Desalination 342 (2014) 75–84

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Desalination

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Modeling the transport of sulfuric acid and its sulfates (MgSO4, ZnSO4,Na2SO4) through an anion-exchange membrane

Stanisław Koter ⁎, Monika Kultys, Barbara Gilewicz-Łukasik, Izabela KoterFaculty of Chemistry, Nicolaus Copernicus University, 7 Gagarina St., 87-100 Toruń, Poland

H I G H L I G H T S

• Selectivity of membrane Selemion AAV in mixtures of H2SO4 and its salts was studied.• Leakage of cations (Mg2+, Zn2+, Na+) on the AAV anode side in electrodialytic process• Model of the H2SO4-sulfate transport through an anion-exchange membrane• Association M·SO4 in the sulfate transport through AAV should be taken into account.

⁎ Corresponding author. Tel.: +48 56 6114318.E-mail address: skoter@umk.pl (S. Koter).

0011-9164/$ – see front matter © 2013 Elsevier B.V. All rihttp://dx.doi.org/10.1016/j.desal.2013.10.025

a b s t r a c t

a r t i c l e i n f o

Article history:Received 29 July 2013Received in revised form 10 October 2013Accepted 24 October 2013Available online 13 November 2013

Keywords:Anion-exchange membraneSulfuric acidSulfatesCation leakageExtended Nernst–Planck equationModelingElectrodialysis

The separation of sulfuric acid from its sulfates (MgSO4, ZnSO4, Na2SO4) by the electrodialytic methodwas inves-tigated. The weak-basic anion-exchange membrane Selemion AAV was used. It was found that the current effi-ciency of anion transport (HSO4

−, SO42−) was independent on the kind of metal cation present on the cathode

side of membrane. Despite the presence of the external electric field acting against the concentration force, thepermeability ofmetal cations to the anode side of AAV in the electrodialytic processwas close to that in a diffusiveone. The developed transport model based on the Nernst–Planck equation and Donnan equilibria satisfactorilydescribed the permeation of acid. To describe suitably the transport of metal sulfates, the assumption abouttheir full dissociation is insufficient and the metal cation–sulfate anion association should be taken into account.

© 2013 Elsevier B.V. All rights reserved.

1. Introduction

In the metal surface treatment for electroplating purposes strongchemical agents (e.g., inorganic acids) and large amounts of waterare used. Taking into account economical and environmental aspects,a recovery of spent solutions and water purification is a very importantissue. For those purposes variousmembrane techniques can be used [1]including the techniqueswith ion-exchangemembranes, like electrodi-alysis (ED) [2–4]. The recovery of acids ormetals fromeffluents can sub-stantially reduce a production cost and a pollution load. The efficiency ofelectrodialytic acid recovery from the mixtures with its metal salts de-pends mostly on the selectivity of an anion-exchange membrane(AM). The electrodialytic transport of strong acids through AMswas in-vestigated experimentally by Gavach and Pourcelly (HCl [5], H2SO4 [6]).The transport of H2SO4 solutions with metal sulfates was mainly inves-tigated in the context of diffusion dialysis (H2SO4 + MgSO4 [7],

ghts reserved.

H2SO4 + ZnSO4 [8], H2SO4 + CuSO4 [9]). Only a few papers discussedthe transport of such mixtures in the processes driven by an electricforce. One of themwas a work by Lorrain et al. [10] where the influenceof cations (Na+, Cr3+, and other) on the anode side of AM (ARA mem-brane) on the proton leakage was studied.

In this work we continue our investigations on the permeation ofacids through a weak-base anion-exchange membrane (SelemionAAV, Asahi Glass Engineering Co.). In the previous works we studiedthe transport of: pure H2SO4 [11], mixture of H2SO4 and H3PO4 [12],and of HCl and H3PO4 [13]. Here, a transport of H2SO4 in the presenceof metal cations (Mg2+, Zn2+, Na+) and their leakage from the cathodeto anode side of anion-exchange membrane in the electrodialytic pro-cesses is investigated from both the experimental and theoreticalpoint of view. The dependence of the membrane selectivity on the cur-rent density, H2SO4 concentration, and the kind of metal cations will bedetermined. The experimental results will be described by the modelbased, as previously, on the extended Nernst–Planck equation and theDonnan equilibria. The different assumptions regarding the model pa-rameters will be checked with respect to the goodness of fit of the ex-perimental data.

76 S. Koter et al. / Desalination 342 (2014) 75–84

2. Theory

The investigated system is shown in Fig. 1. The aqueous solutionof H2SO4 and ZnSO4 mixture on the cathode side of anion-exchangemembrane (AM) is separated from the cathode by a bipolar mem-brane (BM) to prevent a reduction of Zn2+ to Zn on the cathode. Inthe external electric field, the anions HSO4

− and SO42− migrate from

the middle solution through AM to the anode solution, where theirnegative charge is neutralized by protons formed in the anode reac-tion 1/2H2O → H+ + 1/4O2 + e−. In BM, a splitting of water takesplace; the formed protons migrate to the cathode side, whereas thehydroxide anions move to the middle solution where they neutralizeprotons. As a result, the H2SO4 concentration in that solution decreaseswhereas in the anode solution increases. Because of high concentrationdifference ofmetal cations they can diffuse through AM to the anode so-lution against the applied electric field. In the model calculations it willbe assumed that BM works ideally, i.e., the current efficiency of watersplitting is 100%, and no leakage of ions through that membrane takesplace. Thus, only the ion transport through AM will be modeled.

In order to calculate the ion fluxes through AM, in our previous pub-lications [11–13] the steady-state was assumed. However, for the ionmixtures discussed in this paper it was too difficult to find the ion con-centration profiles of ions fulfilling this assumption. Therefore, here itwas necessary to solve the continuity equation for the transport of spe-cies in the membrane pores:

∂ci∂t ¼ −∂ Ji

∂x i ¼ S;4 ð1Þ

where ci is the concentration of i-th species in the pore solution, Ji— fluxof species i through the unit cross-section area of pores,x— x-coordinatealong the pore, and t — time. The subscript 4 denotes metal cations,S — sulfates and bisulfates, i.e., cS ¼ c2 þ c3 and JS ¼ J2 þ J3, where 2is HSO4

− and 3 — SO42−. Eq. (1) was rewritten to the form:

∂ci∂t ¼ − 1

Vplm

∂ Ji∂x i ¼ S;4 ð2Þ

where Ji is the flux referred to the unit membrane area, related to Ji byJi ¼ Xp Ji, Xp is the surface fraction of pores, x is the dimensionless co-ordinate, x ¼ x= θlmð Þ, θ is the pore tortuosity — the ratio of the porelength to the membrane thickness, lm, and Vp is the volume fraction

½H2O

¼O2

H+

e-

H+

½H2

e-

H+

BM

+ H+

OH-

H+

AM

H2O

HSO4-

SO42-

Mz+

ca mc an

H2O H2O

Fig. 1. Scheme of membrane module and the basic ion transport; “ca” — 0.5 M H2SO4 so-lution; “mc” —mixture of H2SO4 and its salt, “an” —H2SO4 solution, AM— anion-exchangemembrane, and BM— bipolar membrane.

of pores (Vp = Xpθ). As Ji is given by the extended Nernst–Planckequation, Ji can be written as [10]:

Ji ¼ Xp Ji ¼ −km;iDidcidx

þ ziciFRT

dφdx

� �þ ci Jv i ¼ 1;…;4 : ð3Þ

The indices 1, …, 4 denote H+, HSO4−, SO4

2−, and metal cation, re-spectively. km,i is a combination of basic membrane parameters in-cluding the diffusivity ratio of i-th ion:

km;i ≡1lm

Vp

θ2Di

Di: ð4Þ

In the above equations, Di and φ are diffusion coefficient of i-thion, electric potential in the pore solution, respectively. Di is a refer-ence diffusion coefficient of i-th ion; it will be calculated from thelimiting ionic conductivity [14]. F and R are Faraday and gas constant,respectively, T is absolute temperature, and Jv is the volume flux. Inthe case of galvanostatic condition, dφ=dx in Eq. (3) can be replacedwith the current density, j, using the relation: j = F∑i = 1

4 zi Ji. ThusEq. (3) is transformed to:

Ji ¼ −km;iDidcidx

− ziciX4k¼1

km;kDkz2kck

jFþ zmcm Jv þ

X4k¼1

km;kDkzkdckdx

!0@ 1Aþ ci Jv:

ð5Þ

In the derivation of Eq. (5) the electroneutrality condition,

∑4i¼1zici þ zmcm ¼ 0, was used; here zm, and cm denote the charge

number and concentration of fixed charges, respectively. The con-centrations of ions H+, HSO4

− and SO42− in the pore solution are the

following functions of cS and c4:

c1 ¼ 12

cS−z4c4−zmcm þ zmcm þ K2;c þ z4c4−cS� �2−4K2;c zmcm þ z4c4−2cSð Þ� �1=2� �

c2 ¼ cSc1= c1 þ K2;c

� �c3 ¼ cS−c2

ð6a;b; cÞ

The Eqs. (6a–c) were derived using the expression for the 2nd con-centration dissociation constant of H2SO4, K2;c ¼ c1c3=c2c ( c =1 mol dm−3) and the abovementioned electroneutrality condition. As-suming the ideal Donnan equilibria, K2;c is equal to K2,c for the free so-lution. In the calculations it is also assumed that K2,c is constant andequal to K2 = 0.0105 [11], whereas the salts are completely dissociated.

The boundary conditions of Eq. (1) (ci 0; tð Þ, ci 1; tð Þ) result from theideal Donnan equilibria [10]:

ci 0; tð Þ=c′i tð Þ� �1=zi ¼ ck 0; tð Þ=c′k tð Þ

� �1=zkci 1; tð Þ=c″i tð Þ� �1=zi ¼ ck 1; tð Þ=c″k tð Þ

� �1=zk ð7a;bÞ

where c′i(t) and c″i(t) are the ion concentrations on the left and rightside of the membrane at time t. Combining Eq. (7) with theelectroneutrality condition inside the membrane, the polynomial ofthe 3-rd (monovalent cation) and 4-th degree (divalent cation)with respect to c1 is obtained. Having c1 , the concentrations ofother ions at the boundaries are calculated from Eq. (7a, b). The pro-file ci x;0ð Þ was taken as that established after a longer time forstarting conditions of experiment. The calculations were performedin the following way: at t = 0 for a given Δt Eq. (2) was solved,JS = J2 + J3 and J4 were obtained from Eq. (5) and then the new

77S. Koter et al. / Desalination 342 (2014) 75–84

concentrations of sulfates and metal cations on both sides of AMwere calculated. Jv in Eq. (5) and the mass changes of solutionswere calculated from the mass fluxes Jm,AM, Jm,BM (details are inAppendix A). Next, the process time was increased by Δt and the cal-culations were repeated until the end of process. The value of Δtwaschosen in such a way that a further decrease of Δt did not change thecalculated concentrations. The block diagram of these calculations isshown in Fig. 2. The model parameters – concentration of fixedcharges including their sign, zmcm , km,1 (H+), km,4 (metal cation)and km,S (common for HSO4

− and SO42−, km,S ≡ km,2 = km,3) – were

Fig. 2. Block diagram of model calculations; m′ denotes the concentration express

found by minimizing the sum of squared residuals using the Simplexmethod [15]:

SSR ¼Xi¼S;4

wi;cm

Xk

ci;cm; exp;k−ci;cm;mod;k

� �2 þwi;an

Xk

ci;an; exp;k−ci;an;mod;k

� �2 !¼ min

ð8Þ

where ci,α,exp,k is the experimental concentration, ci,α,mod,k — modelconcentration of species i in the compartment α at time tk, and wi,α —

ed in moles per kg of solution (mol kg−1), the current density, j, is negative.

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.40.0

0.2

0.4

0.6

0.8

1.0

c a / c

a,m

c,0

c a / c

a,m

c,0

c a / c

a,m

c,0

nF / n

a/2,mc,0

nF / n

a/2,mc,0

nF / n

a/2,mc,0

AAV, H2SO

4 - MgSO

4

E3

E1

E2E4

E3

E2E1

E4

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

0.0

0.2

0.4

0.6

0.8

1.0 AAV, H2SO

4 - ZnSO

4

E2 E1

E3

E3

E1

E4

E4

E2

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.40.0

0.2

0.4

0.6

0.8

1.0AAV, H

2SO

4 - Na

2SO

4

E3

E1E2

E4

E3

E1

E2

E4

a

b

c

78 S. Koter et al. / Desalination 342 (2014) 75–84

weight factor. Because of a very low concentration of metal cations in theanode solution, for those concentrations it was assumedw4,an = 104 andfor other concentrations — wi,α = 1. The calculations were performedusing the program Mathematica® ver.7 (Wolfram). It should benoted that the calculations were time-consuming — the calculationof a one process run lasted even ca. 1 h (PC Procesor Intel® Core™i5 750 @ 2.67 GHz).

3. Experimental

The experiments were performed using the membrane module(FumaTech GmbH) equipped with the weak-basic anion-exchangemembrane Selemion AAV (Asahi Glass Engineering Co., Japan) andthe bipolar membrane Neosepta BP-1 (Tokuyama Corp., Japan). Theexperimental setup was the same as that in [13], only the order ofmembranes in the module was reversed. Here the anion-exchangemembrane (AM) was separated from the cathode by a bipolar mem-brane (BM) to prevent the reduction of metal cations on the cathode(Fig. 1). In all the experiments the catholyte solution (“ca”) was0.5 M H2SO4. On the cathode side of AM (“mc”) a concentrated mix-ture of H2SO4 and its metal salt was circulated. In the anode compart-ment (“an”) a dilute H2SO4 solution (ca. 0.05 M at the beginning ofexperiment) was circulated. The parameters of experiments aregathered in Table 1. The acid concentration was determined by thepH-metric titration of samples (0.3–1 ml) taken during the processfrom each chamber. Another set of samples (each 1 ml) was takenfor the determination of metal cation concentrations by the AAS. Inorder to determine the mass flow through AM (Appendix A), all thesolutions were weighed before and after the experiment. A part ofthe experimental results presented here was published in our previ-ous work [16].

4. Results and discussion

4.1. Current efficiency

The H2SO4 concentration changes during the electrodialysis processare shown in Fig. 3. They are plotted in the coordinates: the H2SO4

concentration divided by its initial concentration in the middle com-partment, ca/ca,mc,0, versus the number of moles of elementarycharge passed through the membrane, nF, divided by the initial num-ber of moles of 1/2H2SO4 in the middle compartment, na/2,mc,0. Theslope of these curves is directly related to the current efficiency ofacid removal from the middle compartment or acid production inthe anode compartment, CEa,α, defined by:

CEa;α ≡ � Vα;0dca=2;αdnF

¼ � Vα;0

Vcm;0

dca;α=ca;mc;0

dnF=na=2;mc;0α ¼ an;mc : ð9Þ

In Eq. (9) the positive sign is for α = an and negative for α = mc.As the relation ca/ca,mc,0 = f(nF/na/2,mc,0) (Fig. 3) is approximately linear,

Table 1Parameters of experiments: j — current density, ca,mc,0, ca,an,0, csalt,mc,0, csalt,an,0 — initialconcentrations of H2SO4, metal sulfate in the middle (“cm”) and anode (“an”)compartments, volumes of circulating solutions: Vca,0 ≈ Vmc,0 ≈ Van,0 ≈ 0.3 dm3; activemembrane area Sm = 49 cm2.

Exp. j [A/m2] ca,mc,0 [M] csalt,mc,0 [M] ca,an,0 [M] csalt,an,0 [M]

D 0 0.5 0.5 0 0E1 1000 0.5 0.5 0.05 0E2 500 0.5 0.5 0.05 0E3 1000 1 0.5 0.05 0E4 500 1 0.5 0.05 0

Fig. 3. Concentration changes of H2SO4 in the middle (filled symbols) and anode (hollowsymbols) compartments during electrodialysis experiment with the AAV membrane inthe presence of MgSO4 (a), ZnSO4 (b), and Na2SO4 (c); dash line represents the modelsimulation.

it means that CEa,an and CEa,cm are almost constant. They are different invalues because of the electroosmotic flow of water changing the con-centrations on both sides of themembrane. However, as these differ-ences are small (within a few percent) only the mean currentefficiency, CEa = (CEa,mc + CEa,an) / 2, is shown in Table 2.

It is seen that AAV shows rather low selectivity towards permeationof protons in the mixtures of H2SO4 and its salts. The kind of cation

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.40.0

0.2

0.4

0.6

0.8

H2SO

4 - MgSO

4

c Mg,

an /1

03 M

nF / n

a/2,an,0

nF / n

a/2,an,0

nF / n

a/2,an,0

E1

E2

E3E4

AAV

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.40.0

0.2

0.4

0.6

0.8

c Zn,

an /1

03 M

E4 E3

E2

E1

AAV

H2SO

4 - ZnSO

4

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.40.0

0.2

0.4

0.6

0.8

1.0

1.2

H2SO

4 - Na

2SO

4

0.5c

Na,

an /1

03 M

E1

E2

E3

E1

AAV

a

b

c

Table 2Mean current efficiency of H2SO4 recovery, CEa = (CEa,mc + CEa,an) / 2, for the AAVmem-brane in the mixtures of H2SO4 and its metal salts.

Exper. CEa

Mg [16] Zn Na [16]

E1 0.53 0.53 0.53E2 0.55 0.55 0.56E3 0.49 0.48 0.49E4 0.51 0.53 0.53

79S. Koter et al. / Desalination 342 (2014) 75–84

(Mg2+, Zn2+, Na+), present on the cathode side of AAV, practicallyhas no influence on CEa. In the experiments of lower current density(E2, E4) CEa is slightly higher than that determined for E1 and E3. Itresults from the longer time of experiments E2 and E4 which in-creases the positive diffusion contribution to CEa. A higher initialacid concentration (E3, E4 vs. E1, E2) results in a lower CEa due tothe increased sorption of protons into the membrane. CEa obtainedin the same conditions (j = 1000 A/m2, ca,mc,0 = 1 M) for theH2SO4 + MSO4 mixture (E3 in Table 2: CEa = 0.48–0.49) is lowerthan that obtained previously for pure H2SO4 solutions (CEa = 0.56[11]). However, according to the Donnan equation the concentrationof protons inside the membrane is lower in the presence of salt,hence one would expect a higher CEa in the case of mixtures. Indeedthemodel calculations confirm this supposition. Thus, for a lower CEaof AAV in the mixtures the membrane aging should be responsible(in both investigations the same sample of AAV was used).

The mass flow during the electrodialytic experiments is shownin Table 3. Similarly as CEa, the mass electroosmotic coefficient,FJm,AM / j, is practically independent on the kind of cation. A lowervalue of FJm,AM / j for E3 coincides with a lower value of CEa forthat experiment. It is interesting to examine what is the main con-tribution of that flow. As CEa is expressed in moles of 1/2H2SO4 per1 F (Eq. (9)), for CEa ≈ 0.5 we get 1/4 mol or 24 g of SO4

2− whichwas transferred through AAV per 1 mol of elementary charge passedthrough the membrane. Thus, the electroosmotic coefficient of waterwould be (40–24) g/mol ≈ 1 mol of H2O per F or 4 mol of H2O per1 mol of SO4

2−. Taking the hydration number of SO42− to be 2 [17] it

means that 1 SO42− drags at least 2 additional H2O. This value should

be increased, because themovement of protons, responsible for the de-creased selectivity of AAV, induces a transfer of water in the opposite di-rection to the movement of SO4

2−. The drag of water by protons is notnegligible, e.g., for proton-conducting membranes it can be 1.4–2(membrane Dow) or 2–3 H2O/H+ (Nafion 117, 30 °C) [18] and in-creases with temperature due to an increasing amount of protonstransported through a vehicle mechanism [19,20].

4.2. Leakage of cations

The salt concentration changes on the anode side of AAV duringthe electrodialytic process (E1–E4) are shown in Fig. 4. The modelsimulation curves are denoted by dash line; they will be discussed

Table 3Mass flow through the anion-exchange membrane AAV.

Exper. Jm,AM [g m−2 s−1] FJm,AM / j [g mol−1]

Mg Zn Na Mg Zn Na

E1 0.42 0.39 0.45 40 38 43E2 0.23 0.23 0.24 44 44 46E3 0.38 0.37 0.37 36 36 36E4 0.21 0.21 0.23 41 41 44D 0.02 0.00 0.03

Fig. 4.Concentration changes ofMgSO4 (a), ZnSO4 (b), andNa2SO4 (c) on the anode side ofAAV during electrodialysis experiments (E1–E4, Table 1); dash line — model calculationswith parameters in Table 7.

later. The data are plotted versus nF/na/2,mc,0, as in Fig. 3. This coordi-nate provides a more transparent data presentation comparing tothe time coordinate. As the current is constant, the character of con-centration changes in both coordinates is the same. Comparing thedata presented in Fig. 4 with those in Fig. 5 it is seen that in both elec-trodialytic (E) and diffusional (D) processes these changes areroughly linear.

0.00

0.01

0.02

0.03

0.04

0 100 200 300 4000.0

0.2

0.4

0.6

0.8

1.0

c salt,

an /1

03 M

t /min

Na

c a,an

/M

AAV, H2SO

4+MSO

4

Mg

Zn

acid(Zn)acid(Mg)acid(Na)

Fig. 5. Time changes of concentrations of MgSO4, ZnSO4, and H2SO4 on the anode side ofAAV during the diffusion experiments in the presence of H2SO4; the salt permeabilitiesare listed in Table 4.

80 S. Koter et al. / Desalination 342 (2014) 75–84

To compare the cation leakage through AAV to the anode side, themean permeability of the i-th cation, ePi , was determined using the ex-pression [13]:

ePi ≡1

Smtp

Ztp0

1Δci

d ci;anVan

� �dt

dt≈ 1Smtp

d ci;anVan

� �dt

Xrj¼1

12

1Δci; j−1

þ 1Δci; j

!Δt j:

ð10Þ

In Eq. (10) Δtj = tj − tj−1 and Δci,j = ci,j,mc − ci,j,an are theconcentration difference through AAV at time tj, tp is the processtime, and r — number of experimental points. It is assumed thatd(ci,anVan) / dt = const. The same formula was used to calculatethe diffusional permeability of H2SO4, ePa . All permeabilities aregathered in Table 4. It is observed that the leakage of monovalentcations Na+ is ca. two times higher than that of divalent ones.One of the reasons is a much higher sorption of Na+ which resultsfrom the two times higher concentration of the Na+ ions in the externalsolution (in the experiments the same concentrations of Na2SO4 andMSO4 were used). Another reason could be a more reduced permeabil-ity of divalent cations through a positively charged polymer network,comparing to the monovalent one.

Permeabilities of cations in the E and D processes are similar de-spite the action of the electric field acting against the concentrationforce. For Na+ ePNa Dð Þ is even significantly lower than ePNa Eð Þ . It isalso observed that at a higher current density and the same acid con-centration (E1 vs. E2, E3 vs. E4) the cation permeability is slightlyhigher. This apparently unexpected behavior could be explained bythe increase of convective part of the ion flux which is mainly causedby the electroosmotic water flow induced by the anion migration

Table 4Permeability coefficients of Mg2+, Zn2+, Na+, and H2SO4 calculated from Eq. (8) for theAAV membrane in the mixtures of H2SO4 and its metal salts.

Exper. ePi [10−8 m s−1]

Mg [16] Zn Na [16]

E1 0.29 0.28 0.80E2 0.25 0.22 0.74E3 0.33 0.31 0.72E4 0.28 0.28 0.60D 0.36 0.31 0.47

ePa [10−8 m s−1]D 11 11 9.7

towards the anode side. Indeed, in the experiments E1 and E3 the ob-served mass flow through the membranes is almost two times higherthan in E2 and E4 (Table 3). The contribution of osmotic flow is negligi-ble because the osmotic mass flux determined in the experiment D isapproximately one order lower than the electroosmotic one and con-trary to the electroosmotic one. The inequality ePNa Eð ÞNePNa Dð Þ could beexplained by the association of Na+ and SO4

2− giving anions NaSO4−

(see the section “Modeling”) which would enhance the sodium trans-port to the anode side.

The diffusional permeability of H2SO4, ePa (Table 4) is one orderhigher than that of salts. There are two basic factors influencing the elec-trolyte permeability — mobility and concentration of ions inside themembrane. According to theDonnan equation the concentration of pro-tons inside the membrane at comparable concentrations of H2SO4 andits salt in the mixture is even lower than that of divalent cations. Thus,this indicates that the relative mobility of protons with respect to themetal cations is much higher inside these membranes than in a freesolution.

4.3. Modeling

There are 4 model parameters — concentration of fixed chargesincluding their sign, zmcm , km,1 (H+), km,4 (metal cation) and km,S

(common for HSO4− and SO4

2−, km,S ≡ km,2 = km,3). It should benoted here that the parameters zmcm , km,1 and km,S can be stronglycorrelated with each other. The zmcm parameter acts in such a waythat with its increase the transport of protons through the mem-brane goes down. On the other hand, the increase of km,1 and de-crease of km,S enhance the proton leakage. Thus, the fit of the H2SO4

concentration changes during electrodialytic experiments (Fig. 3)is possible for a wide range of the fixed charge concentration, evenfor zmcm ¼ 4 M. However, in that case an unreasonably high discrep-ancy between km,H and km,S is obtained. E.g., for the solution H2SO4–

MgSO4 and the experiment E1, the ratio km,H/km,S increases from ca.0.5 for zmcm ¼ −0:25 Mup to 100 for zmcm ¼ 4 M. To check which pa-rameters of the model are important, 3 different assumptions aboutthe model parameters were tested:

1) km,H = km,S = km,M (Table 5),2) km,H = km,S (Table 6), and3) km,H, km,S, km,M are fitted independently (Table 7).

Unfortunately, with the first two assumptions it was not possibleto fit the metal cation concentrations properly. Both approachesyielded the fitting curves of the exponential character, similar tothe model curves obtained for zmcm ¼ 0 or 4 M, seen in Fig. 6a. Inthe 2nd approach, a strong correlation between the parameters km,i

was found. To demonstrate it, for the experiments E1 and E2 twosets of parameters giving a similar fit of data are shown in Table 6.It is seen that lmkm of H+ andmetal cations can be significantly differ-ent for the same value of zmcm, however these discrepancies are lim-ited. zmcm is weakly correlated with km of the metal cations; e.g., forE4-Mg and assuming zmcm ¼ 0:08 M (as for E3) it was not possible tofind the values of km,i giving the same fit as for zmcm ¼ −0:02 M.

The first two approaches yield comparable values of zmcm(Tables 5 and 6). They are smaller than those obtained for pure

Table 5Optimal model parameters for the assumption km,1 = km,S = km,4 (= km); the AAVmem-brane.

Exper. H2SO4–MgSO4 H2SO4–ZnSO4 H2SO4–Na2SO4

zmcm/M lmkm zmcm/M lmkm zmcm/M lmkm

E1 0.13 0.074 0.13 0.074 0.15 0.048E2 0.13 0.040 0.11 0.039 0.16 0.027E3 0.10 0.048 0.11 0.044 0.17 0.026E4 −0.01 0.025 0.01 0.025 0.13 0.015

Table 6Optimal model parameters for the assumption km,1 = km,S; the AAV membrane.

Exper. H2SO4–MgSO4 H2SO4–ZnSO4 H2SO4–Na2SO4

lmkm,i lmkm,i lmkm,i

zmcm/M H+, sulfates Mg2+ zmcm/M H+, sulfates Zn2+ zmcm/M H+, sulfates Na+

E1 0.10 0.034 0.007 0.11 0.050 0.011 0.12 0.034 0.0090.050 0.012 0.065 0.021 0.040 0.013

E2 0.094 0.030 0.008 0.13 0.038 0.022 0.15 0.024 0.0090.035 0.011 0.039 0.028 0.027 0.015

E3 0.080 0.028 0.008 0.12 0.044 0.037 0.15 0.018 0.007E4 −0.022 0.023 0.015 0.041 0.026 0.055 0.12 0.014 0.009

81S. Koter et al. / Desalination 342 (2014) 75–84

H2SO4 – zmcm ¼ 0:1 vs:0:4 M– at the same experimental conditions (inthis work— E3, in [11]— E1). This is in accordance with a lower currentefficiency of acid removal observed for the H2SO4-salt mixtures.

Setting km,H, km,S, km,M, and zmcm as the independent parameters(third approach), the fit of metal cation concentration in the anode so-lution is improved, especially that of Na+, whereas the fit of divalentcation concentrations is not always acceptable (Fig. 4). However, thisimprovement was possible only for negative values of zmcm (Table 7).Obviously, it is in contradiction with the character of the investigatedmembranes. One could guess that the divalent sulfates partially neutral-ize the fixed charges creating negative ionic sites which facilitate themetal cation transport.

In Fig. 6, the influence of zmcm on the fit of cMg,an for the experi-ment E1-Mg is shown. It is seen that for zmcmN0 the model curvecMg,an = f(nF) has an exponential character, and when zmcm de-creases it becomes more and more linear. However, regarding diva-lent cations, for the experiments with a higher initial H2SO4

concentration (cS,mc,0 = 1 M, E3 and E4) it was not possible to obtaina linear dependence by decreasing zmcm . E.g., for E4-Zn it was checkedthat even setting zmcm ¼ −0:7 M no linear dependence cZn,an = f(nF)was obtained. Thus, another reason of the approximately linear changesof cation concentrations should be found. Themost reliable explanation,which does not require the negative values of zmcm, seems to be the dif-fusion of the cation-sulfate associates through the membrane. Accord-ing to Kotrly and Sucha [21] (they cited the work of Smith and Martell[22]), there are following equilibria in the aqueous solutions of sulfates:

Naþ þ SO42−↔Naþ � SO4

2− logKas ;c ¼ 0:7 I ¼ 0ð ÞMg2þ þ SO4

2−↔Mg2þ � SO42− logKas ;c ¼ 1:01 I ¼ 0:7ð Þ

Zn2þ þ νSO42−↔Zn2þ � SO4

2−� �

ν:

In the last case for ν = 1, …, 4 the values of log Kas,c,ν(I = 1) are0.89, 1.2, 1.7, and 1.7, respectively. In parenthesis the value of ionicstrength is given for which a value of logarithm of the concentrationequilibrium constant is valid. Taking these values one can estimatethat in the mixtures H2SO4-sulfate there can be a significant amountof the neutral associates M2+·SO4

2− (M = Mg, Zn) and in the case ofNa2SO4 and ZnSO4 the anions Na+·SO4

2− and Zn2+·(SO42−)ν, ν = 2,

3, 4, will also occur. E.g., for the MgSO4 solution of the concentrationcorresponding to I = 0.7 (0.35 M) the molar fraction of associate

Table 7Optimal model parameters for the independent fit of zmcm , km,1, km,S and km,4; the AAV membr

Exper. H2SO4–MgSO4 H2SO4–ZnSO4

zmcm/M lmkm,i zmcm/M lmkm,i

H+ HSO4− SO4

2− Mg2+ H+

E1 −0.25 0.022 0.042 0.008 −0.20 0.021E2 −0.30 0.019 0.045 0.017 −0.30 0.012E3 −0.26 0.033 0.054 0.042 −0.25 0.021E4 −0.30 0.016 0.024 0.014 −0.30 0.018

Mg2+·SO42− is ca. 0.6, whereas for ZnSO4 (I = 1) the analogous frac-

tion would be 0.4, the fraction of Zn2+·(SO42−)2 − 0.1. Thus, the trans-

port of such species through the membrane would be responsible forthe linear changes of cation concentrations. Regarding the model, oneshould add two more parameters — the association constant, Kas,c, andkm of the associate. Such amodel will fit the experimental data correctlywithout having to change the sign of zmcm . It is also very likely thattaking the associates transport into account, the first model approach(km,H = km,S = km,M) will be able to fit the data satisfactorily. This sup-position is supported by the fact that using the first approach the con-centrations of metal cations were always underestimated. However,because of too lengthy calculations the results for this model modifica-tion are not presented here.

For a better understanding the ion transport through AAV fromthe cathode to anode side, the components of the magnesium andsulfate ion fluxes through AAV were calculated for the experimentE1 (Table 8). According to Eq. (5), one can split Ji into the threeparts — the electric current, diffusion, and convective ones:

Ji ¼ Ji dck=dx ¼ 0; Jv ¼ 0ð Þ þ Ji j ¼ 0; Jv ¼ 0ð Þ þ Ji dck=dx ¼ 0; j ¼ 0ð Þ:ð11Þ

It should be noted that the convective part results mainly from theelectroosmotic flow of water (the osmotic contribution is negligible asit is seen in Table 3 — mass flow) which is induced by the migration ofions; thus, it can also be regarded as the electric current constituent.The contributions expressed as the ratios – xj = Ji(dck / dx = 0,Jv = 0) / Ji, xdif = Ji(j = 0,Jv = 0) / Ji, and xv = Ji(dck / dx = 0,Jv = 0) / Ji – calculated for the cathode and anode sides of AMusing the 2-parameter fit are presented in Table 8.

It is seen that in the case of the total flow of counterions, HSO4−

and SO42−, on both sides of the membrane the dominant component

is always the electric one. One can notice that on the cathode side thediffusional part of the flow is negative, although the concentration ofsulfates on the cathode side is higher than that on the anode side. Itmeans that the concentration profile has a maximum somewhere in-side the membrane. Regarding the flow of coions Mg2+, there aretwo dominant components on the cathode side— electric, xj, and dif-fusion, xdif, ones. They cancel each other to a large extent, as it can bededuced from the high absolute values of xj and xdif. The diffusion andconvection parts together dominate over xj; consequently, the flow

ane.

H2SO4–Na2SO4

zmcm/M lmkm,i

HSO4− SO4

2− Zn2+ H+ HSO4− SO4

2− Na+

0.037 0.006 −0.27 0.018 0.042 0.0120.025 0.005 −0.10 0.015 0.032 0.0370.033 0.010 −0.17 0.017 0.029 0.0390.029 0.054 −0.17 0.010 0.017 0.029

0.0 0.2 0.4 0.6 0.8 1.0 1.20.00

0.05

0.10

0.15

0.20

0.25c M

g,an

/103 M

nF / n

a/2,mc,0

AAV, E1

zmc

m/M:

-0.25

-0.15

0

4

Fig. 6. Influence ofzmcm on thefit of theMgSO4 concentration in the anode solution for theexperiment E1.

82 S. Koter et al. / Desalination 342 (2014) 75–84

of cations is directed towards the anode side of AAV. A high value ofxj on the cathode side results from a high salt absorption into themembrane being in contact with a concentrated solution on thatside. On the opposite (anode) side of AAV the concentration of cat-ions is very low and therefore xj and xv are much smaller than xdif.The model assuming a full dissociation of salts (with 2 parameters)generates a very small flux of cations which drastically increasesduring the process. This is the reason of the exponential characterof the cMg,an = f(nF) curve. Taking the association between cationsand sulfates into account, the model will be able to predict a morestable flow of associate and, consequently, to fit experimental datamore precisely.

5. Conclusions

The selectivity of AAV towards the permeation of protons in themixtures of H2SO4 and its salts is rather low (CEa = 0.50–0.55).The kind of cation (Mg2+, Zn2+, Na+) practically has no influenceon CEa. It is even lower than that obtained previously for the pureH2SO4 solutions. The probable reason of that observation is the neg-ative influence of membrane aging on its selectivity. According to thetransport model based on the Donnan equilibria the opposite shouldbe true.

The diffusional permeability of H2SO4 is one order higher than thatof salts. This indicates that the relative mobility of protons with respecttometal cations is much higher inside the AAVmembrane than in a freesolution.

The permeabilities of cations from the cathode to anode side of AAVin the electrodialytic and diffusive processes are similar despite thepresence of the external electric field acting against the concentrationforce. This apparently unexpected behavior can be explained by the

Table 8The ionflux throughAAVand its electric, xj, diffusional, xdif, and convective, xv, componentson the cathode and anode side of AAV; the 2-parameter fit of the experiment E1.

n Fna=2;mc;0

105Ji [mol/m2s] Cathode side of AAV Anode side of AAV

xj xdif xv xj xdif xv

HSO4− and SO4

2−

0.2 309 1.3 −0.48 0.13 0.79 0.19 0.020.7 259 1.6 −0.73 0.14 0.72 0.23 0.051.2 218 1.8 −0.91 0.14 0.78 0.14 0.09

Mg2+

0.2 0.02 −10400 9226 1175 −0.67 1.6 0.030.7 0.24 −829 741 89 −0.08 1.1 0.011.2 0.72 −280 250 31 −0.13 1.1 0.01

increase of convective part of the ion flux caused by the electroosmoticwater flow induced by the anionmigration towards the anode side. Thecontribution of osmotic flow to the diffusional permeability of ions isnegligible. The diffusional permeability of Na+ is significantly lowerthan that observed in the electrodialytic experiments which can beonly explained by the presence of charged associate Na+·SO4

2− whichmigrates to the anode side.

The model based on the extended Nernst–Planck equation and theDonnan equilibria enables the fit of the H2SO4 concentration changesduring electrodialytic experiments for a wide range of the fixed chargeconcentration, cm. However, a high value of cm implies an unacceptablylow value of the proton diffusion coefficient inside themembrane com-pared to the anion one.

Regarding the metal cation concentration changes, neither the2-parameter (zmcm and km,H = km,S = km,M) nor 3-parameter (zmcm ,km,H = km,S, km,M) fit was satisfactory. Both approaches yielded thefitting curves cM,an = f(nF) of the exponential character. With km,H,km,S, and km,M as independent parameters, the model description ofthe experimental data was improved, especially in the case of Na+.However, this improvement was possible only for negative valuesofzmcm andnot in all the cases of divalent cations. Thus, it can be conclud-ed that apart of the cation flow the diffusion of the cation-sulfate associ-ates and in the case of Namigration of Na+·SO4

2− through themembranetakes place. Introducing these associates would eliminate the unreason-able negative values of zmcm and improve the fit significantly.

Nomenclaturec concentration (mol m−3)ci concentration of i-th species in the pore solution (mol m−3)cm concentration of fixed charges (mol m−3)cS total concentration of HSO4

− and SO42− inside the membrane

(mol m−3)CE current efficiencyDi diffusion coefficient of species i (m2s−1)F Faraday constant (F = 96485 C mol−1)I ionic strength (mol dm−3)Ji flux of i-th species referred to the unit membrane surface

(mol m−2 s−1)JS sum of HSO4

− and SO42−

fluxes (mol m−2 s−1)Ji flux of i-th species through the unit cross-section area of

pores (mol m−2 s−1)Jm mass flux (kg m−2 s−1)Jv volume flux (m s−1)j electric current density (A m−2)Kas,c concentration association constantK2,c second concentration dissociation constant of H2SO4

km,i pore factor of species i defined by Eq. (4)lm thickness of membrane (m)lp pore length (m)Mi molar mass of species i (kg mol−1)m mass (kg)n number of moles (mol)nF number of moles of elementary charge passed through the

system (mol)Ps,i permeability of i-th species (m2s−1)R gas constant (R = 8.314 J K−1 mol−1)rF,α change of mass of solution α, as defined by Eq. (A2a–c)

(kg mol−1)Sm area of membrane (m2)T temperature (K)t time (s)tp time of process (s)V volume of solution (m3)Vp volume fraction of membrane poresXp surface fraction of poresx dimensionless x-coordinate, x ¼ x= θlmð Þ

83S. Koter et al. / Desalination 342 (2014) 75–84

x x-coordinate along the pore (m)xdif diffusional contribution of ion fluxxj electric current contribution of ion fluxxv convective contribution of ion fluxzi charge number of ion izm charge number of fixed chargesθ tortuosity coefficient of membrane poresφ electric potential in the pore solution (V)

Subscripts:1 H+

2 HSO4−

3 SO42−

4, M metal cationS HSO4

− and SO42−

a H2SO4

a/2 1/2H2SO4

an anode solutionca catholyte solutionmc solution on the cathode side of AMsam sample

Abbreviations:AM anion exchange membraneBM bipolar membrane

Acknowledgment

The authors thank Asahi Glass Engineering Co. for supplying thesamples of AAV membranes.

Appendix A

Calculation of the mass fluxes: Jm,AM, and Jm,BM

The change of mass of a solution α resulting only from the masstransport through the membranes is given by:

Δmα ¼ mα;tp þmsam;α þ nFr F;α−mα;0 α ¼ ca;mc; an ðA1Þ

where mα,0 and mα,tp are the mass at the beginning and at the endof process, respectively,msam,α —mass of samples taken for analysis,nF — number of moles of elementary charge passed through the sys-tem, and rF,α — change of mass caused by the electrode reaction orthe water splitting inside the bipolar membrane:

r F;ca ≡ 1−xw;mc

� �Mw; r F;mc ≡ xw;mcMw−MOH− ; r F;an ≡

14MO2

:

ðA2a–cÞ

In the above formulas xw,mc is the molar ratio of water comingfrom the middle compartment and split in BM, Mi is molar mass ofspecies i. Δmca, Δmmc and Δman are related to Jm,AM, and Jm,BM bythe following equations:

Δmca ¼ −tpSm Jm;BM

Δmmc ¼ tpSm Jm;BM− Jm;AM

� �Δman ¼ tpSm Jm;AM

ðA3a–cÞ

where tp is the time of process and Sm — membrane area. In the tensornotation Eqs. (A3a–c) take the form:

ΔmcaΔmmcΔman

0@ 1A ¼ tpSmAJm;AMJm;BM

� �; A ≡

0−11

−110

0@ 1A : ðA4Þ

Multiplying both sides of Eq. (A4) by (ATA)−1A, one gets after rear-rangement the following expressions for Jm,AM, and Jm,BM:

Jm;AM ¼ 13Smtp

2Δman−Δmmc−Δmcað Þ ðA5Þ

Jm;BM ¼ 13Smtp

Δman þ Δmmc−2Δmcað Þ: ðA6Þ

The above formulas are valid under the condition that the masschanges were measured with the same accuracy. The volume fluxthrough AM, Jv needed in the model, was calculated by dividing Jm,AM

through the mean density of solutions separated by AM. Substituting(A1) and (A2) into (A4) it can be noticed that xw,mc disappears in the ex-pression for Jm,AM. Jm,BM depends on the assumed value of xw,mc, howev-er this flux has negligible effect on the model simulations. In thecalculations it was assumed xw,mc = 0.5.

Appendix B. Supplementary data

Supplementary data to this article can be found online at http://dx.doi.org/10.1016/j.desal.2013.10.025.

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