7242
Ind. Eng. Chem. Res. 2008, 47, 7242–7249
Development of an Empirical Model To Predict the Effect of Contaminants in Liquid-Liquid Extraction S. N. Ashrafizadeh,*,† J. Saien,‡ B. Reza,† and M. Nasiri† Research Laboratory for AdVanced Separation Processes, Department of Chemical Engineering, Iran UniVersity of Science & Technology, Narmak, Tehran 16846, Iran, and Department of Applied Chemistry, UniVersity of Bu-Ali Sina, Hamadan, 65174, Iran
In the presence of surface-active reagents, the flux of mass transfer into/from the droplets drastically decreases within the liquid-liquid extraction units. Therefore, particular design equations are required to incorporate the presence of contaminants in the design of industrial-scale extraction columns. In the combinatorial model of Slater, the mass-transfer coefficients for the continuous and dispersed phases are corrected with the aid of a contamination factor. However, since the latter factor in the Slater model is originally obtained from the experimental data, the Slater model is unable to predict the behavior of an extraction system where the experimental data are not available. In this research, the model of Slater, i.e., single-drop single-solvent model, was employed to simulate the experimental data of a water-acetone-toluene system in the presence of three surface-active reagents, i.e., SDS, Triton X-100, and DTMAC. These components exhibit anionic, cationic, and nonionic surfactants, respectively. The experimental data provided by Saien et al. for both directions of the mass transfer, i.e., continuous to dispersed phases and vice versa, were employed. Based on the experimental data along with the theoretical principles, an empirical model for the contamination factor was developed. The provided model predicts the mass-transfer coefficients within (4% of certainty only through the physical properties data of the phases, independent of the surfactant type. Two different mechanisms for the mass transfer within the liquid droplets, depending on the size of the droplets, have been recognized. Simulation of the extraction system by the aid of the present model, along with the combinatorial model of Slater and the terminal velocity model of Grace, predicts the experimental results satisfactorily. Introduction Liquid-liquid extraction is a unit operation that is largely employed by process industries for separation purposes. In this process, the solute is transferred between the droplets of a dispersed phase and the bulk of an immiscible continuous phase. The mass transfer by droplets occurs through three sequential steps of droplet formation, the movement of droplets within the continuous phase, and the coalescence of the droplets. The study of steady-state movement of droplets within a liquid phase is the most appropriate way to grasp a theoretical basis for the liquid-liquid extraction systems. In this regard, numerous researchers have employed the single-drop method as a viable technique to study the influence of various factors on the mass transfer.1-5 In most of the industrial extraction processes, there exist some amount of contaminants of surfactant nature that influence both the droplet characteristics and the flux of mass transfer drastically.2 Such contaminants can reduce the flux of mass transfer through reduction in the internal droplet circulation and increase in the drag formation. The reduction of mass transfer within the single-droplet systems containing surface-active reagents has been attributed to two important factors, i.e., hydrodynamic characteristics of the droplets and physicochemical properties of the contacting phases. The effect of surface-active reagents on the hydrodynamic characteristics of the moving droplet includes the variation in the internal circulation rate and hindering the interfacial movements due to the surface tension gradients of the droplets.4 The influence of surface-active * To whom correspondence should be addressed. E-mail:
[email protected]. † Iran University of Science & Technology. ‡ University of Bu-Ali Sina.
reagents in reducing the flux of mass transfer has been summarized into the following items: (1) blockade of the interface;5 (2) interaction of the contaminant molecules with the other existing molecules;5 (3) hydrodynamic variations including the reduction in circulation and lowering the interfacial movement during the formation, ascension, and coalescence of the droplets;5 (4) prevention of interfacial disturbances and changes in the surface tension, which consequently hinders the bulk transport of the materials;5 (5) reduction in the drop size and thus interfacial area;6 (6) formation of an interfacial tension gradient along the drop surface and exertion of a marangonic stress, which further suppresses the interfacial flow.6 The study of solute mass transfer from/into the single droplet, which surface has been occupied with contaminants, results in a better understanding of the liquid-liquid extraction process as well as a more efficient design of process equipment. Numerous theoretical and experimental research efforts have been conducted in order to investigate the effects of surfaceactive reagents on the mass transfer of single droplets.1,3 Several researchers have also attempted to incorporate the effect of contamination on the mass transfer in liquid-liquid systems. For instance, Locheil incorporated the effect of contamination by using a correction factor for the Sherwood number of the continuous phase. In the latter correction factor, a parameter called a contamination factor was used. Slater9 incorporated the effects of contamination and introduced the term of contamination factor within the continuous phase by employing the same contamination factor into the equation of eddy diffusion coefficient, which was provided previously by Handolos and Baron. The purpose of Slater was to develop a combinatorial model that is capable of incorporating the effect of contamination into the mass-transfer coefficient through only one parameter.7-11 In spite of the attempts made by researchers in
10.1021/ie0715557 CCC: $40.75 2008 American Chemical Society Published on Web 08/29/2008
Ind. Eng. Chem. Res., Vol. 47, No. 19, 2008 7243 7,8
developing theories for the contamination factor, most of them, including Slater, considered this term as an adjustable parameter that is determined from the experimental data for each particular system.7,8 In the current research, a model has been developed to predict the contamination factor for the droplets in the vicinity of circulating drops, using the experimental data provided for a system of toluene-acetone-water, which has been contaminated with one of the three surfactants of different ionic properties. The principal of this model is based on the influence of surfactants in leading the droplets to the rigid form and reducing their internal circulations.
circulations. To consider the effect of surface contaminants, Locheil modified eq 2 with the term R0.5 as a correction factor. The latter factor has been defined as the ratio of the average interfacial velocity to the limiting velocity:
Theoretical Principles
The contamination factor should be determined experimentally.9 The Locheil equation cannot be employed for low Reynolds numbers (Re < 10) by assigning negative values for Shc, and it is valid only for the correction coefficients close to 1. Therefore, Slater used the following assumption:
A. Continuous-Phase Mass-Transfer Coefficients in the Absence of Contaminants. The mass transfer into a single droplet can be regarded as a steady-state process providing that the drop be assumed as a continuous and unlimited media. The latter transfer is however limited to two extreme limits; i.e., mass transfer to the rigid drops, and to those which have fully developed internal circulations. The characteristic of rigid drops reflects the minimum rate of mass transfer, which is determined by creeping flow (κ f ∞). Equation 1 states the same conditions:7 Shc ) 1 + (1 + Pe)
1⁄3
2 1⁄2 Pe (2) √π Besides the above-mentioned theoretical equations, researchers have provided numerous empirical equations for the masstransfer coefficients in the following general format:12 Shc )
(3)
13
Steiner, using experimental data that had incorporated the resistance of the continuous phase, provided a model that describes the mass transfer into the rigid drops: 1⁄3 Shcr ) 2.43 + 0.775 Re1⁄2 Sc1⁄3 c + 0.0103 Re Scc
(4)
Slater provided a model so-called, single-drop single-solute model (SSM), which is viable for the calculation of overall mass-transfer coefficient in the liquid-liquid media containing contamination at both potential and creeping flow conditions.8,14 Due to its ease of application, the Slater model is commonly used by various researchers to discuss their experimental data.6-11 The latter model is briefly reviewed in the following section. B. Continuous-Phase Mass-Transfer Coefficients in the Presence of Contaminants. As mentioned earlier, two ideal extreme limits can be considered for the mass transfer. In the conditions of creeping flow, which is described by eq 4, relatively small and clean drops behave as rigid drops with Re < 10 and Eo < 10. In the case that the interfacial surface has been occupied with surface-active contaminants, even largesize droplets may behave as rigid drops.8 The other extreme limit is defined by the potential flow (eq 2), which describes the mass transfer into the drops with internal
ui (2 + 3κ + m) 1.45 )1ut 1 + (F µ ⁄ F µ )0.5 Re0.5 d d
(5)
c c
where κ is the ratio of the viscosity of the phases, i.e. (µd/µc), and m is the contamination factor, which is defined below:8-10 m ) -β
∂γ ⁄ ∂Γ µc
(6)
R ) 1 - x ≈ (1 + x)-1 (7) In this way, Slater attempted to extend the validity of the equation for larger Reynolds (10 < Re < 1000). As a result, Slater defined the following definition for R:
[
(2 + 3κ + m) 1.45 1 + (Fdµd ⁄ Fcµc)0.5 Re0.5
R) 1+
(1)
where Pee ) Re Sc. On the other hand, the characteristic of drops with developed internal circulations reflects the maximum rate of mass transfer. This characteristic, which has been theoretically developed by Bousinesq for single drops, provides the conditions of potential flow (κ f 0):7
Shc ) 2 + C1ReC2ScCc 3
R)
]
-1
(8)
The last equation, along with the correction that Locheil applied to eq 2, provides the final Sherwood number, which contains only one variable factor, i.e., contamination factor (m) 9,10
Shc )
[ {
(2 + 3κ + m) 2 1+ 1 + (Fdµd ⁄ Fcµc)0.5 √π
}]
-0.5
Re0.5Sc0.5 c
(9)
Meanwhile, Slater suggested that eq 4 should be employed to determine the lower valid limit of the above equation. C. Dispersed-Phase Mass-Transfer Coefficients. The mass transfer within the dispersed phase is totally dependent on the hydrodynamics of the system. For the small droplets who behave similar to rigid drops, the mechanism of diffusive mass transfer governs. As much as the circulation of drops increases, the diffusive mass transfer reaches the eddy mass transfer. In this regard, Slater defined the following relation for the overall effective diffusion coefficient:8-10 Doe ) Dd + De
(10)
Considering the above equation, Slater used the same definition provided by Hanlos and Baron for the eddy diffusion. However, he also employed the same correction factor, R, which he had employed for the continuous phase.7,8,11 dRut (11) 2048(1 + κ) The mass-transfer coefficient for diffusion into the stagnant droplets is calculated from Newman equation, which is valid in the lack of resistance for the continuous phase. In this equation, the overall diffusion coefficient, Doe, is used instead of the molecular diffusion coefficient, Dd,.7,11-14 De )
kd ) -
[
( ) ( )∑ ( ) d 6 ln 6t π2
∞
n)1
(
-4n2π2Doet 1 exp n2 d2
)]
(12)
D. Overall Mass-Transfer Coefficients. Regarding the common practice which the mass transfer within the two phases is considered, the provided models have been combined based
7244 Ind. Eng. Chem. Res., Vol. 47, No. 19, 2008 Table 1. Physical Properties of Toluene-Acetone-Water System at Ambient Temperature8 property
toluene
water
3
866.7 998.2 density (F) (kg/m ) viscosity (µ) (kg · m/s) 0.628 × 10-3 10.0118 × 10-3 diffusion coefficient in water (D) (m2/s) diffusion coefficient in toluene (D) (m2/s)
acetone 790.5 0.3976 × 10-3 1.09 × 10-9 2.55 × 10-9
Table 2. Experimental Variables and Examination Levels variable
level of examination
mass-transfer direction C to D; D to Ca type of contaminant SDS, Triton X-100, DTMAC contaminant concentration (mM) 6.25, 12.5, 25, 50, 100 droplet diameter (mm) 2-5 (produced by 5 different nozzles) a
C, continuous phase; D, dispersed phase.
on the two-film theory of Whiteman to determine the overall mass-transfer coefficient.8-10 1 1 φ ) + Kod kd kc
(13)
In spite of the various provided models, the model of Slater is capable of covering the whole range of mass transfer within the previously mentioned two extreme limits. In the model of Slater, the contamination factor is used to adjust the model with the variable experimental data. Therefore, the contamination factors cannot be used for prediction purposes prior to be determined through experimental data.9 Experimental Data In this research, the experimental data provided by Saien et al.6,15 have been employed. The materials and methods of the experiments have been briefly reviewed in this section while the details on the experimental apparatus, which provides sufficient contact of the single droplets with the continuous phase, and procedures are provided elsewhere.6 A. Chemicals and Experimental Setup. The experimental data for the ternary system of toluene-acetone-water have been investigated. Toluene, water, and acetone have been employed as the dispersed phase, the continuous phase, and the solute, e.g., the transferring component, respectively. Depending on the direction of the mass transfer, the solute has been present only in the origin phase at a specified quantity. The physical properties of the applied system are given in Table 1. Three different types of surfactants including SDS (anionic surfactant), Triton (cationic surfactant), and DTMAC (nonionic surfactant) have been used at various concentrations as the contaminants. The experiments have been conducted at both directions of mass transfer, i.e., from dispersed phase to continuous phase and vice versa, in a two-phase liquid-liquid system for the free movement of single droplets. The experimental variables and the levels at which variables have been examined are given in Table 2. B. Experimental Mass-Transfer Coefficients. Regarding the experimental procedure, which detects the mass transfer only during the ascending of the drops, the continuous phase overall mass-transfer coefficients have been calculated through the following equation: Kod ) -
(
C∗d - Cd,2 d ln ∗ 6∆t Cd - Cd,1
)
(14)
Figure 1. Representation of m values versus Re (Shcr/Sh) for the masstransfer direction from dispersed toward continuous phase.
where Cd,1 and Cd,2 represent the concentrations of the dispersed phase at altitudes of 6.5 and 33 cm from the bottom of the column, respectively. The first location, i.e., 6.5 cm, has been allocated at such an altitude to neglect the mass transfer during the drop formation. The duration of ascending between these two locations is assigned as ∆t. Developed Empirical Model for the Contamination Factor. For each and every experimental datum, i.e., 180 instances, the contamination factor (m) was determined through fitting by SSM model. A quick review of the obtained contamination factors revealed that the latter factor depends on two variables, i.e., drop diameter and the contamination concentration.8,10 Therefore, in the modeling of this research it was attempted to properly consider the effects of these two parameters.16 In this regard, in order to make the drop diameter a dimensionless variable, Reynolds number, which incorporates the drop diameter as well as the hydrodynamic characteristics and the physicochemical properties of the system, was employed. Meanwhile, the ratio of Shc/Shc,cr was used as a criterion for the contamination concentration. As mentioned earlier, with increase of the concentration of contamination, the behavior of droplets with internal circulation becomes similar to that of rigid drops, i.e., the creeping flow. The criterion for such a variation is the comparison of the Sherwood number for the continuous phase (Shc) with that of the creeping flow (eq 4). In order to obtain a definite equation for determination of m, i.e., the developed model in the current research, first the experimental values of m were drawn versus Re (Shc,cr/Shc) in two distinct graphs for the two directions of mass transfer. The results of such drawings are presented in Figures 1 and 2. A notable point that is observed in both of the graphs is the distinction of each set of the experimental data into two particular regions. In other words, the experimental data for each particular direction of the mass transfer were located in two distinct regions of the graph. The distinction character was also the diameter of the droplets; i.e., the data related to droplets with diameters larger than 3.5 mm are located on one side of the graph and those with diameters smaller than 3.5 mm on another side of the graph. Therefore, it was decided to redraw the data for those larger than 3.5 mm and for those smaller than 3.5 mm in two separate graphs. In such an arrangement, the data were scattered in the coordinate plane. It was also noted that when the m values related to drop diameters smaller than 3.5 mm were drawn versus the parameter Re(Shc,cr/Shc)1.8, all of the data were located in a straight line. The same phenomenon was also observed when the m values related to drops of diameters larger than 3.5 mm were drawn versus the Re(Shc,cr/ Shc)1.6 parameter. The
Ind. Eng. Chem. Res., Vol. 47, No. 19, 2008 7245
Figure 2. Representation of m values versus Re(Shcr/Sh) for the masstransfer direction from continuous toward dispersed phase.
Figure 3. Presentation of m for mass-transfer direction of (d f c) and d < 3.5 mm.
Figure 5. Presentation of m for mass-transfer direction of (d f c) and d > 3.5 mm.
Figure 6. Presentation of m for mass-transfer direction of (c f d) and d > 3.5 mm. Table 3. Provided Equations of the Model of This Research Mass-Transfer Direction from Dispersed toward Continuous Phase m ) -62.067 + 4.887Re0.768(Shc/Shc,cr)-1.382 (15) m ) -86.316 + 2.921Re0.804(Shc/ Shc,cr)-1.286 (16)
D < 3.5 mm d > 3.5 mm
Mass-Transfer Direction from Continuous toward Dispersed Phase m ) -21.171 + 0.414Re1.191(Shc/ Shc,cr)-2.143 (17) m ) -20.2533 + 0.0255Re1.595(Shc/ Shc,cr)-2.552 (18)
Figure 4. Presentation of m for mass-transfer direction of (c f d) and d < 3.5 mm.
presentation of the data in this new arrangement is shown in Figures 3-6. With this new arrangement of locating the data in a particular trend, the fitting of the trend with mathematical equations became possible. This fitting was conducted by the aid of Table Curve software. The resulting equations, i.e., eqs 15-18, which are given in Table 3, are the provided model of this research that determine the values of m versus the dimensionless numbers of Re and Sh. The obtained equations, while fitting the experimental values of m, describe two important concepts:
D3.5mm
1. The model prediction is independent of the type of the contaminant. This is obvious since all the values of m for each particular direction of mass transfer are well located in a unique trend, no matter what type of surfactant is used. It is also notable that the parameter Shc/Shc,cr not only incorporates the influence of the contaminant concentration but also is a criterion of the rigidity of the drops due to the presence of each particular contaminant. 2. The data for each particular direction of mass transfer are divided into two distinct groups based on the diameter of the drops. Although due to the break in the experimental data there is an uncertainty in determining the exact value of the distinctive diameter, the diameter of 3.5 mm exhibits a characteristic size in this regard. One may argue the applicability of this model to industrial practice due to its dependence on the drop size parameter. However, as was demonstrated through experimental observations,6 by setting a particular nozzle size, one can produce drops with sizes in a narrow range of desired diameter. As such, the
7246 Ind. Eng. Chem. Res., Vol. 47, No. 19, 2008 Table 4. Calculation Procedure of the Model Provided in Appendix A input data: D, Fc, µc, Fd, µd, Scc, Re, κ ) µd/ µc Shc,cr ) 2.43 + 0.775Re1/2Scc1/3 + 0.0103ReScc1/3 (4) Shc ) 2/π[1 + 2 + 3κ + m/1 + Fdµd/Fcµc]-0.5Re1/2Scc1/2 (9) Shc/Shc,cr ) 2/π[1 + 2 + 3κ + m/1 + Fdµd/Fcµc]-0.5Re1/2Scc1/2/2.43 + 0.775Re 1/2Scc1/3+0.0103Re Scc1/3 (19) m ) A + BRea(2/π[1 + 2 + 3κ + m/1 + Fdµd/Fcµc]-0.5Re1/2Scc1/2/ 2.43 + 0.775Re1/2Scc1/3 + 0.0103ReScc1/3)b A, B, a, b ) Const in eqs 15-18 (20)
average diameter of the drop sizes for a particular tower can be regarded as the drop diameter of this model. Meanwhile, to avoid any ambiguity, the sequence of the model calculations is provided in Table 4 and a sample calculation has been also outlined in Appendix A. As seen, the contamination factor, m, is calculated independent of any prefixed assumption but only based on the input data of the system.
Figure 8. Prediction of the model versus the experimental data for SDS at concentration of 12.5 mg/L and direction of d f c.
Evaluation of the Validity of the Presented Model A. Capability of the Present Model To Predict the Independent Experimental Data. To investigate the capability of the model, its prediction against the experimental data provided by Bosse7 was examined. It is notable that the latter data were prepared from a toluene-water-acetone system for low concentrations of acetone (1 wt %) where the contaminant reagent was SDS and the direction of the mass transfer was from continuous toward the dispersed phase. Bosse also used the SSM model to calculate the contamination factor for each single datum from its relevant experimental value of the overall mass-transfer coefficient. Regarding the direction of the mass transfer and the diameter of the droplets, which were larger than 3.5 mm, eq 18 of the present model should be used to determine the m values. Figure 7 represents the prediction of the present model against the experimental data of Bosse. B. Prediction of the Experimental Mass-Transfer Coefficients Provided in the Literature. The prediction of mass-transfer coefficients through the SSM model and the model of the present research has been conducted using a software program of Matlab. This program calculates the overall masstransfer coefficients by using the physicochemical properties of the system along with the variables including drop diameter, time of ascending, and the concentration of the contaminant.
Figure 9. Prediction of the model versus the experimental data for SDS at concentration of 12.5 mg/L and direction of c f d.
The precision of the predictions through this program for the mass-transfer coefficients is appropriate and has an average uncertainty of (4%. Figures 8 and 9 represent the satisfactory predictions of this model for the experimental values obtained from a typical concentration of 12.5 mg/L SDS at both directions of mass transfer. The same promising results have been also obtained for the other surfactants at various concentrations which for the sake of briefness are not provided in this article. C. Simulation Using the Simultaneous Application of the SSM Model, the Model of this Research, and the Terminal Velocity Model of Grace. Definition of an empirical equation is the most appropriate way to determine the terminal velocity of the contaminated fluids. Grace et al.17 provided such an equation: ut ) (µc ⁄ dFc)[(J - 0.857) ⁄ M0.149]
(21)
Where µc and Fc stand for the viscosity and density of the continuous phase, respectively. The parameter J is also defined as J ) 0.94 H0.757
(2 < H e 59.3)
(22)
J ) 3.42 H
(H > 59.3)
(23)
(µc ⁄ µW)-0.14
(24)
0.757
where H is defined as H)
..
( 34 )EoM
-0.149
Where µW stands for the viscosity of water at 4 °C. For the pure systems containing droplets at low Reynolds numbers, eq 21 becomes Figure 7. Comparison between the predictions of this model with the experimental data of Bosse.
ut,pure ) ut[1 + 1 ⁄ (2 + 3µd ⁄ µc)]
(25)
Ind. Eng. Chem. Res., Vol. 47, No. 19, 2008 7247
Figure 10. Simulation of the data to predict Kod in the presence of Triton at a concentration of 6.25 mg/L and the mass-transfer direction of c f d.
Figure 12. Simulation of the data to predict Kod in the presence of DTMAC at a concentration of 6.25 mg/L and the mass-transfer direction of c f d.
calculating this parameter, which is used for the calculation of overall mass-transfer coefficient. This unique factor incorporates the effects of the contamination on the mass-transfer coefficients. Four empirical correlations were developed for the explicit calculation of the contamination factor, corresponding to the typical high interfacial tension system, studied in this work. The equations of the model are independent of the type of the contaminant and are unique for various types of the surfaceactive reagents. The influence of the contaminants in the blockade of the interface and leading the droplets toward rigidity has been considered in the term Shc/Shc,cr, which is a criterion of the contaminant concentration.
Figure 11. Simulation of the data to predict Kod in the presence of SDS at a concentration of 6.25 mg/L and the mass-transfer direction of c f d.
Regarding the conformity of the above model with the data employed in this research, the attention of the present authors was attracted to simulate the data based on the models of SSM, Grace, and that of the current research, in order to evaluate the validity of the provided model. This simulation was also conducted using the Matlab software. Typical results of this simulation, which are presented in Figures 10-12, are quite promising. Conclusions Addition of very small amounts of each type of the surfactants to the system lowers the mass-transfer coefficient of acetone in the liquid-liquid extraction system. This effect can be attributed to the influence of the surface-active reagents in blockade of the interface, changing the behavior of drops toward rigidity, and lowering the interfacial tension. The experimental results also revealed that the effect of the surfactants in lowering the mass-transfer coefficient is in the order of, Triton X-100 > SDS > DTMAC. Due to the unique variations observed in the contamination factor, m, a consistent model was developed with the aim of
The equations of the model also distinguish the contamination factors versus the diameter of the droplets. In reality, by increasing the drop diameter and passing the region of rigid drops, the internal circulation within the droplets develops. According to the model predictions, it seems that in the region of smaller droplets (d < 3.5 mm) the effect of contaminationdriven rigidity dominates while in the region of larger droplets (d > 3.5 mm) the effect of internal circulation dominates. Although due to the break in the experimental data the exact border of this alteration was not determined, however, the vicinity of 3.5 mm for the drop diameter appeared to be an appropriate distinction diameter. The contribution of the presented model is for its prediction capability in the absence of experimental data. It is notable that the previous models provided based on a single solute along with the combinatorial model of Slater cannot be used for prediction purposes while they are well capable to simulate the experimental data. In this regard, the main attempt of the current research was focused on providing the contamination factor as a tool to predict the mass-transfer coefficients. The results of the model predictions for several available experimental data were promising. The simulation of the model along with the terminal velocity model of Grace proves the validity of the model. Further work is required to provide similar correlations for other chemical systems.
7248 Ind. Eng. Chem. Res., Vol. 47, No. 19, 2008
Appendix A: Sequence of the Model Calculations
Nomenclature
Table A1. Sample Calculation of Contamination Factor
Cd ) concentration in the dispersed phase (kmol/m3) Cd* ) equilibrium concentration (kmol/m3) d ) droplet diameter (m) Dd ) molecular diffusion coefficient in the droplet (m2/s) De ) effective diffusion coefficient in the droplet (m2/s) Doe ) overall diffusion coefficient in the droplet (m2/s) kc ) continuous-phase mass-transfer coefficient (m/s) kd ) dispersed-phase mass-transfer coefficient (m/s) Kod ) dispersed-phase overall mass-transfer coefficient (m/s) m ) contamination factor Pec ) continuous-phase Peclet number Re ) droplet Reynolds number Scc ) continuous-phase Schmidt number Shc ) continuous-phase Sherwood number Shc,cr ) continuous-phase critical Sherwood number t ) time (s) ut ) terminal velocity of the droplet (m/s) Greek Letters R ) ratio of average surface velocity to terminal velocity κ ) ratio of phase viscosities (µd/µc) µc ) viscosity of continuous phase (kg/m · s) µd ) viscosity of dispersed phase (kg/m · s) Fc ) density of continuous phase (kg/m3) Fd ) density of dispersed phase (kg/m3) φ )solute equilibrium distribution coefficient
input data D ) 1.09 × 10-9 m2 s-1 Fc ) 998.2 k g m-3
µc ) 10.0118 × 10-3 k g m-1 s-1
-3
µd ) 0.628 × 10-3 k g m-1 s-1 Scc ) 9200 Re ) 100 (for example) µd κ ) ) 0.0627 µc
Fd ) 866.7 k g m
1
Fdµd ) 0.233(26) Fcµc
1
1
Shc,cr ) 2.43 + 0.775(100) 2 (9200) 3 + 0.0103(100)(9200) 3 ) 186.4 (27) Shc )
)
2 + 3(0.0627) + m 2 1+ 1 + 0.233 √π
[
1
-0.5
]
1082.3 √2.775 + 0.81 1m
(28)
Shc 5.81 ) Shc,cr √2.775 + 0.811m
(
(29)
)
b 5.81 √2.775 + 0.811m A ) -62.067, B ) 4.887, a ) 0.768, b ) -1.382
m ) A + B(100)a
1
(100) 2 (9200) 2
(30)
Substitute into eq (15) m ) -62.067 + 4.887(100)0.768
(
5.81
)
-1.382
√2.775 + 0.811m numerical solution w m ) 8.44(31)
Acknowledgment Department of Applied Chemistry at the University of BuAli Sina is acknowledged for providing the experimental data. This research was financially supported by Iran University of Science and Technology through one of the grants for the university joint projects. Literature Cited
Figure A-1. Flowchart of the Model Calculation Sequence.
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Thesis, Department of Chemical Engineering, Iran University of Science and Engineering, Tehran, Iran, 2005. (17) Thornton, J. D. Science and Practice of Liquid-liquid Extraction, 3rd ed.; Oxford Science Publishers: New York, 1992; Vol. 1.
ReceiVed for reView November 15, 2007 ReVised manuscript receiVed June 26, 2008 Accepted July 15, 2008 IE0715557
