Combined Model of Mass-Transfer Coefficients for Clean and

ARTICLE pubs.acs.org/IECR

Combined Model of Mass-Transfer Coefficients for Clean and Contaminated Liquid-Liquid Systems A. Haghdoost,†,^ Asghar M. Dehkordi,*,† M. Darbandi,‡ M. Shahalami,† and J. Saien§ †

Department of Chemical and Petroleum Engineering, Sharif University of Technology, Tehran, Iran Department of Aerospace Engineering, Sharif University of Technology, Tehran, Iran § Department of Applied Chemistry, Bu-Ali Sina University, Hamedan, Iran ‡

ABSTRACT: Mass-transfer rates to and from drops in liquid-liquid extraction processes are often reduced by the presence of contaminants. To design an industrial extractor, it is essential to consider this contamination effect in a quantitative manner. To achieve this goal, an experimental investigation was conducted on the mass transfer into single drops for n-butanol-succinic acidwater, as the recommended test system by the European Federation of Chemical Engineering (EFCE). The effects of anionic (sodium dodecyl sulfate, SDS), cationic (dodecyl trimethyl ammonium chloride, DTMAC), and nonionic (octylphenol decaethylene glycol ether, Triton X-100) surfactants on the hydrodynamic and mass-transfer parameters such as mean drop size, average velocity, and the overall mass-transfer coefficient were thoroughly investigated. On the basis of experimental results, a model for surfactant effect on the mass-transfer coefficient in a system with unknown type and amount of contaminant was presented. This model is especially applicable in industrial processes in which contaminants have unknown sources and types.

1. INTRODUCTION Generally, in industrial liquid-liquid extraction processes, the feed is not a clean liquid and there are various kinds of contaminants in the feed entering into the extractors. Moreover, the contaminant levels and their types are not fixed and may vary during the operation. The presence of contaminants in masstransfer systems is known to reduce the mass-transfer coefficient markedly by exerting an excess mass-transfer resistance and has been studied by many investigators.1-18 The role of electrostatic potential and the influence of electrolytes on the reduction of the mass-transfer rate in liquid-liquid systems have been investigated by a number of investigators.19-22 The excess mass-transfer resistance exerted by the surfaceactive agents has been attributed to the hydrodynamic effect and/ or to the formation of an interfacial barrier layer. The hydrodynamic effect arises from lowering internal circulation in the presence of surfactants. Two different reasons were proposed in the literature for this retardation. On the basis of the first explanation, surfactants make a quasi-rigid layer at the interface, which retards transmission of shear to the drop and decreases internal circulation.2 The second explanation proposed by most authors is based on the fact that surfactants usually accumulate at the rear of the drop. The nonuniform distribution of the surfactants at the interface exerts a Marangoni stress on the interface of a moving drop due to the gradient of interfacial tension.29 This stress counteracts the viscous force caused by drop movement through the continuous phase. Viscous force is the main reason for the internal circulation and interfacial mobility of drops, which both decrease in the presence of surfactants.3,10,23-25 The Savic stagnant cap model38 considers a stagnant region over the accumulating region of surfactants at the rear of the drop based on the mentioned hydrodynamic effect. Garner and Skelland3 showed that at some surfactant concentrations the experimental values of the extraction fraction fall r 2011 American Chemical Society

below the theoretical values calculated for rigid spheres. This indicates, in addition to hindering drop circulation, another obstruction playing a role in the mass-transfer process in contaminated systems. It can be explained by the formation of an interfacial barrier layer by the contaminants, which results in mechanical obstruction and transfer resistance. The effects of this barrier layer on the mass-transfer process was investigated by several investigators.4,5,26-28 Slater9 proposed a single correction factor for predicting the effects of contaminants on the mass-transfer coefficients of continuous and dispersed phases. This contamination parameter is a ratio of the drop averaged interfacial velocity to its terminal (free stream) velocity. He reported this contamination parameter as a function of the surfactant concentration based on different experimental results. The main objectives of the present investigation were (1) to investigate experimentally the effects of various kinds of surface-active agents on the mass-transfer coefficients in a liquidliquid system and (2) to propose a correlation for the correction factor proposed by Slater, based on a large number of experiments with different surfactants. The correlation can be used regardless of the type and concentration of surfactants; therefore it is applicable in mass-transfer processes with unknown type and amount of contaminants like most industrial masstransfer processes. This article is organized as follows: the theoretical modeling is described in the next section followed by describing experimental procedures and analysis. Experimental results and correlation are explained in section 4 of the article. Received: July 5, 2009 Accepted: February 22, 2011 Revised: September 13, 2010 Published: March 11, 2011 4608

dx.doi.org/10.1021/ie901076f | Ind. Eng. Chem. Res. 2011, 50, 4608–4617

Industrial & Engineering Chemistry Research

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2. THEORETICAL MODEL Mathematical modeling of mass transfer is based on the hydrodynamic condition of the drop: stagnant, circulating, or oscillating. Drops in the oscillating region show deformed or completely damped circulation. They have a considerable level of internal random mixing, and their flow pattern is completely different from circulating drops.39 In clean systems, different criteria are used to determine the onset of drop oscillation based on the thermophysical properties and drop size.30 Skelland et al.31 reported the applicability of these criteria in systems containing surfactants as well. On the basis of their investigation, the Grace et al. criterion seems to be the best with a mean absolute deviation of 20.2%. This criterion is given by37 -0:14 4 -0:149 μc H > 59:3; H ¼ , E€oM 3 μw gμ 4 ΔF gΔFd2 ð1Þ M ¼ c2 3 , E€o ¼ Fc γ γ

Table 1. Specification of n-Butanol

The effect of surfactant type on the onset of drop oscillation was also shown to be small, if any, by Skelland et al.31 On the basis of the Grace et al. criterion, all drops with 2 < H < 59.3 are in the domain of circulating drops. This criterion states that all drops larger than 2.6 mm in diameter are in oscillating region. Since the maximum drop diameter in the present investigation is around 2.5 mm and the parameter H for all generated drops is in the range of 23.84 < H < 53.85, the model concerning circulating drops only is presented. The empirical relation given by Steiner32 was used to verify that none of the organic drops generated in this work behaves like rigid spheres after adding various types of surfactant. This relation shows minimum Sherwood number in the circulating region as follow32

and κ is the viscosity ratio (i.e., = μd/μc). Parameter R represents the effect of surfactant on the mass-transfer coefficient in the dispersed phase. 2.3. Overall Dispersed-Phase Mass-Transfer Coefficient. The overall dispersed-phase mass-transfer coefficient is expressed based on mass-transfer resistances which reside in both continuous and dispersed phases using Whitman theory.

Shcr ¼ 2:43 þ 0:775Re0:5 Scc 0:33 þ 0:0103ReScc 0:33

in which R is a correction factor accounting for the effect of the surfactants. This correction factor is considered as the ratio of the averaged interfacial velocity (U hi) to the terminal velocity of a drop, Ut.9 Ui Ut

ð4Þ

unit

value

purity

ASTM D-5008

wt %

water

ASTM D-1364

wt %

99.5 min 0.1 max

aldehydes

ASTM E-411

wt %

0.05 max

acidity (as acetic acid)

ASTM D-1613

wt %

0.01 max

sulphuric acid color

ASTM E-852

APHA

25 max

∑

where Doe, overall effective diffusivity, is defined by Doe ¼

RdUt 2048ð1 þ kÞ

1 1 φ ¼ þ Kod kd kc

ð6Þ

ð7Þ

in which φ denotes the solute distribution coefficient. On the other hand, the time-averaged overall mass-transfer coefficient can be determined experimentally using following equation: Kod, exp ¼

-d lnð1 - EÞ 6Δt

ð8Þ

where d, Δt, and E are, respectively, the drop diameter, drop contact time, and extraction fraction. The latter is normally defined by E¼

Cdf - Cdi Cd - Cdi

ð9Þ

where Cdi, Cdf, and Cd* denote, respectively, the initial, final, and equilibrium solute concentrations in the dispersed phase. The concentration of the continuous phase is considered to remain constant equal to the concentration of the bulk solution prepared for each experimental run. The present model (i.e., eqs 3-7) is the combined model with one adjustable parameter (i.e., R) accounted for the influence of contaminants. Model prediction of the overall mass-transfer coefficient was fitted to its experimental value (i.e., eq 8) by adjusting this contamination parameter and the adjusted values were correlated as a function of the drop Reynolds and E€otv€os numbers. R ¼ f ðRe, E€oÞ

2.2. Mass-Transfer Coefficients for the Dispersed Phase.

The mass-transfer coefficient for stagnant drops is presented by the Newman model based on mass transfer in rigid spheres with no internal motion. Calderbank and Korchinski34 showed that in the case of mass transfer into the drops, if the overall effective diffusivity is used instead of molecular diffusivity in the Newman expression, then the model can predict the experimental results

test method

for the circulating drops well. According to this model, the dispersed phase mass-transfer coefficient is given by " !# ¥ d 6 1 4n2 π2 Doe t exp ð5Þ kd ¼ - ln 2 6t π n ¼ 1 n2 d2

ð2Þ

In the present investigation, the Sherwood number values for the continuous phase and for all the experiments were greater than the critical value calculated by eq 2. 2.1. Mass-Transfer Coefficients for the Continuous Phase. Despite the negligible resistance of the continuous phase in the present investigation, the following expression for the Sherwood number for the continuous phase was applied33 rffiffiffiffiffiffiffiffiffiffiffi kc d R Pec ð3Þ Shc ¼ ¼2 Dc π

R¼

characteristic

ð10Þ

3. EXPERIMENTAL SECTION 3.1. Chemicals. The chemical system of n-butanol-succinic acid-water (BSW) as a recommended test system by the 4609

dx.doi.org/10.1021/ie901076f |Ind. Eng. Chem. Res. 2011, 50, 4608–4617


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Table 2. Physical Propertiesa A concentration of succinic acid in water (g/L)

Fd (kg/m3)

Fc (kg/m3)

μc (g/(m s))

μd (g/(m s))

γ (mN/m) 1.81

20

0.00

991

843

1.28

3.36

20

6.25

991

843

1.39

3.36

1.78

20

12.5

991

843

1.39

3.36

1.65

20

25.00

991

843

1.39

3.36

1.62

20 20

50.00 100.00

991 991

843 843

1.40 1.41

3.36 3.36

1.59 1.45

concentration of succinic acid in water (g/L)

B DTMAC concentration (mg/L)

Fd (kg/m3)

Fc (kg/m3)

μc (g/(m s))

μd (g/(m s))

γ(mN/m)

20

0.00

991

843

1.28

3.36

1.81

20 20

6.25 12.5

991 991

843 843

1.28 1.29

3.36 3.36

1.79 1.70

20

25.00

991

843

1.32

3.36

1.68

20

50.00

991

843

1.34

3.36

1.65

20

100.00

991

843

1.35

3.36

1.50

concentration of succinic acid in water (g/L)

a

SDS concentration (mg/L)

C Triton X-100 concentration (mg/L)

Fd (kg/m3)

Fc (kg/m3)

μc (g/(m s))

μd (g/(m s))

γ (mN/m)

20

0.00

991

843

1.28

3.36

1.81

20

6.25

991

843

1.28

3.36

1.67

20

12.50

991

843

1.28

3.36

1.59

20

25.00

991

843

1.30

3.36

1.54

20

50.00

991

843

1.31

3.36

1.47

20

100.00

991

843

1.31

3.36

1.30

Diffusivities: succinic acid in water 7.8 10-10 m2/s; succinic acid in n-butanol 2.8 10-10 m2/s.

European Federation of Chemical Engineering (EFCE) for research on liquid-liquid systems with low interfacial tension was chosen.35 Analytical grade succinic acid (Merck product) with purity >99.5% was used as the solute, whereas n-butanol was a technical grade product of Arak petrochemical company whose specifications are summarized in Table 1. Technical grade was chosen to be the same as used in a large-scale pilot work. Since the model discussed in section 4.3 is applicable to find the masstransfer coefficient in industrial applications and is independent of the type and amount of contaminants, it was tried to simulate the same situation as the industrial liquid-liquid processes in the experiments as much as possible. Due to this fact, the term “nominally clean” was used for the system before presenting surfactants. This approach was also used by Slater9 whose combined model was explained earlier. Distilled and deionized water with conductivity 99.5%, >99.5%, and >98%, respectively. The physical properties of the continuous and dispersed phases at different concentrations of the contaminants at 25 C (ambient temperature) are shown in Table 2 (parts a-c). These physical properties were measured using a density meter, a tensiometer (KSV, Sigma 702ET), and an Ostwald viscometer. The tensiometer was cleaned by means of Decon 90 solution and

rinsed with distilled and deionized water several times for each measurement. To evaluate the organic and aqueous molecular diffusivities, the method reported by EFCE35 was applied using average concentrations. The equilibrium distribution of succinic acid between the aqueous and organic phases at 25 C within the concentration range used in the present work (0-20 g/L) was examined and well-correlated by the following linear relation with the correlation coefficient of 0.99

Cd ¼ 1:1105Cc

ð11Þ

where Cd* and Cc are, respectively, the equilibrium concentration of the solute (succinic acid) in the dispersed (organic) and continuous (water) phases both in units of grams per liter. The presence of the surfactants within the concentration range used in the present work shows no profound effect on the equilibrium distribution of the solute. Table 2 also shows values of the interfacial tension as a function of surfactant concentration at 25 C. A higher surfactant concentration results lower interfacial tension in BSW system. In addition, the critical micelle concentrations (CMCs) of SDS, DTMAC, and Triton X-100 were reported to be 201.9, 500.5, and 180.6 mg/L.11 3.2. Experimental Apparatus. A Pyrex glass column used as the single drop liquid-liquid contactor was 10 cm in diameter and 50 cm in height. Drops were introduced into the column by means of different steel nozzles located at the bottom of the contactor. The column contactor was filled with the continuous 4610



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Table 3. Drop Diameter Ranges for BSW d (mm) surface-active agent SDS

DTMAC

Triton X-100

nozzle no.

cfd

1

1.68-1.81

2

2.10-2.16

3

2.28-2.35

4

2.40-2.47

1

1.78-1.81

2

2.09-2.16

3

2.34-2.35

4 1

2.40-2.47 1.79-1.81

2

2.14-2.16

3

2.28-2.35

4

2.40-2.47

phase (i.e., aqueous phase), whereas the dispersed phase (i.e., organic phase) was held in a standard buret connected to the organic-phase reservoir. The latter phase flowed through a tube and came out of the nozzle tip as dispersed drops. An inverted glass funnel attached to a 5 mL glass syringe was used to take in drops at the top of the contactor, 45 cm apart from the nozzle tip. The minimum contact between the aqueous and organic phases in the funnel was maintained by pulling the organic phase immediately from the funnel throat into the syringe. At least three 1-2 mL samples were prepared for each experimental run. To maintain a constant static head, the level of the organic phase in the buret was kept constant at its initial value by adding more organic phase from the reservoir to the buret for each experimental run. To omit effects of the drop formation stage and transient hydrodynamic region, the initial drop concentration (Cdi) was measured 5 cm above the nozzle tip. However, it was observed that the drop motion reached steady state after traveling ∼40 mm. To determine this initial concentration, all experiments were repeated in a short contactor with the same diameter, dispersed flow rate, and nozzles as the long one. 3.3. Experimental Procedures. The glass buret and connecting tube were first filled with the organic phase, and then, the column was filled with the aqueous phase. It should be noted that the organic and aqueous phases were mutually saturated with each other to prevent multicomponent mass transfer. For finding drop diameter, number of drops generated with respect to a specified volume scale on the buret was counted. The volume of the organic phase consumed for generating a counted number of drops, drop volume, and diameter were easily obtained by assuming spherical drops. A variety of drop sizes listed in Table 3 could be obtained by using different steel nozzles. The inside diameters of the steel nozzles were within the range 1.0-2.5 mm. The volumetric flow rate of the dispersed phase was measured before conducting the main experimental run by recording the duration of consumption of a special amount of the dispersed phase. Each experimental run was repeated at least four times, and the formation time of drops were adjusted to 1.8-2.2 s in accordance with the standard formation time of 2-3 s. The contact time of drops from the initial point to the collection point at the funnel throat was measured several times (minimum 10 times) with a stopwatch, and then their average was considered for the calculations. The average velocity was easily determined

Figure 1. Variation of drop diameter with concentration of SDS, DTMAC, and Triton X-100, c f d.

using average contact time and traveling height of the drop. Drops were typically generated at a distance of 60 mm apart from each other to prevent the interaction between two successive moving drops in the column. Skelland and Vasti36 showed that the interaction between the moving drops is negligible for this distance. In each experimental run, 200-600 drops with total volume of 1-2 mL were injected into the apparatus which is negligible in comparison with the volume of the continuous phase (around 4 L). For this reason, the concentration of succinic 4611



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Figure 2. Variation of drop average velocity with concentration of SDS, DTMAC, and Triton X-100, c f d.

acid in the continuous phase was almost constant (20 g/L) in each run. To avoid concentration change, the aqueous phase in the column was replaced by the fresh liquid after each run. In addition, for each nozzle and for each concentration of the contaminants, all parts of the experimental apparatus and nozzles were cleaned with sulfochromic acid solution several times and then rinsed with distilled deionized water three times. 3.4. Analysis. After taking the organic phase by the syringe, a precise volume of 1 mL of the organic sample was transported to a 100 mL volumetric flask using a standard 1 mL pipet. Precise 100 mL aqueous solution was prepared by adding adequate distilled water to the volumetric flask. The

Figure 3. Variation of extraction fraction with concentration of SDS, DTMAC, and Triton X-100, c f d.

concentration of succinic acid in this solution was measured in a titration experiment using 0.1 kmol/m3 sodium hydroxide. The latter’s concentration had been verified by means of a standard solution of 0.1 kmol/m3 hydrochloric acid (Titrisol sample). Phenolphthalein was used as an indicator to show the end point of titration. The titration experiment was repeated at least three times, and the mean value was considered for each sample. The above-mentioned analytical method was first checked by some samples with known concentration, which showed experimental error due to the volumetric titration in the range 1-2%. 4612



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Figure 4. Variation of overall mass-transfer coefficient (experimental) with drop diameter for different concentrations of (a) SDS, (b) DTMAC, (c) Troton X-100.

4. RESULTS AND DISCUSSION A large number of experiments using different nozzle sizes, various kinds of contaminants (SDS, DTMAC, and Triton X-100), short and long columns were conducted for the c f d mass-transfer direction. The initial concentration, final concentration at the funnel throat, drop contact time, and diameter were measured for both nominally clean and contaminated systems. The operating conditions for all experimental runs were as follows: (a) Concentration of succinic acid in the aqueous phase (distilled water) for c f d direction, Cc0 (g/L) = 20 (b) Operating (ambient) temperature (C) = 25 ( 1 (c) Concentration range of SDS, DTMAC, and Triton X-100 in the continuous phase (distilled water) (mg/L) = 0100 (d) Range of drop Reynolds number, Re = 76.02-147.24 (e) Range of drop contact time (s), Δt = 5.1-6.8

(f) Range of H, 23.84-53.85 (g) Range of E€o, 2.37-5.73 Both hydrodynamic and mass-transfer effects of contaminants were investigated in the present work. The former was shown by experimental investigation of the effect of contaminants on drop size and average velocity. On the other hand, the effect of contaminants on the mass transfer was investigated using extraction fraction (E) and time-averaged overall mass-transfer coefficient (Kod). 4.1. Influence of Contaminants on the Hydrodynamics. 4.1.1. Mean Drop Size. Figure 1 shows variation of mean drop size with the concentration of SDS, DTMAC, and Triton X-100. On the basis of this figure, the mean drop size decreases with increasing contamination level from 0-100 mg/L; therefore, contaminants increase the interfacial area for mass transfer in a liquid-liquid apparatus. On the other hand, as explained in section 4.2.2, the overall mass-transfer coefficient is reduced due to the presence of surface-active agents in the system. 4613


Industrial & Engineering Chemistry Research 4.1.2. Average Velocity and Contact Time of Drops. Figure 2 shows the effect of surfactant on the average drop velocity. On the basis of this figure, the average velocity decreases with increasing surfactant concentration. This results in increasing average contact time at higher surfactant concentrations. Although smaller velocity is expected for the drops with smaller size but the effect of surfactants on average velocity is more than that, it can be attributed just to the decrease of the drop diameter in the presence of contaminants. The main reason proposed11 for this trend is based on the fact that in a contaminated system, the stretching of the drop surface sweeps the surfactants to the rear part of a moving drop and causes the surfactant concentration to decrease at the leading part of the rising drop. This results in lower interfacial tension at the rear part of the drop due to the surfactant accumulation. The nonuniform interfacial tension on the drop surface causes Marangoni stress, which pulls the interfacial layer from the low tension region toward the leading part. This Marangoni stress counteracts the viscous force, which drags the interfacial liquid backward and, consequently, retards the interfacial mobility of the drops. This results in more drag force and less terminal velocity of the drop. 4.2. Influence of Contaminants on the Mass Transfer. 4.2.1. Extraction Fraction (E). Equation 9 was used to calculate extraction fraction using the measured values of the initial and final solute concentrations. Figure 3 shows variation of the extraction fraction (E) with the contaminant concentration for each nozzle. On the basis of this figure, the extraction fraction decreases to a minimum value rapidly and then increases slightly with further addition of surfactants. This trend was reported by other researchers as well.2,3,11 The sharp decrease is due to the retardation effect of surfactants on drop circulation and also formation of an interfacial barrier layer. The shape of a moving drop is determined by the viscous stresses that deform the drop and the interfacial tension resisting this deformation. When the interfacial tension is reduced by a contaminant, the shape of the drop becomes less stable and starts to oscillate. The drop oscillation is supposed to enhance the mass-transfer rate at high surfactant concentration and make a minimum as shown in Figure 3. It should be noted that despite this oscillation, all drops are in circulating region based on the Grace et al. criterion (eq 1). Beyond this criterion, there is a considerable level of random mixing inside drops and due to the different flow pattern; formulation is different from the circulating region. In fact, in the oscillating region found by eq 1, the nature of oscillation is completely different from the oscillation due to low interfacial tension at high surfactant concentration. In the latter case, there is not any difference in flow pattern due to the random mixing. 4.2.2. Overall Dispersed-Phase Mass-Transfer Coefficient (Kod). Knowing extraction fraction (E), drop diameter (d), and drop contact time (Δt) between initial and final measurement points, overall mass-transfer coefficient (Kod) was calculated as an average value between initial and final stages using eq 8. Figure 4 shows variation of Kod (experimental) with drop diameter for different concentrations and types of surface-active agents. As may be expected, presence of the contaminants has a profound effect on reducing overall mass-transfer coefficient. Model prediction of overall mass-transfer coefficient (i.e., eqs 37) was fitted to the experimental trends shown in Figure 4 by adjusting contamination parameter (i.e., R). Table 4 summarizes calculated values of R to match model prediction to the experimental values of Kod.

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Table 4. Calculated Values of Contamination Parameter (R) to Match Model Predictions to the Experimental Values of Kod R surfactant nozzle

concentration

no.

(mg L-1)

1

2

3

4

Triton SDS

DTMAC

X-100

0.00

0.081

0.081

0.082

6.25

0.055

0.071

0.065

12.50 25.00

0.039 0.032

0.048 0.028

0.055 0.050

50.00

0.023

0.016

0.038

100.00

0.018

0.013

0.021

0.00

0.091

0.091

0.091

6.25

0.057

0.072

0.066

12.50

0.045

0.049

0.056

25.00

0.032

0.028

0.050

50.00 100.00

0.024 0.018

0.021 0.020

0.043 0.035

0.00

0.094

0.094

0.094

6.25

0.066

0.073

0.066

12.50

0.048

0.052

0.056

25.00

0.032

0.029

0.054

50.00

0.028

0.024

0.043

100.00

0.018

0.022

0.039

0.00 6.25

0.095 0.068

0.095 0.071

0.095 0.079

12.50

0.048

0.052

0.066

25.00

0.035

0.031

0.054

50.00

0.029

0.024

0.043

100.00

0.019

0.023

0.04

Figure 5. Comparison of the effects of three contaminants at a constant concentration on the mass-transfer coefficient.

Figure 5 shows the effect of SDS, Triton X-100, and DTMAC at a constant concentration (i.e., 25 mg/L) on the mass-transfer coefficient. According to this figure, the effect of Triton X-100 on 4614



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decreasing mass-transfer coefficient is considerably less than that of SDS and DTMAC. Due to the large molecular weight of Triton X-100 (602-646) in comparison with SDS and DTMAC (288.38 and 264.4, respectively), at equal concentration, the number of molecules of Triton X-100 in the system is considerably less than two other contaminants and it is considered as the main reason of the trend shown in Figure 5. 4.3. Modeling the Experimental Data. Effect of surfactants on the mass-transfer coefficient can mainly be explained by decreasing interfacial mobility in the presence of surfactants. Due to the important role of the interfacial mobility, contamination parameter, R, is defined based on the averaged interfacial velocity. The effect of the surfactants on the interfacial velocity is not measurable directly, but since surfactants change different measurable aspects of the system like average velocity, mean drop size, interfacial tension, and viscosity, there should be a correlation for the correction factor in terms of the mentioned measurable parameters. The large amount of experimental data in this Table 5. Coefficients of the Combined Model (Equation 12) coefficients

SDS

DTMAC

Triton X-100

β1

0.8893

18.8299

8.7748

β2

-5.6574

-151.9983

-78.9119

β3

0.7525

20.6421

10.3980

r2 (r-squared)

0.96

0.94

0.95

work was used to present the mentioned correlation in terms of the drop Reynolds and E€otv€os numbers as follows: R ¼ β1 þ

β2 β3 þ pffiffiffiffiffi lnðReÞ E€o

ð12Þ

Since eq 12 was correlated based on the results of a large number of experiments with different concentrations and types of contaminants, its general form does not depend on these parameters. On the basis of the experimental data (Table 3), the maximum drop diameter which is applicable in eq 12 is around 2.5 mm. Regarding eq 1, drops larger than 2.6 mm in diameter are in oscillating region; hence, eq 12 covers almost all circulating region. In eq 12, Reynolds and E€otv€os numbers represent the degree of contamination in the system. The effect of the contamination type was considered using three correlation constants (β1, β2, β3) which can be determined by a limited number of single drop experiments on the desired system. These constants do not depend on the drop size and hydrodynamics, so the experiments can be carried out using any size of drops in the circulating region (d < 2.5 mm). Table 5 summarizes values of β1, β2, β3, and correlation coefficient, r2, for different surface active agents used in this work. Although mass-transfer coefficients depend on the amount and type of surfactants but the superiority of the present model is found in using three adjustable constants and offering a limited number of experiments to find them, the model can

Figure 6. Variation of the contamination parameter (R) as a function of Reynolds and E€ otv€os numbers. The experimental (filled symbols) and the correlated values (meshed curves) are presented: (a) SDS; (b) DTMAC; and (c) Triton X-100. 4615



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mass-transfer coefficient in a system with unknown level and type of contaminants.

’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected]. Tel.: þ98 (21) 66165412. Fax: þ98 (21) 66022853. Present Addresses ^

Advanced Materials and Technologies Laboratory, Department of Mechanical Engineering, Virginia Tech, Blacksburg, Virginia 24061-0238, United States.

’ ACKNOWLEDGMENT The authors gratefully acknowledge the financial support provided by Sharif University of Technology. Figure 7. Comparison of the modeling and experimental values of overall dispersed-phase mass-transfer coefficient.

predict mass-transfer coefficient of a system with unknown level and type of contaminant. This makes the model more applicable since different kinds of contaminants may be introduced into an industrial extractor at different levels and from different sources and it is difficult to recognize their types and amounts. Figure 6 shows variations of R with drop Reynolds and E€otv€os numbers for different contaminants. Substituting the contamination parameter obtained from eq 12 in model formulations (i.e., eqs 3-7) resulted in model prediction for overall dispersed-phase mass-transfer coefficient, kod. Moreover, eqs 8 and 9 were used to find the overall dispersed-phase masstransfer coefficient based on experimental results, Kod,exp, which is shown in Figure 4. Figure 7 compares modeling and experimental values of the overall dispersed-phase mass-transfer coefficient. This figure and r2 values presented in Table 4 show good agreement between the correlated model and experimental data.

5. CONCLUSIONS An experimental investigation on BSW as the recommended representative of systems with low interfacial tension for research on liquid-liquid systems was conducted in order to (1) determine the influence of various kinds of surface-active agents on the overall mass-transfer coefficients of liquid-liquid systems and (2) propose a correlation for the correction factor accounted for the influence of contaminants on the time-averaged overall masstransfer coefficient. On the basis of the present investigation, the most important results are as follows: All contaminants reduce the overall masstransfer coefficient, but the reduction amount depends on the type of contaminant. The contamination effect is presented by a single correction factor in the equations of the mass-transfer coefficients in the continuous and dispersed phases, and a unified approach for correlating this correction factor in terms of the Reynolds and E€ otv€os numbers can be used regardless of contamination type and amount. The proposed correlation has three adjustable constants. These constants depend on the type of the contaminants and can be evaluated by a limited number of experiments on the desired system using a single-drop extraction apparatus. This approach makes it possible to model overall

’ NOMENCLATURE C = solute concentration (g/L) Cc0 = concentration of succinic acid in the aqueous phase (distilled water) (g/L) d = drop diameter (mm) D = diffusivity (m2/s) E = extraction fraction defined by eq 9 E€o = E€otv€os number (= gd2ΔF/γ) H = dimensionless group defined by Grace et al.37 k = individual film mass-transfer coefficient (μm/s) Kod = overall dispersed-phase mass-transfer coefficient (μm/s) Pe = Peclet number (= ReSc) Re = drop Reynolds number (= FcUtd/μc) Scc = Schmidt number for continuous phase (= μc/FcDc) Shc = Sherwood number for continuous phase (= kcd/Dc) t = drop contact time (s) U i = average interfacial velocity (m/s) Ut = drop terminal velocity (m/s) U = drop average velocity (m/s) We = drop Weber number (= FcUt2d/γ) Greek Symbols

R = contamination parameter φ = solute distribution coefficient γ = interfacial tension (mN/m) μ = viscosity (kg/(m s)) F = density (kg/m3) κ = viscosity ratio (= μd/μc) Δ = difference operator Subscripts

c = continuous phase cr = critical d = dispersed phase exp = experimental i = initial value f = final value oe = overall effective value w = water Superscript

* = at equilibrium 4616


Industrial & Engineering Chemistry Research Abbreviations

BSW = n-butanol-succinic acid-water

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Combined Model of Mass-Transfer Coefficients for Clean and

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