Computational Fluid Dynamics Simulation and Experimental

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Ind. Eng. Chem. Res. 2010, 49, 14–28

KINETICS, CATALYSIS, AND REACTION ENGINEERING Computational Fluid Dynamics Simulation and Experimental Investigation: Study of Two-Phase Liquid-Liquid Flow in a Vertical Taylor-Couette Contactor Mayur J. Sathe,† Sandesh S. Deshmukh,† Jyeshtharaj B. Joshi,*,† and Sudhir B. Koganti‡ Institute of Chemical Technology, Matunga, Mumbai-400 019, India, and Indira Gandhi Centre for Atomic Research, Kalpakkam, TN-603 102, India

Two-fluid Taylor-vortex flow in coaxial cylinders with a stationary outer cylinder and a rotating inner cylinder has potential advantages over the conventional extraction equipment in chemical and bioprocess industries. The two phase flow behavior in the vertical annular region is either in the form of homogeneous dispersion, banded dispersion, segregated or the stratified flow with both the phases retaining individual integrity. Computational fluid dynamics (CFD) simulations have been carried out for the annular region. The effect of physical properties like density difference, interfacial tension, and the effect of geometrical parameters such as annular gap have been studied. The various regime transmissions have been investigated using CFD as well as flow visualization by particle image velocimetry (PIV) and planar laser induced fluorescence (PLIF). The flow pattern has also been measured and compared with the CFD predictions. An attempt has been made to present the results on transition in the form of a regime map with the Eotvos and Taylor numbers as the two coordinates. 1. Introduction

2. Previous Work

Two-fluid Taylor-vortex flow in coaxial cylinders with a stationary outer cylinder and a rotating inner cylinder (Figure 1, sections 1 and 2, respectively) has potential advantages over the conventional extraction equipment in chemical and bioprocess industries.1-4 Depending upon the orientation of rotating cylinder, the equipment is classified as a horizontal or vertical contactor. In both types of contactors the flow behavior is either in the form of a homogeneous dispersion, banded dispersion, or the stratified flow with both the phases retaining their individual integrity. To design this equipment, it is important to understand a priori the conditions under which the transition takes place from one type of flow regime to the another. This subject needs to be addressed for vertical flows because the industrial scale equipment has vertical orientation. Some published work is available for horizontal orientation.5-7 Although there have been numerous studies on the single-phase Taylor-Couette flow, the effect of the presence of second immiscible phase on the hydrodynamics of this contactor still remains to be delved. The present study has focused on the flow patterns in two-phase flows in a vertical coaxial cylinder system. The various regime transitions have been investigated using computational fluid dynamics (CFD) as well as flow visualization using particle image velocimetry (PIV) and planar laser induced fluorescence (PLIF). At very low tangential velocities a clear interface appears in between the two stratified phases. With increase in tangential velocity, this interface takes paraboloidal shape and the parabola (in 2D) deepens with corresponding increase in the tangential velocity. On further increase in tangential velocity, the interface breaks and the two phases enter in the territories of each other. On further increase in the tangential velocity liquid-liquid dispersion is observed.

The stratification of the liquid-liquid flow in the annulus of the horizontal Taylor-Couette contactor has been reported by several authors who numerically and/or experimentally investigated the conditions of rigid rotation, interface stability, and Taylor instabilities in each fluid layer.8-13 Also, dispersed flows in the annulus of Taylor-Couette contactor with fixed outer cylinder and rotating inner cylinder have been reported relatively recently.14-16 The annular region of horizontal axis coaxial cylinder system filled with equal volumes of various sets of aqueous and organic fluids, gave way to different flow patterns, including layered and banded Couette flows, rollers, coarse and foamy emulsions, and also phase inversions at higher volume fractions of the secondary dispersed phase.5 They focused their study on the hydrodynamic structures and the effects of density ratio, viscosity ratio, feed composition, and azimuthal Reynolds

* To whom correspondence should be addressed. E-mail: [email protected]. Tel.: +91 22 2414 0865. Fax: +91 22 2414 5614. † Institute of Chemical Technology. ‡ Indira Gandhi Centre for Atomic Research.

Figure 1. Schematic of vertical annular domain: (1) rotating inner cylinder, (2) outer stationary cylinder, (3) annular region.

10.1021/ie900185z  2010 American Chemical Society Published on Web 12/03/2009

Ind. Eng. Chem. Res., Vol. 49, No. 1, 2010

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Figure 2. Schematic of experimental set up of PIV and PLIF: (C1) PLIF camera, (C2) PIV camera, (S) synchronizer, (M) dichroic mirror, (T) equipment, (L) laser source, (PC) personal computer.

number (Reθ). Three distinct structures were observed: (i) at large rotation rates, alternating water and organic-rich vortices at low organic-phase volume fractions; (ii) at moderate rotation rates, a spatially homogeneous emulsion with phase inversion at high organic-phase volume fraction; and (iii) at low rotation rates, an axially translating periodic variation between the banded and homogeneous states. Using photography, it was concluded that the banded flow pattern does not consist of separate aqueous and organic-rich vortices. Instead, the banded appearance is caused by disperse-phase droplet migration to vortex cores. A series of experiments with various fluid pairs suggested that the density and the viscosity differences between the fluid phases could not entirely explain the droplet migration to vortex cores. Further, the interfacial tension forms an important factor for the formation of the banded structure. Zhu and Vigil16 reported that the banded flow arises when the less dense dispersed phase migrates to the vortex cores due to the centrifugal forces of the vortices. They developed a scaling relation that describes conditions leading to banded flows based on the characteristic time scales required for the particles to move from periphery to the core and from core to the periphery. Near the transition, the ratio of these characteristic times was found to be close to 1. This criterion depends upon the physical dimensions of the annulus and the operating parameters such as dd, µ, and Ω. However, the criterion fails to explain the appearance of banded patterns when dense phase is dispersed in light phase. For such a case, the CFD simulations showed the accumulation of dispersed phase preferentially at inflow boundaries, where the flow moves from the outer to inner cylinder, between the vortices. Further, the limitations in the experimental methodology could not produce a clear picture of the phenomena. The numerical studies reported the magnitudes of the flow velocities and the hold-up fraction in the flow domain. However, the core velocities and the turbulent parameters cannot be determined, and thus the studies lack in quantitative validation. They also indicated the possibility that the two-phase flow can be considered to be identical to the single-phase Taylor-Couette flow if the average mixture properties are considered, particularly for the homogeneous dispersion. From the foregoing discussion it is clear that very limited information is available on two-phase Taylor-Couette flows with a vertical geometry. Thus it was thought desirable to carry out the experiments in vertical geometry. The operating conditions in the present work spanned the stratified, banded, and

homogeneous dispersion regime. Besides high speed photography, the flow patterns in the two-phase dispersed phase has been quantified with the help of PIV. To have deeper insight into liquid hold-up distribution, PLIF and PIV experiments have been carried out. To characterize the cell structures and different transitions in the liquid-liquid dispersed flow, extensive CFD simulations have been performed. The CFD model has been validated using the velocity data obtained from PIV experiments. An attempt has been made to present the results on transition in the form of regime map with the Eotvos and the Taylor number as the two coordinates. The following sections present the details on experimental, numerical simulations, followed by results and discussion. The present work focuses on the identification of the four regimes of operation: (a) stratified phases, (b) segregated dispersion, (c) banded dispersion, and (d) homogeneous dispersion. The work also tries to establish the conditions under which the transition occurs from one regime to other. Since the actual extraction operation should occur in the regime of homogeneous dispersion, the present work will be useful to the practicing engineers for establishing the homogeneous regime in the proposed design of the annular centrifugal extractors. 3. Experimental Section The experiments were carried out to study the liquid-liquid dispersion in an annular gap between two concentric cylinders of finite height with a rotating inner cylinder. The two-phase flow was analyzed out using PIV, PLIF, and high speed photography. The schematic of experimental setup is shown in Figure 2. The apparatus was fabricated using acrylic (plexiglas). The main advantages of using acrylic are the ease of machining, transparency, and the uniformity of refractive index within the individual part. The setup for the present study consisted of a rotating acrylic cylinder mounted vertically and concentric with the outer cylinder which was stationary. The rotating cylinder was called “rotor” while the stationary cylinder was called “stator”. The rotor OD was 39 mm while the ID of the stator was 49 mm. Thus the annular gap, d, was 5 mm. The height of the annulus was 57 mm. Thus, the aspect ratio, Γ, for the current setup is about 11.5. Kerosene (F ) 790 kg/m3; µ ) 0.0023 Pa s) and NaI solution in water, having refractive index similar to kerosene (F ) 1560 kg/m3; µ ) 0.00145 Pa s) were used as working fluids. In all the measurements, kerosene was observed to be

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the continuous phase and NaI solution was the dispersed phase. The interfacial tension between these two phases has been measured using a capillary tube method by Rashidnia et al.,17 which turned out to be 32 mN/m. A high-resolution PIV and PLIF combined system was employed to measure the liquid velocity, drop size, and dispersed-phase hold-up in the annulus simultaneously. A pulsed Nd:YAG laser (New Wave Research, Inc.) having pulse duration of 6 ns and pulse energy of 120 mJ/pulse was used as a laser source (L). It was synchronized with the camera using a synchronizer (S). The light sheet optics included a combination of cylindrical and spherical lenses, attached in view of creating a thin laser sheet (thickness ≈ 1 mm). The images were captured using two high-resolution 4 M CCD cameras (of 15 Hz frequency): one (C1) was placed perpendicular to the laser sheet, and the other (C2) was in front of a dichroic mirror (M, placed at 45° between the C1 and the laser sheet). Three kinds of optical filters (a dichroic mirror, a narrow band-pass optical filter, and long-wave pass filter) were used to separate the laser-induced fluorescence (LIF) lights from scattered laser lights, which were then recorded separately to obtain PLIF and PIV images simultaneously. A dichroic mirror separated the LIF light (λ ≈ 670 nm) for PLIF and scattered light (λ ) 532 nm) for PIV images. LIF light directly passed to the C1 camera through a high-wave pass filter, while scattered light was reflected to the C2 camera through a narrow band-pass filter. The difference in refractive indices of the vessel wall and the individual liquids causes strong refraction and scattering of the laser sheet. These reflections have few orders of magnitude higher intensity than the scattered laser light from particles, hence making PIV measurements difficult. After several refractions and reflections from interfaces, the laser light sheet no longer remains planar, and hence the precision of velocity magnitudes estimated using cross correlation of PIV images reduces by a significant extent. Very high intensity reflections can also damage the CCD sensor of the camera. To overcome these difficulties, the refractive indices of the two fluids and the cylinder wall need to be equal. Because of the strong curvature of the cylinder, special care has to be taken while designing the setup for a photographic technique like PLIF. The refractive indices (RIs) for water, kerosene, and acrylic are 1.33, 1.437, and 1.47, respectively. Kerosene was used in spite of having a slight RI difference (0.04) because of the simplicity in handling. Sodium iodide solution having an RI 1.437 was used to match the refractive index with kerosene. The entire apparatus was kept in a square tank filled with sodium iodide solution with RI 1.437, similar to that of the liquids used. The outer cylinder wall was kept thin (2.5 mm) in order to minimize the lensing effect due to slightly higher RI of acrylic. It is not possible to match the RI of liquid with air. Thus, utmost care was taken to avoid air induction during the experiments by providing proper sealing near the top surface of annulus. By using this technique, the artifacts of refractive gradient mismatch have been removed to a substantial extent. Rhodamine B has been added to the sodium iodide solution (heavy phase). Hence, only the heavier phase emits the fluorescent light and appears bright. This facilitates the study of planar cross sections of liquid hold-up in axial direction. This is essential to gain insight into the heterogeneity of liquid holdup. The images captured were observed to be sharp even at the boundaries of inner and outer cylinder ensuring negligible artifacts. The liquid hold-upfield was generated by averaging 500 binarized drop images together. Since the sodium iodide droplets

Table 1. Operating Conditions for Experiments N, rpm dp, mm 240 565 750 1050 1300 1440 1570

3.08 0.85 0.56 0.35 0.24 0.21 0.18

Ta 1 309 577 7 257 809 12 788 841 25 066 128 38 423 362 47 144 783 56 041 269

Ta/TaC vortex regime dispersion regime 642 3 556 6 266 12 281 18 826 23 099 27 458

chaotic turbulent turbulent turbulent turbulent turbulent turbulent

stratified segregated segregated banded banded homogeneous homogeneous

contain fluorescent dye, they appear bright in the PLIF images which record the fluorescence intensity. The drops were identified using an image processing program written in MATLAB. The pixels inside the drops detected in individual drops were assigned the hold-up value of 1, while the background was assigned the value of 0. After averaging 300 images, the hold-up contour was obtained as shown in Figure 7. Although, the increase in number of images smoothes the contours, 300 images gave a fair picture of hold-up profile. The image shown in Figure 7 is time averaged using 300 images. This amount of images is not sufficient to predict holdup fluctuations and hence they are not reported. To obtain a good picture of hold-up fluctuations, we need to perform a moving average operation over many more numbers of images; at least 100 times those used to generate the approximation of planar hold-up contour shown in Figure 7. This becomes computationally and experimentally very expensive. Besides, the principal aim of the current work is to identify the flow regimes. Owing to very high tangential velocity compared to the axial and radial velocities, the dispersed-phase hold-up profiles are practically uniform in the tangential direction. Thus, the liquid-liquid dispersion regime can be identified using a single, time averaged, and planar hold-up profile on a vertical plane. While carrying out the experiments, the desired continuous phase was first filled in the apparatus and the rotor was slowly accelerated to the desired rotational speed. The dispersed phase was then injected through the inlet provided at the top using a syringe. After attaining a steady state the measurements were carried out. The kerosene was seeded with silver-coated hollow glass beads of 10 µm size, while a small amount of laser fluorescent dye Rhodamine B (Exciton, Inc.) was added in NaI solution. With traces of dye going into kerosene, it was equilibrated with dyed NaI solution and then used for the measurements. The concentration of the fluorescent dye in sodium iodide was ensured to be significantly higher than that in the kerosene. Thus it was possible to distinctly identify the dispersed phase in PLIF images. The operating conditions for different experiments are summarized in Table 1. Seeding particles adhere to interface because of the improper wetting by either liquid. To alleviate this problem, the particles were first wetted using a very small quantity of nonionic surfactant solution (200 ppm Triton X-100 in distilled water). This wetted seeding was then diluted with distilled water and sonicated for 2-3 min. A uniform suspension of seeding particles was thus achieved in water. Even after taking utmost care for uniform suspension, a very few of the particles still adhered to the drop interface. However, their fraction was found to be very less. The primary reason for this is that the very high tangential velocity and correspondingly higher interface fluctuations ensure suspension of the seeding particles. In addition to the PIV and PLIF experiments, the high speed imaging was also performed for the same geometry with water (F ) 998.2 kg/m3; µ ) 0.001 Pa s) dyed using a dye brown HT and kerosene (F ) 790 kg/m3; µ ) 0.0023 Pa s) as two phases


using a high speed camera (Photron Fastcam super 10 kc). The interfacial tension between the two phases was measured to be 50 mN/m, and an equal amount of the two phases was taken. The images were recorded at a frame rate of 300 frames/sec with the shutter speed up to 1/5000 s. One image was sampled out of every 10 images in order to ensure that different drops are captured in every image to determine the drop size. Thus, the effective recording rate was 30 frames/second. These images were used to give the information about drop sizes. 4. Numerical Simulations The geometry for the Taylor-Couette flow was considered axisymmetric around the axis of rotation. When the geometry is axisymmetric the analysis needs to include swirl or rotation, and these can be modeled in 2D (i.e., solve the axisymmetric problem) and include the prediction of the circumferential (or swirl) velocity. For the numerical simulations of the flow in the annular region, Reynold’s stress model (RSM) was chosen. This was mainly due to its ability to predict the flow behavior in case of anisotropic turbulence, which exists in the annulus. Deshmukh et al.18 have used laser doppler velocimetry (LDV) to obtain the velocity profiles in the annulus. They have found that the numerical solution obtained with RSM is very close to the actual behavior captured with LDV. The CFD simulation was made grid size independent, and a typical minimum grid size was 0.053 × 0.053 mm2. The commercial software FLUENT (version 6.3) was used in all the studies. A pressure based solver was used for solving the momentum equations. The second order upwind scheme was used to discretize the momentum equations as well as turbulence parameters. For the pressure velocity coupling, phase-coupled SIMPLE scheme was used. The multiphase flow was modeled using the Eulerian multiphase model in FLUENT where the phases are considered to be as interpenetrating continua. The simulation also assumes that the droplets are spherical and have uniform and constant diameter. Further, there is no coalescence or breakup occurring in the annulus. The other assumptions are the absence of mass transfer between the phases, negligible virtual mass forces, and the pressure is the same for both the phases. Since both the continuous and the dispersed phases are liquids, they are treated in the same way while solving the governing equations. The continuity and momentum equations for both the phases are given in Table 2. In a fluid-fluid formulation, both phases can be averaged over a fixed volume. The phase fraction, ∈, for the individual phase is calculated as follows: ∈)

1 V

∫ X(r) dV V

r

where V is the averaging volume. This volume is relatively large compared to the volume of individual molecules. A phase indicator function is introduced, X(r), which is unity when the point r is occupied by the given phase, and is zero otherwise. The exchange coefficient, KDC for liquid-liquid mixtures appearing in these equations can be written in the following general form: KDC )

∈D∈CFDf td

(1)

The drag function, f ) CDRe/24 is employed using the model of Schiller and Naumman (Clift et al.19) in which

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for Re e 1000, CD ) for

24 (1 + 0.15Re0.687) Re

Re g 1000, CD ) 0.44

(2) (3)

and, td, relaxation time, is defined as td ) FDdD /(18µC), with µC, FD, and dD being the primary phase viscosity, density, and drop diameter of dispersed phase, respectively. The RSM model equations used in the current work are standard in Fluent. Since we are using the drag law and turbulence model which is available in FLUENT, there was no need to write additional user defined functions to solve the flow equations. The solution of the multiphase model described above for liquid-liquid dispersions is limited by the prior need of the knowledge of drop size of the dispersed phase. The drop size was estimated using the correlation for dispersion of an aqueous-organic liquid pair in Taylor-Couette flow given by Hass20 as 2

( )()

µD dD ) 150(We)-0.65(Reθ)-0.2 D µC

0.5

d ri

0.5

(4)

where We ) D(Ωri)2FC/σ and Reθ ) DΩriFC/(2µC). This correlation has been checked for experimental geometry and conditions and then used in the simulations. It should be noted that this correlation does not account for variation in dispersedphase hold-up, and hence the CFD results are independent of the dispersed-phase hold-up. 5. Results and Discussion As mentioned earlier, liquid-liquid two-phase flow, in a horizontal axis Taylor-Couette contactor, shows different types of flow patterns such as stratified, banded, and homogeneous dispersion. The present work is concerned with a two-phase flow in a vertical axis Taylor-Couette contactor. In addition to the above patterns, the vertical system shows one more type of pattern because of strong gravitational effect. This is called a segregated dispersion and occurs at relatively lower rotational speeds compared to banded and homogeneous dispersion. The photograph of stratified, segregated, banded, and homogeneous dispersion in a vertical geometry is shown in Figure 3. To understand these flow patterns in detail, extensive CFD simulations have been carried out. The flow visualization experiments have been performed using PIV and PLIF. CFD simulations as well as experiments have been carried out in single as well as liquid-liquid two phase flows. The experimental data obtained using PIV have been used for the validation of CFD simulation. 5.1. Drop Size. The drop size is one of the most critical inputs which influence the CFD solution. Mechanistically, the turbulent stresses try to break drops while the elastic stresses generated due to the interfacial tension tend to restore its original shape. The balance between these two forces results in maximum stable drop diameter (dmax). Based on the theory of isotropic turbulence, for liquid-liquid dispersion the maximum stable drop size is related to the Weber number (We) as dmax ∝ We-0.6. In the current work, the systematic study of drop size was carried out by image processing using PLIF and high speed camera photographs. Starting from the lower rotational speed of the inner cylinder, and equal volumes of kerosene and water, the rotational speed was then slowly increased to the desired speed. It was observed that the interface, which was deformed upward initially, after a certain rotational speed, started oscil-

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Figure 3. Photographs of various flow regimes that can be observed in a vertical Taylor-Couette liquid-liquid dispersion (kerosene dyed with iodine in water): (A) stratified flow, (B) segregated dispersion, (C) banded dispersion, (D) homogeneous dispersion. Table 2. Transport Equations for Two-Phase Flows property

equation ∂(∈CFC) ∂(∈CFC〈u1,C〉) ∂(∈CFC〈u2,C〉) ∂(∈CFC〈u3,C〉) + + + )0 ∂t ∂x ∂y ∂z

continuity for continuous phase

x-momentum for continuous phase

y-momentum for continuous phase

z-momentum for continuous phase

∂τ11,C ∂τ21,C ∂p1 ∂(∈CFC〈u1,C〉〈u1,C〉) ∂ (∈ F 〈u 〉) + ) -∈C + + + ∂t C C 1,C ∂x ∂x ∂x ∂y ∂τ31,C + ∈CFCg1 + KDC(〈u1,D〉 - 〈u1,C〉) ∂z ∂τ22,C ∂τ32,C ∂p2 ∂(∈CFC〈u2,C〉〈u2,C〉) ∂ (∈ F 〈u 〉) + ) -∈C + + + ∂t C C 2,C ∂y ∂y ∂y ∂z ∂τ12,C + ∈CFCg2 + KDC(〈u2,D〉 - 〈u2,C〉) ∂x ∂τ13,C ∂τ23,C ∂p3 ∂(∈CFC〈u3,C〉〈u3,C〉) ∂ (∈ F 〈u 〉) + ) -∈C + + + ∂t C C 3,C ∂z ∂z ∂z ∂y ∂τ13,C + ∈CFCg + KDC(〈u3,D〉 - 〈u3,C〉) ∂x ∂(∈DFD) ∂(∈DFD〈u1,D〉) ∂(∈DFD〈u2,D〉) ∂(∈DFD〈u3,D〉) + + + )0 ∂t ∂x ∂y ∂z

continuity for dispersed phase

x-momentum for dispersed phase

y-momentum for dispersed phase

z-momentum for dispersed phase

∂τ11,D ∂τ21,D ∂p1 ∂(∈DFD〈u1,D〉〈u1,D〉) ∂ (∈ F 〈u 〉) + ) -∈D + + + ∂t D D 1,D ∂x ∂x ∂x ∂y ∂τ31,D + ∈DFDg1 + KDC(〈u1,C〉 - 〈u1,D〉) ∂z ∂τ22,D ∂τ32,D ∂p12 ∂(∈DFD〈u2,D〉〈u2,D〉) ∂ (∈ F 〈u 〉) + ) -∈D + + + ∂t D D 2,D ∂y ∂y ∂y ∂z ∂τ12,D + ∈DFDg2 + KDC(〈u2,C〉 - 〈u2,D〉) ∂x ∂τ13,D ∂τ23,D ∂p3 ∂(∈DFD〈u3,D〉〈u3,D〉) ∂ (∈ F 〈u 〉) + ) -∈D + + + ∂t D D 3,D ∂z ∂z ∂z ∂y ∂τ33,D + ∈DFDg + KDC(〈u3,C〉 - 〈u3,D〉) ∂x

lating and finally broke such that drops of individual phases entered into the other phase, and thus there was a coexistence of two dispersions: one was water in kerosene at the top of the contactor, while the other was kerosene in water at the bottom half of the contactor (Figure 4A). Since the water was dyed with brown dye, the bottom half region of the contactor appeared very dark. Hence good drop images could only be obtained for

the dispersion at the top of the contactor, when the rotational speed was below 20 r/s while using white backlight and high speed camera. It was possible to find the drop sizes in the annulus using such images. A typical image is shown in Figure 4B. The area of each drop was marked out and measured from an image. Assuming the drop to be spherical at all the rotational speeds, the number averaged drop diameter was calculated.


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Figure 5. Comparison for the measured and predicted drop size.

Figure 4. (A) High speed image showing presence of two dispersions simultaneously; (B) typical high speed image of Taylor-Couette dispersion (water dispersed in kerosene) for the rotational speed of 19 r/s.

Drop size measurements are reported for Ta ) 2.62 × 107, 1.43 × 107, 1.61 × 107, and 8.64 × 106, which are at least 4000 times the critical Ta for this geometry (TaCr ) 2041). This value of Taylor number clearly indicates that the flow is turbulent. It can be found that the Hass correlation overpredicts the drop size by 16.8%. However, it must be noted that even after using state of the art image processing to determine the drop sizes it is very difficult to avoid artifacts like two touching/ overlapping drops to be detected as a single drop. Also, the drops are randomly distributed, and there are chances of missing small sizes. Hence, the uncertainty in the estimated drop size itself is of the order of 15%. This is comparable to the error in the drop sizes predicted by the Hass correlation. Comparison of the critical Taylor number with those corresponding to the operating conditions in the current work throws some light over the flow patterns prevailing in the annulus. The transitions in Taylor-Couette flow are more popularly repre-

sented in the form of Taylor number. The various transitions of single-phase flows have been discussed by Deshmukh et al.18 The critical Taylor number is the one at which the taylor vortices start to form because of instability, which corresponds to 2040. In the case of the presence of two different liquids in the annulus, the Taylor numbers are different at the same RPM because the Taylor number also depends on the kinematic viscosity of the fluid. Taylor vortices start to form in each individual phases at the respective speeds when the corresponding Taylor number exceeds the critical Taylor number. However, our prime interest lies in the design of this type of devices for liquid-liquid extraction. The two liquids start to disperse into each other only when the axial-radial vortex velocity is high enough, so that the inertial forces overcome the surface tension forces at the liquid-liquid interface. This causes break-up of the interface with subsequent formation of drops. Also, the intercell exchange velocity needs to be high enough to disperse these drops to the entire annulus from the liquid-liquid interface. The different forces acting on the droplets have been summarized in Table 4. It can be observed that, the Taylor numbers required to satisfy the criterion of inertial force being greater than the surface tension force is 2.13 × 106. To summarize this description, the Taylor number required to achieve the dispersion of one liquid into another is more than 1000 times Tacr. Thus, it is clear that the experiments have been carried out in turbulent Taylor-Couette region. This is also confirmed by the instantaneous flow field obtained from PIV experiments. In the current work, the effects of the rotational speed and the aspect ratios on liquid-liquid dispersion have been investigated using CFD and experiments. However, the CFD simulations were carried out for a much wider range of parameters (N and Γ) than the experiments owing to their advantages like time of execution and lower cost. Key CFD simulations were tested against flow patterns obtained using experiments, to validate the parameters like the turbulence model, drop size, and the drag law selected. Thus, it was thought desirable use the current experimental drop size data as a benchmark for the correlations reported in literature to estimate the drop size in Taylor-Couette geometry. Such correlation can then be used for estimating drop size for CFD simulation where the experiments have not been carried out. The correlation reported by Hass20 (eq 6) was found to be the most relevant since it also included the data on drops dispersed in Taylor-Couette geometry. A parity plot for the present measurements against those calculated using eq 6 is shown in Figure 5. It can be found that the correlation overpredicts the drop size by 16.8%. However, it must be noted that even after using state of the art image processing to

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Figure 6. Vectors plot representing flow patterns obtained using PIV at 16.67 r/s with NaI solution dispersed in kerosene (A) ∈D ) 0.025; (B) ∈D ) 0.065.

determine the drop sizes it is very difficult to avoid artifacts like two touching/overlapping drops to be detected as a single drop. Hence, the uncertainty in the estimated drop size is of the order of 15%. This is comparable to the error in the drop sizes predicted by Hass correlation. Henceforth, eq 6 has been used for the estimation of drop size and was used in the CFD simulations. 5.2. PIV and PLIF Measurements. Simultaneous PIV and PLIF measurements have been performed for a two-phase flow in a vertical Taylor-Couette contactor with no net flow across the contactor. Figure 6 shows the continuous phase flow pattern at two different dispersed phase hold-ups of 2.5% and 6.5%, which was measured using PIV at a rotational speed of 16.67 r/s. The corresponding dispersed-phase hold-up obtained from PLIF images using the methodology described in section 4 are shown in Figure 7. The bands of dispersed phase were observed; however, the dispersed phase essentially accumulated at the bottom. This can be attributed to the significant density difference (∆F ) 780 kg/m3) between the two fluids. Hence, sodium iodide with higher density preferably accumulated at the bottom of the device even at lower values of hold-up. Thus, for the current rotational speed the corresponding axial velocities are insufficient to carry the heavier droplets right to top of the annulus and hence are not able to completely disperse one phase into another. Further, it can be seen that the heavy dispersed phase was distributed unevenly along the annulus height. The dispersed-phase hold-up is higher near the bottom, which decreases with an increase in the height. It can also be seen that, the dispersed-phase hold-up is more at the periphery of the rotating vortices than the core; and a large amount of dispersed phase is found near the outer stationary cylinder. This is again caused by the higher density of dispersed phase drops. They experience a corresponding higher magnitude of centrifugal force and are thrown to the periphery of the device, accumulating near the outer wall. The tangential velocity is an order of magnitude higher than the axial-radial vortex velocity;

Figure 7. Hold-up profiles obtained using PLIF at 16.67 r/s with NaI solution dispersed in kerosene: (A) ∈D ) 0.025, (B) ∈D ) 0.065.

and hence the radial component of centrifugal force is also 2 orders of magnitude higher than the force directing droplets away from the vortex core. It should be noted that the CFD result of Zhu and Vigil16 for kerosene dispersed in water also showed significant hold-up of dispersed phase in between the vortices and near the rotating wall rather than at the entire periphery of the vortex. The comparison between Figure 6 and Figure 7 clearly indicates that the dispersed phase (aqueous NaI) not only moves toward the periphery of the vortices but largely accumulates near the inflow region between the vortices. 5.3. Validation of CFD Model. The results of CFD have been compared with the experimental velocity and hold-up data obtained using PIV and PLIF measurements. Also, it was thought interesting to check the predictions of a limiting case of multiphase CFD model when the dispersed-phase hold-up is zero with the single-phase CFD model for the same geometry and grid size. The single-phase model studied by Deshmukh et al.18 was in good agreement with the experimental observations. This model has been used for studying the single-phase flow in the present work. Figure 8 shows the flow patterns obtained using PIV, singlephase CFD model, and as a limiting case of multiphase CFD model. It can be seen that the total number of vortices in the domain are 10. Figure 9 shows the comparison for mean axial and radial velocities at the center of the annulus along the annulus height. A good agreement can be seen between the predictions of mean velocities with both the CFD models. The rms error for radial velocity is 15.3%, while that for axial velocity is 33.5% with respect to the experimental data. It should be noted that, at the center line of the annulus where the profiles in the Figure 9 are shown, the maximum time averaged axial velocity is 0.025 m/s while the maximum time averaged radial velocity is 0.14 m/s. Thus, the errors in predictions of the weaker axial velocity magnitudes by CFD simulations are also expected to be on higher side. For the case mentioned in Figure 9, the Taylor number is 1.43 × 10,7 which corresponds to turbulent Taylor-Couette flow. Hence the simulations are carried out in highly turbulent


21

Figure 10. Predictions of CFD for flow patterns in terms of vector plot and hold-up profiles at 16.67 r/s: (A) ∈D ) 0.025; (B) ∈D ) 0.065.

Figure 8. Flow patterns in terms of vector plot for single-phase flow using kerosene: (A) PIV, (B) single-phase CFD model, (C) Eulerian multiphase model with ∈D ) 0.

Figure 11. Comparison between the CFD predictions and experimental values for mean axial and radial velocities at the center of annulus along the annulus height for ∈D )0.065: (9) experimental mean axial velocity; (∆) experimental mean radial velocity. Figure 9. Comparison between the CFD predictions and experimental values for mean axial and radial velocities at the center of annulus along the annulus height for ∈D ) 0: (2) experimental mean axial velocity; (]) experimental mean radial velocity; (black line) CFD (Eulerian multiphase model); (bold line) CFD (single phase model).

regime. Even if the RSM model is used with due care in making structured grid, the CFD predictions deviate from the experimental time averaged profile for weaker components of velocity, owing to the complex nature of flow. Also, the difference in predicted and experimental drop sizes will affect the CFD simulation result. However, we would like to point out that, for the case of highly nonlinear nature of the flow coupled with the inhomogeneity of the flow pattern in Taylor-vortex flows, the assumption of single drop size throughout the domain is needed for a tractable solution. PLIF and high speed camera images clearly show that the drop size varies from bottom to top of the contactor. Hence the interphase momentum exchange terms used with the current multiphase model formulation is expected to be pretty different than the exact physical values.

Since the magnitude of secondary velocity components (radial and axial, with driving motion being tangential) is strongly affected by this assumption, the poorer prediction of weaker velocity component is expected. Figure 10 shows the hold-up profile and the flow patterns obtained using CFD. The total number of vortices in the annulus is 10 which is in agreement with the experimental measurements. It can be also seen that the dispersed phase mainly accumulates at the inflow region between the vortices. Figure 11 shows the comparison of the mean axial and radial velocities obtained using CFD with the PIV measurements. The comparison between the predictions of the dispersed phase holdup with that of experimental measurement using PLIF at the centerline of the annulus is shown in Figure 12. In agreement with the hold-up profiles obtained from PLIF, the dispersed-phase holdup is the highest at the bottom of the device and it goes on decreasing consistently as we move upward along the annular region. The hold-up profiles obtained from CFD are also not

22


Figure 12. Comparison between the CFD predictions and experimental values for dispersed-phase hold-up at the center of annulus along the annulus height for ∈D ) 0.065: ([) experimental; (black line) CFD; (magenta line) 1-D dispersion model.

uniform but show the accumulation of dispersed phase near the vortex boundaries. This can be seen from periodic behavior of the dispersed-phase hold-up values along the annular region. The comparison of velocities obtained by CFD and experiments is poorer at the top and bottom ends of the contactor. This can be attributed to inherent uncertainties in the experimental values obtained using PIV. At the ends a good cross correlation could not be obtained using PIV because of (i) the light reflections from the top and bottom plates and (ii) heavy unseeded dispersed phase accumulated at the bottom. The difference can be seen in the predictions of hold-up of dispersed phase using the CFD and experimental observations. The experimental results clearly show modest dispersed-phase holdup near the periphery of the vortices which could not be predicted using CFD. The reason behind this would be uniform dispersed phase drop size assumption for the simulation. In reality, size distribution of the drops exists in the flow. The CFD simulations have been also carried out for the various dispersed phase hold-ups (∈D ) 0.15, 0.3, and 0.5). The results of CFD simulations are shown in Figure 13. For all the cases with dispersed-phase hold-up higher than 0.05, the segregation of phases was observed. Thus the qualitative appearance of flow pattern was practically independent of the dispersed-phase hold-up for ∈D > 0.05. The corresponding flow profiles are shown in terms of vector plot. It can be seen that the number of vortices changes from 10 for light phase alone to 6 for heavy phase alone. For the two phase flow the number of vortices lies between these two extremes. This can be seen from the expression of Ta, which is inversely proportional to the square of the kinematic viscosity, ν. Hence, in case of lighter phase, the critical value of Ta is reached earlier than in case of heavier phase. Thus, Taylor-Couette instability occurs more readily in the earlier case. Hence, under similar conditions, the number of Taylor vortices is more for the lighter phase than for the heavier phase. For this reason, an intermediate number of vortices is observed for the two phase dispersion. 5.4. Arrangement of Dispersed Phase. As mentioned earlier, different flow patterns were observed in two-phase flow in Taylor-Couette contactor. When geometry is vertical, the effect of gravity becomes important. Hence for such a geometry,

different types of flow patterns are expected. The different patterns observed are the vertically stratified phases, segregated dispersion, banded dispersion, and homogeneous dispersion. To understand these regimes and their transitions, extensive and systematic CFD simulations were undertaken. It was possible to distinguish the flow from the holdup profiles obtained using CFD. The various flow patterns observed depending upon the operating conditions and the physical properties of the individual phases, can be identified using hold-up profiles. The typical holdup profiles obtained using CFD, are shown in Figure 14. To study the mechanism of the formation of these different patterns, different forces acting on drops in liquid-liquid dispersion were considered. The CFD results were used advantageously to obtain the velocity magnitudes necessary to evaluate different forces acting on the droplets. The major forces which govern the motion of the droplets include the drag, gravity, and centrifugal force. Two other forces are the surface tension force restoring the deformation of liquid-liquid interface and the inertial force exerted by the continuous force resulting from its kinetic energy. The magnitudes of these forces have been evaluated from CFD results using the continuous and dispersed phase velocities, the drop diameter, the density, viscosity, and interfacial tension of both the phases. The formulas used to evaluate these forces are listed below: FS ) σd

(5)

FI ) u1-2,C2dD2FC

(6)

FG )

π 3 d (F - FD)g 6 D C

|uC - uD |(uC - uD) π FD ) CD dD2 4 2FC FCE )

π jr d F ) ( 2πN 60 ) ( 6 2

3

D

D

(7) (8) (9)

The values of these forces are reported in Table 3 for each one of the four regimes observed. The role of these forces in formation of a particular regime of dispersion for given operating conditions has been described as follows: (i) Stratified phases. At very low rotational speeds, each phase retains its integrity (light phase at the top and heavy at the bottom) and remains separated with a clear interface in between. Such a pattern is called as stratified flow (Figure 14A). For the case of stratified flow, the interface is well-defined between the phases. The case shown in Figure 14A corresponds to N ) 300 rpm, which marks the onset of the disturbance of the interface. For N ) 150 rpm, where complete stratification is ensured, the inertial force of the liquid is 7.456 × 10-6 N, the surface tension force is 1 × 10-5 N, and the gravitational force is 3.843 × 10-5 N. At this rotational speed the restoring forces of gravity and surface tension have higher magnitude than the disturbing inertial force. Thus, the two phases are well separated and practically no dispersion is observed. (ii) Segregated flow. When the rotational speed is increased further, the interface between the two phases breaks. From Figure 14B, the segregation of two phases in vertical direction is clearly observed in the annular region. The inertial force of liquid near the vortex boundary is 1.000 × 10-3 N while the surface tension force is 5.000 × 10-6 N. The drag force on the drops is 3.683 × 10-6 N and the gravitational force is 4.396 × 10-7 N, while the centrifugal force is 1.378 × 10-5 N. Thus the stratified interface is totally unstable, and the lighter phase is present as drops rather than accumulating at either end. The centrifugal force on drops is higher than the drag and gravita-


23

Figure 13. CFD predictions for dispersed-phase hold-up and primary phase velocity profiles for the different hold-ups of phases. (A) ∈D ) 0.15; (B) ∈D ) 0.3; (C) ∈D ) 0.5; (D) velocity profiles with heavy phase alone.

tional force at this point. The centrifugal force has a dominant radial component. Thus the radial hold-up pattern is governed by the centrifugal force. Thus, it can be observed that the heavy, continuous phase accumulates near the outer cylinder while the light dispersed phase accumulates near the inner cylinder. The axial dispersion is affected by the convection and turbulent dispersion of the drops along the Taylor vortices. The gravitational force acting in the vertical direction segregates the dispersion. The resultant of these fluxes resulting from these two effects governs the axial hold-up profile. If the dispersion of the droplets is lesser than the gravity segregation effect, each phase retains its identity and the other phase is dispersed. This can be observed in Figure 14 B where two extremes on the color scale are prominent, which signifies the existence of two phases in the lower and upper regions in the annulus. To the check the validity of the explanation, a 1-D dispersion model is formulated for the dispersed phase, following Pandit and Joshi.21 To simplify the model, the following assumptions are made: (a) both the continuous and the dispersed phases are considered in batch mode, with no net axial flow; (b) the turbulent dispersion coefficient is assumed to be constant throughout the annular region; (c) only turbulent dispersion and the gravity effects are considered to affect the motion of the drops in the vertical direction; (d) the effect of the drag force, the centrifugal force, and the turbulent dispersion force is modeled using a single parameter of droplet dispersion coefficient; (e) the hold-up variation in radial and tangential direction is neglected. After taking a material balance over an elementary strip of thickness dz having the same cross sectional area as that of the annulus, we obtain the following differential equation in terms of the dispersed phase hold- up: DD

d2∈D 2

+ uS,D

d∈D )0 dz

dz The boundary conditions are, at z ) 0, ∈D ) ∈D,b

and

(10)

d∈D uS,D )∈ dz z)0 DD D,b (11)

Equation 10 is solved by substituting the boundary conditions in eq 11. The solution is

( )

uS,D ∈D ) exp z ∈D,b DD

(12)

The value of uS,D/DD has been fitted by nonlinear regression of eq 12 to the radially averaged dispersed-phase hold-up profiles obtained from CFD simulations. Figure 15 shows the variation of radial averaged dispersed-phase hold-up as a function of the height. The dispersion model has been fitted to the hold-up variation in segregated and stratified dispersion. The average absolute residual error was found to be 30.5 and 41.2%, respectively. The model was not fitted for the homogeneous case because the hold-up was found to be very uniform implying a very high dispersion coefficient. The model is also not applicable for the banded dispersion case because the drop dispersion coefficient along the vertical axis varies to a much larger extent because of higher values of centrifugal forces involved, and hence the assumption of constant dispersion coefficient does not hold. (iii) Banded dispersion. On further increase in rotational speed from segregated flows for systems having larger drops, banded dispersion (Figure 14C) is observed. In banded dispersion, the dispersed phase accumulates periodically in the flow domain showing an alternate arrangement of the two phases. The banded dispersion also occurs in horizontal Taylor-Couette contactors.16,7 Zhu and Vigil16 reported higher dispersed-phase hold-up at the center of the rotating vortex in the case of horizontal Taylor-Couette contactors. For the case shown in Figure 14C, the inertial force of liquid near the vortex boundary is 5.625 × 10-4 N while the surface tension force is 2.500 × 10-5 N. The drag force on the drops is 2.167 × 10-6 N and the gravitational force is 7.232 × 10-8 N. The centrifugal force due to tangential liquid velocity is 2.041 × 10-5 N, while the centrifugal force caused by the axial-radial vortex velocity is 6.041 × 10-7 N. Zhu and Vigil16 reported both the cases of lighter dispersed phaseandtheheavierdispersedphaseforhorizontalTaylor-Couette contactors. In the case of the lighter dispersed phase, the droplets

24


Figure 14. Typical dispersed-phase hold-up distribution for various types of dispersed flow in a vertical Taylor-Couette flow. ∈D ) 0.3: (A) stratified flow; (B) segregated flow; (C) banded dispersion; (D) homogeneous dispersion. Table 3. Summary of Different Forces Acting on Droplets in Liquid-liquid Dispersion in Different Regimes case

N

dD

F

µ

Ta

u12

u3

FD

FG

FI

FS

FCE

FCE,VOR

stratified segregated banded homogeneous

150 600 1800 1200

0.0032 0.0007 0.0004 0.00003

1000 1000 1000 600

0.001 0.001 0.001 0.001

5.33 ×105 8.52 ×106 7.67 ×107 1.23 ×107

0.017 0.2 0.15 0.42

0.173 0.690 2.150 1.200

5.410 ×10-7 3.683 ×10-6 2.167 ×10-6 2.105 ×10-8

3.843 × 10-5 4.396 × 10-7 7.232 × 10-8 1.445 ×10-11

7.456 ×10-6 1.000 ×10-3 5.625 ×10-4 2.646 ×10-3

1.000 ×10-5 5.000 ×10-6 2.500 ×10-5 1.000 ×10-6

7.533 ×10-5 1.378 ×10-5 2.041 ×10-5 2.217 ×10-9

2.071 × 10-6 4.965 × 10-6 6.041 × 10-7 7.818 ×10-10

were observed to move toward the vortex cores. Further, there is significant hold-up of the dispersed phase in the region, where the flow moves from inner to outer cylinder (outflow region). Whereas, for the case of heavier dispersed phase the maximum concentration of drops was obtained at the location between two vortices where the continuous phase flows from the outer cylinder toward the inner cylinder, rather than around the

periphery of every vortex. This result was overlooked in the discussion provided by Zhu and Vigil.16 The higher dispersed-phase hold-up at the center of the rotating vortex for lighter dispersed phase is primarily caused by the centrifugal forces acting on the lighter drop due to the axial-radial vortex velocity. This forces the droplets to move toward the circumferential axis passing through the vortex


25

Figure 15. Radially averaged dispersed-phase variation in the vertical direction: (squares) stratified phases; (circles) segregated dispersion; (orange line) banded dispersion; (light blue line) homogeneous dispersion; (magenta line) 1-D dispersion model fitted to segregated dispersion; (dark blue line) 1-D dispersion model fitted to stratified phases.

Figure 16. CFD predictions for continuous phase hold-up for the experiments by Zhu and Vigil (2001) with ∈D ) 0.1: (A) kerosene dispersed in water; (B) water dispersed in kerosene.

center. However, the results obtained for the case when heavy phase is dispersed in the light were not in agreement with this hypothesis. The contour plot of dispersed-phase hold-up overlapped by the corresponding vector plot of velocity obtained from CFD simulation for the geometry described by Zhu and Vigil,16 using the Eulerian multiphase model, is shown in Figure 16A. It can be observed for the case of larger drops that the dispersed phase having FD < FC not only concentrates toward the center of vortex, it also preferentially accumulates in the outflow region with more hold-up toward the inner wall. As it is clear from the values of the two centrifugal force values mentioned before, the centrifugal force arising from tangential motion around the axis of geometry is 2 orders of magnitude higher than the centrifugal force arising because of the vortex rotation around its circumferential axis. Hence centrifugal force arising from tangential motion plays a major role in the appearance of banded dispersion. When dispersed drops are heavier than continuous phase, the dispersed phase would move toward the outer cylinder. This is shown in Figure 16B where

Figure 17. Effect of aspect ratio on the flow patterns of banded dispersed flow at Ta ) 2.367 × 107: (A) Γ ) 1, (B) Γ ) 2, (C) Γ ) 4, (D) Γ ) 12.

the heavier phase is preferentially collected in the region where the flow moves from the outer to inner cylinder (inflow). This is in complete agreement with CFD results of Zhu and Vigil.16 From Figure 16, it is clear that the banded flow patterns would appear primarily because of accumulation of dispersed phase in the outflow or inflow region. (iv) Homogeneous dispersion: A very high rotational speed results in the formation of small drops. As shown in Figure 14D, the dispersed phase in this case appears uniformly distributed over entire flow domain. For the case shown in Figure 14D, the inertial force of liquid near the vortex boundary is 2.646 × 10-3 N while the surface tension force is 1 × 10-6

26


Figure 18. Flow map for the two phase flow in a vertical Taylor-Couette flow: (9) stratified phases; (]) segregated dispersion; (2) banded dispersion; (O) homogeneous dispersion; (+) point corresponding to the experimental conditions. Table 4. Range of Operating Parameters and Physical Properties Considered parameter

range

∈D FC FD Ω µC µD σ d

0.1-0.3 600-1500 kg/m3 600-1500 kg/m3 5-50 r/s 0.001-0.006 Pa s 0.001-0.003 Pa s 0.1-80 mN/m 1.5-6.5 mm

N. The drag force on the drops is 2.105 × 10-8 N, and the gravitational force is 1.445 × 10-11 N. The centrifugal force due to the tangential liquid velocity is 2.217 × 10-9 N while the centrifugal force caused by the axial-radial vortex velocity is 7.818 × 10-10 N. In the case of homogeneous dispersion, the drag force on the drop dominates over the centrifugal force. The inertial force is 3 orders of magnitude higher than the surface tension force, causing the small drop size. However, slight variation in the dispersed-phase hold-up exists, and the dispersed phase is arranged in a well-defined pattern along the annulus, primarily caused by the centrifugal forces acting on droplets. It is also important to note that the commencement of homogeneous dispersion is a function of the geometry, in particular the aspect ratio, Γ, as well as the rotational speed and also the dispersed-phase hold-up. For a given geometry, where the aspect ratio is defined, homogeneous dispersion is observed at higher speeds of rotation and/or lower hold-up of dispersed phase.

The end effects and hence the aspect ratio affects the cell structures. For the single-phase flow this subject has been discussed in the literature.18,22 The simulations were carried out for the aspect ratios of 1, 2, 4, 12, and 50. For all these cases the annular gap was kept constant (i.e., 5 mm). The Taylor number was calculated to be 2.367 × 107, and the lighter phase was the dispersed phase. The results are shown in Figure 17 in terms of hold-up contours and velocity vectors. It can be seen that the number of cells increases with an increase in the aspect ratio. For the aspect ratio of 1, only one dominant circulation can be seen. For the smaller (2 and 4) aspect ratios end effects are significant and two counter-rotating cells were observed (longer wavelengths of cell: for Γ ) 2, λ ) 2d, Γ ) 4, λ ) 4d). As the aspect ratio increases further, end effects diminish and the wavelength reduces. For example, the aspect ratio Γ ) 50 λ is 2d corresponding to the wavelength of cells in an infinitely long cylinder, with aspect ratio Γ ) ∞. In all the cases, the vortices at the ends are elongated. These observations are in line with those with single-phase flows. Further from the holdup contours the flow appears to be a banded dispersed flow with lighter phase accumulated mainly at the outflow region on the counter-rotating cell. Zhu and Vigil16 observed banded dispersion for the case with larger interfacial tension (25 mN/m, dd ) 4.1 × 10-4 m). On the other hand, with the lower interfacial tension (0.3 mN/m, dd ) 2.1 × 10-5 m) homogeneous dispersion was observed. For very small drops the slip velocity between the phases is nearly zero. Thus they can be very easily carried by the continuous phase, and homogeneous dispersion was seen. The homogeneous dispersion was also observed when the density difference between the phases was small. It can be observed that the appearance of flow in a Taylor-Couette contactor is mainly decided by the drop size. The various forces acting on the drops are buoyancy and centrifugal forces due to the local rotation within vortex as well as global rotation around axis of geometry. On the other hand the drop size is decided by the inertial and surface tension forces. Estimation of these four forces was thought vital to predict the type of flow. These forces can be considered together using two dimensionless numbers, viz., Taylor number, Ta ) η2/(1 - η2)d4(Ω/ν)2, which is the ratio of centrifugal due to the rotation to viscous forces and Eotvos number Eo ) ∆Fgdd2/σ, the ratio of buoyancy force to surface tension force. The CFD simulations were carried out for wide range of operating conditions and the hold-up profiles were obtained. Each flow pattern was then categorized in one of the above regimes using the hold-up profiles. An attempt has been made to represent them using the plot of Ta vs Eo. To vary these numbers, physical properties and operating conditions were changed. The details of the operational range and the physical properties covered

Table 5. Criteria For the Flow Regimes Obtained Using CFD sr. no.

flow regimes

criteria

1.

stratified flow

Ta < 2 × 106, Eo > 0.046

2.

segregated flow

Eo > 0.046, 7 × 106 > Ta > 2 × 106 or Eo > (1.87 × 10-7Ta - 1.26), Ta > 7 × 106,

3.

banded dispersion

Ta > 7 × 106 and Eo < (1.87 × 10-7Ta - 1.26) for Eo > 0.046 or Eo > (5.04 × 10-2 - 6.23 × 10-11Ta) for Eo < 0.046

4.

homogeneous dispersion

2 × 106 < Ta < 7 × 106, Eo < 0.046 or Ta > 7 × 106, Eo < (5.04 × 10-2 - 6.23 × 10-11Ta)


27

are given in Table 4. These give the variation of Ta and Eo from 1 × 105 to 1 × 109 and from 1 × 10-4 to 10, respectively. The constructed flow map in terms of Eotvos number (Eo) and Taylor number (Ta) is shown in Figure 16. The figure shows that stratified flow occurs at lower Taylor number (Ta < 2 × 106) and higher Eotvos number (Eo > 0.046). At moderate Ta number (Ta > 2 × 106) and high Eo number (Eo > 0.046), segregation of phases is observed. In other words, transition from stratified to segregated flow occurs at Ta ) 2 × 106 when Eo > 0.046. When Ta > 7 × 106 the banded dispersion was seen for higher Eotvos numbers. The homogeneous dispersion can be observed at moderate as well as high Taylor number. From the figure it can be seen that for 2 × 106 < Ta < 7 × 106 the homogeneous dispersion can be observed when Eo < 0.046 as well as for Ta > 7 × 106 depending upon the value of Eo. The criteria for various regimes are summarized in Table 5. The point corresponding to the experimental conditions for which vectors and contours are shown in Figure 6 and 7, respectively, is marked by “+” in Figure 18, which represents a segregated flow pattern and is in agreement with the observations.

(5) At very low rotational speeds (Ta < 2 × 10 , Eo > 0.046), the two phases remain vertically stratified with a clear interface in between them. (6) At moderate rotational speeds, segregated flow can be observed when, Eo > 0.046, and 7 × 106 > Ta > 2 × 106 or Eo > [1.87 × 10-7Ta - 1.26], and Ta > 7 × 106. In such cases the light phase gets dispersed in the heavier phase at the bottom as well as heavy phase is observed to be dispersed in the light phase at the top. (7) When the drops are large, centrifugal force on the drops due to tangential rotation of fluid around the cylinder axis was found to be dominant which gives a banded dispersion such that Ta > 7 × 106 and Eo < [1.87 × 10-7Ta - 1.26] (for Eo > 0.046) or Eo > 5.04 × 10-2 - 6.23 × 10-11Ta (for Eo < 0.046). The dispersed phase in this case was found to mainly accumulate near either outflow or inflow region of Taylor vortices for FC > FD or FC < FD, respectively. (9) When the drops are small and/or the density difference between the phases is small homogeneous dispersion can be observed, such that 2 × 106 < Ta < 7 × 106, and Eo < 0.046 or Ta > 7 × 106, and Eo < [5.04 × 10-2 - 6.23 × 10-11Ta].

6. Conclusions

Nomenclature

(1) The dispersion in a vertical Taylor-Couette contactor has been studied experimentally as well as computationally. The experiments have been also carried out using high speed imaging for getting the information about the drop sizes. The correlation by Hass20 was found to give comparable estimate of the drop sizes. The slight deviation observed was due to the dependence of hold-up on the drop size in the annulus. The correlation by Hass20 does not contain the hold-up of the dispersed phase, whereas the actual drop size depends on it. For this reason, the correlation seems to predict the drop size well at low dispersedphase hold-up values in the annular region. (2) The PIV measurements have been also carried out for the single-phase flow. A very good agreement has been found between the CFD model and experimental data in terms of mean velocity profiles as well as the number of vortices. The CFD simulations for the single-phase flow and also with multiphase flow having ∈D ) 0. The PIV images were processed to obtain the velocity vectors under the similar conditions. It was found that the number of vortices and the spacing between them obtained with the PIV were in good agreement with those shown by the CFD simulations. (3) Simultaneous measurements of flow field and dispersedphase hold-up have been performed using PIV and PLIF techniques, respectively. The experimental data were used to validate the CFD model. A good agreement has been observed for the mean velocities, number of vortices, and hold-up profiles within the acceptable limits. It was also observed that the results obtained with single-phase CFD simulations matched those obtained with the simulations using the Eulerian multiphase model with ∈D ) 0. Thus, it was concluded that Eulerian multiphase model could be extrapolated to single-phase model when the hold-up of the dispersed phase is taken as zero. (4) The effect of physical properties like density difference and interfacial tension and the effect of geometrical parameters such as annular gap on the flow patterns have been studied using CFD. The results have been represented in the form of regime maps with the Eotvos and Taylor number as the two coordinates. The results are independent of dispersed-phase hold-up for ∈D > 0.05.

CD ) drag coefficient d ) annular gap, m dD ) drop diameter, m dmax ) maximum stable drop diameter, m D ) hydraulic diameter () 2 × d), m DD ) drop dispersion coefficient, m2/s Eo ) Eotvos number g ) gravity, m/s2 FCE ) centrifugal force acting on droplet due to rotation along vertical axis, N FCE,VOR ) centrifugal force acting on droplet due to rotation along Taylor vortex, N FD ) drag force acting on droplet, N FG ) gravity force acting on droplet, N FI ) inertial force acting on droplet, N FS ) surface tension force acting on droplet, N KDC ) interphase momentum exchange coefficient, kg/m3 s p ) pressure, N/m2 ri ) radius of inner cylinder, m Re ) drop Reynolds number Reθ ) azimuthal Reynolds number t ) time, s td ) relaxation time, s Ta ) Taylor number We ) Weber number 〈ui,p〉 ) time averaged velocity for component i and phase p, m/s N ) speed of rotation, RPM u12 ) axial-radial vortex velocity, m/s u3 ) tangential velocity, m/s us,D ) settling velocity of drop, m/s Greek Letters ∈ ) hold-up µ ) molecular viscosity, Pa s Ω ) angular velocity, rad/s F ) density, kg/m3 σ ) interfacial tension, mN/m τ ) stress-strain tensor, kg m/s2 Subscripts b ) value at bottom C ) continuous phase D ) dispersed phase

6

28


θ ) azimuthal direction 1,2,3 ) Cartesian coordinates x, y, and z, respectively

Acknowledgment S.S.D. and M.J.S. acknowledge BRNS, Sanction No. 2006/ 34/24-BRNS/2803 and CSIR for the financial support for this work, respectively. Literature Cited (1) Baier, G.; Graham, M. D.; Lightfoot, E. N. Mass transport in a novel two-fluid Taylor vortex extractor. AIChE J. 2000, 46, 2395–2407. (2) Haut, B.; Ben, A. H.; Coulon, L.; Jaquet, A.; Halloin, V. Hydrodynamics and mass transfer in a Couette-Taylor bioreactor for the culture of animal cells. Chem. Eng. Sci. 2003, 58, 774. (3) Gandhi, N. N.; Patil, N. S.; Sawant, S. B.; Joshi, J. B.; Wangikar, P. P.; Mukesh, D. Lipase catalyzed esterification. Catal. ReV., Sci. Eng. 2000, 42, 439–480. (4) Joshi, J. B.; Elias, C. B.; Patole, M. S. Role of hydrodynamic shear in cultivation of animal, plant and microbial cells. Chem. Eng. J. 1996, 62, 121–141. (5) Campro, R. J.; Vigil, R. D. Flow patterns in liquid-liquid TaylorCouette-Poiseuille flow. Ind. Eng. Chem. Res. 1999, 38, 1094–1098. (6) Zhu, X.; Campro, R. J.; Vigil, D. R. Axial mass transport in liquidliquid Taylor-Couette-Poiseuille flow. Chem. Eng. Sci. 2000, 55, 5079– 5087. (7) Vedantam, S.; Joshi, J. B.; Koganti, S. B. Three dimensional CFD simulation of stratified two-fluid Taylor-Couette flow. Can. J. Chem. Eng. 2006, 84, 279–288. (8) Renardy, Y.; Joseph, D. Couette flow of two fluids between cocentric cylinders. J. Fluid Mech. 1985, 150, 381–394. (9) Joseph, D. D.; Preziosi, L. Stability of rigid motions and coating films in bicomponent flows of immiscible liquids. J. Fluid Mech. 1987, 185, 323–329.

(10) King, M. R.; Leighton, D. T.; McCready, M. J. Stability of oscillatory two-phase Couette flow: Theory and experiments. Phys.Fluid 1998, 11, 833–844. (11) Baier, G.; Graham, M. D. Two-fluid Taylor-Couette flow: Experiments and linear theory for immiscible liquids between co-rotating cylinders. Phys. Fluids 1998, 10, 3045–3055. (12) Charru, F.; Hinch, E. J. Phase diagram of interfacial instabilities in a two-layer Couette flow and mechanism of the long wave instability. J. Fluid Mech. 2000, 414, 195–201. (13) Vedantam, S.; Joshi, J. B. Annular centrifugal contactors: A review. Chem. Eng. Res. Des. 2006, 84, 522–542. (14) Joseph, D. D.; Nguyen, K.; Beabers, G. S. Non-uniqueness and stability of the configuration of immiscible fluids with different viscosities. J. Fluid Mech. 1984, 141, 319–345. (15) Campro, R. J.; Vigil, R. D. ; Axial dispersion during low reynolds number Taylor-Couette flow: Intravortex mixing effects. Chem. Eng. Sci. 1997, 52, 3303–3310. (16) Zhu, X.; Vigil, D. R. Banded liquid-liquid Taylor-CouettePoiseullie flow. AIChE J. 2001, 47, 1932–1940. (17) Rashidnia, N.; Balasubramaniam, R.; Del Signore, D. Interfacial tension measurement of immiscible liquids using a capillary tube. AIChE J 1992, 38, 615–618. (18) Deshmukh, S. S.; Vedantam, S.; Joshi, J. B.; Koganti, S. B. Computational flow modeling and visualization in the annular region of annular centrifugal extractor. Ind. Eng. Chem. Res. 2007, 46, 8343–8355. (19) Haas, P. A. Turbulent dispersion of aqueous drops in organic liquids. AIChE J. 1987, 33, 987–995. (20) Clift, R.; Grace, J. R.; Weber, M. E. ; Bubbles, Drops, and Particles; Academic Press: New York, 1978; p 351. (21) Pandit, A. B.; Joshi, J. B. Three phase sparged reactorsssome design aspects. ReV. Chem. Eng. 1984, 2, 1–84. (22) Koschmieder, E. L. Bernard cells and Taylor Vortices; Cambridge University Press: New York, 1993.

ReceiVed for reView February 3, 2009 ReVised manuscript receiVed September 16, 2009 Accepted November 4, 2009 IE900185Z

Computational Fluid Dynamics Simulation and Experimental

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