Modeling extraction separation of Cu(II) in hollow-fiber modules

Chemical Engineering Science 64 (2009) 3455 -- 3465

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Modeling extraction separation of Cu(II) in hollow-fiber modules Jia-Jan Guo, Chii-Dong Ho ∗ , Jr-Wei Tu Department of Chemical and Materials Engineering, Tamkang University, Tamsui, Taipei, Taiwan

A R T I C L E

I N F O

Article history: Received 26 August 2008 Received in revised form 15 April 2009 Accepted 21 April 2009 Available online 3 May 2009 Keywords: Extraction Mass transfer Mathematical modeling Membrane Separations Simulation

A B S T R A C T

The mass transfer problems in the hollow-fiber membrane extractor module with concurrent- and countercurrent-flow were investigated theoretically and experimentally in this study. A two-dimensional mathematical model of the hollow-fiber membrane extractor module was developed theoretically and the shell side flow described by Happel's free surface model was taken into account. The analytical solution is obtained using an eigenfunction expansion in terms of the power series and an orthogonal expansion technique. The theoretical predictions were represented graphically with the mass-transfer Graetz number (Gz), flow pattern and packing density () as parameters and the theoretical results were also compared with those obtained by experimental runs. The highest extraction rate, extraction efficiency and mass transfer efficiency can be achieved by arranging the packing density  = 0.3. The results show that the device performance of the countercurrent-flow device is better than that of the concurrent-flow device. The experiments of the extraction of Cu2+ by using D2EHPA with PVDF hollow fibers is also set up to confirm the accuracy of the theoretical predictions. The accuracy of the theoretical predictions for concurrent- and countercurrent-flow are 5.87 × 10−2 ⱕ E1 ⱕ 6.69 × 10−2 and 2.46 × 10−2 ⱕ E1 ⱕ 3.48 × 10−2 , respectively, for Gza = 40.8. © 2009 Elsevier Ltd. All rights reserved.

1. Introduction The growing demand for high-purity metals, concerns over environmental issues and the need for lower production costs, has made selective metal recovery from raw and waste resources as an important issue. Solvent extraction using organophosphorus compounds such as di(2-ethylhexyl)-phosphoric acid (D2EHPA) has been widely used for selective metal recovery from acidic solutions in the hydrometallurgical and chemical process industries (Cox and Flett, 1983). However, there are the tendencies toward third phase and crude formation at the interface and the solvent loss due to the high aqueous solubility of D2EHPA (Cox and Flett, 1983). Traditional solvent extraction processes are operated in packed towers, mixersettlers, and etc. which seek to maximize the contact area of two immiscible phases for mass transfer. The important disadvantage of above devices includes the interdependence of the two fluid phases in contact, which sometimes leads to difficulties such as intimate mixing taking place in these devices. The intimate mixing often leads to the formation of a stable emulsion, thereby inhibiting the phase separation and product recovery. Independent phase flow rate variations, density difference requirements and inability to handle particulates (Prasad et al., 1986; Prasad and Sirkar, 1987a,b, 1988, 1990;

∗ Corresponding author. Tel.: +886 2 26266632; fax: +886 2 26209887. E-mail address: [email protected] (C.-D. Ho). 0009-2509/$ - see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2009.04.030

Basu and Sirkar, 1991; Gabelman and Hwang, 1999) are the limitations in conventional solvent extraction applications. In the past 20 years, an alternative technology involving dispersion-free membrane solvent extraction has been used to avoid many of the shortcomings associated with solvent extraction (Gabelman and Hwang, 1999; Prasad and Sirkar, 1992). The membrane is in contact with separate miscible fluids on each side, as solvent extraction is operated with a microporous membrane device. The solute is extracted from the raffinate phase through a membrane (Kiani et al., 1984; Lin and Juang, 2002) and then mass transfer occurs by diffusion across the interface of the membrane and extract phase as in traditional contact equipment. One of the most commonly used hollow fiber membrane contactors was designed in a shell-tube configuration with the shell side fluid parallel to the lumen side fluid, which was either in a concurrent or countercurrent pattern. Experimental runs on shell side mass transfer performance in parallel flow hollow fiber membrane modules have been studied widely, and reviewed recently by Lipnizki and Field (2001). Most of the approaches described the shell side mass transfer performance based on experimental work (Yang and Cussler, 1986; Costello et al., 1993; Gawronski and Wrzensinska, 2000). However, the results from these studies varied significantly. This might be caused by the irregularity of the fiber spacing, the fiber diameters, fiber movement during operation, the influence of the module wall, and the inlet and outlet effects. Theoretical treatment on the shell side mass transfer was based on the assumption of an

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ordered fiber arrangement. Miyatake and Iwashita (1991) developed a mathematical model for heat transfer in a parallel flow heat exchanger with different boundary conditions. Furthermore, the fiber distribution and flow distribution in randomly packed fiber bundles with a fluid flowing axially between the hollow fibers were investigated by many researchers (Chen and Hlavacek, 1994; Roger and Long, 1997). The analytical solution and the experimental runs for shell side mass transfer with fluid flowing axially between fibers were studied by Zheng et al. (2003). Two-dimensional dimensionless concentration profiles of the hollow-fiber membrane extractor modules with concurrent- and countercurrent-flow are investigated theoretically and experimentally in this work. The theoretical model is simplified by using the Happel's free surface model to simulate the shell side boundary conditions of hollow fiber module. The two partial differential equations for solute concentration distributions and the outlet concentrations were obtained by using the orthogonal technique and method of separation of variables (Zheng et al., 2003; Nunge and Gill, 1965, 1966; Ho et al., 1998, 2002; Ho and Guo, 2005). The primary aims in this study are to develop the two-dimensional mathematical formulation of hollow fiber membrane extractor, to obtain the analytical solution by using orthogonal expansion method, and to find the dimensionless average outlet concentration for both streams in the hollow-fiber membrane extractor. The theoretical predictions are presented graphically with the packing density (), mass-transfer Gz number (volumetric flow rate), and flow pattern as parameters. The experiments of the extraction of Cu2+ by using D2EHPA with polyvinylidene fluoride (PVDF) hollow fibers as the hydrophobic barriers are also set up in this study. The theoretical results of extraction rate, extraction efficiency and mass transfer efficiency for concurrent- and countercurrent-flow devices were compared with the experimental data to confirm the accuracy of the two-dimensional theoretical model. 2. Theory A cell model was developed to describe the shell side mass transport. To develop the model, the hollow fiber modules were divided into small cells with one fiber in each cell. The free surface appeared in the imaginary outer boundary of the cell (Happel, 1959). The module was assumed to be uniformly packed and the velocity profile across the module radius direction was ignored. Thus, the shell side mass transfer problem was the only issue in the cell. The mathematical formulations of the transport phenomena for this cell belong to the conjugated Graetz problem category and can be solved analytically using the eigenfunction expansion technique with the orthogonality conditions. To solve the mass transfer problem in each cell, the following assumptions were used: (1) steady state and isothermal condition; (2) negligible axial diffusion, entrance length and end effects; (3) using Happel's surface model (Happel, 1959) to characterize the velocity profile in the cell; (4) the physical properties of the fluid were constant; (5) the applicability of thermodynamic equilibrium; (6) purely fully developed laminar flow in each cell, (7) the complex forming reaction occurs at the aqueous-membrane interface; (8) the chemical reaction is very fast and achieves equilibrium immediately which means that the mass transfer rate of copper–D2EHPA complex is controlled by ordinary diffusion not reaction rate; (9) the diffusional resistance through the membrane is small and therefore neglected (Yun et al., 1993). 2.1. Countercurrent-flow device Experimental studies on shell side mass transfer performance in a parallel flow hollow fiber membrane module have been reported widely, as reviewed recently by Zheng et al. (2003). The shell side

ro

rf

Fiber

Free surface

Fig. 1. A scheme for the free surface model.

mass transfer performance was based on the empirical work (Yang and Cussler, 1986; Costello et al., 1993; Gawronski and Wrzensinska, 2000). However, the results from these studies varied because of the different design parameters. According to Happel's free surface model (Happel, 1959; Zheng et al., 2003), the hollow fiber module can be reduced to a circular-tube module, as indicated in Fig. 1. From the continuity and Navier–Stokes' equation, one obtains 

−



 * *v *p r i =− i, r *r *r *z

i = a, b

(1)

and the associated boundary conditions on the subchannel a and b are as follows: va = finite

r=0 r = rf r = rf

(2a)

va = 0 vb = 0 vb = finite

r = rf

(2b)

where  is defined as  =ro /rf and rf is the free surface radius defined as: −0.5

rf = 

ro

(3)

in which ro is the fiber outside radius and  is the packing density of the hollow fiber module. The velocity profiles in tube side and shell side, respectively, are ⎛



va (r) = 2fa ⎝1 −

vb (r) =

r rf

2 ⎞ ⎠

2fb (rf2 − (rf )2 )[(rf )2 − r2 + 2rf2 (ln(r/ rf ))] 4(rf )2 rf2 − (rf )4 − 3rf4 + 4rf4 ln(rf /(rf ))

(4)

(5)

The dimensionless velocity distributions and mass balance equations for each subchannel can be written in terms of the dimensionless variables as   2  va (a ) = 2fa 1 − a (6)



vb (b ) = −

2fb (1 − 2 )[2 − 2b + 2(ln(b / ))] 42 − 4 − 3 + 4 ln(1/ )

(7)

J.-J. Guo et al. / Chemical Engineering Science 64 (2009) 3455 -- 3465

and

va (a )rf2 * ( , ) 1 a a = Da L a * vb (b )rf2 * ( , 0 1 b b = Db L b *



* * ( , ) a a a *a *a





b (b , ) =



* * ( , ) b b b *b *b

Qa /Ntube

(rf )

b =

2

,

fb =

Qb /Ntube

rf2

− (rf )

Cb Kex {Ca,i [(HR)2 ]

z L

= , r b = rf

Gzb =

4fb rf2 Db L

2

,

}/[H+ ]

2

, 2

a =

,

a =

Gza =

r rf

4fa rf2 Da L

(8)  (9)

Ca , Ca,i

The boundary conditions required for solving Eqs. (8) and (9) are

*a (0, ) =0 *a

(11)

*b (1, ) =0 *b

(12) [H+ ]2 b (, ) Kex [(HR)2 ]2

a (a , 0) = ai

(15)

b (b , 1) = bi

(16)

 is the reduced where is the porosity of the membrane and Kex equilibrium constant, such as:

(17)

in which the equilibrium constant Kex in the extraction of Cu2+ from sulfate solutions with D2EHPA dissolved in kerosene can be expressed as follows: (18)

where the over-bar refers to the organic phase, and (HR)2 represents the dimeric form of D2EHPA. The equilibrium constant is Kex = 1.2 × 10−4 at T = 298 K (Juang and Huang, 1999) and is expressed by: (19)

From Eq. (19), one can obtain the CuR2 (HR)2 concentration between the aqueous phase and the membrane surface as: [CuR2 (HR)2 ] = Kex {[Cu2+ ][(HR)2 ]2 }/[H+ ]2

a

 ( ) Fb,m b

b

+

+

va (a )rf2 m Da L vb (b )rf2 m Db L

Fa,m (a ) = 0

(24)

Fb,m (b ) = 0

(25)

(26)

 Fb,m (1) = 0

(27)

 (1) = −Sa,m Fa,m

 (1) = −Sa,m Fa,m

(20)

rf Dc

Da

 Sa,m Fa,m () − Kex

Db [H+ ]2 2Da Kex [(HR)2 ]2

[H+ ]2 Sb,m Fb,m ()

Kex [(HR)2 ]2

 Sb,m Fb,m (1)

(28)

(29)

Without loss of generality, we may assume that the eigenfunctions Fa,m (a ) and Fb,m (b ) are polynomials and expressed in the following forms: ∞ 

dmn na ,

dm,0 = 1 (selected), dm,1 = 0

(30)

emn nb ,

em,0 = 1 (selected), em,1 = 0

(31)

n=0

(13)

(14)

Kex = [CuR2 (HR)2 ][H+ ]2 /{[Cu2+ ][(HR)2 ]2 }

 ( ) Fa,m a

 Fa,m (0) = 0

Fa,m (a ) =

*a (, ) *b (, ) Db [H+ ]2 = 2 * * 2Da Kex [(HR)2 ]

Cu2+ + 2(HR)2 ⇔ CuR2 (HR)2 + 2H+ ; Kex

(23)

and the boundary conditions in Eqs. (11)–(14) can be rewritten as ,

(10)

 = Kex [(HR)2 ]2 /[H+ ]2 Kex

Gm () = e−m 

 (b ) + Fb,m

when 0 < r < rf ,

*a (, ) rf Dc  = Kex a (, ) −

Da *

(22)

Substitution of Eqs. (21) and (22) into Eqs. (8) and (9) leads to

 ( a ) + Fa,m

when rf < r < rf ,



Sb,m Fb,m (b )Gm ()

m=0

where fa =

∞ 

3457

Fb,m (b ) =

∞  n=0

The coefficients dmn and emn can be expressed in terms of eigenvalue m as dm2 =

−fa m 2Da L

dm3 = 0 dmn =

  −2fa d m dmn−2 − mn−4 Da L[n(n − 1) + n] 2

n = 4, 5, 6, . . .

(32)

and em2 =

  −fb m 3T 1− em0 2 Db L 2

em3 =

−4fb m Tem0 9 Db L −fb

m

Db L emn = n(n−1)+n−1 n = 4, 5, 6, . . . where ⎛

 1−

3T 2



   T emn−2 +2Temn−3 − 1+ emn−4 , 2 (33)

⎞

⎜1 −  ⎟ T =⎝ ⎠ 1 ln 2

The analytical solution to this conjugated problem may be obtained using an orthogonal expansion technique in power series. The variable separation leads to the following form

a (a , ) =

∞  m=0

Sa,m Fa,m (a )Gm ()

(21)



Therefore, it is easy to solve all eigenvalues from Eqs. (28) and (29). The eigenfunctions associated with the corresponding eigenvalues are also well defined by Eqs. (30) and (31).

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J.-J. Guo et al. / Chemical Engineering Science 64 (2009) 3455 -- 3465

When m
n , the orthogonality condition in this countercurrentflow liquid-liquid membrane extractor system is Db [H+ ]2



1



R2f vb (b )

The mass transfer equations for two subchannels a and b in concurrent-flow device may also be obtained in the same expressions found in Eqs. (8) and (9) except that the velocity distribution of Eq. (4) and boundary condition of Eq. (16) are replaced by

Sb,m Sb,n Fb,n Fb,m db LDb 2Kex [(HR)2 ]2    R2 v ( ) f a a  + Da Kex Sa,m Sa,n Fa,n Fa,m da = 0 LDa 0

(34) vb (b ) = −

By following the same calculation procedure performed in the previous work (Nunge and Gill, 1965, 1966), the coefficients for Sa,m and Sb,m are as follows. When m = 0 4Gzb em Db bi Da 

  ∞  va (rf )2 = Sa,q · e−q Fa,q da LDa 0 q=0

  D [ F  () + r F ()]  1 v r 2 Kex b f a f a,q a,q − Fb,q db rf Db Fb,q (1) LDb    4Gzb Db  = Sa,0 4Gza +  D Kex a

4Gza ai −

 rf Da Kex

×

q=0

= rf Db Sb,m Fb,m (1) +

 rf Da Kex

e−m

 −

4Gzb Da bi Db

  4Gzb Da = Sa,0 4Gza +  Kex Db   D [ F  () + RF ()]F  (1)  ∞ −  Kex  a a,q a,q e q b,q  + Sa,q · Fa,q () + rf Da Fb,q (1) q q=1

(35)

(42)

 (1) −Db Fb,m





 (1) *Fb,m

*m [H+ ]2

Kex [(HR)2 ]2

 − Fb,m (1)



*Fb,m (1) *m

 (1) Db Fb,m  2Da Fa,m ()

×



Kex [(HR)2 ]2

  (rf )2 va LDa 

0

+ rf Db Sb,m

[H+ ]2



1

m



 [Fa,m ()]



 (1) Db Fb,m  () 2Da Fa,m

  rf Da Kex

×

 (1) −Db Fb,m

 2

 ( ) 2Da Fa,m

*Fa,m ()

Kex [(HR)2 ]2

− Fb,m (1)

*m

Kex [(HR)2 ]

  0

Fa,q (a ) ∞ 

Sb,q

(37)

(38)

=

Sb,q e−q

q=0

×

  0

 ( ) 2Da Fa,m

2

(rf )2 va

e−q

*Fb,m (1) *m

(43)

(39)

+ rf Db

1



Sa,q

q=0

Gq (q ) Gm (m )

Fa,m (a ) da

1

e−m 

∞ 

Fb,q (b ) 

2]

2

⎪ ⎪ ⎪ ⎩

rf2 vb

LDb

−Db F  b,m(1)  () 2Da Fa,m

Fb,m (b ) db

2

 rf Da Kex

e−m





⎧  ⎪ [H+ ]2 ⎪ ⎪ ⎨ Kex [(HR)

Fa,q (a )



 (1) −Db Fb,m

LDa

q=0

∞ 



(rf )2 va

LDa

m=1

∞ 1  −1 S F  (1)e−m  4Gzb m b,m b,m



[H+ ]2

[H+ ]2

+ rf Db

The dimensionless radially averaged concentrations for both streams are

m=1

2 (b ) db Fb,m

LDb



When m
0 and m
q

q=0

∞ 1  −1  a () = Sa,0 + Sa,m Fa,m ()e−m  4Gza m

Sa,m

 ( ) *Fa,m  − Fa,m () *m *m  (1) *Fb,m 

× Fa,m ()





rf2 vb

+ rf Db Sb,m Fb,m (1)

Kex [(HR)2 ]2

 ∞  (R)2 v  a Sa,q × Fa,m (a )Fa,q (a ) da LDa 0 q=0     (1) Db Fb,m 1 [H+ ]2  = rf Da Kex −  () 2Da Fa,m e−m Kex [(HR)2 ]2

  2 (rf ) va × Fa,m (a ) da LDa 0 2  1 ∞ rf vb  e−q + rf Db Sb,q Fb,q (b ) Fb,m (b ) db −  m LDb e k

b () = Sb,0 −

 (1) Db Fb,m  2Da Fa,m ()

2 (a ) da Fa,m



When m
0 and m = q 

1

 = rf Da Kex (Sb,m )

(36)

 rf Da Kex −



[H+ ]2

 rf Da Kex

Kex [(HR)2 ]

  (rf )2 va (a ) Sa,q Fa,m (a )Fa,q (a ) da LDa 0

(41)

The results of all expressions coefficients can be obtained as follows. When m = 0

 () 2Da Fa,m

2

∞ 

(40)

42 − 4 − 3 + 4 ln(1/ )

When m
0 and m = q



[H+ ]2

2fb (1 − 2 )[2 − 2b + 2(ln(b / ))]

b (b , 0) = bi

4Gza ai +

When m
0 and m = q 

2.2. Concurrent-flow device

Fb,q (b )

Fa,m (a ) da

rf2 vb LDb

Fb,m (b ) db

⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭

(44)

J.-J. Guo et al. / Chemical Engineering Science 64 (2009) 3455 -- 3465

The dimensionless average concentrations for both streams are

a () = Sa,0 + b () = Sb,0 +

1 4Gza

∞ 

−1

3.2. Apparatus and procedure

 Sa,m Fa,m ()e−m 

(45)

∞ 1  −1 S F  (1)e−m  4Gzb m b,m b,m

(46)

m=1

m

m=1

2.3. Mass transfer efficiency in the liquid–liquid membrane extraction system The local Sherwood number in the raffinate phase is defined by

Sha =

ka Deq,a Da

∞ 

2 =

∞  m=1

m=1

 (0)e−m  Sam Fam

 Sam Fam (0) −



1 F  (0) 4Gza m am

(47) e−m 

where ka is the local mass transfer coefficient for raffinate phase, and Deq,a is the equivalent diameter of subchannel a ka =

Da *a (0, )/ *a rf a (0, ) − a ()

(48)

So, the average Sherwood number can be obtained as Sha =

 R 0

Sha d

(49)

The total extraction rate M and extraction efficiency  are defined by the total amount of the solute transfer from the raffinate phase to the extraction phase and the ratio of the solute transfer from the raffinate phase to extraction phase to total solute in the initial raffinate phase, respectively, which can be determined using Eqs. (50) and (51) as follows M = Qa (Ca0 − Cae ) = Qb (Cbe − Cb0 )

(50)

Qa (Ca0 − Cae ) Q (C − Cb0 ) = b be Qa Ca0 Qa Ca0

(51)

=

Some equipment parameters and physical properties of the di(2ethylhexyl)phosphoric acid (D2EHPA), Cu(II) solution and the microporous PVDF membrane are given as follows (Juang and Huang, 1999): Da = 3.25 × 10−8 m2 /s;

= 0.67.

3459

Db = Dc = 2.01 × 10−11 m2 /s;

3. Experimental runs 3.1. Chemicals and materials The following chemicals were used in this study: (1) D2EHPA (Merck, 98.5% purity); (2) kerosene (Union Chem. Co., Taiwan) which were used as received. The aqueous solution was prepared by dissolving CuSO4 in deionized water (Millipore Milli-Q) in which the pH was adjusted by adding 0.1 mol/dm3 H2 SO4 or Na2 SO4 . The microporous PVDF hollow fiber (Microza, PALL) with an average pore size of 2 m, porosity of 67% and thickness 1.1 × 10−4 m is a hydrophobic membrane, and the organic solution wets the membrane. The packing density of the hollow fibers is 0.323 and the number of fibers is 15. A schematic diagram of the experimental setup is shown in Fig. 2. Experiments were carried out with a hollow-fiber module as a permeable barrier to extract Cu(II) from aqueous solution (phase a) to organic solution (phase b) by D2EHPA. The temperature of the entire tank was controlled at 298 K.

The aqueous phase consisted of 500 mol/m3 (Na, H, Cu)SO4 , which means that the total sulfate concentration was kept constant. The organic phase was prepared by diluting D2EHPA with kerosene. The initial concentration of Cu2+ in the aqueous phase is 1.0, 2.0 and 3.0 mol/m3 and D2EHPA monomer in the organic phase is 25 mol/m3 . The volumetric flow rate in the aqueous phase is 6.4, 6.9 and 8.2 cm3 /s, respectively, and the volumetric flow rate in the organic phase is 1.0–15 cm3 /s. The valves and the pressure gauges were used to control the flow rates and to ensure that a positive pressure of 18–40 kPa was maintained on the aqueous side of the modules. In this scenario, the aqueous-membrane interface can be stabilized (Prasad and Sirkar, 1990). In a commercial scale process, the pressure drop would actually cause one end of the contactor to approach either zero transmembrane pressure difference or a difference that exceeds the breakthrough pressure. This is obviously not a problem for the short modules used in this work. Under specific operating conditions, outlet stream samples were analyzed at 5 min intervals until steady state was reached, as indicated by no change in the Cu(II) concentration as monitored by a UV detector (Unicam UV 300 UV–visible spectrometer, Unicam UV/VLS, UK) over 40 min. The time to reach steady state was about 30 min. Shorter runs were used at higher flow-rates and duplicate runs were made under identical conditions to ensure reproducibility. A comparison of the experimental runs and the mathematical models is made here. 4. Results and discussions The calculation procedure will be described briefly as follows. First, the eigenvalues in the membrane extractor were solved from Eqs. (28) and (29). Next, the expansion coefficients were estimated from Eqs. (35)–(37) and (42)–(44) for countercurrent- and concurrent-flow device, respectively. Finally, the dimensionless average concentration, mass transfer efficiency, extractive rate and extractive efficiency were calculated from Eqs. (45), (46), (49), (50) and (51), respectively. 4.1. Extended power series and Taylor series convergence Table 1 shows the calculation results of eigenvalues in the concurrent and countercurrent-flow devices and their associated expansion coefficients. The dimensionless outlet concentration in the aqueous phase calculated for [Cu2+ ] = 1.0 mol/m3 , [H+ ] = 0.1 mol/m3 , [(HR)2 ] = 25 mol/m3 , Gza = 50 and Gzb = 50 are also shown in Table 1. Due to the convergence, only six eigenvalues must be considered during the calculation procedure. These eigenfunctions in Eqs. (29) and (30) are expanded in terms of an extended power series. The accuracy of the solution was examined with the results for the extended power series and Taylor series, respectively. A truncated series of n=210 was selected and used for the calculation procedure. 4.2. Dimensionless outlet concentration, extractive rate and mass-transfer efficiency When the solvent is used in the extraction phase, the dimensionless concentrations of the aqueous phase and the organic phase inlet streams are 1 and 0, respectively. Fig. 3 shows the average Cu2+ dimensionless concentration distribution for countercurrent-flow in whole membrane contactor. The theoretical equations are expressed by Eqs. (38) and (39). As indicated in Fig. 3, the average dimensionless concentration distribution in extract phase, b , increases by increasing the packing density . The dimensionless outlet concentration in the extraction phase for concurrent- and countercurrent-flow

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J.-J. Guo et al. / Chemical Engineering Science 64 (2009) 3455 -- 3465

Raffinate outlet Extraction outlet

Raffinate inlet

Flow meter

Extraction inlet

Pump Extraction phase

Raffinate phase

Raffinate inlet Extraction outlet

Raffinate outlet

Flow meter

Extraction inlet

Pump Extraction phase

Raffinate phase

Fig. 2. Schematic diagram of the experimental setup (a) countercurrent-flow device and (b) concurrent-flow device.

Table 1 The dimensionless outlet concentration in the raffinate phase for the eigenvalue number and expansion coefficients with analytical solution for both concurrent- and countercurrent-flow devices.



2

3

4

5

Concurrent-flow device 0.3 5 0.000 −0.011 6 0.000 −0.011 7 0.000 −0.011

−1.755 −1.755 −1.755

−3.679 −3.679 −3.679

−7.889 −7.889 −7.889

– −8.645 −8.645

Countercurrent-flow device 5 0.000 −0.09 6 0.000 −0.09 7 0.000 −0.09

0.035 0.035 0.035

−1.641 −1.641 −1.641

1.474 1.474 1.474

– −2.074 −2.074

m

0

1

Sa,0

Sa,1

Sa,2

Sa,3

Sa,4

Sa,5

Sa,6

¯ ae

−10.905

0.089 0.082 0.082

7.88×10−4 7.68×10−4 7.98×10−4

−3.73×10−4 −2.13×10−4 −2.11×10−4

8.77×10−5 4.67×10−5 4.62×10−5

−8.56×10−5 −3.86×10−5 −3.84×10−5

– 4.35×10−6 4.33×10−6

– – 8.12×10−8

0.8301 0.8809 0.8809

−3.577

0.067 0.065 0.065

7.62×10−4 6.62×10−4 6.48×10−4

−1.12×10−4 −1.32×10−4 −1.33×10−4

8.57×10−5 4.56×10−5 4.55×10−5

−4.88×10−5 −3.76×10−5 −3.74×10−5

– 4.42××10−6 4.44×10−6

– – 4.32×10−8

0.7918 0.8028 0.8028

6 –

n = 210, [Cu2+ ] = 1.0 mol/m3 , [H+ ] = 0.1 mol/m3 , (HR)2 ] = 25 mol/m3 , Gza = 50 and Gzb = 50.

devices are illustrated in Fig. 4. It shows that be increases with increasing mass-transfer Graetz number in the raffinate phase (Gza ) due to the mass-transfer coefficient of the fluid increases with increasing the fluid velocity. Fig. 4 also indicates that the dimensionless outlet concentration in the extraction phase in the countercurrentflow device is larger than that in the concurrent-flow device and the highest be is obtained as packing density  = 0.3. This may be attributed to the channeling flow and dead zone effect on the solute transport become more remarkable with increasing the packing density resulting in the mass transfer efficiency slightly decrease in the moderate packing density (Wu and Chen, 2000; Wang et al., 2003).

Figs. 5 and 6 present the graphical representations of the theoretical extraction rate (M). The result shows that the extraction rate M increases with increasing the packing density, as shown in Fig. 5. It was also found that the extraction rate in the countercurrent-flow device was larger than that in the concurrent-flow device. In Fig. 6, the extraction rate M of the countercurrent-flow device increases with increasing mass-transfer Graetz number and with decreasing the [H+ ] concentration. The extraction efficiency () for concurrentand countercurrent-flow devices is indicated in Fig. 7. Fig. 7 shows that the extraction efficiency increases with increasing the masstransfer Graetz number and [Cu2+ ] concentration. Fig. 8 shows the

J.-J. Guo et al. / Chemical Engineering Science 64 (2009) 3455 -- 3465

3461

1 0.9 0.8

a or b

0.7

[H+] = 0.1 mol/m3

a

[Cu2+] = 1.0 mol/m3

0.6

[HR2] = 25 mol/m3

0.5 0.4

b

 = 0.3  = 0.2  = 0.1 Gza = 50 Gzb = 50

0.3 0.2 0.1 0 0

0.1

0.2

0.3

0.4

0.5 ξ

0.6

0.7

0.8

0.9

1

Fig. 3. Average dimensionless concentration distribution for countercurrent-flow device in whole membrane extractor (Gza = 50 and Gzb = 50).

1.00

12 Countercurrent-flow Concurrent-flow

Countercurrent-flow Concurrent-flow

0.90  = 0.4

 = 0.3

0.80

10

0.70 8

M×107

be

0.60

0.50

6

0.40

 = 0.3  = 0.2

 = 0.2

0.30

4

 = 0.1  = 0.1

0.20

Gzb = 50

2

Gzb = 50

[(HR)2] = 25 mol/m3

0.10

[H+] = 0.1 mol/m3

[H+] = 0.1 mol/m3

[Cu2+] = 1.0 mol/m3 [HR2] = 25 mol/m3

0.00 0

100

200

300 Gza

400

500

600

Fig. 4. Dimensionless outlet concentration of Cu2+ in the extraction phase for concurrent-and countercurrent-flow device with packing density ratio and the H+ concentration as parameters ([[H+ ] = 0.1 mol/m3 , Cu2+ ] = 1.0 mol/m3 , Gzb = 50 and [(HR)2 ] = 25 mol/m3 ).

0 0

100

200

300 Gza

400

500

600

Fig. 5. Extraction rate of Cu2+ for concurrent-and countercurrent-flow device with packing density and the H+ concentration as parameters ([Cu2+ ] = 1.0 mol/m3 , Gza = 50 and [(HR)2 ] = 25 mol/m3 ).

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0.2

25 Countercurrent-flow

+

Countercurrent-flow

3

[H ] = 0.1 mol/m

Concurrent-flow

[HR2] = 25 mol/m3 [HR2] = 25 mol/m3 Gzb = 50

20 [H+] = 0.2 mol/m3

0.15

 = 0.3



M×107

15

0.1 [Cu2+] = 3.0 mol/m3

10

0.05

[Cu2+] = 2.0 mol/m3

5

[Cu2+] = 1.0 mol/m3

[Cu2+] = 3.0 mol/m3

Gzb = 50

[Cu2+] = 2.0 mol/m3

 = 0.3

[Cu2+] = 1.0 mol/m3

0

0 0

100

200

300 Gza

400

500

Fig. 6. Extraction rate of Cu2+ for countercurrent-flow device with H+ concentration and the Cu2+ concentration as parameters ([(HR)2 ]=25 mol/m3 , Gzb =50 and  =0.3).

theoretical results of the average Sherwood number in the raffinate phase (Sha ) versus mass-transfer Graetz number variations in the raffinate phase (Gza ). The larger the mass-transfer Graetz number is operated, the larger Sha is obtained, regardless of the concurrent and countercurrent-flow devices, as demonstrated in Fig. 8. It was also found that Sha increases as the packing density  increases. Moreover, the Sha in the countercurrent-flow device is larger than that in the concurrent-flow device, as shown in Fig. 8. A comparison of the presented theoretical average tube-side Sherwood number and the Sherwood number correlation, reported by Crowder and Cussler (1998) and Soldenhoff et al. (2005), is also illustrated in Fig. 8. The Sherwood number correlation is used to determine the laminar flow in the lumen side of hollow fiber module and is a function of the Graetz number as 

2ro Shc = 1.62 Sc · Re · L

1/3

= 1.62(Gza )

1/3

0

600

(52)

where Sc is the Schmidt number and Re is the Reynolds number. The calculated values of the Sherwood number correlation are relatively lower than those of the present theoretical tube-side Sherwood number. It may be attributed to that the rapidly surface reaction of the copper–D2EHPA complex enhances the mass transfer rate resulting in the higher average Sherwood numbers obtained the two-dimensional model of the present study. The theoretical results of the outlet Cu2+ concentration of the raffinate phase are also shown

100

200

300 Gza

400

500

600

Fig. 7. Extraction efficiency for concurrent-and countercurrent-flow device with Cu2+ concentration as parameters ([(HR)2 ] = 25 mol/m3 , Gzb = 50 and  = 0.3).

in Fig. 9 for comparison with the experimental data. Fig. 9 shows that the theoretical predictions are agree with the experimental results. 4.3. Experimental analysis As referred to Moffat (1988), the precision index of the individual experimental measurements, Cˆ aei , is determined directly from the data set, as follows: SCaei

⎡ ⎤1/2 N ˆ aei − Caei )  ( C ⎦ =⎣ N−1

(53)

i=1

where N is the number of experimental measurements. The resulting uncertainty was associated with the mean value, Caei : SC

aei

SC = √aei N

(54)

The precision index was determined for concurrent and countercurrent-flow device with the different operation parameters. The mean precision index of the experimental measurements in Fig. 9 is 4.27 × 10−5 ⱕ SC ⱕ 2.47 × 10−3 . aei The accuracy of the theoretical predictions may be calculated using the definition E=

N 1  |Caei − Cˆ aei | N Cˆ aei i=1

(55)

J.-J. Guo et al. / Chemical Engineering Science 64 (2009) 3455 -- 3465

3463

1

15 Gzb = 50

Countercurrent-flow Concurrent-flow Crowder and Clussler (1998)

Theo.

Exp. Gza = 52.78 Gza = 44.68 Gza = 40.80 Con. Counter.

[H+] = 0.1 mol/m3

 = 0.323 [Cu2+] = 1.0 mol/m3

[HR2] = 25 mol/m3

0.9

Sha

Cae

10

0.8

5  = 0.3  = 0.2

Gza = 52.78 Gza = 44.68

 = 0.1

Gza = 40.80

0.7

0 0

100

200

300 Gza

400

500

600

Fig. 8. The average Sherwood number in the raffinate phase for concurrent- and countercurrent-flow device with packing density as parameter ([Cu2+ ] = 1.0 mol/m3 , Gzb = 50, [H+ ] = 0.1 mol/m3 and [(HR)2 ] = 25 mol/m3 ).

where Caei denotes the theoretical prediction of Cae and Cˆ aei is the extraction rate experimental data. The accuracy of the theoretical predictions was calculated and shown in Table 2. The accuracy of the theoretical predictions is determined by Eq. (55) for concurrentand countercurrent-flow. The results show that the accuracy of the theoretical predictions for concurrent- and countercurrent-flow are 5.87 × 10−2 ⱕ E1 ⱕ 6.69 × 10−2 and 2.46 × 10−2 ⱕ E1 ⱕ 3.48 × 10−2 , respectively, as Gza = 40.8. 5. Conclusions Mass transfer phenomena of the laminar concurrent and countercurrent-flow hollow-fiber membrane extractor modules were analytically investigated in this study. The velocity profile and mass transfer phenomena in the shell side were described using Happel's free surface model with assuming the uniform arrangement of hollow fibers in this analytical model. The two-dimensional mass transfer mathematical formulation was developed theoretically and solved analytically using the orthogonal technique with eigenfunction expanding in terms of a power series. The influences of extraction and raffinate phase flow rate, flow patterns, and packing density were discussed in this study. The results show that the packing density had significant influence on the mass transfer behavior in the membrane extraction operations, especially for countercurrent-flow devices, and the optimal operation

0

100

200 Gzb

300

400

Fig. 9. The comparison of the extraction rate estimated by the analytical model and experimental data vs. Gzb for both flow patterns ([Cu2+ ] = 1.0 mol/m3 ,  = 0.323, [H+ ] = 0.1 mol/m3 and [(HR)2 ] = 25 mol/m3 ).

value is  = 0.3. It was also seen from the theoretical predictions and experimental runs that the mass transfer efficiency increases with increasing mass-transfer Graetz number in the raffinate phase. The experiments of the extraction of Cu2+ by using D2EHPA with PVDF hollow fibers is set up in this study to confirm the accuracy of the theoretical predictions. The theoretical and the experimental runs showed that increasing the aqueous phase flow-rate, Cu2+ , (HR)2 concentration, and packing density can enhance the Cu2+ extraction rate in the liquid-liquid membrane contactor. The analytical model qualitatively agrees with the experimental results and the dimensionless outlet concentration obtained from the analytical model was in excellent agreement with the experimental data. The accuracy of the theoretical predictions were 5.87×10−2 ⱕ E1 ⱕ 6.69×10−2 and 2.46 × 10−2 ⱕ E1 ⱕ 3.48 × 10−2 for concurrent and countercurrentflow devices, respectively. The results show that the present model can be used to simulate the mass transfer phenomena of a hollow fiber extractor. The advantages and values of the present twodimensional theoretical model are that the extraction rate can be calculated directly without experimental runs, and the dimensionless concentration profile can be obtained in the entire membrane extractor. One can also expect that the present mathematical treatment can be applied to other membrane contactors.

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Table 2 The accuracy of the theoretical predictions. Gza = 40.8, ro = 2.2 × 10−3 m, R = 1.5 × 10−2 m, Ntube = 15,  = 0.323 and L = 0.31 m. E=

Flow pattern

Concurrent-flow device Countercurrent-flow device

1 N

N  i=1

|Caei −Cˆ aei | Cˆ aei

[(HR)2 ] = 25 mol/m3 and [Cu2+ ] = 1.0 mol/m3

[H+ ] = 0.1 mol/m3 and [Cu2+ ] = 1.0 mol/m3

[H+ ] = 0.1 mol/m3

[H+ ] = 0.2 mol/m3

[(HR)2 ] = 25 mol/m3

−2

−2

6.47×10 3.21×10−2

5.87×10 3.48×10−2

Notation C Con. Counter. dmn Da Db Dc emn E Fm f Gz Kex  Kex k L M N Ntube Q r ro rf R Re Sc Sm Sh Sh Shc v f Z

solute concentration, mol/m3 concurrent flow countercurrent flow coefficient in the eigenfunction Fa,m ordinary diffusion coefficient in raffinate phase, m2 /s ordinary diffusion coefficient in extraction phase, m2 /s ordinary diffusion coefficient in the membrane, m2 /s coefficient in the eigenfunction Fb,m the accuracy of the experimental results, defined by Eq. (55) eigenfunction associated with eigenvalue m average velocity, m/s mass-transfer Graetz number equilibrium constant reduced equilibrium constant local mass transfer coefficient of [Cu2+ ], m/s conduit length, m extraction rate defined by Eq. (50), mol/s the number of experimental measurements the tube number of hollow fiber volumetric flow rate of conduit, m3 /s transversal coordinate, m fiber outside radius free surface radius, m shell side diameter, m Reynolds number Schmidt number expansion coefficient associated with eigenvalue m local Sherwood number average Sherwood number Sherwood number correlation velocity distribution of fluid, m/s average velocity of fluid, m/s longitudinal coordinate, m

thickness of the membrane, m porosity of the membrane extraction efficiency, defined by Eq. (51) transversal coordinate, defined by Eq. (10) transversal coordinate, defined by Eq. (10) constant, defined as ro /rf eigenvalue longitudinal coordinate, defined by Eq. (10) density of the fluid, kg/m3 dimensionless concentration, defined by Eq. (10) packing density

Subscripts a b

6.47×10 3.21×10−2

0 e

Greek letters

 a b  m   

−2

in the raffinate phase in the extraction phase

[(HR)2 ] = 100 mol/m3 6.69×10−2 2.46×10−2

at the inlet at the outlet

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