Mathematical Modeling of Carrier-Facilitated Transport in Emulsion

Chapter 8 Mathematical Modeling of Carrier-Facilitated Transport in Emulsion Liquid Membranes

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Ching-Rong Huang, Ken-Chao Wang, and Ding-Wei Zhou Department of Chemical Engineering, Chemistry, and Environmental Science, New Jersey Institute of Technology, University Heights, Newark, NJ 07102

Mass transfer in emulsion liquid membrane (ELM) systems has been modeled by six differential and algebraic equations. Our model takes into account the following: mass transfer of the solute across the film between the external phase and the membrane phase; chemical equilibrium of the extraction reaction at the external phase-membrane interface; simultaneous diffusion of the solute-carrier complex inside globules of the membrane phase and stripping of the complex at the membrane-internal phase interface; and chemical equilibrium of the stripping reaction at the membrane-internal phase interface. Unlike previous ELM models from which solutions were obtained quasi-analytically or numerically, the solution of our model was solved analytically. Arsenic removalfromwater was chosen as our experimental study. Experimental data for the arsenic concentration in the external phase versus time were obtained. From our analytical solution with parameters estimated independently, we were able to obtain an excellent prediction of the experimental data.

Since Norman N. Li (I) introduced the emulsion liquid membrane in 1968, many publications have appeared both on experimental work and on theoretical modeling of such separations. Most of the theoretical work was concentrated on mass transfer in the ELM process in a mixing vessel where the water-in-oil emulsion is mixed with an external aqueous phase and globules of emulsion are formed and suspended in the external phase. The solute, after a series of mass transfer steps, is transferred from the external aqueous phase to the internal aqueous receiving phase through the oil membrane phase. Many mathematical 0097-6156/96/0642-O115$15.00/0 © 1996 American Chemical Society

In Chemical Separations with Liquid Membranes; Bartsch, R., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1996.


116

CHEMICAL SEPARATIONS WITH LIQUID MEMBRANES

models were developed to describe and represent this mass transfer operation. The first model, the advancing front model, was developed by Ho et al. (2). In this model, the solute was assumed to react instaneously and irreversibly with the reagent in the internal phase. The solute concentrations in the membrane phase and in the internal phase were treated as a single term. Fales and Stroeve extended the advancing front model to include external mass transfer resistance (3). The model was further modified independently by Teramoto et al. (4), and by Bunge and Noble (5) by considering the reversibility of chemical reactions. However, their resulting equations were too complicated to develop even quasi-analytical solutions. Lorbach and Marr simplified the modeling equations by assuming constant summation offreeand complexed carrier concentrations (6). In a recently published book by Ho and Sirkar (7), an extensive review is given of the experimental and theoretical work on ELM. We starting working on modeling of the ELM process in 1984 (8). Our model involves the following steps: mass transfer of the solute from the bulk external phase to the external phase-membrane interface; an equilibrium reaction between the solute and the carrier to form the solute-carrier complex at the interface; mass transfer by diffusion of the solute-carrier complex in the membrane phase to the membrane-internal phase interface; another equilibrium reaction of the solute-carrier complex to release the solute at the membraneinternal phase interface into the internal phase. Assumptions used in our model include: a uniform radius of the globules; negligible leakage of the internal phase into the external phase; and negligible mass transfer resistance from the membrane-internal interface into the internal phase. From the model, simultaneous partial differential equations and algebraic equations were formulated. Analytical solutions were obtained by Laplace transform. From these solutions, we were able to predict theoretically: 1) the concentration of the solute in the external phase as a function of time; 2) the concentration profile in a globule at different times; and 3) the change in surface concentration of the solute at the external phase-membrane interface with time. The features of our model are the following: 1) The most significant feature of our model is that an analytical solution was obtained. Previous models gave solutions in quasi-analytical or numerical forms. 2) Most models predicts only the external phase concentration of the solute versus time. Our model allows calculation of the distribution of solute in the external, membrane and internal phases at any time. 3) Two dimensionless groups with physical significance were developed from our model to characterize the ELM systems. One dimensionless group, G, provides the ratio of mass transfer resistance between the film of external phase and the membrane phase. This group is the reciprocal of the Biot number. The other dimensionless group, B, indicates the capability of reduction of the solute concentration in the external phase in an ELM system. A larger value of this dimensionless group gives greater percent reduction of the solute concentration in the external phase.


8.

Mathematical Modeling of Carrier-Facilitated Transport 117

HUANG ETAL.

Mathematical Equations Governing the E L M System. External Phase dCe ,dCm V e ^ = -N(^R )De(^) dt dr 2 2

2

x

(1)

r = R

2


- N ( ^ R ) D e ( ^ ^ ) | r - R= -N(^R )k(Ce-Ce*) dr

(2)

Initial Condition (I.C.): Ce = Ceo, when t = 0 where: Ve = volume of the external phase (liter) Ce = concentration of solute in the external phase (mol/liter) Ν = total number of globules R = radius of the globules (m) De = effective diffusivity of the solute-carrier complex in the globule (m /sec) Cm = concentration of the complex in the membrane phase (mol/liter) k = external mass transfer coefficient (m/sec) k(Ce - Ce*) is the flux for mass transfer across the external resistance film 2

dCm

De(

)| - R is the flux for mass diffusion inside the globules r

dr

At the external phase-membrane interface, chemical equilibrium is reached. Thus, Cm* = pCe* and Ce* φ Ce

when r = R

(3)

where: Cm* = concentration of the solute-carrier complex in the membrane phase at the interface Ce* = concentration of the solute in the external phase at the interface ρ = the extraction partition function at the external phase-membrane interface Membrane Phase A spherical shell material balance is taken inside the globule membrane: Vm*

2

= ( Vi + Vm)* De* [——(r dt r dr

)] - Vi* Rx

(4)

dr

Initial conditions (I.C.): Cm = 0 for all r, when t = 0 Boundary Conditions (B.C.): Cm = finite, when r = 0; Cm = Cm*, when r = R where: Vm = volume of the membrane phase (liter), Vi = volume of the internal phase (liter) Vm+Vi = total volume of the emulsion globules (liter) Rx = rate of release of the solute into the internal phasefromthe membrane-internal phase interface per unit volume of the internal phase (mol/liter/sec.)


118


Internal Phase Rx = M l ^ )

(5)

dt


Initial conditions (I.C.): Ci = 0, when t = 0 where: Ci = concentration of the solute in the internal stripping phase At the membrane-internal phase interface, chemical equilibrium is reached. Thus, at any r Ci = qCm

(6)

where: q = the stripping partition function at the internal phase-membrane interface The partition functions, ρ and q, represent equilibrium chemical reactions for formation of the solute-carrier complex at the external phase-membrane interface and decomposition of the solute-carrier complex at the membraneinternal phase interface, respectively. We have four partial differential equations and two equilibrium equations. Laplace transform is applied to solve four variables: Ce(t), C e(t), Cm(r,t) and Ci(r,t). These equations are combined and changed into dimensionless form as follows: — = -Ko(Ue-Ue*) άτ

(7)

G(^)|

(8)

v = 1

=(Ue-Ue*)

Equilibrium: U*e = U*m Ue * U*e I.C. Ue=l,whenx = 0 ω

dUm

1 dUm =——(ν )

. (9)

x

2

/rk

I.C. B.C.

Um = 0, when τ = 0 Um = finite when ν = 0; Um = Um* when ν = 1 , Ce Det „ 3f -, Vi + Vm where: Ue = τ =—f= f = Ceo R 1-f Vi + Vm+Ve ι /ι χ r __ Cm Vi ω = 1 Vi + Vm (1 + q) v =R— Um =pCeo e =Vm - + Vi TT

2

V i

Ko = B£ De

G

=

P ^ Rk

U e

* = ÇU

Ceo

Um* = - ^ pCeo


8.

HUANG ET AL.


Results and Discussion. Laplace transform is taken for all these dimensionless equations, together with their initial and boundary conditions. The analytical solutions can be found by taking the inverse Laplace transform: Ce _ 3 ^ 2B Ceo~B + 3 t Î 3 B + B +bn +Gbn (Gbn -2B-l) Downloaded by STANFORD UNIV GREEN LIBR on October 11, 2012 | http://pubs.acs.org Publication Date: May 5, 1996 | doi: 10.1021/bk-1996-0642.ch008

+

Ce

2

2

2

2

2

bn ω

eXP

2

3

T )

(10)

2

A 2(B-Gbn ) , bn . = +> ζ ^ —— exp( τ) Ceo B + 3 t i 3 B + B +bn +Gbn (Gbn -2B-l) ω Ί

2

2

2

2

yK

(11)

2

. 2(B-Gbn )(^^) Cm _ 3 γ» v* sin(bn) pCeo ~ Β + 3 £ ί 3B + B + bn + Gbn (Gbn - 2B -1) α

+

2

2

2

bn

(

2

ω

(12)

2

, bn*(B-Gbn ) tan(bn) = -—B + bn (l-G) where bn = eigen values defined by Equation (13), n= 1,2,3— Dimensionless groups are involved in the solution equations. They are _ ^ pDe 1 τ tDe Β = pfco, G = ^—=—— tau = - = —~ Rk Biot ω coR #

2

(13)

2

Experimental Measurements. Removal of arsenic from metallurgical wastewater by emulsion liquid membrane was studied experimentally (9). One set of experimental data is used here to verify the mathematical model. The experiment on emulsion liquid membrane removal of arsenic was conducted as follows. The arsenic solution (feed) was prepared by dissolving arsenic trioxide (AS2O3) in sodium hydroxide solution. The pH was adjusted by adding sulfuric acid solution and the solution was then diluted to approximately 100 ppm. The emulsion consisted of the membrane phase and the internal phase. The membrane phase was formulated with a diluent, such as heptane, a carrier, such as 2ethylhexyl alcohol (2EHA), and a surfactant, such as ECA4360J (Exxon Chemical Company). The emulsion was prepared by adding the internal aqueous phase (1 Ν NaOH solution) to the formulated membrane phase and then emulsifying with Warring Blender for 30 minutes at 10,000 rpm, and cooling down to room temperature. The emulsion was freshly prepared before each permeation experiment. The prepared emulsion was dispersed in an agitated vessel with the feed arsenic solution in a volume ratio of 1/5. The agitation speed was controlled at 300 rpm as monitored by a digital stroboscope (Cole-Parmer). The pH of the external aqueous phase was measured by a pH meter (PHCN-31, OMEGA), and


120



samples were removed periodically for further separation, dilution and analysis. Inductively Coupled Plasma-Mass Spectrometry (ICP-MS, VG PLasma Quad, VG Elemental Limited) was used for quantitative analysis. A set of experimental conditions is listed in Table I under which optimal arsenic removal efficiency was obtained. Table I: One Set of Experimental Conditions of Arsenic Removal by ELM Experimental Conditions Room temperature, agitation speed: 300 rpm External phase: 500 ml, Ceo = 5.51 ppm (initially), [H SO ]=0.2M Membrane phase: 90 ml, 10 vol% 2EHA, 2 vol% ECA4360J, remainder: heptane Internal phase: 10 ml of NaOH (2.0 N) 2

4

Equilibrium data were obtained by independent extraction and stripping experiments conducted in separatory funnels. Extraction was conducted with O/A = 2:1 with an organic phase formulated as 90 vol% of heptane and 10 vol% of 2EHA and an aqueous phase which contained 100 ppm of arsenic in 0.2 M sulfuric acid solution. Stripping was conducted with A/O = 1:1 with a stripping phase of 2.0 Ν NaOH solution. Both extraction and stripping were conducted in closed vessels with magnetic stirring for 24 hours. Evaluation of Parameters. An important feature of our model is that all of the parameters can be evaluated independently. Thus, De, the effective diffusivity of the solute-carrier complex in the membrane phase, is evaluated by the JeffersonWitzell-Sibbert equation (10), Di and Dm are determined by the Wilke-Chang equation (11), the external mass transfer coefficient k is determined by the Skelland and Lee equation (12), and the Sauter mean radius R is determined by the same method as Ohtake (13). The two partition function values ρ and q are obtained from the extraction and stripping experiments conducted in the separatory funnels. For the ELM removal of arsenic, Table II gives the predicted parameters by these methods under the experimental conditions listed on Table I. Table II: List of Model Parameters for Arsenic System Value Model Parameters 1.16*10' Di=Diffusivity of solute in the external phase (m /sec) Dm=Diff. of complex in membrane phase (m /sec) 1.18*10' De=Eff. Diff. of complex in membrane phase (m /sec) 8.73*10 ° f=emulsion volumefraction(Vi+Vm/Vtot) 1.67*10 e=sink phase volumefraction(Vi/(Vi+Vm)) 0.1 R=radius of the globule (m) 5.8*10^ 0.22 p=external distribution coefficient q=internal distribution coefficient 7.0*10 k=external mass transfer coefficient (m/sec) 7.54* 10 9

2

2

9

2

_1

_1

3

-6


8.

HUANG ET AL.



Prediction of the Experimental Data for Arsenic Removal by ELM. Experimental data for the time dependence of the arsenic concentration in the external phase is shown in Figure 1. The theoretical curve predicted from the model is plotted on the same figure. The plot demonstrates that the theoretically predicted curve gives an excellent representation of the experimental data. Engineering Analysis of the ELM System. This experimentally verified mathematical model can be used for engineering analysis of the ELM system. The effect of the two dimensionless groups (B and G) on the permeation rate and removal efficiency are discussed below. Such discussion strengthens our understanding of the ELM operations and facilitates determination of the optimal design of experiments for removal of other species by ELM processes. Steady State Solution. The solutions for Equation 10, 11 and 12 have a non-series term plus a summation of a series. The non-series term of 3/(3+B) is the steady state solution of the partial differential equations when time approaches infinite. As mentioned before that dimensionless group Β could be considered as a indicator of the emulsion capacity. Thus, the bigger the Β number, the smaller is Ce/Ceo, which implies an enhanced removal efficiency. This steady state solution can also be obtained by material balance of the emulsion liquid membrane system as follows: Ceo*Ve=Ce*Ve+Cm*Vm+Ci*Vi with two equilibrium equations: Cm=pCe The steady state solution becomes:

Ceo

Ve + p*(Vm + q*Vi)

(14) and Ci=qCm

B+ 3

^

;

Therefore, a larger Β value gives better performance of the ELM system. It is apparent that a good choice of the extractant and extraction conditions will give a large extraction partition function, p, and a good choice of stripping reagent and stripping conditions will give a large stripping partition function, q. Together with a large emulsion volume ratio f and a large internal volume ratio e, the Β value can be increased. However, the two volume ratios cannot be selected arbitrarily because of the stability of the emulsion. Effect of Β Number on ELM Performance. The effect of B, the emulsion capability dimensionless group, on the external phase concentration versus time is shown in Figure 2 for a fixed Biot number of 22.8. Results for the three curves obtained with different Β number shows that the curve with the largest value of Β gave the best ELM performance in solute removal for this group.



122


. ^)

C

V

0

I

r

I

1

'

5

10

15

20

25

1

30

Time(min) Figure 1. Prediction of the Arsenic ELM Experimental Data by the Model (B=92.5 and Biot=22.8) Experimental conditions: External phase was 500 ml of 5.51 ppm arsenic solution, 0.2 M H2SO4; Emulsion phase was 100 ml offreshemulsion, which contains 90 ml of organic phase (10 vol% 2EHA, 2 vol% ECA4360J, remainder was heptane) and 10 ml of internal phase (2 Ν NaOH solution); Room temperature and 300 rpm of agitation speed.



HUANG ET AL.

Mathematical Modeling of Carrier-Facilitated Transport

2

Dimensionless Time[Tau(De*t/w/R )] Figure 2. Effect of the Β Number on ELM Performance (Biot=22.8)


124


B=19.5, Biot=22.8 0.45 -ι —•—Tau=0

-•-1.11E-03

0.35

—A—2.22Ε-03

-K-4.44E-03

0.3

-*-6.67E-03

-•-8.89E-03

0.25

— I — 1.11E-02

1.33E-02

0.4

ο

£α. 1 ο

0.2

1.55E-02

0.15


0.1 0.05 • 0 0.1

0.2

0.3

0.4

0.5

r/R

Β=147, Biot=22.8 0.45 0.4 0.35 ο φ υα

1

ο

0.3 0.25

Φ

Tau=0

—1.11Ε-03

—Α-2.22Ε-03

—Η-4.44Ε-03

-*-6.67Ε-03

—#-8.89Ε-03

—I—1.11Ε-02

1.33Ε-02

0.2

1.55Ε-02

0.15 0.1 0.05 0 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Dimensionless Radius (r/R)

Figure 3. Effect of the Β Number on the Solute-Carrier Complex Distribution in the Emulsion Globule



8.

HUANG ET AL.


It was interesting to observe theoretically the effect of Β on the concentration profile in a globule at different times. In Figure 3, plots of the globule concentration profiles at three Β values of 19.5, 92.5 and 147 are presented. The results show that at a low value of B=19.5 with a low stripping capability of the internal phase keeps the solute (in the form of the solute-carrier complex) concentration high in the membrane phase. Therefore, the solute-carrier complex in the membrane penetrates deeper toward the center of the globule. On the other hand, at the high value of B=147, most of the solute is stripped into the internal phase and which leaves a small amount in the membrane phase. Since the membrane concentration of the solute-carrier complex is a function of time, the concentration profiles for a globule are plotted at different times from tau=0 to tau=1.55*10" . 2

Effect of the Biot Number on ELM Performance. The Biot number, the reciprocal of the G number, represents the ratio of the internal diffusion transfer resistance to the external mass transfer resistance. The case of an infinite Biot number (G=0) represents no external film resistance. Figure 4 presents the effect of Biot number on ELM performance as calculated from the model. At the fixed value of B=105, the dimensionless external phase concentration is plotted versus time at three different values of the Biot number, namely 7.01, 22.18 and 49.60. It is observed that the smaller the Biot number, the slower is the permeation rate. It is also noted that the curve for an infinite Biot number is exactly the same as the curve for Biot=49.6. This means that a critical Biot number exists so that the external mass transfer resistance may be neglected, if the operative Biot number is greater this critical Biot number. In Figure 5, the external phase concentration of the solute and the external phase-membrane interface concentration of the solute are plotted simultaneously versus time at a constant B=105 and different values of the Biot number. It is seen that a larger film resistance (small Biot number) produces greater difference in the two solute concentrations across the film. Conclusion. The mass transfer of an emulsion liquid membrane process has been modeled mathematically. An analytical solution which allows prediction of concentrations of solutes in the external phase, membrane phase, and external phase-membrane interface was obtained. Experimentally, arsenic was selected as a solute in the external phase to be removed by the ELM process. Our model gives an excellent representation of the experimental data for the external concentration of the arsenic versus time. In addition, the model predicts the concentration distribution in the membrane phase at any time. Thus, the overall distribution of solutes in three phases (external, membrane and internal) at any time of the ELM process can be evaluated. From the model, it was found that the ELM process was characterize by two dimensionless groups. One group for the transport phenomena governs the rate of mass transfer or the Biot number. The other group includes the



126


0.9*

οι 0

, 0.002

0.004

0.006

1 0.008

0.01

0.012

0.014

2

Dimensionless Time [Tau(De*t/w*R )] Figure 4. Effect of the Biot Number on ELM Performance (B=105)


0.016


HUANG ET AL.


Biot=7, B=105

0

0.0005

0.001

0.0015

0.002

0.0025

0.003

0.0035

0.004

0.0035

0.004

2

Tau (De*t/wR)

Biot=22, B=105

0

0.0005

0.001

0.0015

0.002

0.0025

0.003

Tau

Biot=50, B=105

0.004

Tau

Figure 5. Comparison of External and Interfacial Concentrations


128


equilibrium constants for extraction and stripping reactions at both interfaces and expresses the limit or the capability of the separation process.


Literature Cited 1. Li, Ν. N. US Patent, 3,410,794 (Nov. 12,1968). 2. Ho, W. S.; Hatton, Τ. Α.; Lightfoot, Ε. N.; Li, Ν. N. AIChE J. 1982, 28, 662. 3. Fales, J. L.; Strove, P. J. Membr. Sci. 1984, 21, 35. 4. Teramoto, M.; Takihana, H.; Shibutani, M.; Yuasa, T.; Hara, N. Sep. Sci. Technol. 1983, 18, 397. 5. Bunge, A. L.; Noble, R. D. J. Membr. Sci. 1984, 21, 55. 6. Lorbach, D; Marr, R. J. Chem. Eng. Process. 1987, 21, 83. 7. Ho, W. S.; Li, Ν. N. In Membrane Handbook, Ho, W. S., Sirkar, Κ. K., Ed.; Van Nostrand Reinhold: New York, NY, 1992; pp. 597-611. 8. Wang, G.C. Ph.D. Dissertation, New Jersey Institute of Technology, 1984. 9. Huang, C.R.; D.W. Zhou; W.S. Ho; N.N. Li. presented at 1995 AIChE Spring National Meeting, March 19-23, Houston, Texas. 10. Jefferson, T. B.; Witzell, O. W.; Sibbett, W. L. Ing. Eng. Chem. 1958, 50, 1589. 11. Wilke, C. R.; Chang, P. AIChE J. 1955, 1, 264. 12. Skelland, A. H. P.; Lee, J. M. AIChE. J. 1981, 27, 99. 13. Ohtake, T.; Hano, T.; Takagi, K.; Nakashio, F. J. Chem. Eng. Jpn. 1987, 20, 443.


Mathematical Modeling of Carrier-Facilitated Transport in Emulsion

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