0 (80) c o n s is te n t w ith o u r a r b i t r a r y definition of sh e a rin g r a te in tro d u c e d in E q u atio n (67). B. M ean P a r tic le S ite and Shape D e te rm in a tio n The sh ap e of the d is p e r s e d p a rtic le s re m a in e d la rg e ly s p h e ric a l, (F ig u re 6) a p p a re n tly unin fluenced by the high r a te s of s h e a r a s evid en ced by v isu a l m ic ro s c o p ic o b s e rv a tio n s and p h o to g rap h ic m e a s u r e m e n ts p e rfo rm e d im m e d ia te ly at the conclusion of each ru n . A c co rd in g ly , we take the sh ap e fa c to r o r sp h e ric ity , 4r, a s unity. S iz e -fre q u e n c y su m m a tio n (lo g a rith m ic p ro b ab ility ) c u rv e s w e re p r e p a r e d for each te s te d sa m p le to d e te rm in e an a v e ra g e p a rtic le d ia m e te r su itab le for c o rre la tio n w ith the e x p e rim e n ta l r e s u lts . The lo g -g e o m e tric m e a n d ia m e te r or sim p ly , the g e o m e tric m e a n d ia m e te r D^, is the value o btained by re a d in g the 50 p e rc e n t siz e . To e x p r e s s a m e a s u r e of v a ria n c e , the log- s ta n d a rd g e o m e tric dev iation, a is conv en ien tly o b tain ed fro m the p ro b a b ility in te g ra l c u rv e a c c o rd in g to 84. 13 P e r c e n t Size _ 50 P e r c e n t Size g " 50 P e r c e n t S ize " 15, 87 P e r c e n t Size Since fo r m an y s a m p le s the su m m a tio n o r in te g r a l of th e d is trib u - (a) Mineral 011/Water - Sample 1(*276) eft 8 p q § 8^ i * 1 L> (b) n-Hexadecane/Water - Sample 1{x138) Figure 6 . Photomicrographs of Disperse Phase 30 Percent 83 tion d ata did not plot exactly as a stra ig h t line on the lo g a rith m ic - p ro b ab ility p a p e r, it was n e c e s s a ry to use the av erag e value of obtained fro m both ends of the cu rv e. Some typical slz e -fre q u e n c y su m m atio n c u rv e s a r e shown in F ig u re s 7 and 8. The lo g arith m ic d istrib u tio n p a ra m e te rs a re defined m ath e- a m a tic a lly as „ £ n Log x ^ “ g = fcn = ) E l n ( L o g x - L o g D g ) 1 g f £ n w h e re n is the freq u en cy of o b se rv a tio n s of value x and I n is the to tal num b er of o b se rv a tio n s. The lo g -g e o m e tric d istrib u tio n p a r a m e te rs w e re p u rp o sely se le c te d since, by th e m se lv e s, they co m p letely define the siz e -fre q u e n c y d istrib u tio n cu rv e, and a re re a d ily co n v ertib le to co rre sp o n d in g values based on w eight (p rim ed values) ra th e r than on count by sim p ly applying the fo rm u la Log10Dg * L o* l0 Di + ‘ • ’ , L o «n>a g F u rth e rm o re , the s ta tis tic a l p a ra m e te rs D and a a r e re la tiv e ly S B e a sy to c o n v ert to o th er fo rm s of d ia m e te rs such a s the a rith m e tic a v e ra g e , su rfa c e a v e ra g e o r volum e av erag e d ia m e te rs . Thus, Counter Number 30 20 - 95 98 99 99.5 99.9 20 30 40 50 60 70 80 90 5 10 1 2 Percent F iner than S tated Size Figure 7. Size-Frequency Summation Curve fo r 20 Percent cts-D ecalIn/W ater (Sample I) Counter Number kO 30 20 10 3 2 20 30 *0 50 60 70 BO 90 95 98 99 99.5 99.9 1 2 5 10 Percent Finer than S tated S4ze Figure 8. Size-Frequency Summation Curve fo r liO Percent o-Xylene/Water (Sample 1) acco rd in g to [17] we m a y w rite 86 2 A rith m etic m ean d ia m e te r L o g ^ M L ogjgD + 1. I S L o g j ^ ^ 2 Surface a re a a v e ra g e d ia m e te r L o g ^ A = L o g 2. 30 L o g j qO ^ 2 V olum e a v erag e d ia m e te r L og.gD = L o g.gD + 3 .4 5 L o g .g a 8 8 w h ere . End 8 = " s r - [ # ] 4 and n is the nu m b er of p a rtic le s w ith d ia m e te r d, O th er im p o rta n t p a ra m e te r s such as specific su rfa c e , n u m b er of p a rtic le s p e r g ra m , etc. m ay also be obtained w ith c o m p a rab le e a se . C. C o rre la tio n In the p reced in g th e o re tic a l developm ents we have e x p re s s e d the m utual hydrodynam ic d rag o r phase in te ra c tio n by the re la tio n ship given in E quation (24), nam ely 87 (24) o r m o re c o n c is e ly , w h e re X is a c o rre la tio n p a ra m e te r (group) to be d e te rm in e d e m p iric ally . E qually significan t, h ow ev er, is the in te ra c tio n frequency o r k j w hich a p p e a rs in the final e x p re s s io n for v isc o sity and re la te d to *n the m an n e r: A ccordingly, we sh all seek a c o rre la tio n p a ra m e te r X b ased on E quation (82) w hich is equally valid and ap p licab le in the c ase of E quation (24), But f ir s t we m u s t e s ta b lis h that a lin e a r re la tio n sh ip betw een k^ and e x is ts i. e, the d im e n sio n le ss in te ra c tio n fre - q u e n cy (0 2 = — ) constant and independent of ra te of s h e a r if the (82) w hence (83) coefficient of m ix tu re v isc o sity defined by E quation (66) is to be con siste n t w ith the N ew tonian concept of fluids. 1. F unctional D ependence of and R e fe rrin g to the v isc o sity e x p re ss io n in E quation (66), we find that the a n g u la r v elocity a p p e a rs in the final re s u lt. A ccording to N ew ton's h y p o th esis, h ow ever, um m u st not depend on u > q o r m k 2 the ra te of sh e a r. C onsequently, we m u st conclude that — o r sim p ly the d im e n sio n le ss p a ra m e te r constant and independent of u)q. T h e re fo re , a plot of vs m u st n e c e s s a rily yield a stra ig h t line in the range of te s te d speeds. Such a plot is shown in F ig u re 9 for the s y s te m 30 p e rc e n t o -X y le n e /W a te r and c le a rly su b sta n tia te s the contention of lin e a rity betw een the two p a ra m e te rs . E xam ination of F ig u re 10 (30 p e rc e n t M in e ra l O il/W a te r) a lso leads to the sam e conclusion, although in this c a se the influence of p a r tic le size a p p e a rs re v e rs e d . P lo ts such as those depicted in F ig u re s 9 and 10 w e re c o n stru c te d for each sam p le and s im ila r r e su lts w e re obtained but only shown in ta b u la r fo rm (Appendix A). k2 C onsidering now — to be co n stan t and independent of sh e a rin g ra te , lU 0 we thus have e sta b lis h e d that the coefficient of m ix tu re v isc o sity is not in c o n siste n t w ith our fundam ental p re m is e — that of lin e a rity of the s t r e s s - s t r a i n re la tio n sh ip . F in a lly E quation (66) is w ritte n as — I • 0_ > <1.0 alcron* 2.1 M ic r o n s ® 0_ > 1.4 Microns 200 A 0 IS O 100 O ' 4 ) k tk 8 u u « k « C 1 0 40 20 30 Angular V elo city , uq, sec * F igure 9. Functional Dependence of and w q fo r 30 P ercent o-X ylene/W ater, T»20#C c e 3 — Z ' k U w U » N u m k e «i c to • 0 “ 15.0 el crons A D • 3 . 9 0 Ml c r o n s 9 • D > 2 .8 0 M ic ro n s 40 20 0 30 20 10 0 Angular V e lo city , j q , sec * F igure 10. Functional Dependence of k2 and uq for 30 P ercen t Mineral 011/Water, T*20“C 90 U = '"flU. + f-»ll-> Mm l 11 2 2 w h ere (84) and fl-, ^ f(U U q ) a s waa ihow n. 2. Influence of P a rtic le Size Upon fu rth e r ex am ination of the e x p e rim e n ta l re s u lta of F ig u re s 9 and 10, we o b se rv e that the in te ra c tio n frequency and hence the coefficient of v isc o sity u m is stro n g ly influenced by m ean p a rtic le size aa w ell as by the pure com ponents p ro p e rtie s . The effects due to the fo rm e r a re illu s tra te d explicitly in F ig u re s 11 and 12 w h e re the negative of the in te ra c tio n p a ra m e te r is shown to d e c re a s e for o-X ylene /W a te r and in c re a s e for the M in eral O il/ W ater sy s te m w hen plotted as a function of D . M ath em atically , th is is c o n sisten t w ith the sign of the lo g arith m ic ra tio of in divid u al ph ase k in em etic v isc o sitie s (v / v ) th at is a p p a re n t in the gen- 2 1 e ra l e x p re ss io n for m ix tu re v isco sity . P h y sica lly , h ow ever, such tre n d can only be a s c rib e d to som e unique m ic ro sc o p ic (m o lecu lar) c h a ra c te ris tic s of the em ulsion, its co m p lex p h y sic o -c h e m ic a l 200 - & ' 5° c 5 ! O' — V I 1 . u u. « w 100 e o — < N O I « v v _ V t 5° uo-30.5 W q«20. o* ojq® 10 • 2 o— w q “ 6.8 Mean P a r t i c l e Diameter, D , microns 9 Figure I I . Influence of D and tog on In te r a c tio n Frequency, 9 30 Percent o-Xylene/Water > ■ u c 5 v 7 t- u U_ 4) M s - U I m c v e Mr kt - 2C ■ lO g " 6 , 8 Figure t2. Mean P a r t i c l e Diameter, D , microns Influence o f and ug on I n te r a c tio n Frequency, 30 P e r m n t Mineral 011/Water 92 s tru c tu re , and to the h o st of atten d an t su rfa c e and in te r facial phe nom ena p re v a le n t in a m ulticom ponent sy stem . 3, E ffects Due to C oncentration Among the fac to rs studied for th e ir a p p a re n t effects on the rh eo lo g ica l b eh v aio r of two com ponent sy ste m s w as th at of in c re a s e d v o lu m e tric c o n cen tratio n . The re s u lts a re shown g rap h ic ally in F ig u re s 13 through 18 w h e re the co n tro lled v a ria b le effects, f r a c tional volum e in th is c a se , a re am p ly d e m o n stra te d . E ach d ata point on the g rap h s r e p re s e n ts an a v e ra g e of four points of n e a rly id en tica l value c alcu late d fro m the e x p re ss io n for v isc o sity ,E q u a tion (66),and m e a s u re m e n ts th e re o f a t the four te s te d sp eed s. E xcept for the B e n z e n e /W a te r s y s te m w h e re the low er c o n c e n tra tions (not shown) p ro duced e x tre m e ly sm a ll p a rtic le s and n arro w size range we find that, in g e n e ra l, the data points c o rre la te rea so n a b ly w ell, indicating no s e rio u s d isc re p a n c ie s in the o b s e r vational values. It is reco g n ized fro m the plots in F ig u re s 13 through 18 that the c u rv a tu re and slope o f the in te ra c tio n p ro file s a r e quite s im ila r for those s y s te m s w hose org an ic phase p ro p e rtie s and m o le c u la r s tru c tu re a re of the sam e g e n e ra l n a tu re . F o r in sta n c e , the s y s te m s B e n z e n e /W a te r and o-X ylene /W a te r show id e n tic a l tre n d s , yet they a re at v a ria n c e w ith the rem a in in g s y s te m s w hich in th e m se lv e s p o s s e s s s im ila r p ro p e rtie s and hence c o m p a rab le r e su lts. An im p o rta n t consequence of th is phenom enon w ill be ex- Dimension less Interaction Frequency, -82 4.0 3,0 System: Benzene/Water 0 .3 2.0 Mean P a r t i c l e Diameter, D g , microns Figure 13. E ffect of P a r t i c l e Size and Concentration on ft2 D Intension less Interaction Frequency. 9 .0 8.0 System: o-Xylene/Water 0 f, ■ O.k 7.0 5.0 Kean P a r t i c l e Diam eter, D , microns Figure lfc. Effect of P article Size and Concentration on £ 1 2 DImens Ion lets Interaction Frequency System; Mineral 011/Water 2.0 .5 1.0 0.5 16 8 12 0 Mean P a r t i c l e Diameter, 0 , microns Figure 15. Effect of Particle Size and Concentration on 82 'O System: Linseed Oil/W ater 2.0 .5 0 * 0.5 8 k 6 2 0 Mean P a r t i c l e Size, D , microns 9 Figure 16. E ffect o f P a r t i c l e Size and C oncentration on 02 D intension less Interaction Frequency 0.5 System: cls-O ecalIn/W ater -0 .5 8 4 16 0 12 Mean P a r t i c l e Size, D , microns 9 Figure 17. E ffe c t of P a r t i c l e Size and C oncentration on 0^ -j Dlmenslonless Interaction Frequency System: n-Hexadecane/Water 0 .5 0 -0 .5 16 8 12 0 Hean P a r t i c l e S iz e , D , microns 9 Figure 18. E ffe c t of P a r t i c l e Size and Concentration on 1)2 99 pounded in the next section. 4. A S e m i-E m p iric a l C o rre la tio n The p reced in g a rg u m e n ts su g g est th at a g e n e ra lise d c o r r e la tion of m utual in te ra c tio n m ight be possible, a t le a s t for those s y s tem s whose o rg an ic com ponents a re of s im ila r m o le c u la r s tru c tu re . Indeed, we have found w ith the aid of the com p u ter p ro g ra m in Appendix B a c o rre la tio n p a ra m e te r X b ased on E quation (66) which d e s c rib e s clo sely the p e rfo rm a n c e of the sy ste m s M in eral O il/W a te r and L inseed O il/W a te r at a ll c o n ce n tra tio n s, speeds, and m ean p a rtic le d ia m e te rs . The c o rre la tio n p a ra m e te rs for the rem ain in g two p a irs of im m isc ib le sy ste m s re s e m b le d those of the O il/W a te r sy s te m s, d iffering only in the pow ers to which the sy ste m v a ria b le s a r e ra ise d . A ccordingly, we p re s e n t only for the sake of b rev ity the s e m i-e m p iric a l c o rre la tio n for the O il/W a te r s y s te m s shown in F ig u re 19 in which the o rd in ate is plotted as - n 2 > d im e n sio n le ss, and the a b s c is s a X is given by V 5 ' 8 m m u 2 1. 25 Ul 2 » m Again, it is notew orthy to point out that each point r e p re s e n ts an a v erag e of four consecutive m e a s u re m e n ts , a t the v a rio u s ra te s of sh e a r. If, h ow ever, we c o n sid er e v e ry m e a s u re m e n t m ade and, DImens Ionless Interaction Frequency, 100 I I 2.0 O/q 5 0 5 0 80 20 , (microns) 0.5 Figure 19. A C o rre la tio n P lo t fo r the Systems Mineral 011/Water and Linseed 011/Water 101 in stea d , plot as o rd in ate the d im e n sio n al p a ra m e te r k^, then we obtain the lin e a r re la tio n sh ip shown in F ig u re 20. In this c ase the c o rre la tio n p a ra m e te r is w ritte n X = D0- 5 0. 25 Li, ( 86) B e cau se of the efficien cy of the hom ogenization p ro c e ss in obtaining n e a rly m o n o d isp e rse e m u lso id al p a rtic le s , an a tte m p t to include a m e a s u re of v a ria b ility of p a rtic le size d istrib u tio n i. e. ? in the 8 c o rre la tio n p a ra m e te r is , obviously, unm eaningful and th e re fo re w as not p u rsu ed fu rth e r. 5. D e term in a tio n of the In te ra c tio n C oefficients The h y p o th eses in E quations (24) can be used for the c a lc u lation of the in te ra c tio n coefficients a ssu m in g that an e m p iric a l c o rre la tio n s im ila r to F ig u re 20 is obtainable. R e fe rrin g to E qua tion (82), we have It - K ^ 2 - 12 i 2u z w h e re is the coefficient of in te ra c tio n o r the p ro p o rtio n ality "co n stan t" f ir s t in tro d u ced in E quations (24). To illu s tr a te , the slope of the e m p iric a lly d e te rm in e d lin e a r re la tio n sh ip fo r the Interaction Frequency, -k , sec 102 60 60 20 0 0 500 1000 1500 2000 0.25 i t \ 0 . 5 -1 uq. (microns) sec Figure 20. A D etailed C o rre la tio n P lo t for the Systems Mineral 011/W ater and Linseed 011/Water 103 O il/W a te r s y s te m s d e sc rib e d by F ig u re 20 m u st be, acco rd in g to K, E quation (82), 12 2U2 and hence Kjg = t» lo p e] f2u2 (87) To e x p lo re the d im en sio n s of K2 j we p ro c e e d in the following m a n n e r. F ro m e a r l ie r d isc u ssio n s, we have found k 2 to p o s s e s s d i m en sio n s of re c ip ro c a l secon ds. F or the p re s e n t exam ple, X has d im en sio n s c o n siste n t w ith -1 ^0 . 5 0, 25 D g u 2 7 T 7 * *2 “ 1 U ) q, tun, nam ely cm 0. 5 sec Note that the fra c tio n a l volum e f2 is d im e n sio n less and re m a in s so re g a rd le s s of the pow er of the exponent. Using the 2 m e tric units of v isc o sity (i.e . 1 poise = d y n e -s e c /c m = g r a m /c m - sec), we find K j 2 = D im ensions of (• JJ£2£x r ) sec cm Applying this r e s u lt to the a b b re v ia te d fo rm of E quation (24), p, 87, the d im en sio n s of F j 2 w hen r is m e a s u re d in c e n tim e te rs m u st be id en tica l w ith 0. 5 K sec cm cm sec cm 104 i. e. fo rce per unit volum e, and thus in d im en sio n al a g re e m e n t w ith the units of — ^ and ^ in the N av fer-S to k es E quations of r dqp o* or m otion. Since ^ tu rn s out to be a function of the sy ste m v a ria b le s i. e. the p rod uct f^U2 and slope as dem ended by E quation (87), its n u m e ric a l value w ill shift p ro p o rtio n a te ly and in that re s p e c t it c an not be c o n sid e re d constant in the tru e sense. D. Influence of Stabilizing Agent Thus far we have m ade no specific m en tion of the possible effects on flow due to the p re s e n c e of a thin film of e m u lsifie r at the O il/W a te r in te rfa c e . A detailed study of th is phenom enon is c le a rly beyond the intended scope of the p re s e n t d isse rta tio n . How e v e r, we sh a ll give a b rie f account of the lim ited re s e a r c h p e r f o r m ed in th is re s p e c t as m ay have a d ire c t b earin g on the data c o r re la tio n e ffo rts a ttem p ted e a r lie r . It is thus conceivable that failu re to obtain a u n iv e rs a l c o rre la tio n of all o b se rv atio n al values is due, in p a rt, to the m any w ays by w hich a su rfa c e active agent can influence the flow p ro p e rtie s of em u lsio n s. F o re m o s t am ong th e se is the influence of e m u ls ifie r c o n cen tratio n on the in te rfa c ia l tension betw een the o rg an ic and w a ter p h ases and hence on the v is c o sity of the re su ltin g m ix tu re . An exam ple of this is shown in F ig u re s 21 and 22 w h e re sodium o leate co n cen tratio n is v a rie d and the in c re a s e in v isc o sity is d e te rm in e d for v ario u s co n cen tratio n s of the o rg an ic ph ase. The sa m p le s w e re p re p a re d id en tically using IA < • —> S a > IA IA — — o > a a Z »- c 3 a) w u X W 2.0 105 System: Hlnerel O I I A ittr a 30 Percent Disperse Phase A 20 Percent O 10 Percent 8 e .6 .2 0 0.001 0.002 0.003 0.005 Holes Sodium O leate per 100 cc Emulsion Figure 21. Influence o f S t a b i l i z e r C oncentration on V isco sity ia e> 8 r & > — u c L - V 3 O S y itte : Linseed 01l/U eter * 30 Percent Disperse Phase A 20 Percent O 10 Percent ----------- 2.0 0.001 0 0.002 0.003 0.005 Holes Sodium Oleate per 100 cc Emulsion Figure 22. Influence o f S t a b i l i z e r C oncentration on V isco sity 106 the sam e h o m o g en izer o rific e setting. C learly , th e re fo re , the low e rin g of in te rfa c ia l ten sio n as the co n cen tratio n of the s ta b iliz e r is ra is e d m u st be a c u ra te ly known befo re a thorough evaluation of ex p e rim e n ta l re s u lts is a ttem p ted . O ther p e rtin e n t fa c to rs to con s id e r include the extent to which the em ulsifying agent is soluble in the d is p e rs e p h ase, its c o n trib u tio n to e le c tro v isc o u s effects and physical p ro p e rtie s of the film a d so rb e d aroun d the p a rtic le s . A lso, w h e re the em ulsifying agent is soluble in the in te rn a l ph ase, a c o r responding red u ctio n in e x te rn a l phase v isc o sity a c c ru e s due to tra n s fe r of p a rt of the e m u ls ifie r to the d is p e rs e phase, A fu rth e r illu s tra tio n of the effects of die stab ilizin g agent on the rh eo lo g ical p ro p e rtie s of tw o-phase s y s te m s is p re s e n te d in F ig u re s 23a, b and c. Two s y s te m s , one 30 p e rc e n t c is -D e c a lin / W ater and the o th er 30 p e rc e n t n -H e x a d e c a n e /W a te r w e re p re p a re d in the sam e m a n n e r and flow te s te d at constant te m p e ra tu re (20° C), In F ig u re 23a, the p u re com ponents b eh av io r is co m p ared . In the ab sen ce of e m u lsifie r, the s t r e s s - s t r a i n re la tio n sh ip is given by F ig u re 23b, w h e re the a rra n g e m e n t of the sh e a rin g d ia g ra m for each sy ste m follows that of the p u re com ponents. H ow ever, in c o rp o ra tio n of a tr a c e of s ta b iliz e r (0, 001 M sodium oleate p e r 100 cc e m u ls io n ), F ig u re 23c, c a u se s the o rd e r of the sh e a rin g lin es of the two em u lsio n s to r e v e r s e . T his d e m o n s tra te s c le a rly how one sy ste m is influenced d ifferen tly fro m a n o th er by the p r e s ence of sm a ll but equal am ounts of em ulsifying agent, p e rh a p s Shear Stress Shear Stress Shear S tress v c > - ~ o C s l V c > > -o 50 ■ 25 - © n-Hexdecane A c is-D e c a lin 107 500 1000 1500 1 2000 Rate of Shear, sec Figure 23a. Single Component Data, T*20#C 50 o n-Hexadecane/Water A cls-D ecalIn/W ater 25 0 0 1000 1500 500 2000 Rate of Shear, sec * Figure 23b. Shear D)agram Wtthout S ta b iliz in g Agent L > • ■ o 50 & cfs-D ecalIn/W ater O n-Hexadecane/Water 25 0 500 2000 1000 0 -1 Rate of Shear, sec Figure 23c. Shear Diagram w ith 0.001 M S t a b i l i z i n g Agent 108 through the actio n of the red u ced in te rfa c ia l ten sio n o r in d ire c tly throug h p a rtic le size and d istrib u tio n , E. D evelopm ent of V elocity and S tre s s P ro file s We now p ro c e e d to analyze the in itial sta g es of the fluid m o tion a fte r it h as been s ta rte d im p u lsiv ely fro m r e s t. The com pu tatio n al p ro c e d u re is conveniently in c o rp o ra te d in the m ain IBM p ro g ra m and em ploys the n u m e ric a l values of phase in te ra c tio n obtained fro m the solution of the equation for viscosity, E quation (66). A typical output of the co m p u ter p ro g ra m w hich includes velocity d istrib u tio n in the tw o -p h ase fluid and tra n s ie n t fric tio n s t r e s s at the su rfa c e of the ro tatin g c y lin d er in an infinite volum e of em u lsio n is shown in Appendix B. D im en sio n less velocity p ro file s a re c a l culated for sm a ll values of the tim e. In the c a se of la rg e T, the velocity calcu latio n s a r e o m itted b ecau se of the lim ite d rang e of u se fu ln e ss of the expansion solution outside the im m e d ia te reg ion of the w all. At the su rfa c e of the cy lin d er, h ow ev er, the skin fric tio n is calcu lated for all values of the tim e using the unsteady sta te solution d e s c rib e d by E quations (55) through (58), 1. V elocity D istrib u tio n The calcu latio n of velocity p ro file s outside the ro tatin g cyl in d e r a re m ade acco rd in g to the e x p re ssio n s: 109 Vn = -JT T m * ^ [ erfc TTT" + v '4Tn ( t o t - TC ) Zier(C7 r r ] " n ' n n T n [ 4 l2 e rfc ^ T T + ^ 4 T n ( i k " J j 6 i 3e rfC ] , n= 1 ,2 n n and the w eig h ted d im e n s io n le s s v e lo c ity V = f.V, + f,V_ m 1 1 2 2 P lo ts illu s tr a tin g the d ev elo p m e n t of p ro file s (tra n s ie n t re s p o n s e ) c a lc u la te d fro m the p re c e d in g e q u atio n s a r e show n for the s y s te m 3 0 p e rc e n t n -H e x a d e c a n e /W a te r show n in F ig u re s 24 and 25, In a c c o rd a n c e w ith th e e x p e rim e n ta l r e s u l ts , n u m e r ic a l v a lu e s of the p a r a m e te r 0 ^ dep end on m a i y v a ria b le s and c o n seq u e n tly the c a l c u la te d p ro file s tend to d iffe r so m e w h a t fo r e ac h of the s y s te m s . T he ran g e of a p p lic a b ility of the p re c e d in g eq u atio n for v e lo c itie s i. e. the m a x im u m value of T w hich m a y be u se d w a s d e te rm in e d fro m the condition of s t r e s s co n tin u ity as w ill be d is c u s s e d in the follow ing s e c tio n s . 2. W all S h e a r S t r e s s The tr a n s ie n t fric tio n s t r e s s a t the c y lin d e r w a ll w h e re R = 1. 0 is d e te rm in e d a c c o rd in g to the expand ed fo rm of E q u atio n s (56) and (58), n a m e ly 110 i.o W th* c a r M i ■ 09 015 o.t 0.) 0.1 0.1 U,I0* flfo ro 10. HOIfhtoO Avorto* Voloolty OUtrlOutlOA )0 PorooM o*HoMOooono/Walor <fa*«IO O I l l a.; O .i • i n -H ta a O a u n a 0.1 0.1 0.) 0 0.1 F ifu r a I S . In d iv id u a l Phaaa V a lo c lty D I» trlb u i1 o n JO F i r c m t n * W u 4 a w n t/U < t« r I) 112 (55) n = 1,2 for sm a ll values of the tim e , and * „ r 4Ti* l . nn._ _ N2 1 Tn = 2 I T T - + 1 " Log T T T . J 1 (57) n n for la rg e values of the tim e. E quations (57) and (58) can be sim plified fu rth e r if we introdu ce the id en tities ° l f l M l = ' n 2f2M 2 T hus the final e x p re s s io n fo r sh earin g s t r e s s w hen T is la rg e a s s u m e s the fo rm The m eaning of " s m a ll" and " la rg e " tim e is d isc e rn ib le if it is re a liz e d that the a c tu a l s t r e s s tra n s m itte d by both fluids to the cylinder m u st n e c e s s a r ily re m a in continuous a t a ll the tim e s. H ence a plot of t (t) in each tim e dom ain v e rs u s e ith e r T j o r w ill d e te rm in e the valid ran g e of the expansion solution. T his is done in F ig u re 26 for the sy ste m 30 p e rc e n t B e n z e n e /W a te r. It is seen that both equations coincide in the vicinity of T j = 0. 75. A s im ila r d e te rm in a tio n is shown in F ig u re 27 for 30 p e rc e n t n -H e x a d e c a n e / W ater w h ere the " c r itic a l" value of is a p p ro x im a te ly 0. 65. Finally, for a ll the s y s te m s in v estig ated , tra n s ie n t s t r e s s h is to ry at the cylinder w all a re given in F ig u re 2 8. H e re the dashed lines denote the reg io n for each equation beyond which the expansion s o lution is not valid. It is no tew orthy that the s t r e s s c u rv e s s ta r t fro m infinity, a situ atio n analogous to the flow ov er the leading edge of a flat plate, and then d e c re a s e to th e ir final v alu es as shown. The stead y state sh e a rin g s tr e s s t, tin the d e n o m in ato r of the y ' ' ordinate ra tio is the a c tu a l m e a su re d value calcu lated T (ss) fro m E quation (74). Upon close ex am in atio n it is p o ssib le to d is c e r n fro m the g rap h s the ex ce llen t a g re e m e n t betw een this value and that obtained fro m the tr a n s ie n t decay p ro file s p re d ic te d by the proposed m odel for v e ry la rg e T i. e. s t r e s s ra tio being v e ry n e a rly unity. D1mens Ion less Stress R atio % 4.0 System: 30 percent Benzene/Water (Sample I) 0 2.0 0 0.01 0.1 10 100 Dlmensionless Time, Tj Figure 26. T ran sie n t Shearing S tr e s s a t Surface of Cylinder 114 o « * ) « IS ) l/> tft tl c o l< l w 3.0 System: 30 percent n-Hexadecane/Water (Sample 1) 2.0 0 0.01 0.1 10 100 Dlmensionless Time, T| Figure 27. T ran sien t Shearing S tre ss a t Surface of Cylinder U 1 Dlmensionless Stress R atio *.0 3.0 2.0 A Linseed 0 ! 1/Water (Sample 3) B Mineral Oil/Water (Sample 3) C n-Hexadecane/Water(Sample 1) D cis-D ecalin/W ater (Sample 3) E o-Xylene/Water (Sample 3) F Benzene/Water (Sample 3) 1.0 X X 0.001 0.01 10 0.1 1 Dlmensionless Time, Tj Figure 28. T ransient Shearing S tress a t Surface of Cylinder fo r a l l Systems f 2 - 0.3 100 117 It m u tt be m entioned, h ow ever, that the g rap h s in F ig u re s 26 through 28 have been c o rre c te d for the g e o m e try fa c to r to fa c il itate c o m p a riso n . C onversion fro m the infinite (analyzed m odel) to the a ctu al (finite) g e o m e try of the m e a su rin g s y s te m is acc o m p lish e d by using the fo rm u la w h e re 6 is the g e o m e try facto r of the NV m e a su rin g sy s te m in T able 3 and F is given by E quation (84) o r E quation (79). The above re la tio n sh ip is a d ire c t consequence of the hypo th e s is th at the p h y sico m ech an ical c o n stan ts of the s h e a re d sam ple i. i . U and k_ a re tim e and g e o m e try independent. A ccordingly, 1 1 1 b E quation (89) ap p lies to both tra n s ie n t and stead y state conditions. It is d e riv e d in the following way. Since by definition Tm <Finite) = (89) m( Finite) m (® ) M = ----------------- = -----1 —f m j 4 Finite) m (») (90) Hence Tm (F in ite) ” Tm (® ) m( F inite) (91) 118 N°w ^m (F inite) an<* ^m (» ) a re t* lOBe 8*ven W E quations (73) and (78) re sp e c tiv e ly . Com bining th e se with E quation (91)and noting that 2t t N( wo = “T 5 " u = 583. 2 Nrt we obtain finally the e x p re ss io n for Tm {pjnite ) given by E quation (89). CHAPTER V SUMMARY A th e o re tic a l e x p re ss io n for m ix tu re v isc o sity b ased on the u n stead y sta te solution of the N av ier-S to k es equations h as been d e veloped taking into account m utual in te rp h a se d rag. S pecifically, we have an aly sed the flow g e n e ra te d by a c y lin d er of rad iu s im m e rs e d in an infinite, hom ogeneously d istrib u te d tw o-p hase m e dium , w hich at tim e t = 0 s ta r ts rotating with an a n g u lar velocity v 0 — . T his is the analogue for c irc u la r flow of R ay leig h 's problem . r 0 S e p a ra te equations of m o m entum w e re w ritte n for each phase and an im p o rta n t postulate in tro d u ced w as that the m utual h ydrodynam ic d ra g (interaction) is e x p re s s ib le in the fo rm F 12 = K 12 [ y r U2 ,D p m ’ * l ,U,o ] r su b ject to the conditions F 12 + F21 = 0 K 12 = K21 and K j 2 is the coefficient of in te rac tio n . 119 120 The solution of the tim e -d e p e n d e n t d iffe re n tia l equations w hich sa tis fie s the boun dary and in itia l conditions w as obtained in the fo rm of a s e r ie s of c o m p le m e n ta ry e r r o r functions and in te g ra ls by m eans of the a sy m p to tic expansion of the B e s s e l re la tio n s. The tra n s ie n t skin frictio n w as obtained fro m the velocity d istrib u tio n and fra c tio n a l phase volum e for sm a ll and la rg e values of tim e acco rd in g to the fo rm u la Tm fl u l [ p £ ( “ i ^ ) ] + f2U2 [ r *7 ( “ T 6 ) ] r = r = 0 The ap p ro x im ate solution for velocity and s t r e s s fu rn ish ed by the expansion m ethod contained in te ra c tio n te rm s k j and k£ which p o s s e s s the d im en sio n s of re c ip r o c a l tim e and w e re r e f e r r e d to as in te ra c tio n freq u e n cie s. T h ese w e re m ade d lm e n sio n le ss with r e f e re n c e to a suitable constant p a ra m e te r u )q. A s a r e s u lt the non- d im en sio n al in te ra c tio n frequency k n = — n = 1,2 n u,0 a p p e a rs in the final equ ation s, w ith the in trin sic p ro p e rty such that the re c ip ro c ity c rite r io n n i fl u ! = ’ ° 2 f2u 2 121 is alw ays sa tisfie d . A g e n eral e x p re s s io n for the coefficient of m ix tu re v isc o sity li w as obtained fro m the p reced in g equation for sh e arin g s tr e s s and the re la tio n sh ip m = u r ^ wmL dr d r V r= r. w h ere vm ^ a w eighted a v e ra g e velocity defined by v = f , v , + f - V - mcD 1 lqp Z 2cp T h u s ,re g a rd le s s of the sta te of m a tte r of the d isp e rso id , by eq uat ing the above e x p re ss io n s for w all sh e a r s tr e s s we have found when T is larg e 4 T 1+1 . , 4 T 2+1 f2W 2 ° 2 j _ V2 fl u l + f2u2 ~vr2 3 L o g ^ H ere T is a d lm e n sio n le ss tim e v a ria b le given by 122 and Log N = ^ ; Log C = y = 0, 5772. . . is E u le r 's constant. The s u b s c rip ts 1,2 r e f e r to the e x te rn a l and d is p e rs e p h ases r e s p e c tively. By N ew ton's h y p o th esis, the coefficient of m ix tu re v is c o sity w as c o n sid e re d con stant below a c e rta in " c ritic a l" frac tio n a l phase volum e re q u ire d for the o n set of flow d e p a rtu re s th e re fro m , The fact that the tim e p a ra m e te r a p p e a rs in the final equation for v isc o sity m u st not be in te rp re te d as tim e dependency in the con ventional se n se , since this violates the b asic h y p o th esis m ade. T h e re fo re , we can ju stifia b ly in fe r that upon changing T, both n u m e ra to r and d e n o m in ato r of the e x p re ss io n w ill shift p ro p o rtio n a te ly so as to m ain tain a co n stan t o v e ra ll value. When T tends to infinity ,how ever, the tim e dependence can be shown to vanish. In c ase w hen v e ry fine solid p a rtic le s a re contained in a visco u s fluid, the p h y sic o m e ch a n ica l p ro p e rtie s of the solid i. e. the "p ro v isio n al" coefficient of v isc o sity of the solid, u^, and the in te ra c tio n p a ra m e te r 0 ^ a re unknown. Since it is not p o s sible to d e te rm in e th e se d ire c tly fro m the d e riv e d e x p re s s io n for v isc o sity and p h y sic al m e a s u re m e n ts th e re o f (one independent eq u a tio n in two unknowns), we have su g g ested m ethods by w hich th ese can be ob tained in the g e n eral c a se . H ow ever, c o n sid e ra b le m a th e m a tic a l sim p lifica tio n s w e re achieved for the sp e cia l case 123 w hen the d is p e rs e d p a rtic le s a re liquid. H ere |jj is known, being that of the p u re com ponent. F u rth e rm o r e , by e x tra p o la tio n to la rg e values of T , the g e n e ra l e x p re s s io n for v isc o sity b e ca m e V ■ v , r.— * ---------- h r- [L °ev2-S7 w v j A ltern a tiv e ly , if, in analogy w ith the c a se of a cylin der ro tatin g in an infinite single phase m ed iu m , the com bined tw o-phase w all sh e arin g s t r e s s is r e p re s e n te d by the equation Tm 1 2um *0 then the final e x p re s s io n for v isc o sity , a fte r p e rfo rm in g the n e c e s s a ry sim p lific a tio n s, a s s u m e s the fo rm f2u2 n 2 v2 > v = f i * i + f 2 ‘j 2 - - 1 r - i L o * ^ <7 1 > A c o m p u ter p ro g ra m w as w ritte n to calcu late n u m e ric a l values of fro m E quations (66) and (71), identified a s E quation I and E quation 2 re s p e c tiv e ly in the IBM p ro g ra m , using p h y sical m e a s u re m e n ts of v isc o sity of fre s h ly p re p a re d em u lsio n s of the O il/W a te r type. C o n cen tratio n of the second (internal) phase w as 124 g rad u a lly in c re a s e d until the sh e a re d sam p le exhibited e ith e r tim e o r ra te of s h e a r dependency. A naly sis of the e x p e rim e n ta l r e s u lts d e m o n s tra te d c le a rly the m u ltip licity of the in te rr e la te d fac to rs af fecting the flow. N o n eth eless, a s e m ie m p iric a l c o rre la tio n w as p o ssib le for s y s te m s w hose in te rn a l phase p ro p e rtie s and m o le c u la r s tr u c tu r e w e re not w idely d is s im ila r. F o r a ll sy ste m s studied, h o w ev er, the in te ra c tio n frequency fln w as found independent of both tim e and ra te of s h e a r in the range of co n cen tratio n s and sp eed s teste d . In the p a rtic u la r case of the sy s te m s M in e ra l O il/ W ater and L in see d O il/W a te r, calcu lated values of 0 b a sed on The final p h a se s of the c u rre n t r e s e a r c h d ealt with the a n a ly sis of the u n ste ad y state solution for velocity and tra n s m itte d s t r e s s . T hus, having obtained n u m e ric a l v a lu e sfo r the in te ra c tio n fre q u e n c ie s k j and k^, we p ro ceed ed to calcu late the developm ent of velocity p ro file s w ithin the tw o-p hase (infinite) fluid and to ob ta in tr a n s ie n t s h e a r s t r e s s h is to ry at the su rfa c e of the cy lin d er. In o r d e r to lend som e g e n e ra lity to the solution, d lm e n sio n le ss q u a n titie s w e re introduced. All lengths w e re red u c ed w ith the aid of a su itab le re fe re n c e length r^ , the rad iu s of the ro tatin g sylin der; n E quation 1 c o rr e la te d fa irly w ell w ith the p a ra m e te r 125 v e lo c itie s w e re m ade d lm e n sio n le ss with re fe re n c e to a suitable p e rip h e ra l velocity v^ = tt)Q r Q > an^ individual phase sh e a rin g s t r e s s e s w e re divided by a fra c tio n a l value of the equivalent single phase sh e a rin g s t r e s s (2mi)g ). The d lm e n sio n less tim e p a ra m e te r T, in tro d u ce d e a r l ie r , is the analogue of the F o u rie r n u m b er in h eat con duction and an im p o rta n t c rite r io n by which tra n s ie n t sy ste m s r e sp o n se is u su a lly m e a su re d . The p ro c e s s of tra n s itio n fro m one tim e dom ain to a n o th er i. e. sm a ll and large values of the tim e , w as studied in an a ttem p t to ex- ta b lish the applicable range of the d e riv e d re la tio n sh ip s. By e x a m ining the continuity of the s t r e s s p ro file s, we w e re able to d e te rm in e the valid range of the m ethod of solution by noting the reg io n w h ere the two tim e solutions o v erlap . F inally, ex trap o latin g to v ery la rg e values of tim e , and using the p ro p e r g e o m e tric fa c to rs , we m ade c o m p a riso n s of the m e a s u re d values of s tr e s s w ith those p red ic te d by the theory. In the c o u rs e of the th e o re tic a l p h ases of this d is s e rta tio n , c e rta in a ssu m p tio n s and h y p o th eses re g a rd in g the flow phenom ena w e re advanced and la te r exam ined in the light of the e x p e rim e n ta l findings. F o r e x am p le, it b e ca m e c le a rly evident th at m ix tu re s of two im m isc ib le liquids w ith in te rn a l phase co n cen tratio n s up to 40 p e rc e n t can be tre a te d p ro v isio n ally as N ew tonian fluids. The s e r ie s expansion technique em ployed in effecting a solution, r e g a r d le s s of the abandoned h ig h e r o rd e r te r m s , h as proved s a tis fa c to ry as evidenced by tie a b se n c e of p e c u la ritie s o r o th e r unexplained phe 126 nom ena in the co n sid ere d range of expansion. M o re o v e r, p a rtia l s u c c e ss can be claim ed in the a ttem p t to p h y sic ally define the in te r a ctio n p a ra m e te rs and a r r iv e at a c o rre la tio n involving im p o rta n t s y s te m v a ria b le s. In this re s p e c t, the p re s e n c e of a thin film of e m u ls ifie r at the in te rfa c e , its influence on in te rfa c ia l tension , p a r tic le size, etc. and u ltim a te ly on the rh eo lo g ical behavior of the s y s te m m u st be thoroughly c o n sid e re d in the g e n eral c a se . O ther sig nificant o b se rv atio n s m ade include: a. The coefficient of m ix tu re v isc o sity is influenced by both in te rn a l phase c o n ce n tra tio n and m ean p a rtic le size; the com bined influence being of exponential n a tu re . Individ ual effects, h ow ev er, cannot be iso la te d easily . b. The liquid d ro p le ts e x p erien c ed no d e te ctab le lastin g deform atio n b ecau se of s h e a r as evidenced by m ic r o scopic o b se rv atio n s and photographic m e a s u re m e n ts p e rfo rm e d im m e d ia te ly at the conclusion of each e x perim en t, c. C o n trary to conclusions draw n by prev io u s r e s e a r c h w o rk e rs , this study e sta b lish e d the fact th at the co efficient of m ix tu re v isc o sity depends m a rk e d ly on p a rtic le size even at the v e ry low c o n ce n tra tio n s, d. The coefficient of v isc o sity of the m ix tu re in c re a s e s w ith d e c re a sin g p a rtic le size w hen c o n ce n tra tio n is h eld constant. 127 e. At m uch h ig h e r c o n c e n tra tio n ! w h e re the m ix tu re co m m en ces to exhibit non-N ew tonian tre n d s , the effects on v isc o sity of p a rtic le size b ecom e no lo n g er e a s ily predictable. A s a closing re m a r k , it m ay p ro v e b en eficial for fu tu re w o rk on the su b ject to c o n sid e r a m o re g e n e ra l ap p ro ach to the p ro b lem of phase in te ra c tio n . S pecifically, F ^ m ay be c o n sid e re d as the com bined effect of "p u re " hydrodynam ic d ra g (in te r-p h a se friction) an<* f ° rc e s a ris in g fro m su rfa ce and in te rfa c ia l phenom ena ^"^(cy)* Thus F = F + F f 12 12(d) + lZ{rj) A ltern a tiv e ly , a s im ila r ap p ro ach m ay be tak en in the c o rre la tio n w ith the o b se rv a tio n a l data, n am ely ° 2 = ^ 2(d) + n 2(o) * R E F E R E N C E S 1. E in ste in , A. ,Ann. d. F h y sik (4 ), 19, 2 89 (1906); 34, 591 (1911). 2. Je ffe ry , G. B, , F ro c . Roy. Soc. (London), A 102, 161 (1923). 3. T a y lo r, G. I., P ro c . Roy. Soc. (London), A138, 41 (1932). 4. B u rg e rs , J. M. , Second R e p o rt on V isc o sity and P la s tic ity , N orth Holland P u blishing Co. , A m ste rd a m (1938). 5. Sim ha, R. , J. Appl. Phys. , 23, 1020 (1952). 6. H appel, J. , J . Appl. Phys. , 28, 1288 (1957). 7. M ooney, M. , J. Colloid Sci. , ^6, 162 (1951). 8. R ajagopal, E. S ., K ollo id-Z . , 162, 85 (1959); ±64, 1 (1959). 9. R ich a rd so n , E. G. , J . Colloid Sci. , 8, 367 (1953). 10. S h erm an , P . , K o llo id -Z . , 165, 156 (1959). 11. F a izu llae v , D. F. , L a m in a r M otion of M ultiphase M edia in C onduits, C on sultants B u reau , New York (1969). 12. W allis, G. B. , O n e -D e m e n sio n al T w o -P h a se Flow , M cG raw - Hill, Inc. , New York (1959). 13. B e a rm a n , R. J. and K irkw ood, J. G. , J, Chem , Phys, , 28, 136 (1958). 14. R akhm atulin, Kh. A, , " P rin c ip le s of the G as D ynam ics of M utually P e n e tra tin g M edia, " P rik l. M atem . i M ekh, , 20, No. 2 (1956). 15. G oldstein, S., P ro c . London M ath. Soc. (2), 34, 51 (1932). 16. E n d te r, F. and G eb au er, H. , O ptik, 13, 97 (1956). 17. H atch, T. , J . F ra n k lin Inst. . 215, 27 (1933). 128 A PPEND IX A * TABLES O F DATA AND SHEAR DIAGRAMS U nless o th erw ise s ta te d ,sc a le read in g s a r e for die 50 gm cm M easu rin g Head. 129 60 - A n-Hexadecane B c ts-D e c a lln C D i s t i l l e d Water (0.001 M S ta b i l i z e D D is til led Water E o-Xylene F Benzene 40 - 20 SO gm cm Measuring Head Laminar ► - Turbulent 0.2 0 .4 0.6 R eciprocal o f Speed F actor Figure 29a. S ingle Components Data, T * 20*C Linseed 011 Mineral 011 n-Hexadecane cis-D ecal In $00 gm cm Measuring Hea 40 20 0 0 .4 0.5 0.1 0.2 0.3 0 li J ! R eciprocal o f Speed Factor Figure 29b. Single Components Data, T • 20*C TABLE 4 VISCOMETER SCALE DEFLECTION: SINGLE COM PONENT DATA T - 20“C Speed Factor D eflection U ■see- Water Benzene o-Xylene Mineral Oil* Linseed Oil* cis-D ecalin n-Hexadecane 2.5 1.5 1.8 3.1 9 6.0 7.5 9 5.5 3-8 4.3 5.9 9.5 13.8 16.8 6 9-0 6.5 7.5 8.9 14.8 21.0 25.0 3 17.0 12.0 14.5 18.8 29-9 41.5 48.9 2 26.2 18.8 21.8 28.0 45.8 61.6 72.5 Values are fo r 500 gm cm (standard) Measuring Head. To convert to 50 gm cm s ca le m ultiply reading by (-£§2^ . 19. 5) 131 TABLE 5 TGZ3 PARTICLE SIZE ANALYZER COUNTER NUM BER VERSUS SIZE RANGE Counter Size Range Counter Size Range Counter Size Range Counter Size Range Number m m Number m m Number m m Number m m 1 1.22-1.76 13 7.84- 8.39 25 14.46-15.01 34 21.08-21.64 2 1.76-2.32 14 8.29- 8.94 26 15.01-15.56 38 21.64-22.19 3 2.32-2.87 15 8.94- 9.49 27 15-56-16.12 39 22.19-22.74 4 2.87-3.42 16 9.49-10.04 28 16.12-16.67 40 22.74-23.29 5 3-97-4.52 17 10.04-10.60 29 16.67-17.22 41 23.29-23.84 6 3.97-4.52 18 10.60-11.15 30 17.22-17-77 42 23.84-24.40 7 4.52-5.08 19 i 1.15-11.70 31 17.77-18.32 43 24.40-24.95 8 5.08-5.63 20 11.70-12.25 32 18. 32- 18.88 44 24.95-52.50 9 5. 63- 6.18 21 12. 25- 12.80 33 18.88-19.43 45 25.50-26.05 10 6.18-6.73 22 12.80-13.36 34 19.43-19.98 46 26. 05- 26.60 11 6.73-7.28 23 13.36-13.91 35 19.98-20.53 47 26.60- 27.16 12 7-28-7.84 24 13.91-14.46 36 20. 52- 21.08 48 27.16-27.71 u j r s > 133 TABLE 6 VISCOMETER SCALE DEFLECTION AND SHEARING DATA 10 P e rc e n t Benzene/Water T - 20°C Shearing S tress Shearing Rate 2 Speed dyne/cm 1/sec ” ^ 2 Factor D eflectio n Measured Infinite Measured Infinite Sample 1 9 6.1 4.37 0.140 418.7 13.43 0.216 6 9.4 6.73 0.209 628.0 19.55 0.816 3 18.7 13.39 0.419 1256.0 39.34 0.739 2 28.1 20.12 0.629 1864.0 58.89 0.789 Sample 2 9 7.4 5.30 0.137 418.7 10.85 4.436 6 11.0 7.87 0.206 628.0 16.43 4.260 3 22.4 16.04 0.412 1256.0 32.23 4.610 2 33.4 23.91 0.618 1884.0 48.66 4.495 Sample 3 9 7-7 5.51 0.136 418.7 10.39 5.190 6 11.5 8.23 0.205 628.0 15.65 5.109 3 22.9 16.39 0.410 1256.0 31.46 5.028 2 38.5 24.0 0.617 1884.0 48.50 4.552 la 1 TABLE 7 M E A S U ftE O SIZE OISTftlBUTIOM 10 Parcent Benzene/Water Staple 2 S « p l* 3 Countar N t^ a r CtMila- Fra- tlv e quency Count 8 Finer Counter H uber Fre quency Cuewl a- tlve Count (Finer Counter Fra il i«d>er quency Cuwla* tlve Count (F iner 1 U 14 0.0 1 38 38 0.0 1 40 40 0.0 2 28 42 12.4 2 36 74 12.7 2 55 95 13.0 3 20 62 37.2 3 47 121 24.7 3 36 131 32.1 4 13 75 54.9 4 40 161 40.4 4 54 185 42.4 5 10 85 66.4 5 45 206 53.6 5 26 211 60.0 ( 1 93 75.2 6 58 254 68.6 6 53 264 68.4 7 4 97 82.4 7 28 282 84.5 7 20 284 85.5 8 2 99 85-8 8 7 289 94.0 8 15 299 92.0 9 5 104 87.8 9 5 294 96.4 9 4 303 97.0 15 9 113 92.1 14 6 300 98.0 10 4 12 2 307 309 98.2 9 9 5 O rlflnal M agnification *45-0 445.0 445.0 Enlargaaant 4.4 L I L i Seal* Total M agnification 0 » aleroM 9 Standard 1958-0 1.60 1.51 Standard 2225.0 M O 1.51 Standard 2 S B L 1 1.10 I >9 j 1 3 4 0 4 V c VI tt I. w « /> 41 -C to 30 1.6 microns 1.4 microns 1.1 microns 20 10 0 1500 2000 1000 500 0 135 Rate of Shear. & , sec m -1 Figure 30. Shear Diagram fo r 10 percent Benzene/Water T - 20“C (SI « c VI VI t i-t to L. < 0 V -C t o 30 1.7 microns 1.6 microns 1.4 microns 20 10 0 1500 2000 1000 500 0 Rate of Shear. £ , sec m -1 F igure 31. Shear Diagram fo r 20 percent Benzene/Watert T * 20*0 9 6 3 2 9 6 3 2 9 6 3 2 TABLE 8 VISCOMETER SCALE DEFLECTION AND SHEARING DATA 20 Percent Benzene/Water T - 20°C Shearing S tress Shearing Rate dyne/cm I/se c D eflection Measured Infinite Measured Infinite Sample 1 7.4 5-30 0.133 418.7 10.51 11.2 8.02 0.199 628.0 15.62 22.8 16.32 0.398 1256.0 30.64 32.2 23.77 0.599 1884.0 47.47 Sample 2 8.8 6.30 0.131 418.7 8.71 12.5 8.95 0.197 628.0 13.86 26.2 18.76 0.393 1256.0 26.34 39.0 27.92 0.590 1884.0 39.84 Sample 3 9.1 6.52 0.130 418.7 8.40 13.5 9.66 0.196 628.0 12.75 27.0 19.33 0.392 1256.0 25.50 40.6 29.07 0.588 1884.0 38.15 TABLE 9 MEASURED SIZE DISTRIBUTION 20 Percent Im n M / W iU r la I tie 2 sa—u 3 • S l l i Fre quency Cumula tive Count (Finer Countar IhaAir Cumule- Fra- tlve quency Count (Fine r Counter Nwdtor Cu—la- Fra- tlve quency Count (F iner i 19 19 1 30 30 0.0 1 56 56 0.0 2 41 60 2 49 78 11.1 2 51 107 10.9 3 42 102 3 39 117 29.9 3 60 167 36.0 4 43 145 4 50 167 43.2 4 51 219 59.4 9 20 165 5 26 193 61.9 5 15 233 77.4 6 13 179 6 24 217 71.5 6 21 254 92.6 7 9 197 7 20 237 90.3 7 10 264 90.0 • S 192 1 11 249 87.6 9 3 267 93.5 29 • 200 9 1 1 259 91.6 9 6 273 94.6 16 II 270 96.0 12 9 282 97.0 Original M agnification 445.00 445.00 445.00 Enlarge—nt 4.20 4.90 4.50 Sea la Standard Standard St—dard Total M agnification 1969.00 2136.00 2002.50 D .Microns 9 L R 1.62 1.40 •a 1.40 1.46 1.44 9 6 3 2 9 6 3 2 9 6 3 2 TABLE 10 VISCOMETER SCALE DEFLECTION AND SHEARING DATA 30 P ercen t Benzene/Weter T - 20°C Shearing S tre ss Shearing Rate 2 dyne/cm I/s e c D eflection Measured Infinite Measured Infinite Sample I 9.6 6.87 0.125 418.7 7.60 14.2 10.17 0.187 628.0 11.57 28.5 20.40 0.375 1256.0 23.06 43.1 30.86 0.562 1884.0 34.29 Sample 2 11.8 8.45 0 .123 418.7 6.10 17.4 12.46 0.185 628.0 9.32 35.0 25.06 0.369 1256.0 18.53 52.9 37.87 0.554 1884.0 27.57 Sample 3 12.8 9.16 0.122 418.7 5.60 19.0 13.60 0.184 628.0 8.49 39.0 28.21 0.367 1256.0 16.35 58.4 41.81 0.551 1884.0 24.83 TABLE I I MEASURED SIZE DISTRIBUTION 30 N r c m t Banzana/Matar Sanpla 1 Sanpla 2 Sanpla 3 Countar IHinbar Cuaula- fra - tlva quancy Count tfln a r Countar Ntntor Cunula- Fra- tlva quancy Count tF la a r Countar Nwbar Ciaaula- Fra- tlva quancy Count SFInar 1 21 21 0 1 36 36 0.0 1 36 36 0.0 2 29 50 21 2 55 89 16.6 2 28 62 17.0 3 16 6k 50 3 36 125 63.0 3 36 96 31.0 6 5 69 66 6 35 160 60.3 6 39 135 68.0 i 7 76 69 5 13 173 77.2 5 23 158 67.5 6 6 00 76 6 10 >83 63.6 6 18 176 79.0 10 2 82 60 9 12 195 86.0 7 11 167 83.0 13 7 69 62 15 0 203 96.0 9 10 197 93.5 26 6 95 69 36 k 207 98.0 15 3 200 98.5 31 5 100 95 Original M agnification 665.0 665.0 66?.0 Ealarganant L I 6.0 6,8 Sm Ii Total M agnification D^, alcrana Standard 1W.0 L i L i Standard I7>0.0 1 2 L I Standard 2136.0 1 1 1.6 139 9 6 3 2 9 6 3 2 9 6 3 2 TABLE 12 VISCOMETER SCALE DEFLECTION AND SHEARING DATA 40 P e rc e n t Benzene/Water T - 20°C Shearing S tress Shearing Rate 2 dyne/cm 1/s e c D eflection Measured Infinite Measured Infinite Sample I 11.3 8.09 0 . 119 418.7 6.15 16.4 11.74 0.179 628.0 9.55 33.0 23.63 0.357 1256.0 18.99 52.0 37.23 0.534 1884.0 27.03 Sample 2 15.6 11.17 0.117 418.7 4.38 23.5 16.82 0.175 628.0 6.54 47,2 33.79 0.350 1256.0 13.02 70.8 50.69 0.526 1884.0 19.54 Sample 3 20.5 14.68 0.116 418.7 3.29 31.0 22.19 0.173 628.0 4.91 61.4 43.96 0.347 1256.0 9-91 91.0 65.15 0.521 1884.0 15.05 TMU 1 3 Co— t s r W « r l a ^ l * 1 C m — la - F ra - tlv a tw o c y C a a t V I n or MJttlMEO t i n OISTMOVTION 40 F o rc a n t O iw ataa/M atar Sanpla 2 Cm — la - C — n t o r F ra - t lv a N * * a r avancy Count V I — r C a an tar S an p la 3 Cu— l a - F r a - t lv a a— m cf Count V I — r 1 4 4 0 .0 1 4 4 0 .0 i 20 20 0 .0 X 7 13 7 .5 2 10 14 4 1.4 2 13 33 10.0 3 9 22 14.2 3 15 31 11.7 3 14 49 14.5 4 % 27 27.5 4 11 43 2 2.4 4 2» 77 2 4 .5 S 3 30 33-1 5 13 54 3 1 .4 5 11 lo t 30.5 4 II 41 3 7 .4 4 >9 75 4 0.1 4 31 139 5 4 .0 • 9 50 5 1 .1 I 15 90 5 4 .4 • 37 174 4 9 .5 IS 9 59 4 2 .5 10 II 101 4 5 .4 10 14 192 • 2 .0 IX 1 45 73.5 14 14 117 73.4 12 S ■97 9 4 .0 I I 5 70 • l . l *1 7 124 • 5 .4 19 2 199 9 » .5 24 4 74 • 7 .5 21 • 132 9 0 .4 34 1 200 9 3 .5 34 3 77 91-5 34 5 137 9 4 .4 U 3 10 9 4 .2 O r l « l — I i t o p l f t u t l M Sea la T o t a l M a p n l f I c a t t o n 0> t a lc r a — < 1 4 .0 0 * U l 1002.50 i.4o i.o» US. 0 0 4 .5 0 SiaMari M02J0 2.00 L2L C a tl— taO 142 TABLE 14 VISCOMETER SCALE DEFLECTION AND SHEARING DATA 50 P e rc e n t Benzene/Water* T - 20“C Shearing S tre s s Shearing Rate 2 Speed dyne/cm I/s e c Factor D eflection Measured Infinite Measured Infinite Sample I 9 33.0 23.63 418.7 6 42.5 30.41 628.0 3 76.5 54.80 1256.0 2 112.0 80.02 1884.0 Sample 2 9 34.2 24.48 418.7 6 46.0 32.98 628.0 3 86.0 61.60 1256.0 2 126.0 90.03 1884.0 Sample 3 9 36.0 25.80 418.7 6 50.6 36.21 628.0 3 95.4 68.36 1256.0 2 137.1 98.20 1884.0 * Non-Newton I an Sample 143 N . « C * tt i - (/» L. « tt £ l/> □ Sample 1 "coarse" ) A Sample 2 "medium" - O Sample 3 " f in e " ,■ non- Newtonian 0 500 1000 1500 2000 Rate o f Shear, 4„, sec -1 m' Figure 32. Shear Diagram fo r 50 p ercen t Benzene/Water, T » 20°C Data of: System: Benzene/Water T - 20*C o Samples I A Samples 2 O Samples 3 3.0 ( _ > E 2.0 M 4 - > X 0 0.4 0. 1 0.2 0.3 ' Volume F ra c tio n , f j fig u re 33. E f fe c t o f C oncentration (V ariable D ) on Mixture V iscosity * i 9 6 3 2 9 6 3 2 9 6 3 2 TABLE 15 VISCOMETER SCALE DEFLECTION AND SHEARING DATA 10 P e rc e n t o-X ylene/W ater T - 20°C Shearing S tre ss Shearing Rate 2 dyne/cm I/se c D eflection Measured Infinite Measured Infinite Sample 1 7.0 5.01 0.140 418.7 11.72 10.5 7.52 0.211 628.0 17.59 21.8 15.61 0.420 1256.0 33.78 32.5 23.27 0.630 1884.0 51.01 Sample 2 7.8 5.58 0.139 418.7 10.44 11.7 8.38 0.209 628.0 15.65 22, 4 16.04 0.419 1256.0 32.81 33.0 23.64 0.629 1884.0 50.18 Sample 3 8.0 5.73 0.138 418.7 13.56 11.9 8.59 0.208 628.0 20.35 23-3 16.75 0.418 1256.0 31.31 34,5 24,70 9,627 1884.0 47.84 TABLE 1C MEASURED SIZE DISTRIBUTION 10 Percent o-Xy len e/V e ter Saople t Seople 2 (angle 3 Counter H uber Fre quency Cumula tiv e Count (Finer Counter Hinder Fre quency Cumula tive Count (Finer Counter Nunber Fre quency Cumula tiv e Count (F iner 1 *1 *1 0.0 1 0 0 0.0 1 73 73 0.0 2 5* 95 28.* 2 10 10 0.0 2 85 158 36.5 3 15 110 *6.0 3 32 *2 5.0 3 28 186 79.0 * 7 117 76.2 * 3* 7* 21.0 * 10 196 83.0 5 * 121 81.0 5 3* 112 38.0 5 2 198 98.0 C * 127 8*.0 C 2* 138 5*.0 * 1 199 99.0 1 1 135 88.0 8 13 151 69.0 7 1 200 99.5 10 3 138 93.5 10 13 16* 75.5 t3 2 1*0 95.* 12 21 18* 82.0 19 2 1*2 9*.0 17 * 190 93.0 21 1 1*3 98.5 25 * 19* 95.0 31 1 1** 99.0 *0 * 200 98.0 Original M agnificatIon Enlargement Sea la Total M agnification D , alcront 9 212.2 i i Standard JSLi 2.* L I 212.2 375*.0 1.1 1.6 2*70.0 L I I.* CM « l c > * T > M V t V L. Q 4) £. < /) 2.4 microns 1. 1 ml crons 0.8 ml crons 2 0 - 1 0 - 2000 1500 1000 500 0 146 -1 Rate of Shear, sm, sec Figure 34. Shear Diagram fo r 10 p ercen t o-Xylene/W ater, T ■ 20°C 3.5 microns 1.8 microns 1.3 microns C M 20 - E sn « /> 1500 2000 1000 500 -I Rate of Shear, i sec Figure 35. Shear Diagram fo r 20 percen t 0- Xylene/Water, T - 20*C 9 6 3 2 9 6 3 2 9 6 3 2 TABLE 17 VISCOMETER SCALE DEFLECTION AND SHEARING DATA 20 P e rc e n t o -X y len e/U a ter T - 20°C Shearing S tress Shearing Rate 2 dyne/cm 1/s e c D eflection Measured Infinite Measured Infinite Sample 1 8.3 5.9*1 0.136 A18.7 9.59 12.6 9.02 0.20*4 628.0 1*4.21 2 5 .A 18.18 0 .*t08 1256.0 28.19 37-3 26.70 0.613 188*4.0 *•3.25 Sample 2 9.*t 6.73 0.135 *418.7 8.*4l 1*4.2 10.17 0.203 628.0 12.52 27.8 19.90 0.*406 1256.0 25.60 *•1.5 29.71 0.609 188*4.0 38. 6- Sample 3 10.0 7.16 0 . 13*i *•18.7 7.87 15.1 10.81 0.202 628.0 11.73 29.8 21.33 0.*40*t 1256.0 23.79 *4*4.6 31.93 0.606 188*4.0 35.76 TABLE 1ft MEASURED SIZE DISTRIBUTION 30 Parcant o "ly1a na /V a ta r Swplo I Sayli 2 s«vi« i Countar N «bar Fm- quancy Cuaula- tlv a Count SFInar Countar •Un*tr Fra- quancy Cunula- tlva Count tF lnar Countar Nia*ar Fra- quancy Cwwla- ttv a Count RFinar 1 26 26 0.0 1 30 30 0.0 1 26 26 0.0 2 36 60 13-0 2 59 89 10.9 2 53 77 9.6 3 16 76 30.0 3 66 155 32.6 3 69 126 30.6 6 26 9ft 37.0 6 69 226 56.6 6 57 183 50.0 5 10 10ft 69.0 5 26 250 81.6 5 29 212 72.6 6 ft 116 56.0 6 ft 25ft 90.8 6 23 235 86.6 7 16 130 s f t.o 7 7 265 93.6 7 6 261 93.5 ft 15 165 65.0 8 5 270 96.3 8 5 266 96.0 10 17 162 72.5 9 1 271 98.0 9 3 269 97-9 16 2ft 190 •l.O 12 2 273 98.5 11 1 250 99.2 22 6 196 95-0 1ft 1 276 99.1 IS 1 251 99.5 35 6 200 9S. 0 30 1 275 99.5 O rl|lM l H afnlf(cation tllU fj lWHt Seal* Total Magnification < > .nitron* 9 212.2 997.3 h L 2.1 665.0 6.0 Standard 1780.0 l.ft (.3 L I 1.6 1 4 8 9 6 3 2 9 6 3 2 9 6 3 2 TABLE 19 VISCOMETER SCALE DEFLECTION AND SHEARING DATA 30 P e rc e n t o-X ylene/W ater T - 20#C Shearing S tre ss Shearing Rate 2 dyne/cm 1/s e c D eflection Measured Infinite Measured Infinite Sample I 10.8 7.73 0.131 418.7 7.07 16.5 11.61 0.196 628.0 13.56 32.6 23.34 0.392 1256.0 21.08 <(9.6 35.51 0.587 1884.0 31.15 Sample 2 12.5 8.95 0.129 418.7 6.07 18.8 13.46 0.195 628.0 9.07 36.0 27.20 0.389 1256.0 17.95 57-6 41. 24 0.583 1884.0 26.63 Sample 3 14.5 10.38 0.128 418.7 5.19 21.5 15.39 0.193 628.0 7.89 42.5 30.43 0.387 1256.0 15.97 63.6 45.53 0.580 1864.0 24.02 TABLE 20 MEASURED SIZE DISTRIBUTION 30 Farcant o -Iy lu n a /U a to r Sanpla 1 Sanpla 2 Sw pla 3 natter ■ ta r CuMult- Fra- tlva quancy Count tF ln ar Counter Nuritar Cunula* Fro* tlvo quancy Count tF ln ar Counter Nwdtar Cumila- Fra- tlva quancy Count tFtna 1 4 4 0.0 1 4 4 0.0 1 8 8 0.0 2 i 10 2.2 2 3 7 1.8 2 23 31 4.0 J 16 26 5.4 3 26 33 3.1 3 45 76 15.5 4 ID 42 14.0 4 36 69 14.8 4 41 117 38.0 5 39 61 22.6 5 37 106 30.8 5 50 167 58. S 6 3* 115 43.6 6 47 153 47.4 6 23 190 63-5 7 20 135 61.8 7 33 186 68.4 7 4 194 95.0 6 19 156 72.5 8 12 198 83.0 8 4 198 97.0 10 9 163 82.9 to 13 211 88.0 9 1 199 99-0 12 0 171 67.5 1 1 8 219 94.3 12 1 200 99.5 16 9 ISO 92-0 13 2 221 98.0 30 4 184 96.6 15 3 224 98.7 11 2 186 99.0 212.2 *L° 445.0 L i L i L i Standard Standard Standard 1167.1 2002.5 2447.5 4.0 2.1 1.4 1.4 L i L i i J CM. s V. « c > » VI v « /> L. I V V £ < /) 2.4 ml crons 2.1 microns 1.4 ml crons 20 0 2000 1500 500 1000 0 151 Rate of Shear, s , sec fn -i Figure 36. Shear Diagram fo r 30 percent o-Xylene A la te r, T * 20"C 0 " 7*6 microns 9 D *2 .4 microns 9 0 * 1.5 microns e i/l 2000 1500 1000 500 0 -1 Rate of Shear, i . sec is Figure 37. Shear diagram fo r 40 percent o-Xylene/W ater, T ■ 20*C TABLE 21 152 VISCOMETER SCALE DEFLECTION AND SHEARING DATA 40 P e rc e n t o-X ylene/W ater T - 20°C Shearing S tress Shearing Rate 2 Speed dyne/cm I/s e c „ o icto r D eflect Ion Measured 1nfIni te Measured Infinite 2 9 15.5 11.10 Sample 1 0.126 418.7 4.74 5.49 6 23.5 16.82 0.188 628.0 7.03 5.52 3 46.8 33*51 0.377 1256.0 14.13 5.51 2 69.5 49.76 0.566 1884.0 21.41 5.48 Sample 2 9 19.6 14.03 0.125 418.7 3.72 6.19 6 30.0 21.48 0.187 628.0 5.46 6.18 3 59.0 42.24 0.374 1256.0 11.12 6.14 2 88.2 63.14 0.561 1884.0 16.74 6.13 Sample 3 9 21.2 15.18 0.124 418.7 3.43 6.31 6 3K5 22.55 0.186 628.0 5.19 6.29 3 63.6 45.53 0.373 1256.0 10.29 6.31 2 95.0 68.01 0.559 1884.0 15.51 6.30 TABLE 22 MEASUMD SIZE DISTRIBUTION A O P t r t M t o-Xylens/W et«r Saapla 1 Saapla 2 Saapla 3 i Countar Nuabar Fre quency Cumila- tlva Count (Finer Counter Nunbar Fre quency em ula tiv e Count (F iner Counter ttmfcar Fre quency em ula tiv e Count (F iner ; 1 16 16 0.0 1 1 1 0.0 1 2 2 0.0 m 59 7A 5.3 2 A A A S O.A 2 2 A 26 0.8 : 3 72 1A 6 2A.6 3 70 115 18.8 3 12 38 10.A 4 At 132 AB.i A 3A 1A 9 A8.0 A 36 7 A 15.2 J 5 23 221 6A.0 5 26 175 62.0 s A Z 116 29.6 t 10 23) 73-1 6 It 191 73.0 t A A 160 At.A 7 12 2A3 76.9 7 1 1 202 79.5 7 A O 200 6A.0 I 12 255 91.0 8 10 212 8A.0 8 15 215 90.0 10 17 272 85.0 10 8 220 88.5 to 13 228 86.0 12 12 2SA 90.5 12 t 226 91.6 13 IA 2A 2 91.A 15 6 230 3A.5 16 8 23A 9A.0 18 5 2A7 96.5 2} 5 235 96.5 26 6 2 A O 97.5 21 2 2A9 98.6 30 2 297 98. A 26 1 250 99*2 A O 3 300 99.0 Original HegnlfIcatloa InUriMMt Scale Total Magnlf(cation 0 , olcrons 9 SUL h i StanCart A2S.0 L i 1.6 112.2 1AA|.0 L i 2937.0 1.5 1. A U l 154 TABLE 23 VISCOMETER SCALE DEFLECTION AND SHEARING DATA 50 Percent o-Xylene/Water* T - 20®C Shearing S tress Shearing Rate 2 Speed dyne/cm 1/sec Factor D eflection Measured Infinite Measured Infinite Sample I 9 18.9 13.53 '♦18.7 6 24.8 17.78 628.0 3 46.0 33.01 1256.0 2 69.0 49-40 1884.0 Sample 2 9 41,4 29.62 418.7 6 57.7 41.40 628.0 3 103.0 73.70 1256.0 2 150.0 100.02 1884.0 Sample 3 9 36.6 26.20 418.7 6 54.4 38.96 628.0 3 102.0 73.00 1256.0 2 150.0 100.02 1884.0 * Non'Newton 1 an Sample V C > * o in V i / i k i» V JZ m 120 155 □ Sample 1 1 * c i . ■ » * "non- S«"P'« 2 i Newtonian O Sample 3 ' X 500 1000 Rate of Shear, s , sec m 1500 -1 2000 Figure 38. Shear Diagram fo r 50 percent o-Xylene/W ater, T - 20°C System: o-X ylene/V ater T - 20*C Data of: « in Samples 1 Samples 2 Samples 3 £ c « o E * M 2.C- V I o u V I X 0.*t 0.2 0.3 0.1 0 a; Volume F ra c tio n , f^ Figure 39. E ffe ct of C oncentration (V ariable 0 ) on Mixture V isco sity 156 TABLE 24 VISCOMETER SCALE DEFLECTION AND SHEARING DATA 10 P e rc e n t Mineral 011/W ater T - 20°C Shearing S tress Shearing Rate 2 Speed dyne/cm 1/sec _ q Factor D eflection Measured I n f i n i t e Measured I n f i n i t e 2 Sample 1 9 7.0 5.01 0.884 418.7 73.81 1.96 6 10.3 7.37 1.370 628.0 116.8 2.09 3 21.0 15.03 2.650 1256.0 221.4 1.96 2 31.5 22.55 3.980 1884.0 332.1 1.96 Sample 2 9 7.3 5.23 0.824 418.7 66.0 1.71 6 11.2 8.02 1.190 628.0 93.6 1.59 3 22.5 16.11 2.370 1256.0 185.1 1.57 2 33-5 23.98 3.600 1884.0 282.8 1.60 Sample 3 9 7.9 5.66 0.737 418.7 54.6 1.33 6 12.0 8.59 1.090 628.0 79.5 1.28 3 23.0 16.47 2.300 1256.0 175.5 1.46 2 3^.5 34.70 3-450 1884.0 263.3 1.46 TABLE 25 Counter K uO lf Saapla 1 Cimla* Fra- tlv a quency Count SFInar 10 Furcent Counter Ngafcer Mineral Oil/W ater Saapla 2 Cuaula- Fre- tlva quency Count tF ln ar Counter NuMber Saapla 3 Cuaula- Fra- tlva quency Count SFIner 1 7 7 0.0 1 15 15 0.0 1 14 14 0.0 2 7 14 14.0 2 30 45 4.0 2 31 45 6.2 5 10 24 28.0 3 41 86 18.0 3 40 85 19-8 A 9 33 48.0 A 48 134 35.4 4 52 137 37.4 5 2 35 44.0 5 23 157 53.4 5 29 164 60.4 4 3 38 70.0 4 19 174 42.8 4 17 183 73.0 7 3 41 74.0 7 14 192 70.4 7 II 194 80.5 9 4 A S 82.0 8 15 207 74.8 8 8 202 85.5 15 3 A S 90.0 9 II 218 82.8 9 8 210 B9.0 23 2 50 94.0 10 7 225 87.2 to II 211 92.3 II 12 237 90.0 12 10 221 93.0 15 9 244 94.8 14 4 225 97.4 23 2 248 98.4 23 2 227 99.0 33 2 250 99.2 Original N tg n tflc itlM Enlargaaant Seal* Total M agnification D . alcron* 9' 212.2 Standard I.A 1 5 7 158 6 .9 microns 2 .7 microns 1 ,*i microns » /» ^ * - 0“^ - ------------- * — S 0 500 j : IA _ t Rate o f Shear, i , sec m F igure 60. Shear Diagram f o r 10 percent Mineral 011/W ater, T-20*C 2000 1500 1000 8.6 microns 3 .0 microns I .8 microns 2000 1500 1000 500 Rate o f Shear, s , sec F igure 61. Shear Diagram fo r 20 percent Mineral 011/W ater, T»20#C 9 6 3 2 9 6 3 2 9 6 3 2 TABLE 26 159 VISCOMETER SCALE DEFLECTION AND SHEARING DATA 20 P ercent M ineral O il/W ater T - 20®C D eflect ion Shearing S tress dyne/cm^ Measured I n f in ite Shearing Rate l/s e c Measured I n f in ite - 0 Samp 1e 1 7.4 5.30 1.53 11.0 7.87 2.32 21.5 15.39 A.83 32.5 23.27 7.16 Sample 2 9.0 6.44 1.20 13.5 9.66 1.80 26.3 18.83 3.70 39.7 28.42 5.51 Sample 3 10.0 7.16 1.09 15.0 10.74 1.64 29.7 21.26 3.30 44.0 31.86 4.96 418.7 120.9 1.77 628.0 185.6 1.81 1256.0 394.3 1.94 1884.0 579.4 1.89 418.7 78.0 1.06 628.0 117.0 1.06 1256.0 246.8 1.13 1884.0 365.5 1.11 418.7 63.86 0.83 628.0 95.79 0.83 1256.0 195.10 0.84 1884.0 293-30 0.85 TM . E 27 H E A S U ftE O SIZE OISTMBUTION SO N r c w t nlnaral O II/U it«r Saapla 1 ClMul«- Councar Fra- tlv a ir quancv Count t f l i a r Saapla 2 Cuaula- Countar Fra- tlva Ovafcar quency Count tF ln ar Saapla 3 Cuauta- Countar Fra- tlva it ouancv Count Original flagnlfI cation Enlarpaaant Seal# Total HapntfIcatloa 0 , al cram 9 Standard *» •*> $JC 2.21 Standard 1)03.*0 3.00 U 1 Standard i.ao L 2 i tFlnor 1 17 17 0.0 1 3 0.0 1 1 1 II 0.0 2 19 3S 11.3 2 1 29 3.0 2 25 36 6.9 3 17 53 2*.0 3 2 52 29.0 3 17 53 22.5 * 20 73 35.3 * 1 62 52.0 * 17 70 33.2 5 8 81 *8.6 5 69 62.0 5 8 78 *3.8 6 1 1 92 5*.0 7 7* 69.0 6 12 90 *8.8 7 5 97 61.3 9 82 7*.0 7 * 9* 56.* 8 8 105 6*.5 11 86 82.0 9 12 106 58.6 10 5 111 70.0 16 89 86.0 11 * 110 66.2 12 10 121 7*.0 19 9* 89.0 13 13 123 68.2 17 IS 137 80.6 22 96 9*.0 16 12 135 76.9 23 7 US 91.3 26 100 96.0 19 5 1*0 8*.* 28 2 1*6 96.0 2* 7 1*7 87.* 32 2 1*8 97.3 32 * 158 96.2 ** 2 150 98.6 36 2 160 98.9 8*. 80 212.20 **£. 00 L i l 5.20 I J 5 160 9 6 3 2 9 6 3 2 9 6 3 2 TABLE 28 VISCOMETER SCALE DEFLECTION AND SHEARING DATA 30 P ercen t Mineral 011/W ater T - 20°C Shearing S tress Shearing Rate 2 dyne/cm 1/sec D eflection Measured I n f in ite Measured I n f in ite Sample 1 8.2 5.87 1.95 418.7 138.9 N .3 8.09 3.27 628.0 254.0 23-3 16.68 6.27 1256.0 471.8 3*1.7 24.56 9.65 1884.0 740.4 Sample 2 12.0 8.59 1.41 418.7 68.6 17.3 12.39 2.16 628.0 109.8 3*t.O 24.34 4.38 1256.0 225.9 51.0 36.51 6.57 1884.0 338.9 Sample 3 14.0 10.2 1.29 418.7 54.2 20.3 14.53 1.98 628.0 85.5 40.7 29.14 3.95 1256.0 170.4 60.8 43.53 5.94 1884.0 257.1 MCASUftO SIZE OfSTtlMTION S i ^ l i t JO Percent Mineral 01l/U ater Sw pU 2 Counter Fre- Cunule- tlv e Counter Fre- Cmula- tlv e Counter Fre* pie 3 em ula tiv e Umber quency Count (Finer Nu*er quency Count (Finer Nua*er quency Count (Finer 1 S S 0.0 1 7 7 0.0 1 28 28 0.0 2 7 12 4.2 2 10 17 4.7 2 48 76 11.2 3 8 20 10.0 3 27 44 11.3 3 35 111 30.4 4 5 25 16.6 4 22 66 29.3 4 37 148 44.4 5 8 37 24.2 5 16 82 44.0 5 21 169 59.2 7 11 48 30.8 6 17 99 54.6 6 21 190 67.5 10 18 66 40.0 7 10 109 66.0 7 1 1 201 76.0 12 15 81 55.0 8 12 121 72.9 8 6 207 80.5 15 17 38 67.4 10 8 129 80.6 9 8 215 82.8 18 10 108 81.5 12 6 135 86.0 1 1 8 223 86.0 20 4 112 89.6 15 4 139 90.0 13 9 232 89.1 24 6 118 93.0 18 4 >43 92.6 15 2 234 92.6 28 2 120 98.1 22 3 146 95.5 18 1 1 245 93-6 26 2 148 97.5 21 3 248 98.0 37 2 150 99.0 25 2 250 99.2 Original Hegnlf1 catIon 84.80 212.20 212.20 Enlargement h S B . L*£ 5.50 Scale Total Magnification 0 .»! crons 9 Standard 466.40 15-00 U 1 Standard 1114.10 Standard II&7-10 2.80 1.60 < N J c £ tl L. ( /> u « D V £ V) 3.9 microns 2.8 microns 2000 1300 1000 500 0 163 Rate o f Shear, s . sec -1 m Figure 1(2. Shear Diagram fo r 30 percent Mineral 011/Water, T ■ 20#C S e c £ i/ i X O Sample I "co a rse 1 & Sample 2 "medium1 V non- ^ Newton! 0 500 1000 1500 2000 -1 Rate o f Shear, s , sec Figure 43. Shear Diagram fo r 40 percent Mineral Of 1/Water, T - 20*C TABLE 30 164 VISCOMETER SCALE DEFLECTION AND SHEARING DATA 40 P ercen t Mineral 0 ! 1/W ater* T - 20°C Shearing S tress Shearing Rate 2 Speed dyne/on 1/sec Factor D eflection Measured I n f in ite Measured I n f in ite Sample I 9 9.0 6.45 418.7 6 13.2 9.46 628.0 3 24.7 17-70 1256.0 2 36.3 26.00 1884.0 Sample 2 9 24.0 17.20 418.7 6 32.4 23.20 628.0 3 57.0 40.81 1256.0 2 79.0 56.62 1884,0 Sample 3 9 26.0 18.63 418.7 6 37.4 26.80 628.0 3 69.2 49.60 1256.0 2 97.0 69.50 1884.0 * Non-Newtonlan Sample V I A I v u a « k > > VI 8 V I > V L. a X 165 3.or Data of: System: Mineral 01 I/Water T - 20*C o Samples I 2 . Oh A Samples 2 O Samples 3 1.0 0.1 0.2 0.3 x . Volume F ra c tio n , f j Figure A **. E ffe c t of C oncentration (V ariable Dg) on Mixture V iscosity IA i * > u a m >- w 8 £ 3 X r 3.0 2.0 - System: Linseed 011/Watej T - 20*C Data of: Samples 1 Samples 2 Samples 3 0.1 0.2 Volume F ra c tio n , f 2 Figure *t5. E f fe c t of C oncentration (V ariable D^) on Mixture V isco sity TABLE 31 166 VISCOMETER SCALE DEFLECTION AND SHEARING DATA 10 P e rc e n t Linseed O il/W ater T - 20°C Shearing S tress Shearing Rate 2 Speed dyne/cm I /s e c -fig Factor D eflection Measured I n f i n i t e Measured I n f i n i t e Sample 1 9 7.2 5.16 1.49 418.7 120.6 2.23 6 10.4 7.45 2.43 628.0 204.7 2.56 3 21.5 15.39 4.50 1256.0 367.2 2.26 2 32.0 22.91 6.87 1884.0 564.7 2.33 Sample 2 9 7.5 5.37 1.37 418.7 106.7 1.94 6 11.4 8.16 2.00 628.0 154.1 1.85 3 23-3 16.68 3.86 1256.0 290.3 1.71 2 34.3 24.56 5.98 1884.0 458.6 1.84 Sample 3 9 7.9 5.66 1.25 418.7 92.5 1.64 6 12.1 8.66 1.81 628.0 131.4 1.54 3 24.3 17.40 3.60 1256.0 260.0 1.52 2 35.3 25.27 5.69 1884.0 423.9 1.67 TABLE 32 M E A S U R E D SIZE DISTRIBUTION 10 Parcant Llnsaad Oll/Watar O rlflnal Ratification EnlariMMt Seal# Total Ragnlfixation D^, Microns ountar m bar Sanpta 1 Cunula- Fra- tlv a Ruancy Count tF lnar Countar Nuafcar Saapla 2 Ciaaula* Fra- tlva Ruancy Count tF ln ar Countar N«a*ar Saapla 3 Cunula- Fra- tlv a qnancy Count I fiim 1 23 23 0.0 1 12 12 0.0 1 14 14 0.0 2 13 34 23.0 2 85 94 7-0 2 24 42 10.7 3 4 40 34.0 3 33 130 57.0 3 23 45 28.0 4 9 49 40.0 4 13 143 74.5 4 14 81 43.4 5 7 54 49.0 5 3 144 84.0 5 15 94 54.0 4 4 42 54.0 4 3 149 84.0 4 1 1 107 44.0 1 8 70 42.0 9 4 153 87.4 7 1 1 118 71.2 II 4 74 70.0 12 4 159 90.0 8 7 125 78.5 13 9 85 74.0 15 5 144 93.4 10 9 134 83-3 17 8 93 85.0 20 2 144 94.4 12 4 140 89.2 22 2 95 93.0 27 3 149 97.4 »5 5 145 93.0 30 3 100 97.0 35 1 170 99.5 23 3 148 94.5 24 2 150 98.5 42.40 212.20 445.00 7.70 2 J& 7.90 Standard 324.50 5.20 2.43 Standard 1 2 V P . i.4o 1.34 Standard 0.90 1.48 O ' -j CM 6 - s . « c £ v > L. * * V » w v f IA 5.2 microns 1.6 microns 0.9 microns 168 2000 Hate o f Shear, s . sec -1 m' Figure *t6. Shear Diagram fo r 10 percent Linseed 011/Water, T - 20*C 30 - 6 .5 microns 2.0 microns 1.0 mi crons w . 5 £ * * m v * £ (A f c - m « £ V I 20 - 2000 Rate o f Shear, s . sec m - i fig u r e l»7. Shear Diagram fo r 20 percen t Linseed 011/Water, T ■ 20*C 9 6 3 2 9 6 3 2 9 6 3 2 TABLE 33 VISCOMETER SCALE DEFLECTION AND SHEARING DATA 20 P e rc e n t Linseed 011/W ater T - 20°C Shearing S tre s s Shearing Rate 2 dyne/cm l/s e c D eflection Measured I n f i n i t e Measured I n f in ite Sample 1 8.7 6.23 2. 12 418.7 142.8 13.0 9-31 3.20 628.0 216.1 26.0 18.61 6.41 1256.0 432.2 39.0 27.92 9.61 1884.0 648.2 Sample 2 9-5 6.80 1.91 4)8.7 117*7 13-9 9-95 2.95 628.0 186.0 28.5 20.1(0 5-24 1256.0 353.1 l»2.5 30.1(3 8.66 1884.0 536.2 Sample 3 9.6 6.88 1.89 418.7 115.2 14.5 10.38 2.82 628.0 170.3 29.5 21.12 5.53 1256.0 329.0 43.5 31.14 8.45 1884.0 511.0 HUMMED SIZE DISTRIBUTION 20 Parcant U nt«t4 OII/W#t#r Origin*! HagnlfleetIon Enlarqanant Seal* Total M agnification D . nlcrons fl Countar Nwbar Saaqila 1 Cuaula- Fra* tlv a qwncy Count tF lnar 1 7 7 0.0 2 21 28 7.8 3 1 1 39 31.1 4 8 47 43.4 S 4 S O 52.2 4 6 54 55.5 • 7 43 42.0 to 5 48 70.0 1 2 4 74 75.5 18 4 78 82.1 22 4 82 84.5 29 3 85 91.0 34 3 88 94.5 37 2 90 97.4 8 4 Jo 6.50 Standard 551-20 Countar Nuafear Sample 2 Cunula* Fra- tlv a quancy Count tF ln ar 1 5 S 0.0 2 9 14 3.3 3 28 42 9.3 4 34 78 28.0 5 24 104 52.0 4 13 117 49.4 7 7 124 78.0 9 8 132 82.4 1 1 3 135 88.0 IS 5 140 90.0 23 4 144 93.5 24 1 147 94.4 29 2 149 98.1 31 1 150 99.4 ait . 20 US. Standard 1474.40 2.00 Saapla 3 CuMla* Counter Fro* tlv a Nuafear qw rcy Count tF ln a r 1 18 IB 0.0 2 4$ 43 10.0 3 40 103 35.0 4 27 130 57.4 S 10 140 72.2 4 10 150 77.4 7 7 157 83.4 8 3 160 87.4 10 4 164 89.0 13 4 170 91.1 19 4 174 94.S 22 3 177 96.4 33 2 179 98.4 38 1 180 99.5 MS.00 ZJ* Standard 3293.00 1.00 9 6 3 2 9 6 3 2 9 6 3 2 TABLE 35 VISCOMETER SCALE DEFLECTION AND SMEARING DATA 30 P ercen t Linseed 011/W ater T - 20°C Shearing S tress Shearing Rate 2 dyne/cm 1/sec D eflection Measured I n f i n i t e Measured I n f in ite Sample I 10.0 7.16 2.69 418.7 157.2 15. 4 11.03 3.93 628.0 224.1 31.0 22.19 7.82 1256.0 442.6 47.5 34,01 11.51 1684.0 637.5 Sample 2 13.5 9.66 2.15 418.7 92.91 20.4 14.60 3.20 626.0 137*8 40.0 28.64 6.48 1256.0 284.3 59.5 42.60 9.77 1884.0 432.1 Sample 3 15.7 11.24 1.98 418.7 73.9 23.4 16.75 2.98 628.0 111.9 95.5 32.57 6.05 1256.0 333.3 67.0 47.97 9.16 1884.0 359.7 TMLI * 1 1 Saal! 1 Cwlr frm ~ ( t w o ao a a c y Cool KASdNM S I 2 E OISTRiauTIW 3 0 T ikmi llaaa ad 0 1 1 / V a t a r Sa a g l a 2 Cwala* t o w n t a r F r a - t l v a SFIaar Mt^ar gaancy C o w t t t f l n a r C o n n t a r Nwtar Sa nala 1 CMMla- Fm- t l v a Cnaacy C o a n t SF Ina r 1 1 7 1 7 0.0 1 24 24 0.0 1 1 1 I t 0.0 1 I I 21 ■ 1 . 9 2 3 S 5 9 U.O 3 3 9 5 0 1 0 . 5 1 4 3 1 1 1 . 2 3 3 4 93 2 9 - 5 1 10 7 0 1 5 . 0 4 S 1 7 3 5 . 4 4 3 9 1 3 2 4 4 . 5 4 I S 94 3 9 . 0 5 7 4 4 4 1 . 0 S 2 2 1 5 4 44. 0 5 1 1 I I S 4 0 . 0 4 1 47 4S .S 4 7 1 4 1 7 7 . 0 4 1 7 1 3 S 5 9 - 0 « 1 0 S7 5 1 . 1 7 9 1 7 0 •os 7 I I 1 4 4 47. 5 i i 4 4 | 434 < S 1 7 5 • 5 . 0 • 1 5 1 4 1 7 3 . 0 I S 4 47 474 1 0 4 1 7 9 • 7 - 1 9 1 0 1 7 1 So . 5 20 S 7 2 7 4 . S 1 3 7 IK 1 9 . 5 I I 1 4 I I S 0 5 . 5 15 5 7 7 • 0 . 0 1 9 7 1 9 3 93. 0 1 4 1 1 1 9 4 92 .5 3 0 S 1 2 0 5 . 5 24 4 1 9 4 94.5 2 2 2 1 9 S 50 .0 34 1 * 5 91.0 1 1 2 1 9 9 91. 5 27 2 200 99. 0 4 1 1 M 94.S 40 1 200 99.S 40 2 90 97.7 Or ig i n a l Ma gni fication 1 4 . 1 0 21 2 . 2 0 44 5.0 0 ia la r g a a a a t 4J0 Zjo Zjo S e a l * Sta nda rd St and ard St and ard T o t a l Ma gni fication SS I.2 0 1 4 7 4 . 4 0 32 9 3 . 0 0 l (, altnM 1.2 0 LJi S 3.0 0 u a . L±L 172 N V C > - ■ o I/I V l/» m « « /> 173 8.5 microns 2.2 microns 1.4 ml crons 20 0 0 500 1000 1500 2000 Rate of Shear, s , sec m Figure 48. Shear Diagram fo r 30 percent Linseed 011/Water, T * 20°C S c ■ S ’ * £ 4J i / > L . m £ Sample I "co arse" 1 Sample 2 "medium" ^ non" Sample 3 " f i n e 1 60 ■ Newtonian 2 0 - 0 500 1000 2000 Rate o f Shear, s , sec f ll - i Figure 49. Shear Diagram fo r 40 percent Linseed 011/W ater, T - 20*C TABLE 37 174 VISCOMETER SCALE DEFLECTION AND SHEARING DATA *0 P e rc e n t Linseed O il/W ater* T - 20°C Shearing S tress Shearing Rate 2 Speed dyne/cm 1/sec Factor D eflection Measured I n f i n i t e Measured I n f i n i t e Sample 1 9 18.5 13.39 *18.7 6 26.9 19.30 628.0 3 51*0 36,61 1256.0 2 7**0 53.11 188*.0 Sample 2 9 21.5 15*91 *18.7 6 31.* 22.58 628.0 3 59.0 *2.*0 1256.0 2 8A.0 60.36 188*.0 Sample 3 9 23.7 17.00 *18.7 6 33.9 2*.35 628.0 3 63.O *5.23 1256.0 2 91.0 65.37 188*.0 Non-Newton!an Sample 9 6 3 2 9 6 3 2 9 6 3 2 TABLE 38 VISCOM ETER SCALE DEFLECTION A N D SHEARING D A TA 10 Percent cIs-DecaI In/Water T - 20°C Shearing S tre s s Shearing Rate 2 dyne/cm t/s e c D eflection Measured I n f i n i t e Measured I n f in ite Sample I 6.5 *1.65 0.168 *18.7 15.10 10.2 7-30 0.2*7 628.0 21.2* 20.0 1*.32 0 . *98 1256.0 *3.69 30.0 21.*8 0.7*7 188*.0 65.53 Sample 2 7.1 5.08 0.162 *18.7 13.33 10.8 7.73 0.2*1 628.0 19.61 21.5 15.39 0. *8* 1256.0 39. *8 32.5 23.27 0.72* 188*.0 58.59 Sample 3 7.7 5.51 0.157 *18.7 11.93 11.7 8.38 0.235 626.0 17.59 23.0 16.*7 0 . *72 1256.0 36.01 3*. 5 2*. 70 0.708 188*.0 5*.02 TABLE 99 Saapla I MEASURED SIZE DISTRIBUTION 10 Parcant c l*-D *c a lIn /V a tn r U 2 U 3 Countar Ntnfcar Cuwla- Fra- tlv a quancy Count tF ln ar Countar N i^ a r Fre quency Cumla- ttve Count tF lnar Countar N w M bar C<Mlt~ Fra- tlva quancy Count tF ln a r 1 39 39 0.0 1 18 18 0.0 1 25 IS 0.0 2 66 105 19-5 2 35 53 9.8 2 53 78 12.5 3 26 131 52.5 3 29 82 28.6 3 60 118 39.0 6 8 139 65.5 6 27 109 kk.k 6 27 165 59.0 5 10 169 69.5 5 20 129 59.0 5 20 165 72.5 7 12 161 76.5 6 16 165 69.6 7 17 182 82.5 5 7 168 •0.5 7 1 1 156 78.6 10 5 187 91.0 10 5 173 86.0 8 8 166 86.2 13 6 191 93.5 13 12 185 86.5 to 13 177 88.6 19 6 195 95.5 15 6 191 92.5 16 3 180 95.6 26 2 197 97.5 IS 3 196 95.5 22 3 183 97.6 36 2 199 98.5 26 6 198 97.0 31 1 186 99.0 66 1 200 99-5 60 2 200 98.0 62 1 185 99.5 Original Magnification 62.60 212.20 665.00 Enlargaaant 7 J 0 Z JH 7.10 Seal* Total M agnlflcatIon D . a l c r o n s fl Standard Standard 1655.20 2.00 L l i Standard 3*g-?o 1.00 L S k 176 177 7.5 microns 2.0 microns 20 CM 1500 2000 1000 500 Rate o f Shear, s , sec fig u re 50. Shear Diagram fo r 10 p ercen t cls-D ecalin /W ater T - 20*C 8.5 microns 2.2 microns 1.1 microns CM t o 1000 1500 -1 2000 500 Rate of Shear, s , sec ” inr Figure 51. Shear Diagram fo r 20 percen t cIs-D ecaH n/W ater T - 20'C 9 6 3 2 9 6 3 2 9 6 3 2 TABLE 40 VISCOMETER SCALE DEFLECTION AND SHEARING DATA 20 P e rc e n t c l s - D e c a l 1n/W ater T - 20°C Shearing S tress Shearing Rate 2 dyne/cm 1/sec D eflection Measured I n f i n i t e Measured I n f in ite Sample I 8.0 5.73 0.179 418.7 13.14 12.0 8.59 0.269 628.0 19.71 24.5 17.54 0.536 1256.0 38.36 37.5 26.85 0.798 1884.0 56.03 Sample 2 9.5 6.80 0.171 418.7 10.52 14.0 10.02 0.257 628.0 16.13 27.7 19.83 0.516 1256.0 32.71 41.9 30.00 0.773 1884.0 48.55 Sample 3 10.0 7.16 0.169 418.7 9.86 14.9 10.67 0.253 628.0 14.91 30.0 21.48 0.506 1256.0 29.58 45.0 32.22 0.759 1884.0 44.37 TMLI M M M M K I *122 OISTRIOtfTliN M t a r c M t t l f N t i l l i / M m r O rig in a l N atal f I c a tli t a l te«l< 1 Saagla 2 laala 3 C » w » to r C m mIo - fr r tlva taaacy C o u n t VI nor C ountar N w *er fw g u M v e f ( w l r tlva C o u n t trinar Cavttar N M * ar C w m Io * P ro * tl«o gunner C o u n t UImt 1 » 1 1 t.O 1 3 1 3 1 t.O 1 2 2 2 2 0.0 t 3 4 4 9 a.o 2 5 1 1 2 10.4 2 5 9 4 1 12 .9 ) 4 7 lit 14.a 3 4 9 1 5 1 29-4 3 3 4 1 1 9 47.7 4 U 1 1 4 21.2 4 3 7 tat 54.9 4 2 4 1 4 3 70.0 5 1 7 1 1 1 44.1 f 2 2 2 1 0 40.4 S 4 1 4 7 44.0 4 3 7 2 4 1 51.4 4 4 2 1 4 74.) 4 4 1 5 5 44.5 1 3 4 in 40.0 7 4 2 2 0 74.5 9 5 1 4 0 91.0 1 * 7 3 0 3 71.0 9 9 2 2 9 40.0 II 5 1 4 5 9*0 9 1 4 32) 79.2 1 ) 9 2 3 4 4 3 .1 1 2 1 1 4 4 97.0 II M 3 4 1 71.4 1 4 1 4 2 5 4 aa.s 2 2 2 1 4 4 97.5 1 1 it 1 7 9 aa.o 1 9 9 2 4 5 9)0 2 7 1 1 4 9 94.4 1 $ 1 2 3 9 1 92.4 2 2 9 2 7 4 94.4 4 4 1 1 7 0 99.4 IT 1 3 9 9 95.2 2 4 1 2 7 5 99.5 2 0 S 4 0 4 97.1 2 2 4 4 0 1 94.4 2 4 2 4 1 0 99-4 it 42.40 h & 212.20 4.00 445.00 4.10 S eal# T o tal N o t i f i c a t i o n l ( , a lc r a w Stw O arO 1904.00 L £ h i l S u n O a rt Itaw iarO IZliJt 1 .10 h i i 179 9 6 3 2 9 6 3 2 9 6 3 2 TA8LE 42 VISCOMETER SCALE DEFLECTION AND SHEARING DATA 30 P e rc e n t c l s - D e c a l 1n/W ater T - 20°C Shearing S tress Shearing Rate 2 dyne/cm 1/sec D eflection Measured I n f i n i t e Measured I n f i n i t e Sample 1 9.5 6.80 0.193 418.7 11.87 15.0 10.74 0.285 628.0 16.69 30.0 21.48 0.571 1256.0 33.38 45.5 32.57 0.854 1884.0 49.39 Sample 2 11-5 8.23 0.184 418.7 9.37 17.0 12.17 0.277 628.0 14.31 34.5 24.70 0.553 1256.0 28.12 52.0 37.23 0.829 1884.0 41.93 Sample 3 13.5 9.66 0.179 418,7 7.44 20.4 14.60 0.268 628.0 11.52 40.0 28.64 0.537 1256.0 23.57 59.6 43.67 0.807 1884.0 35.64 TM U *3 mmmu tin •isTRiarriM y t r * N M t c la - ia c a l la/W atar O rI f lo a t R afai f I c a t la a Cat it Seal* T o tal I f l c a t l a * « J c n n *1 t a » l * 1 t a * l a 2 Saap la 3 M O r u ^ a r F r*- OMMCV ( M t r tlv a Cauat S F Iaar Cawatar Ouafcar F ta - o a aa er C im la - tlv a Count t F la a r C o aatar N a ^ a r F ra - f a a a c r C f a l a - tlv a Cauat •F ta a 1 u 12 o.o 1 11 11 0 .0 1 5 5 0 .0 t It to io. i 1 19 )0 S.7 1 1 0 IS 3-0 3 20 72 29. S 3 27 S7 IS.5 3 21 37 9.1 0 19 tot 0 0 .1 0 1 ) •0 1 9 .0 0 37 70 « . ) s 1 10) oo.o 5 27 107 01.2 5 00 121 0 0 .5 0 S lot to.o 0 2) 1)0 5S.0 0 20 101 73-5 7 9 117 0 9 .) 7 1 0 100 0 7 .0 • 1 1)1 •3 -2 9 7 lit 7S.0 1 11 ISI 75.1 M 1 ■ S 3 9 1 .0 IS 7 1)1 79-5 9 5 103 1 1 .5 1) 3 150 91-2 IS 7 1)1 to.o II 2 105 to.o IS 0 160 9 0 .0 19 9 107 t o . ) 13 1 17) • 5 .0 17 3 It) 9 0 .S 1 1 1 109 9 0 .2 IS 0 177 ts.o 21 2 its 9 1 1 IS 5 IS O 9S .5 19 7 tto 91.1 21 1 100 9 9 .5 31 1 IS O 9 t .9 1) )0 31 7 1 1 191 19) 190 9 0 .0 9 t-S 9 9 .5 01.00 U i Itarfirt 351-90 1 0 .00 LSI SiL« 7.10 Itw4tr4 3200.00 1.10 1.20 181 182 kO - £ u -V « c ■ e - •n •) L . « -> V I u m v f « /» a A G tO.O microns 2.3 microns 1.2 microns 20 - 500 1000 1500 2000 Rate o f Shear, i , sec -1 m' Figure 52. Shear Diagram fo r 30 percent c ls-D ee alIn /W ate r, T»20°C 5 "s. V c s 1 V ) D ■ 18.0 microns 9 D ■ 3.0 ml crons 9 Sample 3 "non-Newtonian 2 0. 2000 1500 1000 500 0 Rate o f Shear, s . sec H i F igure 53. Shear Diagram fo r kO percent cls-D ec a lI n /Ma t e r , T»20*C | TABLE 44 183 VISCOMETER SCALE DEFLECTION AND SHEARING DATA 40 P e rc e n t c ls-D e c a tin /W a te r T - 20°C Shearing S tress Shearing Rate 2 Speed dyne/cm 1/sec Factor D eflection Measured Infinite Measured Infinite fl2 9 14.0 10.02 Sample 1 0.199 itl8.7 8.30 0.630 6 20. 4 lit.60 0.299 628.0 12.88 0.596 3 <10.0 28.60 0.601 1256.0 26.37 0.572 2 60.0 1|2.96 0.902 1884.0 39.55 0.572 9 19.0 13.60 Sample 2 0.190 418.7 6.78 0.922 6 28. 4 20.33 0.286 628.0 10.17 0.920 3 55.0 39.38 0.573 1256.0 20.35 0.894 2 80.5 57.63 0.863 1884.0 28.21 0.874 9 31.0 22.21 Sample 3* 418.7 6 43.4 31.20 628.0 - - 3 76.0 5^.51 1256.0 - - 2 103.0 71 * .98 1884.0 — - Non-Newtonian Sample T A D L E A S M E A S U R E D SIZE DISTRIDUTION A O Percent ds-D ecal In/Water O riginal M agnification InIargonaut Seal a Total M agnification 0 , Microns 9 Soapla I Cuauta- Saapla 2 Cuaule- Countar Nuafcar Fre quency tlv a Count tFlnar Counter N«a*ar Fre quency tlva Count tF ln ar 1 A A 0.0 1 15 15 0.0 2 29 33 3.2 2 18 33 9.7 3 8 Al 28. A 3 18 51 21.3 A 3 A A 32.8 A 23 7 A 32.9 7 5 59 38.0 5 9 83 87.8 8 A S3 39.2 6 2A 107 53.5 9 5 5* A2.A 7 17 12 A 69.0 10 12 70 SA.A 8 7 131 80.0 II 7 77 58.0 9 8 137 BA.5 IA 13 90 81.5 10 3 IA0 88.A 18 17 107 ?2.0 13 8 lAi 90.3 2A 13 120 85-5 17 2 1A 9 9A.2 31 3 123 98.0 21 3 151 95.5 38 2 125 98.5 28 2 153 97.5 A 2 2 155 98.7 Se^le 3* Cunula- Cowntor Fro- tlv a quancy Coont tF lnar A2.A hi Standard 390 1 H.O L I Mon-Mawtonlan 184 185 TABLE 46 VISCOMETER SCALE DEFLECTION AND SHEARING DATA 50 P e rc e n t cls-D ecal In/Water* T - 20“C Shearing S tress Shearing Rate 2 Speed dyne/cm 1/sec Factor D eflection Measured Infinite Measured Infinite Sample I 9 17.5 12.57 418.7 6 26.4 18.94 628.0 3 51.0 36.60 1256.0 2 75.0 54.82 1884.0 Sample 2 9 24.5 17.59 418.7 6 36.9 26.42 628.0 3 71.0 50.95 1256.0 2 100.0 71.76 1884.0 Sample 3 9 31.5 22.60 418.7 6 48.5 34.90 628.0 3 91.0 65.24 1256.0 2 113.0 81.00 1884.0 Non-Newtontan Sample C M g V C > - T > M l « U u ts> u « « < / > 186 a Sample 1 "co a rse"j A Sample 2 "mediumf non O Sample 3 " f i n e 1 2000 1500 1000 500 0 Rate o f Shear, s . sec m -1 Figure 5^* Shear Diagram fo r 50 percen t cIs-OecaI In/W ater, T«20*C i c V o E > » in 8 « i £ System: d s -D e c a lIn /W a te r T - 20*C 3. Data of: Samples 1 Samples 2 Samples 3 2. 1. Volume F ra c tio n , f Figure 55. E ffe c t of C oncentration (V ariable 0^) on Mixture V isco sity 9 6 3 2 9 6 3 2 9 6 3 2 TABLE 47 VISCOMETER SCALE DEFLECTION AND SHEARING DATA 10 P e rc e n t n-Hexadecane/W ater T - 20°C Shearing S tress Shearing Rate 2 dyne/cm 1/sec D eflection Measured I n f i n i t e Measured I n f in ite Sample I 6.5 4.65 0.178 418.7 16.04 10.0 7-16 0.264 628.0 23.13 19.5 13.96 0.535 1256.0 48.11 29.4 21.05 0.799 1884.0 71.59 Sample 2 7.0 5.01 0.171 418.7 14.31 10.5 7.52 0.257 628.0 21.47 20.8 14.89 0.516 1256.0 43.57 31.2 22.34 0.775 1884.0 65.35 Sample 3 7.4 5.30 0.167 418.7 13.18 11.0 7.87 0.251 628.0 20.04 22.0 15.75 0.502 1256.0 40.07 32.5 23.27 0.759 1864.0 61.48 T U L E tO M C A S I M E B S I Z E IIS T IIM T IO N 1 0 FiK M l a1 H m iE k—/ M lttr ila 2 Sa*la ) •out C Baa t a r l l ^ a r E l f C aaala- tlv a C oait tF ln a r S f l i f r u - quaacy C u a ila - t l va Cauat t f t a a r Cow t a r NuW ar 7 m - auancy C m uI*- 1 1 va Cauat t f l a a r 2 t t 0 .0 1 21 22 0 .0 3 0 • 0 .0 J II IS 1.0 2 *3 OS 13-0 t II 19 3-5 t 2 t t l t . 7 3 90 101 )0 -7 5 20 t5 12.7 5 17 50 10.2 t 20 121 59. t 0 I t t l 30.0 t 10 7* 25.0 5 IS 1)0 7 1.0 7 2 ) Ot t o .7 ? 20 9 t » . o t 1) It9 79.0 0 10 100 50.0 • I t 111 t z .o • 9 I5« • 7 .5 9 9 109 t o .o 10 29 I t l t o o II S I t ) 9 3 .0 10 t 113 72.0 11 II 151 t l . t I t 2 ■ O S 9 5 .0 12 I t 129 75.5 i t 10 I t l 0 7 .5 10 1 I t t 9 7 .0 15 9 D * K .O i t I t 17t 72-0 I t 2 I t* 97.5 19 t U2 9 2 .0 10 10 lOt 70.0 35 1 109 9*.5 12 3 i t s 9 t . t 2 ) 1) 199 •2 .5 a 1 170 99.2 20 I t t 90.0 20 • 207 0 0 .t t l 2 I t s 97.5 32 7 l i t 92.1 to 2 150 91.0 *2 II 225 97.1 O rlflaal • I f I c a t I o n N lftlflcU lW 0 f, at c ro n * 112.20 l . a o S ta n O a r O * •2 7 JO I.to L li M S.0 0 1.00 M a O a c a O lotoo.oo 0.50 1.52 E u > T » V t - *- > « /» L. « £ t/> 6 .5 microns 1.6 microns 0.5 microns 20 10 0 2000 1500 1000 500 0 189 Rate o f Shear, s , sec m Figure 56. Shear Disgram fo r 10 percen t n-Hexadecane/Water, T*20°C § 's. « C -S' « t-t M D » 8,0 microns 30 0 .8 microns 20 10 0 1000 1500 500 Rate of Shear, s , sec m -l Figure 57. Shear Diagram fo r 20 p ercen t n-Hexadecane/Water, T - 20*C 9 6 3 2 9 6 3 2 9 6 3 2 TABLE *9 VISCOMETER SCALE DEFLECTION AND SHEARING DATA 20 P e rc e n t n-Headecane/W ater T - 20°C Shearing S tress Shearing Rate 2 dyne/cm 1/sec D eflection Measured I n f in ite Measured I n f i n i t e Sample 1 7.0 5.01 0.208 418.7 17.38 11.0 7.87 0.305 628.0 24.32 21.5 15.39 0.617 1256.0 50.33 33.2 23.77 0.912 1884.0 72.31 Sample 2 9.0 6.44 0.187 418.7 12.14 13*5 9.66 0.280 628.0 18.21 26.5 18.97 0.56*1 1256.0 37.37 39.5 28.28 0.849 1884.0 56.53 Sample 3 10.0 7.16 0.180 418.7 10.55 15.0 10.74 0.271 628.0 15.83 29.5 21.12 0.542 1256.0 32.36 44.0 31.50 0.818 1884.0 48.91 Tim e so M A S M ft S i n •ISTklW TISII SO f t r ta D I » >w i <n w i / » l » f I * * U 1 Vm » U 2 S « * l o 3 COMWtT OinOar f m - v>Mcir C i m I« - 11 v« C«*Mt t F l M f to w iH r B yuSir I f f C m w ll- I l M Count ■Floor Co m t o r N M to r fn o - fo o n c r tlv o Count SF Inor 1 I 1 0 .0 1 3 3 0 .0 S 1 I 0 .0 3 ( 7 0 .3 1 S I 41 1 .1 s 17 25 3 .4 S » 23 1 .1 3 74 137 2 7.1 4 20 SS 11.1 5 i s S3 • 3 S SO 17/ 4 1 .0 7 IS 53 1 0 .0 * 10 • l 14. S s 14 133 71.4 I 13 70 2 4 .1 7 7 « 2 5 .0 4 7 200 ■5.3 3 IS 33 IS .4 • IS I0S 2 7.1 • 3 203 • 3 .0 11 17 110 S I.S 10 IS I I I 32.0 10 S 201 3 0 .S 13 13 123 * 3 0 11 27 IS5 3 4.0 1) II 213 3 2 .S IS IS ISS S7.S 10 S3 IM SS.4 M 1 221 3 7 .5 I I 35 173 4S .0 13 13 217 s i . o 13 1 123 3S .5 23 13 *33 73-5 23 S3 i / o 4 4 .7 31 2 225 3 3 .2 21 12 210 M .O SO IS i s s n . i 33 1 211 3 3 .* 33 17 312 3 1 .0 31 5 223 3 7 ,0 31 7 313 3 4 .0 S I 2 225 3 3 -0 SI 4 32S 3 3 .2 trlflMl N tfa lfic ttio * b)W |HM t Ic o lo Total I f l O t tl M Y olcraaa 10 .0 0 111.20 1*27.10 s s s .o o 1.00 1.00 L2 0 10410.00 ii2k i .a o 9 6 3 2 9 6 3 2 9 6 3 2 TABLE 51 VISCOMETER SCALE DEFLECTION AND SHEARING DATA 30 P e rc e n t n-H exadecane/W ater T - 20°C Shearing S tr e s s Shearing Rate 2 dyne/cm I /s e c D e fle ctio n Measured I n f i n i t e Measured I n f i n i t e Sample I 8.4 6.01 0.224 418.7 15.58 12.5 8.95 0.337 62B.0 23.63 24.5 17.54 0.679 1256.0 40.69 37.5 26.85 1.010 1884.0 70.90 Sample 2 11.0 7.88 0.205 418.7 16.34 16.5 11.81 0.307 628.0 20.35 32.0 22.91 0.619 1256.0 33.99 48.5 34.72 0.927 1884.0 50.31 Sample 3 13.0 9-31 0.197 418.7 8.84 20.0 14.32 0.293 628.0 12.86 39.5 28.28 0.588 1256.0 26.12 59.2 42.38 0.882 1884.0 39.23 Z9*l ZT7 • ft •ootioi oot oolw •TifTr •rZTii M ilt W l 09* 1 9 • l i t )««) M |1 .MJ i l M IM O ] • l i t C •!*»* l t » ® 3 • • I t • • w t • ! • AM M * J i m JWIIJt *W | J l 1 U M J M |1 I » l* i W * J l | t * I *•!*•»! jr l* J » i » i» j 1 M M * J » |« 3 Miniil s t t ooz t 19 o * H ooz Z £9 S*£S • t l I S9 ®*9S H i 01 09 s 9 t S tl 9 19 0*91 H I 91 9 t S i t H i 9 SC o t t ooz z 9C 0*9£ Z£t O Z I t i d t « l S 9C • IS • t l I oc 0*99 ZSI oz Cl 0 £ | o i l 9 IZ S * H 9 t l t l i S I S Z fl s z I I • H 9£l 01 zz o s t f t l ( Cl S*C9 £01 oz SI S*C£ 991 £ l t l o t t Otl 9 II 0*9f £1 t l Cl s s i m i 91 I I S *£| 911 II t S*9Z 19 SI 01 o n i n S t 91 O i l S£l Cl £ O i l CS £1 t s e t 9 t t z 91 S *££ Z9I SI 9 O t l 9C 01 I S*9l £9 • t II e * ts £91 t z S i n 9Z c £ S I tz t l t 0*19 I I I 9C 9 • *£ t z t 9 S t £1 9 £ 0*9Z Z | 9 t C S '9 91 s S 0 1 (1 II 9 S*£ 09 C C 1 S*t t I 9 0*0 I Z S 0*0 SI SI 1 0*0 £ £ C •mj j u m ) J i m / M I ) ^ H 1 ■ | M 9 J I | K noiifliiusii m is avmsnH It 1 M V 1 194 (lO . 10.5 microns 1.8 microns 1.0 ml crons E tfi 500 1000 1500 2000 0 Rate of Shear, s , sec * ' m Figure 58. Shear Diagram fo r 30 percent n-Hexadecane/Water, T ■ 20“C « c >• ■ o v i l A « U ) I. m « • /> 14*0 microns 2 .0 microns 1 .5 m )cro n s 60 20 0 0 1000 500 1500 2000 Rate of Shear, s .sec"* m' Figure 59. Shear Diagram fo r 1(0 percent n-Hexadecane/Water, T ■ 20*C j TABLE 53 195 VISCOM ETER SCALE DEFLECTION A N D SHEARING D A TA 40 Percent n-Hexadecane/Water T - 20°C Speed Factor D eflection Shearing S tre s s dyne/cm Measured I n f in ite Shearing Rate 1/sec Measured I n f in ite tl2 Sample I 9 12.0 8.59 0.230 4)8.7 11.21 0.212 6 18.5 13.24 0.343 628.0 16.25 0.246 3 36.0 25.77 0.689 1256.0 33.62 0.212 2 54.0 38.66 1.035 Sample 2 1884.0 50.43 0.212 9 14.5 10.38 0.221 418.7 8.89 0.419 6 22.0 15.75 0.330 628.0 13.16 0.430 3 42.5 30.43 0.665 1256.0 27.44 0.397 2 64.5 46.18 0.995 Sample 3 1884.0 40.58 0.409 9 18.2 13.03 0.212 418.7 6.81 0.606 6 27.0 19-33 0.319 628.0 10.35 0.599 3 53.0 37.94 0.639 1256.0 21.15 0.585 2 80.0 57.27 0.958 1884.0 31.50 0.595 TMU S* HU SM CI Sin IIST R IW IiaR M N r t M l r l l i M i n w t O f I n I lM pl« 2 C w w l n - tM>l* 1 CUMlla* O r lf lM l IflC O tll E n la C om* t a r N a O tt F m - f a n c y tl* a CoaM tF ln a r C a tn ta r M * r F ra - tlv a f a n c y Count tF ln a r C ounter N ^ a r F ra - f a n c y t lv a Count tF ln a r 1 9 9 0 .0 1 1 I I 0 .0 5 2 1 0 .0 1 14 23 5 .2 2 3 54 10.3 4 10 11 0 .1 4 25 41 11.1 1 4 99 3 2.0 7 9 2 t 4 .0 S K 04 2 7 .5 4 1 124 5 4.5 9 14 35 1 .4 6 15 99 4 1.0 5 1 134 7 0.6 1 1 40 75 14.0 7 I 107 5 6.5 4 117 7 4.5 14 42 H 7 10.0 • U 119 4 1 .1 a 141 7 1 1 17 19 154 4 4 .1 3 12 H I 4 1 .0 10 I4J •o.5 20 10 144 4 2 .5 11 24 155 75.0 ii ISO 1 1 .6 23 24 190 4 4 .5 14 14 149 U . 6 ii 154 < 5 6 27 23 213 74.0 II S 174 9 4 .4 16 142 u.o 11 21 241 • 5 .3 22 1 175 9 9 .5 11 161 9 2 .5 17 5 144 9 4 .5 21 171 9 4 .0 44 4 2 5 0 9 0 -5 21 173 9 7 .4 29 175 9 1 .6 m 4 2 .4 0 211.20 44 5 .0 0 t 1.00 L3t L3® S e a ls T o tal N a g * !ftc a tla a • g i o l c r a o t ItwNrt 14 .00 LSI S t a M a r l 1476.40 1.00 U 1 MOucaO Miijo > JZ N O O' TABLE 55 197 VISCOMETER SCALE DEFLECTION AND SHEARING DATA 50 P e rc e n t n-Hexadecane/W ater* T - 20*C Shearing S tre ss Shearing Rate 2 Speed dyne/cm 1/sec Factor D eflection Measured I n f i n i t e Measured I n f i n i t e Sample I 9 39.7 28. 40 418.7 6 54.9 39.38 628.0 3 95.0 68.14 1256.0 2 114.0 81.72 1884.0 Sample 2 9 38.0 27.20 418.7 6 55.9 40.05 628.0 3 102.0 73.00 1256.0 2 116.0 83.10 1884.0 Sample 3 9 32.0 22.97 418.7 6 46.9 33.60 628.0 3 86.0 61.62 1256.0 2 112.0 80.24 1884.0 Non-Newton 1 an C M , s 0 c > * ■ o IA 1 / 1 4 > IA < 9 J Z Vt 198 80 60 □ Sample t "co arse" A Sample 2 "medium" oSample 3 " f in e " non- 20 1500 1000 2000 500 0 Rate of Shear, s , sec ID - 1 Figure 60. Shear Diagram fo r 50 percent n-Hexadecane/Water, T “ 20#C u (A o a v u s U 0 1 a > 3 System: n-Hexadecane/Water T - 20*C 3 .0 Data o f: Samples 1 Samples 2 Samples 3 2.0 1.0 0.2 Volume F ra c tio n , f |F igure 61. E ffect of Concentration (V ariable D^) on Mixture V is c o s ity j A P PE N D IX B FO RTR A N PROGRAM FO R D ETERM IN IN G IN T E R A C T IO N , V E L O C ITY D ISTRIBU TIO N , AND STR ESS P R O F IL E S 199 200 tW H k l^ V tW O C t- * ! IMM.I K H Z '* - * I U M « t> l . t .• • nuft1 M M t.ft. II,M,Ml a • u :•* ,* * < i t(D • i t •U i i t . i > i. « £U L F ti.rtn ( M n lw M M m i l to t TM ntr. titt, «i t U ) , t < M > hiM ti • upti «ttii m i • tu * * l . titi UtMNto tl*H* ■ (taal cev imji ,imti*4.a tw v lM .I t'MI • O .t IVMt < tU W l Figure 62. Diagram of Computer Program fo r C alcu la tin g In te ra c tio n and T ran sien t R esponse 0001 0002 0001 000* 0 0 0 1 ooo* 0 0 0 7 0000 0000 0010 0011 0012 001) 0 0 1 * 0 0 1 9 0 0 1 * 201 D E F IN IT IO N S FLUIOS■COMPONENTS UP THE TMU-PHASf SYSTEM BEING STUDIED PH 1 2 a VOLUME F M A C TION OP SECOND (D IS PE R SE ! PHASE NSUMU-TOTAL NUMBER OP MEASUREMENTS PERPORMEO AT ONE CONCENTRATION FOR ALL THREE SAMPLES MSMQl0 * 1 AMPLE NUMBER THE COMPUTER JUST FIN ISH ED CALCULATING NSNPL■CURRENT SAMPLE NUMBER THE COMPUTER ABOUT TO CALCULATE MSP0L0*INDER FOR NSMPL NR'INOEX FOR FQUATI UN NUMBER FR2*FUNCI lilN OF K2 IN THE NENION-RAPMSUN METHOD FK 20*FIkS T DERIVATIVE OF FR2 l?*lTERAT1QM S REQUIRED FOR CONVERGENCE OF THE SOLUTION P S I*SP rlE R IC IT V UR SHAPE FAC TOR* BEING 1*0 FOR SPHERES DG*GEUMEIR1C MEAN DIAMETER A SO I PAOBABILITV-CN TMEAS*STEAI>T STATE HALL SHEAR STRESS FROM MEASURED QUANTITIES DMEAS*ST EAOV STATE SHEAR RATE FROM ROTOR SPEED AND GEOMETRY GAMMA-COEFFICIENT OP RATE OF SHEAR REFLECTING INTERACTION TTNF*STEADY STATE NALL STRESS H U H INTERACTION tIN F IN IT E GEOMETRT CALCULATED HHfN T IS VERY LARGE D!MF«STEAOV STATE SHEAR RATE,INTERACT ION AND IN F IN IT E GEOMETRY I IS VERY LARGE T 1*0IM tN S I UNLESS TIRE PARAMETER FOR PHASE 1 12*01 MENS10NLESS TINE PARAMETER FUR PHASE 2 TN-GEONETMIC MEAN TIME PARAMETER FUR THE TNO-PHASE SYSTEM R-01MENS IUNLESS OISIANCE .VARIES FRUM UNITY TO IN FIN IT Y V1 * 0 1HENS I UNLESS VELOCIIT FOR PHASE I - UNSTEADY STATE V 2>nI MENS I UNLESS VELOCITY FOR PKASF 2 - UNSTEAOV STATE VM*HEIGHTFO OIMFNSIONLESS VELOCITY FOR THE THQ-PHASE SYSTEM TAURAT1Q*RATIU UF TRANSIENT TO STEADY STATE HALL SHEAR STRESS FOR F IN IT E GiUMETRYtAPPRUACHES UNITY AS TIME BECOMES LARGE. RATIO ALSO VALID FOR THE IN F IN IT E GEOMETRY CASE TAUMINF*INSTANTANEUUS TM1-PHASE HALL STRESS .IN F IN IT E GEOMETRY TAU.M*INSTANTANEOUS HALL STRESS FOR NV MEASURING SVSIEM CALCULATED FROM TAUMINF s u n k * d im e n si u n l e s s f r e q u e n c y a v e r a g e o f f o u r s p e e o s > l a a g e o m eg a REAL M U 1,N U2«NU H,M UBAR,M U HAT,R,R1,R2,K2N EH »NUl,N U2,1 0 1 ,1 1 1 ,1 2 1 ,1 ) 1 1,1)1, 102.112,122.1)2 REAL** I TYPE111 DIMENSION R U U .F L U I D S I Y I ,R R I 2 1 I ,T 1 U 2 I CALL E R R S E T (2 0 * ,Q ,- 1 , 1 . 0 , 0 1 DATA I TYPE /'C O A R S E ','M E D IU M * ,'F IN E * / 19 R E A 0 I9 , l.IN O * D IM U l,M U 2 ,R H 0 l,R H 0 2 ,P H I2 ,N S U M U 1 F U R M A T I9FIO .O ,191 R E A 0 I9 .1 0 0 0 1 1 R R I1 1 . 1 * 1 , 2 I I . I T l I J l . J a 1 ,1 2 1 1 0 0 0 FORMAT I 4 F 1 0 .0 /B F 1 0 . 0 / 9 F 1 0 . O /B F 1 0 . 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KO t i l P l i I - U U $ I tO U A T IO N t tKCIFIC S P E C O * 9.000E 00 HUT V ISC O SITY * I.AS A C -02 IN T E R A C T IO N At* l.TO O C 00 IN T E R A C T IO N A 2*-I.AIAC 00 S H E A R ST R E SS- A .01A E « S H E A R h ate - A.II7E 02 P A R T IC L E Oil* l.OSOE-Ol S H A R E F A C T O R * l.O C O E 00 T M IN F * 2.21AE-01 D M 1 M F - 1.J5H 01 C A M M A - l.L A V fc 00 O M E C A * A .T A 2E 00 RIAI- l.SJAE-Ol M (SI—A .V A 2fc-0A RIAI* 1.1A9E 01 RITI- 1.02TE 01 RIAt * *.SkIE-01 RIVI —S.V A V fc-02 Rl1CI* l.TSA fc 02 Rllll* A .R A S E 01 C O U R T IO N 2 SPECIFIC SPE E D - A.0001 00 K IK T V ISC O SIT Y * t.AIAC-02 IN T ER A C T IM Rl—A .SSO E 00 IN T E R A C T IO N R2- A.12SC 00 S H E A R S T R E S S * A .O IA E 00 S M E A R R A T E - A .IA TE 02 P A R T IC L E 01A * l.OSOI-OS S H A P E F A C T O R - l.O O O E 00 TRIM F* 1.A4RE-01 O R IM F * 1.1SA E 01 C A H N f* l.O O O E O C O M E C A - A.f«2E 00 R(AI* 1.S1AE-01 R I SI * 1.2AAF-01 RIAI- l.RA N E 01 RITI- 1.02TE 01 RIRI —l.ATTE-01 A(VI - I.S20E-01 Rl101* 1.2SA C 02 Ri m * a.vase 01 E Q U A T IO N 1 SPECIFIC S P E E D * A .O O O E 00 M IS T V ISC O SIT Y * l.kiSE— 02 IN T E R A C T IO N Rl* 2.V 01C 00 IN T E R A C T IO N K 2 —I.A1CE 00 S H E A R ST R E SS- l.fME 00 S H E A R R A T E a A.2C3E 02 P A R T IC L E OIA* l.OSOE-Ol S H A P E F A C T O R * l.O O O E 00 TM IN F- l.SARt— 01 D R IM F * 2.1AM 01 C A N M A - 1.1A2E 00 O M E C A * 1.01T E 01 RIAI- l.ASOE-OJ RISI— S.SRV I-O A RIAI* l.O A V E 01 RITI- l*C2ri 01 RIRI * T.120C-C2 RIVI —A .AA 2E-02 RI101* 1.R R 1E 02 Rllll- l.O A A E 02 C O U R T IO N 2 SPECIFIC S P E E D * A .O O O E 00 N IRT V ISC O SITY - 1.A2SE-02 IN T E R A C T IO N Rl—T.T12E 00 IN T E R A C T IO N R 2* A .A A A E 00 S H E A R ST R E S S * O .V A V C C O S H E A R R A T E - A.200C 02 P A R T IC L E OIA- l.OSOC-Cl S H A P E F A C T O R - l.O O O E 00 tminf* 2.am -oi O M IM F - 2.O ISE 01 C A M M A - l.O O O E 00 O M E C A * 1.01TF 01 RIAI- 1.AS0E-Q1 RISI* 1.1A 2 E — 01 RIAI* I.SA V E 01 RITI- 1*02T E 01 RIAI — 1.1021-01 R|VI - 1.A11E-01 RltOI- 1.A A 1C 02 Rllll- I.OaS E 02 ! 205 SAMPLE 1 - CONTINUED equation i S P E C IF IC SPEED" l.OOOE 0 0 H U T V1SCUSITY- 1 .1 R A E -0 2 INTERACTION A I " A . 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O B 395 0 .1 6 7 9 0 0 .3 3 5 8 1 0.67161 1 .6 7 9 0 4 2 .5 1 8 5 6 0 .0 0 9 1 6 0 .0 1 8 3 3 0 .0 4 5 8 1 0.0 9 1 6 3 0 .1 8 3 2 5 0 .3 6 6 5 0 0 .9 1 6 2 5 1 .37438 3 .7 0 2 1 4 2 .6 4 8 5 0 2 .1 0 1 9 6 1.73786 1.49594 1.34748 1.27139 1 .2 6 3 4 8 LARGE VALUES OF TIME 2 .5 1 8 5 6 1. 3 .3 5 8 0 7 1. 1 6 .7 9 0 3 6 9 . 3 3 .5 8 0 7 3 18. 134.32297 7 3 . 37438 1.21600 83251 1.16491 16252 1.04230 32504 1.02696 30019 1.01548 TAU M TAURATIO 9 7 .9 3 6 9 2 7 5 . 3 5 4 6 0 5 5 .6 0 5 3 9 4 5 .9 7 3 4 6 39 .5 7 3 7 9 3 5 .6 4 6 4 6 33 .6 3 3 4 4 33.95341 3.64797 2.80682 2 .0 7 1 7 0 1.71243 1.47405 1 .3 2 776 1.25278 1.26470 32.16829 3 0 .8 1 6 7 9 2 7.57320 2 7 .1 6 7 7 7 2 6 .6 6 3 6 6 1.19821 1.14767 1.02705 1.01195 1.00062 211

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University of Southern California Dissertations and Theses

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Creator Kasem, Ahmed (author)

Core Title Investigations On The Flow Behavior Of Disperse Systems

Degree Doctor of Philosophy

Degree Program Chemical Engineering

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag engineering, chemical,OAI-PMH Harvest

Language English

Contributor Digitized by ProQuest (provenance)

Advisor Rebert, Charles J. (committee chair), Escudier, Marcel P. (committee member), Wilcox, William R. (committee member)

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c18-565860

Unique identifier UC11362209

Identifier 7211934.pdf (filename),usctheses-c18-565860 (legacy record id)

Legacy Identifier 7211934.pdf

Dmrecord 565860

Document Type Dissertation

Rights Kasem, Ahmed

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the au...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus, Los Angeles, California 90089, USA