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Three-dimensional CFD simulation of stratified two-fluid Taylor-Couette flow.



Two-fluid Taylor-Couette flow The Taylor-Couette flow consists of a viscous fluid confined in the gap between two rotating cylinders. For low angular velocities, measured by the Reynolds number Re, the flow is steady and purely azimuthal. , with either one or both of the co-axial cylinders rotating, has potential advantages over the conventional process equipment in chemical and bio-process industries. This flow has been investigated using three-dimensional CFD CFD - Computational Fluid Dynamics  simulations. The occurrence of radial stratification, the subsequent onset of centrifugal centrifugal /cen·trif·u·gal/ (sen-trif´ah-gal) efferent (1).

cen·trif·u·gal
adj.
1. Moving or directed away from a center or axis.

2.
 instability in each phase, the cell formation and the dependency on various parameters have been analyzed and discussed. The criteria for the stratification, Taylor cell formation in each phase have been established. It can be stated that the analysis of single-phase flow acts as the base state for the understanding of radial stratification of the two-fluid flows. The extent of interface deformation also has been discussed. A complete energy balance has been established and a very good agreement was found between dissipation rate by CFD predictions and the energy input rate through the cylinder/s rotation.

L'ecoulement biflfluide de Taylor-Couette, avec un des cylindres ou les deux cylindres coaxiaux en rotation, offre un avantage potentiel par rapport au systeme conventionnel utilise dans les industries chimiques et des bio-procedes. Cet ecoulement a ete etudie a l'aide de simulations par CFD tridimensionnelles. On a examine l'occurrence de la stratification radiale, l'apparition subsequente de l'instabilite centrifuge centrifuge (sĕn`trəfyj), device using centrifugal force to separate two or more substances of different density, e.g., two liquids or a liquid and a solid.  dans chaque phase, la formation des cellules et la dependance des divers parametres. Les criteres pour la stratification et la formation des cellules de Taylor dans chaque phase ont ete etablis. On a trouve que l'analyse de l'ecoulement monophasique est a la base de la comprehension de la stratification radiale dans le cas bifluide. Le degre de deformation de l'interface a egalement ete analyse. Un bilan d'energie complet a ete etabli et un tres bon accord Bon Accord may refer to:
  • Bon Accord (motto), the ancient motto of Aberdeen, Scotland
  • Bon Accord, Alberta, Canada
  • Bon Accord, Trinidad and Tobago
  • Shotts Bon Accord F.C.
 a ete trouve entre la vitesse de dissipation par les predictions CFD et le taux d'apport d'energie du a la rotation du ou des cylindres.

Keywords: Taylor-Couette flow, CFD, instability, radial stratification, interface deformation, energy balance

INTRODUCTION

Taylor-Couette flow or flow between two concentric cylinders with either or both cylinders rotating is a classical example of instability. This hydrodynamic hy·dro·dy·nam·ic   also hy·dro·dy·nam·i·cal
adj.
1. Of or relating to hydrodynamics.

2. Of, relating to, or operated by the force of liquid in motion.
 instability termed as the centrifugal instability with a number of secondary variations has led to many chemical process applications, which include emulsion polymerization Emulsion polymerization is a type of radical polymerization that usually starts with an emulsion incorporating water, monomer, and surfactant. The most common type of emulsion polymerization is an oil-in-water emulsion, in which droplets of monomer (the oil) are emulsified (with , synthesis of silica particles, heterogeneous catalytic reactions and liquid-liquid extraction (Imamura et al., 1993; Ogihara et al., 1995; Cohen cohen
 or kohen

(Hebrew: “priest”) Jewish priest descended from Zadok (a descendant of Aaron), priest at the First Temple of Jerusalem. The biblical priesthood was hereditary and male.
 and Maron, 1991; Sczehcowski et al., 1995; Davis and Weber, 1960; Bernstein et al., 1973; Baier et al., 1999). These have also been utilized as bioreactors, filters and also for membrane separation (Holeschovsky and Conney, 1991; Wereley and Lueptow, 1999; Tsao et al., 1994). Taylor-Couette flows offer the advantages of centrifugally accelerated settling, short residence times, low holdup volumes, flexible phase ratios and controlled inventory. These characteristics are desirable in applications where throughput (petroleum and petro-chemical industry), safety (nuclear fuel reprocessing Reprocessing may refer to:
  • Nuclear reprocessing
  • Recycling
), or facilitated settling (bioseparations) are required (Baier et al., 1999). A schematic of such a contactor con·tac·tor  
n.
An electrical relay used to control the flow of power in a circuit.
, with dual cell pattern is shown in Figure 1.

[FIGURE 1 OMITTED]

The flow pattern obtained in the annular annular /an·nu·lar/ (an´u-ler) ring-shaped.

an·nu·lar
adj.
Shaped like or forming a ring.



annular

ring-shaped.
 region could be with two phases retaining their individual integrity and contacting each other at a single well-defined interface (stratified stratified /strat·i·fied/ (strat´i-fid) formed or arranged in layers.

strat·i·fied
adj.
Arranged in the form of layers or strata.
 flow) or the two-phases as dispersion (dispersed flow). The stratified two-fluid Taylor-Couette flow is an interesting variation of one-fluid problem that explores the effect of interface on the vortex flow. The stratified Taylor-Couette flow has two identical layers of vortices vor·ti·ces  
n.
A plural of vortex.
, which fill the annular gap. The liquid interface introduces six additional boundary conditions: velocities and shear stresses at the interface and the normal stress balanced by the interfacial tension Noun 1. interfacial tension - surface tension at the surface separating two non-miscible liquids
interfacial surface tension

surface tension - a phenomenon at the surface of a liquid caused by intermolecular forces
. In addition to this, the interface position is unknown. These interfacial boundary conditions require that the vortex motion in one phase be balanced by the vortex motion in the other phase. High rotation rates are required to first centrifugally stratify strat·i·fy  
v. strat·i·fied, strat·i·fy·ing, strat·i·fies

v.tr.
1. To form, arrange, or deposit in layers.

2.
 the two fluids, and then a subsequent increase in the inner cylinder rotation rate would produce vortices. Of course, alternatively, highly viscous fluids attracted to their respective walls might eliminate the requirement for stratification due to the centrifugal force centrifugal force

Fictitious force, peculiar to circular motion, that is equal but opposite to the centripetal force that keeps a particle on a circular path (see centripetal acceleration).
. The dimensionless groups Dimensionless groups

A dimensionless group is any combination of dimensional or dimensionless quantities possessing zero overall dimensions. Dimensionless groups are frequently encountered in engineering studies of complicated processes or as similarity
 describing the two-fluid Taylor-Couette flow are: Taylor number In fluid dynamics, the Taylor number is a dimensionless quantity that characterizes the importance of centrifugal "forces" or so-called inertial forces due to rotation of a fluid about a vertical axis, relative to viscous forces.  (Ta) for each phase (to signify the centrifugal instability); a Froude number Froude number

The dimensionless quantity U(gL)-1/2, where U is a characteristic velocity of flow, g is the acceleration of gravity, and L is a characteristic length.
 (Fr) (for the gravitational grav·i·ta·tion  
n.
1. Physics
a. The natural phenomenon of attraction between physical objects with mass or energy.

b. The act or process of moving under the influence of this attraction.

2.
 effects in each phase); and the Joseph's factor (J) for the interface stability.

The radially stratified fluid behaviour for the case of either or both the cylinders rotating co-currently or counter-currently so also the rigid rotation of cylinders has been verified experimentally and numerically by few authors (Schneyer and Berger, 1971; Joseph et al., 1985; Renardy and Joseph, 1985; Joseph and Preziosi, 1987; King et al., 1998; Baier and Graham, 1998; Caton et al., 2000; Charru and Hinch, 2000; Albert and Charru, 2000; Zhu and Vigil, 2001). It has also been reported by most authors that the configuration including a stationary outer cylinder and a rotating inner cylinder commonly leads to emulsification of at least a part of the fluid. Schneyer and Berger (1971) report a linear stability analysis for a case of stationary outer cylinder, negligible forces due to interfacial tension and gravity. Though they found two different modes of instability, the spatial structures were not reported. In a study of unbounded two-fluid Couette flow In fluid dynamics, Couette flow refers to the laminar flow of a viscous liquid in the space between two surfaces, one of which is moving relative to the other. The flow is driven by virtue of viscous drag force acting on the fluid. , Hooper and Boyd (1983) demonstrated that in the absence of interfacial tension, the interface between the two fluids is always unstable to short wavelength pertur bations. Renardy and Joseph (1985) investigated theoretically the stability of two-fluid Couette flow with only inner cylinder rotation. They expand the disturbance velocities and pressure in Chebyshev polynomials In mathematics the Chebyshev polynomials, named after Pafnuty Chebyshev, are a sequence of orthogonal polynomials which are related to de Moivre's formula and which are easily defined recursively, like Fibonacci or Lucas numbers.  and numerically solve the linear eigen-value problem for the growth rate of Taylor vortices. They found that a thin layer of less viscous fluid near either cylinder is linearly stable. This does not agree with the theory that viscous dissipation should be minimized. Further, the two-fluid Taylor-Couette flow may be stabilized by the less viscous fluid in a lubrication lubrication, introduction of a substance between the contact surfaces of moving parts to reduce friction and to dissipate heat. A lubricant may be oil, grease, graphite, or any substance—gas, liquid, semisolid, or solid—that permits free action of  layer near the inner cylinder. Also, the denser fluid may be located at the inner cylinder when stabilized by interfacial tension and a favourable viscosity difference. Joseph et al. (1985) predicted a linearly stable rigid interface between the two fluids at rigid rotation when J > 1, with J defined as:

J = ([[rho].sub.o]-[[rho].sub.i])[[OMEGA].sup.2][r.sup.3.sub.in]/[sigma] (1)

Herein, [r.sub.in] refers to the radius of the interface from the inner cylinder and [OMEGA], the rotational speed Rotational speed (sometimes called speed of revolution) indicates, for example, how fast a motor is running. Rotational speed is equivalent to angular speed, but with different units. Rotational speed tells how many complete rotations (i.e.  of either of the cylinders. Further, the condition of rigid rotation refers to the angular velocity of both the cylinders being the same. They predicted a globally stable interface for J>4. This group termed as the Joseph's factor measures the relative importance of centrifugal and interfacial forces. In a similar numerical study of stratified two-phase flow In fluid mechanics, two-phase flow occurs in a system containing gas and liquid with a meniscus separating the two phases.

Historically, probably the most commonly-studied cases of two-phase flow are in large-scale power systems.
 between co-rotating cylinders, Renardy and Joseph (1985) extended the analysis by computationally exploring the stability of the interface for various combinations of relative fluid viscosities and configurations. Toya and Nakamura (1997) studied Taylor-Couette flow of two fluids in a vertical annulus annulus /an·nu·lus/ (an´u-lus) pl. an´nuli   [L.] anulus.

an·nu·lus or an·u·lus
n. pl. an·nu·lus·es or an·nu·li
A circular or ring-shaped structure.
; the fluids were axially stratified. They observed that at the interface, the bottom vortex in the less dense phase could co-rotate with the top vortex in the denser phase; the flow is counter-current at the boundary between the two fluids. Baier and Graham (1998) investigated the centrifugal instability of radially stratified liquids in the annular gap using the linear stability analysis. The experiments carried out by them showed a well-defined interface and vortices in each phase. For fluids with sufficiently low viscosity, they observed instability similar to that of a liquid coating inside the rotating drum. When the two fluids are identically matched, without any counter-current axial flow and a negligible curvature, the linear stability analysis has been shown to give initially counter-rotating vortices as the first mode of instability in the literature. A co-rotating state has been the second mode of instability (Baier et al., 1999).

In the published literature, most of the investigations (numerical as well as experimental) deal with the criteria for stratification as well as transition to Taylor-vortex regime in both the phases. Since, the linear stability analysis is valid only at the onset of two-fluid Taylor-Couette flow; it cannot directly determine the flow behaviour beyond the critical Ta, which may be in viscous, transition or turbulent regime. In the published literature, turbulent mode of transport has not been included in the flow modelling. Hence, it was thought desirable to incorporate a turbulence model and study the flow characteristics. The limited number of CFD simulations in the case of radial stratification (Baier and Graham, 1998; Baier, 2000) is refrained to two-dimensional cases, which in case of higher Ta, may not be valid. Further, the subsequent interface deformation occurring at higher Ta, may not give realistic results with a 2D simulation. Hence, in the present study, three-dimensional simulations have been carried out. In the reported literature, though the linear stability analysis proves useful in understanding the onset of vortex flow and few further transitions, it does not give a clear picture of the underlying physics in further transitions, which has been attempted to be characterized in the present study. In addition, the present study also includes the establishment of the energy balance, the interface deformation and the effect of physical properties and operating conditions on the interface deformation and cell patterns formed therein.

THEORY

Criterion for Stratification

Viscous fluids

In order that the fluid may stratify stably in a Taylor-Couette contactor, the lighter fluid Lighter fluid may refer to:
  • Butane, a highly flammable, colorless, easily liquefied gas used in cigarette lighters
  • Naphtha, a volatile flammable liquid hydrocarbon mixture used in wick type lighters
 needs to move towards the inner cylinder while the heavier phase towards the outer cylinder, which is an indication that the centrifugal forces be dominant enough to overcome the turbulent fluctuations that cause dispersion. This may be explained mathematically making a radial force balance. It would be more apparent that the stratification occurs when the centrifugal force of the outer (heavier) fluid is more than the convective forces of the outer fluid. The criterion may be expressed in terms of the radial velocity radial velocity, in astronomy, the speed with which a star moves toward or away from the sun. It is determined from the red or blue shift in the star's spectrum.  component at the interface and experimentally operable operable /op·er·a·ble/ (op´er-ah-b'l) subject to being operated upon with a reasonable degree of safety; appropriate for surgical removal.

op·er·a·ble
adj.
 parameters as:

[u.sup.2.sub.r,in]/[u.sup.2.sub.[theta Theta

A measure of the rate of decline in the value of an option due to the passage of time. Theta can also be referred to as the time decay on the value of an option. If everything is held constant, then the option will lose value as time moves closer to the maturity of the option.
],in] = ([[rho].sub.o]-[[rho].sub.i]/[[rho].sub.o]+[[rho].sub.i]) [r.sup.2.sub.o]-[r.sup.2.sub.i]/2[r.sup.2.sub.in] (2)

Stratification occurs for this ratio less than 1. This equation holds for the pure circumferential flows for any value of interfacial tension. The simulations considered in the present study are as per the above equation.

Onset of Instability

Inviscid in·vis·cid  
adj.
1. Having no viscosity.

2. Physics & Chemistry Of or relating to a fluid with no viscosity.
 flows

As established by Rayleigh (1916), a radial stratification of angular momentum angular momentum: see momentum.
angular momentum

Property that describes the rotary inertia of a system in motion about an axis. It is a vector quantity, having both magnitude and direction.
 is unstable, if the angular momentum decreases with an increase in the radial distance in the annulus. This was mathematically expressed as:

D[([r.sup.2][OMEGA]).sup.2]/dr < 0 or [[OMEGA].sub.o,i] < [r.sup.2.sub.i,o] (3)

This applied to the Couette flow in the case of two fluids, predicts that for the instability to occur in the inner fluid, the instability criterion becomes

[[OMEGA].sub.in,i] < [r.sup.2.sub.i,in] (4)

and for the outer fluid, the instability criterion becomes:

[[OMEGA].sub.o,in] < [r.sup.2.sub.in,o] (5)

For real fluids, the inviscid instability criterion of Rayleigh may be considered as a necessary condition though not as a sufficient condition. Further, though the Rayleigh's inviscid criterion does not strictly hold for the viscous fluids, it depicts the region of vortex motion in each fluid phase individually.

Viscous flows

In the radially stratified flows, as the rotation of the inner cylinder increases, instability sets in both phases, indicating both the phases to be centrifugally unstable. The critical Ta for the onset of centrifugal instability in both the phases has been calculated as the Ta in each phase. These are:

[Ta.sub.i] = 4[[OMEGA].sup.2.sub.in][[rho].sup.2.sub.i][d.sup.4][[r.sup.2.sub.i,in]- [[OMEGA].sub.in,i]]/[[mu].sup.2.sub.i](1-[r.sup.2.sub.i,in]) (6)

for the inner or the low density fluid and for the outer fluid,

[Ta.sub.o] = 4[[OMEGA].sup.2.sub.in][[rho].sup.2.sub.o][d.sup.4][[r.sup.2.sub.in,2]- [[OMEGA].sub.2,in]]/[[mu].sup.2.sub.o](1-[r.sup.2.sub.in,2]) (7)

wherein, [[r.sup.2.sub.i,in] - [[OMEGA].sub.in,i]] and [[r.sup.2.sub.in,2] - [[OMEGA].sub.2,in]] determine the inviscid instability criteria.

Estimation of Interface Radius and Deformation

The interface deformation has been tracked by determining the interfacial area per unit volume of the deformed interface to that if the interface was cylindrical. This can be expressed for the deformation due to each phase as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII ASCII or American Standard Code for Information Interchange, a set of codes used to represent letters, numbers, a few symbols, and control characters. Originally designed for teletype operations, it has found wide application in computers. ] (8)

a is the interfacial area of the deformed interface, which is dependent on the interface deformed at each axial location. While, [a.sup.*] is the interfacial area, if the interface were cylindrical, which is dependent on [r.sup.*], which is none other than the function of the phase volume fraction in the annulus. The deformation clearly demarcates, the implications of inter vortex mixing between the phases.

MATHEMATICAL MODELLING

The immiscible immiscible /im·mis·ci·ble/ (i-mis´i-b'l) not susceptible to being mixed.

im·mis·ci·ble
adj.
Incapable of being mixed or blended, as oil and water.
 fluid flows mostly consist of a domain of interest with an unknown interface that moves from one location to another and might also undergo deformations, at times leading to breakup. This has been one of the interesting difficulties in the case of two-phase flows. Herein, the interface plays a major role in defining the system and must be determined as a part of the solution.

Model Formulation

In the present case, three-dimensional simulations have been carried out for the case of two immiscible liquid systems. The governing Navier-Stokes equations for the case of flow between two concentric cylinders, for an incompressible in·com·press·i·ble  
adj.
Impossible to compress; resisting compression: mounds of incompressible garbage.



in
, constant viscosity liquid can be written in cylindrical coordinates as:

Continuity:

[partial derivative][rho]/[partial derivative]t + [nabla].([rho][??]) = 0 (9)

The momentum equations may be expressed as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

where p is the static pressure, [??] is the stress tensor For the stress tensor in classical physics, see the article
  • stress (physics).
For the stress tensor in relativistic theories, see
  • stress-energy tensor.
, [rho][??] is the gravitational body forces and [??] corresponds to the external body forces. The stress tensor [??], is given by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

where [mu] is the molecular viscosity, I is the unit tensor tensor, in mathematics, quantity that depends linearly on several vector variables and that varies covariantly with respect to some variables and contravariantly with respect to others when the coordinate axes are rotated (see Cartesian coordinates). .

Multiphase Modelling

The volume of fluids (VOF) model is a surface tracking technique applied to a fixed Eulerian mesh. It is designed for two immiscible fluids where the interface between fluids is of interest. In this, a single set of momentum equations is shared by the fluids, and the volume fraction of each of the fluids in each computational cell is tracked throughout the domain. The following continuity equation for volume fraction is solved in order to accomplish the interface tracking between the two phases.

[partial derivative][[rho].sub.q][[alpha].sub.q]/[partial derivative]t + [??].[nabla][[rho].sub.q][[alpha].sub.q] = 0 (12)

The volume fraction equation will not be solved for the primary phase; the primary phase volume fraction will be computed based on the following constraint:

[n.summation summation n. the final argument of an attorney at the close of a trial in which he/she attempts to convince the judge and/or jury of the virtues of the client's case. (See: closing argument)  over (q=1)][[alpha].sub.q] = 1 (13)

A single momentum equation is solved throughout the domain and the velocity field is shared between the phases. The properties appearing in the transport equations are determined by the presence of the component phases in each control volume. If the volume fraction of phase q is being tracked, the density in each cell is given by:

[rho] = [[alpha].sub.q][[rho].sub.q] + (1 - [[alpha].sub.q])[[rho].sub.p] (14)

This is based on the fact that for an n-phase system,

[rho] = [summation][[alpha].sub.q][[rho].sub.q] (15)

Turbulence Modelling

For turbulence modelling, Reynolds Stress Model (RSM RSM (in Britain) regimental sergeant major ) has been used. In this model, individual Reynolds stresses In fluid dynamics, the Reynolds stresses (or, the Reynolds stress tensor) is the stress tensor in a fluid due to the random turbulent fluctuations in fluid momentum. The stress is obtained from an average (typically in some loosely defined fashion) over these fluctuations.  [u.sub.i][u.sub.j] are computed via a differential transport equation. Thus, the RSM model solves six Reynolds stress transport equations. Along with these, an equation for dissipation rate is also solved. The exact form of Reynolds stress transport equations is derived by taking moments of exact momentum equation. This is a process wherein the exact momentum equations are multiplied by a fluctuating property, the product then being Reynolds averaged. The exact transport equations for the transport of Reynolds stresses [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are given by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)

The turbulent viscosity [[mu].sub.t], is computed as,

[[mu].sub.t] = [rho][c.sub.[mu]] [k.sup.2]/[epsilon] where, [c.sub.[mu]] = 0.09 (17)

The diffusion term is taken as a scalar scalar, quantity or number possessing only sign and magnitude, e.g., the real numbers (see number), in contrast to vectors and tensors; scalars obey the rules of elementary algebra. Many physical quantities have scalar values, e.g.  diffusivity Dif`fu`siv´i`ty

n. 1. Tendency to become diffused; tendency, as of heat, to become equalized by spreading through a conducting medium.
 term as (Launder Launder

To move illegally acquired cash through financial systems so that it appears to be legally acquired.
 et al., 1975):

[partial derivative]/[partial derivative][x.sub.k] ([[mu].sub.t]/[[sigma].sub.k] [partial derivative][u'.sub.i][u'.sub.j]/[partial derivative][x.sub.k], [[sigma].sub.k] = 0.82 (18)

The buoyancy effects and the pressure strain effects have been neglected in the present analysis. The turbulence kinetic energy was obtained by taking the trace of Reynolds stress tensor,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19)

To obtain boundary conditions for Reynolds stresses, the following model equation was used:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20)

Though the above equation is solved globally through the flow domain, the values of k obtained are used only for boundary conditions. In every other case the prior equation is used to obtain k.

In order to model the dissipation rate, the dissipation tensor is modelled as: [[epsilon].sub.ij] = 2/3 [[delta].sub.ij]([rho][epsilon] + [Y.sub.m], where [Y.sub.m] = 2[rho][epsilon][M.sup.2.sub.t] is an additional dilatation dilatation /dil·a·ta·tion/ (dil?ah-ta´shun)
1. the condition, as of an orifice or tubular structure, of being dilated or stretched beyond normal dimensions.

2. the act of dilating or stretching.
 dissipation term. The turbulent Mach number Mach number (mäk) [for E. Mach], ratio between the speed of an object and the speed of sound in the medium in which the object is traveling. An airplane that has the velocity of Mach 3.  is defined as, [M.sub.t] = [square root of (k/[a.sup.2])], with a = [square root of ([gamma]RT)] which is the speed of sound. The scalar dissipation rate [epsilon], is computed with the model transport equation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (21)

The constants are [[sigma].sub.[epsilon]] = 1.0, [c.sub.[epsilon]1] = 1.44, [c.sub.[epsilon]2] = 1.92

Boundary Conditions

The volume flow ratios of both the phases are specified, such that the interface formation could be known. The rotational velocities of the walls are specified. The inner cylinder rotation varied from 10 to 50 rps. The rotation ratio of the cylinders varied from -2.33 to 2.33. Axially, the boundaries are specified as periodic, wherein for purely circumferential flow; the periodic conditions are specified as rotationally cyclic, since the boundaries form an included angle with the rotationally symmetric geometry. Axial lengths of 0.018, 0.072 and 0.18 m have been considered with an annular gap width of 9 mm on each side.

The standard wall functions used in the turbulence model are based on those of Launder and Spalding (1974). At walls, the near wall values of the Reynolds stresses and [epsilon] are computed from wall functions. The explicit wall boundary conditions are applied for Reynolds stresses by using log-law and the assumption of equilibrium, thus disregarding convection and diffusion in the transport equations for stresses.

Method of Solution

With the Finite volume formulation, all the simulations were carried out using three-dimensional grids. Since, the observed stratification involved the global stability, a circular interface has been assumed for the rotational speeds considered. Though two-dimensional simulations have been carried out initially, since it is anticipated that at very high rotational speeds, the flow tends towards asymmetry, three-dimensional simulations have also been carried out and the present work reports results of the three-dimensional simulations. The commercial software FLUENT (version 6.1.2) has been used in all the studies. Uniform grid scheme consisting of 70 000 cells has been employed. The grid structure is shown in Figure 2. As a first step towards selecting this present grid size, simulations were carried out to verify the effect of grid size for the case of single phase flow. The number of grids (structured) was varied over a wide range such as 32 000, 40 000, 54 000, 70 000, 98 000 and 270 000 cells in all the directions. The effect of grid size is shown in the Figure 3 with the axial velocity plotted against the radial distance at an axial location of z/2. It can be seen that the flow pattern is independent of the number of grids beyond a grid size of 70 000 cells. Therefore, the same number has been used in all the simulations. A segregated implicit solver method was used for solving the momentum equations. The momentum equations have been discretized with the first order upwind scheme, and for the pressure velocity coupling, PISO Piso (pī`sō), distinguished family of the ancient Roman gens Calpurnia. One of the best-known members was

Lucius Calpurnius Piso Caesoninus, d. after 43 B.C., father-in-law of Julius Caesar. As consul (58 B.C.
 scheme has been used. The Pressure-Implicit with Splitting of Operators (PISO) pressure-velocity coupling scheme, part of the SIMPLE family of algorithms, is based on the higher degree of the approximate relation between the corrections for pressure and velocity. For the pressure equation, PREssure STaggering Option (PRESTO PRESTO - A parallel language for shared-memory multiprocessors, built on top of C++ by Bershad et al, U Washington 1987. PRESTO provides classes for threads and spinlocks as well as Mesa-style monitors and condition variables.

ftp://cs.washington.edu/pub/presto1.0.tar.Z.
) scheme was used. This uses a discrete continuity balance for a staggered control volume about the face to compute the "staggered" pressure. The Eccentricity eccentricity, in astronomy: see orbit.
Eccentricity
Addams Family

weird family, presented in grotesque domesticity. [TV: Terrace, I, 29]

Boynton, Nanny

travels with set of Encyclopaedia Britannica
 (ratio of offset distance of the cylinder axis (Anat.) See Axis cylinder, under Axis.

See also: Cylinder
 to the average gap width) was assumed to be zero. A segregated solver with implicit linearization In mathematics and its applications, linearization refers to finding the linear approximation to a function at a given point. In the study of dynamical systems, linearization is a method for assessing the local stability of an equilibrium point of a system of nonlinear differential  and an unsteady solution has been applied. A single set of momentum equations has been solved followed by the Reynolds transport equations to account for the stresses and the turbulent kinetic energy Turbulent Kinetic Energy (TKE) is the mean kinetic energy per unit mass associated with eddies in turbulent flow. It is a concept used to assess what contribution to buoyancy is brought by turbulence.  and the energy dissipation rate. Inner iterations were carried out until mass conservation as per the convergence criteria This is an article about European politics, Convergence criteria is also a mathematical term regarding series.

Convergence criteria (also known as the Maastricht criteria) are the criteria for European Union member states to enter the third stage of European Economic and
 (in the present case 10-6 for all the equations). The volume fraction equation has been solved as mentioned in the Multiphase Modelling section. Data was collected at specified points to track the development of the flow and confirm that the asymptotic solution was reached.

[FIGURES 2-3 OMITTED]

RESULTS AND DISCUSSION

As mentioned earlier, three-dimensional simulations have been carried out in order to obtain information on the onset of instability, vortex pattern formation, and the interface radius. The role of physical properties such as interfacial tension and the phase densities has also been investigated. Simulations have been carried out for different rotation ratios (varying from 10? rad/s to 50[pi] rad/s) of both the inner as well as outer cylinders at various interfacial tension values in a coaxial cylinder system with horizontal axis. Three different aspect ratios 2, 4 and 10 have been covered. The holdup of the heavy fluid has been varied as 0.3, 0.4, 0.5, 0.65 and 0.7. Stability aspects have been analyzed in terms of the interface radius. For the geometry and the operating conditions considered in the present simulations, the Joseph's factor (Equation 1) has been slightly modified. Since present study also involved rotation ratios for rigid conditions ([[OMEGA].sub.i]/[[OMEGA].sub.o] = 1) as well as variable rotation ratios ([[OMEGA].sub.i]/[[OMEGA].sub.o] [not equal to] 1), and the instability in each phase arises with an increase in the inner cylinder rotation, the rotational speed term ([OMEGA]) in the Joseph's factor has been replaced by the inner cylinder rotation. Thus, the modified Joseph's factor is:

J = ([[rho].sub.o]-[[rho].sub.i])[[OMEGA].sup.2.sub.i][r.sup.3.sub.in]/[sigma] (22)

The interface radius ([r.sub.in]) used in the above equation is based on the volume fraction of each phase. Since the Joseph's factor is much greater than 4 (of the order of 200) in the present work, which satisfies the condition of global stability, stratification is observed to hold good in all the simulations. The rotational speed term ([[OMEGA].sub.i]) used in Equation (22) is that of the inner cylinder, since above sufficiently high rotation ratios, stratification is stable and vortices start spanning with an increase in the rotation of the inner cylinder. The interface was found to be more stable for J >> 4, calculated with the inner rotational speed, rather than the outer cylinder rotation. The same is shown in the subsequent figures in the following sections. As a first step to establish the validity of the model, validation has been carried out with the help of establishment of energy balance, since there are no experimentally reported flow pattern studies available in the literature.

Energy Balance

The energy balance means that the energy input rate (by the rotation of any one or both the cylinders) must equal the energy dissipation rate (both by the viscous and turbulent modes of dissipation).

The energy input rate is given by the following equation for the case of the inner cylinder rotating and the outer one stationary:

Energy input = [pi]/2 ([[alpha].sub.o][[rho].sub.o] + [[alpha].sub.i][[rho].sub.i])([r.sup.2.sub.o] - [r.sup.2.sub.i])[([[OMEGA].sub.o][r.sub.o] - [[OMEGA].sub.i][r.sub.i]).sup.3] (23)

The predicted value is the volume integral of the energy dissipation rate:

Predicted energy [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (24)

Where, the viscous energy dissipation rate, [[epsilon].sub.v] is given by,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (25)

The turbulent energy dissipation rate, [[epsilon].sub.t] is given by,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (26)

Thus, the dissipation rates are predicted from the CFD simulations. The simulations have been carried out for (d = 0.009 m, and [GAMMA] = 2) and rotational speeds in the range 8 to 16 rps. The heavier phase volume fractions used are 0.3, 0.4, 0.5, 0.65 and 0.7. In the simulations, the multi phase model used is the VOF and turbulence is incorporated by the RSM turbulence model. Table 1 shows a good agreement and validates the simulations. The energy balance is satisfied for the stable radial stratification of the interface when both the cylinders are rotating at sufficiently high speeds. The small deviations in some cases can be attributed to the end effects, which have been neglected in the present study.

Radial Stratification and Flow Pattern

Radial stratification of the two fluids can be visualized in a horizontal axis two-fluid Taylor-Couette contactor with the help of the density stratification shown in Figure 4. This figure shows the contours of the densities of two fluids with the heavier phase towards the outer wall and the lighter phase towards the inner cylinder wall. This being a coarse indication of stratification, high rotation rates are required to first centrifugally stratify the two fluids, and sufficiently high enough rotation rates of the inner cylinder in order for the formation of vortices in both the phases.

[FIGURE 4 OMITTED]

Figure 5 shows the radial velocity contours of centrifugally stratified two-fluids in the annular region for the case of rigid rotation with cylinders rotating in the same direction. This has been observed for the case of an interfacial tension of 0.05 N/m and a viscosity ratio of 0.69. At higher rotational speeds, as the vortices start translating axially along the annulus length, intra vortex mixing occurs, which most likely causes the vortex deformation. Since co-rotating vortex patterns prevailed, it can be said that the intra vortex mixing continued to occur, which could be desirable for separations wherein interface deformation is not desirable. The radial velocity contours refer to the contour plots for the outer fluid holdup of 0.5 at an aspect ratio of 4. As the inner cylinder rotational speed is further increased, the vortex structure deforms towards a helical helical /hel·i·cal/ (hel´i-k'l) spiral (1).

hel·i·cal
adj.
1. Of or having the shape of a helix; spiral.

2. Having a shape approximating that of a helix.
 nature mainly more evident in the inner fluid. Figure 6 depicts the axial velocity profile at a higher rotational ratio of 1.6. This indicates that further transitions in the cell patterns are more likely to occur with an increase in the inner cylinder rotation rate. The variation in the cell patterns can be observed for all the aspect ratios considered in these simulations as shown in the Figure. The turbulent kinetic energy profile is shown in Figure 7 for an aspect ratio of 10, for the above cases depict the turbulent nature of the flow in each phase in the stratification.

[FIGURES 5-7 OMITTED]

Single-Phase Flow--Limiting Condition

In the case of single-phase flow, at higher rotation ratios, a second set of vortices starts spanning in the annular gap along the length of the annulus. This is similar to the two-phase stratified flow asymptotically reaching the single-phase flow solution depicted by the vortex pattern formation in both the phases. This is equivalent to assuming identical density and viscosity fluids and a volume fraction of each phase to be 0.5, the solution is stratified for a two-phase case and a stable solution of two vortex patterns is obtained. Figure 8 shows the radial velocity contours of single-phase flow solution at high rotation ratio, found to be identical to the two-phase stratified solution. It can be said that the secondary bifurcation Bifurcation

A term used in finance that refers to a splitting of something into two separate pieces.

Notes:
Generally, this term is used to refer to the splitting of a security into two separate pieces for the purpose of complex taxation advantages.
 of the mathematical solution for the single-phase flow is identical to the solution of stably stratified two-phase flows.

[FIGURE 8 OMITTED]

Interface Radius and Interface Deformation

For all the simulations carried out, the interface radius separating the two centrifugally stratified fluids has been determined. It has been observed from the simulations that the interface radius depends on the rotation ratios of both the cylinders, the properties of the fluids, the interfacial tension and the volume fraction of the stratified fluid phases. The heavier phase volume fractions have been considered as 0.3, 0.4, 0.5, 0.65 and 0.7.

Effect of Interfacial Tension

Interfacial tension has been found to affect the stability of interfacial disturbances and the vortex pattern formation in the two fluids. Centrifugal forces, in the absence of interfacial tension, at rigid rotation are found to stabilize the disturbance, thus with a uniform interface radius along the length of the annulus. In the conditions other than the rigid rotation, with the incorporation of interfacial tension, the interface was found to be stably stratified. Upon the onset of instability, the vortex formation is found to be dominant initially in the inner fluid, while at increased rotation rates, the vortices started spanning in the outer fluid. As the rotation ratios increased, the vortex patterns obtained also do not show the toroidal form, which is typical of Taylor-Couette flow pattern in the outer, higher dense fluid. This could be attributed to the shear stresses dominant at the interface as the rotation ratio is increased. Figure 8 (shown previously) further establishes the uniform interface for the three holdups of the outer fluid. In all these cases, the vortex patterns also have been non-deforming. These analyses clearly indicate that the interfacial tension plays an important role in deciding the interface deformation that is of prime interest in radially stratified flows. The radial velocity contours observed along a tangential tan·gen·tial   also tan·gen·tal
adj.
1. Of, relating to, or moving along or in the direction of a tangent.

2. Merely touching or slightly connected.

3.
 plane in the length of the annulus for the various volume fractions of the phases showed vortex formation, wherein the interface has been non-deforming in the presence of interfacial tension. For the case of a holdup of 0.5, with negligible curvature and interfacial tension (S = 0.0001N/m), the two phases started to inter-penetrate leading to a dispersion, though the interface radius has been observed to be uniform along the length of the annulus as shown in the Figure 9. The presence of interfacial tension has been shown in Figure 10 (S=0.05 N/m), wherein, a clear stratification is visible in the vortex patterns. In all these cases investigated, though different aspect ratios were considered (2, 4 and 10), there has been no significant difference in the stratification or the vortex pattern formation. This might also be attributed to the assumptions of negligible end effects.

[FIGURES 9-10 OMITTED]

Figure 11 shows the interface deformation for the case of the interchange of lower and higher dense fluids from the inner side to the outer side of the annulus. In this case, the interface was found to be more stable only as the interfacial tension increases (typically 0.05N/m), for the case of dense fluid in the inner side of the annulus. However, with negligible effect of interfacial tension, it is found possible to achieve stability of the interface when the heavier fluid is outside, though the nature of vortex patterns is not clear. This happened for the case of rotation ratio of 1.3 (the centrifugal force is not too large) and gravity is neglected. Under these conditions, if interfacial tension is large enough to stabilize the vortices, then stability is possible for all volume fractions with the denser fluid inside. Further, it has been observed that at high centrifugal forces, the gravity effects become negligible to effect the stratification of the two fluids. This further indicates that the interface stability can be controlled with the interfacial tension.

[FIGURE 11 OMITTED]

CONCLUSIONS

Three-dimensional CFD simulations have been carried out for the case of two fluids stratifying centrifugally in the annulus of the horizontal Taylor-Couette contactor. It has also been noted that the cell patterns obtained in different tangential planes are found to be structurally similar. These can be summarized as follows:

1. Complete energy balance has been established for the circumferential, two-phase radially stratified flow along the length of the annulus. The modes of dissipation are viscous and turbulent and the total dissipation rate has been shown to be equal to the energy input rate. The establishment of energy balance may be considered as the basis for the validation of the model reported in this work.

2. Critical condition for the stratification to occur has been determined and expressed as a function of the mean radial velocity component at the interface of the two fluids.

3. The extent of interface deformation with respect to each phase can be tracked as a function of the interfacial area as the interface deforms relative to the interfacial area if the interface were cylindrical.

4. Considering the case of rigid rotation of cylinders, yielded a stable interface with Taylor vortices in both phases. This signifies intra vortex mixing in each phase rather than inter vortex mixing in both phases leading to dispersion. As the rotation ratio deviated from 1, there occurred vortex deformation, subsequently followed by stretching of vortices in both the phases with an increase in the inner cylinder speed. This is beneficial in determining the axial and radial dispersion characteristics of the phases.

5. At negligible interfacial tension (S = 0.0001 N/m), the phases started to interpenetrate in·ter·pen·e·trate  
v. in·ter·pen·e·trat·ed, in·ter·pen·e·trat·ing, in·ter·pen·e·trates

v.intr.
To become mixed or united by penetration: planes that interpenetrate in a painting.
 leading to dispersion. While, as the interfacial tension was increased, the vortex patterns were more evident. Herein, it is to be noted that the Ta in each phase is independent of S. This has been found to hold good for all the aspect ratios considered (2, 4, 10). There was no significant variation in the extent of interface deformation with aspect ratio.

6. Interface stability for the case of lighter fluid towards the outer wall and heavier fluid towards the inner wall was achieved as the interfacial tension was increased to 0.05 N/m, at high rotation rate of 1.3. At such high centrifugal forces, gravity effects can be neglected.

7. As the inner cylinder rotation is further increased, after the onset of instability, the axial velocity profiles show that the vortex pattern stretches towards a helical pattern. This might be observed as a secondary transition to the onset of instability in each phase.
NOMENCLATURE

[r.sub.i]           inner cylinder radius (m)
[r.sub.o]           outer cylinder radius (m)
[r.sub.o,i]         [r.sub.o]/[r.sub.i]
d                   gap width (m)
e                   eccentricity
v                   average axial velocity (m/s)
Fr                  Froude number [r.sub.o][[OMEGA].sup.2.sub.o]/g
Re                  Reynolds number
[Re.sub.Cr]         critical Reynolds number
Ta                  Taylor number [MATHEMATICAL EXPRESSION NOT
                      REPRODUCIBLE IN ASCII]
[Re.sub.z]          axial Reynolds number (vd/v)
q                   flow rate ([m.sup.3]/s)
l                   annulus height (m)
[Re.sub.[theta]]    azimuthal Reynolds number
                      (([[OMEGA].sub.i]-[[OMEGA].sub.o][r.sub.i]/d))
S                   interfacial tension

Greek Symbols

[OMEGA]             angular velocity (rad/s)
[[OMEGA].sub.o]     angular velocity of outer cylinder (rad/s)
[[OMEGA].sub.i]     angular velocity of inner cylinder (rad/s)
[[OMEGA].sub.i,o]   [[OMEGA].sub.i]/[[OMEGA].sub.o]
[[OMEGA].sub.in]    angular velocity of the interface (rad/s)
[alpha]             holdup
[theta]             azimuthul coordinate
[rho]               density (kg/[m.sup.3])
[mu]                molecular viscosity (kg-m/s)
[eta]               viscosity ratio
[sigma]             interfacial tension (kg/s2)
[lambda]            wavelength (m)
v                   kinematic viscosity (m2/s)
[GAMMA]             aspect ratio (l/d)

Subscripts

i                   inner cylinder
in                  interface
o                   outer cylinder
r                   radial direction
z                   axial direction
[theta]             azimuthal direction
p                   phase


ACKNOWLEDGEMENTS

This work has been part of the project program supported by the Board of Research in Nuclear Sciences (BRNS), Sanction No. 2002/34/7-BRNS/140. The authors also acknowledge L. M. Gantayet for the discussions provided during this study.

Manuscript received May 20, 2005; revised manuscript received February 3, 2006; accepted for publication February 3, 2006.

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in a solid body, the gradient of a shear stress force through the body;
in a fluid, it is the flow induced by such a force gradient - see Viscosity for a fuller treatment.
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Sreepriya Vedantam (1), Jyeshtharaj B. Joshi (1) * and Sudhir B. Koganti (2)

(1.) Institute of Chemical Technology, University of Mumbai Most of the colleges in the city of Mumbai (Bombay) and the districts of Thane, Raigad, Ratnagiri and Sindhudurg are affiliated to the University of Mumbai. The University of Mumbai offers Bachelors, Masters and Doctoral degrees to students. , Matunga, Mumbai-400 019, India

(2.) Indira Gandhi Noun 1. Indira Gandhi - daughter of Nehru who served as prime minister of India from 1966 to 1977 (1917-1984)
Gandhi, Indira Nehru Gandhi, Mrs. Gandhi
 Centre for Atomic Research, Kalpakkam, TN-603 102, India

* Author to whom correspondence may be addressed.

E-mail address: jbj@udct.org
Table 1. Establishment of energy balance (ri=0.042 m, ro = 0.051 m,
[[OMEGA].sub.o]=8 rps, [[rho].sub.o]=1150 kg/[m.sup.3],
[[rho].sub.I]=853 kg/[m.sup.3])

Heavier
phase     Inner cylinder        Energy balance
holdup     speed (rps)     (kg-[m.sup.2]/[s.sup.3])

                                    Input              Predicted

0.3             11                0.20 e-03            0.19 e-03
                12                0.11 e-02            0.99 e-03
                14                7.25 e-03            7.02 e-03
                16                2.29 e-02            2.19 e-02

0.4             11                 0.2 e-02            0.19 e-03
                12                0.11 e-02            1.03 e-03
                14                7.48 e-03            7.37 e-03
                16                2.36 e-02            2.30 e-02

0.5             11                0.21 e-03            0.21 e-03
                12                1.17 e-03             1.2 e-03
                14                7.71 e-03            7.82 e-03
                16                2.43 e-02            2.39 e-02

0.65            11                0.22 e-03             0.2 e-03
                12                1.22 e-03            1.17 e-03
                14                8.05 e-03            7.81 e-03
                16                2.54 e-02            2.50 e-02

0.7             11                0.22 e-03            0.27 e-03
                12                1.24 e-03             1.3 e-03
                14                8.16 e-03            8.41 e-03
                16               2.574 e-02           2.623 e-02
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Date:Jun 1, 2006
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