Desalination 265 (2011) 274–278

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Desalination

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Transport coefficients in desalting processes by electrodialysis

Javier Garrido ⁎Departamento de Física de la Tierra y Termodinámica, Universitat de València, E-46100 Burjassot, Valencia, Spain

⁎ Fax: +34 963543385.E-mail address: garridoa@uv.es.

0011-9164/$ – see front matter © 2010 Elsevier B.V. Adoi:10.1016/j.desal.2010.07.032

a b s t r a c t

a r t i c l e i n f o

Article history:Received 19 May 2010Received in revised form 12 July 2010Accepted 13 July 2010Available online 10 August 2010

Keywords:ElectrodialysisNon-equilibrium thermodynamicsTransport coefficients

In this work a thermodynamic analysis on the transport equations in the processes of electrodiffusion (EF)and electrodialysis (ED) has been developed. The transport equations are classified in two sets according tothe information they contain: i) fundamental and ii) complementary. We determine that there are fourfundamental transport coefficients needed to characterize these membrane systems. We also conclude thatthis number is not reduced to three when the Onsager reciprocal relation (ORR) is assumed. I have alsoobtained a new expression for the concentration rate in EF and ED processes from the mass and volumebalance. This relation provides a new way for evaluating the apparent transport number. This procedure hasbeen applied to two membrane systems using data published in the literature.

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© 2010 Elsevier B.V. All rights reserved.

1. Introduction

One of the techniques used for desalting sea water is theelectrodialysis (ED). An elemental ED stack is constituted by two ion-exchange membranes, one cationic and the other anionic, whichseparate three compartments filled with electrolyte solutions. Theconcentration of the central solution is reducedwhen an electric currentis applied in the correct direction. These processes are characterized bytransport equations with four coefficients [1].

The processes in an ED stack are a consequence of the constituentstransport through the two membranes [2,3]. From the mass andvolume balance we can relate the ED coefficients with those ofelectrodiffusion in membranes (EF). Since there are diverse EFformulations [4,5], there are also different ways of relating thecoefficients of these two processes. A widely accepted EF formulationpostulates the transport equations from the dissipation function [6].

In this paper I propose phenomenological transport equations formembrane processes without deriving them from the dissipationfunction [5,7]. The first step is to choose a set of independent fluxesand forceswhich characterizes thenon-equilibriumstates of the system.I postulate that any flux or force is linearly related to these independentquantities. The transport equations and the coefficients are classified asfundamental or complementary, according to the value of theinformation they provide. Finally, I deduce the relationships amongthe four overall ED coefficients and the four EF fundamental coefficients.

This approach has been satisfactorily used in the description of:i) electrodiffusion in bulk solutions [7], and ii) electrodiffusion inmembranes [5]. These previous works have highlighted severaladvantages of this approach. The electric current characterizes better

the electric equilibrium than the electric potential [5,7]. New relationsbetween the fluxes and the observable quantities have been obtainedfrom the mass and volume balance and it has been shown that theserelations improve the results up to 25% in certain cases [5]. The solventand ionic fluxes in bulk solution electrodiffusion are not independentquantities [7].

I study also the Onsager reciprocal relation (ORR). This one isproposed by the Statistical Mechanics for transport equations derivedfromdissipation function. In this paper I study theway inwhich the newEF approach is affected by the ORR. I discuss also the realmeaning of thereduction in the number of independent transport coefficients due tothe ORR.

2. Electrodiffusion in membranes (EF)

In this section I select firstly two observable quantities to be use inthe study of EF. Then a mass and volume balance relates theseobservables to the membrane fluxes. I discuss also the best set ofindependent variables to describe the non-equilibrium states. Andfinally I postulate the phenomenological equations where fourindependent coefficients characterize the membrane.

Consider a system of uniform temperature and pressure formed by amembrane that separates two compartments (I and II) containingsolutions of the same binary electrolyte Aν+Bν− (component 2). Theelectrolyte dissociates intoν+cationsAz+ andν− anions Bz−, with z+ν++z−ν−=0. Stirrers in both compartments keep the solute concentrationsuniform. The quantityΔc2=c2

II−c2I can bemeasured. An electric current

I is driven through the solution by working electrodes B/Bz−, where theelectrode reaction Bz−→←B + jz−je− is developed. Two horizontalcapillaries at the same level ensure negligible pressure differencebetween the compartments. Fig. 1 shows a sketch of this cell.

The molar fluxes through the membrane of solvent (component 1)J1 and cation J+ can be evaluated from the values measured in two

Fig. 1. Sketch of the cell: M: membrane; WE: working electrodes; RE: reversibleelectrodes;⊗: stirrers; J1, J+, J−:fluxes of solvent, cation, and anion, respectively; I: electriccurrent; I, II: compartments.

275J. Garrido / Desalination 265 (2011) 274–278

observable quantities. These are the apparent volume flow rate thatexits compartment I through the capillary dVc /dt and the electrolyteconcentration rate dc2/dt in the same compartment. I consider ascompartment I that in which a positive volume rate is measured, i.e.dV c /dtN0. In order to simplify the notation I have suppressed thesuperindex I. The fluxes J1 and J+ are defined as positive fromcompartment II to I.

In the control volume delimited by the walls, the membrane andthe capillary, the mass balance is

dn1

dt= J1−c1

dVc

dtð1Þ

dnþdt

= Jþ−νþc2dVc

dtð2Þ

where n1 and n+ are the amounts of solvent and cation, respectively.The volumeof the solutionV s inside this control volume changes due

to the electrode reaction. This one proceeds at a rate dξ /dt= I/z−F andtherefore

dV s

dt= − vPB I

z−Fð3Þ

where vPB is the partial molar volume of species B at the electrode and Fis Faraday's constant.

Since n1=c1Vs, n+=ν+c2Vs, and vP1 dc1 = dtð Þ + vP2 dc2 = dtð Þ = 0,

where vPi is the partialmolar volume of constituents i=1,2, Eqs. (1)–(3)lead to

J1 = c1dVc

dt−V s v

P

2

v�1

dc2dt

−c1vPB Iz−F

ð4Þ

Jþνþ

= c2dV c

dt+ V s dc2

dt−c2

vPB Iz−F

ð5Þ

which provide the fluxes J1 and J+ from the apparent volume flow ratedV c /dt, the concentration rate dc2/dt, and the electric current I.

In this system we found several typical quantities of non-equilibrium states. Among them I can mention: the fluxes throughthemembrane of solvent J1, of cation J+, of anion J−, of electric currentI, etc. And the differences in concentrations Δc1, Δc2, in chemicalpotentials Δμ1, Δμ2, in electrochemical potentials Δ μ̃þ, Δ μ̃−, inobservable electrical potentials Δψ+, Δψ− [5,8], etc. I consider thesedifferences as ΔXi≡Xi

II−XiI. The first ones are usually called thermo-

dynamic fluxes and the latter thermodynamic forces. All thesequantities vanish when the membrane system is at equilibrium. Inorder to analyze the irreversible processes, some of these fluxes andforces must be chosen as independent variables to characterize the

non-equilibrium states. I select Δc2 and I. Then I can postulate thefollowing linear transport equations for the fluxes J1 and J+ [5]

J1 =τ1FI−AP1Δc2 ð6Þ

Jþνþ

=tþ

νþzþFI + AP2Δc2 ð7Þ

where A is the active membrane area, P1 is the permeability of themembrane to the solvent, P2 is the permeability of the membrane tothe electrolyte, τ1 is the solvent transference number in themembrane, and t+ is the cation transport number in the membrane[5]. And for the anion flux we have J− = ν− = t− = ν−z−Fð ÞI + AP2Δc2with t++ t−=1.

Therefore the non-equilibrium state of this membrane system ischaracterizedby the four following fundamental properties: i)mechanicalequilibrium Δp=0; ii) thermal equilibrium ΔT=0; iii) non-equilibriuminmatter distribution expressed by the value ofΔc2; and iv) electric non-equilibriumgiven by I.WhenΔc2=0wehave a process of “pure” electricconduction. And when I=0we have a solute–solvent diffusion process.In this last case the solute flux is given by J2≡ J+/ν+.

From the observable quantities dV c /dt and dc2 /dt the fourcoefficients P1, P2, τ1 and t+ can be determined. And vice versa, fromthe values of the four coefficients P1, P2, τ1 and t+, and Eqs. (4)–(7), thetwo observable quantities

dc2dt

=vP1

V s A c1P2 + c2P1ð ÞΔc2 +c1tþνþzþ

−c2τ1

� �IF

� �ð8Þ

dVc

dt= A vP2 P2− vP1 P1ð ÞΔc2 + vP1 τ1 +

vP2 tþνþzþ

+vPB

z−

� �IF

ð9Þ

can be determined for a process (Δc2, I). In Eq. (8) I can substitutec1tþ = νþzþ−c2τ1� �

for c1 = νþzþ� �

t1þ where t+1 is the apparent

transport number t+1 ≡ t+−(ν+z+c2/c1)τ1. I emphasize the signifi-

cant role which plays the apparent transport number in the EFprocesses.

I consider dV c /dt and dc2/dt as the fundamental observables of theprocess (Δc2, I), because they express the evolution of the membranesystem, and from their values we can determine the matter fluxthrough the membrane [see Eqs. (4), (5)]. But there are otherobservables which provide complementary information. Among theseones I emphasize the observable electric potential difference Δψ−,which is measured between the terminals of two reversible electrodesto the anions [5,8]. For this thermodynamic force I can also postulate aphenomenological equation

Δψ− = α−Δc2 + RI ð10Þ

where α− is a coefficient, and R is the electric resistance. The value ofthese two coefficients can be obtained from electric potentialmeasurements α− = Δψ−ð ÞI=0 =Δc2 and R = Δψ−ð ÞΔc2 =0 = I.

Then I can establish a classification of the coefficients in two kinds:i) fundamental, related with the fundamental observables, e.g. P1, P2,τ1 and t+; and ii) complementary, related with the complementaryobservables, e.g. α− and R.

I discuss now whether the four coefficients P1, P2, τ1 and t+ areindependent quantities. A doubt arises whenwe study the electrodiffu-sion processes in bulk solutions. I consider fluxes in the x direction. Theprocesses are described by the flux equations [7]

J1 =τ1FI + AD1

dc2dx

ð11Þ

Jþνþ

=tþ

νþzþFI−AD2

dc2dx

ð12Þ

Fig. 2. Sketch of the electrodialysis stack: K: cation-exchange membrane; A: anion-exchange membrane; E: electrodes; P: pumps; R: reservoirs; I: electric current; I, II, III:compartments.

276 J. Garrido / Desalination 265 (2011) 274–278

where Di is the diffusion coefficient of the component i=1,2 in thelaboratory reference. From the balance vP1 J1 + vP2 Jþ = νþ + vPB I =z−F = 0 I deduce vP2 D2− vP1 D1 = 0 and νþzþ vP1 τ1 + vP2 tþ = ν− vPB[7]. Then, only two of the four coefficients D1, D2, τ1 and t+ areindependent quantities. However, inside the membrane Eqs. (4) and(5) show that vP1 J1 + vP2 Jþ = νþ + vPB I = z−F = dVc

= dt and I con-clude that the four coefficients D1, D2, τ1 and t+ are independentquantities. A similar conclusion applies to the four fundamentalcoefficients of the membrane system P1, P2, τ1 and t+.

Finally, I comment some advantages of using (Δc2, I) as indepen-dent quantities instead of (Δc2,Δψ−), these ones related withdiffusion–migration flux equations. I emphasize only two: i) theelectric current provides a clear distinction between the diffusionprocesses I = 0ð Þ from electrodiffusion processes I≠0ð Þ; this cannotbe reached using electric potential values Δψ−, and ii) the electriccurrent, and not the electric potential Δψ−, appears explicitly in therelationships among the fluxes J1 and J+ and the observables dVc /dtand dc2/dt [see Eqs. (4) and (5)].

3. The Onsager reciprocal relation

The dissipation function of electrodiffusion processes in mem-branes is given byΨ = J1Δμ1 + JþΔ μ̃þ + J−Δ μ̃− [6]. The observableelectric potential is defined by Δψ− = 1= z−Fð ÞΔ μ̃− and its relation tothe cation electrochemical potential difference is Δ μ̃þ =Δμ2 = νþ + zþFΔψ− [8]. The chemical potential difference mustsatisfy the Gibbs–Duhem relation Δμ1 = − c�2 = c�1ð ÞΔμ2, wherec�i≈ cIi + cIIi

� �= 2 i = 1;2, and hence we obtain Ψ = J1þ = νþ

� �×

Δμ2 + IΔψ− with J1þ = νþ = Jþ = νþ− c�2 = c�1ð ÞJ1 [5]. From this lastform of Ψ I postulate the phenomenological equations

J1þνþ

= Γ11Δμ2 + Γ12Δψ− ð13Þ

I = Γ21Δμ2 + Γ22Δψ− ð14Þ

The Onsager reciprocal relation [4,5] for the transport coefficientsis Γ12=Γ21 and implies the following relation between τ1 and t+

tþ−ðνþzþ cP2 = cP1Þτ1 = −νþzþFΔψ−Δμ2

� �I=0

ð15Þ

where Δψ−ð ÞI=0 is the membrane potential, and t1þ = tþ−ðνþzþ cP2 = cP1Þτ1 the apparent transport number of the cation.

Analyzing Eqs. (13) and (14) it could be concluded that thenumber of independent transport coefficients in membrane electro-diffusion is three, i.e. Γ11, Γ12=Γ21 and Γ22 [9,10]. But this is not thecase because a set of independent coefficients has to be able ofdetermining the process evolution. I have shown before that for theprocess (Δc2, I) the coefficients P1, P2, τ1 and t+ satisfy this condition.Effectively, from their values we can calculate the observables dVc /dtand dc2/dt. However, we cannot evaluate these latter observablesfrom the coefficients Γ11, Γ12=Γ21 and Γ22 in a process (Δμ2,Δψ−).Moreover, I can also affirm that the ORR, which establishes a relationamong τ1, t+ and α−, does not reduce the number of fundamentaltransport coefficients. Therefore the number of independent coeffi-cients in membrane electrodiffusion is four and not three as it isusually assumed.

4. Desalting processes by electrodialysis

An ED stack combines cation-exchange membranes K and anion-exchange membranes A (see Fig. 2). The solution c2

I circulates by thecompartment I and the solution c2

II by the compartments II and III. Inthe solution of compartment I we include also that contained inthe reservoir and in the pipes. I assume mechanical and thermal

equilibrium and that there are not significant concentration differ-ences inside this solution.

For the solvent and solute fluxes which enter the compartment Iwe propose the transport equations

J1 =Δτ1F

I−APKA1 Δc2 ð16Þ

J2 =Δtþ

νþzþFI + APKA

2 Δc2 ð17Þ

where P1KA, P2

KA, Δτ1 and Δt+ are the independent transportcoefficients of electrodialysis [1]. Here J1 and J2 are the fluxes whicharrive to the compartment I. Since themass balance for compartment Iis

J1 = c1dVdt

−VvP2

vP1

dc2dt

ð18Þ

J2 = c2dVdt

+ Vdc2dt

ð19Þ

the measured values of dV /dt and dc2 /dt determine the fluxes J1 andJ2.

The electrodialysis coefficients are related to the electrodiffusioncoefficients P1

α, P2α, τ1α, t+α of the membranes as follows. If JiK and JiA

denote the fluxes through the membranes, we have J1= J1K− J1

A andJ2 = JKþ−JAþ

� �= νþ and thus we conclude P1

KA=P1K+P1

A, P2KA=P2K+P2

A,Δτ1=τ1K−τ1A and Δt+= t+

K − t+A .

From Eqs. (16)–(19) the rate of concentration in compartment I is

dc2dt

=vP1 AV

c1PKA2 + c2P

KA1

� Δc2 +

c1Δt1þ

νþzþ

!iF

" #ð20Þ

where Δt1þ≡ t1þ� �K− t1þ

� �A is the difference in the apparent transportnumbers, and i is the electric current density. A desalting process ischaracterized by a negative value of dc2/dt.

5. Analysis of experimental data

I am now to evaluate these transport coefficients with datapublished in the literature. Firstly I will consider the overall transportcoefficients of an ED processmeasured by Tanaka [1]. Then I will makeuse of the concentration rates dc2 = dtð Þ of an EF process given byKoter and Kultys [11] in order to evaluate the apparent transportnumber and the solute permeability of a membrane system. Andfinally from the concentration rate values dc2 =dtð ÞΔc2 =0 of an EDprocess given by Moresi and Sappino [12] I will evaluate the apparenttransport number of the membranes.

277J. Garrido / Desalination 265 (2011) 274–278

Tanaka [1] used an ED stack formed by Aciplex K-171/A-172membranes in c2=0.50 M NaCl solution at 25 °C (c1=54.9 M,vP1 = 0:0181 M−1 and vP2 = 0:0185 M−1), and the transport equations

j2 = λi + μΔc2 ð21Þ

jV = ϕi−ρΔc2 ð22Þ

where j2= J2/A, and jV = JV = A = 1= Að ÞdV = dt is the volume fluxdensity. Using JV = vP1 J1 + vP2 J2, the relations between Tanaka'stransport coefficients and those used in this work are

PKA1 =

ρ + vP2 μvP1

ð23Þ

PKA2 = μ ð24Þ

Δτ1 =ϕ− vP2 λð ÞF

vP1ð25Þ

Δtþ = νþzþλF ð26Þ

Fromthedata of Ref. [1], the values of theoverall transport coefficientsareλ=9.41 μmol C−1, μ=2.59×10−9 m s−1,ϕ=1.22×10−9m3C−1,andρ=1.08×10−7ms−1M−1. Thesemeasurementshavebeencarriedout in a steady state dc2 /dt=0. Thus, we obtain the valuesP1KA=5.97×10−6 m s−1, P2KA=2.59×10−9 m s−1, Δτ1=5.59, and

Δt+=0.908.With these data I can nowevaluate the desalting rate dc2/dtin theprocessΔc2=0.10 Mand i=−200 Am−2 applyingEq. (20). Thus,we have V = vP1 A

� �dc2 = dtð Þ = −979 mol2 m−5 s−1 and we conclude

that the term c1PKA2 + c2PKA

1

� �Δc2 = −0:284 mol2 m−5 s−1 is almost

negligible.Koter and Kultys [11] study the membrane electrolysis of sulfuric

acid by the batch method using the membrane cell shown in Fig. 3.From the concentration rate in the catholyte and in the anolyte I canevaluate the apparent transport number t−1 in the membrane. In thiscase I can assume mechanical and thermal equilibrium between thetwo solutions Δp=0 and ΔT=0.

The mass balances in this cell are different from those given byEqs. (1) and (2). We have in the catholyte

dn1

dt

� �ca= J1 ð27Þ

dn2

dt

� �ca=

J−ν−

ð28Þ

and in the anolyte

dn1

dt

� �an= − J1 +

I2F

� �ð29Þ

Fig. 3. Sketch of the cell used by Koter and Kultys [11]. Two H2SO4 solutions are pumpedfrom the tanks; M: membrane.

dn2

dt

� �an= − J−

ν−ð30Þ

As cations H+ react at the two electrodes, I have preferred to use J−instead of J+. I assume that the hydrogen and oxygen formed at theelectrodes escape to the environment from the tanks remainingconstant their concentrations.

The concentration rate can be expressed making use of the fourtransport coefficients P1, P2, τ1 and t−. For the catholyte we have

dc2dt

� �ca=

vP1

VA c1P2−c2P1ð ÞΔc2 +

c1t−ν−z−

−c2τ1

� �IF

� �ð31Þ

and for the anolyte

dc2dt

� �an=

vP1

VA c2P1−c1P2ð ÞΔc2−

c1t−ν−z−

−c2τ1−c22

� �IF

� �ð32Þ

As t+1 ≡ t+−(ν+z+c2/c1)τ1, we can evaluate the apparent trans-

port number frommeasurements of the electrolyte rate concentrationwhen Δc2=0, that is

t1− =ν−z−FVc1 v

P

1 Idc2dt

� �ca

Δc2 = 0ð33Þ

t1− = ν−z−c22c1

− FVc1 v

P

1 Idc2dt

� �an

Δc2 = 0

" #ð34Þ

At the experiment four (E4) of Ref. [11] the valueof the electric currentis i=100mA cm−2 and the temperature T=25 °C. The initial conditionsare: c2=1.0 M, Δc2=0, V=0.30 dm3 (c1=53.4 M and vP1 =0:0180 M−1). From the results shown in this reference, I deduce forthe Selemion AAV membrane (Asahi Glass Engineering Co., Japan),dc2 = dtð ÞcaΔc2 = 0 = −0:241 × I = FV a n d t −

1 = 0 . 5 0 , a n ddc2 = dtð ÞanΔc2 = 0 = 0:245 × I = c2FV and t−

1 =0.49. At the same condi-tions, for the Neosepta ACM membrane (Tokuyama Co., Japan) I deducedc2 = dt� �ca

Δc2 = 0 = −0:227 × I = FV a n d t −1 = 0 . 4 7 , a n d

dc2 = dt� �an

Δc2 = 0 = 0:231 × I = c2FV and t−1 =0.46. These results can be

compared with those given by Pourcelly et al. [13], which measure theunidirectional fluxes by radiotracer techniques. They work also with AAVand ACM membranes, sulfuric acid solution c2=1.0 M, Δc2=0 and anelectric current density of 100 mA cm−2. They obtain for the transportnumber the values (see Ref. [13]): t−=0.64 for the AAV membrane andt−=0.47 for the ACM membrane. If I assume t−≈t−

1 , we see a goodagreement between these two quantities in the case of the ACMmembrane.

I can also evaluate the quantity c1P2+c2P1 from dc2/dt data if wedo I=0 and Δc2≠0

c1P2 + c2P1 =V

A vP1 Δc2

dc2dt

� �I=0

ð35Þ

In Ref. [11] the diffusion experiment provides these data. Theinitial conditions are: Δc2=1.0 M, c2=0.0 M, V=0.30 dm3,A=49 cm2 and T=25 °C (c1=55.4 M, c1 v

P

1 = 1). The electric circuitis open. For the AAV membrane we have dc2 =dtð ÞI=0 =1:44 × 10−3 mol m−3 s−1 and P2=8.8×10−8 m s−1. And for theACMmembranewe obtain dc2 =dtð ÞI=0 = 2:91 × 10−3 mol m−3 s−1

and P2=1.78×10−7 m s−1.Moresi and Sappino [12] concentrate citric acid solutions using a

laboratory-scale electrodialyser (Aqualyzer P1, Cornings EIVS, LeVesinet, F), containing 40 cation- and 40 anion-exchange membranes.Instantaneous concentrations of solute in the diluated and concen-trated streams are estimated by refractometry. If I consider negligiblethe mass interchanges between each one of the streams with the

278 J. Garrido / Desalination 265 (2011) 274–278

catholyte and the anolyte, from the molarity rate we can evaluate thedifference between the apparent transport number in membranes.Effectively, from Eq. (20) we have in the concentrated

t1þ� K− t1þ

� A=

νþzþFV dc2 =dtð ÞΔc2 =0

c1 vP

1 Aið36Þ

where dc2 =dtð ÞΔc2 =0 N 0, and in the diluated

t1þ� A− t1þ

� K=

νþzþFV dc2 =dtð ÞΔc2 =0

c1 vP

1 Aið37Þ

where dc2 =dtð ÞΔc2 =0b0. I use the data recorded in Ref. [12], whereν+=1, z+=3, and A=0.55 m2, i=145 A m−2 and T=33 °C. Thevalue c1 v

P

1 = 0:903 has been calculated [14] from data ofsolutions density. The initial conditions are c2=0.52 M and Δc2=0. Inthe concentrating stream we have V=2.1 dm3 and dc2 =dtð ÞΔc2 =0 =

6:3 × 10−2 mol m−3 s−1, and deduce t1þ� K− t1þ

� A= 0:53. And in

the diluating stream we have V=3.0 dm3 and dc2 =dtð ÞΔc2 =0 =

−4:2 × 10−2 mol m−3 s−1, and deduce t1þ� A− t1þ

� K= −0:51.

6. Conclusions

The transport equations of electrodialysis (ED) have been studiedin the framework of a new formulation of electrodiffusion inmembranes (EF). The EF transport equations have been postulatedfrom phenomenological arguments, without reference to the dissipa-tion function. The result obtained shows the following characteristicsand advantages:

i) The variables Δc2 and I have been selected as the bestindependent quantities for characterizing a process in EF.

ii) The observable quantities are classified as fundamental orcomplementary according to the information they provide.

iii) The number of fundamental coefficients is four, and it is notreduced to three by the Onsager reciprocal relation (ORR).

iv) The relationships between the four EF fundamental coefficientsand the four ED overall transport coefficients have beenestablished.

v) From the mass balance I have obtained an expression for theconcentration rate dc2/dt in an ED stack.

vi) The apparent transport number has been evaluated frommeasurements of the electrolyte rate concentration.

SymbolsA active membrane area, m2

c concentration, mol m−3

F Faraday's constant, C mol−1

I electric current, Ai electric current density, A m−2

Ji flux of constituent i=1,2,+,−, mol s−1

ji flux density of constituent i=1,2,+,−, mol m−2 s−1

JV volume flow, m3 s−1

jV volume flow density, m s−1

p pressure, PaPi membrane permeability to constituent i=1,2, m s−1

R electrical resistance, ΩT temperature, Kt time, st transport numbervPB molar volume of species B of the electrode B/Bz− , m3 mol−1

vP partial molar volume, m3 mol−1

V total volume of solution I, m3

Vc volume of solution I inside the capillary, m3

V s volume of solution I submitted to agitation by the stirrers, m3

z charge number

Greekα− transport coefficient, V m3 mol−1

Γ11 phenomenological coefficient, mol2 J−1 s−1

Γ12 phenomenological coefficient, mol V−1 s−1

Γ22 phenomenological coefficient, Sλ overall transport number, mol C−1

μ overall solute permeability coefficient, m s−1

μi chemical potential of constituent i=1,2, J mol−1

μ̃ i electrochemical potential of constituent i=+,−, J mol−1

νi stoichiometric number of constituent i=+,−ξ extent of reaction, molρ overall water permeability, m4 mol−1 s−1

τ1 solvent transference numberϕ overall electro-osmotic coefficient, m3 C−1

Ψ dissipation function, J s−1

ψ− observable electric potential with electrode reversible toanions, V

SubscriptsB electrode constituentV volume1 solvent2 solute+ cation− anion

SuperscriptsA anion-exchange membraneK cation-exchange membrane

References

[1] Y. Tanaka, Irreversible thermodynamics and overall mass transport in ion-exchange membrane electrodialysis, J. Membr. Sci. 281 (2006) 517–531.

[2] Y. Gong, X. Wang, Y. Li-xin, Process simulation of desalination by electrodialysis ofan aqueous solution containing a neutral solute, Desalination 172 (2005)157–172.

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