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Nernst-Plancktransporttheoryfor(reverse)electrodialysis:I.Effectofco-iontransportthroughthemembranesM.Tedescon,H.V.M.Hamelers,P.M.BiesheuvelWetsus,EuropeanCentreofExcellenceforSustainableWaterTechnology,Oostergoweg9,8911MALeeuwarden,TheNetherlandsarticleinfoArticlehistory:Received2October2015Receivedinrevisedform25February2016Accepted5March2016Availableonline10March2016Keywords:Nernst–PlanckequationDonnanpotentialIonexchangemembranesCo-ionexclusionConcentrationpolarizationabstractElectrodialysis(ED)andReverseElectrodialysis(RED)arerelatedtechnologiesforwaterdesalinationandenergyconversion,bothbasedontheselectivetransportofionsthroughionexchangemembranes.Fundamentalunderstandingoftheseprocessesrequiresthedescriptionofiontransferphenomenabothalongandthroughthemembranes.Wedevelopasimple2DmodelvalidforEDandRED,extendingtheapproachbySoninandProbstein(Desalination5,1968,293)byusingtheNernst–Planckequationnotonlyintheflowchannelsbutalsointhemembranes.Thismodelrequiresasonlyinputparametersthegeometricalfeaturesofthesystem,themembranechargedensity,andthediffusioncoefficientsofionsinthechannelandinthemembrane.Theeffectofnon-idealbehaviorofthemembraneduetotheco-iontransportisdiscussed,evaluatingtheperformanceoftheprocessesintermsofcurrentefficiencyandenergyconsumption(forED),andsaltfluxefficiencyandpowerdensity(forRED).Membranepropertiessuchaspermselectivityarecalculatedasoutputsofthemodel,anddependonpositioninthechannel.&2016ElsevierB.V.Allrightsreserved.1.IntroductionElectrodialysis(ED)isatechnologywidelyappliedfordesali-nationofbrackishwatersourcessincethe1950s.Recentdevel-opmentsinrelatedprocesses,suchasreverseelectrodialysis(RED)[1],membranecapacitivedeionization(MCDI)[2],andshockelectrodialysis[3],furtherincreasethenumberofapplicationsandthescientificinterestinelectro-membranetechnologies.TheprincipleofbothEDandREDisbasedontheselectivetrans-portofcationsandanionsthroughionexchangemembranes(IEMs),undereithertheinfluenceofanappliedvoltage(ED)oraconcentra-tiondifference(RED).A(R)EDsystemconsistsofanalternatingseriesofcationandanionexchangemembranes(CEMsandAEMs),whichareseparatedbyspacerchannelsthatallowforwaterflowalongthemembranes(Fig.1).Ideally,eachIEMallowsonlythepassageofcounterions,i.e.,ionswithasignoppositetothatofthefixedchargeinthemembranes(i.e.,cationsforCEMsandanionsforAEMs),whileco-ionsarerejected.InED,anexternalvoltageisappliedbetweentwoend-electrodestogenerateanioniccurrentthroughthemembranestack:therefore,thefeedstreamisdesalinatedandconcentratedinalternatingchannels,andtheconcentrateanddiluatearecollected(Fig.1).Conversely,intheREDprocess,thesystemisfedwithcon-centratedanddilutestreams,exploitingthesaltconcentrationdiffer-enceasdrivingforcetogenerateelectricpower.Thesamedevicecanbeoperated–atleastinprinciple–bothinEDandREDmode,althoughitiscommonlyacceptedthatthecelldesignshouldbeoptimizedforeachprocess.Asanexample,thespacersusedforcommercialEDunitshavetypicallyathick-nessbetween0.3and1mm[4],whileforREDtheuseofathinnerchannel(intherangeof0.1–0.3mm,orevensmaller[5])isgen-erallypreferredtoreducetheelectricalresistanceandenhancethepoweroutput.TransportphenomenainEDandREDinvolveboththecon-vectivefluxthroughspacerchannelsalongionexchangemem-branes(IEMs),andtheelectro-migrativefluxofionstowardsandthroughIEMs.Ideally,onlycounterionscanpassthroughthemembranes.However,becausemembranesdonotbehaveideally,alsoco-ionsandwatercancrossthemembranes,thusreducingtheprocessperformance.Therefore,modelingiontransportphe-nomenainEDandREDisacomplextask,whichrequiresareliabledescriptionofalltransportphenomenawithinchannelsandmembranes,andatthemembrane–solutioninterface.Uptonow,differentmodelingapproacheshavebeenproposedintheliteraturetodescribeiontransportinEDandRED,basedon(i)theNernst–Planckequation[6–13],(ii)Stefan–Maxwelltheory[14,15],(iii)theirreversiblethermodynamicsformalism[16–18],and(iv)semi-empiricalmodelsthatuseexperimentallyestablishedoverallmembraneproperties[19–23].Whenwatertransportisneglected,themostcommonapproachistheNernst–Planck(NP)equation,whichrequiresknowledgeofdiffusioncoefficientsofionsinsolutionsandwithinthemembranes,aswellasinformationofmembraneparameterssuchasfixedchargedensityandporosity.ContentslistsavailableatScienceDirectjournalhomepage:://dx.doi.org/10.1016/j.memsci.2016.03.0120376-7388/&2016ElsevierB.V.Allrightsreserved.nCorrespondingauthor.E-mailaddress:michele.tedesco@wetsus.nl(M.Tedesco).JournalofMembraneScience510(2016)370–381AsimpleyetfundamentalmodelingapproachforafullEDcellwasproposedin1968bySoninandProbstein[7],whereinthechannelstheNernst–Planckequationiscombinedwithasimplifieddescriptionofhydrodynamics.Despitesomestrongsimplifyingassumptions,especiallytheuseofideallypermselectivemembranes,themodeldescribesseveralofthemainaspectsinvolvedintheEDprocessinafundamentalmanner.Inparticular,theSonin–Probsteinapproachisbasedontheexactsteady-statesolutionoftheNP-equationandlocalelectroneutralityintheentiredomainofthespacerchannel,thuswithouttheassumpt