Axial mixing in annular centrifugal extractors

Chemical Engineering Journal 207–208 (2012) 462–472

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Axial mixing in annular centrifugal extractors Tushar V. Tamhane a, Jyeshtharaj B. Joshi a,b,⇑, U. Kamachi Mudali c, R. Natarajan c, R.N. Patil d a

Department of Chemical Engineering, Institute of Chemical Technology, Matunga, Mumbai 400 019, India Homi Bhabha National Institute, Anushakti Nagar, Mumbai 400 094, India c Indira Gandhi Centre for Atomic Research, Kalpakkam, TN 603 102, India d Techno-Force Pvt. Ltd., Ambad D-34, Nashik 422 010, India b

h i g h l i g h t s " Axial mixing in annular centrifugal extractors was studied for different rotor sizes. " Effect of rotor speed and flow rate on axial mixing was quantified. " The effect of helical baffles was studied on axial mixing. " Helical baffles were found to impart a plug flow behavior. " For application in scale-up, correlations were proposed for Peclet number.

a r t i c l e

i n f o

Article history: Available online 20 July 2012 Keywords: Taylor–Couette flow Annular centrifugal extractor Axial mixing Residence time distribution Peclet number Computational fluid dynamics (CFDs) Helical baffles

a b s t r a c t Annular centrifugal extractors (ACEs) based on Taylor–Couette flow, have found various applications, some of which include liquid–liquid extraction and as bio and polymerization reactors. The residence time distribution (RTD) was measured in three size extractors having 50, 150 and 250 mm rotor diameter. The rotational speed was varied in the range of 10–30 r/s so that the power consumption per unit mass in the annular region was in the range of 25–750 W/kg. The extent of axial mixing was substantially reduced by providing helical baffles in the annular region. The effect of pitch was also studied. All the geometries were simulated using computational fluid dynamics (CFDs). A good agreement was obtained between the CFD predictions and the experimental measurements. Also, correlations have been proposed for Peclet numbers in the absence and presence of helical baffles. Ó 2012 Elsevier B.V. All rights reserved.

1. Introduction Annular centrifugal extractors (ACEs) find wide applications in chemical process industries involving liquid–liquid extractions. A schematic diagram of ACE is shown in Fig. 1. It consists of a stationary outer cylinder, concentric to which is a rotating inner cylinder. The annular gap between the two cylinders is kept small. The inner cylinder is rotated at a high speed which creates a large shearing force on the liquid in the annulus. The flow developed inside the annulus is referred to as the Taylor–couette flow. This flow is quantified by a dimensionless number, known as Taylor number (Ta), which is the ratio of centrifugal force to the viscous force and ex2 4  2 pressed as Ta ¼ ð1g g2 Þ d Xm . Annular centrifugal extractors, also called annular centrifugal contactors [1–5] offer several advantages over the other conven⇑ Corresponding author at: Department of Chemical Engineering, Institute of Chemical Technology, Matunga, Mumbai 400 019, India. Tel.: +91 22 33332106, +91 22 25597625; fax: +91 22 33611020. E-mail addresses: [email protected], [email protected] (J.B. Joshi). 1385-8947/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.cej.2012.06.151

tional process equipment such as low hold-up, high process throughput, low residence time, low solvent inventory and high turn down ratio. The equipment provides a very high value of mass transfer coefficient and interfacial area in the annular zone because of the high level of power consumption per unit volume [6,7] and separation inside the rotor due to the high g of centrifugal field. Annular centrifugal extractors find wide applications in nuclear fuel processing where safety is the main concern [2,8,9], in biological operations where controlled shear field and/or facilitated settling is important [10,11] and polymerization [12,13], excellent mixing, heat and mass transfer [14–20]. Annular centrifugal extractors can also be used for a variety of chemical reactions such as synthesis of mono-disperse silica particles, regeneration of spent activated carbon [21,22], esterification and hydrolysis [23], as cavitation reactor [24], and have also been demonstrated for use with ionic liquids [25]. The ACE consists of coaxial cylinders ((1) and (2)) as shown schematically in Fig. 1. The immiscible feed liquids enter tangentially at points (3A) and (3B) into the annular region between the two cylinders. The rotating impeller imparts power (in the range

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463

Nomenclature C(t) Cijk Ce1, Ce2 d Deff Di DL Dm Dt E Eh f g H k L n p P P0 hPi Pe Pk Q r Re Ri Sc Sct

tracer concentration at the outlet at time t (kmol/m3) turbulent diffusive transport (m3/s3) turbulence model constant for the transport equation of e (–) annular gap (m) effective diffusion coefficient (m2/s) diameter of the rotor (m) axial dispersion coefficient (m2/s) molecular diffusion coefficient (m2/s) eddy diffusion coefficient (m2/s) exit age distribution (s1) dimensionless exit age distribution (s1) ratio of VC/VL (–) gravitational acceleration (m/s2) height of the annular region (m) turbulent kinetic energy (m2/s2) path length of fluid in the reactor (m) number of vortices in the annular region (–) pitch of helical baffles (m) pressure (Pa) fluctuating component of pressure (Pa) average pressure (Pa)   Peclet number Pe ¼ VDLLL generation of turbulent kinetic energy due to mean velocity flow rate (m3/s) radial position (m) Reynolds number (–) radius of the rotor (m) Schmidt number (–) turbulent Schmidt number (–)

of 20–600 kW/m3) which results into a very fine dispersion of the two immiscible liquids. The dispersion flows downwards in the annular region (where the mass transfer occurs) and then flows radially inwards in the region below the rotating cylinder (points (4A) and (4B) in Fig. 1) and finally enters the central opening (orifice) of the rotating cylinder (point 5). Baffles (6) are provided in the bottom region which are either attached to the base of the outer cylinder or to the bottom of the rotating cylinder. The dispersion entering the central orifice, gets deflected towards the wall by the horizontal deflecting baffle (7) provided close to the entrance. Above the level of (7) the rotor is provided with vertical baffles (8) so as to create several chambers ranging from 4 to 8. The rotating cylinder imparts the liquid a practically rigid body rotation. The inner surface of the rotating liquid has almost a vertical shape (9) because of high ‘g’ except a small parabolic portion at the bottom. The dispersion entering at the bottom gets separated as it moves upwards. The rate of separation depends upon the drop size distribution, their settling velocities under the centrifugal action (rX2 ), densities, viscosities and coalescing behavior of the two phases. For complete separation (which is considered to be a flagship advantage of ACEs), adequate height needs to be provided for a given level of (rX2 ). After complete separation, the overflow weirs ((10A) and (10B)) are provided in such a way that, only very clean light and heavy phases pass over the weirs. The size and location of the weirs are provided in the hardware according to the relative flow rates of heavy and light phases and their corresponding clean widths (11) and (12). The flow of liquids from points (3A) and (3B) to (13A) and (13B) passes through the steps of extraction and separation. For the reliable design of process equipment, the knowledge of mass transfer rates and reaction

t Ta

time (s) 2 4 Taylor number Ta ¼ ð1g g2 Þ d ðXm Þ2 (–)

Tk ui u0i huii Us Uz VC VL VZR xi z

local mass fraction of the tracer (–) velocity component in ith direction (m/s) fluctuating component of velocity in ith direction (m/s) average velocity in ith direction (m/s) surface velocity of the rotor (m/s) axial velocity (m/s) average circulation velocity within an vortex (m/s) superficial velocity of liquid (m/s) inter-cell recirculation velocity (m/s) Cartesian co-ordinate in the ith direction (m) axial position (m)

Greek Symbols X rotor speed (rad/s) g radius ratio of the inner cylinder to the outer cylinder (–) m kinematic viscosity of the working fluid (m2/s)

r2 2 h

r sij e q s h

re rt lt Pij

variance (s2) ratio of r2/s2 (–) Reynolds stress (N/m2) turbulent energy dissipation rate (m2/s3) density (kg/m3) mean residence time (s) ratio of t/s (–) turbulent Prandtl number for energy dissipation rate (–) turbulent Prandtl number for kinetic energy (–) turbulent viscosity (Pa s) pressure–strain correlation (m2/s3)

rates (when extraction is accompanied by chemical reaction) together with the extent of axial mixing is important. The latter is the subject of present work.

2. Previous work During the past 35 years, some studies have been reported regarding the axial mixing in Taylor Couette flow over a wide range of design and operating parameters. Table 1 gives a summary of the published work. Kataoka et al. [26] performed RTD experiments and found that the Taylor vortex flow with a small constant axial flow can be an ideal plug flow for the range of 51.4 < Ta < 640 and 0 < Re < 90 for the geometry d/Ri = 0.333. Each cellular vortex, considered as a well-mixed batch vessel, was found to move axially with no intermixing over the cell boundary. When Ta exceeded 640, the plug flow could not be maintained owing to the occurrence of longitudinal intermixing over the cell boundary, which was attributed to the transition from singly to doubly periodic flow. The study on axial mixing in Taylor–couette contactors was further carried out by Pudjiono et al. [27]. These authors measured the residence time distribution in a continuous Taylor Couette device having 23.5 mm rotor. They found a near plug flow behavior at Ta = 60 and have described the axial mixing by dispersed plug flow model. Legrand and Coeuret [28] carried out the experimental investigation of circumferential mixing in a Taylor–couette flow. They proposed that each vortex unit be considered as an ideal stirred tank. In single and two phase studies carried out by them, they used tanks in series model. Also, they observed practically no

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Fig. 1. Conceptual diagram of annular centrifugal extractor. (1) Stationary cylinder, (2) rotating cylinder, (3A) light phase inlet, (3B) heavy phase inlet, (4A and B) region below rotating cylinder, (5) central opening for rotating cylinder, (6) radial baffles on the stationary bottom plate, (7) deflecting baffle in the rotor, (8) vertical baffles in the rotor, (9) interface between air and light phase, (10A and B) overflow weirs for lighter and heavier phase respectively, (11) clean width for heavy phase, (12) clean width for light phase, (13A and B) outlets for light and heavy phases respectively, and (14) liquid level in the annulus.

Table 1 Summary of the work on axial dispersion in the annular region of Taylor–Couette flow. Author

Ri/Ro (–)

Annular gap, d (mm)

Axial Reynolds number, Re (–)

Taylor number, Ta (–)

Kataoka et al. [13] Kataoka et al. [26] Desmet et al. [30,31] Pudijiono et al. [27] Ohmura et al. [36] Deshmukh et al. [34] Legrand and Coeuret [28] Croockewit et al. [29] Enokida et al. [35] Richter et al. [37]

0.75 0.5 0.642 0.94 0.703 0.75–0.93 0.548 0.24–0.89 0.593–0.76 0.875

10 20 24 1.5 18.4 1.5–6.5 11.3 5.1–43.2 18–30.5 0.005

0–90 0–35 0–6.3 0.03–5.5 0 17.5–70 0 3–24 60–250 0.03–0.51

50–640 50–640 50–35000 0–118 0–2000 1  106–2.9  108 435–870 170–98400 5148–38220 0–2209

intermixing between the vortices and the presence of second phase increased the circumferential mixing in the continuous phase. Croockewit et al. [29] carried out the studies on longitudinal diffusion through an annulus between stationary outer cylinder and a rotating inner cylinder. They used frequency response technique to measure the residence time distribution in a Taylor–couette contactor. The authors undertook all the measurements in the regime where the rotor speed was greater that the critical rotor speed.

The data was reported for a wide range of fluid viscosities, annular gaps and rotor speeds. It was noted through visual observations that, the height of the Taylor vortices was somewhat greater than the width of the annulus. They also found that, the circumferential speeds of the rotor and of the vortices were proportional to each other. These authors also observed that, there was a relative lowering in the value of average apparent diffusivity with rotor speed which was attributed to the decrease in the scale of turbulence.

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The authors stated that, the conclusions of that work, though not very exact, could certainly provide a basis for estimating the importance of the back-mixing effect of a physical operation or a chemical process was carried out in this apparatus. Thus, this work is certainly a value addition to the development of Taylor vortex flow. Desmet el al. [30,31] have studied the intra-vortex and intervortex axial dispersion in Taylor Couette devices. They refuted the previously published notion that in Taylor–couette flows, the inter-vortex dispersion is absent. Also they showed that the intra-vortex dispersion depended on various factors such as the tangential velocity and the dimensions of the vortices. Also, they showed that a single vortex cannot always be considered as a completely mixed stirred tank and depended upon the concentration gradients within the vortices. Vedantam et al. [5] and Deshmukh et al. [32–34] have carried out CFD simulations for RTD and compared it with the available experimental data in the Taylor vortex regime and the turbulent regime. It has been shown that the plug flow behavior is obtained in small scale equipment. However, substantial axial mixing was found to occur when the size is larger than 50 mm and/or the rotor speed is high. They further found that, the effect of net axial flow on the axial mixing significantly reduced at higher rotational speeds. Effect of axial flow, annular gap width, and rotational speed was investigated on the axial dispersion. Deshmukh et al. [34] performed CFD simulation of ACE with 39 mm rotor and 52 mm stator. They found that the number of vortices decreases with an increase in rotor speed. Further, they made an important observation that the Taylor vortices cannot be considered as non-interacting and substantial intervortex recirculation was found by PIV measurements and CFD simulations. They have proposed a procedure for the estimation of the number of tanks in series based on the number of vortices and the inter-cell circulation velocities. Based on the data obtained from CFD simulations and validation with the experiments, they could quantitatively state the difference between the observed number of vortices and the number of tanks in series. They also carried out an extensive analysis of the global and local dispersion. Also, the mixing time due to turbulent transport was much larger than the mixing time due to convective transport. Thus, it may be emphasized that, RTD and PIV measurements as well as the CFD simulations of Deshmukh et al. [34] (in agreement with Desmet et al. [30,31]) indicated markedly different results as compared with the hitherto well adopted notion that the number of counter rotating vortices can be considered to be equal to the number of completely stirred tanks in series resulting into a plug flow behavior. In view of these important observations, Deshmukh et al. [34] thought it desirable to introduce a geometrical modification in the annular region so that the number of vortices does not reduce with either an increase in the rotor speed or change in the annular gap (which is particularly expected as the size of ACE increases). For this purpose, they introduced radial baffles with a very small clearance between the stationary baffle and the rotating cylinder. It was observed (by PIV) that a pair of counterrotating vortices formed in each space between the two neighboring baffles. Thus, the vortex size could be independent of rotor speed as well as the annular gap. Further, the inter baffle recirculation was found

to be considerably lower than the intervortex recirculation in the absence of baffles. Thus, by introducing radial baffles, Deshmukh et al. [34] could obtain axial mixing behavior equivalent up to five CSTRs against one CSTR behavior in the absence of baffles. Richter et al. [37] studied the micromixing and macromixing in the Taylor-couette device. They used a cylindrical rotor as well as a modified geometry where they used a rotor with ribs. It was shown that, while micromixing, which is useful for multiphase systems was enhanced by this new geometry, macromixing, which is of major concern for most of the processes, reduced significantly for Ta > 130 in presence of ribs and they recommended this modified design. From the foregoing discussion, it is clear that, though a large number of axial mixing studies have been performed in the past, practically all of them have been carried out in small size equipment of the order of 50 mm rotor size. However, in practice, where larger throughputs are required, large size equipment needs to be used. Therefore, it was thought desirable to undertake a systematic investigation of residence time distribution in the annular region for large size (150 and 250 mm) over a wide range of power consumption. A smaller size of 50 mm has also been investigated for understanding the scale-up behavior. An attempt has been made to reduce the axial mixing and to get a practically plug flow behavior. Also, CFD simulations of axial mixing in 50, 150 mm dia. rotor ACEs have been performed and the results have been compared with the experimental measurements of residence time distribution. 3. Experimental 3.1. Equipment In the present work, the residence time distribution experiments have been performed in 50, 150 and 250 mm size (rotor) annular centrifugal extractors. The geometrical details are given in Table 2. The schematic diagram of the experimental set up is shown in Fig. 2. Experiments were also performed by providing baffles in the annular space. For this purpose, helical baffles were incorporated in the annular region of ACE-250. Two different pitches were employed in the present study, viz., 25 mm and 37.5 mm. 3.2. Experimental procedure The present study involved single phase investigations of axial mixing using water (q = 998 kg/m3, l = 0.001 Pa s) as the working fluid. All the experiments were carried out at room temperature. In case of ACE-50, peristaltic pumps were used to pump the fluid. When the experiments were carried out with ACE-150 and ACE250, centrifugal pumps were used. In all the experiments, water was first pumped through the inlet while the rotor was stationary. The flow rate was adjusted to the desired value. Then the rotor was started and its speed was gradually increased to the desired value. The system was allowed to reach steady state by running it for sufficient time. It has to be noted that, in annular centrifugal extractors, the steady state is achieved very quickly.

Table 2 Details of the experimental conditions used for the axial dispersion in the annular centrifugal extractor. Diameter of the inner cylinder (mm)

Height of the bottom baffle (mm)

Bottom gap between bottom baffles and inner cylinder (mm)

Number of baffles at the base

Ta

Mesh size

y+

250 150 50

23 15 8

2 2 2

10 8 6

3.5  109–3.15  1010 4.54  108–4.08  109 5.6  106–5.04  107

43  105 26  105 8.5  105

15–20 14–16 6–7

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r2h ¼ Motor

2DL 2 ¼ V L L Pe

ð7Þ

Thus, knowing the mean residence time and the variance, Pe can be evaluated. 4. Computational fluid dynamics CFD simulations were carried out on three different rotor diameters, viz. ACE-50, ACE-150 and ACE-250. In addition to this, two different baffles were incorporated in ACE-250.

Rotameter Probe position

4.1. Model formulation

Level indication for the annulus Pump

Storage tank

Fig. 2. Schematic diagram of the experimental set up.

Sodium chloride was used as a tracer. It was injected just before the inlet using a syringe. The tracer concentration was measured using a conductivity probe at the outlet of the annular region. The conductivity probe used consisted of graphite electrodes. It was calibrated against the potassium chloride solution of known concentrations. The probe was connected to the data acquisition card. It was in turn connected to the computer with a RS-232 port. The frequency of data acquisition was 10 Hz. It may be pointed out that the time constant of the probe was about 1000 times lower than the minimum residence time in the extractor. 3.3. Data analysis

were obtained over a wide range of flow rates, rotor speeds and baffle dimensions and analyzed. The parameters used to quantify the axial mixing were evaluated stepwise. First, the mean residence time (s) was calculated as:

s ¼ R0t

tCðtÞ dt

CðtÞ dt 0

ð1Þ

In discrete form, if t is the time interval between successive readings, the above expression for s can be re-written as:

P

tCðtÞDt s¼ P CðtÞDt

@ui ¼0 @xi

ð2Þ 2

On similar lines, the second moment, known as the variance (r ) can be evaluated as:

The momentum equation is written as follows:

@ui @ui uj 1 Pi @ ¼ þ þ @t @xj q xi @xj

CðtÞDt

r2 s2

 ð9Þ

@hui i ¼0 @xi

ð11Þ

@hui i @hui i 1 hPi i 1 @ þ huj i ¼ þ @t @xj q xi q @xj



l

@hui i þ sij xj

 ð12Þ

In Eq. (12), the term sij can be modeled using various turbulence models. It has been shown that [32–34] Reynolds Stress Model (RSM) could model the highly turbulent Taylor vortex flow more accurately than the k–e model. Hence, in the present work, RSM has been employed to model turbulence. In this model, individual Reynolds stresses can be computed using the following transport equation:



  @ sij @ sij @huj i @hui i ¼  qðsik þ sjk Þ þ qhuk i @xk @xk @t @xk 

 @ 2 2  C ijk þ Pij  edij þ mr sij xk 3

q

ð13Þ

where

C ijk ¼ hu0i u0j u0k i þ ð4Þ

ð5Þ

During the data analysis, the variance calculated above is made dimensionless by dividing it with the square of mean residence time. Thus,

r2h ¼

@ui xj

Thus, Eqs. (8) and (9) on Reynolds averaging give:

ð3Þ

which can in turn be normalized as:

E h ¼ sE

m

ð10Þ

The exit age distribution is calculated as follows:

CðtÞ E¼P CðtÞDt



The velocity ui can be written as a combination of mean velocity and fluctuating velocity:

P

t 2 CðtÞDt r2 ¼ P  s2

ð8Þ

ui ¼ hui i þ u0i

Axial mixing in the ACE was quantified by the dimensionless   parameter, Peclet number Pe ¼ VDLLL . The respective RTD curves

Rt

In the present case, three dimensional steady-state simulations have been carried out using the open source software OpenFOAM (version 1.6). simpleFoam solver was used in the present study. The following governing equations for continuity and momentum were solved: Continuity:

ðhP0 u0i idjk

þ hP 0 u0j idik Þ . . . Turbulent diffusive transport Y ij

¼

0 0 P @uj

q @xi

þ

@u0i @xj



. . . Pressure —strain correlation

The diffusive transport in Eq. (13) has been modeled  turbulent  @s as @x@ rlt @xij with the value of the constant rk taken as 0.82. Thus, k k k the term lt needs to be modeled. This is done as:

ð6Þ

This quantity can be related to the Dispersion coefficient and hence to Peclet number in the following way:

1

q

2

lt ¼ qC l

k

e

where value of the constant Cl is taken as 0.09.

ð14Þ

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467

The turbulent kinetic energy (k) is calculated as:

k¼

1 ðsii þ sjj þ skk Þ 2

ð15Þ

The turbulent energy dissipation rate, e is computed from the following transport equation:

    @e @e @ 1 l @e e þ C e1 P k ¼þ lþ t þ huj i @xj q @t @xj re r @xj k  C e2 q

e2

ð16Þ

k

where Pk is the generation of turbulent kinetic energy due to mean velocity gradients and is given as:

Pk ¼ sij

@hui i @xj

ð17Þ

The constants in this equation have the following values:

C e1 ¼ 1:44;

C e2 ¼ 1:92 and

re ¼ 1:0:

4.2. Model formulation for species transport The simulation of passive tracer was started only after the flow was converged. The assumptions made for these simulations are: (1) the diffusion process is isothermal, (2) the solute does not undergo any chemical reaction, and (3) the amount of the solute is so small that all the physical properties of the system remain unchanged. A tracer pulse was given at the inlet specifying the mass fraction to be equal to one. After the pulse is given, the mass fraction of tracer at the inlet was again set to zero and the following transport equation was solved:

  @ðqT k Þ @ðqT k Þ @ @T ¼ qDeff k þ huk i @t @xk @xk @xk

ð18Þ

In the above equation, Tk is the local mass fraction of the tracer. The effective diffusion coefficient, Deff is given by:

Deff ¼ Dm þ Dt

ð19Þ

where Dm and Dt are the molecular diffusion coefficient and eddy diffusion coefficient, respectively. Dm and Dt are estimated by:

Dm ¼ Dt ¼

m Sc

mt Sct

ð20Þ ð21Þ

where Sc is the Schmidt number and Sct is the turbulent Schmidt number. 4.3. Boundary conditions Figs. 1 and 3 show schematically the geometry of ACE-50, ACE150 and ACE-250 used for simulations in the present study. It may be noted that, except for the helical baffles incorporated in the annular region of ACE-250, the other geometrical parameters are identical on the respective scales for ACE-50 and ACE-150. As can be seen, tangential inlets were provided at the top. The outlet (or the orifice as it is called) is at the bottom of the rotor. On the bottom surface of the stationary cylinder (stator), curved vanes have been provided which guide the fluid into the rotor suction which is the orifice in the present simulations. At the inlet, the velocity inlet boundary condition was specified. Also, care was taken while specifying the direction of the velocity vector in OpenFOAM. Here, we needed to specify the velocity components for each direction. Thus, the inlet velocity was specified as

Fig. 3. Schematic diagram of the simulation geometry of ACE with parts shown: (1) Stationary cylinder, (2) rotating cylinder, (3A) light phase inlet, (3B) heavy phase inlet, (4) helical baffles, (5) central opening for rotating cylinder, (6) radial baffles on the stationary bottom plate.

a vector. Its value was calculated from the volumetric flow rate of the working fluid. OpenFOAM-1.6 provides the rotatingWallVelocity boundary condition. This was used for giving the desired rotational speed to the inner cylinder. Here, we needed to specify the axis of rotation for the inner cylinder. The rotational speed was specified in rad/s. No-slip condition was specified at all the walls. The mesh was resolved more in the near wall regions (near the stator, rotor, bottom vanes, helical baffles and in the region between the bottom vanes and rotor bottom). In these regions, the gradients for turbulence quantities are high. In order to reduce the excessive computational resources, the standard wall functions were used for Reynolds stresses. At walls, the near-wall values of the Reynolds stresses and e are computed from wall functions. The values of y+ have been given along with the mesh sizes in Table 2. The tracer was initialized by specifying its mass fraction as one at the inlet and the simulations were carried out on the velocity field obtained previously. This was the same as the pulse input in the experiments. Then, the tracer mass fraction was set to zero and the simulations were carried out on the fixed velocity field from the converged steady-state solution. Simulation time for the tracer transport in the frozen flow field was found to be in the range of 4–8 days in the range of all the variables covered in this work. During this time, the tracer concentration at the outlet was recorded continuously at different time intervals. 4.4. Method of solution For the current simulations, simpleFoam solver was used. The momentum equation was discretized using Gauss upwind scheme. Preconditioned bi-conjugate gradient solver (PBiCG) was used for velocity, while preconditioned conjugate gradient solver was used for pressure. Pressure–velocity coupling was done using SIMPLE scheme. After the fully developed flow was obtained, the tracer was injected as a pulse input. Its concentration was calculated using scalarTransportFoam solver in the unsteady state simulation where only the scalar concentration is changing with time. A nonuniform hexahedral mesh was used in all the three extractors. The mesh was finer near the wall regions. In the regions away from the walls, the mesh spacing was more.

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0.1

DIMENSIONLESS AXIAL VELOCITY, Uz/Us

0.04 0.03

0.05

0.02 0.01

0

0 -0.01

-0.05

-0.02 -0.03

-0.1

-0.04 0

0.2

0.4

0.6

0.8

DIMESIONLESS RADIAL DISTANCE,(r-R i)/d

(A)

1

0

0.2

0.4

0.6

0.8

1

DIMESIONLESS RADIAL DISTANCE,(r-R i)/d

(B)

Fig. 4. Grid size dependence of axial velocity profiles for: (A) ACE-50 at z/H = 0.6, (1) Grid size = 8.5  105, (2) Grid size = 10  105, (3) Grid size = 6.2  105, (B) ACE-150 at z/ H = 0.5 (1) 26  105 (2) Grid size = 29.5  105 and (3) Grid size = 21  105.

4.5. Grid independence In case of ACE-50, the total grid size used was 8.5  105. This was arrived at by checking the sensitivity of grid size on the velocity profiles. Initially, a grid coarser than the current grid (6.2  105) was used and the results for the axial velocity profile were obtained. Similarly, a finer grid (106) was also employed. The dependence of velocity profiles on the grid size has been shown at z/H = 0.6. Similarly, in case of ACE-150, the grid size dependence was investigated for the grid sizes 21  105, 26  105 and 29.5  105 and the results of axial velocity profiles have been shown at z/H = 0.5. Thus, a grid-independence test was carried out for different grid sizes. This has been shown in Fig. 4. Similar procedure was followed for ACE-250, where the grid size used was 43  105. The mesh was finer in the near wall regions. In the regions away from walls, a relatively coarse mesh was used. In the near wall regions (near the rotor bottom, in the region between bottom vanes and rotor bottom, regions between the vanes), the mesh spacing was approximately 0.5 mm while in the regions away from the walls, the mesh spacing was of the order of 1 mm. Parallel computing was used for large grid sizes, wherein the total geometry was divided into different parts and the simulations were carried out on different processors simultaneously. The details of the meshing have been provided in Table 2. 5. Results and discussion (1) As has been discussed, most of the data reported so far in the literature has covered laboratory scale rotor sizes. There is a scarcity of data on the axial mixing in annular centrifugal extractors (ACEs) with rotor sizes of the order of 250 mm. (2) The scale-up depends upon our understanding of flow patterns with respect to scale. (3) In spite of a wide range of applications that they are useful in, annular centrifugal extractors have been shown to have a drawback practically complete backmixing. The throughput in ACE can vary between 50 L/h in 50 mm rotor size to 6500 L/h in ACE-250. The flow characteristics certainly cannot be expected to be identical under these two extremes. Moreover, the flow is highly turbulent under the conditions where they are operated. Hence, the backmixing is expected to be high.

(4) Taylor vortices are very peculiar to the annular centrifugal extractors In addition to the bulk tangential motion of the fluid, these vortices also revolve around themselves The extent of backmixing due to these vortices is a complex function of the following variables: (i) the number of vortices, (ii) the dimensionless vortices, (iii) the tangential velocities around the global axis as well as the vortex axis, and (iv) concentration gradients within the vortices. These vortices have been shown to be interacting and substantial intervortex backmixing has been shown to take place. Thus, the Taylor vortices and their attributes are the main cause of the bulk flow behavior in the annular region of annular centrifugal extractors. (5) In the liquid–liquid extraction, the hold-up values of the dispersed phase are typically in the range of 10–25%. Also, the power consumption is in the range of 5–500 kW/m3. At such high power consumption, the drop diameters are of the order of a few microns. These drops have a significant slip velocity and are expected to follow the bulk fluid motion and are not expected to alter the bulk motion to a great extent. The extent of backmixing has been quantified by the Peclet numbers (Pe). The respective values of Pe have been reported in Table 3. Higher values of Pe imply that convection dominates and eventually approaches plug flow behavior at Pe exceeding about 10. Fig. 5 shows the comparison of RTD curves obtained using the experimental measurements and the CFD predictions. 5.1. Effect of rotor speed All the measurements in the present study have been carried out in the turbulent region of Taylor Couette flow. In order to study the effect of rotational motion on the axial mixing, the rotor speed was varied between 10 rps and 30 rps. It was observed that, at a particular flow rate, when the rotor speed was increased, the values of Pe decreased, indicating an increase in the axial mixing. This conclusion holds for all the three extractors covered in this work. It was found that the number of vortex cells in the annular region was reduced with increase in rotor speed. Fig. 6 shows the velocity contours in ACE-50 at two different rotor speeds. Also, the typical profiles of eddy diffusivity have been shown in Fig. 7.

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T.V. Tamhane et al. / Chemical Engineering Journal 207–208 (2012) 462–472 Table 3 Summary of results for axial mixing in annular centrifugal extractors. Baffle pitch, p (mm)

Flow rate, Q (mL/s)

Rotor speed, N (r/s)

VL (mm/s)

Pe (Exp)

50 50 50 50 150 150 150 150 250 250 250 250 250 250 250 250 250 250 250 250

No baffles No baffles No baffles No baffles No baffles No baffles No baffles No baffles No baffles No baffles No baffles No baffles 25 25 25 25 37.5 37.5 37.5 37.5

9 9 14 14 55 55 146 146 278 278 1250 1250 278 278 1250 1250 278 278 1250 1250

10 30 10 30 10 20 10 20 10 30 10 30 10 30 10 30 10 30 10 30

10.4 10.4 16.2 16.2 7.15 7.15 18.8 18.8 12.9 12.9 57.9 57.9 566 566 2548 2548 252 252 1132 1132

2.34 1.16 3.11 1.26 0.81 0.48 1.56 0.91 0.93 0.42 2.78 2.29 7.74 5.78 7.86 6.71 4.71 3.62 5.91 4.41

DIMENSIONLESS EXIT AGE DISTRIBUTION, Eθ

Rotor dia., Di (mm)

1.2

1.2

1.2

1

1

1

0.8

0.8

0.8

0.6

0.6

0.6

0.4

0.4

0.4

0.2

0.2

0.2

0

0 0

2 4 6 8 10 DIMENSIONLESS TIME,θ

n

VZR (mm/s)

f

Pe

2 2 2 2 2 2 2 2 2 2 2 2 6 6 6 6 6 6 6 6

6 18 3 24 12 45 22.5 45 30 120 25 44 193 264 840 1256 232 313 801 1107

0.58 1.73 0.185 1.48 1.68 6.3 1.19 2.39 2.33 9.33 0.43 0.76 0.34 0.47 0.33 0.49 0.92 1.24 0.71 0.98

1.86 0.89 2.92 1.01 0.92 0.29 1.18 0.69 0.7 0.20 2.14 1.58 7.13 6.21 7.23 6.04 4.22 3.44 4.97 4.06

0 0

2 4 6 8 DIMENSIONLESS TIME, θ

(A)

10

0

5 DIMENSIONLESS TIME,θ

10

(C)

(B)

0.9 DIMENSIONLESS EXIT AGE DISTRIBUTION, Eθ

CFD predictions

1

0.8

0.9

0.7

0.8

0.6

0.7 0.6

0.5

0.5 0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0

0 0

1 2 3 4 DIMENSIONLESS TIME,θ

(D)

5

0

1 2 3 4 DIMENSIONLESS TIME,θ

5

(E)

Fig. 5. Sample residence time distribution diagrams for the annular region (. . .) Experimental (–) CFD (A) ACE-50: 9 mL/s and 10 r/s, (B) ACE-150: 146 mL/s and 20 r/s, (C) ACE-250 in the absence of baffles: 278 mL/s and 10 r/s, (D) ACE-250 in the presence of baffles with pitch 25 mm: 278 mL/s and 10 r/s, and (E) ACE-250 in the presence of baffles with pitch 37.5 mm: 1250 mL/s and 30 r/s.

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4.00e-1 3.33e-1 2.67e-1 2.00e-1 1.33e-1 6.67e-2 0 6.67e-2 -1.33e-2 -2.00e-1 --2.67e-2 -3.33e-1 -4.00e-1

(A)

(B)

Fig. 6. Comparison showing the contour plots for axial velocity in ACE-50 at 9 mL/s at (A) 10 r/s (B) 30 r/s.

30

EDDY DIFFUSIVITY, m2/s(X105)

EDDY DIFFUSIVITY, m2/s(X106)

25

20

15

10

5

25 20 15 10 5 0

0 0

0.5

1

0

0.2

0.4

0.6

0.8

1

DIMENSIONLESS RADIAL DISTANCE (r-Ri)/d

DIMENSIONLESS RADIAL DISTANCE (r-Ri)/d

(B)

(A)

Fig. 7. Sample profiles for eddy diffusivity in the annular region of (A) ACE-50 at 9 mL/s and 62.8 rad s. (B) ACE-150 at 146 mL/s and 125.6 rad s.

It was thought desirable to explain this phenomenon on the basis of interstage recirculation velocities. The procedure for analysis was based on the tanks-in-series model with interstage recirculation [38]. This approach has been used by Joshi [39] for the case of bubble columns where multiple recirculation cells prevail. It was shown that, if n is the number of interacting recirculation cells in the column, the Peclet number is given by:

  1 Pe ¼ n= f þ 2

The values of n, f and Pe for the three extractors have been shown in Table 3 over a range of rotor speed and liquid velocity covered in this work. It can be observed that, in the absence of helical baffles, the values of Pe are relatively low, flow closer to backmixed behavior, particularly when the values of Pe are less than 1. 5.2. Effect of flow rate

ð22Þ

where f is the ratio of inter-cell recirculation velocity to the superficial velocity. Joshi [39] has estimated the value of f on the basis of flow pattern at the interface of the two adjacent cells. Further, he has shown excellent agreement between the model predictions and all the experimental data published in the literature over a wide range of column diameter, column height, gas and liquid superficial velocities and the physical properties of the gas and liquid phases. It was shown that the value of interstage recirculation velocity, VZR was equal to 0.3 times the average liquid circulation velocity within the cell, VC. The same approach was adopted for annular centrifugal extractors, where multiple Taylor vortices prevail in the annular region. As has been shown by Desmet et al. [30,31], the Taylor vortices involve inter-vortex dispersion. Thus, the Taylor vortices could be considered as interacting. The inter-cell recirculation velocities were calculated from the flow patterns obtained from CFD simulations.

The flow rate has a direct impact on the hydrodynamics in the annular region. As has been shown in the previous studies [32,33,40], the axial flow stabilizes the Taylor–couette flow and it delays the instability. In other words, occurrence of Taylor vortices in presence of a net axial flow is delayed in comparison with that in the absence of it. This means that, it is the magnitude of axial component of velocity relative to the tangential component that decides the formation of Taylor vortices. Extending the approach suggested by Joshi [39] to this discussion, one is tempted to observe the effect of the flow rate on parameter f. This can be directly observed from Table 3. The superficial velocity VL is a direct indication of the magnitude of axial velocity. Any increase in VL leads to a decrease in the average liquid circulation velocity (VC). Thus, the inter-cell recirculation velocity (VZR) which is directly dependent on VC also decreases with an increase in VL. The values of inter-cell recirculation velocities (VZR) have also been mentioned in this table. It can be seen that, with increase in the axial velocity, there

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7.5e-1 6.25e-1 5e-1 3.75e-1 2.5e-1 1.25e-1 0 -1.25e-1 -2.5e-1 -3.75e-1 -5e-1 -6.25e-1

(A)

-7.5e-1

(B)

Fig. 8. Comparison showing the contour plots for axial velocity in ACE-250 at 1250 mL/s and 20 r/s (A) without helical baffles (B) with helical baffles with 25 mm pitch.

is a decrease in the inter-cell recirculation velocity at a constant rotor speed. Thus, the parameter f (=VZR/VL) decreases. Thus, from Eq. (22), it can be seen that Pe decreases. 5.3. Effect of helical baffles in the annular region of ACE-250 As has been discussed in Section 1, Taylor vortex flow, which is the characteristic of annular centrifugal extractors, has applications in numerous different areas, owing to the high values of mass transfer coefficients, interfacial areas and low residence times compared to conventional devices. However, this feature also imparts a drawback of backmixed behavior [34]. Backmixed behavior is known to result in much larger reactor volumes as compared to the plug flow behavior. For instance, for a first order reaction and 98 per cent conversion, the volume of backmixed reactor is about 400% higher than a plug flow reactor. Thus, a backmixed behavior 3.5

results into high capital as well as operating costs. Therefore, it was thought desirable to make an effort to reduce the extent of backmixing in the annular region. From Eq. (22), it can be seen that the backmixing can be reduces by increasing the number of vortices and by reducing the intervortex recirculation velocity. For this purpose, helical baffle was provided in the annulus. Two different baffles pitches were used, viz. 25 mm and 37.5 mm. In the absence of helical baffles, the fluid was found to be closer to the backmixed behavior. It was observed that, inclusion of helical baffles significantly reduced the extent of backmixing. This is shown by higher values of Pe in Table 3. Fig. 8 shows the comparison between the Taylor vortices obtained in the absence and presence of helical baffles. It can be seem that, each compartment of the helical baffle contains a pair of counter-rotating vortices. For this, the inter-cell recirculation velocities were calculated (Table 3) and these were in turn used to calculate the values of f in Eq. (22). It can be seen that, the values of Pe significantly increase in the presence of helical baffles. This indicates the reduction of backmixing. It can also be seen from this table that, in case of ACE-250, the axial mixing observed at larger baffle pitch was more than that observed at smaller pitch. This can be attributed to the fact that, in case of smaller baffle pitch, the number of vortices formed is more. Thus, according to Eq. (22), the backmixing reduces to a greater extent than that in the case of larger pitch. As has been discussed, axial mixing increases with increase in rotor speed. It can be seen from Table 3 that, there is an agreement in the values of Pe reported by experiments and CFD. It is interesting to note that, the difference in the experimentally measured values and those predicted by CFD is more at higher rotor speeds. This is possibly because, at higher rotor speeds, the pumping capacity of the rotor is high. Hence, the annular region has a reduced liquid height. Also, the contact between the rotor and the fluid is not continuous and the annular liquid height is oscillating [41]. At the typical operating speeds used for the applications using ACE, there is an unavoidable entrainment of air. Therefore, the actual flow in the annulus is not always entirely single phase but has some bubbles trapped in the liquid. This causes some discrepancies in the experimental measurements and CFD predictions. In the experiments, care was taken to ensure that the air entrainment did not largely affect the measurements. The problem was aggravated in larger units (ACE-150 and ACE-250). This was tackled by adjusting the bottom resistance using an adjustable screw and continuously observing the liquid height in the annulus using a transparent tube

10 9

3

Pe (PREDICTED)(-)

8 2.5

7 6

2

5 1.5 4 1

3 2

0.5

1 0

0

1

2

Pe (EXPERIMENTAL)(-)

(A)

3

0

0

2

4

6

8

10

Pe (EXPERIMENTAL)(-)

(B)

Fig. 9. Parity plots for showing comparison between the values of Peclet numbers predicted by experiments and by correlations (A) in the absence of helical baffles (B) in presence of helical baffles.

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attached externally and also through the sight glass provided at the outer cylinder. 5.4. Scale-up aspects of annular centrifugal extractors from the standpoint of current study In this paper, we have undertaken experimental measurements and CFD simulations of the annular centrifugal extractors over a wide range of rotor size, rotor speed and superficial liquid velocity. In previous section, an excellent agreement has been shown between the experimental and predicted (CFD) values of Peclet numbers. Thus, in the exercise of scale-up, one may argue that the tool of CFD can be employed. However, implementation of CFD simulation may sometimes be difficult because of the unavailability of computational power/experience. Therefore, it was thought desirable to propose correlations for Peclet numbers for the two cases: In the absence of helical baffles:

Pe ¼ 16:03

!0:71  0:82 ND2i q Di V L q

l

ð23Þ

l

In presence of helical baffles:

Pe ¼ 139:53

!0:23  0:098 ND2i q Di V L q

l

l

ð1 þ p=dÞ1:51

ð24Þ

The correlation coefficients for the above two equations were found to be 0.94 and 0.97 respectively. The parity plots showing a comparison between the experimental and the predicted values have been shown in Fig. 9. 6. Conclusion (1) Experiments were performed on three different sizes of ACE, viz. ACE-250, ACE-150 and ACE-50. Thus, the RTD studies carried out cover a wide range, from lab scale to the industrial scale spanning the extractor capacity in the range of about 0.1–0.5 m3/h. (2) CFD simulations were undertaken to see their validity in predicting the flow and axial mixing behavior in scaled-up version of annular centrifugal extractors. (3) The approach of inter-cell recirculation was adopted for the analysis of axial mixing behavior. For this, a parameter f (=VZR/VL) was defined and the effect of various operating conditions on f was considered. (4) The annular region was found to be closer to backmixed behavior. It was observed that the extent of backmixing decreases with an increase in the net flow rates. On contrast, increase in the rotor speed caused an increase in the backmixing. (5) Effect of helical baffles in the annular region was also studied. It was found that, incorporating the helical baffles significantly reduced the backmixing. Further, it was observed that the backmixing decreases with a decrease in pitch.

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