A model for metal ion pertraction through supported liquid membranes

A model for metal ion pertraction through supportedliquid membranes

Samira Mohammadia, Tahereh Kaghazchia*, Ali Kargaria,b

aDepartment of Chemical Engineering, Amirkabir University of Technology (Tehran Polytechnic), No. 424,Hafez Ave., PO Box 15875, 4413 Tehran, Iran

Tel. +98 (21) 6649-9066; Fax: +98 (21) 6640-5847; email: [email protected] of Chemical Industries, Iranian Research Organization for Science and Technology (IROST), No. 71,

Mousavi Street, Enghelab Ave., Ferdowsi Sq., PO Box 15815-3538, Tehran, Iran

Received 17 June 2006; Accepted 27 August 2006

Abstract

Transport of metal ions through supported liquid membranes has been investigated. A general dimensionlessmodel for mass transfer coefficient is proposed which consists of diffusional, equilibrium and process parametersalso properties of the membrane. Nine dimensionless equations for mass transfer coefficient have been suggestedand found that the equation in the form of has the best agreement with experimental data. The31 2Re nn n

f f orgk a Sc G=average deviation between experimental and calculated data was found to be 19.93% for this model.

Keywords: Mass transfer model; Supported liquid membrane; Metal ion extraction; Facilitated transport

1. Introduction

The use of liquid membranes containingcarriers has been proposed as an alternative tosolvent extraction processes for the selectiveseparation, purification and concentration ofmetal ions in aqueous solutions, organic acids,bio products and gases [1–9]. In liquid membranetechnology the extraction, stripping and regene-ration operations are combined in a single stage[10–12]. Liquid membranes have also received

*Corresponding author.

considerable attention due to characteristics suchas ease of operation, lower energy consumptionand operational cost, higher selectivity, rapidextraction and higher capacity factors.

Currently, three configurations of liquid mem-branes are being investigated: supported liquidmembranes (SLMs), liquid surfactant membranes(LSMs) and bulk liquid membranes (BLMs). Ineach case, an extraction reagent incorporated intoan organic phase, similar to those used inconventional liquid-liquid extraction, results in asignificant decrease in reagent inventory.

Desalination 219 (2008) 324–334

doi:10.1016/j.desal.2006.08.030

0011-9164/08/$– See front matter © 2008 Published by Elsevier B.V.

S. Mohammadi et al. / Desalination 219 (2008) 324–334 325

The metal extraction chemistry is identical tothose in the conventional liquid-liquid extraction.But the overall process is governed by kineticrather than equilibrium parameters [13]. A sup-ported liquid membrane consists of a thin highlymicroporous film in which the organic phase isadsorbed to separate the feed and strippingphases, represents one of the feasible types of li-quid membranes [2,14]. On the other hand, in theliquid surfactant membrane type an organic mem-brane film is obtained by creating a water-in-oilemulsion with the strip liquor and organic phasewhich is then suspended in the feed solution [15].

Bulk liquid membranes are divided into twoparts: a common part, containing the membraneliquid S and a second part, which is the donorsolution F and the acceptor solution R. The liquidS makes contact with the two other liquids andaffects the transfer between them. All threeliquids are stirred with an appropriate intensity,and avoid mixing the donor and acceptorsolutions.

From the engineering and practical standpoint,SLMs are of particular interest because of theirstability and simplicity. In this case, modeling ofliquid membrane, studies have been performedfor some special metal ions using various types ofextractants such as separation of gold (III) usingcyanex 923 [13], separation of palladium (II)using di-(2-ethylhexyl)thiophosphoric acid [17],and transport of copper(II) using MOC-55TD[18] separation of gold(III) by an emulsion liquidmembrane system [21–24].

A complete review of recovery and separationof organic acids by SLM systems has beenperformed by Schlosser et al. [25]. In this work,the transfer of metal ions from aqueous solutionsthrough SLM has been studied and a dimension-less model for mass transfer coefficient has beenproposed.

2. Permeation mechanismThe transport of metal ions through the

supported liquid membrane system is considered

to be composed of many elementary steps [26–28]. These steps are expressed as follows:C Diffusion of metal ions from the bulk feed

phase to the aqueous stagnant layer on thefeed- membrane side.

C Diffusion of metal ions from stagnant layer tothe membrane surface at the feed side.

C Reaction between metal ions and carrier at thefeed-membrane interface.

C Diffusion of carrier–metal complex from thefeed–membrane interface to the stripping–membrane interface.

C Decomposition reaction of carrier–metal com-plex at the stripping–membrane interface andrelease of second species.

C Diffusion of second species through the mem-brane to the membrane–feed interface.

C Diffusion of second species from the feed–membrane interface through the stagnantlayer.

C Diffusion of second species from the stagnantlayer on the membrane in the feed side to feedbulk.

C Diffusion of second species (in strippingphase) from the bulk of strip phase to thestagnant layer in the stripping–membrane side.

C Diffusion of regenerated carrier back to thefeed membrane interface.

C Diffusion of metal ions from the stripping–membrane interface to the bulk of strippingphase.

Therefore, to model the transport of metal ions, itis necessary to consider diffusion of solutethrough the feed boundary layers, reversiblechemical reaction at the interfaces, diffusion ofthe metal complex species in the membrane,chemical reaction at the stripping interface anddiffusion of metal ions through the stripping sideboundary layer.

2.1. Extraction equilibrium

The extraction equilibrium may be describedby the following reactions:

S. Mohammadi et al. / Desalination 219 (2008) 324–334326

C Hydrolysis:

(1)( ) ( ) ( )-2H O H OH

aq aq aq

+ +

(2)( ) ( ) ( )- +2 32 H O OH H Oorg aq org aq

L L+ → +

C Complexation:

( ) ( ) ( )

( ) ( ) ( )

( ) ( )

2

3

OH OH

For :

OH 2 H O

OH OH H O

n xn

aq org xorg

n x

xorg

n x

n xx aqorg

M x L M L

n x

M L n x

M L

−+ − −

−−

−− − +

−

⎛ ⎞⎡ ⎤+ ↔ ⎜ ⎟⎣ ⎦⎝ ⎠>

⎛ ⎞⎡ ⎤ + − →⎜ ⎟⎣ ⎦⎝ ⎠

⎛ ⎞⎡ ⎤ +⎜ ⎟⎣ ⎦⎝ ⎠

(3a)

For n = x:

( ) ( ) ( ){ }OH OHn

aq org n orgM x L M L+ − −⎡ ⎤+ ⇔ ⎣ ⎦

(3b)C Decomplexation:

( ) ( )( ) ( )2

(OH ) H

H O

n nn org aq aq

org aq

L M n M

n L n

− + + +⎡ ⎤ + →⎣ ⎦

+ + (4)

C Overall:

(5)( ) ( ) ( )

( )

1

22H O

( OH ) H

knorg aqaq k

nn org aq

M n L n

M L n

+

+ − +

⎯⎯→+ + ←⎯⎯

⎡ ⎤ +⎣ ⎦

(6)[ ]

1

2

( OH ) Hnn

n org aqext nn

orgaq

M LkKk M L

+ − +

+

⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦= =⎡ ⎤⎣ ⎦

where L and M refers to the extractant and themetal ion, respectively.

2.2. Assumption of the model

The following assumptions must be taken intoaccount [13,16–20]:C The chemical reaction is very fast relative to

the diffusion processes then local equilibriumat the interface is reached.

C No complex flows into the feed from theorganic phase.

C Fick’s first law is valid for all diffusionalsteps. As a consequence, the concentrationdistribution in the thickness direction isconsidered linear.

C The decomposition reaction takes place onlyat membrane-stripping phase interface.

C The membrane polarity is low enough to makethe concentration of charged species negli-gible with respect to that of uncharged ones.

C The physical properties during the process areconstant.

C The solution flow rate, density, pressure andtemperature of phases are constant.

2.3. Permeation model

The equation describing the diffusion of metalions from bulk to feed membrane interface (Jf) is

(7)[ ] [ ]( )

[ ] [ ]( )

,

1

M ff f if

f

f f if

DJ M M

M M−

= −δ

= Δ −

The following equation describes the diffusion ofmetal ions from the stripping–membrane interfaceto the bulk of stripping phase:

(8)[ ] [ ]( )

[ ] [ ]( )

,

1

M ss is s

s

s is s

DJ M M

M M−

= −δ

= Δ −

This equation describes the diffusion of metalcomplex in the membrane phase (Jorg):


(9)[ ] [ ]( )

[ ] [ ]( )

,2

1

M orgorg if is

org

org if is

DJ ML ML

ML ML−

ε= −

δ τ

= Δ −

If the extraction reaction is fast enough, theconcentration of the complex at the membranephase on the stripping solution side may beconsidered negligible so the distribution coeffi-cient of metal ions between the membrane andstripping phases compared with that on the feedsolution side becomes much lower than thatbetween the feed and the membrane phases. ThenEq. (9) can be rewritten as:

(10),2 [ ]M org

org iforg

DJ MLξ

=δ τ

If the chemical reaction expressed by Eq. (5)is assumed to be fast enough compared to thediffusion rate, then local equilibrium at the (feed–membrane and membrane–stripping) interfaces isachieved and concentrations at the interface arerelated through Eq. (6). Thus, at steady state, Jf =Jorg = Js, and by combination of Eqs (6)–(9), thefollowing expressions are obtained:

[ ]

( )

1

, ,,

2

1 1 1

H

[ ] [ ]

nM f M sext org M org

n sforg

aq

f s

J D DK L D

M M

−

+

⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟= + +⎜ ⎟⎜ ⎟⎜ ⎟⎡ ⎤⎣ ⎦⎝ ⎠

−

εδδ δ τ

(11)

(12)( )[ ] [ ]f sJ k M M= −

[ ]

1

, ,,

2

1 1 1

H


n sforg

aq

k D DK L D

−

+

⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟= + +⎜ ⎟⎜ ⎟⎜ ⎟⎡ ⎤⎣ ⎦⎝ ⎠

εδδ δ τ

(13)

As pointed out before, if the complex con-centration at the membrane phase on the strippingside is considered negligible, Eq. (11) may berewritten as:

(14)

[ ]

1

,,

2+

1 1 [ ]

H

fnM f ext org M org

nforg

aq

J MD K L D

−⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟= +⎜ ⎟⎜ ⎟⎜ ⎟⎡ ⎤⎣ ⎦⎝ ⎠

εδ δ τ

[ ]

[ ]+

[ ]d

H[ ]

H

f

next org

n

aqfn

ext orgorg f n

aq

d M VJt A

K L

MK L

+

= −

⎛ ⎞⎜ ⎟⎜ ⎟⎡ ⎤⎣ ⎦⎜ ⎟=⎜ ⎟⎜ ⎟Δ + Δ⎜ ⎟⎡ ⎤⎣ ⎦⎝ ⎠

(15)

Two extremes are considered:

2.3.1. High feed concentrationIn this case the metal ions concentration in the

feed solution is large enough to make

[ ]H

next org

f orgn

aq

K L+

⎛ ⎞⎜ ⎟Δ < Δ⎜ ⎟⎡ ⎤⎜ ⎟⎣ ⎦⎝ ⎠


in the denominator of Eq. (15):

(16)

[ ]

[ ]

H

H[ ][ ]

d

next org

f orgn

aq

next org

n

f aqb f

org

K L

K L

d M VJ Mt A

+

+

Δ < Δ ⇒⎡ ⎤⎣ ⎦

⎛ ⎞⎜ ⎟⎜ ⎟⎡ ⎤⎣ ⎦⎜ ⎟= − =⎜ ⎟Δ⎜ ⎟⎜ ⎟⎝ ⎠

(17)[ ] ( OH )

H

nnnext org org

n naq aq

M LK LM

+ −

+ +

⎡ ⎤⎣ ⎦=

⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦

(18)[ ]

( OH ) orgnn org

LM L

n+ −⎡ ⎤ =⎣ ⎦

So the membrane ion permeability is entirelycontrolled by membrane diffusion:

(19),02

[ ] [ ]d

f M orgb

org

d M DLVJt A n

⎛ ⎞= − = ⎜ ⎟⎜ ⎟

⎝ ⎠

εδ τ

In this equation [L0] is the initial extractantconcentration and Jb is the mass flux of metalions which refer to this extreme. IntegratingEq. (19):

(20),00 2

[ ][ ] [ ] M orgf t f t

org

DLAM M tV n= − =

εδ τ

(21),00 2

0 0

[ ][ ] [ ]

M orgb

f t f t org

DJ LpM n M= =

= =εδ τ

In this region the plot of vs.0[ ] [ ]f t f tM M= −

time should give a straight line as predicted byEq. (20) allowing the calculation of DM, org and p0.

2.3.2. Low feed concentrationLow metal ions concentration in the feed:

(22)[ ]

0

0

[ ] [ ]

[ ] [ ]

org

nnext orgaq

orgn

aq

L L

K M Ln L L

H

+

+

=

⎡ ⎤⎣ ⎦+ ⇒ =

⎡ ⎤⎣ ⎦

Then Eq. (16) can be integrated to:

(23)0

[ ]ln

[ ]f t

f t

M A ptM V=

= −

where p is time independent and is defined as theion permeability coefficient which expresses:

(24)

[ ]

[ ]1

[ ] H

H

next org

nf

aq

next org

org f n

aq

K LJpM

K L

+

−

+

⎛ ⎞⎜ ⎟= = ⎜ ⎟⎡ ⎤⎜ ⎟⎣ ⎦⎝ ⎠

⎛ ⎞⎜ ⎟Δ + Δ⎜ ⎟⎡ ⎤⎜ ⎟⎣ ⎦⎝ ⎠

The ion permeability coefficient can be calculated

from the slope of vs. time. 0

[ ]ln

[ ]f t

f t

MM =

Between the above discussed extremes, anumerical solution for the integrated flux equa-tions can be obtained:

(25),

, 0

[ ][ ]

M f

M f

C tf

fC t

d M V tM A p=

=∫


2.4. Dimensionless models

Dimensionless models can solve the scale-upproblems and are of particular interest. In thissection a dimensionless model for transport ofmetal ions is derived. For these proposes Eq. (11)is used:

[ ]

( )

1

, ,,

2

1 1 1

H

[ ] [ ]

nM f M sb ext org M org

n sforg

aq

f s

b

JD DJ K L D

M MJ

−

+

⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟= + +⎜ ⎟⎜ ⎟⎜ ⎟⎡ ⎤⎣ ⎦⎝ ⎠

−

εδδ δ τ

(26)

(27)

[ ]

0 ,2

1

, ,,

2

0 0

[ ]

1 1 1

H

[ ] [ ][ ]

M org

org


n sforg

aq

f s

f t

JL D

n

D DK L D

M MM p

−

+

=

=

⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟+ +⎜ ⎟⎜ ⎟⎜ ⎟⎡ ⎤⎣ ⎦⎝ ⎠⎛ ⎞−⎜ ⎟⎜ ⎟⎝ ⎠

εδ τ

εδδ δ τ

[ ]

1

0 0 0

, ,,

2

0

H

[ ] [ ][ ]

nM f M sb ext org M org

n sforg

aq

f s

f t

p p pJD DJ K L D

M MM

−

+

=

⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟= + +⎜ ⎟⎜ ⎟⎜ ⎟⎡ ⎤⎣ ⎦⎝ ⎠

⎛ ⎞−⎜ ⎟⎜ ⎟⎝ ⎠

εδδ δ τ

(28)

[ ]

1

0 0 0

,2

0

Re Re

H

[ ] [ ][ ]

f sn

b f f s sext org M orgn

orgaq

f s

f t

p p pJJ Sc V Sc VK L D

M MM

−

+

=

⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟= + +⎜ ⎟⎜ ⎟⎜ ⎟⎡ ⎤⎣ ⎦⎝ ⎠

⎛ ⎞−⎜ ⎟⎜ ⎟⎝ ⎠

εδ τ

(29)

,Re , Scf f M ff f

f f

V Dv vδ

= =

,Re , Sc M ss ss s

s s

DVv vδ

= =

If the metal–extractant complexes in the mem-brane phase at the stripping solution side incomparison with the one at the feed solution sideare negligible, instead of Eq. (11), Eq. (14) isused:

[ ]

1

,,

2

[ ]1 1

H

fn

M fb bext org M orgnf

orgaq

MJDJ JK L D

−

+

⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟= +⎜ ⎟⎜ ⎟⎜ ⎟⎡ ⎤⎣ ⎦⎝ ⎠

εδ δ τ

(30)

(31)[ ]

1

0 0

0

,2

0

Re

H

[ ][ ]

f

b f fn

ext org M orgn

orgaq

f

f t

p pJpJ Sc V

K L D

MM

−

+

=

⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟= +⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟

⎡ ⎤⎜ ⎟⎣ ⎦⎝ ⎠

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

εδ τ


The relation between k and the system parametersis determined by Eq. (13). But using this equationis a little difficult, so in this section a dimen-sionless algebraic equation for mass transfercoefficient is derived.

As seen from comparing Eqs. (13) and (29),the mass transfer coefficient is a function ofparameters such as Ref, Scf, Gorg, Res and Scs or:

(32)( )Re , , , Re ,f f org s sk f Sc G Sc=

where Gorg is a dimensionless parameter whichcan be written as follows:

[ ]0

,2

H

org next org M org

norg

aq

pGK L D

+

=

⎡ ⎤⎣ ⎦

εδ τ

(33)

Because in the experimental data available[13,16–20] the distribution coefficient of metalions between the membrane phase and strippingphase is considered much lower than that betweenthe feed phase and membrane phase, then insteadof Eq. (13), Eq. (14) is used:

(34)[ ]

1

,,

2

1 1

H

nM f ext org M org

nforg

aq

k D K L D

−

+

⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟= +⎜ ⎟⎜ ⎟⎜ ⎟⎡ ⎤⎣ ⎦⎝ ⎠

εδ δ τ

As seen from comparing Eqs. (31) and (34), themass transfer coefficient is a function of para-meters such as

(35)( )Re , ,f f orgk f Sc G=

Nine different equations have been considered forthe mass transfer coefficient, and k is calculated

Table 1Percent of deviation between the experimental and theproposed models

Kind of equation No. ofpoints

Averagedeviationa

356 47.0331 2Re nn nf f orgk a bSc cG= + +

271 56.4631 2 4Re nn n nf f org fk a Sc bG Sc= +

316 25.7831 2 4Re Renn n nf f org fk a Sc bG= +

316 32.3431 2 4Re nn n nf org org fk a G bG Sc= +

316 28.0731 2Re nn nf org fk a bG Sc= +

316 25.4031 2Re nn nf f orgk aSc b G= +

316 30.5031 2Re nn norg f fk aG b Sc= +

325 19.9331 2Re nn nf f orgk a Sc G=

[ ]

1

2

1 1n

M , f ext org M ,orgnf

orgaq

k D K L D

H

−

+

⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟= +⎜ ⎟⎜ ⎟⎜ ⎟⎡ ⎤⎣ ⎦⎝ ⎠

εδ δ τ

348 33.05

aPercent of deviation = 1

100

nExp. Theo.

i Exp. i

n=

⎛ ⎞−⎜ ⎟⎜ ⎟⎝ ⎠ ×

∑Χ Χ

Χ

from each equation, then is calculated0

[ ][ ]

f

f t

MM =

from following equation:

(36)0

[ ][ ]

f

b f t

MJ kJ M =

=

Then , calculated from the model,0

[ ][ ]

f

f t

MM =

has plotted vs. which were obtained0

[ ][ ]

f

f t

MM =

from experimental data.


Table 2Constants for equation 31 2Re nn n

f f orgk a Sc G=

Parameter Zn2+ [16] Pd2+ [17] Au3+ [13] Cu2+ [18] Zn2+ [19] Au3+ [20]

a 0.264 0.7984 0.328 3.75 1.99 0.00008n1 0.9841 0.222 0.186 0.316 0.6165 0.05n2 0.023 0.339 0.83 !0.022 0.952 0.501n3 !0.662 0.219 0.515 !0.147 !0.0699 !0.267

Fig. 1. Model deviation fromexperimental data for the model of

.31 2 4Re nn n nf f org fk a Sc bG Sc= +

Fig. 2. Model deviation fromexperimental data for the model of .31 2Re nn n

f f orgk a bSc cG= + +


Fig. 3. Model deviation fromexperimental data for the model of .31 2Re nn n

org f fk aG b Sc= +

Fig. 4. Model deviation fromexperimental data for the model of

.31 2Re nn nf f orgk a Sc G=

The deviations of calculated data were ob-tained, and the results are represented in Table 1.As seen from Table 1, the minimum deviation isobserved when equation is31 2Re nn n

f f orgk a Sc G=used for calculating the mass transfer coefficientso that it is better for predicting the equation of k

as follows:

(37)31 2Re nn nf f orgk a Sc G=

where a, n1, n2 and n3 are constants which shouldbe determined using the experimental data.


The constants for are sum-31 2Re nn nf f orgk a Sc G=

marized in Table 2.Figs. 1and 2 show the cases with maximum

deviation and Figs. 3 and 4 show the cases withminimum deviation, respectively.

3. Conclusion

The transport of various metal ions such asZn2+, Au3+, Pd2+ and Cu2+ through supportedliquid membranes containing a carrier was stu-died. The effect of parameters on the transport ofmetal ions was investigated and a permeation rateequation was derived, considering diffusional andequilibrium parameters of the system as well asproperties of the membrane. The mass transfermodeling was performed and a general dimen-sionless model for mass transfer coefficient isproposed. The validity of the developed modelwas evaluated with experimental data and foundto tie in well with theoretical value.

4. Symbols

D — Diffusion coefficient, m/s2

J — Mass flux, mol/(m2s)k — Mass transfer coefficient, mol/(m2 s)Kext — Equilibrium constant[M] — Metal concentration, mol/m3

[ML] — Carrier-metal complex concentra-tion, mol/m3

Re — Reynolds number, Vδ/vSc — Schmidt number, D/v

Greek

Δ — Transport resistance due to diffu-sion, s/m

ξ — Porosity of supportτ — Tortuosity of supportδ — Thickness of diffusional layer, mΧ — Ratio of [Mf]/[Mf 0]

Subscriptsf — Feed bulkif — Feed–membrane interfaceis — Membrane–strip phases interfaceorg — Membrane phases — Stripping phase

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A model for metal ion pertraction through supported liquid membranes

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