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Chapter 1

INTRODUCTION

Laminar liquid jets possess several attractive advantages over othertypes of apparatus for fundamental studies of the mechanism of gasabsorption. To obtain a valid experimental test of unsteady statediffusion theory in a flow system, it is imperative that the fluiddynamics of the system be known accurately.

L. E. Scriven & R. L. Pigford (1959)

Liquid-liquid systems are important to workers in many areas of engineer-

ing, physics, and chemistry. A few examples of current interest are liquid-liquid ex-

traction equipment (Treybal, 1963, 1980; Jeffreys, 1987; Rousseau, 1987, Tsouris

and Tavlarides, 1990), liquid-liquid jets (Richards et al., 1993, 1994a, 1994b),

space applications in propulsion systems, life support, and storage (Kim et al.,

1994), coating flows (Ruschak, 1985; Christodoulou and Scriven, 1989), oil-water

mixtures in pipeline flow (Joseph et al., 1984), extrusion of polymers (Mavridis

et al., 1987), secondary oil recovery (Weidner and Schwartz, 1991), interfacial

tension measurements (Li and Fu, 1992), liquid bridges (Russo and Steen, 1989;

Slobozhanin and Perales, 1993), static menisci (Finn, 1986; Cuvelier and Schulkes,

1990), and interfacial rheology (Edwards et al., 1991).

In this dissertation we seek an improved understanding of the fluid me-

chanics underlying liquid-liquid extraction, a widely used unit operation in which

solutes dissolved in a liquid solution are separated by contact with another, gen-

erally immiscible, liquid. If different solutes in the original solution distribute

themselves differently between the two phases, a certain degree of separation will

develop, and this may be enhanced by multiple contacts in staged operation. If

1


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the liquids are left in contact long enough, mass transfer between the two phases

causes the solute distribution to approach an equilibrium condition. Extractors

can be constructed in many forms, which can be run batch or continuous, e.g.,

mixer-settlers, plate columns, packed columns, as well as more exotic designs.

Many articles and books have been written on the subject, and the details of cur-

rent design techniques are well documented in, for example, Treybal (1963, 1980),

Jeffreys (1987), and Rousseau (1987).

The efficiency and stability of the separation depend on the underlying pro-

cesses taking place in the stage. These are the fluid flow field, the thermodynamic

equilibria, and the mass transfer within and between phases. The research pro-

posed here focuses on the fluid mechanics alone, as once the flow field is known,the mass transfer and thermodynamic effects can, in principle, be added to com-

plete the analysis. The fluid mechanics of the extractor are dependent on the type

of device considered. If it is a sieve-tray extractor, the light fluid rises through

the continuous phase in trains of drops. If the extractor is a mixer-settler, the

dispersed phase is broken into drops by the energy of an impeller, with turbulent

flow commonly found in practice.

1.1 Four Distinct Levels of Analysis

Various levels of analysis can be applied to liquid-liquid extractors, de-

pending on the information sought, and can be classified as corresponding to the

discrete macroscopic (phenomenological), pseudo-continuum, continuum, or the

molecular level. Each level of analysis can be used to answer different questions

to different degrees of accuracy. Most plant design work is typically performed at

the macroscopic level, where little information is required on the internal details

of the system. Equipment design is at the pseudo-continuum level where empir-

ical information is needed to determine appropriate values of model parameters.

This dissertation is not concerned with questions at the macroscopic level such as

the ones related to the overall performance of the extractor, where most of the


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design work has occurred in the past. On the opposite end of the scales of length,

questions addressed at the molecular level are mostly concerned with the physical

properties of the components of the system and the boundary conditions, which

again are fairly well characterized. It remains to examine the extractor at the two

remaining levels.

A more detailed level of analysis than the macroscopic level, sometimes

referred to as the pseudo-continuum level, provides a more elaborate description of

the internal structure of the system and the transport within it. For example, mass

transfer coefficients are often used in equipment design (Treybal, 1980). However,

such descriptions still involve spatial averaging (e.g., over a swarm of droplets or

over random column packing) or temporal averaging (in turbulent flow), so thatquantities such as velocities and concentrations do not represent the true values

at a given point in the system. Further, the transport properties (e.g., turbulent

viscosities) are not true state properties, and their process dependence again makes

empirical information necessary.

These limitations are obviated at the continuum level, where velocities and

concentrations are local quantities and transport properties such as viscosities,

densities, surface tensions, and diffusivities, are true state variables. The obvious

problem is that most actual liquid-liquid contactors are much too complex in

their geometry and operation to permit a full continuum description, even with

the computational capabilities available today. In this dissertation, therefore, we

seek a middle ground, by analyzing at the continuum level idealized liquid-liquid

systems pertinent to actual extractors. We expect these analyses to enhance our

understanding of more realistic systems in two ways:

i. Among the systems we examine are ones that represent key elements

of one or more types of actual contactors; for example, a liquid-liquid jet is

representative of the situation above an orifice on a sieve tray.

ii. The parametric dependence of the behavior seen in the ideal model

systems is expected to be related to that in more complex systems; dimensionless


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parameters that determine the size and the specific surface area of liquid-liquid

jets and drops are investigated.

The object is to model the velocity and pressure fields as well as the interface

shape and location in a two-phase system that has one or more free surfaces.

Various simplified situations can be envisioned. The ones chosen for this work are

the liquid-liquid jet, and the drops formed before and during jetting. Two regions

are considered for the liquid-liquid jets: the steady region near the nozzle, and the

entire region from the nozzle to the breakup of the jet into drops.

1.2 Background

As noted above, most previous analysis and design work on liquid-liquid

contactors has been performed at the macroscopic or the pseudo-continuum level.

Results of such studies, and procedures based on them, may be found in, for

instance, Treybal (1963, 1980), Perry and Green (1984), Jeffreys (1987), and

Rousseau (1987). More detailed analyses at the pseudo-continuum level have

been undertaken using other approaches. For example, Guimaraes et al. (1988,

1990) used the population balance approach to model the effects of drop breakage

and coalescence on the hydrodynamics and mass transfer efficiency of liquid-

liquid continuous-flow stirred tanks at steady-state. However, even in these more

elaborate models it is necessary to estimate parameter values based on empirical

data. It is only at the continuum level, where transport properties are true state

variables, that this problem can be overcome. However, as noted earlier, the

analysis of only relatively idealized systems is possible at the continuum level, even

with the computational capabilities available today. A variety of such systems have

been studied in the past:

Isolated drops. Analytical models of single drops in idealized situations have

a long history. For example, the oscillation of a stationary drop can be described

exactly in the inviscid case (Lamb, 1932), and it has been studied more recently

for small Reynolds numbers by Lundgren and Mansour (1988). More complex


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systems have been examined either numerically or experimentally or both. Oliver

and Chung (1987) investigated the flow of a fluid sphere translating in another fluid

by solving numerically the steady-state Navier-Stokes equations inside and outside

the sphere using a streamfunction formulation. They obtained numerically the

velocity and pressure distributions inside and outside the sphere. Surface tension

acts to keep the drop spherical, but at high shear rates the drop will deform, and

eventually break. Isolated drops have been experimentally investigated during

breakup and coalescence (Clift et al., 1978; Ashgriz and Poo, 1990). The slender

body theory of Acrivos and Lo (1978) can be used for breakup at non-zero Reynolds

numbers, but requires that the drop or bubble that is breaking up must be

elongated. Stone and Leal (1989) have numerically investigated the dynamics ofdrop deformation and breakup at low Reynolds numbers; Bentley and Leal (1986)

studied these effects experimentally. Basaran et al. (1989) studied small amplitude

drop oscillations in liquid-liquid systems both experimentally and theoretically

and Jeelani and Hartland (1991) investigated the collision of two oscillating liquid

drops.

Liquid-liquid jet. When one liquid is injected into another, a jet may

be formed. That this occurs in sieve plate columns has motivated its study in

apparatuses specifically designed for mass transfer and surface tension experiments

with a single jet (Skelland and Huang, 1977, 1979; Skelland and Walker, 1989).

Meister and Scheele (Meister, 1966; Meister and Scheele, 1967, 1969a, 1969b;

Scheele and Meister, 1968) worked to develop an understanding of the jet and

drop formation based on experiments obtained with 15 liquid-liquid systems. They

reported theoretical results based on linear stability analysis for jet breakup and

drop formation, and overall force balances for drop formation below jetting. It was

noted by Meister (1966) that mass transfer coefficients in extraction columns are

generally reported as overall coefficients on a volumetric basis KySs because of the

difficulty of separating the mass transfer coefficient Ky from the specific surface

Ss. To obtain better predictions of the volumetric mass transfer coefficient, and


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hence of the mass transfer rate, better predictions of the specific surface formed in

the injection process are required. More accurate estimates of the mass transfer

coefficients can be obtained once the velocity and concentration fields in the two

phases are known. For an example of mass transfer coefficients calculated using

the concentration and velocity profiles in a liquid jet falling in (and absorbing)

carbon dioxide, see Scriven and Pigford (1959). Richards and Scheele (1985)

measured velocity profiles in liquid jets of xylene injected upwards into water

and found good agreement of the boundary layer theory (Yu and Scheele, 1975;

Gospodinov et al., 1979) with jet radius measurements, but poor agreement with

velocity measurements. Even today, the liquid-liquid jet and drop system is still

not completely understood.Liquid jets into air have been studied extensively (Scriven and Pigford,

1959; Vrentas and Vrentas, 1982), with current research focusing on jet breakup

(Mansour and Lundgren, 1990). Current limitations of jet breakup calculations,

such as those of Mansour and Lundgren (1990), who used a boundary element

method (BEM), are that they are generally inviscid and can only be carried

out only as far as the pinch point, at which time the Lagrangian mesh is highly

distorted. The Eulerian volume of fluid (VOF) technique discussed in section 1.3

can overcome both of these limitations; an example of a simple free liquid cylinder

capillary breakup modeled by this method is given by Kothe et al. (1991).

Rayleigh-Taylor problem. An even simpler situation is the case of a heavier

fluid above a lighter fluid in a gravitational field. It was first investigated in the

linear stability analysis limit by Rayleigh (1883) and by Taylor (1950) and has

since then been the subject of numerous studies. Linear stability analysis can be

used to show when this situation is unstable (Harlow and Welch, 1966; Drazin

and Reid, 1987), but this solution holds only for small interface deformations.

The subsequent evolution of the interface and the flow field is more interesting

for our purposes. This has been examined in both two and three dimensions.

The problem was studied numerically by Daly (1967, 1969a) using a Marker


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and Cell (MAC) type method. Tan (1986) reexamined this unstable situation

assuming incompressible, inviscid and irrotational fluid flow in a bounded three-

dimensional domain. Steady solutions to this problem exist and were derived

in approximate form by using bifurcation theory. It was shown that the surface

develops into a bubbles-and-spikes configuration that can be stable to infinitesimal

disturbances. Tryggvason and Arefs (1983, 1985) numerical investigation of the

Rayleigh-Taylor flow (renamed Taylor-Saffman for Hele-Shaw-cell flow) found that

the evolution of the resulting fingers was affected only by the viscosity ratio. More

recently, Tryggvason and Unverdi (1990) have performed numerical studies in three

dimensions using a front-tracking technique. Both theory and experiment for the

three-dimensional Rayleigh-Taylor instability were studied by Jacobs and Catton(1988).

Kelvin-Helmholtz problem. This problem is a generalization of the previous

one in that the fluids are moving relative to each other. Even when the heavy fluid

is below the light fluid, instability can occur due to shear, as can be shown by a

linear stability analysis (Drazin and Reid, 1987). Surface tension and viscosity

tend to stabilize the flow. Linear stability analyses have also been applied to

more complex situations, e.g., to the Couette flow of two fluids between rotating

concentric cylinders (Renardy and Joseph, 1985a, 1985b), to two-fluid pipe flow

(Joseph et al., 1984), and more recently to air-water flow (Bontozoglou and

Hanratty, 1990). For larger interface deformations, few (numerical) results appear

to have been reported. An example is the Kelvin-Helmholtz analog of the Hele-

Shaw cell studied by Pozrikidis and Higdon (1985), who found different initial

perturbations to result in different final rollups of the finite vortex sheet. However,

many of these and other analyses of the Kelvin-Helmholtz problem are limited to

inviscid, linear regimes, or to the Hele-Shaw approximation (see also Weidner and

Schwartz, 1991).

The Kelvin-Helmholtz problem for two fluids has also been studied exper-

imentally by Thorpe (1968, 1969), using a long rectangular tube containing two


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immiscible fluids. When the tube was tilted away from the horizontal, a uniformly

accelerating flow was produced, with shear at the interface leading to instability

characterized by growing waves. Although these phenomena had already attracted

considerable attention by the end of the last century, a comprehensive parametric

investigation of the effects of different parameters (Reynolds number, Weber num-

ber, etc.) on the development of the mixing of the two phases is still lacking. This

particular example is an inherently unsteady one that is pertinent to the startup

of stirred tank contactors, and may help to shed light on the phenomenon of phase

inversion (Quinn and Sigloh, 1963; Selker and Sleicher, 1965).

Despite the fact that these problems are ones that have been extensively

studied, a significant part of the effort has been devoted to developing solutionmethods, and not much attention has been paid to examining parametric depen-

dence of the solutions. This is a key aspect affecting the relevance of the idealized

continuum problems to more complex systems, as similarities can be expected

at least in the qualitative effects of different dimensionless parameters. Several

different kinds of parameters are important in a real contactor: (i) physical prop-

erty parameters, e.g., density ratio, viscosity ratio; (ii) geometric parameters, e.g.,

aspect ratio; (iii) hydrodynamic parameters, e.g., Reynolds numbers, Weber num-

bers. Each of the idealized problems outlined above is characterized by a smaller

number of parameters, e.g., the classic Rayleigh-Taylor problem contains only

physical property parameters, but by imposing a finite vertical or lateral size, geo-

metric parameters are added. Similarly, the transition from the Rayleigh-Taylor to

the Kelvin-Helmholtz problem introduces one or two hydrodynamic parameters.

The liquid-liquid jet and drop formation problem includes all of the kinds

of parameters seen in the simpler systems already noted. Thus it is particularly

relevant to liquid-liquid contactors. Also, since it is not completely understood at

the present time, it has been chosen as a model problem for this dissertation. The

systematic study of parametric dependence of solutions to idealized problems is

what we expect to shed light on the behavior of more complex systems.


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1.3 Numerical Methods for Multiphase Flow

The simulation of the dynamics of isolated drops, liquid-liquid jets, and

Rayleigh-Taylor and Kelvin-Helmholtz involve the solution of the continuity and

momentum equations for the two fluids with specified boundary conditions. The

description of the free surface offers a particular challenge, which can be dealt

with numerically from either the Eulerian or Lagrangian view. The Eulerian mesh

remains fixed in space with the flow moving through it, while the Lagrangian mesh

is convected with the flow. The major drawback of the Lagrangian approach, e.g.,

Ramaswamys (1990) study of free surface liquid sloshing in a container, is that

only simple nonintersecting interfaces can be represented due to a limitation on the

amount of mesh distortion allowed. The finite element method (FEM) has beenused successfully in calculating steady free jet flows (Reddy and Tanner, 1978;

Omodei, 1980; Georgiou et al., 1988; Cuvelier and Schulkes, 1990). Kheshgi and

Scriven (1984) have reviewed the finite element analysis of unsteady free surface

flows, while the works by Baker (1983) and Zienkiewicz (1988) provide a general

reference for the FEM method. As can be seen by examining these studies, it may

be possible to use the Lagrangian FEM to solve the specific steady liquid-liquid

jet problem considered in this dissertation. However, we are further interested in

predicting jet breakup and subsequent drop formation, past the point of necking

and through pinch-off of drops (with possible subsequent drop coalescence), which

precludes the FEM since it cannot, at the present time, be used to model such

complex breaking and reforming of interfaces.

Two basic Eulerian approaches have been formulated to track the interface

and have been reviewed by Hyman (1984) and Unverdi (1990), namely front

tracking and volume tracking. Front tracking uses the technique of characterizing

the interface by computational elements, such as a string of particles, which are

convected with the flow (Daly 1967, 1969a, 1969b). Tryggvason and Unverdi

(1990) used an additional grid with finite thickness that was convected with the

flow. Volume tracking involves using markers (MAC (marker and cell) method,


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Harlow and Welch, 1966) or a marker function convected by the flow (VOF

(volume of fluid) method, Hirt and Nichols, 1981). Yeung (1982) has presented a

general review of the numerical methods used in free-surface flows, while general

references for computational fluid dynamics (CFD) are Peyret and Taylor (1983)

and Anderson et al. (1984).

A completely different approach is the use of lattice-gas automata (Stock-

man et al., 1990), which are collections of discrete particles constrained to move on

fixed geometric lattices. This approach allows complex interface shapes to evolve

in time, and can be implemented on special parallel architectures. Initially only

fluids with the same density could be accommodated with this method, and an

extremely large number of cells had to be used to obtain accurate results compa-rable to the explicit differential equation methods. More recently, formulation of

lattice-Boltzmann techniques has circumvented most of these deficiencies (Succi

et al. 1991).

Volume of fluid method. The major incentive for using the VOF method is

that the types of problems that can be solved involve highly complex free surface

flows. Reasonable accuracy is attainable and yet the method is relatively simply

implemented. The basic algorithm is available in a two-fluid code called SOLA-

VOF (Nichols et al., 1980; Hirt and Nichols, 1981), and part of our work has been

devoted to modifying this algorithm to adapt it to the problems of interest, which

involve transient free surface flows with two immiscible fluids. More recently,

Kothe et al. (1991) introduced a one-fluid code, RIPPLE, which incorporates

various improvements to the one-fluid VOF algorithm.

The VOF formulation assumes that the flow in each phase is unsteady,

viscous, and incompressible. The equations of motion and continuity are solved in

a manner similar to that used by Nichols et al. (1980) with appropriate boundary

conditions for no-slip, free-slip, continuative, periodic, and contact angle, in two-

dimensional Cartesian or axisymmetric coordinates. The interface surface forces

are incorporated as accelerations in the momentum equations rather than as


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boundary conditions using a marker function field, defined as F = 1 for fluid

2 and F = 0 for fluid 1. This VOF function F is obtained by solving a kinematic

relation. The discontinuous density and viscosity fields are obtained from linear

interpolations using the F function. In our work, we have also included in the

algorithm the calculation of the streamfunction, which can be obtained from a

Poisson equation.

The SOLA-VOF code is well suited for high Reynolds number flows, includ-

ing those involving free surfaces. Among the latter, however, it is better suited for

gas-liquid than for liquid-liquid systems, and relaxing this limitation has been an

important part of our efforts. A one-fluid program, RIPPLE, which implements

the Continuum Surface Force (CSF) algorithm and incorporates various improve-ments in the SOLA-VOF algorithm, has been introduced by Kothe et al. (1991)

and Brackbill, et al. (1992).

1.4 Dissertation Objectives

The goal of this dissertation is to model aspects of complex liquid-liquid flow

behavior, implemented within a robust and efficient code to simulate steady-state

and transient liquid-liquid flows, including high Reynolds number ones. This code

is applied to the realistic simulation of aspects of the complex fluid mechanical

behavior in order to develop quantitative insight into the underlying processes

involved, such as drop and jet formation, size, shape, and breakup. Of course, it is

unavoidable that a limited range of the parameters has been investigated, dictated

primarily by the availability of reliable experimental data.

The two principal barriers to further progress in the area of liquid-liquid

separations are perceived to be the lack of a fundamental understanding of the

fluid mechanical phenomena associated with the development and destruction of

free surfaces separating two immiscible liquid phases, and the lack of a robust

and efficient numerical code that would enable the reliable simulation of key test

flow cases. This dissertation aims to contribute towards the resolution of both the


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above issues, physical and computational.

First, we have focused our research efforts towards the improvement of the

VOF numerical technique that we consider to be the best approach to the numeri-

cal solution of free surface problems that involve two immiscible liquid phases, high

Reynolds number laminar flows and complex, time-dependent interfaces. Consid-

erable progress has recently been made in developing VOF codes for 2-D and 3-D

free surface flows involving liquid-air or liquid-vacuum interfaces (RIPPLE and

FLOW-3D (developed by Flow Science, Los Alamos, NM, Hirt, 1988) are two

examples), and although these indicate the appropriateness of the technique for

complex free surface problems, no commercially available code exists for liquid-

liquid high Reynolds numbers free surface problems beyond the original SOLA-VOF (Hirt and Nichols, 1981). POLYFLOW (Crochet, 1987), one of the best

commercial codes based on FEM that can perform multiple phase calculations,

is best suited for low Reynolds number flows as, for example, the low Reynolds

number free jet die swell problem (Hill and Chenier, 1984). This code, as do other

codes, uses the method of spines for representation of the free surface, so it can

only represent simple, nonintersecting, single valued interfaces. A spine is a line

segment connecting two interface nodes and each interior node must belong to a

single spine. Thus, there is a constraint on the type of meshes that can be formed

with the use of spines. The SOLA-VOF code represented the starting point for our

computational code development. However, due to the age of the original code (it

was written before 1980), substantial further improvement was necessary in order

to transform it to a versatile (robust) and efficient research tool for complex high

Reynolds number free surface flows.

Second, we have contributed to the knowledge base on liquid-liquid interface

issues by investigating a few selected test cases, using the code developed as

discussed above. Specifically, we have examined a liquid jet injected into another

immiscible liquid. As explained in section 1.2, this is a critical problem for the

understanding of liquid-liquid separations. However, and in contrast to the much


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more widely studied liquid-air jet problem, little information is available on the

flow characteristics associated with the jet and the corresponding jet breakup

problem. Most of the available information comes from the previous experimental

work of Meister and Scheele (Meister, 1966; Meister and Scheele, 1967, 1969a,

1969b; Scheele and Meister, 1968) and Richards and Scheele (1985), whereas the

theoretical work, limited to boundary-layer solutions of the problem, seems to

be in considerable disagreement with the experimental data for the established

velocity profiles, despite correctly capturing the jet shape. We have used our

simulations to predict the velocity profiles in an effort to validate the approach

and simultaneously to enable us to do a systematic parametric search. We also

evaluated numerically the breakup conditions as well as the subsequent non-linearevolution of the drops, as a function of time. Thus, this research is expected to have

a significant impact on the way complex liquid-liquid contactors are designed in

the future, not to mention a plethora of other important technological applications

(such as, for example, ink jet printing studied by Croucher and Hair, 1989 or

low-gravity flows in space vehicles, Ostrach, 1982), where the understanding of

interface development is of paramount importance.

1.5 Dissertation Outline

The chapters of this dissertation are organized in the following manner. In

chapter 2 the governing equations that are used throughout this work are derived

from a fundamental viewpoint both to serve as a rigorous basis for the dissertation

and to indicate where future extensions may evolve. Here the Volume of Fluid

(VOF) function is introduced in conjunction with the novel use of the Heaviside

step function, which is an invaluable analytical tool to describe the interface.

The solutions to these problems involve the solution of the equations of

motion and continuity for two fluids with specified boundary conditions. The free

surface offers a particular challenge. During the course of this work we have found

difficulty in producing numerically stable free surface results with the SOLA-VOF


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surface tension algorithm. In chapter 3 we have combined the two-fluid capability

of the SOLA-VOF algorithm with the free surface implementation of the CSF

algorithm to investigate problems that involve transient 2-D free surface flows

with two immiscible fluids with possibly high Reynolds numbers, and found the

results to be much more numerically stable. The numerical method used to solve

the finite difference equations for the velocity and pressure and the advection of

the free surface is described. The surface force calculation is detailed as well as the

boundary conditions used in the free surface problems. Numerical stability issues

are also discussed. Several test problems that have known analytical or numerical

solutions were used to test the accuracy and validate the modified algorithm and

are detailed in section 3.8. The results of these problems indicate reasonableaccuracy, with some difficulty encountered in the capillary tube at small contact

angles and very low Reynolds numbers (Re < 1). It was also found that the

viscosity could not be considered spatially constant across the free surface as it was

in the original SOLA-VOF code, since it was found that the normal stresses were

not equal at the free surface in the problem of two-phase cocurrent flow between

parallel plates. Thus, the code was rewritten in terms of a variable viscosity.

In chapter 4 an efficient and robust method is presented for solving the

Young-Laplace equation that describes the shape of the meniscus in a vertical

cylinder for a constrained liquid volume. This serves as a test problem for future

work and as an introduction to the static free-surface equation. An overview

of this field is covered by Finn (1986). The method of solution proposed here

explicitly incorporates the constraint in the equation, transforming the two-point

boundary value problem into an initial value one with the constraint determining

the unknown centerline height. This allows rapid determination of the solution

families, which are characterized by only centerline height and meniscus arclength.

The method can be generalized to other axisymmetric systems such as spheres,

cones, and ellipsoids with or without axial rotation.

In chapter 5 we investigate the steady liquid-liquid jet. When a liquid


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is injected into another liquid a jet may be formed. This occurs in liquid-

liquid extraction plate columns as well as apparatus specifically designed for mass

transfer experiments with a single jet. The axisymmetric steady-state laminar flow

of a Newtonian liquid jet injected vertically into another immiscible Newtonian

liquid is investigated for various Reynolds numbers. The steady-state solution

was calculated by solving the axisymmetric transient equations of motion and

continuity using a numerical scheme based on the Volume of Fluid (VOF) method

combined with the new Continuum Surface Force (CSF) algorithm. The analysis

takes into account pressure, viscous, inertial, gravitational, and surface tension

forces.

The liquid-liquid jets eventually break up due to the increasing amplitudeof disturbance waves on their surface. In chapter 6 we investigate the length of the

resulting jets, which is dependent on many factors. Meister and Scheele (1969a)

have so far provided the most complete picture of liquid-liquid jet breakup. The

axisymmetric, dynamic breakup of a Newtonian liquid jet injected vertically into

another immiscible Newtonian liquid at various Reynolds numbers is investigated

in chapter 6. The full transient from jet startup to breakup into drops was

simulated numerically by solving the time-dependent axisymmetric equations of

motion and continuity using the code developed in this dissertation. The algorithm

has been further refined here based on its performance on transient problems

such as the solution of the free liquid-liquid capillary jet breakup problem. The

simulation results are compared with previous experimental measurements of jet

length under conditions where all forces, i.e., viscous, inertial, buoyancy, and

surface tension, are important.

The size of the drops produced before and during jetting is important from

an industrial standpoint due to the creation of large, new surface area. In chapter 7

the formation of drops by the vertical injection of a Newtonian liquid into another

stationary immiscible Newtonian liquid, at low to high Reynolds numbers, before

and after jet formation, is investigated. The full transient from before startup


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to breakup into drops was simulated numerically by solving the time-dependent

axisymmetric equations of motion and continuity. The numerical simulations are

compared with experiments and previous simplified analyses of Meister and Scheele

(1968, 1969b) based on drop formation before and after jetting. Prediction of the

sizes of these drops over wide ranges of Reynolds numbers is another important

result from this work, where only approximate theories have existed up until this

time.

Solutions to these difficult problems shed light on the distortion of interfaces

into various shapes and the physical variables that are important in extractors from

a hydrodynamic viewpoint.

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