Mass-Transfer Efficiency of Membrane Extraction in Laminar Flow

7788

Ind. Eng. Chem. Res. 2007, 46, 7788-7801

SEPARATIONS Mass-Transfer Efficiency of Membrane Extraction in Laminar Flow between Parallel-Plate Channels: Theoretical and Experimental Studies Jia J. Guo, Chii D. Ho,* and Ho M. Yeh Department of Chemical and Materials Engineering, Tamkang UniVersity, Tamsui, Taipei, Taiwan 251, Republic of China

The mass-transfer efficiency of the parallel-plate membrane extractor module with concurrent and countercurrent flows was investigated theoretically and experimentally in this study. The analytical solution is obtained using an eigen-function expansion in terms of the power series using an orthogonal expansion technique. The theoretical predictions were represented graphically with the mass-transfer Graetz number (volumetric flow rate), flow pattern, and subchannel thickness ratio (permeable-barrier locations) as parameters, compared with those obtained by numerical approximation and experimental runs. The extractive rate, the extractive efficiency, and the mass-transfer efficiency can be achieved higher for the countercurrent-flow device than those of the concurrent-flow device by setting the barrier location moving away from the centerline. The accuracy of the experimental measurements as compared to both analytical model and approximation model were calculated by 2.11 × 10-2 e E1 e 5.15 × 10-2 and 4.63 × 10-2 e E2 e 8.12 × 10-2, respectively. These operating and design parameters influences on the mass-transfer efficiency enhancement are also discussed. 1. Introduction The liquid or gaseous mixture separation techniques in equilibrium using microporous membranes have been studied in recent years.1-21 The types of microporous membranes used in these membrane extraction studies included the hydrophobic membrane,2-7 the hydrophilic membrane,8-10 and the composite membrane.4,11,12 The traditional solvent extraction processes were applied in devices such as packed columns, mixer-setters, etc., which maximize the mass-transfer rate by producing as much interfacial area as possible. The important disadvantages include the interdependence of the two fluid phases in contact, which sometimes leads to difficulties such as intimate mixing, limitations on independent-phase flow rate variations, requirement of density difference, inability to handle particulates,2-7 emulsions, foaming, unloading, and flooding for conventional solvent extraction applications. Recently, an alternative technology involving dispersion-free membrane solvent extraction can overcome these disadvantages.6,13 Using suitable membrane configurations such as a flat-plate sheet, a concentric tube, a spiral wound, or a hollow fiber, the fluid flow is contacted on the opposite side of the membrane. The membrane is in contact with two miscible fluids (phase a and phase b) at two sides, as solvent extraction is operated with a microporous membrane device. The liquid (phase c), immiscible with these two fluids, fills the membrane pores. The solute is extracted from phase a to phase c and then to phase b.14,15 The mass transfer occurs by diffusion across the interface, just as in traditional contacting equipment. The membrane solvent extraction process includes two phases, an aqueous phase and an organic phase, without the other fluid filled with the microporous membrane.16-21 Under this operation, the hydrophobic membrane is filled only with the organic solution with a distribution coefficient of Hbc ) 1. At the opposite side of the membrane, an aqueous phase which * Corresponding author. Tel.: 886-2-26266632. Fax: 886-226209887. E-mail: [email protected].

does not wet the membrane and is operated at higher pressure than that for the organic phase. However, the pressure must be lower than the pressure needed for the aqueous phase to displace the organic phase in the membrane pores. One the other hand, the hydrophilic membrane, filled only with aqueous solution, has a distribution coefficient of Hac ) 1, respectively. The mass transfer in the membrane extractors could be analogous to the heat transfer in the heat exchangers. The membrane extractor is different in design than the ordinary heat exchangers, but, in principle, the only differences are the relative directions of the two fluids. The mass-transfer efficiency in the membrane extraction through a parallel-plate module has been investigated by using numerical approximation and experimental runs,17-21 under the variant operation and design parameters. In these studies, the solute concentration distributions in extraction and raffinate phases were derived by mass balance. The numerical approximations of the concentration distributions were determined by solving the first-order ordinary differential equations. Therefore, instead of ordinary differential equations, two partial differential equations for solute concentration distributions and the outlet concentrations were obtained by solving these equations using orthogonal technique and variable separated methods.22-27 The purposes of this study are to build up the mathematical formulation, to develop the method of the orthogonal expansion technique in obtaining the analytical solution, and to find the dimensionless average outlet concentrations of both streams in the flat-plate membrane extractor. The two-dimensional dimensionless concentration profile and masstransfer efficiency of the parallel-plate membrane extractor module with concurrent and countercurrent flows were investigated theoretically and experimentally in this work. The theoretical predictions have been presented graphically with the subchannel thickness ratio, the mass-transfer Gz number (volumetric flow rate), and the initial solute concentrations in the extraction and raffinate phases as parameters. The extractive

10.1021/ie0611610 CCC: $37.00 © 2007 American Chemical Society Published on Web 10/04/2007

Ind. Eng. Chem. Res., Vol. 46, No. 23, 2007 7789

∂2ψb(ηb,ξ) ∂ηb2

)

(

)

Wb2Vb(ηb) ∂ψb(ηb,ξ) LDb ∂ξ

(4)

where

Vja )

Figure 1. Schematic diagram of the membrane extractor.

Qa Qb xa xb z , Vjb ) , η a ) , ηb ) , ξ ) , WaB W bB Wa Wb L Ca Cb Wa , ψb ) , ∆ ) , Wb ) (1 - ∆)W, ψa ) Ca0 Ca0 W Qa W QbW , Gzb ) (5) W ) Wa + Wb, Gza ) DaBL DbBL

The boundary conditions required for solving eqs 3 and 4 are rates, the extractive efficiencies, and the mass-transfer efficiencies for concurrent- and countercurrent-flow devices were compared with the approximation solutions in the previous work17,18 and the experimental runs in this study. Experiments are carried with a membrane sheet that was made by microporous cellulose as a hydrophilic barrier to extract acetic acid from phase a by methyl isobutyl ketone (MIBK) to phase b. 2. Theory Consider membrane extraction in a parallel conduit of length L, thickness W, and width B, (B .W) with a hydrophilic microporous membrane of thickness δ inserted between two parallel plates, as shown in Figure 1. The thicknesses of the two subchannels a and b are Wa and Wb, respectively. Extractionand raffinate-phase flows feed in different subchannels when the operation proceeds. The extraction-phase pressure is slightly higher than that of the raffinate phase and lower than the membrane breakthrough pressure. In this process, the total resistance to mass transport consists of the resistances on both sides of the membrane and the resistance within the membrane itself. The mathematical formulations of the transport phenomena for such devices belong to the category of conjugated Graetz problems and can be solved analytically using the eigen-function expansion technique with the orthogonality conditions.22-27 These mathematical formulations are described in this section. The assumptions are as follows: (a) isothermal operation and constant fluid physical properties; (b) purely fully developed laminar flow in each subchannel; (c) negligible axial diffusion as well as entrance length and end effects; (d) the applicability of thermodynamic equilibrium; (e) the membrane being completely raffinate-phase filled; and (f) steady-state operation. 2.1. Countercurrent-Flow Device. The velocity distributions and the dimensionless equations of mass transfer for each subchannel may be written in terms of the dimensionless variables as

-

∂ψa(0,ξ) )0 ∂ηa

(6)

∂ψb(0,ξ) )0 ∂ηb

(7)

( ) ( )( )

∂ψa(1,ξ) WaDc ) [Hacψa(1,ξ) - Hbcψb(1,ξ)] ∂ηa δDa

(8)

∂ψa(1,ξ) Wa Db ∂ψb(1,ξ) ) ∂ηa Wb Da ∂ηb

(9)

-

ψa(ηa,0) ) ψa0

(10)

ψb(ηb,1) ) ψb0

(11)

where is the porosity of membrane and Hac and Hbc are distribution coefficients of solute between two different phases, such as

Hac )

interface solute concentration in phase c interface solute concentration in phase a

Hbc )

interface solute concentration in phase c interface solute concentration in phase b

and

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

Va(ηa) ) Vja(6ηa - 6ηa2)

(1)

ψa(ηa,ξ) )

∑ Sa,m Fa,m(ηa)Gm(ξ)

(12)

m)0

Vb(ηb) ) -Vjb(6ηb - 6ηb2)

(2)

∞

ψb(ηb,ξ) )

and

∂2ψa(ηa,ξ) ∂ηa2

(

)

Wa2Va(ηa) ∂ψa(ηa,ξ) ) LDa ∂ξ

∑ Sb,m Fb,m(ηb)Gm(ξ) m)0

(13)

Substitution of eqs 12 and 13 into eqs 3 and 4 leads to

(3)

Gm(ξ) ) e-λmξ

(14)

7790

Ind. Eng. Chem. Res., Vol. 46, No. 23, 2007

[ [

] ]

λmWa2Va(ηa) Fa,m(ηa) ) 0 LDa

(15)

λmWb2Vb(ηb) Fb,m(ηb) ) 0 F ′′b,m(ηb) + LDb

(16)

F ′′a,m(ηa) +

and the boundary conditions in eqs 6-9 can be rewritten as

-Sa,mF ′a,m(1) )

F ′a,m(0) ) 0

(17)

F′b,m(0) ) 0

(18)

( )

WaDc [HacSa,mFa,m(1) - HbcSb,mFb,m(1)] δDa

-Sa,m F ′a,m(1) )

( )( )

Wa D b S F′ (1) Wb Da b,m b,m

(19)

(20)

Without loss of generality, we may assume the eigen-functions Fa,m(ηa) and Fb,m(ηb) are polynomials and then express them in the following forms: ∞

Fa,m(ηa) )

dmnηan, ∑ n)0

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

(21)

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

(22)

∞

Fb,m(ηb) )

emnηbn, ∑ n)0

All the coefficients dmn and emn can be expressed in terms of eigenvalue λm as

dm2 ) 0 dm3 ) -4Gzaλm dmn )

-24Gzaλm - dm(n - 4)] [d n(n - 1) m(n - 3)

n ) 4, 5, 6, ...

(23)

and

em2 ) 0 em3 ) 4Gzb λm emn )

24Gzb λm - em(n - 4)] [e n(n - 1) m(n - 3)

n ) 4, 5, 6, ...

(24)

Therefore, it is easy to solve all eigen-values from eqs 19 and 20. The eigen-functions associated with the corresponding eigenvalues are also well-defined by eqs 21 and 22. When λm * λn, the orthogonality condition in this countercurrent-flow liquid-liquid membrane extractor system is

WaDbHbc

[

∫01

]

Wb2Vb(ηb) Sb,mSb,n Fb,n Fb,m dηb + DaHacWb LDb

∫01

[

]

Wa2Va(ηa) Sa,mSa,nFa,nFa,m dηa ) 0 LDa

(25)

From eqs 12 and 13, the dimensionless inlet and outlet stream concentrations can be expressed as ∞

ψa(ηa,0) )

∑ Sa,m Fa,m(ηa) m)0

(26)

∞

ψb(ηb,0) )

∑ Sb,m Fb,m(ηb)

m)0

(27)

Ind. Eng. Chem. Res., Vol. 46, No. 23, 2007 7791 ∞

ψa(ηa,1) )

∑ Sa,m Fa,m(ηa) e-λ m)0

m

(28)

∞

ψb(ηb,1) )

Sb,m Fb,m(ηb) e-λ ∑ m)0

According to eqs 25-27, we can get the following relationship:

WbDaHbc

∫0

1

[

]

Va(ηa)Wa2 Sa,mFa,m(ηa)ψa(ηa,0) dηa + WaDbHac LDa

[

∫01

WbDaHbc

∫0

1

]

[

m

(29)

]

Vb(ηb)Wb2 Sb,mFb,m(ηb)ψb(ηb,0) dηb ) LDb

[

Va(ηa)Wa Sa,m2Fa,m2(ηa) dηa + WaDaHac LDa 2

]

Vb(ηb)Wb2 Sb,m2Fb,m2(ηb) dηb (30) LDa

∫01

Equation 19 is rearranged to acquire the relationship between Sa,m and Sb,m as

HbcFb,m(1)

Sa,m ) Sb,m

[( )( ) ]

m*0

δ Da /(∆) F′a,m(1) + HacFa,m(1) W Dc

(31)

Substituting eq 31 into eq 30 and rearranging the equation, one gets

[ ∫[

∫

1

Sa,m )

0

[( )( ) ] [( )( ) ]

δ Da /(∆) F′a,m(1) + HacFa,m(1) W Dc WaDbHac Fa,mψa(ηa,0) dηa + WbDaHbc HbcFb,m(1)

] ]

Va(ηa)Wa LDa

2

Va(ηa)Wa2 Sa,mFa,m2(ηa) dηa + LDa

1

0

[

∫

]

[

1

0

δ Da /(∆) F′a,m(1) + HacFa,m(1) W Dc WaDaHac WbDaHbc HbcFb,m(1) 2

]

Vb(ηb)Wb2 Fb,mψb(ηb,0) dηb LDb ,m * 0

∫

[

1

0

]

(32)

Vb(ηb)Wb2 Fb,m2(ηb) dηb LDa

For the same reason, from eqs 25, 28, and 29, the following equation is established

∫0

1

Sa,m ) eλm

[ ] ∫[ ]

VaWa F (η )ψ (η ,1) dηa + LDa a,m a a a VaWa F 2(η ) dηa + LDa a,m a 2

1

0

[(Wδ )(D )/(∆)]F′ Da

2

[

[ ] ]∫[ ]

+ HacFa,m(1)

a,m(1)

VbWb2 F (η )ψ (η ,1) dηb LDb b,m b b b , m * 0 (33)

∫0

c

HbcFb,m(1)

[( )( ) ]

δ Da /(∆) F′a,m(1) + HacFa,m(1) W Dc

1

2

1

0

HbcFb,m(1)

VbWb2 F 2(η ) dηb LDb b,m b

Equations 32 and 33 are the expansion coefficients in the inlet and outlet streams of the liquid-liquid membrane extractor, respectively. The two equations must be equal, and the following relationship can be expressed.

∫0

1

[ ]

VaWa F (η )ψ (η ,0) dηa + LDa a,m a a a

{

∫01

eλm

[(Wδ )(D )/(∆)]F′ Da

2

a,m(1)

+ HacFa,m(1)

∫

c

[ ]

VaWa2 F (η )ψ (η ,1) dηa + LDa a,m a a a

1

[ ]

VbWb2 F (η )ψ (η ,0) dηb ) LDb b,m b b b

0 HbcFb,m(1) δ Da /(∆) F′a,m(1) + HacFa,m(1) W Dc

[( )( ) ]

∫01

HbcFb,m(1)

[ ]

}

VbWb2 F (η )ψ (η ,1) dηb , m * 0 (34) LDb b,m b b b

Substituting the dimensionless inlet and outlet stream concentrations for both the extraction and raffinate phases into eq 34 produces

ψa0

[ ] {∫ [ ]

∫

1

VaWa2

0

LDa

Fa,m(ηa) dηa +

VaWa

2

λm

e

1

0

LDa

Fa,m(ηa)

[( )( ) ] δ Da

W Dc

/(∆) F′a,m(1) + HacFa,m(1) HbcFb,m(1)

[( )(

) ]

δ Da

∞

∑S q)0

a,qFa,q(ηa)

e

-λq

dηa +

∫

W Dc

[ ]

1

0

VbWb2 LDb

Fb,m(ηb)

∞

∑S

a,qFa,q(ηa)


dηb )

q)0

ψb0

∫

1

0

[ ] VbWb2 LDb

}

Fb,m(ηb) dηb , m * 0 (35)

7792


where the expansion coefficient Sb,q can be expressed the same as in eq 32. Then eq 35 becomes

[ ] V aWa

ψa0

∫0

1

LDa

[( )(

Fa,m(ηa) dηa +

e

{[

λm

∫0

1

) ]

δ Da

eλm

2

W Dc

/(∆) F′a,m(1) + HacFa,m(1) ψb0

HbcFb,m(1)

[ ] VbWb2 LDb

]

∞

∑ Sa,qFa,q(ηa) e

Fa,m(ηa)

-λq

dηa +

Fb,m(ηb) dηb )

[( )( ) ] δ Da

VaWa2 LDa

∫0

1

W Dc


[ ]

q)0

∫0

VbWb2

1

∞

LDb

Fb,m(ηb)

∑ Sa,qFa,q(ηa) dηb, q)0

m * 0 (36)

One can obtain all expressions for the integrals in eq 36 by using eqs 15-18, except λm ) 0. Thus, for λm * 0, eq 34 becomes

ψa0F′a,n(1)

[( )( ) ] δ Da

eλm -

W Dc

λm

/(∆) F′a,m(1) + HacFa,m(1) λmHbcFb,m(1)

{ [( )( ) ] δ Da

W Dc

{[

∞

ψb0 )

∑

q)1

/(∆) F′a,m(1)2 + HacFa,m(1)

}

2

HbcFb,m(1) and

ψa0F′a,m(1)

[( )( ) ]

eλm -

δ Da

W Dc

λm

/(∆) F′a,m(1) + HacFa,m(1) λmHbcFb,m(1) ∞

∑ Sa,q

[[( )( ) ] δ Da

W Dc

q)1

/(∆) F′a,m(1) + HacFa,m(1)

{

ψb0 ) Sa,0

]

∂F′a,m ∂Fa,m (1) (1)F′a,m(1) ∂λm ∂λm

Sa,q Fa,m(1)

{

eλn F(1) λm

[

]

[( )( ) ] δ Da

W Dc

/(∆) F′a,m(1) + HacFa,m(1)

][[( )( ) ] W Dc

F′b,m(1) +

λmHbcFb,m(1)

eλq + λn [F′a,m(1)Fa,q(1) - Fa,m(1)F′a,q(1)] λn + λq

δ Da

]

/(∆) F′a,q(1) + HacFa,q(1)

}

[F′b,m(1)Fb,q(1) - Fb,m(1)F′b,q(1)]

(λn - λq)HbcFb,m(1)HbcFb,q(1)

}

∂F′a,m ∂Fb,m (1) (1)F′a,m(1) , m ) q (37) ∂λm ∂λm

Fb,m(1)

m * q (38)

While the expansion coefficients are obtained, the dimensionless concentrations of inlet and outlet streams from eqs 12 and 13 become ∞

ψa(ηa,ξ) ) Sa,0 +

∑ Sa,mFa,m(ηa)Gm(ξ)

(39)

m)1 ∞

ψb(ηb,ξ) ) Sb,0 +

∑ Sb,mFb,m(ηb)Gm(ξ)

(40)

m)1

Now, we must deal with the situation that λm ) 0. When m ) 0, λ0 ) 0, from eqs 21 and 22 by coupling with the values of dmn and emn, one gets

Fa,m(ηa) ) Fb,m(ηb) ) 1 Using eqs 39 and 40 coupled with eq 41, we can define the dimensionless average concentrations for both streams as

(41)


ψa(ξ) ) Sa,0 +

ψb(ξ) ) Sb,0 +

∞

1

∑ m)1

4Gza

∞

1 4Gzb

∑ m)1

-1

Sa,mF′a,m(1) e-λmξ

(42)

-1 Sb,mF′b,m(1) e-λmξ λm

(43)

λm

and from eq 19, one gets

Hbc S Hac a,0

Sb,0 )

(44)

By substituting the dimensionless inlet and outlet stream concentrations for both phases and eq 44 into the conservation equation, Sa,0 can be obtained

[

Qa ψa0 Sa,0 )

∞

1

] [

a

]

-1 1 ∞ -1 Sa,mF′a,m(1) e-λm + Qb ψb0 Sb,mF′b,m(1) e-λm 4Gz λ m)1 m b m

∑ 4Gz n)1 λ

∑

(45)

Hbc Qa + Qb Hac

2.2. Concurrent-Flow Device. The equations of mass transfer for two subchannels may also be obtained in the same expressions as referred to eqs 3 and 4 except that the velocity distribution of eq 2 and the boundary condition of eq 11 are replaced by

Vb(ηb) ) Vjb(6ηb - 6ηb2)

(46)

ψb(ηb,0) ) ψb0

(47)

The results of all expressions coefficients could be obtained as

F′a,m(1) λm

(

∞

∑

q)1

{[

ψa0 -

Sa,q

Hac Hbc

) {

ψb0 )

e-λq+λm

HacWbDaF′a,q(1)F′a,m(1)

}

[F′b,m(1)Fb,q(1) - Fb,m(1)F′b,q(1)] , m * q HbcWaDbF′b,q(1)F′b,m(1) (48) ∂F′a,m ∂Fa,m HacWbDaF′a,m2(1) ∂F′b,m ∂Fb,m Fa,m(1) (1) (1)F′a,m(1) + Fb,m(1) (1) (1)F′b,m(1) , m ) q ∂λm ∂λm ∂λm ∂λm HbcWaDbF′b,m2(1)

- λm + λq

F′a,m(1)Fa,q(1) - Fa,m(1)F′a,q(1) +

]

[

]

By substituting the expression coefficients in eq 48 into eqs 42, 43, and 45, the average outlet concentrations of the solute in both phases can be determined. 2.3. Mass-Transfer Efficiency in the Liquid-Liquid Membrane Extraction System. The local Sherwood numbers in the raffinate phase and in the extraction phase are defined by ∞

Shaξ )

kaξDeq,a Da

)

∞

m

[

1

∑ Sam Fam(0) - 4Gz λ

m)1

and

SamF′am(0) e-λ ξ ∑ m)1

2

a m

]

(49) -λmξ

F′am(0) e

7794

Shbξ )


kbξDeq,b

)

Db ∞

2

[

∞

SbmF′bm(0) e-λ ξ ∑ m)1 m

1

∑ Sbm Fbm(0) - 4Gz λ

m)1

]

(50) -λmξ

F′bm(0) e

b m

where kaξ and kbξ are the local mass-transfer coefficients for the raffinate phase and the extraction phase, respectively. The equivalent diameters, Deq,a and Deq,b, were defined as follows,

Deq,a )

B∆W 2(∆W + B)

(51)

and

Deq,b )

Figure 2. Schematic diagram of the experimental setup for concurrentand countercurrent-flow devices.

B(1 - ∆)W 2[(1 - ∆)W + B]

(52)

Once the Sherwood numbers, Shaξ and Shbξ, were calculated from eqs 49 and 50, respectively, and the values of Deq,a and Deq,b were obtained from eqs 51 and 52, respectively, the local mass-transfer coefficients, kaξ and kbξ, were determined from eqs 49-52. So, the average Sherwood number is, thus, obtained as

Sha )

∫01 Shaξ dξ

(54)

Qa(Ca0 - Cae) Qb(Cbe - Cb0) ) η) QaCa0 QaCa0

(55)

[ [

Cae ) Ca0ψa(1) ) Ca0 Sa,0 +

Cbe ) Ca0ψb(1) ) Ca0 Sb,0 +

∑ ∞

-1

4Gza m)1 λm 1

∑

4Gzb m)1 λm

Sa,m F′a,m(1) e

-λm

] ]

Sb,mF′b,m(1) e-λm

(56)

(57)

and the average outlet concentrations of the approximation model under the concurrent-flow and countercurrent-flow operation in the previous work17were defined as

Cae ) Ca0 +

[ ] 1 Qa

[ (

) ]

KSHac KSHbc -1 Qa Qb (HacCa0 - HbcCb0) (59) Hac KSHac KSHbc KSHacQb 1exp KSHbcQa Qa Qb exp

(

)

1 Hac 1 Hbc ) + + K ka km kb

(60)

km ) Dc/δτ

(61)

τ)

δreal δ

(62)

and the two mass-transfer coefficients, ka and kb, were calculated using experimental studies17,28,29

The average outlet concentrations of the solute in both phases for the theoretical model in the present study can be obtained from eqs 42 and 43 -1

HacCa0 + Hac

in which

M ) Qa(Ca0 - Cae) ) Qb(Cbe - Cb0)

∞

Cae )

(53)

2.4. Extractive Rate and Extractive Efficiency. The total extractive rate M and the extractive efficiency η 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 the extraction phase to total solute in the initial raffinate phase, respectively, determined by using eqs 54 and 55 as follows

1

and

{ exp(-KS[(Hac/Qa) + Hac Hbc + Qa Qb (Hbc/Qb)]) - 1}(HacCao - HbcCbo) (58)

ka ) 0.816 kb ) 0.816

[

[ ] 6QaDa2

0.33

S(∆W)2 6QbDb2

S((1 - ∆)W)2

(63)

]

0.33

(64)

3. Experimental Runs 3.1. Chemicals and Materials. The following chemicals were used in this study: (i) glacial acetic acid (reagent ACS grade, Fisher); (ii) methyl isobutyl ketone (MIBK; reagent ACS grade, Fisher); (iii) sodium hydroxide (certified ACS grade, Fisher); and (iv) phenolphthalein (loose crystals, Coleman and Bell). A schematic diagram of the experimental setup is shown in Figure 2. Experiments are carried with a membrane sheet as a permeable barrier to extract acetic acid from aqueous solution (phase a) by MIBK to organic solution (phase b). This membrane sheet was made of microporous cellulose (Spectrum Laboratories, Inc.) with an average pore size of 2 µm, porosity of 67%, and thickness 1.8 × 10-4 m. The resulting membrane was immersed in the aqueous phase before use. Since microporous cellulose is a hydrophilic membrane, the aqueous


solution (solute ) acetic acid and solvent ) pure water) wets the membrane; thus, the solution within the membrane (phase c) is the same as that on one side of the membrane (phase a), and thus, Hac ) 1 and Hbc ) 1.90 at 20 °C. 3.2. Apparatus and Procedure. The apparatus used for membrane extraction was almost the same as those shown in Figures 2 and 9 of a previous work,16 expect that the membrane hollow-fiber modules were replaced by the flat-plate microporous membrane modules, as shown in Figure 2. A parallel stainless steel conduit with working dimensions, L ) 0.18 m, B ) 0.12 m, W ) Wa + Wb ) 0.004 m, was constructed for membrane extraction. The membrane area available for mass transfer is 0.0216 m2 (0.18 m × 0.12 m). A membrane sheet with the properties membrane earlier is inserted as a permeable barrier between two parallel plates, with distance from them to divide the conduit into two subchannels (subchannel a and subchannel b) of height Wa and Wb. Since the membrane in this study is hydrophilic, the organic pressure is maintained higher than the aqueous pressure to prevent solvent mixing between phases. The extraction phase (5 × 10-3 m3) and the raffinate phase (5 × 10-3 m3) were poured into the feed tank. Before an experimental run began, the extraction phase was allowed to follow through the bottom half of the extractor for about 3-5 min. Then the raffinate-phase reservoir was pressurized by compressed air and allowed to flow out through the top of the membrane. The extraction-phase flow rate and the raffinate-phase liquid flow rate were controlled by flowmeter between 2.0 × 10-7 to 2.0 × 10-5 m3/s. A sample of the raffinate-phase feed solution was always analyzed before a run, and the initial acetic acid concentration is 5.0 × 102 mol/m3. In general, the raffinate-phase pressure was very close to atmospheric. Thus, any ∆P used for the given experiment was equal to that of the extraction-phase gauge pressure. For high ∆P runs, however, the raffinate-phase pressure had been increased to 2-4 psig in order to maintain the desired raffinatephase flow rate. The ∆P under these conditions was the difference between the gauge pressures of the extraction phase and the raffinate phase. Experiments were carried out by varying the times of the extraction runs to see whether there was any unsteadiness. Under specific operation conditions, samples of the outlet streams were analyzed at 5 min intervals until steady state was reached as indicated by no change in acetic acid concentration (Cae and Cbe) by titration with sodium hydroxide over a period of 40 min. The time for reaching steady state was ∼30 min. At higher flow rates, runs of shorter duration were used. Further, duplicate runs were made under identical conditions to ensure reproducibility. A comparison was made between the experimental runs and the mathematical models in this paper. 4. Results and Discussions The calculation procedure as indicated in Figure 3 will be described briefly as follows. First, the eigen-values in the membrane extractor were solved from eqs 19 and 20. Next, the expansion coefficients were estimated from eqs 37 and 38. Finally, the dimensionless average concentration, the masstransfer efficiency, the extractive rate, and the extractive efficiency were calculated from eqs 31, 32, 51, 54, and 55, respectively. 4.1. Extended Power Series and Taylor Series Convergence. Table 1 shows the calculation results for concurrent- and countercurrent-flow devices with the 5-7 eigen-values and their associated expansion coefficients as well as the dimensionless outlet concentration in the raffinate phase calculated for Ca0 )

Figure 3. Calculation flowchart.

5.0 × 102 mol/m3, Qa ) 2.5 × 10-7 m3/s, and Qb ) 2.5 × 10-7 m3/s. It shows that, because of the convergence, only the 6 eigen-values must be considered during the determined procedure. These eigen-functions, eqs 21 and 22, are expanded in terms of an extended power series. The accuracy of the solution was examined with the results represented for the extended power series and Taylor series, respectively. The truncated series of n ) 170 was selected and employed during the calculation procedure. 4.2. Dimensionless Concentration Distribution in Whole Membrane Extractor. When the solvent was employed as the extraction phase, the dimensionless concentrations of the raffinate phase and the extraction phase inlet streams were 1 and 0, respectively. Figures 4 and 5 show the average acetic acid dimensionless concentration distribution for concurrent and countercurrent flows in a whole membrane contactor, respectively, and the theoretical equations are expressed by eqs 42 and 43. The average dimensionless concentration distribution in the extraction phase, ψb, increases as the subchannel thickness ratio ∆ moves away from 0.5, especially for ∆ < 0.5 (the thickness of subchannel b is larger than that of subchannel a). 4.3. Dimensionless Outlet Concentration, Extractive Rate, and Mass-Transfer Efficiency. The dimensionless outlet concentrations in the extraction phase for concurrent- and countercurrent-flow devices are indicated in Figures 6 and 7. Figure 6 shows that ψbe increase with increasing mass-transfer Graetz number in the raffinate phase (Gza) due to the large masstransfer coefficient of the fluid with Qa ) 2.5 × 10-7 m3/s as the subchannel thickness ratio ∆ moves away from 0.5, especially for ∆ < 0.5 (the thickness of subchannel b is larger than that of subchannel a). It is seen in Figure 7 that the ψbe in concurrent- and countercurrent-flow devices increase as the subchannel thickness ratio ∆ moves away from 0.5, especially for ∆ < 0.5, but decrease as mass-transfer Graetz number in the extraction phase (Gzb) increases. The dimensionless outlet

0.6918 0.7028 0.7028 4.32 × 10-8 4.42 × 10-6 4.44 × 10-6 -4.88 × 10-5 -3.76 × 10-5 -3.74 × 10-5 8.57 × 10-5 4.56 × 10-5 4.55 × 10-5 -7.388 -7.388 -7.388

-7.889 -7.889 -7.889

-1.601 -1.601 -1.601 4 5 6 0.5

0.000 0.000 0.000

-0.015 -0.015 -0.015

-3.679 -3.679 -3.679 -1.745 -1.745 -1.745 -0.016 -0.016 -0.016 0.000 0.000 0.000 4 5 6 0.5

Concurrent-Flow Device

-3.508 -3.508 -3.508

-10.905 -8.645 -8.645

λ6 λ5 λ4 λ3 λ2 λ1 λ0 m ∆

-8.214 -8.214

Countercurrent-Flow Device 0.067 7.62 × 10-4 -1.12 × 10-4 0.065 6.62 × 10-4 -1.32 × 10-4 -10.104 0.065 6.48 × 10-4 -1.33 × 10-4

4.35 × 10-6 4.33 × 10-6 -8.56 × 10-5 -3.86 × 10-5 -3.84 × 10-5 8.77 × 10-5 4.67 × 10-5 4.62 × 10-5 -3.73 × 10-4 -2.13 × 10-4 -2.11 × 10-4 7.88 × 10-4 7.68 × 10-4 7.98 × 10-4 0.089 0.082 0.082

Sa,5 Sa,3 Sa,2 Sa,1 Sa,0

Sa4

Sa,6

8.12 × 10-8

Ψ h ae

0.7301 0.7809 0.7809


Table 1. Dimensionless Outlet Concentration in the Raffinate Phase for the Eigen-value Number and Expansion Coefficients with Analytical Solution for Both Concurrent- and Countercurrent-Flow Devices; n ) 170, Ca0 ) 5 × 102 mol/m3, Qa ) 2.5 × 10-7 m3/s, and Qb ) 2.5 × 10-7 m3/s

7796

Figure 4. Average dimensionless concentration distribution for concurrentflow device in whole membrane extractor (Qa ) 2.5 × 10-7 m3/s and Qb ) 2.5 × 10-7 m3/s).

Figure 5. Average dimensionless concentration distribution for countercurrent-flow device in whole membrane extractor (Qa ) 2.5 × 10-7 m3/s and Qb ) 2.5 × 10-7 m3/s).

concentration in the extraction phase in the countercurrent-flow device is larger than that in the concurrent-flow device. Figure 8 presents a graphical representation of the theoretical extractive rate (M). This shows that both of the extractive rates increase with increasing mass-transfer Graetz number as the subchannel thickness ratio ∆ moves away from 0.5, especially for ∆ < 0.5. It was also found that the extractive rate in the countercurrent-flow device was larger than that in the concurrent-flow device. The extractive efficiencies (η) for concurrent- and countercurrent-flow devices are indicated in Figures 9 and 10, respectively. Figure 9 shows that extractive efficiency increases with increasing mass-transfer Graetz number but decreases the extraction-phase feed concentration as the subchannel thickness ratio ∆ moves away from 0.5, especially for ∆ < 0.5. Figure 10 shows that the extractive efficiency in the countercurrentflow device is larger than that in the concurrent-flow device. Figures 11 and 12 show the theoretical results of the average Sherwood number in the raffinate phase (Sha) versus masstransfer Graetz number variations in the raffinate phase (Gza) and the extraction phase (Gzb), respectively. The larger the masstransfer Graetz number is operated, the larger Sha is obtained, regardless of the raffinate-phase or extraction-phase stream. Increasing the extraction-phase flow rate is more efficacious than increasing the raffinate-phase flow rate for mass-transfer efficiency under the same conditions. Figures 11 and 12 also


Figure 6. Dimensionless outlet concentration of the subchannel b with subchannel thickness ratio and flow pattern as parameters (Qb ) 2.5 × 10-7 m3/s).

Figure 7. Dimensionless outlet concentration of the subchannel b with subchannel thickness ratio, concentration in the raffinate phase, and flow pattern as parameters (Qa ) 2.5 × 10- 7 m3/s).

show that Sha increases as the subchannel thickness ratio ∆ moves away from 0.5, especially for ∆ < 0.5. Also, the Sha in the countercurrent-flow device is larger than that in the

Figure 8. Extractive rate with subchannel thickness ratio, concentration in the raffinate phase, and flow pattern as parameters (Qb ) 2.5 × 10-7 m3/s).

Figure 9. Extractive efficiency for the concurrent-flow device with subchannel thickness ratio and concentration in the raffinate phase as parameters (Qa ) 2.5 × 10-7 m3/s).

concurrent-flow device. Moreover, the theoretical results for the inlet acetic acid concentration of the raffinate phase Ca0 ) 5.0

7798


Figure 10. Extractive efficiency with flow pattern and concentration in the raffinate phase as parameters (∆ ) 0.5 and Qa ) 2.5 × 10-7 m3/s).

Figure 12. Average Sherwood number in the raffinate phase with subchannel thickness ratio and flow pattern as parameters (Qb ) 2.5 × 10-7 m3/s).

Figure 11. Average Sherwood number in the raffinate phase with subchannel thickness ratio and flow pattern as parameters (Qa ) 2.5 × 10-7 m3/s).

Figure 13. Extractive rates estimated by the analytical model and experimental data vs Gza for both flow patterns with the subchannel thickness ratio as a parameter (Ca0 ) 5.0 × 102 mol/m3 and Qb ) 2.5 × 10-7 m3/s).

× 102 mol/m3 are also shown in Figure 13 for comparison with the experimental data. It is found from this figure that the theoretical predictions confirm pretty well with the experimental results.

4.4. Experimental Analysis. As referred to by Moffat,30 the precision index of an individual experimental measurement, M ˆ i, is determined directly from the data set, as follows,

[∑ N

SM i )

(M ˆ i - Mi) N-1

i)1

]


1/2

(65)

and the resulting uncertainty will be associated with the mean value, Mi:

SMh i )

SMi

xN

(66)

The precision index was determined for concurrent- and countercurrent-flow devices with the three subchannel thickness ratios. The mean precision indices of the experimental measurements in Figures 13 and 14 are 1.47 × 10-4 e SMˆ i e 2.41 × 10-3 and 9.51 × 10-5 e SMˆ i e 5.24 × 10-4, respectively. The accuracy of the experimental results may be calculated using the definition

E)

1

N

∑ N i)1

|Mi - M ˆ i| M î

(67)

where Mi denotes the theoretical prediction of M, while M ˆ i and N are the experimental data of extractive rate and the number of experimental measurements, respectively. The accuracy of the experimental results was calculated and shown in Figure 14 and Table 2. The error analyses of the experimental measurements determined by eq 67 for both the analytical model and the approximation model are 2.11 × 10-2 e E1 e 5.15 × 10-2 and 4.63 × 10-2 e E2 e 8.12 × 10-2, respectively, for Qb ) 2.5 × 10-7 m3/s and Ca0 ) 5.0 × 102 mol/m3. It is noted from Table 2 that the estimation of extractive rate compared to experimental runs by the analytical model was better than that by the approximation model. 4.4. Improvement of the Extractive Rate, the Extractive Efficiency, and the Mass-Transfer Efficiency. The improvements of the extractive rate, the extractive efficiency, and the mass-transfer efficiency from concurrent flow to countercurrent flow were defined as follows:

[M(∆, countercurrent flow) IM (%) ) M(∆ ) 0.5, concurrent flow)] ×100% (68) M(∆ ) 0.5, concurrent flow) and

[M(∆,countercurrent flow) M(∆ ) 0.5, concurrent flow)] × 100% (69) Iη (%) ) M(∆ ) 0.5, concurrent flow)

Table 2. Accuracy of the Experimental Results for Both Concurrent- and Countercurrent-Flow Devices ∆ ) 0.5, Ca0 ) 5 × 102 mol/m3, and Qb ) 2.5 × 1- 7 m3/s analytical model flow pattern E1 )

N

∑ N 1

|Mi - M ˆ i|

i)1

concurrent-flow device countercurrent-flow device

2.11 ×

M î

10-2

approximation model E2 )

|Mi - M ˆ i|

i)1

4.63 ×

5.15 × 10-2

N

∑ N 1

M î

10-2

8.12 × 10-2

Table 3. Improvements of the Extractive Rate, the Extractive Efficiency, and the Mass-Transfer Efficiency From Concurrent Flow to Countercurrent Flow with Ca0 ) 5 × 102 mol/m3, Qb ) 4.2 × 10-6 m3/s, and Qb ) 4.2 × 10-6 m3/s IM Iη ISha

∆ ) 0.25

∆ ) 0.5

∆ ) 0.75

114.5 114.5 124.1

26.6 26.6 27.6

97.4 97.4 101.4

in the countercurrent-flow device are higher than those of the concurrent-flow device.

and

[M(∆, countercurrent flow) I Sha (% ) ) M(∆ ) 0.5, concurrent flow)] × 100% M(∆ ) 0.5, concurrent flow)

Figure 14. Comparison of the extractive rates estimated by the approximation model, analytical model, and experimental data vs Gza for both flow patterns (∆ ) 0.5, Ca0 ) 5.0 × 102 mol/m3, and Qb ) 2.5 × 10-7 m3/s).

5. Conclusions

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The improvements of the extractive rate and the extractive efficiency from concurrent flow to countercurrent flow are 114.5, 26.6, and 97.4 for ∆ ) 0.25, ∆ ) 0.5, and ∆ ) 0.75, respectively, while the improvements of the mass-transfer efficiency are 12.41, 27.6, and 101.4 for ∆ ) 0.25, ∆ ) 0.5, and ∆ ) 0.75, respectively. Table 3 shows that the extractive rate, the extractive efficiency, and the mass-transfer efficiency

Mass transfer through laminar concurrent- and countercurrentflow membrane extractors operating with an inserted permeable barrier was analytically investigated by ignoring axial diffusion. The mass-transfer mathematical formulation through the rectangular conduits has been investigated theoretically and solved analytically by using the orthogonal technique with eigenfunction expanding in terms of a power series. There were many parameters that affected the device performance in the membrane extractor. Among these parameters, the extraction- and

7800


raffinate-phase flow rates, the operation devices, the feed concentrations in the extraction and raffinate phases, and the channel thickness ratios may be the most important factors. The influences of extraction- and raffinate-phase flow rates, operation devices, and channel thickness ratios were investigated theoretically in this paper. A good approximation was obtained using six eigen-values, and the optimal operation value is ∆ ) 0.25 (the thickness of subchannel b is three times that of subchannel a) in the rectangular conduits with inserting a permeable barrier. The channel thickness ratio has a significant influence on the masstransfer behavior in the extraction operations, especially for the countercurrent-flow devices. As is also seen from the theoretical predictions and experimental runs, the mass-transfer efficiency increases with increasing mass-transfer Graetz number in the extraction phase and in the raffinate phase. The experimental runs are carried out and compared with the results obtained from the analytical and approximation models under the steady-state operations. The analytical model qualitatively confirmed the experimental results, and the extractive rate obtained from the analytical model is in excellent agreement with the experimental data and the approximation model as well. The accuracy of the experimental measurements as compared to both the analytical model and the approximation model were calculated by 2.11 × 10-2 e E1 e5.15 × 10-2 and 4.63 × 10-2 e E2 e 8.12 × 10-2, respectively. Further, the extractive rate in the approximation model in the previous work18 should be obtained by the overall masstransfer coefficient that was determined by experimental runs or the semiexperimental equations. The advantages and values of the theoretical model used in this study are that one can directly calculate the extractive rate without experimental runs and also obtain the dimensionless concentration profile in the entire membrane extractor. The theoretical model and the experimental runs showed that increasing the raffinate-phase and extraction-phase flow rates and a subchannel thickness ratio moving away from 0.5 under the countercurrent-flow operation can enhance the acetic acid extractive rate, the extractive efficiency, and the mass-transfer efficiency in the liquid-liquid membrane contactor. Acknowledgment The authors thank the National Science Council of the Republic of China for its financial support. Nomenclature B ) conduit width, m C ) solute concentration, mol/m3 Da ) ordinary diffusion coefficient in raffinate phase, m2/s Db ) ordinary diffusion coefficient in extraction phase, m2/s Dc ) ordinary diffusion coefficient in the membrane, m2/s dmn ) coefficient in the eigen-function Fa,m E ) accuracy of the experimental results, defined by eq 67 emn ) coefficient in the eigen-function Fb,m Fm ) eigen-function associated with eigen-value λm Gz ) mass-transfer Graetz number H ) distribution coefficient IM ) improvement of the extractive rate Iη ) improvement of the extractive efficiency ISha ) improvement of the mass-transfer efficiency K ) overall mass-transfer coefficient defined by eq 60, m/s k ) mass-transfer coefficient of acetic acid, m/s kξ ) local mass-transfer coefficient of acetic acid, m/s

L ) conduit length, m M ) extractive rate, mol/s M h i ) experimental data of extractive rate, mol/s M ˆ i ) average experimental data of extractive rate, mol/s Mi ) theoretical prediction of extractive rate, mol/s N ) number of experimental measurements Q ) volumetric flow rate of conduit, m3/s Re ) Reynolds number S ) overall mass-transfer area of a flat-plate membrane module, m2 Sm ) expansion coefficient associated with eigen-value λm Shξ ) local Sherwood number Sh ) average Sherwood number V ) velocity distribution of fluid, m/s V ) average velocity of fluid, m/s W ) distance between two parallel plates, m x ) transversal coordinate, m z ) longitudinal coordinate, m ∆ ) subchannel thickness ratio, defined by eq 5 η ) extractive efficiency, defined by eq 56 ηa ) transversal coordinate, defined by eq 5 ηb ) transversal coordinate, defined by eq 5 ) porosity of the membrane τ ) pore tortuosity of the membrane defined by eq 62 δ ) thickness of the membrane, m δreal ) real thickness of the membrane λm ) eigen-value ξ ) longitudinal coordinate, defined by eq 5 F ) density of the fluid, kg/m3 ψ ) dimensionless concentration, defined by eq 5 Superscripts and Subscripts a ) in the raffinate phase b ) in the extraction phase 0 ) at the inlet e ) at the outlet Literature Cited (1) Gabelman, A.; Hwang, S. T. Hollow Fiber Membrane Contactors. J. Membr. Sci. 1999, 159, 61. (2) Prasad, R.; Kiani, A.; Bhave, R. R.; Sirkar, K. K. Further Studies on Solvent Extraction with Immobilized Interfaces In a Microporous Hydrophobic Membrane. J. Membr. Sci. 1986, 26, 79. (3) Prasad, R.; Sirkar, K. K. Microporous Membrane Solvent Extraction. Sep. Sci. Technol. 1987, 22, 619. (4) Prasad, R.; Sirkar, K. K. Solvent Extraction with Microporous Hydrophilic and Composite Membrane. AIChE J. 1987, 33, 1057. (5) Prasad, R.; Sirkar, K. K. Hollow Fibers Solvent Extraction: Performances and Design. J. Membr. Sci. 1999, 50, 153. (6) Prasad, R.; Sirkar, K. K. Dispersion-Free Solvent Extraction with Microporous Hollow-Fibers Modules. AIChE J. 1988, 34, 177. (7) Prasad, R.; Sirkar, K. K. Hollow Fibers Contained Liquid Membrane Separation of Citric Acid. AIChE J. 1991, 37, 383. (8) Yun, C. H.; Prasad, R.; Sirkar, K. K. Membrane Solvent Extraction Removal of Priority Organic Pollutants from Aqueous Waste Stream. Ind. Eng. Chem. Res. 1992, 31, 1709. (9) Juang, R. S.; Huang, H. C. Non-Dispersive Extraction Separation of Metals Using Hydrophilic Microporous and Cation Exchange Membranes. J. Membr. Sci. 1999, 156, 179. (10) Juang, R. S.; Chen, J. D.; Huan, H. C. Dispersion-Free Membrane Extraction: Case Studies of Metal Ion and Organic Acid Extraction. J. Membr. Sci. 2000, 165, 59. (11) Ding, H.; Cussler, E. L. Fractional Extraction with Hollow Fibers with Hydrogel-Filled Wall. AIChE J. 1991, 37, 855. (12) Prasad, R.; Sirkar, K. K. Microporous Membrane Solvent Extraction. Sep. Sci. Technol. 1987, 22, 619. (13) Prasad, R.; Sirkar, K. K. Hollow Fibers Solvent Extraction: Performances and Design. J. Membr. Sci. 1990, 50, 153.

Ind. Eng. Chem. Res., Vol. 46, No. 23, 2007 7801 (14) Kiani, A.; Bhave, R. R.; Sirkar, K. K. Solvent Extraction with Immobilized Interfaces in a Microporous Hydrophobic Membrane. J. Membr. Sci. 1984, 20, 125. (15) Lin, S. H.; Juang, R. S. Simultaneous Extraction and Stripping of EDTA-Chelated Metallic Anions with Aliquat 336 in Hollow Fiber Contactors. Chem. Eng. Sci. 2002, 57, 143. (16) Yeh, H. M.; Huang, C. M. Solvent Extraction in Multupass ParallelFlow Mass Exchangers of Microporous Hollow-Fiber Modules. J. Membr. Sci. 1995, 103, 135. (17) Yeh, H. M.; Shu, Y. S. Analysis of Membrane Extraction Through Rectangular Mass Exchangers. Chem. Eng. Sci. 1999, 54, 897. (18) Yeh, H. M.; Peng, Y. Y.; Chen, Y. K. Solvent Extraction Through a Double-Pass Parallel-Plate Membrane Channel with Recycle. J. Membr. Sci. 1999, 163, 177. (19) Yeh, H. M.; Chen, Y. K. The Effect of Multipass Arrangement on the Performance in a Membrane Extractor of Fixed Configuration. Chem. Eng. Sci. 2000, 55, 5873. (20) Yeh, H. M.; Chen, C. H. Recycle Effect on Solvent Extraction Through Concurrent-Flow Parallel-plate Membrane Modules. J. Membr. Sci. 2001, 190, 35. (21) Yeh, H. M.; Chen, Y. K. Correction-Factor Analysis of Membrane Extraction in Flat-Plate Modules. J. Chin. Inst. Chem. Eng. 2001, 32, 453. (22) Nunge, R. J.; Gill, W. N. Analysis of Heat Transfer in Some Countercurrent Flows. Int. J. Heat Mass Transfer 1965, 8, 873. (23) Nunge, R. J.; Gill, W. N. An Analytical Study of Laminar Counterflow Double-Pipe Heat Exchangers. AIChE J. 1966, 12, 279.

(24) Ho, C. H.; Yeh, H. M.; Sheu, W. S. An Analytical Study of Heat and Mass Transfer Through a Parallel-Plate Channel with Recycle. Int. J. Heat Mass Transfer 1998, 41, 2589. (25) Ho, C. D. Improvement in Performance of Double-Flow Laminar Countercurrent Mass Exchangers. J. Chem. Eng. Jpn. 2000, 33, 545. (26) Ho, C. D.; Yeh, H. M.; Chiang, S. C. A Study of Mass Transfer Efficiency in a Parallel-Plate Channel with External Refluxes. Chem. Eng. J. 2002, 85, 207. (27) Ho, C. D.; Guo, J. J. An Analytical Study of Separation Efficiency on the Enrichment of Heavy Water in Double-Flow Thermal-Diffusion Columns with Flow-Rate Fraction Variations. Chem. Eng. Commun. 2005, 192, 424. (28) Porter, M. C. Handbook of Industrial Membrane Technology; Noyes Publications: Park Ridge, NJ, 1990. (29) Wike, C. R.; Chang, P. Correlation of Diffusion Coefficients in Dilute Solution. AIChE J. 1955, 1, 264. (30) Moffat, R. J. Describing the Uncertainties in Experimental Results. Exp. Therm. Fluid Sci. 1988, 1, 3.

ReceiVed for reView September 4, 2006 ReVised manuscript receiVed July 19, 2007 Accepted August 7, 2007 IE0611610

Mass-Transfer Efficiency of Membrane Extraction in Laminar Flow

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