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

Two-fluid Taylor-Couette flow, 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 simulations.
The occurrence of radial stratification, the subsequent onset of
centrifugal 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 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 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 instability termed as the centrifugal instability with a number of secondary variations has led to many chemical process applications, which include emulsion polymerization, synthesis of silica particles, heterogeneous catalytic reactions and liquid-liquid extraction (Imamura et al., 1993; Ogihara et al., 1995; Cohen 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), or facilitated settling (bioseparations) are required (Baier et al., 1999). A schematic of such a contactor, with dual cell pattern is shown in Figure 1.

[FIGURE 1 OMITTED]

The flow pattern obtained in the annular region could be with two phases retaining their individual integrity and contacting each other at a single well-defined interface (stratified 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, 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. 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 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. The dimensionless groups describing the two-fluid Taylor-Couette flow are: Taylor number (Ta) for each phase (to signify the centrifugal instability); a Froude number (Fr) (for the gravitational 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, 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 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 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 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 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; 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 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 component at the interface and experimentally operable parameters as:

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

As established by Rayleigh (1916), a radial stratification of angular momentum 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] (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 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, 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, [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.

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 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) has been used. In this model, individual Reynolds stresses [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 diffusivity term as (Launder 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 dissipation term. The turbulent Mach number 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 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) scheme was used. This uses a discrete continuity balance for a staggered control volume about the face to compute the "staggered" pressure. The Eccentricity (ratio of offset distance of the cylinder axis to the average gap width) was assumed to be zero. A segregated solver with implicit linearization 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 and the energy dissipation rate. Inner iterations were carried out until mass conservation as per the convergence criteria (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 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 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 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 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.

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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Ogihara, T., G. Matsuda, T. Yanagawa, N. Ogata, K. Fujita and M. Nomura, "Continuous Synthesis of Mono Dispersed Silica Particles Using Couette-Taylor Vortex Flow," J. Cer. Soc. Jpn. Int. 103, 151. (1995).

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Toya, Y. and I. Nakamura, "Instability of Two-Fluid Taylor Vortex Flow," Trans. Jpn Soc. Mech. Eng. Part B 63(612), 35-43 (1997).

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Wereley, S. T. and R. M. Lueptow, "Inertial Particle Motion in a Taylor Couette Rotating Filter," Phys. Fluids 11, 325 (1999).

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Sreepriya Vedantam (1), Jyeshtharaj B. Joshi (1) * and Sudhir B. Koganti (2)

(1.) Institute of Chemical Technology, University of Mumbai, Matunga, Mumbai-400 019, India

(2.) Indira Gandhi Centre for Atomic Research, Kalpakkam, TN-603 102, India

* Author to whom correspondence may be addressed.

E-mail address: jbj@udct.org

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 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 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 instability termed as the centrifugal instability with a number of secondary variations has led to many chemical process applications, which include emulsion polymerization, synthesis of silica particles, heterogeneous catalytic reactions and liquid-liquid extraction (Imamura et al., 1993; Ogihara et al., 1995; Cohen 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), or facilitated settling (bioseparations) are required (Baier et al., 1999). A schematic of such a contactor, with dual cell pattern is shown in Figure 1.

[FIGURE 1 OMITTED]

The flow pattern obtained in the annular region could be with two phases retaining their individual integrity and contacting each other at a single well-defined interface (stratified 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, 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. 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 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. The dimensionless groups describing the two-fluid Taylor-Couette flow are: Taylor number (Ta) for each phase (to signify the centrifugal instability); a Froude number (Fr) (for the gravitational 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, 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 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 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 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 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; 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 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 component at the interface and experimentally operable parameters as:

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

As established by Rayleigh (1916), a radial stratification of angular momentum 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] (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 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, 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, [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.

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 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) has been used. In this model, individual Reynolds stresses [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 diffusivity term as (Launder 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 dissipation term. The turbulent Mach number 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 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) scheme was used. This uses a discrete continuity balance for a staggered control volume about the face to compute the "staggered" pressure. The Eccentricity (ratio of offset distance of the cylinder axis to the average gap width) was assumed to be zero. A segregated solver with implicit linearization 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 and the energy dissipation rate. Inner iterations were carried out until mass conservation as per the convergence criteria (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 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 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 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 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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Sreepriya Vedantam (1), Jyeshtharaj B. Joshi (1) * and Sudhir B. Koganti (2)

(1.) Institute of Chemical Technology, University of Mumbai, Matunga, Mumbai-400 019, India

(2.) Indira 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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Title Annotation: | chlorofluorocarbons |
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Author: | Vedantam, Sreepriya; Joshi, Jyeshtharaj B.; Koganti, Sudhir B. |

Publication: | Canadian Journal of Chemical Engineering |

Geographic Code: | 1CANA |

Date: | Jun 1, 2006 |

Words: | 7020 |

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