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THESsB --H_——-... 1; LIBRARY Michiganfitate g» University 'r w w This is to certify that the dissertation entitled TWO-PHASE FLOW 0F GAS BUBBLE SUSPENSIONS IN POLYMER LIQUIDS presented by Kai Sun has been accepted towards fulfillment of the requirements for Ph.D. degreein Chemical Engineering mefessor Date Feb. 16. 1983 MS U i: an Affirmative Action/Equal Opportunity Institution 0-12771 MSU LlBRARlES m RETURNING MATERIALS: Place in book drop to remove this checkout from your record. FINES will be charged if book is returned after the date stamped below. TWO-PHASE FLOW OF GAS BUBBLE SUSPENSIONS IN POLYMER LIQUIDS BY KAI SUN A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Chemical Engineering 1982 ABSTRACT TWO-PHASE FLOW OF GAS BUBBLE SUSPENSIONS IN POLYMER LIQUIDS BY Kai Sun This study was undertaken to explain observed trends in apparent viscosity of gas bubble suspensions with viscoelas- tic media flowing through a duct. Experiments were run to study the two-phase structure developing in slow, isothermal flow of such suspensions in aqueous Separan solutions as well as a Newtonian corn syrup (as reference). Bubble growth is negligible under conditions of these experiments and the residence time is much smaller than the characteris- tic time for bubble rise. The variation of bubble volume fraction across a narrow gap between two plates. is recorded at two locations along the flow direction. This is done with a Cesium gamma radiation source focused on a region of area 0.012 cm2 in the flow plane, and a Sodium iodide detec- tor across the channel yielding a resolution of 0.01 over the range of void fraction investigated from 0.02 to 0.08. This measurement allows us to identify conditions which the flow may be described by a two phase model with a uniform bubbly core and a bubble free wall layer. A theoretical third order suspension model was derived for describing the bulk stress of a dilute homogeneous suspension inside the core region, using a second order model for the viscoelastic medium. The bulk viscosity thus derived for the suspension has a shear-thinning factor which is related to the elasti- city of the medium. Using this third order suspension model, in addition to data on core thickness, bubble concentration profile in the two-phase flow structure, the apparent visco- sity can be computed. Such calculations indicate that the observed reduction in apparent viscosity for the two-phase flow may be attributed to a bubbly core which is more shear- thinning than the medium. This two-phase model predicts that the apparent viscosity decreases with increasing wall shear stress and with incresing bulk void fraction con- sistent with experimental observations. To my parents Ping Chang Sun Y. H. WOng Sun and To my wife, Kathleen ACKNOWLEDGMENTS I would like to express my sincerely gratitude and appreciation to Dr. Krishnamurthy Jayaraman for his guidance and inspiration throughout the course of this work. 'I want to thank: ' Dr. Bruce W. Wilkinson for his assistance with the gamma radiation gauge set—up. Mr. Donald Child for his valuable suggestion on experi- mental design and construction. Professors Charles A. Petty, Robert W. Little and Chang- Yi wang for many helpful discussions. Mr. Ekong A. Ekong and Ms. Eden Tan Dionne for their assistance in the experiment. This work was supported in part by NSF Grant No. CP-8110752 and by the Division of Engineering Research of Michigan State University. Lastly, I thank my parents for their encouragement and my wife for her patience and understanding. TABLE OF CONTENTS LIST OF TABLESOOOOOOOOOOO...0.0.0.0....OOOOOOOOOOOOOOOOO Vi LIST OF FIGURESOOOOOOOOOOOOO0.0.0.0...OOOOOOOOOOOOOOO... fli NOMENCLATUREOO0.000.0...COOOOOOOOOOOOOO0.0.00.00.00.00...n '1“ Chapter 1 INTRODUCTION...OOOOOOOOOOOOOOOOOOO0.0.00.0000... 1 1.A Dynamics of Bubble Growth 1.3 Particle Motion in Viscoelastic Liquids 1.C Apparent Viscocity of Polymer Foams in Poiseuille Flow 1.D Suspension Rheology 2 PROBLEM STATEMENTOOOOOOOOO0.000000000000000.0... 23 2. A Problem Statement- 2.3 Flow Structure and Apparent Viscosity 2. C Objectives 3 EXPERIMENTAL ARRANGEMENT AND MATERIALS.......... 29 3.A Apparatus 3.3 Materials 4 RADIATION GAUGE.OOOOOOOOOOOOOOOOOIOO0.0.0.000... 59 4.A Principles of Gamma Attenuation 4.3 Instruments of Radiation Gauge 5 vow FRACTION PROFILES IN VISCOELASTIC FLUIDS... 32 A Profiles in Corn Syrup 3 Profiles in 2.5 wt Percent Separan Solution. C ,Profiles in 3.5 wt Percent Separan Solution D Correlation of Two-Phase Structure with Medium Properties iv Chapter 6 BULK RHEOLOGY OF DILUTE SUSPENSIONS IN VISCOELASTIC LIQUIDS.0.0.000...OOOOOOOOOOOOOOOOO125 6.A Introduction 6.3 Microrheology 6.C Bulk Stress 6.D Third Order Model For Suspension 6.3 Discussion 7 PREDICTION OF APPARENT VISCOSITY IN TWO-PHASE FLWO0.0......OOOOOOOOOOOOOO..OOOOOOOOOOOOOOOOO.149 7.A Third Order Suspension Model 7.3 Power Law Suspension Model 8 CONCLUSIONS AND RECOMMENDATION..................156 8.A Conclusion 8.3 Recommendations APPENDICES A endix pp A SYMMETRY OF VOID FRACTION PROFILE WITH RESPECT ! To FLm AXISOOOO...0...IOOOOOOOOOOOOOOOOOOOOOOOO152 3 DATA ANALYSIS OF VOID FRACTION PROFILES.........133' C FLOW FIELD OF A SPHERICAL, NEWTONIAN DROP IN A SECOND ORDER FLUID.OO...OOOOOOOOOOOOOOOOOOOOOOOO165 D EVALUATION OF EXTRA STRESS TENSORS..............]]0 E COMPUTER PROGRAMS FOR RELATIVE VISCOSITY COMPUTATION.oooooooooooooooooooooooooooooooooooo175 REFERENCES...OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO0.0.179 LIST OF TABLES Table Page 3.1 PROPERTIES OF MEDIAOOOOOOOOOOOOOOOOOOOO0.0.00.0 38 5.1 COMPARISON OF VOID FRACTION PROFILES........... 122 LIST OF FIGURES Figure ' Page 1.1 Schematic of Particle in Poiseuille Flow....... 12 1.2 Observed Trends of Apparent Viscosity.......... 17 2.1 Two-Phase Flow Structure....................... 25 3.1 Schematic of Flow Arrangement.................. 30 3.2 Static Mixer................................... 34 3.3 Transition Channel and Rectangular Channel..... 35 3.4 Viscosity of Corn Syrup........................ 41 3.5 Viscosity and Primary Normal Stress Coefficient of 2.5 wt Percent Aqueous Separan Solution..... 42 3.6 Viscosity and Primary Normal Stress Coefficient of 3.5 wt Percent Aqueous Separan Solution..... 43 3.7 Sensitivity of Pressure Transducer............. 48 3.8 Photograph of Flowing Suspension at Entrance, 5min. after startUPOOO.....OOOOOIOOOOOOOOOOOO. 51 3.9 Photograph of Flowing Suspension at Entrance, 20 min. after startup.OOOOOOOOOOOOOOOOOOOOOOOI. 52 3.10 Photograph of Flowing Suspension at End of Channel, 5 min. after startup.................. 53 3.11 Photograph of Flowing Suspension at End of Channel, 20 min. after startup................. 54 4.1 Schematic of Linear Attenuation Technique...... 62 4.2 ceSium capsu1e...C..........OCOCOOOOCOCOCOOCOCC 66 4.3’ Calibration Curve for Radiation Counter........ 70 5.11 5.13 Radiation Gauge mounting.OOOOOOOOOOOOOOOOOOOOO. Linear Mass Attenuation Coefficients........... Experimental Calibration for the Resolution Of “diation Gauge.00.0.0000...00.00.000.000.0. Vbid Fraction Profile with Corn Syrup ¢A»=.018,'-'.65 cm/sec ............. Vbid Fraction Profile with Corn Syrup W.024,=.54 an/sec ........... Vbid Fraction Profile with Corn Syrup “-0025, -074 all/Sec ooooooooooooooooooooooo oooooooo Vbid Fraction Profile with Corn Syrup “-0036, -.59 Wsec 00.0.00...OIOOOIOOOOOOOOOOOOOOO Vbid Fraction Profile with Corn Syrup new, -.39 cal/sec ...... Vbid Fraction Profile with Corn Syrup (lb-.051, =.62 cm/sec .......... Vbid Fraction Profile with Corn Syrup . “-0058, -077 Wsec OOOOOOOOOOOOOOOOOOOOO 0000000000 Vbid Fraction Profile with 2.5 wt Percent Separan Solution, Oo=.018, =.7 cm/sec ,Vbid Fraction Profile with 2.5 wt Percent Separan Solution, ¢o=.02, =.82 cm/sec .......... Vbid Fraction Profile with 2.5 wt Percent Separan Solution, ¢o‘.022, =.4 cm/sec ........... Vbid Fraction Profile with 2.5 wt Percent Separan Solution, ¢b=.024,=.58 (In/sec .......... Page 72 75 79 87 88 89 90 91 92 93 95 . 96 . 97 Figure Page 5.14 void Fraction Profile with 2.5 wt Percent Separan Solution, ¢b8.025, 8.54 can/sec ......... 99 5.15 Vbid Fraction Profile with 2.5 wt Percent Separan Solution, «0.8.032, 8.78 cm/sec ......... 100 5.16 Vbid Fraction Profile with 2.5 wt Percent Separan Solution, ¢n8.032,81.17 cm/sec 101 5.17 Vbid Fraction Profile with 2.5 wt Percent Separan Solution, $8.031, 81.18 cm/sec . . . . . . . . 102 5.18 Vbid Fraction Profile with 2.5 wt Percent Separan Solution, 4558.034, 8.81 cam/sec .. ....... 103 5.19 Vbid Fraction Profile with 2.5 wt Percent Separan Solution, 458.035, 8.6 (In/sec 104 5.20 void Fraction Profile with 2.5 wt Percent Separan Solution, (0.8.077, 81.38 cm/sec. 105 5.21 Vbid Fraction Profile with 3.5 wt Percent Separan Solution, $8.026, 8.93 cm/sec .. ....... 108 5.22 Vbid Fraction Profile with 3.5 wt Percent Separan Solution, $8.032, 8.32 cm/sec ......... 109 5.23 Vbid Fraction Profile with 3.5 wt Percent Separan Solution, 4958.035, 8.91 can/sec ......... 110 5.24 Vbid Fraction Profile with 3.5 wt Percent Separan Solution, W035, 8l.5 cm/sec ......... 111 5.25 Vbid Fraction Profile with 3.5 wt Percent Separan Solution, ¢b=.04, 8‘.82 cm/sec ......... 112 5.26 void Fraction Profile with 3.5 wt Percent Separan Solution, $8.043, 81.29 an/sec ........ 113 ix Figure 5.27 5.28 5.30 5.31 5.32 6.1 6.3 6.4 .VOid Fraction Profile with 3.5 wt Percent Separan Solution, 43:28.0“. 8.71 an/sec . ......... Vbid Fraction Profile with 3.5 wt Percent Separan Solution, 4298.049, 8.5 cm/sec ..... Vbid Fraction Profile with 3.5 wt Percent Separan Solution, $58.05, 8.4 cm/sec ............ void Fraction Profile with 3.5 wt Percent Separan Solution, 49.8.05, -1.18 can/sec ....... Vbid Fraction Profile with 3.5 wt Percent Separan Solution, cm, 8.056, 8.73 (In/sec . ......... void Fraction Profile with 3.5 wt Percent Separan Solution, cm, 8.057, -.65 cal/sec .. ........ Characteristic Dimension of Dilute Suspension in Simple Steady Shear......................... Relative Viscosity vs Shear Rate for Rigid Sphere suspensions, ¢3002 ......‘C................ Relative Viscosity vs Shear Rate for Rigid Sphere Suspensions, 498.05 .......... Relative Viscosity vs Shear Rate for Rigid Sphere Suspensions, $8.07 Relative Viscosity vs Shear Rate for Rigid Sphere Suspensions, ¢=ul ..... ................. ... Relative Normal Stress Coefficient vs Volume Fraction for Rigid Sphere Suspensions in SCMC.. Appnmnt\HsaxnxylkuioxnzWEU.Shaantnfim .......... Apparait Viscosity Ratio vs Volme Fraction ........... Page 114 115 116 117 118 119 132 142 143 144 145 147 152 155 m In 6) pa 5" H #4 my Filtf k? IH NOMENCLATURE atomic mass number g/g-atom dimensionless Rivlin-Erickson tensors of exterior fluid bubble (or drop) radius, cm number density of bubbles parameter in equation (7.19) norma1.stress components function of x: and 6, (equat1on (6.44)) functions of 5; defined in Appendix C interparticle distance shear-thinning factor, function of K: and 6, (equation (6.44)) nth order hydrodynamic force bulk rate of shear strain, sec'”1 acceleration due to gavity, cm/sec2 length scale (over which suspension is undergoing a simple shear flow) gap height, cm radiation intensity unit tensor summation of surface integrals in equation summation of volume integrals in equation power law parameter Rivlin-Erickson tensors linear dimension of an element volume V N0 Ner2 8 lb .0 0'0 parameter in equation (7.19) Avagadro's number, atoms/g-atom number of drops in an element volume V primary and secondary normal stress dif- ference, dyne/cm2 unit vector in the r direction dimensional pressure dimensionless pressure number of radiation counts isotrOpic pressure Reynolds number, p<0>fl/'( particle Reynolds number, p<0>a/7Z position vector surface of single drop recoverable shear ration, Nl/Z’C nth order hydrodynamic torque bulk stress tensor extra stress tensor non-Newtonian part of stress tensor bubble rise time, sec thickness, cm dimensional velocity vector dimensionless velocity vector element volume volume of single drop axial velocity, cm/sec channel width, cm We Greek Letters filgqubfimhy Rdevxng-QQ £4, 14 14.14. Ac. average quantitites CHAPTER 1 INTRODUCTION The rheology of filled polymer materials has become a subject of considerable scientific and technical interest because of the growing use of composite materials. Gas bubbles, fibers and rigid particles are added to polymer melts or polymer solutions to reduce the production cost, to reinforce the material strength or to facilitate material transport. Among filled polymer systems, polymer foams par- ticularly exhibit unusual rheological behavior, such as a relative viscosity less than one even at low gas bubble con- centrations and a yield stress at moderate concentrations. Polymer foams, suspension of gas bubbles in a polymer matrix, are commonly produced in thermoplastic extrusion, injection molding and polymer devolatilization processes. The gas may be injected directly into polymer melt in the extruder or it may be generated by decomposition of a chemi- cal compound (‘blowing agent“) introduced into the polymer~ before extrusion. In polymer devolatilization, releasing the pressure on molten polymer containing dissolved monomer produces foams with five to ten percent by volume of monomer gas bubbles. The foam quality is determined by the distri- bution of bubbles (“closed cells“) in the polymer. This is obviously affected by any crossflow migration of bubbles in flow of these suspensions. Furthermore, control of these 2 processses requires some knowledge of foam viscosity. The cell size and structure are functions of rheological prop- erties of the polymer. With too high a polymer viscos- elasticity, the bubbles are not able to grow and the optimum yield of gas is not achieved. On the other hand, if the polymer viscosity is too low, the bubbles will burst and leading to open cells. The measurement of foam viscosity must be done in flow through a closed conduit, which is generally complicated by several factors, such as bubble rise, growth and migration transverse to the flow. Usually bubble migration proceeds much faster than the other two processes lending to a nonu- niform suspension. The inhomogeneity of bubble concentration in the flow makes it even more complicated to predict the flow behavior. Bubble growth, bubble migration and bubble rise are all associated with the rheological properties of medium fluid and bubbles. To have a fundamental understand- ing of foam flow in viscoelastic liquids, we start with a study of dilute suspensions, where bubble interactions are not significant, and the modeling is less difficult. The study of dilute polymer foams is of significance for polymer devolatilization.processes where monomer trapped in polymer during emulsion polymerization is removed by extrusion of the polymer melt into a vacuum. In general, the subject of bubbly suspension flow in non-Newtonian media involves three 3 areas - the dynamics of bubble growth, particle migration in non-Newtonian fluid and suspension rheology. l.A Dynamics of Bubble Growth Bubble growth upon pressure release is governed by the diffusivity and solubility of gas in the polymer and by the resultant pressure difference, interfacial tension and hydrodynamic resistance. Numerous theoretical studies have been reported on the growth or collapse of spherical bubble in inviscid, viscous and viscoelastic fluids. Street gt ‘21, (1971) investigate the bubble growth in a power law fluid associated with simultaneous momentum, mass and heat transfer with the assumption that gas concentration gradient is appreciable only in a thin film next to the bubble. The bubble growth rate is controlled by the diffusivity and con- centration of blowing agent, viscosity and the extent of shear-thinning. With higher shear-thinning fluids, a higher initial growth rate is obtained. Yang and Yeh (1972) studied the heat-induced collapse of a spherical bubble in incompressible Power law and Bingham fluids. Their numerical solutions show that in the Bingham fluids, a decrease in the yield stress results in an increase in the collapse rate; a decrease in power law index n results in an increase in the collapse rate for power law fluids. Bubble growth in a viscoelastic fluid was first studied theoretically by Street (1968), using an Oldroyd fluid 4 model, in which the fluid stress on the bubble surface is determined by the history of rate of strain. The driving force in this analysis is the pressure difference between the exterior and interior of gas bubble; diffusion effect and heat conduction are not included. The bubble growth rate predicted is faster initially and then slower compared to that in a Newtonian fluid, thus medium elasticity has a ' significant influence on the growth rate. In addition, Fogler and Goddard (1970) considered inertia effects on collapse of a bubble in a Maxwellian fluid at moderate and high Reynolds numbers. In a fluid with large relaxation times, they showed that bubble may either collapse or under- go oscillation about an equilibrium radius. Its conduct depends on whether the ratio of.ambient pressure to the elastic modulus of Maxwellian fluid exceeds a critical value or not. The bubble collapse process in a fluid with moderate relaxation timed experiences the same oscillation but only over a short period of time. At low Reynolds num- bers, the viscous force dominates the elastic effect, the collapse rate becomes critically damped, and no oscillation is observed. Zana and Leal (1975) incorporated mass transfer into their study on diffusion-induced collapse of a bubble in an Oldroyd 3 type fluid, and examined in detail the roles of viscosity, relaxation time, retardation time, surface tension and Henry's law constant. The bubble collapse rate is significantly influenced by the viscosity - an order of 5 magnitude change in viscosity may triple the collapse rate. The introduction of fluid relaxation and retardation times causes the bubble collapse rate to initially overshoot its corresponding steady state value and then approaches a slower rate later. As the bubble becomes smaller, interfacial tension functions as an additional driving force besides pressure difference, therefore, it enhances the collapse rate. Experimentally, Zana and Leal (1978) investigated the dissolution of carbon dioxide gas bubble in four aqueous Separan solutions. The mass transfer rate was significantly enhanced as a result of viscoelasticity. Pearson and Middleman (1977, 1978) observed isothermal bubble collapse in polymer solutions to study the elongational rheology of viscoelastic liquids of moderate viscosity and high strain rate. Prud'homme and Bird (1977) derived dilatational prop- erties of suspensions of gas bubbles in Newtonian fluid, second order fluid and Goddard-Miller fluid by comparing the equivalent stresses at boundaries of a compressible cell and a second cell containing a dilating bubble. They showed that dilatational properties can be determined from the rheological properties of continuous phase and volume frac- tion of gas phase. 1.3 Particle Motions in Non-Newtonian Fluids Bubble migration in non-Newtonian fluids may result from the gradient of normal stresses around the bubble, 6 inertia, deformation of bubble and hydrodynamic interactions between bubbles and boundaries. The elasticity of polymers plays a principal role among all the factors, in bubble migration and bubble rise in the non-Newtonian fluids. Par- ticle motion in elastic fluids have been investigated, taking into account inertia effect (Caswell and Schwarz (1962)),4 wall effect Caswell (1962, 1970)), and far field hydrodynamic interaction between particles (Caswell (1977), Brunn (1977)). In general, the particle dynamics are solved by a regular perturbation expansion in the weissenberg number starting with known results from a Newtonian medium. Nevertheless, this analytical approach is useful in determining the first order or second order correction due to elasticity if the medium fluid is represented by a so called Rivlin and Ericksen expansion (1955) or order fluids. The stress rela- tion of these order fluids is given by: I - Oéé] +00? +Nadf+013£+o¢(drfia +I4Z'AI) + OH 7‘r 4") .4' + higher farms (x. /) where the Rivlin-Ericksen tensors fiyv are defined by §:-vg+vg’ (42) fin-fig”+éfl-"VS*V-gtfim (43) 7 For a simple shear flow, it can be shown that é”"'° ‘ ' for ”>2 (”4) and the three material functions can be related to 0Q, by 172(7) =(0" (as) N:(>")= 'o,-rn-(2a1+oz,)>>‘ (he) Nah") =- Tu-‘C'33- «27‘ (A 7) there 1: is the stress function, IV.,Ab. are normal stress functions of shear rate;7 with flow direction denoted by’/ and the direction of velocity gradient denoted by 2 . The Rivlin-Ericksen model is a polynomial expansion in velocity gradient so that the perturbation method can be applied to solve steady flow (rheologically slow flow) problems. The dashed-underlined terms, quadratic in the velocity gradients are referred to as the second order fluid model. The mate- rial functions in this model are invariant with strain rate. With all the terms that are cubic in the velocity gradient included, the relation is referred to as third order fluid model. This model is better in that it may be used to represent a shear-thinning fluid, with some elasticity. In general, these order fluids are not suitable for computa- tions involving unsteady flow. The study of particle motion in non-Newtonian fluids can be discussed in two areas - unbounded and bounded flow. Unbounded Flow A study of the deviation from Stokes' law due to visco- elasticity in the unbounded quiescent fluid was first report- ed by Leslie and Tanner (1961) who evaluated the drag force on a sphere in an Oldroyd fluid, and formed the correction to 0(We2). Giesekus (1962) solved for thesimultaneous translation and rotation of a rigid sphere in a third order fluid, his result for the drag force is consistent with Leslie's result. wagner and Slattery (1971) included the effect of inertia and particle deformation on particle shape in the analysis by considering a third order fluid drop in another third order medium. For the motion of a spherical drop in a second order fluid undergoing a steady shear flow, the velocity and pressure fields were derived by Peery (1966). For particle-particle interaction, the hydrodynamic force and torque have to be solved by Oseen method or Green's function method apart from perturbation expansion. Brunn (1977) employed Oseen far field approximation for the particle-particle interaction in a homogeneous second order fluid to predict the relative orientation of spheres. Caswell (1977) used Green's function method, in which the force can be expressed by a single integral equation which contains information of governing equation and boundary con- ditions. The integral equation is manipulated and reduced to a series of auxiliary boundary value equations in an infi- nite domain, the interaction is 0(d'2), where d is the 9 distance between particles. For large d, the viscoelasti- city induces repulsion or attraction between two spheres depending on the normal stress coefficients. For a slender axisymmetric rod-like particle in a simple shear flow, the non-Newtonian force and torque are evaluated by Leal (1975) using reciprocal theorem of Lorentz which allows the calcula- tion of second order forces from the first order (Stokes) flow field. The force correction is found O(We) rather than O(We2) as for sphere. The variation appears to be the characteristic of particle lacking of fore—and-aft symmetry. The motion of a slightly deformed sphere in the second order fluid was studied by Brunn (1979) using direct approach of solving the velocity and pressure fields in the fluid of O(We). Experimentally, Chhabra 35 51. (1979) employed a Boger fluid to examine the effect of fluid elasticity on the drag coefficient alone. For OeWes 0.1, no significant deviation from Stoke's law is observed. For We:>.l, the drag reduces continuously with increasing We and reaches an asympotic reduction of 26% at We >o.7 . Bounded Flow The particle motion in the limit of finite boundaries is more important than in the unbounded flow, because all the experimental observations have to be performed in a closed conduit. The most significant case of the wall 10 effect is the induced particle crossflow migration in the Couette viscometers and in the Poiseuille flows. Caswell (1972) used Green's function method to obtain the first order wall effect, the correction is independent of the par- ticle geometry and its surface boundary condition. Sigli and Coutanceau (1977) conducted experimental investigation of the effect of finite boundaries on a slow laminar isothermal flow of a viscoelastic fluid around a spherical obstacle. The drag reduction is enhanced by the wall effect and'becomes even more significant as the elasticity of the fluid increases. The observation of particle migration was first reported by Segre and Silberberg (1962). They found that rigid spheres in a unidirectional Newtonian flow migrate to an equilibrium position which is about .6 radius away from the wall at moderate Reynolds number. Later, Ho and Leal (1974) showed that in a Newtonian fluid, the occurrence of particle migration is strictly a result of inertia effect. The particle migration in Non-Newtonian fluids was observed by Karnis and Mason (1966), Gauthier st 31, (1971 a, b) and Highgate and Whorlow (1969) and their results are summarized below. In the Couette flow of viscoelastic fluid, neutrally buoyant solid spheres migrate toward the cylinder wall but neutrally buoyant Newtonian drops migrate away from the wall to an equilibrium position in between. In the Poiseuille flow of viscoelastic fluids both spheres and drops migrate 11 away from the wall and settle at the flow axis. However, the particle migration precedes differently in pseudoplastic fluids, i.e. the spheres migrate toward the wall while drops migrate to an intermediate location in between the wall and flow axis in Poiseuille flow. Particle migration is govern- ed by the equation of particle motion and by the reversi- bility condition of Saffman (1956). The reversibility con- dition means the particle motion is reversible regardless of time and flow direction. This property can be destroyed in the presence of inertia effect, wall effect, particle shape and normal stress. Chan and Leal (1979) studied cross flow migration of a deformable second order fluid drop in a Poiseuille flow (Figure 1.1) of another second order fluid with the help of Lorentz reciprocal theorem. To take the bounding walls into account, the method of reflection was used with an asymp- totic expansion for a/h4o< l where h is the gap of flow channel. The dimensional migration velocity for a Newtonian drop was derived as «r (lh U Lg / Us - ( fl?__) “#513.” of) 3/5'(2+5KF{4+k)(/+K)5 {(25-60 "' Io 952 K + 2525’sz + 24606 £3+ /oqqr.<“+ 15751") + e. ( 5920 4- 27588; 4- 6334/ t‘+ 706.26.614- 32940 K+4735k57} a / / _ .. ’ io-l(%’)(‘g') (/+k)’(2+3k) {42(2+3K)(4+/<) {/3 ‘36; - 73K—2afijv-L‘j—g—{5(88k+3&‘)} (/.8) r-—"‘< (N wwncnm H.H moroamnnn 0m mannpowo.»s newmocwwwo mHOt 13 where <0. , 90.: are the primary and secondary normal stress coefficients, 7' is the medium viscosity, 6) is the ratio of normal stress coefficients -4’//2( over the gap and the apparent viscosity may be computed for observed values of 1;”, ¢‘ and 4., to test the bulk stress relation for the core and to predict the trends for apparent viscosity. 2.C Objectives The objectives of present study are: l) to verify the existence of a stationary, uniform, bubbly core and a bubble-free wall layer in the viscoelastic two-phase flow. 2) to correlate the effect of viscoelasticity with the development of bubble concentration profiles. 3) to develop a bulk stress relation for dilute suspension of neutrally buoyant, uniform size, spherical Newtonian drops in viscoelastic liquid medium. 4) to test the bulk stress relation under conditions when the two-phase flow consists of a bubbly core and wall 28 layers. 5) to predict the trends for apparent viscosity from observed values of wall shear stress, core thickness and bubble concentration in the core. To approach the first two objectives, we designed an experimental apparatus which allows us to obtain a slow steady unidirectional and isothermal flow through a slit. A radiation gauge was implemented to measure on-line density variations transverse to the flow. The experimental setup along with procedures for preparing gas bubble suspensions and for obtaining bubble concentration profiles are described in chapter three and four. The profiles measured with two different Separan solutions and Newtonian corn syrup as media are compared and effects of non-Newtonian behavior on the development of concentration profiles are correlated in chapter five. In chapter six, the classical volume average method is utilized to achieve a theoretical relation for the bulk stress of a dilute suspension of neutrally buoyant, spherical Newtonian drops in a second order fluid undergoing a simple shear flow. The effect of medium elasticity on the shear viscosity and normal stress coefficients is discussed. Finally, in chapter seven, with all the knowledge pursued, the predicted trends for apparent viscosity are examined in the light of data. An overall review on the present study and recommendation for future research in the polymeric two- phase flow are discussed in chapter eight. CHAPTER 3 EXPERIMENTAL ARRANGEMENT AND MATERIALS The bulk properties of gas bubble suspensions in visco- elastic fluids are dependent on temperature, viscoelasticity of medium fluids and bubble bulk concentration. In addition, the flow is complicated by bubble growth, bubble migration and bubble rise. This investigation focuses only on the effect of bubble migration on the bulk properties, and thus we limit ourselves to slow, isothermal, uniform size gas bubble suspensions in viscoelastic fluids. Based on these specifications, the apparatus and instrumentation were designed to permit on-line measurement of density variation along the flow. Solutions were chosen so that various fluid properties could be used to achieve low Reynolds numbers and significant migration effects. A new gas bubble generation method was introduced to yield uniformrsize hydrogen gas bubbles at room temperature. The procedure for measuring flow rate and pressure gradient is described in this chapter. The calibration and procedures to obtain the two- dimensional bubble concentration profile are presented in the next chapter. 3.A Appgratus A schematic of the flow arrangement is presented in Figure 3.1 The gas bubble suspension was forced out of a 29 30 Figure 3.1 Schematic of Flow Arrangement The components are labeled with numbers. 1. 2. 3. 4. 5. 6. 7. 8. Tank Pressure regulator Flexible tube Gas release needle Circular static mixer Transition channel Slit section Stand 31 32 tank by compressed nitrogen, through a flexible hose, a ball valve and a static mixer, connected by a tapered transition to a rectangular channel. The flow as well as density measurements were carried out all along the rectangular channel by methods to be described later. The dimension and function of each device are listed as follows: Tank The tank is made of galvanized steel with a 55 liter capacity (72 cm high, 32 cm in diameter), which may be pressurized up to 30 psig. Mixing can be performed by a stirrer propelled by compressed air. The outlet of the tank is at the bottom, the cover on the top can be removed so that the tank can be cleaned at the end of each experiment. The pressure regulator on the cover is used to adjust the tank nitrogen pressure which governs the flow rate. At the beginning of each experiment, the downstream valve of the tank is always closed so that the nitrogen pressure inside the tank can be adjusted. Tygon tubing with 7/8' diameter connects this valve with gas releasing device. The flexi— bility of Tygon tube allows the movement of the flow channel in the flow direction to allow measurement of axial density variations. To ensure a uniform, small-size bubble suspen- sion, a ball valve with small opening is installed to screen off the uneven, large sized bubbles. Bubbles larger than .4 to .6mm gathering behind the throttled valve are drawn out 33 by a syringe needle inserted vertically through a rubber stopper which is tightened in a tee tube into the piping ahead of the valve. This device also eliminates surges in the flow which might occur with occasional bursting of large bubbles. From here on, bubble suspensions enter the static mixer. Flow Channel The flow channel comprises three sections in sequence: a static mixer, a transition channel and a slit channel, all are made of plexiglass. The transparency of plexiglass allows good flow visualization. The static mixer (Figure 3.2) consists of a circular pipe, 30.5 cm long, 3.8 cm in diameter, within which are fixed a series of triangular ele— ments arranging in a spiral fashion. These elements are aligned perpendicularly to the flow direction as suggested by Middleman (1977), to create a rotational flow and there- fore achieve a radial mixing. The mixing destroys the in- homogeneity caused by bubble migration and leads to a homo- geneous suspension before entering the slit section. The transition channel, 8.9 cm long, connecting the static mixer and slit channel, is built in the tapered shape to eliminate secondary flow near the entrance. The cross sectional area of these three sections are the same to retain a constant velocity. These individual flow compartments are bolted together and mounted on a horizontal stand with axial guides 3's aanga 19X!“ 011933 34 «Macao u.u enmsmwnwo: nsmasmw use renewamcpmn nzmsaow 36 to allow longitudinal movement of the flow channel. They can also be taken apart for cleaning. The two most commonly used flow channels for Poisseuille flow structure are circular tube and rectangular channel, the latter corresponds to parallel plates. From instrumental and photographic considerations, the rectangular channel offers a better geometry for two-phase flow study. It provides a uni- form cross section, so that the distance which the gamma rays penetrate through the suspension is constant along the trans- verse direction. The flat walls eliminate optical distor- tion. This rectangular channel (Figure 3.3) is 63.5 cm long, .9 cm wide and 1.27 cm in height. The upper plate is fasten- ed to a U shape open channel, which is removable for cleaning. The thickness of the top and bottom plates is .635 cm and the side walls are 2.03 cm. The channel can sustain pressure up to 25 psig. With aspect ratio (W/h) equal to 7, the flow in- side the channel can be approximated as an ideal Poiseuille flow between parallel plates (cf. Middleman (1977)). The entrance length for the flow to be fully developed is .5 cm (cf. Middleman (1977)), which is 1% of channel length. The entrance effect is negligible. A pressure transducer is located 2.5 downstream from the entrance. 3.3 Materials The gas bubble suspensions were prepared in three dif- ferent media - a Newtonian corn syrup, and two non-Newtonian 37 aqueous solutions of polyacrylamide (Separan AP-30) con- taining 2.5 and 3.5 weight percent respectively of the polymer. The viscosity 7 , the primary normal stress coef- ficient 41, , and recoverable shear ratio, SR: (ad/z (all evaluated at a shear rate of 1 sec“1 for the Separan solu- tions), the density {9 and interfacial tension cr are listed in Table 3.1. The methods and procedures of preparing solu- tions as well as measuring these media properties are described in the following sections. Corn Syrgp: Syrup is a well known Newtonian solution. Crystal syrup, a commercial product of A.E. Staley Manufacturing Co., Illinois, was used to prepare the Newtonian fluid. The contents in crystal syrup are water, corn syrup, sugarand citric acid. The viscosity of this commercial syrup is too low to eliminate the bubble rise. To obtain a higher viscos- ity, some water was boiled off. The color of the syrup changed from clear to brownish transparency. This was caused by the over-heating of carbons in the glucose. How- ever, the heating process did'not change the Newtonian be- havior of the syrup. Aqgeous Separan Solutions Separan AP 30 is a synthetic polyacrylamide, manufac- tured by Dow Chemical Co., with a weight average molecular 38 emUHm w.w mHOtmnnnmm 0n zooms IWWwamnw zonamH mnnmmm nommmwmwman . mcnmmnm mkamlmmnu menocmnmUHm cosmHnK emampos and the bulk void fraction ¢ are specified in each figure. 5.A Profiles in Corn Syrup. Figures 5.1 to 5.9 show that in the Newtonian corn syrup, bubbles are uniformly distributed across the gap at both axial locations with different values of bulk void fraction and flow rate. The scatter of data about the bulk void fraction is due to the statistical error of radiation measurement. The applied wall shear stresses are above 32 N/mz, which is an empirical value reported by Prud'homme (1978), beyond which, he observed bubble migration. The data show that the “tubular pinch effect” due to inertia - observed by Segre SE 51. (1962) does not occur at ReynOlds numbers considered here. The ”tubular pinch effect“ involves crossflow migration of suspended particles toward an equilib- rium position at a distance of 0.6 radius from the flow axis at moderate or high Reynolds number (Ho and Leal (1974)). Reynolds numbers based on gap dimension (Reha 100194. / 7 ) are of order 10"3 and bubble Reynolds numbers (Rep-pa/1( ) are of order 10-4, so that inertia effects are indeed negli- gible. The other factor which might cause crossflow migra- tion of particles suspended in a Newtonian medium is shape distortion, the extent of which is governed by the capillary numberpfiyand the ratio of viscosities {/7 , where 7, 2’ are the viscosities of suspending medium and suspended par- 84 ticle,G-is the average rate of strain, (r is the interfacial tension. When the capillary number is very small, the bub- bles remain spherical irrespective of the viscosity ratio. The observation of uniform void fraction across the gap with corn syrup would suggest that there is very little shape dis- tortion. This is confirmed by photographs taken of the flow- ing suspension in corn syrup. Also, the capillary number is found to be between 0.1 and 0.2 for all runs made with the corn syrup system. Bounded walls also cause crossflow migra- tion, but this wall effect is negligible with the ratio of bubble diameter to slit dimension 2a/h-0.03(< l. The properties presented in Table 3.1 for both polymer solutions indicate the Reynolds numbers as well as the capillary numbers are of comparable magnitudes for all three media so that the factors of inertia and viscous shape dis- tortion of bubbles may be ruled out in all cases. Thus, any crossflow migration observed in the polymer solutions must be due to non-Newtonian effects - viz. the gradient of nor- mal stresses around the bubble (cf. Chan and Lean (1979)) and pseudoplastic effect (cf. Gauthier, st 31., (1971)). The interpretation of the data in the light of these two effects is discussed after presenting the trends in void fraction profiles with the polymer solutions. 85 F .04 - A .02 L ,. .. o o 3 . _ A O l L l I Q 07 .20 32 44 64 yaw?) Figure 5.1 void Fraction Profile With Corn Syrup ¢b '.0/8 , --- .65 cm/sec 86 F .04 - h 8 0 A .LI? - E? o. ‘1 1 1 J 1 Q .07 .20 .32 .44 .64- y (cm) Figure 5.2 Void Fraction Profile With Corn Syrup Cbb =.024 . 8.54 cm/sec 87 ‘i' 04 - "' a —& A 02 .. ° 3 ° Q .07 .20 .32 .44 .64. )Hch Figure 5.3 Vbid Fraction Profile With Corn Syrup 950 ‘025 . 8 .46 cm/sec 88 .06 r" .04 - A A o . “ s O O .02 - Q .07 .20 .32 .44 .64 y(cm) Figure 5.4 Void Fraction Profile With Corn Syrup 9b!) '.033 , - .88 cm/sec 89 .06 r A _ 2 o o .02 - '1. l l l I ¢ .07 .20 .32 .44 .64 yum) Figure 5.5 Vbid Fraction Profile With Corn Syrup ¢b-.034 , 8,74 cm/sec 90 .06 [- .04 - 6 0 6 a 75’ C) .02 - 1 1 l I g .07 .20 .32 .44 .54 . yum) Figure 5.6 void Fraction Profile With Corn Syrup ¢b--036 , 8.59 CID/sec 91 .05 F 6 3 A 04 " A o O .02 - 1 1 1 1 Q .07 .20 .32 .44 .64 yum) Figure 5.7 void Fraction Profile With Corn Syrup Q '.042 , <1!) {39 cm/sec 92 r .06 - A - 0 a 8 fr .04 L .02 ~ @ .07 .20 .32 .44 .64 yam) Figure 5.8 VOid Fraction Profile With Corn Syrup ¢>b =.05/ , -.62 cm/sec 93 .06 - o 6 “5* C) F A :9 .04 - L .02 - F l l 1 L a .07 .20 .32 .44 . .64 y(cm) Figure 5.9 void Fraction Profile With Corn Syrup ([5,, -.058 , -.77cm/sec 94 5.8 Profiles in 2.5 wt. Percent Separan Solution Void fraction profiles measured with the 2.5 wt. per- cent Separan solution are presented in Figures 5.10 to 5.20. The profiles indicate that bubble migration is toward the centerline of flow channel; the number of bubbles decreases next to the wall. The development of the void fraction pro- file is quicker at higher flow rates and higher void frac- tions. For instance, at an average linear velocity of 0.82 cm/sec and a bulk void fraction of 0.02 (Figure 5.11), the downstream profile is sharply peaked at the center while the upstream profile shows little change from the uniform bulk at entrance. As the bulk void fraction is increased to 0.032 with the same average linear velocity (Figure 5.15), both upstream and downstream profiles are seen to be peaked, the downstream one being sharper. With about the same void fraction ‘¢b a 0.034 (Figure 5.18), the upstream profile is nearly uniform as the entrance profile, but downstream pro- file develops into one that may be presented by a uniform bubbly core and a bubble free wall layer. Similar trend can be observed in Figure55.l3, 5.14 and 5.19, where the average linear velocity is about .55 cm/sec. As void fraction in- creases from 0.024 to 0.035, the void fraction profile develops from a peaked one to one that may be presented by a uniform core at downstream locations. With the same void fraction, for instance, .¢% = 0.032 (Figure 5.15), as the 95 .04 r .02 - 2 6 e ‘0 A 1 L L 1 Q .07 .20 .32 .44 .64 ylcm) Figure 5.10 void Fraction Profile With 2.5 wt Percent Separan Solution %-_0/8 . -,7 cm/sec 96 .04 1" 8 .02 o e o - a ' L 1 1 1 Q .07 . 20 .32 .44 .64 y(cm) Figure 5.11 Void Fraction Profile With 2.5 wt Percent Separan Solution ¢b -.02 , - .82 cm/sec 97 Figure 5.12 F .04 b A b 65 ‘0 .02 ' 8 O O 1 1 1 1 20 .32 44 54 )Hcm) void Fraction Profile With 2.5 wt Percent Separan Solution ¢b-.022 , =.4 Cm/Sec 98 F .04 - .. 2 8 .02 - Z 0 A l I L, 1 Q .07 .20 .32 .44 .54 y(cm) void Fraction Profile With 2.5 wt Percent Separan Solution Qty-.024 . -.58 cm/sec Figure 5.13 99 .04 - A T‘ o 8 A .02 - ° 3 l l J l .20 .32 .44 .54 Q .07 Figure 5.14 veid Fraction Profile With 2.5 wt Percent Separan Solution $5 -.025 , -.54 cm/sec 100 .06 {- a 04 - 0 6 r3 " A 8 .02 - Q .07 .20 32 .44 .64 y( cm) Figure 5.15 Vbid Fraction Profile With 2.5 wt - Percent Separan Solution 91% -.032 , -, 78 cm/sec 101 .06 f .04 - A o 8 1— 8 8 .02 ' Q .07 ..20 .32- 44 .64 yum) Figure 5.16 void Fraction Profile With 2.5 wt Percent Separan Solution ¢b -.032 , -/./7 cm/sec 102 Figure 5.17 .06 1' .04 " e 8 a _ o 8 .02 - Q .07 .20 32 .44 .64 yum) void Fraction Profile With 2.5 wt Percent Separan Solution 95b -.03/ - /./8 cm/sec 103 F .06 " .04 - 3 g A ._ O f A .02 - Q .07 .20 32 .44 .64 )Hcm) VOid Fraction Profile With 2.5 wt Percent Separan Solution ¢b -0034 -.8/ cm/sec Figure 5.18 104 F .06 - 8 a q; .04 t 8 o .02 '- 1 1 1 1 ¢ .07 .20 .32 .44 ' .64 y(cm) Figure 5.19 void Fraction Profile With 2.5 wt Percent Separan Solution ¢b -.035 , -.6 M/sec 105 (D e .6 .08 - ° .3 .. A .06 *' .04 - .02 - 1 1 1 1 Q .07 .20 .32 .44 .64 yum) Figure 5.20 Vbid Fraction Profile With 2.5 wt Percent Separan Solution (by-.077 , -/.38 Cm/scc 106 average linear velocity increases from 0.78 cm/sec to 1.18 cm/sec (Figures 5.16, 5.17), the void fraction at both axial locations develops into a uniform bubbly core and a wall layer. With high bulk void fraction and high flow rate, the uniform core develops much faster and is observed at both upstream and downstream locations (e.g. Figure 5.24). The reproducibility of void fraction profile is satisfactory (see Figures 5.13 and 5.14, Figures 5.16 and 5.17), the sha- pes are consistent within the experimental error. 5.C Profiles in 3.5 wt Percent Separan Solution The void fraction profiles of bubble suspensions in 3.5 wt percent Separan solution are presented in Figures 5.21 to 5.32. In the case where the flow rate is very low (Figures 5.29 and 5.28), the bubble migration can still be observed, confirming that even very weak elastic forces can overcome the shear forces and contribute to crossflow migration towards the centerline of the channel. With the same void fraction, as the flow rate increases, the profile develops into a peak one or one with a uniform core. For instance, with bulk void fraction about 0.035, at average linear velo- city 0.32 cm/sec (Figure 5.22), the profiles at upstream and downstream locations do not deviate from the uniform entrance profile much, the migration is very slow. As average linear velocity increases to 0.91 cm/sec (Figure 5.23), the upstream and downstream profiles become peaked, as the velocity in- 107 creases further to 1.5 cm/sec (Figure 5.24), both upstream and downstream profiles indicate core - annular flow. The core void fraction is increased in the downstream location compared to the upstream one, which means the core thickness is narrower with increasing flowrate. Similar trends can be observed in the cases of bulk void fraction of 0.043 (Figures 5.27 and 5.26), with increasing flow rate, the uniform core gets narrower at downstream location. With the same average linear velocity, the formation of uniform core is enhanced by the increasing bulk void fraction. For instance, with average linear velocity about 0.8 cm/sec (Figure 5.23), at bulk void fraction of 0.026. (Figure 5.21), the profile of upstream shows little change from the uniform bulk at en- trance, the profile of downstream is peaked. As bulk void fraction increases to 0.035 (Figure 5.23), profiles at both upstream and down-stream locations are peaked. As bulk void fraction increases further to 0.056 (Figure 5.31), the pro- files at upstream and downstream locations are both uniform near the centerline. The downstream profile is narrower than the upstream one. With similar bulk void fraction and flow rate, the bubble migration effect in 3.5 wt percent Separan solution is more significant than that in 2.5 wt percent Separan solution (cf. Figures 5.16 and 5.23). The core thickness is narrower in the 3.5 wt percent Separan solution (Figure 5.26) than that of 2.5 wt percent Separan solution (Figure 5.17). The void fraction profiles develop 108 04 ” 9b a " (D O 2 72 .02 ’ 1 1 I 1 Q .07 .20 .32 .44 54 y(cm) Figure 5.21 VOid Fraction Profile With 3.5 wt Percent Separan Solution 95b .025 , -, 93 cm/sec 109 .06 F .04 - .02 ~ L 1 1, 1 Q .07 .20 .32 .44 .64 yum) Figure 5.22 void Fraction Profile With 3.5 wt Percent Separan Solution $11 --032. <2» -.32 cm/sec 110 r. .06 - " a O .04 - 3 A e h e .02 - J L l I Q .07 .20 32 .44 .64 yum) Figure 5.23 VOid Fraction Profile With 3.5 wt Percent Separan Solution Q -.035 , < U> -.9/ CID/sec 111 .06 " 1A 1A - C) 04 Q g ‘ o .02 - A Q .07 .20 .32 .44 .64 y(cm) Figure 5.24 Vbid Fraction Profile With 3.5 wt Percent Separan Solution gbb ...035 , - /.5 cm/sec 112 F .06 1- A "' 0 gb .04 ° 2 10 h a .02 - 1— L l l . I Q .07 .20 .32 :44 .64 y(cm) Figure 5.25 VOid Fraction Profile With 3.5 wt Percent Separan Solution @3204 . -.82 cm/sec 113 F .06 - a A 0 - c) C) C) .04 '- O A. .02 _ g .07 .20 .32 .44 .64 yum) Figure 5.26 void Fraction Profile With 3.5 wt Percent Separan Solution 9250 -.043 , - /.29 cm/sec 114 .06 L- A ' (D 65 g; .04 - ° _ ‘A .02 - Q .07 .20 .32 .44 .64 yfcm) Figure 5.27 Void Fraction Profile With 3.5 wt Percent Separan Solution (1):,- 045 -, 7/ cm/sec 115 r .06 - 6 _ e 8 ¢ 8 .04 ~ .02 — l - 1 l 1 Q .07 .20 .32 .44 .64 yum) Figure 5.28 void Fraction Profile With 3.5 wt Percent Separan Solution ¢b-.049 , -.5 cm/sec 116 .06 - A. o 6 ‘ ti C) .04 *- A .02 - Q .07 .20 32 .44 .64 )Hcm) Figure 5.29 Vbid Fraction Profile With 3.5 wt Percent Separan Solution qbb -.05 , -.4 cm/sec 117 Figure 5.30 , .05 '- e 8 8 (D ‘0 .04 ‘ .02 - l l l i .20 .32 .44 .64 y(cm) void Fraction Profile With 3.5 wt Percent Separan Solution ¢b -.05 , -/./8 cm/sec 118 .08 r o .06 " o o O ‘0 — a 925 .04 - .02 *- l l J I Q .07 .20 32 .44 .64 y(cm) Figure 5.31 Void Fraction Profile With 3.5 wt Percent Separan Solution Q-OSG , =. 73 607/566 119 Q .07 Figure 5.32 .08 *- ._ A A 06 _ O O O ‘0 r- 0 . ,Ch? r .02 - 1 1 1 1 .20 32 .44 .54 y(cm) Void Fraction Profile With 3.5 wt Percent Separan Solution 9b.!) -.057 , =65 cm/sec 120 at a lower flow rate. (~0.7) in 3.5 wt percent Separan than 6U>~1.0) in 2.5_wt percent Separan solution. The formation of uniform bubbly core is affected more by flow rate than the bulk void fraction. It would seem that high elasticity of the medium is responsible for greater nonhomogeneity and quicker core formation in this medium. 5.0 Correlation of Two-Phase Structure With Medium Properties Previous studies (Gauthier 35 21° (1971a,b)) have estab- lished that particle migration is toward the flow axis in viscoelastic fluids and toward the wall in pseudoplastic (i.e. only shear-thinning) fluids. Separan solutions are highly elastic but they also have characteristics of a pseudo- plastic fluid. Although normal stresses gradients and shear- thinning effects on bubble migration cannot in general be separated easily, the trends discussed above may be corre- lated by means of a recoverable shear ratio - i.e. N1/21: . where N1 is the primary normal stress difference and 17 is the shear stress. Chan and Leal (1979) derived the migration velocity of a deformable droplet in a second order fluid undergoing a unidirectional flow. For an inviscid drop, the migration velocity can be written as ..- ‘0' .9: .. V (gm-$13)? [ .508( WW2) .58746] (5, I) 121 which indicates the migration velocity is a function of shear rate, the gradient of shear rate, drop size, medium viscosity and elasticity. In general, the secondary normal stress coefficient. #a is usually about 10% of primary nor- mal stress coefficient 4% and is negative. So assuming (A. =- -.1w, , the migration velocity can be simplified to al v - .13 5%452929519) :3 .260( 2—’—V—’)(dd;fi) 2.264153(T‘d;3) (532) The migration velocity is proportional to the recoverable shear ratio SR. This was derived only for a single bubble in a second order fluid. It is worthwhile to note that recoverable shear ratio SR is maximum at the wall, where bubbles are quickly depleted. The value of this recoverable shear ratio at the wall, may be obtained from measurements of the wall shear stress, using the curves for pure polymer solutions of N1 and‘C’ vs 7' in Figures 3.5 and 3.6. Thus, a qualitative correlation between the recoverable shear ratio at the wall SR," and the profile development is attempted in Table 5.1. In cases where a uniform central core is formed, the core thickness COMPARISON OF VOID FRACTION PROFILES 122 TABLE 5.1 2.5 wt Percent Separan Solution r fif I ibdrum Dmmflrom I F 19"" W8¢ ""2 5a,. no em) (47 cm) 5.10 .018 .7 44.26 .329 no verieflon uniform core 5.11 .02 .82 44.57 .329 no variation peaked 5.12 .022 .4 38.2 .292 pnkd pakd 5.13 .624 .50 14.51 .329 poked pukd 5.14 .25 .54 40.1 .268 no verieflon uni form core 5.15 .032 .78 44.0 .276 old pukd 5.16 .032 1.17 49.7 .396 pokd naked 5.17 .031 1.18 49.7 .396 pcfikdf fl uniform our. 5.18 .234 .81 44.4 .396 nomvarieflon uniform core 5.19 .035 .6 40.1 .268 no variefion uniform core 5.20 .077 1.33 50.9 .341 uni formfcore uniform core f 123 TABLE 5 . 1 COMPARISON OF VOID FRACTION PROFILES 3.5 wt Percent Separan Solution ' Upstr; Downstrom Figure cm/ac Him: 5,... no cm) (47 cm) 5.21 .02: .93 51.3 .443 no variation no variation 5.22 .032 .32 20. .27 no variation no variation 5.23 .03: .91 49.7 .441 uniwforvm oora unIform core 5.24 .0351 1.5 49.7 .441 uniform core f uni form oar; 5.25 .04 .82 39.7 .36 poked ‘ poked V 5.26 .043 1.29 43.0 .375 "uni form core uniform cola 5.27 .045 .71 36.4 :345 no variation pakd 5.28 .049 .5 33.1 .328 uniform oore uni form oar; 5.29 .05 .4 24.6 .282 no variation naked 5.30 .051 1.18 38. .35 uniform oora uniform core 5.31 .0515' .73 35. :338 uni forfmfoora uniform core 5.32 .057f .65 39.7 .36 uniform are uniform oora 124 .Ac , may be estimated from the core concentration with the knowledge that expansion during the experiment is negligible. It is to be noted that the residence times corresponding to the two locations are different in different runs, as shown in Table 5.1. The recoverable shear ratios at wall (.28 to .47) of 3.5 wt percent Separan solution are in general higher than those of 2.5 wt percent Separan solution (.26 to .39), which account for quicker formation of void fraction profile in the former solutions. Higher values of SR,w (:> 0.35) lead to quicker formation of uniform central bubbly core, and a narrower core or a thicker bubble free layer. At bulk void fractions of 0.05 or less, the effect of SR,w dominates. At bulk fractions greater than 0.05, the for- mation of uniform core and wall layer is enhanced by the increasing bubble void fraction. .The effect of bulk void fraction on the profile dominates the medium effect in con- centrated suspensions has been observed by Karnis and Mason (1967) which may even change the velocity profile into a blunt one. However, in this study, the bulk void fraction is much less than that in the concentrated system, the effect of elasticity seems to be the predominate one. CHAPTER 6 BULK RHEOLOGY OF DILUTE SUSPENSIONS IN VISCOELASTIC LIQUIDS 6.A Introduction The experimental studies of suspensions of rigid or deformable, neutrally buoyant particles in viscoelastic liquids demonstrate that such suspensions are often macro- scopically nonhomogeneous even in extremely low Reynolds number flow as a consequence of the medium elasticity. Our observations indicate that flow may still be divided into a few macroscopically homogeneous regions so that a bulk stress relation for a homogeneous suspension is useful in each of the subregions. A relative viscosity relation for the suspension is required to explain the increase in - flow rate with increasing void fraction in such situations. The relative shear viscosity of suspensions of rigid, spherical particles in polymer solutions as well as polymer melts has been measured by several investigators including Highgate and Whorlow (1970) and Kataoka gt 2;. (1978). They conclude that for dilute suspensions at low shear rates, the relative viscosity increases with concentration but decreases with shear rate; however the effect of medium properties on this dependence was not clear from these studies. Han and King (1980) report that the relative shear viscosity of sus- pensions containing 10 percent of spherical (Newtonian) poly- 126 butene droplets in a 2 wt. percent aqueous solution of Separan is lower that one and decreases with concentration at shear stresses of 50-100 N/m2 over the low concentration range. The primary normal stress difference is reported to increase with concentration of disperse phase in suspensions of rigid spheres as well as droplets. The object of work presented in this chapter is to drive a theoretical relation for the bulk stress in dilute suspensions of neutrally buoyant, uniform size, spherical inclusions -- rigid or deformable -- with a viscoelastic medium under simple shear. The inclusions are assumed to be small enough to remain spherical, and large enough for exclusion of Brownian motion in the analysis. The visco- elastic medium may be described by the second order fluid model if the fluid motion is 'rheologically slow”, that is the fluid relaxation time is small but finite compared to the time scale of fluid motion. The disturbance velocity and pressure fields interior and exterior to a spherical drop suspended in a second order fluid undergoing steady shear far from the drop were derived by Peery (1966) for small Weissenberg numbers, omitting inertia. These fields are used to derive an average stress over a volume small compared to the dimension along the shear gradient direction, yet con- taining many particles, following the discussion of Batchelor (1972). The result is a third order fluid model for the suspension, which shows that the dependence of relative 127 shear viscosity on shear rate is associated with the elasti- city of the medium. This chapter starts with a statement of the microrheol- ological problem involved here and a summary of Peery's solu- tion to this problem. This is followed by a discussion of the integrals that arise on applying the averaging procedure to dilute suspensions in the second order fluid medium. The resulting third order fluid model is then discussed in the light of available data from viscometers. 6.3 Microrheology The microrheological problem here involves creeping flow of an incompressible, second order fluid around a single spherical drop of an incompressible, Newtonian fluid in the absence of external force or torque, with a steady shear flow imposed far from the drop. ~The relevant equations of continuity and motion are V'g-O , V'Q - O (6.1) 2'- -P5 + 11;: - £1922 wwws-g: “,2; for the continuous phase with constant viscosity I], and constant primary and secondary normal stress coefficients 4’... 4.5,. g, . e: are Rivlin-Erickson tensors given by é.’ .. VI! +1711? - (6.5) as -..q-v.z.-l + any Wye: (...) 128 The fields inside the Newtonian drop obeys the equations V-g’- O . V -g”- 0 (6.5) Q'- -P;z: + 7’: vu'+ vcr’) (...) where the primed variables corresponds to the disperse phase. The above equations may be nondimensionalized using the radius of drop “a” as the length scale, Ga as the velo- city scale, where G is the bulk rate of shear strain, and $6- as the (viscous) stress scale. The resulting dimensionless equations including boundary conditions are summarized below. Exterior flow field V'H-O . V'H-O - (6.7) 71--P£+fi. +ae.4.+ 2114,.47) (6.8) 5\ is the index of the medium elasticity -- a modified Weissenberg number, and 6,-->d) of a representative volume V containing a significant number of drops and over which the microrheological properties vary with simple shear appreciably; 4) H, the length scale over which the suspen- sion is undergoing a simple shear. With a«d , only nonin- teracting spherical drops are considered; with («H , we may evaluate average quantities at a “macroscopic point” -- that is, on the scale of the flow, the neighborhood of averaging may be treated as a point. This is useful in discussion quantities such as (4.4.) and <50 . The dimensionless bulk stress I may be written at very low Reynolds number, following Batchelor (1970) as ;' -VL[L-£wz_rdV+£wa1-’dv] (6.2/) where \b and EL denote the volume and surface area of a single drop. With the help of a tensor identity and the equation of motion, TT’ can be written as Z!"- V-(fl’x) — ( v- 1.1")2“ - V-(ZI’L‘) (6.22) Using the divergence theorem on the second volume integral and incorporating stress continuity at the interface, we 132 wwncnm m.~ AQAAqAAlAAIV nsmnonnmnwmnwn opamsmwos 0m Unwcnm monomomwo: w: mwamwm mnomma muons 133 obtain oth- I . 1% [ [v 7r dv +Zfs. £4 1(2):.6] (6.33) The first volume integral can be related to average quan- tities by the following identities . ._’.. A'dV (d!) V [IV-1V. é: dV 4.2!]. -1 ] - #[fwm 5.4V + {Lugs wands] (6.24). ..., - .. UV... .de + 21. 4:va <4: .51):- #[IV-Lib éo°éldv +wa aflé, dV] Substituting from (6.7). (6.11) and (6.24) into (6.23) and separating the isotropic terms (as qI), we obtain for the bulk deviation stress I + g; .. (5,) + Ae.<4.>i+A<;3:-:I> + 1'4 (6,25) The extra stress due to the disperse phase, I.“ involves surface and volume integrals taken over a drop. 22 --&-z{f..[a-(zr’t-wséM-mww] ds — [VJ Ae:( di- vg’- Vet}: g?) + Adi-4f] av} (6.26) The identity V.(_gé,)- 3.Vfi, +(V'3)fil , with V'S'O 134 has been incorporated into equation (6.26). The terms (5,4,) and (5:) in equation (6.25) deserve some attention here, for example, writing 4. , as the sum of the average <9) and a fluctuation a} , we obtain (Area) - (éo>-.<él>- (6,17) If one looks at these averages at a “macroscopic point”, the fluctuating quantities when averaged, (37- a?) should be of order ¢’ , where ¢ is the volume fraction of disperse phase. Since we are considering dilute suspensions, it should be reasonable to omit (3)-5}) , so that (111-£1): (éo>-<_A'I) (6,18) Similarly, fluctuating quantities involving second order terms in (4;) can be omitted, so that < 4.» «=4 -v<§.>+ <§.>-+ -<5,> (6.17) Substituting'for I; in equation (6.26) and recognizing that the disperse phase volume fraction is related to the number of drops N in the volume V by ¢-($§‘-9-‘)N/v (6.50) we obtain 73 -fi(§+§) (6.3!) 1.35 where £7 is a surface integral ~22 -f5. d$fl[(-P£ +Ao +A€aéa 1' A::-é:)[ - 2.6.559] (6.32) and .2; is a volume integral gs -fv. dv[ A67! aflvg’q- vg141)+Afif-é:] (5.33; Since the velocity and pressure fields occurring in equations (6.32) and (6.33) are available up to 0(a) from Peery (1966), the extra stress tensor 1? in our work in- cludes all terms up to 00“) . ~23 -Las{[21-3;+ALJL-(cso+.eaof, the largest deviations from data occur at shear rate of 2 sec'l, the error is about 6%. The departure at low shear rate may be due to the particle aggregation and disaggregation (cf. Matsumoto (1978)), which 142 ['2 r —--- Einstein Relation _. ----- Highgate _e_t_2_1_. (1970) Data Third Order Suspension Model l.l - ... ~ -" '— — -— rm— I.O ‘- 9 l 1 l l I i l I l J 0 2 4 6 8 I0 ’ G(sec") Figure 6.2 Relative Viscosity vs Shear Rate for Rigid Sphere Suspensions 9Qb'-.CE? 143 /.2 1' l.l 1.0 " t 9 l I l I l l I I l _J 0 2 4 6 8 I 0 Gmec") Figure 6.3 Relative Viscosity vs Shear Rate for Rigid Sphere Suspensions (Pb -.05 144 /.3 ‘r ,0 1 1 1 1 1 L 1 1 l 1 0 2 4 6 8 IO G(sec") Figure 6.4 Relative Viscosity vs Shear Rate for Rigid Sphere Suspensions $12-07 145 ,0 I I J I I I I I I J 0 2 4 6 8 I0 G(sec”) Figure 6.5 Relative Viscosity vs Shear Rate for Rigid Sphere Suspensions ¢b-./0 146 have been neglected in our derivation. The relative primary normal stress difference obtained from our model is only a function of void fraction, their relation is plotted in Figure 6.6, the slope of mm... vs ¢ is the constant coefficient. However, Highgate's data show the relative primary normal stress difference is also a function of shear rate. This can be understood because our model is derived to Owa‘), the shear rate dependent normal stress correction only comes if owe) is taken into con- sideration. Including one higher order of we means doubled terms have to be evaluated, it is not our major concern at present, since our attention is on the apparent viscosity problem. Three experimental primary normal stress ratios at zero shear rate are used to verify our coefficient, the errors are within thirteen percent. Both suspension visco- sity and normal stress different relations are proved to be capable of predicting suspension rheolOgical properties under reasonable circumstances. For viscous droplets, the viscometric data of Newtonian polybutene drops in 2 wt. percent aqueous Separan solution were presented by Man and King (1980). The reported relative viscosity ratio A: is 0.74 for 10 percent of droplets at a ‘shear stress of lON/mz. At this shear stress, the calculated 1”? is 0.8 by using relation (6.48) with component visco- sity ratio k, 0.015 and Weissenberg number, 1. As for nor— mal stress coefficients, the available data have too large 147 2.4 r 2.2- 2.0 t /.4 " /.2 - [.0 I J I L #1 I I I I _I 0 0.2 0.4 0.6 08 I .0 95 Figure 6.6 Relative Normal Stress Coefficient vs volume Fraction for Rigid Sphere A: Highgate 3551.. (1970) Data (at zero shear rate) 148 Weissenberg numbers to be verified by our model. The important aspect of this study is to show that shear-dependent behavior of dilute suspension in non- Newtonian medium can be attributed to the elasticity of medium in the absence of drop deformation. CHAPTER 7 PREDICTION OF APPARENT VISCOSITY IN TWO-PHASE FLOW We turn now to evaluate the utility of bulk stress relations such as those in chapter six, for predicting trends in apparent viscosity of gas bubble suspensions in viscoelastic liquids flowing through a plane slit. This will be done for cases where a uniform bubbly core and a bubble free wall layer are obtained. The velocity profile for such a core annular flow may be obtained with selected expressions for the suspension viscosity and the medium viscosity, using observed values of wall shear stress 12;, core volume frac- tion and core thickness cm. . An average flow rate over the entire gap may then be computed to yield an apparent viscos- ity with equation (2.1). It is appropriate to note here that there is a thin transition region, a bubble diameter, over which the void fraction changes from the core value to ‘that in the outer layer. we shall assume that the two layers are much thicker than the bubble size and carry out the analysis as through two immiscible fluids were flowing together. Some analysis with a linear variation of void fraction in a transition region indicates that the above assumption leads to answers matching within experimental error. In the following discussion, first, for a qualitative understanding, we explore the effect of medium elasticity on 149 150 the apparent viscosity by using the third order suspension model. Then a power law model is used to correlate observed apparent viscosity data. 7.A Third Order Suspension Model The shear viscosity for a dilute, uniform bubbly core with low (particle) Weissenberg numbers, and a constant medium viscosity, 7. is given by 7(.-7.[ I+¢c-F¢c(¢~G/7.)’] (7,/) where (Hg is the constant primary normal stress difference of the medium, ¢k. is the volume fraction of bubbles in the core and F is a number of order 1 depending on the ratio of normal stress coefficients. The computations here were done only for the 2.5 wt percent Separan solution, wherein the particle weissenberg numbers are 0.1 to 0.4 for this solution. ’(c-flx-rsbdl—zfi'fi] (7.2) ?-?0 (7:5) where 7; is 36.8 N/m2 at shear rate of l sec’l. The velocity profile (15(5) in the outer bubble-free layer is immediately obtained with equation (7.3) as m(g)=-_'%L§— —g£-_4_J5%_ (7.4) 151 The other profile U715) in the core region is obtained by solving for 007/33 from the nonlinear equation (7.5). 3 aw] _ 'aw|_ 217.91 =- The roots of this cubic equation may be found in a straight forward manner, using a computer program (MODEL I, given in the Appendix E). The root selected at any y is the one closest to velocity gradient obtained with the pure medium in the entire gap. Throughout the calculation, the other two roots were very different so that the selection was unambig- uous. The results of these calculations are presented in Figure 7.1 and 7.2. Figure 7.1 shows the variation of apparent viscosity with wall shear stress at bulk void frac- tions about 3%. It is seen from this plot that at very low shear rates, the shear-thinning factor is not important and the apparent viscosity is higher for the suspension accord- ing to Taylor's result. At higher shear rates, the shear- thinning factor associated with medium elasticity becomes significant and leads to decreasing apparent viscosity even to lower values than that for the medium. Figure 7.2 pre- sents a comparison of this theory and experimental data over a range of void fractions at higher wall shear stresses. The theoretical study here is qualitative correct; however, 152 /./ - k 770 I0 7700 9 _ o o .8 - o I I I J 30 40 50 60 Wall Shear Slress ( N/m‘) Figure 7.1 Apparent Viscosity Ratio vs Wall Shear Stress A Theoretical Data 0 Experimental 00 la 153 there is a discrepancy of about 10%. It is understandable since the medium used is itself shear-thinning. The results of the computation thus far indicate that the shear-thinning factor in the suspension viscosity associated with the elas- tic parameters of the medium lead to a reduction in apparent viscosity which displays the trends observed in experimental data. 7.3 Power Law Suspension Model In order to obtain a close quantitative fit of the apparent viscosity data, one must take into account the shear-thinning characteristics of the medium used. This is done with the following expressions: 7c- ’Z(/+¢e)(/-b¢bi'"’) (7,6) ?-K7”’ (Z7) It might be noted here that the expression in equation (7.6) is patterned on that in equation (7.2) with the exponent m and the coefficient b in the second bracket being adjustable. In the absence of medium elasticity, m and b are zero, equation (7.6) reduces to Taylor's expression. The parameters in equation (7.7) are measured directly, k a 36.8 N seen-l/m and n . .364 as given in Chapter 3. The velocity profile 05(3) in the outer bubble-free layer is 154 obtained as 2 J5 huh lav-l}; wand—EL) (,7—2, ——)[.9 414) ] (7.8) andtfiqy1 in the core is obtained by solving the shear rate in the following equation " 2m (/+¢¢)II-b¢tl-’-j- W)! ”’-| =0 (7.9) The shear rate can be solved by Newton-Ralphson iteration, using a computer program (MODEL II, given in the Appendix E). The best fit values of the constants b and m. was found to be 2.78 and 0.4, respectively. The resulting ratio of apparent viscosity is also plotted in Figure 7.2. The trends predicted are still the same; but the qualitative fit to data is close. This set of calculations indicates that the choice of power law viscous models for both medium and suspension with an additional shear-thinning factor for the suspension reproduces observed data for the apparent viscos- ity in two-phase flow of these suspensions. 155 /.0 p 6‘ “'~--. ___~ .9 L- O '— ”“"' 777 0 o 7740 o o .8 .- .7 - I I I I _I 0 .O/ .02 .03 .04 .05 <25 Figure 7.2 Apparent Viscosity Ratio vs Volume Fraction ---_. Third Order Suspension Model Power Law Suspension Model 0 Experimental Data canteen 8 CONCLUSIONS AND RECOMMENDATIONS 8.A Conclusion This investigation has focused on the isothermal devel- opment of two-phase flow structure due to bubble migration in a plane gap at Reynolds number of 10‘3 and on the apparent viscosity associated with the flow structure. The bubble concentration profile over a 1.27cm gap containing 0.04cm gas bubble suspensions in both Newtonian and viscoelastic ' fluids, with bulk void fractions of 0.05 or less, is measur- ed by a radiation attenuation technique. Although the effort involved in calibration and averaging of measurements is con- sidered, the radiation gauge yields a resolution of less than 0.01 g/cc, provides more reliable concentration data than techniques of previous studies, i.e. light scanning or counting on photographs (without depth). The Reynolds num- bers and capillary numbers based on bubble size of order 10“ for all three media. Similarly, the capillary number, i.e., the ratio of viscous to surface tension forces is small (about 10-1) for all three media. Measurements with a Newtonian corn syrup as a medium confirm that inertia and viscous deformation are insignificant in these experiments; so any nonhomogeneity observed in the polymer solutions may be attributed to their non-Newtonian behavior - elasticity 156 157 and shear-thinning. Gauthier st 31, (1971 a,b) observed that particles migrated towards the wall of flow channel in 7 pseudoplastic fluids. Our experimental data show that bubbles migrate towards the flow axis in the polymer solu- tions at the shear-thinning region. This observation con- firms that the elastic effect dominates over the shear- thinning effect which leads to particle migration towards the centerline of the channel as predicted by Chan and Leal (1979). A uniform bubbly core and a bubble free layer near the wall was observed in the channel at average velocity . above 1.0 cm/sec. Increasing flow rate and increasing void fraction both lead to a faster development of a uniform bubbly core and bubble free layers. The development of such flow structure is correlated well with the effect of medium elasticity and bulk void fraction. It appears that with higher recoverable shear ratios at the wall, (SR,w), the core develops more quickly and is narrower. Higher values of bulk void fraction lead to a wider central bubbly core or thinner bubble free layers. The two-phase flow may be de- scribed by a two fluid model comprising a bubble free layer (shear-thinning) fluid and a uniform bubbly core for SR,w:> 0.35. Next, to characterize the bubbly core, a theoretical relation was derived for the bulk stress in dilute suspen- sions of neutrally buoyant, uniform size, spherical Newtonian drops in a second order fluid medium. As a result 158 of this derivation including terms up to order wez, the shear viscosity of a suspension of Newtonian droplets in a second order fluid given by °a 71w - 7g[ ”if—:3) ¢+F(k.e.)¢>4-’5-L¢’.F] (a, I) This relation includes in addition to the Taylor result, a shear-thinning factor with F< 0 over all the c's. The pri- mary and secondary normal stress coefficients of the suspen- sion are given by Wasusp = Wm[/+f.(k.etl¢] (8:2) 413...... - w..[ /+ fence-49¢] (8.3) Both normal stress coefficients are increased with volume fraction¢ i.e. f,>o and fz>O for all the _/c's. These trends are borne out qualitatively by available viscometric data or rigid spheres and droplets. This model indicates that the shear-thinning factor for the suspension arises from the elasticity of the medium; however, the normal stress coeffi- cients are shear-rate-dependent if the averaging procedure includes terms up to order W63. The relative shear viscosity expression was tested further indirectly with data from this study on the two-phase flow structure comprising a stationary uniform bubbly core and wall layers. The core thickness and core volume fraction observed for this structure in a 2.5 wt 159 percent Separan solution were used to calculate an apparent viscosity. The apparent viscosity observed for the suspen- sion was up to 20 percent lower than that for the medium at a bulk volume fraction of 0.05. The predicted apparent vis- cosity reduction using relation (8.1) is only 10 percent be- low the medium value. This discrepancy is understandable since the medium used is itself shear-thinning rather than shear-rate-independent like a second order fluid. An empir- ical relation was proposed to correlate data 76.. 7((/+¢)(/+b¢er'") (8.4) 7 - Kim" (8.5) The expression in equation (8.4) is patterned on that in equa- tion (8.1) with the exponent m and the coefficient b being adjustable with the elasticity of the medium. 8.8 Recommendations Although there is some indication that the effects re- ported on two-phase structure as well as apparent viscosity are more strongly influenced by medium elasticity, the reso- lution into separate effects of shear-thinning and elasticity is not available. Therefore, Boger fluids (Chhabra, ggngl., (1980), Boger and Nguyen, (1978)) which has a constant vis- cosity over a wide range of shear rates, 0.5 sec"1 to 50 sec-1, have to be used for future theoretical research in 160 , this area. More viscometric measurements of bulk properties on uniform dilute emulsions with such fluids at low Weissenberg number are needed to confirm relations (8.1) to (8.3). Realistically, most polymer solutions are shear- thinning, derivation of bulk stress relation for such fluids has to use a third order fluid model. The interior and exterior flow fields to a spherical drop suspended is a simple shear flow of a third order fluid can be solved by the same approach that Peery (1966) employed for the second order fluid. By the same token, the effects of deformation and elasticity can also be included into the computation. For the concentrated suspension in viscoelastic liquids, the cell model (Frankel and Acrivos (1967), Goddard (1977)) can be employed to derive the bulk stress relation. The effect of medium elasticity on the rheological functions - shear- thinning or shear thickening (Han and King (1980)) can there- fore be investigated in the nondilute region. The effective viscosity of suspensions in viscoelastic fluids can be predicted by our proposed third order suspen- sion model with data obtained from Couette viscometers. However, it is noted that migration occurs in Couette device in the presence of velocity profile curvature. The inhomo- geneity existing in the wall layer near to the boundary would affect the measured viscosity. This can be correlated by incorporating the presence of the bounding walls. TGzeren 161 and Skalak (1977) have included the wall effect on viscosity of Newtonian suspension by considering the flow field around a sphere near a plane wall and using the surface averaging to evaluate the mean stress throughout the gap. Similar approaches can be carried out to correlate the third order suspension model by taking the wall layer into account. APPENDIX A SYMMETRY OF VOID FRACTION The symmetry of void fraction profile was tested in Newtonian and polymeric suspension flows. One case is pre- sented below. 6 .05- ‘ 1 S O - 9. 11 (a oa- u I t) O C) _ ¢ .05 - I .1 l I - .o/ - 1 1 1 1 1 .us' .355 .195 o .195 .315 .435 .635 W cm) 2.5 wt Percent Separan Solution ¢b"-042 ,—./cm/sec (fr-63 sec) The uniform core is observed around the flow axis, the con- centrations above the centerline are the same as the ones below, within the resolution of radiation gauge. 162 APPENDIX 3 EXAMPLE OF DATA ANALYSIS FOR DENSITY MEASUREMENT The following density calculation is based on the data collected for 3.5 wt percent Separan solution with dab-.035" and -. OI «cm/sec. (Figure 5.23). Radiation counts recorded at entrance at two axial locations are listed in the tables. 9.01 1 SLOT 2 Mr Entrance Llama}. Downstram Entrance Upstrom Dovnstrom 1 7953 81 25 8161 7521 7512 7480 2 791 8 ”53 7872 7434 7505 7492 3 m1 0 m2 81 69 7449 7549 7443 4 7871 7392 81 16 . 7472 7399 7622 5 3525 ”77 7930 7484 7468 7625 6 7897 so 71 8161 7402 7496 7434 7 m 8080 ”60 7494 7479 7484 8 N49 7927 $76 7540 7684 7581 9 7955 81 10 m5 ' 7540 7442 7521 10 81 14 81 47 $33 7534 7471 7534 1 1 7992 ”07 ”44 7614 7805' 7440 12 7994 m9 7962 7552 761 7 7645 13 7924 am 8 81 19 . 7592 7634 7590 14 i 1494 7465 1514 Man 7976 ”37.1 ”52 7507 7537. 5 7528. 9 Standard Deviation 66.7 75.6 ”.7 56.6 110.4 72.4 Mon 7976 N37. 1 $52 7507 751 7 7528.9 * Datum has to be rejected according to Chauvenet criterion 163 164 SLOT 3 $1.07 4 Hub: Entrance Upstram Dovnatr on Entrance _Upstrom Downstr am 1 6880 7102 6941 7750 7854 7810 2 6813 6858 6886 7826 7772 7690 3 6842 6953 6769 7737 7740 7750 4 7017 6883 6997 7731 7869 7830 5 6901 6949 6977 7871 7827 7719 6 6894 671 3 6974 7891 7721 7741 '1 war as9 an: 1mm 1mm 1mm 8 6949 7006 6936 7725 7623 7830 9 6971 6954 6839 7719 7841 7756 10 6882 6842 6861 7914 7554 7820 11 6763 6955 6908 7755 7698 7722 12 6948 6847 6984 7714 7783 7622 13 6853 6&3 6903 7865 7609 7726 Mon 6905 6903 6909 7788.9 7751 . 3 7754. 8 Standard Deviation 30.9 97.3 67.5 73.2 104.3 61.6 From the mean and standard deviation, we can reject the datum deviates too much from the mean. Using equation (4.12) and linear mass attenuation coefficients measured from the calibration, the density variations can be computed from the intensity difference between the downstream loca- tions and entrance. Slot 1 Upstream Apa-InIOOJ'Z l/747‘)/. 5:9 . .. 01/5 Downstream 4p.- -ln( 8052/7976”. 659 - mama Slot 2 Upstream 4,0- -/n( 7517/ 7907”. 5'57 - mecca Downstream 4,0- ../n( 75:23. 9/ ”071/. 7:7 .- - .oos' Slot 3 Upstream apn-Inwwa/eeosvzefl - .aoaa Downstream 4p--/n169o7/e90f)/.65‘5 --.Ool Slot 4 Upstream 4p - -/n(' 779/. a / ”mafia“: - .007 Downstream 4p . -/n( ”sac/77$. 9)/.7ou .. one The void fractions can then be evaluated by equation (4.7) and plotted. APPENDIX c FLOW FIELD OF A NEWTONIAN SPHERICAL DROP IN A SECOND ORDER FLUID UNDERGOING A SIMPLE SHEAR FLOW External Flow Field Zeroth Order Solution I ‘l’ s-[l 3%- - -=r..°.1 News 1.5+(‘5,’1+ I 12;: +(-:;.'-1££] First Order Solution 0;- W[i%‘3%‘ ‘%%';Za%)(§‘££)£+1'i%+3% 3 C -é‘;5+—°-“-H°‘=rr> Summation of 0(2): '92“; m- 35’5‘5 *fs. 4”"2“’5 *1}. dean-eras + 6.18.4...nzds -fs.( nu. +u.n)d.s - 5'I504‘(£°ém)d5 ”Iv. 530.510 dv - 511247.,- vg? + vgc-6.o)dv} A¢ Ur“? [( 5.5-05g“. zest/c - 2.22) + 61(2: aak‘4- [6.3! + 7.75)! 9..." 2,")! + [( £248., I.O7K +0.16) + é.(-o.5'5/c‘-4.36K +1.0?)(5’. 95) + [( -//. 0519— 293‘ -5.9/) +61( 523719-2841: +0.6S'X5- 2.") } 174 Summation of 0 (A‘) 52: {[5} 4‘3" * 3”!“ )1“: “'5 + e’fs. y'flsds - 6'L¢(Mé’° ,, 2.5”)“ -1“ 50.42, +47,.4;)dv- efi 745-5114“ 7954'; “.3512? + 5;. 7g? )dv} . {£7375- [(- 6.747K’- IZWK’v- A0. 341: -/o.lb) + 642.6 (‘4- 1:.w“+36.8k -6.5-6) ... 5,7 1.17184 m.z¢x‘+ 13.97:: + a. 81)] 6‘! 9:” £7 APPENDIX E COMPUTER PROGRAMS FOR SOLVING VISCOSITY RATIO Program Model 1 is for solving shear rate with suspension model(equations 7.7, 7.8) 1003 PROORAN NODELIIINPUTvOUTPUT) IIOg DIHENSION XX(500).V(500) I20? REAL HoNoH 1303C N IS THE POUER LAN INDEX 1408C 050 IS THE POUER LAN PARAHETER. K 1503C PG IS THE PRESSURE GRADIENT IbO=C C0 IS THE DULK VOID FRACTION I70=C D IS THE CORE THICKNESS 1806C 0 IS THE UIDTH OF CHANNEL I90'C H IS THEH HEIGHT OF CHANNEL 2003C R IS THE DUDDLE RADIUS 2IOIC DPoDN ARE THE BOUNDARIES OF CONCENTRATION TRANSITION 220-C DY IS THE INCREHENT IN V DIRECTION 2303C 00 IS THE UOLUNETRIC FLOU RATE UITHOUT DUDDLE SUSPENSIONS 2403C X IS THE SHEAR RATE 250-C V IS THE VELOCITY 260IC VOLUNE IS THE VOLUNETIRC FLOU RATE HITH DUDDLE SUSPENSION. 2708C 230' N-o364 290' 8.0278 300‘ “.04 3I0=I READioPGoCOOD 320* PRI"T.OPBOCOOD 330‘ PRINTIODOH 340' VSO'Jb-B 350' KK'IOI 3603 VEL-O. 3708 “-305.205. 380: VOLUNE'O. 390' ".1027 400' R-.025 4I0= DY'.OI .20. "o I 430' DP-(DORIIZo/H 4403 BH-(B-R112.IH 450' 00.2.IUU(PO/VSOISS(Io/NIIIH/2.)Dl‘(Io§2oIN)$N/(Io§2o3N) 460. D0 IOO I'IIKK ' 470' T'Io-DTIII'I) ‘00. YT‘Y‘“/2o Q90. N'(PODYY/VSO)IS(I.§N) 5003 IFIYoLEoDHIOOTOS 310' IFIYoLToDPIGOTOZ2 520- GOTOQS 330.5 C'CO 540- CPR'O. 5508 GOTOIO 560322 C'o75‘(“(H.Y/2o)‘D3o/3o§D¥(HDT/2o)1I2o§(RI.2o‘D¥IZo)ICHDT/zoI 5708 §+Dttlol3o‘RI!2.¥O“2.DRID3o/3oII(-I./R$I3o) 580= .C'CO 590: CPR'. 75‘I “(HST/2. )‘D'Jo IDlIIDTIRDDZo "0.32. )3( ‘I o/R‘.30, 600= CPR'CPRICO 6I0= GUTOIO 175 940- 176 620-25 C'Oo ‘30. CPR-00 640' OOTOIU 650'I0 CONTINUE 660'C NEUTON RAPHSON ITERATION OF SHEAR RATE 670-9 NEH-COIC3X3‘(H§N)+PGIYY/VSO/I109C,IDCIIo/N) 680- RE-ADSIXEU-X) 69°. IF‘REOLTOIOE’S’GOTONs 700' X'XEU 710' SOTO? 720'IO XEU-(POCTT/VSOIUIIIo/N) .730- x-quu 740.43 CONTINUE 7508 XXIII'XEU 760'IOO CONTINUE 770' VIII'Oo 780' D0 8 I'ZvKK 790' K'I'I OOO' VEL'VELOXXIK)IDTIH/20 SIO' VCIIIVEL 020- VOLUNE'VOLUNEOIV‘I)iVIKII/ZoSDTSHtU 830.9 CONTINUE 840' PRINT 600(V(IIOI'IOKK) OSO'C EVALUATION OF APPARENT VISCOSITY RATIO O60. AVSRIOOIVOLUNE 870. PRINT 7OOVOLUHEOAVSR BOO-I2 FORNATCéIEIZoévZXII . 890.40 FORHAT(6X9STIOIQXIIXDOIIXoDVISCODvIIXO‘STRESS‘OIIXORC!’ 900.50 FORHATIS(EI2-693X)I SIC-6O FORNATIIO(E1206OIX)I 920-70 FORNATCIO‘oDFLOU RATE'IorlzoSoIOXvDAPP VISCOSITY RATIO‘IFIZoSI 930' STOP END 177 Program Model 2 is for solving shear rate with suspension model (equations 7119, 7.20) I00' PROORAH HODEL2IINPUTIOUTRUTI IIO' DIHENSION A(4)92(3’9XX(500IOVI500) I20' REAL H IRO'C VSO IS THE VISCOSITY ISO'C PO IS THE RRESSURE GRADIENT 160'C A'S ARE THE COEFFICIENTS OF POLYNOHIAL EOUATION I70'C Z'S ARE THE ROOTS OF THE POLYNOHIAL EOUATION ISO'C D IS THE CORE THICKNESS I90'C U IS THE UIDTH OF CHANNEL 200'C H IS THEH HEIGHT OF CHANNEL 2I0'C R IS THE DUDDLE RADIUS 220'C DPODH ARE THE DOUNDARIES OE CONCENTRATION TRANSITION 230'C DY IS THE INCREHENT IN T DIRECTION 240'C O0 IS THE VOLUHETRIC ELOH RATE UITHOUT DUDDLE SUSPENSIONS 250'C X IS THE SHEAR RATE 260'C V IS THE VELOCITY 270'C VOLUME IS THE VOLUHETIRC FLOU RATE UITH DUDDLE SUSPENSION. 200'C 290' ACII'Io 300' AC2I'0o 3I0'I READSOPOOCODD 320' PRINTIoPOoC0oD 330' VSO'Sé-O 300' KK'IOI 350' VEL'0. 360' U'3o532054 370' VOLUHE'Oo 38°. “'IoZ’ 390' R'o023 000- ”Y'OOI NIO' v.01 020' DP'(D§RID2o/H 430' DH'ID-RISZo/H 440' O0'UIIRO/VSOIIH 450' DO I00 I'IOKK 460' Y'Io’DYIII‘I) 470' YY'Y‘H/Zo ' 400' X'IROSYY/VSOIDDIIoOH) Q90' IFIYoLEoDH’OOTOS 500' IFIYoLToDPIOOTOZZ 310' OOTOZS 520-5 C'C0 530' CPR'0. 540' OOTOI0 550.22 c.07SDI'IHCT/2oID‘JOISOTODIHDY/zoICCZOTIR‘Dzo'DDDZQIDTH‘Y/ZoI 56°. *TDDDJQIJO'ROCZQ'0‘20080030/30).(’I0’R‘.30) 570' C'C0 580' CRR'075DI-(HSY/20I..2¢§DDHDY*RID2¢‘DDCZ¢)3I’Io/RDISoI 590' CPR'CPR‘C0 600' GOTOIO 610825 6208 6308 640810 178 C80. c9830. OOTOIO CONTINUE 6508C SOLVE FOR SHEAR RATE "6608 6708 AC318T-1olCOlC)/(D1(2.lPG/(8.thVSO)1342.1C0IC) A14381-PGIHtY/(VSOI2.)l(Dt(2.IPG/(8.IH3VSO)1333.8C0IC) 6808C ZPOLR IS THE INSL SUDROUTINE FOR SOLVING ROOTS OF THE NTH ORDER 6908C POLYNONIAL EOUATION 7008 7108 7208 7308 740818 7508 760843 7708 780'100 7908 8008 8108 8208 8308 8408 85088 8608 CALL ZPOLRTA'NDEOOZOIER) PRINT 129(ZCJ10J'193’ XEU'ZCS) GOTO43 XEU'PO‘YDVSO X'XNEH CONTINUE XXTII'XEU CONTINUE VIII'Oo DO 8 I'29KK K'I'I VCL'VEL9XX‘KIIDY8H/2o VIII'VEL VOLUHE'VOLUHE§(V(I)§V(KIIIZoIDYIHtU CONTINUE PRINT 609(V1IIvI'IOKK) 8708C EVALUATION OF APPARENT VISCOSITY RATIO 8808 8908 900812 910840 920850 930860 940870 9508 9608 AVSR800/VOLUNE PRINT 709VOLUNEOAVSR FORNAT‘6CEI2.602X)) FORMAT16X93Y‘914X98X8011XoIVISCOtn11leSTRESSlo11XOICI) FORNAT(5(E12.693X)) FORNATC10¢E12o6v1X11 FORNAT¢IOI¢IFLOU RATE-toF12.5910XotAPP VISCOSITY RATIO81F12c5) STOP END LIST OF REFERENCES LIST OF REFERENCES Adams, F., and R. 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