A model for metal ion pertraction through supported liquid membranes

Desalination 219 (2008) 324–334

A model for metal ion pertraction through supported liquid membranes Samira Mohammadia, Tahereh Kaghazchia*, Ali Kargaria,b a

Department 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] b Department 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 dimensionless model for mass transfer coefficient is proposed which consists of diffusional, equilibrium and process parameters also properties of the membrane. Nine dimensionless equations for mass transfer coefficient have been suggested n n n and found that the equation in the form of k = a Re f1 Sc f 2 Gorg3 has the best agreement with experimental data. The 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 containing carriers has been proposed as an alternative to solvent extraction processes for the selective separation, purification and concentration of metal ions in aqueous solutions, organic acids, bio products and gases [1–9]. In liquid membrane technology the extraction, stripping and regeneration operations are combined in a single stage [10–12]. Liquid membranes have also received *Corresponding author.

considerable attention due to characteristics such as ease of operation, lower energy consumption and operational cost, higher selectivity, rapid extraction and higher capacity factors. Currently, three configurations of liquid membranes are being investigated: supported liquid membranes (SLMs), liquid surfactant membranes (LSMs) and bulk liquid membranes (BLMs). In each case, an extraction reagent incorporated into an organic phase, similar to those used in conventional liquid-liquid extraction, results in a significant decrease in reagent inventory.

0011-9164/08/$– See front matter © 2008 Published by Elsevier B.V. doi:10.1016/j.desal.2006.08.030

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

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The metal extraction chemistry is identical to those in the conventional liquid-liquid extraction. But the overall process is governed by kinetic rather than equilibrium parameters [13]. A supported liquid membrane consists of a thin highly microporous film in which the organic phase is adsorbed to separate the feed and stripping phases, represents one of the feasible types of liquid membranes [2,14]. On the other hand, in the liquid surfactant membrane type an organic membrane film is obtained by creating a water-in-oil emulsion with the strip liquor and organic phase which is then suspended in the feed solution [15]. Bulk liquid membranes are divided into two parts: a common part, containing the membrane liquid S and a second part, which is the donor solution F and the acceptor solution R. The liquid S makes contact with the two other liquids and affects the transfer between them. All three liquids are stirred with an appropriate intensity, and avoid mixing the donor and acceptor solutions. From the engineering and practical standpoint, SLMs are of particular interest because of their stability and simplicity. In this case, modeling of liquid membrane, studies have been performed for some special metal ions using various types of extractants such as separation of gold (III) using cyanex 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 liquid membrane system [21–24]. A complete review of recovery and separation of organic acids by SLM systems has been performed by Schlosser et al. [25]. In this work, the transfer of metal ions from aqueous solutions through SLM has been studied and a dimensionless model for mass transfer coefficient has been proposed.

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 the feed- membrane side. C Diffusion of metal ions from stagnant layer to the membrane surface at the feed side. C Reaction between metal ions and carrier at the feed-membrane interface. C Diffusion of carrier–metal complex from the feed–membrane interface to the stripping– membrane interface. C Decomposition reaction of carrier–metal complex at the stripping–membrane interface and release of second species. C Diffusion of second species through the membrane to the membrane–feed interface. C Diffusion of second species from the feed– membrane interface through the stagnant layer. C Diffusion of second species from the stagnant layer on the membrane in the feed side to feed bulk. C Diffusion of second species (in stripping phase) from the bulk of strip phase to the stagnant layer in the stripping–membrane side. C Diffusion of regenerated carrier back to the feed membrane interface. C Diffusion of metal ions from the stripping– membrane interface to the bulk of stripping phase.

2. Permeation mechanism

2.1. Extraction equilibrium

The transport of metal ions through the supported liquid membrane system is considered

The extraction equilibrium may be described by the following reactions:

Therefore, to model the transport of metal ions, it is necessary to consider diffusion of solute through the feed boundary layers, reversible chemical reaction at the interfaces, diffusion of the metal complex species in the membrane, chemical reaction at the stripping interface and diffusion of metal ions through the stripping side boundary layer.

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C Hydrolysis:

2.2. Assumption of the model

( H 2O )aq R ( H + )aq + ( OH - )aq Lorg + 2 ( H 2 O )aq → ( LOH - )

org

(1)

+ ( H 3O + ) (2) aq

C Complexation:

(M )

+ x ( LOH − )

n+

aq

org

↔ ⎛⎜ ⎡ M ( LOH − ) ⎤ x⎦ ⎝⎣

n− x

For n > x : ⎛ ⎡ M LOH − ⎤ n − x ⎞ + 2 ( n − x )( H O ) → ) x ⎦ ⎟⎠org ⎜⎣ ( 2 ⎝

(3a)

⎛ ⎡ M LOH − ⎤ n − x OH − ⎞ + H O + )x ⎦ ( 3 )aq ⎜⎣ ( n− x ⎟ ⎝ ⎠org For n = x:

(M ) n+

aq

+ x ( L OH − )

org

{

}

⇔ ⎡ M ( L OH − ) ⎤ n⎦ ⎣

org

(3b) C Decomplexation:

(4)

C Overall:

(M ) n+

aq

+ n ( L )org

⎯⎯ → + n ( H 2 O )aq ←⎯ ⎯ k2

(5)

n

K ext

n+ − + k1 ⎡⎣ M ( LOH )n ⎤⎦org ⎡⎣ H ⎤⎦ aq = = n k2 ⎡⎣ M n + ⎤⎦ [ L ]org

Jf =

DM , f δf

= Δ −f 1

k1

⎡⎣ M n + ( LOH − ) n ⎤⎦ + n ( H + ) org aq

2.3. Permeation model The equation describing the diffusion of metal ions from bulk to feed membrane interface (Jf) is

⎡⎣(OHL− ) n M n + ⎤⎦ + n ( H + ) → ( M n + ) org aq aq + n ( L )org + n ( H 2 O )aq

⎞ ⎟ ⎠org

The following assumptions must be taken into account [13,16–20]: C The chemical reaction is very fast relative to the diffusion processes then local equilibrium at the interface is reached. C No complex flows into the feed from the organic phase. C Fick’s first law is valid for all diffusional steps. As a consequence, the concentration distribution in the thickness direction is considered linear. C The decomposition reaction takes place only at membrane-stripping phase interface. C The membrane polarity is low enough to make the concentration of charged species negligible with respect to that of uncharged ones. C The physical properties during the process are constant. C The solution flow rate, density, pressure and temperature of phases are constant.

DM , s

(6)

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

([ M ]

f

f

− [ M ]if

)

− [ M ]if

) (7)

The following equation describes the diffusion of metal ions from the stripping–membrane interface to the bulk of stripping phase:

Js =

aq

([ M ]

=Δ

δs −1 s

([ M ] − [ M ] ) is

s

([ M ] − [ M ] ) is

(8)

s

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

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

J org =

DM ,org ε δorg τ2

−1 = Δ org

([ ML]if − [ ML]is )

([ ML]

if

− [ ML ]is

)

(9)

If the extraction reaction is fast enough, the concentration of the complex at the membrane phase on the stripping solution side may be considered negligible so the distribution coefficient of metal ions between the membrane and stripping phases compared with that on the feed solution side becomes much lower than that between the feed and the membrane phases. Then Eq. (9) can be rewritten as:

J org =

DM ,org ξ [ ML]if δorg τ2

(10)

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

⎛ ⎜ ⎜ 1 1 1 J =⎜ + + n DM , s ⎜ DM , f K ext [ L ] D org M ,org ε ⎜ δ n δs δ org τ 2 ⎜ f ⎡⎣ H + ⎤⎦ aq ⎝

([ M ]

f

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

−1

− [ M ]s ) (11)

J = k ([ M ] f − [ M ]s )

(12)

327

⎛ ⎜ ⎜ 1 1 1 + + k =⎜ n DM , s ⎜ DM , f K ext [ L ] D org M ,org ε ⎜ δ n δs δ org τ 2 ⎜ f ⎡⎣ H + ⎤⎦ aq ⎝

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

−1

(13) As pointed out before, if the complex concentration at the membrane phase on the stripping side is considered negligible, Eq. (11) may be rewritten as:

⎛ ⎜ ⎜ 1 1 + J =⎜ n ⎜ DM , f K ext [ L ] D org M ,org ε ⎜ δ n δ org τ 2 ⎜ f ⎡⎣ H + ⎤⎦ aq ⎝

J =−

−1

⎞ (14) ⎟ ⎟ ⎟ [M ] f ⎟ ⎟ ⎟ ⎠

d [M ] f V dt A

n ⎛ K ext [ L ]org ⎜ n ⎜ ⎡⎣ H + ⎤⎦ aq =⎜ n ⎜ K ext [ L ]org ⎜ Δ org + Δ f n ⎜ ⎡⎣ H + ⎤⎦ aq ⎝

⎞ ⎟ ⎟ ⎟ [M ] f ⎟ ⎟ ⎟ ⎠

(15)

Two extremes are considered: 2.3.1. High feed concentration In this case the metal ions concentration in the feed solution is large enough to make n ⎛ ⎞ ⎜ Δ K ext [ L ]org < Δ ⎟ org ⎜⎜ f ⎡ H + ⎤ n ⎟⎟ ⎣ ⎦ aq ⎝ ⎠

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

in the denominator of Eq. (15): K ext [ L]org n

Δf

n

⎡⎣ H + ⎤⎦ aq

< Δ org ⇒

⎛ K ext [ L ]n org ⎜ + n ⎜ ⎡ ⎤ H d [M ] f V ⎣ ⎦ aq =⎜ Jb = − d t A ⎜ Δ org ⎜ ⎜ ⎝

K ext [ L ]

n org

⎡⎣ H + ⎤⎦ naq

⎞ ⎟ ⎟ ⎟ [M ] f ⎟ ⎟ ⎟ ⎠

[ L]0 = [ L]org +n

K ext ⎡⎣ M n + ⎤⎦

(17)

ln

aq

[ L]org

(18)

n

So the membrane ion permeability is entirely controlled by membrane diffusion:

d [ M ] f V ⎛ [ L]0 DM ,org ε =⎜ d t A ⎜⎝ n δ org τ 2

⎞ ⎟⎟ ⎠

[M ] f [M ] f

n

⎡⎣ H + ⎤⎦ aq

t t =0

=−

−[ M ] f

p0 =

Jb [M ] f

= t =0

A [ L]0 DM ,org ε t V n δ org τ 2

(20)

DM ,org ε [ L]0 n [ M ] f t =0 δ org τ 2

(21)

t

=

In this region the plot of [ M ] f

t =0

⇒ [ L]0 = [ L]org

A pt V

⎛K L n J ext [ ]org p= =⎜ [ M ] f ⎜⎜ ⎡ H + ⎤ n ⎝ ⎣ ⎦ aq

−[ M ] f

(23)

⎞ ⎟ ⎟⎟ ⎠

n ⎛ ⎜ Δ + Δ K ext [ L ]org f n ⎜⎜ org ⎡⎣ H + ⎤⎦ aq ⎝

⎞ ⎟ ⎟⎟ ⎠

−1

(24)

The ion permeability coefficient can be calculated from the slope of ln

t =0

(22)

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

(19)

In this equation [L0] is the initial extractant concentration and Jb is the mass flux of metal ions which refer to this extreme. Integrating Eq. (19):

[M ] f

n [ L]org aq

Then Eq. (16) can be integrated to:

⎡⎣ M n + ( LOH − ) n ⎤⎦ org = n+ ⎡⎣ M ⎤⎦

⎡⎣ M n + ( LOH − ) n ⎤⎦ = org

Jb = −

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

(16)

[M ] f [M ] f

t

vs. time.

t =0

Between the above discussed extremes, a numerical solution for the integrated flux equations can be obtained: CM , f t

t

vs.

d [M ] f V =t [M ] f A p CM , f t = 0

∫

(25)

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

2.4. Dimensionless models Dimensionless models can solve the scale-up problems and are of particular interest. In this section a dimensionless model for transport of metal ions is derived. For these proposes Eq. (11) is used:

⎛ ⎜ ⎜ J ⎜ 1 1 1 = + + n J b ⎜ DM , f K ext [ L ]org DM ,org ε DM , s ⎜ δ n δs δ org τ 2 ⎜ f ⎡⎣ H + ⎤⎦ aq ⎝

([ M ]

f

− [ M ]s )

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

−1

(26)

Jb

J = [ L]0 DM ,org ε nδ org τ 2

(27)

⎛ ⎜ ⎜ 1 1 ⎜ 1 + + n DM , s ⎜ DM , f K ext [ L ] D org M ,org ε ⎜ δ n δs δ orgτ 2 ⎜ f ⎡⎣ H + ⎤⎦ aq ⎝ ⎛ [ M ] f − [ M ]s ⎜⎜ ⎝ [ M ] f t =0 p0

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

−1

⎛ [ M ] f − [ M ]s ⎜⎜ ⎝ [ M ] f t =0

⎞ ⎟⎟ ⎠

⎛ ⎞ ⎜ ⎟ ⎜ Re ⎟ p0 Re s p0 ⎟ J ⎜ f p0 = + + n J b ⎜ Sc f V f K ext [ L ]org DM ,org ε Scs Vs ⎟ ⎜ ⎟ n δ orgτ 2 ⎜ ⎟ ⎡⎣ H + ⎤⎦ aq ⎝ ⎠ ⎛ [ M ] f − [ M ]s ⎜⎜ ⎝ [ M ] f t =0

Re f = Re s =

Vf δ f vf Vs δ s , vs

,

⎞ ⎟⎟ ⎠

Sc f = Sc s =

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

−1

(28)

−1

(29)

DM , f vf

DM , s vs

If the metal–extractant complexes in the membrane phase at the stripping solution side in comparison with the one at the feed solution side are negligible, instead of Eq. (11), Eq. (14) is used:

⎛ ⎜ ⎜ J ⎜ 1 1 = + n J b ⎜ DM , f K ext [ L ]org DM ,org ε ⎜ δ n δ org τ 2 ⎜ f ⎡⎣ H + ⎤⎦ aq ⎝

⎞ ⎟⎟ ⎠

⎛ ⎜ ⎜ p0 p J ⎜ p0 = + + 0 n J b ⎜ DM , f K ext [ L ]org DM ,org ε DM , s ⎜ δ n δs δ orgτ 2 ⎜ f ⎡⎣ H + ⎤⎦ aq ⎝

329

⎛ ⎜ ⎜ ⎜ p0 J ⎜ Re f p0 =⎜ + p0 Jb Sc V ⎜ f f n K ext [ L ]org DM ,org ε ⎜ n ⎜ δ orgτ 2 ⎡⎣ H + ⎤⎦ ⎜ aq ⎝ ⎛ [M ] f ⎞ ⎜⎜ ⎟⎟ M [ ] f t = 0 ⎝ ⎠

−1

⎞ ⎟ ⎟ [M ] f ⎟ ⎟ Jb ⎟ ⎟ (30) ⎠

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

−1

(31)

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

The relation between k and the system parameters is determined by Eq. (13). But using this equation is a little difficult, so in this section a dimensionless algebraic equation for mass transfer coefficient is derived. As seen from comparing Eqs. (13) and (29), the mass transfer coefficient is a function of parameters such as Ref, Scf, Gorg, Res and Scs or: (32)

where Gorg is a dimensionless parameter which can be written as follows:

p0

K ext [ L ]org DM ,org ε n

n

⎡⎣ H ⎤⎦ aq +

(33)

δ org τ 2

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

⎛ ⎜ ⎜ 1 1 k =⎜ + n D ⎜ M , f K ext [ L ] D org M ,org ε ⎜ δ n δ org τ 2 ⎜ f ⎡⎣ H + ⎤⎦ aq ⎝

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

No. of Average points deviationa

n3 k = a Re nf1 + bSc nf 2 + cGorg

356

47.03

n3 k = a Re nf1 Sc nf 2 + bGorg Sc nf 4

271

56.46

k = a Re Sc + bG

316

25.78

n3 n2 k = a Re nf1 Gorg + bGorg Sc nf 4

316

32.34

n2 k = a Re nf1 + bGorg Sc nf 3

316

28.07

n3 k = aSc nf1 + b Re nf2 Gorg

316

25.40

k = aG

316

30.50

325

19.93

348

33.05

n1 org

n3 org

n2 f

+ b Re Sc n2 f

Re

n4 f

n3 f

n3 k = a Re nf1 Sc nf 2 Gorg

⎛ ⎜ ⎜ 1 1 + k =⎜ n D ⎜ M ,f K ext [ L ]org DM ,org ε ⎜ δ n δ org τ 2 ⎜ f ⎡⎣ H + ⎤⎦ aq ⎝

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

−1

⎛ Χ Exp. − Χ Theo. ⎞ ⎟⎟ Χ Exp. i =1 ⎝ ⎠i a × 100 Percent of deviation = n n

∑ ⎜⎜

−1

from each equation, then (34)

Then

[M ] f [M ] f

[M ] f

is calculated

t =0

(36)

, calculated from the model,

t =0

(35)

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

[M ] f

from following equation:

[M ] f J =k Jb [ M ] f t =0

As seen from comparing Eqs. (31) and (34), the mass transfer coefficient is a function of parameters such as

k = f ( Re f , Sc f , Gorg )

Kind of equation

n1 f

k = f ( Re f , Sc f , Gorg , Re s , Scs )

Gorg =

Table 1 Percent of deviation between the experimental and the proposed models

has plotted vs.

[M ] f [M ] f

t =0

from experimental data.

which were obtained

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

331

Table 2 n3 Constants for equation k = a Re nf1 Sc nf 2 Gorg Parameter

Zn2+ [16]

Pd2+ [17]

Au3+ [13]

Cu2+ [18]

Zn2+ [19]

Au3+ [20]

a n1 n2 n3

0.264 0.9841 0.023 !0.662

0.7984 0.222 0.339 0.219

0.328 0.186 0.83 0.515

3.75 0.316 !0.022 !0.147

1.99 0.6165 0.952 !0.0699

0.00008 0.05 0.501 !0.267

Fig. 1. Model deviation from experimental data for the model of n3 k = a Re nf1 Sc nf 2 + bGorg Sc nf 4 .

Fig. 2. Model deviation from experimental data for the model of n3 k = a Renf1 + bSc nf 2 + cGorg .

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

Fig. 3. Model deviation from experimental data for the model of n1 k = aGorg + b Renf2 Sc nf 3 .

Fig. 4. Model deviation from experimental data for the model of k = a Re n1 Sc n2 G n3 . f

The deviations of calculated data were obtained, and the results are represented in Table 1. As seen from Table 1, the minimum deviation is n n n observed when equation k = a Re f1 Sc f 2 Gorg3 is used for calculating the mass transfer coefficient so that it is better for predicting the equation of k

f

org

as follows: n3 k = a Re nf1 Sc nf 2 Gorg

(37)

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

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

The constants for k = a Re f1 Sc f 2 Gorg3 are summarized in Table 2. Figs. 1and 2 show the cases with maximum deviation and Figs. 3 and 4 show the cases with minimum deviation, respectively. n

n

n

333

Subscripts f if is org s

— — — — —

Feed bulk Feed–membrane interface Membrane–strip phases interface Membrane phase Stripping phase

3. Conclusion The transport of various metal ions such as Zn , Au3+, Pd2+ and Cu2+ through supported liquid membranes containing a carrier was studied. The effect of parameters on the transport of metal ions was investigated and a permeation rate equation was derived, considering diffusional and equilibrium parameters of the system as well as properties of the membrane. The mass transfer modeling was performed and a general dimensionless model for mass transfer coefficient is proposed. The validity of the developed model was evaluated with experimental data and found to tie in well with theoretical value.

References

2+

4. Symbols D J k Kext [M] [ML] Re Sc

Diffusion coefficient, m/s2 Mass flux, mol/(m2s) Mass transfer coefficient, mol/(m2 s) Equilibrium constant Metal concentration, mol/m3 Carrier-metal complex concentration, mol/m3 — Reynolds number, Vδ/v — Schmidt number, D/v

— — — — — —

Greek Δ ξ τ δ Χ

— Transport resistance due to diffusion, s/m — Porosity of support — Tortuosity of support — Thickness of diffusional layer, m — Ratio of [Mf]/[Mf 0]

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[21]

[22]

[23]

[24]

[25]

[26]

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