Kertész SPT 05

Separation and Purification Technology 41 (2005) 275–287

Design and simulation of two phase hollow fiber contactors forsimultaneous membrane based solvent extraction and

stripping of organic acids and bases�

Rudolf Kertesz, Stefan Schlosser∗

Department of Chemical and Biochemical Engineering, Slovak University of Technology, Radlinsk´eho 9, 81237 Bratislava, Slovakia

Received 20 October 2003; received in revised form 12 August 2004; accepted 10 September 2004

Abstract

A short-cut method for the design and simulation of two-phase hollow fiber (HF) contactors with variable mass-transfer and distributioncoefficients taking into account reaction kinetics is presented. A parametric study of membrane based solvent extraction (MBSE) and stripping(MBSS) of butyric acid (BA) based on experimental data is given. For the systems with concentration dependent mass-transfer and distributioncsw(aa

atl©

K

1

mtu

bt

P(

1d

oefficients a method dividing the length of fibers into segments with the same inlet to outlet concentration difference in the feed phase isuggested. The mean values of mass-transfer and distribution coefficients are estimated for each segment independently. For the tested systemith BA, convergence of the simulation results was achieved for more than 20 segments. Dependences of the total number of contactors

length of fibers) on the shell Reynolds number and approach to the equilibrium at the raffinate end of the HF contactor exhibit a minimum atbout 0.9 and 0.40, respectively. The overall number of contactors in MBSE and MBSS is very sensitive to the increase in BA yield in MBSEbove 0.85.

For following process parameters: volumetric flow rate 1 m3 h−1, feed concentration 350 mol m−3, yield of BA in MBSE 85%, Reshell = 0.9,pproach to equilibrium at the raffinate end of the HF contactor 0.40, and concentration factor of 10, the number of contactors of Liqui-Celype 10 in. × 28 in. (Celgard) in series, without parallel lines of contactors, is five in MBSE and eight in stripping. In case when two parallelines of contactors are used the number of contactors in series is two in extraction and four in stripping.

2004 Elsevier B.V. All rights reserved.

eywords:Design; Simulation; Hollow fiber contactors; Membrane based solvent extraction; Membrane based solvent stripping; Butyric acid; Organic acids

. Introduction

Hollow fiber (HF) modules are widely used nearly in allembrane processes, especially in ultrafiltration, microfiltra-

ion, gas permeation and pertraction through polymeric or liq-id membranes. Recently, hollow fiber contactors have begun

Abbreviations: BA, butyric acid; HF, hollow fiber; MBSE, membraneased solvent extraction; MBSS, membrane based solvent stripping; P, ex-ractant; TOA, trioctylamine� Presented at the Membrane Science and Technology ConferenceERMEA 2003, Tatranske Matliare, Slovakia, September 7–11, 2003http://www.sschi.chtf.stuba.sk/permea/).∗ Corresponding author. Tel.: +421 2 524 96743; fax: +421 2 524 96920.E-mail address:[email protected] (S. Schlosser).

to be used in membrane based processes as membrane basedabsorption or desorption and membrane based solvent extrac-tion (MBSE) or membrane based solvent stripping (MBSS)in two phase contactors [1–5].

Mass-transfer between two immiscible liquids—solventextraction—is in the classical arrangement realized in con-tactors with dispersed phase (spray or mixed columns, mixer-settlers, etc.) or by combination of the dispersed phase and aliquid film (packed columns). A new approach in contactingtwo fluids is the formation of an immobilized L/L interfaceat the mouths of pores of the microporous wall, throughwhich mass-transfer takes place, as it is schematically shownin Fig. 1. This approach is sometimes called nondispersivesolvent extraction or sorption. Despite the fact that themicroporous wall, “membrane”, does not play an active role

383-5866/$ – see front matter © 2004 Elsevier B.V. All rights reserved.oi:10.1016/j.seppur.2004.09.007

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276 R. Kertesz,S. Schlosser / Separation and Purification Technology 41 (2005) 275–287

Nomenclature

ai effective interfacial area on the inner surfaceof the hydrophobic fibers per unit length of thecontactor, Eq. (26) (m)

A geometric surface area (m2)Aw arithmetic mean value of the inner and outer

geometric surface areas of fibers (m2)c molar concentration of the solute (undissoci-

ated acid) (mol m−3)ca analytical (overall) molar concentration of the

permeant (acid) (mol m−3)d diameter of the fiber (m)dh hydraulic diameter of the shell (4× cross-

sectional area/wetted perimeter) (m)dki inner diameter of the shell (outer diameter of

the central tube) (m)dko outer diameter of the shell (m)� diffusion coefficient (m2 s−1)D distribution coefficientk individual mass-transfer coefficient (m s−1)Ke overall mass-transfer coefficient in the extrac-

tor (m s−1)Ks overall mass-transfer coefficient in the stripper

(m s−1)L length of the fiber (m)Lc,eff effective length of fibers in contactor

(m)n molar flux (mol s−1)Ncp number of parallel contactorsNcs number of contactors in seriesNct total number of contactors (both in parallel and

series)Nf number of fibers in contactorNTU number of transfer unitsre rate constant of the extraction reaction, Eq. (6)

(m s−1)rs rate constant of the stripping reaction, Eq. (7)

(m s−1)R overall mass-transfer resistance (s m−1)Re Reynolds number, Re = du

νReshell Reynolds number for radial flow in the shell,

Eq. (14)Rk individual mass-transfer resistance based on

reaction kinetics (s m−1)Sc Schmidt number, Sc = ν

�Sf cross-section area of fiber lumens (m2)Sh Sherwood number, Eq. (12)u linear velocity of the flow (m s−1)uS mean linear velocity of radial flow in the shell,

defined by Eq. (16) (m s−1)V volumetric flow rate (m3 s−1)

YMA/OA ratio of flux of mineral acid to organic acid,defined by Eq. (47)

Z concentration factor of the solute in the con-centrate (output from the stripper), defined byEq. (37)

Greek lettersβn+1 concentration ratio defined by Eq. (42)δw wall thickness (m)ε porosity of the wallηconv conversion of the reagent in the stripping solu-

tionηe yield of the solute in MBSE defined by Eq. (30)ν kinematic viscosity (m2 s−1)τ tortuosity of the poresτRT residence time in the shell of contactor (s)

Subscriptsb boundary layer in the bulk phasee extractor (MBSE)F feed phase, feed boundary layeri inner surface of the fiber wallloc local valuels logarithmic mean valueMA mineral acidn number of the contactor segmento outside surface of the fiber wallOA organic acidR stripping solution; stripping interfaces stripper (MBSS)S solvent phase; boundary layer in the solventTA total acid (organic plus mineral acid)w fiber wall0 initial value1 feed or stripping solution inlet end of a HF con-

tactor or a series of contactors2 raffinate or stripping solution outlet end of a

HF contactor or a series of contactors

in separation of the given mixture, such systems and relatedcontactors are often termed with the adjective “membrane”.Processes with mass-transfer between two liquids will betermed as membrane based solvent extraction and membranebased solvent stripping, as suggested earlier [2,5,6]. The po-tential for application of MBSE is high and covers recovery orremoval of organics and metals [2,3,5,6] or formation ofintegrated and hybrid systems with MBSE [5,7,8]. The tech-nological scheme of MBSE combined with simultaneousregeneration of the solvent by MBSS, both in HF contactors,and the recirculation of solvent to extraction are shown inFig. 2.

R. Kertesz,S. Schlosser / Separation and Purification Technology 41 (2005) 275–287 277

Fig. 1. Scheme of the two-phase system in the membrane based solventextraction (MBSE) and in the membrane based solvent stripping (MBSS)with immobilized L/L interface in hydrophobic hollow fibers with concen-tration profiles of the transported acid or base. F—feed phase, S—solvent,R—stripping solution.

The types of two phase HF contactors with a microporouswall and their advantages and shortcomings are discussedin references [2,3,5,6,9]. Mass-transfer characteristics ofHF contactors are presented in paper [8,10]. Analysis ofmass-transfer resistances in HF contactors, including theinfluence of the reaction kinetics on them, is shown in papers[10–12]. It has been found that mass-transfer resistancerelated to slower kinetics of the decomposition of the solutecomplex contributes substantially to the overall resistance insome systems [10,11,13–15], e.g., in MBSS of butyric acid(BA) [10,13,16] or heterocyclic carboxylic acid [12,17,18].In latter system also kinetics of the complex formation inMBSE contributes to the overall resistance.

Simulation of integrated extraction and stripping using amathematical model with constant distribution and overallmass-transfer coefficients for valeric acid is presented in pa-per [19], for phenol, lactic acid and phenylglycine in paper[20] and for butyric acid and heterocyclic carboxylic acid inpaper [21].

FMc

The aim of this paper is to present a short-cut methodfor design and simulation of two phase HF contactors witha concentration dependent mass-transfer and distribution co-efficient in the simultaneous MBSE and MBSS with closedcircuit of the solvent. A case study for the design of HF con-tactors in the recovery of butyric acid (BA) by simultaneousMBSE and MBSS will be presented, as well.

2. Theoretical

2.1. Mass-transfer in HF contactors

The schemes of two-phase systems and related concentra-tion profiles of solute in a HF contactor in MBSE and MBSSwith aqueous feed and stripping solutions flowing in the hy-drophobic fiber lumen are schematically shown in Fig. 1. Theoverall reaction between molecules of organic acid (HA) andextractant (P) on the feed/solvent interface can be expressedas

pHA + P → (HA)pP (1)

and for the acid/extractant complex decomposition on thestripping interface as

( − −

wbce[tanpfwmTdtb

ttp

n

wtmtsc

ig. 2. Flow sheet of MBSE with simultaneous regeneration of solvent byBSS in a HF contactors and recirculation of solvent to extraction. In both

ontactors the solvent flows in the contactor shell.

HA)pP + pOH → P + pA + pH2O (2)

here species in the organic phase are marked with an over-ar. A more detail discussion on extraction equilibrium ofarboxylic acids with solvents containing trioctylamine asxtractant and the related mechanism are given in papers22,23]. The extraction mechanism consists of a physical ex-raction by the diluent (n-alkanes) in form of a monomer anddimer of the solute (BA) in the organic phase and simulta-eous reactive extraction of acid with the extractant (in thisaper tertiary amine) when carboxylic acid is extracted inorm of complexes with the extractant. In extraction of BAith trioctylamine (TOA) formation of complexes with 1–7olecules of acid per one amine molecule is assumed [22].he proposed model for BA fits experimental equilibriumata well [22]. Only undissociated acid molecules are ex-racted with tertiary amines. pH of the treated solution has toe adjusted well below the pKa value.

Let us consider steady state or quasi-steady state condi-ions in MBSE. Thus, the solute flux from the aqueous phaseo the bulk organic phase in the extraction, ne, can be ex-ressed by equation

˙e = KeAeiεe(cF − cFS,S) = KeAeiεe

(cF − cS

DS

)(3)

hereDS is the distribution coefficient related to the concen-ration of solute in the solvent, cS. Eq. (4) defines the overall

ass-transfer coefficient in MBSE, Ke, for the effective in-erfacial area. cFS,S is the equilibrium concentration of theolute (undissociated acid) in the feed phase at the interfaceorresponding to its concentration in the bulk of the solvent,


for which holds

cS = DScFS,S (4)

The overall solute flux in stripping from the organic phase(solvent) to the aqueous stripping solution, ns, can be de-scribed by the following relation with simplification for cases,when concentration of the undissociated acid in the strippingsolution is zero. This holds in cases with an excess of reagent,when pHR is well above the pKa value (when cR = 0), and thesimplified form of Eq. (5) can be used

ns = KsAsiεs(cS − cRDR) = KsAsiεscS (5)

The mode of operation of the stripper with an aqueousstripping solution flowing in the lumen of hydrophobic fiberswill be considered. The high stoichiometric excess of reactant(e.g. NaOH for acids) in the stripping solution will result ina zero concentration of undissociated acid in the strippingsolution and consequently in a zero resistance in this phase(RR = 0).

2.1.1. Diffusion-reaction modelResistances in series approach, assuming only diffu-

sion resistances in boundary layers and walls, are usuallyused [2,3]. The discrepancy between the experimental val-u[a[vtotetpfamtc

asBstasacAra

n

Table 1Lumped rate constant of the stripping reaction, Eq. (7), in MBSS and per-traction (PT) of butyric acid [8,25]

Process Solvent rs × 106 m s−1

MBSS in HF contactor 0.4 kmol m−3 TOAin n-alkanes

4.84

PT in HF contactor 5.58PT through layered BLM

(from data in [13])8.35

PT in HF contactor Pure n-alkanes 29.2

ns = rsAsiεscSR (7)

where cFS and cSR are concentrations of acid and/or acid inform of complexes in the solvent close to the extraction orstripping interface, and re and rs are, de facto, lumped pa-rameters reflecting the kinetics of the complex formation ordecomposition interfacial reactions, equilibrium and kineticsof desorption or adsorption of molecules of the complexeson the stripping interface, and desorption of free extractantmolecules from the stripping interface, andcSR is the complexconcentration in the solvent close to the stripping interface.The ratio of the molecules of complexes to free extractantmolecules on the interface can possibly be concentration de-pendent what will be also reflected in the stripping kinetics.For example in pertraction of butyric acid with TOA, themean molar ratio of acid to extractant molecules in the com-plex is about 3, as shown in paper [22]. The values of the rateconstant of the stripping reactions for butyric acid are givenin Table 1 and for other organic acids in papers [8,10].

From this table and paper [25] it is evident that the valueof the rate constant in stripping free butyric acid from puren-alkanes is much higher than in case when the acid/TOAcomplexes are dominant in solvents containing TOA. Thus,the value of the rate constant will depend on the solvent com-position.

Let us consider a system with aqueous phase in the fiberlHfltMlt

wf

R

tic

R

es of the overall mass-transfer coefficient in pertraction11,14,15,24,25], and MBSS [11,14] of carboxylic acids,s well as in both MBSE and MBSS of butyric acid (BA)10,13,16] or heterocyclic carboxylic acid [12,17,18] and thealues calculated according to the diffusion model can be in-erpreted by a slower reaction of the complex decompositionn the stripping interface or complex formation on the ex-raction interface. For lactic acid the stripping is faster thanxtraction [26] and its kinetics will not influence the mass-ransfer rate. However the kinetics of formation of the com-lex may play a role. The influence of reaction kinetics wasound also for metals [27,28]. The kinetics of competitivedsorption and desorption of complexes and free extractantolecules, together with the maximum value of the concen-

ration of these molecules on the interface (independent ofoncentration in the bulk phase) can be important.

Eqs. (1) and (2) can describe the overall reactions for thecid/carrier complex formation and decomposition on thetripping interface. For example in pertraction or MBSE ofA with a solvent containing trioctylamine (TOA) nine acidpecies are in the solvent including also the monomer andhe dimer of BA. The mole fractions of individual complexesre concentration dependent. If two or more surface activeubstances are present in the solution, e.g., free carrier andcid-carrier complexes, the situation is more complex andould not be easily described, as discussed in papers [29,30].s the first approximation, the following simple first order

ate equations for the description of the kinetics of extractionnd stripping reactions will be used [11,12,15,18]

˙ e = reAeiεecFS (6)

umen and organic phase on the shell side, and a hydrophobicF. For the steady state or quasi-steady state conditions, whenuxes in individual liquid layers are equal, the following rela-

ions can be derived for the overall mass-transfer resistance inBSE, taking into account diffusion resistances in boundary

ayers and walls and the resistance connected with kinetics ofhe reaction of formation of the permeant–extractant complex

1

Ke= εe

kF+ Aei

Aw,ekSwDS+ Aeiεe

AeokSbDS+ 1

re(8)

here the overall mass-transfer resistance is composed ofour individual resistances

e = RF + RSw + RSb + Rk,e (9)

In case of the influence of the stripping reaction kinetics onhe overall resistance and for situations with a zero resistancen the stripping solution boundary layer following equationsan be derived for stripping

s = RSb + RSw + Rk,s (10)


1

Ks= Asiεs

AsokSb+ Asi

Aw,skSw+ 1

rs(11)

Eqs. (8)–(11) represent the diffusion-reaction model ofMBSE and MBSS for the mode of operation when the solventflows in the shell. The last terms in these equations representresistances based on the kinetics of the complex formationand stripping reactions. By omitting the last term in Eq. (8)or (11) the classical diffusion model of MBSE and MBSS isobtained.

2.1.2. Estimation of individual mass-transfer coefficientsIdentification of individual mass-transfer coefficients de-

pends on the construction of the module. In order to charac-terize the local mass-transfer coefficients several correlationshave been proposed. For the laminar flow in tubes, correlationof Dahuron and Cussler [31] is often used:

Sh = kdi

�= 1.5

(d2

i u

L�

)1/3

(12)

For mass-transfer in the shell of cross flow modules, thefollowing correlation suggested by Schoner [32] was applied

Sh = 1.76Re0.82shellSc0.33, 0.012 ≤ Re ≤ 2 (13)

where Reynolds number in the shell is defined by relation

R

at

d

u

m

lt

k

eids3

sg3i

of 40% from the interval 2.3 to 2.8 depending on the system.In this work the value of tortuosity of 2.7 will be used.

2.2. Calculation of the length of fibers in HF contactors

In steady state operation of a HF contactor the length offibers can be calculated as a product of the number of transferunits (NTU) and the length of the transfer unit (LTU) based onthe overall mass-transfer coefficient.

L = NTU LTU (18)

Let us consider contactors with hydrophobic fibers with aparallel counter-current flow of phases and an aqueous feedflow in the fiber lumen, which is schematically drawn in Fig. 3[2,5]. A constant volumetric flow rate of phases along the con-tactor will be assumed. For the differential material balanceof the contactor holds

VF dcF = −Ke,loc(πdiNε)

(cF − cS

DS

)dz (19)

where Ds is the distribution coefficient related to cs.Generally, the overall mass-transfer coefficient, the distri-

bution coefficient and also the volumetric flow rate changealong the contactor. To simplify this situation the contactorwlvt(of∫

t

Fp

eshell = dhuSρ

µ(14)

nd the hydraulic diameter and the mean velocity of liquid inhe shell by equations

h = 4Vshell

Ao= d2

ko − d2ki − nd2

o

ndo(15)

¯ S = 2 ¯V S

πleffdshell,ls= 2 ¯V S

πleff

ln(dko/dki)

dko − dki(16)

Other available equations for the estimation of individualass-transfer coefficients are presented in papers [2,3,5].The individual mass-transfer coefficient in the stagnant

iquid filling the pores of the wall can be calculated by equa-ion

w = �

τδw= 2�

τ(do − di)(17)

The diffusion coefficient was calculated by Wilke–Changquation [33]. For example the diffusion coefficient of BAn water at 30 ◦C is 11.01 × 10−10 m2 s−1. The value of theiffusion coefficient of the BA complex with (BA)3TOAtructure in the solvent (0.4 kmol m−3 TOA in n-alkanes) is.55 × 10−10 m2 s−1.

The values of tortuosity reported in papers dealing witholvent extraction with the polypropylene hollow fibers Cel-ard X-10 with a porosity of 30% are ranging from 2 [34] to.5 [35]. Prasad and Sirkar [1] suggested the value of tortuos-ty of the polypropylene fibers Celgard X-20 with a porosity

ill be divided into n segments with the same inlet to out-et concentration difference in the feed phase with a constantalue of the mentioned quantities in the segment defined forhe mean concentration in the segment. By integrating Eq.19) along the length of the nth segment of the contactor onebtains for boundary conditions z= 0, cF = cF,n, cS = cS,n, andor z=Ln, cF = cF,n, cS = cS,n+1

Ln

0Ke,loc dz = KeL = VF

ai

∫ cF,n+1

cF,n

dcF

(cF − (cS/DS))(20)

From the overall material balance of solute in the contac-or, from z= 0 up to the coordinate z=L with the respective

ig. 3. Scheme of the two phase HF contactor and its nth segment witharallel flow of phases in MBSE.


concentrations in aqueous and solvent phases cF and cs fol-lows

VF(cF1 − cF,n+1) = VS(cS1 − cS,n+1) (21)

By inserting cS from this equation into Eq. (20) and afterintegrating for a constant value of the overall mass-transfercoefficient and the distribution coefficient one obtains afterrearrangement

Le,n =¯V F,n

Keai

1

1 − ¯V F,n/DS¯V S,n

ln

(cF,n − cS,n/DS

cF,n+1 − cS,n+1/DS

)(22)

where

¯V i,n = Vi,n + Vi,n+1

2(23)

Considering Eq. (18) for the number of transfer units thefollowing relation is obtained

NTUe,n = 1

1 − ¯V F,n/DS¯V S,n

ln

(cF,n − cS,n/DS

cF,n+1 − cS,n+1/DS

)(24)

and for the length of the transfer unit the following equationholds

L

wtf

a

ad

c

N

wt

eswoNfl

N

L

Expressions for NTUs,n and LTUs,n for different systemsand configurations are presented in papers [2,3,5]. The num-ber of the stripping contactors in series can be calculatedsimilarly as for extraction, see Eq. (27).

More complex is the situation when the distributioncoefficient and/or the volumetric flow rate of phases areconcentration dependent. A more rigorous approach requiresthe formulation of appropriate differential equations formass-transfer in individual layers, which have to be solvednumerically. Relations for the distribution coefficient and/orvolumetric flow rate as functions of concentration should beincluded. This approach was used, e.g., in papers [36–40]. Amodel of MBSE for multicomponent systems is proposed inpaper [41]. In separation of charged species, Fickian diffu-sion approach is not satisfactory and Nernst–Planck equationhas to be considered [42]. Modeling of HF contactors withcross flow of phases is presented in paper [32].

3. Short-cut method for the calculation of two phaseHF contactors

In calculation of simultaneous MBSE and MBSS of so-lutes in HF contactors the typical input process data are:

- flow rate and composition of the feed;-

---

nabolcosmoaiFp

3

t

η

TUe,n =¯V F,n

Ke,nai(25)

here Ke,n is the mean (integral) value of the overall mass-ransfer coefficient in the nth segment of the HF extractor andor ai

i = πdiNfε (26)

i is an effective interfacial area on the inner surface of hy-rophobic fibers per unit length of the contactor.

The number of extraction contactors in series can be cal-ulated by relation

cs,e,n = Le,n

Lc,eff(27)

here Lc,eff is the effective length of fibers in the selectedype of the HF contactor.

For membrane based stripping of the solvent (MBSS) anxcess of reagent in the stripping solution is maintained re-ulting in a zero concentration of undissociated molecules ofeak acids or bases in it or zero concentration of the complexn the strip interface. Under this assumption, equations forTUs,n and LTUs,n can be derived for stripping of the solventowing in the shell

TUs,n = lncS,n

cS,n+1(28)

TUs,n =¯V S,n

Ks,nai(29)

composition of the solvent with related L/L equilibriumdata;concentration factor of solute in the concentrate;yield of solute in MBSE;geometric characteristics of contactor(s) and their mass-transfer data for the given system.

Further quantities, e.g. those characterizing the hydrody-amic conditions in contactors (as linear velocity of the feednd stripping solution in the fiber lumen and Reynolds num-er in the shell), approach to equilibrium at the raffinate endf the extractor, conversion of the reagent in the stripping so-ution, co-extraction of other solutes (e.g. mineral acids), etc.an be estimated only from experimental data for the givenr similar system in a HF contactor. The basic scheme of theimultaneous MBSE and MBSS circuit with notation of theain process parameters is shown in Fig. 2. The flowchart

f a short-cut method for calculation of the length of fibersnd the number of parallel contactors and contactors in seriesn simultaneous MBSE and MBSS processes is presented inig. 4. The basic characteristics of two phase HF contactorsroduced commercially are presented in Table 2.

.1. Material balance of MBSE and MBSS circuits

Calculations are based on the methodology shown in Sec-ion 2.2. The yield of solute in extraction is defined by relation

e = VF1cF1 − VF,n+1cF,n+1

VF1cF1= cF1 − XcF,n+1

cF1(30)


Fig. 4. Flowchart of the short-cut calculation of the fiber length and the number of HF contactors for simultaneous MBSE and MBSS.

For systems with a variable volume of phases, due to themass-transfer of solute, the quantity X is defined by equation

X = VF,n+1

VF1= ρk − MkcF1

ρk − MkcF,n+1(31)

where ρk is the density of the pure acid (solute). Additivityof volumes is supposed. This was experimentally proved forBA, DMCCA and MPCA. For the flow rate of raffinate andthe concentration of solute in it the following equations hold:

VF,n+1 = VF,n − VOA,n (32)

VOA,n = nOA,F,nMOA

ρOA= MOA(VF,ncF,n − VF,n+1cF,n+1)

ρOA(33)

VF,n+1 = VF,n(ρOA − cF,nMOAηe)

ρOA(34)

cF,n+1 = VF,ncF,n(1 − ηe)

VF,n+1(35)

From the overall material balance the flow rate of the con-centrate VR,n+1 can be estimated

VR,n+1 = VF,ncF,n − VF,n+1cF,n+1

cR,n+1

= VF,ncF,n − VF,n+1cF,n+1

ZcF,n

(36)


Table 2Characteristics of HF contactors Liqui-Cel (Celgard) used in simulations

Contactor 2.5 in. × 8 in. 4 in. × 28 in. 10 in. × 28 in.

Number of fibers (cca.) 10000 32000 230000Internal area of fibers (m2) 1.13 15.32 108.38Effective length of fibers (cm) 15.0 63.5 62.5Shell volume (cm3) 195 4200 26100Cross-section area of fibers lumen (cm2) 4.52 14.48 104.05ai (m2/m) 2.82 9.07 65.04Logarithmic mean shell diameter (cm) 3.28 5.29 14.83Hydraulic diameter in the shell (mm) 0.260 0.711 0.546

Characteristics of fibersExternal diameter (�m) 300Internal diameter (�m) 240Wall thickness (�m) 30Porosity (%) 40 40 30Mean pore size (�m) 0.03Tortuosity 2.7

where the concentration cR,n+1 is defined, e.g., by require-ment of the concentration factor of solute in the concentrate(output from the stripper), which is defined by relation

Z = cR,n+1

cF1(37)

The volumetric flow rate of the solvent in the shell isdefined by requirement of proper hydrodynamic conditions,which are characterized by the value of the Reynolds number,defined for example by Eq. (14). On other hand, also dimen-sions of the stripper depend on the value of Reshell (becausethe solvent flows in the shell), what follows from Eq. (29).The mean value of the volumetric flow rate in the shell, ¯V S,is defined for the mean velocity of the radial flow in the shell,us given by Eq. (16). Then,

¯V S = πReshellµSdshell,ls,eleff,eNcp,e

2dh,eρS(38)

dshell,ls = dko − dki

ln (dko/dki)(39)

For the volumetric flow rates VS1 a VS,n+1 following equa-tions can be derived

VS1 = 2 ¯V S + VOA

2(40)

V

t

β

it

c

For the value of the concentration of solute in the solventleaving the extractor, cS,n, and from the material balance re-lation follows

cS,n = VF,ncF,n − VF,n+1cF,n+1 + VS,n+1cS,n+1

VS,n

(44)

The volumetric flow rate of the stripping solution can beestimated from relation

VR0 = VR2 − VOA − VH2O,R (45)

where the flow rate of water formed in the neutralisation re-action of acids (both organic and mineral) can be expressedby equation

VH2O,R,n = nTA,S,nMH2O

ρH2O

= MH2O(VS,ncS,n − VS,n+1cS,n+1)(1 + YMA/OA)

ρH2O(46)

The amount of the transported mineral acid has to be es-timated from experimental data, e.g. from the ratio

YMA/OA = nMA

nOA(47)

oMY

t

c

wlai

˙S,n+1 = VS,n − VOA,n = 2 ¯V S − VOA

2(41)

The approach to equilibrium in MBSE is defined by equa-ion

n+1 = cS,n+1

c∗S,n+1

(42)

This allows the calculation of the concentration of soluten the solvent entering the extractor from its concentration inhe feed phase using equation

S,n+1 = cF,n+1DF,n+1βn+1 (43)

In MBSE of BA from aqueous solutions with a low contentf mineral salts this ratio can be equal to zero. For MBSE ofPCA from sulphate solutions at pH above 2 the value of

MA/OA is about 1 [11,18].The initial concentration of reagent in the stripping solu-

ion can be estimated from equation

reag,R0 = VF,ncF,n − VF,n+1cF,n+1

VR0ηconv= VF,ncF,nηe

VR0ηconv(48)

here ηconv is the conversion of reagent in the stripping so-ution, whose value has to be kept less than 1 (usually atbout 0.7) to guarantee an excess of reagent at the strippingnterface.


3.2. Design of HF contactors for MBSE and MBSS

3.2.1. MBSEThe number of parallel contactors in extraction is calcu-

lated under the consideration of optimal velocity of the feedin the fiber lumens. The number of parallel contactors calcu-lated by equation

Ncp,e = VF1

uFSf(49)

should be rounded off to an integer. Thus, the number ofparallel contactors has to be an integer and the feed velocityhas to be from an optimal interval, which should be estimatedexperimentally.

The length of fibers is estimated from Eq. (22), whichwas derived for constant mass-transfer and distribution co-efficients in the contactor. When this is not the case, thecontactor is divided into segments with the same concen-tration difference in the feed phase, where constant valuesof the mentioned quantities can be supposed. The lengths ofsuch segments are generally unequal. The value of the overallmass-transfer coefficient is estimated from Eq. (8), or anotherequation relevant for the given system. The mean concentra-tion in this segment, usually a logarithmic mean value, is forthe nth segment with inlet and outlet concentrations in thiss

c

cit

cioofipld

3

f

N

(a

N

4. MBSE and MBSS of butyric acid (BA)—a casestudy

4.1. Mass-transfer data

MBSE and MBSS of butyric acid (BA) with TOA as anextractant were studied in papers [10,14,16]. This system andrelated data will be used in the presented case study. Thesystem studied in these papers was follows:

Feed:Aqueous solutions of butyric acid, cF0 = 0.48–1.2 kmol m−3, pHF0 = 2.4–2.6 (without adjustment of pHwith mineral acid).

Solvent: 0.4 kmol m−3 of trioctylamine (TOA) in n-alkanes (dodecane fraction). The density and kinematicviscosity of the solvent at 30 ◦C were 751.9 kg m−3 and2.026 × 10−6 m2 s−1, respectively.

Stripping solution: Aqueous solution of NaOH with a con-centration of 2.5–3.0 kmol m−3.

Temperature: 30 ◦C.For simultaneous extraction and stripping, two hol-

low fiber contactors Liqui-Cel Extra-Flow 2.5 in. × 8 in.(Hoechst-Celanese, USA) with a cross-flow of phases wereused. Contactors were connected to the reservoirs of threephases in the recirculation mode, as described in papers[12,16,18] together with operation modes and analyticalmhaTtiopfi

vicow

FoE(

egment cF,n and cF,n+1 defined by relation

Fls,n = cF,n − cF,n+1

ln(cF,n/cF,n+1)(50)

For the estimation of the individual mass-transfer coeffi-ient in a fiber lumen by Eq. (12) a guess of the fiber lengths used. The calculated length is applied in the next iterationill a satisfactory agreement is reached.

In systems with the concentration dependent distributionoefficient equation(s) describing this dependence should bencluded into the calculation procedure for estimation of theverall mass-transfer coefficient. For a case study with MBSEf BA an appropriate model of L/L equilibrium with a goodt to experimental data, as shown in Fig. 6, is published in pa-er [22]. Another possibility, which usually simplifies calcu-ations, is to correlate experimental or simulated equilibriumata with an empirical interpolation polynomial.

.2.2. MBSSThe number of parallel contactors in stripping is estimated

rom the mean flow rate of the solvent in the shell ¯V S

cp,s = 2 ¯V Sdh,sρS

πReshellµSdshel,ls,sleff,s(51)

The length of fibers in the stripper is calculated from Eqs.28) and (29). The total number of contactors for extractionnd stripping is given by

ct = Ncs,e Ncp,e + Ncs,s Ncp,s (52)

ethods used. Celgard X-30 microporous polypropyleneollow-fibers were assembled in these contactors. The char-cteristics of the contactor and fibers are presented in Table 2.he stripping solution flowed through the fibers lumen and

he solvent in the shell. The overall mass-transfer coefficientsn MBSE and MBSS were evaluated from the concentrationf acid in the feed, solvent and strip reservoirs and in the out-uts from contactors, considering variable distribution coef-cients and experimentally determined driving forces.

The value of the distribution coefficient of BA in the sol-ent with TOA is concentration dependent, as can be seenn Fig. 5. This has to be taken into account in modeling andalculations. The proposed model related to the mechanismf extraction suggested in paper [22] fits experimental dataell. For purposes of the short-cut design following empirical

ig. 5. Distribution coefficient of BA vs. concentration of BA in the aque-us phase for the solvent with 0.4 kmol m−3 of TOA in n-alkanes at 30 ◦C.xperimental equilibrium data from paper [22] are fitted according to Eq.

53) with coefficients presented in Table 3 as shown by lines.


Table 3The values of coefficients of the empirical Eq. (53) for correlation of the distribution coefficient of BA based on aqueous (k= F) or solvent phase (k= S)concentrations for the solvent with 0.40 kmol m−3 TOA in n-alkanes

k A B C D E F G H I

F 0.1578 17.69 279.4 −2802 10434 −20287 21877 −12406 2889S 0.1578 21.25 −74.63 169.6 −227.5 176.1 −77.63 18.14 −1.745

equation was used

D = A + Bck + Cc2k + Dc3

k + Ec4k + Fc5

k + Gc6k

+Hc7k + Ic8

k (53)

which is more simpler than a mentioned physical model. Thisempirical equation fits experimental data well, as shown inFig. 5 by lines. The values of coefficients in Eq. (53) forestimation of the distribution coefficients from the aqueousphase concentration (k= F) or the solvent phase concentra-tions (k= S) are presented in Table 3.

Extraction and stripping were studied at different hydro-dynamic conditions. The experimental data from MBSE ofBA in a HF contactor show that the value of the overall mass-transfer coefficient in MBSE, Ke, depends on concentration(Fig. 6a), what is mostly a result of the concentration de-pendency of the distribution coefficient of BA. The valueof the overall mass-transfer coefficient in MBSS, Ks, practi-cally does not depend on the BA concentration in the solvent(Fig. 6b). With increasing velocity of the feed in the fiberslumen the value of Ke, increases and approaches a steadyvalue at a velocity of about 2.5 cm s−1, as shown in papers[10,14,16]. The main diffusion resistance in the solvent is inthe wall pores. With increasing volumetric flow rate of thesolvent in the extractor shell the value of Ke, increased onlyswttm(

FaBbbV

The experimental values of Ke are in a good agreementwith the values estimated from the diffusion model (Fig. 6a).This means that kinetics of the formation of BA–TOA com-plexes does not contribute to the overall resistance in MBSE.The value of Ks is more or less independent of the BA con-centration (Fig. 6b). The values of Ks calculated from thediffusion model are about two times higher than experimen-tal values (Fig. 6b). This difference is connected with theresistance associated with the kinetics of the stripping pro-cess, i.e., the chemical reaction of the complex decompositionand other interfacial phenomena. The estimated values of thestripping rate constant are presented in Table 1. The kineticmass-transfer resistance plays a decisive role in the overallresistance in MBSS of BA with its participation of about 60%in the overall resistance.

4.2. Calculation of HF contactors and processsimulations

HF contactors 10 in. × 28 in. are supposed in simulationswith characteristics shown in Table 3. The values of theprocess parameters were:cF1 = 450 mol m−3, VF1 = 1 m3 h−1,Reshell = 0.9, ηe = 0.90, βn+1 = 0.40, Z= 10, ηconv = 0.70 if nototherwise stated.

In all simulations no parallel modules are required for theaialmsfi

Fou

lightly and a steady value was approached soon. In case,hen the organic phase circulated in the shell of the stripper,

he value of Ks was slightly greater than in the mode wherehe solvent circulated in the stripper through the fibers lu-

en and is practically independent on the solvent flow rateFig. 7).

ig. 6. Influence of the concentration of BA in the feed on the over-ll mass-transfer coefficient in MBSE, (a) and for the concentration ofA in the solvent on the overall mass-transfer coefficient in the mem-rane based stripping (b). Mode of operation: solvent in the shell inoth contactors. cF0 = 0.453 kmol m−3, uF = 1.78 cm s−1, uR = 0.45 cm s−1,

˙S = 391 cm3 min−1.

ssumed feed flow rate. Thus, the total number of contactorss equal to the number of contactors in series. In extractioncounter-current flow of aqueous feed in the hollow fibers

umen and of the solvent in shell is supposed. The strippingodules are connected in series on the shell side where the

olvent flows co-currently with the stripping solution in theber lumen.

ig. 7. Influence of the volumetric flow rate of solvent and of the mode ofperation of the HF contactor on the overall mass-transfer coefficient of BA.

F = 1.8 cm s−1, cF0 = 0.453 kmol m−3, uR = 0.45 cm s−1.


Fig. 8. Total number of the HF contactors 10 in. × 28 in. in MBSE andMBSS of BA, as well as the overall number of contactors vs. the shellside Reynolds number for a different number of contactor segments withthe same concentration difference in the feed phase. Values of the processparameters were: cF1 = 450 mol m−3, VF1 = 1 m3 h−1, ηe = 0.90, βn+1 = 0.40,Z= 10, ηconv = 0.7.

The assumption of a parallel flow of phases in the extrac-tor, as used in section 0, simplifies the mass-transfer analysis.In reality, in the used modules the solvent flows in every halfof the contactor shell in a radial direction and at the shell walland between halves of the contactor it flows in an axial direc-tion and counter-currently to the aqueous phase flowing inthe fiber lumens, as it is between contactors. Assumption ofa counter-current parallel flow of phases could be a first ap-proximation of the real situation in the described contactorsfor which experimental data and correlations are available.Calculations and simulations were done as described in Sec-tions 3.1 and 3.2 and shown in Fig. 4.

4.3. Discussion of simulation results

The results of simulations are presented in Figs. 8–16.With increasing value of the Reynolds number in the shellthe number of contactors in series needed in extraction de-creases. However, the number of contactors, required in the

Ft(mta

Fig. 10. The value of the overall mass-transfer coefficient in individ-ual segments along MBSE contactors. The values of process parameterswere: cF1 = 450 mol m−3, VF1 = 1 m3 h−1, Reshell = 0.9, Ncp,e = 1, ηe = 0.90,βn+1 = 0.40, Z= 10, ηconv = 0.7.

Fig. 11. Influence of the number of segments, to which was divided theinterval from inlet to outlet concentrations of BA in the feed phase (cF1 tocF2) in calculations of the total number of HF contactors 10 in. × 28 in. inMBSE of BA. The values of process parameters are shown in Fig. 10.

stripping, increases. Thus, the overall number of contactors inMBSE and MBSS exhibits a minimum at about Reshell = 0.9,as shown in Fig. 8. The lengths of the contactor calculated foreach segment were more or less similar (slight increase withdecreasing the mean concentration in the segment). However,in the last segment they jumped to a much higher value dueto a great decrease in the driving force on the raffinate end of

Fig. 12. Total number of contactors 10 in. × 28 in. in MBSE and MBSS ofBA, as well as the overall number of contactors vs. approach to equilibriumon the outlet end of the contactor. The values of process parameters areshown in Fig. 10. The overall number of segments in the feed phase used inc

ig. 9. Influence of the shell side Reynolds number on the number of con-actors 10 in. × 28 in. in MBSE estimated for a selected contactor segmentsa) and on the value of the overall mass-transfer coefficient in individual seg-ents of MBSE contactors (b). The overall number of contactor segments in

he feed phase used in calculations was 40. The values of process parametersre shown in Fig. 8.
alculations was 40.


Fig. 13. Total number of contactors 10 in. × 28 in. in MBSE and MBSS ofBA, as well as the total number of contactors vs. yield of BA in MBSE. Theoverall number of segments in the feed phase used in calculations was 40.The values of process parameters are shown in Fig. 10.

Fig. 14. Influence of the yield of BA on the total number of contactors10 in. × 28 in. in MBSE estimated in selected individual segments in thefeed phase (a) and on the value of the overall mass-transfer coefficient inselected individual segments (b) for the values of process parameters shownin Fig. 10.

the contactor, as shown in Fig. 9a, as well as due to a decreasein the value of the overall mass-transfer coefficient, as shownin Fig. 9b.

The estimated length of fibers or the total number of HFcontactors, Nct is a function of the number of the segments towhich the contactor was divided in the calculation, as shownin Fig. 11. The convergence is good for more than 20 seg-ments in the studied system. The dependences of the number

Fig. 15. The value of the inlet/outlet driving force ratio in individual seg-ments along MBSE contactors for selected yields of BA in extraction (a) anddependence of this ratio on the yield of BA in MBSE in selected segments(b).

Fig. 16. Total number of HF contactors 10 in. × 28 in. in MBSE and MBSSof BA, as well as the overall number of contactors vs. inlet feed concentrationof BA to MBSE. The overall number of the segments in the feed phase used incalculations was 40. The values of process parameters are shown in Fig. 10.

of contactors in MBSE and MBSS are sensitive functions ofthe approach to equilibrium on the raffinate end of the ex-tractor (defined by ratio in Eq. (42)), as shown in Fig. 12,with the optimal value of βn+1 of about 0.4. From simulationpresented in Fig. 13 it is evident that the number of contac-tors in MBSE and MBSS is very sensitive to the increasein BA yield in MBSE above 0.85. A more detailed analysisshowed that the largest part of fiber length required belongsto the last segment with the lowest mean concentration, whatis connected with a decrease of the distribution coefficient ofBA and consequently of Ke and with a decrease of the driv-ing force in MBSE at concentrations below 0.18 kmol m−3,as shown in Figs. 14 and 15.

The total number of HF contactors 10 in. × 28 in. in MBSEand MBSS of BA, as well as the overall number of contac-tors, monotonously decrease with the increasing inlet feedconcentration, as shown in Fig. 16. This is connected with anincrease in the value of the distribution coefficient, as well asof the overall mass-transfer coefficient in the extractor.

For 85% yield of BA in MBSE, which is from the appli-cation point of view reasonable, and for following processparameters: VF1 = 1 m3 h−1, cF1 = 350 mol m−3,Reshell = 0.9,βn+1 = 0.40, Z= 10, YMA/OA = 0, and ηconv = 0.70 the numberof contactors of Liqui-Cel type 10 in. × 28 in. (Celgard) inseries, without parallel lines of contactors, is five in MBSEati

5

tiMph

nd eight in stripping. In case when two parallel lines of con-actors are used the number of contactors in series is 4 (3.91)n extraction and 2 in stripping.

. Conclusions

The presented short-cut method for simulation of HF con-actors in the circuit with simultaneous MBSE and MBSSs a useful tool for the design of simultaneous MBSE and

BSS systems. In the systems with the concentration de-endent mass-transfer and distribution coefficients contactoras to be divided to segments preferably with the same in-


let to outlet concentration difference. For the tested systemwith butyric acid, convergence of the simulation results wasachieved for more than 20 segments. Dependences of the totalnumber of contactors (length of fibers) on the shell Reynoldsnumber and approach to the equilibrium at the raffinate endof the MBSE contactor exhibit a minimum at about 0.9 and0.40, respectively. The number of contactors in MBSE andMBSS is very sensitive to the increase in BA yield in MBSEabove 85%.

Acknowledgements

Support of the grant from the Slovak Grant Agency No.VEGA 1/9136/02 is acknowledged.

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