Open Collections

UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Numerical solution for stratified laminar flow of two immiscible Newtonian liquids in a circular pipe Gemmell, Alan Robert 1961

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-UBC_1961_A7 G3 N8.pdf [ 4.41MB ]
Metadata
JSON: 831-1.0059113.json
JSON-LD: 831-1.0059113-ld.json
RDF/XML (Pretty): 831-1.0059113-rdf.xml
RDF/JSON: 831-1.0059113-rdf.json
Turtle: 831-1.0059113-turtle.txt
N-Triples: 831-1.0059113-rdf-ntriples.txt
Original Record: 831-1.0059113-source.json
Full Text
831-1.0059113-fulltext.txt
Citation
831-1.0059113.ris

Full Text

NUMERICAL SOLUTION FOR STRAT.IFIED LAMINAR FLOW OF TWO IMMISCIBLE NEWTONIAN LIQUIDS IN A CIRCULAR PIPE by ALAN ROBERT GEMMELL B.A.Sc, Un i v e r s i t y of Toronto, 1959 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n the Department of CHEMICAL ENGINEERING We. accept t h i s thesis as conforming to the required standard Members of the Department of Chemical Engineering THE UNIVERSITY OF BRITISH COLUMBIA January 1961 In presenting t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree th a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study. I f u r t h e r agree that permission f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood tha t copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be allowed without my w r i t t e n permission. Department of c w ^ / c w ^ £^G/y</^/zt^JG The U n i v e r s i t y of B r i t i s h Columbia, Vancouver 8, Canada. Date v i ABSTRACT Numerical s o l u t i o n s of the v e l o c i t y p r o f i l e s f o r laminar, s t r a t i f i e d f low of two i m m i s c i b l e , Nev/tonian l i q u i d s i n a c i r c u l a r pipe were determined f o r v i s c o s i t y r a t i o s of 1, 10, 100 and 1000 at v a r i o u s i n t e r f a c e p o s i t i o n s . These r e s u l t s were used to c a l c u l a t e the t h e o r e t i c a l v o l u m e t r i c flow r a t e enhancement f a c t o r s , power r e d u c t i o n f a c t o r s and hold-up r a t i o s , which f o r laminar flow depend only upon the v i s c o s i t y r a t i o and the i n t e r f a c e p o s i t i o n . The maximum v o l u m e t r i c f l o w r a t e enhancen.erit f a c t o r s and maximum power r e d u c t i o n f a c t o r s , and the corresponding input volume r a t i o s , were determined. Dimensionless q u a n t i t i e s were used, making the r e s u l t s a p p l i c a b l e to any pipe diameter, any l i q u i d v i s c o s i t i e s . a n d any pressure g r a d i e n t , p r o v i d i n g laminar flow of both phases p r e v a i l s . The t h e o r e t i c a l r e s u l t s were compared to the experimental r e s u l t s of R u s s e l l , Hodgson and Govier f o r h o r i z o n t a l cocurrent f l o w of a m i n e r a l o i l and water i n a c i r c u l a r pine. As expected, the two sets of r e s u l t s d i f f e r e d c o n s i d e r a b l y i n the r e g i o n of t u r b u l e n t water f l o w . As turbulence decreased however, the d i f f e r e n c e decreased, u n t i l i n the laminar region very good agreement between the t h e o r e t i c a l and experimental r e s u l t s was obtained. v i i ACKMOWLEDGEMENTS I wish to thank Dr. Norman E p s t e i n , under whose d i r e c t i o n t h i s i n v e s t i g a t i o n was conducted, f o r h i s e x c e l l e n t guidance and continued a s s i s t a n c e throughout the study. I am a l s o indebted to the Standard O i l Company of B r i t i s h Columbia L i m i t e d f o r f i n a n c i a l a s s i s t a n c e r e c e i v e d i n the form of t h e i r graduate F e l l o w s h i p , and the N a t i o n a l Research C o u n c i l of Canada f o r a d d i t i o n a l support. i TABLE OP CONTENTS Page ABSTRACT v i ACKNOWLEDGEMENTS v i i NOMENCLATURE v i i i INTRODUCTION 1 COMPUTATIONS 9 A. V e l o c i t y P r o f i l e s 9 a. Theory 9 b 0 Sample C a l c u l a t i o n s 16 c. R e s u l t s 18 B . Vol u m e t r i c Flow Rates 26 a. Theor.y 26 b. Sample C a l c u l a t i o n s 28 c. C a l i b r a t i o n of Numerical Method ,32 d. R e s u l t s 56 C. Power Requirements 4.3 a. Theory 43 b. Sample C a l c u l a t i o n s 45 c. R e s u l t s 4.7 D. Hold-up R a t i o s 52 a. Theory _ 52 b. Sample C a l c u l a t i o n s 53 c. R e s u l t s 53 DISCUSSION OP RESULTS 58 i i Page COMPARISON 0? THEORETICAL AND EXPERIMENT AL RESULTS 65 A. Experimental Data 65 B o Hold-up 67 a. Computational Procedure 67 b. Results 67 c. Discussion of Results 67 C. Pressure Drop 71 a. Computational Procedure 71 b. Sample Calculations and Results 73 c. Discussion of Results 76 D. Enhancement of Volumetric Flow Rate 81 a. Computational Procedure 81 b. Sample Calculations and Results 82 c. Discussion of Results 86 CONCLUSIONS 87 REFERENCES 90 APPENDIX 91 LIST OF FIGURES Figure lo Schematic Diagram of Flow Model 9 2. Relaxation Pattern 12a 3. Relaxation Pattern at the Interface 13 4o Boundary Conditions 15 5o Sample Relaxation Calculations 17 6. Sample of Final Point Velocities for Grid Size Ay' = % 20 I l l Page 7. Sample of Pi n a l Point V e l o c i t i e s f o r Grid Size Ay ' = YA 21 8. Sample of F i n a l Point V e l o c i t i e s f o r Grid Size Ay« = § , 22 9. Horizontal V e l o c i t y P r o f i l e s f o r Interface at j ± = 10 23 10. V e r t i c a l V e l o c i t y P r o f i l e s f o r Interface at 7 ± - § , yU 1 - 10 24 11. Central V e r t i c a l V e l o c i t y P r o f i l e s foryd = 1 . 0 , 10, 100, and 1000, with Interface at j± = | 25 12. Comparison of A n a l y t i c a l and Relaxation Results f o r P a r a l l e l Plate Flow 3 4 13. V a r i a t i o n of Volumetric Flow Rate Factor with Interface P o s i t i o n 40 14. V a r i a t i o n of Volumetric Flow Rate Factor with Flow Area Fraction 1 41 15. Maximum Volumetric Flow Rate Factors and Corresponding Input Volume Fractions f o r Various 42 V i s c o s i t y Ratios 16. V a r i a t i o n of Power Reduction Factor with Interface P o s i t i o n 4 9 17. V a r i a t i o n of Power Reduction Factor with Flow Area Frac t i o n 50 18. Maximum Power Reduction Factors and Corresponding Input Volume Fractions f o r Various V i s c o s i t y Ratios 51 I V Page 19. Hold-up Ratios 55 20. V a r i a t i o n of Volumetric Flow Rate Factor with Input Volume F r a c t i o n 56 21. Vari a t i o n of Power Reduction Factor v/ith Input Volume Fraction 57 22. Comparison of Volumetric Flow Rate Factors f o r Concentric Flow, P a r a l l e l Plate Flow, and S t r a t i f i e d Flow i n a C i r c u l a r Pipe 61 23. Hold-up Ratio Cross Plot 68 24. T h e o r e t i c a l and Experimental Hold-up Ratios f o r JULX =20.1 69 25. V a r i a t i o n of Volumetric Flow Rate Factor with V i s c o s i t y Ratio 74 26. V a r i a t i o n of Volumetric Flow Rate Factor with Flow Area Fraction f o r yLA.' = 20.1 75 2?» T h e o r e t i c a l and Experimental Pressure Gradients f o r V = 0.116 - 0.287 f . s . 77 w ' 28. T h e o r e t i c a l and Fxperimental Pressure Gradients for V = 0.327 - 0.718 f . s . 78 29. T h e o r e t i c a l and Experimental Pressure Gradients for V = 1.08 - 3.55 f . s . 79 iff ' ' 30. T h e o r e t i c a l and Experimental Volumetric Flow . Rate Factors f o r ^ = 20.1 84,85 V LIST OP TABLES Table Page I. Predicted and Observed Pressure Gradient Reduction Factors for Oil-Water Flow 4 II. Volumetric Flow R a t e s , ^ = 1.0 37 III. Volumetric Flow Rates,JUL* = 10 37 IV. Volumetric Flow Rates, JJ} = 100 38 V. Volumetric Flow R a t e s = 1000 38 VI. Maximum Volumetric Flow Rate Factors 39 VII. Power Reduction Factors 48 VIII. Maximum Power Reduction Factors 48 IX. Calculated Hold-up Data 54 X. Interface Positions and Corresponding Flow Area Fractions 93 v i i i NOMENCLATURE a - constant i n polynomial equation b - constant i n polynomial equation c - constant i n polynomial equation d - constant i n polynomial equation D - diameter of pipe e - constant i n polynomial equation f - e x t e r n a l f o r c e S~ ~ g r a v i t a t i o n a l constant h - d i s t a n c e between e q u a l l y spaced p o i n t s i n Douglass-Avakian method k - c o e f f i c i e n t of h L - l e n g t h of pipe p - d i s t a n c e from i n t e r i o r nodal p o i n t to the boundary P - pressure A P - f r i c t i o n a l pressure drop q - d i s t a n c e from e x t e r n a l nodal p o i n t to the boundary Q - v o l u m e t r i c flow r a t e r - r a d i a l d i s t a n c e from centre of the pipe R - J r a d i u s of-'pipe Re - Reynolds number s - h a l f the d i s t a n c e between i n f i n i t e p a r a l l e l p l a t e s t - time u - v e l o c i t y i n the x d i r e c t i o n u' - dimensionless v e l o c i t y , i n x d i r e c t i o n U - average v e l o c i t y i n x d i r e c t i o n v - v e l o c i t y i n the y d i r e c t i o n V - s u p e r f i c i a l v e l o c i t y w - v e l o c i t y i n the z d i r e c t i o n W - power per u n i t l e n g t h x v - C a r t e s i a n c o o r d i n a t e ; h o r i z o n t a l d i s t a n c e along the l e n g t h of the pipe x' - ^, di i n e n s i o n l e s s X - v a r i a b l e i n polynomial equation y - C a r t e s i a n c o o r d i n a t e ; v e r t i c a l d i s t a n c e from bottom of the pipe y' - dimensionless Y - v a r i a b l e i n polynomial equation z - C a r t e s i a n c o o r d i n a t e ; h o r i z o n t a l d i s t a n c e across the pipe z' - •g, dimensionless Z - v a r i a b l e i n polynomial equations, Z = Y - 3h A. - in c r e m e n t a l q u a n t i t y S - sum - v i s c o s i t y yU.' - v i s c o s i t y r a t i o = '^/ju^ Tr _ 3.1416 f> - d e n s i t y © - c e n t r a l angle of s e c t o r , r a d i a n s S u b s c r i p t s A - more v i s c o u s l i q u i d B - l e s s v i s c o u s l i q u i d f u l l - i n d i c a t e s pipe f l o w i n g f u l l of l i q u i d A i - i n t e r f a c e m - h o r i z o n t a l c o o r d i n a t e of r e l a x a t i o n ^ r i d n - v e r t i c a l c o o r d i n a t e of r e l a x a t i o n g r i d 0,1,2,3*4 - p o i n t s i n r e l a x a t i o n p a t t e r n w - water 1 INTRODUCTION The p i p e l i n e transportation of heavy crude o i l i s d i f f i c u l t because of the high v i s c o s i t y of the o i l . Large and closely-spaced pumping stations are required to over-come the high f r i c t i o n a l pressure drop associated with very viscous o i l . I t has been found (1,2,3.4,6) that addition of water to the o i l decreases the resistance to flow and f o r c e r t a i n proportions of water, the same volumetric flow rate of o i l can be maintained at lower pressure gradients and lower power requirements. Clarke (1), i n a private communication to Russell and Charles (2), reported, r e s u l t s using a heavy viscous crude o i l flowing i n a 0.375-inch p i l o t pipeline at Reynolds numbers of 10 to 20. The pressure gradient was reduced by f a c t o r s of 6 to 12 when 7-13% water was introduced. The shape of the i n t e r f a c e was not known but " i t was suggested that the water wetted the insi d e of the pipe p r e f e r e n t i a l l y . " Clark and Shapiro (3) patented a method, described by Russell and Charles (2),whereby they injected water and demulsifying agents into the flowing crude o i l . Using o i l s of v i s c o s i t i e s estimated at 800 to 1000 cp., r e s u l t s were reported f o r laminar flow i n a 6-inch commercial p i p e l i n e 2 3 m i l e s l o n g . They observed pressure g r a d i e n t r e d u c t i o n f a c t o r s r a n g i n g from 7«8 to 10 .5 w i t h the i n j e c t i o n of 7-24 % water, and the maximum r e d u c t i o n f a c t o r at a water i n p u t of 8-10%. C h i l t o n and Handley (4) patented a process i n 1958 which was subsequently mentioned i n a l e t t e r by C h i l t o n ( 5 ) . They observed a pressure drop r e d u c t i o n by adding a f i l m of water a-t the w a l l of a pipe c a r r y i n g extremely h i g h l y v i s c o u s crude o i l . Over a 5 0-foot l e n g t h of approximately one-inch p i p e l i n e there appeared no mixing of the o i l and water, and the water f i l m remained e s s e n t i a l l y i n t a c t over t h i s d i s t a n c e . R u s s e l l , Hodgson, and Govier' (6) s t u d i e d s t r a t i f i e d f l o w of a r e f i n e d m i n e r a l o i l and water i n a 28-foot t r a n s p a r e n t p i p e l i n e w i t h a 0.806-inch diameter. The o i l , which had a v i s c o s i t y of 18 cp., was observed to be f l o w i n g above the water i n the la m i n a r r e g i o n . At a water content of 10%, the pressure g r a d i e n t was reduced by a f a c t o r of 1.2 at Reynolds numbers of 10 to 400. The authors a l s o found t h a t hold-up r a t i o was a f u n c t i o n of the input volume r a t i o i n the laminar r e g i o n , and was a l s o a f u n c t i o n of v e l o c i t y of the l i q u i d s i n the t u r b u l e n t r e g i o n . The experimental r e s u l t s d i s c u s s e d above are 3 summarized in Table I as in Russell and Charles (2). While the maximum pressure gradient reduction factor for an o i l of viscosity 18 cp. was listed as 1.2, which corresponds to 10% by volume of water input, examination of the actual data of Russell, Hodgson and Govier (6,7) showed factors as high as 1.52, which occurred at approximately 40% water input. The existence of an interface between the o i l and the water i s substantiated by the findings of Tipman and Hodgson (8) and Pavlov (9) as discussed by Russell and Charles(2). The former investigators found that the viscosity of an o i l and water emulsion i s almost always greater than that of the pure o i l . Therefore Russell and Charles concluded that a pressure gradient reduction can only occur i f the water flows as a separate phase. Two flow models for s t r a t i f i e d laminar flow of two immiscible liquids were investigated theoretically by Russell and Charles (2). The f i r s t model studied was that of a more viscous l i q u i d , A, flowing above a less., viscous li q u i d , B, between i n f i n i t e parallel plates. Equations relating pressure drop to geometry, flow rates and viscosities were developed. This was done by applying force balances to the two liquids and assuming that the velocities of the two are equal at the interface, which results in expressions for the volumetric flow rates of the more TABLE I PREDICTED AND OBSERVED PRESSURE GRADIENT REDUCTION FACTORS FOR OIL-WATER FLOW (2) Maximum predicted Maximum gradient reduction observed Reference Oil Type Oil gravity, Oil viscosity, factor pressure gradient reduction factor . °API cp. Concentric flow Parallel plates Clarke(l) Crude 7.0 800-1000* 400-500 3-4 12 Clark and Shapiro (3) Crude 13.4 800-1000** 400-S00 3-4 10.5 Russell et a l (6) Refined 38 18 9 2.2 1.2 ) * o Estimated viscosity of McMurray oil-sand o i l at ?0 C , the temperature at which the observations were made. • * Estimated from a general knowledge of the viscosity of heavy crude o i l s at normal pipeline temperatures. 5 viscous l i q u i d , . Q t^ and the less viscous l i q u i d , Q^ . The minimum fractional pressure gradient was found "by differentiating the expression for Q^/("^;) with respect to the i n t e r f a c i a l position, and equating the result to zero. Optimum positions of the interface for the greatest-reduction in the pressure gradient were determined for viscosity ratios greater than one. As reported in Table I, the maximum pressure gradient reduction factor i s 4- for an o i l of viscosity 1000 cp. and 2.2 for an o i l of viscosity 18 cp., flowing with water between parallel plates. It was also shown that the minimum power requirement could be A P computed by differentiating the expression for ~£Y, ^ A + with respect to i n t e r f a c i a l position and equating the result to zero. The second model studied was that of concentric flow of two immiscible liquids in a circular pipe. The less viscous l i q u i d flowed next to the pipe wall as an annulus, with the more viscous liquid flowing inside of i t . Force balances were again employed to obtain expressions for the volumetric flow rates of the two liquids. By differentiating A. P the expression for Q,/( ) with respect to the in t e r f a c i a l A A L position and equating the result to zero, an in t e r f a c i a l position was determined for the maximum pressure gradient reduction. As before, optimum positions were found for viscosity ratios greater than one. Expressions were also obtained for maximum pressure gradient reduction factors, 6 by comparing the pressure drop f o r two-phase flow to the pressure drop i f the pipe were f l o w i n g f u l l of the more v i s c o i i s l i q u i d at i t s same vol u m e t r i c flow r a t e . The pressure g r a d i e n t r e d u c t i o n f a c t o r s f o r c o n c e n t r i c flow are very much gre a t e r than those obtained f o r p a r a l l e l p l a t e f l o w , as i s shown i n Table I . For an o i l of v i s c o s i t y 1 0 0 0 cp., the maximum pressure g r a d i e n t r e d u c t i o n f a c t o r i s 5 0 0 and f o r a v i s c o s i t y of 18 cp., i t i s 9 . Power r e d u c t i o n f a c t o r s were determined i n a manner s i m i l a r to t h a t of determining pressure g r a d i e n t r e d t i c t i o n f a c t o r s . The p o s i t i o n of the i n t e r f a c e f o r maximum power r e d u c t i o n f a c t o r was c l o s e r to the w a l l of the pipe than the i n t e r f a c e p o s i t i o n f o r maximum pressure g r a d i e n t r e d u c t i o n f a c t o r . The r e s u l t s , summarized i n Table I , show t h a t those values d e r i v e d from the two t h e o r e t i c a l models are q u i t e d i f f e r e n t from each other and a l s o quite d i f f e r e n t from the a v a i l a b l e f i e l d data. Maximum pressure gradient r e d u c t i o n f a c t o r s determined e x p e r i m e n t a l l y f o r the two crude o i l s f a l l between the values p r e d i c t e d by the two t h e o r e t i c a l models. However, the v a l u e s of the maximum f a c t o r measured f o r the r e f i n e d o i l , which was observed to be f l o w i n g as a s t r a t i f i e d l a y e r , f e l l below those p r e d i c t e d by the t h e o r e t i c a l models. R u s s e l l and Charles concluded t h a t f o r s t r a t i f i e d flow i n a c i r c u l a r pine, the maximum pressure gradient r e d u c t i o n f a c t o r f a l l s below t h a t 7 p r e d i c t e d f o r p a r a l l e l p l a t e f l o w . Since the measured va l u e s f o r the crude o i l s were 10.5 arid 12, which are above the 3-4 p r e d i c t e d f o r p a r a l l e l p l a t e f l o w , they concluded t h a t i n these cases the f l o w must be i n t e r m e d i a t e between c o n c e n t r i c and s t r a t i f i e d f l o w . I t i s thought t h a t such a c o n c l u s i o n cannot be drawn, because of the absence of the t h e o r e t i c a l s o l u t i o n f o r s t r a t i f i e d f low i n a c i r c u l a r pipe and because of the sparseness of the a v a i l a b l e experimental d a t a . The purpose of the present study was to provide the t h e o r e t i c a l s o l u t i o n f o r the case of s t r a t i f i e d laminar flow of two i m m i s c i b l e Newtonian l i q u i d s i n a c i r c u l a r p ipe. The r e s u l t s c f t h i s i n v e s t i g a t i o n could then be used to p r e d i c t pressure g r a d i e n t and power r e d u c t i o n f a c t o r s f o r t h i s type of f l o w as w e l l as hold-up and optimum inpu t r a t i o s . A l s o , the added r e s u l t s could be used to e i t h e r s u b s t a n t i a t e or d i s p u t e the above c o n c l u s i o n of R u s s e l l and C h a r l e s . The two l i q u i d s i n v o l v e d do not n e c e s s a r i l y have to be o i l and water but can be any two immiscible l i q u i d s , and t h e r e f o r e the model was solved f o r v i s c o s i t y r a t i o s , r a t h e r than absolute v i s c o s i t i e s , of 1.0, 10, 100 and 1000. The cases of v i s c o s i t y r a t i o s of 10,.100 and 1000 were solved f o r 8 d i f f e r e n t i n t e r f a c e p o s i t i o n s . Dimensionless flow equations were solved n u m e r i c a l l y , u s i n g r e l a x a t i o n methods, to o b t a i n v e l o c i t y p r o f i l e s f o r any s i z e of pipe. Volumetric f l o w r a t e s were e a s i l y determined from the v e l o c i 8 p r o f i l e s . From these d a t a , values of the pressure gradient r e d u c t i o n f a c t o r s and power r e d u c t i o n f a c t o r s were determined as w e l l as l o c a t i o n of the i n t e r f a c e f o r the maximum f a c t o r s . The r e l a t i o n s h i p between hold-up r a t i o , i n p u t volume r a t i o and v i s c o s i t y r a t i o were a l s o shown. The t h e o r e t i c a l r e s u l t s were compared to the experimental case of s t r a t i f i e d flow (6) f o r a v i s c o s i t y r a t i o of 20.1, w i t h regards to hold-up r a t i o s , f r i c t i o n a l pressure drop and pressure g r a d i e n t r e d u c t i o n f a c t o r s . COMPUTATIONS A. V e l o c i t y P r o f i l e s a. Theory The present study c o n s i d e r s the s t r a t i f i e d , laminar flow of two i m m i s c i b l e , i n c o m p r e s s i b l e , Newtonian l i q u i d s i n a c i r c u l a r p i p e . A schematic diagram of the fl o w s i t u a t i o n i s shown i n f i g u r e 1 w i t h the co o r d i n a t e a x i s marked. y z Figure 1. Schematic Diagram of Flow Model ( l i q u i d B more dense than l i q u i d A) The b a s i c assumption i s made that the v e l o c i t y a the pipe w a l l i s zero. I t i s f u r t h e r assumed t h a t at the i n t e r f a c e both l i q u i d s d i s p l a y the same v e l o c i t i e s and equal but opposite shear s t r e s s e s with' respect to the 10 i n t e r f a c e . These boundary c o n d i t i o n s can be a p p l i e d to the Navier-Stokes momentum equations, which d e s c r i b e the flow. The Navier-Stokes equation f o r an i n c o m p r e s s i b l e f l u i d f l o w i n g i n the x d i r e c t i o n i s expressed as Dt " u a x ay dz Tt = x" ^  ax T ^ 4 * ^ ^ ( 1 ) Because l a m i n a r , s t r a t i f i e d flow i s assumed in' a conduit of constant c r o s s - s e c t i o n a l area, there i s no f l o w i n the y and z d i r e c t i o n s and the Navier-Stokes equations v a n i s h f o r these two coordinate d i r e c t i o n s . The c o n t i n u i t y equation f o r s t e a d y - s t a t e f l o w of an i n c o m p r e s s i b l e f l u i d i s d5 + 5y + 6z ~ 0 ( 2 ) Because there i s flow o n l y i n the x d i r e c t i o n , v = 0 and w = 0 and t h e r e f o r e i£ - 0 A l s o s i n c e s t e a d y - s t a t e e x i s t s , ^ = 0 at u and t h e r e f o r e Du = 0 Dt (3) W (5) 11 From equation (3) i t follows that » 0 (6) <ix2 As there are no significant external forces in the x direction, f = 0 (7) The Navier-Stokes momentum equations thus reduce to the following; single equation for the case of steady-state, laminar flow of an incompressible liquid in the x direction: Equation (8) applies to each liqu i d , using i t s respective viscosity. At the interface, the condition of equal and opposite shear stresses i s expressed as Equations (8) and (9) restricted by the requirement of no s l i p at the wall and ;at the interface describe completely the flow conditions investigated. The absolute quantities of these equations are transformed to dimensionless quantities, so that the results are applicable to any pressure gradient, pipe diameter and 12 v i s c o s i t y r a t i o of the two phases, r a t h e r than to s p e c i f i c pressure g r a d i e n t s , pipe diameters and v i s c o s i t i e s . This i s achieved by l e t t i n g : u» = / (10) where D A . *L ( - *|) (11) The q u a n t i t y i s the average v e l o c i t y i n a pipe f l o w i n g f u l l of l i q u i d A at the same pressure gradient as i n the two-phase f l o w . The f o l l o w i n g dimensionless d i s t a n c e s are used: r-l, »• - f (is) S u b s t i t u t i o n of equations ( 1 0 ) , (11) and (12) i n t o equations (8) and (9) converts the Navier-Stokes equation to and the shear f o r c e e q u a t i o n to — ± — A = — 2 (14) The f i n i t e d i f f e r e n c e approximation of equation (13) as shown i n M i c k l e y , Sherwood and Reed (10), i s expressed as u ' , - 2u' + u' , U ' , - 2U' + U ' , M* m-l,n m,n m+l,n + m,n-l m,n m,n+l _ _ Q ^ A ( - J . 5 ) ( A y ' ) 2 ( A Z ' ) 2 12 a The relaxation pattern, as shown in figure 2, is obtained by letting z\y' = A z ' and substituting subscripts 1,2,3 and 4- for m,n-l, ra+l,n, m,n+l and m-l,n respectively. These four points are situated on straight lines at right angles to each other and at equal distances from the central point 0, which i s substituted for m^ n. Figure 2. Relaxation Pattern Equation (15) then simplifies to + u' 2 + u' 3 + u'^ - 4u» Q + 8~£( A y ' ) 2 = 0 (16) which i s the general numerical flow equation to be applied to each l i q u i d . This equation i s solved by relaxation methods-which are described b r i e f l y in Appendix I. 13 The general procedure for obtaining a numerical equation applicable at the interface i s found in Allen (11), and was followed in this study. The f i n i t e difference approximation to equation (14) i s ( UV U ' A 5) . < UVU ,B 3) (17) By examining the relaxation pattern at the interface, as in figure 3 below, i t i s seen that the velocities u'. and u '-n A l H3 are f i c t i t i o u s because u'^ i s in liquid B and u'^ i s in liquid A. Figure 3. Relaxation Patterns at the Interface Therefore u'. and u' p must be eliminated by substitution, A l !'3 14 Rewriting equation (16) as i t applies to each liq u i d , A and B , and le t t i n g -—— = zx , a dimensionless viscosity ratio, the following two equations result: u' + u' + u' + u» - 4u' + 8 ( A y ' ) 2 = 0 (18) ' 1 2 3 4 o U ' B I + U ' B 2 + U ' B 5 + U ' B 4 " 4 U * B + 8^c'(Ay') 2= 0 (19) Multiplying equation (18) "by /A.' and subtracting equation (17) from i t eliminates u 1. . Subtraction of equation (19) A l from the resulting equation, eliminates u* n . Since the velocities at the interface are assumed equal, u' A = u' B , o o u'. = U'T, and u'» = u'^ , and therefore the f i n a l A2 B2 4 4 equation at the interface i s Equations (18) and (19) in the main body of each l i q u i d , respectively, equation (20) at the interface, and no s l i p at the wall f u l l y describe the flow conditions. These equations were solved by relaxation methods to obtain point velocities throughout a relaxation grid for viscosity ratios of 10,100 and 1000 at 8 different interface positions. Points of the grid which were outside the curved boundary,on which a l l velocities are zero, were assigned negative values by extrapolating linearly as in figure 4. 15 Figure 4. Boundary C o n d i t i o n s In the case of a v i s c o s i t y r a t i o of one, the two l i q u i d s f l o w as one, t h a t i s , without any d i s c o n t i n u i t y i n the r a t e of shear. Therefore the same v e l o c i t y p r o f i l e e x i s t s f o r a l l i n t e r f a c e p o s i t i o n s . The g r i d p o i n t v e l o c i t i e s were c a l c u l a t e d by r e l a x a t i o n methods and from the F o i s e u i l l e e quation, u U Therefore u* The v e l o c i t y p r o f i l e obtained from the P o i s e u i l l e equation was used to c a l c u l a t e d e r i v a t i v e r e s u l t s f o r seventeen i n t e r f a c e p o s i t i o n s . ( R 2 - r 2 ) R 4/x 2 £ _ 2 ( R 2 - r 2 ) TJ " R 2 (22) (11) (22a) 16 b. Sample Calculations Sample calculations are shown for a viscosity ratio of 10 and an interface situated mid-way between the centre and bottom of the pipe. If y i s the vertical height of the interface from the bottom of the pipe, this position •p corresponds to y^ = T>. Each case was solved for three different grid sizes, progressing from a coarse grid to a fine grid, unless a reasonable estimate of the velocities could be made directly for the finer grid. I n i t i a l grid size = ^  1 ? 1 Therefore Ay» = £ and ( A y ' ) = j j -and the resulting flow equations are for liquid A: u' +u' +u' +u' -4u' +2.00 = 0 (18a) 1 2 3 4 o at interface: 0.182u' +u' +1.82u' +u'. -4u' +3.64 = 0 (20a) 1 2 3 4 o and for liquid B: u 1^ +u' +u' +u'T3 -4u' +20.0 = 0 ( 1 9 a ) E l B2 B 3 B4 Bo These equations were solved by relaxation methods, typical calculations of which are shown in figure 5« Only half of the pipe cross-section was considered because the velocity profile i s symmetrical about the ve r t i c a l axis through the centre of the pipe. This condition of symmetry was imposed when performing the relaxation about the verti c a l axis, as il l u s t r a t e d in figure 7. - 0 . 4 3 0 - 0 . 5 3 1.88 2 .00 1.30 \ 2 .50 1.60 - 1 . 3 0 - 0 . 9 6 - 1 . 9 0 - 1 . 4 6 • 0.10 - 0 . 0 6 - 0 5 0 + 0 . 1 4 - 0 . 0 2 + 0 0 2 • 0 0 3 + 0 0 1 • 0 . 0 2 2.94 2.95 2.19 2 . 9 0 2 . 2 0 3.50 2 .00 - 2 . 8 0 • 0 . 9 0 - 0 . 1 0 • 0 . 3 0 + 0 . 4 0 + 0 - 9 0 - 0 1 0 + 0 . 6 0 + 0 . 3 0 - 0 . 2 0 + 0 . 1 8 - 0 . 0 7 - 0 . 0 2 - 0 0 2 - 0 . 0 1 + 0 . 0 2 - 0 . 0 3 + 0 .01 + 0 0 1 3.51 2.53 3 5 0 2 .40 3 .00 1.80 + 1 . 6 0 + 2 . 3 7 +0 .31 - 0 . 2 7 / + 1.71 • 0 . 2 3 / - 0 . 2 9 + 0 6 9 / - 0 0 3 + 0 0 3 / + 0 0 6 + 0 0 4 / + 0 - 0 2 *0X>2' 0 - 0 . 8 4 - 0 . 8 0 - 0 . 6 0 LIQUID A 0.43 0 .53 fl' = 10 0 8 4 • 0 . 8 0 - 0 . 6 0 INTERFACE LIQUID B Figure 5. Sample Relaxation Calculations (small numbers denote residuals) (large numbers denote values of u'). 18 Intermediate g r i d s i z e = ^ Therefore A y ' = | and ( A y ' ) 2 = ^ and the r e s u l t i n g -How equations are f o r l i q u i d A: u'. +u'i. +u' . +u', -4u'. +0.50 = 0 (18b) A l A 2 A 3 A 4 A o at i n t e r f a c e : 0.182u'„ +u'. +1.82u\ +u' -4u' +0.91 = 0 (20b) *1 A 2 A 3 A 4 A o and f o r l i q u i d B: u' +u'n +u' +u' -4u' +5.00 = 0 (19b) B 1 B 2 B 5 B 4 B Q These equations are sol v e d i n a s i m i l a r manner to the equations f o r the c o a r s e r g r i d . R F i n a l g r i d s i z e = g 1 2 1 Therefore A y ' = g and ( A y 1 ) = and the r e s u l t i n g f l o w equations are f o r l i q u i d A: u' A +u'& +u'fl +u' -4u' A +0.12 = 0 (18c) A l A 2 A 3 4 A o at i n t e r f a c e : 0.182u' +u' +1.82u'A +u\ -4u'. +0.23 = 0 (20c) al A2 A3 A 4 A o and f o r l i q u i d B: u' +u' + U ' , , +U' t i -4u'.r, +1.25 = 0 (19c) B l B 2 B3 B 4 B o These equations are so l v e d as p r e v i o u s l y , c. R e s u l t s F i n a l p o i n t v e l o c i t i e s are shown f o r g r i d s i z e s R R R of ?y, £ and g i n f i g u r e s 6,7 and 8 r e s p e c t i v e l y , f o r the sample case of y i = and JUL X = 10. H o r i z o n t a l and v e r t i c a l 1 9 v e l o c i t y p r o f i l e s f o r t h i s case are shown i n f i g u r e s 9 aud-i o , r e s p e c t i v e l y . In f i g u r e 1 1 , v e l o c i t y p r o f i l e s through the c e n t r a l v e r t i c a l a x i s are compared f o r v i s c o s i t y r a t i o s of 1 0 , 1 0 0 R and 1 0 0 0 , w i t h the i n t e r f a c e again at y±=> Figure 6. Sample of F i n a l P o i n t V e l o c i t i e s f o r G r i d S i z e A y' = | ( s m a l l numbers denote r e s i d u a l s ) Figure 7. Sample of Final Point Velocities for Grid Size A y ' = y^' (small numbers denote residuals) - 0 . 0 5 -0 .13 " 0 . 3 2 0.53 0.49 0.39 - 0 . 0 2 " 0 . 3 3 -0.01 0 -0.01 •0.01 ^ 1.01 0.97 0 .86 0.68 0.43 0.11 - 0 3 3 -0.01 0 •0.01 0 0 - 0 . 0 8 1.44 1.40 1.29 1.10 0 8 3 0 .50 b.io - 0 . 3 0 -0 .01 0 - 0 . 0 2 - 0 . 0 2 + 0.02 +001 +0.02 L82 1.78 1.66 1.46 1.19 0 .85 0 .43 \ - 0 . 0 5 - 0 0 1 - 0 0 1 -0 .01 0 -O.OI - 0 . 0 2 +0.01 \ \ 2.15 2.11 1.98 1.77 1.49 1.14 0.71 \ 0 . 2 2 - 001 - 0 . 0 2 0 +0.01 0 - 0 . 02 -0 .01 - 0 . 0 1 2.43 2 .39 2 . 2 6 2 . 0 4 1.74 1.37 0 . 9 2 0.4 2 \ 0 - 0 . 0 1 - 0 . 0 2 -0.01 0 -0.01 +0.02 - C O 2 \ 2 .67 2 . 6 3 2 . 4 9 2 . 2 6 1.94 1.55 1.08 0 . 5 4 \ •0.01 - 0 . 0 2 -0.01 -0 .01 +0.01 -0 .01 -0 .01 +0.01 2.88 2 .83 2 . 6 8 2 . 4 4 2.10 1.68 1.18 0.61 - 0 . 0 2 -0 .01 0 - 0 . 0 2 o -0 .01 - 0 . 0 2 0 3.05 3 .00 2 . 8 4 2 .58 2 .22 1.76 1.21 0 . 6 0 I 0 -0.01 0 - 0 0 1 - 0 0 2 - 0 . 0 2 +001 - 0 . 0 2 / 3.20 3.15 2 .98 2 .69 2 . 3 0 1.7.9 1.19 0.51 / •0.01 - 0 . 0 2 - 0 . 0 2 +001 -0.01 +0.01 - 0 . 0 2 o / 3 .34 3.28 3.10 2 .79 2 . 3 6 1.80 l.tl 0 . 3 0 - 0 . 0 2 -0.01 - 0 0 2 -0.01 -0.01 -0.01 - 0 . 0 2 - 0 . 0 2 / 3.46 3 . 4 0 3.21 2 . 8 8 2 . 4 2 1.81 1.01 / L •001 0 -0 .01 • 0.01 0 -0 .02 0 / i 4.14 4 . 0 5 3 . 7 7 3 . 2 8 2 . 5 7 1.61 0 . 3 6 - 1 . 0 8 0 -0.01 - 0 0 2 - 0 . 0 2 - 0 . 02 +0.01 / ^ - 0 . 0 3 3.75 3 .63 3 . 2 7 2 . 6 3 1.70 / 0 . 4 6 - 1 . 3 8 - 0 01 -0 .01 - 0 . 0 2 -0.01 0 , - 0 . 0 4 2 . 3 4 2.19 1.78 1.01 ^ ^ 0 .11 -1 .38 +0.02 0 +0.01 - O . O I ^ ' 0 - 0 . 2 4 " ^ - 0 . 5 9 -1 .52 LIQUID A - 0 . 3 3 - 0 . 1 4 - 0 . 0 6 f t ' = | 0 - 0 . 0 7 - 0 . 1 7 - 0 . 4 5 INTERFACE LIQUID B Figure 8. Sample of F i n a l P o i n t V e l o c i t i e s f o r G r i d S i z e A y ' = i ( s m a l l numbers denote r e s i d u a l s ) n-ure 9 . Horizontal Velocity Profiles for Interface 9t Figure 10. Vertical Velocity Profiles for Interface at 26 B. V o l u m e t r i c Flow Rates a. Theory The v o l u m e t r i c f l o w r a t e i s equal to the product of average v e l o c i t y and c r o s s - s e c t i o n a l area. Volumetric f l o w r a t e s f o r the study are approximated by assuming th a t a nodal p o i n t i n the g r i d i s the average v e l o c i t y f o r a square area of dimensions A y ' by A z ( , w i t h the nodal point at i t s c e n t r e . At the boundary, r e c t a n g u l a r - l i k e and t r i a n g u l a r - l i k e areas s m a l l e r than ( A y ' ) ( z 1 ) are l e f t over by t h i s procedure. These areas are approximated by r e c t a n g l e s , and the average v e l o c i t y computed as the a r i t h m e t r i c average of the v e l o c i t i e s at the f o u r corners of the r e c t a n g l e . The v e l o c i t i e s at the corners are found by i n t e r p o l a t i o n between nodal v e l o c i t i e s and boundary v a l u e s . The product of the point v e l o c i t i e s and the corresponding areas, i n c l u d i n g the boundary approximations, i s expressed as 2 u ' A ( A y')( A z 1 ) f o r l i q u i d A and as 2 u' B( A y' ) ( A z ' ) f o r l i q u i d B. Transforming these expressions t o absolute v a l u e s the f o l l o w i n g r e l a t i o n holds: S u ' A ( A y ' ) ( A z » ) - A u •> - ^ (23) A U A R ^ A f u i i / r r where O ^ f ^ i i s the v o l u m e t r i c f l o w r a t e f o r the pipe flowing f u l l of l i q u i d A , under the same pressure g r a d i e n t as f o r the two-phase f l o w . The v o l u m e t r i c f l o w r a t e f a c t o r . 2 7 ^ A ^ A f u l l ' a n e x P r e s s i ° n £°r comparing two-phase to one-phase flow at equal pressure g r a d i e n t s . Factors g r e a t e r than u n i t y c o n s t i t u t e an advantage, because they s i g n i f y t h a t at the same pressure g r a d i e n t , more l i q u i d A can be t r a n s p o r t e d by two-phase flow than by single-phase f l o w . F a c t o r s s m a l l e r than u n i t y s i g n i f y an opposite e f f e c t . The equation f o r e v a l u a t i n g the v o l u m e t r i c flow r a t e f a c t o r i s QA_ ^ 2 u ' A ( ^  y')( A, z') ( 2 3 ) Q A f u l l Tt Prev i o u s authors ( 2 ) have compared pressure g r a d i e n t r e d u c t i o n f a c t o r s at constant v o l u m e t r i c f l o w r a t e s . The f o l l o w i n g steps show the r e l a t i o n s h i p between t h i s f a c t o r and the v o l u m e t r i c f l o w r a t e f a c t o r of the present study. P o i s e u i l l e ' s e q uation f o r one-phase lam i n a r flow can be w r i t t e n as aP ( ~ 3 x " ) f u l l = c l QA f u l l ( 2 4 ) S i m i l a r l y , f o r two-phase f l o w , since equation (8) i n d i c a t e s d i r e c t p r o p o r t i o n a l i t y between pressure g r a d i e n t and v e l o c i t y , ( -H) .= c 2 Q A ( 2 5 ) t h e X 1 5A_ = f l ( 2 6 ) Q A f u l l C 2 28 and i f Q A f u n = QA, t h e n ' ( ~ H > f u l l C l ™ (- ^ ) -V,A V c3x f u l l Therefore — at equal pressure g r a d i e n t s = — r « at equal flow r a t e s (28) Thus the v o l u m e t r i c flow r a t e enhancement f a c t o r and the pressure g r a d i e n t r e d u c t i o n f a c t o r are the same. The maximum v o l u m e t r i c f l o w r a t e f a c t o r i s c a l c u l a t e d u s i n g the Douglass-Avakian method, as d e s c r i b e d p i n Appendix I I I , a p p l i e d to the seven p o i n t s , g a p a r t , between yj. = 0 and yi = jpR. A f o u r t h degree polynomial i s thus obtained, r e l a t i n g QA/Q^fu]_x t 0 i n t e r f a c e p o s i t i o n . T h i s polynomial i s d i f f e r e n t i a t e d and the r e s u l t equated to zero, to give the value and l o c a t i o n of the maximum v o l u m e t r i c flow r a t e f a c t o r . b. Sample C a l c u l a t i o n s Sample c a l c u l a t i o n s f o r v o l u m e t r i c flow r a t e s are based on the case of ytvc' = 10 and an i n t e r f a c e p o s i t i o n of y^ = The f i n a l p o i n t v e l o c i t i e s f o r t h i s case are shown i n f i g u r e 8. For most i n t e r i o r p o i n t s , except those near the boundaries, ( A y ' ) ( A z " ) = ^ and 29 2 U ' A = 0.49 + 0.97 + 1.40 + 1.78 + 2.11 + 2.39 + 2. 63 + 2.8$ + 3.oo + 3.15 + 3.28 + 0.39 + 0.86 + 1. 29 + 1.66 + 1.98 2.26 + 2.49 + 2.68 + 2.84 + 2. 98 + 3.10 + 0.68 1.10 + 1.46 + 1.77 + 2.04 + 2. 26 + 2.44 + 2.58 + 2.69 + 2.79 + 0.43 + 0.83 + 1. 19 + 1.49 + 1.74 + 1.94 + 2.10 + 2.22 + 2.30 + 2. 36 + 0 . 5 0 + 0.85 + 1.14 + 1.37 + 1.55 + 1.68 + 1. 76 + 1.79 + 1.80 + 0.43 + 0 .71 + 0.92 + 1.08 + 1. 18 4 1.21 + 1.19 + 1.11 + 0.42 + 0.54 + 0.61 + 0. 60 + 0 .51 = At the v e r t i c a l a x i s and i n t e r f a c e ( ^ y ' ) ( A z ' ) = j^g and £ u ' A = 0 . 5 3 + 1.01 + 1.44 + 1.82 + 2.15 + 2.43 + 2.67 + 2 .88 + $.05 + 3 . 2 0 + 3-34 + 3.40 + 5.21 + 2.88 + 2.42 + 1.81 + 1.01 = 39.25 At the i n t e r s e c t i o n of the v e r t i c a l a x i s and the i n t e r f a c e ( A y ' ) ( A z ' ) = ^ and u' A = 3-46. For boundary areas approximated by ( A y ' ) ( A z ' ) = •£u' A = 0 .12 + 0 .27 + 0 . 2 2 + 0 .23 + 0 .23 '+ 0.22 + 0 .27 + 0 . 1 3 + 0 . 1 4 + 0 . 3 8 + 0 . 3 6 = 2.57 For boundary areas approximated by ( A y ' ) ( A z ' ) = -g^g 2 u ' A = 0.14 + 0.15 + 0.13 + 0.10 + 0.11 + 0.12 + 0.15 + 0.12 = 1.02 For boundary areas approximated by ( A y ' ) ( A z . t ) = 30 1 5 T 5 2u' A = 0.08 + 0.06 + 0.08 + 0.06 + 0.06 + 0 . 0 9 = 0 . 4 3 Therefore 2 V A ( A y ' ) ( A zl) - 9 or 1.655 ( ^ 2 ^ 2 , 5 2 ) o r 0 > 5 2 6 ? + (3.46,1.02) o r 0 > 0 1 ? 5 + 0j|3 o r 0,00084 = 2.00 This value i s only f o r one h a l f of the pipe; therefore f o r the whole pipe 2 u ' A ( A y ' ) ( A z ' ) = 4 .00 and 4.00 = 1.27 Q A f u l l ^ The volumetric flow rate summation f o r l i q u i d B i s performed i n exactly the same manner. For the whole pipe, ^ ^ ( A y 1 ) O z ' ) = 1.44. Sample c a l c u l a t i o n s f o r l o c a t i n g the maximum volumetric flow rate f a c t o r are i l l u s t r a t e d f o r a v i s c o s i t y r a t i o of 10. The data f o r t h i s case are presented i n Table QA . • III . L e t t i n g X = -gp and Y = i n t e r f a c e p o s i t i o n and ^ A f u l l following the Douglass - Avakian method, described i n Appendix I I I , the c a l c u l a t i o n table below can be set up: 31 Y X Z k kX k 5X k 4X =Y-0.375 0 1 .00 - 0 . 3 7 5 - 3 - 3 . 0 0 9 . 0 0 - 2 7 . 0 0 81 .00 0.125 1.12 . - 0 . 2 5 0 - 2 -2.24 4.48 -8 .96 17.92 0.250 1.20 ' - 0 . 1 2 5 -1 - 1 . 2 0 1.20 - 1 . 2 0 1.20 0.370 1.27 0 0 0 0 0 0 0 .500 1.27 0.125 1 1*27 1.27 1.27 1.27 0.625 1.18 0.250 2 2.36 4 . 72 9.44 18.88 0.750 1.06 0.375 3 3.18 9.54 28.62 85.86 0.37 30.21 2.17 206.13 2 3 - 4 The constants i n the polynomial X = ia+bZ+cZ +dZ +eZ are then c a l c u l a t e d as a - = 524(8.10) - 245(30.21) + 21(206.13) i on = 1.2/ ^ 397(0.37) 7(2.17) n 9 U b = l 5 l 2 ( o : i 2 5 ) " 2 l 6 ( 0 . i 2 5 ) = ° - 2 1 4 c - -840,(8.10) + 679(30.21) - 67(206.13) _ _ 2 0 6 c _ 3168 (0 . 125)2 -A -7(0.37) +2.17; _ N QQR d = 216(0.125)3 =-0.998 e = 72(8 .16) - 67(30.21) +7(206.13) = 2 < 5 9 3168(0125)^ The f o u r t h degree polynomial which f i t s the seven p o i n t s best i s then X = 1.27 + 0.214 Z - 2.06 Z 2 - 0.998 Z 5 + 2.59 Z 4 and d i f f e r e n t i a t i n g , dX = 0 + 0.214 - 4.12 Z - 2 .99 Z 2 + 10.4 7? At dZ Z = 0 . 0 5 0 , ff=o Therefore the maximum v o l u m e t r i c flow r a t e f a c t o r occurs at Y = ( 0 . 0 5 0 + 0.375)R = 0.425R and the maximum value of 52 '-Afull X = 1.27 + 0.011 - 0.005 = 1.28 c. C a l i b r a t i o n of Numerical Method The numerical s o l u t i o n of s t r a t i f i e d laminar flow of two l i q u i d s i n a c i r c u l a r pipe was examined f o r accuracy by two methods. F i r s t , i t was checked against the a n a l y t i c a l r e s u l t s f o r s i n g l e - l i q u i d flow given by P o i s e u i l l e ' s equation. Secondly, s i m i l a r numerical equations were derived f o r s t r a t i f i e d flow of two l i q u i d s between p a r a l l e l plates and solved by r e l a x a t i o n methods. These r e s u l t s were compared to the a n a l y t i c a l r e s u l t s reported by R u s s e l l and Charles (2). The diraensionless, numerical equation f o r single l i q u i d flow i s u ^ + u ' 2 + u' 5 + u\ - 4u' Q + 8 ( A y ' ) 2 = 0 (18) and f o r a g r i d size of A y = g, equation (18) becomes u', + u' 0 + u ' z + u'„ - 4u' + 0.12 = 0 1 2 ^ 4 o This equation was solved by r e l a x a t i o n methods to obtain dimensionless point v e l o c i t i e s on the g r i d . In the f i r s t approximate s o l u t i o n , the res i d u a l s were unbalanced, that i s , there was a predominance of e i t h e r negative or p o s i t i v e r e s i d u a l s . Comparing t h i s unbalanced numerical s o l u t i o n to the a n a l y t i c a l s o l u t i o n of P o i s e u i l l e , the average per cent deviation of the v e l o c i t i e s was 1.25%. The residuals 33 were then balanced so t h a t the sum of a l l the r e s i d u a l s was approximately zero, and the average per cent d e v i a t i o n of the v e l o c i t i e s i n t h i s case was 0.4-5%. The v o l u m e t r i c flow r a t e s were c a l c u l a t e d f o r the two cases of unbalanced and balanced r e s i d u a l s . The a n a l y t i c a l r e s u l t i s 2 u ' ( A y ' ) ( A z l ) whereas f o r the unbalanced numerical s o l u t i o n , 2u* ( A y 1 ) ( A z 1 ) = 3.16 and f o r the balanced numerical s o l u t i o n , 2 u' ( A y ' ) ( A z ') = 3.14. The d e v i a t i o n f o r the unbalanced s o l u t i o n was thus 0.64^- and f o r the balanced case was undetectable i n three s i g n i f i c a n t f i g u r e s . Throughout the i n v e s t i g a t i o n , the r e s i d u a l s were not completely balanced; the above r e s u l t s show t h a t good agreement can n e v e r t h e l e s s be expected. TO For a l a r g e r g r i d s i z e of A y = JJ-, 2 u ' ( A y ' ) ( A z ' ) = 3 . 2 6 , which i s a d e v i a t i o n of 3 .8% from the a n a l y t i c a l r e s u l t . Therefore a g r i d s i z e of A y = -g was chosen, because of the b e t t e r agreement w i t h the a n a l y t i c a l r e s u l t . When the i n t e r f a c e was a d i s t a n c e g above the bottom of the pi p e , there were no g r i d p o i n t s i n the region of flow of l i q u i d B w i t h a g r i d s i z e of A y = g. Therefore a g r i d s i z e of A y = was used. E r r o r s i n the c a l c u l a t i o n of 2 u ' p ( A y ' ) ( A z ' ) u s i n g the l a r g e r g r i d , e s p e c i a l l y at high v i s c o s i t y r a t i o s , were e l i m i n a t e d by u s i n g the s m a l l e r g r i d f o r which point v e l o c i t i e s could be obtained i n the main body of l i q u i d B. 34 A s o l u t i o n f o r two-phase flow with the in t e r f a c e a distance ^ above the bottom of the pipe and a g r i d size of •p A y = ^ was calculated using three-figure accuracy, then rec a l c u l a t e d using two-figure accuracy. As the diffe r e n c e i i i the volumetric flow rate factors was 3'6%> a l l c a l c u l a t i o n s were subsequently performed with three-figure accuracy. t The numerical s o l u t i o n was also checked against an analogous case of s t r a t i f i e d laminar flow of two l i q u i d s between i n f i n i t e p a r a l l e l plates. Numerical equations f o r t h i s model were derived i n a manner s i m i l a r to those derived i n the present i n v e s t i g a t i o n , and are for l i q u i d A: u' A + u' A - 2u' Q + 3 ( ^ y ' ) 2 = 0 (29) 1 3 at i n t e r f a c e : u ' ^ u ' ^ u ' ^ ^ A y ' ) 2 = 0 (30) and f o r l i q u i d B: u ' B + u ' B - 2u' B + 3 M.1 ( A y ' ) 2=0 (3D '1 ~J3 o These dimensionless equations were solved by r e l a x a t i o n methods f o r the case of JLA. 1 = 1000 and the i n t e r f a c e at y i = s, where the plates are a distance 2s apart, using a g r i d size of A y = g. The point v e l o c i t i e s were solved a n a l y t i c a l l y using the expressions given by Russell and Charles (2). The two r e s u l t s are shown i n figure 12, where i t i s seen that a l l corresponding point v e l o c i t i e s match up exactly f o r three-figure accuracy. A N A L Y T I C A L RESULTS R E L A X A T I O N RESULTS 0 0 . 5 4 1.03 1.48 1.88 2 . 2 3 2.53 2 .79 3X)0 _ 167 2 8 4 3 5 4 3 7 7 3 5 3 2 8 2 164 0 f- '=IOOO L I Q U I D A I N T E R F A C E L I Q U I D B 0 0 . 5 4 0 1.03 • 0.01 1.48 0 1.88 0 2 . 2 3 0 2.53 + 0.01 2 . 7 9 0 3 . 0 0 0 167 0 2 8 4 0 3 5 4 0 3 7 7 0 3 5 3 0 2 8 2 0 164 +1 0 Figure 12. Comparison of A n a l y t i c a l and Relaxation Results f o r P a r a l l e l Plate Flow 36 These two checks show that the numerical solution employed i n t h i s study agrees extremely wel l with those a n a l y t i c a l solutions a v a i l a b l e , and can therefore be considered very r e l i a b l e f o r the three s i g n i f i c a n t figures reported. d. Results Calculated values of the numerical flow rate factors f o r d i f f e r e n t interface positions are presented i n Tables I I , I I I , IV and V f o r v i s c o s i t y r a t i o s of 1 , 10,100 and 1000 r e s p e c t i v e l y . In figure 13, the volumetric flow rate f a c t o r i s plotted against the in t e r f a c e p o s i t i o n f o r the four v i s c o s i t y r a t i o s investigated. Flow area f r a c t i o n s were calculated from the interface positions as described i n Appendix I I , and figure 14 i s a plot of volumetric flow rate f a c t o r versus flow area f r a c t i o n of l i q u i d P. for a l l four v i s c o s i t y r a t i o s . The maximum volumetric flow rate f a c t o r s , recorded i n Table VI are plotted i n fig u r e 15 against the v i s c o s i t y r a t i o s . 37 Table I I Volumetric Flow Rates, yx1 = 1 . 0 y i 2 u ' A ( A y ' ) ( A z ' ) 2 U « ( A J - K A Z 1 ) ^ ^Afull 0 3 . 1 6 0 1 . 0 0 R / 8 3 . 1 4 0 . 0 2 0 . 9 9 4 R /4 3 . 0 7 0 . 0 9 0 . 9 7 2 3 R / 8 2 . 9 4 0 . 2 2 0 . 9 3 0 R / 2 2 . 7 5 0 . 4 1 0 . 8 7 0 5 R / 8 2 . 5 1 0 . 6 5 0 . 7 9 4 3 R / 4 2 . 2 2 0 . 9 4 0 . 7 0 3 7 R / 8 1. 9 1 1 . 2 5 0 . 6 0 4 R 1. 5 8 1 . 5 8 0 . 5 0 0 9R /8 1 . 2 5 1 . 9 1 0 . 3 9 6 5 R A 0. 9 4 2 . 2 2 0 . 2 9 7 11 R /8 0 . 6 5 2 . 5 1 0 . 2 0 6 3 R / 2 0. 4 1 2 . 7 5 0 . 1 3 0 13R /8 0 . 2 2 2 . 9 4 0 . 0 7 0 7 R A 0 . 0 9 3 . 0 7 ... ^ 0 . 0 2 6 1 5 R / 8 0 . 0 2 3 . 1 4 0 . 0 0 6 2 R 0 3 . 1 6 0 Table I I I Volumetric Flow Rates, / *' = 1 0 y, 2 u ' U y ' ) U z ' ) 2 u ' ( A y ' ) ( A z ' ) . a ^Afull 0 3 . 1 4 0 1 . 0 0 R /8 3 . 5 3 0 . 0 6 1 4 1 . 1 2 R / 4 ' 3 . 7 6 0 . 2 8 2 1 . 2 0 3 R / 8 3 . 9 9 0 . 7 2 4 1 . 2 7 R /2 4 . 0 0 1 . 4 4 .1.27 5R /8 3.72 2 . 4 5 .. 1 . 1 8 3 R / 4 3 . 3 3 5 . 8 0 1 . 0 6 R 2 . 4 3 7 . 7 8 . : 0 . 7 7 4 3 P / 2 0 . 5 9 0 2 0 . 1 . 0 . 1 8 8 2.R 0 0 38 0 R/8 R/4 3R/8 R/2 5R/8 3R/4 R 3R/2 2R Table IV Volumetric Flow Rates, juJ = liOO S u ' A ( A y ' ) ( A Z ' ) 2 u ' T , ( A y , ) ( A 2 ' ) 3.14 3.85 4.1? 4 . 3 4 4 . 3 1 3 .95 3 .50 2.55 0.615 0 0 0.127 0.801 2.92 7.24 14.6 25.6 61.2 188 Q •A f u l l 1.00 1.23 1.33 1.38 1.37 1.26 1.11 0.812 0.196 0 0 R/8 R/4 3R/8 R/2 5R/8 3 R A R 3R/2 2R Table V Volumetric Flow Rates, /A* = 1000 2 u ' A ( A y ' ) ( A z ' ) 2 u ' B ( A y ' ) ( A z ' . ) 3.14 3.93 4 . 2 7 4.42 4.39 4.02 3.56 2.60 0.619 0 0 0.559 5.47 24.0 64.1 135 242 596 1860 Q, J A Q A f u l l 1.00 1 . 2 5 1 . 3 6 1.41 1.40 1.28 1 . 1 3 0.828 0 . 1 9 6 0 3 9 Table VI Maximum Volumetric Flow Rate Factors Viscosity Ratio, /x' 1,0 10 100 1,000 Maximum 1.00 1.28 1 , 3 8 1 , 4 1 ^ A f u l l Interface Position, j± 0 0.425R 0.390R 0.385R Figure 13 • Variation of Volumetric Flow Rate Factor with Interface Position o 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 . 0 FLOW AREA FRACTION OF LIQUID B Figure 14. V a r i a t i o n of V o l u m e t r i c Flow Rate Factor w i t h Flow Area F r a c t i o n Figure 15 • Maximum Volumetric Flow Rate Factors and Corresponding Input Volume F r a c t i o n s f o r Various V i s c o s i t y R a t i o s 43 C. Power Requirements a. Theory The power requirement per u n i t l e n g t h of pipe i s the product of the pressure drop per u n i t l e n g t h and the t o t a l v o l u m e t r i c f l o w r a t e of both l i q u i d s . In the case of two-phase f l o w , i t i s expressed as Pozer = w = (. | | ) ( Q a + Q b ) (32) and f o r the pipe f l o w i n g f u l l of l i q u i d A, as W f u l l = (~ f l ^ f u l l ^ A f u l l ) (33) i f Q A - Q A f u l l W f u l l W (-^ < i x ; f u l l (- ±E) ^ a x ; f o r constant (34) and from equation (28), i . e , W f u l l W f u l l Q A f o r c o n s t a n t ( - r r ) o X Q A f u l l 1+ Q A Q • A f u l l 2 u ' A ( A y ' ) ( A z ' ) 1 + (35) (36) and t h e r e f o r e the power r e d u c t i o n f a c t o r , 11 ™ , i s , e a s i l y c a l c u l a t e d from the data of the previous s e c t i o n . 44 To obtain the maximum power reduction factor and thus the minimum power requirement for two-phase flow, the Douglass - Avakian method was applied over seven equidistant J? points, at increments of jg. The smaller increments were chosen because the maximum power reduction factors occur at interface positions closer to the bottom of the pipe, and, the plot of power reduction factor versus interface position,has a greater curvature, than in the case of volumetric flow rate factors. Since the previous calculations were for. increments •p p of g, the intermediate values for increments of -jg were obtained by interpolation of the previous data. Prom the Q A , polynomial relating •« and interface position, the y A f u i l volumetric flow rate factors for the intermediate.points were determined and by equation ( 2 3 ) , 2 u ' A ( A y ' ) ( A z 1 ) was calculated. Intermediate values of S u ' B ( A y ' ) ( A z 1 ) were 2 3 interpolated from a polynomial of the form, X = a+bY+cY +dY ,, where X = 2 u , p ( A y ' ) ( A z l ) and Y was the interface position. The constants of the polynomial were evaluated from four pairs of values calculated previously for increments of g. The Douglass - Avakian method was not used with the larger increments because the values of X for seven equidistant . values of Y varied greatly in magnitude, and the resulting relationship, though i t was the best fourth degree polynomial for the seven points, was not accurate enough in the region of the maximum power reduction factor. The Douglass -. Avakian method, followed by differentiation, was, however, a p p l i e d to the s m a l l e r increments to o b t a i n the value and l o c a t i o n of the maximum power r e d u c t i o n f a c t o r s . b. Sample C a l c u l a t i o n s Sample c a l c u l a t i o n s f o r power r e d u c t i o n f a c t o r are based on the case of ^u' = 10 and the i n t e r f a c e at y\ = ^. = 1.27 Q A f u l l S u ' A ( A y ' ) ( A z ' ) * 4.00 S u , B ( A y , ) ( A z l ) = 1.44 and W f u n = (1.27) V T T T T = 0.933 W ' , , 1.44 x + 4.00 A sample d e t e r m i n a t i o n of the maximum power r e d u c t i o n f a c t o r i s based on val u e s f o r a v i s c o s i t y r a t i o of 10. C a l c u l a t i o n s are shown f o r o b t a i n i n g v a l u e s at the in t e r m e d i a t e p o i n t of y = 1§ " ° * 1 8 7 5 R « Considering' f i r s t the l i q u i d A, and l e t t i n g X = ^ and Y = i n t e r f a c e p o s i t i o n such th a t Y = 4 A f u l l ° corresponds to Z = 0, the Douglass - Avakian method gave the f o l l o w i n g polynomial X = 1.27 + 0.214Z - 2.06Z 2 - 0.998Z 3 + 2.59Z 4 At an i n t e r f a c e p o s i t i o n of Y = ^ or Z = -0.1875 X = 1 .27+0.214(-0.1875)-2.06(-0.1875) 2-0 .998(-0.1875) 3 + 2 .59(-0 .1875)Z\ 46 that i s = 1.17 ^Afull and therefore 2 u ' A ( A y ' ) ( A z l ) = TT (1 .17) = 3.68 For liquid B, the following results were obtained: Y 2 u ' B ( A y ' ) ( A Z ' ) 0 0 R/8 0.0614 R/4 0.282 3R/8 0.724 If X = S u ' B ( A y • ) ( A . z 1 ) and Y = interface position, their 2 -5 relationship can be given by the polynomial, X = a+bY+cY +dY . Substitution of the above values and solution for the constants resulted in the polynomial X = 0.012Y + 3.19Y2 + 5.10Y5. At Y = X = 0.012(0.1875)+3.19(0.1875)2+5.10(0.1875)5=0.148 w f 1 , n 1 and therefore ^ = (1 .17) S~l48 = 1 , 1 2 1 + T^T Other intermediate values were obtained for interface positions of a n d The seven equidistant points between Y = 0 and Y = were then used to obtain the value and location of the maximum power reduction factor by the Douglass - Avakian method, followed by differentiation, in a manner identical to that used for determining the maximum 4 7 v o l u m e t r i c f l o w r a t e f a c t o r , c. R e s u l t s R e s u l t s f o r power r e d u c t i o n f a c t o r s are presented i n Table V I I f o r v i s c o s i t y r a t i o s of 10, 100 and 1000. In f i g u r e 16, the power r e d u c t i o n f a c t o r s are p l o t t e d a g a i n s t the i n t e r f a c e p o s i t i o n f o r the three v i s c o s i t y r a t i o s . These f a c t o r s are then r e p l o t t e d a g a i n s t the f r a c t i o n a l flow area i n f i g u r e 1 7 . A graph of the maximum power requirement f a c t o r a g a i n s t the v i s c o s i t y r a t i o i s shown i n f i g u r e 18, based on r e s u l t s recorded i n t a b l e V I I I . 48 Table V I I Power Reduction F a c t o r s W 0 , R/16 R/8 3R/16 R/4 , 5R/16 3R/8 R/2 5R/8 3 R A R 3R/2 2R /J£ = 10 1.00 1.05 1.10 1.12 1.11 1.12 1.08 0.933 0,710 0.495 0.184 0.00536 0 f u l l W ' = 100 1.00 1.10 1.19 1.20 1.12 0.995 0.828 0.5H 0.268 0.133 0.0325 0.000639 0 • = 1000 1.00 1.05 1.10 0.897 0.597 0.366 0.220 0.0897 0.0570 0.0164 0.00435 0.0000656 0 ( - i n t e r p o l a t e d p o s i t i o n ; Table V I I I Maximum Power Reduction F a c t o r s V i s c o s i t y R a t i o , Maximum ' f u l l W I n t e r f a c e P o s i t i o n , y.^  10 1.12 0.267R (0 . 2 7 R ) 100 1000 1.22 0.170R (0.17R) 1.09 (1.1) 0.09R i (0.1R) 0 ^ R j>R f R R 5R I" ? R 2 R INTERFACE POSITION ( Distance f rom B o t t o m of P ipe , R= r a d i u s ) Figure 16. Variation of Power Reduction Factor with Interface Position -p--e- LC = i o — ® - ^ ' =10 ( i n t e r p o l a t e d point ) - © - LC =100 —©—/i.'=100 ( i n t e r p o l a t e d p o i n t ) ^ ' = 1000 / X ' = I000 ( i n t e r p o l a t e d p o i n t ) A . 0.2 0.3 0.4 0.5 0.6 0.7 FLOW AREA FRACTION OF L IQUID B 0.8 0.9 1.0 Figure 17. V a r i a t i o n of Power Reduction F a c t o r w i t h Flow Area F r a c t i o n O Figure 18. Maximum Power Seduction Enactors and Corresponding Input Volume Fractions for Various Viscosity fiatios 52 D. Hold-up Ratios a. Theory The hold-up ratio i s defined as the input volume ratio divided by the in s i t u volume ratio. The input volume ratio i s the volumetric flow rate of liquid A divided by the volumetric flow rate of liquid B, and in this study i t i s evaluated as Su' ( A y ' ) ( A z ' ) input volume ratio = (37) 2 u ' B ( A y ' ) ( A z ' ) The in situ volume ratio i s the ratio of the volumes of A and B inside the pipe, and for steady s t r a t i f i e d flow of incompressible liquids may be expressed as in s i t u volume ratio - °* * ( 58) Therefore •o,, i c A , r n c A „ n /flow area of h o l H „n ~ , i H n A^ ^ ) ( ^ Z } / liquid A ,, Q. hold-up ratio = / -rr* i „ (39) Su' f A Y M f A z ' V flow area of ^ u „iAy ; I A Z y i i q u i d B A plot of hold-up ratio versus input volume ratio can be made for different viscosity ratios, to i l l u s t r a t e the point that the hold-up ratio i s independent of liquid velocities for laminar flow, as realized by Russell, Hodgson and Govier ( 6 ) . 53 b. Sample Calculations Calculations are based on the case of ^  = 10 and * i - 1 -As shown previously, 2 u ' A ( A y * ) ( A z 1 ) = 4.00 and S u " B ( A y ' ) ( A z 1 ) = 1.44 Therefore input volume ratio = = 2 .78 Flow area of liquid A = 2 .53 R 2 Flow area of liquid B = 0.614 R 2 Hence in s i t u volume ratio = = 4 .12 0.614 and therefore hold-up ratio = |-^| = 0 .675 c. Results Table IX contains the results for input volume ratio, in situ volume ratio and hold-up ratio for viscosity ratios of 1, 10, 100 and 1000. In figure 19 hold-up ratio i s plotted against input volume ratio for four viscosity ratios. Graphs of volumetric flow rate factor and power reduction factor versus input volume fraction could now be made and are shown in figure 20 and 2 1 , respectively. Also, input volume fractions for the maximum volumetric flow rate factors and for the maximum power reduction factors could be calculated, and these values are plotted in figures 15 and 18, respectively, for various viscosity ratios. ,Input Volume Ratio Table IX Calculated Hold-up Data In Situ Volume Ratio Hold-up Ratio 1.0 10 100 1000 1.0 10 100 1000 0 O O © o oO oO 0 0 0 0 • R/8 157"""" 57.5 30.3 7 . 0 3 38.3 4.10 1 . 5 0 . . 0.791 0.184 R/4 . 34.1 13.3 5.21 0.781 12.8 2.66 1 . 0 4 ; 0.407 0.0610 5R/8. 13.4 5-51 -1.49 0.184 6.69 2.00 0.834 0 . 2 2 3 0.0275 R/2 6 . 71 2.78 0.595 0.0685 4.12 1.63 0.675 0.144 0.0166 5R/8 3.86 .1 . 5 2 0.271 0.0298 2 . 7 4 1.41 • . 0.555 0.0989 0 . 0 1 0 9 3R/4 2.36 '.' 0.876 0.137 0.0147 1 .91 1.24 b . 4 5 9 0 . 0 7 1 7 0 . 0 0 7 7 0 7R/8 1.53 '•' 1.38 1.11 R 1.00 :•' 0.312 0.0417 0.00436 1.00 1.00 ;0.312 0.0417 0.00436 9R/8 0.654 0.725 0.902 5 R A 0.423 0.524 0.807 11R/8 0 . 2 5 9 0.365 0 . 7 1 0 3R/2 0.149 0.0294 0.00327 0.000333 0.243 0.613 0.121 0.0135 0 . 0 0 1 3 7 13R/8 0.0748 0.149 0.502 7 R A 0.0293 0.0781 0.375 15R/8 0.00637 0.0261 0.244 2R 0 0 0 0 0 0 0 0 0 H O L D - U P RATIO Figure 20„ V a r i a t i o n of Volumetric Flow Rate Factor with Input Volume F r a c t i o n vn 1.8 T 1.6 - O - fJL'= 10 ' - e - fJL'= I O O -CD- /^ ' = I 0 0 0 0.1 0 . 2 0 . 3 0 . 4 0 .5 0 . 6 0 . 7 0 . 8 0 . 9 I N P U T V O L U M E F R A C T I O N O F L I Q U I D B 1.0 Figure 21. Variation of Power Reduction Factor with Input Volume Fraction 58 DISCUSSION OP RESULTS The numerical flow equations were checked by two methods. Comparison of point velocities calculated by relaxation methods with those calculated using Poiseuille's equation, for single-liquid flow, show an average deviation of 1.23% for a numerical solution with unbalanced residuals, and 0.4% f o r a numerical solution with balanced residuals. Finite difference equations were derived for s t r a t i f i e d laminar flow between parall e l plates, in a manner similar to those derived in the present study. Velocities determined numerically using these equations were compared to velocities calculated.analytically (2), for the case of an interface located mid-way between the two plates and a viscosity ratio of 1000. Corresponding velocities were identical. These comparisons show that the numerical method of solution employed 1B accurate. The velocity profiles computed numerically for the round pipe were consistent with those calculated analytically for parallel plates. As the viscosity ratio increased, the point velocities of the less viscous liquid B, as compared to the point velocities of the more viscous liquid A, increased approximately to the same degree as the viscosity ratio. This result can be seen in figure 11. 59 For a solution -with unbalanced residuals, a deviation of 0.64% resulted between the volumetric flow rate calculated by the numerical method and that calculated by Poiseuille's equation, for single-liquid flow. When the residuals were balanced, the deviation could not be detected within three-figure accuracy. This result again shows good agreement between the numerical and analytical solutions. The volumetric flow rate factor at any specific interface position increased with increasing viscosity ratio. This trend was also followed by the maximum volumetric flow rate factor, as seen in figure 15. From the point of view of volumetric flow rate enhancement,, the viscosity ratio of 1000 is prac t i c a l l y equivalent to an i n f i n i t e viscosity ratio. This arises from the fact that equations (18),(19) and (20) produce a constant dimensionless velocity p r o f i l e , within three-figure accuracy, in liquid A for viscosity ratios greater Q. than 1000. It follows that 7? w i l l not increase v % f u l l significantly with an increase of the viscosity ratio above 1000; that i s , i t s value at JULX = 1000 w i l l be within 0.1% of i t s asymptotic value at i n f i n i t e ju}. Similar asymptotic behaviour i s displayed by the concentric flow model, where the factor (/^ + i s safely taken as /^^ at viscosity ratios equal to or exceeding 1000 (2). The interface position of the maximum volumetric flow rate factor moved closer to the bottom of the pipe as 60 the viscosity ratio increased in the range of 10 to 1000. This i s seen i n Table VI and figure 13. As can be seen in figures 15 and 20, the input volume fraction of the less viscous liquid increased with increasing viscosity ratio, for the conditions of maximum volumetric flow rate enhancement. In the case of the viscosity ratio of 1000, the maximum enhancement factor i s achieved at an input of 81% of liquid B, compared to 8.0% B at a viscosity ratio of 10. For specific viscosity ratios, the maximum volumetri flow rate factors for s t r a t i f i e d flow in a circular pipe were smaller than those for s t r a t i f i e d flow between parall e l plates and were very much smaller than those obtained for concentric flow in a circular pipe. This i s seen in figure 22 for a viscosity ratio of 10. The largest of the three computed maximum power reduction factors occured for the case of ^ u' = 100. The maximum factor for JUL • = 10 was slightly larger than that for juS = 1000. This i s easily seen in figures 16, 17 and 18. The case of JUL1 = 1000 resulted in a lowered power reduction factor because for this case a large input of liquid B, which even occurs at small fractional flow areas, more than counteracts the effect of the lowered pressure gradient at a given throughput of liquid A. The fact that power reduction factors were always lower than corresponding pressure reduction factors can be attributed to the mere presence of B, Figure 22 Comparison of Volumetric Flow Rate F.actors for Concentric Flow, Parallel Plate Flow and Stratified Flow in a Circular Pipe 62 which must be incorporated into the calculation of power. The position of the interface for the maximum power reduction factor moved closer to the bottom of the pipe as the viscosity ratio increased in the region of 10 to 1000. This is seen in Table VIII and figure 16. Over the same range, though, the input volume fraction of liquid B remained essentially constant, with approximately 6.0 to 8.0% of liquid B required to produce the maximum power reduction factor, as seen in figure 18. In the laminar region, hold-up ratio i s a function of input volume ratio and viscosity ratio only. At a constant input volume rat i o , the hold-up ratio decreased with increasing viscosity ratios. An oral presentation on the same topic by Redberger and Charles (12) came to the present author's attention after completion of his calculations. Though the two studies are very similar, a number of differences exist which can be noted here. Redberger and Charles solved the flow equations on an electronic computer, which necessitated that the pipe wall be approximated by horizontal and ve r t i c a l straight lines. The present calculations were performed using a true circular boundary, negative values being obtained by extrapolation for grid points lying outside the pipe wall. They performed relaxations at the liquid interface in one-dimension only, while the present study incorporated a two-dimensional , . 63 relaxation method, necessitated by the velocity variation along the interface. The volumetric flow rate factors calculated here are s l i g h t l y larger than those calculated by Redberger and Charles, but these authors admit that their results may be conservative. Finally, their calculations were performed for specific pipe diameters and viscosities of the two phases rather than for the general case, as in the present investigation. 65 COMPARISON OF EXPERIMENTAL AND THEORETICAL RESULTS A. Experimental Data The theoretical results of this study were compared • to the experimental results reported by Russell, Hodgson and Govier (6). The complete tabular data of their investigations are deposited with the American Documentation Institute (7). Their tests were conducted at 77°E in a horizontal, smooth, transparent pipe, 28.18 feet in length, with an inside diameter of 0.8057 inches. The two liquids used were a refined mineral o i l with a specific gravity of 0.834 and a viscosity of 18 c p . , and water with a viscosity of 0.894 c p . , giving a viscosity ratio of 20.1. The two-phase flow was studied at thirteen superficial water velocities and input oil-water volume ratios within the range 0.1-10. These flow rates corresponded to superficial Reynolds numbers ranging from 809 to 24,700 for the water flow and 9.58 to 942 for the o i l flow. Russell et a l measured pressure drop in a l l of their runs and hold-up in some. Most of the pressure drop measurements were made for the whole pipe length, but some were performed on a half-section of pipe. Lockhart and Martinelli (13) have proposed c r i t e r i a 66 f o r d e f i n i n g the re g i o n s of laminar flow and t u r b u l e n t flow of two coc u r r e n t phases. I f the s u p e r f i c i a l Reynolds number of a phase i s l e s s than 1000, the l i q u i d f low i s p o s t u l a t e d to be lam i n a r , w h i l e f o r s u p e r f i c i a l Reynolds numbers g r e a t e r than 2000, the fl o w i s taken as t u r b u l e n t . The exact p o i n t of t r a n s i t i o n i s not known-, the c r i t e r i o n of Re = 2000 f o r turbulence i s considered a c o n s e r v a t i v e one. Values of Reynolds numbers between 1000 and 2000 can be considered to be i n the t r a n s i t i o n r e g i o n . Of the t h i r t e e n s u p e r f i c i a l water v e l o c i t i e s s t u d i e d , o n l y the lowest v e l o c i t y of 0.116 f e e t / s e c . corresponds t o a Reynolds number l e s s than 1000, and i s thus i n the laminar r e g i o n . The next f o u r higher- water v e l o c i t i e s l i e i n the t r a n s i t i o n r e g i o n and the remainder are i n the range of t u r b u l e n t f l o w . A l l the o i l f l o w r a t e s i n v e s t i g a t e d were i n the laminar r e g i o n , the l a r g e s t o i l Reynolds number being 9^2. Therefore, i n comparing the t h e o r e t i c a l and experimental data, the flow regime must be consi d e r e d , s i n c e a l l the present t h e o r e t i c a l r e s u l t s are based on laminar flow of both l i q u i d s . 67 B. Hold-Up a. Computational Procedure A plot of hold-up ratio versus input volume ratio i s shown in figure 19 for viscosity ratios of 1, 10, 100 and 1000. From this graph, a cross-plot of hold-up ratio versus viscosity ratio, with input volume ratio as the parameter was drawn. Values for the viscosity ratio of 20.1 were read from the cross-plot, and the theoretical curve of hold-up ratio versus input volume ratio was drawn for yjt' = 20.1. This curve was compared to the experimental results by direc t l y plotting on the same graph the corresponding tabular data (7) of Russell, Hodgson and Govier. b. Results The cross-plot of hold-up ratio versus viscosity ratio, withiinput volume ratio as the parameter, i s shown in figure 23. From this graph, the theoretical curve of hold-up ratio versus input volume ratio for a viscosity ratio of 20.1 was drawn, as illustrated i n figure 24. The experimental results were plotted on the same graph for the'eight superficial water velocities for which hold-up data were reported. c. Discussion of Results The best agreement between the theoretical and experimental results occurs for the superficial water velocity 68 Figure 23. Hold-up Ratio Cross .Plot 69 0. 3 o X 1.5 1.4 1.2 I.I 1.0 0.9 0.8 0.7 0.5 R e w o 0.116 8 0 9 CD 0.162 1130 e 0 . 2 0 6 1430 © 0 . 2 4 7 1 7 2 0 0 . 2 8 7 2 0 0 0 C 0 .388 2 8 0 0 © 0.718 3 0 0 0 1.79 12800 ( f . s . ) /LL'=20.I © © c € T h e o r e t i c a l I I M i l l 0.02 0.1 1.0 INPUT O I L - W A T E R VOLUME RATIO 10 Figure 24. Theoretical and Experimental Hold-up Ratio for' JJL* = 20.1 70 of 0.116 f t . / s e c , corresponding to a Reynolds number of 8 0 9 . This i s . t h e o n l y water f l o w r a t e t h a t l i e s w i t h i n the laminar r e g i o n d e f i n e d by Lockhart and M a r t i n e l l i (13). Pour of the s i x p o i n t s f o r t h i s v e l o c i t y l i e on or very c l o s e to the t h e o r e t i c a l curve. As the water v e l o c i t i e s i n c r e a s e , the agreement between the t h e o r e t i c a l and experimental r e s u l t s decreases. This i s t r u e even of the data f o r the f o u r s u p e r f i c i a l water v e l o c i t i e s which f a l l w i t h i n the t r a n s i t i o n r e g i o n of Re = 1000-2000. These p o i n t s l i e i n c r e a s i n g l y above the t h e o r e t i c a l curve. The .data f o r the three s u p e r f i c i a l water v e l o c i t i e s of 0.558* 0.718 and 1.79 f t . / s e c , corresponding to Reynolds numbers of 2500, 5000 and 12500 r e s p e c t i v e l y , d e v i a t e even more g r e a t l y from the t h e o r e t i c a l curve, the l a r g e s t s u p e r f i c i a l water v e l o c i t y having the g r e a t e s t d e v i a t i o n from the t h e o r e t i c a l . A l l three v e l o c i t i e s are w e l l w i t h i n the r e g i o n of t u r b u l e n t f l o w . Prom the comparison, i t i s e a s i l y seen that there i s c l o s e agreement between the experimental and t h e o r e t i c a l r e s u l t s i n the laminar r e g i o n . As the flow becomes i n c r e a s i n g l y t u r b u l e n t , however, the experimental hold-up r a t i o develops an i n c r e a s i n g p o s i t i v e d e v i a t i o n from the t h e o r e t i c a l laminar curve. The hold-up r a t i o i s then no longer a f u n c t i o n of i n p u t volume r a t i o and v i s c o s i t y r a t i o alone. 7 1 C. Pressure Drop a. Computational Procedure In order to compare t h e o r e t i c a l and experimental pressure drop data, the t h e o r e t i c a l values were transformed to dimensional quantities and compared with the experimental points at s p e c i f i c s u p e r f i c i a l water v e l o c i t i e s . A plot of Q^/^Afull versus flow area f r a c t i o n f o r yw.' = 20.1 was obtained as the f i r s t step of the c a l c u l a t i o n s . Prom fi g u r e 1 3 , a cross-plot was made of Q^^SAfull v e r s u s v i s c o s i t y r a t i o , with the int e r f a c e p o s i t i o n as the parameter. Prom t h i s c r o s s - p l o t , values of Q^^^Afull f o r / u > = 2 0 , 1 w e r e e a s i l y obtained f o r a l l the i n t e r f a c e p o s i t i o n s . The in t e r f a c e was expressed i n terms of a flow area f r a c t i o n , as i n Appendix I I , and the required plot of Q A ^ A f u l l v e r s u s flow area f r a c t i o n was then drawn. The second step i n the c a l c u l a t i o n s started with the assumption of a s e r i e s of input volume r a t i o s . The pressure drop was calcu l a t e d f o r each input volume r a t i o separately. Prom figur e 24, which i s a plot of hold-up r a t i o versus input volume r a t i o f o r = 20.1, the hold-up r a t i o was found f o r each assumed input volume r a t i o . Since the i n - s i t u volume r a t i o i s equal to the input volume r a t i o divided by the hold-up r a t i o , i t was possible to compute the flow area f r a c t i o n by the r e l a t i o n that 72 flow area fraction - i n 7 s i t u volume ratio ( . I + m-situ volume ratio v ' Knowing the flow area fraction for an assumed input volume ratio, the value of %/Q±fuii a* a 20.1 was easily read from the graph determined in the f i r s t step of the calculations. The third step consisted of choosing a superficial velocity of liquid B which is,equal to one of the superficial water velocities reported in the experimental paper. The volumetric flow rate of the less viscous l i q u i d , QB, i s equal to the product of i t s superficial velocity and the total cross-sectional area of the pipe. The volumetric flow rate of the more viscous l i q u i d , QA, i s given as QA = QB (input volume ratio) (41) ^Afu l l Prom the Poiseuille equation, A l l the values on the right hand side of equation (43) were ZS P then known. Therefore (- could be calculated for an assumed input volume ratio, for a chosen superficial velocity. The calculations were repeated for the several input volume ratios assumed, and a theoretical curve of pressure drop versus input volume ratio was thus drawn for the given 73 s u p e r f i c i a l water v e l o c i t y . This procedure was repeated f o r the other twelve water r a t e s . The experimental data were converted to the same u n i t s as the t h e o r e t i c a l , and were p l o t t e d w i t h the t h e o r e t i c a l curves. b. Sample C a l c u l a t i o n s and Re s u l t s The c r o s s - p l o t s of Q A / Q A f u 2 i v e r s u s v i s c o s i t y r a t i o , w i t h i n t e r f a c e p o s i t i o n as the parameter, are shown i n f i g u r e 25. From these p l o t s , v a l u e s of Q A / Q A f u i i £°T = 20.1 were obtained f o r v a r i o u s ' i n t e r f a c e p o s i t i o n s . Having expressed the i n t e r f a c e p o s i t i o n i n terms of fl o w area f r a c t i o n as i n Table "X, Appendix I I , Q A / Q A f u i i w a s p l o t t e d a g a i n s t flow area f r a c t i o n f o r j u % = 20.1, as i l l u s t r a t e d i n f i g u r e 26. Assume an i n p u t A-B volume r a t i o = 5»00 From f i g u r e 24, the hold-up r a t i o = 0.655 Therefore the i n - s i t u volume r a t i o = ^ '^55 = 7*63 7 6 3 and the f l o w area f r a c t i o n of A = g-j-^ = 0.884 From f i g u r e 26, Q A/Q A f u l l = 1-31 Choose s u p e r f i c i a l water v e l o c i t y of l i q u i d B = 0.116 f t . / s e c . Diameter of pipe = 0.8057 i n . = 0.0671 f e e t . Area of pipe = ^ l S P ^ = ° ' 0 0 ^ f t ' 2 Therefore Q R = (0.116)(0.00354) = 0.000411 f t . 5 / s e c , QA = (0.000411)(5.00) = 0.00206 f t . V s e c , and Q A f u l l = ° i ? 5 l 0 6 = 0.00157 f t . 5 / s e c . Now JU-k = 0.0121 l b s . / f t . s e c . and = 32.174 p o u n d a l s / l t . - f o r c e Figure 25« Variation of Volumetric Flow Rate Factor with Viscosity Ratio FLOW AREA FRACTION OF LIQUID B Figure 26. Variation of Volumetric Flow Rate Factor with Flow Area Fraction for y x * = 20.1 vn 76 m v ^ r ^ ™ ( 128(0.0121)(0.00157) 1 Q 1 v, 0 2/„. Therefore (- — ) = T T O 2 . l 7 4 ) ( 0 . 0 f e 7 1 ) 4 = 1 ' 1 9 l b s - / f t -This c a l c u l a t i o n was repeated f o r s e v e r a l input volume r a t i o s , so t h a t the t h e o r e t i c a l curve could be obtained. Data f o r a t y p i c a l experimental p o i n t are as f o l l o w s : S u p e r f i c i a l water v e l o c i t y = 0.116 f t . / s e c . Experimental input o i l - w a t e r volume r a t i o = 4.21 Average pressure droo = 6.90 inches of water = (6 . 9 0 X 5 . 2 0 2 ) = 35-9 l b s . / f t . 2 Length of t e s t s e c t i o n = 28.18 f e e t and t h e r e f o r e pressure g r a d i e n t (- -^^) = 28* 18 = 1*27 l b s . / f t . 2 / f t o This t a b u l a t i o n was repeated f o r a l l the inp u t o i l - w a t e r volume r a t i o s r e p o r t e d , and the experimental r e s u l t s were p l o t t e d on the same graph as the t h e o r e t i c a l curve. The t h e o r e t i c a l curves and experimental data f o r the t h i r t e e n water r a t e s were p l o t t e d i n f i g u r e s 27, 28, and 29 as pressure g r a d i e n t versus input volume f r a c t i o n . c. D i s c u s s i o n of R e s u l t s In f i g u r e 27, experimental data are shown f o r the s u p e r f i c i a l water v e l o c i t y of 0.116 f t . / s e c , Re =- 809. This i s the only water flow r a t e which d e f i n i t e l y l i e s w i t h i n the laminar r e g i o n , and the experimental and t h e o r e t i c a l r e s u l t s show ve r y c l o s e agreement f o r t h i s case. The d a t a f o r the 77 o V INPUT V O L U M E FRACTION OF L IQUID B Figure 27. Theoretical and Experimental Pressure Gradients for V = 0.116 - 0.287 f . 8 . 78 Exper imenta l V w R e w Theore t i ca l — Points (f.s.) Curves 0 0.5 1.0 INPUT VOLUME F R A C T I O N OF LIQUID B Figure 28. Theoretical and Experimental Pressure Gradients for V = 0.327 - 0.718 f.s. 79 2 4 o < cc o UJ cr CO w UJ cc 0. 2 0 -r 16 12 8 Exp' l Points (f.s.) 1.08 7530 R e w Theoret ical Curves ©^section i.os 7830 — O 1.44 10000 • | section i.4 4 IOOOO — 9 1.79 12800 CD © ^ S e c t i o n 1.79 12800 e 3.55 24700 © ^ s e c t i o n 3.65 24700 CD CD 1 0 0.5 1.0 INPUT VOLUME FRACTION OF L IQUID B Figure 29. Theoretical and Experimental Pressure Gradients for V = 1.08 - 3.55 f.s. 80 other f o u r v e l o c i t i e s p l o t t e d i n f i g u r e 27 l i e in the t r a n s i t i o n a l r e g i o n , and i t i s seen th a t agreement between the experimental and t h e o r e t i c a l r e s u l t s i s not q u i t e as good. .Figures 28 and 29 c o n t a i n experimental data f o r fl o w r a t e s which a l l l i e i n the t u r b u l e n t r e g i o n , and they show i n c r e a s i n g d e v i a t i o n from the t h e o r e t i c a l curve as the v e l o c i t y i n c r e a s e s . At the highest s u p e r f i c i a l water v e l o c i t y of 3.55 f t . / s e c , the d e v i a t i o n i s very g r e a t . Therefore the experimental and t h e o r e t i c a l r e s u l t s agree i n the laminar r e g i o n , but as turbulence i n c r e a s e s , the agreement decreases. In the t u r b u l e n t r e g i o n , the experimental pressure drops are very much g r e a t e r than those p r e d i c t e d t h e o r e t i c a l l y f o r laminar flow. This disagreement i s in the anticipated direction. 81 D. Enhancement of Volumetric Flow Rate a. Computational Procedure The comparison of experimental end t h e o r e t i c a l volumetric flow rate f a c t o r s was made on a plot of Qft/^Afull versus input volume f r a c t i o n of l i q u i d B. The input volume f r a c t i o n was chosen because i t gives a better representation of the data than input volume r a t i o . Figure 24 i s a plot of hold-up r a t i o versus input A-B volume r a t i o f o r yxx = 20.1. From t h i s f i g u r e , the i n - s i t u A-B volume r a t i o f o r a corresponding input A-B volume r a t i o was determined, knowing that the hold-up r a t i o i s equal to the input volume r a t i o divided by the i n - s i t u volume r a t i o . The flow area f r a c t i o n of l i q u i d B was calculated from the i n - s i t u A-B volume r a t i o , and from f i g u r e 26 the corresponding value of Q ^ / ^ f u l l W 8 S o l : > ' t : ; a i r i e d ' ^ n t h i s way the e n t i r e t h e o r e t i c a l curve of Q A/^ Af u]_]_ versus input volume f r a c t i o n of l i q u i d B was pl o t t e d . The experimental input oil-water volume r a t i o was converted to the input volume f r a c t i o n of water as follows: , , „ .. „ , -, input oil-water volume r a t i o input volume f r a c t i o n of water = l - 1 + £ n p u t oil-water volume r a t i . (44) 82 P r e v i o u s l y i t was proved t h a t (- ^ Q ax E f u l 1 at constant Q, = 7^ at constant (- ( 2 8 ) (- -4-^) ^ A f u l l a x Therefore i f the experimental pressure r e d u c t i o n f a c t o r could be c a l c u l a t e d , i t would be equal to the v o l u m e t r i c flow r a t e enhancement f a c t o r . The r a t i o of the pressure g r a d i e n t f o r the pipe f l o w i n g f u l l of o i l to the measured experimental pressure g r a d i e n t f o r two-phase fl o w was expressed as d v ^ o i l V o i l 6JB ™ v d s r f u l l _ D riLt-s (- = (-^ 6 x'exp. v 6 x'exp. The s u p e r f i c i a l o i l v e l o c i t y was c a l c u l a t e d as V ., = V . ( i n p u t volume r a t i o ) (46) o i l water v ' v and then a l l the values on the r i g h t hand side of equation (45) were known. Thus the experimental pressure g r a d i e n t f a c t o r or i t s e q u i v a l e n t , the v o l u m e t r i c flow r a t e f a c t o r , were c a l c u l a t e d f o r a l l the experimental d a t a , and these were p l o t t e d on the same graph as the t h e o r e t i c a l curve. b. Sample C a l c u l a t i o n s and R e s u l t s Consider f i r s t the t h e o r e t i c a l c a l c u l a t i o n s . Assume an inpu t A-B volume r a t i o = 5.00 From f i g u r e 24, hold-up r a t i o = 0.65b Therefore i n - s i t u A-B volume r a t i o = ^ " = 7«6$ 8 3 and the f l o w area f r a c t i o n of l i q u i d R = 1 - - g - j ^ = 0.116 Prom f i g u r e 26, Q A/Q A f u l l =1-31 Corresponding input volume f r a c t i o n of l i q u i d B = 1 - - °- 1 67 This c a l c u l a t i o n was repeated f o r v a r i o u s input volume r a t i o s , and a p l o t of QA/QAfun versus i n p u t volume f r a c t i o n of l i q u i d B i s shown i n f i g u r e 3 0 . Now c o n s i d e r the experimental data. S u p e r f i c i a l water v e l o c i t y = 0.116 f t . / s e c . Input o i l - w a t e r volume r a t i o = 4.21 S u p e r f i c i a l o i l v e l o c i t y = (0.116)(4.21) = 0.488 f t . / s e c . O i l v i s c o s i t y = 0.0121 l b s . / f t . s e c . = 32.174 p o u n d a l s / l b . - f o r c e Pipe diameter, D = 0.0671 f e e t . ^ ^ ^ ^ / P% 52(0.0121)(Q.488) -. x n /„. 2,-. Therefore ( - r r ) f , , n = -— =^ k = 1.30 l b s . / f t . / f t . d x i u i i (32.174)(0.067D Prom the p r e v i o u s s e c t i o n < 4 ! w • x - 2 7 i ^ . / f t . 2 / f t . Hence 7 ^ = T^IS = 1 - 0 2 ^ A f u l l a. p i and the i n p u t volume f r a c t i o n of water = 1 - e^^'i = 0 . 1 9 2 The experimental data f o r a l l t h i r t e e n s u p e r f i c i a l water v e l o c i t i e s are p l o t t e d i n f i g u r e 30. LEGEND FOR FIGURE 30 (On reverse side) 84 LEGEND FOR FIGURE 30 O 0 .116 809 © 0.162 1130 © 0.206 1430 © 0.247 1720 ® 0.287 2000 • 0 .327 2280 © 0.358 2500 O # section 0.358 2500 • 0 .538 3750 © 0.718 5000 3 # section 0.718 5000 m 1.08 7530 B # section 1.08 7530 • 1.44 10000 0 H section 1.44 10000 D 1.79 12500 J 1/? section 1.79 12500 H 3.55 24700 B # section 3-55 24700 85 Figure 30- Theoretical and Experimental Volumetric Flow Rate Factors for yu' = 20.1 86 c. D i s c u s s i o n of R e s u l t s The experimental r e s u l t s i n f i g u r e 30 show a wide s c a t t e r of p o i n t s . A t r e n d i s noted, though, t h a t the values of the v o l u m e t r i c flow r a t e f a c t o r i n the t u r b u l e n t r e g i o n l i e much lower than the t h e o r e t i c a l curve and t h a t values i n the laminar and t r a n s i t i o n a l regions are c l o s e r to the t h e o r e t i c a l curve. A few experimental p o i n t s even f a l l above the t h e o r e t i c a l curve, and i t i s t h e r e f o r e presumed t h a t p e r f e c t h o r i z o n t a l s t r a t i f i c a t i o n d i d not n e c e s s a r i l y occur but t h a t the i n t e r f a c e may have been s l i g h t l y curved. Such a s l i g h t tendency towards c o n c e n t r i c f l o w would r e s u l t i n r a i s i n g the v o l u m e t r i c f l o w r a t e f a c t o r (see f i g u r e 22). R u s s e l l et a l (2,6) reported a pressure g r a d i e n t r e d u c t i o n f a c t o r of 1.2 f o r 10% water i n p u t , and i t i s seen t h a t t h i s agrees w i t h the experimental values p l o t t e d i n f i g u r e 30. L a t e r they reported t h i s as t h e i r maximum f a c t o r , but i t i s seen i n f i g u r e 30 tha t f a c t o r s g r e a t e r than 1.2 were a c t u a l l y obtained, as was mentioned i n the i n t r o d u c t i o n . 87 CONCLUSIONS T h e o r e t i c a l v o l u m e t r i c f l o w r a t e enhancement f a c t o r s , power r e d u c t i o n f a c t o r s and hold-up r a t i o s have been d e r i v e d n u m e r i c a l l y f o r the s t r a t i f i e d laminar flow of two i m m i s c i b l e Newtonian l i q u i d s f l o w i n g i n a c i r c u l a r p i p e , f o r v i s c o s i t y r a t i o s of the two l i q u i d s ranging from 1 to 1 0 0 0 , and f o r v a r i o u s p o s i t i o n s of the l i q u i d - l i q u i d i n t e r f a c e . The v o l u m e t r i c flow r a t e enhancement f a c t o r at any s p e c i f i c i n t e r f a c e p o s i t i o n i n c r e a s e s w i t h i n c r e a s i n g v i s c o s i t y r a t i o . A s i m i l a r trend i s a l s o f o l l o w e d by the maximum vo l u m e t r i c flow r a t e f a c t o r which achieves a maximum asymptotic value at i n f i n i t e v i s c o s i t y r a t i o . T h i s asymptotic value i s p r a c t i c a l l y reached at a v i s c o s i t y r a t i o of 1000. For the range of v i s c o s i t y r a t i o from 10 to 1000, the p o s i t i o n of the i n t e r f a c e f o r the maximum v o l u m e t r i c flow r a t e f a c t o r moves c l o s e r to the bottom of the pipe w i t h i n c r e a s i n g v i s c o s i t y r a t i o . However, the input volume f r a c t i o n of the l e s s v i s c o u s l i q u i d , f o r the c o n d i t i o n of maximum v o l u m e t r i c f l o w r a t e enhancement of the more v i s c o u s l i q u i d , i n c r e a s e s w i t h i n c r e a s i n g v i s c o s i t y r a t i o . At s p e c i f i c v i s c o s i t y r a t i o s , the maximum vo l u m e t r i c f l o w r a t e f a c t o r f o r s t r a t i f i e d f low i n a 88 c i r c u l a r pipe i s s m a l l e r than the corresponding maximum f a c t o r f o r p a r a l l e l p l a t e f l o w , and very much s m a l l e r than the corresponding maximum f a c t o r f o r c o n c e n t r i c flow i n a c i r c u l a r p i p e . Power r e d u c t i o n f a c t o r s are always lower than corresponding v o l u m e t r i c f l o w r a t e f a c t o r s . The p o s i t i o n of the i n t e r f a c e f o r the maximum power r e d u c t i o n f a c t o r moves c l o s e r to the bottom o f the pipe as the v i s c o s i t y r a t i o i n c r e a s e s i n the range of 10 to 1000. In the same range, the in p u t volume f r a c t i o n of the l e s s v i s c o u s l i q u i d necessary to produce the maximum f a c t o r remains e s s e n t i a l l y constant. For laminar f l o w of both l i q u i d s , hold-up r a t i o i s a f u n c t i o n of inpu t volume r a t i o and. v i s c o s i t y r a t i o .only. The present t h e o r e t i c a l r e s u l t s and the experimental r e s u l t s of R u s s e l l , Hodgson and Govier f o r the case of laminar flow of both phases, show good agreement w i t h regard to hold-up r a t i o s , pressure g r a d i e n t s and v o l u m e t r i c f l o w r a t e enhancement f a c t o r s . In the case of experimental r e s u l t s i n the t r a n s i t i o n r e g i o n o f flow of the l e s s v i s c o u s l i q u i d , there i s d i s t i n c t disagreement between these r e s u l t s and the present t h e o r e t i c a l laminar flow r e s u l t s . T h i s expected disagreement i n c r e a s e s as the flo w o f the l e s s v i s c o u s l i q u i d becomes i n c r e a s i n g l y t u r b u l e n t . 8 9 The pressure gradient reduction f a c t o r s measured experimentally f o r the two crude o i l s reported i n table I f a l l between the values determined t h e o r e t i c a l l y f o r ' concentric flow and s t r a t i f i e d flow i n a c i r c u l a r pipe. These r e s u l t s substantiate the statement by Russell and Charles that the flow f o r these experiments was intermediate between the s t r a t i f i e d and concentric models. 90 REFERENCES 1. C l a r k e , K.A., p r i v a t e communication quoted i n reference 2, (1948). 2. R u s s e l l , T.'.V.F., and C h a r l e s , M.E., 3£, 18 (1959). 3. C l a r k , A.F., and Shapiro, A., U.S. Patent no. 2,533,878 (May 1949). 'A. C h i l t o n , E.G., and tfandley, L.R., U.S. Patent No. 2,821,205 (Jan. 1958). 5. C h i l t o n , E.G., O.J.Ch.E., £ 2 , 127 (1959). 6. R u s s e l l , T.W.F., Hodgson, G.W. , and Govier, G.W., C. J.Ch.E., 3£, 9 (1959). 7. Document No. 5772, A.D.I. A u x i l i a r y P u b l i c a t i o n s P r o j e c t , P h o t o d u p l i c a t i o n S e r v i c e , L i b r a r y of Congress, Washington, D. C. 8. Tiproan, E. , and Hodgson, G.W., J . P e t r o l . Tech. 8, No. 9, 91 (1956). 9. Pavlov, P.P., Trudy Azerbaidzahn, Ind. I n s t . im. M. Azizbekova, No. 9, 76 (1955). 10. M i c k l e y , H.S., Sherwood, T.K., and Reed, C.E., A p p l i e d Mathematics i n Chemical E n g i n e e r i n g , 2nd E d i t i o n , McGraw-Hill (1957). 11. A l l e n , D.N. de G., R e l a x a t i o n Methods, McGraw-Hill (1954). 12., Redberger, P.J. and C h a r l e s , M.E., Syposium "Multiphase Flow i n the P r o d u c t i o n and D r i l l i n g of O i l W e l l s " , A.I.Ch.E. - S.P.E. J o i n t Meeting, T u l s a , Oklahoma, Sept. 25-28, I960. 13. Lockhart, R.W., and M a r t i n e l l i , R.C., Chem. Eng. Progress, fL5, 39 (1949). 14. P e r r y , J.H., Chemical Engineers Handbook, 3rd E d i t i o n , McGraw-Hill (1950). 91 APPENDIX I. Relaxation Methods Relaxation methods can i n general be used to solve various tyoes of d i f f e r e n t i a l equations, but f o r t h i s study they were used to solve only l i n e a r second order p a r t i a l d i f f e r e n t i a l equations. This section w i l l describe how r e l a x a t i o n methods are employed to solve t h i s p a r t i c u l a r type of equation. The p a r t i a l d i f f e r e n t i a l equation representing the flow conditions investigated, i s rewritten as a f i n i t e d i fference equation of the form u{ + u 2 + u^ + u^ - 4u£ + 8 /c' O y ' ) 2 = 0 (16) as shown on page 12a The above equation suggests that the v e l o c i t y at the point 0 i s equal to the arithmetric mean of the v e l o c i t y at four surrounding points, plus a constant. The r e l a t i v e oosition of the f i v e points r e f e r r e d to are shown i n figu-re 2. Suppose the values of the f i v e v e l o c i t i e s were guessed at and substituted i n equation (19). I f the guesses were i n c o r r e c t , a remainder or r e s i d u a l would r e s u l t because the equation would not be s a t i s f i e d . The r e s i d u a l can be expressed as u[ + u 2 + u^ + - + 8 ^ ' ( A y 1 ) 2 = Residual (47) 92 I t i s n o t i c e d that by i n c r e a s i n g the value of u^ by 1, the r e s i d u a l i s decreased by 4, whereas i n c r e a s i n g the value of any of the surrounding v e l o c i t i e s by 1, i n c r e a s e s the residua by lo This 4 to 1 r a t i o l e a ds to convergence towards the s o l i r t i o n , which i s a t t a i n e d when the r e s i d u a l i s zero or aoproximately zero. For example, a new guess of u^ g r e a t e r than the previous value by one-quarter of the r e s i d u a l , w i l l e l i m i n a t e the r e s i d u a l and thus achieve a t e n t a t i v e s o l u t i o n . This procedure i s then a p p l i e d to a g r i d of p o i n t s as i n f i g u r e 5» where a l l the p o i n t v e l o c i t i e s are guessed at and the r e s i d u a l s c a l c u l a t e d f o r each point u s i n g equation (1 The p o i n t w i t h the l a r g e s t r e s i d u a l i s r e l a x e d f i r s t by ren d e r i n g the r e s i d u a l equal to zero. This procedure i s repeated u n t i l a l l the r e s i d u a l s are approximately equal to zero, i n which case the d e s i r e d s o l u t i o n has been a r r i v e d a t . There are methods of a i d i n g convergence to the s o l u t i o n , such as o v e r - r e l a x a t i o n and b l o c k r e l a x a t i o n , but these are refinements of the b a s i c method d e s c r i b e d above. K i c k l e y , Sherwood and Reed (10) present a b r i e f d e s c r i p t i o n of r e l a x a t i o n methods, i t s refinements and some examples of i t s use, but a more complete d e s c r i p t i o n i s found i n A l l e n ' s t e x t (11). 93 I I . Plow Area Fraction The r e l a t i o n s h i p between the interface p o s i t i o n and flow area f r a c t i o n i s developed from the expression f o r the area of a segment of a c i r c l e which i s Area = ^  r 2 ( 0 - sin© ) (48) where <9 i s the central angle of radian measure and r i s the radius. Since cos ® = (r-h)/h, Perry (14) has expressed t h i s area i n terms of the height, h, of the segment (which corresponds to the in t e r f a c e p o s i t i o n y i ) and the diameter D of the c i r c l e , and presents a table r e l a t i n g the area of the c i r c u l a r segment to the r a t i o * V D . Thus, knowing the interface p o s i t i o n , the area of the segment i s e a s i l y arrived at. Dividing t h i s area by the t o t a l cross-sectional area of the pipe r e s u l t s i n the flow area f r a c t i o n . The following table was determined using t h i s information. Table X Interface P o s i t i o n Plow Area Fraction of Liquid B 0 0 R/8 0.02602 R/4 0.07214 3R/8 0.1298 R/2 0.1955 5R/8 0.2670 3R/4 0.3425 R 0.5000 3R/2 0.8045 2R 1.0000 93 I I . Flow Area F r a c t i o n The r e l a t i o n s h i p between the i n t e r f a c e p o s i t i o n and flow area f r a c t i o n i s develooed from the expres s i o n f o r the area of a segment of a c i r c l e which i s Area = -| R 2( © - sin© ) (48) where <9 i s the c e n t r a l angle of r a d i a n measure and R i s the r a d i u s . Since cos ® = (R-H)/H, P e r r y (14) has expressed t h i s area i n terms of the h e i g h t , H, of the segment (which corresponds to the i n t e r f a c e p o s i t i o n y^) and the diameter D of the c i r c l e , and presents a t a b l e r e l a t i n g the area of the c i r c u l a r segment to the r a t i o /D. Thus, knowing the i n t e r f a c e p o s i t i o n , the area of the segment i s e a s i l y a r r i v e d a t . D i v i d i n g t h i s area by the t o t a l c r o s s - s e c t i o n a l area of the pipe r e s u l t s i n the flo w area f r a c t i o n . The f o l l o w i n g t a b l e was determined u s i n g t h i s i n f o r m a t i o n . Table X I n t e r f a c e P o s i t i o n Flow Area F r a c t i o n of yj_ L i q u i d B 0 0 R/8 0.02602 R/4 0.07214 3R/8 0.1298 R/2 0.1955 5R/8 0.2670 3R/4 0.3425 R 0.5000 3R/2 0.8045 2R 1.0000 94 I I I . Douglass-Avakian Method The Douglass-Avakian method, which i s d e s c r i b e d by M i c k l e y , Sherwood and Reed (10), employs a f o u r t h degree polynomial which i s f i t t e d to seven e q u i d i s t a n t p o i n t s to give the best curve through them. The polynomial i s The seven p o i n t s must be spaced at equal i n t e r v a l s h of Z, and the c o o r d i n a t e s adjusted so t h a t Z = 0, f o r the c e n t r a l p o i n t . The seven val u e s of the v a r i a b l e Z are then -3h, -2h, -h, o, h, 2h and 3h, and k i s the c o e f f i c i e n t of h i n the values of Z. Thus at Z = -3h, k = -3; at Z = -2h, k = -2 e t c . The v a l u e s of the constants are given by the f o l l o w i n g e x p r e s s i o n s : X = a + bZ + cZ + dZ v + eZ .4 (49) a 524HX - 2 4 5 2 k2 X + 2 l 2 k 4 X (50) ^24" b c (5D (52) d - 7 S k X + Sk^X (53) e 216h^ 72 £X - 67 S k 2 X + 7 S k \ (54) 3168h 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0059113/manifest

Comment

Related Items