CFD Simulation of RTD and Mixing in the Annular Region of a Taylor

Aug 3, 2006 - Flow between two concentric cylinders, with either or both of them rotating, has potential advantages over the conventional process equi...
0 downloads 0 Views 143KB Size
Download PDF
6360

Ind. Eng. Chem. Res. 2006, 45, 6360-6367

CFD Simulation of RTD and Mixing in the Annular Region of a Taylor-Couette Contactor Sreepriya Vedantam and Jyeshtharaj B. Joshi* Institute of Chemical Technology, UniVersity of Mumbai, Matunga, Mumbais400 019, India

Sudhir B. Koganti Indira Gandhi Centre for Atomic Research, Kalpakkam TN-603 102, India

Flow between two concentric cylinders, with either or both of them rotating, has potential advantages over the conventional process equipment. This flow, also termed as Taylor-Couette flow, which reveals a variety of flow regimes, is studied using computational fluid dynamics (CFD) simulations. The vortex motion causes mixing in the annulus, which necessitates the understanding of residence time of the fluid in the annulus and the mixing time. A systematic study of residence time distribution (RTD) and the mixing phenomenon in the annular region of the Taylor-Couette contactor has been carried out. The RTD predictions have been found to be in good agreement with the established experimental results. Axial dispersion in the Taylor vortex regime and the turbulent regime has been predicted, and it can be said that, with an increase in the radial mixing, the effect of velocity profile on the dispersion parameter significantly reduces at higher rotational speeds. Effect of axial flow, annular gap width, and rotational speed has been investigated on the axial dispersion. The dispersion number and the dispersion coefficient have been predicted. 1. Introduction Two concentric cylinders, with either or both rotating, also known as a Taylor-Couette contactor, has attracted attention as efficient process equipment. The duct through which the fluid flows is the annulus between the two cylindrical surfaces, which is small compared to the radii of the coaxial cylinders. The flow is circumferential because of the shearing action of the rotating cylinder. Beyond a certain critical speed of the rotating cylinder, flow results in the formation of a vortex flow regime (Figure 1). This critical condition is expressed as a dimensionless number called the Taylor number (Ta), which is a ratio of the centrifugal forces to the viscous forces. Such a flow pattern is of great interest because it provides high values of heat and mass transfer coefficients. Some important industrial operations that can be carried out to advantage in Taylor-vortex equipment include emulsion polymerization, synthesis of silica particles, heterogeneous catalytic reactions, and liquid-liquid extraction.1-7 Some investigators have also explored the utility of Taylorvortex devices as bioreactors, for membrane separation, and for filtration.8-10 When a small amount of axial flow is added to the TaylorCouette flow, axial motion of the Taylor vortices results in toroidal motion of fluid elements, causing radial mixing. Each vortex can be regarded as an element of a series of well-mixed batch vessels with the same mean residence time. Such a flow system can be considered as a near-ideal plug-flow reactor (PFR).11-13 Many researchers have investigated the axial dispersion in Taylor-Couette flow. The experimental observations established in the literature indicate that the axial dispersion decreases to a minimum near the onset of Taylor vortices and increases as the Taylor number (Ta) increases.14 When the axial flow is absent, the dispersion is proportional to the square root of the diffusion coefficient.15 On the contrary, when the axial flow is present, the dispersion is independent of the diffusion * To whom all the correspondence should be addressed. Phone: 0091-22-2414 0865. Fax: 00-91-22-2414 5614. E-mail: [email protected].

Figure 1. Cross section of concentric cylinders and vortex formation: (a) axisymmetric vortices and (b) asymmetric vortices.

coefficient11 and increases with increasing axial flow rates.16 These experimental results were simulated by a one-dimensional (axial) dispersed plug-flow model16-18 or as a mixing-tanksin-series model.11,13 Desmet et al.15 combined the dispersed plug flow and mixing tank models into a single overall model, thus correlating the axial mass transfer to the position in the vortex. They concluded that convection dominates near the vortex boundaries, while diffusion dominates near the vortex center. The results obtained from previous experimental studies are summarized in Table 1. The corresponding theoretical work for the prediction of axial dispersion in the literature includes asymptotic and full numerical analyses. Rosenbluth et al.19 investigated an infinite annulus in both the high and low Peclet number (Pe) range for an axisymmetric Taylor-Couette flow in the absence of axial flow. They concluded that, for Pe , 1, the vortex has nearly uniform composition. In this case, they find that the axial dispersion increases as the flow velocity increases but decreases as the diffusion coefficient increases. This is similar to the classic study of Taylor-Aris convective dispersion.20,21 When Pe .1, the effect of diffusion is small and boundary layers form along the

10.1021/ie050825n CCC: $33.50 © 2006 American Chemical Society Published on Web 08/03/2006

Ind. Eng. Chem. Res., Vol. 45, No. 18, 2006 6361 Table 1. Experimental Correlations for Axial Dispersion in the Annular Region of Taylor-Couette Flow s. no.

author

ri/ro

al.16

gap width (mm)

Rez

1 2 3 4

Enkoida et Moore and Cooney17 Kataoka et al.11 Tam and Swinney18

0.593-0.76 0.73-0.96 0.75 0.494 -0.875

18-30.5 0.4-3.8 10 3.2-12.85

60-250 0.5-30 0-90 0

5 6 7 8

Kataoka et al.30 Desmet et al.31 Pudijiono et al.14 Ohmura et al.32

2 2.4 0.15 1.84

20 24 1.5 18.4

0-35 0 0.03 -5.5 0

vortex boundaries with little diffusion into the interior of the cell. This is the limit typically observed experimentally. Despite these extensive studies, the agreement between the experiment and theory appears to be fortuitous, since the analysis in these studies has been carried out for creeping flow, while the experiments in most of the reported studies were in the turbulent vortex regime. Under these conditions, it is essential to obtain a numerical solution for determining the flow field that can be used in determining the concentration profiles. Hence, it was thought desirable to undertake a systematic study of residence time distribution (RTD) and the mixing phenomenon in the annular region of the Taylor-Couette contactor where the mixing occurs because of the transport at molecular, eddy, and bulk levels. Since the fluid mechanics in the annular region is complex, computational fluid dynamics (CFD) simulations have been carried out for the RTD as well as the mixing time. 2. Mathematical Modeling 2.1. Model Formulation. In the present case, 2D simulations have been carried out. For an axisymmetric swirling flow between two concentric cylinders, the Navier-Stokes equations for an incompressible, constant viscosity liquid can be written in cylindrical coordinates as follows:

Continuity equation:

correlation

range

model

D* ∝ c1Reθ + c2ReθRez D ∝ Reθ1.05Rez0.17

0.1-0.6 0.01-10

D ∝ ReθR R ) 0.69-0.86 D ∝ Rez1.7

0.8-2.7

1D dispersion- convectio 1D dispersion- convection PFR in laminar vortex regime 1D dispersion- convection

0.81

0.06-0.7

evolves, the form of this radial pressure gradient also changes, driving radial and axial flows in response to highly nonuniform pressures that result therein. 2.2. Turbulence Modeling. For turbulence modeling, the Reynolds stress model (RSM) has been used. In this model, individual Reynolds stresses ui′uj′ have been 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, with the product then being Reynolds averaged. The transport equations for the transport of Reynolds stresses F ui′uj′ are given by

∂(Fui′uj′) ∂(Fukui′uj′) )+ ∂t ∂xk

(1)

hi Ui is divided into a mean part U and a fluctuating part ui so that Ui ) U h i + ui (2) (3)

where the stress tensor is written as

(

)

hj ∂U h i ∂U + - Fuiuj τij ) µ ∂xj ∂xi

(4)

In swirling flows, the conservation of angular momentum tends to create a free vortex flow in which the angular velocity increases as the radius decreases. In an ideal vortex flow, the centrifugal forces created by a circumferential motion are in equilibrium with the radial pressure gradient.

∂P FΩ ) ∂r r

]

(

F ui′uk′

)

∂uj ∂ui + uj′uk′ ∂xk ∂xk

∂ui′ ∂uj′ - 2FΩk(uj′um′ikm + ui′um′jkm) (6) ∂xk ∂xk

The turbulent viscosity (µt) is computed as

Momentum equations:

∂FU hi ∂ ∂P h ∂τij + (FU h iU h j) ) + ∂t ∂xj ∂xi ∂xj

[

∂(ui′uj′) ∂[Fui′uj′uk′ + p(δkjui′ + δikuj′)] ∂ + µ ∂xk ∂xk ∂xk

2µ ∂F ∂ + (FU h j) ) 0 ∂t ∂xj

2.4 0.1-1

D ∝ Reθ2.8 laminar D ∝ Reθ1.5 wavy D ∝ Reθ0.9 turbulent

tanks in series PFR and tanks in series 1D dispersion- convection 1D dispersion- convection

2

(5)

As the distribution of angular momentum in a nonideal vortex

k2 where cµ ) 0.09 µt ) Fcµ 

(7)

The diffusion term is taken as a scalar diffusivity term as22

(

)

∂ µt ∂ui′uj′ , σk ) 0.82 ∂xk σk ∂xk

(8)

The turbulence kinetic energy was obtained by taking the trace of Reynolds stress tensor,

1 k ) ui′ui′ 2

(9)

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

[( ) ] (

)

µt ∂k ∂ui ∂ ∂k ∂(Fkui) ) µ+ + F ui′uk′ - F + ∂t ∂xi ∂xj σk ∂xj ∂xk (10) Though the above equation is solved globally through the flow

6362

Ind. Eng. Chem. Res., Vol. 45, No. 18, 2006

domain, the values of k obtained are used only for boundary conditions. In every other case, the prior equation is used to obtain k. To model the dissipation rate, the dissipation tensor is modeled as

2 ij ) δij(F) 3

(11)

The scalar dissipation rate, , is computed with the model transport equation:

[( ) ] (

)

2.5. Boundary Conditions for Reynolds Stresses. The standard wall functions are based on those of Launder and Spalding.24 At walls, the near-wall values of the Reynolds stresses and  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. 2.6. Boundary Conditions for the Species Transport. The solution must satisfy the following boundary conditions

µt ∂ ∂ui  ∂(F) ∂(Fui) ∂ + ) µ+ + c1F ui′uk′ ∂t ∂xi ∂xj σ ∂xj ∂xk k 2 (12) k

c2F

The constants are σ ) 1.0, c1 ) 1.44, and c2 ) 1.92 2.3. Model Formulation for Species Transport. After fully developed flow field is obtained, mixing of passive tracer is simulated. The following assumptions have been made: (i) the diffusion process is isothermal, (ii) the solute does not undergo any chemical reaction, and (iii) the amount of solute in the system is sufficiently small so that all of the physical properties of the system are effectively constant. Zero tracer concentration was specified to the entire solution domain except for the predetermined computational cells, wherein the tracer concentration was defined as unity. The species conservation equation was then solved to simulate the mixing of tracer in a batch system. A zero flux boundary condition was specified for all the external boundaries. The axial mixing has been studied in a continuous flow annular system with a prespecified imposed axial flow. The transient mass balance expresses the dispersion of the tracer as follows,

(

)

∂(FYi) ∂Yi ∂(FYi) ∂ +U hi )FDeff ∂t ∂xi ∂xi ∂xi

(13)

where Yi is the local mass fraction of the species. The effective diffusion coefficient, Deff, is given by

Deff ) Dm + Dt

(14)

where Dm is the molecular diffusion coefficient and Dt is the eddy diffusion coefficient. The corresponding Schmidt numbers are given by the following equations:

Sc )

υ Dm

(15)

υt Dt

(16)

Sct )

2.4. Boundary Conditions for the Flow Field. The rotational velocities of the walls are specified. The inner cylinder rotation varied from 0 to 15 rps. Across the flow regime, no variation of velocities along the tangential direction is considered (∂/∂θ ) 0). Symmetry boundaries define a flow pattern with mirror symmetry. This assumes zero normal velocity components and zero normal gradients to obey the symmetry condition. For the case of dispersion studies, the Rez was varied in the range of 0.04-6.0. It has been specified that the no-slip condition holds at all the walls. The different lengths of the annuli covered are 0.016 and 0.225 m. The range of gap width used in these simulations has been 1.5-15 mm.

cj(r,z,0) ) C0

(17)

∂cj(0,z,t) ∂cj(a,z,t) ) )0 ∂r ∂r

(18)

for all z and t. 3. Numerical Framework 3.1. Method of Solution. With the Finite volume formulation, all the simulations were carried out using axisymmetric 2D grids. The commercial software FLUENT (version 6.1) has been used in all the studies. The mathematical model was solved in two steps. In the first step, equations of continuity and motion were solved for acquiring velocity profiles, the kinetic energy and dissipation rate, the eddy diffusivity, and the Reynolds stresses. In the second step, the flow and the information of turbulent parameters obtained from step 1 was used for solving the concentration equation. The radial profiles of velocity were assumed to be independent of axial location as the flow is fully developed. A uniform grid scheme consisting of 200 × 100 grids in axial and radial directions was employed. A segregated implicit solver method with implicit linearization was used for the momentum equations. The momentum equations have been discretized with the first-order upwind scheme, and for the pressure velocity coupling, the SIMPLE scheme has been used. For the pressure equation, the 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. Eccentricity (ratio of offset distance of the cylinder axis to the average gap width) was assumed to be zero. After fully developed flow field is obtained, mixing of passive tracer in a batch system and axial mixing in case of a continuous system are simulated. For such a simulation, zero tracer concentration was specified to the entire solution domain except for the predetermined computational cells, wherein the tracer concentration was defined as unity. The species conservation equation was then solved to simulate the mixing of tracer and the axial mixing. For the flow simulations, data were collected at specified points to track the development of the flow and confirm that the asymptotic solution was reached until fully developed flow is attained. Further, species transport equations have been accounted for until the tracer leaves from the outlet of the annulus. 4. Results and Discussions The accuracy with which the dispersion characteristics can be predicted depends on the accuracy in predicting the flow field. To test the accuracy of predictions from the flow simulations, an extensive comparison has been made with the available experimental data. The results have been presented starting with the validation of the model by comparison of the

Ind. Eng. Chem. Res., Vol. 45, No. 18, 2006 6363

Figure 2. Comparison of CFD predictions of axial velocity with the experimental data of Haut et al.25, η ) 0.826, Ωi ) 2.15 rps, and Γ ) 10.6.

Figure 4. Comparison of CFD predictions of axial velocity with the experimental data of Parker and Merati26 at z/d ) 0.66, η ) 0.672, Ωi ) 20 rps, and Γ ) 4.

Figure 5. Comparison of CFD predictions of radial velocity with the experimental data of Parker and Merati26 at z/d ) 0.66, η ) 0.672, Ωi ) 20 rps, and Γ ) 4. Figure 3. Comparison of CFD predictions of axial velocity with the experimental data of Haut et al.25 at an axial flow of 0.167 L/s, η ) 0.826, Ωi ) 4.9 rps, and Γ ) 10.6.

flow field with experimental data, followed by the mixing time and the residence time distribution studies. 4.1. Flow Field. Transient simulations have been carried out for both horizontal and vertical contactors. In the case of the horizontal system, simulations have been carried out for an annular gap width of 8.5 mm and compared with the experimental data of Haut et al.,25 who used particle image velocimetry (PIV). For the case of the wavy vortex flow, beyond the transition from the laminar Taylor flow to the wavy vortex flow at a rotational speed of 0.34 rps, the comparison of the axial velocity at r ) d/2 is shown in Figure 2, which shows very good agreement. For the turbulent regime, at a rotational speed of 0.79 rps and an axial flow rate of 0.167 × 10-3 m3/s, the comparison for the axial velocity at a constant r is shown in Figure 3. Various turbulence-modeling approaches have been analyzed, and it is observed that the RSM predicts values close to the experimental results. The RSM model, hence, has also been used for comparison with the experimental results of Parker and Merati,26 who measured the components of mean velocity, using laser Doppler velocimetry (LDV), in a vertical system. These simulations are for the system with the axis in the vertical direction incorporating the gravitational effects. The axial and radial velocity components at a z/d of 0.66 have been shown in Figures 4 and 5,

respectively. The predictions are observed to be in good agreement with the experimental data. From these simulations, it has been observed from the radial velocity contours that the maximum radial velocity occurs in the radial outflow regions on subsequent vortices. Centrifugal forces act to increase the outward radial velocity in these regions and thus hinder the returning flow. This flow field forms a basis for the mixing time as well as the dispersion characteristics. 4.2. Mixing Time. One of the most commonly used parameters to characterize mixing is the mixing time. Mixing time is usually measured by injecting a pulse of tracer containing a specified amount of tracer and by monitoring tracer concentrations within the reactor at single or multiple locations. Thus, this is the time measured from the instant of tracer addition until the vessel contents reach a specified degree of homogeneity. There are various ways of identifying the degree of homogeneity in the reactor.27 In the present study, mixing time has been considered as the time taken for a specific degree of homogeneity (99% mixing) to occur. An important consideration in simulating the mixing of passive tracer was whether it is necessary to solve all the flow variables simultaneously or if it would suffice to use a frozen flow field and solve the species conservation equation. When only the species equation is solved to simulate the mixing of tracer with a fixed flow field, predicted rates of mixing are lower than when all the equations are solved simultaneously. This difference is anticipated for the two-dimensional axisymmetric cases only. In asymmetric cases, this difference might be

6364

Ind. Eng. Chem. Res., Vol. 45, No. 18, 2006

Table 2. Comparison between Predicted and Experimental Mixing Time Values of Legrand and Coueret13 s. no.

ro/ri

gap width (mm)

annulus height (m)

Ta

location

predicted mixing time (s)

experimental mixing time (s)

1

1.82

11.3

0.1

189 225

1.82

11.3

0.1

525 625

172.6 174.3 173.1 70.4 72.6 71.1

180

2

rotating wall stationary wall top boundary rotating wall stationary wall top boundary

Table 3. Mixing Time Predictions at Varying Rotational Speeds and the Annular Gap Widths, ro/ri ) 1.0625, l ) 0.225 liquid density (kg/m3)

gap width (mm)

rotor speed (rps)

predicted mixing time (s)

DEG 1171.4

1.5

water 998.2

3.75

0.83 5 11.67 0.83 5 11.67 0.83 5 11.67

307 283.4 260.7 436.5 296.0 206.2 442.4 406.2 298.4

DEG 1171.4

significant. Hence, all the simulations involved solving flow as well as tracer conservation equations simultaneously. Simulations have been carried out for the prediction of mixing time in the annulus. As an initial step, a comparison has been made with the experimentally determined mixing time values of Legrand and Coueret.13 The details are given in Table 2. The radius ratio considered here, 1.82, is greater than the value 1.40, which has been reported as the upper limit of appearance of circumferential waves for small aspect ratios.28 The small underprediction (4-9%) in the mixing time may be attributed to the axisymmetric two-dimensional simulations. Considering various interpretations of mixing times, simulations have been carried out to examine the influence of parameters and to carry out a qualitative comparison: with an amount of tracer added in the beginning, the predicted mixing time decreases slightly with an increase in the amount of tracer added. However, for small amounts of tracer added, there was not much difference. The effect of location of tracer addition was investigated by considering the introduction of tracer at three different locations: (1) at the rotating wall, (2) at the stationary wall, and (3) at the top boundary. The details are given in Table 2. It can be seen that the mixing time does not get affected significantly with the tracer location for small amounts of tracer. The effect of rotational speed and the annular gap have been studied. To study these, simulations have been carried out for an annulus of length 0.225 m and a gap width of 15 mm. The test fluids considered are of densities 998 kg/m3 and 1171.4 kg/m3. Table 3 shows the mixing times observed under various operating conditions. It can be seen that, with an increase in the rotational speed of the inner cylinder, the mixing time reduces in a narrow-gap system, thus signifying the effect of radial mixing. In the case of a wide-gap system, the extent of radial mixing decreases. Yet higher rotational speeds lead to a decrease in the mixing time. 4.3. Residence Time Distribution (RTD) Studies. RTD curves were obtained for a range of Ta. These curves have been used for the estimation of the axial dispersion parameter. Once the RTD has been validated with the available experimental data, the effects of various parameters, such as the rotational speed, the axial flow, and the gap width, have been studied. Since the accuracy of the dispersion parameter depends on the accuracy of the predicted velocity profile, it has been thought desirable to quantify the effect of velocity profile over a wide

80

range of Ta covering the laminar Taylor vortex regime and the turbulent regime. The following stepwise procedure has been followed for studying the RTD. The average solute concentration is given by

C)

1 πa2

∫0a 2πcjr dr

(19)

When the reactor is modeled as a closed vessel, the first and second moments of the residence time distribution curve are related to the mean residence time (τ) and the effective dispersion coefficient, D, respectively,29 through

∑tC(t)∆t ∑C(t)∆t

(20)

∑t2C(t)∆t - τ2 ∑C(t)∆t

(21)

σ2 D D )2 (1 - e-UL/D) -2 2 UL UL τ

(22)

UL 2 σ 2 θ

(23)

τ≈

σ2 ≈

σθ2 )

( )

D≈

where L is the working length of the annulus and σ2 represents the variance. This can be used to determine the dispersion number as

σθ2 De ) 2

(24)

Using the above methodology, as an initial step, the RTD curves obtained from CFD simulations have been compared with the experimental data of Pudijiono et al.14 The radius ratio was 0.15, and the gap width was 1.5 mm. Simulations have been carried out for a range of Ta (0-7000) and Rez (0.04-5.49). Figure 6 shows the comparison of the predictions with the experimental data of Pudijiono et al.14 for a Rez of 0.041. The predictions show good agreement with the experimental values. At Ta ) 0, the predicted dispersion number has been compared with the experimental data for varying Rez, as shown in Figure 7a. Along with the predictions, the values indicate that the extent of dispersion decreases rapidly with an increase in the axial flow in the absence of rotational effect. In Figure 7b is shown the variation of dispersion number with the axial flow at different Ta. This shows a gradual decrease in dispersion number with an increase in axial flow. Thus, upon introduction of rotational effect on the axial flow, the dispersion decreases. The variation of dispersion number as a function of Ta at various axial flow rates is shown in Figure 8. Comparison with the experimental values of Pudijiono et al.14 is shown (Figure 8a) for the case of a Re ) 1.8. The increase in rotational effect clearly depicts a decrease in the dispersion number. Figure 8b shows the variation in the values of De.

Ind. Eng. Chem. Res., Vol. 45, No. 18, 2006 6365

Figure 6. Comparison of predicted RTD curve of DEG with the experimental data of Pudijiono et al.14 at ri/ro ) 0.15, Ta ) 703, and Rez) 0.041.

Figure 8. (a) Comparison with the experimental data of Pudijiono et al.14 at ri/ro ) 0.15. (b) Variation of dispersion number at various Rez as a function of Ta: (1) 2.57 and (2) 5.49. Table 4. Dispersion Characteristics in the Annulus, d ) 1.5 mm, l ) 0.225 mm

liquid water density ) 998 kg/m3

axial flow ×10-5 (kg/s) 0.25 5 0.25a

DEG density ) 1171.4 kg/m3

Figure 7. (a) Comparison with the experimental data of Pudijiono et al.14 at ri/ro ) 0.15. (b) Variation of dispersion number at various Ta as a function of Rez ((1) 0.83 rps, (2) 5 rps, and (3) 11.67 rps).

Simulations have been carried out for three rotational speeds of the inner cylinder, and dispersion characteristics have been analyzed. Table 4 shows the characteristics for various rotational speeds and the imposed axial flow as a function of the dispersion number and the dispersion coefficient. Figure 9 shows the RTD curves at various rotational speeds for an axial flow of 0.25 × 10-5 kg/s, for the test fluid with a density of 1171.4 kg/m3. The curves show sharp peaks of maximum values of concentration, and from Table 4 it can be seen that the dispersion

dispersion number (×102)

dispersion coefficient ×107 (m2/s)

0.83 5.00 11.67 0.83 5.00 0.83 5.00

1.06 1.655 4.273 3.393 3.392 1.12 1.879

0.051 0.079 0.2038 3.611 3.609 0.063 0.108

0.83 5.00 11.67 0.83 5.00

0.328 0.416 1.209 3.912 4.016

0.0156 0.019 0.0577 4.16 4.27

0.25

5 a

rotational speed of the rotor (rps)

Gap width ) 3.75 mm.

coefficient increases with an increase in the rotational speed. Figure 10 shows the curves for the test fluid with a density of 998 kg/m3. From the table it can be seen that the dispersion coefficient increases gradually until 5 rps, while a drastic increase is observed as the speed increases to 11.67 rps. This can be attributed to the radial mixing occurring with an increase in the rotational speed. As the rotational speeds are increased, the effect of the velocity profiles on the dispersion parameter (D/dU) decreases, though the overall dispersion coefficient increases. Hence, it can be said that, in the turbulent regime, the dispersion characteristics depend on the properties of the test fluid, unlike the Taylor vortex regime, wherein velocity profile plays a significant role. The effect of increasing the axial

6366

Ind. Eng. Chem. Res., Vol. 45, No. 18, 2006

Figure 9. RTD curves at various rotational speeds, axial flow rate ) 0.25 × 10-5 kg/s, liquid density ) 1171.4 kg/m3: (1) 0.83 rps and (2) 5 rps.

Figure 12. RTD curves at various rotational speeds, axial flow rate ) 5 × 10-5 kg/s, liquid density ) 998 kg/m3: (1) 0.83 rps and (2) 5 rps.

Figure 10. RTD curves at various rotational speeds, axial flow rate ) 0.25 × 10-5 kg/s, liquid density ) 998 kg/m3: (1) 0.83 rps and (2) 5 rps. Figure 13. RTD curves at various gap widths, axial flow rate ) 0.25 × 10-5 kg/s, liquid density ) 1171.4 kg/m3, rotational speed of inner cylinder ) 5 rps: (1) 1.5 mm and (2) 3.75 mm.

m3, as shown in Figure 13. The corresponding values of the dispersion number and the dispersion coefficient can be seen in Table 4. An increase in the dispersion number is observed with an increase in the gap width for a constant rotational speed and axial flow rate. It can be said that, since the critical Taylor number for the Taylor vortex regime only depends on the geometry and the rotational speed, an increase in the gap width leads to a significant variation in the dispersion number along with the variation in the velocity profiles. Since the effect of velocity profiles is significantly lower in the case of the turbulent vortex regime, the variation of dispersion coefficient would be negligible. Figure 11. RTD curves at various rotational speeds, axial flow rate ) 5 × 10-5 kg/s, liquid density ) 1171.4 kg/m3: (1) 0.83 rps and (2) 5 rps.

flow can be observed from Figures 11 and 12. The effect of rotational speed is not very significant in the Taylor vortex regime with an increase in the axial flow. Figure 11 shows the RTD predictions for the test fluid of density 1171.4 kg/m3, while Figure 12 shows the predictions for the test fluid of density 998 kg/m3. The dispersion number as well as the dispersion coefficient does not vary significantly with an increase in rotational speed at higher axial flow rates. The effect of gap width has been studied for the test fluid of density 1171.4 kg/

Conclusions CFD simulations have been carried out to predict the residence time distribution and the mixing time in the annular region of the Taylor-Couette contactor. The conclusions may be summarized as follows: (1) The RTD predictions have been found to be in good agreement with the established experimental results (2) Axial dispersion in the Taylor vortex regime and the turbulent regime has been predicted, and it can be said that, with an increase in the radial mixing, the effect of velocity profile on the dispersion parameter significantly reduces at higher rotational speeds.

Ind. Eng. Chem. Res., Vol. 45, No. 18, 2006 6367

(3) Effect of axial flow, annular gap width, and rotational speed has been investigated on the axial dispersion. The dispersion number and the dispersion coefficient have been predicted. (4) Mixing time has been predicted, and it has been found that there is no effect of tracer location on mixing time. Acknowledgment This work has been part of the project program supported by Board of Research in Nuclear Sciences (BRNS), Sanction No. 2002/34/7-BRNS/140. The authors also acknowledge Dr. Gantayet for the discussions provided during this study. Notation r ) cylinder radius (m) Ω ) rotational speed of cylinder (rev/s) c ) concentration of tracer d ) gap width (m) D ) axial dispersion coefficient (m2/s) e ) eccentricity l ) annulus height (m) Reθ ) azimuthal Reynolds number (ri2Ω/ν) Rez ) axial Reynolds number (duF/µ) Ta ) Taylor number TaCr ) critical Taylor number U ) velocity V ) flow rate (m3/s) Greek Symbols θ ) azimuthal coordinate Ω ) angular velocity (rev/s) Γ ) aspect ratio (l/d) µ ) molecular viscosity (kg‚m/s) η ) radius ratio (ri/ro) F ) density (kg/m3) λ ) wavelength (m) ν ) kinematic viscosity (m2/s) τ ) residence time Subscipts o ) outer cylinder i ) inner cylinder r ) radial direction s ) surface speed z ) axial direction θ ) azimuthal direction Literature Cited (1) Davis, M. W.; Weber, E. J. Liquid-Liquid Extraction. Between rotating concentric cylinders. Ind. Eng. Chem. 1960, 52 (11), 929. (2) Bernstein, G. J.; Grodsvenor, D. E.; Lenc, J. F.; Levitz, N. M. DeVelopment and Performance of a High-Speed Annular Centrifugal Contactor; ANL- 7968; 1973. (3) Cohen, S.; Maron, D. M. Analysis of a rotating annular reactor in the vortex flow regime. Chem. Eng. Sci. 1991, 46 (1), 123-134. (4) Imamura, T.; Saito, K.; Ishikura, S. A New Approach to Continuous Emulsion Polymerization. Polym. Int. 1993, 30, 203-206. (5) Ogihara, T.; Matsuda, G.; Yanagawa, T.; Ogata, N.; Fujita, Nomura, M. Continuous Synthesis of Monodispersed Silica Particles Using Couette-Taylor Vortex Flow. J. Ceram. Soc. Jpn., Int. 1995, 103, 151154.

(6) Sczechowski, J. G.; Koval, C. A.; Noble, R. D. A Taylor Vortex Reactor for Heterogeneous Photocatalysis. Chem. Eng. Sci. 1995, 50 (20), 3163-3173. (7) Baier, G.; Graham, M. D. Two-fluid Taylor-Couette flow: Experiments and linear theory for immiscible liquids between corotating cylinders. Phys. Fluids 1998, 10 (12), 3045-3055. (8) Holeschovsky, U. B.; Cooney, C. L. Quantitative description of ultrafiltration in a rotational filtration device. AIChE J. 1991, 37 (8), 12191226. (9) Tsao, Y. M. D.; Boyd, E.; Spaulding, G. Fluid Dynamics Within a Rotating Bioreactor in Space and Earth Environments. J. Spacecr. Rockets 1994, 31, 937. (10) Wereley, S. T.; Lueptow, R. M. Velocity Field for Taylor-Couette Flow with an Axial Flow. Phys. Fluids 1999, 11 (12), 3637. (11) Kataoka, K.; Doi, H.; Hongo, T.; Futagawa, M. Ideal plug flow properties of Taylor vortex flow. J. Chem. Eng. Jpn. 1975, 8 (6), 472. (12) Legrand, J.; Coueret, F.; Billon, M. Structure dynamique et trasnfert de matiere liquide-paroi dans le cas de lecouelement laminaire tourbillionaire de Couette-Poisuelle. Int. J. Heat Mass Transfer 1983, 26, 1075. (13) Legrand, J.; Coueret, F. Circumferential mixing in one-phase and two-phase Taylor vortex flow. Chem. Eng. Sci. 1986, 41 (1), 47. (14) Pudijiono, P. I.; Tavare, N. S.; Garside, J.; Nigam, K. D. P. Residence time distribution from a continuous Couette flow device. Chem. Eng. J. 1992, 48, 101-110. (15) Desmet, G.; Verelst, H.; Baron, G. V. Local and global dispersion effects in Couette-Taylor flow. I. Description and modeling of the dispersion effects. Chem. Eng. Sci. 1996, 51 (8), 1287-1298. (16) Enkoida, Y.; Nakata, K.; Suzuki, A. Axial turbulent diffusion in fluid between rotating coaxial cylinders. AIChE J. 1989, 35 (7), 12111214. (17) Moore, C.; Cooney, C. Axial dispersion in Taylor-Couette flow. AIChE J. 1995, 41 (3), 723-727. (18) Tam, W.; Swinney, H. Mass transport in turbulent Couette-Taylor flow. Phys. ReV. A. 1987, 36 (3), 1374-1381. (19) Rosenbluth, M. N.; Berk, H. L.; Doxas, I.; Horton, W. Effective diffusion in laminar convective flows. Phys. Fluids 1987, 30 (9), 26362647. (20) Aris, R. On the dispersion of a solute in a fluid flowing through a tube. Proc. R. Soc. London, Ser. A 1956, 237, 67-77. (21) Taylor, G. I. Dispersion of soluble matter in solvent flowing slowly through a tube. Proc. R. Soc. London, Ser. A 1953, 219, 186-203. (22) Launder, B. E.; Reece, G. J.; Rodi, W. Progress in the Development of a Reynolds-stress turbulence closure. J. Fluid Mech. 1975, 68 (3), 537. (23) Reynolds, A. J. The prediction of turbulent Prandtl and Schmidt number. Int. J. Heat Mass Transfer 1975, 18, 1055-1069. (24) Launder, B. E.; Spalding, D. B. The numerical computation of turbulent flows. Comput. Methods Appl. Mech. Eng. 1974, 3, 269. (25) Haut, B.; Ben Amor, H.; Coulon, L.; Jacquet, A.; Halloin, V. Hydrodynamics and mass transfer in a Couette-Taylor bioreactor for the culture of animal cells. Chem. Eng. Sci. 2003, 58, 774. (26) Parker, J.; Merati, P. An Investigation of Turbulent Taylor-Couette Flow using Laser Doppler Velocimetry in a Refractive Index Matched Facility. Trans. ASME 1996, 118, 810-818. (27) Haque, M. W.; Nigam, K. D. P.; Joshi, J. B. Optimum gas sparger design for bubble columns with a low height-to-diameter ratio. Chem. Eng. J. 1986, 33, 63-69. (28) Coles, D. Transition in Circular Couette Flow. J. Fluid Mech. 1965, 21, 385-425. (29) Levenspiel, O. Chemical Reaction Engineering, 2nd ed.; John Wiley and Sons: New York, 1972. (30) Kataoka, K.; Doi, H.; Hongo, T.; Futagawa, M. Ideal plug-flow properties of Taylor-vortex flow. J. Chem. Eng. Jpn. 1975, 8 (6), 472476. (31) Desmet, G.; Verelst, H.; Baron, G. V. Local and global dispersion effects in Couette-Taylor flow. II. Quantitative measurements and discussion of the reactor performance. Chem. Eng. Sci. 1996, 51 (8), 12991309. (32) Ohmura, N.; Kataoka, K.; Shibata, Y.; Makino, T. Effective mass diffusion over cell boundaries in a Taylor-Couette flow system. Chem. Eng. Sci. 1997, 52 (11), 1757-1765.

ReceiVed for reView July 13, 2005 ReVised manuscript receiVed May 2, 2006 Accepted June 13, 2006 IE050825N